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Optical interferometry has become the cornerstone of modern precision metrology with the increasing advancements in science and technology and is considered a powerful and versatile tool employed in a variety of applications, such as Earth observations [1], astronomical explorations [2], light detection and ranging [3], imaging [4], sensing [5], and many others. In particular, ultrafast noncontinuous dynamic detection based on pulse interferometry is extensively applied in the frontiers of emerging fields such as quantum enhanced sensing [6,7], electron dynamics in materials [8], biomedical fingerprinting [9], the study of plentiful light–matter interactions and its dynamic mechanisms, accompanied by unique applications. Optical feedback interference is an intriguing phenomenon observed in laser systems, essentially carrying crucial temporal or spatial information on light. Its working principle is based on the coherent interaction of the intracavity original lasing field and the reinjected light from external reflected or scattered surfaces, resulting in the modulation of laser output characteristics [10,11]. With the remarkable advantages of simplicity, self-alignment, compactness, and cost-effectiveness, the specially developed laser feedback interferometry (LFI) [12–14] is emerging as a desirable and potential noncontact optical diagnostic strategy that outperforms traditional dual-beam interferometers. Most notably, the inherent self-coherent nature of LFI can contribute to high detection sensitivity by suppressing unwanted radiation and perturbations entering the laser cavity, significantly ameliorating the extraction purity of the feedback photons.
Photonics Research, Volume. 13, Issue 3, 671(2025)
Toward exploring noncontinuous-state dynamics based on pulse-modulated frequency-shifted laser feedback interferometry
An exhaustive study of the noncontinuous-state laser dynamics associated with the transient optical process is significant because it reveals the complex physical mechanisms and characteristics in nonlinear laser systems. In this study, in-depth theoretical interpretation and experimental verification of the noncontinuous-state dynamics in laser system are presented, based on developed pulse-modulated frequency-shifted laser feedback interferometry (LFI). By introducing external pulse modulation, we investigate the nonlinear time-of-flight dynamics and related photon behaviors evolution of the pulsed LFI system by observing the changes in effective interference time sequences for interference realization and attainable minimum feedback photon number of the signal under various modulated noncontinuous states. Implementation of the pulse-modulated LFI scheme should exceed the pulse overlapping time window limit of 1.93 μs to effectively extract and preserve the extracavity feedback photon information. Experiments reveal that the minimum feedback photon number of signals successfully measured by the pulsed LFI sensor is 0.067 feedback photons per Doppler cycle, exhibiting high sensitivity for extremely weak signal detection. Further, simultaneous measurement for velocity and distance of the moving object is performed to validate the feasibility and applicability of the pulsed LFI. The system can successfully achieve large-range simultaneous measurements within the velocity range of
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1. INTRODUCTION
Contemporarily, the LFI-based techniques have been widely applied for measurements of various physical quantities of external moving objects, including vibration [15,16], displacement [17,18], velocity [19,20], distance [21,22], and so forth. Furthermore, it is also used as a preferred means for implementations of tomography [23,24], microscopy [25,26], biomedical monitoring [27,28], and other diverse actual applications. Characterization of the laser intrinsic parameters such as linewidth, Henry’s linewidth broadening factor , and optical feedback level factor , is also an indispensable part of the LFI territory by establishing system’s phase-amplitude response transferring function [29,30]. This provides a necessary prerequisite for determination of the intracavity operation state, permission of the extracavity dynamic measurements, and regulation of the mutual coupling action between these effects. Additionally, it also has been developing the study of laser behavior evolution [31,32], and further exploration of complex laser dynamics [33,34], leveraging the LFI effect. This is of vital importance for observing and gaining in-depth understanding of the fascinating experimental manifestations of lasers with different types, including linewidth narrowing or broadening, multistability, single-mode operation, coherence collapse, and chaos. These phenomenological explorations related to the lasing dynamics offer solid theoretical underpinnings for a diverse range of fundamental scientific research and practical applications.
However, the study of the LFI effect associated with diversified applications mainly takes advantage of a continuous-wave (CW) laser as the sensing source, while there are very few studies related to noncontinuous-state-based LFI. Of particular interest is the pulse operation of laser, compared to the CW laser, which has the prominent merits of higher emitted power, better tunability, higher spectral purity, and faster signal acquisition rate over a short timescale [35–37]. It provides a simple effective tool for studying the noncontinuous-state laser dynamics and related properties in LFI systems. Interestingly, the pulsed-mode-based LFI can utilize the time-of-flight (ToF) characteristics and dynamic phase-amplitude variation of the feedback signal to acquire multidimensional motion information, multifeature photon behavior, and multiphysical transformation effects. This is beneficial for the evolution analysis of ToF dynamics behaviors and the study of ultrafast nonlinear transient responses and related mechanisms under different discontinuous states. These are not available in the CW-based LFI system, which is mainly based on the average amplitude and phase information to obtain single portions of sensing information. Furthermore, pulsed LFI also offers advantages in enhancing temporal resolution and sensitivity, reducing system complexity, and enabling good tunability to accommodate various measurement requirements with the advancement and innovation of science and technology. These benefits can effectively mitigate multiple feedback phenomena, improve error noises in discontinuities’ detection, and reduce fluctuations in feedback strengths because of inadequate anti-interference capabilities, all of which are prevalent in CW-based LFI.
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Currently, the mainstream scheme for pulsed LFI generation is based on pulse-periodic excitation on the intracavity active medium, which is subjected to complex intracavity pulse dynamics. The sophisticated government of the entire intracavity pulse dynamics and the trade-off of various interactions, including dispersion, nonlinearity, and dissipation control, remains a considerable challenge. Although mechanical modulation on an optical chopper is also used to achieve the pulsed-mode operation of LFI, there exists a notable defect of insufficient modulation rate at approximately the kilohertz level. Leveraging the regulation of extracavity pulse dynamics is undoubtedly an attractive alternative to producing simple and reliable pulse sequences to meet the urgent needs of manifold research and applications with good flexibility. To our knowledge, there has been a lack of systematic investigation into the noncontinuous-state laser dynamics and underlying physical mechanisms of LFI under high-speed extracavity pulse modulation. Especially, the spatiotemporal dynamics of the pulse-modulated LFI have not been thoroughly examined.
To this end, we are dedicated to investigating the noncontinuous-state-based LFI effect by combining the extracavity frequency-shifted feedback technique [38,39] and external pulse modulation approach. Comprehensive validations of the ToF dynamics characteristics, behavior evolution, and expanded application of the pulse-modulated LFI sensor are conducted through a developed all-fiber system configuration.
2. OPERATING PRINCIPLE AND CHARACTERISTICS
The schematic diagram of the proposed pulse-modulated frequency-shifted LFI is shown in Fig. 1. The external pulse modulator plays a critical role in the system, serving as an effective tool for realizing pulse modulation of the feedback light, complemented by diverse functions such as information encoding and control in a discontinuous state.
Figure 1.Schematic illustration of the pulse-modulated frequency-shifted LFI. An AOM is used as an extracavity pulse modulator.
In this work, a pair of acousto-optic modulators (AOMs) is employed to achieve the functionality of extracavity modulation of light waves. Its fundamental operating principle is based on the acousto-optic interaction, functioning as a diffraction grating, to realize efficient light modulation by fine-tuning the frequency, amplitude, and phase of ultrasonic waves. Consequently, an AOM is employed as a pulse generator to modulate the CW laser into the pulse laser, driven by the external modulation level signal supplied by a radio-frequency (RF) source. Then, the induced pulse modulation light is transmitted to an external moving target, with the feedback photons carrying relevant ToF information and spatiotemporal characteristics returning into the laser cavity to interact with the original lasering field over a certain timescale, yielding the pulsed LFI effect. Especially, note that the short-range action mechanism of the pulsed LFI can suppress most of the average radiation entering the laser cavity, thereby allowing for the capture of extremely weak feedback signals in a noncontinuous state. Additionally, the AOM also acts as a frequency shifter, depicted in Fig. 2(a), concurrently producing the extracavity frequency-shifted feedback enhancement effect when the reinjected light resonates with the relaxation oscillation of the laser. As a result, it can reduce interferences from external low-frequency noises and other parasitic radiation, thus significantly ameliorating the detection capability of the system. This approach combines the pulse-modulated LFI with the extracavity frequency-shifted techniques, providing a solid foundation for the study of the ToF dynamics associated with the spatiotemporal characteristics of the moving targets.
Figure 2.Exploration of the ToF dynamics characteristics in the pulsed LFI system. (a) Schematic illustration of extracavity frequency shift; (b) time-domain waveform envelopes and according Doppler spectra at different pulse overlapping time windows; (c), (d) variation of the velocity signal intensity with pulse overlapping time interval and RF modulation voltage, respectively; (e), (f) minimum feedback photon number of the successfully attained pulsed LFI signal under different overlapping time intervals and modulation voltages, respectively. Here, pulse overlapping time refers to the overlapped time interval between the feedback light signal pulse and the original light pulse, namely, the time window for effective LFI interference occurrence within the pulse modulation period. FWHM, full width at half-maximum; SNR, signal-to-noise ratio. The minimum feedback photon number is defined as the number of feedback photons per Doppler cycle, satisfying the condition of
To demonstrate the dynamical characteristics of the proposed scheme, we vary the overlapping time window and RF voltage of the pulse excitation signal to explore their effects on the LFI signal. When the feedback light-pulse sequence carrying time and spatial information overlaps with the original light pulse under a certain time window, the pulsed LFI effect will produce inside the laser cavity. Within a pulse modulation cycle, when the overlapping time internal to the applied reference level and the temporal LFI signal envelope is exactly 1.93 μs, the feedback light signal at this moment is overwhelmed by the system noises, as shown in Fig. 2(b). As such, the effective sensing time sequence of the pulsed LFI system should be larger than 1.93 μs to ensure the accurate detection of the feedback signal photon. However, the constraints imposed by the intrinsic relaxation oscillation noise of the laser, measurement errors of the system caused by the electronic noises, Doppler signal broadening, and inevitable external disturbances, all comprehensively affect the detection limit of the weak feedback signal. Particularly, the temporal coherence of the LFI system is primarily limited by these influencing factors.
Then, we also explored the dynamic characteristics of the entire pulsed LFI system. The signal-to-noise ratios (SNRs) and local noise levels of Doppler shift-frequency signals under different pulse overlapping time windows and modulation voltages are shown in Figs. 2(c) and 2(d); they correspond to the change of minimum feedback photon number and Doppler signal broadening shown in Figs. 2(e) and 2(f), respectively. We can observe that the larger the pulse overlapping time interval, the higher the SNR of the Doppler frequency signal within the same pulse modulation period, while the FWHM of the Doppler frequency signal exhibits a decreasing trend. Upon reaching a pulse overlapping time window of 899.9 μs, the periodicity of the temporal pulsed LFI signal envelope is gradually blurred, resulting from the intensity fluctuation of the signal itself. For implementations of different pulse modulation, the maintenance of periodicity is related to the pulse overlapping time window and the interval of the adjacent pulse period.
Nevertheless, above the threshold of 2.6 V, the SNR of the Doppler frequency signal gradually deteriorates as the RF modulation voltage increases. Within the adjustable voltage range from 2.6 V to 10 V, the system’s SNR deteriorated by approximately 5 dB, while the FWHM variation of the Doppler frequency signal is not obvious. This is because the diffracted light intensity is modulated by the RF-driven voltage signal, where varying modulation voltage alters the amplitude of the RF circuit, thus affecting the output pulse light intensity. Notably, the broadening of the Doppler frequency signal is mainly attributed to the noise fluctuation caused by the degradation of the system’s SNR. Consequently, the appropriate modulation parameters should be chosen according to the different application requirements by taking advantage of the extracavity pulse modulation. In addition, the minimum feedback photon number necessary for the successful detection of the velocity signal is also analyzed. Accordingly, the sensitivity of the pulse-modulated LFI system reaches 0.067 photons per Doppler cycle of 4.2 μs, demonstrating the superior ability of this system to detect faint feedback signals with high sensitivity.
3. EXPERIMENTAL RESULTS
To verify the sensing feasibility and versatility of the proposed method, we demonstrate the simultaneous motoring of distance and velocity of the moving target based on the pulse-modulated LFI effect, using the established measurement setup (see Appendix A). The detailed description of theoretical interpretation is seen in Appendix B. In the preliminary experimental preparation, the frequency spectra and time-domain waveforms of the pulsed LFI velocity signals are primarily obtained without adding the delay fiber (see Appendix C, Fig. 9). We observe that the introduced appropriate frequency shift can effectively improve the SNR of the system, along with the obvious temporal pulse-modulated signal envelope. Incorporation of the extracavity pulse modulation into measurements is useful for extracting information on the distance and velocity of the external moving target from the ToF characteristics.
Figure 3.Performance characterization of pulsed LFI system at different extracavity velocities. (a) Doppler frequency spectra at different velocities; (b) repeatability test of the sensor in the initial status by five repeated measurements; the target’s velocity is varied from 73.5 to 612.6 mm/s. (c) Mapping of Doppler frequency spectra, under the external distance of 2.0 km; (d) dependence curves of the measured velocity on the actual velocity, under the external distance of 2.0 km. The SNR of the Doppler frequency signal is decreased by 5 dB as the velocity increases.
Further, velocity measurements using the pulsed LFI sensor under the specific external distance of 2.0 km are carried out by introducing the extracavity frequency-shifted technique. Figure 3(c) describes the mapping of average Doppler frequency spectra, while Fig. 3(d) shows the corresponding dependence curve of the measured velocity with different actual velocities. From the obtained experimental results, the relative error of velocity measurement is less than 0.87%, and the linear correlation coefficient of the fitting curve is 0.9999. This means that the pulsed LFI system has good consistency and accuracy in velocity monitoring of the external moving target.
The detection limit of achievable distance is also the core index for characterizing the performance of the current all-fiber pulsed LFI system. We further analyze the SNR, linearity, spectral mapping, and fluctuations of the Doppler frequency signals at various external distances when the turntable velocity was set to 245 mm/s. The measured results are plotted in Fig. 4.
Figure 4.Observations of the pulsed LFI velocity signal characteristics under different extracavity distances. (a) Variation curve of the velocity signal intensity via different distances; (b) dependence of measured distance versus the actual distance; (c) Doppler frequency spectra at different distances; (d) temporal waveform envelopes of the pulsed LFI velocity signals at different distances. Noticeably, the SMF acts as the long-distance transmission platform for the feedback light to carry the effective motion information of the moving target. In the experiment, the length of the SMF is adjusted to measure the target’s velocity at various distances.
With the increment of the external distance, the SNR of the Doppler shift-frequency signal gradually decreases, as shown in Fig. 4(a); this is mainly due to the fact that the extension of the extracavity distance increases the optical transmission loss. When the external distance is 25.5 km, the SNR is 2.2 dB, and the velocity signal can still be observed and distinguished in the spectrum. The maximum relative measurement error of the external distance is no more than 0.85% in the range of 25.5 km, with a linearity of 0.48%, as depicted in Fig. 4(b). This indicates that the proposed pulsed LFI sensor can successfully achieve long-range detection of weak feedback signals of the external noncooperative moving targets. Figure 4(c) gives the spectrograms of the Doppler frequency signal at different distances. Due to the large optical transmission loss induced by ultralong-range transmission, which results in the decrease of the signal intensity, the unsuppressed electronic noises and speckle effect will also bring about the broadening of the Doppler frequency signal. Additionally, the temporal waveform envelopes of the pulsed LFI velocity signal at different external distances are shown in Fig. 4(d). The ToF delay shows linear change trend as the external distance increases. Especially, note that the measurement error of the extracavity distance is primarily related to the signal envelope modulation caused by the random variation of the scattering spots, the long-range transmission optical loss, and the turn-on and turn-off time associated with modulation rate of the AOM (see details in Appendix E). This eventually results in stochastic fluctuation of the pulsed LFI signal, deterioration of the SNR, and reduction of the effectiveness and temporal resolution of the modulated reference level, which in turn affects accurate measurements of the external distances.
To further explore the sensing capability of the all-fiber pulsed LFI system, a specific example of experimental verification is conducted by simultaneously changing the distance and velocity of the turntable with the fixed laser irradiation point. The measurements for different velocities for the cases of 2.0, 5.0, 10.2, 15.2, 20.5, and 25.5 km are given in Fig. 5(a). The fluctuation of velocity measurements at different external distances is relatively small and has good consistency. The relative error of velocity measurement is less than 1.62% over the distance range of , while the relative error of distance measurement is less than 1.81% of the velocity in the range from 73.5 to 612.6 mm/s. The error is mainly caused by the backscattered light inside fiber and its connection section, which will return into the laser cavity to modulate the output light and produce a nonlinear effect, resulting in the emergence of other frequency peaks and affecting the velocity signal detection at an ultralong distance (see Appendix C, Fig. 10). Furthermore, the additional loss introduced by the extension of the extracavity distance, the electronic noises and oscillations of the system itself, and environmental disturbances will also bring measurement errors to the experimental measurements.
Figure 5.Various LFI velocity signals with respect to different extracavity distances. (a) Simultaneous measurement of distance and velocity in the range of
Then, the performance indices of efficiency and feasibility of this pulsed LFI sensor are also analyzed through stability verification. Correspondingly, the extracavity distance measurements under various velocities of the turntable are presented in Fig. 5(b). The relative error of the measured distances for different rotational velocities of the turntable is less than 1.07%, signifying the good stability of this sensor. In contrast, concerning Fig. 5(c), the velocity measurements of the moving target for different extracavity distances are carried out. When the extracavity distance changes from 2.0 to 25.5 km, the relative error of the velocity measurement is less than 1.15%, and the velocity signal is basically maintained at around 245 mm/s, which is well consistent with the preset velocity of the turntable. In a certain sense, this implies that the velocity measurement is independent of the variation of the extracavity distance, and there is no presence of a serious cross-coupling effect.
In summary, these experimental results show the feasibility and practicability of this reported novel all-fiber pulsed LFI sensing strategy for simultaneous detection of extracavity velocity and distance with high accuracy and stability. This study sets a precedent for the pulsed LFI effect based on extracavity pulse modulation, offering a high-performance, cost-effective solution for potential applications such as ultralong-distance multisource site sensing and simultaneous monitoring of multiple physical parameters under diverse complex and harsh environments. In future research, with the improvement of the system and innovation of optical metrology, we will further devote time to studying more complicated noncontinuous-state spatiotemporal dynamics of the pulsed LFI system, including temporal and spatial coherence, to promote its wider applications.
4. CONCLUSION
This study presents what we believe is a novel all-fiber pulse-modulated LFI system to study noncontinuous-state laser feedback dynamics, whereby CW light is converted into pulse light by combining the extracavity frequency-shifted and pulse modulation techniques. The proposed all-fiber pulsed LFI system based on extracavity pulse modulation has a simple and compact structure that is easier to regulate than the pulse-generation strategy relying on the modulation of intracavity pulse dynamics. Primarily, the ToF dynamic characteristics of the pulsed LFI are studied, and the extracavity effective feedback photon information can be acquired within the pulse overlapping time window of more than 1.93 μs and a modulation voltage range of . The detection sensitivity of the pulse-modulated LFI sensor is 0.067 photons per Doppler cycle, revealing the good ability and potential for effective measurement of extremely weak feedback light signals.
The reliability and practicability of the pulsed LFI scheme are further validated by implementing the simultaneous measurement for both distance and velocity of the external moving target. Experimental results demonstrate that the pulsed LFI sensor can realize ultralong-distance simultaneous sensing in the velocity range of within a distance of 25.5 km. The relative error of velocity measurement is less than 1.62%, while the relative error of distance measurement is less than 1.81% under the adjustable measurement range. Our work provides an intriguing photonic platform for investigations of the noncontinuous-state laser dynamics based on the pulsed-modulated LFI, paving the way for a variety of applications, including multiphysical parameter measurement of noncooperative targets, positioning, and remote sensing.
Acknowledgment
Acknowledgment. The authors acknowledge all the members of the “self-mixing” research group of Anhui University for their enthusiastic help.
Author Contributions.L.L. and Y.Z. conceived the idea and designed the experiments. J.L. performed the experiments and processed the data with the help of Y.Z., and J.L. and Y.Z. provided theoretical analysis under the guidance of L.L. All the authors analyzed the data and contributed to the discussion. J.L. and Y.Z. wrote the paper with contributions from all authors. J.L. provided investigative support, and H.L., Q.Y., and J.L. provided formal analysis. L.L. supervised the project.
APPENDIX A: EXPERIMENTAL SETUP
The designed experimental setup of the all-fiber pulsed LFI system is built based on the extracavity frequency-shifted feedback technique [
Figure 6.Experimental system for the pulsed LFI sensor for simultaneous velocity and distance measurement. WDM, wavelength division multiplexer;
In the experiment, the pump current is set to 300 mA, and the pump light with a wavelength of 980 nm travels through a WDM into the laser cavity and then induces stimulated radiation from the -doped gain fiber, resulting in the emission of a 1550 nm wavelength laser. Subsequently, the output light is directed to port 2 of the (), and the incoming light is then output from port 3 and passes through two fiber-based AOMs in series. After transmitting through the and the tunable delay single-mode fiber (SMF) in turn, the emitted light is focused on the turntable’s surface, covered with a white paper, by a collimating lens. Then, the feedback light carrying the movement information of the external moving target is received by the collimator and subsequently returned into the laser cavity through the delay SMF and two CIRs. Ultimately, it can cause modulation on the output frequency and intensity of the laser, leading to the LFI effect. Here, is powered with a direct current voltage of 24 V, and its adjustable frequency range is , serving as a pulse modulator. Nevertheless, the achievable frequency shift by is fixed at 100 MHz. Such a pair of AOMs, utilizing the frequency-heterodyning method, can be used as an extracavity frequency shifter. Accordingly, the generated pulsed LFI velocity signal is detected by the photodetector (PD, New focus, 1811) and is then connected to both the spectrum analyzer (Rohde & Schwarz, FSV13) and the oscilloscope (Teledyne Lecroy, 4104HD) to observe and analyze the measured velocity signals under various lengths of the delay SMF, respectively.
APPENDIX B: THEORETICAL ANALYSIS OF THE PULSED LFI SENSOR
In this work, the pulsed LFI sensing system based on the high-speed extracavity pulse modulation scheme is established by combining the ToF ranging and frequency-shifted optical feedback techniques.
According to the equivalent three-mirror Fabry–Perot (F-P) cavity model [
Figure 7.Schematic of the all-fiber pulsed LFI theoretical model for simultaneous sensing of velocity and distance based on the extracavity frequency-shifted optical feedback effect under pulse modulation. (a) Equivalent three-mirror F-P cavity model of the DFB fiber laser; (b) variation curve of system gain factor with the frequency shift of the external cavity under different normalized pumping coefficients; (c) theoretical attainable maximum attainable gain factor of the LFI system as a function of the normalized pumping coefficient.
Figure 8.Numerical simulation results of simultaneous measurement for the external velocity and distance (
In Fig.
The actual Doppler shift frequency of the measured target’s velocity can be calculated by
Because of the difference of the undergoing optical path between the driving reference level signal [see Fig.
APPENDIX C: EXPERIMENTAL VERIFICATION OF THE PULSED LFI SENSOR
According to the aforementioned well-established theoretical model and current experimental arrangements, the experiments for the observation of ToF dynamics characteristics and the simultaneous measurement of velocity and distance are conducted to verify the sensing performance of the proposed system.
In the preliminary experimental preparation, we obtain the frequency spectra and time-domain waveforms of the pulsed LFI velocity signals under different modulation statuses, without inserting the delay fiber, as shown in Fig.
Figure 9.Spectra and speckle envelopes of the pulsed LFI velocity signals. (a), (b) Spectrum and temporal waveform of the initial LFI velocity signal, respectively; (c), (d) spectrum and temporal waveform of the LFI velocity signal when the Doppler frequency signal moves to the laser relaxation oscillation peak, respectively; (e), (f) spectrum and time-domain waveform of the LFI velocity signal under pulse modulation, respectively.
Figure 10.Spectrum and time-domain waveform of the LFI velocity signal under 2.0 km delay fiber. (a) Pulse-modulated reference level signal; (b) frequency spectrum of the LFI velocity signal; (c) temporal waveform envelope of the pulsed LFI velocity signal; (d) partial enlargement of signal waveform diagram of (c).
Figure
APPENDIX D: ANALYSIS OF THE SOURCES OF SIGNAL BROADENING
In the experiment, we observe that the Doppler frequency signals under different cavity lengths and velocities exhibit a certain broadening. This phenomenon is closely related to the operation state and characteristics of the pulsed LFI measurement system. Next, we aim to analyze the main sources of the broadening of the Doppler frequency signal; the impact of each influencing factor on the spectral broadening is clearly quantified. The comparative details are presented in Table
By differentiating Eq. (
Combining Eq. (
For the DFB fiber laser used in the pulsed LFI measurement system, the output wavelength is 1550 nm, with the linewidth of 4.3 kHz measured by the delayed self-heterodyne method. Therefore, with the Doppler frequency of 152 kHz, the broadening of the Doppler frequency signal caused by the laser spectral widening is , with a velocity measurement uncertainty of . This contribution to the measurement uncertainty of the Doppler frequency signal is relatively small.
Additionally, the velocity distribution inhomogeneity of each point in the light spot will also cause the broadening of the Doppler frequency signal. The linear velocity of the turntable to be measured can be expressed as
According to the actual situation, here we mainly consider the influence of the velocity distribution of each point within the light spot region on the broadening of the Doppler frequency signal. Based on Eq. (
Consequently, the ratio of broadening of the Doppler signal caused by the velocity distribution inhomogeneity of each point in the light spot to the Doppler frequency signal can be described by
For the entire measurement system, the diameter of the output light spot is 0.8 mm, namely, , and the light spot is at the position of , for which the broadening of the Doppler frequency signal caused by the different linear velocities of each point in the light spot is calculated as , which corresponds to the velocity measurement uncertainty of . Notably, the velocity distribution resulting from the uneven light spot area actually contains the real velocity distribution.
The object’s surface can be considered as composed of numerous scattering units with surface roughness. A Doppler frequency shift is introduced in the scattered light when the light spot illuminates these scattering units. Note that the amplitude and phase of this scattered light vary randomly. The random fluctuations of the speckle lead to envelope modulation of the Doppler signal, resulting in the broadening of the Doppler frequency signal. This modulation speckle effect is one of the primary sources of Doppler frequency signal broadening, which can be expressed as
The pulsed LFI sensing system experiences noises generated by the laser source, optical devices, and environmental disturbances, all of which impact the measurement accuracy. Generally, the sources of noise can be categorized as external and internal. The primary external sources include vibration and temperature fluctuation, which can adversely affect the accuracy of weak feedback signal detection. The internal noises mainly arise from the inherent physical processes of photoelectric conversion, including parameters closely related to the device and the shot noise generated by fluctuations due to the random generation of photoelectrons or photogenerated carriers in the laser. In the pulsed LFI measurement system, the dark current noise of the PD is generally much smaller than the quantum noise of the laser. As a result, the main noise source considered in the system is the intensity noises of the laser itself.
The equivalent displacement NED generated by the internal intensity noise in the system can be represented by
In the experiment, the emitted power of the DFB laser is set to and ; the calibrated feedback optical power of the system is . Thereby the effective reflectivity of external cavity is evaluated to be , and the cavity gain coefficient of the system is 114.9. In terms of Eqs. (
In the experiment, the RBW of the spectrum analyzer is set to 100 Hz, the center frequency of the velocity signal spectrum is equal to the Doppler frequency shift, and the corresponding velocity resolution is expressed as
Accordingly, the broadening of the Doppler frequency signal caused by the RBW of the spectrum analyzer is , corresponding to the velocity measurement uncertainty of .
In summary, the total broadening of the Doppler frequency signal can be derived as
Based on Eq. (
Through the above detail analysis, the broadening of the Doppler frequency signal mainly originated from the velocity distribution inhomogeneity at each point within the light spot, the speckle modulation effect, intensity noises, and the RBW limitation of the spectrum analyzer. The spectral broadening will cause the measurement error of the pulsed LFI velocity signal, which affects the accuracy of the practical measurement results. This error cannot be completely eliminated, but the spectral broadening of the Doppler frequency signal can be reduced by optimizing the performance of the laser source, suppressing system noises, enhancing the system’s anti-interference ability, and selecting better detection and analysis devices in further research work.
In the pulsed LFI sensing system, the main influencing factors, including the intrinsic laser relaxation oscillation noise, the system’s measurement errors (such as shot noise, low-frequency noise, and thermal noise), and Doppler frequency signal broadening, collectively affect the detection range and accuracy of the weak feedback signals. Additionally, ambient perturbations from vibration, temperature, humidity, and even airflow floating are also inevitable to bring the certain extra noise to the system, thereby impacting the measurement effects of extracavity distance and velocity. Therefore, in practical applications, it is necessary to reduce these errors by optimizing various aspects, such as the laser source, system structure design, and signal processing. This can be achieved by stabilizing the laser output using temperature and pressure control systems, employing high-precision optical components to reduce spot size spread and speckle effects, introducing noise suppression technologies to reduce noise interference, and setting an appropriate sampling rate to improve signal extraction accuracy.
APPENDIX E: ERROR ANALYSIS OF ToF-BASED DISTANCE MEASUREMENTS
The speckle modulation introduces fluctuations in the strength of the received pulsed LFI signal, which can impact the ranging system’s ability to accurately extract the ToF information from the signal envelope. This fluctuation may cause the measurement system to misjudge the arrival time of the pulsed LFI signal. Additionally, the light beam will be dispersed during propagation in the presence of scattering, resulting in a larger effective light spot size. This enlargement reduces the spatial resolution of the optical system, further compromising the accuracy of the extracavity distance measurements.
Additionally, the characteristics of the extracavity modulation unit will also bring some errors to the actual external distance measurements. Among these, the steepness of the rising edge of the AOM determines the actual width of the pulse, with the turn-on and turn-off time of the AOM used in the experiment being approximately 45 ns. If the rising edge is slow, the effective width of the pulse increases, which leads to uncertainty in the transmission and reception of the signal and affects the ranging accuracy. The steepness of the falling edge affects the speed of the signal transition from “high level” to “low level.” If the falling edge is slow, the signal may remain at a high level for a period of time, which may cause the receiving system to incorrectly calculate the ToF of the feedback signal, affecting the distance measurement. Therefore, the effect of the rising and falling edges of the AOM’s pulse modulation on the distance measurement is mainly reflected in the accuracy of the ToF measurement and signal quality. The faster the rising edge and falling edge are, the higher the temporal resolution is. Furthermore, the cable transmission also causes a certain time delay error in the ToF-based ranging measurements.
[17] D. Guo, Y. Yu, L. Kong. LFI grating interferometer with dual laser diodes for sensing of 2-D dynamic displacement. IEEE J. Quantum Electron., 54, 7500106(2018).
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Jie Li, Yunkun Zhao, Jie Liu, Jianchu Liu, Hongtao Li, Qi Yu, Jialiang Lv, Liang Lu, "Toward exploring noncontinuous-state dynamics based on pulse-modulated frequency-shifted laser feedback interferometry," Photonics Res. 13, 671 (2025)
Paper Information
Category: Instrumentation and Measurements
Received: Nov. 1, 2024
Accepted: Jan. 1, 2025
Published Online: Feb. 24, 2025
The Author Email: Yunkun Zhao (zyk231050@163.com), Liang Lu (lianglu@ahu.edu.cn)
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