Radiant Field Theory: A Transport Approach to Shaped Wave Transmission through Disordered Media

We present a field-theoretic framework to characterize the distribution of transmission eigenvalues for coherent wave propagation through disordered media. The central outcome is a transport equation for a matrix-valued radiance, analogous to the classical radiative transport equation but capable of capturing coherent effects encoded in the transmission matrix. Unlike the Dorokhov-Mello-Pereyra-Kumar (DMPK) theory, our approach does not rely on the isotropy hypothesis, which presumes uniform angular scattering by material slices. As a result, it remains valid beyond the diffusive regime, accurately describing the transmission eigenvalue distribution in the quasiballistic regime as well. Moreover, the framework is more versatile than the DMPK theory, enabling straightforward incorporation of experimental realities such as absorption and incomplete channel control. These factors are frequently encountered in wave experiments on complex media but have lacked an ab initio theoretical treatment until now. We validate our predictions through numerical simulations based on the microscopic wave equation, confirming the accuracy and broad applicability of the theory.