Radiation pressure on a submerged absorptive partial reflector deduced from the Doppler shift

When a light pulse is reflected from a mirror, energy and momentum are exchanged between the electromagnetic field and the material medium. The resulting change in the energy of the reflected photons is directly related to their Doppler shifts arising from the change in the state of motion of the mirror. Similarly, the Doppler shift of photons that enter an absorber is intimately tied to the kinetic energy and momentum acquired by the absorber in its interaction with the incident light. The argument from the Doppler shift yields expressions for the exchanged energy and momentum that are identical with those obtained from Maxwell's equations and the Lorentz law of force, despite the fact that the physical bases of the two methods are fundamentally different. Here, we apply the Doppler-shift argument to a submerged partial reflector (one that absorbs a fraction of the incident light), deducing, in the process, the magnitude of the photon momentum within the submerging medium. We also discuss the case of the submerging medium having a negative refractive index and show the absence of the so-called "reversed" Doppler shift when the reflector is detached from the negative-index medium.