TE and TM Waves

Note well that we have written the mutilated Maxwell Equations so that the $z$ components are all on the right hand side. If they are known functions, and if the only $z$ dependence is the complex exponential (so we can do all the $z$-derivatives and just bring down a $\pm ik$) then the transverse components $\mbox{\boldmath$E$}_\perp$ and $\mbox{\boldmath$B$}_\perp$ are determined!

In fact (for propagation in the $+z$ direction, $e^{+ikz - i\omega t}$):

$$ik\mbox{\boldmath$E$}_\perp + i\omega (\hat{\mbox{\boldmath$z$}}\times\mbox{\boldmath$B$}_\perp) = \mbox{\boldmath$\nabla$}_\perp E_z$$

$$ik(\hat{\mbox{\boldmath$z$}} \times \mbox{\boldmath$E$}_\perp) + ... ...\boldmath$z$}}\times (\hat{\mbox{\boldmath$z$}}\times\mbox{\boldmath$B$}_\perp) = \hat{\mbox{\boldmath$z$}} \times \mbox{\boldmath$\nabla$}_\perp E_z$$

$$ik(\hat{\mbox{\boldmath$z$}} \times \mbox{\boldmath$E$}_\perp) = i\omega \mbox{\boldmath$B$}_\perp + \hat{\mbox{\boldmath$z$}} \times \mbox{\boldmath$\nabla$}_\perp E_z$$ (10.52)

$$\hat{\mbox{\boldmath$z$}}\cdot(\mbox{\boldmath$\nabla$}_\perp \times \mbox{\boldmath$B$}_\perp) = -i\omega\mu\epsilon E_z$$ (10.53)

and

$$ik\mbox{\boldmath$B$}_\perp - i\omega\mu\epsilon (\hat{\mbox{\boldmath$z$}}\times\mbox{\boldmath$E$}_\perp) = \mbox{\boldmath$\nabla$}_\perp B_z$$

$$ik\mbox{\boldmath$B$}_\perp - \mbox{\boldmath$\nabla$}_\perp B_z = i\omega\mu\epsilon (\hat{\mbox{\boldmath$z$}}\times\mbox{\boldmath$E$}_\perp)$$

$$i\frac{k^2}{\omega\mu\epsilon}\mbox{\boldmath$B$}_\perp - \frac{k}{\omega\mu\epsilon}\mbox{\boldmath$\nabla$}_\perp B_z = ik(\hat{\mbox{\boldmath$z$}} \times \mbox{\boldmath$E$}_\perp)$$

$$i\frac{k^2}{\omega\mu\epsilon}\mbox{\boldmath$B$}_\perp - \frac{k}{\omega\mu\epsilon}\mbox{\boldmath$\nabla$}_\perp B_z = i\omega \mbox{\boldmath$B$}_\perp + \hat{\mbox{\boldmath$z$}} \times \mbox{\boldmath$\nabla$}_\perp E_z$$

or

$$\mbox{\boldmath$B$}_\perp = \frac{i}{\mu\epsilon\omega^2 - k^2}\left( k\mbox{\boldmath$\nabla... ...mega (\hat{\mbox{\boldmath$z$}}\times\mbox{\boldmath$\nabla$}_\perp E_z)\right)$$ (10.54)

$$\mbox{\boldmath$E$}_\perp = \frac{i}{\mu\epsilon\omega^2 - k^2}\left( k\mbox{\boldmath$\nabla... ...mega (\hat{\mbox{\boldmath$z$}}\times\mbox{\boldmath$\nabla$}_\perp B_z)\right)$$ (10.55)

(where we started with the second equation and eliminated $\hat{\mbox{\boldmath$z$}}\times\mbox{\boldmath$B$}_\perp$ to get the second equation just like the first).

Now comes the relatively tricky part. Recall the boundary conditions for a perfect conductor:

$$\hat{\mbox{\boldmath$n$}}\times(\mbox{\boldmath$E$} - \mbox{\boldmath$E$}_c) = \hat{\mbox{\boldmath$n$}}\times\mbox{\boldmath$E$} = 0$$

$$\hat{\mbox{\boldmath$n$}}\cdot(\mbox{\boldmath$B$} - \mbox{\boldmath$B$}_c) = \hat{\mbox{\boldmath$n$}}\cdot\mbox{\boldmath$B$} = 0$$

$$\hat{\mbox{\boldmath$n$}}\times\mbox{\boldmath$H$} = \mbox{\boldmath$K$}$$

$$\hat{\mbox{\boldmath$n$}}\cdot \mbox{\boldmath$D$} = \Sigma$$

They tell us basically that $\mbox{\boldmath$E$}$ ( $\mbox{\boldmath$D$}$) is strictly perpendicular to the surface and that $\mbox{\boldmath$B$}$ ( $\mbox{\boldmath$H$}$) is strictly parallel to the surface of the conductor at the surface of the conductor.

This means that it is not necessary for $E_z$ or $B_z$ both to vanish everywhere inside the dielectric (although both can, of course, and result in a TEM wave or no wave at all). All that is strictly required by the boundary conditions is for

$$E_z\vert _S = 0$$ (10.56)

on the conducting surface $S$ (it can only have a normal component so the $z$ component must vanish). The condition on $B_z$ is even weaker. It must lie parallel to the surface and be continuous across the surface (where $\mbox{\boldmath$H$}$ can discontinuously change because of $\mbox{\boldmath$K$}$). That is:

$$\frac{\partial B_z}{\partial n}\vert _S = 0$$ (10.57)

We therefore have two possibilities for non-zero $E_z$ or $B_z$ that can act as source term in the mutilated Maxwell Equations.