Abstract

This paper presents an analytical approach to fast analyzing and designing long-period fiber grating (LPFG) devices with cosine-class apodizations by using the Fourier mode coupling (FMC) theory. The LPFG devices include LPFGs, LPFG-based in-fiber Mach–Zehnder and Michelson interferometers, which are apodized with the cosine-class windows of cosine, raised-cosine, Hamming, and Blackman. The analytic models (AMs) of the apodized LPFG devices are derived from the FMC theory, which are compared with the preferred transfer matrix (TM) method to confirm their efficiencies and accuracies. The AM-based analyses are achieved and verified to be accurate and efficient enough. The AM-based analysis efficiency is improved over 1318 times versus the TM-based one. Based on the analytic models, an analytic design algorithm is proposed and then applied to designing these LPFG devices, which has the complexity of O(N) and is far faster than the existing design methods.

© 2013 Optical Society of America

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  1. X. J. Gu, “Wavelength-division multiplexing isolation fiber filter and light source using cascaded long-period grating,” Opt. Lett. 23, 509–510 (1998).
    [CrossRef]
  2. A. P. Zhang, Z. Guan, and S. He, “Optical low-coherence reflectometry based on long-period grating Mach–Zehnder interferometer,” Appl. Opt. 45, 5733–5739 (2006).
    [CrossRef]
  3. P. L. Swart, “Long-period grating Michelson refractometric sensor,” Meas. Sci. Technol. 15, 1576–1580 (2004).
    [CrossRef]
  4. D. W. Kim, Y. Zhang, K. L. Cooper, and A. Wang, “In-fiber reflection mode interferometer based on a long-period grating for external refractive-index measurement,” Appl. Opt. 44, 5368–5373 (2005).
    [CrossRef]
  5. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
    [CrossRef]
  6. A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, and C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
    [CrossRef]
  7. A. Cusano, A. Iadicicco, P. Pilla, L. Contessa, S. Campopiano, A. Cutolo, and M. Giordano, “Mode transition in high refractive index coated long period grating,” Opt. Express 14, 19–34 (2006).
    [CrossRef]
  8. S. M. Tripathi, W. J. Bock, A. Kumar, and P. Mikulic, “Temperature insensitive high-precision refractive-index sensor using two concatenated dual-resonance long-period gratings,” Opt. Lett. 38, 1666–1668 (2013).
    [CrossRef]
  9. T. Allsop, K. Kalli, K. M. Zhou, G. N. Smith, M. Komodromos, J. Petrovic, D. J. Webb, and I. Bennion, “Spectral characteristics and thermal evolution of long-period gratings in photonic crystal fibers fabricated with a near-IR radiation femtosecond laser using point-by-point inscription,” J. Opt. Soc. Am. B 28, 2105–2114 (2011).
    [CrossRef]
  10. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  11. F. Abrishamian and K. Morishita, “Transfer-matrix method based on a discrete coupling model for analyzing uniform and nonuniform codirectional fiber grating couplers,” Appl. Opt. 51, 2367–2372 (2012).
    [CrossRef]
  12. E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. 15, 1295–1302 (1997).
    [CrossRef]
  13. A. Bouzid and M. A. G. Abushagur, “Scattering analysis of slanted fiber gratings,” Appl. Opt. 36, 558–562 (1997).
    [CrossRef]
  14. E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quant. Electron. 32, 2078–2084 (1996).
  15. R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
    [CrossRef]
  16. L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. 37, 154–156 (2001).
    [CrossRef]
  17. G. Chern and L. A. Wang, “Design of binary long-period fiber grating filters by the inverse-scattering method with genetic algorithm optimization,” J. Opt. Soc. Am. A 19, 772–780 (2002).
    [CrossRef]
  18. A. Buryak, J. Bland-Hawthorn, and V. Steblina, “Comparison of inverse scattering algorithms for designing ultra-broadband fiber Bragg gratings,” Opt. Express 17, 1995–2004 (2009).
    [CrossRef]
  19. X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin. 59, 8607–8614 (2010) (in Chinese).
  20. X. K. Zeng and K. Liang, “Analytic solutions for spectral properties of superstructure, Gaussian-apodized and phase shift gratings with short- or long-period,” Opt. Express 19, 22797–22808 (2011).
    [CrossRef]
  21. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]

2013

2012

2011

2010

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin. 59, 8607–8614 (2010) (in Chinese).

2009

2006

2005

2004

P. L. Swart, “Long-period grating Michelson refractometric sensor,” Meas. Sci. Technol. 15, 1576–1580 (2004).
[CrossRef]

2002

2001

L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. 37, 154–156 (2001).
[CrossRef]

1999

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

1998

1997

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

E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. 15, 1295–1302 (1997).
[CrossRef]

A. Bouzid and M. A. G. Abushagur, “Scattering analysis of slanted fiber gratings,” Appl. Opt. 36, 558–562 (1997).
[CrossRef]

1996

E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quant. Electron. 32, 2078–2084 (1996).

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, and C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Abrishamian, F.

Abushagur, M. A. G.

Allsop, T.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bennion, I.

Bergano, N. S.

Bhatia, V.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Bland-Hawthorn, J.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bock, W. J.

Bouzid, A.

Buryak, A.

Campopiano, S.

Capmany, J.

E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. 15, 1295–1302 (1997).
[CrossRef]

E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quant. Electron. 32, 2078–2084 (1996).

Chern, G.

Contessa, L.

Cooper, K. L.

Cusano, A.

Cutolo, A.

Davidson, C. R.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Erdogan, T.

L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. 37, 154–156 (2001).
[CrossRef]

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

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Feced, R.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Giordano, M.

Gu, X. J.

Guan, Z.

He, S.

Iadicicco, A.

Judkins, J. B.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, and C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Kalli, K.

Kim, D. W.

Komodromos, M.

Kumar, A.

Lemaire, P. J.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, and C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Liang, K.

Marti, J.

E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quant. Electron. 32, 2078–2084 (1996).

Mikulic, P.

Morishita, K.

Muriel, M. A.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Pedrazzani, J. R.

Peral, E.

E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. 15, 1295–1302 (1997).
[CrossRef]

E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quant. Electron. 32, 2078–2084 (1996).

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Petrovic, J.

Pilla, P.

Rao, Y. J.

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin. 59, 8607–8614 (2010) (in Chinese).

Sipe, J. E.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Smith, G. N.

Steblina, V.

Swart, P. L.

P. L. Swart, “Long-period grating Michelson refractometric sensor,” Meas. Sci. Technol. 15, 1576–1580 (2004).
[CrossRef]

Tripathi, S. M.

Vengsarkar, A. M.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, and C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Wang, A.

Wang, L.

L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. 37, 154–156 (2001).
[CrossRef]

Wang, L. A.

Webb, D. J.

Zeng, X. K.

X. K. Zeng and K. Liang, “Analytic solutions for spectral properties of superstructure, Gaussian-apodized and phase shift gratings with short- or long-period,” Opt. Express 19, 22797–22808 (2011).
[CrossRef]

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin. 59, 8607–8614 (2010) (in Chinese).

Zervas, M. N.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Zhang, A. P.

Zhang, Y.

Zhou, K. M.

Acta Phys. Sin.

X. K. Zeng and Y. J. Rao, “Theory of Fourier mode coupling for long-period fiber gratings,” Acta Phys. Sin. 59, 8607–8614 (2010) (in Chinese).

Appl. Opt.

Electron. Lett.

L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. 37, 154–156 (2001).
[CrossRef]

IEEE J. Quant. Electron.

E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel’Fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quant. Electron. 32, 2078–2084 (1996).

IEEE J. Quantum Electron.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

J. Lightwave Technol.

E. Peral and J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. 15, 1295–1302 (1997).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

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

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Meas. Sci. Technol.

P. L. Swart, “Long-period grating Michelson refractometric sensor,” Meas. Sci. Technol. 15, 1576–1580 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Mode couplings in (a) LPFG, (b) LPFG-based MZI, and (c) LPFG-based MI.

Fig. 2.
Fig. 2.

Window functions of the apodizations in cosine-, raised-cosine-, Hamming-, and Blackman-apodized LPFG devices.

Fig. 3.
Fig. 3.

Calculated transmissions of the (a) CA-; (b) RCA-; (c) HA-; and (d) BA-LPFGs along with varying their “DC” index changes and lengths by using the analytic models (solid lines) and the TM method (dotted lines).

Fig. 4.
Fig. 4.

Calculated transmissions of the (a) CA-; (b) RCA-; (c) HA-; and (d) BA-LPFG-based MZIs, according to the analytic models (solid lines) and the TM method (dotted lines).

Fig. 5.
Fig. 5.

Profiles of the index changes in the designed (a) CA-; (b) RCA-; (c) HA-; and (d) BA-LPFGs with the desired transmission specifications.

Fig. 6.
Fig. 6.

Verified transmissions of the designed CA-, RCA-, HA-, and BA-LPFGs by using the TM method.

Fig. 7.
Fig. 7.

Profiles of the index changes in the designed (a) CA-, (b) RCA-, (c) HA-, and (d) BA-LPFG-based MZIs with the expected transmission specifications.

Fig. 8.
Fig. 8.

Transmissions of the designed CA-, RCA-, HA-, and BA-LPFG-based MZIs.

Equations (27)

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dBs(z)dz=mjkBm(z)Δn(z)ej(βmβs)z,
k=0.5ε0ωn0ΩEm(r,φ)Es*(r,φ)dA,
0Bs(L)dBs(z)Bm(z)=jk0LΔn(z)ej(βmβs)zdz,
0LΔn(z)ej2πvLzdz=M(vL)ejϕ(vL),
Bm2(z)=Bm2(0)Bs2(z).
T=cos2[kM(νL)].
TM=cos2{2kM(vL)cos[πvL(P+L)]}.
Δn(z)=δn[1w(z)Vcos(2πz/Λ)],
wc(z)=cos[π(L2z)/(2L)],
wr(z)=0.5+0.5cos[π(L2z)/L],
wH(z)=0.54+0.46cos[π(L2z)/L],
wB(z)=0.42+0.5cos[π(L2z)/L]+0.08cos[2π(L2z)/L].
Mc=δnLVcos(πLσL)2π(14L2σL2),
Mr=δnLVsinc(πLσL)8(1L2σL2),
MH=(0.1350.02L2σL21L2σL2)δnLVsinc(πLσL),
MB=(0.420.045L2σL245L2σL2+L4σL4)δnLVsinc(πLσL),
Tc=cos2[kδnLVcos(πLσL)2π(14L2σL2)],
Tr=cos2[kδnLVsinc(πLσL)8(1L2σL2)],
TH=cos2[kδnLVsinc(πLσL)×(0.1350.02L2σL2)/(1L2σL2)],
TB=cos2[kδnLVsinc(πLσL)×(0.420.045L2σL2)/(45L2σL2+L4σL4)].
TMc=cos2{kδnLVcos[πvL(P+L)]cos(πLσL)π(14L2σL2)},
TMr=cos2{kδnLVcos[πvL(P+L)]sinc(πLσL)4(1L2σL2)},
TMH=cos2{kδnLVcos[πvL(P+L)]sinc(πLσL)×(0.270.04L2σL2)/(1L2σL2)},
TMB=cos2{kδnLVcos[πvL(P+L)]sinc(πLσL)×(0.840.09L2σL2)/(45L2σL2+L4σL4)}.
T0=cos2(ζkδnLV),
W=ηλ0Λ/L,
Δ=λ0Λ/(P+L+Λ).

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