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偏光干涉系统可以对入射光的偏振信息进行检测[1-2]，也可以和光谱技术、成像技术相结合，对目标物体进行多维度探测，提高系统的目标分辨能力。目前三者已经可以实现有机结合，形成偏振光谱成像系统[3-5]，广泛应用于卫星遥感、植被监测、农业灾害预防以及军事等领域[6-9]。偏光干涉系统在偏振光谱成像系统中相对独立，主要包括萨瓦板(Savart)偏光棱镜和格兰棱镜，其中格兰棱镜可以采用格兰-泰勒棱镜、格兰-汤姆逊棱镜等具有起偏和检偏作用的偏光棱镜[10-12]。
Savart偏光棱镜是偏光干涉系统的核心组件，要获得目标物体更多的偏振信息，就需要尽可能地降低Savart偏光棱镜的加工误差[13-14]，然而，由于Savart偏光棱镜由多块偏光晶体部件胶合制作而成，胶合过程中的各组成部分会产生装调误差，每一个组成部分的加工误差都会叠加到Savart偏光棱镜中，造成Savart偏光棱镜总的加工误差很难做得很低。而为了满足偏光干涉系统对Savart偏光棱镜高精度的要求，不仅需要对各个组成部件进行超高精度的加工，还要对各组成部分的相对位置进行精密调整、胶合，装调难度大，成品率低，对于特别高精度的Savart偏光棱镜需要进行筛选，这进一步降低了成品率。
为了满足偏光干涉系统对目标偏振信息的检测需求，同时降低核心器件的加工难度，提出了一种基于单平行分束器(single parallel beam splitter，SPBS)的偏光干涉系统，以SPBS替代Savart偏光棱镜。由于SPBS可以实现将入射光分为具有一定横向剪切量的两束传播方向平行的偏振光，两束光的光矢量振动方向相互垂直，和Savart偏光棱镜具有相同的作用效果，所以SPBS可以实现基于Savart偏光棱镜的偏光干涉效果。作者从理论上对基于SPBS的偏光棱镜系统的结构原理、作用效果进行了分析，得出了与基于Savart偏光棱镜偏光干涉相似的结果，并从实验上获取了干涉条纹，证明了理论分析的正确性。SPBS为单元结构的偏光棱镜，不需要多个部分组合，有效避免了多个部分组合时带来的装调误差，另外, 也可以避免多个组成部分引起的面形等加工误差的叠加，有效降低实际制作的偏光棱镜和理论设计之间的偏离，提高偏光干涉系统对目标判别的准确性。
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基于SPBS的偏光干涉系统结构如图 1所示。包括前置准直系统、偏振棱镜P1、SPBS、偏振棱镜P2、成像物镜L3以及探测器CCD。前置准直系统由透镜L1、光阑M和透镜L2组成。P1和P2的透振方向在x-O-y面内，与x轴正方向的夹角分别是θ1和θ2，SPBS光轴方向在x-O-z面内，光经过前置准直光学系统进入P1，透射光变为光矢量振动方向与x轴成角θ1的线偏振光，经过SPBS后，出射光分为两束光矢量振动方向互相垂直的两束线偏振光，两束光传输方向平行，并且具有一定的横向剪切差，两束光光矢量的振动方向分别沿x轴和y轴，通过检偏器P2(光矢量振动方向为与x轴成角θ2)使两束线偏振光的振动方向相同，最后两束光经过会聚透镜L3后，在其焦平面处形成干涉条纹[15]。
Figure 1. Polarization interference system structural diagram based on a single parallel polarizing beam splitter
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考虑起偏器P1、检偏器P2的透振方向与x轴夹角分别是θ1和θ2，则二者对应的琼斯矩阵分别是JP1和JP2，SPBS可以等效为两个正交线偏振器的组合，两个线偏振器的琼斯矩阵分别为JSPBS(0°)和JSPBS(90°)。若入射光为完全非偏振光，光强为2I0，通过起偏器P1后，成为光强为I0的线偏振光，光矢量振动方向是θ1，此时电场的Jones矢量可以表示为：
$ [\mathit{\boldsymbol{E}}(t)] = \left[ {\begin{array}{*{20}{l}} {{E_x}}\\ {{E_y}} \end{array}} \right] = \left[ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{E}}(t){\rm{cos}}{\theta _1}}\\ {\mathit{\boldsymbol{E}}(t){\rm{sin}}{\theta _1}} \end{array}} \right] $
(1) $ \left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{E}}_1} = {\mathit{\boldsymbol{J}}_{{{\rm{P}}_{\rm{2}}}}}{\mathit{\boldsymbol{J}}_{{\rm{SPBS(}}{{\rm{0}}^{\rm{^\circ }}}{\rm{)}}}}{\rm{exp}}({\rm{i}}{\phi _x})\mathit{\boldsymbol{E}}}\\ {{\mathit{\boldsymbol{E}}_2} = {\mathit{\boldsymbol{J}}_{{{\rm{P}}_{\rm{2}}}}}{\mathit{\boldsymbol{J}}_{{\rm{SPBS(9}}{{\rm{0}}^{\rm{^\circ }}}{\rm{)}}}}{\rm{exp}}({\rm{i}}{\phi _y})\mathit{\boldsymbol{E}}} \end{array}} \right. $
(2) 式中, E1, E2分别为e光和o光经过检偏器P2的复振幅; ϕx, ϕy分别为e光和o光经SPBS后所产生的相位延迟; E为经过起偏器P1的光矢量E(t)的复振幅。将各偏振元器件的Jones矩阵代入上式[16]，可得系统出射两束光的复振幅分别为：
$ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{E}}_1} = {\mathit{\boldsymbol{J}}_{{{\rm{P}}_{\rm{2}}}({\theta _2})}}{\mathit{\boldsymbol{J}}_{{\rm{SPBS(}}{{\rm{0}}^{\rm{^\circ }}}{\rm{)}}}}{\rm{exp}}({\rm{i}}{\phi _x})\mathit{\boldsymbol{E}} = }\\ {\left[ {\begin{array}{*{20}{c}} {{\rm{co}}{{\rm{s}}^2}{\theta _2}}&{{\rm{sin}}{\theta _2}{\rm{cos}}{\theta _2}}\\ {{\rm{sin}}{\theta _2}{\rm{cos}}{\theta _2}}&{{\rm{si}}{{\rm{n}}^2}{\theta _2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\exp (i{\phi _x})}&0\\ 0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{E_x}}\\ {{E_y}} \end{array}} \right] = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{cos}}{\theta _1}{\rm{cos}}{\theta _2}[{E_x}\cos {\theta _1} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {E_y}{\rm{sin}}{\theta _1}]{\rm{exp}}({\rm{i}}{\phi _x})\left[ {\begin{array}{*{20}{c}} {{\rm{cos}}{\theta _2}}\\ {{\rm{sin}}{\theta _2}} \end{array}} \right]} \end{array} $
(3) $ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{E}}_1} = {\mathit{\boldsymbol{J}}_{{{\rm{P}}_{\rm{1}}}({\theta _2})}}{\mathit{\boldsymbol{J}}_{{\rm{SPBS(9}}{{\rm{0}}^{\rm{^\circ }}}{\rm{)}}}}{\rm{exp}}({\rm{i}}{\phi _y})\mathit{\boldsymbol{E}} = }\\ {\left[ {\begin{array}{*{20}{c}} {{\rm{co}}{{\rm{s}}^2}{\theta _2}}&{{\rm{sin}}{\theta _2}{\rm{cos}}{\theta _2}}\\ {{\rm{sin}}{\theta _2}{\rm{cos}}{\theta _2}}&{{\rm{si}}{{\rm{n}}^2}{\theta _2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&0\\ 0&{\exp (i{\phi _y})} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{E_x}}\\ {{E_y}} \end{array}} \right] = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{cos}}{\theta _1}{\rm{cos}}{\theta _2}[{E_x}\cos {\theta _1} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {E_y}{\rm{sin}}{\theta _1}]{\rm{exp}}({\rm{i}}{\phi _y})\left[ {\begin{array}{*{20}{c}} {{\rm{cos}}{\theta _2}}\\ {{\rm{sin}}{\theta _2}} \end{array}} \right]} \end{array} $
(4) E1, E2两束光满足相干条件，发生干涉，在CCD上两束光叠加后的强度为：
$ \begin{array}{*{20}{c}} {I = \langle {\mathit{\boldsymbol{E}}^\dagger }\mathit{\boldsymbol{E}}\rangle = \langle {{({\mathit{\boldsymbol{E}}_1} + {\mathit{\boldsymbol{E}}_2})}^*}({\mathit{\boldsymbol{E}}_1} + {\mathit{\boldsymbol{E}}_2})\rangle = }\\ {[{\rm{co}}{{\rm{s}}^2}{\theta _1}{\rm{co}}{{\rm{s}}^2}{\theta _2} + {\rm{si}}{{\rm{n}}^2}{\theta _1}{\rm{si}}{{\rm{n}}^2}{\theta _2}] \times }\\ {[{I_x}{\rm{co}}{{\rm{s}}^2}{\theta _1} + {I_y}{\rm{si}}{{\rm{n}}^2}{\theta _2}] + 2{\rm{cos}}{\theta _1}{\rm{cos}}{\theta _2} \times }\\ {\quad {\rm{sin}}{\theta _1}{\rm{sin}}{\theta _2}[{I_x}{\rm{co}}{{\rm{s}}^2}{\theta _1} + {I_y}{\rm{co}}{{\rm{s}}^2}{\theta _2}]{\rm{cos}}\phi } \end{array} $
(5) 式中，Ix=〈Ex*Ex〉，Iy=〈Ey*Ey〉，ϕ=ϕx－ϕy是两束光的相位差，上标†表示厄米共轭，上标*表示复共轭。
对于θ1和θ2的取值，由(5)式可知，当θ1和θ2取0°或90°时，CCD上光强分布一致，干涉条纹可见度为0，观测不到干涉条纹，只有当θ1, θ2∈(－π/2, 0)∪(0, π/2)时，CCD上的光强分布不再一致，会呈现出明暗条纹分布，干涉条纹的可见度不再为0，当θ1=θ2=π/4时，明暗条纹之间的光强差别达到最大，对应的干涉条纹可见度也达到最大，在CCD上观察到最清晰的干涉条纹。
基于SPBS和基于Savart偏光棱镜的偏光干涉系统结构相比较，由于Savart偏光棱镜由前后两部分组成，理想情况下，两部分的光轴应该严格呈90°，但在实际加工过程中，不可能做到严格的90°，会存在一定的角度误差[17]，假设棱镜后半部分的光轴和前半部分的不垂直，偏离角度为α，如图 2所示。
Figure 2. Diagram of optical axes deviation from vertical direction between the Savart prism front and back parts
$ {A_{\rm{o}}} = {A_{\rm{e}}} = {A_1} $
(6) 经过第二部分后，Ao和Ae分别向光轴2和垂直于光轴2的方向投影，得到经过第二部分后沿光轴2的光矢量Ae, 2的大小为：
$ \begin{array}{*{20}{c}} {{A_{{\rm{e}}, 2}} = {A_{{\rm{o, e}}}} - {A_{{\rm{e, e}}}} = {A_{\rm{o}}}{\rm{cos}}\alpha - }\\ {{A_{\rm{e}}}{\rm{sin}}\alpha = {A_1}({\rm{cos}}\alpha - {\rm{sin}}\alpha )} \end{array} $
(7) 同样，经过第二部分后沿光轴2的光矢量Ao, 2的大小为：
$ \begin{array}{*{20}{c}} {{A_{{\rm{o, 2}}}} = {A_{{\rm{o, o}}}} + {A_{{\rm{e, o}}}} = {A_{\rm{o}}}{\rm{sin}}\alpha + }\\ {{A_{\rm{e}}}{\rm{cos}}\alpha = {A_1}({\rm{sin}}\alpha + {\rm{cos}}\alpha )} \end{array} $
(8) Ae, 2和Ao, 2在45°方向在进行投影，由于二者不相等，所以干涉后的光强的极大值Ii, max和极小值Ii, min分别表示为：
$ \begin{array}{*{20}{c}} {{I_{{\rm{i, max}}}} = A_1^2[{{({\rm{cos}}\alpha + {\rm{sin}}\alpha )}^2} + {{({\rm{cos}}\alpha - {\rm{sin}}\alpha )}^2}] + }\\ {2A_1^2({\rm{cos}}\alpha + {\rm{sin}}\alpha )({\rm{cos}}\alpha - {\rm{sin}}\alpha ) = }\\ {2A_1^2[{\rm{co}}{{\rm{s}}^2}\alpha + {\rm{si}}{{\rm{n}}^2}\alpha + {\rm{co}}{{\rm{s}}^2}\alpha - }\\ {{\rm{si}}{{\rm{n}}^2}\alpha ] = 4A_1^2{\rm{co}}{{\rm{s}}^2}\alpha } \end{array} $
(9) $ \begin{array}{*{20}{c}} {{I_{{\rm{i, min}}}} = A_1^2[{{({\rm{cos}}\alpha + {\rm{sin}}\alpha )}^2} + {{({\rm{cos}}\alpha - {\rm{sin}}\alpha )}^2}] - }\\ {2A_1^2({\rm{cos}}\alpha + {\rm{sin}}\alpha )({\rm{cos}}\alpha - {\rm{sin}}\alpha ) = }\\ {2A_1^2[{\rm{co}}{{\rm{s}}^2}\alpha + {\rm{si}}{{\rm{n}}^2}\alpha - {\rm{co}}{{\rm{s}}^2}\alpha + }\\ {{\rm{si}}{{\rm{n}}^2}\alpha ] = 4A_1^2{\rm{si}}{{\rm{n}}^2}\alpha } \end{array} $
(10) 所以干涉条纹可见度：
$ V = \frac{{{I_{{\rm{i, max}}}} - {I_{{\rm{i, min}}}}}}{{{I_{{\rm{i, max}}}} + {I_{{\rm{i, min}}}}}} = \frac{{{\rm{co}}{{\rm{s}}^2}\alpha - {\rm{si}}{{\rm{n}}^2}\alpha }}{{{\rm{co}}{{\rm{s}}^2}\alpha + {\rm{si}}{{\rm{n}}^2}\alpha }} = {\rm{cos}}(2\alpha ) $
(11) 可以看到，光轴偏离角α越大，可见度越低，并且α越大，可见度的下降速度越快。当光轴偏离角为1°时，对应的可见度会下降0.02%，而当光轴偏离角为5°时，对应的可见度就会下降0.38%。
以上仅仅分析了Savart偏光棱镜前后两部分光轴在同一平面内但不垂直的情况，二者如果不在同一平面内，情况会更加复杂，并且棱镜的两部分相胶合时，胶合面的面型偏差也会增大系统误差，胶合层的应力不均匀产生的折射率分布不均匀也会影响干涉成像效果，另外，前后两个部分加工和安装误差也会引起光路失配，导致色散现象，特别对于复色光，将产生较大的影响，以上因素在SPBS中均不存在，所以基于SPBS的偏光干涉系统的系统误差要小于基于Savart偏光棱镜的系统。
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采用负单轴冰洲石晶体制作SPBS，冰洲石材料具有较大的双折射率，并且透过率高，具有稳定的物理、化学性质，是制作高性能偏光器件的首选材料[18-20]。假设SPBS主截面在x-O-z平面内，光轴方向与z轴正方向成ψ角，如图 3所示，自然光正入射SPBS后分成o光和e光，经过SPBS后，两束光传播方向平行，光束中心拉开一定的横向距离，形成剪切差d。
Figure 3. Schematic diagram of SPBS optical path
t为SPBS的长度，i为入射角，光进入晶体后o光和e光的折射角分别是φ, φ′，满足：
$ {{\rm{sin}}i = {n_{\rm{o}}}{\rm{sin}}\varphi } $
(12) $ {{\rm{sin}}i = {n^\prime }_{\rm{e}}{\rm{sin}}{\varphi ^\prime }} $
(13) 式中，no为o光的主折射率，ne′为e光的折射率。ne′由Snell定律给出：
$ \frac{1}{{{n_{\rm{e}}}^{\prime 2}}} = \frac{{{\rm{si}}{{\rm{n}}^2}\theta }}{{{n_{\rm{e}}}^2}} + \frac{{{\rm{co}}{{\rm{s}}^2}\theta }}{{{n_{\rm{o}}}^2}} $
(14) 式中，θ为光波法线方向与晶体光轴之间的夹角。
所以由SPBS产生的e光、o光之间的光程差Δ可以表示为：
$ \begin{array}{*{20}{c}} {\Delta = OA \cdot {n_{\rm{e}}}^\prime - (OB \cdot {n_{\rm{o}}} + BC) = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{t}{{{\rm{cos}}{\varphi ^\prime }}} \cdot \frac{{{\rm{sin}}i}}{{{\rm{sin}}{\varphi ^\prime }}} - \frac{t}{{{\rm{cos}}\varphi }} \cdot \frac{{{\rm{sin}}i}}{{{\rm{sin}}\varphi }} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} t{\rm{sin}}i({\rm{tan}}{\varphi ^\prime } - {\rm{tan}}\varphi ) = t{\rm{sin}}i({\rm{cot}}{\varphi ^\prime } - {\rm{cot}}\varphi )} \end{array} $
(15) 式中，OA为e光在SPBS中经过的距离，OB为o光在SPBS中经过的距离，BC为经过SPBS后o光和e光由于出射位置不同带来的光程差。
考虑更一般的光入射情况，如图 4所示。设ABMO为包含光轴Oη的主截面，OCDM为包含e光(ON)的入射面，设入射面与主截面之间的夹角为ω，ψ为光轴与晶体入射端面的法线OM夹角，设$ \mathit{\pmb{{\hat{i}}}}, \mathit{\pmb{{\hat{j}}}}, \mathit{\pmb{{\hat{u}}}}, \mathit{\pmb{{\hat{v}}}}, \mathit{\pmb{{\hat{k}}}} $, 为单位矢量，满足$ \mathit{\pmb{{\hat{i}}}}\bot \mathit{\pmb{{\hat{k}}}} $，$ \mathit{\pmb{{\hat{j}}}}\bot \mathit{\pmb{{\hat{k}}}}$，两个面ABMO和OCDM的夹角(即$ \mathit{\pmb{{\hat{i}}}}$和$ \mathit{\pmb{{\hat{j}}}} $正向夹角)为ω，则：
Figure 4. Schematic diagram when there is an angle ω between the main section and the incident plane
$ \begin{array}{*{20}{c}} {{\rm{cos}}\theta = \mathit{\boldsymbol{\hat u}} \cdot \mathit{\boldsymbol{\hat v}} = (\mathit{\boldsymbol{\hat i}}{\rm{sin}}{\varphi ^\prime } + \mathit{\boldsymbol{\hat k}}{\rm{cos}}{\varphi ^\prime }) \cdot }\\ {(\mathit{\boldsymbol{\hat j}}{\rm{sin}}\psi + \hat k{\rm{cos}}\psi ) = {\rm{cos}}\omega {\rm{sin}}{\varphi ^\prime }{\rm{sin}}{\varphi ^\prime } + {\rm{cos}}{\varphi ^\prime }{\rm{cos}}\psi } \end{array} $
(16) 由(13)式~(16)式可得：
$ \begin{array}{*{20}{c}} {\Delta = {\Delta _{{\rm{SPBS}}}} = t[{\rm{sin}}i{\rm{cot}}{\varphi ^\prime } - {\rm{sin}}i{\rm{cot}}\varphi ] = }\\ {t\left\{ {\frac{1}{{{C_0}}} - \frac{1}{b} + \frac{{({a^2} - {b^2}){\rm{sin}}\psi {\rm{cos}}\psi {\rm{cos}}\omega }}{{C_0^2}}{\rm{sin}}i + } \right.}\\ {\frac{{{\rm{si}}{{\rm{n}}^2}i}}{2}\left[ {\left( {b - \frac{{{a^2}}}{{{C_0}}}} \right){\rm{si}}{{\rm{n}}^2}\omega + \left( {b - \frac{{{a^2}{b^2}}}{{C_0^3}}} \right){\rm{co}}{{\rm{s}}^3}\omega } \right] + }\\ {\left. { {\rm{term}}{\kern 1pt} {\kern 1pt} {\rm{in}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{si}}{{\rm{n}}^4}i, etc } \right\}} \end{array} $
(17) 式中，a2=1/ne2，b2=1/no2，C02=a2sin2ψ+b2cos2ψ，上式中含有常数项，导致了只有在单色光情况下干涉条纹方能被观察到，通常情况下，复色光作为光源产生的干涉条纹看不见。
一般入射角i比较小，略去sini的高次项，得到：
$ \Delta = t\left[ {\frac{{{a^2} - {b^2}}}{{{a^2} + {b^2}}}({\rm{cos}}\omega + {\rm{sin}}\omega ){\rm{sin}}i} \right] $
(18) 式中，sini的系数即为厚度为t的SPBS的横向剪切量d，即：
$ d = t\left[ {\frac{{{a^2} - {b^2}}}{{{a^2}{\rm{si}}{{\rm{n}}^2}\psi + {b^2}{\rm{co}}{{\rm{s}}^2}\psi }}{\rm{sin}}\psi {\rm{cos}}\psi {\rm{cos}}\omega } \right] $
(19) 图 3中e光和o光的剪切量在x-O-z平面内，考虑到最大剪切差及加工难易程度，通常情况下取ψ=45°，并且使入射面与主截面重合(ω=0°)，此时的剪切量d为：
$ d = \frac{{({a^2} - {b^2})t}}{{{a^2} + {b^2}}} $
(20) SPBS长度t=25mm，光程差ΔSPBS随入射角i和入射面与主截面夹角ω的变化如图 5所示。由图中看出，在正入射时，o光和e光的光程差不是最大的，且相位差变化并不关于0°入射对称，提示在调整光路时需注意。图 6a所示为ω=0°、入射角在±6°范围内变化时相位差的变化情况; 图 6b所示为ω=90°、入射角在±6°范围内相位差的变化情况。在SPBS通光孔径足够大的情况下，为增大相位差，可适当调整入射角的大小以满足需求。
Figure 5. Optical path difference ΔSPBS changing with i and ω
Figure 6. a—curve of optical path difference changing with incidence angle when ω=0° b—curve of optical path difference changing with incidence angle when ω=90°
采用单平行分束器，不同于以往Savart板作为分光器件，其优点归纳起来有3个方面：(1)易于加工，在方解石原石上加工成品难度下降，而Savart板由两块单板粘合而成，单板加工精度要一致，并且粘合时光轴对准精度要求高; (2)误差减小，单平行分束器只需光轴与晶体通光面方向严格45°，而Savart板需要两块单板光轴方向同时满足要求，另外粘合时胶水也会增加不确定因素，因而误差会大大增加; (3)有效防止色散影响，两个半块加工误差会导致光路不匹配，引起色散，而单块晶体不存在这一问题。
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实验光源为He-Ne激光(波长为632.8nm)，偏振器P1和P2均为格兰-泰勒棱镜，消光比优于10-5，偏振方向θ1=θ2=45°，SPBS长度t=25mm，室温o光、e光在晶体中折射率no=1.6557，ne=1.4852[21]。会聚透镜焦距f3=54mm，探测器CCD单帧分辨率为1376×1032，像元尺寸大小为6.45μm×6.46μm，有效光谱响应范围400nm~800nm。
图 7a为理论模拟图，出现明暗相见的干涉条纹，图 7b是CCD处于距检偏器54mm处的图像，模拟干涉图与实际成像效果基本一致。由于激光相干性好，出现散斑，对干涉条纹形貌稍有影响。
Figure 7. a—the theoretical interference image b—image taken by CCD
图 8是Savart棱镜的偏振干涉图[18]。因棱镜放置方向不同而干涉条纹方向不同，但成像效果与用SPBS为分光元件的差别不大。
Figure 8. Interference image of Savart polarizer
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采用SPBS构成的偏光干涉系统的工作原理，通过数值计算给出了偏光干涉图像，设计了实验，得到了基于SPBS的偏光干涉系统的干涉图像，和理论计算的进行对比，二者一致，验证了理论分析的正确性，从理论和实验上证明SPBS可以用于偏光干涉系统，替代常用的萨瓦板，并且，SPBS与Savart板相比较，避免了多个双折射晶体组成部分组合带来的误差，另外，减少了多个通光面对入射光的累积误差，有利于提高偏光干涉系统的精度。
基于单平行分束器的偏光干涉系统
Polarization interference system based on single polarization parallel beam splitter
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摘要: 为了克服常规偏光干涉系统中核心器件萨瓦板(Savart)偏光镜制作工艺复杂、装调难度高的缺点，解决由于Savart偏光镜装调、加工误差造成的偏光干涉系统条纹混叠和调制度下降的问题，采用了一种基于单平行分束器(SPBS)的偏光干涉系统的方法，分析了系统的结构原理，采用矩阵传递函数推导了经偏光干涉系统出射光的琼斯矩阵及相干叠加强度，得出了和基于Savart偏光镜的干涉系统类似的干涉结果，分析了系统光程差与入射角及入射面的变化关系，并通过实验验证了理论分析的正确性。结果表明，由于SPBS结构简单，不需要多个单元组合，所以不存在装调误差，并且大幅度降低了加工误差。Abstract: In order to overcome the detects of Savart polarizer, which is the core device of polarizing interference system, such as complex fabrication process and high difficulty in assembling and adjusting, and to solve problems of interference fringe overlying and modulation decline caused by the assembling and processing errors of Savart polarizer, a method of a polarizing interference system based on a single parallel beam splitter (SPBS) was adopted. The structure and principle of this system were analyzed. Jones matrix and coherence intensity of the light exited from the polarizing interferometer system were derived by matrix transfer function. The interference effect is similar to that of the interferometer system based on Savart polarizer. The relationships between the optical path difference of the system and the incident angle and the incident surface were also analyzed. The correctness of the theoretical analysis was verified by experiments. The results show that because SPBS is simple in structure and not required Multiple unit combinations, there is no assembly error, and the processing error will be greatly reduced.
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Key words:
- physical optics /
- polarizer /
- polarization interference /
- single parallel beam splitter
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Figure 1. Polarization interference system structural diagram based on a single parallel polarizing beam splitter
Figure 2. Diagram of optical axes deviation from vertical direction between the Savart prism front and back parts
Figure 4. Schematic diagram when there is an angle ω between the main section and the incident plane
Figure 6. a—curve of optical path difference changing with incidence angle when ω=0° b—curve of optical path difference changing with incidence angle when ω=90°
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