Broadband Coherent Enhancement of Transmission and Absorption in Disordered Media

Spatial modulation of the incident wave front has become a powerful method for controlling the diffusive transport of light in disordered media; however, such interference-based control is intrinsically sensitive to frequency detuning. Here, we show analytically and numerically that certain wave fronts can exhibit strongly enhanced total transmission or absorption across bandwidths that are orders of magnitude broader than the spectral correlation width of the speckles. Such broadband enhancement is possible due to long-range correlations in coherent diffusion, which cause the spectral degrees of freedom to scale as the square root of the bandwidth rather than the bandwidth itself.

Figure 1

Total transmission through a disordered slab for incident light with two discrete frequencies. (a) Schematic setup, with a disordered slab in a multimode waveguide and polychromatic light incident with a shared wave front. (b) Density of the polychromatic transmission eigenvalues as calculated numerically by solving the wave equation (the symbols) and analytically using Eq. (2) from free probability theory (the lines). Numbers indicate the intensity weight $W_1$.

Figure 2

Broadband transmission open channels. (a) Maximal eigenvalue $T_{\max }$ of the broadband flux matrix $A$ and the enhancement $\eta =T_{\max }/\overline{T}$ obtained from numerical simulations (the blue circles) and analytic theory accounting for long-range correlation (the blue line), showing the highest achievable frequency-integrated transmission across bandwidth $\mathrm{\Delta }\omega$. The two lines below show the would-be maximal transmission (the green dashed line) and the transmission of the monochromatic open channel (the orange dot-dashed line) if there were $1+\mathrm{\Delta }\omega /\delta \omega$ uncorrelated frequencies. The black dotted line indicates the average transmission. (b) Transmission as a function of frequency when the input wave front is fixed to the optimal eigenvector with different bandwidths $\mathrm{\Delta }\omega$.

Figure 3

(a) Density of the broadband transmission eigenvalues for various bandwidths, calculated numerically from simulations (the symbols; bandwidths $\mathrm{\Delta }\omega /\delta \omega =0$, 1.0, 2.4, 5.9, 15, 40) and analytically from Eq. (3) with an effective number $M_{\text{eff}}$ of independent frequencies (the lines; $M_{\text{eff}}=1.0$, 1.3, 1.7, 2.4, 3.9, 6.6). (b) $M_{\text{eff}}$ as a function of the square-root bandwidth, evaluated numerically from Eq. (4) (the symbols) and analytically from Eqs. (5) and (6) (the line).

Figure 4

Broadband coherently enhanced absorption (CEA). (a) Maximal frequency-integrated absorption $1-R_{\min }$ and the enhancement $\eta =(1-R_{\min })/(1-\overline{R})$ obtained numerically (the blue circles) and analytically (the blue line) across bandwidth $\mathrm{\Delta }\omega$. The two lines below show the would-be maximal absorption (the green dashed line) and the absorption of the monochromatic CEA mode (the orange dot-dashed line) if there were $1+\mathrm{\Delta }\omega /\delta \omega$ uncorrelated frequencies. The black dotted line indicates the average absorption. (Inset) Schematic setup of the system; a reflecting boundary on the right ensures that the transmission is zero and the absorption is $1-R$. (b) $M_{\text{eff}}$ as a function of the square-root bandwidth, evaluated numerically from Eq. (4) (the symbols) and analytically from Eqs. (5) and (7) (the line).