A numerical investigation of sub-wavelength resonances in polygonal metamaterial cylinders

: The sub-wavelength resonances, known to exist in metamaterial radiators and scatterers of circular cylindrical shape, are investigated with the aim of determining if these resonances also exist for polygonal cylinders and, if so, how they are affected by the shape of the polygon. To this end, a set of polygonal cylinders excited by a nearby electric line current is analyzed numerically and it is shown, through detailed analysis of the near-field distribution and radiation resistance, that these polygonal cylinders do indeed support sub-wavelength resonances similar to those of the circular cylinders. The dispersion and loss, inevitably present in realistic metamaterials, are modeled by the Drude and Lorentz dispersion models to study the bandwidth properties of the resonances. 17. It should be noted that the initial HFSS model utilized radiation boundaries instead of the perfectly matched layers. However, such a model resulted in inconsistent results, in particular with varying side length w , despite the fact that the distance from the perfectly matched layers to the polygonal cylinders and the ELC was larger than 0 / 4 λ as suggested by HFSS, and despite improved discretization along the radiation boundaries. This problem was alleviated by use of perfectly matched layers for which the default discretization options were sufficient to obtain consistent and convergent results. 18. It is important to note that the delta energy, ∆ E , which is the difference in the relative energy error from one adaptive solution to the next, and serves as a stopping criterion for the solution, was set to 0.01 in all cases. This value of ∆ E was targeted and obtained in 3 consecutive adaptive solutions for the 48-, 24-, 12-, and 8-sided PCs, and in 2 consecutive adaptive solutions for the 4-sided PCs. 19. For the 4-sided PC, the MNG shell in the non-dispersive model is described by permeability 2 and a loss all frequencies.

17. It should be noted that the initial HFSS model utilized radiation boundaries instead of the perfectly matched layers. However, such a model resulted in inconsistent results, in particular with varying side length w, despite the fact that the distance from the perfectly matched layers to the polygonal cylinders and the ELC was larger than 0 / 4 λ as suggested by HFSS, and despite improved discretization along the radiation boundaries. This problem was alleviated by use of perfectly matched layers for which the default discretization options were sufficient to obtain consistent and convergent results. 18. It is important to note that the delta energy, ∆E, which is the difference in the relative energy error from one adaptive solution to the next, and serves as a stopping criterion for the solution, was set to 0.01 in all cases. This value of ∆E was targeted and obtained in 3 consecutive adaptive solutions for the 48-, 24-, 12-, and 8-sided PCs, and in 2 consecutive adaptive solutions for the 4-sided PCs. 19. For the 4-sided PC, the MNG shell in the non-dispersive model is described by permeability 2 0 4 µ µ = − and a loss tangent of 0.001 for all frequencies.

Introduction
The past decade has witnessed an enormous activity in the research of a new class of materials, commonly referred to as metamaterials (MTMs), which are typically synthesized by distributing inclusions, each contained inside a unit cell, of various shapes and properties into a host medium, see e.g., [1][2][3][4] and the extensive list of works referenced therein. Very important examples of these materials are double-negative (DNG) materials, characterized by a negative real part of the permittivity and permeability, as well as single-negative (SNG) materials such as epsilon-negative (ENG) materials, which posses a negative real part of the permittivity, and mu-negative (MNG) materials, which possess a negative real part of the permeability. The primary goal of numerous investigations has been the clarification of the fundamental physical properties of these novel materials, as well as their potential applications [1][2][3][4].
Specifically, the potential of DNG and/or SNG materials, as well as combinations of these with normal double-positive (DPS) materials, characterized by a positive real part of the permittivity and permeability, for sub-wavelength waveguides, cavities, scatterers, and radiators of different canonical shapes was demonstrated [3][4][5][6][7][8][9][10][11][12][13][14]. For cylindrical geometries it was shown [10] that a set of MTM concentric circular cylinders excited by a nearby electric line current (ELC) possesses sub-wavelength resonances where the excitation of specific modes leads to e.g., large radiated power for constant ELC.
The purpose of this work is to investigate how the sub-wavelength resonances in MTM cylinders are affected by the shape of the cylinder cross section; in particular, its deviation from the perfect circular shape. From a theoretical point of view such an investigation helps determining the relationship between the shape, on one side, and properties, on the other side, of the material structure. From the practical point of view, an exact circular shape of the MTM structure might be difficult to manufacture due to the shape of the individual unit cells and inclusions forming the MTM. Other geometries might be fitted more accurately by the individual cells of the MTM, and thus it is important to investigate the influence of the deviation of the cylindrical surface on the sub-wavelength resonances. In the present work a set of MTM concentric polygonal cylinders excited by a nearby ELC is analyzed numerically and it is shown that the polygonal cylinders do possess sub-wavelength resonances similar to those of the circular cylinders. The analysis is performed by means of the ANSOFT High Frequency Structural Simulator (HFSS) [15] and includes investigations of the near-field distribution as well as the radiation resistance in the case of simple, but lossy and dispersive, MTMs. This work constitutes an extension of [14] with a full account of the modeling process for the HFSS analysis and additional polygonal configurations.
The manuscript is organized as follows. In Section 2, the investigated configurations are defined and the analysis techniques described. Section 3 presents the numerical results for lossless as well as lossy and dispersive MTM structures. The variation of the radiation resistance with the geometry and the location of the ELC is studied, and the observed subwavelength resonances are further confirmed through analysis of the near-field distributions.

Configuration
The cross-sections of the circular and polygonal cylinder configurations are shown in Fig. 1. In Fig. 1(a), a core circular cylinder (region 1) of radius 1 ρ is covered by a concentric circular shell (region 2) of outer radius 2 ρ . In Fig. 1(b), a core regular polygonal cylinder (region 1), with n sides and a circumscribed circle of radius 1 ρ , is covered by a concentric regular polygonal shell (region 2) with n sides and a circumscribed circle of radius 2 ρ . Although Fig.   1(b) shows just the n = 8-sided polygonal cylinders, this work also investigates 48-, 24-, 12-, and 4-sided polygonal cylinders. Other non-circular configurations could obviously have been employed in the analysis; however, these polygonal cylinder configurations have been selected since they provide a gradual degradation of the perfect circular cylinder.

Methods of analysis
For the CC configuration, see Fig. 1(a), an exact solution is established using the well-known eigenfunction expansion technique, see e.g., [16,Ch. 11]. Below, only the main points of this solution procedure are included while the details and the full analytical solution can be found in [10]. The incident field of the ELC is first expanded in terms of cylindrical wave functions. Then, the unknown fields in the three regions, i.e., the scattered field in the region containing the ELC and the total fields in the remaining regions, are likewise expanded in terms of cylindrical wave functions. The expansions of the incident and unknown fields represent multipole expansions of the fields. The unknown fields contain a set of unknown expansion coefficients: 1 2 3 4 , , , and n n n n C C C C ( 1n C is the expansion coefficient for region 1, 2n C and 3n C are those for region 2, and 4n C is that for region 3), where n designates the mode number in the multipole expansion, i.e., 0 n = is the monopolar mode, 1 n = is the dipolar mode and so on. These unknown coefficients depend on the location of the ELC and are readily determined by enforcing the boundary conditions at the interfaces between the three regions. Upon the determination of the unknown fields, a number of quantities may be derived as was done in [10]; please refer also to Section 2.3 below for specific derived quantities which are of relevance to the present work.
For the PC configuration, see Fig. 1 Fig. 2, where an 8-sided PC configuration is shown, clearly illustrating the details of the employed model. Additional details as well as the values of specific parameters for the HFSS model of the PC structures are given in Section 3.

Derived quantities and resonance condition
For the CC configuration, significant attention was devoted in [10] to the amount of the total power, t r P , radiated by the ELC near-by the circular cylinders for a given constant value of the current e I along the ELC, relative to the power, i r P , radiated by the ELC having the same e I and being situated alone in free space In (1), the quantity Since the total power t r P is proportional to 2 4 | | n C , it is clear that large values of t r P will occur if the amplitude of the expansion coefficients 4n C becomes large. For electrically small, i.e., sub-wavelength structures, these coefficients become large and thus exhibit a resonance when the approximate condition is satisfied [3,4,10]. It was shown in [3,4,10] that in order to satisfy (3), at least one of the parameters, 1 µ or 2 µ , needs to be negative, i.e., DNG and/or MNG materials must be incorporated to obtain a resonant sub-wavelength structure. It is noted that these subwavelength resonances are due to the presence of specific natural modes in the structure [10]. An intuitive explanation for the occurrence of the sub-wavelength resonances is [3,4] that the DNG and /or MNG region acts a tuning element providing the opposite reactance of the DPS region as seen by the ELC, and a condition of zero reactance is thus facilitated by the combination of the two regions. It is clear that possible resonances can be revealed from observations of quantities such as the radiated power (1). However, large values of t r P also occur if the current e I is increased; a scenario which obviously does not correspond to the excitation of sub-wavelength resonances in the structure. In consequence hereof, the so-called total radiation resistance, t r R , which is the radiation resistance of the ELC radiating near-by the circular cylinders, defined by with t r P given by (1), will be considered in the present work . Clearly, the quantity in (4) with i r P given by (2).

Initial remarks
According to Section 2.3 and [10], a dipole ( 1 n = ) mode resonance, leading e.g., to large values of radiated power (and by reciprocity also scattering cross-section), occurs for the CC configuration if region 1 is free space, region 2 is a MNG material with 2  For the first (second) bend radius, the configuration is referred to as the 4-sided(1) PC (4-sided(2) PC). It was verified that the rounding of the sharp corners in the 4-sided polygonal cylinder structure and the inclusion of magnetic loss resulted in consistent results which were discretization independent [18].  It is interesting to observe that the resonance occurs also for the PCs though at a different value of 2 ρ depending on the number of sides n . For large n , i.e., for 48 n = and 24 n = , the outer radius 2 ρ at resonance, and the radiation resistance t r R , is very close to that of the CC. This can be understood in view of the large number of polygon sides. As n decreases, and the PCs thus deviates considerably from the CC, the outer radius 2 ρ at resonance increases slightly for the 12-and 8-sided PCs while the radiation resistance t r R increases more notably. These effects are even more pronounced with the two 4-sided PCs for which both the outer radius 2 ρ at resonance as well as the radiation resistance t r R increases. The values of the outer shell radius 2 ρ at resonance and the associated values of the radiation resistance t r R are summarized in Table 1. For comparison, it is recalled that the ELC in free space has i r R = 0.59 / Ω mm. For the chosen ELC location, it is not only clear to observe that the results for the two 4-sided PCs, although qualitatively the same, differ quantitatively from those of the remaining structures, but also that they differ quantitatively between each other. The reason for the quantitative difference between the results for the two 4-sided PCs is given in Section 3.3.    In all cases, the radiation resistance increases monotonically as the ELC approaches the respective MNG shells. Moreover, the quantitative behaviour of the radiation resistance is rather similar for all configurations for 4 s ρ ≤ mm, though differences occur as the ELC approaches the MNG shell; this is particularly the case with the 8-sided and the two 4-sided PC configurations for which significantly larger values of the radiation resistance are obtained when the ELC is close to the MNG shell. Since the difference between t r R for the circular and polygonal configurations is significant only for s ρ close to 1 ρ , and the qualitative behaviour of the radiation resistance for the investigated ELC locations is the same for all configurations, it is concluded that the shape of the cylindrical structure is not of primary

Near-field distribution
In the following, the above findings related to the sub-wavelength resonances are further illustrated and explained with an analysis of the electric near-field distributions. For all of the following near-field plots, the magnitude of the total electric near field is plotted in the xyplane, and in all cases, the linear dynamic range is set to [150-170000] V/m. As regards the initial examination of the near-field properties of the CC and PC configurations, the ELC is located in region 1 at ( , ) In all cases, a dipolar mode is observed, clearly demonstrating that the resonances shown in Figs. 3 and 4 are due to the excitation of this resonant mode in CC and PC configurations. It is interesting to note that the fields of the PC configurations attain higher values than the fields of the CC configuration. Moreover, the fields of the PCs become more confined near the corners of the structure as the number of sides decreases; once again, this effect is particularly in strong evidence for the 4-sided PC. As to the 4-sided PC configuration, the following additional remarks are in order. As was shown in Fig. 3, the results for the two 4sided PC configurations, although qualitatively the same, differ quantitatively from the remaining structures for the particular ELC location investigated therein. Moreover, they also differ quantitatively from each other for the particular ELC location, although the change in the bend radius from the 4-sided PC(1) to 4-sided PC(2), is not very large. The reason for the latter is straightforward to grasp by observing e.g., Fig. 5(h). Obviously, for the specific ELC location near-by the inner right corner of the 4-sided PC(1) structure, the field attains very high values in the region around the inner right and left corners of the 4-sided PC(1) configuration. Although not shown inhere, as the bend radius decreases, the field values in the said regions increases. As (some of) the rounded corners in the 4-sided PC configurations are located in the region of the high field values, see Fig. 5(h), and as the structures are electrically small, even small changes of the bend radius may lead to different values of the resonances (since the field amplitudes may thus change drastically) as well as different outer radii at which the resonance is attained for the 4-sided PC configuration. This explains the quantitative difference observed between the two 4-sided PCs for the ELC located near-by a corner of these structures.
It is important to note that the sub-wavelength resonances of the PCs are not restricted to locations of the ELC near the corners of the structure. In the following, the location of the ELC is varied along a straight line starting at the point ( , )

Loss and Dispersion
It is well-known that any realizable MNG material is dispersive and lossy [3]. In order to investigate the effects of these realistic MNG material characteristics on the sub-wavelength resonances of the PCs, the Drude and Lorentz dispersion models are employed for the permeability of the MNG shell. These models can in general be expressed as [3] (Fig. 7 (a)), Drude dispersion model ( Fig. 7 (b)), and Lorentz dispersion model (Fig. 7 (c)). The results of the canonical CC are also included, and for both CC as well as PCs, the ELC is located at ( , ) whereas the value of t r R decreases owing to the losses; this is particularly true for the Lorentz model where the losses are more severe than in the Drude model. It is moreover interesting to note that as the number of sides in the PC decreases, the radiation resistance increases. However, it is important to remark that the results for the PCs are qualitatively close to those of the CCs, this implying that it is the dispersion model rather than the geometrical shape that determines the behavior of the resonances for the investigated ELS locations.

Summary and conclusions
In this work a systematic numerical study of circular and polygonal MTM cylinders excited by a nearby ELC was conducted through a detailed investigation of their near-and far-field properties. The purpose of this study was to determine the influence of the deviation from the perfect cylindrical shape on the sub-wavelength resonances. It was found that the sub-wavelength resonances of the circular MTM cylinders are also present in the polygonal MTM cylinders. More specifically, circular and polygonal MTM cylinders supporting dipolar mode resonances were studied, and it was demonstrated that the excitation of this mode leads to large radiation resistances. It is important to emphasize that this effect is independent of the number of polygonal sides, the location of the ELC, as well as the material dispersion model used to analyze the configurations. Hence, the sub-wavelength resonances are not limited to circularly shaped structures but they occur also for other shapes.
The existence of resonances is determined mostly by the material parameters and material models and less by the geometrical shape which has an influence only on the precise values of the resonances.