Diffraction Amplitudes Research Articles

Advances in the Parameter Space Concept towards Picometer Precise Crystal Structure Refinement—A Resolution Study

The Parameter Space Concept (PSC) is an alternative approach to solving and refining (partial) crystal structures from very few pre-chosen X-ray or neutron diffraction amplitudes without the use of Fourier inversion. PSC interprets those amplitudes as piecewise analytic hyper-surfaces, so-called isosurfaces, in the Parameter Space, which is spanned by the spatial coordinates of all atoms of interest. The intersections of all isosurfaces constitute the (possibly degenerate) structure solution. The present feasibility study investigates the La and Sr split position of the potential high-temperature super-conductor (La0.5Sr1.5)MnO4, I4/mmm, with a postulated total displacement between La and Sr of a few pm by theoretical amplitudes of pre-selected 00l reflections (l=2,4,…,20). The revision of 15-year-old results with state-of-the-art computing equipment enhances the former simplified model by varying the scattering power ratio fLa/fSr, as exploitable by means of resonant scattering contrast at synchrotron facilities, and irrevocably reveals one of the two originally proposed solutions as being a “blurred” pseudo-solution. Finally, studying the resolution limits of PSC as a function of intensity errors by means of Monte-Carlo simulations shows both that the split can only be resolved for sufficiently low errors and, particularly for the resonant scattering contrast, a theoretical precision down to ±0.19 pm can be achieved for this specific structural problem.

Manufacture of phase diffraction gratings by acid immersion

In this work, the manufacturing, characterization, and optical testing of phase diffraction gratings is performed. The fabrication is carried out in a very simple way: by immersion of a common amplitude diffraction gratings on acid and a posterior chemical procedure. The process makes it achievable for almost any scientist, since expensive equipment is not required. We have tested three different acids and evaluated the engraving depth (phase retardation) in terms of the immersion time. With this analysis, phase diffraction gratings with the desired phase delay could be manufactured. The quality of the gratings has been evaluated by means of their profiles and histograms of the surface, using confocal microscopy. To conclude, we have measured the near-field behavior of one of the phase gratings and compared it with that obtained analytically, showing high agreement.

Modeling stochastic elastic wave diffraction by the tips of randomly rough defects

Elastic wave scattering from a randomly rough surface of a finite length includes surface reflections and diffractions from the tips. Previous research has focused upon reflection waves with applications in ultrasonic defect detection, seismic wave exploration and phonon boundary transport. However, waves diffracted from the tips/edges have been largely neglected so far for rough defects, despite their importance in engineering applications including ultrasonic defect sizing and imaging for assessment of structural integrity. Currently understanding the statistical nature of elastic wave tip diffraction and the role of roughness is limited due to the lack of theoretical studies. In this article, we develop a statistical geometrical tip diffraction (SGTD) theory to rapidly predict the stochastic properties of tip diffraction amplitude as a function of surface roughness and incident angle. By applying a small slope perturbation to the model, a simplified analytical solution of tip diffraction is obtained. It is found that for defects with small to medium roughness, the diffraction amplitude explicitly follows a Gamma distribution, and its mean and the standard deviation are both proportional to the square of the rms slope. High-fidelity Monte Carlo finite element simulations are then run to evaluate the accuracy of the theoretical model. The range of validity of the analytical solution with respect to the level of roughness and the incident angle is obtained. The SGTD method is accurate when the correlation length is approximately equivalent or larger than one wavelength, for a wide range of angles. It is also applicable for a correlation length as short as half wavelength, but only for small rms values and when the beam angle is larger than 45∘. In addition, at large angles, the tip diffraction is almost not affected by roughness, being very similar to that from a smooth crack. This is explained by the significant dependence on the beam angle factor explicitly shown in the theoretical solution.

Internal mechanism of perfect-reflector-backed dielectric gratings to achieve 100% diffraction efficiency.

The work started 20 years ago [Appl. Opt.42, 6255 (2003)APOPAI0003-693510.1364/AO.42.006255] investigating the physical mechanism of multilayer dielectric reflection gratings to achieve 100% diffraction efficiency is completed to offer much deeper insight than before. How different scattering matrix elements of the top periodic surface corrugation contribute to the -1st-order efficiency of such a compound grating is unveiled analytically using a minimum set of real parameters. The two diffraction amplitudes transmitted through the top corrugation play a dominant role in enabling 100% diffraction efficiency. Simple necessary and sufficient conditions for 100% efficiency are derived. Moreover, the role of the reflection phase of the perfect reflector, including the contribution of the optical path between the top corrugation and the reflector, is emphasized.

Similarity score for screening phase-retrieved maps in X-ray diffraction imaging - characterization in reciprocal space.

X-ray diffraction imaging (XDI) is utilized for visualizing the structures of non-crystalline particles in material sciences and biology. In the structural analysis, phase-retrieval (PR) algorithms are applied to the diffraction amplitude data alone to reconstruct the electron density map of a specimen particle projected along the direction of the incident X-rays. However, PR calculations may not lead to good convergence because of a lack of diffraction patterns in small-angle regions and Poisson noise in X-ray detection. Therefore, the PR calculation is still a bottleneck for the efficient application of XDI in the structural analyses of non-crystalline particles. For screening maps from hundreds of trial PR calculations, we have been using a score and measuring the similarity between a pair of retrieved maps. Empirically, probable maps approximating the particle structures gave a score smaller than a threshold value, but the reasons for the effectiveness of the score are still unclear. In this study, the score is characterized in terms of the phase differences between the structure factors of the retrieved maps, the usefulness of the score in screening the maps retrieved from experimental diffraction patterns is demonstrated, and the effective resolution of similarity-score-selected maps is discussed.

A boundary integration method for calculating diffraction intensity of crystalline materials with planar deformation

Defects and strain states in crystals can be investigated using X-ray and neutron diffraction, two types of penetrating, non-destructive detection techniques. Various approximations are frequently employed to minimize the amount of computation in the practical calculations because the present Fourier transform approach, established by Warren, Averbach et al. based on the Kinematic diffraction theory, has a huge computational cost. This paper develops a numerical method to precisely calculate diffraction intensity by integrating kinematic diffraction amplitude over the boundary, which is applicable to crystals with planar deformation. The precision of this method is assessed by contrasting it with Wilson's analytical equations for the X-ray diffraction results of a single screw dislocation. As an example, the difference in diffraction spots on a single edge dislocation irradiated by different incident X-ray widths are calculated and analyzed. When compared to the X-ray diffraction spots of an ideal material with a dislocation core dug out, the results indicate that if lattice information close to the dislocation core is to be retrieved by X-ray diffraction, the X-ray irradiation area should be restricted.

Evaluation of convolutional neural network for fast extreme ultraviolet lithography simulation using imec 3 nm node mask patterns

BackgroundMask 3D (M3D) effects distort diffraction amplitudes from extreme ultraviolet masks. In our previous work, we developed a convolutional neural network (CNN) that very quickly predicted the distorted diffraction amplitudes from input mask patterns. The mask patterns were restricted to Manhattan patterns.AimWe verify the potentials and the limitations of CNN using imec 3 nm node (iN3) mask patterns.ApproachWe apply the same CNN architecture in the previous work to mask patterns, which mimic iN3 logic metal or via layers. In addition, to study more general mask patterns, we apply the architecture to iN3 metal/via patterns with optical proximity correction (OPC) and curvilinear via patterns. In total, we train five different CNNs: metal patterns w/wo OPC, via patterns w/wo OPC, and curvilinear via patterns. After the training, we validate each CNN using validation data with the above five different characteristics.ResultsWhen we use the training and validation data with the same characteristics, the validation loss becomes very small. Our CNN architecture is flexible enough to be applied to iN3 metal and via layers. The architecture has the capability to recognize curvilinear mask patterns. On the other hand, using the training and validation data with different characteristics will lead to large validation loss. The selection of training data is very important for obtaining high accuracy. We examine the impact of M3D effects on iN3 metal layers. A large difference is observed in the tip to tip (T2T) critical dimension calculated by the thin mask model and thick mask model. This is due to the mask shadowing effect at T2T slits.ConclusionsThe selection of training data is very important for obtaining high accuracy. Our test results suggest that layer specific CNN could be constructed, but further development of CNN architecture could be required.

Structure–function relationship during the early and long‐term hydration of one‐part alkali‐activated slag

AbstractUnderstanding the mechanisms controlling the early (fresh) and long‐term (hardened) hydration of one‐part alkali‐activated slags (AAS) is key to extend their use as low CO2 substitutes for ordinary Portland cement (OPC). Their “just add water” use makes them easier and less hazardous to manipulate than the more studied two‐part ones. This is due to the absence of liquid alkaline activators, which are environmentally and energy demanding. In this work, numerous experimental techniques have been linked to obtain a comprehensive physico‐chemical characterization of a one‐part AAS activated with Na2CO3 and Ca(OH)2 powders at several water to solid ratios (w/s). Calorimetry and pH/conductivity measurements describe the functioning of the activators immediately after contact with water. Early reactivity is characterized through in situ X‐ray powder diffraction (XRPD) and small amplitude oscillatory shear (SAOS) rheology, which reveal a rapid precipitation of nanometric hydration products (nano‐C‐A‐S‐H), which results in a continuous increase in the paste cohesivity until setting. Moreover, SAOS shows that rejuvenating the paste by means of shearing (performed externally to the rheometer in this study) is enough to restore the initial cohesion (i.e., workability) for long time spans until setting occurs. The long‐term hydration is characterized by ex situ XRPD on aged AAS pastes, in parallel with mechanical testing on AAS mortar. A correlation can be observed between the amount of nano‐C‐A‐S‐H and the increase in compressive strength. Overall, this formulation shows satisfactory fresh and solid properties, demonstrating suitability for low‐ and normal‐strength applications.

Airy-type X-ray states generated using 3/2 flat diffractive optics.

X-rays have developed into an essential tool in variety of fields, such as biology, materials, chemistry, and physics etc. Numerous X-ray types, including the orbital angular momentum (OAM), the Laguerre-Gauss, and the Hermite-Gauss states, have been proposed. This greatly enhances the depth of application of X-ray. The X-ray states described above are mostly produced by binary amplitude diffraction elements. In light of this, this paper proposes a flat X-ray diffraction grating based on caustic theory to generate Airy-type X-ray. It is proved by the simulation of multislice method that the proposed grating can generate the Airy beam in the X-ray field. The results show that the generated beams have a secondary parabolic trajectory deflection with the propagation distance, which is consistent with the theory. Inspired by the success of Airy beam in light-sheet microscope, the Airy-type X-ray can be anticipated to enable novel image capability for bio or nanoscience.

Effect of composite vortex beam on a two-dimensional gain assisted atomic grating

We propose an atomic grating based on an electromagnetically induced transparency phenomenon that switches between zeroth-order diffraction to a distinct higher-order diffraction pattern by driving a planar gaseous medium of a four-level tripod () atoms with three laser beams: modulation of standing wave control beam propagating nearly perpendicular to the planar medium, while vortex and weak plane probe beams directed perpendicular to the medium. We numerically investigate the behavior of the amplitude, phase modulations, and probe field diffraction intensities of different orders by the variation of the field detunings and orbital angular momentum number of the composite vortex light beam. Specifically, in the off-resonant case, the interplay between a square lattice of the control and an additional spatial variation of the vortex beam allows the emergence of higher diffraction orders and variable gain due to double transparency windows in this complex optical system. We believe that our proposed scheme might be useful in optical memory devices via the storage of information to diffraction orders of the atomic grating.

Using the Size Strain Plot Method to Specity Lattice Parameters

X-ray diffractometers deliver the best quality diffraction data while being easy to use and adaptable to various applications. When X-ray photons strike electrons in materials, the incident photons scatter in a direction different from the incident beam; if the scattered beams do not change in wavelength, this is known as elastic scattering, which causes amplitude and intensity diffraction, leading to constructive interference. When the incident beam gives some of its energy to the electrons, the scattered beam's wavelength differs from the incident beam's wavelength, causing inelastic scattering, which leads to destructive interference and zero-intensity diffraction. In this study, The modified size-strain plot method was used to examine the pattern of x-ray diffraction lines (101),(121),(202),(042), and (242) for calcium titanate(CaTiO3) nanoparticles in this study. To calculate the new variables, the size strain plot method was created., X-ray line analysis and calculation of crystal size and lattice tension of calcium titanate oxide nanoparticles. It is used to calculate the crystal volume (44.7 nm) and to calculate the determination of network parameters such as the texture modulus (Tc), macro stress (MS), specific surface area (SSA), and dislocation density(η), respectively.

Internal mechanism of perfect-reflector-backed dielectric gratings to achieve high diffraction efficiency

The work started 20 years ago [Appl. Opt. 42, 6255 (2003)] investigating the physical mechanism of multilayer dielectric reflection gratings to achieve 100% diffraction efficiency is extended to offer much deeper insight than before. How different diffraction amplitudes of the top surface corrugation contribute to the −1st-order efficiency of such a grating is shown analytically using a minimum set of real parameters. The two diffraction amplitudes transmitted through the corrugation play a dominant role in enabling DE = 100%. The necessary and sufficient condition for 100% efficiency is derived, and a very simple sufficient condition is also given. Moreover, the role of the reflection phase of the perfect-reflector, including the contribution due to optical path between the corrugation and the reflector, is emphasized.

Generation of multiple-vortex beams in the process of diffraction of cosine-Laguerre–Gaussian laser beam by the fork-shaped grating

We present the theoretical study about Fraunhofer diffraction of cosine-Laguerre–Gaussian laser beam of mode l,n=0 by the fork-shaped grating with a phase singularity of integer order p. The analytical expressions for the diffracted wave field amplitude and intensity distributions are derived. The transverse intensity and phase profiles of the diffracted beam, in a given diffraction order and for different values of l and p, are calculated. They show that the higher diffraction orders are multiple-vortex beams. In addition, the interference patterns of the diffracted wave field with a slightly inclined plane wave are presented, for confirming the presence of the optical vortices and determining the values and signs of their topological charges. We have shown how an azimuthally shaped, non-vortex beam, consisting of 2l bright lattices, can be transformed into a fan of multiple-vortex beams in the higher diffraction orders.

Data augmentation in extreme ultraviolet lithography simulation using convolutional neural network

BackgroundIn the previous work, we developed a convolutional neural network (CNN), which reproduces the results of the rigorous electromagnetic (EM) simulations in a small mask area. The prediction time of CNN was 5000 times faster than the calculation time of EM simulation. We trained the CNN using 200,000 data, which were the results of EM simulation. Although the prediction time of CNN was very short, it took a long time to build a huge amount of the training data. Especially when we enlarge the mask area, the calculation time to prepare the training data becomes unacceptably long.AimReducing the calculation time to prepare the training data.ApproachWe apply data augmentation technique to increase the number of training data using limited original data. The training data of our CNN are the diffraction amplitudes of mask patterns. Assuming a periodic boundary condition, the diffraction amplitudes of the shifted or flipped mask pattern can be easily calculated using the diffraction amplitudes of the original mask pattern.ResultsThe number of training data after the data augmentation is multiplied by 200 from 2500 to 500,000. Using a large amount of training data, the validation loss of CNN was reduced. The accuracy of CNN with augmented data is verified by comparing the CNN predictions with the results of EM simulation.ConclusionsData augmentation technique is applied to the diffraction amplitude of the mask pattern. The data preparation time is reduced by a factor of 200. Our CNN almost reproduces the results of EM simulation. In this work, the mask patterns are restricted to line and space patterns. It is a challenge to build several CNNs for specific mask patterns or ultimately a single CNN for arbitrary mask patterns.

Unveil the unseen: Exploit information hidden in noise

Noise and uncertainty are usually the enemy of machine learning, noise in training data leads to uncertainty and inaccuracy in the predictions. However, we develop a machine learning architecture that extracts crucial information out of the noise itself to improve the predictions. The phenomenology computes and then utilizes uncertainty in one target variable to predict a second target variable. We apply this formalism to PbZr0.7Sn0.3O3 crystal, using the uncertainty in dielectric constant to extrapolate heat capacity, correctly predicting a phase transition that otherwise cannot be extrapolated. For the second example – single-particle diffraction of droplets – we utilize the particle count together with its uncertainty to extrapolate the ground truth diffraction amplitude, delivering better predictions than when we utilize only the particle count. Our generic formalism enables the exploitation of uncertainty in machine learning, which has a broad range of applications in the physical sciences and beyond.

Wigner’s effective mathematics and contradiction

Complex numbers are basic. An inconsistency would question Wigner's unreasonable effectiveness of mathematics. A vehicle to study this question is Kirchoff's scalar diffraction theory. In the paper, an inconsistency in complex phase angle is presented. When this inconsistency is introduced in Kirchoff's theory we can study its influence on the experimental success of this theory. There are no \emph{a priori} reasons to include or exclude phase angles. Referring to Wigner, an experiment can provide more insight. In the experiment a weak intensity, small wavelength source can be employed. When the contradictory phase angle is excluded, a nonzero diffraction amplitude appears physically possible. If it is included, this amplitude vanishes.

Diffractive Deep Neural Network for Optical Orbital Angular Momentum Multiplexing and Demultiplexing

Vortex beams (VBs), characterized by helical phase front and orbital angular momentum (OAM), have shown perspective potential in improving communication capacity density for providing an additional multiplexing dimension. Here, we propose a diffractive deep neural network (D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> NN) method for OAM mode multiplexing and demultiplexing. By designing the D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> NN model and simulating light propagation through multiple diffractive screens, the phase and amplitude values can be automatically adjusted to manipulate the wavefront of light beams. Training the D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> NN model with mode coupler and separator functions, we convert VBs into target light fields with the diffraction efficiency exceeds 97%, and the mode purities are over 97%. Constructing an OAM multiplexing link, we successfully multiplex and demultiplex two OAM channels that carry 16-QAM signals in simulation, and the demodulated bit-error-rates are below 1×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> . It is anticipated that the D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> NN can perform flexible modulation of multiple OAM modes, which may open a new avenue for high-capacity OAM communication and all-optical information processing, etc.

Tailored diffraction asymmetries from spatially odd-symmetric phase gratings

Phase gratings, each period of which consists of even numbers of equal-width and equal-thickness elements, can be devised to attain asymmetric diffraction patterns. We show that engineering of offset refractive indices in different elements exhibiting a spatially odd symmetry leads to the elimination of a single diffraction order (directional elimination), while further manipulations on offset refractive indices may lead to the elimination of all odd or even orders (grouped elimination) or all orders but one or two selected (directional selection). These intriguing effects arise from destructive interference between diffracted amplitudes contributed by paired or successive elements, and violation of Friedel's law in such transparent gratings is an effect of higher order multiple scattering.

Design of a point absorber wave energy converter for an indian coast

Abstract Ocean waves are considered as a potential unharnessed renewable resource being 800 times denser than wind. India has a wide coastal area of nearly 7500 km, providing a huge potential for harnessing ocean wave energy. The article aims to present the parametric optimization of a point absorber wave energy converter (PAWEC) for a location near Ennore port of India. Indian offshore condition such as significant wave height, period, and amplitude was reviewed for their minimum, maximum, and average values for several years. Hydrodynamic coefficients such as Froude-Krylov force, radiation damping, added mass, diffraction, excitation and response amplitude operator are optimized through geometric optimization of PAWEC’s float. In hydrodynamic response analysis, only heaving motion is considered and all motion is neglected for the study. Regular wave is only considered for this study. Output parameters such as structure velocity and structure response are studied for the mentioned geometry. Power-take-off (PTO) device is simulated for maximum efficiency and the float velocity response is observed. Monthly variation in the mean absorbed power and efficiency of Point absorber is calculated with respect to ocean power.

Constraints on pure point diffraction on aperiodic point patterns of finite local complexity* *Dedicated to the memory of Uwe Grimm.

We consider the pure point part of the diffraction on families of aperiodic point sets obeying common local rules. It is shown that imposing such rules results in linear constraints on the partial diffraction amplitudes. These relations can be explicitly derived from the geometry of the prototile space representing the local rules.