Pre-Quantum Electrodynamics

• Gr 9.5.1

A waveguide is a hollow structure in which EM waves can propagate. For simplicity here, we will assume that the boundaries of the waveguide are perfect conductors. Since the electric and magnetic fields vanish inside the walls of the waveguide, the boundary conditions at the inner walls are \[ {\boldsymbol E}^{\parallel} = 0, \hspace{10mm} B^\perp = 0. \] Free charges and currents will circulate on the surfaces in order to satisfy these constraints.

We are interested in monochromatic waves propagating down the waveguide and will thus take the generic ansatz \[ {\boldsymbol E} (x, y, z, t) = {\boldsymbol E}_0 (x,y) e^{i(kz - \omega t)}, \hspace{10mm} {\boldsymbol B} (x, y, z, t) = {\boldsymbol B}_0 (x,y) e^{i(kz - \omega t)}. \] These fields must obey Maxwell's equations. Unlike waves in vacuum, it will turn out that EM waves in waveguides are not purely transverse. We will thus start with a general ansatz for the fields: \[ {\boldsymbol E}_0 = \sum_i E_i ~\hat{\boldsymbol i}, \hspace{10mm} {\boldsymbol B}_0 = \sum_i B_i ~\hat{\boldsymbol i}, \hspace{10mm} i = x, y, z. \] Maxwell's equations (iii) and (iv) then give

\begin{align} \frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y} &= i\omega B_z, &\hspace{10mm} \frac{\partial B_y}{\partial x} - \frac{\partial B_x}{\partial y} &= -i \frac{\omega}{c^2} E_z, \nonumber \\ \frac{\partial E_z}{\partial y} - ik E_y &= i\omega B_x, &\hspace{10mm} \frac{\partial B_z}{\partial y} - ik B_y &= -i \frac{\omega}{c^2} E_x, \nonumber \\ ik E_x - \frac{\partial E_z}{\partial x} &= i\omega B_y, &\hspace{10mm} ik B_x - \frac{\partial B_z}{\partial x} &= -i \frac{\omega}{c^2} E_y. \end{align}

The transverse components can be isolated by simple algebra:

\begin{align} E_x &= \frac{i}{(\omega/c)^2 - k^2} \left( k \frac{\partial E_z}{\partial x} + \omega \frac{\partial B_z}{\partial y} \right), \nonumber \\ E_y &= \frac{i}{(\omega/c)^2 - k^2} \left( k \frac{\partial E_z}{\partial y} - \omega \frac{\partial B_z}{\partial x} \right), \nonumber \\ B_x &= \frac{i}{(\omega/c)^2 - k^2} \left( k \frac{\partial B_z}{\partial x} - \frac{\omega}{c^2} \frac{\partial E_z}{\partial y} \right), \nonumber \\ B_y &= \frac{i}{(\omega/c)^2 - k^2} \left( k \frac{\partial B_z}{\partial y} + \frac{\omega}{c^2} \frac{\partial E_z}{\partial x} \right). \end{align}

Putting these back into Maxwell (i) and (ii) gives the decoupled equations \[ \left[ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \left(\frac{\omega}{c}\right)^2 - k^2 \right] E_z = 0, \] with an identical equation for \(B_z\). If \(E_z = 0\) the waves are called TE waves (for transverse electric), and if \(B_z = 0\) they are called TM (for transverse magnetic}) waves. If \(E_z = 0 = B_z\) they are called TEM waves. The latter cannot occur in a hollow waveguide (simple proof: Gauss + Faraday).