光学仪器  2019, Vol. 41 Issue (5): 17-23 PDF

Depolarization measurement and correction of polarized light normally incident on the corner cube
REN Yuan, HUANG Chunhui
College of Physics and Information Engineering, Fuzhou University, Fuzhou 350116, China
Abstract: In order to solve the depolarization effect of the corner cube, an elliptically polarized light correction scheme with the external wave plates is proposed. The modified formula of the corner cube with a tilted solid ridgeline is summarized, and a polarized light correction model with clockwise and counter-clockwise reflection paths is established. The analysis of the model shows that two half-wave plates and one retarder can keep the polarization state of the incident ray equal to that of the emerging ray. Based on the model, by using a 1 064 nm laser, the correction experiment on the BK7 glass corner cube whose solid ridgeline is placed vertically in clockwise and counter-clockwise reflection paths is carried out. When the angle between the polarized direction and the horizontal direction of the linearly polarized incident light varies from 90° to 0°, the maximum absolute value of the ellipticity of the light in depolarization is corrected from 35° to 5°. When the incident light is with the approximately circular polarization, its ellipticity drops from 44° to 40° after correction.
Key words: corner cube    polarization state    correction

1 正入射角锥的琼斯矩阵

 图 1 定义主坐标系和反射路线 Figure 1 Primary coordinate system and reflection path

 图 2 用不同基偏振态表示的偏振光和全反射基偏振态示意图 Figure 2 Polarized light represented by different base polarization vector and total reflection

 $\left[ {\begin{array}{*{20}{c}} {P'} \\ {S'} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\cos \theta }&{\sin \theta } \\ { - \sin \theta }&{\cos \theta } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} P \\ S \end{array}} \right] = {{ J}_{\rm r}}(\theta )\left[ {\begin{array}{*{20}{c}} P \\ S \end{array}} \right]$ (1)

 ${{ J}_{\rm R}} = \left[ {\begin{array}{*{20}{c}} {\exp (\rm - i{\delta _p})}&0 \\ 0&{\exp (\rm - i{\delta _s})} \end{array}} \right]$ (2)

 ${{ J}_{123}} = {{ J}_{\rm r}}( - \frac{{2\pi }}{3}){{ J}_{\rm R3}}{{ J}_{\rm r}}(\frac{\pi }{3}){{ J}_{\rm R2}}{{ J}_{\rm r}}( - \frac{\pi }{3}){{ J}_{\rm R1}}{{ J}_{\rm r}}( - \frac{\pi }{3})$ (3)
 ${{ J}_{321}} = {{ J}_{\rm r}}( - \frac{\pi }{3}){{ J}_{\rm R1}}{{ J}_{\rm r}}( - \frac{\pi }{3}){{ J}_{\rm R2}}{{ J}_{\rm r}}(\frac{\pi }{3}){{ J}_{\rm R3}}{{ J}_{\rm r}}( - \frac{{2\pi }}{3})$ (4)

 图 3 213反射次序下与角锥旋转时的坐标系夹角 Figure 3 Angle of 213 mode and the angle when the corner cube is inclined

 ${{ J}_{213}} = {{ J}_{\rm r}}(\frac{{2\pi }}{3}){{ J}_{321}}{{ J}_{\rm r}}(\frac{{2\pi }}{3})$ (5)

 ${{ J}_{132}} = {{ J}_{\rm r}}( - \frac{{2\pi }}{3}){{ J}_{321}}{{ J}_{\rm r}}( - \frac{{2\pi }}{3})$ (6)
 ${{ J}_{231}} = {{ J}_{\rm r}}( - \frac{{2\pi }}{3}){{ J}_{123}}{{ J}_{\rm r}}( - \frac{{2\pi }}{3})$ (7)
 ${{ J}_{312}} = {{ J}_{\rm r}}(\frac{{2\pi }}{3}){{ J}_{123}}{{ J}_{\rm r}}(\frac{{2\pi }}{3})$ (8)

 ${{ J}_{123}}(\gamma ) = {{ J}_{\rm r}}(\gamma ){{ J}_{123}}{{ J}_{\rm r}}(\gamma )$ (9)

2 正入射角锥时纠偏

 ${{E}_{\rm o}}={{ J}_{N}} {{ J}_{N-1}} \cdots {{ J}_{2}} {{ J}_{1}} {{E}_{\rm i}}$ (10)

 ${{{ J}}_{N}} {{{ J}}_{N-1}} \cdots {{{ J}}_{2}} {{{ J}}_{1}}={{I}}$ (11)

 ${{ J}_{123}} = \left[ {\begin{array}{*{20}{c}} {{\rm{0}}{\rm{.0088 + 0}}{\rm{.9594i}}}&{{\rm{0}}{\rm{.2338 + 0}}{\rm{.1576i}}} \\ {{\rm{0}}{\rm{.2338 + 0}}{\rm{.1576i}}}&{{\rm{ - 0}}{\rm{.8926 + 0}}{\rm{.3518i}}} \end{array}} \right]$ (12)

 $\begin{split} &{{ E}_{123\_1}} = \left[ {\begin{array}{*{20}{c}} {{{E}_{123{x_1}}}} \\ {{{E}_{123{y_1}}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\rm{0}}{\rm{.97152}}} \\ {{\rm{0}}{\rm{.23698 + 8}}{\rm{.3267}} \times {\rm{1}}{{\rm{0}}^{{\rm{ - 017}}}}{\rm{i}}} \end{array}} \right],\\ &{{ E}_{123\_2}} = \left[ {\begin{array}{*{20}{c}} {{{E}_{123{x_2}}}} \\ {{{E}_{123{y_2}}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\rm{0}}{\rm{.23698 + 1}}{\rm{.9429}} \times {\rm{1}}{{\rm{0}}^{{\rm{ - 016}}}}{\rm{i}}} \\ {{\rm{ - 0}}{\rm{.97152}}} \end{array}} \right]{\text{。}} \end{split}$

 图 4 123反射次序下折射率为1~11时对应的本征矢量 Figure 4 Value of the eigenvector in the 123 mode when the refractive index ranges from 1 to 11

 ${{ V}_\pi } = \left[ {\begin{array}{*{20}{c}} {{\rm{0}}{\rm{.97152}}}&{{\rm{0}}{\rm{.23698}}} \\ {{\rm{0}}{\rm{.23698}}}&{{\rm{ - 0}}{\rm{.97152}}} \end{array}} \right] = \frac{1}{{\sqrt {{\rm{0}}{\rm{.9715}}{{\rm{2}}^2} + {\rm{0}}{\rm{.2369}}{{\rm{8}}^2}} }}\left[ {\begin{array}{*{20}{c}} {{\rm{0}}{\rm{.97152}}}&{{\rm{0}}{\rm{.23698}}} \\ {{\rm{0}}{\rm{.23698}}}&{{\rm{ - 0}}{\rm{.97152}}} \end{array}} \right]$ (13)

 $\begin{split} & {{ J}_{A}}={{ V}_{\pi }} {{ J}_{123}} {{ V}_{\pi }}=\left[ \begin{matrix} {\rm 0}{\rm .06585+0}{\rm .99783i} & {\rm 1}{\rm .1582}\times {\rm 1}{{{\rm 0}}^{{\rm -6}}}{\rm +7}{\rm .8065}\times {\rm 1}{{{\rm 0}}^{{\rm -7}}}{\rm i} \\ {\rm 1}{\rm .1582}\times {\rm 1}{{{\rm 0}}^{{\rm -6}}}{\rm +7}{\rm .8065}\times {\rm 1}{{{\rm 0}}^{{\rm -7}}}{\rm i} & {\rm -0}{\rm .94963+0}{\rm .31337i} \\ \end{matrix} \right] \\ & \quad \approx \left[ \begin{matrix} {\rm 0}{\rm .06585+0}{\rm .99783i} & 0 \\ 0 & {\rm -0}{\rm .94963+0}{\rm .31337{\rm i}} \\ \end{matrix} \right]=\left[ \begin{matrix} {{\rm e}^{1.5049{\rm i}}} & 0 \\ 0 & {{\rm e}^{2.8229{\rm i}}} \\ \end{matrix} \right] \end{split}$ (14)

 $\begin{split} &{{ J}_{A}}{{ D}_{0}}={{ D}_{0}}{{ J}_{A}}={{ D}_{0}}{{ V}_{\pi }}{{ J}_{123}}{{ V}_{\pi }}=\\ &\quad {{ V}_{\pi }}{{ J}_{123}}{{ V}_{\pi }}{{ D}_{0}}\approx {{\rm e}^{2.8229{\rm i}}}{ I}={ I} \end{split}$ (15)

 ${{ J}_\pi }(\theta ) = \left[ {\begin{array}{*{20}{c}} {\cos 2\theta }&{\sin 2\theta } \\ {\sin 2\theta }&{ - \cos 2\theta } \end{array}} \right]$ (16)

 ${{ J}_0}(\delta ) = \left[ {\begin{array}{*{20}{c}} {{{\rm e}^{{\rm i}\delta }}}&0 \\ 0&1 \end{array}} \right]$ (17)

${{ D}_0}$ 形式完全相同，解得 $\delta = 2.822\;9 -1.504\;9 =$ $1.318\;{\rm rad}$

 $\begin{split} {{ J}_\pi}(\theta ) {{ J}_{123}} {{ J}_\pi}(\theta ) {{ J}_{0}}(\delta )=\\ {{ J}_{0}}(\delta ) {{ J}_\pi}(\theta ) {{ J}_{123}} {{ J}_\pi}(\theta )={ I} \end{split}$ (18)

 $\begin{split} &{{ J}_\pi}(\pi -\theta ){{ J}_{321}}{{ J}_\pi}(\pi -\theta ){{ J}_{0}}(\delta )=\\ &\quad {{ J}_{0}}(\delta ){{ J}_\pi}(\pi -\theta ){{ J}_{321}}{{ J}_\pi}(\pi -\theta )={ I} \end{split}$ (19)

 $\begin{split} &{{ J}_\pi}(\frac{2\pi }{3}-\theta ){{ J}_{213}}{{ J}_\pi}(\frac\pi{3}-\theta ) {{ J}_{0}}(\delta )=\\ &\quad {{ J}_{0}}(\delta ) {{ J}_\pi}(\frac{2\pi }{3}-\theta ){{ J}_{213}}{{ J}_\pi}(\frac\pi{3}-\theta )={ I} \end{split}$ (20)

 $\begin{split} &{{ J}_\pi}(\displaystyle\frac\pi{3}-\theta ) {{ J}_{132}} {{ J}_\pi}(\displaystyle\frac{2\pi }{3}-\theta ) {{ J}_{0}}(\delta )=\\ &\quad {{ J}_{0}}(\delta ) {{ J}_\pi}(\frac\pi{3}-\theta ) {{ J}_{132}} {{ J}_\pi}(\frac{2\pi }{3}-\theta )={ I} \end{split}$ (21)
 $\begin{split} &{{ J}_\pi}(\frac\pi{3}+\theta ) {{ J}_{231}} {{ J}_\pi}(\frac{2\pi }{3}+\theta ) {{ J}_{0}}(\delta )=\\ &\quad {{ J}_{0}}(\delta ) {{ J}_\pi}(\frac\pi{3}+\theta ) {{ J}_{231}} {{ J}_\pi}(\frac{2\pi }{3}+\theta )={ I} \end{split}$ (22)
 $\begin{split} &{{ J}_\pi}(\frac{2\pi }{3}+\theta ) {{ J}_{312}} {{ J}_\pi}(\frac\pi{3}+\theta ) {{ J}_{0}}(\delta )=\\ &\quad {{ J}_{0}}(\delta ) {{ J}_\pi}(\frac{2\pi }{3}+\theta ) {{ J}_{312}} {{ J}_\pi}(\frac\pi{3}+\theta )={ I} \end{split}$ (23)

 $\begin{split} &{{ J}_\pi}(\theta -\frac{\gamma }{2}){{ J}_{123}}(\gamma ){{ J}_\pi}(\theta +\frac{\gamma }{2}){{ J}_{0}}(\delta )=\\ &\quad {{ J}_{0}}(\delta ){{ J}_\pi}(\theta -\frac{\gamma }{2}){{ J}_{123}}(\gamma ){{ J}_\pi}(\theta +\frac{\gamma }{2})={ I} \end{split}$ (24)

3 实验验证

 图 5 实验装置（虚线框内对应保偏外加波片） Figure 5 Experimental setup (correction wave plate inside the dotted box)
 ${{ J}_{132}}(\frac\pi{6}) = {{ J}_{\rm r}}(\frac\pi{6}){{ J}_{132}}{{ J}_{\rm r}}(\frac\pi{6})$ (25)
 ${{ J}_{231}}(\frac\pi{6}) = {{ J}_{\rm r}}(\frac\pi{6}){{ J}_{231}}{{ J}_{\rm r}}(\frac\pi{6})$ (26)
 $\begin{split} &{{ J}_\pi}(\frac\pi{4}-\theta ) {{ J}_{132}}(\frac\pi{6}) {{ J}_\pi}(\frac{3\pi }{4}-\theta ) {{ J}_{0}}(\delta )=\\ &\; {{ J}_{0}}(\delta ) {{ J}_\pi}(\frac\pi{4}-\theta ) {{ J}_{132}}(\frac\pi{6}) {{ J}_\pi}(\frac{3\pi }{4}-\theta )={ I} \end{split}$ (27)
 $\begin{split} &{{ J}_\pi}(\frac\pi{4}+\theta ){{ J}_{231}}(\frac\pi{6}){{ J}_\pi}(\frac{3\pi }{4}+\theta ){{ J}_{0}}(\delta )=\\ &\; {{ J}_{0}}(\delta ){{ J}_\pi}(\frac\pi{4}+\theta ){{ J}_{231}}(\frac\pi{6}){{ J}_\pi}(\frac{3\pi }{4}+\theta )={ I} \end{split}$ (28)

 图 6 未纠偏时理论与实验Stokes参数 Figure 6 Theoretical and experimental Stokes parameters without correction

 图 7 纠偏前后修正的132和231反射次序下的椭圆率和方向角 Figure 7 Ellipticity and azimuth of the modified 132/231 path before/after correction
4 结　论

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