Abstract

The propagation of light in a dielectric moving at relativistic velocities is investigated. The analysis shows that, in that case, the medium may support two original propagative modes, analogously to birefringence in crystal optics. The conditions at which the kinematic birefringence occurs and its main features are exhibited and compared with the usual birefringence. In particular, it is shown that this peculiar effect is of second order in relative velocity β=v/c, and the kinematic ordinary refractive index exhibits angular dependence contrary to what happens for the usual ordinary index. Moreover, the existence of negative values for the ordinary and extraordinary indices is manifested in the ultrarelativistic regime.

© 2011 Optical Society of America

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  1. A. J. Fresnel, “Sur l’influence du mouvement terrestre dans quelques phenomenes d’optique: lettre de M. Fresnel a M. Arago,” Ann. Chim. Phys. 9, 57–66 (1818).
  2. A. H. L. Fizeau, “Sur les hypotheses relatives a l’ether lumineux, et sur une experience qui parait demontrer que le mouvement des corps change la vitesse avec laquelle la lumiere se propage dans leur interieur,” C.R. Hebd. Seances Acad. Sci. 33, 349–354 (1851).
  3. H. A. Lorentz, Versuch einer Theorie der Elektrischen und Optischen Erscheinungen von Bewegten Korpern (Brill, 1895).
  4. A. Einstein, “On the electrodynamics of moving bodies,” Ann. Physik 322, 891–921 (1905).
    [CrossRef]
  5. W. Gordon, “Zur Lichtfortpflanzung nach der relativitätstheorie,” Ann. Phys. 377, 421–456 (1923).
    [CrossRef]
  6. A. A. Michelson, H. G. Gale, and F. Pearson, “The effect of the earth’s rotation on the velocity of light. II.,” Astrophys. J. 61, 140–145 (1925).
    [CrossRef]
  7. G. A. Sanders and S. Ezekiel, “Measurement of Fresnel drag in moving media using a ring-resonator technique,” J. Opt. Soc. Am. B 5, 674–678 (1988).
    [CrossRef]
  8. U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
    [CrossRef]
  9. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
    [CrossRef]
  10. R. W. Boyd and D. J. Gauthier, “ “Slow” and “fast” light,” in Progress in Optics 43, E.Wolf, ed. (Elsevier, 2002), Vol.  43, pp. 497–530.
    [CrossRef]
  11. U. Leonhardt and P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
    [CrossRef] [PubMed]
  12. I. Brevik and G. Halnes, “Light rays at optical black holes in moving media,” Phys. Rev. D 65, 024005 (2001).
    [CrossRef]
  13. T. G. Mackay and A. Lakhtakia, “Negative phase velocity in a uniformly moving, homogeneous, isotropic, dielectric magnetic medium,” J. Phys. A: Math. Gen. 37, 5697–5711(2004).
    [CrossRef]
  14. T. M. Grzegorczyk and J. A. Kong, “Electrodynamics of moving media inducing positive and negative refraction,” Phys. Rev. B 74, 033102 (2006).
    [CrossRef]
  15. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
  16. L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).
  17. R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, 1953).
  18. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 2006).
  19. G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312, 895–897 (2006).
    [CrossRef] [PubMed]
  20. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  21. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  22. S. G. Garcia, T. M. Hung-Bao, R. G. Martin, and B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
    [CrossRef]

2006 (2)

T. M. Grzegorczyk and J. A. Kong, “Electrodynamics of moving media inducing positive and negative refraction,” Phys. Rev. B 74, 033102 (2006).
[CrossRef]

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312, 895–897 (2006).
[CrossRef] [PubMed]

2004 (1)

T. G. Mackay and A. Lakhtakia, “Negative phase velocity in a uniformly moving, homogeneous, isotropic, dielectric magnetic medium,” J. Phys. A: Math. Gen. 37, 5697–5711(2004).
[CrossRef]

2001 (1)

I. Brevik and G. Halnes, “Light rays at optical black holes in moving media,” Phys. Rev. D 65, 024005 (2001).
[CrossRef]

2000 (2)

U. Leonhardt and P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

1999 (2)

U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

1996 (1)

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, and B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

1988 (1)

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

1925 (1)

A. A. Michelson, H. G. Gale, and F. Pearson, “The effect of the earth’s rotation on the velocity of light. II.,” Astrophys. J. 61, 140–145 (1925).
[CrossRef]

1923 (1)

W. Gordon, “Zur Lichtfortpflanzung nach der relativitätstheorie,” Ann. Phys. 377, 421–456 (1923).
[CrossRef]

1905 (1)

A. Einstein, “On the electrodynamics of moving bodies,” Ann. Physik 322, 891–921 (1905).
[CrossRef]

1851 (1)

A. H. L. Fizeau, “Sur les hypotheses relatives a l’ether lumineux, et sur une experience qui parait demontrer que le mouvement des corps change la vitesse avec laquelle la lumiere se propage dans leur interieur,” C.R. Hebd. Seances Acad. Sci. 33, 349–354 (1851).

1818 (1)

A. J. Fresnel, “Sur l’influence du mouvement terrestre dans quelques phenomenes d’optique: lettre de M. Fresnel a M. Arago,” Ann. Chim. Phys. 9, 57–66 (1818).

Barsi, C.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312, 895–897 (2006).
[CrossRef] [PubMed]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 2006).

Boyd, R. W.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312, 895–897 (2006).
[CrossRef] [PubMed]

R. W. Boyd and D. J. Gauthier, “ “Slow” and “fast” light,” in Progress in Optics 43, E.Wolf, ed. (Elsevier, 2002), Vol.  43, pp. 497–530.
[CrossRef]

Brevik, I.

I. Brevik and G. Halnes, “Light rays at optical black holes in moving media,” Phys. Rev. D 65, 024005 (2001).
[CrossRef]

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, 1953).

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Einstein, A.

A. Einstein, “On the electrodynamics of moving bodies,” Ann. Physik 322, 891–921 (1905).
[CrossRef]

Ezekiel, S.

Fizeau, A. H. L.

A. H. L. Fizeau, “Sur les hypotheses relatives a l’ether lumineux, et sur une experience qui parait demontrer que le mouvement des corps change la vitesse avec laquelle la lumiere se propage dans leur interieur,” C.R. Hebd. Seances Acad. Sci. 33, 349–354 (1851).

Fresnel, A. J.

A. J. Fresnel, “Sur l’influence du mouvement terrestre dans quelques phenomenes d’optique: lettre de M. Fresnel a M. Arago,” Ann. Chim. Phys. 9, 57–66 (1818).

Gale, H. G.

A. A. Michelson, H. G. Gale, and F. Pearson, “The effect of the earth’s rotation on the velocity of light. II.,” Astrophys. J. 61, 140–145 (1925).
[CrossRef]

Garcia, S. G.

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, and B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Gauthier, D. J.

R. W. Boyd and D. J. Gauthier, “ “Slow” and “fast” light,” in Progress in Optics 43, E.Wolf, ed. (Elsevier, 2002), Vol.  43, pp. 497–530.
[CrossRef]

Gehring, G. M.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312, 895–897 (2006).
[CrossRef] [PubMed]

Gordon, W.

W. Gordon, “Zur Lichtfortpflanzung nach der relativitätstheorie,” Ann. Phys. 377, 421–456 (1923).
[CrossRef]

Grzegorczyk, T. M.

T. M. Grzegorczyk and J. A. Kong, “Electrodynamics of moving media inducing positive and negative refraction,” Phys. Rev. B 74, 033102 (2006).
[CrossRef]

Halnes, G.

I. Brevik and G. Halnes, “Light rays at optical black holes in moving media,” Phys. Rev. D 65, 024005 (2001).
[CrossRef]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, 1953).

Hung-Bao, T. M.

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, and B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Kong, J. A.

T. M. Grzegorczyk and J. A. Kong, “Electrodynamics of moving media inducing positive and negative refraction,” Phys. Rev. B 74, 033102 (2006).
[CrossRef]

Kostinski, N.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312, 895–897 (2006).
[CrossRef] [PubMed]

Lakhtakia, A.

T. G. Mackay and A. Lakhtakia, “Negative phase velocity in a uniformly moving, homogeneous, isotropic, dielectric magnetic medium,” J. Phys. A: Math. Gen. 37, 5697–5711(2004).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

Leonhardt, U.

U. Leonhardt and P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

Lifschitz, E. M.

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

Lorentz, H. A.

H. A. Lorentz, Versuch einer Theorie der Elektrischen und Optischen Erscheinungen von Bewegten Korpern (Brill, 1895).

Mackay, T. G.

T. G. Mackay and A. Lakhtakia, “Negative phase velocity in a uniformly moving, homogeneous, isotropic, dielectric magnetic medium,” J. Phys. A: Math. Gen. 37, 5697–5711(2004).
[CrossRef]

Martin, R. G.

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, and B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Michelson, A. A.

A. A. Michelson, H. G. Gale, and F. Pearson, “The effect of the earth’s rotation on the velocity of light. II.,” Astrophys. J. 61, 140–145 (1925).
[CrossRef]

Olmedo, B. G.

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, and B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Pearson, F.

A. A. Michelson, H. G. Gale, and F. Pearson, “The effect of the earth’s rotation on the velocity of light. II.,” Astrophys. J. 61, 140–145 (1925).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Piwnicki, P.

U. Leonhardt and P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

Sanders, G. A.

Schweinsberg, A.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312, 895–897 (2006).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 2006).

Ann. Chim. Phys. (1)

A. J. Fresnel, “Sur l’influence du mouvement terrestre dans quelques phenomenes d’optique: lettre de M. Fresnel a M. Arago,” Ann. Chim. Phys. 9, 57–66 (1818).

Ann. Phys. (1)

W. Gordon, “Zur Lichtfortpflanzung nach der relativitätstheorie,” Ann. Phys. 377, 421–456 (1923).
[CrossRef]

Ann. Physik (1)

A. Einstein, “On the electrodynamics of moving bodies,” Ann. Physik 322, 891–921 (1905).
[CrossRef]

Astrophys. J. (1)

A. A. Michelson, H. G. Gale, and F. Pearson, “The effect of the earth’s rotation on the velocity of light. II.,” Astrophys. J. 61, 140–145 (1925).
[CrossRef]

C.R. Hebd. Seances Acad. Sci. (1)

A. H. L. Fizeau, “Sur les hypotheses relatives a l’ether lumineux, et sur une experience qui parait demontrer que le mouvement des corps change la vitesse avec laquelle la lumiere se propage dans leur interieur,” C.R. Hebd. Seances Acad. Sci. 33, 349–354 (1851).

IEEE Trans. Microwave Theory Tech. (1)

S. G. Garcia, T. M. Hung-Bao, R. G. Martin, and B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. A: Math. Gen. (1)

T. G. Mackay and A. Lakhtakia, “Negative phase velocity in a uniformly moving, homogeneous, isotropic, dielectric magnetic medium,” J. Phys. A: Math. Gen. 37, 5697–5711(2004).
[CrossRef]

Nature (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Phys. Rev. A (1)

U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

Phys. Rev. B (1)

T. M. Grzegorczyk and J. A. Kong, “Electrodynamics of moving media inducing positive and negative refraction,” Phys. Rev. B 74, 033102 (2006).
[CrossRef]

Phys. Rev. D (1)

I. Brevik and G. Halnes, “Light rays at optical black holes in moving media,” Phys. Rev. D 65, 024005 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

U. Leonhardt and P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

Science (1)

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312, 895–897 (2006).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Other (6)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, 1953).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 2006).

R. W. Boyd and D. J. Gauthier, “ “Slow” and “fast” light,” in Progress in Optics 43, E.Wolf, ed. (Elsevier, 2002), Vol.  43, pp. 497–530.
[CrossRef]

H. A. Lorentz, Versuch einer Theorie der Elektrischen und Optischen Erscheinungen von Bewegten Korpern (Brill, 1895).

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Figures (7)

Fig. 1
Fig. 1

Illustrating the kinematic birefringence. Intersection of the k surface with the k x k z plane for (a) moving dielectrics in the slow regime. (b) Intersection of the k surface with the k x k z plane for a usual uniaxial crystal illustrating the conventional birefringence. (c) Moving dielectrics in the ultrarelativistic regime. Notice that the k surface (sphere, ellipsoid, and hyperboloid of revolution) for β 0 is shifted with respect to the origin k x = k y = k z = 0 , raising an angular dependence for the ordinary index ( n o = n o ( θ ) ) contrary to what happens for the usual birefringence in which the k is centered at the origin and the sphere warrants the radial symmetry for the usual ordinary index ( n o n o ( θ ) ). The sphere depicted by the circular dotted line corresponds to the k surface for β = 0 , corresponding to a linear, homogeneous, and isotropic dielectric of index n . Notice that the kinematic spherical surface is not normal to the wave vector k due to the origin shift. Therefore, even for the ordinary case, the rays and the direction of energy transport are not orthogonal to the wavefronts. Hence, ordinary rays then have the “extraordinary” property of traveling at an oblique angle with their wavefronts.

Fig. 2
Fig. 2

Contour plot for n o showing the dependence of the o index on both the reduced velocity β and the angle θ inside water n = 1.33 in the regime v m < v ph . The broken curve corresponds to n o = n .

Fig. 3
Fig. 3

(a)  n o as a function of θ for several values of β. (b)  n o as a function of β for several values of θ. Both (a) and (b) are in the regime v m < v ph for n = 1.33 .

Fig. 4
Fig. 4

Contour plot for n o in the plane ( θ , β ) for the ultrarelativistic regime v ph < v m c for n = 1.33 . The broken curves correspond to θ = θ c and θ = π θ c and separate the three well-differentiated behaviors for the o index.

Fig. 5
Fig. 5

Contour plot for n e showing the dependence of the e index on both the reduced velocity β and the angle Θ for n = 1.33 in the regime v m < v ph . The broken curve corresponds to n e = n .

Fig. 6
Fig. 6

(a)  n e as a function of Θ for several values of β. (b)  n e as a function of β for several values of Θ. Both (a) and (b) are in the regime v ph < v m < c for n = 1.33 .

Fig. 7
Fig. 7

Contour plot for n e in the plane ( Θ , β ) for the ultrarelativistic regime v ph < v m < c for n = 1.33 . The thick curve corresponds to Θ = Θ s and separates the two propagative modes for the e index.

Equations (65)

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

k × E = ω B ,
k × H = ω D ,
k · D = 0 ,
k · B = 0 ,
D = ϵ E ,
B = μ H ,
v ph = 1 ϵ μ .
E = γ ( E + c β × B ) γ 2 γ + 1 ( E · β ) β ,
B = γ ( B 1 / c β × E ) γ 2 γ + 1 ( B · β ) β ,
D = γ ( D + 1 / c β × H ) γ 2 γ + 1 ( D · β ) β ,
H = γ ( H c β × D ) γ 2 γ + 1 ( H · β ) β ,
( D + 1 / c β × H ) γ γ + 1 ( D · β ) β = ϵ ( E + c β × B ) ϵ γ γ + 1 ( E · β ) β ,
( B 1 / c β × E ) γ γ + 1 ( B · β ) β = μ ( H c β × D ) μ γ γ + 1 ( H · β ) β .
D = α [ ϵ γ 2 E δ β × H + c δ ϵ ( β · E ) β ] ,
B = α [ μ γ 2 H + δ β × E + c δ μ ( β · H ) β ] ,
H · E = c ( β × D ) · E .
H · E = c δ γ 2 ( β · H ) ( β · E ) .
β · ( D × E ) = δ γ 2 ( β · H ) ( β · E ) .
β · ( B × H ) = α δ 1 c δ 1 c α δ ( β · H ) ( β · E ) .
rot E = B t ,
rot H = D t ,
div D = 0 ,
div B = 0.
F ( x μ ) = 1 ( 2 π ) 4 F ( K μ ) exp ( i K μ x μ ) d 4 K μ ,
k × E = ω α [ μ γ 2 H + δ β × E + c δ μ ( β · H ) β ] ,
k × H = ω α [ ϵ γ 2 E δ β × H + c δ ϵ ( β · E ) β ] ,
k · [ ϵ γ 2 E δ β × H + c δ ϵ ( β · E ) β ] = 0 ,
k · [ μ γ 2 H + δ β × E + c δ μ ( β · H ) β ] = 0.
k ˜ × [ k ˜ × E ] + ω 2 α 2 μ ϵ γ 2 [ 1 γ 2 E + c δ ( β · E ) β ] = ω α c δ μ ( β · H ) ( k ˜ × β )
H . β = 0 ,
E . β 0.
( k ˜ y 2 k ˜ z 2 + C 1 k ˜ x k ˜ y k ˜ x k ˜ z k ˜ x k ˜ y k ˜ x 2 k ˜ z 2 + C 1 k ˜ y k ˜ z k ˜ x k ˜ z k ˜ y k ˜ z k ˜ x 2 k ˜ y 2 + C 1 + C 2 β 2 ) ( E ^ x E ^ y E ^ z ) = ( 0 0 0 ) ,
( C 1 k ˜ 2 ) ( C 1 2 C 1 k ˜ 2 C 2 β 2 k ˜ z 2 + C 1 C 2 β 2 ) = 0.
k ˜ 2 = C 1 ,
C 1 k ˜ 2 + C 2 β 2 k ˜ z 2 C 1 C 2 β 2 C 1 2 = 0.
n ˜ x 2 + n ˜ y 2 + n ˜ z 2 = n 2 γ 4 ( 1 n 2 β 2 ) 2 ,
n ˜ x 2 + n ˜ y 2 n 2 γ 2 ( 1 n 2 β 2 ) + n ˜ z 2 n 2 γ 4 ( 1 n 2 β 2 ) 2 = 1 .
ϵ n 2 γ 4 ( 1 n 2 β 2 ) 2 ,
ϵ // n 2 γ 2 ( 1 n 2 β 2 ) .
tan θ = ϵ ϵ // tan θ .
tan θ ˜ = γ 2 ( 1 n 2 β 2 ) tan θ ˜ ,
f o ( k o , ω ) = k o x 2 + k o y 2 + ( k o z ω α δ β ) 2 ω 2 α 2 n 2 c 2 γ 4 = 0.
k o = n o ω c ( sin θ cos ϕ sin θ sin ϕ cos θ ) ,
n o 2 2 n o β ( 1 n 2 ) 1 n 2 β 2 cos θ + ( 1 n 2 ) 2 β 2 ( 1 β 2 ) 2 n 2 ( 1 n 2 β 2 ) 2 = 0.
n o = β ( n 2 1 ) 1 β 2 n 2 [ cos θ + ( ( 1 β 2 ) 2 n 2 ( 1 n 2 ) 2 β 2 1 ) + cos 2 θ ] > 0.
n o = β + n 1 + β n .
v o g = ω k o = f o k o f o ω .
v o g = k ˜ o ( k oz ω α δ β ) α δ β + ω α 2 n 2 c 2 γ 4 .
n o = β ( 1 n 2 ) 1 β 2 n 2 [ cos θ sin 2 ( θ c ) sin 2 ( θ ) ] ,
θ c = arcsin [ ( 1 β 2 ) n ( n 2 1 ) β ] .
k e = n e ω c ( sin Θ cos Φ sin Θ sin Φ cos Θ ) ,
[ sin 2 Θ + 1 n 2 β 2 1 β 2 cos 2 Θ ] n e 2 2 [ 1 n 2 1 β 2 β cos Θ ] n e + ( 1 n 2 ) 2 β 2 ( 1 β 2 ) 2 n 2 ( 1 n 2 β 2 ) ( 1 β 2 ) = 0.
n e = β ( n 2 1 ) cos Θ ( β 2 1 ) [ ( β 2 1 ) n 2 cos 2 Θ + ( β 2 n 2 ) sin 2 Θ ] ( β 2 n 2 1 ) cos 2 Θ + ( β 2 1 ) sin 2 Θ .
n e 2 2 ( 1 n 2 ) β cos Θ n e n 2 0.
Θ s = arccos ( 1 β 2 n 2 β 2 1 )
( β 2 n 2 1 ) cos 2 Θ + ( β 2 1 ) sin 2 Θ = 0.
u = u + v 1 + u v c 0 2 .
n = β + n 1 + β n .
n o 2 2 α β ( 1 n 2 ) cos θ n o + α 2 ( ( 1 n 2 ) 2 β 2 ( 1 β 2 ) 2 n 2 ) = 0.
Δ = α 2 β 2 ( 1 n 2 ) 2 [ g ( β ) g ( n ) sin 2 θ ] ,
g ( x ) = ( 1 x 2 ) 2 x 2 .
n o = α β ( 1 n 2 ) [ cos θ ( 1 β 2 ) 2 n 2 ( 1 n 2 ) 2 β 2 sin 2 θ ] .
θ c = arcsin ( g ( β ) g ( n ) ) .
Δ = α 2 β 2 ( 1 n 2 ) 2 [ sin 2 θ c sin 2 θ ] .
n o = α β ( 1 n 2 ) [ cos θ ± sin 2 θ c sin 2 θ ] .

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