Einstein Coefficients

You have two-state atoms in a thermal radiation field at temperature T. The following three processes take place:

1) Atoms can be promoted from state 1 to state 2 by absorption of a photon according to

(1)

2) Atoms can decay from state 2 to state 1 by spontaneous emission according to

(2)

3) Atoms can decay from state 2 to state 1 by stimulated emission according to

(3)

The populations N₁ and N₂ are in thermal equilibrium, and the radiation density is

(4)

a) What is the ratio N₂/N₁?

b) Calculate the ratios of coefficients A₂₁/B₂₁ and B₂₁/B₁₂.

c) From the ratio of stimulated to spontaneous emission, how does the pump power scale with wavelength when you try to make short wavelength lasers?

SOLUTION

a) At equilibrium the rates of excitation out of and back to state 1 should be equal, so

(5)

Substituting (1), (2), and (3) into (5) gives

(6)

We may find the ratio of the populations from (6) to be

(7)

b) At thermal equilibrium the population of the upper state should be smaller than that of the lower state by the Boltzmann factor e^-hv/τ, so (7) gives

(8)

Substituting the radiation density ρ(v) into (8) gives

(9)

or

(10)

The ratios of coefficients may be found by considering (10) for extreme values of since it must be true for all values of v. For very large values of v, we have

(11)

Substituting (11) back into (10) yields

(12)

or

(13)

which immediately yields

(14)

and so, from (11),

(15)

c) Inspection of (15) shows that the ratio of the spontaneous emission rate to the stimulated emission rate grows as the cube of the frequency, which makes it more difficult to create the population inversion necessary for laser action. The pump power would therefore scale as (1/λ)³.