A condition for simultaneous propagation of coaxial Gaussian electromagnetic beams in a plasma, without convergence or divergence

For coaxial Gaussian electromagnetic beams of equal width, a relation between the weighted sum of axial irradiances and the width of the beams, which corresponds to the propagation in a plasma, without convergence or divergence has been obtained in the paraxial approximation. The collisional (with and without thermal conduction), ponderomotive and relativistic nonlinearities have been separately considered in this investigation. The investigation has relevance to the study and application of multi-wave interaction phenomena such as the growth of combination frequencies, plasma beat wave acceleration and enhanced penetration in an overdense plasma.