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摘要:
为了探究鬼像对成像系统性能的影响，构建了二次反射产生的鬼像影响下的调制传递函数(MTF)的计算模型。本文首先介绍了在近轴近似下的鬼像分析与描述的方法。接着，从调制传递函数的定义出发，考虑鬼像在像面处的照度对像面调制度的影响，构建了鬼像影响下的MTF计算模型。通过对一系统进行实例计算，并与仿真结果进行对比，均方误差最大不超过
0.049373 ，进而验证了该模型的准确性。同时针对误差较大的结果进行了详细分析，明确了该计算方法的适用范围。研究结果表明，利用近轴近似的方法计算鬼像影响下的MTF在大多数情况下是真实准确的，该研究在光学系统的鬼像分析方面做出了有益探索。Abstract:To investigate the impact of ghost images on optical system performance, we develope a Modulation Transfer Function (MTF) calculation model under the influence of ghost images generated by secondary reflection. We first introduce the analyzing and describing method of ghost image using the paraxial approximation. Then, starting from the definition of the MTF, and considering the influence of ghost image irradiance on the modulation of the image plane, an MTF calculation model under the influence of ghost images is constructed. By conducting a case study calculation on a system and comparing it with the simulation results, it is found that the maximum mean square error is less than
0.049373 , which verifies the accuracy of the model. Furthermore, a detailed analysis is conducted, examining cases that exhibited larger errors and clarifying the range in which this calculation method can be applied. The research results indicate that the paraxial approximation method is both accurate and reliable when calculating the MTF under the influence of ghost images is accurate, and is applicable in most cases. This study serves as a valuable exploration in the ghost image analysis of optical systems. -
图 3 4种鬼像路径下两种计算方法的MTF
Figure 3. MTF curves calculated by two methods under four types of ghost image paths
图 4 3_2鬼像路径下的两种计算方法的MTF曲线
Figure 4. MTF curves calculated by two methods under ghost image path 3_2
图 5 非序列光线追迹得到的3_2路径下鬼像光束在像面上的照度分布示意图
Figure 5. Schematic diagram of illuminance distribution of ghost image beams on image plane under 3_2 path calculated by non-sequential ray tracing
图 6 6_1鬼像路径下的MTF曲线示意图
Figure 6. MTF curves calculated by two methods under 6_1 ghost image path
图 7 非序列光线追迹得到的6_1路径下鬼像光束在像面上的照度分布示意图
Figure 7. Schematic diagram of illumination distribution of ghost image on image plane under 6_1 path calculated by non-sequential ray tracing
表 1 E0计算表达式(其中鬼像光束示意图中的蓝色光线代表鬼像光线，黑色代表成像光线)
Table 1. Calculation expressions for E0, where the blue lines in the schematic diagram of the ghost image beam represent the ghost image rays and black ones represent the imaging rays
序号 鬼像光束示意图 E0表达式 1 
$ {E_0} = \dfrac{{{n_k}[{h_k}{u_{kg}} - {h_{kg}}{u_k}]}}{H} $ 2 
$ {E_0} = \dfrac{{{n_k}[{h_k}{u_{kg}} + {h_{kg}}{u_k}]}}{H} $ 3 
$ {E_0} = \dfrac{{{n_k}[{h_{kg}}{u_k} - {h_k}{u_{kg}}]}}{H} $ 
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表 2 所有表面的传输矩阵
Table 2. Transmission matrices for all surfaces
透射矩阵 反射矩阵 前表面 $ {{\boldsymbol{M}}_{{1}}}{{ = }}\left[ {\begin{array}{*{20}{c}} {{1}}&{{0}} \\ {{{ - }}\dfrac{{{{n' - 1}}}}{{{{n'}}{{{r}}_{{1}}}}}}&{\dfrac{{{1}}}{{{{n'}}}}} \end{array}} \right] $ $ {{\boldsymbol{M}}_{{{11}}}}{{ = }}\left[ {\begin{array}{*{20}{c}} {{1}}&{{0}} \\ {{{ - }}\dfrac{{{2}}}{{{{{r}}_{{1}}}}}}&{{{ - 1}}} \end{array}} \right] $ 后表面 $ {{\boldsymbol{M}}_{{2}}}{{ = }}\left[ {\begin{array}{*{20}{c}} {{1}}&{{0}} \\ {{{ - }}\dfrac{{{{1 - n'}}}}{{{{{r}}_{{2}}}}}}&{{{n'}}} \end{array}} \right] $ $ {{\boldsymbol{M}}_{{{22}}}}{{ = }}\left[ {\begin{array}{*{20}{c}} {{1}}&{{0}} \\ {{{ - }}\dfrac{{{2}}}{{{{{r}}_{{2}}}}}}&{{{ - 1}}} \end{array}} \right] $
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表 3 三片式镜头参数数据
Table 3. Triplet lens parameter data (Unit: mm)
表面 曲率半径 厚度 材料 1 3.285 1 N-LASF31A 2 −30.01 0.142 3 −6.769 0.9 SF15 4 2.489 0.673 5 6.31 1 N-LAK21 6 −4.889 10.6713
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表 4 所有路径下的杂光率与鬼像影响下的MTF值
Table 4. The stray light rates and MTF values under the influence of ghost image in all paths
路径 杂光率 $ MTF' $与$ MTF $之间的关系 2_1 0.058858 $ MTF'=0.944\;413MTF $ 3_2 0.877073 $ MTF'=0.532\;744MTF $ 3_1 0.029316 $ MTF'=0.971\;518MTF $ 4_3 0.008263 $ MTF'=0.991\;804MTF $ 4_2 0.005352 $ MTF'=0.994\;676MTF $ 4_1 0.061024 $ MTF'=0.942\;485MTF $ 5_4 0.119049 $ MTF'=0.893\;615MTF $ 5_3 0.006792 $ MTF'=0.993\;252MTF $ 5_2 0.004984 $ MTF'=0.995\;040MTF $ 5_1 0.019154 $ MTF'=0.981\;20MTF $ 6_5 0.067233 $ MTF'=0.937\;001MTF $ 6_4 0.064353 $ MTF'=0.939\;534MTF $ 6_3 0.037417 $ MTF'=0.963\;936MTF $ 6_2 0.018945 $ MTF'=0.981\;406MTF $ 6_1 0.044890 $ MT{F}'=0.957\;038MTF $
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表 5 所有鬼像路径下MSE的值
Table 5. MSE values under all ghost image paths
路径 MSE值 2_1 1.253505 ×10−43_2 1.826590 ×10−23_1 3.627304 ×10−54_3 2.866007 ×10−64_2 2.798693 ×10−54_1 7.217275 ×10−45_4 1.933488 ×10−45_3 1.222153 ×10−55_2 9.061267 ×10−65_1 5.363379 ×10−56_5 4.459207 ×10−46_4 1.627363 ×10−46_3 4.774974 ×10−56_2 1.692015 ×10−56_1 4.937398 ×10−2
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