Sub-Wavelength Resonances in Metamaterial-Based Multi-Cylinder Configurations

Sub-wavelength resonances known to exist in isolated metamaterial-based structures of circular cylindrical shape are investigated with the purpose of determining whether the individual resonances are retained when several of such resonant structures are grouped to form a new structure. To this end, structures consisting of 1, 2 and 4 sets of metamaterial-based concentric cylinders excited by an electric line current are analyzed numerically. It is demonstrated that these structures recover the resonances of the individual structures even when the cylinders are closely spaced and the new structure is thus electrically small. The investigation is conducted through a detailed analysis of the electric near-field distribution as well as the radiation resistance in those cases where the individual structures are made of simple dielectric materials in conjunction with simple, but lossy and dispersive, metamaterials.


Introduction
The field of metamaterials (MTMs) has experienced significant scientific advances in recent years, and numerous applications within the microwave [1][2][3] and the optical [4] frequency regions have been devised. Important examples of MTMs include double-negative (DNG) materials, which possess a negative real part of the permittivity and permeability, as well as epsilon-negative (ENG) and OPEN ACCESS mu-negative (MNG) materials, which possess a negative real part of the permittivity and permeability, respectively. Among the numerous reported applications of these MTMs, specific attention has been devoted to their potential of providing sub-wavelength resonant structures of various canonical shapes [5][6][7][8][9][10][11][12][13][14] either when used alone or in combination with double-positive (DPS) materials, which possess a positive real part of permittivity and permeability. In particular, it was shown in [9] that an isolated set of concentric circular MTM-based cylinders excited by a nearby electric line current (ELC) possesses sub-wavelength resonances where the excitation of specific modes is found to lead to large radiated power for constant ELC.
The purpose of the present work is to investigate how the sub-wavelength resonances of the isolated MTM-based concentric cylinder structures studied in [9] are affected when several of such structures are grouped to form a new structure. To this end, configurations consisting of 1, 2 and 4 sets of MTM-based concentric cylinders, henceforth referred to as 1-, 2-, and 4-cylinder structure, are analyzed. It is shown that these structures recover the resonances of the individual structures even when the cylinders are closely spaced and the configuration is thus electrically small. The analysis is conducted with the ANSOFT High Frequency Structural Simulator (HFSS) [15] and includes detailed investigations of the electric near-field distribution and the radiation resistance in case of simple, but lossy and dispersive, MTMs. A collection of MTM-based objects were studied in [6] with the purpose of devising an effective hybrid MTM, in [16] for cloaking purposes, and in [17] for its scattering properties. The present work is an extension of [18], and in comparison, includes both a full account of the HFSS model as well as additional near-field investigations.
The present manuscript is organized as follows. In Section 2, the investigated structures are defined and the analysis techniques, including the exact method used for the 1-cylinder structure, as well as the numerical method, are described. This section also includes a brief discussion on the conditions for sub-wavelength resonance in the isolated 1-cylinder structures; this is used in conjunction with the exact analytical results to define the electrical and geometrical parameters of a given 1-cylinder structure. In Section 3, the numerical results are presented; in particular, the resonances of the individual structures are studied as the distance between the cylinders is changed. In all cases, the resonant structures are made of simple dielectric materials in conjunction with simple, but lossy and dispersive, MTMs, and the resonant properties of all configurations are analyzed through detailed investigations of their electric field distribution and the radiation resistance. Section 4 includes a summary and conclusion of the present work. The time factor ) exp( t j , with  being angular frequency and t time, is assumed throughout the manuscript. Figure 1 shows the , k th concentric cylinder set (Ck) of the   The 1-cylinder structure consist of a cylinder C1 which has its center at the origin, i.e., )

Configuration
The 2-cylinder structure consists of the previous cylinder C1 and a cylinder C2 having its center at ) 0 , ( . Thus, the cylinder C2 is displaced along the x-axis with a separation distance d to the cylinder C1. The 4-cylinder structure consists of the previous cylinders C1 and C2, and the additional cylinders C3 and C4 with their centers at , respectively.

Analysis Methods
For the 1-cylinder structure, both an exact as well as a numerical solution have been obtained. The exact solution is based on the eigenfunction expansion method, see e.g., [19]. Whereas the details of the exact solution can be found in [9], we emphasize below only the main points. The incident field of the ELC, as well as the unknown fields in the three regions, i.e., the scattered field in the region containing the ELC and total field in the remaining regions, are all expanded in terms of cylindrical wave functions. These expansions represent the multipole expansion of the respective fields, and for the unknown fields, they contain a set of unknown expansion coefficients jn A , to the dipole mode and so on. The unknown expansion coefficients in A depend on the electrical and geometrical parameters of the structure in Figure 1 as well as on the location of the ELC, and they are readily determined by enforcing the boundary conditions at the interfaces between the three regions; once these coefficients are known, the fields in the different regions have been determined. For the 2-and 4-cylinder structures, a numerical solution is established using the ANSOFT HFSS software [15] (the numerical solution was also employed to investigate the 1-cylinder structure and to compare its results with the exact solution in order to confirm the validity of the established HFSS model). Figure 2 shows the HFSS model, where the 4-cylinder structure with the individual cylinders designated as C1, C2, C3, and C4 is depicted. The model consists of the ELC source modeled by a finite length current tube of radius a, current e I , and its axis located at ( s s   , )enlarged in the inset of the figure, and the finite length MTM-based cylinders. The finite length current tube and the MTM-based cylinders are positioned between, and perpendicular to, two parallel, perfectly electrically conducting infinite plates with separation h. Due to image theory [19], these plates model the infinite MTM-based cylinders and the ELC. Between the perfectly conducting plates, uniform perfect matching layers which model free-space radiation, which have thickness d, circumscribe a square of side length w, and have their corners and edges joined, are inserted. The values of the specific parameters of the HFSS model as well additional details are found in Section 3.

Derived Quantities and Resonance Condition
In the present work, the attention is devoted to the radiation resistance, t R , of the ELC for a given constant value of e I radiating in the presence of the material structure, where the quantities t P and i P in the above expressions represent, respectively, the power radiated by the ELC in the presence and absence of the material structure. For the 1-cylinder structure, the exact expressions for t P and i P have been obtained in [9], and are repeated here for the sake of convenience is the Bessel function of order n, when the ELC is outside the 1-cylinder structure. The symbol n  is the Neumann number; thus, 1  n  for 0  n and 2  n  otherwise, and max N is the truncation limit chosen such to ensure the convergence of the cylindrical wave expansion. From (3) and (1), it is clear that large values of the total radiation resistance will result if the amplitude of the expansion coefficients n A 4 becomes large. When a given single concentric cylinder set Ck is electrically small, i.e., when it is sub-wavelength, these expansion coefficients become very large, and thus exhibit a resonance when the condition is satisfied [6,9]. As explained in [6,9], at least one of the regions comprising such a concentric cylinder set Ck must be made of DNG and/or MNG material in order to satisfy the condition in (5), and moreover, the excitation of the sub-wavelength resonances is due to the presence of natural modes in the structure. The resonance condition in (5) has been used in [6,9] to design resonant sub-wavelength 1-cylinder configurations, and is also used next to design the individual concentric cylinder sets of the 1-, 2-, and 4-cylinder structures.

Resonant Configurations and Further Remarks on the HFSS Model
According to Section 2.3 and [6,9], it is possible to design a sub-wavelength 1-cylinder structure capable of exciting a dipole ( 1  n ) mode resonance, which leads to large values of, e.g., radiated power and radiation resistance. In the present section, we investigate whether these resonances of the individual 1-cylinder structures exist, and under which conditions when several cylindrical structures are grouped to form a new structure.   Table 1.  [12], thereby verifying the established HFSS model of the ELC.
In order to assess the frequency behavior of the MNG material of the 1-, 2-, and 4-cylinder structures, the Drude dispersion model [3] has been employed for the permeability

1-Cylinder Structure
The resonances of the 1-cylinder structures are illustrated in Figure 3(a) where the quantity | t R | log 10 RR 10  [dB], where the radiation resistance t R (1) has been normalized by 1 mm /  , is shown as a function of frequency when each of the cylinders is centered at the origin and the ELC is located at ) 0 , mm 5 . represent the exact analytical results while the full lines represent the corresponding HFSS results. The agreement between the exact analytical results and HFSS results is seen to be excellent; a similar agreement was reported in [13]. It is clear that the individual 1-cylinder structures resonate at the desired designed frequencies; moreover, the values of RR are comparable in the four cases and equal to approximately 20 dB, this showing large enhancements of the radiation resistance of the ELC nearby the MTM-based structures relative to the case where the ELC is alone in free space. Figure 3   However, these resonances are still due to the dipole modes in the two cylinders. This is, however, not the case for the separation distance of d = 5 mm where, e.g., the first resonance at f = 241 MHz, which attains higher amplitude than in the case of individual cylinders, is due to a mode characterized by strong coupling between the two cylinders as is illustrated in Figure 5 where the magnitude of the electric field is shown. With the diameter of the individual cylinders being approximately 20 mm, it is thus found that the sub-wavelength resonances of the individual cylinders also occur in 2-cylinder configurations of which the overall size is as small as 20 /   MHz, which is the frequency at which the first resonance appears in Figure 4 Figure 6 shows the results for the 4-cylinder structure (the structure itself is shown in the inset in the right part of the figure). [dB] as a function of frequency for the resonant 4-cylinder structures for different separation distances d. In all cases, the ELC is located

4-Cylinder Structure
. The 4-cylinder structure is shown on the right.
Specifically, Figure 6 shows the quantity | t R | log 10 RR 10  [dB] as a function of frequency for the separation distances d = 50, 40, and 30 mm when the ELC is located at For all separation distances d, four distinct resonances are found, although slightly shifted from the resonant frequencies of the individual cylinders and with lower amplitudes than in the case of the 1-cylinder structures in Figure 3(a). For a given separation d, this shift is larger for the 4-cylinder than for the 2-cylinder configuration and is seen to be largest for the cylinders C1 and C2. The majority of the resonances in Figure 6 are due to the dipole mode excitation in the individual cylinders; this is clear from Figure 7 . This explains why, e.g., large RR values are attained for cylinders C3 and C4 not only at the frequencies f = 266.5 MHz and 282.5 MHz, respectively, but also at the original resonance frequencies of the individual cylinders, whereas this is found not to be the case for cylinders C2 and C1. Moreover, for the separation distance of d = 30 mm, the first resonance occurring at f = 240 MHz is not due to a clear dipole mode in the cylinder C2, but rather to a mode which is due to coupling effects between the four cylinders, as is clearly illustrated by the result in Figure 8, which shows the magnitude of the electric field in this particular case. With the diameter of the individual cylinders being approximately 20 mm, it is thus found that the sub-wavelength resonances of the individual cylinders also occur in 4-cylinder configurations of which the overall size is as small as 5  It is noted that if the individual cylinders are designed such that their resonances are even closer to each other, the coupling becomes more visible than in the case of the presently investigated cylinders. This is supported by the results in Figure 3(a) which suggests that for close enough resonance frequencies, the radiation resistance curves (those parts with significant values of the radiation resistance) for the individual cylinders will considerably overlap each other thus indicating a stronger coupling.   MHz; this is the frequency at which the first resonance appears in Figure 6. The ELC is located at , where 0 f is the design frequency of the respective cylinders, and the radiation resistance was for the 4-cylinder configuration with the separation distance d = 40 mm. The obtained results are reported in Figure 9 in terms of the quantity | t R | log 10 RR 10  [dB] as a function of frequency. This figure also includes the corresponding lossless-case result for comparison purposes. It is observed that resonances occur at the same frequencies as in the lossless case, but that the corresponding amplitudes, as expected, are reduced.

Summary and Conclusions
This work presented a detailed study of resonant properties of a number of sub-wavelength MTM-based structures of circular cylindrical shape. In particular, attention was devoted to sub-wavelength resonances known to exist in isolated MTM-based structures of circular cylindrical shape with the aim of determining whether the individual resonances are retained when several of such resonant structures are grouped to form a new structure. To this end, structures composed of 1, 2 and 4 sets of MTM-based concentric cylinders excited by an ELC were analyzed numerically in ANSOFT HFSS with regard to their near-field properties and radiation resistance. The MTMs of the individual structures were assumed to be simple, but lossy and dispersive, where the effects of the latter were accounted for by the Drude dispersion mode.
It was demonstrated that the sub-wavelength resonances of the isolated MTM-based concentric cylinder structures also occur for the structures composed of 2 and 4 sets of MTM-based concentric cylinders even in the case where the cylinders are closely spaced and the entire structure is thus electrically small. Specifically, overall sizes of about 1/20 and 1/12.5 of the smallest free-space wavelength were found for 2-and 4-cylinder structures, respectively, in which the respective resonances were due to the dipole mode excitation in the constituent cylinders. These MTM-based structures thus offer the possibility for multi-resonant sub-wavelength configurations.