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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
1. Introduction
This investigation is motivated by the discrepancy between a series of my theoretical studies [1, 2, 3, 4, 5] and a previous experimental paper [6] on the localized electromagnetic modes of the Menger sponge fractal [7, 8]. The presence of the localized modes with relatively large quality (Q) factors was insisted by both studies. But the essential details are crucially different from each other. To remove the confusion caused by these contradictory reports and to promote unambiguous experimental studies, I will present in this paper a thorough theoretical investigation on the stage number and refractive index dependence of the eigenfrequency and the Q factor in the spectral range of 0.4 < ωa/2πc < 1.6, where a is the size of the Menger sponge (see Fig. 1) and c is the light velocity in free space. Because the eigenfrequency of the localized mode that was insisted to be found by the microwave transmission and reflection measurements in Ref. [6] is ωa/2πc = 0.576 (12.8 GHz), we can thus make a straightforward comparison between the theory and the experiments.
It will be shown in this paper that the mechanism of the localization is the usual refractive index confinement and the fractal nature of the Menger sponge is irrelevant to the localization in the low frequency region. As was shown by my recent paper [5], the fractal nature appears in the high frequency range such as ωa/2πc ≥ 1.5. It will also be shown that although the first experimental report [6] insisted that a localized mode with a Q factor of 670 was realized in the Menger sponge of stage 3 with a dielectric constant of 2.8, the largest Q factor for such a structure found by this theoretical study is only 13 in the analyzed spectral range. But as I have already shown in another paper [1], such an eigenmode with a large Q factor can actually be realized when the dielectric constant is increased up to 8.8, which can be realized in the microwave frequency range with a mixture of epoxy resin and fine particles of certain metal oxides [9] for example.
Fig. 1. Geometrical structure (top view) of the Menger sponge. The size of the Menger sponge is denoted by 2a.
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.
The Menger sponge is made from a dielectric cube. The initial cube is divided into 27 (= 33) identical cubic pieces, and seven pieces at the body and face centers are removed (stage 1). By repeating the same procedure to the 20 remaining pieces, we obtain the Menger sponge of stage 2. The number of the repetition of the removal procedure is called the stage number (see Fig. 1). The ideal Menger sponge is obtained by repeating the removal infinite times. If one of the 20 smaller cubes of the Menger sponge is magnified three times, it coincides with the larger cube. The property of this kind, which is common to all fractals, is called the self-similarity [7, 8].
In Section 2, the method of calculation will be briefly summarized. In Section 3, the numerical results on the stage number and refractive index dependence of the eigenfrequency and Q factor will be described in detail. The results will be compared with the experimental report. 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 Oh point group [10]. Because Maxwell’s wave equation is thus invariant by the symmetry operations, the electromagnetic eigenmodes of the Menger sponge are irreducible representations of the Oh point group [1, 3]. There are four one-dimensional (non-degenerate) representations (A 1g, A 2g, A 1u, A 2u), two two-dimensional (doubly degenerate) representations (Eg, Eu), and four three-dimensional (triply degenerate) representations (T 1g, T 2g, T 1u, T 2u) [10].
The electromagnetic eigenmodes of the Menger sponge was calculated by the method of dipole radiation with symmetry-adapted boundary conditions [11, 12] based on the FDTD (finite-difference time-domain) calculation [13, 14]. 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. The dipole radiation spectrum was calculated for each irreducible representation by applying the symmetry-adapted boundary conditions. This method reduces the spectral density of resonant peaks originating from localized electromagnetic eigenmodes in each dipole radiation spectrum so that it avoids the overlapping of the resonant peaks and makes it possible to obtain the accurate peak frequencies. In addition to the symmetry-adapted boundary conditions, absorbing boundary conditions of the perfectly matched layer (PML) [13] were imposed on the walls surrounding the Menger sponge and 4a away from its surface. The frequency range in the following analysis is ωa/2πc = 0.4 to 1.6 so that the wavelength range is 0.62a to 2.5a. Thus the absorbing boundary is sufficiently far from the Menger sponge and it has a negligible influence on the properties of the localized modes. An oscillating point dipole was located close to the center of the Menger sponge to excite the localized eigenmodes. The position of the dipole was not exactly the center of the Menger sponge because the amplitude of the electric field of the localized modes is often vanishing in the center due to their spatial symmetry.
The volume of the Menger sponge of stage s, Vs, is given by
where V 0 denotes the volume of the initial cube: V 0 = 8a 3. So, when we denote the dielectric constant of the Menger sponge by ε, the spatially averaged refractive index, nav, may be given by
Here we assumed that the dielectric constant of air is equal to 1.0 and the magnetic permeability is also 1.0. We assumed a very simple form for the averaged refractive index to show qualitatively in the next section that the eigenfrequency of a localized mode is a decreasing function of the refractive index and that we can trace a particular resonance peak in the dipole radiation spectrum even when we change the dielectric constant of the Menger sponge. n av is listed for stage 1 to 4 in Table 1. In the next section, we will assume ε = 2.8 to 12.0. As a representative value, ε = 8.8 is assumed in Table 1.
Because the frequency of the electromagnetic eigenmode is inversely proportional to the refractive index if the geometrical structure is the same. So, from Table 1, our interesting frequency may change by about ±20% when we change the stage number from 1 to 4. On the other hand, when we change the dielectric constant of the Menger sponge (ε) from 2.8 to 12.0, the eigenmode frequency may change by about ±35% for stage 3. Since the frequency of the eigenmode with the largest Q factor found for stage 3 is (ωa/2πc = 1.029 [1], we should cover the frequency range of 0.66 < ωa/2< < 1.39 at least in this study. Actually we covered 0.4 < ωa/2πc < 1.6.
3. Results and Discussion
Most of the numerical analyses given in the following deal with the Eu modes, since the largest Q factor for stage 3 was found for this symmetry in a previous study [1]. First, Fig. 2 shows the dipole radiation spectra for stage 1 to 4 calculated with the Eu boundary condition. The dielectric constant of the Menger sponge was assumed to be 8.8, which is a typical value for a mixture of the photoreactive epoxy resin and fine particles of certain metal oxides [9]. There are many resonant peaks in each spectrum, which implies the presence of so many localized eigenmodes [1].
It is found that the distribution of the eigenfrequency is shifted to the high frequency for larger stage numbers. This is a simple consequence of a smaller averaged refractive index. As shown in Table 1, the volume fraction of the dielectric component decreases with increasing stage number. By the same reason, the confinement of the electromagnetic energy may be stronger for smaller stage numbers. In Ref. [1], the largest Q factor for the stage-3 Menger sponge in the analyzed frequency range was 840. So, we may expect that larger Q factors are realized for stage 1 and 2. This is true, and the case of stage 1 is given in Fig. 3. Those resonant peaks which have relatively large Q factors are denoted by red arrows with their Q factors. The largest Q factor was 1720 for stage 1, which is much larger than that of stage 3.
The Q factor was evaluated by observing the temporal decay of the electromagnetic energy associated with the resonant excitation of the localized mode. Two examples of such calculations are given in Figs. 4 and 5. The localized modes were excited by an oscillating point dipole located close to, but not exactly in the center of the Menger sponge. After 100 cycles of the oscillation, the oscillating point dipole was switched off, and the free decay of the accumulated energy was calculated. As is found in these two figures, the decay is well represented by an exponential fit, and the Q factor was obtained from the rate of the decay. The right panel of the figures is the field distribution of the eigenmodes, which is the z component of the electric field on the x-y plane (see Fig. 4(c)) at time t = 100T, where T is the period of the oscillation of the point dipole. The red square denotes the area where the Menger sponge occupies. The electromagnetic field is mostly distributed in the dielectric region for these two localized modes. So, the confinement of the electromagnetic field by the difference of the refractive indices is reasonably strong, which resulted in the relatively large Q factors. Taking into account the fact that the largest Q factor was obtained for the stage-1 structure, the fractal nature is irrelevant to the electromagnetic localization in this frequency range and we should conclude that the localization is brought about by the usual refractive index confinement.
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. 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. (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. (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. (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. (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.
As for the other symmetries, there are also many localized modes in the analyzed frequency range. An example is given in Fig. 6, which is the dipole radiation spectrum calculated with the A 1u boundary condition. Those resonant peaks with relatively large Q factors are again denoted by red arrows. The largest Q factor was 1053. The Q factor and the field distribution were obtained as before and are shown in Fig. 7. The undulation of the accumulated electromagnetic energy found in the excitation process is caused by a small mismatching between the excitation frequency and the eigenmode frequency and/or by the interference with adjacent eigenmodes. This undulation is, however, not important for the evaluation of the Q factor, since we observe a very clear single exponential decay.
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.
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).
The eigenfrequency and Q factor of the Eu modes for stage 1 to 4 are summarized in Fig. 8. As was described already, the distribution of the eigenfrequency shifts to the high frequency with increasing stage number. On the other hand, the Q factor for lower stage numbers is larger than that of higher stage numbers on average. These two features are consistent with the decrease in the averaged refractive index with increasing stage number.
Next, let us examine the refractive-index dependence of the eigenfrequency and Q factor. Figure 9 shows the dipole radiation spectra for various values of ε calculated for stage 3 with the Eu boundary condition. It is found that the distribution of the resonant peaks shifts to the low frequency and becomes narrower with increasing ε. From the latter feature,we may expect that the Q factor increases with ε.
As for the former feature, the averaged refractive index and the eigenfrequency are compared. The resonant peak that gives the largest Q factor for ε = 8.8 is marked by a red arrow in Fig. 9(d). When we change ε, all peaks shift to the same direction so that we can trace the particular mode. In other panels, the corresponding resonant peaks are also marked by red arrows although the resonance becomes vague for small ε and the correspondence becomes somewhat obscure. The eigenfrequency of the localized mode should be approximately proportional to n -1 av. So, the peak frequency found by the first-principle FDTD calculation, ω FDTD, and the estimated frequency by the proportionality, ω cal, are compared in Table 2, and reasonable agreement was found.
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. 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.
Finally, the refractive-index dependence of the Q factor was examined. The excitation by an oscillating point dipole and the successive free decay of the Eu modes are shown in Fig. 10. From the rate of the decay, we got the Q factor as before. They are summarized in the right column in Table 2. The Q factor increases with the dielectric constant and is saturated for ε > 8.8. However, taking into account the decrease in the eigenfrequency, the decay time (= Q/(ω FDTD) is an increasing function of ε within numerical errors.
For ε = 2.8, the Q factor is extremely small and is only 13. As for other symmetries, I found no distinct resonant peaks with larger quality factors when ε = 2.8. This is a marked contrast to the value reported by the microwave transmission measurement [6]. In Ref. [6], it was claimed that there was a localized eigenmode with the eigenfrequency of ωa/2πc = 0.576 (12.8 GHz) and the Q factor of 670. But such a distinct resonant peak is not found in Fig. 9(a). It was not found in the dipole radiation spectra for other symmetries either. In Fig. 3 of Ref. [6], spectral dips in the transmission and reflection spectra are plotted in a logarithmic scale and look very sharp. However, we should reexamine whether they have such a small full width at half maximum that gives the large Q factor of 670.
In addition, as I reported previously [2], we cannot tell the eigenfrequency or the Q factor from the transmission spectra. Rather the light scattering spectra should be used to obtain those information [3]. Most probably the discrepancy between the theoretical analyses and the experimental studies comes from the misinterpretation of the dips observed in the transmission and reflection spectra.
4. Conclusion
The stage number and refractive index dependence of the eigenfrequency and Q factor of the localized electromagnetic modes in the Menger sponge fractal was investigated by the method of dipole radiation based on the FDTD calculation. In the low frequency range where the first experimental study was done, the largest Q factor was found for the stage-1 structure, and on average the Q factor is larger when the spatially averaged refractive index is larger. These features imply that the fractal nature is irrelevant to the localization of the electromagnetic waves in this frequency range, and rather the localization is brought about by the usual refractive index confinement. To clarify the fractal nature (self-similarity) of the localized waves, higher frequency ranges should be examined.
To resolve the confusion by the discrepancy between the experimental observation and the series of theoretical studies, the width of the spectral dips in the observed transmission and reflection spectra should be reexamined, and rather the light scattering spectra should be measured.
Acknowledgments
The author appreciates the enlightening discussion on the electromagnetic properties of fractals with Prof. Yoshinari Miyamoto and Prof. Soshu Kirihara of Osaka University, and Prof. Mitsuo Wada-Takeda and Prof. Katsuya Honda of Shinshu University. This work was supported by the Grant-in-Aid for Scientific Research (S) (Grant No. 171067010) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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