Essentials for the amplification of electromagnetic radiation

4.2 Essentials for the amplification of electromagnetic radiation

We now investigate a system of atoms in a cavity.
Question: At which conditions do we find an amplification of radiation?
Answer: When the number of emissions is larger than the number of absorption’s! i.e.

$\frac{N_{2} B_{21} ϱ (ω) + N_{2} A_{21}}{N_{1} B_{12} ϱ (ω)} = \frac{N_{2}}{N_{1}} (1 + \frac{A_{21}}{B_{12} ϱ (ω)}) > 1 .$

(4.19)

Independent of $ϱ (ω)$ we get this condition if

$N_{2} > N_{1} .$

(4.20)

This is a direct consequence of

$B_{12} = B_{21} .$

(4.21)

For thermodynamic equilibrium we find at room temperature with $ℏ ω \approx 2$ eV

$\frac{N_{2}}{N_{1}} = exp (- \frac{ℏ ω}{k T}) \approx 1 0^{- 34}$

(4.22)

The condition $N_{2} > N_{1}$ is called inversion.
Only for systems far from thermodynamic equilibrium this inversion state can exist.
$N_{2} > N_{1}$ corresponds to a negative temperature.

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