Abstract

Balancing of Zernike aberrations breaks down if the defocus term is large enough that the condition (z/λ)2/[π(NA)4] is not satisfied. A modified Zernike aberration expansion, based on the Zernike aberrations, is developed that accurately includes axial displacement as a low-order term, even for large displacements. This expansion can be used to analyze aberrations for on-axis illumination of a high numerical aperture system. But more importantly, for systems of moderate numerical aperture it allows balanced aberration coefficients to be determined independent of the assumption of a particular reference point. The approach is applied to the case of a tilted dielectric plate. An exact expression is given for the wave front aberration, valid for both large angles of tilt and high beam convergence angles, that is independent of observation distance. Analytical expressions for the third- and fifth-order aberration coefficients are derived. Expressions are given for expansion of multiple-angle power series terms into Zernike polynomials.

© 2013 Optical Society of America

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  1. C. J. R. Sheppard, “Limitations of the paraxial Debye approximation,” Opt. Lett. 38, 1074–1076 (2013).
    [CrossRef]
  2. H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
    [CrossRef]
  3. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  4. P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  5. L. E. Helseth, “Electromagnetic focusing through a tilted dielectric surface,” Opt. Commun. 215, 247–250 (2003).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics, 1st ed. (Pergamon, 1959).
  7. E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London Sect. B 69, 823–832 (1956).
    [CrossRef]
  8. J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
    [CrossRef]
  9. C. J. R. Sheppard, “Orthogonal aberration functions for high-aperture optical systems,” J. Opt. Soc. Am. A 21, 832–838 (2004).
    [CrossRef]
  10. C. J. R. Sheppard, “Orthogonal aberration functions for high-aperture optical systems: erratum,” J. Opt. Soc. Am. A 21, 2468–2469 (2004).
    [CrossRef]
  11. C. J. R. Sheppard, S. Campbell, and M. D. Hirschhorn, “Zernike expansion of separable function of Cartesian coordinates,” Appl. Opt. 43, 3963–3966 (2004).
    [CrossRef]
  12. C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  13. C. J. R. Sheppard, “Aberrations in high aperture conventional and confocal imaging systems,” Appl. Opt. 27, 4782–4786 (1988).
    [CrossRef]
  14. C. J. R. Sheppard, “Aberrations in high aperture optical systems,” Optik 105, 29–33 (1997).
  15. R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
    [CrossRef]
  16. G. Conforti, “Zernike aberration coefficients from Seidel and higher-order power-series coefficients,” Opt. Lett. 8, 407–408 (1983).
    [CrossRef]
  17. R. K. Tyson, “Conversion of Zernike aberration coefficients to Seidel and higher-order power-series aberration coefficients,” Opt. Lett. 7, 262–264 (1982).
    [CrossRef]
  18. G. N. Lawrence and R. D. Day, “Interferometric characterization of full spheres: data reduction techniques,” Appl. Opt. 26, 4875–4882 (1987).
    [CrossRef]
  19. B. Cuny, “Correction des aberrations introduites par une lame inclinée en lumière convergente,” Rev. Opt. 34, 460–464 (1955).
  20. H. de Lang, “Compensation of aberrations caused by oblique plane parallel plates,” Philips Res. Rep. 12, 181–189 (1957).
  21. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
  22. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1966).
  23. R. Kingslake, Lens Design Fundamentals (Academic, 1978).
  24. W. T. Welford, Aberrations of Optical Systems (Hilger, 1986).
  25. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993).
  26. R. E. Gerber and M. Mansuripur, “Tilt correction in an optical disk system,” Appl. Opt. 35, 7000–7007 (1996).
    [CrossRef]
  27. J. Braat, “Analytical expressions for the wave-front aberration coefficients of a tilted plane-parallel plate,” Appl. Opt. 36, 8459–8467 (1997).
    [CrossRef]
  28. A. Miks and P. Pokorny, “Analytical expressions for the circle of confusion induced by plane-parallel plate,” Opt. Lasers Eng. 50, 1517–1521 (2012).
    [CrossRef]
  29. K. 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]
  30. R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).
  31. J. P. McGuire and R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33, 5080–5100 (1994).
    [CrossRef]
  32. M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high numerical- aperture lenses: theory, measurement, and rectification,” Opt. Eng. 41, 943–954 (2002).
    [CrossRef]
  33. M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
    [CrossRef]

2013 (1)

2012 (1)

A. Miks and P. Pokorny, “Analytical expressions for the circle of confusion induced by plane-parallel plate,” Opt. Lasers Eng. 50, 1517–1521 (2012).
[CrossRef]

2005 (2)

M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

K. 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]

2004 (3)

2003 (1)

L. E. Helseth, “Electromagnetic focusing through a tilted dielectric surface,” Opt. Commun. 215, 247–250 (2003).
[CrossRef]

2002 (1)

M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high numerical- aperture lenses: theory, measurement, and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

1997 (2)

1996 (1)

1995 (1)

1994 (1)

1993 (1)

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

1989 (1)

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

1988 (1)

1987 (2)

1983 (1)

1982 (1)

1969 (1)

J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1957 (1)

H. de Lang, “Compensation of aberrations caused by oblique plane parallel plates,” Philips Res. Rep. 12, 181–189 (1957).

1956 (1)

E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London Sect. B 69, 823–832 (1956).
[CrossRef]

1955 (1)

B. Cuny, “Correction des aberrations introduites par une lame inclinée en lumière convergente,” Rev. Opt. 34, 460–464 (1955).

1943 (1)

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
[CrossRef]

Booker, G. R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 1st ed. (Pergamon, 1959).

Braat, J.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993).

Campbell, S.

Chipman, R. A.

Conforti, G.

Cuny, B.

B. Cuny, “Correction des aberrations introduites par une lame inclinée en lumière convergente,” Rev. Opt. 34, 460–464 (1955).

Dainty, J. C.

J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
[CrossRef]

Day, R. D.

de Lang, H.

H. de Lang, “Compensation of aberrations caused by oblique plane parallel plates,” Philips Res. Rep. 12, 181–189 (1957).

Dittmann, O.

M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Flagello, D.

M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Gerber, R. E.

Göhnermeier, A.

M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Gräupner, P.

M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Heil, T.

M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Helseth, L. E.

L. E. Helseth, “Electromagnetic focusing through a tilted dielectric surface,” Opt. Commun. 215, 247–250 (2003).
[CrossRef]

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

Hirschhorn, M. D.

Hopkins, H. H.

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
[CrossRef]

Inoue, S.

M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high numerical- aperture lenses: theory, measurement, and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

Kamenov, V.

M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Kant, R.

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, 1978).

Krähmer, D.

M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Laczik, Z.

Lawrence, G. N.

Linfoot, E. H.

E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London Sect. B 69, 823–832 (1956).
[CrossRef]

Mansuripur, M.

Matthews, H. J.

McGuire, J. P.

Miks, A.

A. Miks and P. Pokorny, “Analytical expressions for the circle of confusion induced by plane-parallel plate,” Opt. Lasers Eng. 50, 1517–1521 (2012).
[CrossRef]

Oldenbourg, R.

M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high numerical- aperture lenses: theory, measurement, and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

Pokorny, P.

A. Miks and P. Pokorny, “Analytical expressions for the circle of confusion induced by plane-parallel plate,” Opt. Lasers Eng. 50, 1517–1521 (2012).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Ruoff, J.

M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Sheppard, C. J. R.

Shribak, M.

M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high numerical- aperture lenses: theory, measurement, and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1966).

Thompson, K.

Török, P.

Totzeck, M.

M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

Tyson, R. K.

Varga, P.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, 1986).

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London Sect. B 69, 823–832 (1956).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 1st ed. (Pergamon, 1959).

Appl. Opt. (6)

J. Microlith. Microfab. Microsyst. (1)

M. Totzeck, P. Gräupner, T. Heil, A. Göhnermeier, O. Dittmann, D. Krähmer, V. Kamenov, J. Ruoff, and D. Flagello, “Polarization influence on imaging,” J. Microlith. Microfab. Microsyst. 4, 031108 (2005).
[CrossRef]

J. Mod. Opt. (1)

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (2)

L. E. Helseth, “Electromagnetic focusing through a tilted dielectric surface,” Opt. Commun. 215, 247–250 (2003).
[CrossRef]

J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
[CrossRef]

Opt. Eng. (2)

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

M. Shribak, S. Inoue, and R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high numerical- aperture lenses: theory, measurement, and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

Opt. Lasers Eng. (1)

A. Miks and P. Pokorny, “Analytical expressions for the circle of confusion induced by plane-parallel plate,” Opt. Lasers Eng. 50, 1517–1521 (2012).
[CrossRef]

Opt. Lett. (3)

Optik (1)

C. J. R. Sheppard, “Aberrations in high aperture optical systems,” Optik 105, 29–33 (1997).

Philips Res. Rep. (1)

H. de Lang, “Compensation of aberrations caused by oblique plane parallel plates,” Philips Res. Rep. 12, 181–189 (1957).

Proc. Phys. Soc. (1)

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
[CrossRef]

Proc. Phys. Soc. London Sect. B (1)

E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London Sect. B 69, 823–832 (1956).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Rev. Opt. (1)

B. Cuny, “Correction des aberrations introduites par une lame inclinée en lumière convergente,” Rev. Opt. 34, 460–464 (1955).

Other (6)

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1966).

R. Kingslake, Lens Design Fundamentals (Academic, 1978).

W. T. Welford, Aberrations of Optical Systems (Hilger, 1986).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993).

M. Born and E. Wolf, Principles of Optics, 1st ed. (Pergamon, 1959).

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Figures (7)

Fig. 1.
Fig. 1.

Geometry of a wave crossing a parallel-sided plate.

Fig. 2.
Fig. 2.

(a) Contours of the exact wavefront aberration (for d=1). The tilt angle is β=45°, and n=1.5145. Contours of the wavefront aberration, expanded as a power series in r=sinθ to (b) r2, (c) r3, (d) r4, (e) r5, and (f) r6.

Fig. 3.
Fig. 3.

Variation of the aberration coefficients (for d=1) with tilt angle β, for n=1.5145. (a) B20,B22,B40,B42,B60. (b) B11,B31,B33,B51. (c) C40,C51,C60,C62. (d) B44,B53,B62,B55,B64,B66.

Fig. 4.
Fig. 4.

(a) Contours of the exact wavefront aberration. The tilt angle is β=45°, and n=1.5145. Contours of the aberration for the sum of (b) the first-; (c) first- and third-; and (d) first-, third-, and fifth-order aberrations.

Fig. 5.
Fig. 5.

Contours of the exact residual aberration for β=45° and n=1.5145, with (a) the B11; (b) the first-; (c) the first- and third-; and (d) the first-, third-, and fifth-order aberrations subtracted.

Fig. 6.
Fig. 6.

(a) Contours of the exact wavefront aberration. The tilt angle β=45°, and n=1.5145. Contours of the aberration for the sums of (b) the first-, (c) the first- and third-, and (d) first-, third-, and fifth-order hNA aberrations.

Fig. 7.
Fig. 7.

Contours of the exact residual aberration for β=45° and n=1.5145, with (a) the first-, (b) the first- and third-, and (c) the first-, third-, and fifth-order hNA aberrations subtracted.

Tables (1)

Tables Icon

Table 1. Definitions of the Different Aberration Coefficients, where n=(2p+m)

Equations (41)

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

Rnm(ρ)=(1)pρmPp(m,0)(12ρ2),
U(r,ϕ,z)=ikf02π0αeikΨ(θ,ψ)exp[ikrsinθcos(ψϕ)]×exp(ikzcosθ)sinθdθdψ,
kzcosθ=kz(cos2α2sin2α2R200(σ))=kz(cos2α2+sin2α2P1(0,0)(12σ2)),
U(0,z)=i4kfsin2α2exp(ikzcos2α2)×02π01eikΨexp[ikzsin2α2P1(0,0)(12σ2)]σdσdψ.
U(r,ϕ,z)=ikfD2exp(ikziu4)×02π01eikΨexp[iu4P1(0,0)(12σ2)]σdσdψ.
Anm(σ)=(1)pρmPp(m,0)(12σ2)=ρmRnm(σ)/σm=(1+t2)m/2[1(D2σ2)/4]m/2Rnm(σ),
ρ2=σ2(1+t2)σ4t2.
01[Pnm2(m,0)(12σ2)][Pnm2(m,0)(12σ2)]σ2m+1dσ=δnn2(n+1),
01[Anm(σ)][Anm(σ)][1(D2σ2)/4]mσdσ=01[Pnm2(m,0)(12σ2)][Pnm2(m,0)(12σ2)]×ρ2m[1(D2σ2)/4]mσdσ=δnn(1+t2)m2(n+1).
Ψ=a00+12n=2an0Rn0(σ)+n=1m=1nanmρmRnm(σ)σm{cosmϕsinmϕ},(nm)even,=a00+12p=1a2p,0(1)pPp(0,0)(12σ2)+p=0m=1a(2p+m),m(1)pρm{cosmϕsinmϕ}×Pp(m,0)(12σ2).
Ψ=m=0nn=0bnmρn{cosmϕsinmϕ},(nm)even,
Ψ=p=0m=0c2p+m,mσ2pρm{cosmϕsinmϕ}.
c00=b00,c11=b11,c22=b22,c33=b33,c44=b44,,c20=(1+t2)b20,c31=(1+t2)b31,c42=(1+t2)b42,c53=(1+t2)b53,,c40=(1+t2)2b40t2b20,c51=(1+t2)2b51t2b31,c62=(1+t2)2b53t2b42,,c60=(1+t2)3b60t2(1+t2)2b40,c71=(1+t2)3b712t2(1+t2)b51,c80=(1+t2)4b803t2(1+t2)2b60+t4b40,.
R00(ρ)=A00(σ),R11(ρ)=A00(σ),R22(ρ)=A22(σ),R20(ρ)=t23A00(σ)+A20(σ)t23A40(σ),R31(ρ)=t22A11(σ)+(1t25)A31(σ)3t210A51(σ),R40(ρ)=t45A00(σ)+3t25A20(σ)+(12t47)A40(σ)3t25A60(σ)+3t435A80(σ).
bnm=Bnmsinnα=(2t1+t2)nBnm.
c00=B00,c11=2t1+t2B11,c20=4t21+t2B20,c22=4t2(1+t2)2B22,c31=8t3(1+t2)2B31,c40=16t4(1+t2)2(B40B204),c33=8t3(1+t2)3B33,c42=(2t)4(1+t2)3B42,c51=32t5(1+t2)3(B51B314),c60=64t6(1+t2)3(B60B402),c44=B44,c53=32t5(1+t2)4B53,c62=64t6(1+t2)4(B62B424),c71=128t7(1+t2)4(B71B512),c80=256t8(1+t2)4(B803B604+B4016),.
anm1εp!(p+m)!n!cnm,
1cosθ=2sin2θ2=s,
r2=1(1s)2=s(2s),
Ψ=p=0m=0C2p+m,m(2s)prmcosmϕ,
C00=B00,C11=B11,C20=B20,C22=B22,C31=B31,C33=B33,C42=B42,,C40=B40B20/4,C51=B51B31/4,C60=B60B40/2,C62=B62B42/4,
Ψ=pm(2t)n(1+t2)l+1Cnmσ2pρmcosmϕ.
R22(σ)σ2ρ2cos2ϕ[(2t)2(1+t2)2C22+3(2t)44(1+t2)3C42+],R31(σ)σρcosϕ[(2t)33(1+t2)2C31+2(2t)55(1+t2)3C51+],R40(σ)[(2t)46(1+t2)2C40+(2t)64(1+t2)3C60+],
Ψ=p=0m=0(p+m)!p!n!(2t)n(1+t2)p+mCnmRnm(σ)σmρmcosmϕ.
a00=C00,an0=2[(n/2)!]2n!(2t)n(1+t2)n/2Cn0,anm=(nm2)!(n+m2)!n!(2t)n(1+t2)(n+m)/2Cnm,(nm)even,
Ψ=d(ncosθ2cosθ1),
Ψ=nAB+BCAD=d[nsecθ2+(tanθ1tanθ2)sinθ1secθ1],
nsinθ2=sinθ1,
ncosθ2=n21+cos2θ1
Ψ=d(n21+cos2θ1cosθ1).
cosθ1=a^n^,cosθ2=b^n^,
Ψ=d(nb^n^a^n^)=d(n21+(a^n^)2a^n^).
n^=sinβi^+cosβk^,a^=sinθcosϕi^+sinθsinϕj^+cosθk^,
a^n^=sinθcosϕsinβ+cosθcosβ.
Ψ=d({(n2sin2β)[cos2β(a^n^)2]}1/2a^n^)=d([(n2sin2β)+sinθcosθcosϕsin2βsin2θ(cos2βcos2ϕsin2β)]1/2(sinθcosϕsinβ+cosθcosβ).
B00=d[(n2sin2β)1/2cosβ],B11=dsinβ(cosβn2sin2β1),B20=d2[cosβ2n2(3n2+1)sin2β+2sin4β4(n2sin2β)3/2],B22=d(n21)sin2β4(n2sin2β)3/2,B31=d(n21)sin2β(4n2sin2β)16(n2sin2β)5/2,B33=d(n21)sin2βsin2β16(n2sin2β)5/2,B40=d64(n2sin2β)7/2[8(n2sin2β)7/2cosβ8sin8β4(7n2+1)sin6β(35n4+14n21)sin4β8n2(5n21)sin2β+8n4],B42=d(n21)sin2β[2sin4β(7n2+1)sin2β+6n2]16(n2sin2β)7/2,B51=d(n21)sin2β128(n2sin2β)9/2[4sin6β(21n21)sin4β+12n2(5n21)sin2β8n4(n2+3)],B60=d256(n2sin2β)11/2[16(n2sin2β)11/2cosβ16sin12β8(11n2+1)sin10β+2(99n4+22n21)sin8β+(231n6+99n411n2+1)sin6β6n2(63n426n2+3)sin4β+24n4(7n23)sin2β16n6].
B00=d(n1)(1+β22n),B11=dβ(n1)n,B20=d(n1)2n,B22=d(n21)β24n3,B31=d(n21)β2n3,B40=d(n31)8n3,B33=d(n21)β38n5,B42=3d(n21)β28n5,B51=d(n21)(n2+3)β8n5,B60=d(n51)16n5.
C40=d(n21)64(n2sin2β)7/2×[(3n2+1)sin4β+4n2(3n22)sin2β8n4],C51=d(n21)sin2β128(n2sin2β)9/2×[2sin6β(9n21)sin4β+6n2(7n22)sin2β24n4],C60=d(n21)256(n2sin2β)11/2[(35n410n21)sin6β+2n2(35n460n2+9)sin4βn48(10n29)sin2β+16n6].
C40=d(n21)8n3,C51=3d(n21)β8n5,C60=d(n21)16n5.
Ψ=d{[(n2sin2β)+sinθcosθcosϕsin2βsin2θ(cos2βcos2ϕsin2β)]1/2n2sin2β(sinβcosβn2sin2β)sinθcosϕ+[2n2(3n2+1)sin2β+2sin4β4(n2sin2β)3/2](1cosθ)}.
1=R00,ρ=R11,ρ2=12(R20+R00)=R22,ρ3=13(R31+2R11)=R33,ρ4=16(R40+3R20+2R00)=14(R42+3R22)=R44,ρ5=110(R51+4R31+5R11)=15(R53+4R33)=R55,ρ6=120(R60+5R40+9R20+5R00)=115(R62+5R42+9R22)=16(R64+5R44)=R66,ρ7=135(R71+6R51+14R31+14R11)=121(R73+6R53+14R33)=17(R75+6R55)=R77,ρ8=170(R80+7R60+20R40+28R20+14R00)=156(R82+7R62+20R42+28R22)=128(R84+7R64+20R44)=18(R86+7R66)=R88,ρ9=1126(R91+8R71+27R51+48R31+42R11)=184(R93+4R73+27R53+48R33)=136(R95+8R75+27R55)=19(R97+8R77)=R99,ρ10=1252(R100+9R80+35R60+75R40+90R20+42R00)=1210(R102+9R82+15R62+75R42+90R22)=1120(R104+9R84+35R64+75R44)=145(R106+9R86+35R66)=110(R108+9R88)=R1010.

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