For a significant part of my life I have been taught that, if a photon of the "correct" energy meets an excited atom, the atom will then (with a certain probability) undergo transition to a lower energy state, emitting a photon, which 1)has the same energy as the incident photon, 2)propagates in the same direction, 3)and is in phase with that incident photon.
Unfortunately, this statement is rarely, almost never, followed by a discussion about reasons behind the stimulated emission being the way it is. After some searching I have even come up with a few texts claiming that the description above is "not exactly true", meaning that some variation in phase and energy of the stimulated photons is possible. Again followed by no explanation whatsoever.
Currently the most exhaustive discussion I have found is this: http://www.sjsu.edu/faculty/watkins/stimem.htm which is still far from perfect. At least, now I am somewhat convinced, that the two photons should propagate in the same direction. For those not willing to follow that link, the reasoning goes like this: Take two principles (symmetry of physical phenomena under time inversion, and the idea that physical phenomena should be isotropic in an isotropic medium) and apply them to all the variations of the stimulated emission you can think of: 1)the stimulated photon propagates in a random direction. Under time inversion this becomes simple absorption, where one of the spatial directions (in which the former incident photon is propagating) is undesirable; 2)the stimulated photon propagates in the same direction as the incident photon. Under time inversion this becomes completely normal and isotropic absorption (as expected).
However, this argumentation is valid only if we presume that the two photons are completely indistinguishable (identical) in the first place. Therefore a "blue" photon (say, 400nm) stimulating emission of a "red" (600nm) photon in a random direction would still be decent physical process. Another stackexchange question tackling this particular matter (can a photon stimulate emission of a different energy photon) here is apparently still waiting for a satisfactory answer.
Tl;dr / Long story short, why the photon produced in stimulated emission has the same (1)energy; (2)propagation direction and (3)phase as the incident photon? Pointers to books/papers and decent web resources are much appreciated!