Electromagnetic Doppler effect in non-reciprocal medium

Abstract

Reflected light from a moving object, such as a mirror, is frequency-shifted via the Doppler effect. This well-known phenomenon is used to determine the speed and direction of an object due to the red or blue shifting of the received electromagnetic radiation frequency. To date, the Doppler theory has focused on the propagation of electromagnetic radiation through reciprocal media where the transiting photons have the same speed, regardless of direction. However, optical birefringence exists in solid-state, liquid, and gas media, which results in spatially dependent, and vector field-dependent, propagation velocities. When combined with the Faraday effect, which is employed routinely in laboratory setups and present at galactic scales, non-reciprocal Doppler effects may occur. A non-reciprocal Doppler effect theory is derived showing frequency shift dependencies on the object velocity and the directional speed of light in the medium. This general Doppler theory is shown to simplify the canonical relations for the Doppler effect in systems without a directional dependence on light. In non-reciprocal systems, when the mirror velocity is much less than the speed of light, the general frequency shift relation simplifies to a dependency on the roundtrip average speed of light. This theory provides a basis for the application of the Doppler effect to estimate the velocity of moving objects in non-reciprocal systems.