Abstract

We analyze, theoretically, a Fabry-Perot interferometer constructed from superimposed, chirped fiber Bragg gratings. Interference effects between the superimposed gratings play a large role in determining the exact positions of the FP resonances. We give formulae to determine the spatial position of the resonances of the system, and, in certain cases, the profile of their field intensity.

© 2005 Optical Society of America

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References

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  1. S. Doucet, R. Slavik, and S. LaRochelle, “High-Finesse large band Fabry-Perot fiber filter with superimposed chirped Bragg gratings,” Electron. Lett. 38, 402–3 (2002).
    [Crossref]
  2. X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
    [Crossref]
  3. G. Brochu, R. Slavik, and S. LaRochelle, “Ultra-Compact 52 mW 50-GHz spaced 16 channels narrow-line and single polarization fiber laser,” in Optical Fiber Communication Conference (The Optical Society of America, Washington, DC, 2004), postdeadline paper PDP22.
  4. R. Slavik, S. Doucet, and S. LaRochelle, “High-Performance all-fiber Fabry-Perot filters with superimposed chirped Bragg gratings,” J. Lightwave Technol. 21, 1059–65 (2003).
    [Crossref]
  5. C. Sung-Hak, I Yokota, and M. Obara, “Free spectral range variation of a broadband, high-finesse, multi-channel Fabry-Perot filter using chirped fiber Bragg gratings,” Jpn. J. Appl. Phys. Part 1,  36, 6383–7 (1997).
    [Crossref]
  6. G. Town, K. Sugden, J. Williams, I. Bennion, and S. B. Poole, “Wide-Band Fabry-Perot-Like Filters in Optical Fiber,” IEEE Photon. Technol. Letters,  7, 78–80 (1995).
    [Crossref]
  7. A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of a Bragg grating and its applications to Fabry-Perot cavities,” J. Opt. Soc. Ame. A 20, 273–81 (2003).
    [Crossref]
  8. T. Erdogan, “Fiber Grating Spectra,” Journal of Lightwave Technology,  15, 1277–94 (1997).
    [Crossref]
  9. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–67 (1993).
    [Crossref]

2003 (3)

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

R. Slavik, S. Doucet, and S. LaRochelle, “High-Performance all-fiber Fabry-Perot filters with superimposed chirped Bragg gratings,” J. Lightwave Technol. 21, 1059–65 (2003).
[Crossref]

A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of a Bragg grating and its applications to Fabry-Perot cavities,” J. Opt. Soc. Ame. A 20, 273–81 (2003).
[Crossref]

2002 (1)

S. Doucet, R. Slavik, and S. LaRochelle, “High-Finesse large band Fabry-Perot fiber filter with superimposed chirped Bragg gratings,” Electron. Lett. 38, 402–3 (2002).
[Crossref]

1997 (2)

T. Erdogan, “Fiber Grating Spectra,” Journal of Lightwave Technology,  15, 1277–94 (1997).
[Crossref]

C. Sung-Hak, I Yokota, and M. Obara, “Free spectral range variation of a broadband, high-finesse, multi-channel Fabry-Perot filter using chirped fiber Bragg gratings,” Jpn. J. Appl. Phys. Part 1,  36, 6383–7 (1997).
[Crossref]

1995 (1)

G. Town, K. Sugden, J. Williams, I. Bennion, and S. B. Poole, “Wide-Band Fabry-Perot-Like Filters in Optical Fiber,” IEEE Photon. Technol. Letters,  7, 78–80 (1995).
[Crossref]

1993 (1)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–67 (1993).
[Crossref]

Bennion, I.

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

G. Town, K. Sugden, J. Williams, I. Bennion, and S. B. Poole, “Wide-Band Fabry-Perot-Like Filters in Optical Fiber,” IEEE Photon. Technol. Letters,  7, 78–80 (1995).
[Crossref]

Brochu, G.

G. Brochu, R. Slavik, and S. LaRochelle, “Ultra-Compact 52 mW 50-GHz spaced 16 channels narrow-line and single polarization fiber laser,” in Optical Fiber Communication Conference (The Optical Society of America, Washington, DC, 2004), postdeadline paper PDP22.

Byron, K.

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

Doucet, S.

R. Slavik, S. Doucet, and S. LaRochelle, “High-Performance all-fiber Fabry-Perot filters with superimposed chirped Bragg gratings,” J. Lightwave Technol. 21, 1059–65 (2003).
[Crossref]

S. Doucet, R. Slavik, and S. LaRochelle, “High-Finesse large band Fabry-Perot fiber filter with superimposed chirped Bragg gratings,” Electron. Lett. 38, 402–3 (2002).
[Crossref]

Erdogan, T.

T. Erdogan, “Fiber Grating Spectra,” Journal of Lightwave Technology,  15, 1277–94 (1997).
[Crossref]

Felmeri, I.

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

Floridi, M.

A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of a Bragg grating and its applications to Fabry-Perot cavities,” J. Opt. Soc. Ame. A 20, 273–81 (2003).
[Crossref]

Huang, Z.

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

Khrushchev, Igor

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

LaRochelle, S.

R. Slavik, S. Doucet, and S. LaRochelle, “High-Performance all-fiber Fabry-Perot filters with superimposed chirped Bragg gratings,” J. Lightwave Technol. 21, 1059–65 (2003).
[Crossref]

S. Doucet, R. Slavik, and S. LaRochelle, “High-Finesse large band Fabry-Perot fiber filter with superimposed chirped Bragg gratings,” Electron. Lett. 38, 402–3 (2002).
[Crossref]

G. Brochu, R. Slavik, and S. LaRochelle, “Ultra-Compact 52 mW 50-GHz spaced 16 channels narrow-line and single polarization fiber laser,” in Optical Fiber Communication Conference (The Optical Society of America, Washington, DC, 2004), postdeadline paper PDP22.

Lloyd, G.

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

Martinelli, M.

A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of a Bragg grating and its applications to Fabry-Perot cavities,” J. Opt. Soc. Ame. A 20, 273–81 (2003).
[Crossref]

Melloni, A.

A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of a Bragg grating and its applications to Fabry-Perot cavities,” J. Opt. Soc. Ame. A 20, 273–81 (2003).
[Crossref]

Mitchell, J.

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

Morichetti, F.

A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of a Bragg grating and its applications to Fabry-Perot cavities,” J. Opt. Soc. Ame. A 20, 273–81 (2003).
[Crossref]

Obara, M.

C. Sung-Hak, I Yokota, and M. Obara, “Free spectral range variation of a broadband, high-finesse, multi-channel Fabry-Perot filter using chirped fiber Bragg gratings,” Jpn. J. Appl. Phys. Part 1,  36, 6383–7 (1997).
[Crossref]

Poladian, L.

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–67 (1993).
[Crossref]

Poole, S. B.

G. Town, K. Sugden, J. Williams, I. Bennion, and S. B. Poole, “Wide-Band Fabry-Perot-Like Filters in Optical Fiber,” IEEE Photon. Technol. Letters,  7, 78–80 (1995).
[Crossref]

Rhead, P.

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

Shu, X.

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

Slavik, R.

R. Slavik, S. Doucet, and S. LaRochelle, “High-Performance all-fiber Fabry-Perot filters with superimposed chirped Bragg gratings,” J. Lightwave Technol. 21, 1059–65 (2003).
[Crossref]

S. Doucet, R. Slavik, and S. LaRochelle, “High-Finesse large band Fabry-Perot fiber filter with superimposed chirped Bragg gratings,” Electron. Lett. 38, 402–3 (2002).
[Crossref]

G. Brochu, R. Slavik, and S. LaRochelle, “Ultra-Compact 52 mW 50-GHz spaced 16 channels narrow-line and single polarization fiber laser,” in Optical Fiber Communication Conference (The Optical Society of America, Washington, DC, 2004), postdeadline paper PDP22.

Sugden, K.

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

G. Town, K. Sugden, J. Williams, I. Bennion, and S. B. Poole, “Wide-Band Fabry-Perot-Like Filters in Optical Fiber,” IEEE Photon. Technol. Letters,  7, 78–80 (1995).
[Crossref]

Sung-Hak, C.

C. Sung-Hak, I Yokota, and M. Obara, “Free spectral range variation of a broadband, high-finesse, multi-channel Fabry-Perot filter using chirped fiber Bragg gratings,” Jpn. J. Appl. Phys. Part 1,  36, 6383–7 (1997).
[Crossref]

Town, G.

G. Town, K. Sugden, J. Williams, I. Bennion, and S. B. Poole, “Wide-Band Fabry-Perot-Like Filters in Optical Fiber,” IEEE Photon. Technol. Letters,  7, 78–80 (1995).
[Crossref]

Williams, J.

G. Town, K. Sugden, J. Williams, I. Bennion, and S. B. Poole, “Wide-Band Fabry-Perot-Like Filters in Optical Fiber,” IEEE Photon. Technol. Letters,  7, 78–80 (1995).
[Crossref]

Yokota, I

C. Sung-Hak, I Yokota, and M. Obara, “Free spectral range variation of a broadband, high-finesse, multi-channel Fabry-Perot filter using chirped fiber Bragg gratings,” Jpn. J. Appl. Phys. Part 1,  36, 6383–7 (1997).
[Crossref]

Electron. Lett. (1)

S. Doucet, R. Slavik, and S. LaRochelle, “High-Finesse large band Fabry-Perot fiber filter with superimposed chirped Bragg gratings,” Electron. Lett. 38, 402–3 (2002).
[Crossref]

IEEE Photon. Technol. (1)

G. Town, K. Sugden, J. Williams, I. Bennion, and S. B. Poole, “Wide-Band Fabry-Perot-Like Filters in Optical Fiber,” IEEE Photon. Technol. Letters,  7, 78–80 (1995).
[Crossref]

IEEE Photon. Technol. Lett. (1)

X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. 15, 1111–13 (2003).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Ame. A (1)

A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of a Bragg grating and its applications to Fabry-Perot cavities,” J. Opt. Soc. Ame. A 20, 273–81 (2003).
[Crossref]

Journal of Lightwave Technology (1)

T. Erdogan, “Fiber Grating Spectra,” Journal of Lightwave Technology,  15, 1277–94 (1997).
[Crossref]

Jpn. J. Appl. Phys. (1)

C. Sung-Hak, I Yokota, and M. Obara, “Free spectral range variation of a broadband, high-finesse, multi-channel Fabry-Perot filter using chirped fiber Bragg gratings,” Jpn. J. Appl. Phys. Part 1,  36, 6383–7 (1997).
[Crossref]

Phys. Rev. E (1)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–67 (1993).
[Crossref]

Other (1)

G. Brochu, R. Slavik, and S. LaRochelle, “Ultra-Compact 52 mW 50-GHz spaced 16 channels narrow-line and single polarization fiber laser,” in Optical Fiber Communication Conference (The Optical Society of America, Washington, DC, 2004), postdeadline paper PDP22.

Supplementary Material (1)

» Media 1: AVI (1018 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Schematic view of a CFBG-FP the thick grey band to the left (right) is the position-dependent reflection spectrum of the first (second) CFBG. The three wavelengths indicated by solid lines accumulate 2π worth of phase between the two reflection bands, and are thus strongly transmitted. The wavelengths indicated by dotted lines are strongly reflected.

Fig. 2.
Fig. 2.

Transmission through a single CFBG as a function of the grating strength, κ 0, using the CME (solid line) and Eq. (17) in the text (dashed line).

Fig. 3.
Fig. 3.

(a) |κ (z)| (normalized) for the parameters given in the text. (b) Transmission spectrum associated with the index profile in (a). The black dot indicates the wavelength for which the field profile of Figure 4 is calculated.

Fig. 4.
Fig. 4.

Im [keff ] (thick dotted line, normalized) for the wavelength indicated by the large dot in Fig. 3 b. Also shown is the normalized field intensity at resonance |A+(z)|2 (solid line), and the approximation given by Eq. (23) (dotted line).

Fig. 5.
Fig. 5.

Im [keff ] (dotted line) and |A+(z)|2 (both normalized) for a CFBG-FP with a spacing d=3.2mm (thick line). The accompanying movie clip shows the variation in normalized field intensity as d is varied between 1mm and 10mm. [Media 1]

Fig. 6.
Fig. 6.

Resonance wavelength as a function of γ (solid line). The dashed line shows the shift in the resonance wavelength when , and hence the FSR, is held constant. Inset is Im [keff ].

Equations (26)

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n ( z ) = n 0 + δ n [ 1 + cos ( 2 π Λ ( z ) d z ) ] ,
n ( z ) = n 0 + δ n [ 1 + cos ( 2 π Λ 0 z + ϕ ( z ) ) ] ,
ϕ ( z ) = π C h z 2 2 Λ 0 2 .
n ( z ) = n 0 + i = 1 2 δ n i { 1 + cos 2 π Λ ( z z i ) d z }
= n ̅ + δ n mod ( z ) ,
E ( z , t ) = A ̂ + ( z ) e i ( k 0 z ω t ) + A ̂ ( z ) e i ( k 0 z ω t ) + c . c . ,
d A ̂ + ( z ) d z = + i [ Γ A ̂ + ( z ) + κ ̂ ( z ) A ̂ ( z ) ] ,
d A ̂ ( z ) d z = i [ Γ A ̂ ( z ) + κ ̂ * ( z ) A ̂ + ( z ) ] ,
κ i = δ n i π λ
κ ̂ ( z ) = i = l 2 κ i e i [ 2 π Λ 0 z i + ϕ ( z z i ) ] ,
Γ ( ω ) = n ̅ c ( ω ω 0 ) ,
d A ± ( z ) d z = ± i [ δ ( z ; ω ) A ± ( z ) + κ ( z ) A ( z ) ] ,
δ ( z ; ω ) = Γ ( ω ) 1 2 d θ ( z ) d z .
k eff ( z ; ω ) = δ 2 ( z ; ω ) κ 2 ( z ) ,
ζ ( ω ) = k eff ( z ; ω ) d z .
k eff ( z ; ω ) = ( Γ + π C h 2 Λ 2 0 z ) 2 κ 0 2 .
t = exp ( I m [ k eff ( z ; ω ) ] d z )
= exp ( κ 0 2 C h Λ 0 2 ) .
κ ( z ) = κ 1 ( 1 + γ 2 ) + 2 γ cos ( 2 π d Λ 0 + Δ ϕ ( z ) ) ,
1 2 d θ d z = π C h 2 Λ 0 2 { z γ 2 + γ cos ( Δ ϕ ( z ) ) ( 1 + γ 2 ) + 2 γ cos ( Δ ϕ ( z ) ) } ,
Δ ϕ ( z ) = π C h d Λ 0 2 ( z d 2 )
κ ( z ) = 2 κ 0 cos ( π d Λ 0 + Δ ϕ ( z ) 2 ) ,
1 2 d θ ( z ) d z = π C h 2 Λ 0 2 { z d 2 } .
A + ( z ) A + ( z N ) exp ( z N z I m [ k eff ( z ; ω N ) ] d z ) .
z N = d 2 + ( 1 + N Λ 0 d ) 2 Λ 0 C h .
ω N = ω 0 c π n ̅ Λ 0 ( 1 + N Λ 0 d ) .

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