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Influence of the optical axis orientation of the Glan-Taylor prism and the incident angle on the extraordinary ray

According to the propagation characteristic of the plane wave in the Glan-Taylor prism, using the method of the ray tracing, the deflection angle of the e-ray wave vector and its position on the exiting plane as a function of different parameters are calculated for the first time. In order to analyze the influences of the incident angle i together with the angle α of the optical axis of the Glan-Taylor prism and the crystal interface on the position of the extraordinary ray on the exiting plane L and L′, the variations of L and L′ with i and α are simulated, and the effects of α on L and L′ are very small except for i = 90°. To ensure the incident angle having no effects on the position of the e beam on the exiting plane, the incident angle should be controlled in a range of [0°,45°].


Introduction
Since 1808, the polarization light technology has become a specialized and systematic method for the optical information processing [1][2], and the performance and imaging mechanism of the polarizer have also become the research hotspot [3][4].
In 2000, the tempo-spatially modulated polarization interference imaging spectroscopy is proposed by Zhang Chunmin et al. [5], and the static polarization interference imaging spectrometer [6] and static large field of view polarization interference imaging spectrometer [7], based on a Savart polariscope, are developed. These spectrometers use the field diaphragm instead of the slit in the traditional interference imaging spectrometer, thus it has the advantages of high steady state, high flux together with large field of view and high resolution [8][9].
The Glan-Taylor prism is one of the important polarization devices in the polarization interference imaging spectrometer, which is made up of the iceland crystal, and it has the perfect optical performance, wide spectral transmission range, and large birefringence index [10]. Besides, the Glan-Taylor prism adopts the air gap as gluing, which makes its extinction ratio better than 10 -5 , and the spectral range varies from 300 nm to 500 nm [11]. These significant advantages make the Glan-Taylor prism widely used in the national defense, scientific research and teaching equipment.
In this paper, using the method of the ray tracing, the deflection angle of the e -ray wave vector and its position on the exiting plane as a function of different parameters are calculated for the first time.  Ideally, the optical axes of the left-right shear plates for the Glan-Taylor prism are parallel to the incident surface. When the incident ray from the objective source reaches the left shear plate, due to the double refraction effect of the uniaxial crystal, it is divided into two linearly polarized light beams whose intensities are equal and the vibration direction is perpendicular to each other, namely, the ordinary ray o and the extraordinary ray e . After passing through the air gap, the o -ray exits from the bottom of the left shear plate. So, the propagation of the o -ray in the Glan-Taylor prism is neglected. The e -ray enters into the right shear plate, and then split up into two beams, namely, eeray and eo -ray. Because the transmittance of the ee -ray is very small, only the propagation of the eoray is considered.

Ray tracing of the extraordinary light
Because the optical axis is not ensured as the ideal state in the processing of the Glan-Taylor prism, there is a certain machining tolerance existing. In this part, the influences of the tilting error of the optical axis of the Glan-Taylor prism on the deflection angle of the e -ray wave vector and its position on the exiting plane are analyzed and discussed.
On the basis of the Fresnel law and the geometrical relationship in figure 2(a), the following conditions are obtained: and sin arcsin , where  is the angle of the optical axis for the Glan-Taylor prism and its incident interface.
Letting the thickness of the Glan-Taylor prism as d , the exiting position of the e -ray is expressed as follows: When the incident angle is 0 i  , the expression of the exiting position L can be simplified as:   (7) Secondly, assuming that the incident angle is below the normal ray direction, as depicted in figure  2(b), the deflection angle of the e -ray wave vector together with its position on the exiting plane as a function of different parameters are obtained.

Computer simulation
In order to describe the influences of various parameters on the deflection angle of the e -ray wave vector together with its position on the exiting plane, the simulation of a selected design example is carried out in detail. Here, the main refractive indices of the calcite crystal are provided with   On the basis of figure 5, the position of the beam on the exiting plane L and L increase slowly with the increase of i when  remains a constant. In the case of the incident ray above the normal to the incident plane, the influence of i on L is larger for 0   , while the influence of i on L is larger for 30   when the incident ray is below the normal to the incident plane. Besides, the values of L and L have little change when   0 ,45 i    , that is to say, the incident angle has almost no effects on the position of the beam on the exiting plane if it is controlled in a range of   0 ,45  .

Conclusions
According to the propagation characteristic of the plane wave in the Glan-Taylor prism, the exact expressions of the deflection angle of the e -ray wave vector together with its position on the exiting plane as a function of different parameters are obtained when the incident ray is either above or below the normal ray direction. In order to describe the influences of the incident angle i and the angle of the optical axis and the crystal interface  on the positions of the e beam on the exiting plane L and L , the variations of L and L with i and  are simulated, and some important conclusions are obtained. The influences of  on L and L are very small in addition to 90 i  , and the incident angle has almost no effects on the positions of the e beam on the exiting plane if it is controlled in a range of   0 ,45  . Therefore, the incident angle and the optical axis orientation of the Glan-Taylor prism have an important impact on the e -ray tracing.