Supplemental Online Material for ‘ Universal optimal transmission of light through disordered materials ’

In the accompanying Letter we show an experiment where the wavefront of incident light is shaped to excite open eigenchannels, and we demonstrate an increased total transmission through an opaque, strongly scattering sample. Our wavefront shaping algorithm maximizes the intensity in a target point behind the sample. Ideally, the algorithm constructs the wavefront for which the intensity transmission to the target is at a global maximum. This ideal wavefront is given in Eq. 2 of the main document. Here, we give mathematical proof that this wavefront indeed represents the unique (up to an irrelevant overall phase factor) global maximum solution. The goal of the optimization algorithm is to maximize the intensity in the target by changing the incident wavefront, with the constraint that the total incident power should remain constant at unity. The field in the target mode (labelled β) is given by