Dielectric wave guides

We have seen that it is possible to propagate electromagnetic waves down a hollow conductor. However, other types of guiding structures are also possible. The general requirement for a guide of electromagnetic waves is that there be a flow of energy along the axis of the guiding structure but not perpendicular to it. This implies that the electromagnetic fields are appreciable only in the immediate neighbourhood of the guiding structure.

Consider an axisymmetric tube of arbitrary cross section made of some dielectric material and surrounded by a vacuum. This structure can serve as a wave guide provided that the dielectric constant of the material is sufficiently large. Note, however, that the boundary conditions satisfied by the electromagnetic fields are significantly different to those of a conventional wave guide. The transverse fields are governed by two equations; one for the region inside the dielectric, and the other for the vacuum region. Inside the dielectric we have

$$\left[ \nabla_s^{2} +\left(\epsilon_1 \,\frac{\omega^2}{c^2} - k_g^{~2}\right) \right]\!\psi = 0.$$ (1118)

In the vacuum region we have

$$\left[ \nabla_s^{2} +\left( \frac{\omega^2}{c^2} - k_g^{~2}\right) \right]\!\psi = 0.$$ (1119)

Here, $\psi(x,y)\, {\rm e}^{\,{\rm i}\, k_g z}$ stands for either $E_z$ or $H_z$, $\epsilon_1$ is the relative permittivity of the dielectric material, and $k_g$ is the guide propagation constant. The guide propagation constant must be the same both inside and outside the dielectric in order to satisfy the electromagnetic boundary conditions at all points on the surface of the tube.

Inside the dielectric the transverse Laplacian must be negative, so that the constant

$$k_s^{~2} = \epsilon_1\, \frac{\omega^2}{c^2} -k_g^{~2}$$ (1120)

is positive. Outside the cylinder the requirement of no transverse flow of energy can only be satisfied if the fields fall off exponentially (instead of oscillating). Thus,

$$k_t^{~2} = k_g^{~2} - \frac{\omega^2}{c^2}$$ (1121)

must be positive.

The oscillatory solutions (inside) must be matched to the exponentiating solutions (outside). The boundary conditions are the continuity of normal ${\bfm B}$ and ${\bfm D}$ and tangential ${\bfm E}$ and ${\bfm H}$ on the surface of the tube. These boundary conditions are far more complicated than those in a conventional wave guide. For this reason, the normal modes cannot usually be classified as either pure TE or pure TM modes. In general, the normal modes possess both electric and magnetic field components in the transverse plane. However, for the special case of a cylindrical tube of dielectric material the normal modes can have either pure TE or pure TM characteristics. Let us examine this case in detail.

Consider a dielectric cylinder of radius $a$ and dielectric constant $\epsilon_1$. For the sake of simplicity, let us only search for normal modes whose electromagnetic fields have no azimuthal variation. Equations (6.65) and (6.67) yield

$$\left(r^2\,\frac{d^2}{d r^2} + r\,\frac{d}{dr} + r^2 k_s^{~2}\right) \!\psi = 0$$ (1122)

for $r<a$. The general solution to this equation is some linear combination of the Bessel functions $J_0(k_s r)$ and $Y_0(k_s r)$. However, since $Y_0(k_s r)$ is badly behaved at the origin ($r=0$) the physical solution is $\psi\propto J_0(k_s r)$.

Equations (6.66) and (6.68) yield

$$\left(r^2\,\frac{d^2}{d r^2} + r\,\frac{d}{dr} - r^2 k_t^{~2}\right) \psi = 0.$$ (1123)

This can be rewritten

$$\left(z^2\,\frac{d^2}{d z^2} + z\,\frac{d}{dz} - z^2\right) \!\psi = 0,$$ (1124)

where $z=k_t r$. This is type of modified Bessel's equation, whose most general form is

$$\left[z^2\,\frac{d^2}{d z^2} + z\,\frac{d}{dz} - (z^2+m^2)\right] \!\psi = 0.$$ (1125)

The two linearly independent solutions of the above equation are denoted $I_m(z)$ and $K_m(z)$. The asymptotic behaviour of these solutions at small $\vert z\vert$ is as follows:

$$I_m(z) = \left(\frac{z}{2}\right)^m \sum_{k=0}^\infty \frac{(z^2/4)^k}{k! (k+m)!},$$ (1126)

$$K_m(z) = \frac{1}{2} \left(\frac{z}{2}\right)^{-m} \sum_{k=0}^{m-1} \frac{(m-k-1)!}{k!} (-z^2/4)^k + (-1)^{m+1} \ln(z/2) I_m (z)$$

$$+ (-1)^m \frac{1}{2}\left(\frac{z}{2}\right)^m \sum_{k=0}^\infty \left[\psi(k+1)+\psi(m+k+1)\right] \frac{(z^2/4)^m}{k!(m+k)!}.$$ (1127)

Hence, $I_m$ is well behaved in the limit $\vert z\vert\rightarrow 0$, whereas $K_m$ is badly behaved. The asymptotic behaviour at large $\vert z\vert$ is

$$I_m(z) \simeq \frac{{\rm e}^z}{\sqrt{2\pi z}} \left[ 1+O\left(\frac{1}{z}\right)\,\right],$$ (1128)

$$K_m(z) \simeq \sqrt{\frac{\pi}{2z}} \,{\rm e}^{-z} \left[ 1+O\left(\frac{1}{z}\right)\,\right].$$ (1129)

Hence, $I_m$ is badly behaved in the limit $\vert z\vert\rightarrow \infty$, whereas $K_m$ is well behaved. The behaviour of $I_0(z)$ and $K_0(z)$ is shown in Fig. 22. It is clear that the physical solution to Eq. (6.70) (i.e., the one which decays as $\vert r\vert\rightarrow \infty$) is $\psi\propto K_0(k_t r)$.

Figure 22: The Bessel functions $I_0(z)$ (solid line) and $K_0(z)$ (dotted line)

The physical solution is

$$\psi = J_0(k_s r)$$ (1130)

for $r\leq a$, and

$$\psi = A\,K_0(k_t r)$$ (1131)

for $r > a$. Here, $A$ is an arbitrary constant, and $\psi(r) \, {\rm e}^{\,{\rm i}\,k_g z}$ stands for either $E_z$ or $H_z$. It follows from Eqs. (6.28) (using $\partial/\partial\theta = 0$) that

$$H_r = {\rm i}\,\frac{k_g}{k_s^{~2}} \frac{\partial H_z}{\partial r},$$ (1132)

$$E_\theta = - \frac{\omega\mu_0}{k_g} \,H_r,$$ (1133)

$$H_\theta = {\rm i}\, \frac{\omega\epsilon_0\epsilon_1}{k_s^{~2}} \frac{\partial E_z}{\partial r},$$ (1134)

$$E_r = \frac{k_g}{\omega \epsilon_0\epsilon_1} \,H_\theta$$ (1135)

for $r\leq a$. There are an analogous set of relationships for $r > a$. The fact that the field components form two groups; ($H_r$, $E_\theta$), which depend on $H_z$, and ($H_\theta$, $E_r$), which depend on $E_z$; means that the normal modes take the form of either pure TE modes or pure TM modes.

For a TE mode ($E_z=0$) we find that

$$H_z = J_0(k_s r),$$ (1136)

$$H_r = -{\rm i}\,\frac{k_g}{k_s} \,J_1(k_s r),$$ (1137)

$$E_\theta = {\rm i}\, \frac{\omega\mu_0}{k_s}\, J_1(k_s r)$$ (1138)

for $r\leq a$, and

$$H_z = A \,K_0(k_t r),$$ (1139)

$$H_r = {\rm i}\,A\,\frac{ k_g }{k_t}\, K_1(k_t r),$$ (1140)

$$E_\theta = -{\rm i}\,A\, \frac{\omega\mu_0}{k_t} \,K_1(k_t r)$$ (1141)

for $r > a$. Here we have used

$$J_0'(z) = -J_1(z),$$ (1142)

$$K_0'(z) = -K_1(z),$$ (1143)

where $'$ denotes differentiation with respect to $z$. The boundary conditions require $H_z$, $H_r$, and $E_\theta$ to be continuous across $r=a$. Thus, it follows that

$$A\,K_0(k_t a) = J_0(k_s a),$$ (1144)

$$-A\, \frac{K_1(k_t r) }{k_t} = \frac{J_1(k_s a)}{k_s}.$$ (1145)

Eliminating the arbitrary constant $A$ between the above two equations yields the dispersion relation

$$\frac{J_1(k_s a)}{k_s\,J_0(k_s a)} + \frac{K_1(k_t a)}{k_t\,K_0(k_t a)}= 0,$$ (1146)

where

$$k_t^{~2} + k_s^{~2} = (\epsilon_1 -1)\,\frac{\omega^2}{c^2}.$$ (1147)

Figure 23: Graphical solution of the dispersion relation (6.82). The curve $A$ represents $-J_1(k_s/a)/k_s J_0(k_s a)$. The curve $B$ represents $K_1(k_t a) / k_t K_0(k_t a)$.

Figure 23 shows a graphical solution of the above dispersion relation. The roots correspond to the crossing points of the two curves; $-J_1(k_s a)/k_s J_0(k_s a)$ and $K_1(k_t a) / k_t K_0(k_t a)$. The vertical asymptotes of the first curve are given by the roots of $J_0(k_s a)=0$. The vertical asymptote of the second curve occurs when $k_t=0$; i.e., when $k_s^{~2} a^2 = (\epsilon_1-1)\,\omega^2a^2/c^2$. Note from Eq. (6.82) that $k_t$ decreases as $k_s$ increases. In Fig. 23 there are two crossing points, corresponding to two distinct propagating modes of the system. It is evident that if the point $k_t=0$ corresponds to a value of $k_s a$ which is less than the first root of $J_0(k_s a)=0$, then there is no crossing of the two curves, and, hence, there are no propagating modes. Since the first root of $J_0(z)=0$ occurs at $z=2.4048$ (see Table 2) the condition for the existence of propagating modes can be written

$$\omega > \omega_{01} = \frac{2.4048\,c}{\sqrt{\epsilon_1 -1}\, a}.$$ (1148)

In other words, the mode frequency must lie above the cutoff frequency $\omega_{01}$ for the ${\rm TE}_{01}$ mode (here, the 0 corresponds to the number of nodes in the azimuthal direction, and the 1 refers to the 1st root of $J_0(z)=0$). It is also evident that as the mode frequency is gradually increased the point $k_t=0$ eventually crosses the second vertical asymptote of $-J_1(k_s/a)/k_s J_0(k_s a)$, at which point the ${\rm TE}_{02}$ mode can propagate. As $\omega$ is further increased more and more TE modes can propagate. The cutoff frequency for the ${\rm TE}_{0l}$ mode is given by

$$\omega_{0l} = \frac{j_{0l}\,c}{\sqrt{\epsilon_1 -1}\, a},$$ (1149)

where $j_{0l}$ is $l$th root of $J_0(z)=0$ (in order of increasing $z$).

At the cutoff frequency for a particular mode $k_t=0$, which implies from Eq. (6.68) that $k_g = \omega/c$. In other words, the mode propagates along the guide at the velocity of light in vacuum. Immediately below this cutoff frequency the system no longer acts as a guide but as an antenna, with energy being radiated radially. For frequencies well above the cutoff, $k_t$ and $k_g$ are of the same order of magnitude, and are large compared to $k_s$. This implies that the fields do not extend appreciably outside the dielectric cylinder.

For a TM mode ($H_z=0$) we find that

$$E_z = J_0(k_s r),$$ (1150)

$$H_\theta = -{\rm i}\,\frac{\omega\epsilon_0\epsilon_1}{k_s} \,J_1(k_s r),$$ (1151)

$$E_r = -{\rm i}\, \frac{k_g}{k_s}\, J_1(k_s r)$$ (1152)

for $r\leq a$, and

$$E_z = A \,K_0(k_t r),$$ (1153)

$$H_\theta = {\rm i}\,A\,\frac{\omega\epsilon_0}{k_t}\, K_1(k_t r),$$ (1154)

$$E_r = {\rm i}\,A\, \frac{k_g}{k_t} \,K_1(k_t r)$$ (1155)

for $r > a$. The boundary conditions require $E_z$, $H_\theta$, and $D_r$ to be continuous across $r=a$. Thus, it follows that

$$A\,K_0(k_t a) = J_0(k_s a),$$ (1156)

$$-A\, \frac{K_1(k_t r) }{k_t} = \epsilon_1 \frac{J_1(k_s a)}{k_s}.$$ (1157)

Eliminating the arbitrary constant $A$ between the above two equations yields the dispersion relation

$$\epsilon_1 \frac{J_1(k_s a)}{k_s\,J_0(k_s a)} + \frac{K_1(k_t a)}{k_t\,K_0(k_t a)}= 0.$$ (1158)

It is clear from this dispersion relation that the cutoff frequency for the ${\rm TM}_{0l}$ mode is exactly the same as that for the ${\rm TE}_{0l}$ mode. It is also clear that in the limit $\epsilon_1\gg 1$ the propagation constants are determined by the roots of $J_1(k_s a)\simeq 0$. However, this is exactly the same as the determining equation for TE modes in a metallic wave guide of circular cross section (filled with dielectric of relative permittivity $\epsilon_1$).

Modes with azimuthal dependence (i.e., $m>0$) have longitudinal components of both ${\bfm E}$ and ${\bfm H}$. This makes the mathematics somewhat more complicated. However, the basic results are the same as for $m=0$ modes: for frequencies well above the cutoff frequency the modes are localized in the immediate vicinity of the cylinder.