Abstract

A novel device, formed by a widely spaced periodic array of defects in a photonic bandgap crystal, is studied with the goal of designing a waveguide with a prescribed narrow bandwidth. Tunneling of radiation between the defect sites allows wave propagation along the line of the defects. An analytical study based on the weakly coupled cavity model is performed, and the dispersion relation ω(β) of the new waveguide is derived. The frequency shift and the band structure of the periodic defect waveguide are linked by an analytic relationship to the distance between the defect sites and therefore can be tuned by varying the latter. Sections of such waveguides can be employed as ultra-narrow-band filters in optical routing devices. Numerical simulations demonstrate the performances of this new device and support the analytical predictions.

© 2001 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).
  2. X. P. Feng, Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Jpn. J. Appl. Phys. Part 2 36, L120–L123 (1997).
    [CrossRef]
  3. E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
    [CrossRef]
  4. E. Centeno, B. Guizal, D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” Pure Appl. Opt. 1, L10–L13 (1999).
    [CrossRef]
  5. G. Tayeb, D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A 14, 3323–3332 (1997).
    [CrossRef]
  6. A. Boag, M. Gafni, B. Z. Steinberg, “Bandwidth control for photonic bandgap waveguides,” presented at Bianisotropics 2000, the 8th International Conference on Electromagnetics of Complex Media, Lisbon, September 27–29, 2000.
  7. R. E. Peierls, Quantum Theory of Solids (Clarendon, Oxford, UK, 1955).
  8. A. Yariv, Y. Xu, R. K. Lee, A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
    [CrossRef]
  9. Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Antennas Propag. 35, 1119–1127 (1987).
    [CrossRef]
  10. A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
    [CrossRef]

1999 (3)

E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
[CrossRef]

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

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

1997 (2)

G. Tayeb, D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A 14, 3323–3332 (1997).
[CrossRef]

X. P. Feng, Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Jpn. J. Appl. Phys. Part 2 36, L120–L123 (1997).
[CrossRef]

1988 (1)

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

1987 (1)

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Antennas Propag. 35, 1119–1127 (1987).
[CrossRef]

Arakawa, Y.

X. P. Feng, Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Jpn. J. Appl. Phys. Part 2 36, L120–L123 (1997).
[CrossRef]

Boag, A.

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Antennas Propag. 35, 1119–1127 (1987).
[CrossRef]

A. Boag, M. Gafni, B. Z. Steinberg, “Bandwidth control for photonic bandgap waveguides,” presented at Bianisotropics 2000, the 8th International Conference on Electromagnetics of Complex Media, Lisbon, September 27–29, 2000.

Centeno, E.

E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
[CrossRef]

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

Felbacq, D.

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

E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
[CrossRef]

Feng, X. P.

X. P. Feng, Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Jpn. J. Appl. Phys. Part 2 36, L120–L123 (1997).
[CrossRef]

Gafni, M.

A. Boag, M. Gafni, B. Z. Steinberg, “Bandwidth control for photonic bandgap waveguides,” presented at Bianisotropics 2000, the 8th International Conference on Electromagnetics of Complex Media, Lisbon, September 27–29, 2000.

Guizal, B.

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

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).

Lee, R. K.

Leviatan, Y.

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Antennas Propag. 35, 1119–1127 (1987).
[CrossRef]

Maystre, D.

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).

Peierls, R. E.

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

Scherer, A.

Steinberg, B. Z.

A. Boag, M. Gafni, B. Z. Steinberg, “Bandwidth control for photonic bandgap waveguides,” presented at Bianisotropics 2000, the 8th International Conference on Electromagnetics of Complex Media, Lisbon, September 27–29, 2000.

Tayeb, G.

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.

IEEE Trans. Antennas Propag. (1)

Y. Leviatan, A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Antennas Propag. 35, 1119–1127 (1987).
[CrossRef]

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

Jpn. J. Appl. Phys. Part 2 (1)

X. P. Feng, Y. Arakawa, “Defect modes in two-dimensional triangular photonic crystals,” Jpn. J. Appl. Phys. Part 2 36, L120–L123 (1997).
[CrossRef]

Opt. Commun. (1)

E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
[CrossRef]

Opt. Lett. (1)

Pure Appl. Opt. (1)

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

Radio Sci. (1)

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Other (3)

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

A. Boag, M. Gafni, B. Z. Steinberg, “Bandwidth control for photonic bandgap waveguides,” presented at Bianisotropics 2000, the 8th International Conference on Electromagnetics of Complex Media, Lisbon, September 27–29, 2000.

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

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Figures (7)

Fig. 1
Fig. 1

Elements of the microcavity array waveguide. (a) Unperturbed background photonic crystal, with periodic relative permittivity p(r). (b) Basic microcavity, created by a local defect of arbitrary nature. The relative permittivity of the crystal with the basic microcavity is d(r). (c) Linear array formed by periodic repetitions of the basic microcavity by use of an intercavity vector b=2a1. (d) Same as (c) but with an intercavity vector b=2a1+a2.

Fig. 2
Fig. 2

Dispersion curve ω(β) of the microcavity array waveguide (solid curve). We assume here that b=m1a1, with a1 defined in Fig. 1(a) and m1=3. The unperturbed crystal dispersion that corresponds to propagation along the vector a1 (the wave numbers along ΓM in the reciprocal lattice domain) is also shown (dashed curves).

Fig. 3
Fig. 3

Unperturbed crystal transmission curve and transmission of a waveguide obtained by removing a complete row of posts (dashed curve). The dashed curve corresponds to an array waveguide with intercavity vector b=a1.

Fig. 4
Fig. 4

Basic microcavity mode electric field magnitude |E0(r)| (in decibels), corresponding to a local defect obtained by removing the shaded post in Fig. 1(b).

Fig. 5
Fig. 5

Electric field magnitude in the microcavity array waveguides (in decibels): (a) (m1, m2)=(2, 0); (b) (m1, m2)=(3, 0).

Fig. 6
Fig. 6

Waveguide transmission curves for m2=0 and various values of m1 and the microcavity resonance.

Fig. 7
Fig. 7

Computed transmission bandwidth for m2=0 and three values of m1.

Equations (27)

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d(r)1d(r)-1p(r).
b=i=13 miai,
1(r)=1p(r)+n=- d(r-nb),
ΘH(r)=ωc2H(r),
ΘH×1(r)×H.
ωc2=H, ΘHH, H,
F, G= F·G¯dr
(Θper+Θ0)H0(r)=ω0c2H0(r),
Im[H0(r)]=0.
Θ=Θper+n=- Θn,
H(r)=n=- AnHn(r),Hn(r)H0(r-nb),
ωc2=n=-m=- AnA¯mHn, ΘHmn=-m=- AnA¯mHn, Hm.
hn-mHn, Hm=h¯m-n,
tn-mHn, ΘHm=t¯m-n.
tm=ω0c2hm+n=-,n0tmn,
tmnHm, ΘnH0,Im[tmn]=0
k,Ak ωc2=0  m=-tk-m-ωc2hk-mA¯m=0,
k,A¯k ωc2=0  n=-tn-k-ωc2hn-kAn=0,
m=-tm-ωc2hmAm+k=0k.
Am=exp(iβm),
ωc2-ω0c2=mn0tmn exp(iβm)m hm exp(iβm),
ωc2-ω0c2H0-2m=-n0 tmn exp(iβm).
ΘnH0(r)=iω0×[d(r-nb)d(r)E0(r)],
tmn=-iω0 Vn d(r-nb)d(r)×[E¯0(r)×H0(r-mb)]·ds+ω02 Vn d(r-nb)d(r)d(r-mb)×[E¯0(r)·E0(r-mb)]d3r.
ωc2-ω0c22H0-2(t01+t11 cos β),
ω-ω0c2ω0H02(t01+t11 cos β).
Δωω0=2cω0H02t11.

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