1. Introduction

Lidars have been studied widely in recent years for their potential applications in autonomous vehicles, intelligent manufacturing, surveillance, augmented reality and virtual reality (AR/VR), and 3D modeling [1–4]. While sensors such as camera, radar, and ultrasound all play different roles in nowadays imaging and sensing systems, lidars have the unique advantages of providing critical depth information with high accuracy and good resolution for up to a very long distance [5–8].

Lidars based on time-of-flight (TOF) typically send repetitive optical pulses and measure the round-trip time taken for the light to come back from the target. By multiplying the flight time of the pulse with the speed of light, the range of the target can then be determined. While most lidars send out unspecific optical pulses without any designed patterns or codes, mutual interference becomes possible in the scenarios where multiple lidar units are operated at the same time [9–11]. They are also vulnerable to jamming and attack from the malignant hackers [12–14].

To prevent both the interference and jamming, in 2004 a proof-of-concept CW chaos lidar based on nonlinear dynamics of semiconductor lasers was studied [15]. With the aperiodic and uncorrelated chaos waveform transmitted, the range of the target is detected by calculating the lag time of the cross-correlation peak between the received signal backscattered from the target and a transmitted replica. Signals from other sources including interference and jamming are suppressed through the correlation process. To detect common objects with diffuse reflections at standoff distances under the class-1 eye-safe regulation [16], a pulsed chaos lidar using an acousto-optic modulator (AOM) to time-gate the CW chaos into chaos-modulated pulses was proposed in 2018 [17]. With a higher peak power and thus higher signal-to-noise ratio (SNR) than the CW chaos lidar, the pulsed chaos lidar demonstrated can provide depth information with sub-centimeter precision for more than 100 meters.

While loss-modulating with an AOM to generate the chaos-modulated pulses is lossy and energy inefficient, in 2018 we further investigated a gain-modulating configuration based on a gain-switched semiconductor laser subject to optical feedback [18]. By setting a feedback time delay $\tau$ shorter than the pulse repetition interval (PRI) so that each pulse can undergo several feedback iterations, we indeed observed chaos oscillations within each gain-switched pulse. However, only the latter part of the pulse can be effectively utilized for the chaos lidar applications since, within each pulse, it takes several feedback delay loops for the laser to evolve from the transient response, periodic oscillations, and then into the useful chaos oscillations.

Therefore, to generate chaos-modulated pulses with less dwell time and greater portions of chaos oscillations, in this paper we study the nonlinear dynamics of a gain-switched semiconductor laser subject to delay-synchronized optical feedback under the condition of PRI = $\tau$. In this configuration, without the time needed for the pulse to evolve from transient dynamics into chaos, the dynamics of a pulse are induced and affected by the dynamics of the previous pulse in the pulse train through optical feedback. We investigate the correlation properties of the chaos-modulated pulses generated under different feedback strengths and modulation currents of gain-switching. We quantify their quality by analyzing the ratio of chaos oscillations, peak sidelobe levels (PSLs), and cross-correlation peaks. Moreover, we also discuss how the mismatch between PRI and $\tau$ affects the pulses generated.

2. Experimental setup

Figure 1 shows the schematic setup of a gain-switched semiconductor laser subject to delay-synchronized optical feedback for chaos-modulated pulses generation. We use a function generator (FG) (Agilent 81150A) to gain-switch a single-mode DFB semiconductor laser (LD) (Gooch & Housego AA0701) with square-wave modulations, where different modulation currents $I_{\textrm{m}}$, pulsewidths, and pulse repetition intervals (PRI) are adjusted accordingly. A fiber coupler (FC) divides the laser output into two arms. The light in one arm is fed back to the laser by a fiber mirror (FM) through an optical variable attenuator (VA) to control the feedback strength $\xi$. The feedback loop has a time delay $\tau$ of 112.4 ns and a corresponding loop frequency of 8.9 MHz. The light in the other arm is sent to a photodetector (PD) (Newport 1544-B, 12 GHz bandwidth) followed by an oscilloscope (OSC) (Tektronix TDS 6604, 6 GHz bandwidth, 20 GS/s) to analyze the chaos waveforms generated. In this study, the feedback strength $\xi$ is defined as the ratio between the fed back and the emitted fields to and from the laser monitored by the power meter (PM).

 

Fig. 1. Schematic setup of a gain-switched semiconductor laser subject to delay-synchronized optical feedback for chaos-modulated pulses generation. FG: function generator; LD: laser diode; PM: power meter; FC: fiber coupler; VA: variable attenuator; FM: fiber mirror; PD: photodetector; OSC: oscilloscope.

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3. Temporal and spectral characteristics

Figures 2(a) and 2(b) show the chaos-modulated pulses generated by a gain-switched semiconductor laser subject to optical feedback with the operational parameters of ($\xi = 31\%$, $I_{\textrm{m}}$ = 20 mA, pulsewidth = 500 ns, PRI = 1 $\mu$s) and ($\xi = 8\%$, $I_{\textrm{m}}$ = 17 mA, pulsewidth = 80 ns, PRI = 112.4 ns), respectively. The laser has a threshold current $I_{\textrm{th}}$ of 15 mA. Figures 2(c) and 2(d) are their corresponding spectra from the wavelet analyses. When PRI is longer than $\tau$, as shown in Figs. 2(a) and 2(c), the laser experiences multiple iterations of optical feedback in the duration of one gain-switched pulse. The waveform gradually evolves from impulse response (no feedback light has reached the laser yet during the 1$^{\textrm{st}}$ loop delay) to undamped relaxation oscillations (beginning from the 2$^{\textrm{nd}}$ loop) with oscillation frequencies at about 5 GHz, and then into chaos oscillations (beginning from the 3$^{\textrm{rd}}$ loop) with a much broader spread in the spectrum. Depending on the ratio between the PRI and the $\tau$, only a portion of the pulse is oscillating chaotically and can be utilized towards chaos lidar applications. The portion consisting of the impulse response and the relaxation oscillation, in general, has high similarity and correlation between the adjacent pulses and could easily cause ambiguity in ranging and be prone to interference from other lidars. (Please refer to Ref. [18] for detailed analyses of these gain-switched pulses generated under the condition of $\tau <$ PRI.)

 

Fig. 2. Waveforms of the chaos-modulated pulses generated by a gain-switched semiconductor laser subject to optical feedback with the operational parameters of (a)($\xi = 31\%$, $I_{\textrm{m}}$ = 20 mA, pulsewidth = 500 ns, PRI = 1 $\mu$s) and (b)($\xi = 8\%$, $I_{\textrm{m}}$ = 17 mA, pulsewidth = 80 ns, PRI = 112.4 ns), respectively. The red curves are the modulation waveforms of the gain-switching. The numbers above the waveforms denote the loops of delay starting from the onset of the pulses. (c) and (d) are the respective spectra of (a) and (b) from the wavelet analyses, where the colorbar denotes their magnitudes.

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To have the entire pulse oscillating chaotically and make it both energy- and time-efficient for the chaos lidar applications, in this study, we synchronize the PRI to match the $\tau$ so that the chaos oscillations generated in the previous pulse can be fed back and re-injected into the laser starting right at the onset of the next pulse. Figures 2(b) and 2(d) show the chaos-modulated optical pulses generated by the gain-switched semiconductor laser subject to delay-synchronized optical feedback when PRI = $\tau$ = 112.4 ns. As can be seen, by continuously re-injecting the chaos-modulated waveform from the previous pulse through optical feedback, the pulses oscillate chaotically throughout the entire pulse. In this case, contrary to the case when $\tau <$ PRI as shown in Figs. 2(a) and 2(c), no dwell time is needed for the pulse to evolve into chaos and the pulses generated are more efficient and suitable for the chaos lidar applications. Note that, while we only show two periods of the chaos-modulated pulses generated, subsequent pulses followed in the pulse train all have similar characteristics.

For PRI = $\tau$ = 112.4 ns where the pulse repetition interval matches the feedback time delay completely, Figs. 3(a)–3(c) and 3(d)–3(f) show the waveforms and their corresponding spectra of the pulses generated under ($I_{\textrm{m}}$ = 17 mA, pulsewidth = 80 ns) when $\xi$ increases from 2, 8, to 62 $\%$, respectively. At $\xi =2\%$, as shown in Fig. 3(a), the laser experiences very weak feedback that only undamped relaxation oscillations are observed. The spectrum in Fig. 3(d) shows the relaxation oscillation frequency of around 3 GHz. When we increase $\xi$ to $8\%$ (the same operation condition as in Figs. 2(b) and 2(d)), as shown in Figs. 3(b) and 3(e), the laser enters the chaos state that it oscillates chaotically in time and has a broadened spectrum spanning from about 1 to 7 GHz. As we further increase $\xi$ to $62\%$, as can be seen in Figs. 3(c) and 3(f), the laser still possesses some of the chaotic nature but is being intermittently overdamped by the strong feedback.

 

Fig. 3. Waveforms of the pulses generated under ($I_{\textrm{m}}$ = 17 mA, pulsewidth = 80 ns) when $\xi$ increases from (a) 2, (b) 8, to (c) 62 $\%$, respectively. (d)-(f) are the corresponding spectra of (a)-(c) from the wavelet analyses, where the colorbar denotes their magnitudes.

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4. Correlation properties

To evaluate the correlation properties of these pulses, in Figs. 4(a)–4(c) we plot the auto-correlation functions of the first pulses shown in Figs. 3(a)–3(c) and Figs. 4(d)–4(f) we plot the cross-correlation functions between the first pulses and the subsequent pulses followed in the pulse trains, respectively. We quantify these correlations with the values of the PSLs in the auto-correlation functions and the highest cross-correlation peaks in the cross-correlation functions, where the PSL is defined as the ratio of the maximum sidelobe to the peak. Ideally, a delta-like function with minimal sidelobes in the auto-correlation function is essential for chaos lidars to detect targets with good sensitivity. On the other hand, a low cross-correlation prevents possible range ambiguity from self-interference between pulses emitted at a different time or mutual interference between different chaos lidar units [19–21]. Note that, when calculating these correlations, we use a second-order Chebyshev high-pass digital filter with a cut-off frequency of 100 MHz to remove the square-wave modulation in the waveforms from the gain-switching. The span of the correlation window is always the same as the pulsewidth.

 

Fig. 4. (a)-(c) Auto-correlation functions of the first pulses shown in Figs. 3(a)–3(c and (d)-(f) cross-correlation functions between the first pulses and the subsequent pulses followed in the pulse trains, respectively. The numbers denote the loops of delay.

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In Figs. 4(a)–4(c), the PSLs of these auto-correlation functions are 0.741, 0.232, and 0.110, respectively. As can be seen, with weak feedback, the pulses in Fig. 3(a) have higher correlation sidelobes resulting from the periodicity of the undamped oscillations. With moderate feedback, as shown in Fig. 4(b), the sidelobes are lower indicating that the chaos oscillations in the pulses shown in Fig. 3(b) are non-periodic and have little specific frequencies. With strong feedback, as shown in Fig. 4(c), the sidelobes are also very low which is resulting from the intermittent damped oscillations shown in Fig. 3(c).

In Figs. 4(d)–4(f), the values of the highest cross-correlation peaks are 0.801, 0.237, and 0.713, respectively. As shown in Fig. 4(d), having apparent cross-correlation peaks spike at multiple loop delay time, the adjacent pulses in Fig. 3(a) are shown to possess a high similarity. By contrast, as shown in Fig. 4(e), the chaos-modulated pulses from Fig. 3(b) have relatively low cross-correlations indicating that they are more immune to the self-interference or interference from other sources. As for the pulses from the case of strong feedback shown in Fig. 3(c), although they have relatively low PSLs as shown in Fig. 4(c), the adjacent pulses unfavorably posses a high similarity as can be seen from the apparent cross-correlation peaks shown in Fig. 4(f). This correlation between subsequent pulses is mainly due to the reoccurring of the similar intermittent features linked by the delay-synchronized feedback loop. The similarity gradually becomes indistinct as the number of delay loops between two separated pulses becomes larger. From Fig. 4(f), for pulses separated with more than 3 delay loops, the cross-correlation peaks are as low as those obtained in the chaos-modulated pulses shown in Fig. 4(e).

For applications of chaos lidar, chaos-modulated pulses with both low PSLs and low cross-correlation peaks are desired. To have the general view of the correlation properties of the pulses generated based on the gain-switched laser with delay-synchronized optical feedback and to find the optimum operation conditions, Figs. 5(a)–5(c) and 5(d)–5(f) show the mappings of the PSLs of the auto-correlation functions and the highest cross-correlation peaks obtained at different $I_{\textrm{m}}$ and $\xi$ with pulsewidths of 30, 60, and 80 ns, respectively. As can be seen in Figs. 5(a)–5(c), pulses with lower PSLs (blueish regions) can be obtained with $\xi$ higher than about 8$\%$. When the pulsewidth increases from 30, 60, to 80 ns, the PSLs in this region are in general lowered due to the longer correlation windows associated with the longer pulsewidths [22]. For the region of $\xi$ lower than about 8$\%$, represented by the pulses as shown in Fig. 3(a), the feedback is too weak to drive the laser into chaos so that larger PSLs associated with the undamped relaxation oscillations are observed.

 

Fig. 5. Mappings of (a)-(c) the PSLs of the auto-correlation functions and (d)-(f) the highest cross-correlation peaks obtained at different modulation currents and $\xi$ with pulsewidths of 30, 60, and 80 ns, respectively.

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In the mappings of the highest cross-correlation peaks as shown in Figs. 5(d)–5(f), we find regions of high cross-correlation (yellowish and reddish regions) at weak ($\xi \lesssim {8\%}$) and strong ($\xi \gtrsim {14\%}$) feedback corresponding to the pulses having the characteristics of undamped relaxation oscillations and intermittent damped oscillations seen in Figs. 3(a) and 3(c), respectively. With moderate feedback, we find regions of low cross-correlation (blueish regions) around ${8\%}\lesssim {\xi } \lesssim {14\%}$ corresponding to the chaos-modulated pulses shown in Fig. 3(b). As the pulsewidth increases from 30, 60, to 80 ns, the highest cross-correlation peaks are gradually lowered again due to the longer correlation window associated with the longer pulsewidths. Note that, as the modulation current increases, these low cross-correlation regions become slightly broadened and tilt towards larger $\xi$.

5. Mismatch between the PRI and $\tau$

While we have successfully generated chaos-modulated pulses with both low PSLs and low cross-correlations by matching the PRI with the $\tau$ completely, we further investigate the dynamics and the correlation properties of the generated pulses when mismatching between the PRI and the $\tau$ occurs. Figures 6(a)–6(b) and 6(c)–6(d) show the waveforms and spectra of the pulses obtained with negative mismatch when $\Delta$t = PRI-$\tau$ = $-$15 ns (PRI = $\tau -15$ ns = 97.4 ns) and positive mismatch when $\Delta$t = PRI-$\tau$ = 15 ns (PRI = $\tau +15$ ns = 127.4 ns), respectively. Other parameters are the same as those previously used in Fig. 3(b) where $\tau =112.4$ ns, $\xi =8\%$, $I_{\textrm{m}}$ = 17 mA, and pulsewidth = 80 ns.

 

Fig. 6. Waveforms of the pulses obtained with negative mismatch when (a) PRI $=\tau$-15 ns = 97.4 ns and positive mismatch when (b) PRI $=\tau$+15 ns = 127.4 ns, respectively. Other parameters are the same as those previously used in Fig. 3(b) where $\tau =112.4$ ns, $\xi = 8\%$, $I_{\textrm{m}}$ = 17 mA, and pulsewidth = 80 ns. The gray, yellow, and green shaded regions are corresponding to the periods of stable, periodic, and chaos oscillations. (c) and (d) are the respective spectra of (a) and (b) from the wavelet analyses, where the colorbar denotes their magnitudes.

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In the case of a negative mismatch when PRI <$\tau$, as shown in Figs. 6(a) and 6(c), since the previous pulse has not yet reached the laser through the feedback, there is a period at the beginning of each pulse (the gray shaded region) that the laser only experiences the injection of the background noise. Other than the onset transient of the pulse rising from the gain-switched modulation, the laser is quite stable in this 15 ns period that no distinct frequency component is seen in the spectrum. In the next 15 ns period (the yellow shaded region), the laser starts to be affected by the above-mentioned stable region fed back from the previous pulse where periodic oscillations with frequencies around 4 GHz are induced. In the next 50 ns period immediately followed (green shaded region), the laser is affected by the above-mentioned periodic oscillations fed back from the previous pulse where chaos oscillations with broadened spectra are induced. The chaos oscillations last till the end of the pulse when it is abruptly switched off by the gain-switched modulation. As can be seen, resulting from the negative mismatch between the PRI and the $\tau$, for the entire 80 ns pulsewidth there is a 15 ns of the stable region, a 15 ns of the periodic oscillations, and left with 50 ns of the chaos oscillations that can be utilized in the chaos lidar applications.

The case of a positive mismatch when PRI >$\tau$, as shown in Figs. 6(b) and 6(d), is similar to the case of a negative mismatch but with the dynamical sequence in an opposite manner. In the last 15 ns period of the pulse (the gray shaded region), the laser experiences the feedback of the background noise and remains stable without any apparent oscillations. Being injected by the above-mentioned stable region, periodic oscillations with frequencies around 5 GHz are induced in the 15 ns period (the yellow shaded region) before the stable region. Injected by the chaos and periodic oscillations through the feedback from the previous pulse, chaos oscillations with broadened spectra are induced forming the leading part of the pulse. Note that, although being affected by the chaos oscillations from the previous pulse, the amplitudes of the chaos oscillations at the rising edge are relatively small when the laser is just being switched on. In this case, the period of chaos oscillations with notable amplitudes is about 40 ns, which is shorter than the case of negative mismatching as shown in Fig. 6(a).

To quantify the efficiency of chaos generations under different mismatches between the PRI and the $\tau$, Fig. 7(a) shows the ratio of the chaos oscillation period over the entire pulse. As can be seen, when PRI = $\tau$, the ratio of chaos is 0.92 where almost the entire pulse is consists of chaos oscillations. As the mismatch increases both negatively and positively, the ratio decreases accordingly. The ratio drops noticeably lower on the positive mismatch side ($\Delta$t > 0) due to the exclusion of the period with low oscillation amplitude at the rising edge of each pulse as discussed in Fig. 6(b). Figures 7(b) and 7(c) show the PSLs and the cross-correlation peaks of the pulses generated under different mismatches, respectively. As can be seen, the pulses have the best correlation properties when there is no mismatch that both the PSL and the cross-correlation peak are at the lowest. As the mismatch increases, both the PSL and the cross-correlation peak increase in which the positive mismatch side rises higher than the negative side due to the inefficient chaos generation as discussed in Fig. 7(a). In sum, synchronizing the gain-switching modulation with the delayed-feedback is essential in generating the chaos-modulated pulses with the best performance for the chaos lidar applications. Note that, the cases when $\tau$ matches the multiples of PRI have a similar result as the case of PRI = $\tau$ shown here.

 

Fig. 7. (a) Ratios of the chaos oscillation period over the entire pulse, (b) PSLs, and (c) cross-correlation peaks of the pulses generated under different mismatches $\Delta$t.

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6. Conclusion

We successfully generate chaos-modulated pulses based on a gain-switched semiconductor laser subject to delay-synchronized optical feedback. By properly adjusting the feedback strengths and the modulation currents of the gain-switching, pulses with both low PSLs and cross-correlation peaks suitable for the pulsed chaos lidar applications are obtained. To have the highest ratio of chaos oscillation period within a pulse and the lowest PSL and cross-correlation peak that is desired for the chaos lidar applications, we find matching the PRI and the $\tau$ is essential. Under mismatching, we identify the sequences of dynamical periods including stable, periodic, and chaos oscillations evolving within the pulse. Compared to the time-gating configuration using a lossy acousto-optic modulator to time-gate CW chaos into chaos pulses [17], the gain-switching configuration studied and presented in this paper not only eliminates the need of the modulator but also improves the energy utilization efficiency. While we only show the results with the PRI of 112.4 ns and pulsewidth of 80 ns, chaos-modulated pulses with various parameters can be generated to suit the specs and needs of a chaos lidar system.