1. Introduction

Lidars have been widely adopted in applications such as remote sensing, intelligent machinery, 3D imaging, and simultaneous localization and mapping benefited by their advantages of long detection range, high accuracy and precision, high spatial resolution, and less constraints to weather and illumination conditions [1–4]. Conventional pulsed lidars emit repetitive short pulses to measure the time-of-flight (TOF) of the light backscattered from a target [5, 6]. Benefited by their higher peak powers, pulsed lidars are capable of detecting targets at very long ranges. However, with the unspecific waveforms and repetitive pulses emitted, possible interference and range ambiguity make them less practical in modern technology-crowded environment and daily life applications [7–9].

To solve these issues, in 2004, the authors proposed and demonstrated a continuous-wave (cw) chaos lidar based on nonlinear dynamics of semiconductor lasers for the first time [10]. By detecting targets with chaos waveforms that are aperiodic and random in nature, cw chaos lidars have no ambiguity in range and are immune to both interference and jamming [10–14]. Nevertheless, the chaos waveforms used in previous studies were generated by cw semiconductor lasers subject to external perturbations such as optical injection and/or optical feedback [15–20]. With the low average power permitted by the class-1 eye-safe regulation [21], the chaos lidars in their cw forms are inefficient for long-range low-reflectivity target detection [8,22,23]. Recently, the authors demonstrated a 3D pulsed chaos lidar system employing self-homodyning, time-gating, and a master oscillator power amplifier (MOPA) configuration that significantly enhances the energy-utilization efficiency of the chaos lidar [24]. Despite its great performance, the interplay and dynamics between the time-gating and the amplification have to be carefully controlled.

Therefore, to improve the ranging ability of the chaos lidars, pulsed chaos lidars directly generating and emitting chaos-modulated pulses with higher peak powers are of great interest. To generate chaos-modulated pulses with not only higher peak powers but also the aperiodic waveforms desired, in this study, we incorporate an optical feedback scheme in a gain-switched semiconductor laser. In contrast with the ultrashort pulses generated by mode-locking that are typically too short in time, the gain-switched pulses can easily have their pulsewidths adjusted to be comparable or longer than the feedback delay time so that the laser can have enough evolutions with feedback before each pulse is terminated. Characteristics of pulses generated by gain-switched semiconductor lasers subject to external optical perturbations were studied for high-speed optical communications systems [25,26]. Frequency dynamics of injection-locked gain-switched semiconductor lasers were investigated where suppressions of timing jitter and power overshoot were discussed [27]. Impacts of optical injection on the characteristics of gain-switched pulses, including the pulsewidth, frequency chirp, and temporal pedestal suppression, were reported [28]. Moreover, photonic generation of ultrawideband signals based on a gain-switched semiconductor laser with optical feedback was also demonstrated [29]. Nevertheless, generations and analyses of chaos-modulated pulses for pulsed chaos lidar applications have not yet been reported.

In this study, we investigate the characteristics of optical pulses experimentally generated by a gain-switched semiconductor laser subject to optical feedback for pulsed chaos lidar applications. When gain-switching the laser, the transient response induced at the onset of each pulse and the dynamics it generates thereafter are fed back to the laser and recirculated in the delay loop. Evolutionarily interacting between the sustained relaxation oscillations and the dynamics from the delayed feedback, more complex dynamics can progressively be evolved. Unlike its cw counterpart, with a limited duration of each pulse, evolutions can only happen within a time window of a few delay loops before the pulse is abruptly terminated. We therefore emphasize on the temporal and spectral evolutions from the transient to the long-term responses for the complex dynamics of the modulated pulses generated at different feedback and gain-switching conditions. Moreover, by analyzing their cross-correlation properties, we also investigate the feasibility of utilizing the chaos-modulated pulses for pulsed chaos lidar applications.

2. Experimental setup

Figure 1(a) shows the experimental setup of a gain-switched semiconductor laser subject to optical feedback for generating chaos-modulated pulses. We use a signal generator (SG) (Agilent 81150A) to modulate a 1.55 µm single-mode DFB semiconductor laser (LD) (Gooch & Housego AA1408) with a square-wave current to generate gain-switched pulses. The threshold current I_th and the slope efficiency of the LD are 42.7 mA and 0.32 mW/mA in free-running, respectively. By setting the respective modulation repetition rate and duty cycle at 1 MHz and 50%, the laser produces optical pulses with 500 ns duration at different modulation currents I_m. We use a fiber coupler (FC) to divide the laser output into two arms. In one arm, the output light is fed back to the laser by a fiber mirror (FM). We control the feedback strength ξ, defined as the ratio between the emitted field and the fed back field, with a variable attenuator (VA) and monitor it with a power meter (PM). All fibers used are polarization maintaining fibers to ensure the laser is fed back coherently. From the optical path length of the fiber components used, a feedback loop delay of 69.5 ns is measured. In the other arm, we use a photodetector (PD) (Newport 1544-B, 12 GHz bandwidth) to detect and an oscilloscope (OSC) (Tektronix TDS 6604, 6 GHz bandwidth, 20 GS/s) to digitize and record the time series. An optical isolator (ISO) is placed before the PD to prevent any unwanted feedback.

 

Fig. 1 (a) Experimental setup of a gain-switched semiconductor laser subject to optical feedback. (b) The square-wave modulation (red) and the time series of the corresponding modulated pulses generated (blue) with ξ = 0.17 and I_m = 138.5 mA. SG: signal generator; LD: laser diode; PM: power meter; FC: fiber coupler; VA: variable attenuator; FM: fiber mirror; ISO: optical isolator; PD: photodetector; OSC: oscilloscope.

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Figure 1(b) shows the square-wave modulation (red) and the time series of the corresponding modulated pulses (blue) obtained with ξ = 0.17 and I_m = 138.5 mA, where the I_th is lowered to 40.6 mA by the feedback. As can be seen, by incorporating optical feedback to a gain-switched semiconductor laser, chaos-modulated optical pulses are generated. While each modulated pulse in the pulse train is generated under the same operational conditions, governing by the nonlinearity of the system and varying by the noise at the onset of each pulse [30], the modulated waveform on each pulse can be all different and independent.

3. Temporal characteristics of modulated waveforms

By varying the feedback strength ξ and modulation current I_m, we generate modulated pulses with different characteristics. To emphasize on the modulated waveforms of the pulses, we filter the time series of the chaos-modulated pulses with a 10 MHz high-pass digital filter to eliminate the plain pulse components. Figures 2(a)–2(c) show the waveforms of the first pulses in the pulse trains obtained at (ξ, I_m) = (0.87, 84.6 mA), (0.35, 71.6 mA), and (0.17, 120.4 mA), where the corresponding I_th under each feedback strength are 32.0, 38.4, and 40.6 mA, respectively. Here the time is shifted to coincide with the onset of the pulses.

 

Fig. 2 Modulated waveforms filtered from the time series of the modulated pulses obtained at (ξ, I_m) = (a) (0.87, 84.6 mA), (b) (0.35, 71.6 mA), and (c) (0.17, 120.4 mA), respectively. We apply a 10 MHz high-pass digital filter to eliminate the plain components of the pulses. The dashed box in (a) depicts how the wait time and duration of an extracted waveform are defined in the piecewise cross-correlation calculation. The consecutive loop delays with a duration of 69.5 ns each are illustrated for reference. (d) Main oscillation frequencies extracted from the modulated waveforms obtained at different I_m with various ξ. The I_th are 40.6, 38.4, 36.3, 34.1, and 32.0 mA for ξ = 0.17, 0.35, 0.52, 0.69, and 0.87, respectively.

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As can be seen in Fig. 2(a), having a separation in coincidence with the loop delay of 69.5 ns, a series of intensity overshoots initially induced by the transient response occur in the early stage of the pulse. Under this strong feedback condition (feedback regime V) [31], following the overshoots the laser quickly damps to its stable state. While some oscillations are observed in the later stage of the pulse attributed to the perturbation from the delayed feedback, their amplitudes are relatively low under this strongly damped condition. In contrast, by lowering ξ to 0.35 as shown in Fig. 2(b), the overshoots are followed by periodic oscillations in this undamped condition. Further lowering ξ to 0.17 and increasing I_m to 120.4 mA, as shown in Fig. 2(c), the waveform gradually evolves from weakly damped relaxation oscillations into chaos oscillations in a few delay loops. Note that, lowering the feedback strength to a moderate feedback at regime IV together with a stronger modulation, the laser is easier to get excited into instabilities such as chaos as shown in Fig. 2(c) [32].

In Fig. 2(d) we show the main oscillation frequencies extracted from the modulated waveforms obtained at different I_m for various ξ. As can be seen, the main oscillation frequencies of the modulated waveforms are mostly governed by the relaxation oscillation frequency of the laser, which increases with both the feedback strength and the modulation current [18, 32, 33]. To better examine the evolutions of their spectral characteristics throughout different stages of the pulses, we apply the continuous wavelet transform on the modulated waveforms to show their time-frequency analyses.

4. Spectral characteristics and correlation properties of modulated waveforms

By applying the continuous wavelet transform on the modulated waveforms shown in Figs. 2(a)–2(c), we show their time-frequency analyses in Figs. 3(a)–3(c), respectively. The color bars denote the RF amplitudes of their spectra. As shown in Fig. 3(a), we observe mainly the relaxation oscillation frequency and its harmonics induced by the transient responses at the integer multiples of the loop delays. Although we see some frequency components in the later stage of the pulse, their amplitudes are relatively low due to the strong damping under this strong feedback condition (feedback regime V [31]).

 

Fig. 3 (a)–(c) Time-frequency analyses of the respective waveforms shown in Figs. 2(a)–2(c) calculated by the continuous wavelet transformation. The frequency resolution is 10 MHz and the time increment is 0.05 ns. The color bars denote the RF amplitudes of their spectra. (d)–(f) Time-correlation analyses of the respective waveforms shown in Figs. 2(a)–2(c) with different wait times and durations defined in Fig. 2(a). The color bars denote the cross-correlation peaks between the extracted waveforms and the entire waveforms of the next adjacent pulses in the pulse trains. The increments of the time interval in both axes are 2 ns. White dashed grids with spacings of 70 ns (the nearest integer rounded up from the 69.5 ns loop delay) are displayed to aid for clarity.

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In Fig. 3(b), we not only observe the relaxation oscillation frequency (3.3 GHz) and its harmonics at the integer multiples of the loop delay, but also find frequencies of periodic oscillations apparently alternating between 5.25 and 7 GHz (3^rd and 4^th harmonics of a fundamental oscillation frequency at 1.75 GHz) throughout the entire stage of the pulse. As time evolves, the laser gradually transits from periodic states with a single frequency into quasi-periodic states involving two or more frequencies. At one of the weakly damped states, as shown in Fig. 3(c), we find a strong frequency component at the relaxation oscillation frequency (4 GHz) which lasts for a little more than two loop delays. After more evolutions, we observe that the modulated waveform progressively evolves into chaos where continuous spectra with very broad bandwidths are found. The complex dynamics in the later stage of the pulse are attributed to the strong interaction between the sustained relaxation oscillation and the dynamics of the delayed feedback in the laser [18, 34]. As the results, with different feedback and modulation parameters, we generate modulated pulses with different characteristics including stable, periodic, quasi-periodic, and chaos modulations.

Although each modulated pulse in a pulse train is generated under the same operational conditions, varying by the noise at the onset of each pulse, the modulated waveform on each pulse can be all different and independent. For these modulated pulses to be useful in pulsed chaos lidar applications, the modulated waveform on each pulse has to be uncorrelated to one another to achieve detections without ambiguity and be immune from interference and jamming. Therefore, we examine the cross-correlations between the modulated waveforms of each pulse in the pulse trains obtained with the same conditions as those in Figs. 2(a)–2(c) and Figs. 3(a)–3(c) and show their time-correlation analyses in Figs. 3(d)–3(f), respectively. Since a pulsed chaos lidar is not necessary to use the entire waveform of a modulated pulse but may make a time-gated detection with just a portion of it, we extract waveforms with different wait times and durations from different stages of the pulses (depicted with the dashed box in Fig. 2(a) for example) and calculate their piecewise cross-correlations in respect with the entire waveform of the next adjacent pulse in the pulse train. We plot the cross-correlation peaks (in color) obtained with different wait times and durations in a time interval increment of 2 ns. White dashed grids with spacings of 70 ns (the nearest integer rounded up from the 69.5 ns loop delay) are displayed to aid for clarity. Note that, for pulsed chaos lidars to detect unambiguously and have higher energy-utilization efficiency of the pulses, waveforms having lower cross-correlations for longer durations are desired.

As can be seen in Fig. 3(d), we find higher cross-correlations (reddish region) in the early stage of the pulse when the extracted waveforms include more of the relaxation oscillations following the transient responses. Coincided with the overshoots shown in Fig. 2(a), the cross-correlation peaks have a period of the local maxima to be the same as the loop delay. Note that, since the laser quickly damps to its stable state after each overshoot, the low correlations as seen in those small blue triangle regions for duration under 70 ns are merely resulting from the uncorrelated intensity noise of the stable pulses. In the later stage of the pulse, waveforms with longer wait times have lower cross-correlations (bluish region) and bear less similarity to the waveforms from the other pulses.

In almost the entire plot of Fig. 3(e), we find high cross-correlations attributed to the periodic and quasi-periodic oscillations generated from the undamped relaxation oscillations. Having such high similarity between each modulated pulse, states like this are not suitable for the pulsed chaos lidar applications. In contrast, for the weakly damped state as shown in Fig. 3(f), we find a relatively large low cross-correlation region (bluish region) spans from medium to long wait times extending into longer durations. As can be seen in the time series shown in Fig. 2(c) and the spectra shown in Fig. 3(c), the low cross-correlations are attributed to the aperiodic intensity oscillations and the broadened spectra of the chaos-modulated waveforms, respectively. To have cross-correlations lower than the benchmark of 0.19 set by the cross-correlation of the intensity fluctuation on the sole gain-switched pulses without optical feedback, we can extract waveforms with durations up to 218 ns under the current configuration.

Note that although we only show the cross-correlations between the adjacent pulses, similar results are found between any pulses in the pulse train. With the temporal, spectral, and correlation analyses, we show that while all pulses in a pulse train are generated under the same operational conditions including the ξ and I_m, the modulated waveform on each pulse can evolve differently and uncorrelatively governed by the noise at the onset of each pulse and the nonlinearity of the system.

5. Dynamical behaviors of the modulated pulses

In Fig. 4, we examine the cross-correlation peaks with different ξ and I_m to explore the correlation properties of the modulated waveforms in a broader parameter space. To show how the correlation properties evolve with time, as depicted in Fig. 2(a), waveforms from both an earlier stage, i.e. the 2^nd loop, and a later stage, i.e. the 7^th loop, of the pulses are extracted for evaluation. Moreover, in a shorter time scale of each stage, both the transient response covering the intensity overshoot induced at the beginning of each loop delay and the next 10 ns following it, and the long-term response covering another 50 ns of waveform following that of the transient response, are extracted and compared. In Figs. 4(a)–4(d) we show the cross-correlation peaks (in color) of the transient and long-term responses of the 2^nd and 7^th loops, respectively. We mark A, B, and C in the parameter space to show the respective conditions used in Figs. 2(a)–2(c) for reference.

 

Fig. 4 Cross-correlation peaks with different ξ and I_m for the (a) transient and (b) long-term responses of the 2^nd delay loop and the (c) transient and (d) long-term responses of the 7^th delay loop, respectively. The color bars denote the peak values in the cross-correlation traces. The increments of the ξ and I_m are 0.17 and 1.6 mA, respectively. A, B, and C marked in the parameter space show the respective conditions used in Figs. 2(a)–2(c) for reference.

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For the waveforms extracted from the transient responses in the early stage of the pulses, as shown in Fig. 4(a), we observe high cross-correlations in almost the entire region of the parameter space attributed to the similar intensity overshoots and the overdamped relaxation oscillations followed. Relaxing into the long-term responses, as shown in Fig. 4(b), we find low correlations at the strongly damped region around case A where the laser has already damped to its stable state. In contrast, the correlations remain high around the regions of case B and case C revealing the periodic natures of the undamped and the weakly damped relaxation oscillation states.

For the waveforms extracted from the transient responses after progressing into the later stage of the pulses, as shown in Fig. 4(c), we see that the regions around case A and case B still possess high correlations dominated by the similar intensity overshoots and the relaxation oscillations followed. In contrast, for the weakly damped region around case C, the sustained oscillations become more complex and bear less similarity from one another after evolving through several loops of the delayed feedback. As the results, at the later stage of the pulses, we observe low correlations for the extracted waveforms even when they are still within the range of the transient response at the beginning of the integer multiples of the delay loop.

For the waveforms extracted from the long-term responses in the later stage of the pulses, as shown in Fig. 4(d), we find the waveforms from the region around case B still have relatively high correlations while the waveforms from the regions around case A and case C possess relatively low correlations comparable to the benchmark of 0.19 set by the cross-correlation of the intensity fluctuation on the sole gain-switched pulses without feedback. Compared with the waveforms from the region around case A, waveforms from the region around case C are more suitable for the pulsed chaos lidar applications owing to their higher amplitudes and longer windows of durations bearing low cross-correlations. After all, the laser is easier to get excited into instabilities such as chaos with a stronger modulation and a moderate feedback as those in the region around case C (feedback regime IV [31]).

6. Conclusion

In this study, we experimentally generate and analyze chaos-modulated pulses for pulsed chaos lidar applications based on gain-switched semiconductor lasers subject to optical feedback. We investigate the temporal, spectral, and cross-correlation characteristics of the modulated pulses obtained with different feedback strengths and modulation currents. Induced by the transient response and evolving with the delayed feedback, pulses exhibiting periodic oscillations and complex dynamics such as chaos are observed. Under a weakly damped condition with large modulation current and moderate feedback strength, we successfully generate the uncorrelated chaos-modulated pulses suitable for the pulsed chaos lidar applications. With the current configuration, for cross-correlations comparable to the benchmark of 0.19 set by the cross-correlation of the intensity fluctuation on the sole gain-switched pulses without feedback, uncorrelated waveforms with durations up to 218 ns in a 500 ns modulated pulse can be utilized.

While the 69.5 ns loop delay is determined by the optical path length of the fiber components used in our experiment, it can be significantly shortened by using an integrated semiconductor laser with optical feedback [35] to expedite the dynamical evolution. It is more complicated, however, in choosing an optimal duration of the pulse for the pulsed chaos lidar applications since that, on one hand it needs to be long enough for the laser to have enough dynamical evolutions to enter into chaos and last an extended period of time, and on the other hand it has to be short enough so that higher peak power (thus higher signal-to-noise ratio) can be emitted for the same average output power permitted by the eye-safe regulation. As part of our future work, we plan to investigate and compare the chaos-modulated pulses generated with different loop delays, duty cycles, and pulse durations and optimize both their correlation properties and peak powers at the same time.