Dynamic and spectral properties of transmission eigenchannels in random media

The eigenvalues of the transmission matrix provide the basis for a full description of the statistics of steady-state transmission and conductance and for control of transmitted waves. However, the nature of pulsed transmission of the transmission eigenchannels and their spectral correlation, which would permit control of propagation in the time domain, has not yet been discussed. Here we report the dramatic variation of the dynamic properties of transmission with incident waveform. Computer simulations show that lower-transmission eigenchannels respond more promptly to an incident pulse and are correlated over a wide frequency range. We explain these results together with the puzzlingly large dynamic range of transmission eigenvalues in terms of the way the quasinormal modes of the medium combine to form specific transmission eigenchannels. Key factors are the closeness of the illuminating waves to resonance with the modes forming an eigenchannel and the strong anticorrelation of low-transmission eigenchannels with resonant modes. In addition, the spectral range of modes comprising the transmission eigenchannel increases with decreasing transmission, which leads to an additional substantial suppression of transmission due to destructive interference.

Figure 1

(a)–(c) Simulation of responses (blue curves) to an incident Gaussian pulse (red dotted curves) with waveform $V_n$ of the transmission eigenchannels $n=1,2,5$ for the central frequency of the pulse in a single random configuration for a sample with $N=16$ channels drawn from an ensemble with $g=0.26$. The central frequency of the pulse is indicated by the red dashed line in the frames. The peak of the response moves towards the peak of the incident pulse as the net transmitted flux decreases and $n$ increases. (d)–(f) Spectra of transmission eigenchannels ${\tau }_n$ for $n=1,2,5$. The intensity of the incident Gaussian pulse in the frequency domain is shown as the red curve centered above the red dashed line. The width of the spectral features increases as the transmission decreases and $n$ increases.

Figure 2

(a) Variation of average delay of peak of transmitted pulse with incident waveform of the $n\mathrm{th}$ transmission eigenchannel at the center of the pulse vs an eigenchannel index $n$ in an ensemble with $g=0.26$. (b) Normalized cumulant correlation function of transmission eigenchannel ${\tau }_n$ vs a frequency shift. The inset shows the increase of the correlation frequency with $n$. The correlation frequency is taken as the frequency shift at which the correlation function falls to half its value.

Figure 3

Normalized correlation function of the square of the overlap integral of the waveform of the transmission eigenchannel on the output surface, $U_n$, with the waveform of $U_n$ at a shifted frequency $\mathrm{\Delta }\nu$.

Figure 4

Response of a Lorentzian mode to excitation by a Gaussian pulse for resonant excitation and for the central frequency of the pulse shifted one and two linewidths from the center of a Lorentzian line with linewidth ${\mathrm{\Gamma }}_m$. Bandwidth $\sigma$ of the Gaussian pulse is 1/3 of the linewidth ${\mathrm{\Gamma }}_m$ of the Lorentzian mode. The peak in the response is seen to move towards the time of the incident pulse.

Figure 5

Spectral correlation of the field patterns of modes and the eigenchannels. The square of the correlation function $C_n^m\left(\mathrm{\Delta }\nu \right)=(U_n^{*},u_1^m)$ between the normalized output field pattern of the $m\mathrm{th}$ mode, $u_1^m$, and the $n\mathrm{th}$ eigenchannel, $U_n$, for a wave shifted from the central frequency of the mode by $\mathrm{\Delta }\nu =\nu -{\nu }_m$.

Figure 6

The factors that when taken together yield the transmission eigenvalues for the first four eigenchannels are shown. The upper set of green circles gives the average of the incoherent sum of transmitted flux in all modes excited by an incident waveform a transmission eigenchannel $V_n$. The middle set of red triangles is the average of the incoherent sum of modal contributions to the transmission for an incident wave $V_n$ with the output wave projected onto the 2D space of $U_n$ and $i{U}_n$. The lower set of blue square gives the flux associated with the projections onto $U_n$ from all modes and is equal to ${\tau }_n$.

Figure 7

The projection from the 2D space with unit vectors in- and out-of-phase with $U_1$ onto the $U_1$ axis for the incident wave of the first transmission eigenchannel $V_1$ in a single sample configuration at a particular frequency. The resultant is ${\lambda }_1{U}_1$. The same as (a) but for the third transmission eigenchannel with incident wave $V_3$ and projection is onto the $U_3$ axis. Many modes with vectors of random phase contribute.