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Impact of fiber design on polarization dependence in microbend gratings

Steven E. Golowich and Siddharth Ramachandran

Open Access Open Access

Abstract

Polarization dependence in microbend gratings is an inherent problem. We formulate simple analytical expressions to describe it, and demonstrate their effectiveness via a comparison with experimental results on a standard transmission fiber. The ability to control polarization dependence with fiber design potentially enables replacing UV-LPGs within low-cost, tunable microbend gratings.

©2005 Optical Society of America

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Figures (3)
Fig. 1.
Fig. 1. Measurements of transmitted power as a function of wavelength, for two orthogonal input SOPs. The fiber under test is TWRS™.

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Fig. 2.
Fig. 2. (a) Phase matching curves. The lines denote the predictions from theory; the symbols are the experimentally measured values. (b)Polarization splitting: δλ ℓ,m is the maximal difference in resonance wavelength between the fundamental and LP ℓm modes obtained by varying the input polarization.

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Fig. 3.
Fig. 3. (a) Fiber index profile plotted along with radial mode fields for the LP12 to LP15 modes; (b) index profile near the cladding-coating boundary, again with mode fields.

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Equations (32)

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δλ = β 12 β ˙ 20 ∣ λ = λ 0 .
E t x t z t = e i ( βz − ωt ) e t ( x t )
( − ∇ t 2 − k 2 n 2 + V ( vect ) + β 2 ) e t = 0
V ( vect ) e t = − ∇ t ( e t · ∇ t ln n 2 ) .
n 2 ( x t ) = n 0 2 ( r ) + n ell 2 r θ
n 0 2 ( r ) = n co 2 ( 1 − 2 Δ f ( r ) )
n ell 2 ( r , θ ) = c ell r cos ( 2 θ ) f ′ ( r )
c ell = 1 2 e 2 Δ n co 2
( − ∇ t 2 − k 2 n 0 2 + V ( ell ) + V ( vect ) + β 2 ) e t = 0
( ∇ t 2 + k 2 n 0 2 − β 2 ) e t = 0 .
φ ℓ , m , ν , p ̂ r θ = F ℓ , m ( r ) ν ( ℓθ ) p ̂ ,
− F ℓ , m ″ − 1 r F ℓ , m ′ + ( ℓ 2 r 2 − k 2 n 0 2 ) F ℓ , m = − β 2 F ℓ , m .
I 1 = ∫ rdrdθF ( r ) F ′ ( r ) f ′ ( r )
I 2 ( ℓ ) = ℓ ∫ rdrdθ 1 r F ( r ) 2 f ′ ( r )
I 3 = ∫ rdrdθrF ( r ) 2 f ′ ( r ) .
〈 ℓ = 1 , m , · , · ∣ V ( ell ) ∣ ℓ = 1 , m , · , · 〉 = π 2 k 2 c ell I 3 ( − 1 1 − 1 1 ) .
〈 ℓ = 0 , m , · , · ∣ V ( vect ) ∣ ℓ = 0 , m , · , · 〉 = − 2 Δπ ( I 1 I 1 )
〈 ℓ = 0 , m , · , · ∣ V ( vect ) ∣ ℓ = 0 , m , · , · 〉 =
− 2 Δ π 4 ( 3 I 1 + I 2 ( 1 ) I 1 + 3 I 2 ( 1 ) I 1 − I 2 ( 1 ) I 1 − I 2 ( 1 ) I 1 − I 2 ( 1 ) I 1 − I 2 ( 1 ) I 1 + 3 I 2 ( 1 ) 3 I 1 + I 1 ( 1 ) )
〈 ℓ > 1 , m , · , · ∣ V ( vect ) ∣ ℓ > 1 , m , · , · 〉 = − 2 Δ π 4 ( I 1 I 2 ( ℓ ) I 1 − I 2 ( ℓ ) − I 2 ( ℓ ) I 1 I 2 ( ℓ ) I 1 ) .
δ β TE 2 = 0 e t ( TE ) = ∣ 1 , m , sin , x ̂ 〉 + ∣ 1 , m , cos , y ̂ 〉
δ β TM 2 = 2 πΔ ( I 1 + I 2 ( 1 ) ) e t ( TM ) = ∣ 1 , m , cos , x ̂ 〉 + ∣ 1 , m , sin , y ̂ 〉
δ β HE 2 = πΔ ( I 1 − I 2 ( 1 ) ) e t ( HE ) = ∣ 1 , m , cos , x ̂ 〉 - ∣ 1 , m , sin , y ̂ 〉 ,
∣ 1 , m , sin , x ̂ 〉 + ∣ 1 , m , cos , y ̂ 〉 .
n gr 2 x t z ∝ r cos ( θ − θ 0 ) cos ( 2 π Λ z )
da 0 dz = iK 0,1 e iαz a 1
da 1 dz = iK 1,0 e − iαz a 0 ,
K i , j = κ ∫ rdrdθ e t ( i ) r cos ( θ − θ 0 ) e t ( j ) .
∣ a 1 ( L ) ∣ 2 = ( 1 + x ) − 1 sin 2 ( π 2 ( 1 + x ) 1 2 )
x = L 2 π 2 ( β 01 − β Λ ) 2 .
δλ ( ε ) ≃ π L β ˙ 01 ( 0 ) ε ,
β ˙ 01 ( 0 ) = dβ 01 dλ ∣ λ = λ 0 .
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