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临近空间是指距离地面20~100 km的空域,包括平流层(18~55 km空域)、大气中间层(55~85 km空域)和小部分增温层区域(85~800 km空域)。将大气温度随高度变化的分布作为主要依据,可将大气分为四个层:对流层、平流层、中间层和热层。
平流层飞艇一般航行高度位于20 km,不同大气模型在此处的温度预测值不同[7],标准大气模型的温度预测值为216.65 K;极热大气模型的温度预测值为234.87 K;极冷大气模型的温度预测值为192.28 K。上述大气模型的温度随高度的变化趋势如图1所示。
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反射镜的焦距主要受其曲率半径的影响,两者成正比关系。由于离轴抛物面反射镜的特性,反射镜背板与镜面曲率中心的距离减少,背板的厚度也随之减小。在镜面工作温度为−60℃时,根据热膨胀公式可知其厚端的变形量大于薄端的变形量,反射镜在不同位置发生的形变随着厚度的增加而增大,导致曲率半径增大,反射面上各点的变形量也不同,进而使得曲率半径增大,焦点远离镜面。
$$ \Delta L = L \times \alpha \times \Delta T $$ (1) 式中:ΔL为物体热变形后的长度;L为初始温度下的长度;α为线膨胀系数;ΔT为温度的改变量。
光学元件微小变形会导致反射镜产生形状复杂的波面误差,用通常的函数难以表示,但镜面波前的变化总是趋于光滑和连续的。因此,可以使用一个完备的基函数的线性组合,或者一个线性无关的基函数系的组合来表示镜面的形状变化[8]。Fringe Zernike构造出的Zernike多项式[9−10]已应用在许多科研领域中。一般使用Zernike多项式来描述干涉图的波前像差,其线性组合能够表示任意k阶波面ω(ρ,θ),如公式(2)所示:
$$ \omega (\rho ,\theta ) = \sum\limits_{i = 1}^N {{q_i}{Z_i}(\rho ,\theta )} $$ (2) 式中:Zi为Zernike多项式的第i项;qi为第i项的系数;N为最大项数。极坐标形式的Zernike多项式具体表达式为[11]:
$$ Z_n^l(\rho ,\theta ) = \left\{ \begin{gathered} R_n^l(\rho )\cos (l\theta ),(l \leqslant 0) \\ R_n^l(\rho )\sin (l\theta ),(l > 0) \\ \end{gathered} \right. $$ (3) 式中:n取自然数,为多项式的阶数;l为与n有关的序号,其值恒与n同奇偶,且$ \left|l\right|\leqslant n $。现假设正整数m定义为$ (n-l)/2 $,则:
$$ R_n^l(\rho ) = \sum\limits_{s = 0}^m {{{( - 1)}^s}\frac{{(n - s)!}}{{s!(m - s)!(n - m - s)!}}{\rho ^{n - 2s}}} $$ (4) 确定基函数后,干涉条纹的级数分布函数如公式(5)所示[9]:
$$ F(\rho ,\theta )={\displaystyle \sum _{{i}=0}^{\infty }{q}_{i}{Z}_{n}^{n-2m}(\rho ,\theta )={q}^{{\rm{T}}}\cdot Z} $$ (5) 式中:qi为Zernike第i项系数;qT为Zernike系数即qi组成的列向量转置;Z为由Zernike多项式项组成的列向量。
在使用Zernike多项式拟合变形反射面时,需要反射面面形的离散数据。通过实验测量或结构分析方法可以获得反射镜的离散数据,例如通过有限元软件进行结构分析,导出的变形输出文件即可用作反射面的离散数据[12]。通过得到的离散变形数据和反射面镜面参数即可进行Zernike多项式拟合。通过Zernike多项式拟合可以获得各项多项式系数,从而得到以Zernike多项式表示的面形误差,并且能够快速拟合出连续光滑的变形形面。反射镜镜面拟合即用Zernike多项式作为连续基函数,用于拟合离散的波相差W(xi, yi),连续函数W(x, y)用于表征被测面形的波相差函数。常用的Zernike多项式为Fringe Zernike多项式和标准Zernike多项式[13−14],选择使用前28项的Fringe Zernike多项式拟合非球面反射镜。
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在光纤模式相关理论的分析中,光纤主要分为渐变型光纤和阶跃型光纤,分类的原则为光纤的折射率分布方式。纤芯区域的折射率会变化的为渐变型光纤,光纤包层材料的折射率大于纤芯材料折射率,且两者折射率为定值的光纤为阶跃型光纤。
光纤的归一化频率(Normalized Frequency)是决定光纤可以传输多少个模式的光场的参数,归一化频率越大,光纤可以传输的模式数就越多[15]。通过公式(6)可求得归一化频率为:
$$ \begin{gathered} {{V = }}\frac{{2\pi }}{\lambda }\omega \sqrt {n_{ce}^2 - n_{ca}^2} \approx \\ \frac{{2\pi }}{\lambda }\omega {n_{ce}}\sqrt {\frac{{2(n_{ce}^2 - n_{ca}^2)}}{{2n_{ce}^2}}} \approx \frac{{2\pi }}{\lambda }\omega {n_{ce}}\sqrt {2\Delta } \\ \end{gathered} $$ (6) 式中:ω为光纤的纤芯半径;λ为波长;nce为纤芯材料的折射率;nca为包层材料的折射率。
使用多模光纤作为空间激光耦合器件时,其具有更大的纤芯半径,相较于单模光纤和少模光纤,多模光纤具有更高的空间激光-光纤耦合效率,因此文中的激光雷达采用阶跃型多模光纤作为激光接收端。多模光纤纤芯直径为62.5 μm,纤芯材料折射率nce为1.48,包层材料折射率nca为1.46。当空间光波长λ为1.064 μm时,通过公式(6)即可求出归一化频率为44.748。多模光纤对空间激光的耦合效率为其所有线偏振模式的耦合效率之和,文中以基模模式的耦合效率为基准,其耦合效率可以表示为:
$$ {\eta }_{m.1}=\frac{{\left|{\displaystyle \iint {U}_{of}^{\ast }(r)\cdot{U}_{mf.1}(r)\text{d}s}\right|}^{2}}{{\displaystyle \iint \left|{U}_{of}{(r)}^{2}\right|\text{d}s\cdot{\displaystyle \iint \left|{U}_{mf.1}{(r)}^{2}\right|\text{d}s}}} $$ (7) 式中:Uof(r)为空间激光的入射光场经过反射镜后焦平面处的场分布,简化后只保留振幅项;Umf.1(r)为基模对应的场分布。
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设计分析的对象为离轴抛物面的反射镜系统。入射光波长λ为1.064 μm,主镜面形为正六边形,口径为650 mm的内切圆,曲率半径为2600 mm,焦距为1300 mm;平面支撑背板、后开式结构;支撑方式为背部六点式。要求在标准地球重力下镜面PV值<λ/10,耦合效率在−60 ℃时高于60%。
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针对几种常见的反射镜材料——碳化硅(SiC)、铝基碳化硅(Al/SiC)、K9、微晶玻璃(Zerodur)、熔石英(Fused silica)和背板金属材料——钛合金TC4、ZTC4、超因瓦合金4J32进行组合配对,对每种反射镜材料和背板材料在临近空间工作时的反射镜热变形和光学性能进行比较,分析其在−60 ℃下、原焦点处的RMS半径的变化量。通过热仿真分析得出了不同材料组合后的结果,不同组合下的RMS值如图2所示,其中SiC+4J32组合的RMS值为21.75 μm,是所有组合中的最小值,其在−60 ℃下的影响最小。
图 2 不同材料组合在−60 ℃下的光学性能对比
Figure 2. Comparison of optical properties of different material combinations at −60 ℃
综合考虑反射镜的机械性能、物理性能、加工工艺以及成本等因素,选定主镜材料为SiC,其具有热畸变小、比刚度大的优点且稳定性好。背板选择4J32,该材料的线膨胀系数与SiC基本一致,可减小热变形不一致带来的影响。桁架及遮光板采用碳纤维复合材料(Carbon Fiber Reinforced Polymer/Plastic, CFRP),其具有高强度、低密度的优点,应用于桁架及遮光板时可以减小自身重力带来的影响,材料的性能参数如表1所示。
表 1 反射镜常用材料属性
Table 1. Properties of commonly used materials for mirrors
Material Density
ρ/kg·m–3Elasticity
modulus
E/GPaThermal
conductivity
Kcc/W·m–1·℃–1Thermal
Expansion
α/K−1Poisson
ratio
νSiC 3200 400 270 2.5×10−6 0.18 Al/SiC 3010 215 210 7.9×10−6 0.2 K9 2510 81 1.21 7.5×10−6 0.21 Zerodur 2500 92 1.46 0.05×10−6 0.24 Fused Silica 2201 74 1.38 5.6×10−7 0.17 TC4 4440 114 6.8 9.1×10−6 0.34 ZTC4 4400 112 8.8 8.9×10−6 0.29 4J32 810 138.2 14.7 2.4×10−6 0.25 CFRP 1480 9-91.82 9.68 0.5×10−6 0.05-0.3 -
为建立有限元分析仿真,先创建天线的3D CAD模型,包括离轴抛物面反射镜、支撑结构、调焦结构和附加组件。反射镜焦距为1300 mm,口径为650 mm,y偏心为330 mm。
文中的抛物面反射镜口径为650 mm,使用背部3点支撑即可使反射镜在工作时具有较好的稳定性。但在实际工作时,由于探测角度可以进行变化,光轴存在接近竖直的工作状态,在该工作状态时该支撑方式难以满足要求。要保证光轴竖直状态的面形精度,背部支撑点的最少个数可由Friedman[16]给出的最少支撑点数计算经验公式来推算:
$$ N = \left(\frac{{1.5{r^2}}}{d}\right){\left(\frac{\rho }{{E\delta }}\right)^{{1}/{2}}} $$ (8) 式中:r为反射镜半径;d为反射镜厚度;E为弹性模量;δ为反射镜PV值。面形设计指标要求及面形PV约小于λ/10,入射光波长为1.064 μm,因此PV值要小于100 nm。由公式(8)可知,背部支撑点至少需要6个。考虑背部轻量化孔的大小,最终确定6个支撑点分布于镜体背部Φ580 mm的圆周上,呈60°均匀分布。
激光雷达系统采用全反射光学系统,不会有光束透过,反射镜的背部不参与光束传输,因此反射镜可使用刚度较强的背部支撑结构。背部支撑方式的选择首先要考虑镜面本身重力的影响,需先确定镜面厚度,设镜面在重力下变形量为δ,通过经验公式[17]:
$$ \delta = \frac{{3\rho g{r^4}}}{{16E{f_t}^2}} = \frac{{3\rho gd{r^2}{D^2}}}{{256E}} $$ (9) 式中:δ为镜面面形PV值;ρ为反射镜材料的密度;g为重力加速度;r为反射镜半径;E为材料的弹性模量;ft为反射镜厚度。由公式(9)即可得到反射镜厚度ft为58 mm,径厚比为11.2∶1。考虑镜坯制作工艺,选择三角形轻量化孔和背部半封闭式结构以满足镜体强度,并对反射镜进行轻量化。反射镜如图3所示。
引入了一定的柔性支撑结构,以此来抵消反射镜由于温度变化产生的热应力和微小变形[18]。柔性支撑结构通过在某一方向上切开一个柔性槽,以降低该方向上的刚度,体现其柔性,使其能够产生微小变形,释放热应力,只存在一个柔性槽的柔性支撑结构被称为单向柔性支撑结构。而在一般情况下,往往将多个柔性槽成组使用,即可实现在多方向上的柔性,达到释放多个自由度的目的,将其称之为多层柔性支撑结构。经过电火花线切割加工出对称分布的“L”型切口柔性槽,形成三个相对中心的空槽,进而产生能向三个方向转动的柔性环节;向上偏置一定距离后旋转90°,再加工三个对称分布的“L”型切口柔性槽,使其产生能够缓解从柔性支撑结构侧面产生的位移。使用的柔性支撑结构如图4所示。
通过上述设计确定了激光雷达主镜的结构,将桁架、遮光板、光纤座等组件加入模型后,确定了其整体结构。为了简化计算量,对光学组件模型做了部分简化,去除了信号接收端的电子元件和其外部细节。简化后的研究模型如图5所示。
Design of automatic out-of-focus correction system for near-space lidar receivers
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摘要: 针对临近空间激光雷达所使用的离轴抛物面反射镜在低温下会发生形变导致反射光耦合进光纤时效率下降的问题,通过研究自调焦技术,设计了自动补偿激光雷达组件,用于抵消温度对系统的影响。利用有限元法分析了接收系统在热载荷作用下的变形,获得镜面离散变形数据,使用Zernike多项式拟合镜面变形后的面形,通过光学设计软件模拟得出该激光雷达接收系统优化后的焦距变化与温度呈线性关系,经过光学设计软件确定补偿位置。使用温度自适应调整机构降低热变形带来的离焦量影响。分析结果表明,补偿后均方根半径从26.495 μm下降至15.93 μm,光斑半径减少39.9%,耦合效率提升至80%以上。Abstract:
Objective Lidar is the main way to obtain three-dimensional geographic information within the military, and the data results obtained through this way are also widely used in resource exploration, land use, environmental monitoring and national key construction projects, providing extremely important original information for the national economy, social development and scientific research, and has achieved significant economic benefits, showing good application prospects. The lower temperature of the near space can reach –60 ℃, the optical antenna as the core component of LiDAR, its optical components have strict requirements for temperature changes. Temperature variations can lead to thermal deformation of the element, resulting in problems of defocusing and focal plane translation, which reduces the coupling efficiency. Improving the coupling efficiency can increase the detection rate, and the off-axis reflective optical system can be realized without obstruction, which can improve the energy utilization. However, unlike conventional refractive or coaxial reflection systems, each optical element in the off-axis reflection system does not have rotational symmetry, and its temperature deformation after temperature change is not uniform, so solving the effect of temperature change on the focus of the off-axis parabolic mirror is the key to improve the coupling efficiency of LiDAR. Methods In this paper, the LiDAR is modeled, and the temperature field is simulated by the finite element analysis method for the model. The Zernike polynomials are used to fit the surface shape data obtained after the analysis to obtain the shape and position change of the reflector after temperature deformation, and the optical design software is used to obtain the optimal focal point position. Finally, the optical fiber position is adjusted by the focusing device to achieve the effect of focusing. Results and Discussions The PV value of the mirror of the designed LiDAR is less than 10/λ. Its optimal focus position curve is obtained by optical design software, and a temperature-adaptive focusing structure is designed according to this curve, through which the RMS radius of the mirror compensated by the focusing structure decreases from 26.495 μm to 15.93 μm, the spot radius is reduced by 39.9%, and the coupling efficiency is improved from 15.8% to 91%. This focusing method does not need to keep track of the changes in the temperature of the reflector and reduces a certain amount of weight and cost compared to focusing with a motor. However, the method requires a certain temperature response time, can not adjust the focus to the best position at the first time after the temperature change. If the temperature changes frequently, the motor should be used to quickly adjust. Due to the working height of the blimp is more fixed, and its ambient temperature does not change much, the method is feasible. Conclusions Aiming at the problem that the off-axis parabolic mirrors used in the near-space lidar will deform at low temperatures which leads to a decrease in the efficiency of the reflected light when it is coupled into the optical fiber, the self-focusing technique is investigated, and the auto-compensating lidar assembly is designed to offset the effect of temperature on the system. The deformation of the receiving system under thermal load is analyzed using the finite element method to obtain the discrete deformation data of the mirror surface, and the Zernike polynomials are used to fit the surface shape after the deformation of the mirror surface, and the simulation of the optical design software concludes that the change of the focal length of this LiDAR receiving system after optimization has a linear relationship with the temperature, and the compensation position is determined by the optical design software. A temperature adaptive adjustment mechanism is used to reduce the effect of out-of-focus amount caused by thermal deformation, which improves the coupling efficiency by more than 80%. -
表 1 反射镜常用材料属性
Table 1. Properties of commonly used materials for mirrors
Material Density
ρ/kg·m–3Elasticity
modulus
E/GPaThermal
conductivity
Kcc/W·m–1·℃–1Thermal
Expansion
α/K−1Poisson
ratio
νSiC 3200 400 270 2.5×10−6 0.18 Al/SiC 3010 215 210 7.9×10−6 0.2 K9 2510 81 1.21 7.5×10−6 0.21 Zerodur 2500 92 1.46 0.05×10−6 0.24 Fused Silica 2201 74 1.38 5.6×10−7 0.17 TC4 4440 114 6.8 9.1×10−6 0.34 ZTC4 4400 112 8.8 8.9×10−6 0.29 4J32 810 138.2 14.7 2.4×10−6 0.25 CFRP 1480 9-91.82 9.68 0.5×10−6 0.05-0.3 -
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