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

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

H29(ρ,θ)=(15.56917599ρ+130.07864353ρ3291.15952742ρ5+190.97455178ρ7)sinθ+1.41366362ρ5sin5θ,
H30(ρ,θ)=(15.56917599ρ+130.07864353ρ3291.15952742ρ5+190.97455178ρ7)cosθ1.41366362ρ5cos5θ,
H33(ρ,θ)=(3.87525156ρ+41.84243767ρ3117.56342978ρ5+94.71450820ρ7)sinθ+(38.04631430ρ5+54.80141514ρ7)sin5θ,
H34(ρ,θ)=(3.87525156ρ41.84243767ρ3+117.56342978ρ594.71450820ρ7)cosθ+(38.04631430ρ5+54.80141514ρ7)cos5θ,
H35(ρ,θ)=(3.10311187ρ34.93479698ρ3+102.08124605ρ585.32630533ρ7)sinθ+(6.01202622ρ510.14399046ρ7)sin5θ+8.97812952ρ7sin7θ,
H36(ρ,θ)=(3.10311187ρ34.93479698ρ3+102.08124605ρ585.32630533ρ7)cosθ+(6.01202622ρ5+10.14399046ρ7)cos5θ+8.97812952ρ7cos7θ,
H38(ρ,θ)=(42.96232789+287.78381063ρ2565.13651608ρ4+339.98298180ρ6)ρ2cos2θ+(8.4978641413.58537785ρ2)ρ4cos4θ,
H39(ρ,θ)=(42.96232789+287.78381063ρ2565.13651608ρ4+339.98298180ρ6)ρ2sin2θ(8.4978641413.58537785ρ2)ρ4sin4θ,
H20(x,y)=(2.17600247+13.23551876ρ2+13.64110699ρ4)x119.18577680ρ2x3+95.34862128x5,
H21(x,y)=(2.1760024713.23551876ρ2+45.95178131ρ4)y119.18577680ρ2y3+95.34862128y5,
H29(x,y)=(15.56917599+130.07864353ρ2284.09120931ρ4+190.97455178ρ6)y28.2732724ρ2y3+22.61861792y5,
H30(x,y)=(15.56917599+130.07864353ρ2298.22784553ρ4+190.97455178ρ6)x+28.27327243ρ2x322.61861792x5,
H21(30°)=0.71499593Z30.72488884Z70.46636441Z17+1.72029850Z21,
E6(x,y)=[6/b232b2+3b4][b2(1b2)+b2(3b21)x2(3b2)y2],
E11(ρ,θ)=(5/α)[3+2b2+3b424(1+b2)ρ2+48ρ412(1b2)ρ2cos2θ],
E12=5/8b2(195475b2+558b4422b6+159b815b10)β1Z115/8b2(105205b2+194b4114b6+5b8+15b10)β1Z4+(1/2)15b2(75155b2+174b4134b6+55b815b10)β1Z6102b2(32b2+2b63b8)β1Z11+b2αγ1Z12,
where
β=αγ,
E14=(5/2/4)(1b2)2b4(3510b2b4)γ1Z1+(515/2/8)(1b2)2b4(7+2b2b4)γ1Z415/8b4(3570b2+56b426b6+5b8)γ1Z6+(5/82)(1b2)2b4(7+10b2+7b4)γ1Z11(5/8)b4(76b2+6b67b8)γ1Z12+(γ/8)b4Z14,
E15=(15/4)b3(58b2+3b4)δ1Z5(5/4)(1b4)b3δ1Z13+b3(δ/2)Z15,
R12=(3μ/16a2νη){(105550a2+1559a42836a6+2695a81078a10)Z1+53(1474a2+205a4360a6+335a8134a10)Z4+(5/2)3/2(35156a2+421a4530a6+265a8)Z6+215(14a2+6a44a6)Z11+[(7/2)5/2η/(1a2)]Z12},
where
η=945a2+139a4237a6+201a867a10=(1a2)μ2,
and
S37(x,y)=2.3447555855.32128002ρ2+283.78448194ρ4532.71123567ρ6+332.94452229ρ8+8(12.75329096ρ220.75498320ρ4)x2+8(12.75329096+20.75498320ρ2)x4.

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

cj,k=14a1a2aadx1a21a2ZjRkdy
and
14a1a2aadx1a21a2RjRjdy=δjj.

Section 8 should start with “Letting a1”. There is a minor error in Table 16 in that the last item in the first column should be σbs.

The interferograms and PSFs for the astigmatism and spherical aberration polynomials E6, E11, R6, and R11 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 1/2.”

 

Fig. 1. Interferogram and PSF for polynomials E6 (astigmatism), E11 (spherical), R6 (astigmatism), and R11 (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.

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 (1)

J. Opt. Soc. Am. A (1)

Other (1)

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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