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/2 πc = 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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  1. K. Sakoda, "Electromagnetic eigenmodes of a three-dimensional photonic fractal," Phys. Rev. B 72, Art. No. 184201 (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]
  3. K. Sakoda, "90-degree light scattering by the Menger sponge fractal," Opt. Express 13, 9585-9597 (2005).
    [CrossRef]
  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]
  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).
  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).
  10. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer-Verlag, Berlin 1990).
  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).

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]

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]

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).

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).

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).

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).

Sakoda, K.

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

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

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]

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).

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).

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).

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).

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

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

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).

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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