Nonlinear Abbe theory

All limitations commonly associated with imaging, such as resolution, field of view and depth of field, arise from linear theory¹. Nonlinear optics can break these limits by exploiting the presence and interaction of many photons at once. To date, nearly all nonlinear imaging techniques have relied on point processes, such as two-photon fluorescence² or harmonic effects³, in which the temporal frequency is the relevant parameter. These methods ignore the spatial content of the object, typically require scanning to record a whole image, and remain restricted by linear propagation from the sample to the detector. Spatial nonlinearity can overcome these issues by mixing modes with high and low spatial frequencies. Here, we generalize Abbe's theory of diffraction¹ to include nonlinear propagation and show that wave mixing can be treated as a self-induced structured illumination. We demonstrate this experimentally by nonlinearly enhancing a standard imaging system beyond its conventional limits.