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35 pages, 1470 KiB  
Article
Tight Focusing of Light
by Colin J. R. Sheppard
Photonics 2024, 11(10), 913; https://doi.org/10.3390/photonics11100913 - 27 Sep 2024
Viewed by 331
Abstract
The effects of various properties on the tight focusing of light are considered. In particular, polarization of the incident field is an important consideration. Plots are presented for the variations in the focal intensity, and the area and volume of the focal spot, [...] Read more.
The effects of various properties on the tight focusing of light are considered. In particular, polarization of the incident field is an important consideration. Plots are presented for the variations in the focal intensity, and the area and volume of the focal spot, with numerical aperture. We consider Bessel beams, focusing with a system of circular pupil, and 4Pi focusing by a pair of opposing high-numerical-aperture lenses or a single paraboloidal mirror. Full article
(This article belongs to the Special Issue Structured Light Beams: Science and Applications)
Show Figures

Figure 1

Figure 1
<p>Different polarization distributions on the reference sphere. The left column shows transverse electric (<math display="inline"><semantics> <msub> <mi>TE</mi> <mn>1</mn> </msub> </semantics></math>), electric dipole, mixed-dipole, magnetic dipole, and transverse magnetic (<math display="inline"><semantics> <msub> <mi>TM</mi> <mn>1</mn> </msub> </semantics></math>) fields. Electric field is shown in red, and magnetic field in blue. For the mixed-dipole case, as the fields are not spherically symmetric, the view from the front of the sphere (left sub-image, incident polarization for focused linearly polarized light) and the back of the sphere (right sub-image, focused linearly polarized light for outgoing polarization, or forthe input for a deep paraboloidal mirror) are shown. The back of the sphere exhibits a polarization singularity of order 2 on the axis. The right column shows radial (<math display="inline"><semantics> <msub> <mi>TM</mi> <mn>0</mn> </msub> </semantics></math>, axial electric dipole) and azimuthal (<math display="inline"><semantics> <msub> <mi>TE</mi> <mn>0</mn> </msub> </semantics></math>, axial magnetic dipole) polarizations.</p>
Full article ">Figure 2
<p>A model for a high-NA lens. The lens is described as a black box. The input is specified by the field in the front focal plane <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>B</mi> <msup> <mi>A</mi> <mo>′</mo> </msup> </mrow> </semantics></math> of the lens, situated a distance <span class="html-italic">f</span> before the lens, and truncated by the aperture stop. Rays are refracted by the lens, so that they converge onto the focal point <span class="html-italic">F</span>. The intersection of the parallel input rays <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>Q</mi> </mrow> </semantics></math> and the convergent focused rays <math display="inline"><semantics> <mrow> <mi>Q</mi> <mi>F</mi> </mrow> </semantics></math> defines an equivalent refractive locus which forms a surface in 3D space. For a system satisfying the sine condition, this surface is a sphere of radius <span class="html-italic">f</span>. In general, the equivalent refractive locus generates a surface where the length <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>F</mi> </mrow> </semantics></math> is <span class="html-italic">f</span>.</p>
Full article ">Figure 3
<p>The amplitude variation <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>(</mo> <mi>sin</mi> <mi>θ</mi> <mo>)</mo> </mrow> </semantics></math> of the illumination wave that when incident on an aplanatic optical system gives the same effect for different focusing conditions.</p>
Full article ">Figure 4
<p>The variation in the parameters with NA for illumination with a linearly polarized wave: (<b>a</b>) <span class="html-italic">F</span>, the normalized electric energy density at the focus for a given power input; (<b>b</b>) <math display="inline"><semantics> <msub> <mi>F</mi> <mi>I</mi> </msub> </semantics></math>, the normalized electric energy density for a given integrated intensity.</p>
Full article ">Figure 5
<p>The area (<b>a</b>) and volume (<b>b</b>) of the focal spot for illumination of a high-NA lens with linearly or circularly polarized light. Solid lines denote linear polarization and dashed lines denote circular polarization.</p>
Full article ">Figure 6
<p>The area of the parabolic central lobe of a Bessel beam generated by focusing different polarized illuminations. Dashed lines refer to rotating fields (circular polarized, rotating dipoles, or azimuthal vortex <math display="inline"><semantics> <mrow> <mo>=</mo> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </semantics></math>-<math display="inline"><semantics> <msub> <mi>TE</mi> <mn>1</mn> </msub> </semantics></math>).</p>
Full article ">Figure 7
<p>The variation in the parameters <span class="html-italic">F</span> (<b>a</b>) and <math display="inline"><semantics> <msub> <mi>F</mi> <mi>I</mi> </msub> </semantics></math> (<b>b</b>) with NA for different illumination polarizations. The curves also apply to the rotating cases, i.e., elec. dipole is also <span class="html-italic">rot.</span>-elec. dipole, and <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math> is also azimuthal vortex.</p>
Full article ">Figure 8
<p>For focusing by a high-NA lens, the absolute values of the ratios of the strengths in a multipole expansion of the magnetic dipole term <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>/</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> (solid lines), electric quadrupole term <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>/</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> (dashed lines), and magnetic quadrupole term <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>/</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> (chained lines), to the electric dipole term, plotted against NA for different polarization conditions.</p>
Full article ">Figure 9
<p>The area and volume of the focal spot produced by a high-NA lens for different illumination polarizations. <math display="inline"><semantics> <mrow> <mi>r</mi> <mi>o</mi> <mi>t</mi> <mo>.</mo> </mrow> </semantics></math>-<math display="inline"><semantics> <msub> <mi>TE</mi> <mn>1</mn> </msub> </semantics></math> is labelled azimuthal vortex.</p>
Full article ">Figure 10
<p>The area of the focal spot produced by a high-NA lens for rotating electric dipole polarization with a pupil weighting <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <msup> <mi>cos</mi> <mi>j</mi> </msup> <mi>θ</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (solid green line line), <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (dashed line), and <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (chained line), and for radial electric dipole polarization with a pupil weighting <math display="inline"><semantics> <mrow> <msup> <mi>sin</mi> <mi>j</mi> </msup> <mi>θ</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (solid black line), <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (dashed line), and <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (chained line). The area for azimuthal vortex polarization (<math display="inline"><semantics> <msub> <mi>TE</mi> <mn>1</mn> </msub> </semantics></math>) is also shown in blue for comparison.</p>
Full article ">Figure 11
<p>The variation in the parameter <span class="html-italic">F</span> for a high-NA lens for electric dipole polarization with a pupil weighting <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <msup> <mi>cos</mi> <mi>j</mi> </msup> <mi>θ</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (solid green line), <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (dashed line), and <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (chained line), and for radial electric dipole polarization with a pupil weighting <math display="inline"><semantics> <mrow> <msup> <mi>sin</mi> <mi>n</mi> </msup> <mi>θ</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (solid black line), <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (dashed line), and <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (chained line). The variation in <span class="html-italic">F</span> for azimuthalvortex polarization (<math display="inline"><semantics> <msub> <mi>TE</mi> <mn>1</mn> </msub> </semantics></math>) is also shown in blue for comparison. The curves also apply to the rotating cases.</p>
Full article ">Figure 12
<p>The variation in normalized electric energy density <math display="inline"><semantics> <msub> <mi>W</mi> <mi>E</mi> </msub> </semantics></math> in the focal plane of a high-NA lens, <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>A</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>, for rotating electric dipole polarization with a pupil weighting <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <msup> <mi>cos</mi> <mi>n</mi> </msup> <mi>θ</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (solid line), <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (dashed line), and <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (chained line). The variation for azimuthal vortex polarization (<span class="html-italic">rot.</span>-<math display="inline"><semantics> <msub> <mi>TE</mi> <mn>1</mn> </msub> </semantics></math>) is also shown for comparison.</p>
Full article ">Figure 13
<p>The fraction of power propagating in different directions after scattering of an electric dipole wave by a point-like dipole, as a function of the angular semi-aperture, <math display="inline"><semantics> <mi>α</mi> </semantics></math>. For <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>≤</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>, the power scattered other than in the forward direction is broken up into backscattering and sideways scattering.</p>
Full article ">Figure 14
<p>The variation in the area of a 4Pi Bessel beam with NA, for different polarization distributions.</p>
Full article ">Figure 15
<p>The variation in <math display="inline"><semantics> <msub> <mi>F</mi> <mi>I</mi> </msub> </semantics></math> with NA of each lens for 4Pi focusing for different apodizations of linearly or circularly polarized input (optimized mixed dipole, perfect mixed dipole, aplanatic, mixed dipole, Herschel, parabolic, Helmholtz), and for different polarization distributions (magnetic dipole, optimized radial electric dipole, perfect radial electric dipole, optimized electric dipole, electric dipole, <math display="inline"><semantics> <msub> <mi>TE</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>TM</mi> <mn>1</mn> </msub> </semantics></math>, perfect <math display="inline"><semantics> <msub> <mi>TE</mi> <mn>1</mn> </msub> </semantics></math>).</p>
Full article ">Figure 16
<p>The area of the focal spot for 4Pi focusing with a pair of high-NA lenses illuminated by linearly (solid lines) and circularly (dashed lines) polarized waves for different apodization conditions. The difference between linear and circular polarization is very small.</p>
Full article ">Figure 17
<p>The area (<b>a</b>) and volume (<b>b</b>) of the focal spot for 4Pi focusing with a pair of high-NA lenses, for different polarization conditions. <span class="html-italic">rot.</span> cases are shown as dashed lines. The volumes for radial vortex, mag. dipole, and <span class="html-italic">rot</span>-mag. dipole are identical (red-orange dashed line). <span class="html-italic">l</span>-aplanatic and <span class="html-italic">c</span>-aplanatic (green) are slightly smaller.</p>
Full article ">Figure 18
<p>The variation in <span class="html-italic">F</span> with <math display="inline"><semantics> <mi>β</mi> </semantics></math> for a 4Pi system based on a single paraboloidal mirror for different input waves. The inputs are (1) a radially polarized ED wave (grey); (2) a uniform radially polarized plane wave (magenta); and (3) a radially polarized plane wave with amplitude varying proportional to cylindrical radius (blue). The curve for direct illumination by a radial electric dipole wave is shown for comparison (cyan).</p>
Full article ">Figure 19
<p>The strength of the electric field vector in the input plane to achieve different orientations of the electric field at the focal point. For all cases the direction cosine <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>A</mi> <mo>/</mo> <mi>n</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>.</p>
Full article ">
25 pages, 10028 KiB  
Review
Ultrafast Laser Processing for High-Aspect-Ratio Structures
by Muyang Qin, Xinjing Zhao, Hanyue Fan, Ruizhe Leng, Yanhao Yu, Aiwu Li and Bingrong Gao
Nanomaterials 2024, 14(17), 1428; https://doi.org/10.3390/nano14171428 - 31 Aug 2024
Viewed by 649
Abstract
Over the past few decades, remarkable breakthroughs and progress have been achieved in ultrafast laser processing technology. Notably, the remarkable high-aspect-ratio processing capabilities of ultrafast lasers have garnered significant attention to meet the stringent performance and structural requirements of materials in specific applications. [...] Read more.
Over the past few decades, remarkable breakthroughs and progress have been achieved in ultrafast laser processing technology. Notably, the remarkable high-aspect-ratio processing capabilities of ultrafast lasers have garnered significant attention to meet the stringent performance and structural requirements of materials in specific applications. Consequently, high-aspect-ratio microstructure processing relying on nonlinear effects constitutes an indispensable aspect of this field. In the paper, we review the new features and physical mechanisms underlying ultrafast laser processing technology. It delves into the principles and research achievements of ultrafast laser-based high-aspect-ratio microstructure processing, with a particular emphasis on two pivotal technologies: filamentation processing and Bessel-like beam processing. Furthermore, the current challenges and future prospects for achieving both high precision and high aspect ratios simultaneously are discussed, aiming to provide insights and directions for the further advancement of high-aspect-ratio processing. Full article
Show Figures

Figure 1

Figure 1
<p>The principle of ultrafast laser processing. (<b>a</b>) Microbull sculptures produced by two-photon photopolymerization technology [<a href="#B68-nanomaterials-14-01428" class="html-bibr">68</a>]; (<b>b</b>) Spatial intensity distribution of far-field focusing laser spots [<a href="#B69-nanomaterials-14-01428" class="html-bibr">69</a>]; (<b>c</b>) Schematic diagram of two-photon absorption; (<b>d</b>) Schematic illustration of spatial distributions of laser energy intensity based on various absorption mechanisms: single-photon absorption (blue line), two-photon absorption (green line), and three-photon absorption (red line).</p>
Full article ">Figure 2
<p>The principle of filamentation. (<b>a</b>) Laser intensity and refractive index in the case of self-focusing [<a href="#B88-nanomaterials-14-01428" class="html-bibr">88</a>]; (<b>b</b>) Laser intensity and refractive index in the case of defocusing [<a href="#B88-nanomaterials-14-01428" class="html-bibr">88</a>]; (<b>c</b>) The focusing–defocusing cycles undergone by the intense core of the beam during the formation of the filament [<a href="#B88-nanomaterials-14-01428" class="html-bibr">88</a>]; (<b>d</b>) Physical scene of femtosecond laser filamentation in a solid medium [<a href="#B86-nanomaterials-14-01428" class="html-bibr">86</a>].</p>
Full article ">Figure 3
<p>The filament induced modification and ablation. (<b>a</b>) The filament structure in fused silica induced by light intensity originates proximate to the laser’s geometric focus and extends along the laser pulse’s propagation [<a href="#B90-nanomaterials-14-01428" class="html-bibr">90</a>]; (<b>b</b>) A side-view illustration showcasing a series of voids within a fused silica filament resulting from the application of a 250-fs laser pulse with a pulse energy of 80 microjoules [<a href="#B90-nanomaterials-14-01428" class="html-bibr">90</a>]; (<b>c</b>) Enhancing the thickness of the soda-lime glass plate placed behind the objective lens will hinder the radial propagation of the pulse and amplify the power density at the focal point [<a href="#B92-nanomaterials-14-01428" class="html-bibr">92</a>]; (<b>d</b>) Subjected to identical laser parameters, the depth of processing achieved in a vacuum environment is twice as deep compared to that in air [<a href="#B93-nanomaterials-14-01428" class="html-bibr">93</a>].</p>
Full article ">Figure 4
<p>The ultrafast evolution of filaments. (<b>a</b>) An image showcases the extended filamentary track length (1180 μm) achieved within sapphire with 107 µJ pulse energy, 300 kHz repetition rate, and 10,000 pulses [<a href="#B95-nanomaterials-14-01428" class="html-bibr">95</a>]; (<b>b</b>) A series of depictions of the evolution of filaments within sapphire and silica glass on different femtosecond and picosecond timescales with pump energy of 50 µJ and a relative position D = 60 µm [<a href="#B98-nanomaterials-14-01428" class="html-bibr">98</a>].</p>
Full article ">Figure 5
<p>(<b>a</b>) The demonstration of using an axicon illuminated with a Gaussian beam to create a Bessel beam [<a href="#B17-nanomaterials-14-01428" class="html-bibr">17</a>]; (<b>b</b>) The above SEM image depicts a through channel in glass with an aspect ratio approaching 100, achieved using a single shot from a Bessel beam, while the bottom SEM image displays an array of channels with each channel having a length of 10 μm, a diameter of 230 nm, and a pitch of 1.6 μm [<a href="#B105-nanomaterials-14-01428" class="html-bibr">105</a>]; (<b>c</b>) The SEM images, respectively, represent a channel cut and structures at the top and back surfaces of the microchannels formed with single pulses [<a href="#B106-nanomaterials-14-01428" class="html-bibr">106</a>]; (<b>d</b>) The SEM image of the nanochannel structure when the sample surface is put 0.9 μm above the focal plane of the objective lens by a single-shot femtosecond Bessel beam with 1.0 μJ [<a href="#B107-nanomaterials-14-01428" class="html-bibr">107</a>].</p>
Full article ">Figure 6
<p>The generation and measurement of segmented Bessel beams [<a href="#B122-nanomaterials-14-01428" class="html-bibr">122</a>]. (<b>a</b>) A schematic diagram of the segmented Bessel beam region ∆z generated by placing a ring aperture with aperture size d behind an axicon with apex angle γ; (<b>b</b>) With an axicon angle γ = 170◦ and wavelength λ = 632.8 nm, the illustration depicting the position and length of the segmented Bessel beam with different optimal apertures <span class="html-italic">d</span><sub>min</sub> and radial positions: (i) <span class="html-italic">r</span> = 1 mm, <span class="html-italic">d</span><sub>min</sub> = 197.85 µm, (ii) <span class="html-italic">r</span> = 2 mm, <span class="html-italic">d</span><sub>min</sub> = 281.58 µm and (iii) <span class="html-italic">r</span> = 3 mm, <span class="html-italic">d</span><sub>min</sub> = 344.90 µm.</p>
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<p>Analysis of the nanochannel characteristics generated by the skillfully shaped Bessel beam [<a href="#B124-nanomaterials-14-01428" class="html-bibr">124</a>]. (<b>a</b>) Side-view trans-illumination images and epi-illumination images of a series of 10 channels in fused silica, fabricated by single-shot ablation with the energy ≈ 0.5 μJ and pulse duration of, respectively, 1 ps, 200 fs, 30 fs; (<b>b</b>) An SEM image of channels is also shown for the case 1 ps; (<b>c</b>) Longitudinal profiles of a channel extracted from images for series 1 ps and 30 fs.</p>
Full article ">Figure 8
<p>Comparisons between Bessel beams and Bessel-like beams [<a href="#B128-nanomaterials-14-01428" class="html-bibr">128</a>]. (<b>a</b>) Theoretical holograms of the axicon to create Bessel beam and the simplified arc axicon to create Bessel-like beam; (<b>b</b>) Intensity distribution of the Bessel beam and Bessel-like beam; (<b>c</b>) On-axis intensity distribution of the Bessel beam and Bessel-like beam; (<b>d</b>) Variation in ablation length with different Bessel-like beams (<span class="html-italic">k</span> = 1–22) and Bessel beam (<span class="html-italic">k</span> = ∞).</p>
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<p>Super-stealth dicing (SSD) by ultrafast laser back-scattered interference crawling [<a href="#B139-nanomaterials-14-01428" class="html-bibr">139</a>]. (<b>a</b>) Experimental setup for laser processing; (<b>b</b>) SEM images and close-up-view of silica nanogratings with a period of 300 nm etched in 2% hydrofluoric acid for 300 s and using 250 μm<sup>−1</sup> pulse density; (<b>c</b>) SEM images of drilling of specific shapes on silica glass etched in 5% hydrofluoric acid for 300 s and optical microscope image of microscale silica cuboids collected after wet etching.</p>
Full article ">
21 pages, 20069 KiB  
Article
Tunable Photonic Hook Design Based on Anisotropic Cutting Liquid Crystal Microcylinder
by Renxian Li, Huan Tang, Mingyu Zhang, Fengbei Liu, Ruiping Yang, Naila Khaleel, Muhammad Arfan, Muhammad Asif, Igor V. Minin and Oleg V. Minin
Photonics 2024, 11(8), 736; https://doi.org/10.3390/photonics11080736 - 7 Aug 2024
Viewed by 532
Abstract
The selective control and manipulation of nanoparticles require developing and researching new methods for designing optical tweeters, mainly based on a photonic hooks (PHs) effect. This paper first proposes a tunable PH in which a structured beam illuminates an anisotropic cutting liquid crystal [...] Read more.
The selective control and manipulation of nanoparticles require developing and researching new methods for designing optical tweeters, mainly based on a photonic hooks (PHs) effect. This paper first proposes a tunable PH in which a structured beam illuminates an anisotropic cutting liquid crystal microcylinder based on the Finite-DifferenceTime-Domain (FDTD) method. The PHs generated by plane wave, Gaussian, and Bessel beam are analyzed and compared. The impact of beams and LC particle parameters on the PHs are discussed. Where the influence of the extraordinary refractive index (ne) on PHs is emphasized. Our results reveal that introducing birefringence can change the bending direction of PH. Besides, the maximum intensity of the PHs increases as ne increases regardless of the beam type. The PH generated by a plane wave has a higher maximum intensity and smaller FWHM than that generated by the Gaussian and Bessel beams. The smallest FWHM and maximum intensity of the PHs generated by the Gaussian falls between that generated by the plane wave and the Bessel beam. The PH generated by a Bessel beam has the minor maximum intensity and the largest FWHM. Still, it exceeds the diffraction limit and exhibits bending twice due to its self-recovery property. This paper provides a new way to modulate PH. This work offers novel theoretical models and the degree of freedom for the design of PHs, which is beneficial for the selective manipulation of nanoparticles. It has promising applications in Mesotronics and biomedicine. Full article
(This article belongs to the Special Issue Vortex Beams: Transmission, Scattering and Application)
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Figure 1

Figure 1
<p>The FDTD model that waves illuminate an anisotropic cutting liquid crystal microcylinder in water.</p>
Full article ">Figure 2
<p>Schematic diagram of the photonic hook. The start point, inflection point, and end point are colored green, purple, and red. The inflection point corresponds to the maximum electric field intensity <math display="inline"><semantics> <msub> <mi>I</mi> <mo movablelimits="true" form="prefix">max</mo> </msub> </semantics></math>.</p>
Full article ">Figure 3
<p>The PH is generated by an anisotropic cutting microcylinder with different extraordinary refractive indices irradiated with plane waves. The first and second columns are the distribution figures of electric field intensity. The third and fourth columns represent the field intensities in <span class="html-italic">y</span> and <span class="html-italic">x</span> directions, respectively.</p>
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<p>Energy distribution of a plane wave incident on the cutting LC microcylinder.</p>
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<p>Upon incidence of a plane wave, the bending angle, electric field intensity, and FWHM exhibit variations in response to changes in the <math display="inline"><semantics> <msub> <mi>n</mi> <mi mathvariant="normal">e</mi> </msub> </semantics></math>. (<b>a</b>) Bending angle (<b>b</b>) Electric field intensity (<b>c</b>) FWHM.</p>
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<p>The bimodal phenomenon when a beam illuminates the cutting LC microcylinder.</p>
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<p>Schematic diagram of PH generation by Gaussian beam illumination on cutting microcylinder particles.</p>
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<p>The PH generated by anisotropic cutting microcylinder particles with different extraordinary refractive indices irradiated with a Gaussian beam. The first and second columns are the distribution figures of electric field intensity. The third and fourth columns represent the field intensities in <span class="html-italic">y</span> and <span class="html-italic">x</span> directions, respectively.</p>
Full article ">Figure 9
<p>Upon incidence of a Gaussian beam, the bending angle, electric field intensity, and FWHM exhibit variations in response to changes in the <math display="inline"><semantics> <msub> <mi>n</mi> <mi mathvariant="normal">e</mi> </msub> </semantics></math>. (<b>a</b>) Bending angle (<b>b</b>) Electric field intensity (<b>c</b>) FWHM.</p>
Full article ">Figure 10
<p>Schematic diagram of PH generation by Bessel beam illumination on cutting microcylinder particles.</p>
Full article ">Figure 11
<p>The PH is generated by anisotropic cutting microcylinder particles with different extraordinary refractive indices irradiated with a Bessel beam. The first and second columns are the distribution figures of electric field intensity. The third and fourth columns represent the field intensities in <span class="html-italic">y</span> and <span class="html-italic">x</span> directions, respectively.</p>
Full article ">Figure 12
<p>When a Bessel beam is incident, the bending angle, electric field intensity, and FWHM change in response to variations in the <math display="inline"><semantics> <msub> <mi>n</mi> <mi mathvariant="normal">e</mi> </msub> </semantics></math>. (<b>a</b>) Bending angle (<b>b</b>) Electric field intensity (<b>c</b>) FWHM.</p>
Full article ">
21 pages, 2611 KiB  
Article
Scattering of a Bessel Pincer Light-Sheet Beam on a Charged Particle at Arbitrary Size
by Shu Zhang, Shiguo Chen, Qun Wei, Renxian Li, Bing Wei and Ningning Song
Micromachines 2024, 15(8), 975; https://doi.org/10.3390/mi15080975 - 29 Jul 2024
Viewed by 519
Abstract
Electromagnetic scattering is a routine tool for rapid, non-contact characterization of particle media. In previous work, the interaction targets of scattering intensity, scattering efficiency, and extinction efficiency of Bessel pincer light-sheet beams were all aimed at dielectric spheres. However, most particles in nature [...] Read more.
Electromagnetic scattering is a routine tool for rapid, non-contact characterization of particle media. In previous work, the interaction targets of scattering intensity, scattering efficiency, and extinction efficiency of Bessel pincer light-sheet beams were all aimed at dielectric spheres. However, most particles in nature are charged. Considering the boundary condition on a charged sphere, the beam shape coefficients (BSCs) (pmn,qmn) of the charged spherical particle illuminated by a Bessel pincer light-sheet beam are obtained. The extinction, scattering, and absorption efficiencies are derived under the generalized Lorenz–Mie theory (GLMT) framework. This study reveals the significant differences in scattering characteristics of Bessel pincer light-sheet beams on a charged particle compared to traditional beams. The simulations show a few apparent differences in the far-field scattering intensity and efficiencies between charged and natural spheres under the influence of dimensionless size parameters. As dimensionless parameters increase, the difference between the charged and neutral spheres decreases. The effects of refractive index and beam parameters on scattering, extinction, and absorption coefficients are different but tend to converge with increasing dimensionless parameters. When applied to charged spheres with different refractive indices, the scattering, extinction, and absorption efficiencies of Bessel pincer light-sheet beams change with variations in surface charge. However, once the surface charge reaches saturation, these efficiencies become stable. This study is significant for understanding optical manipulation and super-resolution imaging in single-molecule microbiology. Full article
Show Figures

Figure 1

Figure 1
<p>Distribution of the propagation of a non-paraxial Bessel pincer light-sheet on a charged sphere along <span class="html-italic">z</span>-direction (<math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0.6328 <math display="inline"><semantics> <mo>μ</mo> </semantics></math>m, the radius <span class="html-italic">a</span> = 10 <math display="inline"><semantics> <mo>μ</mo> </semantics></math>m, ł = 10, <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>).</p>
Full article ">Figure 2
<p>Scattered light intensity distribution varies with the scattering angle for a given particle size ka ([0.6, 5, 25]), in which <math display="inline"><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>1.33</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>i</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> S. The wavelength of incident beam is <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.6328</mn> </mrow> </semantics></math> μm, beam scaling parameter is <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, beam order is <math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and the azimuth angle is <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>Far-field scattering intensity varies with the scattering angle for given charged particle (<math display="inline"><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>1.33</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>i</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> S) size parameters of 25, 5, and 0.6 under different beam wavelengths (<math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <msub> <mi>v</mi> <mi>c</mi> </msub> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math>; <math display="inline"><semantics> <msub> <mi>v</mi> <mi>c</mi> </msub> </semantics></math> is the speed of light in vacuum). The frequency of incident beam is <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mo>[</mo> <mn>0.24</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>2.0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>24</mn> <mo>]</mo> </mrow> </semantics></math> THz, beam scaling parameter is <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, beam order is <math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and the azimuth angle is <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Far-field scattering intensity varies with the scattering angle for given charged particle (<math display="inline"><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>1.33</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>i</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math> S) size parameters of 25, 5, and 0.6 under different azimuth angles (<math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mo>[</mo> <mi>π</mi> <mo>/</mo> <mn>64</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>π</mi> <mo>/</mo> <mn>4</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>]</mo> </mrow> </semantics></math>), respectively. The incident wavelength is <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.6328</mn> </mrow> </semantics></math> μm, beam scaling parameter is <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, and beam order is <math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>Far-field scattering intensity varies with the scattering angle for given charged particle (<math display="inline"><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>1.33</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>i</mi> </mrow> </semantics></math>) size parameters of 25, 5, and 0.6 under different surface charges (<math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>s</mi> </msub> </semantics></math> = [<math display="inline"><semantics> <mrow> <mn>5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>11</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>13</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math>, 0]), respectively. The incident wavelength is <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.6328</mn> </mrow> </semantics></math> μm, beam scaling parameter is <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, beam order is <math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, and the azimuth angle is <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>64</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 5 Cont.
<p>Far-field scattering intensity varies with the scattering angle for given charged particle (<math display="inline"><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>1.33</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>i</mi> </mrow> </semantics></math>) size parameters of 25, 5, and 0.6 under different surface charges (<math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>s</mi> </msub> </semantics></math> = [<math display="inline"><semantics> <mrow> <mn>5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>11</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>13</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math>, 0]), respectively. The incident wavelength is <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.6328</mn> </mrow> </semantics></math> μm, beam scaling parameter is <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, beam order is <math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, and the azimuth angle is <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>64</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>Far-field scattering intensity varies with the scattering angle for given charged particle size parameters of 25, 5, and 0.6 under different refractive indexes (<math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.33</mn> <mo>+</mo> <mi>n</mi> <mi>p</mi> <mo>∗</mo> <mi>i</mi> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mspace width="0.166667em"/> <msub> <mi>n</mi> <mi>p</mi> </msub> <mo>=</mo> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.5</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>), respectively. The incident wavelength is <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.6328</mn> </mrow> </semantics></math> μm, beam scaling parameter is <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, beam order is <math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, and the azimuth angle is <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 6 Cont.
<p>Far-field scattering intensity varies with the scattering angle for given charged particle size parameters of 25, 5, and 0.6 under different refractive indexes (<math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.33</mn> <mo>+</mo> <mi>n</mi> <mi>p</mi> <mo>∗</mo> <mi>i</mi> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mspace width="0.166667em"/> <msub> <mi>n</mi> <mi>p</mi> </msub> <mo>=</mo> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.5</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>), respectively. The incident wavelength is <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.6328</mn> </mrow> </semantics></math> μm, beam scaling parameter is <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, beam order is <math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, and the azimuth angle is <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 7
<p>Far-field scattering intensity varies with the scattering angle for given charged particle (<math display="inline"><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>1.33</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>i</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>5.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mspace width="0.166667em"/> <mi>S</mi> </mrow> </semantics></math>) size parameters of 25, 5, and 0.6 under different beam orders (<math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mo>[</mo> <mn>10</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>20</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>30</mn> <mo>]</mo> </mrow> </semantics></math>), respectively. The incident wavelength is <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.6328</mn> </mrow> </semantics></math> μm, beam scaling parameter is <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, and the azimuth angle is <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>Far-field scattering intensity varies with the scattering angle for given charged particle (<math display="inline"><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>1.33</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>i</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>5.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mspace width="0.166667em"/> <mi>S</mi> </mrow> </semantics></math>) size parameters of 25, 5, and 0.6 under different beam scaling parameter values (<math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mrow> <mo>[</mo> <mn>0.4</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.6</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>), respectively. The incident wavelength is <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.6328</mn> </mrow> </semantics></math> μm, beam order is <math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, and the azimuth angle is <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 9
<p><math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mrow> <mi>s</mi> <mi>c</mi> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> on a charged particle illuminated by a Bessel pincer light-sheet beam (beam order <math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, beam scaling parameter <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, and wavelength <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.6328</mn> </mrow> </semantics></math> μm) under different refractive indexes <math display="inline"><semantics> <msub> <mi>n</mi> <mi>p</mi> </msub> </semantics></math> (in which <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.33</mn> <mo>+</mo> <mi>j</mi> <mo>∗</mo> <msub> <mi>n</mi> <mi>p</mi> </msub> </mrow> </semantics></math>).</p>
Full article ">Figure 10
<p>The same as <a href="#micromachines-15-00975-f009" class="html-fig">Figure 9</a> but with different beam scaling parameter <math display="inline"><semantics> <msub> <mi>α</mi> <mn>0</mn> </msub> </semantics></math> values (<math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mrow> <mo>[</mo> <mn>0.4</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.6</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>).</p>
Full article ">Figure 11
<p>The same as <a href="#micromachines-15-00975-f009" class="html-fig">Figure 9</a> but with different beam order ł (<math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mo>[</mo> <mn>10</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>11</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>12</mn> <mo>]</mo> </mrow> </semantics></math>).</p>
Full article ">Figure 12
<p><math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>s</mi> <mi>c</mi> <mi>a</mi> </mrow> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> under the Bessel pincer light-sheet beam on a charged particle (with different imaginary <math display="inline"><semantics> <msub> <mi>n</mi> <mi>p</mi> </msub> </semantics></math> of refractive index) varying with the varying surface charge <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>s</mi> </msub> </semantics></math>. The refractive index of particle is <math display="inline"><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>1.33</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>i</mi> </mrow> </semantics></math>, the beam scaling parameter is <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>, and the beam order is <math display="inline"><semantics> <mrow> <mi>ł</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 13
<p><math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>s</mi> <mi>c</mi> <mi>a</mi> </mrow> </msub> </semantics></math>m and <math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> under different beam scaling parameter <math display="inline"><semantics> <msub> <mi>α</mi> <mn>0</mn> </msub> </semantics></math> values of a Bessel pincer light-sheet beam on a charged particle varying with the varying Sigma <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>s</mi> </msub> </semantics></math>. The refractive index of particle is <math display="inline"><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>1.33</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>i</mi> </mrow> </semantics></math>, and the beam order is ł = 10.</p>
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<p><math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>s</mi> <mi>c</mi> <mi>a</mi> </mrow> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> under different beam order ł of a Bessel pincer light-sheet beam on a charged particle varying with the varying Sigma <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>s</mi> </msub> </semantics></math>. The refractive index of particle is <math display="inline"><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>1.33</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>i</mi> </mrow> </semantics></math>, and the beam scaling parameter is <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> </mrow> </semantics></math> 0.8.</p>
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<p><math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>s</mi> <mi>c</mi> <mi>a</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>Q</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> of a Bessel pincer light-sheet beam on a charged particle with different particle radius (radius = <span class="html-italic">a</span>), varying with the varying surface charge <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>s</mi> </msub> </semantics></math>. The refractive index of particle is <math display="inline"><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>1.33</mn> <mo>+</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>i</mi> </mrow> </semantics></math>, and the beam scaling parameter is <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> </mrow> </semantics></math> 0.8.</p>
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8 pages, 7013 KiB  
Article
Bessel-Beam Single-Photon High-Resolution Imaging in Time and Space
by Huiyu Qi, Zhaohui Li, Yurong Wang, Xiuliang Chen, Haifeng Pan, E Wu and Guang Wu
Photonics 2024, 11(8), 704; https://doi.org/10.3390/photonics11080704 - 29 Jul 2024
Viewed by 481
Abstract
Synchronous laser beam scanning is a common technique used in single-photon imaging where the spatial resolution is primarily determined by the beam divergence angle. In this context, Bessel beams have been investigated as they can overcome the diffraction limit associated with traditional Gaussian [...] Read more.
Synchronous laser beam scanning is a common technique used in single-photon imaging where the spatial resolution is primarily determined by the beam divergence angle. In this context, Bessel beams have been investigated as they can overcome the diffraction limit associated with traditional Gaussian beams. Notably, the central spot of a Bessel beam retains its size almost unchanged within a non-diffractive distance. However, the presence of sidelobes in the Bessel beam can negatively impact spatial resolution. To address this challenge, we have developed a single-photon imaging system with high-depth resolution, which allows for the suppression of echo photons from the sidelobe light in the depth image, particularly when their flight time differs from that of the central spot. In our LiDAR setup, we successfully achieved high-resolution scanning imaging with a spatial resolution of approximately 0.5 mm while also demonstrating a high-depth resolution of 12 mm. Full article
(This article belongs to the Special Issue Photonics: 10th Anniversary)
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Figure 1

Figure 1
<p>(<b>a</b>) A schematic diagram of the Bessel-beam single-photon imaging system. Laser: picosecond pulsed laser with a central wavelength at 532 nm and a repetition rate of 10 kHz (G-10P-C, Shanghai Buchuang Laser Technology Co., Ltd., Shanghai, China); M: high-reflection mirror; PBS: polarization beam splitter cube (PBS22-532, LBTEK, Changsha, China); BE: 20× beam expander (BE20-532-20X UVFS high-power beam expander, Thorlabs, America); PIN: PIN photodiode; DOE: diffraction optical element (Sichuan Jiuguang Technology Co., Ltd., Chengdu, China); GM-x: X-axis galvanometer scanner (S-9210, SUNNY, Beijing, China); GM-y: Y-axis galvanometer scanner (S-9210, SUNNY, Beijing, China); L: lens with the focal length of 15 mm and the diameter of 18 mm (ACL1815U-A, Thorlabs, America); SPAD: silica single-photon avalanche photodiode-based single-photon detector (Homebuilt); TDC: time-to-digital converter (HydraHarp 400, PicoQuant, Germany); SG: signal generator; BF: bandpass filter (FLH532-10, Thorlabs, America). (<b>b</b>) The intensity distribution of the Bessel beam spot at 14.3 m. (<b>c</b>) The photograph of the target.</p>
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<p>The time distribution histogram of the echo photons when the Bessel beam illuminated the targets as shown in the inset picture, where C1 and C3 are the echo photon counts of the side-lobe light, and C2 is the echo photon count of the central spot.</p>
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<p>The depth image with different denoising thresholds: (<b>a</b>) original image; (<b>b</b>) 4; (<b>c</b>) 10; (<b>d</b>) 20; (<b>e</b>) 40; (<b>f</b>) 60.</p>
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<p>The depth images with varying measurement times: (<b>a</b>) 1000 s, (<b>b</b>) 100 s, (<b>c</b>) 10 s, (<b>d</b>) 2 s.</p>
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<p>Comparison of the photo and grayscale image of the target with dimensions: (<b>a</b>) the photo of the target; (<b>b</b>) the grayscale image of the target.</p>
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<p>Comparison of the depth image with sidelobes eliminated and sidelobes retained: (<b>a</b>) depth image with sidelobes retained; (<b>b</b>) depth image with sidelobes eliminated.</p>
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<p>Comparison of target size errors extracted by eliminating and retaining sidelobes in grayscale images with different measurement areas and actual size: (<b>a</b>) 2.0 mm; (<b>b</b>) 4.0 mm; (<b>c</b>) 8.4 mm.</p>
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18 pages, 1985 KiB  
Article
Optical Halo: A Proof of Concept for a New Broadband Microrheology Tool
by Jorge Ramírez, Graham M. Gibson and Manlio Tassieri
Micromachines 2024, 15(7), 889; https://doi.org/10.3390/mi15070889 - 7 Jul 2024
Cited by 1 | Viewed by 948
Abstract
Microrheology, the study of material flow at micron scales, has advanced significantly since Robert Brown’s discovery of Brownian motion in 1827. Mason and Weitz’s seminal work in 1995 established the foundation for microrheology techniques, enabling the measurement of viscoelastic properties of complex fluids [...] Read more.
Microrheology, the study of material flow at micron scales, has advanced significantly since Robert Brown’s discovery of Brownian motion in 1827. Mason and Weitz’s seminal work in 1995 established the foundation for microrheology techniques, enabling the measurement of viscoelastic properties of complex fluids using light-scattering particles. However, existing techniques face limitations in exploring very slow dynamics, crucial for understanding biological systems. Here, we present a proof of concept for a novel microrheology technique called “Optical Halo”, which utilises a ring-shaped Bessel beam created by optical tweezers to overcome existing limitations. Through numerical simulations and theoretical analysis, we demonstrate the efficacy of the Optical Halo in probing viscoelastic properties across a wide frequency range, including low-frequency regimes inaccessible to conventional methods. This innovative approach holds promise for elucidating the mechanical behaviour of complex biological fluids. Full article
(This article belongs to the Special Issue Optical Tools for Biomedical Applications)
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Figure 1
<p>Two equivalent representations of Jeffreys model of a viscoelastic fluid: (<b>a</b>) is made of a dashpot (of viscosity <math display="inline"><semantics> <msub> <mi>η</mi> <mn>1</mn> </msub> </semantics></math>) connected in series with a Kelvin–Voight element, which is made of a dashpot (of viscosity <math display="inline"><semantics> <msub> <mi>η</mi> <mn>2</mn> </msub> </semantics></math>) and a spring (of modulus <span class="html-italic">G</span>) placed in parallel, whereas (<b>b</b>) is made of a dashpot (of viscosity <math display="inline"><semantics> <msub> <mi>η</mi> <mn>1</mn> </msub> </semantics></math>) connected i parallel with a Maxwell element, which is made of a dashpot (of viscosity <math display="inline"><semantics> <msub> <mi>η</mi> <mn>2</mn> </msub> </semantics></math>) and a spring (of modulus <span class="html-italic">G</span>) placed in series. Both models are connected to a material point of mass <span class="html-italic">m</span>, whose contribution to the dynamics of the system is neglected in this work.</p>
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<p>(<b>a</b>) A proposed experimental configuration of a ring-based optical trap. The setup is based on the configurations reported by Shao et al. [<a href="#B58-micromachines-15-00889" class="html-bibr">58</a>], where an axicon is used to create a smooth optical ring. This can range from a simple fixed design to a design that allows the size of the ring traps to be adjusted [<a href="#B59-micromachines-15-00889" class="html-bibr">59</a>]. (<b>b</b>) Geometrical representation of the torus-shaped optical halo. The torus has major radius <span class="html-italic">R</span> and minor radius <span class="html-italic">b</span> which are related to the stiffness of the halo in the radial direction by <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>r</mi> </msub> <mo>=</mo> <msub> <mi>k</mi> <mi>B</mi> </msub> <mi>T</mi> <mo>/</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The stiffness of the optical halo in the <span class="html-italic">z</span> direction is <math display="inline"><semantics> <msub> <mi>κ</mi> <mi>z</mi> </msub> </semantics></math>.</p>
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<p>(<b>a</b>–<b>d</b>) Trajectory of a colloidal particle suspended in a Newtonian fluid and constrained by a toroidal optical trap with major radius <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, small radius <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>b</mi> <mo>ˇ</mo> </mover> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </semantics></math>. (<b>e</b>–<b>h</b>) Similar simulation conditions to (<b>a</b>–<b>d</b>), but exploring the effects of varying <math display="inline"><semantics> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> </semantics></math> on the MSD in (<b>e</b>), and on the normalised position autocorrelation function in the radial <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<b>g</b>). The inset in (<b>g</b>) shows the steady-state value of <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> </semantics></math>. (<b>f</b>,<b>h</b>) show the effects of varying the ratio <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> </mrow> </semantics></math> on the MSD in (<b>f</b>), and on the normalised position autocorrelation function <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, both only in the axial direction. The insets in (<b>f</b>,<b>h</b>) show the master curves for MSD<sub><span class="html-italic">z</span></sub> and <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> when the same data shown in the main are plotted against <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>z</mi> </msub> <mi>t</mi> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> </mrow> </semantics></math>, respectively, and the MSD<sub><span class="html-italic">z</span></sub> is normalised by the variance of the optical trap in the <span class="html-italic">z</span> direction [<a href="#B26-micromachines-15-00889" class="html-bibr">26</a>].</p>
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<p>The mean squared displacement (<b>a</b>,<b>c</b>) and the normalized position autocorrelation function (<b>b</b>,<b>d</b>), evaluated along the three main directions <span class="html-italic">r</span>, <span class="html-italic">z</span>, and <math display="inline"><semantics> <mi>θ</mi> </semantics></math> of a toroidal optical trap from Brownian dynamics simulations of an ensemble of 1000 particles on a torus with <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> <mo>=</mo> <mn>150</mn> </mrow> </semantics></math>, a ratio of axial to radial stiffness of <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math>; these are parametric (colour coded) using <math display="inline"><semantics> <mi>De</mi> </semantics></math>, as depicted in the legend. The simulations were performed by assuming two different Jeffreys fluids: (i) a fluid with an elastic component stiffer than that of the optical tweezers, i.e., <math display="inline"><semantics> <mrow> <msup> <mi>κ</mi> <mo>∗</mo> </msup> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>16</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) and (ii) a fluid with an elastic component softer than that of the optical tweezers, i.e., <math display="inline"><semantics> <mrow> <msup> <mi>κ</mi> <mo>∗</mo> </msup> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>). Note that the normalized position autocorrelation function for the azimuthal component is not reported in order to maintain clarity in the diagrams. This is because the data would consistently hover near a value of unity throughout almost the entire time window, falling to zero only at large lag times.</p>
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<p>The mean squared displacement (<b>a</b>,<b>c</b>) and the normalised position autocorrelation function (<b>b</b>,<b>d</b>), evaluated along the three main directions <span class="html-italic">r</span>, <span class="html-italic">z</span> and <math display="inline"><semantics> <mi>θ</mi> </semantics></math> of a toroidal optical trap, from Brownian dynamics simulations of an ensemble of 1000 particles on a torus with <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> <mo>=</mo> <mn>150</mn> </mrow> </semantics></math>. The outcomes are parametric (colour coded) using the ratio of axial to radial stiffness <math display="inline"><semantics> <mrow> <msup> <mi>κ</mi> <mo>∗</mo> </msup> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> </mrow> </semantics></math> as depicted in the legend. They show two cases: <math display="inline"><semantics> <mrow> <mi>De</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) and <math display="inline"><semantics> <mrow> <mi>De</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>). Note that the normalised position autocorrelation function for the azimuthal component is not reported in order to maintain clarity in the diagrams. This is because the data would consistently hover near a value of unity throughout almost the entire time window, falling to zero only at large lag times.</p>
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<p>(<b>a</b>) The MSD along the three main directions <span class="html-italic">r</span>, <span class="html-italic">z</span>, and <math display="inline"><semantics> <mi>θ</mi> </semantics></math>. (<b>b</b>) The viscoelastic moduli <math display="inline"><semantics> <mrow> <msup> <mi>G</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>ω</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msup> <mi>G</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>ω</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> derived from each of the three MSD functions shown on the left. The MSD data have been obtained from Brownian dynamics simulations of an ensemble of 1000 particles on a torus with <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> <mo>=</mo> <mn>150</mn> </mrow> </semantics></math> and by assuming a value of the fluid’s plateau modulus of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> Pa, a relaxation time of <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math> s, and the solvent to have a viscosity of <math display="inline"><semantics> <mrow> <mn>0.001</mn> </mrow> </semantics></math> Pa·s (e.g., water).</p>
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13 pages, 5014 KiB  
Article
Thick Glass High-Quality Cutting by Ultrafast Laser Bessel Beam Perforation-Assisted Separation
by Suwan Chen, Yuxuan Luo, Xinhu Fan, Congyi Wu, Guojun Zhang, Yu Huang, Youmin Rong and Long Chen
Micromachines 2024, 15(7), 854; https://doi.org/10.3390/mi15070854 - 29 Jun 2024
Viewed by 843
Abstract
The cutting of thick glass is extensively employed in aerospace, optical, and other fields. Although ultrafast laser Bessel beams are heavily used for glass cutting, the cutting thickness and cutting quality need to be further improved. In this research, the high-quality cutting of [...] Read more.
The cutting of thick glass is extensively employed in aerospace, optical, and other fields. Although ultrafast laser Bessel beams are heavily used for glass cutting, the cutting thickness and cutting quality need to be further improved. In this research, the high-quality cutting of thick glass was realized for the first time using ultrafast laser perforation assisted by CO2 laser separation. Initially, an infrared picosecond laser Bessel beam was employed to ablate the soda-lime glass and generate a perforated structure. Subsequently, a CO2 laser was employed to induce crack propagation along the path of the perforated structure, resulting in the separation of the glass. This study investigates the influence of hole spacing, pulse energy, and the defocusing distance of the picosecond laser Bessel beam on the average surface roughness of the glass sample cutting surface. The optimal combination of cutting parameters for 6 mm thick glass results in a minimum surface roughness of 343 nm in the cross-section. Full article
(This article belongs to the Section D:Materials and Processing)
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Figure 1
<p>Schematic diagrams of the experimental setup for picosecond laser Bessel beam ablation: (<b>A</b>) schematic diagram of the light path and devices, (<b>B</b>) schematic diagram of the Bessel beam generation, and (<b>C</b>) glass crack propagation control and separation method.</p>
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<p>Typical microscope image of each sample.</p>
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<p>Estimated marginal means of the effect of influence parameters on the average surface roughness of glass-cutting sample cross-sections.</p>
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<p>Interaction between each parameter for average surface roughness.</p>
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<p>SEM images of the cross-section of the optimal sample (<b>A</b>) near the front surface, (<b>B</b>) the middle, and (<b>C</b>) near the rear surface.</p>
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<p>Demonstrated roughness against thickness of glass cutting utilizing pulsed Bessel beams [<a href="#B9-micromachines-15-00854" class="html-bibr">9</a>,<a href="#B13-micromachines-15-00854" class="html-bibr">13</a>,<a href="#B14-micromachines-15-00854" class="html-bibr">14</a>,<a href="#B26-micromachines-15-00854" class="html-bibr">26</a>,<a href="#B27-micromachines-15-00854" class="html-bibr">27</a>,<a href="#B28-micromachines-15-00854" class="html-bibr">28</a>,<a href="#B29-micromachines-15-00854" class="html-bibr">29</a>].</p>
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9 pages, 2037 KiB  
Article
Subsurface Spectroscopy in Heterogeneous Materials Using Self-Healing Laser Beams
by Benjamin R. Anderson, Natalie Gese and Hergen Eilers
Optics 2024, 5(2), 310-318; https://doi.org/10.3390/opt5020022 - 20 Jun 2024
Viewed by 623
Abstract
Self-healing optical beams are a class of propagation modes that can recover their beam shapes after distortion or partial blockage. This self-healing property makes them attractive for use in applications involving turbid media as they can—in theory—penetrate further into these materials than standard [...] Read more.
Self-healing optical beams are a class of propagation modes that can recover their beam shapes after distortion or partial blockage. This self-healing property makes them attractive for use in applications involving turbid media as they can—in theory—penetrate further into these materials than standard Gaussian beams. In this paper, we characterize the propagation of two different self-healing beams (Bessel and Airy) through a solid scattering material with different scatterer concentrations and find that both beams do recover after scattering for samples below a threshold scatterer concentration. Additionally, we test the applicability of both beam shapes for improved sub-surface spectroscopy in heterogeneous materials using fluorescent particles and find that there is an average fluorescence intensity enhancement of 1.3× using self-healing beams versus a standard Gaussian beam. Full article
(This article belongs to the Section Laser Sciences and Technology)
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Figure 1
<p>Layered sample structure used in this study.</p>
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<p>Schematic of experimental setup used to test self-healing beam propagation and spectroscopy. HWP: halfwave plate; L1/L2/L3: lenses; PH: pinhole; P: polarizer; BS: beam splitter; BD: beam dump; Obj 1/2: objectives; SPF: short-pass filter; LPF: long-pass filter.</p>
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<p>Example Airy and Bessel phase masks (<b>a</b>,<b>b</b>) and their corresponding beam profiles (<b>c</b>,<b>d</b>). Note that for the phase masks the Airy shape parameter is <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> and the Bessel shape parameter is <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math>.</p>
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<p>Peak fluorescence intensity as a function of beam parameter for an Airy (<b>a</b>) and Bessel (<b>b</b>) beam using a 50 wt% sugar concentration sample.</p>
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<p>Example fluorescence spectra measured from the 25 wt% concentration sample for the three beam types (Airy, Bessel, and Gaussian) (<b>a</b>) and the sample-averaged intensity ratios as a function of sugar concentration (<b>b</b>).</p>
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<p>Fluorescence images of EYAD particles for the three different beam types for the 50 wt% sample.</p>
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<p>Example images of beams transmitted through heterogeneous samples containing different sugar concentrations for Gaussian, Airy, and Bessel beams.</p>
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<p>Line profiles across transmitted beam images for each beam type and concentration.</p>
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<p>Correlation coefficient between 0 wt% beam and different sugar concentrations for each beam type.</p>
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18 pages, 2319 KiB  
Article
Propagation of a Partially Coherent Bessel–Gaussian Beam in a Uniform Medium and Turbulent Atmosphere
by Igor Lukin and Vladimir Lukin
Photonics 2024, 11(6), 562; https://doi.org/10.3390/photonics11060562 - 14 Jun 2024
Viewed by 1225
Abstract
In this paper, the coherent properties of partially coherent Bessel–Gaussian optical beams propagating through a uniform medium (free space) or a turbulent atmosphere are examined theoretically. The consideration is based on the analytical solution of the equation for the transverse second-order mutual coherence [...] Read more.
In this paper, the coherent properties of partially coherent Bessel–Gaussian optical beams propagating through a uniform medium (free space) or a turbulent atmosphere are examined theoretically. The consideration is based on the analytical solution of the equation for the transverse second-order mutual coherence function of the field of partially coherent optical radiation in a turbulent atmosphere. For the partially coherent Bessel–Gaussian beam, the second-order mutual coherence function of the source field is taken as a Gaussian–Schell model. In this approximation, we analyze the behavior of the coherence degree and the integral coherence scale of these beams as a function of the propagation pathlength, propagation conditions, and beam parameters, such as the radius of the Gauss factor of the beam, parameter of the Bessel factor of the beam, topological charge, and correlation width of the source field of partially coherent radiation. It was found that, as a partially coherent vortex Bessel–Gaussian beam propagates through a turbulent atmosphere, there appear not two (as might be expected: one due to atmospheric turbulence and another due to the partial coherence of the source field), but only one ring dislocation of the coherence degree (due to the simultaneous effect of both these factors on the optical radiation). In addition, it is shown that the dislocation of the coherence degree that significantly affects the beam coherence level is formed only for beams, for which the coherence width of the source field is larger than the diameter of the first Fresnel zone. Full article
(This article belongs to the Special Issue Recent Advances in Diffractive Optics)
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Figure 1
<p>Coherence degree <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mrow> <mi>vbb</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>ρ</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> of partially coherent vortex Bessel beams propagating in a uniform medium at <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>β</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> for different values of the topological charge of the vortex beam <math display="inline"><semantics> <mi>m</mi> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>. The white curves represent the coordinates of a ring dislocation.</p>
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<p>Coherence degree <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mrow> <mi>vbb</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>ρ</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> of coherent vortex Bessel beams propagating in a turbulent atmosphere at <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>β</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> for different values of the topological charge of the vortex beam <math display="inline"><semantics> <mi>m</mi> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>. The white curves represent the coordinates of a ring dislocation.</p>
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<p>Coherence degree <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mrow> <mi>vbb</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>ρ</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> of partially coherent vortex Bessel beams propagating in a turbulent atmosphere at <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>β</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> for different values of the dimensionless parameter <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> </mrow> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>10.0</mn> </mrow> </semantics></math>.</p>
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<p>Integral coherence scale <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <mi>m</mi> <mo> </mo> <mi>vbb</mi> </mrow> </msub> </mrow> </semantics></math> of partially coherent vortex Bessel beams propagating in a turbulent atmosphere at different source coherence levels <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> </mrow> </semantics></math> for four values of the topological charge of the vortex beam <math display="inline"><semantics> <mi>m</mi> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>.</p>
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<p>Integral coherence scale <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <mi>m</mi> <mo> </mo> <mi>vbb</mi> </mrow> </msub> </mrow> </semantics></math> of partially coherent vortex Bessel beams propagating in a turbulent atmosphere at different turbulence level <math display="inline"><semantics> <mi>q</mi> </semantics></math> in the propagation medium for four values of the topological charge of vortex beam <math display="inline"><semantics> <mi>m</mi> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>.</p>
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<p>Integral coherence scale <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <mi>m</mi> <mo> </mo> <mi>vbb</mi> </mrow> </msub> </mrow> </semantics></math> of partially coherent vortex Bessel beams (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>) propagating in a turbulent atmosphere at the different source coherence <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> </mrow> </semantics></math> for four values of the Bessel beam parameter <math display="inline"><semantics> <mover accent="true"> <mi>β</mi> <mo>˜</mo> </mover> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>β</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>β</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>β</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>2.0</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>β</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>4.0</mn> </mrow> </semantics></math>.</p>
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<p>Ratio of the integral coherence scales of partially coherent vortex Bessel–Gaussian <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <mi>m</mi> <mo> </mo> <mi>vbgb</mi> </mrow> </msub> </mrow> </semantics></math> and Gaussian <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <mi>m</mi> <mo> </mo> <mi>gb</mi> </mrow> </msub> </mrow> </semantics></math> beams with <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>β</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> at different values of the Fresnel number of the transmitting aperture <math display="inline"><semantics> <mrow> <msub> <mo>Ω</mo> <mn>0</mn> </msub> </mrow> </semantics></math> for three source coherence levels <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> </mrow> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>10.0</mn> </mrow> </semantics></math>.</p>
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12 pages, 3930 KiB  
Article
Nanosecond Laser Fabrication of Dammann Grating-like Structure on Glass for Bessel-Beam Array Generation
by Prasenjit Praharaj and Manoj Kumar Bhuyan
Photonics 2024, 11(5), 473; https://doi.org/10.3390/photonics11050473 - 18 May 2024
Viewed by 944
Abstract
The generation of optical beam arrays with prospective uses within the realms of microscopy, photonics, non-linear optics, and material processing often requires Dammann gratings. Here, we report the direct fabrication of one- and two-dimensional Dammann grating-like structures on soda lime glass using a [...] Read more.
The generation of optical beam arrays with prospective uses within the realms of microscopy, photonics, non-linear optics, and material processing often requires Dammann gratings. Here, we report the direct fabrication of one- and two-dimensional Dammann grating-like structures on soda lime glass using a nanosecond pulsed laser beam with a 1064 nm wavelength. Using the fabricated grating, an axicon lens, and an optical magnification system, we propose a scheme of generation of a diverging array of zero-order Bessel beams with a sub-micron-size central core, extending longitudinally over several hundred microns. Two different grating fabrication strategies are also proposed to control the number of Bessel beams in an array. It was demonstrated that Bessel beams of 12 degrees conical half-angle in an array of up to [5 × 5] dimensions can be generated using a suitable combination of Dammann grating, axicon lens and focusing optics. Full article
(This article belongs to the Special Issue Laser Processing and Modification of Materials)
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<p>Schematic diagram of nanosecond pulsed laser micromachining setup.</p>
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<p>(<b>a</b>) Optical micrograph of trenches machined (on a single-pass basis) on the front surface of glass using nanosecond laser pulses of the following fluence levels: 191, 239, 335, 406, 478, 645 and 789 J/cm<sup>2</sup>. (<b>b</b>) As described above, 3D profilometric images of trenches. (<b>c</b>) Plot showing the variation of trench width with respect to laser fluence.</p>
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<p>Schematics of Bessel beam generation and characterisation setups are shown in (<b>a</b>,<b>b</b>), respectively. (<b>c</b>) Cross-sectional profile image of generated Bessel beams of 12-degree conical half-angle. (<b>d</b>) Typical radial profile of generated Bessel beams showing concentric rings.</p>
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<p>Cross-sectional images of zero-order Bessel beams in an array format. Bessel beams in array format were generated using axicon lens, telescope and different Dammann gratings (machined with a laser fluence of 335 J/cm<sup>2</sup>) of fixed period (P) of 200 µm, with various combinations of transparent zone (TZ) and opaque zone (OZ): (<b>a</b>) TZ = 180 µm, OZ = 20 µm; (<b>b</b>) TZ = 150 µm, OZ = 50 µm; and (<b>c</b>) TZ = 100 µm, OZ = 100 µm. Bessel beam arrays were also generated using Dammann gratings of defined parameters, i.e., TZ = OZ = 100 µm, which were machined with various laser fluence levels: (<b>d</b>) 335 J/cm<sup>2</sup>, (<b>e</b>) 478 J/cm<sup>2</sup>, and (<b>f</b>) 645 J/cm<sup>2</sup>. The insets show the optical micrographs and 3D profiles of Dammann gratings in each case.</p>
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<p>Cross-sectional images of arrayed Bessel beams generated using an axicon lens, telescope, and Dammann gratings (machined with a laser fluence of 335 J/cm<sup>2</sup>) of the following parameters: (<b>a</b>) TZ = OZ = 60 µm, (<b>b</b>) TZ = OZ = 100 µm, (<b>c</b>) TZ = OZ = 200 µm, and (<b>d</b>) TZ = OZ = 400 µm. (<b>e</b>–<b>j</b>) Also shown here are the radial profiles of the Bessel beam array captured at three longitudinal distances, i.e., Z = 120 µm, 240 µm and 430 µm, respectively, corresponding to the cases (<b>b</b>,<b>d</b>). These longitudinal positions are marked as Z1, Z2, and Z3 on the images. The radial intensity profiles corresponding to images (<b>f</b>,<b>i</b>) are insets. (<b>k</b>) A table indicating the inter-Bessel beam separation (with respect to the central Bessel beam) at longitudinal distances of 120 µm, 240 µm, and 430 µm, as associated with two distinct gratings. (<b>l</b>) Plot showing the variation of diverging half-angle θ1 of Bessel beam arrays as a function of the grating period.</p>
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<p>Optical micrographs of 2D Dammann gratings with periods of 200 μm fabricated using a laser fluence of 335 J/cm<sup>2</sup> with (<b>a</b>) line-scanning and (<b>b</b>) patch-scanning strategies. The corresponding 3D profiles of gratings are shown as insets. (<b>c</b>,<b>d</b>) The typical radial profiles of 2D Bessel beam arrays are generated using an axicon lens and gratings machined with both considered scanning strategies. The insets to (<b>c</b>,<b>d</b>) show the enlarged size of Bessel beams placed at the centre of the array.</p>
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<p>(<b>a</b>) Optical micrograph of laser fabricated (using laser fluence of 335 J/cm<sup>2</sup>) 1D Dammann grating for the generation of a [1 × 5] Bessel beam array. (<b>b</b>–<b>d</b>) Also shown here are the radial profiles of the Bessel beam array captured at three longitudinal distances, i.e., Z = 190 µm, 295 µm, and 335 µm, respectively. The inset to (<b>d</b>) shows the central portion of the corresponding saturated image.</p>
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13 pages, 8405 KiB  
Article
Rapid Fabrication of Yttrium Aluminum Garnet Microhole Array Based on Femtosecond Bessel Beam
by Heng Yang, Yuan Yu, Tong Zhang, Shufang Ma, Lin Chen, Bingshe Xu and Zhiyong Wang
Photonics 2024, 11(5), 408; https://doi.org/10.3390/photonics11050408 - 27 Apr 2024
Viewed by 961
Abstract
High-aspect-ratio microholes, the fundamental building blocks for microfluidics, optical waveguides, and other devices, find wide applications in aerospace, biomedical, and photonics fields. Yttrium aluminum garnet (YAG) crystals are commonly used in optical devices due to their low stress, hardness, and excellent chemical stability. [...] Read more.
High-aspect-ratio microholes, the fundamental building blocks for microfluidics, optical waveguides, and other devices, find wide applications in aerospace, biomedical, and photonics fields. Yttrium aluminum garnet (YAG) crystals are commonly used in optical devices due to their low stress, hardness, and excellent chemical stability. Therefore, finding efficient fabrication methods to produce high-quality microholes within YAG crystals is crucial. The Bessel beam, characterized by a uniform energy distribution along its axis and an ultra-long depth of focus, is highly suitable for creating high-aspect-ratio structures. In this study, an axicon lens was used to shape the spatial profile of a femtosecond laser into a Bessel beam. Experimental verification showed a significant improvement in the high aspect ratio of the microholes produced in YAG crystals using the femtosecond Bessel beam. This study investigated the effects of the power and defocus parameters of single-pulse Bessel beams on microhole morphology and size, and microhole units with a maximum aspect ratio of more than 384:1 were obtained. Based on these findings, single-pulse femtosecond Bessel processing parameters were optimized, and an array of 181 × 181 microholes in a 400 μm thick YAG crystal was created in approximately 13.5 min. The microhole array had a periodicity of 5 μm and a unit aspect ratio of 315:1, with near-circular top and subface apertures and high repeatability. Full article
(This article belongs to the Special Issue Laser Processing and Modification of Materials)
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<p>Schematic diagram of the femtosecond laser processing system (HWP: half-wave plate; PBS: polarizing beam splitter; NDF: neutral density filter; MS: mechanical shutter; MO: microscope objective; DM: dichroic mirror; CCD: charge-coupled device image sensor; Axicon: axicon lens: L1 and L2: lenses; M1, M2: mirrors).</p>
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<p>(<b>a</b>) Optical photographs of Gaussian pulse processing at different repetition rates, with SEM images as insets. (<b>b</b>) Variation trend of damage diameter with power for Gaussian pulses at different repetition rates. (<b>c</b>) Optical photographs and damage depth of Gaussian beam processing with a single pulse of 100 mW. (<b>d</b>) Variation trend of the ratio of processing diameter to depth with power for single-pulse processing.</p>
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<p>(<b>a</b>) Intensity distribution in the propagation direction of the initial Bessel beam. (<b>b</b>) Transverse intensity distribution of the initial Bessel beam. (<b>c</b>) Intensity distribution in the propagation direction of the Gaussian beam. (<b>d</b>) Transverse intensity distribution of the Gaussian beam.</p>
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<p>(<b>a</b>) Schematic diagram of the Bessel beam generated by the axicon lens. (<b>b</b>) Parameters of the initial Bessel beam generated by different apex-angle axicon lenses: “No diffraction distance”. (<b>c</b>) Size of the central main lobe (ω = 4 mm λ = 800 nm) of the initial Bessel beam.</p>
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<p>SEM image of femtosecond Bessel beam preparation for microhole (<b>a</b>) at the beam entrance and (<b>b</b>) at the beam exit. (<b>c</b>) Sectional profile of micropores fabricated by Gaussian beam and Bessel beam.</p>
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<p>The variation trend of damage diameter and surface morphology in microhole processing with different single-pulse energies.</p>
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<p>(<b>a</b>) Schematic diagram of the focal length of the processing beam. (<b>b</b>) Schematic diagram of the interaction area of the Bessel beam and the sample with the amount of defocusing. (<b>c</b>) Machining results from changing the defocus amount parameters.</p>
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<p>Schematic diagram of the microhole array fabrication method.</p>
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<p>(<b>a</b>) Morphology images of microhole top surface and subface fabricated with different defocus amounts in a 400 μm thick sample. (<b>b</b>) Microscopic photograph of the microhole array.</p>
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0 pages, 1913 KiB  
Article
The Helicity of Magnetic Fields Associated with Relativistic Electron Vortex Beams
by Norah Alsaawi and Vasileios E. Lembessis
Symmetry 2024, 16(4), 496; https://doi.org/10.3390/sym16040496 - 19 Apr 2024
Viewed by 1034
Abstract
For radially extended Bessel modes, the helicity density distributions of magnetic fields associated with relativistic electron vortex beams are investigated for first time in the literature. The form of the distribution is defined by the electron beam’s cylindrically symmetric density flux, which varies [...] Read more.
For radially extended Bessel modes, the helicity density distributions of magnetic fields associated with relativistic electron vortex beams are investigated for first time in the literature. The form of the distribution is defined by the electron beam’s cylindrically symmetric density flux, which varies with the winding number and the electron spin. Different helicity distributions are obtained for different signs of the winding number ±, confirming the chiral nature of the magnetic fields associated with the electron vortex beam. The total current helicity for the spin-down state is smaller than that of the spin-up state. The different fields and helicities associated with opposite winding numbers and/or spin values will play an important role in the investigation of the interaction of relativistic electron vortices with matter and especially chiral matter. A comparison of the calculated quantities with the corresponding ones in the case of non-relativistic spin-polarized electron beams is performed. Full article
(This article belongs to the Section Physics)
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<p>The current helicity density of an infinite relativistic electron Bessel beam with (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>, (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>, (<b>c</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics> </math>, (<b>d</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics> </math>, and (<b>e</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>−</mo> <mn>10</mn> </mrow> </semantics> </math>. The radial oscillatory behaviour dominates at larger radial distances for a higher winding number associated with a lower current helicity value.</p>
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<p>The in-plane current helicity density distribution for an infinite relativistic electron Bessel beam with (<b>a</b>) <span class="html-italic">ℓ</span> = 0, (<b>c</b>) <span class="html-italic">ℓ</span> = 1, (<b>e</b>) <span class="html-italic">ℓ</span> = −1, (<b>g</b>) <span class="html-italic">ℓ</span> = 10, and (<b>i</b>) <span class="html-italic">ℓ</span> = −10 in the case of spin-up. The corresponding plots in the case of spin-down are illustrated for the winding number with (<b>b</b>) <span class="html-italic">ℓ</span> = 0, (<b>d</b>) <span class="html-italic">ℓ</span> = 1, (<b>f</b>) <span class="html-italic">ℓ</span> = −1, (<b>h</b>) <span class="html-italic">ℓ</span> = 10, and (<b>j</b>) <span class="html-italic">ℓ</span> = −10.</p>
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<p>The total current helicity for an infinite relativistic electron Bessel beam from <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>−</mo> <mn>10</mn> </mrow> </semantics> </math> to <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics> </math> in spin-up and spin-down states.</p>
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<p>The magnetic helicity density of an infinite relativistic electron Bessel beam with (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>, (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>, (<b>c</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics> </math>, (<b>d</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics> </math>, and (<b>e</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>−</mo> <mn>10</mn> </mrow> </semantics> </math>. The difference between the magnetic helicity densities of the two spin states is clearly demonstrated. The height of the first peak increases for higher values of <span class="html-italic">ℓ</span>.</p>
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<p>The in-plane magnetic helicity density distribution for an infinite relativistic electron Bessel beam with (<b>a</b>) <span class="html-italic">ℓ</span> = 0, (<b>c</b>) <span class="html-italic">ℓ</span> = 1, (<b>e</b>) <span class="html-italic">ℓ</span> = −1, (<b>g</b>) <span class="html-italic">ℓ</span> = 10, and (<b>i</b>) <span class="html-italic">ℓ</span> = −10 in the case of spin-up. The corresponding plots in the case of spin-down are illustrated for the winding number with (<b>b</b>) <span class="html-italic">ℓ</span> = 0, (<b>d</b>) <span class="html-italic">ℓ</span> = 1, (<b>f</b>) <span class="html-italic">ℓ</span> = −1, (<b>h</b>) <span class="html-italic">ℓ</span> = 10, and (<b>j</b>) <span class="html-italic">ℓ</span> = −10.</p>
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<p>The current helicity density of an infinite relativistic electron Bessel beam with (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>, (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>, and (<b>c</b>) <math display="inline"> <semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics> </math> within the limit of small velocities.</p>
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19 pages, 12490 KiB  
Article
Determining Topological Charge of Bessel-Gaussian Beams Using Modified Mach-Zehnder Interferometer
by Mansi Baliyan and Naveen K. Nishchal
Photonics 2024, 11(3), 263; https://doi.org/10.3390/photonics11030263 - 14 Mar 2024
Viewed by 1069
Abstract
The orbital angular momentum (OAM) associated with structured singular beams carries vital information crucial for studying various properties and applications of light. Determining OAM through the interference of light is an efficient method. The interferogram serves as a valuable tool for analyzing the [...] Read more.
The orbital angular momentum (OAM) associated with structured singular beams carries vital information crucial for studying various properties and applications of light. Determining OAM through the interference of light is an efficient method. The interferogram serves as a valuable tool for analyzing the wavefront of structured beams, especially identifying the order of singularity. In this study, we propose a modified Mach–Zehnder interferometer architecture to effectively determine the topological charge of Bessel–Gaussian (BG) beams. Several numerically generated self-referenced interferograms have been used for analysis. Moreover, this study examines the propagation property and phase distribution within BG beams after they are obstructed by an aperture in the interferometer setup. Full article
(This article belongs to the Special Issue Structured Light Beams: Science and Applications)
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Figure 1

Figure 1
<p>Schematic diagram of the experimental set-up for BG beam generation. Dotted red represent modulation within component of light field.</p>
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<p>Schematic representing in-line interference of BG beams with a reference beam. Green dotted represent reference beam after reflection from mirror while red represent interfered reference and BG beam.</p>
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<p>Simulation results of an in-line interferogram obtained after interference of BG beams with a Gaussian beam. Column (i): intensity distributions of BG beams of different orders <span class="html-italic">l</span> = 1, 3, 5, and 10 as depicted in rows (a–d). Column (ii): intensity distributions of Gaussian beams. Column (iii): intensity distributions of fringe patterns.</p>
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<p>Schematic diagram representing off-axis interference of BG beams with a Gaussian beam. Green dotted represent misaligned reference beam after reflection from mirror while red represent interfered reference and BG beam.</p>
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<p>Simulation results of an off-axis interferogram obtained after interference of BG beams with a Gaussian beam. Column (i): intensity distributions of BG beams of different orders <span class="html-italic">l</span> = 1, 3, 5, and 10 as depicted in rows (a–d). Column (ii): intensity distributions of Gaussian beams. Column (iii): intensity distributions of fringe patterns.</p>
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<p>Simulation results of an in-line interferogram obtained after interference of BG beams with a spherical beam. Column (i): intensity distributions of BG beams of different orders <span class="html-italic">l</span> = 1, 3, 5, and 10 as depicted in rows (a–d). Column (ii): intensity distributions of spherical beams. Column (iii): intensity distributions of fringe patterns.</p>
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<p>Schematic diagram of the modified MZI. BS1, BS2: beam splitter; M1, M2: mirror; RAP: right angle prism; CMOS camera: complementary metal-oxide semiconductor camera. Dotted red represent tilt introduced in the mirror.</p>
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<p>Simulation results of an inline interferogram obtained after interference of BG beams with their conjugate copies. Columns (i, ii) of rows (a–f): intensity distributions of BG beams of different orders <span class="html-italic">l</span> = 1, 3, 5, 10, and 20 and their conjugate copies. Column (iii): intensity distributions of the desired fringe patterns.</p>
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<p>Simulation results of in-line interference obtained after interference of BG beams with their conjugate copies. Columns (i, ii) of rows (a–c): intensity distributions of BG beams of different orders <span class="html-italic">l</span> = −1, −3, −5 and their conjugate copies. Column (iii): intensity distributions of the desired fringe patterns.</p>
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<p>Simulation results of the interference of BG beams with theirtilted conjugate copies. Columns (i, ii) of rows (a–f): intensity distributions of BG beams of different orders <span class="html-italic">l</span> = 1, 3, 5, 10, 20, 30 and their tilted conjugate copies. Column (iii): intensity distributions of the desired fringe patterns.</p>
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<p>Simulation results of the interference of BG beams with their tilted conjugate copies. Columns (i, ii) of rows (a–c): intensity distributions of BG beams of different orders <span class="html-italic">l</span> = −1, −3, −5 and and their tilted conjugate copies. Column (iii): intensity distributions of the desired fringe patterns.</p>
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<p>Schematic diagram of the modified MZI. Red dotted represent tilt and lateral displacement introduced in mirror M1 and M2.</p>
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<p>Simulation results of the interference of BG beams with their laterally displaced copies in the −<span class="html-italic">y</span> direction and tilted in the same direction. Rows (a–c):intensity distributions of BG beams of different orders <span class="html-italic">l</span> = 1, 3, 5. Columns (i, ii): intensity distributions of BG beams of different orders and their laterally displaced and tilted copies. Column (iii): intensity distributions of the desired fringe patterns.</p>
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<p>Simulation results of the interference of BG beams with their laterally displaced copies in the−<span class="html-italic">y</span> direction and tilted in the orthogonal –<span class="html-italic">x</span> direction. Rows (a–c):intensity distributions of BG beams of different orders <span class="html-italic">l</span> = 1, 3, 5. Columns (i, ii): intensity distributions of BG beams of different orders and their laterally displaced and tilted copies. Column (iii): intensity distributions of the desired fringe patterns.</p>
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<p>Simulation results of the intensity distribution of truncated-phase BG beams of order <span class="html-italic">l</span> = 0 at different propagation distances. Rows (a, c): intensity distributions of BG beams obstructed by a circular aperture of radii <span class="html-italic">r</span> = 0.1, 0.15 mm at different propagation distances <span class="html-italic">d</span> = 30, 50, 80, 100, 200, and 300 cm. Column (i): intensity of BG beams of order <span class="html-italic">l</span> = 0 without obstacles. Column (ii): intensity of BG beams of order <span class="html-italic">l</span> = 0 obstructed by a circular aperture of radii <span class="html-italic">r</span> = 0.1, 0.13 mm. rows (b, d): theoretical absolute values of intensity plots carried out when the BG beamspropagated at different propagation distances <span class="html-italic">d</span> = 30, 50, 80, 100, 200, and 300 cm as depicted in column (iii–viii). Green represents the intensity profile for the truncated BG beam and the red pattern represents the intensity profile of the theoretical absolute value of the BG beam.</p>
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<p>Simulation results of the intensity distribution of truncated-phase BG beams of order <span class="html-italic">l</span> = 1 and 5 at different propagation distances, respectively. Rows (a, c): intensity distributions of BG beams of order <span class="html-italic">l</span> = 1 and 5 obstructed by a circular aperture of radii <span class="html-italic">r</span> = 0.15, 0.3 mm at different propagation distances <span class="html-italic">d</span> = 30, 50, 80, 100, 200, 300, and 500 cm. Rows (a, c) of column (i): intensity of BG beams of order <span class="html-italic">l</span> = 1 and 5 without obstacles, Row (b) of column (i): theoretical absolute values of intensity of BG beams of order <span class="html-italic">l</span> = 1 and 5 without obstacles. Rows (a, c) of Column (ii): intensity of BG beams of order <span class="html-italic">l</span> = 1 and 5 obstructed by a circular aperture of radii <span class="html-italic">r</span> = 0.15, 0.3 mm. Rows (b) of column (i): theoretical absolute values of intensity of BG beams of order <span class="html-italic">l</span> = 1 and 5 without obstacles. Rows (b, d): theoretical absolute values of intensity plots carried out when BG beams of order <span class="html-italic">l</span> = 1 and 5 propagated at different propagation distances <span class="html-italic">d</span> = 30, 50, 80, 100, 200, 300, and 500 cm as depicted in columns (iii–viii).</p>
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<p>Interference patterns ofBG beams of different orders, after being obstructed by an aperture of varying radii. Rows (a, b): off-axis and in-lineinterference of obstructed BG beams as shown in <a href="#photonics-11-00263-f015" class="html-fig">Figure 15</a> and <a href="#photonics-11-00263-f016" class="html-fig">Figure 16</a> with their conjugate copies. Columns (i, ii): interference patterns of BG beamsof order <span class="html-italic">l</span> = 0 when obstructed by a circular aperture of radii <span class="html-italic">r</span> = 0.1, 0.15 mm. Column (iii, iv): interference patterns of BG beams of order <span class="html-italic">l</span> = 1, 5 when obstructed by a circular aperture of radii <span class="html-italic">r</span> = 0.15, 0.3 mm, respectively.</p>
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13 pages, 11362 KiB  
Article
High-Quality Cutting of Soda–Lime Glass with Bessel Beam Picosecond Laser: Optimization of Processing Point Spacing, Incident Power, and Burst Mode
by Jiaxuan Liu, Jianjun Yang, Hui Chen, Jinxuan Li, Decheng Zhang, Jian Zhong and Xinjian Pan
Appl. Sci. 2024, 14(5), 1885; https://doi.org/10.3390/app14051885 - 25 Feb 2024
Viewed by 890
Abstract
Soda–lime glass has a wide range of applications in the fields of smart electronics, optical components, and precision originals. In order to investigate the effect of processing parameters on picosecond Bessel laser cutting of soda–lime glass and to achieve high-quality soda–lime glass cutting, [...] Read more.
Soda–lime glass has a wide range of applications in the fields of smart electronics, optical components, and precision originals. In order to investigate the effect of processing parameters on picosecond Bessel laser cutting of soda–lime glass and to achieve high-quality soda–lime glass cutting, a series of cutting experiments were conducted in this study. In this study, it was found that the machining point spacing, the incident laser energy, and the number of burst modes had a significant effect on the machining of the samples. The atomic force microscope (AFM) showed a better quality of roughness of the machined cross-section when the spacing of the machining points was 1 μm, a locally optimal solution was obtained when the number of burst modes was 2, and a locally optimal solution was also obtained when the incident laser power was 11.5 W. In this study, better machining quality was achieved for soda–lime glass of 1 mm thickness, with an average roughness of 158 nm and a local optimum of 141 nm. Full article
(This article belongs to the Special Issue Applied Laser Processing, Manufacturing, and Materials Science)
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Figure 1
<p>Schematic diagram of picosecond laser with Bessel beam cutting soda–lime glass experimental system. (<b>a</b>) The structure diagram of the cutting system. (<b>b</b>) The physical diagram of the system.</p>
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<p>Schemes of Bessel beam simulation results. Picosecond laser with Bessel beam cutting soda–lime glass experimental system. (<b>a</b>) The structure diagram of the cutting system. (<b>b</b>) The physical diagram of the system.</p>
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<p>Schemes of sample separation and sample cross-section. (<b>a</b>) Scheme of sample separation force application. (<b>b</b>) Scheme of sample cross-section and the yellow boxes in the figure indicate the area tested by AFM.</p>
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<p>Diagrams of the samples with different processing point spacings and local SEM scans at a laser re-frequency of 100 kHz, an incident laser power of 11.5 W, and burst mode of 2. (<b>a</b>–<b>d</b>) represent the cross-section and the upper surface of the samples with processing point spacing of 1–4 μm, respectively, and (<b>e</b>,<b>f</b>) represent the SEM schemes of the edges of the modified layer at the processing point spacing of 4 μm.</p>
Full article ">Figure 4 Cont.
<p>Diagrams of the samples with different processing point spacings and local SEM scans at a laser re-frequency of 100 kHz, an incident laser power of 11.5 W, and burst mode of 2. (<b>a</b>–<b>d</b>) represent the cross-section and the upper surface of the samples with processing point spacing of 1–4 μm, respectively, and (<b>e</b>,<b>f</b>) represent the SEM schemes of the edges of the modified layer at the processing point spacing of 4 μm.</p>
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<p>(<b>a</b>) Scheme of variation in cross-section roughness with processing point spacing with an incident laser power of 11.5 W and burst mode of 2. (<b>b</b>) Scheme of variation in breaking strength with processing point spacing with an incident laser power of 11.5 W and burst mode of 2.</p>
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<p>Diagrams of the samples at different laser incident powers when the processing point spacing was 1 μm and the burst mode was 2. (<b>a</b>–<b>e</b>) represent the cross-sections at different laser incident energies, and (<b>f</b>) represents the enlarged pattern of the location of the local ablation crystals when the laser incident power was 19.01 W.</p>
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<p>(<b>a</b>) Schemes of the variation in the sample cross-section roughness with the laser incident power for a processing point spacing of 1 μm and burst mode of 2. (<b>b</b>) Schemes of the variation in the breaking strength with the laser incident power for a processing point spacing of 1 μm and burst mode of 2.</p>
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<p>Diagrams of the sample cross-section for different burst mode numbers with 1 μm spacing and 11.5 W laser incident power. (<b>a</b>–<b>e</b>) represent cross-sections of the samples for burst mode numbers of 1–5, respectively.</p>
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<p>Schemes of the local AFM morphology of the sample cross-section at different number of burst mode with a processing point spacing of 1 μm and a laser incident power of 11.5 W. (<b>a</b>–<b>e</b>) represent the local AFM morphology of the samples for burst mode numbers of 1–5, respectively.</p>
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<p>(<b>a</b>) Scheme of the local AFM morphology of the sample cross-section with different numbers of burst modes with a processing point spacing of 1 μm and laser incident power of 11.5 W. (<b>b</b>) Scheme of breaking strength at different numbers of burst modes with a processing point spacing of 1 μm and laser incident power of 11.5 W.</p>
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15 pages, 3725 KiB  
Article
Creating an Array of Parallel Vortical Optical Needles
by Paulius Šlevas and Sergej Orlov
Photonics 2024, 11(3), 203; https://doi.org/10.3390/photonics11030203 - 24 Feb 2024
Cited by 3 | Viewed by 1286
Abstract
We propose a method for creating parallel Bessel-like vortical optical needles with an arbitrary axial intensity distribution via the superposition of different cone-angle Bessel vortices. We analyzed the interplay between the separation of individual optical vortical needles and their respective lengths and introduce [...] Read more.
We propose a method for creating parallel Bessel-like vortical optical needles with an arbitrary axial intensity distribution via the superposition of different cone-angle Bessel vortices. We analyzed the interplay between the separation of individual optical vortical needles and their respective lengths and introduce a super-Gaussian function as their axial profile. We also analyzed the physical limitations to observe well-separated optical needles, as they are influenced by the mutual interference of the individual beams. To verify our theoretical and numerical results, we generated controllable spatial arrays of individual Bessel beams with various numbers and spatial separations by altering the spectrum of the incoming laser beam via the spatial light modulator. We demonstrate experimentally how to implement such beams using a diffractive mask. The presented method facilitates the creation of diverse spatial intensity distributions in three dimensions, potentially finding applications in specific microfabrication tasks or other contexts. These beams may have benefits in laser material processing applications such as nanochannel machining, glass via production, modification of glass refractive indices, and glass dicing. Full article
(This article belongs to the Special Issue Recent Advances in Diffractive Optics)
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Figure 1
<p>Examples of axial profiles used for both numerical simulations and experimental measurements; the order of the super-Gaussian function is <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> </mrow> </semantics></math> in (<b>a</b>–<b>c</b>), respectively. The lengths <span class="html-italic">L</span> of the axial profile function <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math> are 1 mm (black), 4 mm (green), and 8 mm (red).</p>
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<p>Depiction of the original and translated cylindrical coordinates. <span class="html-italic">O</span> is the original coordinate (<math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>ρ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mn>1</mn> </msub> </mrow> </semantics></math>), and <math display="inline"><semantics> <msub> <mi>O</mi> <mn>1</mn> </msub> </semantics></math> is the center of the translated coordinate (<math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>ρ</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mn>2</mn> </msub> </mrow> </semantics></math>).</p>
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<p>Optical setup of the experiments. HWP is the half-wave plate; BP is the Brewster polarizer; MO1, MO2 are the microscope objectives; L1, L2, L3, L4, and L5 are the lenses; A1 and A2 are the apertures; SLM is a spatial light modulator; BS is a beam splitter; M1 is a mirror; CCD is a charge-coupled device camera.</p>
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<p>Transverse intensity distributions for vortical needles with topological charges <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (<b>a</b>–<b>c</b>), <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (<b>e</b>–<b>g</b>) and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (<b>i</b>–<b>k</b>) and lengths of the axial super-Gaussian profile <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> mm (<b>a</b>,<b>e</b>,<b>i</b>), <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> mm (<b>b</b>,<b>f</b>,<b>j</b>), and <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math> mm (<b>c</b>,<b>g</b>,<b>k</b>). Longitudinal axial profiles of vortical optical needles, measured in the brightest ring (or central lobe) for topological charges <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (<b>d</b>), <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (<b>h</b>), and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (<b>l</b>) and different lengths <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math> mm (red), <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> mm (green), and <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> mm (black). Intensity distributions (<b>m</b>–<b>o</b>) of cross-sections marked by a red line in (<b>i</b>–<b>k</b>), respectively. Longitudinal intensity distributions for optical needles with topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (<b>p</b>).</p>
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<p>Transverse intensity distributions for an array of three optical needles with topological charges <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and the length of the axial super-Gaussian profile <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> mm. The spatial separation of the individual optical needles is <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>20</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>a</b>), <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>40</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>b</b>), <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>60</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>c</b>), and <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>80</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>d</b>).</p>
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<p>Transverse intensity distributions for an array of three optical needles with topological charges <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and the length of the axial super-Gaussian profile <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> mm (<b>a</b>–<b>d</b>), <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> mm (<b>e</b>–<b>h</b>), and <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> mm (<b>i</b>–<b>l</b>). The spatial separation of individual optical needles is <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>20</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>a</b>,<b>e</b>,<b>i</b>), <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>40</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>b</b>,<b>f</b>,<b>j</b>), <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>60</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>c</b>,<b>g</b>,<b>k</b>), and <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>80</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>d</b>,<b>h</b>,<b>l</b>).</p>
Full article ">Figure 7
<p>Transverse intensity distributions for an array of three optical needles with topological charges <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and the length of the axial super-Gaussian profile <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> mm. The spatial separation of individual optical needles was <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>20</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>a</b>), <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>40</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>b</b>), <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>60</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>c</b>), and <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>80</mn> <mi>λ</mi> </mrow> </semantics></math> (<b>d</b>).</p>
Full article ">Figure 8
<p>Phase (<b>a</b>) and amplitude (<b>b</b>) of the spatial spectra of a complex array of vortical needles. Longitudinal intensity distribution of an array of seven vortical optical needles with individual positions and topological charges (<b>c</b>). Transverse intensity distributions of an array and the particular positions in (<b>c</b>), marked green; see (<b>d</b>–<b>g</b>).</p>
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