Exact and approximate solutions of Maxwell's equations for a confocal cavity

The scalar Gaussian wave, which is the usual solution of the eigenfunction problem for an open laser resonator, does not satisfy Maxwell's equations. We obtain the exact solution by means of the Hertz vector instead of the more usual electric field. The exact solution (that satisfies Maxwell's equations) is paraxially approximated, and the approximation fails to satisfy Maxwell's equations. It is shown that for the Gaussian wave the electric and the magnetic vectors are not interchangeable in coordinates, which is a consequence of the inhomogeneous boundary condition.