Hertz vector

It can be shown that Maxwell Eqs. (35) are satisfied if the Hertz vectors have the form... 

As the refractive index within the slice depends on the vertical coordinate only, TE and TM modes can propagate independently in the slices, and Eqs (35) and (36) are fully applicable to this geometry without any change, too. Since we are interested in the propagation of waves in the azimuthal direction, it is advantageous to decompose the Hertz vectors into the eigenmodes of the slice as follows ... 

Let us carefully note that the emergence of Beltrami field relations from the Hillion-Quinnez model, is not due to an ad hoc rendering of the EM field relations, but is a logically consistent result that follows from recognizing that any electromagnetic field in a homogeneous, isotropic medium in free space, away from charges and currents, can be expressed either in terms of the spinors fa, or in terms of the complex Hertz-vector II. 

Substituting (3.A. 11) into (3.A. 10), we see that the Hertz vector satisfies a wave equation, with a simple source term — (<5e) 4,... 

By this means the problem of evaluating the radiation field originating from the wave (3.56) is reduced to the problem of evaluating the radiation field of a polarization wave with a constant amplitude. This field can be found far from the surface in terms of the Hertz vector which satisfies a nonuniform wave equation with the polarization vector as a source (Bom and Wolf 1975). 

It is known that the complete electromagnetic field in each slice s can be expressed in terms of Hertz electric and magnetic vectors rf and rf that are both parallel with the x coordinate axis. (It is a consequence of the invariance of the refractive index distribution of the slice with respect of the coordinates y and x). The field components in the slice s are given by the expressions... 

In utilizing a complex three-vector (self-dual tensor) rather than a real antisymmetric tensor to describe the electromagnetic field, Hillion and Quinnez discussed the equivalence between the 2-spinor field and the complex electromagnetic field [63]. Using a Hertz potential [64] instead of the standard 4-vector potential in this model, they derived an energy momentum tensor out of which Beltrami-type field relations emerged. This development proceeded from the Maxwell equations in free homogeneous isotropic space... 

Figure A.8 shows the time evolution of product operators as a result of /-coupling evolution alone. In-phase coherence continuously progresses into antiphase and back to in-phase as the two component vectors (a and /3) counterrotate in the x -y plane. The angle of rotation in all cases is nJt in radians (where / is in hertz). This corresponds to tt/2 or one fourth of a full rotation for time equal to 1/(27). To write out the general time...

It will prove useful in the sequel to use the hertz potentials (Born and Wolf, 1980, pp. 76-84) to describe the electromagnetic field. The hertz potentials also satisfy the homogeneous vector Helmholtz equation in free space. The advantage of the hertz potentials is that they display much higher symmetry than the conventional vector and scalar potentials. Furthermore, they may be written in a form that displays the paraxial approximation of quasioptics directly. 

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