1. Introduction

Durnin introduced the term Bessel beam into optics in 1987 [1]. As a representative of the so-called nondiffracting beams, the Bessel beam maintains an invariant transverse intensity distribution during propagation, even in the presence of obstacles. This phenomenon, known as the self-healing effect, refers to the beam’s ability to reconstruct itself after encountering an obstruction [2]. These properties are primarily effective within a specific region along the propagation axis, referred to as the depth of focus (DOF) or line focus distance. It is widely used in practical applications such as optical tweezers [3], laser processing [4,5], and particle manipulation [6,7]. With the advancement of structured optical fields, the focus has shifted to three-dimensional spatial field structure manipulation [8,9]. The Bessel beam with controllable emission angle and uniform light intensity distribution along the propagation direction has essential applications on many occasions [10–13], particularly in laser processing, like the oblique hole process of metal components [14]. Therefore, developing a technique that combines the Bessel beam’s angle and axial light intensity regulation is crucial.

Researchers have deeply explored nonmechanical beam steering technologies [15–17]. In 1996, McManamon achieved beam deflection of ±4° by driving a liquid crystal to generate a periodic blazed grating [18]. However, the liquid crystal optical phased array could only achieve discrete scanning angles without controlling the axial intensity. In 2008, Jihwan Kim combined a coarse and fine steering module to amplify the scanning angle through stitched scanning ranges [19]. The control range of the fine steering module can reach ±3.125°. In 2016, James R. Lindle utilized a spatial light modulator (SLM) to fabricate Fresnel zone plates (FZP) for laser beam control by shifting the central region to achieve beam steering [20]. In 2018, Sage Doshay fabricated efficient axicon lenses from high-contrast, multilayer silicon gratings, demonstrating high performance and field profiles consistent with theoretical predictions at deflection angles up to 40° [21]. In 2023, Tianyang Fu used a direct binary search inverse design method to design non-uniform gratings for one-dimensional beam steering on an insulating silicon and achieved beam control through wavelength tuning [22].

Although the methods mentioned above can control the angle of the beam, they have yet to achieve control of the axial light intensity uniformity of the Bessel beam. Bessel beam axial intensity uniformity refers to the constancy of its light intensity distribution along the propagation direction, a characteristic that allows it to maintain stable intensity over long distances. In 1993, Z. Jaroszewicz demonstrated that the apodized annular-aperture logarithmic axicon maintains excellent uniformity in on-axis intensity, energy flow, and lateral resolution [23]. In 2000, Pu Ji-Xiong’s research on lens axicons illuminated by Gaussian beams unveiled a method to achieve uniform axial intensity, efficiently mimicking logarithmic axicon effects with both backward and forward-type configurations [24]. In 2009, Yanzhong Yu enhanced the uniformity of the axial intensity distribution of Bessel beams by designing an anamorphic binary axicon. His method concentrated on improving axial intensity uniformity through axicon design, utilizing genetic algorithms (GA) in conjunction with two-dimensional finite-difference time-domain (2-D FDTD) methods for optimization [25]. However, the article needs to provide specific metrics to demonstrate the effectiveness of its claimed uniformity, and researchers need to expand the range of its application scenarios. In the same year, Tomáš ?i?már presented a method for generating Bessel beams with adjustable axial intensity and controllable propagation, enabling constant intensity along the path and tuneable lateral dimensions [26]. However, the technique is impeded by suboptimal power efficiency and the constraints of single-plane modulation, potentially restricting its wider adaptability. In 2016, Ismail Ouadghiri-Idrissi introduced a non-iterative beam shaping technique for creating Bessel beams with high energy throughput from direct space using a single phase-only spatial light modulator, a method constrained in practical application by the inherent physical limitations of numerical aperture and wavelength [27]. In 2018, Raghu Dharmavarapu devised a technique employing two diffractive optical elements to control the axial intensity of Bessel beams without expanding the incident laser, utilizing a binary axicon lens for propagation adjustment [28]. The mentioned literature managed to control the axial intensity uniformity of Bessel beams, underscoring the need to investigate the generation of high-uniformity Bessel beams with adjustable angles.

Due to the remarkable advantages of liquid crystal optical phased arrays in scanning accuracy and resolution, liquid-crystal spatial light modulators (LC-SLMs) have become the mainstream devices for beam deflection control [29]. This study explores beam steering by modulating the angle of Bessel beams through precise adjustments to the central region of the Fresnel Zone Plate (FZP). A controllable angle Bessel beam with programmable characteristics is designed using the SLM. This study established an improved axicon model with high-order surfaces to generate a uniform intensity distribution along the beam’s propagation direction. The GA is used to search for the optimal radial phase expression and further optimized using an unconstrained nonlinear minimization algorithm (UNMA) to reduce the computation time. The phase function is continuously modified to approach the target function. Finally, the hologram obtained is loaded onto the SLM for experimental validation, taking advantage of the phase delay characteristics of the SLM. The results show that this method can effectively control the Gaussian beam. It enables the generation of Bessel beams with uniform axial intensity, controllable angle, and selectable focal range. These results open new avenues for generating Bessel beams with multiple controllable angles, holding significant potential for multibeam control and precision measurement applications.

2. Methodology

2.1 Angle control

For an FZP with focal length f, this study approximated the region near the optical axis with a quadratic phase function due to the less pronounced step-like features. It approximately gave corresponding central phase distribution as:

Therefore, the minimum focal length constraint when the SLM carries the FZP is as follows [30]:

Figure 1 illustrates the phase hologram of the FZP: (a) represents the original pattern without any changes to the central region, and (b) shifts the central 2 mm to the left. After the central region is changed, the incident beam on the SLM is deflected.

 

Fig. 1. Schematic of the modulation of the FZP. (a) (x_m, y_m) = (0, 0) mm. (b) (x_m, y_m) = (-2, 0) mm.

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Theoretical considerations suggest that lateral displacement of an FZP may introduce asymmetries in generating Bessel beams. However, the practical impact of such asymmetries on beam formation is typically limited. This asymmetry is predominantly because the central region of the SLM plays a pivotal role in the beam-shaping process, and this region exhibits relative insensitivity to the lateral shifts of the FZP. Consequently, prioritizing the SLM’s central area significantly mitigates the potential asymmetrical influences introduced by FZP displacement. In most practical applications, the high-quality beam performance from the SLM’s central region ensures that any existing lateral displacement does not substantially compromise the overall symmetry and functionality of the beam.

The FZP method can achieve more precise angle control than altering the grating period or spacing. The position of the central region determines the deflection angle of the beam with angle resolution:

The angle resolution can be achieved up to 0.0008° when the pixel pitch is 12.5µm, and the focal length of the FZP is 900 mm. The diffraction pattern corresponding to the hologram in Fig. 1 is obtained by the angular spectrum diffraction method, as shown in Fig. 2. With an offset of 160 pixels, equivalent to 2 mm, the off-axis FZP achieves a control of 0.1273° on the incident beam compared to the standard FZP.

 

Fig. 2. 3D diffraction pattern (left) and 2D schematic diagram (right). (a) (x_m, y_m) = (0, 0) mm FZP. (b) (x_m, y_m) = (-2, 0) mm FZP.

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When a light beam is typically incident on an SLM or other diffractive element with a periodic pixel structure, it diffracts at specific angles, with the wavelength of the light and the pixel size of the SLM influencing the diffraction angle. The SLM expresses the maximum diffraction angle of the beam as [20]:

The maximum diffraction angle θ_max sets the Bessel beam’s angular width and focal spot size, increasing the beam’s non-diffractive propagation distance.

2.2 Bessel beam control

The axicon, also known as a conical lens or cylindrical lens, can be either a refractive or diffractive optical element and is the simplest and most efficient means of generating a Bessel beam. Programmable axicons have long been implemented on LC-SLMs by encoding phase profiles that vary linearly along the radial coordinate [31,32]. The phase distribution function of an axicon with a refractive index of n and bottom cone angle of α is expressed as:

The incident beam passes through optical elements, which, due to the transmittance function of the optical components, will generate corresponding phase differences, resulting in different phase delays. According to the Fresnel approximation and Fresnel diffraction calculation formula:

E(x, y, z) is determined jointly by the incident beam’s complex amplitude and the diffractive element’s transmittance function. When the diffraction angle is ψ, the Bessel function exhibits the following properties:

Figure 3 illustrates the holographic patterns and their associated axial intensity distributions derived from the amalgamation of phase contours inherent to traditional axicons with those of FZPs. This composite approach underpins the control efficiency of both standard and off-axis FZPs over the angular trajectory of Bessel beams. In stark contrast to the strategies involving direct manipulation of incident Gaussian beams, as delineated in Fig. 2, the presented method maintains the extended depth of focus — a defining feature of Bessel beams — and enhances the precision in steering the beam’s angular disposition. Figure 4 depicts the maximum deflection angle achieved at a wavelength of 1064?nm, demonstrating the optical steering capabilities at this particular spectral point.

 

Fig. 3. By combining the hologram of the traditional axicon with the FZP at (x_m, y_m) = (+2, 0) mm. (a) hologram. (b) 3D diffraction pattern. (c) 2D schematic diagram.

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Fig. 4. The maximum deflection angle θ_max = 2θ = 4.0742°. (a) (x_m, y_m) = (-32, 0) mm. (b) (x_m, y_m) = (+32, 0) mm.

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2.3 Axial light intensity of Bessel beams for optimization

Bessel beams are widely used due to their self-healing effect in propagation. In practical applications, the axial intensity distribution of a Bessel beam typically follows an increasing-decreasing pattern, irrespective of the incident light’s shape. Therefore, achieving uniform axial intensity in a Bessel beam is crucial. Traditional axicons possess rotational symmetry, and their profiles are first-order functions about radius r, as shown in Fig. 5(a).

 

Fig. 5. Integrated axicon-FZP design for uniform axial intensity distribution.

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Theoretically, an axicon with an ideally fitted profile exists, and combined with the FZP, it achieves a uniform distribution of axial intensity, as shown in Fig. 5(b). The composite phase function in polar coordinates can be rewritten as:

A high-order polynomial curve about r represents the profile of the axicon, and D/2 is the integration radius of the aperture plane. Then Eq. (10) can be formulated as:

Since the optical axis is invariant and the Bessel beam’s radial center contains the most energy, only the axial intensity distribution is considered. Setting r₁= 0, Eq. (12) is simplified to an integration concerning z.

In Eq. (13), the coefficients a, b, c, …, m of the axicon profile function $\varphi$(r) are implicitly involved. This paper employs a dual algorithm approach combining the GA and UNMA to seek the optimal coefficient solution. Considering the computational time and the accuracy of axicon profile fitting, a fourth-order polynomial form is adopted for Eq. (11). Figure 6 shows the flowchart of the algorithm [33]. The least mean square error rule evaluates the optimization process, and the evaluated function is expressed as:

Q represents the desired uniform intensity value, and m is the number of sampling points along the z-axis. The optimized parameters can be generalized to the off-axis FZP, as depicted in Fig. 5(c).

 

Fig. 6. Process of coefficient optimization algorithm based on GA and UNMA.

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3. Simulation

3.1 Combination of a standard FZP and axicon

Based on the above theories and equations, this study designed an iterative optimization program using the angular spectrum diffraction and Fresnel diffraction theories. The study set the parameters used in the numerical simulation as follows: laser wavelength λ = 1064?nm, Gaussian beam radius ω = 3.2?nm, FZP focal length f = 900 mm, refractive index of the traditional axicon n = 1.5, and bottom cone angle α = 0.5°. Table 1. shows that the study obtained the improved fourth-order axicon profile fitting coefficients without changing the center of the FZP. Figure 7 shows the diffraction pattern with these coefficients. Figure 8(a-b) compares the profiles before and after optimization and the numerical simulation comparison of axial intensity. Compared to the traditional profile, the optimized profile exhibits a significant improvement in the uniformity of the axial intensity distribution.

 

Fig. 7. Simulated axial intensity distribution of the SLM. (a) 3D (left) and 2D diffraction patterns (right) for the traditional profile. (b) 3D (left) and 2D diffraction patterns (right) for the optimized profile.

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Fig. 8. The axial intensity distribution and the profile fitting curve. (a) the traditional profile (red), the profile optimized by GA (green), and the profile optimized by GA + UNMA (blue). (b) the axial intensity distribution of the beam passing through the profile corresponds to (a).

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3.2 Composite of an off-axis FZP and axicon

Table 2 applies the optimized results of the standard FZP to control the off-axis F and demonstrates the effects of angle control of the Bessel beam at different offset distances.

Figure 9 compares axial intensity profiles at various offset distances as listed in Table 2, and it includes additional profiles for offsets of (+16, 0) and (+32, 0). An analysis of the simulation data from Table 2 and the axial intensity profiles depicted in Fig. 9 reveals that the off-axis displacement of the FZP has a minimal impact on the uniformity of its intensity distribution, which is consistent with that of a standard FZP. These findings confirm that the performance of the FZP remains stable across different off-axis displacement conditions.

 

Fig. 9. A single beam’s axial intensity distribution profiles at different offset distances fit curves.

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This method can extend the control effect of a single-component FZP on a Bessel beam to dual-component and even multi-component systems without changing the optimization coefficients. Table 3 shows the control effects of a dual-component FZP on the incident beam at different offset distances. Figure 10 compares the axial intensity profiles at different offset distances presented in Table 3 and adds axial intensity curves for (±16,0) and (±32,0) offset distances. The results indicate that when the FZP is closely offset, the sidelobes of the Bessel beam impact the central core energy during the initial propagation. Compared to a single component, there is a difference in the axial intensity distribution after the beam splits. However, this influence gradually diminishes and stabilizes as the offset distance increases.

 

Fig. 10. The axial intensity distribution profiles of dual beams at different offset distances fit curves.

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4. Experiment

4.1 Experimental setup

Figure 11 shows the schematic diagram of the experimental setup. The laser beam (with a diameter of d = 6.4 mm and a wavelength of λ = 1064?nm) passes through a beam splitter, attenuator, and two mirrors before entering the SLM (Hamamatsu X13138-02) at an incident angle of 10°. Two positive lenses (with focal lengths f1 = 500 mm and f2 = 150 mm) form a 4f system. A CCD is positioned at the back focal plane of lens 2, and the CCD is moved along the optical axis to observe the diffraction patterns at different axial distances.

 

Fig. 11. Experimental setup for optimization combining FZP and axicon.

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4.2 Experimental results

According to the analysis in section 2, the FZP is set at (x_m, y_m) = (+4, 0) mm. The traditional and modified composite holograms are loaded onto the SLM, and the energy values at the central core are collected under the same conditions. The fitted curve is shown in Fig. 12. Evaluate the uniformity of Bessel beams using the formula mentioned in Ref. [33]. Compared to the axial intensity distribution of the FZP combined with the traditional axicon (blue), the improved uniformity (red) within the optimization range of 200-1800?mm is enhanced from 54.65% to 94.66%.

 

Fig. 12. The axial intensity distribution.

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Figure 13 shows the diffraction patterns at different axial distances, with an angle resolution of 0.0008°. The range of angle deviation can reach up to 4.0742°.

 

Fig. 13. The diffraction patterns at different axial distances.

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When the FZP is superimposed with (x_m, y_m) = (±4, 0) mm and phase combined with the axicon, the axial intensity contrast results are shown in Fig. 14. The split beams at the same diffraction distance have the same intensity values. The modified(red) uniformity is improved from 56.67% to 96.15%.

 

Fig. 14. The axial intensity distribution.

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Fig. 15. The diffraction patterns at different axial distances.

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5. Discussions

Typically, a Bessel beam’s extended depth of focus is the primary reason for its use as an optical tool. In the model presented in this paper, introducing an FZP may alter the maximum axial length of the Bessel beam (see Fig. 15). The following analysis will examine this impact.

When analyzing the impact of introducing an FZP on the focal depth of a Bessel beam, the Gaussian beam can be approximated as collimated light incident, as shown in Fig. 16. The collimated light E₀ passes through an axicon with a diameter D and generates a diffraction field at z. Obtaining an exact analytical solution for this diffraction field is challenging; therefore, the stationary phase method is used to obtain an approximate solution [34]. The phase obtained is:

 

Fig. 16. Schematic diagram of collimated light incident on an axicon.

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The polar radius r is given by $r = \sqrt {{x^2} + {y^2}}$, and $r^{\prime}$ is the polar coordinate at z. Since the first derivative of $f(r)$, $f^{\prime}(r) = 0$, the maximum transmission distance z_max is obtained as follows:

The introduction of an FZP is equivalent to the addition of a lens. Figure 17 depicts the light path after introducing the FZP. Let us assume that the collimated light E₀ becomes a converging spherical wave E₁ after passing through the lens and illuminates the axicon. Under ideal lens conditions, we have:

 

Fig. 17. Schematic diagram of the light path after introducing FZP.

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In Eq. (17), z’ represents the convergence radius, and f is the lens’s focal length. Thus, z’ can be expressed as follows:

When collimated light is incident, as $l \to \infty$, the combination model of the lens and axicon makes $s = 0$, Therefore, $z^{\prime} \approx f$. according to the Fresnel diffraction formula, the diffraction field at z after the axicon is:

The stationary phase method obtains the maximum transmission distance ${z^{\prime}_{max}}$ as follows:

Substituting $z^{\prime} \approx f$ into Eq. (20), we obtain:

By comparing Eq. (16) and Eq. (21), it is evident that after introducing the FZP, the focal depth formula has an additional term D/f in the denominator. This indicates that the shorter the focal length f of the FZP, the more the focal depth of the Bessel beam decreases. Therefore, when designing the combination of FZP and axicon, the reduction in focal depth can be effectively minimized by appropriately adjusting the values of D and f. This paper employs GA and UNMA to ensure design optimization and optimize the phase modulation curve of the axicon lens. This approach allows us to precisely adjust the geometry of the axicon lens to match the parallelism adjusted by FZP, thereby achieving optimal focal depth control.

Although adding an FZP affects the focal depth of the Bessel beam, simulation results indicate that this impact is not significant in the studied application scenarios. This demonstrates the feasibility and effectiveness of introducing an FZP to adjust the beam parameters. For applications that do not require extremely long focal depths, such as certain imaging techniques or specific material processing procedures, a Bessel beam adjusted by an FZP can provide sufficient focal depth and better control and match the actual needs, thereby improving overall efficiency and effectiveness.

Additionally, the method can theoretically be expanded to multiple Bessel beams. However, while more FZPs could create numerous beams, each beam’s diameter would decrease, reducing the degrees of freedom and potentially limiting the number of effectively generated beams. This aspect warrants further investigation to assess the trade-off between the number of beams and their operational quality when scaled up.

6. Conclusion

The proposed method generates Bessel beams with high uniformity and controllable angles by using a combined FZP and axicon phase. The off-axis FZP achieves beam steering control, resulting in a Bessel beam with an extended focal depth due to the combined phase with the axicon. We use a Gaussian beam as the light source and design an ideally contoured axicon to achieve uniform axial intensity distribution when integrated with the FZP. We optimize the fitted profile using GA and UNMA. Our findings indicate that this method can control the angle to 4.0742° with a resolution of 0.0008° and achieve a uniformity of 94.66%. The intensities of the beams at different axial distances are nearly identical, achieving an overall uniformity of 96.15%.