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“ If I were again beginning my studies, I would follow the advice of Plato and start with mathematics. ”
- Galileo Galilei
Feb 23, '15
Transmission eigenchannels and the densities of states of random media
M. Davy, Z. Shi, J. Wang, X. Cheng and A. Z. Genack
Phys. Rev. Lett. 114, 033901 (2015).
Natural channels of transmission and resonances of disordered systems
Recently, considerable excitement has focused on the power of the field transmission matrix to mold the flow of waves through random samples by manipulating the waveform of the incident electromagnetic or acoustic wave. Such work has exploited the description of propagation in terms of natural channels with specific values of transmission. Here we report the discovery of a complementary set of parameters that makes it possible to probe and control the energy deposited inside the material and the delay in transmission. The new eigenchannel delay parameters are the rate of change of the average phase of natural transmission channels with frequency shift. This makes it possible to select the delay time in transmission and the extent to which a sample is excited for applications to energy deposition and low-threshold lasing. Summing up the eigenchannel delay times over all eigenchannels gives the strength of all the resonances within a medium at each frequency, which is the density of states. Fluctuations in the density of states in disordered materials provide a fundamental measure of the degree to which waves are localized spatially within the sample.
Figure. Variation of the density of states in a random sample with angular  frequency at microwave  frequencies. (a) Contributions of the individual modes to the density of states. (b) Comparison of the density of states determined from the transmission matrix (red curve) by summing spectra of the eigenchannel delay time and from a determination of the modes found from spectra of the field at the output (black curve) and from spectra of the field inside the sample (grey dashed curve).