Probability Distribution for Coherent Transport of Random Waves

Abstract

We establish a comprehensive probability theory for coherent transport of random waves through arbitrary linear media. The transmissivity distribution for random coherent waves is a fundamental B-spline with knots at the transmission eigenvalues. We analyze the distribution's shape, bounds, moments, and asymptotic behaviors. In the large n limit, the distribution converges to a Gaussian whose mean and variance depend solely on those of the eigenvalues. This result resolves the apparent paradox between bimodal eigenvalue distribution and unimodal transmissivity distribution.

Figures (6)

• Figure 1: (a) Central problem: What is the probability distribution $p(t)$ of transmissivity $t$ for random coherent waves incident on a scattering medium? (b,c) Transmission eigenvalue distribution $p(\lambda_t)$ (red) versus Monte Carlo-simulated transmissivity distribution $p(t)$ (blue) for (b) fully chaotic and (c) diffusive systems. (d,e) Corresponding results for Bernoulli-distributed eigenvalues with (d) $q=0.5$ and (e) $q=0.8$.
• Figure 2: (a) An $(n+m)$-port linear system transforms a coherent input wave $\bm{a}$ into a transmitted wave $\bm{b} = \tau \bm{a}$. (b-d) Transmissivity distribution $p_n(t)$ for $n=2, 3, 4$ input ports. Histograms show Monte Carlo results from $10^5$ random coherent inputs, blue curves show analytical B-spline predictions, and red dots and lines mark transmission eigenvalues $\bm{\lambda}(\tau^\dagger \tau)$.
• Figure 3: Transmissivity distribution $p_n(t)$ for $n=5$ input ports. The blue curve shows the analytical result, with red dots and lines marking the transmission eigenvalues. The blue triangle indicates the mode $t_M$ (peak position), which is close to the Greville abscissa $\xi$ (red triangle). Orange- and purple-shaded regions indicate tail probabilities for extreme events.
• Figure 4: Effects of eigenvalue degeneracy and asymptotic behavior. (a,b) Transmissivity distribution $p_4(t)$ with twofold and threefold eigenvalue degeneracy exhibits reduced smoothness at the degeneracy point. (c,d) Distributions $p_4(t)$ and $p_5(t)$ compared with their Gaussian approximations (orange dashed curves) demonstrate the central limit theorem.
• Figure 5: Numerical demonstration of the theory. (a) A slab waveguide with permittivities $\varepsilon_i = 12.1$ (core), $\varepsilon = 2.1$ (cladding), and $\varepsilon_s = 2.0+0.86i$ (lossy scatterers), supporting $n=4$ TE modes. (b-d) Probability distributions for transmissivity $p_4(t)$, reflectivity $p_4(r)$, and absorptivity $p_4(\alpha)$. Histograms show Monte Carlo results from $10^5$ random coherent inputs, and solid curves show theoretical B-spline predictions.

Theorems & Definitions (4)

• proof
• Theorem : Curry and Schoenberg, 1966 curry1966
• proof
• Proposition : Curry and Schoenberg, 1966 curry1966