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【AIGC One Sentence Reading】：This study explores noncontinuous laser dynamics using pulsed LFI, achieving high-sensitivity signal detection and validating large-range simultaneous measurements.AI Voice 

【AIGC Short Abstract】：This study explores noncontinuous-state laser dynamics using pulse-modulated frequency-shifted laser feedback interferometry. It reveals high sensitivity in weak signal detection, achieving simultaneous velocity and distance measurements over long ranges. The work paves the way for new applications in integrated sensing, remote monitoring, and navigation.AI Voice 

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1. INTRODUCTION

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2. OPERATING PRINCIPLE AND CHARACTERISTICS

Figure 1.Schematic illustration of the pulse-modulated frequency-shifted LFI. An AOM is used as an extracavity pulse modulator. $T$, Doppler cycle; $\mathrm{\Delta }t$, time delay.

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 $\mathrm{SNR}=1$, which corresponds to the achievable sensitivity of the system under different modulation parameters.

3. EXPERIMENTAL RESULTS

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.

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.

Figure 5.Various LFI velocity signals with respect to different extracavity distances. (a) Simultaneous measurement of distance and velocity in the range of $73.5-612.6\mathrm{mm}/\mathrm{s}$, and the distance ranges from 2.0 to 25.5 km. The horizontal and vertical error bars indicate the SD of the distance and velocity measurements, respectively. (b) Distance measurements at different velocities, under the external distance of 5.0 km; (c) velocity measurements with the external distances when the turntable velocity was set to 245 mm/s.

4. CONCLUSION

Acknowledgment

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 [38,40,41] and external pulse modulation method [42,43]. Further, a particular application example of the pulsed LFI sensor is validated by combining the ToF dynamics characteristics for simultaneous measurement of the external-cavity distance and velocity of the moving target in combination with theoretical modeling and experimental verifications. This experimental system employs a DFB laser as the light source; the specific systematic architecture is illustrated in Fig. 6.

Figure 6.Experimental system for the pulsed LFI sensor for simultaneous velocity and distance measurement. WDM, wavelength division multiplexer; ${\mathrm{AOM}}_{1,2}$, acousto-optic modulators; ${\mathrm{CIR}}_{1,2}$, circulators; PD, photodetector; SMF, single-mode fiber.

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 ${\mathrm{Er}}^{3+}$-doped gain fiber, resulting in the emission of a 1550 nm wavelength laser. Subsequently, the output light is directed to port 2 of the ${\text{circulator}}_1$ (${\mathrm{CIR}}_1$), and the incoming light is then output from port 3 and passes through two fiber-based AOMs in series. After transmitting through the ${\mathrm{CIR}}_2$ 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, ${\mathrm{AOM}}_1$ is powered with a direct current voltage of 24 V, and its adjustable frequency range is $100\pm 5\mathrm{MHz}$, serving as a pulse modulator. Nevertheless, the achievable frequency shift by ${\mathrm{AOM}}_2$ 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 [11,12] of the DFB fiber laser and in combination with the Lang–Kobayashi (L-K) rate equations [11,12] and the extracavity frequency-shifted theory [38,44], an all-fiber pulse-modulated frequency-shifted LFI theoretical model is developed for multiple-site simultaneous sensing of velocity and distance of the extracavity noncooperative moving object, as shown in Fig. 7.

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 ($\nu =0.03\mathrm{m}/\mathrm{s}$, $\theta =60°$). (a) Without frequency shift of the external cavity; (b) frequency shift amount of $f_{\mathrm{AOM}}=60\mathrm{kHz}$, (c) $f_{\mathrm{AOM}}=120\mathrm{kHz}$, and (d) $f_{\mathrm{AOM}}=180\mathrm{kHz}$; (e) applied external pulse modulation level; (f) without adding the external sensing fiber, (g) with the sensing fiber length of $\mathrm{\Delta }{L}_{\mathrm{ext}}=5\mathrm{km}$, and (h) $\mathrm{\Delta }{L}_{\mathrm{ext}}=10\mathrm{km}$.

In Fig. 8, the output light intensities of the pulsed LFI velocity signal waveforms under various frequency shifts are shown in Figs. 8(a)–8(d). When the generated LFI velocity signal is gradually shifted to the vicinity of the inherent relaxation oscillation peak of the laser system, the interaction strength between the feedback light and the gain medium inside the laser cavity will be significantly enhanced. This resonance effect reveals a harmonious interaction between the laser and the LFI signal, amplifying the signal’s intensity within the specific frequency match. It is especially worthwhile to note that when the frequency of the frequency-shift-modulated LFI velocity signal is exactly equal to the laser relaxation oscillation frequency, the amplitude of the LFI velocity signal reaches the maximum value. Therefore, we can improve the modulation signal strength of the feedback light in the intracavity of the DFB fiber laser by properly adjusting the frequency shift of the LFI velocity signal to realize very weak light detection. Simultaneously, such a simple operation is effective in avoiding the influence of the low-frequency noises, significantly improving the SNR of the pulsed LFI sensing system.

The actual Doppler shift frequency of the measured target’s velocity can be calculated by $$f_d=\frac{2\nu \cos \theta }{\lambda }.$$(B8)

Because of the difference of the undergoing optical path between the driving reference level signal [see Fig. 8(e)] and the pulse-modulated LFI velocity signal [see Figs. 8(f)–8(h)], it will generate the ToF delay $\mathrm{\Delta }t$ between them. Notice that the time delay gradually increases with the increment of the external distance, and the distance of the external moving target to be measured can be obtained from the time delay $\mathrm{\Delta }t$ of the original pulse modulation reference level and the LFI velocity signal. Therefore, the pulse-modulated frequency-shifted LFI can be implemented for multiple-site simultaneous sensing of velocity and distance of the external moving object by combining the extracavity frequency-shifted technique and active pulse modulation scheme. Accordingly, numerical simulations verify the feasibility and applicability of the proposed pulse-modulated frequency-shifted LFI, which provides solid theoretical guidance for the subsequent experiments.

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. 9.

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 10(a) represents the pulse-modulated reference level signal applied to ${\mathrm{AOM}}_2$ with a period of 1 ms and a duty cycle of 50%. In Fig. 10(b), we can observe the obvious frequency peak $f_{\mathrm{AOM}}$, which is the applied frequency shift of AOMs during the round trip of the light. When the Doppler frequency $f_d+f_{\mathrm{AOM}}$ is shifted to near the relaxation oscillation frequency, the nonlinear dynamical effects are gradually induced by the resonant enhancement of relaxation oscillation originating from frequency-shifted optical feedback. Thus, a series of harmonic peaks ($2f_d+2f_{\mathrm{AOM}},3f_d+3f_{\mathrm{AOM}}$) and parametric peaks ($f_d+2f_{\mathrm{AOM}}$) are excited in the laser power spectrum due to nonlinear coherent excitation response. The Doppler frequency of the pulsed LFI velocity signal after the adjustment of extracavity frequency shift is 233 kHz, nearly approaching the relaxation oscillation frequency of the laser. Consequently, the motion velocity of the external object can be determined by detecting the Doppler frequency shift. Figures 10(c) and 10(d) depict the time-domain waveform envelope of the pulsed LFI velocity signal and the according partial amplification, respectively. It is worth noting that the external distance of the moving target to be measured can be obtained by comparing the time interval of the initial reference level and the pulse LFI velocity speckle envelope signal. Consequently, it is possible to realize the real-time effective information acquisition of the distance and velocity of the external moving target based on the pulse-modulated frequency-shifted LFI sensor.

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 1.

By differentiating Eq. (B8), we can obtain $$\mathrm{\Delta }{f}_d=\frac{2\nu \cos \theta }{{\lambda }^2}\mathrm{\Delta }\lambda .$$(D1)

Combining Eq. (B8) and Eq. (D1), the following dependence of the laser linewidth and the Doppler frequency shift can be further acquired and yields $$\frac{\mathrm{\Delta }{f}_{d(\mathrm{width})}}{f_d}=\frac{\mathrm{\Delta }\lambda }{\lambda }.$$(D2)

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 $\mathrm{\Delta }{f}_{d(\mathrm{width})}=3.42\times {10}^{-6}\mathrm{Hz}$, with a velocity measurement uncertainty of $2.25\times {10}^{-11}$. 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 $$\nu =r\omega ,$$(D3)where $r$ is the radius of the light spot relative to the center on the turntable, and $\omega$ is the angular velocity of the moving object. Through differential calculation of Eq. (D3), we can achieve the following expression: $$\mathrm{d}\nu =\omega \mathrm{d}r+r\mathrm{d}\omega .$$(D4)

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. (B8), we can obtain $$\mathrm{\Delta }{f}_d=\frac{2\cos \theta }{\lambda }\mathrm{d}\nu =\frac{2\cos \theta }{\lambda }\omega \mathrm{d}r.$$(D5)

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 $$\frac{\mathrm{\Delta }{f}_{d(\nu )}}{f_d}=\frac{\mathrm{d}r}{r}.$$(D6)

For the entire measurement system, the diameter of the output light spot is 0.8 mm, namely, $\mathrm{d}r=0.8\mathrm{mm}$, and the light spot is at the position of $r=58\mathrm{mm}$, 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 $\mathrm{\Delta }{f}_{d(\nu )}=2.10\times {10}^3\mathrm{Hz}$, which corresponds to the velocity measurement uncertainty of $1.38\times {10}^{-2}$. 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 $$\mathrm{\Delta }{f}_{d(s)}\approx \frac{2}{\pi \tau \sqrt{N}},$$(D7)where $\tau$ is the average time length of the pulsed LFI velocity signal envelope, and $N$ is the ratio of the sampling time to the average time length of the signal envelope. In the actual experimental measurement, the sampling time of the system is set as 5 ms, and the average time length of the pulsed LFI velocity signal envelope is 0.35 ms. Consequently, the broadening of the Doppler frequency signal caused by the speckle effect is roughly obtained as $\mathrm{\Delta }{f}_{d(s)}=481.24\mathrm{Hz}$, corresponding to the velocity measurement uncertainty of $3.17\times {10}^{-3}$.

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 $$\mathrm{NED}=\frac{(\lambda /2\pi ){(2eB/I_0)}^{1/2}}{\mathit{RM}},$$(D8)with $$M=\sqrt{R_{\mathrm{eff}}}\times G(\mathrm{\Omega }),$$(D9)$$\mathrm{\Delta }{f}_{d(\mathrm{NED})}=\frac{2\cos \theta }{\lambda }{d}_{\nu (\mathrm{NED})}=\frac{2\cos \theta }{\lambda }\frac{\mathrm{NED}}{\mathrm{\Delta }{t}_s},$$(D10)where $e=1.62\times {10}^{-19}$ is the elementary charge, $B$ is the scanning RBW of the spectrum analyzer, $R=1.0\mathrm{A}/\mathrm{W}$ is the responsivity of the PD at the wavelength of 1550 nm, and $I_0=R{P}_1$ is the induced photocurrent of the output laser traveling through the PD, in which $P_1$ is the output optical power of the PD end. $M$ is the modulation contrast of the system, and $\mathrm{\Delta }{t}_s=4.74\mathrm{\mu s}$ is the scanning time between two adjacent components in the spectrum.

In the experiment, the emitted power of the DFB laser is set to $P_0=120\mathrm{\mu W}$ and $P_1=15\mathrm{\mu W}$; the calibrated feedback optical power of the system is $P_{\mathrm{feedback}}=0.1\mathrm{\mu W}$. Thereby the effective reflectivity of external cavity is evaluated to be $R_{\mathrm{eff}}=P_{\mathrm{feedback}}/P_0=8.33\times {10}^{-4}$, and the cavity gain coefficient $G$ of the system is 114.9. In terms of Eqs. (D8)–(D10), the broadening of the Doppler frequency signal resulting from the inherent intensity noise of the system is achieved to be $\mathrm{\Delta }{f}_{d(\mathrm{NED})}=1.48\times {10}^{-2}\mathrm{Hz}$, with a velocity measurement uncertainty of $2.29\times {10}^{-8}$. Additionally, other system noise sources and its own instability as well as environmental disturbances including thermal creeping, vibration, humidity, and airflow will also cause certain measurement errors.

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 $$\frac{\mathrm{\Delta }{f}_{d(\mathrm{res})}}{f_d}=\frac{\mathrm{RBW}}{f_c}.$$(D11)

Accordingly, the broadening of the Doppler frequency signal caused by the RBW of the spectrum analyzer is $\mathrm{\Delta }{f}_{d(\mathrm{res})}=100\mathrm{Hz}$, corresponding to the velocity measurement uncertainty of $1.55\times {10}^{-4}$.

In summary, the total broadening of the Doppler frequency signal can be derived as $$\mathrm{\Delta }{f}_{d(\mathrm{total})}=\sqrt{\mathrm{\Delta }{f}_{d(\mathrm{width})}^2+\mathrm{\Delta }{f}_{d(\nu )}^2+\mathrm{\Delta }{f}_{d(s)}^2+\mathrm{\Delta }{f}_{d(\mathrm{N}\mathrm{E}\mathrm{D})}^2+f_{d(\mathrm{res})}^2}.$$(D12)

Based on Eq. (D12), we can obtain the overall broadening $\mathrm{\Delta }{f}_{d(\mathrm{total})}$ of the Doppler frequency signal to $2.15\times {10}^3\mathrm{Hz}$, which is also in good agreement with the deviation of the measured Doppler frequency signal.

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 $1/f$ 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.