Understanding Energy Partition in Laser Cavity: Molecules vs. Radiation

[Phrak]

A laser cavity consists of a couple mirrors, a bunch of metastable molecules and some radiation. What's the partition of energy between molecules and radiation at equilibrium?

 

[Cthugha]

That depends on a lot of stuff.
I suppose you want to take the fraction of molecules, which are in some excited state, into account for calculating the amount of energy you assign to the molecules.

Still there are a lot of parameters: Are you operating near, at, above or far above threshold? How high is the Q factor (how good are the mirrors). On what order is the beta factor. How many modes of the radiation field do you need to take into account? How many molecules are there inside the laser? Do you have a three level laser or a four level laser? Is there mode competition?
And so on and so on...

The easiest way might be to formulate laser rate equations for the laser design you are interested in and to solve them using mathematica or some other program for scientific computing.

 

[Phrak]

That depends on a lot of stuff.
I suppose you want to take the fraction of molecules, which are in some excited state, into account for calculating the amount of energy you assign to the molecules.

Still there are a lot of parameters: Are you operating near, at, above or far above threshold? How high is the Q factor (how good are the mirrors). On what order is the beta factor. How many modes of the radiation field do you need to take into account? How many molecules are there inside the laser? Do you have a three level laser or a four level laser? Is there mode competition?
And so on and so on...

The easiest way might be to formulate laser rate equations for the laser design you are interested in and to solve them using mathematica or some other program for scientific computing.

Thanks, Cthugha. After some punching around, I found the keywords I needed; "population inversion." So in the simplest cases, at least, the relationship is quite different than I had thought, where the ratio of excited state molecules vs. ground state molecules is the interesting number, and is independent of the radiation intensity.

 

[Cthugha]

So in the simplest cases, at least, the relationship is quite different than I had thought, where the ratio of excited state molecules vs. ground state molecules is the interesting number, and is independent of the radiation intensity.

The population inversion is usually not independent of the radiation intensity. But I suppose in the case of many emitters, fast pumping and a not too high Q factor, you will always end up in a steady state regime, where relaxation oscillations will always guide you back to this steady state for a wide range of the distortions of the photon number inside the laser cavity.

 

[Phrak]

The population inversion is usually not independent of the radiation intensity. But I suppose in the case of many emitters, fast pumping and a not too high Q factor, you will always end up in a steady state regime, where relaxation oscillations will always guide you back to this steady state for a wide range of the distortions of the photon number inside the laser cavity.

hmm. Perhaps I scanned the relevant portion of the Wiki article too quickly.

For a two state system they give the equation

$$\frac{N_2}{N_1} = exp \left( \frac{-(E_2-E_1)}{kT} \right)$$

where no dependence on the radiation is contained.

 

[Cthugha]

hmm. Perhaps I scanned the relevant portion of the Wiki article too quickly.

For a two state system they give the equation

$$\frac{N_2}{N_1} = exp \left( \frac{-(E_2-E_1)}{kT} \right)$$

where no dependence on the radiation is contained.

Right, this is the ratio of the populations of the two states in thermal equilibrium without additional radiation. As you can see, N2 will always be smaller than N1 or at most equal to N1 at extrely high temperatures, so you will have no population inversion in thermal equilibrium, which is a sensible result. However to achieve lasing, you need population inversion and therefore some nonequilibrium situation - usually some 3- or 4-level optical pumping scheme.

 

[Phrak]

That makes sense. What I don't see is how the temperature is involved.

 

[Cthugha]

Well, as usual the temperature becomes important if the thermal energy corresponding to that temperature becomes comparable to the difference between the energy levels.

 

[Phrak]

Well, as usual the temperature becomes important if the thermal energy corresponding to that temperature becomes comparable to the difference between the energy levels.

I'm not sure what you mean by that. Isn't there plenty of radiation at the difference frequency in the beam itself?

 

[Cthugha]

I'm not sure what you mean by that. Isn't there plenty of radiation at the difference frequency in the beam itself?

Well, yes of course. I thought you are still referring to the equation you posted above as that equation is usually just taken to show that there is no inversion in equilibrium independent of how high the temperature is. As soon as you have steady state lasing emission, the rolle of temperature is usually not that crucial anymore.

The role temperature plays in lasers depends strongly on the lasers. For example you might change the band gap of the gain medium in semiconductor lasers so that you move the main emission frequency out of resonance with the laser cavity. However the influence of temperature depends very strongly on the kind of laser you look at.

 

[Phrak]

Well, yes of course. I thought you are still referring to the equation you posted above as that equation is usually just taken to show that there is no inversion in equilibrium independent of how high the temperature is. As soon as you have steady state lasing emission, the rolle of temperature is usually not that crucial anymore.

The role temperature plays in lasers depends strongly on the lasers. For example you might change the band gap of the gain medium in semiconductor lasers so that you move the main emission frequency out of resonance with the laser cavity. However the influence of temperature depends very strongly on the kind of laser you look at.

That's a very practicle point of veiw. From what I've been reading, the temperature would only have a significant effect, at perhaps 10K to 20K degrees, for a beam in the visible spectrum.

From a less practical point of veiw, there are two questions I find interesting about lasers. Why is there stimulated emission, and how did Einstein predict it? And how does temperature play a role in the partition of energy between bosons and fermions?

 

[Cthugha]

That's a very practicle point of veiw. From what I've been reading, the temperature would only have a significant effect, at perhaps 10K to 20K degrees, for a beam in the visible spectrum.

As I said before, there is no general answer. In VCSELs you will be able to change the coupling of the gain medium to the cavity if you change the temperature, while other lasers might not react that strongly to temperature changes. Also the linewidth of your gain medium might change with increasing temperature, which is for example important in QD lasers. However in most cases temperature is not the most important aspect in a laser.

From a less practical point of veiw, there are two questions I find interesting about lasers. Why is there stimulated emission, and how did Einstein predict it?

Well, from a more modern point of view, you can apply Fermi's golden rule to the laser transition and will find out, that the process is symmetric. This means that the probabilities of stimulated absorption and stimulated emission are equal if the densities of states and occupations are equal.Einstein used a different formal treatment, but had bsically the same idea. This is also the reason, why 2 level laser schemes do not work.

And how does temperature play a role in the partition of energy between bosons and fermions?

In the steady state lasing regime with lots of resonant photons present? If the laser is below the threshold temperature has some effect. For example the ratio of radiative transitions from the excited state to the ground state to nonradiative transitions (by phonon scattering) depends strongly on temperature, especially if the laser is pumped nonresonantly and needs some spontaneous emission to turn on. But generally speaking the effect of temperature is just really important if you do not have lasing and stimulated emission present. In steady state lasing it does not matter much.