Correlation of spontaneous emission in a one-dimensional random medium with Anderson localization

Abstract

Under Anderson localization, two types of correlation of spontaneous emission in one-dimensional random media are investigated: i.e., time-domain field correlation ${C}_{t}^{E}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\tau})$ and energy-spectrum correlation ${C}_{\ensuremath{\omega}}^{P}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\Delta}\ensuremath{\omega})$. The results show that the spatial correlation length of ${C}_{t}^{E}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\tau}=0)$ is unrelated to the localization length; however, the increase of the correlation length of ${\mathrm{max}}_{\ensuremath{\tau}}\ensuremath{\mid}{C}_{t}^{E}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\tau})\ensuremath{\mid}$ with the localization length is sensitive and monotonous. In particular, we find that the fields at the different locations keep on exchanging with each other in a certain fixed speed by investigation of time-domain field correlation. The speed is almost not affected by the random strength and almost equal to the group velocity of the corresponding periodic structure. Therefore the localized mode is really a dynamic equilibrium state though the energy of the localized mode is localized. In addition, because it is not convenient to characterize the localization length by ${C}_{t}^{E}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\tau})$, another correlation---energy-spectrum correlation ${C}_{\ensuremath{\omega}}^{P}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\Delta}\ensuremath{\omega})$---is proposed. By investigation of the energy-spectrum correlation for $\ensuremath{\Delta}\ensuremath{\omega}=0$, we obtain that there is an approximately linear relation between the spatial correlation length of energy-spectrum and the localization length. Obviously, in the aspect of characterizing the localization length, the energy-spectrum correlation is more convenient than the time-domain field correlation.