In this paper, we propose a combined system of heterodyne detection with laser pulse and photon counting based on Geiger-mode avalanche photodiode (GM-APD) that is designed to achieve the range of remote non-cooperative target. Based on the heterodyne principle and assuming that the creation of primary electrons in GM-APD is Poisson-distributed, the range accuracy model is established. The factors that influence the range accuracy, namely pulse width, echo intensity, local oscillator (LO) intensity, noise, echo position, and beat frequency, are discussed. The results show that these six factors have significant influence on the range accuracy when the echo intensity is extremely weak. In case that the primary electrons of the echo signal are beyond 4, the pulse width and echo intensity are the main influence factors. It is also shown that the stronger echo intensity, narrower pulse width, low noise, large echo position, and small beat frequency produce higher range accuracy in a pulsed photon heterodyne detection system based on GM-APD.
©2013 Optical Society of America
In order to detect the extremely weak echo signal diffused or reflected from remote non-cooperative target, the heterodyne laser radar with photon counting based on GM-APD has been developed . Heterodyne detection with a high detection sensitivity and high anti-interference performance can detect the intensity and phase of echo signal synchronously, thus the range and velocity of the target can be acquired simultaneously [2,3]. To operate the APD in the Geiger-mode, the APD is reverse-biased beyond its breakdown voltage, when a primary electron is generated it will be accelerated and by impact ionization creates a large and sharp current surge, which makes GM-APD has extremely high sensitivity and high timing resolution [4–6]. The output of GM-APD can directly drive the subsequent digital signal processing circuits, the benefits of using GM-APD are its ability to minimize the integrated circuit size and enable a growth path to large-format arrays which is the key device for the scannerless three-dimensional laser radar [4,6–10].
Due to the advantage of the heterodyne detection based on GM-APD, in recent years several authors have published the photon heterodyne detection theory. J. X. Luu et al.  reported the signal-to-noise ratio (SNR) theory that accounted for saturation effects in the photon counting array. L. A. Jiang et al.  presented and tested the photon heterodyne detection theory in a limit of a few LO photons. L. Liu et al.  established the theory obtaining the beat frequency and estimating the mixing efficiency by using the probability density function (PDF) of the photon counting time-interval in the laser heterodyne signal. In these analysis, the range accuracy of the photon heterodyne detection with laser pulse using GM-APD as detector has never been done, whereas only a direct detection system based on GM-APD has been studied [13,14].
In this paper, a designed photon heterodyne detection system is introduced, and based on the heterodyne operating principle and statistical detection theory of GM-APD, the theoretical expressions of the detection probability, detection PDF, mean and variation of arrival time are derived. Then, the theoretical model of range accuracy of the photon heterodyne detection is established. Finally, neglecting the variation of the echo pulse waveform and utilizing the tentative parameters, and by numerical method, how the factors, namely the pulse width, echo intensity, LO intensity, echo position, noise generated by the background light, laser backscatter and dark counts, and beat frequency, influence the range accuracy is discussed.
2. Heterodyne detection system
Heterodyne detection is a powerful detection technique that is used to measure the round-trip time of the emitted laser pulse and the target velocity owing to its noise-reduction capabilities, its high sensitivity as well as its high spectral resolution [15,16]. Compared with the direct detection, the photon heterodyne detection system is much more complex. However, since the GM-APD can directly digitize the echo signal without the need for bulky RF mixers, amplifiers, and digitizer elements, the photon heterodyne detection system is often much less complex than a conventional heterodyne detection system. At the same time, due to using the range gate, the performance of the photon heterodyne detection system can also be further improved.
2.1 System structure
The system structure of photon heterodyne detection with laser pulse based on GM-APD is shown in Fig. 1. The emitted continuous wave (CW) laser is split into two by the beam splitter (BS1), one of the light beams is modulated by an acousto-optic modulator (AOM) to shift the beat frequency and is used as the LO, another light beam is modulated by the electro-optic modulator (EOM) to produce a pulse train. By the BS2, the laser pulse energy is split into two, one fires the triggered APD and generates a start signal for initiating the time-to-digital convertor (TDC) circuit to begin timing, and the other one is emitted to the remote target. During the range gate generated by the control & signal processing system, photons diffused in the field-of-view (FOV) of the photon heterodyne detection system mix with LO photons and generate the primary electrons in GM-APD, which are amplified and provide a large and sharp current surge as a stop signal for TDC circuit to stop counting, and the count data in correspondence with the round-trip time of the laser pulse is achieved. Then the histogram of arrival time of the pulse train is acquired in the photon counting system. The analysis of the detected data in the histogram is to obtain the round-trip time, i.e. the target distance.
2.2 Time sequence of pulse heterodyne detection
Three-dimensional laser radar acquires the range information by measuring the time-of-flight (TOF) of the laser pulse [17,18], the arrival time of the echo photons is randomly distributed during the detection time, as a consequence, the detection range is randomly distributed. The difference of the actual distance and the detection range is defined as range accuracy, which is a function of the waveform and energy of the laser pulse diffused or reflected from the target [19,20]. Range gate  is used to gate the detector for limiting laser backscatter from such obscures as fog, camouflage or water and eliminating the contribution of ambient light because the detector only works during the range gate is on. The time sequence of the system is shown in Fig. 2. is the location of the starting time of the range gate, is the gate width, and the scattered echo pulse occurs at time which is relative to the starting position of the range gate.
3. Detection theory
In this section we develop the detection theory that describes the photon heterodyne detection with laser pulse. According to the detection time sequence in the range gate, the rate function of the primary electrons in GM-APD is presented. To an arbitrary input mixing photon flux, assuming Poissonian mixing signal and noise statistics, the detection probability of the photon arrival time is derived. Then, using the statistical theory, the range accuracy model is established.
3.1 Heterodyne signal
The emitted instantaneous field variables of the continuous LO and laser pulse is given by19]:
Let us assume that the echo signal pulse and LO field mix coherently with the same polarized direction in the detector plane, and the instantaneously varying photon mixing current is given by the expression [22,23]:
For simplicity, the average number of the noise primary electrons is assumed to be constant during the integration time, then, the rate function of the primary electrons in the range gate is presented by :
Thus the average number of primary electrons generated at the detector during the detection time can be expressed as
3.2 Statistics theory of GM-APD
In an extremely weak light detection condition, the number of primary electrons generated in GM-APD obeys a negative-binomial distribution (NBD) from a diffuse target and a Poisson distribution from a specular target . If the number of photons received is much less than the mode parameter which represents the “diffuseness” of the target or, more quantitatively, the number of degrees of freedom of the intensity included within the measurement interval, Poisson distribution is a good approximation to the NBD [24,25]. Since and also obey a Poisson distribution, with the result that the total number of the primary electrons generated in GM-APD is assumed to be Poisson-distributed. For a Poisson distribution in a GM-APD, the probability that primary electrons are detected during times and is given by [26,27]:
The probability that no primary electrons are detected between times and is , and it is simply the probability that at least one photon event occurs is . Then, in the integration time, the detection probability is given by
The target detection probability is a conditional probability which makes it a condition that a detecting event has generated in the range gate. The target detection probability, which arrives at a full probability of 1 during the range gate, is given by
3.3 Range accuracy model
According to the statistical theory, the detection PDF is
Then the mean and variation of the arrival time are
Range accuracy is the mean of range error, and higher range accuracy is achieved when the detection distance is closer to the real range. The real arrival time is assumed to be , then, the range accuracy is given by
The variance of the arrival time is representative of the dispersion degree of the detected data vs. the mean of the arrival time, and the square root of the variance is defined as the standard deviation. Therefore, the standard deviation of the detection range which indicates the deviation degree of the detection range vs. the mean of detection range can be used to represent the range precision of the system, and the standard deviation of the detection range is expressed as
4. Range accuracy analysis
4.1 Detection probability and range accuracy
The detection probability was measured as a function of the mixing light incident on the detector. The rectangular laser pulse is used to simplify the analysis, in other words, during the laser pulse width , . Then with the help of Eq. (7), the average primary electron is given by
Then, with Eq. (11), the detection PDF is presented by
Equations (18) and (19) are the ultimate conclusion in this paper, unfortunately which have no simple analytical expression. Therefore, by substituting Eqs. (18) and (19) into Eqs. (14) and (15) respectively, the range accuracy can be plotted by using numerical method.
4.2 Computational results and discussion
According to Eqs. (18) and (19), the factors that influence on the range accuracy are the pulse width , echo intensity , LO intensity , beat frequency , noise and echo position. For a GM-APD, the target detection probability varies only within ; this fact will be used for range accuracy analysis. At the same time, as a simplification, is assumed.
According to Figs. 4–8, it is shown that the range accuracy and range standard deviation result in a better situation as the echo intensity increases. This is because as the echo intensity becomes stronger, more primary electrons are excited, and then the probability of target detection becomes larger. Therefore, the average arrival time is closer to the real time. As a consequence, the higher range accuracy and smaller range standard deviation are achieved, and the detection range is closer to the real distance.
The range accuracy with different pulse width is shown in Fig. 4. The results show that the higher range accuracy and smaller range standard deviation are acquired with the narrower pulse width. Owing to the fact that the narrower pulse width with more concentrated pulse energy will produce a higher target detection probability, at a higher echo intensity, the influence of the pulse width on the range accuracy is smaller than at an extremely weak echo intensity, i.e. , and the narrower pulse width brings higher range accuracy. However, at low echo intensity, the range standard deviation becomes larger when the pulse width is narrower. Thus the narrow echo pulse has a few benefits for heterodyne detection system at low echo intensity. When is beyond 4, with a noise level of and 10 ns pulse width, the range accuracy can arrive at less than 50 cm. So, for photon heterodyne detection system with laser pulse, stronger echo intensity and narrower pulse width help to acquire higher range accuracy.
The influence of the noise on the range accuracy is shown in Fig. 5(a). It is shown that the range accuracy and the range standard deviation become more excellent with the noise decreases. If the echo intensity is weak, the fired probability by the noise in GM-APD increases, therefore, the influence of noise on the accuracy is very intensive. When the echo intensity is stronger, the detection probability fired by the heterodyne mixing signal becomes larger, thus the influence obeys the same change when is beyond 4.
Due to the different target distance, the echo position varies in the range gate. At the same time, the noise uniformly exists in the range gate, thus GM-APD can be fired by the noise before , too. The influence of echo position on the range accuracy is shown in Fig. 5(b). The results show that, owing to the fact that the detector is more likely to be triggered by the noise photons at extremely weak echo intensity, the influence of the echo position on the accuracy at low echo intensity is larger than at strong echo intensity, and the range accuracy is a little smaller than at higher intensity level. However, in the case of , due to the great fluctuation of the detection data, the range standard deviation, i.e. the range precision, becomes worse for a larger . At the same time, when is large, the fired probability by the noise increases, too, and the fired time position by the noise is closer to . Furthermore, taking account of the detection characteristic of GM-APD, the triggered probability by the echo pulse will be prevented by a higher noise fired probability. Therefore, the average arrival time is closer to , as a consequence, the better range accuracy can be acquired when increases, and the range precision becomes better with increases when is less than 2.
A conventional heterodyne detection system assumes an LO strength large enough to overcome the thermal and circuit noise. However, according to the previous studies [11,28], in the heterodyne detection system, large LO strength leads to a large shot-noise. Then, taking account of the acceptable operation safe value of GM-APD and shot-noise-limited sensitivity, it is essential to mix the echo pulse with a weak LO. The influence of LO on the range accuracy is shown in Fig. 6(a). In the case that the echo is at a low strength, owing to the fact that the detection probability becomes higher for stronger LO, the better range accuracy is achieved with the LO intensity increases. When is beyond 4, the range accuracy lower than 50 cm can be arrived at, and the influence of different intensity of LO on the accuracy and range precision obeys the same change.
The beat frequency is the difference between the local oscillator and the signal frequency, the Doppler frequency caused by the target velocity is added into the signal frequency. Therefore, the beat frequency varies with the target velocity. The influence on the accuracy at four different beat frequencies is represented in Fig. 6(b). The results show that the influence of different beat frequency is a little larger at the weak echo intensity than at the strong echo intensity, and the detection system obtains a high range accuracy and better range standard deviation when the beat frequency decreases. Furthermore, when is beyond 4, alike the previous consequence, the influence obeys the same change.
In this paper, we present the detection performance and the theoretical model of the range accuracy of the pulsed photon heterodyne detection, and investigate six main factors which have an influence on the range accuracy. In the present investigation, the tentative parameters used in our numerical studies were established according to the previous experimental researches [7,12,14]. The numerical results show that the range accuracy results in a better degree with strong echo intensity. In the case of weak echo intensity, the pulse width, LO, noise, echo position and beat frequency have a significant influence on the range accuracy. When the echo intensity is strong, the pulse width and echo intensity are the most important influence factors, and the influence of other four factors almost obeys the same trend. In general, among the factors discussed in this work, the echo intensity and pulse width are more important than others. Higher range accuracy can be obtained with the stronger echo strength and narrower pulse width.
This analysis effort was supported by National Natural Science Foundation of China (No. 61275081).
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