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76 2. Signal Processing with Optics
2.3, FRESNEL-KIRCHHOFF AND FOURIER TRANSFORMATION
2.3.1, FREE SPACE IMPULSE RESPONSE
To understand the basic concept of optical Fourier transformation, we begin
our discussion with the development of the Fresnel-Kirchhoff integral. Let us
start from the Huygens principle, in which the complex amplitude observed at
the point p' of a coordinate system <r(a, /?), due to a monochromatic light field
located in another coordinate system p(x, y), as shown in Fig. 2.6, can be
calculated by assuming that each point of light source is an infinitesimal
spherical radiator. Thus, the complex light amplitude /i ((a, ^; k) contributed by
a point p in the (x, y) coordinate system can be considered to be that from an
unpolarized monochromatic point source, such as
,= — — exp[z(lcr (2.J8)
/r
where /I, k, and to are the wavelengths, wave number, and angular frequency of
the point source, respectively, and r is the distance between the point source
and the point observation.
direction of wave propagation
f(x,y)
Fig. 2.6. Fresnel-Kirchhoff theory.
