Page 293 - Introduction to Information Optics

278 5. Transformation with Optics
r
formulated as the fractional Fourier transform. Let p t(x) and p 2(x ) denote the
complex amplitude distributions on the two spherical surfaces of the radii, R^
and R 2 ' respectively. The complex field amplitude in the planes x and x' are
expressed with

/(x) = p t (
2
/(x') = p 2 (x')exp(mx' //K 2 ),
respectively, where /I is the wavelength. One introduces the dimensionless
variables, v = x/Sj and v' = x'/s 2, where s l and s 2 are scale factors. In order to
describe the Fresnel diffraction by the relationship between the complex field
amplitudes in the two spherical surfaces, we rewrite Eq. (5.29) as

exp(i27tz//l) . / x
p 2(V) = r^— Si p,(v)exp 2.s 1s 2vv

where
0, = 1 + z/JR,

Comparing this Fresnel integral formula with the definition of the fractional
Fourier transform, Eqs. (5.27) and (5.28), we conclude that the complex
amplitude distribution on a spherical surface of radius R 2 in the output, p 2(v'),
is the fractional Fourier transform of that on a spherical surface of radius R l
in the input, /^(v), i.e.,

, exp(z'27rz//l) exp|J7t((/> — a)/4J
P-I(V ) = —
J/.Z

provided that
0 2s 2/Az = t/jsf/'/z = cot(a7r/2) (5.30)

and
Sls 2/lz = 1 /sin(«7t/2). (5.31)

Given the reference spherical surfaces R ls R 2, and z we can compute g v and
0 2 and the scales s t and s 2 and the fractional order a, according to the
preceding relations.
A special case concerns the Fresnel diffraction of an aperture with a unit
amplitude illumination and the aperture complex amplitude transmittance f(x).