Abstract

We present an accurate numerical method based on the Floquet–Bloch formalism to analyze the propagation properties and the radiation loss in infinitely long uniform fiber Bragg gratings. The model allows us to find all the propagation characteristics including the propagation constants, the space harmonics and the total field distribution, the guided and radiated power, and the modal loss induced by the periodic structure. The influence of the geometrical and physical parameters on the performance of the Bragg gratings has been established. A clear explanation of the physical phenomena related to the index modulation amplitude changes is presented, including the photonic bandgap effect, which is not easily described by the finite-difference time-domain method and cannot be described by the widely used coupled-mode theory.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
    [CrossRef]
  2. A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
    [CrossRef]
  3. J. Jung, N. Park, B. Le, “Simultaneous measurement of strain and temperature by use of a single fiber Bragg grating written in an erbium:ytterbium-doped fiber,” Appl. Opt. 39, 1118–1120 (2000).
    [CrossRef]
  4. Y. Yu, H. Tam, W. Chung, M. S. Demokan, “Fiber Bragg grating sensor for simultaneous measurement of displacement and temperature,” Opt. Lett. 25, 1141–1143 (2000).
    [CrossRef]
  5. V. M. Murukeshan, P. Y. Chan, Lin Seng Ong, A. Asundi, “Intracore fiber Bragg gratings for strain measurement in embedded composite structures,” Appl. Opt. 40, 145–149 (2001).
    [CrossRef]
  6. C. R. Giles, “Lightwave applications of fiber Bragg gratings,” J. Lightwave Technol. 15, 1391–1404 (1997).
    [CrossRef]
  7. N. L. Litchinitser, B. J. Eggleton, G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1998).
    [CrossRef]
  8. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
  9. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  10. M. Yamada, K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
    [CrossRef] [PubMed]
  11. L. A. Weller-Brophy, D. G. Hall, “Analysis of waveguide gratings: application of Rouard’s method,” J. Opt. Soc. Am. A 2, 863–871 (1985).
    [CrossRef]
  12. J. Skaar, “Synthesis of fiber Bragg gratings for use in transmission,” J. Opt. Soc. Am. A 18, 557–564 (2001).
    [CrossRef]
  13. E. Peral, J. Capmany, “Generalized Bloch wave analysis for fiber and waveguide gratings,” J. Lightwave Technol. 15, 1295–1302 (1997).
    [CrossRef]
  14. K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–812 (1980).
    [CrossRef]
  15. W. P. Huang, “Coupled-mode theory for optical waveguide: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [CrossRef]
  16. T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
    [CrossRef]
  17. V. M. N. Passaro, M. N. Armenise, “Analysis of radiation loss in grating-assisted codirectional couplers,” IEEE J. Quantum Electron. 31, 1691–1697 (1995).
    [CrossRef]
  18. V. M. N. Passaro, “Optimal design of grating-assisted directional couplers,” J. Lightwave Technol. 18, 973–984 (2000).
    [CrossRef]
  19. A. Giorgio, A. G. Perri, M. N. Armenise, “Very fast and accurate modeling of multilayer waveguiding photonic bandgap structures,” J. Lightwave Technol. 19, 1598–1613 (2001).
    [CrossRef]
  20. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–219.

2001 (3)

2000 (3)

1998 (1)

1997 (5)

C. R. Giles, “Lightwave applications of fiber Bragg gratings,” J. Lightwave Technol. 15, 1391–1404 (1997).
[CrossRef]

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

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

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

1996 (1)

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

1995 (1)

V. M. N. Passaro, M. N. Armenise, “Analysis of radiation loss in grating-assisted codirectional couplers,” IEEE J. Quantum Electron. 31, 1691–1697 (1995).
[CrossRef]

1994 (1)

1987 (1)

1985 (1)

1980 (1)

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Agrawal, G. P.

Armenise, M. N.

A. Giorgio, A. G. Perri, M. N. Armenise, “Very fast and accurate modeling of multilayer waveguiding photonic bandgap structures,” J. Lightwave Technol. 19, 1598–1613 (2001).
[CrossRef]

V. M. N. Passaro, M. N. Armenise, “Analysis of radiation loss in grating-assisted codirectional couplers,” IEEE J. Quantum Electron. 31, 1691–1697 (1995).
[CrossRef]

Askins, C. G.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Asundi, A.

Capmany, J.

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

Chan, P. Y.

Chang, K. C.

Chung, W.

Davis, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Demokan, M. S.

Eggleton, B. J.

Erdogan, T.

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

Friebele, E. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Giles, C. R.

C. R. Giles, “Lightwave applications of fiber Bragg gratings,” J. Lightwave Technol. 15, 1391–1404 (1997).
[CrossRef]

Giorgio, A.

Hall, D. G.

Hill, K. O.

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Huang, W. P.

Jung, J.

Kersey, A. D.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Koo, K. P.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Le, B.

LeBlanc, M.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Litchinitser, N. L.

Meltz, G.

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Murukeshan, V. M.

Ong, Lin Seng

Park, N.

Passaro, V. M. N.

V. M. N. Passaro, “Optimal design of grating-assisted directional couplers,” J. Lightwave Technol. 18, 973–984 (2000).
[CrossRef]

V. M. N. Passaro, M. N. Armenise, “Analysis of radiation loss in grating-assisted codirectional couplers,” IEEE J. Quantum Electron. 31, 1691–1697 (1995).
[CrossRef]

Patrick, H. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Peral, E.

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

Perri, A. G.

Putnam, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Sakuda, K.

Shah, V.

Skaar, J.

Tam, H.

Tamir, T.

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–812 (1980).
[CrossRef]

Weller-Brophy, L. A.

Yamada, M.

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–219.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–219.

Yu, Y.

Zhang, S.

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (2)

V. M. N. Passaro, M. N. Armenise, “Analysis of radiation loss in grating-assisted codirectional couplers,” IEEE J. Quantum Electron. 31, 1691–1697 (1995).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

J. Lightwave Technol. (9)

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

C. R. Giles, “Lightwave applications of fiber Bragg gratings,” J. Lightwave Technol. 15, 1391–1404 (1997).
[CrossRef]

N. L. Litchinitser, B. J. Eggleton, G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1998).
[CrossRef]

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

V. M. N. Passaro, “Optimal design of grating-assisted directional couplers,” J. Lightwave Technol. 18, 973–984 (2000).
[CrossRef]

A. Giorgio, A. G. Perri, M. N. Armenise, “Very fast and accurate modeling of multilayer waveguiding photonic bandgap structures,” J. Lightwave Technol. 19, 1598–1613 (2001).
[CrossRef]

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

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (1)

Other (1)

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–219.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Geometry of the infinitely long fiber grating.

Fig. 2
Fig. 2

Optical fiber sections.

Fig. 3
Fig. 3

Leakage factor α (μm-1) versus wavelength for a sinusoidal fiber grating with different index changes Δn.

Fig. 4
Fig. 4

Real part of the propagation constant kz0 versus wavelength for n=0 (solid curves) and n=-1 (dashed curves) space harmonics for a sinusoidal fiber grating with different index changes Δn. Dotted lines are relevant to the fundamental and first diffracted modes of the unperturbed fiber.

Fig. 5
Fig. 5

Real part of the propagation constant kz0 versus wavelength for n=0 (solid curves) and n=-1 (dashed curves) space harmonics with different index changes Δn, where S refers to sinusoidal profiles and R refers to squared profiles. Dotted lines are relevant to the fundamental and first diffracted modes of the unperturbed fiber.

Fig. 6
Fig. 6

Group to phase velocity ratio versus wavelength (μm) for rectangular profiles (dashed curves) and sinusoidal profiles (solid curves) for different index changes Δn.

Fig. 7
Fig. 7

Guided and scattered power percentages (%) and normalized leakage factor versus wavelength for a FBG with Δn=10-2, Λ=0.5364 µm, and a sinusoidal profile at different z values.

Fig. 8
Fig. 8

Leakage factor/Δn ratio (μm-1) at the center of the PBG region versus normalized grating radius for different wavelengths λ0 and index changes Δn=10-4 and 10-2 (dotted curves) in the case of sinusoidal profiles (solid curves) and rectangular profiles (dashed curves).

Fig. 9
Fig. 9

Leakage factor (μm-1) at the center of the PBG region versus index modulation amplitude for λ0 in the case of sinusoidal profiles (solid curves) and rectangular profiles (dotted curves).

Fig. 10
Fig. 10

Leakage factor (μm-1) versus normalized wavelength (λ/λ0) for a sinusoidal fiber grating with different index changes Δn for different central wavelengths λ0.

Equations (63)

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

n(z)=ncore+Δnf2πΛz=ncore+Δnf(Kz),
rtg,
nr(r)=next,r>bnclad,arbncore,tgra.
ϵi(r, z)=nϵni(r)exp(jnKz),
ϵni(r)=1Λ -Λ/2Λ/2ϵi(r, z)exp(-jnKz)dz,i=±1,
Ψξ=exp(jνϑ)nφξn(r)exp(jkznz),
Ψξ=Eξ,Hξ;ξ=z,ϑ,r
kzn=kz0+nK=k0neff+jα+n 2πΛ,n=0,±1,±2, ,
dv(r)dr=M(r)v(r),
v(r)=ez(r)hz(r)eϑ(r)hϑ(r),
M(r)=O-jνωϵ0r kzηOjωμ0kzηkzk02-Ijνωμ0r kzOjωϵ0ϵ-kz2k02 IOOjωμ0I-ν2k02r2 η-1r Ijνωϵ0r ηkz-jωϵ0ϵ-ν2k02r2 IO-jνωμ0r kz-1r I,
ϵ(n,m)=ϵn-m1,η(n,m)=ϵn-m-1.
kz(n,m)=kznδnm,n, m=0,±1,±2, ,
d2φn(r)dr2+1r dφn(r)dr-[-jkrn(core)]2+ν2r2φn(r)+k02mnϵn-mφm(r)=0,n,m=0,k±1,±2, ,
φn(r)=pn+Iν(-jkrn(core)r),
limr0 φm(r)=0,m=0,±1,±2, .
ϵn-m1,nm.
ezn(r)=Iν(-jkrn(core)r)Iν(-jkrn(core)γ)ezn(γ)rγνezn(γ),
hzn(r)=Iν(-jkrn(core)r)Iν(-jkrn(core)γ)hzn(γ)rγνhzn(γ),rγ,
vϑ(γ)=eϑ(γ)hϑ(γ)=νγ Q-1OOC-1-kz-jωμ0I-jωϵ0Ikzηez(γ)hz(γ)=νγ Q˜vz(γ),
Q˜=Q-1OOC-1-kz-jωμ0I-jωϵ0Ikzη
cnm=kznηn-mkzm-δnmk02,
qnm=ϵn-mk02-kzm2δnm.
ezn(r)=Afn(+) Iν(-jkrn(f)r)Iν(-jkrn(f)tbot)+Afn(-) Kν(-jkrn(f)r)Kν(-jkrn(f)tbot),
hzn(r)=Bfn(+) Iν(-jkrn(f)r)Iν(-jkrn(f)tbot)+Bfn(-) Kν(-jkrn(f)r)Kν(-jkrn(f)tbot).
v(r)=MOm(f)(r)Af(+)Af(-)Bf(+)Bf(-),
MOm(f)(r)=W(r)M˜(r),
W(r)=R(r)OOOOR(r)OOOOR(r)OOOOR(r),
M˜(r)=IS(r)OOOOIS(r)P(r)P(r)S(r)-ωμ0U(r)-ωμ0Y(r)ωϵ0nf2U(r)ωϵ0nf2Y(r)P(r)P(r)S(r),
rnm(r)=Iν(-jkrn(f)r)Iν(-jkrn(f)tbot)δnm,
snm(r)=Iν(-jkrn(f)tbot)Iν(-jkrn(f)r) Kν(-jkrn(f)r)Kν(-jkrn(f)tbot)δnm,
pnm(r)=-νr kzn[krn(f)]2δnm,
unm(r)=1krn(f) Iν(-jkrn(f)r)Iν(-jkrn(f)r)δnm,
ynm(r)=1krn(f) Iν(-jkrn(f)tbot)Iν(-jkrn(f)r) Kν(-jkrn(f)r)Kν(-jkrn(f)tbot)δnm,
v(ttop)=NOm(f)(tbot, ttop)v(tbot),
NOm(f)(tbot, ttop)=MOm(f)(ttop)[MOm(f)(tbot)]-1.
ezn(r)=Ainf,n(-)Kν(-jkrn(inf)r),
hzn(r)=Binf,n(-)Kν(-jkrn(inf)r),
eϑn(r)=-1[krn(inf)]2 ωμ0krn(inf)Binf,n(-)×Kν(-jkrn(inf)r)+kznνrezn(r),
hϑn(r)=+1[krn(inf)]2 ωϵ0ninf2krn(inf)Ainf,n(-)×Kν(-jkrn(inf)r)-kznνrhzn(r),
krn(inf)=±(k02ninf2-kzn2)1/2
eϑ(b)hϑ(b)=G-ωμ0Lωϵ0ninf2LGez(b)hz(b),
gnm=-kznνb[krn(inf)]2δnm,
lnm=1krn(inf) Kν(-jkrn(inf)b)Kν(-jkrn(inf)b)δnm.
k02ninf2=kzn2+[krn(inf)]2=(βn+jα)2+{Re[krn(inf)]+j Im[krn(inf)]}2=βn2+2jβnα-α2+Re(krn(inf))2-Im(krn(inf))2+2j Re(krn(inf))Im(krn(inf)),
Re(krn(inf))Im(krn(inf))=-βnα.
Kν(-jkrn(inf)r)π exp[jkrn(inf)r][-2πjkrn(inf)r]1/2.
λBragg(q)=2neffΛq=λBragg(1)q,
kzn=k0neff1+2nq+jα.
Re{[krn(inf)]2}=k02ninf2-k02neff21+2nq2>0,
1+2nq2<ninf2neff2.
kzn=k0neff+jα+2πn 2neffqλBragg(q)=k0neff1+2nχq+jα,
χ=λ0λBragg(q).
-qχ<n<0.
v(tg)=Ngr(γ, tg)v(γ),
v(a)=NOm(f)(tg, a)v(tg),
v(b)=NOm(f)(a, b)v(a).
v(b)=W(b)M˜(b)[MOm(f)(a)]-1NOm(f)(tg, a)Ngr(γ, tg)v(γ)=W(b)Ω(tg, a, b)v(γ),
vz(b)vϑ(b)=W(b)Ω11Ω12Ω21Ω22vz(γ)vϑ(γ).
Tvz(γ)=O,
T=G-ωμ0Lωϵ0ninf2LGγν Ω11+Ω12Q˜-γν Ω21+Ω22Q˜,
det(T)=0.
Δλλ=|ϵ1|ϵ0,

Metrics