Abstract

The angular spectrum of plane waves representation is obtained for the electromagnetic field radiated by a localized current source in a uniaxially anisotropic medium. The linear medium is homogeneous throughout all space. It may be absorbing and temporally dispersive but is not spatially dispersive. The conductivity and permeability are scalar constants. The time behavior of the source is arbitrary except that its magnitude is always bounded and it begins radiating at a finite time. The representation of monochromatic fields radiated by time-harmonic sources is included also in the results. The radiated field is expressed as a superposition of monochromatic ordinary and extraordinary plane waves propagating in various directions. The spectral amplitudes of the plane waves are determined explicitly in terms of the source. The orientation of the guide axis of the representation with respect to the optic axis of the medium is arbitrary. The form of the solution is particularly appropriate for the study of diffraction, reflection, and refraction at plane boundaries.

© 1976 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Chap. XIV.
  2. G. N. Ramachandran and S. Ramaseshan, “Crystal Optics,” in Handbuch der Physik, Vol. XXV/1, edited by S. Flügge, (Springer, Berlin, 1961).
    [Crossref]
  3. G. F. Carrier, Quart. Appl. Math. 4, 160 (1945).
  4. F. V. Bunkin, Sov. Phys.- JETP 5, 277 (1957).
  5. M. J. Lighthill, Philos. Trans. R. Soc. Lond. A 252, 397 (1960).
    [Crossref]
  6. N. R. Ogg, J. Phys. A 4, 382 (1971).
    [Crossref]
  7. V. T. Buchwald, Proc. R. Soc. Lond. A 253, 563 (1959).
    [Crossref]
  8. S. D. Nigam and P. D. Nigam, Proc. R. Soc. Lond. A 266, 247 (1962).
    [Crossref]
  9. G. A. Deschamps and O. B. Kesler, Radio Sci. 2, 757 (1967).
  10. K. S. Lee, Radio Sci. 3, 1098 (1968).
  11. M. Lax and D. F. Nelson, Coherence and Quantum Optics, edited by L. Mandel and E. Wolf, (Plenum, New York, 1973), p. 415.
    [Crossref]
  12. For example, see Sec. B in Electromagnetic Theory and Antennas, edited by E. C. Jordan (Pergamon, New York, 1963), Pt. 1.
  13. Radio Sci. 2, No. 8 (1967). This whole issue is devoted to anisotropic media.
  14. A. G. Sitenko and A. A. Kolomenskii, Sov. Phys. -JETP 30, 511 (1956).
  15. K. A. Barsukov, Sov. Phys. -JETP 36, 1485 (1959).
  16. G. A. Begiashville and E. V. Gedalin, Sov. Phys.- JETP 36, 1939 (1959).
  17. A. Wünsche, Ann. Phys. (Leipz.) 25, 179 (1970).
    [Crossref]
  18. M. Lax and D. F. Nelson, Phys. Rev. B 4, 3694 (1971).
    [Crossref]
  19. D. F. Nelson, P. D. Lazay, and M. Lax, Phys. Rev. B 6, 3109 (1972).
    [Crossref]
  20. L. Bergstein and T. Zachos, J. Opt. Soc. Am. 56, 931 (1966); J. Opt. Soc. Am. 61, 1477 (1971).
    [Crossref]
  21. A. Kujawski and J. Petykievicz, Opt. Commun. 3, 23 (1971).
    [Crossref]
  22. É. Lalor, J. Math. Phys. 13, 437 (1972); J. Math. Phys. 13, 449 (1972).
    [Crossref]
  23. P. C. Clemmow, Proc. IEE 110, 101 (1963); Proc. IEE 110, 107 (1963).
  24. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).
  25. L. B. Felsen, IEEE Trans. Antennas Propagation AP-12, 469 (1963); and Proc. IEE 111, 445 (1964).
    [Crossref]
  26. E. Arbel and L. B. Felsen, in Ref. 12, p. 391.
  27. L. B. Felsen, Radio Sci. 69D, 155 (1965).
  28. S. Gross and L. B. Felsen, Radio Sci. 69D, 333 (1965).
  29. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973).
  30. A. J. Devaney, Ph. D. thesis (University of Rochester, 1971) (unpublished).
  31. J. J. Stamnes, Ph.D. thesis (University of Rochester, 1974) (unpublished).
  32. A. Wünsche, Ann. Phys. (Leipz.) 25, 201 (1970).
    [Crossref]

1972 (2)

D. F. Nelson, P. D. Lazay, and M. Lax, Phys. Rev. B 6, 3109 (1972).
[Crossref]

É. Lalor, J. Math. Phys. 13, 437 (1972); J. Math. Phys. 13, 449 (1972).
[Crossref]

1971 (3)

A. Kujawski and J. Petykievicz, Opt. Commun. 3, 23 (1971).
[Crossref]

N. R. Ogg, J. Phys. A 4, 382 (1971).
[Crossref]

M. Lax and D. F. Nelson, Phys. Rev. B 4, 3694 (1971).
[Crossref]

1970 (2)

A. Wünsche, Ann. Phys. (Leipz.) 25, 201 (1970).
[Crossref]

A. Wünsche, Ann. Phys. (Leipz.) 25, 179 (1970).
[Crossref]

1968 (1)

K. S. Lee, Radio Sci. 3, 1098 (1968).

1967 (2)

Radio Sci. 2, No. 8 (1967). This whole issue is devoted to anisotropic media.

G. A. Deschamps and O. B. Kesler, Radio Sci. 2, 757 (1967).

1966 (1)

1965 (2)

L. B. Felsen, Radio Sci. 69D, 155 (1965).

S. Gross and L. B. Felsen, Radio Sci. 69D, 333 (1965).

1963 (2)

P. C. Clemmow, Proc. IEE 110, 101 (1963); Proc. IEE 110, 107 (1963).

L. B. Felsen, IEEE Trans. Antennas Propagation AP-12, 469 (1963); and Proc. IEE 111, 445 (1964).
[Crossref]

1962 (1)

S. D. Nigam and P. D. Nigam, Proc. R. Soc. Lond. A 266, 247 (1962).
[Crossref]

1960 (1)

M. J. Lighthill, Philos. Trans. R. Soc. Lond. A 252, 397 (1960).
[Crossref]

1959 (3)

V. T. Buchwald, Proc. R. Soc. Lond. A 253, 563 (1959).
[Crossref]

K. A. Barsukov, Sov. Phys. -JETP 36, 1485 (1959).

G. A. Begiashville and E. V. Gedalin, Sov. Phys.- JETP 36, 1939 (1959).

1957 (1)

F. V. Bunkin, Sov. Phys.- JETP 5, 277 (1957).

1956 (1)

A. G. Sitenko and A. A. Kolomenskii, Sov. Phys. -JETP 30, 511 (1956).

1945 (1)

G. F. Carrier, Quart. Appl. Math. 4, 160 (1945).

Arbel, E.

E. Arbel and L. B. Felsen, in Ref. 12, p. 391.

Barsukov, K. A.

K. A. Barsukov, Sov. Phys. -JETP 36, 1485 (1959).

Begiashville, G. A.

G. A. Begiashville and E. V. Gedalin, Sov. Phys.- JETP 36, 1939 (1959).

Bergstein, L.

Born, M.

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

Buchwald, V. T.

V. T. Buchwald, Proc. R. Soc. Lond. A 253, 563 (1959).
[Crossref]

Bunkin, F. V.

F. V. Bunkin, Sov. Phys.- JETP 5, 277 (1957).

Carrier, G. F.

G. F. Carrier, Quart. Appl. Math. 4, 160 (1945).

Clemmow, P. C.

P. C. Clemmow, Proc. IEE 110, 101 (1963); Proc. IEE 110, 107 (1963).

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

Deschamps, G. A.

G. A. Deschamps and O. B. Kesler, Radio Sci. 2, 757 (1967).

Devaney, A. J.

A. J. Devaney, Ph. D. thesis (University of Rochester, 1971) (unpublished).

Felsen, L. B.

S. Gross and L. B. Felsen, Radio Sci. 69D, 333 (1965).

L. B. Felsen, Radio Sci. 69D, 155 (1965).

L. B. Felsen, IEEE Trans. Antennas Propagation AP-12, 469 (1963); and Proc. IEE 111, 445 (1964).
[Crossref]

E. Arbel and L. B. Felsen, in Ref. 12, p. 391.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973).

Gedalin, E. V.

G. A. Begiashville and E. V. Gedalin, Sov. Phys.- JETP 36, 1939 (1959).

Gross, S.

S. Gross and L. B. Felsen, Radio Sci. 69D, 333 (1965).

Kesler, O. B.

G. A. Deschamps and O. B. Kesler, Radio Sci. 2, 757 (1967).

Kolomenskii, A. A.

A. G. Sitenko and A. A. Kolomenskii, Sov. Phys. -JETP 30, 511 (1956).

Kujawski, A.

A. Kujawski and J. Petykievicz, Opt. Commun. 3, 23 (1971).
[Crossref]

Lalor, É.

É. Lalor, J. Math. Phys. 13, 437 (1972); J. Math. Phys. 13, 449 (1972).
[Crossref]

Lax, M.

D. F. Nelson, P. D. Lazay, and M. Lax, Phys. Rev. B 6, 3109 (1972).
[Crossref]

M. Lax and D. F. Nelson, Phys. Rev. B 4, 3694 (1971).
[Crossref]

M. Lax and D. F. Nelson, Coherence and Quantum Optics, edited by L. Mandel and E. Wolf, (Plenum, New York, 1973), p. 415.
[Crossref]

Lazay, P. D.

D. F. Nelson, P. D. Lazay, and M. Lax, Phys. Rev. B 6, 3109 (1972).
[Crossref]

Lee, K. S.

K. S. Lee, Radio Sci. 3, 1098 (1968).

Lighthill, M. J.

M. J. Lighthill, Philos. Trans. R. Soc. Lond. A 252, 397 (1960).
[Crossref]

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973).

Nelson, D. F.

D. F. Nelson, P. D. Lazay, and M. Lax, Phys. Rev. B 6, 3109 (1972).
[Crossref]

M. Lax and D. F. Nelson, Phys. Rev. B 4, 3694 (1971).
[Crossref]

M. Lax and D. F. Nelson, Coherence and Quantum Optics, edited by L. Mandel and E. Wolf, (Plenum, New York, 1973), p. 415.
[Crossref]

Nigam, P. D.

S. D. Nigam and P. D. Nigam, Proc. R. Soc. Lond. A 266, 247 (1962).
[Crossref]

Nigam, S. D.

S. D. Nigam and P. D. Nigam, Proc. R. Soc. Lond. A 266, 247 (1962).
[Crossref]

Ogg, N. R.

N. R. Ogg, J. Phys. A 4, 382 (1971).
[Crossref]

Petykievicz, J.

A. Kujawski and J. Petykievicz, Opt. Commun. 3, 23 (1971).
[Crossref]

Ramachandran, G. N.

G. N. Ramachandran and S. Ramaseshan, “Crystal Optics,” in Handbuch der Physik, Vol. XXV/1, edited by S. Flügge, (Springer, Berlin, 1961).
[Crossref]

Ramaseshan, S.

G. N. Ramachandran and S. Ramaseshan, “Crystal Optics,” in Handbuch der Physik, Vol. XXV/1, edited by S. Flügge, (Springer, Berlin, 1961).
[Crossref]

Sitenko, A. G.

A. G. Sitenko and A. A. Kolomenskii, Sov. Phys. -JETP 30, 511 (1956).

Stamnes, J. J.

J. J. Stamnes, Ph.D. thesis (University of Rochester, 1974) (unpublished).

Wolf, E.

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

Wünsche, A.

A. Wünsche, Ann. Phys. (Leipz.) 25, 179 (1970).
[Crossref]

A. Wünsche, Ann. Phys. (Leipz.) 25, 201 (1970).
[Crossref]

Zachos, T.

Ann. Phys. (Leipz.) (2)

A. Wünsche, Ann. Phys. (Leipz.) 25, 179 (1970).
[Crossref]

A. Wünsche, Ann. Phys. (Leipz.) 25, 201 (1970).
[Crossref]

IEEE Trans. Antennas Propagation (1)

L. B. Felsen, IEEE Trans. Antennas Propagation AP-12, 469 (1963); and Proc. IEE 111, 445 (1964).
[Crossref]

J. Math. Phys. (1)

É. Lalor, J. Math. Phys. 13, 437 (1972); J. Math. Phys. 13, 449 (1972).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. A (1)

N. R. Ogg, J. Phys. A 4, 382 (1971).
[Crossref]

Opt. Commun. (1)

A. Kujawski and J. Petykievicz, Opt. Commun. 3, 23 (1971).
[Crossref]

Philos. Trans. R. Soc. Lond. A (1)

M. J. Lighthill, Philos. Trans. R. Soc. Lond. A 252, 397 (1960).
[Crossref]

Phys. Rev. B (2)

M. Lax and D. F. Nelson, Phys. Rev. B 4, 3694 (1971).
[Crossref]

D. F. Nelson, P. D. Lazay, and M. Lax, Phys. Rev. B 6, 3109 (1972).
[Crossref]

Proc. IEE (1)

P. C. Clemmow, Proc. IEE 110, 101 (1963); Proc. IEE 110, 107 (1963).

Proc. R. Soc. Lond. A (2)

V. T. Buchwald, Proc. R. Soc. Lond. A 253, 563 (1959).
[Crossref]

S. D. Nigam and P. D. Nigam, Proc. R. Soc. Lond. A 266, 247 (1962).
[Crossref]

Quart. Appl. Math. (1)

G. F. Carrier, Quart. Appl. Math. 4, 160 (1945).

Radio Sci. (5)

G. A. Deschamps and O. B. Kesler, Radio Sci. 2, 757 (1967).

K. S. Lee, Radio Sci. 3, 1098 (1968).

Radio Sci. 2, No. 8 (1967). This whole issue is devoted to anisotropic media.

L. B. Felsen, Radio Sci. 69D, 155 (1965).

S. Gross and L. B. Felsen, Radio Sci. 69D, 333 (1965).

Sov. Phys. -JETP (2)

A. G. Sitenko and A. A. Kolomenskii, Sov. Phys. -JETP 30, 511 (1956).

K. A. Barsukov, Sov. Phys. -JETP 36, 1485 (1959).

Sov. Phys.- JETP (2)

G. A. Begiashville and E. V. Gedalin, Sov. Phys.- JETP 36, 1939 (1959).

F. V. Bunkin, Sov. Phys.- JETP 5, 277 (1957).

Other (9)

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

G. N. Ramachandran and S. Ramaseshan, “Crystal Optics,” in Handbuch der Physik, Vol. XXV/1, edited by S. Flügge, (Springer, Berlin, 1961).
[Crossref]

M. Lax and D. F. Nelson, Coherence and Quantum Optics, edited by L. Mandel and E. Wolf, (Plenum, New York, 1973), p. 415.
[Crossref]

For example, see Sec. B in Electromagnetic Theory and Antennas, edited by E. C. Jordan (Pergamon, New York, 1963), Pt. 1.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973).

A. J. Devaney, Ph. D. thesis (University of Rochester, 1971) (unpublished).

J. J. Stamnes, Ph.D. thesis (University of Rochester, 1974) (unpublished).

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

E. Arbel and L. B. Felsen, in Ref. 12, p. 391.

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Figures (2)

FIG. 1
FIG. 1

(a) Backward waves for χ > 0. (b) Backward waves for χ < 0.

FIG. 2
FIG. 2

Poynting vector and wave vector (χ < 0).

Equations (128)

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· D ( r , t ) = 4 π ρ ( r , t ) ,
c × E ( r , t ) = - B ˙ ( r , t ) ,
· B ( r , t ) = 0 ,
c × B ( r , t ) = μ D ˙ ( r , t ) + 4 π μ [ J o ( r , t ) + σ E ( r , t ) ] ,
A ( k , ω ) = 0 - A ( r , t ) exp [ - i ( k · r - ω t ) ] d r d t .
A ( r , t ) = ( 2 π ) - 4 i a - i a + - + A ( k , ω ) exp [ i ( k · r - ω t ) ] d k d ω ,
A ( r , t ) = 2 Re ( 2 π ) - 4 i a i a + - A ( k , ω ) exp [ i ( k · r - ω t ) ] d k d ω ,
D ( k , ω ) = ( ω ) · E ( k , ω ) ,
ρ ˙ ( r , t ) = - · J ( r , t ) ,
ρ ( k , ω ) = [ k · J o ( k , ω ) + σ k · E ( k , ω ) ] / ω .
ω k · [ c ( ω ) · E ( k , ω ) ] = - 4 π i k · J o ( k , ω ) ,
c k × E ( k , ω ) = ω B ( k , ω ) ,
k · B ( k , ω ) = 0 ,
c k × B ( k , ω ) = - ω μ c · E ( k , ω ) - 4 π i μ J o ( k , ω ) ,
c = ( ω ) + 4 π i σ I / ω ,
E ( k , ω ) = 4 π i ω μ Λ - 1 · J o ( k , ω ) / c 2 ,
Λ = k 2 I - kk - ω 2 μ c ( ω ) / c 2 .
= o I + o ( e / o - 1 ) ŝ ŝ .
= o ê x ê x + o ê y ê y + e ê z ê z ,
Λ = ( k 2 - ξ ) I - ξ χ ŝ ŝ - kk ,
ξ = ω 2 μ ( o + 4 π i σ / ω ) / c 2 ,
η = ω 2 μ ( e + 4 π i σ / ω ) / c 2 ,
χ = η / ξ - 1.
Λ - 1 = ( k × ŝ ) ( k × ŝ ) ( k 2 - ξ ) ( k × ŝ ) 2 + Q ( k ) Q ( k ) ( k × ŝ ) 2 [ k 2 - ( k × ŝ ) 2 ( η - ξ ) / η ] [ k 2 - ξ - ( k × ŝ ) 2 ( η - ξ ) / η ] - kk η [ k 2 - ( k × ŝ ) 2 ( η - ξ ) / η ] ,
Q ( k ) = k × ( k × ŝ ) + ŝ ( k × ŝ ) 2 ( η - ξ ) / η .
E ( r , t ) = μ 2 π 3 c 2 Re i i a i a + - ω Λ - 1 J o ( k , ω ) × exp [ i ( k · r - ω t ) ] d k d ω ,
B ( r , t ) = μ 2 π 3 c Re i i a i a + - k × { Λ - 1 · J o ( k , ω ) } × exp [ i ( k · r - ω t ) ] d k d ω ,
J o ( k , ω ) = - Z Z I ( k x , k y , ω , z ) exp ( - i k z z ) d z ,
I ( k x , k y , ω , z ) = 0 - J o ( r , t ) × exp [ - i ( k x x + k y y - ω t ) ] d x d y d t .
J o i ( k , ω ) K exp ( k z Z ) ,
F = k 2 - ( k × ŝ ) 2 ( η - ξ ) / η
Q ( k ) Q ( k ) F ( F - ξ ) = Q ( k ) Q ( k ) ξ ( F - ξ ) - Q ( k ) Q ( k ) ξ F .
Q ( k ) Q ( k ) = ( k · ŝ ) 2 kk - ( k · ŝ ) F ( k ŝ + ŝ k ) + F 2 ŝ ŝ .
- ( k · ŝ ) 2 kk ξ F - ( k × ŝ ) 2 kk η F = - kk ξ ,
( k · ŝ ) 2 + ( k × ŝ ) 2 = k 2 .
Λ - 1 = ξ - 1 ( k × ŝ ) - 2 ( ξ ( k × ŝ ) ( k × ŝ ) k 2 - ξ + Q ( k ) Q ( k ) F - ξ - kk + ( k · ŝ ) ( k ŝ + ŝ k ) - F ŝ ŝ ) .
k z 2 + k t 2 - ξ = 0 ,
k z 2 + M k z + N = 0 ,
M = 2 n χ ( k t · ŝ ) / ( 1 + χ n 2 ) ,
N = [ χ ( k t · s t ) 2 + ( k t 2 - η ) ] / ( 1 + χ n 2 ) .
k z o = ( ξ - k t 2 ) 1 / 2 ,
k z e ± = α ± β 1 / 2 ,
α = - χ n k t · s t / ( 1 + χ n 2 ) ,
β = ( η - k t 2 ) / ( 1 + χ n 2 ) - α 2 / ( χ n 2 ) .
E ± ( r , t ) = ( μ ( π c ) 2 ) Re i a i a + - ( R o ± + R e ± ) d k x d k y d ω ,
k 2 - ξ = ( k z - k z o ) ( k z + k z o ) ,
F - ξ = ( 1 + χ n 2 ) ξ ( k z - k z e + ) ( k z - k z e - ) / η .
R o ± = ± ω ( k o ± × ŝ ) · J o ( k o ± , ω ) 2 k z o ( k o ± × ŝ ) 2 ( k o ± × ŝ ) exp [ i ( k o ± · r - ω t ) ] ,
R e ± = ± ω η Q ( k e ± ) · J o ( k e ± , ω ) ( 1 + χ n 2 ) ( k z e + - k z e - ) ξ 2 ( k e ± × ŝ ) 2 × Q ( k e ± ) exp [ i ( k e ± · r - ω t ) ] ,
k o ± = k x ê x + k y ê y ± k z o ê z ,
k e ± = k x ê x + k y ê y + k z e ± ê z .
R e ± = ± ω η [ k e ± × ( k e ± + ŝ ) · J o ( k e ± , ω ) + ŝ · J o ( k e ± , ω ) ( k e ± × ŝ ) 2 ( η - ξ ) / η ] ( 1 + χ n 2 ) 2 β 1 / 2 ξ 2 ( k e ± × ŝ ) 2 ( k e ± × ( k e ± × ŝ ) + ŝ ( k e ± × ŝ ) 2 ( η - ξ ) η ) exp [ i ( k e ± · r - ω t ) ] .
( k e ± ) 2 - ( k e ± × ŝ ) 2 ( η - ξ ) / η = ξ .
R e ± = ± ω η { ( k e ± · ŝ ) [ k e ± · J o ( k e ± , ω ) ] - ξ ŝ · J o ( k e ± , ω ) } ( 1 + χ n 2 ) 2 β 1 / 2 ξ 2 ( k e ± × ŝ ) 2 × [ ( k e ± · ŝ ) k e ± - ξ ŝ ] exp [ i ( k e ± · r - ω t ) ] .
E ± ( r , t ) = E o ± ( r , t ) + E e ± ( r , t ) ,
B ± ( r , t ) = B o ± ( r , t ) + B e ± ( r , t ) ,
V p ± ( r , t ) = Re i a i a + - v p ( k p ± , ω ) × exp [ i ( k p ± · r - ω t ) ] d k x d k y d ω ,
e o ( k , ω ) = ω μ [ k × J o ( k , ω ) ] · ŝ 2 π 2 c 2 k z o ( k × ŝ ) 2 ( k × ŝ ) ,
e e ( k , ω ) = - ω μ η { [ k · J o ( k , ω ) ] k - ξ J o ( k , ω ) } · ŝ 2 π 2 c 2 ξ 2 ( 1 + χ n 2 ) β 1 / 2 ( k × ŝ ) 2 × [ ( k · ŝ ) k - ξ ŝ ] ,
b p ( k , ω ) = c k × e p ( k , ω ) / ω ,
k p ± = k x ê x + k y ê y + k z p ± ê z ,
k z o ± = ± ( ξ - k t 2 ) 1 / 2 ,
k z e ± = α ± β 1 / 2 ,
α = - χ n k t · s t / ( 1 + χ n 2 ) ,
β = ( η - k t 2 ) / ( 1 + χ n 2 ) - α 2 / ( χ n 2 ) ,
χ = η / ξ - 1 ,
ξ = ω 2 μ ( o + 4 π i σ / ω ) / c 2 ,
η = ω 2 μ ( e + 4 π i σ / ω ) / c 2 ,
k t = k x ê x + k y ê y ,
k t = k t .
V p ± ( r , t ) = Re i a i a + V p ± ( r , ω ) exp ( - i ω t ) d ω ,
· · = ξ [ 2 + χ ( ŝ · ) 2 ] ,
[ · · + ξ ( χ + 1 ) ] V e ± ( r , ω ) = 0.
( 2 + ξ ) V o ± ( r , ω ) = 0 ,
e o ( k o ± , ω ) = ω μ [ k o ± × J o ( k o ± , ω ) ] · ê z 2 π 2 c 2 k z o k t 2 ( k o ± × ê z ) ,
e e ( k e ± , ω ) = - ω μ { [ k e ± · J o ( k e ± , ω ) ] k e ± - ξ J o ( k e ± , ω ) } · ê z 2 π 2 c 2 ξ k z e k t 2 × [ k z e ± k e ± - ξ ê z ] ,
b p ( k p ± , ω ) = c k p ± × e p ( k p ± , ω ) / ω ,
k p ± = k x ê x + k y ê y ± k z p ê z ,
k z o = ( ξ - k t 2 ) 1 / 2 ,
k z e = ( ξ - ξ k t 2 / η ) 1 / 2 .
σ = σ o ê x ê x + σ o ê y ê y + σ e ê z ê z .
k t 2 ξ < 1 + χ 1 + χ ( 1 - n 2 ) cos 2 ϕ / ( 1 + χ n 2 ) ,
cos ϕ = k t · s t k t s t = k t · s t k t ( 1 - n 2 ) 1 / 2
α 2 > β
k t 2 ξ > 1 + χ 1 + χ ( 1 - n 2 ) cos 2 ϕ .
1 + χ 1 + χ ( 1 - n 2 ) cos 2 ϕ < k t 2 ξ < 1 + χ 1 + χ ( 1 - n 2 ) cos 2 ϕ / ( 1 + χ n 2 ) .
ρ 2 = ( 1 + χ ) / [ 1 + χ ( 1 - n 2 ) cos 2 ϕ ]
a 1 = { ( χ + 1 ) / [ 1 + χ ( 1 - n 2 ) ] } 1 / 2 ,
b 1 = ( 1 + χ ) 1 / 2 .
ρ 2 = 1 + χ 1 + χ ( 1 - n 2 ) cos 2 ϕ / ( 1 + χ n 2 )
a 2 = ( 1 + χ n 2 ) 1 / 2 ,
b = b 1 .
S ± = c Re [ E ± × ( B ± ) * ] / 8 π μ ,
E ± = α e ± g ± exp [ i ( k e ± - ω t ) ] ,
B ± = c k e ± + E ± / ω ,
g ± = ξ ŝ - ( k e ± · ŝ ) k e ± ,
α e ± = g ± · e e ( k e ± , ω ) / g ± 2 ,
S ± = c 2 α e ± 2 Re [ g ± × ( k e ± × g ± ) * ] / 8 π μ ω .
S ± = c 2 α e ± 2 ξ { k e ± [ ξ - ( k e ± · ŝ ) 2 ] - ŝ ( k e ± · g ± ) } / 8 π μ ω .
k e ± 2 - η = - χ ( k e ± · ŝ ) 2 ,
V × U 2 = V 2 U 2 - ( V · U ) 2
S ± = ξ 2 c 2 α e ± 2 k e ± × ŝ 2 8 π μ ω η [ k e ± + χ ( k e ± · ŝ ) ŝ ] .
ê z · S ± = C ± [ k z e ± + χ n ( k t · s t + n k z e ± ) ] ,
ê z · S ± = ± C ± ( 1 + χ n 2 ) β ,
C ± = ξ 2 c 2 α e ± 2 k e ± × ŝ 2 8 π μ ω η > 0.
k e ± · S ± = C ± [ k e ± 2 + χ ( k e ± · ŝ ) 2 ] = C ± [ k e ± 2 - ( k e ± · ŝ ) 2 + ( χ + 1 ) ( k e ± · ŝ ) 2 ] ,
cos Φ ± = k e ± · S ± k e ± S ± = 1 + χ cos 2 δ ± [ 1 + χ ( χ + 2 ) cos 2 δ ± ] 1 / 2 ,
k e ± · ŝ = k e ± cos δ ± .
tan Φ ± = χ sin δ ± cos δ ± 1 + χ cos 2 δ ± .
P N 2 2 + M 2 2 N 1 - M 1 M 2 N 2 > 0.
M 1 = 2 A n ( χ 1 + n 2 χ 2 ) k t ζ ,
M 2 = 2 A n χ 2 k t ζ ,
N 1 = A { k t 2 [ χ 1 ( ζ 2 + n 2 ) + 1 + n 2 χ 2 ζ 2 ] - η 1 - n 2 ( χ 1 η 1 + χ 2 η 2 ) } ,
N 2 = A { k t 2 χ 2 ( ζ 2 - n 2 ) - η 2 - n 2 ( χ 1 η 2 - χ 2 η 1 ) } ,
A = 1 + χ n 2 - 2 ,
ζ = ( k t · s t ) / k t ,
k t = k t .
P = ( u k t 2 + v ) 2 + δ k t 2 ,
u = ( ζ 2 + n 2 ) χ 2 ,
v = n 2 ( χ 1 η 2 - χ 2 η 1 ) + η 2 ,
δ = - 4 ζ 2 χ 2 η 2 .
P = u 2 k t 2 + ( 2 u v + δ ) k t 2 + v 2 .
k t 2 = { - ( 2 u v + δ ) ± [ δ ( 4 u v + δ ) ] 1 / 2 } / 2 u 2 .
P ˜ 4 u v + δ = 4 n 2 [ ( ζ 2 + n 2 ) ( χ 1 χ 2 η 2 - χ 2 2 η 1 ) + χ 2 η 2 ]
P ˆ χ 2 η 2 + χ 1 χ 2 η 2 - χ 2 2 η 1
χ 1 = [ ξ 1 ( η 1 - ξ 1 ) + ξ 2 ( η 2 - ξ 2 ) ] / ξ 2 ,
χ 2 = ( ξ 1 η 2 - η 1 ξ 2 ) / ξ 2 .
P ˆ = χ 2 ξ 2 η 2 / ξ 2 .