Spoof plasmon radiation using sinusoidally modulated corrugated reactance surfaces

Anastasios H. Panaretos; Douglas H. Werner

doi:10.1364/OE.24.002443

1. Introduction

Reactance surfaces (RSs) are well known for their ability to support TM surface waves (SWs) [1]. Plasmons are a very prominent class of SWs excited along a RS defined at the interface between two dielectric media with opposite sign in their permittivity [2]. This type of SWs, also referred to as surface plasmon polaritons (SPPs), have played a pivotal role in the advancement of modern plasmonics/nanophotonics, since they allow the confinement and manipulation of light at deep subwavelength scales [3–5]. The existence of optical SPPs is a direct consequence of the fact that at these frequencies noble metals are characterized by negative dielectric permittivity, therefore SPPs may be excited along their surface.

More recently, researchers have also examined and successfully demonstrated the possibility of scaling the theory and techniques originally developed for optical plasmonics to the THz and microwave frequency ranges. The main difficulty in accomplishing this task has been the fact that noble metals could not be utilized since at these lower frequencies they behave more like good conductors. The alternative was to utilize RSs realized with metallic (good/perfect conductor) corrugated structures [6–9]. A planar corrugated surface consists of a ground-plane with a periodic arrangement of transversely infinite vertical grooves defined along its length. Similarly, a corrugated cylindrical surface consists of a metallic rod with a periodic arrangement of axial grooves defined along its symmetry axis. Such surfaces can support SPP-like surface waves usually referred to as spoof-plasmons (SPs). It should be noted that among the electromagnetics community the properties of such structures have been thoroughly studied as early as the 1950’s, and their applicability to either guide or radiate electromagnetic waves has been well demonstrated [10–15]. As expected, the revived interest in corrugated surfaces has led to a plethora of research publications demonstrating novel electromagnetic devices that exploit the appealing properties of SPs [16–29].

As mentioned previously, SPs are inherently guided waves therefore they can transfer information between two distant points connected through an engineered corrugated surface. In this paper a design methodology is described that allows a corrugated surface to be converted into a leaky wave structure, thereby enabling the efficient coupling of SPs into free-space radiating modes. The proposed approach is based on the fact that any SW can be converted to a leaky wave if a periodic perturbation is introduced along the guiding structure [30]. More specifically, Oliner and Hessel in [31] showed that if a sinusoidally modulated reactance perturbation is introduced along a RS, then under certain conditions the latter can support leaky wave modes. This scenario is directly applicable in the case of a corrugated surface since the latter is a RS. Therefore, it is expected that if a sinusoidal reactance perturbation is introduced along a corrugated surface (which can be effectively realized by judiciously varying the groove depths) then the supported SPs can be converted into radiating modes.

Sinusoidally modulated reactance surfaces (SMRSs) provide a compact design methodology to create leaky wave antennas (LWAs). Their popularity stems from the fact that there is a closed form expression which can be exploited to characterize their dispersion properties (see Appendix). Therefore, the designer can control not only the angle of the LWA’s major radiation lobe, but also its beamwidth as well as its leakage rate. For these reasons, several sophisticated and versatile LWA designs utilizing SMRSs have been recently proposed, which operate in the microwave as well as in the THz frequency range [32–38]. In this paper we contribute to the existing SMRS-LWA designs by demonstrating how this design methodology can be extended in the case of planar and cylindrical corrugated surfaces. It should be emphasized that although the analysis presented herein is laid out for structures operating in the microwave regime, it is directly applicable in the THz frequency range, where it is expected that the proposed LWAs will provide attractive alternatives for the realization of highly efficient THz sensors. Throughout this paper the $e^{j ω t}$ time convention has been adopted. All the numerical results reported herein have been obtained using ANSYS-HFSS.

2. Planar corrugated surfaces

A. Sinusoidally modulated LWAs

In this section the operating principles of LWAs that utilize SMRSs are briefly presented. As mentioned previously, it is well-known that spatially uniform RSs can support TM guided SWs, therefore they are characterized by a real valued wavenumber $β_{u} > k_{0}$ , where $k_{0}$ is the free space wavenumber. If we introduce a periodic reactance perturbation of period $p$ along this surface, then according to Floquet theory the resulting perturbed reactance surface can be characterized by a modified wavenumber $κ$ such that [2]

κ \equiv β_{u} - j α + \frac{2 π n}{p} = β_{n} - j α

In the preceding expression, the

2 π n / p

term represents the adjustment in the propagation constant due to the n-th Floquet spatial harmonic;

n

is an integer number indicating the order of the corresponding Floquet mode. Note here that the realization of a leaky wave structure requires that this is characterized by a propagation wavenumber that attains values smaller than that of free space. In the case of a periodic reactance surface, such a scenario becomes feasible since for some negative value of n the condition

β_{u} - 2 π n / p < k_{0}

may be satisfied (loosely speaking a positive quantity needs to be subtracted from

β_{u}

so that the latter becomes less than

k_{0}

). Finally, the imaginary part of the wavenumber corresponding to

α

denotes the leakage rate of the potentially radiating periodic RS.

A subset of the periodic RS based LWAs are those that utilize periodic SMRSs. The latter are characterized by a non-uniform periodic impedance profile given by

Z (y) = j X_{u} [1 + M \cos (\frac{2 π y}{p})]

In particular, the impedance value of the surface fluctuates sinusoidally with a spatial period

p

around a mean or un-modulated impedance value

j X_{u}

(which is characterized by a purely real wavenumber

β_{u} > k_{0}

). In Eq. (2)

M

denotes the modulation factor which in principle can attain any value lower than one. For such structures it has been shown [30] that the n = −1 Floquet mode is responsible for their radiation, while the elevation angle

θ_{s}

of their major radiating lobe is determined by

Re {κ} = β_{- 1} \Rightarrow k_{0} \sin (θ_{s}) = β_{u} - \frac{2 π}{p}

B. Dispersion analysis

In this study the RS of the LWA is realized through a planar metallic corrugated structure. The geometrical details of such a structure are shown in Fig. 1. The length of a single corrugation is equal to G + T, while the thickness of a tooth is equal to T and the depth of the grooves is equal to $l$ . The material constitution in the grooves as well in the surrounding region is defined as free space.

Fig. 1 Planar corrugated surface and equivalent transmission line representation. Wave guidance is assumed along the y-direction. The structure is infinite along the x-direction.

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With respect to the structure shown in Fig. 1, the guided wavenumber $k_{y}$ supported by such a structure can be determined in the following way. We begin by considering the input impedance of a corrugated surface, which is given by [39]

Z_{i n} = j Z_{0} \frac{G}{G + T} \tan (k_{0} l)

Note here that from the preceding expression (which is basically that of a short-circuited TEM transmission line) it can be concluded that the upper frequency for which the corrugated surface can support SPs corresponds to the λ/4 resonance of the groove. Now, again with respect to Fig. 1, the free space TM input impedance is given by

Z_{u p} \equiv Z_{0}^{T M} = \frac{k_{z}}{ω ε_{0}}

where

k_{z}

is the transverse wavenumber in the space over the corrugated surface. Since we are considering only guided waves along the corrugated surface,

k_{y}

is purely real and greater than

k_{0}

. For this reason

k_{z}

is purely imaginary or

k_{z} = - j \sqrt{k_{y}^{2} - k_{0}^{2}}

Given the previous set of definitions, the objective becomes to determine the longitudinal guided wavenumber

k_{y}

. This is achieved by numerically solving the non-linear equation that the transverse resonance method yields as applied at the interface between the corrugated surface and free-space [40], or

Z_{i n} + Z_{u p} = 0 \Rightarrow Z_{0} \frac{G}{G + T} \tan (k_{0} l) - \frac{\sqrt{k_{y}^{2} - k_{0}^{2}}}{ω ε_{0}} = 0

Once

k_{y}

has been determined, the goal is to design a corrugated impedance surface that exhibits the properties of some desired SMRS. This task basically entails judiciously choosing the parameters of the impedance model in Eq. (2) that essentially determine the radiation properties of the LWA.

In this design we set the unit-cell periodicity equal to $p$ = 30 mm, and at 9 GHz it is desired to direct the main radiating lobe at an angle $θ_{s}$ = 30°. Given these specifications, the propagation constant along the unmodulated surface can be computed from Eq. (3) and set equal to $β_{u} = 1.61 k_{0}$ . Now, from Eq. (2) it becomes evident that within a period $p$ there should be a continuous sinusoidal distribution of reactance values. However, with a corrugated surface the varying impedance values can be represented only at a finite number of discrete spatial points. After some numerical experimentation it was concluded that 15 sub-unit-cells suffice to accurately represent the SMRS variation. In particular, the length of each sub-unit-cell is set equal to $T + G$ = 2 mm, while the depth of the groove is set equal to $T$ = 0.5 mm.

Subsequently Eq. (7) was numerically solved to determine the groove depth that yields a wavenumber equal to that for an un-modulated planar corrugated surface. In Fig. 2, indicated by the solid line, the longitudinal wavenumber variation at 9 GHz is plotted for different tooth lengths. The dashed line in the plot indicates the desired wavenumber value, which according to the map of Fig. 2, can be realized with a groove depth equal to l = 5.49 mm. From Eq. (4) this length corresponds to a mean surface reactance (or to an un-modulated surface reactance) value equal to X_u = 1.26Z₀.

Fig. 2 Longitudinal wavenumber along a corrugated surface at 9 GHz as a function of tooth length when T = 0.5 mm and G + T = 2 mm. The dash-line indicates the desired un-modulated wavenumber value.

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The next step in the design process is to determine the length of the teeth for the 15 sub-unit-cells that constitute the discrete realization of one SMRS period. In this study we set the modulation factor to $M = 0.3$ . In principle, $M$ can assume any value less than one, while its magnitude determines the beamwidth of the main radiating lobe. Subsequently, we evaluate the impedance in Eq. (2) at the sequence of points: $y_{m} = (p / 15) m$ , with m = 0:1:14. These reactance values are then mapped to a sequence of tooth lengths as dictated by Eq. (4). The set of tooth lengths required to realize the desired SMRS profile across the fifteen sub-unit-cells, in order to achieve radiation at 30°, were determined to be: [5.0932, 4.8005, 4.6207, 4.6207, 4.8005, 5.0932, 5.4107, 5.6874, 5.8916, 6.0141, 6.0546, 6.0141, 5.8916, 5.6874, 5.4107] × 1 mm. Note here that all these lengths are smaller than the one corresponding to the λ/4 resonance of the corrugation, which is equal to l = 8.33 mm.

The spatial distribution of the groove depths along two periods is shown in Fig. 3(a). Similarly, in Figs. 3(b) and 3(c) the longitudinal wavenumber and the surface reactance variation are displayed, respectively. It should be emphasized that locally the longitudinal wavenumber corresponds to purely guided waves, since its value is always greater than that of free space. Note that in all three plots of Fig. 3 the LW sections follow a waveguide (WG) section. This corresponds to a uniform corrugated surface characterized by the un-modulated longitudinal wavenumber $β_{u} = 1.61 k_{0}$ . More details about this WG section will be provided in the next section. In this study the reactance modulation was realized by varying the depth of the grooves. In principle, however, the desired non-uniform reactance could be achieved by varying any of the parameters in Eq. (4), namely the thickness or the filling material of the grooves.

Fig. 3 Planar SMRS. (a) Groove depth variation as a function of length. (b) Longitudinal wavenumber variation as a function of length. (c) Reactance variation as a function of length. Note that only two periods of the SMRS are shown. The LW section follows a 10 mm long WG section that is characterized by a longitudinal wavenumber equal to that corresponding to the un-modulated case.

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C. Planar LWA performance

The computational HFSS model of the planar LW structure is shown in Figs. 4(a) and 4(b). It is comprised of 25 unit-cells (25 sinusoid periods) while there are waveguide sections located at its two ends, each with a length of 10 mm. These sections are designed so that they exhibit a longitudinal wavenumber equal to that of the un-modulated impedance surface, or $β_{u} = 1.61 k_{0}$ . The corrugated surface is excited at its two ends by open TEM waveguides.

Fig. 4 Planar corrugated LWA utilizing a SMRS. (a) Perspective view. (b) Source detail. Red color indicates the open TEM waveguide source. Green color indicates the waveguide section comprised of an un-modulated corrugated surface. Blue color corresponds to the leaky wave section of the antenna.

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Also, the surface along the transverse direction appears infinite, therefore the two side-walls of the computational domain are defined as perfectly magnetic conductors (PMCs). The radiating performance of the structure is summarized in Fig. 5. In particular, Fig. 5(b) displays the realized gain of the leaky wave structure for three different frequencies as a function of elevation angle. First, it can be seen that at 9 GHz the main radiating lobe is directed at 33°. Also, it can be seen that as the operating frequency sweeps from 8.5 GHz to 9.5 GHz, the major lobe of the antenna scans from 20° to 55°. The $S_{11}$ and $S_{21}$ performance for this LWA are plotted in Fig. 5(a), while Fig. 5(c) reports its radiation efficiency. For completeness it should be noted here that the parameter |S₁₁|² indicates the ratio of the power reflected back to Port #1 over the power supplied to it. Similarly |S₂₁|² is the ratio of the power delivered to Port #2 over the power supplied to Port #1. Moreover, the radiation efficiency is the ratio of the power radiated by the antenna over the power supplied to the antenna.

Fig. 5 (a) Reflection and transmission coefficients. (b) Realized gain. (c) Radiation efficiency. (d) Leakage rate.

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Finally, Fig. 5(d) shows the leakage rate of the LWA as computed (a) by directly solving the dispersion relation (see Appendix) and (b) by utilizing the approximate perturbation based formula reported in [30, Eq. (18)]. The two approaches for the frequency range of interest yield the same leakage rate, while the later remains relatively constant within the frequency range under consideration.

3. Cylindrical corrugated surfaces

A. Dispersion analysis

A cylindrical corrugated surface consists of a metallic cylindrical rod with radius $R_{o u t}$ where radial grooves with depth $l = R_{o u t} - R_{i n}$ are defined along its surface. A longitudinal cross-section of such a structure is shown in Fig. 6. Similar to the planar case, the objective is to define a sinusoidally modulated impedance profile along the surface of the rod so that leaky waves are supported. We begin our analysis by determining the longitudinal wavenumber $k_{z}$ along the surface of the rod. With respect to Fig. 6, the transverse resonance condition on the surface of the structure, along the radial direction $ρ$ , is given by:

Z_{ρ}^{+} + Z_{ρ}^{-} = 0

Assuming that the TM₀₁ mode suffices to describe the electromagnetic field configuration created by the corrugated rod, the impedance expressions in Eq. (8) are represented by [39]

Z_{ρ}^{+} = j \frac{k_{ρ}}{ω ε_{0}} \frac{H_{0}^{(2)} (k_{ρ} R_{o u t})}{H_{1}^{(2)} (k_{ρ} R_{o u t})}

and

Z_{ρ}^{-} = - j Z_{0} \frac{G}{G + T} \frac{J_{0} (k_{0} R_{o u t}) Y_{0} (k_{0} R_{i n}) - J_{0} (k_{0} R_{i n}) Y_{0} (k_{0} R_{o u t})}{J_{1} (k_{0} R_{o u t}) Y_{0} (k_{0} R_{i n}) - J_{1} (k_{0} R_{i n}) Y_{0} (k_{0} R_{o u t})}

In the preceding expressions

H_{n}^{(2)} (•)

is the n-th order Hankel function of the second kind, while

J_{n} (•)

and

Y_{n} (•)

are the Bessel functions of the first and second kind of order n. Also,

k_{ρ}

is the radial wavenumber and, since we are interested in guided surface waves along the axis of the rod, this quantity should be purely imaginary, such that

k_{ρ} = - j \sqrt{k_{z}^{2} - k_{0}^{2}}

where

k_{z} \in ℜ

and

k_{z} > k_{0}

.

Fig. 6 Cylindrical corrugated impedance surface. (a) Perspective view. (b) Longitudinal cross-section. Guidance is assumed along the z-direction. The structure is axi-symmetric with respect to the z-axis.

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The dashed line in Fig. 7(a) shows the response of the longitudinal wavenumber as a function of frequency. In this example we have set T = 0.5 mm and G + T = 2 mm. Also R_in = 4 mm and R_out = 10 mm. The same figure shows the response (solid line) of the longitudinal wavenumber along a planar corrugated surface with the same geometrical characteristics as the cylindrical one (T, G, and l). Although the two responses are different, it is straightforward to show that the values of the wavenumber along the cylindrical rod converge to those of the planar corrugated surface when $R_{o u t}$ becomes significantly larger than the groove depth $l$ .

Fig. 7 (a) Longitudinal wavenumber response along a planar and a cylindrical corrugated surface with the same geometrical characteristics. (b) Longitudinal wavenumber response along a cylindrical corrugated surface as a function of the groove depth. Dashed line indicates the desired un-modulated wavenumber.

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As in the planar case it is desired to achieve directive radiation at $θ_{s}$ = 30°. As before from Eq. (3), the propagation constant along the un-modulated surface is computed when set equal to $β_{u} = 1.61 k_{0}$ . This wavenumber corresponds to a groove length l = 4.16 mm as indicated in Fig. 7(b). The wavenumber response shown in Fig. 7 was derived by numerically solving Eq. (8).

The next step in the design process is to determine the length of the teeth for the sequence of the 15 cylindrical corrugated sub-unit-cells. As in the planar case, we set the modulation factor $M = 0.3$ , and we evaluate the impedance in Eq. (2) at the sequence of points: $y_{m} = (p / 15) m$ , with m = 0:1:14. Then these reactance values are mapped to tooth lengths as dictated by Eq. (10). The set of tooth lengths required to realize the desired SMRS profile across the 15 sub-unit-cells in order to achieve radiation at 30° were determined to be: [4.1038, 3.8573, 3.6335, 3.4974, 3.4974, 3.6335, 3.8573, 4.1038, 4.3224, 4.4863, 4.5858, 4.6190, 4.5858, 4.4863, 4.3224] × 1 mm. The spatial distribution of the cylindrical groove depths across a length of two periods is shown in Fig. 8(a). Similarly, in Figs. 8(b) and 8(c) the longitudinal wavenumber and the surface reactance variations are plotted, respectively. Again, it can be seen that locally the longitudinal wavenumber corresponds to purely guided waves, since its value is always greater than that of free space

Fig. 8 Cylindrical SMRS. (a) Groove depth variation as a function of length. (b) Longitudinal wavenumber variation as a function of length. (c) Reactance variation as a function of length. Note that only two periods of the SMRS are shown. The LW section follows a 10 mm long WG section that is characterized by a longitudinal wavenumber equal to the corresponding un-modulated case.

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B. Cylindrical LWA performance

Figures 9(a) and 9(b) illustrate the computational HFSS model of the cylindrical SMRS LWA. It is comprised of 30 unit-cells (30 sinusoid periods) while SPP waveguide sections are located at its two ends, each with a length of 10 mm. These sections are designed so that they exhibit a longitudinal wavenumber equal to that of the un-modulated impedance surface, or $β_{u} = 1.61 k_{0}$ . The outer radius of the corrugated rod is set equal to R_out = 10 mm. The corrugated rod is excited at its two ends by open coaxial waveguides. Axial symmetry is imposed in the computational model by terminating one quadrant of the computational domain with PMC walls as shown in Fig. 9(b). The radiating performance of the LWA is summarized in Fig. 10. In particular, Figs. 10(c) and 10(d) display the realized gain of the LWA for three different frequencies, in the E- and H-planes, respectively.

Fig. 9 Cylindrical corrugated LWA utilizing a SMRS. (a) Perspective view. (b) Source detail. Red color indicates the open coaxial waveguide source. Green color indicates the waveguide section comprised of an un-modulated corrugated surface. Blue color corresponds to the leaky wave section of the antenna.

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Fig. 10 (a) Reflection and transmission coefficients. (b) Radiation efficiency. (c) Realized E-plane gain. (d) Realized H-plane gain.

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In particular, from Fig. 10(c) it can be seen that at 9 GHz the main radiating lobe is directed at 61°. This is the complementary of the desired radiating angle because it corresponds to the elevation angle which is defined with respect to the z-axis, and not with respect to an axis perpendicular to the surface. Also, it can be seen that as the operating frequency sweeps from 8.5 GHz to 9.5 GHz, the major lobe of the antenna scans from 70° to 50° (20° to 40°). Due to the structure’s cylindrical symmetry and coaxial waveguide excitation, the E-plane radiation patterns are symmetric with respect to the z-axis (θ = 0°). Figure 10(d) shows the realized gain in the H-plane for the previous three frequencies, and it can be clearly seen that on this plane the radiation pattern is circularly symmetric. Finally, the $S_{11}$ and $S_{21}$ performance for this LWA are shown in Fig. 10(a), while Fig. 10(b) reports its radiation efficiency.

4. Conclusion

We have numerically demonstrated the feasibility of devising LWAs that utilize sinusoidally modulated corrugated surfaces. The proposed methodology relies on the effective mapping of a sinusoidal reactance variation to a sinusoidal groove depth variation. When this sinusoidal groove depth variation is applied along a corrugated surface the resulting structure supports leaky waves and therefore it can efficiently couple guided spoof plasmons into radiating modes. The design methodology was demonstrated in the case of planar as well as cylindrical corrugated surfaces, where the theoretically expected radiating characteristics for both configurations were numerically verified. The proposed methodology provides a compact approach towards the synthesis of structurally simple LWA antennas that can operate either in the microwave regime or in the THz frequency range. It should be mentioned here that in the microwave frequency range corrugated surfaces that support spoof plasmons have been fabricated and their performance has been experimentally verified in several cases such as reported in [25]. In the THz regime, however, the fabrication of such structures becomes more intricate. Nevertheless, several studies have been performed that clearly demonstrate the successful excitation of THz spoof plasmons [41–44].

Appendix A

The dispersion relation of a SMRS with the non-uniform impedance profile described by Eq. (2) can be expressed in the following continued fraction expansion

D (n, 0) - \frac{1}{D (n, + 1) - \frac{1}{D (n, + 2) - \dots}} - \frac{1}{D (n, - 1) - \frac{1}{D (n, - 2) - \dots}} = 0

where

D (n, m) \equiv \frac{2}{M} [1 - \frac{j}{X_{u}} \sqrt{1 - {(\frac{γ}{k_{0}} + \frac{2 π (n - m)}{k_{0} p})}^{2}}]

In the preceding expression

γ \equiv β - j α

is the complex propagation constant. The dispersion relation is solved numerically with respect to

γ

using Davidenko’s method, after setting

n = 0

and

m = 5

. Figure 11(a) displays the propagation constant

β_{n}

for modes

n = 0, - 1, - 2

. It can be clearly seen that part of modes

n = - 1

and

n = - 2

lie within the light triangle. Also, with the horizontal magenta lines we indicate the frequency range of interest (8 GHz to 10 GHz), while the square and diamond markers indicate the propagation constant (and thus the radiating angle) of the two radiating modes corresponding to

n = - 1

and

n = - 2

, at 9 GHz.

Fig. 11 Dispersion diagram of a SMRS with M = 0.3. (a) Propagation constant β_n for modes n = 0, −1, and −2. (b) Attenuation constant.

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Finally, the attenuation constant response is plotted in Fig. 11(b). Around $k_{0} p \approx 2$ the structure exhibits a bandgap and therefore the values of $α p$ are increased dramatically. Also, note that around $k_{0} p \approx 4$ , where broadside radiation occurs, $α p$ is discontinuous thus revealing the poor radiation characteristics of the LWA. Again, in Fig. 11(b) we indicate with the horizontal magenta lines the frequency range of interest. In this frequency range, the attenuation constant response, computed using the approximate formula given in [30, Eq. (18)], is also depicted. A zoomed-in version of this plot is shown is Fig. 5(d).

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Spoof plasmon radiation using sinusoidally modulated corrugated reactance surfaces

Abstract

1. Introduction

2. Planar corrugated surfaces

A. Sinusoidally modulated LWAs

B. Dispersion analysis

C. Planar LWA performance

3. Cylindrical corrugated surfaces

A. Dispersion analysis

B. Cylindrical LWA performance

4. Conclusion

Appendix A

References and links

Cited By

Figures (11)

Equations (13)

Optics Express