Theory and operational rules for the discrete Hankel transform

Pump-probe X-ray holographic imaging of laser-induced cavitation bubbles with femtosecond FEL pulses

Cavitation bubbles can be seeded from a plasma following optical breakdown, by focusing an intense laser in water. The fast dynamics are associated with extreme states of gas and liquid, especially in the nascent state. This offers a unique setting to probe water and water vapor far-from equilibrium. However, current optical techniques cannot quantify these early states due to contrast and resolution limitations. X-ray holography with single X-ray free-electron laser pulses has now enabled a quasi-instantaneous high resolution structural probe with contrast proportional to the electron density of the object. In this work, we demonstrate cone-beam holographic flash imaging of laser-induced cavitation bubbles in water with nanofocused X-ray free-electron laser pulses. We quantify the spatial and temporal pressure distribution of the shockwave surrounding the expanding cavitation bubble at time delays shortly after seeding and compare the results to numerical simulations.

Correlation between geometric parametric instability sidebands in graded-index multimode fibers

The spectral analysis of the light propagating in normally dispersive graded-index multimode fibers is performed under initial noisy conditions. Based on the obtained spectra with multiple simulations in the presence of noise, we investigate the correlation in energy between the well-separated spectral sidebands through both the scattergrams and the frequency-dependent energy correlation map and find that conjugate couples are highly correlated while cross-combinations exhibit a very poor degree of correlation. These results reveal that the geometric parametric instability processes associated with each sideband pair occur independently from each other, which can provide significant insights into the fundamental dynamical effect of the geometric parametric instability and facilitate the future implementation of high-efficiency photon pair sources with reduced Raman decorrelations.

Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform

The theory of the continuous two-dimensional (2D) Fourier Transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In the first part of this two-paper series, we proposed and evaluated the theory of the 2D Discrete Fourier Transform (DFT) in polar coordinates. The theory of the actual manipulated quantities was shown, including the standard set of shift, modulation, multiplication, and convolution rules. In this second part of the series, we address the computational aspects of the 2D DFT in polar coordinates. Specifically, we demonstrate how the decomposition of the 2D DFT as a DFT, Discrete Hankel Transform and inverse DFT sequence can be exploited for coding. We also demonstrate how the proposed 2D DFT can be used to approximate the continuous forward and inverse Fourier transform in polar coordinates in the same manner that the 1D DFT can be used to approximate its continuous counterpart.

Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules

The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT.

Optimal transmission modes under atmosphere turbulence with transmitter/receiver aperture size constraint

In this paper, an optimal mode set is proposed to maximize the receiving power for free space transmission under atmosphere turbulence with transmitter/receiver aperture size constraint. The optimal beam profiles are evaluated through eigenmode analysis of the Fredholm integral equation, which is mathematically equivalent to the eigen vector analysis of an infinitely large matrix. The matrix is formed by orthonormal basis expansion, and its element is the overlap integral of the orthonormal basis functions and the Fredholm kernel. If circular aperture is implemented, then it is rigorously proven in this work that the eigenmodes possess certain topological charges (i.e., they are the OAM modes). These OAM modes have specific radial beam profiles, which have been optimized to minimize the power loss and the inter-mode crosstalk. While the traditional OAM beams, such as the Laguerre-Gauss (LG) beams, suffer significant energy loss and inter-radial-mode crosstalk, the optimized beam profiles will remarkably reduce the penalties.

Optimal Alignment of Structures for Finite and Periodic Systems

Finding the optimal alignment between two structures is important for identifying the minimum root-mean-square distance (RMSD) between them and as a starting point for calculating pathways. Most current algorithms for aligning structures are stochastic, scale exponentially with the size of structure, and the performance can be unreliable. We present two complementary methods for aligning structures corresponding to isolated clusters of atoms and to condensed matter described by a periodic cubic supercell. The first method (Go-PERMDIST), a branch and bound algorithm, locates the global minimum RMSD deterministically in polynomial time. The run time increases for larger RMSDs. The second method (FASTOVERLAP) is a heuristic algorithm that aligns structures by finding the global maximum kernel correlation between them using fast Fourier transforms (FFTs) and fast SO(3) transforms (SOFTs). For periodic systems, FASTOVERLAP scales with the square of the number of identical atoms in the system, reliably finds the best alignment between structures that are not too distant, and shows significantly better performance than existing algorithms. The expected run time for Go-PERMDIST is longer than FASTOVERLAP for periodic systems. For finite clusters, the FASTOVERLAP algorithm is competitive with existing algorithms. The expected run time for Go-PERMDIST to find the global RMSD between two structures deterministically is generally longer than for existing stochastic algorithms. However, with an earlier exit condition, Go-PERMDIST exhibits similar or better performance.

Rotationally symmetric formulation of the wave propagation method-application to the straylight analysis of diffractive lenses

We introduce a modified formulation of the wave propagation method for the efficient simulation of rotationally symmetric micro-optical components. The reformulated algorithm provides an increased computational performance of approximately two orders of magnitude and strongly reduced memory requirements, in comparison to the original formulation. This enables the efficient wave optical simulation of extended micro-optical structures beyond the common thin-element approximation. As a prototypical example, we assess the modified algorithm for the evaluation of straylight induced by diffractive lenses. We find an excellent accuracy, while comparing to rigorous simulations, which justifies the ability to overcome the limitations of the thin-element approximation.