Abstract

An equivalent circuit of both uniform and apodized symmetrical Bragg gratings is proposed, consisting of an ideal partially reflecting mirror placed between two wavelength-dependent uniform propagating sections. The model is simple, exact, and valid for every wavelength; it can be combined with other port-based models and is of great aid in the analysis and design of devices containing two or more Bragg gratings. As an application, the synthesis of dispersive Bragg gratings based Fabry–Pérot cavities is demonstrated. The conditions under which a Bragg-grating-based Fabry–Pérot cavity behaves exactly as an ideal dispersive Fabry–Pérot cavity are discussed. Experimental results that confirm the theoretical spectral response calculated by using the equivalent circuit are reported.

© 2003 Optical Society of America

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References

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  1. K. Raman, Fiber Bragg Gratings, Optics and Photonics Series (Academic, London, 1999).
  2. H.-P. Nolting, R. März, “Results of benchmark tests for different numerical BPM algorithms,” J. Lightwave Technol. 13, 216–224 (1995).
    [CrossRef]
  3. W. P. Wang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
    [CrossRef]
  4. H. Rao, R. Scarmozzino, R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett. 11, 830–832 (1999).
    [CrossRef]
  5. A. Sudbo, “Numerically stable formulation of the transverse resonance method for vector mode-field calculations in dielectric waveguides,” IEEE Photonics Technol. Lett. 5, 342–344 (1993).
    [CrossRef]
  6. R. Costa, A. Melloni, M. Martinelli, “Bandpass resonant filters in photonic crystal waveguides,” IEEE Photon Technol. Lett. (to be published).
  7. T. Kudou, K. Shimizu, K. Harada, T. Ozeki, “Synthesis of grating lattice circuits,” J. Lightwave Technol. 17, 347–353 (1999).
    [CrossRef]
  8. C. K. Madsen, J. H. Zhao, Optical Filter Design and Analysis. A Signal Processing Approach (Wiley, New York, 1999).
  9. R. Zengerle, O. Leminger, “Phase-shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
    [CrossRef]
  10. H. A. Haus, Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
    [CrossRef]
  11. F. Bakhti, P. Sansonetti, “Design and realization of multiple quarter-wave phase-shifts UV-written bandpass filters in optical fibers,” J. Lightwave Technol. 15, 1433–1437 (1997).
    [CrossRef]
  12. A. Melloni, M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM Systems,” J. Lightwave Technol. 20, 296–303 (2002).
    [CrossRef]
  13. A. Melloni, F. Morichetti, M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave structures,” Opt. Quantum Electron. (to be published).
  14. M. B. Steer, J. W. Bandler, C. M. Snowden, “Computer-aided design of RF and microwave circuits and systems,” IEEE Trans. Microwave Theory Tech. 50, 996–1005 (2002).
    [CrossRef]
  15. H. Haus, “Mirrors and interferometers” in Waves and Fields in Optoelectronics, N. Holonyak, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  16. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  17. S. Legoubin, M. Douay, P. Bernage, P. Niay, S. Boj, E. Delevaque, “Free spectral range variations of grating-based Fabry–Pérot filters photowritten in optical fiber,” J. Opt. Soc. Am. A 12, 1687–1694 (1995).
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    [CrossRef] [PubMed]

2002 (2)

A. Melloni, M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM Systems,” J. Lightwave Technol. 20, 296–303 (2002).
[CrossRef]

M. B. Steer, J. W. Bandler, C. M. Snowden, “Computer-aided design of RF and microwave circuits and systems,” IEEE Trans. Microwave Theory Tech. 50, 996–1005 (2002).
[CrossRef]

1999 (2)

H. Rao, R. Scarmozzino, R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett. 11, 830–832 (1999).
[CrossRef]

T. Kudou, K. Shimizu, K. Harada, T. Ozeki, “Synthesis of grating lattice circuits,” J. Lightwave Technol. 17, 347–353 (1999).
[CrossRef]

1997 (2)

F. Bakhti, P. Sansonetti, “Design and realization of multiple quarter-wave phase-shifts UV-written bandpass filters in optical fibers,” J. Lightwave Technol. 15, 1433–1437 (1997).
[CrossRef]

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

1995 (3)

S. Legoubin, M. Douay, P. Bernage, P. Niay, S. Boj, E. Delevaque, “Free spectral range variations of grating-based Fabry–Pérot filters photowritten in optical fiber,” J. Opt. Soc. Am. A 12, 1687–1694 (1995).
[CrossRef]

H.-P. Nolting, R. März, “Results of benchmark tests for different numerical BPM algorithms,” J. Lightwave Technol. 13, 216–224 (1995).
[CrossRef]

R. Zengerle, O. Leminger, “Phase-shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
[CrossRef]

1993 (2)

W. P. Wang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

A. Sudbo, “Numerically stable formulation of the transverse resonance method for vector mode-field calculations in dielectric waveguides,” IEEE Photonics Technol. Lett. 5, 342–344 (1993).
[CrossRef]

1992 (1)

H. A. Haus, Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

1987 (1)

Bakhti, F.

F. Bakhti, P. Sansonetti, “Design and realization of multiple quarter-wave phase-shifts UV-written bandpass filters in optical fibers,” J. Lightwave Technol. 15, 1433–1437 (1997).
[CrossRef]

Bandler, J. W.

M. B. Steer, J. W. Bandler, C. M. Snowden, “Computer-aided design of RF and microwave circuits and systems,” IEEE Trans. Microwave Theory Tech. 50, 996–1005 (2002).
[CrossRef]

Bernage, P.

Boj, S.

Costa, R.

R. Costa, A. Melloni, M. Martinelli, “Bandpass resonant filters in photonic crystal waveguides,” IEEE Photon Technol. Lett. (to be published).

Delevaque, E.

Douay, M.

Erdogan, T.

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

Harada, K.

Haus, H.

H. Haus, “Mirrors and interferometers” in Waves and Fields in Optoelectronics, N. Holonyak, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Haus, H. A.

H. A. Haus, Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

Kudou, T.

Lai, Y.

H. A. Haus, Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

Legoubin, S.

Leminger, O.

R. Zengerle, O. Leminger, “Phase-shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
[CrossRef]

Madsen, C. K.

C. K. Madsen, J. H. Zhao, Optical Filter Design and Analysis. A Signal Processing Approach (Wiley, New York, 1999).

Martinelli, M.

A. Melloni, M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM Systems,” J. Lightwave Technol. 20, 296–303 (2002).
[CrossRef]

A. Melloni, F. Morichetti, M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave structures,” Opt. Quantum Electron. (to be published).

R. Costa, A. Melloni, M. Martinelli, “Bandpass resonant filters in photonic crystal waveguides,” IEEE Photon Technol. Lett. (to be published).

März, R.

H.-P. Nolting, R. März, “Results of benchmark tests for different numerical BPM algorithms,” J. Lightwave Technol. 13, 216–224 (1995).
[CrossRef]

Melloni, A.

A. Melloni, M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM Systems,” J. Lightwave Technol. 20, 296–303 (2002).
[CrossRef]

A. Melloni, F. Morichetti, M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave structures,” Opt. Quantum Electron. (to be published).

R. Costa, A. Melloni, M. Martinelli, “Bandpass resonant filters in photonic crystal waveguides,” IEEE Photon Technol. Lett. (to be published).

Morichetti, F.

A. Melloni, F. Morichetti, M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave structures,” Opt. Quantum Electron. (to be published).

Niay, P.

Nolting, H.-P.

H.-P. Nolting, R. März, “Results of benchmark tests for different numerical BPM algorithms,” J. Lightwave Technol. 13, 216–224 (1995).
[CrossRef]

Osgood, R. M.

H. Rao, R. Scarmozzino, R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett. 11, 830–832 (1999).
[CrossRef]

Ozeki, T.

Raman, K.

K. Raman, Fiber Bragg Gratings, Optics and Photonics Series (Academic, London, 1999).

Rao, H.

H. Rao, R. Scarmozzino, R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett. 11, 830–832 (1999).
[CrossRef]

Sakuda, K.

Sansonetti, P.

F. Bakhti, P. Sansonetti, “Design and realization of multiple quarter-wave phase-shifts UV-written bandpass filters in optical fibers,” J. Lightwave Technol. 15, 1433–1437 (1997).
[CrossRef]

Scarmozzino, R.

H. Rao, R. Scarmozzino, R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett. 11, 830–832 (1999).
[CrossRef]

Shimizu, K.

Snowden, C. M.

M. B. Steer, J. W. Bandler, C. M. Snowden, “Computer-aided design of RF and microwave circuits and systems,” IEEE Trans. Microwave Theory Tech. 50, 996–1005 (2002).
[CrossRef]

Steer, M. B.

M. B. Steer, J. W. Bandler, C. M. Snowden, “Computer-aided design of RF and microwave circuits and systems,” IEEE Trans. Microwave Theory Tech. 50, 996–1005 (2002).
[CrossRef]

Sudbo, A.

A. Sudbo, “Numerically stable formulation of the transverse resonance method for vector mode-field calculations in dielectric waveguides,” IEEE Photonics Technol. Lett. 5, 342–344 (1993).
[CrossRef]

Wang, W. P.

W. P. Wang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

Xu, C. L.

W. P. Wang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

Yamada, M.

Zengerle, R.

R. Zengerle, O. Leminger, “Phase-shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
[CrossRef]

Zhao, J. H.

C. K. Madsen, J. H. Zhao, Optical Filter Design and Analysis. A Signal Processing Approach (Wiley, New York, 1999).

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

W. P. Wang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

H. A. Haus, Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

H. Rao, R. Scarmozzino, R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett. 11, 830–832 (1999).
[CrossRef]

A. Sudbo, “Numerically stable formulation of the transverse resonance method for vector mode-field calculations in dielectric waveguides,” IEEE Photonics Technol. Lett. 5, 342–344 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. B. Steer, J. W. Bandler, C. M. Snowden, “Computer-aided design of RF and microwave circuits and systems,” IEEE Trans. Microwave Theory Tech. 50, 996–1005 (2002).
[CrossRef]

J. Lightwave Technol. (6)

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

F. Bakhti, P. Sansonetti, “Design and realization of multiple quarter-wave phase-shifts UV-written bandpass filters in optical fibers,” J. Lightwave Technol. 15, 1433–1437 (1997).
[CrossRef]

A. Melloni, M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM Systems,” J. Lightwave Technol. 20, 296–303 (2002).
[CrossRef]

H.-P. Nolting, R. März, “Results of benchmark tests for different numerical BPM algorithms,” J. Lightwave Technol. 13, 216–224 (1995).
[CrossRef]

T. Kudou, K. Shimizu, K. Harada, T. Ozeki, “Synthesis of grating lattice circuits,” J. Lightwave Technol. 17, 347–353 (1999).
[CrossRef]

R. Zengerle, O. Leminger, “Phase-shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
[CrossRef]

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

Other (5)

H. Haus, “Mirrors and interferometers” in Waves and Fields in Optoelectronics, N. Holonyak, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1984).

A. Melloni, F. Morichetti, M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave structures,” Opt. Quantum Electron. (to be published).

C. K. Madsen, J. H. Zhao, Optical Filter Design and Analysis. A Signal Processing Approach (Wiley, New York, 1999).

K. Raman, Fiber Bragg Gratings, Optics and Photonics Series (Academic, London, 1999).

R. Costa, A. Melloni, M. Martinelli, “Bandpass resonant filters in photonic crystal waveguides,” IEEE Photon Technol. Lett. (to be published).

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Figures (9)

Fig. 1
Fig. 1

(a) Symmetrical Bragg grating and (b) its equivalent circuit.

Fig. 2
Fig. 2

(a) Bragg-grating-based FP and (b) its equivalent circuit.

Fig. 3
Fig. 3

Typical spectral response of a uniform Bragg grating near λB: magnitude and equivalent length (δn=10-2; L=100 μm).

Fig. 4
Fig. 4

Bandwidth of uniform (solid curves) and Gaussian apodized [Eq. (20), dashed curves] Bragg gratings. In the bandwidth B near λB the reflectivity remains within 10% of its maximum value.

Fig. 5
Fig. 5

Equivalent-length slope normalized to λB of uniform (solid curves) and Gaussian apodized [Eq. (20), dashed curves] Bragg gratings.

Fig. 6
Fig. 6

Transmissivity of IFP1a (dashed curve), BGFP1 (dashed–dotted curve), and BGFP2 (solid curve) structures reported in Table 1.

Fig. 7
Fig. 7

Transmissivity of IFP1a (dashed curve) and BGFP2 (solid curve) structures reported in Table 1.

Fig. 8
Fig. 8

Experimental (solid curve) and calculated (dashed curve) transmission spectral behavior of the BGFP realized in an optical fiber.

Fig. 9
Fig. 9

Definition of the initial phase of the grating.

Tables (3)

Tables Icon

Table 1 IFP and BGFP Parameters with λB =λ0 =1550 nm and FSR =100 GHz

Tables Icon

Table 2 IFP and BGFP Parameters with λB =λ0 =1550 nm and FSR =5000 GHz

Tables Icon

Table 3 FSRs of the BGFP Realized in an Optical Fiber (GHz)

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

n(λ)=nB1+aλ-λBλB,
a(0)b(0)=jt-exp(j2φe)r-rexp(-j2φe)a(L)b(L)=Tea(L)b(L),
φe=2πλnBLe.
r=Tg(1,2)Tg(1,1)=κ sinh(γL)[γ2 cosh2(γL)+Δβ2 sinh2(γL)]1/2,
|t|=1|Tg(1,1)|=γ[γ2 cosh2(γL)+Δβ2 sinh2(γL)]1/2;
Le=λλBL2±Λ4+Λ2πtan-1Δβγtanh(γL),
Le=LeB+ΔLeλ-λBλB,
LeB=L/2±Λ/4
ΔLe=LeB-rM2κ(1-a),
ΔLe=LeB+Λ2πλ(λφg11),
Lp=rMλB2πδn(1-a).
Lr=Le1+Ld+Le2=LrB+ΔLrλ-λBλB,
2πλ1nBLrB+ΔLrλ1-λBλB=Qπ,
2πλ2nBLrB+ΔLrλ2-λBλB=(Q+1)π,
FSR=c2nB(LrB-ΔLr)=c2nBLF.
LF=Ld(1-a)-Λ2πλ(λφg11(1)+λφg11(2)),
LF=Ld+rM12κ1+rM22κ2(1-a)=Ld(1-a)+Lp1+Lp2.
δn=λBπrM-tanh-1(rM)Lri(1-ai)/(1-a)-QΛ±Λ/2,
a=1-Lri(1-ai)Ld+rM/κ.
δn(z)=δn exp-4 ln 2z-L/2L/32,
L=Λm+12-Φπ,
Tg(1, 1)=Tg(2, 2)*=cosh(γL)+jΔβγsinh(γL)exp(jπL/Λ),
Tg(1, 2)=Tg(2, 1)*=-κγsinh(γL)exp[-j(πL/Λ+Φ)],
κ=δnπλ,
Δβ=2πλn-πΛ,
γ2=κ2-Δβ2.

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