טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentMutzafi Maor
SubjectSparsity-Based Recovery of Lost Optical Information and
Quantum Wave Functions Shaping
DepartmentDepartment of Physics
Supervisor ? 18? Mordechai Segev
Full Thesis textFull thesis text - English Version


Abstract

My PhD research consist of two unrelated topics: "Sparse optical imaging systems”, which deals with algorithmic recovery of lost information in optical systems by embedding prior knowledge on the structure of the information, and "Quantum wave functions shaping" - controlling the dynamics of quantum EBeams (Electron Beams) by shaping their initial wavefunctions.

Sparse optical imaging systems:

Many problems that arise in optics are inverse problems - i.e. the recovery of an unknown optical signal from measurements related to that object. However, many of the inverse problems in optics are ill posed. Inverting ill-posed problem means finding the 'true' object out of the infinitely many possible objects that would yield the same measurements. The only way to select the true object out of all possible solutions is by using some prior information on the object.

In this topic, I present my work describing the use of signal sparsity as such prior knowledge. A signal is said to be sparse, when it can be represented compactly in a known mathematical basis. Practically, this means that this signal has some characteristic structure. Here, I studied this concept in two projects:

                     The first is "Sparsity-based Ankylography - Recovering 3D molecular structures from single-shot 2D scattered light intensity". This project demonstrates that using prior knowledge of sparsity enables overcoming both the dimension deficiency and the lack of optical phase measurement.

                     The second is "Sparsity-Based Single-Shot Super-Resolution Fluorescence Imaging Using Dictionary Learning". Showing that a sub-wavelength object that is sparse in some specific basis (a dictionary) containing the characteristic features of in-kind biological images, can be recovered from a single diffraction-limited exposure. The dictionary is trained in a learning procedure that uses a database of high-resolution images, available in advance. The dictionary learning step also employs sparsity.

Deciphering the three-dimensional (3D) structure of complex molecules is of major importance, typically accomplished with X-ray crystallography. Unfortunately, many important molecules cannot be crystallized, hence their 3D structure is unknown. Ankylography presents an alternative, relying on scattering an ultrashort X-ray pulse off a single molecule before it disintegrates, measuring the far-field intensity on a two-dimensional surface, followed by computation. However, significant information is absent due to lower dimensionality of the measurements and the inability to measure the phase. Recent Ankylography experiments attracted much interest, but it was counter-argued that Ankylography is valid only for objects containing a small number of volume pixels. Here, we propose a sparsity-based approach to reconstruct the 3D structure of molecules. Sparsity is natural for Ankylography, because molecules can be represented compactly in stoichiometric basis. Utilizing sparsity, we surpass current limits on recoverable information by orders of magnitude, paving the way for deciphering the 3D structure of macromolecules.Quantum wave functions shaping: 

Beam shaping in optics is well established, and there are quite a few interesting phenomena employing wavefront shaping. For example, generating a beam that carries OAM (Orbital Angular Momentum) with a helical wavefront, creating diffraction-free beams (e.g. Bessel beam). According to quantum mechanics, elementary particles are also waves. For example, it is known from the 20s that electrons have wavelike nature. In spite the fact that the wavelike nature of electrons is known for almost a century, only recently shaping EBeams has become experimentally accessible. The wavelength of an electron with accessible energies is orders of magnitude shorter than the optical wavelength. Now with the ability to shape its wavefront, the EBeam could reveal new physical phenomena at a new resolution scale.

Here, I show that interference effects of the quantum wavefunction describing multiple electrons can exactly balance both the repulsion among the electrons and their diffraction-broadening, which can also carry OAM. Such beams are multi-electron non-diffracting vortex beams. This wavefunction shaping facilitates the use of EBeams of higher current in numerous applications, thereby improving the SNR in electron microscopy and related systems without compromising on the spatial resolution. This scheme potentially applies for any beams of charged particles, such as protons, muons and ion beams.