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Exact optical path difference and complete performance analysis of a spectral zooming imaging spectrometer

Xiangzhe Zhang, Liqing Huang, Jingping Zhu, Ning Zhang, Kang Zong, Lipeng Zhai, Yu Zhang, Yakun Cai, and Huimin Wang

Open Access Open Access

Abstract

The optical path difference (OPD) equations of the dual Wollaston prisms (DWP) with an adjustable air gap (AG) are derived by the wave normal tracing method, which is suitable for arbitrary incidence plane and angle. The spatial distribution of the OPD for various AG is presented. The validity of the OPD equation is verified by comparing the calculated interferograms with experimentally observed one. The performance of a novel static birefringent Fourier transform imaging spectrometer (SBFTIS) based on the DWP is investigated. The spectral resolution can be adjusted by changing the AG and the field of view can reach 10.0°, which is much larger than that predicted by our previous work. The results obtained in this article provide a theoretical basis for completely describing the optical transmission characteristic of the DWP and developing the high-performance birefringent spectral zooming imaging spectrometer.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The static birefringent Fourier transform imaging spectrometer (SBFTIS) [1–13] is highly regarded for its high robustness and precision. However, the traditional SBFTIS has a limited field of view (FOV) and a fixed spectral resolution, which greatly limits its application.

In previous work, our research group proposed a novel SBFTIS with a zoomable spectral resolution based on two identical Wollaston prisms with an adjustable air gap (AG) [14,15], which greatly reduces the size of the spectral image data to adapt to different application requirements [16,17]. The optical path difference (OPD) of the dual Wollaston prisms (DWP) at an arbitrary incident angle in the principal section was calculated by a quasi-parallel-plate approximation [18,19] scheme and the spectral resolution, and FOV of the novel SBFTIS were analyzed based on the calculated principal section OPD. However, our previous work still has limitations as it is only an approximate OPD in the main section and thus cannot properly analyze the performance of SBFTIS. To know the exact performance of novel SBFTIS, it is necessary to exactly calculate the OPD in an arbitrary incident plane and angle.

In this paper, the OPD equations of the DWP with an adjustable AG for arbitrary incidence plane and angle are derived by the wave normal tracing method [20–22]. The exact dependence of the OPD on incident angle and AG is analyzed and verified by the interferogram observed by the experiment. The spectral zooming and FOV performance of the novel SBFTIS are completely investigated by making the non-uniform Fourier transform (NUFT) of the exactly calculated interferogram, and the comparison of FOV predicted based on the exact calculated OPD and the quasi-parallel-plate approximated OPD, respectively, is presented.

2. Wave normal transmission characteristics in the DWP

In the birefringent uniaxial crystal, one incident wave splits up into a pair of orthogonally polarized waves, an ordinary (o) wave and an extraordinary (e) wave. According to Snell’s laws, the double refraction that happened on the interfaces can be expressly described by the following relationships in a general case of non-normal incidence into the birefringent crystal with an arbitrary optic axis:

$${n_i}\sin i\textrm{ = }n({r_e})\sin {r_e} = {n_o}\sin {r_o}$$
$$n({r_e}) = \frac{{{n_o}{n_e}}}{{\sqrt {n_e^2{{\cos }^2}{\theta _e} + n_o^2{{\sin }^2}{\theta _e}} }}$$
$$\cos {\kern 1pt} {\theta _e} = \sin \psi \cos \omega \sin {r_e} + \cos \psi \cos {r_e}$$

Here i is the incident angle, ${r_o}$ and ${r_e}$ are the refracted angles for the ordinary and extraordinary wave normal, respectively. $n({r_e})$ is the refractive index for the extraordinary wave which is determined by the angle ${\theta _e}$ between the extraordinary wave normal and the optical axis. $\psi$ is the angle between the optical axis and the crystal surface normal. $\omega$ is the incident plane angle, which is the angle between the plane of incidence and the principal section of the crystal.

2.1 Dual Wollaston prisms

The schematic of the SBFTIS with a zoomable spectral resolution is shown in Fig. 1(a), which is based on a DWP. A DWP is formed by two identical Wollaston prisms WP1 and WP2 with an adjustable AG of width s between them, as shown in Fig. 1(b). A Wollaston prism (WP) is composed of two uniaxial wedge-shaped crystals with mutually perpendicular optical axes, characterized by the structure angle β and the plate thickness t.

 figure: Fig. 1.

Fig. 1. (a) The schematic of the SBFTIS. (b) Layout of the DWP.(c) Schematic of the path of waves passing through the DWP.

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A wave separates into two waves with orthogonal polarization directions and parallel propagation directions after passing through the DWP and the schematic of the path of waves passing through the DWP is shown in Fig. 1(c). Due to the mutually perpendicular optic axes of the two wedge-shaped crystals of WP, their principal sections are at right angles to each other. Thus, the wave which is ordinary in the first crystal becomes extraordinary in the second and vice versa, the two waves are designated as oeeo and eooe wave, respectively. The OPD of oeeo and eooe wave generated by the DWP can be separated into two parts, the interior OPD ($\Delta {\,_{in}}$) generated inside the DWP, and the exterior OPD (${\Delta _{ex}}$) generated outside the DWP.

2.2 OPD of wave normal generated by the DWP

The OPD calculated by the ray is equivalent to the OPD calculated by the wave normal [20]. To exactly calculate the OPD of oeeo and eooe wave generated by the DWP, the explicit coordinates of the intersection of the wave normal on each interface must be derived through the wave normal equation. Therefore, we first determine the direction vector and the corresponding wave normal equation of both waves in each crystal according to the Eq. (1) to (3). Then, the explicit coordinates of the intersection of the wave normal on each interface can be obtained based on wave normal equations. And the OPD of the two waves can be calculated according to the obtained explicit coordinates of the intersection.

2.2.1 Wave normal direction vector in each wedge-shaped crystal

As shown in Fig. 1, the geometry configuration of the DWP determines that the wave normal direction in crystals 1 and 4 is the same and so is in crystals 2 and 3. Therefore, it is necessary to determine the direction of wave normal in crystal 1, crystal 2, and the AG.

The first coordinate system is formed at interface 1, with the optical axis of the crystal 1 ${{\mathbf w}_{1}}$ parallel to the z-axis, and the normal vector of the interface 1 ${{\mathbf n}_{1}}$ parallel to the x-axis (as shown in Fig. 2).

 figure: Fig. 2.

Fig. 2. Refraction at interface 1 in the first coordinate system.

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The wave enters the crystal from the air at interface 1, and the direction vector of the incident wave normal ${{\mathbf k}_{0}}$ can be expressed as:

$${{\mathbf k}_{\mathbf 0}} = (\cos {i_1},\sin {\omega _1}\sin {i_1},\cos {\omega _1}\sin {i_1})$$
${i_1}$ is the incident angle, and ${\omega _1}$ is the incident plane angle. Because the wave normal of the refracted light and the incident light are in the same plane, the direction vector of the eooe and oeeo wave normal in the crystal 1 can be expressed as:
$${{\mathbf k}_{{\mathbf {1e}}}} = (\cos {r_{1e}},\sin {\omega _1}\sin {r_{1e}},\cos {\omega _1}\sin {r_{1e}})$$
$${{\mathbf k}_{{\mathbf {1o}}}} = (\cos {r_{1o}},\sin {\omega _1}\sin {r_{1o}},\cos {\omega _1}\sin {r_{1o}})$$
Where ${r_{1e}}$ is the refraction angle of eooe wave normal and be determined by the following relationships:
$$\sin {i_1}\textrm{ = }n({r_{1e}})\sin {r_{1e}}$$
$$n({r_{1e}}) = \frac{{{n_o}{n_e}}}{{\sqrt {n_e^2{{\cos }^2}{\theta _e} + n_o^2{{\sin }^2}{\theta _e}} }}$$
$$\cos {\theta _e}\textrm{ = }\cos {\omega _1}\sin {r_{1e}}$$
and ${r_{1o}}$ is the refraction angle of oeeo wave normal and be determined by:
$$\sin {i_1} = {n_o}\sin {r_{1o}}$$

The wave normal in crystal 2 in the first coordinate system is shown in Fig. 3. The normal vector of interface 2 ${{\mathbf n}_{2}}$ is:

$${{\mathbf n}_{2}} = (\cos \beta ,0, - \sin \beta )$$
and the optical axis direction vector of the crystal 2 ${{\mathbf w}_{2}}$ is:
$${{\mathbf w}_{2}}\textrm{ = (0,1,0)}$$

 figure: Fig. 3.

Fig. 3. Wave normal in crystal 2 in the first coordinate system.

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The refraction relationship in the first coordinate system is too complicated to calculate. To simplify the calculation, the appropriate coordinate conversion is made and the second coordinate system is shown in Fig. 4, with the normal direction of interface 2 parallel to the x-axis and the optical axis of crystal 2 parallel to the y-axis.

 figure: Fig. 4.

Fig. 4. Refraction at interface 2 in the second coordinate system.

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Then the incident angle of eooe wave normal and oeeo wave normal at interface 2 can be determined by:

$$\cos {i^{\prime}_{2e}} = \frac{{{{\mathbf k}_{{\mathbf {1e}}}} \cdot {{\mathbf n}_{2}}}}{{|{{{\mathbf k}_{{\mathbf {1e}}}}} ||{{{\mathbf n}_{2}}} |}} = \cos {r_{1e}}\cos \beta + \cos {\omega _1}\sin {r_{1e}}\sin \beta$$
$$\cos {i^{\prime}_{2o}} = \frac{{{{\mathbf k}_{{\mathbf {1o}}}} \cdot {{\mathbf n}_{2}}}}{{|{{{\mathbf k}_{{\mathbf {1o}}}}} ||{{{\mathbf n}_{2}}} |}} = \cos {r_{1o}}\cos \beta + \cos {\omega _1}\sin {r_{1o}}\sin \beta$$

The normal vector of the incident plane is:

$$\begin{aligned} {{{\mathbf n^{\prime}}}_{{{\mathbf \omega }_{\mathbf 2}}{\mathbf e}}} &= {{\mathbf k}_{{\mathbf {1e}}}} \times {{\mathbf n}_{\mathbf 2}}\\ {}\textrm{ } =&{-} \sin \omega \sin {r_{1e}}\sin \beta \textrm{ }{\mathbf i}\\ \textrm{ } &+ \cos {r_{1e}}\sin \beta + \cos \omega \sin {r_{1e}}\cos \beta \textrm{ }{\mathbf j}\\& \textrm{ } - \sin \omega \sin {r_{1e}}\cos \beta \textrm{ }{\mathbf k} \end{aligned}$$
$$\begin{aligned} {{{\mathbf n^{\prime}}}_{{{\mathbf \omega }_{\mathbf 2}}{\mathbf o}}} &= {{\mathbf k}_{{\mathbf {1o}}}} \times {{\mathbf n}_{\mathbf 2}}\\ {}\textrm{ } =&{-} \sin \omega \sin {r_{1o}}\sin \beta \textrm{ }{\mathbf i}\\ \textrm{ } &+ \cos {r_{1o}}\sin \beta + \cos \omega \sin {r_{1o}}\cos \beta \textrm{ }{\mathbf j}\\& \textrm{ } - \sin \omega \sin {r_{1o}}\cos \beta \textrm{ }{\mathbf k} \end{aligned}$$

Then the direction vector of the two wave normal emitted at interface 2 is:

$${{\mathbf k^{\prime}}_{{\mathbf {2e}}}} = (\cos {r^{\prime}_{2e}},\cos {\omega ^{\prime}_{2e}}\sin {r^{\prime}_{2e}},\sin {\omega ^{\prime}_{2e}}\sin {r^{\prime}_{2e}})$$
$${{\mathbf k^{\prime}}_{{\mathbf {2o}}}} = (\cos {r^{\prime}_{2o}},\cos {\omega ^{\prime}_{2o}}\sin {r^{\prime}_{2o}},\sin {\omega ^{\prime}_{2o}}\sin {r^{\prime}_{2o}})$$

According to Snell's law, in the second coordinate system, the incident plane angle and incident angle of eooe wave normal at interface 2 are determined as follows:

$$\cos {\omega ^{\prime}_{2e}} = \frac{{{{{\mathbf n^{\prime}}}_{{\mathbf \omega \mathbf {2e}}}} \cdot {{\mathbf w}_{\mathbf 2}}}}{{|{{{{\mathbf n^{\prime}}}_{{\mathbf \omega \mathbf {2e}}}}} ||{{{\mathbf w}_{\mathbf 2}}} |}}$$
$$n({r_{1e}})\sin {i^{\prime}_{2e}}\textrm{ = }{n_o}\sin {r^{\prime}_{2e}}$$

For oeeo wave normal, the incident plane angle and incident angle at interface 2 are determined as follows:

$$\cos {\omega ^{\prime}_{2o}} = \frac{{{{{\mathbf n^{\prime}}}_{{\mathbf \omega \mathbf {2o}}}} \cdot {{\mathbf w}_{\mathbf 2}}}}{{|{{{{\mathbf n^{\prime}}}_{{\mathbf \omega \mathbf {2o}}}}} ||{{{\mathbf w}_{\mathbf 2}}} |}}$$
$${n_o}\sin {i^{\prime}_{2o}}\textrm{ = }n({r^{\prime}_{2o}})\sin {r^{\prime}_{2o}}$$
$$n({r^{\prime}_{2o}}) = \frac{{{n_o}{n_e}}}{{\sqrt {n_e^2{{\cos }^2}{\theta _o} + n_o^2{{\sin }^2}{\theta _o}} }}$$
$$\cos {\theta _o}\textrm{ = }{{\mathbf k}_{{\mathbf {1o}}}} \cdot {{\mathbf w}_{\mathbf 1}} = \cos {\omega _1}\sin {r_{1o}}$$

The y-axis component remains unchanged during the process of coordinate transformation, the following relationships can be used to make coordinate inverse transformation:

$$\cos {\omega _{2e}}\sin {r_{2e}} = \cos {\omega ^{\prime}_{2e}}\sin {r^{\prime}_{2e}}$$
$$\cos {\omega _{2o}}\sin {r_{2o}} = \cos {\omega ^{\prime}_{2o}}\sin {r^{\prime}_{2o}}$$
$$n({r_{2o}})\textrm{ = }n({r^{\prime}_{2o}})$$

The direction vectors of the two wave normal in the first coordinate system are determined as follows:

$${{\mathbf k}_{{\mathbf {e2}}}} = (\cos {r_{2e}},\cos {\omega _{2e}}\sin {r_{2e}},\sin {\omega _{2e}}\sin {r_{2e}})$$
$${{\mathbf k}_{{\mathbf {o2}}}} = (\cos {r_{2o}},\cos {\omega _{2o}}\sin {r_{2o}},\sin {\omega _{2o}}\sin {r_{2o}})$$

In the first coordinate system, the refraction at interface 3 is shown in Fig. 5, the refraction angles of the two waves at the interface 3 can be determined as,

$${n_i}\sin {r_{3e}}\textrm{ = }{n_o}\sin {i_{3e}}$$
$${n_i}\sin {r_{3o}} = n({r_{2o}})\sin {i_{3o}}$$

 figure: Fig. 5.

Fig. 5. Refraction at interface 3 in the first coordinate system.

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In the second coordinate system, the normal vector of interface 3 is:

$${{\mathbf n^{\prime}}_{3}} = (\cos \beta ,0,\sin \beta )$$

Then the incident angle of eooe wave normal and oeeo wave normal at interface 3 can be determined by

$$\cos {i_{3e}} = \frac{{{{{\mathbf k^{\prime}}}_{{\mathbf {2e}}}} \cdot {{{\mathbf n^{\prime}}}_{3}}}}{{|{{{{\mathbf k^{\prime}}}_{{\mathbf {2e}}}}} ||{{{{\mathbf n^{\prime}}}_{3}}} |}} = \cos {r^{\prime}_{2e}}\cos \beta - \cos {\omega _1}\sin {r^{\prime}_{2e}}\sin \beta$$
$$\cos {i_{3o}} = \frac{{{{{\mathbf k^{\prime}}}_{{\mathbf 2o}}} \cdot {{{\mathbf n^{\prime}}}_{\mathbf 3}}}}{{|{{{{\mathbf k^{\prime}}}_{{\mathbf {2e}}}}} ||{{{{\mathbf n^{\prime}}}_{\mathbf 3}}} |}} = \cos {r^{\prime}_{2o}}\cos \beta - \cos {\omega _1}\sin {r^{\prime}_{2o}}\sin \beta$$

According to the geometric relationship, ${i_{3e}} = {r_{2e}}\textrm{, }{i_{3o}} = {r_{2o}}$.

Therefore, the direction vectors of eooe wave and oeeo wave in the AG are:

$${{\mathbf k}_{{\mathbf {3e}}}} = (\cos {r_{3e}},\cos {\omega _{2e}}\sin {r_{3e}},\sin {\omega _{2e}}\sin {r_{3e}})$$
$${{\mathbf k}_{{\mathbf {3o}}}} = (\cos {r_{3o}},\cos {\omega _{2o}}\sin {r_{3o}},\sin {\omega _{2o}}\sin {r_{3o}})$$

2.2.2 Calculation of the OPD

The paths of eooe and oeeo wave passing through the DWP are depicted in detail in Fig. 6, where O $({x_1},{y_1},{z_1})$ denotes the incident point, A, B, C, D and E denote the intersection points of eooe wave at each interface, A’, B’, C’, D’ and E’ denote the intersection points of oeeo wave at each interface, and $({x_i},{y_i},{z_i})$ denotes the coordinates of the intersection points.

 figure: Fig. 6.

Fig. 6. The path of waves passing through the DWP (for calcite WPs, projected in the OXZ plane).

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The coordinates of intersection points at each interface can be calculated according to the wave normal equation, which can be obtained based on the calculated direction vectors The calculated coordinates of the intersection points at each interface are as follows:

Points A and A’:

$$\quad \quad \left\{ \begin{array}{l} {x_{2e}} = \frac{{t\sin {\omega_{2e}}\tan {r_{2e}} - {z_{3e}}}}{{\sin {\omega_{2e}}\tan {r_{2e}} - \cot \beta }}\\ {y_{2e}} = {x_{2e}}\sin {\omega_1}\tan {r_{1e}} + {y_1}\\ {z_{2e}} = {x_{2e}}\cos {\omega_1}\tan {r_{1e}} + {z_1} \end{array} \right. \quad \quad \quad \quad \quad \;\left\{ \begin{array}{l} {x_{2o}} = \frac{{t\sin {\omega_{2o}}\tan {r_{2o}} - {z_{3o}}}}{{\sin {\omega_{2o}}\tan {r_{2o}} - \cot \beta }}\\ {y_{2o}} = {x_{2o}}\sin {\omega_1}\tan {r_{1o}} + {y_1}\\ {z_{2o}} = {x_{2o}}\cos {\omega_1}\tan {r_{1o}} + {z_1} \end{array} \right.\quad \;$$

Points B and B’:

$$\quad \quad \left\{ \begin{array}{l} {x_{3e}} = t\\ {y_{3e}} = (t - {x_{2e}})\cos {\omega_{2e}}\tan {r_{2e}} + {y_{2e}}\\ {z_{3e}} = (t - {x_{2e}})\sin {\omega_{2e}}\tan {r_{2e}} + {z_{2e}} \end{array} \right. \quad \quad \,\,{\kern 1pt} \,\,\left\{ \begin{array}{l} {x_{3o}} = t\\ {y_{3o}} = (t - {x_{2o}})\cos {\omega_{2o}}\tan {r_{2o}} + {y_{2o}}\\ {z_{3o}} = (t - {x_{2o}})\sin {\omega_{2o}}\tan {r_{2o}} + {z_{2o}} \end{array} \right.$$

Points C and C’:

$$\quad \quad \left\{ \begin{array}{l} {x_{4e}} = t + s\\ {y_{4e}} = s\cos {\omega_{2e}}\tan {r_{3e}} + {y_{3e}}\\ {z_{4e}} = s\sin {\omega_{2e}}\tan {r_{3e}} + {z_{3e}} \end{array} \right. \,\quad {\kern 1pt} \quad \quad \quad \left\{ \begin{array}{l} {x_{4o}} = t + s\\ {y_{4o}} = s\cos {\omega_{2o}}\tan {r_{3o}} + {y_{3o}}\\ {z_{4o}} = s\sin {\omega_{2o}}\tan {r_{3o}} + {z_{3o}} \end{array} \right.$$

Points D and D’:

$$\quad \quad \left\{ \begin{array}{l} {x_{5e}} = \frac{{ - {z_{3e}} - s\sin {\omega_{2e}}\tan {r_{2e}}}}{{\sin {\omega_{2e}}\tan {r_{2e}} - \cot \beta }} + t + s\\ {y_{5e}} = ({x_{5e}} - {x_{4e}})\cos {\omega_{2e}}\tan {r_{2e}} + {y_{4e}}\\ {z_{5e}} = ({x_{5e}} - {x_{4e}})\sin {\omega_{2e}}\tan {r_{2e}} + {z_{4e}} \end{array} \right.\,\quad \quad \,\,\left\{ \begin{array}{l} {x_{5o}} = \frac{{ - {z_{3o}} - s\sin {\omega_{2o}}\tan {r_{2o}}}}{{\sin {\omega_{2o}}\tan {r_{2o}} - \cot \beta }} + t + s\\ {y_{5o}} = ({x_{5o}} - {x_{4o}})\cos {\omega_{2o}}\tan {r_{2o}} + {y_{4o}}\\ {z_{5o}} = ({x_{5o}} - {x_{4o}})\sin {\omega_{2o}}\tan {r_{2o}} + {z_{4o}} \end{array} \right.$$

Points E and E’:

$$\quad \quad \left\{ \begin{array}{l} {x_{6e}} = 2t + s\\ {y_{6e}} = (2t + s - {x_{5e}})\sin {\omega_1}\tan {r_{1e}} + {y_{5e}}\\ {z_{6e}} = (2t + s - {x_{5e}})\cos {\omega_1}\tan {r_{1e}} + {z_{5e}} \end{array} \right.\quad \quad \left\{ \begin{array}{l} {x_{6o}} = 2t + s\\ {y_{6o}} = (2t + s - {x_{5o}})\sin {\omega_1}\tan {r_{1o}} + {y_{5o}}\\ {z_{6o}} = (2t + s - {x_{5o}})\cos {\omega_1}\tan {r_{1o}} + {z_{5o}} \end{array} \right.$$

Therefore, the interior OPD (${\Delta _{in}}$) and the exterior OPD (${\Delta _{ex}}$) between eooe wave and oeeo wave can be calculated as follows:

$$\begin{array}{l} {\Delta _{in}} = n({r_{1e}})OA + n({r_{1e}})DE + {n_o}AB + {n_o}CD + {n_i}BC\\ \textrm{ } - ({n_o}OA^{\prime} + {n_o}D^{\prime}E^{\prime} + n({r_{2o}})A^{\prime}B^{\prime} + n({r_{2o}})C^{\prime}D^{\prime} + {n_i}B^{\prime}C^{\prime})\\ = n({r_{1e}})\left[ {\sqrt {{{({x_{2e}} - {x_1})}^2} + {{({y_{2e}} - {y_1})}^2} + {{({z_{2e}} - {z_1})}^2}} + \sqrt {{{({x_{6e}} - {x_{5e}})}^2} + {{({y_{6e}} - {y_{5e}})}^2} + {{({z_{6e}} - {z_{5e}})}^2}} } \right]\\ \textrm{ } + {n_o}\left[ {\sqrt {{{({x_{5e}} - {x_{4e}})}^2} + {{({y_{5e}} - {y_{4e}})}^2} + {{({z_{5e}} - {z_{4e}})}^2}} + \sqrt {{{({x_{3e}} - {x_{2e}})}^2} + {{({y_{3e}} - {y_{2e}})}^2} + {{({z_{3e}} - {z_{2e}})}^2}} } \right]\\ \textrm{ } + {n_i}\left[ {\sqrt {{{({x_{4e}} - {x_{3e}})}^2} + {{({y_{4e}} - {y_{3e}})}^2} + {{({z_{4e}} - {z_{3e}})}^2}} - \sqrt {{{({x_{4o}} - {x_{3o}})}^2} - {{({y_{4o}} - {y_{3o}})}^2} - {{({z_{4o}} - {z_{3o}})}^2}} } \right]\\ \textrm{ } - n({r_{2o}})\left[ {\sqrt {{{({x_{5o}} - {x_{4o}})}^2} + {{({y_{5o}} - {y_{4o}})}^2} + {{({z_{5o}} - {z_{4o}})}^2}} + \sqrt {{{({x_{3o}} - {x_{2o}})}^2} + {{({y_{3o}} - {y_{2o}})}^2} + {{({z_{3o}} - {z_{2o}})}^2}} } \right]\\ \textrm{ } - {n_o}\left[ {\sqrt {{{({x_{2o}} - {x_1})}^2} + {{({y_{2o}} - {y_1})}^2} + {{({z_{2o}} - {z_1})}^2}} + \sqrt {{{({x_{6o}} - {x_{5o}})}^2} + {{({y_{6o}} - {y_{5o}})}^2} + {{({z_{6o}} - {z_{5o}})}^2}} } \right]\\ \textrm{ = }\frac{{n({r_{1e}})}}{{\cos {r_{1e}}}}(t - {l_e}) + \frac{{{n_o}}}{{\cos {r_{2e}}}}(t + {l_e}) + \frac{{{n_i}}}{{\cos {r_{3e}}}}s - \frac{{{n_o}}}{{\cos {r_{1o}}}}(t - {l_o}) - \frac{{n({r_{2o}})}}{{\cos {r_{2o}}}}(t + {l_o}) - \frac{{{n_i}}}{{\cos {r_{3o}}}}s \end{array}$$
$$\begin{aligned} {\Delta _{ex}} &= ({y_{6o}} - {y_{6e}})\sin {\omega _1}\sin {i_1} + ({z_{6o}} - {z_{6e}})\cos {\omega _1}\sin {i_1}\\ \textrm{ } &= \sin {\omega _1}\sin {i_1}[\sin {\omega _1}\tan {r_{1o}}(t - {l_o}) + \sin {\omega _{2o}}\tan {r_{2o}}(t + {l_e}) + s\sin {\omega _{2o}}\tan {r_{2o}})\\ &\textrm{ } - \sin {\omega _1}\tan {r_{1e}}(t - {l_e}) - \sin {\omega _{2e}}\tan {r_{2e}}(t + {l_e}) - s\sin {\omega _{2e}}\tan {r_{2e}}]\\& \textrm{ } + \cos {\omega _1}\sin {i_1}[\cos {\omega _1}\tan {r_{1o}}(t - {l_o}) + \cos {\omega _{2o}}\tan {r_{2o}}(t + {l_e}) + s\cos {\omega _{2o}}\tan {r_{2o}}\\ \;\quad \,\;\;\, &- \cos {\omega _1}\tan {r_{1e}}(t - {l_e}) - \cos {\omega _{2e}}\tan {r_{2e}}(t + {l_e}) - s\cos {\omega _{2e}}\tan {r_{2e}}] \end{aligned}$$
Where
$${l_e} = \frac{{t\cos {\omega _{2e}}\tan {r_{2e}} + s\cos {\omega _{2e}}\tan {r_{3e}}}}{{\cot \beta - \cos {\omega _{2e}}\tan {r_{2e}}}}$$
$${l_o} = \frac{{t\cos {\omega _{2o}}\tan {r_{2o}} + s\cos {\omega _{2o}}\tan {r_{3o}}}}{{\cot \beta - \cos {\omega _{2o}}\tan {r_{2o}}}}$$

The total OPD (${\Delta _{total}}$) generated by the DWP is :

$${\Delta _{total}} = {\Delta _{in}} + {\Delta _{out}}$$

The Eq. (26) to (28) do not contain the position coordinate of the incident point, which means that the total OPD generated by the DWP is only related to the incident direction of the waves but not the incident position.

Compared with the OPD equations obtained by the quasi-parallel plate approximation method in our previous work [19], which can only calculate the OPD in the principal section, the most significant improvement of the OPD Eq. (26)-(29) is that they can completely describe the OPD in arbitrary incidence plane and angle.

3. Analysis of optical transmission characteristics

3.1 Spatial distribution characteristics of OPD and experimental verification

The interior, exterior and total OPD (${\Delta _{in}}$, ${\Delta _{ex}}$ and ${\Delta _{total}}$) in arbitrary incidence angles for various AG (s = 0 mm, 5 mm, and 10 mm) are calculated according to the Eqs. (26)-(29) and the corresponding spatial distributions are shown in Fig. 7. Here, the different incident angles correspond to the different row and column units. The calcite prism with a structural angle of 27.15° and a thickness of 10.2 mm was employed in the calculation, the FOV used in the simulation is 10°.

 figure: Fig. 7.

Fig. 7. The spatial distributions of the OPD.

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Figure 7(a) shows that ${\Delta _{in}}$ presents an irregular curved surface distribution and the maximum ${\Delta _{in}}$ increase slightly with the increase of s. The ${\Delta _{in}}$ along the row and column direction present an asymmetrical curve line and a symmetrical concave arc line distribution, respectively. Figure 7(b) shows that ${\Delta _{ex}}$ presents a slightly convex plane distribution and the maximum ${\Delta _{ex}}$ increase significantly with the increase of s. The ${\Delta _{ex}}$ along the row and column direction present an asymmetrical curve line and a symmetrical convex arc line distribution, respectively. ${\Delta _{total}}$ is the sum of ${\Delta _{in}}$ and ${\Delta _{ex}}$, and the maximum ${\Delta _{total}}$ increase significantly with the increase of s. ${\Delta _{total}}$ is of spatial distribution characteristic of both ${\Delta _{in}}$ and ${\Delta _{ex}}$, as shown in Fig. 7(c). The proportion of ${\Delta _{in}}$ and ${\Delta _{ex}}$ in ${\Delta _{total}}$ is changing with the increase of s. The spatial distribution of contour lines is shown in Fig. 7(d), which indicates that the contour lines are slightly concave arcs with unequal intervals, and the interval of contour lines decreases obviously and the bend decreases slightly with the increase of s.

To verify the validity of the calculated OPD, the simulated interferogram based on the calculated OPD is compared with that obtained by the experiments.

The experimental setup is shown in Fig. 8(c), which mainly consists of a polarizer, the DWP (the same parameters as the simulation calculation), an analyzer, and a CCD camera with a 50 mm lens and 4024 × 3036 pixels. A 632.8 nm He-Ne laser and a deuterium/halogen light source (DH-2000-BAL) are used as monochromatic and polychromatic light sources, respectively. The wavelength range of DH-2000-BAL is 230-2500 nm, but the wavelength range of the fiber used in the experiment is 400-1000 nm, so the effective wavelength range of the experiment is 400-1000 nm.

 figure: Fig. 8.

Fig. 8. Interference experiment: (a) Simulated interferogram; (b) Experiment interferogram; (c) Experiment system.

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The monochromatic and polychromatic interferogram simulated based on the calculated ${\Delta _{total}}$ are shown in Fig. 8(a). The monochromatic interferograms with FOV of 1°for various AG (s = 0 mm, 5 mm, and 10 mm) indicate that the fringe interval decreases obviously and the bent decreases slightly with the increase of AG. The polychromatic interferograms with various FOV (1°, 5°, and 10°) indicate that the curvature of the fringes increases slightly with the increase of the FOV.

The monochromatic and polychromatic interferograms obtained by experiments are shown in Fig. 8(b), which match well with simulated interferograms and confirm the validity of the calculated ${\Delta _{total}}$.

3.2 Comparison with the quasi-parallel plate method

The OPD equations obtained by the quasi-parallel plate method in our previous work [19] are as follows:

$$\begin{aligned} {{\Delta ^{\prime}}_{total.}} &= (2t + s)({n_o}\cos {r_{o1}} - n({r_{e1}})\cos {r_{e1}})\\& \textrm{ } + (t + s)\cos \theta ({n_e}\cos {r_{o2}} + n({r_{e1}})\cos {i_{e2}} - {n_o}\cos {r_{e2}} - {n_o}\cos {i_{o2}})\\& \textrm{ } + s({n_i}\cos {r_{o3}} - {n_i}\cos {r_{e3}} - {n_e}\cos {i_{e3}} + {n_o}\cos {i_{o3}}) \end{aligned}$$
which is the approximation of the OPD in the principal section (i.e. the OPD corresponding to Row = 0 in Fig. 7). The ${\Delta ^{\prime}_{total}}$ and exact ${\Delta _{total}}$ in the principal section for various AG is shown by the dash and solid lines, respectively, in Fig. 9(a), which indicate that even in the principal section, the ${\Delta ^{\prime}_{total}}$ deviates from the ${\Delta _{total}}$. As shown in Fig. 9(b), the deviation (${\Delta ^{\prime}_{total}} - {\Delta _{total}}$) increases with the increase of incidence angle and s. Also, the differential (the change rate of the OPD with incident angle) of ${\Delta ^{\prime}_{total}}$ also deviates from that of ${\Delta _{total}}$, and the differential deviation increases with the increase of incident angle and AG. All of the above deviations of ${\Delta ^{\prime}_{total}}$ with respect to ${\Delta _{total}}$ prove that the ${\Delta ^{\prime}_{total}}$ is the approximate OPD in the principal section.

 figure: Fig. 9.

Fig. 9. Comparison between the approximate OPD and the exact OPD.

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The interferograms simulated based on the ${\Delta _{total}}$ and ${\Delta ^{\prime}_{total}}$ are shown in Fig. 10(a) and (b), respectively. It shows clearly that the interference fringes predicted by ${\Delta _{total}}$ are slightly curved but that by ${\Delta ^{\prime}_{total}}$ are almost straight and the fringe intervals predicated by ${\Delta _{total}}$ and ${\Delta ^{\prime}_{total}}$ are different.

 figure: Fig. 10.

Fig. 10. Polychromatic simulation interferogram: (a) Exact simulation; (b) Approximate simulation.

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4. Performance analysis of the SBFTIS

To know the spectral zooming and FOV performance of the novel SBFTIS, it needs to perform Fourier transform on the calculated interferogram. However, as can be seen from the spatial distribution of contour lines in Fig. 7 (d), the interferogram is a group of slightly curved fringes with unequal intervals. Therefore, the non-uniform Fourier transform (NUFT), instead of the fast Fourier transform, needs to be used to restore the spectra.

4.1 Performance of spectral zooming

To investigate the spectral zooming performance, we perform the NUFT on the exact simulated monochromatic (632.8 nm) interferograms for various AG (s = 0 mm, 5 mm and 10 mm) and the restored spectra are shown in Fig. 11(the intensity coordinates are offset).

 figure: Fig. 11.

Fig. 11. The restored spectra of monochromatic light.

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The full width at half maxima (FWHM) of the peak decreases from 17.62 to 7.61nm as s increases from 0 nm to 10 nm, which indicates that the novel SBFTIS is of the spectrum zooming performance.

4.2 Performance of FOV

To investigate the FOV performance, we perform the NUFT on the exact simulated polychromatic interferogram with various FOV (1.0°, 2.5°, 5.0°, 10.0°). The raw spectra are shown by the black dash lines in Fig. 12. The spectra restored based on ${\Delta _{total}}$ for various FOV are shown by red solid lines, which match very well with the raw spectra. For comparison, the spectra restored based on ${\Delta ^{\prime}_{total}}$ are also shown by blue short dash-dot lines. When the FOV is less than 2.5°, the restored spectra still match the raw spectrum. However, as the FOV is larger than 5.0°, the restored spectra deviate from the raw spectrum completely.

 figure: Fig. 12.

Fig. 12. The restored spectra of polychromatic light: (a) FOV = 1.0°; (b) FOV = 2.5°; (c) FOV = 5.0°; (d) FOV = 10.0°.

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The above results show that the exact OPD calculation is very important for designing the novel SBFTIS with a large FOV.

5. Conclusions

The exact OPD equations of the DWP at arbitrary incidence plane and angle are derived by the wave normal tracing method. The spatial distribution characteristic of the OPD for various AG is analyzed. The interferograms for various AG calculated based on the equations matched well with those obtained by experiments, which verify the validity of the derived OPD equations.

The performance of the novel SBFTIS is investigated by performing the NUFT on the exactly calculated interferograms. The full width at half maxima (FWHM) of the monochromatic spectra peak changes from 17.62 to 7.61nm as AG varies from s = 0 nm to 10 nm, which means that the novel SBFTIS is of spectral zooming capability. The restored spectra matched very well with the raw spectra as the FOV varies from 1.0 to 10.0°, and the FOV is much larger than that predicted by reference.

The results obtained in this article provide a theoretical basis for completely describing the optical transmission characteristic of the DWP and developing the high-performance birefringent spectral zooming imaging spectrometer.

Funding

National Natural Science Foundation of China (61890961); Key R&D project in the Shaanxi Province of China (2020GY-274); National Key Research and Development Program of China (2018JM6008); National Major Scientific Instruments and Equipments Development Project of National Natural Science Foundation of China (62127813).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. P. Griffiths and J. D. Haseth, Fourier Transform Infrared Spectrometry (John Wiley & Sons, Inc., 1986).

2. J. Courtial, B. Patterson, A. Harvey, W. Sibbett, and M. Padgett, “Design of a static Fourier-transform spectrometer with increased field of view,” Appl. Opt. 35(34), 6698–6702 (1996). [CrossRef]  .

3. C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203(1-2), 21–26 (2002). [CrossRef]  .

4. A. R. Harvey and D. W. Fletcher-Holmes, “Birefringent Fourier-transform imaging spectrometer,” Opt. Express 12(22), 5368–5374 (2004). [CrossRef]  .

5. J. Li, J. Zhu, and H. Wu, “Compact static Fourier transform imaging spectropolarimeter based on channeled polarimetry,” Opt. Lett. 35(22), 3784–3786 (2010). [CrossRef]  .

6. T. Mu, C. Zhang, C. Jia, and W. Ren, “Static hyperspectral imaging polarimeter for full linear Stokes parameters,” Opt. Express 20(16), 18194–18201 (2012). [CrossRef]  .

7. C. Bai, J. Li, Y. Shen, and J. Zhou, “Birefringent Fourier transform imaging spectrometer with a rotating retroreflector,” Opt. Lett. 41(15), 3647–3650 (2016). [CrossRef]  .

8. Q. Liu, C. Bai, J. Liu, J. He, and J. Li, “Fourier transform imaging spectropolarimeter using ferroelectric liquid crystals and Wollaston interferometer,” Opt. Express 25(17), 19904–19922 (2017). [CrossRef]  .

9. C. Bai, J. Li, Y. Xu, H. Yuan, and J. Liu, “Compact birefringent interferometer for Fourier transform hyperspectral imaging,” Opt. Express 26(2), 1703 (2018). [CrossRef]  .

10. Q. Li, F. Lu, X. Wang, and C. Zhu, “Low crosstalk polarization-difference channeled imaging spectropolarimeter using double-Wollaston prism,” Opt. Express 27(8), 11734 (2019). [CrossRef]  .

11. J. Li, C. Qu, H. Wu, and C. Qi, “Spectral resolution enhanced static Fourier transform spectrometer based on a birefringent retarder array,” Opt. Express 27(11), 15505 (2019). [CrossRef]  .

12. C. Bai, J. Li, W. Zhang, Y. Xu, and Y. Feng, “Static full-Stokes Fourier transform imaging spectropolarimeter capturing spectral, polarization, and spatial characteristics,” Opt. Express 29(23), 38623 (2021). [CrossRef]  .

13. C. Zhang, T. Yan, C. Jia, and W. E. Ward, “Tempo-spatially modulated imaging spectropolarimetry based on polarization modulation array,” J. Quant. Spectrosc. Radiat. Transfer 261, 107448 (2021). [CrossRef]  .

14. J. Li, J. Zhu, C. Qi, C. Zheng, B. Gao, Y. Zhang, and X. Hou, “Compact static imaging spectrometer combining spectral zooming capability with a birefringent interferometer,” Opt. Express 21(8), 10182 (2013). [CrossRef]  .

15. J. Li, J. Zhu, Y. Zhang, H. Liu, and X. Hou, “Spectral zooming birefringent imaging spectrometer,” Acta. Phys. Sin. 62(2), 261–268 (2013).

16. R. G. Sellar and G. D. Boreman, “Classification of imaging spectrometers for remote sensing applications,” Opt. Eng. 44(1), 13602–13603 (2004).

17. B. Chen, M. R. Wang, Z. Liu, and J. J. Yang, “Dynamic spectral imaging with spectral zooming capability,” Opt. Lett. 32(11), 1518–1520 (2007). [CrossRef]  .

18. S. P. Prunet, “Exact calculation of the optical path difference and description of a new birefringent interferometer,” Opt. Eng. 38(6), 983 (1999). [CrossRef]  .

19. J. Li, H. Wu, and C. Qi, “Complete description of the optical path difference of a novel spectral zooming imaging spectrometer,” Opt. Commun. 410, 598–603 (2018). [CrossRef]  .

20. M. Françon and S. Mallick, Polarization Interferometers: Applications in Microscopy and Macroscopy (Wiley-Interscience, 1971).

21. M. Lv and P. Wang, “Ray tracing in Rochon prisms with absorption,” Opt. Express 25(13), 14676 (2017). [CrossRef]  .

22. X. Hu, J. Ai, Z. Kong, P. Gao, S. Zhang, and X. Wang, “Research on the optical axis of the birefringent crystal rotating in the three dimensional space,” Optik 181, 786–795 (2019). [CrossRef]  .

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (12)
Fig. 1.
Fig. 1. (a) The schematic of the SBFTIS. (b) Layout of the DWP.(c) Schematic of the path of waves passing through the DWP.

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Fig. 2.
Fig. 2. Refraction at interface 1 in the first coordinate system.

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Fig. 3.
Fig. 3. Wave normal in crystal 2 in the first coordinate system.

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Fig. 4.
Fig. 4. Refraction at interface 2 in the second coordinate system.

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Fig. 5.
Fig. 5. Refraction at interface 3 in the first coordinate system.

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Fig. 6.
Fig. 6. The path of waves passing through the DWP (for calcite WPs, projected in the OXZ plane).

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Fig. 7.
Fig. 7. The spatial distributions of the OPD.

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Fig. 8.
Fig. 8. Interference experiment: (a) Simulated interferogram; (b) Experiment interferogram; (c) Experiment system.

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Fig. 9.
Fig. 9. Comparison between the approximate OPD and the exact OPD.

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Fig. 10.
Fig. 10. Polychromatic simulation interferogram: (a) Exact simulation; (b) Approximate simulation.

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Fig. 11.
Fig. 11. The restored spectra of monochromatic light.

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Fig. 12.
Fig. 12. The restored spectra of polychromatic light: (a) FOV = 1.0°; (b) FOV = 2.5°; (c) FOV = 5.0°; (d) FOV = 10.0°.

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Equations (47)

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n i sin ⁡ i  =  n ( r e ) sin ⁡ r e = n o sin ⁡ r o
n ( r e ) = n o n e n e 2 cos 2 θ e + n o 2 sin 2 θ e
cos ⁡ θ e = sin ⁡ ψ cos ⁡ ω sin ⁡ r e + cos ⁡ ψ cos ⁡ r e
k 0 = ( cos ⁡ i 1 , sin ⁡ ω 1 sin ⁡ i 1 , cos ⁡ ω 1 sin ⁡ i 1 )
k 1 e = ( cos ⁡ r 1 e , sin ⁡ ω 1 sin ⁡ r 1 e , cos ⁡ ω 1 sin ⁡ r 1 e )
k 1 o = ( cos ⁡ r 1 o , sin ⁡ ω 1 sin ⁡ r 1 o , cos ⁡ ω 1 sin ⁡ r 1 o )
sin ⁡ i 1  =  n ( r 1 e ) sin ⁡ r 1 e
n ( r 1 e ) = n o n e n e 2 cos 2 θ e + n o 2 sin 2 θ e
cos ⁡ θ e  =  cos ⁡ ω 1 sin ⁡ r 1 e
sin ⁡ i 1 = n o sin ⁡ r 1 o
n 2 = ( cos ⁡ β , 0 , − sin ⁡ β )
w 2  = (0,1,0)
cos ⁡ i 2 e ′ = k 1 e ⋅ n 2 | k 1 e | | n 2 | = cos ⁡ r 1 e cos ⁡ β + cos ⁡ ω 1 sin ⁡ r 1 e sin ⁡ β
cos ⁡ i 2 o ′ = k 1 o ⋅ n 2 | k 1 o | | n 2 | = cos ⁡ r 1 o cos ⁡ β + cos ⁡ ω 1 sin ⁡ r 1 o sin ⁡ β
n ′ ω 2 e = k 1 e × n 2   = − sin ⁡ ω sin ⁡ r 1 e sin ⁡ β   i   + cos ⁡ r 1 e sin ⁡ β + cos ⁡ ω sin ⁡ r 1 e cos ⁡ β   j   − sin ⁡ ω sin ⁡ r 1 e cos ⁡ β   k
n ′ ω 2 o = k 1 o × n 2   = − sin ⁡ ω sin ⁡ r 1 o sin ⁡ β   i   + cos ⁡ r 1 o sin ⁡ β + cos ⁡ ω sin ⁡ r 1 o cos ⁡ β   j   − sin ⁡ ω sin ⁡ r 1 o cos ⁡ β   k
k ′ 2 e = ( cos ⁡ r 2 e ′ , cos ⁡ ω 2 e ′ sin ⁡ r 2 e ′ , sin ⁡ ω 2 e ′ sin ⁡ r 2 e ′ )
k ′ 2 o = ( cos ⁡ r 2 o ′ , cos ⁡ ω 2 o ′ sin ⁡ r 2 o ′ , sin ⁡ ω 2 o ′ sin ⁡ r 2 o ′ )
cos ⁡ ω 2 e ′ = n ′ ω 2 e ⋅ w 2 | n ′ ω 2 e | | w 2 |
n ( r 1 e ) sin ⁡ i 2 e ′  =  n o sin ⁡ r 2 e ′
cos ⁡ ω 2 o ′ = n ′ ω 2 o ⋅ w 2 | n ′ ω 2 o | | w 2 |
n o sin ⁡ i 2 o ′  =  n ( r 2 o ′ ) sin ⁡ r 2 o ′
n ( r 2 o ′ ) = n o n e n e 2 cos 2 θ o + n o 2 sin 2 θ o
cos ⁡ θ o  =  k 1 o ⋅ w 1 = cos ⁡ ω 1 sin ⁡ r 1 o
cos ⁡ ω 2 e sin ⁡ r 2 e = cos ⁡ ω 2 e ′ sin ⁡ r 2 e ′
cos ⁡ ω 2 o sin ⁡ r 2 o = cos ⁡ ω 2 o ′ sin ⁡ r 2 o ′
n ( r 2 o )  =  n ( r 2 o ′ )
k e 2 = ( cos ⁡ r 2 e , cos ⁡ ω 2 e sin ⁡ r 2 e , sin ⁡ ω 2 e sin ⁡ r 2 e )
k o 2 = ( cos ⁡ r 2 o , cos ⁡ ω 2 o sin ⁡ r 2 o , sin ⁡ ω 2 o sin ⁡ r 2 o )
n i sin ⁡ r 3 e  =  n o sin ⁡ i 3 e
n i sin ⁡ r 3 o = n ( r 2 o ) sin ⁡ i 3 o
n ′ 3 = ( cos ⁡ β , 0 , sin ⁡ β )
cos ⁡ i 3 e = k ′ 2 e ⋅ n ′ 3 | k ′ 2 e | | n ′ 3 | = cos ⁡ r 2 e ′ cos ⁡ β − cos ⁡ ω 1 sin ⁡ r 2 e ′ sin ⁡ β
cos ⁡ i 3 o = k ′ 2 o ⋅ n ′ 3 | k ′ 2 e | | n ′ 3 | = cos ⁡ r 2 o ′ cos ⁡ β − cos ⁡ ω 1 sin ⁡ r 2 o ′ sin ⁡ β
k 3 e = ( cos ⁡ r 3 e , cos ⁡ ω 2 e sin ⁡ r 3 e , sin ⁡ ω 2 e sin ⁡ r 3 e )
k 3 o = ( cos ⁡ r 3 o , cos ⁡ ω 2 o sin ⁡ r 3 o , sin ⁡ ω 2 o sin ⁡ r 3 o )
{ x 2 e = t sin ⁡ ω 2 e tan ⁡ r 2 e − z 3 e sin ⁡ ω 2 e tan ⁡ r 2 e − cot ⁡ β y 2 e = x 2 e sin ⁡ ω 1 tan ⁡ r 1 e + y 1 z 2 e = x 2 e cos ⁡ ω 1 tan ⁡ r 1 e + z 1 { x 2 o = t sin ⁡ ω 2 o tan ⁡ r 2 o − z 3 o sin ⁡ ω 2 o tan ⁡ r 2 o − cot ⁡ β y 2 o = x 2 o sin ⁡ ω 1 tan ⁡ r 1 o + y 1 z 2 o = x 2 o cos ⁡ ω 1 tan ⁡ r 1 o + z 1
{ x 3 e = t y 3 e = ( t − x 2 e ) cos ⁡ ω 2 e tan ⁡ r 2 e + y 2 e z 3 e = ( t − x 2 e ) sin ⁡ ω 2 e tan ⁡ r 2 e + z 2 e { x 3 o = t y 3 o = ( t − x 2 o ) cos ⁡ ω 2 o tan ⁡ r 2 o + y 2 o z 3 o = ( t − x 2 o ) sin ⁡ ω 2 o tan ⁡ r 2 o + z 2 o
{ x 4 e = t + s y 4 e = s cos ⁡ ω 2 e tan ⁡ r 3 e + y 3 e z 4 e = s sin ⁡ ω 2 e tan ⁡ r 3 e + z 3 e { x 4 o = t + s y 4 o = s cos ⁡ ω 2 o tan ⁡ r 3 o + y 3 o z 4 o = s sin ⁡ ω 2 o tan ⁡ r 3 o + z 3 o
{ x 5 e = − z 3 e − s sin ⁡ ω 2 e tan ⁡ r 2 e sin ⁡ ω 2 e tan ⁡ r 2 e − cot ⁡ β + t + s y 5 e = ( x 5 e − x 4 e ) cos ⁡ ω 2 e tan ⁡ r 2 e + y 4 e z 5 e = ( x 5 e − x 4 e ) sin ⁡ ω 2 e tan ⁡ r 2 e + z 4 e { x 5 o = − z 3 o − s sin ⁡ ω 2 o tan ⁡ r 2 o sin ⁡ ω 2 o tan ⁡ r 2 o − cot ⁡ β + t + s y 5 o = ( x 5 o − x 4 o ) cos ⁡ ω 2 o tan ⁡ r 2 o + y 4 o z 5 o = ( x 5 o − x 4 o ) sin ⁡ ω 2 o tan ⁡ r 2 o + z 4 o
{ x 6 e = 2 t + s y 6 e = ( 2 t + s − x 5 e ) sin ⁡ ω 1 tan ⁡ r 1 e + y 5 e z 6 e = ( 2 t + s − x 5 e ) cos ⁡ ω 1 tan ⁡ r 1 e + z 5 e { x 6 o = 2 t + s y 6 o = ( 2 t + s − x 5 o ) sin ⁡ ω 1 tan ⁡ r 1 o + y 5 o z 6 o = ( 2 t + s − x 5 o ) cos ⁡ ω 1 tan ⁡ r 1 o + z 5 o
Δ i n = n ( r 1 e ) O A + n ( r 1 e ) D E + n o A B + n o C D + n i B C   − ( n o O A ′ + n o D ′ E ′ + n ( r 2 o ) A ′ B ′ + n ( r 2 o ) C ′ D ′ + n i B ′ C ′ ) = n ( r 1 e ) [ ( x 2 e − x 1 ) 2 + ( y 2 e − y 1 ) 2 + ( z 2 e − z 1 ) 2 + ( x 6 e − x 5 e ) 2 + ( y 6 e − y 5 e ) 2 + ( z 6 e − z 5 e ) 2 ]   + n o [ ( x 5 e − x 4 e ) 2 + ( y 5 e − y 4 e ) 2 + ( z 5 e − z 4 e ) 2 + ( x 3 e − x 2 e ) 2 + ( y 3 e − y 2 e ) 2 + ( z 3 e − z 2 e ) 2 ]   + n i [ ( x 4 e − x 3 e ) 2 + ( y 4 e − y 3 e ) 2 + ( z 4 e − z 3 e ) 2 − ( x 4 o − x 3 o ) 2 − ( y 4 o − y 3 o ) 2 − ( z 4 o − z 3 o ) 2 ]   − n ( r 2 o ) [ ( x 5 o − x 4 o ) 2 + ( y 5 o − y 4 o ) 2 + ( z 5 o − z 4 o ) 2 + ( x 3 o − x 2 o ) 2 + ( y 3 o − y 2 o ) 2 + ( z 3 o − z 2 o ) 2 ]   − n o [ ( x 2 o − x 1 ) 2 + ( y 2 o − y 1 ) 2 + ( z 2 o − z 1 ) 2 + ( x 6 o − x 5 o ) 2 + ( y 6 o − y 5 o ) 2 + ( z 6 o − z 5 o ) 2 ]  =  n ( r 1 e ) cos ⁡ r 1 e ( t − l e ) + n o cos ⁡ r 2 e ( t + l e ) + n i cos ⁡ r 3 e s − n o cos ⁡ r 1 o ( t − l o ) − n ( r 2 o ) cos ⁡ r 2 o ( t + l o ) − n i cos ⁡ r 3 o s
Δ e x = ( y 6 o − y 6 e ) sin ⁡ ω 1 sin ⁡ i 1 + ( z 6 o − z 6 e ) cos ⁡ ω 1 sin ⁡ i 1   = sin ⁡ ω 1 sin ⁡ i 1 [ sin ⁡ ω 1 tan ⁡ r 1 o ( t − l o ) + sin ⁡ ω 2 o tan ⁡ r 2 o ( t + l e ) + s sin ⁡ ω 2 o tan ⁡ r 2 o )   − sin ⁡ ω 1 tan ⁡ r 1 e ( t − l e ) − sin ⁡ ω 2 e tan ⁡ r 2 e ( t + l e ) − s sin ⁡ ω 2 e tan ⁡ r 2 e ]   + cos ⁡ ω 1 sin ⁡ i 1 [ cos ⁡ ω 1 tan ⁡ r 1 o ( t − l o ) + cos ⁡ ω 2 o tan ⁡ r 2 o ( t + l e ) + s cos ⁡ ω 2 o tan ⁡ r 2 o − cos ⁡ ω 1 tan ⁡ r 1 e ( t − l e ) − cos ⁡ ω 2 e tan ⁡ r 2 e ( t + l e ) − s cos ⁡ ω 2 e tan ⁡ r 2 e ]
l e = t cos ⁡ ω 2 e tan ⁡ r 2 e + s cos ⁡ ω 2 e tan ⁡ r 3 e cot ⁡ β − cos ⁡ ω 2 e tan ⁡ r 2 e
l o = t cos ⁡ ω 2 o tan ⁡ r 2 o + s cos ⁡ ω 2 o tan ⁡ r 3 o cot ⁡ β − cos ⁡ ω 2 o tan ⁡ r 2 o
Δ t o t a l = Δ i n + Δ o u t
Δ ′ t o t a l . = ( 2 t + s ) ( n o cos ⁡ r o 1 − n ( r e 1 ) cos ⁡ r e 1 )   + ( t + s ) cos ⁡ θ ( n e cos ⁡ r o 2 + n ( r e 1 ) cos ⁡ i e 2 − n o cos ⁡ r e 2 − n o cos ⁡ i o 2 )   + s ( n i cos ⁡ r o 3 − n i cos ⁡ r e 3 − n e cos ⁡ i e 3 + n o cos ⁡ i o 3 )
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