Transition Probability for a Laser system

In summary, the first W is the probability of transitioning between levels 1 and 2 in a laser. The second W is the probability of transitioning between levels 2 and 1 in a laser.
  • #1
Angelos K
48
0
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

[tex]E_2 = E_1 +h\nu[/tex]

it is supposed to be given by:

[tex]W = \frac{1}{\tau VD(\nu)\Delta\nu}[/tex]

where [tex] \tau [/tex] is the lifetime of the level 2 (probably for the case of spontaneous emission making the only important contribution), [tex] D(\nu)d\nu[/tex] is the number of modes of the field in the intervall [tex] (\nu,\nu+d\nu) [/tex] per unit volume of the laser substance and [tex] \Delta\nu [/tex] 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:

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

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

[tex] A_{21} = \frac{1}{\tau} [/tex]

and

[tex] D(\nu) = \frac{8\pi{\nu}^2}{c^3} [/tex]

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

Thanks for any help,

Angelos
 

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  • #2
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?
 
  • #3
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.
 
  • #4
Definition of W

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

My textbook gets rate equations of the form:

[tex]\frac{dn}{dt} = W(N_2 -N_1)n + ...[/tex]

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:

[tex]\frac{dI}{dt} = W(N_2 -N_1)I + ...[/tex]

wheras the W from the second formula would yield:

[tex]\frac{dI}{dt} = W(N_2 -N_1) + ...[/tex]

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.
 
Last edited:
  • #5
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 ...
 

1. What is transition probability in a laser system?

The transition probability in a laser system refers to the likelihood of an atom or molecule in the laser medium undergoing a transition between energy levels, resulting in the emission of a photon. It is a critical factor in determining the efficiency and output of a laser system.

2. How is transition probability calculated in a laser system?

Transition probability is calculated by taking the square of the transition dipole moment, which is a measure of the strength of the interaction between the atom or molecule and the laser field. This value is then multiplied by the density of states and a statistical factor to obtain the transition probability.

3. Why is transition probability important in laser technology?

Transition probability is important in laser technology because it determines the probability of emitting a photon and the rate at which this occurs. This, in turn, affects the efficiency and output power of the laser, making it a crucial factor in designing and optimizing laser systems for various applications.

4. How does the transition probability affect the properties of a laser beam?

The transition probability directly impacts the properties of a laser beam, such as its wavelength, coherence, and polarization. A higher transition probability results in a stronger and more stable laser beam, while a lower transition probability can lead to reduced coherence and fluctuations in the beam intensity.

5. Can the transition probability in a laser system be controlled?

Yes, the transition probability in a laser system can be controlled by adjusting the laser parameters, such as the intensity and frequency of the laser field, as well as the properties of the laser medium. This allows for fine-tuning of the laser output and optimization for specific applications.

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