Abstract

We consider defect modes created in complete gaps of 2D photonic crystals by perturbing the dielectric constant in some region. We study their evolution from a band edge with increasing perturbation using an asymptotic method that approximates the Green function by its dominant component which is associated with the bulk mode at the band edge. From this, we derive a simple exponential law which links the frequency difference between the defect mode and the band edge to the relative change in the electric energy. We present numerical results which demonstrate the accuracy of the exponential law, for TE and TM polarizations, hexagonal and square arrays, and in each of the first and second band gaps.

© 2007 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light (Princeton University Press, Princeton, 1995).
  2. M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92, 063903 (2004).
    [CrossRef] [PubMed]
  3. A. Figotin and I. Vitebskiy, "Slow light in photonic crystals," Waves Random Complex Media 16, 293-382 (2006).
    [CrossRef]
  4. E. N. Economou, Green’s functions in quantum physics, 2nd ed. (Springer-Verlag, Berlin, 1983).
  5. R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, "Density of states functions for photonic crystals," Phys. Rev. E 69, 016609 (2004).
    [CrossRef]
  6. S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E 71, 056606 (2005).
    [CrossRef]
  7. L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave and Optical Technol. 1, 133-145 (2006).
  8. D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, "Two-dimensional treatment of the level shift and decay rate in photonic crystals," Phys. Rev. E 72, 046605 (2005).
    [CrossRef]
  9. K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comput. Phys. 219, 120-143 (2006).
    [CrossRef]
  10. W. Kohn and J. Luttinger, "Theory of donor states in silicon," Phys. Rev. 98, 915 - 922 (1955).
    [CrossRef]

2006

A. Figotin and I. Vitebskiy, "Slow light in photonic crystals," Waves Random Complex Media 16, 293-382 (2006).
[CrossRef]

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave and Optical Technol. 1, 133-145 (2006).

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comput. Phys. 219, 120-143 (2006).
[CrossRef]

2005

D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, "Two-dimensional treatment of the level shift and decay rate in photonic crystals," Phys. Rev. E 72, 046605 (2005).
[CrossRef]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E 71, 056606 (2005).
[CrossRef]

2004

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, "Density of states functions for photonic crystals," Phys. Rev. E 69, 016609 (2004).
[CrossRef]

1955

W. Kohn and J. Luttinger, "Theory of donor states in silicon," Phys. Rev. 98, 915 - 922 (1955).
[CrossRef]

Asatryan, A. A.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave and Optical Technol. 1, 133-145 (2006).

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, "Density of states functions for photonic crystals," Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Botten, L. C.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave and Optical Technol. 1, 133-145 (2006).

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comput. Phys. 219, 120-143 (2006).
[CrossRef]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E 71, 056606 (2005).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, "Density of states functions for photonic crystals," Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Byrne, M. A.

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comput. Phys. 219, 120-143 (2006).
[CrossRef]

de Sterke, C. M.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave and Optical Technol. 1, 133-145 (2006).

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E 71, 056606 (2005).
[CrossRef]

D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, "Two-dimensional treatment of the level shift and decay rate in photonic crystals," Phys. Rev. E 72, 046605 (2005).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, "Density of states functions for photonic crystals," Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Dossou, K.

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comput. Phys. 219, 120-143 (2006).
[CrossRef]

Dossou, K. B.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave and Optical Technol. 1, 133-145 (2006).

Figotin, A.

A. Figotin and I. Vitebskiy, "Slow light in photonic crystals," Waves Random Complex Media 16, 293-382 (2006).
[CrossRef]

Fink, Y.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

Fussell, D. P.

D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, "Two-dimensional treatment of the level shift and decay rate in photonic crystals," Phys. Rev. E 72, 046605 (2005).
[CrossRef]

Ibanescu, M.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

Joannopoulos, J. D.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

Johnson, S. G.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

Kohn, W.

W. Kohn and J. Luttinger, "Theory of donor states in silicon," Phys. Rev. 98, 915 - 922 (1955).
[CrossRef]

Luo, C.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

Luttinger, J.

W. Kohn and J. Luttinger, "Theory of donor states in silicon," Phys. Rev. 98, 915 - 922 (1955).
[CrossRef]

McOrist, J.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, "Density of states functions for photonic crystals," Phys. Rev. E 69, 016609 (2004).
[CrossRef]

McPhedran, R. C.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave and Optical Technol. 1, 133-145 (2006).

D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, "Two-dimensional treatment of the level shift and decay rate in photonic crystals," Phys. Rev. E 72, 046605 (2005).
[CrossRef]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E 71, 056606 (2005).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, "Density of states functions for photonic crystals," Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Nicorovici, N. A.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave and Optical Technol. 1, 133-145 (2006).

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, "Density of states functions for photonic crystals," Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Poulton, C. G.

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E 71, 056606 (2005).
[CrossRef]

Roundy, D.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

Vitebskiy, I.

A. Figotin and I. Vitebskiy, "Slow light in photonic crystals," Waves Random Complex Media 16, 293-382 (2006).
[CrossRef]

Wilcox, S.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave and Optical Technol. 1, 133-145 (2006).

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E 71, 056606 (2005).
[CrossRef]

J. Comput. Phys.

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comput. Phys. 219, 120-143 (2006).
[CrossRef]

J. Microwave and Optical Technol.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave and Optical Technol. 1, 133-145 (2006).

Phys. Rev.

W. Kohn and J. Luttinger, "Theory of donor states in silicon," Phys. Rev. 98, 915 - 922 (1955).
[CrossRef]

Phys. Rev. E

D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, "Two-dimensional treatment of the level shift and decay rate in photonic crystals," Phys. Rev. E 72, 046605 (2005).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, "Density of states functions for photonic crystals," Phys. Rev. E 69, 016609 (2004).
[CrossRef]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E 71, 056606 (2005).
[CrossRef]

Phys. Rev. Lett.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

Waves Random Complex Media

A. Figotin and I. Vitebskiy, "Slow light in photonic crystals," Waves Random Complex Media 16, 293-382 (2006).
[CrossRef]

Other

E. N. Economou, Green’s functions in quantum physics, 2nd ed. (Springer-Verlag, Berlin, 1983).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light (Princeton University Press, Princeton, 1995).

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Figures (5)

Fig. 1.
Fig. 1.

Band surface for TE polarization for a hole-type hexagonal lattice with holes of normalized radius a/d = 0.2 and refractive index nc = 1, in a background of refractive index nb = 3. The frequency range corresponds to the lower edge of the first band gap shown in Fig. 3 (b), and shows the maxima (in red) at the six equivalent K points that characterize the band edge.

Fig. 2.
Fig. 2.

For a square rod-type PC with nb = 1, nc = 3 and a/d = 0.3 operated in TM polarization: (a) Band diagram showing the first two band gaps. (b) Evolution of the defect mode in each of the gaps with changing defect refractive index nd . Thick, red curves show the defect mode frequency computed by the FSS method. The dashed black curves indicate analytic result (18), with prefactor ��, obtained as described in the text. The vertical dashed lines indicate the refractive index nc of unperturbed cylinders, while the horizontal dashed lines show the band gap edges. The dotted blue curves are for other defect modes.

Fig. 3.
Fig. 3.

Band diagrams for the hexagonal lattices for (a) TM polarization in a rod-type PC with nb = 1, nc = 3, and a/d = 0.2, and (b) TE polarization in a hole-type PC with nb = 3, nc = 1, and a/d = 0.2.

Fig. 4.
Fig. 4.

(a) Similar to Fig. 2(b), but for a hexagonal lattice with TM polarization in a rod type structure. (b) Replotting of the data in (a) showing |ω̃ -ω̃ L | for nd > nc for both the numerical (red dots) and the analytic (curve) results. Panels (c) and (d) are similar to (a) and (b) respectively, but for TE polarization and a hole-type structure.

Fig. 5.
Fig. 5.

Normalized electric energy distribution (on a base 10 logarithmic scale) of the bulk modes for the hexagonal lattice (a/d = 0.2, νb = 3, νc = 1) in TE polarization respectively at the (a) lower and (b) upper edges of the band gap. Panels (c) and (d) are horizontal and vertical section plots through the centre of the defect for the normalized energy density for modes at the lower (red) and upper (blue) edge of the band gap.

Tables (2)

Tables Icon

Table 1. Overview of the results for the structure in Fig. 2. For the four lowest band edges we show the normalized frequency, the model fit parameters ��FSS and ��FSS, the analytic value ��asymp, and the relative difference between the latter.

Tables Icon

Table 2. Similar to Table 1, but for the lowest band gap of the hexagonal lattices from Fig. 4 for TE and TM polarization.

Equations (22)

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ψ m k 0 r = ω m 2 c 2 ψ m k 0 r ,
ψ k 0 r = 1 s ( r ) ( p ( r ) ψ k 0 r ) ,
ψ , ϕ = 1 A WSC WSC s ( r ) ψ k 0 r ϕ * k 0 r d 2 r ,
ψ n , ψ m = M n δ nm ,
( p ( r ) G r r ; ω ) + ω 2 c 2 s ( r ) G r r ; ω = δ ( r r )
G r r ; ω = c 2 4 π 2 l 1 M l BZ ψ l ( k 0 , r ) ψ l * k 0 r ω 2 ω l 2 ( k 0 ) d 2 k 0 .
ω L ( k 0 ) = ω L + 1 2 C L k 0 k L 2 + ,
BZ ψ L k 0 r ψ L * k 0 r ω ω L 1 2 C L k 0 k L 2 d 2 k 0 ψ L k L r ψ L * k L r BZ e i ( k 0 k L ) ( r r ) ω ω L 1 2 C L k 0 k L 2 d 2 k 0 2 π C L ln ω ω L ψ L k L r ψ L * k L r + O ( ω ω L 0 ) .
G r r ; ω G L r r ; ω c 2 C L 4 π ω L M L ψ L k L r ψ L * ( k L , r ) ln ω ω L .
ψ ( r ) = C 0 ( p ( r′ ) p ˜ ( r′ ) ) ψ ( r′ ) G ( r , r′ ; ω ) d 2 r′ +
+ ω 2 c 2 C 0 ( s ( r′ ) s ˜ ( r′ ) ) ψ ( r′ ) G ( r , r′ ; ω ) d 2 r′ .
ψ ( r ) j θ L , j ψ L ( k L , j , r ) ln ω ω L c 2 C L , j 4 π ω L M L , j ×
× C 0 ε ˜ ( r ) ε ( r ) ε ˜ ( r ) ε ( r ) ψ ( r ) ψ L * ( k L , j r ) d 2 r ,
1 M L C 0 ε ˜ ( r ) ε ( r ) ε ˜ ( r ) ε ( r ) ψ L , 1 ψ L , j * d 2 r = A WSC ω L 2 c 2 C 0 ( ε ˜ ( r ) ε ( r ) ) E ( k L , 1 r ) 2 d 2 r WSC ε ( r ) E ( k L , 1 r ) 2 d 2 r ,
M L = 1 A WSC WSC ψ L , j 2 d 2 r = c 2 A WSC ω L 2 WCS 1 ε ( r ) ψ L , j 2 d 2 r ,
1 ln ω ω L = 1 S δ𝓔 WSC 𝓔 WSC = 1 S WSC δε ( r ) E ( k L , 1 , r ) 2 d 2 r WSC ε ( r ) E ( k L , 1 , r ) 2 d 2 r ,
S = 4 π A WSC ω L C L f L = 2 ω L N L ,
ω ω L = 𝓐 exp ( S δ𝓔 WSC 𝓔 WSC ) ,
ψ ( r ) j θ L , j ψ L ( k L , j , r ) ln ω ω L c 2 C L , j 4 π ω L M L , j ×
× C 0 ( ε ˜ ( r ) ε ( r ) ) ψ ( r ) ψ L * ( k L , j , r ) d 2 r
ω ω L = 𝓐 exp ( 𝓢 δε δε C 0 ε C 0 ) ,
𝓢 δε = S𝓔 WSC 𝓔 C 0

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