Search Results (89)

Figure 1
<p>Flow Chart showing research methodology.</p> Full article ">Figure 2
<p>CS structure of a four-layered FGM with ring support.</p> Full article ">Figure 3
<p>Natural frequency (NF) variation with varying power exponent laws in opposition to circumferential wave number for four-layered CS without ring support, when (<span class="html-italic">m</span> = 3, <span class="html-italic">R</span> = 1, <span class="html-italic">N</span> = 3, <span class="html-italic">d</span> = 0.1, <span class="html-italic">L/R</span> = 24, and <span class="html-italic">H</span> = 0.05 <span class="html-italic">m</span>, 0.07 <span class="html-italic">m</span>).</p> Full article ">Figure 4
<p>Natural frequency (NF) variation with varying power exponent laws in opposition to circumferential wave number for the four-layered cylindrical shell, when (<span class="html-italic">m</span> = 2, <span class="html-italic">d</span> = 0.5, <span class="html-italic">L/R</span> = 20 <span class="html-italic">m</span>, and <span class="html-italic">H</span> = 0.02 <span class="html-italic">m</span>).</p> Full article ">Figure 5
<p>Natural frequency (NF) variation with varying power exponent laws in opposition to the circumferential wave number for four-layered CS (<span class="html-italic">m</span> = 1, <span class="html-italic">H</span> = 0.006 <span class="html-italic">m</span>, <span class="html-italic">N</span> = 2, <span class="html-italic">R</span> = 1, <span class="html-italic">L/R</span> = 10 <span class="html-italic">m</span> to 30 <span class="html-italic">m</span> and <span class="html-italic">d</span> = 0.2).</p> Full article ">Figure 6
<p>Natural frequency (NF) variation with ring support ‘<span class="html-italic">d</span>’ at different <span class="html-italic">L/R</span> ratios for four-layered CS without ring support, when (<span class="html-italic">m</span> = 1, <span class="html-italic">R</span> = 1, <span class="html-italic">N</span> = 1, <span class="html-italic">L/R</span> = 30 <span class="html-italic">m</span>, and <span class="html-italic">H</span> = 0.04 <span class="html-italic">m</span>).</p> Full article ">

Figure 1
<p>Comparison of Ti:Sa and CO₂ direct laser acceleration efficiency. Column titles refer to the electron beam diameter. The term “high-energy fraction" refers to the fraction of the beam that has gained at least 10% of the theoretical gain limit and is used here as a measure of acceleration efficiency. The high-energy fraction of the CO₂ laser is at least 2 times higher than for the Ti:Sa laser at all simulated powers and electron beam diameters, and is almost always 5–10 times higher. The average energy gain is correspondingly higher for the CO₂ laser as well.</p> Full article ">Figure 2
<p>Energy distributions for some selected peak powers and electron beam diameters. Plots in the same row have the same electron beam diameter and plots in the same column have the same laser peak power. It is clear that, for the Ti:Sa laser, a large portion of the electron beam essentially does not participate in the acceleration interaction, while for the CO₂ laser, the majority of the beam is accelerated to some degree.</p> Full article ">Figure 3
<p>Energy distribution of a 2 <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m initial diameter electron beam after interacting with a 10 PW Ti:Sa with <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>23</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>86</mn> </mrow> </semantics></math> fs. The electrons experience essentially no acceleration (note that the units on the x-axis are electron volts, not megaelectron volts).</p> Full article ">Figure 4
<p>Final beam parameters of an accelerated electron beam with an initial energy of 7 MeV. (<b>a</b>) Electron energy distribution. (<b>b</b>) Electron beam divergence. (<b>c</b>–<b>f</b>) Electron beam parameters as a function of beam fraction (defined in the text). The peak energy gain is around a MeV for both laser powers used. The 0.5 TW laser, while providing around half the energy gain, allows for better beam quality.</p> Full article ">Figure 5
<p>Final beam parameters of an accelerated electron beam with an initial energy of 20 MeV. (<b>a</b>) Electron energy distribution. (<b>b</b>) Electron beam divergence. (<b>c</b>–<b>f</b>) Electron beam parameters as a function of beam fraction (defined in the text). Compared to <a href="#photonics-11-01066-f004" class="html-fig">Figure 4</a>, we see substantially improved acceleration and beam quality, and the 1 TW laser provides around 3 times the peak energy gain as the 0.5 TW laser.</p> Full article ">Figure A1
<p>Comparison of the simulations conducted in [<a href="#B15-photonics-11-01066" class="html-bibr">15</a>] and a simulated Ti:Sa and CO₂ laser in ASTRA. The laser parameters are listed in the text. We see good agreement between all three sets of simulations, and, notably, the energy gain of the Ti:Sa laser and the CO₂ laser are identical, as predicted.</p> Full article ">Figure A2
<p>Comparison of a linearly polarized beam and a radially polarized beam. The linearly polarized beam shows noticeably lower energy gain and much higher divergence. The plot of energy evolution corresponds to the particle with the highest final energy. Inset: zoomed-in view of energy oscillations near the focus.</p> Full article ">

Figure 1
<p>Artificial random reflectors of different types fs-inscribed inside the GRIN fiber core: (<b>a</b>) 1D in-line point reflector written along the fiber axis by the direct P-b-P technique, (<b>b</b>) similar 1D in-line reflector shifted off the axis; (<b>c</b>) 3D ring reflector written by SLM-assisted L-b-L technique, meaning that circular lines of overlapping points are inscribed in different planes with the average distance between the planes Δ<span class="html-italic">L</span> and integral length of the structure <span class="html-italic">L</span>.</p> Full article ">Figure 2
<p>Optical scheme for SLM-assisted writing of random reflective structures using a 4f system with a focus length of 20 cm.</p> Full article ">Figure 3
<p>Rayleigh backscattering level (<b>a</b>) and reflection spectrum (<b>b</b>) of ring-shaped random structure with length <span class="html-italic">L</span> = 2 mm inscribed in GRIN fiber.</p> Full article ">Figure 4
<p>Scheme of the Raman fiber laser with a random distributed reflector.</p> Full article ">Figure 5
<p>Output Stokes power together with a residual pump (<b>a</b>), spectra at different input pump powers (<b>b</b>) and output beam quality and intensity profile (color corresponds to intensity) in the waist shown in the inset, (<b>c</b>) of the MM RFL with an OC in-line random reflector (<span class="html-italic">L</span> = 120 mm, Δ<span class="html-italic">L</span> = 25 µm).</p> Full article ">Figure 6
<p>Output Stokes power together with a residual pump (<b>a</b>), spectra at different input pump powers (<b>b</b>) and output beam quality and intensity profile (color corresponds to intensity) in the waist shown in the inset (<b>c</b>) of the MM RFL with an OC in-line random reflector (<span class="html-italic">L</span> = 60 mm, Δ<span class="html-italic">L</span> = 50 µm).</p> Full article ">Figure 7
<p>Output Stokes power together with a residual pump (<b>a</b>), spectra at different input pump powers (<b>b</b>) and output beam quality and intensity profile (color corresponds to intensity) in the waist shown in the inset (<b>c</b>) of the MM RFL with an OC in-line random reflector (<span class="html-italic">L</span> = 2 × 120 mm, Δ<span class="html-italic">L</span> = 25 µm) with a relative shift Δ<span class="html-italic">y</span> ~ 3 μm.</p> Full article ">Figure 8
<p>(<b>a</b>) Output power of laser with different samples of 3D random distributed reflectors (OC). (<b>b</b>) Generated Stokes power shown in larger scale.</p> Full article ">Figure 9
<p>Output laser spectra in comparison with HR FBG reflectance spectra for different 3D random reflectors.</p> Full article ">Figure 10
<p>Measured beam quality parameter M² for lasers with different random reflectors at maximum RFL power (from left to right): 2.75, 3, 3.2 μJ. Inset: beam intensity (marked by different colors) profile in the waist.</p> Full article ">Figure 11
<p>Beam intensity profile at 5.3 W output power captured in the plane of the fiber end face (OC 3.2 μJ).</p> Full article ">

Figure 1
<p>Schematic diagram of the RFA structure.</p> Full article ">Figure 2
<p>The change in TMI threshold and output signal LP₀₁ power with varying seed purities. (<b>a</b>) When the LP₀₁ power in seed is fixed at 5 W; (<b>b</b>) When the seed power is fixed at 50 W.</p> Full article ">Figure 3
<p>The TMI threshold under different seed power.</p> Full article ">Figure 4
<p>The TMI threshold under different pump purities.</p> Full article ">Figure 5
<p>(<b>a</b>) The TMI threshold and (<b>b</b>) the LP₀₁ content varies with different seed and pump purities.</p> Full article ">Figure 6
<p>The pump power, total signal power, and LP₀₁ power of the signal under different LP₁₁ mode losses when TMI occurs.</p> Full article ">Figure 7
<p>(<b>a</b>) Power evolution in RFA; (<b>b</b>) Power evolution and proportion of Stokes frequency-shifted LP₁₁ mode at different positions along the fiber under a LP₁₁ mode loss of 4.34 dB/m.</p> Full article ">Figure 8
<p>(<b>a</b>) Total output signal power and LP₀₁ power of different fiber lengths; (<b>b</b>) LP₀₁ mode purity at TMI thresholds of different fiber lengths.</p> Full article ">

Figure 1
<p>Plots of <italic>x</italic>₆(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>β</italic>, <italic>λ</italic>) = (1.9, 1.9, 0.3) (solid line), (2.1, 1.8, 0.4) (dotted line), and (2.2, 1.7, 0.5) (dashed line).</p> Full article ">Figure 2
<p>Plots of <italic>h</italic>₆(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>β</italic>, <italic>λ</italic>) = (1.9, 1.9, 0.3) (solid line), (2.1, 1.8, 0.4) (dotted line), and (2.2, 1.7, 0.5) (dashed line).</p> Full article ">Figure 3
<p>Plots of <italic>g</italic>₆(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>β</italic>, <italic>λ</italic>) = (1.9, 1.9, 0.3) (solid line), (2.1, 1.8, 0.4) (dotted line), and (2.2, 1.7, 0.5) (dashed line).</p> Full article ">Figure 4
<p>Plots of <italic>x</italic>_6zs(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>β</italic>, <italic>λ</italic>) = (1.9, 1.9, 0.3) (solid line), (2.1, 1.8, 0.4) (dotted line), and (2.2, 1.7, 0.5) (dashed line).</p> Full article ">Figure 5
<p>Plots of |<italic>H</italic>₆(<italic>ω</italic>)| and <italic>φ</italic>₆(<italic>ω</italic>) for <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>β</italic>, <italic>λ</italic>) = (1.9, 1.0, 0) (solid line), (2.1, 0.8, 0.1) (dotted line), and (2.5, 0.5, 0.6) (dashed line). (<bold>a</bold>) |<italic>H</italic>₆(<italic>ω</italic>)|; (<bold>b</bold>) <italic>φ</italic>₆(<italic>ω</italic>); (<bold>c</bold>) log<italic>φ</italic>₆(<italic>ω</italic>).</p> Full article ">Figure 6
<p>Plots of <italic>x</italic>₁(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when <italic>α</italic> = 1.5 (solid line), 1.7 (dotted line), and 1.9 (dashed line).</p> Full article ">Figure 7
<p>Plots of <italic>h</italic>₁(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when <italic>α</italic> = 1.5 (solid line), 1.7 (dotted line), and 1.9 (dashed line).</p> Full article ">Figure 8
<p>Plots of <italic>g</italic>₁(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when <italic>α</italic> = 1.5 (solid line), 1.7 (dotted line), and 1.9 (dashed line).</p> Full article ">Figure 9
<p>Plots of <italic>x</italic>_1zs(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when <italic>α</italic> = 1.5 (solid line), 1.7 (dotted line), and 1.9 (dashed line).</p> Full article ">Figure 10
<p>Plots of |<italic>H</italic>₁(<italic>ω</italic>)| and <italic>φ</italic>₁(<italic>ω</italic>) for <italic>m</italic> = 1 and <italic>k</italic> = 1 when <italic>α</italic> = 1.7 (solid line), 1.9 (dotted line), and 2.2 (dashed line). (<bold>a</bold>) |<italic>H</italic>₁(<italic>ω</italic>)|; (<bold>b</bold>) log|<italic>H</italic>₁(<italic>ω</italic>)|; (<bold>c</bold>) <italic>φ</italic>₁(<italic>ω</italic>).</p> Full article ">Figure 11
<p>Plots of <italic>x</italic>₂(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when <italic>β</italic> = 0.9 (solid line), 1.4 (dotted line), and 1.9 (dashed line).</p> Full article ">Figure 12
<p>Plots of <italic>h</italic>₂(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when <italic>β</italic> = 0.9 (solid line), 1.4 (dotted line), and 1.9 (dashed line).</p> Full article ">Figure 13
<p>Plots of <italic>g</italic>₂(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when <italic>β</italic> = 0.9 (solid line), 1.4 (dotted line), and 1.9 (dashed line).</p> Full article ">Figure 14
<p>Plots of <italic>x</italic>_2zs(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when <italic>β</italic> = 0.9 (solid line), 1.4 (dotted line), and 1.9 (dashed line).</p> Full article ">Figure 15
<p>Plots of |<italic>H</italic>₂(<italic>ω</italic>)| and <italic>φ</italic>₂(<italic>ω</italic>) for <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when <italic>β</italic> = 0.9 (solid line), 1.5 (dotted line), and 1.9 (dashed line). (<bold>a</bold>) |<italic>H</italic>₂(<italic>ω</italic>)|; (<bold>b</bold>) log|<italic>H</italic>₂(<italic>ω</italic>)|; (<bold>c</bold>). <italic>φ</italic>₂(<italic>ω</italic>).</p> Full article ">Figure 16
<p>Plots of <italic>x</italic>₃(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>β</italic>) = (1.3, 1.9) (solid line), (1.6, 1.8) (dotted line), and (1.9, 1.7) (dashed line).</p> Full article ">Figure 17
<p>Plots of <italic>h</italic>₃(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>β</italic>) = (1.3, 1.9) (solid line), (1.6, 1.8) (dotted line), and (1.9, 1.7) (dashed line).</p> Full article ">Figure 18
<p>Plots of <italic>g</italic>₃(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>β</italic>) = (1.3, 1.9) (solid line), (1.6, 1.8) (dotted line), and (1.9, 1.7) (dashed line).</p> Full article ">Figure 19
<p>Plots of <italic>x</italic>_3zs(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>β</italic>) = (1.3, 1.9) (solid line), (1.6, 1.8) (dotted line), and (1.9, 1.7) (dashed line).</p> Full article ">Figure 20
<p>Plots of |<italic>H</italic>₃(<italic>ω</italic>)| and <italic>φ</italic>₃(<italic>ω</italic>) for <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>β</italic>) = (1.3, 1.9) (solid line), (1.6, 1.8) (dotted line), and (1.9, 1.7) (dashed line). (<bold>a</bold>) |<italic>H</italic>₃(<italic>ω</italic>)|; (<bold>b</bold>) log|<italic>H</italic>₃(<italic>ω</italic>)|; (<bold>c</bold>) <italic>φ</italic>₃(<italic>ω</italic>); (<bold>d</bold>) log<italic>φ</italic>₃(<italic>ω</italic>).</p> Full article ">Figure 21
<p>Plots of <italic>x</italic>₄(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>λ</italic>) = (1.3, 0.2) (solid line), (1.6, 0.4) (dotted line), and (1.9, 0.6) (dashed line).</p> Full article ">Figure 22
<p>Plots of <italic>h</italic>₄(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>λ</italic>) = (1.3, 0.2) (solid line), (1.6, 0.4) (dotted line), and (1.9, 0.6) (dashed line).</p> Full article ">Figure 23
<p>Plots of <italic>g</italic>₄(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>λ</italic>) = (1.3, 0.2) (solid line), (1.6, 0.4) (dotted line), and (1.9, 0.6) (dashed line).</p> Full article ">Figure 24
<p>Plots of <italic>x</italic>_4zs(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>λ</italic>) = (1.3, 0.2) (solid line), (1.6, 0.5) (dotted line), and (1.9, 0.8) (dashed line).</p> Full article ">Figure 25
<p>Plots of |<italic>H</italic>₄(<italic>ω</italic>)| and <italic>φ</italic>₄(<italic>ω</italic>) for <italic>m</italic> = 1 and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>λ</italic>) = (1.9, 0.1) (solid line), (2.1, 0.2) (dotted line), and (2.5, 0.3) (dashed line). (<bold>a</bold>) |<italic>H</italic>₄(<italic>ω</italic>)|; (<bold>b</bold>) log|<italic>H</italic>₄(<italic>ω</italic>)|; (<bold>c</bold>) <italic>φ</italic>₄(<italic>ω</italic>); (<bold>d</bold>) log<italic>φ</italic>₄(<italic>ω</italic>).</p> Full article ">Figure 25 Cont.
<p>Plots of |<italic>H</italic>₄(<italic>ω</italic>)| and <italic>φ</italic>₄(<italic>ω</italic>) for <italic>m</italic> = 1 and <italic>k</italic> = 1 when (<italic>α</italic>, <italic>λ</italic>) = (1.9, 0.1) (solid line), (2.1, 0.2) (dotted line), and (2.5, 0.3) (dashed line). (<bold>a</bold>) |<italic>H</italic>₄(<italic>ω</italic>)|; (<bold>b</bold>) log|<italic>H</italic>₄(<italic>ω</italic>)|; (<bold>c</bold>) <italic>φ</italic>₄(<italic>ω</italic>); (<bold>d</bold>) log<italic>φ</italic>₄(<italic>ω</italic>).</p> Full article ">Figure 26
<p>Plots of <italic>x</italic>₅(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when <italic>λ</italic> = 0.3 (solid line), 0.4 (dot line), and 0.5 (dash line).</p> Full article ">Figure 27
<p>Plots of <italic>h</italic>₅(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when <italic>λ</italic> = 0.3 (solid line), 0.4 (dotted line), and 0.5 (dashed line).</p> Full article ">Figure 28
<p>Plots of <italic>g</italic>₅(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when <italic>λ</italic> = 0.3 (solid line), 0.4 (dotted line), and 0.5 (dashed line).</p> Full article ">Figure 29
<p>Plots of <italic>x</italic>_5zs(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, and <italic>k</italic> = 1 when <italic>λ</italic> = 0.3 (solid line), 0.4 (dotted line), and 0.5 (dashed line).</p> Full article ">Figure 30
<p>Plots of |<italic>H</italic>₅(<italic>ω</italic>)| and <italic>φ</italic>₅(<italic>ω</italic>) for <italic>m</italic> = 1 and <italic>k</italic> = 1 when <italic>λ</italic> = 0.1 (solid line), 0.2 (dotted line), and 0.3 (dashed line). (<bold>a</bold>) |<italic>H</italic>₅(<italic>ω</italic>)|; (<bold>b</bold>) log|<italic>H</italic>₅(<italic>ω</italic>)|; (<bold>c</bold>) <italic>φ</italic>₅(<italic>ω</italic>).</p> Full article ">Figure 30 Cont.
<p>Plots of |<italic>H</italic>₅(<italic>ω</italic>)| and <italic>φ</italic>₅(<italic>ω</italic>) for <italic>m</italic> = 1 and <italic>k</italic> = 1 when <italic>λ</italic> = 0.1 (solid line), 0.2 (dotted line), and 0.3 (dashed line). (<bold>a</bold>) |<italic>H</italic>₅(<italic>ω</italic>)|; (<bold>b</bold>) log|<italic>H</italic>₅(<italic>ω</italic>)|; (<bold>c</bold>) <italic>φ</italic>₅(<italic>ω</italic>).</p> Full article ">Figure 31
<p>Plots of <italic>x</italic>₅(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>β</italic>, <italic>λ</italic>) = (1.9, 0.3) (solid line), (1.8, 0.4) (dotted line), and (1.7, 0.5) (dashed line).</p> Full article ">Figure 32
<p>Plots of <italic>h</italic>₇(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>β</italic>, <italic>λ</italic>) = (1.9, 0.3) (solid line), (1.8, 0.4) (dotted line), and (1.7, 0.5) (dashed line).</p> Full article ">Figure 33
<p>Plots of <italic>g</italic>₇(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>β</italic>, <italic>λ</italic>) = (1.9, 0.3) (solid line), (1.8, 0.4) (dotted line), and (1.7, 0.5) (dashed line).</p> Full article ">Figure 34
<p>Plots of <italic>x</italic>_7zs(<italic>t</italic>) for <italic>ω</italic> = 1.3, <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>β</italic>, <italic>λ</italic>) = (1.9, 0.3) (solid line), (1.8, 0.4) (dotted line), and (1.7, 0.5) (dashed line).</p> Full article ">Figure 35
<p>Plots of |<italic>H</italic>₇(<italic>ω</italic>)| and <italic>φ</italic>₇(<italic>ω</italic>) for <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>β</italic>, <italic>λ</italic>) = (1.9, 0.05) (solid line), (1.8, 0.10) (dotted line), and (1.7, 0.15) (dashed line). (<bold>a</bold>) |<italic>H</italic>₇(<italic>ω</italic>)|; (<bold>b</bold>) log|<italic>H</italic>₇(<italic>ω</italic>)|; (<bold>c</bold>) <italic>φ</italic>₇(<italic>ω</italic>); (<bold>d</bold>) log<italic>φ</italic>₇(<italic>ω</italic>).</p> Full article ">Figure 35 Cont.
<p>Plots of |<italic>H</italic>₇(<italic>ω</italic>)| and <italic>φ</italic>₇(<italic>ω</italic>) for <italic>m</italic> = 1, <italic>c</italic> = 0.2, and <italic>k</italic> = 1 when (<italic>β</italic>, <italic>λ</italic>) = (1.9, 0.05) (solid line), (1.8, 0.10) (dotted line), and (1.7, 0.15) (dashed line). (<bold>a</bold>) |<italic>H</italic>₇(<italic>ω</italic>)|; (<bold>b</bold>) log|<italic>H</italic>₇(<italic>ω</italic>)|; (<bold>c</bold>) <italic>φ</italic>₇(<italic>ω</italic>); (<bold>d</bold>) log<italic>φ</italic>₇(<italic>ω</italic>).</p> Full article ">Figure 36
<p>Plots of <italic>w</italic>₁(<italic>x</italic>, <italic>t</italic>) = w1 of a free-free damped Euler–Bernoulli beam when <italic>f_j</italic>(<italic>t</italic>) = <italic>u</italic>(<italic>t</italic>) and <italic>l</italic> = 10 for <italic>ω</italic> = 0.0314, <italic>a</italic> = 1, <italic>b</italic> = 1, <italic>E</italic> = 1, <italic>x</italic> = 0, …, 10, and <italic>t</italic> = 0, …, 60. (<bold>a</bold>) w1 for <italic>α</italic> = 1.8; <italic>β</italic> = <italic>λ</italic> = 1.2. (<bold>b</bold>) w1 for <italic>α</italic> = 1.8, <italic>β</italic> = 1.2, and <italic>λ</italic> = 0.1.</p> Full article ">Figure 37
<p>Plot of P-M spectrum.</p> Full article ">Figure 38
<p>Plots of |<italic>H</italic>₁(<italic>ω</italic>)|² with <italic>α</italic> = 1.8 (solid), 2.4 (dotted), and 2.8 (dashed) when <italic>m</italic> = 1 and <italic>k</italic> = 36. (<bold>a</bold>) |<italic>H</italic>₁(<italic>ω</italic>)|²; (<bold>b</bold>) log|<italic>H</italic>₁(<italic>ω</italic>)|².</p> Full article ">Figure 39
<p>Plots of response PSD <italic>S_xx</italic>₁(<italic>ω</italic>) with <italic>α</italic> = 1.8 (solid), 2.4 (dotted), and 2.8 (dashed) when <italic>m</italic> = 1, <italic>k</italic> = 36, and <italic>V</italic> = 15. (<bold>a</bold>) <italic>S_xx</italic>₁(<italic>ω</italic>); (<bold>b</bold>) log<italic>S_xx</italic>₁(<italic>ω</italic>).</p> Full article ">Figure 40
<p>Resonance when with <italic>α</italic> = 2, <italic>m</italic> = 1, <italic>k</italic> = 36, and <italic>V</italic> = 15. (<bold>a</bold>) Plot of log|<italic>H</italic>₁(<italic>ω</italic>)|²; (<bold>b</bold>) plot of response PSD log<italic>S_xx</italic>₁(<italic>ω</italic>).</p> Full article ">Figure 41
<p>Illustrations of cross-PSD response <italic>S_fx</italic>₁(<italic>ω</italic>) with <italic>α</italic> = 1.8 (solid), 2.4 (dotted), and 2.8 (dashed) when <italic>m</italic> = 1, <italic>k</italic> = 36, and <italic>V</italic> = 15. (<bold>a</bold>) |<italic>S_fx</italic>₁(<italic>ω</italic>)|; (<bold>b</bold>) log|<italic>S_fx</italic>₁(<italic>ω</italic>)|; (<bold>c</bold>) Phase of <italic>S_fx</italic>₁(<italic>ω</italic>). (<bold>d</bold>). Phase of <italic>S_fx</italic>₁(<italic>ω</italic>) in log scale.</p> Full article ">Figure 42
<p>Plots of random series of driven signal and response signals when <italic>m</italic> = 1, <italic>k</italic> = 36, and <italic>V</italic> = 15: (<bold>a</bold>) Driven signal. (<bold>b</bold>) Response <italic>x</italic>₁(<italic>t</italic>) for <italic>α</italic> = 1.8. (<bold>c</bold>) Response <italic>x</italic>₁(<italic>t</italic>) for <italic>α</italic> = 2.4. (<bold>d</bold>) Response <italic>x</italic>₁(<italic>t</italic>) for <italic>α</italic> = 2.4 in log scale. (<bold>e</bold>) Response <italic>x</italic>₁(<italic>t</italic>) for <italic>α</italic> = 2.8. (<bold>f</bold>) Response <italic>x</italic>₁(<italic>t</italic>) for <italic>α</italic> = 2.8 in log scale.</p> Full article ">Figure 42 Cont.
<p>Plots of random series of driven signal and response signals when <italic>m</italic> = 1, <italic>k</italic> = 36, and <italic>V</italic> = 15: (<bold>a</bold>) Driven signal. (<bold>b</bold>) Response <italic>x</italic>₁(<italic>t</italic>) for <italic>α</italic> = 1.8. (<bold>c</bold>) Response <italic>x</italic>₁(<italic>t</italic>) for <italic>α</italic> = 2.4. (<bold>d</bold>) Response <italic>x</italic>₁(<italic>t</italic>) for <italic>α</italic> = 2.4 in log scale. (<bold>e</bold>) Response <italic>x</italic>₁(<italic>t</italic>) for <italic>α</italic> = 2.8. (<bold>f</bold>) Response <italic>x</italic>₁(<italic>t</italic>) for <italic>α</italic> = 2.8 in log scale.</p> Full article ">Figure 43
<p>Plots of |<italic>H</italic>₂(<italic>ω</italic>)|² (log) with <italic>β</italic> = 0.4 (solid) and 1.8 (dotted) when <italic>m</italic> = 1, <italic>c</italic> = 0.1, and <italic>k</italic> = 36.</p> Full article ">Figure 44
<p>Plots of PSD response <italic>S_xx</italic>₂(<italic>ω</italic>) with <italic>β</italic> = 1.8 when <italic>m</italic> = 1, <italic>c</italic> = 0.1, <italic>k</italic> = 36, and <italic>V</italic> = 15: (<bold>a</bold>) <italic>S_xx</italic>₂(<italic>ω</italic>). (<bold>b</bold>) log<italic>S_xx</italic>₂(<italic>ω</italic>).</p> Full article ">Figure 45
<p>Resonance when with <italic>β</italic> = 1, <italic>m</italic> = 1, <italic>c</italic> = 0.1, <italic>k</italic> = 36, and <italic>V</italic> = 15: (<bold>a</bold>) log|<italic>H</italic>₂(<italic>ω</italic>)|². (<bold>b</bold>) log<italic>S_xx</italic>₂(<italic>ω</italic>).</p> Full article ">Figure 46
<p>Plots of cross-PSD response <italic>S_fx</italic>₂(<italic>ω</italic>) with <italic>β</italic> = 0.4 (solid) and 1.8 (dot) when <italic>m</italic> = 1, <italic>c</italic> = 0.1, <italic>k</italic> = 36, and <italic>V</italic> = 15: (<bold>a</bold>) log|<italic>S_fx</italic>₂(<italic>ω</italic>)|. (<bold>b</bold>) Phase of <italic>S_fx</italic>₂(<italic>ω</italic>) in log scale.</p> Full article ">Figure 47
<p>Simulated response series <italic>x</italic>₂(<italic>t</italic>) when <italic>m</italic> = 1, <italic>c</italic> = 0.1, <italic>k</italic> = 36, and <italic>V</italic> = 15: (<bold>a</bold>) <italic>x</italic>₂(<italic>t</italic>) for <italic>β</italic> = 0.4. (<bold>b</bold>) <italic>x</italic>₂(<italic>t</italic>) for <italic>β</italic> = 1. (<bold>c</bold>) <italic>x</italic>₂(<italic>t</italic>) for <italic>β</italic> = 1.4. (<bold>d</bold>) <italic>x</italic>₂(<italic>t</italic>) for <italic>β</italic> = 1.8.</p> Full article ">Figure 48
<p>Plots of |<italic>H</italic>₃(<italic>ω</italic>)|² when <italic>m</italic> = 1, <italic>c</italic> = 0.1, and <italic>k</italic> = 36 for (<italic>α</italic>, <italic>β</italic>) = (1.6, 0.8) (solid), (1.6, 1.8) (dotted), (2.5, 0.8) (dashed), (2.8, 1.8) (dashed–dotted). (<bold>a</bold>) |<italic>H</italic>₃(<italic>ω</italic>)|²; (<bold>b</bold>) log|<italic>H</italic>₃(<italic>ω</italic>)|².</p> Full article ">Figure 49
<p>Plots of response PSD <italic>S_xx</italic>₃(<italic>ω</italic>) when <italic>m</italic> = 1, <italic>c</italic> = 0.1, <italic>k</italic> = 36, and <italic>V</italic> = 15 for (<italic>α</italic>, <italic>β</italic>) = (1.6, 0.8) (solid), (1.6, 1.8) (dotted), (2.5, 0.8) (dashed), and (2.8, 1.8) (dashed–dotted). (<bold>a</bold>) <italic>S_xx</italic>₃(<italic>ω</italic>); (<bold>b</bold>) log<italic>S_xx</italic>₃(<italic>ω</italic>).</p> Full article ">Figure 50
<p>Plots of cross-PSD response <italic>S_fx</italic>₃(<italic>ω</italic>) when <italic>m</italic> = 1, <italic>c</italic> = 0.1, <italic>k</italic> = 36, and <italic>V</italic> = 15 for (<italic>α</italic>, <italic>β</italic>) = (1.6, 0.8) (solid), (1.6, 1.8) (dotted), (2.5, 0.8) (dashed), and (2.8, 1.8) (dashed–dotted). (<bold>a</bold>) log|<italic>S_fx</italic>₃(<italic>ω</italic>)|; (<bold>b</bold>) Phase of |<italic>S_fx</italic>₃(<italic>ω</italic>)|.</p> Full article ">Figure 51
<p>Plots of |<italic>H</italic>₄(<italic>ω</italic>)|² when <italic>m</italic> = 1, <italic>c</italic> = 0, and <italic>k</italic> = 36 for (<italic>α</italic>, <italic>λ</italic>) = (1.6, 0.2) (solid), (1.6, 0.4) (dotted), (2.5, 0.2) (dashed), and (2.5, 0.4) (dashed–dotted). (<bold>a</bold>) |<italic>H</italic>₄(<italic>ω</italic>)|²; (<bold>b</bold>) log|<italic>H</italic>₄(<italic>ω</italic>)|².</p> Full article ">Figure 52
<p>Plots of response PSD <italic>S_xx</italic>₄(<italic>ω</italic>) when <italic>m</italic> = 1, <italic>c</italic> = 0, <italic>k</italic> = 36, and <italic>V</italic> = 15 for (<italic>α</italic>, <italic>λ</italic>) = (1.6, 0.2) (solid), (1.6, 0.4) (dotted), (2.5, 0.2) (dashed), and (2.5, 0.4) (dashed–dotted). (<bold>a</bold>) <italic>S_xx</italic>₄(<italic>ω</italic>); (<bold>b</bold>) log<italic>S_xx</italic>₄(<italic>ω</italic>).</p> Full article ">Figure 53
<p>Plots of cross-PSD response <italic>S_fx</italic>₄(<italic>ω</italic>) when <italic>m</italic> = 1, <italic>c</italic> = 0, <italic>k</italic> = 36, and <italic>V</italic> = 15 for (<italic>α</italic>, <italic>λ</italic>) = (1.6, 0.2) (solid), (1.6, 0.4) (dotted), (2.5, 0.2) (dashed), and (2.5, 0.4) (dashed–dotted). (<bold>a</bold>) log|<italic>S_fx</italic>₄(<italic>ω</italic>)|; (<bold>b</bold>) phase of |<italic>S_fx</italic>₄(<italic>ω</italic>)|.</p> Full article ">Figure 54
<p>Plots of |<italic>H</italic>₅(<italic>ω</italic>)|² (log) when <italic>m</italic> = 1, <italic>c</italic> = 0, and <italic>k</italic> = 36 for <italic>λ</italic> = 0.2 (solid), 0.4 (dotted), 0.6 (dashed), and 0.8 (dashed–dotted).</p> Full article ">Figure 55
<p>Plots of response PSD <italic>S_xx</italic>₅(<italic>ω</italic>) when <italic>m</italic> = 1, <italic>c</italic> = 0, <italic>k</italic> = 36, and <italic>V</italic> = 15 for <italic>λ</italic> = 0.2 (solid), 0.4 (dotted), 0.6 (dashed), and 0.8 (dashed–dotted): (<bold>a</bold>) <italic>S_xx</italic>₅(<italic>ω</italic>); (<bold>b</bold>) log<italic>S_xx</italic>₅(<italic>ω</italic>).</p> Full article ">Figure 56
<p>Plots of cross-PSD response <italic>S_fx</italic>₅(<italic>ω</italic>) when <italic>m</italic> = 1, <italic>c</italic> = 0, <italic>k</italic> = 36, and <italic>V</italic> = 15 for <italic>λ</italic> = 0.2 (solid), 0.4 (dotted), 0.6) (dashed), and 0.8 (dashed–dotted): (<bold>a</bold>) log|<italic>S_fx</italic>₅(<italic>ω</italic>)|; (<bold>b</bold>) phase of |<italic>S_fx</italic>₅(<italic>ω</italic>)|.</p> Full article ">Figure 57
<p>Plots of |<italic>H</italic>₆(<italic>ω</italic>)|² (log) when <italic>m</italic> = 1, <italic>c</italic> = 0.1, and <italic>k</italic> = 36 for (<italic>α</italic>, <italic>β</italic>, <italic>λ</italic>) = (1.6, 0.8, 0.2) (solid), (1.6, 1.8, 0.4) (dotted), (2.5, 0.4, 0.2) (dashed), and (2.5, 0.8, 0.4) (dashed–dotted).</p> Full article ">Figure 58
<p>Plots of response PSD <italic>S_xx</italic>₆(<italic>ω</italic>) in log scale when <italic>m</italic> = 1, <italic>c</italic> = 0.1, <italic>k</italic> = 36, and <italic>V</italic> = 15 for (<italic>α</italic>, <italic>β</italic>, <italic>λ</italic>) = (1.6, 0.8, 0.2) (solid), (1.6, 1.8, 0.4) (dotted), (2.5, 0.4, 0.2) (dashed), and (2.5, 0.8, 0.4) (dashed–dotted).</p> Full article ">Figure 59
<p>Plots of cross-PSD response <italic>S_fx</italic>₆(<italic>ω</italic>) when <italic>m</italic> = 1, <italic>c</italic> = 0.1, <italic>k</italic> = 36, and <italic>V</italic> = 15 for (<italic>α</italic>, <italic>β</italic>, <italic>λ</italic>) = (1.6, 0.8, 0.2) (solid), (1.6, 1.8, 0.4) (dotted), (2.5, 0.4, 0.2) (dashed), and (2.5, 0.8, 0.4) (dashed–dotted): (<bold>a</bold>) log|<italic>S_fx</italic>₆(<italic>ω</italic>)|; (<bold>b</bold>) phase of |<italic>S_fx</italic>₆(<italic>ω</italic>)|.</p> Full article ">Figure 60
<p>Plots of |<italic>H</italic>₇(<italic>ω</italic>)|² (log) when <italic>m</italic> = 1, <italic>c</italic> = 0.1, and <italic>k</italic> = 36, for (<italic>β</italic>, <italic>λ</italic>) = (0.5, 0.2) (solid), (0.5, 0.3) (dot), (1.5, 0.4) (dash), (1.5, 0.6) (dash dot).</p> Full article ">Figure 61
<p>Plots of response PSD <italic>S_xx</italic>₇(<italic>ω</italic>) in log scale when <italic>m</italic> = 1, <italic>c</italic> = 0.1, <italic>k</italic> = 36, and <italic>V</italic> = 15 for (<italic>β</italic>, <italic>λ</italic>) = (0.5, 0.2) (solid), (0.5, 0.3) (dot), (1.5, 0.4) (dash), (1.5, 0.6) (dash dot).</p> Full article ">Figure 62
<p>Plots of cross-PSD response <italic>S_fx</italic>₇(<italic>ω</italic>) when <italic>m</italic> = 1, <italic>c</italic> = 0.1, <italic>k</italic> = 36, and <italic>V</italic> = 15 for (<italic>β</italic>, <italic>λ</italic>) = (0.5, 0.2) (solid), (0.5, 0.3) (dot), (1.5, 0.4) (dash), (1.5, 0.6) (dash dot). (<bold>a</bold>). log|<italic>S_fx</italic>₇(<italic>ω</italic>)|. (<bold>b</bold>). Phase of |<italic>S_fx</italic>₇(<italic>ω</italic>)| in log scale.</p> Full article ">

Figure 1
<p>Experimental set-up of the aqueous drop at the end of the capillary tube, this itself mounted on a vertical translation stage. The bottom air–aqueous drop interface and volume are illuminated with the fundamental beam of a femtosecond Ti-Sa laser at 810 nm, and the output SHG light is captured with a photomultiplier tube placed behind a monochromator. The light beam is focused and captured through the same objective (see the text for further details). Red line indicates path of the fundamental beam whereas blue line indicates path of harmonic beam.</p> Full article ">Figure 2
<p>Extinction spectrum for a 2 nM aqueous solution of 16 nm diameter gold nanoparticles (AuNPs), synthesized using Turkevich’s method. Insert: Transmission electron microscopy (TEM) of sample; the scale bar is 20 nm.</p> Full article ">Figure 3
<p>SHG intensity profile, collected at 405 nm, of the aqueous solution–air interface as the position of the laser focal point was scanned from within and outside of a water droplet in the absence of AuNPs. The disks represent the experimental data and the red line represents the fit to the model (see <a href="#sec3dot3-photonics-11-00789" class="html-sec">Section 3.3</a>). The parameter obtained from the model is <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>; see below.</p> Full article ">Figure 4
<p>SHG intensity profile, collected at 405 nm, of the AuNP-modified aqueous solution–air interface as the position of the laser focal point was scanned from within and outside of a water droplet containing a 1:4 <span class="html-italic">v</span>/<span class="html-italic">v</span> dilution of a 2 nM aqueous solution measuring 16 nm in diameter AuNPs. At z &gt; 0.40 mm, the laser beam waist was positioned in the bulk aqueous phase, and at z &lt; 0.40 mm, it was located in air. The disks represent the experimental data, and the red line represents the fit to the model (see <a href="#sec3dot3-photonics-11-00789" class="html-sec">Section 3.3</a>). The parameter obtained from the model is <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1.65</mn> </mrow> </semantics></math>; see below.</p> Full article ">Figure 5
<p>SHG intensity, obtained at 405 nm, of the AuNP-modified aqueous solution–air interface as the position of the laser focal point was scanned in and out of a water droplet containing a 2 nM aqueous solution of 16 nm diameter AuNPs. At <span class="html-italic">z</span> &gt; 0.40 mm, the laser beam waist was positioned in the bulk aqueous phase and at <span class="html-italic">z</span> &lt; 0.40 mm, it was located in air. The disks represent the experimental data, and the red line represents the theoretical model fitting (see <a href="#sec3dot3-photonics-11-00789" class="html-sec">Section 3.3</a>). The parameter obtained from the model is <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>7.9</mn> </mrow> </semantics></math>; see below.</p> Full article ">Figure 6
<p>Plot of the <span class="html-italic">α</span> parameter as a function of the concentration of the aqueous 16 nm diameter AuNP solution. The disks represent the experimental data, and the red line represents the linear adjustment.</p> Full article ">

Figure 1
<p>Comparison of Experimental Data and Curve Fitting Representation of Storage and Loss Moduli vs. Magnetic Flux Densities at 2 Hz and 15% Shear Strain [<a href="#B16-vibration-07-00032" class="html-bibr">16</a>].</p> Full article ">Figure 2
<p>Demagnetization Curve for N52 Permanent Magnet.</p> Full article ">Figure 3
<p>Electromagnet Layout with Parametric Dimensions.</p> Full article ">Figure 4
<p>(<b>a</b>) FEMM Results for Input Current of −3 A; (<b>b</b>) FEMM Results for Input Current of 3 A.</p> Full article ">Figure 5
<p>Comparison of the FEMM Results with the Curve Fitted 4th-Order Polynomial of the Magnetic Flux Density vs. Input Current Curve.</p> Full article ">Figure 6
<p>(<b>a</b>) Overhang Beam with Spring at Overhang Support; (<b>b</b>) Equivalent Beam Model with MRE in Direct shear.</p> Full article ">Figure 7
<p>Shear and Bending Moment Shown in Red of (<b>a</b>) Cantilever Beam and (<b>b</b>) Overhanging Beam.</p> Full article ">Figure 8
<p>Equivalent Single-Degree-of-Freedom System.</p> Full article ">Figure 9
<p>Curve Fit for Equivalent Stiffness Storage and Loss Components, Equivalent Mass, and Equivalent Damping.</p> Full article ">Figure 10
<p>Comparison of Transient Response and Input Current between Passive and Semi-active Systems due to different Control Strategies.</p> Full article ">Figure 11
<p>Steady-State Time Response and Control Current for Different Controllers under Harmonic Input at (<b>a</b>) 5.14 Hz and (<b>b</b>) at 7.06 Hz.</p> Full article ">Figure 12
<p>Time Response and Controller Input for Random Input over (<b>a</b>) 50 Seconds and (<b>b</b>) 5 Seconds.</p> Full article ">Figure A1
<p>Static and Dynamic Shear Strain Representations on Deflected MRE.</p> Full article ">Figure A2
<p>Free-Body Diagram of Proposed Beam Model under Static Loading.</p> Full article ">

Figure 1
<p>Schematic diagram of focusing system geometry.</p> Full article ">Figure 2
<p>Polarization distributions of incident vector beam with different orders: (<b>a</b>) <span class="html-italic">m</span> = 0, <span class="html-italic">ϕ</span>₀ = 0, (<b>b</b>) <span class="html-italic">m</span> = 1, <span class="html-italic">ϕ</span>₀ = 0, (<b>c</b>) <span class="html-italic">m</span> = −1, <span class="html-italic">ϕ</span>₀ = 0, and (<b>d</b>) <span class="html-italic">m</span> = 2, <span class="html-italic">ϕ</span>₀ = 0. Arrows indicate the direction of polarization at the arrow location.</p> Full article ">Figure 3
<p>The intensity distribution on the focal plane of vector beams <span class="html-italic">E_x</span>, <span class="html-italic">E_y</span>, <span class="html-italic">E_z</span> and <span class="html-italic">E_total</span> with different orders: (<b>a</b>–<b>d</b>) <span class="html-italic">m</span> = 0, (<b>e</b>–<b>h</b>) <span class="html-italic">m</span> = 1, (<b>i</b>–<b>l</b>) <span class="html-italic">m</span> = −1, (<b>m</b>–<b>p</b>) <span class="html-italic">m</span> = 2.</p> Full article ">Figure 4
<p>The intensity distribution on the focal plane of vector beams <span class="html-italic">H_x</span>, <span class="html-italic">H_y</span>, <span class="html-italic">H_z</span> and <span class="html-italic">H_total</span> with different orders: (<b>a</b>–<b>d</b>) <span class="html-italic">m</span> = 0, (<b>e</b>–<b>h</b>) <span class="html-italic">m</span> = 1, (<b>i</b>–<b>l</b>) <span class="html-italic">m</span> = −1, (<b>m</b>–<b>p</b>) <span class="html-italic">m</span> = 2.</p> Full article ">Figure 5
<p>The iso-surface intensity distribution of the electric and magnetic fields in the tightly focused region with <span class="html-italic">E</span> = 0.5<span class="html-italic">E_max</span> and <span class="html-italic">H</span> = 0.5<span class="html-italic">H_max</span>.</p> Full article ">Figure 6
<p>The intensity contour of the electric and magnetic fields in the focal plane for the vector beams: (<b>a</b>,<b>b</b>) m = 0, <span class="html-italic">ϕ</span>₀ = π/4, (<b>c</b>,<b>d</b>) m = 0, <span class="html-italic">ϕ</span>₀ = π/2, (<b>e</b>,<b>f</b>) m = 1, <span class="html-italic">ϕ</span>₀ = π/4, (<b>g</b>,<b>h</b>) m = 1, <span class="html-italic">ϕ</span>₀ = π/2, (<b>i</b>,<b>j</b>) m = −1, <span class="html-italic">ϕ</span>₀ = π/4, (<b>k</b>,<b>l</b>) m = −1, <span class="html-italic">ϕ</span>₀ =π/2, (<b>m</b>,<b>n</b>) m = 2, <span class="html-italic">ϕ</span>₀ = π/4, (<b>o</b>,<b>p</b>) m = 2, <span class="html-italic">ϕ</span>₀ = π/2. The short lines indicate the polarization distributions.</p> Full article ">Figure 7
<p>The gradient force distributions produced by highly focused CV beams with <span class="html-italic">m</span> equal to 0, 1, −1, and 2. (<b>a</b>–<b>d</b>) represent the transverse gradient force distributions. (<b>e</b>–<b>h</b>) represent the longitudinal gradient force distributions. Arrows denote the direction and magnitude of the gradient force.</p> Full article ">Figure 8
<p>The longitudinal scattering force in the focal plane for different orders <span class="html-italic">m</span> of CV beam: (<b>a</b>) <span class="html-italic">m</span> = 0, (<b>b</b>) <span class="html-italic">m</span> = 1, (<b>c</b>) <span class="html-italic">m</span> = −1, (<b>d</b>) <span class="html-italic">m</span> = 2. Arrows denote the direction and magnitude of the scattering force.</p> Full article ">Figure 9
<p>The longitudinal curl-spin force for different orders <span class="html-italic">m</span> of CV beam: (<b>a</b>) <span class="html-italic">m</span> = 0, (<b>b</b>) <span class="html-italic">m</span> = 1, (<b>c</b>) <span class="html-italic">m</span> = −1, (<b>d</b>) <span class="html-italic">m</span> = 2. Arrows denote the direction and magnitude of the curl-spin force.</p> Full article ">

Figure 1
<p>Contours of <span class="html-italic">u</span> (blue) and <span class="html-italic">v</span> (red) for FrA1 for diffraction of a plane wave by a circular aperture.</p> Full article ">Figure 2
<p>Contours of <span class="html-italic">u</span> (blue) and <span class="html-italic">v</span> (red) for FrA2 for diffraction of a plane wave by a circular aperture.</p> Full article ">Figure 3
<p>Contours of <span class="html-italic">u</span> (blue) and <span class="html-italic">v</span> (red) for the generalized Fresnel approximation for diffraction of a plane wave by a circular aperture.</p> Full article ">Figure 4
<p>Contours of <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>=</mo> <mi>k</mi> <mi>z</mi> <mo>+</mo> <msup> <mi>v</mi> <mn>2</mn> </msup> <mo>/</mo> <mn>2</mn> <mi>u</mi> </mrow> </semantics></math> for the generalized Fresnel approximation for diffraction of a plane wave by a circular aperture.</p> Full article ">Figure 5
<p>Schematic diagram of a spherical wave diffracted by a circular aperture of radius <span class="html-italic">a</span> in an opaque screen at distance <span class="html-italic">d</span> from the geometrical focal point <span class="html-italic">F</span>, with sagitta <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mi>f</mi> <mo>−</mo> <mi>d</mi> </mrow> </semantics></math>; the angle <math display="inline"><semantics> <mi>α</mi> </semantics></math> is the semi-angular aperture, and <math display="inline"><semantics> <mi>θ</mi> </semantics></math> is the angle subtended at the optical axis by a ray from a general point <span class="html-italic">W</span> on the wavefront to <span class="html-italic">F</span>. The points <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> on the aperture edge correspond to the maximum and minimum distances <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>g</mi> <mn>2</mn> </msub> </mrow> </semantics></math> from the observation point <span class="html-italic">P</span>, with cylindrical coordinates <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>,</mo> <mi>ϕ</mi> <mo>,</mo> <mi>z</mi> </mrow> </semantics></math> relative to <span class="html-italic">F</span> and distant <math display="inline"><semantics> <msub> <mi>r</mi> <mi>P</mi> </msub> </semantics></math> from <span class="html-italic">F</span> and <span class="html-italic">r</span> from <span class="html-italic">W</span>. The triangle <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>F</mi> <mi>P</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> has an exterior angle <math display="inline"><semantics> <mrow> <mo>∡</mo> <msub> <mi>E</mi> <mn>2</mn> </msub> <mi>F</mi> <mi>A</mi> <mo>=</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </semantics></math> and interior angles <math display="inline"><semantics> <mrow> <mo>∡</mo> <mi>F</mi> <mi>P</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>∡</mo> <mi>F</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> <mi>P</mi> <mo>=</mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> </mrow> </semantics></math>. The angle rays between rays from point <span class="html-italic">W</span> to <span class="html-italic">F</span> and <span class="html-italic">P</span> are <math display="inline"><semantics> <mrow> <mo>∡</mo> <mi>F</mi> <mi>W</mi> <mi>P</mi> <mo>=</mo> <mi>ψ</mi> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The behaviour of (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>/</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>N</mi> <mi>f</mi> </msub> </mrow> </semantics></math> (from Ks) and (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>N</mi> <mi>f</mi> </msub> </mrow> </semantics></math> (from aKs), with <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> for different semi-angular apertures. Lines are coloured red for small <math display="inline"><semantics> <mi>α</mi> </semantics></math>, orange for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, green for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, blue for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and purple for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. For small <math display="inline"><semantics> <mi>α</mi> </semantics></math>, the behaviour of <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>/</mo> <mn>2</mn> <mi>π</mi> <mi>N</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>N</mi> <mi>f</mi> </msub> </mrow> </semantics></math> agrees with that of Li and Wolf [<a href="#B50-photonics-11-00346" class="html-bibr">50</a>]. Along the optical axis, <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>/</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>N</mi> <mi>f</mi> </msub> <mo>=</mo> <mi>Z</mi> <mo>/</mo> <mi>f</mi> <mo>=</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>/</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, with <span class="html-italic">Z</span> as in Equation (<a href="#FD42-photonics-11-00346" class="html-disp-formula">42</a>) and <span class="html-italic">D</span> as in Equation (<a href="#FD46-photonics-11-00346" class="html-disp-formula">46</a>).</p> Full article ">Figure 7
<p>The fractional focal shift as a function of the Fresnel number <span class="html-italic">N</span> for different NAs: (<b>a</b>) for the Ks model (Equations (<a href="#FD46-photonics-11-00346" class="html-disp-formula">46</a>) and (<a href="#FD55-photonics-11-00346" class="html-disp-formula">55</a>)) and (<b>b</b>) from the analytic aKs expression in Equation (<a href="#FD59-photonics-11-00346" class="html-disp-formula">59</a>). Lines are coloured red for small <math display="inline"><semantics> <mi>α</mi> </semantics></math>, orange for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, green for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, blue for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and purple for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. For small <math display="inline"><semantics> <mi>α</mi> </semantics></math>, the behaviour agrees with that of Li and Wolf [<a href="#B50-photonics-11-00346" class="html-bibr">50</a>]. The curves for small <math display="inline"><semantics> <mi>α</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> are very close to each other. The small circles indicate where <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>λ</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>: for small <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>λ</mi> </mrow> </semantics></math>, the Kirchhoff boundary conditions tend to break down.</p> Full article ">Figure 8
<p>Contours of constant intensity in the focal region of a focused scalar wave diffracted by a circular aperture in the Debye approximation: (<b>a</b>) for a small semi-angular aperture <math display="inline"><semantics> <mi>α</mi> </semantics></math>, (<b>b</b>) for the nonparaxial Debye case for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>c</b>) for the nonparaxial Debye case for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and (<b>d</b>) for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, computed from the analytic expression in Equation (<a href="#FD65-photonics-11-00346" class="html-disp-formula">65</a>). The intensity at the focal point is normalized to unity. An analytic plot for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> was first shown in [<a href="#B86-photonics-11-00346" class="html-bibr">86</a>].</p> Full article ">Figure 9
<p>Contours of <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (in red) and <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (in green for <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> and blue for <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>) for the scaled Debye–Wolf theory for diffraction of a spherical wave by a circular aperture: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The black contour is for <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>/</mo> <mi>f</mi> <mo>=</mo> <mo>[</mo> <mi>W</mi> <mo>−</mo> <mn>2</mn> <mi>Z</mi> <msup> <mo form="prefix">sin</mo> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>α</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>]</mo> <mo>/</mo> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The shadow edge is also indicated in black.</p> Full article ">Figure 10
<p>Contours of <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (in green for <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>/</mo> <mi>f</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> and blue for <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>/</mo> <mi>f</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>) for the scaled Debye–Wolf theory for diffraction of a spherical wave by a circular aperture: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The black contour is for <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>/</mo> <mi>f</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>A magnified view of the contours of <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (in red) and <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (in green for <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> <mo>≤</mo> <mn>0</mn> </mrow> </semantics></math> and blue for <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>) for the scaled Debye–Wolf theory for diffraction of a spherical wave by a circular aperture: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The shadow edge is indicated in black. The contours for <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> provided by the two alternative expressions in Equation (<a href="#FD74-photonics-11-00346" class="html-disp-formula">74</a>) are almost identical at this scale.</p> Full article ">Figure 12
<p>The variations in (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>/</mo> <mi>f</mi> <mo>=</mo> <mo>[</mo> <mi>W</mi> <mo>−</mo> <mn>2</mn> <mi>Z</mi> <msup> <mo form="prefix">sin</mo> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>α</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>]</mo> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> along the shadow edge. Here, <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>P</mi> </msub> <mo>=</mo> <msqrt> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>ρ</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </semantics></math>. Lines are coloured orange for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, green for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, blue for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and purple for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 13
<p>Contours of <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>/</mo> <mi>f</mi> <mo>=</mo> <mo>[</mo> <mi>W</mi> <mo>−</mo> <mn>2</mn> <mi>Z</mi> <msup> <mo form="prefix">sin</mo> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>α</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>]</mo> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> for the scaled Debye–Wolf theory for diffraction of a spherical wave by a circular aperture: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 14
<p>Plots of <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (<b>top row</b>), <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (<b>middle row</b>), and <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (<b>bottom row</b>) along lines of constant <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> ((<b>left column</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mn>0.2</mn> <mi>f</mi> </mrow> </semantics></math>, (<b>middle column</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, and (<b>right column</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0.2</mn> <mi>f</mi> </mrow> </semantics></math>), shown for different values of <math display="inline"><semantics> <mi>α</mi> </semantics></math> (indicated by colour, orange <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, green <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, blue <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and purple <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>) Break points are indicated by small circles.</p> Full article ">Figure 15
<p>Plots showing contours of constant <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (<b>left column: a and c</b>) and <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (<b>right column: b and d</b>) for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (<b>top row: a and b</b>) and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (<b>bottom row: c and d</b>) for the csDW theory. Colours: illuminated region, blue <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, green <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>; shadow region, purple.</p> Full article ">Figure 16
<p>Plots showing contours of constant <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (in green) (<b>left column: a and c</b>) and <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (in blue) (<b>right column: b and d</b>) for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (<b>top row: a and b</b>) and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (<b>bottom row: c and d</b>) for the asDW theory.</p> Full article ">Figure 17
<p>Contours of <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (in red), <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> (in green for <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> <mo>,</mo> <mi>W</mi> <mo>/</mo> <mi>f</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> and blue for <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> <mo>,</mo> <mi>W</mi> <mo>/</mo> <mi>f</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>) for the pseudo-paraxial theory for diffraction of a spherical wave by a circular aperture: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The black contours in (<b>c</b>,<b>d</b>) are for <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>/</mo> <mi>f</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 18
<p>(<b>a</b>) The variation in <math display="inline"><semantics> <msup> <mi>C</mi> <mn>2</mn> </msup> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> for RSI (red), RSII (blue), and Kp (green). Curves are shown for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (solid line), <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (dashed line), and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>75</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (chained line). The variation in <math display="inline"><semantics> <mrow> <msup> <mi>D</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mn>2</mn> <mi>f</mi> <mo>/</mo> <mrow> <mo>(</mo> <mi>f</mi> <mo>+</mo> <mi>z</mi> <mo>+</mo> <mi>g</mi> <mo>)</mo> </mrow> <mo>]</mo> </mrow> <mn>2</mn> </msup> </mrow> </semantics></math> is shown in purple. (<b>b</b>) The variation in <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>C</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> for RSI (red), RSII (blue), and Kp (green). Curves are shown for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (solid line), <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (dashed line), and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>75</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (chained line).</p> Full article ">

Graphical abstract
Full article ">Figure 1
<p>Transmission spectrum lines of edge channels under different incident beam divergence angles.</p> Full article ">Figure 2
<p>(<b>a</b>) The case of beam with divergence angle obliquely incident on FPI. (<b>b</b>) The geometric position of any light ray in the beam.</p> Full article ">Figure 3
<p>The geometric relationship within the incident surface.</p> Full article ">Figure 4
<p>The transmission spectrum lines of the FPI correspond to different incident angles.</p> Full article ">Figure 5
<p>The transmission spectrum lines of the FPI when multiple beams are incident. (<b>a</b>) Spectral components in the model (<b>b</b>) Measured transmission spectrum line, Airy fitting spectrum line, and calculated spectrum line for multiple beam incidence.</p> Full article ">Figure 6
<p>Principle of Doppler shift determination using a triple channel FPI.</p> Full article ">Figure 7
<p>The response curves of Rayleigh and Mie signals that derived from the actual FPI transmission spectral line and the designed FPI transmission spectral line.</p> Full article ">Figure 8
<p>The input horizontal (<b>a</b>) wind velocity and (<b>b</b>) wind direction profiles.</p> Full article ">Figure 9
<p>The input (solid lines) and retrieved (dashed lines) LOS wind velocity profiles.</p> Full article ">Figure 10
<p>The system error in the retrieved horizontal (<b>a</b>) wind velocity and (<b>b</b>) wind direction.</p> Full article ">Figure 11
<p>The designed and actual response sensitivities for Rayleigh and Mie signals.</p> Full article ">Figure 12
<p>Calculation of LOS wind velocity errors caused by aerosols with different transmission spectral lines. (<b>a</b>) design FPI spectral lines (<b>b</b>) actual FPI spectral lines.</p> Full article ">

Figure 1
<p>Intensity (<b>a</b>) and SAM density (<b>b</b>) distributions of the light field (2) at a distance <span class="html-italic">z</span> = 2<span class="html-italic">z</span>₀ for the following computation parameters: wavelength λ = 0.532 μm, waist radius of the Gaussian envelope <span class="html-italic">w</span>₀ = 1 mm, beam orders <span class="html-italic">m</span> = 2 and <span class="html-italic">n</span> = 3, and initial phases <span class="html-italic">α</span> = 0, <span class="html-italic">β</span> = −π/2. Scale mark in each figure denotes 1 mm. Black and yellow color (<b>a</b>) denote zero and maximal intensity. Blue and red color (<b>b</b>) denote positive and negative SAM density. Pink and cyan ellipses (<b>b</b>) denote right- and left-handed elliptic polarization. White lines (<b>b</b>) denote lines with zero SAM density. Dashed circles (<b>a</b>) illustrate the circles with maximal SAM density (green circle) and with maximal intensity of the components <span class="html-italic">E_x</span> and <span class="html-italic">E_y</span> (white circles) obtained using Equations (12) and (13).</p> Full article ">Figure 2
<p>Intensity (<b>a</b>–<b>d</b>) and SAM density (<b>e</b>–<b>h</b>) distributions of the light field (2) in several transverse planes for the following computation parameters: wavelength λ = 0.532 μm, waist radius of the Gaussian envelope <span class="html-italic">w</span>₀ = 1 mm, beams orders <span class="html-italic">m</span> = 7 and <span class="html-italic">n</span> = 14, initial phases <span class="html-italic">α</span> = 0, <span class="html-italic">β</span> = −π/2, and propagation distances <span class="html-italic">z</span> = 0 (<b>a</b>,<b>e</b>), <span class="html-italic">z</span> = <span class="html-italic">z</span>₀/2 (<b>b</b>,<b>f</b>), <span class="html-italic">z</span> = <span class="html-italic">z</span>₀ (<b>b</b>,<b>f</b>), and <span class="html-italic">z</span> = 2<span class="html-italic">z</span>₀ (<b>b</b>,<b>f</b>). The scale mark in each figure denotes 1 mm. Dashed circles (d,h) illustrate the circles with maximal SAM density (green circle) and with the maximal intensity of the components <span class="html-italic">E_x</span> and <span class="html-italic">E_y</span> (white circles) obtained using Equations (12) and (13). Numbers near the color bar under each figure denote minimal and maximal values.</p> Full article ">Figure 3
<p>Intensity (<b>a</b>–<b>d</b>) and SAM density (<b>e</b>–<b>h</b>) distributions of the light field (2) in several transverse planes for the following computation parameters: wavelength λ = 0.532 μm, waist radius of the Gaussian envelope <span class="html-italic">w</span>₀ = 1 mm, beams orders <span class="html-italic">m</span> = 3 and <span class="html-italic">n</span> = 9, initial phases <span class="html-italic">α</span> = <span class="html-italic">β</span> = 0, and propagation distances <span class="html-italic">z</span> = 0 (<b>a</b>,<b>e</b>), <span class="html-italic">z</span> = <span class="html-italic">z</span>₀/2 (<b>b</b>,<b>f</b>), <span class="html-italic">z</span> = <span class="html-italic">z</span>₀ (<b>b</b>,<b>f</b>), and <span class="html-italic">z</span> = 2<span class="html-italic">z</span>₀ (<b>b</b>,<b>f</b>). Scale mark in each figure denotes 1 mm. Numbers near the color bar under each figure denote minimal and maximal values.</p> Full article ">Figure 4
<p>Dependence of the maximal SAM density of the light field (2), with the same parameters as in <a href="#photonics-10-01288-f003" class="html-fig">Figure 3</a>, on the propagation distance <span class="html-italic">z</span>/<span class="html-italic">z</span>₀. Green dots correspond to the maximal SAM densities from <a href="#photonics-10-01288-f003" class="html-fig">Figure 3</a>.</p> Full article ">Figure 5
<p>Intensity (<b>a</b>,<b>c</b>) and SAM density (<b>b</b>,<b>d</b>) distributions of the light field (2) for the following computation parameters: wavelength λ = 0.532 μm, waist radius of the Gaussian envelope <span class="html-italic">w</span>₀ = 1 mm, beams orders <span class="html-italic">m</span> = 6 and <span class="html-italic">n</span> = 14 (<b>a</b>,<b>b</b>), <span class="html-italic">m</span> = 7 and <span class="html-italic">n</span> = 14 (<b>c</b>,<b>d</b>), initial phases <span class="html-italic">α</span> = 0, <span class="html-italic">β</span> = −π/2, and propagation distance <span class="html-italic">z</span> = 2<span class="html-italic">z</span>₀. Scale mark in each figure denotes 1 mm. Numbers near the color bar under each figure denote minimal and maximal values.</p> Full article ">

Figure 1
<p>Geometry of elastically restrained tapered beam with concentrated mass and axial loading.</p> Full article ">Figure 2
<p>Beam types utilized to consider various conditions: (<b>a</b>) Type A, (<b>b</b>) Type B, (<b>c</b>) Type C, and (<b>d</b>) Type D.</p> Full article ">Figure 3
<p>Variation of nondimensional natural frequency with respect to increase of spring value.</p> Full article ">Figure 4
<p>Effects of the concentrated mass on the first three natural frequencies of single tapered beams with the cantilevered end condition: (<b>a</b>) results for entire concentrated masses, and (<b>b</b>) results for <math display="inline"><semantics> <mrow> <mover accent="true"> <mrow> <mi>m</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math> = 0, 0.1, 0.5, and 1.</p> Full article ">Figure 5
<p>Effects of the concentrated mass on the first three natural frequencies of double tapered beams with the cantilevered end condition: (<b>a</b>) results for entire concentrated masses, and (<b>b</b>) results for <math display="inline"><semantics> <mrow> <mover accent="true"> <mrow> <mi>m</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math> = 0, 0.1, 0.5, and 1.</p> Full article ">Figure 6
<p>Comparison of natural frequencies for <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> = 1 and <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> = 2 when having <math display="inline"><semantics> <mrow> <mover accent="true"> <mrow> <mi>m</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math> = 0.1 at the arbitrary location.</p> Full article ">Figure 7
<p>Effects of the concentrated mass and taper ratios on the first three natural frequencies of <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> = 1: Cases A1–A4.</p> Full article ">Figure 8
<p>Effects of the concentrated mass and taper ratios on the first three natural frequencies of <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> = 1 with respect to Cases A5–A8.</p> Full article ">Figure 9
<p>Effects of the concentrated mass and taper ratios on the first three natural frequencies of <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> = 1 with respect to Cases A9–A12.</p> Full article ">Figure 10
<p>Effects of the concentrated mass and taper ratios on the first three natural frequencies of <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> = 2 with respect to Cases A1–A4.</p> Full article ">Figure 11
<p>Effects of the concentrated mass and taper ratios on the first three natural frequencies of <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> = 2 with respect to Cases A5–A8.</p> Full article ">Figure 12
<p>Effects of the concentrated mass and taper ratios on the first three natural frequencies of <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> = 2: Cases A9–A12.</p> Full article ">Figure 13
<p>Comparison of <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> = 1 and <math display="inline"><semantics> <mrow> <mi>n</mi> </mrow> </semantics></math> = 2 for effects of the concentrated mass on the first three natural frequencies when <span class="html-italic">c</span> = 0.5: (<b>a</b>) Cases A3 and A4, (<b>b</b>) Cases A7 and A8, and (<b>c</b>) Cases A11 and A12.</p> Full article ">Figure 14
<p>Effects of the axial loading on the first three natural frequencies of elastically restrained tapered beam.</p> Full article ">Figure 15
<p>Relationships between compressive loadings and values of elastic springs at which the first natural frequency vanished.</p> Full article ">Figure 16
<p>Variation in the first three mode shapes of tapered beams for Type A.</p> Full article ">Figure 17
<p>Effects of the elastic supports on the first three mode shapes of uniform beams and single and double tapered beams having <math display="inline"><semantics> <mrow> <mi>c</mi> </mrow> </semantics></math> = 0.5.</p> Full article ">

Figure 1
<p>An embedded continuum-based tube-like nanostructure acted upon by the excitations of both elastic medium and MNO.</p> Full article ">Figure 2
<p>(<b>a</b>) Plots of <math display="inline"><semantics> <msub> <mi>W</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </semantics></math>-<math display="inline"><semantics> <mrow> <msub> <mi>l</mi> <mi>b</mi> </msub> <mo>/</mo> <mi>d</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <msub> <mi>r</mi> <mi>m</mi> </msub> </semantics></math> = 0.325 nm, <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi>b</mi> </msub> </semantics></math> = 2300 kg/m<math display="inline"><semantics> <msup> <mrow/> <mn>3</mn> </msup> </semantics></math>, <math display="inline"><semantics> <msub> <mi>t</mi> <mi>b</mi> </msub> </semantics></math> = 0.35 nm, <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.1, <math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> = 0; (∘) <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mn>0</mn> </msub> <mi>a</mi> </mrow> </semantics></math> = 0, <math display="inline"><semantics> <mfenced open="(" close=")"> <mo>▵</mo> </mfenced> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mn>0</mn> </msub> <mi>a</mi> </mrow> </semantics></math> = 1 nm, (⋄) <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mn>0</mn> </msub> <mi>a</mi> </mrow> </semantics></math> = 2 nm; (—) Simsek [<a href="#B108-symmetry-15-01827" class="html-bibr">108</a>], and (...) present study). (<b>b</b>) Plots of <math display="inline"><semantics> <msub> <mi>W</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </semantics></math>-<math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> (<math display="inline"><semantics> <msub> <mi>r</mi> <mi>m</mi> </msub> </semantics></math> = 3 nm, <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi>b</mi> </msub> </semantics></math> = 2500 kg/m<math display="inline"><semantics> <msup> <mrow/> <mn>3</mn> </msup> </semantics></math>, <math display="inline"><semantics> <msub> <mi>M</mi> <mi>N</mi> </msub> </semantics></math> = 0, <math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> = 0.01<math display="inline"><semantics> <msub> <mi>t</mi> <mi>b</mi> </msub> </semantics></math>, <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 30; (—) Kiani [<a href="#B103-symmetry-15-01827" class="html-bibr">103</a>], (...) present study; (∘) NRABT, and <math display="inline"><semantics> <mfenced open="(" close=")"> <mo>▵</mo> </mfenced> </semantics></math> NREBT) retained.</p> Full article ">Figure 3
<p>Plots of <math display="inline"><semantics> <msub> <mi>W</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>m</mi> <mi>i</mi> <mi>d</mi> </mrow> </msub> </semantics></math>-<math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>/</mo> <msub> <mi>τ</mi> <mi>f</mi> </msub> </mrow> </semantics></math> based on the NREBT: (<b>a</b>) <math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> = 0, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> <msub> <mi>l</mi> <mi>b</mi> </msub> </mrow> </semantics></math>, and (<b>c</b>) <math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> <msub> <mi>l</mi> <mi>b</mi> </msub> </mrow> </semantics></math> (<math display="inline"><semantics> <mi>β</mi> </semantics></math> = 0.6, <math display="inline"><semantics> <msub> <mi>M</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>r</mi> <mi>R</mi> </msubsup> </semantics></math> = <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>t</mi> <mi>R</mi> </msubsup> </semantics></math> = 10, <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 20; (...) MNFA, (—) MNMA; (□) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.03, and <math display="inline"><semantics> <mfenced open="(" close=")"> <mo>▵</mo> </mfenced> </semantics></math> <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, and (⋄) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.6).</p> Full article ">Figure 4
<p>Three-dimensional plots of <math display="inline"><semantics> <msub> <mi>W</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>m</mi> <mi>i</mi> <mi>d</mi> </mrow> </msub> </semantics></math>-<math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math>-<math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>/</mo> <msub> <mi>τ</mi> <mi>f</mi> </msub> </mrow> </semantics></math> based on the NRABT: (<b>a</b>) in the absence of medium excitation (<math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> = 0); (<b>b</b>) in the presence of medium excitation (<math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> <msub> <mi>l</mi> <mi>b</mi> </msub> </mrow> </semantics></math>); (<math display="inline"><semantics> <mi>β</mi> </semantics></math> = 0.6, <math display="inline"><semantics> <msub> <mi>M</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>r</mi> <mi>R</mi> </msubsup> </semantics></math> = <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>t</mi> <mi>R</mi> </msubsup> </semantics></math> = 10, and <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 20).</p> Full article ">Figure 5
<p>Plots of <math display="inline"><semantics> <msub> <mi>W</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </semantics></math>-<math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> for various MNO’s speeds: (<b>a</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.6, and (<b>c</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.9; ((∘) MNFA, <math display="inline"><semantics> <mfenced open="(" close=")"> <mo>▵</mo> </mfenced> </semantics></math> MNMA; (...) NRABT, and (—) NREBT; <math display="inline"><semantics> <mi>β</mi> </semantics></math> = 0.6, <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>g</mi> </msub> <mo>/</mo> <msub> <mi>l</mi> <mi>b</mi> </msub> </mrow> </semantics></math> = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> </semantics></math>, <math display="inline"><semantics> <msub> <mi>M</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>t</mi> <mi>R</mi> </msubsup> </semantics></math> = 10, and <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>r</mi> <mi>R</mi> </msubsup> </semantics></math> = 10).</p> Full article ">Figure 6
<p>Plots of <math display="inline"><semantics> <msub> <mi>W</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </semantics></math>-<math display="inline"><semantics> <mi>β</mi> </semantics></math> based on the MNMA for various MNO’s speeds: (<b>a</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.6, and (<b>c</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.9; ((...) NRABT, (—) NREBT; <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>g</mi> </msub> <mo>/</mo> <msub> <mi>l</mi> <mi>b</mi> </msub> </mrow> </semantics></math> = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> </semantics></math>, <math display="inline"><semantics> <msub> <mi>M</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>t</mi> <mi>R</mi> </msubsup> </semantics></math> = 10, <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>r</mi> <mi>R</mi> </msubsup> </semantics></math> = 10; and <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mi>β</mi> </mstyle> </semantics></math> = <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mover> <mi>ω</mi> <mo>¯</mo> </mover> <msub> <mi>ω</mi> <mn>1</mn> </msub> </mfrac> </mstyle> </semantics></math>).</p> Full article ">Figure 7
<p>Plots of <math display="inline"><semantics> <msub> <mi>W</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </semantics></math>-MNO’s mass for various MNO’s speeds: (<b>a</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.6, and (<b>c</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.9; ((∘) MNFA, <math display="inline"><semantics> <mfenced open="(" close=")"> <mo>▵</mo> </mfenced> </semantics></math> MNMA; (...) NRABT, and (—) NREBT; <math display="inline"><semantics> <mi>β</mi> </semantics></math> = 0.6, <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>g</mi> </msub> <mo>/</mo> <msub> <mi>l</mi> <mi>b</mi> </msub> </mrow> </semantics></math> = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>t</mi> <mi>R</mi> </msubsup> </semantics></math> = 10, <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>r</mi> <mi>R</mi> </msubsup> </semantics></math> = 10).</p> Full article ">Figure 8
<p>Plots of <math display="inline"><semantics> <msub> <mi>W</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </semantics></math>-MNO’s speed for three levels of the excitation amplitude of the elastic medium: (<b>a</b>) <math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> <msub> <mi>l</mi> <mi>b</mi> </msub> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> <msub> <mi>l</mi> <mi>b</mi> </msub> </mrow> </semantics></math>, and (<b>c</b>) <math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> <msub> <mi>l</mi> <mi>b</mi> </msub> </mrow> </semantics></math>; ((∘) MNFA, <math display="inline"><semantics> <mfenced open="(" close=")"> <mo>▵</mo> </mfenced> </semantics></math> MNMA; (...) NRABT, and (—) NREBT; <math display="inline"><semantics> <msub> <mi>M</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, and <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>t</mi> <mi>R</mi> </msubsup> </semantics></math> = <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>r</mi> <mi>R</mi> </msubsup> </semantics></math> = 10).</p> Full article ">Figure 9
<p>Graphs of <math display="inline"><semantics> <msub> <mi>W</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </semantics></math>-<math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>r</mi> <mi>R</mi> </msubsup> </semantics></math> for various MNO’s speeds: (<b>a</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.6, and (<b>c</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.9; ((∘) MNFA, <math display="inline"><semantics> <mfenced open="(" close=")"> <mo>▵</mo> </mfenced> </semantics></math> MNMA; (...) NRABT, and (—) NREBT; <math display="inline"><semantics> <mi>β</mi> </semantics></math> = 0.6, <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>g</mi> </msub> <mo>/</mo> <msub> <mi>l</mi> <mi>b</mi> </msub> </mrow> </semantics></math> = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> </semantics></math>, <math display="inline"><semantics> <msub> <mi>M</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, and <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>t</mi> <mi>R</mi> </msubsup> </semantics></math> = 10).</p> Full article ">Figure 10
<p>Graphs of <math display="inline"><semantics> <msub> <mi>W</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </semantics></math>-<math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>t</mi> <mi>R</mi> </msubsup> </semantics></math> for various MNO’s speeds: (<b>a</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.6, and (<b>c</b>) <math display="inline"><semantics> <msub> <mi>V</mi> <mi>N</mi> </msub> </semantics></math> = 0.9; ((∘) MNFA, <math display="inline"><semantics> <mfenced open="(" close=")"> <mo>▵</mo> </mfenced> </semantics></math> MNMA; (...) NRABT, and (—) NREBT; <math display="inline"><semantics> <mi>β</mi> </semantics></math> = 0.6, <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>g</mi> </msub> <mo>/</mo> <msub> <mi>l</mi> <mi>b</mi> </msub> </mrow> </semantics></math> = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> </semantics></math>, <math display="inline"><semantics> <msub> <mi>M</mi> <mi>N</mi> </msub> </semantics></math> = 0.3, and <math display="inline"><semantics> <msubsup> <mover> <mi>k</mi> <mo>¯</mo> </mover> <mi>r</mi> <mi>R</mi> </msubsup> </semantics></math> = 10).</p> Full article ">

Figure 1
<p>A schematic illustration of the water-soil-viscoelastic beam coupling system under a moving load.</p> Full article ">Figure 2
<p>Schematic diagram of the immersed tunnel structure.</p> Full article ">Figure 3
<p>Comparison with reference results. (<b>a</b>) horizontal displacement, (<b>b</b>) vertical displacement. The black square refers to Ref. [<a href="#B26-applsci-13-10319" class="html-bibr">26</a>].</p> Full article ">Figure 4
<p>Vertical displacement change of the structural beam of the underwater immersed tunnel during different riverbed water depths. (<b>a</b>) <span class="html-italic">v</span>_c = 0.2 <span class="html-italic">v_s</span>, (<b>b</b>) <span class="html-italic">v_c</span> = 0.5 <span class="html-italic">v_s</span>, (<b>c</b>) <span class="html-italic">v_c</span> = 1.2 <span class="html-italic">v_s</span>.</p> Full article ">Figure 4 Cont.
<p>Vertical displacement change of the structural beam of the underwater immersed tunnel during different riverbed water depths. (<b>a</b>) <span class="html-italic">v</span>_c = 0.2 <span class="html-italic">v_s</span>, (<b>b</b>) <span class="html-italic">v_c</span> = 0.5 <span class="html-italic">v_s</span>, (<b>c</b>) <span class="html-italic">v_c</span> = 1.2 <span class="html-italic">v_s</span>.</p> Full article ">Figure 5
<p>Shear force change of the structural beam of the underwater immersed tunnel during different riverbed water depths. (<b>a</b>) v_c = 0.2 <span class="html-italic">v_s</span>, (<b>b</b>) v_c = 0.5 <span class="html-italic">v_s</span>, (<b>c</b>) v_c = 1.2 <span class="html-italic">v_s</span>.</p> Full article ">Figure 6
<p>Change of bending moment of the underwater immersed tunnel during different riverbed water depths. (<b>a</b>) v_c = 0.2 <span class="html-italic">v_s</span>, (<b>b</b>) v_c = 0.5 <span class="html-italic">v_s</span>, (<b>c</b>) v_c = 1.2 <span class="html-italic">v_s</span>.</p> Full article ">Figure 7
<p>Change of vertical displacement of riverbed foundation observation point during different riverbed water depths. (<b>a</b>) v_c = 0.2 <span class="html-italic">v_s</span>, (<b>b</b>) v_c = 0.5 <span class="html-italic">v_s</span>, (<b>c</b>) v_c = 1.2 <span class="html-italic">v_s</span>.</p> Full article ">Figure 7 Cont.
<p>Change of vertical displacement of riverbed foundation observation point during different riverbed water depths. (<b>a</b>) v_c = 0.2 <span class="html-italic">v_s</span>, (<b>b</b>) v_c = 0.5 <span class="html-italic">v_s</span>, (<b>c</b>) v_c = 1.2 <span class="html-italic">v_s</span>.</p> Full article ">Figure 8
<p>Horizontal displacement change of riverbed foundation observation point A (0.0 m, 2.0 m) at different riverbed water depths. (<b>a</b>) v_c = 0.2 <span class="html-italic">v_s</span>, (<b>b</b>) v_c = 0.5 <span class="html-italic">v_s</span>, (<b>c</b>) v_c = 1.2 <span class="html-italic">v_s</span>.</p> Full article ">Figure 9
<p>Change of hole pressure at the observation point of the riverbed foundation during different riverbed water depths. (<b>a</b>) v_c = 0.2 <span class="html-italic">v_s</span>, (<b>b</b>) v_c = 0.5 <span class="html-italic">v_s</span>, (<b>c</b>) v_c = 1.2 <span class="html-italic">v_s</span>.</p> Full article ">Figure 10
<p>Influence of different riverbed foundation permeability characteristics on the vibration of an underwater immersed tunnel structure beam. (<b>a</b>) vertical displacement, (<b>b</b>) shear force, (<b>c</b>) bending moment.</p> Full article ">Figure 10 Cont.
<p>Influence of different riverbed foundation permeability characteristics on the vibration of an underwater immersed tunnel structure beam. (<b>a</b>) vertical displacement, (<b>b</b>) shear force, (<b>c</b>) bending moment.</p> Full article ">Figure 11
<p>Influence of the permeability characteristics of different riverbed foundations on the vibration of the observation points in the foundation. (<b>a</b>) vertical displacement, (<b>b</b>) Horizontal displacement, (<b>c</b>) Hole pressure.</p> Full article ">Figure 11 Cont.
<p>Influence of the permeability characteristics of different riverbed foundations on the vibration of the observation points in the foundation. (<b>a</b>) vertical displacement, (<b>b</b>) Horizontal displacement, (<b>c</b>) Hole pressure.</p> Full article ">Figure 12
<p>Effect of the viscosity coefficient characteristics of different viscoelastic beams on the vibration of underwater immersed tunnel structure beams. (<b>a</b>) vertical displacement, (<b>b</b>) shear force, (<b>c</b>) bending moment.</p> Full article ">Figure 12 Cont.
<p>Effect of the viscosity coefficient characteristics of different viscoelastic beams on the vibration of underwater immersed tunnel structure beams. (<b>a</b>) vertical displacement, (<b>b</b>) shear force, (<b>c</b>) bending moment.</p> Full article ">Figure 13
<p>Effect of the viscosity coefficient characteristics of different viscoelastic beams on the vibration of the observation points in the foundation. (<b>a</b>) vertical displacement, (<b>b</b>) Horizontal displacement, (<b>c</b>) Hole pressure.</p> Full article ">Figure 13 Cont.
<p>Effect of the viscosity coefficient characteristics of different viscoelastic beams on the vibration of the observation points in the foundation. (<b>a</b>) vertical displacement, (<b>b</b>) Horizontal displacement, (<b>c</b>) Hole pressure.</p> Full article ">

Figure 1
<p>Transverse intensity distributions of typical structured light beams under the paraxial approximation at different propagation distances. (<b>a1</b>–<b>a5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>4</mn> <mi>λ</mi> </mrow> </semantics></math>, (<b>b1</b>–<b>b5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>8</mn> <mi>λ</mi> </mrow> </semantics></math>, and (<b>c1</b>–<b>c5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>12</mn> <mi>λ</mi> </mrow> </semantics></math>. Shown from left to right are the cases of the fundamental Gaussian beam, the Hermite–Gaussian beam, the Laguerre–Gaussian beam, the Bessel beam, and the Airy beam, respectively.</p> Full article ">Figure 2
<p>Transverse intensity distributions of typical structured light beams beyond the paraxial approximation at different propagation distances. (<b>a1</b>–<b>a5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>4</mn> <mi>λ</mi> </mrow> </semantics></math>, (<b>b1</b>–<b>b5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>8</mn> <mi>λ</mi> </mrow> </semantics></math>, and (<b>c1</b>–<b>c5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>12</mn> <mi>λ</mi> </mrow> </semantics></math>. Shown from left to right are the cases of the fundamental Gaussian beam, the Hermite–Gaussian beam, the Laguerre–Gaussian beam, the Bessel beam, and the Airy beam, respectively.</p> Full article ">Figure 3
<p>Illustrations of the propagation of typical structured light beams in a gradient-index medium. (<b>a</b>) Fundamental Gaussian beam, (<b>b</b>) Hermite–Gaussian beam, (<b>c</b>) Laguerre–Gaussian beam, (<b>d</b>) Bessel beam, and (<b>e</b>) Airy beam.</p> Full article ">