Vacuum Fluctuation Noise and Dissipation

An electron beam interacting with the fields in an enclosed region is considered. The induced noise caused by the vacuum fluctuations and the thermal fluctuations is calculated in terms of the dissipation function $R\left(\omega \right)$ and the electron transit time $\tau$. The observable mean squared electromotive force is given by ${〈V^2〉}_{\mathrm{Av}}=\frac{2}{\pi }\int 0^{{\omega }_{max}}R\left(\omega \right)\left[\frac{\hslash \omega }{2}+\frac{\hslash \omega }{\exp \left(\frac{\hslash \omega }{\mathrm{kT}}\right)-1\mathrm{}}\right]{\left[\frac{sin\frac{1}{2}\omega \tau }{\frac{1}{2}\omega \tau }\right]}^{2}d\omega +\frac{1}{\pi }\int {{\omega }_{max}}^{\infty }\left[\frac{R\left(\omega \right)\hslash \omega }{\exp \left(\frac{\hslash \omega }{\mathrm{kT}}\right)-1\mathrm{}}\right]{\left[\frac{sin\frac{1}{2}\omega \tau }{\frac{1}{2}\omega \tau }\right]}^{2}d\omega .$ ${\omega }_{max}$ is defined by $\hslash {\omega }_{max}=U$, where $U$ is the energy of the incident electrons. The effect of the fluctuations of the vacuum is given by the temperature-independent part of the first integral.