JEOS RP ISSN01 | Page 16

J. Eur. Opt. Society-Rapid Publ. 21, 1( 2025) 11
Table A. 1. Zernike integer indices, mode orders, and polynomials, with wavefront aberration and symmetry.
Integer indices nm
Mode order( Noll)
00
J = 1
0
11
J = 2
2
1 – 1
J = 3
1
20
J = 4
4
2 – 2
J = 5
3
22
J = 6
5
3 – 1
J = 7
7
31
J = 8
8
3 – 3
J = 9
6
33
J = 10
9
40
J = 11
12
42
J = 12
13
4 – 2
J = 13
11
44
J = 14
14
4 – 4
J = 15
10
51
J = 16
18
Mode order( OSA / ANSI)
Normalized Zernike polynomials, Z m n pffiffiffiffiffiffiffiffi p 1 ffiffiffiffiffiffiffiffi = p p 4 ffiffiffiffiffiffiffiffi
= p pffiffiffiffiffiffiffiffi
4 = p p 3 ffiffiffiffiffiffiffiffi
= p pffiffiffiffiffiffiffiffi
6 = p p 6 ffiffiffiffiffiffiffiffi
= p pffiffiffiffiffiffiffiffi
8 = p p 8 ffiffiffiffiffiffiffiffi = p p 8 ffiffiffiffiffiffiffiffi = p p 8 ffiffiffiffiffiffiffiffi = p p 5 ffiffiffiffiffiffiffiffiffiffi = p p 10 ffiffiffiffiffiffiffiffiffiffi
= p pffiffiffiffiffiffiffiffiffiffi
10 = p p 10 ffiffiffiffiffiffiffiffiffiffi
= p p 10 ffiffiffiffiffiffiffiffiffiffi
= p 12 = p
Wavefront aberration
Symmetry( x i, y i)
1
Piston
( Even, Even)
q cosðhÞ
Tilt( along x i)
( Odd, Even)
q sinðhÞ
Tip( along y i)
( Even, Odd)
ð2q 2 � 1Þ
Defocus
( Even, Even)
q 2 sinð2hÞ
Primary astigmatism( at 45 °)
( Odd, Odd)
q 2 cosð2hÞ
Primary astigmatism( at 0 °)
( Even, Even)
ð3q 3 � 2qÞ sinðhÞ
Primary coma( along y i)
( Even, Odd)
ð3q 3 � 2qÞ cosðhÞ
Primary coma( along x i)
( Odd, Even)
q 3 sinð3hÞ
Trefoil( at 30 °)
( Even, Odd)
q 3 cosð3hÞ
Trefoil( at 0 °)
( Odd, Even)
ð6q 4 � 6q 2 þ 1Þ
Primary spherical aberration
( Even, Even)
ð4q 4 � 3q 2 Þ cosð2hÞ
Secondary astigmatism( at 0 °)
( Even, Even)
ð4q 4 � 3q 2 Þ sinð2hÞ
Secondary astigmatism( at 45 °)
( Odd, Odd)
q 4 cosð4hÞ
Tetrafoil( at 0 °)
( Even, Even)
q 4 sinð4hÞ
Tetrafoil( at 22.5 °)
( Odd, Odd)
ð10q 5 � 12q 3 þ 3qÞ cosðhÞ
Secondary coma( along x i)
( Odd, Even)