Soliton model of a photon propagating in dielectrics

Abstract

The previously proposed three-dimensional soliton model of a photon propagating in vacuum is modified to describe its propagation in a homogeneous linear dielectric medium. The one-soliton solution of the derived nonlinear equation correctly predicts the energy and the Abraham and Minkowski momenta of the photon in dielectrics. A new nonlinear equation is proposed, which has a one-soliton solution that at every point oscillates with the same frequency and falls exponentially in the longitudinal, as well as in the transverse direction from the centre of the soliton.

Model of compact 3D electromagnetic solitons

“…The soliton in our 3D model propagates with the velocity of light and the direction of propagation is defined by the vector k  in the variable w = u t kr.   Expression (10) describes the compact 3D electromagnetic soliton, which propagates with the speed of light and its every point oscillates with the frequency w. Two independent parameters m and m determine the size of the soliton: the field on the symmetry axis decreases as , another constraint based on the energy expression was used [20,21]. The energy density of the electromagnetic wave in the Maxwell theory is defined as…”

“…where r-the polar radius, x 0 and y 0 define the centre of the field in the xy plane. The equation (1) was constructed [20,21] on the basis of the Maxwell equations for vector potential. The essential proposal was that a vacuum is a medium where light induces the polarization and magnetization components along the direction of the light propagation.…”

“…It was proposed that the field is axially symmetric, so instead of two variables x and y we have only one variable r-the polar radius, given by (4), and the problem is reduced to the construction of the operator T ˆin the equation r ¶ ¶ + = A TA 0. / Some mathematical, physical and dimensional arguments for this construction are given in [20,21], where equation (1) is proposed. If the dimensionless constant m in equation (1) goes to zero, we obtain the traditional plane wave solution.…”

“…This model is called a power model. The variation of the model (exponential model) is proposed [21] by introducing the additional term…”

“…The field of the soliton in this model decreases exponentially in the transverse direction too and the frequency of oscillations changes very little with r changing (especially for a small value of the constant m). The obtained 3D solitons were proposed [20,21] as a model of the photons. The field of the solitons looks physically more or less suitable but the applicability of this model for the photons is, of course, an open question.…”