0(0, 0)

1(1, 1)

m=0

1(1,−1)

3(2, 2)

n=1

m=−1

2(1, 1)

2(2, 1)

m=1

3(2,−2)

6(3, 2)

n=2

m=−2

4(2, 0)

4(2, 3)

5(2, 2)

5(3, 1)

m=2

6(3,−3)

11(4, 2)

n=3

m=−3

7(3,−1)

8(3, 4)

8(3, 1)

7(3, 3)

9(3, 3)

10(4, 1)

m=3

10(4,−4)

18(5, 2)

n=4

m=−4

11(4,−2)

13(4, 4)

12(4, 0)

9(3, 5)

13(4, 2)

12(4, 3)

14(4, 4)

17(5, 1)

m=4

15(5,−5)

27(6, 2)

n=5

m=−5

16(5,−3)

20(5, 4)

17(5,−1)

15(4, 6)

18(5, 1)

14(4, 5)

19(5, 3)

19(5, 3)

20(5, 5)

26(6, 1)

m=5

21(6,−6)

38(7, 2)

n=6

m=−6

22(6,−4)

29(6, 4)

23(6,−2)

22(5, 6)

24(6, 0)

16(4, 7)

25(6, 2)

21(5, 5)

26(6, 4)

28(6, 3)

27(6, 6)

37(7, 1)

m=6

28(7,−7)

51(8, 2)

n=7

m=−7

29(7,−5)

40(7, 4)

30(7,−3)

31(6, 6)

31(7,−1)

24(5, 8)

32(7, 1)

23(5, 7)

33(7, 3)

30(6, 5)

34(7, 5)

39(7, 3)

35(7, 7)

50(8, 1)

m=7

36(8,−8)

66(9, 2)

n=8

m=−8

37(8,−6)

53(8, 4)

38(8,−4)

42(7, 6)

39(8,−2)

33(6, 8)

40(8, 0)

25(5, 9)

41(8, 2)

32(6, 7)

42(8, 4)

41(7, 5)

43(8, 6)

52(8, 3)

44(8, 8)

65(9, 1)

m=8

45(9,−9)

83(10, 2)

n=9

m=−9

46(9,−7)

68(9, 4)

47(9,−5)

55(8, 6)

48(9,−3)

44(7, 8)

49(9,−1)

35(6, 10)

50(9, 1)

34(6, 9)

51(9, 3)

43(7, 7)

52(9, 5)

54(8, 5)

53(9, 7)

67(9, 3)

54(9, 9)

82(10, 1)

m=9

55(10,−10)

102(11, 2)

n=10

m=−10

56(10,−8)

85(10, 4)

57(10,−6)

70(9, 6)

58(10,−4)

57(8, 8)

59(10,−2)

46(7, 10)

60(10, 0)

36(6, 11)

61(10, 2)

45(7, 9)

62(10, 4)

56(8, 7)

63(10, 6)

69(9, 5)

64(10, 8)

84(10, 3)

65(10, 10)

101(11, 1)

m=10

66(11,−11)

123(12, 2)

n=11

m=−11

67(11,−9)

104(11, 4)

68(11,−7)

87(10, 6)

69(11,−5)

72(9, 8)

70(11,−3)

59(8, 10)

71(11,−1)

48(7, 12)

72(11, 1)

47(7, 11)

73(11, 3)

58(8, 9)

74(11, 5)

71(9, 7)

75(11, 7)

86(10, 5)

76(11, 9)

103(11, 3)

77(11, 11)

122(12, 1)

m=11

78(12,−12)

146(13, 2)

n=12

m=−12

79(12,−10)

125(12, 4)

80(12,−8)

106(11, 6)

81(12,−6)

89(10, 8)

82(12,−4)

74(9, 10)

83(12,−2)

61(8, 12)

84(12, 0)

49 37(7, 13)

85(12, 2)

60(8, 11)

86(12, 4)

73(9, 9)

87(12, 6)

88(10, 7)

88(12, 8)

105(11, 5)

89(12, 10)

124(12, 3)

90(12, 12)

145(13, 1)

m=12

j(n,m)

A(g, f)

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

+0.5

+1.0

+1.5

+2.0

+2.5

+3.0

The Recursion relationship between three radial polynomials,

R|m|n (ρ) =

1

2ρ(n+ 1)

[

(n+ |m|+ 2)R|m|+1n+1 (ρ) + (n− |m|)R|m|+1

n−1 (ρ)]

connects the following indexes shown in their relative location form > 0 on the Zernike tree. For m < 0 the mirror image of thefigure below is used. Any one of the polynomials can be written interms of the other two.

(n− 1, |m|+ 1)

(n, |m|)

(n+ 1, |m|+ 1)

Zernike polynomials are often used to describe wavefront error on an optical system. Mathematically they are an infinite set of orthogonalfunctions over the unit disk. Any arbitrary function defined on a unit disk can be described as a linear combination of Zernike polynomials.On the left are the definitions for the Zernike polynomials that follow the Optical Society of America (OSA) standard. On the right are theFringe Zernike polynomials. Below is a tree of the index values for the Zernike polynomials. In the top of each box are the index values forthe OSA standard Zernike polynomials. In the bottom of each box are the index values of the Fringe Zernike polynomials. Rows of constantn are shown in blue and light blue. Groups of Fringe Zernike polynomials with constant g are red and light red. Below each index box isa contour plot of the Zernike polynomial and a logarithmically scaled point spread function (PSF) that would result from an optical systemwith wavefront error of the given Zernike polynomial. All the PSF plots have the same spatial extent.

Zernike PolynomialsZernike Polynomial Definitions:

Zj (ρ, θ) = Zmn (ρ, θ) = Nm

n R|m|n (ρ)

{

sin (|m|θ) m < 0

cos (mθ) m ≥ 0where Nm

n =

√

2(n+ 1)

1 + δm,0

The index numbers j and the pair n, m have the following properties:

j ∈ (0, 1, 2, · · · ,∞); n ∈ (0, 1, 2, · · · ,∞); m ∈ (−n,−n+2, · · · , n−2, n);

j =n(n + 2) +m

2; n =

⌈−3 +√9 + 8j

2

⌉

; m = 2j − n(n+ 2);

Relationships for the radial polynomial R|m|n (ρ):

R|m|n (ρ) =

n−|m|2∑

s=0

(−1)s(n− s)!

s![

12(n+m)− s

]

![

12(n−m)− s

]

!ρn−2s

R|m|n (ρ) =

Γ (n+ 1) 2F1

(

−12(|m|+ n) , 1

2(|m| − n) ;−n; ρ−2

)

Γ(

12(2 + n +m)

)

Γ(

12(2 + n−m)

) ρn

R|m|n (ρ) = (−1)(n−|m|)/2ρ|m|P

{|m|,0}(n−|m|)/2(1− 2ρ2)

Where P {α,β}n (x) is a Jacobi polynomial

P {α,β}n (x) =

(−1)n

2nn!(1− x)−α (1 + x)−β

dn

dxn

[

(1− x)α+n (1 + x)β+n]

Orthogonality and integral relationships for Zernike polynomials:

∫ 1

0

R|m|n (ρ)R

|m|n′ (ρ) ρ dρ =

1

2 (n+ 1)δnn′

∫ 1

0

R|m|n (ρ) J|m| (νρ) ρ dρ = (−1)(n−|m|)/2Jn+1 (ν)

ν∫ 2π

0

∫ 1

0

Zmn (ρ, θ)Zm′

n′ (ρ, θ) ρ dρ dθ = πδn,n′δm,m′

Jn(z) =i−n

π

∫ π

0

eiz cos θ cos(nθ)dθ

Jn(−z) = (−z)nz−nJn(z)

Table of OSA Zernike Polynomials:

The indexes, j, n and m are shown below along withthe main index for the Fringe Zernikes, A and thestandard Code V Zernikes, jCV.

j n m A jCV OSA Zernikes

0 0 0 1 1 1

1 1 -1 3 3 2ρ sin(θ)2 1 1 2 2 2ρ cos(θ)

3 2 -2 6 6√6ρ2 sin(2θ)

4 2 0 4 5√3 (2ρ2 − 1)

5 2 2 5 4√6ρ2 cos(2θ)

6 3 -3 11 10 2√2ρ3 sin(3θ)

7 3 -1 8 9 2√2 (3ρ3 − 2ρ) sin(θ)

8 3 1 7 8 2√2 (3ρ3 − 2ρ) cos(θ)

9 3 3 10 7 2√2ρ3 cos(3θ)

10 4 -4 18 15√10ρ4 sin(4θ)

11 4 -2 13 14√10 (4ρ4 − 3ρ2) sin(2θ)

12 4 0 9 13√5 (6ρ4 − 6ρ2 + 1)

13 4 2 12 12√10 (4ρ4 − 3ρ2) cos(2θ)

14 4 4 17 11√10ρ4 cos(4θ)

15 5 -5 27 21 2√3ρ5 sin(5θ)

16 5 -3 20 20 2√3 (5ρ5 − 4ρ3) sin(3θ)

17 5 -1 15 19 2√3 (10ρ5 − 12ρ3 + 3ρ) sin(θ)

18 5 1 14 18 2√3 (10ρ5 − 12ρ3 + 3ρ) cos(θ)

19 5 3 19 17 2√3 (5ρ5 − 4ρ3) cos(3θ)

20 5 5 26 16 2√3ρ5 cos(5θ)

21 6 -6 38 28√14ρ6 sin(6θ)

22 6 -4 29 27√14 (6ρ6 − 5ρ4) sin(4θ)

23 6 -2 22 26√14 (15ρ6 − 20ρ4 + 6ρ2) sin(2θ)

24 6 0 16 25√7 (20ρ6 − 30ρ4 + 12ρ2 − 1)

25 6 2 21 24√14 (15ρ6 − 20ρ4 + 6ρ2) cos(2θ)

26 6 4 28 23√14 (6ρ6 − 5ρ4) cos(4θ)

27 6 6 37 22√14ρ6 cos(6θ)

A g f j Fringe Zernikes Aberration Name∗

1 1 1 0 1 Piston

2 2 1 2 ρ cos(θ) x-Tilt3 2 2 1 ρ sin(θ) y-Tilt4 2 3 4 2ρ2 − 1 Defocus

5 3 1 5 ρ2 cos(2θ) Astigmatism 0◦ or 90◦

6 3 2 3 ρ2 sin(2θ) Astigmatism ±45◦

7 3 3 8 (3ρ3 − 2ρ) cos(θ) x-Coma8 3 4 7 (3ρ3 − 2ρ) sin(θ) y-Coma9 3 5 12 6ρ4 − 6ρ2 + 1 Primary Spherical

10 4 1 9 ρ3 cos(3θ)11 4 2 6 ρ3 sin(3θ)12 4 3 13 (4ρ4 − 3ρ2) cos(2θ) Secondary Astigmatism 0◦ or 90◦

13 4 4 11 (4ρ4 − 3ρ2) sin(2θ) Secondary Astigmatism ±45◦

14 4 5 18 (10ρ5 − 12ρ3 + 3ρ) cos(θ) Secondary x-Coma15 4 6 17 (10ρ5 − 12ρ3 + 3ρ) sin(θ) Secondary y-Coma16 4 7 24 20ρ6 − 30ρ4 + 12ρ2 − 1 Secondary Spherical

17 5 1 14 ρ4 cos(4θ)18 5 2 10 ρ4 sin(4θ)19 5 3 19 (5ρ5 − 4ρ3) cos(3θ)20 5 4 16 (5ρ5 − 4ρ3) sin(3θ)21 5 5 25 (15ρ6 − 20ρ4 + 6ρ2) cos(2θ) Tertiary Astigmatism 0◦ or 90◦

22 5 6 23 (15ρ6 − 20ρ4 + 6ρ2) sin(2θ) Tertiary Astigmatism ±45◦

23 5 7 32 (35ρ7 − 60ρ5 + 30ρ3 − 4ρ) cos(θ) Tertiary x-Coma24 5 8 31 (35ρ7 − 60ρ5 + 30ρ3 − 4ρ) sin(θ) Tertiary y-Coma25 5 9 40 70ρ8 − 140ρ6 + 90ρ4 − 20ρ2 + 1 Tertiary Spherical

26 6 1 20 ρ5 cos(5θ)27 6 2 15 ρ5 sin(5θ)28 6 3 26 (6ρ6 − 5ρ4) cos(4θ)29 6 4 22 (6ρ6 − 5ρ4) sin(4θ)30 6 5 33 (21ρ7 − 30ρ5 + 10ρ3) cos(3θ)31 6 6 30 (21ρ7 − 30ρ5 + 10ρ3) sin(3θ)32 6 7 41 (56ρ8 − 105ρ6 + 60ρ4 − 10ρ2) cos(2θ)33 6 8 39 (56ρ8 − 105ρ6 + 60ρ4 − 10ρ2) sin(2θ)34 6 9 50 (126ρ9 − 280ρ7 + 210ρ5 − 60ρ3 + 5ρ) cos(θ)35 6 10 49 (126ρ9 − 280ρ7 + 210ρ5 − 60ρ3 + 5ρ) sin(θ)36 6 11 60 252ρ10 − 630ρ8 + 560ρ6 − 210ρ4 + 30ρ2 − 1

37 7 13 84 924ρ12 − 2772ρ10 + 3150ρ8 − 1680ρ6 + 420ρ4 − 42ρ2 + 1∗The words “orthonormal Zernike” are to be associated with these names.

Fringe Zernike Polynomials:The index numbers A and the pair g, f have the following properties:

A = (g − 1)2 + f ; g =

⌊

√

A− 1

2

⌋

+ 1; f = A− (g − 1)2

Index conversion between the Fringe Zernikes and the OSA Zernikes:

n = (g − 1) +

⌊

f − 1

2

⌋

m = (−1)f+1

(

(g − 1)−⌊

f − 1

2

⌋)

g =n+ |m|

2+ 1

f = n− |m|+ 1 +1

2

(

1− 2m+ 1

|2m+ 1|

)

The Fourier transform of the Zernike polynomials defined as

Sj (κ, φ) = Smn (~κ) = F [Zmn (~ρ)] =

∫

Zmn (~ρ) e−2πi~κ·~ρd~ρ

is as follows where Jn(2πκ) is a Bessel function of the first kind

Smn (κ, φ) =

√

2(n + 1)i|m|(−1)(n+|m|)/2(

Jn+1(2πκ)κ

)

sin(|m|φ) m < 0

√n+ 1(−1)n/2

(

Jn+1(2πκ)κ

)

m = 0

√

2(n + 1)im(−1)(n+m)/2(

Jn+1(2πκ)κ

)

cos(mφ) m > 0

Therefore:

F [f(ρ, θ)] =∞∑

j=0

ajSj(κ, φ)

were

f(ρ, θ) =∞∑

j=0

ajZj(ρ, θ)

The Zernike polynomials are complete over the unit disk,

limN→∞

∫ 1

0

∫ 2π

0

(

f(ρ, θ)−N∑

j=0

ajZj(ρ, θ)

)2

ρdρdθ = 0

where f(ρ, θ) is an arbitrary function defined on the unitdisk and the expansion coefficients aj are

aj =1

π

∫ 1

0

∫ 2π

0

f(ρ, θ)Zj(ρ, θ)ρdρdθ

Therefore:

f(ρ, θ) =

∞∑

j=0

ajZj(ρ, θ)

The Fringe Zernike polynomials follow an indexing pattern that isdistinctly different than the indexing of the OSA standard Zernikepolynomials. The Fringe Zernike polynomials are only the 37that are shown in the table above. The numbering of the FringeZernike polynomial number 37 does not follow the same rulesas the rest of them. This 37th Fringe Zernike polynomialis shown in the tree as the red box on the bottom cen-ter. If one follows the indexing rule set up by the first36 Fringe Zernike polynomials the generalized FringeZernike polynomials can be defined and are shownin the index tree as the green and light greenboxes. The Fringe Zernike polynomials with aconstant g form “V” shapes in the index tree.

Made by Brad Paul (bradpaul@rochester.rr.com)with Asymptote and LATEX 2010

Pupil Function

D

y

x

w

y

z

f

w

−w2

w2−w

n

wn

D

y′

−nλf2w

nλf2w

− λfw

nλfw

λfw

PSFy′

x′

nλfw

MTF

PTF

wλf

wnλf

− w2λf

w2λf

− wnλf

fy

The Root Mean Square (RMS) of an arbitrary two dimensionalfunction f(~x) over the domain D is given by

RMS [f(~x)] =

√

∫

Df(~x)2d~x∫

Dd~x

In the special case where the domain of the function is a disk ofradius R then the RMS of the function can be written as

RMS [f(~r)] =

√

√

√

√

∞∑

j=0

hja2j where f(~r) = f(ρR, θ) =

∞∑

j=0

ajZj(ρR, θ)

Where the orthogonality relationship for Zj(ρ, θ) is

∫ 2π

0

∫ 1

0

Zj(ρ, θ)Zk(ρ, θ)ρdρdθ = πhjδj,k

The Pupil function P (x, y;λ, t) is a complex function of the form

P (x, y;λ, t) = T (x, y;λ, t)e−iφ(x,y;λ,t) = T (x, y;λ, t)e−2πi

λψ(x,y;λ,t) = T (x, y;λ, t)e−2πiW (x,y;λ,t)

where T (x, y;λ, t) is a real function representing the optical transmission of the aperture,0 ≤ T (x, y;λ, t) ≤ 1 and φ(x, y;λ, t) is the wavefront relative to the Gaussian referencesphere in radians, ψ(x, y;λ, t) in length or W (x, y;λ, t) in waves. The PSF is the scaledmodules squared of the Fourier transform of the pupil function.

PSF(x′, y′;λ, t) =∣

∣F{x,y} [P (λfx, λfy;λ, t)] (x′, y′)

∣

∣

2

The OTF can be calculated from the PSF via a Fourier transform

OTF(fx, fy;λ, t) = F{x′,y′} [PSF(x′, y′;λ, t)] (fx, fy)

When using FFT’s to perform the Fourier transforms one must ensure that the width ofthe pupil sampling grid is at least twice the size as the enclosing diameter of the pupil,w ≥ D. The size and sampling of the axis are shown in the figure to the far right.

PSF for a diffraction-limited incoherent imaging system with a clear circular aperture is

PSF(r) =

(

πD2

4λf

)2

2J1

(

πDrλf

)

πDrλf

2

where D is the diameter, λ is the wavelength of light, and f is the focal length of thesystem. The MTF of the system is

MTF(ρn) =

{

2π

[

cos−1(ρn)− ρn√

1− ρ2n

]

0 ≤ ρn ≤ 1

0 ρn > 1

where

ρn = ρλf

D

and ρ is the spacial frequency that produces non-zero MTF values over the range of0 ≤ ρ ≤ D

λf.

The pupil grid is sampled from −w2to w

2− w

nat steps of w

nas shown on the y axis. The wavefront

is relative to the Gaussian reference sphere. The horizontal axis of the PSF is sampled as shownon the y′ axis. The Fourier transform of the PSF is the OTF which is Hermitian because the PSFis real. The MTF and PTF are shown at the far right. The horizontal axis of the OTF is sampledas shown on the fy axis.