  Nonlinear imaging systems can surpass the limits of linear optics, but to
date they have all relied on physical media (e.g. crystals) to work. These
materials are all constrained by their physical properties, such as frequency
selectivity, environmental sensitivity, time behavior, and fixed nonlinear
response. Here, we show that electro-optic spatial light modulators (SLMs) can
take the place of traditional nonlinear media, provided that there is a
feedback between the shape of the object and the pattern on the modulator. This
feedback creates a designer illumination that generalizes the field of adaptive
optics to include object-dependent patterns. Unlike physical media, the SLM
response can provide arbitrary mathematical functions, operate over broad
bandwidths at high speeds, and work equally well at high power and
single-photon levels. We demonstrate the method experimentally for both
coherent and incoherent light.
