Abstract

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.

© 2007 Optical Society of America

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Errata

Virendra N. Mahajan, "Orthonormal polynomials in wavefront analysis: analytical solution: errata," J. Opt. Soc. Am. A 29, 1673-1674 (2012)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-29-8-1673

References

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    [CrossRef]
  6. V. N. Mahajan, "Zernike polynomials and wavefront fitting," in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007) pp. 498-546.
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 1, 685 (1984).
    [CrossRef]
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    [CrossRef]
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2007

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

G.-m. Dai and V. N. Mahajan, "Nonrecursive orthonormal polynomials with matrix formulation," Opt. Lett. 32, 74-76 (2007).
[CrossRef]

2006

2004

M. Bray, "Orthogonal polynomials: a set for square areas," Proc. SPIE 5252, 314-320 (2004).
[CrossRef]

2003

V. N. Mahajan, "Zernike polynomials and aberration balancing," Proc. SPIE 5173, 1-17 (2003).
[CrossRef]

1996

1995

1994

1992

1986

1984

V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 1, 685 (1984).
[CrossRef]

1982

1981

1976

1969

H. Sumita, "Orthogonal expansion of the aberration difference function and its application to image evaluation," Jpn. J. Appl. Phys. 8, 1027-1036 (1969).
[CrossRef]

1968

1965

1934

F. Zernike, "Diffraction theory of knife-edge test and its improved form, the phase contrast method," Mon. Not. R. Astron. Soc. 94, 377-384 (1934).

Barakat, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

Brase, J. M.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Bray, M.

M. Bray, "Orthogonal polynomials: a set for square areas," Proc. SPIE 5252, 314-320 (2004).
[CrossRef]

Combs, R. L.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Dai, G.-m.

Fochs, S. N.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Harbers, G.

Hurd, R. L.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

King, W. B.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Kunst, P. J.

LaFortune, K. N.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Leibbrandt, G. W. R.

Mahajan, V. N.

Noll, R. J.

Olivier, S. S.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Pax, P. H.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Rayces, J. L.

Riseberg, L.

Rotter, M. D.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Sumita, H.

H. Sumita, "Orthogonal expansion of the aberration difference function and its application to image evaluation," Jpn. J. Appl. Phys. 8, 1027-1036 (1969).
[CrossRef]

Szapiel, S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

Yamamoto, R. M.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Zernike, F.

F. Zernike, "Diffraction theory of knife-edge test and its improved form, the phase contrast method," Mon. Not. R. Astron. Soc. 94, 377-384 (1934).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

H. Sumita, "Orthogonal expansion of the aberration difference function and its application to image evaluation," Jpn. J. Appl. Phys. 8, 1027-1036 (1969).
[CrossRef]

Mon. Not. R. Astron. Soc.

F. Zernike, "Diffraction theory of knife-edge test and its improved form, the phase contrast method," Mon. Not. R. Astron. Soc. 94, 377-384 (1934).

Opt. Lett.

Proc. SPIE

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

M. Bray, "Orthogonal polynomials: a set for square areas," Proc. SPIE 5252, 314-320 (2004).
[CrossRef]

V. N. Mahajan, "Zernike polynomials and aberration balancing," Proc. SPIE 5173, 1-17 (2003).
[CrossRef]

Other

V. N. Mahajan, "Zernike polynomials and wavefront fitting," in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007) pp. 498-546.
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics (SPIE, 2004).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

http://scikits.com/KFacts.html.

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Figures (12)

Fig. 1
Fig. 1

Unit hexagon inscribed inside a unit circle showing the coordinates of its corners. Each side of the hexagon has a length of unity. The x axis passes through the corners D and A of the hexagon, and the y axis bisects its parallel sides E F and C B .

Fig. 2
Fig. 2

Unit hexagon inscribed inside a unit circle rotated clockwise by 30 ° with respect to that in Fig. 1, showing the coordinates of its corners. Each side of the hexagon has a length of unity. The x axis bisects the parallel sides F A and D C of the hexagon, and, the y axis passes through its corners E and B .

Fig. 3
Fig. 3

Unit ellipse of aspect ratio b inscribed inside a unit circle. Its semimajor axis is unity along the x axis.

Fig. 4
Fig. 4

Unit rectangle inscribed inside a unit circle. Its corner points, such as A, lie at a distance of unity from its center.

Fig. 5
Fig. 5

Unit square inscribed inside a unit circle. Its corner points lie at a distance of unity from its center. Each side of the square has a length of 1 2 .

Fig. 6
Fig. 6

Unit slit pupil along the x axis inscribed inside a unit circle.

Fig. 7
Fig. 7

Variation of standard deviation of a primary or Seidel aberration as a function of aspect ratio b of a unit elliptical pupil. Subscript d, defocus; a, astigmatism; c, coma, s, spherical aberration.

Fig. 8
Fig. 8

Variation of standard deviation of a balanced primary aberration as a function of aspect ratio b of a unit elliptical pupil. Subscripts ba, balanced astigmatism, bc, balanced coma; and bs, balanced spherical aberration.

Fig. 9
Fig. 9

Variation of standard deviation of a primary or Seidel aberration as a function of half-width a of a unit rectangular pupil.

Fig. 10
Fig. 10

Variation of standard deviation of a balanced primary aberration as a function of half-width a of a unit rectangular pupil.

Fig. 11
Fig. 11

Aberration-free PSFs for unit pupils. (a) Hexagonal, (b) elliptical with b = 0.85 , (c) rectangular with a = 0.8 , (d) square.

Fig. 12
Fig. 12

Isometric plot, interferogram, and PSF for orthonormal polynomials with a sigma of one wave corresponding to a Seidel aberration. For example, H 4 , defocus; H 6 , astigmatism; H 8 , coma; H 11 , spherical aberration. (a) Hexagonal, (b) elliptical, (c) rectangular, (d) square polynomials.

Tables (17)

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Table 1 Relationship among the Zernike Circle Polynominal Indices n, m, and j

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Table 2 Orthonormal Hexagonal Polynomials in Terms of Zernike Circle Polynomials

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Table 3 Orthonormal Hexagonal Polynomials in Polar Coordinates

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Table 4 Orthonormal Hexagonal Polynomials in Cartesian Coordinates, Where ρ 2 = x 2 + y 2

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Table 5 Orthonormal Hexagonal Polynomials with 30° Rotation of the Hexagon, as in Fig. 2

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Table 6 Orthonormal Elliptical Polynomials in Terms of Zernike Circle Polynomials a

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Table 7 Orthonormal Elliptical Polynomials in Polar Coordinates a

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Table 8 Orthonormal Elliptical Polynomials in Cartesian Coordinates, Where ρ 2 = x 2 + y 2 a

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Table 9 Orthonormal Rectangular Polynomials in Terms of Zernike Circle Polynomials a

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Table 10 Orthonormal Rectangular Polynomials in Polar Coordinates a

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Table 11 Orthonormal Rectangular Polynomials in Cartesian Coordinates, Where ρ 2 = x 2 + y 2 a

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Table 12 Orthonormal Square Polynomials in Terms of Zernike Circle Polynomials

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Table 13 Orthonormal Square Polynomials in Polar Coordinates

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Table 14 Orthonormal Square Polynomials in Cartesian Coordinates, Where ρ 2 = x 2 + y 2

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Table 15 Orthonormal Polynomials for a Unit Slit Pupil

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Table 16 Standard Deviation or Sigma of a Primary and a Balanced Primary Aberration for Elliptical and Rectangular Pupils

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Table 17 Standard Deviation or Sigma of a Primary and a Balanced Primary Aberration for Circular, Hexagonal, Square, and Slit Pupils

Equations (38)

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Z e v e n j ( ρ , θ ) = 2 ( n + 1 ) R n m ( ρ ) cos m θ , m 0 ,
Z o d d j ( ρ , θ ) = 2 ( n + 1 ) R n m ( ρ ) sin m θ , m 0 ,
Z j ( ρ , θ ) = n + 1 R n 0 ( ρ ) , m = 0 ,
R n m ( ρ ) = s = 0 ( n m ) 2 ( 1 ) s ( n s ) ! s ! ( n + m 2 s ) ! ( n m 2 s ) ! ρ n 2 s ,
0 1 R n m ( ρ ) R n m ( ρ ) ρ d ρ = 1 2 ( n + 1 ) δ n n ,
0 1 0 2 π Z j ( ρ , θ ) Z j ( ρ , θ ) ρ d ρ d θ 0 1 0 2 π ρ d ρ d θ = δ j j .
N n = ( n + 1 ) ( n + 2 ) 2 .
W ( x , y ) = j a j F j ( x , y ) ,
1 A pupil F j ( x , y ) F j ( x , y ) d x d y = δ j j ,
a j = 1 A pupil W ( x , y ) F j ( x , y ) d x d y ,
σ 2 = j a j 2 , j 1 ,
G 1 = Z 1 = 1 ,
G j + 1 = k = 1 j c j + 1 , k F k + Z j + 1 ,
F j + 1 = G j + 1 G j + 1 = G j + 1 [ 1 A pupil G j + 1 2 d x d y ] 1 2 ,
c j + 1 , k = 1 A pupil Z j + 1 F k d x d y .
F l ( x , y ) = i = 1 l M l i Z i ( x , y ) with M l l = 1 G l .
M = ( Q T ) 1 ,
Q T Q = C .
C i j = pupil Z i Z j d x d y .
W ̂ ( x , y ) = j = 1 J b j Z j ( x , y ) ,
W ( x , y ) = j = 1 J a j i = 1 j M j i Z i ( x , y ) = j = 1 J i = j J a i M i j Z j ( x , y ) .
b j = i = j J a i M i j .
c j , k = 2 3 3 hexagon Z j H k d x d y .
2 3 3 hexagon H j H j d x d y = δ j j .
x 2 + y 2 b 2 = 1
x = ± 1 y 2 b 2 .
c j , k = 1 π b b b d y 1 y 2 b 2 1 y 2 b 2 Z j E k d x .
1 π b b b d y 1 y 2 b 2 1 y 2 b 2 E j E j d x = δ j j .
c j , k = 1 4 a 1 a 2 a a d y 1 a 2 1 a 2 Z j R k d x .
1 4 a 1 a 2 a a d y 1 a 2 1 a 2 R j R j d x d y = δ j j .
c j , k = 1 2 1 2 1 2 d y 1 2 1 2 Z j S k d x .
1 2 1 2 1 2 d y 1 2 1 2 S j S j d x = δ j j .
1 2 1 1 P j P j d x = δ i j .
E 6 = [ 6 ( 3 2 b 2 + 3 b 4 ) ] 1 2 b 2 [ ρ 2 cos 2 θ 3 b 2 ( 3 2 b 2 + 3 b 4 ) 1 2 ρ 2 ] + const.
W b a ( ρ , θ ) = ρ 2 cos 2 θ 3 b 2 ( 3 2 b 2 + 3 b 4 ) 1 2 ρ 2 .
σ b a = b 2 [ 6 ( 3 2 b 2 + 3 b 4 ) ] 1 2 .
W a ( ρ , θ ) = ρ 2 cos θ = b 2 [ 6 ( 3 2 b 2 + 3 b 4 ) ] 1 2 E 6 + 3 b 2 4 [ 3 ( 3 2 b 2 + 3 b 4 ) ] 1 2 E 4 + c E 1 ,
σ a = 1 4 .

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