Page 198 - Phase Space Optics Fundamentals and Applications

Imaging Systems: Phase-Space Representations 179

phase mask that produces this type of AF was discovered by Dowski
and Cathey. 39
Now, we follow the treatment in Ref. 40. The complex amplitude
transmittance of the pupil aperture is

2
W 2,0
Q( ) = T( ) exp i2 (5.33)

In Eq. (5.33) we denote as T( ) the complex amplitude transmittance
ofthemasklocatedatthepupilaperture.Now,asdiscussedpreviously
except for a normalization factor, the OTF of T( ) with variable focus
error is
∞

! "
H( ; W 2,0 ) = T + T ∗ − exp i2 2W 2,0 d
2 2 2
−∞
(5.34)
Next, we note that if T( ) is a continuous function in , then the OTF
H( ; W 2,0 ) is also continuous in both and W 2,0 . Consequently, we
can express Eq. (5.34) as a Maclaurin power series expansion. That is,

2 2
∂ H W 2,0 ∂ H
H( ; W 2,0 ) = H( ;0) + W 2,0 + 2 + ···
∂W 2,0 2! ∂W
2,0
(5.35)
In the above power series, the nth coefficient is
n
∞
i2 n
n
∂ H/∂W = T + T ∗ − d

2,0 2
2 2
−∞
n
∞
i2 n
= 2 P T ( , ) d (5.36)

−∞
In Eq. (5.36), P T ( , ) is the product spectrum representation of the
mask T( ). Here, it is relevant to recognize the following. If the
complex amplitude transmittance of the pupil mask is a Hermitian
function T( ) = T (− ), then the product spectrum representa-
∗
tion P T ( , ) is an even function in the integrating variable . That
is,

P T ( , − ) = P T ( , ) = T + T ∗ − (5.37)
2 2
Hence, the integrand in Eq. (5.36) is an odd function, provided that
the power order n is an odd integer number (n = 2s + 1). And con-
sequently, the odd-order coefficients are equal to zero for Hermitian