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Abstract
With the advent of serial production lidars, single photon lidar faces an increasingly severe threat of crosstalk. In this paper, we first propose the concept of Chaos Single Photon (CSP) lidar and establish the theoretical model. In CSP lidar system, chaos laser replaces pulsed laser, and the physical random sequence generated by a Geiger mode avalanche photodiode (GM-APD) responding to chaos laser substitutes the traditional pseudo-random sequence. The mean density of ‘1’ code of the CSP lidar system can exceed 10 million counts per second (Mcps) with a dead time immunity. The theoretical models of detection probability and false alarm rate are derived and demonstrated based on Poisson distribution. The bit error rate (BER) is introduced into the CSP lidar system for evaluating the range walk error intuitively. Additionally, the simulation results indicate that the CSP lidar system has a robust anti-crosstalk capability. Compared with the traditional pseudo-random single photon (PRSP) lidar system, the CSP lidar system not only overcomes range ambiguity but also has a signal to noise rate (SNR) of 60 times, reaching 10000 when the mean echo photoelectron number is 10 per nanosecond. Benefited from large-scale arrays and extremely high sensitivity properties of GM-APDs, we are looking forward to the applications of the CSP lidar in weak signal detection, remote mapping, autonomous driving, etc.
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1. Introduction
In recent years, benefited from extremely high sensitivity and digital output, single photon lidar has been gaining great interest in autonomous vehicle, long-range imaging, underwater detection, and obscurant objects imaging [1–4]. On the other hand, the dead time and the feature with vulnerable to noise are inevitable for the single photon lidar because of unique circuits designing [5]. The multi-pulse accumulation is a common method to distinguish signals and noise at the expense of processing time and imaging speed. To accelerate the process of detection, higher repetition frequency pulsed lasers are usually employed. However, there is a tradeoff between repetition frequency and ambiguous distance [6,7]. More importantly, single photon lidar system faces growing threats of crosstalk with the rapid popularity of lidar sensors and lighting sources [2,8].
To mitigate the possibility of crosstalk and reduce ambiguous distance, many researchers proposed and improved the pseudo-random single photon (PRSP) lidar [9–13], which combines pseudo-random coded method and single photon counting technique. Compared with the pulse accumulation single photon lidar, the PRSP lidar has greatly improved on detection speed and efficiency, anti-crosstalk capability and unambiguous distance [6,10,13].
However, with more lidars growing in a technology-crowded environment, identical pseudo-random coded sequence (PRCS) from other lidars becomes inevitable, where a PRSP lidar still can produce ghost images or result in false alarms [2,14]. Especially, a malicious jammer can easily record the PRCS and then re-emit it to act fake echoes due to periodicity and regularity of the PRCS. Additionally, the distance ambiguity still exists for the finite codes of pseudo-random sequence. Consequently, the long-range detection is hampered when detection range exceeds the maximal unambiguous distance. To overcome those issues, many schemes have been proposed, including chaotic pulse position modulation method [2] and true random coded method [15,16].
Recently, chaos laser has drawn extensive attention on secret communication [17], physical random number generation [18,19], lidar [14] and fiber sensing [20], etc. As a physical random source, chaos laser behaves ultra-bandwidth and noise-like characteristics. Therefore, the lidar system based on chaos laser has excellent anti-crosstalk and confidentiality capabilities by taking the advantages of the aperiodic and unpredictable nature of chaos laser [21]. As early as 2004, Lin F. Y. et al. first proposed the concept of chaos lidar [22]. In 2018, Lin F. Y. developed a 3D pulsed chaos lidar system [14]. However, it is difficult for traditional chaos lidars to realize long range detection constrained with the power of chaos lasers and the sensitivity of linear avalanche photodiodes. An insight occurs, combining chaos laser and single photon technique.
In this study, we utilize chaos laser to generate reference sequence through a Geiger mode avalanche photodiode (GM-APD) and transmit the chaos laser modulated by the reference sequence. Meanwhile, the echo chaos signal is detected by another GM-APD for generating echo sequence. It’s well known that chaos signal has excellent auto-correlation characteristics but noise not [22]. Therefore, it has a good cross-correlation between the reference random sequence and the echo random sequence for ranging detection based on the premise that the dead time of GM-APD of transmitting terminal (GM-APD 1) is not less than GM-APD of receiving terminal (GM-APD 2). We call this system Chaos Single Photon (CSP) lidar. The mean density of ‘1’ code of the reference sequence can exceed 10 million counts per second (Mcps) with a dead time immunity. The detection probability and false alarm rate of the CSP lidar system are derived and demonstrated using Monte Carlo simulation. The bit error rate (BER) is introduced to evaluate the range walk error of the CSP lidar system more intuitively. The simulation result indicates that the range walk error is strongly associated with the BER. The anti-interference and anti-crosstalk capabilities of the CSP lidar system are well verified. Compared with the PRSP lidar system, the CSP lidar system has a signal to noise rate (SNR) of up to 60 times when mean echo photoelectron number is 10 per nanosecond (10/ns).
Generally, the CSP lidar system has some substantial advantages over the PRSP lidar system. Firstly, in terms of modulation method, the traditional pseudo-random modulation or chaotic pulse position modulation method usually adopts approximately Gb/s random sequence to modulate laser for accelerating detection speed [12], which puts forward extremely demanding requirements on laser source and random sequence generator. By contrast, chaos laser itself is a physical random source with bandwidth of excess several GHz. What’s more, monolithically integrated chaos lasers are also readily available [23,24].
Secondly, the equivalent repetition frequency of laser pulse modulated by pseudo-random sequence is much higher than the saturation counting rate of GM-APD, which leads to serious ‘1’ codes leakage detection. Fortunately, the ‘1’ codes generated by GM-APD 1 responding to the chaos laser are mutually independent and almost immune to the dead time of GM-APD 2 due to the premise that the dead time of GM-APD 1 is not less than GM-APD 2. In addition, the mean density of ‘1’ code generated by a GM-APD responding to the chaos laser far exceeds that of the GM-APD responding to the background noise (ambient light and dark count) and is controllable.
Thirdly, pseudo-random sequence behaves repeated periodicity, but the chaos laser is physically random and never repeated. Consequently, the CSP lidar has nature anti-crosstalk and confidentiality performances and removes the range ambiguity in principle. It’s believed that the CSP lidar is a huge potential solution for safe and reliable autonomous driving.
2. CSP lidar and ranging principle
In this part, the system structure and ranging principle of the CSP lidar is described. The structure of the CSP lidar system is depicted in Fig. 1. The emitted chaos laser is split into two parts, the reference signal and transmitting signal. The reference chaos signal attenuated by variable optical attenuator (VOA) is injected into GM-APD 1. The random sequence yielded by GM-APD 1 is shaped by the pulse shaping circuits (PSC 1) to adjust the code width, as reference sequence a(n). Meanwhile, the transmitting signal modulated by a(n) through a modulator, and modulated signal is emitted to illuminate targets via the optical transmitter system (OTS). The echo signal diffused by the target is focused on GM-APD 2 through the optical receiver system (ORS). The GM-APD 2 generates random sequence with $\tau$ time delay referring to reference random sequence a(n). The sequence generated by GM-APD 2 is shaped to the same code width as a(n) by PSC 2, which severs as echo random sequence b(n). The reference and the echo random sequences generated by GM-APDs are recorded and stored by channel 1(Ch1) and channel 2 (Ch2) of time-correlation single photon counting (TCSPC), respectively. Ultimately, the target detection is accomplished by correlation operation between the reference random sequence a(n) and the echo random sequence b(n).
Fig. 1. Structure of CSP lidar. FC: fiber coupler; VOA: variable optical attenuator; OTS: optical transmitter system; ORS: optical receiver system; PSC: pulse shaping circuits; Ch: channel; TCSPC: time-correlation single photon counting.
The ranging principle of CSP lidar system is clearly shown in Fig. 2. The reference chaos signal is randomly quantized to 0/1 code according to signal intensity. The transmitting chaos signal is modulated by reference random sequence a(n), which has similar intensity fluctuations to the reference chaos signal when a(n) is 1 code. Theoretically, the reference random sequence a(n) can be reconstructed by GM-APD 2 with only time delay difference as long as the power of echo signal is higher than that of reference signal. The reconstructed random sequence is recorded as echo random sequence b(n). A cross-correlation function $g(\tau )$ is operated between the reference random sequence and the echo random sequence, from which the round-trip time can be extracted [25].
where a(n) is the reference random sequence code, n = 1, 2, …, N, N is the number of codes in one sequence period. $\tau$ corresponds to the time of flight (TOF), b(n) is the echo random sequence. $g(\tau )$ is the cross-correlation function, which likes a delta function. $\Delta t$ is the time duration of a code.3. Theoretical model and analysis
In this study, it is assumed that the statistics model of weak chaos laser signal responded by GM-APDs follows Poisson distribution according to Goodman’s statistical optics theory [26].
The Poisson probability density function (PDF) describes the probability that exactly k signal photon events occur in a time bin (time resolution of TCSPC) when the mean number of signal photon counts is $\overline m$. The PDF follows as [27]
3.1 Generation and properties of physical random sequence
It is well known that chaos laser is a physical random source and the outputs of GM-APD are digital with ‘0’ or ‘1’ code. The combination of chaos laser and GM-APD introduces new random factors into the physical random sequence generation process. The amplitude fluctuations of chaos laser are extensively large and behave physical randomness. Therefore, in the peak or high energy timestamp of chaos signal, the detection probability of signal photon events is much higher, corresponding to increasing the probability of ‘1’ code. On the contrary, the probability of ‘0’ code increases. Additionally, the dead time of GM-APD will interfere with the generation of ‘1’ code and result in that the ‘0’ code fill between adjacent ‘1’ codes.
The structure of physical random sequence generation is shown in Fig. 3. The attenuated chaos laser is detected by GM-APD 1, and the detection probability Pd within the sampling time interval of $(t,t + {\tau _{bin}})$ can be derived by Eq. (3). Considering the dead time ${\tau _{d1}}$ of GM-APD 1, the detection probability is modified as [27]
In order to analyze the properties of physical random sequence more literally, 10000 Monte Carlo simulations are implemented. The main simulation parameters are shown in Table 1. The specific simulation parameters of chaos laser refer to Ref. [28]. By the way, the InGaAs GM-APDs are taken as a sample in this paper and can be substituted by other single photon detectors.
Table 1. Simulation parameters for generating physical random sequence [29]
Figure 4 describes the properties of physical random sequence generated by GM-APD. Figure 4(a) zooms the part data of the inset figure, where the red curve represents chaos laser, and the black curve represents ambient light. Compared with ambient light (the level of ambient is 8 MHz), the photon number of chaos laser behaves considerably fluctuating in different time bin intervals when mean photon number of chaos laser is 0.1/ns, obviously shown in Fig. 4(a). This is a crucial reason that GM-APD responding to weak chaos laser can yield physical random sequence. Figure 4(b) shows superb auto-correlation property of physical random sequence generated by GM-APD, which demonstrates the ranging feasibility of the CSP lidar. The inset figure of Fig. 4(b) displays the physical random codes generated by GM-APD. Figure 4(c) and Fig. 4(d) depict the mean density of ‘1’ code and the maximum side lobe of auto-correlation curve with the mean photon number of chaos signal increasing, respectively. The mean density of ‘1’ code is defined as the mean counting number of ‘1’ code per second. For example, 1Mcps represents 1 million count per second (Mcps). The maximum side lobe is the largest value of auto-correlation curve besides the main peak, which evaluates the quality of auto-correlation property. The smaller value of maximum side lobe, the better auto-correlation property. When considering only ambient light and dark counts, the mean density of ‘1’ code generated by GM-APD is about 1.5 Mcps. Normally, it is not convenient to adjust and control the level of ambient light. By contrast, the mean photon number of chaos laser is simple and direct to change. The mean density of ‘1’ code can be changed through adjusting mean photon number of chaos signal impinging on GM-APD (assuming the level of ambient light is zero at this time), illustrated in Fig. 4(c). With the mean photon number increasing, the mean density of ‘1’ code gradually increases and approaches saturation, far beyond that of background noise. However, the maximum side lobe of auto-correlation curve suffers a mild fall in and then increases with the mean photon number of chaos signal increasing. This can be explained that the sparse mean density of ‘1’ code leads to poor auto-correlation property when the mean photon number is relatively low. The auto-correlation property is improved gradually with the mean photon number increasing, but the auto-correlation property deteriorates gradually as the mean photon number continues to increase due to photon pileup effect, displayed in Fig. 4(d). In the following, the mean photon number of reference chaos signal is set to 0.1/ns, corresponding to 12-Mcps mean density of ‘1’ code. Additionally, the auto-correlation property is also closely related to the dead time and time bin. The auto-correlation property improves as the dead time or time bin reduces.
Fig. 4. Properties of physical random sequence generated by GM-APD. (a) The photon number of chaos laser (red) and ambient light (black) at the ith time bin when the mean photon number of chaos laser is 0.1/ns and the level of ambient light is 8 MHz; (b) The auto-correlation property of physical random sequence; (c) The mean density of ‘1’ code generated by GM-APD under different mean photon number of chaos signal (blue) and the 8 MHz ambient light (black); (d) The maximum side lobe of auto-correlation curve under different mean photon number of chaos signal.
3.2 Detection probability and false alarm rate of CSP lidar
Both detection probability and false alarm rate are key parameters to evaluate single photon lidar system. In this section, the detection probability and false alarm rate models of the CSP lidar system are built and demonstrated using Monte Carlo simulation.
The detection probability and false alarm rate are decided by cross-correlation function $g(\tau )$ between a(n) generated by GM-APD 1 and b(n) detection by GM-APD 2. However, the theoretical models of detection probability and false alarm rate are difficult to be expressed intuitively by mathematical expressions. Fortunately, the transmitting signal intensity of each ‘1’ code in physical random sequence is independent. Therefore, it makes sense that the models can be built on one of the detected ‘1’ codes as an example in this section.
In fact, the code width $\Delta t$ of physical random sequence generated GM-APD 1 is larger than the time bin ${\tau _{bin}}$ of TCSPC. The shaped code width is usually a few of nanoseconds, but the time bin τ is a very short time interval that ranges from dozens of picoseconds to hundreds of picoseconds. In other words, the code width is composed of multiple time bins. Hence, the detection probability that at least one photon event occurs in the ith time bin within the code width can be expressed as
Because the dead time of GM-APD 2 is more than the width of code, the probability of each ‘1’ code detection by GM-APD 2 can be expressed as [30]
The false alarm rate of GM-APD 2 is defined that noise photon events occur in time interval $\Delta T$ between two adjacent ‘1’ codes. The false alarm rate can be expressed as
It is noted that the dead time of GM-APD 1 is not less than that of GM-APD 2 in the CSP lidar system. That is to say, the time intervals between two adjacent ‘1’ codes of the transmitting signal are not less than the dead time of GM-APD 2, i.e., $\Delta T \ge {\tau _{d2}}$.
As can be seen in Eq. (5), Eq. (7) and Eq. (8), the detection probability and false alarm rate are related to code width, dead time, time bin, and the length of sequence. The detection probability and false alarm rate of the CSP lidar system are demonstrated using Monte Carlo simulation. The simulation parameters for CSP lidar ranging system are shown in Table 2. The parameters of GM-APD 1 are identical to that of GM-APD 2. The ambient light of GM-APD 1 is assumed to be zero for highlighting the background noise effect on the detection probability and false alarm rate. The mean photon number of preference chaos signal is set to always 0.1/ns. Except special instructions, the dead times of both GM-APDs are 40 ns, the time bins of both GM-APDs are 200 ps, the code width is 1 ns, the sequence length is 3.8 us and the ambient light of GM-APD 2 is 8 MHz in the following simulations. To demonstrate the detection probability and false alarm rate, each parameter point is performed 10000 Monte Carlo simulations.
Table 2. Simulation parameters for CSP lidar ranging system [29]
The information of target location is extracted by cross-correlation peak. Therefore, setting a reasonable threshold is crucial to achieve desired detection probability and false alarm rate. In this paper, the threshold is set to be 3 dB to guarantee the false alarm rate lower than 1%. The cross-correlation peak of expected target location is substituted to the following formula [21]
where ${\sigma _{noise}}$ is the standard deviation of the background noise of cross-correlation curve. When G is more than 3 dB, the target location is considered correctly identified. In the same way, the false target location is triggered when the G value of a false target is higher than 3 dB.Figure 5 illustrates the detection probability and false alarm rate of the CSP lidar ranging system with mean echo photon increasing under different code widths, different dead times, different time bins and different ambient light. As seen from Fig. 5, the code width has the most significant effect on the detection probability compared with other parameters. Generally, the detection probability increases and converges to saturate rapidly as the mean echo photon number increases. By contrast, the ambient light has relatively little effect on the detection probability, which indicates the CSP lidar system has a robust capability of anti-interference. Meanwhile, all false alarm rates are limited to less than 1%.
Fig. 5. Detection probability (sphere) and false alarm rate (triangle) of the CSP lidar under different parameters. (a) The influence of code width on detection probability and false alarm rate; (b) The influence of dead time on detection probability and false alarm rate; (c) The influence of time bin on detection probability and false alarm rate; (d) The influence of ambient light on detection probability and false alarm rate.
3.3 Range resolution and range accuracy of CSP lidar
Range resolution and range accuracy are two other important indicators for evaluating ranging performance. The range resolution is strongly associated with the code width. In theory, once the code width is determined, the range resolution is determined.
The range accuracy is also called range walk error in single photon lidar. Many researchers have studied the range walk error of the single photon lidar system with GM-APD [25,30,31]. The range walk error mainly originates from signal intensity variation. Normally, the width of the code is larger than the time bin of TCSPC. The echo photon number is different at each time bin within the ‘1’ code width for chaos laser, which induces the range walk error. However, the range walk error is hard to be expressed intuitively with mathematical formulas whether it is the CSP lidar or the PRSP lidar. It is common to take one of the detected codes as an example. The average TOF of code width is determined by the method of centroid detection based on Eq. (10) [30].
To evaluate the rang walk error of the CSP lidar more intuitively, the BER is introduced learning from the spread spectrum communication system. The BER is acknowledged to be defined as the rate of the number of different codes between the reference random sequence and the echo random sequence responded by GM-APDs to the total code number of the sequence. The expression of BER follows the formula
where $\Delta N$ is the number of different codes between the reference random sequence and the echo random sequence, and N is the total code number of the sequence.The range walk error and BER of the CSP lidar system are illustrated in Fig. 6. It is found that the trend of range walk error is almost consistent with that of the BER. As theoretical analysis, when the number of mean photoelectrons is identical, both range walk error and BER reduce as the code width decreases. In general, the range walk error and BER reduce gradually with the mean photoelectron number increasing. When the mean photoelectron number is more than 4/ns, the walk error and BER both saturate. The physical mechanism can be explained as the probability of photon events occurring in the 1th time bin of code gradually increases and saturates with the mean photoelectron number increasing. As can be seen in Fig. 6(a), the range walk error is on the order of centimeters and even smaller when the mean photoelectron number is more than 2/ns.
3.4 Anti-crosstalk capability of CSP lidar
With the increasing demands of the intelligent transportation system and unmanned vehicles especially in complex application scenarios, crosstalk between lidar systems must be taken into consideration [2]. In this part, the anti-crosstalk capability of the CSP lidar system is discussed. The capability of crosstalk resistance derives from the randomness performance of random sequence. Therefore, the cross-correlation function can be used to evaluate the anti-crosstalk performance of the CSP lidar system. It is obvious in Fig. 7(b) and Fig. 7(c) that cross-correlation is almost none between reference random sequence and M sequence, as well as between reference random sequence and other physical sequence. By contrast, an obvious cross-correlation peak presents at 120-m location between reference random sequence and echo random sequence in Fig. 7(a), which means the CSP lidar system has a good anti-crosstalk capability.
Fig. 7. Anti-crosstalk capability of CSP lidar system. (a) Cross-correlation property between reference random sequence and echo random sequence; (b) Cross-correlation property between reference random sequence and M sequence; (c) Cross-correlation property between reference random sequence and other physical sequence.
In practice, the anti-crosstalk capability of physical random sequence has been demonstrated [2,8,21]. Chaos laser is acknowledged to behave naturally physical randomness. Hence, the anti-crosstalk capability of the CSP lidar system overwhelms that of the PRSP lidar system.
3.5 Comparison with PRSP lidar
In this section, two critical metrics, the SNR and detection probability of the CSP lidar system, are compared to that of the PRSP lidar system for validating the superiority of the CSP lidar. The SNR of the CSP lidar system is defined as the ratio of the square of the cross-correlation peak to the variance of noise floor of cross-correlation function based on Ref. [25,27]. Hence, the SNR of the CSP lidar system can be expressed as [25]
As the most widely used pseudo-random code in the spread spectrum communication system, M sequence is implemented in the PRSP lidar. In simulation, the transmitting pulse waveform is assumed to be a square wave. The simulation parameters of two lidar systems are identical in Monte Carlo simulation. Figure 8 shows the SNR and detection probability of the CSP lidar and PRSP lidar systems under different mean echo photoelectron numbers. In Fig. 8(a), it can be found that the SNR of CSP lidar system is lower than that of PRSP lidar system when the mean echo photoelectron number is lower than 0.12/ns. During this period, the detection probabilities of the CSP lidar and the PRSP lidar systems are relatively low, displayed in Fig. 8(b). With the mean echo photoelectron number increasing, the SNR of the CSP lidar system increases significantly, far exceeding that of the PRSP lidar system. The SNR of the CSP lidar system is about 60 times of the PRSP lidar system when the mean echo photoelectron number is 10/ns, where the SNRs of two lidar systems converge to saturate. Correspondingly, the detection probability of the CSP lidar system is higher than that of the PRSP lidar system and approaches 100% with the mean echo photoelectron number increasing. In general, the SNR and detection probability of the CSP lidar system are higher than that of the PRSP lidar system when the mean echo photoelectron number is more than 0.12/ns. The underlying reason is that the physical random sequence generated by GM-APD1 is almost immune to the dead time of GM-APD 2, while many ‘1’ codes of M sequence are missed detection due to the dead time. It’s noted that the threshold is set as the same as section 3.2.
Fig. 8. SNR and detection probability statistical results of the CSP lidar system and the PRSP lidar system (a) The SNR statistical characteristics of the CSP lidar system (blue sphere) and the PRSP lidar system (red sphere) with mean echo photoelectron number increasing; (b) The detection probability statistical characteristics of the CSP lidar system (blue sphere) and the PRSP lidar system (red sphere) with the mean echo photoelectron number increasing.
4. Discussion
In the above analysis, the theoretical and simulation models of the CSP lidar system are based on the Poisson distribution. Although the negative binomial distribution converges to the Poisson distribution when mean echo photon number is sparse, the discrepancy between the negative binomial distribution and Poisson distribution has some effect on theoretical and simulation models [32]. In the follow-up study, we will investigate the effect of this discrepancy on the CSP lidar system.
To simplify simulation, the dead time of GM-APD 1 equals that of GM-APD 2 in the paper. In practice, it has hardly any impact on the CSP lidar system when the dead time of GM-APD 2 is not more than that of GM-APD 1. The device condition is available that the dead time of GM-APD 1 is not less than that of GM-APD 2. Therefore, it is reasonable that the dead time of GM-APD 1 is set to equal that of GM-APD 2 in the above simulations.
5. Conclusion
In this paper, we first develop and demonstrate the concept of CSP lidar. The developed CSP lidar ranging feasibility is demonstrated by auto-correlation operation. The theoretical models of the detection probability and the false alarm rate are demonstrated using Monte Carlo simulation. It is found that the code width has the most significant effect on the detection probability. The range walk error (range accuracy) is analyzed and validated. The simulation results indicate that the BER can characterize the range walk error, and the lower BER, the smaller range walk error. More importantly, the anti-crosstalk capability of the CSP lidar system is investigated. Compared with the PRSP lidar system, the SNR of the CSP lidar overwhelms and is up to 60 times when the mean echo photoelectron number is 10/ns. Benefited from extremely high sensitivity of GM-APD, the emission energy of CSP lidar is much lower than that of traditional chaos lidar in the case of the same detection distance in theory. Constrained with the current hardware conditions, the demonstrated experiment of the CSP lidar will be implemented in the follow-up study.
In summary, the CSP lidar has the characteristics of exceeding 10-Mcps mean density of ‘1’ code, no range ambiguity, ultra-high SNR, robust anti-crosstalk capabilities et al. The GM-APD can be substituted by other single photon detectors. It’s believed that the CSP lidar has promising potentials in future intelligent technologies, including unmanned driving, long-distance ranging, and weak signal detection and so on.
Funding
Key special projects of the Ministry of Science and Technology (2018YFC0825602).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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