Yeganeh Mohammad, Rasouli Saifollah
Appl Opt. 2018 Nov 20;57(33):9777-9788. doi: 10.1364/AO.57.009777.
This work presents a very simple and comprehensive approach for classification of the combinational spatial frequencies of the superimposed periodic or quasi-periodic structures. The reciprocal vectors of the structures are used to express their respective spectral components, and a unique reciprocal vectors equation is introduced for presenting the corresponding combinational frequencies. By the aid of the reciprocal vectors equation we classify moiré patterns of combinational frequencies into four classes: the conventional moiré pattern, moiré fringes of higher-order harmonics, higher-order moiré patterns, and pseudo-moiré patterns. The difference between the moiré fringes of higher-order harmonics and higher-order moiré patterns is expressed in the formulas. By some typical examples, conditions for simultaneous formation of moiré patterns of different harmonics of the superimposed gratings are investigated. We show that in the superimposition of two gratings, where at least one has a varying period and another has a non-sinusoidal profile, different moiré patterns are formed over different parts of the superimposed area, where a distinct pair of spatial frequencies of the superimposed structures contributes to the formation of each of the patterns. We use the same procedure in the analysis of simultaneously produced defected moiré patterns in the superimposition of a linear grating and a zone plate, where one or both consist of some topological defects at specific locations and at least one of the gratings has a non-sinusoidal profile. The topological defects of resulting moiré fringes are similar to those appearing in the interference patterns of optical vortices. It is shown that the defect number of resulting moiré fringes depends on the defect numbers and order of frequency harmonics of the gratings. The dependency of the defect number of the moiré fringes and its sign to the defect numbers of the gratings and their contributed frequency harmonics is derived for both additive and subtractive terms of moiré fringes, and the results are verified with several examples based on computational simulations.
这项工作提出了一种非常简单且全面的方法，用于对叠加的周期性或准周期性结构的组合空间频率进行分类。利用结构的倒易矢量来表示其各自的光谱分量，并引入一个独特的倒易矢量方程来表示相应的组合频率。借助倒易矢量方程，我们将组合频率的莫尔条纹分为四类：传统莫尔条纹、高阶谐波莫尔条纹、高阶莫尔条纹和赝莫尔条纹。文中公式给出了高阶谐波莫尔条纹与高阶莫尔条纹之间的区别。通过一些典型例子，研究了叠加光栅不同谐波莫尔条纹同时形成的条件。我们表明，在两个光栅的叠加中，其中至少一个具有变化的周期且另一个具有非正弦轮廓，在叠加区域的不同部分会形成不同的莫尔条纹，其中叠加结构的一对独特空间频率促成了每种条纹的形成。我们在分析线性光栅和波带片叠加时同时产生的缺陷莫尔条纹时采用相同的方法，其中一个或两者在特定位置由一些拓扑缺陷组成，并且至少有一个光栅具有非正弦轮廓。所得莫尔条纹的拓扑缺陷与光学涡旋干涉图案中出现的缺陷相似。结果表明，所得莫尔条纹的缺陷数取决于光栅的缺陷数和频率谐波阶数。针对莫尔条纹的相加和相减项，推导了莫尔条纹缺陷数及其符号与光栅缺陷数及其贡献的频率谐波之间的依赖关系，并通过基于计算模拟的几个例子对结果进行了验证。
