TE-wave propagation in graded waveguide structures

Mariana Dalarsson; Sven Nordebo

doi:10.1364/OSAC.379847

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),

(1)$$\epsilon \left( \omega, z \right) = \epsilon_0 \epsilon_{\mathrm{R}} \left( z \right)= \epsilon_0 \left\{ \epsilon_{\mathrm{L}} \left( \omega \right) - \left[\epsilon_{\mathrm{L}}\left( \omega \right) - \epsilon_{\mathrm{G}}\left( \omega \right) \right] \tanh^2 \left( \frac{z}{z_0} \right) \right\} \hspace{1mm} ,$$

where $\epsilon _{\mathrm {R}}(z)$ denotes the relative permittivity, and $2 z_0$ determines the size of the inserted layer about the plane $z$ = 0, as indicated in Fig. 1. Far away from the layer in both directions ($z \to \pm \infty$), we have

(2)$$\tanh^2 \left( \frac{z}{z_0} \right) \to 1 \hspace{1mm} \Rightarrow \hspace{1mm} \epsilon\left(\omega, \pm \infty\right) = \epsilon_0 \epsilon_{\mathrm{G}}\left(\omega\right) \hspace{1mm} ,$$

while at the layer ($z \to 0$), we have

(3)$$\tanh^2 \left( \frac{z}{z_0} \right) \to 0 \hspace{1mm} \Rightarrow \hspace{1mm} \epsilon\left(\omega, 0\right) = \epsilon_0 \epsilon_{\mathrm{L}}\left(\omega\right) \hspace{1mm} ,$$

as required by the geometry of the problem. A geometry with a very thin single layer with rapid smooth transition from $\epsilon _{\mathrm {G}}\left (\omega \right )$ to $\epsilon _{\mathrm {L}}\left (\omega \right )$ and back to $\epsilon _{\mathrm {G}}\left (\omega \right )$ is then obtained in the limit $z_0 \to 0$. A few examples of permittivity functions for different values of $z_0$, are shown in Fig. 2.

Fig. 1. Hollow waveguide with a dielectric layer

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Fig. 2. Three examples of permittivity functions changing from $\Re [\epsilon _{\mathrm {G}}] = 2$ to $\Re [\epsilon _{\mathrm {L}}] = 4$ and back for $z_0 = 0.1$ (black line), $z_0 = 0.2$ (red line) and $z_0 = 0.3$ (blue line). Here $\Re [\epsilon _{\mathrm {L}}\left (\omega \right )]\;>\;\Re [\epsilon _{\mathrm {G}}\left (\omega \right )]$. Note, however, that this assumption is not essential for the present approach, and is used only for graphical illustration.

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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

(4)$$\nabla \times \boldsymbol{E} = - \textrm{j} \omega \mu_0 \boldsymbol{H} \hspace{2mm} , \hspace{2mm} \nabla \cdot \left[ \epsilon\left(z\right) \boldsymbol{E} \right] = 0 \hspace{2mm} , \hspace{2mm} \nabla \times \boldsymbol{H} = \textrm{j} \omega \epsilon\left(z\right) \boldsymbol{E} \hspace{2mm} , \hspace{2mm} \nabla \cdot \boldsymbol{H} = 0 \hspace{1mm} .$$

The Maxwell Eqs. (4) give rise to the following wave equations for the electric and magnetic fields $\boldsymbol {E}$ and $\boldsymbol {H}$, respectively,

(5)$$\nabla^2 \boldsymbol{E} + \nabla \left( \frac{1}{\epsilon_{\mathrm{R}}} \frac{\mathrm{d} \epsilon_{\mathrm{R}}}{\mathrm{d}z} E_z \right) + k^2 \epsilon_{\mathrm{R}}\left(z\right) \boldsymbol{E} = 0 \hspace{1mm} ,$$

(6)$$\nabla^2 \boldsymbol{H} + \frac{1}{\epsilon_{\mathrm{R}}} \frac{\mathrm{d} \epsilon_{\mathrm{R}}}{\mathrm{d}z} \left( \nabla H_z - \frac{\partial \boldsymbol{H}}{\partial z} \right) + k^2 \epsilon_{\mathrm{R}}\left(z\right) \boldsymbol{H} = 0 \hspace{1mm} ,$$

where $k^2 = \omega ^2 \epsilon _0 \mu _0 = \omega ^2 / c^2$. For TE-waves with $E_z = 0$, the wave Eqs. (5)–(6) become simply

(7)$$\nabla^2 \boldsymbol{E} + k^2 \epsilon_{\mathrm{R}}\left(z\right) \boldsymbol{E} = 0 \hspace{2mm} , \hspace{2mm} \nabla^2 H_z + k^2 \epsilon_{\mathrm{R}}\left(z\right) H_z = 0 \hspace{1mm}.$$

It is possible to solve the second of the Eqs. (7) for the longitudinal component of the magnetic field $H_z$, and from that solution obtain all the other field components using standard waveguide analysis techniques. On the other hand, it is also possible to solve the first of the Eqs. (7) for the electric field $\boldsymbol {E}$, whereby the magnetic field $\boldsymbol {H}$ is readily obtained from the first of Maxwell’s Eqs. (4), i.e. using

(8)$$\boldsymbol{H} = \frac{\textrm{j}}{\omega \mu_0} \nabla \times \boldsymbol{E} \hspace{1mm} .$$

Thus, the first of the Eqs. (7) with $E_z = 0$ is equivalent to two scalar equations, both of which are of the form

(9)$$\nabla^2 E_j + k^2 \epsilon_{\mathrm{R}}\left(z\right) E_j = 0 \hspace{1mm} , \hspace{1mm} j \in \{ x , y \} \hspace{1mm} .$$

By means of standard separation of variables $E_j = F_j\left (x,y\right ) Z\left (z\right )$, any of these two differential equations can be split into an equation for $F_j\left (x,y\right )$ and an equation for $Z\left (z\right )$

(10)$$\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) F_j + k_T^2 F_j = 0 \hspace{1mm} , \hspace{1mm} j \in \{ x , y \} \hspace{1mm} ,$$

(11)$$\frac{\mathrm{d}^2 Z}{\mathrm{d}z^2} + \left[k^2 \epsilon_{\mathrm{R}}\left(z\right) - k_T^2\right] Z = 0 \hspace{1mm} ,$$

where $k_T$ denotes the transverse wave number of the waveguide. The solutions of the Eqs. (10) are the standard solutions obtained for a particular waveguide cross sectional shape, and are unaffected by the graded material transition in the $z$-direction. Let us, as an example, consider the rectangular waveguide, with a cross section shown in Fig. 3. Eq. (10) then has the well-known solutions for a rectangular waveguide

(12)$$F_x = A \left( \frac{n \pi}{b} \right) \cos \left( \frac{m \pi x}{a} \right) \sin \left( \frac{n \pi y}{b} \right) \hspace{1mm} ,$$

(13)$$F_y = - A \left( \frac{m \pi}{a} \right) \sin \left( \frac{m \pi x}{a} \right) \cos \left( \frac{n \pi y}{b} \right) \hspace{1mm} ,$$

where the well-known result for the square of the transverse wave number is

(14)$$k_T^2 = \left( \frac{m \pi}{a} \right)^2 + \left( \frac{n \pi}{b} \right)^2 \hspace{1mm} ,$$

and the constant $A$ can be determined from the incoming field intensity $E_0 = E\left (- \infty \right )$. The Eqs. (10) are quite general, and upon choosing any other hollow waveguide (e.g. parallel-plate waveguide, coaxial waveguide etc.) we can simply reuse the existing results for the transverse functions, whenever they are available in closed form. The waves can propagate in the waveguide only if

(15)$$k^2 \Re\{\epsilon_{\mathrm{R}}\left(z\right) \} - k_T^2\;>\;0 \hspace{1mm} \Rightarrow \hspace{1mm} k^2 [\Re\{\epsilon_{\mathrm{R}}\left(z\right) \}]_\textrm{min} - k_T^2\;>\;0 \hspace{1mm} .$$

In the particular case where we have assumed $\Re [\epsilon _{\mathrm {L}}\left (\omega \right )]\;>\;\Re [\epsilon _{\mathrm {G}}\left (\omega \right )]\;>\;0$, (15) yields the following condition

(16)$$k^2 \Re\{\epsilon_{\mathrm{G}} \} - k_T^2\;>\;0 \hspace{0.7mm} \Rightarrow \hspace{0.7mm} k^2 \Re\{\epsilon_{\mathrm{G}}\}\;>\;k_T^2 \hspace{0.7mm} \Rightarrow \hspace{0.7mm} \omega^2\;>\;\frac{k_T^2 c^2}{\Re\{\epsilon_{\mathrm{G}} \}} \hspace{0.7mm} ,$$

or

(17)$$f\;>\;\frac{c}{2 \pi} \frac{k_T}{\sqrt{\Re\{\epsilon_{\mathrm{G}} \}}} = f_{c,\textrm{max}} \hspace{1mm} .$$

Thus, with our assumptions, the waves can only propagate if their frequency is higher than the maximum cutoff frequency $f_{c,\textrm {max}}$ defined in Eq. (17) above. It is interesting to note that in the stratified media model employed here, the cutoff frequency is a function of the spatial $z$-coordinate, i.e. we can write $f_c\left (z\right )=\frac {c}{2 \pi } \cdot k_T/\sqrt {\Re \{\epsilon _{\mathrm {R}}\left (z\right )\}}$. Furthermore, with our choice of permittivities ($\Re [\epsilon _{\mathrm {L}}\left (\omega \right )]\;>\;\Re [\epsilon _{\mathrm {G}}\left (\omega \right )]$ in the entire operating frequency range), waves with lower frequencies than $f_{c,\textrm {max}}$ could propagate in the layer region about $z = 0$. However, such waves could only be trapped in the layer region, and would never be able to propagate either to or from the layer region. Thus, the waves that can propagate through the entire waveguide must necessarily have frequencies higher than $f_{c,\textrm {max}}$.

Fig. 3. Cross section of a rectangular waveguide with dimensions $a$ and $b$ such that $a\;>\;b$.

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3. Solution of the longitudinal equation

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

(18)$$\frac{\mathrm{d}^2 Z}{\mathrm{d}w^2} + \left( D - B \tanh^2 w \right) Z = 0 \hspace{1mm} ,$$

where we introduced a dimensionless variable $w =\frac {z}{z_0}$ and the two dimensionless functions

(19)$$D = (k^2 \epsilon_{\mathrm{L}} - k_T^2) z_0^2 \hspace{1mm} , \hspace{1mm} B = k^2 z_0^2 \left(\epsilon_{\mathrm{L}} - \epsilon_{\mathrm{G}}\right) \hspace{1mm} .$$

Using analogous solution procedures to the ones used in our previous work on graded metamaterials [15–21], we readily obtain

Z\left(z\right) = T \exp \left( 2 p \hspace{1mm} \frac{z}{z_0} \right) \left[ 1 + \exp \left( 2 \hspace{1mm} \frac{z}{z_0} \right) \right]^{-2p} \cdot

(20)$$\cdot_2 F_1 \left\{2p + \frac{1}{2} + \sqrt{r^2 + \frac{1}{4}} \hspace{1mm} , \hspace{1mm} 2p + \frac{1}{2} - \sqrt{r^2 + \frac{1}{4}} \hspace{1mm} , \hspace{1mm} 2p + 1 ; \hspace{1mm} \left[ 1 + \exp \left( 2 \hspace{1mm} \frac{z}{z_0} \right) \right]^{-1} \right\} ,$$

where $T$ is a constant to be determined from the asymptotic behavior of the solution (20) far away from the layer ($z \to \pm \infty$) $\hspace {1mm} \textrm {and}_{\hspace {2mm} 2}F_1\left (a,b,c;u\right ) = F\left (a,b,c;u\right )$ is the ordinary Gaussian hypergeometric function defined by Gauss hypergeometric series [22]

(21)$$F\left(a,b,c;u\right) = \frac{\Gamma\left(c\right)}{\Gamma\left(a\right) \Gamma\left(b\right)} \sum_{n=0}^{\infty} \frac{\Gamma\left(a+n\right) \Gamma\left(b+n\right)}{\Gamma\left(c+n\right)} \hspace{1mm} \frac{u^n}{n!} \hspace{1mm} ,$$

$\Gamma$ is the Gamma function [22], and we define two dimensionless constants $p$ and $r$, as follows

(22)$$p = \textrm{j} \hspace{1mm} \frac{z_0}{2} \hspace{1mm} \sqrt{k^2 \epsilon_{\mathrm{G}} - k_T^2} = \textrm{j} \hspace{1mm} \frac{k_{z\mathrm{G}} z_0}{2} \hspace{1mm} , \hspace{1mm} r = k \hspace{0.2mm} z_0 \hspace{0.2mm} \sqrt{\epsilon_{\mathrm{L}} - \epsilon_{\mathrm{G}}} \hspace{1mm} ,$$

with $k_{z\mathrm {G}} = \sqrt {k^2 \epsilon _{\mathrm {G}} - k_T^2}$ being the $z$-component of the wave vector of the asymptotic waves for $z \to \pm \infty$ . On the other hand, the $z$-component of the wave vector about the origin, where the thin dielectric layer is situated, is denoted by $k_{z\mathrm {L}} = \sqrt {k^2 \epsilon _{\mathrm {L}} - k_T^2}$. Let us now investigate the asymptotic behavior of the solution (20) for $z \to + \infty$, when the argument of the hypergeometric function becomes zero, i.e.

(23)$$u = \left[ 1 + \exp \left( 2 \hspace{1mm} \frac{z}{z_0} \right) \right]^{-1}\to 0 \hspace{1mm} \textrm{for} \hspace{1mm} z \to + \infty \hspace{1mm} .$$

Using the series (21), we see that $F\left (a,b,c;0\right ) = 1$, and we obtain from (20),

(24)$$Z\left(z\right) \to T \exp \left( - \textrm{j} \hspace{1mm} k_{z\mathrm{G}} \hspace{1mm} z \right) \hspace{1mm} \textrm{for} \hspace{1mm} z \to + \infty \hspace{1mm} ,$$

being a transmitted forward-propagating wave with amplitude equal to one, as required. Next, we investigate the asymptotic behavior of the solution (20) for $z \to - \infty$, when the argument of the hypergeometric function becomes equal to one, i.e.

(25)$$u = \left[ 1 + \exp \left( 2 \hspace{1mm} \frac{z}{z_0} \right) \right]^{-1} \to 1 \hspace{1mm} \textrm{for} \hspace{1mm} z \to - \infty \hspace{1mm} .$$

In order to investigate the asymptotic behavior of the solution (20) for $z \to - \infty$, it is convenient to use the following transformation formula for hypergeometric functions [22]

F\left(a,b,c;u\right) \hspace{-0.5mm}=\hspace{-0.5mm} \frac{\Gamma\left(c\right) \Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right) \Gamma\left(c-b\right)} F\left(a,b,a+b-c+\hspace{-0.2mm}1;\hspace{-0.5mm}1-u\right)

(26)$$+ \left(1-u\right)^{c-a-b} \hspace{0.5mm} \cdot \frac{\Gamma\left(c\right) \Gamma\left(a+b-c\right)}{\Gamma\left(a\right) \Gamma\left(b\right)} F\left(c-a,c-b,c-a-b+1;1-u\right) ,$$

such that in the limit $z \to - \infty$, with $c-a-b = - 2p$, we obtain

Z\left(z\right) \to T \frac{\Gamma\left(c\right) \Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right) \Gamma\left(c-b\right)} \exp \left( + \textrm{j} \hspace{1mm} k_{z\mathrm{G}} \hspace{1mm} z \right)

(27)$$+ T \frac{\Gamma\left(c\right) \Gamma\left(a+b-c\right)}{\Gamma\left(a\right) \Gamma\left(b\right)} \exp \left( - \textrm{j} \hspace{1mm} k_{z\mathrm{G}} \hspace{1mm} z \right) \hspace{1mm} \textrm{for} \hspace{1mm} z \to - \infty \hspace{1mm} .$$

where we require the solution to be a combination of an incident TE-wave and a reflected TE-wave with amplitude equal to one, as follows

(28)$$Z\left(z\right) \to \exp \left( - \textrm{j} \hspace{1mm} k_{z\mathrm{G}} \hspace{1mm} z \right) + R \exp \left( + \textrm{j} \hspace{1mm} k_{z\mathrm{G}} \hspace{1mm} z \right) \hspace{0.7mm} \textrm{for} \hspace{0.8mm} z \to - \infty \hspace{0.7mm} ,$$

with the notation

(29)$$a = 2p + \frac{1}{2} + \sqrt{r^2 + \frac{1}{4}} \hspace{0.7mm} , \hspace{0.7mm} b = 2p + \frac{1}{2} - \sqrt{r^2 + \frac{1}{4}} \hspace{0.7mm} , \hspace{0.7mm} c = 2p +1 \hspace{0.7mm} .$$

Comparing the Eqs. (27) and (28), we readily obtain the general expressions for the transmission coefficient ($T$) and the reflection coefficient ($R$), in the form

(30)$$T = \frac{\Gamma\left(a\right) \Gamma\left(b\right)}{\Gamma\left(c\right) \Gamma\left(a+b-c\right)} \hspace{2mm} , \hspace{2mm} R = \frac{\Gamma\left(a\right) \Gamma\left(b\right)}{\Gamma\left(c-a\right) \Gamma\left(c-b\right)} \frac{\Gamma\left(c-a-b\right)}{\Gamma\left(a+b-c\right)} \hspace{1mm} .$$

The formulae (30) are the most general exact analytic results for transmission and reflection coefficients over a graded dielectric layer in a straight hollow waveguide, valid for waveguides with any cross sectional shape. In the special case of a rectangular waveguide, we then obtain the overall expressions for the two electric field components in the form

E_x = A \hspace{1mm} T \left( \frac{n \pi}{b} \right) \cos \left( \frac{m \pi x}{a} \right) \sin \left( \frac{n \pi y}{b} \right) \exp \left( 2 p \hspace{1mm} \frac{z}{z_0} \right) \cdot \left[ 1 + \exp \left( 2 \hspace{1mm} \frac{z}{z_0} \right) \right]^{-2p} \cdot

(31)$$\cdot_2 F_1 \left\{2p + \frac{1}{2} + \sqrt{r^2 + \frac{1}{4}} \hspace{1mm} , \hspace{1mm} 2p + \frac{1}{2} - \sqrt{r^2 + \frac{1}{4}} \hspace{1mm} , \hspace{1mm} 2p + 1 ; \hspace{1mm} \left[ 1 + \exp \left( 2 \hspace{1mm} \frac{z}{z_0} \right) \right]^{-1} \right\} ,$$

E_y = - A \hspace{1mm} T \left( \frac{m \pi}{a} \right) \sin \left( \frac{m \pi x}{a} \right) \cos \left( \frac{n \pi y}{b} \right) \exp \left( 2 p \hspace{1mm} \frac{z}{z_0} \right) \left[ 1 + \exp \left( 2 \hspace{1mm} \frac{z}{z_0} \right) \right]^{-2p} \cdot

(32)$$\cdot_2 F_1 \left\{2p + \frac{1}{2} + \sqrt{r^2 + \frac{1}{4}} \hspace{1mm} , \hspace{1mm} 2p + \frac{1}{2} - \sqrt{r^2 + \frac{1}{4}} \hspace{1mm} , \hspace{1mm} 2p + 1 ; \hspace{1mm} \left[ 1 + \exp \left( 2 \hspace{1mm} \frac{z}{z_0} \right) \right]^{-1} \right\} ,$$

where $A$ is a constant proportional to the incident electric field amplitude $E_0$. The magnetic field components in a rectangular waveguide are then readily obtained using the Maxwell Eq. (8).

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)

(33)$$T = 1 + \textrm{j} k_{z\mathrm{L}} z_0 \hspace{1mm} \frac{\epsilon_{\mathrm{L}} - \epsilon_{\mathrm{G}}}{\sqrt{\epsilon_{\mathrm{L}} - k_T^2/k^2} \sqrt{\epsilon_{\mathrm{G}} - k_T^2/k^2}} + {\cal O}\{z_0^2\}\hspace{1mm} ,$$

(34)$$R = \textrm{j} k_{z\mathrm{L}} z_0 \hspace{1mm} \frac{\epsilon_{\mathrm{L}} - \epsilon_{\mathrm{G}}}{\sqrt{\epsilon_{\mathrm{L}} - k_T^2/k^2} \sqrt{\epsilon_{\mathrm{G}} - k_T^2/k^2}} +{\cal O}\{z_0^2\} \hspace{1mm} ,$$

where $T=1+R$ as required. From the results (33–34), we readily see that there is no reflection ($R = 0$) whenever the two materials have the same relative permittivity $\epsilon _{\mathrm {L}}\left (\omega \right ) = \epsilon _{\mathrm {G}}\left (\omega \right )$, e.g. the two materials are the same. Furthermore, we see that for $z_0 = 0$, which implies that the dielectric layer is removed, there is no reflection either ($R = 0$), as expected.

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

(35)$$T_{11}^{(2)} = - 2 \textrm{j} k_{z\mathrm{L}} \left(2 z_0\right) \frac{S_{11}^{(1)}}{1 - \left( S_{11}^{(1)} \right)^2} +{\cal O}\{z_0^2\}\hspace{1mm} ,$$

where $2 z_0 = d_{\mathrm {L}}$ is the thickness of the thin dielectric layer, $\mu _{\mathrm {G}} = \mu _{\mathrm {L}} = 1$ and

(36)$$S_{11}^{(1)} = \frac{k_{z\mathrm{G}} - k_{z\mathrm{L}}}{k_{z\mathrm{G}} + k_{z\mathrm{L}}} \hspace{1mm} .$$

Substituting (36) into the result (35), after some algebra, we obtain

(37)$$T_{11}^{(2)} = - 2 \hspace{1mm} \textrm{j} k_{z\mathrm{L}} \left(2 z_0\right) \frac{k_{z\mathrm{G}}^2 - k_{z\mathrm{L}}^2}{4 \hspace{1mm} k_{z\mathrm{G}} k_{z\mathrm{L}}} +{\cal O}\{z_0^2\}\hspace{1mm} ,$$

Using the definitions of $k_{z\mathrm {G}}$ and $k_{z\mathrm {L}}$ stated earlier in this paper, and inserting them into Eq. (35), we obtain

(38)$$T_{11}^{(2)} = R = \textrm{j} k_{z\mathrm{L}} z_0 \hspace{0.5mm} \frac{\epsilon_{\mathrm{L}} - \epsilon_{\mathrm{G}}}{\sqrt{\epsilon_{\mathrm{L}} - k_T^2/k^2} \sqrt{\epsilon_{\mathrm{G}} - k_T^2/k^2}} +{\cal O}\{z_0^2\}\hspace{0.7mm} .$$

From the results (38) and (34), we see that the scattering matrix parameters reported in [14] for a homogeneous dielectric layer in a hollow waveguide structure, have the same thin layer asymptotics as the scattering parameters obtained using the graded dielectric layer based on (1). This concludes the asymptotic analysis, and illustrates that the graded permittivity function (1) can be employed to obtain useful scattering parameters for the waveguide without any need of mode matching and cascading techniques. Furthermore, the proposed technique gives the flexibility to model realistic, smooth transitions.

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.

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TE-wave propagation in graded waveguide structures

Abstract

1. Introduction

2. Problem formulation

3. Solution of the longitudinal equation

4. Asymptotic analysis

5. Conclusions

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (3)

Equations (43)

OSA Continuum