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A Method of Range Walk Error Correction in SiPM LiDAR with Photon Threshold Detection

A Method of Range Walk Error Correction in SiPM LiDAR with Photon Threshold Detection hv photonics Article A Method of Range Walk Error Correction in SiPM LiDAR with Photon Threshold Detection 1 1 2 1 , 1 Runze Yang , Yumei Tang , Zeyu Fu , Jian Qiu * and Kefu Liu School of Information Science and Engineering, Fudan University, 220 Handan Road, Shanghai 200433, China; 19210720059@fudan.edu.cn (R.Y.); 19110720076@fudan.edu.cn (Y.T.); kfliu@fudan.edu.cn (K.L.) Academy for Engineering and Technology, Fudan University, 220 Handan Road, Shanghai 200433, China; 20210860034@fudan.edu.cn * Correspondence: jqiu@fudan.edu.cn Abstract: A silicon photomultiplier (SiPM) LiDAR with photon threshold detection can achieve high dynamic performance. However, the number fluctuations of echo signal photons lead to the range walk error (RWE) in SiPM LIDARs. This paper derives the RWE model of SiPM LiDAR by using the LiDAR equation and statistical property of SiPM’s response. Based on the LiDAR system parameters and the echo signal intensity, which is obtained through the SiPM’s photon-number- resolving capability, the RWE is calculated through the proposed model. After that, we carry out experiments to verify its effectiveness. The result shows that the method reduces the RWE in TOF measurements using photon threshold detection from 36.57 cm to the mean deviation of 1.95 cm, with the number of detected photons fluctuating from 1.3 to 46.5. Keywords: LiDAR; SiPM; TOF; range walk error; threshold detection 1. Introduction A single-photon avalanche diode (SPAD) has been widely used in remote laser ranging Citation: Yang, R.; Tang, Y.; Fu, Z.; systems due to its single-photon detection sensitivity and high time resolution [1–3]. SPAD Qiu, J.; Liu, K. A Method of Range functions as a binary device with a deadtime [4]. The time-to-digital converter (TDC) for Walk Error Correction in SiPM TOF computation can only timestamp the first detected photon in a SPAD pixel. Therefore, LiDAR with Photon Threshold a SPAD LiDAR system encounters the problem of a high false alarm rate, which becomes Detection. Photonics 2022, 9, 24. https://doi.org/10.3390/ prominent when the background light is strong. To solve this problem, References [5–7] photonics9010024 utilize the time-correlated single-photon counting (TCSPC) technique to calculate the flight time by counting the trigger time distribution. However, TCSPC requires thousands of Received: 17 November 2021 successive measurements to improve the signal-to-noise ratio (SNR) and calculate the time Accepted: 23 December 2021 of flight (TOF). The long acquisition time restricts the application of SPAD LiDAR in areas Published: 1 January 2022 such as autonomous driving, given the requirements for high frame rate imaging and Publisher’s Note: MDPI stays neutral fast-moving target tracking [8]. with regard to jurisdictional claims in SiPMs are large photosensitive area detectors that consist of parallel SPAD pixels, and published maps and institutional affil- each pixel has an independent quenching circuit [9]. In analog SiPM, the summation of iations. each pixel’s response constitutes the analog output signal of SiPM [10–12]. As a result, the signal of SiPM is proportional to the number of incident photons. In digital SiPM, the SPAD frontend is an active circuit that sets a fixed and well-defined deadtime after each ignition and provides a digital pulse when a photon is detected. The output of all microcells is Copyright: © 2022 by the authors. combined through logic gates. The digital output is directly the number of fired cells [11,12]. Licensee MDPI, Basel, Switzerland. Regarding SiPM’s photon-number-resolving capability, compared to SPAD LiDAR, SiPM This article is an open access article LiDAR can utilize the photon threshold to filter out low photon numbers. The threshold distributed under the terms and can lower the false alarm rate and improve the SNR of the LiDAR system [13]. In addition conditions of the Creative Commons to this, setting the photon threshold helps suppress the ranging uncertainty caused by the Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ pulse width of the laser. These advantages help the SiPM LiDAR system achieve a better 4.0/). dynamic range. Photonics 2022, 9, 24. https://doi.org/10.3390/photonics9010024 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 24 2 of 15 For wide-dynamic-range applications, SiPM LiDAR requires high accuracy when measuring targets with extreme variations in reflectivity and angles. Moreover, the RWE caused by the number fluctuation of the received photons is the dominating influential factor for accurate ranging. References [14–17] establish several methods to reduce RWE emerging in SPAD LiDAR systems. Reference [18] proposes a correction method for RWE in LiDAR with a photomultiplier tube. References [19–21] discuss the accuracy of linear APD LiDAR using a fixed threshold discriminator, the constant fraction discriminator (CFD) method and the peak detection method (PKD). However, to date, there have been few related studies focusing on the RWE reduction in SiPM LiDAR with threshold detection. In this article, an RWE correction method of an SiPM LiDAR system with threshold detection is proposed. First, we establish a theoretical RWE model containing the response characteristics of SiPM and the photon threshold method. Thereafter, the RWE is calculated through the above model. In the proposed RWE model using the photon threshold method, we derive the relationship between RWE and the number of detected photons, which can be obtained via either analog SiPM or digital SiPM. Therefore, the proposed method is available in the two types of SiPM LiDAR systems. Groups of experiments with different echo light intensities are performed to measure the distance to a fixed target. In each group, we conduct the correction method to offset the RWE emerging in the result of the fixed photon threshold detection. Meanwhile, the CFD and PKD methods are utilized for comparison with the proposed method. Finally, we develop a method that utilizes the detection probability of the photon threshold detection to estimate the mean number of detected photons instead of using the signal of SiPM, which may simplify the system. As proved by the experiments, this method can work at the few-photon level. 2. Theoretical Analysis 2.1. Laser Propagation Equation The D-TOF LiDAR system delivers a narrow-width laser pulse to the target. The power of an emitted Gaussian pulse [22] can be described with the average single-pulse energy E as follows: 2s P (t) = E p e (1) Emit T 2ps where P (t) is the power of the emitted pulse, and s is the emitted pulse width. When the Emit f pulse is scattered from a Lambertian target at distance R, the received number of photons, N (t), can be derived from Equation (1) [16] as Photon 1 2 2s N t = N p e + N (2) ( ) S Noise Photon 2ps where N is the background noise rate, and s is the received pulse width. We assume Noise s  s in this manuscript because the transmitted pulse width is in the order of nanosec- S f onds [23]; N is the mean number of received signal photons, which can be calculated as [24] E h h h r A T T R R N = (3) p R hv where h is the transmission of the transmitter; h is the transmission of the receiver optic; T R h is the one-way transmission of the atmosphere; r is the reflectivity of the target; A is A R the area of the aperture of the receiver; h is the Plank constant; and v is the frequency of the laser. Photonics 2022, 9, 24 3 of 15 2.2. SiPM Response Model Assuming that the number of arriving photons on each pixel is evenly distributed, then, in any time interval (t , t ), the mean number of photons hitting each pixel, N, can be 1 2 derived from Equation (2) as N (t)dt Photon N t , t = (4) ( ) 1 2 Cell where N is the number of pixels in SiPM. Poisson distribution is an appropriate model Cell to describe the response of a pixel [25]. The probability that a pixel responds in the ith time bin is derived from Equation (4) [26]: N(tbin(i1),tbini)h P (i 1, i) = 1 e (5) D_bin where h is the photon detection efficiency of SiPM; i is an integer larger than 0; and t is q bin the length of a time bin, which corresponds to the timing resolution of the TDC. Therefore, the mean number of fired pixels in one measurement, N , can be calculated as N h Total ( ) Cell N = N 1 e (6) D Cell here, 0.5T Signal N = N (t)dt = N + N  T (7) S Noise Total Photon Signal 0.5T Signal Therefore, N can also be calculated through Equations (6) and (7) as N N Cell Cell N =  ln N  T (8) S Noise Signal h N N Cell D where N is the number of arrived photons (including ambient noise) in a signal dura- Total tion. T is the effective signal duration, usually T > 6s . As seen in Equation (3), Signal Signal environment parameters, such as the one-way transmission of the atmosphere and the reflectivity of the target, are needed to calculate Ns. However, according to Equation (8), we can estimate N by the number of fired pixels, which can be obtained from the signal of Photonics 2022, 9, x FOR PEER REVIEW 4 of 15 SiPM. N has a linear relation with Ns when N >> N as seen in Figure 1. D Cell D Figure 1. The mean number of fired pixels, N , versus the mean number of received photons, N , Figure 1. The mean number of fired pixels, N D D, versus the mean number of received photons, N SS, with different number of pixels in SiPM, N . with different number of pixels in SiPM, NCell. Cell Considering that the response of each pixel is independent and that the probability of k pixel responses in a time interval PCell obeys the binomial distribution [25], then the probability is derived from Equation (5) as NCell Nk −  Cell Pi−= 1,i ,k P i−1,i 1−P i−1,i () () () () () (9) Cell  D__ bin D bin  In fact, if more than one photon hits a pixel, only one can be detected, which causes non-linear loss. In Equation (9), the nonlinear loss is not considered, because we limited our signal to a few photons where nonlinear loss has a low effect on the results [13]. This is due to the relatively low photon number compared to the number of pixels, which is about two orders of magnitude less in the experiment. For light detection at the few-photon level, the number of fired pixels is very small compared to thousands or even tens of thousands of pixels in the SiPM. Meanwhile, the probability of detecting a photon in a single pixel is minimal; therefore, Equation (9) can be approximated as a Poisson distribution [27]: −⋅ NP () i−1,i Cell D_ bin NP⋅−i 1,i ⋅e () () Cell D_ bin (10) Pi−= 1,i ,k () () Cell k! In the fixed threshold detection method for APD LiDAR, the arrival time of the light is determined when the voltage of the detector reaches a given voltage amplitude. The method for the SiPM LiDAR is similar. After the light is detected by SiPM, the photon number is then thresholded by a one-bit comparator, which stops the timer when the volt- age of SiPM, caused by the detection of the arriving photons, exceeds the given threshold. The threshold detection probability in the ith time bin P(i) is calculated by summing the probability of all possible events that the accumulative number of fired pixels exceeds as the threshold photon number k in the ith time bin. Therefore, P(i) can be derived from Equation (10) as kk −−11u−   Pi ()=− P (0,i 1),u⋅ 1− P () i−1,i ,v () () (11)  Cell Cell  uv ==00  where u is the number of fired pixels before the ith time bin, and v is the fired pixels within the ith time bin. The event where the accumulative number of fired pixels exceeds k in the ith time bin must satisfy that u + v ≥ k and v ≥ 1. Then, the whole detection probability PD of one shot can be obtained from Equation (11) as Photonics 2022, 9, 24 4 of 15 Considering that the response of each pixel is independent and that the probability of k pixel responses in a time interval P obeys the binomial distribution [25], then the Cell probability is derived from Equation (5) as N k Cell k Cell P ((i 1, i), k) = P (i 1, i)(1 P (i 1, i)) (9) Cell D_bin D_bin In fact, if more than one photon hits a pixel, only one can be detected, which causes non-linear loss. In Equation (9), the nonlinear loss is not considered, because we limited our signal to a few photons where nonlinear loss has a low effect on the results [13]. This is due to the relatively low photon number compared to the number of pixels, which is about two orders of magnitude less in the experiment. For light detection at the few-photon level, the number of fired pixels is very small compared to thousands or even tens of thousands of pixels in the SiPM. Meanwhile, the probability of detecting a photon in a single pixel is minimal; therefore, Equation (9) can be approximated as a Poisson distribution [27]: N P (i1,i) Cell D_bin (N  P (i 1, i))  e Cell D_bin P i 1, i , k = (10) (( ) ) Cell k! In the fixed threshold detection method for APD LiDAR, the arrival time of the light is determined when the voltage of the detector reaches a given voltage amplitude. The method for the SiPM LiDAR is similar. After the light is detected by SiPM, the photon number is then thresholded by a one-bit comparator, which stops the timer when the voltage of SiPM, caused by the detection of the arriving photons, exceeds the given threshold. The threshold detection probability in the ith time bin P(i) is calculated by summing the probability of all possible events that the accumulative number of fired pixels exceeds as the threshold photon number k in the ith time bin. Therefore, P(i) can be derived from Equation (10) as !! k1 ku1 P(i) = P ((0, i 1), u) 1 P ((i 1, i), v) (11) å Cell å Cell u=0 v=0 where u is the number of fired pixels before the ith time bin, and v is the fired pixels within the ith time bin. The event where the accumulative number of fired pixels exceeds k in the ith time bin must satisfy that u + v  k and v  1. Then, the whole detection probability P of one shot can be obtained from Equation (11) as P = P(i) (12) D å signalbins where signalbins is a collection of intervals (0.5T , 0.5T ) divided into cells with a Signal Signal length of t . bin Figure 2 is the simulation result based on Equation (11). The time bin is of the interval (0.5T , 0.5T ). The origin of the time axis is the center of the echo signal. The Signal Signal photon threshold k = 3. The s is 1.02 ns, the number of pixels of SiPM is N = 2688, S Cell and the mean number of fired pixels is N = 7, 5, 3, 2, 1. Figure 2a shows the detection probability based on Equation (11) under different echo light intensities. It indicates that, as the intensity of the echo decreases, the center of the detection probability distribution shifts to the right, and the full width at half maximum (FWHM) increases. Figure 2b shows the detection probability under different laser pulses (N = 7, FWHM = 1, 2, 3, 5 ns). Figure 2b indicates that, as the FWHM increases, the probability distribution center shifts to the right, and the ranging uncertainty increases. The centroid algorithm can calculate the triggered time using the following equation: Photonics 2022, 9, x FOR PEER REVIEW 5 of 15 PP = ()i D  (12) signalbins where signalbins is a collection of intervals (−0.5TSignal, 0.5TSignal) divided into cells with a length of τbin. Figure 2 is the simulation result based on Equation (11). The time bin is of the interval (−0.5TSignal, 0.5TSignal). The origin of the time axis is the center of the echo signal. The photon threshold k = 3. The σS is 1.02 ns, the number of pixels of SiPM is NCell = 2688, and the mean number of fired pixels is ND = 7, 5, 3, 2, 1. Figure 2a shows the detection probability based on Equation (11) under different echo light intensities. It indicates that, as the intensity of the echo decreases, the center of the detection probability distribution shifts to the right, and the full width at half maximum (FWHM) increases. Figure 2b shows the detection probability under different laser pulses (ND = 7, FWHM = 1, 2, 3, 5 ns). Figure 2b indicates that, as the FWHM increases, the probability distribution center shifts to the right, and the Photonics 2022, 9, 24 5 of 15 ranging uncertainty increases. The centroid algorithm can calculate the triggered time us- ing the following equation: Pi()⋅⋅i τ  bin P(i) i t signalbins bin T = (13) Sim signalbins Pi () T =  (13) Sim signalbins P(i) signalbins (a) (b) Figure 2. Detection probability under different circumstances: (a) echo light intensities and (b) Figure 2. Detection probability under different circumstances: (a) echo light intensities and (b) FWHM FWHM of the laser pulse. of the laser pulse. Therefore, the range walk error can be calculated using the following equation: Therefore, the range walk error can be calculated using the following equation: RWE=−() T T (14) Sim Ref RW E = (T T ) (14) Sim Re f where TRef is the reference time of flight. In the following experiment, TSim (ND = 46.5) is where T is the reference time of flight. In the following experiment, T (N = 46.5) is Ref Sim D recognized as TRef because the experimental ND varies from 1.3 to 46.5, and RWE is almost recognized as T because the experimental N varies from 1.3 to 46.5, and RWE is almost Ref D unchanged when ND becomes large. Figure 3 depicts the simulation between RWE, the unchanged when N becomes large. Figure 3 depicts the simulation between RWE, the detection probability and ND with a different photon threshold k. Figure 3a shows that the detection probability and N with a different photon threshold k. Figure 3a shows that the RWE is large when ND varies from 1 and 10, while the rate of change in RWE is small RWE is large when N varies from 1 and 10, while the rate of change in RWE is small when when ND becomes larger than 10. We also find that setting a higher threshold fails to im- N becomes larger than 10. We also find that setting a higher threshold fails to improve prove measurement accuracy, although it may help reduce false alarms and improve sys- measurement accuracy, although it may help reduce false alarms and improve system tem precision. In addition, as seen in Figure 3b, setting the photon threshold may lower precision. In addition, as seen in Figure 3b, setting the photon threshold may lower the Photonics 2022, 9, x FOR PEER REVIEW 6 of 15 the detection probability when few photons are received, as the threshold may filter out detection probability when few photons are received, as the threshold may filter out some some signal photons, thus causing a loss in detection. signal photons, thus causing a loss in detection. (a) (b) Figure 3. (a) Simulation result between RWE and mean number of fired pixels with different thresh- Figure 3. (a) Simulation result between RWE and mean number of fired pixels with different thresh- olds. (b) Detection probability between fired pixels with different thresholds. olds. (b) Detection probability between fired pixels with different thresholds. 3. Experiments and Analysis Figure 4 shows the system structure diagram and the experimental platform. The MCU generates the trigger signal divided into two channels. One channel triggers the laser driver to generate the pulsed laser, and the other one is transmitted to the oscillo- scope. The emitted laser with a certain divergence angle is diffusely reflected by the flat target, and, finally, only the reflected light in a specific direction is received by the detec- tor. Then, the SiPM signal is transmitted to the oscilloscope, and functions are utilized to measure the time interval between the trigger and the echo signal. A multimode semicon- ductor fiber laser (LSPLD905-50W) with a wavelength center of 905 nm is used as the light source. The FWHM of the laser pulse is 2.40 ns, and the peak power is adjustable within 50 W. The laser pulse emitted by the fiber passes through the collimating lens and is re- flected by the target. Thereafter, it hits the detector through the attenuator, the receiving lens and the 905 nm narrowband filter. The analog SiPM detector used is the MPPC mod- ule (C14193-1325SA, Hamamatsu) with 2668 Si-APD pixel SiPM (S13720-1325CS, Hama- matsu) inside. The PDE (photon detection efficiency) of 7% is used for photons with a wavelength of 905 nm. The oscilloscope used is the Infinium (DSO90604A, Angilent) with a 6 GHz bandwidth and 20 GSa/s, which means that the time resolution is 50 ps. The power meter used is PM101 from Thorlabs. (a) Photonics 2022, 9, x FOR PEER REVIEW 6 of 15 Photonics 2022, 9, 24 6 of 15 3. Experiments and Analysis Figure 4 shows the system structure diagram and the experimental platform. The MCU generates the trigger signal divided into two channels. One channel triggers the laser driver to generate the pulsed laser, and the other one is transmitted to the oscilloscope. The emitted laser with a certain divergence angle is diffusely reflected by the flat target, and, finally, only the reflected light in a specific direction is received by the detector. Then, the SiPM signal is transmitted to the oscilloscope, and functions are utilized to measure the time interval between the trigger and the echo signal. A multimode semiconductor fiber laser (LSPLD905-50W) with a wavelength center of 905 nm is used as the light source. The FWHM of the laser pulse is 2.40 ns, and the peak power is adjustable within 50 W. The laser pulse emitted by the fiber passes through the collimating lens and is reflected by the target. Thereafter, it hits the detector through the attenuator, the receiving lens and the 905 nm narr (aowband) ( filter. The analog SiPM detector used b) is the MPPC module (C14193-1325SA, Hamamatsu) with 2668 Si-APD pixel SiPM (S13720-1325CS, Hamamatsu) Figure 3. (a) Simulation result between RWE and mean number of fired pixels with different thresh- inside. The PDE (photon detection efficiency) of 7% is used for photons with a wavelength olds. (b) Detection probability between fired pixels with different thresholds. of 905 nm. The oscilloscope used is the Infinium (DSO90604A, Angilent) with a 6 GHz bandwidth and 20 GSa/s, which means that the time resolution is 50 ps. The power meter 3. Experiments and Analysis used is PM101 from Thorlabs. Figure 4 shows the system structure diagram and the experimental platform. The MCU generates the trigger signal divided into two channels. One channel triggers the 3.1. SiPM Response Measurement laser driver to generate the pulsed laser, and the other one is transmitted to the oscillo- The voltage of the analog SiPM signal is proportional to the number of fired pixels. scope. The emitted laser with a certain divergence angle is diffusely reflected by the flat Figure 5 is obtained by setting the oscilloscope to the glow mode under the condition target, and, finally, only the reflected light in a specific direction is received by the detec- that the echo signal is a few photons. The brighter the color, the higher the frequency tor. Then, the SiPM signal is transmitted to the oscilloscope, and functions are utilized to of the signal voltage distribution in the area. This indicates that the voltage of SiPM is measure the time interval between the trigger and the echo signal. A multimode semicon- discontinuous but discrete. The voltage difference between adjacent bright lines V is Cell ductor fiber laser (LSPLD905-50W) with a wavelength center of 905 nm is used as the light the peak voltage of the signal that a fired pixel generates. The mean number of fired source. The FWHM of the laser pulse is 2.40 ns, and the peak power is adjustable within pixels N , therefore, can be measured through the peak voltage V generated by SiPM D Peak 50 W. The laser pulse emitted by the fiber passes through the collimating lens and is re- as N = V /V . Then, N is calculated based on Equation (8). The background noise D S Peak Cell flected by the target. Thereafter, it hits the detector through the attenuator, the receiving rate N in the experiment is about 0.005 counts/ns. The noise level is measured in the Noise lens and the 905 nm narrowband filter. The analog SiPM detector used is the MPPC mod- environment only with background noise. The statistic function of the oscilloscope can ule (C14193-1325SA, Hamamatsu) with 2668 Si-APD pixel SiPM (S13720-1325CS, Hama- count the number of pulses on the screen. By using this function, we count the mean matsu) inside. The PDE (photon detection efficiency) of 7% is used for photons with a number of false alarm pulses appearing on the screen N. Then, the noise rate is calculated wavelength of 905 nm. The oscilloscope used is the Infinium (DSO90604A, Angilent) with as N/T, where T is the time span of the screen. By adjusting the laser power, the echo a 6 GHz bandwidth and 20 GSa/s, which means that the time resolution is 50 ps. The light is attenuated to the order of a few photons, and the voltage of SiPM is measured by power meter used is PM101 from Thorlabs. the oscilloscope. (a) Figure 4. Cont. Photonics 2022, 9, x FOR PEER REVIEW 7 of 15 Photonics 2022, 9, 24 7 of 15 Photonics 2022, 9, x FOR PEER REVIEW 7 of 15 (b) Figure 4. (a) Schematic of the LiDAR system based on SiPM. (b) Experimental platform. 3.1. SiPM Response Measurement The voltage of the analog SiPM signal is proportional to the number of fired pixels. Figure 5 is obtained by setting the oscilloscope to the glow mode under the condition that the echo signal is a few photons. The brighter the color, the higher the frequency of the signal voltage distribution in the area. This indicates that the voltage of SiPM is discontin- uous but discrete. The voltage difference between adjacent bright lines VCell is the peak voltage of the signal that a fired pixel generates. The mean number of fired pixels ND, therefore, can be measured through the peak voltage VPeak generated by SiPM as ND = VPeak/VCell. Then, NS is calculated based on Equation (8). The background noise rate NNoise in the experiment is about 0.005 counts/ns. The noise level is measured in the environment only with background noise. The statistic function of the oscilloscope can count the num- ber of pulses on the screen. By using this function, we count the mean number of false alarm pulses appearing on the screen N. Then, the noise rate is calculated as N/T, where (b) T is the time span of the screen. By adjusting the laser power, the echo light is attenuated Figure 4. (a) Schematic of the LiDAR system based on SiPM. (b) Experimental platform. Figure 4. (a) Schematic of the LiDAR system based on SiPM. (b) Experimental platform. to the order of a few photons, and the voltage of SiPM is measured by the oscilloscope. 3.1. SiPM Response Measurement The voltage of the analog SiPM signal is proportional to the number of fired pixels. Figure 5 is obtained by setting the oscilloscope to the glow mode under the condition that the echo signal is a few photons. The brighter the color, the higher the frequency of the signal voltage distribution in the area. This indicates that the voltage of SiPM is discontin- uous but discrete. The voltage difference between adjacent bright lines VCell is the peak voltage of the signal that a fired pixel generates. The mean number of fired pixels ND, therefore, can be measured through the peak voltage VPeak generated by SiPM as ND = VPeak/VCell. Then, NS is calculated based on Equation (8). The background noise rate NNoise in the experiment is about 0.005 counts/ns. The noise level is measured in the environment only with background noise. The statistic function of the oscilloscope can count the num- ber of pulses on the screen. By using this function, we count the mean number of false alarm pulses appearing on the screen N. Then, the noise rate is calculated as N/T, where T is the time span of the screen. By adjusting the laser power, the echo light is attenuated to the order of a few photons, and the voltage of SiPM is measured by the oscilloscope. Figure 5. SiPM output signal measured by the oscilloscope in glow mode. Figure 5. SiPM output signal measured by the oscilloscope in glow mode. 3.2. Background Noise Measurement 3.2. Background Noise Measurement As the SiPM’s signal is the summation of each pixel, an appropriate photon threshold can effectively avoid the false alarms caused by ambient photons and dark counts. The threshold voltage determines the number of photons that activate the TDC. Figure 6a shows the triggered time tag distribution when setting the detection threshold to one-photon and three-photon levels in the laboratory environment. Compared with Figure 6a, the false alarm rate of Figure 6b is significantly reduced, as the background noise mainly causes <3 pixels response [28]. Figure 6c shows that the detection precision is positively correlated with the threshold. However, in practice, choosing the threshold photon number creates a trade-off, with a higher threshold improving the SNR while lowering the probability of successful detection [13]. In the following experiment, we set the threshold photon number Figure 5. SiPM output signal measured by the oscilloscope in glow mode. as 3 to avoid most of the false alarms and to guarantee the detection probability. 3.2. Background Noise Measurement Photonics 2022, 9, x FOR PEER REVIEW 8 of 15 As the SiPM’s signal is the summation of each pixel, an appropriate photon threshold can effectively avoid the false alarms caused by ambient photons and dark counts. The threshold voltage determines the number of photons that activate the TDC. Figure 6a shows the triggered time tag distribution when setting the detection threshold to one- photon and three-photon levels in the laboratory environment. Compared with Figure 6a, the false alarm rate of Figure 6b is significantly reduced, as the background noise mainly causes <3 pixels response [28]. Figure 6c shows that the detection precision is positively correlated with the threshold. However, in practice, choosing the threshold photon num- ber creates a trade-off, with a higher threshold improving the SNR while lowering the probability of successful detection [13]. In the following experiment, we set the threshold Photonics 2022, 9, 24 8 of 15 photon number as 3 to avoid most of the false alarms and to guarantee the detection prob- ability. Figure 6. (a,b) Triggered time tag distribution of 3 p.e. and 1 p.e. detection threshold. (c) Relation- Figure 6. (a,b) Triggered time tag distribution of 3 p.e. and 1 p.e. detection threshold. (c) Relationship ship between system precision and threshold. between system precision and threshold. 3.3. 3.3. T TOF OF Mea Measur suremen ementt The SiPM LiDAR using photon threshold detection suffers from RWE when the The SiPM LiDAR using photon threshold detection suffers from RWE when the num- number of received photons fluctuates as seen in Figure 3. Experiments are performed to ber of received photons fluctuates as seen in Figure 3. Experiments are performed to verify verify the effectiveness of the RWE correction method we proposed. the effectiveness of the RWE correction method we proposed. Fourteen groups of distance measurement data are obtained under different echo Fourteen groups of distance measurement data are obtained under different echo signal intensities, with the N varying from 1.30 to 46.45. Here, we adjust the power of the signal intensities, with the ND varying from 1.30 to 46.45. Here, we adjust the power of the semiconductor laser diode to achieve the number fluctuation of the received photons. The semiconductor laser diode to achieve the number fluctuation of the received photons. The experiments are performed for the attenuation rates of 1/1000. The distance between the experiments are performed for the attenuation rates of 1/1000. The distance between the target and the system is 5.61 m, measured by a phase laser rangefinder (GLM-25, Bosch target and the system is 5.61 m, measured by a phase laser rangefinder (GLM-25, Bosch Shanghai, China) with a precision of 1.5 mm. Shanghai, China) with a precision of 1.5 mm. In each group, the fixed threshold with k = 3, the CFD method and the PKD method In each group, the fixed threshold with k = 3, the CFD method and the PKD method are separately utilized to measure the distance to the same target. As seen in Figure 7a, the are separately utilized to measure the distance to the same target. As seen in Figure 7a, three methods are realized through the measurement functions in the oscilloscope, which the three methods are realized through the measurement functions in the oscilloscope, is controlled by python script through VISA COM [29]. The flight time using the threshold which is controlled by python script through VISA COM [29]. The flight time using the method, T , is obtained through the function TVOLT, which measures the time interval TH threshold method, TTH, is obtained through the function TVOLT, which measures the time between the trigger event and the 2.5 p.e. voltage level on the SiPM signal. The flight time using the CFD method, T , is obtained through the function DELTatime, which measures CFD the time interval from the first specified edge on the trigger event to the next specified edge on the SiPM signal. The flight time using the PKD method, T , is obtained through PKD the function TMAX, which measures the first time at which the maximum voltage of the source waveform occurs. The distance is calculated by aggregating 8000 measurements for each method. Meanwhile, within the measurement using the fixed threshold method, the peak voltage of SiPM is measured, and the detection probability P is calculated. The relationship between the peak voltage, detection probability and the power of the laser is shown in Figure 7b. It indicates that the detection probability is positively correlated with the emitted power of the laser. Moreover, the peak voltage of SiPM has an approximately linear relationship with the laser power, which is consistent with Equations (3) and (8). When the optical power continues to increase, the detection probability approaches 1. N D Photonics 2022, 9, x FOR PEER REVIEW 9 of 15 interval between the trigger event and the 2.5 p.e. voltage level on the SiPM signal. The flight time using the CFD method, TCFD, is obtained through the function DELTatime, which measures the time interval from the first specified edge on the trigger event to the next specified edge on the SiPM signal. The flight time using the PKD method, TPKD, is obtained through the function TMAX, which measures the first time at which the maxi- mum voltage of the source waveform occurs. The distance is calculated by aggregating 8000 measurements for each method. Meanwhile, within the measurement using the fixed threshold method, the peak voltage of SiPM is measured, and the detection probability PD is calculated. The relationship between the peak voltage, detection probability and the power of the laser is shown in Figure 7b. It indicates that the detection probability is pos- Photonics 2022, 9, 24 9 of 15 itively correlated with the emitted power of the laser. Moreover, the peak voltage of SiPM has an approximately linear relationship with the laser power, which is consistent with Equations (3) and (8). When the optical power continues to increase, the detection proba- is calculated through the peak voltage of SiPM as N = V /V . Then, N is obtained D Peak Cell S bility approaches 1. ND is calculated through the peak voltage of SiPM as ND = VPeak/VCell. through Equation (8). Then, NS is obtained through Equation (8). (a) (b) (c) Figure 7. (a) The oscilloscope waveform of time-of-flight measurement with the markers of the Figure 7. (a) The oscilloscope waveform of time-of-flight measurement with the markers of the measured time using fixed threshold, CFD and PKD methods. (b) Peak voltage of SiPM and detec- measured time using fixed threshold, CFD and PKD methods. (b) Peak voltage of SiPM and detection tion probability versus optical power. (c) Schematic diagram of range walk error restraint. probability versus optical power. (c) Schematic diagram of range walk error restraint. After obtaining the value of ND, a theoretical RWE based on Equation (14) is calcu- After obtaining the value of N , a theoretical RWE based on Equation (14) is calculated lated with TRef = TSim (ND = 46.5). Figure 7c shows the overall RWE correction process. Fig- with T = T (N = 46.5). Figure 7c shows the overall RWE correction process. Figure 8 Ref Sim ure 8 shows the experiment and simulation results of the 14 groups. The results indicate shows the experiment and simulation results of the 14 groups. The results indicate that that the RWE between ND = 1.30 and ND = 46.45 is 36.57 cm using the fixed threshold the RWE between N = 1.30 and N = 46.45 is 36.57 cm using the fixed threshold method. D D As the triggered time of N = 46.45 is chosen as the reference time, the measurements with a weaker echo signal have a longer triggered time. Therefore, the RWE is positive using the fixed threshold method. By subtracting the simulation results from the experimental results using the fixed threshold method, the RWE is reduced as presented in Figure 9c. The mean RWE after correction is 1.95 cm compared with 7.91 cm using the CFD method and 18.91 cm using the PKD method. As seen in Table 1 below, the proposed RWE correction method shows better performance in improving system accuracy compared with the CFD method and the PKD method. To verify that the proposed technique is valid with different types of SiPMs, the Appendix A includes a set of TOF measurement experiments using the latest detector with high PDE. Photonics 2022, 9, x FOR PEER REVIEW 10 of 15 method. As the triggered time of ND = 46.45 is chosen as the reference time, the measure- ments with a weaker echo signal have a longer triggered time. Therefore, the RWE is pos- itive using the fixed threshold method. By subtracting the simulation results from the ex- perimental results using the fixed threshold method, the RWE is reduced as presented in Figure 9c. The mean RWE after correction is 1.95 cm compared with 7.91 cm using the CFD method and 18.91 cm using the PKD method. As seen in Table 1 below, the proposed RWE correction method shows better performance in improving system accuracy com- pared with the CFD method and the PKD method. To verify that the proposed technique is valid with different types of SiPMs, the Appendix A includes a set of TOF measurement experiments using the latest detector with high PDE. Table 1. Accuracy result and precision of different echo signal intensities with 100 repetitive meas- urements (part of). ND 2.88 4.79 7.98 18.14 33.65 ETH (cm) 31.52 26.86 16.45 5.65 0.97 ECFD (cm) 15.74 17.21 12.80 5.83 0.79 EPKD (cm) −19.54 −13.44 −15.83 −16.23 −9.02 ETHC (cm) 4.34 5.30 0.51 −1.85 −0.91 Precision (cm) 1.19 1.11 1.31 0.27 0.21 Photonics 2022, 9, 24 10 of 15 ETH: RWE of threshold method. ECFD: RWE of CFD method. EPKD: RWE of PKD method. ETHC: Cor- rected RWE. (a) (b) Figure 8. (a) Simulation and experiment results of detection probability versus mean number of Figure 8. (a) Simulation and experiment results of detection probability versus mean number of fired fired pixels based on Equation (13). (b) ND derived from peak voltage and detection probability ver- pixels based on Equation (13). (b) N derived from peak voltage and detection probability versus sus optical power. optical power. Table 1. Accuracy result and precision of different echo signal intensities with 100 repetitive measure- ments (part of). N 2.88 4.79 7.98 18.14 33.65 E (cm) 31.52 26.86 16.45 5.65 0.97 TH E (cm) 15.74 17.21 12.80 5.83 0.79 CFD E (cm) 19.54 13.44 15.83 16.23 9.02 PKD E (cm) 4.34 5.30 0.51 1.85 0.91 THC Precision (cm) 1.19 1.11 1.31 0.27 0.21 E : RWE of threshold method. E : RWE of CFD method. E : RWE of PKD method. E : Corrected RWE. TH CFD PKD THC The precision in Table 1 shows that the precision of N with aggregate 100 repetitive measurements is within 0.25 counts, and the ranging precision with aggregate 100 repetitive measurements is within 1.4 cm at the few-photon level. The results indicate that the method does not require thousands of repeated measurements. Therefore, the method is available in wide dynamic LIDAR applications, such as autonomous driving and fast- moving target detection. Moreover, as the peak voltage measurement requires an extra circuit, considering the simplicity of the LiDAR system, there is another method to calculate N with no extra system design required. According to Equation (12), the whole detection probability of one shot with the threshold photon number k can be calculated. The detection probability P is only related to the intensity of echo signal photons under certain instrument parameters. The simulation between N and P is performed based on Equation (12) with the same D D parameters of the above experiments. Meanwhile, we calculate the P of the above 14 group measurements using the fixed threshold method. P is the ratio of the number of successful detections to the number of total measurements. The result is as shown in Figure 5. The black dots are the experiment results. It indicates that the simulation can predict the relationship between N and P well. Then, N can be estimated by the simulation D D D function we obtained in Figure 8a. The N obtained from SiPM’s peak voltage is expressed as N , and the N obtained from the detection probability based on Equation (12) is N . DP D DR Figure 8a depicts the trend of N and N with the optical power. As seen in Figure 8b, DP DR when the N is less than 12, the result of N is close to that of N . However, when N D DR DP D becomes larger, the P approaches 1 according to Figure 8a. The DN /DP in Figure 8a D DR D becomes very large, which means that the error when measuring P may induce a huge error when estimating N . Considering the restriction of the number of times repeated DR Photonics 2022, 9, 24 11 of 15 measurements can be taken, the experimental P may deviate from the true P ; thus, it is D D Photonics 2022, 9, x FOR PEER REVIEW 11 of 15 not appropriate to estimate N when the number of echo photons is large. As a result, N D D can be calculated through P in few-photon detection. (a) (b) (c) Figure 9. (a) Triggered time tag distribution using fixed threshold method with different mean num- Figure 9. (a) Triggered time tag distribution using fixed threshold method with different mean ber of fired pixels, (left), and triggered time tag distribution using three methods with ND number of fired pixels, N (left), and triggered time tag distribution using three methods with ND = 2.48 (right). (b) Experiment and simulation result of RWE. The dashed line is the simulation N = 2.48 (right). (b) Experiment and simulation result of RWE. The dashed line is the simulation result; the blue triangles are the experiment results of fixed threshold method. (c) Correction results result; the blue triangles are the experiment results of fixed threshold method. (c) Correction results of the three methods. The blue triangles are the error after applying the correction method; the black of the three methods. The blue triangles are the error after applying the correction method; the black squares are the CFD results; the red spots are the PKD results. squares are the CFD results; the red spots are the PKD results. The precision in Table 1 shows that the precision of ND with aggregate 100 repetitive 4. Conclusions measurements is within 0.25 counts, and the ranging precision with aggregate 100 repeti- For the challenge of RWE, which SiPM LiDAR encounters in wide-dynamic-range tive measurements is within 1.4 cm at the few-photon level. The results indicate that the measurement applications, we propose an RWE correction method based on the Poisson method does not require thousands of repeated measurements. Therefore, the method is distribution of arrival photons and the photon number resolvability of SiPM. The only available in wide dynamic LIDAR applications, such as autonomous driving and fast- input parameter of this method is the mean number of fired pixels, N , which can be moving target detection. obtained from the signal of SiPM or the detection probability. Experiments are carried out Moreover, as the peak voltage measurement requires an extra circuit, considering the simplicity of the LiDAR system, there is another method to calculate ND with no extra system design required. According to Equation (12), the whole detection probability of one shot with the threshold photon number k can be calculated. The detection probabil- ity PD is only related to the intensity of echo signal photons under certain instrument Photonics 2022, 9, 24 12 of 15 to verify our method. According to the experiment result, the RWE is almost 37 cm when N varies from 1.3 to 46.5. The mean residual RWE after complementing the correction method is 1.95 cm, which shows better performance compared with the CFD method and the PKD method. The ranging precision with aggregate 100 repetitive measurements is within 1.2 cm at the few-photon level. The above result indicates that the method enables RWE correction in photon threshold LiDAR systems based on SiPM. In practice, the peak voltage measurement for the picosecond width pulse generated from SiPM requires a complex system design. Considering the simplicity of the correction method, the method based on detection probability to estimate the light intensity requires no additional measuring system and can improve the accuracy within 4 cm when the mean number of fired pixels fluctuates from 1.31 to 11.22. Therefore, this method may be applicable in LiDAR for detection at the few-photon level. Author Contributions: Conceptualization, R.Y. and Y.T.; software, R.Y.; validation, R.Y., Y.T. and Z.F.; formal analysis, R.Y.; investigation, K.L., J.Q.; resources, K.L., Z.F.; data curation, R.Y.; writing— original draft preparation, R.Y.; writing—review and editing, R.Y., K.L.; visualization, R.Y.; supervi- sion, K.L., J.Q.; project administration, R.Y.; funding acquisition, K.L., J.Q. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Fudan University and Changchun Institute of Optics Joint Foundation: Grant FC2017-002; Fudan University Innovation Foundation: Grant Y18112. Conflicts of Interest: The authors declare no conflict of interest. Appendix A TOF Measurement with 9% PDE (Photon Detection Efficiency) SiPM To verify that the proposed technique is valid with different types of SiPMs, the Appendix Section includes a set of TOF measurement experiments using the latest detector with high PDE. We change the detector from SiPM (S13720-1325SA, HAMAMATSU) to SiPM (S15639-1325PS, HAMAMATSU, SiPM module from Gudatek, Hefei, China). The comparison of the two types of SiPM is as shown in Table A1, and their photos are included in Figure A1a. Table A1. Comparison of the two types of SiPM. Parameter S15639-1325PS S13720-1325CS PDE (905 nm) 9% 7% Dark count rate 0.7 MHz 0.5 MHz Crosstalk probability 6% 6% Rise time 500 ps 500 ps 6 6 Gain 1.7  10 1.1  10 Number of microcells 2120 2668 The experimental platform is as shown in Figure A1b. The schematic of the system and the experimental devices is introduced in the manuscript. Eight groups of distance measurement data are obtained under different echo signal intensities, with the N varying from 1.1 to 16.6. Here, we adjust the power of the semiconductor laser diode from 0.63 W to 13.83 W to achieve the number fluctuation of received photons. Meanwhile, within the mea- surement using the fixed threshold method, the peak voltage of SiPM is measured as seen in Figure A2a. After obtaining the value of N , a theoretical RWE based on Equation (14) is calculated, with N = 2120, h = 9% and T = T (N = 16.68). Figure A2b shows the q D Cell Ref Sim experiment and simulation results of RWE. Photonics 2022, 9, x FOR PEER REVIEW 13 of 15 Table A1. Comparison of the two types of SiPM. Parameter S15639-1325PS S13720-1325CS PDE (905 nm) 9% 7% Dark count rate 0.7 MHz 0.5 MHz Crosstalk probability 6% 6% Rise time 500 ps 500 ps 6 6 Photonics 2022, 9, 24 13 of 15 Gain 1.7 × 10 1.1 × 10 Number of microcells 2120 2668 (a) (b) Photonics 2022, 9, x FOR PEER REVIEW 14 of 15 Figure A1. (a) The two SiPM modules with S13720-1325CS (left) and S15639-1325PS (right) inside. Figure A1. (a) The two SiPM modules with S13720-1325CS (left) and S15639-1325PS (right) inside. (b) Experimental platform. (b) Experimental platform. The experimental platform is as shown in Figure A1b. The schematic of the system and the experimental devices is introduced in the manuscript. Eight groups of distance measurement data are obtained under different echo signal intensities, with the ND vary- ing from 1.1 to 16.6. Here, we adjust the power of the semiconductor laser diode from 0.63 W to 13.83 W to achieve the number fluctuation of received photons. Meanwhile, within the measurement using the fixed threshold method, the peak voltage of SiPM is measured as seen in Figure A2a. After obtaining the value of ND, a theoretical RWE based on Equa- tion (14) is calculated, with NCell = 2120, ηq = 9% and TRef = TSim (ND = 16.68). Figure A2b shows the experiment and simulation results of RWE. (a) (b) Figure A2. (a) Peak voltage of SiPM distribution at ND = 1.91. (b) Experiment and simulation results Figure A2. (a) Peak voltage of SiPM distribution at N = 1.91. (b) Experiment and simulation results of RWE. of RWE. In our current work, the proposed RWE models take the PDE of the detector into In our current work, the proposed RWE models take the PDE of the detector into account according to Equations (5) and (6). The result of the TOF measurements using the account according to Equations (5) and (6). The result of the TOF measurements using the 9% PDE SiPM in Table A2 shows that the method reduces the RWE from 29.07 cm to the 9% PDE SiPM in Table A2 shows that the method reduces the RWE from 29.07 cm to the mean mean d deviation eviation o off 2 2.33 .33 c cm, m, w with ith t the he n number umber of of detec detected ted photons fluctuating photons fluctuating fro from m 1.1 to 1.1 to 16.6. 16.6. Th Ther erefo eforr e, e, the the method method is is proved proved to to be be ef effective fective for for reducing reducing the the RWE RW emer E emerging ging in the in SiPM LiDAR with 9% PDE, and the proposed model can accurately predict the response the SiPM LiDAR with 9% PDE, and the proposed model can accurately predict the re- characteristics sponse characteristics of differ of ent di SiPMs. fferent SiPMs. Table A2. Accuracy results with different echo signal intensities. Table A2. Accuracy results with different echo signal intensities. N 1.13 1.91 4.88 7.72 8.40 11.00 14.22 1.13 1.91 4.88 7.72 8.40 11.00 14.22 E (cm) 29.07 26.00 18.03 11.43 9.81 6.26 2.42 TH E (cm) 29.07 26.00 18.03 11.43 9.81 6.26 2.42 TH E (cm) 4.07 2.86 2.54 2.15 1.84 1.75 1.08 THC E (cm) 4.07 2.86 2.54 2.15 1.84 1.75 1.08 THC E : RWE of threshold method. E : Corrected RWE. TH THC ETH: RWE of threshold method. ETHC: Corrected RWE. References 1. 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A Method of Range Walk Error Correction in SiPM LiDAR with Photon Threshold Detection

Photonics , Volume 9 (1) – Jan 1, 2022

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Abstract

hv photonics Article A Method of Range Walk Error Correction in SiPM LiDAR with Photon Threshold Detection 1 1 2 1 , 1 Runze Yang , Yumei Tang , Zeyu Fu , Jian Qiu * and Kefu Liu School of Information Science and Engineering, Fudan University, 220 Handan Road, Shanghai 200433, China; 19210720059@fudan.edu.cn (R.Y.); 19110720076@fudan.edu.cn (Y.T.); kfliu@fudan.edu.cn (K.L.) Academy for Engineering and Technology, Fudan University, 220 Handan Road, Shanghai 200433, China; 20210860034@fudan.edu.cn * Correspondence: jqiu@fudan.edu.cn Abstract: A silicon photomultiplier (SiPM) LiDAR with photon threshold detection can achieve high dynamic performance. However, the number fluctuations of echo signal photons lead to the range walk error (RWE) in SiPM LIDARs. This paper derives the RWE model of SiPM LiDAR by using the LiDAR equation and statistical property of SiPM’s response. Based on the LiDAR system parameters and the echo signal intensity, which is obtained through the SiPM’s photon-number- resolving capability, the RWE is calculated through the proposed model. After that, we carry out experiments to verify its effectiveness. The result shows that the method reduces the RWE in TOF measurements using photon threshold detection from 36.57 cm to the mean deviation of 1.95 cm, with the number of detected photons fluctuating from 1.3 to 46.5. Keywords: LiDAR; SiPM; TOF; range walk error; threshold detection 1. Introduction A single-photon avalanche diode (SPAD) has been widely used in remote laser ranging Citation: Yang, R.; Tang, Y.; Fu, Z.; systems due to its single-photon detection sensitivity and high time resolution [1–3]. SPAD Qiu, J.; Liu, K. A Method of Range functions as a binary device with a deadtime [4]. The time-to-digital converter (TDC) for Walk Error Correction in SiPM TOF computation can only timestamp the first detected photon in a SPAD pixel. Therefore, LiDAR with Photon Threshold a SPAD LiDAR system encounters the problem of a high false alarm rate, which becomes Detection. Photonics 2022, 9, 24. https://doi.org/10.3390/ prominent when the background light is strong. To solve this problem, References [5–7] photonics9010024 utilize the time-correlated single-photon counting (TCSPC) technique to calculate the flight time by counting the trigger time distribution. However, TCSPC requires thousands of Received: 17 November 2021 successive measurements to improve the signal-to-noise ratio (SNR) and calculate the time Accepted: 23 December 2021 of flight (TOF). The long acquisition time restricts the application of SPAD LiDAR in areas Published: 1 January 2022 such as autonomous driving, given the requirements for high frame rate imaging and Publisher’s Note: MDPI stays neutral fast-moving target tracking [8]. with regard to jurisdictional claims in SiPMs are large photosensitive area detectors that consist of parallel SPAD pixels, and published maps and institutional affil- each pixel has an independent quenching circuit [9]. In analog SiPM, the summation of iations. each pixel’s response constitutes the analog output signal of SiPM [10–12]. As a result, the signal of SiPM is proportional to the number of incident photons. In digital SiPM, the SPAD frontend is an active circuit that sets a fixed and well-defined deadtime after each ignition and provides a digital pulse when a photon is detected. The output of all microcells is Copyright: © 2022 by the authors. combined through logic gates. The digital output is directly the number of fired cells [11,12]. Licensee MDPI, Basel, Switzerland. Regarding SiPM’s photon-number-resolving capability, compared to SPAD LiDAR, SiPM This article is an open access article LiDAR can utilize the photon threshold to filter out low photon numbers. The threshold distributed under the terms and can lower the false alarm rate and improve the SNR of the LiDAR system [13]. In addition conditions of the Creative Commons to this, setting the photon threshold helps suppress the ranging uncertainty caused by the Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ pulse width of the laser. These advantages help the SiPM LiDAR system achieve a better 4.0/). dynamic range. Photonics 2022, 9, 24. https://doi.org/10.3390/photonics9010024 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 24 2 of 15 For wide-dynamic-range applications, SiPM LiDAR requires high accuracy when measuring targets with extreme variations in reflectivity and angles. Moreover, the RWE caused by the number fluctuation of the received photons is the dominating influential factor for accurate ranging. References [14–17] establish several methods to reduce RWE emerging in SPAD LiDAR systems. Reference [18] proposes a correction method for RWE in LiDAR with a photomultiplier tube. References [19–21] discuss the accuracy of linear APD LiDAR using a fixed threshold discriminator, the constant fraction discriminator (CFD) method and the peak detection method (PKD). However, to date, there have been few related studies focusing on the RWE reduction in SiPM LiDAR with threshold detection. In this article, an RWE correction method of an SiPM LiDAR system with threshold detection is proposed. First, we establish a theoretical RWE model containing the response characteristics of SiPM and the photon threshold method. Thereafter, the RWE is calculated through the above model. In the proposed RWE model using the photon threshold method, we derive the relationship between RWE and the number of detected photons, which can be obtained via either analog SiPM or digital SiPM. Therefore, the proposed method is available in the two types of SiPM LiDAR systems. Groups of experiments with different echo light intensities are performed to measure the distance to a fixed target. In each group, we conduct the correction method to offset the RWE emerging in the result of the fixed photon threshold detection. Meanwhile, the CFD and PKD methods are utilized for comparison with the proposed method. Finally, we develop a method that utilizes the detection probability of the photon threshold detection to estimate the mean number of detected photons instead of using the signal of SiPM, which may simplify the system. As proved by the experiments, this method can work at the few-photon level. 2. Theoretical Analysis 2.1. Laser Propagation Equation The D-TOF LiDAR system delivers a narrow-width laser pulse to the target. The power of an emitted Gaussian pulse [22] can be described with the average single-pulse energy E as follows: 2s P (t) = E p e (1) Emit T 2ps where P (t) is the power of the emitted pulse, and s is the emitted pulse width. When the Emit f pulse is scattered from a Lambertian target at distance R, the received number of photons, N (t), can be derived from Equation (1) [16] as Photon 1 2 2s N t = N p e + N (2) ( ) S Noise Photon 2ps where N is the background noise rate, and s is the received pulse width. We assume Noise s  s in this manuscript because the transmitted pulse width is in the order of nanosec- S f onds [23]; N is the mean number of received signal photons, which can be calculated as [24] E h h h r A T T R R N = (3) p R hv where h is the transmission of the transmitter; h is the transmission of the receiver optic; T R h is the one-way transmission of the atmosphere; r is the reflectivity of the target; A is A R the area of the aperture of the receiver; h is the Plank constant; and v is the frequency of the laser. Photonics 2022, 9, 24 3 of 15 2.2. SiPM Response Model Assuming that the number of arriving photons on each pixel is evenly distributed, then, in any time interval (t , t ), the mean number of photons hitting each pixel, N, can be 1 2 derived from Equation (2) as N (t)dt Photon N t , t = (4) ( ) 1 2 Cell where N is the number of pixels in SiPM. Poisson distribution is an appropriate model Cell to describe the response of a pixel [25]. The probability that a pixel responds in the ith time bin is derived from Equation (4) [26]: N(tbin(i1),tbini)h P (i 1, i) = 1 e (5) D_bin where h is the photon detection efficiency of SiPM; i is an integer larger than 0; and t is q bin the length of a time bin, which corresponds to the timing resolution of the TDC. Therefore, the mean number of fired pixels in one measurement, N , can be calculated as N h Total ( ) Cell N = N 1 e (6) D Cell here, 0.5T Signal N = N (t)dt = N + N  T (7) S Noise Total Photon Signal 0.5T Signal Therefore, N can also be calculated through Equations (6) and (7) as N N Cell Cell N =  ln N  T (8) S Noise Signal h N N Cell D where N is the number of arrived photons (including ambient noise) in a signal dura- Total tion. T is the effective signal duration, usually T > 6s . As seen in Equation (3), Signal Signal environment parameters, such as the one-way transmission of the atmosphere and the reflectivity of the target, are needed to calculate Ns. However, according to Equation (8), we can estimate N by the number of fired pixels, which can be obtained from the signal of Photonics 2022, 9, x FOR PEER REVIEW 4 of 15 SiPM. N has a linear relation with Ns when N >> N as seen in Figure 1. D Cell D Figure 1. The mean number of fired pixels, N , versus the mean number of received photons, N , Figure 1. The mean number of fired pixels, N D D, versus the mean number of received photons, N SS, with different number of pixels in SiPM, N . with different number of pixels in SiPM, NCell. Cell Considering that the response of each pixel is independent and that the probability of k pixel responses in a time interval PCell obeys the binomial distribution [25], then the probability is derived from Equation (5) as NCell Nk −  Cell Pi−= 1,i ,k P i−1,i 1−P i−1,i () () () () () (9) Cell  D__ bin D bin  In fact, if more than one photon hits a pixel, only one can be detected, which causes non-linear loss. In Equation (9), the nonlinear loss is not considered, because we limited our signal to a few photons where nonlinear loss has a low effect on the results [13]. This is due to the relatively low photon number compared to the number of pixels, which is about two orders of magnitude less in the experiment. For light detection at the few-photon level, the number of fired pixels is very small compared to thousands or even tens of thousands of pixels in the SiPM. Meanwhile, the probability of detecting a photon in a single pixel is minimal; therefore, Equation (9) can be approximated as a Poisson distribution [27]: −⋅ NP () i−1,i Cell D_ bin NP⋅−i 1,i ⋅e () () Cell D_ bin (10) Pi−= 1,i ,k () () Cell k! In the fixed threshold detection method for APD LiDAR, the arrival time of the light is determined when the voltage of the detector reaches a given voltage amplitude. The method for the SiPM LiDAR is similar. After the light is detected by SiPM, the photon number is then thresholded by a one-bit comparator, which stops the timer when the volt- age of SiPM, caused by the detection of the arriving photons, exceeds the given threshold. The threshold detection probability in the ith time bin P(i) is calculated by summing the probability of all possible events that the accumulative number of fired pixels exceeds as the threshold photon number k in the ith time bin. Therefore, P(i) can be derived from Equation (10) as kk −−11u−   Pi ()=− P (0,i 1),u⋅ 1− P () i−1,i ,v () () (11)  Cell Cell  uv ==00  where u is the number of fired pixels before the ith time bin, and v is the fired pixels within the ith time bin. The event where the accumulative number of fired pixels exceeds k in the ith time bin must satisfy that u + v ≥ k and v ≥ 1. Then, the whole detection probability PD of one shot can be obtained from Equation (11) as Photonics 2022, 9, 24 4 of 15 Considering that the response of each pixel is independent and that the probability of k pixel responses in a time interval P obeys the binomial distribution [25], then the Cell probability is derived from Equation (5) as N k Cell k Cell P ((i 1, i), k) = P (i 1, i)(1 P (i 1, i)) (9) Cell D_bin D_bin In fact, if more than one photon hits a pixel, only one can be detected, which causes non-linear loss. In Equation (9), the nonlinear loss is not considered, because we limited our signal to a few photons where nonlinear loss has a low effect on the results [13]. This is due to the relatively low photon number compared to the number of pixels, which is about two orders of magnitude less in the experiment. For light detection at the few-photon level, the number of fired pixels is very small compared to thousands or even tens of thousands of pixels in the SiPM. Meanwhile, the probability of detecting a photon in a single pixel is minimal; therefore, Equation (9) can be approximated as a Poisson distribution [27]: N P (i1,i) Cell D_bin (N  P (i 1, i))  e Cell D_bin P i 1, i , k = (10) (( ) ) Cell k! In the fixed threshold detection method for APD LiDAR, the arrival time of the light is determined when the voltage of the detector reaches a given voltage amplitude. The method for the SiPM LiDAR is similar. After the light is detected by SiPM, the photon number is then thresholded by a one-bit comparator, which stops the timer when the voltage of SiPM, caused by the detection of the arriving photons, exceeds the given threshold. The threshold detection probability in the ith time bin P(i) is calculated by summing the probability of all possible events that the accumulative number of fired pixels exceeds as the threshold photon number k in the ith time bin. Therefore, P(i) can be derived from Equation (10) as !! k1 ku1 P(i) = P ((0, i 1), u) 1 P ((i 1, i), v) (11) å Cell å Cell u=0 v=0 where u is the number of fired pixels before the ith time bin, and v is the fired pixels within the ith time bin. The event where the accumulative number of fired pixels exceeds k in the ith time bin must satisfy that u + v  k and v  1. Then, the whole detection probability P of one shot can be obtained from Equation (11) as P = P(i) (12) D å signalbins where signalbins is a collection of intervals (0.5T , 0.5T ) divided into cells with a Signal Signal length of t . bin Figure 2 is the simulation result based on Equation (11). The time bin is of the interval (0.5T , 0.5T ). The origin of the time axis is the center of the echo signal. The Signal Signal photon threshold k = 3. The s is 1.02 ns, the number of pixels of SiPM is N = 2688, S Cell and the mean number of fired pixels is N = 7, 5, 3, 2, 1. Figure 2a shows the detection probability based on Equation (11) under different echo light intensities. It indicates that, as the intensity of the echo decreases, the center of the detection probability distribution shifts to the right, and the full width at half maximum (FWHM) increases. Figure 2b shows the detection probability under different laser pulses (N = 7, FWHM = 1, 2, 3, 5 ns). Figure 2b indicates that, as the FWHM increases, the probability distribution center shifts to the right, and the ranging uncertainty increases. The centroid algorithm can calculate the triggered time using the following equation: Photonics 2022, 9, x FOR PEER REVIEW 5 of 15 PP = ()i D  (12) signalbins where signalbins is a collection of intervals (−0.5TSignal, 0.5TSignal) divided into cells with a length of τbin. Figure 2 is the simulation result based on Equation (11). The time bin is of the interval (−0.5TSignal, 0.5TSignal). The origin of the time axis is the center of the echo signal. The photon threshold k = 3. The σS is 1.02 ns, the number of pixels of SiPM is NCell = 2688, and the mean number of fired pixels is ND = 7, 5, 3, 2, 1. Figure 2a shows the detection probability based on Equation (11) under different echo light intensities. It indicates that, as the intensity of the echo decreases, the center of the detection probability distribution shifts to the right, and the full width at half maximum (FWHM) increases. Figure 2b shows the detection probability under different laser pulses (ND = 7, FWHM = 1, 2, 3, 5 ns). Figure 2b indicates that, as the FWHM increases, the probability distribution center shifts to the right, and the Photonics 2022, 9, 24 5 of 15 ranging uncertainty increases. The centroid algorithm can calculate the triggered time us- ing the following equation: Pi()⋅⋅i τ  bin P(i) i t signalbins bin T = (13) Sim signalbins Pi () T =  (13) Sim signalbins P(i) signalbins (a) (b) Figure 2. Detection probability under different circumstances: (a) echo light intensities and (b) Figure 2. Detection probability under different circumstances: (a) echo light intensities and (b) FWHM FWHM of the laser pulse. of the laser pulse. Therefore, the range walk error can be calculated using the following equation: Therefore, the range walk error can be calculated using the following equation: RWE=−() T T (14) Sim Ref RW E = (T T ) (14) Sim Re f where TRef is the reference time of flight. In the following experiment, TSim (ND = 46.5) is where T is the reference time of flight. In the following experiment, T (N = 46.5) is Ref Sim D recognized as TRef because the experimental ND varies from 1.3 to 46.5, and RWE is almost recognized as T because the experimental N varies from 1.3 to 46.5, and RWE is almost Ref D unchanged when ND becomes large. Figure 3 depicts the simulation between RWE, the unchanged when N becomes large. Figure 3 depicts the simulation between RWE, the detection probability and ND with a different photon threshold k. Figure 3a shows that the detection probability and N with a different photon threshold k. Figure 3a shows that the RWE is large when ND varies from 1 and 10, while the rate of change in RWE is small RWE is large when N varies from 1 and 10, while the rate of change in RWE is small when when ND becomes larger than 10. We also find that setting a higher threshold fails to im- N becomes larger than 10. We also find that setting a higher threshold fails to improve prove measurement accuracy, although it may help reduce false alarms and improve sys- measurement accuracy, although it may help reduce false alarms and improve system tem precision. In addition, as seen in Figure 3b, setting the photon threshold may lower precision. In addition, as seen in Figure 3b, setting the photon threshold may lower the Photonics 2022, 9, x FOR PEER REVIEW 6 of 15 the detection probability when few photons are received, as the threshold may filter out detection probability when few photons are received, as the threshold may filter out some some signal photons, thus causing a loss in detection. signal photons, thus causing a loss in detection. (a) (b) Figure 3. (a) Simulation result between RWE and mean number of fired pixels with different thresh- Figure 3. (a) Simulation result between RWE and mean number of fired pixels with different thresh- olds. (b) Detection probability between fired pixels with different thresholds. olds. (b) Detection probability between fired pixels with different thresholds. 3. Experiments and Analysis Figure 4 shows the system structure diagram and the experimental platform. The MCU generates the trigger signal divided into two channels. One channel triggers the laser driver to generate the pulsed laser, and the other one is transmitted to the oscillo- scope. The emitted laser with a certain divergence angle is diffusely reflected by the flat target, and, finally, only the reflected light in a specific direction is received by the detec- tor. Then, the SiPM signal is transmitted to the oscilloscope, and functions are utilized to measure the time interval between the trigger and the echo signal. A multimode semicon- ductor fiber laser (LSPLD905-50W) with a wavelength center of 905 nm is used as the light source. The FWHM of the laser pulse is 2.40 ns, and the peak power is adjustable within 50 W. The laser pulse emitted by the fiber passes through the collimating lens and is re- flected by the target. Thereafter, it hits the detector through the attenuator, the receiving lens and the 905 nm narrowband filter. The analog SiPM detector used is the MPPC mod- ule (C14193-1325SA, Hamamatsu) with 2668 Si-APD pixel SiPM (S13720-1325CS, Hama- matsu) inside. The PDE (photon detection efficiency) of 7% is used for photons with a wavelength of 905 nm. The oscilloscope used is the Infinium (DSO90604A, Angilent) with a 6 GHz bandwidth and 20 GSa/s, which means that the time resolution is 50 ps. The power meter used is PM101 from Thorlabs. (a) Photonics 2022, 9, x FOR PEER REVIEW 6 of 15 Photonics 2022, 9, 24 6 of 15 3. Experiments and Analysis Figure 4 shows the system structure diagram and the experimental platform. The MCU generates the trigger signal divided into two channels. One channel triggers the laser driver to generate the pulsed laser, and the other one is transmitted to the oscilloscope. The emitted laser with a certain divergence angle is diffusely reflected by the flat target, and, finally, only the reflected light in a specific direction is received by the detector. Then, the SiPM signal is transmitted to the oscilloscope, and functions are utilized to measure the time interval between the trigger and the echo signal. A multimode semiconductor fiber laser (LSPLD905-50W) with a wavelength center of 905 nm is used as the light source. The FWHM of the laser pulse is 2.40 ns, and the peak power is adjustable within 50 W. The laser pulse emitted by the fiber passes through the collimating lens and is reflected by the target. Thereafter, it hits the detector through the attenuator, the receiving lens and the 905 nm narr (aowband) ( filter. The analog SiPM detector used b) is the MPPC module (C14193-1325SA, Hamamatsu) with 2668 Si-APD pixel SiPM (S13720-1325CS, Hamamatsu) Figure 3. (a) Simulation result between RWE and mean number of fired pixels with different thresh- inside. The PDE (photon detection efficiency) of 7% is used for photons with a wavelength olds. (b) Detection probability between fired pixels with different thresholds. of 905 nm. The oscilloscope used is the Infinium (DSO90604A, Angilent) with a 6 GHz bandwidth and 20 GSa/s, which means that the time resolution is 50 ps. The power meter 3. Experiments and Analysis used is PM101 from Thorlabs. Figure 4 shows the system structure diagram and the experimental platform. The MCU generates the trigger signal divided into two channels. One channel triggers the 3.1. SiPM Response Measurement laser driver to generate the pulsed laser, and the other one is transmitted to the oscillo- The voltage of the analog SiPM signal is proportional to the number of fired pixels. scope. The emitted laser with a certain divergence angle is diffusely reflected by the flat Figure 5 is obtained by setting the oscilloscope to the glow mode under the condition target, and, finally, only the reflected light in a specific direction is received by the detec- that the echo signal is a few photons. The brighter the color, the higher the frequency tor. Then, the SiPM signal is transmitted to the oscilloscope, and functions are utilized to of the signal voltage distribution in the area. This indicates that the voltage of SiPM is measure the time interval between the trigger and the echo signal. A multimode semicon- discontinuous but discrete. The voltage difference between adjacent bright lines V is Cell ductor fiber laser (LSPLD905-50W) with a wavelength center of 905 nm is used as the light the peak voltage of the signal that a fired pixel generates. The mean number of fired source. The FWHM of the laser pulse is 2.40 ns, and the peak power is adjustable within pixels N , therefore, can be measured through the peak voltage V generated by SiPM D Peak 50 W. The laser pulse emitted by the fiber passes through the collimating lens and is re- as N = V /V . Then, N is calculated based on Equation (8). The background noise D S Peak Cell flected by the target. Thereafter, it hits the detector through the attenuator, the receiving rate N in the experiment is about 0.005 counts/ns. The noise level is measured in the Noise lens and the 905 nm narrowband filter. The analog SiPM detector used is the MPPC mod- environment only with background noise. The statistic function of the oscilloscope can ule (C14193-1325SA, Hamamatsu) with 2668 Si-APD pixel SiPM (S13720-1325CS, Hama- count the number of pulses on the screen. By using this function, we count the mean matsu) inside. The PDE (photon detection efficiency) of 7% is used for photons with a number of false alarm pulses appearing on the screen N. Then, the noise rate is calculated wavelength of 905 nm. The oscilloscope used is the Infinium (DSO90604A, Angilent) with as N/T, where T is the time span of the screen. By adjusting the laser power, the echo a 6 GHz bandwidth and 20 GSa/s, which means that the time resolution is 50 ps. The light is attenuated to the order of a few photons, and the voltage of SiPM is measured by power meter used is PM101 from Thorlabs. the oscilloscope. (a) Figure 4. Cont. Photonics 2022, 9, x FOR PEER REVIEW 7 of 15 Photonics 2022, 9, 24 7 of 15 Photonics 2022, 9, x FOR PEER REVIEW 7 of 15 (b) Figure 4. (a) Schematic of the LiDAR system based on SiPM. (b) Experimental platform. 3.1. SiPM Response Measurement The voltage of the analog SiPM signal is proportional to the number of fired pixels. Figure 5 is obtained by setting the oscilloscope to the glow mode under the condition that the echo signal is a few photons. The brighter the color, the higher the frequency of the signal voltage distribution in the area. This indicates that the voltage of SiPM is discontin- uous but discrete. The voltage difference between adjacent bright lines VCell is the peak voltage of the signal that a fired pixel generates. The mean number of fired pixels ND, therefore, can be measured through the peak voltage VPeak generated by SiPM as ND = VPeak/VCell. Then, NS is calculated based on Equation (8). The background noise rate NNoise in the experiment is about 0.005 counts/ns. The noise level is measured in the environment only with background noise. The statistic function of the oscilloscope can count the num- ber of pulses on the screen. By using this function, we count the mean number of false alarm pulses appearing on the screen N. Then, the noise rate is calculated as N/T, where (b) T is the time span of the screen. By adjusting the laser power, the echo light is attenuated Figure 4. (a) Schematic of the LiDAR system based on SiPM. (b) Experimental platform. Figure 4. (a) Schematic of the LiDAR system based on SiPM. (b) Experimental platform. to the order of a few photons, and the voltage of SiPM is measured by the oscilloscope. 3.1. SiPM Response Measurement The voltage of the analog SiPM signal is proportional to the number of fired pixels. Figure 5 is obtained by setting the oscilloscope to the glow mode under the condition that the echo signal is a few photons. The brighter the color, the higher the frequency of the signal voltage distribution in the area. This indicates that the voltage of SiPM is discontin- uous but discrete. The voltage difference between adjacent bright lines VCell is the peak voltage of the signal that a fired pixel generates. The mean number of fired pixels ND, therefore, can be measured through the peak voltage VPeak generated by SiPM as ND = VPeak/VCell. Then, NS is calculated based on Equation (8). The background noise rate NNoise in the experiment is about 0.005 counts/ns. The noise level is measured in the environment only with background noise. The statistic function of the oscilloscope can count the num- ber of pulses on the screen. By using this function, we count the mean number of false alarm pulses appearing on the screen N. Then, the noise rate is calculated as N/T, where T is the time span of the screen. By adjusting the laser power, the echo light is attenuated to the order of a few photons, and the voltage of SiPM is measured by the oscilloscope. Figure 5. SiPM output signal measured by the oscilloscope in glow mode. Figure 5. SiPM output signal measured by the oscilloscope in glow mode. 3.2. Background Noise Measurement 3.2. Background Noise Measurement As the SiPM’s signal is the summation of each pixel, an appropriate photon threshold can effectively avoid the false alarms caused by ambient photons and dark counts. The threshold voltage determines the number of photons that activate the TDC. Figure 6a shows the triggered time tag distribution when setting the detection threshold to one-photon and three-photon levels in the laboratory environment. Compared with Figure 6a, the false alarm rate of Figure 6b is significantly reduced, as the background noise mainly causes <3 pixels response [28]. Figure 6c shows that the detection precision is positively correlated with the threshold. However, in practice, choosing the threshold photon number creates a trade-off, with a higher threshold improving the SNR while lowering the probability of successful detection [13]. In the following experiment, we set the threshold photon number Figure 5. SiPM output signal measured by the oscilloscope in glow mode. as 3 to avoid most of the false alarms and to guarantee the detection probability. 3.2. Background Noise Measurement Photonics 2022, 9, x FOR PEER REVIEW 8 of 15 As the SiPM’s signal is the summation of each pixel, an appropriate photon threshold can effectively avoid the false alarms caused by ambient photons and dark counts. The threshold voltage determines the number of photons that activate the TDC. Figure 6a shows the triggered time tag distribution when setting the detection threshold to one- photon and three-photon levels in the laboratory environment. Compared with Figure 6a, the false alarm rate of Figure 6b is significantly reduced, as the background noise mainly causes <3 pixels response [28]. Figure 6c shows that the detection precision is positively correlated with the threshold. However, in practice, choosing the threshold photon num- ber creates a trade-off, with a higher threshold improving the SNR while lowering the probability of successful detection [13]. In the following experiment, we set the threshold Photonics 2022, 9, 24 8 of 15 photon number as 3 to avoid most of the false alarms and to guarantee the detection prob- ability. Figure 6. (a,b) Triggered time tag distribution of 3 p.e. and 1 p.e. detection threshold. (c) Relation- Figure 6. (a,b) Triggered time tag distribution of 3 p.e. and 1 p.e. detection threshold. (c) Relationship ship between system precision and threshold. between system precision and threshold. 3.3. 3.3. T TOF OF Mea Measur suremen ementt The SiPM LiDAR using photon threshold detection suffers from RWE when the The SiPM LiDAR using photon threshold detection suffers from RWE when the num- number of received photons fluctuates as seen in Figure 3. Experiments are performed to ber of received photons fluctuates as seen in Figure 3. Experiments are performed to verify verify the effectiveness of the RWE correction method we proposed. the effectiveness of the RWE correction method we proposed. Fourteen groups of distance measurement data are obtained under different echo Fourteen groups of distance measurement data are obtained under different echo signal intensities, with the N varying from 1.30 to 46.45. Here, we adjust the power of the signal intensities, with the ND varying from 1.30 to 46.45. Here, we adjust the power of the semiconductor laser diode to achieve the number fluctuation of the received photons. The semiconductor laser diode to achieve the number fluctuation of the received photons. The experiments are performed for the attenuation rates of 1/1000. The distance between the experiments are performed for the attenuation rates of 1/1000. The distance between the target and the system is 5.61 m, measured by a phase laser rangefinder (GLM-25, Bosch target and the system is 5.61 m, measured by a phase laser rangefinder (GLM-25, Bosch Shanghai, China) with a precision of 1.5 mm. Shanghai, China) with a precision of 1.5 mm. In each group, the fixed threshold with k = 3, the CFD method and the PKD method In each group, the fixed threshold with k = 3, the CFD method and the PKD method are separately utilized to measure the distance to the same target. As seen in Figure 7a, the are separately utilized to measure the distance to the same target. As seen in Figure 7a, three methods are realized through the measurement functions in the oscilloscope, which the three methods are realized through the measurement functions in the oscilloscope, is controlled by python script through VISA COM [29]. The flight time using the threshold which is controlled by python script through VISA COM [29]. The flight time using the method, T , is obtained through the function TVOLT, which measures the time interval TH threshold method, TTH, is obtained through the function TVOLT, which measures the time between the trigger event and the 2.5 p.e. voltage level on the SiPM signal. The flight time using the CFD method, T , is obtained through the function DELTatime, which measures CFD the time interval from the first specified edge on the trigger event to the next specified edge on the SiPM signal. The flight time using the PKD method, T , is obtained through PKD the function TMAX, which measures the first time at which the maximum voltage of the source waveform occurs. The distance is calculated by aggregating 8000 measurements for each method. Meanwhile, within the measurement using the fixed threshold method, the peak voltage of SiPM is measured, and the detection probability P is calculated. The relationship between the peak voltage, detection probability and the power of the laser is shown in Figure 7b. It indicates that the detection probability is positively correlated with the emitted power of the laser. Moreover, the peak voltage of SiPM has an approximately linear relationship with the laser power, which is consistent with Equations (3) and (8). When the optical power continues to increase, the detection probability approaches 1. N D Photonics 2022, 9, x FOR PEER REVIEW 9 of 15 interval between the trigger event and the 2.5 p.e. voltage level on the SiPM signal. The flight time using the CFD method, TCFD, is obtained through the function DELTatime, which measures the time interval from the first specified edge on the trigger event to the next specified edge on the SiPM signal. The flight time using the PKD method, TPKD, is obtained through the function TMAX, which measures the first time at which the maxi- mum voltage of the source waveform occurs. The distance is calculated by aggregating 8000 measurements for each method. Meanwhile, within the measurement using the fixed threshold method, the peak voltage of SiPM is measured, and the detection probability PD is calculated. The relationship between the peak voltage, detection probability and the power of the laser is shown in Figure 7b. It indicates that the detection probability is pos- Photonics 2022, 9, 24 9 of 15 itively correlated with the emitted power of the laser. Moreover, the peak voltage of SiPM has an approximately linear relationship with the laser power, which is consistent with Equations (3) and (8). When the optical power continues to increase, the detection proba- is calculated through the peak voltage of SiPM as N = V /V . Then, N is obtained D Peak Cell S bility approaches 1. ND is calculated through the peak voltage of SiPM as ND = VPeak/VCell. through Equation (8). Then, NS is obtained through Equation (8). (a) (b) (c) Figure 7. (a) The oscilloscope waveform of time-of-flight measurement with the markers of the Figure 7. (a) The oscilloscope waveform of time-of-flight measurement with the markers of the measured time using fixed threshold, CFD and PKD methods. (b) Peak voltage of SiPM and detec- measured time using fixed threshold, CFD and PKD methods. (b) Peak voltage of SiPM and detection tion probability versus optical power. (c) Schematic diagram of range walk error restraint. probability versus optical power. (c) Schematic diagram of range walk error restraint. After obtaining the value of ND, a theoretical RWE based on Equation (14) is calcu- After obtaining the value of N , a theoretical RWE based on Equation (14) is calculated lated with TRef = TSim (ND = 46.5). Figure 7c shows the overall RWE correction process. Fig- with T = T (N = 46.5). Figure 7c shows the overall RWE correction process. Figure 8 Ref Sim ure 8 shows the experiment and simulation results of the 14 groups. The results indicate shows the experiment and simulation results of the 14 groups. The results indicate that that the RWE between ND = 1.30 and ND = 46.45 is 36.57 cm using the fixed threshold the RWE between N = 1.30 and N = 46.45 is 36.57 cm using the fixed threshold method. D D As the triggered time of N = 46.45 is chosen as the reference time, the measurements with a weaker echo signal have a longer triggered time. Therefore, the RWE is positive using the fixed threshold method. By subtracting the simulation results from the experimental results using the fixed threshold method, the RWE is reduced as presented in Figure 9c. The mean RWE after correction is 1.95 cm compared with 7.91 cm using the CFD method and 18.91 cm using the PKD method. As seen in Table 1 below, the proposed RWE correction method shows better performance in improving system accuracy compared with the CFD method and the PKD method. To verify that the proposed technique is valid with different types of SiPMs, the Appendix A includes a set of TOF measurement experiments using the latest detector with high PDE. Photonics 2022, 9, x FOR PEER REVIEW 10 of 15 method. As the triggered time of ND = 46.45 is chosen as the reference time, the measure- ments with a weaker echo signal have a longer triggered time. Therefore, the RWE is pos- itive using the fixed threshold method. By subtracting the simulation results from the ex- perimental results using the fixed threshold method, the RWE is reduced as presented in Figure 9c. The mean RWE after correction is 1.95 cm compared with 7.91 cm using the CFD method and 18.91 cm using the PKD method. As seen in Table 1 below, the proposed RWE correction method shows better performance in improving system accuracy com- pared with the CFD method and the PKD method. To verify that the proposed technique is valid with different types of SiPMs, the Appendix A includes a set of TOF measurement experiments using the latest detector with high PDE. Table 1. Accuracy result and precision of different echo signal intensities with 100 repetitive meas- urements (part of). ND 2.88 4.79 7.98 18.14 33.65 ETH (cm) 31.52 26.86 16.45 5.65 0.97 ECFD (cm) 15.74 17.21 12.80 5.83 0.79 EPKD (cm) −19.54 −13.44 −15.83 −16.23 −9.02 ETHC (cm) 4.34 5.30 0.51 −1.85 −0.91 Precision (cm) 1.19 1.11 1.31 0.27 0.21 Photonics 2022, 9, 24 10 of 15 ETH: RWE of threshold method. ECFD: RWE of CFD method. EPKD: RWE of PKD method. ETHC: Cor- rected RWE. (a) (b) Figure 8. (a) Simulation and experiment results of detection probability versus mean number of Figure 8. (a) Simulation and experiment results of detection probability versus mean number of fired fired pixels based on Equation (13). (b) ND derived from peak voltage and detection probability ver- pixels based on Equation (13). (b) N derived from peak voltage and detection probability versus sus optical power. optical power. Table 1. Accuracy result and precision of different echo signal intensities with 100 repetitive measure- ments (part of). N 2.88 4.79 7.98 18.14 33.65 E (cm) 31.52 26.86 16.45 5.65 0.97 TH E (cm) 15.74 17.21 12.80 5.83 0.79 CFD E (cm) 19.54 13.44 15.83 16.23 9.02 PKD E (cm) 4.34 5.30 0.51 1.85 0.91 THC Precision (cm) 1.19 1.11 1.31 0.27 0.21 E : RWE of threshold method. E : RWE of CFD method. E : RWE of PKD method. E : Corrected RWE. TH CFD PKD THC The precision in Table 1 shows that the precision of N with aggregate 100 repetitive measurements is within 0.25 counts, and the ranging precision with aggregate 100 repetitive measurements is within 1.4 cm at the few-photon level. The results indicate that the method does not require thousands of repeated measurements. Therefore, the method is available in wide dynamic LIDAR applications, such as autonomous driving and fast- moving target detection. Moreover, as the peak voltage measurement requires an extra circuit, considering the simplicity of the LiDAR system, there is another method to calculate N with no extra system design required. According to Equation (12), the whole detection probability of one shot with the threshold photon number k can be calculated. The detection probability P is only related to the intensity of echo signal photons under certain instrument parameters. The simulation between N and P is performed based on Equation (12) with the same D D parameters of the above experiments. Meanwhile, we calculate the P of the above 14 group measurements using the fixed threshold method. P is the ratio of the number of successful detections to the number of total measurements. The result is as shown in Figure 5. The black dots are the experiment results. It indicates that the simulation can predict the relationship between N and P well. Then, N can be estimated by the simulation D D D function we obtained in Figure 8a. The N obtained from SiPM’s peak voltage is expressed as N , and the N obtained from the detection probability based on Equation (12) is N . DP D DR Figure 8a depicts the trend of N and N with the optical power. As seen in Figure 8b, DP DR when the N is less than 12, the result of N is close to that of N . However, when N D DR DP D becomes larger, the P approaches 1 according to Figure 8a. The DN /DP in Figure 8a D DR D becomes very large, which means that the error when measuring P may induce a huge error when estimating N . Considering the restriction of the number of times repeated DR Photonics 2022, 9, 24 11 of 15 measurements can be taken, the experimental P may deviate from the true P ; thus, it is D D Photonics 2022, 9, x FOR PEER REVIEW 11 of 15 not appropriate to estimate N when the number of echo photons is large. As a result, N D D can be calculated through P in few-photon detection. (a) (b) (c) Figure 9. (a) Triggered time tag distribution using fixed threshold method with different mean num- Figure 9. (a) Triggered time tag distribution using fixed threshold method with different mean ber of fired pixels, (left), and triggered time tag distribution using three methods with ND number of fired pixels, N (left), and triggered time tag distribution using three methods with ND = 2.48 (right). (b) Experiment and simulation result of RWE. The dashed line is the simulation N = 2.48 (right). (b) Experiment and simulation result of RWE. The dashed line is the simulation result; the blue triangles are the experiment results of fixed threshold method. (c) Correction results result; the blue triangles are the experiment results of fixed threshold method. (c) Correction results of the three methods. The blue triangles are the error after applying the correction method; the black of the three methods. The blue triangles are the error after applying the correction method; the black squares are the CFD results; the red spots are the PKD results. squares are the CFD results; the red spots are the PKD results. The precision in Table 1 shows that the precision of ND with aggregate 100 repetitive 4. Conclusions measurements is within 0.25 counts, and the ranging precision with aggregate 100 repeti- For the challenge of RWE, which SiPM LiDAR encounters in wide-dynamic-range tive measurements is within 1.4 cm at the few-photon level. The results indicate that the measurement applications, we propose an RWE correction method based on the Poisson method does not require thousands of repeated measurements. Therefore, the method is distribution of arrival photons and the photon number resolvability of SiPM. The only available in wide dynamic LIDAR applications, such as autonomous driving and fast- input parameter of this method is the mean number of fired pixels, N , which can be moving target detection. obtained from the signal of SiPM or the detection probability. Experiments are carried out Moreover, as the peak voltage measurement requires an extra circuit, considering the simplicity of the LiDAR system, there is another method to calculate ND with no extra system design required. According to Equation (12), the whole detection probability of one shot with the threshold photon number k can be calculated. The detection probabil- ity PD is only related to the intensity of echo signal photons under certain instrument Photonics 2022, 9, 24 12 of 15 to verify our method. According to the experiment result, the RWE is almost 37 cm when N varies from 1.3 to 46.5. The mean residual RWE after complementing the correction method is 1.95 cm, which shows better performance compared with the CFD method and the PKD method. The ranging precision with aggregate 100 repetitive measurements is within 1.2 cm at the few-photon level. The above result indicates that the method enables RWE correction in photon threshold LiDAR systems based on SiPM. In practice, the peak voltage measurement for the picosecond width pulse generated from SiPM requires a complex system design. Considering the simplicity of the correction method, the method based on detection probability to estimate the light intensity requires no additional measuring system and can improve the accuracy within 4 cm when the mean number of fired pixels fluctuates from 1.31 to 11.22. Therefore, this method may be applicable in LiDAR for detection at the few-photon level. Author Contributions: Conceptualization, R.Y. and Y.T.; software, R.Y.; validation, R.Y., Y.T. and Z.F.; formal analysis, R.Y.; investigation, K.L., J.Q.; resources, K.L., Z.F.; data curation, R.Y.; writing— original draft preparation, R.Y.; writing—review and editing, R.Y., K.L.; visualization, R.Y.; supervi- sion, K.L., J.Q.; project administration, R.Y.; funding acquisition, K.L., J.Q. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Fudan University and Changchun Institute of Optics Joint Foundation: Grant FC2017-002; Fudan University Innovation Foundation: Grant Y18112. Conflicts of Interest: The authors declare no conflict of interest. Appendix A TOF Measurement with 9% PDE (Photon Detection Efficiency) SiPM To verify that the proposed technique is valid with different types of SiPMs, the Appendix Section includes a set of TOF measurement experiments using the latest detector with high PDE. We change the detector from SiPM (S13720-1325SA, HAMAMATSU) to SiPM (S15639-1325PS, HAMAMATSU, SiPM module from Gudatek, Hefei, China). The comparison of the two types of SiPM is as shown in Table A1, and their photos are included in Figure A1a. Table A1. Comparison of the two types of SiPM. Parameter S15639-1325PS S13720-1325CS PDE (905 nm) 9% 7% Dark count rate 0.7 MHz 0.5 MHz Crosstalk probability 6% 6% Rise time 500 ps 500 ps 6 6 Gain 1.7  10 1.1  10 Number of microcells 2120 2668 The experimental platform is as shown in Figure A1b. The schematic of the system and the experimental devices is introduced in the manuscript. Eight groups of distance measurement data are obtained under different echo signal intensities, with the N varying from 1.1 to 16.6. Here, we adjust the power of the semiconductor laser diode from 0.63 W to 13.83 W to achieve the number fluctuation of received photons. Meanwhile, within the mea- surement using the fixed threshold method, the peak voltage of SiPM is measured as seen in Figure A2a. After obtaining the value of N , a theoretical RWE based on Equation (14) is calculated, with N = 2120, h = 9% and T = T (N = 16.68). Figure A2b shows the q D Cell Ref Sim experiment and simulation results of RWE. Photonics 2022, 9, x FOR PEER REVIEW 13 of 15 Table A1. Comparison of the two types of SiPM. Parameter S15639-1325PS S13720-1325CS PDE (905 nm) 9% 7% Dark count rate 0.7 MHz 0.5 MHz Crosstalk probability 6% 6% Rise time 500 ps 500 ps 6 6 Photonics 2022, 9, 24 13 of 15 Gain 1.7 × 10 1.1 × 10 Number of microcells 2120 2668 (a) (b) Photonics 2022, 9, x FOR PEER REVIEW 14 of 15 Figure A1. (a) The two SiPM modules with S13720-1325CS (left) and S15639-1325PS (right) inside. Figure A1. (a) The two SiPM modules with S13720-1325CS (left) and S15639-1325PS (right) inside. (b) Experimental platform. (b) Experimental platform. The experimental platform is as shown in Figure A1b. The schematic of the system and the experimental devices is introduced in the manuscript. Eight groups of distance measurement data are obtained under different echo signal intensities, with the ND vary- ing from 1.1 to 16.6. Here, we adjust the power of the semiconductor laser diode from 0.63 W to 13.83 W to achieve the number fluctuation of received photons. Meanwhile, within the measurement using the fixed threshold method, the peak voltage of SiPM is measured as seen in Figure A2a. After obtaining the value of ND, a theoretical RWE based on Equa- tion (14) is calculated, with NCell = 2120, ηq = 9% and TRef = TSim (ND = 16.68). Figure A2b shows the experiment and simulation results of RWE. (a) (b) Figure A2. (a) Peak voltage of SiPM distribution at ND = 1.91. (b) Experiment and simulation results Figure A2. (a) Peak voltage of SiPM distribution at N = 1.91. (b) Experiment and simulation results of RWE. of RWE. In our current work, the proposed RWE models take the PDE of the detector into In our current work, the proposed RWE models take the PDE of the detector into account according to Equations (5) and (6). The result of the TOF measurements using the account according to Equations (5) and (6). The result of the TOF measurements using the 9% PDE SiPM in Table A2 shows that the method reduces the RWE from 29.07 cm to the 9% PDE SiPM in Table A2 shows that the method reduces the RWE from 29.07 cm to the mean mean d deviation eviation o off 2 2.33 .33 c cm, m, w with ith t the he n number umber of of detec detected ted photons fluctuating photons fluctuating fro from m 1.1 to 1.1 to 16.6. 16.6. Th Ther erefo eforr e, e, the the method method is is proved proved to to be be ef effective fective for for reducing reducing the the RWE RW emer E emerging ging in the in SiPM LiDAR with 9% PDE, and the proposed model can accurately predict the response the SiPM LiDAR with 9% PDE, and the proposed model can accurately predict the re- characteristics sponse characteristics of differ of ent di SiPMs. fferent SiPMs. Table A2. Accuracy results with different echo signal intensities. Table A2. Accuracy results with different echo signal intensities. N 1.13 1.91 4.88 7.72 8.40 11.00 14.22 1.13 1.91 4.88 7.72 8.40 11.00 14.22 E (cm) 29.07 26.00 18.03 11.43 9.81 6.26 2.42 TH E (cm) 29.07 26.00 18.03 11.43 9.81 6.26 2.42 TH E (cm) 4.07 2.86 2.54 2.15 1.84 1.75 1.08 THC E (cm) 4.07 2.86 2.54 2.15 1.84 1.75 1.08 THC E : RWE of threshold method. E : Corrected RWE. TH THC ETH: RWE of threshold method. ETHC: Corrected RWE. References 1. 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Journal

PhotonicsMultidisciplinary Digital Publishing Institute

Published: Jan 1, 2022

Keywords: LiDAR; SiPM; TOF; range walk error; threshold detection

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