On the Excitation of Magnetic Current Surface Waves in Truncated Periodic Arrays of Slots Under Extraordinary Transmission Conditions

In this paper, we explore the excitation of magnetic current surface waves in truncated periodic arrays of slots in a conducting screen. A specialized Method of Moments (MoM) implementation is presented, which makes it possible to efficiently solve the scattering problem involving truncated arrays of several thousands of slots. By making use of the dispersion diagrams of surface waves propagating along infinite periodic arrays of slots, we are able to explain the absence of magnetic current surface waves in the arrays at frequencies in the neighborhood of the transmission peak associated to the slots natural resonances (length roughly equal to half the wavelength), while they are present when the arrays are excited under extraordinary transmission (EOT) conditions. In order to experimentally check this different behavior, an aluminium plate periodically perforated with slots has been fabricated and fed by means of a pyramidal horn, and the electric field behind the plate has been measured with a planar near-field system at a few centimeters from the plate. Our experimental results and MoM simulations agree, demonstrating the presence of a standing wave pattern of magnetic current surface waves at the EOT frequency, and the absence of surface waves at the slots natural resonant frequency.


I. INTRODUCTION
In a seminal paper published in 1998, Ebbesen et al. discovered that periodic arrays of small apertures present a transmission coefficient as large as unity at a narrow band of frequencies well-below the expected resonance associated with the size of the apertures [1]. This phenomenon was called Extraordinary Optical Transmission (EOT) since it was first detected at optical frequencies. The first theories for EOT relied on the plasmonic behaviour of metals at optical frequencies [2], although it was not long before several researchers found similarly remarkable EOT for hole arrays at millimeter wave frequencies, where metals can be treated as perfect conductors [3], [4]. Finally, it was accepted that EOT can be explained in terms of the excitation of surface waves (not necessarily surface plasmons) in periodically structured The associate editor coordinating the review of this manuscript and approving it for publication was Santi C. Pavone . conductor surfaces [5], which is a concept well-known by antenna engineers [6].
In 2000 Neto et al. studied the effects introduced by truncations on the distribution of magnetic currents on arrays of slots [7]. At the frequency for which the slots were resonant (length equal to a half a wavelength), the authors found the magnetic currents to be not very different from those obtained in the case of infinite arrays of slots (see [7, Fig. 4.a]), except for the slots located at the boundaries of the truncated arrays where edge effects were stronger. Similar results were obtained for truncated arrays of resonant dipoles in [8, Fig. 1.3.b], and again for truncated arrays of resonant slots in [9,Fig.11]. However, in a series of papers published in 2001 and 2002 [10]- [12], and in a book published in 2003 [8], Munk and collaborators found that bound current surface waves may propagate along truncated metallic arrays of dipoles. These surfaces waves exist at frequencies well-below the resonant frequency of the dipoles VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ (typically between 20% and 30% below). Recently, Camacho et al. [9] found that magnetic current surface waves, similar to those observed by Munk, also appear in truncated arrays of slots at the frequency of EOT, and in this latter case, the wavelength of the surface waves is extremely large [9,Fig. 13]. Also, whereas the electric current surface waves observed in [8], [10]- [12] appear for oblique incidence under plane wave illumination, the magnetic current surface waves of [9] are excited even for normal incidence.
In this paper we study the excitation of magnetic current surface waves in a truncated periodic array of slots under both plane wave and pyramidal horn illuminations. Since the MoM analysis of truncated arrays of slots under plane wave illumination has been described in detail in [9, Section III], in Section II we briefly describe the extension of the MoM analysis of [9] for the case in which the feed is a pyramidal horn, this being a much more realistic scenario. In Section III we use the MoM analysis to show that magnetic current surface waves appear at frequencies well below the resonant frequency of the slots but do not appear at the resonant frequencies of the slots [8], [10]- [12], and that they are present at the frequency of the EOT transmission peak [9]. This difference in behavior is explained by using the dispersion relation of the surfaces waves propagating along the periodic array when it is assumed to be infinite. Also, in Section III we validate our MoM results for pyramidal horn illumination with measurements, and these results show the absence of magnetic current surface waves at the resonant frequency of the slots and the presence of these waves at the frequency of the EOT transmission peak.
The results presented here can pave the way for novel focusing platforms with application in lens and transmitarray antennas and could also be extended for their application at optical frequencies, where the large electric field enhancement achieved on the surface of the array could be exploited for non-linear phenomena.

II. MoM ANALYSIS FOR PYRAMIDAL HORN FEED
Let us consider the problem depicted in Fig.1, which shows a conducting screen of negligible thickness containing a truncated periodic array of M = N × N rectangular cells of dimensions a × b. The conducting screen will be assumed to be a perfect electric conductor (PEC). Each of the screen periodic cells includes a rectangular slot of dimensions w×l occupying a surface η i (i = 1, . . . , M ). The conducting screen is illuminated by a pyramidal horn antenna with rectangular aperture of dimensions a F × b F (a F > b F ), the distance between the phase center of the horn antenna and the screen being d cF as shown in Fig.1. If we assume a time dependence of the type e jωt that will be suppressed throughout, the electric field radiated by the horn in the far-field region can be modelled as [13]  where k 0 = ω √ µ 0 0 = 2π/λ 0 , and where The coordinates {r F , θ F , φ F } of (1) to (3) are spherical coordinates relative to the coordinate system with origin at the phase center of the horn (see Fig. 1). The value of q E can be estimated for a particular pyramidal horn by carrying out a logarithmic least squares fit of the main beam of the normalized radiation pattern of the horn to (2) in the E plane. A similar fit of the main beam of the normalized radiation pattern of the horn to (3) in the H plane enables the determination of q H . The magnetic field radiated by the horn in the far field can be obtained as H inc = r F × E inc /Z 0 (Z 0 = √ µ 0 /ε 0 ). Let E sc t (x, y, z = 0) be the tangential electric field excited in the M slots of the conducting screen of Fig.1 by the feeding horn (E sc t (x, y, z = 0) = 0 outside the slots since the screen is assumed to be a PEC), and let M sc (x, y) =ẑ×E sc t (x, y, z = 0) be the magnetic current density in the slots. As shown in [9], in order to obtain E sc where the dyadic Green's function G M (x, y) is defined in [9,Eqns. (2)-(3)], and J as (x, y) is the electric current density that would be generated on the conducting screen by the feeding horn in the absence of the slots (which is generated by the horn itself and by its image through the conducting screen). Since J as (x, y) does not appreciably vary within η i (i = 1, . . . , M ), we can assume that J as (x, y) is constant within η i and equal to the value of this function at the center point of η i of coordinates (x = x ci , y = y ci , z = 0). Therefore, if J as (x = x ci , y = y ci ) = J as i , (4) can be approximately rewritten as It can be shown that J as i is given by In order to solve the set of M integral equations of (5), E sc t (x, y, z = 0) has to be expressed as a linear combination of known basis functions d jl (x, y) in the M slots as shown below In this paper Chebyshev polynomials weighted by an edge condition term have been chosen for the basis functions d jl (x, y) (see [9, Eqns. (29)-(39) & Eqns. (52)-(56)]). After substituting (6) into (5) and applying Galerkin's version of MoM, a system of linear equations for the unknown coefficients e jl is obtained. The solution of this system of equations makes it possible to determine the magnetic current density, M sc (x, y), in the slots. As shown in [9], the use of basis functions that account for the singular behavior of the fields at the slot edges guarantees a fast convergence of MoM with respect to the number of basis functions (see [9,Fig. 6]). Also, very sophisticated analytical techniques have been developed for the efficient computation of the MoM matrix entries when these basis functions are employed (see [9, Section III]), which enables the analysis of screens with several thousands of slots within a few seconds of CPU time in a laptop computer.
A dimensionless transmission coefficient through the perforated screen, T , can be defined as the ratio between the power radiated by the slots into the z > 0 half plane, P rad , and the power radiated by the horn that is available at the surface occupied by the M slots, P av , i. e., where After some manipulations, it can be shown that P rad can be determined as where the coefficients e jl have been introduced in (7), and  (8), the transmission coefficient T will always be smaller (or at most, equal) than one in natural units.
Once the set of integral equations of (4) has been solved and the tangential electric field at the conducting screen, E sc t (x, y, z = 0), is known, the tangential electric field at a plane parallel to the screen, placed at a distance h p of the screen in the region z > 0, can be obtained as   Fig. 2 shows the transmission coefficient of two periodic truncated arrays of slots when the array is illuminated by a plane wave under normal incidence. This transmission coefficient can be obtained, either by introducing the results of the MoM formulation of [9] into Eqns. (8) to (10), or by using the MoM formulation of Section II and taking a sufficiently large value of d cF . The dashed line corresponds to an array with a slot length of l = 15 mm, i. e., l > a/2, which is within the range of values usually employed in frequency selective surfaces (FSSs) to allocate a band pass filter before the onset of grating lobes [4]. Note the presence of a transmission peak at roughly 9.3 GHz when the slots are resonant (length roughly equal to half the wavelength). The solid line corresponds to an array with a slot length of l = 9 mm, i. e., l < a/2. In this latter case, there is an EOT peak at 12.4 GHz below the Wood's anomaly (onset of grating lobes) at 13.3 GHZ, and the slots resonate individually at a frequency above the onset of grating lobes [4].

Figs.3(a) and (b)
show the distribution of magnetic currents obtained for the truncated periodic array with longer slots studied in Fig.2 at 9.3 GHz (resonant frequency of the slots) and 6 GHz respectively. Note that the distribution is very smooth at the resonant frequency of the slots, 9 GHz, except for some edge effects of the type predicted in [7]. To find the excitation of a couple of surface waves of magnetic currents along the x direction, one needs to go as low as 6 GHz, well below the resonant frequency of the slots, as predicted by Munk [8]. In contrast, Figs.4(a) and (b) show the distribution of magnetic currents for the case of a truncated periodic array with shorter slots studied in Fig. 2. The results of Figs.4(a) and (b) are obtained at 12.4 GHz (EOT frequency) and 11.4 GHz. At both frequencies, we find a standing surface wave pattern of magnetic currents along the x direction, but the wavelength of the surface waves excited is substantially larger at the EOT frequency, in agreement with the results of [9]. Following [11] and [12], we have computed the discrete spatial Fourier transform of the magnetic currents in Figs. 3(b) and 4(b) and we have obtained three peaks, one corresponding to the so-called Floquet current (with zero wavenumber in this case since normal incidence is assumed), and the other two corresponding to surface waves propagating along the x axis in oppposite directions (see [11,Fig. 1] and [12, Fig.2]). When these three currents are isolated and combined, one obtains a sinusoidal function on a pedestal (the pedestal being due to the Floquet current, i.e. non-truncated solution), and the distance between maxima provides the wavelength of the magnetic current surface waves. This implies that the wavelength (distance between maxima) of the surface waves shown in Fig.3(b) is roughly 47.5 mm, and that the wavelenght of the surface waves shown in Fig.4(b) is roughly 180 mm.
Although here we restrict our study to the excitation of the aforementioned magnetic current surface waves at normal incidence, one would expect that an obliquely incident plane wave would break the mirror symmetry of the excitation of the pair of surface waves, i.e., the two counter-propagating surface waves would be excited with different amplitudes, plus the non-truncated solution would also consist on a Floquet-wave solution with phasing constant along the surface different than zero. In sum, one would obtain a non-symmetric electric field distribution on the surface of the array, which could lead to interesting field concentrations and which will be subject of further study elsewhere.
In order to justify the results of Figs. 3 and 4, we have obtained the complex propagation constants for the surface waves that can propagate along the x direction of the arrays studied in Fig. 2 when they are assumed to be infinite. These propagation constants can be obtained as the zeros of the determinant of the MoM matrix of the infinite periodic problem for a prescribed frequency. For details concerning the numerical computation of these propagation constants, please refer to [14]. Figs. 5(a) and (b) show the phase constant (real part of the propagation constant) versus frequency (the so-called dispersion diagram) for the arrays studied in Figs.3 and4 respectively. In the case of Fig.5(a), note that surface waves can propagate for frequencies below 6.5 GHz VOLUME 8, 2020 but not beyond that frequency, which justifies why surface waves are not present in Fig. 3(a) at 9.3 GHz, whereas a standing surface wave pattern is clearly present in Fig. 3(a) at 6 GHz. In fact, Fig.5(a) shows the phase constant at 6 GHz is roughly k x a/π = 0.95, which corresponds to a bound mode in the infinite periodic structure with wavelength of λ x = a/0.475 ≈ 47.5 mm, in agreement with the distance between maxima of Fig. 3(b). Concerning Fig. 5(b), one can see the surface waves can propagate for frequencies below 12.5 GHz, which justifies the appearance of a standing wave pattern in both Figs. 4(a) and (b). At the EOT transmission peak, 12.4 GHz, the phase constant is close to zero, which justifies the large wavelength observed in the standing surface wave pattern of Fig. 4(a). This phase constant in the neighborhood of zero indicates the surface wave has become leaky in the infinite periodic structure and radiates in the broadside direction, which is an alternative explanation for the appearance of the EOT phenomenon. Fig.5(b) shows that the phase constant of the surface waves becomes larger as we decrease frequency, and at a frequency of 11.4 GHz, it is roughly k x a/π = 0.25, which corresponds to a wavelength of λ x = a/0.125 = 180 mm, again in agreement with the distance between maxima of Fig.4(b).
To validate the results obtained from the MoM analysis of the truncated periodic arrays, we have performed an experiment in which we have built a periodic array composed of 40×40 slots of length l = 9 mm and width w = 3 mm, which are cut into a 1.5 mm-thick aluminium plate. This plate is illuminated with a standard-gain horn antenna Narda 639 for Ku-band. To accurately results for such experimental setup, the illumination is introduced into the model as shown in Section II. The experimental setup is shown in Fig. 6(a). A NSI planar near field measurement scanner has been used to measure the tangential electric field at a distance roughly h p ≈ 4λ 0 behind the screen. Numerical simulations have shown that the electric field distribution at this position can be used to reliably represent the magnetic currents in the array. Fig.6(b) shows the simulated values of the transmission coefficient defined in (8) as a function of frequency in the Ku-band under the conditions of the experiment for several values of the distance between the horn phase center and the conducting screen, d cF . The results of Fig.6(b) for d cF = 20 m practically mimic those of Fig.2 when l = 9 mm, showing the EOT peak at 12.4 GHz, and the resonant frequency of the slots at roughly 16 GHz (l ≈ λ 0 /2). As the horn approaches the screen, the less uniform the illumination is (the stronger the phase gradient), and the less the transmission is at the EOT peak. Figs. 7(a) and (b) show the experimental and numerical results obtained for the electric field distribution at the EOT peak within the plane z = h p = 9 cm behind the screen. Good agreement is found between the two sets of results. These results indicate the existence of a standing wave pattern with nodes and antinodes along the x direction, which is in agreement with the results of Fig.4

IV. CONCLUSION
In this paper we have explored the remarkable differences between the distribution of magnetic currents in truncated periodic arrays of slots in conducting screens at the resonant frequency of the slots, where only edge effects are apparent, and at the EOT frequency, where magnetic current surface waves are excited. For that purpose, a very efficient MoM code has been implemented for the determination of the magnetic currents in the slots when the conducting screen is illuminated by a pyramidal horn antenna. The dispersion relations of the surfaces waves propagating along the arrays when they are assumed to be infinite allow us to show the conditions for the appearance of the magnetic current surface waves in the truncated arrays. The absence of surface waves at the resonant frequency of the slots and the presence of surface waves at the EOT frequency peak have booth been experimentally demonstrated by measuring the electric field distribution behind a slotted conducting screen with a near-field system. Good agreement has been found between the MoM results and experimental results. Since 1986, he has been with the Electronics and Electromagnetism Department, University of Seville, where he became a Tenured Professor, in 2010. His research interests include efficient numerical analysis of periodic planar multilayered structures with applications to the design of frequency selective surfaces, and reflectarray antennas.
FRANCISCO MEDINA (Fellow, IEEE) was born in Puerto Real, Cádiz, Spain, in 1960. He received the Licenciado and Ph.D. degrees in physics from the Universidad de Sevilla, Seville, Spain, in 1983 and 1987, respectively. He is currently a Professor of electromagnetism with the Department of Electronics and Electromagnetism, Universidad de Sevilla, where he is also the Head of the Microwaves Group. His current research interests include analytical and numerical methods for planar structures, anisotropic materials, artificial media modeling, and planar microwave circuits. He has published a number of book chapters, journal articles, and conference papers on these topics. He is the Editor-in-Chief of the International Journal of Microwave and Wireless Technologies and a member of the editorial board of the International Journal of RF and Microwave Computer-Aided Engineering.