Summary |
We focus on the study the classical wave localization and amplification in a one-dimensional random layered medium. In our work a simple criterion is proposed to separates the physical solutions from the unphysical one in the framework of time-independent equation so that the threshold gain can be obtained in the framework of time-independent equation. We have tested the validity of this criterion in three different layered media by comparing the lasing threshold obtained from both time-dependent and time-independent equations. Base on this criterion, we have reinvestigated the statistical properties of wave amplification in random layered media by selecting only the physical solutions. Our results show that the probability distribution of physical solutions decay exponentially as a function of Λ≡L/ξo, where L is the sample size and ξo is the localization length without gain. The decay constant depends only on the dimensionless parameter q≡ξo/lg, where lg, is the gain length. Furthermore, the probability distribution of reflection P(R) for the physical solutions obeys the two-parameter scaling of q and Λ. At critical length Λc, the function P(R) is proportional to an analytical distribution derived previously from a stochastic equation. We also find that Λc∝q-x with x=2/3. This result is different from the previous result of x=1/2 and is able to explain the recent experimental finding of Pth∝lt0.5 in semiconductor powder, where Pth is the threshold pumping power and lt is the transport mean free path. Finally, our new results show that the localization length in a gain medium ξ(L) is always larger than previous one. It approaches ξo in the limit of large sample size. |