Complex Waves in Dielectric Layer

Abstract

The propagation of monochromatic TE-polarized waves in a partially shielded dielectric layer is considered. The existence of infinitely many complex leaky waves is proved as well as the absence of complex surface waves.

Appendix

The results presented below is well-known in general [2–6, 10, 11, 24]. We only refine the localization of solutions of DE. All results can be confirmed using graphical analysis of functions.

Propagating surface waves. Let $\lambda=\alpha$, $\alpha\in\mathbb{R}.$ Then equation (14) takes the form

$$\tan{\alpha h}+\dfrac{\alpha}{\!\!\!\!\!\!\phantom{x}{}_{\phantom{x}{}_{+}}\!\sqrt{\epsilon^{2}-\alpha^{2}}}=0.$$

(21)

Here and below sign $\phantom{x}{}_{\phantom{x}{}_{+}}\!\sqrt{\,}$ denotes the arithmetic root.

Lemma 1. Equation (21) has no solution for $\alpha^{2}>\epsilon^{2}$ . If $m-1/2\leq\epsilon h/\pi\leq m+1/2$ then there exist $m\geq 1$ nonzero positive (and $m$ nonzero negative) roots of equation (21). The $j$ th positive root $\alpha_{j}$ is located so that

$$\dfrac{\pi}{2h}(2j-1)<\alpha_{j}<\dfrac{\pi j}{h}\quad(j=1,\dots,m-1),\quad\dfrac{\pi}{2h}(2m-1)<\alpha_{m}<\epsilon.$$

If $0<\epsilon h/\pi\leq 1/2$ then there are no roots of equation (21).

Let $\lambda=i\beta$, $\beta\in\mathbb{R}.$ Then equation (14) takes the form

$$\tanh{\beta h}+\dfrac{\beta}{\!\!\!\!\!\!\phantom{x}{}_{\phantom{x}{}_{+}}\!\sqrt{\epsilon^{2}+\beta^{2}}}=0.$$

(22)

Lemma 2. Equation (22) has no nontrivial solution.

Propagating leaky waves. Let $\lambda=\alpha$, $\alpha\in\mathbb{R}.$ Taking into account (18) we can write (14) as

$$\tan{\alpha h}-\dfrac{\alpha}{\!\!\!\!\!\!\phantom{x}{}_{\phantom{x}{}_{+}}\!\sqrt{\epsilon^{2}-\alpha^{2}}}=0.$$

(23)

Equation (23) leads to the analysis of solutions of the equations $\sin(z)=\pm z/(\epsilon h)$ under the conditions $\tan{z}>0$ and $0<z<\epsilon h$, which can be fulfilled graphically.

Lemma 3. Equation (23) has no solution for $\alpha^{2}>\epsilon^{2}$ . Let $z_{0}^{(j)}$ be the $j$ th positive root of equation $\tan{z}=z$ such that $z_{0}^{(j)}<z_{0}^{(j+1)}$ , $j=1,2,\dots$ and $b_{0}^{(j)}=\sqrt{(z_{0}^{(j)})^{2}+1}$ . There are no roots of (23) for $0<\epsilon h\leq 1$ and $\pi/2\leq\epsilon h\leq b_{0}^{(1)}.$ If $\epsilon h=b_{0}^{(n)}$ then there exist $n\geq 1$ nonzero positive roots (and $n$ nonzero negative roots) of equation (23); the $j$ th root $\alpha_{j}$ is located so that

$$\dfrac{\pi j}{h}<\alpha_{j}<\dfrac{\pi}{2h}(2j+1),\quad j=1,\dots,n.$$

If $b_{0}^{(j)}<\epsilon h<\pi\left(2n+1\right)/2$ then there exist $n+1$ , $n\geq 1$ nonzero positive roots (and $n+1$ nonzero negative roots) of (23); the $j$ th root $\alpha_{j}$ is located so that

$$\dfrac{\pi j}{h}<\alpha_{j}<\dfrac{\pi}{2h}(2j+1),\quad j=1,\dots,n-1,$$

and

$$\dfrac{\pi}{h}<\alpha_{n}<\dfrac{\pi}{2h}(2n+1),\quad\dfrac{\pi}{h}<\alpha_{n+1}<\dfrac{\pi}{2h}(2n+1).$$

If $\pi\left(2n+1\right)/2\leq\epsilon h<b_{0}^{(n+1)}$ then there exist $n\geq 1$ nonzero positive roots (and $n$ nonzero negative roots) of equation (23); the $j$ th root $\alpha_{j}$ is located so that

$$\dfrac{\pi j}{h}<\alpha_{j}<\dfrac{\pi}{2h}(2j+1),\quad j=1,\dots,n.$$

Let $\lambda=i\beta$, $\beta\in\mathbb{R}.$ Then equation (14) takes the form

$$\tanh{\beta h}-\dfrac{\beta}{\!\!\!\!\!\!\phantom{x}{}_{\phantom{x}{}_{+}}\!\sqrt{\epsilon^{2}+\beta^{2}}}=0.$$

(24)

Lemma 4. Equation (24) has two nontrivial solutions $\pm i\beta_{0}$ if $\epsilon h<1$ . If $\epsilon h>1$ then there are no nontrivial solutions of (24).