Abstract

Zernike circle polynomials, their numbering scheme, and relationship to balanced optical aberrations of systems with circular pupils are discussed.

© 1994 Optical Society of America

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References

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  1. 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);“Beugungs-theorie des Schneidenverfahrens und Seiner Verbes-serten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
  2. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, The Netherlands, 1942).Also, Physica 23, 605–620 (1947.
  3. M. Born, E. Wolf, Principles of Optics, 5th ed. Pergamon, New York, N.Y. (1975), Chapter 9.
  4. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  5. Bill Swantner, “Comparison of Zernike polynomial sets in commercial software,” Opt & Phot. News 9 (1992), pp. 42–43.
  6. R. K. Tyson, “Using Zernike polynomials,” Opt. & Phot. News 12, 3 (1992).
  7. J. R. Rogers, “Zernike polynomials,” Opt. & Phot. News 8, 2–3 (1993).
  8. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85, 1408 (1981),J. Opt. Soc. Am.A1, 685 (1984).
    [CrossRef]
  9. V. N. Mahajan, “Uniform versus Gaussian beams: A comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A3, 470–485 (1986).
    [CrossRef]
  10. C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, Vol. X, pp. 193–221, R. R. Shannon, J. Wyant eds. Academic Press, New York, N.Y. (1987).
  11. S. Zhang, R. R. Shannon, “Catalog of spot diagrams,” in Applied Optics and Optical Engineering, Vol. XI, pp. 201–238, R. R. Shannon, J. Wyant eds. Academic Press, New York, N.Y. (1992).
  12. J. Wyant, K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, Vol. XI, pp. 1–53, R. R. Shannon, J. Wyant Eds. Academic Press, New York, N.Y. (1992).

1993

J. R. Rogers, “Zernike polynomials,” Opt. & Phot. News 8, 2–3 (1993).

1992

Bill Swantner, “Comparison of Zernike polynomial sets in commercial software,” Opt & Phot. News 9 (1992), pp. 42–43.

R. K. Tyson, “Using Zernike polynomials,” Opt. & Phot. News 12, 3 (1992).

1986

V. N. Mahajan, “Uniform versus Gaussian beams: A comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A3, 470–485 (1986).
[CrossRef]

1981

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85, 1408 (1981),J. Opt. Soc. Am.A1, 685 (1984).
[CrossRef]

1976

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);“Beugungs-theorie des Schneidenverfahrens und Seiner Verbes-serten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. Pergamon, New York, N.Y. (1975), Chapter 9.

Creath, K.

J. Wyant, K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, Vol. XI, pp. 1–53, R. R. Shannon, J. Wyant Eds. Academic Press, New York, N.Y. (1992).

Kim, C.-J.

C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, Vol. X, pp. 193–221, R. R. Shannon, J. Wyant eds. Academic Press, New York, N.Y. (1987).

Mahajan, V. N.

V. N. Mahajan, “Uniform versus Gaussian beams: A comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A3, 470–485 (1986).
[CrossRef]

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85, 1408 (1981),J. Opt. Soc. Am.A1, 685 (1984).
[CrossRef]

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, The Netherlands, 1942).Also, Physica 23, 605–620 (1947.

Noll, R. J.

Rogers, J. R.

J. R. Rogers, “Zernike polynomials,” Opt. & Phot. News 8, 2–3 (1993).

Shannon, R. R.

C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, Vol. X, pp. 193–221, R. R. Shannon, J. Wyant eds. Academic Press, New York, N.Y. (1987).

S. Zhang, R. R. Shannon, “Catalog of spot diagrams,” in Applied Optics and Optical Engineering, Vol. XI, pp. 201–238, R. R. Shannon, J. Wyant eds. Academic Press, New York, N.Y. (1992).

Swantner, Bill

Bill Swantner, “Comparison of Zernike polynomial sets in commercial software,” Opt & Phot. News 9 (1992), pp. 42–43.

Tyson, R. K.

R. K. Tyson, “Using Zernike polynomials,” Opt. & Phot. News 12, 3 (1992).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. Pergamon, New York, N.Y. (1975), Chapter 9.

Wyant, J.

J. Wyant, K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, Vol. XI, pp. 1–53, R. R. Shannon, J. Wyant Eds. Academic Press, New York, N.Y. (1992).

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);“Beugungs-theorie des Schneidenverfahrens und Seiner Verbes-serten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).

Zhang, S.

S. Zhang, R. R. Shannon, “Catalog of spot diagrams,” in Applied Optics and Optical Engineering, Vol. XI, pp. 201–238, R. R. Shannon, J. Wyant eds. Academic Press, New York, N.Y. (1992).

J. Opt. Soc. Am.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
[CrossRef]

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85, 1408 (1981),J. Opt. Soc. Am.A1, 685 (1984).
[CrossRef]

V. N. Mahajan, “Uniform versus Gaussian beams: A comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A3, 470–485 (1986).
[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);“Beugungs-theorie des Schneidenverfahrens und Seiner Verbes-serten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).

Opt & Phot. News

Bill Swantner, “Comparison of Zernike polynomial sets in commercial software,” Opt & Phot. News 9 (1992), pp. 42–43.

Opt. & Phot. News

R. K. Tyson, “Using Zernike polynomials,” Opt. & Phot. News 12, 3 (1992).

J. R. Rogers, “Zernike polynomials,” Opt. & Phot. News 8, 2–3 (1993).

Other

C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, Vol. X, pp. 193–221, R. R. Shannon, J. Wyant eds. Academic Press, New York, N.Y. (1987).

S. Zhang, R. R. Shannon, “Catalog of spot diagrams,” in Applied Optics and Optical Engineering, Vol. XI, pp. 201–238, R. R. Shannon, J. Wyant eds. Academic Press, New York, N.Y. (1992).

J. Wyant, K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, Vol. XI, pp. 1–53, R. R. Shannon, J. Wyant Eds. Academic Press, New York, N.Y. (1992).

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, The Netherlands, 1942).Also, Physica 23, 605–620 (1947.

M. Born, E. Wolf, Principles of Optics, 5th ed. Pergamon, New York, N.Y. (1975), Chapter 9.

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Tables (1)

Tables Icon

Table 1 Orthonormal Zernike circle polynomials Zj(ρ, θ). The indices j, n, and m are defined as the polynomial number, radial degree, and azimuthal frequency, respectively. The polynomials Zj are ordered such that even j corresponds to a symmetric polynomial defined by cosmθ, while odd j corresponds to an antisymmetric polynomial given by sin. For a given n, a polynomial with a lower value of m is ordered first, x = ρ cosθ, y = ρ sinθ, 0 ≤ ρ ≤ 1, 0 ≤ θ < 2π

Equations (47)

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W ( ρ , θ ) = n = 0 m = 0 n [ 2 ( n + 1 ) / ( 1 + δ m 0 ) ] 1 / 2 R n m ( ρ ) · ( c nm cos m θ + s nm sin m θ ) ,
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
R n m ( 0 ) = { δ m 0 , n / 2 even δ m 0 , n / 2 odd , R n m ( 1 ) = 1 , R n n ( ρ ) = ρ n .
0 1 R n m ( ρ ) R n m ( ρ ) ρ d ρ = 1 2 ( n + 1 ) δ n n ,
0 2 π cos m θ cos m θ d θ = π ( 1 + δ m 0 ) δ m m ,
0 2 π cos m θ sin m θ d θ = 0 ,
0 2 π sin m θ sin m θ d θ = π δ m m .
( c nm , s nm ) = ( 1 / π ) [ 2 ( n + 1 ) / ( 1 + δ m 0 ) ] 1 / 2 · 0 1 0 2 π W ( ρ , θ ) R n m ( ρ ) ( cos m θ , sin m θ ) ρ d ρ d θ ,
N n = ( n + 1 ) ( n + 2 ) / 2 .
W ( ρ , θ ) = j = 1 a j Z j ( ρ , θ ) ,
Z even j ( ρ , θ ) = 2 ( n + 1 ) R n m ( ρ ) cos m θ , m 0 ,
Z odd 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 .
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 .
a j = π 1 0 1 0 2 π W ( ρ , θ ) Z j ( ρ , θ ) ρ d ρ d θ .
W ( ρ , θ ) = 0 1 0 2 π W ( ρ , θ ) ρ d ρ d θ / 0 1 0 2 π ρ d ρ d θ = c 00
W 2 ( ρ , θ ) = 0 1 0 2 π W 2 ( ρ , θ ) ρ d ρ d θ / 0 1 0 2 π ρ d ρ d θ = n = 0 m = 0 n ( c nm 2 + s nm 2 ) ,
σ w 2 = W 2 ( ρ , θ ) W ( ρ , θ ) 2 = n = 0 m = 0 n ( c nm 2 + s nm 2 ) ,
W ( ρ , θ ) = a 1
W 2 ( ρ , θ ) = j = 1 a j 2 .
σ w 2 = j = 2 a j 2 .
N n = ( n + 2 ) ( n + 4 ) / 8 .
3 ( 2 ρ 2 1 )
6 ρ 2 sin 2 θ
6 ρ 2 cos 2 θ
8 ( 3 ρ 3 2 ρ ) sin θ
8 ( 3 ρ 3 2 ρ ) cos θ
8 ρ 3 sin 3 θ
8 ρ 3 cos 3 θ
5 ( 6 ρ 4 6 ρ 2 + 1 )
10 ( 4 ρ 4 3 ρ 2 ) cos 2 θ
10 ( 4 ρ 4 3 ρ 2 ) sin 2 θ
10 ρ 4 cos 4 θ
10 ρ 4 sin 4 θ
12 ( 10 ρ 5 12 ρ 3 + 3 ρ ) cos θ
12 ( 10 ρ 5 12 ρ 3 + 3 ρ ) sin θ
12 ( 5 ρ 5 4 ρ 3 ) cos 3 θ
12 ( 5 ρ 5 4 ρ 3 ) sin 3 θ
12 ρ 5 cos 5 θ
12 ρ 5 sin 5 θ
7 ( 20 ρ 6 30 ρ 4 + 12 ρ 2 1 )
14 ( 15 ρ 6 20 ρ 4 + 6 ρ 2 ) sin 2 θ
14 ( 15 ρ 6 20 ρ 4 + 6 ρ 2 ) cos 2 θ
14 ( 6 ρ 6 5 ρ 4 ) sin 4 θ
14 ( 6 ρ 6 5 ρ 4 ) cos 4 θ
14 ρ 6 sin 6 θ
14 ρ 6 cos 6 θ

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