# I Ratio between spontaneous/stimulated emission

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1. Jan 4, 2017

### greypilgrim

Hi.

In thermodynamic equilibrium, the ratio between spontaneous and stimulated emission is
$$\frac{A_{21}\cdot N_2}{B_{21}\cdot N_2\cdot u(f)}=e^{\frac{hf}{k_B T}}-1$$
where $A_{21}$ and $B_{21}$ are Einstein coefficients. This means, that there's always much more spontaneous than stimulated emission, and it's getting worse for higher frequencies.

What can be said about this ratio for lasers, i.e. if the population of energy levels is inverted (no thermal equilibrium)? Obviously this depends on the type of laser, I'm just asking about a qualitative statement: Is there still more spontaneous emission, more or less the same or far more stimulated emission? My guess would be the latter, but I'm not sure.

I also saw above calculation as an argument for the difficulty of building Röntgen or gamma lasers, saying that incoherent noise increases for higher $f$. However this seems flawed to me since it's based on the assumption of thermodynamic equilibrium (by using $u(f)$), and laser systems operate outside this equilibrium.

2. Jan 4, 2017

### Staff: Mentor

You don't need a laser. Any sufficiently intense source of radiation can lead to a predominance of stimulated emission, giving Rabi oscillations. This is especially true for forbidden transitions, where the rate of spontaneous emission is very low to start with.

I would agree with you that the above equation only applies to the thermal case, and lasing is definitely an out-of-equilibrium process. But the Einstein A coefficient is proportional to $f^3$, so the argument is valid, but not on the grounds of that equation.