Diffraction Phenomena on Extended Asymmetric Slit with Absolutely Absorbing Inner Faces

Abstract

Fraunhofer diffraction patterns (spectra) of extended (in depth) asymmetric slit holes with absolutely absorbing inner faces and with different input ($D$) and output ($D_{1}$) apertures are calculated in the far field based on the model of equivalent diaphragms. The behavior of the spectrum of an extended object is studied analytically in the case of aperture differences $2|\Delta|=|D_{1}-D|$ that are noticeably smaller than the size of the Fresnel zone $\delta_{d}=\sqrt{\lambda d}$ ($\lambda$ is the light wavelength and $d$ is the hole depth). It is shown that in the range of angles $|\theta|\ll\theta_{\text{cr}}=\sqrt{\lambda/d}$ the observed diffraction pattern of an extended object is equivalent to the diffraction of light on a flat slit ($d=0$) with an effective width $D_{\text{eff}}=D+\Delta-\theta_{d}/(\sqrt{2}\pi)$. Based on a constructive approximation of the integral Fresnel function, the features of light diffraction on volumetric holes with noticeably different apertures are studied analytically: $2|\Delta|\gg\delta_{d}$. Calculations show that in the cases of expanding ($D_{1}>D$) and narrowing ($D_{1}<D$) apertures, the behavior of the minima of the observed diffraction patterns in the ranges of angles $|\theta|<|\theta_{\Delta}|=|\Delta|/d$ differs little from the equidistant behavior for a flat slit ($d=0$) of width $D$ and $D_{1}$, respectfully. The results can be used in the development of optoelectronic systems for dimensional inspection of perforated plates.