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Abstract
Kerker effect is one of the unique phenomena in modern electrodynamics. Due to overlapping of electric and magnetic dipole moments, all-dielectric particles can be invisible in forward or backward directions. In our paper we propose new conditions between resonantly excited electric dipole and magnetic quadrupole in ceramic high index spheroidal particles for demonstrating transverse Kerker effect. Moreover, we perform proof-of-concept microwave experiment and demonstrate dumbbell radiation pattern with suppressed scattering in both forward and backward directions and enhanced scattering in lateral directions. Our concept is promising for future planar lasers, nonreflected metasurface and laterally excited waveguides and nanoantennas.
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Complex Inverse Design of Meta-optics by Segmented Hierarchical Evolutionary Algorithm. ACS NANO 2019; 13:821-829. [PMID: 30615418 DOI: 10.1021/acsnano.8b08333] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
With the recent burgeoning advances in nano-optics, ultracompact, miniaturized photonic devices with high-quality and spectacular functionalities are highly desired. Such devices' design paradigms often call for the solution of a complex inverse nonanalytical/semianalytical problem. However, currently reported strategies dealing with amplitude-controlled meta-optics devices achieved limited functionalities mainly due to restricted search space and demanding computational schemes. Here, we established a segmented hierarchical evolutionary algorithm, aiming to solve large-pixelated, complex inverse meta-optics design and fully demonstrate the targeted performance. This paradigm allows significantly extended search space at a rapid converging speed. As typical complex proof-of-concept examples, large-pixelated meta-holograms are chosen to demonstrate the validity of our design paradigm. An improved fitness function is proposed to reinforce the performance balance among image pixels, so that the image quality is improved and computing speed is further accelerated. Broadband and full-color meta-holograms with high image fidelities using binary amplitude control are demonstrated experimentally. Our work may find important applications in the advanced design of future nanoscale high-quality optical devices.
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Planar Diffractive Lenses: Fundamentals, Functionalities, and Applications. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2018; 30:e1704556. [PMID: 29672949 DOI: 10.1002/adma.201704556] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/11/2017] [Revised: 12/17/2017] [Indexed: 05/09/2023]
Abstract
Traditional objective lenses in modern microscopy, based on the refraction of light, are restricted by the Rayleigh diffraction limit. The existing methods to overcome this limit can be categorized into near-field (e.g., scanning near-field optical microscopy, superlens, microsphere lens) and far-field (e.g., stimulated emission depletion microscopy, photoactivated localization microscopy, stochastic optical reconstruction microscopy) approaches. However, they either operate in the challenging near-field mode or there is the need to label samples in biology. Recently, through manipulation of the diffraction of light with binary masks or gradient metasurfaces, some miniaturized and planar lenses have been reported with intriguing functionalities such as ultrahigh numerical aperture, large depth of focus, and subdiffraction-limit focusing in far-field, which provides a viable solution for the label-free superresolution imaging. Here, the recent advances in planar diffractive lenses (PDLs) are reviewed from a united theoretical account on diffraction-based focusing optics, and the underlying physics of nanofocusing via constructive or destructive interference is revealed. Various approaches of realizing PDLs are introduced in terms of their unique performances and interpreted by using optical aberration theory. Furthermore, a detailed tutorial about applying these planar lenses in nanoimaging is provided, followed by an outlook regarding future development toward practical applications.
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4
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Dielectric Meta-Holograms Enabled with Dual Magnetic Resonances in Visible Light. ACS NANO 2017; 11:9382-9389. [PMID: 28898048 DOI: 10.1021/acsnano.7b04868] [Citation(s) in RCA: 73] [Impact Index Per Article: 10.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
Efficient transmission-type meta-holograms have been demonstrated using high-index dielectric nanostructures based on Huygens' principle. It is crucial that the geometry size of building blocks be judiciously optimized individually for spectral overlap of electric and magnetic dipoles. In contrast, reflection-type meta-holograms using the metal/insulator/metal scheme and geometric phase can be readily achieved with high efficiency and small thickness. Here, we demonstrate a general platform for design of dual magnetic resonance based meta-holograms based on the geometric phase using silicon nanostructures that are quarter wavelength thick for visible light. Significantly, the projected holographic image can be unambiguously observed without a receiving screen even under the illumination of natural light. Within the well-developed semiconductor industry, our ultrathin magnetic resonance-based meta-holograms may have promising applications in anticounterfeiting and information security.
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Abstract
Rapid progress in nanophotonics is driven by the ability of optically resonant nanostructures to enhance near-field effects controlling far-field scattering through intermodal interference. A majority of such effects are usually associated with plasmonic nanostructures. Recently, a new branch of nanophotonics has emerged that seeks to manipulate the strong, optically induced electric and magnetic Mie resonances in dielectric nanoparticles with high refractive index. In the design of optical nanoantennas and metasurfaces, dielectric nanoparticles offer the opportunity for reducing dissipative losses and achieving large resonant enhancement of both electric and magnetic fields. We review this rapidly developing field and demonstrate that the magnetic response of dielectric nanostructures can lead to novel physical effects and applications.
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Abstract
Subwavelength confined waveguiding is experimentally demonstrated with high refractive index dielectric nanoparticles with photon energy propagation at distances beyond 500 μm. These particles have naturally occurring electric and magnetic dipole resonances. When they are placed in a 1D chain, the magnetic resonances of adjacent elements couple to each other, providing a means to transport energy at visible or NIR wavelengths in a confined mode. Chains of nanoparticles made of silicon were fabricated and guided waves were measured with near-field scanning optical microscopy. Propagation loss is quantified at 34 dB/mm for 720 nm and 5.5 dB/mm for 960 nm wavelengths with 150 and 220 nm diameter particles, respectively. Simulations confirm the unique properties of this waveguiding in comparison with photonic crystals. The resonant nature of the waveguide lays a foundation for integrated photonics beyond nanowire waveguides of silicon and silicon nitride. This technology is promising for more compact and deeper photonic integration such as right angle bends, more compact modulators, slow light and interfacing with single photon emitters for photonic integrated circuits, quantum communications, and biosensing.
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Suppression of scattering for small dielectric particles: anapole mode and invisibility. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2017; 375:rsta.2016.0069. [PMID: 28220000 PMCID: PMC5321830 DOI: 10.1098/rsta.2016.0069] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 09/05/2016] [Indexed: 05/26/2023]
Abstract
We reveal that an isotropic, homogeneous, subwavelength particle with high refractive index can produce ultra-small total scattering. This effect, which follows from the inhibition of the electric dipole radiation, can be identified as a Fano resonance in the scattering efficiency and is associated with the excitation of an anapole mode in the particle. This anapole mode is non-radiative and emerges from the destructive interference of electric and toroidal dipoles. The invisibility effect could be useful for the design of highly transparent optical materials.This article is part of the themed issue 'New horizons for nanophotonics'.
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Topological effects in anisotropy-induced nano-fano resonance of a cylinder. OPTICS LETTERS 2015; 40:4162-4165. [PMID: 26368737 DOI: 10.1364/ol.40.004162] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We demonstrate that optical Fano resonance can be induced by the anisotropy of a cylinder rather than frequency selection under the resonant condition. A tiny perturbation in anisotropy can result in a giant switch in the principal optic axis near plasmon resonance. Such anisotropy-induced Fano resonance shows fast reversion between forward and backward scattering at the lowest-energy interference. The near and far fields of the particle change dramatically around Fano resonance. The topology of optical singular points and the trajectory of energy flux distinctly reveal the interaction between the incident wave and the localized surface plasmons, which also determine the far-field scattering pattern. The anisotropy-induced Fano resonance and its high sensitivity open new perspectives on light-matter interactions and promise potential applications in biological sensors, optical switches, and optomechanics.
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Split-ball resonator as a three-dimensional analogue of planar split-rings. Nat Commun 2015; 5:3104. [PMID: 24430506 DOI: 10.1038/ncomms4104] [Citation(s) in RCA: 46] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2013] [Accepted: 12/13/2013] [Indexed: 11/09/2022] Open
Abstract
Split-ring resonators are basic elements of metamaterials, which can induce a magnetic response in metallic nanosctructures. Tunability of such response up to the visible frequency range is still a challenge. Here we introduce the concept of the split-ball resonator and demonstrate the strong magnetic response in the visible for both gold and silver spherical plasmonic nanoparticles with nanometre scale cuts. We realize this concept experimentally by employing the laser-induced transfer method to produce near-perfect metallic spheres and helium ion beam milling to make cuts with the clean straight sidewalls and nanometre resolution. The magnetic resonance is observed at 600 nm in gold and at 565 nm in silver nanoparticles. This method can be applied to the structuring of arbitrary three-dimensional features on the surface of nanoscale resonators. It provides new ways for engineering hybrid resonant modes and ultra-high near-field enhancement.
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10
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Abstract
The study of the resonant behavior of silicon nanostructures provides a new route for achieving efficient control of both electric and magnetic components of light. We demonstrate experimentally and numerically that enhancement of localized electric and magnetic fields can be achieved in a silicon nanodimer. For the first time, we experimentally observe hotspots of the magnetic field at visible wavelengths for light polarized across the nanodimer's primary axis, using near-field scanning optical microscopy.
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Beyond the hybridization effects in plasmonic nanoclusters: diffraction-induced enhanced absorption and scattering. SMALL (WEINHEIM AN DER BERGSTRASSE, GERMANY) 2014; 10:576-583. [PMID: 24000095 DOI: 10.1002/smll.201301419] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2013] [Revised: 06/25/2013] [Indexed: 06/02/2023]
Abstract
It is demonstrated herein both theoretically and experimentally that Young's interference can be observed in plasmonic structures when two or three nanoparticles with separation on the order of the wavelength are illuminated simultaneously by a plane wave. This effect leads to the formation of intermediate-field hybridized modes with a character distinct of those mediated by near-field and/or far-field radiative effects. The physical mechanism for the enhancement of absorption and scattering of light due to plasmonic Young's interference is revealed, which we explain through a redistribution of the Poynting vector field and the formation of near-field subwavelength optical vortices.
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12
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Locomotion of microspheres for super-resolution imaging. Sci Rep 2013; 3:3501. [PMID: 24336231 PMCID: PMC3863983 DOI: 10.1038/srep03501] [Citation(s) in RCA: 86] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2013] [Accepted: 11/26/2013] [Indexed: 11/12/2022] Open
Abstract
Super-resolution virtual imaging by micron sized transparent beads (microspheres) was recently demonstrated by Wang et al. Practical applications in microscopy require control over the positioning of the microspheres. Here we present a method of positioning and controllable movement of a microsphere by using a fine glass micropipette. This allows sub-diffraction imaging at arbitrary points in three dimensions, as well as the ability to track moving objects. The results are relevant to a broad scope of applications, including sample inspection, microfabrication, and bio-imaging.
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Hybrid phase-change plasmonic crystals for active tuning of lattice resonances. OPTICS EXPRESS 2013; 21:13691-13698. [PMID: 23736622 DOI: 10.1364/oe.21.013691] [Citation(s) in RCA: 60] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
Tunable lattice resonances are demonstrated in a hybrid plasmonic crystal incorporating the phase-change material Ge2Sb2Te5 (GST) as a 20-nm-thick layer sandwiched between a gold nanodisk array and a quartz substrate. Non-volatile tuning of lattice resonances over a range Δλ of about 500 nm (1.89 µm to 2.27 µm) is achieved experimentally via intermediate phase states of the GST layer. This work demonstrates the efficacy and ease of resonance tuning via GST in the near infrared, suggesting the possibility to design broadband non-volatile tunable devices for optical modulation, switching, sensing and nonlinear optical devices.
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Abstract
Spherical silicon nanoparticles with sizes of a few hundreds of nanometers represent a unique optical system. According to theoretical predictions based on Mie theory they can exhibit strong magnetic resonances in the visible spectral range. The basic mechanism of excitation of such modes inside the nanoparticles is very similar to that of split-ring resonators, but with one important difference that silicon nanoparticles have much smaller losses and are able to shift the magnetic resonance wavelength down to visible frequencies. We experimentally demonstrate for the first time that these nanoparticles have strong magnetic dipole resonance, which can be continuously tuned throughout the whole visible spectrum varying particle size and visually observed by means of dark-field optical microscopy. These optical systems open up new perspectives for fabrication of low-loss optical metamaterials and nanophotonic devices.
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Generating and manipulating higher order Fano resonances in dual-disk ring plasmonic nanostructures. ACS NANO 2012; 6:5130-7. [PMID: 22577794 DOI: 10.1021/nn3007898] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
In this article, we investigate higher order (quadrupolar, octupolar, hexadecapolar, and triakontadipolar) Fano resonances generated in disk ring (DR) silver plasmonic nanostructures. We find that the higher order Fano resonances are generated when the size of the disk is reduced and falls into a certain range. With dual-disk ring (DDR) nanostructures, a rich set of tunable Fano line shapes is provided. More specifically, we report our observations on the optical behavior of the DDRs including asymmetric cases either in two disks with different sizes or their asymmetric locations inside the ring. In the case of symmetric dual-disk ring (SDDR) nanostructures, we demonstrate that the quadrupolar and the hexadecapolar Fano resonances are suppressed, which can reduce the cross-talk in spectroscopic measurements, while the octupolar and the triakontadipolar Fano resonances are enhanced. The potential of using the studied plasmonic nanostructures as biochemical sensors is evaluated with the figure of merit (FOM) and the contrast ratio (CR). The values of the FOM and the CR achieved using the triakontadipolar Fano resonance in the SDDR are 17 and 57%, respectively. These results indicate that the SDDRs could be developed into a high-performance biochemical sensor in the visible wavelength range.
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Abstract
We propose a novel type of hybrid metal-dielectric structures composed of silicon nanoparticles and split-ring resonators for advanced control of optically induced magnetic response. We reveal that a hybrid "metamolecule" may exhibit a strong distance-dependent magnetic interaction that may flip the magnetization orientation and support "antiferromagnetic" ordering in a hybrid metamaterial created by a periodic lattice of such metamolecules. The propagation of magnetization waves in the hybrid structures opens new ways for manipulating artificial "antiferromagnetic" ordering at high frequencies.
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The near field properties of colloidal polystyrene microspheres on silicon. JOURNAL OF NANOSCIENCE AND NANOTECHNOLOGY 2011; 11:10981-10985. [PMID: 22409039 DOI: 10.1166/jnn.2011.4081] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We have investigated the optical resonance and near field inside and under absorptive polystyrene (PS) microspheres on Si wafers. Near field flat plane images of PS microspheres were numerically simulated. Nanostructures were prepared on Si substrates using the regular two dimensional (2D) arrays by a single pulsed laser irradiation (KrF, lambda = 248 nm). Periodical PS nanoparticle, PS nanoflowers and Si nanobumps were fabricated by different laser fluence. Mechanisms for PS particle size reduction and Si nanobump formation by laser irradiation were discussed on the base of the theoretical calculations. Good agreement between theoretical calculations and experimental results has been observed.
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Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope. Nat Commun 2011; 2:218. [PMID: 21364557 DOI: 10.1038/ncomms1211] [Citation(s) in RCA: 261] [Impact Index Per Article: 20.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/16/2010] [Accepted: 01/26/2011] [Indexed: 11/09/2022] Open
Abstract
The imaging resolution of a conventional optical microscope is limited by diffraction to ~200 nm in the visible spectrum. Efforts to overcome such limits have stimulated the development of optical nanoscopes using metamaterial superlenses, nanoscale solid immersion lenses and molecular fluorescence microscopy. These techniques either require an illuminating laser beam to resolve to 70 nm in the visible spectrum or have limited imaging resolution above 100 nm for a white-light source. Here we report a new 50-nm-resolution nanoscope that uses optically transparent microspheres (for example, SiO₂, with 2 μm<diameter<9 μm) as far-field superlenses (FSL) to overcome the white-light diffraction limit. The microsphere nanoscope operates in both transmission and reflection modes, and generates magnified virtual images with a magnification up to ×8. It may provide new opportunities to image viruses and biomolecules in real time.
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Abstract
AbstractFemtosecond (fs) laser application in three-dimensional (3D) optical recording is introduced. The laser irradiation on transparent glass and polymer matrix doped with fluorescent material is carried out, which changes the physical or chemical properties of the recording media and records information bits. With the change of the focusing positions inside the transparent substrates, 3D optical recording can be available for ultrahigh capacity data storage. Feasibility on fs laser drilling of poly-caprolactone (PCL) thin films for tissue engineering is investigated. It is found that precisely defined micro-hole arrays can be formed on the sample surfaces. Hydrophilic property of the processed samples is much improved, which provides good conditions for tissue cells to anchor on the man-made skin. Fs laser applications to form nanostructures on substrate surfaces are studied. Fs laser combination with near-field scanning optical microscopy (NSOM) to induce surface property modification in the sub 50-nm under NSOM tip and nanoparticles is also discussed.
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Huge light scattering from active anisotropic spherical particles. OPTICS EXPRESS 2010; 18:24868-24880. [PMID: 21164832 DOI: 10.1364/oe.18.024868] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
The light scattering by a spherical particle with radial anisotropic permittivity ε and permeability μ are discussed in detail by expanding Mie theory. With the modified vector potential formulation, the electric anisotropy effects on scattering efficiency are addressed by studying the extinction, scattering, absorption and radar cross sections following the change of the transverse permittivity ε(t), the longitudinal permittivity ε(r) and the particle size q. The huge scattering cross sections are shown by considering the possible coupling between active medium and plasmon polaritons and this will be possible to result in spaser from the active plasmons of small particle.
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21
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The Fano resonance in plasmonic nanostructures and metamaterials. NATURE MATERIALS 2010; 9:707-15. [PMID: 20733610 DOI: 10.1038/nmat2810] [Citation(s) in RCA: 1230] [Impact Index Per Article: 87.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
Since its discovery, the asymmetric Fano resonance has been a characteristic feature of interacting quantum systems. The shape of this resonance is distinctively different from that of conventional symmetric resonance curves. Recently, the Fano resonance has been found in plasmonic nanoparticles, photonic crystals, and electromagnetic metamaterials. The steep dispersion of the Fano resonance profile promises applications in sensors, lasing, switching, and nonlinear and slow-light devices.
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Hybrid metamaterial design and fabrication for terahertz resonance response enhancement. OPTICS EXPRESS 2010; 18:12421-12429. [PMID: 20588369 DOI: 10.1364/oe.18.012421] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
Planar hybrid metamaterial with different split ring resonators (SRR) structure dimensions are fabricated on silicon substrates by femtosecond (fs) laser micro-lens array (MLA) lithography and lift-off process. The fabricated metamaterial structures consist of: (a) uniform metamaterial with 4 SRRs at same design and dimension as a unit cell and (b) hybrid metamaterial with 4 SRRs at same design but different dimensions as a unit cell. The electromagnetic field responses of these hybrid and single dimension metamaterial structures are characterized using a terahertz (THz) time-domain spectroscopy. Transmission spectra of these metamaterial show that a broader resonance peak is formed when 2 SRRs are close to each other. FDTD simulation proves that there is a strong mutual coupling between 2 SRRs besides a strong localized electric field at the split gap, which can enhance the electric field up to 364 times for tunable, broad band and high sensitivity THz sensing. Meanwhile, the strong coupling effect could lead to the formation of an additional resonance peak at approximately 0.2 THz in the THz spectra regime.
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Chemical-assisted laser parallel nanostructuring of silicon in optical near fields. NANOTECHNOLOGY 2008; 19:455302. [PMID: 21832767 DOI: 10.1088/0957-4484/19/45/455302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
The authors present a simple and efficient technique for producing hexagonal arrays of nanostructures on silicon surfaces in chemical solutions. It utilizes the effect of optical near-field enhancement by self-assembled particle-lens arrays and a thermally induced chemical reaction with an alkaline solution. About 10(8) features can be produced simultaneously by one single laser pulse. Furthermore, the shape of the structures was found to be controllable, from concave holes to convex bumps, by means of a post-etching process, in the same chemical solution.
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Peculiarities in light scattering by spherical particles with radial anisotropy. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2008; 25:1623-1628. [PMID: 18594617 DOI: 10.1364/josaa.25.001623] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
Light scattering by a spherical particle with radial anisotropy is discussed by extending Mie theory to diffraction by an anisotropic sphere, including both the electric and the magnetic anisotropy ratio. It is shown that radial anisotropy may lead to great modifications in scattering efficiencies and field enhancement, elucidating the importance of anisotropies in the control of scattering. The capacity for nondissipating damping is demonstrated for anisotropic spheres with different signs in radial and transversal permittivities.
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25
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Cross talk in photochemical three-dimensional optical recording. OPTICS LETTERS 2004; 29:2055-2057. [PMID: 15455778 DOI: 10.1364/ol.29.002055] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
It is shown that the cross talk in three-dimensional optical photochemical recording results in ineffectiveness of single-photon recording. For two-photon recording for a given allowed cross-talk level there are optimal distances between spots that provide the maximum memory density. Estimations show that real restrictions here are connected with the reading of information.
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