Abstract

The general polarization behavior of almost-plane waves, in which the electric field varies slowly over a circular pupil, is considered, on the basis of an axial Hertz potential treatment and expansion in Zernike polynomials. The resultant modes of a circular aperture are compared with the well-known waveguide (or optical fiber) modes and Gaussian beam modes. The wave can be decomposed into partial waves of electric and magnetic types. The modes for a square pupil are also considered. The particular application of the effect on polarization of focusing the waves is discussed. Another application discussed is the Fresnel reflection from a dielectric interface, it being shown that the Fresnel reflection alters the relative strength of the electric and magnetic components.

© 2000 Optical Society of America

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References

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    [CrossRef]
  3. B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. 9, pp. 311–407.
  4. J. F. Nye, M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  5. S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).
  6. J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
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    [CrossRef]
  8. H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. London Sect. A 66, 1129–1137 (1953).
    [CrossRef]
  9. E. Wolf, “A scalar representation of electromagnetic fields. II,” Proc. Phys. Soc. London Sect. A 74, 269–280 (1959).
    [CrossRef]
  10. E. T. Whittaker, “On an expression of the electromagnetic field due to electrons by means of two scalar potential functions,” Proc. London Math. Soc. 1, 367–372 (1904).
    [CrossRef]
  11. D. Pattanayak, G. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
    [CrossRef]
  12. L. W. Davis, G. Patsakos, “Comment on ‘Representation of vector electromagnetic beams’” Phys. Rev. A 26, 3702–3703 (1982).
    [CrossRef]
  13. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  14. J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London Ser. A 387, 105–132 (1983).
    [CrossRef]
  15. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London Ser. A 389, 279–290 (1983).
    [CrossRef]
  16. J. F. Nye, J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. London Ser. A 409, 21–36 (1987).
    [CrossRef]
  17. J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. 1. Theory,” Proc. R. Soc. London Ser. A 414, 433–446 (1987).
    [CrossRef]
  18. J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. 2. Observations on the electric field,” Proc. R. Soc. London Ser. A 414, 447–468 (1987).
    [CrossRef]
  19. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Maxwell beams,” J. Opt. Soc. Am. A 3, 536–540 (1986).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. 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]
  27. C. J. R. Sheppard, P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik (Stuttgart) 104, 175–177 (1997).
  28. C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 2, 163–166 (1978).
    [CrossRef]
  29. C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
    [CrossRef]
  30. C. J. R. Sheppard, M. Gu, “Axial imaging through an aberrating layer of water in confocal microscopy,” Opt. Commun. 88, 180–190 (1992).
    [CrossRef]
  31. P. Török, P. Varga, Z. Laczik, 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]
  32. S. Hell, Wijnaendts-van-Resandt, “The application of polarized confocal microscopy for size measurement of resist structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. SPIE1139, 92–98 (1989).
    [CrossRef]

1997

C. J. R. Sheppard, P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik (Stuttgart) 104, 175–177 (1997).

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

1995

1992

C. J. R. Sheppard, M. Gu, “Axial imaging through an aberrating layer of water in confocal microscopy,” Opt. Commun. 88, 180–190 (1992).
[CrossRef]

1987

J. F. Nye, J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. London Ser. A 409, 21–36 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. 1. Theory,” Proc. R. Soc. London Ser. A 414, 433–446 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. 2. Observations on the electric field,” Proc. R. Soc. London Ser. A 414, 447–468 (1987).
[CrossRef]

1986

1983

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London Ser. A 387, 105–132 (1983).
[CrossRef]

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London Ser. A 389, 279–290 (1983).
[CrossRef]

J. M. Vaughan, D. V. Willetts, “Temporal and interference fringe analysis of TEM01* laser modes,” J. Opt. Soc. Am. 73, 1018–1021 (1983).
[CrossRef]

1982

L. W. Davis, G. Patsakos, “Comment on ‘Representation of vector electromagnetic beams’” Phys. Rev. A 26, 3702–3703 (1982).
[CrossRef]

1980

D. Pattanayak, G. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

1979

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

1978

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

1974

J. F. Nye, M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

1971

1970

1966

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1961

1959

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]

E. Wolf, “A scalar representation of electromagnetic fields. II,” Proc. Phys. Soc. London Sect. A 74, 269–280 (1959).
[CrossRef]

1955

A. Nisbet, “Hertzian electromagnetic potentials and associated gauge transformations,” Proc. R. Soc. London Ser. A 231, 250–263 (1955).
[CrossRef]

1953

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. London Sect. A 66, 1129–1137 (1953).
[CrossRef]

1904

E. T. Whittaker, “On an expression of the electromagnetic field due to electrons by means of two scalar potential functions,” Proc. London Math. Soc. 1, 367–372 (1904).
[CrossRef]

Agrawal, G.

D. Pattanayak, G. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

Berry, M.

J. F. Nye, M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Booker, G. R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).

Davis, L. W.

L. W. Davis, G. Patsakos, “Comment on ‘Representation of vector electromagnetic beams’” Phys. Rev. A 26, 3702–3703 (1982).
[CrossRef]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Frieden, B. R.

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. 9, pp. 311–407.

Gloge, D.

Green, H. S.

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. London Sect. A 66, 1129–1137 (1953).
[CrossRef]

Gu, M.

C. J. R. Sheppard, M. Gu, “Axial imaging through an aberrating layer of water in confocal microscopy,” Opt. Commun. 88, 180–190 (1992).
[CrossRef]

Hajnal, J. V.

J. F. Nye, J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. London Ser. A 409, 21–36 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. 2. Observations on the electric field,” Proc. R. Soc. London Ser. A 414, 447–468 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. 1. Theory,” Proc. R. Soc. London Ser. A 414, 433–446 (1987).
[CrossRef]

Hell, S.

S. Hell, Wijnaendts-van-Resandt, “The application of polarized confocal microscopy for size measurement of resist structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. SPIE1139, 92–98 (1989).
[CrossRef]

Itoh, Y.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Laczik, Z.

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Mukunda, N.

Nisbet, A.

A. Nisbet, “Hertzian electromagnetic potentials and associated gauge transformations,” Proc. R. Soc. London Ser. A 231, 250–263 (1955).
[CrossRef]

Nye, J. F.

J. F. Nye, J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. London Ser. A 409, 21–36 (1987).
[CrossRef]

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London Ser. A 387, 105–132 (1983).
[CrossRef]

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London Ser. A 389, 279–290 (1983).
[CrossRef]

J. F. Nye, M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Osterberg, H.

Patsakos, G.

L. W. Davis, G. Patsakos, “Comment on ‘Representation of vector electromagnetic beams’” Phys. Rev. A 26, 3702–3703 (1982).
[CrossRef]

Pattanayak, D.

D. Pattanayak, G. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

Ramo, S.

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communications Electronics, 2nd ed. (Wiley, New York, 1984).

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]

Schelkunoff, S. A.

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).

Sheppard, C. J. R.

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

C. J. R. Sheppard, P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik (Stuttgart) 104, 175–177 (1997).

C. J. R. Sheppard, M. Gu, “Axial imaging through an aberrating layer of water in confocal microscopy,” Opt. Commun. 88, 180–190 (1992).
[CrossRef]

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

Simon, R.

Snitzer, E.

Stratton, J.

J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Sudarshan, E. C. G.

Török, P.

C. J. R. Sheppard, P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik (Stuttgart) 104, 175–177 (1997).

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

P. Török, P. Varga, Z. Laczik, 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]

Van Duzer, T.

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communications Electronics, 2nd ed. (Wiley, New York, 1984).

Varga, P.

Vaughan, J. M.

Whinnery, J. R.

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communications Electronics, 2nd ed. (Wiley, New York, 1984).

Whittaker, E. T.

E. T. Whittaker, “On an expression of the electromagnetic field due to electrons by means of two scalar potential functions,” Proc. London Math. Soc. 1, 367–372 (1904).
[CrossRef]

Wijnaendts-van-Resandt,

S. Hell, Wijnaendts-van-Resandt, “The application of polarized confocal microscopy for size measurement of resist structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. SPIE1139, 92–98 (1989).
[CrossRef]

Willetts, D. V.

Wolf, E.

E. Wolf, “A scalar representation of electromagnetic fields. II,” Proc. Phys. Soc. London Sect. A 74, 269–280 (1959).
[CrossRef]

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]

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. London Sect. A 66, 1129–1137 (1953).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).

Appl. Opt.

IEE J. Microwaves, Opt. Acoust.

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

J. Mod. Opt.

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

C. J. R. Sheppard, M. Gu, “Axial imaging through an aberrating layer of water in confocal microscopy,” Opt. Commun. 88, 180–190 (1992).
[CrossRef]

Optik (Stuttgart)

C. J. R. Sheppard, P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik (Stuttgart) 104, 175–177 (1997).

Phys. Rev. A

D. Pattanayak, G. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

L. W. Davis, G. Patsakos, “Comment on ‘Representation of vector electromagnetic beams’” Phys. Rev. A 26, 3702–3703 (1982).
[CrossRef]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Proc. IEEE

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Proc. London Math. Soc.

E. T. Whittaker, “On an expression of the electromagnetic field due to electrons by means of two scalar potential functions,” Proc. London Math. Soc. 1, 367–372 (1904).
[CrossRef]

Proc. Phys. Soc. London Sect. A

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. London Sect. A 66, 1129–1137 (1953).
[CrossRef]

E. Wolf, “A scalar representation of electromagnetic fields. II,” Proc. Phys. Soc. London Sect. A 74, 269–280 (1959).
[CrossRef]

Proc. R. Soc. London Ser. A

J. F. Nye, M. Berry, “Dislocations of wave-fronts,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

A. Nisbet, “Hertzian electromagnetic potentials and associated gauge transformations,” Proc. R. Soc. London Ser. A 231, 250–263 (1955).
[CrossRef]

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London Ser. A 387, 105–132 (1983).
[CrossRef]

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London Ser. A 389, 279–290 (1983).
[CrossRef]

J. F. Nye, J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. London Ser. A 409, 21–36 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. 1. Theory,” Proc. R. Soc. London Ser. A 414, 433–446 (1987).
[CrossRef]

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. 2. Observations on the electric field,” Proc. R. Soc. London Ser. A 414, 447–468 (1987).
[CrossRef]

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]

Other

S. Hell, Wijnaendts-van-Resandt, “The application of polarized confocal microscopy for size measurement of resist structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. SPIE1139, 92–98 (1989).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. 9, pp. 311–407.

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).

J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communications Electronics, 2nd ed. (Wiley, New York, 1984).

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Figures (4)

Fig. 1
Fig. 1

Behavior of the electric field for the low-order modes of a square aperture.

Fig. 2
Fig. 2

Behavior of the electric field for the low-order modes of a circular aperture.

Fig. 3
Fig. 3

Electric and magnetic Hertz potentials for an electric dipole wave, numerical aperture of unity. The constant c is taken as zero.

Fig. 4
Fig. 4

Electric and magnetic Hertz potentials for an electric dipole wave, numerical aperture of (a) unity and (b) 0.85. The constant c is taken to make the TE11 component zero. For (b), θ is redefined, so that ρ sin α=sin θ, where sin α is the numerical aperture, 0.85.

Tables (2)

Tables Icon

Table 1 Modes for the Field over a Square Aperture in Cartesian Coordinates

Tables Icon

Table 2 Modes of the Field over a Circular Aperture in Cylindrical Coordinates

Equations (40)

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

2P+k2P=0,
2M+k2M=0,
E=    P+iωμ0  M,
H=    M-iω0  P.
P=ψe exp(ikz)k,
M=ψm exp(ikz)k,
E=2ψexz+iωμ0ψmyi+2ψeyz-iωμ0ψmxj+2ikzψez+2ψe2zkeikz.
ψey=0,ψmx=0.
E=2ψeρz+iωμ01ρψmϕaˆρ+1ρ2ψeϕz-iωμ0ψmρaˆϕ+2ikzψez+2ψe2zkexp(ikz),
l=k,m=0,l=k+1,m0,
k=n-m2,
LP0,k+1=TE1,k+1+TM1,k+1=LG0,k,k0,
Ex=dRn1(ρ)dρ+Rn1(ρ)ρ,
P=R1(ρ)cos ϕf(z)exp(ikz)k,
M=R2(ρ)cos ϕg(z)iωμ0exp(ikz)k,
dR(ρ)dρ+R(ρ)ρ=b(ρ),
R(ρ)=1ρb(ρ)ρdρ.
E=12{[S1(ρ)+S2(ρ)]i+[D1(ρ)-D2(ρ)]×(cos 2ϕi+sin 2ϕj)},
S1,2=dR1,2(ρ)dρ+R1,2(ρ)ρ=1ρd[ρR1,2(ρ)]dρ,
D1,2=dR1,2(ρ)dρ-R1,2(ρ)ρ.
E=ρ2(cos 2ϕi+sin 2ϕj),
E=ρ|m|(cos mϕi+sin mϕj),
m=|p-1|,
n=1+|p|.
E=a(θ)[(sin2 ϕ+cos θ cos2 ϕ)i-(1-cos θ)sin ϕ cos ϕj-sin θ cos ϕk],
Ep=a(θ)1+cos θ[(1-sin2 θ cos2 ϕ)i-sin2 θ sin ϕ cos ϕj-sin θ cos θ cos ϕk],
Em=a(θ)1+cos θ(cos θi-sin θ cos ϕk).
Ep=a(θ)1+cos θ(cos θ cos ϕaˆθ-sin ϕaˆϕ),
Em=a(θ)1+cos θ(cos ϕaˆθ-cos θ sin ϕaˆϕ).
E=a(θ)(cos ϕaˆθ-sin ϕaˆϕ),
Ep=12i-tan2θ2(cos 2ϕi+sin 2ϕj),
Em=12i+tan2θ2(cos 2ϕi+sin 2ϕj),
Ep=cos θ cos ϕ1+cos θaˆρ-sin ϕ1+cos θaˆϕ,
Em=cos ϕ1+cos θaˆρ-cos θ sin ϕ1+cos θaˆϕ.
R1=sin θ14+c+ln cosθ2+14sec2θ2,
R2=sin θ14-c-ln cosθ2-14sec2θ2,
01Rnm(ρ)Rnm(ρ)ρdρ=12(n+1)δnn.
E=b(ρ)2[(Rp-Rs)i+(Rp+Rs)(cos 2ϕi+sin 2ϕj)].
R1+R2=1ρb(ρ)(Rp-Rs)ρdρ,
R1-R2=ρb(ρ)(Rp+Rs)ρdρ.

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