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Eigen generalized Jones matrix method

Song Dongsheng Zheng Yuanlin Liu Hu Hu Weixing Zhang Zhiyun Chen Xianfeng

宋东升, 郑远林, 刘虎, 胡维星, 张志云, 陈险峰. 本征广义琼斯矩阵方法[J]. 中国光学. doi: 10.3788/CO.2019-0163
引用本文: 宋东升, 郑远林, 刘虎, 胡维星, 张志云, 陈险峰. 本征广义琼斯矩阵方法[J]. 中国光学. doi: 10.3788/CO.2019-0163
Song Dongsheng, Zheng Yuanlin, Liu Hu, Hu Weixing, Zhang Zhiyun, Chen Xianfeng. Eigen generalized Jones matrix method[J]. Chinese Optics. doi: 10.3788/CO.2019-0163
Citation: Song Dongsheng, Zheng Yuanlin, Liu Hu, Hu Weixing, Zhang Zhiyun, Chen Xianfeng. Eigen generalized Jones matrix method[J]. Chinese Optics. doi: 10.3788/CO.2019-0163

本征广义琼斯矩阵方法

doi: 10.3788/CO.2019-0163
详细信息
  • 中图分类号: O436.3

Eigen generalized Jones matrix method

More Information
    Author Bio:

    SONG Dong-sheng (1985—), Male, born in Zhengzhou City, Henan Province. M.Sc., Graduated from Shanghai Jiao Tong University in 2018. Engineer, Luoyang Electronic Equipment Test Center of China. His research interests are on nonlinear optics, frequency conversion and light field regulation. E-mail: sds0754@alumni.sjtu.edu.cn

    Corresponding author: E-mail: sds0754@alumni.sjtu.edu.cn
  • 摘要: 为了描述完全偏振光在非线性晶体中传播时偏振态及相位的变化,我们基于Ortega-Quijano等人在推导非线性晶体的广义琼斯矩阵时的微分广义琼斯矩阵方法,提出了本征广义琼斯矩阵方法。与微分广义琼斯矩阵方法相比,本征广义琼斯矩阵方法使用了更精确的数学技巧,在描述光在非线性晶体中传播的物理过程上更为严谨。解决了微分广义琼斯矩阵不能计算斜入射光或者光轴与实验室坐标不重合时的光的偏振变化的问题。首先,根据折射率椭球方程和光的入射方向,计算出非线性晶体中本征光的传播方向和折射率。然后,写出本征光的本征广义琼斯矩阵。最后,计算本征光的偏振态和相位的变化。我们使用本征广义琼斯矩阵对带有一个奇点的涡旋光在KDP晶体中的传播进行模拟计算,计算结果表明本征广义琼斯矩阵方法能够描述完全偏振光以任意角度入射光轴方向任意的非线性晶体的传播过程。
  • Figure  1.  Schematic of the three coordinate systems. The black, blue, and red axes represent the laboratory, principal, and eigen coordinates, respectively. z1 and z2 are the optical axes.

    Figure  2.  Spatial distribution of the polarization state. (a) Original linear polarization. (b) Left(right) polarization. (c) Circular polarization. (d) Right(left) polarization. (e) Opposite Linear polarization.

    Figure  3.  Change in walk-off angle with θz.

    Figure  4.  Spatial distribution of the polarization state with a right direction walk-off effect. (a) Original linear polarization. (b) Left(right) polarization. (c) Circular polarization. (d) Right(left) polarization. (e) Opposite Linear polarization.

    Figure  5.  Change in walk-off angle with (θ, φ).

    Figure  6.  Spatial distributions of the polarization state with a right up direction walk-off effect. (a) Original linear polarization. (b) Left(right) polarization. (c) Circular polarization. (d) Right(left) polarization. (e) Opposite Linear polarization.

    Figure  7.  Phase difference and polarization. (a) Phase difference of refraction light beam in birefringent crystals. (b) Polarization of reflection and refraction light beam at the interface in birefringent crystals.

    Figure  8.  (a) Experimental image and our (b) simulation.

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Eigen generalized Jones matrix method

doi: 10.3788/CO.2019-0163
    通讯作者: E-mail: sds0754@alumni.sjtu.edu.cn
  • 中图分类号: O436.3

摘要: 为了描述完全偏振光在非线性晶体中传播时偏振态及相位的变化,我们基于Ortega-Quijano等人在推导非线性晶体的广义琼斯矩阵时的微分广义琼斯矩阵方法,提出了本征广义琼斯矩阵方法。与微分广义琼斯矩阵方法相比,本征广义琼斯矩阵方法使用了更精确的数学技巧,在描述光在非线性晶体中传播的物理过程上更为严谨。解决了微分广义琼斯矩阵不能计算斜入射光或者光轴与实验室坐标不重合时的光的偏振变化的问题。首先,根据折射率椭球方程和光的入射方向,计算出非线性晶体中本征光的传播方向和折射率。然后,写出本征光的本征广义琼斯矩阵。最后,计算本征光的偏振态和相位的变化。我们使用本征广义琼斯矩阵对带有一个奇点的涡旋光在KDP晶体中的传播进行模拟计算,计算结果表明本征广义琼斯矩阵方法能够描述完全偏振光以任意角度入射光轴方向任意的非线性晶体的传播过程。

English Abstract

宋东升, 郑远林, 刘虎, 胡维星, 张志云, 陈险峰. 本征广义琼斯矩阵方法[J]. 中国光学. doi: 10.3788/CO.2019-0163
引用本文: 宋东升, 郑远林, 刘虎, 胡维星, 张志云, 陈险峰. 本征广义琼斯矩阵方法[J]. 中国光学. doi: 10.3788/CO.2019-0163
Song Dongsheng, Zheng Yuanlin, Liu Hu, Hu Weixing, Zhang Zhiyun, Chen Xianfeng. Eigen generalized Jones matrix method[J]. Chinese Optics. doi: 10.3788/CO.2019-0163
Citation: Song Dongsheng, Zheng Yuanlin, Liu Hu, Hu Weixing, Zhang Zhiyun, Chen Xianfeng. Eigen generalized Jones matrix method[J]. Chinese Optics. doi: 10.3788/CO.2019-0163
    • Jones calculus is a simple and general method for modelling several optical phenomena, such as liquid crystal displays [1,2], diffraction gratings [3], Šolc filters [4-6], holographic imaging [7,8], quantum communication [9] in classical and quantum optical fields, radio telescope image calibrators [10], radio polarimeters [11] in astronomical observation, human retinal imaging [12], human brain tissues [13], and biological specimens [14] in the biomedical imaging field. Moreover, in three dimensions, the Jones vector changes into the generalized Jones vector [15] and can be used to describe light propagating through a high-numerical-aperture focus [16], light interacting with nanoparticles [17], and optical coherence tomography [18].

      Jones matrix calculus was first proposed by R. Clark Jones in the 1940s to describe the change in phase and polarization in matrix or vector forms for media or light, respectively [19]. It is a basic and widely used calculus method for describing the polarization of light transmitting in media. However, it has only been applied to normally or paraxially incident light. Zhang et al. introduced a generalized Jones vector (GJV), also called 3D Jones vector to describe the polarization effect of light and optical media or systems [20-23]. Yeh et al. extended the method to treat the transmission of off-axis light through an anisotropic medium with an arbitrary optical axis orientation [24]. Azzam et al. invented the generalized Jones matrix (GJM) to describe the interaction between the fully polarized beam and its linear transformations in three dimensions [25]. Recently, Ortega-Quijano and colleagues proposed the differential generalized Jones matrix (dGJM) method to derive the GJM to model uniaxial and biaxial crystals with arbitrary orientations [26,27]. However, our repeatedly and precisely calculation showed that the dGJM method is not applicable for samples with an arbitrary optical axis orientation or when the light is oblique incidence. The reason leads to this limitation is that, the dGJM method tries to get the GJM for an anisotropic crystal with arbitrary orientation in the laboratory coordinate system through the rotation of the GJM consist of the principle index in principle coordinate system. But when the light is oblique incidence, the principle index should be replaced by the eigen refraction index which can be calculated with the n-face equation of the crystal and the direction of the beam in the principle coordinate. Meanwhile, the eigen refraction index can be used to calculate the phase difference of the two eigen polarization lights.

      In this paper, we propose a new method to calculate the phase and polarization of fully polarized light propagating in an anisotropic crystal with an arbitrary orientation. The method overcomes the limitations of the dGJM method. In Sec. 2, the eigen generalized Jones matrix (eGJM) is derived to address the problem and can be used in uniaxial and biaxial crystals. In Sec. 3, the eGJM is extended to contain the light refraction in the crystal interface. Then, we use the method to simulate the polarization distribution of the cross-section for a light beam with a vortex and compare the results to an experimental image [28,29]. The results demonstrate that our method is effective.

    • To overcome the limitation of the dGJM method, three coordinate systems are necessary: the laboratory coordinate system (S), describing the position of the crystal; the principal axis coordinate system (Z), describing the orientation of the optical axis; and the eigen coordinate system (B), describing the polarized beam propagation direction. In addition, only one eigen coordinate system is required when the light beam transfers in the crystal without any refraction, or two eigen coordinate systems for the two different wave vectors. These coordinates are illustrated in Fig. 1. We define the rotation relationship between them as Z=TZS and B=TBS, where TZ and TB are the rotation matrices, which can be calculated using Euler rotation matrix theory.

      Figure 1.  Schematic of the three coordinate systems. The black, blue, and red axes represent the laboratory, principal, and eigen coordinates, respectively. z1 and z2 are the optical axes.

      To obtain the eigen dGJM, we first calculate the eigen indices n1 and n2 from Eq. (1) and Eq. (2) in principal coordinates:

      $${\vec K_i} = {T_Z}\vec k$$ (1)
      $$\begin{split} & n_{}^4({K_{ix}}^2 + {K_{iy}}^2 + {K_{iz}}^2)({K_{ix}}^2{n_x}^2 + {K_{iy}}^2{n_y}^2 + {K_{iz}}^2{n_z}^2) \\ & - n_{}^2{K_{ix}}^2{n_x}^2({n_y}^2 + {n_z}^2) \\ & - n_{}^2{K_{iy}}^2{n_y}^2({n_z}^2 + {n_x}^2) - n_{}^2{K_{iz}}^2{n_z}^2({n_x}^2 + {n_y}^2)\\ & + {n_x}^2{n_y}^2{n_z}^2 = 0 \end{split} $$ (2)

      where n is the refractive index, Ki is the principal wave vector, and nx, ny and nz are the principal indices of the crystal. Eq. (1) is used to rewrite the transporting direction of the eigen light beam in Z, which can directly be used in Eq. (2), the index face equation in Z.

      Second, we can directly write the eGJM in B:

      $${G_B} = \left[ {\begin{array}{*{20}{c}} {\exp ( - i\delta /2)}&0&0 \\ 0&{\exp (i\delta /2)}&0 \\ 0&0&0 \end{array}} \right]$$ (3)

      where δ=2π(n1d1−n2d2)/λ describes the phase difference. d1 and d2 are the propagation path lengths of the wave vector for the two eigen lights in the crystal. They are different and require calculation with refraction at oblique incidence and identical at normal incidence.

      According to the relationship between B and S, the GJM in S is

      $${G_S} = T_B^{ - 1}{G_B}{T_B}$$ (4)

      where TB is the transfer matrix between the eigen coordinates and laboratory coordinates. The electric displacement vector D of the output light beam can be expressed as

      $$D' = {G_S}{D_i} = T_B^{ - 1}{G_B}{T_B}{D_i}$$ (5)

      The electric field vector E can be expressed as

      $$E' = {(T_{\rm{Z}}^{ - 1}{\varepsilon _{\rm{Z}}}{T_{\rm{Z}}})^{ - 1}}T_B^{ - 1}{G_B}{T_B}(T_{\rm{Z}}^{ - 1}{\varepsilon _{\rm{Z}}}{T_{\rm{Z}}}){E_i}$$ (6)

      where TZ is the transfer matrix between the principal coordinates and laboratory coordinates and εZ is the polarizability tensor in principal coordinates and can be written as

      $${\varepsilon _Z} = \left[ {\begin{array}{*{20}{c}} {{n_x}}&{}&{} \\ {}&{{n_y}}&{} \\ {}&{}&{{n_z}} \end{array}} \right]$$ (7)

      The physical meaning of Eq. (6) is easily understood. Ei represents the electric field vector of a light beam in laboratory coordinates and will be transferred to the electric displacement vector D in the same coordinates by the left multiplication of the factor (TZ−1εZTZ). Then, TB will transfer D to eigen coordinates. GB will change the phase of light, which will finally be reversed to an electric field vector form in laboratory coordinates.

    • We use the eGJM method to calculate the polarization distribution of the light beam in anisotropic crystals.

      a) Beam direction perpendicular to the optical axis.

      Consider the situation that the direction of the beam is perpendicular to the optical axis. The principal coordinate system is then the superposition of the eigen coordinate system; thus, TB=TZ. The refractive indices for the eigen beams are exactly the principal index, no and ne. Then, the eGJM is

      $$ \begin{split} {G_S} &= {({T_Z}^{ - 1}{\varepsilon _Z}{T_Z})^{ - 1}}{T_{Bu1}}^{ - 1}{G_B}{T_B}({T_Z}^{ - 1}{\varepsilon _Z}{T_Z}) \\ &= {T_Z}^{ - 1}{\varepsilon _Z}^{ - 1}{G_{Bu1}}{\varepsilon _Z}{T_Z} \end{split} $$ (8)

      where

      $${G_{Bu1}} = \left[ {\begin{array}{*{20}{c}} {\!\!\!\!\exp ( - i\delta /2)}&\!\!0\!\!&\!\!0\!\!\!\\ \!\!\!0&\!\!{\exp (i\delta /2)}\!\!&\!\!0\!\!\!\\\!\!\! 0\!\!&\!\!0\!\!&\!\!0 \end{array}} \!\!\!\!\right],\;\delta = 2{\text π} ({n_o} - {n_e})d/\lambda $$ (9)

      In this case, there is no walk-off angle between the two eigen beams. Assuming the initial polarization direction is 45° off the x-axis, according to the eGJM method, we can calculate the polarization after a length d. The polarization distribution of the cross-section of the beam is presented in Fig. 2, which shows the change in polarization from the original direction to the opposite polarization direction.

      Figure 2.  Spatial distribution of the polarization state. (a) Original linear polarization. (b) Left(right) polarization. (c) Circular polarization. (d) Right(left) polarization. (e) Opposite Linear polarization.

      b) Arbitrary angle between the beam and optical axis

      When the angle between the beam and optical axis is arbitrary, the eigen refractive indices for the extraordinary ray will no longer be ne, but they should be calculated from Eq. (10) [30].

      $${n_e}(\theta ) = \frac{{{n_o}{n_e}}}{{{{[n_o^2{{\sin }^2}\theta + n_e^2{{\cos }^2}\theta ]}^{1/2}}}}$$ (10)

      Then, the eGJM for the electric displacement vector D is

      $${G_{SD}} = {({T_Z}^{ - 1}{\varepsilon _Z}{T_Z})^{ - 1}}(T_B^{ - 1}{G_{Bu2}}{T_B})({T_Z}^{ - 1}{\varepsilon _Z}{T_Z})$$ (11)

      where

      $${G_{Bu2}} = \left[\!\!\!\! {\begin{array}{*{20}{c}} {\exp ( - i\delta /2)}\!\!&\!\!0\!\!&\!\!0\\ 0\!\!&\!\!{\exp (i\delta /2)}\!\!&\!\!0\\ 0\!\!&\!\!0\!\!&\!\!0 \end{array}}\!\!\!\! \right],\;\delta = 2{\text π} ({n_o} - {n_{e\theta }})d/\lambda $$ (12)

      For the extraordinary beam, the array direction is not the same as the wave vector direction, so the eGJM for the electric field vector E should be changed to

      $${G_{SE}} = {({T_Z}^{ - 1}{\varepsilon _Z}{T_Z})^{ - 1}}(T_{Bo}^{ - 1}{G_{Bo}}{T_{Bo}} + T_{Be}^{ - 1}{G_{Be}}{T_{Be}})({T_Z}^{ - 1}{\varepsilon _Z}{T_Z})$$ (13)

      where

      $${G_{Bo}} = \left[ {\begin{array}{*{20}{c}} {\exp ( - i{\delta _1})}&0&0\\ 0&0&0\\ 0&0&0 \end{array}} \right],\;{\delta _1} = 2{\text π} {n_o}{d_1}/\lambda $$ (14)
      $${G_{Be}} = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&{\exp ( - i{\delta _2})}&0\\ 0&0&0 \end{array}} \right],\;{\delta _2} = 2{\text π} {n_{e\theta }}{d_2}/\lambda $$ (15)

      The walk-off angle should be calculated before we obtain the polarization distribution of the cross-section of the beam. For a potassium dideuterium phosphate (KDDP) crystal, no=1.494 2, ne=1.460 3. The change in walk-off angle with θz from 0 to π/2 is shown in Fig. 3, where the angle between the beam direction and optical axis direction is θz.

      Figure 3.  Change in walk-off angle with θz.

      Figure 3 indicates that the maximum value of the walk-off angle is 0.023 3 rad, equal to 1.335°, and the corresponding refractive index is ne(θ)=1.476 96. The walk-off distance is 0.023 3 cm if the beam transfers a length of 1 cm. Thus, there will be an overlap region if the size of the beam cross-section is larger than 0.023 3 cm. The polarization distribution for the overlap region is presented in Fig. 4. The light polarization for the overlap region could be elliptical, circular, or linear; meanwhile, the overlap region decreases in size with the transfer length.

      Figure 4.  Spatial distribution of the polarization state with a right direction walk-off effect. (a) Original linear polarization. (b) Left(right) polarization. (c) Circular polarization. (d) Right(left) polarization. (e) Opposite Linear polarization.

    • In biaxial crystals, there is always a walk-off effect for the light beam, so the light transfer along the optical axis direction for the conical refraction effect is not considered a special situation. The eigen refractive indices for the two eigen linear polarization light beams can be calculated from Eq. (1) and Eq. (2). The eGJM for the electric field vector can immediately be written as

      $${G_{SE}} = {({T_Z}^{ - 1}{\varepsilon _Z}{T_Z})^{ - 1}}(T_{B1}^{ - 1}{G_{B1}}{T_{B1}} + T_{B2}^{ - 1}{G_{B2}}{T_{B2}})({T_Z}^{ - 1}{\varepsilon _Z}{T_Z})$$ (16)

      where

      $${G_{B1}} = \left[ {\begin{array}{*{20}{c}} {\exp ( - i{\delta _1})}&0&0\\ 0&0&0\\ 0&0&0 \end{array}} \right],\;{\delta _1} = 2{\text π} {n_1}{d_{\rm{1}}}/\lambda $$ (17)
      $${G_{B2}} = \left[ {\begin{array}{*{20}{c}} {\rm{0}}&0&0\\ 0&{\exp ( - i{\delta _2})}&0\\ 0&0&0 \end{array}} \right],\;{\delta _2} = 2{\text π} {n_2}{d_{\rm{2}}}/\lambda $$ (18)

      The eGJM for the electric displacement vector can be written as

      $${G_{SD}} = {({T_Z}^{ - 1}{\varepsilon _Z}{T_Z})^{ - 1}}(T_B^{ - 1}{G_{Bb}}{T_B})({T_Z}^{ - 1}{\varepsilon _Z}{T_Z})$$ (19)

      where

      $${G_{Bb}} = \left[\!\!\!\! {\begin{array}{*{20}{c}} {\exp ( - i\delta /2)}\!\!&\!\!0\!\!&\!\!0\\ 0\!\!&\!\!{\exp (i\delta /2)}\!\!&\!\!0\\ 0\!\!&\!\!0\!\!&\!\!0 \end{array}}\!\!\!\! \right],\;\delta = 2{\text π} ({n_{\rm{1}}} - {n_{\rm{2}}})d/\lambda $$ (20)

      To calculate the polarization distribution of the cross-section of the light beam, we also need to calculate the walk-off angle. We define the light direction as (θ, φ) in principal coordinates, and the change in walk-off angle with direction is shown in Fig. 5.

      Figure 5.  Change in walk-off angle with (θ, φ).

      In Fig 5, θ is the polar angle and φ is the azimuth angle. Figures 5(a) and 5(b) are related to the walk-off angle for the light beam with the smaller and larger eigen refractive indices, respectively. There is no walk-off effect for any eigen light when the light beam transfers along the axis direction, corresponding to θ=0, or θ=π/2 and φ=0, π/2, π, 3π/2. Only one eigen light beam exhibits a walk-off effect when φ=0, π/2, π, 3π/2 while θ is arbitrary, or when θ=π/2 while φ is arbitrary. There are two singularity points when θ=0.304 and φ=0, π, corresponding to the optical axis direction. Similar to the uniaxial crystal, the polarization distribution of the cross-section of the light beam is presented in Fig. 6. The movement of the different polarizations toward the upper-right quarter represents the array direction.

      Figure 6.  Spatial distributions of the polarization state with a right up direction walk-off effect. (a) Original linear polarization. (b) Left(right) polarization. (c) Circular polarization. (d) Right(left) polarization. (e) Opposite Linear polarization.

    • We extend the eGJM to a more general case of the refraction on the interface. Figure 7(a) shows the phase difference when the light beam transfers through the anisotropic crystals, where the blue line represents ordinary light, the red line represents the direction of energy flow of extraordinary light, and the pink line represents the wave vector direction. Let us calculate the phase difference:

      Figure 7.  Phase difference and polarization. (a) Phase difference of refraction light beam in birefringent crystals. (b) Polarization of reflection and refraction light beam at the interface in birefringent crystals.

      $$\begin{split} {\delta _e} - {\delta _o} = & {n_e} \cdot d/\cos {\theta _e} + n \cdot EF \cdot \tan {\theta _i} - {n_o} \cdot d/ \cos {\theta _o}\\ =& ({n_e}\cos {\theta _e} - {n_o}\cos {\theta _o})d \\ \end{split} $$ (21)
      $$\begin{split} {\delta _s} - {\delta _o} =& {n_e} \cdot d\cos ({\theta _e} - {\theta _s})/\cos {\theta _s} + n \cdot AF \cdot \tan {\theta _i}\\ &- {n_o} \cdot d/\cos {\theta _o} = ({n_e}\cos {\theta _e} - {n_o}\cos {\theta _o})d \\ \end{split} $$ (22)

      The results indicate that the two phases differences are identical. They are equivalent to either the energy flow direction or the wave vector direction for the extraordinary light. Figure 7(b) shows the different polarization directions at the interface. For the anisotropic crystal, the birefringence must be considered. Yeh [24] already provided a method to calculate the polarization of the output light beam. Assuming the thickness of the crystal is d, we know the optical distance difference is (necosθe−nocosθo) from Eq. (21) and Eq. (22); thus, the output light beam can be expressed as

      $$\begin{split} {{A'}_s} =& \left( {{t_{os}}{T_o}{e^{ - i{\delta _{\rm{1}}}{\rm{/2}}}} + {t_{es}}{T_e}{e^{i{\delta _1}{\rm{/2}}}}} \right){e^{ - i{\delta _{\rm{2}}}{\rm{/2}}}}\\ {{A'}_p} =& \left( {{t_{ps}}{T_o}{e^{ - i{\delta _{\rm{1}}}{\rm{/2}}}} + {t_{ps}}{T_e}{e^{i{\delta _1}{\rm{/2}}}}} \right){e^{ - i{\delta _{\rm{2}}}{\rm{/2}}}} \end{split}$$ (23)

      where δ1=(necosθe−nocosθo)dω/c and δ2=(ne/cosθe+no/cosθo)dω/c. Ignoring the phase factor e−iδ2/2, we have the following matrix form:

      $$\left(\!\! {\begin{array}{*{20}{c}} {{{A'}_s}} \\ {{{A'}_p}} \end{array}}\!\! \right) = \left(\!\! {\begin{array}{*{20}{c}} {{t_{os}}}&{{t_{es}}} \\ {{t_{op}}}&{{t_{ep}}} \end{array}}\!\! \right)\left( \!\!{\begin{array}{*{20}{c}} {{e^{ - i{\delta _{\rm{1}}}{\rm{/2}}}}}&0 \\ 0&{{e^{i{\delta _{\rm{1}}}{\rm{/2}}}}} \end{array}}\!\! \right)\left(\!\! {\begin{array}{*{20}{c}} {{t_{so}}}&{{t_{po}}} \\ {{t_{se}}}&{{t_{pe}}} \end{array}}\!\! \right)\left(\!\! {\begin{array}{*{20}{c}} {{A_s}} \\ {{A_p}} \end{array}} \!\!\right)$$ (24)

      If we extend Eq. (24) to three dimensions, we have

      $$\begin{split} & \left(\!\! \begin{array}{l} \begin{array}{*{20}{c}} {{{E'}_x}} \\ {{{E'}_y}} \end{array} \\ {{E'}_z} \\ \end{array} \right) = T\left( \begin{array}{l} \begin{array}{*{20}{c}} {{{A'}_s}} \\ {{{A'}_p}} \end{array} \\ 0 \\ \end{array} \!\!\right) = T\left( \!\!{\begin{array}{*{20}{c}} {{t_{os}}}&{{t_{es}}} \\ \begin{array}{l} {t_{op}} \\ 0 \\ \end{array} &\begin{array}{l} {t_{ep}} \\ 0 \\ \end{array} \end{array}\begin{array}{*{20}{c}} {} \\ {} \\ {} \end{array}\begin{array}{*{20}{c}} 0 \\ 0 \\ 0 \end{array}} \right)\\ &\left( {\begin{array}{*{20}{c}} {{e^{ - i{\delta _1}/2}}}&0 \\ \begin{array}{l} 0 \\ 0 \\ \end{array} &\begin{array}{l} {e^{i{\delta _1}/2}} \\ 0 \\ \end{array} \end{array}\begin{array}{*{20}{c}} {} \\ {} \\ {} \end{array}\begin{array}{*{20}{c}} 0 \\ 0 \\ 0 \end{array}}\!\! \right)\left(\!\! {\begin{array}{*{20}{c}} {{t_{so}}}&{{t_{po}}} \\ \begin{array}{l} {t_{se}} \\ 0 \\ \end{array} &\begin{array}{l} {t_{pe}} \\ 0 \\ \end{array} \end{array}\begin{array}{*{20}{c}} {} \\ {} \\ {} \end{array}\begin{array}{*{20}{c}} 0 \\ 0 \\ 0 \end{array}} \!\!\right)\\ &{T^{ - 1}}T\left(\!\!{\begin{array}{*{20}{c}} {{A_s}} \\ \begin{array}{l} {A_p} \\ 0 \\ \end{array} \end{array}} \!\!\right) = {G_S}\left(\!\! {\begin{array}{*{20}{c}} {{E_x}} \\ \begin{array}{l} {E_y} \\ {E_z} \\ \end{array} \end{array}}\!\! \right) \\ \end{split} $$ (25)

      As an application, we calculate the phase distribution for a vector vortex light beam with a singularity transferring through the KDP crystal and compare the simulation results to the experimental results by Flossmann [28], as shown in Fig. 8.

      Figure 8.  (a) Experimental image and our (b) simulation.

      The black squares indicate the singularities of the light beam. The colored circles represent the circular polarization state points and the yellow lines represent the linear polarization states in the cross-section of the output vector beam. There is a small difference in the bottom and middle areas between these two images because of the experimental error and simulation method. However, the polarization distribution and positions of the special points are almost identical, which clearly indicates that the eGJM method is practical.

    • In this study, we analyzed the GJM method, which provides a convenient way to establish the Jones matrix for anisotropic crystals with their optical axis oriented arbitrarily in three-dimensional space. We proposed the eGJM method to overcome the limitation of the dGJM, which is effective only when the light is perpendicular incidence and the optical axis is perpendicular or parallel to the incidence face. The calculation results indicate that our method can be used to construct the Jones matrix when the directions of the light beam and optical axis are both arbitrary. The eGJM can also be extended to contain the light refraction on the interface when light travels through the crystal, to precisely calculate the polarization and phase. Finally, we use this method to simulate the polarization distribution of the cross-section for a fully polarized light beam with a vortex transferred through an anisotropic crystal, and we compare the results to an experiment. The results demonstrate that our method is effective. Thus, the eGJM method has a potential application to simulate the space evolution of the vector beams, on the contrary, the optional optical crystal instruments can be calculated based on the requirement beams. Interesting aspects like electro-photon effect, magnetic-photon effect and optical rotation should be further studied to fully develop the eGJM method for applications like light propagation in crystal in electromagnetic field.

    • National Natural Science Foundation of China (11734011); Foundation for Development of Science and Technology of Shanghai (17JC1400402).

      国家自然科学基金(11734011); 上海科技发展基金(17JC1400402).

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