# Browsing by Subject "Phase retrieval"

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Item Open Access Experimental Study of Structured Light Using a Free-electron Laser Oscillator(2021) Liu, PeifanOver the past three decades, laser beams with complex amplitude and phase structures, especially orbital angular momentum (OAM) beams, have been extensively investigated. Researchers have found a wide range of applications for OAM beams spanning a vast range of distance scales, from fundamental physics at the atomic level with modified selection rules, to macroscopic use such as optical tweezers, to probing of the universe such as detection of rotating black holes.

While structured light beams in the visible and longer wavelength regimes can be generated using many techniques, at shorter wavelengths, from vacuum ultraviolet to x-rays to gamma rays, it is much more challenging to produce such light beams. In recent years, to generate structured light in the shorter wavelengths, particle accelerator-based light sources, such as magnetic undulators and free-electron lasers (FELs), have been explored as a promising candidate. While the FEL work was mostly limited to single-pass FELs, we recognized that the oscillator FEL is very attractive for producing high-quality OAM beams with high intracavity power. In this work, we report the first experimental generation of a particular kind of structured light, a coherently mixed (CM) OAM beam, using the Duke storage ring FEL. The coherently mixed OAM beams have been generated up to the fourth order. This was made possible by modifying the FEL cavity to obtain cylindrical symmetry, while suppressing the low-order transverse modes. The cavity modification was implemented using a set of specially developed masks, including an annulus mask and a disk mask.

On the other hand, a reliable and rapid assessment of the structured light has a wide range of applications in the laser development, including high-quality OAM beam generation, optical characterization of beam quality and mode contents, and manipulation and correction of distorted OAM beams. While the diagnostic methods for structured light have been widely investigated in long wavelengths during recent years, they are not available for the short-wavelength regimes due to wavelength limitations of optics used. We report here two general diagnostic techniques for structured light: a phase retrieval method for wavefront reconstruction; and a modal analysis method for assessing the mode contents and beam quality of a structured laser beam. These newly developed methods involve very few optics, and in principle, can be used in a wide range of wavelengths, from infrared to visible to UV and x-ray.

The produced coherently mixed OAM FEL beams are found to possess good beam quality, excellent stability and reproducibility, and substantial intracavity power. Using the aforementioned diagnostic techniques, we have analyzed the measured FEL beam images to retrieve the complex wavefront and mode content. These beams have been found to have good mode quality, dominated by two degenerate OAM modes of the same order but opposite helicities. A pulsed mode operation of the OAM FEL beam has also been developed using an external drive, in which the OAM beams exhibit a highly reproducible temporal structure when the pulsing frequency is varied from 1 Hz to 30 Hz.

The development of OAM FEL beams using the storage ring FEL has paved the way for short-wavelength OAM laser beam generation using future FEL oscillators operating in the extreme ultraviolet and x-ray regimes. The operation of the storage ring FEL also paves the way for the generation of OAM gamma-ray beams via Compton scattering.

Item Open Access High Dimensional Signal Representation(2017) Yin, RujieIn this thesis we explore the efficiency of signal representations and their robustness in signal reconstruction in three subfields of signal and image processing.

The first result concerns regularity limitation in the construction of directional wavelet bases due to redundancy constraint on the scheme, in an effort to construct “optimal” directional bases with multiresolution and perfect reconstruction proper- ties. We showed that for orthonormal and biorthogonal bases with dilated quincunx downsampling, the wavelets cannot be well localized; however, this regularity limit can be circumvented in a tight frame with dyadic downsampling and a redundancy factor smaller than 2.

The second result introduces a novel framework for patch-based image models combining local structure of patches and nonlocal information in image domain. In particular, we built convolution framelets from local and nonlocal bases, which form a tight frame of the image space and has energy concentration when the local and nonlocal bases are coherent. We applied this framework to reinterpret and improve state-of-the-art low dimensional manifold model.

The final result proposes a new paradigm of phase retrieval, considering signal reconstruction up to a larger equivalence class than a uniform phase shift. It is known that in the classical setting, phase retrieval in infinite or high dimension is inherently unstable. We showed that stability can be achieved, however, for frames of Gabor wavelets or Cauchy wavelets in this new paradigm.

Item Open Access On Lipschitz analysis and Lipschitz synthesis for the phase retrieval problem(Linear Algebra and Its Applications, 2016-05-01) Balan, R; Zou, D© 2016 Elsevier Inc. All rights reserved. We prove two results with regard to reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem"). First we show that phase retrievable nonlinear maps are bi-Lipschitz with respect to appropriate metrics on the quotient space. Specifically, if nonlinear analysis maps α,β:H→→ℝm are injective, with α(x)=(||)km=1 and β(x)=(||2)km=1, where {f1,...,fm} is a frame for a Hilbert space H and H=H/T1, then α is bi-Lipschitz with respect to the class of "natural metrics" Dp(x,y)=minφ||x-eiφy||p, whereas β is bi-Lipschitz with respect to the class of matrix-norm induced metrics dp(x,y)=||xx∗-yy∗||p. Second we prove that reconstruction can be performed using Lipschitz continuous maps. That is, there exist left inverse maps (synthesis maps) ω,ψ:ℝm→H of α and β respectively, that are Lipschitz continuous with respect to appropriate metrics. Additionally, we obtain the Lipschitz constants of ω and ψ in terms of the lower Lipschitz constants of α and β, respectively. Surprisingly, the increase in both Lipschitz constants is a relatively small factor, independent of the space dimension or the frame redundancy.