227
3.3. Diffraction
Z
0
2 = 2 ρ
2 − 1
defocus
Z
2
2 = ρ
2 cos (2φ)
astigmatism
Z
−3
3
= ρ
3 sin (3φ)
trefoil
Z
−1
3
= ρ (3 ρ
2 − 2) sin φ
coma
Z
1
3 = ρ (3 ρ
2 − 2) cos φ
coma
Z
3
3 = ρ
3 cos (3φ)
trefoil
Z
−4
4
= ρ
4 sin (4φ)
Z
−2
4
= ρ
2 (4 ρ
2 − 3) sin (2φ)
Z
0
4 = 6 ρ
4 − 6 ρ
2 + 1
spherical
Z
2
4 = ρ
2 (4 ρ
2 − 3) cos (2φ)
Z
4
4 = ρ
4 cos (4φ),
(3.331)
where we recall that 0 ≤ ρ ≤ 1 and 0 ≤ φ ≤ 2π.
It is useful in some cases to express these in Cartesian coordinates, again confined to the unit disk. To do this we first expand
the trigonometric functions according to the well-known relations
as follows:
sin (2φ) = 2 sin φ cos φ

cos (2φ) = cos
2 φ − sin
2 φ

sin (3φ) = sin φ (3 cos
2 φ − sin
2 φ)

cos (3φ) = cos φ (cos
2 φ − 3 sin
2 φ)

sin (4φ) = 4 sin φ cos φ (cos
2 φ − sin
2 φ)

cos (4φ) = cos
4 φ − 6 sin
2 φ cos
2 φ + sin
4 φ. (3.332)

Substituting the Cartesian coordinates x = ρ cos φ and y = ρ sin φ
we immediately obtain
Z
0 = 1
0
Z
−1 = y
1
Z
1 = x
1
Z
−2
2
= 2 x y
Z
0
2
2 = 2 (x + y
2 ) − 1