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Simple model of errors in chirped fiber gratings

M. R. Matthews, J. Porque, C. D. Hoyle, M. J. Vos, and T. L. Smith

Open Access Open Access

Abstract

A simple etalon based model is presented to show the origin of the wavelength-dependent ripples in the group delay and phase, and in the intensity of optical signals reflected from chirped fiber gratings. The simplicity of the model allows intuitive understanding of the effects, and quantitative predictions. We derive accurate scaling laws that allow the experimenter to make quantitative connections between the grating writing process parameters and grating performance.

©2004 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. (a) Reflection spectrum of a grating written with intentional periodic errors to generate sidebands. (b) The GDR and phase ripple of the same grating, measured from the long wavelength side.

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Fig. 2.
Fig. 2. (left) Diagram of reflections to first order in sideband strength. Light enters from the right (E 0) and is reflected from the grating at various points. The highlighted regions of the fiber grating A,B,C represent the near sideband, main band, and far sideband respectively, for a particular wavelength. (right) Time domain representation of a single pulse after reflection from the band structure, showing early and late echoes.

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Fig. 3.
Fig. 3. Peak to peak phase ripple versus relative sideband size for two different grating strengths. Lines show sideband model and points show CM simulation. Right axis is peak-to-peak GDR for the given phase ripple based on a grating with C 0=0.079 nm/cm and sideband spacing Δλ=0.72 nm.

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Fig. 4.
Fig. 4. Peak to peak GDR versus grating strength. Line is from Eq. (7). Solid points are data from gratings as shown in Fig.1. Open points are the same gratings after annealing.

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Equations (18)

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n ( z ) = n 0 + Δ n ( z ) 2 ( 1 + m ( z ) ) cos ( p ( z ) z + ϕ ( z ) )
m ( z ) = ∫ ( W ( g ) cos gz + X ( g ) sin gz ) dg
ϕ ( z ) = ∫ ( Y ( g ) cos gz + Z ( g ) sin gz ) dg
n ( z ) = n 0 + Δ n 2 Re [ e ipz + 1 2 ∫ ( N + ( g ) e i ( p + g ) g + N − ( g ) e i ( p − g ) z ) dg ]
E r ( k , t ) = e i φ ( AE 0 ( t − τ ) + BE 0 ( t ) + ( − AB 2 + C ( 1 − B 2 ) ) E 0 ( t + τ ) + O ( A 2 , B 2 , AB ) … )
A ≡ f ⌊ Re ( N + ( g ) ) Δ n ( k − g ⁄ 2 n 0 ) ⁄ 4 ⌋
B ≡ f [ Δ n ( k ) ⁄ 2 ]
C ≡ f ⌊ Re ( N − ( g ) ) Δ n ( k + g ⁄ 2 n 0 ) ⁄ 4 ⌋
I = E r * E r ≈ ( B 2 + 2 B ( A + C ) ( 1 − B 2 ) cos ( ϕ ) ) E 0 2
α = tan − 1 Im ( E r ) Re ( E r ) ≈ φ − ( 1 + B 2 ) A B sin ( ϕ ) + C B ( 1 − B 2 ) sin ( ϕ ) .
Δ τ = 2 nL c ( − ( 1 + B 2 ) A B + ( 1 − B 2 ) C B ) cos ( 2 knL )
γ = Δ n sb Δ n mb = A tanh − 1 B .
n ( z ) = n 0 + Δ n ( z ) 2 cos ( p ( z ) z ) + η ( z )
p eff ( z ) = p ( z ) ( 1 + η ( z ) n 0 ) .
n eff ( z ) ≈ n 0 + Δ n ( z ) 2 cos ( p ( z ) z + p 0 n 0 ∫ 0 z η ( x ) dx ) .
n eff ( z ) = n 0 + Δ n ( z ) 2 cos ( p ( z ) z ) + Δ n ( z ) 4 p 0 δ n g n 0 [ cos ( ( p ( z ) + g ) z ) − cos ( ( p ( z ) − g ) z ) ]
A , − C = 1 2 p 0 g δ n n 0 tanh − 1 ( B )
l tr = 2 π ⁄ g tr = λ 2 ⁄ 2 nc τ C 0 .
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