Page 143 - Phase Space Optics Fundamentals and Applications

124 Chapter Four

where the bar denotes the polar coordinate expression for the corre-
sponding function and where we have split out the generalized pupil
P(x ) to explicitly show the dependence on the amplitude pupil vari-

ations p(x ) and the aberration function W(r , ) of the system. The
N
classic defocus coefficient has also been introduced in this equation,
namely,
za 2
W 20 (z) =− (4.51)
2 f ( f + z)
In many practical situations the most important contribution to the
aberration function is the primary spherical aberration (SA), whose de-
pendence on the pupil coordinates is given by

W 40 (r , ) = W 40 r 4 (4.52)
N
N
where W 40 is the SA coefficient design constant. In the following rea-
soning, we will consider this term explicitly, splitting the generalized
pupil of the system as follows:

4
i2 W(r , ) i2 W 40 r N

N
¯ p(r , ) exp = Q(r , ) exp (4.53)

N
N

Thus Q(r , ) includes the amplitude variations on the pupil plane

N
and the aberration effects except for SA. Note that if no aberrations
different from SA are present in the system, Q(r , ) reduces simply

N

to the pupil mask ¯p(r , ).

N
By substituting Eq. (4.53) into Eq. (4.50), we finally obtain
¯ I(r N , ,z)

2 1 4 2
1 i2 W 40 r N i2 W 20 (z) r N

= Q(r , ) exp exp
N
2
( f + z) 2

0 0

2
−i2
× exp r r N cos( − ) r dr d (4.54)

N
N
N
( f + z)
Let us now consider explicitly the angular integration in this equation,
namely,
2

−i2

Q(r ,r N , ,z) = Q(r , ) exp r r N cos( − ) d
N
N
N
( f + z)
0
(4.55)