Page 92 - Introduction to Information Optics

2.3. Fresnel-Kirchhoff and Fourier Transformation 77

If the separation / of the two coordinate systems is assumed to be large
compared to the regions of interest in the (x, y) and fa, /?) coordinate systems,
r can be approximated by

._, , «- * OS-jO 3
t — I ~r (2.19)
21 21
which is known as paraxial approximation. By substituting into Eq. (2.18) we
have

(a (P-y?
h {(a — p\ k) — —— exp < ik 1 + (2.20)
21 21
which is known as the spatial impulse response, where the time-dependent
exponent has been dropped for convenience. Thus, we see that the complex
light field produced at the (a, ft} coordinate system by the monochromatic
wavefield f(x, y) can be written as

y(a, /?) = I I f(x, y)h t(a — p;k) dx dy, (2.21)
)(x,y)

which is the well-known Kirchhoff's integral. In view of the preceding equation,
we see that the Kirchhoff's integral is, in fact, representing a convolution
integral which can be written as

g(ct, P) = f(x,y)*h l (x,y), (2.22)

where the asterisk denotes the convolution operation,

fV , -« -i
/j,(x, y) — C exp (2.23)

and C is a complex constant. Consequently, Eq. (2.22) can be represented by
a block box diagram, as shown in Fig. 2.7. In other words, the complex wave
field distributed over the (a,/?) coordinate plane can be evaluated by the
convolution integral of Eq. (2.21).

2.3.2. FOURIER TRANSFORMATION BY LENSES

It is well known that a two-dimensional Fourier transformation can be
obtained with a positive lens. Fourier transform operations usually require