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Investigation of high-precision algorithm for the spot position detection for four-quadrant detector
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Investigation of high-precision algorithm for the spot position detection for four-quadrant detector
T
Xuan Wanga,b,c,*, Xiuqin Sua, Guizhong Liub, Junfeng Hana, Rui Wanga,c a b c
Xi'an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an, 710119, China School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi’an, 710049, China University of Chinese Academy of Sciences, Beijing 100049, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Four-quadrant detector Polynomial fitting Gaussian spot
In this paper, we propose a new polynomial fitting algorithm to improve the spot position detection accuracy based on four-Quadrant Detector (4QD) when the circular spot with Gaussian energy is used as the incident light model. The traditional polynomial fitting method is difficult to ensure high spot position detection accuracy in a wide detection range. To solve this problem, we analyze and compare the characteristics of different algorithms for spot position detection, and consider the influence of the 4QD gap size in the model. Based on the initial solution of the geometric approximation method, we introduce the error compensation factor function, a new spot position detection model is designed. The results of simulation and experiment show that the new algorithm can greatly reduce the position detection error of 4QD for Gaussian spot. When the radius of incident spot is 0.5 mm and within the detection range of [-0.5 mm∼0.5 mm], the maximum error is 0.001353 mm and the root-mean-square error is 0.0004596 mm with the new five-order polynomial fitting algorithm which are reduced 56.7% and 69.7% than traditional nine-order polynomial fitting algorithm. Moreover, the computational complexity of the new algorithm is much less than traditional algorithm and the new algorithm also has good prospects in laser communication, high energy laser weapons or others.
1. Introduction Nowadays, laser communication system has been widely used because it has better characteristics than wireless communication [1,2]. In laser communication system, the laser beam needs to be transmitted point-to-point, and the laser would be affected by many factors, such as atmospheric turbulence, mechanical vibration, etc. after long-distance transmission. Therefore, it is necessary to monitor the position of the spot by the sensor to perform closed-loop control the jitter error [3–6]. In other words, the accuracy of spot position detection largely determines the performance of laser communication system [7]. The four-quadrant detector(4QD) has excellent characteristics such as high position resolution, low natural noise and short response time. Therefore, it is widely used in laser communication, high-energy laser weapons and precision measurement to detect laser spot position [8,9]. The relationship between the output signal of the 4QD and the actual spot position is rather complicated, and there is a gap between the four quadrants on the 4QD photo-sensitive surface. So many scholars are studying how to improve the accuracy of 4QD spot position detection and expand the range of linear measurement through detection algorithm [10–12]. Jun Zhang et al. studied the establishment and error analysis of 4QD detection model related to non-uniform spot and blind area
⁎
Corresponding author at: Xi'an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an, 710119, China. E-mail address: [email protected] (W. Xuan).
https://doi.org/10.1016/j.ijleo.2019.163941 Received 29 September 2019; Accepted 29 November 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
W. Xuan, et al.
[13]. Mengwei Chen et al. used a square spot with uniform light intensity to enhance the linear measurement range of 4QD [14]. Wu Jiabin et al. proposed an innovative infinite integral model to reduce position detection errors at different radii [15]. Quangsang Vo et al. proposed a method based on the combination of central approximation method and polynomial fitting algorithm to improve the linear measurement range of 4QD in micro-displacement measurement system [16]. Song Cu and Yeng Chai Soh proposed a new estimation formula for estimating the spot position on 4QD and improving the linearity of 4QD for different spot sizes [17,18]. These algorithms can effectively improve the detection accuracy or linear measurement range of 4QD. However, how to improve the spot position detection accuracy while ensuring a wide linear measurement range is a challenge for laser communication system. Therefore, it is necessary to find a better way to solve this problem. In this paper, we propose a new polynomial fitting algorithm, and consider the influence of 4QD photo-sensitive surface gap in the model, which effectively improves the detection accuracy and linear measurement range of 4QD for Gaussian spot. We analyze the solution relationship between the QD output signal and the actual position of the spot. Then, based on the preliminary solution of the geometric approximation method, we introduce an error compensation factor function that includes the detector gap and spot size information in the initial solution value, get a new spot position detection model. The new algorithm effectively improves the positional accuracy. Simulation and experimental results show that this method has high detection accuracy and wide detection range. Moreover, this new algorithm effectively overcomes the shortcomings of the original algorithm. This paper is arranged as follows: Section 2 analyzes the structure of 4QD and the basic principle of spot detection, considering the detection model under Gaussian spot. Section 3 outlines the algorithm principle and model establishment process. Section 4 introduces the optical measurement system and demonstrates the effectiveness of the algorithm with experimental results. Section 5 concludes the paper. 2. Principle of measurement based on 4QD Nowadays, 4QD has been widely used as a sensor for laser spot position detection in many fields. The structural composition of the 4QD and the laser spot position measurement principle are shown in Fig. 1a. and Fig. 1b. The photo-sensitive surface of 4QD is composed of four photodiodes with the same properties and constitutes four quadrants of 4QD. There is a gap between each quadrant that cannot be photoelectric conversion, also called blind area. As shown in Fig. 1b., the size d of the gap affects the detection accuracy of the 4QD to the spot. It has been proved that the existence of gap can bring errors to position detection results [10,19]. The laser is condensed by a coupling lens, and the concentrated spot is irradiated on the 4QD photo-sensitive surface. A laser spot with Gaussian distribution is irradiate on the photo-sensitive surface, and each quadrant receives a different amount of light energy. Therefore, each quadrant on the 4QD photo-sensitive surface will produce different sizes of photocurrent due to the photoelectric effect. Assume that the light energy received by the four quadrants are PA , PB , PC and PD , the corresponding photocurrents are IA , IB , IC and ID . When the optical path is affected by the jitter, the position of the spot on the photo-sensitive surface will also jump, and the photocurrent of each quadrant also changes in real time. At this point, the position of the centroid of the spot can be judged by the magnitude of the current generated by each quadrant. So, the position of the centroid of the spot can be presented as [20]:
(IA + ID ) − (IB + IC ) (P + PD ) − (PB + PC ) = A IA + IB + IC + ID PA + PB + PC + PD (IA + IB ) − (IC + ID ) (PA + PB ) − (PC + PD ) Ey = = IA + IB + IC + ID PA + PB + PC + PD
Ex =
(1)
Where Ex and Ey are the estimated positions of the spot on the x-axis and the y-axis in the coordinate system, which can be used to determine the approximate position of the spot on the 4QD photo-sensitive surface. However, the estimated value is not the actual position of the spot centroid. We need to construct a spot position detection system based on 4QD to calculate the actual position through the algorithm. The model of the system constructed is shown in Fig. 2. The system is divided into two parts, one part is the optical part consists of laser and optical system, and the other part is the electronics part consists of 4QD and processing circuit. After
Fig. 1. Principle of measurement based on 4QD (a) 4QD structure; (b) principle of position detection. 2
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
W. Xuan, et al.
Fig. 2. Schematic of 4QD detection system.
the electrical signal is processed by a certain detection algorithm, we can obtain the spot position which is closer to the actual value.
2.1. Detection algorithm based on Gaussian spot Based on the distribution model of the spot energy, the actual position can be solved by the estimated value of the position of the centroid of the spot. In practical applications, the laser is imaged on the 4QD photo-sensitive surface after being transformed by the optical system. The energy distribution of the spot is usually a circular point with Gauss distribution, which can be expressed as follows:
2((x − X )2 + (y − Y ))2 ⎤ 2P0 h ⎛⎜x , y ⎞⎟ = exp ⎡− 2 ⎢ ⎥ πω ω2 ⎣ ⎦ ⎝ ⎠
(2)
Where P0 is the total energy of the spot, ω is the waist radius of the Gaussian beam, and (X, Y) is the coordinate position of the centroid of the spot. Ignore the effects of light energy outside the 4QD photo-surface and the gap, the integration interval becomes infinite, and we can get the spot energy distribution model: +∞
PA + PD =
SA+ SD
−∞ +∞
PB + PC =
⎛
+∞
⎞
∬ h (x, y) dxdy = ∫ ⎜ ∫ h (x, y) dx ⎟ dy ⎝
0
⎠
0
⎛ ⎞ h (x , y ) dx ⎟ dy ⎜ ⎝ −∞ ⎠
∬ h (x, y) dxdy = ∫ ∫ SB + SC
−∞
(3)
Since the positions of the spots on the x-axis and the y-axis are independent of each other, We can use the algorithm to analyze the spot position on the x-axis separately [21]. Bring (3) into (1) to get: +∞
Ex =
2(y − y0 )2 4 ⎛ dy πω2 ⎜−∞ ω2 ⎝
∫
∞
∫ − 2(x ω−2 x 0) 0
2
2 x0 ⎞ ⎞ ⎟ dx ⎟ = erf ⎛⎜ ⎝ ω ⎠ ⎠
2 2 Where erf ⎛⎜⎟⎞ = π ∫ e−t dt is the error function, from above formula, the solution value can be obtained. ⎝⎠ erf −1 (Ex ) ω x0 ≈ 2
(4)
(5)
In this way, we obtain the approximate solution value of the spot position with the Gaussian spot as the incident light.
3. Model derivation and discussion of improved algorithm In this section, we establish a spot detection model that takes into account the effects of blind area. Then, in order to further improve the detection accuracy and linearity of the algorithm, we introduce an error compensation factor function into the model, and discuss the influence of the size of the gap on spot detection. Considering the Gaussian distribution of the spot on the photo-sensitive surface, the blind area cannot produce the corresponding photocurrent, and it is difficult to express the photocurrent by (3). Therefore, it is necessary to use (6) to represent the photocurrents of each quadrant considering the influence of the blind area: 3
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
W. Xuan, et al.
PA + PD =
∬ h (x, y) dxdy − ∬ h (x, y) dxdy SA+ SD
PB + PC =
Sgap
∬ h (x, y) dxdy − ∬ h (x, y) dxdy SB + SC
(6)
Sgap
After obtaining the currents generated by each quadrant, we can still obtain the solution value by position detection algorithm based on the Gaussian spot. The solution value is the approximate expression of the centroid position, which is called the preliminary solution value because of its inadequate accuracy. In order to further improve the detection accuracy and the linear measurement range, an error compensation factor function is introduced. So, the preliminary solution value of the centroid position of the spot can be rewritten as follows:
x = g (Ex )•ω•λ (x 0)
(7)
erf −1 (E
Where g (Ex ) = x )/ 2 represents the spot position function of the approximate solution value, which determines the overall change trend of the function curve. The incident radius ω of the spot can be regarded as the ratio coefficient between the centroid position of spot and the calculated value function of the detector. Since different incident radii lead to different error curves, the latter two items are combined into a θc = ω•λ (x 0) to characterize the contribution of the spot radius to the position error. In order to obtain the error compensation factor function, a Gaussian spot with a radius ω can be moved from (-0.5 mm, 0) to (0.5 mm, 0) at a very small distance intervals . We record the actual position X of each spot and the output current values IA 、 IB 、 IC 、 ID at s interval and calculate the solution value Ex of each point according to the solution formula, then obtain a theoretical one-toone correspondence value. The (Xi , Exi ) value of each point is also obtained. According to the least square method, the mathematical model of position residual is established. N
I (θc ) = || δx ||2 =
∑ [x (Exi, θc ) − Xi ]2
(8)
i=1
Find the first derivative of θc in the above formula, and make the first derivative be 0, we can get the optimalθc . N
θc =
∑i = 1 g (x 0i )•Xi N
∑i = 1 g 2 (x 0i )
(9)
In practical applications, the incident spot radius is a constant value, so the error compensation factor term is only a function of the preliminary solution value x 0 . The expression of the θc can be obtained by the method of polynomial fitting. n
θc = A0 +
∑ Ai x 0i
(10)
i=1
Therefore, the improved polynomial fitting algorithm expression for the spot position detection result is:
x=
erf −1 (σx ) ⎛ •ω•⎜A0 + 2 ⎝
n
∑ Ai x 0i ⎞⎟ i=1
(11)
⎠
At this point, we have obtained a model of the modified algorithm by considering the influence of the blind area on the accuracy and introducing the error compensation factor function. Compared with the traditional polynomial fitting algorithm, the new model effectively improves the spot position detection accuracy while ensuring a wide linear measurement range, which can be proved in the experimental results. 4. Experiment analysis In order to verify the validity of the new model, we design the QD detection system as shown in Fig. 3. The system includes a fiber laser with wavelength 1550 nm and a InGaAs QD detector (HAMAMATSU G6849-01) with Photo-sensitive surface diameter 1 mm and gap size 0.03 mm. The laser beam passes through a collimating lens to become a parallel light, which is attenuated by an attenuator and then focused by a coupling lens. The spot falls on the QD photo-surface, and its size is changed by adjusting the spacing between the QD and the coupling lens. The QD is mounted on a micro-displacement platform to achieve um-level displacement accuracy. By moving the micro-displacement platform, the spot can move along the x-axis on the detector. We move the spot with a radius of 0.5 mm from (-0.5 mm, 0) to (0.5 mm, 0) and record the actual position of the spot and the output current value of each quadrant every 20um. The solution value Ex of each point is calculated according to (1). Fig. 4a. shows the correspondence between the actual position detected by the detector and the solution value Ex , while Fig. 4b. shows the correspondence between the actual position and the approximate solution value g (Exi ) of each point. It can be seen from the figure that there is good detection accuracy only in a small range near the center of the detector, and the position detection accuracy is poor on a wide range. Therefore, it needs to be further processed by the detection algorithm. Then we compare the detection results of traditional detection algorithms. The experimental results are shown in Fig. 5. The results show that the central approximation method and the geometric approximation method also have better detection accuracy in the center range. The geometric approximation shows a better detection effect than the central approximation in a wide range. 4
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
W. Xuan, et al.
Fig. 3. QD detection system.
Fig. 4. (a) Correspondence between actual position andEx , (b) Correspondence between actual position and g (Exi ) .
Fig. 5. Comparison of experimental results between two traditional algorithms: geometric approximation (red) and central approximation (blue) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
However, the linear measurement range of the two algorithms is small. In practical applications, once the spot exceeds this range, the position information will be wrong. In some high-precision applications, these two algorithms will fail. The proposed algorithm model was obtained by fitting the experimental data using MATLAB toolbox. By comparison, we choose the fifth-degree polynomial fitting expression, and the evaluation parameter of the linear fitting result are the sum of squares for error (SSE) = 0.0377, R-square = 0.9908, adjusted R-square = 0.9896 and root-mean-square error (RMSE) = 0.0307. Curve fitting results 5
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
W. Xuan, et al.
Fig. 6. New fifth degree polynomial curve.
are shown in Fig. 6.
x = −0.5597 × x 05 − 0.001697 × x 04 − 0.2894 × x 03 − 0.0005438 × x 02 + 0.9585 × x + 0.0009817
(12)
Fig. 7. shows the error between fitting results and actual position. It can be seen from the experimental results that the position detection accuracy is kept high in a wide detection range. e also obtain the new fifth-degree polynomial fitting position detection model without considering the influence of the blind area in the model, and show the contribution of the gap in the model by comparing with the improved model. The comparison results of the detection errors between the two detection models are shown in Fig. 8. The results show that the detection accuracy of the algorithm can be greatly improved when considering the influence of the blind area in the model. In order to verify the advancement of the new detection algorithm over traditional detection algorithms, we compare the improved model with the traditional polynomial algorithm. Fig. 9. shows the spot position detection error of these models. It can be seen from the experimental results that the improved model not only greatly improves the detection accuracy and detection range, but also has advantages in the complexity of the algorithm. The results of the detection accuracy of these algorithms are shown in Table 1. 5. Conclusions This paper introduces the principle of spot position measurement of QD detector, derives the position detection algorithm model by Gaussian spot as the incident light model, get a new modified algorithm. The improved polynomial fitting detection model is obtained by introducing the error compensation factor function and considering the influence of the blind area on the accuracy in the model. Based on the analysis and comparison of traditional detection algorithms, we summarize their shortcomings. The simulation and experimental results show that considering the blind area in the model will greatly improve the system detection accuracy. In addition, the experimental results show that the improved detection algorithm has higher detection accuracy than the traditional algorithm. The linearity of the improved polynomial detection algorithm over a wide detection range shows better results than the traditional algorithm, which means that the detector has a wide effective measurement range, which is exactly what we expect. Our model improves the linear measurement rang and accuracy without increasing the algorithm complexity. Therefore, our algorithm has good practicability in applications such as laser communication.
Fig. 7. New fifth degree polynomial algorithm detection error. 6
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
W. Xuan, et al.
Fig. 8. Comparison of detection error between model with gap and model without gap.
Fig. 9. Detection error result. Table 1 Comparison of model detection accuracy. An example of a column heading
Maximum error (mm)
RMSE (mm)
central approximation model geometric approximation model Traditional ninth degree polynomial Improved model
0.1081 0.2648 0.003060 0.001353
0.04303 0.07434 0.001518 0.0004596
Acknowledgements The authors express sincere thanks for the experiments provided by the Photoelectric Tracking and Measurement Technology Laboratory, Xi'an Institute of Optics and Precision Mechanics, CAS. Laser communication project of Xi'an Institute of Optics and Precision Mechanics, (Grant No.Y655811213), China References [1] T. Nagatsuma, G. Ducournau, C.C. Renaud, Advances in terahertz communications accelerated by photonics, Nat. Photonics 10 (6) (2016) 371–379. [2] L. Jiang, L.Z. Zhang, C. Wang, Optical multiaccess free-space laser communication system, Opt. Eng. 55 (8) (2016) 086102. [3] Y. Ni, J. Wu, X. San, S. Gao, S. Ding, J. Wang, T. Wang, Deflection angle detecting system for the large-angle and high-linearity fast steering mirror using quadrant detector, Opt. Eng. 57 (2) (2018) 024110. [4] N.O. Perez-Arancibia, J.S. Gibson, T.C. Tsao, Frequency-weighted minimum-variance adaptive control of laser beam jitter, IEEE. ASME 14 (3) (2009) 337–348. [5] N. Tsuchiya, S. Gibson, T.C. Tsao, M. Verhaegen, Receding-horizon adaptive control of laser beam jitter, IEEE. ASME 21 (1) (2015) 227–237. [6] H. Yoon, B.E. Bateman, B.N. Agrawal, Laser beam jitter control using recursive-least-squares adaptive filters, J. Dyn. Syst-T. ASME. 133 (4) (2011) 041001. [7] Q. Li, S. Xu, J. Yu, L. Yan, Y. Huang, An Improved Method for the Position Detection of a Quadrant Detector for Free Space Optical Communication, Sensors. 19 (1) (2019) 175. [8] X. Hao, C. Kuang, Y. Ku, X. Liu, Y. Li, A quadrant detector based laser alignment method with higher sensitivity, Optik. 123 (24) (2012) 2238–2240. [9] C. Lu, Y.S. Zhai, X.J. Wang, Y.Y. Guo, Y.X. Du, G.S. Yang, A novel method to improve detecting sensitivity of quadrant detector, Optik. 125 (14) (2014)
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3519–3523. C. Lu, Y.S. Zhai, X.J. Wang, A novel method to improve detecting sensitivity of quadrant detector, Optik. 125 (14) (2014) 3519–3523. J. Wu, Y. Chen, S. Gao, et al., Improved measurement accuracy of spot position on an InGaAs quadrant detector, Appl. Opt. 54 (27) (2015) 8049. Z.J. Gao, L.L. Dong, L.Y. Meng, Study for high-precision positioning algorithm of four-quadrant detector, J. Optoelectronics, Laser 24 (12) (2013) 2314–2321. J. Zhang, W. Qian, G. Gu, K. Ren, Q. Chen, C. Mao, L. Xu, Quadrant response model and error analysis of four-quadrant detectors related to the non-uniform spot and blind area, Appl. Opt. 57 (24) (2018) 6898–6905. M. Chen, Y. Yang, X. Jia, H. Gao, Investigation of positioning algorithm and method for increasing the linear measurement range for four-quadrant detector, Optik. 124 (24) (2013) 6806–6809. J.B. Wu, Y.S. Chen, S.J. Gao, Z.Y. Wu, High precision spot position detection model for the near infrared light, Infrared. Laser. Eng. 45 (7) (2016) 0717001-1. Q. Vo, X. Zhang, F. Fang, Extended the linear measurement range of four-quadrant detector by using modified polynomial fitting algorithm in micro-displacement measuring system, Opt. Laser Technol. 112 (2019) 332–338. S. Cui, Y.C. Soh, Improved measurement accuracy of the quadrant detector through improvement of linearity index, Appl. Phys. Lett. 96 (8) (2010) 081102. S. Cui, Y.C. Soh, Analysis and improvement of Laguerre–Gaussian beam position estimation using quadrant detectors, Opt. Lett. 36 (9) (2011) 1692–1694. J. Zhang, W. Qian, G. Gu, K. Ren, Q. Chen, C. Mao, L. Xu, Quadrant response model and error analysis of four-quadrant detectors related to the non-uniform spot and blind area, Appl. Opt. 57 (24) (2018) 6898–6905. A.J. Makynen, J.T. Kostamovaara, R.A. Myllyla, A high-resolution lateral displacement sensing method using active illumination of a cooperative target and a focused four-quadrant position-sensitive detector, IEEE. T. Instrum. Meas. 44 (1) (1995) 46–52. J. Narag, N. Hermosa, Response of quadrant detectors to structured beams via convolution integrals, JOSA A. 34 (7) (2017) 1212–1216.
8
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Investigation of high-precision algorithm for the spot position detection for four-quadrant detector
T
Xuan Wanga,b,c,*, Xiuqin Sua, Guizhong Liub, Junfeng Hana, Rui Wanga,c a b c
Xi'an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an, 710119, China School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi’an, 710049, China University of Chinese Academy of Sciences, Beijing 100049, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Four-quadrant detector Polynomial fitting Gaussian spot
In this paper, we propose a new polynomial fitting algorithm to improve the spot position detection accuracy based on four-Quadrant Detector (4QD) when the circular spot with Gaussian energy is used as the incident light model. The traditional polynomial fitting method is difficult to ensure high spot position detection accuracy in a wide detection range. To solve this problem, we analyze and compare the characteristics of different algorithms for spot position detection, and consider the influence of the 4QD gap size in the model. Based on the initial solution of the geometric approximation method, we introduce the error compensation factor function, a new spot position detection model is designed. The results of simulation and experiment show that the new algorithm can greatly reduce the position detection error of 4QD for Gaussian spot. When the radius of incident spot is 0.5 mm and within the detection range of [-0.5 mm∼0.5 mm], the maximum error is 0.001353 mm and the root-mean-square error is 0.0004596 mm with the new five-order polynomial fitting algorithm which are reduced 56.7% and 69.7% than traditional nine-order polynomial fitting algorithm. Moreover, the computational complexity of the new algorithm is much less than traditional algorithm and the new algorithm also has good prospects in laser communication, high energy laser weapons or others.
1. Introduction Nowadays, laser communication system has been widely used because it has better characteristics than wireless communication [1,2]. In laser communication system, the laser beam needs to be transmitted point-to-point, and the laser would be affected by many factors, such as atmospheric turbulence, mechanical vibration, etc. after long-distance transmission. Therefore, it is necessary to monitor the position of the spot by the sensor to perform closed-loop control the jitter error [3–6]. In other words, the accuracy of spot position detection largely determines the performance of laser communication system [7]. The four-quadrant detector(4QD) has excellent characteristics such as high position resolution, low natural noise and short response time. Therefore, it is widely used in laser communication, high-energy laser weapons and precision measurement to detect laser spot position [8,9]. The relationship between the output signal of the 4QD and the actual spot position is rather complicated, and there is a gap between the four quadrants on the 4QD photo-sensitive surface. So many scholars are studying how to improve the accuracy of 4QD spot position detection and expand the range of linear measurement through detection algorithm [10–12]. Jun Zhang et al. studied the establishment and error analysis of 4QD detection model related to non-uniform spot and blind area
⁎
Corresponding author at: Xi'an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an, 710119, China. E-mail address: [email protected] (W. Xuan).
https://doi.org/10.1016/j.ijleo.2019.163941 Received 29 September 2019; Accepted 29 November 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
W. Xuan, et al.
[13]. Mengwei Chen et al. used a square spot with uniform light intensity to enhance the linear measurement range of 4QD [14]. Wu Jiabin et al. proposed an innovative infinite integral model to reduce position detection errors at different radii [15]. Quangsang Vo et al. proposed a method based on the combination of central approximation method and polynomial fitting algorithm to improve the linear measurement range of 4QD in micro-displacement measurement system [16]. Song Cu and Yeng Chai Soh proposed a new estimation formula for estimating the spot position on 4QD and improving the linearity of 4QD for different spot sizes [17,18]. These algorithms can effectively improve the detection accuracy or linear measurement range of 4QD. However, how to improve the spot position detection accuracy while ensuring a wide linear measurement range is a challenge for laser communication system. Therefore, it is necessary to find a better way to solve this problem. In this paper, we propose a new polynomial fitting algorithm, and consider the influence of 4QD photo-sensitive surface gap in the model, which effectively improves the detection accuracy and linear measurement range of 4QD for Gaussian spot. We analyze the solution relationship between the QD output signal and the actual position of the spot. Then, based on the preliminary solution of the geometric approximation method, we introduce an error compensation factor function that includes the detector gap and spot size information in the initial solution value, get a new spot position detection model. The new algorithm effectively improves the positional accuracy. Simulation and experimental results show that this method has high detection accuracy and wide detection range. Moreover, this new algorithm effectively overcomes the shortcomings of the original algorithm. This paper is arranged as follows: Section 2 analyzes the structure of 4QD and the basic principle of spot detection, considering the detection model under Gaussian spot. Section 3 outlines the algorithm principle and model establishment process. Section 4 introduces the optical measurement system and demonstrates the effectiveness of the algorithm with experimental results. Section 5 concludes the paper. 2. Principle of measurement based on 4QD Nowadays, 4QD has been widely used as a sensor for laser spot position detection in many fields. The structural composition of the 4QD and the laser spot position measurement principle are shown in Fig. 1a. and Fig. 1b. The photo-sensitive surface of 4QD is composed of four photodiodes with the same properties and constitutes four quadrants of 4QD. There is a gap between each quadrant that cannot be photoelectric conversion, also called blind area. As shown in Fig. 1b., the size d of the gap affects the detection accuracy of the 4QD to the spot. It has been proved that the existence of gap can bring errors to position detection results [10,19]. The laser is condensed by a coupling lens, and the concentrated spot is irradiated on the 4QD photo-sensitive surface. A laser spot with Gaussian distribution is irradiate on the photo-sensitive surface, and each quadrant receives a different amount of light energy. Therefore, each quadrant on the 4QD photo-sensitive surface will produce different sizes of photocurrent due to the photoelectric effect. Assume that the light energy received by the four quadrants are PA , PB , PC and PD , the corresponding photocurrents are IA , IB , IC and ID . When the optical path is affected by the jitter, the position of the spot on the photo-sensitive surface will also jump, and the photocurrent of each quadrant also changes in real time. At this point, the position of the centroid of the spot can be judged by the magnitude of the current generated by each quadrant. So, the position of the centroid of the spot can be presented as [20]:
(IA + ID ) − (IB + IC ) (P + PD ) − (PB + PC ) = A IA + IB + IC + ID PA + PB + PC + PD (IA + IB ) − (IC + ID ) (PA + PB ) − (PC + PD ) Ey = = IA + IB + IC + ID PA + PB + PC + PD
Ex =
(1)
Where Ex and Ey are the estimated positions of the spot on the x-axis and the y-axis in the coordinate system, which can be used to determine the approximate position of the spot on the 4QD photo-sensitive surface. However, the estimated value is not the actual position of the spot centroid. We need to construct a spot position detection system based on 4QD to calculate the actual position through the algorithm. The model of the system constructed is shown in Fig. 2. The system is divided into two parts, one part is the optical part consists of laser and optical system, and the other part is the electronics part consists of 4QD and processing circuit. After
Fig. 1. Principle of measurement based on 4QD (a) 4QD structure; (b) principle of position detection. 2
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
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Fig. 2. Schematic of 4QD detection system.
the electrical signal is processed by a certain detection algorithm, we can obtain the spot position which is closer to the actual value.
2.1. Detection algorithm based on Gaussian spot Based on the distribution model of the spot energy, the actual position can be solved by the estimated value of the position of the centroid of the spot. In practical applications, the laser is imaged on the 4QD photo-sensitive surface after being transformed by the optical system. The energy distribution of the spot is usually a circular point with Gauss distribution, which can be expressed as follows:
2((x − X )2 + (y − Y ))2 ⎤ 2P0 h ⎛⎜x , y ⎞⎟ = exp ⎡− 2 ⎢ ⎥ πω ω2 ⎣ ⎦ ⎝ ⎠
(2)
Where P0 is the total energy of the spot, ω is the waist radius of the Gaussian beam, and (X, Y) is the coordinate position of the centroid of the spot. Ignore the effects of light energy outside the 4QD photo-surface and the gap, the integration interval becomes infinite, and we can get the spot energy distribution model: +∞
PA + PD =
SA+ SD
−∞ +∞
PB + PC =
⎛
+∞
⎞
∬ h (x, y) dxdy = ∫ ⎜ ∫ h (x, y) dx ⎟ dy ⎝
0
⎠
0
⎛ ⎞ h (x , y ) dx ⎟ dy ⎜ ⎝ −∞ ⎠
∬ h (x, y) dxdy = ∫ ∫ SB + SC
−∞
(3)
Since the positions of the spots on the x-axis and the y-axis are independent of each other, We can use the algorithm to analyze the spot position on the x-axis separately [21]. Bring (3) into (1) to get: +∞
Ex =
2(y − y0 )2 4 ⎛ dy πω2 ⎜−∞ ω2 ⎝
∫
∞
∫ − 2(x ω−2 x 0) 0
2
2 x0 ⎞ ⎞ ⎟ dx ⎟ = erf ⎛⎜ ⎝ ω ⎠ ⎠
2 2 Where erf ⎛⎜⎟⎞ = π ∫ e−t dt is the error function, from above formula, the solution value can be obtained. ⎝⎠ erf −1 (Ex ) ω x0 ≈ 2
(4)
(5)
In this way, we obtain the approximate solution value of the spot position with the Gaussian spot as the incident light.
3. Model derivation and discussion of improved algorithm In this section, we establish a spot detection model that takes into account the effects of blind area. Then, in order to further improve the detection accuracy and linearity of the algorithm, we introduce an error compensation factor function into the model, and discuss the influence of the size of the gap on spot detection. Considering the Gaussian distribution of the spot on the photo-sensitive surface, the blind area cannot produce the corresponding photocurrent, and it is difficult to express the photocurrent by (3). Therefore, it is necessary to use (6) to represent the photocurrents of each quadrant considering the influence of the blind area: 3
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
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PA + PD =
∬ h (x, y) dxdy − ∬ h (x, y) dxdy SA+ SD
PB + PC =
Sgap
∬ h (x, y) dxdy − ∬ h (x, y) dxdy SB + SC
(6)
Sgap
After obtaining the currents generated by each quadrant, we can still obtain the solution value by position detection algorithm based on the Gaussian spot. The solution value is the approximate expression of the centroid position, which is called the preliminary solution value because of its inadequate accuracy. In order to further improve the detection accuracy and the linear measurement range, an error compensation factor function is introduced. So, the preliminary solution value of the centroid position of the spot can be rewritten as follows:
x = g (Ex )•ω•λ (x 0)
(7)
erf −1 (E
Where g (Ex ) = x )/ 2 represents the spot position function of the approximate solution value, which determines the overall change trend of the function curve. The incident radius ω of the spot can be regarded as the ratio coefficient between the centroid position of spot and the calculated value function of the detector. Since different incident radii lead to different error curves, the latter two items are combined into a θc = ω•λ (x 0) to characterize the contribution of the spot radius to the position error. In order to obtain the error compensation factor function, a Gaussian spot with a radius ω can be moved from (-0.5 mm, 0) to (0.5 mm, 0) at a very small distance intervals . We record the actual position X of each spot and the output current values IA 、 IB 、 IC 、 ID at s interval and calculate the solution value Ex of each point according to the solution formula, then obtain a theoretical one-toone correspondence value. The (Xi , Exi ) value of each point is also obtained. According to the least square method, the mathematical model of position residual is established. N
I (θc ) = || δx ||2 =
∑ [x (Exi, θc ) − Xi ]2
(8)
i=1
Find the first derivative of θc in the above formula, and make the first derivative be 0, we can get the optimalθc . N
θc =
∑i = 1 g (x 0i )•Xi N
∑i = 1 g 2 (x 0i )
(9)
In practical applications, the incident spot radius is a constant value, so the error compensation factor term is only a function of the preliminary solution value x 0 . The expression of the θc can be obtained by the method of polynomial fitting. n
θc = A0 +
∑ Ai x 0i
(10)
i=1
Therefore, the improved polynomial fitting algorithm expression for the spot position detection result is:
x=
erf −1 (σx ) ⎛ •ω•⎜A0 + 2 ⎝
n
∑ Ai x 0i ⎞⎟ i=1
(11)
⎠
At this point, we have obtained a model of the modified algorithm by considering the influence of the blind area on the accuracy and introducing the error compensation factor function. Compared with the traditional polynomial fitting algorithm, the new model effectively improves the spot position detection accuracy while ensuring a wide linear measurement range, which can be proved in the experimental results. 4. Experiment analysis In order to verify the validity of the new model, we design the QD detection system as shown in Fig. 3. The system includes a fiber laser with wavelength 1550 nm and a InGaAs QD detector (HAMAMATSU G6849-01) with Photo-sensitive surface diameter 1 mm and gap size 0.03 mm. The laser beam passes through a collimating lens to become a parallel light, which is attenuated by an attenuator and then focused by a coupling lens. The spot falls on the QD photo-surface, and its size is changed by adjusting the spacing between the QD and the coupling lens. The QD is mounted on a micro-displacement platform to achieve um-level displacement accuracy. By moving the micro-displacement platform, the spot can move along the x-axis on the detector. We move the spot with a radius of 0.5 mm from (-0.5 mm, 0) to (0.5 mm, 0) and record the actual position of the spot and the output current value of each quadrant every 20um. The solution value Ex of each point is calculated according to (1). Fig. 4a. shows the correspondence between the actual position detected by the detector and the solution value Ex , while Fig. 4b. shows the correspondence between the actual position and the approximate solution value g (Exi ) of each point. It can be seen from the figure that there is good detection accuracy only in a small range near the center of the detector, and the position detection accuracy is poor on a wide range. Therefore, it needs to be further processed by the detection algorithm. Then we compare the detection results of traditional detection algorithms. The experimental results are shown in Fig. 5. The results show that the central approximation method and the geometric approximation method also have better detection accuracy in the center range. The geometric approximation shows a better detection effect than the central approximation in a wide range. 4
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
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Fig. 3. QD detection system.
Fig. 4. (a) Correspondence between actual position andEx , (b) Correspondence between actual position and g (Exi ) .
Fig. 5. Comparison of experimental results between two traditional algorithms: geometric approximation (red) and central approximation (blue) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
However, the linear measurement range of the two algorithms is small. In practical applications, once the spot exceeds this range, the position information will be wrong. In some high-precision applications, these two algorithms will fail. The proposed algorithm model was obtained by fitting the experimental data using MATLAB toolbox. By comparison, we choose the fifth-degree polynomial fitting expression, and the evaluation parameter of the linear fitting result are the sum of squares for error (SSE) = 0.0377, R-square = 0.9908, adjusted R-square = 0.9896 and root-mean-square error (RMSE) = 0.0307. Curve fitting results 5
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W. Xuan, et al.
Fig. 6. New fifth degree polynomial curve.
are shown in Fig. 6.
x = −0.5597 × x 05 − 0.001697 × x 04 − 0.2894 × x 03 − 0.0005438 × x 02 + 0.9585 × x + 0.0009817
(12)
Fig. 7. shows the error between fitting results and actual position. It can be seen from the experimental results that the position detection accuracy is kept high in a wide detection range. e also obtain the new fifth-degree polynomial fitting position detection model without considering the influence of the blind area in the model, and show the contribution of the gap in the model by comparing with the improved model. The comparison results of the detection errors between the two detection models are shown in Fig. 8. The results show that the detection accuracy of the algorithm can be greatly improved when considering the influence of the blind area in the model. In order to verify the advancement of the new detection algorithm over traditional detection algorithms, we compare the improved model with the traditional polynomial algorithm. Fig. 9. shows the spot position detection error of these models. It can be seen from the experimental results that the improved model not only greatly improves the detection accuracy and detection range, but also has advantages in the complexity of the algorithm. The results of the detection accuracy of these algorithms are shown in Table 1. 5. Conclusions This paper introduces the principle of spot position measurement of QD detector, derives the position detection algorithm model by Gaussian spot as the incident light model, get a new modified algorithm. The improved polynomial fitting detection model is obtained by introducing the error compensation factor function and considering the influence of the blind area on the accuracy in the model. Based on the analysis and comparison of traditional detection algorithms, we summarize their shortcomings. The simulation and experimental results show that considering the blind area in the model will greatly improve the system detection accuracy. In addition, the experimental results show that the improved detection algorithm has higher detection accuracy than the traditional algorithm. The linearity of the improved polynomial detection algorithm over a wide detection range shows better results than the traditional algorithm, which means that the detector has a wide effective measurement range, which is exactly what we expect. Our model improves the linear measurement rang and accuracy without increasing the algorithm complexity. Therefore, our algorithm has good practicability in applications such as laser communication.
Fig. 7. New fifth degree polynomial algorithm detection error. 6
Optik - International Journal for Light and Electron Optics 203 (2020) 163941
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Fig. 8. Comparison of detection error between model with gap and model without gap.
Fig. 9. Detection error result. Table 1 Comparison of model detection accuracy. An example of a column heading
Maximum error (mm)
RMSE (mm)
central approximation model geometric approximation model Traditional ninth degree polynomial Improved model
0.1081 0.2648 0.003060 0.001353
0.04303 0.07434 0.001518 0.0004596
Acknowledgements The authors express sincere thanks for the experiments provided by the Photoelectric Tracking and Measurement Technology Laboratory, Xi'an Institute of Optics and Precision Mechanics, CAS. Laser communication project of Xi'an Institute of Optics and Precision Mechanics, (Grant No.Y655811213), China References [1] T. Nagatsuma, G. Ducournau, C.C. Renaud, Advances in terahertz communications accelerated by photonics, Nat. Photonics 10 (6) (2016) 371–379. [2] L. Jiang, L.Z. Zhang, C. Wang, Optical multiaccess free-space laser communication system, Opt. Eng. 55 (8) (2016) 086102. [3] Y. Ni, J. Wu, X. San, S. Gao, S. Ding, J. Wang, T. Wang, Deflection angle detecting system for the large-angle and high-linearity fast steering mirror using quadrant detector, Opt. Eng. 57 (2) (2018) 024110. [4] N.O. Perez-Arancibia, J.S. Gibson, T.C. Tsao, Frequency-weighted minimum-variance adaptive control of laser beam jitter, IEEE. ASME 14 (3) (2009) 337–348. [5] N. Tsuchiya, S. Gibson, T.C. Tsao, M. Verhaegen, Receding-horizon adaptive control of laser beam jitter, IEEE. ASME 21 (1) (2015) 227–237. [6] H. Yoon, B.E. Bateman, B.N. Agrawal, Laser beam jitter control using recursive-least-squares adaptive filters, J. Dyn. Syst-T. ASME. 133 (4) (2011) 041001. [7] Q. Li, S. Xu, J. Yu, L. Yan, Y. Huang, An Improved Method for the Position Detection of a Quadrant Detector for Free Space Optical Communication, Sensors. 19 (1) (2019) 175. [8] X. Hao, C. Kuang, Y. Ku, X. Liu, Y. Li, A quadrant detector based laser alignment method with higher sensitivity, Optik. 123 (24) (2012) 2238–2240. [9] C. Lu, Y.S. Zhai, X.J. Wang, Y.Y. Guo, Y.X. Du, G.S. Yang, A novel method to improve detecting sensitivity of quadrant detector, Optik. 125 (14) (2014)
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