## Abstract

Inspections of moiré fringe characteristics, such as period and orientation, conventionally are done by two approaches; namely,
parametric equation and Fourier analysis methods. In some cases these methods yield different results. This inconsistency is removed by revising the derivation of the indicial equation for moiré fringes by the parametric equation method.

© 2007 Optical Society of America

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### Equations (56)

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(3)
$${d}_{1}\approx M{d}_{2};M\in Z\mathrm{.}$$
(4)
$${T}_{1}(x,y)=\sum _{m=-\infty}^{\infty}{a}_{m}\text{\hspace{0.17em}}\mathrm{exp}\left(im\text{\hspace{0.17em}}\frac{2\pi}{{d}_{1}}\text{\hspace{0.17em}}x\right),$$
(5)
$${T}_{2}(x,y)=\sum _{n=-\infty}^{\infty}{b}_{n}\text{\hspace{0.17em}}\mathrm{exp}\left[in\text{\hspace{0.17em}}\frac{2\pi}{{d}_{2}}\left(x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta -y\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta \right)\right],$$
(10)
$${T}_{M}={T}_{1}(x,y){T}_{2}(x,y)$$
(11)
$$=\sum _{m=-\infty}^{\infty}\text{\hspace{0.17em}}\sum _{n=-\infty}^{\infty}{a}_{m}{b}_{n}\text{\hspace{0.17em}}\mathrm{exp}\left\{i2\pi \left[\left(\frac{m}{{d}_{1}}+\frac{n\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta}{{d}_{2}}\right)x-\frac{n\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta}{{d}_{2}}\text{\hspace{0.17em}}y\right]\right\}\text{.}$$
(12)
\mathrm{exp}[i2\pi ({f}_{X}x+{f}_{Y}y)]
(14)
$$f={\left({f}_{X}^{2}+{f}_{Y}^{2}\right)}^{1/2}\mathrm{.}$$
(16)
$${f}_{mn}={\left(\frac{{m}^{2}}{{d}_{1}^{2}}+\frac{{n}^{2}}{{d}_{2}^{2}}+\frac{2mn\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta}{{d}_{1}{d}_{2}}\right)}^{1/2}\mathrm{.}$$
(22)
{f}_{{m}_{0}{n}_{0}}
(23)
$${d}_{M}=\frac{1}{{f}_{{m}_{0}{n}_{0}}}=\frac{{d}_{1}{d}_{2}}{\sqrt{{n}_{0}^{2}{d}_{1}^{2}+{m}_{0}^{2}{d}_{2}^{2}+2{m}_{0}{n}_{0}{d}_{1}{d}_{2}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta}}\mathrm{.}$$
(26)
$${d}_{M}=\frac{{d}_{1}{d}_{2}}{\sqrt{{d}_{1}^{2}+{M}^{2}{d}_{2}^{2}-2M{d}_{1}{d}_{2}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta}}\mathrm{.}$$
(27)
$$\frac{x}{{d}_{1}}=h\text{;}h\in Z,$$
(28)
$$\frac{x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta -y\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta}{{d}_{2}}=k\text{;}k\in Z\mathrm{.}$$
(31)
$$h-k=p\text{;}p\in Z\mathrm{.}$$
(32)
$$\frac{{d}_{1}{d}_{2}}{\sqrt{{d}_{1}^{2}+{d}_{2}^{2}-2{d}_{1}{d}_{2}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta}}\text{,}$$
(35)
$$M\prime h+N\prime k=p\text{;}p\in Z,$$
(39)
(-M\prime ,-N\prime )
(42)
$$N\prime \text{\hspace{0.17em}}\frac{x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta -y\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta}{{d}_{2}}+M\prime \text{\hspace{0.17em}}\frac{x}{{d}_{1}}=p\text{;}p\in Z\mathrm{.}$$
(43)
$$\frac{{d}_{1}{d}_{2}}{\sqrt{{N\prime}^{2}{d}_{1}^{2}+{M\prime}^{2}{d}_{2}^{2}+2M\prime N\prime {d}_{1}{d}_{2}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta}}\mathrm{.}$$