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3.3. Diffraction
image; i.e., how closely the image approximates an ideal replica
of the object. In order to be useful, this procedure must account
for aberrations, defocus, and diffraction with a finite pupil, all of
which tend to degrade the image.
We begin our analysis by considering a specific hypothetical optical configuration, consisting of two ideal lenses of focal lengths f 1
and f 2 , respectively. This is shown schematically in Figure 3.17.
By assumption, a physical aperture is located in the Fourier plane
z A of the first lens. Furthermore, the image plane is assumed to
lie in the Fourier plane of the second lens. The back focal plane
of the first lens thus coincides with the front focal plane of the
second lens. By inspection, it is easy to see that a real image of
the original object in the plane z O is formed in the plane z I . The
magnification is given by
f 2
M = − ,
(3.280)
f 1
where the minus sign indicates that the image is inverted with
respect to the object. From (3.250) the amplitude u A (r A ) is expressed in terms of the object u O (r O ) in the Fraunhofer approximation by
1
ik r O · r A
u A (r A ) =
d
2 r O u O (r O ) exp −
.
(3.281)
i λ f 1
f 1
Similarly, the amplitude u I (r I ) is expressed in terms of u A (r A ) by
1
ik r A · r I
u I (r I ) =
d
2 r A u A (r A ) P (r A ) exp −
, (3.282)
i λ f 2
f 2
where P (r A ) is the pupil function. Substituting the first of these
equations into the second, and interchanging the order of integrations, we obtain
−1
d
2
d
2
u I (r I ) =
r O u O (r O )
r A P (r A )
λ 2 f 1 f 2
ik
· exp − r A · (r I − M r O ) ,
(3.283)
f 2