�
�
�
�
� �
�
−iK·ξ
u ˜ I (K) =
d
2 ξ u I (ξ) e
−iK·ξ
u ˜ G (K) =
d
2 ξ u G (ξ) e
−iK·ξ
H ˜ (K) =
d
2 ξ H(ξ) e
,
(3.289)
where K is the two-vector transform variable, and the integration
variable ξ is a two-vector position having mathematical significance, but no particular physical significance. The physical significance of K can be understood by considering a sinusoidal object,
with spatial period Λ I in the image plane. In this case,
2π
K =
(3.290)
Λ I
in one Cartesian axis. Thus K is 2π times the spatial frequency
1/Λ I . Applying the convolution theorem (see Appendix A) to
(3.287) it follows immediately that
u ˜ I (K) = u ˜ G (K) H ˜ (K).
(3.291)
Thus, the spatial frequency spectrum of the ideal image is modu˜
lated by H to yield the spatial frequency spectrum of the actual
image. For this reason, H ˜ is called the amplitude transfer function
or ATF.
To understand the physical significance of this, we substitute in
the expression for the kernel H. This gives
˜
d
2
H(K) = d
2 ξ e
−iK·ξ
1
r A P (r A ) exp −
ik r A · ξ
.
λ 2 f
2
2
f 2
(3.292)
Interchanging the order of integrations, this gives
1
kr A
H ˜ (K) = d
2 r A P (r A ) ·
d
2 ξ exp −iξ · K +
.
λ 2 f
2
2
f 2
(3.293)
We define a new integration variable η by the substitution
2
f 2
f 2
ξ =
η,
d
2 ξ =
d
2 η.
(3.294)
k
k
213
3.3. Diffraction