Page 292 - Introduction to Information Optics

5.7. Fractional Fourier Transform

ijs n(x) obey the eigenvalue equations of the Fourier transform. Using the
expression for A n, we have

BJ(x, u)f(x) </x,

with the octh order fractional Fourier transform kernel functions

OC
B a(x, w) =

After a number of algebraic calculations, one obtains

exp[ — in((j> — a)/4J
B.(x, M) s
x/sin(a7r/2)
2
2
x exp[m(x cot(a7r/2) - 2xM/sin(arc/2)) + u cot(a7i/2)], (5.28)
where the fractional order 0 < ja| < 2 and 0 = sgn(sin(a7r/2)). When a = 0 and
a = ±2, the kernel function is defined separately from Eq. (5.28) as

B 0(x, u) = d(x •— u) and J3 ±2(x, u) = d(x + u),
respectively.

5.7.2. FRACTIONAL FOURIER TRANSFORM AND
FRESNEL DIFFRACTION

Let us consider now the Fresnel integral, Eq. (5.5), which describes propa-
gation of the complex amplitude /(x) under the Fresnel paraxial approxi-
mation

ikz f*
2 2
f(x') = £_ f( x) exp[-m(x - 27ixx' + x' )//z] dx, (5.29)
'
where /(x) and /(x') are the complex amplitude distribution in the input and
output plane, respectively, which are separated by a distance z.
One now can compare the Fresnel diffraction, Eq. (5.29), and the fractional
Fourier transform, Eq. (5.27), and find out that the Fresnel diffraction can be