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At least 397 records · Page 22
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Strange Metals as Ersatz Fermi Liquids

Here, a long-standing mystery of fundamental importance in correlated electron physics is to understand strange non-Fermi liquid metals that are seen in diverse quantum materials. A striking experimental feature of these metals is a resistivity that is linear in temperature (T). In this Letter we ask what it takes to obtain such non-Fermi liquid physics down to zero temperature in a translation invariant metal. If in addition the full frequency (ω) dependent conductivity satisfies ω/T scaling, we argue that the T-linear resistivity must come from the intrinsic physics of the low energy fixed point. Combining with earlier arguments that compressible translation invariant metals are “ersatz Fermi liquids” with an infinite number of emergent conserved quantities, we obtain powerful and practical conclusions. We show that there is necessarily a diverging susceptibility for an operator that is odd under inversion and time reversal symmetries, and has zero crystal momentum. We discuss a few other experimental consequences of our arguments, as well as potential loopholes, which necessarily imply other exotic phenomena.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Electro-Optic Modulation Using Metal-Free Perovskites

Electro-optic (EO) modulation is of interest to impart information onto an optical carrier. Inorganic crystals—most notably LiNbO 3 and BaTiO 3 —exhibit EO modulation and good stability, but are difficult to integrate with silicon photonic technology. Solution-processed organic EO materials are readily integrated but suffer from thermal degradation at the temperatures required in operating conditions for accelerated reliability studies. Hybrid organic–inorganic metal halide perovskites have the potential to overcome these limitations; however, these have so far relied on heavy metals such as lead and cadmium. Here, we report linear EO modulation using metal-free perovskites, which maintain the crystalline features of the inorganic EO materials and incorporate the flexible functionality of organic EO chromophores. We find that, by introducing a deficiency of cations, we reduce the symmetry in the perovskite crystal and produce thereby an increased EO response. The best-engineered perovskites reported herein showcase an EO coefficient of 14 pm V –1 at a modulation frequency of 80 kHz, an order of magnitude higher than in the nondefective materials. Furthermore, we observe split peaks in the X-ray diffraction and neutron diffraction patterns of the defective sample, indicating that the crystalline structure has been distorted and the symmetry reduced. Density functional theory (DFT) studies link this decreased symmetry to NH 4 + deficiencies. This demonstration of EO from metal-free perovskites highlights their potential in next-generation optical information transmission.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Phase-space entropy cascade and irreversibility of stochastic heating in nearly collisionless plasma turbulence

We consider a nearly collisionless plasma consisting of a species of “test particles” in one spatial and one velocity dimension, stirred by an externally imposed stochastic electric field—a kinetic analog of the Kraichnan model of passive advection. The mean effect on the particle distribution function is turbulent diffusion in velocity space—known as stochastic heating. Accompanying this heating is the generation of fine-scale structure in the distribution function, which we characterize with the collisionless (Casimir) invariant C 2 ∝ ∫ ∫ d x d v 〈 f 2 〉 —a quantity that here plays the role of (negative) entropy of the distribution function. We find that C 2 is transferred from large scales to small scales in both position and velocity space via a phase-space cascade enabled by both particle streaming and nonlinear interactions between particles and the stochastic electric field. We compute the steady-state fluxes and spectrum of C 2 in Fourier space, with k and s denoting spatial and velocity wave numbers, respectively. In our model, the nonlinearity in the evolution equation for the spectrum turns into a fractional Laplacian operator in k space, leading to anomalous diffusion. Whereas even the linear phase mixing alone would lead to a constant flux of C 2 to high s (towards the collisional dissipation range) at every k , the nonlinearity accelerates this cascade by intertwining velocity and position space so that the flux of C 2 is to both high k and high s simultaneously. Integrating over velocity (spatial) wave numbers, the k -space ( s -space) flux of C 2 is constant down to a dissipation length (velocity) scale that tends to zero as the collision frequency does, even though the rate of collisional dissipation remains finite. The resulting spectrum in the inertial range is a self-similar function in the ( k , s ) plane, with power-law asymptotics at large k and s . Our model is fully analytically solvable, but the asymptotic scalings of the spectrum can also be found via a simple phenomenological theory whose key assumption is that the cascade is governed by a “critical balance” in phase space between the linear and nonlinear timescales. We argue that stochastic heating is made irreversible by this entropy cascade and that, while collisional dissipation accessed via phase mixing occurs only at small spatial scales rather than at every scale as it would in a linear system, the cascade makes phase mixing even more effective overall in the nonlinear regime than in the linear one. Published by the American Physical Society 2024

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Compositional Reasoning for Hierarchical State Machines

Harel statecharts and its derivatives are popular graphical languages for specifying discrete control systems via hierarchical state machines. Separately, there has been a long line of work on specifying concurrent systems with process calculi which come equipped with an algebraic theory, the ability reason compositionally about various temporal properties, and strong type systems. While these two approaches to modeling systems are tantalizingly similar, the integrated reasoning principles that exist for process calculi have not been demonstrated in hierarchical state machines. A key issue is that operational theories for process calculi do not behave like control systems, and thus, there is virtually no tool support for modeling control systems with such languages. For a control system designer, bringing the integrated, more scalable reasoning from the process calculi to state-machine languages would enable the specification of more complex systems and a more modular systems development process. Our insight is that we can recover many important results from the process calculi in hierarchical state machines with local scope. We employ a structural operational semantics, which is ubiquitous in process and 𝜆-calculi but uncommon in hierarchical statemachine formalizations, to enable inductive reasoning about behavior. Taking inspiration from the structure of process calculi metatheories, we define a calculus of refinement and equivalence that we prove sound with respect to local notion of (bi)simulation. Furthermore, we prove that the calculus preserves the behavioral properties of reactivity, observational determinism, traces, and linear temporal properties. Our results are mechanized in the Rocq proof assistant.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2013

Model-Based Control of a Nonlinear Aircraft Engine Simulation using an Optimal Tuner Kalman Filter Approach

This paper covers the development of a model-based engine control (MBEC) methodology featuring a self tuning on-board model applied to an aircraft turbofan engine simulation. Here, the Commercial Modular Aero-Propulsion System Simulation 40,000 (CMAPSS40k) serves as the MBEC application engine. CMAPSS40k is capable of modeling realistic engine performance, allowing for a verification of the MBEC over a wide range of operating points. The on-board model is a piece-wise linear model derived from CMAPSS40k and updated using an optimal tuner Kalman Filter (OTKF) estimation routine, which enables the on-board model to self-tune to account for engine performance variations. The focus here is on developing a methodology for MBEC with direct control of estimated parameters of interest such as thrust and stall margins. Investigations using the MBEC to provide a stall margin limit for the controller protection logic are presented that could provide benefits over a simple acceleration schedule that is currently used in traditional engine control architectures.

Control Theory↗
DOE OSTI2023

Theory of glide symmetry protected helical edge states in a WTe 2 monolayer

Helical edge states in quantum spin Hall (QSH) materials are central building blocks of topological matter design and engineering. Despite their principal topological protection against elastic backscattering, the level of operational stability depends on manifold parameters such as the band gap of the given semiconductor system in the “inverted” regime, temperature, disorder, and crystal orientation. Here, we theoretically investigate electronic and transport properties of QSH edge states in large gap 1-T' WTe 2 monolayers. We explore the impact of edge termination, disorder, temperature, and interactions on experimentally addressable edge state observables, such as local density of states and conductance. We show that conductance quantization can remain surprisingly robust even for heavily disordered samples because of an anomalously small edge state decay length and additional protection related to the large direct gap allowed by glide symmetry. From the simulation of temperature-dependent resistance, we find that moderate disorder enhances the stability of conductance by localizing bulk states. We evaluate the edge state velocity and Luttinger liquid parameter as functions of the chemical potential, finding prospects for physics beyond linear helical Luttinger liquids in samples with ultraclean and well-defined edges.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Modifying the Asynchronous Jacobi Method for Data Corruption Resilience

Moving scientific computation from high-performance computing (HPC) and cloud computing (CC) environments to devices on the edge, i.e., physically near instruments of interest, has received tremendous interest in recent years. Such edge computing environments can operate on data in situ, offering enticing benefits over data aggregation to HPC and CC facilities that include avoiding costs of transmission, increased data privacy, and real-time data analysis. Because of the inherent unreliability of edge computing environments, new fault-tolerant approaches must be developed before the benefits of edge computing can be realized. Motivated by algorithm-based fault tolerance, a variant of the asynchronous Jacobi (ASJ) method is developed that achieves resilience to data corruption by rejecting solution approximations from neighbor devices according to a bound derived from convergence theory. Numerical results on a two-dimensional Poisson problem show that the new rejection criterion, along with a novel approximation to the shortest path length on which the criterion depends, restores convergence for the ASJ variant in the presence of certain types data corruption. Numerical results are obtained for when the singular values in the analytic bound are approximated. Additional linear systems are also explored, one with a more dense sparsity pattern and one that includes advection. All results indicate that successful resilience to data corruption depends on whether the bound tightens fast enough to reject corrupted data before the iteration evolution deviates significantly from that predicted by the convergence theory defining the bound. This observation generalizes to future work on algorithm-based fault tolerance for other asynchronous algorithms, including upcoming approaches that leverage Krylov subspaces.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2006

Development of a Nonlinear Probability of Collision Tool for the Earth Observing System

The Earth Observing System (EOS) spacecraft Terra, Aqua, and Aura fly in constellation with several other spacecraft in 705-kilometer mean altitude sun-synchronous orbits. All three spacecraft are operated by the Earth Science Mission Operations (ESMO) Project at Goddard Space Flight Center (GSFC). In 2004, the ESMO project began assessing the probability of collision of the EOS spacecraft with other space objects. In addition to conjunctions with high relative velocities, the collision assessment method for the EOS spacecraft must address conjunctions with low relative velocities during potential collisions between constellation members. Probability of Collision algorithms that are based on assumptions of high relative velocities and linear relative trajectories are not suitable for these situations; therefore an algorithm for handling the nonlinear relative trajectories was developed. This paper describes this algorithm and presents results from its validation for operational use. The probability of collision is typically calculated by integrating a Gaussian probability distribution over the volume swept out by a sphere representing the size of the space objects involved in the conjunction. This sphere is defined as the Hard Body Radius. With the assumption of linear relative trajectories, this volume is a cylinder, which translates into simple limits of integration for the probability calculation. For the case of nonlinear relative trajectories, the volume becomes a complex geometry. However, with an appropriate choice of coordinate systems, the new algorithm breaks down the complex geometry into a series of simple cylinders that have simple limits of integration. This nonlinear algorithm will be discussed in detail in the paper. The nonlinear Probability of Collision algorithm was first verified by showing that, when used in high relative velocity cases, it yields similar answers to existing high relative velocity linear relative trajectory algorithms. The comparison with the existing high velocity/linear theory will also be used to determine at what relative velocity the analysis should use the new nonlinear theory in place of the existing linear theory. The nonlinear algorithm was also compared to a known exact solution for the probability of collision between two objects when the relative motion is strictly circular and the error covariance is spherically symmetric. Figure I shows preliminary results from this comparison by plotting the probabilities calculated from the new algorithm and those from the exact solution versus the Hard Body Radius to Covariance ratio. These results show about 5% error when the Hard Body Radius is equal to one half the spherical covariance magnitude. The algorithm was then combined with a high fidelity orbit state and error covariance propagator into a useful tool for analyzing low relative velocity nonlinear relative trajectories. The high fidelity propagator is capable of using atmospheric drag, central body gravitational, solar radiation, and third body forces to provide accurate prediction of the relative trajectories and covariance evolution. The covariance propagator also includes a process noise model to ensure realistic evolutions of the error covariance. This paper will describe the integration of the nonlinear probability algorithm and the propagators into a useful collision assessment tool. Finally, a hypothetical case study involving a low relative velocity conjunction between members of the Earth Observation System constellation will be presented.

McKinley, David P.↗
NASA NTRS1991

Systems engineering studies of on-orbit assembly operation

While the practice of construction has a long history, the underlying theory of construction is relatively young. Very little has been documented as to techniques of logistic support, construction planning, construction scheduling, construction testing, and inspection. The lack of 'systems approaches' to construction processes is certainly one of the most serious roadblocks to the construction of space structures. System engineering research efforts at CSC are aimed at developing concepts and tools which contribute to a systems theory of space construction. The research is also aimed at providing means for trade-offs of design parameters for other research areas in CSC. Systems engineering activity at CSC has divided space construction into the areas of orbital assembly, lunar base construction, interplanetary transport vehicle construction, and Mars base construction. A brief summary of recent results is given. Several models for 'launch-on-time' were developed. Launch-on-time is a critical concept to the assembly of such Earth-orbiting structures as the Space Station Freedom, and to planetary orbiters such as the Mars transfer vehicle. CSC has developed a launch vehicle selection model which uses linear programming to find optimal combinations of launch vehicles of various sizes (Atlas, Titan, Shuttles, HLLV's) to support SEI missions. Recently, the Center developed a cost trade-off model for studying on orbit assembly logistics. With this model it was determined that the most effective size of the HLLV would be in the range of 120 to 200 metric tons to LEO, which is consistent with the choices of General Stafford's Synthesis Group Report. A second-generation Dynamic Construction Activities Model ('DYCAM') process model has been under development, based on our past results in interruptability and our initial DYCAM model. This second-generation model is built on the paradigm of knowledge-based expert systems. It is aimed at providing answers to two questions: (1) what are some necessary or sufficient conditions for judging conceptual designs of spacecraft?, and (2) can a methodology be formulated such that these conditions may be used to provide computer-aided tools for evaluating conceptual designs and planning for space assembly sequences? Early simulation results indicate that the DYCAM model has a clear ability to emulate and simulate human orbital construction processes.

Morgenthaler, George W.↗
DOE OSTI2025

Non-invertible defects from the Conway SCFT to K3 sigma models. Part I. General results

We initiate the study of supersymmetry-preserving topological defect lines (TDLs) in the Conway moonshine module V f♮ . We show that the tensor category of such defects, under suitable assumptions, admits a surjective but non-injective ring homomorphism into the ring of $\mathbb{Z}$-linear maps of the Leech lattice into itself. This puts strong constraints on possible defects and their quantum dimensions. We describe a simple construction of non-invertible TDLs from orbifolds of holomorphic (super)vertex operator algebras, which yields non-trivial examples of TDLs satisfying our main theorem. We conjecture a correspondence between four-plane-preserving TDLs in V f♮ and supersymmetry-preserving TDLs in K3 non-linear sigma models, which extends the correspondence between symmetry groups to the level of tensor category symmetry. We establish evidence for this conjecture by constructing non-invertible TDLs in special K3 non-linear sigma models.

Conformal Field Models in String Theory↗
DOE OSTI2024

Weighted Composition Operators for Learning Nonlinear Dynamics

Operator theoretic methods in dynamical system have been dominated by the use of Koopman operators and their continuous time counterparts, such as Koopman Generators and Liouville Operators. The advantage gained from their use primarily stems from the ability to extract subspaces and eigenfunctions within a space of observables that are invariant with respect to the Koopman operator over that space. When this occurs, a dynamic mode decomposition of the systems state provides a linear model for the dynamical system. Not all Koopman operators have eigenfunctions that may be exploited in this manner. However, the framework can still be leveraged for approximations using other operators. In this setting, we present a different operator for the study of dynamical systems, the weighted composition operator. These operators are compact for a wide range of dynamics and spaces, and through their interactions with occupation kernels and vector valued kernels, they admit an estimation of the underlying dynamics. Here, this manuscript presents a new algorithm for the data driven study of dynamical systems from data, and also provides two numerical experiments where convergence is achieved as a proof of concept.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1980

Application of functional analysis to perturbation theory of differential equations

The deviation of the solution of the differential equation y' = f(t, y), y(O) = y sub O from the solution of the perturbed system z' = f(t, z) + g(t, z), z(O) = z sub O was investigated for the case where f and g are continuous functions on I x R sup n into R sup n, where I = (o, a) or I = (o, infinity). These functions are assumed to satisfy the Lipschitz condition in the variable z. The space Lip(I) of all such functions with suitable norms forms a Banach space. By introducing a suitable norm in the space of continuous functions C(I), introducing the problem can be reduced to an equivalent problem in terminology of operators in such spaces. A theorem on existence and uniqueness of the solution is presented by means of Banach space technique. Norm estimates on the rate of growth of such solutions are found. As a consequence, estimates of deviation of a solution due to perturbation are obtained. Continuity of the solution on the initial data and on the perturbation is established. A nonlinear perturbation of the harmonic oscillator is considered a perturbation of equations of the restricted three body problem linearized at libration point.

Bogdan, V. M.↗
DOE OSTI2020

A Lagrangian perturbation theory in the presence of massive neutrinos

We develop a Lagrangian Perturbation Theory (LPT) framework to study the clustering of cold dark matter (CDM) in cosmologies with massive neutrinos. We follow the trajectories of CDM particles with Lagrangian displacements fields up to third order in perturbation theory. Once the neutrinos become non-relativistic, their density fluctuations are modeled as being proportional to the CDM density fluctuations, with a scale-dependent proportionality factor. This yields a gravitational back-reaction that introduces additional scales to the linear growth function, which is accounted for in the higher order LPT kernels. Through non-linear mappings from Eulerian to Lagrangian frames, we ensure that our theory has a well behaved large scale behavior free of unwanted UV divergences, which are common when neutrino and CDM densities are not treated on an equal footing, and in resummation schemes that manifestly break Galilean invariance. We use our theory to construct correlation functions for both the underlying matter field, as well as for biased tracers using ConvolutionLPT. Redshift-space distortions effects are modeled using the Gaussian Streaming Model. When comparing our analytical results to simulated data from the Quijote 1 simulation suite, we find good accuracy down to r = 20 Mpc h -1 at redshift z = 0.5, for the real space and redshift space monopole particle correlation functions with no free parameters. The same accuracy is reached for the redshift space quadrupole if we additionally consider an effective field theory parameter that shifts the pairwise velocity dispersion. For modeling the correlation functions of tracers we adopt a simple Lagrangian biasing scheme with only density and curvature operators, which we find sufficient to reach down to r = 20 Mpc h -1 when comparing to simulated halos.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2002

NASA Tech Briefs, April 2002

The contents include: 1) Application Briefs; 2) Sneak Preview of Sensors Expo; 3) The Complexity of the Diagnosis Problem; 4) Design Concepts for the ISS TransHab Module; 5) Characteristics of Supercritical Transitional Mixing Layers; 6) Electrometer for Triboelectric Evaluation of Materials; 7) Infrared CO2 Sensor With Built-In Calibration Chambers; 8) Solid-State Potentiometric CO Sensor; 9) Planetary Rover Absolute Heading Detection Using a Sun Sensor; 10) Concept for Utilizing Full Areas of STJ Photodetector Arrays; 11) Development of Cognitive Sensors; 12) Enabling Higher-Voltage Operation of SOl CMOS Transistors; 13) Estimating Antenna-Pointing Errors From Beam Squints; 14) Advanced-Fatigue-Crack-Growth and Fracture- Mechanics Program; 15) Software for Sequencing Spacecraft Actions; 16) Program Distributes and Tracks Organizational Memoranda; 16) Flat Membrane Device for Dehumidification of Air; 17) Inverted Hindle Mount Reduces Sag of a Large, Precise Mirror; 18) Heart-Pump-Outlet/Cannula Coupling; 19) Externally Triggered Microcapsules Release Drugs In Situ; 20) Combinatorial Drug Design Augmented by Information Theory; 21) Multiple-Path-Length Optical Absorbance Cell; 22) Model of a Fluidized Bed Containing a Mixture of Particles; 23) Refractive Secondary Concentrators for Solar Thermal Systems; 24) Cold Flow Calorimeter; 25) Methodology for Tracking Hazards and Predicting Failures; 26) Estimating Heterodyne-Interferometer Polarization Leakage; 27) An Efficient Algorithm for Propagation of Temporal- Constraint Networks; 28) Software for Continuous Replanning During Execution; 29) Surface-Launched Explorers for Reconnaissance/Scouting; 30) Firmware for a Small Motion-Control Processor; 31) Gear Bearings and Gear-Bearing Transmissions; and 32) Linear Dynamometer With Variable Stroke and Frequency.

Source record↗
NASA NTRS2000

Solar Spicules: A Review of Recent Models and Targets for Future Observations

Since their discovery over 100 years ago, there have been many suggestions for the origin and development of solar spicules. Because the velocities of spicules are comparable to the sound and Alfven speeds of the low chromosphere, linear theory cannot fully describe them. Consequently, detailed tests of theoretical ideas had to await the development of computing power that only became available during the 1970s. This work reviews theories for spicules and spicule-like features over approximately the past 25 years, with an emphasis on the models based on nonlinear numerical simulations. These models have given us physical insight into wave propagation in the solar atmosphere, and have helped elucidate how such waves, and associated shock waves, may be capable of creating motions and structures on magnetic flux tubes in the lower solar atmosphere. So far, however, it has been difficult to reproduce the most- commonly -quoted parameters for spicules with these models, using what appears to be the most suitable input parameters. A key impediment to developing satisfactory models has been the lack of reliable observational information, which is a consequence of the small angular size and transient lifetime of spicules. I close with a list of key observational questions to be addressed with space-based satellites, such as the currently operating TRACE satellite, and especially the upcoming Solar-B mission. Answers to these questions will help determine which, if any, of the current models correctly explains spicules.

Sterling, Alphonse C.↗
NASA NTRS2000

Solar Spicules: A Review of Recent Models and Targets for Future Observations

Since their discovery over 100 years ago, there have been many suggestions for the origin and development of solar spicules. Because the velocities of spicules are comparable to the sound and Alfven speeds of the low chromosphere, linear theory cannot fully describe them. Consequently, detailed tests of theoretical ideas had to await the development of computing power that only became available during the 1970s. This work reviews theories for spicules and spicule-like features over approximately the past 25 years, with an emphasis an the models based on nonlinear numerical simulations. These models have given us physical insight into wave propagation in the solar atmosphere and have helped elucidate how such waves, and associated shock waves, may be capable of creating motions and structures on magnetic flux tubes in the lower solar atmosphere. So far, however, it has been difficult to reproduce the most commonly quoted parameters for spicules with these models, using what appears to be the most suitable input parameters. A key impediment to developing satisfactory models has been the lack of reliable observational information, which is a consequence of the small angular size and transient lifetime of spicules. I close with a list of key observational questions to be addressed with space-based satellites, such as the currently operating Transition Region and Coronal Explorer (TRACE) satellite, and especially the upcoming Solar-B mission. Answers to these questions will help determine which, if any, of the current models correctly explains spicules.

Sterling, Alphonse C.↗
NASA NTRS2005

Multipurpose Fiber Injected-micro-spherical LIDAR System

A technological revolution is occurring in the field of fiber lasers. Over the past two years, the level of power has increased from approx. 100 watts to nearly 1 kilowatt. We are developing a novel fiber laser system, which is a satellite-based LIDAR transmitter of multi-lines. The system is made of a hollow fiber filled with micro-spheres doped with lasing materials. Each sphere has its inherent optical cavity, which makes the system a cavity free and in the same time, emits multi-laser lines for simultaneous multi-task operations. The system is also rugged, compact, lightweight, and durable. Our earlier studies on micro-spheres doped with different laser dyes demonstrated the emission of extremely fine laser lines of less than 3 A line-width, which are of interest for spectroscopic applications, sensing, imaging, and optical communications. Individual dye-doped micro-spheres demonstrated a lasing resonance peaks phenomenon in their fluorescence spectra of linear and nonlinear features that do not exist in the bulk dye solutions. Each individual micro-sphere acts as a laser system with inherent cavity, where the fluorescence line suffers multiple internal reflections within the micro-sphere and gains enough energy to become a laser line. Such resonance peaks are dependent on the sphere's morphology, size, shape, and its refractive index. These resonance peaks are named structural resonance, whispering modes or whispering gallery modes, creeping waves, circumferential waves, surfaces modes, and virtual modes. All of these names refer to the same phenomenon of morphology dependent resonance (MDR), which has already been described and predicted precisely by electromagnetic theory and Lorentz-Mie theory since 1908. The resonance peaks become more obvious when the particle size approaches and exceeds the wavelength of the laser used and the relative index of the particle is greater than that of the surrounding medium. Additional information is included in the original extended abstract.

Abdelayem, Hossin↗