Population Inversion

EP410: Chapter 9 Popula3on Inversion

Dinesh Kabra, Phys. IITB

²  In thermal equilibrium Nu<Nl ²  Pumping is required to overcome the Boltzmann distribu3on à Nu>Nl ²  Necessary condi3on (but not sufficient) ²  Two level system can not generate popula3on inversion ²  Inclusion of extra level is must à results in reduc3on in efficiency

Contents •  Inversions and 2-‐level systems •  Radia3ve decay rates – radia3ve vs collisional •  Steady-‐state inversions in 3 and 4-‐level systems

–  3-‐level where middle level is Eu –  3-‐level where highest level in Eu –  4-‐level system

•  Transient popula3on inversions •  Inversion destroying processes

–  Radia3on trapping –  Electron collisional thermaliza3on –  Absorp3on within gain medium –  Excited state absorp3on –  Ground state absorp3on

Inversions and two-‐level Systems Consider a two-‐level atomic system Ø  Gain length = L Ø  Atomic species = N Ø  Total decay rate = Υul Ø  Radia3ve decay rate = Aul Ø  By defini3on à Υul ≥ Aul Ø  Emission cross sec3on = σul(ν)

(valid for both homogeneous & inhomogeneous broadenings) At room temperature all the atoms are in lower level as per Boltzmann distribu3on Nl ≈ N and Nu ≈ 0 On op3cal excita3on with an intensity of I0 and frequency νul we find light gets absorb as it travels through the medium: For weak input intensity, light beam gets absorbed with an absorp3on coefficient of σul(ν)Nl

•  Our assump3on prior to this chapter was that we increase the pump intensity I0 to very high value that eventually all of the popula3on is transferred from level l to level u by absorp3on of the input beam.

•  This is not possible, because, as soon as a frac3on of the popula3on is pumped to level u, Nu becomes greater than zero. Since Nu = N-‐Nl, we can rewrite previous equa3on as

•  As the popula3on leaves level l, the ra3o Nl/N begins to drop below 1. When it reaches 0.5, no more energy will be absorbed, because the value of exponent is zero. The gain medium at this point is actually transparent at the frequency νul! Nu can never exceed Nl.

•  Hence in two level system it is impossible to get popula3on inversion and we will see how can we overcome from this J

Inversions and two-‐level Systems……con3nue

Radia3ve Decay Rates-‐ Radia3ve vs Collisional

Before we move to 3 and 4-‐level laser system, we must introduce a very important property of decay rates. From Chapter 4 we know Radia?ve decay rate α νul2 Aul α ΔEul2

Furthermore, it has been shown empirically that collisional decay rate has the property: kul α 1/ΔEul

Steady-‐state Inversions in 3-‐ and 4-‐ Level Systems

v  ΔNul > 1

v  dN/dt = 0

v  Pumping flux is constant

v  Constant flow of energy among the levels.

v What would be the expression for Nu/Nl, i.e., ΔNul

3-‐ Level Laser with Intermediate Level as Upper Laser Level

Ø  Aul>> γul

Ø  ΔEli & ΔEul >> kT

Ø  Principle of detailed balance Ø  Nuγul = Nlγlu

Ø  à Very small at room temp

for values of ΔEul associated with visible lasers

Ø  No external pumping occurs from level l to level u


Rate equa3ons for Ns

Also

On solving for Nl and Nu


All γ terms are constants – physical proper3es of laser material. However, Γli is a variable, indica3ng how hard we are pumping the gain medium to achieve popula3on inversion.

Highlights: ü  Ra3o of γil/ γiu to be as small as possible

(atoms goes to level u rather than back to level l)

ü  γul should be small (long lived u level is referred as metastable level)

The last point tends to confuse, don’t we want radia3ve decay from u to l? What can be gained by suppressing this component? Decay from level u to level l is predominantly radia3ve, i.e., & Inversion is produced if

3-‐ Level Laser with Highest Level as Upper Laser Level

The steady-‐state equa3on keeping in mind with previous assump3on of thermal equilibrium role.

Solving it for Nu and Nl gives:


In order to have net gain, we require that

We are including degeneracy factors here, as this type of 3-‐level system is generally used for gas lasers. All together, the condi3on for gain becomes:

In gases collisional decay processes are negligible compared to radia3ve decat due to much lower atomic density in comparison to solids, hence Thus we have


Apparently, a popula3on inversion can be obtained in this system if Alo is significantly greater than Auo, provided as well that Γol is not highly favoured over Γou. The rela3onship between Alo and Auo is quite common. Recall from our discussion of selec3on rules in chapter 4 that the laser transi3ons is in most cases a dipole transi3on, requiring that u and l have opposite parity. If Alo is large, this implies that l and o have opposite parity as well, meaning that u and o have same parity-‐ and this means that Auo must be very small!! In this case, we can rewrite the previous equa3on in approximate form as


For popula3on inversion: High Alo/Aul Low Γol/Γou

For example, if Γol = Γou then Alo must be greater than 2Aul (assuming for simplicity that the degeneracy factors are equal). In any case, it should be clear that a fast decay out of the lower laser level and high pumping flux to the upper laser level are desirable. We cannot force Γol to be zero, because gas lasers must be electrically pumped rather than op3cally pumped. With op3cal pumping, we can tune the pumping frequency to exactly match the energy difference between u and o. in contrast, electrical pumping is a broadband; the kine3c energy of the electrons follows a broad sta3s3cal distribu3on; a significant Γou implies a significant Γol.

4-‐ Level Laser

•  A 4-‐level laser is a combina3on of already discussed two 3-‐level systems, see figure à

•  For reasons iden3cal to those stated for 3-‐level systems we neglect the upward excita3on processes γil, γoi, γlu, γli and γui.

•  We can also neglect the decay processes γil, γlo, and γuo. The reasoning behind this simplifica3on if that the 4-‐level laser the generally used only with solid-‐state laser materials, for which collisional decay rates are inversely propor3onal to energy separa3on.

•  Γul = Aul

•  Note that levels l and o are generally very closely spaced, so that γlo and γol cannot be considered negligible.

4-‐ Level Laser The steady-‐state rate equa3ons are wrisen as follows:

And because No = N-‐ Ni – Nu – Nl, we can write:

Solving for the two levels of interest, we obtain

4-‐ Level Laser We thus find the ra3o:

Assuming that gu = gl (solid-‐state lasers) we find

Note as well that typically γlo >> γul, since the energy separa3on between l and 0 is very small. In that case,

We have used the Boltzmann rela3onship for the ra3o γ0l/ γl0.

Overall effect is that there is a rapid collisional decay (on the order of 10-‐13 s) from I to u and from l to 0 – this ensures that u is highly populated while level l is sparsely populated (but not too sparsely…the energy separa3on ΔΕl0 is very small, so it would be incorrect to assume Nl<< N0).

Population Inversion

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