Diagonal relations between elegant Hermite-Gaussian and Laguerre-Gaussian beam fields

Standard Hermite-Gaussian and standard Laguerre-Gaussian diagonal beams can be expressed by similar summation of non-diagonal standard Hermite-Gaussian beams with expansion coefficients that are equal in magnitude in both cases. We show theoretically and numerically that the same relations hold also for elegant Hennite-Gaussian beams and elegant Laguerre-Gaussian beams. On the other hand, our theoretical estimations and numerical simulations show substantial differences between standard and elegant beams in their phase transverse distribution and evolution of this distribution during beam propagation.