Abstract

Various authors have presented the aberration function of an optical system as a power series expansion with respect to the ray coordinates in the exit pupil and the coordinates of the intersection point with the image field of the optical system. In practical applications, for reasons of efficiency and accuracy, an expansion with the aid of orthogonal polynomials is preferred for which, since the 1980s, orthogonal Zernike polynomials have become the reference. In the literature, some conversion schemes of power series coefficients to coefficients for the corresponding Zernike polynomial expansion have been given. In this paper we present an analytic solution for the conversion problem from a power series expansion in three or four dimensions to a double Zernike polynomial expansion. The solution pertains to a general optical system with four independent pupil and field coordinates and to a system with rotational symmetry in which case three independent coordinate combinations have to be considered. The conversion of the coefficients is analytically in closed form and the result is independent of a specific sampling scheme or sampling density as this is the case for the commonly used least squares fitting techniques. Computation schemes are given that allow the evaluation of coefficients of arbitrarily high order in pupil and field coordinates.

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  1. L. Seidel, “Über die Entwicklung der Glieder 3ter Ordnung welche den Weg eines ausserhalb der Ebene der Axe gelegene Lichtstrahles durch ein System brechender Medien bestimmen,” Astr. Nach. 43, 289–304 (1856).
  2. K. Schwarzschild, “Untersuchungen zur geometrischen Optik, I-II,” Abh. Königl. Ges. Wiss. Göttingen, Math. Phys. Kl, Neue Folge 4, 1–54 (1905).
  3. F. Wachendorf, “Bestimmung der Bildfehler fünfter Ordnung in zentrierten optischen Systemen,” Optik (Jena) 5, 80–122 (1949).
  4. H. A. Buchdahl, Aberrations of Optical Systems (Dover Publications, 1968).
  5. M. R. Rimmer, “Optical aberration coefficients,” Ph.D. thesis (University of Rochester, 1963).
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    [CrossRef]
  7. F. Bociort, T. B. Andersen, and L. H. J. F. Beckmann, “High-order optical aberration coefficients: extension to finite objects and to telecentricity in object space,” Appl. Opt. 47, 5691–5700 (2008).
    [CrossRef]
  8. J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method, Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge University, 1937).
  9. T. Smith, “The changes in aberrations when the object and stop are moved,” Trans. Opt. Soc. 23, 139–153 (1922).
    [CrossRef]
  10. G. C. Steward, “Aberration diffraction effects,” Phil. Trans. R. Soc. A 225, 131–198 (1926).
    [CrossRef]
  11. C. Carathéodory, “Geometrische Optik,” in Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer-Verlag, 1937), Vol. 4, no. 5.
  12. C. H. F. Velzel and J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. I. Transverse aberrations when the eikonal is given,” J. Opt. Soc. Am. A 5, 246–250 (1988).
    [CrossRef]
  13. C. H. F. Velzel and J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. II. Addition of aberrations,” J. Opt. Soc. Am. A 5, 251–256 (1988).
    [CrossRef]
  14. C. H. F. Velzel and J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. III. Calculation of eikonal coefficients,” J. Opt. Soc. Am. A 5, 1237–1243 (1988).
    [CrossRef]
  15. H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, 1950).
  16. A. E. Conrady, Applied Optics and Optical Design [Dover Publications, Part I (1957), Part II (1960)].
  17. J. Sasián, “Theory of sixth-order wave aberrations,” Appl. Opt. 49, D69–D95 (2010).
    [CrossRef]
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    [CrossRef]
  19. G. Conforti, “Zernike aberration coefficients from Seidel and higher-order power-series coefficients,” Opt. Lett. 8, 407–408 (1983).
    [CrossRef]
  20. I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2, 21–32 (1993).
    [CrossRef]
  21. I. Agurok, “Double expansion of wavefront deformation in Zernike polynomials over the pupil and field-of-view of optical systems: lens design, testing, and alignment,” Proc. SPIE 3430, 80–87 (1998).
    [CrossRef]
  22. T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev. 11, 199–207 (2004).
    [CrossRef]
  23. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. thesis (The University of Arizona, 1980).
  24. I. Agurok, “Aberrations of perturbed and unobscured optical systems,” Proc. SPIE 3779, 166–177 (1999).
    [CrossRef]
  25. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389–1401 (2005).
    [CrossRef]
  26. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26, 1090–1100 (2009).
    [CrossRef]
  27. T. Matsuzawa, “Image field distribution model of wavefront aberration and models of distortion and field curvature,” J. Opt. Soc. Am. A 28, 96–110 (2011).
    [CrossRef]
  28. R. W. Gray, C. Dunn, K. P. Thompson, and J. P. Rolland, “An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems,” Opt. Express 20, 16436–16449 (2012).
    [CrossRef]
  29. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (Rijksuniversiteit Groningen, 1942), J. B. Wolters, Groningen, downloadable from www.nijboerzernike.nl .
  30. A. J. E. M. Janssen and P. Dirksen, “Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform,” J. Eur. Opt. Soc. Rapid Pub. 2, 07012 (2007).
    [CrossRef]
  31. V. N. Mahajan, Optical Imaging and Aberrations: Part I, Ray Geometrical Optics, (SPIE Press, 1998), p. 156.
  32. J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, “Extended Nijboer–Zernike representation of the field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
    [CrossRef]
  33. M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge University, 1999), pp. 909–910.
  34. A. J. E. M. Janssen, “Extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002).
    [CrossRef]
  35. J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
    [CrossRef]

2012 (1)

2011 (1)

2010 (1)

2009 (1)

2008 (1)

2007 (1)

A. J. E. M. Janssen and P. Dirksen, “Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform,” J. Eur. Opt. Soc. Rapid Pub. 2, 07012 (2007).
[CrossRef]

2005 (1)

2004 (1)

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev. 11, 199–207 (2004).
[CrossRef]

2003 (1)

2002 (2)

1999 (1)

I. Agurok, “Aberrations of perturbed and unobscured optical systems,” Proc. SPIE 3779, 166–177 (1999).
[CrossRef]

1998 (1)

I. Agurok, “Double expansion of wavefront deformation in Zernike polynomials over the pupil and field-of-view of optical systems: lens design, testing, and alignment,” Proc. SPIE 3430, 80–87 (1998).
[CrossRef]

1993 (1)

I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2, 21–32 (1993).
[CrossRef]

1988 (3)

1983 (1)

1982 (1)

1980 (1)

1949 (1)

F. Wachendorf, “Bestimmung der Bildfehler fünfter Ordnung in zentrierten optischen Systemen,” Optik (Jena) 5, 80–122 (1949).

1926 (1)

G. C. Steward, “Aberration diffraction effects,” Phil. Trans. R. Soc. A 225, 131–198 (1926).
[CrossRef]

1922 (1)

T. Smith, “The changes in aberrations when the object and stop are moved,” Trans. Opt. Soc. 23, 139–153 (1922).
[CrossRef]

1905 (1)

K. Schwarzschild, “Untersuchungen zur geometrischen Optik, I-II,” Abh. Königl. Ges. Wiss. Göttingen, Math. Phys. Kl, Neue Folge 4, 1–54 (1905).

1856 (1)

L. Seidel, “Über die Entwicklung der Glieder 3ter Ordnung welche den Weg eines ausserhalb der Ebene der Axe gelegene Lichtstrahles durch ein System brechender Medien bestimmen,” Astr. Nach. 43, 289–304 (1856).

Agurok, I.

I. Agurok, “Aberrations of perturbed and unobscured optical systems,” Proc. SPIE 3779, 166–177 (1999).
[CrossRef]

I. Agurok, “Double expansion of wavefront deformation in Zernike polynomials over the pupil and field-of-view of optical systems: lens design, testing, and alignment,” Proc. SPIE 3430, 80–87 (1998).
[CrossRef]

Andersen, T. B.

Beckmann, L. H. J. F.

Bociort, F.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge University, 1999), pp. 909–910.

Braat, J. J. M.

Buchdahl, H. A.

H. A. Buchdahl, Aberrations of Optical Systems (Dover Publications, 1968).

Carathéodory, C.

C. Carathéodory, “Geometrische Optik,” in Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer-Verlag, 1937), Vol. 4, no. 5.

Conforti, G.

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design [Dover Publications, Part I (1957), Part II (1960)].

de Meijere, J. L. F.

Dirksen, P.

Dunn, C.

Gray, R. W.

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, 1950).

Janssen, A. J. E. M.

Kwee, I. W.

I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2, 21–32 (1993).
[CrossRef]

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations: Part I, Ray Geometrical Optics, (SPIE Press, 1998), p. 156.

Matsuyama, T.

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev. 11, 199–207 (2004).
[CrossRef]

Matsuzawa, T.

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (Rijksuniversiteit Groningen, 1942), J. B. Wolters, Groningen, downloadable from www.nijboerzernike.nl .

Rimmer, M. R.

M. R. Rimmer, “Optical aberration coefficients,” Ph.D. thesis (University of Rochester, 1963).

Rolland, J. P.

Sasián, J.

Schwarzschild, K.

K. Schwarzschild, “Untersuchungen zur geometrischen Optik, I-II,” Abh. Königl. Ges. Wiss. Göttingen, Math. Phys. Kl, Neue Folge 4, 1–54 (1905).

Seidel, L.

L. Seidel, “Über die Entwicklung der Glieder 3ter Ordnung welche den Weg eines ausserhalb der Ebene der Axe gelegene Lichtstrahles durch ein System brechender Medien bestimmen,” Astr. Nach. 43, 289–304 (1856).

Smith, T.

T. Smith, “The changes in aberrations when the object and stop are moved,” Trans. Opt. Soc. 23, 139–153 (1922).
[CrossRef]

Steward, G. C.

G. C. Steward, “Aberration diffraction effects,” Phil. Trans. R. Soc. A 225, 131–198 (1926).
[CrossRef]

Synge, J. L.

J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method, Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge University, 1937).

Thompson, K. P.

Tyson, R. K.

Ujike, T.

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev. 11, 199–207 (2004).
[CrossRef]

van de Nes, A. S.

Velzel, C. H. F.

Wachendorf, F.

F. Wachendorf, “Bestimmung der Bildfehler fünfter Ordnung in zentrierten optischen Systemen,” Optik (Jena) 5, 80–122 (1949).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge University, 1999), pp. 909–910.

Abh. Königl. Ges. Wiss. Göttingen, Math. Phys. Kl, Neue Folge (1)

K. Schwarzschild, “Untersuchungen zur geometrischen Optik, I-II,” Abh. Königl. Ges. Wiss. Göttingen, Math. Phys. Kl, Neue Folge 4, 1–54 (1905).

Appl. Opt. (3)

Astr. Nach. (1)

L. Seidel, “Über die Entwicklung der Glieder 3ter Ordnung welche den Weg eines ausserhalb der Ebene der Axe gelegene Lichtstrahles durch ein System brechender Medien bestimmen,” Astr. Nach. 43, 289–304 (1856).

J. Eur. Opt. Soc. Rapid Pub. (1)

A. J. E. M. Janssen and P. Dirksen, “Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform,” J. Eur. Opt. Soc. Rapid Pub. 2, 07012 (2007).
[CrossRef]

J. Opt. Soc. Am. A (9)

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26, 1090–1100 (2009).
[CrossRef]

T. Matsuzawa, “Image field distribution model of wavefront aberration and models of distortion and field curvature,” J. Opt. Soc. Am. A 28, 96–110 (2011).
[CrossRef]

A. J. E. M. Janssen, “Extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002).
[CrossRef]

J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
[CrossRef]

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, “Extended Nijboer–Zernike representation of the field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
[CrossRef]

K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389–1401 (2005).
[CrossRef]

C. H. F. Velzel and J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. I. Transverse aberrations when the eikonal is given,” J. Opt. Soc. Am. A 5, 246–250 (1988).
[CrossRef]

C. H. F. Velzel and J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. II. Addition of aberrations,” J. Opt. Soc. Am. A 5, 251–256 (1988).
[CrossRef]

C. H. F. Velzel and J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. III. Calculation of eikonal coefficients,” J. Opt. Soc. Am. A 5, 1237–1243 (1988).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Opt. Rev. (1)

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev. 11, 199–207 (2004).
[CrossRef]

Optik (Jena) (1)

F. Wachendorf, “Bestimmung der Bildfehler fünfter Ordnung in zentrierten optischen Systemen,” Optik (Jena) 5, 80–122 (1949).

Phil. Trans. R. Soc. A (1)

G. C. Steward, “Aberration diffraction effects,” Phil. Trans. R. Soc. A 225, 131–198 (1926).
[CrossRef]

Proc. SPIE (2)

I. Agurok, “Double expansion of wavefront deformation in Zernike polynomials over the pupil and field-of-view of optical systems: lens design, testing, and alignment,” Proc. SPIE 3430, 80–87 (1998).
[CrossRef]

I. Agurok, “Aberrations of perturbed and unobscured optical systems,” Proc. SPIE 3779, 166–177 (1999).
[CrossRef]

Pure Appl. Opt. (1)

I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2, 21–32 (1993).
[CrossRef]

Trans. Opt. Soc. (1)

T. Smith, “The changes in aberrations when the object and stop are moved,” Trans. Opt. Soc. 23, 139–153 (1922).
[CrossRef]

Other (10)

C. Carathéodory, “Geometrische Optik,” in Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer-Verlag, 1937), Vol. 4, no. 5.

H. A. Buchdahl, Aberrations of Optical Systems (Dover Publications, 1968).

M. R. Rimmer, “Optical aberration coefficients,” Ph.D. thesis (University of Rochester, 1963).

J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method, Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge University, 1937).

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. thesis (The University of Arizona, 1980).

H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, 1950).

A. E. Conrady, Applied Optics and Optical Design [Dover Publications, Part I (1957), Part II (1960)].

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (Rijksuniversiteit Groningen, 1942), J. B. Wolters, Groningen, downloadable from www.nijboerzernike.nl .

V. N. Mahajan, Optical Imaging and Aberrations: Part I, Ray Geometrical Optics, (SPIE Press, 1998), p. 156.

M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge University, 1999), pp. 909–910.

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Figures (3)

Fig. 1.
Fig. 1.

Ray propagation from the exit pupil plane to the image plane. An aberrated aperture ray intersects the reference sphere through E1 in the exit pupil in the point P with coordinates (XP,YP,ZP(XP,YP)). The reference ray (dashed in the figure) intersects the reference sphere in Q. The position of the perfect image point is A1. The aberrated ray intersects the pupil plane in the point P1(X,Y) and the image plane in the point A1, with coordinates (xA+δxA,yA+δyA). The distance from the center O1 of the image plane to the center E1 of the exit pupil is R1, negative in the figure. PA1=E1A1 is the radius of the reference sphere S, centered on A1, that is associated with the particular oblique imaging pencil issued from an object point A0 (not shown in the figure).

Fig. 2.
Fig. 2.

Real part of the exponential function f(ρ,θ). The parameter values are u=2.5 and v=1.2. (a) R{f(X,Y)} at the rim of the unit circle (ρ=1 and 0θ<2π). (b) Solid curve: log|R{ffp40}|10 on the unit circle rim; dashed curve: log|R{fp40fpZ40}|10; dot–dashed curve: log|R{ffZ40}|10.

Fig. 3.
Fig. 3.

Residual rms errors δ1 and δ2 for the representation of the exponential test function [u=2.5, v=1.2 in Eq. (27)] according to Eqs. (33) and (34), respectively, as a function of NNmax=30, 50, and 70 in (a), (b), and (c), respectively. Furthermore, the residual rms error δ3 for the representation of the same test function according to Eq. (36) with N=Nmax and N1=0,1,,Nmax.

Equations (58)

Equations on this page are rendered with MathJax. Learn more.

W(X,Y,x,y)=[A0P][A0Q].
W(X,Y;x,y)=anmlkXnYmxlyk=anmlkρP1n+mrfl+kcosnθsinmθcoslϕsinkϕ
W{ρP12,rf2,ρP1rfcos(θϕ)}=anml(ρP12)n(rf2)l[rfρP1cos(θϕ)]m.
Nns=N(N+1)(N+5)6,Nrs=N(N+6)8
f(ρ,θ)=n,mcnmRn|m|(ρ)exp(imθ)
02π01|Znm(ρ,θ)|2ρdρdθ=πn+1.
cnm=n+1π02π01f(ρ,θ)Znm*(ρ,θ)ρdρdθ.
ac=(cnm+cn,m),as=+i(cnmcn,m).
W(ρ,r;θ,ϕ)=n1,n2;m1,m2cn1n2m1m2Rn1|m1|(ρ)Rn2|m2|(r)exp[i(m1θ+m2ϕ)]
Inlmn1n2m1m2=010102π02πρ2n+mr2l+mcosm(θϕ)Rn1|m1|(ρ)Rn2|m2|(r)×exp[i(m1θ+m2ϕ)]ρrdρdrdθdϕ.
cosm(θϕ)=12mj=0m(mj)exp[i(m2j)(θϕ)].
02π02πcosm(θϕ)exp[i(m1θ+m2ϕ)]dθdϕ=12mj=0m(mj)02π02πexp[+i(mm12j)θ]exp[i(m+m22j)ϕ)]dθdϕ=4π22mj=0m(mj)δmm12jδm+m22j
{4π22m(mm|m1|2)m1=m2;m|m1,2|even and nonnegative,0otherwise.
Inlmn1n2m1m2=4π22m(mm|m1|2)01ρ2n+mRn1|m1|(ρ)ρdρ01r2l+mRn2|m1|(r)rdr.
Jnmn1m1=01ρ2n+mRn1|m1|(ρ)ρdρ=12(n+m|m1|2)!(n+m+|m1|2)!(n+m|m1|2p1)!(n+m+|m1|2+p1+1)!,
ρ2n+mr2l+mcosm(θϕ)=n1=02n+mn2=02l+mm1=mmbnmln1n2m1Rn1|m1|(ρ)Rn2|m1|(r)exp[im1(θϕ)]
bnmln1n2m1=4(n1+1)(n2+1)2m(mm|m1|2)Jnmn1m1Jlmn2m1=(n1+1)(n2+1)2m(mm|m1|2)×(n+m|m1|2)!(n+m+|m1|2)!(n+m|m1|2p1)!(n+m+|m1|2+p1+1)!(l+m|m1|2)!(l+m+|m1|2)!(l+m|m1|2p2)!(l+m+|m1|2+p2+1)!.
nlmanlmρ2n+mr2l+mcosm(θϕ)=n1=02Np+Mpn2=02Lp+Mpm1=MpMp{nlmanlmbnmln1n2m1}Rn1|m1|(ρ)Rn2|m1|(r)exp[im1(θϕ)]=n1n2m1cn1n2m1Rn1|m1|(ρ)Rn2|m1|(r)exp[im1(θϕ)]
Inlmn1n2m1m2=010102π02πρn+mrl+kcosnθsinmθcoslϕsinkϕRn1|m1|(ρ)Rn2|m2|(r)exp[i(m1θ+m2ϕ)]ρrdρdrdθdϕ.
Inmm1=02πcosnθsinmθexp(im1θ)dθ=12n+mimj1=0nj2=0m(nj1)(mj2)(1)j202πexp[i(n2j1)θ+i(m2j2)θ]exp(im1θ)dθ=2π2n+mimj1=0nj2=0m(nj1)(mj2)(1)j2δn+mm12j12j2.
Inmm1=2π2n+mimj(1)j(mj)(nn+mm12j)
Knmn1m1=01ρn+mRn1|m1|(ρ)ρdρ=12(n+m|m1|2)!(n+m+|m1|2)!(n+m|m1|2p1)!(n+m+|m1|2+p1+1)!
ρn+mrl+kcosnθsinmθcoslϕsinkϕ=n1=0Mn2=0Km1=MMm2=KKbnmlkn1n2m1m2Rn1|m1|(ρ)Rn2|m2|(r)exp[i(m1θ+m2ϕ)]
bnmlkn1n2m1m2=(n1+1)(n2+1)π2Inmlkn1n2m1m2=(n1+1)(n2+1)π2Inmm1Ilkm2Knmn1m1Klkn2m2.
nmlkanmlkρn+mrl+kcosnθsinmθcoslϕsinkϕ=n1n2m1m2{nmlkanmlkbnmlkn1n2m1m2}Rn1|m1|(ρ)Rn2|m2|(r)exp[i(m1θ+m2ϕ)]=n1n2m1m2cn1n2m1m2Rn1|m1|(ρ)Rn2|m2|(r)exp[i(m1θ+m2ϕ)],
n1n2m1m2cn1n2m1m200000.1428571428571429=1/720000.3214285714285714=9/2840000.2976190476190476=25/8460000.1666666666666667=1/680000.0584415584415584=9/154100000.0119047619047619=1/84120000.0010822510822511=1/924.
W(ρ,θ,r,ϕ)=ρ3r2(12cos3θ+sin3θ)cos2ϕ+ρ4r3cos3θsinθcos2ϕsinϕ,
f(X,Y)=exp{2πi(uX+vY)}=exp{2πiρwcos(θψ)},
fp(X,Y)=n,m=0anmXnYm,withanm=(2πiu)nn!(2πiv)mm!.
fZ(ρ,θ)=n,mcnmZnm(ρ,θ),
cnm=2(n+1)inJn+1(2πw)2πwexp(imψ)
02πexp[2πiρwcos(θψ)]exp(imθ)dθ=2πimJm(2πρw)exp(imψ),
01Rn|m|(ρ)Jm(bρ)ρdρ=(1)nm2Jn+1(b)b.
fpN(X,Y)=n+mNanmXnYm,
fZN=|m|nNcnmZnm,
fpZN=n1,m1cn10m10NZn1m1
fpZN,N1=|m1|n1N1cn10m10NZn1m1,
δ=(1Jj=1J|f{(X,Y)j}fapp{(X,Y)j}|2)1/2,
I|ffapp|2=1πX2+Y21|f(X,Y)fapp(X,Y)|2dXdY.
ffpZN,N1=n,m,n>N1cnmZnm+n1,m1,n1N1(cn1m1cn10m10N)Zn1m1.
I|ffpZN,N11|2I|ffpZN,N1|2=1N1+1(n,m,n=N1|cnm|2n1,m1,n1=N1|cn1m1cn10m10N|2),
Rnm(ρ)=ρm(nm2)!{dd(ρ2)}nm2{(ρ2)n+m2(ρ21)nm2}
I=01ρaRnm(ρ)ρdρ.
I=12(p!)01x(am)/2{ddx}p[xq(x1)p]dx.
I=12(p!)(am2)01x(am)/21{ddx}p1[xq(x1)p]dx.
I=(1)p2(p!)(am2)(am2p+1)01x(a+m)/2(x1)pdx.
I=(1)2p2(am2)(am2p+1)(a+m2+1)(a+m2+p)01x(a+m)/2+pdx.
I=12(am2)(am2p+1)(a+m2+1)(a+m2+p+1).
I=12Γ(am2+1)Γ(a+m2+1)Γ(am2p+1)Γ(a+m2+p+2),
I=1a+m+2p+2j=0p1am2ja+m+2j+2.
f(θ)=m2=+am2exp(im2θ).
am1=12π02πf(θ)exp(im1θ)dθ,integerm1.
1Ss=0S1f(2πsS)exp(2πim1sS)=1Ss=0S1m2=+am2exp[2πi(m2m1)sS]=r=+am1+rS,
s=0S1exp(2πitsS)={S,tmultipleofS0,otherwise.
f(θ)=cosnθsinmθ,
am1=1Ss=0S1f(2πsS)exp(2πim1sS).
Inmm1=02πcosnθsinmθexp(im1θ)dθ=2πam1=2πSs=0S1cosn(2πsS)sinm(2πsS)exp(2πim1sS),
2πδm1m202πcosmθexp(im1θ)dθ,

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