Single photon detectors have many advantages, such as the single photon sensitivity and extremely high timing precision.^[1–4] Single photon detectors have been widely used in many weak signal detection fields, such as remote sensing, biological imaging through highly scattering and absorbing tissues, weak fluorescence imaging, and so on.^[5–11] One of the leading applications is remote ranging and imaging with high accuracy. A Geiger mode Avalanche Photodiode Detector (Gm-APD), as one of the single photon detectors, was applied in a lidar system for the first time by MIT Lincoln Laboratories in the JIGSAW project.^[12,13] Subsequently, Buller et al. used a series of improved techniques based on TCSPC (Time Correlated Single Photon Counting) techniques to demonstrate highly accurate remote detection.^[5,7,14,15] Yan et al. improved the TCSPC technique-based TOF ranging system, employing pulsed lasers at 1550 nm with multiple repetition rates to offer a robust and convenient method to decrease the range ambiguity.^[16] Li et al. used the 1.5-GHz sine-wave gated Geiger mode to improve the ranging accuracy of a photon-counting chirped amplitude modulation (CAM) Lidar.^[17] Remote ranging and imaging with high accuracy has been implemented by Korean^[18] and Chinese groups.^[19]

However, Gm-APD has a drawback that the intensity information of the signal is lost. Bao et al. used photon number resolving (PNR) detectors and recognized two remote targets with different reflection coefficients at the few-photon level.^[20] Liang et al. researched a high-speed photon-number resolving detector based on a sinusoidally gated multi-pixel photon counter, which could identify the photon at an average of 7.22 photons/pulse with a repetition frequency of 50 MHz before saturation at room temperature.^[21] Besides, Diagne et al. used the average firing rate of pixels to obtain the pixel brightness. With a human hand in front of the diffused target illuminated by a 2- wavelength pulse signal, an intensity image was acquired using a 32× 32 Sb-based Gm-APD array.^[22] He et al. used the number of triggered pulses within the duration of the pixel dwell time to obtain the optical signal intensity and achieved the passive optical imaging with ultimate sensitivity.^[23] Liu et al. used the response possibility estimation of laser echo to acquire the intensity.^[24] Aongus et al. obtained intensity images of a target by calculating the number of detected echo photons in the histogram peak for each individual pixel.^[25] Researchers were trying to resolve the loss of the intensity information with single-photon detectors. These existing methods using the photon counting probability of single-photon detectors to gain the intensity information are called the Photon Counting Probability Statistic (PCPS) method.

In this paper, an alternative technique called the Photon Counting Probability Density Statistic (PCPDS) method is proposed. This new method not only counts the triggered number of sampling gates, but also records the triggered time position of photon counting pulses. According to a series of triggered time positions, the photon counting probability density is obtained and used to estimate the echo signal intensity. This new method is capable of improving the estimation accuracy of signal intensity, compared with the existing PCPS method. The rest of this paper is arranged as follows. We first describe the conventional PCPS method and the new proposed PCPDS method. Then, a proof-of-principle laboratory system using Gm-APD is established, and the relative intensity accuracy is compared and analyzed. Finally, a high precision intensity image is acquired under the condition of a few photons per nanosecond.

In order to characterize the performance of the proposed method in this paper, a proof-of-principle system is established in the laboratory, as shown in Fig. 2. A 532-nm semiconductor continuous laser, whose output power can be adjustable from 0 W to 2 W according to the external driving voltage, in order to simulate different signal intensities. Two galvanometer mirrors are used to control high precision scanning, and the angle range is and accuracy is . The echo signal is received by the Gm-APD module which consists of a narrow-band filter (the center wavelength is 532 nm, the band width is 8 nm, and the center transmittance is 65%), optical lenses (used to gather the echo signal photons to the photosensitive surface of Gm-APD, and the effective aperture is 25 mm), and a Gm-APD (LASER COMPONENTS COUNT-50C; the photon detection efficiency at 532 nm is 60% approximately; the dead time of the Gm-APD is 50 ns; the timing jitter is 100 ps). The control signal of unified equidistant sampling gates is generated by the signal generator (Tektronix AFG3252), and this control signal is divided into two channels: one channel is used to control the Gm-APD sampling detection and the other channel as the starting signal is transmitted to an oscilloscope. The leading edge of each sampling gate starts timing, and the avalanche pulse of Gm-APD ends timing. A Tektronix DPO4032 oscilloscope is used to record a series of triggers. Finally, all data are collected to the computer for data processing. 3

Fig. 2.

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	Fig. 2. (color online) (a) Photo of the laboratory system for the proof-of-principle experiment. (b) Experimental setup sketch.

Fig. 3.

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	Fig. 3. (color online) The relative intensity accuracy comparison between the proposed PCPDS method and the conventional PCPS method with the number of sampling gates N from to (circle marks are for the conventional PCPS method and square marks are for the PCPDS method).

The parameters of the system are as follows: the time width of sampling gates is set as , and the interval of sampling gates is set as (it is longer than the dead time of , in order to be sure of the effectiveness of each sampling gate). The statistical segment width is set as . In this paper, the relative intensity accuracy is obtained by 100 repeated experiments, and the relative intensity accuracy is defined as the ratio of the standard deviation to the mean value , namely .

First, the photon counting probability density statistics (PCPDS) method proposed in this paper and the conventional photon counting probability statistics (PCPS) method are compared with the number of sampling gates N from to . The relative intensity accuracy of two methods are improved with the increase of the number of sampling gates N. However, under the same conditions, the relative intensity accuracy of the PCPDS method of this paper is superior to that of the conventional PCPS method. For example, when sampling gates are employed in both methods, the conventional PCPS method can achieve the relative intensity accuracy of 1.89% (see point A), and the PCPDS method of this paper can improve the relative intensity accuracy to 0.96% (see point B). If the demand is that the relative intensity accuracy must be less than 1%, the conventional PCPS method needs accumulating about sampling gates (see point C), while the PCPDS method only needs about sampling gates(see point B). This experimental result demonstrates that the PCPDS method of this paper using the photon counting probability density distribution is better than the conventional PCPS method.

The relative intensity accuracies with different intensities of the echo signal are shown in Fig. 4(a). It is clear that the relative intensity accuracy of the PCPDS method can meet the certain accuracy requirement in a specific range, which is called the effective range. When the signal intensity is stronger than the effective range, Gm-APD tends to be saturated. All sampling gates are almost triggered as soon as the sampling gates just open, and the photon counting probability density becomes insensitive to the change of the signal intensity. Therefore, the relative intensity accuracy becomes worse. When the signal intensity is weaker than the effective range, the effect of noise becomes significant, and thus the relative intensity accuracy becomes worse. For example, considering the accuracy requirement of 1%, the effective range is approximately from 0.15 ns⁻¹ to 1.3 ns⁻¹ (corresponding to the count rate of from 4.51 MHz to 9.94 MHz); considering the accuracy requirement of 2%, the effective range is approximately from 0.05 ns⁻¹ to 1.9 ns⁻¹ (corresponding to the count rate of from 1.81 MHz to 9.99 MHz).

Fig. 4.

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	Fig. 4. (color online) (a) The relative intensity accuracy with different signal intensities from 0 to 2.4 ns−1 with . (b) The relative intensity accuracy with different noise intensities , , and .

Figure 4(b) shows the relative intensity accuracies with different noise intensities of , , and , respectively. In the experiment, a daylight lamp is used to simulate the different noise intensities and illuminate the target. The sun background noise intensity for our experiment system is approximately 1 MHz during a sunny day (it is measured as the number of Gm-APD triggered by noises per second with no signal). In Fig. 4, it can be seen that the relative intensity accuracy becomes worse with the increase of the noise intensity. Besides, the smaller the signal intensity is, the more remarkable the influence of the noise intensity is. This is because the noise intensity becomes the main influence factor when the signal intensity reduces to a certain degree. Therefore, noise reduction is especially important in weak signal intensity detection. Besides, under the condition of the equivalent noise, the PCPDS method of this paper shows an overall improvement compared with the conventional PCPS method.

When the intensity of the echo signal is not in the effective range, a measured intensity value is also obtained, only with no satisfaction of the accuracy. This case will only add an adjustment measurement for satisfying the accuracy requirement. We will use the signal intensity adjustment of a known ratio K to transform the intensity of the echo signal into the effective range. Thus, the measured intensity value after adjustment will be in the effective range and the accuracy requirement is also satisfied. Through this intensity measured value divided by the known adjustment ratio K, the intensity beyond the effective range can be measured satisfying the accuracy requirement.

Finally, the imaging experiment is performed in the laboratory, using the PCPDS method proposed in this paper. As shown in Fig. 5, the imaged target is a piece of white paperboard with three letters H, I, and T with different reflectivities with .

Fig. 5.

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	Fig. 5. (color online) Photograph of the imaged target. The dashed box is the detection area.

Figures 6(a) and 6(b) are the imaging results and the corresponding intensity data statistics with the conventional PCPS method. Figures 6(c) and 6(d) are the imaging results and the corresponding intensity data statistics with our PCPDS method. In Fig. 6, different colors represent different intensities of echo signals from 0.3 ns⁻¹ to 2 ns⁻¹. Comparing Figs. 6(a) and 6(c) directly, the PCPDS method of this paper obtains a better image with small fluctuations in the image. Then the intensity data statistical results of two methods are compared. At the edge of three letter targets, the laser spot may cover both the target and background plate, therefore causing great errors. Intensity data statistics in Figs. 6(b) and 6(d) have got rid of the edge data. Figure 6(b) shows statistical results of the conventional PCPS method, the mean intensities of three letters H, I, and T are 0.4103, 0.7723, and 1.2094 and the standard deviations are 0.0081, 0.0135, and 0.0258, respectively. Thus, the relative intensity accuracy for three letters (H, I, and T) can be obtained as 1.97%, 1.75%, and 2.13%, respectively. While figure 6(d) shows statistical results of the PCPDS method of this paper, the mean intensities of three letters H, I, and T are 0.4011, 0.7677, and 1.2166 and the standard deviations are 0.0034, 0.0051, and 0.0116, respectively. Thus, the relative intensity accuracy for three letters (H, I, and T) can be obtained as 0.85%, 0.67%, and 0.95%, respectively. It is clear that the PCPDS method of this paper improves the relative intensity accuracy efficiently, compared with the conventional PCPS method.

Fig. 6.

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	Fig. 6. (color online) The imaging results and analysis. (a) The imaging results 60 × 100 pixels with the conventional PCPS method. (b) Intensity data statistics for letters H, I, and T in panel (a). (c) The imaging results 60 × 100 pixels with the PCPDS method proposed in this paper. (d) Intensity data statistics for letters H, I, and T in panel (c).

In this paper, a PCPDS method for retrieving the ultra-weak signal intensity based on single-photon detectors has been demonstrated. This method uses photon counting probability density statistics to estimate the signal intensity, which improves the estimation accuracy of signal intensity compared with the conventional PCPS method. A proof-of-principle laboratory system is established, and the relative intensity accuracy is analyzed. There is an effective range of signal intensity for a certain accuracy requirement, and the intensity estimation accuracy becomes worse with the intensity out of the effective range. Finally, a high accuracy intensity image is acquired under low-light level environments successfully.