Classical Electrodynamics · Problem 10.11

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · diffraction-half-plane-fresnel

Problem

A perfectly conducting flat screen occupies half of the

x

y

plane (i.e.,

x < 0

). A plane wave of intensity

I_0

and wave number

k

is incident along the

z

axis from the region

z < 0

. Discuss the values of the diffracted fields in the plane parallel to the

x

y

plane defined by

z = Z > 0

. Let the coordinates of the observation point be

(X, 0, Z)

. (a) Show that, for the usual scalar Kirchhoff approximation and in the limit

Z \gg X

and

\sqrt{kZ} \gg 1

, the diffracted field is

\psi(X, 0, Z, t) \approx I_0^{1/2} e^{ikZ - i\omega t}\left(\frac{1+i}{2i}\right)\sqrt{\frac{2}{\pi}}\int_{-\xi}^{\infty} e^{it^2}\,dt

where

\xi = (k/2Z)^{1/2}X

. (b) Show that the intensity can be written

I = |\psi|^2 = \frac{I_0}{2}\left[(C(\xi) + \tfrac{1}{2})^2 + (S(\xi) + \tfrac{1}{2})^2\right]

where

C(\xi)

and

S(\xi)

are the so-called Fresnel integrals. Determine the asymptotic behavior of

I

for

\xi

large and positive (illuminated region) and

\xi

large and negative (shadow region). What is the value of

I

X = 0

? Make a sketch of

I

as a function of

X

for fixed

Z

. (c) Use the vector formula (10.101) to obtain a result equivalent to that of part a. Compare the two expressions.