Invariants of 3D Transformation for Point Rotation Coordinate Frames

TL;DR

This paper explores the invariants of 3D transformation in point rotation coordinate frames, which have a rotating axis at each point and are relevant in optical and electrooptical applications, especially in modulators.

Contribution

It introduces the concept of invariants in 3D transformations specific to point rotation frames, extending understanding beyond Cartesian coordinate systems.

Findings

Identifies invariants of 3D point rotation transformations.

Links the mathematical framework to optical indicatrix rotation.

Discusses potential applications in frequency modulation and shifting.

Abstract

Recently the general linear transformation for point rotation coordinate frames was considered. A distinguishing feature of the frame, in contrast to the Cartesian one, is the existence of the rotation axis at every point. The frame coordinates are an angle and time, the frequency of rotation is a parameter. The concept of the frame originated from the optical indicatrix (index ellipsoid). Rotation of the optical indicatrix arises in three-fold electrooptical crystals under the action of the rotating electric field applied perpendicular to the optical axis \cite% {pat}. Such a rotation is possible as in the Pockels as Kerr crystals and also in the isotropic Kerr medium. The rotation is used in single-sideband modulators. The single-sideband modulation has very interesting features from the theoretical viewpoint. In applications it may be used for the frequency modulation and frequency shifting. In contrast to usual modulation such a shifting is "100 persent transformation" of the initial into output frequency. However at present the modulation practically is not in use. It is connected with the high controlling voltage of bulk modulators; creating waveguide single-sideband modulators calls for considerable technological efforts.