Plasma modification of spoof plasmon propagation along metamaterial-air interfaces

The control of electromagnetic (EM) waves at surface interfaces is of particular interest for various applications including microwave electronics and devices,^1–3 photonics,^4–8 metamaterial-based switches,⁹ and metamaterial-integrated devices.^10–14 These surface waves, known as surface plasmon polaritons (SPPs), travel at metal-dielectric interfaces with propagation dependent on the dielectric constants of the interface materials.¹⁵ SPPs are especially of interest because of their highly localized nature with sub-wavelength confinement at the interface. The unique way that SPP devices manipulate EM waves enables their application for optoelectronic circuits.¹⁴ Since evanescent fields extend beyond the interfaces, SPPs have also been used for chemical and biological sensing. In this paper, we demonstrate the manipulation of SPPs propagating along a metamaterial-air interface using plasma discharges positioned near the interface.

SPPs propagate because of the high negative permittivity of metals, allowing for surface plasmon resonance frequencies within the optical regime.^5,16 Microwave (<100 GHz) SPPs cannot be naturally excited at metal-dielectric interfaces due to the near perfect conducting behavior of metals at these lower frequencies.¹⁶ Patterned metal-dielectric metamaterial structures, sometimes referred to as slow-wave structures, have been designed to support interface propagation at lower frequencies. These “spoof” surface plasmons exhibit wave characteristics and dispersion similar to optical SPPs.⁵ The structures generally consist of patterned arrays of sub-wavelength scale combs that contribute capacitive and inductive elements that exhibit behavior mimicking coaxial transmission lines.^1,2 Due to the sub-wavelength structure, surface waves propagating along the metamaterial-air interface have reduced phase velocity that varies as a function of frequency and structure geometry.

The EM wave propagation characteristics of the metamaterial described here are determined entirely by the metal's periodic structure and surrounding dielectric properties. The SPP propagation can be manipulated by dynamically altering the device geometry; however, changes in geometry are not practical for high frequency applications. Propagation can also be manipulated by incorporating materials of tunable dielectric constants. Gaseous discharge plasmas are of particular interest as potentially tunable elements for manipulating spoof SPPs. The dielectric constant of a plasma varies significantly in the vicinity of its plasma frequency, which is determined by the electron density. Gas discharge plasmas have plasma frequencies that lie conveniently in the range of 0.1–100 GHz or higher, well in the range of microwave frequencies relevant to many engineering applications. In this letter, we describe the use of plasma elements in shifting the dispersion of spoof SPPs that propagate on a slow wave structure with resonance frequency in the C-band (4–8 GHz) of the EM spectrum.

The metamaterial device studied here is a 2D metal-on-dielectric comb structure designed with periodic copper teeth that contribute capacitances and inductances to the EM wave impedance, as illustrated in Fig. 1(a). This structure has been shown to propagate spoof SPPs with high confinement [see ANSYS simulations in Fig. 1(b)] on both rigid and thin film dielectric substrates.^12,14

Transmission line circuit theory provides a good initial approximation of the dispersion of the comb microstrip structure.¹⁷ Periodicity of the microstrip allows for the modeling of the entire strip with a simplified single unit cell with repeating boundary conditions. Each “U-shaped” unit cell corresponds to two halves of adjacent teeth and the spine of the comb structure. The device was modeled and simulated using Keysight Advanced Design Systems (ADS) 2017 as a unit cell with corresponding inductances and capacitances calculated from methods described by Kianinejad et al.¹⁷ Unit cell impedances, Z₁₁ and Z₂₁, were determined from S parameters simulated in ADS. The propagation constant β was calculated assuming a lossless network and from components of the transmission ABCD matrix¹⁸ 

The unit cell is symmetric, and so, A = D and the propagation constant is

A graph of β as a function of frequency gives the dispersion for the transmission line, shown in Fig. 2, with real solutions of the dispersion terminating after $βd/π=1$⁠. A low density plasma with plasma frequency less than the SPP resonance frequency has a dielectric constant $0<εp<1$⁠. If positioned near the surface of the structure, it displaces the air (⁠$ε=1$⁠) with a fluid of lower dielectric constant (thereby reducing the capacitance). The effect of this plasma can be captured, qualitatively, as a scalar coefficient $α<1$ multiplied with all capacitances shown in Fig. 2. This coefficient causes an overall decrease in unit cell capacitance, which shifts the dispersion curve to higher frequencies. However, a high density plasma with a plasma frequency greater than the SPP resonance frequency will exhibit a negative dielectric constant $εp<0$⁠. The high density plasma will exist in a regime of greater inductive impedance than capacitive impedance, thereby affecting the microstrip's inductive elements in addition to its capacitive elements. With a conductive plasma nearby, each unit cell will experience some mutual inductance with the resonant eddy currents in the plasma. A transformer model derived from Maxwell's equations can be applied to simulate the coupling between a low pressure plasma and induction coils. This model involves connecting the plasma's inductive impedance to the driving circuit via a single loop transformer.^19,20 Since the plasma is externally driven in our case, we modified the model to include the plasma's inherent inductance instead of the dissipative impedance. To model this resonance, an ideal transformer with a turn ratio, T, is introduced in parallel with L_mid and tuned to a ratio of 0 < T < 1 to simulate the dynamic coupling to the plasma as it approaches the unit cell. Connected to the other side of the transformer is the plasma, with estimated inductance L_plasma = 1 nH. In Fig. 2, the blue line represents the calculated dispersion without the plasma (α = 1 and T = 0), which has a spoof plasmon resonance frequency of 6.17 GHz. The green line is the dispersion with low density plasma (α = 0.95 and T = 0), with a resonance frequency of 6.33 GHz. The red line is the dispersion with high density plasma (α = 0.95 and T = 0.3) and has a resonance frequency of 6.41 GHz. We see that by decreasing the overall capacitance of the unit cell by 5%, the resonance frequency increases by 160 MHz, and by increasing the transformer ratio to 0.3, the resonance frequency increases further by 80 MHz.

Preliminary ANSYS HFSS simulations (discussed in more detail below) reveal that plasma close to the surface of the microstrip causes similar upward shifts in the spoof SPP resonance, agreeing with ADS simulations. They also show that the Fabry-Pérot (FP) modes of the finite length microstrip shift to higher frequencies with increased plasma density as well. Below, we describe an experiment that verifies this upward shift in resonance frequency.

The metamaterial microstrip transmission line of study consists of 33 periodic teeth that form the copper comb structure etched from a double sided copper plated FR-4 board. A copper ground plane covers the bottom of the device. As shown in Fig. 1(a), the 163 mm long comb structure is designed with a repeating structure scale of d = 5 mm, a strip width of w = d, a tooth height of $h=0.8d$⁠, and a separation of $a=0.4d$⁠, similar to the device parameters used by Shen.¹² The thicknesses of copper and FR-4 are 0.03 mm and 1.39 mm, respectively. We recorded the microstrip's S parameters using a vector network analyzer (VNA HP8722D) connected via SMA ports with center pins soldered to the top structure and grounded with the microstrip's ground plane.

Figure 3(a) shows a schematic of the experimental setup comprising the discharge ballasts, plasma lamps, and VNA. The plasma layer is represented by an array of six 15 mm diameter quartz-enveloped discharge lamps positioned approximately 0.5 mm above the comb structure, with the entire array arranged perpendicularly to the microstrip. The quartz (⁠$ε=3.82$⁠) tube has an inner wall thickness of 1 mm and a length of 290 mm, containing mercury and filled with argon to a pressure of 250 Pa. Each discharge was individually driven by an AC ballast to output a triangular voltage waveform with a peak to peak voltage of 160 V. To control the discharge parameters, we changed the ballast output current (also close to triangular in waveform) by varying the input AC voltage from the power supply. The ballast frequency decreases linearly from 55.0 kHz to 37.0 kHz for peak currents from 24.8 mA to 51.2 mA and increases from 32.2 kHz to 33.8 kHz for peak currents from 54.4 mA to 111.2 mA. Figure 3(b) shows a top-down image of the microstrip and discharge array with the plasmas on. The entire device was placed on a block of microwave-transparent foam.

Simulations of the device and its performance were carried out using the commercially available finite element method solver ANSYS HFSS 16.1. The plasma was represented as a uniform rectangular volume, with a height of z = 22 mm, a length of x = 162 mm, and a width of y = 22 mm centered 2 mm above the microstrip. A 1 mm quartz layer (⁠$ε=3.82$⁠) was positioned between the plasma and the microstrip to model the effect of the bulbs' quartz envelope. To account for the complex nature of the EM response of the plasma slab, we use a Drude model for the frequency-dependent plasma dielectric constant

Here, γ is the plasma electron collision frequency (taken to be 1 GHz, reasonable for a discharge plasma in predominantly argon at a pressure of 250 Pa) and ω_p is the plasma frequency

In Eq. (4), e is the electron charge, m_e is the electron mass, and $εo$ is the permittivity of free space. Simulations were carried out for two plasma densities: low $(ne=3.1×1011 cm−3)$ and high $(ne=7.8×1011 cm−3)$⁠, corresponding to plasma frequencies of $ωp=3.1×1010 rad/s$ (5 GHz) and $5.0×1010 rad/s$ (7.93 GHz), respectively, and completed using the first order basis function solver with convergence criteria of maximum 1% change in S-parameters between adaptive passes. The low plasma density simulation gives a representation of the effect with a positive permittivity plasma $(ε<1)$ of similar density to the experimental setup, and the high plasma density case simulates the effect of a negative permittivity plasma $(ε<0)$⁠.

The simulated S₁₁ and S₂₁ and plasma densities considered are shown in Fig. 4. Apparent in the S₁₁ graphs (top panel) are the longitudinal Fabry-Pérot (FP) resonances below the spoof plasmon resonance frequency (computed to be approximately 7.7 GHz in these ANSYS simulations). These FP modes are associated with the finite length of the microstrip, with a spacing between modes (no plasma case) of approximately 0.33 GHz at low frequency (note that $εp<0$ below 5 GHz for the low density case and 7.93 GHz for the high density case), increasing slightly for higher frequencies, indicative of a slowing down of the phase velocity as the plasmon resonance is approached. It is noteworthy that the frequency of these modes increases with increasing plasma density, consistent with an upward shift in the resonance frequency. The upward shift in frequency is most clearly seen in the simulated S₂₁ (middle panel) where the resonance (transmission cut-off) appears at approximately 7.74 GHz for the no plasma condition, 7.81 GHz for $ne=3.1×1011 cm−3$⁠, and 7.91 GHz for $ne=7.8×1011cm−3$⁠. An expanded scale of the region near the spoof SPP resonance is given in the bottom panel of the figure.

The simulated S₂₁ at frequencies below 4.5 GHz indicates the presence of some attenuation due to absorption caused by finite plasma collisionality, as it is not seen as enhanced reflection in the S₁₁ signal. Also noticeable in the S₂₁ panel is the strong and sharp attenuation band at 3.9 GHz for the high plasma density case. This is not seen in the experiments simply because the simulated plasma and quartz layers uniformly extend across the whole length of the microstrip, which causes excitation of surface plasmon propagation at the plasma-quartz interface. At these attenuated frequencies, the plasma dielectric constant is negative, while the quartz dielectric constant is positive, allowing for SPP excitation. The surface plasmon resonance for propagation at a semi-infinite planar interface between collisionless plasma and dielectric (of dielectric constant ε) is given by

Substituting values of the properties of the plasma and quartz, we find that this theory predicts surface plasmon resonance at 3.61 GHz, right before the drop in S₂₁ computed in the simulations. By examining the electric field distributions in the ANSYS simulations, we confirmed that low frequency waves below 4 GHz couple into surface plasmon modes of the plasma-quartz interface and attenuate before reaching the receiving wave port. After resonance where there are no real propagation modes and S₂₁ dips, the waves couple more strongly to the plasma-quartz layer than to the microstrip but decay soon along the interface. S₂₁ recovers after 3.93 GHz when the phase velocity reaches zero and allows the plasma-quartz surface to sustain a standing wave, forcing all wave propagation through the microstrip. This unexpected narrow band of 50 dB attenuation demands further study as it can be easily applied to microwave transmission lines as a non-reflecting power limiter.

Figure 5 shows the experimental reflection and transmission spectra and the effect of the added plasma elements on the surface of the SPP microstrip. There is good overall qualitative agreement between the measured S₁₁ and S₂₁ and those computed using ANSYS HFSS. The top panel of Fig. 5 also shows the presence of the characteristic FP modes, albeit less pronounced. With the plasma on, the shifts to higher frequency in the reflection minima are very small, indicative of a very low plasma density in the lamps (⁠$ne<3.1×1011 cm−3$⁠). The transmission spectra shown in the middle panel confirm some enhanced attenuation with the plasma below 5 GHz, attributed to dissipation due to electron collisionality. The middle panel of Fig. 5 shows the measured S₂₁ and depicts the upward shift in resonance frequency, while the lower panel is an expanded scale showing a clearer shift in resonance. We see that the shift in frequency is approximately 100 MHz, further confirming that plasma densities in the range of the evanescent fields are much lower than those used for the simulations in Fig. 4.

The plasma discharge operation was chosen to reproduce the highest plasma density conditions in the study by Wang and Cappelli.²¹ In that study, a single discharge plasma tube was used to detune a photonic crystal defect. The measured detuning, when compared to ANSYS simulations, suggested that the lamp's cross-sectional average plasma density was $ne≈3.5×1011 cm−3$⁠, similar to the lower values used in our simulations of Fig. 4. We expect that the plasma density within the discharge tube is strongly radially varying, with a substantial drop in density near the quartz envelope. The low shifts seen in the experimentally measured reflection and transmission may be a consequence of the short EM field penetration depth of only 1 mm into the bulk of the plasma beyond the quartz layer. Nevertheless, the measured shifts are at least a weak confirmation of the anticipated role played by the plasma elements in the surface wave propagation. Although our experiment featured plasma bulbs that are relatively large compared to the microstrip device, plasmas of even higher densities can be generated in much smaller microdischarges that cover only targeted regions of the metamaterial transmission line. Such discharges, if designed to be in direct contact with the microstrip, would result in a more robust manipulation of the plasmon resonance feature as it would eliminate the quartz envelope which limits the plasmas in proximity to the structures.

This work was supported in part by a Multidisciplinary University Research Initiative from the Air Force Office of Scientific Research with Dr. Mitat Birkan as the program manager. B.W. was also supported by a National Defense Science and Engineering Graduate Fellowship.