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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)
Show Figures

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>
Full article ">Figure 2
<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>
Full article ">Figure 3
<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>
Full article ">Figure 4
<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>
Full article ">Figure 5
<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>
Full article ">Figure 6
<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>
Full article ">Figure 7
<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>
Full article ">
7 pages, 178 KiB  
Editorial
Non-Diffractive Beams for State-of-the-Art Applications
by Muhammad A. Butt and Svetlana N. Khonina
Micromachines 2024, 15(6), 771; https://doi.org/10.3390/mi15060771 - 9 Jun 2024
Viewed by 912
Abstract
Non-diffractive beams, also known as diffraction-free beams, are a class of optical beams that maintain their intensity profile over a long distance without spreading out due to diffraction [...] Full article
(This article belongs to the Special Issue Non-diffractive Beams for the State of the Art Applications)
23 pages, 11568 KiB  
Article
Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
by Klemensas Laurinavičius, Sergej Orlov and Ada Gajauskaitė
Appl. Sci. 2024, 14(10), 3984; https://doi.org/10.3390/app14103984 - 8 May 2024
Viewed by 786
Abstract
Theoretical studies on the generation of nondiffracting and nondispersive light pulses and their experimental implementation are one of the renowned problems within electromagnetics. Current technologies enable the creation of short-duration pulses of a few cycles with high power and fluency. An application of [...] Read more.
Theoretical studies on the generation of nondiffracting and nondispersive light pulses and their experimental implementation are one of the renowned problems within electromagnetics. Current technologies enable the creation of short-duration pulses of a few cycles with high power and fluency. An application of these techniques to the field of nondiffracting and nondispersive pulses requires a proper mathematical description of highly focused vector pulses. In this work, we study vector optical bullets in a dielectric medium with different polarization structures: linear, azimuthal, and radial. We report the differences caused by the vector model compared to the scalar model. We analyze effects caused by superluminal, subluminal, or even negative group velocity on the properties of vector optical bullets inside a dielectric material. Full article
(This article belongs to the Section Optics and Lasers)
Show Figures

Figure 1

Figure 1
<p>(<b>a</b>) Angular dispersion of the optical bullet inside the BK7 glass, when <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1.25</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mn>0.7</mn> </mrow> </semantics></math> (1), <math display="inline"><semantics> <mrow> <mn>0.75</mn> </mrow> </semantics></math> (2), <math display="inline"><semantics> <mrow> <mn>0.85</mn> </mrow> </semantics></math> (3), 1 (4), <math display="inline"><semantics> <mrow> <mn>1.4</mn> </mrow> </semantics></math> (5). (<b>b</b>) Angular dispersion of the optical bullet within the BK7 glass, when <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mn>0.629</mn> </mrow> </semantics></math> (1), <math display="inline"><semantics> <mrow> <mn>0.63685</mn> </mrow> </semantics></math> (2), <math display="inline"><semantics> <mrow> <mn>0.645</mn> </mrow> </semantics></math> (3), <math display="inline"><semantics> <mrow> <mn>0.65495</mn> </mrow> </semantics></math> (4), <math display="inline"><semantics> <mrow> <mn>0.685</mn> </mrow> </semantics></math> (5), (<b>c</b>) Angular dispersion of the optical bullet within the BK7 glass, when <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mi>γ</mi> </semantics></math>: <math display="inline"><semantics> <mrow> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (1), <math display="inline"><semantics> <mrow> <mn>7</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (2), <math display="inline"><semantics> <mrow> <mn>8</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (3), <math display="inline"><semantics> <mrow> <mn>9</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (4), <math display="inline"><semantics> <mrow> <mn>10</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (5). The frequency is normalized to the value of <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.7716</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 2
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse electric (TE) linearly polarized optical bullets and their individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. The frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 3
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse magnetic (TM) linearly polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)). The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 4
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of azimuthally polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 5
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of radially polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 6
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse electric (TE) linearly polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. The frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 7
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse magnetic (TM) linearly polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. The frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 8
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of azimuthally polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 9
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of radially polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 10
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of higher polarization order optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. The frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 11
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of higher polarization order optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)) in the transverse plane. The white arrows in (<b>b</b>) represent the orientation of the electric field. The frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 12
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse electric (TE) linearly polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)). The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 13
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of transverse magnetic (TM) linearly polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)). The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 14
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of azimuthally polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)). The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 15
<p>Intensity distributions in the longitudinal (<b>a</b>) and transverse (<b>b</b>) planes of radially polarized optical bullets and its individual components (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>, (<b>c</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>, (<b>d</b>), <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>, (<b>e</b>)). The white arrows in (<b>b</b>) represent the orientation of the electric field. Frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>fs</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs, <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 16
<p>Dependencies of FWHM (<b>a</b>) and second-moment (<b>b</b>) pulse widths in BK7 glass for different values of frequency <math display="inline"><semantics> <msub> <mi>ω</mi> <mi>c</mi> </msub> </semantics></math> for linear, azimuthal and radial polarizations. The red color represents <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and the blue color <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs.</p>
Full article ">Figure 17
<p>Normalized intensities of individual components of FWM pulses (<math display="inline"><semantics> <msub> <mi>E</mi> <mi>x</mi> </msub> </semantics></math>—blue, <math display="inline"><semantics> <msub> <mi>E</mi> <mi>y</mi> </msub> </semantics></math>—red, <math display="inline"><semantics> <msub> <mi>E</mi> <mi>z</mi> </msub> </semantics></math>—orange). Topological charge <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, pulse duration <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> fs. For the cases: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mn>0.63685</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>0.628</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>1.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. The solid line represents the linear polarization of TE, the dashed line represents the linear polarization of TM, the dotted line represents the azimuthal polarization, and the dashed-dotted line represents the radial polarization.</p>
Full article ">
10 pages, 2092 KiB  
Article
Exact and Paraxial Broadband Airy Wave Packets in Free Space and a Temporally Dispersive Medium
by Ioannis M. Besieris and Peeter Saari
Photonics 2024, 11(1), 94; https://doi.org/10.3390/photonics11010094 - 21 Jan 2024
Cited by 1 | Viewed by 897
Abstract
A question of physical importance is whether finite-energy spatiotemporally localized (i.e., pulsed) generalizations of monochromatic accelerating Airy beams are feasible. For luminal solutions, this question has been answered within the framework of paraxial geometry. The time-diffraction technique that has been motivated by the [...] Read more.
A question of physical importance is whether finite-energy spatiotemporally localized (i.e., pulsed) generalizations of monochromatic accelerating Airy beams are feasible. For luminal solutions, this question has been answered within the framework of paraxial geometry. The time-diffraction technique that has been motivated by the Lorentz invariance of the equation governing the narrow angular spectrum and narrowband temporal spectrum paraxial approximation has been used to derive finite-energy spatiotemporally confined subluminal, luminal, and superluminal Airy wave packets. The goal in this article is to provide novel exact finite-energy broadband spatio-temporally localized Airy solutions (a) to the scalar wave equation in free space; (b) in a dielectric medium moving at its phase velocity; and (c) in a lossless second-order temporally dispersive medium. Such solutions can be useful in practical applications involving broadband (few-cycle) wave packets. Full article
(This article belongs to the Section Lasers, Light Sources and Sensors)
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Figure 1

Figure 1
<p>Surface plots of the modulus of <math display="inline"><semantics> <mrow> <msub> <mi>ψ</mi> <mn>2</mn> </msub> <mfenced> <mrow> <mi>R</mi> <mo>,</mo> <mi>π</mi> <mo>/</mo> <mn>4</mn> <mo>,</mo> <msub> <mo>Λ</mo> <mo>+</mo> </msub> <mo>,</mo> <msub> <mo>Λ</mo> <mo>−</mo> </msub> </mrow> </mfenced> </mrow> </semantics></math> versus <math display="inline"><semantics> <mrow> <msub> <mo>Λ</mo> <mo>+</mo> </msub> <mo>∈</mo> <mfenced close="]" open="["> <mrow> <mo>−</mo> <mn>4</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>4</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>∈</mo> <mfenced close="]" open="["> <mrow> <mo>−</mo> <mn>6</mn> <mo>,</mo> <mn>6</mn> </mrow> </mfenced> </mrow> </semantics></math> for three values of <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>,</mo> </mrow> </semantics></math> with the latter defined by the relationship <math display="inline"><semantics> <mrow> <msub> <mo>Λ</mo> <mo>−</mo> </msub> <mo>=</mo> <msub> <mo>Λ</mo> <mo>+</mo> </msub> <mo>+</mo> <mn>2</mn> <mi>T</mi> <mo>.</mo> </mrow> </semantics></math> The parameters <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mn>2</mn> </msub> </mrow> </semantics></math> have the values <math display="inline"><semantics> <mrow> <mn>8</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>10</mn> <mo>,</mo> </mrow> </semantics></math> respectively, with the speed of light in vacuum normalized to unity.</p>
Full article ">Figure 2
<p>Surface plots of the modulus of <math display="inline"><semantics> <mrow> <mi>ψ</mi> <mfenced> <mrow> <mi>X</mi> <mo>,</mo> <msub> <mo>Λ</mo> <mo>+</mo> </msub> <mo>,</mo> <msub> <mo>Λ</mo> <mo>−</mo> </msub> </mrow> </mfenced> </mrow> </semantics></math> versus <math display="inline"><semantics> <mrow> <msub> <mo>Λ</mo> <mo>+</mo> </msub> <mo>∈</mo> <mfenced close="]" open="["> <mrow> <mo>−</mo> <mn>10</mn> <mo>,</mo> <mn>10</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∈</mo> <mfenced close="]" open="["> <mrow> <mn>0</mn> <mo>,</mo> <mn>60</mn> </mrow> </mfenced> </mrow> </semantics></math> for three values of <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>,</mo> </mrow> </semantics></math> with the latter defined by the relationship <math display="inline"><semantics> <mrow> <msub> <mo>Λ</mo> <mo>−</mo> </msub> <mo>=</mo> <msub> <mo>Λ</mo> <mo>+</mo> </msub> <mo>+</mo> <mn>2</mn> <mi>T</mi> <mo>.</mo> </mrow> </semantics></math> The parameters <math display="inline"><semantics> <mi>a</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> </semantics></math> have the values <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> and 100, respectively, with the speed of light in vacuum normalized to unity.</p>
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<p>Surface plot of the modulus of <math display="inline"><semantics> <mrow> <mi>φ</mi> <mfenced> <mrow> <mi>ρ</mi> <mo>,</mo> <mi>τ</mi> </mrow> </mfenced> </mrow> </semantics></math> versus <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>∈</mo> <mfenced close="]" open="["> <mrow> <mo>−</mo> <mn>15</mn> <mo>,</mo> <mn>15</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>∈</mo> <mfenced close="]" open="["> <mrow> <mo>−</mo> <mn>10</mn> <mo>,</mo> <mn>10</mn> </mrow> </mfenced> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Surface plot of the modulus of the azimuthally symmetric wave packet versus <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>∈</mo> <mfenced close="]" open="["> <mrow> <mo>−</mo> <mn>10</mn> <mo>,</mo> <mn>15</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>∈</mo> <mfenced close="]" open="["> <mrow> <mn>0</mn> <mo>,</mo> <mn>80</mn> </mrow> </mfenced> </mrow> </semantics></math> for three values of <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>.</mo> </mrow> </semantics></math> The dimensionless parameters are as follows: <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>10</mn> <mo>,</mo> <mo> </mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>β</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>5</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
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<p>Plot of the modulus of <math display="inline"><semantics> <mrow> <mi>u</mi> <mfenced> <mrow> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> </mrow> </mfenced> </mrow> </semantics></math> versus <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mi>z</mi> <mfenced> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>β</mi> <mn>2</mn> </msup> </mrow> </mfenced> <mo>/</mo> <mfenced> <mrow> <mn>2</mn> <msub> <mi>v</mi> <mn>0</mn> </msub> </mrow> </mfenced> <mo> </mo> <mo>∈</mo> <mfenced close="]" open="["> <mrow> <mo>−</mo> <mn>2</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>2</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>∈</mo> <mfenced close="]" open="["> <mrow> <mn>0</mn> <mo>,</mo> <mn>4</mn> </mrow> </mfenced> </mrow> </semantics></math> for four values of <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>.</mo> </mrow> </semantics></math> The parameter values are <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>5</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo> </mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.9</mn> <mo>,</mo> </mrow> </semantics></math> with the speed of light in vacuum normalized to unity.</p>
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16 pages, 1666 KiB  
Review
The Arago–Poisson Spot: New Applications for an Old Concept
by Olivier Emile and Janine Emile
Photonics 2024, 11(1), 55; https://doi.org/10.3390/photonics11010055 - 4 Jan 2024
Cited by 1 | Viewed by 2089
Abstract
Herein, we report some specific properties and applications of the so-called Arago–Poisson spot in optics. This spot results from the diffraction of a plane wave by an occulting disk that leads to a small bright spot in its shadow. We discuss some of [...] Read more.
Herein, we report some specific properties and applications of the so-called Arago–Poisson spot in optics. This spot results from the diffraction of a plane wave by an occulting disk that leads to a small bright spot in its shadow. We discuss some of the properties of such beams. In particular, we focus on the ultimate size that can be reached for these beams, which depends on the diameter of the disk, the wavelength, and the distance from the disk. We also highlight self-healing and faster-than-light properties. Applications are then proposed. The applications mainly deal with new traps with nanometer sizes dedicated to the trapping of nanoparticles. We also discuss beams that change frequency during propagation and their application for signal delivery in a precise and determined area. Full article
(This article belongs to the Section Lasers, Light Sources and Sensors)
Show Figures

Figure 1

Figure 1
<p>Principle of the Arago–Poisson spot observation. (<b>a</b>) A light beam impinges on an occulting disk (diameter <span class="html-italic">d</span>). The diffracted light interferes in the shadow of the disk on a screen at a distance <span class="html-italic">z</span> from the occulting disk. (<b>b</b>) Example of a picture of the Arago–Poisson spot at a distance of <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> cm from the disk, for <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> cm and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>633</mn> </mrow> </semantics></math> nm.</p>
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<p>Experimental intensity distribution (in arbitrary units) of the dark Arago–Poisson spot for <math display="inline"><semantics> <mrow> <mo mathvariant="sans-serif">ℓ</mo> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m and a disk diameter of <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>600</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m. <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>633</mn> </mrow> </semantics></math> nm.</p>
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<p>Principle of the experiment: a collimated laser beam impinges on a transparent window with an occulting disk. The light is diffracted by the disk and forms a bright spot in the shadow of the disk. The light is collected with an inverted microscope with a <math display="inline"><semantics> <mrow> <mo>×</mo> <mn>100</mn> </mrow> </semantics></math> objective. <span class="html-italic">z</span>: distance from the disk, <span class="html-italic">d</span>: diameter of the disk, <math display="inline"><semantics> <mi>θ</mi> </semantics></math>: maximum angle corresponding to the numerical aperture of the objective. We cannot observe a spot at a distance shorter than 100 <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m for a <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>600</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m disk.</p>
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<p>Variation in the size of the Arago–Poisson spot with the distance from the occulting disk (diameter <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> mm. As the distances reduce, the size tends to 176 nm. The experimental uncertainty reads in the error bars. The solid line corresponds to the theoretical curve of Equation (<a href="#FD1-photonics-11-00055" class="html-disp-formula">1</a>). Inserts, pictures of the Arago–Poisson spots corresponding to the experimental points. Note that the scale changes for each insert.</p>
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<p>Variation in the size of the Arago–Poisson spot with the laser wavelength. The experimental uncertainty reads in the error bars. The solid line corresponds to the theoretical curve of Equation (<a href="#FD1-photonics-11-00055" class="html-disp-formula">1</a>), with <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>600</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> mm.</p>
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<p>Variation in the size of the Arago–Poisson spot with diameter of the occulting disk <span class="html-italic">d</span>. The experimental uncertainty reads in the error bars. The solid line corresponds to the theoretical curve of Equation (<a href="#FD1-photonics-11-00055" class="html-disp-formula">1</a>), with <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>532</mn> </mrow> </semantics></math> nm and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> mm.</p>
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<p>Principle of self-healing for Arago–Poisson spot. (<b>Top</b>): usual situation leading to an Arago–Poisson spot. (<b>Bottom</b>): as a perturbing obstacle (blue diamond) is inserted, in its immediate shadow, the spot disappears. But further on, it reconstructs itself with the same characteristics. The obstacle does not need to be exactly on axis or even symmetric.</p>
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<p>The wavelets emitted by the edges of the occulting disk propagate at a velocity <span class="html-italic">c</span>. During a time interval <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </semantics></math>, they travel from <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>d</mi> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>. However, on the axis, the spot travels from <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>d</mi> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> over a distance that is greater than the distance from <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>d</mi> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>, thus at an apparent faster-than-light velocity.</p>
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<p>Experimental set up: (<b>a</b>) an Arago–Poisson spot originates from the diffraction of a 488 nm collimated laser beam diffracted by a chromium occulting disk deposited on a glass lamella. First, 200 nm diameter fluorescent colloids flowing within the liquid, are exited with a 470 nm diode. The fluorescence is collected with a X20 microscope objective and a camera. DM: Dichroic Mirror. (<b>b</b>) Details of the experimental set-up. The Arago–Poisson spot is under total internal reflection on the glass/liquid interface. The particles are trapped by the evanescent Arago–Poisson spot.</p>
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<p>Principle of the trapping mechanism. The particle is pushed to the upper side of the channel by the radiation pressure from the 470 nm diode that excites the particle. Then, the particle reemits light in a stimulated way at 488 nm in the outgoing evanescent wave. The particle is thus pushed towards the center of the trap by the force <span class="html-italic">F</span>. Insert shows an example of a trapped particle.</p>
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<p>(<b>Left</b>): experimental situation for the realization of an evanescent trap. The Arago–Poisson spot is under total internal reflection. As the size of the occulting disk increases, the angle of incidence of the spot on the surface increases. (<b>Right</b>): variation of the radius of the trap size versus the angle of incidence. The diameter of the occulting disk is recalled for each size. The solid line corresponds to the variation in the Goos–Hänchen shift (<math display="inline"><semantics> <msub> <mi>δ</mi> <mrow> <mi>G</mi> <mo>−</mo> <mi>H</mi> </mrow> </msub> </semantics></math>) with the angle of incidence.</p>
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<p>Experimental set-up. The infrared light from a 1550 nm impinges on a dissymmetric object. The diffracted Arago–Poisson spot on the axis of the light, close to the occulting dissymmetric object, has a donut-shaped structure (see insets) and carries OAM. <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>: focusing lens. The occulting object can be rotated at a given frequency <math display="inline"><semantics> <msub> <mi>ν</mi> <mi>r</mi> </msub> </semantics></math>. Inserts: picture of the spot intensity along the axis for different positions <span class="html-italic">z</span>. The scale is different for each insert.</p>
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<p>Frequency shift of the diffracted beam as a function of the distance from the occulting disk <span class="html-italic">z</span>. The object rotates at a frequency <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math> Hz. The frequency shift is measured with a self-heterodyne technique. The reference beam is frequency shifted with acousto-optic modulators.</p>
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25 pages, 8379 KiB  
Article
An Intriguing Interpretation of 1D and 2D Non-Diffracting Modes in Cosine Profile
by Allam Srinivasa Rao
Photonics 2023, 10(12), 1358; https://doi.org/10.3390/photonics10121358 - 8 Dec 2023
Cited by 1 | Viewed by 945
Abstract
We provide a simple analysis based on ray optics and Dirac notation for 1D (one-dimensional) and 2D (two-dimensional) non-diffracting modes in the cosine profile, which are often called Cosine beams. We explore various kinds of structured modes formed by the superposition of two [...] Read more.
We provide a simple analysis based on ray optics and Dirac notation for 1D (one-dimensional) and 2D (two-dimensional) non-diffracting modes in the cosine profile, which are often called Cosine beams. We explore various kinds of structured modes formed by the superposition of two 1D Cosine beams. We then went on to understand the properties of the Bessel beams in terms of Cosine beams. For the first time, we report on the generation of three-dimensional tunable needle structures based on the interference of 1D Cosine beams. These size-tunable optical needles can have multiple advantages in material processing. Also, we report, for the first time, on the Talbot effect in Cosine beams. Straightforward mathematical calculations are used to derive analytical expressions for Cosine beams. The present method of demonstrating Cosine beams may be utilized to understand other structured modes. The Dirac notation-based interference explanation used here can provide new researchers with an easy way to understand the wave nature of light in a fundamental aspect of interferometric experiments as well as in advanced-level experiments such as beam engineering technology, imaging, particle manipulation, light sheet microscopy, and light–matter interaction. We also provide an in-depth analysis of similarities among Cosine, Bessel, and Hermite–Gaussian beams. Full article
(This article belongs to the Special Issue Research in Computational Optics)
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Figure 1

Figure 1
<p>Ray optics representation of Cosine beam: (<b>a</b>) formation of the Cosine beam through the interference of two plane waves, (<b>b</b>) pictorial interpretation of optical waves’ amplitude and propagation vector (dashed arrows are projections), and (<b>c</b>) projection of the optical field along the three axes of the Cartesian coordinate system.</p>
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<p>Schematic experimental diagrams for Cosine beam generation: (<b>a</b>) beam splitter and mirror configuration for Cosine beam generation and (<b>b</b>) Fresnel biprism-based Cosine beam generation.</p>
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<p>Properties of the Cosine beam generated with a Fresnel biprism of base angle <span class="html-italic">α</span> = 1° at 640 nm wavelength: (<b>a</b>) phase introduced by the biprism to the incident laser beam to produce Cosine beam, (<b>b</b>) transverse intensity distribution of Cosine beam, and (<b>c</b>) line profiles of the biprism and the Cosine beam (amplitude and intensity of Cosine beam normalized to one).</p>
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<p>Cross-sectional intensity components of the Cosine beam in the longitudinal and transverse directions are presented in the first row, and corresponding line profiles are plotted in the second row.</p>
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<p>Visualization of self-healing in the 1D Cosine beam through ray representation.</p>
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<p>1D Cosine beam formed as a result of cross-interference of two beams provided by the experimental configuration shown in <a href="#photonics-10-01358-f002" class="html-fig">Figure 2</a>a: (<b>a</b>) Cosine–Gaussian beam in the presence of superposition of two Gaussian beams, and (<b>b</b>) Cosine–Hermite–Gauss beam under the superposition of two HG<sub>01</sub> modes (here, <span class="html-italic">θ</span> = 5°).</p>
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<p>(<b>a</b>) Intensity distribution and (<b>b</b>) phase of 2D Cosine beam (here, <span class="html-italic">θ<sub>x</sub></span> = <span class="html-italic">θ<sub>y</sub></span> = 5°).</p>
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<p>The transverse intensity distribution of superposed 1D Cosine beams in the <span class="html-italic">xy</span>-plane for <span class="html-italic">ϕ</span> = 0°, 90°, and 45°. Fractional intensities of interference along <span class="html-italic">x</span>, <span class="html-italic">y</span>, and <span class="html-italic">z</span> directions are provided in Green font in the respective images (here, <span class="html-italic">θ<sub>x</sub></span> = <span class="html-italic">θ<sub>y</sub></span> = 45°).</p>
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<p>Optical needle array in series created by two one-dimensional plane wave Cosine beams in the presence of <span class="html-italic">θ<sub>x</sub></span> − <span class="html-italic">θ<sub>y</sub></span> = 10°. The first row and first two images of the second row are transverse intensity distributions of the optical needle array at various longitudinal positions. The last two-dimensional image is a longitudinal cross-section of the needle beam in the <span class="html-italic">xz</span>-plane with <span class="html-italic">y</span> = 0. The dependence of periodicity or size of optical needles on <span class="html-italic">θ<sub>x</sub></span> − <span class="html-italic">θ<sub>y</sub></span> is shown in the line plot.</p>
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<p>Generation of various kinds of Cosine–Gauss beams by the superposition of two one-dimensional Cosine–Gauss beams with orthogonal polarization. The first and second rows are the intensity and phase of the Cosine–Gauss beam, respectively, for different states of polarization. Arrows in the ket notation indicate the polarization state of the corresponding Cosine–Gauss beam (here, <span class="html-italic">θ<sub>x</sub></span> = <span class="html-italic">θ<sub>y</sub></span> = 5°). The state of superposed beam (<b>a</b>,<b>e</b>) without any polarizer, (<b>b</b>,<b>f</b>) horizontally polarized, (<b>c</b>,<b>g</b>) vertically polarized, and (<b>d</b>,<b>h</b>) polarized at either 45<sup>o</sup> or 135<sup>o</sup>.</p>
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<p>Properties of Bessel beam generated with an axicon of base angle <span class="html-italic">α</span> = 1° at 640 nm wavelength: (<b>a</b>) phase introduced by the axicon to the incident laser beam for Bessel beam generation, (<b>b</b>) transverse intensity distribution of the Bessel beam, and (<b>c</b>) line profiles of the axicon and Cosine beam (amplitude and intensity of the Cosine beam are normalized to one).</p>
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15 pages, 16631 KiB  
Article
Full-Space Wavefront Shaping of Broadband Vortex Beam with Switchable Terahertz Metasurface Based on Vanadium Dioxide
by Xueying Li, Ying Zhang, Jiuxing Jiang, Yongtao Yao and Xunjun He
Nanomaterials 2023, 13(23), 3023; https://doi.org/10.3390/nano13233023 - 26 Nov 2023
Cited by 4 | Viewed by 1371
Abstract
Currently, vortex beams are extensively utilized in the information transmission and storage of communication systems due to their additional degree of freedom. However, traditional terahertz metasurfaces only focus on the generation of narrowband vortex beams in reflection or transmission mode, which is unbeneficial [...] Read more.
Currently, vortex beams are extensively utilized in the information transmission and storage of communication systems due to their additional degree of freedom. However, traditional terahertz metasurfaces only focus on the generation of narrowband vortex beams in reflection or transmission mode, which is unbeneficial for practical applications. Here, we propose and design terahertz metasurface unit cells composed of anisotropic Z-shaped metal structures, two dielectric layers, and a VO2 film layer. By utilizing the Pancharatnam–Berry phase theory, independent control of a full 2π phase over a wide frequency range can be achieved by rotating the unit cell. Moreover, the full-space mode (transmission and reflection) can also be implemented by utilizing the phase transition of VO2 film. Based on the convolution operation, three different terahertz metasurfaces are created to generate vortex beams with different wavefronts in full-space, such as deflected vortex beams, focused vortex beams, and non-diffraction vortex beams. Additionally, the divergences of these vortex beams are also analyzed. Therefore, our designed metasurfaces are capable of efficiently shaping the wavefronts of broadband vortex beams in full-space, making them promising applications for long-distance transmission, high integration, and large capacity in 6G terahertz communications. Full article
(This article belongs to the Special Issue Nanomaterials for Terahertz Technology Applications)
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Figure 1

Figure 1
<p>Schematic diagram of terahertz metasurfaces working in full-space: (<b>a</b>) functional illumination of metasurfaces; (<b>b</b>) structure and parameters of metasurface unit cell.</p>
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<p>The change in conductivity of VO<sub>2</sub> at different temperatures.</p>
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<p>Terahertz responses of the designed unit cell under the RCP incident waves: (<b>a</b>) PCR and transmission amplitudes of the cross-polarized and co-polarized output waves; (<b>b</b>) PCR and reflection amplitudes of the cross-polarized and co-polarized output waves.</p>
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<p>Surface current distributions of the unit cell for different states of VO<sub>2</sub> under RCP incident wave: (<b>a</b>) top view and (<b>b</b>) sectional view at the insulating state; (<b>c</b>) top view and (<b>d</b>) sectional view at the metallic state.</p>
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<p>Amplitude and phase of the cross-polarized output wave under different rotation angle <span class="html-italic">θ</span>: (<b>a</b>) transmission intensity; (<b>b</b>) transmission phase; (<b>c</b>) reflection intensity; and (<b>d</b>) reflection phase.</p>
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<p>Amplitudes and phases of the cross-polarized wave for eight unit cells at different working modes: (<b>left</b>) transmission intensity and phase at 3.6 THz for insulating state; (<b>right</b>) reflection intensity and phase at 3.2 THz for the metallic state.</p>
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<p>Convolutional phase process of the deflected vortex beam at insulating state under RCP wave illumination: (<b>a</b>) phase distribution of the deflected beam along the negative <span class="html-italic">x</span>-axis; (<b>b</b>) phase distribution of the vortex beam with <span class="html-italic">l</span> = 1; and (<b>c</b>) phase distribution of the vortex beam with <span class="html-italic">l</span> = 1 deflected along the negative <span class="html-italic">x</span>-axis.</p>
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<p>3D far-field radiation patterns of the deflected vortex beam at different working states: (<b>a</b>) insulating state (transmission mode); (<b>b</b>) metallic state (reflection mode).</p>
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<p>Convolutional phase process of the focused vortex beam at insulating state under RCP wave illumination: (<b>a</b>) phase distribution of the focused beam with <span class="html-italic">F</span> = 300 μm; (<b>b</b>) phase distribution of the vortex beam with <span class="html-italic">l</span> = 1; and (<b>c</b>) phase distribution of the focused vortex beam with <span class="html-italic">F</span> = 300 μm and with <span class="html-italic">l</span> = 1.</p>
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<p>Energy and phase distributions of the focused vortex beam at different working states: (<b>a</b>) insulating state (transmission mode); (<b>b</b>) metallic state (reflection mode).</p>
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<p>Superposition phase distribution of the non-diffraction vortex beam at insulating state under RCP wave illumination: (<b>a</b>) phase distribution of the zero-order Bessel beam with <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> </semantics></math> = 22°; (<b>b</b>) phase distribution of the vortex beam with <span class="html-italic">l</span> = 1; and (<b>c</b>) phase distribution of the non-diffraction vortex beam with <span class="html-italic">l</span> = 1 and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> </semantics></math> = 22°.</p>
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<p>Energy and phase distributions of the non-diffractive vortex beam under different working states: (<b>a</b>) insulating state (transmission mode); (<b>b</b>) metallic state (reflection mode).</p>
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<p>Normalized reflection energy intensity of the vortex beam, focused vortex beam, and non-diffracting vortex beam in different <span class="html-italic">xoy</span> planes: (<b>a</b>) z = 500 μm; (<b>b</b>) z = 800 μm.</p>
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14 pages, 5661 KiB  
Article
Illustrations of Bessel Beams in s-Polarization, p-Polarization, Transverse Polarization, and Longitudinal Polarization
by A. Srinivasa Rao
Photonics 2023, 10(10), 1092; https://doi.org/10.3390/photonics10101092 - 29 Sep 2023
Cited by 2 | Viewed by 1266
Abstract
The generation of Bessel beams (BBs) and their characterization in a wide range of the electromagnetic spectrum are well established. The unique properties of BBs, including their non-diffracting and self-healing nature, make them efficient for use in material science and engineering technology. Here, [...] Read more.
The generation of Bessel beams (BBs) and their characterization in a wide range of the electromagnetic spectrum are well established. The unique properties of BBs, including their non-diffracting and self-healing nature, make them efficient for use in material science and engineering technology. Here, I investigate the polarization components (s-polarization, p-polarization, transverse polarization, and longitudinal polarization) created in scalar BBs owing to their conical wave front. For emphasis, I provide a theoretical analysis to characterize potential experimental artifacts created in the four polarization components. Further, I provide a brief discussion on how to prevent these artifacts in scalar BBs. To my knowledge, for the first time, I can generate vector BBs in s-polarization and p-polarization via the superposition of two orthogonally polarized scalar BBs. This method of generation can provide the four well-known types of vector modes categorized in the V-point phase singularity vector modes. I suggest a suitable experimental configuration for realizing my theoretical results experimentally. The present analysis is very practical and beneficial for young researchers who seek to utilize BBs in light applications of modern science and technology. Full article
(This article belongs to the Special Issue Research in Computational Optics)
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<p>Vector diagram of a Bessel beam whose polarization is in <span class="html-italic">yz</span>-plane: (<b>a</b>) Propagation vector and electric field vector of Bessel beam along the <span class="html-italic">x</span>-, <span class="html-italic">y</span>-, and <span class="html-italic">z</span>-axes. (<b>b</b>) Optical field distribution of the Bessel beam formed by pumping a vertically polarized plane wave to an axicon. The red arrows correspond to the <span class="html-italic">k</span>-vector and the blue color double arrows represent electric field oscillation.</p>
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<p>The total intensity of the Bessel beam and its intensity (<b>first row</b>) in <span class="html-italic">s</span>-polarization and <span class="html-italic">p</span>-polarization and (<b>second row</b>) in transverse polarization and longitudinal polarization (cone angle of Bessel beam used is 40°). Here, superscript V on intensity <span class="html-italic">I</span> is used for the representation of vertical polarization.</p>
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<p>The intensity of longitudinal and transverse components of vertically and horizontally polarized Bessel beams.</p>
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<p>Different types of vector Bessel beams are generated in <span class="html-italic">s</span>-polarization and <span class="html-italic">p</span>-polarization by the superposition of two orthogonally linear polarized zero-order scalar Bessel beams. The four types of vector modes created in <span class="html-italic">s</span>-polarization and <span class="html-italic">p</span>-polarization are given in white insets. Also, the central part of the vector BB is enlarged and given in the left-side inset for all the images.</p>
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<p>Various types of vector Bessel beams are generated in <span class="html-italic">s</span>-polarization and <span class="html-italic">p</span>-polarization by the superposition of two orthogonally linear polarized first-order scalar Bessel beams. The central part of the vector BB is enlarged and given in the left-side inset for all the images.</p>
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<p>Optical field distribution in <span class="html-italic">p</span>-polarization and <span class="html-italic">s</span>-polarization in the azimuthal direction.</p>
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<p>The intensity modulation in the Bessel beam while it is propagating through the interface formed by two mediums of relative refractive index <span class="html-italic">n</span><sub>1</sub>/<span class="html-italic">n</span><sub>2</sub> = 1/1.5: (<b>a</b>) transmittance and (<b>b</b>) reflectance of Bessel beam at the interface. The transverse intensity distribution of the Bessel beam while it is (<b>c</b>) transmitted and (<b>d</b>) reflected at the interface. Here, the intensity distribution in the beam cross-section is self-normalized with its peak intensity.</p>
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<p>Experimental configuration can be used to generate various cylindrical vector modes in <span class="html-italic">s</span>-polarization and <span class="html-italic">p</span>-polarization. Here, PBS<span class="html-italic"><sub>i</sub></span> is the polarizing beam splitter, M<span class="html-italic"><sub>i</sub></span> is the mirror, and λ/2 is the half-wave plate. The double arrows in the experimental setup represent the polarization direction.</p>
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9 pages, 4899 KiB  
Communication
Plasmonic Metalens to Generate an Airy Beam
by Citlalli T. Sosa-Sánchez and Ricardo Téllez-Limón
Nanomaterials 2023, 13(18), 2576; https://doi.org/10.3390/nano13182576 - 17 Sep 2023
Cited by 2 | Viewed by 1299
Abstract
Airy beams represent an important type of non-diffracting beams—they are the only non-diffracting wave in one dimension, and thus they can be produced with a cylindrical geometry that modifies a wavefront in one dimension. In this paper, we show the design of a [...] Read more.
Airy beams represent an important type of non-diffracting beams—they are the only non-diffracting wave in one dimension, and thus they can be produced with a cylindrical geometry that modifies a wavefront in one dimension. In this paper, we show the design of a cylindrical plasmonic metalens consisting of an array of nanoslits in a gold thin layer that modulates the phase of a Gaussian beam to generate an airy beam propagating in free space. Based on the numerical results, we show that it is possible to generate an airy beam by only matching the phase of wavefronts coming out from the array of gold nanoslits to the airy beam phase at plane z=0. We numerically demonstrate that the airy beam exhibits bending over propagation and self-healing properties. The transmission efficiency is around 60%. The simplicity of the proposed structure open new perspectives in the design of flat metasurfaces for light-focusing applications. Full article
(This article belongs to the Section Nanophotonics Materials and Devices)
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<p>Schematic of airy beam generation with a plasmonic metalens. A p-polarized Gaussian beam illuminates the metasurface (nanoslits array), and the nanoslits are infinite and invariant along y-axis. A proper distribution of the nanoslit widths leads to light structuration and generates an airy beam.The metalens is not shown at true scale.</p>
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<p>Calculated distribution of the slit width to match the phase of an airy Beam. The red plot shows the behavior of the airy beam phase at plane <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, which is the plane where the PM was located. The purple squared dots illustratively show the position distribution along the <span class="html-italic">x</span>-axis and the widths of every nanoslit necessary to generate the phase of an airy beam, when the PM array is composed of 50 nanoslits. The intensity distribution is scaled to the intensity of the incident beam.</p>
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<p>Simulation of an airy beam produced by the PM and its corresponding fit. (<b>a</b>) The plot shows the intensity pattern of the transmitted light (propagated through vacuum) that results from the incident gaussian beam on the PM array. The PM is composed of 101 gold nanoslits of thickness <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>210</mn> </mrow> </semantics></math> nm, whose widths are <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> nm and <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> nm; it is designed for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>784</mn> </mrow> </semantics></math> nm, <math display="inline"><semantics> <mrow> <msub> <mi>ϵ</mi> <mrow> <mi>g</mi> <mi>o</mi> <mi>l</mi> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mn>0.14860</mn> <mo>+</mo> <mi>i</mi> <mn>4.7747</mn> </mrow> </semantics></math>. The bending parameter of the airy beam is <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.9</mn> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">μ</mi> </mrow> </semantics></math>m, truncation factor <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.04</mn> </mrow> </semantics></math>, and the waist of the incident gaussian beam is <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>=</mo> <mn>12</mn> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">μ</mi> </mrow> </semantics></math>m. (<b>b</b>) The plot shows the fit associated with the region of maximum intensity, between 0–48<math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">μ</mi> </mrow> </semantics></math>m. This demonstrates the parabolic trajectory of the maximum. Intensity distribution is scaled to the intensity of the incident beam.</p>
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<p>Airy beam self-healing comparison for different obstacle positions. (<b>a</b>) PM without obstacle, (<b>b</b>) with Au circular obstacle (<math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>0.5</mn> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">μ</mi> </mrow> </semantics></math>m) placed at <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>2</mn> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> <mo>,</mo> <mn>5</mn> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, and (<b>c</b>) with the same Au circular obstacle placed at <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mo>−</mo> <mn>2</mn> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> <mo>,</mo> <mn>5</mn> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>d</b>) The plot shows three different fits for the cases: (1) red curve, when there is no obstacle, (2) vlack curve, when there is an <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>u</mi> </mrow> </semantics></math> circular obstacle at <math display="inline"><semantics> <mrow> <mi>P</mi> <mn>1</mn> </mrow> </semantics></math> and (3) blue curve, <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>u</mi> </mrow> </semantics></math> circular obstacle at <math display="inline"><semantics> <mrow> <mi>P</mi> <mn>2</mn> </mrow> </semantics></math>. Intensity distribution is scaled to intensity of the incident beam.</p>
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<p>Comparison of transmission efficiency for different waist values of the incident Gaussian beam on PM. (<b>a</b>) Intensity map when the PM is illuminated with a Gaussian beam of waist <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>=</mo> <mn>12</mn> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">μ</mi> </mrow> </semantics></math>m, and the transmission efficiency is <math display="inline"><semantics> <mrow> <mn>62</mn> <mo>%</mo> </mrow> </semantics></math>. (<b>b</b>) Intensity map when the PM is illuminated with a Gaussian beam of waist <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>B</mi> <mi>w</mi> </msub> <mo>=</mo> <mn>18</mn> <mspace width="3.33333pt"/> <mi mathvariant="sans-serif">μ</mi> </mrow> </semantics></math>m, and the transmission efficiency is <math display="inline"><semantics> <mrow> <mn>47</mn> <mo>%</mo> </mrow> </semantics></math>.</p>
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<p>Chromatic dependence of the PM for airy beam generation. Intensity maps for PM designed for an operating wavelength <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>784</mn> </mrow> </semantics></math> nm when illuminated with a Gaussian beam at (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>784</mn> </mrow> </semantics></math> nm with transmission efficiency <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi>T</mi> </msub> <mo>=</mo> <mn>62</mn> <mo>%</mo> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>633</mn> </mrow> </semantics></math> nm with transmission efficiency <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi>T</mi> </msub> <mo>=</mo> <mn>40</mn> <mo>%</mo> </mrow> </semantics></math>.</p>
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17 pages, 2434 KiB  
Article
A Systematic Summary and Comparison of Scalar Diffraction Theories for Structured Light Beams
by Fuping Wu, Yi Luo and Zhiwei Cui
Photonics 2023, 10(9), 1041; https://doi.org/10.3390/photonics10091041 - 13 Sep 2023
Cited by 1 | Viewed by 1377
Abstract
Structured light beams have recently attracted enormous research interest for their unique properties and potential applications in optical communications, imaging, sensing, etc. Since most of these applications involve the propagation of structured light beams, which is accompanied by the phenomenon of diffraction, it [...] Read more.
Structured light beams have recently attracted enormous research interest for their unique properties and potential applications in optical communications, imaging, sensing, etc. Since most of these applications involve the propagation of structured light beams, which is accompanied by the phenomenon of diffraction, it is very necessary to employ diffraction theories to analyze the obstacle effects on structured light beams during propagation. The aim of this work is to provide a systematic summary and comparison of the scalar diffraction theories for structured light beams. We first present the scalar fields of typical structured light beams in the source plane, including the fundamental Gaussian beams, higher-order Hermite–Gaussian beams, Laguerre–Gaussian vortex beams, non-diffracting Bessel beams, and self-accelerating Airy beams. Then, we summarize and compare the main scalar diffraction theories of structured light beams, including the Fresnel diffraction integral, Collins formula, angular spectrum representation, and Rayleigh–Sommerfeld diffraction integral. Finally, based on these theories, we derive in detail the analytical propagation expressions of typical structured light beams under different conditions. In addition, the propagation of typical structured light beams is simulated. We hope this work can be helpful for the efficient study of the propagation of structured light beams. Full article
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<p>Transverse intensity distributions of typical structured light beams under the paraxial approximation at different propagation distances. (<b>a1</b>–<b>a5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>4</mn> <mi>λ</mi> </mrow> </semantics></math>, (<b>b1</b>–<b>b5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>8</mn> <mi>λ</mi> </mrow> </semantics></math>, and (<b>c1</b>–<b>c5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>12</mn> <mi>λ</mi> </mrow> </semantics></math>. Shown from left to right are the cases of the fundamental Gaussian beam, the Hermite–Gaussian beam, the Laguerre–Gaussian beam, the Bessel beam, and the Airy beam, respectively.</p>
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<p>Transverse intensity distributions of typical structured light beams beyond the paraxial approximation at different propagation distances. (<b>a1</b>–<b>a5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>4</mn> <mi>λ</mi> </mrow> </semantics></math>, (<b>b1</b>–<b>b5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>8</mn> <mi>λ</mi> </mrow> </semantics></math>, and (<b>c1</b>–<b>c5</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>12</mn> <mi>λ</mi> </mrow> </semantics></math>. Shown from left to right are the cases of the fundamental Gaussian beam, the Hermite–Gaussian beam, the Laguerre–Gaussian beam, the Bessel beam, and the Airy beam, respectively.</p>
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<p>Illustrations of the propagation of typical structured light beams in a gradient-index medium. (<b>a</b>) Fundamental Gaussian beam, (<b>b</b>) Hermite–Gaussian beam, (<b>c</b>) Laguerre–Gaussian beam, (<b>d</b>) Bessel beam, and (<b>e</b>) Airy beam.</p>
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16 pages, 18183 KiB  
Article
Dual-Functional Tunable Metasurface for Meta-Axicon with a Variable Depth of Focus and Continuous-Zoom Metalens
by Chang Wang, Yan Sun, Zeqing Yu, Xinyu Liu, Bingliang Chen, Yang Zhang and Zhenrong Zheng
Nanomaterials 2023, 13(18), 2530; https://doi.org/10.3390/nano13182530 - 10 Sep 2023
Cited by 1 | Viewed by 1803
Abstract
Optical metasurfaces have been widely investigated for their versatile ability to manipulate wavefront and miniaturize traditional optical components into ultrathin planar devices. The integration of metasurfaces with multifunctionality and tunability has fundamentally transformed optics with unprecedented control over light propagation and manipulation. This [...] Read more.
Optical metasurfaces have been widely investigated for their versatile ability to manipulate wavefront and miniaturize traditional optical components into ultrathin planar devices. The integration of metasurfaces with multifunctionality and tunability has fundamentally transformed optics with unprecedented control over light propagation and manipulation. This study introduces a pioneering framework for the development of tunable metasurfaces with multifunctionality, and an example of a tunable metasurface of dual functionalities is proposed and numerically verified as one of the tunable meta-axicon for generating Bessel beams with a variable depth of focus (DOF) and a continuous-zoom metalens. Specifically, this design achieves dual-functional phase modulation by helicity-multiplexing from the combination of the geometric phase as well as the propagation phase and realizes tunability for both functionalities through rotational actuation between double metasurface layers. As a result, dual functionalities with continuous tunability of the proposed TiO2 metasurface are enabled independently for the left and right circularly polarized (LCP and RCP) incidences at 532 nm. Specifically, LCP light triggers the metasurface to function as a tunable axicon, generating non-diffracting Bessel beams with variable numerical apertures (NA) and DOFs. Conversely, the RCP incidence induces it to operate as a continuous-zoom metalens and generates variable spherical wavefront focusing on diverse focal lengths. This study not only initially implements the design of tunable meta-axicon, but also achieves the integration of such a tunable meta-axicon and continuous-zoom metalens within a single metasurface configuration. The proposed device could find potential applications in biological imaging, microscopic measurement, laser fabrication, optical manipulation, multi-plane imaging, depth estimation, optical data storage, etc. Full article
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<p>Schematic representation of the DFTM. The nanostructures on both metasurface layers are arranged face to face with a small gap. For LCP incidence, the metasurface functions as a tunable axicon that generates Bessel beams with adjustable DOFs, while for RCP incidence, it operates as a continuous-zoom metalens with changeable focal length.</p>
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<p>(<b>a</b>) Schematic of the working principle of the proposed bilayer metasurface. The TiO<sub>2</sub> nanostructures on quartz substrates are spaced with a small gap distance <span class="html-italic">g</span>, featuring distinct rotational angles for inducing geometric phases and diverse structural parameters for producing propagation phases at various positions on both metasurface layers. A normally incident circularly polarized light interacts initially with the first layer and is converted to an output with opposite helicity. After passing the second layer, the final output light will be in the same polarization state as the incidence, with the overall phase distributions being continuously changed. (<b>b</b>) Polarization conversion efficiencies and the propagation phases of the 24-step nanostructures simulated by the commercial package of Lumerical FDTD Solutions. The dimensions of the nanostructures, spanning from 1 to 24, are optimized as follows: lengths (<span class="html-italic">L</span>) of 268, 220, 235, 262, 265, 274, 250, 247, 241, 60, 66, 69, 72, 90, 90, 90, 96, 99, 108, 114, 120, 241, 232, and 253 nm; and widths (<span class="html-italic">W</span>) of 72, 90, 90, 90, 96, 99, 108, 114, 120, 241, 232, 253, 268, 220, 235, 262, 265, 274, 250, 247, 241, 60, 66, and 69 nm, respectively.</p>
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<p>(<b>a</b>) Top views of TiO<sub>2</sub> nanostructures on both metasurface layers within the designed DFTM. The variation in nanostructure arrangements generates diverse combinations of propagation and geometric phases. (<b>b</b>) Phase profiles of <span class="html-italic">φ</span><sub>DFTM+ =</sub> <span class="html-italic">φ</span><sub>1+</sub> + <span class="html-italic">φ</span><sub>2−</sub> illustrating the phase manipulation mechanism for meta-axicon functionality under LCP incidence. (<b>c</b>) Phase profiles of <span class="html-italic">φ</span><sub>DFTM− =</sub> <span class="html-italic">φ</span><sub>1−</sub> + <span class="html-italic">φ</span><sub>2+</sub> for continuous-zoom metalens functionality under RCP incidence.</p>
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<p>Tunability of the meta-axicon functionality in DFTM under LCP incidence. (<b>a1</b>–<b>a7</b>) Normalized electric field intensity profiles of Bessel beams generated by the meta-axicon with different DOFs under various rotation angles from 0–90° in the <span class="html-italic">x</span>–-<span class="html-italic">z</span> plane. (<b>b1</b>–<b>b7</b>) Zoomed-in results of (<b>a1</b>–<b>a7</b>) showcasing finer details. (<b>c1</b>–<b>c7</b>) Corresponding intensity profiles in the <span class="html-italic">x</span>–<span class="html-italic">y</span> plane provide insights into the lateral distributions of the beams. (<b>d1</b>–<b>d7</b>) Corresponding intensity profiles along the <span class="html-italic">x</span>-axis.</p>
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<p>(<b>a</b>) The graph showcasing a comparison between the theoretical and measured DOF results for Bessel beams created under LCP incidence while varying the rotation angles at 0°, 15°, 30°, 45°, 60°, 75° and 90°. (<b>b</b>) The comparison between the corresponding theoretical and simulated FWHM values for the same rotation angles offers insights into the beam characteristics.</p>
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<p>(<b>a</b>) Diagram illustrating the far-field ring-shaped beam of the meta-axicon. (<b>b</b>–<b>h</b>) The FDTD-simulated outcomes of the far-field projection, displayed as images of the far-field intensity on a hemisphere surface with a radius of 1 m, were observed from above. These results correspond to the rotation of the top metasurface layer at angles of 0°, 15°, 30°, 45°, 60°, 75° and 90°, respectively, providing a comprehensive view of the far-field focusing patterns under different rotation conditions.</p>
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<p>The simulated electronic field intensity distributions of the varifocal performance of the DFTM under RCP incidence. (<b>a1</b>–<b>a7</b>) Electric field intensity distributions in the <span class="html-italic">x–z</span> plane under 532 nm RCP incidence when the top metasurface layer is rotationally actuated by 0°, 15°, 30°, 45°, 60°, 75° and 90°, respectively. (<b>b1</b>–<b>b7</b>) The corresponding zoom-in intensity profiles of (<b>a1</b>–<b>a7</b>) reveal the transverse distribution in the <span class="html-italic">x–y</span> plane. (<b>c1</b>–<b>c7</b>) The corresponding relative intensity profiles of (<b>b1</b>–<b>b7</b>) along the <span class="html-italic">z</span>-axis.</p>
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<p>(<b>a</b>) The comparison of theoretical and measured focal length results for continuous-zoom metalens functionality under RCP incidence at rotation angles of 0°, 15°, 30°, 45°, 60°, 75° and 90°, respectively. (<b>b</b>) The comparison between the simulated FWHM values and the theoretical diffraction limits for the same rotation angles (0°, 15°, 30°, 45°, 60°, 75° and 90°) provides insights into the achievable spot sizes and their proximity to the diffraction limit.</p>
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16 pages, 3135 KiB  
Review
Bessel Beams in Ophthalmology: A Review
by C. S. Suchand Sandeep, Ahmad Khairyanto, Tin Aung and Murukeshan Vadakke Matham
Micromachines 2023, 14(9), 1672; https://doi.org/10.3390/mi14091672 - 27 Aug 2023
Cited by 4 | Viewed by 1922
Abstract
The achievable resolution of a conventional imaging system is inevitably limited due to diffraction. Dealing with precise imaging in scattering media, such as in the case of biomedical imaging, is even more difficult owing to the weak signal-to-noise ratios. Recent developments in non-diffractive [...] Read more.
The achievable resolution of a conventional imaging system is inevitably limited due to diffraction. Dealing with precise imaging in scattering media, such as in the case of biomedical imaging, is even more difficult owing to the weak signal-to-noise ratios. Recent developments in non-diffractive beams such as Bessel beams, Airy beams, vortex beams, and Mathieu beams have paved the way to tackle some of these challenges. This review specifically focuses on non-diffractive Bessel beams for ophthalmological applications. The theoretical foundation of the non-diffractive Bessel beam is discussed first followed by a review of various ophthalmological applications utilizing Bessel beams. The advantages and disadvantages of these techniques in comparison to those of existing state-of-the-art ophthalmological systems are discussed. The review concludes with an overview of the current developments and the future perspectives of non-diffractive beams in ophthalmology. Full article
(This article belongs to the Special Issue Non-diffractive Beams for the State of the Art Applications)
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<p>The first five orders of the Bessel function of the first kind.</p>
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<p>(<b>a</b>) Generation of Bessel beam using a refractive axicon. (<b>b</b>) Pseudo-colored image of the generated Bessel beam.</p>
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<p>Simulations demonstrating the self-reconstruction properties of a micro-axicon (with a base diameter of 100 μm)-generated Bessel beam. Ray traces showing the (<b>a</b>) Gaussian beam and (<b>b</b>) Bessel beam encountering micro-spheres. (<b>c</b>) Gaussian beam interacting with a micro-sphere (<span class="html-italic">d</span> = 20 µm, <span class="html-italic">n</span> = 1.37, <span class="html-italic">dev</span> = 0 µm). (<b>d</b>–<b>f</b>) Bessel beam interacting with micro spheres of diameters of 10, 20, and 30 µm (<span class="html-italic">n</span> = 1.37, <span class="html-italic">dev</span> = 0 µm). (<b>g</b>,<b>h</b>) Bessel beam interacting with micro-spheres (<span class="html-italic">d</span> = 20 µm) with refractive indices of 1.46 and 1.76. (<b>i</b>,<b>j</b>) Bessel beam interacting with micro-spheres (<span class="html-italic">d</span> = 20 µm) displaced 10 µm and 20 µm from the axis. Self-reconstruction distances with respect to the (<b>k</b>) micro-sphere’s diameter, (<b>l</b>) micro-sphere’s refractive index, and (<b>m</b>) micro-sphere’s deviation distance from the beam axis. Reproduced from reference [<a href="#B64-micromachines-14-01672" class="html-bibr">64</a>] with permission from Elsevier.</p>
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<p>Schematic illustration of Purkinje images from ocular surfaces. Adapted from reference [<a href="#B70-micromachines-14-01672" class="html-bibr">70</a>] with permission from MDPI.</p>
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<p>Schematic of the sequential imaging system for the high-resolution imaging of the ICA and the cornea. Reproduced from reference [<a href="#B72-micromachines-14-01672" class="html-bibr">72</a>] with permission from ARVO.</p>
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<p>Three-dimensional visualization of the TM in an intact porcine eye. (<b>a</b>) Extended-depth-of-focus (EDF) image of the TM generated from the optical sections. (<b>b</b>) Color-coded optical sections. (<b>c</b>) Three-dimensional visualization of the TM. Reproduced from reference [<a href="#B59-micromachines-14-01672" class="html-bibr">59</a>] with permission from Wiley WCH.</p>
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<p>In vivo images of the TM in Wistar rats obtained using the Bessel beam-based LSFM system. S denotes the sclera, A denotes the ICA, and I denotes the iris. Reproduced from reference [<a href="#B74-micromachines-14-01672" class="html-bibr">74</a>] with permission from the Optical Society.</p>
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<p>Schematic of the experimental adaptive optics (AO) system used to perform eye fixation measurements. Reproduced from reference [<a href="#B78-micromachines-14-01672" class="html-bibr">78</a>] with permission from Wiley WCH.</p>
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15 pages, 5344 KiB  
Article
Bessel Beam Dielectrics Cutting with Femtosecond Laser in GHz-Burst Mode
by Pierre Balage, Théo Guilberteau, Manon Lafargue, Guillaume Bonamis, Clemens Hönninger, John Lopez and Inka Manek-Hönninger
Micromachines 2023, 14(9), 1650; https://doi.org/10.3390/mi14091650 - 22 Aug 2023
Cited by 4 | Viewed by 2043
Abstract
We report, for the first time to the best of our knowledge, Bessel beam dielectrics cutting with a femtosecond laser in GHz-burst mode. The non-diffractive beam shaping is based on the use of an axicon and allows for cutting glasses up to 1 [...] Read more.
We report, for the first time to the best of our knowledge, Bessel beam dielectrics cutting with a femtosecond laser in GHz-burst mode. The non-diffractive beam shaping is based on the use of an axicon and allows for cutting glasses up to 1 mm thickness with an excellent cutting quality. Moreover, we present a comparison of the cutting results with the state-of-the-art method, consisting of short MHz-bursts of femtosecond pulses. We further illustrate the influence of the laser beam parameters such as the burst energy and the pitch between consecutive Bessel beams on the machining quality of the cutting plane and provide process windows for both regimes. Full article
(This article belongs to the Special Issue Laser Micro/Nano Fabrication)
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<p>Schematic of the Bessel beam generation using an axicon and a set of lenses.</p>
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<p>Schematic of the measurement method for the primary Bessel beam.</p>
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<p>Measurement of the primary Bessel beam diameter (<b>a</b>). Intensity profile measured at two different planes (<b>b</b>,<b>c</b>). All measurements are made using a WinCamD beam analyzer.</p>
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<p>Characterization of the Bessel beam by sideview imaging of the luminescence of the Bessel beam and of a laser induced modification in a glass sample. (<b>a</b>) Top view image of the crack orientation using a phase mask upstream the axicon with a pitch of 20 µm (<b>b</b>).</p>
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<p>Topography measurements on sidewalls after singulation corresponding to the lowest surface roughness obtained in our study in 1 mm-thick sodalime for a burst energy of 253 µJ and a pitch of 0.04 µm for a resulting Sa of 0.47 µm (<b>a</b>), in 200 µm-thick fused silica for a burst energy of 294 µJ and a pitch of 0.1 µm for a resulting Sa of 0.75 µm (<b>c</b>), and in 430 µm-thick sapphire for a burst energy of 337 µJ and a pitch of 0.04 µm for a resulting Sa of 1.14 µm (<b>e</b>). Surface roughness as a function of the pitch in sodalime for burst energies in a range from 253 µJ to 383 µJ (<b>b</b>), in fused silica for burst energies in a range from 294 µJ to 383 µJ (<b>d</b>), and in sapphire for burst energies in a range from 294 µJ to 383 µJ (<b>f</b>). Note the pitch scale differences, especially in sapphire. Laser comes from the top.</p>
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<p>Graphic representation of the surface roughness as a function of the pitch between two consecutive Bessel beams in sodalime and AF32 for burst energies of 200 µJ (<b>a</b>) and 215 µJ (<b>b</b>). The images corresponding to the bests results obtained are displayed on the left for GHz-bursts and on the right for MHz-bursts. Laser comes from the bottom.</p>
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<p>Profilometer images of the results obtained in sodalime for a pitch of 1 µm in MHz-burst mode for a burst energy of 147 µJ with a resulting Sa of 0.58 µm (<b>a</b>), in sodalime for a pitch of 0.04 µm in GHz-burst mode for a burst energy of 194 µJ with a resulting Sa of 0.46 µm (<b>b</b>), in AF32 for a pitch of 1 µm in MHz-burst mode for a burst energy of 127 µJ with a resulting Sa of 0.42 µm (<b>c</b>), obtained in AF32 for a pitch of 0.025 µm in GHz-burst mode for a burst energy of 194 µJ with a resulting Sa of 0.27 µm (<b>d</b>).</p>
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<p>Schematic representation of the different operating windows that appeared during the cutting study. The values of burst energies, pitch, and crosses correspond to experimental data in sodalime. The green zone (1) corresponds to the optimum process window, the orange zone (2) represents a process window for which cutting is possible but with a lowered surface quality of the cutting plane, the red zone (3) represents the parameters for which cutting is not feasible, and the grey zone (4) corresponds to very low pitches and/or too high burst energy leading to the thermal cutting regime.</p>
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<p>Schematic view of the HAZ appearing with high overlapping (<b>top</b>) with the corresponding microscope sideview image of the cutting plane (<b>middle</b>), and 3D representation of the surface measured with the profilometer (<b>bottom</b>).</p>
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13 pages, 8910 KiB  
Article
Variable Bessel Beam Profiles Generated through Refraction by Liquid Media
by Dina C. Palangyos and Raphael A. Guerrero
Micromachines 2023, 14(8), 1609; https://doi.org/10.3390/mi14081609 - 15 Aug 2023
Cited by 1 | Viewed by 1083
Abstract
Various methods have been employed to produce Bessel beams (BBs), with axicon-based techniques remaining the most efficient. Among the limitations of axicons are manufacturing defects such as oblate tips and difficulty in tuning the generated BBs. In this work, we combine the effect [...] Read more.
Various methods have been employed to produce Bessel beams (BBs), with axicon-based techniques remaining the most efficient. Among the limitations of axicons are manufacturing defects such as oblate tips and difficulty in tuning the generated BBs. In this work, we combine the effect of a blunt-tip axicon with refraction using various combinations of liquid media to generate variable BB intensity profiles. The output BBs from the axicon are made to pass through a custom-built fluid chamber and magnified using a telescope system. When traversing an empty chamber, the Bessel beam core diameter is measured to be 773.8 µm at propagation distance z’ = 30 cm. The core diameter increases as the beam passes through a chamber containing different liquids as a result of an effective axicon–telescope distance produced by the indices of refraction of the pertinent fluids. Bessel beams modified by the fluid chamber maintain the properties of non-diffraction and self-healing. Full article
(This article belongs to the Section A:Physics)
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<p>Refracting axicon with apex angle <math display="inline"><semantics> <mrow> <mi>γ</mi> </mrow> </semantics></math>: interference of waves along the focal line leads to a Bessel beam propagating for distance <span class="html-italic">z<sub>max</sub></span>.</p>
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<p>An axicon with a rounded tip described by radius of curvature <math display="inline"><semantics> <mrow> <mi>R</mi> </mrow> </semantics></math> and base angle <math display="inline"><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math>.</p>
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<p>Schematic diagram of a Bessel beam wave vector being deflected through a fluid chamber: axicon–telescope distance <span class="html-italic">D</span> is reduced to <span class="html-italic">D</span>’ due to refraction through walls of thickness <span class="html-italic">t</span> and fluid sections of length <span class="html-italic">l</span>.</p>
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<p>Optical system for generating variable Bessel beams using refraction through a fluid chamber. <span class="html-italic">L</span> = lens, <span class="html-italic">M</span> = mirror, NDF = neutral density filter.</p>
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<p>Images of Bessel beams from a blunt-tip axicon at (<b>a</b>) <span class="html-italic">z</span> = 16 cm, (<b>b</b>) <span class="html-italic">z</span> = 20 cm, (<b>c</b>) <span class="html-italic">z</span> = 30 cm, and (<b>d</b>) <span class="html-italic">z</span> = 40 cm. Core diameters become smaller, while the number of rings increases with the increase in distance.</p>
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<p>Dependence of output Bessel beams, generated without the liquid chamber, on the distance between axicon and telescope: intensity profiles recorded at <span class="html-italic">z</span>’ = 25 cm with (<b>a</b>) <span class="html-italic">D</span> = 20 cm, (<b>b</b>) <span class="html-italic">D</span> = 25 cm, (<b>c</b>) <span class="html-italic">D</span> = 30 cm, (<b>d</b>) 35 cm, and (<b>e</b>) <span class="html-italic">D</span> = 40 cm. Core diameters are observed to decrease with <span class="html-italic">D</span>.</p>
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<p>Intensity profiles (left) of Bessel beams are altered using refraction through different liquid combinations: (<b>a</b>) empty chamber, (<b>b</b>) H<sub>2</sub>O–H<sub>2</sub>O, (<b>c</b>) Si oil–H<sub>2</sub>O, (<b>d</b>) Mi oil–H<sub>2</sub>O, (<b>e</b>) Si oil–Si oil, (<b>f</b>) Mi oil–Si oil, and (<b>g</b>) Mi oil–Mi oil. Line scans (right) across the centers of the beams show how the core diameter widens as the fluid refractive index becomes higher.</p>
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<p>Images of Bessel beams at <span class="html-italic">z</span>’ = 20 cm and <span class="html-italic">z</span>’ = 80 cm showing propagation invariance: (<b>a</b>) empty chamber, (<b>b</b>) H<sub>2</sub>O–H<sub>2</sub>O, (<b>c</b>) Si oil–H<sub>2</sub>O, (<b>d</b>) H<sub>2</sub>O–Mi oil, (<b>e</b>) Si oil–Si oil, (<b>f</b>) Si oil–Mi oil, and (<b>g</b>) Mi oil–Mi oil.</p>
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<p>The self-healing of Bessel beams refracted by different liquid combinations, with reconstructed intensity profiles at (<b>a</b>) <span class="html-italic">z</span>’ = 7.5 cm, (<b>b</b>) <span class="html-italic">z</span>’ = 30 cm, and (<b>c</b>) <span class="html-italic">z</span>’ = 60 cm. A wire obstruction with a diameter of 118 µm is placed at <span class="html-italic">z</span>’ = 5 cm.</p>
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<p>Dependence of Bessel beam core diameter on effective axicon–telescope distance due to different fluid refractive indices.</p>
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9 pages, 222 KiB  
Editorial
Preface: International Conference on Holography Meets Advanced Manufacturing (HMAM2)
by Vijayakumar Anand, Amudhavel Jayavel, Viktor Palm, Shivasubramanian Gopinath, Andrei Bleahu, Aravind Simon John Francis Rajeswary, Kaupo Kukli, Vinoth Balasubramani, Daniel Smith, Soon Hock Ng and Saulius Juodkazis
Eng. Proc. 2023, 34(1), 29; https://doi.org/10.3390/engproc2023034029 - 24 Jul 2023
Viewed by 893
Abstract
The CIPHR group, Institute of Physics, University of Tartu, Estonia, and Optical Sciences Center, Swinburne University of Technology, Australia, jointly organized the interdisciplinary online conference “Holography Meets Advanced Manufacturing” during 20–22 February 2023. Full article
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