Abstract

We formulated a novel method to calculate the dispersion relations of arbitrary photonic crystals with frequency-dependent dielectric constants based on the numerical simulation of dipole radiation. As an example, we applied this method to a two-dimensional square lattice of metallic cylinders and obtained a good agreement with the previous result by means of the plane-wave expansion method by Kuzmiak et al. [Phys. Rev. B 50, 16 835 (1994)]. In addition to the dispersion relations, we could obtain the symmetries and the wave functions of the eigenmodes.

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, (Princeton University Press, Princeton, 1995).
  2. Photonic Band Gaps and Localization, edited by C. M. Soukoulis (Plenum, New York, 1993).
  3. Photonic Band Gap Materials, edited by C. M. Soukoulis (Kluwer, Dordrecht, 1996).
  4. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, "Microwave propagation in two-dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991).
    [CrossRef] [PubMed]
  5. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, "Donor and acceptor modes in photonic band structure," Phys. Rev. Lett. 67, 3380-3383 (1991).
    [CrossRef] [PubMed]
  6. E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Am. B 10, 283-295 (1993).
    [CrossRef]
  7. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, "Photonic band structures of two-dimensional systems containing metallic components," Phys. Rev. B 50, 16 835-16 844 (1994).
    [CrossRef]
  8. K. Sakoda and H. Shiroma, "Numerical method for localized defect modes in photonic lattices," Phys. Rev. B 56, 4830-4835 (1997).
    [CrossRef]
  9. K. Sakoda, T. Ueta, and K. Ohtaka, "Numerical analysis of eigenmodes localized at line defects in photonic lattices," Phys. Rev. B 56, 14 905-14 908 (1997).
    [CrossRef]
  10. K. Sakoda and K. Ohtaka, "Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwell's equations and their applications," Phys. Rev. B 54, 5732-5741 (1996).
    [CrossRef]

Other

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

Photonic Band Gaps and Localization, edited by C. M. Soukoulis (Plenum, New York, 1993).

Photonic Band Gap Materials, edited by C. M. Soukoulis (Kluwer, Dordrecht, 1996).

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, "Microwave propagation in two-dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, "Donor and acceptor modes in photonic band structure," Phys. Rev. Lett. 67, 3380-3383 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Am. B 10, 283-295 (1993).
[CrossRef]

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, "Photonic band structures of two-dimensional systems containing metallic components," Phys. Rev. B 50, 16 835-16 844 (1994).
[CrossRef]

K. Sakoda and H. Shiroma, "Numerical method for localized defect modes in photonic lattices," Phys. Rev. B 56, 4830-4835 (1997).
[CrossRef]

K. Sakoda, T. Ueta, and K. Ohtaka, "Numerical analysis of eigenmodes localized at line defects in photonic lattices," Phys. Rev. B 56, 14 905-14 908 (1997).
[CrossRef]

K. Sakoda and K. Ohtaka, "Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwell's equations and their applications," Phys. Rev. B 54, 5732-5741 (1996).
[CrossRef]

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Figures (4)

Figure 1.
Figure 1.

The electromagnetic energy radiated by an oscillating dipole moment located at r 0/a = (0.3,0) as a function of the oscillation frequency for k = 0. The electric field parallel to the cylinder axis (E polarization) was calculated. □, ○, ◇, and • denote the accumulated electromagnetic energy after 10, 20, 50, and 100 cycles of the oscillation, respectively. The abscissa represents the normalized frequency. According to the previous calculation by Kuzmiak et al. (Ref. [7]), the following parameters were used: R/a = 0.472, ω = 1.0, ϵ ωp a/2πc = 1.0. We also assumed that τ-1 = 0.01 × ωp . A resonance at ωa/2πc = 0.745 is clearly observed.

Figure 2.
Figure 2.

The dispersion relations in the (1, 0) direction (or the Γ-X direction) of the 2D photonic crystal composed of the metallic cylinders. The ordinate is the normalized frequency and the abscissa is the normalized wave vactor. The same parameters as for Fig. 1 were used.

Figure 3.
Figure 3.

The distribution of the electric fields of the (a) A 1(1), (b) A 1(2), (c) B 1, and (d) (e) E modes at the Γ point that has the C 4v symmetry. The maximum of each electric field is normalized to unity. For all eigenmodes, the eigenfunctions show their peculiar symmetries. For doubly degenerate E mode, one eigenfunction is a replica of the other given by a 90° rotation.

Figure 4.
Figure 4.

The distribution of the electric fields of the (a) B 1(1), (b) A 1(1), (c) A 2, (d) B 1(2), (e) A 1(2), and (f) B 2 modes at the X point that has the C 2v symmetry. The maximum of each electric field is normalized to unity as before.

Equations (15)

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P d r t = μ δ ( r r 0 ) exp ( iωt ) ,
E r t 2 π ω d { μ · E d * ( r 0 ) } E d ( r ) exp ( iωt ) V ( ω ω d + ) ,
U π ω d 2 γ μ · E d ( r 0 ) 2 V { ( ω ω d ) 2 + γ 2 } ,
V ϵ ( r ) E d ( r ) 2 d r = V .
× E r t = 1 c t H r t ,
× H r t = 1 c t { D 0 r t + 4 π P d r t } ,
D 0 r t = dt Φ r t t E r t .
Φ r t = 1 2 π dωϵ r ω exp ( iωt ) .
ϵ r ω = ϵ [ 1 ω p 2 τ ω ( ωτ + i ) ] ,
Φ r t = ϵ δ ( t ) + ϵ ω p 2 τ [ 1 exp ( t τ ) ] θ ( t ) ,
× { × E r t }
= 1 c 2 [ ϵ 2 t 2 E r t + ϵ ω p 2 E r t ϵ ω p 2 τ 0 dt exp ( t τ ) E r t t
4 π ω 2 μ δ ( r r 0 ) exp ( iωt ) ] .
× { × E r t } = 1 c 2 [ 2 t 2 E r t 4 π ω 2 μ δ ( r r 0 ) exp ( iωt ) ] .
E ( r + a , t ) = exp ( i k · a ) E r t ,

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