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Short question: Why is the light emitted by a laser coherent?

Respectively, why is stimulated emission coherent?

edit: How does a laser emit light in a coherent state? seems to be connected to the question but is, as far as I understand, not really aimed at it

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    $\begingroup$ Thanks, but on the provided link is no explanation why it is coherent. jFYI: For sure I tried to find an answer already somwhere else but I didn't. $\endgroup$ – Ben Jan 19 '18 at 14:53
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    $\begingroup$ I admit I am at the edge of my understanding with the answer cited, but +1 as I would like a bit more explanation myself. :) $\endgroup$ – user181180 Jan 19 '18 at 16:02
  • $\begingroup$ Is it too simplistic to say that each new emission is triggered by an existing photon and so is in-phase with it, and so on? After that it is an engineering problem to ensure that the cavity remains the same and only one population of triggering photons gets in $\endgroup$ – Martin Beckett Jan 19 '18 at 16:30
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    $\begingroup$ Per the Bose-Einstein statistical distribution for bosons, photons are more likely to be found in the same state than in different states. For this reason, a photon passing by an exited atom triggers a stimulated emission of another photon. So the BE distribution here is responsible for two things: (1) the simulated emission (the second photon "wants" to fly along with the first photon) and (2) the fact that the emitted photon has the same phase and polarization (the second photon "wants" to "look" like the first one). Two parallel mirrors then do the rest allowing only one state to win. $\endgroup$ – safesphere Jan 19 '18 at 17:33
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First of all, note that there are two relevant forms of coherence for a laser: temporal (where the phase of the field is steady with time, i.e. the field is single-frequency) and spatial (the phase of the field is consistent across the width of the beam).

Assuming you're thinking of temporal coherence, perhaps try thinking about this classically and starting with a related question:

Why is the light that exits a clear piece of glass coherent (same frequency, phase, and direction) with the light entering?

One way to approach this is to imagine the light field as a time-varying perturbation on the atoms with which it is interacting. Classically, the E-field sinusoidally accelerates the electrons, getting absorbed into their motion. When those accelerating charges re-emit the radiation, it will be with the same phase, frequency, and polarization as the exciting field because that's just how the charges happen to be moving--they were driven that way by the field. Thus the light emerges the same color and phase it had going in, and no one is surprised.

Stimulated emission is a similar situation. While (obviously) more complex in certain ways, again one can add a sinusoidally-varying field to the electron Hamiltonian. In a population-inverted gain medium, this will have the effect of causing the electrons to evolve to a lower energy state, emitting light. What color and phase will the light have? Well, the same as the excitation light since that's just how the electrons happen to be moving. There is simply no other phase and frequency for them to have, because there is no intermediary between them and the driving field; they are directly driven.

All of this is to say that any process where the emission is directly driven by an external field will preserve the phase information. These are the coherent processes, including, for example, nonlinear second-harmonic or difference-frequency generation. These are in contrast with incoherent processes, such as photoluminescence, which are basically forms of spontaneous emission. Incidentally as well, preserving the frequency information of the exciting field is a matter of conservation of energy, since photon energy equals frequency times Planck's constant.

So this just describes stimulated emission. In a laser, you have feedback, the dynamics of which add another layer of complexity and result in the formation of a single dominant state existing in the cavity. A topic for another day, I suppose.

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    $\begingroup$ I like the explanation, thanks! Nevertheless it's (of course ?) more visual than exact? I struggle with the circumstance, even if excited atom/electron is driven by the electromagnetic field of the incident photon, that this leads to an deexcitation of the same phase and so on. This is probably now going in to theoretical details but do you know the mathematic connection/description between the incident and "both" outgoing photons? Btw: Will the incident photon be influenced in this scenario? $\endgroup$ – Ben Jan 22 '18 at 13:29
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There are three characteristics for lasers:

  1. The laser emits photons of nearly the same wavelength. This has to do with the used gain medium and its purity.
  2. The electric field component of photon has nodes at the mirrors of the lasers cavity. This are called longitudinal modes. Only these photons will get reflected from the mirrors with nearly the same frequency, the others fade out to heat.
  3. The mirrors of the cavity could have different shapes. From the shape depends the transversal modes of the laser. This and the use of half-reflecting mirrors could make the laser beam polarized.

(The links are going to the German Wikipedia. For that I’m sorry but on the English wiki this information is not available.)

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