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四象限探测器是利用集成电路光刻蚀技术制作的一种光电器件,如图1所示,将一个光敏面等分成为四个形状相同的区域,每个区域相当于一个独立的光电器件,而四个象限则具有相同的性能参数[6]。由于四象限探测器具有分辨率高,响应时间短,灵敏度高,体积和结电容较小,抗干扰能力强等特点,因而被广泛应用于光学定位[7-8]、激光定准[9-10]、信号传输[11-12]等诸多领域。
四象限探测器进行光斑定位时,光斑落在光敏面上,后经过光电转换和电信号放大等过程,得到四个象限光照强度所对应的电压值,由于四个象限的处理电路在理论上完全相同,因而放大系数一致。四象限探测器光斑中心坐标常采用比幅归一算法[13],将光斑近似看作均匀分布可得到:
$$\begin{split} & {{{\Delta x}} = \frac{{\left( {{U_A} + {U_D}} \right) - \left( {{U_B} + {U_C}} \right)}}{{\left( {{U_A} + {U_B}} \right) + \left( {{U_C} + {U_D}} \right)}}}\\ & {{{\Delta y}} = \frac{{\left( {{U_A} + {U_B}} \right) - \left( {{U_C} + {U_D}} \right)}}{{\left( {{U_A} + {U_B}} \right) + \left( {{U_C} + {U_D}} \right)}}} \end{split}$$ (1) 式中:光斑中心位置为
$({{\Delta x}},{{\Delta y}})$ ;A、B、C、D四个象限对应的输出电压值为UA、UB、UC、UD。该方法采用如图2所示的中心开孔型四象限探测器,在探测器光敏面上利用刻蚀技术进行打孔处理,再与光纤端口进行耦合,在接收光斑信号的同时四象限探测器利用孔外的余晖实现星象和光纤的耦合情况的实时监测。
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天文观测中所用光纤芯径通常在几十微米至几百微米之间,比如LAMOST中通常使用320 μm的光纤,最粗的光纤达440 μm[14]。将四象限探测器应用于光纤定位系统,在星象光斑完全落入四象限中心孔时,在光斑覆盖中心小孔时,孔内四个象限的光通量近似地相等,四个象限内光通量分别是孔内光通量总量的1/4,如公式(2)所示:
$$\begin{split} & {{U_A} = {U_{AO}} + 1/4{U_H}}\\ & {{U_{B = }}{U_{BO}} + 1/4{U_H}}\\ & {{U_C} = {U_{CO}} + 1/4{U_H}}\\ & {{U_D} + {U_{DO}} + 1/4{U_H}} \end{split}$$ (2) 式中:UH为孔内的光辐射能量对应的电压值。进而,将公式(2)代入公式(1)中,化简可以得到打孔之后光斑中心位置的比幅归一公式,如公式(3)所示:
$$\begin{split} & {{{\Delta x}} = \frac{{\left( {{U_{A0}} + {U_{D0}}} \right) - \left( {{U_{B0}} + {U_{C0}}} \right)}}{{{U_{A0}} + {U_{B0}} + {U_{C0}} + {U_{D0}} + {U_H}}}}\\ & {{{\Delta y}} = \frac{{\left( {{U_{A0}} + {U_{B0}}} \right) - \left( {{U_{C0}} + {U_{D0}}} \right)}}{{{U_{A0}} + {U_{B0}} + {U_{C0}} + {U_{D0}} + {U_H}}}} \end{split}$$ (3) 望远镜星象轮廓可近似为二维高斯分布,星象落在四象限探测器上的光斑中心位置坐标为
$({{\Delta x}},\;{{\Delta y}})$ ,I(x,y)表示(x,y)处的光强值,则二维高斯分布的能量分布概率密度函数如公式(4)所示[15]:$${I_{(x,y)}} = \frac{{{I_0}}}{{2{\text{π}} {\sigma ^2}}}\exp [ - \frac{{{{(x - \Delta x)}^2} + {{(y - \Delta y)}^2}}}{{2{\sigma ^2}}}]$$ (4) 式中:
$\sigma $ 为高斯分布的束腰半径,它反映了高斯函数的衰减程度。四象限探测器焦面上的电流和电压值可以分为孔内(IH)和孔外(Ii)两个部分,即可得到:$$\begin{split} & {{I_0} = {I_i} + {I_o} = {I_{AO}} + {I_{BO}} + {I_{CO}} + {I_{DO}} + {I_H}}\\ & {U = k{I_0} = k\left( {{I_i} + {I_H}} \right) = {U_{AO}} + {U_{BO}} + {U_{CO}} + {U_{DO}} + {U_H}} \end{split}$$ (5) 在四象限探测器中,直角坐标系下,可以得到各象限电压与偏移量对应关系:
$$\begin{split} & {U_A} = \frac{{k{I_0}}}{{2{\text{π}} {\sigma ^2}}}\mathop \smallint \nolimits_0^\infty \mathop \smallint \nolimits_0^\infty {\rm{exp}}\left[ { - \frac{{{{\left( {x - \Delta x} \right)}^2} + {{\left( {y + \Delta y} \right)}^2}}}{{2{\sigma ^2}}}} \right]{\rm{d}}x{\rm{d}}y\\ & {U_B} = \frac{{k{I_0}}}{{2{\text{π}} {\sigma ^2}}}\mathop \smallint \nolimits_0^\infty \mathop \smallint \nolimits_{ - \infty }^0 {\rm{exp}}\left[ { - \frac{{{{\left( {x - \Delta x} \right)}^2} + {{\left( {y + \Delta y} \right)}^2}}}{{2{\sigma ^2}}}} \right]{\rm{d}}x{\rm{d}}y\\ & {U_C} = \frac{{k{I_0}}}{{2{\text{π}} {\sigma ^2}}}\mathop \smallint \nolimits_{ - \infty }^0 \mathop \smallint \nolimits_{ - \infty }^0 {\rm{exp}}\left[ { - \frac{{{{\left( {x - \Delta x} \right)}^2} + {{\left( {y + \Delta y} \right)}^2}}}{{2{\sigma ^2}}}} \right]{\rm{d}}x{\rm{d}}y\\ & {U_D} = \frac{{k{I_0}}}{{2{\text{π}} {\sigma ^2}}}\mathop \smallint \nolimits_{ - \infty }^0 \mathop \smallint \nolimits_0^\infty {\rm{exp}}[ - \frac{{{{\left( {x - \Delta x} \right)}^2} + {{\left( {y + \Delta y} \right)}^2}}}{{2{\sigma ^2}}}]{\rm{d}}x{\rm{d}}y \end{split}$$ (6) 采用比幅归一算法计算,则需要得到四象限探测器上、下、左、右四个半区的电压值公式。利用公式(6)可以得到公式(7):
$$\begin{split} {U_A} + {U_D} =\;& \frac{{k{I_0}}}{{2{\text{π}} {\sigma ^2}}}\mathop \smallint \nolimits_0^\infty \mathop \smallint \nolimits_{ - \infty }^\infty \exp \left[ { - \frac{{{{\left( {x - {\rm{\Delta }}x} \right)}^2} + {{\left( {y - {\rm{\Delta }}y} \right)}^2}}}{{2{\sigma ^2}}}} \right]{\rm{d}}x{\rm{d}}y =\\ & \frac{{k{I_0}}}{{\sqrt {2{\text{π}} } \sigma }}\mathop \smallint \nolimits_0^\infty \exp \left[ { - \frac{{{{\left( {x - {\rm{\Delta }}x} \right)}^2}}}{{2{\sigma ^2}}}} \right]{\rm{d}}x\\ {U_A} + {U_B} =\;& \frac{{k{I_0}}}{{2{\text{π}} {\sigma ^2}}}\mathop \smallint \nolimits_{ - \infty }^\infty \mathop \smallint \nolimits_0^\infty \exp \left[ { - \frac{{{{\left( {x - {\rm{\Delta }}x} \right)}^2} + {{\left( {y - {\rm{\Delta }}y} \right)}^2}}}{{2{\sigma ^2}}}} \right]{\rm{d}}x{\rm{d}}y =\\ & \frac{{k{I_0}}}{{\sqrt {2{\text{π}} } \sigma }}\mathop \smallint \nolimits_0^\infty \exp \left[ { - \frac{{{{\left( {y - {\rm{\Delta }}y} \right)}^2}}}{{2{\sigma ^2}}}} \right]{\rm{d}}y \end{split}$$ (7) 代入公式(3)即可得到中心开孔型四象限探测器的高斯中心定位公式,如公式(8)所示:
$$\begin{split} & \frac{{{U_{AO}} + {U_{DO}} + 1/2{U_H}}}{{{U_{AO}} + {U_{BO}} + {U_{CO}} + {U_{DO}} + {U_H}}} =\\ & \frac{1}{{\sqrt {2{\text{π}} } \sigma }}\mathop \smallint \nolimits_0^\infty \exp \left[ { - \frac{{{{(x - \Delta x)}^2}}}{{2{\sigma ^2}}}} \right]{\rm{d}}x\\ & \frac{{{U_{AO}} + {U_{BO}} + 1/2{U_H}}}{{{U_{AO}} + {U_{BO}} + {U_{CO}} + {U_{DO}} + {U_H}}} = \\ & \frac{1}{{\sqrt {2{\text{π}} } \sigma }}\mathop \smallint \nolimits_0^\infty \exp \left[ { - \frac{{{{\left( {y - \Delta y} \right)}^2}}}{{2{\sigma ^2}}}} \right]{\rm{d}}y \end{split}$$ (8) 在公式(8)中,有束腰半径σ和偏移量坐标(∆x,∆y)三个变量,通过定准(可利用四个象限的电压值数据进行比对验证定准)实现偏移量置零可以解得束腰半径σ。束腰半径对于单个单元是一个常量,因此同一个单元的偏移量坐标(∆x,∆y)可近似地采用恒定的σ进行计算。
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实验中使用如图3所示的系统,以模拟中心开孔型四象限探测器与光纤的耦合体在实际天文观测中的性能。氦氖激光器在实验中常用来模拟星光,此实验中使用的激光器波长为632.8 nm。鉴于实验中所用激光器功率远超实际星光辐照功率,实验中使用由扩束镜、准直透镜、光阑、会聚透镜组成的光学系统来模拟望远镜成像系统。
实验中使用的四象限探测器采样频率为2.5 kHz,
四象限探测器单象限取样电压范围为(−1 V,1 V),四象限探测器零点偏置范围为(−2 mm,2 mm),四象限探测器测量单位为mm,以VR-L表示四象限探测器右半焦面与左半焦面的电压之差,VT-B表示四象限探测器上半焦面与下半焦面的电压之差,VSUM表示四象限探测器整个焦面的电压之和,(X,Y)为实际的光斑中心坐标位置。由中心开孔型四象限探测器定位原理可以得到: $$\begin{split} & {{V_R} = 1/2\left( {{V_{SUM}} + {V_{R - L}}} \right)}\\ & {{V_T} = 1/2\left( {{V_{SUM}} + {V_{T - B}}} \right)} \end{split}$$ (9) 式中:VR表示右半焦面的电压值;VT表示上半焦面的电压值。
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在单方向上实现定准,使得光斑关于沟道对称,由VR、VT、X、Y四个值代入公式(8)即可得到光斑的束腰半径大小,光斑束腰半径大小测定获得的部分数据如表1所示。为减小误差,如表2所示,本实验中进行三次重复实验分别获得束腰半径的值,得到的结果取平均值,得到光斑束腰半径为3.09 mm。
表 1 光斑束腰半径的测定数据
Table 1. Measurement data of the beam waist radius
No. VR-L VT-B VSUM/V X/mm Y/mm A0001 −0.113 −0.002 0.662 −0.681 −0.015 A0002 −0.112 −0.004 0.674 −0.666 −0.021 A0003 −0.113 −0.005 0.670 −0.676 −0.027 … … … … … … A1500 −0.111 −0.002 0.665 −0.669 −0.012 表 2 束腰半径确定的实验结果
Table 2. Experimental results of waist radius determination
VR-L VT-B VSUM/V X/mm Y/mm VR/VSUM R/mm −0.112 0.003 0.667 −0.673 −0.018 0.416 3.17 −0.278 0 0.679 −1.639 0.003 0.295 3.04 0.255 0 0.672 1.517 −0.003 0.690 3.05 -
得到光斑束腰半径之后,即可进入定位阶段,利用图3中所示的光学实验系统,即可采集到如表3所示的实验数据,利用中心开孔型四象限探测器的高斯中心定位算法可得到光斑中心偏移量。由于实验中采用的是未打孔的四象限探测器,模拟中心开孔型四象限探测器需要对采样数据进行预处理。
表 3 中心开孔型四象限探测器模拟采样数据
Table 3. Center-opening 4-Q detector simulation sampling data
No. VR-L VT-B VSUM/V X/mm Y/mm D0001 0.003 −0.004 0.652 0.017 −0.021 D0002 0.003 −0.004 0.650 0.020 −0.022 D0003 0.002 −0.002 0.650 0.014 −0.015 … … … … … … D2000 0.001 −0.004 0.648 0.007 −0.022 图4中,在焦面上黑色区域内即为中心开孔区域,即
图 4 四象限探测器上中心开孔虚拟区域示意图
Figure 4. Schematic diagram of the virtual area of the center-drilling on the 4-Q detector
$${X^2} + {Y^2} \le {R^2}$$ (10) 区域内的采样数据在实际的中心开孔型四象限探测器之中无法得到需要剔除,其中R表示圆孔半径大小。
天文观测中所用光纤芯径通常在几十微米至几百微米之间,比如LAMOST中通常使用320 μm的光纤,最粗的光纤达440 μm[14]。本模拟装置中四象限探测器的零点偏置直径为4 mm,据此,取光纤截面积在1 000 μm2范围内进行计算,得到绝对误差,计算数据如表4所示,其中(X’,Y’)为计算得到的理论中心坐标值,R2=0时即为高斯拟合的未开孔四象限探测器中心结果,Er(X)、Er(Y)为实验定位绝对误差,光斑实际中心位置坐标为(0.015 0, −0.020 1),单位为mm。
表 4 中心开孔型四象限探测器模拟结果
Table 4. Simulation results of the center-drilled 4-Q detector
R2/μm2 VL/VSUM VT/VSUM X'/mm Y'/mm Er(X)/mm Er(Y)/mm 0 0.501 88 0.497 38 0.014 5 −0.020 1 −0.000 5 0 200 0.501 88 0.497 36 0.014 5 −0.020 1 −0.000 5 0 300 0.501 89 0.497 32 0.014 6 −0.020 4 −0.000 4 −0.000 3 400 0.501 93 0.497 32 0.015 0 −0.020 6 0 −0.000 5 500 0.501 99 0.497 23 0.015 5 −0.021 2 0.000 5 −0.001 1 600 0.502 06 0.497 08 0.016 0 −0.022 3 0.001 0 −0.002 2 700 0.502 18 0.496 90 0.016 9 −0.023 7 0.001 9 −0.003 6 800 0.502 20 0.496 76 0.017 1 −0.024 8 0.002 1 −0.004 7 900 0.502 33 0.496 42 0.018 2 −0.025 5 0.003 2 −0.005 4 1 000 0.502 39 0.496 22 0.018 8 −0.026 0 0.003 8 −0.005 9 光纤光谱仪望远镜光纤定位系统中对定位精度有较高要求,例如在LAMOST光纤定位系统中,要求误差控制在40 μm之内,即相对误差需控制在10%之内。从模拟结果可以看出,随光纤直径的增大,两个方向上的定位精度均有下降,但即使在光纤截面积达到1 000 μm2情况下,误差仍可以控制在6 μm之内,相对误差可以控制在0.15%,可以满足光纤定位的需要。
Closed-loop control method of optical fiber positioning of center-opening four-quadrant detector
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摘要: 光纤定位技术是多目标光纤光谱望远镜中的关键技术,光纤定位精度是影响望远镜观测效率的重要因素,随着光谱巡天项目的开展,光纤定位单元的小型化、高密度化、集成化和高精度定位要求成为普遍趋势,这对光纤定位系统提出了更高技术要求和挑战。光纤定位技术也期望实现高精度的实时监测和反馈系统,形成有效的闭环控制。基于此提出了一种中心开孔型四象限探测器光纤定位技术,并利用二维高斯模型对中心开孔型四象限探测器定位算法进行了设计,该算法对单元光斑束腰单次标定,可实现高精度的多次实时光斑位置确定和光纤位置调整。利用光纤光谱仪望远镜原理搭建了模拟实验对此装置和算法的性能进行了模拟,应用此闭环控制方法,在四象限探测器零点偏置直径为4 mm、光纤截面积达到1 000 μm2情况下,绝对定位误差可以控制在6 μm之内,相对误差可控制在0.15%范围内,可以有效提高望远镜星象和光纤的耦合效率。Abstract: Optical fiber positioning technology is a key technology in multi-target optical fiber spectroscopes. The accuracy of optical fiber positioning is an important factor affecting the observation efficiency of telescopes. With the development of the spectral survey project, the requirements for the optical fiber positioning unit to be miniaturized, high density, integrated and high precision positioning have become a general trend, which poses higher technical requirements and challenges for optical fiber positioning systems. Optical fiber positioning technology was also expected to achieve a high-precision real-time monitoring and feedback system, forming an effective closed-loop control. Based on the requirements, a center-opening four-quadrant(4-Q) detector fiber positioning technology was proposed, and a two-dimensional Gaussian model was used to design the center-opening four-quadrant detector positioning algorithm. The algorithm performs a single calibration of the unit beam spot waist, can achieve high precision multiple real-time spot position determination and fiber position adjustment. The performance of the device and algorithm was simulated with the experiment set up based on the principle of the optical fiber spectroscope telescope. With this closed-loop control method, the absolute positioning error was obtained when the four-quadrant detector had a zero offset diameter of 40 mm and a fiber cross-sectional area of 1 000 μm2. It can be controlled within 6 μm, and the relative error can be controlled within 0.15% ,which meets the requirements of the optical fiber spectroscope telescope fiber positioning technology. This device can be applied as an optical fiber positioning system.
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Key words:
- fiber positioning /
- four-quadrant detector /
- astronomy observation
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表 1 光斑束腰半径的测定数据
Table 1. Measurement data of the beam waist radius
No. VR-L VT-B VSUM/V X/mm Y/mm A0001 −0.113 −0.002 0.662 −0.681 −0.015 A0002 −0.112 −0.004 0.674 −0.666 −0.021 A0003 −0.113 −0.005 0.670 −0.676 −0.027 … … … … … … A1500 −0.111 −0.002 0.665 −0.669 −0.012 表 2 束腰半径确定的实验结果
Table 2. Experimental results of waist radius determination
VR-L VT-B VSUM/V X/mm Y/mm VR/VSUM R/mm −0.112 0.003 0.667 −0.673 −0.018 0.416 3.17 −0.278 0 0.679 −1.639 0.003 0.295 3.04 0.255 0 0.672 1.517 −0.003 0.690 3.05 表 3 中心开孔型四象限探测器模拟采样数据
Table 3. Center-opening 4-Q detector simulation sampling data
No. VR-L VT-B VSUM/V X/mm Y/mm D0001 0.003 −0.004 0.652 0.017 −0.021 D0002 0.003 −0.004 0.650 0.020 −0.022 D0003 0.002 −0.002 0.650 0.014 −0.015 … … … … … … D2000 0.001 −0.004 0.648 0.007 −0.022 表 4 中心开孔型四象限探测器模拟结果
Table 4. Simulation results of the center-drilled 4-Q detector
R2/μm2 VL/VSUM VT/VSUM X'/mm Y'/mm Er(X)/mm Er(Y)/mm 0 0.501 88 0.497 38 0.014 5 −0.020 1 −0.000 5 0 200 0.501 88 0.497 36 0.014 5 −0.020 1 −0.000 5 0 300 0.501 89 0.497 32 0.014 6 −0.020 4 −0.000 4 −0.000 3 400 0.501 93 0.497 32 0.015 0 −0.020 6 0 −0.000 5 500 0.501 99 0.497 23 0.015 5 −0.021 2 0.000 5 −0.001 1 600 0.502 06 0.497 08 0.016 0 −0.022 3 0.001 0 −0.002 2 700 0.502 18 0.496 90 0.016 9 −0.023 7 0.001 9 −0.003 6 800 0.502 20 0.496 76 0.017 1 −0.024 8 0.002 1 −0.004 7 900 0.502 33 0.496 42 0.018 2 −0.025 5 0.003 2 −0.005 4 1 000 0.502 39 0.496 22 0.018 8 −0.026 0 0.003 8 −0.005 9 -
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