Abstract

A model of both uniform finite-length optical fiber Bragg gratings and grating arrays is presented. The model is based on the Floquet–Bloch formalism and allows rigorous investigation of all the physical aspects in either single- or multiple-periodic structures realized on the core of a monomodal fiber. Analytical expressions of reflectivity and transmittivity for both single gratings and grating arrays are derived. The influence of the grating length and the index modulation amplitude on the reflected and transmitted optical power for both sinusoidal and rectangular profiles is evaluated. Good agreement between our method and the well-known coupled-mode theory (CMT) approach has been observed for both single gratings and grating arrays only in the case of weak index perturbation. Significant discrepancies exist there in cases of strong index contrast because of the increasing approximation of the CMT approach. The effects of intragrating phase shift are also shown and discussed.

© 2002 Optical Society of America

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References

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  1. 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]
  2. 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]
  3. X. Shu, B. A. L. Gwandu, Y. Liu, L. Zhang, I. Bennion, “Sampled fiber Bragg grating for simultaneous refractive-index and temperature measurement,” Opt. Lett. 26, 774–776 (2001).
    [CrossRef]
  4. C. R. Giles, “Lightwave applications of fiber Bragg gratings,” J. Lightwave Technol. 15, 1391–1404 (1997).
    [CrossRef]
  5. Z. Wei, H. M. H. Shalaby, H. Ghafouri-Shiraz, “Modified quadratic congruence codes for fiber Bragg-grating-based spectral-amplitude-coding optical CDMA systems,” J. Lightwave Technol. 19, 1274–1281 (2001).
    [CrossRef]
  6. L. R. Chen, “Flexible fiber Bragg grating encoder/decoder for hybrid wavelength-time optical CDMA,” IEEE Photon. Technol. Lett. 13, 1233–1235 (2001).
    [CrossRef]
  7. A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
    [CrossRef]
  8. P. Petruzzi, C. Lowry, P. Sivanesan, “Dispersion compensation using only fiber Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 5, 1339–1344 (1999).
    [CrossRef]
  9. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [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. M. N. Armenise, V. M. N. Passaro, M. Armenise, R. Diana, “Recent advances in guided-wave devices for optical signal processing,” in Optical Information Processing, Y. V. Gulyaev, ed., Proc. SPIE3900, 42–53 (1999).
    [CrossRef]
  15. V. M. N. Passaro, R. Diana, M. N. Armenise, “Optical fiber Bragg gratings. Part I. Modeling of infinitely long gratings,” J. Opt. Soc. Am. A 19, 1844–1854 (2002).
    [CrossRef]
  16. M. McCall, “On the application of coupled mode theory for modeling fiber Bragg gratings,” J. Lightwave Technol. 18, 236–242 (2000).
    [CrossRef]

2002

2001

2000

1999

A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
[CrossRef]

P. Petruzzi, C. Lowry, P. Sivanesan, “Dispersion compensation using only fiber Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 5, 1339–1344 (1999).
[CrossRef]

1997

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]

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

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

1987

1985

1973

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

Armenise, M.

M. N. Armenise, V. M. N. Passaro, M. Armenise, R. Diana, “Recent advances in guided-wave devices for optical signal processing,” in Optical Information Processing, Y. V. Gulyaev, ed., Proc. SPIE3900, 42–53 (1999).
[CrossRef]

Armenise, M. N.

V. M. N. Passaro, R. Diana, M. N. Armenise, “Optical fiber Bragg gratings. Part I. Modeling of infinitely long gratings,” J. Opt. Soc. Am. A 19, 1844–1854 (2002).
[CrossRef]

M. N. Armenise, V. M. N. Passaro, M. Armenise, R. Diana, “Recent advances in guided-wave devices for optical signal processing,” in Optical Information Processing, Y. V. Gulyaev, ed., Proc. SPIE3900, 42–53 (1999).
[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]

Azaña, J.

A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
[CrossRef]

Bennion, I.

Capmany, J.

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

Carballar, A.

A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
[CrossRef]

Chen, L. R.

L. R. Chen, “Flexible fiber Bragg grating encoder/decoder for hybrid wavelength-time optical CDMA,” IEEE Photon. Technol. Lett. 13, 1233–1235 (2001).
[CrossRef]

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.

Diana, R.

V. M. N. Passaro, R. Diana, M. N. Armenise, “Optical fiber Bragg gratings. Part I. Modeling of infinitely long gratings,” J. Opt. Soc. Am. A 19, 1844–1854 (2002).
[CrossRef]

M. N. Armenise, V. M. N. Passaro, M. Armenise, R. Diana, “Recent advances in guided-wave devices for optical signal processing,” in Optical Information Processing, Y. V. Gulyaev, ed., Proc. SPIE3900, 42–53 (1999).
[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]

Ghafouri-Shiraz, H.

Giles, C. R.

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

Gwandu, B. A. L.

Hall, D. G.

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]

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]

Liu, Y.

Lowry, C.

P. Petruzzi, C. Lowry, P. Sivanesan, “Dispersion compensation using only fiber Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 5, 1339–1344 (1999).
[CrossRef]

McCall, M.

Muriel, M. A.

A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
[CrossRef]

Passaro, V. M. N.

V. M. N. Passaro, R. Diana, M. N. Armenise, “Optical fiber Bragg gratings. Part I. Modeling of infinitely long gratings,” J. Opt. Soc. Am. A 19, 1844–1854 (2002).
[CrossRef]

M. N. Armenise, V. M. N. Passaro, M. Armenise, R. Diana, “Recent advances in guided-wave devices for optical signal processing,” in Optical Information Processing, Y. V. Gulyaev, ed., Proc. SPIE3900, 42–53 (1999).
[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]

Petruzzi, P.

P. Petruzzi, C. Lowry, P. Sivanesan, “Dispersion compensation using only fiber Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 5, 1339–1344 (1999).
[CrossRef]

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.

Shalaby, H. M. H.

Shu, X.

Sivanesan, P.

P. Petruzzi, C. Lowry, P. Sivanesan, “Dispersion compensation using only fiber Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 5, 1339–1344 (1999).
[CrossRef]

Skaar, J.

Tam, H.

Wei, Z.

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]

Yu, Y.

Zhang, L.

Appl. Opt.

IEEE J. Quantum Electron.

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

IEEE J. Sel. Top. Quantum Electron.

P. Petruzzi, C. Lowry, P. Sivanesan, “Dispersion compensation using only fiber Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 5, 1339–1344 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

L. R. Chen, “Flexible fiber Bragg grating encoder/decoder for hybrid wavelength-time optical CDMA,” IEEE Photon. Technol. Lett. 13, 1233–1235 (2001).
[CrossRef]

A. Carballar, M. A. Muriel, J. Azaña, “Fiber grating filter for WDM systems: an improved design,” IEEE Photon. Technol. Lett. 11, 694–696 (1999).
[CrossRef]

J. Lightwave Technol.

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]

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

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

M. McCall, “On the application of coupled mode theory for modeling fiber Bragg gratings,” J. Lightwave Technol. 18, 236–242 (2000).
[CrossRef]

Z. Wei, H. M. H. Shalaby, H. Ghafouri-Shiraz, “Modified quadratic congruence codes for fiber Bragg-grating-based spectral-amplitude-coding optical CDMA systems,” J. Lightwave Technol. 19, 1274–1281 (2001).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Other

M. N. Armenise, V. M. N. Passaro, M. Armenise, R. Diana, “Recent advances in guided-wave devices for optical signal processing,” in Optical Information Processing, Y. V. Gulyaev, ed., Proc. SPIE3900, 42–53 (1999).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Scheme of a finite-length grating.

Fig. 2
Fig. 2

Two-grating array.

Fig. 3
Fig. 3

CMT and LMP power reflectivity versus wavelength (μm) for sinusoidal (S) and rectangular (R) profiles and different index changes, Δn=10-4, 2×10-4, and 10-3.

Fig. 4
Fig. 4

Power reflectivity versus grating length (μm) for both the LMP and CMT approaches at different index changes, i.e., Δn=10-4, 2×10-4, and 10-3, at λ0=1.55 μm.

Fig. 5
Fig. 5

Reflection bandwidth (nm) versus grating length (μm) for sinusoidal (solid curves) and rectangular (dashed curves) profiles with different index changes, i.e., Δn=10-4, 2×10-4, and 10-3.

Fig. 6
Fig. 6

ca and cb parameters versus index change for sinusoidal (S) and rectangular (R) profiles.

Fig. 7
Fig. 7

Power reflectivity, transmittivity, and normalized power loss evaluated by LMP and CMT approaches versus wavelength (μm) for Δn=10-2, Λ=0.5364 μm, L=400Λ=214.6 μm, and a sinusoidal profile.

Fig. 8
Fig. 8

LMP and CMT power reflectivity and transmittivity versus grating length for different values of wavelength.

Fig. 9
Fig. 9

Spectra of LMP and CMT power reflectivity versus wavelength (μm) of an array formed by two identical gratings with period Λ=0.5364 μm, length L=107.3 μm (200 periods), and index change Δn=10-2 for two values of distances, i.e., s=0.2682 μm (Δϕ=π) and s=0.5364 μm (Δϕ=π/2).

Fig. 10
Fig. 10

Power reflectivity (LMP, CMT), transmittivity (LMP, CMT), and normalized modal power loss (LMP) versus wavelength (μm) for an array of two strong gratings centered at λ1=1.55 μm and λ2=1.6 μm, with Δn1,2=10-2, periods Λ1=0.5364 μm and Λ2=0.5417 μm, and lengths L1=400Λ1=214.6 μm and L2=216.7 μm.

Equations (65)

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n(z)=ncore+Δnf(Kz),rtg,
ν r(rEϑ)-ωμ0r2 Hϑz=j[ν2-ϵ(r, z)k02r2]Er,
ν r(rHϑ)+ωϵ0ϵ(r, z)r2 Eϑz=j[ν2-ϵ(r, z)k02r2]Hr,
Ψi(I)(r, z, ϑ)=Ψi(r)exp(jβhz)exp(jνϑ),i=1,, 6.
ν r(rEϑ)-jωμ0r2βhHϑ=j(ν2-nr2k02r2)Er,
Ψi(R)(r, z, ϑ)=ρiΨi(r)exp(-jβhz)exp(jνϑ),i=1, , 6,
ρ1ν r(rEϑ)+jωμ0r2βhρ2Hϑ=jρ3(ν2-nr2k02r2)Er;
ωμ0r2βh(ρ1+ρ2)Hϑ=(ρ3-ρ1)(ν2-nr2k02r2)Er,
ρ1=-ρ2=ρ3=ρ.
ρ4=-ρ.
ϵ(r, z)=nϵn(r)exp(jnKz),n=0, ±1, ±2, ,
ϵn(r)=1Λ -Λ/2Λ/2ϵ(r, z)exp(-jnKz)dz
Ψi,l(r, z, ϑ)=exp(jνϑ)nψin(r)exp(±jkznz),i=1, , 6,l=1, 2,
kzn=β+jα+n 2πΛ=kz0+nK, n=0, ±1, ±2, ,
ΨiT(r, z, ϑ)=ci,1Ψi,1(r, z, ϑ)+ci,2Ψi,2(r, z, ϑ),i=1, , 6,
ν r(reϑn)jωμ0kznr2hϑn-jm(ν2δmn-k02r2ϵn-m)erm=0,
nc1,1ν r(reϑn)-jωμ0kznr2c2,1hϑn-jc3,1m(ν2δmn-k02r2ϵn-m)ermexp(jkznz)+nc1,2ν r(reϑn)+jωμ0kznr2c2,2hϑn-jc3,2m(ν2δmn-k02r2ϵn-m)ermexp(-jkznz)=0.
c1,1=c2,1=c3,1=c1,c1,2=-c2,2=c3,2=c2
c4,1=c5,1=c6,1=c1,c4,2=c5,2=-c6,2=-c2.
Hϑ(I)=Hϑ(r)exp(jβhz)=Hϑ(r)exp(-jδhz)exp(jKz),
Hϑ(R)=-ρHϑ(r)exp(jδhz)exp(-jKz),
Hϑ(T)=τHϑ(r)exp(-jδhz)exp(jKz),
Hϑ(r)=c1 exp(-αz)nhϑn(r)exp[j(β+nK)z]-c2 exp(αz)nhϑn(r)exp[-j(β+nK)z]=c1 exp(-αz)n+hϑn(r)exp(jBnz)-c2 exp(αz)n-hϑn(r)exp(jBnz)exp(jKz)+c1 exp(-αz)n-hϑn(r)exp(-jBnz)-c2 exp(αz)n+hϑn(r)exp(-jBnz)exp(-jKz),
Hϑ(r)=c1n+hϑn(r)-c2n-hϑn(r),
-ρHϑ(r)=c1n-hϑn(r)-c2n+hϑn(r),
τHϑ(r)exp(-jδhL)=c1 exp(-αL)n+hϑn(r)exp(jBnL)-c2 exp(αL)n-hϑn(r)exp(jBnL),
c1 exp(-αL)n-hϑn(r)exp(-jBnL)
-c2 exp(αL)n+hϑn(r)exp(-jBnL)=0.
0+(ErHϑ*-EϑHr*)rdr=c1*n+in-c2*n-in,
-ρ*0+(ErHϑ*-EϑHr*)rdr=c1*n-in-c2*n+in,
τ* exp(jδhL)0+(ErHϑ*-EϑHr*)rdr
=c1* exp(-αL)n+in exp(-jBnL)-c2* exp(αL)n-in exp(-jBnL),
c1* exp(-αL)n-in exp(jBnL)-c2* exp(αL)
×n+in exp(jBnL)=0,
in=0+(Erhϑn*-Eϑhrn*)rdr
ξ(±)(z)=n-in exp(±jBnz)n+in exp(±jBnz),
ρ=ξ(+)(L)exp(-2αL)-ξ(+)(0)1-ξ(+)(0)ξ(+)(L)exp(-2αL)*,
τ=exp(-jδhL) n+in exp(-jBnL)n+in×1-ξ(+)(L)ξ(-)(L)exp(αL)-ξ(+)(L)ξ(+)(0)exp(-αL)*,
c1=1Γ 11-ξ(+)(L)ξ(+)(0)exp(-2αL)*,
c2=1Γ ξ(+)(L)exp(-2αL)1-ξ(+)(L)ξ(+)(0)exp(-2αL)*,
Γ=n+0+(Erhϑn*-Eϑhrn*)rdr0+(ErHϑ*-EϑHr*)rdr
g1=exp(jβhs1),
g2=τ1 exp[jβh(s1+s2)],
gm=τ1τ2τm-1 exp[jβh(s1++sm)]
=τm-1gm-1 exp(jβhsm),m=2, , N,
gm*0+(ErHϑ*-EϑHr*)rdr
=c1(m)*nm+in(m)-c2(m)*nm-in(m),
-ρm*gm*0+(ErHϑ*-EϑHr*)rdr
=c1(m)*nm-in(m)-c2(m)*nm+in(m),
τm*gm*0+(ErHϑ*-EϑHr*)rdr
=c1(m)* exp(-αmLm)nm+in(m) exp[-jBn(m)Lm]
-c2(m)* exp(αmLm)nm-in(m) exp[-jBn(m)Lm],
-ρm+1*gm+1* exp(-jβhsm+1)0+(ErHϑ*-EϑHr*)rdr
=c1(m)* exp(-αmLm)nm-in(m) exp[jBn(m)Lm]
-c2(m)* exp(αmLm)nm+in(m) exp[jBn(m)Lm],
ρm=ζm(Lm)exp(-2αmLm)-ξm(+)(0)1-ξm(+)(0)ζm(Lm)exp(-2αmLm)*,
ζm(Lm)=ζm(c)(Lm)ξm(+)(Lm)+ρm+1* exp(-j2βhsm+1)ρm+1*ξm(-)(Lm)exp(-j2βhsm+1)+ξm(c)(Lm),
ξm(±)(z)=nm-in(m) exp(±jBn(m)z)nm+in(m) exp[±jBn(m)z],
ξm(c)(z)=nm+in(m) exp[jBn(m)z]nm+in(m) exp[-jBn(m)z].
τm=nm+in(m) exp[-jBn(m)Lm]nm+in(m)×1-ζm(Lm)ξm(-)(Lm)exp(αmLm)-exp(-αmLm)ζm(Lm)ξm(+)(0)*,
c1(m)=1Γm gm*1-ζm(Lm)ξm(+)(0)exp(-2αmLm)*,
c2(m)=1Γm gm*ζm(Lm)exp(-2αmLm)1-ζm(Lm)ξm(+)(0)exp(-2αmLm)*,
Γm=nm+0+[Erhϑn(m)*-Eϑhrn(m)*]rdr0+[ErHϑ*-EϑHr*]rdr.
FWHM=ca(Δn)1+cb(Δn)L21/2,

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