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 and E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. I, p. 837.
  5. L.-A. Wu, H. J. Kimble, J. L. Hall, and 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 and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
    [Crossref]
  9. C. W. Gardiner and C. M. Savage, Opt. Commun. 50, 173 (1984).
    [Crossref]
  10. M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
    [Crossref] [PubMed]
  11. Y. R. Shen, Phys. Rev. 155, 921 (1967).
    [Crossref]
  12. J. Tucker and D. F. Walls, Phys. Rev. 178, 2036 (1969).
    [Crossref]
  13. M. Hillery and L. D. Mlodinow, Phys. Rev. A 30, 1860 (1984).
    [Crossref]
  14. M. Hillery and 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 and B. L. Schumaker, Phys. Rev. A 31, 3068 (1985).
    [Crossref] [PubMed]
  17. K. Wódkiewicz and M. S. Zubairy, Phys. Rev. A 27, 2003 (1983).
    [Crossref]
  18. G. Scharf and D. F. Walls, Opt. Commun. 50, 245 (1984).
    [Crossref]
  19. A. Lane, P. Tombesi, H. J. Carmichael, and D. F. Walls, Opt. Commun. 48, 155 (1983).
    [Crossref]
  20. C. M. Caves, Phys. Rev. D 26, 1817 (1982).
    [Crossref]
  21. H. P. Yuen and 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 and B. L. Schumaker, in Quantum Optics IV, J. D. Harvey and D. F. Walls, eds. (Springer, Berlin, 1986), p. 20.
    [Crossref]
  24. M. J. Potasek and B. Yurke, Phys. Rev. A 35, 3974 (1987).
    [Crossref] [PubMed]

1987 (1)

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

1986 (1)

L.-A. Wu, H. J. Kimble, J. L. Hall, and 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 and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[Crossref] [PubMed]

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

1984 (7)

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

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

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

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

C. W. Gardiner and 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 and V. W. S. Chan, Opt. Lett. 8, 177 (1983).
[Crossref] [PubMed]

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

K. Wódkiewicz and 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 and 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, and D. F. Walls, Opt. Commun. 48, 155 (1983).
[Crossref]

Caves, C. M.

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

Chan, V. W. S.

Collett, M. J.

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

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

Gardiner, C. W.

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

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

Hall, J. L.

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

Hillery, M.

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

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

Kimble, H. J.

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

Lane, A.

A. Lane, P. Tombesi, H. J. Carmichael, and 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 and L. D. Mlodinow, Phys. Rev. A 30, 1860 (1984).
[Crossref]

Potasek, M. J.

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

Savage, C. M.

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

Scharf, G.

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

Schumaker, B. L.

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

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

C. M. Caves and B. L. Schumaker, in Quantum Optics IV, J. D. Harvey and 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 and 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, and D. F. Walls, Opt. Commun. 48, 155 (1983).
[Crossref]

Tucker, J.

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

Walls, D. F.

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

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

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

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

Wódkiewicz, K.

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

Wu, H.

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

Wu, L.-A.

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

Yuen, H. P.

Yurke, B.

M. J. Potasek and 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 and M. S. Zubairy, Phys. Rev. A 29, 1275 (1984).
[Crossref]

K. Wódkiewicz and 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 and C. M. Savage, Opt. Commun. 50, 173 (1984).
[Crossref]

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

A. Lane, P. Tombesi, H. J. Carmichael, and 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 and D. F. Walls, Phys. Rev. 178, 2036 (1969).
[Crossref]

Phys. Rev. A (9)

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

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

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

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

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

M. J. Potasek and 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, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Other (2)

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

R. G. Smith, in Laser Handbook, F. T. Arecchi and 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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