Why is the photon emitted in the same direction as incoming radiation in Laser?

When an atom “lases” it always gives up its energy in the same direction and phase as the incoming light. Why does this happen? How can this be explained? How does the photon generated because of stimulated emission, know which direction to take? What are the factors leading to this?

Answer

The word "stimulated" means that the emission of the photon is "encouraged" by the existence of photons in the same state as the state where the new photon may be added. The "same state" is one that has the same frequency, the same polarization, and the same direction of motion. Such a one-photon state may be described by the wave vector and the polarization vector, e.g. $|\vec k,\lambda\rangle$.

The physical reason why photons like to be emitted in the very same state as other photons is that they are bosons obeying the Bose-Einstein statistics. The probability amplitude for a new, $(N-1)$-st photon to be added into a one-photon state which already has $N$ photons in it is proportional to the matrix element of the raising operator $$ \langle N+1| a^\dagger|N\rangle = \sqrt{N+1}$$ of the harmonic oscillator between the $N$ times and $(N+1)$ times excited levels. Because the probability amplitude scales like $\sqrt{N+1}$, the probability for the photon to be emitted into the state goes like the squared amplitude i.e. as $N+1$. Recall that $N$ is the number of photons that were already in that state.

This coefficient $N+1$ may be divided to $1$ plus $N$. The term $1$ describes the probability of a spontaneous emission – that occurs even if no other photons were present in the state to start with – while the term $N$ is the stimulated emission whose probabilities scales with the number of photons that are already present.

But in all cases, we must talk about "exactly the same one-photon state" which also means that the direction of the motion is the same. It's because quantum field theory associates one quantum harmonic oscillator with each state i.e. with each information $\vec k$ about the direction of motion and wavelength; combined with a binary information about $\lambda$, the polarization (e.g.left-handed vs right-handed).