Proceedings. Mathematical, Physical, and Engineering Sciences

The Royal Society Publishing

Dielectric-induced surface wave radiation loss

Volume:
476,
Issue: 2236

DOI 10.1098/rspa.2019.0859

Abstract

We investigate a model which shows how the introduction of a perturbing dielectric close to an electromagnetic surface wave leads to radiation away from the surface through the dielectric. This resembles a surface waveguide passing through a wall or being deployed underground. Our theory, which is based on the mode-matching technique, allows quantitative determination of losses from a bound surface wave mode up to the point of its complete extinction. For a surface wave supported by a coated, conducting sheet the attenuation due to the perturbing dielectric is calculated for a number of frequencies, permittivities of the perturbation and separations between the sheet and the perturbing dielectric. The accuracy of our results is verified by simulation of the system with a full-wave numerical solution. Finally, we report experimental data of perturbed surface waves on a cable, which are in qualitative agreement with our model.

Keywords

Electromagnetic surface waves in open waveguide structures are a means of transmitting signals in the GHz and THz range with low loss and dispersion compared with other closed waveguide structures [1–3]. Surface waveguides can, for instance, be single wires with finite conductivity [4] or perfectly conducting wires that have a dielectric sheath or are corrugated [5]. However, because these systems are open, the surface waves carried on them are susceptible to disturbances by the surrounding environment.

A number of previous works have studied the effect of different coatings and media surrounding a coated wire or related systems [6,7]. John and Chatterjee [8] reported that if the dielectric constant of the surrounding medium is higher than the dielectric constant of the coating the surface wave solution ceases to exist. However, if we imagine that the dielectric medium is moved very far away from the conductor and the opening gap is filled with vacuum or air, we would expect the surface wave to propagate again.

The question of how a neighbouring dielectric affects a propagating surface wave at intermediate distances between the dielectric and the surface is investigated in this paper. An approach based on the mode-matching technique [9] is developed and employed. To simplify the calculations, a one-dimensional system, i.e. an infinite, perfectly conducting plane coated with a lossless dielectric as discussed by Attwood [10], is considered as an example. However, qualitatively similar results would be expected for a cylindrical wire and other geometries. This is supported by experimental data using a cylindrical cable at varying distances from the ground. Hence, the conclusions of this paper can be used to model and understand the behaviour of a surface waveguide passing through a wall or being used underground.

We first recapitulate the mode-matching method and state the approximations used in our theory. Consider a general system of two dissimilar waveguides (1 and 2) on either side of the *z* = 0 plane separated by a discontinuity. In this system the *z*-axis is the axis of wave propagation. The transverse components of any electric and magnetic field on either side of the discontinuity can be expanded in terms of incoming and outgoing eigenmodes of the relevant waveguide in the form [9]

${\mathit{E}}_{t}^{(i)}=\sum _{m}{a}_{m}^{(i)}{\mathit{e}}_{m}^{(i)}\mathrm{exp}(-j{\beta}_{m}z)+{b}_{m}^{(i)}{\mathit{e}}_{m}^{(i)}\mathrm{exp}(j{\beta}_{m}z)$

and
${\mathit{H}}_{t}^{(i)}=\sum _{m}{a}_{m}^{(i)}{\mathit{h}}_{m}^{(i)}\mathrm{exp}(-j{\beta}_{m}z)-{b}_{m}^{(i)}{\mathit{h}}_{m}^{(i)}\mathrm{exp}(j{\beta}_{m}z).$

Here, At the discontinuity at *z* = 0, the transverse fields must be continuous

${\mathit{E}}_{t}^{(1)}{|}_{z=0}={\mathit{E}}_{t}^{(2)}{|}_{z=0}$

and
${\mathit{H}}_{t}^{(1)}{|}_{z=0}={\mathit{H}}_{t}^{(2)}{|}_{z=0}.$

Let ${\hspace{0.17em}}^{\ast}$ denote complex conjugation. Then, integrating the vector product of equation (2.3) with ${\mathit{h}}_{n}^{(2)\ast}$ and of ${\mathit{e}}_{n}^{(2)\ast}$ with equation (2.4) over the entire waveguide cross section at $\sum _{m}2{P}_{mn}^{(2)}({a}_{m}^{(2)}+{b}_{m}^{(2)})=2\sum _{m}({a}_{m}^{(1)}+{b}_{m}^{(1)}){I}_{mn}$

and
$\sum _{m}2{P}_{mn}^{(2)\ast}({a}_{m}^{(2)}-{b}_{m}^{(2)})=2\sum _{m}({a}_{m}^{(1)}-{b}_{m}^{(1)}){J}_{mn},$

where we used the eigenmode expansion of the fields. The complex power ${P}_{mn}^{(i)}$ and the interaction integrals $\begin{array}{rl}& {P}_{mn}^{(i)}=\frac{1}{2}\int \hspace{0.17em}\text{d}S({\mathit{e}}_{m}^{(i)}\times {\mathit{h}}_{n}^{(i)\ast})\hat{z},\end{array}$

$\begin{array}{rl}& {I}_{mn}=\frac{1}{2}\int \hspace{0.17em}\text{d}S({\mathit{e}}_{m}^{(1)}\times {\mathit{h}}_{n}^{(2)\ast})\hat{z}\end{array}$

$\begin{array}{rl}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}& {J}_{mn}=\frac{1}{2}\int \hspace{0.17em}\text{d}S({\mathit{e}}_{n}^{(2)\ast}\times {\mathit{h}}_{m}^{(1)})\hat{z},\end{array}$

where $\hat{z}$ is the unit vector in the Using the orthogonality of modes assuming no contribution of complex modes, ${P}_{mn}^{(i)}$ reduces to a constant multiplied by a Kronecker delta in the discrete and a delta function in the continuous case [11,12]. We may write this as

${P}_{mn}^{(i)}={\delta}_{mn}{P}_{n}^{(i)},$

with ${P}_{n}^{(2)}{a}_{n}^{(2)}={a}_{1}^{(1)}{I}_{1n}+\sum _{m}{b}_{m}^{(1)}{I}_{mn}$

and
${P}_{n}^{(2)\ast}{a}_{n}^{(2)}={a}_{1}^{(1)}{J}_{1n}-\sum _{m}{b}_{m}^{(1)}{J}_{mn}.$

This means that even for a single mode input we end up with an infinite set of equations as both In the next step, we apply the developed method to the one-dimensional surface waveguide described by Attwood: a perfectly conducting plane covered by a dielectric coating of thickness *y*_{1} and permittivity *ϵ*_{1} surrounded by a medium with dielectric constant *ϵ*_{2}. This waveguide is perturbed over a length *L* by another dielectric with dielectric constant *ϵ*_{3} positioned at a distance *y*_{2} from the conductor. A sketch of the system can be seen in figure 1. We will assume that the unperturbed waveguide is operated at a frequency where only a single bound mode (denoted by subscript 1) can propagate and are interested in the amplitude of this mode beyond the perturbed region. Using the method described above at both discontinuities between the perturbed and unperturbed waveguide, we find that the amplitudes of the bound mode before and after the perturbation, ${a}_{1}^{\text{in}}$ and ${a}_{1}^{\text{out}}$, are related by

${a}_{1}^{\text{out}}\approx \sum _{m}{a}_{1}^{\text{in}}\frac{{I}_{1m}{J}_{1m}^{\ast}}{{P}_{1}^{(1)}{P}_{m}^{(2)}}\mathrm{exp}(-j{\beta}_{m}L).$

It is easy to show that for the case The bound modes of the unperturbed guide have been discussed by Attwood [10]. However, if we introduce the perturbing dielectric with *ϵ*_{3} > *ϵ*_{1} there are no lossless, bound TM modes, as we show in appendix A. Thus, we only need to consider radiating and evanescent modes. Omitting the wave propagation factor of exp (*j*(*ωt* − *βz*)) these are given by the following fields:

$\begin{array}{rl}{E}_{z}& =\{\begin{array}{ll}{E}_{1}\mathrm{sin}({h}_{1}y)\phantom{\rule{1em}{0ex}}& 0\le y\le {y}_{1},\\ {E}_{2}\mathrm{sin}({h}_{2}y+\phi )& {y}_{1}\le y\le {y}_{2},\\ {E}_{3}\mathrm{sin}({h}_{3}y+\psi )& {y}_{2}\le y,\end{array}\end{array}$

$\begin{array}{rl}{H}_{x}& =j\omega \{\begin{array}{ll}{E}_{1}\frac{{\u03f5}_{1}}{{h}_{1}}\mathrm{cos}({h}_{1}y)& 0\le y\le {y}_{1},\\ {E}_{2}\frac{{\u03f5}_{2}}{{h}_{2}}\mathrm{cos}({h}_{2}y+\phi )& {y}_{1}\le y\le {y}_{2},\\ {E}_{3}\frac{{\u03f5}_{3}}{{h}_{3}}\mathrm{cos}({h}_{3}y+\psi )& {y}_{2}\le y,\end{array}\end{array}$

$\begin{array}{rl}{E}_{y}& =-j\beta \{\begin{array}{ll}{E}_{1}\frac{1}{{h}_{1}}\mathrm{cos}({h}_{1}y)& 0\le y\le {y}_{1},\\ {E}_{2}\frac{1}{{h}_{2}}\mathrm{cos}({h}_{2}y+\phi )& {y}_{1}\le y\le {y}_{2},\\ {E}_{3}\frac{1}{{h}_{3}}\mathrm{cos}({h}_{3}y+\psi )& {y}_{2}\le y\end{array}\end{array}$

$\begin{array}{rl}\text{and}\phantom{\rule{1em}{0ex}}{h}_{i}& =\sqrt{{\omega}^{2}{\u03f5}_{i}{\mu}_{0}-{\beta}^{2}},\end{array}$

where φ and $\frac{1}{2}{\int}_{0}^{\mathrm{\infty}}({\mathit{E}}_{{h}_{3}}\times {\mathit{H}}_{{h}_{3}^{\prime}}^{\ast})\hat{z}\hspace{0.17em}\text{d}y={E}_{3}{E}_{3}^{\prime \ast}\frac{\pi \beta \omega {\u03f5}_{3}}{4{h}_{3}{h}_{3}^{\prime \ast}}\delta ({h}_{3}-{h}_{3}^{\prime}),$

where all primed quantities correspond to ${\mathit{H}}_{{h}_{3}^{\prime}}^{\ast}$.In the case where *ϵ*_{1} > *ϵ*_{3} there can be a bound mode in addition to the radiation and evanescent modes described above. The existence of a bound mode depends on whether there is a solution to the characteristic equation (A 5) derived in appendix A. The fields are given there as well.

With the mode structure described here and in [10] the interaction integrals in equations (2.8) and (2.9) can be calculated analytically. As we have shown, the modes of the perturbed guide are completely defined for a given value of *h*_{3}. So, the output amplitude can be calculated by expressing the sum in equation (2.13) as an integral over *h*_{3} for the radiation and evanescent modes and by adding the contribution of any bound modes if they exist. Because of the increasingly fast decay of the evanescent modes in the *z*-direction with increasing *h*_{3}, the integral can be truncated at a sufficiently large value of *h*_{3}. Thus, we only need to calculate a finite number of modes and can then interpolate in between them to find the behaviour of our system. In our calculations we used of the order of $\mathcal{O}({10}^{5})$ to $\mathcal{O}({10}^{6})$ excited modes. By calculating the power contained in the bound surface wave mode after the perturbing dielectric, it can be established how strongly the perturbation affects the bound mode. This allows us to assess if the bound mode may be used for signal transmission in more realistic scenarios.

To obtain quantitative understanding of the energy losses in a real system, we consider the attenuation of a surface wave propagating along a waveguide consisting of a conductor coated with Teflon in air perturbed by glass. The dielectric constants of Teflon, air and glass relative to the vacuum permittivity *ϵ*_{0} are taken as *ϵ*_{1}/*ϵ*_{0} = 2.1, *ϵ*_{2}/*ϵ*_{0} = 1 and *ϵ*_{3}/*ϵ*_{0} = 5.5, respectively. The conductor and the dielectrics are assumed to be lossless throughout. Surface wave frequencies in the GHz range were considered.

First, we investigate the effect of the perturbing dielectric at different distances *y*_{2} from the conductor at a fixed surface wave frequency of 10 GHz. The results are presented in figure 2. As we would expect, at very large distances of the perturbing dielectric from the conductor there is little to no attenuation of the surface wave. However, as the dielectric gets closer, energy is lost from the surface wave to radiation modes in the dielectric. This can lead to complete attenuation of the surface wave. We can think of the perturbing dielectric as opening a radiation channel allowing energy to propagate away from the surface. Hence, the surface wave loses energy. In addition to the average decrease of surface wave energy, we see a modulation on the output power with the length of the perturbation *L*. This can be attributed to the phase differences accrued by the multitude of excited modes while propagating along the perturbed waveguide section as each mode has a unique propagation constant *β*(*h*_{3}). More specifically, the excited radiation modes have different phases *β*(*h*_{3}) *L* due to propagation causing interference between these modes. This affects the amount of power that can return into the bound surface wave after the perturbation.

In a next example, we keep the perturbation at a constant distance of *y*_{2} = 2 cm from the waveguide and vary the frequency of the incident surface wave. We expect a weak influence at high frequencies when the surface wave is tightly bound to the conductor and a stronger influence with decreasing frequency. This is exactly the behaviour predicted by our theory, as can be seen in figure 3. To validate our results and, hence, the accuracy of the approximations introduced to solve equations (2.5) and (2.6), we calculated the transmission characteristics of the same system with the finite element solver Ansys^{®} HFSS^{TM} [13]. Details of the HFSS model can be found at the end of this paper. We see good agreement between the simulations for high frequencies. As the frequency decreases, our approximation of negligible reflection becomes less valid and we observe stronger deviations between our results and those obtained using HFSS. In particular, the sharp peaks seen in the HFSS simulation are not found in our calculation.

Finally, we look at the influence of the dielectric constant of the perturbing medium on the loss, leaving both the surface wave frequency and distance of the perturbation *y*_{2} constant at 15 GHz and 2 cm, respectively. Under these conditions, a bound mode in the perturbed system is possible if *ϵ*_{3}/*ϵ*_{0} < 1.13. In figure 4, we show the transmitted power for different dielectric constants. The results indicate that at high dielectric constant *ϵ*_{3} the average loss of the surface wave depends only weakly on the exact value of the dielectric constant. Only the interference pattern and the loss for short perturbations is influenced. At lower values of the permittivity *ϵ*_{3}, close to the onset of the bound mode and below, loss is reduced to zero as *ϵ*_{3} approaches the dielectric constant of the surrounding medium *ϵ*_{2}.

To experimentally verify some predictions of this model, a 20 m long cable was suspended above ground at a height of 100 mm [14]. The cable was a Dropwire 11 with a single, 0.5 mm diameter annealed copper twisted pair with polyethylene insulation of thickness 0.215 mm for each conductor [15]. The twisted pair was contained in a polyethylene sheath of nominal diameter 5.3 mm and the steel strength members usually contained in the cable were removed. Surface waves were launched and received using horn launchers of 5 cm depth and 5 cm maximum cross-section. A signal generator was used as the source and a spectrum analyser as the receiver. The midpoint of the cable was brought close to the ground at distances of 100 mm (straight cable), 20 mm and 5 mm. As a benchmark the cable was also entirely suspended at 1.5 m above ground. The received power in these measurements is displayed in figure 5. The input power for all measurements was 0 dBm. As the horn launchers were not altered during the measurement, we believe their losses and launching efficiency to be constant throughout. The results in figure 5 show that at a given frequency the received power is reduced when the cable is brought close to the ground. Second, the results indicate that as the frequency increases the behaviour of a cable near the ground approaches the behaviour of a cable further from the ground. Both of these results are in qualitative agreement with the results presented in figures 2 and 3. Quantitative agreement cannot be established owing to the different geometries of the considered systems. Note that, because of the size of the experiment, measurements were carried out outside where the set-up was subjected to sources of radiation such as Wi-Fi, TV and cellular communications. As the observed spikes at low frequencies in figure 5 are at standard network carrier frequencies, we attribute them to our cable picking up some of this radiation.

As we have shown, surface waves are susceptible to disturbances of the surrounding medium. In fact, such disturbances can lead to the complete attenuation of a propagating wave. This occurs because of the perturbing dielectric opening a radiation channel into which the surface wave loses energy. Our model assumes negligible reflections, which should be accurate under the conditions that the perturbing dielectric is far away from the conductor. However, if the perturbation is close to the conductor, reflection can play an increasingly important role.

Quantifying *a priori* when reflections are small is not easy. One way to approach the problem is to consider the transverse decay constant $\kappa =\sqrt{{\beta}_{1}^{2}-{\omega}^{2}{\u03f5}_{2}{\mu}_{0}}$ of the unperturbed, bound mode, which describes how fast the fields decay away from the surface of the waveguide (*β*_{1} is the propagation constant of the bound mode [10]). In cases where *κy*_{2} ≫ 1 the fields have essentially decayed at the position of the perturbing dielectric and, so, there should not be any reflections. Even for values of *κy*_{2} ≈ 1 only about 13% of the power transmitted outside the coating is in the region *y* > *y*_{2}. Thus, a sensible condition seems to be *κy*_{2} ≥ 1. In the examples seen in figure 3, which compares our model with a numerical solution, values of *κy*_{2} rounded to the second decimal place are 0.94 and 2.14 for 10 GHz and 15 GHz, respectively. Finally, it should be stressed that *κy*_{2} ≥ 1 is indicative of weak reflections but does not necessitate them.

Another important factor is mode-matching, meaning that both the perturbed and the unperturbed waveguide support modes with similar propagation constants. In our case the incident, unperturbed wave will have a propagation constant between $\omega \sqrt{{\u03f5}_{2}{\mu}_{0}}$ and $\omega \sqrt{{\u03f5}_{1}{\mu}_{0}}$. When *ϵ*_{3} > *ϵ*_{1} the perturbed waveguide always supports radiation modes in that region because all propagation constants below $\omega \sqrt{{\u03f5}_{3}{\mu}_{0}}$ belong to the radiating spectrum. However, in the case where *ϵ*_{3} < *ϵ*_{1} there is the possibility of poor matching. In particular, if *ϵ*_{3} is close to the threshold where the bound mode ceases to exist, a gap between the unperturbed propagation constant and the propagation constants of the perturbed bound and radiating modes is found to exist. Therefore, matching in this region of the dielectric constant *ϵ*_{3} might be poor, reflections can be strong and our model may produce inaccurate results.

As discussed, the developed model is valid under the condition of no or weak reflections. So, deviations from it are bound to increase as more power gets reflected at the discontinuities of the system. More specifically in the case considered above, the excited mode structure inside the perturbing dielectric region will deviate from the one calculated under the assumption of no reflections. Additionally, resonances which are due to re-reflections in the perturbing dielectric interfering constructively or destructively with the transmitted light are not included either. Within our approximation of negligible reflections their effect should be small. But, as reflections increase, they will play an important role for the exact description of the resulting transmission. This may explain the deviations seen for the 10 GHz wave in figure 3 between the presented model and numerical calculations.

Despite these restrictions, the measured data are in support of the presented model showing the predicted behaviour with frequency and distance from the perturbation. However, we acknowledge that losses may also increase simply because of increased dielectric loss inside the ground when the surface wave is brought into close proximity. Furthermore, the loss associated with the sagging of the line has not been considered separately. Thus, further measurements including radiation tests with different perturbing dielectrics are necessary for the verification of the model.

In summary, our results show a potential limitation of using surface waves for data transmission along cables. Specifically, use of the surface wave either underground or close to the ground seems impractical for large propagation distances unless the wave decays quickly enough in the transverse direction such that no perturbing medium nearby can influence it. On the one hand, our model shows that most dielectrics will cause some loss to the wave. On the other hand, it emphasizes that at increased frequencies these effects become weaker because the bound mode is more confined. Another way to enhance confinement is to increase the thickness of the dielectric coating. Therefore, careful design considerations need to be made when deploying surface waves as transmission lines.

In conclusion, we have derived a method to calculate the loss a perturbing dielectric induces on a bound surface wave by opening a radiation channel. The method is based on the approximation that reflections can be neglected and otherwise uses the well-established mode-matching technique. Having derived the mode structure of the perturbed, one-dimensional, coated, conducting sheet, we used our equations to simulate the effect of such a perturbation. We predict losses up to total attenuation depending on the exact geometry and frequency used. Independent quantitative verification of our results using a finite element solver is established. Furthermore, qualitative agreement with measurement is reported.

The HFSS model consisted of a solution domain (*x*,*y*,*z*; see figure 1) of dimensions 20 × 7 × 15 cm at 10 GHz and 15 × 3.5 × 15 cm at 15 GHz. The extension of the domain in the *x*-direction was chosen to be much larger than the wavelength to approximate the infinite model. Excitation in the *x*–*y*-planes was achieved with perfectly matched wave ports with 100% excitation efficiency. The conductor was modelled as a perfect electrical conductor boundary and all other boundaries were implemented as radiation boundaries. A dielectric box was implemented as the insulating sheet above the conductor. The adaptive meshing algorithm of HFSS was used to create the solution mesh with the convergence criterion Δ*S* < 0.03. It was verified that the height in the *y*-direction allowed for propagation losses without perturbation lower than 1%. The perturbing dielectric was centred in the *x*-direction touching the radiation boundaries in the *y*–*z*-planes and the *x*–*z*-plane but not in contact with the wave ports.

The authors thank Michael Fitch and Amelia Winterburn for their help and valuable time.

The experimental data, simulation data and source code may be obtained from: https://doi.org/10.5061/dryad.sbcc2fr2r [14].

T.S. conceived the theoretical model including the necessary simplifications. He computed the presented results and simulations and drafted the manuscript. A.A.R. gave the initial incentive for the theory, carried out experimental research and critically revised the manuscript. T.M. collected the experimental data and revised the manuscript. M.P. helped conceive the theoretical model and critically helped to draft the manuscript. All authors gave final approval for publication and agree to be held accountable for the work performed herein.

This work was supported by the Royal Society(grant nos. IF170002 and INF/PHD/180021). Additional funds were provided by BT plc.

The continuous fields of a bound TM mode of the perturbed guide are, omitting a factor exp (*j*(*ωt* − *βz*)), in general given by

$\begin{array}{rl}{E}_{z}& =\{\begin{array}{ll}{E}_{1}\mathrm{sin}({h}_{1}y)& 0\le y\le {y}_{1},\\ {E}_{2}\mathrm{cos}({h}_{2}y+\varphi )& {y}_{1}\le y\le {y}_{2},\\ {E}_{3}{\text{e}}^{-\gamma y}& {y}_{2}\le y,\end{array}\end{array}$

$\begin{array}{rl}{H}_{x}& =\{\begin{array}{ll}{E}_{1}\frac{j\omega {\u03f5}_{1}}{{h}_{1}}\mathrm{cos}({h}_{1}y)& y\le {y}_{1},\\ -{E}_{2}\frac{j\omega {\u03f5}_{2}}{{h}_{2}}\mathrm{sin}({h}_{2}y+\varphi )& {y}_{1}\le y\le {y}_{2},\\ {E}_{3}\frac{j\omega {\u03f5}_{3}}{\gamma}{\text{e}}^{-\gamma y}& {y}_{2}\le y,\end{array}\end{array}$

$\begin{array}{rl}{E}_{y}& =\{\begin{array}{ll}-{E}_{1}\frac{j\beta}{{h}_{1}}\mathrm{cos}({h}_{1}y)& y\le {y}_{1},\\ {E}_{2}\frac{j\beta}{{h}_{2}}\mathrm{sin}({h}_{2}y+\varphi )& {y}_{1}\le y\le {y}_{2},\\ -{E}_{3}\frac{j\beta}{\gamma}{\text{e}}^{-\gamma y}& {y}_{2}\le y\end{array}\end{array}$

$\begin{array}{rl}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{h}_{i}& =\sqrt{{\omega}^{2}{\u03f5}_{i}{\mu}_{0}-{\beta}^{2}}\phantom{\rule{1em}{0ex}}i=1,2,3,\end{array}$

where we have used the definitions as in §3 and we define $-\frac{{\u03f5}_{2}}{{h}_{2}}\mathrm{tan}({h}_{2}({y}_{1}-{y}_{2})+{\mathrm{tan}}^{-1}(-\frac{{h}_{2}{\u03f5}_{3}}{{\u03f5}_{2}\gamma}))=\frac{{\u03f5}_{1}}{{h}_{1}}\mathrm{cot}({h}_{1}{y}_{1}).$

We now show that there is no solution to equation (A 5) in the case For the left-hand side, we first look at the argument of the tangent function. Because *h*_{2} is imaginary, we may write the argument as

$-j(|{h}_{2}|({y}_{2}-{y}_{1})+{\mathrm{tanh}}^{-1}\left(\frac{|{h}_{2}|{\u03f5}_{3}}{{\u03f5}_{2}\gamma}\right)).$

The argument in the inverse hyperbolic tangent is always greater than 1, so we get a factor of −$\frac{{\u03f5}_{2}}{|{h}_{2}|}\mathrm{coth}(|{h}_{2}|({y}_{2}-{y}_{1})+\mathrm{\Re}({\mathrm{tanh}}^{-1}\left(\frac{|{h}_{2}|{\u03f5}_{3}}{{\u03f5}_{2}\gamma}\right)))>0,$

where $\mathrm{\Re}$ symbolizes taking the real part of the function. This expression is greater than zero as the argument of the hyperbolic cotangent is purely positive and real. We have shown beforehand that the right-hand side of equation (A 5) is negative. Therefore, there are no bound modes.Citing articles via

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