Rotational Doppler shift of a phase-conjugated photon

The rotational Doppler shift of a photon with orbital angular momentum $\pm \ell \hbar$ is shown to be an even multiple of the angular frequency $\Omega$ of the reference frame rotation when photon is reflected from the phase-conjugating mirror. We consider the one-arm phase-conjugating interferometer which contains $N$ Dove prisms or other angular momentum altering elements rotating in opposite directions. When such interferometer is placed in the rotating vehicle the $\delta \omega=4 (N+1/2) \ell \cdot \Omega$ rotational Doppler shift appears and rotation of the helical interference pattern with angular frequency $\delta \omega /{2 \ell}$ occurs. The accumulation of angular Doppler shift via successive passage through the $N$ image-inverting prisms is due to the phase conjugation, for conventional parabolic retroreflector the accumulation is absent. The features of such a vortex phase conjugating interferometry at the single photon level are discussed.