The T.M. Field

For the T.M. degrees of freedom the source and the electromagnetic field are also derived from a solution to the same inhomogeneous scalar wave Eq.(5). However, the difference from the T.E. case is that the four-vector components of the source and the vector potential lie in the Lorentz $(t,z)$-plane. Thus, instead of Eqs.(9) and (10), one has the T.M. source

$$( S_{t}, S_{z}, S_{x}, S_{y})=\left( \frac{\partial S}{\partial z}, \frac{\partial S}{\partial t},0,0 \right)$$ (11)

and the T.M. vector potential

$$( A_{t}, A_{z}, A_{x}, A_{y})=\left( \frac{\partial \psi}{\partial z}, \frac{\partial \psi}{\partial t} ,0,0 \right)~.$$ (12)

All the corresponding T.M. field components are derived from the scalar $\psi(t,z,x,y)$:

$$\begin{array}{c} \begin{tabular}[t]{\vert l\vert c\vert} \h... ...ial \psi}{\partial t}$\\ [3mm]\hline \end{tabular}\end{array}$$ (13)

These components are guaranteed to satisfy all the Maxwell field equations with the T.M. source, Eq.(11), whenever $\psi$ satisfies the inhomogeneous scalar wave equation, Eq.(5).