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Yue X, Wang T, Cai Y, Wang H, Lv E, Yuan X, Zeng J, Zhao W, Wang J. Composite metamaterial of hyperbolic nanoridges and gold nanoparticles for biosensing. NANOSCALE 2025; 17:7271-7280. [PMID: 39973344 DOI: 10.1039/d4nr05517a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/21/2025]
Abstract
Hyperbolic metamaterials have gained considerable attention in the field of optical biosensing due to their ability to support highly sensitive plasmonic modes. The high sensitivity of these electromagnetic modes mainly manifests as a strong response to variations in the bulk refractive index of the surrounding environment. However, its capability to detect low-concentration biochemical molecules near the surface of the metamaterial still requires further enhancement. In this work, we developed a composite metamaterial of gold nanoparticles and nanoridge hyperbolic metamaterials. The hyperbolic nanoridges are high-periodicity metamaterial arrays fabricated by combining electron beam lithography and electroplating. By exciting the high-sensitivity coupling modes formed by the bulk plasmon-polariton and localized surface plasmon resonance in this composite metamaterial, we achieved an improvement of over one order of magnitude in the detection limit for biomolecules, while maintaining the high bulk sensitivity of 23 333 nm RIU-1. Our research not only plays a key role in advancing the field of real-time, high-precision plasmonic biosensing but also offers substantial promise for improving early disease detection and monitoring.
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Affiliation(s)
- Xinzhao Yue
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074, China.
| | - Tao Wang
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074, China.
| | - Yaohua Cai
- Institute of Hematology, Union Hospital, Tongji Medical College, Huazhong University of Science and Technology, Wuhan, 430030, China
| | - Huimin Wang
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074, China.
| | - Enze Lv
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074, China.
| | - Xuyang Yuan
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074, China.
| | - Jinwei Zeng
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074, China.
| | - Wenyu Zhao
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074, China.
| | - Jian Wang
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074, China.
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Lv J, Wu Y, Liu J, Gong Y, Si G, Hu G, Zhang Q, Zhang Y, Tang JX, Fuhrer MS, Chen H, Maier SA, Qiu CW, Ou Q. Hyperbolic polaritonic crystals with configurable low-symmetry Bloch modes. Nat Commun 2023; 14:3894. [PMID: 37393303 DOI: 10.1038/s41467-023-39543-w] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/05/2022] [Accepted: 06/17/2023] [Indexed: 07/03/2023] Open
Abstract
Photonic crystals (PhCs) are a kind of artificial structures that can mold the flow of light at will. Polaritonic crystals (PoCs) made from polaritonic media offer a promising route to controlling nano-light at the subwavelength scale. Conventional bulk PhCs and recent van der Waals PoCs mainly show highly symmetric excitation of Bloch modes that closely rely on lattice orders. Here, we experimentally demonstrate a type of hyperbolic PoCs with configurable and low-symmetry deep-subwavelength Bloch modes that are robust against lattice rearrangement in certain directions. This is achieved by periodically perforating a natural crystal α-MoO3 that hosts in-plane hyperbolic phonon polaritons. The mode excitation and symmetry are controlled by the momentum matching between reciprocal lattice vectors and hyperbolic dispersions. We show that the Bloch modes and Bragg resonances of hyperbolic PoCs can be tuned through lattice scales and orientations while exhibiting robust properties immune to lattice rearrangement in the hyperbolic forbidden directions. Our findings provide insights into the physics of hyperbolic PoCs and expand the categories of PhCs, with potential applications in waveguiding, energy transfer, biosensing and quantum nano-optics.
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Affiliation(s)
- Jiangtao Lv
- College of Information Science and Engineering, Northeastern University, Shenyang, 110004, China
- School of Control Engineering, Hebei Key Laboratory of Micro-Nano Precision Optical Sensing and Measurement Technology, Northeastern University at Qinhuangdao, Qinhuangdao, 066004, China
| | - Yingjie Wu
- ZJU-Hangzhou Global Scientific and Technological Innovation Center, Zhejiang University, Hangzhou, 311215, China.
| | - Jingying Liu
- Macao Institute of Materials Science and Engineering (MIMSE), Faculty of Innovation Engineering, Macau University of Science and Technology, Taipa, Macao, 999078, China
- Department of Materials Science and Engineering, Monash University, Clayton, Victoria, 3800, Australia
| | - Youning Gong
- State Key Laboratory of Radio Frequency Heterogeneous Integration, College of Electronics and Information Engineering, Shenzhen University, Shenzhen, 518060, China
| | - Guangyuan Si
- Melbourne Centre for Nanofabrication, Victorian Node of the Australian National Fabrication Facility, Clayton, 3168, VIC, Australia
| | - Guangwei Hu
- School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798, Singapore
| | - Qing Zhang
- School of Physics, University of Electronic Science and Technology of China, Chengdu, 611731, China
| | - Yupeng Zhang
- State Key Laboratory of Radio Frequency Heterogeneous Integration, College of Electronics and Information Engineering, Shenzhen University, Shenzhen, 518060, China
| | - Jian-Xin Tang
- Macao Institute of Materials Science and Engineering (MIMSE), Faculty of Innovation Engineering, Macau University of Science and Technology, Taipa, Macao, 999078, China
- Jiangsu Key Laboratory for Carbon-Based Functional Materials & Devices, Institute of Functional Nano & Soft Materials (FUNSOM), Soochow University, Jiangsu, 215123, China
| | - Michael S Fuhrer
- ARC Centre of Excellence in Future Low-Energy Electronics Technologies, Monash University, Clayton, VIC, 3800, Australia
- School of Physics and Astronomy, Monash University, Clayton, VIC, 3800, Australia
| | - Hongsheng Chen
- ZJU-Hangzhou Global Scientific and Technological Innovation Center, Zhejiang University, Hangzhou, 311215, China
| | - Stefan A Maier
- ARC Centre of Excellence in Future Low-Energy Electronics Technologies, Monash University, Clayton, VIC, 3800, Australia
- School of Physics and Astronomy, Monash University, Clayton, VIC, 3800, Australia
- Department of Physics, Imperial College London, London, SW7 2AZ, UK
| | - Cheng-Wei Qiu
- Department of Electrical and Computer Engineering, National University of Singapore, Singapore, 117583, Singapore.
| | - Qingdong Ou
- Macao Institute of Materials Science and Engineering (MIMSE), Faculty of Innovation Engineering, Macau University of Science and Technology, Taipa, Macao, 999078, China.
- Department of Materials Science and Engineering, Monash University, Clayton, Victoria, 3800, Australia.
- ARC Centre of Excellence in Future Low-Energy Electronics Technologies, Monash University, Clayton, VIC, 3800, Australia.
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Wang H, Wang T, Zhong S, Zhang J, Yan R, Xu P, Zhang YH, Yue X, Wang L, Wang Y, Yuan X, Zeng J, Wang J. Sensitivity investigation of a biosensor with resonant coupling of propagating surface plasmons to localized surface plasmons in the near infrared region. NANOSCALE 2023. [PMID: 37334909 DOI: 10.1039/d3nr01076g] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/21/2023]
Abstract
Gold nanoparticles (AuNPs) can be used to improve the performance of propagating surface plasmon resonance (PSPR) refractive index sensors. The resonant coupling effect between PSPR and localized surface plasmon resonance (LSPR) supported by AuNPs on sensitivity remains to be elucidated in terms of evanescent field intensity and distribution. In this study, we directly compare the sensitivity of the PSPR sensor and the resonant coupling mode between the PSPR and LSPR sensors in the wavelength scanning mode. The sensitivity of PSPR can be significantly improved in the near-infrared region excitation wavelength. 1,6-Hexanedithiol was used to achieve a AuNP modified gold film (GF-AuNP). The PSPR excited by the prism coupling mechanism can effectively stimulate LSPR supported by AuNPs in the GF-AuNP, and then resonant coupling is generated. Compared with PSPR, the resonant coupling mode shows a decrease in penetration depth by 28 times and an increase in the surface electric field intensity by 4.6 times in the numerical simulations. The decrease in the penetration depth in the GF-AuNP is made at the expense of bulk sensitivity. The biosensing sensitivity of the GF-AuNP shows up to 7-fold improvement in the carcinoembryonic antigen immunoassay and the GF-AuNP is proven to be a better biosensor. The experimental measurements are in excellent agreement with the theoretical model. This study can be also considered as a guide for the design of plasmonic sensors for detecting multiple substances at different scales, such as cells and proteins.
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Affiliation(s)
- Huimin Wang
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Tao Wang
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Simei Zhong
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Jinyan Zhang
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Ruoqin Yan
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Peng Xu
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Yu-Hui Zhang
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Xinzhao Yue
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Lu Wang
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Yuandong Wang
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Xuyang Yuan
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Jinwei Zeng
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
| | - Jian Wang
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, 430074.
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Li Z, Zhang Y, Guo X, Tong C, Chen X, Zeng Y, Shen J, Li C. Highly sensitive short-range mode resonance sensor with multilayer structured hyperbolic metamaterials. OPTICS EXPRESS 2023; 31:3520-3535. [PMID: 36785343 DOI: 10.1364/oe.477697] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2022] [Accepted: 12/31/2022] [Indexed: 06/18/2023]
Abstract
Hyperbolic metamaterial (HMM) based sensors can achieve superior sensing performance than conventional surface plasmon resonance sensors. In this work, the operator approach to effective medium approximation (OEMA) is used to characterize the HMM dielectric constant properties of metal-dielectric multilayer structures, which are classified into short-range (SR) mode and long-range (LR) mode according to the propagation length of the bulk high K waves in HMM. The dispersion relations of SR modes are derived, and a high-sensitivity refractive index sensor is designed for the near-infrared SR mode resonance. The effects of the number of periods, cell thickness, metal fill rate and incidence angle on the SR mode resonance were analyzed for the multilayer structured HMM. Our designed sensing structure achieves a maximum sensitivity of 330 µm/RIU in the near-infrared band with a quality factor of 492 RIU-1. In addition, the simulations show that the SR mode resonance wavelength is flexible and tunable. We believe that the study of HMM-based SR mode resonance sensors offers potential applications for high-sensitivity biochemical detection.
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Wang P, Nasir ME, Krasavin AV, Dickson W, Zayats AV. Optoelectronic Synapses Based on Hot-Electron-Induced Chemical Processes. NANO LETTERS 2020; 20:1536-1541. [PMID: 32013449 DOI: 10.1021/acs.nanolett.9b03871] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/05/2023]
Abstract
Highly efficient information processing in the brain is based on processing and memory components called synapses, whose output is dependent on the history of the signals passed through them. Here, we have developed an artificial synapse with both electrical and optical memory effects using chemical transformations in plasmonic tunnel junctions. In an electronic implementation, the electrons tunneled into plasmonic nanorods under a low bias voltage are harvested to write information into the tunnel junctions via hot-electron-mediated chemical reactions with the environment. In an optical realization, the information can be written by an external light illumination to excite hot electrons in the plasmonic nanorods. The stored information is nonvolatile and can be read either electrically or optically by measuring the resistance or inelastic-tunneling-induced light emission, respectively. The described architecture provides a high density (∼1010 cm-2) of memristive optoelectronic devices which can be used as multilevel nonvolatile memory, logic units, or artificial synapses in future electronic, optoelectronic, and artificial neural networks.
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Affiliation(s)
- Pan Wang
- Department of Physics and London Centre for Nanotechnology, King's College London, Strand, London WC2R 2LS, United Kingdom
| | - Mazhar E Nasir
- Department of Physics and London Centre for Nanotechnology, King's College London, Strand, London WC2R 2LS, United Kingdom
| | - Alexey V Krasavin
- Department of Physics and London Centre for Nanotechnology, King's College London, Strand, London WC2R 2LS, United Kingdom
| | - Wayne Dickson
- Department of Physics and London Centre for Nanotechnology, King's College London, Strand, London WC2R 2LS, United Kingdom
| | - Anatoly V Zayats
- Department of Physics and London Centre for Nanotechnology, King's College London, Strand, London WC2R 2LS, United Kingdom
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Zang R, Hu H, Li X, Gao J, Liang L, Zhang H, Zhuge F, Luo J, Cao H. Broadband hyperbolic metamaterial covering the whole visible-light region. OPTICS LETTERS 2019; 44:2970-2973. [PMID: 31199358 DOI: 10.1364/ol.44.002970] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/26/2019] [Accepted: 05/08/2019] [Indexed: 06/09/2023]
Abstract
Nanowire-based hyperbolic metamaterials (HMMs) with rich optical dispersion engineering capabilities are promising for use in miniaturization devices, such as nanophotonic chips and circuits. Herein, based on a one-step and template-free sputtering method, we are capable of precisely tuning the microstructural parameters of Ag nanowires (with a diameter <10 nm) in silica matrix, offering plenty of opportunities to perform hyperbolic dispersion engineering. Thus, the effective plasma frequency of the designed HMMs was shifted into the near-ultraviolet region (∼350 nm), leading to a broadband hyperbolic dispersion feature covering the whole visible-light region. This demonstration could pave the way for the development of metamaterial-based flat lenses, deep-subwavelength waveguiding, and broadband perfect absorbers and sensing, etc.
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Alfaro-Mozaz FJ, Rodrigo SG, Alonso-González P, Vélez S, Dolado I, Casanova F, Hueso LE, Martín-Moreno L, Hillenbrand R, Nikitin AY. Deeply subwavelength phonon-polaritonic crystal made of a van der Waals material. Nat Commun 2019; 10:42. [PMID: 30604741 PMCID: PMC6318287 DOI: 10.1038/s41467-018-07795-6] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2018] [Accepted: 11/27/2018] [Indexed: 12/02/2022] Open
Abstract
Photonic crystals (PCs) are periodically patterned dielectrics providing opportunities to shape and slow down the light for processing of optical signals, lasing and spontaneous emission control. Unit cells of conventional PCs are comparable to the wavelength of light and are not suitable for subwavelength scale applications. We engineer a nanoscale hole array in a van der Waals material (h-BN) supporting ultra-confined phonon polaritons (PhPs)-atomic lattice vibrations coupled to electromagnetic fields. Such a hole array represents a polaritonic crystal for mid-infrared frequencies having a unit cell volume of [Formula: see text] (with λ0 being the free-space wavelength), where PhPs form ultra-confined Bloch modes with a remarkably flat dispersion band. The latter leads to both angle- and polarization-independent sharp Bragg resonances, as verified by far-field spectroscopy and near-field optical microscopy. Our findings could lead to novel miniaturized angle- and polarization-independent infrared narrow-band couplers, absorbers and thermal emitters based on van der Waals materials and other thin polar materials.
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Affiliation(s)
| | - S G Rodrigo
- Instituto de Ciencia de Materiales de Aragón and Departamento de Física de la Materia Condensada, CSIC-Universidad de Zaragoza, 50009, Zaragoza, Spain
- Centro Universitario de la Defensa, Ctra. de Huesca s/n, 50090, Zaragoza, Spain
| | | | - S Vélez
- CIC nanoGUNE, 20018, Donostia-San Sebastián, Spain
- Department of Materials, ETH Zürich, 8093, Zürich, Switzerland
| | - I Dolado
- CIC nanoGUNE, 20018, Donostia-San Sebastián, Spain
| | - F Casanova
- CIC nanoGUNE, 20018, Donostia-San Sebastián, Spain
- IKERBASQUE, Basque Foundation for Science, 48013, Bilbao, Spain
| | - L E Hueso
- CIC nanoGUNE, 20018, Donostia-San Sebastián, Spain
- IKERBASQUE, Basque Foundation for Science, 48013, Bilbao, Spain
| | - L Martín-Moreno
- Instituto de Ciencia de Materiales de Aragón and Departamento de Física de la Materia Condensada, CSIC-Universidad de Zaragoza, 50009, Zaragoza, Spain
| | - R Hillenbrand
- IKERBASQUE, Basque Foundation for Science, 48013, Bilbao, Spain.
- CIC nanoGUNE and UPV/EHU, 20018, Donostia-San Sebastián, Spain.
| | - A Y Nikitin
- IKERBASQUE, Basque Foundation for Science, 48013, Bilbao, Spain.
- Donostia International Physics Center (DIPC), 20018, Donostia-San Sebastián, Spain.
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Shen KC, Ku CT, Hsieh C, Kuo HC, Cheng YJ, Tsai DP. Deep-Ultraviolet Hyperbolic Metacavity Laser. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2018; 30:e1706918. [PMID: 29633385 DOI: 10.1002/adma.201706918] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/26/2017] [Revised: 02/11/2018] [Indexed: 06/08/2023]
Abstract
Given the high demand for miniaturized optoelectronic circuits, plasmonic devices with the capability of generating coherent radiation at deep subwavelength scales have attracted great interest for diverse applications such as nanoantennas, single photon sources, and nanosensors. However, the design of such lasing devices remains a challenging issue because of the long structure requirements for producing strong radiation feedback. Here, a plasmonic laser made by using a nanoscale hyperbolic metamaterial cube, called hyperbolic metacavity, on a multiple quantum-well (MQW), deep-ultraviolet emitter is presented. The specifically designed metacavity merges plasmon resonant modes within the cube and provides a unique resonant radiation feedback to the MQW. This unique plasmon field allows the dipoles of the MQW with various orientations into radiative emission, achieving enhancement of spontaneous emission rate by a factor of 33 and of quantum efficiency by a factor of 2.5, which is beneficial for coherent laser action. The hyperbolic metacavity laser shows a clear clamping of spontaneous emission above the threshold, which demonstrates a near complete radiation coupling of the MQW with the metacavity. This approach shown here can greatly simplify the requirements of plasmonic nanolaser with a long plasmonic structure, and the metacavity effect can be extended to many other material systems.
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Affiliation(s)
- Kun-Ching Shen
- Research Center for Applied Sciences, Academia Sinica, Taipei, 11529, Taiwan
| | - Chen-Ta Ku
- Research Center for Applied Sciences, Academia Sinica, Taipei, 11529, Taiwan
| | - Chiieh Hsieh
- Research Center for Applied Sciences, Academia Sinica, Taipei, 11529, Taiwan
| | - Hao-Chung Kuo
- Department of Photonics and Institute of Electro-Optical Engineering, National Chiao Tung University, Hsinchu, 30010, Taiwan
| | - Yuh-Jen Cheng
- Research Center for Applied Sciences, Academia Sinica, Taipei, 11529, Taiwan
| | - Din Ping Tsai
- Research Center for Applied Sciences, Academia Sinica, Taipei, 11529, Taiwan
- Department of Physics, National Taiwan University, Taipei, 10617, Taiwan
- College of Engineering, Chang Gung University, Taoyuan, 33302, Taiwan
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Superluminal and stopped light due to mode coupling in confined hyperbolic metamaterial waveguides. Sci Rep 2015; 5:17678. [PMID: 26643503 PMCID: PMC4672343 DOI: 10.1038/srep17678] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2015] [Accepted: 11/02/2015] [Indexed: 12/02/2022] Open
Abstract
Anisotropic metamaterials with hyperbolic dispersion can be used to design waveguides with unusual properties. We show that, in contrast to planar waveguides, geometric confinement leads to coupling of ordinary (forward) and extraordinary (backward) modes and formation of hybrid waveguided modes, which near the crossing point may exhibit slow, stopped or superluminal behavior accompanied by very strong group velocity dispersion. These modes can be used for designing stopped-light nanolasers for nanophotonic applications and dispersion-facilitated signal reshaping in telecom applications.
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Segovia P, Marino G, Krasavin AV, Olivier N, Wurtz GA, Belov PA, Ginzburg P, Zayats AV. Hyperbolic metamaterial antenna for second-harmonic generation tomography. OPTICS EXPRESS 2015; 23:30730-30738. [PMID: 26698705 DOI: 10.1364/oe.23.030730] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
The detection and processing of information carried by evanescent field components are key elements for subwavelength optical microscopy as well as single molecule sensing applications. Here, we numerically demonstrate the potential of a hyperbolic medium in the design of an efficient metamaterial antenna enabling detection and tracking of a nonlinear object, with an otherwise hidden second-harmonic signature. The presence of the antenna provides 103-fold intensity enhancement of the second harmonic generation (SHG) from a nanoparticle through a metamaterial-assisted access to evanescent second-harmonic fields. Alternatively, the observation of SHG from the metamaterial itself can be used to detect and track a nanoparticle without a nonlinear response. The antenna allows an optical resolution of several nanometers in tracking the nanoparticle's location via observations of the far-field second-harmonic radiation pattern.
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