Three-dimensional transformation for point rotation coordinate frames

TL;DR

This paper develops a three-dimensional transformation for point rotation coordinate frames to analyze light modulation in electrooptical crystals, showing that certain wave interactions do not produce frequency shifts, contrary to previous claims.

Contribution

It introduces a novel three-dimensional transformation for point rotation frames and clarifies conditions under which optical frequency shifts occur or do not occur.

Findings

Light propagating through 3-fold electrooptical crystals does not produce an extra optical frequency shift.

Modulation by a rotating electric field can produce a frequency shift.

The transformation properties are discussed and formalized.

Abstract

We consider the transformation for the point rotation frames with the angle, spatial coordinate along the axis of rotation and time as variables. The problem arises when light, propagating through 3-fold electrooptical crystal, is modulated by the circularly polarized electromagnetic wave traveling along the optical axis of the crystal. With help of the transformation we show that such a wave cannot produce an extra optical frequency shift discussed earlier. In contrast to that the modulation by the rotating spatially invariable electric field produces the shift. The formal change to this case may be carried out by reducing the velocity of the traveling wave to zero. Some properties of the three-dimensional transformation are discussed.