Optical leptons

Abstract

We obtain optical vortices with classical orbital momentum ℓ=1 and spin j=±1/2 as exact solutions of a system of nonlinear Maxwell equations (NMEs). Two kinds of Kerr-type media, namely, those with and without linear dispersion of the electric and the magnet susceptibility, are investigated. The electric and magnetic fields are represented as sums of circular and linear components. This allows us to reduce the NME to a set of nonlinear Dirac equations (NDEs). The vortex solutions in the case of media with dispersion admit finite energy, while the solutions in case of media without dispersion admit infinite energy. The amplitude equations are obtained from equations of nonstationary optical and magnetic response (dispersion). This includes also the optical pulses with time duration of order of and less than the time of relaxation of the media (femtosecond pulses). The possible generalization of NME to a higher number of optical components and a higher number of ℓ and j is discussed

Dark states and vortex solitary wave with finite energy of Maxwell-Dirac nonlinear equations

“…In this paper we use a new kind of complex presentation, where the electrical and magnetic components are separated [10,11]. This allow us to obtain two kind of Pauli spinors, the first one associated with the first Maxwell equation and the linear and nonlinear polarization of the electrical field, and the second one associated with second Maxwell equation and the magnetization of the media.…”

“…In this way the well-known effect as linear and nonlinear polarization and magnetization of the media are mixed and there are no clear physical interpretation in this situation. This is the reason in our investigation to look for new kind of complex presentation, where the linear and nonlinear polarization effects are separated form magnetization of the media [10,11]. This new kind of complex presentation allow us to obtain two kind of Pauli spinors, where the first one is associated with the first Maxwell equation and the polarization of the electrical field, while the second one is associated with second Maxwell equation and the magnetization of the media.…”

“…In particular, they keeps on the difference between the linear     and nonlinear       3 ˆ dispersion of the media, and the influence of the magnetic permeability 0  . After a special complex transformation of the amplitude functions of NMEs right way we obtain nonlinear Dirac equations with nonlinear terms in first Pauli spinor of the Dirac field only [11,12]. This asymmetry is due of certain physical principlesin arbitrary media, including the nonlinear vacuum, the polarization of the media dominate with 5-6 order of magnitudes than magnetization.…”

“…In this section, we derive the amplitude approximation of Maxwell equations (1-5) (NME) with no-stationary linear and nonlinear electric responses (6, 7) in a standard way [11]. We define the electric and magnetic fields amplitudes by the following relations:…”