Quantenmechanische Berechung der Photonenzahl und Amplitudenfluktuation einer Laserstrahlung

From the quantum mechanical equations describing the laser we derive a system of two coupled equations which contains only b + b , b + bb + b and the fluctuation operators. If we introduce a distribution function for the eigenvalues of b + b which depends on the parameters n 0 = 〈 b + b 〉 (average photon number) and \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{\alpha ^2 }}{{12}} = \langle b + bb + b\rangle - \langle b + b\rangle ^2 $\end{document} (amplitude fluctuation) and neglecting noise terms of higher order, then it is easily possible to calculate the quantum mechanical expection values given above and derive two equations which determine n 0 and α for the stationary case. Below the threshold of laser action these equations then describes thermal light with an amplitude fluctuation \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{\alpha ^2 }}{{12}} = n_0^2 + n_0 $\end{document} and a photon number n 0 . These results agree with that of other authors. The same holds also above threshold for the photon number, if we neglect correction terms of the order n   0 −1 . For the amplitude fluctuation we obtain \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{\alpha ^2 }}{{12}} \sim n_0 $\end{document} and it is shown, that far above the laser threshold the relative amplitude fluctuation decreases with increasing pumping intensity and therefore increasing photon number. However, the relative amplitude fluctuation remains approximately constant also in the case of increasing photon number if the latter is achieved by varying such experimental parameters which also increase the number of spontaneous quanta.