Volume 1: Chapters 1-10 by Brilliant Light Power

192

Chapter 4

Fi gu re 4.1 . The stationary Cartesian coordinate system xyz wherein the first great circle magnetic field line lies initially in

the xz-plane, and the second great circle electric field line lies initially in the yz-plane. The rotated coordinates are primed. z

 H



Consider a point on each of the two orthogonal great-circle field lines, one and two, in the basis-set reference frame at time zero wherein initially the first loop lies in the xz-plane, and the second loop lies in the yz-plane. Point one is at x '  rn ,

y '  0 , and z '  0 and point two is at x '  0 , y '  0 , and z '  rn . Let point one move clockwise on the great circle in the x'z'plane toward the positive z'-axis, and let point two move counterclockwise on the great circle in the y'z'-plane toward the negative y'-axis, as shown in Figure 4.1. The equations of motion, in the sub-basis-set reference frame are given by point one (H FIELD): x'  r cos( t ) 2

y2'  0

z2'  rn sin(nt )

(4.4)

point two (E FIELD): x'  0

(4.5) y1'   rn sin(n t ) z1'  rn cos(n t ) The right-handed-circularly-polarized photon electric and magnetic vector field (RHCP photon-e&mvf) and the lefthanded-circularly-polarized photon electric and magnetic vector field (LHCP photon-e&mvf) are generated by rotating the great 1

circles about the  i x , i y , 0i z  -axis or the  i x , i y , 0i z  -axis by



, respectively. The corresponding primed Cartesian coordinate 2 system refers to the axes that rotate with the great circles relative to the xyz-system and determines the basis-element reference frame. The fields are continuous on the spherical surface, but they can be visualized by a discrete-element representation wherein each element of the field-line density function is obtained with each incremental rotation of a series over the span of



. 2 Thus, the two points, one and two, are on the first member pair of the orthogonal great circles of an infinite series that comprises a representation of a photon. The right-handed-circularly-polarized photon electric and magnetic vector field (RHCP photon-e&mvf) shown in Figure 4.2 is generated by the rotation of the basis elements comprising the great circle magnetic field line in the xz-plane and the great circle electric field line in the yz-plane about the  i x , i y , 0i z  -axis by



corresponding to the output of the matrix given by Eq.

(4.6). RHCP PHOTON E FIELD and H FIELD:

 1 cos 2  2  x'      1 cos  y '   2  2  z '    sin   2

1 cos  2 2 1 cos  2 2 sin   2

sin   2   0    rn cos    sin           rn cos    0    2    rn sin    rn sin    Red Blue    cos  



(4.6)

The left-handed-circularly-polarized photon electric and magnetic vector field (LHCP photon-e&mvf) is generated by the rotation of the basis elements comprising the great circle magnetic field line in the xz-plane and the great circle electric field line in the yz-plane about the  i x , i y , 0i z  -axis by



corresponding to the output of the matrix given by Eq. (4.7). The mirror 2 image of the RHCP photon-e&mvf, the left-handed circularly polarized photon-e&mvf, is shown with three orthogonal views in Figure 4.3.