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.

\begin{equation*} \frac{N_2 B_{21} \varrho(\omega) + N_2 A_{21}}{N_1 B_{12} \varrho(\omega)} = \frac{N_2}{N_1}\left(1 + \frac{A_{21}}{B_{12}\varrho(\omega)}\right) \gt 1 \qquad . \end{equation*}

(4.19)

Independent of \(\varrho(\omega)\) we get this condition if

\begin{equation*} N_2 \gt N_1 \qquad . \end{equation*}

(4.20)

This is a direct consequence of

\begin{equation*} B_{12} = B_{21} \qquad . \end{equation*}

(4.21)

For thermodynamic equilibrium we find at room temperature with \(\hbar \omega \approx 2\)eV

\begin{equation*} \frac{N_2}{N_1}=\exp\left(-\frac{\hbar \omega}{k T} \right) \approx 10^{-34} \end{equation*}

(4.22)

The condition \(N_2 \gt N_1\) is called inversion.
Only for systems far from thermodynamic equilibrium this inversion state can exist.
\(N_2 \gt N_1\) corresponds to a negative temperature.