Page 200 - Phase Space Optics Fundamentals and Applications

Imaging Systems: Phase-Space Representations 181

In Eq. (5.40) we denote as sign( ) the signum function. We use
for denoting the maximum phase delay, at the edge of the pupil
aperture. And n is any integer number. However, we note that n can
also represent a real number. For further discussion of this topic, see
Refs. 30, 31, and 40 to 43.
Next, we discuss how these results are also useful for reducing
the impact of spherical aberration in two-dimensional optical sys-
tems with radial symmetry. To that end, we discuss a simplified ver-
sion of McCutchen theorem 44 that is useful for analyzing the Strehl
ratio of rotationally symmetric systems, which suffer from wave
aberrations. 45−50
The generalized pupil function of an optical system, with rotation-
ally symmetry, that suffers from focus error W 2,0 and from spherical
aberration W 4,0 is

2 4
W 2,0 W 4,0
S( ; W 2,0 ; W 4,0 ) = T( ) exp i2 +

(5.41)
In Eq. (5.41) the complex amplitude transmittance of the pupil mask
is T( ). We denote as the radial spatial frequency, and its maximum
value is the cutoff spatial frequency . In polar coordinates, the im-
pulse response of the optical system is obtained by taking the two-
dimensional Fourier transform of Eq. (5.41). That is,

s(r; W 2,0 ; W 4,0 ) = 2 S( ; W 2,0 ; W 4,0 )J 0 (2 r ) d (5.42)
0
At the optical axis r = 0, Eq. (5.42) reduces to

s(0; W 2,0 ; W 4,0 ) = 2 S( ; W 2,0 ; W 4,0 ) d (5.43)
0
Now, we map the radially symmetric, two-dimensional pupil into a
one-dimensional pupil aperture, by using the geometrical transfor-
mation
1/2
2
= − 0.5; Q( ) = T( ( + 0.5) ) (5.44)

From Eq. (5.44), we observe that the geometrical transformation
defines an effective pupil function Q( ) from the physical pupil
mask T( ). And therefore, by using Eq. (5.43) at the optical axis, we
can define an effective impulse response as the square modulus of