Abstract

We introduce a power spectral density matrix formalism that incorporates both the pulse shape and the field polarization and can therefore easily describe averages over random fluctuations of the local birefringence vector. We demonstrate that quantities such as the differential time delay, power diffusion, and decoherence effects can be obtained directly from the equations of motion for the power density matrix. This approach can be applied to pulses with arbitrary frequency-dependent polarization and intensity distributions and in particular makes possible the minimization of the eye-opening penalty through the proper choice of the initial pulse profile.

© 2005 Optical Society of America

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References

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  1. C. D. Poole, R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
    [CrossRef]
  2. P. K. A. Wai, C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (2001).
    [CrossRef]
  3. S. J. Savory, F. P. Payne, “Pulse propagation in fibers with polarization-mode dispersion,” J. Lightwave Technol. 19, 350–357 (2001).
    [CrossRef]
  4. R. F. Fox, “Critique of the generalized cumulant expansion method,” J. Math. Phys. 17, 1148–1152 (1976).
    [CrossRef]

2001 (2)

P. K. A. Wai, C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (2001).
[CrossRef]

S. J. Savory, F. P. Payne, “Pulse propagation in fibers with polarization-mode dispersion,” J. Lightwave Technol. 19, 350–357 (2001).
[CrossRef]

1986 (1)

C. D. Poole, R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

1976 (1)

R. F. Fox, “Critique of the generalized cumulant expansion method,” J. Math. Phys. 17, 1148–1152 (1976).
[CrossRef]

Fox, R. F.

R. F. Fox, “Critique of the generalized cumulant expansion method,” J. Math. Phys. 17, 1148–1152 (1976).
[CrossRef]

Menyuk, C. R.

P. K. A. Wai, C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (2001).
[CrossRef]

Payne, F. P.

Poole, C. D.

C. D. Poole, R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

Savory, S. J.

Wagner, R. E.

C. D. Poole, R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

Wai, P. K. A.

P. K. A. Wai, C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (2001).
[CrossRef]

Electron. Lett. (1)

C. D. Poole, R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

J. Lightwave Technol. (2)

P. K. A. Wai, C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (2001).
[CrossRef]

S. J. Savory, F. P. Payne, “Pulse propagation in fibers with polarization-mode dispersion,” J. Lightwave Technol. 19, 350–357 (2001).
[CrossRef]

J. Math. Phys. (1)

R. F. Fox, “Critique of the generalized cumulant expansion method,” J. Math. Phys. 17, 1148–1152 (1976).
[CrossRef]

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Equations (55)

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2 z 2 E ( x , y , z ) + 2 E ( x , y , z ) + ϵ ( x , y , z , ω ) k 2 E ( x , y , z ) = 0 ,
E α ( x , y , z ) = exp [ j β α ( ω ) z ] e α ( x , y , ω ) , α = 1 , 2 ,
2 e α ( x , y , ω ) + [ ϵ 0 ( x , y , ω ) k 2 β α 2 ( ω ) ] e α ( x , y , ω ) = 0 .
E α ( x , y , z , ω ) = α = 1 , 2 A α ( z , ω ) exp [ j β α ( ω 0 ) z ] e α ( x , y , ω 0 ) .
γ [ 2 j β α ( ω 0 ) δ α γ z A α + δ α γ ( β α 2 ( ω ) β α 2 ( ω 0 ) ) A α + δ ϵ α γ ( z , ω ) k 2 A γ ] = 0 ,
z A ( z , ω ) = j 2 β 0 ( β 2 ( ω ) β 0 2 ) A ( z , ω ) j B ̂ A ( z , ω ) .
z A ( z , ω ) = j 2 β 0 ( β 2 ( ω ) β 0 2 ) A ( z , ω ) + j A ( z , ω ) B ̂ .
ρ ̃ ( z , ω 1 , ω 2 ) = A ( z , ω 1 ) A ( z , ω 2 )
z ρ ̃ ( z , ω 1 , ω 2 ) = j 2 β 0 ( β 2 ( ω 1 ) β 2 ( ω 2 ) ) ρ ̃ ( z , ω 1 , ω 2 ) + j B ̂ ( ω 1 ) ρ ̃ ( z , ω 1 , ω 2 ) + j ρ ̃ ( z , ω 1 , ω 2 ) B ̂ ( ω 2 ) .
A ( z , ω ) = U ̂ ( z , z 0 , ω ) A ( z 0 , ω ) ,
ρ ̃ ( z , ω 1 , ω 2 ) = U ̂ ( z , z 0 , ω 1 ) ρ ̃ ( z 0 , ω 1 , ω 2 ) U ̂ ( z , z 0 , ω 2 ) .
ρ ̃ ( z , t 1 , t 2 ) = d ω 1 2 π d ω 2 2 π exp ( j ω 1 t 1 j ω 2 t 2 ) ρ ̃ ( z , ω 1 , ω 2 )
t ( z ) = d t t A ( z , t ) 2 d t A ( z , t ) 2 = d t Tr { t ρ ̃ ( z , t , t ) } d t Tr { ρ ̃ ( z , t , t ) } .
t ( z ) = d t d Ω 2 π d ω 2 π Tr { t ρ ̃ ( z , ω + Ω 2 , ω Ω 2 ) } exp ( j Ω t ) d t d Ω 2 π d ω 2 π Tr { ρ ̃ ( z , ω + Ω 2 , ω Ω 2 ) } exp ( j Ω t ) = d ω 2 π Tr { j Ω ρ ̃ ( z , ω + Ω 2 , ω Ω 2 ) Ω = 0 } d ω 2 π Tr { ρ ̃ ( z , ω , ω ) } .
ρ ̂ ( z , ω , Ω ) ρ ̃ ( z , ω + Ω 2 , ω Ω 2 ) ,
t ( z ) = d ω 2 π Tr { j Ω ρ ̂ ( z , ω , Ω ) Ω = 0 } .
t ( z ) = d ω 2 π j Tr { ( ω U ̂ ) U ̂ ρ ̂ ( z , ω , 0 ) } = d ω 2 π j Tr { U ̂ ( ω U ̂ ) ρ ̂ ( z 0 , ω , 0 ) } .
1 2 τ s σ = j U ̂ ( ω U ̂ ) t av σ 0 ,
1 2 τ t σ = j ( ω U ̂ ) U ̂ t av σ 0 .
z ρ ̂ ( z , ω , Ω ) = j 2 β 0 ( β 2 ( ω Ω 2 ) β 2 ( ω + Ω 2 ) ) ρ ̂ ( z , ω , Ω ) j B ̂ ( ω + Ω 2 ) ρ ̂ ( z , ω , Ω ) + j ρ ̂ ( z , ω , Ω ) B ̂ ( ω Ω 2 ) .
z ρ ̂ ( z , ω , Ω ) j β ( ω ) Ω ρ ̂ ( z , ω , Ω ) j [ B ̂ ( ω ) , ρ ̂ ( z , ω , Ω ) ] j 2 Ω [ B ̂ ( ω ) , ρ ̂ ( z , ω , Ω ) ] + ,
m ̂ = 1 2 ( m 0 σ 0 + m σ ) .
z ρ ( z , ω , Ω ) = j Ω ( β ( ω ) + 1 2 B 0 ) ρ ( z , ω , Ω ) + B ( ω ) × ρ ( z , ω , Ω ) j 2 Ω B ( ω ) ρ 0 ( z , ω , Ω ) ,
z ρ 0 ( z , ω , Ω ) = j Ω ( β ( ω ) + 1 2 B 0 ) ρ 0 ( z , ω , Ω ) j 2 Ω B ( ω ) ρ ( z , ω , Ω ) .
ρ ( z , ω , Ω ) = exp [ j Ω ( β ( ω ) + 1 2 B 0 ) z ] ρ I ( z , ω , Ω )
z ρ I ( z , ω , Ω ) = B ( ω ) × ρ I ( z , ω , Ω ) j 2 Ω B ( ω ) ρ I 0 ( z , ω , Ω ) ,
z ρ I 0 ( z , ω , Ω ) = j 2 Ω B ( ω ) ρ I ( z , ω , Ω ) .
2 β 0 B ̂ ( ω ) = δ ϵ ̂ ( ω ) k 2 = β 2 ( ω ) [ Δ 11 Δ 12 Δ 21 Δ 22 ] ,
B ̂ = 1 2 β 0 β 2 ( ω ) ( Δ m σ ̂ 0 + Δ 21 σ ̂ 1 + Δ d σ ̂ 3 ) ,
v = ( Δ 21 , 0 , Δ d ) and v 0 = Δ m ,
z ρ I ( z , ω , Ω ) = β ( ω ) v × ρ I ( z , ω , Ω ) j Ω β ( ω ) v ρ I 0 ( z , ω , Ω ) ,
z ρ I 0 ( z , ω , Ω ) = j Ω β ( ω ) v ρ I ( z , ω , Ω ) ,
z R = M ̂ R ,
M ̂ = [ β ( ω ) v ̂ × j Ω β ( ω ) v j Ω β ( ω ) v T 0 ]
R = ( ρ I ρ I 0 ) .
v ̂ × = [ 0 v 3 v 2 v 3 0 v 1 v 2 v 1 0 ] .
R ( z ) = T z exp [ 0 z M ̂ ( z ) d z ] R ( 0 ) .
R ( z ) = T z exp [ m = 1 0 z G ̂ m ( z ) d z ] R ( 0 ) ,
T z exp [ 0 z M ̂ ( z ) d z ] = m = 0 1 m ! T z ( 0 z M ̂ ( z ) d z ) m = m = 0 0 z A ̂ ( m ) ( z ) d z
0 z G ̂ ( 1 ) ( z ) d z = 0 z A ̂ ( 1 ) ( z ) d z ,
0 z G ̂ ( 2 ) ( z ) d z = 0 z A ̂ ( 2 ) ( z ) d z 1 2 T z ( 0 z A ̂ ( 1 ) ( z ) d z ) 2 .
0 z G ̂ ( 1 ) ( z ) d z = 0 ,
0 z G ̂ ( 2 ) ( z ) d z = 1 2 T z ( 0 z M ̂ ( z ) d z ) 2 = 0 z 0 z 1 M ̂ ( z 1 ) M ̂ ( z 2 ) d z 1 d z 2 .
Γ i = 0 z v i ( z ) v i ( z 1 ) d z 1 0 v i ( 0 ) v i ( z 1 ) d z 1 , i = 1 , 2 , 3 .
( ρ I ρ I 0 ) = [ D 1 0 + D 1 1 Ω 2 0 0 0 0 D 2 0 + D 2 1 Ω 2 0 0 0 0 D 3 0 + D 3 1 Ω 2 0 0 0 0 D 0 1 Ω 2 ] ( ρ I ρ I 0 ) .
D k 0 = β 0 2 ( Γ l + Γ m ) ,
D k 1 = β 2 Γ k ,
D 0 1 = β 2 ( Γ 1 + Γ 2 + Γ 3 ) ,
ρ ̂ ( z , ω , T ) = d Ω 2 π exp ( j Ω T ) ρ ̂ ( z , ω , Ω ) .
P e ( T ) = d ω 2 π e ρ ̂ ( z , ω , T ) e .
P ( T ) = d ω 2 π Tr [ ρ ̂ ( z , ω , T ) ] = d ω 2 π ρ 0 ( z , ω , T )
( ρ ρ 0 ) = [ D 1 0 D 1 1 T 2 0 0 0 0 D 2 0 D 2 1 T 2 0 0 0 0 D 3 0 D 3 1 T 2 0 0 0 0 D 0 1 T 2 ] ( ρ ρ 0 ) .
ρ 1 ( z ) = exp ( D 1 0 z ) ρ 1 ( 0 ) ,
Δ d ( z ) Δ d ( 0 ) = Δ d 2 ( 0 ) exp ( z h fiber ) ,
D 1 0 = β 0 2 Δ d 2 ( 0 ) h fiber .

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