Single speckle SRS threshold as determined by electron trapping, collisions and speckle duration

Speckle SRS intensity threshold has been shown\footnote{L. Yin \textit{et al}., \textit{Physics of Plasmas} \textbf{15}, 013109 (2008).} to increase with spatial dimension, D, because both diffraction and trapped electron escape rate increase with D, though the net effect is to substantially decrease\footnote{D. S. Montgomery \textit{et al}., \textbf{9}, 2311(2002).} the threshold compared to 1D linear gain calculations. On the other hand, the apparent threshold appears to decrease\footnote{Bruce Langdon \textit{et al}., 38$^{th}$ Anomalous Absorption Conference (2008).} with integration time in PIC simulations. We present an optimum\footnote{Harvey A. Rose, \textit{Physics of Plasmas} \textbf{10}, 1468 (2003).}$^,$\footnote{Harvey A. Rose and L. Yin, \textit{Physics of Plasmas} \textbf{15}, 042311 (2008).}, nonlinearly resonant\footnote{Harvey A. Rose and David A. Russell, \textit{Phys. Plasma} \textbf{8}, 4784 (2001).} calculation of the SRS threshold, taking into account large fluctuations of the SRS seed reflectivity, R$_{0}$. Such fluctuations, absent in 1D, are caused by a gap in the linear reflectivity gain spectrum which leads to an exponential probability distribution for R$_{0}$. While the SRS threshold intensity is of course finite, these fluctuations lead to a decrease of apparent threshold with increasing speckle lifetime.