﻿ 基于光学衍射神经网络的拉盖尔–高斯光束识别
 光学仪器  2024, Vol. 46 Issue (2): 77-85 PDF

1. 上海理工大学 光子芯片研究院，上海 200093;
2. 上海理工大学 光电信息与计算机工程学院，上海 200093

 图 1 LG光束的DNN识别概念图 Figure 1 DNN identification concept diagram of LG beam
1 基于LG光束识别的DNN网络 1.1 拉盖尔–高斯光束

LG 光束是亥姆霍兹方程在缓变振幅近似下的一个特解，可以表示为

 $\begin{split} L{G}_{pl}\left(r,\phi , {\textit{z}}\right)= & \dfrac{{C}_{pl}}{{w}_{0}}{\left(\dfrac{\sqrt{2}r}{w\left( {\textit{z}}\right)}\right)}^{l}{L}_{p}^{\left|l\right|}\left(\dfrac{2{r}^{2}}{w{\left( {\textit{z}}\right)}^{2}}\right)\mathrm{exp}\left(-\dfrac{{r}^{2}}{w\left( {\textit{z}}\right)}\right)\\ & \mathrm{exp}\left(\mathrm{i}l\varphi \right)\mathrm{exp}\left(\mathrm{i}\phi \right) \end{split}$ (1)

 $\begin{array}{c}w\left( {\textit{z}}\right)={w}_{0}\sqrt{1+{\left(\dfrac{ {\textit{z}}}{f}\right)}^{2}} \end{array}$ (2)
 $\begin{array}{c}\phi =\left(l+2p+1\right)\mathrm{arctan}\dfrac{ {\textit{z}}}{f}-k\left( {\textit{z}}+\dfrac{{r}^{2}}{2R}\right) \end{array}$ (3)

${L}_{p}^{\left|l\right|}\left(\vartheta \right)$为缔合拉盖尔多项式

 $\begin{array}{c}{L}_{p}^{\left|l\right|}\left(\vartheta \right)=\displaystyle\sum _{m=0}^{p}\dfrac{\left(p+\left|l\right|\right)!{\left(-\vartheta \right)}^{m}}{\left(\left|l\right|+m\right)!m!\left(p-m\right)!} \end{array}$ (4)

 图 2 LG光束模拟图 Figure 2 LG beam simulation diagram
1.2 大气湍流模型及其影响

Davis[26]根据大气折射率结构常数${C}_{n}^{2}$数值的大小，将大气湍流划分成3种：强湍流，${C}_{n}^{2} > 2.5\times {10}^{-13}\;{{\rm{m}}}^{-\frac{2}{3}}$；中湍流，${6.4\times {10}^{-17}\;{{\rm{m}}}^{-\frac{2}{3}} <} C_{n}^{2} < 2.5\times {10}^{-13}\;{{\rm{m}}}^{-\frac{2}{3}}$；弱湍流，${C}_{n}^{2} < 6.4\times {10}^{-17}\; {{\rm{m}}}^{-\frac{2}{3}}$

 $\begin{split} {\phi }_{n}\left(k\right)=& 0.033{C}_{n}^{2}\mathrm{exp}\left(\dfrac{{k}_{x}^{2}+{k}_{y}^{2}}{{k}_{l}^{2}}\right){\left({k}_{x}^{2}+{k}_{y}^{2}+\dfrac{1}{{L}_{0}}\right)}^{\frac{11}{6}} \times \\ & \left(1+1.802\sqrt{\dfrac{{k}_{x}^{2}+{k}_{y}^{2}}{{k}_{l}^{2}}}-0.254\dfrac{{k}_{x}^{2}+{k}_{y}^{2}}{{k}_{l}^{2}}\right) \end{split}$ (5)

 $\begin{array}{c}{\varphi }_{n}\left(x,y\right)={\mathcal{F}}^{-1}\left\{{{\boldsymbol{C}}}\times \dfrac{2{\text{π}}}{N\Delta x}\sqrt{\varphi \left({k}_{x},{k}_{y}\right)}\right\} \end{array}$ (6)

 图 3 不同强度下的AT对相位的影响 Figure 3 Phase effects under different ATs
1.3 用于LG光束识别的DNN网络

 图 4 不同LG模式的DNN识别图 Figure 4 DNN identification diagram of different LG modes
 $\begin{array}{c}U\left(x,y,z\right)={\mathcal{F}}^{-1}\left\{{A}_{0}\left({f}_{x},{f}_{y},0\right)H\left({f}_{x},{f}_{y}\right)\right\}\end{array}$ (7)

 $\begin{array}{c}E\left({\varphi }_{i}^{q}\right)=\dfrac{1}{K}{{\displaystyle\sum }_{K}\left({s}_{K}^{M+1}-{g}_{K}^{M+1}\right)}^{2} \end{array}$ (8)

 $\begin{split} \dfrac{\partial E\left({\varphi }_{i}^{q}\right)}{\partial {\varphi }_{i}^{q}}=& \dfrac{4}{k}{\displaystyle\sum }_{k} \left ({s}_{k}^{M+1}-{g}_{k}^{M+1}\right)\cdot \Biggr.\\ & {\rm{Real}}\left\{({m}_{k}^{M+1}{)}^{\mathrm{*}}\dfrac{\partial {m}_{k}^{M+1}}{\partial {\varphi }_{i}^{q}}\right\} \end{split}$ (9)

1.4 用于AT影响下LG光束识别的DNN网络

 图 5 不同大气湍流强度下的LG模式识别 Figure 5 LG pattern recognition under different ATs

 图 6 不同ATs强度下训练识别准确率曲线 Figure 6 Training recognition accuracy curve under different ATs
2 讨　论 2.1 相位层个数影响

2.2 相位层对准误差

 图 7 引入偏移训练的DNN模型 Figure 7 DNN model with the grating
3 结　论

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