Why the emitted photon has exactly the same energy,phase and direction as the incident photon in stimulated emission?

My textbook says:-

When an atom emits a photon due to its interaction with a photon incident on it,the process is called stimulated emission.the emitted photon has exactly the same energy,phase and direction as the incident photon.

Why the emitted photon has exactly the same energy,phase and direction as the incident photon in stimulated emission?Do we have a mathematical proof?

• This answer by @knznou gives some intuition. – Chiral Anomaly Nov 6 at 14:31
• @Chiral Anomaly Hi,I got some intuition but couldn't understand how it is related to stimulated emission? – VK_fan Nov 6 at 15:36
• Photons are bosons, so the statistics apply. But the analysis of the coherence of a laser is a challenging business. There's a remarkably short description here. But an intuitive understanding is elusive. – garyp Nov 6 at 21:08

1. Direction

You need a background photon field, and population inversion. Basically spontaneous emission happens in a random direction, and stimulated emission happens in the direction of the background photon field.

The background field creates extra photons, helping the transition from higher energy levels to lower levels

Stimulated emission is the inverse process of absorption, both effect happen at the same time, it is just that because of the background photon field, overproduction happens, and more are emitted in the same direction as absorbed.

Coming back to lasers specifically. It is essential that you have a population inversion. That means you must have a configuration such that the number of electrons Nk around a certain energy Ek is bigger than that of another mode Ni where Ei

https://en.wikipedia.org/wiki/Stimulated_emission

1. Frequency

You need an optical resonator. This helps the production of a certain frequency of photons.

https://physics.stackexchange.com/a/445279/132371

Its my understanding that stimulated emission is a resonant process. It only occurs if the emitting electron is an excited state in which is is ready to emit a photon which is identical to the stimulating photon. To produce a laser beam, electrons in the source are “pumped” to a selected excited state. When one of those emits a photon in the right direction, a cascade begins which is enhanced and shaped by multiple reflections.