Abstract

Taking the classical Ignatowsky/Richards and Wolf formulas as our starting point, we present expressions for the electric field components in the focal region in the case of a high-numerical-aperture optical system. The transmission function, the aberrations, and the spatially varying state of polarization of the wave exiting the optical system are represented in terms of a Zernike polynomial expansion over the exit pupil of the system; a set of generally complex coefficients is needed for a full description of the field in the exit pupil. The field components in the focal region are obtained by means of the evaluation of a set of basic integrals that all allow an analytic treatment; the expressions for the field components show an explicit dependence on the complex coefficients that characterize the optical system. The electric energy density and the power flow in the aberrated three-dimensional distribution in the focal region are obtained with the expressions for the electric and magnetic field components. Some examples of aberrated focal distributions are presented, and some basic characteristics are discussed.

© 2003 Optical Society of America

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References

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  1. B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, H. van Houten, “Optical pickup for blue optical recording at NA=0.85,” Opt. Rev. 8, 211–213 (2001).
    [CrossRef]
  2. H. P. Urbach, D. A. Bernard, “Modeling latent-image formation in the photolithography, using the Helmholtz equation,” J. Opt. Soc. Am. A 6, 1343–1356 (1989).
    [CrossRef]
  3. V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Tr. Opt. Inst. 1 (4), 1–36 (1919).
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    [CrossRef]
  5. 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]
  6. J. J. M. Braat, P. Dirksen, 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]
  7. P. Dirksen, J. J. M. Braat, A. J. E. M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr. Microfabr. Microsyst. 2, 61–68 (2003).
  8. J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Retrieval of aberrations from intensity measurements in the focal region using an extended Nijboer–Zernike approach,” manuscript available from the author, j.j.m.braat@tnw.tudelft.nl.
  9. N. R. Heckenberg, T. A. Nieminen, M. E. J. Friese, H. Rubinsztein-Dunlop, “Trapping microscopic particles with singular beams,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 46–53 (1998).
    [CrossRef]
  10. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  11. L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
    [CrossRef]
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  13. S. Stallinga, “Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001).
    [CrossRef]
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    [CrossRef]
  17. A. J. E. M. Janssen, “Stable representation of Zernike polynomials,” (Philips Research Laboratories, Eindhoven, The Netherlands, 2002).
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  19. G. Szegő, Orthogonal Polynomials, 4th ed. (American Mathematical Society, Providence, 1975).
  20. F. G. Tricomi, Vorlesungen über Orthogonalreihen (Springer, Berlin, 1955).
  21. G. E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge U. Press, Cambridge, UK, 1999).

2003 (1)

P. Dirksen, J. J. M. Braat, A. J. E. M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr. Microfabr. Microsyst. 2, 61–68 (2003).

2002 (2)

2001 (2)

S. Stallinga, “Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001).
[CrossRef]

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, H. van Houten, “Optical pickup for blue optical recording at NA=0.85,” Opt. Rev. 8, 211–213 (2001).
[CrossRef]

1999 (1)

L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
[CrossRef]

1995 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1989 (1)

1959 (1)

B. Richards, 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]

1919 (1)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Tr. Opt. Inst. 1 (4), 1–36 (1919).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Allen, L.

L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Andrews, G. E.

G. E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge U. Press, Cambridge, UK, 1999).

Askey, R.

G. E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge U. Press, Cambridge, UK, 1999).

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Bernard, D. A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Braat, J. J. M.

P. Dirksen, J. J. M. Braat, A. J. E. M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr. Microfabr. Microsyst. 2, 61–68 (2003).

J. J. M. Braat, P. Dirksen, 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]

Courtial, J.

L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
[CrossRef]

Dirksen, P.

P. Dirksen, J. J. M. Braat, A. J. E. M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr. Microfabr. Microsyst. 2, 61–68 (2003).

J. J. M. Braat, P. Dirksen, 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]

Friese, M. E. J.

N. R. Heckenberg, T. A. Nieminen, M. E. J. Friese, H. Rubinsztein-Dunlop, “Trapping microscopic particles with singular beams,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 46–53 (1998).
[CrossRef]

Heckenberg, N. R.

N. R. Heckenberg, T. A. Nieminen, M. E. J. Friese, H. Rubinsztein-Dunlop, “Trapping microscopic particles with singular beams,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 46–53 (1998).
[CrossRef]

Hendriks, B. H. W.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, H. van Houten, “Optical pickup for blue optical recording at NA=0.85,” Opt. Rev. 8, 211–213 (2001).
[CrossRef]

Ignatowsky, V. S.

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Tr. Opt. Inst. 1 (4), 1–36 (1919).

Janssen, A. J. E. M.

P. Dirksen, J. J. M. Braat, A. J. E. M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr. Microfabr. Microsyst. 2, 61–68 (2003).

J. J. M. Braat, P. Dirksen, 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]

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]

A. J. E. M. Janssen, “Stable representation of Zernike polynomials,” (Philips Research Laboratories, Eindhoven, The Netherlands, 2002).

Juffermans, C.

P. Dirksen, J. J. M. Braat, A. J. E. M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr. Microfabr. Microsyst. 2, 61–68 (2003).

Nemeth, G.

Nieminen, T. A.

N. R. Heckenberg, T. A. Nieminen, M. E. J. Friese, H. Rubinsztein-Dunlop, “Trapping microscopic particles with singular beams,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 46–53 (1998).
[CrossRef]

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, Groningen, The Netherlands, 1942).

Padgett, M. J.

L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
[CrossRef]

Richards, B.

B. Richards, 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]

Roy, R.

G. E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge U. Press, Cambridge, UK, 1999).

Rubinsztein-Dunlop, H.

N. R. Heckenberg, T. A. Nieminen, M. E. J. Friese, H. Rubinsztein-Dunlop, “Trapping microscopic particles with singular beams,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 46–53 (1998).
[CrossRef]

Schleipen, J. J. H. B.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, H. van Houten, “Optical pickup for blue optical recording at NA=0.85,” Opt. Rev. 8, 211–213 (2001).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Stallinga, S.

S. Stallinga, “Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001).
[CrossRef]

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, H. van Houten, “Optical pickup for blue optical recording at NA=0.85,” Opt. Rev. 8, 211–213 (2001).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Szego, G.

G. Szegő, Orthogonal Polynomials, 4th ed. (American Mathematical Society, Providence, 1975).

Török, P.

Tricomi, F. G.

F. G. Tricomi, Vorlesungen über Orthogonalreihen (Springer, Berlin, 1955).

Urbach, H. P.

van Houten, H.

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, H. van Houten, “Optical pickup for blue optical recording at NA=0.85,” Opt. Rev. 8, 211–213 (2001).
[CrossRef]

Varga, P.

Welford, W.

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

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Wolf, E.

B. Richards, 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]

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

J. Microlithogr. Microfabr. Microsyst. (1)

P. Dirksen, J. J. M. Braat, A. J. E. M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr. Microfabr. Microsyst. 2, 61–68 (2003).

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

Opt. Rev. (1)

B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, H. van Houten, “Optical pickup for blue optical recording at NA=0.85,” Opt. Rev. 8, 211–213 (2001).
[CrossRef]

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (1)

L. Allen, J. Courtial, M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
[CrossRef]

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

B. Richards, 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]

Tr. Opt. Inst. (1)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Tr. Opt. Inst. 1 (4), 1–36 (1919).

Other (10)

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Retrieval of aberrations from intensity measurements in the focal region using an extended Nijboer–Zernike approach,” manuscript available from the author, j.j.m.braat@tnw.tudelft.nl.

N. R. Heckenberg, T. A. Nieminen, M. E. J. Friese, H. Rubinsztein-Dunlop, “Trapping microscopic particles with singular beams,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 46–53 (1998).
[CrossRef]

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

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, Groningen, The Netherlands, 1942).

A. J. E. M. Janssen, “Stable representation of Zernike polynomials,” (Philips Research Laboratories, Eindhoven, The Netherlands, 2002).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

G. Szegő, Orthogonal Polynomials, 4th ed. (American Mathematical Society, Providence, 1975).

F. G. Tricomi, Vorlesungen über Orthogonalreihen (Springer, Berlin, 1955).

G. E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge U. Press, Cambridge, UK, 1999).

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Figures (9)

Fig. 1
Fig. 1

Propagation of the incident wave from the entrance pupil S0 through the optical system toward the exit pupil S1 and the focal region at the image plane PI. The incident wave has a planar wave front. The unit propagation vector has been denoted by s0, and the meridional and tangential field components are directed along, respectively, the unit vectors e0 and g0. After propagation through the optical system, the field components in the exit pupil are projected onto the unit vectors e1 and g1, which form an orthogonal basis with the local propagation vector s1. The position on the exit pupil sphere is defined by means of the cylindrical coordinates (ρ, θ); the position in the image plane region is defined by the cylindrical coordinate system (r, ϕ, f). The maximum aperture (NA) of the imaging pencil is represented by s0=sin αmax.

Fig. 2
Fig. 2

Distribution of polarization states over, e.g., the entrance pupil for a certain value of the coordinate ρ. The angle is constant over the pupil and equals zero; θ0=13π.

Fig. 3
Fig. 3

Comparison of the energy density function in the high-NA focus (f=0) and the scalar Airy distribution as a function of the radial coordinate r. The energy density function in the cross section θ=0 is represented by the dotted curve, and the one in the cross section θ=π/2 by the dashed curve. The scalar Airy distribution is shown by the solid curve. The fact that the dashed high-aperture curve does not reach very low levels is due to the sampling density used in plotting this curve.

Fig. 4
Fig. 4

Modulus of the x-polarization component of the electric field distribution in the focal region along the x axis (θ=0). The amplitude of the field component is indicated by a gray level on a logarithmic scale, and the contours denote equiphase lines. The radial coordinate r has been normalized with respect to the diffraction unit λ/s0, and the axial coordinate with respect to the quantity u0 [see Eq. (15)].

Fig. 5
Fig. 5

Modulus of the x-polarization component of the electric field distribution in the focal region along the y axis (θ=π/2). The amplitude of the field component is indicated by a gray level on a logarithmic scale, and the contours denote equiphase lines.

Fig. 6
Fig. 6

Modulus of the y-polarization component of the electric field distribution in the focal region along the diagonal x=y (θ=π/4).

Fig. 7
Fig. 7

Modulus of the z-polarization component of the electric field distribution in the focal region along the x axis (θ=0).

Fig. 8
Fig. 8

A radially polarized entrance pupil distribution yields a radially symmetric Poynting vector distribution in the focal region. The absolute value of the Poynting vector is indicated by a gray level on a logarithmic scale, and its direction is given by a set of arrows. Note that the value in the geometrical focus equals zero.

Fig. 9
Fig. 9

A helical phase distribution in the presence of a linear polarization state in the entrance pupil yields a rotating Poynting vector field in the focal region. The absolute value of the Poynting vector is indicated by a gray level on a logarithmic scale, and its direction is given by a set of arrows.

Equations (75)

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

ρ0=R sin α,
Bx(ρ, θ)=Ax(ρ, θ)exp[i2πW(ρ, θ)],
By(ρ, θ)=Ay(ρ, θ)exp[i2πW(ρ, θ)+i(ρ, θ)],
Bx=A0 cos(θ+θ0),
By=A0 sin(θ+θ0)exp(i)
Bx(ρ, θ)(1-s02ρ2)1/4=Ax(ρ, θ)(1-s02ρ2)1/4exp[i2πW(ρ, θ)]=n,mβnmxRn|m|(ρ)exp(imθ),
βnmx=n+1π0102π Bx(ρ, θ)(1-s02ρ2)1/4Rn|m|(ρ)×exp(-imθ)ρdρdθ,
01Rn|m|(ρ)Rn|m|(ρ)ρdρ=δnn2(n+1).
βnmx=βnmy=(n+1)A0π01 Rn|m|(ρ)(1-s02ρ2)1/4ρdρ02π exp(-imθ)dθ=2(n+1)A0δm001 Rn0(ρ)(1-s02ρ2)1/4ρdρ.
βn,±1x=iβn,±1y=(n+1)A001 Rn1(ρ)(1-s02ρ2)1/4ρdρ,
Bx(ρ, θ)=By(ρ, θ)=A0 exp(iaθ).
βnmx=βnmy=(n+1)A0 exp(i2πa)-1iπ(a-m)01 Rn|m|(ρ)(1-s02ρ2)1/4ρdρ,
Exx(r, ϕ, f)=-iγs02 exp-ifu0n,mimβnmx exp(imϕ)×01Rn|m|(ρ)expifu0(1-1-s02ρ2)×{(1+1-s02ρ2)Jm(2πrρ)+12(1-1-s02ρ2)×[Jm+2(2πrρ)exp(2iϕ)+Jm-2(2πrρ)exp(-2iϕ)]}ρdρ,
Eyx(r, ϕ, f)=-iγs02 exp-ifu0n,mimβnmx exp(imϕ)×01Rn|m|(ρ)-i2expifu0(1-1-s02ρ2)×(1-1-s02ρ2)×[Jm+2(2πrρ)exp(2iϕ)-Jm-2(2πrρ)exp(-2iϕ)]ρdρ,
Ezx(r, ϕ, f)=-iγs02 exp-ifu0n,mimβnmx exp(imϕ)×01Rn|m|(ρ)-is0ρ expifu0(1-1-s02ρ2)×[Jm+1(2πrρ)exp(iϕ)-Jm-1(2πrρ)exp(-iϕ)]ρdρ,
f=-2πu0λz,
u0=1-1-s02.
Ex(r, ϕ, f)=-iγs02 exp-ifu0n,mimβnmx exp(imϕ)Vnm,0+s022Vnm,2 exp(2iϕ)+s022Vnm,-2 exp(-2iϕ)-is022Vnm,2 exp(2iϕ)+is022Vnm,-2 exp(-2iϕ)-is0Vnm,1 exp(iϕ)+is0Vnm,-1 exp(-iϕ).
Vnm,j=01ρ|j|(1+1-s02ρ2)-|j|+1×expifu0(1-1-s02ρ2)×Rn|m|(ρ)Jm+j(2πrρ)ρdρ.
Vnm(r, f)=01 exp(ifρ2)Rn|m|(ρ)Jm(2πrρ)ρdρ,
Vnm(r, f)=m exp(if)l=1(-2if)l-1×j=0pvlj J|m|+l+2j(2πr)l(2πr)l.
vlj=(-1)p(|m|+l+2j)|m|+j+l-1l-1j+l-1l-1×l-1p-jq+l+jl
(1+1-s02ρ2)-|j|+1 expifu0(1-1-s02ρ2)=exp(gj+ifjρ2)k=0hkjR2k0(ρ),
ρ|j|Rn|m|(ρ)=s=0|j|cn|m|jsRn+|j|-2s|m|+j(ρ).
R2k0(ρ)Rn+|j|-2s|m|+j(ρ)=t=0dn|m|jsktRn+|j|-2s+2t|m|+j(ρ),
Ey(r, ϕ, f)=-iγs02 exp-ifu0n,mimβnmy exp(imϕ)-is022[Vnm,2 exp(2iϕ)-Vnm,-2 exp(-2iϕ)]Vnm,0-s022[Vnm,2 exp(2iϕ)+Vnm,-2 exp(-2iϕ)]-s0[Vnm,1 exp(iϕ)+Vnm,-1 exp(-iϕ)].
B=s×Ev,
B(r, ϕ, f)=-inrγs02cexp-ifu0n,mim exp(imϕ)×-βnmyVnm,0-s022(βnmy+iβnmx)Vnm,2 exp(2iϕ)-s022(βnmy-iβnmx)Vnm,-2 exp(-2iϕ)βnmxVnm,0-s022(βnmx-iβnmy)Vnm,2 exp(2iϕ)-s022(βnmx+iβnmy)Vnm,-2 exp(-2iϕ)-s0(βnmx-iβnmy)Vnm,1 exp(iϕ)-s0(βnmx+iβnmy)Vnm,-1 exp(-iϕ),
We=04nr2|E|2.
S=0c22Re(E×B*).
Im+2pm=01 Rm+2pm(ρ)(1-s02ρ2)1/4ρdρ,m,p=0,1,,
01ραRm+2pm(ρ)ρdρ=12(-1)p (12m-12α)p(12m+12α+1)p+1,
(a)0=1,(a)n=a(a+1)××(a+n-1), n=1,2,.
Rm+2pm(ρ)=ρmPp(0,m)(2ρ2-1),
Pp(0,m)(z)=(-1)p2pp!ddzp[(1-z)p(1+z)p+m],
(1-x)-1/4=k=0 (14)kk!xk,
Im+2pm=k=0 (14)kk!s02k01ρ2kRm+2pm(ρ)ρdρ=12(-1)pk=0 (14)kk!(12m-k)p(12m+k+1)p+1s02k.
0(14)kk!1Γ(14)k3/4,k=1,2,,
(12m-k)p(12m+k+1)p+11k+12m+p+1,k,m,p=0,1,.
1-s02ρ2=-12u0n=0d0n-12n-1-d0n+12n+3×R2n0(ρ),
ln(1+1-s02ρ2)=12d0+lnu0d0-12n=1d0nn-d0n+1n+1R2n0(ρ),
u0=1-1-s02,
d0=u0s02.
n=0znR2n0(ρ)=1(1+z)2-4zρ2,
n=0znR2n+11(ρ)=2ρ(1+z)2-4zρ2[1+z+(1+z)2-4zρ2]
4z(1+z)2=s02,
ρ1-s02ρ2=(1+d0)n=0d0nρR2n0(ρ),
ddρ[ln(1+1-s02ρ2)]=-s02ρ1-s02ρ2(1+1-s02ρ2)=-2d0n=0d0nR2n+11(ρ).
1s02(1-1-s02ρ2)=(1+d0)n=0d0n0ρϱR2n0(ϱ)dϱ,
ln(1+1-s02ρ2)-lnu0d0=2d0n=0d0nρ1R2n+11(ϱ)dϱ.
0ρϱR2n0(ϱ)dϱ,ρ1R2n+11(ϱ)dϱ.
(2n+1)Pn(x)=Pn+1(x)-Pn-1(x),n=1,2,.
0ρϱR2n0(ϱ)dϱ=14(2n+1)[R2n+20(ρ)-R2n-20(ρ)]
0ρϱR00(ϱ)dϱ=12ρ2=14[R20(ρ)+R00(ρ)].
R2n+11(ρ)=14(n+1)ddρ[R2n+20(ρ)-R2n0(ρ)]
ρ1R2n+11(ϱ)dϱ=-14(n+1)[R2n+20(ρ)-R2n0(ρ)]
(-|j|+1)ln(1+1-s02ρ2)+ifu0(1-1-s02ρ2)=(-|j|+1)n=0bnR2n0(ρ)+ifn=0anR2n0(ρ)=gj+ifjρ2+n=2τnjR2n0(ρ).
a0=12-16d0,an=12d0n-12n-1-d0n+12n+3,n=1,2,,
b0=12d0+lnu0d0,bn=-12d0nn-d0n+1n+1,n=1,2,,
gj=(-|j|+1)(b0-b1)+if(a0-a1),
fj=2fa1-2i(-|j|+1)b1,
τnj=(-|j|+1)bn+ifan,n=2,3,.
(1+1-s02ρ2)-|j|+1 expifu0(1-1-s02ρ2)=exp(gj+ifjρ2)Gj(ρ),
Gj(ρ)=expn=2τnjR2n0(ρ).
Gj(ρ)1+n=2NτnjR2n0(ρ),N=2,3,or4.
12n=2NτnjR2n0(ρ)2
R2n0R2n0=r=0n An-rArAn-rAn+n-r2n+2n-4r+12n+2n-2r+1R2n+2n-4r0,
An=1n![1×3××(2n-1)]=12n2nn,
ρRnm(ρ)=q+1n+1Rn+1m+1(ρ)+pn+1Rn-1m+1(ρ),
ρRnm(ρ)=p+1n+1Rn+1m-1(ρ)+qn+1Rn-1m-1(ρ),
ρ2Rnm(ρ)=(p+1)(p+2)(n+1)(n+2)Rn+2m-2(ρ)+2(p+1)qn(n+2)Rnm-2(ρ)+q(q-1)n(n+1)Rn-2m-2(ρ),
ρ2Rnm(ρ)=(q+1)(q+2)(n+1)(n+2)Rn+2m+2(ρ)+2p(q+1)n(n+2)Rnm+2(ρ)+p(p-1)n(n+1)Rn-2m+2(ρ).
R20(ρ)Rnm(ρ)=2 (p+1)(q+1)(n+1)(n+2)Rn+2m(ρ)+m2n(n+2)Rnm(ρ)+2 pqn(n+1)Rn-2m(ρ),
R2k0(ρ)=Pk[R20(ρ)]=l=0[k/2] (-1)l2kkl2k-2lk×[R20(ρ)]k-2l.
R2k0(ρ)Rn+|j|-2s|m|+j(ρ)=tdn|m|jsktRn+|j|-2s+2t|m|+j(ρ).

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