Can one detect a single photon through measuring its impulse/momentum on a mirror? Can one detect a single photon through measuring its impulse/momentum on a mirror?
If the answer if YES or theoretically possible, photon path and interference fringes can be detected simultaneously in interference experiments, such as     https://en.wikipedia.org/wiki/Lloyd%27s_mirror
Thus, particle-wave complementarity principle will be violated.
If the answer is NO, what is the theory behind it? 
 A: This paradox is an artifact of describing the mirror classically. Quantum mechanics is itself free from paradoxes, and it is easy to see that this paradox vanishes the moment you describe the mirror as a quantum mechanical object. A simple way to see what goes wrong is to apply the uncertainty relation. Suppose that we have a freely floating mirror and we can measure it's change in momentum to extract the "which way information". You can then only detect the change in the momentum of the mirror if the uncertainty of the mirror's momentum is smaller than the change of the momentum of the photon, which is of the order of $\frac{h}{\lambda}$. This means that the uncertainty in the mirror's position must be of the order of at least $\lambda$. But, of course, if the uncertainty in the mirror's position should be larger than $\lambda$ then you'll have a hard time detecting any fringes.
You can then ask what a realistic uncertainty in the mirror's momentum should be. It can be shown that an object of mass $M$ in a heat bath at temperature $T$ will have a localized wavefunction due to interactions with the heath bath. The typical quantum mechanical uncertainty is of the order of $\sqrt{M k T}$.
