Open channels and radiation trapping eigenstates in complex resonant media

We present a statistical study of the transmission and dwell-time matrices in disordered media composed of resonators, focusing on how frequency detuning influences their eigenvalue distributions. Our analysis reveals that the distribution of transmission eigenvalues undergoes a transition from a monomodal to a bimodal profile, and back to monomodal, as the frequency approaches the resonant frequency of the particles. Moreover, the distribution of dwell-time eigenvalues broadens significantly near resonance, with the longest lifetimes exceeding the median by several orders of magnitude. These results are explained by examining how frequency <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>ω</a:mi></a:math> affects the transport mean free path of light <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>ℓ</b:mi><b:mo>(</b:mo><b:mi>ω</b:mi><b:mo>)</b:mo></b:mrow></b:math> and the energy transport velocity <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msub><c:mi>v</c:mi><c:mi>E</c:mi></c:msub><c:mrow><c:mo>(</c:mo><c:mi>ω</c:mi><c:mo>)</c:mo></c:mrow></c:mrow></c:math>, which in turn shape the observed distributions. We demonstrate the strong potential of wavefront shaping to synthesize wavefronts associated with eigenstates that enhance transmission and energy storage (or radiation trapping) in resonant disordered media. In the diffusive regime, where the system thickness <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>L</d:mi></d:math> exceeds the mean free path, both transmission and dwell time can be enhanced by a factor <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mo>∝</e:mo><e:mi>L</e:mi><e:mo>/</e:mo><e:mi>ℓ</e:mi><e:mo>(</e:mo><e:mi>ω</e:mi><e:mo>)</e:mo><e:mo>≫</e:mo><e:mn>1</e:mn></e:mrow></e:math> when using wavefronts associated with the largest eigenvalues instead of plane waves. In the localized regime, the enhancements become <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mo>∝</f:mo><f:mi>N</f:mi><f:msup><f:mi>e</f:mi><f:mrow><f:mn>2</f:mn><f:mi>L</f:mi><f:mo>/</f:mo><f:mi>ξ</f:mi></f:mrow></f:msup></f:mrow></f:math> for transmission and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mo>∝</g:mo><g:mi>N</g:mi><g:mi>ξ</g:mi><g:mo>/</g:mo><g:mi>L</g:mi></g:mrow></g:math> for dwell time, where <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>ξ</h:mi></h:math> is the localization length and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>N</i:mi></i:math> is the number of controlled scattering channels. Finally, we show that employing high-<j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>Q</j:mi></j:math> resonators instead of low-<k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mi>Q</k:mi></k:math> ones increases energy storage within the medium by a factor of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mrow><l:mo>∝</l:mo><l:mi>Q</l:mi><l:mo>/</l:mo><l:mi>k</l:mi><l:mi>ℓ</l:mi><l:mo>(</l:mo><l:mi>ω</l:mi><l:mo>)</l:mo></l:mrow></l:math> in both the diffusive and localized regimes.