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10 pages, 816 KiB  
Article
Theoretical Investigation on Vortex Electron Impact Excitation of a Mg Atom Confined in a Solid-State Environment
by Sophia Strnat, Aloka K. Sahoo, Lalita Sharma, Jonas Sommerfeldt, Daesung Park, Christian Bick and Andrey Surzhykov
Atoms 2025, 13(3), 23; https://doi.org/10.3390/atoms13030023 - 24 Feb 2025
Viewed by 238
Abstract
We present a theoretical investigation of the inelastic scattering of vortex electrons by many-electron atoms embedded in a solid-state environment. Special emphasis is placed on the probability of exciting a target atom and on the relative population of its magnetic substates as described [...] Read more.
We present a theoretical investigation of the inelastic scattering of vortex electrons by many-electron atoms embedded in a solid-state environment. Special emphasis is placed on the probability of exciting a target atom and on the relative population of its magnetic substates as described by the set of alignment parameters. These parameters are directly related to the angular distribution of the subsequent radiative decay. To demonstrate the application of the developed theoretical approach, we present calculations for the 3s2 S013s3p P13 excitation of a Mg atom and its subsequent 3s3p P133s2 S01 radiative decay. Our results highlight the significance of the orbital angular momentum (OAM) projection as well as the relative position of the vortex electron with respect to the target atom. Full article
(This article belongs to the Special Issue 21st International Conference on the Physics of Highly Charged Ions)
Show Figures

Figure 1

Figure 1
<p>The geometry of inelastic scattering of a vortex electron beam by a target Mg atom. The magnesium atom is confined within a fullerene molecule. The incident electron has a well-defined longitudinal momentum <math display="inline"><semantics> <msub> <mi>p</mi> <mi>z</mi> </msub> </semantics></math> and projection of orbital angular momentum <math display="inline"><semantics> <msub> <mi>m</mi> <mo>ℓ</mo> </msub> </semantics></math> on the propagation axis. This axis is shifted from the coordinate origin (position of the target atom) by the vector <math display="inline"><semantics> <mi mathvariant="bold-italic">b</mi> </semantics></math>. The photon detector is positioned at the polar angle <math display="inline"><semantics> <mi>θ</mi> </semantics></math> and azimuthal angle <math display="inline"><semantics> <mi>φ</mi> </semantics></math>.</p>
Full article ">Figure 2
<p>(<b>Upper row</b>): Relative total excitation probability of the <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>S</mi> <mn>0</mn> <none/> <mprescripts/> <none/> <mn>1</mn> </mmultiscripts> <mo>→</mo> <mmultiscripts> <mi>P</mi> <mn>1</mn> <none/> <mprescripts/> <none/> <mn>3</mn> </mmultiscripts> </mrow> </semantics></math> excitation of a Mg atom as a function of the impact parameter <span class="html-italic">b</span>. The incident vortex electron has a kinetic energy of 20 eV, an opening angle <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mi mathvariant="bold-italic">p</mi> </msub> <mo>=</mo> <msup> <mn>15</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and a spin projection <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>. (<b>Lower row</b>): Alignment parameter <math display="inline"><semantics> <msub> <mi>A</mi> <mn>20</mn> </msub> </semantics></math> of the excited atomic state, again as a function of <span class="html-italic">b</span> in units of Bohr radii <math display="inline"><semantics> <msub> <mi>a</mi> <mn>0</mn> </msub> </semantics></math>. The columns correspond to the scattering of a free Mg atom and Mg confined within a fullerene molecule. The different curves represent the OAM projections <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mo>ℓ</mo> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (red dashed line), <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mo>ℓ</mo> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (blue dash-dotted line), and <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mo>ℓ</mo> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (orange dotted line), while the non-vortex values are displayed as black solid lines as references.</p>
Full article ">Figure 3
<p>Angular distribution of the radiative decay of the <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>P</mi> <mn>1</mn> <none/> <mprescripts/> <none/> <mn>3</mn> </mmultiscripts> <mo>→</mo> <mmultiscripts> <mi>S</mi> <mn>0</mn> <none/> <mprescripts/> <none/> <mn>1</mn> </mmultiscripts> </mrow> </semantics></math> transition in confined Mg. The polar angle is fixed to <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <msup> <mn>120</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> and the <math display="inline"><semantics> <mi>φ</mi> </semantics></math> dependence of the emitted photons is shown. The incident vortex electron has a kinetic energy of 20 eV, an opening angle <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mi mathvariant="bold-italic">p</mi> </msub> <mo>=</mo> <msup> <mn>15</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and a TAM projection <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mrow> <mi>T</mi> <mi>A</mi> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>. The curves belong to different impact parameters: <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (red solid line), <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>1</mn> <mspace width="0.166667em"/> <msub> <mi>a</mi> <mn>0</mn> </msub> </mrow> </semantics></math> (blue dashed line), and <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mspace width="0.166667em"/> <msub> <mi>a</mi> <mn>0</mn> </msub> </mrow> </semantics></math> (orange dotted line).</p>
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14 pages, 7661 KiB  
Article
Single Scattering Dynamics of Vector Bessel–Gaussian Beams in Winter Haze Conditions
by Yixiang Yang, Yuancong Cao, Wenjie Jiang, Lixin Guo and Mingjian Cheng
Photonics 2025, 12(3), 182; https://doi.org/10.3390/photonics12030182 - 22 Feb 2025
Viewed by 229
Abstract
This study investigates the scattering dynamics of vector Bessel–Gaussian (BG) beams in winter haze environments, with a particular emphasis on the influence of ice-coated haze particles on light propagation. Employing the Generalized Lorenz–Mie Theory (GLMT), we analyze the scattering coefficients of particles transitioning [...] Read more.
This study investigates the scattering dynamics of vector Bessel–Gaussian (BG) beams in winter haze environments, with a particular emphasis on the influence of ice-coated haze particles on light propagation. Employing the Generalized Lorenz–Mie Theory (GLMT), we analyze the scattering coefficients of particles transitioning from water to ice coatings under varying atmospheric conditions. Our results demonstrate that the presence of ice coatings significantly alters the scattering and extinction efficiencies of BG beams, revealing distinct differences compared to particles coated with water. Furthermore, the study examines the role of Orbital Angular Momentum (OAM) modes in shaping scattering behavior. We show that higher OAM modes, characterized by broader energy distributions and larger beam spot sizes, induce weaker localized interactions with individual particles, leading to diminished scattering and attenuation. In contrast, lower OAM modes, with energy concentrated in smaller regions, exhibit stronger interactions with particles, thereby enhancing scattering and attenuation. These findings align with the Beer–Lambert law in the single scattering regime, where beam intensity attenuation is influenced by the spatial distribution of radiation, while overall power attenuation follows the standard exponential decay with respect to propagation distance. The transmission attenuation of BG beams through haze-laden atmospheres is further explored, emphasizing the critical roles of particle concentration and humidity. This study provides valuable insights into the interactions between vector BG beams and atmospheric haze, advancing the understanding of optical communication and environmental monitoring in hazy conditions. Full article
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Figure 1

Figure 1
<p>(<b>a</b>) Transmission schematic of BG beams in haze environment. (<b>b</b>) Ice-coated haze particles corrected model and schematic of BG beam particle scattering.</p>
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<p>RCS of various haze particles with ice and water coatings under X-polarized BG beam irradiation: (<b>a</b>) meteoric-type particles, (<b>b</b>) dust-like particles, (<b>c</b>) soot particles, and (<b>d</b>) sulfate particles.</p>
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<p>RCS of soot-type haze particles with ice and water coatings under linearly polarized BG beam irradiation featuring OAM modes <span class="html-italic">s</span> = 1 and <span class="html-italic">s</span> = 3: (<b>a</b>) <span class="html-italic">λ</span> = 0.86 μm, (<b>b</b>) <span class="html-italic">λ</span> = 1.06 μm, (<b>c</b>) <span class="html-italic">λ</span> = 2.0 μm, (<b>d</b>) <span class="html-italic">λ</span> = 5 μm.</p>
Full article ">Figure 4
<p>RCS of soot-type haze particles with ice and water coatings for various inner-to-outer diameter ratios (<span class="html-italic">a</span>/<span class="html-italic">b</span> = 0.7 and <span class="html-italic">a</span>/<span class="html-italic">b</span> = 0.9) under vector BG beam irradiation with different polarization states: (<b>a</b>) X-polarized, (<b>b</b>) Y-polarized, (<b>c</b>) left circularly polarized, (<b>d</b>) right circularly polarized, (<b>e</b>) azimuthally polarized, and (<b>f</b>) radially polarized.</p>
Full article ">Figure 5
<p>Efficiency factors of soot-type haze particles with ice and water coatings under X-polarized BG beam irradiation at wavelengths <span class="html-italic">λ</span> = 0.86 μm and <span class="html-italic">λ</span> = 5.0 μm: (<b>a</b>) absorption efficiency factor, (<b>b</b>) scattering efficiency factor, and (<b>c</b>) extinction efficiency factor.</p>
Full article ">Figure 6
<p>Transmittance of X-polarized BG beam at wavelengths λ = 0.86 μm (<b>a</b>), λ = 1.06 μm (<b>b</b>), λ = 2.0 μm (<b>c</b>), and λ = 5.0 μm (<b>d</b>) in soot-type haze environments with varying concentrations <span class="html-italic">N</span> = 5000, 10,000, 15,000, 20,000 cm<sup>−3</sup>.</p>
Full article ">Figure 7
<p>Transmittance of X-polarized BG beam with OAM modes <span class="html-italic">m</span> = 0 (<b>a</b>), <span class="html-italic">m</span> = 1 (<b>b</b>), <span class="html-italic">m</span> = 2 (<b>c</b>), and <span class="html-italic">m</span> = 3 (<b>d</b>) in haze environments containing varying components: meteoric, dust-type, soot, and sulfate particles.</p>
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19 pages, 8765 KiB  
Article
Spatial Multiplexing Holography for Multi-User Visible Light Communication
by Chaoxu Chen, Yuan Wei, Haoyu Zhang, Ziyi Zhuang, Ziwei Li, Chao Shen, Junwen Zhang, Haiwen Cai, Nan Chi and Jianyang Shi
Photonics 2025, 12(2), 160; https://doi.org/10.3390/photonics12020160 - 17 Feb 2025
Viewed by 194
Abstract
Given the burgeoning necessity for high-speed, efficient, and secure wireless communication in 6G, visible light communication (VLC) has emerged as a fervent subject of discourse within academic and industrial circles alike. Among these considerations, it is imperative to construct scalable multi-user VLC systems, [...] Read more.
Given the burgeoning necessity for high-speed, efficient, and secure wireless communication in 6G, visible light communication (VLC) has emerged as a fervent subject of discourse within academic and industrial circles alike. Among these considerations, it is imperative to construct scalable multi-user VLC systems, meticulously addressing pivotal issues such as power dissipation, alignment errors, and the safeguarding of user privacy. However, traditional methods like multiplexing holography (MPH) and multiple focal (MF) phase plates have shown limitations in meeting these diverse requirements. Here, we propose a novel spatial multiplexing holography (SMH) theory, a comprehensive solution that overcomes existing hurdles by enabling precise power allocation, self-designed power coverage, and secure communication through orbital angular momentum (OAM). The transformative potential of SMH is demonstrated through simulations and experimental studies, showcasing its effectiveness in power distribution within systems of VR glasses users, computer users, and smartphone users; enhancing power coverage with an 11.6 dB improvement at coverage edges; and securing data transmission, evidenced by error-free 1080P video playback under correct OAM keys. Our findings illustrate the superior performance of SMH in facilitating seamless multi-user communication, thereby establishing a new benchmark for future VLC systems in the 6G landscape. Full article
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Figure 1

Figure 1
<p>(<b>a</b>) Scenario of multi-user system in VLC; (<b>b</b>) Sketch of power allocation; (<b>c</b>) Sketch of power coverage; (<b>d</b>) Sketch of OAM secure communication.</p>
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<p>Application of power allocation holography in projection.</p>
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<p>(<b>a</b>) Process of generating SMH for power allocation; (<b>b</b>) Process of generating SMH for power coverage; (<b>c</b>) Process of generating SMH for OAM secure communication.</p>
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<p>Schematic diagram of precise transmission space sampling. (<b>a</b>) Transmission process without pre-control; (<b>b</b>) Transmission process with pre-control.</p>
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<p>(<b>a</b>) Simulation results for power allocation; (<b>b</b>) Convergence line of GA; (<b>c</b>) Simulation results for power coverage under MF method; (<b>d</b>) Simulation results for power coverage under SMH.</p>
Full article ">Figure 6
<p>(<b>a</b>) Simulation of optical field at receiver plane under SMH and conventional method; (<b>b</b>) Power trend of user encoded with OAM modes <span class="html-italic">l</span> = 20 under different OAM keys; (<b>c</b>) Simulation results for OAM secret communication.</p>
Full article ">Figure 7
<p>Experimental setup. (<b>a</b>) Experiment platform; (<b>b</b>) Signal processing.</p>
Full article ">Figure 8
<p>Experimental results for power allocation. (<b>a</b>) Data rate curve with ROP; (<b>b</b>) One frame of 4K video received under MPH and SMH; (<b>c</b>) BER of each row in the first frame; (<b>d</b>) Mean BER of each second of the video; (<b>e</b>) Working point test for User 1; (<b>f</b>) Working point test for User 2; (<b>g</b>) Working point test for User 3.</p>
Full article ">Figure 9
<p>Experimental results for power coverage. (<b>a</b>) User 1’s ROP trend with XoY offset; (<b>b</b>) User 2’s ROP trend with XoY offset; (<b>c</b>) User 3’s ROP trend with XoY offset; (<b>d</b>) Coverage area diameter of 3 users under MF and SMH; (<b>e</b>) Data rate trend with XoY offset of three users; (<b>f</b>) CCD recording at User 1’s plate under different methods; (<b>g</b>) CCD recording at User 2’s plate under different methods; (<b>h</b>) CCD recording at User 3’s plate under different methods; (<b>i</b>) MSE and SSIM of recordings under different methods.</p>
Full article ">Figure 10
<p>Experimental results for OAM secure communication. (<b>a</b>) ROP of three users with different OAM keys; (<b>b</b>) Data rate of three users with different OAM keys; (<b>c</b>) CCD recording of three users; (<b>d</b>) One frame of a 1080P video received by User 3 under different OAM keys.</p>
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25 pages, 9252 KiB  
Article
Extensions of the Variational Method with an Explicit Energy Functional for Nuclear Matter with Spin-Orbit Force
by Kento Kitanaka, Toshiya Osuka, Tetsu Sato, Hayate Ichikawa and Masatoshi Takano
Particles 2025, 8(1), 11; https://doi.org/10.3390/particles8010011 - 7 Feb 2025
Viewed by 363
Abstract
Two extensions of the variational method with explicit energy functionals (EEFs) with respect to the spin-orbit force were performed. In this method, the energy per nucleon of nuclear matter is explicitly expressed as a functional of various two-body distribution functions, starting from realistic [...] Read more.
Two extensions of the variational method with explicit energy functionals (EEFs) with respect to the spin-orbit force were performed. In this method, the energy per nucleon of nuclear matter is explicitly expressed as a functional of various two-body distribution functions, starting from realistic nuclear forces. The energy was then minimized by solving the Euler–Lagrange equation for the distribution functions derived from the EEF. In the first extension, an EEF of symmetric nuclear matter at zero temperature was constructed using the two-body central, tensor, and spin-orbit nuclear forces. The energy per nucleon calculated using the Argonne v8’ two-body nuclear potential was found to be lower than those calculated using other many-body methods, implying that the energy contribution caused by the spin-orbit correlation, whose relative orbital angular momentum operator acts on other correlations, is necessary. In a subsequent extension, the EEF of neutron matter at zero temperature, including the spin-orbit force, was extended to neutron matter at finite temperatures using the method by Schmidt and Pandharipande. The thermodynamic quantities of neutron matter calculated using the Argonne v8’ nuclear potential were found to be reasonable and self-consistent. Full article
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Figure 1

Figure 1
<p>Energy per nucleon of SNM with the Argonne v8’ potential as a function of the density <math display="inline"><semantics> <mi>ρ</mi> </semantics></math>. The results obtained with the BHF, SCGF, BBG, and FHNC methods are also shown in black, dark green, purple, and light green lines, respectively. The open red circles represent the solutions where the Mayer condition was violated, while the filled red circles represent the case where the Mayer condition was satisfied. The open blue squares represent the case using the Argonne v6’ potential. The small red square represents the empirical saturation point [<a href="#B26-particles-08-00011" class="html-bibr">26</a>].</p>
Full article ">Figure 2
<p>Spin-isospin-dependent radial distribution functions <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> as functions of the distance between nucleons <span class="html-italic">r</span> at <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> </mrow> </semantics></math> 0.08, 0.56, and 0.96 <math display="inline"><semantics> <mrow> <mspace width="0.166667em"/> <msup> <mi>fm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>Isospin-dependent tensor distribution functions <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi mathvariant="normal">T</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and spin-orbit distribution functions <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>SO</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> as functions of the distance between nucleons <span class="html-italic">r</span> at <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> = 0.08, 0.56, and 0.96 <math display="inline"><semantics> <mrow> <mspace width="0.166667em"/> <msup> <mi>fm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Structure functions <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mi mathvariant="normal">c</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> as functions of the wave number <span class="html-italic">k</span> at various densities <math display="inline"><semantics> <mi>ρ</mi> </semantics></math>.</p>
Full article ">Figure 5
<p>Tensor structure functions <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mi>cT</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> as functions of the wave number <span class="html-italic">k</span>.</p>
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<p>Free energies per neutron <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>/</mo> <mi>N</mi> </mrow> </semantics></math> of PNM as a function of the neutron number density <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> with the Argonne v8’ potential. The black, blue, green, and red lines represent <span class="html-italic">T</span> = 0, 10, 20, and 30 MeV, respectively.</p>
Full article ">Figure 7
<p>Approximate entropies per neutron <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mn>0</mn> </msub> <mo>/</mo> <mi>N</mi> </mrow> </semantics></math> of PNM as a function of the neutron number density <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> with the Argonne v8’ potential. Entropies per neutron <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>/</mo> <mi>N</mi> </mrow> </semantics></math> derived from the free energy per neutron <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>/</mo> <mi>N</mi> </mrow> </semantics></math> by the thermodynamic relation are also shown. Solid lines represent the entropies, while dashed lines represent the approximated entropies. The black, blue, and red lines represent the cases of <span class="html-italic">T</span> = 10, 20, and 30 MeV, respectively.</p>
Full article ">Figure 8
<p>Approximate internal energy per neutron <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mrow> <mi mathvariant="normal">T</mi> <mn>0</mn> </mrow> </msub> <mo>/</mo> <mi>N</mi> </mrow> </semantics></math> of PNM as a function of the neutron number density <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> with the Argonne v8’ potential. The internal energy per neutron <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi mathvariant="normal">T</mi> </msub> <mo>/</mo> <mi>N</mi> </mrow> </semantics></math>, derived from the free energy per neutron <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>/</mo> <mi>N</mi> </mrow> </semantics></math> by the thermodynamic relation, is also shown. The solid lines represent the internal energies, while the dashed lines represent the approximated internal energies. The black, blue, and red lines represent the cases of <span class="html-italic">T</span> = 10, 20, and 30 MeV, respectively.</p>
Full article ">Figure 9
<p>Neutron effective masses normalized by the bare neutron mass as a function of the neutron number density <math display="inline"><semantics> <mi>ρ</mi> </semantics></math>. The blue, green, and red lines represent the results at <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> </mrow> </semantics></math> 10, 20, and 30 MeV, respectively.</p>
Full article ">Figure 10
<p>Spin-dependent radial distribution functions <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for PNM at zero (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> MeV) and finite temperatures (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> MeV). The results at <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.08</mn> <mspace width="0.166667em"/> <msup> <mi>fm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.6</mn> <mspace width="0.166667em"/> <msup> <mi>fm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> are shown.</p>
Full article ">Figure 11
<p>Tensor distribution functions <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mi mathvariant="normal">T</mi> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and spin-orbit distribution functions <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mi>SO</mi> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for PNM at zero (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> MeV) and finite temperatures (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> MeV). The results at <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> = 0.08 and <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> = 0.6 <math display="inline"><semantics> <msup> <mi>fm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math> are shown.</p>
Full article ">Figure 12
<p>Structure functions <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mi mathvariant="normal">c</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for PNM at zero (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> MeV) and finite temperatures (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> MeV). The results at <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.08</mn> <mspace width="0.166667em"/> <msup> <mi>fm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.6</mn> <mspace width="0.166667em"/> <msup> <mi>fm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> are shown.</p>
Full article ">Figure 13
<p>Structure functions <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mi>cT</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for PNM at zero (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> MeV) and finite temperatures (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> MeV). The results at <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.08</mn> <mspace width="0.166667em"/> <msup> <mi>fm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.6</mn> <mspace width="0.166667em"/> <msup> <mi>fm</mi> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> are shown.</p>
Full article ">Figure A1
<p>Energy per neutron of PNM with the Argonne v8’ potential as a function of the density <math display="inline"><semantics> <mi>ρ</mi> </semantics></math>. Also shown is the energy per neutron of PNM with the Argonne v8’ potential and the repulsive part of the Urbana IX three-body nuclear potential. They were compared with the results using the AFDMC method. The blue open circles represents the energy per nucleon of PNM with the Argonne v6’ potential [<a href="#B22-particles-08-00011" class="html-bibr">22</a>].</p>
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18 pages, 852 KiB  
Article
Non-Keplerian Charged Accretion Disk Orbiting a Black Hole Pulsar
by Audrey Trova and Eva Hackmann
Universe 2025, 11(2), 45; https://doi.org/10.3390/universe11020045 - 1 Feb 2025
Viewed by 365
Abstract
Recent studies have focused on how spinning black holes (BHs) within a binary system containing a strongly magnetized neutron star, then immersed in external magnetic fields, can acquire charge through mechanisms like the Wald process and how this charge could power pulsar-like electromagnetic [...] Read more.
Recent studies have focused on how spinning black holes (BHs) within a binary system containing a strongly magnetized neutron star, then immersed in external magnetic fields, can acquire charge through mechanisms like the Wald process and how this charge could power pulsar-like electromagnetic radiation. Those objects called “Black hole pulsar” mimic the behaviour of a traditional pulsar, and they can generate electromagnetic fields, such as magnetic dipoles. Charged particles within an accretion disk around the black hole would then be influenced not only by the gravitational forces but also by electromagnetic forces, leading to different geometries and dynamics. In this context, we focus here on the interplay of the magnetic dipole and the accretion disk. We construct the equilibrium structures of non-conducting charged perfect fluids orbiting Kerr black holes under the influence of a dipole magnetic field aligned with the rotation axis of the BH. The dynamics of the accretion disk in such a system are shaped by a complex interplay between the non-uniform, non-Keplerian angular momentum distribution, the black hole’s induced magnetic dipole, and the fluid’s charge. We show how these factors jointly influence key properties of the disk, such as its geometry, aspect ratio, size, and rest mass density. Full article
(This article belongs to the Special Issue Universe: Feature Papers 2024 – Compact Objects)
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Figure 1

Figure 1
<p>Variation in the ISCO (<b>left panel</b>) and the disk’s inner edge (<b>right panel</b>) with respect to the magnetic/charge parameter <math display="inline"><semantics> <mi>μ</mi> </semantics></math> for various values of spin parameter <span class="html-italic">a</span>. The blue crosses show the values of the ISCO in the neutral case (<math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>).</p>
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<p>Contour maps of the variation in the radius of the center of the disk with respect to the angular momentum parameter <math display="inline"><semantics> <mi>β</mi> </semantics></math> and the magnetic/charge parameter <math display="inline"><semantics> <mi>μ</mi> </semantics></math> for prograde and retrograde spin.</p>
Full article ">Figure 3
<p>Shape of the charged accretion disk, defined as the zero-pressure surface for various combinations of the parameters. The (<b>leftmost</b>) panel shows the <math display="inline"><semantics> <msub> <mi>C</mi> <mi>a</mi> </msub> </semantics></math> configuration. The (<b>middle</b>) panel presents the <math display="inline"><semantics> <msub> <mi>C</mi> <mi>β</mi> </msub> </semantics></math> one. Finally, the (<b>rightmost</b>) panel shows the <math display="inline"><semantics> <msub> <mi>C</mi> <mi>δ</mi> </msub> </semantics></math> configuration.</p>
Full article ">Figure 4
<p>Same as <a href="#universe-11-00045-f003" class="html-fig">Figure 3</a> but for configurations <math display="inline"><semantics> <msub> <mi>C</mi> <mrow> <mi>μ</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <msub> <mi>C</mi> <mrow> <mi>μ</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </semantics></math> (<b>right</b>).</p>
Full article ">Figure 5
<p>Variation in the <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>r</mi> </mrow> </semantics></math> ratio for the various parameters of the model. Configurations <math display="inline"><semantics> <msub> <mi>C</mi> <mi>a</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>C</mi> <mi>β</mi> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>C</mi> <mi>δ</mi> </msub> </semantics></math> are depicted on the (<b>left</b>). The (<b>right</b>) panel presents configurations <math display="inline"><semantics> <msub> <mi>C</mi> <mrow> <mi>μ</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>C</mi> <mrow> <mi>μ</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </semantics></math> together.</p>
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<p>Same as <a href="#universe-11-00045-f006" class="html-fig">Figure 6</a> but for the disk vertical scale height to radius.</p>
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<p>Density distribution in the entire space for some notable combination of the parameters.</p>
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<p>Variation in the density distribution with <span class="html-italic">r</span> for configuration <math display="inline"><semantics> <msub> <mi>C</mi> <mi>a</mi> </msub> </semantics></math> (<b>leftmost plot</b>), for negative <math display="inline"><semantics> <mi>μ</mi> </semantics></math> in the (<b>middle plot</b>), and for positive <math display="inline"><semantics> <mi>μ</mi> </semantics></math> in the (<b>rightmost plot</b>).</p>
Full article ">Figure 9
<p>Variation in the density distribution, normalized to <math display="inline"><semantics> <msub> <mi>ρ</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </semantics></math>, with respect to <span class="html-italic">r</span> for configurations <math display="inline"><semantics> <msub> <mi>C</mi> <mi>β</mi> </msub> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <msub> <mi>C</mi> <mi>δ</mi> </msub> </semantics></math> (<b>right</b>).</p>
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7 pages, 1822 KiB  
Communication
Grating Pair Wavepacket Shaper for Crafting Spatiotemporal Optical Vortices with Arbitrary Tilt Angles
by Jordan Adams and Andy Chong
Photonics 2025, 12(2), 126; https://doi.org/10.3390/photonics12020126 - 31 Jan 2025
Viewed by 399
Abstract
Spatiotemporal optical vortices with arbitrary tilt angles can be generated by adjusting spatial chirp and beam size at a phase modulation plane in a pulse shaper setup. A grating pair setup is proposed to generate variable spatial chirp independent of the beam profile. [...] Read more.
Spatiotemporal optical vortices with arbitrary tilt angles can be generated by adjusting spatial chirp and beam size at a phase modulation plane in a pulse shaper setup. A grating pair setup is proposed to generate variable spatial chirp independent of the beam profile. The initial dispersion of the pulse allows for the independent control of the vortex orientation. By adjusting the beam size, spatial chirp, and initial dispersion, arbitrary vortex orientation across all the possible angles can be achieved. The ability to achieve arbitrary vortex orientations at long propagation distances could offer significant advantages for long-distance communication applications. Full article
(This article belongs to the Special Issue Progress in OAM Beams: Recent Innovations and Future Perspectives)
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Figure 1

Figure 1
<p>A grating pair enables high spatial chirp across the SLM plane without spatial focusing. This generates a tilted vortex outside the grating pair shaper.</p>
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<p>Vortex orientation of different spatial chirps plotted over variable <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>λ</mi> <mi>z</mi> </mrow> <mrow> <mi>π</mi> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mrow> <mi>o</mi> </mrow> </msubsup> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> values (<b>a</b>) for <math display="inline"><semantics> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>l</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>, where the dashed lines are constant <math display="inline"><semantics> <mrow> <mi>r</mi> </mrow> </semantics></math> values. The hemisphere with blue and red coloring on the top right shows the relative orientation of the positive and negative topological charges. Iso-intensity plots showing (<b>c</b>–<b>e</b>) entire wavepacket and (<b>f</b>–<b>h</b>) internal vortex in a fixed volume. The inset on (<b>f</b>–<b>h</b>) shows an alternative perspective.</p>
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<p>Vortex orientation at <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math> for different spatial chirps plotted over variable initial dispersions <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>D</mi> </mrow> <mrow> <mi>o</mi> </mrow> </msub> </mrow> </semantics></math> with a (<b>a</b>) top view and (<b>b</b>) side view. (<b>c</b>–<b>e</b>) The iso-intensity plots of vortex inside wavepacket for different initial dispersion values at <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> mm/THz.</p>
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<p>Telescope systems for increasing beam size and controlling vortex orientation. (<b>a</b>) Two positive focal length lenses that switch the signs of the <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="normal">V</mi> </mrow> <mrow> <mi mathvariant="normal">x</mi> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="normal">V</mi> </mrow> <mrow> <mi mathvariant="normal">y</mi> </mrow> </msub> </mrow> </semantics></math> components. (<b>b</b>) A combination of negative and positive focal length lenses. (<b>c</b>) Vortex orientation at <math display="inline"><semantics> <mrow> <mi mathvariant="normal">z</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> km for various magnifications <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="normal">f</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>/</mo> <msub> <mrow> <mi mathvariant="normal">f</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> with an input beam size of <math display="inline"><semantics> <mrow> <mn>1</mn> </mrow> </semantics></math> mm and spatial chirp <math display="inline"><semantics> <mrow> <mi mathvariant="normal">s</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> mm/THz is given with the black line and a few magnifications are highlighted with markers.</p>
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21 pages, 15250 KiB  
Review
Plasmonic Vortices: A Promising Tool Utilizing Plasmonic Orbital Angular Momentum
by Zhi Gao, Dmitri V. Voronine and Alexei V. Sokolov
Photonics 2025, 12(2), 125; https://doi.org/10.3390/photonics12020125 - 31 Jan 2025
Viewed by 607
Abstract
An optical vortex (OV) beam is an important type of spatially structured beam. However, the diffraction limit for light with orbital angular momentum (OAM) remains a challenge for certain applications. Surface plasmon polaritons (SPPs) can confine light to nanoscale dimensions and enhance light–matter [...] Read more.
An optical vortex (OV) beam is an important type of spatially structured beam. However, the diffraction limit for light with orbital angular momentum (OAM) remains a challenge for certain applications. Surface plasmon polaritons (SPPs) can confine light to nanoscale dimensions and enhance light–matter interactions. Over the past two decades, researchers have begun to explore the imparting of OAM onto SPPs to generate plasmonic vortices (PVs). Since the discovery of PVs, significant efforts have been made in this field, leading to considerable progress. This article reviews these studies in three key areas: (a) the generation and manipulation of PVs, (b) the characterization of PVs, and (c) the application of PVs. We believe that PVs represent a promising tool utilizing plasmonic OAM for both fundamental research and practical applications and hold great potential for the future with continued dedicated efforts. Full article
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Figure 1

Figure 1
<p>Schematic of OAM beams with different topological charge <math display="inline"><semantics> <mrow> <mi mathvariant="script">l</mi> </mrow> </semantics></math> values. They exhibit different wavefronts, intensity distributions, and phase profiles. Reprinted from [<a href="#B2-photonics-12-00125" class="html-bibr">2</a>]. Creative Commons Attribution (CC BY) license.</p>
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<p>Schematic of a nanolens and its field confinement for OAM light. Adapted with permission from [<a href="#B35-photonics-12-00125" class="html-bibr">35</a>]. © 2022 Astro Ltd.</p>
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<p>Archimedes spirals and modified ASs. (<b>a</b>) Schematic of an AS and the excitation of PVs. Reprinted from [<a href="#B37-photonics-12-00125" class="html-bibr">37</a>]. Copyright © 2016, The Author(s). Creative Commons Attribution 4.0 International License. (<b>b</b>) Schematic of a segmented spiral metasurface, whose rotation angle of the slits is <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mfenced separators="|"> <mrow> <mi>θ</mi> </mrow> </mfenced> <mo>=</mo> <mi>q</mi> <mi>θ</mi> <mo>+</mo> <msub> <mrow> <mi>α</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> and the radius of the spiral is <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mi>n</mi> <mi>i</mi> <mi>t</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mo> </mo> <mi>r</mi> <mi>a</mi> <mi>d</mi> <mi>i</mi> <mi>u</mi> <mi>s</mi> <mo>+</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>s</mi> <mi>p</mi> <mi>p</mi> </mrow> </msub> <mo>×</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>m</mi> <mi>o</mi> <mi>d</mi> <mo>(</mo> <mi>m</mi> <mi>θ</mi> <mo>,</mo> <mo> </mo> <mn>2</mn> <mi>π</mi> <mo>)</mo> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mstyle> </mrow> </semantics></math>, where <math display="inline"><semantics> <mrow> <mi>θ</mi> </mrow> </semantics></math> is the azimuthal angle, <math display="inline"><semantics> <mrow> <mi>q</mi> </mrow> </semantics></math> is the rotation order of the slits, <math display="inline"><semantics> <mrow> <mi>m</mi> </mrow> </semantics></math> is the geometric order of the spiral, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>α</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> is the initial angle of the slits, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>s</mi> <mi>p</mi> <mi>p</mi> </mrow> </msub> </mrow> </semantics></math> is the SPP wavelength, and <math display="inline"><semantics> <mrow> <mi>m</mi> <mi>o</mi> <mi>d</mi> <mo>(</mo> <mi>m</mi> <mi>θ</mi> <mo>,</mo> <mo> </mo> <mn>2</mn> <mi>π</mi> <mo>)</mo> </mrow> </semantics></math> represents the remainder of the division of <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>m</mi> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mstyle> </mrow> </semantics></math>. (<b>c</b>) Near-field intensity measurements for the metasurface samples A–H interacting with right circularly polarized light, varying three parameters: <math display="inline"><semantics> <mrow> <mi>q</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>α</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math>. Adapted with permission from [<a href="#B32-photonics-12-00125" class="html-bibr">32</a>]. © 2019 WILEY-VCH Verlag GmbH &amp; Co. KGaA, Weinheim.</p>
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<p>Schematic of various plasmonic vortex generation structures. (<b>a</b>) A six-long-spiral metaparticle and four snapshots showing the real-time evolution of the near-field distribution of the <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math> component, excited by a linearly polarized plane wave at 8.5 GHz. Adapted with permission from [<a href="#B42-photonics-12-00125" class="html-bibr">42</a>]. © 2021 Wiley-VCH GmbH. (<b>b</b>) A metasurface of ring-shaped paired resonators. Reprinted with permission from [<a href="#B43-photonics-12-00125" class="html-bibr">43</a>]. © 2019 WILEY-VCH Verlag GmbH &amp; Co. KGaA, Weinheim. (<b>c</b>) A meta-atom comprising four rectangular slits arranged along a circular contour. Reprinted from [<a href="#B44-photonics-12-00125" class="html-bibr">44</a>]. © 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement. (<b>d</b>) A metal–insulator–metal structure featuring a circular array of elliptical holes embedded in the bottom gold film and a central circular hole in the top gold film, along with its normalized electric field distribution in the XY plane and along the X-axis. Adapted from [<a href="#B46-photonics-12-00125" class="html-bibr">46</a>]. Copyright © 2023, The Author(s). Creative Commons CC BY license.</p>
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<p>Spatiotemporal dynamics of plasmonic vortices. (<b>a</b>) Revolution stages of a vortex, from its generation to after the first and second reflections at the boundary. The bottom lines show azimuthal and radial fitting of the diameter and the number of lobes in the main vortex signal. Adapted from [<a href="#B48-photonics-12-00125" class="html-bibr">48</a>]. Copyright © 2021, The Authors, some rights reserved. Creative Commons Attribution License 4.0 (CC BY). (<b>b</b>) Schematic of distinct generation and evolution behaviors of vortices with the same topological charge, produced by different couplers. Reprinted from [<a href="#B49-photonics-12-00125" class="html-bibr">49</a>]. © The Author(s) 2023. Creative Commons Attribution 4.0 International License.</p>
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<p>Plasmonic vortex detection. (<b>a</b>) Structure of a plasmonic lens with four nanorods at its center. The “on” and “off” states of these rods under the SNOM scan indicate the OAM states of PVs. Adapted with permission from [<a href="#B58-photonics-12-00125" class="html-bibr">58</a>]. © 2014 Optical Society of America. (<b>b</b>) Schemes of measuring dispersion in PVs. The cathodoluminescence of PVs, generated by the interaction of a fast electron beam of a scanning transmission electron microscope, enables the simultaneous measurement of their phase and amplitude across a broad spectral range. Adapted from [<a href="#B59-photonics-12-00125" class="html-bibr">59</a>]. © The Author(s), 2019. Creative Commons Attribution 4.0 International License.</p>
Full article ">Figure 6 Cont.
<p>Plasmonic vortex detection. (<b>a</b>) Structure of a plasmonic lens with four nanorods at its center. The “on” and “off” states of these rods under the SNOM scan indicate the OAM states of PVs. Adapted with permission from [<a href="#B58-photonics-12-00125" class="html-bibr">58</a>]. © 2014 Optical Society of America. (<b>b</b>) Schemes of measuring dispersion in PVs. The cathodoluminescence of PVs, generated by the interaction of a fast electron beam of a scanning transmission electron microscope, enables the simultaneous measurement of their phase and amplitude across a broad spectral range. Adapted from [<a href="#B59-photonics-12-00125" class="html-bibr">59</a>]. © The Author(s), 2019. Creative Commons Attribution 4.0 International License.</p>
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<p>Dynamics detection of PVs. (<b>a</b>) Schematic of time-resolved, two-photon photoemission electron microscopy for detecting the revolution stage of PVs within a single optical cycle of ~2.67 fs. Reprinted with permission from [<a href="#B60-photonics-12-00125" class="html-bibr">60</a>]. © 2017, American Association for the Advancement of Science. (<b>b</b>) Simulated and experimental results showing spin–orbit mixing of light with PVs. Reprinted from [<a href="#B61-photonics-12-00125" class="html-bibr">61</a>]. Creative Commons Attribution 4.0 International License. (<b>c</b>) Attosecond-resolved videos depicting the spatial evolution of vortex fields, captured using ultrafast nonlinear coherent photoelectron microscopy (top) and simulated (bottom). The color scale represents photoelectron counts. The white scale bars are SPP wavelength, 530 nm. Adapted with permission from [<a href="#B62-photonics-12-00125" class="html-bibr">62</a>]. © The Author(s), under exclusive license to Springer Nature Limited 2020.</p>
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<p>Chirality detection using optical or plasmonic vortices. (<b>a</b>) Helical dichroism through optical vortices interacting with enantiomers via strong electric quadrupole fields. Reprinted with permission from [<a href="#B67-photonics-12-00125" class="html-bibr">67</a>]. Copyright © 2016, The Authors. (<b>b</b>) Helical dichroism in fenchone using linearly polarized light, where <math display="inline"><semantics> <mrow> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> represents Gaussian beams and <math display="inline"><semantics> <mrow> <mi>l</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> </mrow> </semantics></math> represents vortex beams. Reprinted with permission from [<a href="#B68-photonics-12-00125" class="html-bibr">68</a>]. © The Author(s), under exclusive license to Springer Nature Limited 2022. (<b>c</b>) Chirality detection with plasmonic vortices, showing that like-chirality nanospirals exhibit stronger cathodoluminescence intensities than unlike-chirality nanospirals. Reprinted with permission from [<a href="#B69-photonics-12-00125" class="html-bibr">69</a>]. © 2017 Optical Society of America.</p>
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<p>SNOM images of a (<b>a</b>) left-handed and (<b>b</b>) right-handed single Archimedean spiral slot (LHS) under left-handed circular (LHC) and right-handed circular (RHC) polarization illumination, showing that a specific spiral has different focusing behaviors for two circular polarizations. Reprinted with permission from [<a href="#B71-photonics-12-00125" class="html-bibr">71</a>]. Copyright © 2010 American Chemical Society.</p>
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<p>Generation of optical OAM. (<b>a</b>) Structure of a tapered gold tip surrounded by a spiral slit for generating optical vortices. The right two |E| maps in the middle row show that the same tip with different basis curvature radii <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>c</mi> </mrow> </msub> </mrow> </semantics></math> has different far-field coupling effects. The |E| maps at the bottom show the far-field coupling effect of the same tip for PVs with different topological charges <math display="inline"><semantics> <mrow> <mi>l</mi> </mrow> </semantics></math>. Reprinted from [<a href="#B76-photonics-12-00125" class="html-bibr">76</a>]. © 2016 American Chemical Society. Creative Commons license CC BY-NC-ND 4.0. (<b>b</b>) Metal–insulator–metal holey plasmonic vortex lens used to generate optical vortices. Reprinted from [<a href="#B80-photonics-12-00125" class="html-bibr">80</a>]. Copyright © 2016, The Author(s). Creative Commons CC BY license.</p>
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<p>Readout of optical OAM. (<b>a</b>) Positions and sizes of PVs in an elliptical AS reveal both the spin and orbital angular momentum of light. Reprinted from [<a href="#B82-photonics-12-00125" class="html-bibr">82</a>]. IEEE Open Access Publishing Agreement (OAPA), 1943-0655 © 2017 IEEE. (<b>b</b>) Semi-ring plasmonic nanoslits capable of focusing various OAM modes of light onto spatially distinct positions. The right figure depicts intensity profiles along the white dashed lines in the left figure. Adapted with permission from [<a href="#B84-photonics-12-00125" class="html-bibr">84</a>]. © The Royal Society of Chemistry 2016. (<b>c</b>) Circular plasmonic lens that utilizes interference between plasmon and transmitted light to identify OAM states of light, causing different intensity distributions, shown experimentally (top) and computationally (bottom). Reprinted with permission from [<a href="#B83-photonics-12-00125" class="html-bibr">83</a>]. Copyright © 2013, The Author(s).</p>
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<p>Plasmonic vortex interferometers (PVIs). Interferometers PVI-1 and PVI-2 have different interferograms under the left-handed and right-handed circular polarization incidences. Adapted with permission from [<a href="#B85-photonics-12-00125" class="html-bibr">85</a>]. © 2022 Wiley-VCH GmbH.</p>
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<p>Plasmonic and optical spanner. (<b>a</b>) Schematic of the spanner setup and its conceptually operational principles, capable of stably trapping and dynamically rotating particles. (<b>b</b>) Video recordings illustrating the movement of gold particles in the OV field (top two rows) and in the PV field (bottom row). The black arrows denote motion direction of particles. Reprinted from [<a href="#B87-photonics-12-00125" class="html-bibr">87</a>]. Copyright © 2015, The Author(s). Creative Commons Attribution 4.0 International License.</p>
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<p>Plasmonic and optical spanner. (<b>a</b>) Schematic of the spanner setup and its conceptually operational principles, capable of stably trapping and dynamically rotating particles. (<b>b</b>) Video recordings illustrating the movement of gold particles in the OV field (top two rows) and in the PV field (bottom row). The black arrows denote motion direction of particles. Reprinted from [<a href="#B87-photonics-12-00125" class="html-bibr">87</a>]. Copyright © 2015, The Author(s). Creative Commons Attribution 4.0 International License.</p>
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<p>Free-electron spatial amplitude modulation. (<b>a</b>) Conceptual illustration of the modulation mechanism. (<b>b</b>) The spatial modulation of free electrons can be adjusted via altering the SPP field boundary conditions by changing the incident laser polarization. The upper scale bar, 10 <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>; the lower scale bar, 0.5 <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>. Reprinted with permission from [<a href="#B93-photonics-12-00125" class="html-bibr">93</a>]. Copyright © 2023, The Author(s), under exclusive license to Springer Nature Limited.</p>
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12 pages, 784 KiB  
Article
Thermal Profile of Accretion Disk Around Black Hole in 4D Einstein–Gauss–Bonnet Gravity
by Odilbek Kholmuminov, Bakhtiyor Narzilloev and Bobomurat Ahmedov
Universe 2025, 11(2), 38; https://doi.org/10.3390/universe11020038 - 26 Jan 2025
Viewed by 495
Abstract
In this study, we investigate the properties of a thin accretion disk around a static spherically symmetric black hole in 4D Einstein–Gauss–Bonnet gravity, with an additional coupling constant, α, appearing in the spacetime metric. Using the Novikov–Thorne accretion disk model, we examine [...] Read more.
In this study, we investigate the properties of a thin accretion disk around a static spherically symmetric black hole in 4D Einstein–Gauss–Bonnet gravity, with an additional coupling constant, α, appearing in the spacetime metric. Using the Novikov–Thorne accretion disk model, we examine the thermal properties of the disk, finding that increasing α reduces the energy, angular momentum, and effective potential of a test particle orbiting the black hole. We demonstrate that α can mimic the spin of a Kerr black hole in general relativity up to a 0.23 M for the maximum value of α. Our analysis of the thermal radiation flux shows that larger α values increase the flux and shift its maximum towards the central black hole, while far from the black hole, the solution recovers the Schwarzschild limit. The impact of α on the radiative efficiency of the disk is weak but can slightly alter it. Assuming black-body radiation, we observe that the disk’s temperature peaks near its inner edge and is higher for larger α values. Lastly, the electromagnetic spectra reveal that the disk’s luminosity is lower in Einstein–Gauss–Bonnet gravity compared to general relativity, with the peak luminosity shifting toward higher frequencies, corresponding to the soft X-ray band as α increases. Full article
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<p>Radial variation of the metric function for the chosen values of the Gauss–Bonnet coupling constant.</p>
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<p>Variation of the event horizon radius (represented by the black solid line) and the re-scaled Innermost Stable Circular Orbit (ISCO) radius (red dashed line) on the Gauss–Bonnet coupling constant (see also [<a href="#B34-universe-11-00038" class="html-bibr">34</a>]).</p>
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<p>Radial dependence of <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>L</mi> <mo>,</mo> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mrow> </semantics></math> for selected values of Gauss–Bonnet coupling constant.</p>
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<p>The degeneracy between the Gauss–Bonnet coupling constant <math display="inline"><semantics> <mi>α</mi> </semantics></math> and the black hole rotation parameter <span class="html-italic">a</span> as reflected in the ISCO location.</p>
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<p>The radial distribution of the electromagnetic radiation flux emitted by the disk for various values of <math display="inline"><semantics> <mi>α</mi> </semantics></math>.</p>
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<p>Continuous increase in radiation efficiency from disk as <math display="inline"><semantics> <mi>α</mi> </semantics></math> increases.</p>
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<p>Temperature distribution across the disk for various values of the spacetime parameter <math display="inline"><semantics> <mi>α</mi> </semantics></math>. On the left side, the temperature is expressed in Kelvin (K), and on the right, in energy units (erg).</p>
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<p>Temperature distribution in density-plot format. On right, X and Y represent Cartesian coordinates, while X-Y plane is situated on equatorial plane.</p>
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<p>Spectral properties of the accretion disk surrounding a static black hole in EGB gravity. The inclination angle is set to be <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>. The graph is displayed on a standard scale near the region where the lines reach their maxima.</p>
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9 pages, 4080 KiB  
Article
Giant Vortex Dichroism in Simplified-Chiral-Double-Elliptical Metamaterials
by Shiqi Luo, Kangzhun Peng, Zhi-Yuan Li and Wenyao Liang
Nanomaterials 2025, 15(3), 189; https://doi.org/10.3390/nano15030189 - 25 Jan 2025
Viewed by 466
Abstract
Vortex dichroism in chiral metamaterials is of great significance to the study of photoelectric detection, optical communication, and the interaction between light and matter. Here we propose a compact chiral metamaterials structure composed of two elliptical SiO2 rods covered with a Au [...] Read more.
Vortex dichroism in chiral metamaterials is of great significance to the study of photoelectric detection, optical communication, and the interaction between light and matter. Here we propose a compact chiral metamaterials structure composed of two elliptical SiO2 rods covered with a Au film on a substrate to achieve a significant vortex-dichroism effect. Such a structure has different responses to a Laguerre-Gaussian beam carrying opposite-orbital angular momentum, resulting in giant vortex dichroism. The influences of various structural parameters are analyzed, and the optimal parameters are obtained to realize a remarkable vortex dichroism of about 58.5%. The simplicity and giant VD effect of the proposed metamaterials make it a promising candidate for advancing chiral optical applications such as optical communication and sensing. Full article
(This article belongs to the Special Issue Nanoscale Photonics and Metamaterials)
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<p>(<b>a</b>) Schematic diagram for Laguerre-Gaussian beams with LHW and RHW incident on the designed CMM; (<b>b</b>) intensity distributions of the vortex beams with <span class="html-italic">l</span> = ±10; and (<b>c</b>) phase distributions of the vortex beams with <span class="html-italic">l</span> = ±10.</p>
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<p>(<b>a</b>) Normalized reflection spectra of LHW and RHW for the designed CMM; (<b>b</b>) the corresponding VD curve.</p>
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<p>Electric field distributions measured at different locations: (<b>a</b>) schematic of the detector positioned within the Au layer, with two detection planes labeled as P<sub>1</sub> and P<sub>2</sub>; (<b>b</b>,<b>c</b>) field distributions corresponding to <span class="html-italic">l</span> of +12 and −12 at P<sub>1</sub>, respectively; and (<b>d</b>,<b>e</b>) field distributions corresponding to <span class="html-italic">l</span> of +12 and −12 at P<sub>2</sub>, respectively.</p>
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<p>VD peak curve (red) and the corresponding topological charge curve |<span class="html-italic">l</span>| (orange) obtained by adjusting different geometric parameters. (<b>a</b>) The case of changing major axis of the elliptical rod; (<b>b</b>) the case of changing <span class="html-italic">ρ</span>; (<b>c</b>) the case of changing the coordinate positions of the two elliptical rods; and (<b>d</b>) the case of changing the rotation angle.</p>
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<p>(<b>a</b>) Top view of designed CMM; (<b>b</b>) top view of the enantiomer of designed CMM; (<b>c</b>) top view of achiral metamaterials; and (<b>d</b>) VD response of three metamaterials.</p>
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16 pages, 323 KiB  
Article
Derivation of Meson Masses in SU(3) and SU(4) Extended Linear Sigma Model at Finite Temperature
by Abdel Nasser Tawfik, Azar I. Ahmadov, Alexandra Friesen, Yuriy Kalinovsky, Alexey Aparin and Mahmoud Hanafy
Particles 2025, 8(1), 9; https://doi.org/10.3390/particles8010009 - 22 Jan 2025
Viewed by 600
Abstract
The present study focused on the mesonic potential contributions to the Lagrangian of the extended linear sigma model (eLSM) for scalar and pseudoscalar meson fields across various quark flavors. The present study focused on the low-energy phenomenology associated with quantum chromodynamics (QCD), where [...] Read more.
The present study focused on the mesonic potential contributions to the Lagrangian of the extended linear sigma model (eLSM) for scalar and pseudoscalar meson fields across various quark flavors. The present study focused on the low-energy phenomenology associated with quantum chromodynamics (QCD), where mesons and their interactions serve as the pertinent degrees of freedom, rather than the fundamental constituents of quarks and gluons. Given that SU(4) configurations are completely based on SU(3) configurations, the possible relationships between meson states in SU(3) and those in SU(4) were explored at finite temperature. Meson states, which are defined by distinct chiral properties, were grouped according to their orbital angular momentum J, parity P, and charge conjugation C. Consequently, this organization yielded scalar mesons with quantum numbers JPC=0++, pseudoscalar mesons with JPC=0+, vector mesons with JPC=1, and axial vector mesons with JPC=1++. We accomplished the derivation of analytical expressions for a total of seventeen noncharmed meson states and twenty-nine charmed meson states so that an analytical comparison of the noncharmed and charmed meson states at different temperatures became feasible and the SU(3) and SU(4) configurations could be analytically estimated. Full article
(This article belongs to the Special Issue Infinite and Finite Nuclear Matter (INFINUM))
37 pages, 4504 KiB  
Review
Singularities in Computational Optics
by S. Deepa, Kedar Khare and Senthilkumaran Paramasivam
Photonics 2025, 12(2), 96; https://doi.org/10.3390/photonics12020096 - 22 Jan 2025
Viewed by 633
Abstract
Phase singularities in optical fields are associated with a non-vanishing curl component of phase gradients. Huygen’s diverging spherical wavefronts that primary/secondary point sources emit, during propagation, a have zero curl component. Therefore, the propagation of waves that contain phase singularities exhibits new exciting [...] Read more.
Phase singularities in optical fields are associated with a non-vanishing curl component of phase gradients. Huygen’s diverging spherical wavefronts that primary/secondary point sources emit, during propagation, a have zero curl component. Therefore, the propagation of waves that contain phase singularities exhibits new exciting features. Their effect is also felt in computational optics. These singularities provide orbital angular momentum and robustness to beams and remove degeneracies in interferometry and diffractive optics. Recently, the improvisations in a variety of computation algorithms have resulted in the vortices leaving their footprint in fast-expanding realms such as diffractive optics design, multiplexing, signal processing, communication, imaging and microscopy, holography, biological fields, deep learning, and ptychography. This review aims at giving a gist of the advancements that have been reported in multiple fields to enable readers to understand the significance of the singularities in computation optics. Full article
(This article belongs to the Special Issue Structured Light Beams: Science and Applications)
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<p>(<b>a</b>) Intensity object for the CGH. (<b>b</b>) Numerically simulated reconstructed object from the CGH that is phase-randomized. Phase randomization increases the redundancy at the cost of speckles. Therefore, diffusers that improve the redundancy without introducing speckles are synthesized by iterative procedures.</p>
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<p>(<b>a</b>) Schematic setup for the reconstruction of CGH and (<b>b</b>) phase retrieval setup. In CGH, the reconstructed object that appears at the location C is an intensity object. To have redundancy in CGH, the intensity object at C is provided random phase, and the required complex field at the hologram plane is computed through IFTA. With the computed field, CGH is made. However, in (<b>b</b>), for the measured intensity in the detection plane D, a random phase is added as an initial guess, and the unknown phase that is received by the lens is computed (phase retrieved) through IFTA.</p>
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<p>Simulateddata for phase imaging using spiral phase diversity: (<b>a</b>,<b>b</b>) Fourier intensity data corresponding to “boat” object for plane wave and vortex illumination, (<b>c</b>) reconstruction of “boat” object, (<b>d</b>,<b>e</b>) Fourier intensity data corresponding to aspheric phase object for plane wave and vortex illumination, and (<b>f</b>) reconstruction of aspheric phase.</p>
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<p>Illustration of vortex illumination-based phase retrieval for “A” phase object using HIO algorithm. (<b>a</b>) Phase retrieval using HIO method where Fourier magnitude data are recorded with plane wave illumination. The phase recovery clearly shows twin stagnation. (<b>b</b>) Phase retrieval using HIO method where Fourier magnitude data are recorded with charge 1 vortex illumination. The phase recovery does not show any twin stagnation. (<b>c</b>) Phase reconstruction after subtracting out the extra vortex phase in (<b>b</b>). The circular region shows the support window used.</p>
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<p>A rawand improved underwater image using a vortex optimization algorithm. Reprinted with permission from the authors of [<a href="#B73-photonics-12-00096" class="html-bibr">73</a>].</p>
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<p>Mode purity as a function of the number of iterations for GSA and HIOA: (<b>a</b>) intensity distribution of the distorted signal beam before compensation; (<b>b</b>–<b>d</b>) intensity distributions of the corrected signal beam with HIOA-based AO compensation in the cases of 20, 50, and 100 iterations, respectively; and (<b>e</b>) intensity distribution in the case of 100 iterations with GSA-based AO compensation. Reprinted with permission from [<a href="#B74-photonics-12-00096" class="html-bibr">74</a>] © Optical Society of America.</p>
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<p>HHD of a random vortex field added to a diverging spherical beam: (<b>a</b>) transverse phase profile. (<b>b</b>) phase gradient field lines of the beam superimposed on the phase profile, (<b>c</b>) flow lines of the solenoidal component of the Hodge decomposed field, and (<b>d</b>) irrotational component with diverging field lines. This is the vortex free field. Reprinted with permission from [<a href="#B49-photonics-12-00096" class="html-bibr">49</a>] © Optical Society of America.</p>
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<p>The edge enhancement reconstructions from the holograms by using vortex-BP (Spiral) and amp-vortex BP (LG, Bessel, Airy, and Sinc). The contrast C and intersecting lines are shown in the results. Reprinted with permission from [<a href="#B125-photonics-12-00096" class="html-bibr">125</a>] © Optica Publishing Group.</p>
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<p>Simulated results of an input Chinese dragon pattern shown in (<b>a</b>). The outputs shown in figures (<b>b</b>–<b>d</b>) correspond to on-axis vortex filtering, off-axis vortex filtering, and polychromatic filtering, respectively. (<b>c1</b>–<b>d2</b>) are two orthogonal polarized components of (<b>c</b>,<b>d</b>). Insets on the upper-right corner of each figure of (<b>c1</b>–<b>d2</b>) illustrate the polarization direction of the analyzer. Reprinted with permission from [<a href="#B120-photonics-12-00096" class="html-bibr">120</a>] Copyright IOP publishing.</p>
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<p>Experimental result of the enhancement of an onion cell, which is a complex phase amplitude object (PAO) (<b>a</b>) without the Q-plate, (<b>b</b>) VVF image, (<b>c</b>) SVF image, and (<b>d</b>,<b>e</b>) surface plot of the phase edge of a wrapped portion (<b>b</b>,<b>c</b>). Reprinted with permission from [<a href="#B139-photonics-12-00096" class="html-bibr">139</a>] © Optica Publishing Group.</p>
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<p>Resolution enhancement experimental results. (<b>A</b>–<b>C</b>) Confocal image, cFED image (r = 0.65), FED image (r = 0.65) of nanoparticles. (<b>D</b>,<b>E</b>) Intensity profiles of the dashed and solid white lines in (<b>A</b>–<b>C</b>), respectively. All scales are the same as (<b>A</b>). Pixel size of 20 nm and dwell time of 6 <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>s. (<b>F</b>–<b>H</b>) Confocal image, cFED image (r = 0.6), FED image (r = 0.65) of microtubules. (<b>I</b>) FRC (Fourier ring correlation) of (<b>F</b>–<b>H</b>). Note: r is a subtraction factor. Reprinted with permission from [<a href="#B142-photonics-12-00096" class="html-bibr">142</a>] Copyright: John Wiley &amp; Sons, Inc.</p>
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<p>Vortex beams carrying phase singularities in nanophotonics. (<b>A</b>) Schematic illustration of vortex beams with helical wavefronts and unlimited topological charge. (<b>B</b>) Metasurfaces with a geometric phase provide phase singularities for generating vortex beams. (<b>C</b>) Vortex beams extracted in unidirectional WGM microcavities. (<b>D</b>) Phase singularities in momentum space generated by photonic crystal slabs. (<b>E</b>) Planar Archimedean nanostructures’ excited phase singularities in spatiotemporal domain. (<b>F</b>) On-chip discrimination of OAM beams by metallic gratings. (<b>G</b>) Nanoring slits for both SAM and OAM multiplexing of broadband light. (<b>H</b>) Photocurrent measurements for discriminating OAM of vortex beams. (<b>I</b>) Transmission of OAM beams in on-chip waveguides. From [<a href="#B148-photonics-12-00096" class="html-bibr">148</a>]. Reprinted with permission from AAAS.</p>
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<p>Illustration of vortex diversity incoherent imaging: (<b>a</b>) test object and blurred versions of the test object with (<b>b</b>) open aperture and (<b>c</b>) vortex aperture and (<b>d</b>) reconstructed image using (<b>b</b>,<b>c</b>) as data and generalized Wiener filter. The insets in (<b>b</b>,<b>c</b>) show blurring point spread functions (PSFs), and the inset in (<b>d</b>) shows effective computational PSF due to generalized Wiener filter processing. The reconstruction (<b>d</b>) shows significant contrast enhancement with respect to the diffraction-limited open aperture image in (<b>b</b>).</p>
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<p>The trajectory of the pseudo phase singularities inside the fugu’s body at different instants of time (<b>a</b>) t = 0.067 s, (<b>b</b>) t = 0.699 s, (<b>c</b>) t =1.665 s, and (<b>d</b>) t = 3.33 s. The locations for the positive and negative pseudophase singularities have been indicated by red and green points, respectively. Reprinted with permission from [<a href="#B151-photonics-12-00096" class="html-bibr">151</a>] © Optical Society of America.</p>
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<p>Recorded images (<b>a</b>,<b>b</b>) of the floating tea leaves on the water surface at different instants of time separated by 8 ms and the corresponding L–G signals (<b>c</b>,<b>d</b>) with positive and negative pseudo phase singularities indicated by red and green dots. Reprinted with permission from the authors of [<a href="#B154-photonics-12-00096" class="html-bibr">154</a>].</p>
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<p>The experimental results for phase-only object. (<b>a</b>) Target phase map. (<b>b</b>) The single-shot captured triple images via a metasurface. (<b>c</b>) The reconstructed intensity image based on the TTIE algorithm. (<b>d</b>) The reconstructed phase image based on the TTIE algorithm. (<b>e</b>) The second target phase map. The phase values are marked, unit: rad. (<b>f</b>) The second single-shot captured triple images via a metasurface. (<b>g</b>) The reconstructed intensity picture based on the TTIE algorithm. (<b>h</b>) The reconstructed phase image based on the TTIE algorithm. Reprinted from [<a href="#B174-photonics-12-00096" class="html-bibr">174</a>] with the permission of AIP Publishing.</p>
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<p>OAM multiplexing transmission principle based on multimode fiber. Reprinted from [<a href="#B180-photonics-12-00096" class="html-bibr">180</a>], Copyright (2024), with permission from Elsevier.</p>
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<p>Schematic of the vortex modulation method. (<b>a</b>) Phase and amplitude of the complex-amplitude object and its Fourier spectrum, which are shown in grayscale images with yellow hue; (<b>b</b>) vortex phase with topological charge l; (<b>c</b>) the complex-amplitude object modulated with the vortex phase and its Fourier spectrum with topological modulation. Reprinted with permission from [<a href="#B189-photonics-12-00096" class="html-bibr">189</a>] copyright Nature.</p>
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<p>Robust and reliable ptychographic imaging of highly periodic structures. (<b>a</b>) Schematic showing HHG ptychographic imaging of a periodic structure using conventional Gaussian–HHG illumination. The resulting diffraction orders are isolated (see zoomed-in green circle), where the white circles indicate the edges of each diffraction order. This leads to a complete loss of the relative phase information between the orders in the far-field diffraction, which subsequently leads to the failure of the ptychographic reconstruction in (<b>b</b>). (<b>c</b>) OAM-HHG illumination intrinsically has a larger source divergence and a ring-shaped intensity profile to support the overlap and interference between diffraction orders (see zoomed-in blue circle), in which the yellow circles indicate the edges of each diffraction order. This interference converts the relative phase between the diffraction orders into measurable intensity modulation, enabling fast and robust ptychographic reconstruction of the 2D periodic structure in (<b>d</b>). In (<b>b</b>,<b>d</b>), the complex-valued amplitude and phase are plotted as brightness and hue, respectively. Reprinted with permission from [<a href="#B198-photonics-12-00096" class="html-bibr">198</a>] © Optica Publishing Group.</p>
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<p>(<b>a</b>–<b>g</b>) Ptychographic reconstructions of the Siemens star for <span class="html-italic">ℓ</span> = 0, ±1, ±2, and ±3. For each panel, the main illumination amplitude, the main illumination phase, and the object amplitude are shown. Reprinted with permission from [<a href="#B199-photonics-12-00096" class="html-bibr">199</a>] © Optica Publishing Group.</p>
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<p>(<b>a</b>–<b>d</b>) is phase representation of radial Hilbert mask with the topological charge l = 10, 14, 17, and 20. (<b>e</b>) represents the phase pattern of plane of phase with some specific parameters (given in [<a href="#B207-photonics-12-00096" class="html-bibr">207</a>]) and (<b>f</b>–<b>i</b>) are sum of phase pattern of RHT and plane of phase (a + e = f, b + e = g, c + e = h and d + e = i). Reprinted from [<a href="#B207-photonics-12-00096" class="html-bibr">207</a>] with permission from Taylor and Francis Ltd.</p>
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<p>Simulation results: (<b>a</b>) plaintext, (<b>b</b>) first encryption key, (<b>c</b>) second encryption key, (<b>d</b>) first decryption key, (<b>e</b>) second decryption key, (<b>f</b>) PVD (<b>g</b>) I(00), (<b>h</b>) I(450), (<b>i</b>) I(900), (<b>j</b>) I(1350), (<b>k</b>) Stokes’ parameter S1 (final decryption key), (<b>l</b>) Stokes’ parameter S2 (final encrypted image), and (<b>m</b>) retrieved image obtained after applying all correct keys. Reprinted from [<a href="#B214-photonics-12-00096" class="html-bibr">214</a>], Copyright (2024), with permission from Elsevier.</p>
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<p>Simulation and experimental reconstruction results of the T-n-encrypted OAM multiplexed holography. (<b>a</b>) Schematic diagram of the hologram design process based on PEHC-OAM selectivity. (<b>b</b>–<b>e</b>) Simulation reconstruction results. (<b>b1</b>–<b>e1</b>) Experimental reconstruction results. Reprinted from [<a href="#B215-photonics-12-00096" class="html-bibr">215</a>], Copyright (2024), with permission from Elsevier.</p>
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<p>Schematic diagram of a chiral vortex holographic encryption. Reprinted with permission from [<a href="#B216-photonics-12-00096" class="html-bibr">216</a>] Copyright 2024 American Chemical Society.</p>
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<p>(<b>a</b>) The coding rule between grayscale values of image pixels and both the l and m recombination of CVVBs; (<b>b</b>) an 8-bit grayscale cat image transmission application. Reprinted from [<a href="#B220-photonics-12-00096" class="html-bibr">220</a>], Copyright (2024), with permission from Elsevier.</p>
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12 pages, 2296 KiB  
Article
Effects of Homogeneous Doping on Electron–Phonon Coupling in SrTiO3
by Minwoo Park and Suk Bum Chung
Nanomaterials 2025, 15(2), 137; https://doi.org/10.3390/nano15020137 - 17 Jan 2025
Viewed by 622
Abstract
Bulk n-type SrTiO3 (STO) has long been known to possess a superconducting ground state at an exceptionally dilute carrier density. This has raised questions about the applicability of the BCS-Eliashberg paradigm with its underlying adiabatic assumption. However, recent experimental reports have set [...] Read more.
Bulk n-type SrTiO3 (STO) has long been known to possess a superconducting ground state at an exceptionally dilute carrier density. This has raised questions about the applicability of the BCS-Eliashberg paradigm with its underlying adiabatic assumption. However, recent experimental reports have set the pairing gap to the critical temperature (Tc) ratio at the BCS value for superconductivity in Nb-doped STO, even though the adiabaticity condition the BCS pairing requires is satisfied over the entire superconducting dome only by the lowest branch of optical phonons. In spite of the strong implications these reports have on specifying the pairing glue, they have not proved sufficient in explaining the magnitude of the optimal doping. This motivated us to apply density functional theory to Nb-doped STO to analyze how the phonon band structures and the electron–phonon coupling evolve with doping. To describe the very low doping concentration, we tuned the homogeneous background charge, from which we obtained a first-principles result on the doping-dependent phonon frequency that is in good agreement with experimental data for Nb-doped STO. Using the EPW code, we obtain the doping-dependent phonon dispersion and the electron–phonon coupling strength. Within the framework of our calculation, we found that the electron–phonon coupling forms a dome in a doping range lower than the experimentally observed superconducting dome of the Nb-doped STO. Additionally, we examined the doping dependence of both the orbital angular momentum quenching in the electron–phonon coupling and the phonon displacement correlation length and found the former to have a strong correlation with our electron–phonon coupling in the overdoped region. Full article
(This article belongs to the Special Issue Low-Dimensional Perovskite Materials and Devices)
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<p>Doping dependence of the polar soft mode energy at <math display="inline"><semantics> <mo>Γ</mo> </semantics></math> (black dots) and the Fermi energy (red dots). The black dotted curve shows the <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>ω</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>γ</mi> <mi>n</mi> </msub> <mi>n</mi> </mrow> </semantics></math> fitting and the red dotted curve <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi>F</mi> </msub> <mo>=</mo> <msup> <mo>ℏ</mo> <mn>2</mn> </msup> <msup> <mrow> <mo>(</mo> <mn>3</mn> <msup> <mi>π</mi> <mn>2</mn> </msup> <mi>n</mi> <mo>)</mo> </mrow> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mo>/</mo> <mn>2</mn> <msup> <mi>m</mi> <mo>*</mo> </msup> </mrow> </semantics></math>, with <math display="inline"><semantics> <msup> <mi>m</mi> <mo>*</mo> </msup> </semantics></math> being approximately 1.65 times the free electron mass.</p>
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<p>Superconducting critical temperatures and pairing gaps as a function of doping concentration for Nb-doped STO from Ref. [<a href="#B13-nanomaterials-15-00137" class="html-bibr">13</a>]. The top plot shows the resistive transitions observed by Yoon et al. [<a href="#B13-nanomaterials-15-00137" class="html-bibr">13</a>], Koonce et al. [<a href="#B20-nanomaterials-15-00137" class="html-bibr">20</a>], Thiemann et al. [<a href="#B22-nanomaterials-15-00137" class="html-bibr">22</a>], Lin et al. [<a href="#B41-nanomaterials-15-00137" class="html-bibr">41</a>], and Collignon et al. [<a href="#B42-nanomaterials-15-00137" class="html-bibr">42</a>], while the horizontal dashed red line of the bottom plot indicates the BCS value 2<math display="inline"><semantics> <msub> <mo>Δ</mo> <mn>0</mn> </msub> </semantics></math>/k<sub>B</sub>T<sub>c</sub> = 3.53.</p>
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<p>Phonon-mode-resolved electron–phonon coupling <math display="inline"><semantics> <msub> <mi>λ</mi> <mrow> <mi mathvariant="bold">q</mi> <mo>,</mo> <mi>ν</mi> </mrow> </msub> </semantics></math> along the high-symmetry lines of the first BZ at doping concentrations ranging from 0.0001 e/u.c. to 0.02e/u.c., i.e., approximately 1.7 ×<math display="inline"><semantics> <msup> <mn>10</mn> <mn>18</mn> </msup> </semantics></math> cm<sup>−3</sup> to 3.4 ×<math display="inline"><semantics> <msup> <mn>10</mn> <mn>20</mn> </msup> </semantics></math> cm<sup>−3</sup>, plotted on top of phonon frequencies (drawn as narrow black curves); note that the frequency 1 cm<sup>−1</sup> corresponds to approximately 0.124 meV. All plots are on the same scale and share the same colorbar for the dimensionless <math display="inline"><semantics> <msub> <mi>λ</mi> <mrow> <mi mathvariant="bold">q</mi> <mo>,</mo> <mi>ν</mi> </mrow> </msub> </semantics></math>. The insets in the top right corners of each subplot show a zoomed-in part of the area around <math display="inline"><semantics> <mo>Γ</mo> </semantics></math> towards the X, M and R points.</p>
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<p>The doping evolution of the total electron–phonon coupling strength <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <msub> <mo>∑</mo> <mrow> <mi mathvariant="bold">q</mi> <mo>,</mo> <mi>ν</mi> </mrow> </msub> <msub> <mi>λ</mi> <mrow> <mi mathvariant="bold">q</mi> <mo>,</mo> <mi>ν</mi> </mrow> </msub> </mrow> </semantics></math>, with the summation carried out over all phonon modes and the entire 1st BZ.</p>
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<p>The BCS <math display="inline"><semantics> <mi>λ</mi> </semantics></math> that includes the orbital angular momentum quenching effect, marked in red, is compared to the <math display="inline"><semantics> <mi>λ</mi> </semantics></math> obtained from our first-principles calculation.</p>
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<p>The doping evolution of <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>F</mi> </msub> <msub> <mo>ℓ</mo> <mn>0</mn> </msub> </mrow> </semantics></math>, where <math display="inline"><semantics> <mrow> <msub> <mo>ℓ</mo> <mn>0</mn> </msub> <mo>≡</mo> <msub> <mi>v</mi> <mi>p</mi> </msub> <mo>/</mo> <mi>ω</mi> </mrow> </semantics></math> is the polar soft mode correlation length.</p>
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10 pages, 2579 KiB  
Article
Optical Vortex-Pumped KTiOAsO4 Narrow-Linewidth Picosecond-Pulsed Parametric Oscillator
by Xiazhuo Jiao, Jianqiang Ye, Mailikeguli Aihemaiti, Yuxia Zhou, Sujian Niu and Xining Yang
Appl. Sci. 2025, 15(2), 539; https://doi.org/10.3390/app15020539 - 8 Jan 2025
Viewed by 478
Abstract
Herein, we present a picosecond-pulsed optical vortex parametric oscillator capable of generating high-power, narrow-linewidth near- and mid-infrared optical vortex outputs. The optical parametric oscillator (OPO), consisting of a KTiOAsO4 (KTA) crystal and a Z-shaped standing wave cavity formed by five mirrors, transferred [...] Read more.
Herein, we present a picosecond-pulsed optical vortex parametric oscillator capable of generating high-power, narrow-linewidth near- and mid-infrared optical vortex outputs. The optical parametric oscillator (OPO), consisting of a KTiOAsO4 (KTA) crystal and a Z-shaped standing wave cavity formed by five mirrors, transferred the orbital angular momentum (OAM) of the pump field to the signal and idler fields. The transmission mechanism of the OAM within the signal singly resonantsingly-resonant KTA-OPO was investigated, and the OAM was controlled and selectively transferred among the pump, signal, and idler fields by adjusting the focus position of the pump beam on the KTA crystal. With an incident pump power of 17 W, the maximum average output power was 2.14 W at 1535 nm (signal vortex field) and 0.95 W at 3468 nm (idler vortex field), respectively, corresponding to optical conversion efficiencies of 20.8% and 9.2%. The spectral linewidths of the signal and idler vortex fields were 0.502 nm and 1.216 nm, respectively. To the best of our knowledge, this is the first instance of a picosecond-pulsed optical vortex parametric oscillator with a KTA crystal. Full article
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<p>(<b>a</b>) Experimental configuration of the optical vortex-pumped KTA picosecond-pulsed parametric oscillator. λ/2, HWP; (<b>b</b>,<b>c</b>) show the spatial intensity profile of the pump vortex beam after the SPP conversion and when passing through a tilted lens.</p>
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<p>Images (<b>a</b>–<b>f</b>) depicting the spatial intensity profiles of the signal and idler beams as generated from the KTA-OPO for a variety of pump beam focal positions; images (<b>a1</b>–<b>f1</b>) showing the corresponding spatial intensity profiles of the signal and idler beams as generated when employing a Mach–Zehnder interferometer or a tilted lens.</p>
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<p>Plot showing the spatial overlap efficiency between the pump field and the signal or idler fields as a function of the pump beam radius (<span class="html-italic">ω<sub>p</sub></span>).</p>
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<p>(<b>a</b>) The beam quality of the output idler vortex; (<b>b</b>) the output power of the signal and idler vortices; (<b>c</b>,<b>d</b>) the spectral of the output signal and idler vortices. Insets show the spectral bandwidths of the output signal and idler vortices, respectively.</p>
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17 pages, 4791 KiB  
Article
Photoreconfigurable Metasurface for Independent Full-Space Control of Terahertz Waves
by Zhengxuan Jiang, Guowen Ding, Xinyao Luo and Shenyun Wang
Sensors 2025, 25(1), 119; https://doi.org/10.3390/s25010119 - 27 Dec 2024
Viewed by 873
Abstract
We present a novel photoreconfigurable metasurface designed for independent and efficient control of electromagnetic waves with identical incident polarization and frequency across the entire spatial domain. The proposed metasurface features a three-layer architecture: a top layer incorporating a gold circular split ring resonator [...] Read more.
We present a novel photoreconfigurable metasurface designed for independent and efficient control of electromagnetic waves with identical incident polarization and frequency across the entire spatial domain. The proposed metasurface features a three-layer architecture: a top layer incorporating a gold circular split ring resonator (CSRR) filled with perovskite material and dual C-shaped perovskite resonators; a middle layer of polyimide dielectric; and a bottom layer comprising a perovskite substrate with an oppositely oriented circular split ring resonator filled with gold. By modulating the intensity of a laser beam, we achieve autonomous manipulation of incident circularly polarized terahertz waves in both transmission and reflection modes. Simulation results demonstrate that the metasurface achieves a cross-polarized transmission coefficient of 0.82 without laser illumination and a co-polarization reflection coefficient of 0.8 under laser illumination. Leveraging the geometric phase principle, adjustments to the rotational orientation of the reverse split ring and dual C-shaped perovskite structures enable independent control of transmission and reflection phases. Furthermore, the proposed metasurface induces a +1 order orbital angular momentum in transmission and +2 order in reflection, facilitating beam deflection through metasurface convolution principles. Imaging using metasurface digital imaging technology showcases patterns “NUIST” in reflection and “LOONG” in transmission, illustrating the metasurface design principles via the proposed metasurface. The proposed metasurface’s capability for full-space control and reconfigurability presents promising applications in advanced imaging systems, dynamic beam steering, and tunable terahertz devices, highlighting its potential for future technological advancements. Full article
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<p>(<b>a</b>) OAM Mode 1 in transmission mode under high laser beam illumination for Metasurface I. (<b>b</b>) OAM Mode 2 in reflection mode without laser beam illumination for Metasurface I. (<b>c</b>) Holographic imaging with high laser beam illumination, generating a holographic image of the letters “NUIST” in reflection mode for Metasurface II. (<b>d</b>) Holographic imaging without laser beam illumination, generating a holographic image of the letters “LOONG” in transmission mode for Metasurface II.</p>
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<p>(<b>a</b>) Schematic of the proposed three-layer metasurface. (<b>b</b>) Top layer with CSRR and dual <span class="html-italic">C</span>-shaped resonators. (<b>c</b>) Bottom layer with circular split ring resonator.</p>
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<p>(<b>a</b>) Phase of co-polarized reflection coefficients (<span class="html-italic">r<sub>xx</sub></span> and <span class="html-italic">r<sub>yy</sub></span>) and co-polarized transmission coefficients (<span class="html-italic">t<sub>xx</sub></span> and <span class="html-italic">t<sub>yy</sub></span>) under laser beam illumination. (<b>b</b>) Phase of co-polarized reflection coefficients (<span class="html-italic">r<sub>xx</sub></span> and <span class="html-italic">r<sub>yy</sub></span>) and co-polarized transmission coefficients (<span class="html-italic">t<sub>xx</sub></span> and <span class="html-italic">t<sub>yy</sub></span>) without laser beam illumination. (<b>c</b>) Amplitude of co-polarized transmission coefficients (<span class="html-italic">t<sub>xx</sub></span> and <span class="html-italic">t<sub>yy</sub></span>) and co-polarized transmission coefficients (<span class="html-italic">t<sub>xx</sub></span> and <span class="html-italic">t<sub>yy</sub></span>) under laser beam illumination. (<b>d</b>) Amplitude of co-polarized transmission coefficients (<span class="html-italic">t<sub>xx</sub></span> and <span class="html-italic">t<sub>yy</sub></span>) and co-polarized transmission coefficients (<span class="html-italic">t<sub>xx</sub></span> and <span class="html-italic">t<sub>yy</sub></span>) without laser beam illumination.</p>
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<p>(<b>a</b>) Reflection amplitude and transmission amplitude at different rotation angles when unit is under laser beam illumination. (<b>b</b>) Transmission amplitude and reflection amplitude at different rotation angles when unit is without laser beam illumination. (<b>c</b>) Reflection phase at different rotation angles when unit is under laser beam illumination. (<b>d</b>) Transmission phase at different rotation angles when unit is without laser beam illumination.</p>
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<p>(<b>a</b>) Surface current distribution on the top layer under strong laser beam illumination, showing time-varying current distribution within the period. (<b>b</b>) Surface current distribution on the bottom layer under strong laser beam illumination, showing time-varying current distribution within the period. (<b>c</b>) Surface current distribution on the top layer without laser beam illumination, showing time-varying current distribution within the period. (<b>d</b>) Surface current distribution on the bottom layer without laser beam illumination, showing time-varying current distribution within the period.</p>
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<p>(<b>a</b>) Phase gradients along the +<span class="html-italic">x</span> direction. (<b>b</b>) Phase gradients along the −<span class="html-italic">x</span> direction. (<b>c</b>) Far-field distribution of a +1 order vortex beam deflected by 30° in the +<span class="html-italic">x</span> direction under strong laser beam illumination. (<b>d</b>) Far-field distribution of a −2 order vortex beam deflected by 30° in the −<span class="html-italic">x</span> direction without laser beam illumination. (<b>e</b>) Planar electric field intensity and phase distribution of the +1 order vortex beam under strong laser beam illumination, perpendicular to the 30° direction. (<b>f</b>) Planar electric field intensity and phase distribution of the −2 order vortex beam without laser beam illumination, perpendicular to the −30° direction.</p>
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<p>(<b>a</b>) Target images: “Zhu” in reflection mode, “Long” in transmission mode. (<b>b</b>) Phase distribution for holographic images, with phase changes from 0° to 360°. (<b>c</b>) Metasurface design layout with 50 × 50 unit structures in top and bottom layers. (<b>d</b>) Simulation results under different laser beam illumination: “Zhu” in reflection mode under laser beam illumination; “Long” in transmission mode without laser beam illumination.</p>
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<p>(<b>a</b>) Target images in reflection mode. (<b>b</b>) Phase distribution for reflection mode calculated using the GS algorithm. (<b>c</b>) Reproduced image of “NUIST” in reflection mode. (<b>d</b>) Target images in transmission mode. (<b>e</b>) Phase distribution for transmission mode calculated using the GS algorithm. (<b>f</b>) Reproduced image of “LOOGN” in transmission mode.</p>
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<p>(<b>a</b>) Near-field imaging results. (<b>b</b>) Far-field electric field distribution in reflection mode. (<b>c</b>) Far-field electric field distribution in transmission mode.</p>
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7 pages, 2383 KiB  
Proceeding Paper
Dual-Band Shared-Aperture Multimode OAM-Multiplexing Antenna Based on Reflective Metasurface
by Shuaicheng Li and Jie Cui
Phys. Sci. Forum 2024, 10(1), 6; https://doi.org/10.3390/psf2024010006 - 26 Dec 2024
Viewed by 389
Abstract
In this paper, a novel single-layer dual-band orbital angular momentum (OAM) multiplexed reflective metasurface array antenna is proposed, which can independently generate OAM beams with different modes in the C-band and Ku-band, and complete flexible beam control in each operating band, achieving the [...] Read more.
In this paper, a novel single-layer dual-band orbital angular momentum (OAM) multiplexed reflective metasurface array antenna is proposed, which can independently generate OAM beams with different modes in the C-band and Ku-band, and complete flexible beam control in each operating band, achieving the generation of an OAM beam with mode l = −1 under oblique incidence at 7G with 94.4% mode purity, and having a wider usable operating bandwidth at 12G with a wide operating bandwidth, and an OAM beam with mode l = +2 is generated under oblique incidence, achieving 82.5% mode purity, which verifies the performance of the unit, makes preparations for the next research, and provides new possibilities for communication in more transmission bands and larger channel capacity. Full article
(This article belongs to the Proceedings of The 1st International Online Conference on Photonics)
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<p>Unit structure and dimensions: P = 12 mm, L1 = 6 mm, L2 = 4 mm, H = 1.5 mm, W1 = 0.5 mm, and W3 = 0.5 mm. (<b>a</b>) the oblique view, (<b>b</b>) the top view.</p>
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<p>Reflective magnitude and phases (<b>a</b>) at 7 GHz and (<b>b</b>) at 12 GHz; (<b>c</b>,<b>d</b>) curves of 16 reflection phase simulation results at 7 and 12 GHz for the 2-bit cell.</p>
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<p>(<b>a</b>,<b>b</b>) are the phase distributions of the array antenna in two modes, l = −1 and l = +2, for positive incidence of the horn, and (<b>c</b>,<b>d</b>) are the phase distributions of the horn at 25° oblique incidence; (<b>e</b>) is the model of the array.</p>
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<p>(<b>a</b>,<b>b</b>) are the phase distributions of the array antenna in two modes, l = −1 and l = +2, for positive incidence of the horn, and (<b>c</b>,<b>d</b>) are the phase distributions of the horn at 25° oblique incidence; (<b>e</b>) is the model of the array.</p>
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<p>(<b>a</b>,<b>b</b>) are the l = mode l = +2 OAM beam phase distribution and amplitude for x-polarization, and (<b>c</b>,<b>d</b>) are the l = −1 mode OAM beam phase distribution and amplitude for y-polarization. (<b>e</b>) Histograms of an OAM spectrum weight l = −1; (<b>f</b>) histograms of an OAM spectrum weight l = +2.</p>
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