Classical Electrodynamics · Problem 10.18

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · diffraction-small-circular-hole-bethe

Problem

Discuss the diffraction due to a small, circular hole of radius

a

in a flat, perfectly conducting sheet, assuming that

ka \ll 1

. (a) If the fields near the screen on the incident side are normal

\mathbf{E}_0 e^{-i\omega t}

and tangential

\mathbf{B}_0 e^{-i\omega t}

, show that the diffracted electric field in the Fraunhofer zone is

\mathbf{E} = \frac{e^{ikr-i\omega t}}{3\pi r}\,k^2 a^3\left[2c\frac{\mathbf{k}}{k}\times\mathbf{B}_0 + \frac{\mathbf{k}}{k}\times\left(\mathbf{E}_0\times\frac{\mathbf{k}}{k}\right)\right]

where

\mathbf{k}

is the wave vector in the direction of observation. (b) Determine the angular distribution of the diffracted radiation and show that the total power transmitted through the hole is

P = \frac{2}{27\pi Z_0}\,k^4 a^6(4c^2 B_0^2 + E_0^2)

Related theory

Diffraction By A Circular Aperture

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