Abstract

We study finite-size bidimensional photonic crystals doped by a microcavity. We characterize the defect modes by developing a simple model of an infinitely conducting cavity. This model allows the prediction of the number of defect modes and their evolution versus the conicity angle. Finally, we present a method for the computation of the quality factor of the microcavity by means of the poles of the scattering matrix. We demonstrate numerically that the quality factor grows exponentially with respect to the size of the crystal. These results obtained in the frequency domain are checked by solving Maxwell’s equations in the time domain.

© 1999 Optical Society of America

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References

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  1. 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]
  2. J. D. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).
  3. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1996).
    [CrossRef]
  4. P. R. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
    [CrossRef]
  5. D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  6. D. Felbacq, G. Tayeb, D. Maystre, “Localization of light by a set of parallel cylinders,” J. Mod. Opt. 42, 473–482 (1995).
    [CrossRef]
  7. D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–256 (1997).
    [CrossRef]
  8. E. Centeno, D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional crystals,” submitted to J. Opt. Soc. Am. A.
  9. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
  10. A. Figotin, A. Klein, “Localization of light in lossless inhomogeneous dielectrics,” J. Opt. Soc. Am. A 15, 1423–1435 (1998).
    [CrossRef]
  11. A. Figotin, A. Klein, “Localized classical waves created by defects,” J. Stat. Phys. 86, 165–177 (1997).
    [CrossRef]
  12. D. Felbacq, “Defect modes in one-dimensional periodic media,” (in preparation; available from the author at the address on the title page).
  13. M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1970).
  14. R. Feynman, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1965), Vol. 2.
  15. E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
    [CrossRef]

1999 (1)

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

1998 (1)

1997 (3)

A. Figotin, A. Klein, “Localized classical waves created by defects,” J. Stat. Phys. 86, 165–177 (1997).
[CrossRef]

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–256 (1997).
[CrossRef]

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]

1996 (2)

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1996).
[CrossRef]

P. R. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

1995 (1)

D. Felbacq, G. Tayeb, D. Maystre, “Localization of light by a set of parallel cylinders,” J. Mod. Opt. 42, 473–482 (1995).
[CrossRef]

1994 (1)

Bouchitté, G.

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–256 (1997).
[CrossRef]

Centeno, E.

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

E. Centeno, D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional crystals,” submitted to J. Opt. Soc. Am. A.

Chan, C. T.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1996).
[CrossRef]

Economou, E. N.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1996).
[CrossRef]

Fan, S.

P. R. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

Felbacq, D.

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

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–256 (1997).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Localization of light by a set of parallel cylinders,” J. Mod. Opt. 42, 473–482 (1995).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

E. Centeno, D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional crystals,” submitted to J. Opt. Soc. Am. A.

D. Felbacq, “Defect modes in one-dimensional periodic media,” (in preparation; available from the author at the address on the title page).

Feynman, R.

R. Feynman, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1965), Vol. 2.

Figotin, A.

A. Figotin, A. Klein, “Localization of light in lossless inhomogeneous dielectrics,” J. Opt. Soc. Am. A 15, 1423–1435 (1998).
[CrossRef]

A. Figotin, A. Klein, “Localized classical waves created by defects,” J. Stat. Phys. 86, 165–177 (1997).
[CrossRef]

Ho, K. M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1996).
[CrossRef]

Joannopoulos, J. D.

P. R. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

J. D. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Klein, A.

A. Figotin, A. Klein, “Localization of light in lossless inhomogeneous dielectrics,” J. Opt. Soc. Am. A 15, 1423–1435 (1998).
[CrossRef]

A. Figotin, A. Klein, “Localized classical waves created by defects,” J. Stat. Phys. 86, 165–177 (1997).
[CrossRef]

Maystre, D.

Meade, R.

J. D. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Sigalas, M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1996).
[CrossRef]

Soukoulis, C. M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1996).
[CrossRef]

Tayeb, G.

Villeneuve, P. R.

P. R. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

Winn, J.

J. D. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

J. Mod. Opt. (1)

D. Felbacq, G. Tayeb, D. Maystre, “Localization of light by a set of parallel cylinders,” J. Mod. Opt. 42, 473–482 (1995).
[CrossRef]

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

J. Stat. Phys. (1)

A. Figotin, A. Klein, “Localized classical waves created by defects,” J. Stat. Phys. 86, 165–177 (1997).
[CrossRef]

Opt. Commun. (1)

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

Phys. Rev. B (2)

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1996).
[CrossRef]

P. R. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

Waves Random Media (1)

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–256 (1997).
[CrossRef]

Other (6)

E. Centeno, D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional crystals,” submitted to J. Opt. Soc. Am. A.

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).

J. D. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

D. Felbacq, “Defect modes in one-dimensional periodic media,” (in preparation; available from the author at the address on the title page).

M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1970).

R. Feynman, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1965), Vol. 2.

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Figures (11)

Fig. 1
Fig. 1

Definition of incident angle φ0, polarization angle δ0, and conicity angle θ0 in the Cartesian system (O, x, y, z).

Fig. 2
Fig. 2

Hexagonal 9×9 photonic crystal doped by a microcavity of radius R=3.4. The central rod is removed. The optical index of the fibers is equal to 2.9, and the filling ratio is ρ=1.2/4=30%. The segment below the structure is used for computation of the transmission coefficient T.

Fig. 3
Fig. 3

Logarithm of the transmission T versus the wavelength λ in normal incidence (θ0=90°) in TM polarization (δ0=90°) for the crystal of Fig. 2 (solid curve) and the same crystal with the central rod (dotted–dashed curve). The monomode microcavity creates a defect mode for the resonant wavelength λr=9.056.

Fig. 4
Fig. 4

Map of the electric modulus mode inside the crystal defined in Fig. 2 for the pole wavelength λp=9.0566-0.0001i.

Fig. 5
Fig. 5

Hexagonal 11×13 photonic crystal doped by a multimode microcavity of radius Rtop6.32. The seven central rods are removed. The filling ratio is ρ=30%, and the optical index of the fibers is equal to 2.9.

Fig. 6
Fig. 6

Logarithm of the transmission T versus the wavelength λ in normal incidence (θ0=90°) in TM polarization (δ0=90°) for the crystal of Fig. 5 (solid curve) and the same crystal with the central rods (dotted–dashed curve). The multimode microcavity creates three defect modes for the resonant wavelengths λ1=10.207, λ2=8.592, and λ3=8.023.

Fig. 7
Fig. 7

Logarithm of the transmission T versus the wavelength λ and the conicity angle θ0 in TM polarization (δ0=90°) for the direct crystal of Fig. 2. The crosses represent the predicted resonant wavelengths.

Fig. 8
Fig. 8

Logarithm of the transmission T versus the wavelength λ in normal incidence (θ0=90°) in TE polarization (δ0=0°) for the inverted-contrast crystal doped by a monomode microcavity defined in Subsection 4.B.2 (solid curve) and the same crystal with the central rod (dotted–dashed curve). The resonant wavelength is λr=19.206.

Fig. 9
Fig. 9

Map of the magnetic modulus mode inside the inverted crystal for the resonant wavelength λr=19.206. The photonic crystal is represented by the disks. In TE polarization the defect mode is not very well localized in the microcavity.

Fig. 10
Fig. 10

Transmission coefficient T versus wavelength λ and conicity angle θ0 in TE polarization (δ0=0°) for the inverse crystal defined in Subsection 4.B.3. The crosses represent the predicted resonant wavelengths.

Fig. 11
Fig. 11

Decimal logarithm of the quality factor of the monomode microcavity defined in Fig. 2 versus the size of the crystal given by Eq. (25). The circles represent the value of the quality factor given by the computation in the time domain.

Tables (1)

Tables Icon

Table 1 Comparison of Resonant Wavelengths and Solutions Given by the Model of the Infinitely Conducting Cavity

Equations (32)

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k0=k0 sin θ0 cos φ0k0 sin θ0 sin φ0k0 cos θ0.
γ=k0 cos θ0,
χ02=k02-γ2.
F=EzHz,
Fd=F-Fi.
S-1Fd=Fi.
F(P)=Fi(P)+exp[-iγz(P)]×j=1Nm=-+Fj,mdHm(2)(χrj(P))×exp[imφj(P)].
S-1Fd=0.
det(S-1(λp))=0.
U=-+UnJn(χr)exp(imφ)exp(-iγz),
TMcase:Jn2πνλ Req sin θ0=0,
TEcase:Jn2πνλ Req sin θ0=0.
TMcase:λr=2πνReqYn,p sin θ0,
TEcase:λr=2πνReqYn,p sin θ0.
IY=2πνRtopλsup sin θ0, 2πνRtopλinf sin θ0.
TMcase:λr2πνRtopYn,p sin θ0,
TEcase:λr2πνRtopYn,p sin θ0.
SY={Y0,2=5.520;Y1,1=3.832;Y2,1=5.136}.
Sλ={λ1=10.207, λ2=8.592, λ3=8.023};
Req=λrYn,p2π.
λr(θ0)=λr(90°)sin θ0.
Req=λrYn,p2 π ν
Y0,2=3.832Req=3.38,
Y3,1=4.201Req=3.71.
-ω2-iω0Q ω+ω02=0,
λp=λ01-14Q21/2-i2Q=λ0 exp(iϕ),
|λp| =λ0Q.
det(S-1(λp))=0.
Q=12 1+1tan2 ϕ1/2,
λ0=|λp|.
Ui(r, t)=-+A(ω)H0(2)(χ(ω)r-r0)exp(iωt)dω.
A(ω)=τ2π exp-τ24 (ω-ω0)2.

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