## Abstract

Defects in rectangular *x-y* axis decomposable periodic patterns are found to be detected by an omnidirectional (*r-θ* axis decomposable) spatial filter optical system, where use is made of the spectral difference between periodic patterns and defects. A novel omnidirectional spatial filter, which has bandpass characteristics, is designed to block all the repetitive periodic regular pattern spectra and pass the defect information carrying spectra. According to a computer simulation, the minimum detectable defect size using this optical system is about one-twentieth that of the periodic pattern pitch. This novel optical system is characterized by its ability to detect defects at any location and rotation (shift and rotation immunity of the optical system) and also to be insensitive to object magnification by the frequency domain operation characteristics. This filter application is not limited to 2-D rectangular periodic-pattern-defects detection, but defects in 1-D or skew periodic patterns are also detected by the same filter.

© 1980 Optical Society of America

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### Equations (7)

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(1)
$$(m\mathrm{\lambda}f/{p}_{x})\phantom{\rule{0.1em}{0ex}}\times \phantom{\rule{0.1em}{0ex}}(n\mathrm{\lambda}f/{p}_{y})\phantom{\rule{0.1em}{0ex}},$$
(2)
$${d}_{m,n}=\mathrm{\lambda}f\phantom{\rule{0.2em}{0ex}}{[{\left(\frac{\mathit{\text{m}}}{{p}_{x}}\right)}^{2}+{\left(\frac{\mathit{\text{n}}}{{p}_{y}}\right)}^{2}]}^{1/2}\phantom{\rule{0.2em}{0ex}}.$$
(3)
$${d}_{m,n}\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}\frac{\mathrm{\lambda}f}{p}\phantom{\rule{0.2em}{0ex}}{({m}^{2}+{n}^{2})}^{1/2}\phantom{\rule{0.2em}{0ex}}.$$
(4)
$$\text{\u2211}_{i\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}1}^{2n\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}1}\phantom{\rule{0.2em}{0ex}}\frac{{r}_{i\phantom{\rule{0.1em}{0ex}}+\phantom{\rule{0.1em}{0ex}}1}\phantom{\rule{0.1em}{0ex}}{J}_{1}\phantom{\rule{0.1em}{0ex}}(2\phantom{\rule{0.1em}{0ex}}\pi {r}_{i\phantom{\rule{0.1em}{0ex}}+\phantom{\rule{0.1em}{0ex}}1}\phantom{\rule{0.1em}{0ex}}{\xi}_{\rho}\phantom{\rule{0.1em}{0ex}})\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{r}_{i}{J}_{1}\phantom{\rule{0.1em}{0ex}}(2{r}_{i}\phantom{\rule{0.1em}{0ex}}{\xi}_{\rho}\phantom{\rule{0.1em}{0ex}})}{{\xi}_{\rho}}\phantom{\rule{0.2em}{0ex}},$$
(5)
$$\frac{{r}_{2}{J}_{1}\phantom{\rule{0.1em}{0ex}}(2\phantom{\rule{0.1em}{0ex}}\pi {r}_{2}{\xi}_{\rho}\phantom{\rule{0.1em}{0ex}})}{{\xi}_{\rho}}\phantom{\rule{0.3em}{0ex}}.$$
(6)
$$\text{\u2211}_{i\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}3}^{2n\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}1}2\phantom{\rule{0.1em}{0ex}}\pi \phantom{\rule{0.1em}{0ex}}({r}_{i\phantom{\rule{0.1em}{0ex}}+\phantom{\rule{0.1em}{0ex}}1}\phantom{\rule{0.1em}{0ex}}-\phantom{\rule{0.1em}{0ex}}{r}_{i}\phantom{\rule{0.1em}{0ex}})\phantom{\rule{0.1em}{0ex}}{r}_{i}{J}_{0}(2\phantom{\rule{0.1em}{0ex}}\pi {r}_{1}{\xi}_{\rho}\phantom{\rule{0.1em}{0ex}})\phantom{\rule{0.2em}{0ex}},$$
(7)
$${r}_{2}{J}_{1}(2\phantom{\rule{0.1em}{0ex}}\pi {r}_{2}{\xi}_{\rho}\phantom{\rule{0.1em}{0ex}})/{\xi}_{\rho}\phantom{\rule{0.1em}{0ex}}+\phantom{\rule{0.1em}{0ex}}\text{\u2211}_{i\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}3}^{2n\phantom{\rule{0.1em}{0ex}}+\phantom{\rule{0.1em}{0ex}}1}\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.1em}{0ex}}\pi \phantom{\rule{0.1em}{0ex}}({r}_{i\phantom{\rule{0.1em}{0ex}}+\phantom{\rule{0.1em}{0ex}}1}\phantom{\rule{0.1em}{0ex}}-\phantom{\rule{0.1em}{0ex}}{r}_{i}\phantom{\rule{0.1em}{0ex}})\phantom{\rule{0.1em}{0ex}}{r}_{i}{J}_{0}(2\phantom{\rule{0.1em}{0ex}}\pi {r}_{i}{\xi}_{\rho}\phantom{\rule{0.1em}{0ex}})\phantom{\rule{0.2em}{0ex}},$$