Abstract

We consider an effective-medium description of Bragg gratings. Though this method can be used to obtain exact results that agree with coupled-mode theory for counterpropagating modes, we show that it is particularly useful for obtaining simple approximate results, for example for uniform gratings and for gratings with point defects.

© 1997 Optical Society of America

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References

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  1. C. M. Ragdale, D. Reid, I. Bennion, “Fibre grating devices,” in Fiber Laser Sources and Amplifiers, M. J. Digonnet, ed., Proc. SPIE1171, 148–156 (1990).
    [CrossRef]
  2. R. J. Campbell, R. Kashyap, “The properties and applications of photosensitive germanosilicate fiber,” Int. J. Opt. 9, 33–57 (1994).
  3. K. Hill, “A periodic distributed-parameter waveguide for integrated optics,” Appl. Opt. 13, 1853–1856 (1974).
    [CrossRef] [PubMed]
  4. F. Ouellette, “Dispersion cancellation using linearly chirped Bragg grating filters in optical waveguides,” Opt. Lett. 12, 847–849 (1987).
    [CrossRef] [PubMed]
  5. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
    [CrossRef]
  6. J. E. Sipe, L. Poladian, C. Martijn de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
    [CrossRef]
  7. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
    [CrossRef]
  8. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), Sec. 8.
  9. W. H. Loh, R. I. Laming, “1.55 µm phase-shifted distributed feedback fiber laser,” Electron. Lett. 31, 1440–1441 (1995).
    [CrossRef]
  10. J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
    [CrossRef]
  11. K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “λ/4-shifted InGaAsP/InP DFN lasers by simultaneous holographic exposure of positive and negative photoresists,” Electron. Lett. 20, 1008–1010 (1984).
    [CrossRef]
  12. See, e.g., C. M. de Sterke, “Simulations of gap soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
    [CrossRef] [PubMed]
  13. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 1, p. 38.
  14. H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Hilger, Birmingham, 1984), Chap. 3, p. 67.
  15. Ref. 13, Chap. 7, p. 323.
  16. See, e.g., P. Sheng, Scattering and Localization of Classical Waves in Random Media (World Scientific, Singapore, 1990).
  17. See, e.g., J. M. Ziman, Models of Disorder (Cambridge U. Press, Cambridge, 1979), Chap. 9, p. 332.
  18. P. St. J. Russell, “Bragg resonance of light in optical superlattices,” Phys. Rev. Lett. 56, 596–599 (1986).
    [CrossRef] [PubMed]
  19. V. Jayaraman, D. A. Cohen, L. A. Coldren, “Demonstration of broadband tunability in a semiconductor laser using sampled gratings,” Appl. Phys. Lett. 60, 2321–2323 (1992).
    [CrossRef]
  20. C. M. de Sterke, N. G. R. Broderick, “Coupled mode equations for superstructure Bragg gratings,” Opt. Lett. 20, 2039–2041 (1995).
    [CrossRef] [PubMed]
  21. J. Canning, M. G. Sceats, “π-phase-shifted periodic distributed structures in optical fibers,” Electron. Lett. 30, 1344–1345 (1994).
    [CrossRef]
  22. See, e.g., W. A. Harrison, Solid State Theory (Dover, New York, 1979), p. 176.

1995

W. H. Loh, R. I. Laming, “1.55 µm phase-shifted distributed feedback fiber laser,” Electron. Lett. 31, 1440–1441 (1995).
[CrossRef]

C. M. de Sterke, N. G. R. Broderick, “Coupled mode equations for superstructure Bragg gratings,” Opt. Lett. 20, 2039–2041 (1995).
[CrossRef] [PubMed]

1994

J. Canning, M. G. Sceats, “π-phase-shifted periodic distributed structures in optical fibers,” Electron. Lett. 30, 1344–1345 (1994).
[CrossRef]

J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
[CrossRef]

J. E. Sipe, L. Poladian, C. Martijn de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

R. J. Campbell, R. Kashyap, “The properties and applications of photosensitive germanosilicate fiber,” Int. J. Opt. 9, 33–57 (1994).

1993

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[CrossRef]

1992

See, e.g., C. M. de Sterke, “Simulations of gap soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
[CrossRef] [PubMed]

V. Jayaraman, D. A. Cohen, L. A. Coldren, “Demonstration of broadband tunability in a semiconductor laser using sampled gratings,” Appl. Phys. Lett. 60, 2321–2323 (1992).
[CrossRef]

1987

1986

P. St. J. Russell, “Bragg resonance of light in optical superlattices,” Phys. Rev. Lett. 56, 596–599 (1986).
[CrossRef] [PubMed]

1984

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “λ/4-shifted InGaAsP/InP DFN lasers by simultaneous holographic exposure of positive and negative photoresists,” Electron. Lett. 20, 1008–1010 (1984).
[CrossRef]

1976

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
[CrossRef]

1974

Akiba, S.

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “λ/4-shifted InGaAsP/InP DFN lasers by simultaneous holographic exposure of positive and negative photoresists,” Electron. Lett. 20, 1008–1010 (1984).
[CrossRef]

Bennion, I.

C. M. Ragdale, D. Reid, I. Bennion, “Fibre grating devices,” in Fiber Laser Sources and Amplifiers, M. J. Digonnet, ed., Proc. SPIE1171, 148–156 (1990).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 1, p. 38.

Broderick, N. G. R.

Campbell, R. J.

R. J. Campbell, R. Kashyap, “The properties and applications of photosensitive germanosilicate fiber,” Int. J. Opt. 9, 33–57 (1994).

Canning, J.

J. Canning, M. G. Sceats, “π-phase-shifted periodic distributed structures in optical fibers,” Electron. Lett. 30, 1344–1345 (1994).
[CrossRef]

Carroll, J. E.

J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
[CrossRef]

Cohen, D. A.

V. Jayaraman, D. A. Cohen, L. A. Coldren, “Demonstration of broadband tunability in a semiconductor laser using sampled gratings,” Appl. Phys. Lett. 60, 2321–2323 (1992).
[CrossRef]

Coldren, L. A.

V. Jayaraman, D. A. Cohen, L. A. Coldren, “Demonstration of broadband tunability in a semiconductor laser using sampled gratings,” Appl. Phys. Lett. 60, 2321–2323 (1992).
[CrossRef]

de Sterke, C. M.

Garret, B.

J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
[CrossRef]

Harrison, W. A.

See, e.g., W. A. Harrison, Solid State Theory (Dover, New York, 1979), p. 176.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), Sec. 8.

Hill, K.

Jayaraman, V.

V. Jayaraman, D. A. Cohen, L. A. Coldren, “Demonstration of broadband tunability in a semiconductor laser using sampled gratings,” Appl. Phys. Lett. 60, 2321–2323 (1992).
[CrossRef]

Kashyap, R.

R. J. Campbell, R. Kashyap, “The properties and applications of photosensitive germanosilicate fiber,” Int. J. Opt. 9, 33–57 (1994).

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
[CrossRef]

Laming, R. I.

W. H. Loh, R. I. Laming, “1.55 µm phase-shifted distributed feedback fiber laser,” Electron. Lett. 31, 1440–1441 (1995).
[CrossRef]

Loh, W. H.

W. H. Loh, R. I. Laming, “1.55 µm phase-shifted distributed feedback fiber laser,” Electron. Lett. 31, 1440–1441 (1995).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Hilger, Birmingham, 1984), Chap. 3, p. 67.

Martijn de Sterke, C.

Matsushima, Y.

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “λ/4-shifted InGaAsP/InP DFN lasers by simultaneous holographic exposure of positive and negative photoresists,” Electron. Lett. 20, 1008–1010 (1984).
[CrossRef]

Ouellette, F.

Poladian, L.

J. E. Sipe, L. Poladian, C. Martijn de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[CrossRef]

Ragdale, C. M.

C. M. Ragdale, D. Reid, I. Bennion, “Fibre grating devices,” in Fiber Laser Sources and Amplifiers, M. J. Digonnet, ed., Proc. SPIE1171, 148–156 (1990).
[CrossRef]

Reid, D.

C. M. Ragdale, D. Reid, I. Bennion, “Fibre grating devices,” in Fiber Laser Sources and Amplifiers, M. J. Digonnet, ed., Proc. SPIE1171, 148–156 (1990).
[CrossRef]

Russell, P. St. J.

P. St. J. Russell, “Bragg resonance of light in optical superlattices,” Phys. Rev. Lett. 56, 596–599 (1986).
[CrossRef] [PubMed]

Sakai, K.

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “λ/4-shifted InGaAsP/InP DFN lasers by simultaneous holographic exposure of positive and negative photoresists,” Electron. Lett. 20, 1008–1010 (1984).
[CrossRef]

Sceats, M. G.

J. Canning, M. G. Sceats, “π-phase-shifted periodic distributed structures in optical fibers,” Electron. Lett. 30, 1344–1345 (1994).
[CrossRef]

Sheng, P.

See, e.g., P. Sheng, Scattering and Localization of Classical Waves in Random Media (World Scientific, Singapore, 1990).

Sipe, J. E.

Thompson, G. H. B.

J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
[CrossRef]

Tsang, C. F.

J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
[CrossRef]

Utaka, K.

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “λ/4-shifted InGaAsP/InP DFN lasers by simultaneous holographic exposure of positive and negative photoresists,” Electron. Lett. 20, 1008–1010 (1984).
[CrossRef]

White, I. H.

J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
[CrossRef]

Whiteaway, J. E. A.

J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
[CrossRef]

Williams, K. A.

J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 1, p. 38.

Wright, A. P.

J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
[CrossRef]

Zhang, L. M.

J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
[CrossRef]

Ziman, J. M.

See, e.g., J. M. Ziman, Models of Disorder (Cambridge U. Press, Cambridge, 1979), Chap. 9, p. 332.

Appl. Opt.

Appl. Phys. Lett.

V. Jayaraman, D. A. Cohen, L. A. Coldren, “Demonstration of broadband tunability in a semiconductor laser using sampled gratings,” Appl. Phys. Lett. 60, 2321–2323 (1992).
[CrossRef]

Bell Syst. Tech. J.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
[CrossRef]

Electron. Lett.

W. H. Loh, R. I. Laming, “1.55 µm phase-shifted distributed feedback fiber laser,” Electron. Lett. 31, 1440–1441 (1995).
[CrossRef]

K. Utaka, S. Akiba, K. Sakai, Y. Matsushima, “λ/4-shifted InGaAsP/InP DFN lasers by simultaneous holographic exposure of positive and negative photoresists,” Electron. Lett. 20, 1008–1010 (1984).
[CrossRef]

J. Canning, M. G. Sceats, “π-phase-shifted periodic distributed structures in optical fibers,” Electron. Lett. 30, 1344–1345 (1994).
[CrossRef]

Int. J. Opt.

R. J. Campbell, R. Kashyap, “The properties and applications of photosensitive germanosilicate fiber,” Int. J. Opt. 9, 33–57 (1994).

J. Opt. Soc. Am. A

Opt. Lett.

Opt. Quantum Electron.

J. E. A. Whiteaway, A. P. Wright, B. Garret, G. H. B. Thompson, J. E. Carroll, L. M. Zhang, C. F. Tsang, I. H. White, K. A. Williams, “Detailed large-signal dynamic modelling of DFB laser structures and comparison with experiment,” Opt. Quantum Electron. 26, 817–842 (1994).
[CrossRef]

Phys. Rev. A

See, e.g., C. M. de Sterke, “Simulations of gap soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
[CrossRef] [PubMed]

Phys. Rev. E

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[CrossRef]

Phys. Rev. Lett.

P. St. J. Russell, “Bragg resonance of light in optical superlattices,” Phys. Rev. Lett. 56, 596–599 (1986).
[CrossRef] [PubMed]

Other

See, e.g., W. A. Harrison, Solid State Theory (Dover, New York, 1979), p. 176.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), Sec. 8.

C. M. Ragdale, D. Reid, I. Bennion, “Fibre grating devices,” in Fiber Laser Sources and Amplifiers, M. J. Digonnet, ed., Proc. SPIE1171, 148–156 (1990).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 1, p. 38.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Hilger, Birmingham, 1984), Chap. 3, p. 67.

Ref. 13, Chap. 7, p. 323.

See, e.g., P. Sheng, Scattering and Localization of Classical Waves in Random Media (World Scientific, Singapore, 1990).

See, e.g., J. M. Ziman, Models of Disorder (Cambridge U. Press, Cambridge, 1979), Chap. 9, p. 332.

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Figures (3)

Fig. 1
Fig. 1

Values of μeff [long-dashed line, Eqs. (8)] eff [short-dashed line, Eqs. (8)] and (n eff)2 [dotted curve, the square of Eq. (9)] of the effective medium as a function of δ/κ (left-hand scale) for a grating with κL = 3. Also shown is the reflection spectrum (solid curve) of the grating (right-hand scale).

Fig. 2
Fig. 2

Exact (solid curve) and approximate (dashed curve) results for the reflectivity of a uniform Bragg grating with κL = 8. The approximation is valid only outside the photonic band gap (|δ| > κ) and is given by relation (21).

Fig. 3
Fig. 3

Exact reflectivity (solid curve) and average reflectivity according to relation (27) (dashed curve) for a grating with κL = 8 and a single defect with Δϕ = π/2, located in the center of the structure.

Equations (35)

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

n(z)=n¯+Δn cos(2πz/d+ϕ),
E(x, y, z, t)=xˆE(z)exp(-iωt)f(x, y)+c.c.,
=xˆ[E+(z)exp(+ikBz)+E-(z)exp(-ikBz)]×exp(-iωt)f(x, y)+c.c.,
+i dE+(z)dz+δ(z)E+(z)+κ(z)E-(z)exp(+iϕ)=0,
-i dE-(z)dz+δ(z)E-(z)+κ(z)E+(z)exp(-iϕ)=0,
δ=k-kB=k-π/d,
κ=πΔn/λB,
Eeff(z)=E+(z)exp(-i ϕ2)+E-(z)exp(i ϕ2),
Heff(z)=1Z0[E+(z)exp(-i ϕ2)-E-(z)exp(i ϕ2)],
dEeff/dz=iωeffμ0μeffHeff,dHeff/dz=iωeff0effEeff,
μeff=(δ-κ)/keff,eff=(δ+κ)/keff
neff=μeffeff=δ2-κ2/κ,
Zeffμeff/eff=(δ-κ)/(δ+κ).
Eeff=A+ exp(+ikeffneffz)+A- exp(-ikeffneffz),
Heff=1Z0Zeff[A+ exp(+ikeffneffz)-A- exp(-ikeffneffz)],
E=A+21+1Zeffexpi ϕ2expi πdz+1-1Zeffexp-i ϕ2exp-i πdzexpikeffneffz+A-21-1Zeffexpi ϕ2expi πdz+1+1Zeffexp-i ϕ2exp-i πdzexp-ikeffneffz.
H=n¯Z0A+21+1Zeffexpi ϕ2expi πdz-1-1Zeffexp-i ϕ2exp-i πdzexpikeffneffz+A-21-1Zeffexpi ϕ2expi πdz-1+1Zeffexp-i ϕ2exp-i πdzexp-ikeffneffz.
A+(1)A-(1)=exp(-ik1effn1effz0)00exp(+ik1effn1effz0)×I0×exp(+ik2effn2effz0)00exp(-ik2effn2effz0)×A+(2)A-(2),
1/teffreff/teff-reff/teffteff-reffreff/teff
reff=(Z2eff-Z1eff)cos(Δϕ/2)-i(1-Z1effZ2eff)sin(Δϕ/2)(Z2eff+Z1eff)cos(Δϕ/2)-i(1+Z1effZ2eff)sin(Δϕ/2),
reff=(Z1eff-Z2eff)cos(Δϕ/2)-i(1-Z1effZ2eff)sin(Δϕ/2)(Z2eff+Z1eff)cos(Δϕ/2)-i(1+Z1effZ2eff)sin(Δϕ/2),
teff=2Z2eff(Z2eff+Z1eff)cos(Δϕ/2)-i(1+Z1effZ2eff)sin(Δϕ/2),
teff=2Z1eff(Z2eff+Z1eff)cos(Δϕ/2)-i(1+Z1effZ2eff)sin(Δϕ/2),
Δϕ(z0)=ϕ1-ϕ2+2πz0(1/d1-1/d2).
reff=1/Z1eff-1/Z2eff1/Z1eff+1/Z2eff,reff=1/Z2eff-1/Z1eff1/Z1eff+1/Z2eff,
teff=1/Z1eff1/Z1eff+1/Z2eff,teff=1/Z2eff1/Z1eff+1/Z2eff,
r=Z-1Z+1=δ-κ-δ+κδ-κ+δ+κ-κ2δ,
Rκ2δ2sin2(keffneffL)=κ2δ2sin2(δ2-κ2L).
Ravg12κδ2.
Eeff(z)exp(±κ2-δd2z),
2Zeff cos(Δϕ/2)-i[1+(Zeff)2]sin(Δϕ/2)=0,
δd(Δϕ)=±κ cos(Δϕ/2),
exp(-κ2-δd2z0), exp[-κ2-δd2(L-z0)]1
R=κ2 sin2(Δϕ/2)δ2-κ2 cos2(Δϕ/2)κ2δ2sin2(Δϕ/2),
Ravgκ2δ212+sin2(Δϕ/2).

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