The T.E. Field

For the T.E. degrees of freedom the components of the charge-current four-vector are

$$( S_{t}, S_{z}, S_{x}, S_{y})=\left(0,0, \frac{\partial S}{\partial y}, -\frac{\partial S}{\partial x}\right)~.$$ (9)

The components of the T.E. vector potential are

$$( A_{t}, A_{z}, A_{x}, A_{y})=\left(0,0, \frac{\partial \psi}{\partial y}, -\frac{\partial \psi}{\partial x}\right)~,$$ (10)

and those of the e.m. field are

$E_{long.}:$	$F_{zt}=0$
$E_x:$	$$F_{xt}= -\frac{\partial }{\partial y}\frac{\partial \psi}{\partial t }$$
$E_y:$	$$F_{yt}= \frac{\partial }{\partial x}\frac{\partial \psi}{\partial t}$$
$B_{long.}:$	$$F_{xy}= -\left( \frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} \right) \psi$$
$B_x:$	$$F_{yz}= \frac{\partial }{\partial x} \frac{\partial \psi }{\partial z}$$
$B_y:$	$$F_{zx}= \frac{\partial }{\partial y} \frac{\partial \psi}{\partial z}$$

These components are guaranteed to satisfy all the Maxwell field equations with T.E. source, Eq.(9), whenever $\psi$ satisfies the inhomogeneous scalar wave equation, Eq.(5).