Zernike polynomials $Z_n^l$

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Polynomials

$n$

$l$

form

$R_n^{|l|}(t)$ or $Z_n^l(x,y)$

$R_n^{|l|}(t)$:

2*t^2 - 1

comment: Radial part for Noll $j=4$, OSA/ANSI $j=4$, Fringe $j=4$, Wyant $j=3$; defocus.

$Z_n^l(x,y)$:

2*x^2 + 2*y^2 - 1

comment: Noll $j=4$, OSA/ANSI $j=4$, Fringe $j=4$, Wyant $j=3$; defocus.

$Z_n^l(x,y)$:

x^2 - y^2

comment: Noll $j=6$, OSA/ANSI $j=5$, Fringe $j=5$, Wyant $j=4$; vertical astigmatism.

-2

$Z_n^l(x,y)$:

2*x*y

comment: Noll $j=5$, OSA/ANSI $j=3$, Fringe $j=6$, Wyant $j=5$; oblique astigmatism.

$R_n^{|l|}(t)$:

3*t^3 - 2*t

comment: Radial part for $l=1$ (Noll $j=8$, OSA/ANSI $j=8$, Fringe $j=7$, Wyant $j=6$; horizontal coma) and $l=-1$ (Noll $j=7$, OSA/ANSI $j=7$, Fringe $j=8$, Wyant $j=7$; vertical coma).

$Z_n^l(x,y)$:

3*x^3 + 3*x*y^2 - 2*x

comment: Noll $j=8$, OSA/ANSI $j=8$, Fringe $j=7$, Wyant $j=6$; horizontal coma.

-1

$Z_n^l(x,y)$:

3*x^2*y + 3*y^3 - 2*y

comment: Noll $j=7$, OSA/ANSI $j=7$, Fringe $j=8$, Wyant $j=7$; vertical coma.

$Z_n^l(x,y)$:

x^3 - 3*x*y^2

comment: Noll $j=10$, OSA/ANSI $j=9$, Fringe $j=10$, Wyant $j=9$; oblique trefoil.

-3

$Z_n^l(x,y)$:

3*x^2*y - y^3

comment: Noll $j=9$, OSA/ANSI $j=6$, Fringe $j=11$, Wyant $j=10$; vertical trefoil.

$R_n^{|l|}(t)$:

6*t^4 - 6*t^2 + 1

comment: Radial part for Noll $j=11$, OSA/ANSI $j=12$, Fringe $j=9$, Wyant $j=8$; primary spherical aberration.

$Z_n^l(x,y)$:

6*x^4 + 12*x^2*y^2 + 6*y^4 - 6*x^2 - 6*y^2 + 1

comment: Noll $j=11$, OSA/ANSI $j=12$, Fringe $j=9$, Wyant $j=8$; primary spherical aberration.

$R_n^{|l|}(t)$:

4*t^4 - 3*t^2

comment: Radial part for $l=2$ (Noll $j=12$, OSA/ANSI $j=13$, Fringe $j=12$, Wyant $j=11$; vertical secondary astigmatism) and $l=-2$ (Noll $j=13$, OSA/ANSI $j=11$, Fringe $j=13$, Wyant $j=12$; oblique secondary astigmatism).

$Z_n^l(x,y)$:

4*x^4 - 4*y^4 - 3*x^2 + 3*y^2

comment: Noll $j=12$, OSA/ANSI $j=13$, Fringe $j=12$, Wyant $j=11$; vertical secondary astigmatism.

-2

$Z_n^l(x,y)$:

8*x^3*y + 8*x*y^3 - 6*x*y

comment: Noll $j=13$, OSA/ANSI $j=11$, Fringe $j=13$, Wyant $j=12$; oblique secondary astigmatism.

$Z_n^l(x,y)$:

x^4 - 6*x^2*y^2 + y^4

comment: Noll $j=14$, OSA/ANSI $j=14$, Fringe $j=17$, Wyant $j=16$; vertical quadrafoil.

-4

$Z_n^l(x,y)$:

4*x^3*y - 4*x*y^3

comment: Noll $j=15$, OSA/ANSI $j=10$, Fringe $j=18$, Wyant $j=17$; oblique quadrafoil.

$R_n^{|l|}(t)$:

10*t^5 - 12*t^3 + 3*t

comment: Radial part for $l=1$ (Noll $j=16$, OSA/ANSI $j=18$, Fringe $j=14$, Wyant $j=13$) and $l=-1$ (Noll $j=17$, OSA/ANSI $j=17$, Fringe $j=15$, Wyant $j=14$).

$Z_n^l(x,y)$:

10*x^5 + 20*x^3*y^2 + 10*x*y^4 - 12*x^3 - 12*x*y^2 + 3*x

comment: Noll $j=16$, OSA/ANSI $j=18$, Fringe $j=14$, Wyant $j=13$.

-1

$Z_n^l(x,y)$:

10*x^4*y + 20*x^2*y^3 + 10*y^5 - 12*x^2*y - 12*y^3 + 3*y

comment: Noll $j=17$, OSA/ANSI $j=17$, Fringe $j=15$, Wyant $j=14$.

$R_n^{|l|}(t)$:

5*t^5 - 4*t^3

comment: Radial part for $l=3$ (Noll $j=18$, OSA/ANSI $j=19$, Fringe $j=19$, Wyant $j=18$) and $l=-3$ (Noll $j=19$, OSA/ANSI $j=16$, Fringe $j=20$, Wyant $j=19$).

$Z_n^l(x,y)$:

5*x^5 - 10*x^3*y^2 - 15*x*y^4 - 4*x^3 + 12*x*y^2

comment: Noll $j=18$, OSA/ANSI $j=19$, Fringe $j=19$, Wyant $j=18$.

-3

$Z_n^l(x,y)$:

15*x^4*y + 10*x^2*y^3 - 5*y^5 - 12*x^2*y + 4*y^3

comment: Noll $j=19$, OSA/ANSI $j=16$, Fringe $j=20$, Wyant $j=19$.

$Z_n^l(x,y)$:

x^5 - 10*x^3*y^2 + 5*x*y^4

comment: Noll $j=20$, OSA/ANSI $j=20$, Fringe $j=26$, Wyant $j=25$.

-5

$Z_n^l(x,y)$:

5*x^4*y - 10*x^2*y^3 + y^5

comment: Noll $j=21$, OSA/ANSI $j=15$, Fringe $j=27$, Wyant $j=26$.

$R_n^{|l|}(t)$:

20*t^6 - 30*t^4 + 12*t^2 - 1

comment: Radial part for Noll $j=22$, OSA/ANSI $j=24$, Fringe $j=16$, Wyant $j=15$.

$Z_n^l(x,y)$:

20*x^6 + 60*x^4*y^2 + 60*x^2*y^4 + 20*y^6 - 30*x^4 - 60*x^2*y^2 - 30*y^4 + 12*x^2 + 12*y^2 - 1

comment: Noll $j=22$, OSA/ANSI $j=24$, Fringe $j=16$, Wyant $j=15$.

$R_n^{|l|}(t)$:

15*t^6 - 20*t^4 + 6*t^2

comment: Radial part for $l=2$ (Noll $j=24$, OSA/ANSI $j=25$, Fringe $j=21$, Wyant $j=20$) and $l=-2$ (Noll $j=23$, OSA/ANSI $j=23$, Fringe $j=22$, Wyant $j=21$).

$Z_n^l(x,y)$:

15*x^6 + 15*x^4*y^2 - 15*x^2*y^4 - 15*y^6 - 20*x^4 + 20*y^4 + 6*x^2 - 6*y^2

comment: Noll $j=24$, OSA/ANSI $j=25$, Fringe $j=21$, Wyant $j=20$.

-2

$Z_n^l(x,y)$:

30*x^5*y + 60*x^3*y^3 + 30*x*y^5 - 40*x^3*y - 40*x*y^3 + 12*x*y

comment: Noll $j=23$, OSA/ANSI $j=23$, Fringe $j=22$, Wyant $j=21$.

$R_n^{|l|}(t)$:

6*t^6 - 5*t^4

comment: Radial part for $l=4$ (Noll $j=26$, OSA/ANSI $j=26$, Fringe $j=28$, Wyant $j=27$) and $l=-4$ (Noll $j=25$, OSA/ANSI $j=22$, Fringe $j=29$, Wyant $j=28$).

$Z_n^l(x,y)$:

6*x^6 - 30*x^4*y^2 - 30*x^2*y^4 + 6*y^6 - 5*x^4 + 30*x^2*y^2 - 5*y^4

comment: Noll $j=26$, OSA/ANSI $j=26$, Fringe $j=28$, Wyant $j=27$.

-4

$Z_n^l(x,y)$:

24*x^5*y - 24*x*y^5 - 20*x^3*y + 20*x*y^3

comment: Noll $j=25$, OSA/ANSI $j=22$, Fringe $j=29$, Wyant $j=28$.

$Z_n^l(x,y)$:

x^6 - 15*x^4*y^2 + 15*x^2*y^4 - y^6

comment: Noll $j=28$, OSA/ANSI $j=27$, Fringe $j=37$, Wyant $j=36$.

-6

$Z_n^l(x,y)$:

6*x^5*y - 20*x^3*y^3 + 6*x*y^5

comment: Noll $j=27$, OSA/ANSI $j=21$, Fringe $j=38$, Wyant $j=37$.

$R_n^{|l|}(t)$:

35*t^7 - 60*t^5 + 30*t^3 - 4*t

comment: Radial part for $l=1$ (Noll $j=30$, OSA/ANSI $j=32$, Fringe $j=23$, Wyant $j=22$) and $l=-1$ (Noll $j=29$, OSA/ANSI $j=31$, Fringe $j=24$, Wyant $j=23$).

$Z_n^l(x,y)$:

35*x^7 + 105*x^5*y^2 + 105*x^3*y^4 + 35*x*y^6 - 60*x^5 - 120*x^3*y^2 - 60*x*y^4 + 30*x^3 + 30*x*y^2 - 4*x

comment: Noll $j=30$, OSA/ANSI $j=32$, Fringe $j=23$, Wyant $j=22$.

-1

$Z_n^l(x,y)$:

35*x^6*y + 105*x^4*y^3 + 105*x^2*y^5 + 35*y^7 - 60*x^4*y - 120*x^2*y^3 - 60*y^5 + 30*x^2*y + 30*y^3 - 4*y

comment: Noll $j=29$, OSA/ANSI $j=31$, Fringe $j=24$, Wyant $j=23$.

$R_n^{|l|}(t)$:

21*t^7 - 30*t^5 + 10*t^3

comment: Radial part for $l=3$ (Noll $j=32$, OSA/ANSI $j=33$, Fringe $j=30$, Wyant $j=29$) and $l=-3$ (Noll $j=31$, OSA/ANSI $j=30$, Fringe $j=31$, Wyant $j=30$).

$Z_n^l(x,y)$:

21*x^7 - 21*x^5*y^2 - 105*x^3*y^4 - 63*x*y^6 - 30*x^5 + 60*x^3*y^2 + 90*x*y^4 + 10*x^3 - 30*x*y^2

comment: Noll $j=32$, OSA/ANSI $j=33$, Fringe $j=30$, Wyant $j=29$.

-3

$Z_n^l(x,y)$:

63*x^6*y + 105*x^4*y^3 + 21*x^2*y^5 - 21*y^7 - 90*x^4*y - 60*x^2*y^3 + 30*y^5 + 30*x^2*y - 10*y^3

comment: Noll $j=31$, OSA/ANSI $j=30$, Fringe $j=31$, Wyant $j=30$.

$R_n^{|l|}(t)$:

7*t^7 - 6*t^5

comment: Radial part for $l=5$ (Noll $j=34$, OSA/ANSI $j=34$, Fringe $j=39$, Wyant $j=38$) and $l=-5$ (Noll $j=33$, OSA/ANSI $j=29$, Fringe $j=40$, Wyant $j=39$).

$Z_n^l(x,y)$:

7*x^7 - 63*x^5*y^2 - 35*x^3*y^4 + 35*x*y^6 - 6*x^5 + 60*x^3*y^2 - 30*x*y^4

comment: Noll $j=34$, OSA/ANSI $j=34$, Fringe $j=39$, Wyant $j=38$.

-5

$Z_n^l(x,y)$:

35*x^6*y - 35*x^4*y^3 - 63*x^2*y^5 + 7*y^7 - 30*x^4*y + 60*x^2*y^3 - 6*y^5

comment: Noll $j=33$, OSA/ANSI $j=29$, Fringe $j=40$, Wyant $j=39$.

$Z_n^l(x,y)$:

x^7 - 21*x^5*y^2 + 35*x^3*y^4 - 7*x*y^6

comment: Noll $j=36$, OSA/ANSI $j=35$, Fringe $j=50$, Wyant $j=49$.

-7

$Z_n^l(x,y)$:

7*x^6*y - 35*x^4*y^3 + 21*x^2*y^5 - y^7

comment: Noll $j=35$, OSA/ANSI $j=28$, Fringe $j=51$, Wyant $j=50$.

$R_n^{|l|}(t)$:

70*t^8 - 140*t^6 + 90*t^4 - 20*t^2 + 1

comment: Radial part for Noll $j=37$, OSA/ANSI $j=40$, Fringe $j=25$, Wyant $j=24$.

$Z_n^l(x,y)$:

70*x^8 + 280*x^6*y^2 + 420*x^4*y^4 + 280*x^2*y^6 + 70*y^8 - 140*x^6 - 420*x^4*y^2 - 420*x^2*y^4 - 140*y^6 + 90*x^4 + 180*x^2*y^2 + 90*y^4 - 20*x^2 - 20*y^2 + 1

comment: Noll $j=37$, OSA/ANSI $j=40$, Fringe $j=25$, Wyant $j=24$.

$R_n^{|l|}(t)$:

56*t^8 - 105*t^6 + 60*t^4 - 10*t^2

comment: Radial part for $l=2$ (Noll $j=38$, OSA/ANSI $j=41$, Fringe $j=32$, Wyant $j=31$) and $l=-2$ (Noll $j=39$, OSA/ANSI $j=39$, Fringe $j=33$, Wyant $j=32$).

$Z_n^l(x,y)$:

56*x^8 + 112*x^6*y^2 - 112*x^2*y^6 - 56*y^8 - 105*x^6 - 105*x^4*y^2 + 105*x^2*y^4 + 105*y^6 + 60*x^4 - 60*y^4 - 10*x^2 + 10*y^2

comment: Noll $j=38$, OSA/ANSI $j=41$, Fringe $j=32$, Wyant $j=31$.

-2

$Z_n^l(x,y)$:

112*x^7*y + 336*x^5*y^3 + 336*x^3*y^5 + 112*x*y^7 - 210*x^5*y - 420*x^3*y^3 - 210*x*y^5 + 120*x^3*y + 120*x*y^3 - 20*x*y

comment: Noll $j=39$, OSA/ANSI $j=39$, Fringe $j=33$, Wyant $j=32$.

$R_n^{|l|}(t)$:

28*t^8 - 42*t^6 + 15*t^4

comment: Radial part for $l=4$ (Noll $j=40$, OSA/ANSI $j=42$, Fringe $j=41$, Wyant $j=40$) and $l=-4$ (Noll $j=41$, OSA/ANSI $j=38$, Fringe $j=42$, Wyant $j=41$).

$Z_n^l(x,y)$:

28*x^8 - 112*x^6*y^2 - 280*x^4*y^4 - 112*x^2*y^6 + 28*y^8 - 42*x^6 + 210*x^4*y^2 + 210*x^2*y^4 - 42*y^6 + 15*x^4 - 90*x^2*y^2 + 15*y^4

comment: Noll $j=40$, OSA/ANSI $j=42$, Fringe $j=41$, Wyant $j=40$.

-4

$Z_n^l(x,y)$:

112*x^7*y + 112*x^5*y^3 - 112*x^3*y^5 - 112*x*y^7 - 168*x^5*y + 168*x*y^5 + 60*x^3*y - 60*x*y^3

comment: Noll $j=41$, OSA/ANSI $j=38$, Fringe $j=42$, Wyant $j=41$.

$R_n^{|l|}(t)$:

8*t^8 - 7*t^6

comment: Radial part for $l=6$ (Noll $j=42$, OSA/ANSI $j=43$, Fringe $j=52$, Wyant $j=51$) and $l=-6$ (Noll $j=43$, OSA/ANSI $j=37$, Fringe $j=53$, Wyant $j=52$).

$Z_n^l(x,y)$:

8*x^8 - 112*x^6*y^2 + 112*x^2*y^6 - 8*y^8 - 7*x^6 + 105*x^4*y^2 - 105*x^2*y^4 + 7*y^6

comment: Noll $j=42$, OSA/ANSI $j=43$, Fringe $j=52$, Wyant $j=51$.

-6

$Z_n^l(x,y)$:

48*x^7*y - 112*x^5*y^3 - 112*x^3*y^5 + 48*x*y^7 - 42*x^5*y + 140*x^3*y^3 - 42*x*y^5

comment: Noll $j=43$, OSA/ANSI $j=37$, Fringe $j=53$, Wyant $j=52$.

$Z_n^l(x,y)$:

x^8 - 28*x^6*y^2 + 70*x^4*y^4 - 28*x^2*y^6 + y^8

comment: Noll $j=44$, OSA/ANSI $j=44$, Fringe $j=65$, Wyant $j=64$.

-8

$Z_n^l(x,y)$:

8*x^7*y - 56*x^5*y^3 + 56*x^3*y^5 - 8*x*y^7

comment: Noll $j=45$, OSA/ANSI $j=36$, Fringe $j=66$, Wyant $j=65$.

$R_n^{|l|}(t)$:

126*t^9 - 280*t^7 + 210*t^5 - 60*t^3 + 5*t

comment: Radial part for $l=1$ (Noll $j=46$, OSA/ANSI $j=50$, Fringe $j=34$, Wyant $j=33$) and $l=-1$ (Noll $j=47$, OSA/ANSI $j=49$, Fringe $j=35$, Wyant $j=34$).

$Z_n^l(x,y)$:

126*x^9 + 504*x^7*y^2 + 756*x^5*y^4 + 504*x^3*y^6 + 126*x*y^8 - 280*x^7 - 840*x^5*y^2 - 840*x^3*y^4 - 280*x*y^6 + 210*x^5 + 420*x^3*y^2 + 210*x*y^4 - 60*x^3 - 60*x*y^2 + 5*x

comment: Noll $j=46$, OSA/ANSI $j=50$, Fringe $j=34$, Wyant $j=33$.

-1

$Z_n^l(x,y)$:

126*x^8*y + 504*x^6*y^3 + 756*x^4*y^5 + 504*x^2*y^7 + 126*y^9 - 280*x^6*y - 840*x^4*y^3 - 840*x^2*y^5 - 280*y^7 + 210*x^4*y + 420*x^2*y^3 + 210*y^5 - 60*x^2*y - 60*y^3 + 5*y

comment: Noll $j=47$, OSA/ANSI $j=49$, Fringe $j=35$, Wyant $j=34$.

$R_n^{|l|}(t)$:

84*t^9 - 168*t^7 + 105*t^5 - 20*t^3

comment: Radial part for $l=3$ (Noll $j=48$, OSA/ANSI $j=51$, Fringe $j=43$, Wyant $j=42$) and $l=-3$ (Noll $j=49$, OSA/ANSI $j=48$, Fringe $j=44$, Wyant $j=43$).

$Z_n^l(x,y)$:

84*x^9 - 504*x^5*y^4 - 672*x^3*y^6 - 252*x*y^8 - 168*x^7 + 168*x^5*y^2 + 840*x^3*y^4 + 504*x*y^6 + 105*x^5 - 210*x^3*y^2 - 315*x*y^4 - 20*x^3 + 60*x*y^2

comment: Noll $j=48$, OSA/ANSI $j=51$, Fringe $j=43$, Wyant $j=42$.

-3

$Z_n^l(x,y)$:

252*x^8*y + 672*x^6*y^3 + 504*x^4*y^5 - 84*y^9 - 504*x^6*y - 840*x^4*y^3 - 168*x^2*y^5 + 168*y^7 + 315*x^4*y + 210*x^2*y^3 - 105*y^5 - 60*x^2*y + 20*y^3

comment: Noll $j=49$, OSA/ANSI $j=48$, Fringe $j=44$, Wyant $j=43$.

$R_n^{|l|}(t)$:

36*t^9 - 56*t^7 + 21*t^5

comment: Radial part for $l=5$ (Noll $j=50$, OSA/ANSI $j=52$, Fringe $j=54$, Wyant $j=53$) and $l=-5$ (Noll $j=51$, OSA/ANSI $j=47$, Fringe $j=55$, Wyant $j=54$).

$Z_n^l(x,y)$:

36*x^9 - 288*x^7*y^2 - 504*x^5*y^4 + 180*x*y^8 - 56*x^7 + 504*x^5*y^2 + 280*x^3*y^4 - 280*x*y^6 + 21*x^5 - 210*x^3*y^2 + 105*x*y^4

comment: Noll $j=50$, OSA/ANSI $j=52$, Fringe $j=54$, Wyant $j=53$.

-5

$Z_n^l(x,y)$:

180*x^8*y - 504*x^4*y^5 - 288*x^2*y^7 + 36*y^9 - 280*x^6*y + 280*x^4*y^3 + 504*x^2*y^5 - 56*y^7 + 105*x^4*y - 210*x^2*y^3 + 21*y^5

comment: Noll $j=51$, OSA/ANSI $j=47$, Fringe $j=55$, Wyant $j=54$.

$R_n^{|l|}(t)$:

9*t^9 - 8*t^7

comment: Radial part for $l=7$ (Noll $j=52$, OSA/ANSI $j=53$, Fringe $j=67$, Wyant $j=66$) and $l=-7$ (Noll $j=53$, OSA/ANSI $j=46$, Fringe $j=68$, Wyant $j=67$).

$Z_n^l(x,y)$:

9*x^9 - 180*x^7*y^2 + 126*x^5*y^4 + 252*x^3*y^6 - 63*x*y^8 - 8*x^7 + 168*x^5*y^2 - 280*x^3*y^4 + 56*x*y^6

comment: Noll $j=52$, OSA/ANSI $j=53$, Fringe $j=67$, Wyant $j=66$.

-7

$Z_n^l(x,y)$:

63*x^8*y - 252*x^6*y^3 - 126*x^4*y^5 + 180*x^2*y^7 - 9*y^9 - 56*x^6*y + 280*x^4*y^3 - 168*x^2*y^5 + 8*y^7

comment: Noll $j=53$, OSA/ANSI $j=46$, Fringe $j=68$, Wyant $j=67$.

$Z_n^l(x,y)$:

x^9 - 36*x^7*y^2 + 126*x^5*y^4 - 84*x^3*y^6 + 9*x*y^8

comment: Noll $j=54$, OSA/ANSI $j=54$, Fringe $j=82$, Wyant $j=81$.

-9

$Z_n^l(x,y)$:

9*x^8*y - 84*x^6*y^3 + 126*x^4*y^5 - 36*x^2*y^7 + y^9

comment: Noll $j=55$, OSA/ANSI $j=45$, Fringe $j=83$, Wyant $j=82$.

$R_n^{|l|}(t)$:

252*t^10 - 630*t^8 + 560*t^6 - 210*t^4 + 30*t^2 - 1

comment: Radial part for Noll $j=56$, OSA/ANSI $j=60$, Fringe $j=36$, Wyant $j=35$.

$Z_n^l(x,y)$:

252*x^10 + 1260*x^8*y^2 + 2520*x^6*y^4 + 2520*x^4*y^6 + 1260*x^2*y^8 + 252*y^10 - 630*x^8 - 2520*x^6*y^2 - 3780*x^4*y^4 - 2520*x^2*y^6 - 630*y^8 + 560*x^6 + 1680*x^4*y^2 + 1680*x^2*y^4 + 560*y^6 - 210*x^4 - 420*x^2*y^2 - 210*y^4 + 30*x^2 + 30*y^2 - 1

comment: Noll $j=56$, OSA/ANSI $j=60$, Fringe $j=36$, Wyant $j=35$.

$R_n^{|l|}(t)$:

210*t^10 - 504*t^8 + 420*t^6 - 140*t^4 + 15*t^2

comment: Radial part for $l=2$ (Noll $j=58$, OSA/ANSI $j=61$, Fringe $j=45$, Wyant $j=44$) and $l=-2$ (Noll $j=57$, OSA/ANSI $j=59$, Fringe $j=46$, Wyant $j=45$).

$Z_n^l(x,y)$:

210*x^10 + 630*x^8*y^2 + 420*x^6*y^4 - 420*x^4*y^6 - 630*x^2*y^8 - 210*y^10 - 504*x^8 - 1008*x^6*y^2 + 1008*x^2*y^6 + 504*y^8 + 420*x^6 + 420*x^4*y^2 - 420*x^2*y^4 - 420*y^6 - 140*x^4 + 140*y^4 + 15*x^2 - 15*y^2

comment: Noll $j=58$, OSA/ANSI $j=61$, Fringe $j=45$, Wyant $j=44$.

-2

$Z_n^l(x,y)$:

420*x^9*y + 1680*x^7*y^3 + 2520*x^5*y^5 + 1680*x^3*y^7 + 420*x*y^9 - 1008*x^7*y - 3024*x^5*y^3 - 3024*x^3*y^5 - 1008*x*y^7 + 840*x^5*y + 1680*x^3*y^3 + 840*x*y^5 - 280*x^3*y - 280*x*y^3 + 30*x*y

comment: Noll $j=57$, OSA/ANSI $j=59$, Fringe $j=46$, Wyant $j=45$.

$R_n^{|l|}(t)$:

120*t^10 - 252*t^8 + 168*t^6 - 35*t^4

comment: Radial part for $l=4$ (Noll $j=60$, OSA/ANSI $j=62$, Fringe $j=56$, Wyant $j=55$) and $l=-4$ (Noll $j=59$, OSA/ANSI $j=58$, Fringe $j=57$, Wyant $j=56$).

$Z_n^l(x,y)$:

120*x^10 - 360*x^8*y^2 - 1680*x^6*y^4 - 1680*x^4*y^6 - 360*x^2*y^8 + 120*y^10 - 252*x^8 + 1008*x^6*y^2 + 2520*x^4*y^4 + 1008*x^2*y^6 - 252*y^8 + 168*x^6 - 840*x^4*y^2 - 840*x^2*y^4 + 168*y^6 - 35*x^4 + 210*x^2*y^2 - 35*y^4

comment: Noll $j=60$, OSA/ANSI $j=62$, Fringe $j=56$, Wyant $j=55$.

-4

$Z_n^l(x,y)$:

480*x^9*y + 960*x^7*y^3 - 960*x^3*y^7 - 480*x*y^9 - 1008*x^7*y - 1008*x^5*y^3 + 1008*x^3*y^5 + 1008*x*y^7 + 672*x^5*y - 672*x*y^5 - 140*x^3*y + 140*x*y^3

comment: Noll $j=59$, OSA/ANSI $j=58$, Fringe $j=57$, Wyant $j=56$.

$R_n^{|l|}(t)$:

45*t^10 - 72*t^8 + 28*t^6

comment: Radial part for $l=6$ (Noll $j=62$, OSA/ANSI $j=63$, Fringe $j=69$, Wyant $j=68$) and $l=-6$ (Noll $j=61$, OSA/ANSI $j=57$, Fringe $j=70$, Wyant $j=69$).

$Z_n^l(x,y)$:

45*x^10 - 585*x^8*y^2 - 630*x^6*y^4 + 630*x^4*y^6 + 585*x^2*y^8 - 45*y^10 - 72*x^8 + 1008*x^6*y^2 - 1008*x^2*y^6 + 72*y^8 + 28*x^6 - 420*x^4*y^2 + 420*x^2*y^4 - 28*y^6

comment: Noll $j=62$, OSA/ANSI $j=63$, Fringe $j=69$, Wyant $j=68$.

-6

$Z_n^l(x,y)$:

270*x^9*y - 360*x^7*y^3 - 1260*x^5*y^5 - 360*x^3*y^7 + 270*x*y^9 - 432*x^7*y + 1008*x^5*y^3 + 1008*x^3*y^5 - 432*x*y^7 + 168*x^5*y - 560*x^3*y^3 + 168*x*y^5

comment: Noll $j=61$, OSA/ANSI $j=57$, Fringe $j=70$, Wyant $j=69$.

$R_n^{|l|}(t)$:

10*t^10 - 9*t^8

comment: Radial part for $l=8$ (Noll $j=64$, OSA/ANSI $j=64$, Fringe $j=84$, Wyant $j=83$) and $l=-8$ (Noll $j=63$, OSA/ANSI $j=56$, Fringe $j=85$, Wyant $j=84$).

$Z_n^l(x,y)$:

10*x^10 - 270*x^8*y^2 + 420*x^6*y^4 + 420*x^4*y^6 - 270*x^2*y^8 + 10*y^10 - 9*x^8 + 252*x^6*y^2 - 630*x^4*y^4 + 252*x^2*y^6 - 9*y^8

comment: Noll $j=64$, OSA/ANSI $j=64$, Fringe $j=84$, Wyant $j=83$.

-8

$Z_n^l(x,y)$:

80*x^9*y - 480*x^7*y^3 + 480*x^3*y^7 - 80*x*y^9 - 72*x^7*y + 504*x^5*y^3 - 504*x^3*y^5 + 72*x*y^7

comment: Noll $j=63$, OSA/ANSI $j=56$, Fringe $j=85$, Wyant $j=84$.

$Z_n^l(x,y)$:

x^10 - 45*x^8*y^2 + 210*x^6*y^4 - 210*x^4*y^6 + 45*x^2*y^8 - y^10

comment: Noll $j=66$, OSA/ANSI $j=65$, Fringe $j=101$, Wyant $j=100$.

-10

$Z_n^l(x,y)$:

10*x^9*y - 120*x^7*y^3 + 252*x^5*y^5 - 120*x^3*y^7 + 10*x*y^9

comment: Noll $j=65$, OSA/ANSI $j=55$, Fringe $j=102$, Wyant $j=101$.

$R_n^{|l|}(t)$:

462*t^11 - 1260*t^9 + 1260*t^7 - 560*t^5 + 105*t^3 - 6*t

comment: Radial part for $l=1$ (Noll $j=68$, OSA/ANSI $j=72$, Fringe $j=47$, Wyant $j=46$) and $l=-1$ (Noll $j=67$, OSA/ANSI $j=71$, Fringe $j=48$, Wyant $j=47$).

$Z_n^l(x,y)$:

462*x^11 + 2310*x^9*y^2 + 4620*x^7*y^4 + 4620*x^5*y^6 + 2310*x^3*y^8 + 462*x*y^10 - 1260*x^9 - 5040*x^7*y^2 - 7560*x^5*y^4 - 5040*x^3*y^6 - 1260*x*y^8 + 1260*x^7 + 3780*x^5*y^2 + 3780*x^3*y^4 + 1260*x*y^6 - 560*x^5 - 1120*x^3*y^2 - 560*x*y^4 + 105*x^3 + 105*x*y^2 - 6*x

comment: Noll $j=68$, OSA/ANSI $j=72$, Fringe $j=47$, Wyant $j=46$.

-1

$Z_n^l(x,y)$:

462*x^10*y + 2310*x^8*y^3 + 4620*x^6*y^5 + 4620*x^4*y^7 + 2310*x^2*y^9 + 462*y^11 - 1260*x^8*y - 5040*x^6*y^3 - 7560*x^4*y^5 - 5040*x^2*y^7 - 1260*y^9 + 1260*x^6*y + 3780*x^4*y^3 + 3780*x^2*y^5 + 1260*y^7 - 560*x^4*y - 1120*x^2*y^3 - 560*y^5 + 105*x^2*y + 105*y^3 - 6*y

comment: Noll $j=67$, OSA/ANSI $j=71$, Fringe $j=48$, Wyant $j=47$.

$R_n^{|l|}(t)$:

330*t^11 - 840*t^9 + 756*t^7 - 280*t^5 + 35*t^3

comment: Radial part for $l=3$ (Noll $j=70$, OSA/ANSI $j=73$, Fringe $j=58$, Wyant $j=57$) and $l=-3$ (Noll $j=69$, OSA/ANSI $j=70$, Fringe $j=59$, Wyant $j=58$).

$Z_n^l(x,y)$:

330*x^11 + 330*x^9*y^2 - 1980*x^7*y^4 - 4620*x^5*y^6 - 3630*x^3*y^8 - 990*x*y^10 - 840*x^9 + 5040*x^5*y^4 + 6720*x^3*y^6 + 2520*x*y^8 + 756*x^7 - 756*x^5*y^2 - 3780*x^3*y^4 - 2268*x*y^6 - 280*x^5 + 560*x^3*y^2 + 840*x*y^4 + 35*x^3 - 105*x*y^2

comment: Noll $j=70$, OSA/ANSI $j=73$, Fringe $j=58$, Wyant $j=57$.

-3

$Z_n^l(x,y)$:

990*x^10*y + 3630*x^8*y^3 + 4620*x^6*y^5 + 1980*x^4*y^7 - 330*x^2*y^9 - 330*y^11 - 2520*x^8*y - 6720*x^6*y^3 - 5040*x^4*y^5 + 840*y^9 + 2268*x^6*y + 3780*x^4*y^3 + 756*x^2*y^5 - 756*y^7 - 840*x^4*y - 560*x^2*y^3 + 280*y^5 + 105*x^2*y - 35*y^3

comment: Noll $j=69$, OSA/ANSI $j=70$, Fringe $j=59$, Wyant $j=58$.

$R_n^{|l|}(t)$:

165*t^11 - 360*t^9 + 252*t^7 - 56*t^5

comment: Radial part for $l=5$ (Noll $j=72$, OSA/ANSI $j=74$, Fringe $j=71$, Wyant $j=70$) and $l=-5$ (Noll $j=71$, OSA/ANSI $j=69$, Fringe $j=72$, Wyant $j=71$).

$Z_n^l(x,y)$:

165*x^11 - 1155*x^9*y^2 - 3630*x^7*y^4 - 2310*x^5*y^6 + 825*x^3*y^8 + 825*x*y^10 - 360*x^9 + 2880*x^7*y^2 + 5040*x^5*y^4 - 1800*x*y^8 + 252*x^7 - 2268*x^5*y^2 - 1260*x^3*y^4 + 1260*x*y^6 - 56*x^5 + 560*x^3*y^2 - 280*x*y^4

comment: Noll $j=72$, OSA/ANSI $j=74$, Fringe $j=71$, Wyant $j=70$.

-5

$Z_n^l(x,y)$:

825*x^10*y + 825*x^8*y^3 - 2310*x^6*y^5 - 3630*x^4*y^7 - 1155*x^2*y^9 + 165*y^11 - 1800*x^8*y + 5040*x^4*y^5 + 2880*x^2*y^7 - 360*y^9 + 1260*x^6*y - 1260*x^4*y^3 - 2268*x^2*y^5 + 252*y^7 - 280*x^4*y + 560*x^2*y^3 - 56*y^5

comment: Noll $j=71$, OSA/ANSI $j=69$, Fringe $j=72$, Wyant $j=71$.

$R_n^{|l|}(t)$:

55*t^11 - 90*t^9 + 36*t^7

comment: Radial part for $l=7$ (Noll $j=74$, OSA/ANSI $j=75$, Fringe $j=86$, Wyant $j=85$) and $l=-7$ (Noll $j=73$, OSA/ANSI $j=68$, Fringe $j=87$, Wyant $j=86$).

$Z_n^l(x,y)$:

55*x^11 - 1045*x^9*y^2 - 330*x^7*y^4 + 2310*x^5*y^6 + 1155*x^3*y^8 - 385*x*y^10 - 90*x^9 + 1800*x^7*y^2 - 1260*x^5*y^4 - 2520*x^3*y^6 + 630*x*y^8 + 36*x^7 - 756*x^5*y^2 + 1260*x^3*y^4 - 252*x*y^6

comment: Noll $j=74$, OSA/ANSI $j=75$, Fringe $j=86$, Wyant $j=85$.

-7

$Z_n^l(x,y)$:

385*x^10*y - 1155*x^8*y^3 - 2310*x^6*y^5 + 330*x^4*y^7 + 1045*x^2*y^9 - 55*y^11 - 630*x^8*y + 2520*x^6*y^3 + 1260*x^4*y^5 - 1800*x^2*y^7 + 90*y^9 + 252*x^6*y - 1260*x^4*y^3 + 756*x^2*y^5 - 36*y^7

comment: Noll $j=73$, OSA/ANSI $j=68$, Fringe $j=87$, Wyant $j=86$.

$R_n^{|l|}(t)$:

11*t^11 - 10*t^9

comment: Radial part for $l=9$ (Noll $j=76$, OSA/ANSI $j=76$, Fringe $j=103$, Wyant $j=102$) and $l=-9$ (Noll $j=75$, OSA/ANSI $j=67$, Fringe $j=104$, Wyant $j=103$).

$Z_n^l(x,y)$:

11*x^11 - 385*x^9*y^2 + 990*x^7*y^4 + 462*x^5*y^6 - 825*x^3*y^8 + 99*x*y^10 - 10*x^9 + 360*x^7*y^2 - 1260*x^5*y^4 + 840*x^3*y^6 - 90*x*y^8

comment: Noll $j=76$, OSA/ANSI $j=76$, Fringe $j=103$, Wyant $j=102$.

-9

$Z_n^l(x,y)$:

99*x^10*y - 825*x^8*y^3 + 462*x^6*y^5 + 990*x^4*y^7 - 385*x^2*y^9 + 11*y^11 - 90*x^8*y + 840*x^6*y^3 - 1260*x^4*y^5 + 360*x^2*y^7 - 10*y^9

comment: Noll $j=75$, OSA/ANSI $j=67$, Fringe $j=104$, Wyant $j=103$.

$Z_n^l(x,y)$:

x^11 - 55*x^9*y^2 + 330*x^7*y^4 - 462*x^5*y^6 + 165*x^3*y^8 - 11*x*y^10

comment: Noll $j=78$, OSA/ANSI $j=77$, Fringe $j=122$, Wyant $j=121$.

-11

$Z_n^l(x,y)$:

11*x^10*y - 165*x^8*y^3 + 462*x^6*y^5 - 330*x^4*y^7 + 55*x^2*y^9 - y^11

comment: Noll $j=77$, OSA/ANSI $j=66$, Fringe $j=123$, Wyant $j=122$.

$R_n^{|l|}(t)$:

924*t^12 - 2772*t^10 + 3150*t^8 - 1680*t^6 + 420*t^4 - 42*t^2 + 1

comment: Radial part for Noll $j=79$, OSA/ANSI $j=84$, Fringe $j=49$, Wyant $j=48$.

$Z_n^l(x,y)$:

924*x^12 + 5544*x^10*y^2 + 13860*x^8*y^4 + 18480*x^6*y^6 + 13860*x^4*y^8 + 5544*x^2*y^10 + 924*y^12 - 2772*x^10 - 13860*x^8*y^2 - 27720*x^6*y^4 - 27720*x^4*y^6 - 13860*x^2*y^8 - 2772*y^10 + 3150*x^8 + 12600*x^6*y^2 + 18900*x^4*y^4 + 12600*x^2*y^6 + 3150*y^8 - 1680*x^6 - 5040*x^4*y^2 - 5040*x^2*y^4 - 1680*y^6 + 420*x^4 + 840*x^2*y^2 + 420*y^4 - 42*x^2 - 42*y^2 + 1

comment: Noll $j=79$, OSA/ANSI $j=84$, Fringe $j=49$, Wyant $j=48$.

$R_n^{|l|}(t)$:

792*t^12 - 2310*t^10 + 2520*t^8 - 1260*t^6 + 280*t^4 - 21*t^2

comment: Radial part for $l=2$ (Noll $j=80$, OSA/ANSI $j=85$, Fringe $j=60$, Wyant $j=59$) and $l=-2$ (Noll $j=81$, OSA/ANSI $j=83$, Fringe $j=61$, Wyant $j=60$).

$Z_n^l(x,y)$:

792*x^12 + 3168*x^10*y^2 + 3960*x^8*y^4 - 3960*x^4*y^8 - 3168*x^2*y^10 - 792*y^12 - 2310*x^10 - 6930*x^8*y^2 - 4620*x^6*y^4 + 4620*x^4*y^6 + 6930*x^2*y^8 + 2310*y^10 + 2520*x^8 + 5040*x^6*y^2 - 5040*x^2*y^6 - 2520*y^8 - 1260*x^6 - 1260*x^4*y^2 + 1260*x^2*y^4 + 1260*y^6 + 280*x^4 - 280*y^4 - 21*x^2 + 21*y^2

comment: Noll $j=80$, OSA/ANSI $j=85$, Fringe $j=60$, Wyant $j=59$.

-2

$Z_n^l(x,y)$:

1584*x^11*y + 7920*x^9*y^3 + 15840*x^7*y^5 + 15840*x^5*y^7 + 7920*x^3*y^9 + 1584*x*y^11 - 4620*x^9*y - 18480*x^7*y^3 - 27720*x^5*y^5 - 18480*x^3*y^7 - 4620*x*y^9 + 5040*x^7*y + 15120*x^5*y^3 + 15120*x^3*y^5 + 5040*x*y^7 - 2520*x^5*y - 5040*x^3*y^3 - 2520*x*y^5 + 560*x^3*y + 560*x*y^3 - 42*x*y

comment: Noll $j=81$, OSA/ANSI $j=83$, Fringe $j=61$, Wyant $j=60$.

$R_n^{|l|}(t)$:

495*t^12 - 1320*t^10 + 1260*t^8 - 504*t^6 + 70*t^4

comment: Radial part for $l=4$ (Noll $j=82$, OSA/ANSI $j=86$, Fringe $j=73$, Wyant $j=72$) and $l=-4$ (Noll $j=83$, OSA/ANSI $j=82$, Fringe $j=74$, Wyant $j=73$).

$Z_n^l(x,y)$:

495*x^12 - 990*x^10*y^2 - 8415*x^8*y^4 - 13860*x^6*y^6 - 8415*x^4*y^8 - 990*x^2*y^10 + 495*y^12 - 1320*x^10 + 3960*x^8*y^2 + 18480*x^6*y^4 + 18480*x^4*y^6 + 3960*x^2*y^8 - 1320*y^10 + 1260*x^8 - 5040*x^6*y^2 - 12600*x^4*y^4 - 5040*x^2*y^6 + 1260*y^8 - 504*x^6 + 2520*x^4*y^2 + 2520*x^2*y^4 - 504*y^6 + 70*x^4 - 420*x^2*y^2 + 70*y^4

comment: Noll $j=82$, OSA/ANSI $j=86$, Fringe $j=73$, Wyant $j=72$.

-4

$Z_n^l(x,y)$:

1980*x^11*y + 5940*x^9*y^3 + 3960*x^7*y^5 - 3960*x^5*y^7 - 5940*x^3*y^9 - 1980*x*y^11 - 5280*x^9*y - 10560*x^7*y^3 + 10560*x^3*y^7 + 5280*x*y^9 + 5040*x^7*y + 5040*x^5*y^3 - 5040*x^3*y^5 - 5040*x*y^7 - 2016*x^5*y + 2016*x*y^5 + 280*x^3*y - 280*x*y^3

comment: Noll $j=83$, OSA/ANSI $j=82$, Fringe $j=74$, Wyant $j=73$.

$R_n^{|l|}(t)$:

220*t^12 - 495*t^10 + 360*t^8 - 84*t^6

comment: Radial part for $l=6$ (Noll $j=84$, OSA/ANSI $j=87$, Fringe $j=88$, Wyant $j=87$) and $l=-6$ (Noll $j=85$, OSA/ANSI $j=81$, Fringe $j=89$, Wyant $j=88$).

$Z_n^l(x,y)$:

220*x^12 - 2640*x^10*y^2 - 5940*x^8*y^4 + 5940*x^4*y^8 + 2640*x^2*y^10 - 220*y^12 - 495*x^10 + 6435*x^8*y^2 + 6930*x^6*y^4 - 6930*x^4*y^6 - 6435*x^2*y^8 + 495*y^10 + 360*x^8 - 5040*x^6*y^2 + 5040*x^2*y^6 - 360*y^8 - 84*x^6 + 1260*x^4*y^2 - 1260*x^2*y^4 + 84*y^6

comment: Noll $j=84$, OSA/ANSI $j=87$, Fringe $j=88$, Wyant $j=87$.

-6

$Z_n^l(x,y)$:

1320*x^11*y - 440*x^9*y^3 - 7920*x^7*y^5 - 7920*x^5*y^7 - 440*x^3*y^9 + 1320*x*y^11 - 2970*x^9*y + 3960*x^7*y^3 + 13860*x^5*y^5 + 3960*x^3*y^7 - 2970*x*y^9 + 2160*x^7*y - 5040*x^5*y^3 - 5040*x^3*y^5 + 2160*x*y^7 - 504*x^5*y + 1680*x^3*y^3 - 504*x*y^5

comment: Noll $j=85$, OSA/ANSI $j=81$, Fringe $j=89$, Wyant $j=88$.

$R_n^{|l|}(t)$:

66*t^12 - 110*t^10 + 45*t^8

comment: Radial part for $l=8$ (Noll $j=86$, OSA/ANSI $j=88$, Fringe $j=105$, Wyant $j=104$) and $l=-8$ (Noll $j=87$, OSA/ANSI $j=80$, Fringe $j=106$, Wyant $j=105$).

$Z_n^l(x,y)$:

66*x^12 - 1716*x^10*y^2 + 990*x^8*y^4 + 5544*x^6*y^6 + 990*x^4*y^8 - 1716*x^2*y^10 + 66*y^12 - 110*x^10 + 2970*x^8*y^2 - 4620*x^6*y^4 - 4620*x^4*y^6 + 2970*x^2*y^8 - 110*y^10 + 45*x^8 - 1260*x^6*y^2 + 3150*x^4*y^4 - 1260*x^2*y^6 + 45*y^8

comment: Noll $j=86$, OSA/ANSI $j=88$, Fringe $j=105$, Wyant $j=104$.

-8

$Z_n^l(x,y)$:

528*x^11*y - 2640*x^9*y^3 - 3168*x^7*y^5 + 3168*x^5*y^7 + 2640*x^3*y^9 - 528*x*y^11 - 880*x^9*y + 5280*x^7*y^3 - 5280*x^3*y^7 + 880*x*y^9 + 360*x^7*y - 2520*x^5*y^3 + 2520*x^3*y^5 - 360*x*y^7

comment: Noll $j=87$, OSA/ANSI $j=80$, Fringe $j=106$, Wyant $j=105$.

$R_n^{|l|}(t)$:

12*t^12 - 11*t^10

comment: Radial part for $l=10$ (Noll $j=88$, OSA/ANSI $j=89$, Fringe $j=124$, Wyant $j=123$) and $l=-10$ (Noll $j=89$, OSA/ANSI $j=79$, Fringe $j=125$, Wyant $j=124$).

$Z_n^l(x,y)$:

12*x^12 - 528*x^10*y^2 + 1980*x^8*y^4 - 1980*x^4*y^8 + 528*x^2*y^10 - 12*y^12 - 11*x^10 + 495*x^8*y^2 - 2310*x^6*y^4 + 2310*x^4*y^6 - 495*x^2*y^8 + 11*y^10

comment: Noll $j=88$, OSA/ANSI $j=89$, Fringe $j=124$, Wyant $j=123$.

-10

$Z_n^l(x,y)$:

120*x^11*y - 1320*x^9*y^3 + 1584*x^7*y^5 + 1584*x^5*y^7 - 1320*x^3*y^9 + 120*x*y^11 - 110*x^9*y + 1320*x^7*y^3 - 2772*x^5*y^5 + 1320*x^3*y^7 - 110*x*y^9

comment: Noll $j=89$, OSA/ANSI $j=79$, Fringe $j=125$, Wyant $j=124$.

$Z_n^l(x,y)$:

x^12 - 66*x^10*y^2 + 495*x^8*y^4 - 924*x^6*y^6 + 495*x^4*y^8 - 66*x^2*y^10 + y^12

comment: Noll $j=90$, OSA/ANSI $j=90$, Fringe $j=145$, Wyant $j=144$.

-12

$Z_n^l(x,y)$:

12*x^11*y - 220*x^9*y^3 + 792*x^7*y^5 - 792*x^5*y^7 + 220*x^3*y^9 - 12*x*y^11

comment: Noll $j=91$, OSA/ANSI $j=78$, Fringe $j=146$, Wyant $j=145$.

$R_n^{|l|}(t)$:

1716*t^13 - 5544*t^11 + 6930*t^9 - 4200*t^7 + 1260*t^5 - 168*t^3 + 7*t

comment: Radial part for $l=1$ (Noll $j=92$, OSA/ANSI $j=98$, Fringe $j=62$, Wyant $j=61$) and $l=-1$ (Noll $j=93$, OSA/ANSI $j=97$, Fringe $j=63$, Wyant $j=62$).

$R_n^{|l|}(t)$:

1287*t^13 - 3960*t^11 + 4620*t^9 - 2520*t^7 + 630*t^5 - 56*t^3

comment: Radial part for $l=3$ (Noll $j=94$, OSA/ANSI $j=99$, Fringe $j=75$, Wyant $j=74$) and $l=-3$ (Noll $j=95$, OSA/ANSI $j=96$, Fringe $j=76$, Wyant $j=75$).

$R_n^{|l|}(t)$:

715*t^13 - 1980*t^11 + 1980*t^9 - 840*t^7 + 126*t^5

comment: Radial part for $l=5$ (Noll $j=96$, OSA/ANSI $j=100$, Fringe $j=90$, Wyant $j=89$) and $l=-5$ (Noll $j=97$, OSA/ANSI $j=95$, Fringe $j=91$, Wyant $j=90$).

$R_n^{|l|}(t)$:

286*t^13 - 660*t^11 + 495*t^9 - 120*t^7

comment: Radial part for $l=7$ (Noll $j=98$, OSA/ANSI $j=101$, Fringe $j=107$, Wyant $j=106$) and $l=-7$ (Noll $j=99$, OSA/ANSI $j=94$, Fringe $j=108$, Wyant $j=107$).

$R_n^{|l|}(t)$:

78*t^13 - 132*t^11 + 55*t^9

comment: Radial part for $l=9$ (Noll $j=100$, OSA/ANSI $j=102$, Fringe $j=126$, Wyant $j=125$) and $l=-9$ (Noll $j=101$, OSA/ANSI $j=93$, Fringe $j=127$, Wyant $j=126$).

$R_n^{|l|}(t)$:

13*t^13 - 12*t^11

comment: Radial part for $l=11$ (Noll $j=102$, OSA/ANSI $j=103$, Fringe $j=147$, Wyant $j=146$) and $l=-11$ (Noll $j=103$, OSA/ANSI $j=92$, Fringe $j=148$, Wyant $j=147$).

$R_n^{|l|}(t)$:

3432*t^14 - 12012*t^12 + 16632*t^10 - 11550*t^8 + 4200*t^6 - 756*t^4 + 56*t^2 - 1

comment: Radial part for Noll $j=106$, OSA/ANSI $j=112$, Fringe $j=64$, Wyant $j=63$.

$R_n^{|l|}(t)$:

3003*t^14 - 10296*t^12 + 13860*t^10 - 9240*t^8 + 3150*t^6 - 504*t^4 + 28*t^2

comment: Radial part for $l=2$ (Noll $j=108$, OSA/ANSI $j=113$, Fringe $j=77$, Wyant $j=76$) and $l=-2$ (Noll $j=107$, OSA/ANSI $j=111$, Fringe $j=78$, Wyant $j=77$).

$R_n^{|l|}(t)$:

2002*t^14 - 6435*t^12 + 7920*t^10 - 4620*t^8 + 1260*t^6 - 126*t^4

comment: Radial part for $l=4$ (Noll $j=110$, OSA/ANSI $j=114$, Fringe $j=92$, Wyant $j=91$) and $l=-4$ (Noll $j=109$, OSA/ANSI $j=110$, Fringe $j=93$, Wyant $j=92$).

$R_n^{|l|}(t)$:

1001*t^14 - 2860*t^12 + 2970*t^10 - 1320*t^8 + 210*t^6

comment: Radial part for $l=6$ (Noll $j=112$, OSA/ANSI $j=115$, Fringe $j=109$, Wyant $j=108$) and $l=-6$ (Noll $j=111$, OSA/ANSI $j=109$, Fringe $j=110$, Wyant $j=109$).

$R_n^{|l|}(t)$:

364*t^14 - 858*t^12 + 660*t^10 - 165*t^8

comment: Radial part for $l=8$ (Noll $j=114$, OSA/ANSI $j=116$, Fringe $j=128$, Wyant $j=127$) and $l=-8$ (Noll $j=113$, OSA/ANSI $j=108$, Fringe $j=129$, Wyant $j=128$).

$R_n^{|l|}(t)$:

91*t^14 - 156*t^12 + 66*t^10

comment: Radial part for $l=10$ (Noll $j=116$, OSA/ANSI $j=117$, Fringe $j=149$, Wyant $j=148$) and $l=-10$ (Noll $j=115$, OSA/ANSI $j=107$, Fringe $j=150$, Wyant $j=149$).

$R_n^{|l|}(t)$:

14*t^14 - 13*t^12

comment: Radial part for $l=12$ (Noll $j=118$, OSA/ANSI $j=118$, Fringe $j=172$, Wyant $j=171$) and $l=-12$ (Noll $j=117$, OSA/ANSI $j=106$, Fringe $j=173$, Wyant $j=172$).

$R_n^{|l|}(t)$:

6435*t^15 - 24024*t^13 + 36036*t^11 - 27720*t^9 + 11550*t^7 - 2520*t^5 + 252*t^3 - 8*t

comment: Radial part for $l=1$ (Noll $j=122$, OSA/ANSI $j=128$, Fringe $j=79$, Wyant $j=78$) and $l=-1$ (Noll $j=121$, OSA/ANSI $j=127$, Fringe $j=80$, Wyant $j=79$).

$R_n^{|l|}(t)$:

5005*t^15 - 18018*t^13 + 25740*t^11 - 18480*t^9 + 6930*t^7 - 1260*t^5 + 84*t^3

comment: Radial part for $l=3$ (Noll $j=124$, OSA/ANSI $j=129$, Fringe $j=94$, Wyant $j=93$) and $l=-3$ (Noll $j=123$, OSA/ANSI $j=126$, Fringe $j=95$, Wyant $j=94$).

$R_n^{|l|}(t)$:

3003*t^15 - 10010*t^13 + 12870*t^11 - 7920*t^9 + 2310*t^7 - 252*t^5

comment: Radial part for $l=5$ (Noll $j=126$, OSA/ANSI $j=130$, Fringe $j=111$, Wyant $j=110$) and $l=-5$ (Noll $j=125$, OSA/ANSI $j=125$, Fringe $j=112$, Wyant $j=111$).

$R_n^{|l|}(t)$:

1365*t^15 - 4004*t^13 + 4290*t^11 - 1980*t^9 + 330*t^7

comment: Radial part for $l=7$ (Noll $j=128$, OSA/ANSI $j=131$, Fringe $j=130$, Wyant $j=129$) and $l=-7$ (Noll $j=127$, OSA/ANSI $j=124$, Fringe $j=131$, Wyant $j=130$).

$R_n^{|l|}(t)$:

455*t^15 - 1092*t^13 + 858*t^11 - 220*t^9

comment: Radial part for $l=9$ (Noll $j=130$, OSA/ANSI $j=132$, Fringe $j=151$, Wyant $j=150$) and $l=-9$ (Noll $j=129$, OSA/ANSI $j=123$, Fringe $j=152$, Wyant $j=151$).

$R_n^{|l|}(t)$:

105*t^15 - 182*t^13 + 78*t^11

comment: Radial part for $l=11$ (Noll $j=132$, OSA/ANSI $j=133$, Fringe $j=174$, Wyant $j=173$) and $l=-11$ (Noll $j=131$, OSA/ANSI $j=122$, Fringe $j=175$, Wyant $j=174$).

$R_n^{|l|}(t)$:

15*t^15 - 14*t^13

comment: Radial part for $l=13$ (Noll $j=134$, OSA/ANSI $j=134$, Fringe $j=199$, Wyant $j=198$) and $l=-13$ (Noll $j=133$, OSA/ANSI $j=121$, Fringe $j=200$, Wyant $j=199$).

$R_n^{|l|}(t)$:

12870*t^16 - 51480*t^14 + 84084*t^12 - 72072*t^10 + 34650*t^8 - 9240*t^6 + 1260*t^4 - 72*t^2 + 1

comment: Radial part for Noll $j=137$, OSA/ANSI $j=144$, Fringe $j=81$, Wyant $j=80$.

$R_n^{|l|}(t)$:

11440*t^16 - 45045*t^14 + 72072*t^12 - 60060*t^10 + 27720*t^8 - 6930*t^6 + 840*t^4 - 36*t^2

comment: Radial part for $l=2$ (Noll $j=138$, OSA/ANSI $j=145$, Fringe $j=96$, Wyant $j=95$) and $l=-2$ (Noll $j=139$, OSA/ANSI $j=143$, Fringe $j=97$, Wyant $j=96$).

$R_n^{|l|}(t)$:

8008*t^16 - 30030*t^14 + 45045*t^12 - 34320*t^10 + 13860*t^8 - 2772*t^6 + 210*t^4

comment: Radial part for $l=4$ (Noll $j=140$, OSA/ANSI $j=146$, Fringe $j=113$, Wyant $j=112$) and $l=-4$ (Noll $j=141$, OSA/ANSI $j=142$, Fringe $j=114$, Wyant $j=113$).

$R_n^{|l|}(t)$:

4368*t^16 - 15015*t^14 + 20020*t^12 - 12870*t^10 + 3960*t^8 - 462*t^6

comment: Radial part for $l=6$ (Noll $j=142$, OSA/ANSI $j=147$, Fringe $j=132$, Wyant $j=131$) and $l=-6$ (Noll $j=143$, OSA/ANSI $j=141$, Fringe $j=133$, Wyant $j=132$).

$R_n^{|l|}(t)$:

1820*t^16 - 5460*t^14 + 6006*t^12 - 2860*t^10 + 495*t^8

comment: Radial part for $l=8$ (Noll $j=144$, OSA/ANSI $j=148$, Fringe $j=153$, Wyant $j=152$) and $l=-8$ (Noll $j=145$, OSA/ANSI $j=140$, Fringe $j=154$, Wyant $j=153$).

$R_n^{|l|}(t)$:

560*t^16 - 1365*t^14 + 1092*t^12 - 286*t^10

comment: Radial part for $l=10$ (Noll $j=146$, OSA/ANSI $j=149$, Fringe $j=176$, Wyant $j=175$) and $l=-10$ (Noll $j=147$, OSA/ANSI $j=139$, Fringe $j=177$, Wyant $j=176$).

$R_n^{|l|}(t)$:

120*t^16 - 210*t^14 + 91*t^12

comment: Radial part for $l=12$ (Noll $j=148$, OSA/ANSI $j=150$, Fringe $j=201$, Wyant $j=200$) and $l=-12$ (Noll $j=149$, OSA/ANSI $j=138$, Fringe $j=202$, Wyant $j=201$).

$R_n^{|l|}(t)$:

16*t^16 - 15*t^14

comment: Radial part for $l=14$ (Noll $j=150$, OSA/ANSI $j=151$, Fringe $j=228$, Wyant $j=227$) and $l=-14$ (Noll $j=151$, OSA/ANSI $j=137$, Fringe $j=229$, Wyant $j=228$).

$R_n^{|l|}(t)$:

24310*t^17 - 102960*t^15 + 180180*t^13 - 168168*t^11 + 90090*t^9 - 27720*t^7 + 4620*t^5 - 360*t^3 + 9*t

comment: Radial part for $l=1$ (Noll $j=154$, OSA/ANSI $j=162$, Fringe $j=98$, Wyant $j=97$) and $l=-1$ (Noll $j=155$, OSA/ANSI $j=161$, Fringe $j=99$, Wyant $j=98$).

$R_n^{|l|}(t)$:

19448*t^17 - 80080*t^15 + 135135*t^13 - 120120*t^11 + 60060*t^9 - 16632*t^7 + 2310*t^5 - 120*t^3

comment: Radial part for $l=3$ (Noll $j=156$, OSA/ANSI $j=163$, Fringe $j=115$, Wyant $j=114$) and $l=-3$ (Noll $j=157$, OSA/ANSI $j=160$, Fringe $j=116$, Wyant $j=115$).

$R_n^{|l|}(t)$:

12376*t^17 - 48048*t^15 + 75075*t^13 - 60060*t^11 + 25740*t^9 - 5544*t^7 + 462*t^5

comment: Radial part for $l=5$ (Noll $j=158$, OSA/ANSI $j=164$, Fringe $j=134$, Wyant $j=133$) and $l=-5$ (Noll $j=159$, OSA/ANSI $j=159$, Fringe $j=135$, Wyant $j=134$).

$R_n^{|l|}(t)$:

6188*t^17 - 21840*t^15 + 30030*t^13 - 20020*t^11 + 6435*t^9 - 792*t^7

comment: Radial part for $l=7$ (Noll $j=160$, OSA/ANSI $j=165$, Fringe $j=155$, Wyant $j=154$) and $l=-7$ (Noll $j=161$, OSA/ANSI $j=158$, Fringe $j=156$, Wyant $j=155$).

$R_n^{|l|}(t)$:

2380*t^17 - 7280*t^15 + 8190*t^13 - 4004*t^11 + 715*t^9

comment: Radial part for $l=9$ (Noll $j=162$, OSA/ANSI $j=166$, Fringe $j=178$, Wyant $j=177$) and $l=-9$ (Noll $j=163$, OSA/ANSI $j=157$, Fringe $j=179$, Wyant $j=178$).

$R_n^{|l|}(t)$:

680*t^17 - 1680*t^15 + 1365*t^13 - 364*t^11

comment: Radial part for $l=11$ (Noll $j=164$, OSA/ANSI $j=167$, Fringe $j=203$, Wyant $j=202$) and $l=-11$ (Noll $j=165$, OSA/ANSI $j=156$, Fringe $j=204$, Wyant $j=203$).

$R_n^{|l|}(t)$:

136*t^17 - 240*t^15 + 105*t^13

comment: Radial part for $l=13$ (Noll $j=166$, OSA/ANSI $j=168$, Fringe $j=230$, Wyant $j=229$) and $l=-13$ (Noll $j=167$, OSA/ANSI $j=155$, Fringe $j=231$, Wyant $j=230$).

$R_n^{|l|}(t)$:

17*t^17 - 16*t^15

comment: Radial part for $l=15$ (Noll $j=168$, OSA/ANSI $j=169$, Fringe $j=259$, Wyant $j=258$) and $l=-15$ (Noll $j=169$, OSA/ANSI $j=154$, Fringe $j=260$, Wyant $j=259$).

$R_n^{|l|}(t)$:

48620*t^18 - 218790*t^16 + 411840*t^14 - 420420*t^12 + 252252*t^10 - 90090*t^8 + 18480*t^6 - 1980*t^4 + 90*t^2 - 1

comment: Radial part for Noll $j=172$, OSA/ANSI $j=180$, Fringe $j=100$, Wyant $j=99$.

$R_n^{|l|}(t)$:

43758*t^18 - 194480*t^16 + 360360*t^14 - 360360*t^12 + 210210*t^10 - 72072*t^8 + 13860*t^6 - 1320*t^4 + 45*t^2

comment: Radial part for $l=2$ (Noll $j=174$, OSA/ANSI $j=181$, Fringe $j=117$, Wyant $j=116$) and $l=-2$ (Noll $j=173$, OSA/ANSI $j=179$, Fringe $j=118$, Wyant $j=117$).

$R_n^{|l|}(t)$:

31824*t^18 - 136136*t^16 + 240240*t^14 - 225225*t^12 + 120120*t^10 - 36036*t^8 + 5544*t^6 - 330*t^4

comment: Radial part for $l=4$ (Noll $j=176$, OSA/ANSI $j=182$, Fringe $j=136$, Wyant $j=135$) and $l=-4$ (Noll $j=175$, OSA/ANSI $j=178$, Fringe $j=137$, Wyant $j=136$).

$R_n^{|l|}(t)$:

18564*t^18 - 74256*t^16 + 120120*t^14 - 100100*t^12 + 45045*t^10 - 10296*t^8 + 924*t^6

comment: Radial part for $l=6$ (Noll $j=178$, OSA/ANSI $j=183$, Fringe $j=157$, Wyant $j=156$) and $l=-6$ (Noll $j=177$, OSA/ANSI $j=177$, Fringe $j=158$, Wyant $j=157$).

$R_n^{|l|}(t)$:

8568*t^18 - 30940*t^16 + 43680*t^14 - 30030*t^12 + 10010*t^10 - 1287*t^8

comment: Radial part for $l=8$ (Noll $j=180$, OSA/ANSI $j=184$, Fringe $j=180$, Wyant $j=179$) and $l=-8$ (Noll $j=179$, OSA/ANSI $j=176$, Fringe $j=181$, Wyant $j=180$).

$R_n^{|l|}(t)$:

3060*t^18 - 9520*t^16 + 10920*t^14 - 5460*t^12 + 1001*t^10

comment: Radial part for $l=10$ (Noll $j=182$, OSA/ANSI $j=185$, Fringe $j=205$, Wyant $j=204$) and $l=-10$ (Noll $j=181$, OSA/ANSI $j=175$, Fringe $j=206$, Wyant $j=205$).

$R_n^{|l|}(t)$:

816*t^18 - 2040*t^16 + 1680*t^14 - 455*t^12

comment: Radial part for $l=12$ (Noll $j=184$, OSA/ANSI $j=186$, Fringe $j=232$, Wyant $j=231$) and $l=-12$ (Noll $j=183$, OSA/ANSI $j=174$, Fringe $j=233$, Wyant $j=232$).

$R_n^{|l|}(t)$:

153*t^18 - 272*t^16 + 120*t^14

comment: Radial part for $l=14$ (Noll $j=186$, OSA/ANSI $j=187$, Fringe $j=261$, Wyant $j=260$) and $l=-14$ (Noll $j=185$, OSA/ANSI $j=173$, Fringe $j=262$, Wyant $j=261$).

$R_n^{|l|}(t)$:

18*t^18 - 17*t^16

comment: Radial part for $l=16$ (Noll $j=188$, OSA/ANSI $j=188$, Fringe $j=292$, Wyant $j=291$) and $l=-16$ (Noll $j=187$, OSA/ANSI $j=172$, Fringe $j=293$, Wyant $j=292$).

$R_n^{|l|}(t)$:

92378*t^19 - 437580*t^17 + 875160*t^15 - 960960*t^13 + 630630*t^11 - 252252*t^9 + 60060*t^7 - 7920*t^5 + 495*t^3 - 10*t

comment: Radial part for $l=1$ (Noll $j=192$, OSA/ANSI $j=200$, Fringe $j=119$, Wyant $j=118$) and $l=-1$ (Noll $j=191$, OSA/ANSI $j=199$, Fringe $j=120$, Wyant $j=119$).

$R_n^{|l|}(t)$:

75582*t^19 - 350064*t^17 + 680680*t^15 - 720720*t^13 + 450450*t^11 - 168168*t^9 + 36036*t^7 - 3960*t^5 + 165*t^3

comment: Radial part for $l=3$ (Noll $j=194$, OSA/ANSI $j=201$, Fringe $j=138$, Wyant $j=137$) and $l=-3$ (Noll $j=193$, OSA/ANSI $j=198$, Fringe $j=139$, Wyant $j=138$).

$R_n^{|l|}(t)$:

50388*t^19 - 222768*t^17 + 408408*t^15 - 400400*t^13 + 225225*t^11 - 72072*t^9 + 12012*t^7 - 792*t^5

comment: Radial part for $l=5$ (Noll $j=196$, OSA/ANSI $j=202$, Fringe $j=159$, Wyant $j=158$) and $l=-5$ (Noll $j=195$, OSA/ANSI $j=197$, Fringe $j=160$, Wyant $j=159$).

$R_n^{|l|}(t)$:

27132*t^19 - 111384*t^17 + 185640*t^15 - 160160*t^13 + 75075*t^11 - 18018*t^9 + 1716*t^7

comment: Radial part for $l=7$ (Noll $j=198$, OSA/ANSI $j=203$, Fringe $j=182$, Wyant $j=181$) and $l=-7$ (Noll $j=197$, OSA/ANSI $j=196$, Fringe $j=183$, Wyant $j=182$).

$R_n^{|l|}(t)$:

11628*t^19 - 42840*t^17 + 61880*t^15 - 43680*t^13 + 15015*t^11 - 2002*t^9

comment: Radial part for $l=9$ (Noll $j=200$, OSA/ANSI $j=204$, Fringe $j=207$, Wyant $j=206$) and $l=-9$ (Noll $j=199$, OSA/ANSI $j=195$, Fringe $j=208$, Wyant $j=207$).

$R_n^{|l|}(t)$:

3876*t^19 - 12240*t^17 + 14280*t^15 - 7280*t^13 + 1365*t^11

comment: Radial part for $l=11$ (Noll $j=202$, OSA/ANSI $j=205$, Fringe $j=234$, Wyant $j=233$) and $l=-11$ (Noll $j=201$, OSA/ANSI $j=194$, Fringe $j=235$, Wyant $j=234$).

$R_n^{|l|}(t)$:

969*t^19 - 2448*t^17 + 2040*t^15 - 560*t^13

comment: Radial part for $l=13$ (Noll $j=204$, OSA/ANSI $j=206$, Fringe $j=263$, Wyant $j=262$) and $l=-13$ (Noll $j=203$, OSA/ANSI $j=193$, Fringe $j=264$, Wyant $j=263$).

$R_n^{|l|}(t)$:

171*t^19 - 306*t^17 + 136*t^15

comment: Radial part for $l=15$ (Noll $j=206$, OSA/ANSI $j=207$, Fringe $j=294$, Wyant $j=293$) and $l=-15$ (Noll $j=205$, OSA/ANSI $j=192$, Fringe $j=295$, Wyant $j=294$).

$R_n^{|l|}(t)$:

19*t^19 - 18*t^17

comment: Radial part for $l=17$ (Noll $j=208$, OSA/ANSI $j=208$, Fringe $j=327$, Wyant $j=326$) and $l=-17$ (Noll $j=207$, OSA/ANSI $j=191$, Fringe $j=328$, Wyant $j=327$).

$R_n^{|l|}(t)$:

184756*t^20 - 923780*t^18 + 1969110*t^16 - 2333760*t^14 + 1681680*t^12 - 756756*t^10 + 210210*t^8 - 34320*t^6 + 2970*t^4 - 110*t^2 + 1

comment: Radial part for Noll $j=211$, OSA/ANSI $j=220$, Fringe $j=121$, Wyant $j=120$.

$R_n^{|l|}(t)$:

167960*t^20 - 831402*t^18 + 1750320*t^16 - 2042040*t^14 + 1441440*t^12 - 630630*t^10 + 168168*t^8 - 25740*t^6 + 1980*t^4 - 55*t^2

comment: Radial part for $l=2$ (Noll $j=212$, OSA/ANSI $j=221$, Fringe $j=140$, Wyant $j=139$) and $l=-2$ (Noll $j=213$, OSA/ANSI $j=219$, Fringe $j=141$, Wyant $j=140$).

$R_n^{|l|}(t)$:

125970*t^20 - 604656*t^18 + 1225224*t^16 - 1361360*t^14 + 900900*t^12 - 360360*t^10 + 84084*t^8 - 10296*t^6 + 495*t^4

comment: Radial part for $l=4$ (Noll $j=214$, OSA/ANSI $j=222$, Fringe $j=161$, Wyant $j=160$) and $l=-4$ (Noll $j=215$, OSA/ANSI $j=218$, Fringe $j=162$, Wyant $j=161$).

$R_n^{|l|}(t)$:

77520*t^20 - 352716*t^18 + 668304*t^16 - 680680*t^14 + 400400*t^12 - 135135*t^10 + 24024*t^8 - 1716*t^6

comment: Radial part for $l=6$ (Noll $j=216$, OSA/ANSI $j=223$, Fringe $j=184$, Wyant $j=183$) and $l=-6$ (Noll $j=217$, OSA/ANSI $j=217$, Fringe $j=185$, Wyant $j=184$).

$R_n^{|l|}(t)$:

38760*t^20 - 162792*t^18 + 278460*t^16 - 247520*t^14 + 120120*t^12 - 30030*t^10 + 3003*t^8

comment: Radial part for $l=8$ (Noll $j=218$, OSA/ANSI $j=224$, Fringe $j=209$, Wyant $j=208$) and $l=-8$ (Noll $j=219$, OSA/ANSI $j=216$, Fringe $j=210$, Wyant $j=209$).

$R_n^{|l|}(t)$:

15504*t^20 - 58140*t^18 + 85680*t^16 - 61880*t^14 + 21840*t^12 - 3003*t^10

comment: Radial part for $l=10$ (Noll $j=220$, OSA/ANSI $j=225$, Fringe $j=236$, Wyant $j=235$) and $l=-10$ (Noll $j=221$, OSA/ANSI $j=215$, Fringe $j=237$, Wyant $j=236$).

$R_n^{|l|}(t)$:

4845*t^20 - 15504*t^18 + 18360*t^16 - 9520*t^14 + 1820*t^12

comment: Radial part for $l=12$ (Noll $j=222$, OSA/ANSI $j=226$, Fringe $j=265$, Wyant $j=264$) and $l=-12$ (Noll $j=223$, OSA/ANSI $j=214$, Fringe $j=266$, Wyant $j=265$).

$R_n^{|l|}(t)$:

1140*t^20 - 2907*t^18 + 2448*t^16 - 680*t^14

comment: Radial part for $l=14$ (Noll $j=224$, OSA/ANSI $j=227$, Fringe $j=296$, Wyant $j=295$) and $l=-14$ (Noll $j=225$, OSA/ANSI $j=213$, Fringe $j=297$, Wyant $j=296$).

$R_n^{|l|}(t)$:

190*t^20 - 342*t^18 + 153*t^16

comment: Radial part for $l=16$ (Noll $j=226$, OSA/ANSI $j=228$, Fringe $j=329$, Wyant $j=328$) and $l=-16$ (Noll $j=227$, OSA/ANSI $j=212$, Fringe $j=330$, Wyant $j=329$).

$R_n^{|l|}(t)$:

20*t^20 - 19*t^18

comment: Radial part for $l=18$ (Noll $j=228$, OSA/ANSI $j=229$, Fringe $j=364$, Wyant $j=363$) and $l=-18$ (Noll $j=229$, OSA/ANSI $j=211$, Fringe $j=365$, Wyant $j=364$).

Definition

The Zernike polynomials $Z_n^l$ are the orthogonal disk polynomials with radial part $R_n^{|l|}$ and a sine or cosine angular part [1].

Parameters

$n$

— radial degree ($n\geq0$)

$l$

— signed azimuthal degree ($|l|\leq n$ and $n-l$ is even)

form

— stored representation (either radial or cartesian)

Formulas

(1)

$Z_n^l(\rho,\varphi)=R_n^{|l|}(\rho)\cos(l\varphi)$ for $l\geq0$, and $Z_n^l(\rho,\varphi)=R_n^{|l|}(\rho)\sin(|l|\varphi)$ for $l<0$. The Cartesian entries use $x=\rho\cos\varphi$ and $y=\rho\sin\varphi$.

(2)

$R_n^m(\rho)=\sum_{k=0}^{(n-m)/2}(-1)^k \binom{n-k}{k}\binom{n-2k}{(n-m)/2-k}\rho^{n-2k}$ for $0\leq m\leq n$ and $n-m$ even [1].

(3)

If $R_n^m(\rho)=\sum_s c_s\rho^{m+2s}$, then $Z_n^l(x,y)=\sum_s c_s(x^2+y^2)^s\operatorname{Re}\left[(x+iy)^m\right]$ for $l=m>0$, $Z_n^l(x,y)=\sum_s c_s(x^2+y^2)^s\operatorname{Im}\left[(x+iy)^m\right]$ for $l=-m<0$, and $Z_n^0(x,y)=R_n^0(\sqrt{x^2+y^2})$.

(4)

$R_n^m(\rho)=(-1)^{(n-m)/2}\rho^m P_{(n-m)/2}^{(m,0)}(1-2\rho^2)$, where $P_k^{(\alpha,\beta)}$ is the Jacobi polynomial [1] [2]. In particular $R_{2k}^0(\rho)=P_k(2\rho^2-1)$ for the Legendre polynomials.

(5)

$\int_0^1 R_n^m(\rho)R_{n'}^m(\rho)\rho\,d\rho =\delta_{n,n'}/(2n+2)$, and $\int Z_n^l Z_{n'}^{l'}\,d^2r =\epsilon_l\pi\,\delta_{n,n'}\delta_{l,l'}/(2n+2)$ on the unit disk, with $\epsilon_0=2$ and $\epsilon_l=1$ for $l\ne0$ [1].

(6)

The Noll index is $j_N=n(n+1)/2+|l|+\epsilon$, where $\epsilon=0$ for $l>0$ and $n\equiv0,1\pmod4$, or for $l<0$ and $n\equiv2,3\pmod4$, and $\epsilon=1$ otherwise [1].

(7)

The OSA/ANSI index is $j_{\mathrm{OSA}}=(n(n+2)+l)/2$ [1].

(8)

The Fringe index is $j_F=(1+(n+|l|)/2)^2-2|l|+\lfloor(1-\operatorname{sgn}l)/2\rfloor$, with $\operatorname{sgn}0=0$; the Wyant index is $j_F-1$ [1].

(9)

$\int_0^1 R_n^m(\rho)J_m(v\rho)\rho\,d\rho =(-1)^{(n-m)/2}J_{n+1}(v)/v$ [2], relating the radial polynomials to the zeros of Bessel functions of the first kind.

Comments

(10)

The radial entries write the radial coordinate as $t$, and use only $l\geq0$, since $R_n^{|-l|}=R_n^l$. The cartesian entries use the signed upper index $l$: nonnegative $l$ gives the cosine mode and negative $l$ gives the sine mode. The angle $\varphi$ is measured anticlockwise from the positive $x$-axis.

(11)

The entries are unnormalised, scaled so that $R_n^m(1)=1$. The OSA/ANSI and Noll normalised modes multiply these polynomials by $\sqrt{n+1}$ when $l=0$ and by $\sqrt{2(n+1)}$ when $l\ne0$ [1].

(12)

The monomial rows are omitted: $R_n^n(t)=t^n$ for the radial form, and the Cartesian rows $Z_0^0=1$, $Z_1^1=x$ and $Z_1^{-1}=y$. The non-monomial edge modes such as $Z_2^2=x^2-y^2$ and $Z_2^{-2}=2xy$ are included.

(13)

Entry comments give the Noll, OSA/ANSI, Fringe and Wyant single indices for the corresponding Cartesian mode. For a radial entry with $l>0$ the comment gives both signs of $l$, because the same radial polynomial is shared by the sine and cosine modes.

(14)

Zernike polynomials are used for wavefront aberrations on circular pupils, Zernike moments in image analysis and related expansions on the unit disk [1] [2]. The non-monomial entries with $n\leq4$ carry the classical aberration name, in both forms; the table gives no aberration names for $n>4$.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.arith.misc import binomial
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

R = PolynomialRing(ZZ, 't')
t = R.gen()
XY = PolynomialRing(ZZ, ('x', 'y'))
x, y = XY.gens()

def radial_terms(n, m):
    return [((-1)**k * ZZ(binomial(n - k, k)) *
             ZZ(binomial(n - 2*k, (n - m)//2 - k)),
             n - 2*k)
            for k in range((n - m)//2 + 1)]

def radial(n, m):
    return sum(c * t**e for c, e in radial_terms(n, m))

def real_part_power(m):
    return sum((-1)**(j//2) * ZZ(binomial(m, j)) * x**(m - j) * y**j
               for j in range(0, m + 1, 2))

def imag_part_power(m):
    return sum((-1)**((j - 1)//2) * ZZ(binomial(m, j)) * x**(m - j) * y**j
               for j in range(1, m + 1, 2))

def cartesian(n, l):
    m = abs(l)
    if l > 0:
        angular = real_part_power(m)
    elif l < 0:
        angular = imag_part_power(m)
    else:
        angular = XY.one()
    r2 = x**2 + y**2
    return sum(c * r2**((e - m)//2) * angular for c, e in radial_terms(n, m))

radial(22, 0)                       # the next even radial row after this table
cartesian(13, 1)                    # the next odd Cartesian row after this table

Links

[1]

Wikipedia: Zernike polynomials

[2]

MathWorld: Zernike Polynomial

Similar tables

Jacobi polynomials — the radial polynomials are Jacobi polynomials after the change of variable in (4)

Legendre polynomials — the $m=0$ radial polynomials are Legendre polynomials evaluated at $2\rho^2-1$

Zeros of Bessel functions of the first kind — the Hankel transform in (9) has zeros at the positive zeros of $J_{n+1}$

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every non-monomial radial polynomial with $n\leq20$ and every non-monomial Cartesian polynomial with $n\leq12$, where the Cartesian rows stop sooner because they grow faster; $Z_{12}^0$ is 365 characters, while $R_{12}^0$ is 65 characters)

How they were obtained:

Every entry is exact, a polynomial with integer coefficients. The generator computes (2) over Sage's integer polynomial rings and forms the Cartesian entries with (3).

Before an entry is returned, the radial polynomial must agree with the Jacobi-polynomial identity (4); the radial polynomials must satisfy $R_n^m(1)=1$ and the exact orthogonality relation (5); and the single-index formulas must reproduce the four single indices tabulated for $n\leq4$ in [1].