Page 272 - Introduction to Information Optics

5,2. Fresnel Transform 2 5 7

the angle between the normal of the wavefront at the aperture and the direction
of the radiation.
The Huygens-Fresnel formula was found to be the solution of the Helm-
hotz equation with the Rayleigh-Sommerfeld's Green function [It]. The
Helmhotz equation results from the Maxwell equations for a monochromatic
component of the light field.
In Eq. ( 5.1) one considers only one scalar component E(r) of the complex
amplitude of the vector electric field. When the aperture dimension is much
larger than the wavelength and the observation is far from the aperture, one
can consider the components E x, E y, and E z of the vector electric field as
independent; they can then be computed independently by the scalar Helmhotz
equation. The Huy gens-Fresnel formula is still the foundation of optical
diffraction theory.

5.2, FRESNEL TRANSFORM

5.2.1. DEFINITION

Mathematically the Fresnel transform is defined as
f
2
F(u) = /(x) exp[(i7t(M - x) \dx, (5.2)
v
where the notation J() represents the integrals with the limits from — oo to
+ oo. The inverse Fresnel transform is given by

f
2
f(x) = F(u] exp[-m(M - x) \du. (5.3)
v

5.2.2. OPTICAL FRESNEL TRANSFORM

In relation to optics, the Fresnel transform describes paraxial light propa-
gation and diffraction under the Fresnel approximation. We now describe the
optical field in an input and an output plane, which are normal to the optical
axis. Let x denote the position in the aperture plane, which is considered as the
input plane, and x' denote the position in the output plane, where the diffracted
pattern is observed. Let z denote the distance from the aperture to the output
plane. (Although both the input and output planes of an optical system are 2D,
we use 1D notations throughout this chapter for the sake of simplicity of the
formula. Generalization of the results to 2D is straightforward.)