Abstract

The electromagnetic eigenmodes of a three-dimensional fractal called the Menger sponge were analyzed by the LCAO (linear combination of atomic orbitals) approximation and a first-principle calculation based on the FDTD (finite-difference time-domain) method. Due to the localized nature of the eigenmodes, the LCAO approximation gives a good guiding principle to find scaled eigenfunctions and to observe the approximate self-similarity in the spectrum of the localized eigenmodes.

© 2006 Optical Society of America

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References

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  1. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman & Company, San Francisco, 1982).
  2. J. Feder, Fractals (Plenum Press, New York, 1988).
  3. X. Sun and D. L. Jaggard, "Wave interactions with generalized Cantor bar fractal multilayers," J. Appl. Phys. 70, 2500-2507 (1991).
    [CrossRef]
  4. M. Bertolotti, P. Masciulli, and C. Sibilia, "Spectral transmission properties of a self-similar optical Fabry-Perot resonator," Opt. Lett. 19, 777-779 (1994).
    [CrossRef] [PubMed]
  5. S. Alexander and R. Orbach, "Density of states on fractals-fractons," J. Phys. (Paris), Lett. 43, L625-L631 (1982).
    [CrossRef]
  6. J. W. Kantelhardt, A. Bunde, and L. Schweitzer, "Extended fractons and localized phonons on percolation clusters," Phys. Rev. Lett. 81, 4907-4910 (1998).
    [CrossRef]
  7. W. J. Wen, L. Zhou, J.S. Li,W. K. Ge, C. T. Chan, and P. Sheng, "Subwavelength photonic band gaps from planar fractals," Phys. Rev. Lett. 89, Art. No. 223901 (2002).
    [CrossRef] [PubMed]
  8. 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).
  9. K. Sakoda, "Electromagnetic eigenmodes of a three-dimensional photonic fractal," Phys. Rev. B 72, Art. No. 184201 (2005).
    [CrossRef]
  10. K. Sakoda, "90-degree light scattering by the Menger sponge fractal," Opt. Express 13, 9585 (2005).
    [CrossRef] [PubMed]
  11. K. Sakoda, "Localized electromagnetic eigenmodes in three-dimensional metallic photonic fractals," Laser Phys. 16897-901 (2006).
    [CrossRef]
  12. 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]
  13. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer-Verlag, Berlin 1990).
    [CrossRef]
  14. K. Sakoda and H. Shiroma, "Numerical method for localized defect modes in photonic lattices," Phys. Rev. B 56, 4830-4835 (1997).
    [CrossRef]
  15. K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed., (Springer-Verlag, Berlin, 2004) Chap. 6.
  16. See for example A. Taflove, Computational Electrodynamics (Artech House, Boston, 1995).

2006 (1)

K. Sakoda, "Localized electromagnetic eigenmodes in three-dimensional metallic photonic fractals," Laser Phys. 16897-901 (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 (2005).
[CrossRef] [PubMed]

1998 (1)

J. W. Kantelhardt, A. Bunde, and L. Schweitzer, "Extended fractons and localized phonons on percolation clusters," Phys. Rev. Lett. 81, 4907-4910 (1998).
[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]

1994 (1)

1991 (1)

X. Sun and D. L. Jaggard, "Wave interactions with generalized Cantor bar fractal multilayers," J. Appl. Phys. 70, 2500-2507 (1991).
[CrossRef]

1982 (1)

S. Alexander and R. Orbach, "Density of states on fractals-fractons," J. Phys. (Paris), Lett. 43, L625-L631 (1982).
[CrossRef]

Alexander, S.

S. Alexander and R. Orbach, "Density of states on fractals-fractons," J. Phys. (Paris), Lett. 43, L625-L631 (1982).
[CrossRef]

Bertolotti, M.

Bunde, A.

J. W. Kantelhardt, A. Bunde, and L. Schweitzer, "Extended fractons and localized phonons on percolation clusters," Phys. Rev. Lett. 81, 4907-4910 (1998).
[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]

Jaggard, D. L.

X. Sun and D. L. Jaggard, "Wave interactions with generalized Cantor bar fractal multilayers," J. Appl. Phys. 70, 2500-2507 (1991).
[CrossRef]

Kantelhardt, J. W.

J. W. Kantelhardt, A. Bunde, and L. Schweitzer, "Extended fractons and localized phonons on percolation clusters," Phys. Rev. Lett. 81, 4907-4910 (1998).
[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]

Masciulli, P.

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]

Orbach, R.

S. Alexander and R. Orbach, "Density of states on fractals-fractons," J. Phys. (Paris), Lett. 43, L625-L631 (1982).
[CrossRef]

Sakoda, K.

K. Sakoda, "Localized electromagnetic eigenmodes in three-dimensional metallic photonic fractals," Laser Phys. 16897-901 (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 (2005).
[CrossRef] [PubMed]

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

Schweitzer, L.

J. W. Kantelhardt, A. Bunde, and L. Schweitzer, "Extended fractons and localized phonons on percolation clusters," Phys. Rev. Lett. 81, 4907-4910 (1998).
[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]

Sibilia, C.

Sun, X.

X. Sun and D. L. Jaggard, "Wave interactions with generalized Cantor bar fractal multilayers," J. Appl. Phys. 70, 2500-2507 (1991).
[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. Appl. Phys. (1)

X. Sun and D. L. Jaggard, "Wave interactions with generalized Cantor bar fractal multilayers," J. Appl. Phys. 70, 2500-2507 (1991).
[CrossRef]

Laser Phys. (1)

K. Sakoda, "Localized electromagnetic eigenmodes in three-dimensional metallic photonic fractals," Laser Phys. 16897-901 (2006).
[CrossRef]

Lett. (1)

S. Alexander and R. Orbach, "Density of states on fractals-fractons," J. Phys. (Paris), Lett. 43, L625-L631 (1982).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

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]

Phys. Rev. Lett. (1)

J. W. Kantelhardt, A. Bunde, and L. Schweitzer, "Extended fractons and localized phonons on percolation clusters," Phys. Rev. Lett. 81, 4907-4910 (1998).
[CrossRef]

Other (8)

W. J. Wen, L. Zhou, J.S. Li,W. K. Ge, C. T. Chan, and P. Sheng, "Subwavelength photonic band gaps from planar fractals," Phys. Rev. Lett. 89, Art. No. 223901 (2002).
[CrossRef] [PubMed]

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).

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]

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

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

K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed., (Springer-Verlag, Berlin, 2004) Chap. 6.

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

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Figures (10)

Fig. 1.
Fig. 1.

Geometrical structure (top view) of the Menger sponge: (a) stage 0, (b) stage 1, (c) stage 2, and (d) stage 3. The size of the Menger sponge is commonly denoted by 2a.

Fig. 2.
Fig. 2.

Dipole radiation intensity calculated for the metallic Menger sponge of stage 1 with the T 1u boundary conditions. Accumulated electromagnetic energy after 50 cycles of oscillation 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.

(a) The x-y plane that intersects the center of the Menger sponge of stage 1 whose field distribution is shown. (b) Field distribution of the T 1u mode at ωa/2πc=0.6147. The z component of the electric field on the x-y plane is shown. The electric field is normalized by its maximum amplitude on the x-y plane. It is mostly confined in the fractal structure (|x/a|, |y/a|<1) denoted by a red square. (c) Free decay of the electromagnetic energy after excitation by a dipole moment oscillating at ωa/2πc=0.6146 for 100T, where T denotes one period of oscillation.

Fig. 4.
Fig. 4.

Illustration of approximate field distribution of the T 1u modes for stage 2 obtained by the projection operator method described by the molecular orbital ψ in Eq. (12). Red arrows denote the electric field of ψ and black lines are a guide for the eyes. Three types of field distributions were obtained according to Eq. (12) by first putting an atomic orbital ϕ (e z) on (a) a corner cube in the top layer, (b) a corner cube in the middle layer, and (c) a middle cube in the top layer, respectively.

Fig. 5.
Fig. 5.

Dipole radiation spectrum for stage 2 with the T 1u boundary conditions. The oscillating dipole pointed perpendicular to the x-y plane was located at the center of a corner cube in the top layer in order to excite the eigenmodes of type A.

Fig. 6.
Fig. 6.

(a) The plane that intersects the center of the top layer for which the field distribution is shown. Field distribution of the T 1u mode of type A for stage 2 (a) at ωa/2πc = 1.6278 and (b) at ωa/2πc = 1.6502. The z component of the electric field is shown. The field is mostly confined in the fractal structure (|x/a|, |y/a| < 1 denoted by the biggest red square. The electric field in each corner cube is quite similar to the T 1u mode at ωa/2πc=0.6147 in stage 1, which is shown in Fig. 3(b).

Fig. 7.
Fig. 7.

Dipole radiation spectrum for stage 2 with the T 1u boundary conditions. The oscillating dipole pointed perpendicular to the x-y plane was located at the center of a corner cube in the middle layer in order to excite the eigenmodes of type B.

Fig. 8.
Fig. 8.

(a) The x-y plane that intersects the center of the Menger sponge of stage 2 whose field distribution is shown. (b) Field distribution of the T 1u mode of type B at ωa/2πc=1.6912 in the Menger sponge of stage 2. The z component of the electric field on the x-y plane is shown. The electric field is mostly confined in the fractal structure (|x/a|, |y/a|<1) denoted by the biggest red square. The electric field in each fundamental unit (the smallest red square) is quite similar to the T 1u mode at ωa/2πc=0.6147 in the Menger sponge of stage 1, which is shown in Fig. 3(b).

Fig. 9.
Fig. 9.

Dipole radiation spectrum for stage 2 with the T 1u boundary conditions. The oscillating dipole pointed perpendicular to the x-y plane was located at the center of a middle cube in the top layer in order to excite the eigenmodes of type C.

Fig. 10.
Fig. 10.

Field distribution of the T 1u mode of type C for stage 2 (a) at ωa/2πc=1.6253 and (b) at ωa/2πc=1.6551. The z component of the electric field on the plane denoted in Fig. 6(a) is shown. The field is mostly confined in the fractal structure (|x/a|, |y/a| < 1) denoted by the biggest red square. The electric field in each corner cube is quite similar to the T 1u mode at ωa/2πc=0.6147 in stage 1, which is shown in Fig. 3(b).

Tables (5)

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Table 1. Character table of the Oh point group.

Tables Icon

Table 2. Calculation of the character for the molecular orbital composed of the linear combination of 20 T 1u modes.

Tables Icon

Table 3. Reduction of the linear combination of 20 eigenmodes.

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Table 4. Calculation of the character for the molecular orbital composed of the linear combination of 400 A 1u modes.

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Table 5. Reduction of the linear combination of 400 eigenmodes.

Equations (12)

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O h = { E , 6 C 4 , 3 C 2 , 6 C 2 , 8 C 3 , I , 6 I C 4 , 3 σ h , 6 σ d , 8 I C 3 } ,
[ R E ( α ) ] ( r ) R E ( α ) ( R 1 r ) = χ ( α ) ( R ) E ( α ) ( r ) ,
( [ R E 1 ( α ) ] ( r ) [ R E 2 ( α ) ] ( r ) ) = ( A 11 , A 12 A 21 , A 22 ) ( E 1 ( α ) ( r ) E 2 ( α ) ( r ) ) ,
i = 1 2 A ii = χ ( α ) ( R ) .
( [ R E 1 ( α ) ] ( r ) [ R E 2 ( α ) ] ( r ) [ R E 3 ( α ) ] ( r ) ) = ( A 11 , A 12 , A 13 A 21 , A 22 , A 23 A 31 , A 32 , A 33 ) ( E 1 ( α ) ( r ) E 2 ( α ) ( r ) E 3 ( α ) ( r ) ) ,
i = 1 3 A ii = χ ( α ) ( R ) .
χ ( R ) = α q α χ ( α ) ( R ) .
R χ ( α ) ( R ) * χ ( β ) ( R ) = g δ α β ,
q α = 1 g R χ ( α ) ( R ) * χ ( R ) .
20 × T 1 u 2 A 1 g + A 2 g + 3 E g + 3 T 1 g + 4 T 2 g + 2 A 2 u + 2 E u + 5 T 1 u + 3 T 2 u .
400 × A 1 u 4 A 1 g + 12 A 2 g + 14 E g + 30 T 1 g + 22 T 2 g + 15 A 1 u + 5 A 2 u + 18 E u + 19 T 1 u + 29 T 2 u .
ψ = R χ ( T 1 u ) ( R ) R ϕ .

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