Selectively exciting quasi-normal modes in open disordered systems

Transmission through disordered samples can be controlled by illuminating a sample with waveforms corresponding to the eigenchannels of the transmission matrix (TM). But can the TM be exploited to selectively excite quasi-normal modes and so control the spatial profile and dwell time inside the medium? We show in microwave and numerical studies that spectra of the TM can be analyzed into modal transmission matrices of rank unity. This makes it possible to enhance the energy within a sample by a factor equal to the number of channels. Limits to modal selectivity arise, however, from correlation in the speckle patterns of neighboring modes. In accord with an effective Hamiltonian model, the degree of modal speckle correlation grows with increasing modal spectral overlap and non-orthogonality of the modes of non-Hermitian systems. This is observed when the coupling of a sample to its surroundings increases, as in the crossover from localized to diffusive waves.


Supplementary Note 1 -Decomposition into modal components.
We confirm in recursive Green's function simulations that the modal transmission matrices (MTMs) are of unit rank. The MTM is obtained from spectra of the TM for two samples with different values of the conductance ; = 0.1 ( = 16) and = 1.1 ( = 33). Excellent agreement is found in both cases in Supplementary Figure 1. The contribution of each mode to ( ), ( ), is also shown. We also compare the second and first eigenvalues of ϯ to the transmission eigenvalues, which are eigenvalues of ( ) ϯ ( ). The ratio of the two first eigenvalues of the MTM is found to be typically 10 −8 in strongly localized samples and 10 −6 in diffusive samples with greater modal overlap. Figure 1| Decomposition of simulated spectra into MTMs. (a) Transmittance (blue curve) and its reconstruction found using HI (dashed red curve) for a sample with = 0.1 and = 16. Further confirmation that the MTM is of rank unity is found in measurements in a weakly disordered system with moderate spectral overlap. The TM is measured between 8.8-9.25 GHz, which is a frequency range in which the antennas are weakly coupled to the sample. The cavity contains 150 randomly placed Teflon disks of index of refraction, = 1.44. The coupling strength of the antennas is ̃∼ 0.15. The leakage from the cavity is therefore small and resonances are narrow. = 0.6 in this case and modes extend throughout the sample even though the degree of overlap is weak. The reconstruction of ( ) is presented in Supplementary Figure 2

Supplementary Note 2 -Coupling between eigenfunctions in finite-element simulations.
To further explore the impact of the mixing of wavefunctions of the closed system upon the degree of modal selectivity for an open disordered system, we carry out two-dimensional simulations in COMSOL to solve Maxwell's equations and compute i) the eigenfrequencies and the associated field patterns and ii) the spectrum of the transmission matrix ( ). The scattering disks are included in a waveguide with perfectly reflecting side walls. The waveguide supports = 9 channels (see Fig. 5). Two different configurations are considered. The two samples are each a collection of 220 disks of radius = 0.18 0 randomly positioned inside a waveguide of width = 4.5 0 . The wavelength 0 is at the frequency 0 = 10 GHz. The sample lengths and relative permittivities are = 35 0 and = 2.3 for the first sample and = 25 0 and = 4.8 for the second sample.
The eigenfrequencies are found using the Eigenfrequency solver. Open boundary conditions are simulated using Perfectly Matched Layers (PML) at the left and right sides of the waveguide. The TM is then simulated using the frequency domain solver over the range 9.7-10 GHz. Boundary conditions are transverse electric rectangular ports.
For the first sample, we first consider a case of two modes with modal overlap = 0.55. The square of the real and imaginary parts of the field patterns 1 and 2 found in first step normalized so that 〈 * | 〉 = 1, are shown in Fig. SM3 and are seen to extend throughout the sample. Here, 〈Im( ) 2 〉 ≪ 〈Re( ) 2 〉, so that the degree of complexness is small for both modes, 1 2 = 0.07 and 2 2 = 0.03. The two eigenfunctions are seen to be very different. Transmission spectra corresponding to maximal coupling for each of the two modes are shown in Fig. SM3(b,d). Transmission is close to unity at resonance with the chosen mode while the contribution of the other mode is small. Because of the small correlation between the modes, it is possible to discriminate between the two modes.
However, when two modes overlap spectrally, the ability to exclusively select one of the modes is reduced. Two hybridized modes, which are double-peaked inside the sample, are found using the second sample and are shown in Fig. SM3(e,g). The degree of spectral overlap between the modes is = 1.13. Re( 1 ) and Im( 2 ), and Im( 1 ) and Re ( 2 ), are similar as expected from Eq. (10) of the main text for ≠ 0. The eigenfunctions give 1 2 = 0.17 and 2 2 = 0.16. The two values are close but not precisely equal because of the influence of other modes that overlap weakly. The extended modes in the localized regime are 'necklace states' which exhibit multiple peaks within the sample due to the hybridization of spectrally overlapping localized states [1][2][3][4][5]. Such states with high transmission and broad linewidth are rare but contribute substantially to average transmission. We observe in Figs. SM3(f,h) that the transmission spectra and the contribution of the two modes maximally coupled to the first or second mode are nearly the same so that it is not possible to discriminate the two modes . In this case, the correlation between the incident waveforms 1 and 2 , | 1 ϯ