Abstract

Photonic crystal microcavities, formed by local defects within an otherwise perfectly periodic structure, can be used as narrowband optical resonators and filters. The coupled-cavity waveguide (CCW) is a linear array of equally spaced identical microcavities. Tunneling of light between microcavities forms a guiding effect, with a central frequency and bandwidth controlled by the local defects’ parameters and spacing, respectively. We employ cavity perturbation theory to investigate the sensitivity of microcavities and CCWs to random structure inaccuracies. For the microcavity, we predict a frequency shift that is due to random changes in the lattice structure and show an approximate linear dependence between the standard deviation of the structure inaccuracy and that of the resonant frequency. The effect of structural inaccuracy on the CCW devices, however, is different; it has practically no effect on the CCW performance if it is below a certain threshold but may destroy the CCW if this threshold is exceeded.

© 2003 Optical Society of America

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References

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  1. E. Centeno, B. Guizal, D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L10–L13 (1999).
    [CrossRef]
  2. N. Stefanou, A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
    [CrossRef]
  3. M. Bayindir, B. Temelkuran, E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. B 61, 11855–11858 (2000).
    [CrossRef]
  4. A. Yariv, Y. Xu, R. K. Lee, A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
    [CrossRef]
  5. A. Boag, M. Gafni, B. Z. Steinberg, “Bandwidth control for photonic bandgap waveguides,” in Proceedings of Bianisotropics 2000, the Eighth International Conference on Electromagnetics of Complex Media (Instituto Superior Tecnico, Lisbon, 2000), pp. 321–324.
  6. A. Boag, B. Z. Steinberg, “Narrow-band microcavity waveguides in photonic crystals,” J. Opt. Soc. Am. A 18, 2799–2805 (2001).
    [CrossRef]
  7. R. E. Peierls, Quantum Theory of Solids (Oxford Clarendon Press, Oxford, UK, 1955.
  8. R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw Hill, New York, 1961).
  9. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N. J.1995).
  10. P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, Orlando, Fla., 1985).

2001 (1)

2000 (1)

M. Bayindir, B. Temelkuran, E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. B 61, 11855–11858 (2000).
[CrossRef]

1999 (2)

A. Yariv, Y. Xu, R. K. Lee, A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
[CrossRef]

E. Centeno, B. Guizal, D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L10–L13 (1999).
[CrossRef]

1998 (1)

N. Stefanou, A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

Bayindir, M.

M. Bayindir, B. Temelkuran, E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. B 61, 11855–11858 (2000).
[CrossRef]

Boag, A.

A. Boag, B. Z. Steinberg, “Narrow-band microcavity waveguides in photonic crystals,” J. Opt. Soc. Am. A 18, 2799–2805 (2001).
[CrossRef]

A. Boag, M. Gafni, B. Z. Steinberg, “Bandwidth control for photonic bandgap waveguides,” in Proceedings of Bianisotropics 2000, the Eighth International Conference on Electromagnetics of Complex Media (Instituto Superior Tecnico, Lisbon, 2000), pp. 321–324.

Centeno, E.

E. Centeno, B. Guizal, D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L10–L13 (1999).
[CrossRef]

Felbacq, D.

E. Centeno, B. Guizal, D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L10–L13 (1999).
[CrossRef]

Gafni, M.

A. Boag, M. Gafni, B. Z. Steinberg, “Bandwidth control for photonic bandgap waveguides,” in Proceedings of Bianisotropics 2000, the Eighth International Conference on Electromagnetics of Complex Media (Instituto Superior Tecnico, Lisbon, 2000), pp. 321–324.

Guizal, B.

E. Centeno, B. Guizal, D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L10–L13 (1999).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw Hill, New York, 1961).

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N. J.1995).

Lancaster, P.

P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, Orlando, Fla., 1985).

Lee, R. K.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N. J.1995).

Modinos, A.

N. Stefanou, A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

Ozbay, E.

M. Bayindir, B. Temelkuran, E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. B 61, 11855–11858 (2000).
[CrossRef]

Peierls, R. E.

R. E. Peierls, Quantum Theory of Solids (Oxford Clarendon Press, Oxford, UK, 1955.

Scherer, A.

Stefanou, N.

N. Stefanou, A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

Steinberg, B. Z.

A. Boag, B. Z. Steinberg, “Narrow-band microcavity waveguides in photonic crystals,” J. Opt. Soc. Am. A 18, 2799–2805 (2001).
[CrossRef]

A. Boag, M. Gafni, B. Z. Steinberg, “Bandwidth control for photonic bandgap waveguides,” in Proceedings of Bianisotropics 2000, the Eighth International Conference on Electromagnetics of Complex Media (Instituto Superior Tecnico, Lisbon, 2000), pp. 321–324.

Temelkuran, B.

M. Bayindir, B. Temelkuran, E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. B 61, 11855–11858 (2000).
[CrossRef]

Tismenetsky, M.

P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, Orlando, Fla., 1985).

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N. J.1995).

Xu, Y.

Yariv, A.

J. Opt. A Pure Appl. Opt. (1)

E. Centeno, B. Guizal, D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L10–L13 (1999).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Phys. Rev. B (2)

N. Stefanou, A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

M. Bayindir, B. Temelkuran, E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. B 61, 11855–11858 (2000).
[CrossRef]

Other (5)

A. Boag, M. Gafni, B. Z. Steinberg, “Bandwidth control for photonic bandgap waveguides,” in Proceedings of Bianisotropics 2000, the Eighth International Conference on Electromagnetics of Complex Media (Instituto Superior Tecnico, Lisbon, 2000), pp. 321–324.

R. E. Peierls, Quantum Theory of Solids (Oxford Clarendon Press, Oxford, UK, 1955.

R. F. Harrington, Time Harmonic Electromagnetic Fields (McGraw Hill, New York, 1961).

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N. J.1995).

P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, Orlando, Fla., 1985).

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Figures (5)

Fig. 1
Fig. 1

CCW (or array waveguide) and its components. (a) Schematic description of the CCW in an otherwise perfect 2D hexagonal crystal. Solid circles, microcavities created by local defects. The intercavity distance is b. (b) Example of a local defect microcavity, created by removing a single post. (c) Isolated microcavity modal field E0.

Fig. 2
Fig. 2

Nearest neighbors radii variation.

Fig. 3
Fig. 3

Resonance wavelength versus radii. The reference point is the point for which δϵr=0 and for which the mode E0 is computed.

Fig. 4
Fig. 4

Resonance wavelength standard deviation versus maximal magnitude of radii perturbations.

Fig. 5
Fig. 5

Transmission of the perfect CCW (dashed–dotted curve) and the randomly perturbed CCW (solid curve) for post radii random inaccuracy of (a) 2%, (b) 5%, (c) 10%.

Equations (59)

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ω-ω0ωs+Δω cos β.
ΘH(r)=λH(r),λ=ωc2,
ΘH×1ϵr(r) ×H.
ΘHn(r)=λnHn(r),
Θ*Hm*(r)=λm*Hm*(r),
HnHn |Hn|2d3r-1/2,
EnEn |Hn|2d3r-1/2,
λnλm*mn.
Hn, Hm Hn·Hm¯d3r=δmn,
Enϵr/η02, Em (ϵr/η02)En·Em¯d3r=δmn,
ωnωm*ϵ*Em*, ϵ0En-ωnωm*ϵEn, ϵ0Em*=Hn, Hm*(λn-λm*).
λm*-λnϵ02ωnωm* Hn, Hm*=En, δϵrEm*=δϵrEn, Em*,
m,mn|Hn, Hm*|2=η0-4m,mnωnωm*ωm*2-ωn22|(δϵr/ϵr*)En, ϵr*Em*|2αn2m|(δϵr/ϵr*)En, (ϵr*/η02)Em*|2,
m,mn|Hn, Hm*|2αn2η0-2(δϵr/ϵr*)En2αn2η0-2δϵr/ϵr*2 maxr|En|2,
Hm*=Hm+δHm,δHmO(δϵr/ϵr*)0asδϵr/ϵr*0.
ωn*2-ωn2ωn*ωnEn, δϵrEnη02Hn2.
δωnωn2 En, δϵrEnη02Hn2,
Θi=1NaiHn(i)(r)=λni=1NaiHn(i)(r),
Hn(i), Hn(j)=δij,i, j=1  N.
i=1NaiδϵrEm*, En(i)=λn-λm*ϵ02ωnωm* i=1NaiHn(i), Hm*.
Em*=j=1NajEn(j).
λn-λm*ϵ02wnωm*=ΣiΣjaia¯jδϵrEn(j), En(i)ΣiΣjaia¯jHn(i), Hn(j).
[A][a]=λn-λm*ϵ02ωnωm* [a],
Ai,k=δϵrEn(k), En(i).
δω0ω0(η0H0)-2πa(ϵr-1)Σi=16δai|E0|i2¯,
δω0ω0(η0H0)-2πa(ϵr-1)|E0|2¯i=1Nδai.
(δω0)2[ω0(η0H0)-2πa(ϵr-1)|E0|2¯]2×N(δa)2.
dn(r)=1ϵdn(r)-1ϵb(r),
1ϵr(r)=1ϵb(r)+ndn(r).
Θ×1ϵr(r) ×,Θb×1ϵb(r) ×,Θn×dn(r)×.
(Θb+Θn)Hn(r)=ωnc2Hn(r),
ΘH(r)=Θb+nΘnH(r)=ωc2H(r).
H(r)=mAmHm(r),
ωc2=H, ΘHH, H.
ωc2=ΣmΣnAnAm¯TnmΣmΣnAnAm¯Inm,
TnmHn, ΘHm=ωmc2Inm+Tnm,
InmHn, Hm,
TnmHn, jmΘjHm.
nωkc2-ωc2Ink+TnkAn=0,k.
Hn(r)H0(r-nb).
InkI0n-k=Iˆn-k,IˆmH0(r),H0(r-mb),
TnkT0n-k=τn-k,τmH0(r-mb),j0ΘjH0(r).
ωk=ω0+δωk,ωk2ω02+2ω0δωk.
n(Ω2Iˆn-k+2c-2ω0δωkIˆn-k+τn-k)An=0,k,
Ω2c-2(ω02-ω2).
[τ_+Δ_]A=-Ω2H02A,
Δk=2c-2H02ω0δωk,
τ_mn=τm-n.
τ_+Δ_=0τ1τ0+Δk-1τ100τ1τ0+Δkτ100τ1τ0+Δk+1τ100τ1τ0+Δk+2τ10.
Ωn2|δωk=0=H0-2{τ0-2τ1 cos[nπ/(N+1)]},
n=1  N.
H0-2[τ0-2τ1, τ0+2τ1],
Δωω0=2cω0H02τ1.
Ωn2=H0-2{τ0-2τ1 cos[nπ/(N+1)]+Δn},n=1  N.
δωn=O(Δω|perfectwaveguide)forsomen.
fn=mamn(δ)gmδ,amn(δ)=fn, gmδ.
fn-ann(δ)gnδ2=m,mn|fn, gmδ|2.
fn-ann(δ)gnδ0asδ0.
fn-ann(δ)gnδO(δ)asδ0,

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