Dynamical Imaging Using Spatial Nonlinearity

Abstract

The limitations common to linear imaging techniques were formalized succinctly by Ernst Abbe in 1873. In his theory, each Fourier component of a signal propagates independently from the source to the detector, and the corresponding feature can be detected only if the modal wavenumber lies within the spatial bandwidth of the imaging system. For systems with small numerical apertures, only those modes with low wavenumbers can be detected, so that the resulting image suffers in quality.

On the other hand, Abbe's theory does not take into account the dynamical propagation of a signal. It is true that nonlinear optics has been used in imaging methods by exploiting the presence and interaction of many photons at once. However, to date, all nonlinear methods have utilized only temporal frequency mixing, which are typically point processes that circumvent linear limits by generating shorter wavelengths, tighter focal spots, and less unwanted scattering. Beam propagation from the sample to detector is still linear, so that observations are still restricted by the numerical aperture of the system.

In this work, Abbe's theory of image formation is generalized to accommodate spatial nonlinearity. In this case, spatial nonlinearity acts as a mechanism to transfer energy among the different spatial modes. Thus, the high-frequency content couples to the low-frequency content and then scatters into the field of view. Numerical processing can reverse the scattering and reconstruct the object to provide increased field of view and super-resolution effects. To accomplish this task, insight into dynamical propagation is sought first by studying nonlinear propagation of optical shock waves. Shock propagation is complex, but well-understood, and provides knowledge of the properties of unknown material responses. This information is then used to develop a computational algorithm for reconstructing (unknown) objects with only measured nonlinear output information. As a final step, spatial nonlinearity is introduced into diffraction limited systems and is shown to surpass these very limits. This new, dynamical method of imaging provides a new degree of freedom in system design and invites reexamination of all linear limits, and methods to overcome them, in light of spatial wave mixing.