Page 199 - Phase Space Optics Fundamentals and Applications

180 Chapter Five

pupilmasks.Ifthis symmetry conditionis fulfilled, then theMaclaurin
series expansion becomes

∞
, W 2n ∂ H
2n
2,0
H( ; W 2,0 ) = H( ;0) + 2n (5.38)
2n! ∂W
n=1 2,0
The result in Eq. (5.38) has two powerful consequences. First, the OTF
exhibits a symmetrical behavior before and after the in-focus image
W 2,0 = 0. Second, the optical system can have a large tolerance to
focus error W 2,0 , provided that the second-order coefficient has small
values.
We note that if the pupil mask is a phase-only filter, then the con-
dition for Hermitan symmetry implies that the phase profile must
exhibit odd symmetry. That is, if ( ) =− (− ), then

T (− ) = exp [−i (− )] = T( ) (5.39)
∗
In Fig. 5.9, we display numerically simulated images of a spoke pat-
tern. Except for the clear pupil aperture, along the columns of Fig. 5.9,
the phase masks obey the relationship
n
! "

exp [i ( )] = exp isign ( )
(5.40)

Clear
pupil n = 3 n = 4 n = 5 n = 6

W 20 = 0

W 20 = l

W 20 = 2l

FIGURE 5.9 Numerical simulations of in-focus and out-of-focus images of a
spoke pattern, when using a clear pupil aperture, and a mask with phase
variations that exhibits odd symmetry.