Deducing radiation pressure on a submerged mirror from the Doppler shift

Radiation pressure on a flat mirror submerged in a transparent liquid depends not only on the refractive index n of the liquid but also on the phase angle ψ ₀ of the Fresnel reflection coefficient of the mirror, which could be anywhere between ⁰ and ¹⁸⁰. Depending on the value of ψ ₀, the momentum per incident photon picked up by the mirror covers the range between the Abraham and Minkowski values, i.e., the interval (2ω ₀/nc, 2nω ₀/c). Here is the reduced Planck constant, ω ₀ is the frequency of the incident photon, and c is the speed of light in vacuum. We argue that a simple experimental setup involving a dielectric slab of refractive index n, a vibrating mirror placed a short distance behind the slab, a collimated, monochromatic light beam illuminating the mirror through the slab, and an interferometer to measure the phase of the reflected beam is all that is needed to deduce the precise magnitude of the radiation pressure on a submerged mirror. In the proposed experiment, the transparent slab plays the role of the submerging liquid (even though it remains detached from the mirror at all times), and the adjustable gap between the mirror and the slab simulates the variable phase angle ψ ₀. The phase of the reflected beam, measured as a function of time during one oscillation period of the mirror, then provides the information needed to determine the gap dependence of the reflected beam's Doppler shift and, consequently, the radiation pressure experienced by the mirror.