Noise suppressions in synchronized chaos lidars

Wen-Ting Wu; Yi-Huan Liao; Fan-Yi Lin

doi:10.1364/OE.18.026155

1. Introduction

Chaos lidar (CLIDAR) utilizing laser chaos was proposed and studied [1] due to its advantages of high resolution and long unambiguous range in ranging. The unique characteristics of CLIDAR were mostly benefited by the broadband chaotic waveforms used, which can be generated with nonlinear laser dynamics in an optical feedback (OF) [2] [3], an optoelectronic feedback (OEF) [4] [5] [6], or an optical injection (OI) [7] [8] [9] scheme. Similar to other conventional modulated continuous-wave lidars [10] [11] [12], target detections in the CLIDAR system are realized by correlating the signal waveform backscattered from the target with a delayed replica from the transmitter laser. Although the proof-of-concept experiments have been demonstrated previously [1] where the range resolutions and peak-to-sidelobe levels have been investigated, the effect of channel noise on the detection performance has not yet been discussed.

To understand the behavior of the CLIDAR system in a real environment with atmosphere disturbance, perturbations of additive white Gaussian noise and random phase noise on the amplitude and phase, respectively, of the signal waveforms are considered in this study. To mitigate the degradation in detection due to the undesired noise and to take a step further to fully exploit the advantages of laser chaos, a modified synchronized-CLIDAR (S-CLIDAR) system using a receiver laser to synchronize with the transmitter laser are proposed and studied. Instead of detecting the noise-contaminated signal with a photodetector directly, the signal waveform is coupled into a receiver laser and drives it into synchronization [13] where the receiver laser can reproduce the original chaotic waveform from the transmitter laser without being distorted by the channel noise. In this paper, the detection performances of the CLIDAR and the S-CLIDAR systems under the OEF and the OF schemes are compared and investigated numerically. With the CLIDAR system as the benchmark, noise suppressions of the S-CLIDAR systems under different levels of noise are studied. The optimized conditions to have the highest possible correlation coefficients are also discussed and given.

2. Model

Figures 1(a)–(d) show the schematic setups of the original CLIDAR and the newly proposed S-CLIDAR systems with the OEF and the OF schemes. The yellow and blue dots denote the transmitter and receiver outputs. For the CLIDAR systems as shown in Figs. 1(a) and (b), the range delays of the targets are determined directly from the correlations of the receiver outputs (the backscattered signals detected by the photodetectors) (blue dots) and the transmitter outputs (the reference signals from the transmitter lasers (Tx) (yellow dots). On the other hand, for the S-CLIDAR systems as shown in Figs. 1(c) and (d), the backscattered signals are instead coupled into the receiver lasers (Rx), electrically for the OEF and optically for the OI schemes, to drive the Rx to synchronize with the Tx. The receiver outputs (the synchronized waveforms) (blue dots) are then used to correlate with the transmitter output (yellow dots). Note that, the Rx has its own feedback loop governed by the feedback strength η_R and the delay time τ_R, where η_R = 0 represents cases when the Rx is in an open-loop configuration and η_R ≠ 0 represents the cases when the Rx is in a close-loop configuration.

Fig. 1 (Color online) Schematic setups of the CLIDAR and the S-CLIDAR systems with the OEF and the OF schemes. Tx and Rx are the transmitter and the receiver lasers. η_T and η_R are the feedback strengths of the Tx and Rx and η_C is the coupling strength from the channel to the Rx, respectively. τ_T and τ_R are the feedback delay times of the Tx and Rx and τ_C is the target range delay in the channel, respectively.

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The nonlinear dynamics of the Tx and Rx for the CLIDAR and the S-CLIDAR systems can be modelled by the following coupled rate equations:

\begin{array}{l} \frac{{da}_{T}}{dt} = \frac{1}{2} [\frac{γ_{c} γ_{n}}{γ_{s} {\tilde{J}}_{T}} {\tilde{n}}_{T} - γ_{p} (2 a_{T} + a_{T}^{2})] (1 + a_{T}) \\ + η_{OF, T} (1 + a_{T} (t - τ_{T})) cos (ϕ_{T} (t - τ_{T}) - ϕ_{T} (t) + ω_{T} τ_{T}) \end{array}

\begin{array}{l} \frac{d ϕ_{T}}{d t} = - \frac{b}{2} [\frac{γ_{c} γ_{n}}{γ_{s} {\tilde{J}}_{T}} {\tilde{n}}_{T} - γ_{p} (2 a_{T} + a_{T}^{2})] \\ + η_{OF, T} \frac{(1 + a_{T} (t - τ_{T}))}{1 + a_{T}} sin (ϕ_{T} (t - τ_{T}) - ϕ_{T} (t) + ω_{T} τ_{T}) \end{array}

\begin{array}{l} \frac{d {\tilde{n}}_{T}}{dt} = - γ_{s} {\tilde{n}}_{T} - γ_{n} {(1 + a_{T})}^{2} {\tilde{n}}_{T} - γ_{s} {\tilde{J}}_{T} (2 a_{T} + a_{T}^{2}) + \frac{γ_{s} γ_{p}}{γ_{c}} {\tilde{J}}_{T} (2 a_{T} + a_{T}^{2} {(1 + a_{T})}^{2} \\ + η_{OEF, T} γ_{s} ({\tilde{J}}_{T} + 1) (1 + 2 a_{T} (t - τ_{T})) + a_{T} {(t - τ_{T})}^{2} \end{array}

\begin{array}{l} \frac{d a_{R}}{dt} = \frac{1}{2} [\frac{γ_{c} γ_{n}}{γ_{s} {\tilde{J}}_{R}} {\tilde{n}}_{R} - γ_{p} (2 a_{R} + a_{R}^{2})] (1 + a_{R}) \\ + η_{OF, R} (1 + a_{R} (t - τ_{R})) cos (ϕ_{R} (t - τ_{R}) - ϕ_{R} (t) + ω_{R} τ_{R}) \\ + η_{OF, C} (1 + a_{C} (t - τ_{C})) cos (ϕ_{C} (t - τ_{C}) - ϕ_{R} (t) + ω_{T} τ_{C} - Δ ω t) \end{array}

\begin{array}{l} \frac{d ϕ_{R}}{dt} = - \frac{b}{2} [\frac{γ_{c} γ_{n}}{γ_{s} {\tilde{J}}_{R}} {\tilde{n}}_{R} - γ_{p} (2 a_{R} + a_{R}^{2})] \\ + η_{OF, R} \frac{(1 + a_{R} (t - τ_{R}))}{1 + a_{R}} sin (ϕ_{R} (t - τ_{R}) - ϕ_{R} (t) + ω_{R} τ_{R}) \\ + η_{OF, C} \frac{(1 + a_{C} (t - τ_{C}))}{1 + a_{R}} sin (ϕ_{C} (t - τ_{C}) - ϕ_{R} (t) + ω_{T} τ_{C} - Δ ω t) \end{array}

\begin{array}{l} \frac{d {\tilde{n}}_{R}}{dt} = - γ_{s} {\tilde{n}}_{R} - γ_{n} {(1 + a_{R})}^{2} {\tilde{n}}_{R} - γ_{s} {\tilde{J}}_{R} (2 a_{R} + a_{R}^{2}) + \frac{γ_{s} γ_{p}}{γ_{c}} {\tilde{J}}_{R} (2 a_{R} + a_{R}^{2} {(1 + a_{R})}^{2} \\ + η_{OEF, R} γ_{s} ({\tilde{J}}_{R} + 1) (1 + 2 a_{R} (t - τ_{R})) + a_{R} {(t - τ_{R})}^{2} \\ + η_{OEF, C} γ_{s} ({\tilde{J}}_{R} + 1) (1 + 2 a_{C} (t - τ_{C})) + a_{C} {(t - τ_{C})}^{2}, \end{array}

where a is the normalized optical field, ϕ is the optical phase, ñ is the normalized carrier density, J̃ is the normalized dimensionless injection current parameter, γ_c is the cavity decay rate, γ_n is the differential carrier relaxation rate, γ_p is the nonlinear carrier relaxation rate, γ_s is the spontaneous carrier relaxation rate, b is the linewidth enhancement factor, η is the coupling rate, τ is the delay time, and Δω is the angular frequency detuning between the Tx and Rx. The subscripts T, R, and C denote the Tx, Rx, and channel, respectively. The laser parameters used here are b = 4, γ_n = 0.667 × 10⁹s⁻¹, γ_p = 1.2 × 10⁹s⁻¹, γ_s = 1.458 × 10⁹s⁻¹, γ_c = 2.4 × 10¹¹s⁻¹, Δω = 0, and J̃ = 0.333 [14].

To simulate the disturbance in the transmission channel and study the degradation in detection affected by the noise, amplitude and phase noises are added to the backscattered signals before receiving by the photodetector in the CLIDAR and by the Rx in the S-CLIDAR systems, respectively. Modelled by an additive white Gaussian noise RN1 with zero mean and variance of $σ_{RN1}^{2}$ and a random phase noise RN2 evenly distributed in the range between [−π, π], the amplitude and phase noise are added to the respective amplitude a_C(t) and the phase ϕ_C(t) of the received signals,

a_{C} (t) = a_{T} (t) + RN 1 (t)

ϕ_{C} (t) = ϕ_{T} (t) + mRN 2 (t), 0 \leq m \leq 1

The relative strength of the amplitude noise is defined by the signal-to-noise ratio (SNR)

SNR = 10 log \frac{P_{s}}{σ_{RN1}^{2}},

where P_s is the average power of the transmitted signal. The influence of the phase noise is controlled by m, where m = 0 is the case when no phase noise is considered.

To quantify the performance of target detection in different schemes, the correlation coefficients under different noise levels are calculated as

ρ (Δ τ) = \frac{〈 [S_{T} (t) - 〈 S_{T} (t) 〉] [S_{R} (t + Δ τ) - 〈 S_{R} (t) 〉] 〉}{{〈 {| S_{T} (t) - 〈 S_{T} (t) 〉 |}^{2} 〉}^{\frac{1}{2}} {〈 {| S_{R} (t) - 〈 S_{R} (t) 〉 |}^{2} 〉}^{\frac{1}{2}}},

where S_T(t) and S_R(t) are the intensity outputs of the transmitter (yellow dot) and the receiver (blue dot), 〈·〉 denotes the time average, and Δτ is the relative time difference between the transmitter and the receiver, respectively. The correlation coefficient is bounded with −1 ≤ ρ ≤ 1, where a larger value of |ρ| indicates better quality of detection.

3. Results and discussions

Figures 2(a) and (b) show the time series and autocorrelation of the transmitted waveform from the Tx of the OEF scheme with a delay time τ_T = 9.5 ns and a feedback strength η_OEF,T = 0.123, respectively. For the S-CLIDAR system with the OEF scheme, since the backscattered light is converted to the electric signal before coupling to the Rx, little effect of the phase noise from the channel is found. Under the influence of the amplitude noise with SNR = 0 dB, the detected waveforms in the receivers and their corresponding correlations to the transmitted waveform for the CLIDAR and the S-CLIDAR systems are shown in Figs. 2(c)–(f), respectively. As can be seen, compared with the received waveform in the CLIDAR system that is noisy and severely distorted from the transmitted waveform, the waveform reproduced by the Rx through synchronization in the S-CLIDAR system has little distortion. Correlation coefficients of 0.73 and 0.97 are obtained for the CLIDAR and the S-CLIDAR systems at a delay of 15.5 ns, which is the range delay of the target in the transmission channel τ_C = 15.5 ns under the generalized synchronized condition. Note that, to have the best synchronization for the highest possible correlation coefficient, the coupling strength η_OEF,C and the feedback strength η_OEF,R of the S-CLIDAR system have to be optimized with different levels of noise presented in the channel. Through simulations, for SNR = 0 dB, optimized coupling strength and feedback strength of η_OEF,C = 1.3 and η_OEF,R = 0 are found showing that the S-CLIDAR system has a better synchronization performance under a generalized synchronization condition with an open-loop configuration. Detailed investigations on the optimized coupling strengths under different SNRs for the S-CLIDAR systems with the OEF and the OE schemes will be discussed and given in Fig. 6.

Fig. 2 (a) Time series and (b) autocorrelation of the transmitted waveform from the Tx of the OEF scheme with a delay time τ_T = 9.5 and a feedback strength η_OEF,T = 0.123. (c)–(f) The detected waveforms in the receivers and their corresponding correlations to the transmitted waveform for the CLIDAR and the S-CLIDAR systems, respectively.

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Fig. 6 (Color online) The optimized coupling strengths of η_OEF,C and η_OF,C for the S-CLIDAR system with (a) the OEF and (b) the OF schemes under different levels of noise

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Figures 3(a) and (b) show the time series and autocorrelation of the transmitted waveform from the Tx of the OF scheme with a delay time τ_T = 9.5 ns and a feedback strength η_OF,T = 0.2, respectively. Unlike in the OEF scheme, the phase noise is found to affect the synchronization significantly in the Rx for the S-CLIDAR system with the OF scheme. With SNR = 0 dB, Figs. 3(c)–(j) show the time series of the detected waveforms and their corresponding correlations to the transmitted waveform for the CLIDAR and the S-CLIDAR systems with the phase noise levels of m = 0, 0.5, and 0.75, respectively. As can be seen, although not being affected by the phase noise, the detected waveform of the CLIDAR system as shown in Fig. 3(c) is distorted severely from the transmitted waveform solely because of the influence of the amplitude noise where a correlation coefficient of only 0.36 is found. On the contrary, the S-CLIDAR system shows good ability in filtering both the amplitude and the phase noises, where correlation coefficients of 0.88, 0.82, and 0.53 are achieved for phase noise levels of m = 0, 0.5, and 0.75, respectively.

Fig. 3 (a) Time series and (b) autocorrelation of the transmitted waveform from the Tx of the OF scheme with a delay time τ_T = 9.5 ns and a feedback strength η_OEF,T = 0.2. (c)–(j) The detected waveforms in the receivers and their corresponding correlations to the transmitted waveform for the CLIDAR and the S-CLIDAR systems with the phase noise levels of m = 0, 0.5, and 0.75, respectively.

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Figure 4 shows the correlation coefficients of the CLIDAR and the S-CLIDAR systems with the OEF and the OF schemes for different levels of noise. As shown in Fig. 4(a), both the CLIDAR and the S-CLIDAR systems with the OEF scheme show excellent performance with correlation coefficients close to unity for SNR > 15 dB. As for −17 dB < SNR < 15 dB, the S-CLIDAR shows better performance benefitted from the noise filtering through synchronization. After the amplitude noise increases to a level of SNR < −17 dB, the synchronization condition is broken and both the CLIDAR and the S-CLIDAR systems cannot determined the range delay from the correlation coefficients.

Fig. 4 (Color online) Correlation coefficients of the CLIDAR and the S-CLIDAR systems with (a) the OEF and (b) the OF schemes for different levels of noise.

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For the OF scheme as shown in Fig. 4(b), the correlation coefficients of the CLIDAR and the S-CLDAR systems maintain at constant levels for SNR > 15 dB. While the S-CLIDAR system has the same correlation coefficient of a value close to unity as in the CLIDAR system for m = 0, the performance degrades as the level of phase noise increases. For −17 dB < SNR < 15 dB, the correlation coefficient of the CLIDAR system drops quickly even when only affecting by the amplitude noise. Meanwhile, the correlation coefficients of the S-CLIDAR system stay at higher levels benefitted by the synchronization process.

Using the CLIDAR system as the benchmark, the capabilities of the S-CLIDAR system in noise suppression with the OEF and the OF schemes under different noise levels are calculated and shown in Fig. 5. Figure 5 shows the differences of the correlation coefficients between the S-CLIDAR and the CLIDAR systems

Δ ρ = ρ_{S - CLIDAR} - ρ_{CLIDAR} .

As can be seen, a range of suppression of about 32 dB (−17 dB < SNR < 15 dB) is found for the OEF scheme where the S-CLIDAR system outperforms the CLIDAR system. Compared with the OEF scheme that is inherently not influenced by the phase noise, the OF scheme shows better performance when the phase noise is not presented (m = 0). While the range of suppression gradually decreases as the level of the phase noise increases, suppression ranges of 29.1 and 22.4 dB are still obtained for m = 0.5 and 0.75 in a low SNR regime similar to practical scenarios [15] [16].

Fig. 5 (Color online) The differences of the correlation coefficients Δρ between the S-CLIDAR and the CLIDAR systems for different levels of noise

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In this study, the coupling and the feedback strengths of the Rx in the S-CLIDAR system are optimized for the highest possible correlation coefficients. With all levels of noise, an open-loop configuration and a generalized synchronization condition are found to have the best performance in general. Figures 6(a) and (b) show the optimized η_OEF,C and η_OF,C for different noise levels, respectively. For the OEF scheme in a low noise regime with SNR > 15 dB, a larger η_OEF,C is desired for better synchronization. As the level of noise increases to SNR < 15 dB, the optimized coupling strength decreases as the noise level increases. In this regime, a strong coupling couples too much noise into the Rx and causes the degradation in synchronization. For the OF scheme, a larger η_OF,C is also desired in the low noise regime when the phase noise is not presented (m = 0). When the phase noise is notable (m = 0.5 and 0.75) or when the amplitude noise increases (lower SNR), lower the coupling strengths are required for optimal synchronization and target detections.

4. Conclusions

In conclusion, the noise suppressions of the CLIDAR and the S-CLIDAR systems with the OEF and the OF schemes are numerically studied and compared. With the capability of noise filtering through synchronization, the S-CLIDAR system with the OEF scheme shows better detection performance for SNR < 15 dB. The S-CLIDAR system with the OF scheme also shows better detections in the low SNR regime, where the range outperforming the CLIDAR system gradually decreases when the phase noise in the channel increases. The conditions for the highest possible correlation coefficients are also given, where an open-loop configuration under a generalized synchronization condition along with an optimized coupling strength depending on the noise level are desired. Experimental characterizations on the noise suppressions and detection performance of the S-CLIDAR system will be carried out and reported separately.

Acknowledgments

This work is supported by the National Science Council of Taiwan under contract NSC-97-2112-M-007-017-MY3.

References and links

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Noise suppressions in synchronized chaos lidars

Abstract

1. Introduction

2. Model

3. Results and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (11)

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