Z
2
2 = x
2 − y
2
Z
−3
3
= y (3 x
2 − y
2 )
Z
−1
3
= y [ 3 (x
2 + y
2 ) − 2 ]
Z
1
3 = x [ 3 (x
2 + y
2 ) − 2 ]
Z
3
3 = x (x
2 − 3 y
2 )
Z
−4
4
= 4 x y (x
2 + y
2 )
Z
−2
4
= 2 x y [ 4 (x
2 + y
2 ) − 3 ]
Z
0
4 = 6 (x
2 + y
2 )
2 − 6 (x
2 + y
2 ) + 1
Z
2
4 = (x
2 − y
2 ) [ 4 (x
2 + y
2 ) − 3 ]
2 2
4
Z 4
4 = x
4 − 6 x y + y .
(3.333)
Our goal is to express an arbitrary phase shift kχ(x A , y A ) in terms
of a corresponding set of Zernike coefficients (a mn , b mn ), where
k = 2π/λ is the wave number. To this end we define a set of
coordinate transformations as follows:
x = ρ cos φ = x A /R A
y = ρ sin φ = y A /R A
ρ =
x 2 + y 2 ,
(3.334)
where R A is the radius of the aperture in the equivalent confocal
system. Based on this, we define the formal functional substitutions
G(x, y) = F (ρ, φ)
kχ(x A , y A ) = G(x, y).
(3.335)
Finally, we define the correspondence between the physical system
and the equivalent confocal system according to
x I
�
= x A /f 2
y I
�
= y A /f 2
M = −f 2 /f 1 ,
(3.336)
228
Chapter 3. Wave optics