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14 pages, 8579 KiB  
Article
Fano and Electromagnetically Induced Transparency Resonances in Dual Side-Coupled Photonic Crystal Nanobeam Cavities
by Yong Zhao, Yuxuan Chen and Lijun Hao
Materials 2024, 17(24), 6213; https://doi.org/10.3390/ma17246213 - 19 Dec 2024
Abstract
We propose two types of structures to achieve the control of Fano and electromagnetically induced transparency (EIT) line shapes, in which dual one-dimensional (1D) photonic crystal nanobeam cavities (PCNCs) are side-coupled to a bus waveguide with different gaps. For the proposed type Ⅰ [...] Read more.
We propose two types of structures to achieve the control of Fano and electromagnetically induced transparency (EIT) line shapes, in which dual one-dimensional (1D) photonic crystal nanobeam cavities (PCNCs) are side-coupled to a bus waveguide with different gaps. For the proposed type Ⅰ and type Ⅱ systems, the phase differences between the nanobeam periodic structures of the two cavities are π and 0, respectively. The whole structures are theoretically analyzed via the coupled mode theory and numerically demonstrated using the three-dimensional finite-difference time-domain (3D FDTD) method. The simulation results show that the proposed structure can achieve several kinds of spectra, including Fano, EIT and asymmetric EIT line shapes, which is dependent on the width of the bus waveguide. Compared to the previously proposed Fano resonator with 1D PCNCs, the proposed structures have the advantages of high transmission at the resonant peak, low insertion loss at non-resonant wavelengths, a wide free spectral range (FSR) and a high roll-off rate. Therefore, we believe the proposed structure can find broad applications in optical switches, modulators and sensors. Full article
Show Figures

Figure 1

Figure 1
<p>Simplified model of two side-coupled standing-wave cavities (<span class="html-italic">C</span><sub>1</sub> and <span class="html-italic">C</span><sub>2</sub>).</p>
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<p>The calculated transmission spectra of the (<b>a</b>) type Ⅰ and (<b>b</b>) type Ⅱ systems based on the temporary CMT. The coupling Q-factors are <span class="html-italic">Q<sub>w</sub></span><sub>1</sub> = 500 and <span class="html-italic">Q<sub>w</sub></span><sub>2</sub> = 2000, and the intrinsic Q-factors are <span class="html-italic">Q<sub>i</sub></span><sub>1</sub> = <span class="html-italic">Q<sub>i</sub></span><sub>2</sub> = 1 × 10<sup>5</sup>. The resonant wavelengths of <span class="html-italic">C</span><sub>1</sub> and <span class="html-italic">C</span><sub>2</sub> are <span class="html-italic">λ</span><sub>1</sub> = <span class="html-italic">λ</span><sub>2</sub> = 1550 nm in (<b>a</b>) and <span class="html-italic">λ</span><sub>1</sub> = 1550 nm and <span class="html-italic">λ</span><sub>2</sub> = 1549 nm in (<b>b</b>).</p>
Full article ">Figure 3
<p>Models of two side-coupled cavities with phase-shifted periodic structure under weak perturbation conditions: (<b>a</b>) Type Ⅰ system with same resonant wavelengths and grating phase difference of <span class="html-italic">π</span> between <span class="html-italic">C</span><sub>1</sub> and <span class="html-italic">C</span><sub>2</sub>. (<b>b</b>) Type Ⅱ system with different resonant wavelengths and same grating phase between <span class="html-italic">C</span><sub>1</sub> and <span class="html-italic">C</span><sub>2</sub>.</p>
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<p>The calculated transmission spectra of the (<b>a</b>) type Ⅰ and (<b>b</b>) type Ⅱ systems under weak perturbation conditions based on the coupled mode equations of Equations (6)–(8).</p>
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<p>Schematics of proposed (<b>a</b>) type Ⅰ and (<b>b</b>) type Ⅱ dual-PCNC systems on SOI platform.</p>
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<p>The reflection spectra of the designed Bragg reflector with 4 tapered holes and 9 uniform holes and the Bragg reflector with 13 uniform holes.</p>
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<p>A schematic of the calculation of the effective index of the strip waveguide with a hole.</p>
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<p>Simulated (<b>a</b>) transmission and (<b>b</b>) reflection spectra when only <span class="html-italic">C</span><sub>1</sub> or <span class="html-italic">C</span><sub>2</sub> is side-coupled to the bus waveguide.</p>
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<p>(<b>a</b>) The transmission and reflection spectra of the type Ⅰ dual-PCNC system with an EIT-like line shape, and the |<span class="html-italic">H<sub>y</sub></span>| profile at the (<b>b</b>) transmission peak point <span class="html-italic">A</span>, (<b>c</b>) transmission dip point <span class="html-italic">B</span> and (<b>d</b>) transmission dip point <span class="html-italic">C</span> calculated by the 3D FDTD method.</p>
Full article ">Figure 10
<p>(<b>a</b>) Transmission spectra and (<b>b</b>) phase of type Ⅰ dual-PCNC system with different bus waveguide widths calculated by 3D FDTD method. Red dot is point of maximum slope change in phase relative to wavelength.</p>
Full article ">Figure 11
<p>(<b>a</b>) The transmission and reflection spectra of the type Ⅰ dual-PCNC system with a Fano line shape (<span class="html-italic">w</span><sub>0</sub> = 435 nm), and the |<span class="html-italic">H<sub>y</sub></span>| profile at (<b>b</b>) point <span class="html-italic">A</span>, (<b>c</b>) point <span class="html-italic">B</span> and (<b>d</b>) point <span class="html-italic">C</span> calculated by the 3D FDTD method.</p>
Full article ">Figure 12
<p>(<b>a</b>) The transmission and reflection spectra of the type Ⅱ dual-PCNC system with an EIT-like line shape, and the |<span class="html-italic">H<sub>y</sub></span>| profile at the (<b>b</b>) transmission peak point <span class="html-italic">A</span>, (<b>c</b>) transmission dip point <span class="html-italic">B</span> and (<b>d</b>) transmission dip point <span class="html-italic">C</span> calculated by the 3D FDTD method.</p>
Full article ">Figure 13
<p>(<b>a</b>) The transmission spectra and (<b>b</b>) phase of the type Ⅱ dual-PCNC system with different bus waveguide widths calculated by the 3D FDTD method. The red dot is the point of the maximum slope change in phase relative to the wavelength.</p>
Full article ">Figure 14
<p>(<b>a</b>) The transmission and reflection spectra of the type Ⅱ dual-PCNC system with an asymmetric EIT line shape (<span class="html-italic">w</span><sub>0</sub> = 380) nm, and the |<span class="html-italic">H<sub>y</sub></span>| profile at (<b>b</b>) point <span class="html-italic">A</span>, (<b>c</b>) point <span class="html-italic">B</span> and (<b>d</b>) point <span class="html-italic">C</span> calculated by the 3D FDTD method.</p>
Full article ">Figure 15
<p>(<b>a</b>) The transmission and reflection spectra of the type Ⅱ dual-PCNC system with a Fano line shape and (<span class="html-italic">w</span><sub>0</sub> = 460), and the |<span class="html-italic">H<sub>y</sub></span>| profile at (<b>b</b>) point <span class="html-italic">A</span>, (<b>c</b>) point <span class="html-italic">B</span> and (<b>d</b>) point <span class="html-italic">C</span> calculated by the 3D FDTD method.</p>
Full article ">Figure 16
<p>The transmission spectra of the (<b>a</b>) type Ⅰ and (<b>b</b>) type Ⅱ dual-PCNC systems with EIT line shapes with different waveguide width deviations (∆<span class="html-italic">w</span>).</p>
Full article ">Figure 17
<p>The transmission spectra of the (<b>a</b>) type Ⅰ and (<b>b</b>) type Ⅱ dual-PCNC systems with EIT line shapes with different hole radius deviations (∆<span class="html-italic">r</span>).</p>
Full article ">
21 pages, 2778 KiB  
Article
Research on the Mechanical Parameter Identification and Controller Performance of Permanent Magnet Motors Based on Sensorless Control
by Mingchen Luan, Yun Zhang, Jiuhong Ruan, Yongwu Guo, Long Wang and Huihui Min
Actuators 2024, 13(12), 525; https://doi.org/10.3390/act13120525 - 19 Dec 2024
Abstract
In order to improve the control performance of the position sensorless control system of permanent magnet synchronous motors and to reduce the influence of external uncertainties on the control system, such as inertia ingestion and load disturbance, this paper proposes a novel position [...] Read more.
In order to improve the control performance of the position sensorless control system of permanent magnet synchronous motors and to reduce the influence of external uncertainties on the control system, such as inertia ingestion and load disturbance, this paper proposes a novel position sensorless control algorithm for permanent magnet synchronous motors based on an interleaved parallel extended sliding mode observer. Firstly, in order to identify the time-varying moment of inertia, load torque and viscous friction coefficient of the system, a novel interleaved parallel extended sliding mode observer based on a single-observer model is proposed, and a robust activator is designed to reduce the coupling between the parameters to be measured. Then, a new predefined-time sliding mode controller is designed for the face-mounted permanent magnet synchronous motor using sliding film control theory, which improves the response speed and control accuracy of the system. Then, the proposed novel interleaved parallel extended sliding mode observer and predefined-time sliding mode controller are used to design the permanent magnet synchronous motor control system, and the stability of the system is proved using the Lyapunov stability theorem. Finally, through simulation analysis and experimental tests, it is verified that the control strategy proposed in this paper can improve the identification accuracy of the motor parameters, reduce the time of identification, and improve the control accuracy and tracking speed. Full article
(This article belongs to the Special Issue Power Electronics and Actuators—Second Edition)
Show Figures

Figure 1

Figure 1
<p>Voltage equivalent circuit diagram.</p>
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<p>Block diagram of permanent magnet synchronous motor control system.</p>
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<p>Block diagram of the observer principle.</p>
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<p>Convergence of the error system for different control schemes.</p>
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<p>Curves at convergence of sliding mode surface for different control schemes.</p>
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<p>Single-state observer load torque observation results.</p>
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<p>Multi- state observer load torque observation results.</p>
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<p>Multi-state observer rotational inertia observation results.</p>
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<p>Observations of <math display="inline"><semantics> <mi mathvariant="italic">B</mi> </semantics></math> when using the IPESMO algorithm.</p>
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<p>Experimental control platform.</p>
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<p>Experimental logic diagram.</p>
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<p>Fitting of the friction coefficients on the test bench. (<b>A</b>) Motor reversal. (<b>B</b>) Motor forward.</p>
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<p>Actual and estimated motor speeds obtained using the SMC and NFTSMC algorithms, respectively, when the desired speeds are different. (<b>A</b>) 1000 rpm. (<b>B</b>) 1500 rpm. (<b>C</b>) 2000 rpm. (<b>D</b>) 3000 rpm.</p>
Full article ">Figure 14
<p>Comparison of estimation results with and without Coulomb friction torque compensation. (<b>A</b>) With compensation <math display="inline"><semantics> <mover accent="true"> <mi>J</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>. (<b>B</b>) With compensation <math display="inline"><semantics> <msub> <mover accent="true"> <mi>T</mi> <mo stretchy="false">^</mo> </mover> <mi>L</mi> </msub> </semantics></math>. (<b>C</b>) With compensation <math display="inline"><semantics> <mover accent="true"> <mi>B</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>. (<b>D</b>) Without compensation <math display="inline"><semantics> <mover accent="true"> <mi>J</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>. (<b>E</b>) Without compensation <math display="inline"><semantics> <msub> <mover accent="true"> <mi>T</mi> <mo stretchy="false">^</mo> </mover> <mi>L</mi> </msub> </semantics></math>. (<b>F</b>) Without compensation <math display="inline"><semantics> <mover accent="true"> <mi>B</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>.</p>
Full article ">Figure 15
<p>Comparison of errors with and without Cullen friction torque compensation.</p>
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<p>(<b>A</b>) IPESMO. (<b>B</b>) DESMO. (<b>C</b>) ESO.</p>
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<p>Comparison of the errors in the observations of the three algorithms.</p>
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<p>Comparison of observed values of each mechanical parameter at different times of gain. (<b>A</b>) Observation value of J. (<b>B</b>) Observation value of <math display="inline"><semantics> <msub> <mi>T</mi> <mi>L</mi> </msub> </semantics></math>. (<b>C</b>) Observation value of B.</p>
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<p>Multi-load multi-algorithm comparison experiments. (<b>A</b>) IPESMO −0.1 N·m. (<b>B</b>) ESMO −0.1 N·m. (<b>C</b>) DESMO −0.1 N·m. (<b>D</b>) IPESMO −0.2 N·m. (<b>E</b>) ESMO −0.2 N·m. (<b>F</b>) DESMO −0.2 N·m.</p>
Full article ">
28 pages, 8434 KiB  
Article
Extending Generalized Explicit Terms and Applying Euler–Bernoulli Beam Theory to Enhance Dynamic Response Prediction in Receptance Coupling Method
by Behzad Hamedi and Saied Taheri
Appl. Sci. 2024, 14(24), 11841; https://doi.org/10.3390/app142411841 - 18 Dec 2024
Viewed by 193
Abstract
This paper presents a theoretical framework to enhance the prediction of dynamic responses in complex mechanical systems, such as vehicle structures, by incorporating both translational and rotational degrees of freedom. Traditional receptance coupling methods often neglect rotational effects, leading to significant inaccuracies at [...] Read more.
This paper presents a theoretical framework to enhance the prediction of dynamic responses in complex mechanical systems, such as vehicle structures, by incorporating both translational and rotational degrees of freedom. Traditional receptance coupling methods often neglect rotational effects, leading to significant inaccuracies at higher frequencies. Additionally, approaches that implicitly include full dynamics frequently result in redundancy of generalized coordinates, especially at connection points. To address these limitations, the generalized receptance coupling method using Frequency-Based Substructuring is extended to explicitly account for rotational dynamics resulting in a refined GRCFBS approach. This extension enhances both the understanding and prediction of system responses, which are represented through the receptance matrix or Frequency Response Function. Building on Jetmundsen’s foundational work, the proposed framework introduces a practical, generalized formulation that explicitly incorporates full translational and rotational dynamics at each substructure node. This explicit definition provides deeper insights into system behavior, particularly for complex interactions between substructures under weak and strong coupling scenarios at interface points. The Euler–Bernoulli beam theory is employed to model rotational behavior at critical points, yielding reduced-order and explicit receptance matrices for substructures in the coupling process. The methodology’s accuracy and applicability in capturing resonance and anti-resonance modes are validated through two case studies: the coupling of two flexible subsystems and the integration of flexible and rigid components. Results are benchmarked against numerical finite element analysis, and all limitations and potential improvements are discussed. By directly incorporating rotational dynamics directly, this approach enables more reliable dynamic response predictions under multi-directional loading conditions, particularly for vehicle and machinery system design. The GRCFBS method offers a versatile and reliable tool for dynamic system analysis, with significant potential for vibration analysis over a broad frequency range. Full article
(This article belongs to the Section Acoustics and Vibrations)
Show Figures

Figure 1

Figure 1
<p>Schematic of system AB for the 1st case study.</p>
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<p>Decomposition of system AB into substructures A and B for case study 1.</p>
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<p>Excitation force and moment on substructures A for case study 1.</p>
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<p>Excitation force and moment on substructures B for case study 1.</p>
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<p>Schematic of system AC, consisting of substructures A and C.</p>
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<p>Decomposition of system AC into substructures A and C using substructuring strategy.</p>
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<p>Excitation loads (force and moment) applied to substructure C.</p>
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<p>Receptance component (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>): translational disp. at the tip due to excitation force at the same point (case study 1).</p>
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<p>Receptance component (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>): translational displacement at tip due to couple at tip (case study 1).</p>
Full article ">Figure 10
<p>Receptance component (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>): rotational displacement at the tip due to couple at the same point (case study 1).</p>
Full article ">Figure 11
<p>Comparison between direct/cross-receptance components at tip (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <mo>,</mo> <mtext> </mtext> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <mo>,</mo> <mtext> </mtext> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>) (case study 1).</p>
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<p>Receptance (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>): translational disp. at connection due to excitation force at same point (case study 1).</p>
Full article ">Figure 13
<p>Receptance (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>): translational disp. at connection point due to excitation couple at same point (case study 1).</p>
Full article ">Figure 14
<p>Receptance (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>): Rotational disp. at connection point due to excitation couple at the same point (case study 1).</p>
Full article ">Figure 15
<p>Comparison between cross-receptance at connection point (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <mo>,</mo> <mtext> </mtext> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <mo>,</mo> <mtext> </mtext> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>) (case study 1).</p>
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<p>Cross-receptance (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>): translational disp. between tip and connection points (case study 1).</p>
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<p>Cross-receptance <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mtext> </mtext> <mo>(</mo> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>): translational disp. at the connection node due to the moment excitation at the tip (case study 1).</p>
Full article ">Figure 18
<p>Cross-receptance (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>): rotational disp. at the connection due to the excitation couple at the tip (case study 1).</p>
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<p>Comparison between cross-receptance components (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <mo>,</mo> <mtext> </mtext> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> </mrow> </semantics></math>) (case study 1).</p>
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<p>Receptance component (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>C</mi> </mrow> </msubsup> </mrow> </semantics></math>): translational displacement to excitation force (2nd case study).</p>
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<p>Receptance component <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mo>(</mo> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>C</mi> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>: translational displacement to excitation moment (2nd case study).</p>
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<p>Receptance component (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>C</mi> </mrow> </msubsup> </mrow> </semantics></math>): rotational disp. to excitation couple (2nd case study).</p>
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<p>Comparison between cross-receptance components (<math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>C</mi> </mrow> </msubsup> <mo>,</mo> <mtext> </mtext> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>C</mi> </mrow> </msubsup> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>A</mi> <mi>C</mi> </mrow> </msubsup> </mrow> </semantics></math>) for 2nd case study.</p>
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<p>Comparison between receptance component at the tip: translational displacement to excitation force for two cases.</p>
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15 pages, 5530 KiB  
Article
Regulation and Liquid Sensing of Electromagnetically Induced Transparency-like Phenomena Implemented in a SNAP Microresonator
by Chenxiang Liu, Minggang Chai, Chenglong Zheng, Chengfeng Xie, Chuanming Sun, Jiulin Shi, Xingdao He and Mengyu Wang
Sensors 2024, 24(21), 7069; https://doi.org/10.3390/s24217069 - 2 Nov 2024
Viewed by 638
Abstract
Optical microresonators supporting whispering-gallery modes (WGMs) have become a versatile platform for achieving electromagnetically induced transparency-like (EIT-like) phenomena. We theoretically and experimentally demonstrated the tunable coupled-mode induced transparency based on the surface nanoscale axial photonics (SNAP) microresonator. Single-EIT-like and double-EIT-like (DEIT-like) effects with [...] Read more.
Optical microresonators supporting whispering-gallery modes (WGMs) have become a versatile platform for achieving electromagnetically induced transparency-like (EIT-like) phenomena. We theoretically and experimentally demonstrated the tunable coupled-mode induced transparency based on the surface nanoscale axial photonics (SNAP) microresonator. Single-EIT-like and double-EIT-like (DEIT-like) effects with one or more transparent windows are achieved due to dense mode families and tunable resonant frequencies. The experimental results can be well-fitted by the coupled mode theory. An automatically adjustable EIT-like effect is discovered by immersing the sensing region of the SNAP microresonator into an aqueous environment. The sharp lineshape and high slope of the transparent window allow us to achieve a liquid refractive index sensitivity of 2058.8 pm/RIU. Furthermore, we investigated a displacement sensing phenomenon by monitoring changes in the slope of the transparent window. We believe that the above results pave the way for multi-channel all-optical switching devices, multi-channel optical communications, and biochemical sensing processing. Full article
(This article belongs to the Special Issue Research Progress in Optical Microcavity-Based Sensing)
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Figure 1

Figure 1
<p>(<b>a</b>) Normalized electric field distribution for the fifth order axial mode. (<b>b</b>) Normalized electric field distribution for the third order radial mode. (<b>c</b>) Cross-sectional views of the normalized electric field distributions with (<span class="html-italic">p</span> = 1, <span class="html-italic">m</span> = 365, <span class="html-italic">q</span> = 0), (<span class="html-italic">p</span> = 2, <span class="html-italic">q</span> = 2, <span class="html-italic">m</span> = 337), and (<span class="html-italic">p</span> = 3, <span class="html-italic">m</span> = 319, <span class="html-italic">q</span> = 4). (<b>d</b>) Resonant wavelengths as a function of axial mode numbers <span class="html-italic">q</span> with measurement size.</p>
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<p>(<b>a</b>) Schematic diagram for the microresonator–taper coupling system. (<b>b</b>) Schematic diagram for the three-mode-coupling microcapillary system. (<b>c</b>) Illustration for the three-pathway interference effect induced by the three kinds of WGMs.</p>
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<p>(<b>a</b>) Simulation spectrum varies with <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math>. The resonant frequency difference between high-<span class="html-italic">Q</span> mode and low-<span class="html-italic">Q</span> mode is not zero and induces the Fano resonance. (<b>b</b>) Transparent window spectrum varies with <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math>. The resonant frequency difference between the two modes is zero and induces the standard transparent window. (<b>c</b>,<b>d</b>) show the derivatives of the spectra with respect to <math display="inline"><semantics> <mrow> <mi>ω</mi> </mrow> </semantics></math> in (<b>a</b>,<b>b</b>), respectively. (<b>e</b>) Transparent window lineshape (blue line) and Fano lineshape (red line) with <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> = 150. (<b>f</b>) Slope curves of transparent window and Fano resonance in case of (<b>e</b>).</p>
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<p>Dual-mode-coupling and the corresponding Fano/EIT–like effects realized in the SNAP microresonator. (<b>a</b>–<b>e</b>) Experimental normalized transmission spectra at different coupling positions, <math display="inline"><semantics> <mrow> <mo>∆</mo> <mi>z</mi> </mrow> </semantics></math> = 5 μm, 10 μm, 15 μm, 20 μm, and 25 μm. (<b>f</b>–<b>j</b>) Fitting experimental data corresponding to (<b>a</b>–<b>e</b>). The simulation data are set to: <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>κ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>κ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>∆</mo> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>∆</mo> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>]</mo> </mrow> </semantics></math> = [783.2, 26.4, 183.2, 12.8, 295, 772.7, 0.28, 8.9], [643.2, 1, 263.2, 74.8, 195, 564.7, 0.52, 8.76], [373.2, 5, 263.2, 124.8, −15, 394.7, 0.73, 8.91], [293.2, 10, 240.2, 140.8, −75, 204.7, 0.71, 9], [153.2, 10, 290.2, 130.8, −100, −19.7, 0.69, 12.25].</p>
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<p>Multi-mode-coupling and DEIT effect realized in the SNAP microresonator. (<b>a</b>–<b>f</b>) Experimental normalized transmission spectra at different coupling positions. (<b>g</b>–<b>l</b>) Theoretical transmission lineshapes for fitting experimental data. The simulation data are set to: <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mi>κ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>κ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>κ</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mo>∆</mo> <mi>ω</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>∆</mo> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>∆</mo> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>]</mo> </mrow> </semantics></math> = [403.2, 576.4, 77.4, −3403.2, −290.8, −12, 170, −5, 740.7, 0.17, 8.82, 8.6], [1253.2, 585.4, 77.4, −3203.2, −287.8, −12, 60, −15, 690.7, 0.17, 8.82, 8.7], [1753.2, 625.4, 22.4, −4003.2, −287.8, −1, 20, −90, 590.7, 0.12, 8.82, 9.6], [1753.2, 625.4, 32.4, −4003.2, −297.8, −1, −50, −110, 565.7], [1753.2, 625.4, 32.4, −4003.2, −304.8, −5, −50, −110, 535.7, 0.12, 8.82, 8.4], [1753.2, 625.4, 32.4, −3503.2, −294.8, −5, −180, −180, 485.7, 0.12, 8.82, 8.7].</p>
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<p>(<b>a</b>) Experimental apparatus for the axial separation sensing. (<b>b</b>) Schematic diagram of the axial separation sensing technology for detecting the aqueous environment. (<b>c</b>) The physical device corresponding to (<b>a</b>). (<b>d</b>) Automatically adjustable EIT-like effect when the sensing region is immersed in deionized water. (<b>e</b>) Theoretical transmission lineshapes for fitting experimental data. The simulation data are set to: <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>γ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>κ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>κ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mo>∆</mo> <mi>ω</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>∆</mo> <msub> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>]</mo> </mrow> </semantics></math> = [94.2, 11.3, 31.4, 0.063, 375, 562, 0.24, 8.78], [94.2, 7.54, 31.4, 0.063, 187, 312, 0.24, 8.85], [94.2, 6.91, 31.4, 0.063, 62.5, 137.4, 0.44, 8.91], [94.2, 3.14, 31.4, 0, 0, 0, 0.58, 9.23], [94.2, 3.14, 31.4, 0, −50, −125, 0.52, 10.88], [94.2, 3.14, 31.4, 0, −250, −400, 0.61, 11.91], [94.2, 3.14, 31.4, 0, −324.6, −500, 0.61, 11.97].</p>
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<p>(<b>a</b>) Cross-sectional WGMs distribution with increasing air refractive indices. (<b>b</b>) The resonant wavelength shifts towards longer wavelengths as <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> continues to increase. (<b>c</b>) The electric field intensity distribution decreases as <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> continues to increase. (<b>d</b>) Evolution of the transmission spectrum with increasing liquid refractive index. The green arrow indicates the wavelength shift direction. (<b>e</b>) The resonant wavelength shifts as a function of the liquid refractive index.</p>
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<p>(<b>a</b>) Simulation model for the electric field distribution. One stem of the microresonator is immersed in liquid. Axial mode with <span class="html-italic">q</span> = 6 is excited in the model. The liquid surfaces are set to −10 μm, −5 μm, and 0 μm; the corresponding electric field distribution models are shown in (<b>i</b>), (<b>ii</b>), and (<b>iii</b>), respectively. The electric field intensity in liquid and air along the radial section line direction is shown in (<b>iv</b>). (<b>b</b>) Axial electric field intensity distribution with different liquid surfaces along the axial section line direction. (<b>i</b>), (<b>ii</b>), and (<b>iii</b>) are electric field distribution models as the interaction area between the liquid and the sensing region of the SNAP microresonator is increased. (<b>c</b>) Representative transmission spectra corresponding to each liquid lifting height; red (0 μm), green (100 μm), and blue (200 μm). (<b>d</b>) Localized magnification for the spectra in (<b>c</b>). (<b>e</b>) The slope values of the transparent window with MD = 0 μm, 100 μm, 200 μm, respectively. The slope data are divided with three different color intervals from left to right, corresponding to each MD. The mean values of the slopes as a function of the MD are displayed in the inset. (<b>f</b>) Probability distribution histogram of the slope values corresponding to various MD.</p>
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32 pages, 9532 KiB  
Article
Considering Carbon–Hydrogen Coupled Integrated Energy Systems: A Pathway to Sustainable Energy Transition in China Under Uncertainty
by Anjie Lu, Jianguo Zhou, Minglei Qin and Danchen Liu
Sustainability 2024, 16(21), 9256; https://doi.org/10.3390/su16219256 - 24 Oct 2024
Viewed by 1243
Abstract
The low-carbon construction of integrated energy systems is a crucial path to achieving dual carbon goals, with the power-generation side having the greatest potential for emissions reduction and the most direct means of reduction, which is a current research focus. However, existing studies [...] Read more.
The low-carbon construction of integrated energy systems is a crucial path to achieving dual carbon goals, with the power-generation side having the greatest potential for emissions reduction and the most direct means of reduction, which is a current research focus. However, existing studies lack the precise modeling of carbon capture devices and the cascaded utilization of hydrogen energy. Therefore, this paper establishes a carbon capture power plant model based on a comprehensive, flexible operational mode and a coupled model of a two-stage P2G (Power-to-Gas) device, exploring the “energy time-shift” characteristics of the coupled system. IGDT (Information Gap Decision Theory) is used to discuss the impact of uncertainties on the power generation side system. The results show that by promoting the consumption of clean energy and utilizing the high energy efficiency of hydrogen while reducing reliance on fossil fuels, the proposed system not only meets current energy demands but also achieves a more efficient emission reduction, laying a solid foundation for a sustainable future. By considering the impact of uncertainties, the system ensures resilience and adaptability under fluctuating renewable energy supply conditions, making a significant contribution to the field of sustainable energy transition. Full article
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<p>Liquid storage carbon capture power plant CO<sub>2</sub> flow diagram.</p>
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<p>Comparison of net generations from carbon capture power plants.</p>
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<p>Energy time shifting in carbon capture power plants.</p>
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<p>Two-stage energy bus diagram for Power-to-Gas.</p>
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<p>Dispatch structure of the system.</p>
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<p>Day-ahead forecast load in case studies.</p>
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<p>Carbon capture power plant outputs.</p>
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<p>Outputs of other conventional thermal power units.</p>
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<p>Scenario 3: composition of electrical power.</p>
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<p>Output power of the PV Unit.</p>
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<p>Net generation power of the carbon capture unit.</p>
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<p>Total carbon emissions of the system.</p>
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14 pages, 1458 KiB  
Article
An Energy Approach to the Modal Identification of a Variable Thickness Quartz Crystal Plate
by Zhe Wang, Bin Huang, Yan Guo, Yanan Jiang and Asif Khan
Sensors 2024, 24(20), 6707; https://doi.org/10.3390/s24206707 - 18 Oct 2024
Viewed by 648
Abstract
The primary objective of modal identification for variable thickness quartz plates is to ascertain their dominant operating mode, which is essential for examining the vibration of beveled quartz resonators. These beveled resonators are plate structures with varying thicknesses. While the beveling process mitigates [...] Read more.
The primary objective of modal identification for variable thickness quartz plates is to ascertain their dominant operating mode, which is essential for examining the vibration of beveled quartz resonators. These beveled resonators are plate structures with varying thicknesses. While the beveling process mitigates some spurious modes, it still presents challenges for modal identification. In this work, we introduce a modal identification technique based on the energy method. When a plate with variable thickness is in a resonant state of thickness–shear vibration, the proportions of strain energy and kinetic energy associated with the thickness–shear mode in the total energy reach their peak values. Near this frequency, their proportions are the highest, aiding in identifying the dominant mode. Our research was based on the Mindlin plate theory, and appropriate modal truncation were conducted by retaining three modes for the coupled vibration analysis. The governing equation of the coupled vibration was solved for eigenvalue problem, and the modal energy proportions were calculated based on the determined modal displacement and frequency. Finally, we computed the eigenvalue problems at different beveling time, as well as the modal energies associated with each mode. By calculating the energy proportions, we could clearly identify the dominant mode at each frequency. Our proposed method can effectively assist engineers in identifying vibration modes, facilitating the design and optimization of variable thickness quartz resonators for sensing applications. Full article
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<p>Geometry and coordinate system of a rectangular plate with variable thickness.</p>
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<p>Strain energy density proportion of the three modes at the center line and different beveling times.</p>
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<p>Kinetic energy density proportion of the three modes at the center line and different beveling times.</p>
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<p>Kinetic energy density proportion of the three modes at the center line and different beveling times.</p>
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24 pages, 7307 KiB  
Article
Vibration Analysis of Multilayered Quasicrystal Annular Plates, Cylindrical Shells, and Truncated Conical Shells Filled with Fluid
by Xin Feng, Han Zhang and Yang Gao
J. Compos. Sci. 2024, 8(10), 433; https://doi.org/10.3390/jcs8100433 - 18 Oct 2024
Viewed by 671
Abstract
An approach to estimate the dynamic characteristic of multilayered three-dimensional cubic quasicrystal cylindrical shells, annular plates, and truncated conical shells with different boundary conditions is presented. These investigated structures can be in a vacuum, totally filled with quiescent fluid, and subjected to internal [...] Read more.
An approach to estimate the dynamic characteristic of multilayered three-dimensional cubic quasicrystal cylindrical shells, annular plates, and truncated conical shells with different boundary conditions is presented. These investigated structures can be in a vacuum, totally filled with quiescent fluid, and subjected to internal flowing fluid where the fluid is incompressible and inviscid. The velocity potential, Bernoulli’s equation, and the impermeability condition have been applied to the shell–fluid interface to obtain an explicit expression, from which the fluid pressure can be converted into the coupled differential equations in terms of displacement functions. The state-space method is formulated to quasicrystal linear elastic theory to derive the state equations for the three structures along the radial direction. The mixed supported boundary conditions are represented by means of the differential quadrature technique and Fourier series expansions. A global propagator matrix, which connects the field variables at the internal interface to those at the external interface for the whole structure, is further completed by joint coupling matrices to overcome the numerical instabilities. Numerical examples show the correctness of the proposed method and the influence of the semi-vertical angle, different boundary conditions, and the fluid debit on the natural frequencies and mode shapes for various geometries and boundary conditions. Full article
(This article belongs to the Special Issue Feature Papers in Journal of Composites Science in 2024)
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<p>Geometry of the truncated conical shell, cylindrical shell, and annular shell.</p>
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<p>Summary of the solution procedure followed in this study.</p>
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<p>Normalized displacement mode shapes: (<b>a</b>) comparison of results between SSM and SS-DQM; (<b>b</b>) comparison of results between pseudo-Stroh formalism [<a href="#B30-jcs-08-00433" class="html-bibr">30</a>] and SS-DQM.</p>
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<p>First-order dimensionless displacement mode shapes: (<b>a</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>u</mi> </mrow> <mrow> <mi>r</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>u</mi> </mrow> <mrow> <mi>θ</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>r</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, and (<b>d</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>z</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>.</p>
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<p>First-order dimensionless stress mode shapes: (<b>a</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">σ</mi> </mrow> <mrow> <mi mathvariant="italic">rr</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">σ</mi> </mrow> <mrow> <mrow> <mi>r</mi> <mi>θ</mi> </mrow> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <mi mathvariant="italic">rr</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, and (<b>d</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <mi mathvariant="italic">rz</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>.</p>
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<p>First-order dimensionless stress mode shapes: (<b>a</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">σ</mi> </mrow> <mrow> <mi mathvariant="italic">rr</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">σ</mi> </mrow> <mrow> <mrow> <mi>r</mi> <mi>θ</mi> </mrow> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <mi mathvariant="italic">rr</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, and (<b>d</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <mi mathvariant="italic">rz</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>.</p>
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<p>The contour plots of the phonon displacement mode shape <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>u</mi> </mrow> <mrow> <mi>r</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> (10<sup>−2</sup>): (<b>a</b>) <span class="html-italic">α</span><sub>0</sub> = <span class="html-italic">π</span>/6, (<b>b</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/6, (<b>c</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/4, and (<b>d</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/3.</p>
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<p>The contour plots of the phonon displacement mode shape <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>u</mi> </mrow> <mrow> <mi>θ</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> (10<sup>−3</sup>): (<b>a</b>) <span class="html-italic">α</span><sub>0</sub> = <span class="html-italic">π</span>/6, (<b>b</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/6, (<b>c</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/4, and (<b>d</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/3.</p>
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<p>The contour plots of the phason displacement mode shape <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>θ</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> (10<sup>−3</sup>): (<b>a</b>) <span class="html-italic">α</span><sub>0</sub> = <span class="html-italic">π</span>/6, (<b>b</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/6, (<b>c</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/4, and (<b>d</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/3.</p>
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<p>The contour plots of the phason displacement mode shape <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>z</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> (10<sup>−3</sup>): (<b>a</b>) <span class="html-italic">α</span><sub>0</sub> = <span class="html-italic">π</span>/6, (<b>b</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/6, (<b>c</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/4, and (<b>d</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/3.</p>
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<p>The contour plots of the phason displacement mode shape <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>z</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> (10<sup>−3</sup>): (<b>a</b>) <span class="html-italic">α</span><sub>0</sub> = <span class="html-italic">π</span>/6, (<b>b</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/6, (<b>c</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/4, and (<b>d</b>) <span class="html-italic">α</span> = <span class="html-italic">π</span>/3.</p>
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<p>First-order dimensionless displacement mode shapes: (<b>a</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>u</mi> </mrow> <mrow> <mi>r</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>u</mi> </mrow> <mrow> <mi>θ</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>u</mi> </mrow> <mrow> <mi>z</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, and (<b>d</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>r</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>.</p>
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<p>First-order dimensionless stress mode shapes: (<b>a</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">σ</mi> </mrow> <mrow> <mi mathvariant="italic">rr</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">σ</mi> </mrow> <mrow> <mrow> <mi>r</mi> <mi>θ</mi> </mrow> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="sans-serif">σ</mi> </mrow> <mrow> <mi mathvariant="italic">rz</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>, and (<b>d</b>) <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <mi mathvariant="italic">rz</mi> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>.</p>
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12 pages, 3538 KiB  
Article
A Nonlinear Adaptive Control and Robustness Analysis for Autonomous Landing of UAVs
by Yue Feng, Quanwen Hu, Weihan Wu, Liaoni Wu, Qiuquan Guo and Haitao Zhang
Drones 2024, 8(10), 587; https://doi.org/10.3390/drones8100587 - 17 Oct 2024
Viewed by 729
Abstract
The UAV landing process has higher requirements for automatic flight control systems due to factors such as wind disturbances and strong constraints. Considering the proven effective adaptation of the out-of-loop L1 adaptive control (OLAC) system proposed in previous studies, this paper applies it [...] Read more.
The UAV landing process has higher requirements for automatic flight control systems due to factors such as wind disturbances and strong constraints. Considering the proven effective adaptation of the out-of-loop L1 adaptive control (OLAC) system proposed in previous studies, this paper applies it to landing control to enhance robustness and control accuracy in the presence of complex uncertainties. Based on modern control theory, an LQR-based OLAC algorithm for multi-input–multi-output (MIMO) systems is proposed, which is conducive to the coupling control of the flight attitude mode. To evaluate the robustness of the designed system, an equivalence stability margin analysis method for nonlinear systems is proposed based on parameter linearization. Along with a detailed autonomous landing strategy, including trajectory planning, control, and guidance, the effectiveness of the proposed methods is verified on a high-fidelity simulation platform. The Monte–Carlo simulation is implemented in the time domain, and the results demonstrate that OLAC exhibits strong robustness and ensures the state variables strictly meet the flight safety constraints. Full article
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<p>UAV automatic landing contrail.</p>
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<p>Control structure for autonomous landing.</p>
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<p>Structure of the OLAC system.</p>
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<p>The impact of adaptive parameters on stability margin. (<b>a</b>) Phase Margin, (<b>b</b>) Delay Margin.</p>
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<p>Error bars for the touchdown index under single parameter deviation. (<b>a</b>) Speed, (<b>b</b>) Pitch Angle, (<b>c</b>) Forward Distance, (<b>d</b>) Sink Rate.</p>
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<p>Time-domain curves for Monte–Carlo simulation. (<b>a</b>) Height-Forward Distance, (<b>b</b>) Pitch Tracking Error, (<b>c</b>) Speed, (<b>d</b>) Sink Rate.</p>
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<p>Time-domain curves for Monte–Carlo simulation. (<b>a</b>) Height-Forward Distance, (<b>b</b>) Pitch Tracking Error, (<b>c</b>) Speed, (<b>d</b>) Sink Rate.</p>
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<p>Density map of landing index for Monte–Carlo simulation. (<b>a</b>) Landing Point, (<b>b</b>) Speed-Pitch Angle.</p>
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16 pages, 29393 KiB  
Article
Switchable Dual-Wavelength Fiber Laser with Narrow-Linewidth Output Based on Parity-Time Symmetry System and the Cascaded FBG
by Kaiwen Wang, Bin Yin, Chao Lv, Yanzhi Lv, Yiming Wang, Hao Liang, Qun Wang, Shiyang Wang, Fengjie Yu, Zhong Zhang, Ziwang Li and Songhua Wu
Photonics 2024, 11(10), 946; https://doi.org/10.3390/photonics11100946 - 8 Oct 2024
Viewed by 989
Abstract
In this paper, a dual-wavelength narrow-linewidth fiber laser based on parity-time (PT) symmetry theory is proposed and experimentally demonstrated. The PT-symmetric filter system consists of two optical couplers (OCs), four polarization controllers (PCs), a polarization beam splitter (PBS), and cascaded fiber Bragg gratings [...] Read more.
In this paper, a dual-wavelength narrow-linewidth fiber laser based on parity-time (PT) symmetry theory is proposed and experimentally demonstrated. The PT-symmetric filter system consists of two optical couplers (OCs), four polarization controllers (PCs), a polarization beam splitter (PBS), and cascaded fiber Bragg gratings (FBGs), enabling stable switchable dual-wavelength output and single longitudinal-mode (SLM) operation. The realization of single-frequency oscillation requires precise tuning of the PCs to match gain, loss, and coupling coefficients to ensure that the PT-broken phase occurs. During single-wavelength operation at 1548.71 nm (λ1) over a 60-min period, power and wavelength fluctuations were observed to be 0.94 dB and 0.01 nm, respectively, while for the other wavelength at 1550.91 nm (λ2), fluctuations were measured at 0.76 dB and 0.01 nm. The linewidths of each wavelength were 1.01 kHz and 0.89 kHz, with a relative intensity noise (RIN) lower than −117 dB/Hz. Under dual-wavelength operation, the maximum wavelength fluctuations for λ1 and λ2 were 0.03 nm and 0.01 nm, respectively, with maximum power fluctuations of 3.23 dB and 2.38 dB. The SLM laser source is suitable for applications in long-distance fiber-optic sensing and coherent LiDAR detection. Full article
(This article belongs to the Special Issue Single Frequency Fiber Lasers and Their Applications)
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Figure 1
<p>Experimental structure of the proposed PT-symmetric fiber laser.</p>
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<p>Schematic diagram of non-reciprocal light transmission in a dual-physical ring PT-symmetric structure.</p>
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<p>Trend of the enhancement factor G with gain ratio variation.</p>
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<p>(<b>a</b>) Dual wavelength spectrum of laser output. (<b>b</b>) The output spectra of the laser’s dual wavelength at 6-minute intervals over 60 min; (<b>c</b>,<b>d</b>) the wavelength and power stability of λ<sub>1</sub> and λ<sub>2</sub>.</p>
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<p>Output laser spectra. (<b>a</b>,<b>b</b>) Output laser spectra; (<b>c</b>,<b>d</b>) spectral fluctuations within 60 min. (<b>e</b>,<b>f</b>) wavelength and power fluctuations for λ<sub>1</sub> and λ<sub>2</sub>.</p>
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<p>Corresponding relationship between pump power and laser output of (<b>a</b>) λ<sub>1</sub>, (<b>b</b>) λ<sub>2</sub>, and (<b>c</b>) a dual-wavelength test.</p>
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<p>Spectra of laser output under different conditions. (<b>a</b>,<b>b</b>) Output spectra without the PT-symmetric system; (<b>c</b>,<b>d</b>) output spectra without the 2 × 2 coupled sub-cavity; (<b>e</b>,<b>f</b>) spectra when PT symmetry was unbroken; (<b>g</b>,<b>h</b>) spectra when λ<sub>1</sub> and λ<sub>2</sub> were in the PT-broken phase.</p>
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<p>Frequency spectra stability of stable SLM operation at λ<sub>1</sub> and λ<sub>2</sub>.</p>
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<p>Measurement of the DW-EDFL linewidth: measurement results for (<b>a</b>) λ<sub>1</sub> and (<b>b</b>) λ<sub>2</sub>.</p>
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<p>Laser intensity noise. (<b>a</b>) Detector baseline noise; (<b>b</b>–<b>d</b>) intensity noise of λ<sub>1</sub>, λ<sub>2</sub>, and dual-wavelength operation.</p>
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<p>Phase noise of output laser (<b>a</b>) λ<sub>1</sub> and (<b>b</b>) λ<sub>2</sub>.</p>
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36 pages, 18495 KiB  
Article
Size-Dependent Mechanical Properties and Excavation Responses of Basalt with Hidden Cracks at Baihetan Hydropower Station through DFN–FDEM Modeling
by Changdong Ding, Zhenjiang Liu, Xiancheng Mei and Shaoming Ouyang
Appl. Sci. 2024, 14(19), 9069; https://doi.org/10.3390/app14199069 - 8 Oct 2024
Viewed by 1030
Abstract
Basalt is an important geotechnical material for engineering construction in Southwest China. However, it has complicated structural features due to its special origin, particularly the widespread occurrence of hidden cracks. Such discontinuities significantly affect the mechanical properties and engineering stability of basalt, and [...] Read more.
Basalt is an important geotechnical material for engineering construction in Southwest China. However, it has complicated structural features due to its special origin, particularly the widespread occurrence of hidden cracks. Such discontinuities significantly affect the mechanical properties and engineering stability of basalt, and related research is lacking and unsystematic. In this work, taking the underground caverns in the Baihetan Hydropower Station as the engineering background, the size-dependent mechanical behaviors and excavation responses of basalt with hidden cracks were systematically explored based on a synthetic rock mass (SRM) model combining the finite-discrete element method (FDEM) and discrete fracture network (DFN) method. The results showed that: (1) The DFN–FDEM model generated based on the statistical characteristics of the geometric parameters of hidden cracks can consider the real structural characteristics of basalt, whereby the mechanical behaviors found in laboratory tests and at the engineering site could be exactly reproduced. (2) The representative elementary volume (REV) size of basalt blocks containing hidden cracks was 0.5 m, and the mechanical properties obtained at this size were considered equivalent continuum properties. With an increase in the sample dimensions, the mechanical properties reflected in the stress–strain curves changed from elastic–brittle to elastic–plastic or ductile, the strength failure criterion changed from linear to nonlinear, and the failure modes changed from fragmentation failure to local structure-controlled failure and then to splitting failure. (3) The surrounding rock mass near the excavation face of underground caverns typically showed a spalling failure mode, mainly affected by the complex structural characteristics and high in situ stresses, i.e., a tensile fracture mechanism characterized by stress–structure coupling. The research findings not only shed new light on the failure mechanisms and size-dependent mechanical behaviors of hard brittle rocks represented by basalt but also further enrich the basic theory and technical methods for multi-scale analyses in geotechnical engineering, which could provide a reference for the design optimization, construction scheme formulation, and disaster prevention of deep engineering projects. Full article
(This article belongs to the Special Issue Advances in Failure Mechanism and Numerical Methods for Geomaterials)
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Figure 1

Figure 1
<p>Layout of the underground caverns at Baihetan Hydropower Station, where 1–4 represent the main powerhouse, main transformer chamber, headgate chamber, and tailrace surge chamber, respectively, and I–X represent the excavation sequences of the cross section of the underground powerhouse from top to bottom [<a href="#B39-applsci-14-09069" class="html-bibr">39</a>]. (<b>a</b>) Location of the study area; (<b>b</b>) layout of underground caverns; (<b>c</b>) 3D visualization of left-bank underground caverns; (<b>d</b>) right-bank underground caverns.</p>
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<p>Structural characteristics of basalt and in situ failure features of surrounding rock mass in the underground caverns at Baihetan Hydropower Station [<a href="#B40-applsci-14-09069" class="html-bibr">40</a>]. (<b>a</b>) Geological longitudinal section of the left-bank underground powerhouse; (<b>b</b>) basalt block with hidden cracks; (<b>c</b>) fracture morphologies at engineering site.</p>
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<p>Meso-structural characteristics of cryptocrystalline basalt. (<b>a</b>) Polarizing microscopy image of a thin section; (<b>b</b>) CT scanning images.</p>
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<p>Physical forms and sketches of basalt blocks with hidden cracks obtained from the underground caverns at Baihetan Hydropower Station.</p>
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<p>Statistical distribution characteristics of the geometric parameters of hidden cracks in a basalt block. (<b>a</b>) Crack length; (<b>b</b>) crack orientation.</p>
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<p>Element mesh and constitutive models of crack elements in an FDEM model [<a href="#B41-applsci-14-09069" class="html-bibr">41</a>]. (<b>a</b>) Element mesh in an FDEM model; (<b>b</b>) cohesive model for mode I; (<b>c</b>) slip-weakening model for mode II; (<b>d</b>) I–II mixed mode.</p>
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<p>SRM (DFN–FDEM) models of a basalt block of different sizes with hidden cracks. (<b>a</b>) DFN models of different sizes; (<b>b</b>) SRM model with dimensions of 300 mm × 600 mm.</p>
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<p>Numerical basalt samples with a standard size for meso-parameter calibration. (<b>a</b>) DFN models under uniaxial and triaxial compression; (<b>b</b>) DFN models under Brazilian disc splitting; (<b>c</b>) SRM models under uniaxial and triaxial compression (<b>left</b>) and Brazilian disc splitting (<b>right</b>).</p>
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<p>FDEM model configuration of the left-bank underground powerhouse at Baihetan Hydropower Station.</p>
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<p>Effects of the sample size on the mechanical parameters of basalt with hidden cracks. (<b>a</b>) Uniaxial compression strength, UCS; (<b>b</b>) tensile strength, BDS; (<b>c</b>) cohesive strength, <span class="html-italic">c</span>; (<b>d</b>) internal friction angle, <span class="html-italic">φ</span>.</p>
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<p>Effects of the sample size on the stress–strain curve characteristics of basalt with hidden cracks under different confining pressures. (<b>a</b>) Intact specimen at a size of 50 mm; (<b>b</b>) specimen with local cracks at a size of 100 mm; (<b>c</b>) specimen with many cracks at a size of 300 mm; (<b>d</b>) specimen with sufficient cracks at a size of 500 mm.</p>
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<p>Compressive strength of different-sized basalt samples with hidden cracks under different confining pressures. (<b>a</b>) Relationship between sample size and compressive strength; (<b>b</b>) relationship between confining pressure and compressive strength.</p>
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<p>Fracture evolution and failure modes of basalt samples of different sizes with hidden cracks under uniaxial compression. (<b>a</b>) Fragmentation failure of intact basalt; (<b>b</b>) structure-controlled failure of medium-sized basalt with local fractures; (<b>c</b>) splitting failure of REV-sized basalt with sufficient fractures.</p>
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<p>Typical failure characteristics of a intact basalt rock mass of the underground powerhouse during excavation at Baihetan Hydropower Station [<a href="#B45-applsci-14-09069" class="html-bibr">45</a>].</p>
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<p>Typical failure characteristics of a jointed basalt rock mass of the underground powerhouse during excavation at Baihetan Hydropower Station [<a href="#B45-applsci-14-09069" class="html-bibr">45</a>]. (<b>a</b>) Spatial form of spalling failure; (<b>b</b>) Sketch of spalling failure; (<b>c</b>) Enlarged diagram of failure modes; (<b>d</b>) Thin slab after spalling failure.</p>
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<p>Fracture evolution process of the intact basalt rock mass during different excavation stages of each layer of the left-bank underground powerhouse: the maximum principal stress fields (denoted by <span class="html-italic">σ</span><sub>1</sub>), displacement fields (denoted by <span class="html-italic">S</span>), crack evolutions (S, T-S, and T represent shear, tensile–shear, and tensile cracks, respectively), and in situ monitoring or existing research results, respectively [<a href="#B46-applsci-14-09069" class="html-bibr">46</a>]. (<b>a</b>) Excavation of the middle pilot tunnel of the first layer; (<b>b</b>) first expansion excavation of both sides of the first layer; (<b>c</b>) expansion excavation of the floor of the first layer; (<b>d</b>) second expansion excavation of both sides of the first layer; (<b>e</b>) excavation of the fourth layer; (<b>f</b>) excavation of the seventh layer; (<b>g</b>) excavation completion.</p>
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<p>Fracture evolution process of the intact basalt rock mass during different excavation stages of each layer of the left-bank underground powerhouse: the maximum principal stress fields (denoted by <span class="html-italic">σ</span><sub>1</sub>), displacement fields (denoted by <span class="html-italic">S</span>), crack evolutions (S, T-S, and T represent shear, tensile–shear, and tensile cracks, respectively), and in situ monitoring or existing research results, respectively [<a href="#B46-applsci-14-09069" class="html-bibr">46</a>]. (<b>a</b>) Excavation of the middle pilot tunnel of the first layer; (<b>b</b>) first expansion excavation of both sides of the first layer; (<b>c</b>) expansion excavation of the floor of the first layer; (<b>d</b>) second expansion excavation of both sides of the first layer; (<b>e</b>) excavation of the fourth layer; (<b>f</b>) excavation of the seventh layer; (<b>g</b>) excavation completion.</p>
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<p>Progressive fracture process and failure mechanisms of the intact basalt rock mass during excavation in underground caverns. (<b>a</b>) Stress state change within the surrounding rock mass near excavation surface; (<b>b</b>) Crack evolution process within the surrounding rock mass near excavation face.</p>
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<p>Influence of specimen size on the UCS [<a href="#B47-applsci-14-09069" class="html-bibr">47</a>,<a href="#B48-applsci-14-09069" class="html-bibr">48</a>,<a href="#B49-applsci-14-09069" class="html-bibr">49</a>].</p>
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<p>Structural characteristics and failure modes of several different types of hard brittle rocks, including basalt (<b>left</b>), marble (<b>middle</b>), and granite (<b>right</b>) [<a href="#B50-applsci-14-09069" class="html-bibr">50</a>,<a href="#B51-applsci-14-09069" class="html-bibr">51</a>,<a href="#B52-applsci-14-09069" class="html-bibr">52</a>]. (<b>a</b>) Polarized microscopy images of a thin section of different types of hard brittle rocks; (<b>b</b>) failure modes of different types of hard brittle rocks under uniaxial compression; (<b>c</b>) failure modes of different types of hard brittle rock mass on site.</p>
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12 pages, 846 KiB  
Article
Undamped Higgs Modes in Strongly Interacting Superconductors
by José Lorenzana and Götz Seibold
Condens. Matter 2024, 9(4), 38; https://doi.org/10.3390/condmat9040038 - 30 Sep 2024
Viewed by 577
Abstract
In superconductors, gauge U(1) symmetry is spontaneously broken. According to Goldstone’s theorem, this breaking of a continuous symmetry establishes the existence of the Bogoliubov phase mode while the gauge-invariant response also includes the amplitude fluctuations of the order parameter. The [...] Read more.
In superconductors, gauge U(1) symmetry is spontaneously broken. According to Goldstone’s theorem, this breaking of a continuous symmetry establishes the existence of the Bogoliubov phase mode while the gauge-invariant response also includes the amplitude fluctuations of the order parameter. The latter, which are also termed ‘Higgs’ modes in analogy with the standard model, appear at the energy of the spectral gap 2Δ, when the superconducting ground state is evaluated within the weak-coupling BCS theory, and, therefore, are damped. Previously, we have shown that, within the time-dependent Gutzwiller approximation (TDGA), Higgs modes appear inside the gap with a finite binding energy relative to the quasiparticle continuum. Here, we show that the binding energy of the Higgs mode becomes exponentially small in the weak-coupling limit converging to the BCS solution. On the other hand, well-defined undamped amplitude modes exist in strongly coupled superconductors when the interaction energy becomes of the order of the bandwidth. Full article
(This article belongs to the Special Issue Superstripes Physics, 3rd Edition)
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<p>(<b>a</b>) Ginzburg–Landau (GL) potential landscape of a superconductor. Bogoljubov phase excitations (<math display="inline"><semantics> <mo>Φ</mo> </semantics></math>) and amplitude ‘Higgs’ (H) modes are indicated by yellow and blue arrows, respectively. (<b>b</b>) Evolution of the spectral gap and the energy of the Higgs mode as a function of the interaction parameter <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>U</mi> <mo>|</mo> </mrow> </semantics></math> (in terms of the bandwidth) for a two-dimensional system. In the regime <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>U</mi> <mo>|</mo> <mo>&gt;</mo> <mi>B</mi> </mrow> </semantics></math>, the Higgs mode is significantly split off inside the gap, cf. inset.</p>
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<p>The black line shows the amplitude correlation function Equation (<a href="#FD23-condensedmatter-09-00038" class="html-disp-formula">23</a>) times <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>U</mi> <mo>|</mo> </mrow> </semantics></math>. The intersection of the horizontal lines with the black curve yields the position of the Higgs pole in Equation (<a href="#FD22-condensedmatter-09-00038" class="html-disp-formula">22</a>). In the BCS + RPA approximation, quasiparticles interact with an effective matrix element <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>=</mo> <mo>−</mo> <mo>|</mo> <mi>U</mi> <mo>|</mo> </mrow> </semantics></math> [Equation (19)] (red dashed line). This produces a pole in the denominator of Equation (<a href="#FD22-condensedmatter-09-00038" class="html-disp-formula">22</a>) at <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mo>Δ</mo> </mrow> </semantics></math>. In the TDGA, it turns out that the effective interaction (exemplified by the blue dashed line) produces a pole within the spectral gap.</p>
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<p>The individual terms 1, 2, 3, and 4 which are defined in Equation (<a href="#FD36-condensedmatter-09-00038" class="html-disp-formula">36</a>).</p>
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16 pages, 2548 KiB  
Article
Fault Diagnosis of Pumped Storage Units—A Novel Data-Model Hybrid-Driven Strategy
by Jie Bai, Chuanqiang Che, Xuan Liu, Lixin Wang, Zhiqiang He, Fucai Xie, Bingjie Dou, Haonan Guo, Ruida Ma and Hongbo Zou
Processes 2024, 12(10), 2127; https://doi.org/10.3390/pr12102127 - 30 Sep 2024
Viewed by 670
Abstract
Pumped storage units serve as a crucial support for power systems to adapt to large-scale and high-proportion renewable energy sources by providing a stable and flexible energy supply. However, due to the coupling effects of electric power load demands and the complex multi-source [...] Read more.
Pumped storage units serve as a crucial support for power systems to adapt to large-scale and high-proportion renewable energy sources by providing a stable and flexible energy supply. However, due to the coupling effects of electric power load demands and the complex multi-source factors within the water–mechanical–electrical system, the interrelationship between unit parameters becomes more intricate, posing significant threats to the operational reliability and health status of the units. The complexity of fault diagnosis is further aggravated by the intricate and varied nature of fault characteristics, as well as the challenges in signal extraction under conditions of strong electromagnetic interference and high noise levels. To address these issues, this paper proposes a novel data-model hybrid-driven strategy that analyzes vibration signals to achieve rapid and accurate fault diagnosis of the units. Firstly, the spectral kurtosis theory is employed to enhance the traditional empirical mode decomposition, achieving optimal decomposition and noise reduction effects for vibration signals. Secondly, the intrinsic mode functions (IMFs) obtained from the decomposition are reconstructed, and the entropy values of effective IMFs are calculated as fault feature vectors. Subsequently, the CNN-LSTM model is utilized for fault diagnosis. The effectiveness and feasibility of the proposed method are verified through actual operational data from pumped storage units in a specific region. Through analysis, the fault diagnosis accuracy of the method proposed in this paper can be maintained above 95%, demonstrating robustness in complex engineering environments and effectively ensuring the safe and stable operation of pumped storage units. Full article
(This article belongs to the Section Process Control and Monitoring)
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<p>The structure diagram of the CNN.</p>
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<p>The structure diagram of the designed CNN-LSTM.</p>
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<p>Flowchart of fault diagnosis technology for the pumped storage unit.</p>
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<p>Schematic diagram of the testing system structure.</p>
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<p>The collected typical vibration signals.</p>
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<p>Spectrum analysis results of the traditional method.</p>
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<p>Spectrum analysis results of the proposed method.</p>
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14 pages, 1669 KiB  
Article
Bidentate Substrate Binding Mode in Oxalate Decarboxylase
by Alvaro Montoya, Megan Wisniewski, Justin L. Goodsell and Alexander Angerhofer
Molecules 2024, 29(18), 4414; https://doi.org/10.3390/molecules29184414 - 17 Sep 2024
Viewed by 709
Abstract
Oxalate decarboxylase is an Mn- and O2-dependent enzyme in the bicupin superfamily that catalyzes the redox-neutral disproportionation of the oxalate monoanion to form carbon dioxide and formate. Its best-studied isozyme is from Bacillus subtilis where it is stress-induced under low pH [...] Read more.
Oxalate decarboxylase is an Mn- and O2-dependent enzyme in the bicupin superfamily that catalyzes the redox-neutral disproportionation of the oxalate monoanion to form carbon dioxide and formate. Its best-studied isozyme is from Bacillus subtilis where it is stress-induced under low pH conditions. Current mechanistic schemes assume a monodentate binding mode of the substrate to the N-terminal active site Mn ion to make space for a presumed O2 molecule, despite the fact that oxalate generally prefers to bind bidentate to Mn. We report on X-band 13C-electron nuclear double resonance (ENDOR) experiments on 13C-labeled oxalate bound to the active-site Mn(II) in wild-type oxalate decarboxylase at high pH, the catalytically impaired W96F mutant enzyme at low pH, and Mn(II) in aqueous solution. The ENDOR spectra of these samples are practically identical, which shows that the substrate binds bidentate (κO, κO’) to the active site Mn(II) ion. Domain-based local pair natural orbital coupled cluster singles and doubles (DLPNO-CCSD) calculations of the expected 13C hyperfine coupling constants for bidentate bound oxalate predict ENDOR spectra in good agreement with the experiment, supporting bidentate bound substrate. Geometry optimization of a substrate-bound minimal active site model by density functional theory shows two possible substrate coordination geometries, bidentate and monodentate. The bidentate structure is energetically preferred by ~4.7 kcal/mol. Our results revise a long-standing hypothesis regarding substrate binding in the enzyme and suggest that dioxygen does not bind to the active site Mn ion after substrate binds. The results are in agreement with our recent mechanistic hypothesis of substrate activation via a long-range electron transfer process involving the C-terminal Mn ion. Full article
(This article belongs to the Section Chemical Biology)
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<p>X-band <sup>13</sup>C Mims ENDOR spectra. Experimental parameters: Microwave (MW) frequency = 9.746 GHz, MW pulse length (π/2) = 16 ns, τ = 540 ns, 521 points per spectrum. Radio frequency (RF) pulse width, <span class="html-italic">T</span> = 20 µs representing a π-pulse. The sample temperature was 5 K in all cases. (<b>A</b>) WT OxDC in Tris buffer pH 8.5 (blue), OxDC mutant W96F in citrate buffer pH 5.0 (red), and 1 mM MnCl<sub>2</sub> aqueous solution (green). All samples contained 50 mM <sup>13</sup>C-oxalate, and 50% glycerol as a glassing agent. The protein samples also contained 10 mM ascorbate and 5 μM DTPA. (<b>B</b>) X-band <sup>13</sup>C Mims ENDOR spectra of 1 mM MnCl<sub>2</sub> in 50:50 water/glycerol mixture with 50 mM <sup>13</sup>C-oxalate, showing the loss of the <sup>13</sup>C-ENDOR signal with increasing DTPA concentration. The molar ratio of DTPA:Mn(II) is given in the box, i.e., the DTPA concentrations are: (black) no DTPA, (red) 50 µM, (green) 100 µM, (cyan) 200 µM, (magenta) 400 µM, and (violet) 1000 µM.</p>
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<p>Results of geometry optimization of the substrate bound to the active site N-terminal Mn(II) ion. (<b>A</b>) Bidentate binding conformation. (<b>B</b>) Monodentate binding conformation. Distances between C and Mn, as well as coordinating O and Mn, are indicated by dashed lines and given in units of Ångstrom. Atom colors follow a modified CPK scheme where carbon atoms appear in cyan.</p>
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<p>Simulations of the <sup>13</sup>C ENDOR spectra using a bidentate (<b>A</b>) and a monodentate (<b>B</b>) model for substrate binding to Mn(II). Green: Experimental ENDOR spectrum. Pink and orange: individual contributions of C<sub>1</sub> and C<sub>2</sub> (<b>A</b>) or C<sub>3</sub> and C<sub>4</sub> (<b>B</b>) to the simulated ENDOR spectrum. Blue: sum of the individual contributions from the two carbon atoms. <a href="#app1-molecules-29-04414" class="html-app">Supporting information, Figure S3c</a>, shows the structure on which the calculations were based and the carbon numbering scheme.</p>
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11 pages, 4690 KiB  
Communication
Inter-Mode Crosstalk Estimation between Cores for LPmn Modes in Weakly Coupled Few-Mode Multicore Fiber with Perturbations
by Shuangmeng Liu and Lian Xiang
Sensors 2024, 24(18), 5969; https://doi.org/10.3390/s24185969 - 14 Sep 2024
Viewed by 616
Abstract
A novel inter-mode crosstalk (IMXT) model of LPmn mode for weakly coupled few-mode multicore fiber is proposed based on the coupled mode theory (CMT) with bending and twisting perturbations. A universal expression of the mode coupling coefficient (MCC) between [...] Read more.
A novel inter-mode crosstalk (IMXT) model of LPmn mode for weakly coupled few-mode multicore fiber is proposed based on the coupled mode theory (CMT) with bending and twisting perturbations. A universal expression of the mode coupling coefficient (MCC) between LPmn modes is derived. By employing this MCC, the universal semi-analytical model (USAM) of inter-core crosstalk (ICXT) can be applied to calculate the IMXT. Simulation results show that our model is generally consistent with previous theories when stochastic perturbations are absent. Moreover, our model can work effectively when stochastic perturbations are present, where former theories are not able to work properly. It has been theoretically found that the MCC has an intimate relationship with core pitch. Our model, based on the CMT, can provide physical characteristics in detail, which has not been reported clearly by former theories. In addition, our model is applicable to phase-matching and non-phase-matching regions of both real homogeneous and heterogeneous few-mode multicore fibers (FM-MCFs) with a wider range of applications. Full article
(This article belongs to the Special Issue Novel Technology in Optical Communications)
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<p>Schematic diagram of few-mode two-core fiber.</p>
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<p>Schematic of mode coupling in few-mode two-core fiber.</p>
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<p>Schematic diagram of few-mode four-core fiber.</p>
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<p>IMXT as a function of (<b>a</b>) FM−MCF length without stochastic perturbations, (<b>b</b>) bending radius, and (<b>c</b>) FM−MCF length with bending and twisting perturbation.</p>
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<p>IMXT as a function of (<b>a</b>) core pitch, (<b>b</b>) optical wavelength, and (<b>c</b>) twisting rate for IMXT and analytical expression.</p>
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<p>Estimation of crosstalk from the <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> <mi>P</mi> </mrow> <mrow> <mn>01</mn> </mrow> </msub> </mrow> </semantics></math> to the <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> <mi>P</mi> </mrow> <mrow> <mn>11</mn> </mrow> </msub> <mo>(</mo> <msub> <mrow> <mi>L</mi> <mi>P</mi> </mrow> <mrow> <mn>21</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>L</mi> <mi>P</mi> </mrow> <mrow> <mn>31</mn> </mrow> </msub> </mrow> </semantics></math>) as a function of FM−MCF length (<b>a</b>) without stochastic perturbations and (<b>b</b>) with stochastic perturbation.</p>
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<p>Estimation of crosstalk from <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> <mi>P</mi> </mrow> <mrow> <mn>01</mn> </mrow> </msub> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>L</mi> <mi>P</mi> </mrow> <mrow> <mn>11</mn> <mo> </mo> </mrow> </msub> <mo>(</mo> <msub> <mrow> <mi>L</mi> <mi>P</mi> </mrow> <mrow> <mn>21</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>L</mi> <mi>P</mi> </mrow> <mrow> <mn>31</mn> </mrow> </msub> </mrow> </semantics></math><span class="html-italic">)</span> as a function of (<b>a</b>) core pitch, (<b>b</b>) bending radius in homogeneous FM−MCF, and (<b>c</b>) bending radius in heterogeneous FM−MCF.</p>
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9 pages, 2198 KiB  
Communication
Ultra-Broadband Mode (De)Multiplexer on Thin-Film Lithium Niobate Platform Adopting Phase Control Theory
by Kun Yin, Wenting Jiao, Lin Wang and Shiqiang Zhu
Micromachines 2024, 15(9), 1084; https://doi.org/10.3390/mi15091084 - 28 Aug 2024
Viewed by 622
Abstract
Mode (de)multiplexers (MDMs) serve as critical foundational elements within systems for facilitating high-capacity communication, relying on mode conversions achieved through directional coupler (DC) structures. However, DC structures are challenged by dispersion issues for broadband mode coupling, particularly for high-order modes. In this work, [...] Read more.
Mode (de)multiplexers (MDMs) serve as critical foundational elements within systems for facilitating high-capacity communication, relying on mode conversions achieved through directional coupler (DC) structures. However, DC structures are challenged by dispersion issues for broadband mode coupling, particularly for high-order modes. In this work, based on the principles of phase control theory, we have devised an approach to mitigate the dispersion challenges, focusing on a thin-film lithium niobate-on-onsulator (LNOI) platform. This solution involves integrating a customized inverse-dispersion section into the device architecture, offsetting minor phase shifts encountered during the mode coupling process. By employing this approach, we have achieved broadband mode conversion from TE0 to TE1 and TE0 to TE2 within a 300 nm wavelength range, and the maximum deviations were maintained below −0.68 dB and −0.78 dB, respectively. Furthermore, the device exhibited remarkably low crosstalk, reaching down to −26 dB. Full article
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<p>(<b>a</b>) Schematic of the proposed broadband mode converter. (<b>b</b>) Cross-sectional views of the asymmetric directional couplers and the phase control unit. (<b>c</b>) Mode field distribution of <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math> mode. (<b>d</b>) Mode field distribution of <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> mode.</p>
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<p>(<b>a</b>) Effective indices for <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> mode, <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math> mode, and <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> mode at different waveguide widths. (<b>b</b>) Effective indices as a function of wavelength when waveguide widths are equal to 1.2 <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m, 2.5 <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m, and 4.29 <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m, respectively, for <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> mode, <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math> mode, and <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> mode.</p>
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<p>Contour maps of <math display="inline"><semantics> <msub> <mi>ν</mi> <mi>c</mi> </msub> </semantics></math> as functions of <math display="inline"><semantics> <msub> <mi>L</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>L</mi> <mi>p</mi> </msub> </semantics></math> at <math display="inline"><semantics> <mi>λ</mi> </semantics></math> =1550 nm for the mode converter of (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> mode to <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math> mode and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> mode to <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> mode. Contour maps of <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msub> <mi>ν</mi> <mi>c</mi> </msub> </mrow> </semantics></math> as functions of <math display="inline"><semantics> <msub> <mi>L</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>L</mi> <mi>p</mi> </msub> </semantics></math> in the wavelength range from 1400 nm to 1700 nm for the mode converter of (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> mode to <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math> mode and (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> mode to <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> mode.</p>
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<p>Power distributions in the plane of <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> of the broadband mode converters for (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> mode to <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math> mode and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> mode to <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> mode at different wavelengths.</p>
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<p>The calculated crosstalk of the broadband mode converters for (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> mode to <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math> mode and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> mode to <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> mode. The influence of gap change between thin waveguide and wide waveguide for (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> − <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </semantics></math> − <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math>.</p>
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<p>Illustrative of the <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>T</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>+</mo> <mi>T</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> (de)multiplexer concept.</p>
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