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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 559 records · Page 31

Room-temperature polariton quantum fluids in halide perovskites

Abstract Quantum fluids exhibit quantum mechanical effects at the macroscopic level, which contrast strongly with classical fluids. Gain-dissipative solid-state exciton-polaritons systems are promising emulation platforms for complex quantum fluid studies at elevated temperatures. Recently, halide perovskite polariton systems have emerged as materials with distinctive advantages over other room-temperature systems for future studies of topological physics, non-Abelian gauge fields, and spin-orbit interactions. However, the demonstration of nonlinear quantum hydrodynamics, such as superfluidity and Čerenkov flow, which is a consequence of the renormalized elementary excitation spectrum, remains elusive in halide perovskites. Here, using homogenous halide perovskites single crystals, we report, in both one- and two-dimensional cases, the complete set of quantum fluid phase transitions from normal classical fluids to scatterless polariton superfluids and supersonic fluids—all at room temperature, clear consequences of the Landau criterion. Specifically, the supersonic Čerenkov wave pattern was observed at room temperature. The experimental results are also in quantitative agreement with theoretical predictions from the dissipative Gross-Pitaevskii equation. Our results set the stage for exploring the rich non-equilibrium quantum fluid many-body physics at room temperature and also pave the way for important polaritonic device applications.

42 ENGINEERING↗

Modeling and Understanding of Rear Junction Double-Side Passivated Contact Solar Cells with Selective Area TOPCon on Front

Device modeling is performed to propose > 25% efficient industry-compatible rear junction double-side passivated contacts solar cell structure with full area p-TOPCon on the rear and selective area n-TOPCon under the front grid pattern (selective TOPCon). Here, this design enables the use of thicker TOPCon (>100nm) on the front for traditional screen-printed contacts without incurring metal-induced damage, high parasitic absorption loss, and compromise in lateral transport or carrier collection on the front. Rear junction design with appropriate bulk lifetime and resistivity combination eliminate the need for heavy doping in the front field region because carriers can flow through the bulk Si without appreciable FF loss. High VOC is maintained because high-quality Si surface passivation in the field region by Al 2 O 3 /SiN gives J 0 comparable to the TOPCon. Our device modeling specifies the practically achievable properties and parameters for each region, including full area rear p-TOPCon, selective area front n-TOPCon, bulk and contacts, to achieve 25.4% efficiency screen-printed bifacial rear junction selective TOPCon cells.

14 SOLAR ENERGY↗

Post-hazard condition assessment of nuclear piping-equipment systems: Novel approach to feature extraction and deep learning

Over the past decade, the use of artificial intelligence techniques in the field of health-monitoring has gained significant interest, especially for structures such as building and bridges. However, applications to industrial systems such as equipment-piping systems in nuclear plants have not been explored. Here, in this paper, it is shown that the existing techniques developed for buildings and bridges cannot be extended directly to equipment-piping systems as the response of such systems is governed by multiple localized modes unlike that in buildings and bridges. This paper proposes a new approach that consists of three key aspects: (i) a novel vector of degradation-sensitive features extracted from measured data, (ii) using a deep Artificial Neural Network (ANN) for diagnosis of degradation location and degradation severity, and (iii) consideration of uncertainty in degradation severity when training the ANN. Degradation in piping-equipment systems can occur due to flow-accelerated erosion and corrosion. These locations can potentially exhibit damage such as localized yielding or initiation of cracking due to an external event such as an earthquake. Moreover, such locations can at times go undetected by current inspection techniques. Therefore, a robust framework is needed for detection of degradation after a seismic event. This manuscript proposes a proof-of-concept framework, which utilizes data collected from sensors to generate a deep ANN database for predicting degraded locations and severity in a piping-equipment system. Degradation severity is classified as minor, moderate, and severe. In the suggested methodology, a novel vector of degradation-sensitive features is extracted from the sensor data to train the ANN. A simple piping-equipment system is selected to demonstrate feature extraction as a means to simplify pattern recognition, explore the design and parameters of an ANN, and develop a sensor placement strategy. The effectiveness of the proposed framework is demonstrated on a realistic primary safety system of a two-loop nuclear reactor. It is shown that the proposed post-hazard condition assessment framework is able to detect degraded locations along with the severity levels, including minor degradation, with considerably higher accuracy.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

The role of driver head pose dynamics and instantaneous driving in safety critical events: Application of computer vision in naturalistic driving

This paper investigates the role of driver behavior especially head pose dynamics in safety–critical events (SCEs). Using a large dataset collected in a naturalistic driving study, this paper analyzes the head pose dynamics and driving behavior in moments leading up to crashes or near-crashes. The study uses advanced computer vision and mixed logit modeling techniques to identify patterns and relationships between drivers’ head pose dynamics and crash involvement. The results suggest that driver-head pose dynamics, especially poses that indicate distraction and movement volatility, are important factors that can contribute to undesirable safety outcomes. Marginal effects show that angular deviation for head pose dynamics indicated by yaw, pitch and roll increase the likelihood of crash intensity by 4.56%, 4.92% and 8.26% respectively. Furthermore, traffic flow and lane changing also contribute to increase in likelihood of crash intensity. These findings provide new insights into pre-crash factors, especially human factors and safety–critical events. The study highlights the importance of considering human factors in designing driver assistance systems and developing safer vehicles. This research contributes by examining naturalistic driving data at the microscopic level with early detection of behaviors that lead to SCEs and provides a basis for future research on automation.

33 ADVANCED PROPULSION SYSTEMS↗

Chemical and Structural Alterations in the Amorphous Structure of Obsidian due to Nanolites

Obsidian is volcanic glass that results from the rapid cooling of silica-rich melt. Nanoscale crystallites precipitate out of the melt prior to solidification and remain embedded in the amorphous matrix. These crystallites provide information on the flow kinetics and composition of the melt. Due to the sparsity and size of nanolites, studies often focus on supramicron crystallites. This research takes advantage of the conchoidal fracture of obsidian by knapping samples with nanometer-thin edges for transmission electron microscopy characterization. Nanolites in the amorphous matrix are studied using energy-dispersive spectroscopy (EDS) and electron diffraction. Certain alkali and alkaline-earth cations exhibit patterns of depletion near Fe-oxide nanolites. EDS is used to identify nanolites and variations in the composition of the matrix. Parallel beam diffraction and radial distribution function analysis of nearest-neighbor distances determine average bond lengths in the matrix near nanolites, showing that nanolites influence the nearby short-range ordering and atomic character of the matrix. Analysis reveals decreased mean nearest-neighbor distances in the matrix adjacent to nanolites compared to the bulk. Our methods exhibit the required sensitivity to detect variations in the composition and structure near nanolites, and our findings indicate that obsidian nanolites contribute to quantifiable localized changes in the amorphous structure.

36 MATERIALS SCIENCE↗

PISCES-RF: a liquid-cooled high-power steady-state helicon plasma device

Radio-frequency (RF) driven helicon plasma sources can produce relatively high-density plasmas (n > 1019 m -3 ) at relatively moderate powers (<2 kW) in argon. However, to produce similar high-density plasmas for fusion relevant gases such as hydrogen (H), deuterium (D) and helium (He), much higher RF powers are needed. For very high RF powers, thermal issues of the RF-transparent dielectric window, used in the RF source design, limit the plasma operation timescales. To mitigate this constraint, we have designed, built and tested a novel helicon plasma source assembly with a fully liquid-cooled RF-transparent window which allows steady state operations at high power (up to 20 kW) and successfully produces high-density plasma with both argon and H. Deionized (DI) water, flowing between two concentric dielectric RF windows, is used as the coolant. We show that a full azimuthal blanket of DI water does not prevent high-density plasma production. From calorimetry on the DI water, we measure the net heat removed by the coolant at steady state conditions. Using infra-red imaging, we calculate the constant plasma heat deposition and measure the final steady state temperature distribution patterns on the inner surface of the ceramic layer. The heat deposition pattern follows the helical shape of the antenna. We also show the consistency between the heat absorbed by the DI water, as measured by calorimetry, and the total heat due to the combined effect of the plasma heating and the absorbed RF. These results are being used to answer critical engineering questions for the 200 kW RF device materials plasma exposure experiment being designed at the Oak Ridge National Laboratory as a next generation plasma material interaction device.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Alkali-silica reaction products and cracks: X-ray micro-tomography-based analysis of their spatial-temporal evolution at a mesoscale

In this study, we propose a laboratory-scale methodology, based on X-ray micro-tomography and caesium (Cs) as a contrast agent, to advance the understanding of cracking due to alkali-silica reaction (ASR) in concrete. The methodology allows achieving a completely non-destructive and time-lapse characterization of the spatial-temporal patterns of both the cracks and the ASR products. While Cs addition slightly accelerated the ASR kinetics, the crack patterns, with and without Cs addition, were statistically equivalent. Cracks with ASR products appeared first in the aggregates, close to the interface with the cement paste. They propagated afterwards towards the aggregates interior. Some products were then extruded for several mm into air voids and cracks in the cement paste. This process suggests that, in the early stage, the ASR products may be a low-viscosity gel that can flow away from the source aggregate and may settle later elsewhere as a rigid phase, upon calcium uptake.

36 MATERIALS SCIENCE↗

A detailed study of interpretability of deep neural network based top taggers

Abstract Recent developments in the methods of explainable artificial intelligence (XAI) allow researchers to explore the inner workings of deep neural networks (DNNs), revealing crucial information about input–output relationships and realizing how data connects with machine learning models. In this paper we explore interpretability of DNN models designed to identify jets coming from top quark decay in high energy proton–proton collisions at the Large Hadron Collider. We review a subset of existing top tagger models and explore different quantitative methods to identify which features play the most important roles in identifying the top jets. We also investigate how and why feature importance varies across different XAI metrics, how correlations among features impact their explainability, and how latent space representations encode information as well as correlate with physically meaningful quantities. Our studies uncover some major pitfalls of existing XAI methods and illustrate how they can be overcome to obtain consistent and meaningful interpretation of these models. We additionally illustrate the activity of hidden layers as neural activation pattern diagrams and demonstrate how they can be used to understand how DNNs relay information across the layers and how this understanding can help to make such models significantly simpler by allowing effective model reoptimization and hyperparameter tuning. These studies not only facilitate a methodological approach to interpreting models but also unveil new insights about what these models learn. Incorporating these observations into augmented model design, we propose the particle flow interaction network model and demonstrate how interpretability-inspired model augmentation can improve top tagging performance.

97 MATHEMATICS AND COMPUTING↗

Validation of Manufacturing Process Designs for Stamped Microchannel Recuperators

Microchannel recuperators can achieve high-efficiency power cycles and chemical reactions by recovering waste heat from hot turbine exhaust and chemical reaction streams. However, the industrial application of microchannel recuperators are restricted by the high cost of patterning and bonding. The objective of this study was thus to investigate new methods for producing microchannel recuperators for emerging markets. Based on greenfield cost estimates, process designs based on stamping, laser welding and vacuum brazing of 316L stainless steel laminae were found to be more economical than conventional photochemical machining and diffusion bonding. Subsequently, validation experiments were conducted to produce stamped microchannel components using cost model parameters. At a mass flow rate of 0.99 g/s, the test article made with laser welding was found to provide higher effectiveness than the test article produced using vacuum brazing. Preliminary investigations showed that residual stresses during laser welding of the laminae led to significant warpage, collapsing channels and making lamina-to-lamina fit up challenging. In contrast, the test article produced by vacuum brazing of stamped laminae showed no clogging and 4.6% and 10.5% channel height standard deviation for the cold and hot channels, respectively. Channel variation in the vacuum brazing sample was determined to be caused by misregistration of laminae during brazing. In future work, efforts are needed to choose cheaper, more ductile brazing foils as well as design a brazing fixture to improve lamina-to-lamina registration.

42 ENGINEERING↗

Linking Synoptic Patterns to Cloud Properties and Local Circulations Over Southeastern Texas

This report classifies meteorological regimes in the southeastern Texas region to identify environmental conditions that favor sea-breeze induced convection. The classification is accomplished using a Self-Organizing Map (SOM) approach. We applied SOM to 10 years of 700-hPa geopotential height anomalies during the summer months from reanalysis data to distinguish three dominant synoptic regimes, with a continuum of transitional states between those. The primary regimes include: (a) a pre-trough regime associated with a synoptic trough, (b) a post-trough regime with upper-level northerly flow, and (c) an anticyclonic regime within the westward extent of the Bermuda High. We project the data from the Geostationary Operational Environmental Satellite and the Next Generation Weather Radar system onto each SOM node to investigate the characteristics of cloud and precipitation properties in different regimes. When southeastern Texas is positioned to the southwest quadrant of a maritime high pressure system, an increased cloud frequency is observed over the region during the afternoon hours due to significant moisture advection. A confluence of synoptic southerly flow and sea-breeze circulation commonly occurs in this regime. When a high pressure system is over southeastern Texas, the area is dominated by large-scale subsidence with weak pressure gradients and moderate precipitable water vapor. This weak synoptic forcing is favorable for the formation of a sea-breeze circulation. This is confirmed by an enhanced onshore flow and a decreased temperature at the surface in the early afternoon, as well as a sharp increase in radar echo top height.

54 ENVIRONMENTAL SCIENCES↗

Quasi-periodic travelling gravity–capillary waves

We present a numerical study of spatially quasi-periodic travelling waves on the surface of an ideal fluid of infinite depth. This is a generalization of the classic Wilton ripple problem to the case when the ratio of wavenumbers satisfying the dispersion relation is irrational. We propose a conformal mapping formulation of the water wave equations that employs a quasi-periodic variant of the Hilbert transform to compute the normal velocity of the fluid from its velocity potential on the free surface. We develop a Fourier pseudo-spectral discretization of the travelling water wave equations in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on the torus. This leads to an overdetermined nonlinear least-squares problem that we solve using a variant of the Levenberg–Marquardt method. We investigate various properties of quasi-periodic travelling waves, including Fourier resonances, time evolution in conformal space on the torus, asymmetric wave crests, capillary wave patterns that change from one gravity wave trough to the next without repeating and the dependence of wave speed and surface tension on the amplitude parameters that describe a two-parameter family of waves.

97 MATHEMATICS AND COMPUTING↗

Phylogenomic analysis of the hemp family (Cannabaceae) reveals deep cyto‐nuclear discordance and provides new insights into generic relationships

Abstract Cannabaceae are a relatively small family of angiosperms, but they include several species of huge economic and cultural significance: marijuana or hemp ( Cannabis sativa ) and hops ( Humulus lupulus ). Previous phylogenetic studies have clarified the most deep relationships in Cannabaceae, but relationships remain ambiguous among several major lineages. Here, we sampled 82 species representing all genera of Cannabaceae and utilized a new dataset of 90 nuclear genes and 82 chloroplast loci from Hyb‐Seq to investigate the phylogenomics of Cannabaceae. Nuclear phylogenetic analyses revealed a robust and consistent backbone for Cannabaceae. We observed nuclear gene‐tree conflict at several deep nodes in inferred species trees, also cyto‐nuclear discordance concerning the relationship between Gironniera and Lozanella and the relationships among Trema s.l. (including Parasponia ), Cannabis + Humulus , and Chaetachme + Pteroceltis . Coalescent simulations and network analyses suggest that observed deep cyto‐nuclear discordances were most likely to stem from incomplete lineage sorting (ILS); nuclear gene‐tree conflict might be caused by both ILS and gene flow between species. All genera of Cannabaceae were recovered as monophyletic, except for Celtis , which consisted of two distinct clades: Celtis I (including most Celtis species) and Celtis II (including Celtis gomphophylla and Celtis schippii ). We suggest that Celtis II should be recognized as the independent genus Sparrea based on both molecular and morphological evidence. Our work provides the most comprehensive and reliable phylogeny to date for Cannabaceae, enabling further exploration of evolutionary patterns across this family and highlighting the necessity of comparing nuclear with chloroplast data to examine the evolutionary history of plant groups.

Plant Sciences↗

Maximizing oil production from water alternating gas (CO 2 ) injection into residual oil zones: The impact of oil saturation and heterogeneity

Residual oil zones (ROZs) are widespread reservoirs, characterized by oil at residual saturation, either underlying oil fields (brownfield) or lateral (greenfield) to such fields. These reservoirs have the potential to produce volumes of oil sufficiently significant to make appreciable impacts on the US’s oil reserves and associated incidental CO 2 sequestration. The objective of this study is to improve our understanding the impact of heterogeneous and low oil saturations, in brownfield ROZs, on the effectiveness of water alternating gas (WAG) injection strategies. ROZs occur in the Permian Basin and elsewhere, and operators are using CO 2 injection for enhanced oil recovery (EOR) in these zones. The consensus model for the formation of ROZs is that they were formed by the effect of faster regional aquifer flow, acting over millions of years. Both the magnitude of oil saturation and the spatial distribution of oil differ from water-flooded main pay zones (MPZs). To explore the most effective injection strategies, we conducted simulations of CO 2 injection into synthetic geologic reservoirs. These simulations focused on injection into reservoirs subject to either man-made waterflooding or long-term natural waterflooding. By exploring the impact of varying: oil saturation; well patterns; reservoir heterogeneity; and permeability anisotropy, we attempt to quantify the factors that most influence the effectiveness of WAG injection. WAG ratios (the ratio of injected water and CO 2 , in reservoir volumes) of interest are those that either minimize the net CO 2 utilization ratios or maximize oil production rates. In general, the most effective WAG ratios for ROZs, are consistently less than those observed undergoing CO 2 injection in the same geologic reservoir models after traditional (man-made) waterflooding. This work demonstrates that most favorable WAG ratios for oil production in ROZs are different from those in traditional MPZs because of oil saturation differences. Therefore, CO 2 injection into both zones or directly copying WAG injection designs from MPZs to ROZs might not maximize oil production.

04 OIL SHALES AND TAR SANDS↗

Modeling the Effects of Artificial Drainage on Agriculture-dominated Watersheds using a Fully Distributed Integrated Hydrology Model: Datasets, scripts, model files

This model-data archive supports the research paper that demonstrates the integration of agricultural drainage features—specifically, narrow engineered ditches and tile drains—into a fully distributed, basin-scale integrated surface-subsurface hydrology model (ISSHM), Amanzi-ATS. The model employs innovative computational meshes aligned with agricultural ditches and incorporates the physically based Hooghoudt's drainage equation to simulate tile drainage, offering a novel strategy that enhances the accuracy of hydrological simulations.The archived dataset includes input parameters, model configurations, and select simulation outputs for the Amanzi-ATS model that successfully captured the streamflow patterns in the Portage River Watershed as validated by USGS gauge readings. Jupyter notebook for the preparation of model inputs and post-processing of outputs are also included. The model's predictive performance achieved a normalized Kling-Gupta Efficiency (KGE) of 0.81, surpassing SWAT without the necessity for site-specific calibration.The Amanzi-ATS model presented in this modeL-data archive allows for numerical experiments to explore the shifts in the flow structure under different drainage scenarios. As a tool for advancing the understanding of distributed hydrological responses and nutrient cycling, this archived model provides valuable insights for researchers, modelers, and decision-makers involved in watershed management and environmental modeling.The Watershed Workflow package is implemented in Python3. The Jupyter notebooks can be executed through multiple open-source tools, for example, Anaconda Jupyter Lab, VS Studio Code, etc. Other data files include CSV and HDF5 files, which can be read through Python scripts. The input files for the ATS model, open-source integrated hydrology, and transport model, are in XML format and can be edited in any commonly used text editors.

54 ENVIRONMENTAL SCIENCES↗

Numerical Modeling of CO2 Sequestration within a Five-Spot Well Pattern in the Morrow B Sandstone of the Farnsworth Hydrocarbon Field: Comparison of the TOUGHREACT, STOMP-EOR, and GEM Simulators

The objectives of this study were (1) to assess the fate and impact of CO2 injected into the Morrow B Sandstone in the Farnsworth Unit (FWU) through numerical non-isothermal reactive transport modeling, and (2) to compare the performance of three major reactive solute transport simulators, TOUGHREACT, STOMP-EOR, and GEM, under the same input conditions. The models were based on a quarter of a five-spot well pattern where CO2 was injected on a water-alternating-gas schedule for the first 25 years of the 1000 year simulation. The reservoir pore fluid consisted of water with or without petroleum. The results of the models have numerous broad similarities, such as the pattern of reservoir cooling caused by the injected fluids, a large initial pH drop followed by gradual pH neutralization, the long-term persistence of an immiscible CO2 gas phase, the continuous dissolution of calcite, very small decreases in porosity, and the increasing importance over time of carbonate mineral CO2 sequestration. The models differed in their predicted fluid pressure evolutions; amounts of mineral precipitation and dissolution; and distribution of CO2 among immiscible gas, petroleum, formation water, and carbonate minerals. The results of the study show the usefulness of numerical simulations in identifying broad patterns of behavior associated with CO2 injection, but also point to significant uncertainties in the numerical values of many model output parameters.

STOMP↗

Module-Fluidics: Building Blocks for Spatio-Temporal Microenvironment Control

Generating the desired solute concentration signal in micro-environments is vital to many applications ranging from micromixing to analyzing cellular response to a dynamic microenvironment. We propose a new modular design to generate targeted temporally varying concentration signals in microfluidic systems while minimizing perturbations to the flow field. The modularized design, here referred to as module-fluidics, similar in principle to interlocking toy bricks, is constructed from a combination of two building blocks and allows one to achieve versatility and flexibility in dynamically controlling input concentration. The building blocks are an oscillator and an integrator, and their combination enables the creation of controlled and complex concentration signals, with different user-defined time-scales. We show two basic connection patterns, in-series and in-parallel, to test the generation, integration, sampling and superposition of temporally-varying signals. All such signals can be fully characterized by analytic functions, in analogy with electric circuits, and allow one to perform design and optimization before fabrication. Such modularization offers a versatile and promising platform that allows one to create highly customizable time-dependent concentration inputs which can be targeted to the specific application of interest.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Nuclear speed and cycle length co-vary with local density during syncytial blastoderm formation in a cricket

Abstract The blastoderm is a broadly conserved stage of early animal development, wherein cells form a layer at the embryo’s periphery. The cellular behaviors underlying blastoderm formation are varied and poorly understood. In most insects, the pre-blastoderm embryo is a syncytium: nuclei divide and move throughout the shared cytoplasm, ultimately reaching the cortex. In Drosophila melanogaster , some early nuclear movements result from pulsed cytoplasmic flows that are coupled to synchronous divisions. Here, we show that the cricket Gryllus bimaculatus has a different solution to the problem of creating a blastoderm. We quantified nuclear dynamics during blastoderm formation in G. bimaculatus embryos, finding that: (1) cytoplasmic flows are unimportant for nuclear movement, and (2) division cycles, nuclear speeds, and the directions of nuclear movement are not synchronized, instead being heterogeneous in space and time. Moreover, nuclear divisions and movements co-vary with local nuclear density. We show that several previously proposed models for nuclear movements in D. melanogaster cannot explain the dynamics of G. bimaculatus nuclei. We introduce a geometric model based on asymmetric pulling forces on nuclei, which recapitulates the patterns of nuclear speeds and orientations of both unperturbed G. bimaculatus embryos, and of embryos physically manipulated to have atypical nuclear densities.

59 BASIC BIOLOGICAL SCIENCES↗

Arrays of glass wedges for multi-dimensional optical diagnostics

There is a common need in the advancement of optical diagnostic techniques to increase the dimensionality of measurements. For example, point measurements could be improved to multi-point, line, planar, volumetric, or time-resolved volumetric measurements. In this work, a unique optical element is presented to enable multi-dimensional measurements, namely, an array of glass wedges. A light source is passed through the wedges, and different portions of the illumination are refracted by different amounts depending on the glass wedge angle. Subsequent optics can be used to focus the light to multiple points, lines, or planes. Basic characterization of a glass wedge array is presented. Additional wedge-array configurations are discussed, including the use of a periodic intensity mask for multi-planar measurements via structured illumination. The utility of this optical element is briefly demonstrated in (a) multi-planar flame particulate measurements, (b) multi-point femtosecond-laser electronic excitation tagging for flow velocimetry, and (c) multi-line nitric oxide molecular tagging velocimetry in a hypersonic shock-tunnel. One significant advantage of this optical component is its compatibility with high-energy laser sources, which may be a limiting factor with other beam-splitting or beam-forming elements such as some diffractive optics. Additionally, an array of glass wedges is simple and easily customizable compared to other methods for forming multiple closely spaced illumination patterns. In conclusion, suggestions for further development and applications are discussed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗