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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

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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