Field profiles and spectral properties of chirped Bragg grating Fabry-Perot interferometers

Suresh Pereira; Sophie LaRochelle

doi:10.1364/OPEX.13.001906

Field profiles and spectral properties of chirped Bragg grating Fabry-Perot interferometers

Suresh Pereira and Sophie LaRochelle

Optics Express
Vol. 13,
Issue 6,
pp. 1906-1915
(2005)
•https://doi.org/10.1364/OPEX.13.001906

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Suresh Pereira and Sophie LaRochelle, "Field profiles and spectral properties of chirped Bragg grating Fabry-Perot interferometers," Opt. Express 13, 1906-1915 (2005)

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- Research Papers
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fabry-Perot (050.2230)
- Gratings (050.2770)

About this Article
History
- Original Manuscript: February 8, 2005
- Revised Manuscript: March 3, 2005
- Manuscript Accepted: March 3, 2005
- Published: March 21, 2005

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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.

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References

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|
Author
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Publication

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

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.

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.

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.

Khrushchev, Igor

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]

Lloyd, G.

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.

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.

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.

Shu, X.

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]

Sugden, K.

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.

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

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)

J. Lightwave Technol. (1)

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]

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)

Phys. Rev. E (1)

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

Other (1)

Supplementary Material (1)

» Media 1: AVI (1018 KB)

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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.

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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).

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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.

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Fig. 4. Im [k_eff ] (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).

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Fig. 5. Im [k_eff ] (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]

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Fig. 6. Resonance wavelength as a function of γ (solid line). The dashed line shows the shift in the resonance wavelength when n̄, and hence the FSR, is held constant. Inset is Im [k_eff ].

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Equations (26)

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n (z) = n_{0} + δ n [1 + \cos (\int \frac{2 π}{Λ (z)} d z)],

n (z) = n_{0} + δ n [1 + \cos (\frac{2 π}{Λ_{0}} z + ϕ (z))],

ϕ (z) = - \frac{π C_{h} z^{2}}{2 Λ_{0}^{2}} .

n (z) = n_{0} + \sum_{i = 1}^{2} δ n_{i} {1 + \cos \int \frac{2 π}{Λ (z - z_{i})} d z}

= \overset{̅}{n} + δ n_{\mod} (z),

E (z, t) = {\hat{A}}_{+} (z) e^{i (k_{0} z - ω t)} + {\hat{A}}_{-} (z) e^{i (- k_{0} z - ω t)} + c . c .,

\frac{d \hat{A} + (z)}{d z} = + i [Γ {\hat{A}}_{+} (z) + \hat{κ} (z) {\hat{A}}_{-} (z)],

\frac{{d \hat{A}}_{-} (z)}{d z} = - i [Γ {\hat{A}}_{-} (z) + {\hat{κ}}^{*} (z) {\hat{A}}_{+} (z)],

κ_{i} = δ n_{i} \frac{π}{λ}

\hat{κ} (z) = \sum_{i = l}^{2} κ_{i} e^{i} [- \frac{2 π}{Λ_{0}} z_{i} + ϕ (z - z_{i})],

Γ (ω) = \frac{\overset{̅}{n}}{c} (ω - ω_{0}),

\frac{d A_{\pm} (z)}{d z} = \pm i [δ (z; ω) A_{\pm} (z) + κ (z) A_{\mp} (z)],

δ (z; ω) = Γ (ω) - \frac{1}{2} \frac{d θ (z)}{d z} .

k_{eff} (z; ω) = \sqrt{δ^{2} (z; ω) - κ^{2} (z)},

ζ (ω) = \int k_{eff} (z; ω) d z .

k_{eff} (z; ω) = \sqrt{{(Γ + \frac{π C_{h}}{2 {Λ^{2}}_{0}} z)}^{2} - κ_{0}^{2}} .

∣ t ∣ = \exp (- \int I m [k_{eff} (z; ω)] d z)

= \exp (- \frac{κ_{0}^{2}}{C_{h}} Λ_{0}^{2}) .

κ (z) = κ_{1} \sqrt{(1 + γ^{2}) + 2 γ \cos (- \frac{2 π d}{Λ_{0}} + Δ ϕ (z))},

\frac{1}{2} \frac{d θ}{d z} = - \frac{π C_{h}}{2 Λ_{0}^{2}} {z - \frac{γ^{2} + γ \cos (Δ ϕ (z))}{(1 + γ^{2}) + 2 γ \cos (Δ ϕ (z))}},

Δ ϕ (z) = - \frac{π C_{h} d}{Λ_{0}^{2}} (z - \frac{d}{2})

κ (z) = 2 κ_{0} ∣ \cos (- \frac{π d}{Λ_{0}} + \frac{Δ ϕ (z)}{2}) ∣,

\frac{1}{2} \frac{d θ (z)}{d z} = - \frac{π C_{h}}{2 Λ_{0}^{2}} {z - \frac{d}{2}} .

\frac{∣ A_{+} (z) ∣}{∣ A_{+} (z_{N}) ∣} ≃ \exp (- \int_{z_{N}}^{z} I m [k_{eff} (z; ω_{N})] d z) .

z_{N} = \frac{d}{2} + (1 + \frac{N Λ_{0}}{d}) \frac{2 Λ_{0}}{C_{h}} .

ω_{N} = ω_{0} - \frac{c π}{\overset{̅}{n} Λ_{0}} (1 + \frac{N Λ_{0}}{d}) .

Abstract

References

2003 (3)

2002 (1)

1997 (2)

1995 (1)

1993 (1)

Bennion, I.

Brochu, G.

Byron, K.

Doucet, S.

Erdogan, T.

Felmeri, I.

Floridi, M.

Huang, Z.

Khrushchev, Igor

LaRochelle, S.

Lloyd, G.

Martinelli, M.

Melloni, A.

Mitchell, J.

Morichetti, F.

Obara, M.

Poladian, L.

Poole, S. B.

Rhead, P.

Shu, X.

Slavik, R.

Sugden, K.

Sung-Hak, C.

Town, G.

Williams, J.

Yokota, I

Electron. Lett. (1)

IEEE Photon. Technol. (1)

IEEE Photon. Technol. Lett. (1)

J. Lightwave Technol. (1)

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

Journal of Lightwave Technology (1)

Jpn. J. Appl. Phys. (1)

Phys. Rev. E (1)

Other (1)

Supplementary Material (1)

Cited By

Figures (6)

Equations (26)

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