Abstract

Static optical transmission is restudied by postulation of the optical path as the proper element in a three-dimensional Riemannian manifold (no torsion); this postulation can be applied to describe the light–medium interactive system. On the basis of the postulation, the behaviors of light transmitting through the medium with refractive index n are investigated, the investigation covering the realms of both geometrical optics and wave optics. The wave equation of light in static transmission is studied modally, the postulation being employed to derive the exact form of the optical field equation in a medium (in which the light is viewed as a single-component field). Correspondingly, the relationships concerning the conservation of optical fluid and the dynamic properties are given, and some simple applications of the theories mentioned are presented.

© 1995 Optical Society of America

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References

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  1. A. Gahtak, Optics (McGraw-Hill, New Delhi, 1977) Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).
    [Crossref]
  2. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  3. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [Crossref]
  4. T. Eguchi, P. B. Gilkey, A. J. Hanson, “Gravitation, gauge theories and differential geometry,” Phys. Rep. 66, 241–393 (1980).
    [Crossref]
  5. C. N. Yang, R. L. Mills, “Conservation of isotopic spin and isotopic gauge invariance,” Phys. Rev. 96, 191–195 (1954).
    [Crossref]
  6. R. Utiyama, “Invariant theoretical interpretation of interaction,” Phys. Rev. 101, 1597–1607 (1956).
    [Crossref]
  7. J. Durnin, “Exact solution for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  8. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  9. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  10. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).
  11. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
    [Crossref]
  12. E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, “Effects of nonlinear propagation on laser focusing properties,” IEEE J. Quantum Electron. QE-12, 402–406 (1976).
    [Crossref]

1987 (2)

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

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

1980 (1)

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

1976 (1)

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, “Effects of nonlinear propagation on laser focusing properties,” IEEE J. Quantum Electron. QE-12, 402–406 (1976).
[Crossref]

1970 (1)

1965 (1)

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

1956 (1)

R. Utiyama, “Invariant theoretical interpretation of interaction,” Phys. Rev. 101, 1597–1607 (1956).
[Crossref]

1954 (1)

C. N. Yang, R. L. Mills, “Conservation of isotopic spin and isotopic gauge invariance,” Phys. Rev. 96, 191–195 (1954).
[Crossref]

Bliss, E. S.

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, “Effects of nonlinear propagation on laser focusing properties,” IEEE J. Quantum Electron. QE-12, 402–406 (1976).
[Crossref]

Born, M.

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

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

Collins, S. A.

Durnin, J.

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

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

Eberly, J. H.

J. Durnin, J. J. Miceli, 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]

Gahtak, A.

A. Gahtak, Optics (McGraw-Hill, New Delhi, 1977) Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).
[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]

Hanson, A. J.

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

Hunt, J. T.

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, “Effects of nonlinear propagation on laser focusing properties,” IEEE J. Quantum Electron. QE-12, 402–406 (1976).
[Crossref]

Kelley, P. L.

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

Love, J. D.

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

Miceli, J. J.

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

Mills, R. L.

C. N. Yang, R. L. Mills, “Conservation of isotopic spin and isotopic gauge invariance,” Phys. Rev. 96, 191–195 (1954).
[Crossref]

Renard, P. A.

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, “Effects of nonlinear propagation on laser focusing properties,” IEEE J. Quantum Electron. QE-12, 402–406 (1976).
[Crossref]

Snyder, A. W.

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

Sommargren, G. E.

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, “Effects of nonlinear propagation on laser focusing properties,” IEEE J. Quantum Electron. QE-12, 402–406 (1976).
[Crossref]

Utiyama, R.

R. Utiyama, “Invariant theoretical interpretation of interaction,” Phys. Rev. 101, 1597–1607 (1956).
[Crossref]

Weaver, H. J.

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, “Effects of nonlinear propagation on laser focusing properties,” IEEE J. Quantum Electron. QE-12, 402–406 (1976).
[Crossref]

Wolf, E.

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

Yang, C. N.

C. N. Yang, R. L. Mills, “Conservation of isotopic spin and isotopic gauge invariance,” Phys. Rev. 96, 191–195 (1954).
[Crossref]

IEEE J. Quantum Electron. (1)

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, “Effects of nonlinear propagation on laser focusing properties,” IEEE J. Quantum Electron. QE-12, 402–406 (1976).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (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. (2)

C. N. Yang, R. L. Mills, “Conservation of isotopic spin and isotopic gauge invariance,” Phys. Rev. 96, 191–195 (1954).
[Crossref]

R. Utiyama, “Invariant theoretical interpretation of interaction,” Phys. Rev. 101, 1597–1607 (1956).
[Crossref]

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, 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 (4)

A. Gahtak, Optics (McGraw-Hill, New Delhi, 1977) Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).
[Crossref]

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

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

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

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Equations (94)

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d s 2 = n 2 d l 2 = g i j ( x ) d x i d x j             ( i , j = 1 , 2 , 3 ) ,
g i j = n 2 δ i j ,
d 2 x k d s 2 + Γ i j k d x i d s d x j d s = 0 ,
d s 2 = n 2 ( d x 2 + d y 2 + d z 2 ) = ( e 1 ) 2 + ( e 2 ) 2 + ( e 3 ) 2 ,
e 1 = n d x ,             e 2 = n d y ,             e 3 = n d z .
d e a + ω a b e b = 0 d ω a b + ω a c ω c b = - ½ R a b }             ( Cartan ' s structure equation ) ,
d e a + ω a b e b = 0             ( geodesic equation ) ,
R a b = R a b c d e c e d             ( a , b , c , d = 1 , 2 , 3 ) ,
R 212 1 = n - 3 ( n x x + n y y ) - n - 4 ( n x 2 + n y 2 - n z 2 ) , R 313 1 = n - 3 ( n x x + n z z ) - n - 4 ( n x 2 + n z 2 - n y 2 ) , R 323 2 = n - 3 ( n y y + n z z ) - n - 4 ( n y 2 + n z 2 - n x 2 ) ,
R = R a b a b = 4 n 3 [ 2 n - 1 2 n ( n ) 2 ] ,
ω 1 2 = - ω 2 1 = n - 1 ( n y d x - n x d y ) , ω 1 3 = - ω 3 1 = n - 1 ( n z d x - n x d z ) , ω 2 3 = - ω 3 2 = n - 1 ( n z d y - n y d z ) , ω a b = 0             ( others ) .
d d l ( n d x d l ) = n ,
2 ϕ + k 2 ϕ = 0 ,
i = ( x 1 , x 2 , x 3 )             d x i = ( d x 1 , d x 2 , d x 3 )
g = det ( g i j ) = n 6 ,             g i j = n - 2 δ i j ,
i i ϕ + k 2 ϕ = 0             ( i = 1 , 2 , 3 ) ,
i i ϕ = 1 g 1 / 2 i ( g 1 / 2 g i j j ϕ ) ,
2 ϕ + n 2 k 2 ϕ + log n · ϕ = 0.
k j k = 0 ,
j k = ϕ * k ϕ - ϕ k ϕ *             ( k = 1 , 2 , 3 ) .
j k = ϕ 0 2 k L = ϕ 0 2 n 2 k L ,
j = ϕ 0 2 n 2 L .
k L k L = 1 + 1 k 2 k k ϕ 0 ϕ 0 ,
( L ) 2 = n 2 + 1 k 2 2 ϕ 0 ϕ 0 + 1 k 2 log n · log ϕ 0 .
( L ) 2 = n 2 ,
n G 2 = ( L ) 2
= n 2 + 1 k 2 2 ϕ 0 ϕ 0 + 1 k 2 log n · ϕ 0 ,
ρ ( x ) = ϕ 0 2 ( x )             V k = k L .
T i j = ρ V i V j ,             ( i , j = 1 , 2 , 3 ) ,
f j = i T i j ,
i T i j = 0
f j = ϕ 0 2 n G 2 [ L · - 3 log n G · L ] V j             ( j = 1 , 2 , 3 ) ,
f = - 5 ϕ 0 2 n G · L n G 5 L + ϕ 0 2 n G 3 n G .
ϕ 0 = a 0 σ exp [ - ( r / σ ) 2 ] ,
σ ( z ) = ( k 2 σ 0 4 + 4 z 2 ) 1 / 2 k σ 0 ,
2 ϕ 0 ϕ 0 = - 4 σ 2 ( 1 - r 2 σ 2 ) + 1 4 ( 16 r 4 σ 2 σ 6 - 32 r 2 σ 2 σ 4 + 3 σ 2 σ 2 + 8 r 2 σ σ 3 - 2 σ σ ) ,
σ = 4 k σ 0 z ( k 2 σ 0 4 + 4 z 2 ) 1 / 2 ,
σ = 4 k σ [ 1 ( k 2 σ 0 4 + 4 z 2 ) 1 / 2 + z 2 1 ( k 2 σ 0 4 + 4 z 2 ) 3 / 2 ] ,
σ ( 0 ) = σ 0 ,             σ ( 0 ) = 0 ,             σ ( 0 ) = 4 k 2 σ 0 3 .
2 ϕ 0 ϕ 0 | z = 0 = - 4 σ 0 2 ( 1 - r 2 σ 0 2 ) ,
R z = 0 1 k 2 ( r σ 0 4 - 1 k 2 r 2 σ 0 8 ) ~ 1 k 2 σ 0 4 > 0 ,
R = 0.
2 ϕ 0 ϕ 0 = const .
ϕ 0 = 0 2 π A ( 0 ) exp [ i α ( x cos θ + y sin θ ) ] d θ
ϕ = ϕ exp ( i β z ) ,
( 2 + α 2 ) ϕ 0 = 0 ,
( 2 + α 2 ) ϕ 0 = 0 ,
2 ϕ 0 ϕ 0 = - α 2 = const .
ϕ 0 = J n ( c r )             or N n ( c r ) ,
n 2 + 1 k 2 2 ϕ 0 ϕ = const .
n = n 0 - r 2 k σ 0 3 ,
n = n 0 + n 2 I ,
n G = n 0 + n 2 I + 1 2 n 0 k 2 2 ϕ 0 ϕ 0 ,
n G = n 0 + n 2 I + 1 4 n 0 k 2 [ 2 I I - 1 2 ( log I ) 2 ] ;
n 2 I - 1 4 n 0 k 2 [ 2 I I - 1 2 ( log I ) 2 ] ,
R ~ 2 n G - 1 2 n G ( n G ) 2 ,
R = n 2 2 I - 1 2 n 0 ( n 2 I ) 2 ,
R n 2 2 I ,
2 I ~ - 8 σ 2 ( 1 - 2 r 2 σ 2 ) I < 0             ( r < σ ) ,
R < 0.
n 2 I - 1 4 n 0 k 2 [ 2 I I - 1 2 ( log I ) 2 ] ,
R > 0.
n 2 I = - 1 4 n 0 k 2 [ 2 I I - 1 2 ( log I ) 2 ] ,
n G = n 0 ;
R ~ 2 n 0 - 1 2 n 0 ( n 0 ) 2 = 0 ,
d 2 λ π ( 2 n 0 n 2 I ) - 1 / 2 ~ 0.62 λ ( 2 n 0 n 2 I ) - 1 / 2 ,
P cr = π 4 d 2 I ~ π ( 0.62 λ ) 2 8 n 0 n 2 ,
g i j = n G 2 δ i j ,
k L k L = 1 ,
V k V k = 1 ,
d s 2 = g i j d x i d x j = n G 2 d l 2 ,
V k = d x k d s ,             V k = d x k d s .
0 i T i j = i ( ρ V i V j ) , i ( ρ V i ) = i j i 0 ,
V i i V j = 0 ;
D V k D s = 0 ,
d 2 x k d s 2 + Γ i j k d x i d s d x j d s = 0 ,
V = L / n G 2 = 1 n G d x d l ;
n G 2 = n 2 + 1 k 2 2 ϕ 0 ϕ 0 .
n G 2 = 1 k 2 ( 2 ϕ 0 ϕ 0 ) ,
f = 0 ,             R = 0 ,
f 0 ,             R 0 ,
d s 2 = g i j d x i d x j ,
d s 2 = η a b e a i e b j d x i d x j ,
d s 2 = η a b e a e b ,
e a = e a i d x i             ( Cartan s frame ) ,
η a b = g i j e a i e b j             ( Cartan s metric ) ,
d e a + ω a b e b = 1 / 2 T a ,
d ω a b + ω a c ω c b = - 1 / 2 R a b ,
d e a + ω a b e b = 0 ,
Γ i j k = ½ g k l ( g i l , j + g j l , i - g i j , l ) ,
R j k l i = Γ j k , l i - Γ j l , k i + Γ j k m Γ l m i - Γ j l m Γ k m i .
d 2 x k d s 2 + Γ i j k d x i d s d x j d s = 0.
i ϕ = i ϕ ,
i i ϕ = 1 g i ( g g i j j ϕ ) .

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