Abstract

An approach is presented to the design of binary long-period fiber grating (LPFG) filters based on the Gel’fand–Levitan–Marchenko (GLM) inverse-scattering method and genetic algorithm optimization. The nonuniform coupling strength of the binary grating can be realized by varying the local duty ratio. A coupled-mode theory combined with the Poisson sum formula for treating the binary index perturbation is developed for the application of the GLM synthesis method. Since the coupled-mode theory, which smears out the discrete coupling nature, can be regarded only as an approximation to the modeling of a binary LPFG, we use instead the transfer-matrix model to analyze the coupling behavior of a nonuniform binary LPFG. Based on the synthesized grating patterns from the GLM method, a real-coded genetic algorithm with the transfer-matrix model is used to compensate for the discrepancies resulting from use of the coupled-mode theory and to optimize the design. We exemplify the above procedure by designing a flatband LPFG filter and a high-visibility all-fiber Mach–Zehnder filter.

© 2002 Optical Society of America

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  1. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
    [CrossRef]
  2. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
    [CrossRef]
  3. A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
    [CrossRef] [PubMed]
  4. J. R. Qian, H. F. Chen, “Gain flattening fibre filters using phase-shifted long period fibre gratings,” Electron. Lett. 34, 1132–1133 (1998).
    [CrossRef]
  5. J. Bae, J. Chun, S. B. Lee, “Equalization of the non-flat erbium gain spectrum using the multiport lattice filter model,” in Optical Fiber Communications Conference, Vol. 2 of 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 80–83.
  6. W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, R. L. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
    [CrossRef] [PubMed]
  7. G. W. Chern, L. A. Wang, “Transfer-matrix method based on perturbation expansion for periodic and quasi-periodic binary long-period gratings,” J. Opt. Soc. Am. A 16, 2675–2689 (1999).
    [CrossRef]
  8. H. Sakata, “Sidelobe suppression in grating-assisted wavelength-selective couplers,” Opt. Lett. 17, 463–465 (1992).
    [CrossRef] [PubMed]
  9. Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP-InGaAsP vertical grating-assisted co-directional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
    [CrossRef]
  10. G. W. Chern, L. A. Wang, “Analysis and design of almost-periodic vertical-grating-assisted codirectional coupler filters with nonuniform duty ratios,” Appl. Opt. 39, 4629–4637 (2000).
    [CrossRef]
  11. G. H. Song, S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
    [CrossRef]
  12. L. Poladian, “Simple grating synthesis algorithm,” Opt. Lett. 25, 787–789 (2000).
    [CrossRef]
  13. R. Feced, M. N. Zervas, 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]
  14. A. Ishimaru, “Theory of unequally spaced arrays,” IRE Trans. Antennas Propag. 10, 691–702 (1963).
    [CrossRef]
  15. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991).
  16. K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. 9, 1481–1492 (1991).
    [CrossRef]
  17. X. J. Gu, “Wavelength-division multiplexing isolation fiber filter and light source using cascaded long-period fiber gratings,” Opt. Lett. 23, 509–510 (1998).
    [CrossRef]
  18. B. H. Lee, J. Nishii, “Dependence of fringe spacing on the grating separation in a long-period fiber grating pair,” Appl. Opt. 38, 3450–3459 (1999).
    [CrossRef]
  19. J. Skaar, K. M. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Lightwave Technol. 16, 1928–1932 (1998).
    [CrossRef]
  20. E. Michielssen, S. Ranjithan, R. Mittra, “Optimal multilayer filter design using real coded genetic algorithm,” IEE Proc. J Optoelectron. 139, 413–420 (1992).
    [CrossRef]
  21. P. L. Swart, A. P. Kotze, B. M. Lacquet, “Effects of the nature of the starting population on the properties of rugate filters designed with the genetic algorithm,” J. Lightwave Technol. 18, 853–859 (2000).
    [CrossRef]

2000 (3)

1999 (3)

1998 (3)

1997 (2)

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP-InGaAsP vertical grating-assisted co-directional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
[CrossRef]

1996 (2)

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

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

1995 (1)

1992 (2)

E. Michielssen, S. Ranjithan, R. Mittra, “Optimal multilayer filter design using real coded genetic algorithm,” IEE Proc. J Optoelectron. 139, 413–420 (1992).
[CrossRef]

H. Sakata, “Sidelobe suppression in grating-assisted wavelength-selective couplers,” Opt. Lett. 17, 463–465 (1992).
[CrossRef] [PubMed]

1991 (1)

K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. 9, 1481–1492 (1991).
[CrossRef]

1985 (1)

1963 (1)

A. Ishimaru, “Theory of unequally spaced arrays,” IRE Trans. Antennas Propag. 10, 691–702 (1963).
[CrossRef]

Bae, J.

J. Bae, J. Chun, S. B. Lee, “Equalization of the non-flat erbium gain spectrum using the multiport lattice filter model,” in Optical Fiber Communications Conference, Vol. 2 of 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 80–83.

Barcelos, S.

Bergano, N. S.

Bhatia, V.

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

Chen, H. F.

J. R. Qian, H. F. Chen, “Gain flattening fibre filters using phase-shifted long period fibre gratings,” Electron. Lett. 34, 1132–1133 (1998).
[CrossRef]

Chern, G. W.

Chun, J.

J. Bae, J. Chun, S. B. Lee, “Equalization of the non-flat erbium gain spectrum using the multiport lattice filter model,” in Optical Fiber Communications Conference, Vol. 2 of 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 80–83.

Coldren, L. A.

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP-InGaAsP vertical grating-assisted co-directional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

Cole, M. J.

Davidson, C. R.

DenBaars, S. P.

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP-InGaAsP vertical grating-assisted co-directional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

Erdogan, T.

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, 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, 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]

Fish, G. A.

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP-InGaAsP vertical grating-assisted co-directional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

Gu, X. J.

Ishimaru, A.

A. Ishimaru, “Theory of unequally spaced arrays,” IRE Trans. Antennas Propag. 10, 691–702 (1963).
[CrossRef]

Jan, Y. H.

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP-InGaAsP vertical grating-assisted co-directional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

Judkins, J. B.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, 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, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef] [PubMed]

Kotze, A. P.

Lacquet, B. M.

Laming, R. L.

Lee, B. H.

Lee, S. B.

J. Bae, J. Chun, S. B. Lee, “Equalization of the non-flat erbium gain spectrum using the multiport lattice filter model,” in Optical Fiber Communications Conference, Vol. 2 of 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 80–83.

Lemaire, P. J.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, 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, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef] [PubMed]

Loh, W. H.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991).

Michielssen, E.

E. Michielssen, S. Ranjithan, R. Mittra, “Optimal multilayer filter design using real coded genetic algorithm,” IEE Proc. J Optoelectron. 139, 413–420 (1992).
[CrossRef]

Mittra, R.

E. Michielssen, S. Ranjithan, R. Mittra, “Optimal multilayer filter design using real coded genetic algorithm,” IEE Proc. J Optoelectron. 139, 413–420 (1992).
[CrossRef]

Muriel, M. A.

R. Feced, M. N. Zervas, 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]

Nishii, J.

Pedrazzani, J. R.

Poladian, L.

Qian, J. R.

J. R. Qian, H. F. Chen, “Gain flattening fibre filters using phase-shifted long period fibre gratings,” Electron. Lett. 34, 1132–1133 (1998).
[CrossRef]

Ranjithan, S.

E. Michielssen, S. Ranjithan, R. Mittra, “Optimal multilayer filter design using real coded genetic algorithm,” IEE Proc. J Optoelectron. 139, 413–420 (1992).
[CrossRef]

Risvik, K. M.

Sakata, H.

Shin, S. Y.

Sipe, J. E.

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

Skaar, J.

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991).

Song, G. H.

Swart, P. L.

Vengsarkar, A. M.

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

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

Wang, L. A.

Winick, K. A.

K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. 9, 1481–1492 (1991).
[CrossRef]

Zervas, M. N.

R. Feced, M. N. Zervas, 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]

W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, R. L. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
[CrossRef] [PubMed]

Appl. Opt. (2)

Electron. Lett. (1)

J. R. Qian, H. F. Chen, “Gain flattening fibre filters using phase-shifted long period fibre gratings,” Electron. Lett. 34, 1132–1133 (1998).
[CrossRef]

IEE Proc. J Optoelectron. (1)

E. Michielssen, S. Ranjithan, R. Mittra, “Optimal multilayer filter design using real coded genetic algorithm,” IEE Proc. J Optoelectron. 139, 413–420 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. Feced, M. N. Zervas, 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]

IEEE Photon. Technol. Lett. (1)

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP-InGaAsP vertical grating-assisted co-directional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

IRE Trans. Antennas Propag. (1)

A. Ishimaru, “Theory of unequally spaced arrays,” IRE Trans. Antennas Propag. 10, 691–702 (1963).
[CrossRef]

J. Lightwave Technol. (4)

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

P. L. Swart, A. P. Kotze, B. M. Lacquet, “Effects of the nature of the starting population on the properties of rugate filters designed with the genetic algorithm,” J. Lightwave Technol. 18, 853–859 (2000).
[CrossRef]

J. Skaar, K. M. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Lightwave Technol. 16, 1928–1932 (1998).
[CrossRef]

K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. 9, 1481–1492 (1991).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Lett. (5)

Other (2)

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991).

J. Bae, J. Chun, S. B. Lee, “Equalization of the non-flat erbium gain spectrum using the multiport lattice filter model,” in Optical Fiber Communications Conference, Vol. 2 of 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 80–83.

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of a binary LPFG. The shadowed fiber core corresponds to region 1, while the dotted one is region 0. Also shown is the index perturbation along the longitudinal distance.

Fig. 2
Fig. 2

Schematic diagram and corresponding parameters of a unit period. Also shown are couplings of the core mode to the phase-matched cladding mode through the two heterointerfaces.

Fig. 3
Fig. 3

Equivalent coupling to that in Fig. 2 from the core mode to the cladding mode; here Kn is the equivalent coupling coefficient of the period.

Fig. 4
Fig. 4

Variation of the normalized local width wn/Λ¯ obtained from the GLM inverse-scattering method.

Fig. 5
Fig. 5

Same as Fig. 4 but for the normalized local period Λn/Λ¯.

Fig. 6
Fig. 6

Cross-transmission spectrum of the grating parameters in Figs. 4 and 5 calculated by the transfer-matrix model. The dotted lines represent the desired flatband spectrum.

Fig. 7
Fig. 7

Variation of the normalized local width wn/Λ¯ obtained from the GLM inverse-scattering method with genetic algorithm optimization. Also shown for comparison is the result obtained by using only the GLM method.

Fig. 8
Fig. 8

Same as Fig. 7 but for the normalized local period Λn/Λ¯.

Fig. 9
Fig. 9

Cross-transmission spectrum of the grating parameters in Figs. 7 and 8 optimized by genetic algorithm. The dotted lines represent the desired flatband spectrum.

Fig. 10
Fig. 10

Transmission spectrum of the synthesized flatband binary LPFG.

Fig. 11
Fig. 11

Transmission spectrum of the all-fiber MZ filter composed of two symmetrically cascaded flatband LPFGs.

Equations (52)

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Δn2(x, y, z)=n=1NΔng2(x, y)uz-znwn,
n=-f(n)=m=--f(n)exp(i2m πn)dn,
Δn2(x, y, z)=Δng2(x, y)m=--uz-z(n)w(n)×exp(i2m πn)dn.
n=n(z)=zΛ¯+ν(z),
1Λ(z)dndz=1Λ¯+dνdz.
μ(z)w(z)Λ(z).
Δn2(x, y, z)=m=-Δnˆm2(x, y; z)×expim0z 2πΛ(z) dz,
Δnˆm2(x, y; z)=Δng2(x, y)μ(z),m=0Δng2(x, y) sin[μ(z)mπ]mπ,m0.
E(r)=j=1,2Aj(z)ej(x, y)=j=1,2aj(z)exp(iβjz)ej(x, y),
12 Aco[ej×hk]·z dA=δjk(j, k=1, 2).
da1dz=iΔβ1(z)a1+iK(z)a2×exp-i0zβ1-β2-2πΛ(z)dz,
da2dz=iΔβ2(z)a2+iK*(z)a1×expi0zβ1-β2-2πΛ(z)dz,
Δβj(z)κjjμ(z)(j=1, 2),
K(z)κ12 sin[μ(z)π]π.
κjk=ωε04 AΔng2etj*·etk+n¯2n2 ezj*ezkdA,
aˆj(z)=aj(z)exp-i0zΔβj(z)dz(j=1, 2),
daˆ1dz=iK(z)aˆ2 exp-2i0zδ(z)dz,
daˆ2dz=iK*(z)aˆ1 exp2i0zδ(z)dz,
δ(z)=12 β1-β2-2πΛ(z)+Δβ1(z)-Δβ2(z)=ξ+12 Δβ1(z)-Δβ2(z)-2π dνdz.
ξ(λ)=12 β1(λ)-β2(λ)-2πΛ¯.
ν1(z, ξ)=aˆ1(z)exp(iξz),
ν2(z, ξ)=aˆ2(z)exp(-iξz).
dν1(z, ξ)/dz-iξν1(z, ξ)=q(z)ν2(z, ξ),
dν2(z, ξ)/dz+iξν2(z, ξ)=-q*(z)ν1(z, ξ),
q(z)=i{κ12 sin[μ(z)π]/π}exp[iθ(z)]
θ(z)=2πν(z)-(κ11-κ22)0zμ(z)dz.
F(1|0)=α1γ-γ*α2.
F(0|1)=α1-γγ*α2,
γ=κ12β1-β2.
αj=1-Dj/2(j=1, 2),
α1=α2=1-|γ|2α.
P(1)(z)=exp[i(β1+κ11)z]00exp[i(β2+κ22)z].
P(0)(z)=exp(iβ1z)00exp(iβ2z).
Λn=(1/2)[(zn+1-zn)+(zn-zn-1)].
ln=(1/2)[(zn+1-zn)-wn],
ln=(1/2)[(zn-zn-1)-wn].
Fn=P(0)(ln)F(0|1)P(1)(wn)F(1|0)P(0)(ln)=exp(iθn)AnΓn-Γn*An*,
An=α2 expi (Δn+Δ¯n)+σμn2+|γ|2 exp-i (Δn-Δ¯n)+σμn2,
Γn=exp[i(β1-β2)(ln-ln)]×(2iαγ)sinΔn+σμn2.
θn=βΛ¯n+(κ11+κ22)wn/2
Δn=(β1-β2)wn,Δ¯n=(β1-β2)(ln+ln)
σ=(κ11-κ22)Λ¯.
exp(iθn)Γn*=expiβ1ln+i(β1+κ11) wn2(iΛK¯n)×expi(β2+κ22) wn2+iβ2ln;
Kn=α κ12π+ξ˜ sin[μn(π+σ+ξ˜)],
r(ξ)=ν2(L, ξ)/ν1(L, ξ),
μ(z)=1π sin-1π |q(z)|κ12,
1Λ(z)=1Λ¯+12π dθdz+σμ(z)2πΛ¯.
n(z)=0z dzΛ(z),
wn=wn(GLM)+Δwn,Λn=Λn(GLM)+ΔΛn,
x={Δw1, ΔΛ1; Δw2, ΔΛ2 , Δwn, ΔΛN}.
F(x)=1M j=1MT(λj;x)-Ttarget(λj)dTj21/2,
FMZ=(FLPFG)TP(0)(LMZ)FLPFG,

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