Universal structure of transmission eigenchannels inside opaque media
Controlling the distribution of energy inside disordered materials for applications such as depth profiling or enhanced nonlinear interactions is a tantalizing prospect that has been frustrated by multiple scattering. There has been considerable recent progress in controlling net transmission and focusing through scattering media by utilizing measurements of transmission, but corresponding measurements of the field inside a random sample are generally not possible. Only the result of the diffusion equation is known. Here we find a universal expression for the average energy density profile inside the medium in specific eigenchannels of the transmission matrix. In 1984 Dorokhov gave a complete description of conductance and transmission in disordered systems in terms of the eigenvalues of the transmission matrix that gather together the field transmission coefficients between input and output modes of the sample. However, this result is mute regarding the energy distribution inside the sample.
[if gte vml 1]><v:shapetype id="_x0000_t202" coordsize="21600,21600" o:spt="202" path="m,l,21600r21600,l21600,xe"> <v:stroke joinstyle="miter"></v:stroke> <v:path gradientshapeok="t" o:connecttype="rect"></v:path> </v:shapetype><v:shape id="Zone_x0020_de_x0020_texte_x0020_2" o:spid="_x0000_s1026" type="#_x0000_t202" style='position:absolute;left:0;text-align:left; margin-left:221.55pt;margin-top:0;width:228.25pt;height:206.7pt;z-index:251658752; visibility:visible;mso-width-percent:400;mso-height-percent:200; mso-width-percent:400;mso-height-percent:200;mso-width-relative:margin; mso-height-relative:margin'> <v:textbox style='mso-fit-shape-to-text:t'> <![if !mso]> <table cellpadding=0 cellspacing=0 width="100%"> <tr> <td><![endif]> <div> <p class=Standard><span lang=EN-GB><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"></v:stroke> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"></v:f> <v:f eqn="sum @0 1 0"></v:f> <v:f eqn="sum 0 0 @1"></v:f> <v:f eqn="prod @2 1 2"></v:f> <v:f eqn="prod @3 21600 pixelWidth"></v:f> <v:f eqn="prod @3 21600 pixelHeight"></v:f> <v:f eqn="sum @0 0 1"></v:f> <v:f eqn="prod @6 1 2"></v:f> <v:f eqn="prod @7 21600 pixelWidth"></v:f> <v:f eqn="sum @8 21600 0"></v:f> <v:f eqn="prod @7 21600 pixelHeight"></v:f> <v:f eqn="sum @10 21600 0"></v:f> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"></v:path> <o:lock v:ext="edit" aspectratio="t"></o:lock> </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:194.25pt; height:138.75pt'> <v:imagedata src="file:///C:\Users\mdavy\AppData\Local\Temp\msohtmlclip1\01\clip_image001.emz" o:title=""></v:imagedata> </v:shape><o:p></o:p></span></p> <p class=Standard><span lang=EN-GB>Ensemble averages of the eigenchannel energy density profiles for eigenvalues τ=1, 0.5, 0.1 and 0.001 for samples with channel number N=66 and L/ξ=66, where ξ is the localization length.<o:p></o:p></span></p> </div> <![if !mso]></td> </tr> </table> <![endif]></v:textbox> <w:wrap type="square"></w:wrap> </v:shape><![endif][if !vml][endif]We find that waves in different eigenchannels inside random systems propagate independently with an average energy density profile, Wτ(x), for an eigenchannel with eigenvalue τ given in terms of the corresponding auxiliary localization length. The results of recursive Green’s function simulations are shown as solid curves in the attached figure. These results are compared to an expression shown as dashed curves which are obtained from numerical simulations and analytic theory. We find that the profiles can be factorized as the product of the strength of an universal source term in a generalized diffusion equation and the profile of the perfectly transmitting eigenchannel.This last is a symmetrical function peaked in the middle of the sample which the sum of a unit background and the probability of return of the wave to a cross section of the sample at depth x. These results reveal the rich structure of transmission eigenchannels and enable the control of energy deposition inside random media. Such control may be utilized in a wide range of applications such as resource exploration and low-threshold random lasing.
Davy, M., Shi, Z., Park, J., Tian, C. & Genack, A. Z. Universal structure of transmission eigenchannels inside opaque media. Nat. Commun. 6 (2015).
