Page 291 - Introduction to Information Optics

276 5. Transformation with Optics

where H n is the Hermite polynomial of order n. Hence, the Hermite -Gaussian
function is an eigenfunction of the F"ourier transform [21] satisfying the
eigenvalue equation such that

FT(ilt H(x)) = exp(-z>m/2)OA „(«)), (3.26)

n
where u is the frequency. The exp( — inn/2) = r is the eigenvalue of the
Fourier transform operator.
In order to define the fractional Fourier transform one expands the validity
of Eq. (5.26) to meet a real value a, which can be inserted in the power of the
eigenvalue, such that the fractional Fourier transform of the eigenfunction
« x ca n b e written a s

In this condition, the additive fractional order property, Eq. (5.24), can be
observed.
One can define the fractional Fourier transform as an operator that has the
Hermite- Gaussian function as an eigenfunction, with the eigenvalue of
exp( ' — inctn/2). The Hermite-Gaussian functions form a complete set of orthog-
onal polynomials [20]. Therefore, an arbitrary function can be expressed in
terms of these eigenfunctions,

4,<M*),

where the expansion coefficients of f(x) on the Hermite-Gaussian function
basis are

The definition of the fractional Fourier transform can then be cast in the form
of a general linear transform

The fractional Fourier transform applied to an arbitrary function f(x) is
expressed as a sum of the fractional Fourier transforms applied to the
X
eigenfunctions «A«( ) multiplied with a coefficient A n, which is the expansion
coefficient of f(x) on a Hermite-Gaussian function basis. The eigenfunctions