This was the given question:
A light beam of intensity $I$ and frequency $f$, directed along the
positive $z$-axis, is reflected perpendicularly from a perfect mirror which
itself is moving along the positive $z$-axis with a constant velocity $v$.
Find the reflected intensity $I'$ and the frequency $f'$ of the reflected light.
If a light beam is directed towards a mirror which is moving towards us with a constant velocity $v$, the apparent frequency of the reflected beam would be higher. However, I would like to know if there is any change in the observed intensity of the light beam.
Also, just rechecking, does the intensity of a light beam depend only on the amplitude?
Thanks in advance
Best Answer
As in User58220's answer, if the source is a point or small (i.e. diverging beam) source, the intensity varies as a real source placed at the virtual source's position - almost. There is a further increase in intensity owing to the Doppler shift alone.
So if your source is a collimated beam at right angles to the mirror's surface, all the frequency components in the beam are going to be blue shifted by the square $\frac{c+v}{c-v}$ of the more wonted Doppler frequency scale factor $\sqrt{\frac{c+v}{c-v}}$ that holds if the source is real (i.e. not a virtual one made by a mirror) moving towards you at speed $v$.
At the same time, the light's power intensity is scaled by this same factor $\frac{c+v}{c-v}$. You can think of this intensity scaling as arising from a conservation of photon number, but the reflected photons are now more energetic as the mirror inputs work to the beam. The result can also be derived classically using Lorentz transormations of the relevant electromagnetic boundary value problem and you can see details of this calculation in my answer to the Physics SE question "Can Planck's constant be derived from Maxwell's equations?". Indeed one can use this result (equality of Doppler and power scales) to motivate the idea that a photon's energy must be proportional to its frequency, so you can in a sense derive the quantum reasoning from the classical, although this reasoning won't tell you the value (i.e. $h$) of the scaling constant.