Prospects of coherent control in turbid media: Bounds on focusing broadband laser pulses

We study the prospects of controlling transmission of broadband and bi-chromatic laser pulses through turbid samples. The ability to focus transmitted broadband light is limited via both the scattering properties of the medium, and the technical characteristics of the experimental set-up. There are two time scales, given by pulse stretching in the near- and far-field regions, which define the maximum bandwidth of a pulse amenable to focusing. In the geometric optics regime of wave propagation in the medium, a single set-up can be optimal for focusing light at frequencies w and nw simultaneously, providing the basis for the 1 + n coherent quantum control. Beyond the regime of geometric optics, we discuss a simple solution for the shaping, which provides the figure of merit for one's ability to focus simultaneously several transmission modes.