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.

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1. Introduction

Self-similar structures known as fractals have unique properties that are absent in regular structures [1, 2]. One of the most interesting features is their ability to localize various waves that are propagated in the fractals. Localization of electromagnetic waves [3, 4] as well as vibrational waves [5, 6] has been of theoretical and practical interest. The scaling properties due to the self-similarity were clearly demonstrated for the reflection spectra of two-dimensional fractals [7]. Dimensionality of the fractal structure is an important factor for the formation of the localized electromagnetic eigenmodes. Generally speaking, localization takes place more easily in low dimensions because of the small number of scattering channels. In three dimensions, electromagnetic waves may be scattered into all solid angles, which decreases their lifetime, and hence, prevents the formation of well-defined eigenmodes. In recent studies, stereolithography with photoreactive epoxy resin was used to fabricate three-dimensional (3D) photonic fractals called the Menger sponge and it was insisted that localized electromagnetic eigenmodes with quality (Q) factors as large as several hundreds were observed in the microwave frequency range [8].

The Menger sponge is the 3D version of the Cantor bar fractal. The Cantor bar fractal is made by removing the center segment from three equivalent segments obtained by dividing an initial bar, and repeating this procedure to the two remaining segments. Similarly, the Menger sponge may be made from a dielectric or metallic cube. The initial cube is divided into 27 (=3³) identical cubic pieces, and seven pieces at the body and face centers are removed. By repeating the same procedure to the 20 remaining pieces, we obtain the Menger sponge. The number of the repetition of the removal procedure is called the stage number (see Fig. 1). The idealMenger sponge is obtained by repeating the removal infinite times. If one of the 20 smaller cubes of theMenger sponge is magnified three times, it coincides perfectly with the larger cube. The property of this kind, which is common to all fractals, is called the self-similarity.

The first report on the experimental study of the localized electromagnetic modes in the Menger sponge [8], which was composed of epoxy resin with a dielectric constant of 2.8, had a misinterpretation of dips in the transmission and reflection spectra, and apparently overestimated the Q factor. Later, I calculated the eigenfrequency, field distribution, and Q factor of the localized modes in the frequency range comparable to the experimental study and found that Q factors as high as 840 can be realized by increasing the dielectric constant to 8.8, which can be attained by using a mixture of fine particles of metal oxides and the epoxy resin [9]. I further derived the selection rules of the 90-degree light scattering by the Menger sponge and showed by numerical analysis that the light scattering spectra have such desirable features that the eigenfrequency and the Q factor can be obtained from the peak frequency and the spectral width [10]. Then I also analyzed the metallic Menger sponge to find similar localized eigenmodes with relatively small Q factors [11]. In addition, I compared the transmission and light scattering spectra to clearly show that we cannot obtain the eigenfrequency or the Q factor from the transmission spectra [12]. This fact implies that we should measure the light scattering rather than the transmission or reflection to characterize the localized electromagnetic eigenmodes of the Menger sponge.

In this paper, I will report on the theoretical analysis of the approximate self-similarity found for the electromagnetic eigenmodes of the Menger sponge. In Section 2, I will first summarize the symmetry of the Menger sponge and the consequence in the symmetry of its electromagnetic eigenmodes based on the group theory. Then I will describe the LCAO (linear combination of atomic orbitals) approximation for the eigenmodes of higher stage numbers starting with a localized mode of a lower stage number. In Section 3, I will present a first-principle calculation of the eigenmodes based on the FDTD (finite-difference time-domain) method and show that the LCAO approximation gives a good guiding principle to find scaled (self-similar) eigenfunctions because of the localized nature of the eigenmodes. A brief summary of this paper will be given in Section 4.

2. Theory

The Menger sponge has the octahedral symmetry and it is invariant by any symmetry operation of the O_h point group:

where the standard notations of the symmetry operations are used [13]. Because Maxwell’s wave equation is thus invariant by the symmetry operations, the electromagnetic eigenmodes of the Menger sponge are irreducible representations of the O_h point group [9, 10]. There are four one-dimensional (non-degenerate) representations (A _1g, A _2g, A _1u, A _2u), two two-dimensional (doubly degenerate) representations (E_g , E_u ), and four three-dimensional (triply degenerate) representations (T _1g, T _2g, T _1u, T _2u).

Each eigenmode is transformed according to its spatial symmetry when a symmetry operation, R(∈O_h ), is applied. The way of transformation is described by the character, χ ^(α)(R), where α stands for the irreducible representation. For the one-dimensional representation, the electric field of the eigenmode, E ^(α)(r), is transformed such that

where r is the position vector and R denotes the 3×3 matrix representation of symmetry operation R. In the case of the two-dimensional representation, there are two independent and mutually orthogonal eigenfunctions for each eigenfrequency. We denote them by ${\mathbf{E}}_1^{\left(\mathrm{\alpha }\right)}$(r) and ${\mathbf{E}}_2^{\left(\mathrm{\alpha }\right)}$(r). They are transformed such that

and the trace of the orthogonal transformation matrix on the right-hand side of this equation is given by the character of the two-dimensional representation:

Table 1. Character table of the Oh point group.

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Similarly, three orthogonal eigenfunctions of a triply degenerate eigenmode (${\mathbf{E}}_1^{\left(\mathrm{\alpha }\right)}$(r), ${\mathbf{E}}_2^{\left(\mathrm{\alpha }\right)}$(r), and ${\mathbf{E}}_3^{\left(\mathrm{\alpha }\right)}$(r)) are transformed such that

and

The character of the Oh point group is listed in Table 1.

The electromagnetic eigenmodes of the Menger sponge can be calculated accurately by a standard method, that is, the method of dipole radiation with symmetry-adapted boundary conditions [14, 15] based on the FDTD calculation [16]. This method was applied to the Menger sponge for the first time in Ref. [9], and eigenmodes of all possible symmetries were found. For the Menger sponge of stage 3 made of a dielectric material with a dielectric constant of 8.8, I found a large number of eigenmodes in the analyzed frequency range of 0.4≤ωa/2πc≤ 1.6, where a is the size of the Menger sponge (see Fig. 1) and c denotes the light velocity in free space. The maximum quality factor was 840. As for the metallic Menger sponge, the number of eigenmodes was relatively small since the freedom of the electromagnetic field distribution is limited by the presence of the conducting metallic region [11]. The quality factors of its eigenmodes were generally small because the electromagnetic field is distributed only in the air region and the confinement mechanism that is present for the dielectric Menger sponge, that is, the index confinement is absent for the metallic structures. However, for the purpose of this paper, the small number of eigenmodes is preferable because that makes it easy for us to distinguish certain molecular orbitals, which will be described later, from other eigenmodes. Thus let us consider the metallic Menger sponge hereafter. Particularly, let us examine the T _1u symmetry, since it gives the smallest eigenfrequency so that we can perform the most accurate numerical calculation with a relatively small computational labor. But note that the same method is, of course, applicable to other symmetries.

 

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.

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Figure 2 is the dipole radiation spectrum calculated for the metallic Menger sponge of stage 1 with the T _1u boundary conditions. Accumulated electromagnetic energy after 50 cycles of oscillation of an electric point dipole located in the center of the Menger sponge is shown. The abscissa is the dimensionless frequency of the oscillating dipole. We can see some peaks in the analyzed frequency range. Each resonant peak gives the eigenfrequency of an eigenmode and the emitted field at the peak frequency is proportional to the eigenfunction [9].

Let us examine the sharp resonance at ωa/2πc=0.6147 in more detail. This mode had the smallest eigenfrequency among those found for all possible symmetries. Figure 3(b) is the field distribution (eigenfunction) of this mode, where the z component of the electric field on the x-y plane is shown. As can be clearly seen, the field is well confined in the Menger sponge denoted by a red square. It certainly has a spatial symmetry expected for the T 1_u mode. Figure 3(c) is the free decay of the electromagnetic energy accumulated after 100 cycles of oscillation of an electric dipole at the eigenfrequency. The energy shows an exponential decay, and from the decay rate we obtain the quality factor, Q, of the resonance. Q is 43.8 for this eigenmode, which is not large at all as was mentioned already.

Now let us proceed to the analysis of the scaling properties. When we go from stage 1 to stage 2, the size of the unit structure is reduced to one third so that the wavelength of the eigenmode that is resonant to the unit structure (each of the 20 smaller cubes) is also reduced to one third whereas its eigenfrequency increases three times. Thus we can expect that we should find eigenmodes of stage 2 in a frequency range three times higher than the eigenfrequencies of stage 1. This expectation is true in principle. However, the eigenfrequencies of stage 2 are not exactly equal to those of stage 1 times three due to the interaction between adjacent unit structures.

Nevertheless, we can use this picture as the guiding principle to construct the eigenfunctions of stage 2. Let me describe the detail with the T _1u mode presented in Fig. 3 as an example. Because the eigenfunction is well localized in the unit structure, it may be regarded as an atomic orbital like an electronic wave function of an isolated atom. The Menger sponge of stage 2 is composed of 20 such unit structures so that it may be regarded as a molecule composed of 20 atoms. It is established in the field of quantum chemistry that the molecular orbital can be well represented by the linear combination of atomic orbitals when the latter are sufficiently localized around each atom.

 

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.

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If we know the spatial symmetry of the molecule and the symmetry of the individual atomic orbital, we can tell the symmetry of themolecular orbital by the group-theoretical consideration. Following the general prescription of the group theory [13], we first count the number of unit structures of stage 2 that do not change their positions, N_R , when symmetry operation R of O_h is applied. Next, we calculate the character of the molecular orbitals, χ(R), by multiplying N_R by the character of the atomic orbital. An example is given in Table 2, where I assumed that the atomic orbital has the T _1u symmetry.

In this case, each atomic eigenmode localized in the 20 unit structures has three independent eigenfunctions because the T _1u mode is triply degenerate. So, the total number of molecular orbitals given by the linear combination of atomic orbitals is 60 (=20×3). Since the Menger sponge of stage 2 is also invariant by any symmetry operation of O_h , each molecular orbital should be its irreducible representation. Because the character χ(R) is the trace of the transformation matrix of whole 60 orbitals, it is given by the sum of the trace of each molecular orbital, χ ^(α)(R):

Table 2. Calculation of the character for the molecular orbital composed of the linear combination of 20 T_1u modes.

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

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The coefficients q_α , which denotes the number of eigenmodes that have the symmetry of the irreducible representation α, depend on the symmetry of the atomic orbital. Now we use the orthogonality of the characters [13]:

where g is the number of elements of the point group and δ is Kronecker’s delta. For O_h , g is 48. From Eqs. (7) and (8), we obtain

According to this procedure that is called reduction, we obtain the number of molecular eigenmodes of each symmetry. For the 20 T _1u atomic orbitals shown in Table 2, we obtain the following molecular orbitals:

This method can, of course, be applied to atomic orbitals of other symmetries. The results are summarized in Table 3.

This method can be generalized to higher stage numbers. Table 4 gives an example for which the stage number is three and the symmetry of the atomic orbital is A _1u. The symmetry of molecular orbitals can again be calculated according to Eq. (9). Thus we obtain

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.

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The results for stage 3 are summarized in Table 5.

The eigenfunction of the molecular orbital may be calculated by various numerical methods. In the next section, I will show the numerical results given by the method of dipole radiation based on the FDTD calculation. On the other hand, the group theory also provides us with solutions within the LCAO approximation. For example, we may use the method of the projection operator [13]. When we denote one of the atomic orbitals in the Menger sponge of stage 2 by ϕ, the molecular orbital ψ of the T _1u symmetry, for example, is given as follows:

From this equation, we can obtain the molecular orbitals very easily although they are not necessarily mutually orthogonal when we start from ϕ’s on different unit structures or from another eigenfunction of a degenerate eigenmode on the same unit structure.Molecular orbitals of other symmetries can, of course, be obtained by using their character χ(α)(R) in Eq. (12). For example, we may obtain the molecular orbitals derived from the linear combination of the T _1u mode shown in Fig. 3(b). As I mentioned previously, there are three independent eigenfunctions for each eigenmode of the T _1u symmetry. The transformation of each eigenfunction by symmetry operations can be described by the polynomial representations [13]. It is known that for the T _1u mode, {x, y, z} gives the simplest polynomial representation. This means that each eigenfunction is transformed like the x, y, or z coordinate when symmetry operations are applied. In the present case, I assumed boundary conditions compatible with the {z} representation. So, the electric field of the eigenfunction is antisymmetric for the x-y plane, and symmetric for the x-z and y-z planes. Thus the x and y components of the electric field of the eigenfunction on the x-y plane are exactly equal to zero. Actually the main component of the electric field of this eigenfunction found by the numerical calculation is pointed in the z direction.

So, let us simply denote this atomic eigenfunction by a unit vector pointed in the z direction. When we take this function of a unit structure on the corner of the top layer of the Menger sponge of stage 2 as ϕ in Eq. (12), we obtain the field distribution shown in Fig. 4(a) as the molecular orbital ψ. We call the field distribution of this kind type A hereafter. When we take ϕ on a corner cube in the middle layer, we obtain the field distribution shown in Fig. 4(b) and call it type B. Finally, when we take ϕ on a middle cube in the top layer, we obtain Fig. 4(c) and call it type C. Here we should note that if we take ϕ in the bottom layer instead of the top layer, we simply obtain field distributions similar to type A and C but antisymmetric for the x-y plane.

 

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.

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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.

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3. Numerical Results and Discussion

Now let us proceed to the numerical results. Figure 5 shows the dipole radiation spectrum for theMenger sponge of stage 2 calculated by the FDTDmethod with the T _1u boundary conditions. To obtain difference equations from Maxwell’s equations, the space and time were discretized such that the length a (see Fig. 1) was divided into 40 parts and one cycle of the oscillation of the dipole was divided into 1024 parts. In addition to the symmetry-adapted boundary conditions, absorbing boundary conditions of the perfectly matched layer (PML) [16] were imposed on the walls surrounding the Menger sponge and 4a away from its surface. Because the important frequency range in the following analysis is ωa/2πc=1.5 - 1.8, the wavelength of the relevant electromagnetic waves is about 0.6a. Thus the absorbing boundary is sufficiently far from the Menger sponge and it has a negligible influence on the properties of the localized modes. The metallic region was regarded as a perfect conductor, which is a good approximation for the microwave frequency range, so that the electromagnetic field in it was assumed to be vanishing. An oscillating point dipole was located in the center of the corner cube in the top layer and polarized in the z direction in order to find scaled eigenmodes of type A. Two adjacent peaks at ωa/2πc = 1.6278 and 1.6502 are clearly observed. Their field distributions (the z component of the electric field) on the horizontal plane denoted in Fig. 6(a) are shown in Figs. 6(b) and 6(c). It is apparent that these field distributions are very close to the combinations of the atomic orbital presented in Fig. 3(b). The field intensity in the middle layer was small as is expected from Fig. 4(a). Here we should note that the eigenfrequencies of the molecular orbitals for stage 2 are appreciably different from that of stage 1 (ωa/2πc = 0.6147) times three due to the interaction between adjacent unit structures. This is true for arbitrary stage numbers. So, the idea of the self-similarity holds only approximately for the spectrum of the localized electromagnetic eigenmodes of the Menger sponge.

 

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).

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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.

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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).

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Next let us examine the eigenmode of type B. For this purpose, I placed an oscillating dipole in the center of the corner cube in the middle layer and calculated the dipole radiation spectrum. Figure 7 shows the result of the FDTD calculation, and we find one distinct peak at ωa/2πc=1.6912, which is again reasonably close to the eigenfrequency of stage 1 (ωa/2πc=0.6147) times three. Its field distribution (the z component of the electric field) on the x-y plane is shown in Fig. 8(b). It is again very close to what we expect for an eigenmode of type B (see Fig. 4(b)). Finally, Figs. 9 and 10 show the dipole radiation spectrum and the field distributions for type C. There are two eigenmodes and their field distributions are mostly concentrated in the middle cube in the top and bottom layers as was expected from Fig. 4(c).

 

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.

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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).

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Let us conclude this section by giving five remarks. First, we should note that the total number of scaled eigenmodes of the T _1u symmetry that we found for stage 2 by the numerical calculation is five (two for type A, one for type B, and two for type C) and it agrees with the prediction of the group theory given in Eq. (10). Second, scaled eigenmodes of other symmetries can also be found both by Eq. (12) with their character in the LCAO approximation and by the FDTD calculation assuming appropriate boundary conditions that match their symmetries. Third, the oscillating dipole to excite the eigenmodes should be placed at a right position where the target eigenmodes have a large amplitude, for which the knowledge of the symmetry and approximate field distribution of the scaled eigenmodes given by the LCAO approximation such as Eq. (10) and Fig. 4 is very useful. Fourth, the present method can be applied to stage 3 in principle as shown in Tables 4 and 5, although the number of scaled eigenfrequencies is 20 times larger than stage 2 so that the detailed analysis of their eigenfrequencies and eigenfunctions may meet difficulties due to the dense distribution of the eigenfrequency and hybridization of scaled eigenmodes of the same symmetry with close eigenfrequencies. Finally, the limit of stage number → ∞ is an interesting problem. In this limit, the volume of the metallic region goes down to zero whereas the area of its surface goes up to infinity. So, if we can regard the metallic region as a perfect conductor, the Menger sponge of the infinite stage number can confine the electromagnetic waves in unit structures of various sizes although the thickness of the metallic walls is infinitesimally small. In the case of an ordinary metal with a finite conductivity or a dielectric material, the spatially averaged conductivity or the averaged dielectric constant goes down to zero or one with increasing stage number, and thus, the confinement of the electromagnetic waves is expected to become less effective. So, there should be an optimized stage number for the observation of the approximate self-similarity in the spectrum of the localized electromagnetic eigenmodes. How much is it? What determines it? How close is the approximate self-similarity to the ideal self-similarity? These are open questions to be solved in the future.

4. Conclusion

To analyze the scaling (self-similar) properties of the electromagnetic eigenmodes of the Menger sponge, I first used the localized nature of the eigenmodes and described the LCAO approximation for molecular orbitals of higher stage numbers starting from atomic orbitals of a lower stage number. Then I presented numerical results of the method of dipole radiation with symmetry-adapted boundary conditions based on the FDTD calculation for the eigenfrequency, eigenfunction, and Q factor. From those examples shown in Figs. 5–10, we can conclude that the LCAO approximation gives a good guiding principle for the scaling properties. Particularly, the scaled eigenfunctions of stage 2 keep the original features found for stage 1. The same method can, in principle, be applied to higher stage numbers, although the detailed analysis of the eigenfrequencies and eigenfunctions may meet difficulties due to the dense distribution of the eigenfrequency and hybridization of scaled eigenmodes.