ObjectiveA composite controller can achieve stable suppression of high-bandwidth beam jitter of large-aperture tilt mirrors. The composite control method comprises multiple control links in series or parallel, and the controller is large-scale. There are many jitter analysis results for a single high-frequency narrowband; however, the correction of complex-frequency beam jitter requires further consideration of the mutual amplification and traction between multiple control links in the composite controller. Therefore, parameter matching is a challenging problem in the design of composite controllers. Accordingly, in this study, a parameter optimization method is proposed based on stochastic parallel gradient descent assistance to achieve high-bandwidth and high-precision control of large-aperture tilt mirrors using a composite controller.MethodsTo minimize the beam jitter residual variance, the power spectrum characteristics of the open-loop beam jitter to be corrected are first analyzed, and a composite controller is designed accordingly, including a proportional integral (PI) controller and dual two-order filter. Subsequently, the residual variance of the beam jitter is selected as the cost function to optimize the parameters of the composite controller using the stochastic parallel gradient descent method. Finally, after several iterations, optimal adjustment of the control performance of the composite controller is achieved. After parameter optimization, the correction and residual variance suppression abilities of the composite controller are verified using the Bode diagram and power spectrum density.Results and DiscussionsFor the beam jitter signal of the multifrequency narrowband power spectral density function (Fig. 4), the design of multiple controller links includes a PI controller, 30-Hz, 70- Hz, 146-Hz dual two-order filters, 70-Hz advanced phase compensation, and resonance elimination model. The initial parameters of each link of the composite controller are designed accordingly (Table 1). The composite controller corresponding to the initial parameters suppresses the residual variance from 117.48 μrad2 to 84.58 μrad2, with a residual variance suppression ratio of only 28%, and has a significant amplification effect on the high-frequency jitter. After 1000 iterations using the stochastic gradient descent method, the transfer function of beam jitter suppression in multiple controller links with the optimized parameters is obtained (Table 2). The jitter residual variance of a single tilting mirror closed-loop beam is suppressed from 117.48 μrad2 to 44.35 μrad2, and the residual variance suppression ratio reaches 62.2%. Furthermore, jitter is significantly suppressed near the low-frequency broadband and 30 Hz, 70 Hz, and 146 Hz narrowbands (Fig. 9). An optical test platform is set up (Fig. 10) to verify the effectiveness of the optimization parameters. The beam jitter composite controller with optimized parameters can effectively suppress the jitter of 30 Hz and 70 Hz and low-frequency broadband. The jitter at 146 Hz is partially suppressed, and there is a certain amplification at high frequency. The magnification factor is small and does not exceed 6 dB. The beam jitter residual variance is suppressed from 200.05 μrad2 to 93.93 μrad2. Compared with the composite control algorithm with initial parameters, under the same correction conditions, the beam jitter suppression performance of a single large-aperture tilt mirror is improved by 34.2 percentage points after optimization by the proposed stochastic gradient parallel descent auxiliary method.ConclusionsThe experimental results show that the residual variance-suppression ratio of the high-bandwidth beam jitter of a single large-aperture tilt mirror is 53.1%, which is comparable to that of the traditional method based on two tilt mirrors. This method can effectively reduce the complexity of the system and contribute to the high-bandwidth and high-precision control of beam jitter by large-aperture tilt mirrors. Furthermore, it can contribute to an optimization and evaluation method for the design of beam jitter controllers.

ObjectiveIn this study, technologies including beam combination structures, uncoupled two-dimensional adjustment, optical axis detection, and consistency control in fiber laser phased arrays are investigated. During the experiment implementation, seven fiber laser beams are placed in a regular hexagonal arrangement by adopting a “6 +1” array. Beam expansion is achieved by placing an expanding lens behind the fiber rod for each fiber beam, and two-dimensional adjustment is obtained by adopting high-precision uncoupled adaptive fiber optic collimator (AFOC) adjustment devices. Intelligent cameras are used to detect the reflected beam and control the optical axis in the closed loop, which provides an important guarantee of the coherent combination of the seven fiber lasers. The dynamic range of the AFOC is larger than ±230 μrad, the optical axis pointing accuracy is better than 1 μrad, the effective spot size of a single beam laser in the beam combination device is 85 mm, and the area duty cycle is 56.2%. The capability of beam combination, dynamic adjustment, and optical axis consistency control of the seven-beam laser combination device is fully verified by a number of full power experiments, and the average power-in-the-bucket after the phase close loop is 4.8 times that of the open loop.MethodsThe combination device adopts an optical fiber rod and a beam-expanding lens to collimate and expand a single laser beam. The reflective far-field detection system has three built-in charge coupled devices (CCDs) to detect the centroids of the seven fiber lasers. After calculating the optical axis shift, the control parameters are amplified by a proportional-integral-derivative (PID) control algorithm and loaded onto the AFOC for translation control to realize the optical axis pointing consistency control of multiple lasers, as shown in Figs. 2?4. By adopting large-thrust piezoelectric ceramics for high-precision adjustment, the AFOC can realize displacement amplification, and a two-dimensional uncoupled design can achieve high-precision control over a large dynamic range. The modal frequency and structural stability of the AFOC are improved through simulation optimization, as shown in Figs. 5?10. The beam combination device adopts a frame structure that integrates the AFOC, built-in CCD, beam expanding lens, weak mirror, and other components, and carries out an integrated lightweight design. Its layout is “6+1”, that is, a center light beam is surrounded by six light beams arranged in a regular hexagon. Invar rods are used for series-fixing and stiffening the plates to enhance their structural strength. The stability of the device is improved through simulations and design optimization. The structural diagram and simulation results are shown in Figs. 11?15.Results and DiscussionsThe effective output aperture of a single laser beam is 85 mm, the area duty cycle of seven lasers is 56.2% ( Fig. 15), and the dynamic range of the AFOC is greater than ±230 μrad (Figs. 20 and 21). Prior to the experiment, the zero point of the beam combination device is calibrated using a far-field sensing device. The calibrated device is used in multiple rounds of experiments. During the experiment, the AFOC is closed to ensure pointing consistency of the seven fiber lasers. After the closed loop, the optical axis pointing accuracy is better than 1 μrad. During coherent combining, the average power-in-the-bucket after the phase closed loop is 4.8 times that of the open loop (Fig. 23), which verifies the dynamic adjustment ability and optical axis pointing control ability of the beam combination device.ConclusionsA “6+1” layout is used to assemble seven fiber lasers with a phased array. A built-in CCD is used to detect the optical axis in different regions, and the consistency of optical axis pointing is adjusted in real time through a low-coupling and high-thrust AFOC. The performance test shows that the dynamic range of the AFOC is more than 230 μrad, and the resolution is better than 10 μrad. Finally, the experiment verifies the beam combination and optical axis consistency control abilities of the seven-fiber laser phased array combination device. The average power-in-the-bucket after the phase closed loop is 4.8 times that of the open loop.

ObjectiveThe output energy of regenerative amplifiers and its stability are crucial performance indicators. With the rapid advancement of high-power laser technology, there is a growing demand for higher output energy and power in regenerative amplifiers. However, the peak pulse energy in these amplifiers is limited by the damage threshold of the optical elements in the resonator. When the pulse width and peak power are fixed, increasing the output energy requires expanding the beam diameter in the resonator. Consequently, regenerative amplifier resonators are typically designed in the second stability region to achieve a sufficiently large beam diameter. However, resonators in this region are highly sensitive to misalignment, and longer resonator length exacerbates this sensitivity. Therefore, maintaining beam pointing stability in regenerative amplifiers is critical for ensuring stable output power. To achieve a long resonator length within a limited space, mirrors are commonly used to fold the resonator, and their arrangement influences beam pointing stability.MethodsThis study analyzed the impact of mirror arrangements on beam pointing stability within the resonator to ensure stable output power. Using modal analysis and microvibration theory, we examined the spatial distribution of disturbances on optical elements. Two mirror arrangement models were developed: a large-mirror configuration (model A) and a mirror-array configuration (model B). We used a regenerative amplifier in a high-power laser system as a case study, conducting Monte Carlo simulations and a over continuous 22-hour energy output experiment to compare the effects of these mirror arrangements on the output energy stability, thereby indirectly characterizing beam pointing stability.Results and DiscussionsExperimental and simulation results indicate that the mirror-array configuration (model B) significantly outperforms the large-mirror configuration (model A) in terms of beam pointing stability, even under the same vibration amplitude. During a over continuous 22-hour energy output test, model A exhibited a peak-to-valley (PV) value of 60.704% and an RMS (root-mean-square) value of 14.729% in average energy output stability. In contrast, model B achieved a much lower PV value of 2.325% and an RMS value of 0.429%. The mirror-array configuration effectively minimizes the impact of individual mirror disturbances by averaging errors across multiple elements, thereby enhancing the overall system stability. Conversely, the large-mirror configuration amplifies the influence of a single mirror’s stability owing to the multiple reflections required by each large mirror, which can degrade system performance. Although the large-mirror setup theoretically reduces the number of optical elements and potential error sources, it demands higher stability from each mirror. When mirror stability is consistent, the mirror-array configuration demonstrates stronger resistance to disturbances, leading to significantly improved beam pointing and energy output stability compared to the large-mirror configuration.ConclusionsThis study uses structural modal analysis and examines the influence of microvibrations on beam pointing to establish a kinematic model for the microvibration of individual optical elements. Additionally, it analyzes the spatial distribution of disturbances on these elements. By comparing the large-mirror configuration (model A) and mirror-array configuration (model B), the study explores how mirror arrangements affect beam pointing stability. Theoretical analysis shows that the mirror-array configuration (model B) outperforms the large-mirror configuration (model A), regardless of whether the mirrors are correlated or uncorrelated. In tests in which only the mirror arrangement was changed, a over continuous 22-hour energy output experiment showed that model A had an average energy output stability with a PV value of 60.704% and an RMS value of 14.729%. In contrast, model B achieved a PV value of 2.325% and an RMS value of 0.429%. These results highlight the clear advantages of the mirror-array configuration over the large-mirror setup. The experimental results indicate that the mirror arrangement significantly affects the output energy stability of regenerative amplifiers. In summary, the mirror arrangement affects beam pointing stability in laser systems, thereby influencing energy output stability. The study provides theoretical support and practical guidance for the design of precision laser systems.

ObjectiveIn optical fiber communication, the collimator is a fundamental component with substantial market demand. However, its manufacturing process has long been mired in manual or semiautomatic stages, resulting in low efficiency, high costs, and poor product consistency. Existing semiautomatic systems employ closed-loop feedback control based on the deviation of production data from predefined targets for global optimization, which leads to issues such as high computational burden, lengthy processing time, and sensitivity to production parameters. This study proposes a methodology that combines classical Gaussian beam propagation theory with nonlinear fitting, to swiftly achieve the target spot diameter during collimator fabrication.MethodsClassical Gaussian beam propagation theory was employed to perform nonlinear fitting on production process data. The independent variable was the distance between the optical fiber and lens, and the dependent variable was the detected collimator output spot diameter at the measurement position. Parameters such as lens length (L), fiber mode field radius (ω0), lens curvature radius (R), and lens refractive index (n) of the collimator were determined through fitting, deriving the distance between the optical fiber and lens corresponding to the target spot diameter in a single step. The nonlinear fitting of production data utilized the Levenberg?Marquardt algorithm. Considering the adaptability issues of different collimator models and the difficulty of obtaining ideal target values in practice, this study incorporates the actual measured spot diameters into subsequent rounds of nonlinear fitting to progressively approach the target, thereby enhancing the adaptability and efficiency of the algorithm.Results and DiscussionsClassical Gaussian beam propagation theory was employed for the nonlinear fitting of the production process data. Comparative testing with commercial systems, as depicted in Fig. 3, shows that the rolling fitting LM system optimizes within 1?2 steps on average to achieve the target spot diameter size, except for the fixed search points, thereby reducing the number of optimization steps by an average of 57% compared to commercial systems. As shown in Fig. 4, when faced with different collimator parameters, the rolling fitting LM system stabilizes the production of fiber collimators at an average of 18.3 s after switching the fiber parameters, with a standard error of 1.2 s, reducing the production time by 43.7% compared to commercial systems. The average spot diameter error is -0.14%, for a spot diameter reduction error of 1.0%. As illustrated in Fig. 5, even with changes in detection distance z, the algorithm maintains stable production time of 18?19 s, with the average accuracy of optimized spot diameter fluctuating within ±0.3% and the overall error fluctuation not exceeding ±2%. These results demonstrate the robustness and high consistency of the algorithm, indicating its strong resilience and minimal error susceptibility to product nonuniformity.ConclusionsIn this study, a system of nonlinear equations was established based on classical Gaussian beam formulas and actual measurement data, thereby formulating an objective function using the least-squares method and introducing a rolling optimization feedback mechanism to enhance the traditional LM algorithm. This allows the system to rapidly converge while intelligently searching for the target spot diameter, which differs significantly from traditional methods in imparting clear physical significance rather than a simple point-by-point search or abstract mathematical polynomial approximations, thereby significantly reducing the production time of the fiber collimators. The rolling fitting LM system required an average of five optimization steps with an average production time of 18.3 s per collimator. The average accuracy of optimized spot diameter was -0.2%, with an error fluctuation of ±1.1%. Compared to commercial systems, the spot diameter error range was reduced by 1.0%, ensuring higher quality assurance for fiber collimators. More importantly, the rolling fitting LM system reduced the production time for fiber collimators by over 43.7% compared to commercial systems, significantly enhancing automation efficiency. The system demonstrated robustness in consistently optimizing the steps, optimization time, and spot diameter error rates across different parameter configurations of fiber collimators, providing a method of significant reference value for optimization in fiber collimator industry.

ObjectiveAdaptive optics technology provides core assurance for high-resolution optical systems by real-time detecting and correcting dynamic wavefront distortions. Since the concept of adaptive optics was first proposed by Babcock in 1953, this technology has been successfully extended to fields such as astronomical observation, laser beam purification, and assisted medical treatment, achieving a significant improvement—approximately 2-fold—in indicators like the concentration of far-field spot energy (β factor). However, constrained by high-order residual aberrations, existing systems still struggle to achieve near-diffraction-limited performance. Research indicates that such residual aberrations mainly stem from high-frequency, high-order aberrations induced by vibrations of the optical platform—vibrations caused by components such as coolers, fans, etc. For instance, Rousset et al. found that high-frequency narrowband aberrations induced by vibrations of electronic components in the very large telescope (VLT) lead to a decrease in the Strehl ratio of imaging spots; Kulcsár et al. further revealed that 55 Hz narrowband disturbances excited by the cooling system of the Gemini South Telescope (GST) exhibit significant coupling effects in Zernike high-order modes (e.g., defocus and coma), severely impairing the system’s imaging quality. Effectively suppressing such high-frequency, high-order aberrations has become a key challenge in enhancing the performance of adaptive optical systems.MethodsTo achieve effective correction of high-frequency, high-order aberrations, this paper proposes a linear quadratic Gaussian (LQG) control architecture based on full-mode prediction. This method innovatively introduces a full-mode wavefront aberration coefficient prediction model. Firstly, dynamic equations for Zernike modes are established through online spectral analysis and parameter identification. Secondly, the Kalman filter is employed to perform multi-step-ahead prediction of full-mode coefficients; when combined with optimal voltage calculation, this forms a closed-loop correction mechanism, which can enhance the adaptive optical system’s capability to correct high-frequency, high-order aberrations. It is expected to achieve near-diffraction-limited correction for such aberrations.Results and DiscussionsTo verify the superiority of the proposed method over proportional-integral (PI) control in correcting high-frequency, high-order aberrations, comparative experiments between the LQG and PI control methods were conducted on 25 Hz defocus aberrations (Fig. 3). When the PI control method was used to correct 25 Hz defocus aberrations, the correction effect was not significant: the root mean square (RMS) value of the input aberration only decreased from 0.371 μm (before correction) to 0.189 μm, and the peak-to-valley (PV) value decreased from 1.949 μm to 0.850 μm. In contrast, the LQG control method exhibited excellent correction performance for 25 Hz defocus aberrations: the RMS value of the input aberration significantly decreased from 0.554 μm (before correction) to 0.056 μm, and the PV value decreased from 2.271 μm to 0.295 μm (Table 1).To further validate the method’s effectiveness in correcting higher-frequency and higher-order aberrations, correction experiments were conducted on 180 Hz defocus and coma (Fig. 4). Under a sampling frequency of 500 Hz, using the proposed method to control deformable mirror (DM2) reduced the input aberration to nearly 1/10 of its initial value (Tables 2 and 3).To confirm the LQG method's correction effect on mixed high-frequency, high-order aberrations, experiments were performed on mixed high-order aberrations (180 Hz defocus, 180 Hz astigmatism, and 180 Hz coma) (Fig. 5). The LQG method achieved a spectral suppression amplitude of more than 23 dB for all mixed high-frequency, high-order aberrations (defocus, astigmatism, and coma) (Table 4).ConclusionsExperimental results demonstrate that the LQG modal coefficient prediction correction method exhibits excellent suppression effects on both single-order and mixed-mode aberrations. It can extend the control bandwidth of adaptive optical systems to more than 1/3 the sampling frequency, overcoming the traditional limitation of 1/20 the sampling frequency. This holds promise for achieving near-diffraction-limited correction of high-frequency, high-order aberrations in adaptive optical systems.

ObjectiveHigh-power fiber lasers are widely used in national defense and scientific research because of their good beam quality, small size, and excellent thermal management. However, increasing the output power of single fiber lasers is challenging because of the limitations of nonlinear effects, thermal effects, and mode instability. Spectral beam combining (SBC) has emerged as a promising technology to boost the output power while maintaining good beam quality. However, as laser arrays expand, SBC systems become increasingly large, limiting their practical applications. Therefore, miniaturizing SBC systems is crucial for enhancing their applicability. SBC based on conical diffraction offers a viable solution—by directing the incident beam at an oblique angle onto the diffraction grating surface. The diffracted beams are separated in different planes, spatially separating the incident and diffracted beams. This approach facilitates the miniaturization of SBC systems and enables novel optical path designs. This study systematically investigated beam quality degradation in SBC systems based on conical diffraction and experimentally examined the grating diffraction efficiency under conical diffraction conditions. We hope that our investigations and findings will aid in the design and study of SBC systems based on conical diffraction.MethodsBased on the fundamental principles of conical diffraction and the grating equations, we parameterized the diffraction angles using the incident azimuth and polar angles. We established a dispersion model for conical grating diffraction by discretizing a laser beam with a finite linewidth into monochromatic sub-beams and applying the conical diffraction grating equation. Numerical simulations were performed to investigate the diffracted beam characteristics after single-beam conical diffraction through the grating, focusing on the evolution of beam spot patterns and beam quality degradation. For experimental validation, a diffraction efficiency measurement setup was developed using an 1170 line/mm multilayer dielectric diffraction grating and a 1083 nm fiber laser. The grating rotation enabled precise control of the conical diffraction conditions. The experiment examined the variation in the grating diffraction efficiency with the incident polar angle at different azimuth angles and the effect of the incident azimuth angle on the diffraction efficiency under near-Littrow-angle incidence conditions.Results and DiscussionsThe incident azimuth angle affects the diffraction efficiency of the grating. However, high efficiency is maintained within a certain range. The diffraction efficiency begins to decline as the azimuth angle exceeds a specific threshold. When the azimuth angle is maintained within 17.5°, the conical diffraction grating maintains a high efficiency over 95% (Fig. 11), which slightly affects the beam combining system. In contrast, the incident polar angle exhibits a more significant effect on the diffraction efficiency. Under nonconical diffraction conditions, the grating efficiency drops below 90% (Fig. 7) when the incident angle is 42.64° (only 3.33° deviation from the Littrow angle). Numerical simulations reveal that when the azimuth angle is within 20°, the effect on the beam quality is relatively small, resulting in a beam quality degradation factor of less than 1.07 (Fig. 16). For a 5 mm beam waist radius at a 20° azimuth angle, the spectral width must remain below 25 GHz to maintain M2 factor below 1.5 (Fig. 15). When the azimuth angle is less than 20°, the spectral width (10?100 GHz) and beam waist radius (2?30 mm) exhibit a more significant impact on beam quality. Therefore, in conical diffraction based SBC systems, a properly selected azimuth angle does not significantly affect either the combined efficiency or beam quality.ConclusionsIn this study, we systematically examine the impact of conical diffraction on SBC systems, focusing on combining efficiency and beam quality through experimental and theoretical approaches. The experimental results demonstrate that polarization-independent gratings achieve maximum diffraction efficiency near the Littrow angle for conical and nonconical diffraction configurations, exhibiting similar efficiency trends across different polar angles. However, as the azimuth angle increases, the peak efficiency shifts toward angles smaller than Littrow angle. Under conical diffraction conditions, the diffraction efficiency remains above 95% when the azimuth angle remains below 17.5°. In contrast, nonconical diffraction decreases below 90% diffraction efficiency, with a polar angle deviation of only 3.33° from the Littrow angle. This indicates that the azimuth angle variation has less influence on diffraction efficiency than the polar angle deviation from the Littrow angle. The experimental results further demonstrate that conical diffraction configurations achieve a slightly higher maximum diffraction efficiency than nonconical diffraction configurations when the incident angle is near the Littrow angle. To examine this effect, we established a conical diffraction dispersion model by analyzing the rotation of far-field elliptical spots with varying azimuth angles. Through irradiance distribution fitting and systematic simulations, we characterized the beam quality degradation after single-beam diffraction and evaluated the effects of the spectral width, beam radius, and azimuth angle on the beam quality. The results show that azimuth angles within 20° have a minimal effect on beam quality, with a degradation factor below 1.07. In conical-diffraction-based SBC systems, the appropriate selection of the azimuth angle not only avoids significant effects on the combined diffraction efficiency and beam quality but also introduces new degrees-of-freedom for optical path design. These findings help develop compact and lightweight SBC systems.

ObjectiveIn a fiber array spectral beam combining (SBC) system, imperfections such as installation deviation of the fiber array, linewidth broadening, and thermal effects of the optical elements and internal optical channel coexist and interact with each other, leading to degradation of the combined beam quality. Specifically, the displacement deviation, rotation angle, linewidth broadening, and divergence angle of the fiber laser emitters independently affect the combined beam, leading to non-common-path phase perturbations in each sub-beam. In addition, thermal effects of the diffraction grating and internal optical path introduce aberrations, collectively referred to as common-path thermal aberrations. As the non-common-path phase perturbations and common-path thermal aberrations increase, the degradation of the combined beam quality inevitably becomes more severe. To date, an in-depth analysis of the comprehensive degradation effects and correction characteristics of the combined beam quality has been insufficient. Therefore, studying the integrated degradation and correction characteristics of the combined beam quality under these imperfections, particularly with adaptive optics (AO), is crucial for the effective management and control of beam quality in SBC systems.MethodsThis study presents a theoretical and numerical analysis aimed at improving the beam quality of spectrally combined fiber lasers using adaptive optics. First, a physical model is established for the fiber SBC system for the imperfect factors. It consists of four components: an AO-based SBC system, an optical transmission model, a physical model for the thermal deformation of the multilayer dielectric grating (MDG), and a multifield coupling interaction model that describes the light-fluid-solid interactions (LFS-MFCI) within the internal optical path. The AO-based SBC system is primarily composed of three elements: the SBC, expanding laser beam, and AO systems. Ray tracing and diffraction integral methods are employed to develop and solve the optical transmission model for the SBC system. Additionally, finite element models are constructed for the MDG and LFS-MFCI. This enables us to simulate and analyze the temperature distribution and thermal deformation in the MDG and the internal optical path after 60 s of irradiation at an initial temperature of 20 ℃ and a power density of 1000 kW/cm2. Subsequently, the degradation mechanisms and characteristics of the combined beam quality are investigated by categorizing the aberrations within the SBC system into two types: noncommon-path phase perturbations and common-path thermal aberrations. Finally, we discuss strategies for enhancing the spectrally combined beam quality using adaptive optics and simulations.Results and DiscussionsIn the presence of imperfections such as displacement deviation, rotation angle, linewidth broadening, and divergence angle, the intensity distribution of the combined beam irradiating the multilayer dielectric grating displays a Gaussian-like profile that varies with the effect degree of these imperfections. Assuming an initial temperature of 20 ℃ and a power density of 1000 kW/cm2 for the incident laser, both the temperature and thermal deformation gradually decrease from the center toward the edge of the MDG after 60 s irradiation, approximating a Gaussian-like distribution. Moreover, the maximum temperature and thermal deformation of the MDG demonstrate a nearly linear increase with increasing incident laser power density (Fig. 5). In the axial cross section of the optical transmission, the temperature distribution and heat source distribution of the flow field within the internal optical path align with the Gaussian-like distribution of the laser irradiation. The peak values of the temperature and heat source are situated at the center of the optical path and reflector, respectively (Fig.7). The accumulated optical path difference induced by thermal effects in the gas is the primary contributor to the optical path differences, which significantly exceeds the thermal deformation on the surface of the reflector (Fig.8). Under the influence of non-common path phase perturbations and common path thermal aberrations, the far-field β factor of the combined beam increases, indicating deteriorating beam quality. After correction with the AO system, the far-field β factor decreases and approaches 1, demonstrating a significant improvement in the quality of the combined beam (Fig.10). Under the specific boundary conditions considered, the rotation angle of the fiber array is identified as the primary factor affecting output beam quality. When the variance of the rotation angle exceeds 1.5 mrad, the corrected far-field β factor remains higher than 1.5, indicating that the beam quality is still inadequate. Additionally, the degradation of the beam quality caused by the common-path thermal effects and optical path is comparatively less severe than that caused by non-common-path phase perturbations. The thermal effects are nearly uniform across each subbeam in the SBC system, making the correction through AO more manageable. The AO system exhibits superior correction capabilities for low-order common path aberrations resulting from thermal effects as opposed to high-order aberrations, thereby significantly enhancing the combined beam quality of the SBC system (Fig.12).ConclusionsThis study investigates the degradation and correction characteristics of the fiber array SBC system based on the established physical model of the fiber array with imperfect factors. The results indicate that both non-common-path phase perturbations and common-path thermal aberrations degrade the combined beam quality, which can be corrected by the AO system. Notably, AO demonstrates better correction capabilities for common-path thermal aberrations than for non-common-path phase perturbations. However, the combined beam quality after correction with the AO is still not sufficient for practical applications, particularly when the degradation of the combined beam quality is severe because of aberrations in the SBC system. Therefore, the control and management of the fiber array and other optical elements are crucial for reducing noncommon-path phase perturbations and alleviating the effects of common-path thermal aberrations within the SBC system.

ObjectiveVector vortex beams (VVBs), as a new type of structured light with anisotropic spatial polarization and phase, carry orbital angular momentum related to the phase distribution and possess potential application values in fields such as optical manipulation, high-resolution imaging, and optical information transmission. Currently, there are scarce studies on the correlation between the polarization and phase topological charges of VVBs. Additionally, in contrast to scalar vortex beams, research on the interference and diffraction measurement of the orbital angular momentum of VVBs is relatively rare. This paper explores the correlation between the polarization and phase topological charges of VVBs, realizes the determination of topological charges through polarization methods, and investigates the interference and diffraction measurement approaches of the orbital angular momentum of VVBs, which is anticipated to broaden the thinking for the measurement and application of the orbital angular momentum of VVBs.MethodsBased on the field distribution characteristics of VVBs with different polarization orders and phase topological charges, this paper studies the correlation issue between the polarization order and phase topological charge values of VVBs and analyzes the influence of changes in the azimuthal angle and ellipticity angle on the polarization state and orbital angular momentum of VVBs. The orbital angular momentum measurement method of VVBs is analyzed by exploiting the characteristics of double-slit interference fringes, and the diffraction effects of VVBs with different aperture diaphragms are simulated and calculated. By comparing the diffraction spot shapes of different apertures, parameters including the phase topological charge and polarization order are derived, and the orbital angular momentum measurement method of VVBs is studied.Results and DiscussionsThe innovative results of this paper are as follows:1) According to the intensity and phase distributions of VVBs with different polarization orders and phase topological charges, the relationship of the light intensity and phase with the polarization order and phase topological charge values is analyzed. It is discovered that the phase topological charge and polarization order of VVBs respectively affect the sizes of the phase singularity and polarization singularity of the beams. The numbers of peripheral and central bifurcations of the beam phase distribution correspond to the polarization order and phase topological charge (Figs. 1?3). The ellipticity angle and azimuthal angle in the polarization matrix term of the vector vortex field have an impact on the polarization distribution of the field, the ellipticity angle affects the distribution range of light intensity, and the azimuthal angle causes a related rotation in the distribution direction of polarization state (Figs. 4?7).2) A double-slit interference simulation experiment of VVBs is carried out. It is found that when the polarization order and the absolute value of the phase topological charge are equal, the central dark core disappears. When the phase topological charge is l, the width of the interference fringe distortion is l stripe, and the direction of the interference fringe distortion is also related to the phase topological charge. When the phase topological charge is positive, the fringes twist to the right, and when the phase topological charge is negative, the fringes twist to the left. Moreover, the double-slit interference of VVBs with different polarization orders changes the size of the polarization singularity (Fig. 8).3) Through the diffraction calculation of VVBs after passing through single slit, right-angle triangular aperture, etc., the relationship of the diffraction pattern morphology with the phase topological charge value and polarization order of VVBs is compared and analyzed. It is found that when the polarization orders and phase topological charges of VVBs after passing through right-angle triangular hole diaphragms are simultaneously opposite numbers, the diffraction spot undergoes a phase shift related to the absolute value of the polarization order (Figs. 9?14).ConclusionsThe research findings indicate that the polarization order and the phase topological charge value jointly determine the intensity and phase distributions of VVBs. During the transmission process of VVBs under different polarization conditions, the vortex phase distribution remains relatively stable, and the alteration of the ellipticity angle leads to the difference in the polarization state distribution. Based on the regular changes in the fringe distribution after interference, the polarization order and the phase topological charge value of VVBs can be determined. When the VVBs pass through the right-angle triangular aperture diaphragm, the diffraction spot undergoes a phase shift related to the absolute value of the polarization order only when the polarization orders are opposite to each other, which suggests that single-slit diffraction is more conducive to the determination of the polarization order value, while right-angle triangular aperture diffraction is more effective for the determination of the phase topological charge.

ObjectiveLaser beam propagating in atmosphere is affected by many effects such as turbulence, diffraction, self-focusing, and thermal blooming, which is an extremely complex physical process. Although the numerical simulation method can be used to predict the effects of different atmospheric conditions and beam parameters on laser propagation, its computational efficiency is insufficient for practical applications. Therefore, the study of scaling law becomes essential. However, on the one hand, most studies on the scaling law of laser atmospheric propagation focus on beam spreading. On the other hand, the thermal blooming effect causes the focus shift of a focused Gaussian beam, and the actual focus point is shifted to the position in front of the target plane, which severely degrades the beam quality at the target plane. In this paper, to achieve the maximum peak intensity at the target plane, the scaling law for the optimal focal length of a focused Gaussian beam under thermal blooming effect is investigated.MethodsThermal blooming effect of a laser beam propagating through atmosphere can be described by the paraxial wave equation and the hydrodynamic equation. A time-dependent four-dimensional code to simulate the focused Gaussian beam propagating in atmosphere is developed by using the multi-phase screen method, fast Fourier transform method, and difference method. In this paper, the method of pre-defocusing is employed to obtain the maximum peak intensity at the target plane. To predict the optimal focal length fopt under varying atmospheric conditions and beam parameters, this paper employs the controlled variable method to numerically analyze the effects of six scaling factors (initial laser power P, atmospheric wind speed v, atmospheric absorption coefficient α, initial beam width w0, target plane distance z0, and wavelength λ)on fopt. Through physical analysis and extensive numerical calculations, a formula of the optimal focal length fopt relating with the six scaling factors is established.Results and DiscussionsIn this paper, the method of pre-defocusing is employed to determine the optimal focal length of the lens, thereby achieving the maximum peak intensity at the target plane (Figs. 1 and 2). First, three scaling factors (P, v, and α) are considered , which significantly affect the intensity of thermal blooming effect. The generalized thermal distortion parameter N is a physical quantity used to describe the intensity of thermal blooming effect. A scaling law between fopt and N is investigated [Fig. 3 and Eq. (10)]. In contrast, when the other three scaling factors (z0,w0,andλ) are changed, both the intensity of the thermal blooming effect and the intensity of the diffraction effect are altered (Figs. 4?6). Therefore, it is necessary to modify Eq. (10). The scaling law of fopt relating with N, z0 and w0 is modified on the basis of Eq. (10) [Fig. 7 and Eq. (13)]. This scaling law can predict the optimal focal length under various atmospheric conditions and beam parameters. When N<11 and Fresnel number NF>4, this scaling law is applicable, and the mean relative error is 5.2% [Fig. 8 and Eq. (13)]. Additionally, it is shown that the increase of peak intensity at the target plane after focusing is greater when N is larger (Fig. 9).ConclusionsIn this paper, a scaling law for the optimal focal length of a focused Gaussian beam considering atmospheric thermal blooming effect is studied. The method of pre-defocusing is employed to determine the optimal focal length of the lens, thereby achieving the maximum peak intensity at the target plane. Using the controlled variable method, the influence of six scaling factors (P, v, α, z0, w0, andλ) on the optimal focal length fopt is obtained by varying them, respectively. Then, the scaling law of fopt relating with the generalized thermal distortion parameter N, the target plane distance z0 and the initial beam width w0 is established. This scaling law is applicable when N<11 and Fresnel number NF>4. This scaling law can predict the optimal focal length under various atmospheric conditions and beam parameters, and the mean relative error is 5.2%. Additionally, it is shown that the increase of peak intensity at the target plane after focusing is greater when N is larger. It is worth noting that laser atmospheric propagation is a very complex physical process. In practical application, the beam quality at the target plane is also related to the beam quality at the source plane, beam truncation ratio, tracking accuracy, and atmospheric turbulence. The results obtained in this paper are useful for the application of laser atmospheric propagation.

ObjectiveHigh-power fiber lasers and their sub-beam combining technology are effective for achieving high-brightness and high-power laser outputs. With the continuous improvement of output laser power and energy concentration in the research and development of high-power laser system engineering, multiple sub-beam lasers are required to operate simultaneously for system testing. The output sub-beam laser exhibits high power density, a small divergence angle, and strong destructiveness. To prevent damage to the test environment and ensure personnel safety, a laser absorption device is necessary to effectively absorb and control multi-channel high-power-density laser energy. This ensures a high-efficiency, pollution-free, and safe testing process. When a multi-channel high-power-density laser is incident on the absorber, light-field coupling can cause excessive local temperature rise inside the absorber, leading to laser melting, structural deformation, and debris attachment. These issues pollute the optical environment and damage both the absorber and the laser output end. Additionally, excessive reinjection of laser return power into the laser causes the laser to burn out. To address the effective control requirements of multi-channel high-energy lasers in high-power laser system development, a laser absorption device is designed for the test system, and its thermo-optical characteristics are analyzed through simulations and experimental studies.MethodsA novel three-channel high-power-density laser beam absorption device is designed. Unlike single-aperture absorption devices, multi-beam lasers exhibit a small divergence angle, a small spot size, high power density, and independent distribution. A discrete light-cone arrangement is employed for single-channel reflection beam expansion and multiple coupling absorption. Combined with an inner surface absorption coating and a surrounding extinction microstructure, the multi-aperture structure can independently absorb sub-beam laser energy simultaneously. Additionally, the fully sealed design of the absorption cavity allows for internal gas replacement through charging and exhaust mechanisms, ensuring the cleanliness and purity of the internal medium atmosphere while maintaining the safety of the light output. Based on beam tracing analysis, combined with the parameters of the beam-expanding cone, the absorber substrate material, the coating absorption coefficient, and the surface microstructure, the light field distribution on the internal absorption surface after coupling and superposition of the three-beam laser fields is simulated. The distribution pattern of the peak intensity and position of the internal light field, as well as the influence of the cone-tip fillet on internal light field distribution, is analyzed. The temperature rise in various regions inside and outside the absorber is quantitatively calculated using laser irradiation at 10.4 kW for 135 s. A three-beam high-power light output test is conducted to verify absorption temperature rise, anti-damage performance, and return power.Results and DiscussionsThe new laser absorber is studied using simulation analysis and experimental testing. The simulation results show that the sub-beam laser power is 3.5 kW, and the three-beams emit light simultaneously at a total power of 10.5 kW. After passing through the expanding light cone, the optical power density on each absorption surface inside the absorber is effectively attenuated. Following beam coupling, the light field is superimposed onto the absorption area. The absorbed power in the light cone area is 5418.2 W, while the sidewall absorbs 3495 W (Fig. 4). The fiber end cap installation surface absorbs 5.05 W, and the optical aperture of the fiber end cap absorbs 0.06 W (Fig. 7). The radius of the cone tip significantly influences the laser power distribution on the bottom and side surfaces but has little effect on the return power (Fig. 8). At an ambient temperature of 20 ℃, when the three-beam laser operates at 10.4 W@135 s, the highest internal temperature of 131.667 ℃ is observed near the light cone (Fig. 5). The highest external temperature of 74.4 ℃ is recorded outside the heat insulation board in the top absorption area (Fig. 6). A high-power light output test is conducted. For a single laser output of 3.5 kW@135 s, the maximum temperature rise at the end cap is 5 ℃, and the maximum temperature rise at the front end is 15.8 ℃ (Fig. 12). When the three-beam laser operates at 10.4 W@135 s, the maximum external surface temperature of the absorber reaches 79.1 ℃, with a temperature rise of 59.1 ℃, occurring on the sidewall of the absorber ring. The highest temperature location relatively aligns with the simulation results (Fig. 13). The detected return light power is 130 mW, and the return light throughout the entire system remains within the safety threshold (Fig. 11).ConclusionsTo address the challenges of multi-beam output lasers with small beam diameters, high power densities, and small divergence angles, a new three-channel high-power-density laser absorption device is designed. The device incorporates optical cone beam expansion, cavity multiple coupling absorption, and stray light suppression. Using the beam-tracing method, simulations and experimental studies are conducted to analyze the laser field distribution inside the absorber, the influence of return power, the effect of cone tip radius, and the temperature rise distribution during the light output process. The temperature distribution of the absorber aligns with the simulation results. The full-link laser return power remains below the safety threshold, and the absorption temperature rise, anti-damage capability, and laser return power suppression effects are successfully verified. This research provides valuable insights for the efficient absorption of multi-channel high-power-density lasers, the design of measurement devices, and the studies on internal anti-damage mechanisms and return-light suppression. The findings can be extended to the development of similar applications in high-power laser systems.

ObjectiveTo further decrease lithographic resolution, specific illumination modes are required for different mask patterns. The freeform pupil-shaping module is a standard component of lithography illumination systems at the 28 nm node and below. To ensure stable and uniform energy distribution at the optical pupil, a homogenization unit is used before the freeform pupil-shaping module. The current mainstream solution involves using a microlens array (MLA) for homogenization. The MLA divides the incident beam into a multitude of sub-beams, which are superimposed on the back focal plane of the condenser to obtain a homogeneous illumination field. Although a low-spatial-coherence excimer laser is used in lithography, it still produces interference patterns in the back focal plane of the condenser, significantly impacting pupil performance. In this paper, we propose a decoherence method based on the deformation design of MLAs that improves the uniformity without introducing additional elements.MethodsIn this paper, we propose a new design method for decoherent MLAs. First, considering the partial coherence of a laser, transmission of partially coherent light in the homogenization unit is modeled using mutual intensity theory. Based on the mutual intensity distribution in the back focal plane of the condenser, it can be concluded that the intensity distribution of the light field is modulated by the coherence length and related to the aperture function P(?,η). Changing this function not only affects the initial phase φ(m,n) of each sub-beam but also changes the frequency of sinc function. As a consequence, φ(m,n) affects the interference results and sinc function affects the amplitude modulation. Based on the aforementioned model, an optimized design of the mean value of the pitch size of the MLA was carried out. To increase the phase difference between the microlenses within the coherence length, degrees of freedom for pitch variation and border line tilting were introduced based on the optimized design described above.Results and DiscussionsFirst, the mean value of the pitch size of the MLA was optimized. The design results (Figs. 5 and 6) show that when the microlens pitch size is 0.5 mm, non-uniformity is at a minimum of 43.87% and energy utilization is 85.39%. While this optimization solution satisfies the energy utilization requirement of the freeform pupil-shaping module, it is not able to achieve a non-uniformity less than 40%. Introducing a small random pitch variable in the pitch size of the microlens, the design results (Figs. 8 and 9) show that non-uniformity is minimum at 41.53% and energy utilization is 84.16% at the amplitude of the pitch variation of 0.05 mm, which also satisfies the requirement for energy utilization. However, non-uniformity does not fulfill the aforementioned requirement yet. Making the microlens at the cylindrical border line in the xy plane introduces a tilt factor k, which is a random small quantity. The design results (Figs. 11 and 12) show that when the amplitude of the tilt factor is 5 μm, non-uniformity is at a minimum of 38.67%, and energy utilization is 82.74%, thereby fulfilling specifications. Combining the optimization solutions of pitch size variation and border line tilting, non-uniformity of 36.51% (Fig. 14) and energy utilization of 83.90% are achieved when the amplitude of the pitch variation is 0.03 mm and that of the tilting factor is 4 μm. This co-optimization approach results in a more significant reduction in non-uniformity, along with improved energy utilization.ConclusionsIn present paper, we propose a decoherence method based on the deformation design of MLA to improve homogeneity without introducing additional elements. First, the mean value of the pitch size of the MLA was optimally designed. Subsequently, degrees of freedom for pitch variation and border line tilting were introduced. The simulation results show that with the optimal combination of suitable pitch variation and border line tilting, non-uniformity and energy utilization reach 36.51% and 83.90%, respectively, which satisfy the requirements for the use of a freeform pupil-shaping module.

ObjectiveHigh-power lasers are widely used in various fields for industrial, scientific, and military applications. Generally, the intensity distribution of a laser beam is not uniform and generally exhibits a Gaussian distribution, which may lead to material damage during laser processing owing to the uneven energy distribution. Different application fields have different demands for the spot shape and intensity distribution of laser beams. Recently, flat-topped laser beams with a uniform distribution of beam intensity have become commonly used, with a wide range of applications in material processing, semiconductor substrate annealing, optical holography, and laser lighting. A flat-topped beam can be obtained using beam shaping technology, and common beam homogenization technologies include the light field mapping method and beam integration method. Light field mapping is realized using an aspheric lens group, a birefringent lens group, and diffractive optical elements, which are suitable for single-mode laser light sources. Beam integration is mainly performed using mirror arrays, prism arrays, and microlens arrays, which are particularly suitable for excimer lasers, multi-mode lasers, or laser light sources with irregular light intensity distribution. The microlens array homogenization system is generally wavelength insensitive, and the output spot shape is modulated by the sub-lenses. It is widely used owing to its simple structure, high damage threshold, and low transmission loss.MethodsBased on the superior properties of microlens arrays, a beam homogenization and shaping system based on cylindrical microlens arrays was designed. The microlens arrays were placed orthogonally to homogenize and shape the vertical and horizontal directions of a Gaussian circular beam, respectively, achieving a square beam output with a near-flat-top intensity distribution. Based on the theories of matrix optics and Fourier optics, the light transmission mode was analyzed, the structural parameters of the microlens array were optimized using Zemax software, and a simulation model was constructed to shape the homogenization effect of the system. The research system was established using an experimental platform. First, one pair or two pairs of orthogonal microlenses were employed to compare the Gaussian beam homogenization. With two pairs of cylindrical microlens arrays, the effect of the incident Gaussian beam diameter on the uniformity and size of the homogenized light spot was investigated experimentally and theoretically. When the input laser beam had a diameter of 8 mm, the relationship between the output spot shape and the interval distance of the microlens arrays was analyzed. Moreover, the effect of laser beam quality on homogenization and shaping was examined for the cylindrical microlens array system.Results and DiscussionsBased on the simulation result of Zemax software, the structural parameters of the microlens array are optimally designed with a sub-lens aperture size of 500 μm and a focal length of 5.4 mm. Compared with the use of one pair of cylindrical microlens arrays to homogenize and shape the horizontal and vertical directions, the experimental and theoretical results show that the Gaussian circular beam can be better homogenized with a more uniform energy distribution by utilizing two pairs of cylindrical microlens arrays placed orthogonally. As the size of the incident Gaussian beam increases, the uniformity of the homogenized spot increases and the sharpness of the spot edges decreases (Figs. 2 and 3). By controlling the interval distance between the microlens arrays, square beams with adjustable spot shapes and sizes and a near-flat-top distribution of light intensity can be obtained. With an incident spot diameter of 8 mm, homogenized output spots with adjustable beam aspect ratios such as 100 mm×100 mm squares, 100 mm×130 mm rectangles, and 130 mm×130 mm squares were successfully obtained (Fig. 4). In our case, the size of the spots increases with the transmission distance of the homogenized beams; however, the corresponding uniformity shows little change. The homogenization shaping system is flexible and versatile on a spatial scale, and it can better meet practical applications in scientific research and production. The microlens array system is insensitive to the beam quality of the incident laser, which makes it especially suitable for homogenizing and shaping excimer lasers, laser diode arrays, multimode light fields, or laser sources with irregular intensity distributions.ConclusionsIn this study, the physical mechanism and homogenization process of a cylindrical microlens array homogenization system are investigated in depth using a combination of theory and experiments. A discrete structure with two pairs of orthogonally placed cylindrical microlens arrays is designed for beam homogenization and shaping of a circular Gaussian beam in both the horizontal and vertical directions. By controlling the distance between the microlens arrays, a homogenized beam with an adjustable shape and size is obtained, and spot uniformity is maintained. These results open a novel way to realize a uniform square spot with an adjustable spot size and high flexibility in space utilization, which is suitable for practical applications in scientific research and industrial fields.

ObjectiveTarget-tracking control technology is extensively employed in several fields, including aerospace, satellite remote sensing, and laser communication. The Risley-prism system enables the direction of the beam to be altered or the visual axis to be adjusted by controlling the rotation angle of the prism. Compared with alternative mechanical beam-pointing mechanisms, such as gimbal and fast steering mirror mechanisms, the target tracking control system based on Risley-prisms exhibits several advantageous characteristics. These features include a compact structure, high reliability, and high pointing accuracy, which collectively provide the system with a wide range of potential applications. In the case of a traditional beam-pointing mechanism, the relationship between the pose adjustment of the actuator and target motion trajectory is characterized by intuitive linearity. However, there is a nonlinear relationship and strong coupling between the prism rotation angle and visual axis orientation of the Risley prism system, which makes it challenging to accurately determine the prism rotation angle via analytical means. Furthermore, conventional numerical methods have limitations in terms of accuracy and efficiency, which impede the advancement of research and practical applications of the Risley-prism system in target-tracking scenarios. In this study, we report a target-tracking method based on the Risley-prism system using a virtual system, whereby the spatial direction of the outgoing beam of the actual Risley-prism system is mapped. Our basic approach and discoveries provide useful insights into the design of pointing and tracking control systems based on Risley-prisms for time-varying optical targets.MethodsA particle swarm-optimized target tracking method based on a virtual system was employed in this study. First, based on the nonparaxial ray-tracing method, a virtual Risley-prism system was constructed to map the spatial direction of the outgoing beam of the actual Risley-prism system. Subsequently, by combining the virtual system model projected by the actual two-prism system with the particle swarm algorithm, multiple possible prism rotation angles (particles) and their corresponding virtual pointing targets were calculated in parallel. Subsequently, if the estimation error between the outgoing beam-pointing of the virtual Risley-prism system and the target to be tracked was less than the actual error, then the actual prism rotation angle was replaced by the estimated prism rotation angle and applied to the actual Risley-prism system. In the next step, the prism rotation angle that best matches the target to be pointed at and tracked was selected based on the interoperability and information-sharing mechanism of the particle swarm algorithm. In addition, the prism angles of the experimental prototype Risley-prism system were adjusted to realize dynamic target tracking.Results and DiscussionsThe prepared Risley-prism system based on the virtual system with the RPSO algorithm presents comparable performance for static target pointing in numerous simulations, and the final convergence accuracy of the proposed RPSO-based Risley-prism system approaches 5?10 mm (Fig. 6). In addition, when tracking a moving target, the RPSO-based Risley-prism system can converge to the global optimum more quickly than can the PSO-based method, exhibiting a faster convergence speed and higher convergence accuracy (Fig. 8). The results of the simulation analysis show the effect of particle population size on virtual system-based target tracking methods: larger particle populations lead to faster convergence but increased computation (Fig. 9). In the simulation of continuously tracking target points, the estimation error of the virtual system and real error of the Risley-prism system can still converge, indicating that the proposed algorithm still has a stable tracking effect when tracking continuously changing dynamic targets (Fig. 10). The pixel deviation distribution of the 60 target pointing tests demonstrates the excellent performance of the proposed method: the mean pointing error and standard deviation are 9.43 and 10.14 pixel, respectively (Fig. 13). In the static target pointing experiments, the proposed method demonstrates better pointing performance. The fitted circle radius of the pointing error distribution of the proposed method is smaller than that of the two-step method, and the average pointing error, root mean square error, and maximum pointing error of the proposed method are all smaller than those of the two-step method. During the dynamic tracking experiments, the Risley-prism system sequentially achieved the tracking of three targets with a final pixel error of approximately 13.04 pixel, thus demonstrating the excellent performance of the proposed target-tracking method in the application of continuous target tracking. The performance difference in dynamic target pointing tracking shows that the performance of the proposed RPSO-based algorithm is superior to that of the two-step method. The average tracking errors (in pixel) and the root-mean-square (RMS) tracking errors of the two algorithms are as follows: 10.64 pixel and 11.22 pixel (two-step method) and 8.113 pixel and 9.429 pixel (proposed method), respectively.ConclusionsThis study successfully develops a new Risley-prism system-based target tracking method by introducing a combination of particle swarm optimization and a virtual system into an actual Risley-prism system. The particle swarm method is used to adjust the Risley prism angle and achieve target tracking in the Risley-prism system. To maintain a certain degree of correlation between the virtual and actual systems, a virtual target is constructed based on the deviation of the center of the camera field-of-view from the center of the actual target to be tracked in the x- and y-directions. The error feedback information used to estimate the prism angle in the virtual system is consistent with the tracking error fed back from the actual system, and the prism angle is calculated based on the dynamic changes of the target to be tracked. The simulation and experimental results demonstrate the feasibility of the method for achieving target tracking. In the static target experiments, the average pointing error and standard deviation are 9.43 pixel and 10.14 pixel, respectively, whereas in the dynamic target tracking experiments, the average tracking error is approximately 16 pixel at the three key positions. The proposed method provides a promising method for realizing the target pointing and dynamic target tracking of rotating Risley-prism systems with a wide range of applications.

ObjectiveThermal blooming severely reduces beam quality and limits the efficiency of high-energy laser transmission in the atmosphere. Therefore, it is important to study the thermal blooming of high-energy laser propagation in the atmosphere systematically. Numerical simulation of thermal blooming is beneficial for the application of high-energy lasers. Numerical simulation methods for thermal blooming include perturbation, integral, and phase screen methods. However, most existing studies used only one of them for simulation, which led to obvious errors under some conditions. In this study, we compared the results of three methods and selected an appropriate numerical simulation range for each method based on the reported experimental results. In addition, field experiments for measuring the transmission of high-energy lasers in the atmosphere are complicated, and the experimental conditions are uncontrollable. Therefore, using a liquid crystal spatial light modulator (LC-SLM) based on the transmission characteristics of high-power lasers in the atmosphere is a valuable and cost-saving method for laboratory simulations of thermal blooming. We extracted the thermal blooming distortion phase based on the principle of the numerical simulation method and then used an LC-SLM to simulate thermal blooming in the laboratory. The experimental simulation results are consistent with the numerical simulation results.MethodsIn this study, numerical simulations of thermal blooming were performed using the perturbation, integral, and phase screen methods. First, the results of different numerical simulation methods for different transmission distances were compared. The results obtained using the different numerical simulation methods were significantly different, even under identical conditions. The relative peak intensity IREL (ratio of maximum received light intensity to maximum initial light intensity) was measured as a function of the generalized distortion parameter N in the reported reference. A reasonably fitted curve was then selected as the reference data. The reasonable ranges for each method were determined by comparing the three numerical simulation results with the reference results. Subsequently, the thermal blooming distortion phase was extracted according to the applicable numerical simulation method under the conditions set for the laboratory experiments. Finally, laboratory experiments on thermal blooming were conducted using LC-SLM.Results and DiscussionsThe relationship between the normalized peak intensity IREL generated by the three numerical simulation methods and the generalized distortion parameter N is compared with the reference data at different transmission distances (Fig. 3). The errors of the perturbation and integral methods are small when N<4.8 (the errors of the two methods are approximate), and the error of the phase screen method is small when N>4.8. The relationship between the normalized peak intensity IREL generated by the three numerical simulation methods and the generalized distortion parameter N is compared with the reference line at different initial laser powers (Fig. 4). The error of the perturbation method is small when N<3, that of the integral method is small when 3<N<4.8, and that of the phase screen method is small when N>4.8. The relationship between the normalized peak intensity IREL generated by the three numerical simulation methods and the generalized distortion parameter N is compared with the reference line at different wind speeds (Fig. 5). The error of the perturbation method is small when N<2.8, that of the integral method is small when 2.8<N<4.8, and that of the phase screen method is small when N>4.8.ConclusionsIn this study, various ranges of generalized distortion parameter N applicable to each numerical simulation method are selected by comparing the error between the normalized peak intensity IREL generated by the three numerical simulation methods and the reference line under different setting conditions. When the generalized distortion parameter N is less than 3, the error of the integral method is the smallest. When the generalized distortion parameter N ranges from 3 to 4.8, the error of the perturbation method is the smallest. When the generalized distortion parameter N is greater than 4.8, the error of the phase screen method is the smallest. Additionally, laboratory experiments are performed using LC-SLM. The phase of the thermal blooming distortion is accurately extracted and added to the phase modulator, and its effect of thermal blooming distortion is recorded using a CCD. The experimental results are in good agreement with the simulation results, confirming the feasibility of this experimental method. This work proposes a quantitative, accurate, programmable, and easily repeatable laboratory simulation device that provides an effective means for laboratory evaluations of the atmospheric transmission of high-energy lasers.

ObjectiveIt is well known that there are two typical phase singularities in the fully coherent beams, i.e., the optical vortex and the edge dislocation. Although much of research has explored properties of the fully coherent beams, there are practical uses of the partially coherent beams because they are more resistant to degradation with propagation through turbulent medium than the former. The propagation of the partially coherent beams carrying coherence singularities in oceanic turbulence has attracted much attention due to its application in underwater wireless communication. It is interesting to ask how oceanic turbulence can affect the interaction of coherence vortex and edge dislocation carried by partially coherent beams. Because the Gaussian Schell-model beam is a typical example of partially coherent beams, the interaction of the coherence vortex and edge dislocation carried by the Gaussian Schell-model beams in oceanic turbulence is studied in detail.MathodsBy making an analogy with definition of the edge dislocation in coherent beams, the coherence edge dislocation is shown to be in existence in partially coherent beams. Based on the extended Huygens-Fresnel principle, the analytical expression of the cross-spectral density for the Gaussian Schell-model beams carrying the coherence vortex and edge dislocation propagating through oceanic turbulence is derived, which is used to study the interaction of them in oceanic turbulence. The position of correlation singularities of the partially coherent beams at the z plane can be determined by the curves of the real component and imaginary component, as well as phase distribution of the spectral degree of coherence of the Gaussian Schell-model beams.Results and DiscussionsThere should exist another type of coherence singularities, namely the coherence edge dislocation with π-phase jump located along a line in the transverse plane of the correlation function, which is different from the edge dislocation in fully coherent beams (Fig.1), because the transverse edge dislocation with π-phase shift is located along a line in the transverse plane. The coherence edge dislocation is split into two optical vortices by the coherence vortex if the edge dislocation is off-axis, while it is broken into one optical vortex as it is on-axis. The result is similar to the interaction of the phase vortex and edge dislocation in free space. The coherence edge dislocation is translated into one coherence vortex or two vortices with propagation of the beams in oceanic turbulence (Fig.3). The total topological charge is not conserved with propagation of the initial beams with the coherence vortex and off-axis edge dislocation in oceanic turbulence, because appearance or disappearance of a coherent vortex may take place with propagation. The result is different from the interaction of a phase vortex and an off-axis edge dislocation in free space, because the total topological charge is conserved in the latter case. The evolution of the coherence singularities speeds up with increasing the value of the rate of dissipation of mean-square temperature χT and the relative strength of salinity and temperature fluctuationω, as well as decreasing the rate of dissipation of turbulent kinetic energy per unit mass ε (Fig.4). The physical reason can be explained by the theoretical expression of the strength of oceanic turbulence. It is seen that the strength of the oceanic turbulence becomes stronger with increasing the rate of dissipation of mean-square temperature and the relative strength of salinity and temperature fluctuation, as well as decreasing the rate of dissipation of turbulent kinetic energy per unit mass. When the initial beam parameters, such as the spatial correlation length δ0, the off-axis distance and the slope of the edge dislocation of the coherence edge dislocation change, the changes of positions and number of coherence singularities in the fields take place with propagation of the beams. It is found that not only creation and annihilation of a pair of coherent vortices, but also appearance and disappearance of a vortex take place with varying the initial beams parameters (Figs.5‒7).ConclusionsIn the present study, we have firstly introduced the definition of the coherence edge dislocation in accordance with previous researches. Then, the analytical expression of the cross-spectral density for the Gaussian Schell-model beams carrying the coherence vortex and edge dislocation propagating through oceanic turbulence is derived, which is then used to study the interaction of them in oceanic turbulence. It has been shown that the interaction depends on propagation distance, oceanic turbulence parameters, and the beam parameters such as the spatial correlation length and the slope and off-axis distance of the coherent edge dislocation. The creation and annihilation of pairs of coherence vortices occur and the appearance and disappearance of a coherent vortex may also take place by changing these influencing factors. The total topological charge is not generally conserved with propagation of the initial beams. Furthermore, the stronger the oceanic turbulence is, the faster the decrease of the distance for the conservation of the topological charges is.

ObjectiveWith the rapid development of laser technology, it has been widely applied in important fields such as medicine, biology, materials and national defense. The amplitude of a laser beam generally has a Gaussian distribution, and such an uneven energy limits its further application. Thus, beam shaping techniques have been proposed to transform Gaussian beams into flat top beams with a uniform energy distribution. Researchers have proposed various beam shaping methods, among which shaping using liquid crystal spatial light modulators has been widely investigated for its controllable transmittance function, good flexibility and real-time performance. Traditional phase distribution algorithms suffer from the problems of being easily trapped in local extrema, being sensitive to the initial value of the phase, and not being able to obtain high utilization of energy and high beam top uniformity at the same time. In this paper, the phase distribution function algorithm where beam is shaped using liquid crystal spatial light modulators is optimized by using the combination of lowliest place elimination (LPE), genetic algorithm (GA) and Gerchberg-Saxton (GS) algorithm. The hybrid method is called LPE-GSGA algorithm, which further improves the output beam top uniformity without sacrificing the utilization of energy, or even improving it. Meanwhile, it reduces the dependence of conventional algorithms on initial values to a certain extent and has important applications in flat top beam shaping with high utilization of energy and high beam top uniformity.MethodsThe LPE-GSGA algorithm designed in this paper uses the strong global search capability of the GA algorithm to help the GS algorithm to jump out of local extrema. Also, LPE is introduced to retain individuals with good phase points and accelerate convergence. Sum of squares for error ess and fitting coefficient η are used as evaluating indicators to describe the quality of output beams. The algorithm can be divided into two processes: the first is the iterations of all initial phase groups using GS algorithm, and the second is the calculation of the comprehensive evaluation index where some phase individuals with good indexes are selected to enter the next generation phase group directly and the remaining phase individuals experience selection, crossover (Fig. 1), mutation and LPE to enter the next generation phase population until the number of individuals in the phase population is 1. The flow chart of the process is shown in Fig. 2.Results and DiscussionsWe calculate the output beam's information use LPE-GSGA algorithm through simulation, show its iterative process (Figs. 3 and 4) and further compare it with those of the GS, generalized adaptive additive (GAA), weighted Gerchberg-Saxton (GSW) and GSGA algorithms under the same input and evaluation metrics (Table 1). The ess and η calculated by LPE-GSGA algorithm are superior to those obtained with other algorithms. Compared with GS algorithm, the LPE-GSGA algorithm shows great advantages with 10.1% reduction in ess and 0.85% improvement in fitting coefficient η. From the point of initial value dependence, the variances of ess and η of 50 sets of results figured by LPE-GSGA algorithm are much lower than those of the other algorithms, with the variance of ess being about 74% lower than that of the GS algorithm, and a nearly one order of magnitude reduction of variance of η. The role of each process is also discussed: process 1 makes use of the fast convergence ability of the GS algorithm to obtain the local extrema quickly, and process 2 uses the screening of the LPE and the global search ability of the GA algorithm to help the GS algorithm obtain better iterative initial phase values, reduce its dependence on the initial values, and thus obtain better phase distributions.ConclusionsThe LPE-GSGA phase distribution algorithm based on the LPE, GS algorithm and genetic algorithm is proposed in this paper. Based on the algorithm, we get the quality of the output beam by simulation which is superior to those of the GS, GAA, GSW and GSGA algorithms, and solve the problem of initial values dependence. Additionally, the improved algorithm diminishes the number of intensity abrupt change points on the top of output beam, the number of sidelobes, and the sidelobe amplitude. In a word, we demonstrate the effectiveness of the LPE-GSGA algorithm in improving the quality of the output flat top beam and getting a flat top beam with high utilization of energy and high beam top uniformity.

ObjectiveAdaptive optics (AO) are commonly utilized for real-time wavefront detection and aberration correction. Nevertheless, a typical AO system often experiences a time delay equivalent to at least 2?3 sampling periods due to wavefront sensor readout data delay and control calculation delay. Hence, existing AO systems do not offer truly real-time corrections. When the frequency of turbulence alterations surpasses the closed-loop bandwidth of the AO system, this delay leads to the compensation wavefront on the deformable mirror (DM) trailing the changes in the distortion wavefront. This lag substantially hampers the correction efficacy of the AO technology. Consequently, predicting future atmospheric turbulence information in real-time bears significant research importance and practical value in AO technology.MethodsIn this paper, a spatiotemporal prediction network was proposed for AO wavefronts based on the attention mechanism. This network simultaneously considered the temporal and spatial characteristics of atmospheric turbulence and selected the slope data, which was easier to acquire in the AO system and had a lower dimension, as both the input and output of the network. With only six frames of previous wavefront slope information, the wavefront slope of the second frame at a subsequent moment was determined. The network initially employed the spatial attention mechanism to capture similar target features in each frame of the distorted wavefront. It then utilized the residual learning strategy to remove redundant information between the input wavefronts of two consecutive frames, producing refined target features. Additionally, recognizing that the degree of target feature information in each frame of the distorted wavefront varied, the channel attention mechanism was further employed. This mechanism emphasized distorted wavefronts containing a richer set of target feature values rather than evenly weighting the wavefront from each frame. Following these steps, the final prediction of the wavefront slope was realized.Results and DiscussionsThe generalizability of the distorted wavefront dataset is tested under different atmospheric turbulence intensities. As the turbulence intensity increases, the structural similarity (SSIM) between the predicted and true wavefronts stabilizes above 0.9300, and the ratio of the root-mean-square wavefront error (RMSe) between the predicted and true wavefronts to the true RMS is stable at approximately 5.00% (Table 2). In the two-frame delay system, when compared to the non-prediction method, the performance improvement of the proposed prediction method is stable at approximately 40% (Fig. 4). Extended tests are performed on the three-frame-delay AO system under the assumption that the RMSe increases when compared with the two-frame delay prediction; however, the performance of the proposed prediction method increases by more than 43% when compared with that of the non-predicted method (Fig. 5). Additionally, a set of ablation comparison experiments are conducted. Under the assumption of a two-frame delay system, compared to the non-prediction method, the RMSe based on the attention mechanism and residual learning prediction methods reduce by approximately 29% and 16%, respectively, while that based on the proposed method reduces by approximately 40% (Fig. 7). To further verify the performance of the proposed prediction network, we test the one-time open-loop data collected by an actual 1 km laser atmospheric transmission system (Table 3); the average ground truth RMS of the experimental data is 0.9488λ, the average prediction RMS is approximately 0.9557λ, the average RMSe is approximately 0.0675λ, and the RMSe is approximately 7.1% of the real true RMS (Fig. 9). Furthermore, to demonstrate the prediction capability of the network for the experimental data, we extend the simulation of the three-frame delay open-loop correction process again. Compared to the non-predicted method, in two-frame delay and three-frame delay systems, the average performance improvements of the proposed prediction method are 41.2% and 42.9%, respectively (Fig. 10). This reiterates the effectiveness of the proposed open-loop slope prediction network and the feasibility of applying this network model to open-loop correction systems.ConclusionsIn this study, an AO open-loop slope-prediction-network-based attention mechanism is proposed to achieve high-precision wavefront slope prediction. The effectiveness of the network is experimentally proven, which can effectively overcome the inherent delay problem of real AO systems. Additionally, the network uses only six consecutive frames of wavefront slope prior information to achieve a high-precision actual wavefront for correction. This input and output configuration reduces the hardware system's load capacity and enhances the network model's running speed, offering crucial guidance and application value for the subsequent deployment of an actual AO delay system.

ObjectiveGround-based laser systems can remove centimeter-scale space debris in the low-Earth orbit region. However, as the high-power laser beam propagates through the atmosphere, it encounters significant challenges. When the beam’s power exceeds the atmosphere’s critical power for the self-focusing effect, the beam quality at the target diminishes due to this nonlinear effect.Interestingly, Airy beams exhibit self-accelerating characteristics, making them potentially advantageous for bypassing obstacles. However, in homogeneous self-focusing media, an Airy beam can lose its self-accelerating traits if its power is exceedingly high. This leads to pressing inquiries: Does the nonlinear self-focusing effect in an inhomogeneous atmosphere disrupt the self-accelerating nature of Airy beams? Is an Airy beam better suited than a Gaussian beam for ground-based laser space debris removal? How to enhance the target quality of Airy beams? Hence, analyzing the influence of nonlinear self-focusing on the attributes and quality of upward-propagating Airy beams in the atmosphere becomes crucial.MethodsUnder the paraxial approximation, the beam characteristics of diffraction and self-focusing nonlinearity were described via a nonlinear Schr?dinger equation. However, solving the nonlinear Schr?dinger equation analytically for an Airy beam propagating in the atmosphere is challenging. In this study, the nonlinear Schr?dinger equation was solved numerically using the multiphase screen method. As the altitude increased, the nonlinear refractive index decreased, and the nonlinear self-focusing effect became negligible at sufficiently high altitudes. Consequently, an Airy beam that propagated upward in the atmosphere experienced two stages: inhomogeneous atmospheric propagation (comprising both diffraction and self-focusing effects) and free space propagation (with only the diffraction effect).Results and DiscussionsAs the exponential truncation factor of the Airy beams increases, the value of the B integral also increases (Fig.1), indicating a strengthening of the nonlinear self-focusing effect. The real focus of Airy beams shifts to the target due to self-focusing in an inhomogeneous atmosphere, a behavior distinct from that of Gaussian beams (Fig.2). By employing the preliminary defocusing method, an Airy beam maintains its Airy profile at the target even when the beam power significantly exceeds the critical power of the self-focusing effect in the atmosphere, and the intensity at the target notably increases (Fig.5). Specifically, a formula for the focal length of the preliminary defocusing of the Airy beams is obtained, and this is also confirmed (Fig.4). With the preliminary defocusing method, the self-accelerating characteristics of the Airy beams remain unaffected by the nonlinear self-focusing effect in an inhomogeneous atmosphere, even when the beam power significantly surpasses the critical power (Figs.6 and 7). This differs from the behavior of Airy beams in a homogeneous atmosphere. Given the same beam power, the intensity of the Airy beam at the target surpasses that of the Gaussian beam (Fig.8). Additionally, the Airy beam’s resistance to the nonlinear self-focusing effect in an inhomogeneous atmosphere exceeds that of the Gaussian beam (Fig.8).ConclusionsIn this study, the influence of nonlinear self-focusing on the characteristics and quality of Airy beams, as they are propagated from the ground through the atmosphere to space orbit, is numerically investigated. The strengthening of the self-focusing effect with the increasing exponential truncation factor of the Airy beams is observed. It is found that the Airy profile can be maintained at the target, even when the beam power significantly exceeds the critical power of the self-focusing effect, when the preliminary defocusing method is used, leading to a significant increase in target intensity. Furthermore, a formula for the focal length of the preliminary defocusing of Airy beams is derived. The self-accelerating characteristic of Airy beams is shown to be preserved with the preliminary defocusing method, proving beneficial for avoiding obstacles in the path. Under the same beam power, the target intensity of the Airy beam is found to be significantly higher than that of the Gaussian beam, suggesting that Airy beams are deemed more suitable than Gaussian beams for ground-based laser space debris removal.

ObjectiveSensing technology based on Rydberg atoms overcomes the physical limitations of traditional electromagnetic sensing systems and offers many advantages such as small size, high sensitivity, and a broad measurement frequency range. The system typically requires two beams of light, with different wavelengths (such as 510 nm and 852 nm), transmitted in opposite directions to excite the atoms. Using optical fibers, rather than space optical links, to collimate and control dual-wavelength beams for constructing optical fiber-integrated atomic antenna probes is an effective method for practical application. However, collimated coupling elements are large and prone to scattering, which causes serious dispersion effects on dual-wavelength light with significant wavelength differences. In this study, a collimation metasurface with wavelengths of 510 nm and 852 nm was designed based on the working principle of achromatic metalenses. The simulation results indicate that the structure can achieve high efficiency and confocal collimation within the bandwidth of 500?1200 nm, which can enhance the coupling efficiency and level of miniaturization, thus promoting the practical development of portable atomic sensing probes.MethodsThe proposed structure is investigated using COMSOL Multiphysics software. Perfectly matched layers (PMLs) are employed along the incident direction to eliminate boundary scattering, and periodic boundary conditions (PBCs) are applied to the lateral boundaries of the unit cell. A cross-shaped dielectric column is selected as the phase-control unit structure to ensure effective total polarization control. Compared with conventional dielectric column structures, such as circular, elliptical, and rectangular, the cross-shaped structure offers more structural parameter variables, which can achieve a wider range of phase adjustment functions, and a single structure can meet the 0-π transmission phase requirements. The substrate is SiO2 with a refractive index of 1.47, and the dielectric column material is Si3N4 with a refractive index of approximately 2 at wavelengths of 510 nm and 852 nm. The period of the designed metasurface unit is 250 nm, with a height of 1300 nm, cross-arm length of 250 nm, and width of 90 nm.Results and DiscussionsThe dual-wavelength collimated metasurface structure was simulated. The focus for both wavelengths is set at F=20.0 μm, and the number of unit cells is N=21 in the x-direction. The distribution of the focused electric field is shown in Fig. 5(a). Theoretically, the beams of both wavelengths should focus at F=20.0 μm after passing through the metasurface. However, in the actual structure, due to a certain deviation between the simulated and theoretical phase values, the focal points for the wavelengths are at F=20.2 μm and F=17.5 μm, respectively. The focus of the 510 nm wavelength beam is almost the same as the theoretical value, while the focus of the 852 nm wavelength beam deviates by 2.5 μm. This deviation occurs because the optical aperture for the 852 nm wavelength beam is smaller than that for the 510 nm wavelength beam under the same metasurface size; hence, the focus deviation of the focusing field is larger, and its half-height full width becomes wider. The focusing error can be reduced by increasing the size of the optical aperture. Increasing the number of x-direction elements to N=27 and N=33 showed that the focus deviations decrease with an increase in the optical aperture, as illustrated in Figs. 6(a) and 6(b).ConclusionsTo address the issues of low efficiency and large volume in the dual-wavelength laser collimation module of a fiber-integrated Rydberg atomic electromagnetic sensing system, a collimated metasurface suitable for wavelengths of 510 nm and 852 nm was designed. The phase conditions for dual-wavelength confocal collimation were calculated using the metalens analysis method, and the values for dual-wavelength laser dispersion compensation were determined. A cross-shaped dielectric metasurface element was designed using a dielectric waveguide structure sensitive to geometric parameters. By varying the width of the cross-shaped structure from 150 nm to 30 nm, the transmission phase could cover 0 to π, and the average transmission amplitude exceeded 90%, meeting the design requirements for dual-wavelength arbitrary focal length collimating lenses. The average phase deviation of the designed metasurface was less than 10°, and the focal length deviation was less than 6%. The metasurface structure designed in this study supports highly sensitive and miniaturized Rydberg atomic electromagnetic sensing systems.

ObjectiveLaser scanning technologies are applied into numerous fields, such as free space optical communication, LIDAR, laser processing, and remote imaging. Therefore, it is important in military equipment and industrial manufacturing. Currently, most laser scanning technologies are realized by silicon-based photo-electronics, liquid crystal spatial light modulator, micro electro mechanical system (MEMS), and coherent laser arrays, etc. Among them, coherent laser arrays are proved to be an efficient method for generating laser with high power, brightness, and beam quality. Owing to designable coherent laser arrays and flexible phase controlling algorithm, remarkable progress in scanning technologies has been realized. In 2022, Zhou et al. utilized constant piston phase differences between -π and π to control the positions of far-field light spots, with limited maximum scanning angle and relatively low diffraction efficiency. In 2021, construction of quasi-continuous scanning system with the combination of micro-lens arrays and adaptive fiber optics collimators (AFOCs) was proposed. Such a system realizes controllable tilting-phase mainly by AFOCs. However, only experimental results of one-dimensional quasi-continuous scanning patterns are provided. Thus, it is urgent to study more possibilities in the customization of any light field patterns and determine the detailed scanning characteristics under the condition of huge coherent laser array to satisfy additional application requirements.MethodsThe method of two-dimensional continuous scanning is mainly based on the regular hexagonal arrangement of coherent laser arrays. Then, the phase modulation mode is set as the sawtooth titling phase corresponding to maximum optical path differences, which belongs to a type of blazed grating phase-controlling mode. When the maximum optical path differences of adjacent sub-apertures increase, their phase differences increase, i.e., the tilting phase in single sub-aperture can sustain periodic change compared to constant piston phase. Therefore, the beam will deflect an angle of θduring transmission as the wavefront iso-phase surface tilts at a certain angle of θ.Results and DiscussionsUsing typical coherent laser arrays with 19, 127, and 919 sub-apertures shown in Fig. 6, simulated results of single scanning point locating at γ=0, γ=π/2, γ=π/4, and γ=-π/4 are displayed in Figs. 7‒9, respectively. Utilizing these single scanning points, two-dimensional quasi-continuous scanning can be realized along x, y, y=x, and y=-x axes, as illustrated in Fig. 10. All patterns show clear outlines, evenly distributed energy, and a smooth curved effect. Owing to the advantages of this tilting phase-controlling model, specific scanning patterns (S, B, and W) are constructed by switching the distributed phase calculated in advance (Fig. 11). Barring the scanning patterns achieved by coherent laser arrays, spatial scanning characteristics are further studied. Owing to the linear relationships between the tilting phase and the scanning angles, the steering angles of far-field beams continuously increase as the tilting phase experiences more periods. Thus, the scanning angles have no limitations under ideal conditions. Moreover, the scanning straight lines with average distributed energy indicate that near-unity diffraction efficiency can be achieved by tilting phase-controlled coherent laser arrays. Most importantly, the number of sub-apertures shows no influence on the diffraction efficiency, energy distribution, and scanning scope. With increasing number of sub-apertures, the scanning precision is improved owing to the larger caliber of coherent laser arrays. Although the grating lobes exist near the central bright spots, the increased sub-apertures can avoid the interferences to some extent because of long distances among them. Notably, the focused energy of far-field spots is higher as the number of sub-apertures increases, which is beneficial in obtaining scanning pattens with better performance.ConclusionsWith the regulation of tilting wavefront, coherent laser arrays can realize periodical phase change within a single sub-aperture to achieve single scanning points at any position, quasi-continuous scanning, and customized specific optical field patterns in a two-dimensional plane. Compared to the piston phase-controlling model, the scanning characteristics of coherent laser array with the controlled tilting phase are optimized. First, the diffraction efficiency can reach one theoretically. Second, the scanning range is not limited under the ideal condition. Last, the far-field spot energy and scanning precision can be further improved by increasing the number of sub-apertures. This work can provide significant guidance in terms of fast optical field coverage and target tracking, and scanning technologies will develop towards direction of non-mechanical mode, large steering angle, high precision, and anti-interference. In future, more studies will be performed in this regard, including reducing the influence of gate lobe, realizing the customization of arbitrary optical field pattern, and expanding the function of coherent laser arrays.

ObjectivePhotoinjector lasers, which stimulate the photocathode surfaces for electron-bunch generation via the photoelectric effect, are crucial components of free-electron laser (FEL) facilities. The Dalian advanced light source (DALS), which is a newly proposed light source operating in the extreme ultraviolet spectrum, is a continuous-wave FEL with a maximum repetition rate of 1 MHz and is designed for chemical physics research. To achieve the optimal brightness of the DALS, one must ensure that the electron bunches exhibit a sufficiently low emittance. An effective method for reducing emittance is by shaping the photoinjector laser three dimensionally, both transversely and longitudinally. This study specifically addresses longitudinal shaping. Current longitudinal shaping techniques include manipulating the light spectrum using devices such as spatial light modulators or acousto-optic modulators, or employing pulse stacking in the time domain. However, the direct adaptation of these techniques to our system poses significant challenges. In this study, based on the fundamental principles of pulse stacking, we utilize a grating pair, two interferometers, and a sequence of birefringent crystals to generate flat-top pulses with durations exceeding 40 ps. Results of electron-beam dynamics simulations show that the generated flat-top pulse significantly improved the emittance of electron beams, thus demonstrating its potential for optimizing the performance of FELs.MethodsWe directed infrared laser pulses generated by a Yb-doped fiber amplifier into a fourth-harmonic generator to generate ultraviolet laser pulses with a repetition rate of 1 MHz, a wavelength of 257.5 nm, and a pulse duration of approximately 200 fs. Subsequently, the ultraviolet light was stretched to 1.5 ps by passing it through a grating pair, which introduced the appropriate second-order dispersion. The stretched single pulse was segmented into four equally spaced sub-pulses using two interferometers. Finally, three birefringent crystals were used to further segregate the light, which resulted in overlapping pulse sequences that formed a pulse with a relatively flat top. The shaping results were measured using optical cross-correlation by mixing the shaped ultraviolet laser pulses with fundamental infrared pulses emitted from the amplifier. The intensity distribution was obtained by varying the temporal delay between two light paths. To evaluate the pulse-shaping performance, electron-beam dynamics simulations were conducted using a DALS injector. The simulations yielded four initial electron-beam distributions: two based on actual laser measurements and two theoretical distributions (ideal flat top and Gaussian), with almost identical pulse base widths. In the simulations, a multi-objective genetic algorithm based on the electromagnetic simulation code ASTRA was used, which optimized for a Pareto front comprising the bunch length and normalized emittance at the injector exit. The results were compared and evaluated.Results and DiscussionsThe temporal distributions of the ultraviolet laser pulses generated by the fourth-harmonic generator after undergoing stretching, interferometer splitting, and birefringent crystal manipulation are presented in Fig. 3. The post-stretching pulse duration is approximately 1.5 ps, which is consistent with the calculated value. After passing through the two interferometers, the interval between the four sub-pulses is approximately 11 ps, with almost equal light intensities. The final flat-top distribution achieves through three α-BBO crystals exhibits rising and falling edges of 2.0 ps and 1.9 ps, respectively, with the middle flat-top region spanning approximately 42 ps. The intensity fluctuation in the flat-top region is 5.7% (Root mean square, RMS). The spectrum remained unchanged after shaping. Because of the aperture limitations of the optical components, the intensity at the edges of the shaped light spot is truncated. However, considering the subsequent transverse shaping through aperture truncation, this does not significantly affect the final results. The overall transmission efficiency of the shaping process is approximately 45%, and the obtained energy satisfies the requirements of an electron gun. The beam-dynamics simulation results show that regardless of whether the laser distributions are measured or ideal, the Pareto front for the flat distribution consistently performs better than that for the Gaussian distribution, with the minimum emittance improved by approximately 20%. For each distribution type (flat or Gaussian), the minimum emittances of the measured and ideal lasers are almost identical, with a variance of less than 5%. These results indicate that lasers with flat temporal distributions, instead of Gaussian distributions, are significantly better for DALS injectors.ConclusionsThis study demonstrates the longitudinal flat-top pulse shaping of the photoinjector laser for the DALS and validates the effect of the longitudinal distribution of the photoinjector laser on electron-beam emittance via electron-beam dynamics simulations. Specifically, we employ the pulse-stacking method, where we initially calculate the parameters and design the layout of the optical components based on the requirements for a flat-top pulse. Subsequently, we conduct shaping experiments. The experimental results show that the appropriate combination of gratings, interferometers, and birefringent crystals can effectively transform femtosecond Gaussian pulses into flat-top pulses with durations exceeding 40 ps. While the incident power is increased gradually, the transmission efficiency of the pulse-shaping optical path remained consistently above 45%, thus exhibiting stable operational performance without significant nonlinear effects such as two-photon absorption. The electron-beam dynamics simulations indicate that, under almost identical base durations, the experimentally obtained flat-top pulses can reduce the emittance by 20% compared with Gaussian pulses, thus significantly improving the beam quality. This provides substantial support for the construction of a DALS injector with improved performance. Additionally, the results suggest that the shaping method employed in this study can be broadly applied to the development and construction of photoinjector-driven laser systems.

In the subsequent time step, the heat source distribution is updated based on the newly calculated optical field, and the optical field as well as the temperature, velocity, and deformation in the next moment can be obtained by repeating the iteration process. Based on this, we propose a novel approach to suppress the thermal effect of a laser via a rotating beam, which is generated by the coherent superposition of two Laguerre-Gaussian (LG) subbeams with different wavelengths and opposite topological charges. The effects of the rotation rate, topological charge number, and gas absorption coefficient on the propagation characteristics of the rotating beam in the inner channel are quantitatively analyzed.ObjectiveThe absorption of laser energy by gases and optical components in inner channels of high-power laser systems leads to complex interactions among laser, fluid, and solids. This interaction causes uneven optical path differences as laser beams propagate, significantly degrading beam quality. As laser power continuously increases, this degradation, driven by thermal effects in the inner channel, becomes more severe. Understanding the mechanism of thermal effect and developing strategies to reduce beam quality degradation are essential. However, most existing studies on mitigating the thermal effect of the inner channel focus primarily on the laser-fluid interaction, often overlooking the optical-structure interaction. In this study, a physical model of the multi-field coupling interaction of the laser-fluid-solid in the inner channel is established to reveal the propagation characteristics of a rotating beam in the inner channel and mitigation methods are proposed for the degradation of beam quality caused by the thermal effect.MethodsIn this study, a physical model is established to investigate the propagation characteristics of a rotating beam in an inner channel, considering the laser-fluid-solid multifield coupling effect. To achieve this, a split-step Fourier algorithm is used to simulate the optical field in the inner channel at each time step. The resulting heat source distributions of the optical elements and fluid are then integrated into the calculations of the flow field and solid mechanics using the finite-element-method (FEM) method. By employing ray tracing and power spectrum inversion method, the aberrations due to the nonuniform distribution of temperature and velocity within the fluid, as well as the deformation distribution of the solid, can be accurately calculated. These results are subsequently incorporated into the analysis of the laser propagation, enabling the calculation and analysis of the propagation characteristics of the rotating beam within the inner channel.Results and DiscussionsDuring the evolution of the laser beam within the inner channel, heat in the fluid primarily accumulates along the beam path, resulting in the highest temperature near the surfaces of the mirrors (Fig. 4). Owing to the influence of natural convection, a gas with a higher temperature tends to flow in the opposite direction to gravity, leading to centroid drifting of the phase screen. Additionally, the centroids of the thermal deformation distributions of the optical elements exhibit different degrees of deviation because the optical elements exhibit different angles between the normal line and gravity direction (Figs. 4 and 5). It is worth noting that the optical path difference induced by the gas thermal effects is several micrometers, whereas the thermal deformations of the window mirror and reflection mirror are several sub-nanometers and tens of nanometers, respectively (Figs. 4 and 5), which are significantly smaller than those caused by the gas thermal effects (Fig. 6). The rotating beam effectively mitigates the laser thermal effect in the inner channel. The optical path differences caused by the thermal effect of the gas and optical elements heated by the rotated beam is more uniform than those heated by the unrotated beam (Figs. 7 and 8). Moreover, the peak-valley (PV) and root mean square (RMS) values of the optical path differences and deformations induced by the gas thermal effects are minimized when compared with those of the LG beam (Table 1). Furthermore, increasing the rotating beam angular rate and topological charge number and decreasing the gas absorption coefficient can lead to reductions in the Zernike coefficients of the additional phase. This phase is induced by the combined effects of gas thermal effects and mirror thermal deformations. Furthermore, it leads to improvements in the beam quality (β) of the output beam (Figs. 10, 11, and 12).ConclusionsBased on the physical model of the multi-field coupling interaction of the laser-fluid-solid in the inner channel, in this study, the propagation characteristics of the rotating beam are investigated in the inner channel. Under the specific boundary conditions considered in this study, the primary factor affecting the output beam quality is the nonuniform optical path difference induced by the thermal effects in the gas, which is significantly more pronounced than the thermal deformations on the mirror surface. The use of a rotating beam has been proven to be an effective method for suppressing these thermal effects during laser propagation in the inner channel, particularly for mitigating beam astigmatism and comet-like distortions. Additionally, increasing the rotation rate and topological charge can significantly improve the ability to suppress thermal effects. Reducing the gas absorption coefficient can further improve the quality of the output beam.

ObjectiveCoherent detection lidar, a pivotal optical sensing technology, is widely used in various fields, including meteorological forecasting, wind energy generation, and other fields. However, the performance of coherent-detection lidar is significantly affected by atmospheric turbulence in practical applications. Turbulence induces random variations in the optical path, resulting in wavefront distortion that adversely affects the quality of the received beam. Wavefront distortion correction, achieved through adaptive optics technology, has been proved to be an effective solution. The core of this method involves the use of optimization algorithms to control a deformable mirror, generating a phase that is conjugate to the wavefront distortion, thereby compensating for wavefront aberrations. The stochastic parallel-gradient descent (SPGD) algorithm is widely used for this purpose. However, because of the introduction of random perturbations, it exhibits a slow convergence speed. The particle swarm optimization (PSO) algorithm, proposed by Kennedy and Eberhart, is favored owing to its rapid convergence, simplicity, independence from derivative information, and parallel computation capabilities. However, both algorithms are susceptible to becoming trapped in local optima, particularly when addressing large and complex problem spaces. To address this challenge, we propose an improved PSO algorithm for distortion spot correction.MethodsThe improved PSO algorithm introduces the Metropolis criterion to probabilistically accept solutions with relatively low performance, which aids in escaping local optima, thereby achieving a higher convergence limit. The application of this algorithm to wavefront distortion correction further enhances the correction capabilities. First, we simulated the laser transmission through atmospheric turbulence based on the multi-phase screen propagation principle, resulting in the generation of distorted spots. Subsequently, we optimized the inertial parameters in both the PSO and improved PSO algorithms as well as the gain coefficients and perturbation amplitudes in the SPGD algorithm. This is because different parameter values can significantly influence the optimization performance. Hence, these parameters were adjusted to ensure that the algorithms operated at their peak efficiencies. Finally, we conducted a comprehensive comparative analysis of the correction results achieved by the SPGD, PSO, and improved PSO algorithms under medium and strong turbulence conditions, using the Strehl ratio (SR) as the evaluation function.Results and DiscussionsThe improved PSO algorithm exhibited remarkable insensitivity to the inertial parameters (Fig. 9), indicating its superior robustness. All three algorithms were employed to correct the distorted spots under medium and strong turbulence conditions (Figs. 10 and 11). Based on the correction results, the convergence speed and limit were analyzed. Table 2 lists the convergence iterations and the time required by each of the three algorithms to achieve convergence. Under similar conditions, SPGD converges the slowest, followed by PSO, and the improved PSO converges the fastest. The reason for this discrepancy is the pronounced stochasticity of the SPGD algorithm during the optimization process, resulting in a longer convergence time. Additionally, the improved PSO algorithm concentrated the energy of the corrected distorted spot and achieved a higher SR because it increased the probability of accepting bad solutions (Fig. 12). Under strong turbulence conditions, the SPGD, PSO, and improved PSO algorithms contributed to SR improvements of 1.2, 2.6, and 3.2 times, respectively. Strong turbulence can result in severe spot distortion. When local optima are present during optimization, the advantages of the improved PSO algorithm become particularly prominent, enabling it to attain a higher convergence limit. This is advantageous for enhancing the system coupling efficiency, thereby effectively improving the performance of coherent detection lidar.ConclusionsCoherent detection lidar is affected by atmospheric turbulence. Turbulence results in spot distortion, which reduces the detection performance. AO technology is an effective method for mitigating this distortion, and the selection of an intelligent optimization algorithm is crucial in this process. The SPGD algorithm exhibits parallel processing capabilities; its incorporation of random voltage perturbations results in slow convergence, whereas the PSO algorithm not only offers parallel processing and simplicity but also achieves rapid convergence without the need for derivative information. Nonetheless, both algorithms easily fall into the local optima. To address this problem, this study proposes an improved PSO algorithm that introduces the Metropolis criterion to escape local optima and reach a higher convergence limit. This algorithm is insensitive to the inertial parameters and exhibits better robustness. In comparison with the SPGD and PSO algorithms, the improved PSO algorithm enhances the convergence speed and convergence limit. In summary, the improved PSO algorithm demonstrates a more advantageous capacity for improving the performance of coherent detection lidar, particularly for strong turbulence.

Results and Discussions The Fourier synthesis illumination device composed of MEMS and off-axis ellipsoidal mirrors can achieve various illumination patterns such as disk, dipole, quadrupole, and annular and the illumination area size (representing the partial coherent factor) and spacing of the illumination area can be adjusted. The tested illumination profile has no distortion and the illuminating intensity distribution is relatively uniform. When the MEMS scanning angle is ±1° and the magnification of the ellipsoidal condenser is 10 (i.e., with an object distance of 1 m), the maximum illumination diameter can reach >30 mm on the condenser, and the 4×NA on the ellipsoid focal surface can reach >0.6. Moreover, the illumination area on the surface to be detected located at the imaging focal point of the ellipsoidal mirror was tested, and all scanning rays were concentrated in the same area on the surface to be detected. Neither the scanning mode nor scanning angle influences the position of the overlapping area. Ellipsoidal mirrors with different magnifications can be used to adjust the size of the illumination area on the surface, and the actual magnification of the illumination area is basically consistent with the theoretical value.ObjectiveOff-axis illumination is an important resolution enhancement technology in lithography, and it can effectively enhance both resolution and the focal depth of the lithography tool. Conventional off-axis illumination methods, such as those using pupil filters, have the disadvantage of serious energy loss. Moreover, realizing a few special illumination patterns is difficult using transmission elements represented by an axicon, and diffractive elements have the problem that a single diffractive element corresponds to only one illumination pattern. In the EUV spectral band, because optical materials intensely absorb EUV radiation, the transmission elements and transmission type diffraction optical elements cannot be used. In the present study, we investigate an illumination system based on Fourier synthesis technology. It has advantages of realizing any off-axis illumination patterns, increasing imaging numerical aperture (NA), high energy efficiency, and wide applicability in various spectral bands, especially for applications in the EUV spectrum. We hope that our research results will improve the understanding of Fourier synthesis technology and achieve an illumination technology that limits illumination divergence and can easily provide uniform illumination, making it useful in applications such as lithography projection exposure and mask defect detection.MethodsWe use a micro-electro-mechanical systems (MEMS) mirror combined with an off-axis ellipsoidal mirror to construct our Fourier synthesis illumination device. The surfaces of the MEMS and ellipsoidal mirrors are coated with a high reflectivity film for working wavelength. Based on the characteristic high-frequency two-dimensional rotation of the MEMS mirror, with the support of an optimization scan program, we set ray-scanning paths of the MEMS mirror in the x and y directions, achieving various illumination patterns such as disk, dipole, quadrupole, and annular, and adjust the partial coherent factor. The scanning ray is then collected and imaged by using the ellipsoidal mirror with two imaging focal points, whose surface to be detected (such as the mask) is located at the focal point. The Fourier synthesis illumination device provides uniform illumination with the required illumination pattern and illumination divergence to the surface to be detected. In this study, the imaging characteristics of two ellipsoidal mirrors with different magnifications of M=10 and M=2.5 are verified, and the simulation results are found to be basically consistent with the experimental test results.ConclusionsThe Fourier synthesis technique based on MEMS and off-axis ellipsoidal mirror is studied and an experimental confirmatory device is set up. The feasibility of Fourier synthesis technology is verified, and it can achieve various illumination patterns and illumination size by adjusting the pupil and partial coherent factor. The experiment demonstrates that Fourier synthetic illumination technology can meet the requirements of off-axis illumination and illumination divergence of imaging systems. Our research shows that Fourier synthesis technology is an illumination method that can be easily meet illumination requirements. There are only two main reflection elements needed to minimize energy loss, and their reflection characteristics are widely applicable over a wide spectral range.

Results and Discussions A simulation is used to validate the performance of the project-constraint decoupling method for the aberration correction, coupling error elimination, stability, and computation complexity. It can compensate the aberration better than the traditional method to decoupling method for the wave front sensor-free system (Fig. 2). Additionally, it can effectively eliminate the decoupling error between the Woofer and the Tweeter (Fig. 3), as the aberration is broken down into low order Zernike modes and high order modes before being corrected by the Woofer and Tweeter. During the decoupling operation, it is more stable than the conventional approach, and the advancements have improved its performance in the control process. Finally, the decoupling method suggests in the research has a lower computational complexity than the conventional method (Fig. 4). An experimental system was built to evaluate the effectiveness of the method. The experiment demonstrates that the decoupling algorithm can effectively compensate for phase distortions (Fig. 6 and Fig. 7) and significantly suppress the coupling error between the dual deformable mirrors and decompose the aberration accurately (Fig. 8).ObjectiveDual deformable mirrors are often used to create wave front-sensor-free adaptive optics systems that can be used to correct aberrations with broad strokes and high spatial frequencies. The Woofer, which has big amplitude and is used to correct low order aberrations, and the Tweeter, which has a high spatial resolution and is used to correct high order aberrations, are two examples of dual deformable mirrors. However, without the decoupling process, it is difficult to avoid the coupling error, which would cause the deformable mirrors to generate an opposite surface shape and waste the ability of aberration correction in the dual deformable mirror adaptive optics system. To solve this problem and make the Woofer and Tweeter could work efficiently together; a decoupling method must be developed. Even the decoupling algorithms are the subject of considerable study, most of them focus on dual deformable mirror adaptive optics systems with wave front sensors. These techniques frequently employ the data from the wave front sensor to aid in decoupling. A few decoupling methods are used for the wave front sensor-free adaptive optics system, and their performances are usually not satisfactory for the engineering project. To improve the performance in the aberration correction, coupling error reduction, stability, and computation complexity for the wave front sensor-free adaptive optics system, a new decoupling technique must be developed. This might lead to further applications for the adaptive optics technology in things like large-scale telescopes, vision equipment, and laser beam cleanup.MethodTo make the dual deformable mirrors in the system work together to correct the aberration, a straightforward but effective decoupling method based on the mode project-constraint is proposed. The Woofer is controlled by a low order Zernike mode coefficient to avoid correcting the high order modes, and the Tweeter is constrained by the project-constraint method to eliminate the low order modes in its corrections. Obtaining the related matrix of the mode coefficients to the Woofer control signal is essential to the decoupling control process. It can be obtained through the Woofer’s influence functions and the low order Zernike modes which will be corrected by the Woofer. The project-constraint, which requires the following processes, can also limit the use of low order modes in the Tweeter. To start, a relationship matrix between the Zernike mode coefficients and the Tweeter’s control signals needs to be produced. Then, the component of the signals in the Tweeter-induced coupling error can be solved by the relation matrix. Finally, by subtracting the component-induced coupling error from the initial Tweeter control signals, the signals free of coupling error can be obtained. These techniques result in the realization of the Woofer and Tweeter’s decoupling.ConclusionsIn this paper, a simple and effective method was proposed based on project-constraint to restrict the coupling error and eliminate the aberration in a wave front sensor less adaptive optics system with a dual deformable mirror. This method can outperform the conventional method in terms of aberration correction, coupling error elimination, stability, and processing complexity. It can be used to make the Woofer and Tweeter cooperate efficiently to correct the aberration by Zernike mode decomposition. Then the low order Zernike modes of the aberration can be compensated by the Woofer, and the other Zernike modes of the aberration can be corrected by the Tweeter.

Objective A space laser communication terminal generally comprises two basic systemsa laser communication system and an optical tracking system. The former is for information transmission between two satellites, and the latter is for pointing, acquisition, and tracking (PAT). The space laser communication system is advancing toward miniaturization and is lightweight. However, traditional optical tracking and sighting systems usually use gimbal turrets and gimbal turning mirrors to attain significant beam angles. Furthermore, such structures are large in size, large in inertia, poor in dynamic performance, slow in response time, and sensitive to vibration, which is not conducive to the installation of the carrier platform and the balance of the carrier posture. Compared with the traditional structure, Risley prisms are small in size, have excellent viewing axis adjustment function, and can realize large-angle deflection of the beam; therefore, the rotating biprism is more suitable for space laser communication. However, since Risley prisms are composed of two coaxial wedge prisms, there is no linear relationship between the outgoing light and the wedge prism s rotation angle, making it challenging to solve the outgoing beam of Risley prisms. Additionally, there are several error sources of Risley prisms, and the pointing is not sufficiently accurate. Therefore, it should be corrected to obtain a more precise pointing, which can be used in space laser communication.MethodsA new method of correcting the pointing deviation of Risley prisms is proposed to aim at the problem of poor pointing accuracies, including the significant pointing error of Risley prisms and more error sources. This study uses a non-paraxial ray tracing method to establish a Risley prism pointing model and a two-dimensional turntable pointing model. Many points are uniformly chosen in the entire field of view, and the deviation between the rotating double prism s theoretical and actual output beams is compared. The Levenberg-Marquardt iterative algorithm corrects the rotation angle error, wedge angle, and refractive index of the front and back mirrors of Risley prisms. Higher-precision pointing is achieved by correcting the inaccuracy of the initial incident beam relative to the ideal optical axis and separately correcting the region with a small pitch angle to address the issue of low pointing accuracy in the area with a big pitch angle.Results and DiscussionsFrom the simulation findings of the final convergence of the Levenberg-Marquardt algorithm with various initial values, different initial values have little impact on the final optimization results of this experiment (Fig. 2). The entire field of view of Risley prisms is pitch angle 0°-29.22°, azimuth angle 0°-360°. After optimizing the whole area of view by the Levenberg-Marquardt algorithm, the maximum pointing deviation is 5.33 mrad and the average pointing deviation is 1.82 mrad (Fig. 6). It can be observed that the pointing error of the initial incident beam relative to the ideal optical axis will have a relatively large impact on the pointing accuracy of Risley prisms from the effect of the simulation error on the pointing deviation between the actual outgoing beam and the theoretical outgoing beam (Fig. 7). After adding the correction of the error of the initial incident beam relative to the ideal optical axis by the Levenberg-Marquardt algorithm, the maximum pointing deviation is 3.75 mrad and the average pointing deviation is 1.38 mrad (Fig. 8). After using the Levenberg-Marquardt algorithm to correct the points with a pitch angle of less than 15°, the maximum pointing deviation is 1.51 mrad and the average deviation is 0.84 mrad (Fig. 9).ConclusionsIn this study, the non-paraxial ray tracing method is used to develop the pointing model of Risley prisms. Numerous points are evenly chosen in the entire area, and a two-dimensional turntable pointing model is shown to accurately measure the actual outgoing beam of the Risley prisms. Comparison is made between the deviation of the theoretical and real output beams of Risley prisms. The rotation angle error, wedge angle, and refractive index of the front and rear mirrors of Risley prisms are corrected via the Levenberg-Marquardt iterative procedure. After correction in the entire field of view, the maximum pointing deviation changes from 8.37 mrad to 3.75 mrad, and the average pointing deviation changes from 4.00 mrad to 1.38 mrad. Moreover, the correction effect is better when the pitch angle is small. For example, after individually correcting the field of view area with the pitch angle less than 15°, the maximum pointing deviation becomes 1.51 mrad and the average deviation becomes 0.84 mrad. This method improves the pointing accuracy of Risley prisms, and it has a particular reference value for correcting the pointing deviation of Risley prisms.

On the other hand, the propagation of light beams in anisotropic media has always been of interest. In 2001, Ciattoni A discovered that when a circularly polarized (CP) beam propagates along the optical axis of a uniaxial crystal, a portion of the light beam acquires a topological charge vortex phase of ±2 due to spin reversal. In 2020, Ling X H et al. found that the conversion efficiency of spin angular momentum (SAM) to orbital angular momentum (OAM) is related to the anisotropy of the crystal and shape of the beam. To improve the “abruptly autofocusing effect” of the CAB and improve the conversion efficiency of SAM to OAM, this study investigates the propagation characteristics of a modified CAB (MCAB) propagating along the optical axis of a uniaxial crystal.Results adn Discussions In our numerical study, the incident light is a left-hand CP (LHCP) MCAB without a vortex. During the propagation, a right-hand CP (RHCP) component is generated. First, we investigate the intensity, phase, and polarization distributions of the MCAB at z=100 mm. Due to the “abruptly autofocusing effect,” the radii of the first rings for the LHCP and RHCP components become smaller [Figs. 2(c),(d)]. The phase distribution shows that the LHCP component has no vortex, whereas the RHCP component has a vortex phase with a topological charge number of 2 [Figs. 2(e),(f)]. This is the singularity of the central phase that causes the RHCP component to be a hollow beam throughout the propagation. The polarization distribution shows that the beam is no longer a uniformly CP beam (Fig. 3). Due to the anisotropy of a uniaxial crystal, the abruptly autofocusing positions of the two components differ. The “abruptly autofocusing effect” of the MCAB is approximately 3.4 times as strong as that of an ordinary CAB (Fig. 4). Furthermore, we investigate the propagation dynamics of the two components. The results show that both the LHCP and RHCP components exhibit an “abruptly autofocusing effect”. The LHCP component without a vortex forms a solid beam at the focus, whereas the RHCP with a vortex forms a hollow beam at the focus (Fig. 5). For a 10 cm long crystal, the efficiency of conversion from the LHCP component to the RHCP component with a vortex can reach 43.28%, which is approximately 10% higher than that of an ordinary CAB (Fig. 6).ObjectiveThe circular Airy beam (CAB) has received significant attention because of its peculiar “abruptly autofocusing effect”. The “abruptly autofocusing effect” has shown significant advantages in biomedical treatment, laser cutting, and other applications because the CAB can be applied solely to the target without damaging other areas. Various schemes have been designed to improve the “abruptly autofocusing effect”. For example, direct blocking of the first few rings of the CAB and modulation of the CAB’s angular spectrum can significantly enhance its “abruptly autofocusing effect”.MethodsThe method proposed by Ciattoni A is adopted to deal with the propagation of light beams along the optical axis of a uniaxial crystal. According to the results of Ciattoni A, a light field propagating along the optical axis of a uniaxial crystal can be treated as a linear superposition of ordinary and extraordinary components. Based on the angular spectrum theory, the propagation dynamics of these two components can be obtained by the Fourier transform of the MCAB’s angular spectrum. A closed-form approximation of the CAB’s angular spectrum with a suitable plane wave angular spectrum representation has been reported by Chremmos I et al. A modulation function is introduced to modulate the CAB’s angular spectrum. The “abruptly autofocusing effect” of the MCAB is superior to that of the ordinary CAB. Following the approach proposed by Ciattoni A, the propagation characteristics of the MCAB in a uniaxial crystal can be obtained.ConclusionsSimilar to other ordinary beams, when an LHCP MCAB propagates along the optical axis in a uniaxial crystal, an RHCP vortex MCAB with a topological charge number of 2 is generated. With a proper modulation function, the “abruptly autofocusing effect” of the MCAB is much stronger than that of an ordinary CAB, and the efficiency of conversion from the LHCP component to the RHCP component with a vortex is also improved.

Results and Discussions The far-field normalized light intensity (NLI) distribution of the double-beam is obtained using MATLAB simulation (Fig. 3, Fig. 4). The phase depression between adjacent pixels results in diffraction side lobes. The maximum NLI of the diffraction side lobes generated using the composite phase, sub-aperture, and IFT methods are 14%, 8%, and 5%, respectively. The IFT method can suppress the diffraction side lobes more effectively than the composite phase and sub-aperture methods. The camera records the far-field light intensity distribution of the double-beam during the experiment (Fig. 5). The experimentally measured far-field light intensity distribution is consistent with the simulated far-field light intensity distribution. The beam tracks and aims the double-target moving at different speeds. The trajectories of the double-target positions approximately coincide with those of their corresponding spot position estimations when the double-target moves at a speed of 2 mm/s (Fig. 9), indicating that the synchronicity of beam tracking and aiming is better. As the moving speed of the double-target increases, the lag between the trajectories becomes increasingly evident (Fig. 10, Fig. 11), and the error of beam tracking and aiming also gradually increases. In addition, both the measured root mean square error (RMSE) and estimated RMSE progressively increases (Table 1).ObjectiveAcquisition, tracking, and pointing (ATP) technology is a core technology for establishing stable physical links in the field of laser communications. Research on ATP technology focuses on beam deflection. Traditional beam deflection techniques are typically implemented using mechanical devices such as gimbals and mechanical mirrors. However, mechanical devices have the disadvantages of large mass, high energy consumption, and mechanical inertia that result in a slow response to beam deflection and unstable beam control. Therefore, new nonmechanical beam deflection devices have been widely used in recent years, such as acousto-optic modulators, electro-optic modulators, and liquid crystal spatial light modulators (LCSLMs). LCSLMs can overcome the defects of mechanical inertia; therefore, they are widely used for beam tracking. In addition, LCSLMs can overcome the defects of traditional mechanical beam deflection techniques, which require multiple devices to achieve multibeam deflection. The deflection direction of multiple beams can be simultaneously controlled using a single LCSLM. The methods for generating multibeams based on the LCSLM mainly include the composite phase, sub-aperture, and iterative Fourier transform (IFT) methods. Currently, double-target tracking can be achieved using the spatial polarization division method that employs a polarizing beam splitter to generate two beams with perpendicular polarization directions to track two targets. However, this method requires two LCSLMs. To use a single LCSLM to deflect multiple beams for synchronously tracking multiple targets, a scheme of synchronous beam tracking and aiming for multiple targets is proposed by combining LCSLM-based beam tracking technology and the multibeam generation method. This scheme is expected to be applicable to the multibeam tracking mechanism of laser communication networks. We also construct an experimental system of synchronous beam tracking for double-target that can communicate with two mobile target terminals.MethodsIn this experiment, a camera was employed as the position detector, and an LCSLM was employed as the beam deflection device. The tracking system mainly consisted of an LCSLM, a laser, collimator, polarizer, nonpolarizing beam splitter (NPBS), an angle magnifier, a camera, and stepper motor. First, the stepper motor controlled the movement of the two targets within the field of view. The two targets were imaged in the camera using scattered light, and the images were processed using the feature matching method to obtain the initial and current positions of the two targets. Subsequently, the offsets of the initial and current positions were converted into pre-deflection angles that were substituted into the grating equation to calculate the period of the corresponding blazed grating. Next, a phase grayscale map was generated using the sub-aperture method and loaded onto the phase screen of the LCSLM. Finally, the NPBS spatially divided the beam into two mutually perpendicular beams, one of which was incident perpendicular to the LCSLM. The LCSLM controlled the deflection direction of the double-beam based on the phase grayscale map, enabling passive tracking and aiming of two mobile targets.ConclusionsThe results show that the double-target tracking system can achieve synchronous tracking of two targets using a single LCSLM to deflect two beams. Moreover, the double-target tracking algorithm is also applicable to beam tracking and aiming for multiple targets that verifies the feasibility of the multitarget synchronous beam tracking and aiming scheme. This tracking system can achieve beam tracking of a target within a field of view of ±57.9 mrad. The tracking error of the system is less than 20 μrad that meets the tracking error requirement. This multitarget tracking scheme has promising applications in the multibeam tracking mechanism of laser communication networks. However, improved synchronicity of beam tracking and aiming can be obtained using the filter prediction technique in the tracking algorithm.

ObjectiveLaser beams propagating in the atmosphere suffer from adverse effects due to the atmospheric optical characteristics and laser system features, which broaden the beam radius and weaken the encircled mean intensity. The wave-optics-based four-dimensional codes work with redundant inputs and slow speed, failing to meet the requirements of rapid assessment for practical applications. Researchers have made efforts to develop new methods, holding reasonable accuracy, calculating quickly and easily, without consideration of the mesh size and computational stability as wave optics programs. Integrated with characteristic parameters of laser system and atmosphere, the scale law has received much attention and is widely used in system design and applications with lots of computation.Current laser beam propagation scale law is based on radius-square-sum (RSS) assumption, meaning that the resulting far-field radius is the root of the sum of radii squared of the individual effect contributions. The RSS assumption lacks scientific foundation and may bring some errors in use. Besides, though the accuracy of scale law is crucial for reliable analysis, few reports on the accuracy of scale models have been released. Furthermore, previous attention was focused mainly on flat-top source, thus the effect of new features of Gaussian source, such as truncating extent, on far-field spot has not been well studied.MethodsTheoretical analysis and numerical simulations are used to build the scale model. Analytical expression of 63.2% encircled power radius in the far-field of infinite Gaussian source is deduced on the basis of Huygens-Fresnel principle, showing that the radius is a function of wavelength, distance and aperture. When the Gaussian source is truncated, split-step wave optics simulations are used to obtain the far-field radii corresponding to 63.2% and 86.5% encircled power. Referring to the analytical expression of infinite Gaussian source, a radius scale function for truncated Gaussian source is built, and the scale exponents are given for different truncating factors. On the basis of established turbulent spread radius expression of infinite Gaussian beam, a radius scale model is given for truncated Gaussian source propagating through turbulence, showing that the scale exponent varies with the value of truncating factor.When the mutual interaction among diffraction, beam quality, jitter of platform and optical turbulence is considered, the generally used RSS assumption is improved to a modified version which is named MRSS method. This new method introduces three scale exponents and an exponent term which consists of the ratio of two different characteristic radii in order to promote the model's applicability. For Gaussian source with truncating factor of 22 propagating in vacuum, the split-step wave optics simulations are operated in a wide range of parameter space shown in Table 2, with Fresnel number changing from 1.0 to 6003.4. The far-field radius scale models based on RSS assumption and MRSS method are built respectively, and the exponents are fixed with the help of genetic algorithm. Comparison with numerical simulations shows that the mean relative errors of the results from the model based on MRSS method are smaller than those based on RSS assumption.A similar process is conducted to build the scale model of far-field radius and encircled mean intensity for the Gaussian source with truncating factor of 22 propagating in turbulent atmosphere. The numerical simulations are conducted with the Hufnagel-Valley optical turbulence profile, and with the propagating distance and other parameters varying in a wide range shown in Table 3. Comparison with numerical simulations shows that the accuracy of the model based on MRSS method is higher than that based on RSS assumption.Results and DiscussionsWhen the Gaussian source is truncated, the far-field radius of free diffraction in vacuum and turbulent spread in atmosphere is affected by the truncating factor, as the scale exponents vary with Fa, as shown in Fig. 1 and Fig. 2(b), respectively. For the scale models based on RSS assumption, aVR gives a mean relative error of 3.12%, as shown in Fig. 4(c), while aLR gives a mean relative error of 4.15%, as shown in Fig. 6(c). For the scale models based on MRSS method, aV gives a mean relative error of 1.55%, as shown in Fig. 4(c), while aL gives a mean relative error of 1.92%, as shown in Fig. 7(c). The mean relative error of mean intensity is 8.33% based on RSS assumption, and 3.80% based on MRSS method. In summary, the accuracy of the models based on MRSS method is higher than those based on RSS assumption.The expression of aL based on MRSS method is equivalent to ad for ideal Gaussian beam propagating in vacuum, and to aV when the interaction among diffraction, beam quality and jitter of platform is considered. When only turbulence spread is considered, the optical quality of aL works well with the optical quality of turbulence spread radius, as shown in Fig. 8.ConclusionsThe scale models of far-field radius and encircled mean intensity for truncated Gaussian source are built in vacuum and turbulent atmosphere. Comparison with split-step wave optics simulations shows that the proposed MRSS method is able to improve the accuracy and applicability of scale models. The results are discussed for Gaussian source with truncating factor of 22 and far-field radius of 63.2% encircled power ratio. However, scale exponents and accuracy for other conditions need more research.

ObjectiveHigh coherence is one of the characteristics of laser, which brings adverse effects to some applications of laser, so it is necessary to suppress the high coherence of laser. Rotating ground glass is the most commonly used coherence suppression method, but when the laser passes through the ground glass, the beam will be seriously scattered and the spot distribution cannot be controlled, which results in low utilization rate of light energy. Different from ground glass, random microlens array is a kind of diffuser with a structured surface. The laser is concentrated in a certain divergence angle after passing through a random microlens array, so it has high energy utilization rate. Previous studies have demonstrated that rotating random microlens arrays can be used for laser coherence suppression. However, different application scenarios have different requirements for laser divergence angle and coherence, so it is necessary to analyze the influence of rotating random microlens array parameters on laser divergence angle and coherence. In this paper, the problems are analyzed and discussed.MethodsFirstly, this paper applies the region division method of Thiessen polygon to the design of microlens array, which obtains the random microlens array with high filling ratio and random variation of sublens unit apertures. Secondly, we analyze the influence of the mean aperture and curvature radius of the sublens element on the divergence angle of the random microlens array by changing the parameters of the sublens element. Finally, we establish a laser complex coherence regulation model based on random microlens array. The modulus of complex coherence is used as the criterion of coherence intensity. We use the modulus of complex coherence as the criterion to evaluate the intensity of coherence and analyze the influence of rotation speed of rotating random microlens array on the complex coherence modulus of laser field.Results and DiscussionsThe simulation and experimental results show that in terms of divergence angle, the average aperture and curvature radius of the sublens element affect the divergence angle of the random microlens array (Fig. 5). In the simulation, the average aperture range of the sublens element is set to be 50?141.14 μm, and the radius of curvature is set to be 1?5 mm. The divergence angle of the random microlens array is 9.1?148.1 mrad. The divergence angle of the random microlens array decreases with the decrease of the mean aperture and the increase of the curvature radius of the sublens unit. In terms of complex coherence modulus, the interference fringe diagrams (Figs. 7, 8, 12, and 13) at different rotation speeds (speed range 0?4800 r/min) are obtained in simulation and experiment in this paper. The complex coherence moduli at different rotation speeds are obtained by calculating the contrast of interference fringe and measuring the ratio of double-hole aperture light intensity (Fig. 15). The modulus of complex coherence decreases with increasing rotation speed of the random microlens array. When the rotation speed increases from 0 to 4800 r/min, the total modulus reduction of complex coherence is about 96.67%. We further analyze the modulus decline trend of complex coherence by calculating the modulus decline percentage of complex coherence at every 60 r/min increase in different speed ranges (Table 2). The modulus decline percentage of complex coherence decreases from 61.52% to 1.17% when the rotation speed increases by 60 r/min, with the decline trend gradually slowing down.ConclusionsIn this paper, we design a random microlens array based on Thiessen polygon arrangement. We establish a laser complex coherence control model based on rotating random microlens array, and analyze the effects of rotating random microlens array parameters on laser divergence angle and complex coherence modulus of laser field. The simulation and experimental results show that, the average aperture and curvature radius of the sublens element affect the divergence angle of the random microlens array. The divergence angle of the random microlens array decreases with the decrease of the mean aperture and the increase of the curvature radius of the sublens unit. The average aperture of the random microlens array used in the experiment is 50 μm and the curvature radius is 1 mm. The divergence angle obtained by simulation based on these parameters is basically consistent with the measured divergence angle. In terms of complex coherence modulus, the rotation speed of the random microlens array affects the modulus of complex coherence of laser light field: the higher the rotation speed, the lower the modulus of complex coherence of laser light field. In the experiment, when the rotation speed of the random microlens array increases from 0 to 4800 r/min, the modulus of complex coherence of the laser light field decreases continuously and the total amplitude of the decline of the complex coherence modulus is about 96.67%. The modulus decline percentage of complex coherence decreases from 61.52% to 1.17% when the rotation speed increases by 60 r/min in different speed ranges. The decline trend of the complex coherence modulus gradually slows down with the increase of the rotation speed.

ObjectiveTo explore and exploit ocean resources, it is necessary to establish wireless communication networks between underwater and air platforms. In these wireless networks, data should be transmitted efficiently across the water-to-air (W2A) interface; reliable W2A communication links play a significant role in such data transmission. Although acoustic waves are the primary means for communication in water because of their long propagation distance (up to several kilometers), they are mostly reflected off when crossing the water surface. Moreover, the transmission rate of an acoustic communication system is relatively low (on the order of kb/s), which limits its application. Radio frequency waves are suitable for long distances (up to tens of kilometers) and high transmission rates (up to hundreds of Mb/s) of wireless communication in air, but they can only travel a few meters through water because of their high absorption and attenuation in underwater environments. Compared with acoustic and RF waves, optical waves can achieve long-distance wireless transmission in both water and air media; they provide a very high bandwidth, high transmission rate, and low latency and enable the use of advanced transceiver devices. Thus, the use of optical waves is a potential solution for communication across the W2A interface. However, when a light beam passes across the W2A interface, the propagating photons experience an unpredictable path deviation owing to the dynamic nature of the waves. Therefore, it is necessary to obtain the statistical properties of the physical responses of photons passing across different W2A interfaces, which can be used to characterize the correlation between the light beam drift and water wave dynamics.MethodsThis study focused on water-to-air visible light communication (W2A-VLC) through regular and random waves. The physical response of the propagating photons and corresponding link performances were evaluated by combining laboratory experiments with theoretical simulations. First, we built a laser diode (LD) transmission experiment and captured laser spots at the receiving end using a high-speed camera. The physical response of the propagating photons could be visualized by extracting the centroids of the laser spots, and a Monte Carlo simulation of the photon transmission was performed for comparative analysis. Second, by numerically fitting the centroid distribution, we further obtained the statistical properties of the photon responses under regular and random waves conditions. The inner dynamic processes of the statistical properties were also analyzed. Finally, we validated the narrow-beam characteristics from the perspective of wide-beam transmission through both theoretical simulations and experimental measurements. The statistical laws of the LD narrow beam were validated from the perspective of the LED wide beam.Results and DiscussionsThe physical response of the propagating photons was first theoretically predicted based on Monte Carlo simulations. In the case of a calm water surface, the photons are mainly distributed around the coordinate center of the receiving end and present a circular structure. In the case of regular waves, the photons are distributed in a strip shape at the receiving end, whereas in the case of random waves, the received photons diffuse from the center to the periphery, and the distribution range significantly increases [Fig. 3(a)?(c)]. The experimental results of the photon responses are consistent with the Monte Carlo simulation patterns. The corresponding statistical features were analyzed further. For regular waves, the centroid points on the x-axis ( perpendicular to the wind) obey a normal distribution, whereas those on the y-axis (wind direction) obey a negatively skewed distribution with a skewed parameter of λ′=-2.5. For random waves, the distribution of the centroid points presents an approximately normal distribution (Fig. 4). We also justify the LD link characterization based on the simulation and real test of an LED transmitter. A Monte Carlo simulation of the LED wide-beam link was performed to obtain the light spot at the receiving end. The light spot on the calm water surface is a regular circle, and its brightness gradually decreases from the center. In the case of regular waves, the pattern of the light spot is elliptical. Conversely, in the case of random waves, the light spot still exhibits a circular outline, but the bright and dark areas in the light spot are irregularly distributed (Fig. 5). An experimental verification system for the LED link was designed to verify the simulation and extend the general statistical laws of the LD narrow beam (Fig. 6). The experimental results reveal that the photon diffusion and beam drift are mainly along the wind direction, consistent with the conclusion obtained for the LD narrow-beam link. Furthermore, the spatial distribution of the link gain values is consistent with the simulation pattern (Fig. 7).ConclusionsIn this study, narrow-beam light transmission through a wavy water-to-air (W2A) surface was evaluated. The physical response of the propagating photons and corresponding statistical characteristics were determined through a combination of lab experiments and theoretical simulations. We experimentally tested the LD narrow-beam link and obtained the photon-response characteristics. The test experiment reveals that, for regular waves, the photon response presents a negatively skewed distribution in the wind direction, whereas for random waves, the photon response shows a normal distribution. These statistical features imply an intrinsic dynamic correlation of the photon response with the wavy W2A surface and its driving forces. Because the LED transmitter can be treated as the integration of infinite LD lights over space, the narrow-beam link characteristics were validated using a wide-beam transmitter perspective. The simulation and real test of the LED transmitter confirm the characterization of the narrow-beam link under both regular and random waves.

ObjectiveLaser technology has gained widespread applications in various fields, such as communication, guidance, and precision machining. However, the accuracy of these applications is often compromised by beam pointing instability caused by uneven internal laser temperature and external environmental changes. Traditional methods of improving beam pointing stability, including using materials with low thermal expansion coefficients, adopting cooling systems and deformation mirrors, and reducing vibration, have proven to be limited in their effectiveness. More advanced lasers have internal beam pointing correction systems, but their performance is limited by their size, and they cannot correct pointing deviations caused by external factors. In view of these limitations, the use of an external beam pointing deviation correction system has emerged as a promising solution. The beam pointing deviation correction system based on fast steering mirrors (FSMs) is considered to be the most mature and effective approach. However, the Z-shaped beam path commonly used in these systems changes the propagation direction of the original beam, resulting in poor expansion performance. In this paper, we present a U-shaped beam pointing deviation correction system based on FSMs that does not change the original beam propagation direction. The system is modeled using geometric optics, including the mapping model from FSMs to four-term beam pointing deviations and the FSMs control model. To further enhance the system's performance, we propose a predictive control model, simplifying the model operation and improving the system responsiveness. Our results demonstrate that the constructed system exhibits excellent beam pointing deviation correction performance.MethodsThe beam pointing deviation correction system consists of three main components: detector, controller, and actuator. The detector consists of a uniform beam splitter and two vertically distributed high-accuracy position-sensitive detectors (PSDs). The non-uniform beam splitter diverts a small portion of the beam from the main beam into the detector, where deviations in the beam's position from the center of the PSDs can be detected and used to determine beam pointing deviations. The controller is a computer that implements four distinct models: a beam pointing deviation detection model, a FSM attitude control model, a beam pointing deviation prediction model, and a simplified FSM attitude control model. The beam pointing deviation detection model calculates the four beam pointing deviations from the position deviations of PSDs. The FSM attitude control model determines the control angles of the FSMs from the four beam pointing deviations to realize the correction of beam pointing deviations. The latter two models, i.e., the beam pointing deviation prediction model and the simplified FSM attitude control model, are constructed to achieve predictive correction of the beam pointing deviation. The beam pointing deviation prediction model is based on the mean deviation correction method and predicts future beam pointing deviations, while the simplified FSM attitude control model has a simpler operation and can rapidly obtain the control angle of the FSMs. The FSMs act as actuators, adjusting their attitude based on control signals to correct the beam pointing deviations and improve the beam pointing stability.Results and DiscussionsThe constructed beam pointing deviation prediction model has a slight lag in the prediction of beam pointing deviations, but it still has a high accuracy and can filter the high-frequency signals in the pointing deviations (Fig. 4). Therefore, the attitude of the corrected FSMs can be calculated based on the predicted pointing deviations. The simplified FSM attitude control model has a reliable accuracy (Fig. 5). The error between the FSMs' control angles obtained from the simplified model and the results calculated based on geometrical optics is within 1 μrad. Therefore, the simplified model can be used to correct the beam pointing deviations. The experiments show that the constructed beam pointing deviation detection model and FSM attitude control model also have high accuracy. The beam pointing deviation correction system can effectively reduce the beam pointing deviations. Although the beam pointing deviations are not completely corrected due to the open-loop control without additional feedback, the errors of the beam pointing deviations in X and Y directions are reduced by 78.08% and 70.28%, respectively.ConclusionsA U-shaped beam pointing deviation correction system is designed, and a beam pointing deviation model and a FSM attitude control model are constructed based on geometric optics. Two PSDs are used to detect the beam pointing deviations, and two FSMs are used to correct the beam pointing deviations. A predictive beam pointing deviation correction model is proposed. A beam pointing deviation prediction model based on the average deviation correction method is constructed to correct the attitudes of FSMs based on the predicted beam pointing deviations rather than the real-time detections. The calculation of the control angles of the FSMs is simplified. A mapping model is constructed to calculate the control angle, which avoids solving the control angle in the original FSM attitude control model and effectively improves the response performance of the system. At last, experiments demonstrate that the developed system and model can effectively reduce the beam pointing deviations. The pointing deviations of the beam are reduced by 78.08% and 70.28% in the X and Y directions, respectively.

ObjectiveIn diode laser array (DLA) grating-external cavity spectral beam combining (SBC) systems, the combining efficiency and beam quality are very important indicators. However, the beam crosstalk caused by imperfect factors including the divergence angle and deflection angle of DLA emitters results in the degradation of the output beam quality and combining efficiency. Therefore, to understand the physical mechanism of crosstalk in the round-trip propagation of the beams in a DLA grating-external cavity SBC system and further analyze its influence on the performance of the SBC system, the relationship between the crosstalk and the combining efficiency or beam quality is established. In addition, the influences of key factors, including the DLA spacing, focal length of the lens, and line density of the grating, on the performance of the SBC system are analyzed.MethodsTo study the behavior of beam crosstalk and its influence on the performance of an SBC system, a round-trip propagation model of a DLA is established. On this basis, by taking the semiconductor laser rate equation involving beam crosstalk injection into consideration, a physical model of combining efficiency of the SBC system is also developed. Furthermore, the influence of the divergence angle and deflection angle on the beam combining performance is studied using numerical calculations and statistical analysis.Results and DiscussionsWith only the divergence angle considered, there is no obvious crosstalk (Fig. 4). This is because the beam emitted by each emitter can be fed back to itself after being reflected by the external cavity. Even when the divergence angle increases to 12 mrad, the beam quality factor M2, beam combining efficiency η, and feedback intensity κ are not strongly affected by the divergence angle. With only the deflection angle considered, the feedback beam from the external cavity exhibits different degrees of deviation (Fig. 5). With an increase in the deflection angle, the feedback beam reflected to its own emitter decreases in intensity, whereas the feedback intensity from the other emitters increases, resulting in crosstalk. When the effects of the divergence and deflection angles are considered comprehensively, the feedback beam to the emitters deviates significantly, resulting in an obvious degradation of the output beam quality and combining efficiency (Fig. 6). For a given maximum divergence angle, the combining efficiency remains almost unchanged as the maximum deflection angle increases and then decreases sharply beyond a certain angle. The output beam quality exhibits the same trend. For a given maximum deflection angle, the beam combining efficiency remains almost unchanged up to a divergence angle of 12 mrad, whereas the beam quality decreases significantly. In addition, the DLA spacing, focal length of the lens, and line density of the grating have almost no influence on the combining efficiency with an increase in the maximum deflection angle up to a certain angle, after which the efficiency decreases dramatically (Fig. 7).ConclusionsWe established a round-trip propagation model of a DLA that can reveal the behavior of beam crosstalk and its influence on the performance of the SBC system. By taking the semiconductor laser rate equation involving beam crosstalk injection into consideration, we established a physical model for the combining efficiency of an SBC system. Based on numerical calculations and statistical analysis, we found that the influence of the deflection angle on the beam combining performance is greater than that of the divergence angle. In addition, we found that the focal length of the lens and the DLA spacing have obvious effects on the critical maximum deflection angle, whereas the line density of the grating has little effect on the critical maximum deflection angle. In conclusion, it is necessary to appropriately reduce the focal length of the lens and increase the DLA spacing in practical applications.

ObjectiveThe stochastic parallel gradient descent (SPGD) algorithm is one of the most commonly used control algorithms for wavefront sensorless adaptive optics (AO) systems. This method usually uses the driving voltages of deformable mirrors as control parameters, and the number of actuators is equal to the number of dimensions of the control parameters. It is simple and suitable for AO systems that have a small number of actuators and do not have any requirement for the convergence speed. With the increasing applications of AO, the number of actuators required has gradually increased.In wavefront sensorless AO systems, when taking the driving voltages of the deformable mirrors as control parameters, an increase in the number of actuators leads to a greater number of dimensions of the control parameters and a larger optimization space of the algorithm, which will lead to slower convergence of the algorithm. Various modal coefficients are often used as control parameters to reduce the dimensions of control parameters. When the modal coefficients are modeled as control parameters, the optimization space of the algorithm can be reduced, and the convergence speed can be improved.MethodsThe Zernike polynomial, which was introduced by Zernike to represent the diffraction effects on concave mirrors, is often used to describe optical wavefront aberrations. When using the Zernike mode to calculate the covariance matrix of the amplitude in Kolmogorov turbulence, namely, the Noll matrix, there are nonzero elements outside the diagonal. This inherent modal crosstalk indicates the statistical dependence between modes, which limits the correction ability of AO systems based on these modes. In this study, the Karhunen–Loève (K-L) modal coefficients derived from the Zernike mode are used as the control parameter of a wavefront sensorless AO system. First, the rationality of the K-L mode is analyzed. The aberration-fitting ability of the deformable mirror (DM) to the K-L and Zernike modes is then discussed. Finally, the convergence speed and correction effect of the AO system are compared when the driving voltages of the actuators, K-L modes, and Zernike modes are used as control parameters.Results and DiscussionsGenerally, the order of the mode needs to be determined based on the fitting ability of the deformable mirror, so that the dimension of control parameters can be relatively small while ensuring the correction ability of the deformable mirror. The Zernike modes and K-L modes are fitted with several deformable mirrors with 32, 61, 97, and 127 actuators, respectively. The results show that the fitting ability is relatively stable for K-L modes while fluctuations appear for Zernike modes (Fig.3). We use the error rate (η) as the evaluation standard. The fitting is effective if η<1. The lower the error rate, the better the fitting ability. Notably, 32-element, 61-element, 97-element, and 127-element deformable mirrors can fit the first 22-order, 55-order, 79-order, and 91-order K-L modes, respectively, while they can fit the first 20-order, 36-order, 54-order, and 68-order Zernike modes (Fig.4), respectively. It can be seen from the above data that the ability to fit K-L modes for the deformable mirror is greater than that to fit Zernike modes. A greater number of modes indicates better correction ability, which implies that the correction capability and convergence speed of AO systems can be improved when K-L modal coefficients are used as the control parameters.The two modal methods only need 20 modes as control parameters when the atmospheric turbulence strength (D/r0) is 5, and the convergence of the conventional SPGD is used as the reference index. When the Strehl ratio (SR) is up to 0.8, the K-L modal method, Zernike modal method, and conventional SPGD require 122, 139, and 180 iterations, respectively. The convergence speed of the K-L modal method and Zernike modal method is 47.5% and 29.4% greater than that of the conventional SPGD control algorithm, respectively (Fig.5). The correction results also show that when the D/r0 is 10 (Fig. 6), 15 (Fig.7), and 20 (Fig.8), the correction performance and convergence speed obtained using K-L modal coefficients are better than those obtained using Zernike modal coefficients as control parameters (Table 1).ConclusionsThe SPGD control algorithm, based on optimizing the actuator voltages, is widely used as a control algorithm for wavefront sensorless AO systems. The number of actuators in the deformable mirror determines the dimensions of the control parameters. Generally, the greater the number of actuators, the better the correction effect. Moreover, the more actuators tend to reduce the convergence speed of the AO system. The SPGD algorithm, which is based on optimizing the modal coefficients, can effectively resolve this contradiction. When the control parameters are modal coefficients, the optimization space of the algorithm can be reduced, and the convergence speed can be improved.The fitting capability of the DM to the aberrations of K-L modes and Zernike modes is compared and analyzed. The convergence speed and correction performance of the AO system are investigated when the voltages of the actuators, K-L modes, and Zernike modes are used as control parameters under various turbulence strengths. The convergence speed of the K-L modal method and Zernike modal method is 47.5% and 29.4% greater than that of the conventional SPGD control algorithm, respectively. The results under several turbulence strengths also show that the correction performance and convergence speed of the K-L modal method are better than those of the Zernike modal method. The results of the study can provide a reference for the practical application of the SPGD control algorithm based on K-L modes.

ObjectiveLithography is currently an essential tool in the production of integrated circuits (ICs) and other micro- and nano-scale elements used widely in the electronics industry. The resolutions of deep-ultraviolet step-and-scan dry lithography machines in the international market range from 57 nm to 350 nm. The illumination system is an important component of the lithography exposure system. The illumination mode is typically adjustable according to the mask pattern, including a full circle, annular rings, or poles, to improve resolution, imaging contrast, and focal depth. During the IC manufacturing process, the exposure field is scanned using a narrower illumination field. Therefore, illumination-integrated nonuniformity is a key factor in determining the resolution and critical dimension uniformity (CDU), which are crucial to the performance of advanced lithography systems. To obtain higher resolutions and better CDU, the exposure dose must be maintained as uniformly as possible in the scanning direction. The illumination uniformity unit is the primary component for obtaining a uniform illumination field. Currently, the homogenizer elements used in lithography illumination systems include diffractive optical elements (DOEs), microlens arrays (MLAs), and integrator rods. The adoption of DOEs is limited to small angles and reduces transmission efficiency owing to typical diffraction losses. MLAs are refractive optical elements (ROEs) suitable for large numerical apertures (NAs) with minimal energy loss and no effect on laser beam polarization. However, MLAs remain expensive and difficult to process, install, and adjust. Although integrator glass rods are simple in structure, easy to process, and inexpensive, their length-to-width ratio becomes extremely large when small NA and sufficient reflections are required, making them unsuitable for space conservation.This research intends to be beneficial in terms of reducing structural complexity and production costs, as well as improving the effectiveness of the illumination uniformity unit's automatic optimization design.MethodsConsidering the advantages and disadvantages of the aforementioned homogenizer elements, an illumination uniformity unit comprising a plano-convex microcylindrical lens array (PCMCLA), condenser lens group, and a rectangular integrator rod is proposed in this study to obtain the uniform rectangular illumination field required in dry lithography. Compared with the current common uniformity unit structure based on double MLAs, the proposed structure reduces the MLA and the number of lens pieces in the condenser lens group. An MLA is typically made up of multiple microlenses of the same size arranged side by side. The aperture of each microlens divides incident light into different channels, indicating differentiation. The beam in each channel is refracted by a lens called an integrator or condenser lens. Subsequently, all channels are superimposed on the rear focal plane of the condenser lens, which is equivalent to integration. The operating principle of an integrator glass rod is based on multiple reflections that mix the incident light distribution and produce a homogenizing effect. Hence, the combination of these two elements can achieve illumination uniformity in the X and Y directions, respectively, which can be suitable for the small NA case triggered by a small partial coherence factor σ. Then, via geometric analysis, we inferred that in the X-direction, the spatial distribution of the incident beam of the integrator is linear with the direction cosine distribution of the outgoing light from the microlens. Accordingly, to obtain the optical optimization design of the plano-convex microcylindrical lens and condenser lens group in the CODE V software, the weighted square sum of the difference between the cosine value of each relative pupil ray's exit direction and its ideal value was adopted as an evaluation function. Taking the KrF lithography exposure system as an example, an illumination uniformity unit was designed via automatic optimization based on the evaluation function and conventional evaluation function of the diffusion spot root-mean-square (RMS). Subsequently, the obtained structural model of the uniformity unit was simulated using the LightTools software.Results and DiscussionsThe simulation results (Table 3) indicate that the non-uniformity of all types of output illumination in the simulation [illumination-integrated nonuniformity (IINU) of scheme A in Table 3] is less than 0.60% under different coherence factors, which is better than the contrasted optimization results [IINU of scheme B in Table 3] based on the conventional evaluation function of the diffusion spot RMS. For conventional illumination, IINU increases with the decrease in σ because this decrease implies a decrease in the light source area and NA. This decrease triggers a decrease in the number of microlenses involved and the reflection times of the extreme ray in the rod, which affects the homogenizing effect to some extent. When the outer ring diameter remains constant in annular illumination, IINU is getting smaller with increasing ring width. This is because the larger the ring width, the more similar the illumination pattern is to that of conventional illumination. In addition, when the ring width is constant, IINU is worse when the outer ring is larger than that when the outer ring is smaller. This is because the performance of the condenser lens with a large relative pupil is inferior to that with a small relative pupil. To further improve the performance of the proposed structure, different grey filters with specific transmittance distributions are added at the output end of the integrator rod to correct the illuminance distribution of the illumination fields with poor uniformity. Subsequently, the simulation results [corrected IINU of scheme A in Table 3] reach the IINU target of less than 0.43%. The corresponding normalized integrated irradiance profiles derived from the proposed design are presented.ConclusionsThe proposed structure of the illumination uniformity unit in lithography exposure systems comprises a PCMCLA, three-piece condenser lens group, and a rectangular integrator rod. To obtain the automatic optimization design of the microlens and condenser lens groups, an evaluation function based on the angle cosine distribution of light rays is proposed to characterize illumination uniformity. The simulation results indicate that the designed structure achieves the requirement of IINU (≤0.46%), and verify that the proposed evaluation function is more effective than the conventional evaluation function of diffusion spot RMS, thereby providing a simply constructed and economical reference for related engineering.

ObjectiveWhen laser propagates in gas medium, the gas absorbs the laser energy and causes the refractive index to change, forming the gas thermal effect and reducing the beam quality. The laser has a high power density in the inner channel of the system, so the attenuation of Gaussian beam transmitted in the inner channel is far greater than that transmitted in the outer atmosphere. In addition, the thermal blooming effect of high-energy laser is not only closely related to the beam shape, but also very complex due to the interaction between the air flow and the absorption of laser energy. We hope to establish a more comprehensive thermal coupling effect model of laser transmission, which will provide strong support for the design and performance evaluation of the high-energy laser system. In this paper, the theory of optical-fluid-thermal coupling effect is introduced, and the simulation model of the thermal coupling effect of combined transmission of the elliptical Gaussian laser beams is established.MethodsTo solve the thermal effect of laser transmission in gas medium, flow field calculation, optical transmission calculation and optical-thermal coupling calculation are required. The optical transmission satisfies Maxwell equations. When simulating the electromagnetic field propagation of optical structures with long distance in a large size space, ray tracing is performed by solving the ray position and wave vector. Fluid flow and heat transfer follow the three conservation laws of mass, momentum and energy. The energy attenuation of laser transmission can be calculated according to Beer’s law, while the refractive index change of gas medium due to temperature and density changes satisfies Gladstone-Dale relationship. The wavefront aberration of the laser propagating in the enclosed space of the system is obtained by solving the above theoretical model with the finite element method, and the beam quality is calculated by comparing the results with those of the ideal Gaussian beam. The established model of laser thermal coupling effect is in good agreement with the experimental law, and can predict the degradation of beam quality caused by the thermal effect of gas medium.Results and DiscussionsThrough numerical simulation, the natural convection phenomenon induced by laser heating in a cuboid enclosed space and the change process of the influence of the flow field on the beam propagation are analyzed, and the influential factors of the thermal effect are studied. After the light comes out, the surface temperature of the optical elements will continue to rise, while the air temperature will not change within a few seconds (Figs. 4 and 5). The optical path difference (OPD) distribution is elliptical Gaussian type at first, with obvious defocusing aberration, and its peak-to-valley (PV) value and beam quality show a trend of first increasing, then decreasing, and then stabilizing (Figs. 7 and 9). With the increase of the absorption coefficient, the OPD PV value caused by the thermal effect of gas increases (Fig. 12), but the absorption of the element surface has a low contribution to the thermal effect of gas medium (Fig. 13). The gas thermal effect can be suppressed and the beam quality can be improved by certain gas replacement methods (Fig. 15). The flow field distribution of each region in the spectral beam combining system is different, and the path of each sub-beam is different, so the aberration is also different. The number and power of sub-beams, the shape and distribution, the combining method, the transmission distance and the system layout will all affect the thermal aberration difference of the medium in the process of combined beam transmission.ConclusionsIn this paper, the theory of optical-fluid-thermal coupling effect is introduced, and the thermal coupling effect model of laser beam combining transmission in an enclosed space is established. The absorption of laser energy by the gas medium will affect the laser transmission, resulting in optical axis deflection, beam quality degradation, and changes in the shape of far-field spot. In this study, the influences of medium absorption, optical element absorption and gas replacement on the thermal effect of internal transport gas are analyzed. In the spectral beam combining system, all sub-beams will interfere with each other, the wavefront distribution will be different, and every sub-beam will have a dispersion trend. The model can analyze the thermal effects of different systems according to the actual situation, and the influence of structural parts and electronic devices on the flow field and laser transmission can also be considered, providing an effective reference for the design and performance evaluation of high-energy laser systems.

ObjectiveTo provide precise tip-tilt correction, the tip-tilt mirror (TTM) control with high bandwidth and high dynamic response is crucial for adaptive optics (AO) systems. The time delay of the system, which is a significant factor restricting the error attenuation bandwidth of the system, is 2-3 times the sampling period. Additionally, the dynamic response performance of TTM will be decreased by the vibration of the experimental device, the unstable operating environment, and other factors brought on by external system disturbances, as well as the time-delay parameters of the system model brought on by internal troubles. While strategies such as the linear quadratic Gaussian (LQG) control method and hybrid control can suppress the high frequency jitter of the beam, and the Smith predictor control method can compensate for the time delay in the system. Nevertheless, the common problem with these methods is that the control effectiveness depends on the accuracy of the model, which results in the inability to suppress disturbances. In order to achieve high bandwidth and high dynamic response TTM control in AO systems, a control method utilizing the Smith predictor and filter-based linear active disturbance rejection (FLADRC-Smith) is proposed to synchronously compensate for the effects of time delay and internal and external disturbances on the tip-tilt correction performance.MethodsIn order to increase error attenuation bandwidth, the FLADRC-Smith control method employs a Smith predictor to modify the linear active disturbance rejection control (LADRC) method. The AO system is considered a pure time-delay system. Therefore, a filter is designed to modify the control amount of TTM in order to achieve LADRC that is resistant to high frequency disturbance. In this paper, characteristics of the error attenuation transfer function of the control system are analyzed from a frequency domain perspective, and the control system is optimized and made simpler. Meanwhile, the stability of the control system when the time-delay parameters of the system model are varied is ensured by parameter constraints, and the connection between the error attenuation bandwidth and the system performance in suppressing internal and external disturbances is analyzed. The formula for the error attenuation bandwidth is supplied in the analysis, along with a straightforward method for tuning the parameters.Results and DiscussionsMATLAB/Simulink is used to establish the simulation model of AO tip-tilt correction to verify the effectiveness of the FLADRC-Smith control method. Firstly, it is confirmed that FLADRC-Smith can increase the error attenuation bandwidth of the system. The error attenuation bandwidth is 25 Hz with proportional-integral (PI) control and 114 Hz with FLADRC-Smith control, showing a 4.56 times improvement with FLADRC-Smith control, according to simulation results after properly setting the controller parameters (Fig. 6). Then, to verify the disturbance rejection ability of FLADRC-Smith control method, it is compared with the PI-Smith control method. The controller parameters are set properly so that both control methods result in the same system error attenuation bandwidth. When the Smith parameter and time-delay parameter of the system model are both set to 0.0025 s, both control strategies can quickly and stably track a constant value signal. However, when the time-delay parameter of the system model is mismatched to 0.008 s, the dynamic performance index is the transient time required to achieve and maintain the response of TTM at ±5% of the input signal amplitude. FLADRC-Smith control, compared with PI-Smith control, improves the dynamic response of TTM by 23.9% (Fig. 8). Finally, the FLADRC-Smith control method is compared with the PI-Smith control method in a second-order oscillatory time-delay system. The controller parameters are reasonably tuned so that the system error attenuation bandwidth is the same for both control methods (Fig. 9). Both control methods can track the constant value signal quickly and steadily when the Smith parameter and time-delay parameter of the system model are set to 0.0025 s. But when the time-delay parameter of the system model is mismatched to 0.008 s, the response transient time of TTM with PI-Smith control is 0.0564 s, while the FLADRC-Smith overshoot does not exceed ±5% of the input signal amplitude. It indicates that the required transient time is 0 when using the FLADRC-Smith control method, which significantly improves the dynamic response performance of TTM (Fig. 10).ConclusionsThe FLADRC-Smith control method can improve the bandwidth and the dynamic response performance of TTM effectively. In a pure time-delay system, this method improves the error attenuation bandwidth by 4.56 times compared with PI control. Under the same error attenuation bandwidth condition, the method improves the dynamic response performance of TTM by more than 20% with stronger suppression of internal and external disturbances compared with the PI-Smith control method in a pure time-delay system and a second-order oscillatory time-delay system.

ObjectiveUltrafast laser direct writing processing technology with a single focus has a small processing area and low energy utilization and is not suitable for applications in large-area processing, volume processing, and the single forming of structures. To solve these problems, researchers generally adopt multibeam parallel processing method. The most important aspect of parallel processing technology is to realize three-dimensional (3D) multibeams and form 3D multi-focus with the same energy. Therefore, a method for generating 3D multi-focus is urgently required. In industrial applications, laser focusing requires precise control to obtain high quality results. Moreover, in the actual optical path, setup and manufacturing errors exist in the optical elements and have a significant impact on the final machining results; however, the existing iterative algorithms rarely compensate for such errors. Therefore, a 3D multi-focus control method based on a feedback-weighted 3D-GS algorithm is proposed in this study.MethodsA spatial light modulator (SLM) can adjust optical parameters such as amplitude, phase, and polarization of a laser beam by loading computer-generated holograms. With the help of SLM, the light intensity distribution in the target region can be easily controlled. The key to obtaining uniform 3D multiple beams with SLM is to obtain the corresponding computer-generated holograms, which can then be used to flexibly control the number, position, and focus energy distribution of the outgoing laser beam. In this study, to improve the uniformity of the energy distribution of multiple beams, a feedback method was used to improve the traditional 3D-GS algorithm. Each beam was set using a weight coefficient. The feedback-weighted 3D-GS algorithm collected the energy and position information of multiple foci through a CCD camera in real time and fed the information back to the controlling terminal to dynamically adjust the weight coefficient of each beam. After a couple of iterations, 3D multi-focus with high uniformity were obtained.Results and DiscussionsThe coordinate parameters related to the desired 3D structures in different Z-axis planes are set, and the energy uniformity of the 3D multi-focus is calculated using the feedback-weighted 3D-GS algorithm and traditional 3D-GS algorithm. For the expected 3D structure of “HBUT,” the feedback-weighted 3D-GS algorithm improved the uniformity of 47 diffraction points on four different planes from 47% to more than 96% after 20 iterative feedback calculations. Using the feedback-weighted 3D-GS algorithm, the homogeneity of the 3D multi-focus energy distribution is significantly improved, as shown in Table 1. Another experiment was performed for the 3D spiral structure, and the feedback-weighted 3D-GS algorithm improved the uniformity of 15 diffraction points on 15 different planes from 47% to 94% after five iterative feedback calculations. The spots at the corresponding positions were filled according to the 3D model of the spiral structure (Fig.9). From a comparison of spot energy distribution calculated by different algorithms, it can be found that the presented feedback-weighted 3D-GS method can effectively improve the uniformity of 3D multi foci.ConclusionsBased on the feedback-weighted 3D-GS algorithm and programmable SLM, this study proposes a method to generate 3D multi-focus with high uniformity to compensate for the fabrication and setup errors of devices in real optical paths. The designed pattern of “HBUT” and helical structures are used to prove the validity of the method. Through the analysis and calculation of feedback parameters in iterative feedback calculations, the uniformity of the 3D multibeam obtained by this method is verified to be 95%. The number of multifocal points in the 3D structure has a greater influence on the stability of the reconstructed 3D multi-focus light field than the number of Z-planes in the 3D structure during the feedback iterative calculation. Additionally, the laser high-uniformity 3D multi-focus optical field reconstruction technique proposed in this study can be used for 3D structure machining.

ObjectiveThe citrus industry, one of the most important fruit industries in China, is currently focusing on nondestructive internal quality grading. Near-infrared (NIR) spectroscopy has been widely used to provide accurate analysis of the material components of agricultural products, owing to its advantages of convenience and efficiency. Among different measurement setups, optical fiber probes are frequently used as a key optics accessory for collecting the NIR spectrum. The Monte Carlo (MC) method offers an accurate description of light propagation in the fruit tissue using a multilayer sample model, which provides the theoretical basis for the design of a more effective optical fiber probe in fruit-quality inspection. However, in the MC model, the target sample is usually described as a combination of multiple semi-infinite turbid mediums, which simplifies the structural characteristics of the tissue. Moreover, the structure and geometry of optical fiber probes are also ignored in most cases, which may lead to deviations in the simulation results. This study develops a more detailed description of light propagation in citrus tissues by analyzing the photon transmission near the optical fiber probe as well as the surface boundary of the sphere sample and suggests a probe design for reflectance-based detection.MethodsBased on the characteristics of citrus fruits, a three-layered optical model in the shape of a sphere is established, which includes flavedo, albedo, and vesicle layers with different absorption coefficients, scatter coefficients, refractive indices, and anisotropy factors. A simulation system of citrus-quality detection is developed, comprehensively considering the injection near the source fiber, the photon track near the surface boundary, and the information collected by the detector fiber. The corresponding changes are implemented in the general MC code for the simulation of transmission characteristics in the citrus tissue, including normalized relative diffuse reflectance, average motion pathlength of photons, and the percentage of effective photons in vesicles. Then, MC models with specific parameters of the optical fiber probe, including the numerical aperture and radius of both the source and detector fibers, and the source–detector distance are established for MC simulation.Results and DiscussionsBased on the simulation results at a wavelength of 800 nm, the guiding principle of the optical fiber probe design suitable for citrus spectrum acquisition is determined. As shown in Fig. 4, the normalized relative diffuse reflectance increases with an increase in the radius of the detector fiber and decreases with an increase of source-detector distance, while other parameters show no significant influence. The average motion pathlength of the received photons increases with increasing source-detector distance (Fig. 5). In this case, the acquired spectrum carries more internal information about the citrus sample. For the percentage of effective photons in vesicles, a large detector fiber radius of 250 μm or more is recommended to achieve a stable result (Fig. 6) as well as a longer source-detector distance. Based on the simulation results, a new structure of “4 in 9 out” coaxial fiber probe is designed to direct light emitted by the light source and to receive the feedback signal (Fig.7). This versatile fiber probe achieves a collecting efficiency of 3.24% and a percentage of effective photons of 0.07% in MC simulation.ConclusionsNondestructive inner quality inspection techniques play an important role in the fruit industry. In this study, light propagation through the citrus tissue is simulated by the MC method to determine the relationships between fiber probe geometry parameters and the detected optical signal. A three-layered media model in the shape of sphere is established for further simulation and analysis. The optical fiber probe parameters that are closely related to NIR spectrum measurement are explained and introduced into the general MC model, and corresponding simulations are performed to determine the basic designing principles of the fiber probe in citrus tissues. Based on the simulation results, the radius of the detector fiber probe and the source-detector distance can be optimized to obtain more information of interest, while other parameters including the radius of the source fiber and the numerical aperture show limited impact on the simulation results. To achieve higher collection efficiency, a larger detector radius and smaller source-detector distance are recommended. Moreover, the photons received by the detector fiber are found to carry more information on the inner citrus tissue when the source-detector distance increases. A versatile optical fiber detection structure is designed with a large detector fiber radius and multiple source-detector distances to increase the level of received information of the citrus sample. The MC simulation result of the fiber probe indicates that the photons from the vesicle layer can be efficiently collected and the sensitivity of citrus inner quality detection is further improved, which provides a theoretical reference for designing a new detection accessory of nondestructive quality evaluation by NIR spectroscopy.

ConclusionsIn this article, the influence of pump symmetry on the dynamic drift of the beam in a large-aperture chip amplifier is introduced. The beam dynamic drift characteristics of two large-aperture slap amplifiers and amplification configurations are compared by the numerical simulation calculations, and a large-aperture high-profile pump based on a pulsed xenon lamp with a diameter of 37 mm and asymmetrical arrangement of multiple lamps is studied experimentally. The dynamic drift characteristics of the beam caused by the pump of the gain slap amplifier are studied. When the symmetry of the pump field of the neodymium glass sheet is about 1.038∶1, the pump-induced beam dynamic wavefront tilt PV value is about 0.98λ (6 sheets accumulation), and the dynamic drift angle of a single laser passing through a single neodymium glass plate is about 0.32 μrad in 375 mm beam aperture, which is well consistent with the result of the theoretical model. This can provide reference for the design of slap amplifiers with different pump configurations as well as the design of multi-path optical paths of laser devices and the design of spatial filter apertures.

ConclusionsIn this study, a phase modulating metasurface based on ITO film is designed. The combination of the ENZ characteristics of ITO and the plasma resonance effect of the upper metal electrode significantly enhances the light-matter interaction. By designing the metal-insulator-metal (MIM) capacitor structure, a phase modulation range of >330° is achieved under the gate voltage -2-5 V. Consequently, the optical phase array and tunable focal length lens are successfully realized. The designed phase modulating metasurface is significant for applications, such as optical detection, imaging, and communication.

ConclusionsThis paper proposes an ultrafast beam smoothing scheme based on the rotation arrangement of phase plates. By rotationally arranging the phase plates with a rotational asymmetric distribution in the laser quad, different spatial phase modulations are first provided for each sub-beam in the laser quad. Then the dynamic interference of the sub-beams with a certain wavelength difference on the target plane makes the speckles within the focal spot sweep rapidly in multiple directions and multiple dimensions, so as to achieve the goal of improving the uniformity of the focal spot in the picosecond time scale. On this basis, the effects of parameters of phase plates, beam arrangement, spatial wavefront distortion, and spatial deviation of laser beams on beam smoothing are analyzed. The results show that in the ultrafast beam smoothing scheme based on the rotation arrangement of phase plates, only the same phase plates with a rotational asymmetric distribution are required to be processed at the same time, which can lower the design and processing difficulty of phase plates. In addition, this scheme is little affected by spatial wavefront distortion, beam arrangement, and spatial deviation of laser beams. Combining the beam smoothing scheme based on the rotation arrangement of phase plates proposed in this paper with the traditional beam smoothing schemes can significantly improve the uniformity of focal spots within a few picoseconds, which can be an effective supplement to the traditional beam smoothing schemes.

ConclusionsThe propagation model for the laser quads in the hohlraum based on broadband laser beam smoothed using ISI and de-DLA has been built in this study, aiming at the optical path arrangement scheme and cylindrical target hohlraum structure of the NIF, and then the irradiation characteristics of the laser quads on the hohlraum wall have been analyzed and optimized. The results show that the mismatch between the focal length of the primary lens and parameters of the cylindrical target hohlraum will cause varying degrees of damage to the envelope of the laser spots on the hohlraum wall for beams with different incident angles, as a result of the laser spots overlapping on the hohlraum wall, the irradiation uniformity on the hohlraum wall greatly diminishes. The effect of the principal lens focal length, the number of DLA sub-lens, and the long and short axes ratio of the sub-lens on the intensity distribution of the laser quads on the hohlraum wall are discussed on this basis. The optimization of these parameters, such as the principal lens focal length, number of DLA sub-lens, the long and short axes ratio of the sub-lens, and incident angle of the laser quads, achieves the improvement of the irradiation uniformity on the hohlraum wall. The results show that increasing the focal length of the primary lens appropriately can effectively retain the envelope of the laser spots on the hohlraum wall for beams with different incident angles, minimizing the overlap of the laser spots on the hohlraum wall. Besides, the duty ratio of laser spots on the hohlraum wall increases, and the time required for beam smoothing reduces when the number of DLA sub-lens and the long and short axes ratio of the sub-lens are optimized. Furthermore, after optimizing the incident angle of the inner cone laser quads, the overlap of the inner and outer cones laser quads on the hohlraum wall is eliminated.

ConclusionsIn this paper, the beam quality optimization of an elliptical Gaussian beam under wind-dominated thermal blooming is studied analytically and numerically. The expression of distortion parameter of an elliptical Gaussian beam propagating in the atmosphere is derived, and its correctness is proved. Due to the astigmatism of an elliptical Gaussian beam, the influence of atmospheric thermal blooming on its propagation depends on the wind direction. Thermal blooming can be weakened by making the short axis of a focused EGB along the general wind direction. In the atmosphere, as the beam width in the windward direction of the source plane is large (the same spot area), the thermal blooming effect on the EGB is weak, which results in a better symmetrical spot and energy focus ability at the target, i.e. better beam quality at the target. The time required to achieve steady-state thermal blooming for a focused elliptical Gaussian beam is proportional to the beam width along the wind direction in the source plane.

ConclusionsIn a two-dimensional plane, the characteristics of a light field near the focus of a 1D convergent GB are analyzed using a 1D diffraction integral formula. As the normalized inclination factor that is expressed by the square root of the cosine of the observation side inclination angle is engaged, the characteristics of the focal-plane light field of the 1D nonparaxial convergent GB that is expressed by a new diffraction integral formula conform to the law of conservation of radiation energy in traveling wave field, the characteristics of a light field near the focal line of the 1D nonparaxial convergent GB are reasonable. Then, the propagation characteristics of the GB are extended from the paraxial field to the nonparaxial field using the reasonable observation side inclination factor. Furthermore, the rationality of the observation side inclination factor expressed by the square root of the cosine of the observation side inclination angle is verified.

ObjectiveOptical integrated aperture imaging involves an array of objects in a certain form based on multiple small aperture sub-mirrors arranged in a certain form. Its imaging resolution can be equivalent to that of a large-aperture lens, which is widely used in astronomy, medicine, commerce, military scientific research, and other fields. Resolving the problems of loading error and environmental engineering requires independent static or dynamic phase compensatory adjustment for each sub-mirror to meet the phase modulation accuracy requirement of 0.1λ (λ is the wavelength of incident light)or the higher requirement. Liquid optical devices have potential application prospects owing to their compact structure, light weight, and low price, and they do not require mechanical devices. A piezoelectric ceramic tube has the advantages of high sensitivity, good linearity, strong integration, and easy control. Furthermore, the tube can be filled with clear liquid and its length is adjustable, providing a new approach for the preparation of liquid optical phase modulators. Much research has been conducted on magnesium alloy liquid optical devices and good results have been achieved. However, further research is necessary to make practical engineering applications possible and new liquid optical phase modulators must be developed.MethodsTransparent liquid is filled into the cavity of a piezoelectric ceramic tube. The length of the liquid produces a micrometer-level displacement change because of the inverse piezoelectric effect. Then, encapsulation and illumination are performed in the cavity length direction and the optical phase produces little change. A piezoelectric ceramic tube with an inner diameter of 15 mm, outer diameter of 20 mm, and height of 12 mm is filled with methyl silicone oil and encapsulated with a gasket, an upper cover sheet, and an lower cover sheet. Finally, a Michelson interferometer is used to observe and analyze the accuracy, range, response time, and other performance characteristics of the phaser.Results and DiscussionsFirst, integer fringes are detected using fixed-line grayscale values. Alignment gray value detection involves making a horizontal reference line in the middle of the interference image and recording all gray values on the reference line. These gray values can reflect the intensity change trend of the interference fringes. Then, the curve is detected and recorded once at a certain voltage interval, and a series of curves of the relationship between the gray values and the position of the fringes are obtained. The differences between the positions of the dark and bright fringes are compared; as a result, the phase shift can reflect the relationship between the shift of phase and the voltage (Fig. 6). The interference fringes are recorded, and an extreme value of the grayscale curve appears every 7.0 V interval, which can be understood as a first-order shift of the interference fringe, indicating that the optical path is adjusted by λ/2 (Fig. 7). Moreover, the fixed-line grayscale values are used to detect the fractional fringes, and the interference fringes can move in one direction under the driving voltage. The fractional counting of interference fringe can be achieved by comparing the movement of the fringes in a single cycle and calculating the movement of a fringe relative to the previous position (Fig. 8). The variation curve of the optical path of phase modulator with the applied voltage measured at 0-28 V shows the good linear relationship (Fig. 9). The measured phaser response time is 7 ms (Fig. 10).ConclusionsTo meet the high-precision requirements and achieve a large adjustment range, a liquid optical phase modulator is constructed by filling the transparent liquid into a piezoelectric ceramic tube. Optical phase adjustment is performed through a small electric displacement generated by the piezoelectric effect in the light-transmitting direction. The piezoelectric ceramic tube filled with transparent liquid is used as the core structure of transmissive optical phase modulator. The optical phase modulation devices in this study are not only compact and significantly reduced in cost but also have high precision and a wide range. An experiment is conducted using the fixed-line grayscale values and a Michelson interferometer with a wavelength of 632.8 nm. Under a voltage of 0-28 V, the phase modulator can reach a modulation range of 0-4π and an accuracy of λ/36. When the applied voltage is 150 V, the modulation range can be expanded to 20π. This meets the requirements of optical synthetic aperture subaperture phase modulation.

ObjectiveCurrently, the approaches to achieving uniform illumination largely involve adjusting the array arrangement of light-emitting diode (LED) plant light sources, designing the structure of an LED light source board, and adding optical features such as diffusion plate or free-form surface baseplate. These approaches cannot effectively enhance the uniform illumination of LED plant light because the optical efficiency is not high, and it is not conducive to large-area lighting of LED plant lamps. The red and blue LED array light sources widely used in plant manufacturing have been tuned to provide spectral uniformity based on the aforementioned issues. The scale lens is then designed, and the light output by the LED is uniform by employing the scale radian of the scale lens, which improves the optical efficiency and uniformity of the light surface by increasing the light coupling degree. It has some practical utility and serves as a guide for consistent illumination of large-area plants in practical engineering.MethodsFirst, 14 pieces of 1 W Samsung 3535, type LH351H 450 nm L4A4B blue LEDs and 70 pieces of 1.6 W Samsung 3535, type LH351H 660 nm V2 E7W410 red LEDs were used as plant lamp light sources, divided into seven rows with 12 LEDs in each row with a size of 380 mm×220 mm of LED plant lamp; the ratio and way of arrangement of the red and blue LEDs were studied. After comparing the spectra of two distinct LED array arrangement methods, the nine-point method was used to measure the spectrum. Second, a scale lens was designed according to optical and nonimaging optical principles. The scale radian of the scale lens ensures that the light output by the LED is uniform. Then, to find the best scale lens size, the impacts of different bulge heights, scales width, and the spacing between two LEDs center columns on photosynthetic photon flux density (PPFD) uniformity and optical efficiency were analyzed. Finally, the results were verified by experiment.Results and DiscussionsIt is the first time that a scale lens has been applied to an LED plant lamp to improve the illumination spectral uniformity and optical efficiency of the LED plant lamp. The homogeneous arrangement of red and blue LEDs was used to evaluate the LED plant light source. The experimental spectrum with uniform arrangement of red and blue LEDs in the LED plant light is essentially consistent with the theoretical simulation spectrum, and the spectral uniformity is good (Fig. 17). The scale lens was designed according to optical and nonimaging optical principles. By comparing the effects of the width and convex height of the scale lens on the PPFD uniformity and optical efficiency of the plant lamp, the optimized convex height H of the scale lens is 0.1 mm, the optimized scale width D is 0.35 mm, the optimized center column spacing B between two LEDs is 25 mm, the ray-tracing simulation with the TracePro software revealed that the light distribution curve of the whole LED plant lamp exhibits batwing light distribution. The experiment and simulation were well-accorded (Fig. 19). The test results show that the scale lens module has a PPFD uniformity of 93.12% in the illumination area within 500 mm×500 mm (Fig. 20), and optical efficiency of 90%, which is higher than the technical indexes reported in the references, thereby demonstrating the effectiveness of the scale lens.ConclusionsThe software simulation demonstrates that when the photon number ratio of red light to blue light is 8∶1, and 14 pieces of 1 W Samsung 3535, type LH351H 450 nm L4A4B blue LEDs, and 70 pieces of 1.6 W Samsung 3535, type LH351H 660 nm V2 E7W410 red LEDs are divided into seven rows with twelve LEDs in each row with a size of 380 mm×220 mm of LED plant lamp, the spectral uniformity of uniform arrangement of red LED and blue LED is better than that of uniform arrangement of blue LED in two columns. The experimental findings reveal that when the length of the scale lens is 22 mm, the width is 16 mm, the maximum height is 6.68 mm, the diameter of the bottom surface of the inner surface is 6.26 mm, the height is 2.98 mm, the height of the scale protrusion is 0.1 mm, the width of the scale is 0.35 mm, and the distance between the center columns of the two LEDs is 25 mm, the beam angle of the plant lamp is 90°, the receiving surface is 0.5 m away from the light-emitting surface of the plant lamp, and the 500 mm×500 mm plant illumination surface light source with PPFD uniformity of 93.12% and optical efficiency of 90% is obtained in the illumination area.

ObjectiveLocalized hollow beam optical tweezers techniques have been widely used in life and nanosciences as physical tools to generate microforces without direct contact and realize the precise manipulation of microparticles. The expanding application needs have created more requirements for the optical tweezers technology, necessitating new optical field regulation techniques for producing various tunable optical traps. In conventional axial cone-lens optical systems, closed localized hollow beams can be generated to capture particles. However, once the localized hollow beams form, it is difficult for the particles to pass through the light wall into the light trap. Furthermore, the size cannot be freely regulated after the conventional localized hollow beams are produced. Therefore, this paper proposes the production of a localized hollow beam that freely opens and closes, allowing the particles to enter through the gap and controlling the particle capture and escape using the change in the gap. This is important for capturing particles and more conducive to such an operation.MethodsWe added a rectangular appendix in a conventional axial cone-lens optical system. We found that by modulating the incident circular Gaussian beam using the axial-cone optical appendix, the localized hollow beam generated after the axion-lens optical system could regulate and had opening properties. We simulated the distribution properties of the local rectangular Gaussian beam by simulating the beam passing through the axis cone and the beam after focusing on the incident lens. Next, we analyzed the causes and influences of localized hollow beam defects and experimentally verified the localized hollow beam from generation to closure. The particles were first analyzed using the beam properties in both longitudinal and transverse directions in localized hollow beams. The longitudinal gradient and scattering forces were calculated in the beam propagation direction, and gravity was claculated in the transverse direction to analyze the particle process from the gap into the stable confinement.Results and DiscussionsThe incident circular Gaussian beam is modulated using a rectangular aperture, and the size and orientation of the gap are adjusted by changing the length-width ratio of the rectangular aperture. This ensures that the localized hollow beam generated after passing the axial cone-lensing optical system freely regulates the size. Furthermore, we increase the localized hollow beam light field gradient using a high numerical aperture lens, and analyze the gradient force, scattering force, and gravity of the particles in a liquid environment. In this paper, the process of particles from entering the bottle beam to stable trapping is recorded by analyzing the lateral and longitudinal forces (Fig.7).ConclusionsWe designed a size-tunable bottle beam. To be more specific, the size of the bottle beam can be changed by adjusting the distance between the axicon and the lens. Besides, the gap can be generated by adjusting the length-width ratio of the rectangular aperture. We use MATLAB to simulate the size change of the bottle beam in the adjustment and the Bessel beam change of the beam after the axicon. Furthermore, we demonstrated the process of the bottle beam from forming to closing with the help of simulation and experiments. Experimental results show that the simulation of gradient, scattering, and resultant forces in size-adjusted localized hollow beams creates a particle passage through the gap and controls particle capture and escape according to the change of the gap.

ConclusionsIn this study, a flat-topped beam with good uniformity is obtained using a spatial light modulator and designing a phase hologram composed of a geometric mask and gratings inside and outside of the mask. Among these, the internal grating can be gradient orthogonal grating or gradient binary grating, which can diffract the incident Gaussian beam into multiple nonzero-order beam to change the energy distribution and obtain a flat-topped beam with high uniformity. At the same time, the grating methods are compared to the GS method. Although GS method can obtain high energy efficiency, the uniformity is poor, so it is not suitable for areas requiring high uniformity. The method proposed in this paper can obtain a shaping beam with steep edge and high uniformity, with an energy efficiency of 43.71% and a minimum nonuniformity of 2.12%. It can be used in areas where high uniformity and steepness edge are required, such as scribing and removing films from coating materials (thin films without harming the substrate), and grooving in semiconductor materials (such as Low-K). This method is simple and easy to use, and it will broaden the scope of laser applications in the field of material processing.

ConclusionsIn present study, aiming to the temporal and spatial evolutional characteristics of SSD beam focal spots and based on the optical imaging system of a streak camera, we carry out the beam test experiment with time resolution on the high-power laser device. Experiments demonstrate the influence of SSD beam control parameters on the dynamic evolutional characteristics of focal spots. The focal spots along the SSD scanning direction have typical CPP shaped speckle patterns. The dynamic evolutional modes presented by different modulation frequencies are different. The overall spatial-temporal distribution of intensity modulated by high frequency is more uniform, and a better smoothing effect can be achieved in short smoothing time. The spatial distribution and temporal evolution of the focal spot intensity in the vertical SSD sweep direction become very chaotic, showing the specific differences of SSD beams in two-dimensional space. The focal spot of the SSD beam has strong amplitude modulation, and the modulation has an obvious regional effect. Modulation depth is generally related to the SSD beam control parameters. Our results provide a better understanding of the dynamic evolution of the spot details of SSD beams, which can be combined with the mechanism analysis of LPI to support the optimization of SSD smoothing performances. The inconsistency of SSD beams in two dimensions can be used to optimize the sweeping direction of SSD, and the modulation structure of SSD beams can assist in the selection of modulation frequencies. An accurate description of focal characteristics provides important boundary conditions for optimizing the location of focal trajectory points.

Objective Inertial confinement fusion (ICF) is important to achieve controlled nuclear fusion. To solve the problem of low energy coupling efficiency in the traditional ignition scheme in a laser-driven ICF experiment, Zhang Jie and other researchers proposed double-cone ignition (DCI) as a new solution for ICF ignition. In DCI experiments, using laser irradiation, the surface area of the target material filled in the metal cone decreases sharply as the laser irradiates fusion targets; therefore, the focal spot size of the irradiated beam must be dynamically reduced accordingly. In this process, the control of shooting laser beamlines in the laser driver must be flexible. Thus, to meet the aforementioned demand, the dynamic focusing of shooting beams has been proposed. In 2013, to mitigate cross-beam energy transfer during low-adiabat cryogenic experiments on the OMEGA laser facility, researchers in the Laboratory for Laser Energetics of the University of Rochester in the United States proposed a technical solution of two-state focal zooming. The dynamic focusing process of the proposed technique is similar to that of the two-state focal zooming. Two combined spatiotemporal laser pulses are amplified by coaxial propagation. Then, a specially designed continuous phase plate is used to modulate the wavefronts of these two beams (denoted as beams 1 and 2) and smooth the focal spot separately. Next, the target material is irradiated by laser spots after focusing. In this study, to realize beam control using dynamic focusing technology, a new method for achieving dynamic focus using a combined spatiotemporal beam based on a preamplifier system is proposed to provide support for the subsequent research of DCI.Methods Herein, near-field beam quality during the generation and propagation of the combined spatiotemporal beam is analyzed. First, based on the Fresnel diffraction propagation theory and the numerical simulation method of fast Fourier transform, a propagating model of the combined spatiotemporal beam is established. Then, the influences of the filter pinhole diameter, softening factors, extinction ratio, and phase difference on the propagation of the combined spatiotemporal beams are discussed. Finally, based on the overall plan of the spatiotemporal beam combination, an experimental laser system is constructed to verify the accuracy of the numerical simulation.Results and Discussions In the analysis of the generation and propagation of the combined spatiotemporal beam, the outer size of beam 1 is 12 mm×12 mm, the outer edge softening factor is 0.1, the inner circle diameter is 6 mm, and the diameter of beam 2 is 6.6 mm. When the softening factor of the inner edge of beam 1 is Q3≥0.0625 and that of the outer edge of beam 2 is Q2≥0.055, and the pinhole of the spatial filter is greater than or equal to 32DL. After the propagation of laser in the 4f system, the near-field modulations of beams 1 and 2 are less than 5% at the positions of the image plane and 15 cm away from the image plane (Fig. 5). These results meet the demand for the near-field beam quality of the combined spatiotemporal beam. Furthermore, considering the perspective of risk prevention, when the time-domain extinction ratio is higher than 30 dB and the spatial extinction ratio is higher or equal to 20 dB, the influence of near-field intensity fluctuation on the system can be negligible [Fig. 6(b)]. Presently, the time-domain extinction ratio of the system can be greater than 40 dB and the spatial extinction ratio is ~20 dB, meeting the requirements of intensity stability of combined spatiotemporal beams in the propagation.Conclusions Herein, to meet the physical experimental requirements of dynamic focus in the scheme of DCI using a high-power laser facility, a method based on the spatiotemporal beam combination laser preamplifier system in the laser driver is proposed. Based on Fresnel diffraction propagation theory, the laser propagation model of the combined spatiotemporal beam is established using the numerical simulation method of fast Fourier transform. The effects of the filter pinhole diameter, softening factor, extinction ratio, and phase difference on the propagation of the combined spatiotemporal beams are analyzed in the simulation, and reasonable softening factors and spatial filter pinhole diameters are obtained. The preliminary experimental results agree well with the simulation results. Moreover, the parameters of the facility can meet the intensity stability requirements of the propagation of the combined spatiotemporal beam from a risk prevention viewpoint. The numerical model can be used for optimizing other parameters, which can guide the propagation and amplification of the combined spatiotemporal beams in future research. The proposed method will be used in the spatiotemporal beam combination system of the preamplifier of the high-power laser facility. In the future, experimental research will be conducted on dynamic focus to meet target physics requirements.

Objective The solid-state slab laser has become one of the most reliable, promising and potential lasers among current high-power lasers due to its small size, lightweight and high conversion efficiency. It is commonly used in various fields, including scientific research, industry and medical treatment. High output power and good beam quality are two constant goals in the development of high-power solid-state slab lasers. As laser output power increased, the edge effects became more severe and distorted the wavefront of the solid-state slab laser output beam, resulting in a non-linear drop in laser beam quality. Several methods are present to compensate for the distortions, including the use of static compensation components, non-linear optical compensation methods, and adaptive optics. Adaptive optics is a promising method to compensate for wavefront distortion in the laser output beam. The direct slope reconstruction method is commonly used in the research of solid-state slab laser beam clean-up, and the least-squares algorithm is used to solve the system’s optimal solution.Although adaptive optics can considerably improve the beam quality of laser output beams, some issues still need to be addressed. The least-squares reconstruction method’s criteria are to minimise the sum of slope residual squares. The adaptive optics system’s correction capability is limited by factors such as materials and cost. When the adaptive optics system can completely compensate for wavefront distortion, the least-squares reconstruction method can be used to obtain the system’s optimal solution. However, if a portion of the distortions is beyond the capability of adaptive optics system, the wavefront distortions of the laser beam cannot be fully compensated and a considerable amount of wavefront residual is still present after compensation; the minimum sum of slope residuals squares is not equivalent to the best beam quality at this time. When the solid-state slab laser operates at high gain, the wavefront distortions are very likely to exceed the adaptive optics system’s correction capability. Under these conditions, the least-squares reconstruction method cannot produce the optimal system solution.Methods To solve the abovementioned problems, the most straightforward and effective method is to increase the number of actuators or even cascade multiple deformable mirrors with compatible wavefront sensors to improve an adaptive optics system’s inherent correction capability. However, as the number of actuators in deformable mirrors increases, their size, weight and cost also increase. Another approach is to optimise the adaptive optics system’s correction method, and a weighted least-squares reconstruction method has been proposed to improve the beam quality by assigning low weights to the uncorrectable wavefront components in the least-squares method. Unfortunately, determining the weights in a practical adaptive optics system is difficult. The edge effect remains a challenge, particularly when the number of actuators is limited because of the beam size or cost.We proposed a novel adaptive optics correction method to further improve beam quality when wavefront distortions exceed the adaptive optics system’s correction capability. In this method, we used the idea of optimal correction in the wavefront sensor-less adaptive optics system and combined it with the traditional adaptive optics system, with the improvement of beam quality as the optimisation goal, and the optimisation algorithm is used to optimise the calibration position of the wavefront sensor according to wavefront distortions and the correction capability of the deformable mirror, and it then uses the traditional adaptive optics system for aberration compensation.Results and Discussions We used simulation to validate the proposed method. First, we combined the two-dimensional Legendre polynomials based on the characteristics of the solid-state slab laser’s output wavefront to obtain a wavefront with severe edge distortion, and the corresponding beam quality is β=5.1 (Fig. 6). Then, the optimisation algorithm is used to find the best solution; the best beam quality that can be obtained after correction is β=1.8 (Fig. 7). Finally, the proposed method and the traditional correct method are used to compensate for the distortions (Fig. 8). After correction using the traditional method, the beam quality improves to β=3.7, whereas correction with the proposed method improves the beam quality to β=2.0, which is closer to the optimal solution. Analysing the wavefront slope distribution using a different method reveals that after processing using the proposed method, the effect of uncorrectable large distortions on adaptive optics systems is reduced (Fig. 9).Conclusions When the correction capability of the deformable mirror is limited, the correction results obtained using different methods show that, when compared with the traditional adaptive optics system without calibration optimization the method proposed in this paper can effectively improve the correction effect of the adaptive optics system.

Objective Recent studies have shown that the optical system that produces the bottle beam is gradually diversifying. However, most optical systems are more complex and the resulting bottle beams are larger in size. This article uses the metasurface to generate a bottle beam. The ultrasurface system is straightforward and well-integrated. The lateral and longitudinal inner diameters of the generated bottle beam are considerably reduced, and the capture of tiny particles is more accurate. This has potential research and application value for multiparticle capture and precise capture.Methods In this study, to generate multiple bottle beams, an opaque annular obstacle is added to the metasurface with the hyperbolic phase distribution (PB phase). The metasurface is constructed with titanium dioxide (TiO2) nanopillars arranged on a silicon dioxide (SiO2) substrate. To design the working wavelength of the nanopillars to 632.8 nm, the length, width, and height of the super surface nanopillars are designed to be 377, 87, and 600 nm, respectively. Moreover, by varying the relative aperture value (RA) of the metasurface, the lateral and longitudinal inner diameters of the bottle beam are altered. The number of bottle beams produced can be altered by changing the size of the annular obstacle on the metasurface.Results and Discussions This article produced four micron-level bottle beams [Fig.6(a)]. Further, by increasing the RA value of the metasurface, the lateral and longitudinal inner diameters of the bottle beams are reduced [Fig.7(a)]. Thus, the RA value of the metasurface can be changed to alter the size of the generated light field. We select one of the bottle beams and observe that its transverse and radial full width at half maximum (FWHM) are roughly linear with RA [Fig.7(b)--(c)]. In this paper, the number of bottle beams produced is changed by changing the size of the annular obstacle on the metasurface [Fig.8(a)--(c)].Conclusions The metasurface method used in this paper generates four bottle beams. Two of the bottle beams are selected. The measured FWHMs of the two bottle beams are 0.47 and 0.61 μm in the transverse direction, respectively, and 0.9 and 1.2 μm in the longitudinal direction. Simultaneously, this paper finds that by changing the RA value of the metasurface, the inner diameter of the multiple bottle beams is variable, and its transverse FWHM and radial FWHM are roughly linear with RA. Therefore, if particles with a specific size need to be captured, the metasurface with a specific RA can be designed to generate a bottle beam with the required size. This paper also found that the size of the outer ring that controls the annular obstacle remains unchanged, and when its size is changed, the number of the bottle beams is changed to 2, 4, and 5, respectively. The size of the multiple bottle beams produced in this paper is considerably reduced, and capturing tiny particles is more accurate, which is of great significance to the study of particle capture.

Objective In the implosion process of laser inertial confinement fusion (ICF), several requirements are proposed for the uniformity of the target irradiation and various beam-smoothing techniques, including spectral dispersion smoothing (SSD), have been developed and applied. According to different physical processes, different physical parameters are adopted in different periods to further improve the focal spot smoothing effect. Considering OMEGA as an example, the main pulse generated by its front end is divided into two parts, corresponding to SSD. In the initial part of the pulse, high-frequency multifrequency phase modulation is used to achieve a wide bandwidth output to obtain a better smoothing effect. The relatively narrow spectral width output can smooth the main pulse and suppress the stimulated Brillouin scattering. Finally, the initial part of the pulse and the main pulse form a complete pulse output through the beam combination. The modulation requirements of the pulse light in the main pulse front end of the OMEGA device differ at different time points. However, the pulse light in the main pulse front end of the OMEGA device is realized by splicing time and space; the optical system is relatively complex. Therefore, it is necessary to implement a scheme that can achieve different modulations at different time points, changing the corresponding spectral width. The scheme can meet different spectral width requirements of different time points in the pulse light without beam-splitting modulation.Methods This study investigates the specific phase modulation technology. The specific phase modulation function is obtained by integrating original phase change function of the target. It is essential that the modulation depth of the specific phase modulation function is a time-varying function. The spectrum width can be changed at any time using a specific phase modulation function to modulate the phase of the pulse light. Based on phase modulation spectrum theory, the spectrum characteristics of the laser with a specific phase modulation are analyzed. Using an arbitrary waveform generator (AWG) for digital pulse shaping, two output channels of AWG are used to output specific phase-modulated electrical signal and pulse shaping signal, respectively. After amplifying the two signals of two output channels of AWG using two electric amplifiers, the pulse shaping signal is connected to the bias modulator to shape the output of the continuous wave (CW) laser into the pulse optical input phase modulator. Additionally, a specific phase-modulated electrical signal is loaded onto the phase modulator. The modulation spectrum of 250-ps signal light at different time points in 3 ns is obtained by changing the relative time difference of two electrical signal outputs using AWG. The experimental results are consistent with the theoretical simulation.Results and Discussions To obtain a modulation output with gradually increasing spectral width, phase modulation function f1(t) is obtained by integrating the target phase change function φ'1(t) (Fig. 1). Under the modulation of f1(t), the spectrum (Fig. 6) of the pulse light increases slowly at first. Then, the spectrum width significantly increases as the modulation depth increases, vibrating in a zigzag pattern. The oscillation amplitude of the experimental results is relatively large due to the influence of the measurement accuracy of the spectrometer. Since the signal light has a certain time width, the broadening of the spectrum appears delay and tailing at the beginning and end of the experiment, respectively. If the sampling interval is constant and the signal pulse width is reduced, the oscillation amplitude of the spectrum width with time will increase. However, if the pulse width of the signal light remains unchanged and the sampling interval is reduced, the zigzag oscillation will be smoother. If the pulse width of the signal light is reduced and the sampling interval is reduced, the spectrum width of the signal light changes with increasing oscillation, and it is closer to the absolute value of the phase change function φ' 1(t). To obtain a modulation output with gradually decreasing spectral width, the phase modulation function f2(t) is obtained by integrating the target phase change function φ'2(t)(Fig. 2). Under the modulation of f2(t), the spectral width (Fig. 7) of the signal light increases rapidly at first, reaches the maximum value, and then decreases in zigzag oscillation. The experimental results are consistent with the results of the simulation. At the tail of the f2(t) modulation function, the modulation is smaller, and the modulation change is weaker in the pulse width of the signal light. The analysis and experiment show that the spectrum width decreases rapidly in tail time.Conclusions This study proposes a method of specific phase modulation function. The corresponding time-varying phase modulation function is designed, and its physical process is analyzed. The phase modulation under the function can effectively realize the change in signal light spectral width with time. The experimental results show that under the effect of the designed phase modulation function, the spectral width of the signal light increases or decreases with time. It is consistent with the results of the simulation and verifies the feasibility of the real-time dynamic control of the spectrum. In practical applications, a higher power amplifier can achieve higher modulation voltage output and wider spectrum width adjustment. The research of the specific phase modulation can provide theoretical reference to improve the time domain-smoothing performance, realize dynamic control of spectral width, and improve the control ability of high-power lasers.

Objective Diffraction can affect beam propagation. To reduce its influence, people have tried many means to search for a nondiffracted beam. Airy beam, featuring self-acceleration and self-healing, is a typical nondiffracted beam discovered in exploration. Unfortunately, it carries infinite energy. Later, an Airy beam with finite energy is obtained through truncation in practice. Airy-Gaussian beam is modulated by finite energy Airy beam, ranging from Airy beam to Gaussian beam by adjusting the distribution factor, which is convenient for research. Based on the nonlinear Schrödinger equation, researchers have studied the propagation of Airy-Gaussian beams in various nonlinear media such as Kerr medium, strongly nonlocal nonlinear media, photorefractive media, and obtained many intriguing phenomena.Following the discovery of the nonlinear Schrödinger equation, the fractional Schrödinger equation, which is proposed in the category of quantum mechanics, is discovered. Longhi introduces it into optics, which sparks widespread interest and promptes a series of researches. The propagation and interaction characteristics of Airy and Gaussian beams have been extensively studied within the framework of the fractional Schrödinger equation. However, little research has been conducted on the Airy-Gaussian beam. Therefore, studying the propagation of the Airy-Gaussian beam modulated using the fractional Schrödinger equation is necessary.The split-step Fourier method, considering that diffraction and nonlinearity act independently when the transmission distance is very small, is one of the most common methods to solve the nonlinear Schrödinger equation. Therefore, the transmission process is calculated in two steps, the influence of diffraction effect and nonlinearity effect is considered respectively, and the transmission result of the beam is obtained finally.Methods The fractional Schrödinger equation model is used in this paper to study the interaction of dual Airy-Gaussian beams in the Gaussian potential and the effect of various parameters on the propagation process, including distribution factors, Lévy index, and barrier parameters, is thoroughly examined. The interaction process of dual Airy-Gaussian beams in the Gaussian potential is periodic.Results and Discussions We first consider the effect of the Lévy index and the potential barrier’s position x0 on the interaction process. With the increase of α, the diffraction effect becomes stronger, and the splitting phenomenon gradually disappears, accompanied by the larger angle between the two main lobes and the decreasing transmission period. When α takes a certain value, the energy exchange of the splitting sub-beams occurs after the collision. The position of the potential barrier will affect the propagation period, performing that when x0 increases, the evolution period of the beam becomes larger (Fig. 1). When χ0 is small, the interval parameter B has a certain influence on the period, and with its increment, the effect of the Airy-Gaussian beam on the period decreases, even disappears finally. And the number of peak points varies with the value of B at the same transmission distance (Fig. 2). In addition, when B is assigned different values, with the increase of the transmission distance, the number of peaks changes accordingly, whose number and intensity are always completely symmetric with x=0 in the transmission process (Fig. 3). In the case of in-phase and out-phase, the energy distributions of the beam are symmetric about the center axis. While energy transfer occurs in other phase conditions, the energy distributions are no longer symmetric (Fig. 4). When retaining other parameters unchanged, transmission/reflection ratio and period of beam interaction can be controlled by potential barrier depth p and potential barrier width d0, both of which can decrease the transmission period. Simultaneously, the transmission of the beam in the wall of the potential barrier is weakened, whereas the reflection is enhanced (Fig. 5).Conclusions The interaction of dual Airy-Gaussian beams in the Gaussian potential is studied using the split-step Fourier method in this paper, which is based on the fractional Schrödinger equation. The results show that the interaction between the two Airy-Gaussian beams in the Gaussian potential is periodic and the period can be changed by adjusting the potential barrier parameters. The increase of the potential barrier width and potential depth leads to the decrease of the period, whereas the change of the barrier position leads to the increase of the period. The depth and width of the potential barrier not only affect the periodic variation, but also affect the reflection and transmission of beams. With the increase of the depth and width of the potential barrier, the Gaussian potential’s reflection effect on the beam is enhanced, the transmission is weakened, and the beam reaches total reflection. The Lévy index mainly affects the splitting and diffraction of the beam. With the increase of the Lévy index, the splitting phenomenon gradually disappears, and the diffraction effect is strengthened. When the Lévy index increases to a certain value, chaos will appear after a propagation period. The distribution characteristics of the Airy-Gaussian beam can be adjusted by changing the distribution factor. When the distribution factor is small, the interval parameter can affect the beam period. In addition, the interaction between interval parameter and relative phase will affect the intensity distribution of beam in the transmission process. In the case of in-phase and out-phase, the energy of two beams remains symmetrical in the propagation process, but when a relative phase is π/2 or -π/2, the energy is no longer symmetrical and the energy transfer phenomenon appears. Changing the interval parameter also affects the direction of energy transfer. Our findings can be used to control the propagation direction of a light beam and the number of light beams generated. They have potential applications in optical switches, splitters, and other fields.

Objective The step-and-scan photolithography machine is the main equipment for manufacturing integrated circuits used in the key layer preparation. The exposure system of the photolithography machine consists of the illumination system and projection objective. The illumination system is the core of the subsystems, therefore, it is used for several purposes, and it generates an illumination field with a specific intensity profile. The critical dimension uniformity is determined using the uniformity of the dose energy in the scanned exposure field. In and below 28 nm node, the top-Gaussian illumination is an important technology for reducing the influence of the pulse quantization error on dose energy. Considering the matching and consistency of the photolithography machine and the moving range of the wafer platform, the requirements and tolerances of the top-Gaussian illumination field are relatively strict. However, owing to limitations in design, manufacturing, and installation for the microlens arrays, diffuser, and Fourier transform lens (FTL), the dimensions of the top-Gaussian illumination field in scan direction are relatively difficult to meet the requirements directly. The performance of optical elements, such as coating transmittance, may degrade the performance of the top-Gaussian illumination field after long-term use. The general correction methods are focused on the uniformity correction of the illumination field. Therefore, in this study, a correction technology for the intensity profile of the illumination field is proposed. A corrector designed and optimized by this technology can correct the intensity profile and integral uniformity of the illumination field simultaneously. The energy loss can be decreased by considering the dimension tolerances of the intensity profile in the scan direction as boundary conditions during the optimization.Methods The design process of the corrector includes the intensity profile and integral uniformity corrections. The corrector is located at the front of the rear focal plane of FTL with a defocus distance. The illumination field on the rear focal plane of FTL should be transferred to the defocus plane during the design and optimization process. For the conventional illumination mode, the intensity distribution of a field point on the rear focal plane of FTL at the corrector plane is shown in Fig. 5. The intensity profile correction in the scan direction is coupled with the integral uniformity correction in the nonscan direction. Conducting a few iterations of the optimization during the design of the corrector is necessary. To reduce the number of iterations, the integral uniformity of the illumination field in the nonscan direction is corrected. Then, the intensity profile of the illumination field in the scan direction is corrected. Additionally, the dimension tolerances of the intensity profile in the scan direction should be included in optimizing the corrector to reduce the energy loss of the correction of the illumination field.Results and Discussions To verify the feasibility of this technology, correctors are designed for three unsatisfied top-Gaussian illumination fields. Moreover, the transmittance distributions of the correctors are optimized through the simulated annealing algorithm. Fig. 10 shows the transmittance distributions of the designed and optimized correctors. The correctors’ abilities are tested through simulations in LightTools software. The integral uniformities of the three top-Gaussian illumination fields in the nonscan direction are less than 0.29% after correction (Table 6 and Table 7). The dimensions DY_97, DY_50, DY_003, and DY_25~75 of the three illumination fields in the scan direction are [3.79 mm, 4.39 mm], [13.18 mm, 13.58 mm], [23.3 mm, 23.4 mm], and [3.05 mm, 3.44 mm] after corrections, respectively. The correction results show that the integral uniformities and intensity profiles are meet the requirements. The energy loss introduced by the correction is reduced by an average of 4.09%. Furthermore, the time taken by algorithm is less than 10 s (CPU: Intel Core i7-10750H, SDRAM: 16 GB). The consistency of the algorithm and simulation results shows the feasibility and veracity of the proposed technology.Conclusions In this study, a correction technology for the intensity profile of the illumination field in a photolithography machine is proposed. To reduce the number of iterations, the integral uniformity in the nonscan direction of the illumination field should be corrected. Then, the intensity distribution of the illumination field in the scan direction is corrected by the proposed technology. The tolerances of the dimensions of the intensity distribution in the scan direction are taken as the boundary conditions during the optimization to reduce the energy loss of the illumination field. According to three unsatisfied top-Gaussian illumination fields, the correctors are designed and optimized in a relatively short time. The integral uniformities in the nonscan direction and the dimensions of the intensity profiles in the scan direction are satisfied with the requirements after correction.

Objective In laser applications, the propagation characteristics of the beam directly affect its application quality. Accordingly, various standards for measuring the laser beam quality have been proposed to better evaluate the laser beam quality. The laser beam quality factor M2 is the product of the beam waist diameter and the far-field divergence angle, which does not change with the optical system. Therefore, using M2 for the beam quality measurement is stricter and more comprehensive. The representative measurement methods of the M2 factor are knife-edge and array detection, among others. However, the measurement process of these methods is slow and requires multiple captures, exhibiting high requirements on the beam stability. A pulsed beam (e.g., laser output from a high-power driver) shows a certain degree of instability; hence, a simple pulsed beam quality measurement method is required. Methods The algorithm of coherent modulation imaging based on amplitude coding(CAMI), which uses a binary random amplitude plate to modulate the incident beam. A single-shot method based on coherent modulation imaging is presented for the measuring of the beam quality. The laser beam to be measured first illuminates a highly random phase plate with a known structure and subsequently the intensity of the resulting diffraction pattern is recorded by a charge-coupled device positioned behind the phase plate. Intensity distribution of the laser beam is accurately reconstructed with the coherent modulation imaging method, then the scalar diffraction theory is used to perform numerical inversion, the beam intensity distribution of any plane can be obtained by calculation. According to the standard beam quality analysis algorithm, the quality of the laser beam is calculated. In addition, since the CAMI method adopts an amplitude modulation structure and does not require calibration, in theory, this method is applicable to any wavelength. Therefore, compared with the existing method, the structure is simpler, suitable for single exposure measurement, and theoretically can be used as a brand-new beam quality analysis technology.Results and Discussions First, the feasibility of using CAMI algorithm to realize beam quality parameters was simulated and verified. It is assumed that the incident beam is an ideal Gaussian beam with a wavelength of 351nm. Considering the diffraction pattern saturation error, uniform random background noise (0~1) and quantization noise, the reconstruction results are shown in Fig.3. The incident beam at the amplitude plate recovered by CAMI is transmitted through the angular spectrum, and the beam parameters are calculated using the calculation method described in section 2.2. The maximum error is 2.12%, and all errors are within acceptable limits. For further verification, the CAMI optical path diagram shown in Fig.1 and the Ophir-Spiricon beam quality analyzer (model: BSQ-SP920) were used to measure the beam. Ophir-Spiricon beam quality analyzer measured the He-Ne laser beam quality factor Mx2=1.044,My2=1.042, CAMI method calculated Mx2=1.090,My2=1.044, the relative error along x direction and y direction was 4% and 0.2%. Finally, using the CAMI 351nm pulsed beam algorithm actual measurement, the beam path diagram is shown in Fig.6(a). After 300 iterations, the saturated area of the diffraction spot is restored, and the reconstruction results are shown in Fig.6(b)--(e). Through wavefront inversion, the beam intensity distribution of other vertical sections along the optical axis can be calculated. The beam width expands outward along the transmission direction in accordance with the hyperbolic law, and the coefficients of the hyperbola are fitted by multiple sets of beam intensity data, thereby calculating the beam quality factor Mx2=1.4746,My2=1.2101.Conclusions Compared with the far-field divergence angle and focal spot size, the laser beam quality factor M2 is a technical evaluation that can strictly characterize the laser beam quality. A real-time complex amplitude reconstruction method based on the coherent amplitude modulation imaging algorithm is proposed to determine the laser beam quality factor M2. CCD is used as an image sensor to directly detect the laser beam distribution, and the wavefront distributions at different positions are obtained by numerical calculation. Laser beam quality measurement is based on the theory of second-order moments, and the M2 is measured by the method of propagation trajectory curve fitting. Compared with the traditional mobile CCD method to obtain the wavefront distribution at different positions, the automatic measurement is more convenient and faster, and the wavefront distribution information of laser beam can be accurately obtained, which is suitable for measuring the quality of pulsed laser beam. Simulations and experiments have proved the effectiveness of proposed method.

Objective As a displacement measurement tool, laser triangulation displacement sensors (LTDS) are widely used in industrial detection because of their noncontact nature and high accuracy. The optical path of LTDS is illustrated as follows: a collimated laser beam is projected onto the detected object and the diffuse reflected light is focused by a receiver lens onto a charge-coupled device (CCD) detector. When the detected object is moved along the direction perpendicular to the optical axis of the source laser beam, the reflected light beam spot (image spot) focused on the CCD will move correspondingly. Thus, the displacement over which the detected object moves can be calculated using a geometric optical model that is related to the image spot displacement. Laser-beam dithering is considered one of the major error sources in laser applications. To solve this problem, many averaging methods have been proposed. However, methods that can image two image spots on one CCD with a single image system are ineffective at improving the error factor of the laser beam directional instability. In other methods, structures comprising prisms or reflectors for producing two differential optical paths have been proposed. With those methods, the positional average of two image spots remains constant irrespective of the angle by which the source laser beam dithers. However, an installation error might be introduced, for which calibration is not possible because it is related to the laser dithering angle.Methods In this paper, a laser beam pointing control-based dual-view for laser triangulation displacement sensors (DVLTDS) is proposed. The structure comprises a collimated laser, two receiver lenses, and two CCDs, where the two receiver lenses and two CCDs are symmetrically arranged around to the optical axis of the source laser beam. DVLTDS generates two beam intensity distributions (BIDS) on two CCDs simultaneously. It converts the centroid movement of each BID into the detected object displacements through averaging. Hence, if the relationship among the optical parameters such as object distance, image distance, view angle, image angle, working distance, and the dithering angle satisfies certain constraint conditions (Eq. 6), then at the angle at which the source laser beam dithers, the average positions of the two laser spots imaged on the two detectors are equal within the permissible error range.Results and Discussions For validating DVLTDS, an experimental setup was built and various tests, including calibration, repeatability, and nonlinearity, were conducted. To satisfy the constraint condition, the optical parameters, including the focus of the receiver lens, working distance, object distance, image distance, view angle, and image angle, were designed in Zemax software in the nonsequential mode (Fig. 2). The experimental platform of DVLTDS was built according to the design data (Table 1, Fig. 3).DVLTDS was calibrated with a dual-frequency laser interferometer (RENISHAW XL-80)with a linear resolution of 1 nm. The relative positions of DVLTDS were calibrated. The target object (a ceramic block) was driven point-by-point along the optical axis of the source laser beam using a stepper motor with an increment of 0.1 mm within 10 mm. At each point, the collimated red laser was rotated with an increment of 0.1° within ±0.4°. Results show that the calibration curve of DVLTDS was coincident with that of the laser interferometer. Moreover, the standard deviation (STD) was found to be 0.2532 μm with DVLTDS [Fig. 5(a)]. In comparison, the STD was found to be 28.53 μm with LTDS [Fig. 5(b)].In repeatability tests, the ceramic block was fixed at a closer point, zero point, and a farther point. The collimated red laser was rotated within ±0.4° with an increment of 0.1°. Using DVLTDS, the repeatability accuracy was within ±5 μm and the STD was within 0.0035 mm. In comparison, with LTDS, the repeatability accuracy was ±1.4 mm and the STD was more than 0.5 mm (Table 3).The nonlinearity is expressed as the ratio of lr to (xt-xr), where xt is the tested value of DVLTDS, xr is the tested value of the XL-80 interferometer, and lr is the tested range. In this experiment, measurements were performed by moving objects from a closer point to a farther point with an increment of 0.1 mm for three runs. The results reveal that the nonlinearity of DVLTDS is within ±0.04% full scale (F.S.) [Fig. 7(a)]. In comparison, the nonlinearity without differential dual view is within ±0.8% F.S. [Fig. 7(b)].Conclusions In conclusion, experimental results indicate that with DVLTDS, the estimated STD of the fitting error decreases from 28.53 to 0.2532 μm, the repeatability accuracy can be reduced from ±1.4 to ±5 μm, and the nonlinear error can be reduced from ±0.8% F.S. to ±0.04% F.S. These results verify that laser beam pointing control-based dual-view for laser displacement sensors can suppress the effects of laser beam dithering.

Objective Fast and accurate beam scanning is the key technology in free space laser application. An optical phased array (OPA) based on electro-optics or thermo-optics overcomes the limitation of mechanical steering and can achieve noninertial beam steering with flexible beam pointing. It has demonstrated applications in free-space laser, such as light detection and ranging (LIDAR), free-space optical (FSO) communication, and optical imaging. The spatial light modulator (SLM) and microelectromechanical mirror array (MEMS) have realized tens of kHz and 1-MHz beam steering, respectively. Chip-scale OPA can achieve ultrawide beam steering and it has been demonstrated at kHz to GHz. However, chip-scale OPA suffers serious loss and cannot be used in remote detection. The optical fiber phased array (OFPA) based on a lithium niobate (LiNbO3) phase modulator can achieve beam directional and fast steering at GHz while realizing high power laser synthesis output, but the phase noises make each beam phase fluctuate strongly, which seriously affects the output beam quality and cannot guarantee its steering angle accuracy. Compared with the coherent combination of high power fiber lasers, the fast beam steering of OFPA not only involves phase control compensation but also ensures the accuracy of beam steering angle and higher steering speed, which improves the difficulty of phase control.Methods Based on LiNbO3 phase modulators and the stochastic parallel gradient descent (SPGD) algorithm, the phase noises of 1×16 channel OFPAs are compensated, and the multibeam fast steering is achieved by the “steering after correction” method. First, a narrow-linewidth (Results and Discussions The phase control system developed in this study effectively achieves phase noise compensation, the performance metric (on-axis intensity) increases from 0.43 to 0.94, the quality of coherent combined beams is improved (Fig. 6), and the convergence time of the SPGD algorithm is 1.2 ms [Fig. 7(a)]. Using peak-to-side lobe ratio (PSLR) as the quality evaluation index of beam optimization, the PSLR is 24.7 dB after phase noise compensation, which is close to the theoretical limit of 26.4 dB (Fig. 8). In this paper, the beam steering angle is set to be -0.30°, -0.20°, -0.10°, 0.10°, 0.20°, and 0.30°. The quality of the coherent combined beam after steering maintains a good intensity distribution state (Fig. 9). The results show that the actual steering angles are -0.31°, -0.20°, -0.11°, 0.12°, 0.22°, and 0.29°. Three factors lead to errors: 1) the half-wave voltage of each LiNbO3 is different from the theoretical value; 2) the actual output voltage of the phase control system has errors compared with the theory values; 3) owing to the processing error, the spacing parameters of adjacent bare fibers in the fiber array are inaccurate. The pixel size of the short-wave infrared camera (SWIC) is 15 μm×15 μm, and its FOV is 0.008°; the maximum error of the experimental results is 0.02°, which shows that the systematic error caused by the resolution of the SWIC cannot be ignored. The beam steering speed of the system (defined by the switching speed of the beam between any two angles) measured by a single point detector is 500 kHz.Conclusions An experimental system containing an OFPA of 1×16 channels is built, and the experiments for coherent beam combining and multibeam steering are conducted. Experimental results show that the algorithm is of high efficiency, taking only 10 μs in a single iteration. Besides, the PSLR reaches 24.7 dB, with the theoretical limit being 26.4 dB. The steering range is in good consistency with the theoretical prediction range. The OFPA developed in this study permits high-quality coherent beam combining and allows high-speed (500 kHz) beam steering, and the scan angle range is -0.70°--0.70°. Finally, the feasibility of fast multibeam steering of 1×16 channel OFPA is verified. In the future, the bandwidth of the phase control circuit system will be further improved, and higher precision beam steering will be achieved, which will lay a foundation for the applications of laser detection and imaging.

Herein, the generation of Hermite-Gaussian beams using a spatial light modulator was experimentally investigated and the relation between mode conversion efficiency and transverse distribution of the input beam was theoretically analyzed. High-quality Hermit-Gaussian beams with HG6,0, HG8,0, and HG10,0 modes, whose purity was 96.2%, 94.9%, and 93.4%, respectively, were experimentally generated using the optimum input beam with an elliptical transverse distribution. Moreover, the 217-mW HG10,0 mode was obtained and the mode conversion efficiency reached 14.45%, which is 5.6 times higher than that of the traditional input beam with fundamental Gaussian mode. The generation method of Hermite-Gaussian beam with high quality and efficiency has promising application potentials in the precision measurement of small displacement and preparation of a high-order spatial squeezed light field.

Aiming at the requirement of illumination uniformity and backscattering suppression in laser-driven inertial confinement fusion (ICF) facilities, we propose a polarization rotation (PR) smoothing scheme based on the interference of circularly polarized vortex beamlets, which makes the simultaneous control of intensity and polarization of the focal spot. The basic mechanism is that a conjugate spiral phase plate is first used to transform the beamlets with certain wavelength difference into vortex beams with conjugate helical charges, and then the polarization control plate is used to change the polarization states of the beamlets into counter-rotating ones. Finally, the rapid rotation of both the local intensity and polarization of the focal spot can be realized by means of the interference of the circularly polarized vortex beamlets in the target plane. The physical model of the PR scheme based on the interference of circularly polarized vortex beamlets is established and the variations of illumination uniformity and polarization of the focal spot with the beamlet wavelength, the helical charge of the spiral phase plate, and the parameters of the incident beams are analyzed. The results indicate that the proposed PR scheme can be used to realize the intensity and polarization rotation at a specific rotation frequency by selecting the suitable wavelength combination. When this novel scheme is implemented together with the conventional spectral dispersion smoothing scheme, the smoothing performance can be further improved and the backscattering suppression can be effectively achieved.

A new optical transmission network structure is proposed based on the two-dimensional magneto-optical photonic crystal and topological photonics theory, which is composed of a series of rectangular photonic crystals and air. To create coding units, the three conditions of applying a positive magnetic field, a negative magnetic field, and an air area are marked as “+1”, “-1”, and “0”, respectively. Then, three types of optical path matrix networks of 2×2, 3×3, and 4×4 are constructed. Finally, COMSOL software package is employed to simulate the transmission path of light waves. The simulation results prove that through different coding sequences, flexible and diverse control of the optical path can be achieved. The proposed control fulfils the requirements of high density optical path transmission and large-capacity information processing in the photonic integrated circuits to emerge in the future.

Aiming at the requirements of high-power large-blocking ratio narrow-ring thin-tube lasers for ring aberration correction, a new ring-shaped edge-driven deformable mirror-based method for ring aberration correction is proposed herein. An adaptive optics closed-loop control system is constructed using the ring-shaped edge-driven deformable mirror to correct the wavefront of the high-power, large-blocking, thin-tube laser. A novel ring-shaped edge-driven deformable mirror is used to verify the correction ability of the proposed method for single-circular low-order aberrations, and analyze its correction effect on thin tube lasers with a large obstruction ratio. Experimental results reveal that the proposed ring aberration correction method can effectively correct the wavefront distortion of a thin tube laser with a large obstruction ratio and a narrow ring, significantly improving the beam quality.

Diffraction can cause beam divergence. In this study, a new type of waveguide exit port structure was constructed for a photonic crystal by adopting a two-dimensional square-lattice photonic crystal to reduce the divergence and deal with the short radiation distance and low radiation efficiency of the waveguide exit port. First, a Y-shaped channel structure was used for improving the transmittance at the output end; then, multi-branch structures was coupled for improving the convergence of the emitted light to increase the radiation distance of the light wave. Finally, the exit port period was optimized. In addition, analyses and simulations were conducted using the plane wave expansion method and the finite-difference time-domain method. The simulation result denoted that the divergence angle of the emitted light was approximately 3° and that the radiation efficiency exceeded 25% when the radiation distance of the light was 110 μm. Thus, a square-lattice photonic crystal with a Y-shaped defect, 16 exit ports, and 4 exit port periods can achieve well-directed radiation.

Freeform pupil illumination technology is an important photolithographic resolution enhancement technology for immersion photolithography machines locating in nodes at 28 nm and below. Arbitrary illumination mode could be realized by adjusting the angular spectrum of the beam by micro mirror array (MMA). The angular position distribution of MMA is of great significance to the application of the freeform pupil illumination technology. An angular position distribution algorithm of MMA based on genetic algorithm is proposed in this paper. Compared with the angular position distribution algorithm of MMA based on the simulated annealing algorithm, the iteration speed of the proposed algorithm is increased by more than 10 times, and the MMA angular position distribution obtained by the proposed algorithm can accurately reproduce the distribution of the target pupil intensity. Results of the photolithography performance simulation show that the root mean square (RMS) values of the asymmetry distribution of the photoresist exposure patterns of the algorithm pupil and the target pupil are basically the same, and the RMS values of critical dimension difference distribution are less than 0.5 nm.

Herein, a deformable lens that can directly generate tunable Airy beams was proposed. Compared with other methods, this deformable lens can be embedded in the optical path to directly generate Airy beams. A prototype deformable lens was fabricated, and an optics system based on the wavefront sensor was established to generate Airy beams. Using the fabricated deformable lens, high-quality Airy beams with an adjustable cubic phase were generated. The residual error of generated cubic phase was less than 3.4% of the corresponding target amplitude. The intensity distribution and propagation properties of the generated Airy beams were consistent with the theoretical findings, thereby demonstrating the potential application of the deformable lens in directly generating tunable Airy beams.

To solve the problems of the difficulty in adjusting parameters in real time and the convergence speed being slow in the traditional stochastic parallel gradient descent (SPGD) algorithm, this study proposes an efficient SPGD algorithm based on adaptive gain and joint index optimization and establishes a numerical simulation model of this algorithm. The proposed algorithm is used for the beam cleaning of kilowatt-class slab lasers. Simulation results show that compared with the traditional SPGD algorithm, the proposed algorithm does not require a parameter adjustment, and the convergence speed and convergence effect are significantly improved. Furthermore, in the beam purification experiment of the kilowatt-class slab laser, the laser beam quality β is optimized from 7.89 to 1.95 herein.

In this study, a microwave photon up and down-conversion signal generation scheme is proposed to realize phase tuning. The main component of this scheme is a polarization multiplexing dual-drive Mach-Zehnder modulator (PDM-DMZM). In the proposed scheme, up and down-conversion signals switching can be realized by adjusting the direct current bias point of the modulator. Further, the phase is continuously tuned by adjusting the polarization controller. The proposed scheme can be extended to a multichannel independent phase tuning system. Simulation results show that the radio frequency signal with a frequency of 10GHz can be converted into a down-conversion signal (1GHz) and an up-conversion signal (19GHz). Its phase can be continuously tuned in the range of 0°--360° and generated under different phases. The maximum power fluctuation of the signal is less than 0.3dB, the spurious signal suppression ratio can be maintained above 20dB, the maximum input frequency of the system can be adjusted from 0.5 to 65.0GHz, and the frequency of the generated phase-shifted signal can range from several GHz to 130GHz.

In order to meet the control requirements of the surface accuracy and structural stability of large aperture mirrors for laser devices, a multi-point mounting method with decoupling of multiple degrees of freedom of the mirror is proposed. The control of multiple degrees of freedom of mirror is realized by limiting position to avoid the additional surface shape caused by mounting. The effectiveness of the proposed method has been analyzed by finite element method, and the feasibility of the analysis method and the mounting method has been verified through experiments. The results show that the additional surface shape brought by the mounting method of the mirror is small, which meets the requirement of the low-stress additional mounting surface of the mirror. On the basis, the surface shape of the mirror placed at a tilt angle of 45° is simulated, and the influences of the position distributions of different mounting points on the surface accuracy of the mirror are explored. Simulation results show that in order to ensure the surface accuracy of the mirror, at least one mounting point should be located on the longer side of the mirror. These results have important guiding significance for the mounting design of large aperture mirrors.

The basic working principle and mathematical model for a fast-steering mirror driven by PZT, which is a type of mirror widely used in optical-axis control systems, are analyzed, and the identification process of genetic algorithms in the model is described. A system for measuring the frequency response of a large-aperture fast-steering mirror driven by PZT is set up to realize parameter identification for the transfer function. The frequency response of the identified transfer function and the actual measured data are compared, and the identification accuracy of multiple resonance points in the system is analyzed. The experimental results show that at each resonance point, the frequency response of the transfer function obtained by the genetic algorithm is consistent with the measured data. The results also show that the transfer function obtained by this algorithm can analyze the dynamic characteristics of optical axis control system more accurately, and we can design advanced control algorithm, thus improving the performance of the optical-axis control system.

In numerical and experimental analyses, we studied the effect of misaligning the cylindrical mirror rotation in shaping system on the far-field spot and beam quality of the laser during beam shaping based on Huygens--Fresnel diffraction theory. Increasing the rotation angle not only rotated and tilted the far-field spot of the shaping beam, but also elongated and defocused the spot. This aberration derives from the superposition of astigmatism and defocusing caused by rotating the cylindrical mirror. The results show that misaligned rotation also defocuses the shaping system with cylindrical mirror. We then quantitatively analyzed the correlation among the normalized phase coefficient and focal spot, in which the normalized phase coefficient is related with the center wavelength of the laser, size of the shaping beam, distance of the cylindrical mirrors and magnification factor. From the correlation coefficient, we can evaluate the sensitivity of the shaping system to the cylinder-mirror rotation disorder. Finally, we verified our calculated results using a corresponding experimental device. The experimental results agreed well with the calculated results.

Fast steering mirrors (FSM) are widely used in aeronautical optoelectronic stabilization platform for line-of-sight stabilization. The stability of the FSM is affected by various disturbances especially the vibration in the aviation environment. Traditional anti-disturbance methods, such as proportion integration differentiation controller (PID) and disturbance observer (DOB), have a little effect on suppressing disturbance in FSM. To solve these problems, a fast anti-disturbance strategy based on adaptive robust control (ARC) is proposed. The experimental results show that the steady-state root mean square error of FSM in vibration environment is reduced by about 80% compared with that of the PID control strategy and about 60% compared with that of the DOB control strategy after the introduction of the ARC. It shows that ARC has remarkable effect on improving the anti-interference ability and stability of FSM, and has large engineering application value.

A chromatic aberration pre-compensation scheme with dynamic adjustment ability is proposed to address the problem that chromatic aberration introduced by transmission-based spatial filters in ultrashort high-power laser systems strongly affects the focused intensity. In the pre-compensation scheme, the confocal transfer system comprises of a group of concave lenses and a reflecting system. The proposed scheme can realize dynamic precise compensation for chromatic aberration in the total system. Based on the proposed scheme, an optical path of chromatic aberration compensation is designed and established for the Shen-Guang-II 5 PW (SG-II 5 PW) laser system. The experimental results verify that chromatic aberration in the laser system can be fully corrected by accurately adjusting the pre-compensator, and the peak power density is improved significantly. A proton energy greater than 16 MeV is obtained in the proton acceleration experiment with the SG-II 5 PW laser system after chromatic aberration compensation.

In order to realize the decoupling control for the Woofer-Tweeter adaptive optics system, we propose a decoupling control algorithm based on Laplacian eigenfunction. By solving Laplacian eigen equations under different homogeneous Neumann boundary conditions, we can obtain Laplacian eigenfunctions which are orthogonal to themselves and whose first-order partial derivatives are orthogonal under different pupil regions. Using the Laplacian eigenfunctions under different pupil regions, we achieve the decoupling control for the Woofer-Tweeter adaptive optics system in different pupil regions. In addition, the first-order partial derivatives of the Laplacian eigenfunction is orthogonality, so that it is not necessary to measure the response function surface shape of Tweeter actuator to construct a constraint matrix for Tweeter. A Woofer-Tweeter adaptive optics system is used to verify the validity of this algorithm. The experimental results show that the decoupling control algorithm based on Laplacian eigenfunctions can synchronously control Woofer and Tweeter, and effectively suppress the coupling error between Woofer and Tweeter.

Based on the influence functions of the deformable mirrors (DMs), the wavefront correction process is simulated by the finite element method and the influence of the fatigue characteristics of piezoelectric ceramics (PZT) actuators on the correction ability of DMs is analyzed. The research results show that the correction ability of DMs decreases when the phenomenon of the fatigue occurs in PZT actuators. The larger the peak-valley value of the distorted wavefront to be corrected or the higher the proportion of the spatial high-frequency components is, the more obvious of the decrease of the correction ability of DMs after the occurrence of fatigue is. In addition, the influence from the former is relatively larger.

Based on 3D particle-in-cell (PIC) numerical simulation method, the high quality, high energy proton beam is transmitted and focused on the far end via a pulse current solenoid. Simulation results show that proton beam with peak energy of 250 MeV, energy spread of 10% and a spatial divergence angle less than 15 mrad can be focused on a spot 2.5 m away from the proton source, after transported in a 760-mm pulse power solenoid under magnetic field strength of 10.87 T. The focal spot cross section diameter is 1.2 mm, less than the initial proton beam spot size; meanwhile, the number loss of proton beam is less than 3%. We conclude that it is feasible to use a powered solenoid to transmit and regulate a high energy proton beam. This scheme can be used to optimize the proton beam quality and promote the laser-driven proton acceleration to be applied early in the fields such as cancer treatment, where high proton beam unipotency and small divergence angle are required.

To adapt to compact space requirements of high power laser system, a far-field and near-field coupling scheme based on second order ghost image is used in high power laser collimation technology. A sensitive small-size collimation optical system is designed and built. We present complete test scheme and verify its feasibility. The near-field measurement sensitivity is 7.04 μm/pixel and the far-field measurement sensitivity is about 18.14 (″)/pixel. The influence of far-field change on near-field is researched and used to optimize beam collimation. Compared with traditional collimation system of far- and near-field separation, this scheme sharply reduces the number of light paths and devices under the condition of imaging quality and resolution assured. It is more convenient to feedback the interaction effect and collimation of far-field and near-field.

The generating process of self-accelerating Airy beams is researched from the viewpoint of geometric optics, and the relationship between the phase distribution on the incident plane and the self-accelerating trajectory of Airy beams is analyzed based on the concept of tangent clusters. A iterated algorithm for calculating the phase distribution of the airy beam incidence surface based on the self-acceleration trajectory distribution function is proposed, and the formation principle of the self-healing property of Airy beams is also discussed. Two-dimensional Airy beams are generated by collimated laser modulated with spatial light modulator, and the features of these beams such as non-diffraction, self-healing and self-accelerating are verified. In addition, the one-dimensional Airy beams are modulated to generate autofocusing beams and Bessel-like beams.

In view of the complexity of underwater laser transmission channel caused by oceanic suspended particles, the effect of oceanic suspended particles on the underwater optical communication link is studied based on equivalent spherical particle Mie scattering theory and Monte Carlo simulation method. The characteristics of suspended particles and the relationship between the incident wavelength and the optical coefficients are analyzed. The effects of particle size and complex refractive index on the received normalized energy, received light intensity, channel transmission length and channel delay are investigated. Theoretical analysis and simulation results indicate that optical coefficients of the particles increase with the increase of particle size, thus the received normalized energy under same channel length decreases, received light intensity decreases, and the channel time delay increases. The smaller the imaginary part of the complex refractive index of the particles is, the stronger the received normalized energy and the greater the peak intensity of the received light are. However, when the imaginary part of the complex refractive index is the same but the real part is different, the magnitude of the received intensity peak depends on the albedo. The larger the albedo is, the stronger the received intensity is.

Stray light of transmission type Mie scattering lidar system is analyzed by ZEMAX software, and the reflection of optical lens surface in the lidar system and multiple reflection and scattering of the tube wall system are simulated.The power of multiple reflection light imaging on the detector surface source reaches 3×10-8 W, and the power of lens barrel light scattering imaging on the detector surface source reaches 2.8×10-9 W. Under the same measurement conditions, we analyze the continuous observation results of two kinds of laser lidars, and conclude that the echo signal strength converted from near-field stray light in the transmission type Mie scattering lidar can reach 1000 mV, which is nearly 17 times that of the reflection type lidar system. Attenuation piece is used to attenuate the two lidar signals respectively. The results show that when the signal attenuation is 80%, the near field signal of the transmission type lidar system is turned up. It can be seen from the results of nonlinear signal processing that the reflection type lidar system is better than the transmission type lidar system.

The random phase screens in anisotropic non-Kolmogorov turbulence are generated by the non-uniform sampling and power spectral inversion method, and the spatial optical modulator is adopted to simulate the drift characteristics and the changes of the orbital angular momentum of ring Airy Gaussian vortex beams in atmospheric turbulence. Numerical simulation and optical experiment results show that the drift values of ring Airy Gaussian vortex beams increase with the increasing beam attenuation coefficients, radius of the primary ring, outer scale of the atmospheric turbulence, and transmission distance, and decrease with the increasing turbulent anisotropic coefficients and topological charge of the beams. There exists a maximum drift value near the turbulent power-law value of 3.3. Moreover, comparing the interference fringe patterns of ring Airy Gaussian bortex beams before and after propagating through the atmospheric turbulence, we find that the smaller the topological charge is, the better the stability of topological charge after beam propagating in the turbulence is.

We obtain a bottle beam by focusing a Bessel beam with the semiconductor laser as the light source to realize the tunability of a bottle beam. The plano-convex cylindrical lens and the gradient index lens are used for beam shaping to obtain a laser beam with a variable divergence angle, and thus a size tunable bottle beam is obtained. The numerical simulation results show that, when the divergence angle of the beam incident on the axial cone varies continuously in the range of 0°-1.5°, the maximum radial size of the bottle beam can be tunable in the range of 90.23-64.05 μm, however the length of the bottle beam varies from 1.85 mm to 1.47 mm. A tunable bottle beam makes optical tweezers more flexible.

The steady-state temperature distribution of high-power semiconductor lasers is simulated, the thermal lens focal length and the variation of the slow axis divergence angle are calculated with the temperature distribution under different thermal power conditions. Results show that, under the same conditions the thermal power and the thermal lens focal length are approximately inversely proportional, and the slow axis divergence angle is approximately linear with thermal power. When the thermal power reaches 10 W, the thermal lens focal length is 568 μm, and the slow axis divergence angle increases about 10°. The slow axis divergence angle of the laser under different working current conditions is measured and the results show that the simulated values are consistent with the experimental values.

For the spatial periodic modulation that affects the output performance of high power laser systems, a novel method of compensating and controlling the spatial periodic modulation based on phase carrier is proposed and analyzed. First, the theoretical analysis shows that this method is able to control the spatial frequency of the spatial periodic modulation. The intensity of the spatial frequency is modulated by changing the magnitude of the phase carrier, and for the amplitude-type spatial periodic modulation, the period of the phase carrier can change the position of the maximum intensity of the spatial frequency. Then, the experiment on the effect of phase carrier on the amplitude-type spatial periodic modulation is carried out. The feasibility of the method is verified by experimental results and numerical simulations. The output near-field beam and the corresponding one-dimensional average power spectral density curve before and after the phase carrier modulation are compared. It is found that the peak value of the spatial frequency drops by an order of magnitude and decreases to near the background value after phase carrier modulation in the experiment. This method provides a new way to compensate and control the sensitive spatial periodic modulation in high power laser systems.

In the study of narrow linewidth laser light echol detection under strong background, we utilize spectral filtering method to filter out background light and improve the signal-to-noise ratio. For the grating filter system, the transmission of signal light through the atmospheric channel will cause the wavefront phase distortion, and this will have a certain effect on the filtering performance of the system, so it is necessary to study it further. Aiming at the application of grating spectral filtering in the laser atmospheric transmission detection direction, based on the laser atmospheric transmission theory and the grating diffraction principle, we establish the simulation model of the spectral intensity distribution of the grating when the incident light field is atmospheric disturbance light field. The influences of atmospheric coherent length and system structural parameters on system performance are analyzed. The system of the grating spectral filtering technology and the atmospheric applicable conditions are given. When the atmospheric coherent length r0 is greater than 0.05 m, sub-nanometer level spectral filter linewidth (full width at half maximum is 0.3 nm) is obtained, and the transmission of the effective spectrum exceeds 0.90. The results are verified by a large number of simulation experiments.

In order to reduce the carrier frequency error of the processed interferogram by using the Fourier transform method, and accurately recover the laser phase to be measured, we propose a phase retrieval method based on the propagation property of Gaussian beam. On the cross section of the Gaussian beam along its direction of propagation, the phase of the beam is a symmetrical convex or concave surface. The curvature of the convex or concave surface is related to the propagation distance. According to this property, the size of the carrier frequency is calculated and the phase to be measured is finally restored. In theory, the size of the carrier frequency can be calculated accurately, and the phase to be measured can be restored with high precision. Both numerical simulation and experimental results show that the proposed method can accurately recover the phase to be measured.

Aimed at the application of smoothing of multi-color spectral dispersion in inertial confinement fusion facilities, an independent design and optimization method for main unit components, including electro-optic modulators, dispersion gratings and continuous phase plates (CPP) and so forth, is proposed. The effects of parameters of electro-optic phase modulators, dispersion gratings and the CPP on the laser-quad irradiation characteristics are simulated and analyzed. The parameters of optimized smoothing of multi-color spectral dispersion are proposed, and the effect of the smoothing parameters on the characteristics of longitudinal filaments of lasers is simulated. Results indicate that the irradiation uniformity on the target surface can be further improved when we optimize the parameters of electro-optical modulator and dispersion grating of different sub-beams. When different CPPs are implemented for different sub-beams in the laser-quad, the focal-spot uniformity can be improved effectively and the proportion of hot spots can be decreased significantly. Moreover, the beam quality of the laser beam is analyzed by the power spectral density. The simulation results show that the smoothing of multi-color spectral dispersion can effectively decrease the peak intensity and make the light intensity distribution more uniform, which is benefit to suppress filaments of laser beams.

A method based on Mach-Zehnder interferometer (MZI) to generate arbitrary vectorial vortex beams in a higher-order Poincare sphere is proposed. The combination of half-wave plate and polarization beam splitter prism is used to adjust the amplitude of branch beam, and the two common half-wave plates are used to adjust the phase of the composite beam in order to generate and transform the vector vortex beam. The traditional experimental optical path is optimized, the energy loss of the beam is reduced, and the transformation of the vector vortex beam on the different high-order Poincare spheres is realized by the quarter-wave plate. Compared with the existing Mach-Zehnder interferometer-based vector beam generation method, the optical path structure is simple and the beam conversion efficiency is improved. Theoretically, the polarization state of each vector vortex beam on Poincare sphere with topological charge m=±1 is obtained by Jones matrix calculation. According to this method, a set of experimental light path of vector vortex beam is set up. The experimental results agree well with the theoretical analysis, which proves the practicability of this method.

The problems of the amplitude spatial light modulator(SLM) by electrical addressing in high power laser systems are analyzed. Firstly, the effect of the black gate of the SLM by electrical addressing on the near-field beam shaping is analyzed theoretically. The technical scheme of obtaining the best shaping effect by optimizing the aperture size of the spatial filter is proposed, and the energy efficiency of the filter is calculated. Meanwhile, the effect of liquid crystal SLM′s opening ratio on beam shaping is analyzed. In order to enhance the near-field shaping accuracy, the opening ratio of the SLM should be better than 64%. Finally, the phase distortion introduced by the pure amplitude spatial light modulator is studied. The additional phase is obtained from the theoretical analysis and experimental verification. The calculated maximum additional phase is 0.135λ, which means that the influence on the laser device is within acceptable range.

A two-pulse bidirectional propagating amplification configuration, which employs two laser pulses to propagate along a main amplifier in two directions of symmetric space and amplifies the two-pulse, can effectively increase the extraction efficiency for the stored energy in the amplifier, and has great application value in the field of inertial confinement fusion laser driver. When we build the equivalent optical path, the wave-front distortion superposition process of the two laser pulses in the process of propagation amplification is derived, and the numerical simulation is carried out with the data measured by the Shenguang-Ⅲ host device. The results indicate that the wave-front shapes of the two laser pulses in the bidirectional propagating amplification configuration are different. The wave-front aberration mainly concentrates in low frequency range. Moreover, the wave-front distortions of the output laser pulse and the laser pulse from the pinhole of spatial filter which is with three-pass and four-pass amplification are large. Thus, the wave-front aberration must be compensated and controlled to ensure the quality of output beam and the beam pass through the spatial filter pinhole. These conclusions will provide theoretical guidance for the design of wave-front control scheme of two-pulse bidirectional propagating amplification configuration.