## Abstract

We investigate TE-wave propagation in a hollow waveguide with a graded dielectric layer, described using a hyperbolic tangent function. General formulae for the electric field components of the TE-waves, applicable to hollow waveguides with arbitrary cross sectional shapes, are presented. We illustrate the exact analytical results for the electric field components in the special case of a rectangular waveguide. Furthermore, we derive exact analytical results for the reflection and transmission coefficients valid for waveguides of arbitrary cross sectional shapes. Finally, we show that the obtained reflection and transmission coefficients are in exact asymptotic agreement with those obtained for a very thin homogeneous dielectric layer using mode-matching and cascading. The proposed method gives analytical results that are directly applicable without the need of mode-matching, and it has the ability to model realistic, smooth transitions.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

New theory and applications of plasmonics and artificial materials are constantly being explored in technology, biology and medicine [1]. This includes interesting applications in microwave and terahertz technologies, and nanophotonics such as plasmonic nanoparticle enhancement of light absorption for solar cells, etc., see e.g. [1]. The topic also includes studying surface plasmonic resonances in small structures of various shapes, embedded in different media, see e.g. [2–4]. A new and very interesting application of plasmonic resonance phenomena [1] is within the electrophoretic heating of gold nanoparticle suspensions as a radiotherapeutic hyperthermia based method to treat cancer [5–10]. In particular, gold nanoparticles (GNPs) can be coated with ligands that target specific cancer cells, as well as provide a net electronic charge of the GNPs [7–9]. The localized charged GNP suspension will then facilitate an electrophoretic current that can destroy the cancer under radio or microwave radiation, without causing damage to the normal tissue [7,8,10]. It is important to recognize the complexity of this clinical application with many possible physical and biophysical phenomena to take into account, including cellular properties and their influence on the dielectric spectrum [5,11], as well as thermodynamics and heat transfer, see e.g. [6]. The above mentioned results [6–10] indicate that electromagnetic heating mechanisms such as Joule heating and inductive heating, most likely can be disregarded for this application. However, the potential remains with using radio- or microwave radiation to achieve plasmonic (electrophoretic) resonances inside the GNP-targeted cells.

Recent studies of radio frequency absorption and optimal plasmonic resonances in gold nanoparticle (GNP) suspensions [10,12,13] have given rise to an interest in direct and inverse waveguide problems involving thin samples or surfaces of materials having extraordinary electromagnetic properties.

In particular, plasmonic resonances in layered waveguide structures involving scattering on a single thin layer, modeled as a thin dielectric layer in a straight waveguide, with perfectly electrically conducting (PEC) boundaries and a homogeneous cross section with material parameters $\epsilon$ and $\mu$, is reported in [14].

The ultimate goal behind the work in [14] and in the present paper, is to establish methods for identifying the model parameters associated with thin GNP treated cell line substrates inserted in a waveguide. The identified parameters can then be used to assess the feasibility of the medical application towards achieving a localized electrophoretic heating of an electrically charged GNP suspension as a radiotherapeutic hyperthermia based method to treat cancer. However, this can only be achieved provided that the suspension of GNPs can be designed to be plasmonically resonant and have a sufficiently large absorption cross-section in contrast to the surrounding medium.

Following a number of previous studies by one of the present authors [15–21], in this paper the surrounding homogeneous straight waveguide medium with a single thin layer is described as a stratified medium with frequency-dependent permittivity $\epsilon = \epsilon \left (\omega , z\right )$ being a function of the waveguide axis direction (chosen to be the $z$-direction). One important feature of the present approach to the TE-wave scattering on a thin dielectric layer in a hollow waveguide, is that it is possible to obtain the total scattering matrix parameters in the entire waveguide structure without any need to use boundary conditions, mode matching and cascading techniques. The waveguide is treated as filled with a single composite material with stratified frequency-dependent permittivity. Thus, a single solution of Maxwell’s equations in such a material replaces partial solutions in different materials, while at the same time asymptotically approaching such partial solutions in different materials. Furthermore, the boundary conditions between the materials are built in into the stratified permittivity function, and are hence not needed.

In order to avoid misunderstanding, we would like to emphasize that there is a number of important differences between the work presented here and the work presented in the previous studies by one of the present authors [15–21]. These differences concern both the fundamental nature of the studied physical systems and the properties of the mathematical models used to describe them.

One major difference is that in all the previous work reported in [15–21], the physical systems were spatially unlimited. The wave propagation was envisioned as plane wave propagation through a stratified medium as a model of the inhomogeneous space without any boundary conditions. To the best of our knowledge, the study presented in the present article is the first attempt to treat guided wave propagation in stratified media embedded in a straight hollow waveguide, where the space is spatially limited in one or two transverse directions, thus imposing waveguide boundary conditions.

Furthermore, in the previous work [15–21] the wave propagation was assumed to take place between right-handed media (RHM) and left-handed media (LHM). Thus, the magnetic properties of the stratified medium were an essential feature to describe the wave propagation. Unlike these studies, here we model a single lossy non-magnetic thin layer with complex relative permittivity $\epsilon _{\mathrm {L}}\left (\omega \right )$, inserted about the plane $z$ = 0. At a glance, it may seem as a simpler problem, but in some cases for realistic stratified models, the wave equations become more complex, in particular when there is no impedance matching as in several previous studies [15–20]. Thus the present model poses a new challenge in obtaining exact analytical results for the fields and transmission parameters.

It should also be emphasized that for the sake of notational and mathematical standardization, we have consistently reduced very different wave equations to the same differential equation having Gaussian hypergeometric functions as solutions. We have also used the same parameter symbols $(a,b,c)$ in the resulting Gaussian hypergeometric functions, although these parameters were both physically and mathematically very different for various physical situations. It is well known that, for different values of parameters $(a,b,c)$, the Gaussian hypergeometric functions represent very different special functions [22], and the mathematical structure of the solutions discussed here is not the same as in the previous work reported in [15–21]. Thus, for a reader not directly involved in these studies, the solutions of the differential equations and the expression for transmission parameters may appear to be the same in several papers. It is, however, not the case. It is just a consequence of our choice to standardize the presentation.

Last, but not least, we would also like to emphasize a very interesting new physical feature of the proposed model. To the best of our knowledge, no previous studies have reported the property of continuous expansion and reduction of the transverse wave pattern along the wave propagation direction, while at the same time keeping the same shape. Furthermore, there is also a property of trapped attenuated standing waves within the inserted thin layer. In the model proposed here, these phenomena can be studied without complicated mode-matching and cascading techniques.

Regarding notation and conventions, we consider classical electrodynamics where the electric and magnetic fields $\boldsymbol {E}$ and $\boldsymbol {H}$, respectively, are given in SI-units. The time convention for time harmonic fields (phasors) is given by $\exp \left (\mathrm {j} \omega t\right )$ where $\omega$ is the angular frequency and $t$ the time. We assume time-harmonic fields in a non-magnetic ($\mu = \mu _0 \mu _R$ with $\mu _R = 1$) inhomogeneous isotropic waveguide material.

## 2. Problem formulation

The geometry of the problem is illustrated in Fig. 1. In the surrounding non-magnetic lossy homogeneous straight waveguide medium with complex relative permittivity $\epsilon _{\mathrm {G}}\left (\omega \right )$, a single lossy non-magnetic thin layer with complex relative permittivity $\epsilon _{\mathrm {L}}\left (\omega \right )$ is inserted about the plane $z$ = 0, as shown in Fig. 1. The proposed model is applicable to any complex permittivities of the two media, including negative values in chiral metamaterials, as long as they satisfy the Kramers-Kronig relations. Mathematically, the media in the waveguide can be described as a single stratified medium with frequency-dependent permittivity $\epsilon = \epsilon \left (\omega , z\right )$, given by the following function of the waveguide axis direction (chosen to be the $z$-direction),

The new feature of the model described by (1) is that a single mathematical function is used to describe two material transitions. In our previous work [15–21], the hyperbolic tangent function has only been used to model a single transition between two different media. In the present paper, we have chosen a symmetric situation with two different materials, where one material is embedded within an otherwise homogeneous space consisting of the other material. However, the model (1) offers a possibility of exact analytical solutions even for the case where the wave propagates through three different materials. Such a possibility did not exist in any of the previous studies [15–21]. Wave propagation in a waveguide, with no field sources inside ($\rho = 0$ , $\boldsymbol {J} = 0$), is governed by Maxwell’s equations

## 3. Solution of the longitudinal equation

The next objective is to find the solutions of the longitudinal Eq. (11), which has the form

where we introduced a dimensionless variable $w =\frac {z}{z_0}$ and the two dimensionless functions## 4. Asymptotic analysis

It is now of interest to study the transmission and reflection coefficients (30) in the case of a thin dielectric layer ($z_0 \to 0$), when both constants $p$ and $r$ approach zero. Using the properties of the Gamma function [22] and (29), with the assumption $z_0 \to 0$, we obtain from the results (30)

Finally, it is of interest to compare the asymptotic formula (34) to the corresponding result obtained in [14] for a non-graded layered waveguide structure using mode-matching and cascading methods for hollow waveguides. The reflection coefficient reported in [14] is denoted by $T_{11}^{(2)}$, and using the notation employed in the present paper, has the form

## 5. Conclusions

We have investigated TE-wave propagation in a hollow waveguide with a graded dielectric layer, described using a hyperbolic tangent function. General formulae for the electric field components of TE-waves, as well as exact analytical results for the reflection and transmission coefficients, were obtained. The results are applicable to hollow waveguides with arbitrary cross sectional shapes. We illustrated the obtained general results on the special case of a rectangular waveguide. Finally, we showed that the obtained reflection and transmission coefficients are in exact asymptotic agreement with those obtained in [14] for a very thin homogeneous dielectric layer using mode-matching and cascading. The proposed method is tractable since it gives analytical results that are directly applicable without the need of mode-matching. At the same time, our method has the ability to model realistic, smooth transitions. The importance of the results presented here is that they indicate a clear possibility to use continuous analytical models as input to direct and inverse waveguide problems involving thin samples or surfaces of materials having extraordinary electromagnetic properties. The existence of the exact analytical results may provide a way to simplify the numerical approach to these problems, and reduce the number of degrees of freedom in the required algorithms.

## Funding

Stiftelsen för Strategisk Forskning.

## Acknowledgments

The work of S. N. was supported by the Swedish Foundation for Strategic Research (SSF) under the program Applied Mathematics and the project “Complex analysis and convex optimization for EM design”.

## Disclosures

The authors declare no conflicts of interest.

## References

**1. **S. A. Maier, “* Plasmonics: Fundamentals and Applications*,” (Springer-Verlag, 2007).

**2. **S. Link and M. A. El-Sayed, “Shape and size dependence of radiative non-radiative and photothermal properties of gold nanocrystals,” Int. Rev. Phys. Chem. **19**(3), 409–453 (2000). [CrossRef]

**3. **N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. **85**(21), 5040–5042 (2004). [CrossRef]

**4. **O. D. Miller, A. G. Polimeridis, M. T. H. Reid, C. W. Hsu, B. G. DeLacy, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Fundamental limits to optical response in absorptive systems,” Opt. Express **24**(4), 3329–3364 (2016). [CrossRef]

**5. **T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials **32**(36), 9776–9784 (2011). [CrossRef]

**6. **G. W. Hanson, R. C. Monreal, and S. P. Apell, “Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles,” J. Appl. Phys. **109**(12), 124306 (2011). [CrossRef]

**7. **S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C **116**(45), 24380–24389 (2012). [CrossRef]

**8. **E. Sassaroli, K. C. P. Li, and B. E. O’Neil, “Radio frequency absorption in gold nanoparticle suspensions: a phenomenological study,” J. Phys. D: Appl. Phys. **45**(7), 075303 (2012). [CrossRef]

**9. **C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale **6**(15), 8459–8472 (2014). [CrossRef]

**10. **S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. **50**(15), 155401 (2017). [CrossRef]

**11. **S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol. **41**(11), 2251–2269 (1996). [CrossRef]

**12. **M. Dalarsson, S. Nordebo, D. Sjöberg, and R. Bayford, “Absorption and optimal plasmonic resonances for small ellipsoidal particles in lossy media,” J. Phys. D: Appl. Phys. **50**(34), 345401 (2017). [CrossRef]

**13. **Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S. Nordebo, and J. Toft, “Passive approximation and optimization using B-splines,” arXiv 1711.07937 (2017).

**14. **Y. Ivanenko, M. Dalarsson, S. Nordebo, and R. Bayford, “On the Plasmonic Resonances in a Layered Waveguide Structure,” in Proceedings of The 12th International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials’ 2018 (2018).

**15. **M. Dalarsson and P. Tassin, “Analytical solution for wave propagation through a graded index interface between a right-handed and a left-handed material,” Opt. Express **17**(8), 6747 (2009). [CrossRef]

**16. **M. Dalarsson, M. Norgren, and Z. Jakšić, “Lossy gradient index metamaterial with sinusoidal periodicity of refractive index: case of constant impedance throughout the structure,” J. Nanophotonics **5**(1), 051804 (2011). [CrossRef]

**17. **M. Dalarsson, M. Norgren, N. Dončov, and Z. Jakšić, “Lossy gradient index transmission optics with arbitrary periodic permittivity and permeability and constant impedance throughout the structure,” J. Opt. **14**(6), 065102 (2012). [CrossRef]

**18. **M. Dalarsson, M. Norgren, T. Asenov, N. Dončov, and Z. Jakšić, “Exact analytical solution for fields in gradient index metamaterials with different loss factors in negative and positive refractive index segments,” J. Nanophotonics **7**(1), 073086 (2013). [CrossRef]

**19. **M. Dalarsson, M. Norgren, and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with linear gradient index metamaterials,” Prog. Electromagn. Res. **151**, 109–117 (2015). [CrossRef]

**20. **M. Dalarsson and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with hyperbolic tangent gradient index metamaterials,” Opt. Quantum Electron. **48**(3), 209 (2016). [CrossRef]

**21. **M. Dalarsson, “General theory of wave propagation through graded interfaces between positive- and negative-refractive-index media,” Phys. Rev. A **96**(4), 043848 (2017). [CrossRef]

**22. **M. Abramowitz and I. A. Stegun, “* Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables*,” (Dover Books, New York, 1965).