Abstract

Selection rules for the 90° light scattering by the Menger sponge fractal were derived by the group theory based on the symmetry of its localized electromagnetic eigenmodes. The light scattering spectra were calculated by the finite-difference time-domain method and compared with the eigenfrequencies of the localized modes obtained from the dipole radiation spectra. Their correspondence is quite good and supports the accuracy of the numerical calculation and the correctness of the selection rules. It was also shown that the scattering spectra are background free and the quality factor of the localized modes can be obtained from the spectral width of the scattering peaks. Thus the 90° light scattering is a powerful method for the investigation of the Menger sponge fractal.

© 2005 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. W. J. Wen , L. Zhou , J.S. Li , W. K. Ge , C. T. Chen , and P. Sheng , “ Subwavelength photonic band gaps from planar fractals ,” Phys. Rev. Lett.   89 , Art. No. 223901 ( 2002 ).
    [Crossref] [PubMed]
  6. S. Alexander and R. Orbach , “ Density of states on fractals - fractons ,” J. Phys. (Paris), Lett.   43 , L625 – L631 ( 1982 ).
    [Crossref]
  7. J. W. Kantelhardt , A. Bunde , and L. Schweitzer , “ Extended fractons and localized phonons on percolation clusters ,” Phys. Rev. Lett.   81 , 4907 – 4910 ( 1998 ).
    [Crossref]
  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 in press.
  10. See, for example, T. Inui , Y. Tanabe , and Y. Onodera , Group Theory and Its Applications in Physics ( Springer-Verlag, Berlin 1990 ).
    [Crossref]
  11. K. Sakoda , “ Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices ,” Phys. Rev. B   52 , 7982 – 7986 ( 1995 ).
    [Crossref]
  12. Z. Yuan , J. W. Haus , and K. Sakoda , “ Eigenmode symmetry for simple cubic lattices and the transmission spectra ,” Opt. Express   3 , 19 – 27 ( 1998 ).
    [Crossref] [PubMed]
  13. K. Sakoda , Optical Properties of Photonic Crystals , 2nd Ed. ( Springer-Verlag, Berlin , 2004 ).
  14. See for example A. Taflove , Computational Electrodynamics ( Artech House, Boston , 1995 ).
  15. K. Sakoda and H. Shiroma , “ Numerical method for localized defect modes in photonic lattices ,” Phys. Rev. B   56 , 4830 – 4835 ( 1997 ).
    [Crossref]

2004 (1)

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 ).

2002 (1)

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

1998 (2)

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

Z. Yuan , J. W. Haus , and K. Sakoda , “ Eigenmode symmetry for simple cubic lattices and the transmission spectra ,” Opt. Express   3 , 19 – 27 ( 1998 ).
[Crossref] [PubMed]

1997 (1)

K. Sakoda and H. Shiroma , “ Numerical method for localized defect modes in photonic lattices ,” Phys. Rev. B   56 , 4830 – 4835 ( 1997 ).
[Crossref]

1995 (1)

K. Sakoda , “ Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices ,” Phys. Rev. B   52 , 7982 – 7986 ( 1995 ).
[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]

Chen, C. T.

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

Feder, J.

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

Ge, W. K.

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

Haus, J. W.

Honda, K.

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 ).

Inui, T.

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

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 ).

Li, J.S.

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

Mandelbrot, B. B.

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

Masciulli, P.

Miyamoto, Y.

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 ).

Onodera, Y.

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

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 ).

Z. Yuan , J. W. Haus , and K. Sakoda , “ Eigenmode symmetry for simple cubic lattices and the transmission spectra ,” Opt. Express   3 , 19 – 27 ( 1998 ).
[Crossref] [PubMed]

K. Sakoda and H. Shiroma , “ Numerical method for localized defect modes in photonic lattices ,” Phys. Rev. B   56 , 4830 – 4835 ( 1997 ).
[Crossref]

K. Sakoda , “ Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices ,” Phys. Rev. B   52 , 7982 – 7986 ( 1995 ).
[Crossref]

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

K. Sakoda , “ Electromagnetic eigenmodes of a three-dimensional photonic fractal ,” Phys. Rev. B in press.

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]

Sheng, P.

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

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]

Taflove, A.

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

Tanabe, Y.

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

Wada-Takeda, M.

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 ).

Wen, W. J.

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

Yuan, Z.

Zhou, L.

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

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]

J. Phys. (Paris), 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 (3)

K. Sakoda and H. Shiroma , “ Numerical method for localized defect modes in photonic lattices ,” Phys. Rev. B   56 , 4830 – 4835 ( 1997 ).
[Crossref]

K. Sakoda , “ Electromagnetic eigenmodes of a three-dimensional photonic fractal ,” Phys. Rev. B in press.

K. Sakoda , “ Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices ,” Phys. Rev. B   52 , 7982 – 7986 ( 1995 ).
[Crossref]

Phys. Rev. Lett. (3)

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

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

Other (5)

See, for example, 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 ).

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

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Figures (11)

Fig. 1.
Fig. 1.

Illustration of the EM wave travelling in the (1,0,0) direction of the Menger sponge. k denotes the wave vector of the EM wave. e 1 and e 2 denote its two possible polarization components, which we assume perpendicular to the side surface of the Menger sponge without a loss of generality.

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Fig. 2.
Fig. 2.

Configuration for the calculation of the light scattering intensity. The center of the Menger sponge whose size was 2a was located at the origin of the coordinates. Its surface was perpendicular to the x, y, and z axes. An oscillating dipole was assumed as the light source S and the electric field at point D was examined. The distance b was 3.65a. Both polarizations, perpendicular (denoted by ⊙) and parallel (denoted by ↦) to the x-y plane, were analysed.

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Fig. 3.
Fig. 3.

The second configuration for the calculation of light scattering. Notations are the same as Fig. 2.

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Fig. 4.
Fig. 4.

Dipole radiation intensity calculated with the T 1u and T 2u boundary conditions. Accumulated EM 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 Figs. 2 and 3), and the light velocity in free space, c. Peaks in the spectra are denoted by arrows.

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Fig. 5.
Fig. 5.

Light scattering spectrum in the geometry shown in Fig. 2. An oscillating dipole moment pointed to the z direction (⊙) was assumed as a light source S. The intensity of the electric field at D is poltted as a function of the normalized frequency. Eigenfrequencies of the localized modes found in the dipole radiation spectra (Fig. 4) are denoted by arrows. Their agreement with the peaks of the light scattering intensity is observed.

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Fig. 6.
Fig. 6.

Dipole radiation intensity calculated with the T 1g and T 2g boundary conditions. Accumulated EM energy after 50 cycles of oscillation is shown. Peaks in the spectra are denoted by arrows.

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Fig. 7.
Fig. 7.

Light scattering spectrum in the geometry shown in Fig. 2. An oscillating dipole moment pointed parallel to the x-y plane (↔) was assumed as a light source S. The intensity of the electric field at D is poltted as a function of the normalized frequency. Eigenfrequencies of the localized modes found in the dipole radiation spectra (Fig. 6) are denoted by arrows.

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Fig. 8.
Fig. 8.

Dipole radiation intensity calculated with the A 2g and A 2u boundary conditions. Accumulated EM energy after 50 cycles of oscillation of the dipole moment is shown. Peaks in the spectra are denoted by arrows.

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Fig. 9.
Fig. 9.

Dipole radiation intensity calculated with the Eg and Eu boundary conditions. Accumulated EM energy after 50 cycles of oscillation of the dipole moment is shown. Peaks in the spectra are denoted by arrows.

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Fig. 10.
Fig. 10.

Light scattering spectrum in the geometry shown in Fig. 3. An oscillating dipole moment pointed perpendicular to the x-y plane (⊙) was assumed as a light source S. The intensity of the electric field at D is poltted as a function of the normalized frequency. Eigenfrequencies of the relevant localized modes found in the dipole radiation spectra are denoted by arrows.

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Fig. 11.
Fig. 11.

Light scattering spectrum in the geometry shown in Fig. 3. An oscillating dipole moment pointed parallel to the x-y plane (↔) was assumed as a light source S. The intensity of the electric field at D is poltted as a function of the normalized frequency. Eigenfrequencies of the relevant localized modes found in the dipole radiation spectra are denoted by arrows.

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Tables (3)

Tables Icon

Table 1. Reduction of the irreducible representations of the Oh point group. The irreducible representations that have the same symmetry as the plane wave travelling in each direction are underlined.

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Table 2. Symmetry of triply degenerate eigenmodes and their coupling to the incident and scattered light waves in the (1,0,0) direction illustrated in Fig. 2. ↔ and ⊙ denote coupling to light waves with the horizontal and vertical polarization, respectively. x shows the absence of coupling.

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Table 3. Symmetry of eigenmodes and their coupling to the incident and scattered light waves in the (1,1,0) direction. ↔ and ⊙ denote coupling to light waves with the horizontal and vertical polarization, respectively. x shows the absence of coupling. σz changes the sign of the z coordinate. σx,y (σx,-y ) interchanges x and y (-y).

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Equations (10)

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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 } ,
C 4 v = { E , 2 C 4 , C 2 , 2 σ v , 2 σ d } ,
C 4 ( E 1 E 2 ) = ( 0 , 1 − 1 , 0 ) ( E 1 E 2 ) .
χ ( C 4 ) = 0 .
C 2 ( E 1 E 2 ) = ( − 1 , 0 0 , − 1 ) ( E 1 E 2 ) .
χ ( C 2 ) = − 2 .
χ ( σ v ) = χ ( σ d ) = 0 .
χ ( E ) = 2 .
C 2 v = { E , C 2 , σ y , σ x } ,
C 3 v = { E , 2 C 3 , 3 σ v } .

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