This paper describes a theory of transverse coherence properties of Undulator Radiation. Our study is of very practical relevance, because it yields specific predictions of Undulator Radiation cross-spectral density in various parts of the beamline. On the contrary, usual estimations of coherence properties assume that the undulator source is quasi-homogeneous, like thermal sources, and rely on the application of van Cittert-Zernike (VCZ) theorem, in its original or generalized form, for calculating transverse coherence length in the far-field approximation. The VCZ theorem is derived in the frame of Statistical Optics using a number of restrictive assumptions: in particular, the quasi-homogeneous assumption is demonstrated to be inaccurate in many practical situations regarding undulator sources. We propose a technique to calculate the cross-spectral density from undulator sources in the most general case. Also, we find the region of applicability of the quasi-homogeneous model and we present an analytical expression for the cross-spectral density which is valid up to the exit of the undulator. For the case of more general undulator sources, simple formulas for the transverse coherence length, interpolated from numerical calculations and suitable for beamline design applications are found. Finally, using a simple vertical slit, we show how transverse coherence properties of an X-ray beam can be manipulated to obtain a larger coherent spot-size on a sample. This invention was devised almost entirely on the basis of theoretical ideas developed throughout this paper.
Theory and measurement of spatial coherence of synchrotron radiation beams are briefly reviewed. Emphasis is given to simple relationships between electron beam characteristics and far field properties of the light beam.
An analytical approach describing properties of focused partially coherent X-ray beams is presented. The method is based on the results of statistical optics and gives both the beam size and transverse coherence length at any distance behind an optical element. In particular, here we consider Gaussian Schell-model beams and thin optical elements. Limiting cases of incoherent and fully coherent illumination of the focusing element are discussed. The effect of the beam defining aperture, typically used in combination with focusing elements at synchrotron sources to improve transverse coherence, …
A general theoretical approach based on the results of statistical optics is used for the analysis of the transverse coherence properties of 3-rd generation synchrotron sources and x-ray free-electron lasers (XFEL). Correlation properties of the wavefields are calculated at different distances from an equivalent Gaussian Schell-model source. This model is used to describe coherence properties of the five meter undulator source at the synchrotron storage ring PETRA III. In the case of XFEL sources the decomposition of the statistical fields into a sum of independently propagating transverse mod…
We performed an analysis of coherence properties of the 4-th generation high-energy storage rings with emittance values of 10 pmrad. It is presently expected that a storage ring with these low emittance values will reach diffraction limit at hard X-rays. Simulations of coherence properties were performed with the XRT software and analytical approach for different photon energies from 500 eV to 50 keV. It was demonstrated that a minimum photon emittance (diffraction limit) reached at such storage rings is λ/2π. Using mode decomposition we showed that at the parameters of the storage ring consid…
This article describes a complete theory of spatial coherence for undulator radiation sources. Current estimations of coherence properties often assume that undulator sources are quasi-homogeneous, like thermal sources, and rely on the application of the van Cittert-Zernike theorem for calculating the degree of transverse coherence. Such assumption is not adequate when treating third generation light sources, because the vertical(geometrical) emittance of the electron beam is comparable or even much smaller than the radiation wavelength in a very wide spectral interval that spans over four ord…
We present a method to characterize the spatial coherence of soft X-ray radiation from a single diffraction pattern. The technique is based on scattering from non-redundant arrays (NRA) of slits and records the degree of spatial coherence at several relative separations from one to 15 microns, simultaneously. Using NRAs we measured the transverse coherence of the X-ray beam at the XUV X-ray beamline P04 of the PETRA III synchrotron storage ring as a function of different beam parameters. To verify the results obtained with the NRAs additional Young's double pinhole experiments were conducted a…
We present a comprehensive analysis of coherence properties of the radiation from X-ray free electron laser (XFEL). We consider practically important case when XFEL is optimized for maximum gain. Such an optimization allows to reduce significantly parameter space. Application of similarity techniques to the results of numerical simulations allows to present all output characteristics of the optimized XFEL as functions of the only parameter, ratio of the emittance to the radiation wavelength, $ \hatε = 2πε/λ$. Our studies show that optimum performance of the XFEL in terms of transverse coherenc…
The evolution from 3rd to 4th generation synchrotron radiation (SR) sources significantly enhanced their coherence, making coherent scattering techniques such as coherent X-ray diffraction imaging (CXDI) and X-ray photon correlation spectroscopy (XPCS) more accessible. We take advantage of a coherent modes decomposition and propagation model to carry out coherence analysis and simulations of CXDI experiments for the coherent Hard X-ray Coherent Scattering (HXCS) beamline at High Energy Photon Source.
We study the interference between two parallel-propagating Gaussian beams, originated from the same source, as their transverse separation is tuned. The interference pattern as a function of such separation lead us to determine the spatial coherence length of the original beam, in a similar way that a Michelson-Morley interferometer can be employed to measure the temporal coherence of a transform limited pulse. Moreover, performing a Fourier transform of the two-beam transverse plane, we observe an intensity modulation in the transverse momentum variable. This observation resembles the Alford …
The paper describes a new concept of visible pump/X-ray probe/slow detector experiments that could be performed at third-generation synchrotron light sources. We propose a technique that would allow time resolution up to femtosecond capabilities to be recovered from a long (100 ps) X-ray probe pulse. The visible pump pulse must be as short as the desired time resolution. The principle of operation of the proposed pump-probe scheme is essentially based on the statistical properties of the synchrotron radiation. These properties are well known in statistical optics as properties of completely ch…
In this paper we analyze the image formation problem for undulator radiation through an optical system, accounting for the influence of the electron beam emittance. On the one hand, image formation with Synchrotron Radiation is governed by the laws of Statistical Optics. On the other hand, the widely used Gaussian-Shell model cannot be applied to describe the coherence properties of X-ray beams from third generation Synchrotron Radiation sources. As a result, a more rigorous analysis of coherence properties is required. We propose a technique to explicitly calculate the cross-spectral density …
It is known that a spatially partially coherent light field produces better imaging contrast compared to a spatially coherent field and that the contrast increases as the spatial coherence length of the field becomes smaller. The transverse spatial coherence length of most spatially partially coherent fields increases upon propagation. As a result, the field produces progressively decreasing image quality at subsequent transverse planes. By controlling the propagation of spatial coherence, we demonstrate enhanced image quality at different transverse planes along the propagation direction thro…
In parametric down conversion, a nonlinear crystal is pumped by a laser and spontaneous emission takes place in signal and idler modes according to the phase matching conditions. A seed laser can stimulate the emission in the signal beam if there is mode overlap between them. This also enhances the emission in the idler beam, affecting its coherence properties. While the degree of coherence of the idler field as a function of the seed power has already been studied, the transverse coherence length has not yet been properly investigated. The transverse coherence length is a key parameter of opt…
Recently, a spatiotemporal optical vortex (STOV) with a transverse orbital angular momentum (OAM) has been generated from coherent ultrafast pulses using mode-locked lasers. In contrast, we demonstrate theoretically and experimentally that a STOV can be generated from a light source with partial temporal coherence with fluctuating temporal phase. By eliminating the need of mode-locked laser sources, the partially coherent STOV will serve as a convenient and cost-effective transverse OAM source.
Optical coherence tomography angiography (OCTA) requires high transverse sampling density for visualizing retinal and choroidal capillaries. Low transverse sampling causes resolution degradation, such as the angiograms in wide-field OCTA. In this paper, we propose to address this problem using deep learning. We conducted extensive experiments on converting the centrally cropped 3 x 3 mm2 field of view (FOV) of the 8 x 8 mm2 foveal OCTA images (a sampling density of 22.9 $μ$m) to the native 3 x 3 mm2 en face OCTA images (a sampling density of 12.2 $μ$m). We employed a cycle-consistent adversari…
High harmonic generation (HHG) is an established means of producing coherent, short wavelength, ultrafast pulses from a compact set-up. Table-top high-harmonic sources are increasingly being used to image physical and biological systems using emerging techniques such as coherent diffraction imaging and ptychography. These novel imaging methods require coherent illumination, and it is therefore important to both characterize the spatial coherence of high-harmonic beams and understand the processes which limit this property. Here we investigate the near- and far-field spatial properties of high-…
We report on measurements of second-order intensity correlations at the high brilliance storage ring PETRA III using a prototype of the newly developed Adaptive Gain Integrating Pixel Detector (AGIPD). The detector recorded individual synchrotron radiation pulses with an x-ray photon energy of 14.4 keV and repetition rate of about 5 MHz. The second-order intensity correlation function was measured simultaneously at different spatial separations that allowed to determine the transverse coherence length at these x-ray energies. The measured values are in a good agreement with theoretical simulat…
When a coherent laser beam impinges on a random sample (e.g. a colloidal suspension), the scattered light exhibits characteristic speckles. If the temporal coherence of the light source is too short, then the speckles disappear, along with the possibility of performing homodyne or heterodyne scattering detection or photon correlation spectroscopy. Here we investigate the scattering of a so-called "skewed coherence beam", i.e., a short-coherence beam modified such that the field is coherent within slabs that are skewed with respect to the wave fronts. We show that such a beam generates speckles…
Using three-dimensional particle-in-cell simulations we show that a strongly nonlinear plasma wave excited by an ultrahigh intensity laser pulse works as a compact high-brightness source of X-ray radiation. It has been recently suggested by A. Pukhov and J. Meyer-ter-Vehn, Appl. Phys. B \textbf{74}, 355 (2002), that in a strongly nonlinear regime the plasma wave transforms to a ``bubble'', which is almost free from background electrons. Inside the bubble, a dense bunch of relativistic electrons is produced. These accelerated electrons make betatron oscillations in the transverse fields of the …
Proposed, justified and tested measuring of beam spatial coherence, based on the detection of an interference visibility of the equal to intensity of beam replicas emerging under reflection from the rotated plane-parallel plate. The method consists in measuring the interference contrast at the simultaneous change of the longitudinal path difference and transverse shift of named replicas. Accurate measurement of the angle of rotated plate and the amplitude distribution of interferential minimaxes allows to determine the width of the beam spatial coherence. Spatial coherence of some types of mul…
Measurements of the spatial and temporal coherence of single, femtosecond x-ray pulses generated by the first hard x-ray free-electron laser (FEL), the Linac Coherent Light Source (LCLS), are presented. Single shot measurements were performed at 780 eV x-ray photon energy using apertures containing double pinholes in "diffract and destroy" mode. We determined a coherence length of 17 micrometers in the vertical direction, which is approximately the size of the focused LCLS beam in the same direction. The analysis of the diffraction patterns produced by the pinholes with the largest separation …
For paraxial propagation of scalar waves the classic electromagnetic theory definition of transverse linear (TLM) and orbital angular (OAM) momenta of the beam wave are represented in terms of the coherence function. We show in examples that neither the presence of optical vortices is necessary for the intrinsic OAM, nor does the presence of optical vortices warrant the non-zero intrinsic OAM. The OAM is analyzed for homogeneously coherent and twisted partially coherent beam waves. A twisted Gaussian beam has an intrinsic OAM with a per-unit power value that can be continuously changed by vary…
Eight models for the coherence of the quasi-monochromatic light from spherical incoherent sources are constructed by placing incoherent monopole and dipole sources on the surface of a sphere, inside a ball and on a plane circular disk. All models allow relatively simple numerical calculations of coherence functions for arbitrary source sizes and positions of observation points. We show that the far-field regime for transverse coherence is formed at the distances larger than the source size, regardless of the wavelength. Wide angle far field coherences have simple analytical representations in …
We demonstrate theoretically and experimentally coherence-induced polarization changes in generic and higher-order vector vortex beams with polarization singularity. The prominent depolarization on decreasing transverse correlation-width in focused partially coherent vector vortex beam provides a means to shape the intensity profile and the degree of polarization (DOP) while preserving the polarization distribution. The intensity variation and DOP-dip are found to be dependent on the polarization singularity index of the beam. Our results may provide an additional degree of freedom in the myri…
We describe statistical and coherence properties of the radiation from x-ray free electron lasers (XFEL). It is shown that the X-ray FEL radiation before saturation is described with gaussian statistics. Particularly important is the case of the optimized X-ray FEL, studied in detail. Applying similarity techniques to the results of numerical simulations allowed us to find universal scaling relations for the main characteristics of an X-ray FEL operating in the saturation regime: efficiency, coherence time and degree of transverse coherence. We find that with an appropriate normalization of th…
We demonstrate that the shift of the stop band position with increasing oblique angle in periodic structures results in a wide transverse exponential field distribution corresponding to strong angular confinement of the radiation. The beam expansion follows an effective diffusive equation depending only upon the spectral mode width. In the presence of gain, the beam cross section is limited only by the size of the gain area. As an example of an active periodic photonic medium, we calculate and measure laser emission from a dye-doped cholesteric liquid crystal film.
We consider a realistic model for calculating the cross-spectral density of partially coherent beams from an x-ray undulator in a modern storage ring. This two-point coherence function is seen to have a speckled structure associated with the presence of x-ray coherence vortices and domain walls. Such cross-spectral density speckle is associated with a network of spatial pairs of points for which there is zero correlation. X-ray coherence vortices and domain walls are seen to emerge naturally as the number of coherent modes required increases. An understanding of the existence and nature of suc…
The investigation of the static and dynamic structural properties of colloidal systems relies on techniques capable of atomic resolution in real space and femtosecond resolution in time. Recently, the cross-correlation function (CCF) analysis of both X-rays and electron diffraction patterns from dilute and dense aggregates has demonstrated the ability to retrieve information on the sample's local order and symmetry. Open questions remain regarding the role of the beam coherence in the formation of the diffraction pattern and the properties of the CCF, especially in dense systems. Here, we simu…
We consider an inverse source problem for partially coherent light propagating in the Fresnel regime. The data is the coherence of the field measured away from the source. The reconstruction is based on a minimum residue formulation, which uses the authors' recent closed-form approximation formula for the coherence of the propagated field. The developed algorithms require a small data sample for convergence and yield stable inversion by exploiting information in the coherence as opposed to intensity-only measurements. Examples with both simulated and experimental data demonstrate the ability o…
When a disordered array of coupled waveguides is illuminated with an extended coherent optical field, discrete speckle develops: partially coherent light with a granular intensity distribution on the lattice sites. The same paradigm applies to a variety of other settings in photonics, such as imperfectly coupled resonators or fibers with randomly coupled cores. Through numerical simulations and analytical modeling, we uncover a set of surprising features that characterize discrete speckle in one- and two-dimensional lattices known to exhibit transverse Anderson localization. Firstly, the finge…
Transverse quantum-state characterization of programmable electron optics
Friedrich-Alexander-Universität Erlangen-Nürnberg（弗里德里希·亚历山大埃尔朗根-纽伦堡大学）；CNR Istituto Nanoscienze（意大利国家研究委员会纳米科学研究所）；CNR Istituto per lo Studio dei Materiali Nanostrutturati（意大利国家研究委员会纳米结构材料研究所）；Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons, Forschungszentrum Jülich（于利希研究中心恩斯特·鲁斯卡电子显微镜与光谱学中心）
