Izdebski Marek
Institute of Physics, Technical University of Łódź, Poland.
Appl Opt. 2006 Nov 10;45(32):8262-72. doi: 10.1364/ao.45.008262.
An analytical approach is presented for studying the convergence of the general Jacobi method applied to diagonalizing the second-rank tensors that describe the optical properties of a medium subjected to an external field. This approach utilizes the fact that the components of such tensors are usually given in field-free principal axes as power series in the field strength, neglecting terms beyond a chosen power of the field. It is shown that for a biaxial or uniaxial medium, the finite number of iterations, which guarantees exact reduction of all the initial terms up to the required power in the series expansions of all off-diagonal elements, can always be found. Moreover, a fixed sequence of rotations in the Jacobi algorithm can be predicted. These findings allow one to derive analytical formulas in noniterative form for a given highest order of the effects being considered and also to optimize numerical iterative diagonalization procedures. Formulas for eigenvalues and eigenvectors applicable to biaxial and uniaxial mediums perturbed by the linear and quadratic effects are presented. Illustrations are given of the electro-optic and piezo-optic effects for the point group 3m. Conditions for biaxial and uniaxial perturbation of a uniaxial crystal are discussed.
本文提出了一种分析方法，用于研究将一般雅可比方法应用于对角化描述受外场作用介质光学性质的二阶张量时的收敛性。该方法利用了这样一个事实：此类张量的分量通常在无场主轴系中表示为场强的幂级数，忽略高于所选场幂次的项。结果表明，对于双轴或单轴介质，总能找到有限次数的迭代，以确保在所有非对角元素的级数展开中，将所有初始项精确约化到所需的幂次。此外，还可以预测雅可比算法中固定的旋转序列。这些发现使得对于给定的所考虑效应的最高阶数，可以以非迭代形式导出解析公式，并且还能优化数值迭代对角化过程。给出了适用于受线性和二次效应扰动的双轴和单轴介质的特征值和特征向量公式。针对点群3m给出了电光和压光效应的示例。讨论了单轴晶体双轴和单轴扰动的条件。
