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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References

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  1. R. Kashyap and M. deLacerda-Rocha, “On the group delay of chirped fibre Bragg gratings,” Opt. Commun. 153, 19–22 (1998).
    [CrossRef]
  2. L. Poladian, “Understanding profile-induced group-delay ripple in Bragg gratings,” Appl. Opt. 39, 1920–1923 (2000).
    [CrossRef]
  3. R. Feced and M.N. Zervas, “Effects of Random Phase and Amplitude Errors in Optical Fiber Bragg Gratings,” J. Lightwave Technol. 18, 90–101 (2000).
    [CrossRef]
  4. M. Sumetsky, B.J. Eggleton, and C. Martijn de Sterke, “Theory of group delay ripple generated by chirped fiber gratings,” Opt. Express 10, 332–340, 2002. http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-332.
    [CrossRef] [PubMed]
  5. D. Garthe, G. Milner, and Y. Cai, “System performance of broadband dispersion compensating gratings,” Electron. Lett. 19, 582–583 (1998).
    [CrossRef]
  6. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
    [CrossRef]
  7. J.F. Brennan and D.L. LaBrake, “Realization of >10-m-long chirped fiber Bragg gratings,” OSA, Bragg Gratings, Photosensitivity, and Poling, Stuart, FL, ThD2, pp. 35–37 (September 1999).
  8. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474 (1987).
    [CrossRef] [PubMed]
  9. T. Erdogan, “Fiber grating spectra,” J. Lightwave Techno. 15, 1277–1294 (1997).
    [CrossRef]
  10. T. Erdogan, private communication. We have also verified this scaling against coupled-mode simulations.
  11. X. Fan, D.L. LaBrake, and J.F. Brennan, “Chirped fiber grating characterization with phase ripples,” OSA Optical Fiber Communications (Optical Society of America, Washington, D.C., 2003), FC2.
  12. M. Eiselt, C.B. Clausen, and R.W. Tkach, “Performance characterization of components with group delay fluctuations,” Symposium on Optical Fiber Measurements (NIST, Boulder, Colorado, 2002), Session III.
  13. C. Sheerer, C. Glingener, G. Fisher, M. Bohn, and W. Rosenkranz, “Influence of filter group delay ripples on system performance,” European Conf. Opt. Commun. (Nice, France, 1999), I-410.
  14. M. Sumetsky, P.I. Reyes, P.S. Westbrook, N.M. Litchinitser, and B.J. Eggleton, “Group-delay ripple correction in chirped fiber Bragg gratings,” Opt. Lett. 28, 777–779, (2003).
    [CrossRef] [PubMed]

2003 (1)

2002 (1)

2000 (2)

1998 (2)

R. Kashyap and M. deLacerda-Rocha, “On the group delay of chirped fibre Bragg gratings,” Opt. Commun. 153, 19–22 (1998).
[CrossRef]

D. Garthe, G. Milner, and Y. Cai, “System performance of broadband dispersion compensating gratings,” Electron. Lett. 19, 582–583 (1998).
[CrossRef]

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Techno. 15, 1277–1294 (1997).
[CrossRef]

1993 (1)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[CrossRef]

1987 (1)

Bohn, M.

C. Sheerer, C. Glingener, G. Fisher, M. Bohn, and W. Rosenkranz, “Influence of filter group delay ripples on system performance,” European Conf. Opt. Commun. (Nice, France, 1999), I-410.

Brennan, J.F.

J.F. Brennan and D.L. LaBrake, “Realization of >10-m-long chirped fiber Bragg gratings,” OSA, Bragg Gratings, Photosensitivity, and Poling, Stuart, FL, ThD2, pp. 35–37 (September 1999).

X. Fan, D.L. LaBrake, and J.F. Brennan, “Chirped fiber grating characterization with phase ripples,” OSA Optical Fiber Communications (Optical Society of America, Washington, D.C., 2003), FC2.

Cai, Y.

D. Garthe, G. Milner, and Y. Cai, “System performance of broadband dispersion compensating gratings,” Electron. Lett. 19, 582–583 (1998).
[CrossRef]

Clausen, C.B.

M. Eiselt, C.B. Clausen, and R.W. Tkach, “Performance characterization of components with group delay fluctuations,” Symposium on Optical Fiber Measurements (NIST, Boulder, Colorado, 2002), Session III.

deLacerda-Rocha, M.

R. Kashyap and M. deLacerda-Rocha, “On the group delay of chirped fibre Bragg gratings,” Opt. Commun. 153, 19–22 (1998).
[CrossRef]

Eggleton, B.J.

Eiselt, M.

M. Eiselt, C.B. Clausen, and R.W. Tkach, “Performance characterization of components with group delay fluctuations,” Symposium on Optical Fiber Measurements (NIST, Boulder, Colorado, 2002), Session III.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Techno. 15, 1277–1294 (1997).
[CrossRef]

T. Erdogan, private communication. We have also verified this scaling against coupled-mode simulations.

Fan, X.

X. Fan, D.L. LaBrake, and J.F. Brennan, “Chirped fiber grating characterization with phase ripples,” OSA Optical Fiber Communications (Optical Society of America, Washington, D.C., 2003), FC2.

Feced, R.

Fisher, G.

C. Sheerer, C. Glingener, G. Fisher, M. Bohn, and W. Rosenkranz, “Influence of filter group delay ripples on system performance,” European Conf. Opt. Commun. (Nice, France, 1999), I-410.

Garthe, D.

D. Garthe, G. Milner, and Y. Cai, “System performance of broadband dispersion compensating gratings,” Electron. Lett. 19, 582–583 (1998).
[CrossRef]

Glingener, C.

C. Sheerer, C. Glingener, G. Fisher, M. Bohn, and W. Rosenkranz, “Influence of filter group delay ripples on system performance,” European Conf. Opt. Commun. (Nice, France, 1999), I-410.

Kashyap, R.

R. Kashyap and M. deLacerda-Rocha, “On the group delay of chirped fibre Bragg gratings,” Opt. Commun. 153, 19–22 (1998).
[CrossRef]

LaBrake, D.L.

X. Fan, D.L. LaBrake, and J.F. Brennan, “Chirped fiber grating characterization with phase ripples,” OSA Optical Fiber Communications (Optical Society of America, Washington, D.C., 2003), FC2.

J.F. Brennan and D.L. LaBrake, “Realization of >10-m-long chirped fiber Bragg gratings,” OSA, Bragg Gratings, Photosensitivity, and Poling, Stuart, FL, ThD2, pp. 35–37 (September 1999).

Litchinitser, N.M.

Martijn de Sterke, C.

Milner, G.

D. Garthe, G. Milner, and Y. Cai, “System performance of broadband dispersion compensating gratings,” Electron. Lett. 19, 582–583 (1998).
[CrossRef]

Poladian, L.

L. Poladian, “Understanding profile-induced group-delay ripple in Bragg gratings,” Appl. Opt. 39, 1920–1923 (2000).
[CrossRef]

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[CrossRef]

Reyes, P.I.

Rosenkranz, W.

C. Sheerer, C. Glingener, G. Fisher, M. Bohn, and W. Rosenkranz, “Influence of filter group delay ripples on system performance,” European Conf. Opt. Commun. (Nice, France, 1999), I-410.

Sakuda, K.

Sheerer, C.

C. Sheerer, C. Glingener, G. Fisher, M. Bohn, and W. Rosenkranz, “Influence of filter group delay ripples on system performance,” European Conf. Opt. Commun. (Nice, France, 1999), I-410.

Sumetsky, M.

Tkach, R.W.

M. Eiselt, C.B. Clausen, and R.W. Tkach, “Performance characterization of components with group delay fluctuations,” Symposium on Optical Fiber Measurements (NIST, Boulder, Colorado, 2002), Session III.

Westbrook, P.S.

Yamada, M.

Zervas, M.N.

Appl. Opt. (2)

Electron. Lett. (1)

D. Garthe, G. Milner, and Y. Cai, “System performance of broadband dispersion compensating gratings,” Electron. Lett. 19, 582–583 (1998).
[CrossRef]

J. Lightwave Techno. (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Techno. 15, 1277–1294 (1997).
[CrossRef]

J. Lightwave Technol. (1)

Opt. Commun. (1)

R. Kashyap and M. deLacerda-Rocha, “On the group delay of chirped fibre Bragg gratings,” Opt. Commun. 153, 19–22 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[CrossRef]

Other (5)

J.F. Brennan and D.L. LaBrake, “Realization of >10-m-long chirped fiber Bragg gratings,” OSA, Bragg Gratings, Photosensitivity, and Poling, Stuart, FL, ThD2, pp. 35–37 (September 1999).

T. Erdogan, private communication. We have also verified this scaling against coupled-mode simulations.

X. Fan, D.L. LaBrake, and J.F. Brennan, “Chirped fiber grating characterization with phase ripples,” OSA Optical Fiber Communications (Optical Society of America, Washington, D.C., 2003), FC2.

M. Eiselt, C.B. Clausen, and R.W. Tkach, “Performance characterization of components with group delay fluctuations,” Symposium on Optical Fiber Measurements (NIST, Boulder, Colorado, 2002), Session III.

C. Sheerer, C. Glingener, G. Fisher, M. Bohn, and W. Rosenkranz, “Influence of filter group delay ripples on system performance,” European Conf. Opt. Commun. (Nice, France, 1999), I-410.

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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.

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.

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.

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.

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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