Abstract

We develop a wideband traveling-wave formalism for analyzing quantum mechanically a degenerate parametric amplifier. The formalism is based on spatial differential equations—spatial Langevin equations—that propagate temporal Fourier components of the field operators through the nonlinear medium. In addition to the parametric nonlinearity, the Langevin equations include absorption and associated fluctuations, dispersion (phase mismatching), and pump quantum fluctuations. We analyze the dominant effects of phase mismatching and pump quantum fluctuations on the squeezing produced by a degenerate parametric amplifier.

© 1987 Optical Society of America

Full Article  |  PDF Article

Corrections

Carlton M. Caves and David D. Crouch, "Quantum wideband traveling-wave analysis of a degenerate parametric amplifier: erratum," J. Opt. Soc. Am. B 5, 1343-1343 (1988)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-5-6-1343

References

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  1. H. Takahashi, Adv. Commun. Syst. 1, 227 (1965), especially Sec. XI.
  2. E. Y. C. Lu, Lett. Nuovo Cimento 3, 585 (1972).
    [CrossRef]
  3. D. Stoler, Phys. Rev. Lett. 33, 1397 (1974).
    [CrossRef]
  4. R. G. Smith, in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. I, p. 837.
  5. L.-A. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
    [CrossRef] [PubMed]
  6. B. Yurke, Phys. Rev. A 29, 408 (1984).
    [CrossRef]
  7. B. Yurke, Phys. Rev. A 32, 300 (1985).
    [CrossRef] [PubMed]
  8. M. J. Collett, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
    [CrossRef]
  9. C. W. Gardiner, C. M. Savage, Opt. Commun. 50, 173 (1984).
    [CrossRef]
  10. M. J. Collett, D. F. Walls, Phys. Rev. A 32, 2887 (1985).
    [CrossRef] [PubMed]
  11. Y. R. Shen, Phys. Rev. 155, 921 (1967).
    [CrossRef]
  12. J. Tucker, D. F. Walls, Phys. Rev. 178, 2036 (1969).
    [CrossRef]
  13. M. Hillery, L. D. Mlodinow, Phys. Rev. A 30, 1860 (1984).
    [CrossRef]
  14. M. Hillery, M. S. Zubairy, Phys. Rev. A 29, 1275 (1984).
    [CrossRef]
  15. C. M. Caves, Phys. Rev. D 23, 1693 (1981).
    [CrossRef]
  16. C. M. Caves, B. L. Schumaker, Phys. Rev. A 31, 3068 (1985).
    [CrossRef] [PubMed]
  17. K. Wódkiewicz, M. S. Zubairy, Phys. Rev. A 27, 2003 (1983).
    [CrossRef]
  18. G. Scharf, D. F. Walls, Opt. Commun. 50, 245 (1984).
    [CrossRef]
  19. A. Lane, P. Tombesi, H. J. Carmichael, D. F. Walls, Opt. Commun. 48, 155 (1983).
    [CrossRef]
  20. C. M. Caves, Phys. Rev. D 26, 1817 (1982).
    [CrossRef]
  21. H. P. Yuen, V. W. S. Chan, Opt. Lett. 8, 177 (1983).
    [CrossRef] [PubMed]
  22. B. L. Schumaker, Opt. Lett. 9, 189 (1984).
    [CrossRef] [PubMed]
  23. C. M. Caves, B. L. Schumaker, in Quantum Optics IV, J. D. Harvey, D. F. Walls, eds. (Springer, Berlin, 1986), p. 20.
    [CrossRef]
  24. M. J. Potasek, B. Yurke, Phys. Rev. A 35, 3974 (1987).
    [CrossRef] [PubMed]

1987 (1)

M. J. Potasek, B. Yurke, Phys. Rev. A 35, 3974 (1987).
[CrossRef] [PubMed]

1986 (1)

L.-A. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

1985 (3)

B. Yurke, Phys. Rev. A 32, 300 (1985).
[CrossRef] [PubMed]

M. J. Collett, D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[CrossRef] [PubMed]

C. M. Caves, B. L. Schumaker, Phys. Rev. A 31, 3068 (1985).
[CrossRef] [PubMed]

1984 (7)

G. Scharf, D. F. Walls, Opt. Commun. 50, 245 (1984).
[CrossRef]

M. Hillery, L. D. Mlodinow, Phys. Rev. A 30, 1860 (1984).
[CrossRef]

M. Hillery, M. S. Zubairy, Phys. Rev. A 29, 1275 (1984).
[CrossRef]

M. J. Collett, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[CrossRef]

C. W. Gardiner, C. M. Savage, Opt. Commun. 50, 173 (1984).
[CrossRef]

B. Yurke, Phys. Rev. A 29, 408 (1984).
[CrossRef]

B. L. Schumaker, Opt. Lett. 9, 189 (1984).
[CrossRef] [PubMed]

1983 (3)

H. P. Yuen, V. W. S. Chan, Opt. Lett. 8, 177 (1983).
[CrossRef] [PubMed]

A. Lane, P. Tombesi, H. J. Carmichael, D. F. Walls, Opt. Commun. 48, 155 (1983).
[CrossRef]

K. Wódkiewicz, M. S. Zubairy, Phys. Rev. A 27, 2003 (1983).
[CrossRef]

1982 (1)

C. M. Caves, Phys. Rev. D 26, 1817 (1982).
[CrossRef]

1981 (1)

C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[CrossRef]

1974 (1)

D. Stoler, Phys. Rev. Lett. 33, 1397 (1974).
[CrossRef]

1972 (1)

E. Y. C. Lu, Lett. Nuovo Cimento 3, 585 (1972).
[CrossRef]

1969 (1)

J. Tucker, D. F. Walls, Phys. Rev. 178, 2036 (1969).
[CrossRef]

1967 (1)

Y. R. Shen, Phys. Rev. 155, 921 (1967).
[CrossRef]

1965 (1)

H. Takahashi, Adv. Commun. Syst. 1, 227 (1965), especially Sec. XI.

Carmichael, H. J.

A. Lane, P. Tombesi, H. J. Carmichael, D. F. Walls, Opt. Commun. 48, 155 (1983).
[CrossRef]

Caves, C. M.

C. M. Caves, B. L. Schumaker, Phys. Rev. A 31, 3068 (1985).
[CrossRef] [PubMed]

C. M. Caves, Phys. Rev. D 26, 1817 (1982).
[CrossRef]

C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[CrossRef]

C. M. Caves, B. L. Schumaker, in Quantum Optics IV, J. D. Harvey, D. F. Walls, eds. (Springer, Berlin, 1986), p. 20.
[CrossRef]

Chan, V. W. S.

Collett, M. J.

M. J. Collett, D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[CrossRef] [PubMed]

M. J. Collett, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[CrossRef]

Gardiner, C. W.

M. J. Collett, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[CrossRef]

C. W. Gardiner, C. M. Savage, Opt. Commun. 50, 173 (1984).
[CrossRef]

Hall, J. L.

L.-A. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Hillery, M.

M. Hillery, L. D. Mlodinow, Phys. Rev. A 30, 1860 (1984).
[CrossRef]

M. Hillery, M. S. Zubairy, Phys. Rev. A 29, 1275 (1984).
[CrossRef]

Kimble, H. J.

L.-A. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Lane, A.

A. Lane, P. Tombesi, H. J. Carmichael, D. F. Walls, Opt. Commun. 48, 155 (1983).
[CrossRef]

Lu, E. Y. C.

E. Y. C. Lu, Lett. Nuovo Cimento 3, 585 (1972).
[CrossRef]

Mlodinow, L. D.

M. Hillery, L. D. Mlodinow, Phys. Rev. A 30, 1860 (1984).
[CrossRef]

Potasek, M. J.

M. J. Potasek, B. Yurke, Phys. Rev. A 35, 3974 (1987).
[CrossRef] [PubMed]

Savage, C. M.

C. W. Gardiner, C. M. Savage, Opt. Commun. 50, 173 (1984).
[CrossRef]

Scharf, G.

G. Scharf, D. F. Walls, Opt. Commun. 50, 245 (1984).
[CrossRef]

Schumaker, B. L.

C. M. Caves, B. L. Schumaker, Phys. Rev. A 31, 3068 (1985).
[CrossRef] [PubMed]

B. L. Schumaker, Opt. Lett. 9, 189 (1984).
[CrossRef] [PubMed]

C. M. Caves, B. L. Schumaker, in Quantum Optics IV, J. D. Harvey, D. F. Walls, eds. (Springer, Berlin, 1986), p. 20.
[CrossRef]

Shen, Y. R.

Y. R. Shen, Phys. Rev. 155, 921 (1967).
[CrossRef]

Smith, R. G.

R. G. Smith, in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. I, p. 837.

Stoler, D.

D. Stoler, Phys. Rev. Lett. 33, 1397 (1974).
[CrossRef]

Takahashi, H.

H. Takahashi, Adv. Commun. Syst. 1, 227 (1965), especially Sec. XI.

Tombesi, P.

A. Lane, P. Tombesi, H. J. Carmichael, D. F. Walls, Opt. Commun. 48, 155 (1983).
[CrossRef]

Tucker, J.

J. Tucker, D. F. Walls, Phys. Rev. 178, 2036 (1969).
[CrossRef]

Walls, D. F.

M. J. Collett, D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[CrossRef] [PubMed]

G. Scharf, D. F. Walls, Opt. Commun. 50, 245 (1984).
[CrossRef]

A. Lane, P. Tombesi, H. J. Carmichael, D. F. Walls, Opt. Commun. 48, 155 (1983).
[CrossRef]

J. Tucker, D. F. Walls, Phys. Rev. 178, 2036 (1969).
[CrossRef]

Wódkiewicz, K.

K. Wódkiewicz, M. S. Zubairy, Phys. Rev. A 27, 2003 (1983).
[CrossRef]

Wu, H.

L.-A. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Wu, L.-A.

L.-A. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Yuen, H. P.

Yurke, B.

M. J. Potasek, B. Yurke, Phys. Rev. A 35, 3974 (1987).
[CrossRef] [PubMed]

B. Yurke, Phys. Rev. A 32, 300 (1985).
[CrossRef] [PubMed]

B. Yurke, Phys. Rev. A 29, 408 (1984).
[CrossRef]

Zubairy, M. S.

M. Hillery, M. S. Zubairy, Phys. Rev. A 29, 1275 (1984).
[CrossRef]

K. Wódkiewicz, M. S. Zubairy, Phys. Rev. A 27, 2003 (1983).
[CrossRef]

Adv. Commun. Syst. (1)

H. Takahashi, Adv. Commun. Syst. 1, 227 (1965), especially Sec. XI.

Lett. Nuovo Cimento (1)

E. Y. C. Lu, Lett. Nuovo Cimento 3, 585 (1972).
[CrossRef]

Opt. Commun. (3)

C. W. Gardiner, C. M. Savage, Opt. Commun. 50, 173 (1984).
[CrossRef]

G. Scharf, D. F. Walls, Opt. Commun. 50, 245 (1984).
[CrossRef]

A. Lane, P. Tombesi, H. J. Carmichael, D. F. Walls, Opt. Commun. 48, 155 (1983).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (2)

Y. R. Shen, Phys. Rev. 155, 921 (1967).
[CrossRef]

J. Tucker, D. F. Walls, Phys. Rev. 178, 2036 (1969).
[CrossRef]

Phys. Rev. A (9)

M. Hillery, L. D. Mlodinow, Phys. Rev. A 30, 1860 (1984).
[CrossRef]

M. Hillery, M. S. Zubairy, Phys. Rev. A 29, 1275 (1984).
[CrossRef]

C. M. Caves, B. L. Schumaker, Phys. Rev. A 31, 3068 (1985).
[CrossRef] [PubMed]

K. Wódkiewicz, M. S. Zubairy, Phys. Rev. A 27, 2003 (1983).
[CrossRef]

M. J. Collett, D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[CrossRef] [PubMed]

B. Yurke, Phys. Rev. A 29, 408 (1984).
[CrossRef]

B. Yurke, Phys. Rev. A 32, 300 (1985).
[CrossRef] [PubMed]

M. J. Collett, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[CrossRef]

M. J. Potasek, B. Yurke, Phys. Rev. A 35, 3974 (1987).
[CrossRef] [PubMed]

Phys. Rev. D (2)

C. M. Caves, Phys. Rev. D 26, 1817 (1982).
[CrossRef]

C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[CrossRef]

Phys. Rev. Lett. (2)

D. Stoler, Phys. Rev. Lett. 33, 1397 (1974).
[CrossRef]

L.-A. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Other (2)

C. M. Caves, B. L. Schumaker, in Quantum Optics IV, J. D. Harvey, D. F. Walls, eds. (Springer, Berlin, 1986), p. 20.
[CrossRef]

R. G. Smith, in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. I, p. 837.

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Figures (2)

Fig. 1
Fig. 1

Effect of pump quantum fluctuations on squeezing. Ideal squeezing is represented by the ellipse with solid lines. Pump phase fluctuations cause the orientation of the ellipse to fluctuate through a characteristic angle Δϕ = 1/4 A p, as indicated schematically by the dotted ellipse. These fluctuations feed noise from the amplified signal quadrature into the squeezed signal quadrature.

Fig. 2
Fig. 2

Trick for introducing absorption and dispersion (phase mismatching). The actual nonlinear medium between z and z + Δz is replaced by a slab of ideal (lossless, dispersionless) nonlinear medium preceded by a beam splitter. Reflection at the beam splitter accounts for losses, and frequency-dependent phase shifts at the beam splitter introduce dispersion.

Equations (104)

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α 0 2 π χ ( 2 ) A p n 0 2 = 2 π χ ( 2 ) n 0 3 / 2 ( 8 π P p c σ ) 1 / 2 ,
g 0 α 0 ( Ω / c ) = α 0 ( K / n 0 ) .
g 0 , γ K .
Δ k ( ) K p - k ( Ω + ) - k ( Ω - ) = - ( Ω + ) Δ n ( Ω + ) c - ( Ω - ) Δ n ( Ω - ) c .
Δ n ( Ω ± ) = ± n + ½ n 2 ,
Δ k ( ) = - p 2 / Ω c ,             p 2 Ω n + Ω 2 n ,
Δ k ( 1 ) L = 1 1 = p - 1 / 2 ( Ω c / L ) 1 / 2 ,
Δ k ( 2 ) / 2 g 0 = 1 2 = p - 1 / 2 ( 2 Ω c g 0 ) 1 / 2 .
Δ / 2 π π - 1 min ( 1 , 2 ) Ω / 2 π .
Δ ϕ e g 0 L = e g 0 L / 4 A p e - g 0 L A p ¼ e 2 α 0 ( Ω L / c ) .
N p P p Ω p Δ p / 2 π c σ 4 n 0 Ω p Δ p A p 2 = ( A p A vac ) 2 A p 2 .
A vac ( 4 n 0 Ω p Δ p / c σ ) 1 / 2
( Δ ϕ p ) 2 = ¼ N p = ( Ω p / 4 P p ) ( Δ p / 2 π ) .
α vac α 0 A p = 2 π χ ( 2 ) A vac n 0 2 ,
A p ¼ exp [ 2 α vac ( Ω L / c ) A p ] .
A max = ¼ exp [ 2 α vac ( Ω L / c ) A max ] ;
P p 1 16 Ω p Δ p 2 π exp [ 8 π χ ( 2 ) n 0 3 / 2 ( Ω L c ) ( 8 π P p c σ ) 1 / 2 ] .
Δ p / 2 π = c g 0 / Ω n ,
B in ( + ) = D in ( + ) = E in ( + ) = B s d ω 2 π ( 2 π ω c σ ) 1 / 2 × a in ( ω ) exp [ i ω ( z / c - t ) ] ,             z 0 ,
[ a in ( ω ) , a in ( ω ) ] = 2 π δ ( ω - ω ) .
B s d ω 2 π ω a in ( ω ) a in ( ω ) .
B s ( + ) = B s d ω 2 π B s ( ω , z ) exp [ i ( k z - ω t ) ] ,             k = ω n ( ω ) / c ,             0 z L ,
B s ( ω , z ) = [ c n ( ω ) v g ( ω ) ] 1 / 2 [ 2 π n ( ω ) ω c σ ] 1 / 2 a s ( ω , z ) .
B s d ω 2 π ω a s ( ω ) a s ( ω )
B s d ω 2 π ω a s ( ω , z ) a s ( ω , z ) .
a s ( ω , 0 ) = a in ( ω ) ,             a out ( ω ) = a s ( ω , L ) .
[ a s ( ω , z ) , a s ( ω , z ) ] = 2 π δ ( ω - ω ) .
B p ( + ) = B p d ω 2 π B p ( ω , z ) exp [ i ( k 0 z - ω t ) ] ,             k 0 = ω n 0 / c ,             0 z L ,
B p ( ω , z ) = 1 2 i A p e i ϕ p 2 π δ ( ω - Ω p ) + ( 2 π n 0 ω c σ ) 1 / 2 a p ( ω , z ) .
B 0 ( + ) = B 0 s ( + ) + B 0 p ( + ) .
B 0 s ( + ) = B s d ω 2 π B 0 s ( ω , ξ ) exp [ i ( k 0 ξ - ω t ) ] , k 0 = ω n 0 / c ,
B 0 s ( ω , ξ ) = ( 2 π n 0 ω c σ ) 1 / 2 a 0 s ( ω , ξ ) ,
B 0 p ( + ) = B p d ω 2 π B 0 p ( ω , ξ ) exp [ i ( k 0 ξ - ω t ) ] ,
B 0 p ( ω , ξ ) = 1 2 i A p e i ϕ p 2 π ξ ( ω - Ω p ) + ( 2 π n 0 ω c σ ) 1 / 2 a 0 p ( ω , ξ )
a 0 p ( ω , z ) = a p ( ω , z ) .
[ b s ( ω ) , b s ( ω ) ] = 2 π δ ( ω - ω ) .
a 0 s ( ω , z ) e i k 0 z = exp [ i ω Δ n ( ω ) Δ z / c ] { [ 1 - γ ( ω ) Δ z ] 1 / 2 a s ( ω , z ) e i k z + [ γ ( ω ) Δ z ] 1 / 2 b s ( ω ) e i k z } .
b s ( ω ) = ( Δ z ) - 1 / 2 z z + Δ z d ξ b s ( ω , ξ ) .
[ b s ( ω , ξ ) , b s ( ω , ξ ) ] = 2 π δ ( ω - ω ) δ ( ξ - ξ ) .
a p ( ω , z + Δ z ) = a 0 p ( ω , z + Δ z ) ,
a s ( ω , z + Δ z ) exp [ i k ( z + Δ z ) ] = a 0 s ( ω , z + Δ z ) × exp [ i k 0 ( z + Δ z ) ] .
E 0 s ( + ) = n 0 - 2 D 0 s ( + ) - 8 π η ( 2 ) D 0 p ( + ) D 0 s ( - ) ,
E 0 p ( + ) = n 0 - 2 D 0 p ( + ) - 4 π η ( 2 ) [ D 0 s ( + ) ] 2 .
d a 0 s ( ω , ξ ) d ξ = - g 0 [ ω ( Ω p - ω ) Ω 2 ] 1 / 2 e 2 i ϕ a 0 s ( Ω p - ω , ξ ) + i g 0 A p B p d ω 2 π [ ω ω ( ω - ω ) Ω 2 Ω p ] 1 / 2 × a 0 p ( ω , ξ ) ( Δ p / 2 π ) 1 / 2 a 0 s ( ω - ω , ξ ) ;
d a 0 p ( ω , ξ ) d ξ = i 2 g 0 A p B s d ω 2 π [ ω ω ( ω - ω ) Ω 2 Ω p ] 1 / 2 × a 0 s ( ω , ξ ) a 0 s ( ω - ω , ξ ) ( Δ p / 2 π ) 1 / 2 .
a 0 s ( ω , z + Δ z ) = a 0 s ( ω , z ) + d a 0 s ( ω , ξ ) d ξ | ξ = z Δ z ,
a 0 p ( ω , z + Δ z ) = a 0 p ( ω , z ) + d a 0 p ( ω , ξ ) d ξ | ξ = z Δ z .
d a s ( ω , z ) d z = - 1 2 γ ( ω ) a s ( ω , z ) - g 0 [ ω ( Ω p - ω ) Ω 2 ] 1 / 2 e 2 i ϕ exp [ i Δ K ( Ω p , ω ) z ] × a s ( Ω p - ω , z ) + [ γ ( ω ) ] 1 / 2 b s ( ω , z ) + i g 0 A p B p d ω 2 π [ ω ω ( ω - ω ) Ω 2 Ω p ] 1 / 2 exp [ i Δ K ( ω , ω ) z ] × a p ( ω , z ) ( Δ p / 2 π ) 1 / 2 a s ( ω , - ω , z ) ;
d a p ( ω , z ) d z = i 2 g 0 A p B s d ω 2 π [ ω ω ( ω - ω ) Ω 2 Ω p ] 1 / 2 × exp [ - i Δ K ( ω , ω ) z ] a s ( ω , z ) a s ( ω - ω , z ) ( Δ p / 2 π ) 1 / 2 .
Δ K ( ω , ω ) ω n 0 c - k ( ω ) - k ( ω - ω ) = - ω Δ n ( ω ) c - ( ω - ω ) Δ n ( ω - ω ) c
d a s ( Ω + , z ) d z = - ½ γ a s ( Ω + , z ) - g 0 exp [ i Δ k ( ) z ] a s ( Ω - , z ) + γ 1 / 2 b s ( Ω + , z ) + P ( , z ) ,
d a p ( Ω p + , z ) d z = i 2 g 0 A p - d 2 π exp [ - i Δ k ( , - ) z ] × a s ( Ω + , z ) a s ( Ω + - , z ) ( Δ p / 2 π ) 1 / 2 .
P ( , z ) i g 0 A p - d 2 π exp [ i Δ k ( , ) z ] a p ( Ω p + - , z ) ( Δ p / 2 π ) 1 / 2 × a s ( Ω - , z ) ,
Δ k ( , ) Δ K ( Ω p + - , Ω + ) = - ( Ω + ) Δ n ( Ω + ) c - ( Ω - ) Δ n ( Ω - ) c ,
Δ k ( ) Δ k ( , ) = - ( Ω + ) Δ n ( Ω + ) c - ( Ω - ) ( Δ n ) ( Ω - ) c
α 1 ( , z ) / 2 i [ a s ( Ω + , z ) + a s ( Ω - , z ) ] ,
α 2 ( , z ) - / 2 i [ a s ( Ω + , z ) - a s ( Ω - , z ) ] .
α ¯ 1 ( , z ) ½ { exp [ - i Δ k ( ) z / 2 ] a s ( Ω + , z ) + exp [ i Δ k ( ) z / 2 ] a s ( Ω - , z ) } ,
α ¯ 2 ( , z ) - / 2 i { exp [ - i Δ k ( ) z / 2 ] a s ( Ω + , z ) - exp [ i Δ k ( ) z / 2 ] a s ( Ω - , z ) } ,
α ¯ 1 ( , z ) = α 1 ( , z ) cos [ Δ k ( ) z / 2 ] + α 2 ( , z ) sin [ Δ k ( ) z / 2 ] ,
α ¯ 2 ( , z ) = - α 1 ( , z ) sin [ Δ k ( ) z / 2 ] + α 2 ( , z ) cos [ Δ k ( ) z / 2 ] .
d α ¯ 1 ( , z ) d z = - ( g 0 + 1 2 γ ) α ¯ 1 ( , z ) + 1 2 Δ k ( ) α ¯ 2 ( , z ) + γ 1 / 2 β ¯ 1 ( , z ) + P ¯ 1 ( , z ) ,
d α ¯ 2 ( , z ) d z = + ( g 0 - 1 2 γ ) α ¯ 2 ( , z ) - ½ Δ k ( ) α ¯ 1 ( , z ) + γ 1 / 2 β ¯ 2 ( , z ) + P ¯ 2 ( , z ) .
β ¯ 1 ( , z ) ½ { exp [ - i Δ k ( ) z / 2 ] b s ( Ω + , z ) + exp [ i Δ k ( ) z / 2 ] b s ( Ω - , z ) } ,
β ¯ 2 ( , z ) - / 2 i { exp [ - i Δ k ( ) z / 2 ] b s ( Ω + , z ) - exp [ i Δ k ( ) z / 2 ] b s ( Ω - , z ) }
P ¯ 1 ( , z ) ½ { exp [ - i Δ k ( ) z / 2 ] P ( + , z ) + exp [ i Δ k ( ) z / 2 ] P ( - , z ) } ,
P ¯ 2 ( , z ) - / 2 i { exp [ - i Δ k ( ) z / 2 ] P ( + , z ) - exp [ i Δ k ( ) z / 2 ] P ( - , z ) ] .
α ¯ m ( , z ) = n = 1 , 2 { G ¯ m n ( , z ) α ¯ n ( , 0 ) + 0 z d z G ¯ m n ( , z - z ) [ γ 1 / 2 β ¯ n ( , z ) + P ¯ n ( , z ) ] } ,             m = 1 , 2.
G ¯ 11 ( , z ) e - γ z / 2 e - g z - μ 2 e g z 1 - μ 2 ,
G ¯ 22 ( , z ) e - γ z / 2 e g z - μ 2 e - g z 1 - μ 2 ,
G ¯ 12 ( , z ) = - G ¯ 21 ( , z ) = μ e - γ z / 2 e g z - e - g z 1 - μ 2 ,
g = g ( ) { g 0 2 - [ Δ k ( ) / 2 ] 2 } 1 / 2 ,
μ = μ ( ) Δ k ( ) / 2 g 0 + g ( ) .
G ¯ 11 ( , z ) = e - γ z / 2 e - g 0 z ,             G ¯ 22 ( , z ) = e - γ z / 2 e g 0 z .
Δ α m ( , L ) Δ α n ( , L ) sym = π S m n ( ) δ ( - ) ,             m , n = 1 , 2.
Δ α ¯ m ( , L ) Δ α ¯ n ( , L ) sym = π S ¯ m n ( ) δ ( - ) .
S 11 = S ¯ 11 cos 2 ( Δ k L / 2 ) + S ¯ 22 sin 2 ( Δ k L / 2 ) - ( S ¯ 12 + S ¯ 21 ) cos ( Δ k L / 2 ) sin ( Δ k L / 2 ) ,
S 22 = S ¯ 11 sin 2 ( Δ k L / 2 ) + S ¯ 22 cos 2 ( Δ k L / 2 ) + ( S ¯ 12 + S ¯ 21 ) cos ( Δ k L / 2 ) sin ( Δ k L / 2 ) ,
S 12 = S 21 * = ( S ¯ 11 - S ¯ 22 ) cos ( Δ k L / 2 ) sin ( Δ k L / 2 ) + S ¯ 12 cos 2 ( Δ k L / 2 ) - S ¯ 21 sin 2 ( Δ k L / 2 ) ,
α m ( , 0 ) α n ( , 0 ) sym = α ¯ m ( , 0 ) α ¯ n ( , 0 ) sym = ½ π δ m n δ ( - ) ,
β ¯ m ( , z ) β ¯ n ( , z ) sym = ½ π δ m n δ ( - ) δ ( z - z ) .
S ¯ m n ( ) = 1 2 p = 1 , 2 [ G ¯ m p * ( , L ) G ¯ n p ( , L ) + γ 0 L d z G ¯ m p * ( , L - z ) G ¯ n p ( , L - z ) ] .
S ¯ 11 ( ) = S 11 ( ) = 1 2 γ + 2 g 0 exp [ - ( γ + 2 g 0 ) L ] γ + 2 g 0 ,
S ¯ 22 ( ) = S 22 ( ) = 1 2 γ - 2 g 0 exp [ - ( γ - 2 g 0 ) L ] γ - 2 g 0 .
S 11 ( ) = ½ e - 2 g 0 L ,             S 22 ( ) = ½ e 2 g 0 L .
S ¯ 11 ( ) = ½ [ G ¯ 11 ( , L ) 2 + G ¯ 12 ( , L ) 2 ] .
Δ k ( ) / 2 g 0 = ( p p ) ( / 2 ) 2
μ ( ) = Δ k ( ) / 4 g 0 = - ½ ( p / p ) ( / 2 ) 2 .
G ¯ 11 ( , L ) = e - g 0 L ,             G ¯ 12 ( , L ) = μ ( ) e g 0 L ,
S ¯ 11 ( ) = ½ { e - 2 g 0 L + [ μ ( ) ] 2 e 2 g 0 L } = ½ [ e - 2 g 0 L + ¼ ( / 2 ) 4 e 2 g 0 L ] .
μ ( ) e g 0 L e - g 0 L 2 1 / 2 2 e - g 0 L .
Δ k ( ) L = - ( p / p ) ( / 1 ) 2
S 11 ( ) = ½ { e - 2 g 0 L + [ μ ( ) - ½ Δ k ( ) L ] 2 e 2 g 0 L } .
G ¯ 1 n ( , z ) = δ 1 n e - g 0 z ,
α ¯ 1 ( , L ) = e - g 0 L α ¯ 1 ( , 0 ) + 0 L d z exp [ - g 0 ( L - z ) ] P ¯ 1 ( , z ) .
P ¯ 1 ( , z ) = ½ g 0 A p 0 d 2 π { exp [ i Ω n ( - ) z / c ] × a p ( Ω p + - , z ) + a p ( Ω p - + , z ) ( Δ p / 2 π ) 1 / 2 α ¯ 2 ( , z ) + exp [ - i Ω n ( + ) z / c ] × a p ( Ω p + + , z ) + a p ( Ω p - - , z ) ( Δ p / 2 π ) 1 / 2 α ¯ 2 ( , z ) } ,
Ω n 3 / c 2 g 0 = 1 3 = 2 c g 0 / Ω n .
α ¯ 1 ( , L ) = e - g 0 L α ¯ 1 ( , 0 ) + e g 0 L 4 A p 0 d 2 π { exp [ i Ω n ( - ) L / c ] 1 + i Ω n 2 c g 0 ( - ) a p ( Ω p + - ) + a p ( Ω p - + ) ( Δ p / 2 π ) 1 / 2 α ¯ 2 ( , 0 ) + exp [ - i Ω n ( + ) L / c ] 1 - i Ω n 2 c g 0 ( + ) a p ( Ω p + + ) + a p ( Ω p - - ) ( Δ p / 2 π ) 1 / 2 α ¯ 2 ( , 0 ) } .
α p 1 ( , z ) ½ [ a p ( Ω p + , z ) + a p ( Ω p - , z ) ] ,
α p 2 ( , z ) - / 2 i [ a p ( Ω p + , z ) - a p ( Ω p - , z ) ] .
S ¯ 11 ( ) = 1 2 ( e 2 g 0 L + e + 2 g 0 L 16 A p 2 ) ,             2 e - g 0 L ,
Δ p 2 π - d 2 π 1 1 + ( / 3 ) 2 = 1 2 3 = c g 0 Ω n
S ˜ 11 ( ) = ½ e - 2 g 0 L { 1 + 4 [ μ ( ) ] 2 g 0 L } = ½ e - 2 g 0 L [ 1 + ( / 2 ) 4 g 0 L ] .
2 μ ( ) ( g 0 L ) 1 / 2 1 2 ( g 0 L ) - 1 / 4

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