Abstract

We use a rigorous method for diffraction by a finite set of parallel cylinders to study the influence of defects in a photonic crystal. The method allows us to give an accurate description of all the characteristics of the electromagnetic field (near-field map, scattered field, and energy flow). The localized resonant modes can also be computed. We show some of their symmetry properties and the influence of coupling between two neighboring defects. Finally, an example is given, which shows that a slight local change in the crystal period can be used for the realization of devices that radiate energy in a very narrow angular range.

© 1997 Optical Society of America

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  1. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
    [CrossRef]
  2. J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).
  3. J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
    [CrossRef] [PubMed]
  4. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
    [CrossRef]
  5. P. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
    [CrossRef]
  6. E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
    [CrossRef]
  7. D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
    [CrossRef]
  8. T. Baba, T. Matsuzaki, “GaInAsP/InP 2-dimensional photonic crystals,” in Microcavities and Photonic Bandgaps: Physics and Applications, Vol. 324 of NATO Advanced Scientific Institutes SeriesE. J. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  9. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  10. D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  11. D. Felbacq, D. Maystre, G. Tayeb, “Localization of light by a set of parallel cylinders,” J. Mod. Opt. 42, 473–482 (1995).
    [CrossRef]
  12. M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  13. N. R. Hill, V. Celli, “Limits of convergence of the Rayleigh method for surface scattering,” Phys. Rev. B 17, 2478–2481 (1978).
    [CrossRef]
  14. P. M. Van den Berg, J. T. Fokkema, “The Rayleigh hypothesis in the theory of reflection by a grating,” J. Opt. Soc. Am. 69, 27–31 (1979).
    [CrossRef]
  15. D. Maystre, M. Cadilhac, “Singularities of the continuation of the fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
    [CrossRef]
  16. D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
    [CrossRef]

1996 (1)

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

1994 (4)

E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
[CrossRef]

D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
[CrossRef]

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[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]

1993 (2)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

1992 (1)

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

1985 (1)

D. Maystre, M. Cadilhac, “Singularities of the continuation of the fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

1981 (1)

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
[CrossRef]

1979 (1)

1978 (1)

N. R. Hill, V. Celli, “Limits of convergence of the Rayleigh method for surface scattering,” Phys. Rev. B 17, 2478–2481 (1978).
[CrossRef]

Abramovitz, M.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Baba, T.

T. Baba, T. Matsuzaki, “GaInAsP/InP 2-dimensional photonic crystals,” in Microcavities and Photonic Bandgaps: Physics and Applications, Vol. 324 of NATO Advanced Scientific Institutes SeriesE. J. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Cadilhac, M.

D. Maystre, M. Cadilhac, “Singularities of the continuation of the fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

Celli, V.

N. R. Hill, V. Celli, “Limits of convergence of the Rayleigh method for surface scattering,” Phys. Rev. B 17, 2478–2481 (1978).
[CrossRef]

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 the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[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 the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Fan, S.

P. 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.

D. Felbacq, D. Maystre, G. Tayeb, “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]

Fokkema, J. T.

Hill, N. R.

N. R. Hill, V. Celli, “Limits of convergence of the Rayleigh method for surface scattering,” Phys. Rev. B 17, 2478–2481 (1978).
[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 the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Joannopoulos, J.

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

Joannopoulos, J. D.

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

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Kleppner, D.

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
[CrossRef]

MacKinnon, A.

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Matsuzaki, T.

T. Baba, T. Matsuzaki, “GaInAsP/InP 2-dimensional photonic crystals,” in Microcavities and Photonic Bandgaps: Physics and Applications, Vol. 324 of NATO Advanced Scientific Institutes SeriesE. J. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Maystre, D.

D. Felbacq, D. Maystre, G. Tayeb, “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]

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

D. Maystre, M. Cadilhac, “Singularities of the continuation of the fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

McCall, S. L.

D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
[CrossRef]

Meade, R.

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

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Pendry, J. B.

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Platzmann, P. M.

D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
[CrossRef]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
[CrossRef]

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 the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Smith, D. R.

D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
[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 the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Stegun, I.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Tayeb, G.

D. Felbacq, D. Maystre, G. Tayeb, “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]

Van den Berg, P. M.

Villeneuve, P.

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

Yablonovitch, E.

E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
[CrossRef]

J. Math. Phys. (1)

D. Maystre, M. Cadilhac, “Singularities of the continuation of the fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26, 2201–2204 (1985).
[CrossRef]

J. Mod. Opt. (3)

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

E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173–194 (1994).
[CrossRef]

D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Phys. Rev. B (4)

N. R. Hill, V. Celli, “Limits of convergence of the Rayleigh method for surface scattering,” Phys. Rev. B 17, 2478–2481 (1978).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

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

Phys. Rev. Lett. (2)

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
[CrossRef]

Pure Appl. Opt. (1)

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

Other (3)

T. Baba, T. Matsuzaki, “GaInAsP/InP 2-dimensional photonic crystals,” in Microcavities and Photonic Bandgaps: Physics and Applications, Vol. 324 of NATO Advanced Scientific Institutes SeriesE. J. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

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

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

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Figures (25)

Fig. 1
Fig. 1

(a) Finite crystal (11 layers of six or seven cylinders) with two defects, (b) associated periodic crystal dealt by our integral grating code (11 grids that are infinite in the horizontal direction).

Fig. 2
Fig. 2

Scattering by a set of parallel cylinders of arbitrary shape, index, and position.

Fig. 3
Fig. 3

The circles containing each cylinder must have no intersection.

Fig. 4
Fig. 4

Crystal with one defect. The segment below the structure is the one used for the computation of the transmission.

Fig. 5
Fig. 5

(a) Decimal logarithm of the transmission T versus the wavelength for the crystal of Fig. 4 (solid curve) and for the same crystal, but with no defect (dashed curve). (b) Dashed curve: same meaning as that of the dashed curve in (a) (finite crystal of Fig. 4 with no defect); solid curve: curve obtained for a crystal with infinite extension in the horizontal direction. The wavelength sampling pitch is equal to 0.05, with some extra points near the central peak of (a).

Fig. 6
Fig. 6

Modulus of the field associated with the resonant mode for λp=9.0572±i0.00092. The crystal is the one shown in Fig. 4.

Fig. 7
Fig. 7

Poynting vector map for the crystal of Fig. 4, illuminated by a plane wave with λ=9.0572 in normal incidence.

Fig. 8
Fig. 8

Crystal with two distant defects. The segment below the structure is the one used for the computation of the transmission.

Fig. 9
Fig. 9

Crystal with two close defects. The segment below the structure is the one used for the computation of the transmission.

Fig. 10
Fig. 10

Decimal logarithm of the transmission T versus the wavelength for the crystals of Figs. 4 (solid curve), 8 (dashed curve), and 9 (dotted curve). Note that the wavelength sampling pitch is equal to 0.05, with some extra points near the peaks.

Fig. 11
Fig. 11

Crystal with two defects. The segment below the structure is the one used for the computation of the transmission.

Fig. 12
Fig. 12

Same crystal as that of Fig. 11, but rotated 90°. The segment below the structure is the one used for the computation of the transmission.

Fig. 13
Fig. 13

Decimal logarithm of the transmission T versus the wavelength for the crystals of Figs. 11 (solid curve) and 12 (dashed curve).

Fig. 14
Fig. 14

Real part of the complex field for the mode associated with λp1=8.8335+i0.0162 (the crystal is placed as in Fig. 11).

Fig. 15
Fig. 15

Real part of the complex field for the mode associated with λp2=9.3210+i0.0159 (the crystal is placed as in Fig. 11).

Fig. 16
Fig. 16

Wire source located in a cavity close to the limits of the crystal.

Fig. 17
Fig. 17

Poynting vector map for the crystal of Fig. 16. The wavelength of the wire source is λ=Re(λp)=9.0575.

Fig. 18
Fig. 18

Radiation pattern at infinity for the crystal of Fig. 16 (polar plot of the intensity at infinity versus the emitting angle). The wavelength of the wire source is λ=Re(λp)=9.0575.

Fig. 19
Fig. 19

Superposition of a hexagonal photonic crystal (0<y<20.8) and of a vertically expanded crystal (20.8<y<48.1). The cavity is obtained by removing four cylinders.

Fig. 20
Fig. 20

Total electric-field modulus for the crystal of Fig. 19. The wavelength of the wire source is λ=7.934.

Fig. 21
Fig. 21

Poynting vector map for the crystal of Fig. 19. The wavelength of the wire source is λ=7.934.

Fig. 22
Fig. 22

Radiation pattern at infinity for the crystal of Fig. 19 (polar plot of the intensity at infinity versus the emitting angle). The wavelength of the wire source is λ=7.934.

Fig. 23
Fig. 23

Decimal logarithm of the transmission T versus the wavelength for the regularly spaced crystal (dashed curve) and for the vertically expanded crystal (solid curve) for normal incidence. The vertical line corresponds to λ=7.934.

Fig. 24
Fig. 24

Decimal logarithm of the transmission T versus the wavelength for the vertically expanded crystal for different incidences. The vertical line corresponds to λ=7.934.

Fig. 25
Fig. 25

Decimal logarithm of the transmission T versus the wavelength for the crystal of Fig. 19 for normal incidence. The marked peak corresponds to the wavelength λ=7.934.

Equations (13)

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

Ez(P)=m=-+[aj,mJm(krj(P))+bj,mHm(1)(krj(P))]exp[imθj(P)].
aj=Qj+kjTj,kbk,
bj=Sjaj.
I-S1T1,2-S1T1,N-S2T2,1I-S2T2,N-SNTN,1-SNTN,2Ib1b2bN
=S1Q1S2Q2SNQN,
S-1B=A
B=SA.
Ez(P)=Ezinc(P)+j=1Nm=-+bj,mHm(1)(krj(P))exp[imθj(P)].
Ezinc(x, y)=exp[ik(x sin θinc-y cos θinc)].
Ezinc(r)=H0(1)(kr-r0).
S-1B=0.
det[S-1(λp)]=0
det[S(λp)]=.

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