Polaritonic slot waveguides have been explored as a means of manipulating nanoscale fields to compete in the race for the sub-diffractional confinement of light. Hexagonal boron nitride (h-BN), when incorporated into hyperbolic-insulator-hyperbolic (HIH) configurations, is a strong contender, with its naturally occurring anisotropy allowing it to strongly confine and enhance local fields. However, while the volumetric phonon polaritons of h-BN have been widely used for these means, its hyperbolic surface phonon polaritons (HSPhPs) or D’yakonov polaritons contain untapped potential and are widely unused. In this paper, we qualitatively discuss the hybridization of fundamental hyperbolic surface phonon polariton modes in an HIH slot waveguide. The resulting symmetric dark, or lower mode, is then used to design a patch antenn, which shows possibilities for applying the familiar microstrip transmission-line approach of antenna design to this HSPhP antenna.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Among the various quaesita in the field of nanophotonics, sub-diffractional confinement of light has been one of the most coveted. The ability to squeeze light into spaces far smaller than its wavelength allows for interesting applications and further enhances the ability to study other systems such as those in nanotechnology and biomedical imaging [1–5]. Metamaterials that are inhomogeneous at subwavelength dimensions provide the easiest way to manipulate light at sub-diffractional scale. Spearheading one class of such heterostructures is the hybrid waveguide. Hybrid waveguides have been engineered using natural polaritonic materials in the past to achieve high field confinement while maintaining relatively high propagation lengths [6–9]. These hybrid waveguides leverage the inherent properties of plasmon/phonon polaritons to confine and enhance optical energy at sub-diffractional volumes making them attractive in sensing and imaging applications. The advent of hexagonal-boron nitride (h-BN) heralded further improvement with h-BN’s low-loss and anisotropic phonon polaritons (PhPs) allowing for longer propagation with much higher confinement [10–12].
The h-BN crystal is a van der Waals (vdW) material that is naturally hyperbolic. It exhibits two infrared resonances which are classified as the Type I or lower reststrahlen band resonance (ωTO = 760cm−1, ωLO = 825cm−1) and the Type II or upper reststrahlen band resonance (ωTO = 1370cm−1, ωLO = 1614cm−1). While the Type I resonance is associated with an out-of-plane phonon resonance, the Type II resonance is in-plane where it has negative in-plane permittivity and positive out-of-plane permittivity. Here the plane is perpendicular to the optical axis of the h-BN crystal. Implementation of h-BN within the hybrid-waveguide paradigm typically involves arranging h-BN (H) and dielectric insulator (I) components in various combinations including but not limited to HII, IHI and HIH configurations [10–13]. The HIH configuration represents a slot waveguide where the field is confined within the low-index core which is surrounded by high-index cladding, or in this case, high-index hyperbolic waveguides. Conventional dielectric single-slot and multi-slot waveguides are considered superior to traditional high-index wire/slab or rib-strip dielectric waveguides that use total internal reflection as the propagating mechanism [14,15]. Dielectric slot waveguides are amongst the most attractive solutions in applications such as high-speed optical switches [16,17] and high-sensitivity biochemical sensors [18,19] due to high confinement and field enhancement in the low-index slot. HIH slot waveguides, moreover, are amongst the best candidates for boosting this confinement and are analytically and numerically well studied [12,13].
Previous studies, however, primarily used h-BN slabs as the waveguides and considered only the hyperbolic volumetric PhP (HPhP) mode of the hyperbolic waveguides, while the more uncommon hyperbolic surface PhPs (HSPhPs) haven’t seen use. Although not commonly studied, the HSPhPs which are also called D’yakonov polaritons , aren’t elusive and were reported as the dominant resonances associated with finite h-BN waveguides . These are essentially waves that are supported on the interfaces between the hyperbolic anisotropic faces of h-BN waveguides and the surrounding isotropic medium (air). Hillenbrand et al. showed the existence of the fundamental symmetric (SM0-S) and antisymmetric (SM0-A) modes resulting due to the hybridization of the SM0 HSPhP modes on two opposing faces of an h-BN waveguide. In this paper we aim to define the field characteristics of these modes and explain their interactions in an HIH slot-waveguide scenario. After understanding the key features of the various dominant hybrid modes in the HIH slot waveguide, we select the most suitable mode in order to design a novel slot-waveguide based mid-infrared patch antenna that radiates using the traditional TM10 patch mode. In section 2, we present a qualitative understanding of the inherent interactions between the fields of two SM0-S and two SM0-A modes that occur to produce the four hybrid HIH slot-waveguide modes. In section 3, we then present the design of the novel HIH slot waveguide using h-BN that feeds the HIH patch antenna for the purpose of radiating at 1400cm−1. Finally we summarize and present the conclusions in section 4.
2. SM0-S lower mode of h-BN slot waveguide
Due to the uniaxial anisotropy of h-BN, owing to its layered structure, the diagonal permittivity tensor in the type II reststrahlen band of h-BN is given by ɛ̿ = diag(ɛxx, ɛyy, ɛzz) where, in our frame of reference, ɛxx = ɛzz = ɛ⊥ is the permittivity perpendicular to the optical axis and ɛyy = ɛ|| is the permittivity parallel to the optical axis. For the purposes of this paper, the dispersion of h-BN is as calculated in  and the dielectric constant of h-BN is ɛ⊥ = −36.7 and ɛ|| = 2.7 at 1400cm−1. At the surfaces exhibiting in-plane anisotropy, hyperbolic vdW materials such as h-BN support HSPhPs known as D’yakonov surface waves. The existence of HSPhPs in h-BN has been experimentally shown in . Both, the volume-confined hyperbolic polaritons (HPs), and hyperbolic surface polaritons (HSPs) exhibit highly confined fields in vdW materials, where HSPs are confined to the interface between the anisotropic surface and the surrounding isotropic dielectric medium [22–24]. The propagation of HPs in vdW materials depends on the permittivity of the material itself. In contrast to HPs, the propagation of HSPs is not only dependent on the permittivity of the material, but also on the surrounding material . HSPs propagating in hyperbolic material must satisfy the following dispersion relation shown in Eq. (1) [20,25–27].28]. The extraordinary ray propagation-wavevector ke refers to the HSP wavevector.
In  it was shown that a finite width waveguide/antenna of h-BN under illumination with a near-field probe, when observed in the near-field, produced peaks that corresponded to the Fabry-Perot resonances associated with the HSPhP modes of the structure, specifically the fundamental surface mode SM0. The fundamental volumetric waveguide mode M0-W isn’t easily observable, especially at the lower end of the reststrahlen band at 1400cm−1 which is attributed to strong damping of the M0-W modes which propagates less than one polariton wavelength. As the wavenumber increases, the propagation length of the M0-W mode relative to the SM0 mode increases. The SM0 mode describes the fundamental surface mode of a finite-width slab of h-BN. These HSPhPs, or D’yakonov polaritons, exhibit stronger field confinement compared to the volumetric modes of the same mode order . The HSPhPs, due to the finite thickness and width of the h-BN slab, propagate along the Z direction on the YZ surfaces of the h-BN waveguide in the SM0 mode at 1400cm−1 with electric field along the Y-axis on the YZ interfaces. The vertical Ey field is relevant when designing a TM10 patch antenna seen in section 3. The y-component of electric field on the XY cross-section of a semi-infinite h-BN waveguide is shown in the inset of Fig. 1(a). Here, a ‘semi-infinite’ slab refers to an h-BN waveguide of sufficiently large width such that the index of the waveguide remains constant as the width keeps increasing. In Fig. 1(a), the width of the ‘semi-infinite’ h-BN slab is 20µm. The dispersion of the SM0 mode of a semi-infinite h-BN slab with 0.6µm thickness is shown in Fig. 1(a).
Due to electromagnetic coupling of the two SM0 surface modes on the opposing YZ edges of the finite-width slab, hybridized symmetric (SM0-S) and antisymmetric (SM0-A) modes manifest, with the top and bottom corners of the YZ faces showing field enhancement as seen in Fig. 1(inset). Interestingly, the zig-zag rays present on the XY cross-section are due to the hyperbolic polaritonic rays launched from the field-enhanced YZ corners that undergo total internal reflection at the top and bottom of the waveguide [21,24], with the angle θ given by Eq. (2).Fig. 1(b). Figures 2(a) and 2(b) illustrate the normalized spatial distribution of the real part of the y-component of electric field, Re(Ey), for the SM0-S and SM0-A on the XY cross-section of an h-BN waveguide, respectively. The width and thickness of the h-BN waveguide shown in Fig. 2 are W = 1.6µm and H = 0.6µm respectively and the operating wavenumber is 1400cm−1. Note that the finite element mesh inside the h-BN waveguide was made extremely fine in order to clearly show the hyperbolic polaritonic rays on the XY cross-section.
In a conventional dielectric slot waveguide, the enhancement of electric field occurs in the low index slot region of the surrounding high index waveguides, as a result of the discontinuity of the normal displacement-field at the high-low index contrast interfaces, as can be shown using Maxwell’s equations [14,15]. Previous work has been done in designing slot waveguides using hyperbolic metamaterials and h-BN where the volumetric mode is manipulated to form the hybrid slot-waveguide mode in the low-index dielectric core of the waveguide [13,29,30]. As a result, the hyperbolic slot waveguide can support large wavevectors and therefore ultrahigh effective refractive index [31–35]. Here we use the SM0-S mode of the h-BN waveguide due to its ability to confine relatively larger amounts of energy within the slot and due to the fact that the M0-W mode isn’t easily accessible . The results in this paper are computed using ANSYS HFSS finite-element method software. In order to determine the dispersion characteristics and field distribution of the slot-waveguide modes presented in section 2, the Eigen Modal solver of HFSS was used where a unit cell of the waveguide was simulated with periodic master and slave boundaries placed on either end of the waveguide. The slot waveguide considered in this paper consists of two identical h-BN waveguides placed on either side of a low-index dielectric spacer SiO2 with ɛr = 1.15 + j0.011 . This configuration is commonly referred to as a HIH waveguide.
In order to investigate the mode coupling behavior between the fundamental surface modes of the two h-BN waveguides that constitute the hybrid HIH slot-waveguide mode, we first observe the constituent modes in a nearly-isolated scenario where they are barely coupled together. Figure 3 shows Re(Ey) of a slot waveguide with a large separation between the constituent h-BN waveguides. In this case, we selected a large enough separation to display the four possible combinations of the barely-coupled fundamental surface modes. The separation filled with SiO2 shown in Fig. 3 with dashed rectangles. Akin to the previously mentioned symmetric and antisymmetric hybridization of the SM0 modes along the XY and XZ edges of the h-BN waveguide, the SM0-S and SM0-A modes of the two h-BN waveguides also interact with each other undergoing symmetric and antisymmetric hybridization. This phenomenon is analogous to that described in  where the coupling between two plasmonic nanodimers was shown. They showed that for coupling between two on-axis particles, configurations corresponding to symmetric electric fields or bonding modes (positive phase parity) and those corresponding to antisymmetric electric fields or anti-bonding modes (negative phase parity) exist. They also highlighted an interesting feature where, as the separation between the two particles grows larger, the nth order mode energies (in eV) or in other words the nth order mode frequency of the bonding and anti-bonding modes are not split as far apart compared with smaller separations between the particles. This behavior is also seen in the HIH slot waveguide considered in this paper. In other words, for very small separation, the indexes of the modes seen in Fig. 4 are much further apart than in the same four modes seen in Fig. 3. In various works dealing with the coupling of two resonances, the resulting symmetric and antisymmetric modes have previously been called bright and dark, bonding and anti-bonding, and even and odd, respectively. Here we choose to refer to the hybrid symmetric and antisymmetric slot-waveguide modes as upper and lower modes, respectively, because they correspond to upper and lower mode frequencies (smaller and larger indexes), respectively. Figures 3(a) and 3(b) show interactions leading to the lower mode and upper mode involving the SM0-S mode, respectively, while Figs. 3(c) and 3(d) show the lower mode and upper mode involving the SM0-A mode, respectively.
Decreasing the separation to 50nm increases the coupling between the h-BN fundamental surface modes giving rise to the four modes known as SM0-S lower, SM0-S upper, SM0-A lower and SM0-A upper as shown in Figs. 4(a), 4(b), 4(c) and 4(d), respectively. As was previously seen in , for an h-BN waveguide of a given size, the SM0-S mode has a higher index than the SM0-A mode. In Fig. 4, for a slot thickness = 50nm, H = 0.6µm and W = 1.6µm, the indexes of the hybrid slot-waveguide modes follow the trend SM0-S lower = 1.69 > SM0-S upper = 1.08 > SM0-A lower = 0.81 > SM0-A upper = 0.43. The propagation lengths of each mode are SM0-S lower = 14.8µm, SM0-S upper = 75.1µm, SM0-A lower = 659µm and SM0-A upper = 4022µm. The attenuation constants of each mode are SM0-S lower = 33629Np/m, SM0-S upper = 6652Np/m, SM0-A lower = 757Np/m and SM0-A upper = 124Np/m. The insets of Fig. 4 show the magnetic field vectors (red arrows) and electric field vectors (blue arrows) of each hybrid mode. As observed from the inset of Fig. 4(a), majority of the electric field vectors of the SM0-S lower mode reside within the SiO2 slot, aligned parallel to the Y-axis with relatively small fringe electric fields external to the slot. The magnetic field vectors only exist in the XY-plane and within the slot they are aligned parallel to the X-axis. The characteristics of both the electric and magnetic field vectors of the SM0-S lower mode in the slot resemble a transverse magnetic mode. This mode is the ideal candidate in order to couple into a patch-cavity mode which will be presented in the next section. In contrast to the electric field distribution of the SM0-S lower mode, the SM0-A lower mode has electric field of opposite phase on either end of the slot waveguide as seen in Fig. 4(c). This impedes a patch antenna’s ability to radiate as it produces destructively interfering fields on the radiating edges (edges parallel to XY plane) of the patch and is an unsuitable candidate mode for the slot waveguide. Meanwhile, it is clear that the SM0-S upper and SM0-A upper hybrid slot-waveguide modes are also unsuitable due to the phase cancellation of the electric field within the slot, despite the high propagation lengths and low attenuation losses of these modes.
As shown in Fig. 4(a), the electric field of the hybrid SM0-S lower mode is highly confined in the slot with modal areas on the order of 10−2λ2 0. Figure 5(a) shows the normalized energy density of the hybrid SM0-S lower mode (red plot) compared with the hybrid volumetric slot-waveguide mode (black plot) along y, x = 0 (dashed line in inset) for T = 50nm and H = 0.6µm.The shaded orange and blue areas represent the SiO2 and h-BN regions, respectively. When comparing the energy density of the hybrid SM0-S lower mode present in the SiO2 slot versus that in the h-BN volume, the electromagnetic energy density stored in the h-BN volume is almost 80% less than that stored in the SiO2 layer. For hybrid SM0-S lower mode, the fields existing in the h-BN volume are due to the penetration from the field enhancement on the interface between h-BN and the medium containing the source [24,28] which is different from traditional plasmonic slot waveguides . In our case, the field penetration into the h-BN volume (blue area) is due to the enhancement produced in the slot.
Compared with the hybrid SM0-S upper, SM0-A upper and SM0-A lower slot-waveguide modes, the hybrid SM0-S lower mode shows the largest field confinement where the confined mode in the slot is a transverse magnetic mode with electric field aligned vertically (Y direction) and magnetic field aligned horizontally (X direction) shown in Fig. 5(b). As seen in previous works [13,29], the HIH slot waveguide with hybrid volumetric modes has field-vector alignment similar to the slot shown in Fig. 5(c), in contrast to that of the hybrid SM0-S lower mode. However, the field confinement in the slot for the case of the hybrid SM0-S lower mode (red curve) is up to 70% larger than in the case of the hybrid volumetric mode (black curve) for a given set of slot-waveguide geometry, as seen in Fig. 5(a), owing to a larger distribution of the mode energy into the h-BN volume in the latter case as well as due to the fact that HSPhPs in h-BN show larger field-confinement than that of the h-BN HPhPs .
In order to determine the appropriate h-BN slot waveguide dimensions, Fig. 6(a) provides the dispersion of a semi-infinite h-BN slot waveguide when H = 0.6µm with the hybrid SM0-S lower mode from 1380cm−1 to 1520cm−1. The inset shows normalized Re(Ey) distribution of the SM0-S lower mode for the semi-infinite h-BN slot waveguide at 1400cm−1. Figure 6(b) shows the real effective index, neff, when the thickness of SiO2 layer is set as T = 50nm. The thickness of h-BN, H is then varied from 0.6µm to 1.6µm as shown by the different colored lines, versus the width of the slot waveguide, W1 varied from 1µm to 5µm. For a single h-BN waveguide of a given thickness, as the width is increased, the index of the volumetric M0-W mode, increases and converges to the index of an infinite-width h-BN slab mode M0 of the same thickness. Similarly, as the width of the h-BN slot waveguide increases, the index of the SM0-S lower slot-waveguide mode, neff converges to that of a semi-infinite slot waveguide of h-BN capable of supporting the SM0-S lower mode, nsemi-inf. Equation (3) describes the resulting relationship between nsemi-inf and neff. Function f(H, W1) is the geometric fitting equation calculated from the data points in Fig. 8(b) by using the curve fitting tool in MATLAB. In function f(H, W1), parameters W1 and H are in µm. As observed from the curve when H = 0.6µm, the index converges to the value of semi-infinite slot waveguide as the width increases.
3. H-BN slot-waveguide fed hyperbolic surface phonon polariton patch antenna
Here we propose a novel h-BN slot-waveguide fed hybrid HSPhP patch antenna operating at 1400cm−1. In this section, we used the Driven Modal solution-type in ANSYS HFSS wherein a waveport was used to feed the slot waveguide and radiation boundaries were placed at around a quarter of the free-space wavelength away from the entire structure. The goal is to produce broadside or near-broadside radiation from a planar SPhP antenna in the mid-infrared spectrum. Here we choose to couple the hybrid h-BN slot waveguide into an HIH-type h-BN patch antenna i.e. both the slot waveguide and the patch antenna consist of a thin SiO2 layer (orange) sandwiched between two h-BN waveguides (blue) with optical axis (OA, indicated by a red arrow) as shown in Fig. 7. In a conventional rectangular patch antenna on a grounded substrate, the antenna assumes a cavity model in which the substrate “cavity” is bounded by electric walls at the top and bottom and four magnetic walls along the edges of the patch . This cavity model isn’t unlike the “cavity” formed by the SiO2 slot bounded at the top and bottom by the h-BN waveguides. A traditional microstrip patch antenna with an in-plane feed radiates due to the fringing fields at the discontinuities formed by the patch in the feeding direction (edges parallel to the X axis) due to the resonant wave formed along the longitudinal axis of the patch (Z axis). These discontinuities give rise to “radiating slots” that are formed between the top and bottom walls of the cavity and the mode is a TM10 mode . In the cavity model, the waveguide/transmission-line feed must thus provide a transverse magnetic mode in order to transform into the radiating TM10 mode. Similarly, our HIH h-BN patch antenna supports the radiating TM10 mode as shown in Fig. 9(a) and the feeding h-BN slot waveguide also supports a transverse magnetic field distribution provided by the hybrid SM0-S lower mode as shown in Fig. 4(a).
The thicknesses of both the slot waveguide and the HSPhP patch antenna are H = 0.6µm. The widths of the slot waveguide and patch antenna are W1 = 1µm and W2 = 1.9µm, respectively, while the length of the patch antenna is L = 2.12µm. The thickness of SiO2 slot is set to T = 50nm. Figure 8 compares the normalized real part of the electric field in the Y direction, Re(Ey), and in the Z direction, Re(Ez), at the center of the SiO2 layer i.e. along x = 0, y = 0, z, as a function of z, i.e. from the feed towards the patch antenna along the feeding slot waveguide. The magnitude of Re[Ez(z)] is close to zero along the propagation direction (Z axis). When plotting along one wavelength of the SM0-S lower mode, i.e. λeff = 4.24µm, it can be seen that the magnitude of Re[Ez(z)] is much smaller than that of Re[Ey(z)] as shown in Fig. 8. Therefore, the mode propagating down the slot in the slot waveguide can be approximated as a TEM mode . The TEM mode in the SiO2 layer validates the use of a transmission line model for designing and analyzing the h-BN slot waveguide and h-BN patch antenna.
The mode propagating in the slot is determined to be a TEM mode, with the characteristic impedance of the waveguide in TEM mode calculated analytically. If Z0 is the characteristic impedance of air, and neff = 1.67 is the effective index of the slot waveguide for H = 0.6µm, W1 = 1µm and T = 50nm, (from Eqs. (3) and (4)), then the input impedance of the slot waveguide is calculated to be ZWaveguide = 225.7Ω using the equation :
Applying transmission line theory at the resonant frequency 42THz (1400cm−1), the input impedance of the patch antenna when fed from an in-plane transmission-line type feed, operating in the dominant TM10 mode is calculated in Eqs. (6) and (7) :43]. By selecting W2 = 1.9µm and T = 50nm, the matched input impedance of the patch antenna is calculated to be 225.5Ω. The required half-wave resonant length, L = 2.12µm with neff = 1.67 is found from the effective wavelength in Fig. 6(b). Figure 9 shows normalized Re(Ey) distribution in the slot of the hybrid SM0-S lower waveguide fed h-BN patch antenna. It can be seen that the electric field Ey(z) along the two transverse edges of the patch are of opposite phase. The electric field vectors shown in the inset of Fig. 9 describe the typical cavity fields of a radiating patch with fringing fields that give rise to TM10 radiation.
Figure 10(a) shows the normalized far-field radiation pattern field of the h-BN slot-waveguide fed h-BN patch antenna for T = 50nm, H = 0.6µm, W1 = 1µm, W2 = 1.9µm and L = 2.12µm at 1400cm−1. In contrast to the conventional microstrip patch antenna, there is no ground plane underneath the proposed h-BN patch antenna which results in a radiation pattern that appears in both + Y and –Y directions. The peaks of the upward and downward major lobes in the radiation pattern are shifted slightly off the Y axis due to the antisymmetric structure and fringing fields in Z direction. The absence of the ground plane results in a bidirectional ( + Y and -Y) radiation pattern that enables multilayer radiation transfer and emission .
Figure 10(b) illustrates the return loss, S11 as a function of frequency for the h-BN slot waveguide fed h-BN patch antenna for T = 50nm, H = 0.6µm, W1 = 1µm, W2 = 1.9µm and L = 2.12µm. S11 appears to be lowest at 1400cm−1 which indicates good impedance matching between the h-BN slot waveguide and the h-BN patch antenna at the resonant wavenumber 1400cm−1. The radiation efficiency of the antenna is 61.2%. This validates that the proposed analytical impedance model is suitable for designing a hybrid SM0-S lower h-BN slot-waveguide fed h-BN patch antenna radiating in the TM10 mode.
A novel hybrid h-BN slot-waveguide fed mid-infrared (HIH-type) HSPhP patch antenna is designed. The hybrid h-BN slot waveguide and the patch antenna consist of an SiO2 layer sandwiched between two h-BN waveguides. The fundamental symmetric and antisymmetric hyperbolic surface phonon polariton modes of the h-BN waveguides, SM0-S and SM0-A formed on the anisotropic interfaces of the h-BN waveguides are used as the ingredients of the hybrid slot-waveguide modes. Combinations of two SM0-S or two SM0-A modes give rise to upper and lower slot-waveguide modes viz. the SM0-S upper, SM0-S lower, SM0-A upper and SM0-A lower modes. By “upper” we refer to modes with positive phase parity while “lower” refers to negative phase parity of the electric fields of the combining SM0-S or SM0-A modes. Moreover the upper modes are formed at higher frequencies than the lower modes or conversely, the lower modes have larger wavevectors than the upper modes. The mode indexes follow the trend: SM0-S lower > SM0-S upper > SM0-A lower > SM0-A upper. It was also shown that, in order to achieve radiation from the HSPhP patch antenna, the SM0-S lower mode is the ideal candidate due to the relatively highly-confined transverse magnetic field within the slot. Additionally, the more commonly used [13,29], hybrid volumetric slot-waveguide mode was shown to confine up to 70% less electric field in the slot when compared with the SM0-S lower slot-waveguide mode. Finally, this paper proposes an analytical approach to selecting the geometry of the h-BN slot-waveguide fed HSPhP patch antenna by providing an analytical impedance matching approach between the h-BN slot waveguide and the patch antenna. The analytical approach uses microstrip transmission line theory to calculate the input impedance of the h-BN slot waveguide and h-BN patch antenna. A radiation efficiency of 61.2% with an S11 of −13.39dB is achieved at 1400cm−1. While section 2 introduced the various hybrid HIH slot-waveguide modes using h-BN, which are highly attractive in applications such as optical switches [16,17] and biochemical sensors [18,19], section 3 exploited one of these modes to design and tune a novel mid-infrared patch antenna while at the same time plying the traditional microstrip design methods.
Publication of this article was funded in part by the Open Access Subvention Fund and the Florida Tech Libraries.
The authors acknowledge Michael F. Finch and Edriss Mirnia of the Applied and Computational Electromagnetics Lab at Florida Institute of Technology and Dr. Joshua Caldwell of Vanderbilt University for their valuable insight and discussions.
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