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According to many sources, the minimum power required to maintain a population N in the upper level of a two level laser system is $$P_{min}=fA_{21}Nh\nu$$ where $f$ accounts for the lack of 100% efficiency, $A_{21}$ is the Einstein A coefficient and $\nu$ is the frequency of the transition. This is just the energy required to counteract the spontaneous decay in the system.

However I don't really understand why this is a minimum energy? A possibility is that if some external radiation field is applied we should also get an imbalance in the rates of spontaneous emission and absorption and so is this the extra contribution that we are missing in the above calculation (if not, why not)? Thanks in advance!

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  • $\begingroup$ Are you suggesting that the additional field will raise or lower the minimum power due to the effect of stimulated emission? $\endgroup$ – garyp Jan 6 '16 at 14:03
  • $\begingroup$ @garyp well you want more stimulated emission than absorption in a laser so the required power would be larger than the minimum power (the minimum power wouldn't change though because it by definition is the minimum) $\endgroup$ – Watw Jan 6 '16 at 14:08
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    $\begingroup$ Can you provide a source for your first statement? I didn't think a two level system could have a population inversion... $\endgroup$ – Floris Jan 6 '16 at 14:51
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    $\begingroup$ It seems your expression is valid for maintaining the population inversion between the two levels of a four-level laser between which laser action occurs. $\endgroup$ – Floris Jan 6 '16 at 14:53
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It is not possible to have a population inversion in a system with just two states.

The trick with any laser is that you need three possible states - let's call them 1, 2 and 3 (in increasing order of energy - see image from wikipedia):

enter image description here

If we can make it so that the 2->1 transition is much slower than the 3->2 transition, then you can create a population inversion, but you need quite a lot of power (since you are exciting from the ground state, you need to excite more than half of all the atoms).

For this reason, in practice, most lasers actually are four-level: this means that the two levels between which transition occurs contain a minority of the atoms, and it's possible to create a population inversion between these two levels with minimal power. At that point, "minimal" is however much power is needed to counter the spontaneous decay taking place (which is the expression given in your question) multiplied by some factor describing losses (for example, this will include the energy lost in going from state 4->3 and from 2->1)

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