Abstract

The change in the polarization plane of light vector beams retaining their shape (beam modes) under propagation through gradient-index media is evaluated as a topological phase acquired by cyclic and noncyclic evolutions of these beams on their projective Hilbert space (momentum sphere). The polarization changes are evaluated by means of the characteristic parameters of the light beam selected.

© 1994 Optical Society of America

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References

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  1. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
    [CrossRef]
  2. Y. Aharonov, J. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
    [CrossRef] [PubMed]
  3. J. Samuel, R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
    [CrossRef] [PubMed]
  4. A. Tomita, R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
    [CrossRef] [PubMed]
  5. G. S. Argarwal, “Quantum theory of partially polarizing devices and SU(1,1) Berry’s phase in polarization optics,” Opt. Commun. 82, 213–217 (1991).
    [CrossRef]
  6. T. H. Chyba, L. J. Wang, L. Mandel, R. Simon, “Measurement of the Pancharatnam phase factor for a light beam,” Opt. Lett. 13, 562–564 (1988).
    [CrossRef] [PubMed]
  7. J. Liñares, M.C. Nistal, “Geometric phases in multidirectional electromagnetic theory,” Phys. Lett. A 162, 7–14 (1992).
    [CrossRef]
  8. J. N. Ross, “The rotation of polarization in low birefringence monomode optical fibers due to geometric effects,” Opt. Quant. Electron. 16, 455–461 (1984).
    [CrossRef]
  9. R. Y. Chiao, Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
    [CrossRef] [PubMed]
  10. O. J. Kwon, H. T. Lee, S. B. Lee, S. C. Choi, “Observation of a topological phase in a noncyclic case by use of a half-turn optical fiber,” Opt. Lett. 16, 223–225 (1991).
    [CrossRef] [PubMed]
  11. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), pp. 41–61.
  12. L. Lewin, Theory of Waveguides (Wiley, New York, 1975), pp. 90–111.
  13. O. Costaños, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 159–182; J. R. Klauder, “Wave theoy of imaging systems,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 183–191; V. I. Man’ko, K. B. Wolf, “The influence of spherical aberration on Gaussian Beam propagation,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 206–225; V. I. Man’ko, “Invariants and coherent states in fiber optics,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 193–205.
    [CrossRef]
  14. S. G. Krivoshlykov, I. N. Sissakian, “Optical beam and pulse propagation in inhomogeneous media. Application to multimode parabolic-index waveguides,” Opt. Quantum Electron. 12, 463–475 (1980).
    [CrossRef]
  15. W. K. Kahn, S. Yang, “Hamiltonian analysis of beams in an optical slab guide,” J. Opt. Soc. Am. 73, 684–689 (1983).
    [CrossRef]
  16. C. Gomez-Reino, J. Liñares, “Optical path integrals in graded-index media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
    [CrossRef]
  17. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 100–111.
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 109–119.
  19. M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature (London) 326, 277–278 (1987).
    [CrossRef]
  20. S. M. Rytov, “Transition from wave to geometrical optics,” Dokl. Akad. Nauk. SSSR 18, 263–267 (1938); V. V. Vladimirskii, “The rotation of polarization plane for curved light ray,” Dokl. Akad. Nauk. SSSR 21, 222–226 (1941).
  21. J. Liñares, M. C. Nistal, “Geometric phases in chiral waveguiding structures,” J. Mod. Opt. 40, 889–900 (1993).
    [CrossRef]

1993

J. Liñares, M. C. Nistal, “Geometric phases in chiral waveguiding structures,” J. Mod. Opt. 40, 889–900 (1993).
[CrossRef]

1992

J. Liñares, M.C. Nistal, “Geometric phases in multidirectional electromagnetic theory,” Phys. Lett. A 162, 7–14 (1992).
[CrossRef]

1991

O. J. Kwon, H. T. Lee, S. B. Lee, S. C. Choi, “Observation of a topological phase in a noncyclic case by use of a half-turn optical fiber,” Opt. Lett. 16, 223–225 (1991).
[CrossRef] [PubMed]

G. S. Argarwal, “Quantum theory of partially polarizing devices and SU(1,1) Berry’s phase in polarization optics,” Opt. Commun. 82, 213–217 (1991).
[CrossRef]

1988

1987

Y. Aharonov, J. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
[CrossRef] [PubMed]

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature (London) 326, 277–278 (1987).
[CrossRef]

C. Gomez-Reino, J. Liñares, “Optical path integrals in graded-index media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
[CrossRef]

1986

A. Tomita, R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[CrossRef] [PubMed]

R. Y. Chiao, Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[CrossRef] [PubMed]

1984

J. N. Ross, “The rotation of polarization in low birefringence monomode optical fibers due to geometric effects,” Opt. Quant. Electron. 16, 455–461 (1984).
[CrossRef]

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
[CrossRef]

1983

1980

S. G. Krivoshlykov, I. N. Sissakian, “Optical beam and pulse propagation in inhomogeneous media. Application to multimode parabolic-index waveguides,” Opt. Quantum Electron. 12, 463–475 (1980).
[CrossRef]

1938

S. M. Rytov, “Transition from wave to geometrical optics,” Dokl. Akad. Nauk. SSSR 18, 263–267 (1938); V. V. Vladimirskii, “The rotation of polarization plane for curved light ray,” Dokl. Akad. Nauk. SSSR 21, 222–226 (1941).

Aharonov, Y.

Y. Aharonov, J. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
[CrossRef] [PubMed]

Anandan, J.

Y. Aharonov, J. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
[CrossRef] [PubMed]

Argarwal, G. S.

G. S. Argarwal, “Quantum theory of partially polarizing devices and SU(1,1) Berry’s phase in polarization optics,” Opt. Commun. 82, 213–217 (1991).
[CrossRef]

Berry, M. V.

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature (London) 326, 277–278 (1987).
[CrossRef]

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
[CrossRef]

Bhandari, R.

J. Samuel, R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 109–119.

Chiao, R. Y.

A. Tomita, R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[CrossRef] [PubMed]

R. Y. Chiao, Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[CrossRef] [PubMed]

Choi, S. C.

Chyba, T. H.

Costaños, O.

O. Costaños, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 159–182; J. R. Klauder, “Wave theoy of imaging systems,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 183–191; V. I. Man’ko, K. B. Wolf, “The influence of spherical aberration on Gaussian Beam propagation,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 206–225; V. I. Man’ko, “Invariants and coherent states in fiber optics,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 193–205.
[CrossRef]

Gomez-Reino, C.

Kahn, W. K.

Krivoshlykov, S. G.

S. G. Krivoshlykov, I. N. Sissakian, “Optical beam and pulse propagation in inhomogeneous media. Application to multimode parabolic-index waveguides,” Opt. Quantum Electron. 12, 463–475 (1980).
[CrossRef]

Kwon, O. J.

Lee, H. T.

Lee, S. B.

Lewin, L.

L. Lewin, Theory of Waveguides (Wiley, New York, 1975), pp. 90–111.

Liñares, J.

J. Liñares, M. C. Nistal, “Geometric phases in chiral waveguiding structures,” J. Mod. Opt. 40, 889–900 (1993).
[CrossRef]

J. Liñares, M.C. Nistal, “Geometric phases in multidirectional electromagnetic theory,” Phys. Lett. A 162, 7–14 (1992).
[CrossRef]

C. Gomez-Reino, J. Liñares, “Optical path integrals in graded-index media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
[CrossRef]

Lopez-Moreno, E.

O. Costaños, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 159–182; J. R. Klauder, “Wave theoy of imaging systems,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 183–191; V. I. Man’ko, K. B. Wolf, “The influence of spherical aberration on Gaussian Beam propagation,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 206–225; V. I. Man’ko, “Invariants and coherent states in fiber optics,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 193–205.
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), pp. 41–61.

Mandel, L.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 100–111.

Nistal, M. C.

J. Liñares, M. C. Nistal, “Geometric phases in chiral waveguiding structures,” J. Mod. Opt. 40, 889–900 (1993).
[CrossRef]

Nistal, M.C.

J. Liñares, M.C. Nistal, “Geometric phases in multidirectional electromagnetic theory,” Phys. Lett. A 162, 7–14 (1992).
[CrossRef]

Ross, J. N.

J. N. Ross, “The rotation of polarization in low birefringence monomode optical fibers due to geometric effects,” Opt. Quant. Electron. 16, 455–461 (1984).
[CrossRef]

Rytov, S. M.

S. M. Rytov, “Transition from wave to geometrical optics,” Dokl. Akad. Nauk. SSSR 18, 263–267 (1938); V. V. Vladimirskii, “The rotation of polarization plane for curved light ray,” Dokl. Akad. Nauk. SSSR 21, 222–226 (1941).

Samuel, J.

J. Samuel, R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

Simon, R.

Sissakian, I. N.

S. G. Krivoshlykov, I. N. Sissakian, “Optical beam and pulse propagation in inhomogeneous media. Application to multimode parabolic-index waveguides,” Opt. Quantum Electron. 12, 463–475 (1980).
[CrossRef]

Tomita, A.

A. Tomita, R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[CrossRef] [PubMed]

Wang, L. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 109–119.

Wolf, K. B.

O. Costaños, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 159–182; J. R. Klauder, “Wave theoy of imaging systems,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 183–191; V. I. Man’ko, K. B. Wolf, “The influence of spherical aberration on Gaussian Beam propagation,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 206–225; V. I. Man’ko, “Invariants and coherent states in fiber optics,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 193–205.
[CrossRef]

Wu, Y. S.

R. Y. Chiao, Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[CrossRef] [PubMed]

Yang, S.

Dokl. Akad. Nauk. SSSR

S. M. Rytov, “Transition from wave to geometrical optics,” Dokl. Akad. Nauk. SSSR 18, 263–267 (1938); V. V. Vladimirskii, “The rotation of polarization plane for curved light ray,” Dokl. Akad. Nauk. SSSR 21, 222–226 (1941).

J. Mod. Opt.

J. Liñares, M. C. Nistal, “Geometric phases in chiral waveguiding structures,” J. Mod. Opt. 40, 889–900 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature (London)

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature (London) 326, 277–278 (1987).
[CrossRef]

Opt. Commun.

G. S. Argarwal, “Quantum theory of partially polarizing devices and SU(1,1) Berry’s phase in polarization optics,” Opt. Commun. 82, 213–217 (1991).
[CrossRef]

Opt. Lett.

Opt. Quant. Electron.

J. N. Ross, “The rotation of polarization in low birefringence monomode optical fibers due to geometric effects,” Opt. Quant. Electron. 16, 455–461 (1984).
[CrossRef]

Opt. Quantum Electron.

S. G. Krivoshlykov, I. N. Sissakian, “Optical beam and pulse propagation in inhomogeneous media. Application to multimode parabolic-index waveguides,” Opt. Quantum Electron. 12, 463–475 (1980).
[CrossRef]

Phys. Lett. A

J. Liñares, M.C. Nistal, “Geometric phases in multidirectional electromagnetic theory,” Phys. Lett. A 162, 7–14 (1992).
[CrossRef]

Phys. Rev. Lett.

Y. Aharonov, J. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
[CrossRef] [PubMed]

J. Samuel, R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

A. Tomita, R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[CrossRef] [PubMed]

R. Y. Chiao, Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
[CrossRef]

Other

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 100–111.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 109–119.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), pp. 41–61.

L. Lewin, Theory of Waveguides (Wiley, New York, 1975), pp. 90–111.

O. Costaños, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 159–182; J. R. Klauder, “Wave theoy of imaging systems,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 183–191; V. I. Man’ko, K. B. Wolf, “The influence of spherical aberration on Gaussian Beam propagation,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 206–225; V. I. Man’ko, “Invariants and coherent states in fiber optics,” in Lie Methods in Optics, J. S. Mondragn, K. B. Wolf, eds., Vol. 250 of Springer Series in Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 193–205.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

k sphere showing a circle circuit corresponding to the evolution of a coherent state in a GRIN medium. The solid angle subtended by the circuit gives the topological phase.

Equations (56)

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n 2 = n 0 2 [ 1 - g 2 ( z ) r 2 ] ,
ϕ ( x 0 , y 0 ; 0 ) = A exp [ - ( x 0 - a ) 2 2 ω 2 ] × exp ( - y 0 2 2 ω 2 ) exp ( i k α y ) ,
ϕ ( x , y , x 0 , y 0 ; z ) = G ( x , y , x 0 , y 0 ; z ) ϕ ( x 0 , y 0 ; 0 ) d x 0 d y 0 ,
G ( x , y , x 0 , y 0 ; z ) = - i k n 0 g 0 2 π sin ( g 0 z ) exp [ i k n 0 g 0 2 sin ( g 0 z ) × ( x 2 + y 2 + x 0 2 + y 0 2 ) cos ( g 0 z ) ] × exp [ - i k n 0 g 0 2 sin ( g 0 z ) ( 2 x x 0 + 2 y y 0 ) ] × exp ( i k n 0 z ) ,
ϕ ( x , y , z ) = A 2 ω 0 2 ( α x α y ) 1 / 2 sin ( g 0 z ) × exp [ i 2 ω 0 2 ( x 2 + y 2 ) cos ( g 0 z ) ] × exp - ( β x 2 4 α x + β y 2 4 α y ) × exp ( i k n 0 z - a 2 2 ω 0 2 ) ,
α x = α y = i 2 ω 0 2 cos g 0 z sin g 0 z - 1 2 ω 2 ,
β x = - i x ω 0 2 sin ( g 0 z ) + a ω 2 ,
β y = - i y ω 0 2 sin ( g 0 z ) + i k α ,
ω 2 = ( k n 0 g 0 ) - 1 = ω 0 2 ,             α = n 0 a g 0 .
ϕ ( x , y ; z ) = A exp [ i ( k n 0 z - g 0 z ) ] × exp [ - ( x - a cos g 0 z ) 2 - ( y - a sin g 0 z ) 2 2 ω 2 ] × exp { i k α [ x sin ( g 0 z ) + y cos ( g 0 z ) ] } .
x = ϕ * x ϕ d x d y ϕ * ϕ d x d y = a cos g 0 z , y = ϕ * y ϕ d x d y ϕ * ϕ d x d y = a sin g 0 z .
r = [ a cos ( g 0 z ) , a sin ( g 0 z ) , z ] .
k = k [ α sin ( g 0 z ) , α cos ( g 0 z ) , n 0 ] ,
E = E 0 ϕ ( x , y , z ) = [ E n ( s ) + E b ( s ) ] ϕ ( x , y , z ) ,
E 0 = c n ( s ) · n ( s ) + c b ( s ) · b ( s ) .
i s E 0 = H τ E 0 = τ σ 2 E 0 ,
E 0 + ( E 0 · r ) r = 0 ,
r = [ x ( s ) , y ( s ) , z ( s ) ] .
t = κ n ,             n = - κ t + τ b ,             b = - τ n ,
τ ( s ) = r ( s ) [ r ( s ) r ( s ) ] r ( s ) 2 ,
κ ( s ) = r ( s ) .
s ( ξ ) = 0 ξ | d r d ξ | d ξ .
E 0 + κ ( n · E 0 ) · t = 0.
( c n , c b ) T = i τ σ 2 ( c n , c b ) T ,
H κ = ( κ 2 2 β 0 0 0 ) ,
i s E 0 = H E = ( H τ + H κ ) E 0
i s E 0 = ( τ · σ 2 + κ 2 4 β · Π + κ 2 4 β · σ 3 ) E 0 ,
τ = g 0 ( 1 + a 2 g 0 2 ) ,
κ = a g 0 2 ( 1 + a 2 g 0 2 ) .
β k n 0 ( 1 + a 2 g 0 2 ) 1 / 2 .
γ = κ 2 4 β = a 2 g 0 4 4 k n 0 ( 1 + a 2 g 0 2 ) 5 / 2 .
E 0 ( s ) = [ exp ( i δ s ) 0 0 exp ( - i δ s ) ] E ( 0 ) ,
δ = ( τ 2 + γ 2 4 ) 1 / 2 ,
E 0 ( s ) = [ c 1 ( 0 ) V 1 , c 2 ( 0 ) V 2 ] ,
V 1 = [ 1 , - i ( a + b ) ] ,
V 2 = [ 1 , - i ( a - b ) ] ,
a = γ 2 τ ,
b = ( 4 τ 2 + γ 2 ) 1 / 2 2 τ .
δ τ + γ 2 8 τ
[ c 1 ( s ) c 2 ( s ) ] = [ exp ( i τ s ) 0 0 exp ( - i τ s ) ] ( i γ 2 8 τ s 0 0 - i γ 2 8 τ s ) [ c 1 ( 0 ) c 2 ( 0 ) ] = M T M R [ c 1 ( 0 ) c 2 ( 0 ) ] .
ϕ G = 2 π - τ s = 2 π - τ [ ( 2 π g 0 ) 2 + ( 2 π a ) 2 ] 1 / 2 ,
ϕ G = 2 π ( 1 - cos θ ) ,
cos θ = τ s 2 π ,
ϕ G ( 1 / 2 ) = π ( 1 - cos θ ) ,
ϕ G ( 1 / 4 ) = π - τ s 4 - 2 arccos { τ s [ 4 π 2 + ( τ s ) 2 ] 1 / 2 } .
ϕ G = π 2 - ɛ ,
ɛ = cos - 1 ( κ 2 2 β δ ) .
d σ 2 = d n 2 + d b 2 + ( 1 - κ n ) 2 d s 2 .
L = N a d σ = N a [ d n 2 + d b 2 + ( 1 - κ n ) 2 d s 2 ] 1 / 2 ,
L p N a ( 1 - κ n ) ( 1 + n 2 + b 2 2 ) d s ,
N N a ( 1 - κ n ) .
( 2 n 2 + k 0 2 M ) ϕ = β 2 ϕ + P 2 ϕ ,
P 2 ( n ) = 3 4 ( 1 M d M d N ) 2 - 1 2 M ( 2 M κ 2 ) ,
P 2 ( n ) = κ 2 ,
β n β + κ 2 2 β .
β b = β .

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