Abstract

The theory of the group delay ripple generated by apodized chirped fiber gratings is developed using the analogy between noisy gratings and superstructure Bragg gratings. It predicts the fundamental cutoff of the high frequency spatial noise of grating parameters in excellent agreement with the experimental data. We find simple general relationship between the high-frequency ripple in the grating period and the group delay ripple. In particular, we show that the amplitude of a single-frequency group delay ripple component changes with grating period chirp, C, as C -3/2 and is proportional to the grating index modulation, while its phase shift and period changes as C -1 .

© 2002 Optical Society of America

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References

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  1. B. J. Eggleton, A. Ahuja, P. S. Westbrook, J. A. Rogers, P. Kuo, T. N. Nielsen, and B. Mikkelsen, �Integrated tunable fiber gratings for dispersion management in high-bit rate systems,� J. Lightwave Technol. 18, 1418-1432 (2000)
    [CrossRef]
  2. R. Kashyap, Fiber Bragg Gratings, (Academic Press, 1999).
  3. M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R.I. Laming, �Dispersion compensating fibre Bragg gratings,� in Active and Passive Optical Components for WDM Communication, Proc. SPIE 4532, 540-551 (2001).
    [CrossRef]
  4. F. Ouellette, �The effect of profile noise on the spectral response of fiber gratings,� in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, (Optical Society of America Williamsburg 1997) Paper BMG13-2.
  5. R. Feced, M. N. Zervas, �Effect of random phase and amplitude errors in optical fiber gratings,� J. Lightwave Technol. 18, 90-101 (2000).
    [CrossRef]
  6. R. Feced, J. A. J. Fells, S. E. Kanellopoulos, P. J. Bennett, and H. F. M. Priddle, �Impact of random phase errors on the performance of fiber grating dispersion compensators,� in Optical Fiber Communication Conference, (Optical Sosiety of America, Washington, D.C., 2001) Paper WDD89.
  7. L. Poladian, �Graphical andWKB analysis of nonuniform Bragg gratings,� Phys. Rev. E 48, 4758-4767 (1993).
    [CrossRef]
  8. N. G. R. Broderick and C. M. de Sterke, �Theory of grating superstructures,� Phys. Rev. E 55, 3634-3646 (1997).
    [CrossRef]
  9. A. B. Migdal, Qualitative Methods in Quantum Theory, (W.A. Benjamin, Inc, 1977).

J. Lightwave Technol.

Optical Fiber Communication Conference

R. Feced, J. A. J. Fells, S. E. Kanellopoulos, P. J. Bennett, and H. F. M. Priddle, �Impact of random phase errors on the performance of fiber grating dispersion compensators,� in Optical Fiber Communication Conference, (Optical Sosiety of America, Washington, D.C., 2001) Paper WDD89.

Phys. Rev. E

L. Poladian, �Graphical andWKB analysis of nonuniform Bragg gratings,� Phys. Rev. E 48, 4758-4767 (1993).
[CrossRef]

N. G. R. Broderick and C. M. de Sterke, �Theory of grating superstructures,� Phys. Rev. E 55, 3634-3646 (1997).
[CrossRef]

Proc. SPIE

M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R.I. Laming, �Dispersion compensating fibre Bragg gratings,� in Active and Passive Optical Components for WDM Communication, Proc. SPIE 4532, 540-551 (2001).
[CrossRef]

TOPS

F. Ouellette, �The effect of profile noise on the spectral response of fiber gratings,� in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, (Optical Society of America Williamsburg 1997) Paper BMG13-2.

Other

A. B. Migdal, Qualitative Methods in Quantum Theory, (W.A. Benjamin, Inc, 1977).

R. Kashyap, Fiber Bragg Gratings, (Academic Press, 1999).

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Figures (5)

Fig 1.
Fig 1.

Reflection of light from chirped Bragg fiber grating

Fig. 2.
Fig. 2.

Physical picture of the GDR cutoff effect

Fig 3.
Fig 3.

a – comparison of the single-harmonic GDR amplitude calculated from Eq. (4) with numerical calculations and with classical ray approximation Eq. (2) for the amplitude of spatial ripple 0.0025 nm; b – fitting the numerically calculated GDR amplitude vs. chirp dependence (squares) by C -3/2 power law (solid lines) for the spatial period ripple 5 mm and 20 mm.

Fig. 4.
Fig. 4.

Ripple in grating parameters and corresponding GDR

Fig.5.
Fig.5.

a - Experimentally measured group delay and GDR of a typical CFBG; b - Fourier spectrum of this GDR calculated for different bandwidths ∆λ measured from the high-wavelength edge of the reflection band and demonstrating the cutoff frequencies coincident with the ones predicted by Eq. (4).

Equations (21)

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Λ ( z ) = Λ 0 + C ( z z 0 ) + Δ Λ ( z ) ,
Δ Λ ( z ) = q > 0 Δ Λ q exp [ i q ( z z 0 ) ] + c . c . ,
Δ τ ( λ ) = 2 n 0 c 0 C Δ Λ ( z t ) , z t = z 0 + ( λ 2 n 0 Λ 0 ) 2 n 0 C ,
2 k eff ( z st ) = q , k eff ( z ) = π 2 n 0 Λ 0 2 [ Δ λ 2 n 0 C ( z z 0 ) ] 2 Λ 0 2 Δ n 2 ( z ) .
q c = π Δ λ n 0 Λ 0 2 , ν c = π Δ λ 2 C n 0 2 Λ 0 2 .
Δ τ ( Δ λ ) = 0 < q < q c Δ τ q exp [ i q Δ λ 2 n 0 C ] + c . c . ,
Δ τ q = 2 i π Δ n c 0 ( 2 C ) 3 / 2 exp ( i q 2 Λ 0 2 4 π C ) Δ Λ q .
n ( z ) = n 0 + Δ n cos ( 2 π Λ 0 z + 2 π Λ 0 2 Z d z ( Λ ( z ) ) Λ 0 )
u ( z ) = + i [ δ ( z ) u ( z ) + κ ( z ) v ( z ) ]
v ( z ) = i [ δ ( z ) v ( z ) + κ ( z ) u ( z ) ]
δ ( z ) = β π Λ 0 2 ( Λ ( z ) Λ 0 ) , κ ( z ) = π Δ n ( z ) 2 n 0 Λ 0 ,
β = 2 π n 0 λ π Λ 0 π Δ λ 2 n 0 Λ 0 2
Δ τ = Δ τ 1 + Δ τ 2 , Δ τ i = 2 π n 0 c 0 Λ 0 2 d d β Re [ z 0 d x Δ Λ ( z ) G i ( z ) ]
G 1 = u 0 + ( u 0 + ) * + u 0 ( u 0 ) * , G 2 = r 0 1 u 0 + ( u 0 ) * + r 0 u 0 ( u 0 + ) *
( u 0 ± ( z ) v 0 ± ( z ) ) = e ± i z 0 z k eff ( z ) d z , 2 Q ( z ) ( Q ( z ) ± 1 Q ( z ) ± 1 ) ,
Q ( z ) = δ 0 ( z ) κ ( z ) δ 0 ( z ) + κ ( z ) , k eff ( z ) = δ 0 2 ( z ) κ 2 ( z )
Δ τ = Δ τ 1 + Δ τ 2
Δ τ 1 = π n 0 c 0 Λ 0 2 q > 0 Δ Λ q d d β z 0 z t d z δ 0 ( z ) k eff ( z ) e i q ( z z 0 ) + c . c .
Δ τ 2 = π n 0 2 c 0 Λ 0 2 q > 0 Δ Λ q d d β z 0 z t d z κ ( z ) k eff ( z ) ( r 0 1 e i q ( z z 0 ) + 2 i z 0 z k eff ( z ) d z + r 0 e i q ( z z 0 ) 2 i z 0 z k eff ( z ) d z ) + c . c .
2 k eff ( z st ) = q .
z t = Λ 0 2 β π C , z st = Λ 0 2 π C ( β q 2 ) .

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