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At least 289 records · Page 16

Hybrid X-ray telescope systems

It has been known for some time that glancing incidence X-ray optics have sharply peaked point response functions. The spatial resolution of Wolter X-ray telescope systems is frequently limited by the performance characteristics of the X-ray detector rather than by the optics, even when high resolution detectors such as photographic emulsions are used. This problem becomes even more severe when it is desirable to take advantage of the high sensitivity and broad wavelength response afforded by modern solid state detectors such as charge coupled devices or multi-anode microchannel arrays. Results of theoretical investigations of methods for increasing the telescope plate scale by the use of glancing incidence hyperboloid/ellipsoid X-ray microscope optics and normal incidence contoured layered synthetic microstructure optics for coupling the Wolter mirrors to the detector are reported. The relative merits of the different hybrid X-ray telescope systems which result will be discussed. Considerations will be made concerning applications of these systems to high resolution imagery and spectroscopy of astronomical X-ray/XUV sources.

Shealy, David L.↗

The AERONET Version 3 Aerosol Retrieval Algorithm, Associated Uncertainties and Comparisons to Version 2

The Aerosol Robotic Network (AERONET) Version 3 (V3) aerosol retrieval algorithm is described, which is based on the Version 2 (V2) algorithm with numerous updates. Comparisons of V3 aerosol retrievals to those of V2 are presented, along with a new approach to estimate uncertainties in many of the retrieved aerosol parameters. Changes in the V3 aerosol retrieval algorithm include (1) a new polarized radiative transfer code (RTC), which replaced the scalar RTC of V2, (2) detailed characterization of gas absorption by adding NO2 and H2O to specify total gas absorption in the atmospheric column, specification of vertical profiles of all the atmospheric species, (3) new bidirectional reflectance distribution function (BRDF) parameters for land sites adopted from the MODIS BRDF/Albedo product, (4) a new version of the extraterrestrial solar flux spectrum, and (5) a new temperature correction procedure of both direct Sun and sky radiance measurements. The potential effect of each change in V3 on single scattering albedo (SSA) retrievals was analyzed. The operational almucantar retrievals of V2 versus V3 were compared for four AERONET sites: GSFC, Mezaira, Mongu, and Kanpur. Analysis showed very good agreement in retrieved parameters of the size distributions. Comparisons of SSA retrievals for dust aerosols (Mezaira) showed a good agreement in 440 nm SSA, while for longer wavelengths V3 SSAs are systematically higher than those of V2, with the largest mean difference at 675 nm due to cumulative effects of both extraterrestrial solar flux and BRDF changes. For non-dust aerosols, the largest SSA deviation is at 675 nm due to differences in extraterrestrial solar flux spectrums used in each version. Further, the SSA 675 nm mean differences are very different for weakly (GSFC) and strongly (Mongu) absorbing aerosols, which is explained by the lower sensitivity to a bias in aerosol scattering optical depth by less absorbing aerosols. A new hybrid (HYB) sky radiance measurement scan is introduced and discussed. The HYB combines features of scans in two different planes to maximize the range of scattering angles and achieve scan symmetry, thereby allowing for cloud screening and spatial averaging, which is an advantage over the principal plane scan that lacks robust symmetry. We show that due to an extended range of scattering angles, HYB SSA retrievals for dust aerosols exhibit smaller variability with solar zenith angles (SZAs) than those of almucantar (ALM), which allows extension of HYB SSA retrievals to SZAs less than 50∘ to as small as 25∘. The comparison of SSA retrievals from closely time-matched HYB and ALM scans in the 50 to 75∘ SZA range showed good agreement with the differences below ∼0.005. We also present an approach to estimate retrieval uncertainties which utilizes the variability in retrieved parameters generated by perturbing both measurements and auxiliary input parameters as a proxy for retrieval uncertainty. The perturbations in measurements and auxiliary inputs are assumed as estimated biases in aerosol optical depth (AOD), radiometric calibration of sky radiances combined with solar spectral irradiance, and surface reflectance. For each set of Level 2 Sun/sky radiometer observations, 27 inputs corresponding to 27 combinations of biases were produced and separately inverted to generate the following statistics of the inversion results: average, standard deviation, minimum and maximum values. From these statistics, standard deviation (labeled U27) is used as a proxy for estimated uncertainty, and a lookup table (LUT) approach was implemented to reduce the computational time. The U27 climatological LUT was generated from the entire AERONET almucantar (1993–2018) and hybrid (2014–2018) scan databases by binning U27s in AOD (440 nm), Angström exponent (AE, 440–870 nm), and SSA (440, 675, 870, 1020 nm). Using this LUT approach, the uncertainty estimates U27 for each individual V3 Level 2 retrieval can be obtained by interpolation using the corresponding measured and inverted combination of AOD, AE, and SSA.

Aliaksandr Siniuk↗

The starburst ring around the Seyfert nucleus in NGC 7469

High-resolution radio continuum, optical emission line, and optical continuum images of the luminous Seyfert plus circumnuclear starburst hybrid galaxy NGC 7469 are presented. The radio emission from the starburst is largely confined to a ring of diameter 3 arcsec, or 1.0 kpc. There is clear evidence for optical continuum emission and probably optical line emission from the ring. The disks of NGC 1068 and NGC 7469 exhibit a strong similarity in almost all observed properties. Current star formation models can account for the luminous far-infrared emission and the strength of the nonthermal radio emission in NGC 7469, with an expected supernova rate of about one per year. The starburst ring is probably a result of concentration of gas into a ring via resonances between orbital motion and a rotating barlike or ovally distorted potential.

Wilson, A. S.↗

Core-shell metallic alloy nanopillars-in-dielectric hybrid metamaterials with magneto-plasmonic coupling

Combining plasmonic and magnetic properties, namely magneto-plasmonic coupling, inspires great research interest and the search for magneto-plasmonic nanostructure becomes considerably critical. In this work, we designed a nanopillar-in-matrix structure with core–shell alloyed nanopillars for both BaTiO 3 (BTO)-Au 0.5 Co 0.5 (AuCo) and BTO-Au 0.25 Cu 0.25 Co 0.25 Ni 0.25 (AuCuCoNi) hybrid systems, i.e., ferromagnetic alloy cores (e.g., Co or CoNi) with plasmonic shells (e.g., Au or Au/Cu). These core–shell alloy nanopillars are uniformly embedded into a dielectric BTO matrix to form a vertically aligned nanocomposite (VAN) structure. Both hybrid systems present excellent epitaxial quality and interesting multi-functionality, e.g., high magnetic anisotropy, magneto-optical coupling response, tailorable plasmonic resonance wavelength, tunable hyperbolic properties and strong optical anisotropy. These alloyed nanopillars-in-matrix designs provide enormous potential for complex hybrid material designs with multi-functionality and demonstrate strong interface enabled magneto-plasmonic coupling along with plasmonic and magnetic performance.

36 MATERIALS SCIENCE↗

High-frequency asymptotic methods for analyzing the EM scattering by open-ended waveguide cavities

Four high-frequency methods are described for analyzing the electromagnetic (EM) scattering by electrically large open-ended cavities. They are: (1) a hybrid combination of waveguide modal analysis and high-frequency asymptotics, (2) geometrical optics (GO) ray shooting, (3) Gaussian beam (GB) shooting, and (4) the generalized ray expansion (GRE) method. The hybrid modal method gives very accurate results but is limited to cavities which are made up of sections of uniform waveguides for which the modal fields are known. The GO ray shooting method can be applied to much more arbitrary cavity geometries and can handle absorber treated interior walls, but it generally only predicts the major trends of the RCS pattern and not the details. Also, a very large number of rays need to be tracked for each new incidence angle. Like the GO ray shooting method, the GB shooting method can handle more arbitrary cavities, but it is much more efficient and generally more accurate than the GO method because it includes the fields diffracted by the rim at the open end which enter the cavity. However, due to beam divergence effects the GB method is limited to cavities which are not very long compared to their width. The GRE method overcomes the length-to-width limitation of the GB method by replacing the GB's with GO ray tubes which are launched in the same manner as the GB's to include the interior rim diffracted field. This method gives good accuracy and is generally more efficient than the GO method, but a large number of ray tubes needs to be tracked.

Burkholder, R. J.↗

Water-Stable 1D Hybrid Tin(II) Iodide Emits Broad Light with 36% Photoluminescence Quantum Efficiency

The optical and light emission properties of tin and lead halide perovskites are remarkable because of the robust room-temperature (RT) performance, broad wavelength tunability, high efficiency, and good quenching resistance to defects. These highly desirable attributes promise to transform current light-emitting devices, phosphors, and lasers. One disadvantage in most of these materials is the sensitivity to moisture. Here, we report a new air-stable one-dimensional (1D) hybrid lead-free halide material (DAO)Sn 2 I 6 (DAO, 1,8-octyldiammonium) that is resistant to water for more than 15 h. The material exhibits a sharp optical absorption edge at 2.70 eV and a strong broad orange light emission centered at 634 nm, with a full width at half-maximum (fwhm) of 142 nm (0.44 eV). The emission has a long photoluminescence (PL) lifetime of 582 ns, while the intensity is constant over a very broad temperature range (145–415 K) with a photoluminescence quantum yield (PLQY) of at least 20.3% at RT. Above 415 K the material undergoes a structural phase transition from monoclinic (C2/c) to orthorhombic (Ibam) accompanied by a red shift in the band gap and a quench in the photoluminescence emission. Density functional theory calculations support the trend in the optical properties and the 1D electronic nature of the structure, where the calculated carrier effective masses along the inorganic chain are significantly lower than those perpendicular to the chain. As a result, thin films of the compound readily fabricated from solutions exhibit the same optical properties, but with improved PLQY of 36%, for a 60 nm thick film, among the highest reported for lead-free low-dimensional 2D and 1D perovskites and metal halides.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Data-driven design of novel halide perovskite alloys

The great tunability of the properties of halide perovskites presents new opportunities for optoelectronic applications as well as significant challenges associated with exploring combinatorial chemical spaces. Here, in this work, we develop a framework powered by high-throughput computations and machine learning for the design and prediction of mixed cation halide perovskite alloys. In a chemical space of ABX 3 perovskites with a selected set of options for A, B, and X species, pseudo-cubic structures with B-site mixing are simulated using density functional theory (DFT) and several properties are computed, including stability, lattice constant, band gap, vacancy formation energy, refractive index, and optical absorption spectrum, using both semi-local and hybrid functionals. Neural networks (NN) are used to train predictive models for every property using tabulated elemental properties of A, B, and X site species as descriptors. Starting from a DFT dataset of 229 points, we use the trained NN models to predict the structural, energetic, electronic, and optical properties of an extensive dataset of 17955 compounds, and perform high-throughput screening in terms of stability, band gap, and defect tolerance, to obtain 392 promising compounds that are ranked as potential absorbers according to their photovoltaic figure of merit. Compositional trends in the screened set of attractive mixed cation halide perovskites are revealed and additional computations are performed on selected compounds. The data-driven design framework developed here is promising for designing novel mixed compositions and can be extended to a wider perovskite chemical space in terms of A, B, and X species, different kinds of mixing at the A, B, or X sites, non-cubic phases, and other properties of interest.

36 MATERIALS SCIENCE↗

DOE Early Career: Elucidating Chirality-induced Magnetism and Magnetoelectric Functionalities in Layered Chiral Hybrid Metal Halide Perovskites

This final report summarizes the research conducted under the DOE Early Career Award DE SC0020992, which aimed to elucidate the fundamental origins of chirality-induced spin selectivity and magnetoelectric functionalities in 2D chiral hybrid metal halide systems. By integrating spin-orbitronics with magneto-optics, this project successfully pioneered new paradigms for controlling spin and orbital degrees of freedom via chirality in solution-processed hybrid organic-inorganic semiconductors.

36 MATERIALS SCIENCE↗

TiN‐Au/HfO 2 ‐Au Multilayer Thin Films with Tunable Hyperbolic Optical Response

Abstract Hyperbolic metamaterials (HMM) possess significant anisotropic physical properties and tunability and thus find many applications in integrated photonic devices. HMMs consisting of metal and dielectric phases in either multilayer or vertically aligned nanocomposites (VAN) form are demonstrated with different hyperbolic properties. Herein, self‐assembled HfO 2 ‐Au/TiN‐Au multilayer thin films, combining both the multilayer and VAN designs, are demonstrated. Specifically, Au nanopillars embedded in HfO 2 and TiN layers forming the alternative layers of HfO 2 ‐Au VAN and TiN‐Au VAN. The HfO 2 and TiN layer thickness is carefully controlled by varying laser pulses during pulsed laser deposition (PLD). Interestingly, tunable anisotropic physical properties can be achieved by adjusting the bi‐layer thickness and the number of the bi‐layers. Type II optical hyperbolic dispersion can be obtained from high layer thickness structure (e.g., 20 nm), while it can be transformed into Type I optical hyperbolic dispersion by reducing the thickness to a proper value (e.g., 4 nm). This new nanoscale hybrid metamaterial structure with the three‐phase VAN design shows great potential for tailorable optical components in future integrated devices.

Chemistry↗

Pre-Finishing of SiC for Optical Applications

13 Manufacturing & Prototyping A method is based on two unique processing steps that are both based on deterministic machining processes using a single-point diamond turning (SPDT) machine. In the first step, a high-MRR (material removal rate) process is used to machine the part within several microns of the final geometry. In the second step, a low-MRR process is used to machine the part to near optical quality using a novel ductile regime machining (DRM) process. DRM is a deterministic machining process associated with conditions under high hydrostatic pressures and very small depths of cut. Under such conditions, using high negative-rake angle cutting tools, the high-pressure region near the tool corresponds to a plastic zone, where even a brittle material will behave in a ductile manner. In the high-MRR processing step, the objective is to remove material with a sufficiently high rate such that the process is economical, without inducing large-scale subsurface damage. A laser-assisted machining approach was evaluated whereby a CO2 laser was focused in advance of the cutting tool. While CVD (chemical vapor deposition) SiC was successfully machined with this approach, the cutting forces were substantially higher than cuts at room temperature under the same machining conditions. During the experiments, the expansion of the part and the tool due to the heating was carefully accounted for. The higher cutting forces are most likely due to a small reduction in the shear strength of the material compared with a larger increase in friction forces due to the thermal softening effect. The key advantage is that the hybrid machine approach has the potential to achieve optical quality without the need for a separate optical finishing step. Also, this method is scalable, so one can easily progress from machining 50-mm-diameter samples to the 250-mm-diameter mirror that NASA desires.

Rozzi, Jay↗

Ultra-compact hybrid silicon:chalcogenide waveguide temperature sensor

We demonstrate a real-time, reusable, and reversible integrated optical sensor for temperature monitoring within harsh environments. The sensor architecture combines the phase change property of chalcogenide glasses (ChG) with the high-density integration advantages of high index silicon waveguides. To demonstrate sensor feasibility, ChG composition Ge 40 S 60 , which is characterized by a sharp phase transition from amorphous to crystalline phase around 415 °C, is deposited over a 50 µ m section of a single mode optical waveguide. The phase transition changes the behavior of Ge 40 S 60 from a low loss to high loss material, thus significantly affecting the hybrid waveguide loss around the phase transition temperature. A transmission power drop of over 40dB in the crystalline phase compared to the amorphous phase is experimentally measured. Moreover, we recover the amorphous phase through the application of an electrical pulse, thus showing the reversible nature of our compact temperature sensor. Through integrating multiple compositions of ChG with well-defined phases transition temperatures over a silicon waveguide array, it is possible to determine, in real-time, the temperature evolution within a harsh environment, such as within a nuclear reactor cladding.

Badamchi, Bahareh (ORCID:0000000193105300)↗

Transient superconductivity at nano- and meso-scales

High-T c superconductivity is surpassed by few, if any, other unsolved problems in contemporary physics in terms of its richness, complexity, impact on other fields, and potential technological importance in energy applications. Recent discoveries reveal that the highest transition temperatures emerge under extraordinary experimental conditions such as the strong perturbation of a single atomic layer material by the underlying substrate, application of ~200 GPa pressure or ~MV/cm static electric fields, or irradiation by intense femtosecond optical fields. Research in superconductivity under extreme experimental conditions is challenging and requires the development of novel experimental methods and new theoretical tools suitable to tackle the problem. Our team has approached the challenges of transient superconductivity under intense optical fields by rethinking the types of experimental observables that are best suited to elucidate the physics of light-induced effects. This exercise has identified the need for: i) ultra-fast spatio-temporal probes enabling access to plasmonic properties and also nano-scale inhomogeneities that are ubiquitous in unconventional superconductors; ii) meso-scale structures and hybrid meta-surfaces imperative for strong enhancement of optical fields and iii) theoretical approaches suitable to explain and predict the response of superconductors under ultra-fast photo-excitation. Co-PIs have already made major advances with developing capabilities i)-iii). Therefore, our team is poised to make significant progress in transient superconductivity by exploring novel spatio-temporal effects that currently remain unattainable. Co-investigators will search for light-induced superconductivity in the high-T c cuprates to build upon spectacular recent results still lacking independent confirmations. Novel data acquisition methods combined with nano-plasmonic imaging will allow our team to test the hypothesis of photo-induced superconducting pairing. Spatio-temporal experiments will provide insights into the interplay between superconductivity and competing orders in the cuprates. Co-PIs will also investigate strong light-matter interaction and superconductivity in FeSe monolayers. Finally, we will study spatio-temporal electrodynamics of planar Josephson junctions. This latter direction will allow our team to thoroughly characterize one of the most fundamental examples of inhomogeneity in all of unconventional superconductivity. The proposed experiments in combination with theoretical analysis will provide information that is difficult to obtain using alternative methods. Basov and Averitt will carry out pump-probe spectroscopy and nano-imaging experiments. Averitt and Hone will design and fabricate state-of-the-art meta-surfaces. Theoretical and computational studies of ultrafast transient phenomena will be carried out by Millis and Fogler. Modeling of time-resolved nano-optical effects will be done by Fogler. The bulk of the requested budget will be used to support graduate students at Columbia & UCSD that will be co-supervised by co-PIs. The proposed program is transformative, since it will enable an entire suite of experiments previously either impossible or technically implausible. It will deliver critically important insights not only into mechanisms of unconventional superconductivity but also to many other correlated quantum materials.

36 MATERIALS SCIENCE↗

Altermagnetic polar metallic phase in ultrathin epitaxially strained RuO 2 films

Altermagnetism refers to a wide class of magnetic orders featuring magnetic sublattices with opposite spins related by rotational symmetries, resulting in nontrivial spin splitting and magnetic multipoles. However, the direct observation of the altermagnetic transition remains elusive. Here, by combining theoretical analysis, electrical transport, X-ray, and optical spectroscopies, we establish a phase diagram in hybrid molecular beam epitaxy-grown RuO 2 /TiO 2 (110) films, mapping symmetries along with altermagnetic/electronic/structural phase transitions as functions of film thickness and temperature. This features an altermagnetic metallic polar phase in epitaxially strained 2 nm films, suggesting a potential link between polar metals and altermagnetic materials. Such a clear signature of a magnetic phase transition at ~500 K is observed exclusively in ultrathin strained films, unlike in bulk RuO 2 single crystals. These results highlight the power of epitaxial heterostructure engineering to induce altermagnetism in systems initially nonmagnetic, opening avenues for realizing emergent quantum phases with multifunctional properties.

RuO2 thin film↗

Self-Assembled BaTiO 3 –Au x Ag 1– x Low-Loss Hybrid Plasmonic Metamaterials with an Ordered “Nano-Domino-like” Microstructure

Metallic plasmonic hybrid nanostructures have attracted enormous research interest due to the combined physical properties coming from different material components and the broad range of applications in nanophotonic and electronic devices. However, the high loss and narrow range of property tunability of the metallic hybrid materials have limited their practical applications. In this study, a metallic alloy-based self-assembled plasmonic hybrid nanostructure, i.e., a BaTiO 3 –Au x Ag 1– x (BTO) vertically aligned nanocomposite, has been integrated by a templated growth method for low-loss plasmonic systems. Comprehensive microstructural characterizations including high-resolution scanning transmission electron microscopy (HRSTEM), energy-dispersive X-ray spectroscopy (EDS), and three-dimensional (3D) electron tomography demonstrate the formation of an ordered “nano-domino-like” morphology with Au 0.4 Ag 0.6 nanopillars as cylindrical cores and BTO as square shells. By comparing with the BTO–Au hybrid thin film, the BTO–Au 0.4 Ag 0.6 alloyed film exhibits much broader plasmon resonance, hyperbolic dispersion, low-loss, and thermally robust features in the UV–vis–NIR wavelength region. This study provides a feasible platform for a complex alloyed plasmonic hybrid material design with low-loss and highly tunable optical properties toward all-optical integrated devices.

36 MATERIALS SCIENCE↗

Long-range optical coupling with epsilon-near-zero materials

Long-range resonant quantum tunneling of electrons happens across potential barriers when the wavefunction interferes constructively outside the barrier. Here we demonstrate an analogy in optical systems based on epsilon-near-zero materials, achieving phase-modulated, long-range optical interactions between transparent semiconducting oxide layers beyond the evanescent photonic coupling. Distinct from weak thin-film interference, intense electromagnetic fields confined within the epsilon-near-zero thin films show anti-correlated intensity oscillations as a function of interlayer separation up to hundreds of microns. The oscillatory, anti-correlated electromagnetic field intensities were probed by second harmonic generation from wedged indium tin oxide multilayers. Such a system that hosts subwavelength mode footprint and simultaneously long-range radiative coupling offers prospects for long-distance optical communication, large-scale photonic circuits, and hybrid quantum photonic systems.

Wang, Danqing [Fudan University, Shanghai (China);↗

Unusual electronic transport in (1 - x )Cu 2 Se-( x )CuInSe 2 hierarchical composites

The ability to control the relative density of electronic point defects as well as their energy distribution in semiconductors could afford a systematic modulation of their electronic, optical, and optoelectronic properties. Using a model binary hybrid system Cu 2 Se–CuInSe 2 , we have investigated the correlation between phase composition, microstructure, and electronic transport behavior in the synthesized composites. We found that both Cu 2 Se and CuInSe 2 phases coexist at multiple length scales, ranging from sub-ten nanometer to several micrometers, leading to the formation of a hybrid hierarchical microstructure. Astonishingly, the electronic phase diagram of the (1 - x)Cu 2 Se–(x)CuInSe 2 (15% ≤ x ≤ 100%) hierarchical composites remarkably deviates from the trend normally expected for composites between a heavily doped semiconductor (Cu 2 Se) and a poorly conducting phase (CuInSe 2 ). A sudden 3-fold increase in the electrical conductivity and carrier concentration along with a marginal increase in the carrier mobility is observed for composites at the vicinity of equimolar composition (48% ≤ x ≤ 52%). The carrier concentration increases from ~1.5 × 10 20 cm -3 for the composites with x ≤ 45% to 5.0 × 10 20 cm -3 for x = 50%, and remains constant at 4.5 × 10 20 cm -3 with x value in the range of 52% < x ≤ 90%, then quickly drops to 8 × 10 18 cm -3 for pristine CuInSe 2 phase (x = 100%). The atypical electronic behavior was rationalized in the light of the formation of an inter-band (IB) within the band gap, which arises from the hybridization of native electronic point defects from both Cu 2 Se and CuInSe 2 phases in the resulting hierarchical composites. The result points to a new strategy to modulate the electronic structure of semiconductor composites to maximize interaction and coupling between two fundamentally contrasting properties enabling access to electronic hybrid systems with potential applications as interactive and stimuli-responsive multifunctional materials.

36 MATERIALS SCIENCE↗

Design and development of optical modules for the BUTTON-30 detector

BUTTON-30 is a neutrino detector demonstrator located in the STFC Boulby underground facility in the north-east of England. The main goal of the project is to deploy and test the performance of the gadolinium-loaded water-based liquid scintillator for neutrino detection in an underground environment. This will pave the way for a future large-volume neutrino observatory that can also perform remote monitoring of nuclear reactors for nonproliferation. This paper describes the design and construction of the watertight optical modules of the experiment.

COMSOL simulations↗

Anisotropic elliptic optical fibers

The exact characteristic equation for an anisotropic elliptic optical fiber is obtained for odd and even hybrid modes in terms of infinite determinants utilizing Mathieu and modified Mathieu functions. A simplified characteristic equation is obtained by applying the weakly guiding approximation such that the difference in the refractive indices of the core and the cladding is small. The simplified characteristic equation is used to compute the normalized guide wavelength for an elliptical fiber. When the anisotropic parameter is equal to unity, the results are compared with the previous research and they are in close agreement. For a fixed value normalized cross-section area or major axis, the normalized guide wavelength lambda/lambda(sub 0) for an anisotropic elliptic fiber is small for the larger value of anisotropy. This condition indicates that more energy is carried inside of the fiber. However, the geometry and anisotropy of the fiber have a smaller effect when the normalized cross-section area is very small or very large.

Kang, Soon Ahm↗