Abstract

The eigenfrequency and quality factor of the localized electromagnetic modes of the dielectric Menger sponge fractal were investigated theoretically for stage number 1 to 4 with a dielectric constant of 2.8 to 12.0 in the normalized frequency range of ωa/2πc = 0.4 to 1.6, where a is the size of the Menger sponge and c is the light speed in free space. It was found that the quality factor of the eigenmode is larger on average when the spatially averaged dielectric constant of the fractal structure is larger, which is consistent with the mechanism of the usual refractive index confinement. Particularly the largest quality factor of 1720 was found for stage 1. These features imply that the fractal nature is irrelevant to the localization in this frequency range. The theoretical results are compared with previous experimental observation and the reason for their discrepancy is discussed.

© 2007 Optical Society of America

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References

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  1. K. Sakoda, "Electromagnetic eigenmodes of a three-dimensional photonic fractal," Phys. Rev. B 72, Art. No. 184201 (2005).
    [CrossRef]
  2. K. Sakoda, S. Kirihara, Y. Miyamoto, M. Wada-Takeda, and K. Honda, "Light scattering and transmission spectra of the Menger sponge," Appl. Phys. B 81, 321-324 (2005).
    [CrossRef]
  3. K. Sakoda, "90-degree light scattering by the Menger sponge fractal," Opt. Express 13, 9585-9597 (2005).
    [CrossRef] [PubMed]
  4. K. Sakoda, "Localized electromagnetic eigenmodes in three-dimensional metallic photonic fractals," Laser Phys. 16, 897-901 (2006).
    [CrossRef]
  5. K. Sakoda, "LCAO approximation for scaling properties of the Menger sponge fractal," Opt. Express 14, 11372-11384 (2006).
    [CrossRef] [PubMed]
  6. M. Wada-Takeda, S. Kirihara, Y. Miyamoto, K. Sakoda, and K. Honda, "Localization of electromagnetic waves in three-dimensional photonic fractal cavities," Phys. Rev. Lett. 92, Art. No. 093902 (2004).
    [CrossRef]
  7. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman & Company, San Francisco, 1982).
  8. J. Feder, Fractals (Plenum Press, New York, 1988).
  9. S. Kanehira, S. Kirihara, Y. Miyamoto, K. Sakoda, and M. Takeda, "Microwave properties of photonic crystals composed of ceramic/polymer with lattice defects," J. Soc. Mater. Sci. Jpn. 53, 975-980 (2004).
    [CrossRef]
  10. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer-Verlag, Berlin 1990).
    [CrossRef]
  11. K. Sakoda and H. Shiroma, "Numerical method for localized defect modes in photonic lattices," Phys. Rev. B 56, 4830-4835 (1997).
    [CrossRef]
  12. K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, Berlin, 2004).
  13. A. Taflove, Computational Electrodynamics (Artech House, Boston, 1995).
  14. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE Press, Piscataway, 2000).
    [CrossRef]

2006 (2)

K. Sakoda, "Localized electromagnetic eigenmodes in three-dimensional metallic photonic fractals," Laser Phys. 16, 897-901 (2006).
[CrossRef]

K. Sakoda, "LCAO approximation for scaling properties of the Menger sponge fractal," Opt. Express 14, 11372-11384 (2006).
[CrossRef] [PubMed]

2005 (2)

K. Sakoda, S. Kirihara, Y. Miyamoto, M. Wada-Takeda, and K. Honda, "Light scattering and transmission spectra of the Menger sponge," Appl. Phys. B 81, 321-324 (2005).
[CrossRef]

K. Sakoda, "90-degree light scattering by the Menger sponge fractal," Opt. Express 13, 9585-9597 (2005).
[CrossRef] [PubMed]

2004 (1)

S. Kanehira, S. Kirihara, Y. Miyamoto, K. Sakoda, and M. Takeda, "Microwave properties of photonic crystals composed of ceramic/polymer with lattice defects," J. Soc. Mater. Sci. Jpn. 53, 975-980 (2004).
[CrossRef]

1997 (1)

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

Honda, K.

K. Sakoda, S. Kirihara, Y. Miyamoto, M. Wada-Takeda, and K. Honda, "Light scattering and transmission spectra of the Menger sponge," Appl. Phys. B 81, 321-324 (2005).
[CrossRef]

Kanehira, S.

S. Kanehira, S. Kirihara, Y. Miyamoto, K. Sakoda, and M. Takeda, "Microwave properties of photonic crystals composed of ceramic/polymer with lattice defects," J. Soc. Mater. Sci. Jpn. 53, 975-980 (2004).
[CrossRef]

Kirihara, S.

K. Sakoda, S. Kirihara, Y. Miyamoto, M. Wada-Takeda, and K. Honda, "Light scattering and transmission spectra of the Menger sponge," Appl. Phys. B 81, 321-324 (2005).
[CrossRef]

S. Kanehira, S. Kirihara, Y. Miyamoto, K. Sakoda, and M. Takeda, "Microwave properties of photonic crystals composed of ceramic/polymer with lattice defects," J. Soc. Mater. Sci. Jpn. 53, 975-980 (2004).
[CrossRef]

Miyamoto, Y.

K. Sakoda, S. Kirihara, Y. Miyamoto, M. Wada-Takeda, and K. Honda, "Light scattering and transmission spectra of the Menger sponge," Appl. Phys. B 81, 321-324 (2005).
[CrossRef]

S. Kanehira, S. Kirihara, Y. Miyamoto, K. Sakoda, and M. Takeda, "Microwave properties of photonic crystals composed of ceramic/polymer with lattice defects," J. Soc. Mater. Sci. Jpn. 53, 975-980 (2004).
[CrossRef]

Sakoda, K.

K. Sakoda, "Localized electromagnetic eigenmodes in three-dimensional metallic photonic fractals," Laser Phys. 16, 897-901 (2006).
[CrossRef]

K. Sakoda, "LCAO approximation for scaling properties of the Menger sponge fractal," Opt. Express 14, 11372-11384 (2006).
[CrossRef] [PubMed]

K. Sakoda, S. Kirihara, Y. Miyamoto, M. Wada-Takeda, and K. Honda, "Light scattering and transmission spectra of the Menger sponge," Appl. Phys. B 81, 321-324 (2005).
[CrossRef]

K. Sakoda, "90-degree light scattering by the Menger sponge fractal," Opt. Express 13, 9585-9597 (2005).
[CrossRef] [PubMed]

S. Kanehira, S. Kirihara, Y. Miyamoto, K. Sakoda, and M. Takeda, "Microwave properties of photonic crystals composed of ceramic/polymer with lattice defects," J. Soc. Mater. Sci. Jpn. 53, 975-980 (2004).
[CrossRef]

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

Shiroma, H.

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

Takeda, M.

S. Kanehira, S. Kirihara, Y. Miyamoto, K. Sakoda, and M. Takeda, "Microwave properties of photonic crystals composed of ceramic/polymer with lattice defects," J. Soc. Mater. Sci. Jpn. 53, 975-980 (2004).
[CrossRef]

Wada-Takeda, M.

K. Sakoda, S. Kirihara, Y. Miyamoto, M. Wada-Takeda, and K. Honda, "Light scattering and transmission spectra of the Menger sponge," Appl. Phys. B 81, 321-324 (2005).
[CrossRef]

Appl. Phys. B (1)

K. Sakoda, S. Kirihara, Y. Miyamoto, M. Wada-Takeda, and K. Honda, "Light scattering and transmission spectra of the Menger sponge," Appl. Phys. B 81, 321-324 (2005).
[CrossRef]

J. Soc. Mater. Sci. Jpn. (1)

S. Kanehira, S. Kirihara, Y. Miyamoto, K. Sakoda, and M. Takeda, "Microwave properties of photonic crystals composed of ceramic/polymer with lattice defects," J. Soc. Mater. Sci. Jpn. 53, 975-980 (2004).
[CrossRef]

Laser Phys. (1)

K. Sakoda, "Localized electromagnetic eigenmodes in three-dimensional metallic photonic fractals," Laser Phys. 16, 897-901 (2006).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (1)

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

Other (8)

K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, Berlin, 2004).

A. Taflove, Computational Electrodynamics (Artech House, Boston, 1995).

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE Press, Piscataway, 2000).
[CrossRef]

K. Sakoda, "Electromagnetic eigenmodes of a three-dimensional photonic fractal," Phys. Rev. B 72, Art. No. 184201 (2005).
[CrossRef]

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer-Verlag, Berlin 1990).
[CrossRef]

M. Wada-Takeda, S. Kirihara, Y. Miyamoto, K. Sakoda, and K. Honda, "Localization of electromagnetic waves in three-dimensional photonic fractal cavities," Phys. Rev. Lett. 92, Art. No. 093902 (2004).
[CrossRef]

B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman & Company, San Francisco, 1982).

J. Feder, Fractals (Plenum Press, New York, 1988).

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Figures (10)

Fig. 1.
Fig. 1.

Geometrical structure (top view) of the Menger sponge. The size of the Menger sponge is denoted by 2a.

Fig. 2.
Fig. 2.

Stage number dependence of the dipole radiation intensity with the Eu boundary condition calculated for the Menger sponge with a dielectric constant of 8.8. Accumulated electromagnetic energy after 50 cycles of oscillation of the dipole is shown. The abscissa is the frequency of the dipole oscillation normalized with the size of the Menger sponge, a (see Fig. 1), and the light velocity in free space, c.

Fig. 3.
Fig. 3.

Dipole radiation intensity calculated for the Menger sponge of stage 1 with a dielectric constant of 8.8 and the Eu boundary condition. Accumulated electromagnetic energy after 10 (black), 20 (green), 35 (red), and 50 (blue) cycles of oscillation is shown. Some resonant peaks are accompanied by the Q factor of the eigenmodes.

Fig. 4.
Fig. 4.

(a) Excitation and the successive decay of the localized mode at ωa/2πc = 0.7365 with the Eu symmetry in the stage-1 Menger sponge with a dielectric constant of 8.8. The ordinate is the electromagnetic energy accumulated in the cubic volume with the dimension of 4a surrounding the Menger sponge. The abscissa is the time normalized with the period of the oscillation of the dipole, T. (b) Field distribution of the same mode. The z component of the electric field on the x-y plane is shown where the maximum amplitude of the electric field is normalized to unity. The electric field is localized in the volume of the Menger sponge that is denoted by a red square. (c) The x-y plane that intersects the center of the Menger sponge on which the field distribution was evaluated.

Fig. 5.
Fig. 5.

(a) Excitation and the successive decay of the localized mode at ωa/2πc = 0.5755 with the Eu symmetry in the stage-1 Menger sponge with a dielectric constant of 8.8. (b) Field distribution of the same mode. The z component of the electric field on the x-y plane is shown.

Fig. 6.
Fig. 6.

(a) Dipole radiation intensity calculated for the Menger sponge of stage 1 with the A 1u boundary condition and a dielectric constant of 8.8. Accumulated electromagnetic energy after 10 (black), 20 (green), 35 (red), and 50 (blue) cycles of oscillation is shown. Some resonant peaks are associated with the Q factor of the eigenmodes. (b) Frequency range from ωa/2πc = 0.6 to 0.8 is magnified.

Fig. 7.
Fig. 7.

(a) Excitation and the successive decay of the localized mode at ωa/2πc = 0.7136 with the A 1u symmetry in the stage-1 Menger sponge with a dielectric constant of 8.8. (b) Field distribution of the same mode. The z component of the electric field on the x-y plane is shown.

Fig. 8.
Fig. 8.

Q factor of the Eu modes of stage 1 to 4. The dielectric constant of the Menger sponge was assumed to be 8.8.

Fig. 9.
Fig. 9.

Dipole radiation intensity calculated for the Menger sponge of stage 3 with the Eu boundary condition. The dielectric constant of the Menger sponge is (a) 2.8, (b) 5.8, (c) 7.3, (d) 8.8, (e) 10.4, and (f) 12.0. Accumulated electromagnetic energy after 10 (black), 20 (green), 35 (red), and 50 (blue) cycles of oscillation is shown. The eigenmode with the largest Q factor (see text) is denoted by a red arrow in each figure.

Fig. 10.
Fig. 10.

Excitation and the successive decay of the Eu mode with the largest Q factor in the stage-3 Menger sponge (see text) with a dielectric constant of (a) 2.8 (Q = 13), (b) 5.8 (Q = 46), (c) 7.3 (Q = 290), (d) 8.8 (Q = 840), (e) 10.4 (Q = 830), and (f) 12.0 (Q = 770). The origins of the curves are shifted so that they do not overlap each other.

Tables (2)

Tables Icon

Table 1. Averaged refractive index (nav ) of the Menger sponge of stage 1 to 4. The dielectric constant (ε) is assumed to be 8.8.

Tables Icon

Table 2. Comparison between the eigenfrequency of the Eu mode with the largest Q factor in the stage-3 Menger sponge (see text) estimated from the averaged refractive index (ω cal) and that obtained by the FDTD method (ω FDTD).

Equations (2)

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V s V 0 = ( 1 7 27 ) s ,
n av = ε × V s + 1.0 × ( V 0 V s ) V 0 .

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