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Abstract
To enhance the performance of single photon lidar in remote sensing applications, an adaptive single photon detection method under fluctuating background noise is proposed in this work. Given a desired false alarm probability, an adaptive threshold can be determined according to the background noise in real time. As a result, the single photon lidar operating under fluctuating background noise can implement high sensitivity and low false alarm rate simultaneously. Experiments are performed to verify the proposed method, and the experimental results are in accordance well within the theoretical values. In addition, the mean number of echo photons can be estimated in a fluctuating background noise environment, and the estimation accuracy is less than 1 photon.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Single photon avalanche diode (SPAD) can offer the ability to detect single photon events, which made widely applications across diverse fields, including remote sensing [1,2,3], quantum communication [4,5], biological imaging [6], and so on. In general, photons incident on the photosensitive surface of the SPAD can trigger an avalanche event. By repeatedly recording the avalanche events against the emission time of associated pulsed laser, a histogram can be produced [7,8]. Meanwhile, applying statistical measurement technique (Time-correlated single photon counting, TCSPC), the sensitivity of such devices can be increased far beyond classical devices (Linear mode APD or PIN diode) and the needed photon flux has significant lower intensities [9,10,11]. In this case, The SPAD will allow lower power laser sources to be used or permit time-of-flight (TOF) data to be measured from significantly longer ranges [12,13,14]. Therefore, the SPAD devices enable the development of novel sensing methods and technologies, and open laser ranging and imaging to new fields of application.
One inherent disadvantage of the SPAD is that there is a dead time following each triggered event. The detector can’t detect any received photon during the dead time, which will limit the dynamic range of the detector [1,15,16]. This fact will result in the SPAD being unable to handle extremely strong background noise. In the past few years, implementation of the single photon lidar operating under strong background noise has been reported. A single photon lidar at 1550 nm is demonstrated, where an InGaAs/InP avalanche photodiode (APD) was operated in 1.5-GHz sine-wave gated Geiger mode to reduce the dead time from 1us to 6.4 ns [17,18]. Owing to the short dead time, the Geiger mode APD achieves robustness to background noise by electronically gating out all but a narrow temporal slice [19,20,21]. To avoid pileup distortion and improve the accuracy of depth, sub-picosecond photon-efficient 3D imaging is demonstrated [22]. Using a sub-picosecond timing accuracy for pulsed light sources with a full width at half maximum (FWHM) wider than 50 ps, the system was proved to be operated across a wide dynamic range, from low-flux to high-flux measurements. Recently, a QPMS-based 3D imager is reported with exceptional detection sensitivity and noise tolerance [23], where sub-picosecond timing resolution and quantum parametric mode sorting (QPMS) allow to perform 3D imaging with weak returning signal at 0.0006 mean photon detection per pulse despite stronger background noise. However, these methods using sub-picosecond timing resolution is unsuitable in dynamic applications. The motion of target or platform may cause the echo photons to be discretely distributed in multiple timing bins, which will make the echo photons submerged in the background noise. Filtering out the noise remains a challenging work since it is difficult to distinguish whether the photon event is generated by the echo or noise [24]. To suppress the noise, a returned laser pulse is divided into two Gm-APD (Geiger mode avalanche photodiode) arrays, from which an AND gate is used to compare the arrival time of the electrical signals. The false alarm probability is drastically decreased, because noise distributed randomly in time domain is filtered out [25]. Based on the fact that the echo signal of adjacent pixels will arrive at the photosensitive surface of the detector at almost the same time, a Gm-APD array is divided into many elementary units and a proper threshold is used to decide whether there exists echo signal or not. In this case, a clear 3D image is taken from the experimental system in spite of strong background noise in sunny day [26].
In remote sensing applications, the single photon lidar may work in diverse environments, including sunny day, cloudy day and night, which correspond to different levels of the background noise. Even in the daytime, the background noise also fluctuates along with the angle between the line of sight (LOS) and the sunlight. In order to handle such complicated background noise, an adaptive single photon detection method based on Si-based Geiger mode APD is proposed in this paper, where the histogram is compared with an adaptive threshold to implement a desired false alarm probability. In this case, the single photon lidar can be operated under fluctuating background noise. Moreover, the mean number of echo photons can be estimated by the proposed method in spite of the background noise fluctuation.
2. Adaptive single photon detection
With the TCSPC technique, M pulse echoes are accumulated to produce a histogram. The histogram is compared to a threshold, which is set to achieve a desired false alarm probability. If the photon events accumulated exceeds the threshold, the detection is declared. A higher threshold will reduce the chance that a false target is declared at the expense of a lower detection probability. When at least one signal or noise photon appears and causes a photon event, the detection probability of single pulse echo (${P_{DS}}$) is given by [27,28]
where ${\eta _{qe}}$ is the quantum efficiency of the SPAD detector, ${n_s}$ and ${n_n}$ are the mean number of signal and noise photons in a timing unit, respectively. ${P_A}$ is the arm probability (blocking loss factor) and it is a function of the mean number of photoelectrons in a dead time (${t_d}$). Even though the background noise fluctuates continuously, a low photon flux is still required to ensure that the SPAD is not saturated. Besides, the ranging duration is much longer than the dead time. In this case, the incident photon flux can reach a steady-state constant level during M pulse accumulation. Ultimately, the arm probability can be approximately expressed as [15,27,29] where ${T_{RES}}$ is the timing resolution of the histogram. Specially, when the pulsed laser is shut down, no echo photon is collected. In this case, the false alarm probability of single pulse echo (${P_{FAS}}$) can be obtained by simplifying Eq. (1), which is given by${P_{FAS}}$ is the probability that at least one noise photon appears. In fact, the echo photon event can’t be distinguished from the noise photon events by single pulse echo. Hence, the TCSPC technique is applied to improve the sensitivity in this paper. Assuming that M pulse echoes are accumulated to produce a histogram, the detection probability of M pulse echoes (${P_D}$), which is in contrast to the detection probability of single pulse echo (${P_{DS}}$), can be expresses as
According to Eqs. (4) and (5), both the detection probability and the false alarm probability of M pulse echoes are determined by several factors, including the threshold (${k_{thr}}$), quantum efficiency of the detector (${\eta _{qe}}$), mean number of the noise photons (${n_n}$), the number of pulse accumulation ($M$), and the dead time (${t_d}$). In addition, the detection probability is still determined by the mean number of echo photons (${n_s}$). It should be mentioned that the false alarm probability is approximately independent of the echo photons, while the detection probability depends on both the echo photons and the noise photons. As the echo photons increase, the detection probability increases accordingly. For a definite system, most of the parameters (such as ${\eta _{qe}}$, M and ${t_d}$) have already been obtained while the mean number of the echo and noise photons (${n_s}$ and ${n_n}$) is unknown. Fortunately, we can approximately derive the mean number of the noise photons from the following expression.
where ${N_{CR}}$ and ${N_{DK}}$ are the total count rate and the dark count rate of the SPAD, respectively. The total count rate contains the dark count rate, the echo count rate and the background count rate. The dark count rate is considered as a constant, which can be measured in advance and then deducted from the total count rate. However, it’s difficult to obtain the expression about the mean number of echo photons (${n_s}$). Hence, several operations need to be performed. Firstly, calculate the total count rate of the SPAD using a frame of histogram, so as to adapt to the occasions that the background noise varies rapidly. Secondly, solve the arm probability (${P_A}$) by combining Eqs. (2) and (6). Thirdly, set a desired false alarm probability, and then an adaptive threshold is deduced according to Eq. (5). Afterwards, the false alarm probability is determined in real time. Finally, the mean number of echo photons (${n_s}$) is derived from Eq. (4). When the background noise fluctuates, the total count rate will make a change accordingly. Subsequently, the above operations need to be performed repeatedly to handle the background noise fluctuation. In order to show the adaptive single photon detection method more clearly, a flow chart is given in Fig. 1.
Fig. 1. Flow chart of the proposed method. Here, the statistical result of ${P_{FA}}$ is obtained from single frame histogram while that of ${P_D}$ is obtained from 100 frame histograms.
The signal-to-noise ratio (SNR) is defined as the ratio of the mean number of signal detections squared to the variance of the total number of noise and signal detections [27]. It’s well known that the mean and variance are equal for Poissonian signals. In addition, the SNR is degraded by the arm probability (${P_A}$) for the SPAD detector. Consequently, the SNR can be given by
3. Experimental setup
To evaluate the performance of the proposed method, a single photon lidar system is established in this paper. As shown in Fig. 2, the pulsed laser works in external triggering mode. When a pulsed light is emitted from the laser, a synchronous signal is generated simultaneously by a PIN diode, which is embedded in the pulsed laser and connected to the TCSPC module (quTAG made by quTools). The echo and noise photons are collected by the receiver together and then coupled to the SPAD detector (SPCM-AQRH made by Excelitas). A narrow bandpass filter (NBF) with 1 nm bandwidth is used to remove most of background noise from sunlight and other sources.
Fig. 2. Experimental setup of the single photon lidar system. (a) a block diagram; (b) the experimental optical system, where the PIN diode is embedded in the pulsed laser and a CW laser combining with an optical attenuator is applied to simulate the fluctuating background noise.
Here, a continuous-wave (CW) laser combining with an optical attenuator is applied to simulate the fluctuating background noise. By adjusting optical power of the CW laser or coefficient of the optical attenuator, the fluctuating background noise is produced and then injected into the receiving optical path, which is mixed with the echo photons and finally collected by the SPAD detector. Subsequently, the TCSPC technique is applied to extract the echo photons from the background light and other noise. The TCSPC module measures the time interval between the emitted photons and their corresponding echo photons. The “start” is triggered by the emitted photons, and the “stop” is triggered by the arrival of an echo photon at the SPAD detector. Repeating these “start-stop” measurements with 20 cycles, a histogram can be produced, from which the distance between the lidar and the target can be measured.
The main parameters of the single photon lidar system is given in Table 1. It should be noted that the quTAG photon counting module can provide excellent performance with a time resolution of 1ps. Nevertheless, a time resolution of 5 ns is sufficient because the pulse duration of the laser is 3.5 ns and the time jitter of the SPAD detector is about 350ps. Otherwise, the echo photons may be discretely distributed in multiple timing units, which will be obstructive to extracting the distance information from the histogram.
Table 1. The main parameters of the single photon lidar system.
4. Results and discussions
During measurements, the background count rate is changed by adjusting optical power of the CW laser or coefficient of the optical attenuator, while the dark count rate and the echo count rate remain constant. When the total count rate varies by means of changing the background count rate, an appropriate threshold is adjusted in order to obtain a desired false alarm probability, which is set to 0.1% in this paper. When the total count rate fluctuates between 10 kcps (cps, counts per second) and 465 kcps, the threshold is adapted to 2. The density of the point clouds increases gradually in pace with the increase of total count rate, which indicates that the false alarm probability increases accordingly. With the total count rate rising from 465kcps to 2Mcps, the threshold is adapted to 3 so that the desired false alarm probability is obtained. The situation where the total count rate fluctuates above 2Mcps is similar with that below 2Mcps. The point clouds shown in Fig. 3(a) are obtained from adaptively thresholding a large number of histograms, where each histogram is produced by 20 cycles. Apparently, when the total count rate fluctuates from 100kcps to 7.8Mcps, the density of the point clouds shows an alternating trend. The main reason is that the threshold increases when the false alarm probability reaches the desired value. Subsequently, the false alarm probability is limited to the desired value. To observe the distance information of the target more clearly, three selected areas in Fig. 3(a) is zoomed in and shown in Fig. 3(b). It can be concluded from the figure that the target is located approximately 1.2 km away from the single photon lidar system. With the total count rate increasing, the point clouds representing the target become more and more sparse. This phenomenon indicates that detection probability of the echo photons decreases gradually when the background noise becomes stronger and stronger.
Fig. 3. The point clouds (a) obtained from adaptively thresholding a large number of histograms, where the total count rate fluctuates from 100kcps to 7.8 Mcps, and (b) Zoom in on the selected areas (1), (2) and (3).
Experimental results of the false alarm probability can be obtained by performing statistical analysis on the point clouds dataset. According to Eq. (5), the false alarm probability is affected by the total count rate (background noise). Therefore, when the background noise becomes stronger and stronger, the threshold will be adjusted to adapt to the fluctuations of the background noise, so as to obtain the desired false alarm probability. As shown in Fig. 4(a), the experimental results may exceed the desired false alarm probability, even though the desired vaule is set to 0.1%. Because the desired value of 0.1% is only a statistical result for a large number of samples, the actual value may fluctuate around the desired value (between 0% and 0.2%). Smoothing the experimental results by median filtering, the filtering results are limited within the desired value of 0.1%, which is consistent well with the theoretical results (see Fig. 4(b)).
Fig. 4. The false alarm probability of the single photon lidar, where the desired false alarm probability is set to 0.1%. (a) The false alarm probability is limited by the desired value; (b) Zoom in on the selected area, where the runtime ranges from 15s to 18s.
Similarly, experimental results of the detection probability can also be obtained by performing statistical analysis on the point clouds dataset. As shown in Fig. 5(a), the experimental results are roughly consistent with the theoretical results. Nevertheless, the number of statistical samples used to evaluate the detection probability is a little small so that the stochastic behavior seems to be more significant than that of the false alarm probability. Besides, the signal-to-noise ratio (SNR) is given in Fig. 5(b), which decreases from 1.4 to 0.9 when the total count rate rises from 10 kcps to 8Mcps.
Fig. 5. Experimental results about the detection probability and the signal-to-noise ratio (SNR). (a) The detection probability shows a stepwise decrease; and (b) The SNR shows a parabolic decrease.
As shown in Fig. 6(a), the mean number of echo photons can be estimated from the false alarm probability (Fig. 4(a)) and the detection probability (Fig. 5(a)), regardless of the background noise fluctuation. To evaluate the estimation accuracy of echo photons, additional method should be compared with the proposed method. Unfortunately, it’s difficult to accurately estimate the mean number of echo photons from lidar equation since some parameters are not so easy to obtain accurately, such as the reflectivity of target and the coefficient of atmospheric transmission, which may degrade the estimation accuracy of echo photons. However, additional information (photon detection efficiency of the SPCM-AQRH detector) could be utilized to estimate the mean number of echo photons. As shown in Fig. 6(b), (a) histogram is derived by 20000 pulses accumulation under background noise-free environment (${n_n} \approx 0$ and ${P_A} \approx 1$). Then, the mean number of echo photons can be solved from Eq. (1). It can be seen from the Fig. 6 that the mean number of echo photons derived from the histogram method is 3.6 while that derived from the proposed method is approximately 3.1.
Fig. 6. Comparison of the echo photon estimation between the proposed method and the histogram method. (a) Results of the echo photon estimation using the proposed method; and (b) results of the echo photon estimation using the histogram method, where the normalized histogram is derived under the background noise-free environment.
Additional experiments are implemented to improve the statistical accuracy, where the dataset is divided into eight sections and each section corresponds to different count rates but same echo photons. When the total count rate rises from 0.11Mcps to 6.35Mcps step by step, the point clouds with different densities are shown in Fig. 7(a). Diverse densities of the point clouds mean that the false alarm probability is also different. Specifically, the point clouds in the first step is relatively sparse since the false alarm probability is merely 0.01%, while that in the seventh step is dense since the false alarm probability has reached 0.08% (see Fig. 7(b)). However, the false alarm probability fluctuates greatly in the third step, which is zoomed in and shown in Fig. 7(c). The main reason is that the third step (count rate: 0.56Mcps) is just locates at the boundary of the threshold adaptation (see Fig. 3(a)). Small fluctuation in the count rate may cause the threshold to adapt to such fluctuation back and forth. When the count rate increases slightly, the threshold adaptively increases and the false alarm probability decreases. On the contrary, when the count rate decreases slightly, the threshold decreases and the false alarm probability increases accordingly.
Fig. 7. The experimental results consist of eight sections, corresponding to the total count rate of 0.11Mcps, 0.23Mcps, 0.56Mcps, 1.17Mcps, 2.36Mcps, 3.55Mcps, 4.89Mcps and 6.35Mcps, respectively. (a) The point clouds; (b) the false alarm probability is limited by the desired value; (c) zoom in on the selected area, where the runtime ranges from 17s to 23s and (d) the detection probability also shows a stepwise decrease.
As shown in Fig. 7(d), the experimental results of the detection probability are approximately in accordance with the theoretical results, which is obtained from 100 frame histograms. Apparently, the number of histograms seems insufficient to ensure the statistical accuracy of the detection probability. To improve the statistical accuracy, all histograms in each step is used to perform statistical analysis to obtain an experimental result of the detection probability and the false alarm probability. As a result, the number of histograms to be analyzed will reach about 5000 frames.
Comparison of the experimental and the theoretical results is given in Table 2. When the total count rate rises from 0.11Mcps to 6.35Mcps, the detection probability decreases gradually and the false alarm probability is limited to the desired value of 0.1%. With the threshold rising from 2 to 5, the detection probability shows a stepped downward trend. But within the same threshold, the detection probability at high count rate is a litter higher than that at low count rate. Because the noise photons as well as the echo photons in the same timing unit are treated as the echo photons together. This phenomenon can also be seen from Eq. (4), where the detection probability is not only related to the echo photons, but also to the noise photons. It’s also indicated from Table 2 that the experimental results are consistent well with the theoretical results.
Table 2. The comparison of the experimental and theoretical results.
Even though the background noise becomes stronger and stronger, the estimation result of echo photons is consistent with that in Fig. 6(b), which is approximately 3.2. Hence, the estimation accuracy of echo photons is less than 1 photon.
The threshold of each histogram is determined in real time according to the current background noise, which is independent to the number of targets and the intensity of echo. Therefore, the proposed method is suitable for multi-targets detection. Besides, the pulse width may be broadened by the unknown targets. In this case, binning operation of the timing units is required to reduce the timing resolution, otherwise the echo will be distributed in multiple timing units and be submerged in the background noise. However, redefining a new desired false alarm probability, the proposed method is also suitable for the applications with low timing resolution.
5. Conclusion
An adaptive single photon detection method under fluctuating background noise is proposed in this paper. Given a desired false alarm probability, an adaptive threshold of the histogram can be determined in real time according to the current background noise, which is independent to the number of targets and the intensity of echo. Therefore, the proposed method is suitable for multi-targets detection. Moreover, weaker background noise corresponds to a lower threshold while stronger background noise corresponds to a higher threshold, and therefore both high sensitivity and low false alarm can be implemented simultaneously. It is proved that the single photon lidar can be operated across a wide dynamic range, from weak to strong background noise. In general, extremely low background noise is required to estimate the mean number of echo photons. Using the proposed method, the estimation of echo photons can be performed under fluctuating background noise. Besides, the proposed method can be used in many applications with diverse timing resolution, regardless of picosecond or nanosecond, which is more suitable for dynamic applications.
Funding
National Natural Science Foundation of China (61805249); Youth Innovation Promotion Association (2019369).
Disclosures
The authors declare no conflicts of interest.
References
1. Paul F. McManamon, “Review of ladar: a historic, yet emerging, sensor technology with rich phenomenology,” Opt. Eng. 51(6), 060901 (2012). [CrossRef]
2. Agata M. Pawlikowska, Abderrahim Halimi, Robert A. Lamb, and Gerald S. Buller, “Single-photon three-dimensional imaging at up to 10 kilometers range,” Opt. Express 25(10), 11919–11931 (2017). [CrossRef]
3. Zheng-Ping Li, Xin Huang, Peng-Yu Jiang, Yu Hong, Chao Yu, Yuan Cao, Jun Zhang, Feihu Xu, and Jian-Wei Pan, “Super-resolution single-photon imaging at 8.2 kilometers,” Opt. Express 28(3), 4076–4087 (2020). [CrossRef]
4. R. Hadfield, “single-photon detectors for optical quantum information applications,” Nat. Photonics 3(12), 696–705 (2009). [CrossRef]
5. Jun Zhang, Mark A Itzler, Hugo Zbinden, and J.-W. Pan, “Advances in InGaAs/InP single-photon detector systems for quantum communication,” Light: Sci. Appl. 4(5), e286 (2015). [CrossRef]
6. Lei Dai, Jian Liu, Kun Liang, Ru Yang, Dejun Han, and B. Lu, “Realization of a time-correlated photon counting technique for fluorescence analysis,” Biomed. Opt. Express 11(4), 2205–2212 (2020). [CrossRef]
7. W. Becker, Advanced Time-Correlated Single-Photon Counting Techniques (Springer, 2005).
8. W. Becker and A. Bergmann, “Multi-dimensional time-correlated single-photon counting,” Proc. SPIE 6771, 677102 (2007). [CrossRef]
9. Martin Laurenzis, “Single photon range, intensity and photon flux imaging with kilohertz frame rate and high dynamic range,” Opt. Express 27(26), 38391–38403 (2019). [CrossRef]
10. Martin Laurenzis, Jonathan Klein, Emmanuel Bacher, and Nicolas Metzger, “Multiple-return single-photon counting of light in flight and sensing of non-line-of-sight objects at shortwave infrared wavelengths,” Opt. Lett. 40(20), 4815–4818 (2015). [CrossRef]
11. Martin Laurenzis, Jonathan Klein, and Emmanuel Bacher, “Relativistic effects in imaging of light in flight with arbitrary paths,” Opt. Lett. 41(9), 2001–2004 (2016). [CrossRef]
12. Xiaoli Sun, J. Bryan Blair, Jack L. Bufton, Marcela Faina, Sigrid Dahl, Philippe Bérard, and Richard J. Seymour, “Advanced silicon avalanche photodiodes on NASA's Global Ecosystem Dynamics Investigation (GEDI) mission,” Proc. SPIE 11287, 1128713 (2020).
13. Ralph Dubayah, James Bryan Blair, Scott Goetz, Lola Fatoyinbo, Matthew Hansen, Sean Healey, Michelle Hofton, George Hurtt, James Kellner, Scott Luthcke, John Armston, Hao Tang, Laura Duncanson, Steven Hancock, Patrick Jantz, Suzanne Marselis, Paul L. Patterson, Wenlu Qi, and Carlos Silva, “The Global Ecosystem Dynamics Investigation: High-resolution laser ranging of the Earth’s forests and topography,” Sci. Remote Sens. 1, 100002 (2020). [CrossRef]
14. Bingcheng Du, Di Wu Chengkai Pang, Zhaohui Li, Huan Peng, Yuliang Tao, E. Wu, and G. Wu, “High-speed photon-counting laser ranging for broad range of distances,” Sci. Rep. 8(1), 4198 (2018). [CrossRef]
15. Phillip Gatt, Steven Johnson, and Terry Nichols, “Dead-time effects on geiger-mode APD performance,” Proc. SPIE 6550, 65500I (2007). [CrossRef]
16. Zhijian Li, Jiancheng Lai, Chunyong Wang, Wei Yan, and Zhenhua Li, “Influence of dead-time on detection efficiency and range performance of photon-counting laser radar that uses a Geiger-mode avalanche photodiode,” Appl. Opt. 56(23), 6680–6687 (2017). [CrossRef]
17. Zhaohui Li, Zeyu Bao, Yafan Shi, Baicheng Feng, E. Wu, Guang Wu, and Heping Zeng, “Photon-counting chirped amplitude modulation lidar with 1.5-GHz gated InGaAs/InP APD,” IEEE Photonics Technol. Lett. 27(6), 616–619 (2015). [CrossRef]
18. Yan Liang, Jianhua Huang, Min Ren, Baicheng Feng, Xiuliang Chen, E Wu, Guang Wu, and Heping Zeng, “2014-nm time-of-flight ranging system employing laser with multiple repetition rates for reducing the range ambiguity,” Opt. Express 22(4), 4662–4670 (2014). [CrossRef]
19. A. Tosi, A. Dalla Mora, F. Zappa, S. Cova, D. Contini, A. Pifferi, L. Spinelli, A. Torricelli, and R. Cubeddu, “Fast-gated single-photon avalanche diode for extremely wide dynamic-range applications,” Proc. SPIE 7170, 71700K (2009). [CrossRef]
20. Alberto Tosi, Alberto Dalla Mora, Franco Zappa, Angelo Gulinatti, Davide Contini, Antonio Pifferi, Lorenzo Spinelli, Alessandro Torricelli, and Rinaldo Cubeddu, “Fast-gated single-photon counting technique widens dynamic range and speeds up acquisition time in time-resolved measurements,” Opt. Express 19(11), 10735–10746 (2011). [CrossRef]
21. Tommaso Lunghi, Enrico Pomarico, Claudio Barreiro, Damien Stucki, Bruno Sanguinetti, and Hugo Zbinden, “Advantages of gated silicon single-photon detectors,” Appl. Opt. 51(35), 8455–8459 (2012). [CrossRef]
22. Felix Heide, Steven Diamond, David B. Lindell, and G. Wetzstein, “Sub-picosecond photon-efficient 3D imaging using single-photon sensors,” Sci. Rep. 8(1), 17726 (2018). [CrossRef]
23. Patrick Rehain, Yong Meng Sua, Shenyu Zhu, Ivan Dickson, Bharathwaj Muthuswamy, Jeevanandha Ramanathan, Amin Shahverdi, and Yu-Ping Huang, “Noise-tolerant single photon sensitive three-dimensional imager,” Nat. Commun. 11(1), 921 (2020). [CrossRef]
24. Cheng Wu, Wenge Xing, Linghao Xia, Huixin Huang, and Chi Xu, “Receiver performance characteristics of single-photon lidar in a strong background environment,” Appl. Opt. 58(1), 102–108 (2019). [CrossRef]
25. Hong Jin Kong, Tae Hoon Kim, Sung Eun Jo, and Min Seok Oh, “Smart three-dimensional imaging ladar using two Geiger-mode avalanche photodiodes,” Opt. Express 19(20), 19323–19329 (2011). [CrossRef]
26. Zijing Zhang, Yuan Zhao, Yong Zhang, Long Wu, and Jianzhong Su, “A real-time noise filtering strategy for photon counting 3D imaging lidar,” Opt. Express 21(8), 9247–9254 (2013). [CrossRef]
27. Philip Gatt, Steven Johnson, and Terry Nichols, “Geiger-mode avalanche photodiode ladar receiver performance characteristics and detection statistics,” Appl. Opt. 48(17), 3261–3276 (2009). [CrossRef]
28. Daniel G. Fouche, “Detection and false-alarm probabilities for laser radars that use Geiger-mode detectors,” Appl. Opt. 42(27), 5388–5398 (2003). [CrossRef]
29. Steven E. Johnson, “Target detection with randomized thresholds for lidar applications,” Appl. Opt. 51(18), 4139–4150 (2012). [CrossRef]
