Transmission Eigenchannels and the Densities of States of Random Media
The transmission matrix (TM) is the basis of a powerful approach to quantum and classical wave propagation that is able to explain the statistics of conductance and transmission and prescribe the degree to which the transmitted wavefront can be manipulated. For a waveguide that support N propagating channels, the elements of thee TM, t, tba, are the field transmission coefficients through the sample between the incident channels a and outgoing channels b. The probability distribution of the transmission eigenvalues τn of the matrix product determines the statistics of transmission. Recently, considerable attention has focused on the power of the TM to mold the flow of waves through random samples and to modify the energy density inside the medium. The possibility of sharp focusing and enhanced transmission has been demonstrated for sound, elastic waves, light and microwave radiation. However static transmission parameters cannot explain the dynamics of transmission or provide the density of states (DOS) whose statistics control emission, absorption and wave localization and give the proclivity of a medium to emit radiation and store energy. In this Letter, we show that it is possible to measure the dynamics and stored energy in addition to the transmission of each transmission eigenchannel. The DOS can be determined from measurements of spectra of the TM. The contribution of each eigenchannel to the DOS, the EDOS, is the derivative with angular frequency of a composite phase of the eigenchannel. Summing the contributions from all eigenchannels provides the first direct measurement of the DOS of a multiply scattering medium as a whole. This result is of importance because the usual expression of the DOS involves both the TM and the reflection matrices that can hardly be measured simultaneously.
Measurements of the TM for which the impact of absorption is removed are carried out in a copper waveguide containing randomly positioned alumina spheres. The DOS determined from the eigenchannels is found to be in excellent agreement with the DOS found from a decomposition of the transmitted field into modes for localized waves (see the following picture). The probability distribution of the DOS broadens substantially in the crossover to Anderson localization reflecting the increasing spectral isolation of long-lived localized modes. An algebraic tail is found in accord with theoretical investigations for 1D samples.
We also carry out two-dimensional numerical simulations using the recursive Green’s function method to determine the impact of an incomplete measurement of the TM on estimates of the DOS. Furthermore we show that the EDOS, which is equal to the delay time in transmission, increases with τn. Delay times are thus lengthened in coherent eigenchannels with high transmission. When normalized by the average delay time, the eigenchannel delay time versus τn for diffusive samples of different length is found to fall on universal curve, suggesting a universal behaviour of the energy density inside random media. These results illuminate the relationships between scattering, stored energy, and dynamics in complex media.
Davy, M., Shi, Z., Wang, J., Cheng, X. & Genack, A. Z. Transmission Eigenchannels and the Densities of States of Random Media. Physical Review Letters 114, 033901 (2015).
