Wavefront Propagation Simulations Research Articles

Evaluation of the X-ray/EUV Nanolithography Facility at AS through wavefront propagation simulations.

Synchrotron light sources can provide the required spatial coherence, stability and control to support the development of advanced lithography at the extreme ultraviolet and soft X-ray wavelengths that are relevant to current and future fabricating technologies. Here an evaluation of the optical performance of thesoft X-ray (SXR) beamline of the Australian Synchrotron (AS) and its suitability for developing interference lithography using radiation in the 91.8 eV (13.5 nm) to 300 eV (4.13 nm) range are presented. A comprehensive physical optics model of the APPLE-II undulator source and SXR beamline was constructed to simulate the properties of the illumination at the proposed location of a photomask, as a function of photon energy, collimation and monochromator parameters. The model is validated using a combination of experimental measurements of the photon intensity distribution of the undulator harmonics. It is shown that the undulator harmonics intensity ratio can be accurately measured using an imaging detector and controlled using beamline optics. Finally, the photomask geometric constraints and achievable performance for the limiting case of fully spatially coherent illumination are evaluated.

Towards a wavefront-preservation X-ray crystal monochromator for high-repetition-rate FELs.

The wavefront preservation of coherent X-ray free-electron laser beams is pushing the requirement on the quality and performance of X-ray optics to an unprecedented level. The Strehl ratio can be used to quantify this requirement. In this paper, the criteria for thermal deformation of the X-ray optics are formulated, especially for crystal monochromators. To preserve the X-ray wavefront, the standard deviation of the height error should be sub-nm for mirrors and less than 25 pm for crystal monochromators. Cryocooled silicon crystals combined with two techniques can be used to achieve this level of performance for monochromator crystals: (1) using a focusing element to compensate the second-order component of the thermal deformation; (2)introducing a cooling pad between the cooling block and silicon crystal and optimizing the effective cooling temperature. Each of these techniques allows the thermal deformation in standard deviation of the height error to be reduced by an order of magnitude. As an example, for the LCLS-II-HE Dynamic X-ray Scattering instrument, the criteria on thermal deformation of a high-heat-load monochromator crystal can be achieved for a 100 W SASE FEL beam. Wavefront propagation simulations confirm that the reflected beam intensity profile is satisfactory on both the peak power density and focused beam size.

A WISER way for simulating the performance of gratings.

Soft X-ray monochromators for synchrotron radiation sources have been continuously developed over the years, improving energy resolution and stability. Great effort has been made in improving the surface quality of the optics involved, reaching values permitting diffraction-limited images. Still, one problem has not been solved, nor fully understood, yet: groove placing errors. Nowadays, these are one of the major factors responsible for resolving the power reduction of diffraction-gratings-based X-ray monochromators. Despite decades of use of gratings, there is not yet a universally established method for predicting and simulating the effect of groove placing errors on monochromator performance. This is especially important in the new scenario of high-coherent X-ray sources, i.e. diffraction-limited storage rings and free-electron lasers. To address this problem, in this article an approach based on WISER (Wavefront propagatIon Simulation codE libRary) is presented. WISER is a physical optics simulation package, also available in the user-friendly Orange Synchrotron Radiation Suite - OASYS. Even though it was originally conceived to assess the focusing performance of X-ray mirrors in the presence of height defects, it perfectly simulates the performance of a periodic (or quasi-periodic) structure like a diffraction grating. In this article, the way to use WISER and its application to a specific case, e.g. the design of a monochromator for the upgrade of a beamline at the Advanced Light Source, are shown. A simple rule for estimating how well the grooves are placed on a grating, based on calculation of the Strehl ratio, is also presented.

Memory and CPU efficient coherent mode decomposition of partially coherent synchrotron radiation with subtraction of common quadratic phase terms.

Application examples of a memory and CPU efficient coherent mode decomposition (CMD) method for wave-optics based simulation of the partially coherent undulator radiation propagation through a hard X-ray beamline in a 3rd generation synchrotron radiation source are presented. The high efficiency of the method is achieved thanks to the analytical treatment of the common quadratic phase terms that are developed in the phase of cross-spectral density (CSD) of partially coherent radiation at a distance from source. This treatment allows for a considerable, several orders of magnitude, reduction of the 4D CSD mesh density (and the memory occupied by the CSD) required for ensuring sufficient accuracies of wavefront propagation simulations with the modes produced by the CMD at a beamline entrance. This method, implemented in the "Synchrotron Radiation Workshop" open-source software, dramatically increases the feasibility of the CMD of 4D CSD for producing 2D coherent modes for a large variety of applications at storage rings and other types of radiation sources.

Impact of real mirror profiles inside a split-and-delay unit on the spatial intensity profile in pump/probe experiments at the European XFEL.

For the High-Energy-Density (HED) beamline at the SASE2 undulator of theEuropean XFEL, a hard X-ray split-and-delay unit (SDU) has been built enabling time-resolved pump/probe experiments with photon energies between 5 keV and 24 keV. The optical layout of the SDU is based on geometrical wavefront splitting and multilayer Bragg mirrors. Maximum delays between Δτ= ±1 ps at 24 keV and Δτ = ±23 ps at 5 keV will be possible. Time-dependent wavefront propagation simulations were performed by means of theSynchrotron Radiation Workshop (SRW) software in order to investigate theimpact of the optical layout, including diffraction on the beam splitter and recombiner edges and the three-dimensional topography of all eight mirrors, on the spatio-temporal properties of the XFEL pulses. The radiation is generated from noise by the code FAST which simulates the self-amplified spontaneous emission (SASE) process. A fast Fourier transformation evaluation of the disturbed interference pattern yields for ideal mirror surfaces a coherence time of τc = 0.23 fs and deduces one of τc = 0.21 fs for the real mirrors, thus with an error of Δτ = 0.02 fs which is smaller than the deviation resulting from shot-to-shot fluctuations of SASE2 pulses. The wavefronts are focused by means of compound refractive lenses in order to achieve fluences of a few hundred mJ mm-2 within a spot width of 20 µm (FWHM) diameter. Coherence effects and optics imperfections increase the peak intensity between 200 and 400% for pulse delays within the coherence time. Additionally, the influence of two off-set mirrors in the HED beamline are discussed. Further, we show the fluence distribution for Δz = ±3 mm around the focal spot along the optical axis. The simulations show that the topographies of the mirrors of the SDU are good enough to support X-ray pump/X-ray probe experiments.

Polarization aberration analysis for the PICTURE-C exoplanetary coronagraph

The Planetary Imaging Concept Testbed Using a Recoverable Experiment - Coronagraph (PICTURE-C) mission will directly image debris disks and exozodiacal dust around nearby stars from a high-altitude balloon using a vector vortex coronagraph. One of the many starlight leakage sources that can degrade the performance of the coronagraph is polarization aberration induced by the reflective optical coatings. A polarization ray trace of the PICTURE-C telescope and coronagraph is combined with a physical optics wavefront propagation simulation to quantify the expected amount of coronagraph leakage due to polarization aberration. The simulations show the leakage is below the budgeted contrast of 1 × 10 − 8.

Wavefront-propagation simulations supporting the design of a time-delay compensating monochromator beamline at FLASH2.

Wavefront-propagation simulations have been performed to complete the design of a monochromator beamline for FLASH2, the variable-gap undulator line at the soft X-ray free-electron laser in Hamburg (FLASH). Prior to propagation through the beamline optical elements, the parameters of the photon source were generated using the GENESIS code which includes the free-electron laser experimental data. Threshold tolerances for the misalignment of mirror angles are calculated and, since diffraction effects were included in the simulations, the minimum quality with respect to the slope errors required for the optics is determined.

Image Simulation and Analysis

AbstractNo image is perfect, except for paintings.We have to deal with various noise sources and image sensor artifacts as well as with diffraction and other geometrical effects. The simulation of such artifacts is a great help during a design process and a perfect method for proof of concepts: Without access to any real hardware it is possible to predict and visualize the performance of opto‐electronic systems.Algorithms for correction or analysis of image data can be developed and evaluated a long time before the imaging hardware is ready to use. The statistical methods used during the simulation process can also be used reversely to provide a deep statistical analysis of already existing images. Such an analysis as well as the comparison with simulated images can help to identify the source of artifacts and design mistakes or even be used during forensic work.Last but not least, the reverse usage of the simulation of diffraction and wave front propagation is a very effective method for the generation of computer generated holograms or DOEs.

The coherent radiation fraction of low-emittance synchrotrons.

In this work the coherence properties of the synchrotron radiation beam from an X-ray undulator in a fourth-generation storage ring are analyzed. A slightly focused X-ray beam is simulated using a wavefront propagation through a non-redundant array of slits and the mutual coherence function is directly obtained and compared with the Gaussian-Schell approximation. The numerical wave propagation and the approximate analytical approaches are shown to agree qualitatively, and it is also shown that, when the coherent fraction is selected by a finite aperture before the focusing element, even achromatic focusing systems like total reflection mirrors become slightly chromatic. This effect is also well accounted for in the Gaussian-Schell model. The wavefront propagation simulation through the non-redundant array was repeated with an imperfect mirror demonstrating that, although the wavefront is distorted, its coherent length is practically unchanged.

WavePropaGator: interactive framework for X-ray free-electron laser optics design and simulations.

This article describes the WavePropaGator (WPG) package, a new interactive software framework for coherent and partially coherent X-ray wavefront propagation simulations. The package has been developed at European XFEL for users at the existing and emerging free-electron laser (FEL) facilities, as well as at the third-generation synchrotron sources and future diffraction-limited storage rings. The WPG addresses the needs of beamline scientists and user groups to facilitate the design, optimization and improvement of X-ray optics to meet their experimental requirements. The package uses the Synchrotron Radiation Workshop (SRW) C/C++ library and its Python binding for numerical wavefront propagation simulations. The framework runs reliably under Linux, Microsoft Windows 7 and Apple Mac OS X and is distributed under an open-source license. The available tools allow for varying source parameters and optics layouts and visualizing the results interactively. The wavefront history structure can be used for tracking changes in every particular wavefront during propagation. The batch propagation mode enables processing of multiple wavefronts in workflow mode. The paper presents a general description of the package and gives some recent application examples, including modeling of full X-ray FEL beamlines and start-to-end simulation of experiments.

Time-dependent wave front propagation simulation of a hard x-ray split-and-delay unit: Towards a measurement of the temporal coherence properties of x-ray free electron lasers

For the European x-ray free electron laser (XFEL) a split-and-delay unit based on geometrical wavefront beam splitting and multilayer mirrors is built which covers the range of photon energies from 5 keV up to 20 keV. Maximum delays between $\mathrm{\ensuremath{\Delta}}\ensuremath{\tau}=\ifmmode\pm\else\textpm\fi{}2.5\text{ }\text{ }\mathrm{ps}$ at $h\ensuremath{\nu}=20\text{ }\mathrm{keV}$ and up to $\mathrm{\ensuremath{\Delta}}\ensuremath{\tau}=\ifmmode\pm\else\textpm\fi{}23\text{ }\text{ }\mathrm{ps}$ at $h\ensuremath{\nu}=5\text{ }\text{ }\mathrm{keV}$ will be possible. Time-dependent wave-optics simulations have been performed by means of Synchrotron Radiation Workshop software for XFEL pulses at $h\ensuremath{\nu}=5\text{ }\text{ }\mathrm{keV}$. The XFEL radiation was simulated using results of time-dependent simulations applying the self-amplified spontaneous emission code FAST. Main features of the optical layout, including diffraction on the beam splitter edge and optics imperfections measured with a nanometer optic component measuring machine slope measuring profiler, were taken into account. The impact of these effects on the characterization of the temporal properties of XFEL pulses is analyzed. An approach based on fast Fourier transformation allows for the evaluation of the temporal coherence despite large wavefront distortions caused by the optics imperfections. In this way, the fringes resulting from time-dependent two-beam interference can be filtered and evaluated yielding a coherence time of ${\ensuremath{\tau}}_{c}=0.187\text{ }\text{ }\mathrm{fs}$ (HWHM) for real, nonperfect mirrors, while for ideal mirrors a coherence time of ${\ensuremath{\tau}}_{c}=0.191\text{ }\text{ }\mathrm{fs}$ (HWHM) is expected.

Partially coherent X-ray wavefront propagation simulations including grazing-incidence focusing optics.

X-ray beamlines in modern synchrotron radiation sources make extensive use ofgrazing-incidence reflective optics, in particular Kirkpatrick-Baez elliptical mirror systems. These systems can focus the incoming X-rays down to nanometer-scale spot sizes while maintaining relatively large acceptance apertures and high flux in the focused radiation spots. In low-emittance storage rings and in free-electron lasers such systems are used with partially or even nearly fully coherent X-ray beams and often target diffraction-limited resolution. Therefore, their accurate simulation and modeling has to be performed within the framework of wave optics. Here the implementation and benchmarking of a wave-optics method for the simulation of grazing-incidence mirrors based on the local stationary-phase approximation or, in other words, the local propagation of the radiation electric field along geometrical rays, is described. The proposed method is CPU-efficient and fully compatible with the numerical methods of Fourier optics. It has been implemented in the Synchrotron Radiation Workshop (SRW) computer code and extensively tested against the geometrical ray-tracing code SHADOW. The test simulations have been performed for cases without and with diffraction at mirror apertures, including cases where the grazing-incidence mirrors can be hardly approximated by ideal lenses. Good agreement between the SRW and SHADOW simulation results is observed in the cases without diffraction. The differences between the simulation results obtained by the two codes in diffraction-dominated cases for illumination with fully or partially coherent radiation are analyzed and interpreted. The application of the new method for the simulation of wavefront propagation through a high-resolution X-ray microspectroscopy beamline at the National Synchrotron Light Source II (Brookhaven National Laboratory, USA) is demonstrated.

Probing transverse coherence of x-ray beam with 2-D phase grating interferometer.

Transverse coherence of the x-ray beam from a bending magnet source was studied along multiple directions using a 2-D π/2 phase grating by measuring interferogram visibilities at different distances behind the grating. These measurements suggest that the preferred measuring orientation of a 2-D checkerboard grating is along the diagonal directions of the square blocks, where the interferograms have higher visibility and are not sensitive to the deviation of the duty cycle of the grating period. These observations are verified by thorough wavefront propagation simulations. The accuracy of the measured coherence values was also validated by the simulation and analytical results obtained from the source parameters. In addition, capability of the technique in probing spatially resolved local transverse coherence is demonstrated.

Wavefront propagation simulations for beamlines and experiments with "Synchrotron Radiation Workshop"

An overview of the basic principles, the approach to fully- and partially-coherent synchrotron radiation wavefront propagation simulations, implemented in the "Synchrotron Radiation Workshop" (SRW) computer code, and simulation examples for different types of sources, including undulator and wiggler, different beamlines and experiments, are presented. Recent developments, such as the support of parallel calculations, which considerably increase the efficiency of partially-coherent wavefront propagation calculations, are described.

Experimental characterization of X-ray transverse coherence in the presence of beam transport optics

A simple Boron fiber based interference scheme [1] and other similar schemes are currently routinely used for X-ray coherence estimation at 3rd generation synchrotron radiation sources. If such a scheme is applied after a perfect monochromator and without any focusing / transport optics in the optical path, the interpretation of the measured interference pattern is relatively straightforward and can be done in terms of the basic parameters of the source [2]. However, if the interference scheme is used after some focusing optics, e.g. close to the X-ray beam waist, the visibility of fringes can be significantly affected by the new shape of the focused beam phase-space. At the same time, optical element imperfections still have a negative impact on the transverse coherence. In such situations, which are frequently encountered in experiments at beamlines, the quantitative interpretation of a measured interference pattern is not straightforward. Here we show that this can nevertheless be done by using partially-coherent synchrotron radiation wavefront propagation simulations. The results obtained from measurements, performed at the 32-ID undulator beamline of the Advanced Photon Source, and wavefront propagation based simulations show, in particular, that new generation 1D Beryllium Compound Refractive Lenses [3, 4] do not reduce the X-ray transverse coherence in any significant manner.

Simulation of planar ionization wave front propagation on an unstructured adaptive grid

The paper deals with the numerical solution of a basic 2D model of the propagation of an ionization wave. The system of equations describing this propagation consists of a coupled set of reaction–diffusion-convection equations and a Poissons equation. The transport equations are solved by a finite volume method on an unstructured triangular adaptive grid. The upwind scheme and the diamond scheme are used for the discretization of the convection and diffusion fluxes, respectively. The Poisson equation is also discretized by the diamond scheme. Numerical results are presented. We deal in more detail with numerical tests of the grid adaptation technique and its influence on the numerical results. An original behavior is observed. The grid refinement is not sufficient to obtain accurate results for this particular phenomenon. Using a second order scheme for convection is necessary.

Calculation for computer generated hologram using ray-sampling plane

We introduce a new algorithm for calculating computer generated hologram (CGH) using ray-sampling (RS) plane. RS plane is set at near the object and the light-rays emitted by the object are sampled at the plane. Then the light-rays are transformed into the wavefront with using the Fourier transforms. The wavefront on the CGH plane is calculated by wavefront propagation simulation from RS plane to CGH plane. The proposed method enables to reproduce high resolution image for deep 3D scene with angular reflection properties such as gloss appearance.

Rendering Wave Effects with Augmented Light Field

AbstractRay–based representations can model complex light transport but are limited in modeling diffraction effects that require the simulation of wavefront propagation. This paper provides a new paradigm that has the simplicity of light path tracing and yet provides an accurate characterization of both Fresnel and Fraunhofer diffraction. We introduce the concept of a light field transformer at the interface of transmissive occluders. This generates mathematically sound, virtual, and possibly negative‐valued light sources after the occluder. From a rendering perspective the only simple change is that radiance can be temporarily negative. We demonstrate the correctness of our approach both analytically, as well by comparing values with standard experiments in physics such as the Young's double slit. Our implementation is a shader program in OpenGL that can generate wave effects on arbitrary surfaces.

Stability of a short Rayleigh length laser resonator

Motivated by the prospect of constructing a short Rayleigh length free-electron laser in a high-vibration environment, we demonstrate the use of a collection of rays to study the effect of mirror vibration and distortion on the behavior of the fundamental optical mode of a cold-cavity resonator. We find that the ray collection accurately describes both on-axis and off-axis optical beams. We show that a tilt or transverse shift of a mirror causes the optical mode to rock about the original resonator axis, while a longitudinal mirror shift or a change in the mirror's radius of curvature causes the beam diameter at a mirror to successively dilate and contract on the mirror. Results are in excellent agreement with analytic calculations and wave front propagation simulations as long as the mirrors remain large with respect to the beam diameter.

Modeling impulse propagation and extracellular potential distributions in anisotropic cardiac tissue using a finite volume element discretization

This paper describes a physically-based spatial discretization scheme for the bidomain equations used for modeling wavefront conduction in the heart. The discretization method uses unstructured finite volumes, or the boxes, that are formed as a secondary geometric structure from an underlying hexaderal mesh. The method enforces local flux conservation and thus is ideally suited for dealing with the spatially varying material properties and complex geometries that characterize realistic cardiac muscle. Several examples are presented to demonstrate the application of the method to large scale simulation of wavefront propagation in the heart.