Can a photon increase its frequency by bouncing against a mirror moving toward it? As discussed in preceding questions, photons can lower their frequency after a reflection happens and radiation pressure sets the mirror in motion.
I wonder if the opposite can happen, i.e. increasing a photon frequency by transfering momentum to it.
And if so, does it have to do with the blue shift of Doppler effect?
 A: Conserning the first question, yes. The reasoning is the same, as in that post. Write the conservation of energy and momentum:
$$
\begin{align}
k + m v &= k^{'} + mv^{'} \\
k + \frac{1}{2} m v^2 &= k^{'} + \frac{1}{2} m v^{'2}
\end{align}
$$
Where unprimed indices denote the initial momentum of the photon and velocity of the mirror, and primed - after the reflection. Now, part of the kinetic energy of the mirror is transferred to the photon.
Concerning the relationship with the Doppler effect, also you may have such an interpretation. In the frame of reference, moving with the mirror, you now think of the photon, as being emitted from the source, moving towards the mirror with relative velocity $v$. In this frame, after reflection, momentum would be same in magnitude, opposite direction after reflection. Now return to the laboratory frame, and the mirror is a moving emitter. So the Doppler effect occurs twice from this point of view.
A: If the mirror is moving towards the (source) beam of photons at relativistic speeds, then yes, the laser's frequency will get blue-shifted with respect to the source's emitted frequency after it bounces off the mirror. Similar effect happens with sound waves as well.
