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A pulsed-laser range finding based on differential optical-path is proposed, and the mathematical models are developed and verified. Based on the method, some simulations are carried out and important conclusions are deduced. (1) Background power is suppressed effectively. (2) Compared with signal-to-noise ratio (SNR) of traditional method, SNR of the proposed method is more suitable than traditional method in long-range finding and large tilt angle of target. (3) No matter what the tilt angle of target is, it always has optimal sensitivity of zero cross as long as the differential distance is equal to the light speed multiplied by the received pulse length and there is an overlap between two echoes.
© 2014 Optical Society of America
1. Introduction
Pulsed-laser range finding is an active remote sensing technique. The target is illuminated by short pulse emitted from a pulsed-laser source, and the distance is obtained by calculating time of flight (TOF) of echo from target. Supposing the light speed is c, and the round-trip time is t, the distance can be obtained with R = ct/2 [1, 2]. Since this method has advantages of simplicity, high accuracy and utility, it is widely used in the fields of both military and civilian [3–5]. Laser source with short pulse duration of nanosecond, picosecond or even femtosecond increase the range finding accuracy greatly [6–9].
The key of acquiring TOF is discriminating the exact arrival time of the echo pulse. Some methods have been used in acquiring TOF. Leading edge discriminator (LED) is a simple approach, and the leading edge of the received pulse is detected as the signal crosses a certain threshold. LED is easy to carry out because of its simple electrical structure, but it produces large walk error of a few of nanosecond [10]. Hansang Lim et al used four kinds of amplitude parameters, including slew rates, peak values, signal widths, as well as charge amounts [11], to correct time error in the leading edge discriminator respectively. Constant fraction discriminator (CFD) is used to enhance timing accuracy, and it reduces the walk error to less than picosecond [12, 13]. Peak discriminator is an approach to decrease walk error, but the disadvantage of this method is that echo is easily affected by both background power and targets tilt angle, which results into poor performance in the condition of low SNR [14]. Recently, Hong Jin Kong et al proposed a method using two Geiger-mode avalanche photodiodes (GmAPD) to realize acquisition of TOF. The echo was divided into two beams by splitter and arrival time of the electrical signals from the GmAPDs were compared. Though intensity of echo is decreased by half, the false alarm was decreased and detection probability was increased because the noise was filtered out [15].
Actually, the sources of noise of range finding include speckle, thermal and background, etc. Among of them, background power is prime noise for long rang detecting and exists in most of optical system [16–18]. Meanwhile, it is constant and does not change with time. Other noises distribute randomly [19]. In general, the method of differential optical-path has the ability to suppress background power. The method uses two receiving channels, of which one is reference channel, and the other is signal channel. Background power is suppressed by subtracting them. For example, the structure of differential optical-path suppresses common-mode noise and improves measurement accuracy of the confocal microscopy [20].
In order to suppress background power and increase SNR of pulsed-laser range finding, a method based on differential optical-path is proposed. The principle and theoretical analysis are illustrated in Section 2. Simulations based on the method are carried out in section 3. Conclusions are listed in the last section, which suggest the proposed pulsed-laser range finding can suppress background power and gain high SNR at long range.
2. Methods and materials
2.1 Principle
The system based on differential optical-path is shown as Fig. 1. A short pulse is triggered by field programmable gate array (FPGA) and collimated by TL. Then, the pulse is divided into two parts by beam splitter BS1. One is focused by CL and detected by PD, which generates the electrical start signal. The other is used to illuminate the target. The scattered or reflected light from the target is reflected by BS1 and divided into two beams by BS2. There are two same receivers consisting of an APD and the corresponding lens RL, respectively. APD A is illuminated by reflected light from BS2 and RL1, and generates echo signal Pr2. Similarly, APD B generates echo signal Pr1. The differential echo signal Pr2-Pr1 is obtained by subtraction in FPGA. The differential echo signal has a point of zero power, i.e. zero cross, and it is set as the stop signal. TOF is determined between the start and stop signal.
Fig. 1 Pulsed-laser range finding system structure based on differential optical path. BS-beam splitter, TL-transmitting lens, RL-receiving lens, PD-photo detector, APD-avalanche photo diode, CL-convergent lens, SC- stage controller.
The temporal change rate of the power at the zero cross in the differential echo signal is defined as the sensitivity of zero cross. It can be changed by adjusting differential distance between optical-path OA and OB. FPGA controls the stage using SC to change differential distance until the system acquire highest sensitivity of zero cross. Supposing the distance between BS1 and CL is l0, the distance between BS1 and BS2 is l1, the distance between BS2 and RLA is l2, the distance between BS2 and RLB is l3, differential distance between OA and OB is 2d. The positions of two receivers can be described as
Figure 2 shows the difference of echo signal between proposed method based on differential optical path and peak discriminator. On the left side is the start signal from PD. The peak is easy to detect because of the high intensity and low background power, so it is set as the start timing moment. Traditional system includes only a single APD. In the long-range measurement, peak position of the echo is difficult to discriminate because of the background power and echo broadening. Meanwhile, the temporal change rate of echo power near the peak is very small, i.e. the sensitivity near peak is very low, which increases the difficulty of discriminating peak position. Different from peak discriminator, differential optical-path method uses two APDs. Two receivers are moved by a distance of d symmetrically near or far from BS2. Compared with traditional method, although echo power is reduced by half in each detector, the background power can be suppressed by subtracting between two echo signals, Meanwhile, there is a zero cross in the differential echo signal and the sensitivity of zero cross is obviously higher than that of the peak in the original echo signal.
2.2 Echo analysis
Schematic diagram of laser range finding is shown in Fig. 3, left side is an system of transmitter and receiver, and right side is a target plane, which has a tilt angle of θ. R is the range between the system and the target. Laser pulse is with a temporal function of Gaussian model, which is written as [1]
where Et is the original pulse energy, τ is transmitting pulse width. Traditional receiver includes a single detector and echo signal power expression is written as [1, 20]where Pr(t) is the received power on the detector, Ta is the one way atmospheric transmission, To is the receiver optics transmission efficiency, ηD is quantum efficiency, ρr is the reflectance of the target, τr is the received pulse width, w0 is the waist radius of the laser, w(z) is the beam radius, and λ is the wavelength.According to Fig. 1, Eqs. (1) and (3), the echo signal power expressions of APD A and APD B are written as
Equation (3) is under the ideal condition with no effects of background power. In fact, the background power should be concerned in the system, especially in the long-range measurement. The background power is written as [1, 21]
where ρr is the reflectance of target, hsun is the background solar irradiance, Ar is the area of the receiver, α is the field of view, and △λ is the optical bandwidth. According to Eq. (5), PB is constant and does not change with time. The echo signal power from APD A and APD B with background power are described as following.The differential echo power is obtained by subtraction and is written asEquation (7) shows that the background power can be suppressed by using the structure of differential optical-path. Furthermore, at the zero crossing temporal point when Prd = 0, we can obtain Eq. (8), which demonstrates that the TOF of zero cross is the same as peak discriminator.2.3 Sensitivity of zero cross
Sensitivity of zero cross kd = dPrd(t)/dt is defined as the temporal change rate of echo power at zero cross, and it is written as
where kd is sensitivity of zero cross, u1 = t-[(2R-d)/c], u2 = t-[(2R + d)/c] and other symbols are described above. In order to obtain the crossing point of the trailing edge and the leading edge of echo signals, the differential distance should be less than [20], i.e. there is an overlap between two echoes. When t = 2R/c, the sensitivity of zero cross isDenoting f = kd|t = 2R/c, we can obtain Eq. (11) for the derivative term d from Eq. (10).
The highest sensitivity of zero cross is deduced by f’d = 0, yielding
The highest sensitivity of zero cross is acquired when d = cτr, and the differential distance should be changed with τr which results from tilt angle of target [22]. Therefore, stage is used to adjust the position of the receivers to meet Eq. (12) in different conditions.
According to analysis above, the method based on differential optical-path has two notable advantages. (1) Background power is suppressed because of the subtraction of two echo signals. (2) The sensitivity of zero cross is much higher than that of the peak. The proposed method transforms discriminating time of peak into detecting zero cross. .
2.4 SNR analysis
SNR is one of most important parameters since it affects the measurement range and accuracy. Under the same conditions of transmitting system and target, the range and accuracy of system increase with improvement of SNR. SNR of traditional method is written as [4]
where ρD is the detector current responsivity, ρD = ηDe/hf, <Pr> is effective value, i.e. root mean square (RMS) of the echo power at the detector, <σn> is RMS of total noise, B is the receiver bandwidth, h is the Planck’s constant, f = c/λ is the frequency of the received signal, e is the electron charge, <iDK> is RMS of the dark current of detector, k is the Boltzmann’s constant, T is the temperature in Kelvin, and <RTH> is RMS of the effective load resistance that creates the same thermal noise spectral density as the receiver electronics.The background power PB is constant and suppressed by subtracting the two echoes. Therefore, the total noise of system based on differential optical-path can be written as
According to the definition of SNR, the method based on differential optical-path can be written as3. Simulations and results
3.1 System model verification
According to echo analysis discussed above, in long-range finding system, echo of traditional method is mainly affected by background power. According to parameters of typical ranging finding and target, simulation parameters are set as following: Et = 0.4nJ, ρr = 0.5, To = 0.8, ηD = 0.6, τ = 2ps, d = 250mm, θ = 10°, 20°, 30°, 40°, d = 50mm, hsun = 500W/m2/μm (λ = 905nm) [23], α = 5°, △λ = 10nm, R = 18km, 19km, 20km, 21km. Ta is difficult to determine because of the complex environment conditions. In order to simple the question, the approximate relation among γ(λ), visibility and λ is used in the simulation, shown in Eq. (16), where Rv is visibility, q is correction factor which is depended on different visibility, shown in Table 1. Rv = 10km is chosen in the simulation.
Table 1. Correction Factor under Different Visibility
In order to illustrate conditions that echo signal is affected by background power and verify model, according to the conditions set above, the results are shown in Fig. 4. Figure 4 shows the echo power and background power at different range directly. In the conditions of 18km and 19km, shown in Fig. 4(a) and 4(b), the echo signal power is higher than background power. Therefore, signal may be distinguished and peak position can be detected with peak discriminator correctly. However, when echo signal power is lower than background power, shown in Fig. 4(c) and 4(d), it is hard to detect peak of echo signal. For such situations, the method of differential optic-path is very useful because background power is suppressed, as shown in Fig. 5. Figure 5(a) and 5(b), corresponding to Fig. 4(c) and 4(d) respectively, are differential echo signals obtained by subtracting echo signal of APD A and APD B. Background power is suppressed and peak detection is transform into zero cross.
Fig. 4 Comparison between background power and echo signal power. (a) is under the condition that range is 18km. (b) is under the condition that range is 19km. (c) is under the condition that range is 20km. (d) is under the condition that range is 21km.
Fig. 5 Differential echo signal at different ranges. (a) Range between system and target is 20km. (b) Range between system and target is 21km.
Figure 6 shows comparison between echo signal from single detector and differential echo signal at different range, under the conditions that the range is 20km and tilt angle of target are θ = 10°, 20°, 30°, 40°. In the traditional method, as shown in the Fig. 6(a), echo is broadened because of the tilt angle of target. Echo broadening increases with the growth of tilt angle. For instance, with tilt angle of target changing from 10° to 40°, receiving echo width is broadened from 3ns to 16ns. Echo broadening and low peak sensitivity may lead to greater error in peak position detection. Compared with traditional method, differential echo signal under the same condition is shown in the Fig. 6(b). It is clear that no matter what the angle of target is, the position of zero cross is not affected by pulse broadening.
Fig. 6 Comparision of echo signal under different tilt angle of target. (a) The echo pulse is broadened with growth of tilt angle. (b) The position of the zero cross is not affected by pulse broadening.
3.2 Zero crossing sensitivity affected by differential distance
Based on analysis on sensitivity of zero cross and differential distance, some simulations are carried out under the condition that target is R = 20km, θ = 10°and other parameters unchanged. The results is shown in Fig. 7, from 7(a) to 7(d) are the differential echo signals from APD A and APD B under the condition of d = cτr/3, cτr/2, cτr, 3cτr. According to Fig. 7, we can find: (1) There is a zero cross in differential echo signal; (2) The relative positions of echo signal from APD A and APD B are changed with different differential distances, which results in different sensitivity of zero cross.
Fig. 7 Echo signals from APD A, APD B and differential echo signal under the different differential distance. (a) Differential distance is cτr/3. (b) Differential distance is cτr/2. (c) Differential distance is cτr. (d) Differential distance is 3cτr.
In order to quantify the sensitivity of zero cross kd affected by differential distance d, the relation between sensitivity and differential distance is simulated. Under the conditions of d = cτr/3, cτr/2, cτr, , and 3cτr, the corresponding differential distance are 0.34m, 0.51m, 1.02m, 2.4m, and 3.06m, respectively. The results are shown in Fig. 8 in which five differential echo signals are acquired. It is clear that sensitivity of zero crossing is different because of different differential distance. For example, kd is 410.5W/s under the condition that d is cτr/2, and kd is 564.3W/s under the condition that d = cτr. According to analysis above, the differential distance should be less than to realize the crossing point of the trailing edge and the leading edge of echo signals. If d is larger than that value, the slop of zero cross decrease quickly. For instance, under the condition of d = 3cτr, shown in Fig. 8, sensitivity of zero cross kd is low and is only 31W/s.
Figure 9 shows the relation between sensitivity of zero cross and differential distance. We can find: (1) with the increase of differential distance, the trend of sensitivity of zero cross first increases and then decreases. Meanwhile, keeping the differential distance unchanged, zero crossing sensitivity decreases as the tilt angle of target increases. For example, holding d = cτr, kd is 564.3W/s under the condition of θ = 10°. As θ is 30°, kd decreases to 52.6W/s. (2) There is an optimal differential distance, i.e. d = cτr. No matter what the tilt angle of target is, the optimal differential distance is still satisfying d = cτr. As shown in Fig. 9, under different tilt angle (10°, 20°, 30°, 40°), cτr is 1.02m, 2.1m, 3.33m, 4.83m, respectively. Although the sensitivity of zero cross is different, the position of highest sensitivity is always cτr, which help to keep the system obtain highest sensitivity of zero cross.
3.3 SNR comparison
Many factors affect SNR, including the features of detectors, transmitting or receiving lens and noise source. Meanwhile, the intensity of echo signal is related to the range, tilt angle and reflectance of target. Therefore, it is difficult to consider so many conditions at the same time. Usually, for a range finding system, the parameters of target are unknown, and parameters of system are given. Therefore, unknown parameters can be seen as variant, such as range and angle of target, and designed systematic parameters are kept as constant, such as the optical or electrical parameters of transmitting or receiving system.
The typical parameters are set as following: R = 1km~25km, θ = 0°~50°, <iDK> = <iDK1> = <iDK2> = 10pA [24], <RTH> = <RTH1> = <RTH2> = 10kΩ, h = 6.63 × 10−34Js, e = 1.602 × 10−19C, B = 1GHz. The results of SNR of traditional and proposed method are shown in the Fig. 10(a) and 10(b) respectively. The data can be analyzed from two aspects: fixing range and changing angle, fixing angle and changing range. First, setting range unchanged (R = 10km) and θ increasing from 10° to 50°, shown in Fig. 10(a), SNR decreases from 185.4dB to 15.9dB. Under the same situation, SNR of proposed method, shown in the Fig. 10(b), decreases from 192.2dB to 27.6dB. Second, setting angle unchanged (θ = 40°), SNR of traditional method decreases from 187dB to 26.7dB with the increase of range from 0.1km to 24km, while SNR of the proposed method decreases from 198dB to 39.7dB.
Fig. 10 Comparison of SNR between the traditional method and the proposed method. (a) Relation between SNR, range and tilt angle in the traditional method. (b) Relation between SNR, range and tilt angle in the proposed method.
In order to compare the SNR of two methods clearly, relative increment percentage of SNR is used, i.e. △SNR = [(SNRd-SNR)/SNR] × 100%. The result is shown in Fig. 11, in which △SNR is growing with the increase of range and angle of target, especially in the conditions of long range or large tilted angle of target. Taking θ = 50° as an example, the range of the target is from 0.1km to 24km, and △SNR increases from 3.6% to 317.1%. Under the condition that R is 24km, △SNR increases from 2.3% to 317.1% as θ changes from 0° to 50°. According to analysis above, although SNR of two methods decrease with the increase of the range and angle, the SNR of differential optical-path is better than that of the traditional method, especially in long range finding.
4. Discussion
The system based on differential optical-path transforms peak position into zero-crossing. However, the position of zero-crossing can’t correspond to peak position precisely due to inconsistent beam of splitting ratio, difference of photo detectors, and difference of optical alignment in the two paths. An example is shown in Fig. 12, the echo of Pr2 is stronger than Pr1, so the zero-crossing position deviates from t1 to t2, and △t is the amount of deviation. Actually, the system should be calibrated both of the inconsistence of the two receivers and system errors before use.
4.1 Calibrating consistence of two receivers
The key of deviation is the different amplitude of echo signals from two receivers. Therefore, gain control circuit (GCC) is used in the system, shown in Fig. 13. First, a uniform diffuse reflector is placed in front of the system as the target. Second, differential distance is located precisely by precision stage. Meanwhile, amplitude of the echo signals from two receivers can be obtained. Third, the gain of detectors are adjusted by GCC until the two echo signals are the same. The calibration of two receivers is finished. According to the process, the error resulting from different echo signal amplitude is avoided.
4.2 Calibrating systematic error
Systematic errors include delay of inner optical-path, delayed-response of start signal and stop signal, etc., should be calibrated. Firstly, put a target plane in front of the system and the range R0 is measured by another range finder with higher accuracy than the system based on differential optical-path. Secondly, according to the nominal range of target R0, the time-of-flight (TOF) is obtain by t = 2R0/c. Meanwhile, the system based on differential optical-path acquires the TOF t’. Finally, the system error can be calculated by t’- t, and calibration of system error is finished. In regular work condition, the system error should be subtracted from the TOF acquired by the system based on differential optical-path.
5. Conclusion
In this paper, a novel pulsed-laser range finding system based on differential optical-path is proposed, and the mathematical models are developed and verified. Two receivers are used to form differential optical-path, and background power is suppressed effectively by subtracting both echo signals. Meanwhile, the position of zero cross is not affected by pulse broadening. Based on the proposed system, some simulations are carried out, including modeling verification, factors affecting sensitivity of zero cross, comparison of SNR between two methods. Some conclusions are achieved. (1) The model based on differential optical-path is verified, in which peak discrimination transform into detecting zero cross. (2) Effects from background power can be suppressed. (3) Zero cross is not affected by echo broadening. (4) Compared with the SNR of two methods, proposed method is more suitable than traditional method in long-range finding and large tilt angle of target. (5) No matter what the tilt angle of target is, it always has optimal sensitivity of zero cross as long as d = cτr and there is an overlap between two echoes.
Acknowledgments
This research was supported by the grant from the National Natural Science Foundation of China (No. 61275003 and No.51327005), Research Fund for the Doctoral Program of Higher Education of China (20101101110016).
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