Orthonormal polynomials in wavefront analysis: analytical solution: errata

Virendra N. Mahajan

doi:10.1364/JOSAA.29.001673

We have found errors in some polynomials and figures in our paper [1]. The correct polynomials are listed below:

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) = [\sqrt{6} / b^{2} \sqrt{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) = (\sqrt{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} = - \sqrt{5 / 8} b^{- 2} (195 - 475 b^{2} + 558 b^{4} - 422 b^{6} + 159 b^{8} - 15 b^{10}) β^{- 1} Z_{1} - \sqrt{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) \sqrt{15} b^{- 2} (75 - 155 b^{2} + 174 b^{4} - 134 b^{6} + 55 b^{8} - 15 b^{10}) β^{- 1} Z_{6} - 10 \sqrt{2} b^{- 2} (3 - 2 b^{2} + 2 b^{6} - 3 b^{8}) β^{- 1} Z_{11} + b^{- 2} α γ^{- 1} Z_{12},

where

β = α γ,

E_{14} = - (\sqrt{5 / 2} / 4) {(1 - b^{2})}^{2} b^{- 4} (35 - 10 b^{2} - b^{4}) γ^{- 1} Z_{1} + (5 \sqrt{15 / 2} / 8) {(1 - b^{2})}^{2} b^{- 4} (7 + 2 b^{2} - b^{4}) γ^{- 1} Z_{4} - \sqrt{15 / 8} b^{- 4} (35 - 70 b^{2} + 56 b^{4} - 26 b^{6} + 5 b^{8}) γ^{- 1} Z_{6} + (5 / 8 \sqrt{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} = - (\sqrt{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) {\begin{matrix} (105 - 550 a^{2} + 1559 a^{4} - 2836 a^{6} + 2695 a^{8} - 1078 a^{10}) Z_{1} \\ + 5 \sqrt{3} (14 - 74 a^{2} + 205 a^{4} - 360 a^{6} + 335 a^{8} - 134 a^{10}) Z_{4} \\ + (5 / 2) \sqrt{3 / 2} (35 - 156 a^{2} + 421 a^{4} - 530 a^{6} + 265 a^{8}) Z_{6} \\ + 21 \sqrt{5} (1 - 4 a^{2} + 6 a^{4} - 4 a^{6}) Z_{11} + [(7 / 2) \sqrt{5 / 2} η / (1 - a^{2})] Z_{12} \end{matrix}},

where

η = 9 - 45 a^{2} + 139 a^{4} - 237 a^{6} + 201 a^{8} - 67 a^{10} = (1 - a^{2}) μ^{2},

and

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} .

Note that 201 in the definition of $η$ was written incorrectly as 210. The polynomial corrections also apply to Chapter 11 on “Orthonormal polynomials in wavefront analysis” in the Handbook of Optics [2].

Equations (23) and (24) are in error. They should read as

c_{j, k} = - \frac{1}{4 a \sqrt{1 - a^{2}}} \int_{- a}^{a} d x \int_{- \sqrt{1 - a^{2}}}^{\sqrt{1 - a^{2}}} Z_{j} R_{k} d y

and

\frac{1}{4 a \sqrt{1 - a^{2}}} \int_{- a}^{a} d x \int_{- \sqrt{1 - a^{2}}}^{\sqrt{1 - a^{2}}} R_{j} R_{j^{'}} d y = δ_{j j^{'}} .

Section 8 should start with “Letting $a \to 1 \dots$ ”. There is a minor error in Table 16 in that the last item in the first column should be $σ_{b s}$ .

The interferograms and PSFs for the astigmatism and spherical aberration polynomials $E_{6}$ , $E_{11}$ , $R_{6}$ , and $R_{11}$ given in Fig. 12 are also in error. Their correct form is given here in Fig. 1. The second sentence of the caption of Fig. 5 should read “Half width of the square is $1 / \sqrt{2}$ .”

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.

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ACKNOWLEDGMENTS

The author gratefully acknowledges help from José A. Díaz, Robert W. Gray, and William H. Swantner for identifying the errors and helping with their corrections.

REFERENCES

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.

Orthonormal polynomials in wavefront analysis: analytical solution: errata

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Journal of the Optical Society of America A