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Abstract
Detectors based on single-photon avalanche diodes (SPADs) operating in free-running mode surfer from distorted detection signals due to the impact of afterpulse, dead time, and the non-linear detection efficiency response. This study presents a correction method based on conditional probability. In the experiments with high temporal resolution and huge dynamic range conditions, this method’s residual sum of squares is near 68 times smaller than the uncorrected received data of SPAD and near 50 times smaller than deconvolution method. This method is applied to polarization lidar and CO2 lidar, and the performance shows significant improvement. This method effectively mitigates the impact of SPAD afterpulse, dead time, and detection efficiency non-linear response, making it suitable for all SPADs. Especially, our method is primarily employed for atmospheric detection.
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1. Introduction
Single photon detection technology is currently the most precise technology for photon detection. It is widely used in high-resolution spectral detection and analysis [1], high-energy physics [2], bioluminescence [3], astronomical observation [4], optical time domain reflection (OTDR) [5,6], laser radar [7–9], optical communication [10–13], quantum information [14–16] and other weak light detection fields. Commonly used single-photon detectors include photomultiplier tubes (PMTs), superconducting nanowire single-photon detectors (SNSPDs), quantum field effect transistors, and upconversion single-photon detectors (UCSPDs) [17] based on single-photon avalanche photodiodes and upconversion technology. Among these, detectors based on single photon avalanche diodes (SPADs) are more mature and have numerous advantages, such as low cost, lightweight, high reliability, good comprehensive performance, and no need for ultra-low temperature [18–22]. Consequently, they are the most widely used single photon detectors.
InGaAs/InP single-photon detector (InGaAs/InP SPD) is a near-infrared photodetector based on SPAD. In the near-infrared band, although the quantum efficiency of InGaAs/InP SPD is not as high as that of SNSPD and UCSPD, above all, there is a more serious afterpulse problem [20]. However, considering factors such as weight, cost, lifespan, portability, reliability and ease of use, it exhibits acceptable dark count rate (DCR), quantum detection efficiency (PDE), and lower timing jitter. Combined with the advantages mentioned above, it has become the most widely used detector in this waveband range. In this work, we use InGaAs/InP SPD as an example for reserch.
The InGaAs/InP SPD currently operates in two modes [23,24]: gated and free-running. In free-running mode, the detector is always in the detection state, generating an electrical pulse count only when a photon, dark count, or afterpulse is detected, and is only not ready for detection in the subsequent dead time. It is convenient for applications where the photon's arrival time is unknown. However, free-running mode requires a longer dead time to suppress afterpulse effects, resulting in a limited saturation count rate for single-photon detectors. Correspondingly, gated mode only allows the InGaAs/InP SPD to be in the detection state for a short period, and the short period is the gate. The detector only counts when the gate is open. If the gate is closed or in a dead-time zone, no photon can be detected, and no electrical pulse is generated. To reduce afterpulses and improve count rates, SPADs can use the self-differencing technique [25], sine-wave gating technique [26] or dual-anode SPAD technique [27]. Gated mode is commonly used in applications where the arrival time of photons is known.
In free-running mode, the accumulated signal output of the InGaAs/InP SPD is susceptible to distortion from afterpulse and dead time effects. Lunghi T and Barreiro C et al. reported an InGaAs SPD based on a negative feedback avalanche photodiode that effectively reduced the afterpulse [28]. Ma et al [29] and Liu et al [30] used an active-quenching technique to reduce the afterpulse. Liu et al optimized the active-quenching circuit and packaging, decreasing the avalanche pulse discrimination level to 2.4 mV, and the full-width at half-maximum of the avalanche pulse is as short as 500 ps, which greatly lessens afterpulse effects [31]. Liu et al [32] reduced afterpulsing in InGaAs(P) single-photon detectors with hybrid quenching. All of the aforementioned methods involve the development of integrated quenching circuits, even though they mitigate the afterpluse, but they still limit the application area of InGaAs/InP SPD. Rapp et al. proposed a way based on nonconvex optimization of an objective derived from an analysis of the Markov chain formed by the detection times to correct for dead time effects in free-running SPADs [33]. Felix Heide et al. develop a probabilistic image formation model that accurately models pileup, they devise inverse methods to efficiently and robustly estimate scene depth and reflectance from recorded photon counts using the proposed model along with statistical priors [34]. But the methods of Rapp et al. and Felix Heide et al. have high time resolution and are suitable for high-precision ranging at short-distance. Yu et al. combined an active quenching circuit with a specific afterpulsing and dead time correction algorithm and employed the NFAD SPD in a LiDAR system, which improved the NFAD SPD’s performance [9,24]. But under the high temporal resolution applications, Yu’s deconvolution method remains with large distortion. Additionally, the detector response exhibits nonlinearity, meaning that the quantum efficiency is nonlinear with incident light intensity [35]. This characteristic can cause signal distortion in applications where optical signals change considerably. To address signal distortion caused by afterpulse, dead time, and nonlinear detection efficiency response, a conditional probability correction(CPC) method is proposed to mitigate them. This method is applied to InGaAs/InP SPD-based CO2 and polarization lidars, showing an improved effective detection range and accuracy for both polarization lidar and CO2 lidar, respectively. These improvements confirm the feasibility of CPC method.
2. Principle and instrument
The InGaAs/InP SPD detector is used for asynchronous photon detection with unknown photon arrival time in free-running mode. As the detection mechanism of InGaAs/InP SPD can only count one photon at a time, it is typically used to detect optical signals with periodic changes. To describe changes in the received signal over time, the period is divided into several equal histogram bins within which the number of detected photons is counted. The variation in the number of photons across histogram bins within a single period represents the change in the received signal for that period. However, due to the low signal-to-noise ratio (SNR) in one period, it may not accurately reflect the actual signal characteristics. To address this issue, multiple adjacent periodic signals are superimposed to count the number of photons in one bin, which provides a more precise indication of changes in the received optical signal over a short time. As an example, we take the single photon atmospheric lidar, in which the InGaAs/InP SPD detector is widely used. The detector receives atmospheric backscattering signals (Fig. 1(a)) and generates electrical pulses that record the photon arrival time (Fig. 1(b)). As the pulsed light emitted in the lidar system is periodic, the atmospheric backscattering signal is also periodic. To improve the SNR of the received signal and recover the backscattering signal, the electric pulse signal output of the detector is collected by the acquisition card and accumulated periodically (Fig. 1(c)). However, due to the problems such as afterpulse, dead time, and nonlinear response of detector efficiency, the signal may be distorted to varying degrees. This distortion is particularly evident when the received signal is strong.
Fig. 1. The principle of the lidar signal recorded by the SPAD. (a) lidar echo signal. (b) The electric pulse signal generated by the SPAD when receiving the lidar echo signal. (c) The number of accumulated electric pulse signals periodically.
To accurately retrieve the atmospheric signal, our work shows CPC method to correct the InGaAs/InP SPD detection signal. The specific process is as follows:
First, the afterpulse probability density distribution curve and detection efficiency curve are calibrated based on the optical setup shown in Fig. 4. The calibration process is described in detail below. The data of the two calibration experiments are collected by the following steps. The electrical pulse signal output by the InGaAs/InP SPD is accumulated by the acquisition card according to the step of the laser pulse period, and the accumulated signal is divided into multiple histogram bins ($\Delta {t_1},\Delta {t_2},\ldots ,\Delta {t_i},\ldots ,\Delta {t_n}$).The histogram composed of $\Delta {t_i}$ reflects the statistical average change in the light signal detected by the SPAD during a certain period of time within the laser pulse period. For the acquisition of the afterpulse probability density distribution curve, we first remove the dark noise generated by the SPD from the collected signal data. Then, make $P(i )= N(i )/N(p )$ ($N(i )$is the photon counts of $\Delta {t_i}$, $N(p )$ is the photon counts of $\Delta {t_p}$, $\Delta {t_p}$ represents the bin where the pulse is located), $P(i )$ is the afterpulse probability estimation at $\Delta {t_i}$ (after calculation, the data $P(p )$ needs to be set to zero), and then use the multiexponential function to fit $P(i )$. In fact, if the cumulative time is long enough, the calculated $P(i )$ can accurately represent the afterpulse probability density distribution; Therefore, the afterpulse probability density distribution curve can also be written as ${P_{ap}}(i )= P(i )$. To acquire the detection efficiency curve, we use the UCSPD with high linearity to calibrate the InGaAs/InP SPD. The basic experimental idea is that we utilize a beam splitter to divide the narrow pulse laser signal emitted by the laser. Dual beams are simultaneously detected by the UCSPD and InGaAs/InP SPD detectors. For the data received by both SPADs, we also first eliminated the dark noise generated by the two SPADs. The two SPADs both have good linearity under weak signal input. multiple groups of photon data within the linearity are used to calculate proportional coefficient $\alpha $ between the two SPADs, the detection efficiency curve ${f_{p,c}}(\chi )$ can obtained as follows:
Fig. 2. (a) Calibration of the InGaAs/InP SPD afterpulse probability density distribution curve. (b) The detection efficiency correction curve.
The calibration of the afterpulse probability density distribution curve and the detection efficiency correction curve belong to the preprocessing process, which is the first step of our algorithm. In our work, we have obtained the afterpulse probability density distribution function ${P_{ap}}(i )$ through experiments, as shown in Fig. 2, where i represents the serial number of $\Delta {t_i}$, and it has been verified that ${P_{ap}}(i )$ does not change with the incident light power. Next, we use ${P_{ap}}(i )$ to correct the afterpulse in the detector.
2.1 Afterpulse correction
For the original cumulative signal $N(i )$ detected by the detector, $N(i )$ is the result of multiple accumulations of detection signals in a single period. $N(i )$ is divided into n bins ($\Delta {t_1},\Delta {t_2},\ldots ,\Delta {t_i},\ldots ,\Delta {t_n}$), and the probability estimation of the afterpulse generated by $\Delta {t_i}$ in $\Delta {t_j}$ can be expressed as:
2.2 Dead time correction
The dead time $\Delta {t_{DT}}$ can cause a distorted detection signal when the SPAD operates in free running mode. The distortion caused by $\Delta {t_{DT}}$ follows a certain rule and can be restored mathematically. Here, $\Delta {t_{DT}}$ is the time period after the SPAD generates an avalanche signal during which the detector enters a self-recovery period and cannot continue to detect. In each single period, if the SPAD detects a photon signal at time t, the detector will be in a state of self-recovery within $\Delta {t_{DT}}$ after t, so the detection count cannot be generated. This means that for $N(i )$, the detection count at $\Delta {t_i}$ will lead to a decrease in the detection count in the subsequent $\Delta {t_{DT}}$ region (because $\Delta {t_i}$ is the time period, $\Delta {t_i}$ itself will also be affected). Similarly, the detection count at $\Delta {t_i}$ will be affected by the photon counts detected in the $\Delta {t_{DT}}$ region before $\Delta {t_i}$ and $\Delta {t_i}$ itself. The signal distortion caused by $\Delta {t_{DT}}$ is more obvious when the detection count is strong. After afterpulse correction, we make ${P_{ap,c}}(i )= {N_{ap,c}}(i )/(m\lceil{{{\Delta t} / {\Delta {t_{DT}}}}} \rceil )$, it represents the detection count probability estimation at $\Delta {t_i}$, then, the dead time correction for ${N_{ap,c\,}}\,(i)$ is performed as follows:
2.3 Detection efficiency nonlinear correction
As mentioned before, the SPAD detection efficiency is related to the intensity of the incident light, it changes with the intensity of the incident light, which will also lead to a distorted detection signal. Therefore, the detection efficiency needs to be corrected by experiments. Benefiting from the linear detection efficiency of UCSPD with weak signal, it is used as a reference in this work [36], and the detection efficiency curve ${f_{p,c}}(\chi )$ is calibrated by the UCSPD. The specific calibration schematic is shown in Fig. 4. The detection efficiency can be corrected by using ${f_{p,c}}(\chi )$ combined with ${N_{ap,DT,c}}(i )$ as follows:
${N_{ap,DT,p,c}}(i )$ is the photon number data after the afterpulse, dead time and detection efficiency nonlinear correction. The reason why ${N_{ap,DT,c}}(i )$ is used here instead of $N(i )$ is that the data when calibrating the detection efficiency nonlinear curve are almost unaffected by the afterpulse and dead time.
To verify the feasibility of this method, the continuous light emitted by the laser is modulated into periodic analog signal light by replacing the AOM in Fig. 4 with EOM and received simultaneously by UCSPD and InGaAs/InP SPD(the period cumulative time is 1 sec and the perod is 100us). By using the CPC method and Yu’s method [9,24] to correct the detection data of InGaAs / InP SPD, compared with UCSPD, because UCSPD is also affected by dead time, the data of UCSPD are also corrected by the dead time correction method proposed in this paper. The comparison results are as follows:
It can be seen that from Fig. 3(b), In high temporal resolution and huge dynamic range conditions, CPC method’s residual sum of squares is near 68 times smaller than The uncorrected received data of InGaAs/InP SPD and near 50 times smaller than Yu’s method.
Fig. 3. The comparison results of InGaAs/InP SPD and UCSPD. (a) Comparison of before and after afterpulse correction (narrow-pulse light detection). (b) Comparison of before and after dead time correction (pulse light detection). (c) Comparison of before and after detection efficiency nonlinear correction (narrow-pulse light detection). (d) The uncorrected received data of InGaAs/InP SPD(green line), the corrected received data of InGaAs/InP SPD by Yu’s method(blue line), the corrected received data of InGaAs/InP SPD by CPC method(red line), and the received data of UCSPD that corrected dead time (black line). (e) The horizontal axis represents the normalized data of UCSPD-corrected dead time, while the vertical axis(left) represents the normalized data of UCSPD-corrected dead time, the normalized data of InGaAs/InP SPD corrected with CPC method and with Yu’s method and uncorrected. The vertical axis(right) are the residuals A-D, B-D and C-D respectively. ${\alpha _1} = \frac{{RS{S_{A - D}}}}{{RS{S_{B - D}}}} = \frac{{{{({A - D} )}^2}}}{{{{({B - D} )}^2}}} \approx 68.06$, ${\alpha _2} = \frac{{RS{S_{C - D}}}}{{RS{S_{B - D}}}} = \frac{{{{({C - D} )}^2}}}{{{{({B - D} )}^2}}} \approx 49.98$.
3. Optical layout and instruments
The calibration principle diagram of the afterpulse probability density curve is shown in Fig. 4. After the continuous fiber laser with a wavelength of 1548.51 nm is emitted, the emitted laser is attenuated by the attenuator, modulated by AOM into pulsed light (pulse width of 10 ns, frequency of 10 kHz), and then detected by InGaAs/InP SPD. The pulse sequence signal detected by InGaAs/InP SPD is collected by Multiscaler at a step size of 100 µs (pulse laser period) for 10 minutes (6 × 10^6 times) and transmitted to the computer for processing. The reason why the pulse width of the pulsed laser here is 10 ns is that during data processing, we divide the signals with a step length of 100 µs accumulated for 10 minutes into 1000 bins, each of which is 100 ns. The pulse width of 10 ns can basically ensure that the counts detected by the detector will only fall into one bin. The accumulation of 10 minutes is to obtain a more accurate probability density curve of the afterpulse.
Fig. 4. The calibration principle diagram of the afterpulse probability density curve and the detection efficiency correction curve. CW, continuous wave, AWG, arbitrary waveform generator; OA, optical attenuator; AOM, acoustic optical modulator; BS, beam splitter; PC personal computer.
The calibration schematic of the detection efficiency correction curve is also shown in Fig. 4. After the continuous fiber laser with a wavelength of 1548.51 nm is emitted, the emitted laser is attenuated by the attenuator, modulated by AOM into pulsed light (pulse width of 10 ns, frequency of 10 kHz), and then detected simultaneously by InGaAs/InP SPD and UCSPD after beam splitting. The pulse sequence signal detected simultaneously by InGaAs/InP SPD and UCSPD is collected by Multiscaler at a step size of 100 µs (pulse laser period) for 3 minutes (1.8 × 10^6 times) and transmitted to the computer for processing. By adjusting the attenuator OA, the optical power incident on the InGaAs/InP SPD and UCSPD can be controlled. The reason why the pulse width of the pulsed laser is $\tau = 10ns$ is that $\Delta {T_{DT,UCSPD}} = 28ns$, $\Delta {t_{DT,InGaAs/InP\; \; SPD}} = 250ns$, $\tau < \Delta {t_{DT,UCSPD}},\Delta {t_{DT,InGaAs/InP\; \; SPD}}$, so the influence of the dead time can be ignored. The accumulation of 3 minutes is used to obtain a more accurate detection efficiency curve.
In the methods section, we introduce the specific correction process, and through experimental verification and comparison, we can see that CPC method has a good correction effect on SPAD detection signals. To test the effect of CPC method in practical application scenarios, we apply CPC method to applications with high accuracy requirements. In this work, polarization lidar and CO2 lidar data with distance resolution are selected. Both systems use InGaAs/InP SPD for detection.
4. Experiments and results
This section contains two experimental results. The first is the verification of the results of polarization lidar experiments. Then, the experimental verification of CO2 lidar. By applying the CPC method to two types of lidar detection data, the afterpulse, dead time and detection efficiency nonlinear effect can be alleviated, and the detection accuracy and effective detection range of polarization lidar and CO2 lidar can be improved.
5. Polarization lidar experiments
The original polarization lidar data were measured and observed horizontally by the polarization lidar system built in Hefei, China, on October 10, 2021. The detector used is InGaAs/InP SPD, and the specific optical path diagram is shown in Fig. 5. The polarization lidar system has been making long-term observations since it was built. We selected the data on June 5, 2022 for processing. To highlight the effect of our algorithm, we selected two observations with the most obvious comparison. After processing by CPC method, the calculated polarization results are shown in Fig. 6.
Fig. 5. Optical path diagram of polarization lidar and CO2 lidar. (a) Polarization lidar. (b) CO2 lidar. PFL, Pulse fiber laser; OS, optical switch; EDFA, erbium doped fiber amplifier.
Fig. 6. The polarization data of the polarization lidar. (a) and (c) are the uncorrected data. (b) and (d) are the corrected data.
It can be seen from Fig. 6 that there is a air mass with large depolarization ratio floating by, conventional air pollution does not produce more than 0.5 atmospheric depolarization, combined with the actual weather conditions at that time, if the data are not processed by CPC method, less than 1.5 km is not trusted data [37].
The calculated polarization value completely deviates from the actual situation due to the influence of the detector's dead time, afterpulse and nonlinear response. After the collected data are processed by CPC method, compared with the polarization data directly calculated, it can be seen that through this method, the polarization data of 0.5 km ∼ 1.5 km can be reliably recovered, and the polarization data in this range can be restored to normal. Some contours that cannot be clearly seen in the range can be clearly displayed. Meanwhile, some details outside this range can be shown more clearly, improving the detection range and accuracy of the entire polarization lidar.
6. CO2 lidar experiment
The original lidar measurement data were measured using a CO2 lidar system built in Hefei, China, in June 2022. The system uses InGaAs/InP SPD detectors for detection, and the specific optical path diagram is shown in Fig. 5. We selected 9 hours of data from the system from October 29 to 30, 2022 for processing. The CO2 concentration calculated directly and the CO2 concentration calculated after processing by CPC method are shown in Fig. 7.
As shown in Fig. 7, when the data were not processed using CPC method, a large part of the calculated value of the CO2 concentration within 1.5 km-2.2 km was below 400 ppm, which seriously deviated from the CO2 concentration in the Hefei urban area [38]. After processing by CPC method, the calculated data change with time and distance near 418 ppm. CPC method restores the calculated CO2 concentration data to normal after correcting the effect of the detector's afterpulse and dead time and detecting nonlinear response, thus improving the accuracy of the CO2 lidar system.
7. Summary
In this paper, a CPC method is proposed to correct the SPAD’s afterpulse, dead time and detection efficiency nonlinear response, which realizes the correction of SPAD signal by precalibrating the afterpulse probability curve and detection efficiency curve (the detector dead time is known). In the comparison experiment of InGaAs/InP SPD and UCSPD, in high temporal resolution conditions and huge dynamic range, CPC method’s residual sum of squares is near 68 times smaller than the uncorrected received data of InGaAs/InP SPD and near 50 times smaller than Yu’s method, which means that CPC method can greatly alleviate the signal distortion caused by the afterpulse, dead time and detection efficiency nonlinear response and exhibits a remarkable ability to improve the performance of InGaAs/InP SPD. At the same time, we apply CPC method to two practical application scenarios of polarization lidar and CO2 lidar, which require high accuracy of the detection signal. It is found that the near-field signal that cannot be used by polarization lidar 0.5 ∼ 1.5 km can be recovered. The untrustworthy near-field data of CO2 lidar at 1.5-2.2 km can be used continuously.
CPC method is universal and applicable not only to SPAD but also to UCSPD, superconducting nanowire detector, PMT and other SPDs that are affected by any one or more of the three effects of afterpulse, dead time and detection efficiency nonlinearity, which can be corrected by CPC method after acquiring the corresponding calibration curve and the dead time. Therefore, this method can be applied to the SPDs mentioned above in all free-running modes. Therefore, it can be applied in high-resolution spectral detection and analysis, high-energy physics, bioluminescence, astronomical observation, optical time domain reflection (OTDR), lidar, long-distance single-photon three-dimensional imaging, quantum information and other weak light detection fields.
Funding
National Key Research and Development Program of China (2023YFC3081100); National Natural Science Foundation of China (42125402, 42188101, 42304165, 42374185); Innovation Program for Quantum Science and Technology (2021ZD0300302); Natural Science Foundation of Anhui Province (AHY140000, 2208085QD118); Fundamental Research Funds for the Central Universities; the Ground-based Space Environment Monitoring Network; State Key Laboratory of Pulsed Power Laser Technology Foundation.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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