Page 273 - Introduction to Information Optics

258 5. Transformation with Optics

Under the paraxial condition, the distance z is much larger than the size of
the aperture Z, z » max(x), so that the direction factor in the Huygens--Fresnel
formula in Eq. (5.1) may be neglected and the denominator \r ~ r' in the
right-hand side of Eq. (5.1) is approximated by the distance z. The phase factor
in the integrand on the right-hand side of Eq. (5.1) is approximated, in the
ID notation for the input plane x and output plane x', by (r — r'}—
2
2
z(l +(x — x') /2z ); this results in
,,ikz
2
E(u) = — I E(x) //.z) dx, (5.4)
./'- J
where the phase shift, exp(//cz), and the amplitude factor \/z in the front of the
integral in Eq. (5.4) are associated with wave propagation over the distance z,
and are constant over the output plane at a given propagation distance z. The
paraxial approximation that leads to Eqs. (5.1) to (5.4) is referred to as the
Fresnel approximation.
Hence, the diffracted field E(u] is the Fresnel transform of the complex
amplitude E(x) at the aperture, where u — x'/AZ is the spatial frequency. The
optical diffraction under the Fresnel paraxial approximation is described by the
Fresnel transform. On the other side, we can consider light propagation through
an aperture as an implementation of the mathematical Fresnel transform.
One can expand the quadratic phase on the right-hand side of Eq. (5.4) and
rewrite Eq. (5.4) as

2
E(u] = -7- exp(/7rx' /xz) E(x) exp( — i2nxx'/Az dx. (5.5)
'
2
2
In Eq. (5.5) the quadratic phase terms, Qxp(inx' /Az) and exp(mx /Xz), describe
the spherical wavefronts of radius z in the input and output planes, respec-
tively, as shown in Fig. 5.1. A spherical wavefront passing through an aperture

Fig. 5.1. Optical Fresnel transform.