Embedding the Photon with Its Relativistic Mass As a Particle into the Electromagnetic Wave Explains the Gouy Phase Shift as an Energetic Effect

In the paper “Embedding the photon with its relativistic mass as a particle into the electromagnetic wave” a newaspect concerning the relationship between photon and electromagnetic wave has been developed by considering thequestion why the energy and the mass density of an electromagnetic wave are propagating in the same direction [1].For instance, in optical resonators the energy density usually propagates along curved lines. However, accordingto Newton’s first law the mass density should propagate along a straight line, if no force is exerted it. To solve thisproblem, the assumption has been made that a transverse force is exerted on the mass density and in consequenceon the mass of the photons which forces them to follow the propagating energy density. This leads to the result thatthe photon is moving within a transverse potential which allows describing the transverse quantum mechanicalmotion of the photon by a Schrödinger equation. These results are used to show that in case of a Gaussian wave theeffective axial propagation constant kz, nm (z) can be expressed as kz, nm (z) = [Eph ‒ Enm (z)] /ћc where Eph is the totalenergy of the photon, and the Enm (z) are the energy eigenvalues of the transverse quantum mechanical motion ofthe photon. Since according to this result ћckz, nm (z)] represents a real energy, it has been concluded that also theeffective axial propagation constant represents a real propagation constant. This leads to the conclusion that λnm(z) =2π/ kz, nm (z)=hc/(Eph ‒ Enm (z) represents the real local wave length of the electromagnetic wave at the position z.According to this conclusion, λnm (z) increases inversely proportional to the energy difference Eph-Enm (z), which decreaseswith decreasing z, and therefore describes the Gouy phase shift in agreement with wave optics. This shows that the deeperphysical reason for Gouy phase shift consists in the fact that the energy of the photon is increasingly converted into itstransverse quantum mechanical motion when the photon approaches the focus. This explains the Gouy phase shift as anenergetic effect.