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Polarization mode dispersion in chirped fiber Bragg gratings

Ding Wang, Michael R. Matthews, and James F. Brennan III

Open Access Open Access

Abstract

We clarify the relationship between group delay ripple and differential group delay in birefringent, chirped fiber Bragg gratings and relate this information to polarization mode dispersion. We illustrate that a grating can be characterized completely by measuring the grating phase ripple and fiber birefringence with careful selection of measurement system parameters. The impact of these imperfections on device performance as dispersion compensators in optical communications systems is explored with system testbed simulations and measurements.

©2004 Optical Society of America

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Figures (9)
Fig. 1.
Fig. 1. (a) Low-bandwidth measurement of DGD across the bandwidth of a chirped fiber grating in reflection. The thick line corresponds to a 20 GHz moving-window average. (b) The corresponding position of the PMD vector in Poincaré space. The solid diamond is the angle θ on the sphere horizontal plane; the open square, the azimuth angle φ. (c) Q-penalty due to the grating in a 10 Gb/s communications system. Error bars indicate the penalty variation with launch polarization at each wavelength.

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Fig. 2.
Fig. 2. Orientation of the PMD vector across the bandwidth of the grating in Poincaré space.

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Fig. 3.
Fig. 3. GDR along each birefringent axes of a grating (upper curves), and the lower curve is the difference between them, i.e. DGD with sign.

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Fig. 4.
Fig. 4. The angles θ and φ (a) and magnitude (b) of the 1st-order PMD vector in Poincaré space measured with high-frequency MPS methods. The thick line in (b) is the 1st-order PMD calculated with the fiber birefringence and grating dispersion.

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Fig. 5.
Fig. 5. Comparison of PMD measurements made with MPS (black solid line) and Jones eigen-analysis (grey dashed line) methods when care is taken to use equivalent bandwidths.

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Fig. 6.
Fig. 6. Grating DGD measured with the MPS method at a 10 GHz modulation frequency (diamond demarcated line) compared with that calculated by taking the vector average of the data in Fig. 1(a)

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Fig. 7.
Fig. 7. Simulations of pulse distortion when the PMD is modeled as a vector, the input pulse (solid line) is not distorted after propagating through the grating (open circles), but the pulse is distorted significantly when the scalar average is used (filled circles).

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Fig. 8.
Fig. 8. Pulse width within the original time slot as a function of the ripple frequency normalized to the standard deviation of the pulse spectrum.

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Fig. 9.
Fig. 9. (a) Simulations of Q-penalty induced by rapidly varying DGD when the input signal polarization is aligned with X (solid line) or Y birefringence axis (dotted line). (b) Penalty variation when the polarization alignment of the input signal changes gradually from the X to the Y birefringence axis. The penalty offset is caused by noise added to the simulations.

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Equations (18)

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ϕ x ( ω ) = β x · L x ( ω ) = n x ω c · L x ( ω ) ,
2 · ( n + Δ n ) Λ 2 · n · Λ = λ + Δ λ λ = ω ω + Δ ω
Δ n n = Δ λ λ ≈ − Δ ω ω .
ϕ y ( ω ) = n y ω c · L y ( ω ) = ( n x + Δ n ) ω c · L x ( ω + Δ ω )
= ( n x + Δ n ) · ω c · { L x ( ω ) + Δ ω · L ˙ x ( ω ) + 1 2 Δ ω 2 · L ̈ x ( ω ) + ⋯ }
= ϕ x ( ω ) + Δ ϕ
Δ ϕ ≡ Δ n · ω c · L x ( ω ) + ( n x + Δ n ) · ω c · { Δ ω · L ˙ x ( ω ) + 1 2 Δ ω 2 · L x ̈ ( ω ) + ⋯ }
≡ Δ n · ω c · L x ( ω ) + ( n x + Δ n ) · ω c · Δ L ( ω )
ϕ y ( ω ) = ( n x + Δ n n x ) · ( ω ω + Δ ω ) · ϕ x ( ω + Δ ω )
= ( n x + Δ n n x ) 2 · ϕ x ( ω + Δ ω )
ϕ y ( ω ) ≈ ϕ x ( ω + Δ ω )
t g = L v g ≡ L · ( ∂ β ⁄ ∂ ω ) ,
Δ t ≈ Δ n c · L x ( ω ) + n x c · Δ L ( ω ) ,
∂ Δ t ∂ ω ≈ Δ n c · L ˙ x ( ω ) + n x c · Δ L ˙ ( ω ) .
e i ϕ x ( ω ) [ 1 0 0 e i Δ ϕ ] [ e i ϕ x R ( ω ) 0 0 e i ϕ x R ( ω + Δ ω ) ] ,
e i { ϕ x ( ω ) + ϕ x R ( ω ) } [ 1 0 0 e i { Δ ϕ + Δ ϕ R } ]
DGD ( ω ) ≈ ∣ D · Δ λ + GDR x ( ω + Δ ω ) − GDR x ( ω ) ∣ ,
Ω → ≈ 〈 Δ τ x ( ω ) 〉 · i ̂ x + 〈 Δ τ y ( ω ) 〉 · i ̂ y + 〈 Δ τ z ( ω ) 〉 · i ̂ z
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