How does stimulated emission occur in 3 level lasers?

Question

My understanding is that in 3-level lasers, we pump up a collection of atoms from the ground state of energy $E_1$ to some excited state $E_3$ with the help of monochromatic radiation of Energy $E_3-E_1$. Then the atoms in the excited state make a fast and preferably non-radiative transition to a metastable state $E_2$, after which the atoms return to the ground state emitting photons of energy $E_2-E_1$. Is this emission spontaneous or stimulated? If this is really only spontaneous then the photons must come out in different directions which is not seen in laser, so stimulated emission must be occuring somehow...But for getting stimulated radiation we need to stimulate the atoms in metastable level 2, with radiation of energy $E_2-E_1$ which we don't do.(We only pump from 1 to 3, nothing from 1 to 2). So if stimulated emission occurs at all, how does it occur without any incoming stimulating radiation?

My guess is that since the metastable states are taken to be very close to the excited state(for other reasons), so $E_3-E_1\approx E_2-E_1$.So we can pump the atoms with a nearly monochromatic light(with a small dispersion), so that the appropriate frequency of the incident radiation can stimulate too alongside pumping. Is this correct or stimulated emission occurs in some other way in 3 level lasers?

Answer 1

All you need to do is to start stimulated emission in the correct direction. Once it starts it will be self-sustaining if the gain (population in level 2) is high enough.

The question then becomes: how does it start?

The answer is spontaneous emission. It is true that spontaneous emission light is emitted in all directions, but one of those directions is right through the cavity in the correct direction. And if the population of 2 is high, which it will be if we expect to have enough gain to support lasing, there will be a lot of spontaneous emission and the likelihood of one photon going in the correct direction is very high.

In principle, you need only one spontaneous photon.

By the way, you suggest that pumping from 1 to 2 would produce the stimulated emission you need to get things started. On the contrary. Pumping from 1 to 2 produces stimulated absorption which is a loss mechanism, not a gain mechanism.

Answer 2

The image I am showing here can be found in my thesis and the references for the data are therein (thesis here). The left image shows the gain cross section of Yb:YAG, typically a quasi-three or three-level system, with the energy structure on the right. On the left the different curves are for a system in the ground state at room temperature ($\beta=0$) up to full inversion ($\beta=1$). If you pump the system at 969 nm (Low1 to Up1) you have a true 3-level system, pumped at 940 nm, a quasi-three-level system

So, you are confusing some terms and concepts of lasers, and your question would apply to both four- and three-level laser systems.

First: as Garyp explained, you need to understand that stimulated emission only occurs at the exact energy difference, as in your case $E2-E1$ for example and that the device built around your medium is what creates a positive feedback for the amplification to happen: one photon is spontaneously emitted in the direction of your laser cavity and is reflected back into the medium, allowing for this photon to now induce stimulated emission of the same energy.

Secondly, if you can clear that confusion that stimulated emission indeed occurs for a single energy, than you understand that your second question is a product of that confusion.

You see, you do not need the pump or any other mechanism to start the seeding process for stimulated amplification, because you build a resonator around your medium to allow for the positive feedback, to catch any spontaneously emitting photon that happens to be emitted in the direction of the laser cavity.

In a system you want to decouple the absorption of the pump, through stimulated absorption, as Garyp mentioned, from the actual laser at a higher wavelength (for several technical reasons). See the case of the image I showed you. I pump at 969nm, a wavelength (or frequency, if you convert it) on a line that is narrow and absorbs most of the light... Now the system gets a population inversion: I take electrons from the Low1 level to the Up1 level. This creates an imbalance between the populations of Up1 and Low3 (the wavelength at which I was operating my laser), with Up1 having more electrons than Low3. Now spontaneous emission, between those levels starts occurring, if any photon happens to be in the direction of my cavity then it gets amplified as this photon will be passed many times through the gain medium, taking electrons from the Up1 to the Low3 and increasing the number of present photons with 1030 nm.

Some interesting technicalities of Yb:YAG, operating at 1030 nm and pumped at 969nm:

-You can only achieve 50% inversion between Low1 and Up1, which at that point the medium becomes transparent: for every absorbed photon, on average, one is also stimulated.

-When pumped at this wavelength, the difference between the energies of the pump photons to those of the laser is only 6%, making it one of the lowest differences in all current solid-state lasers. This ensures low thermal energy deposition in the system after the Low3 level decays into the Low1 level.

Hope this helped clear some more confusion.