Transition Probability for a Laser system

[Angelos K]

Hello!

My textbook quotes the probability W of a transition between the levels 1 and 2 of a laser that appears in the rate equations. For

$$E_2 = E_1 +h\nu$$

it is supposed to be given by:

$$W = \frac{1}{\tau VD(\nu)\Delta\nu}$$

where $$\tau$$ is the lifetime of the level 2 (probably for the case of spontaneous emission making the only important contribution), $$D(\nu)d\nu$$ is the number of modes of the field in the intervall $$(\nu,\nu+d\nu)$$ per unit volume of the laser substance and $$\Delta\nu$$ is the broadness of the spectral line corresponding to transitions between states 2 and 1.

There are no comments on how to prove this. I would appreciate help, since many important conclusions are driven from that formula.

I have also discovered the attached document, which derives a more complex formula:

$$W = g(\nu) \frac{A_{21}c^{2}I(\nu)}{8\pi h {\nu}^3}$$

containing the Einstein coefficient for spontaneous emission, the radiation Intensity $$I(\nu)$$ and the line shape $$g(\nu)$$. The formulas are fairly similar if we remember the equalities:

$$A_{21} = \frac{1}{\tau}$$

and

$$D(\nu) = \frac{8\pi{\nu}^2}{c^3}$$

It would be sufficient if you could explain how to go from the second expression for W to the first one. It is the $$\Delta\nu$$ in particular that I do not see how to obtain!

Thanks for any help,

Angelos

 

[Redbelly98]

Something is puzzling me about your formulas.

The first formula does not depend on the laser intensity, while the second one does. This makes me wonder if they are really expressing the same quantity or not. I.e., perhaps the first expression refers to spontaneous emission, while the second one is referring to stimulated emission?

 

[Angelos K]

You are right.

You are right. That is very strange.

Yet both sources state that the corresponding formulae give W for stimulated emission! I will check again wether that textbook uses any anusual definition of W that is not a probability per unit time.

The second formula is prooven in the pdf that I attached, but for the first one my textbook ( Haken, Wolf Atom- und Quantenphysik doesn't give any hint for it's proof. It might also be wrong :-(

Thanks for the comment. I have been having trouble with that equation for several days.

 

[Angelos K]

Definition of W

I suspect that the definitions of W utilized defer in the following sense:

My textbook gets rate equations of the form:

$$\frac{dn}{dt} = W(N_2 -N_1)n + ...$$

for the number of (axial) photons in the material. This number n should now be some scaled intensity. I assume that in the Intensity picture this corresponds too:

$$\frac{dI}{dt} = W(N_2 -N_1)I + ...$$

wheras the W from the second formula would yield:

$$\frac{dI}{dt} = W(N_2 -N_1) + ...$$

In other words I suspect, that the first expression uses a W that does not contain I per definition, wheras the second one does. In a photon picture, where I coresponds to n, it is clear that both Ws have the same units. In the wave picture I find this still a bit confusing.

 

[Redbelly98]

Okay, so the two W's are similar but not quite the same. Looks like the first W is to be multiplied by I or n (or some measure of intensity) in order to get the second W.

I'm noticing the second W expression, after accounting for the terms equating to D(nu), contains the factors
I/(h*nu*c)

I is intensity
h*nu is the energy per photon
c is c

So
I/(h*nu) is the number of photons, per second, crossing per unit area.
Divide that by c and you get the number of photons per unit volume.

Don't know if that helps any more ...