Abstract

By generalization of the methods presented in Part I of the study [ J. Opt. Soc. Am. A 12, 600 ( 1994)] to the four-dimensional (4D) Riemannian manifold case, the time-dependent behavior of light transmitting in a medium is investigated theoretically by the geodesic equation and curvature in a 4D manifold. In addition, the field equation is restudied, and the 4D conserved current of the optical fluid and its conservation equation are derived and applied to deduce the time-dependent general refractive index. On this basis the forces acting on the fluid are dynamically analyzed and the self-consistency analysis is given.

© 1995 Optical Society of America

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References

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  1. See, for example, A. Gahtak, Optics (McGraw-Hill, New Delhi, 1977); Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin1990).
    [Crossref]
  2. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  3. A. G. Kostenbauder, “Ray–pulse matrices: a rational treatment for dispersive optical systems,” IEEE. J. Quantum Electron. 26, 1148–1157 (1990).
    [Crossref]
  4. Z. Zhang, D. Fan, “Temporal diffraction integration of optical systems and its applications,” Acta Opt. Sinica 12, 179–182 (1992) (in Chinese).
  5. G. Ding, B. Lü, “Canonical operator representations of temporal diffraction integration,” Acta Opt. Sinica (to be published) (in Chinese).
  6. H. Guo, X. Deng, “Differential geometrical methods in the study of optical transmission (scalar theory). I. Static transmission case,” J. Opt. Soc. Am. A 12, 600–606 (1995).
    [Crossref]
  7. T. Eguchi, P. B. Gilkey, A. J. Hanson, “Gravitation, gauge theories and differential geometry,” Phys. Rep. 66, 241–393 (1980).
    [Crossref]
  8. See, for example, P. Roman, Theory of Elementary Particles (North-Holland, Amsterdam, 1960).
  9. A. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  10. See, for example, I. M. Besieris, “Wave-kinetic methods, phase–space path integrals, and stochastic wave propagation,” J. Opt. Soc. Am. A 2, 2092–2098 (1985).
    [Crossref]
  11. J. Shamir, “Cylindrical lens systems described by operator algebra,” Appl. Opt. 18, 4195–4202 (1969).
    [Crossref]
  12. J. Durnin, “Exact solution for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  13. J. Durnin, J. J. Meceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  14. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
    [Crossref]
  15. O. Svelto, “Self-focusing, self-trapping, and self-phase modulation of laser beams,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. XII.
    [Crossref]
  16. R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [Crossref]
  17. S. Zhu, W. Shen, “General relativistic ponderomotive force in a moving medium,” J. Opt. Soc. Am. B 4, 739–742 (1987).
    [Crossref]

1995 (1)

1992 (1)

Z. Zhang, D. Fan, “Temporal diffraction integration of optical systems and its applications,” Acta Opt. Sinica 12, 179–182 (1992) (in Chinese).

1990 (1)

A. G. Kostenbauder, “Ray–pulse matrices: a rational treatment for dispersive optical systems,” IEEE. J. Quantum Electron. 26, 1148–1157 (1990).
[Crossref]

1987 (3)

1985 (1)

1980 (1)

T. Eguchi, P. B. Gilkey, A. J. Hanson, “Gravitation, gauge theories and differential geometry,” Phys. Rep. 66, 241–393 (1980).
[Crossref]

1969 (1)

1965 (1)

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[Crossref]

1964 (1)

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Besieris, I. M.

Born, M.

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

Chiao, R. Y.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Deng, X.

Ding, G.

G. Ding, B. Lü, “Canonical operator representations of temporal diffraction integration,” Acta Opt. Sinica (to be published) (in Chinese).

Durnin, J.

J. Durnin, J. J. Meceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

J. Durnin, “Exact solution for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Meceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Eguchi, T.

T. Eguchi, P. B. Gilkey, A. J. Hanson, “Gravitation, gauge theories and differential geometry,” Phys. Rep. 66, 241–393 (1980).
[Crossref]

Fan, D.

Z. Zhang, D. Fan, “Temporal diffraction integration of optical systems and its applications,” Acta Opt. Sinica 12, 179–182 (1992) (in Chinese).

Gahtak, A.

See, for example, A. Gahtak, Optics (McGraw-Hill, New Delhi, 1977); Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin1990).
[Crossref]

Garmire, E.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Gilkey, P. B.

T. Eguchi, P. B. Gilkey, A. J. Hanson, “Gravitation, gauge theories and differential geometry,” Phys. Rep. 66, 241–393 (1980).
[Crossref]

Guo, H.

Hanson, A. J.

T. Eguchi, P. B. Gilkey, A. J. Hanson, “Gravitation, gauge theories and differential geometry,” Phys. Rep. 66, 241–393 (1980).
[Crossref]

Kelley, P. L.

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[Crossref]

Kostenbauder, A. G.

A. G. Kostenbauder, “Ray–pulse matrices: a rational treatment for dispersive optical systems,” IEEE. J. Quantum Electron. 26, 1148–1157 (1990).
[Crossref]

Love, J. D.

A. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Lü, B.

G. Ding, B. Lü, “Canonical operator representations of temporal diffraction integration,” Acta Opt. Sinica (to be published) (in Chinese).

Meceli, J. J.

J. Durnin, J. J. Meceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Roman, P.

See, for example, P. Roman, Theory of Elementary Particles (North-Holland, Amsterdam, 1960).

Shamir, J.

Shen, W.

Snyder, A.

A. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Svelto, O.

O. Svelto, “Self-focusing, self-trapping, and self-phase modulation of laser beams,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. XII.
[Crossref]

Townes, C. H.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Wolf, E.

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

Zhang, Z.

Z. Zhang, D. Fan, “Temporal diffraction integration of optical systems and its applications,” Acta Opt. Sinica 12, 179–182 (1992) (in Chinese).

Zhu, S.

Acta Opt. Sinica (1)

Z. Zhang, D. Fan, “Temporal diffraction integration of optical systems and its applications,” Acta Opt. Sinica 12, 179–182 (1992) (in Chinese).

Appl. Opt. (1)

IEEE. J. Quantum Electron. (1)

A. G. Kostenbauder, “Ray–pulse matrices: a rational treatment for dispersive optical systems,” IEEE. J. Quantum Electron. 26, 1148–1157 (1990).
[Crossref]

J. Opt. Soc. Am. A (3)

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

Phys. Rep. (1)

T. Eguchi, P. B. Gilkey, A. J. Hanson, “Gravitation, gauge theories and differential geometry,” Phys. Rep. 66, 241–393 (1980).
[Crossref]

Phys. Rev. Lett. (3)

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

J. Durnin, J. J. Meceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[Crossref]

Other (6)

O. Svelto, “Self-focusing, self-trapping, and self-phase modulation of laser beams,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. XII.
[Crossref]

See, for example, P. Roman, Theory of Elementary Particles (North-Holland, Amsterdam, 1960).

A. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

G. Ding, B. Lü, “Canonical operator representations of temporal diffraction integration,” Acta Opt. Sinica (to be published) (in Chinese).

See, for example, A. Gahtak, Optics (McGraw-Hill, New Delhi, 1977); Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin1990).
[Crossref]

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

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Equations (45)

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g μ ν = [ - 1 n 2 n 2 n 2 ] ,
d τ 2 = n 2 d l 2 - d t 2 .
x μ = ( x 0 , x 1 , x 2 , x 3 ) , μ = ( x 0 , x 1 , x 2 , x 3 ) = ( t , ) .
g 00 = - 1 ,             g i j = n 2 ( x , t ) δ i j             ( i , j = 1 , 2 , 3 ) ,
d τ 2 = n 2 d l 2 - d t 2 ,
d τ = 0 ;
d τ 2 = ( e 1 ) 2 + ( e 2 ) 2 + ( e 3 ) 2 - ( e 0 ) 2 ,
e 0 = d t ,             e a = n d x a             ( a = 1 , 2 , 3 ) .
ω 0 a = ω a 0 = n t d x a             ( a = 1 , 2 , 3 ) ,
ω a b = - ω b a = 1 n ( n b d x a - n a d x b )             ( a , b = 1 , 2 , 3 ) ,
R 0 101 = R 0 202 = R 0 303 = - n t t n ,
R a b a b = n - 3 ( n a a + n b b ) - n - 4 ( n a 2 + n b 2 - n c 2 ) - ( n t n ) 2 ,
R = R a b a b ,
R = 2 ( R 0101 + R 0202 + R 0303 + R 1212 + R 1313 + R 2323 ) ,
R = 4 n 3 [ 2 n - 1 2 n ( n ) 2 ] + n 6 [ 2 n t 2 - 1 n ( n t ) 2 ] .
d 2 t = - n n t d l 2
d d l ( n d x d l ) = n ,
d l 2 = d x 2 + d y 2 + d z 2 ,             d n = n t d t + n · d x
2 ϕ - 2 ϕ / t 2 = 0 ;
λ λ ϕ = 0             ( λ = 0 , 1 , 2 , 3 ) ,
λ λ ϕ = 0.
λ λ ϕ = ( - g ) - 1 / 2 λ [ ( - g ) 1 / 2 g λ η η ϕ ] ,
2 ϕ - n 2 2 ϕ t 2 + log n · ϕ - 3 n n t ϕ t = 0 ,
ϕ ( x , t ) = ϕ 0 ( x , t ) exp [ i k L ( x , t ) ] ,
λ j λ = 0 ,
j λ = ϕ 0 2 λ L ;
j 0 = - ϕ 0 2 L t = ρ V 0 ,             j = ϕ 0 2 n 2 L = ρ V ,
ρ ( x , t ) = ϕ 0 2 ,             V 0 = - L t ,             V = L / n 2 .
λ L λ L = 1 k 2 λ λ ϕ 0 ϕ 0 ,             λ λ L = - 2 λ L λ ϕ 0 ϕ 0 .
( L ) 2 = n 2 ( L t ) 2 + 1 k 2 ϕ 0 ( 2 ϕ 0 - n 2 2 ϕ 0 t 2 + log n · ϕ 0 - 3 n n t ϕ 0 t ) .
( L ) 2 = n 2 + 1 k 2 ( 2 ϕ 0 ϕ 0 - n 2 ϕ 0 - 1 2 ϕ 0 t 2 + log n · log ϕ 0 - 3 n n t log ϕ 0 t ) ,
n G 2 = ( L ) 2 ,
g 00 = - 1 ,             g i j = n G 2 δ i j             ( i , j = 1 , 2 , 3 ) .
λ L λ L = 0 ,
V λ V λ = 0 ;
d x λ d s d x λ d s = 0.
0 d τ 2 = n G 2 d l 2 - d t 2 = g μ ν d x μ d x ν ,
V 0 = 1 ,             V = 1 n G d x d l .
T μ ν = ρ V μ V ν ,
f ν = μ T μ ν ,
μ T μ ν 0.
f 0 = - 3 ϕ 0 2 [ t ( log n G ) + 1 n G 3 ( n G · L ) ] ,
f = ϕ 0 2 [ - 5 ( n G / t ) n G 3 L - 5 n G · L n G 5 L + n G n G 3 ] .
f = 5 3 f 0 n G 2 L + ϕ 0 2 n G 3 n G ,
j = j 0 n G 2 L

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