Page 274 - Introduction to Information Optics

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5.3. Fourier Transform ~:, 9
2
would have the complex amplitude E(x) e\p(inx /lz) in the aperture plane,
According to Eq. (5.5), this complex amplitude field is exactly Fourier
transformed, as described by the integral on the right-hand side of Eq. (5,5).
2
and then multiplied by a quadratic phase factor, exp(z'7Lx' //,z), resulting in E(n)
2
in the output plane x'. The E(u] exp(i7ix' //z) is the representation in the output
plane of a spherical wavefront with the radius of z. In this sense one can say
that the exact Fourier transform exists between the complex amplitudes on two
spherical surfaces in the input and output planes in the Fresnel transform
scheme.

53. FOURIER TRANSFORM

When the diffraction distance from the aperture to the diffracted pattern, z,
tends toward infinity, the Fresnel diffraction becomes the Fraunhofer diffrac-
tion. When the condition
2
z » x /A (5.6)
2
is satisfied, the quadratic phase factor exp(mx //lz) in the integral on the
right-hand side of Eq. (5.5) can be removed. Then, the Fresnel diffraction
formula becomes the Fourier transform

f
E(u) = E(x)QXp(-2niux/Az)dx. (5.7)

For instance, when / = 0.6 [im and the aperture is of radius of 25 mm, to satisfy
the condition in Eq. (5.6), z should be much longer than 1000 m, which is not
practical.
The Fraunhofer diffraction at an infinite distance is implemented usually by
means of a converging lens, which brings the Fraunhofer diffraction from
infinity to its focal plane. There are basically three schemes for the implemen-
tation of the optical Fourier transform:
1. The input is placed in the front of a converging lens. The quadratic phase
2
introduced by the lens cancels the quadratic phase factor Gxp(mx //.f) in
Eq. (5.5) and we obtain the Fourier transform in the back focal plane of
the lens at a distance z = /, where / is the focal length of the lens, with
2
a quadratic phase factor exp(mx' /Af). This phase factor has no effect, if
we are only interested in the intensity of the Fourier transform.
2. The input is placed in the front focal plane of a converging lens, and the
exact Fourier transform is obtained in the back focal plane of the lens.
(For more details see [9]).