Abstract

We address and correct errors that we found in the polynomials and figures in our paper [V. N. Mahajan and G.-m. Dai, J. Opt. Soc. Am. A 24, 2994 (2007) [CrossRef]  ].

© 2012 Optical Society of America

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References

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  1. V. N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
    [CrossRef]
  2. V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” Handbook of Optics, 3rd ed., V. N. Mahajan, ed., Vol. II (McGraw Hill, 2009), pp. 11.3–11.41.

2007

J. Opt. Soc. Am. A

Other

V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” Handbook of Optics, 3rd ed., V. N. Mahajan, ed., Vol. II (McGraw Hill, 2009), pp. 11.3–11.41.

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Figures (1)

Fig. 1.
Fig. 1.

Interferogram and PSF for polynomials E 6 (astigmatism), E 11 (spherical), R 6 (astigmatism), and R 11 (spherical) for a sigma value of one wavelength.

Equations (24)

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H 29 ( ρ , θ ) = ( 15.56917599 ρ + 130.07864353 ρ 3 291.15952742 ρ 5 + 190.97455178 ρ 7 ) sin θ + 1.41366362 ρ 5 sin 5 θ ,
H 30 ( ρ , θ ) = ( 15.56917599 ρ + 130.07864353 ρ 3 291.15952742 ρ 5 + 190.97455178 ρ 7 ) cos θ 1.41366362 ρ 5 cos 5 θ ,
H 33 ( ρ , θ ) = ( 3.87525156 ρ + 41.84243767 ρ 3 117.56342978 ρ 5 + 94.71450820 ρ 7 ) sin θ + ( 38.04631430 ρ 5 + 54.80141514 ρ 7 ) sin 5 θ ,
H 34 ( ρ , θ ) = ( 3.87525156 ρ 41.84243767 ρ 3 + 117.56342978 ρ 5 94.71450820 ρ 7 ) cos θ + ( 38.04631430 ρ 5 + 54.80141514 ρ 7 ) cos 5 θ ,
H 35 ( ρ , θ ) = ( 3.10311187 ρ 34.93479698 ρ 3 + 102.08124605 ρ 5 85.32630533 ρ 7 ) sin θ + ( 6.01202622 ρ 5 10.14399046 ρ 7 ) sin 5 θ + 8.97812952 ρ 7 sin 7 θ ,
H 36 ( ρ , θ ) = ( 3.10311187 ρ 34.93479698 ρ 3 + 102.08124605 ρ 5 85.32630533 ρ 7 ) cos θ + ( 6.01202622 ρ 5 + 10.14399046 ρ 7 ) cos 5 θ + 8.97812952 ρ 7 cos 7 θ ,
H 38 ( ρ , θ ) = ( 42.96232789 + 287.78381063 ρ 2 565.13651608 ρ 4 + 339.98298180 ρ 6 ) ρ 2 cos 2 θ + ( 8.49786414 13.58537785 ρ 2 ) ρ 4 cos 4 θ ,
H 39 ( ρ , θ ) = ( 42.96232789 + 287.78381063 ρ 2 565.13651608 ρ 4 + 339.98298180 ρ 6 ) ρ 2 sin 2 θ ( 8.49786414 13.58537785 ρ 2 ) ρ 4 sin 4 θ ,
H 20 ( x , y ) = ( 2.17600247 + 13.23551876 ρ 2 + 13.64110699 ρ 4 ) x 119.18577680 ρ 2 x 3 + 95.34862128 x 5 ,
H 21 ( x , y ) = ( 2.17600247 13.23551876 ρ 2 + 45.95178131 ρ 4 ) y 119.18577680 ρ 2 y 3 + 95.34862128 y 5 ,
H 29 ( x , y ) = ( 15.56917599 + 130.07864353 ρ 2 284.09120931 ρ 4 + 190.97455178 ρ 6 ) y 28.2732724 ρ 2 y 3 + 22.61861792 y 5 ,
H 30 ( x , y ) = ( 15.56917599 + 130.07864353 ρ 2 298.22784553 ρ 4 + 190.97455178 ρ 6 ) x + 28.27327243 ρ 2 x 3 22.61861792 x 5 ,
H 21 ( 30 ° ) = 0.71499593 Z 3 0.72488884 Z 7 0.46636441 Z 17 + 1.72029850 Z 21 ,
E 6 ( x , y ) = [ 6 / b 2 3 2 b 2 + 3 b 4 ] [ b 2 ( 1 b 2 ) + b 2 ( 3 b 2 1 ) x 2 ( 3 b 2 ) y 2 ] ,
E 11 ( ρ , θ ) = ( 5 / α ) [ 3 + 2 b 2 + 3 b 4 24 ( 1 + b 2 ) ρ 2 + 48 ρ 4 12 ( 1 b 2 ) ρ 2 cos 2 θ ] ,
E 12 = 5 / 8 b 2 ( 195 475 b 2 + 558 b 4 422 b 6 + 159 b 8 15 b 10 ) β 1 Z 1 15 / 8 b 2 ( 105 205 b 2 + 194 b 4 114 b 6 + 5 b 8 + 15 b 10 ) β 1 Z 4 + ( 1 / 2 ) 15 b 2 ( 75 155 b 2 + 174 b 4 134 b 6 + 55 b 8 15 b 10 ) β 1 Z 6 10 2 b 2 ( 3 2 b 2 + 2 b 6 3 b 8 ) β 1 Z 11 + b 2 α γ 1 Z 12 ,
β = α γ ,
E 14 = ( 5 / 2 / 4 ) ( 1 b 2 ) 2 b 4 ( 35 10 b 2 b 4 ) γ 1 Z 1 + ( 5 15 / 2 / 8 ) ( 1 b 2 ) 2 b 4 ( 7 + 2 b 2 b 4 ) γ 1 Z 4 15 / 8 b 4 ( 35 70 b 2 + 56 b 4 26 b 6 + 5 b 8 ) γ 1 Z 6 + ( 5 / 8 2 ) ( 1 b 2 ) 2 b 4 ( 7 + 10 b 2 + 7 b 4 ) γ 1 Z 11 ( 5 / 8 ) b 4 ( 7 6 b 2 + 6 b 6 7 b 8 ) γ 1 Z 12 + ( γ / 8 ) b 4 Z 14 ,
E 15 = ( 15 / 4 ) b 3 ( 5 8 b 2 + 3 b 4 ) δ 1 Z 5 ( 5 / 4 ) ( 1 b 4 ) b 3 δ 1 Z 13 + b 3 ( δ / 2 ) Z 15 ,
R 12 = ( 3 μ / 16 a 2 ν η ) { ( 105 550 a 2 + 1559 a 4 2836 a 6 + 2695 a 8 1078 a 10 ) Z 1 + 5 3 ( 14 74 a 2 + 205 a 4 360 a 6 + 335 a 8 134 a 10 ) Z 4 + ( 5 / 2 ) 3 / 2 ( 35 156 a 2 + 421 a 4 530 a 6 + 265 a 8 ) Z 6 + 21 5 ( 1 4 a 2 + 6 a 4 4 a 6 ) Z 11 + [ ( 7 / 2 ) 5 / 2 η / ( 1 a 2 ) ] Z 12 } ,
η = 9 45 a 2 + 139 a 4 237 a 6 + 201 a 8 67 a 10 = ( 1 a 2 ) μ 2 ,
S 37 ( x , y ) = 2.34475558 55.32128002 ρ 2 + 283.78448194 ρ 4 532.71123567 ρ 6 + 332.94452229 ρ 8 + 8 ( 12.75329096 ρ 2 20.75498320 ρ 4 ) x 2 + 8 ( 12.75329096 + 20.75498320 ρ 2 ) x 4 .
c j , k = 1 4 a 1 a 2 a a d x 1 a 2 1 a 2 Z j R k d y
1 4 a 1 a 2 a a d x 1 a 2 1 a 2 R j R j d y = δ j j .

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