An explicit representation for the axisymmetric solutions of the free Maxwell equations
Résumé
Garay-Avenda\~no \& Zamboni-Rached (2014) define some time-harmonic axisymmetric free Maxwell fields in explicit form by starting from a ``wave vector spectrum" $S(k)$, which allows one first to define a solution $\Psi $ of the scalar wave equation as a sum of Bessel beams. To that scalar wave, they associate two distinct EM fields: the first one derives from the vector potential ${\bf A}=\Psi {\bf e}_z$ (with ${\bf e}_z$ the unit vector along the symmetry axis $z$), and the second one is deduced from the first one by EM duality. We show that actually this method (with its two variants) allows one to obtain all time-harmonic axisymmetric free Maxwell fields, and hence, by summation on frequencies, all axisymmetric free Maxwell fields. It provides an explicit representation for these fields. This will be important, e.g., to have the interstellar radiation field in a disc galaxy modelled as an exact solution of the free Maxwell equations.
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