Rotational Doppler shift of a phase-conjugated photon

The rotational Doppler shift of a photon with orbital angular momentum ±ℓℏ is shown to be an even multiple of the angular frequency Ω of the reference frame rotation when the photon is reflected from the phase-conjugating mirror. The one-arm phase-conjugating interferometer is considered. It contains N Dove prisms or other angular-momentum-altering elements rotating in opposite directions. When such interferometer is placed in the rotating vehicle, the δω=4(N+1/2)ℓ·Ω rotational Doppler shift appears. As a result, the helical interference pattern will rotate with angular frequency δω/2ℓ. The accumulation of angular Doppler shift via successive passages through the N image-inverting prisms is due to the phase conjugation; for a conventional parabolic retroreflector, the accumulation is absent. The features of such a vortex phase-conjugating interferometry at the single-photon level are discussed.