Addressing the Inverse Problem of Far-Field Imaging: A Noniterative Exact Solution for Phase in Imaging

Abstract

This talk describes an approach to a solution of inverse problems in far-field imaging with application to optical, electron/ion beam, x-ray and other imaging modalities. Two aspects of the inverse problem in optical imaging are phase and resolution. Whereas techniques such as STED, PALM, STORM etc address the latter, they lack phase information as does near-field optics. Furthermore, all of these techniques are serial. We introduce an approach based on a controllable nanoscopic far-field optical point source integrated into parallel imaging. In the first demonstration of this development we focus on the inverse problem of the phase of an image which has never been solved exactly and can only be approached through iterative methods with all their problems of nonconvergence, slow convergence, convergence to local minima, and stagnation. We now show that it is possible to obtain, with such a method, an exact solution to the inverse problem of phase both experimentally and theoretically. Our method is based on the breakthrough that crystallography experienced in phase retrieval for large molecular entities by Max Perutz's introduction of “heavy atoms” using the method of isomorphous replacement. Scanning probe microscopy and its full integration with optical microscopy allows us to apply these X-ray concepts to implement “heavy atom” restoration of phase in optical phase retrieval. In analogy to the heavy atom method, we acquire Fourier intensities in place of an X-ray diffraction pattern, and in place of the heavy atom, we utilize a nanometrically translatable point source of light, coherently related to the far-field illumination this leads to 3D optical imaging. The methodology has super-resolution potential, and thus, heavy atom restoration of phase with super-resolution (HARPS) has the potential to provide super-resolution 3D images in real time.