Abstract

Light experiences matter as an effective curved space–time. With such a geometrization of the underlying background of electromagnetic waves, I derive and discuss the general form of the radiative-transfer equation in any weakly absorbing, linear or not, static or in motion, participating media. The role played by the geometry of the effective space–time on the energetic balance is highlighted, and, in particular, it is demonstrated that the curvature can give rise to an amazing amplifying effect on the radiant intensity even in the presence of absorption. An application on the problem of expanding dielectric envelopes shows the simplicity of such an approach for solving numerous radiation hydrodynamic problems.

© 2002 Optical Society of America

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References

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  1. F. Dalfovo, S. Giorgini, L. P. Pitaevski, and S. Stringari, “Theory of Bose–Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
    [CrossRef]
  2. W. G. Unruh, “Sonic analogue of black holes and the effects of high frequencies on black hole evaporation,” Phys. Rev. D 51, 2827–2838 (1995).
    [CrossRef]
  3. N. B. Kopnin and G. E. Volovik, “Critical velocity and event horizon in pair-correlated systems with relativistic fermionic quasiparticles,” JETP Lett. 67, 140–145 (1998).
    [CrossRef]
  4. M. Visser, “Acoustic black holes: horizon, ergospheres, and Hawking radiation,” Class. Quantum Grav. 15, 1767–1792 (1998).
    [CrossRef]
  5. L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic black holes in dilute Bose–Einstein condensates,” Phys. Rev. A 63, 023611 (2001).
    [CrossRef]
  6. L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
    [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. W. Dittrich and H. Gies, “Light propagation in nontrivial QED vacua,” Phys. Rev. D 58, 025004 (1998).
    [CrossRef]
  10. S. Liberati, S. Sonego, and M. Visser, “Scharnhorst effect at oblique incidence,” Phys. Rev. D 63, 085003 (2001).
    [CrossRef]
  11. M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspect of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 045001 (2000).
    [CrossRef]
  12. M. Novello and J. M. Salim, “Effective electromagnetic geometry,” Phys. Rev. D 63, 083511 (2001).
    [CrossRef]
  13. U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
    [CrossRef]
  14. Ulf Leonhardt, “Space–time geometry of quantum dielectrics,” Phys. Rev. A 62, 012111 (2000).
    [CrossRef]
  15. T. A. Jacobson and G. E. Volovik, “Effective spacetime and Hawking radiation from a moving domain wall in a thin film of 3He-A,” JETP Lett. 68, 874–880 (1998).
    [CrossRef]
  16. G. Befeki, Radiation Processes in Plasmas (Wiley, New York, 1966).
  17. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).
  18. P. M. Alsing, “The optical-mechanical analogy for stationary metrics in general relativity,” Am. J. Phys. 66, 779–790 (1998).
    [CrossRef]
  19. M. Born and E. Wolf, Principles of Optics (Cambridge University, Cambridge, UK, 1997).
  20. D. Mihalas and B. W. Mihalas, Foundations of Radiation Hydrodynamics (Dover, New York, 1999).
  21. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  22. P. Ben-Abdallah, “Radiative transfer in static and spherically symmetric distorting media: from the effective geometry to a kinetic theory,” J. Quant. Spectrosc. Radiat. Transf. 73, 69–90 (2002).
    [CrossRef]
  23. L. Landau and E. Lifshitz, Electrodynamique des Milieux Continues (Mir, Moscow, 1981).
  24. V. A. De Lorenci and M. A. Souza, “Electromagnetic wave propagation inside a material medium: an effective geometry interpretation,” Phys. Lett. B 512, 417–422 (2001).
    [CrossRef]
  25. C. H. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, New York, 1973).
  26. A. Einstein, H. Lorentz, A. Minkowsky, and H. Weyl, The Principle of Relativity (collected papers) (Dover, New York, 1952).
  27. I. T. Drummond and S. J. Hathrell, “Quantum vacuum polarization in a background gravitational field and its effect on the velocity of photons,” Phys. Rev. D 22, 343–355 (1980).
    [CrossRef]
  28. L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
    [CrossRef] [PubMed]

2002 (1)

P. Ben-Abdallah, “Radiative transfer in static and spherically symmetric distorting media: from the effective geometry to a kinetic theory,” J. Quant. Spectrosc. Radiat. Transf. 73, 69–90 (2002).
[CrossRef]

2001 (4)

V. A. De Lorenci and M. A. Souza, “Electromagnetic wave propagation inside a material medium: an effective geometry interpretation,” Phys. Lett. B 512, 417–422 (2001).
[CrossRef]

S. Liberati, S. Sonego, and M. Visser, “Scharnhorst effect at oblique incidence,” Phys. Rev. D 63, 085003 (2001).
[CrossRef]

M. Novello and J. M. Salim, “Effective electromagnetic geometry,” Phys. Rev. D 63, 083511 (2001).
[CrossRef]

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic black holes in dilute Bose–Einstein condensates,” Phys. Rev. A 63, 023611 (2001).
[CrossRef]

2000 (5)

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (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]

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspect of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 045001 (2000).
[CrossRef]

Ulf Leonhardt, “Space–time geometry of quantum dielectrics,” Phys. Rev. A 62, 012111 (2000).
[CrossRef]

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

1999 (3)

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]

F. Dalfovo, S. Giorgini, L. P. Pitaevski, and S. Stringari, “Theory of Bose–Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

1998 (5)

N. B. Kopnin and G. E. Volovik, “Critical velocity and event horizon in pair-correlated systems with relativistic fermionic quasiparticles,” JETP Lett. 67, 140–145 (1998).
[CrossRef]

M. Visser, “Acoustic black holes: horizon, ergospheres, and Hawking radiation,” Class. Quantum Grav. 15, 1767–1792 (1998).
[CrossRef]

W. Dittrich and H. Gies, “Light propagation in nontrivial QED vacua,” Phys. Rev. D 58, 025004 (1998).
[CrossRef]

T. A. Jacobson and G. E. Volovik, “Effective spacetime and Hawking radiation from a moving domain wall in a thin film of 3He-A,” JETP Lett. 68, 874–880 (1998).
[CrossRef]

P. M. Alsing, “The optical-mechanical analogy for stationary metrics in general relativity,” Am. J. Phys. 66, 779–790 (1998).
[CrossRef]

1995 (1)

W. G. Unruh, “Sonic analogue of black holes and the effects of high frequencies on black hole evaporation,” Phys. Rev. D 51, 2827–2838 (1995).
[CrossRef]

1980 (1)

I. T. Drummond and S. J. Hathrell, “Quantum vacuum polarization in a background gravitational field and its effect on the velocity of photons,” Phys. Rev. D 22, 343–355 (1980).
[CrossRef]

Alsing, P. M.

P. M. Alsing, “The optical-mechanical analogy for stationary metrics in general relativity,” Am. J. Phys. 66, 779–790 (1998).
[CrossRef]

Anglin, J. R.

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic black holes in dilute Bose–Einstein condensates,” Phys. Rev. A 63, 023611 (2001).
[CrossRef]

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[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]

Ben-Abdallah, P.

P. Ben-Abdallah, “Radiative transfer in static and spherically symmetric distorting media: from the effective geometry to a kinetic theory,” J. Quant. Spectrosc. Radiat. Transf. 73, 69–90 (2002).
[CrossRef]

Cirac, J. I.

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic black holes in dilute Bose–Einstein condensates,” Phys. Rev. A 63, 023611 (2001).
[CrossRef]

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[CrossRef] [PubMed]

Dalfovo, F.

F. Dalfovo, S. Giorgini, L. P. Pitaevski, and S. Stringari, “Theory of Bose–Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

De Lorenci, V. A.

V. A. De Lorenci and M. A. Souza, “Electromagnetic wave propagation inside a material medium: an effective geometry interpretation,” Phys. Lett. B 512, 417–422 (2001).
[CrossRef]

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspect of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 045001 (2000).
[CrossRef]

Dittrich, W.

W. Dittrich and H. Gies, “Light propagation in nontrivial QED vacua,” Phys. Rev. D 58, 025004 (1998).
[CrossRef]

Dogariu, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Drummond, I. T.

I. T. Drummond and S. J. Hathrell, “Quantum vacuum polarization in a background gravitational field and its effect on the velocity of photons,” Phys. Rev. D 22, 343–355 (1980).
[CrossRef]

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]

Garay, L. J.

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic black holes in dilute Bose–Einstein condensates,” Phys. Rev. A 63, 023611 (2001).
[CrossRef]

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[CrossRef] [PubMed]

Gies, H.

W. Dittrich and H. Gies, “Light propagation in nontrivial QED vacua,” Phys. Rev. D 58, 025004 (1998).
[CrossRef]

Giorgini, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevski, and S. Stringari, “Theory of Bose–Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[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]

Hathrell, S. J.

I. T. Drummond and S. J. Hathrell, “Quantum vacuum polarization in a background gravitational field and its effect on the velocity of photons,” Phys. Rev. D 22, 343–355 (1980).
[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]

Jacobson, T. A.

T. A. Jacobson and G. E. Volovik, “Effective spacetime and Hawking radiation from a moving domain wall in a thin film of 3He-A,” JETP Lett. 68, 874–880 (1998).
[CrossRef]

Klippert, R.

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspect of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 045001 (2000).
[CrossRef]

Kopnin, N. B.

N. B. Kopnin and G. E. Volovik, “Critical velocity and event horizon in pair-correlated systems with relativistic fermionic quasiparticles,” JETP Lett. 67, 140–145 (1998).
[CrossRef]

Kuzmich, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

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]

Leonhardt, Ulf

Ulf Leonhardt, “Space–time geometry of quantum dielectrics,” Phys. Rev. A 62, 012111 (2000).
[CrossRef]

Liberati, S.

S. Liberati, S. Sonego, and M. Visser, “Scharnhorst effect at oblique incidence,” Phys. Rev. D 63, 085003 (2001).
[CrossRef]

Novello, M.

M. Novello and J. M. Salim, “Effective electromagnetic geometry,” Phys. Rev. D 63, 083511 (2001).
[CrossRef]

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspect of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 045001 (2000).
[CrossRef]

Pitaevski, L. P.

F. Dalfovo, S. Giorgini, L. P. Pitaevski, and S. Stringari, “Theory of Bose–Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

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]

Salim, J. M.

M. Novello and J. M. Salim, “Effective electromagnetic geometry,” Phys. Rev. D 63, 083511 (2001).
[CrossRef]

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspect of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 045001 (2000).
[CrossRef]

Sonego, S.

S. Liberati, S. Sonego, and M. Visser, “Scharnhorst effect at oblique incidence,” Phys. Rev. D 63, 085003 (2001).
[CrossRef]

Souza, M. A.

V. A. De Lorenci and M. A. Souza, “Electromagnetic wave propagation inside a material medium: an effective geometry interpretation,” Phys. Lett. B 512, 417–422 (2001).
[CrossRef]

Stringari, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevski, and S. Stringari, “Theory of Bose–Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Unruh, W. G.

W. G. Unruh, “Sonic analogue of black holes and the effects of high frequencies on black hole evaporation,” Phys. Rev. D 51, 2827–2838 (1995).
[CrossRef]

Visser, M.

S. Liberati, S. Sonego, and M. Visser, “Scharnhorst effect at oblique incidence,” Phys. Rev. D 63, 085003 (2001).
[CrossRef]

M. Visser, “Acoustic black holes: horizon, ergospheres, and Hawking radiation,” Class. Quantum Grav. 15, 1767–1792 (1998).
[CrossRef]

Volovik, G. E.

T. A. Jacobson and G. E. Volovik, “Effective spacetime and Hawking radiation from a moving domain wall in a thin film of 3He-A,” JETP Lett. 68, 874–880 (1998).
[CrossRef]

N. B. Kopnin and G. E. Volovik, “Critical velocity and event horizon in pair-correlated systems with relativistic fermionic quasiparticles,” JETP Lett. 67, 140–145 (1998).
[CrossRef]

Wang, L. J.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Zoller, P.

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic black holes in dilute Bose–Einstein condensates,” Phys. Rev. A 63, 023611 (2001).
[CrossRef]

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[CrossRef] [PubMed]

Am. J. Phys. (1)

P. M. Alsing, “The optical-mechanical analogy for stationary metrics in general relativity,” Am. J. Phys. 66, 779–790 (1998).
[CrossRef]

Class. Quantum Grav. (1)

M. Visser, “Acoustic black holes: horizon, ergospheres, and Hawking radiation,” Class. Quantum Grav. 15, 1767–1792 (1998).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (1)

P. Ben-Abdallah, “Radiative transfer in static and spherically symmetric distorting media: from the effective geometry to a kinetic theory,” J. Quant. Spectrosc. Radiat. Transf. 73, 69–90 (2002).
[CrossRef]

JETP Lett. (2)

T. A. Jacobson and G. E. Volovik, “Effective spacetime and Hawking radiation from a moving domain wall in a thin film of 3He-A,” JETP Lett. 68, 874–880 (1998).
[CrossRef]

N. B. Kopnin and G. E. Volovik, “Critical velocity and event horizon in pair-correlated systems with relativistic fermionic quasiparticles,” JETP Lett. 67, 140–145 (1998).
[CrossRef]

Nature (2)

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

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. Lett. B (1)

V. A. De Lorenci and M. A. Souza, “Electromagnetic wave propagation inside a material medium: an effective geometry interpretation,” Phys. Lett. B 512, 417–422 (2001).
[CrossRef]

Phys. Rev. A (3)

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

Ulf Leonhardt, “Space–time geometry of quantum dielectrics,” Phys. Rev. A 62, 012111 (2000).
[CrossRef]

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic black holes in dilute Bose–Einstein condensates,” Phys. Rev. A 63, 023611 (2001).
[CrossRef]

Phys. Rev. D (6)

W. G. Unruh, “Sonic analogue of black holes and the effects of high frequencies on black hole evaporation,” Phys. Rev. D 51, 2827–2838 (1995).
[CrossRef]

W. Dittrich and H. Gies, “Light propagation in nontrivial QED vacua,” Phys. Rev. D 58, 025004 (1998).
[CrossRef]

S. Liberati, S. Sonego, and M. Visser, “Scharnhorst effect at oblique incidence,” Phys. Rev. D 63, 085003 (2001).
[CrossRef]

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspect of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 045001 (2000).
[CrossRef]

M. Novello and J. M. Salim, “Effective electromagnetic geometry,” Phys. Rev. D 63, 083511 (2001).
[CrossRef]

I. T. Drummond and S. J. Hathrell, “Quantum vacuum polarization in a background gravitational field and its effect on the velocity of photons,” Phys. Rev. D 22, 343–355 (1980).
[CrossRef]

Phys. Rev. Lett. (2)

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (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]

Rev. Mod. Phys. (1)

F. Dalfovo, S. Giorgini, L. P. Pitaevski, and S. Stringari, “Theory of Bose–Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Other (8)

M. Born and E. Wolf, Principles of Optics (Cambridge University, Cambridge, UK, 1997).

D. Mihalas and B. W. Mihalas, Foundations of Radiation Hydrodynamics (Dover, New York, 1999).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

C. H. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, New York, 1973).

A. Einstein, H. Lorentz, A. Minkowsky, and H. Weyl, The Principle of Relativity (collected papers) (Dover, New York, 1952).

G. Befeki, Radiation Processes in Plasmas (Wiley, New York, 1966).

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).

L. Landau and E. Lifshitz, Electrodynamique des Milieux Continues (Mir, Moscow, 1981).

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Figures (1)

Fig. 1
Fig. 1

Plot of the phase and group velocity within an expanding dielectric: (a) The phase velocity becomes negative at countercurrent when the expansion velocity is closed to the light velocity in vacuum. (b) The group velocity shows the presence of ultrafast phenomena in a moving dielectric for ultrarelativistic expansion in cones around the radial directions.

Equations (67)

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

Fμν=Eμνν-Eννμ+ημνρσνρHσ,
Pμν=Dμνν-Dννμ+ημνρσνρBσ,
Dν=ε(E, H)Eν,
Bν=μHν,
νFμν*=0,νPμν=0,
Fμν*=12ημντσFτσ.
Fμν=Fμν exp(iS),
S=-Kνdxν,
dxν=(dt, dx)
Kν(ω,-K)=-νS
gμν(μS)(νS)=0.
gμν=γμν-1-μ (εE)E+1(KV)H εHητναβHτKνVαEβVμVν-1εE εEEμEν,
δF(λ, xμ, x˙μ)dλ=δgμνx˙μ2x˙ν2dλ=0,
dFx˙μdλ-Fxμ=0foranyxμ.
g00ω2+2g0αωKα+gαβKαKβKβ=0,α,β=1,2,3.
νphase=ω|K|
g00νphase2-2g0αkανphase+gαβkαkβ=0,
neff=1νphase.
νphase2(1-A)+2A(V·k)νphase
-1+A(V·k)2+1εE εE(E·k)2=0,
A=γ21-μ (εE)E+1(KV)H εHητναβHτKνVαEβ.
νphase=-A(V·k)+1+A[(V·k)2-1]1-A.
vg=dωdK=dνphasedk+νphasek,
vg=A1-A V·k{1+A[(V·k)2-1]}1/2-1V+νphasek,
k=(sin ψ cos θ, sin ψ sin θ, cos ψ),
V=V(sin α cos β, sin α sin β, cos β).
ng=1νgroup,
dds Iνν3ng2=0
dds Iν(x, t, Ω)ν3ng2(x, t, Ω)+κν(x, t) Iν(x, t, Ω)ν3ng2(x, t, Ω)
=Sν(x, t, Ω),
Sν(x, t, Ω)=Qν(x, t)+0+ 1ν3 4πσin(x, t, νν, ΩΩ) Iν(x, t, Ω)ng2(x, t, Ω) dΩdν
dIν(x, t, Ω)ds+κν(x, t)-d log(ν3ng2)dsIν(x, t, Ω)
=ν3ng2Sν(x, t, Ω),
drds=Hp=t,dpds=-Hr.
t=Ω-np=Ω- ln(n)Ω,dpds=n.
dpds=n dΩds+Ω dnds=n.
dnds=n drds+nΩ dΩds=nΩ- ln(n)Ω+nΩ dΩds.
= ln(n)-Ω[ ln(n)Ω]=Ω×[ ln(n)×Ω].
dΩds=ρN,
ρ=Ω×[ ln(n)×Ω]+ ln(n)Ω  ln(n)Ω1+Ω  ln(n)Ω.
ρ=|Ω×[ ln(n)×Ω]|=| ln(n)×Ω|=| ln(n)·N|.
G(s)=Iν(s)Iν(0)=GaGg,
Ga=exp-0sκνdτ<1,
Gg=ν3ng2ν03ng02=exp0s d log(ng2ν3)ds˜dτ.
gμν=ημν+(εμ-1)VμVν,
Vμγ(1, V(r), 0, 0),
δ[gttτ˙2+2grtr˙τ˙+grrr˙2-r2(θ˙2+sin2 θφ˙2)]dλ=0,
gtt=1+1εμ-1γ2,
grt=-γ1εμ-1V(r),
grr=-1+1εμ-1γ2V2(r).
dφdr=±(-det g)1/2r(ηr2-gtt)1/2,
νphase=(εμ-1)γ2V cos φ+(εμ-1)γ2(1-V2 cos2 φ)+11+(εμ-1)γ2,
νg2=νphase2+ξV(2νphase cos φ+ξV),
ξ=A1-A V cos φ1+A(V2 cos2 φ-1)-1.
dφdr=±1r(ηng2r2-1)1/2.
η-1=ng2r2 sin2 ζ=ng2(r0)r02 sin2 ζ0,
ds=dMdrdr=1+r2dφdr21/2dr.
ds=1+ng2(r0)r02 sin2 ζ0r2[ng2(r)r2-ng2(r0)r02 sin2 ζ0]1/2dr,
ng2(r)r2 sin2 ζ=ng2(r0)r02 sin2 ζ0ng2(r)r2,
h(r)=ng2(r˜)r˜2-ng2(r0)r02 sin2 ζ0.
r˙2=E02-Veff(r),
Veff(r)=E02[1-ng2(r)]+h02r2.
r2φ˙=h0,τ˙ng2=E0.
E02=Veff(r˜).
η=E02h02.
ν0=1+V cos φ1-V2 ν.
Gg=(1+V2)3/2(1+V cos φ)3 r0 sin ζ0r sin ζ2.

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