Of the many topics generally taught in undergraduate or graduate optics courses, the propagation of light through crystals with electrical anisotropy is one of the most difficult topics for students and for teachers who do not work in this field. In particular, the mathematics and equations are complex vectorial equations, and this complicates the visualization of the implications of the theory. Also, the standard textbooks and papers on the subject have led to some misconceptions on this problem by presenting results only for specific cases of the theory in graphical form. For example, it is a general belief that after refraction of an incident wave on the crystal, the polarization state of the two waves propagating inside the crystal, given by their displacement vectors D, are always orthogonal to each other. However, this is not the case, as explained and illustrated for uniaxial crystals by Alemán-Castañeda and Rosete-Aguilar [J. Opt. Soc. Am. A 33(4), 677–682 (2016)]. The purpose of this paper is to help students and teachers visualize the solution to Maxwell's equations in electrical anisotropic crystals. An algorithm has been written in mathematica so the reader can use it to visualize the normal surface in uniaxial and biaxial crystals. This algorithm can be used as the basis of computational projects for students to better understand the equations involved, or as a visualization tool for teachers in class.

中文翻译：

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可视化各向异性晶体中3D法向表面以及单轴晶体中o和e波的偏振态的算法

在本科或研究生的光学课程中通常教授的许多主题中，光通过具有各向异性的晶体的传播对于学生和不在这一领域工作的老师来说是最困难的主题之一。特别地，数学和方程是复杂的矢量方程，这使该理论的含义的可视化变得复杂。此外，有关该主题的标准教科书和论文仅通过以图形形式展示该理论的特定案例的结果，就引起了对该问题的一些误解。例如，一般认为，在入射波在晶体上折射之后，由其位移矢量D给出的在晶体内传播的两个波的偏振态总是彼此正交的。然而，这种情况并非如此，如Alemán-Castañeda和Rosete-Aguilar对单轴晶体的解释和说明。选择。Soc。是。一种33（4），677–682（2016）]。本文的目的是帮助学生和教师形象化电各向异性晶体中麦克斯韦方程的解。已经在mathematica中编写了一种算法，因此读者可以使用它来可视化单轴和双轴晶体中的法线表面。该算法可用作学生更好地理解所涉及方程式的计算项目的基础，或用作课堂教师的可视化工具。