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15 pages, 879 KiB  
Article
Solar Energetic Particles Propagation under 3D Corotating Interaction Regions with Different Characteristic Parameters
by Yuji Zhu and Fang Shen
Universe 2024, 10(8), 315; https://doi.org/10.3390/universe10080315 - 2 Aug 2024
Viewed by 289
Abstract
Solar energetic particles (SEPs) are bursts of high-energy particles that originate from the Sun and can last for hours or even days. The aim of this study is to understand how the characteristics of energetic particles ware affected by the characteristic parameters of [...] Read more.
Solar energetic particles (SEPs) are bursts of high-energy particles that originate from the Sun and can last for hours or even days. The aim of this study is to understand how the characteristics of energetic particles ware affected by the characteristic parameters of corotating interaction regions (CIRs). In particular, the particle intensity distribution with time and space in CIRs with different characteristics were studied. The propagation and acceleration of particles were described by the focused transport equation (FTE). We used a three-dimensional magnetohydrodynamic (MHD) model to simulate the background solar wind with CIRs. By changing the inner boundary conditions, we constructed CIRs with different solar wind speeds, angles between the polar axis and rotation axis, and the azimuthal widths of the fast streams. Particles were impulsively injected at the inner boundary of the MHD model. We then studied the particle propagation and compression acceleration in different background solar wind. The results showed that the CIR widths are related to the solar wind speed, tilt angles, and the azimuthal widths of the fast stream. The acceleration of particles in the reverse and forward compression regions are mainly influenced by the solar wind speed difference and the slow solar wind speed, respectively. Particles with lower energy (sub-MeV) are more sensitive to the solar wind speed difference and the tilt angle. The particle intensity variation with time and the radial distance is mainly influenced by the solar wind speed. The longitudinal distribution of particle intensity is affected by the solar wind speed, tilt angles, and the azimuthal widths of the fast stream. Full article
(This article belongs to the Section Space Science)
Show Figures

Figure 1

Figure 1
<p>The radial solar wind speed at the inner boundary of the cases in <b>Set A</b>–<b>D</b>. The cases in <b>Set A</b> have the same <math display="inline"><semantics> <msubsup> <mi>V</mi> <mrow> <mi>b</mi> </mrow> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> </semantics></math>. The cases in <b>Set B</b> have the same <math display="inline"><semantics> <mrow> <msubsup> <mi>V</mi> <mrow> <mi>b</mi> </mrow> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>V</mi> <mrow> <mi>b</mi> </mrow> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> </mrow> </semantics></math>. The cases in <b>Set C</b> have the same <math display="inline"><semantics> <msubsup> <mi>V</mi> <mrow> <mi>b</mi> </mrow> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>V</mi> <mrow> <mi>b</mi> </mrow> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msubsup> </semantics></math>, but different <math display="inline"><semantics> <mi>α</mi> </semantics></math>. The cases in <b>Set D</b> have the same <math display="inline"><semantics> <mi>α</mi> </semantics></math>, <math display="inline"><semantics> <msubsup> <mi>V</mi> <mrow> <mi>b</mi> </mrow> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> </semantics></math>, and <math display="inline"><semantics> <msubsup> <mi>V</mi> <mrow> <mi>b</mi> </mrow> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msubsup> </semantics></math>, but different <math display="inline"><semantics> <mrow> <mi>δ</mi> <msub> <mi>ϕ</mi> <mrow> <mi>s</mi> <mi>t</mi> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>m</mi> </mrow> </msub> </mrow> </semantics></math>. From panel (<b>D1</b>–<b>D3</b>), <math display="inline"><semantics> <mrow> <mi>δ</mi> <msub> <mi>ϕ</mi> <mrow> <mi>s</mi> <mi>t</mi> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>m</mi> </mrow> </msub> </mrow> </semantics></math> is <math display="inline"><semantics> <mrow> <mn>136</mn> <mo>°</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>45</mn> <mo>°</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mn>18</mn> <mo>°</mo> </mrow> </semantics></math>, respectively.</p>
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<p>The CIR width of the selected cases. The blue, red and yellow asterisks indicate the CIR radial extent, width projected into the solar equatorial plane and CIR width considering 3D geometry, respectively. Panels (<b>A</b>–<b>D</b>) are for cases in <b>Set A</b>–<b>D</b>, respectively.</p>
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<p>The ratio of the maximum energy to the original energy (<math display="inline"><semantics> <msub> <mi>E</mi> <mn>0</mn> </msub> </semantics></math> = 5 MeV) at 1 AU after 60 h versus the CIR parameters. The red asterisks and blue circles are for the reverse and the forward compression regions, respectively. Panels (<b>A</b>–<b>D</b>) are for cases in <b>Set A</b>–<b>D</b>, respectively.</p>
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<p>The divergence of the background solar wind velocity <math display="inline"><semantics> <mrow> <mo>∇</mo> <mo>·</mo> <mi>V</mi> </mrow> </semantics></math> of Case 1 (panel (<b>a</b>)), Case 3 (panel (<b>b</b>)), and Case 5 (panel (<b>c</b>)) in Set B.</p>
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<p>The maximum ratio of the maximum energy in the reversed compression region to that in the approximated Parker region at 1 AU in the first 60 h versus the CIR parameters. The blue asterisks, red circles, and yellow squares are for protons with original energy of 0.5 Mev, 5 MeV, and 20 MeV, respectively. Panels (<b>A</b>–<b>D</b>) are for cases in <b>Set A</b>–<b>D</b>, respectively.</p>
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<p>The peak intensity of particles versus the CIR radial extent <math display="inline"><semantics> <msub> <mi>D</mi> <mi>c</mi> </msub> </semantics></math> at 1 AU. The blue asterisks, red circles, yellow squares, and purple crosses indicate cases in <b>Set A</b>, <b>Set B</b>, <b>Set C</b>, and <b>Set D</b>, respectively.</p>
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<p>The variations of particle intensity (6.7–11.4 MeV) with time, longitude, and radial distance for different cases. The rows from top to bottom display the cases in <b>Set A</b>–<b>D</b>, respectively. The columns from left to right represent the variations of particle intensity with time, longitude, and radial distance, respectively. Panels <b>A3(a)</b>–<b>D3(a)</b> and panels <b>A3(b)</b>–<b>D3(b)</b> show the distribution after 30 h and 120 h since the injection, respectively.</p>
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15 pages, 4718 KiB  
Article
A Miniaturized and Highly Stable Frequency-Selective Rasorber Incorporating an Embedded Transmission Window
by Yi Li, Yuxi Zhong, Minrui Wang, Keqing Chen, Peng Ren and Zheng Xiang
Micromachines 2024, 15(8), 980; https://doi.org/10.3390/mi15080980 - 30 Jul 2024
Viewed by 260
Abstract
In this article, a miniaturized and highly stable frequency-selective rasorber (FSR) incorporating an embedded transmission window is designed. This FSR consists of a lossy layer loaded with resistors, an air layer, and a bandpass layer. The lossy layer is provided with a rectangular, [...] Read more.
In this article, a miniaturized and highly stable frequency-selective rasorber (FSR) incorporating an embedded transmission window is designed. This FSR consists of a lossy layer loaded with resistors, an air layer, and a bandpass layer. The lossy layer is provided with a rectangular, square ring structure loaded with four 180 Ω resistors and four quadrilateral metal plates. The four metal plates are connected to the four corners of the inner ring around the square ring and are radially distributed along the diagonal. The bandpass layer is a square metal patch that a cross-ring slot structure is loaded inside of, and the cross points lie in the direction along the diagonal of the unit. The inner boundary of the cross-ring is composed of two mutually perpendicular and long rectangular elements. This FSR shows an embedded transmission window from 3.63 GHz to 3.80 GHz and has a transmission rate of 93% at 3.72 GHz. Moreover, both sides of the transmission band, namely, 1.86–3.35 GHz and 3.99–8.28 GHz, have an absorption rate of more than 80% and bilateral relative bandwidth of more than 50%. In addition, this structure exhibits excellent miniaturization performance, polarization insensitivity, and angular stability. Finally, a prototype of the designed FSR is processed and measured. The measured results are basically consistent with the simulation results. Full article
(This article belongs to the Section D:Materials and Processing)
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Figure 1

Figure 1
<p>Structure diagram of the FSR.</p>
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<p>Equivalent circuit of a general FSR.</p>
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<p>Equivalent circuit under FSR total transmission condition.</p>
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<p>Equivalent circuit under FSR total absorption condition.</p>
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<p>Equivalent circuit model of the proposed FSR in this article.</p>
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<p>The three-dimensional unit structure diagram of the proposed FSR.</p>
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<p>Detailed dimensions of the lossy layer in the FSR unit cells.</p>
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<p>Detailed dimensions of the bandpass layer in the FSR unit cells.</p>
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<p>Simulation results for the S-parameters of the proposed FSR.</p>
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<p>Simulation results for the absorbance, transmittance, and reflectance of the proposed FSR.</p>
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<p>Impact of different air layer thicknesses on FSR filtering performance.</p>
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<p>Simulation results for polarization stability for the proposed FSR.</p>
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<p>Simulation results for angular stability in TE-polarization for the proposed FSR.</p>
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<p>Simulation results for angular stability in TM-polarization for the proposed FSR.</p>
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<p>The prototype of the proposed FSR.</p>
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<p>Test environment.</p>
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<p>Comparison of simulation results and test results under TE-polarization in normal incidence.</p>
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<p>Comparison of simulation results and test results under TE-polarization in a 30° oblique incidence.</p>
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<p>Comparison of the simulation results and test results under TM-polarization in normal incidence.</p>
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<p>Comparison of simulation results and test results under TM-polarization in a 30° oblique incidence.</p>
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15 pages, 5061 KiB  
Article
Spatially Variable Ripple and Groove Formation on Gallium Arsenide Using Linear, Radial, and Azimuthal Polarizations of Laser Beam
by Kalvis Kalnins, Vyacheslav V. Kim, Andra Naresh Kumar Reddy, Anatolijs Sarakovskis and Rashid A. Ganeev
Photonics 2024, 11(8), 710; https://doi.org/10.3390/photonics11080710 - 30 Jul 2024
Viewed by 315
Abstract
We demonstrated the linear, radial, and annular ripple formation on the surface of GaAs. The formation of linear ripples was optimized by the number of shots and the fluence of 30 ps, 532 nm pulses. The radial and annular nanoripples were produced under [...] Read more.
We demonstrated the linear, radial, and annular ripple formation on the surface of GaAs. The formation of linear ripples was optimized by the number of shots and the fluence of 30 ps, 532 nm pulses. The radial and annular nanoripples were produced under the ablation using doughnut-like beams possessing azimuthal and radial polarizations, respectively. We compare the ripples and grooves formed by a linearly polarized Gaussian beam relative to an annular vector beam. The joint overlap of sub-wavelength grooves with ripples formed by azimuthally and radially polarized beams was reported. The conditions under which the shape of radial and ring-like nano- or micro-relief on the GaAs surface can be modified by modulating the polarization of laser pulse were determined. The resultant surface processing of GaAs using a laser beam with different polarization modes is useful for exploring valuable insights and benefits in different applications. Full article
(This article belongs to the Section Lasers, Light Sources and Sensors)
Show Figures

Figure 1

Figure 1
<p>(<b>a</b>) Experimental scheme for LIPSS formation on the surface of GaAs using 532 nm, 30 ps pulses. The bottom insets show the spatial shapes of (<b>b</b>) a linearly polarized Gaussian beam, (<b>c</b>) a radially polarized annular beam, and (<b>d</b>) an azimuthally polarized annular beam and (<b>e</b>–<b>g</b>) their corresponding line-outs of spatial distribution. White arrows are the directions of polarization.</p>
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<p>SEM images of ablated GaAs using (<b>a</b>) 5 shots, (<b>b</b>) 15 shots, and (<b>c</b>) 25 shots of 532 nm, 30 ps pulses at the fluence 0.18 J cm<sup>−2</sup> on the target surface. The bottom panels of (<b>a</b>–<b>c</b>) correspond to the enlarged parts of the corresponding areas marked in red. White lines correspond to 10 μm. Blue arrows show the direction of polarization of the Gaussian beam.</p>
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<p>(<b>a</b>) Appearance of orthogonally directed lines (HSFL) with a smaller spatial period at a larger number of shots (30) on the same place of GaAs. (<b>b</b>) Enlarged part of <a href="#photonics-11-00710-f003" class="html-fig">Figure 3</a>a. The blue arrow shows the direction of polarization of the laser beam. The white line corresponds to 1 μm. The average spatial periods of orthogonal (LSFL and HSFL) ripples were 480 and 270 nm, respectively.</p>
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<p>(<b>a</b>) Spatial shape of the ablated area on the surface of GaAs by the doughnut-like beam with a radial distribution of polarization (white arrows). (<b>b</b>) The enlarged square of <a href="#photonics-11-00710-f004" class="html-fig">Figure 4</a>a shows the part of the ablated ring dominated by the presence of the grooves growing parallel to the polarization. (<b>c</b>) An enlarged part of <a href="#photonics-11-00710-f004" class="html-fig">Figure 4</a>b shows the inner region of the ring. One can see the rings of ripples, followed by the grooves. These ripples remain almost unchanged on the right side of <a href="#photonics-11-00710-f004" class="html-fig">Figure 4</a>c while appearing under the grooves. (<b>d</b>) The enlarged part of <a href="#photonics-11-00710-f004" class="html-fig">Figure 4</a>b shows the outer region of the ring. One can see the LSFL at the smallest fluence of the laser beam, followed by the appearance of the grooves above the rings on the left side of <a href="#photonics-11-00710-f004" class="html-fig">Figure 4</a>d. The white bars correspond to 40 μm (<a href="#photonics-11-00710-f004" class="html-fig">Figure 4</a>a) and 10 μm (<a href="#photonics-11-00710-f004" class="html-fig">Figure 4</a>b–d).</p>
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<p>(<b>a</b>) The shape of the ablated area using the radially polarized beam at F = 0.18 J cm<sup>−2</sup> and N = 5. (<b>b</b>) Enlarged part of <a href="#photonics-11-00710-f005" class="html-fig">Figure 5</a>a showing the ring-like LSFL throughout the ablation area. The white bars correspond to 60 μm (<a href="#photonics-11-00710-f005" class="html-fig">Figure 5</a>a) and 10 μm (<a href="#photonics-11-00710-f005" class="html-fig">Figure 5</a>b). While arrows show the direction of polarization of the laser beam.</p>
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<p>(<b>a</b>) SEM image of the whole area of ablation using 20 shots of the azimuthally polarized beam. Almost the whole ablated area was filled in with grooves. (<b>b</b>,<b>c</b>) Enlarged parts of the ablated area and outer border of the ablated area, respectively. The ripples were observed only in the area of the outer border of the ablation. (<b>d</b>) SEM image of the whole area of ablation using 5 shots of the azimuthally polarized beam at the same fluence of heating radiation (F = 0.22 J cm<sup>−2</sup>) as in the case shown in <a href="#photonics-11-00710-f006" class="html-fig">Figure 6</a>a. (<b>e</b>,<b>f</b>) Enlarged parts of the ablated area and inner border of the ablated area, respectively. LSFL, in that case, dominated along the whole ablated area. The white bars correspond to 80 μm (<a href="#photonics-11-00710-f006" class="html-fig">Figure 6</a>a,d) and 10 μm (<a href="#photonics-11-00710-f006" class="html-fig">Figure 6</a>c,f). While arrows show the direction of polarization of the laser beam.</p>
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<p>(<b>a</b>,<b>d</b>) The intensity profiles of the linearly polarized Gaussian beam and radially polarized beam, respectively. (<b>b</b>,<b>e</b>) The corresponding images of ablated areas. (<b>c</b>,<b>f</b>) The enlarged parts of the red squares are marked in <a href="#photonics-11-00710-f007" class="html-fig">Figure 7</a>b and <a href="#photonics-11-00710-f007" class="html-fig">Figure 7</a>e, respectively. Blue arrows show the direction of polarization of the laser beam.</p>
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<p>Left panels of <a href="#photonics-11-00710-f008" class="html-fig">Figure 8</a>a,b: the images of the ablation areas after laser–matter interaction in the case of the rotation of the S-waveplate by (<b>a</b>) 22.5° and (<b>b</b>) 45° from the position corresponding to the pure radial polarization of the annular beam. The enlarged images of the ablation areas in these two cases are shown in the right panels. Red curved lines show the leaned directions of polarizations at different points of the ablation. The grooves follow these leaned directions of polarization.</p>
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18 pages, 11970 KiB  
Article
Contrasting the Effects of X-Band Phased Array Radar and S-Band Doppler Radar Data Assimilation on Rainstorm Forecasting in the Pearl River Delta
by Liangtao He, Jinzhong Min, Gangjie Yang and Yujie Cao
Remote Sens. 2024, 16(14), 2655; https://doi.org/10.3390/rs16142655 - 20 Jul 2024
Viewed by 334
Abstract
Contrasting the X-band phased array radar (XPAR) with the conventional S-Band dual-polarization mechanical scanning radar (SMSR), the XPAR offers superior temporal and spatial resolution, enabling a more refined depiction of the internal dynamics within convective systems. While both SMSR and XPAR data are [...] Read more.
Contrasting the X-band phased array radar (XPAR) with the conventional S-Band dual-polarization mechanical scanning radar (SMSR), the XPAR offers superior temporal and spatial resolution, enabling a more refined depiction of the internal dynamics within convective systems. While both SMSR and XPAR data are extensively used in monitoring and alerting for severe convective weather, their comparative application in numerical weather prediction through data assimilation remains a relatively unexplored area. This study harnesses the Weather Research and Forecasting Model (WRF) and its data assimilation system (WRFDA) to integrate radial velocity and reflectivity from the Guangzhou SMSR and nine XPARs across Guangdong Province. Utilizing a three-dimensional variational approach at a 1 km convective-scale grid, the assimilated data are applied to forecast a rainstorm event in the Pearl River Delta (PRD) on 6 June 2022. Through a comparative analysis of the results from assimilating SMSR and XPAR data, it was observed that the assimilation of SMSR data led to more extensive adjustments in the lower- and middle-level wind fields compared to XPAR data assimilation. This resulted in an enlarged convergence area at lower levels, prompting an overdevelopment of convective systems and an excessive concentration of internal hydrometeor particles, which in turn led to spurious precipitation forecasts. However, the sequential assimilation of both SMSR and XPAR data effectively reduced the excessive adjustments in the wind fields that were evident when only SMSR data were used. This approach diminished the generation of false echoes and enhanced the precision of quantitative precipitation forecasts. Additionally, the lower spectral width of XPAR data indicates its superior detection accuracy. Assimilating XPAR data alone yields more reasonable adjustments to the low- to middle-level wind fields, leading to the formation of small-to-medium-scale horizontal convergence lines in the lower levels of the analysis field. This enhancement significantly improves the model’s forecasts of composite reflectivity and radar echoes, aligning them more closely with actual observations. Consequently, the Threat Score (TS) and Equitable Threat Score (ETS) for heavy-rain forecasts (>10 mm/h) over the next 5 h are markedly enhanced. This study underscores the necessity of incorporating XPAR data assimilation in numerical weather prediction practices and lays the groundwork for the future joint assimilation of SMSR and XPAR data. Full article
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Figure 1

Figure 1
<p>(<b>a</b>) Location of Guangzhou SMSR (red pentagram) and nine XPARs in Guangdong Province (blue triangles), with the radar-coverage circles in 230 (solid red circle) and 60/40 (blue dashed circles) km ranges for SMSR and XPAR, respectively. The black dashed frame delineates the D03. (<b>b</b>) The position of automatic weather stations across Guangdong Province (green scatters).</p>
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<p>The potential geopotential height field (black contours, units: dagpm), temperature field (red contours, units: °C), wind field (wind barbs, units: m/s), and relative humidity field (shaded areas) from the ERA5 reanalysis data at 18:00 UTC on 6 June 2022 are depicted at different pressure levels: (<b>a</b>) 500 hPa; (<b>b</b>) 700 hPa; (<b>c</b>) 850 hPa; (<b>d</b>) 925 hPa. “D” in red represents the center of the cyclone. The brown line represents the wind shear line.</p>
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<p>Model grid configuration and topography (shaded). (<b>a</b>) Domain configuration; (<b>b</b>) D03 configuration.</p>
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<p>The flow chart for the DA experiments.</p>
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<p>Velocity spectrum width (SW) at 1.5° elevation angle for (<b>a</b>) SMSR and (<b>b</b>) XPARs. And (<b>c</b>) the spatial average velocity SW from the lowest to the highest of the first 9 elevation angles for both at 18:00 UTC on 6 June 2022. Unit: m/s.</p>
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<p>The average analysis increment in each model layer for the first assimilation cycle in the D03 region: (<b>a</b>) u (units: m/s), (<b>b</b>) v (units: m/s), (<b>c</b>) T (units: 10<sup>−3</sup> K), (<b>d</b>) q (units: g/kg).</p>
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<p>The first row shows the 850hPa horizontal wind field (vector) and wind speed (shaded, units: m/s) at 18:00 UTC on 6 June 2022, for (<b>a</b>) CTRL, (<b>b</b>) DA_S, (<b>c</b>) DA_X, and (<b>d</b>) DA_S_X. The second row depicts the incremental field of the horizontal wind field relative to CTRL, with wind speed greater than 5 m/s indicated by a red vector, for (<b>e</b>) DA_S, (<b>f</b>) DA_X, and (<b>g</b>) DA_S_X.</p>
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<p>Radar composite reflectivity-analysis field at 18:00 UTC in D03 on 6 June 2022, for (<b>a</b>) OBS-SMSR, (<b>b</b>) CTRL, (<b>c</b>) DA_S, (<b>d</b>) DA_X, and (<b>e</b>) DA_S_X; line AB is the profile in <a href="#remotesensing-16-02655-f009" class="html-fig">Figure 9</a> and <a href="#remotesensing-16-02655-f010" class="html-fig">Figure 10</a>.</p>
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<p>Vertical cross-sections of RF (shaded, units: dBZ) along the black solid line A (112.7°E, 21.6°N) and B (114.6°E, 23.2°N) in <a href="#remotesensing-16-02655-f008" class="html-fig">Figure 8</a> at 18:00 UTC on 6 June 2022. (<b>a</b>) OBS-SMSR, (<b>b</b>) CTRL, (<b>c</b>) DA_S, (<b>d</b>) DA_X, (<b>e</b>) DA_S_X. The height is from sea level.</p>
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<p>Vertical cross-sections of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>q</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mo> </mo> </mrow> </semantics></math>(first row), <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>q</mi> </mrow> <mrow> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> (second row), and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>q</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> (third row) along line AB for each experiment at 18:00 UTC on 6 June 2022. (<b>a</b>,<b>e</b>,<b>i</b>) CTRL, (<b>b</b>,<b>f</b>,<b>j</b>) DA_S, (<b>c</b>,<b>g</b>,<b>k</b>) DA_X, (<b>d</b>,<b>h</b>,<b>l</b>) DA_S_X. The black contours represent the distribution of hydrometeors in CTRL. The height is from sea level.</p>
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<p>Composite reflectivity at 19:00 UTC (first row), 20:00 UTC (second row), and 21:00 UTC (third row) on 6 June 2022 (units: dBZ). (<b>a</b>,<b>f</b>,<b>k</b>) OBS-SMSR, (<b>b</b>,<b>g</b>,<b>l</b>) CTRL, (<b>c</b>,<b>h</b>,<b>m</b>) DA_S, (<b>d</b>,<b>i</b>,<b>n</b>) DA_X, (<b>e</b>,<b>j</b>,<b>o</b>) DA_S_X.</p>
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<p>Vertical cross-sections of RF (shaded, units: dBZ) and wind (vector) along the black solid line A (113.0°E, 21.6°N) and B (114.5°E, 23.2°N) in <a href="#remotesensing-16-02655-f011" class="html-fig">Figure 11</a> at 19:00 UTC on 6 June 2022. (<b>a</b>) OBS-SMSR, (<b>b</b>) CTRL, (<b>c</b>) DA_S, (<b>d</b>) DA_X, (<b>e</b>) DA_S_X. The height is from sea level.</p>
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<p>Hourly precipitation from 18:00 UTC (first column) to 22:00 (last column) on 6 June 2022 (units: mm). (<b>a</b>–<b>d</b>) OBS, (<b>e</b>–<b>h</b>) CTRL, (<b>i</b>–<b>l</b>) DA_S, (<b>m</b>–<b>p</b>) DA_X, (<b>q</b>–<b>t</b>) DA_S_X.</p>
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<p>Threat Score (TS) (first row), False Alarm Rate (FAR) (second row) and Equitable Threat Score (ETS) (third row) for hourly precipitation from 18:00 to 22:00 on 6 June 2022 (<b>a</b>,<b>d,g</b>) for &gt;1 mm, (<b>b</b>,<b>e,h</b>) for &gt;5 mm, (<b>c</b>,<b>f,i</b>) for &gt;10 mm.</p>
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14 pages, 9065 KiB  
Article
A Modified High-Selective Frequency Selective Surface Designed by Multilevel Green’s Function Interpolation Method
by Ze Huang, Rongrong Sun, Peng Zhao, Kanglong Zhang, Yanyang Wang, Zhimin Guan and Gaofeng Wang
Electronics 2024, 13(13), 2453; https://doi.org/10.3390/electronics13132453 - 22 Jun 2024
Viewed by 431
Abstract
A compact high-selective band-pass frequency selective surface (FSS) with the unit cell less than λ/7 is presented. For the simulation of the structure, the multilevel Green’s function interpolation method (MLGFIM) using Floquet theory is adopted to accelerate the calculation of the complex [...] Read more.
A compact high-selective band-pass frequency selective surface (FSS) with the unit cell less than λ/7 is presented. For the simulation of the structure, the multilevel Green’s function interpolation method (MLGFIM) using Floquet theory is adopted to accelerate the calculation of the complex unit cell. The radial basis function (RBF)-QR method is used in the interpolation, which makes the shape parameter in the RBF function not required to be retested for different periodicity. In this design, with an aperture coupling structure between the top and bottom layers patterned by triangular patches and meander lines, the FSS has two transmission zeros (TZs) on both sides of the pass-band and achieves a steep roll-off rate of 192 dB/GHz. Consequently, the FSS has high selectivity and out-of-band suppression, besides profiting from the low profile and symmetric geometry, this FSS exhibits good angular and polarization stabilities. The prototype of the proposed FSS is fabricated and good performance is obtained. Full article
(This article belongs to the Special Issue RF/Microwave Device and Circuit Integration Technology)
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<p>Topology of the proposed FSS. (<b>a</b>) <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> unit cell structure of the proposed FSS. (<b>b</b>) Unit cell of the FSS. (<b>c</b>) aperture coupling interlayer.</p>
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<p>Equivalent circuit mode of the proposed FSS.</p>
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<p>2-D pictorial representation of a periodic FSS.</p>
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<p>Maximum interpolation error for periodic Green’s function interpolation.</p>
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<p>Transmission responses of the proposed FSS with different coupling aperture dimensions.</p>
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<p>Comparison of frequency response for different characteristic impedances <span class="html-italic">Z</span><sub>T</sub>.</p>
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<p>Comparison of the transmission coefficient between the equivalent circuit model and full-wave simulation.</p>
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<p>Current distribution in (<b>a</b>) 4.74 GHz, (<b>b</b>) 4.86 GHz, and (<b>c</b>) 5.46 GHz (The arrow represents the direction of current flow).</p>
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<p>Transmission responses of the proposed FSS under different incident angles: (<b>a</b>) TE mode; (<b>b</b>) TM mode.</p>
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<p>The fabrication photograph of the proposed three-metal-layers FSS prototype.</p>
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<p>Measurement setup.</p>
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<p>Measurement versus simulation of the FSS.</p>
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<p>Measured transmission curves with different polarizations under 0 degree and 45 degree incidence.</p>
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23 pages, 1001 KiB  
Article
A Fast Method for the Off-Boundary Evaluation of Laplace Layer Potentials by Convolution Sums
by Wenchao Guan, Zhicheng Wang, Leqi Xue and Yueen Hou
Symmetry 2024, 16(6), 764; https://doi.org/10.3390/sym16060764 - 18 Jun 2024
Viewed by 737
Abstract
In off-boundary computations of layer potentials, the near-singularities in integrals near the boundary presents challenges for conventional quadrature methods in achieving high precision. Additionally, the significant complexity of O(n2) interactions between n targets and n sources reduces the efficiency [...] Read more.
In off-boundary computations of layer potentials, the near-singularities in integrals near the boundary presents challenges for conventional quadrature methods in achieving high precision. Additionally, the significant complexity of O(n2) interactions between n targets and n sources reduces the efficiency of these methods. A fast and accurate numerical algorithm is presented for computing the Laplace layer potentials on a circle with a boundary described by a polar curve. This method can maintain high precision even when evaluating targets located at a close distance from the boundary. The radial symmetry of the integral kernels simplifies their description. By exploiting the polar form of the boundary and applying a one-dimensional exponential sum approximation along the radial direction, an approximation of layer potentials by the convolution sum is obtained. The algorithm uses FFT convolution to accelerate computation and employs a local quadrature to maintain accuracy for nearly singular terms. Consequently, it achieves spectral accuracy in regions outside of a sufficiently small neighborhood of the boundary and requires O(nlogn) arithmetic operations. With the help of this algorithm, layer potentials can be efficiently evaluated on a computational domain. Full article
(This article belongs to the Special Issue Computational Mathematics and Its Applications in Numerical Analysis)
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<p>The Fourier coefficients of <math display="inline"><semantics> <msub> <mi>μ</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>β</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>ω</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>β</mi> </mrow> </msub> </semantics></math>. The radius function of the boundary <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Γ</mi> <mn>1</mn> </msub> </semantics></math> is <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>0.3</mn> <mo form="prefix">cos</mo> <mo>(</mo> <mn>5</mn> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>, and the target point is located at <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0.66</mn> <mo>,</mo> <mn>0.37</mn> <mo>)</mo> </mrow> </semantics></math>, which is a distance of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> from the boundary. Top row: <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. Bottom row: <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>a</b>) The Fourier coefficients of <math display="inline"><semantics> <msub> <mi>μ</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>β</mi> </mrow> </msub> </semantics></math>. (<b>b</b>) The Fourier coefficients of <math display="inline"><semantics> <msub> <mi>ω</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>β</mi> </mrow> </msub> </semantics></math>.</p>
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<p>The values of <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>σ</mi> </msub> <mo>∗</mo> <msub> <mi>ω</mi> <mi>σ</mi> </msub> <mrow> <mo>(</mo> <mn>0.66</mn> <mo>,</mo> <mn>0.37</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and the relative error of the Gauss–Chebyshev quadrature against <math display="inline"><semantics> <mi>σ</mi> </semantics></math>. The radius function of the boundary <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Γ</mi> <mn>1</mn> </msub> </semantics></math> is <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>0.3</mn> <mo form="prefix">cos</mo> <mo>(</mo> <mn>5</mn> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>, and the target point is at a distance of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> from the boundary. (<b>a</b>) The values of <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>σ</mi> </msub> <mo>∗</mo> <msub> <mi>ω</mi> <mi>σ</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">x</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi mathvariant="bold">x</mi> <mo>=</mo> <mrow> <mo>(</mo> <mn>0.66</mn> <mo>,</mo> <mn>0.37</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>b</b>) The relative error of the 25-point Gauss quadrature. (<b>c</b>) The relative error of the 40-point Gauss quadrature. (<b>d</b>) The relative error of the 55-point Gauss quadrature.</p>
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<p>Test curves <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Γ</mi> <mn>1</mn> </msub> </semantics></math> (black) and <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Γ</mi> <mn>2</mn> </msub> </semantics></math> (blue). (<b>a</b>) A starfish curve (black) and a unit circle (blue). (<b>b</b>) A limaçon (black) and a unit circle (blue).</p>
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<p>Double-layer potential evaluation at <math display="inline"><semantics> <msup> <mn>10</mn> <mn>4</mn> </msup> </semantics></math> equispaced points on a unit circle with the starfish boundary described by (<a href="#FD15-symmetry-16-00764" class="html-disp-formula">15</a>). (<b>a</b>) The graph of <math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="script">D</mi> <mo stretchy="false">˜</mo> </mover> <msub> <mi>f</mi> <mi>d</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0.09</mn> <mi>π</mi> <mo>,</mo> <mn>0.11</mn> <mi>π</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>b</b>) The <math display="inline"><semantics> <msup> <mn>10</mn> <mn>4</mn> </msup> </semantics></math>-point PTR. (<b>c</b>) Algorithm 1 with <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>1.54</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>312</mn> </mrow> </semantics></math>. (<b>d</b>) Algorithm 2 with <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>55</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>1.54</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>312</mn> <mo>,</mo> <msub> <mi>σ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>10</mn> <mn>5</mn> </msup> </mrow> </semantics></math>.</p>
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<p>Double-layer potentials evaluation at <math display="inline"><semantics> <msup> <mn>10</mn> <mn>4</mn> </msup> </semantics></math> equispaced points on a unit circle with the limaçon boundary described by (<a href="#FD16-symmetry-16-00764" class="html-disp-formula">16</a>). (<b>a</b>) The graph of <math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="script">D</mi> <mo stretchy="false">˜</mo> </mover> <msub> <mi>f</mi> <mi>d</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mi>π</mi> <mo>/</mo> <mn>10</mn> <mo>,</mo> <mi>π</mi> <mo>/</mo> <mn>10</mn> <mo>)</mo> </mrow> </semantics></math>. (<b>b</b>) A <math display="inline"><semantics> <msup> <mn>10</mn> <mn>4</mn> </msup> </semantics></math>-point PTR. (<b>c</b>) Algorithm 1 with <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>1.54</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>312</mn> </mrow> </semantics></math>. (<b>d</b>) Algorithm 2 with <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>55</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>1.54</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>312</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>σ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>10</mn> <mn>5</mn> </msup> </mrow> </semantics></math>.</p>
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<p>Double-layer potential evaluations at <math display="inline"><semantics> <msup> <mn>10</mn> <mn>4</mn> </msup> </semantics></math> equispaced points on a circle located at a distance of <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </semantics></math> from the starfish boundary described by (<a href="#FD15-symmetry-16-00764" class="html-disp-formula">15</a>). (<b>a</b>) A starfish curve (black) and a circle of radius <math display="inline"><semantics> <mrow> <mn>1.3001</mn> </mrow> </semantics></math> (blue). (<b>b</b>) A <math display="inline"><semantics> <msup> <mn>10</mn> <mn>4</mn> </msup> </semantics></math>-point PTR. (<b>c</b>) Algorithm 1 with <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>1.18</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>10</mn> </mrow> </msup> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>397</mn> </mrow> </semantics></math>. (<b>d</b>) Algorithm 2 with <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>55</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> <mo>,</mo> <mrow/> <mi>β</mi> <mo>=</mo> <mn>1.18</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>10</mn> </mrow> </msup> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>397</mn> <mo>,</mo> <msub> <mi>σ</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mn>10</mn> <mn>5</mn> </msup> </mrow> </semantics></math>.</p>
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<p>Absolute errors of computing single-layer potentials at targets on <math display="inline"><semantics> <mrow> <mi mathvariant="double-struck">T</mi> <mo>∖</mo> <mo>Ω</mo> </mrow> </semantics></math> with a minimum distance of <math display="inline"><semantics> <mrow> <mn>1.4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> from the exterior Laplace Neumann BVPs. The first column shows the errors of the 256-point PTR. The last two columns present the errors of Algorithms 1 and 2 using the proposed parameter selection scheme with <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>256</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo>{</mo> <mi>R</mi> <mo>−</mo> <msub> <mi>r</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> <mo>,</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mo>}</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>55</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>. The bottom row shows the errors relative to the points parameterized by angle <math display="inline"><semantics> <mi>η</mi> </semantics></math> and radius <span class="html-italic">R</span>. (<b>a</b>) PTR: <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mo>∞</mo> </msub> <mo>=</mo> <mn>6.4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>b</b>) Algorithm 1: <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mo>∞</mo> </msub> <mo>=</mo> <mn>6.4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>c</b>) Algorithm 2: <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mo>∞</mo> </msub> <mo>=</mo> <mn>1.4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>9</mn> </mrow> </msup> </mrow> </semantics></math>. (<b>d</b>) PTR. (<b>e</b>) Algorithm 1. (<b>f</b>) Algorithm 2.</p>
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13 pages, 6374 KiB  
Article
Construction and Validation of Surface Soil Moisture Inversion Model Based on Remote Sensing and Neural Network
by Rencai Lin, Zheng Wei, Rongxiang Hu, He Chen, Yinong Li, Baozhong Zhang, Fengjing Wang and Dongxia Hu
Atmosphere 2024, 15(6), 647; https://doi.org/10.3390/atmos15060647 - 28 May 2024
Viewed by 595
Abstract
Surface soil moisture (SSM) reflects the dry and wet states of soil. Microwave remote sensing technology can accurately obtain regional SSM in real time and effectively improve the level of agricultural drought monitoring, and it is of great significance for agricultural precision irrigation [...] Read more.
Surface soil moisture (SSM) reflects the dry and wet states of soil. Microwave remote sensing technology can accurately obtain regional SSM in real time and effectively improve the level of agricultural drought monitoring, and it is of great significance for agricultural precision irrigation and smart agriculture construction. Based on Sentinel-1, Sentinel-2, and Landsat-8 images, the effect of vegetation was removed by the water cloud model (WCM), and SSM was retrieved and validated by a radial basis function (RBF) neural network model in bare soil and vegetated areas, respectively. The normalized difference vegetation index (NDVI) calculated by Landsat-8 (NDVI_Landsat-8) had a better effect on removing the influence the of vegetation layer than that of NDVI_Sentinel-2. The RBF network model, established in a bare area (R = 0.796; RMSE = 0.029 cm3/cm3), and the RBF neural network model, established in vegetated areas (R = 0.855; RMSE = 0.024 cm3/cm3), have better simulation effects on SSM than a linear SSM inversion model with single polarization. The introduction of surface parameters to the RBF neural network model can improve the accuracy of the model and realize the high-accuracy inversion of SSM in the study area. Full article
(This article belongs to the Section Biosphere/Hydrosphere/Land–Atmosphere Interactions)
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<p>Geographical location of Daxing District, Beijing, China.</p>
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<p>Results of NDVI removal of vegetation backscattering calculated from different data sources in VV polarization: (<b>a</b>) Sentinel-2; (<b>b</b>) Landsat-8.</p>
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<p>Results of NDWI removal of vegetation backscattering calculated from different data sources in VV polarization: (<b>a</b>) Sentinel-2; (<b>b</b>) Landsat-8.</p>
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<p>Results of NDVI removal of vegetation backscattering calculated from different data sources in VH polarization: (<b>a</b>) Sentinel-2; (<b>b</b>) Landsat-8.</p>
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<p>Comparison of effects of VH polarization and NDWI removal from different data sources: (<b>a</b>) Sentinel-2; (<b>b</b>) Landsat-8.</p>
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<p>Validation of inversion model of SSM in vegetated areas.</p>
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<p>Validation of inversion model of SSM in bare soil.</p>
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<p>Establishment and validation of VV polarization linear model: (<b>a</b>) Model establishment; (<b>b</b>) Model validation.</p>
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<p>Establishment and validation of VH polarization linear model: (<b>a</b>) Model establishment; (<b>b</b>) Model validation.</p>
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13 pages, 2320 KiB  
Article
Identification and Functional Analysis of the Flower Development-Related TCP Genes in Erycina pusilla
by Yu-Huan Tang, Ying-Yin Zhong and Xia Huang
Horticulturae 2024, 10(6), 534; https://doi.org/10.3390/horticulturae10060534 - 21 May 2024
Viewed by 510
Abstract
Orchid flowers have evolved in concert with pollinators to form highly specialized structures resulting in zygomorphy. In dicotyledons, it is widely accepted that CYC-like genes are involved in the dorsoventral polarity establishment of flowers, which determines the development of zygomorphic flowers. However, [...] Read more.
Orchid flowers have evolved in concert with pollinators to form highly specialized structures resulting in zygomorphy. In dicotyledons, it is widely accepted that CYC-like genes are involved in the dorsoventral polarity establishment of flowers, which determines the development of zygomorphic flowers. However, the function of TCP transcription factors involved in orchid floral development is rarely known. Here, we found 15 unigenes with TCP domain (EpTCPs) from the previously reported Erycina pusilla unigene database. The expression patterns of EpTCPs in various tissues and different floral organs were successively detected by quantitative real-time PCR. The results revealed that the CYC-like gene (EpTCP25) and CIN-like genes (EpTCP11 and EpTCP26) were highly expressed in inflorescences but lowly expressed in leaves and roots. What is more, these three genes were expressed relatively high in the dorsal labellum, and EpTCP26 showed differential expression along the dorsoventral polarity of tepals, which was high in the dorsal and low in the ventral. Ectopic expression of EpTCP25 in Arabidopsis repressed primary root growth and delayed flowering. EpTCP26 overexpression in Arabidopsis promoted primary root growth and leaf growth. In contrast, EpTCP11 overexpression repressed primary root growth and changed the radially symmetric flower to a bilaterally symmetric flower by inhibiting the elongation of one or two adjacent petals. In addition, the homeotic transition of floral organs is generated when these genes are ectopically expressed in Arabidopsis, suggesting their roles in floral morphogenesis. Altogether, our results indicate that CIN-like genes would be associated with the unique flower pattern development of Erycina pusilla. Full article
(This article belongs to the Section Floriculture, Nursery and Landscape, and Turf)
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<p>Phylogenetic tree of TCP transcription factors from <span class="html-italic">Erycina pusilla</span> and <span class="html-italic">Arabidopsis</span>. Different colored bands represent different subgroups. The accession numbers of sequences used for phylogeny from GenBank are as follows: AtTCP1 (NP_001077781.1); AtTCP2 (NP_001078407.1); AtTCP3 (NP_001322492.1); AtTCP4 (NP_001189896.1); AtTCP5 (NP_200905.1); AtTCP7 (NP_197719.1); AtTCP8 (NP_176107.1); AtTCP9 (NP_182092.1); AtTCP10 (NP_565712.1); AtTCP12 (NP_177047.2); AtTCP13 (NP_850501.1); AtTCP14 (NP_190346.2); AtTCP15 (NP_564973.1); AtTCP16 (NP_190101.1); AtTCP17 (NP_001318505.1); AtTCP19 (NP_200004.1); AtTCP20 (NP_001327814.1); AtTCP21 (NP_196450.1); AtTCP22 (NP_177346.1); AtTCP23 (NP_174789.1); AtTCP24 (NP_564351.1).</p>
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<p>Relative expression of <span class="html-italic">EpTCPs</span> in different tissue materials of <span class="html-italic">Erycina pusilla</span>. (<b>A</b>) Plantlet, different periods of inflorescences and floral buds; bar = 20 mm. (<b>B</b>) qPCR assay. F, flower; L, leaf; R, root; I1, inflorescence length of 1~2 mm; I2, inflorescence length of 3~4 mm; B1, floral bud length of 3~4 mm; B2, floral bud length of 5~7 mm.</p>
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<p>Relative expression of <span class="html-italic">EpTCPs</span> in different floral organs of <span class="html-italic">Erycina pusilla</span> floral bud about 4 mm. (<b>A</b>) Floral organ pattern of floral bud; bar = 20 mm. (<b>B</b>) qPCR assay. OT-D, dorsal outer tepal; OT-V, ventral outer tepal; L-D, dorsal labellum; IT-V, ventral inner tepal; C, column.</p>
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<p>Root growth of 7-day-old Col-0, three independent lines of <span class="html-italic">35S:EpTCP11</span> (<b>A</b>), <span class="html-italic">35S:EpTCP25</span> (<b>B</b>) and <span class="html-italic">35S:EpTCP26</span> (<b>C</b>) transgenic <span class="html-italic">Arabidopsis</span>; bar = 10 mm. (<b>D</b>) Root lengths of Col-0 and transgenic <span class="html-italic">Arabidopsis</span> plants. Asterisks indicated significant differences between transgenic <span class="html-italic">Arabidopsis</span> and Col-0 (* <span class="html-italic">p</span> &lt; 0.05, ** <span class="html-italic">p</span> &lt; 0.01).</p>
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<p>Growth of Col-0 (<b>A</b>), three independent lines of <span class="html-italic">35S:EpTCP11</span> (<b>B</b>–<b>D</b>), <span class="html-italic">35S:EpTCP25</span> (<b>E</b>–<b>G</b>) and <span class="html-italic">35S:EpTCP26</span> (<b>H</b>–<b>J</b>) transgenic <span class="html-italic">Arabidopsis</span> plants after 15 days of transplant; bar = 10 mm. (<b>K</b>) Statistical rosette leaf numbers of Col-0 and transgenic <span class="html-italic">Arabidopsis</span> when flowering. Asterisks indicated significant different between transgenic <span class="html-italic">Arabidopsis</span> and Col-0 (* <span class="html-italic">p</span> &lt; 0.05).</p>
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<p>Flower mutations of <span class="html-italic">35S:EpTCP11</span>, <span class="html-italic">35S:EpTCP25</span> and <span class="html-italic">35S:EpTCP26</span> transgenic <span class="html-italic">Arabidopsis</span> plants. (<b>A</b>) Wild-type flower. (<b>B</b>) Flower with phenotypic variation 1; yellow arrow markers short petal. (<b>C</b>) Flower with phenotypic variation 2. (<b>D</b>) The short and long petal lengths of flowers with phenotypic variation 1 and 2 in <span class="html-italic">35S:EpTCP11</span> transgenic <span class="html-italic">Arabidopsis</span> plants; P, petals of Col-0; SP, short petals; LP, long petals. Different letters indicated significant differences in petal length (<span class="html-italic">p</span> &lt; 0.05). (<b>E</b>) Side view of wild-type flower. (<b>F</b>) Side view of flower with phenotypic variation 3; red arrow markers the mutant stamen. (<b>G</b>) Side view of flower with phenotypic variation 4; red arrow markers the mutant petal. Bar = 1 mm.</p>
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19 pages, 3597 KiB  
Article
Surface Deformation Calculation Method Based on Displacement Monitoring Data
by Lin He and Yibin Yao
Buildings 2024, 14(5), 1417; https://doi.org/10.3390/buildings14051417 - 14 May 2024
Viewed by 572
Abstract
Considering the importance of calculating surface deformation based on monitoring data, this paper proposes a method for calculating horizontal deformation based on horizontal displacement monitoring data. This study first analyzes the characteristics of horizontal displacement monitoring data, then proposes a scheme for obtaining [...] Read more.
Considering the importance of calculating surface deformation based on monitoring data, this paper proposes a method for calculating horizontal deformation based on horizontal displacement monitoring data. This study first analyzes the characteristics of horizontal displacement monitoring data, then proposes a scheme for obtaining the surface horizontal displacement field through corresponding discrete point interpolation. Subsequently, the calculation method for surface horizontal strain is introduced, along with relevant examples. The study also systematically summarizes the calculation methods for surface curvature and surface tilt deformation values, forming a set of surface deformation calculation methods based on monitoring data. The research results indicate that when there is a large number of on-site monitoring points, effective monitoring points can be selected based on the direction of horizontal displacement. When interpolating the surface horizontal displacement field, the interpolation accuracy of the radial basis function method is slightly higher than that of ordinary Kriging. The form of coordinate expression has a significant impact on interpolation accuracy. The accuracy of interpolation using horizontal displacement vectors expressed in polar coordinates is higher than that using vectors expressed in Cartesian coordinates. The calculated surface horizontal strain has effective upper and lower limits, with lower-limit strain on the contour line conforming to the typical surface deformation patterns around mined-out areas. Full article
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Figure 1

Figure 1
<p>Vector diagram of horizontal displacement at monitoring point.</p>
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<p>Cumulative <span class="html-italic">μ<sub>x</sub></span>, <span class="html-italic">μ<sub>y</sub></span>, ‖<b><span class="html-italic">μ</span></b>‖ and <span class="html-italic">θ</span> values at monitoring points.</p>
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<p>The semi-variance function corresponding to the Gaussian model.</p>
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<p>Optimal <span class="html-italic">R</span><sup>2</sup> value determination curve.</p>
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<p>Calculation steps for surface horizontal strain.</p>
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<p>Principal curvature and principal directions.</p>
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<p>Calculation method for surface deformation based on monitoring data.</p>
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<p>Isoline map of major principal strain.</p>
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<p>A polar plot representing horizontal strains on equipotential lines of 0.2 cm/m maximum principal strain.</p>
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23 pages, 11568 KiB  
Article
Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
by Klemensas Laurinavičius, Sergej Orlov and Ada Gajauskaitė
Appl. Sci. 2024, 14(10), 3984; https://doi.org/10.3390/app14103984 - 8 May 2024
Viewed by 681
Abstract
Theoretical studies on the generation of nondiffracting and nondispersive light pulses and their experimental implementation are one of the renowned problems within electromagnetics. Current technologies enable the creation of short-duration pulses of a few cycles with high power and fluency. An application of [...] Read more.
Theoretical studies on the generation of nondiffracting and nondispersive light pulses and their experimental implementation are one of the renowned problems within electromagnetics. Current technologies enable the creation of short-duration pulses of a few cycles with high power and fluency. An application of these techniques to the field of nondiffracting and nondispersive pulses requires a proper mathematical description of highly focused vector pulses. In this work, we study vector optical bullets in a dielectric medium with different polarization structures: linear, azimuthal, and radial. We report the differences caused by the vector model compared to the scalar model. We analyze effects caused by superluminal, subluminal, or even negative group velocity on the properties of vector optical bullets inside a dielectric material. Full article
(This article belongs to the Section Optics and Lasers)
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Figure 1

Figure 1
<p>(<b>a</b>) Angular dispersion of the optical bullet inside the BK7 glass, when <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1.25</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mn>0.7</mn> </mrow> </semantics></math> (1), <math display="inline"><semantics> <mrow> <mn>0.75</mn> </mrow> </semantics></math> (2), <math display="inline"><semantics> <mrow> <mn>0.85</mn> </mrow> </semantics></math> (3), 1 (4), <math display="inline"><semantics> <mrow> <mn>1.4</mn> </mrow> </semantics></math> (5). (<b>b</b>) Angular dispersion of the optical bullet within the BK7 glass, when <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mn>0.629</mn> </mrow> </semantics></math> (1), <math display="inline"><semantics> <mrow> <mn>0.63685</mn> </mrow> </semantics></math> (2), <math display="inline"><semantics> <mrow> <mn>0.645</mn> </mrow> </semantics></math> (3), <math display="inline"><semantics> <mrow> <mn>0.65495</mn> </mrow> </semantics></math> (4), <math display="inline"><semantics> <mrow> <mn>0.685</mn> </mrow> </semantics></math> (5), (<b>c</b>) Angular dispersion of the optical bullet within the BK7 glass, when <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mi>γ</mi> </semantics></math>: <math display="inline"><semantics> <mrow> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (1), <math display="inline"><semantics> <mrow> <mn>7</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (2), <math display="inline"><semantics> <mrow> <mn>8</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (3), <math display="inline"><semantics> <mrow> <mn>9</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (4), <math display="inline"><semantics> <mrow> <mn>10</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (5). The frequency is normalized to the value of <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.7716</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
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<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse electric (TE) linearly polarized optical bullets and their individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. The frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
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<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse magnetic (TM) linearly polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)). The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
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<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of azimuthally polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
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<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of radially polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
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<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse electric (TE) linearly polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. The frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 7
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse magnetic (TM) linearly polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. The frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 8
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of azimuthally polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 9
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of radially polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 10
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of higher polarization order optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. The frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 11
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of higher polarization order optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. The frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 12
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse electric (TE) linearly polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)). The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 13
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse magnetic (TM) linearly polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)). The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 14
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of azimuthally polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)). The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 15
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of radially polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)). The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 16
<p>Dependencies of FWHM (<b>a</b>) and second-moment (<b>b</b>) pulse widths in BK7 glass for different values of frequency <math display="inline"><semantics> <msub> <mi>ω</mi> <mi>c</mi> </msub> </semantics></math> for linear, azimuthal and radial polarizations. The red color represents <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and the blue color <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs.</p>
Full article ">Figure 17
<p>Normalized intensities of individual components of FWM pulses (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>—blue, <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>—red, <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>—orange). Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs. For the cases: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. The solid line represents the linear polarization of TE, the dashed line represents the linear polarization of TM, the dotted line represents the azimuthal polarization, and the dashed-dotted line represents the radial polarization.</p>
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0 pages, 1913 KiB  
Article
The Helicity of Magnetic Fields Associated with Relativistic Electron Vortex Beams
by Norah Alsaawi and Vasileios E. Lembessis
Symmetry 2024, 16(4), 496; https://doi.org/10.3390/sym16040496 - 19 Apr 2024
Viewed by 920
Abstract
For radially extended Bessel modes, the helicity density distributions of magnetic fields associated with relativistic electron vortex beams are investigated for first time in the literature. The form of the distribution is defined by the electron beam’s cylindrically symmetric density flux, which varies [...] Read more.
For radially extended Bessel modes, the helicity density distributions of magnetic fields associated with relativistic electron vortex beams are investigated for first time in the literature. The form of the distribution is defined by the electron beam’s cylindrically symmetric density flux, which varies with the winding number and the electron spin. Different helicity distributions are obtained for different signs of the winding number ±, confirming the chiral nature of the magnetic fields associated with the electron vortex beam. The total current helicity for the spin-down state is smaller than that of the spin-up state. The different fields and helicities associated with opposite winding numbers and/or spin values will play an important role in the investigation of the interaction of relativistic electron vortices with matter and especially chiral matter. A comparison of the calculated quantities with the corresponding ones in the case of non-relativistic spin-polarized electron beams is performed. Full article
(This article belongs to the Section Physics)
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Figure 1

Figure 1
<p>The current helicity density of an infinite relativistic electron Bessel beam with (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>, (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>, (<b>c</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics> </math>, (<b>d</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics> </math>, and (<b>e</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>−</mo> <mn>10</mn> </mrow> </semantics> </math>. The radial oscillatory behaviour dominates at larger radial distances for a higher winding number associated with a lower current helicity value.</p>
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<p>The in-plane current helicity density distribution for an infinite relativistic electron Bessel beam with (<b>a</b>) <span class="html-italic">ℓ</span> = 0, (<b>c</b>) <span class="html-italic">ℓ</span> = 1, (<b>e</b>) <span class="html-italic">ℓ</span> = −1, (<b>g</b>) <span class="html-italic">ℓ</span> = 10, and (<b>i</b>) <span class="html-italic">ℓ</span> = −10 in the case of spin-up. The corresponding plots in the case of spin-down are illustrated for the winding number with (<b>b</b>) <span class="html-italic">ℓ</span> = 0, (<b>d</b>) <span class="html-italic">ℓ</span> = 1, (<b>f</b>) <span class="html-italic">ℓ</span> = −1, (<b>h</b>) <span class="html-italic">ℓ</span> = 10, and (<b>j</b>) <span class="html-italic">ℓ</span> = −10.</p>
Full article ">Figure 3
<p>The total current helicity for an infinite relativistic electron Bessel beam from <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>−</mo> <mn>10</mn> </mrow> </semantics> </math> to <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics> </math> in spin-up and spin-down states.</p>
Full article ">Figure 4
<p>The magnetic helicity density of an infinite relativistic electron Bessel beam with (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>, (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>, (<b>c</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics> </math>, (<b>d</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics> </math>, and (<b>e</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>−</mo> <mn>10</mn> </mrow> </semantics> </math>. The difference between the magnetic helicity densities of the two spin states is clearly demonstrated. The height of the first peak increases for higher values of <span class="html-italic">ℓ</span>.</p>
Full article ">Figure 5
<p>The in-plane magnetic helicity density distribution for an infinite relativistic electron Bessel beam with (<b>a</b>) <span class="html-italic">ℓ</span> = 0, (<b>c</b>) <span class="html-italic">ℓ</span> = 1, (<b>e</b>) <span class="html-italic">ℓ</span> = −1, (<b>g</b>) <span class="html-italic">ℓ</span> = 10, and (<b>i</b>) <span class="html-italic">ℓ</span> = −10 in the case of spin-up. The corresponding plots in the case of spin-down are illustrated for the winding number with (<b>b</b>) <span class="html-italic">ℓ</span> = 0, (<b>d</b>) <span class="html-italic">ℓ</span> = 1, (<b>f</b>) <span class="html-italic">ℓ</span> = −1, (<b>h</b>) <span class="html-italic">ℓ</span> = 10, and (<b>j</b>) <span class="html-italic">ℓ</span> = −10.</p>
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<p>The current helicity density of an infinite relativistic electron Bessel beam with (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>, (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>, and (<b>c</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics> </math> within the limit of small velocities.</p>
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17 pages, 7522 KiB  
Article
Exploring Solvation Properties of Protic Ionic Liquids by Employing Solvatochromic Dyes and Molecular Dynamics Simulation Analysis
by Stuart J. Brown, Andrew J. Christofferson, Calum J. Drummond, Qi Han and Tamar L. Greaves
Liquids 2024, 4(1), 288-304; https://doi.org/10.3390/liquids4010014 - 20 Mar 2024
Cited by 1 | Viewed by 850
Abstract
Solvation properties are key for understanding the interactions between solvents and solutes, making them critical for optimizing chemical synthesis and biochemical applications. Designable solvents for targeted optimization of these end-uses could, therefore, play a big role in the future of the relevant industries. [...] Read more.
Solvation properties are key for understanding the interactions between solvents and solutes, making them critical for optimizing chemical synthesis and biochemical applications. Designable solvents for targeted optimization of these end-uses could, therefore, play a big role in the future of the relevant industries. The tailorable nature of protic ionic liquids (PILs) as designable solvents makes them ideal candidates. By alteration of their constituent structural groups, their solvation properties can be tuned as required. The solvation properties are determined by the polar and non-polar interactions of the PIL, but they remain relatively unknown for PILs as compared to aprotic ILs and their characterization is non-trivial. Here, we use solvatochromic dyes as probe molecules to investigate the solvation properties of nine previously uncharacterized alkyl- and dialkylammonium PILs. These properties include the Kamlet–Aboud–Taft (KAT) parameters: π* (dipolarity/polarizability), α (H-bond acidity) and β (H-bond basicity), along with the ET(30) scale (electrophilicity/polarizability). We then used molecular dynamics simulations to calculate the radial distribution functions (RDF) of 21 PILs, which were correlated to their solvation properties and liquid nanostructure. It was identified that the hydroxyl groups on the PIL cation increase α, π* and ET(30), and correspondingly increase the cation–anion distance in their RDF plots. The hydroxyl group, therefore, reduces the strength of the ionic interaction but increases the polarizability of the ions. An increase in the alkyl chain length on the cation led to a decrease in the distances between cations, while also increasing the β value. The effect of the anion on the PIL solvation properties was found to be variable, with the nitrate anion greatly increasing π*, α and anion–anion distances. The research presented herein advances the understanding of PIL structure–property relationships while also showcasing the complimentary use of molecular dynamics simulations and solvatochromic analysis together. Full article
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<p>The chemical structures, names and abbreviations of (<b>a</b>) the solvatochromic dyes used for calculation of KAT parameters and the E<sub>T</sub>(30) scale, (<b>b</b>) previously uncharacterized PILs that have been characterized for their solvation properties via solvatochromic dye absorbance analysis in this study (in the red frame), and (<b>c</b>) PILs from the literature previously characterized for their solvation properties and have been analyzed in this study via MD simulations for their respective RDF plots.</p>
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<p>The calculated E<sub>T</sub>(30) values of PILs separated by anion. The experimental values determined in this study are denoted by a circle (●) and the reported values from Yalcin et al. [<a href="#B38-liquids-04-00014" class="html-bibr">38</a>] are denoted by a triangle (<span style="color:blue">▲</span>). The error bars are smaller than the symbols, and the values are provided in <a href="#liquids-04-00014-t001" class="html-table">Table 1</a>.</p>
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<p>The calculated π* values of PILs separated by anion. The experimental values determined in this study are denoted by a circle (●) and the reported values from Yalcin et al. [<a href="#B38-liquids-04-00014" class="html-bibr">38</a>] are denoted by a triangle (<span style="color:blue">▲</span>). The error bars are smaller than the symbols, and the values are provided in <a href="#liquids-04-00014-t001" class="html-table">Table 1</a>.</p>
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<p>The calculated α values of PILs separated by anion. The experimental values determined in this study are denoted by a circle (●) and the reported values from Yalcin et al. [<a href="#B38-liquids-04-00014" class="html-bibr">38</a>] are denoted by a triangle (<span style="color:blue">▲</span>). The error bars are smaller than the symbols, and the values are provided in <a href="#liquids-04-00014-t001" class="html-table">Table 1</a>.</p>
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<p>The calculated β values of PILs separated by anion. The experimental values determined in this study are denoted by a circle (●) and the reported values from Yalcin et al. [<a href="#B38-liquids-04-00014" class="html-bibr">38</a>] are denoted by a triangle (<span style="color:blue">▲</span>). The error bars are smaller than the symbols, and the values are provided in <a href="#liquids-04-00014-t001" class="html-table">Table 1</a>.</p>
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<p>The radial distribution functions of the formate series of PILs (<b>a</b>) EAF, (<b>b</b>) PAF, (<b>c</b>) BAF, (<b>d</b>) PeAF, (<b>e</b>) EtAF and (<b>f</b>) DEtAF. The RDF plots are inset with the molecular structure of each PIL, where the ions are highlighted, N<sup>+</sup> in blue with the cation subscript e.g., N<sub>EA</sub> is the N<sup>+</sup> atom on ethylammonium, O<sup>−</sup> in red with the anion subscript e.g., O<sub>F</sub> is the O<sup>−</sup> atoms of formate. Additional functional groups are highlighted in grey. The terminal carbon in the cation alkyl chain e.g., C<sub>PeA</sub> is the terminal carbon of the pentylammonium cation, and hydroxyl groups e.g., O<sub>EtA</sub> is the oxygen atom of the hydroxyl group in the ethanolammonium cation.</p>
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<p>Solvation maps developed based on the RDF plots presented in <a href="#liquids-04-00014-f006" class="html-fig">Figure 6</a> (not to scale). The effect of hydroxyl groups on interatomic distances is presented in (<b>a</b>) for EAF, EtAF and DEtAF. (<b>b</b>) The effect of the alkyl chain on N–N distances is inversely proportional to its length, where increasing alkyl chain length decreases N–N distance.</p>
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<p>The RDF plots of (<b>a</b>) PAA, (<b>b</b>) PAG, (<b>c</b>) PAL and (<b>d</b>) PAN inlayed with its molecular structure with N<sup>+</sup> ions highlighted in blue and O<sup>−</sup> ions in red. Each RDF plot (<b>left</b>) has its corresponding solvation map adjacent (<b>right</b>) with a visualization of the peaks seen in each plot. The solvation maps are not to scale.</p>
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12 pages, 732 KiB  
Article
Image Processing in L1-Norm-Based Discrete Cartesian and Polar Coordinates
by Geunmin Lee and Wonha Kim
Electronics 2024, 13(6), 1088; https://doi.org/10.3390/electronics13061088 - 15 Mar 2024
Viewed by 762
Abstract
This paper proposes a radial image processing method performed in an L1-norm-based discrete polar coordinate system. For this purpose, we address the problem that polar coordinates based on the L2-norm cannot exist in discrete systems and then develop a [...] Read more.
This paper proposes a radial image processing method performed in an L1-norm-based discrete polar coordinate system. For this purpose, we address the problem that polar coordinates based on the L2-norm cannot exist in discrete systems and then develop a method for converting Cartesian coordinates to L1-norm-based discrete polar coordinates. The proposed method greatly reduces the directional variance occurring in the Cartesian coordinate system and so processes radial directional images along the directions of the local image signal flows. To verify the usages of the proposed method, it was applied to the stabilization of mass-type breast cancer images, a segmentation of extremely deformable objects such as biomedical objects. In all cases, the proposed method produced superior results compared to the processing in the Cartesian coordinate systems. The proposed method is useful for processing or analyzing diffusing and deformable images such as bio-cell and smoke images. Full article
(This article belongs to the Section Electronic Multimedia)
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<p>Examples of radial image signals. (<b>a</b>) Mass-type breast cancer, (<b>b</b>) iris, (<b>c</b>) plume, (<b>d</b>) hippocampus.</p>
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<p><math display="inline"><semantics> <msup> <mi>Z</mi> <mn>2</mn> </msup> </semantics></math>-lattice structures of pixel areas. Distribution of pixels belonging to (<b>a</b>) circles using <math display="inline"><semantics> <msup> <mi>L</mi> <mn>2</mn> </msup> </semantics></math>-norm distance and (<b>b</b>) pyramids using <math display="inline"><semantics> <msup> <mi>L</mi> <mn>1</mn> </msup> </semantics></math>-norm distance.</p>
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<p>An example of image reconstruction using <math display="inline"><semantics> <msup> <mi>L</mi> <mn>2</mn> </msup> </semantics></math>-norm: (<b>a</b>) Original image and (<b>b</b>) image converted to polar coordinates and then converted back to Cartesian coordinates. The PSNR between two images is about 35 dB.</p>
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<p>A lattice mapping from a continuous Cartesian coordinate to an <math display="inline"><semantics> <msup> <mi>L</mi> <mn>1</mn> </msup> </semantics></math>-norm-based discrete polar coordinate.</p>
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<p>Lattice points on the line drawn at <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <msup> <mo form="prefix">tan</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mn>5</mn> <mo>/</mo> <mn>4</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mo>≃</mo> <mn>51</mn> <mo>.</mo> <msup> <mn>34</mn> <mo>∘</mo> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math> with distance <span class="html-italic">r</span> from 0 to 7. Grey areas are the lattices that each lattice point covers.</p>
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<p>An example of image reconstruction using <math display="inline"><semantics> <msup> <mi>L</mi> <mn>1</mn> </msup> </semantics></math>-norm: (<b>a</b>) original image and (<b>b</b>) image converted to the polar coordinate system and then converted back to the Cartesian coordinate system. The two images are exactly the same.</p>
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<p>Image conversion between Cartesian and polar coordinates. (<b>a</b>) Image in the discrete Cartesian coordinate. (<b>b</b>) Image in the discrete polar coordinate.</p>
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<p>Procedure of the proposed radial smoothing method. (<b>a</b>) Original mass image; (<b>b</b>) smoothed image.</p>
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<p>Comparison of smoothed mass images: (<b>a</b>) original mass image, (<b>b</b>) image smoothed by the proposed method, and (<b>c</b>) image smoothed by Gaussian filtering.</p>
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<p>Illustration of the proposed biomedical object segmentation method.</p>
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<p>Segmentation results of biomedical objects. (<b>a</b>) Bio-cell images. From the top: benign tumor, hippocampus, and spiculated cancer. (<b>b</b>) Segmentation in Cartesian coordinate; (<b>c</b>) proposed segmentation in polar coordinate.</p>
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20 pages, 7930 KiB  
Article
An Improved One-Line Evolution Formulation for the Dynamic Shoreline Planforms of Embayed Beaches
by Hung-Cheng Tao, Tai-Wen Hsu and Chia-Ming Fan
Water 2024, 16(5), 774; https://doi.org/10.3390/w16050774 - 5 Mar 2024
Viewed by 874
Abstract
In this paper, an improved one-line evolution formulation is proposed and derived for the dynamic shoreline planforms of embayed beaches. Although embayed sandy beaches can perform several functions, serving as leisure spots and areas of coastal protection, shoreline advances and retreats occur continuously [...] Read more.
In this paper, an improved one-line evolution formulation is proposed and derived for the dynamic shoreline planforms of embayed beaches. Although embayed sandy beaches can perform several functions, serving as leisure spots and areas of coastal protection, shoreline advances and retreats occur continuously as a result of many natural forces, such as winds, waves, currents, tides, etc. The one-line evolution formulation for dynamic shoreline planforms based on the polar coordinate can be adopted to simulate high-planform-curvature shorelines and achieve better stability and simplicity in comparison with other description coordinates. While the polar coordinate and rectangular control volume are adopted to derive the one-line evolution formulation for dynamic shoreline planforms, the difference between the radial direction of the polar coordinate and the normal direction of the shoreline segment may result in inaccurate predictions of shoreline movements. In this study, a correction coefficient, which can adjust the influence of these two misaligned directions, is derived and included in the one-line evolution formulation, which is based on the polar coordinate. Thus, by considering the correction coefficient, an improved one-line evolution formulation for dynamic shoreline planforms of crenulate-shaped bays is proposed in this paper. Some numerical examples are provided to verify the merits of the proposed improved one-line evolution formulation. Moreover, the proposed numerical approach is applied to simulate the dynamic movements of the shoreline in Taitung—the southeastern part of Taiwan—and the effectiveness of the proposed formulation in solving realistic engineering applications is evidently verified. Full article
(This article belongs to the Special Issue Advanced Research in Civil, Hydraulic, and Ocean Engineering)
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<p>A schematic diagram of the shoreline planform of crenulate-shaped bay.</p>
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<p>The control volume between two sequential time steps.</p>
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<p>The control volume defined in this study and its dimension.</p>
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<p>A schematic diagram of the different angles adopted in this study.</p>
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<p>A schematic of the numerical example for verification of consistency and stability.</p>
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<p>Numerical results in different specific time frames when using the proposed improved one-line model (<math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>=</mo> <mn>223</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>∆</mo> <mi mathvariant="normal">t</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>).</p>
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<p>Numerical results and comparisons with Ref. [<a href="#B14-water-16-00774" class="html-bibr">14</a>] when using different ∆t for (<b>a</b>) m = 109, (<b>b</b>) m = 179 and (<b>c</b>) m = 223.</p>
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<p>Numerical results and comparisons with Ref. [<a href="#B14-water-16-00774" class="html-bibr">14</a>] when using different ∆t for (<b>a</b>) m = 109, (<b>b</b>) m = 179 and (<b>c</b>) m = 223.</p>
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<p>Numerical results and comparison of Example 2 [<a href="#B14-water-16-00774" class="html-bibr">14</a>,<a href="#B25-water-16-00774" class="html-bibr">25</a>].</p>
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<p>Changes in the shoreline near JinZum Island in the past years (2012~2021, TWD 97).</p>
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<p>The spatial position of the nine shoreline positions over the past years.</p>
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<p>Numerical solution from EEMSE for the wave field in nearshore area.</p>
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<p>Geometric definitions of the DEP in JinZum Island.</p>
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<p>Numerical results of DEP, GenCade, Tao et al. (2022) [<a href="#B14-water-16-00774" class="html-bibr">14</a>] and the present formulation for the shoreline planform in August 2013.</p>
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<p>Comparisons of numerical solutions along the shoreline portion between the control point and the JinZum Island [<a href="#B14-water-16-00774" class="html-bibr">14</a>].</p>
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<p>Numerical comparisons for the shoreline planform in the southern coastal area of the control point [<a href="#B14-water-16-00774" class="html-bibr">14</a>].</p>
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11 pages, 829 KiB  
Article
Search for Wormhole Candidates: Accreting Wormholes with Monopole Magnetic Fields
by Mikhail Piotrovich, Serguei Krasnikov, Stanislava Buliga and Tinatin Natsvlishvili
Universe 2024, 10(3), 108; https://doi.org/10.3390/universe10030108 - 27 Feb 2024
Cited by 1 | Viewed by 1120
Abstract
The existence of even the simplest magnetized wormholes may lead to observable consequences. In the case where both the wormhole and the magnetic field around its mouths are static and spherically symmetric, and gas in the region near the wormhole falls radially into [...] Read more.
The existence of even the simplest magnetized wormholes may lead to observable consequences. In the case where both the wormhole and the magnetic field around its mouths are static and spherically symmetric, and gas in the region near the wormhole falls radially into it, the former’s spectrum contains bright cyclotron or synchrotron lines due to the interaction of charged plasma particles with the magnetic field. At the same time, due to spherical symmetry, the radiation is non-polarized. The emission of this just-described exotic type (non-thermal, but non-polarized) may be a wormhole signature. Also, in this scenario, the formation of an accretion disk is still quite possible at some distance from the wormhole, but a monopole magnetic field could complicate this process and lead to the emergence of asymmetrical and one-sided relativistic jets. Full article
(This article belongs to the Special Issue The Friedmann Cosmology: A Century Later)
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<p>Trajectories (black line) of a proton near a point-like gravitating object (black sphere) in the fictional space with the mass of the Sun and a monopole magnetic field for different starting parameter values. <math display="inline"><semantics> <msub> <mi>B</mi> <mn>10</mn> </msub> </semantics></math> is the magnetic field strength at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> km, and <math display="inline"><semantics> <msub> <mi>R</mi> <mi>st</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>V</mi> <mi>st</mi> </msub> </semantics></math> are the starting position and speed of a particle in km and km/s, respectively.</p>
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