Page 201 - Phase Space Optics Fundamentals and Applications

182 Chapter Five

s(0; W 2,0 ; W 4,0 ). That is,
h(W 2,0 ; W 4,0 )

0.5

W 4,0 2
2
=|s(0; W 2,0 ; W 4,0 )| = ( ) 2 Q( ) exp i2

−0,5
2

+ (W 4,0 + W 2,0 ) d (5.45)

Of course, the effective impulse response, in Eq. (5.45), can be related
to an effective OTF
∞

(W 4,0 + W 2,0 )
h(W 2,0 ; W 4,0 ) = H( ; W 4,0 ) exp i2 d

−∞
(5.46)
Except for a normalization factor, the effective OTF is

∞

2W 4,0

H( ; W 4,0 ) = Q + Q ∗ − exp i2 d
2 2
−∞

= A q ;2W 4,0 (5.47)

Now, in a similar fashion to the discussion in Sec. 5.5, it is apparent
from Eq. (5.47) that the effective OTF is related to the AF of the effective
pupil function Q( ). Hence, if the effective pupil mask has complex
amplitude transmittance with Hermitian symmetry (odd phase distri-
bution for phase-only mask), then the effective OTF has low sensitivity
to spherical aberration.
Of course, after the effective pupil function is selected, it is necessary
to find the complex amplitude transmittance of the physical mask
T( ), by using the inverse of the geometrical mapping in Eq. (5.44).
The effective OTF is not to be confused with the OTF of the physical
mask. We illustrate this result with the following simple example.
We consider an effective pupil mask that has a cubic phase profile.
That is,
3
Q( ) = exp (i2
) rect ( ) (5.48)
The effective pupil function has an odd phase profile. However, as ex-
pected,thephysicalpupilmaskhasacomplexamplitudetransmission