Symposium Organizers
Shih-Yuan (SY) Wang Hewlett-Packard Laboratories
Nicholas Xuanlai Fang University of Illinois, Urbana-Champaign
Lars Thylen Royal Institute of Technology (KTH)
M. Saif Islam University of California-Davis
J1: Novel Physics of NIMS
Session Chairs
Tuesday PM, April 18, 2006
Room 3018 (Moscone West)
9:45 AM - **J1.2
A DoD Perspective on Negative Index Materials.
Valerie Browning 1
1 , DARPA/DSO, Arlington, Virginia, United States
Show AbstractIn the quest for ever smaller, lighter weight, and conformal components and devices for radar and communication applications, researchers in the rf community have increasingly turned to artificially engineered, composite structures (or “metamaterials”) in order to exploit the extraordinary electromagnetic response these materials offer. One particularly promising class of metamaterials that has recently received a great deal of attention are “left-handed” or negative index materials. Because these metamaterials exhibit the unique ability to bend and focus light in ways no other conventional materials can, they hold great potential for enabling a number of innovative lens and antenna structures for a broad range of DoD relevant applications. Exploring the possible implementation of negative index materials for DoD applications will require significant enhancements in the properties of existing NIM (bandwidth, loss, operational frequency, etc.), as well as improved understanding of the physics of their electromagnetic transport properties. For this reason the Defense Advanced Research Project Agency (DARPA) has initiated a program that seeks to further develop and demonstrate NIM for future DoD missions including, but not limited to, the following: 1) lightweight, compact lenses with improved optics; 2) sub wavelength/high resolution imaging across the electromagnetic spectrum; 3) novel approaches to beam steering for radar, rf, and/or optical communications; and 4) novel approaches for integrating optics with semiconductor electronics. A brief overview of several efforts being funded under the DARPA NIM program will be presented and implications for future DoD applications will be discussed.
10:15 AM - **J1.3
Magnetic Response and Negative Index of Refraction at THz Frequencies.
Costas Soukoulis 1 2
1 Ames Lab and Dept of Physics, Iowa State University, Ames, Iowa, United States, 2 , Research Center of Crete - FORTH, Heraklion, Crete Greece
Show Abstract10:45 AM - J1.4
Surface Photon Drag on Plasmonic Metamaterials: A Fizeau-Doppler Shift Study.
Leilei Yin 1 , Nicholas Fang 1
1 MIE, University of Illinois, Urbana, Illinois, United States
Show AbstractMetamaterials have opened an exciting gateway to create unprecedented physical properties and functionality unattainable from naturally existing materials. The “atoms” and “molecules” in metamaterials can be tailored in shape and size, the lattice constant and inter-atomic interaction can be artificially tuned, and “defects” can be designed and placed at desired locations. The recent discovery of negative index of refraction is an excellent example of the new physics from metamaterials. The objective of this work is to provide some new insight on the anomalous Doppler effect [1-3], another class of unusual wave propagation phenomena that stem from the engineered metamaterials. A light beam incident upon a moving dielectric medium is, in general, subject to a frequency shift due to a Fizeau-Doppler light drag effect. In most normal materials, the magnitude of this shift is small and not easy to detect. However, such effect is dramatically enhanced when a slow group velocity is introduced at a proper surface plasmon resonance condition of the metamaterials. Based on an optical interferometry to measure the frequency shift in the near vicinity of resonance condition, our theoretical study indicates that both appreciable upstream and downstream photon drag could be achieved. Possible experimental configurations are also discussed.[1] N. Seddon, T. Bearpark, Science 302, 1537 (2003);[2] A. Kozyrev, D. van der Weide, Phys. Rev. Lett. 94, 203902 (2005);[3] E. Reed, M. Soljacic, J. Joannopoulos, Phys. Rev. Lett. 91, 133901 (2003).
11:00 AM - J1: NovP
BREAK
11:30 AM - **J1.5
Veselago Lens and Plasmonic Nanostructures: Towards Optical Frequencies.
A Bratkovsky 1 , E. Ponizovskaya 1
1 , Hewlett-Packard Laboratories, Palo Alto, California, United States
Show AbstractTuesday, April 18New Abstract10:30 am *J1.5Veselago Lens and Plasmonic Nanostructures: Towards Optical Frequencies. A. M. Bratkovsky, E. Ponizovskaya, Hewlett-Packard Labs, Palo Alto, California. We review various suggestions to design metamaterials for Veselago lens that may potentially perform at optical frequencies. Plasmonic metallic nanostructures present one interesting possibility for both 2D and 3D negative index medium (NIM) systems. Systems with strong spatial dispersion e.g. photonic dielectric bandgap crystals) are known to exhibit negative refraction. Same is true of metallic 2D and 3D metamaterials but places them in a complementary class of systems, since they support plasmon excitations. In particular, surface plasmons are known to enable local field enhancement and sub-wavelength imaging. The behavior of metallic metamaterials may be strongly dispersive and accompanied by considerable losses. Sub-wavelength imaging is adversely affected by losses and spatial dispersion. One way of mitigating losses is to use gain medium. There are potential mechanisms of modulating signals with the use of compact NIM slabs.
12:00 PM - **J1.6
Applications of Metamaterials to Realize Efficient Electrically Small Antennas
Richard Ziolkowski 1
1 Electrical and Computer Engineering, University of Arizona, Tuscon, Arizona, United States
Show AbstractMetamaterials are engineered media whose electromagnetic responses are different from those of their constituent components. They are often generated by incorporating in a periodic manner various types of artificially fabricated, extrinsic, low dimensional inhomogeneities in some background substrate. Metamaterials that mimic known material responses or that qualitatively have new response functions that do not occur in nature have been realized. We are investigating the use of epsilon (permittivity) negative (ENG), mu (permeability) negative (MNG), and double negative (DNG) (both epsilon and mu are negative) metamaterials to realize dipole and loop antenna systems that are electrically small, efficient, and have significant bandwidth. By surrounding the electrically small antennas with a correspondingly small metamaterial spherical shell, we have demonstrated the existence of resonant states that can be used to match these antennas to sources so that the resulting system efficiently radiates the power applied to it.Several metamaterial issues will be discussed. These will include the unit cell size and design to achieve the requisite electromagnetic properties and the spherical shell shape. Most metamaterials to date have been realized as planar slabs. The challenges of realizing spherical, non-planar structures will be described. The properties of the dispersion of the metamaterials have a significant impact on the bandwidth of the antenna system. Both passive and active metamaterials have been considered for these electrically small radiating systems. Their roles in achieving significant bandwidth will also be described. The advantages and disadvantages of the ENG, MNG, and DNG shell configurations will be discussed, particularly from the point of view of which metamaterial choice may have an advantage over the others for a proof-of-concept experiment.
12:30 PM - J1.7
Design of Acoustic Metamaterials With Effective Negative Material Properties.
Muralidhar Ambati 1 , Vladimir Fokin 1 , Cheng Sun 1 , Xiang Zhang 1
1 , University of California, Berkeley, Berkeley, California, United States
Show AbstractMultiple scattering in periodic structures with strong modulation of elastic constants leads to phononic band structures. Experimental and theoretical studies of such periodic structures have been carried out extensively with applications ranging from waveguides to band gap structures. According to the Bragg’s theory, the spatial modulation of the periodic structure - phononic crystal - must be of the same order as the wavelength of considered elastic waves. Hence these phononic crystals are not suitable for some practical application. A new class of materials - acoustic metamaterials - is proposed. In these materials respective wavelength is about two orders of magnitude larger than lattice constant. Each unit cell consists of a locally resonant structural unit in contrast to simple acoustic scatterers in phononic crystals. Recently, some interesting experimental and theoretical studies on locally resonant sonic materials are reported. In this paper, we will address the design of locally resonant sonic materials. These materials offer dynamic effective negative material properties (mass density ρ and bulk modulus K) around resonance frequency. Analysis of band structure may be not enough for prediction of effective properties of a finite slab of these metamaterials. Locally resonant sonic crystals - such as an array of Helmholtz resonators and cylinders or spheres coated with acoustically soft material - were analyzed in the literature. But existing literature lacks systematic study of the dependence of effective acoustic metamaterial both on the geometry and the properties of its constituent materials. Design of resonant acoustic metamaterial requires analysis of these dependencies. In this report, we present a robust method for obtaining effective properties - effective mass density and bulk modulus - of a thin slab of acoustic metamaterial. In addition, we will consider a method to determine the thickness of the effective slab of the metamaterial. The dependence of effective acoustic properties of a metamaterial from the geometrical and the acoustic properties of the constituent materials will be discussed.
12:45 PM - J1.8
Perfect Transmission of Electromagnetic Waves through a Stack of Positive and Negative Index Materials
Ye Lu 1 , Ru-Wen Peng 1 , L. S Cao 1 , De Li 1 , Mu Wang 1
1 , National Laboratory of Solid State Microstructures , Nanjing China
Show AbstractJ2: Novel Fabrication Technology for NIMS
Session Chairs
Tuesday PM, April 18, 2006
Room 3018 (Moscone West)
2:30 PM - J2.1
Realization of 3D Isotropic Negative Index Materials using Massively Parallel and Manufacturable Microfabrication and Micromachining Technology
Logeeswaran VJ 1 , M. Saif Islam 1 , Mei-Lin Chan 2 , David Horsley 2 , Wei Wu 3 , Shih-Yuan Wang 3 , R.Stanley Williams 3
1 Electrical & Computer Engineering, University of California-Davis, DAVIS, California, United States, 2 Mechanical and Aeronautical Engineering, University of California-Davis, DAVIS, California, United States, 3 Quantum Science Research Group, Hewlett Packard Laboratories, Palo Alto, California, United States
Show AbstractWe report the first three dimensional (3D) homogeneous and isotropic negative index materials (3D-NIMs) fabricated using a low cost and massively parallel manufacturable microfabrication and microassembly technique. The construction of self-assembled 3D-NIM array was realized through two dimensional (2-D) planar microfabrication techniques exploiting the as-deposited residual stress imbalance between a bi-layer consisting of e-beam evaporated metal (650nm of chromium) and a structural layer of 500nm of low stress silicon nitride deposited by LPCVD on a silicon substrate. A periodic continuation of a single rectangular unit cell consisting of split-ring resonators (SRR) and wires were fabricated to generate a 3D assembly by orienting them along all three Cartesian axes. The thin chromium and silicon nitride bi-layer is formed as hinges. The strain mismatch between the two layers curls the structural layer (flap) containing the SRR upwards. The self-assembled out-of-plane angular position depends on the thickness and material composing the bi-layer. This built-in stress-actuated assembly method is suitable for applications requiring a thin dielectric layer for the SRR. The split-ring resonators and other structures are created on the membrane which is then assembled into the 3-D configuration. Efforts are currently underway to develop a parallel microfabrication and self-assembly process using a thicker released holding plate (~5 to 10 microns) with deformable hinges for the SRR and wires. Unlike the research-based approach of fabricating a single structure for characterizing the unique properties of NIM, our mass-manufacturable process may offer opportunities for reproducible fabrication of 3D-NIM materials with frequencies from microwave to optical domain.
2:45 PM - **J2.2
Negative Refraction in 2D Photonic Crystals and Structured Metal Surfaces.
Min Qiu 1
1 Department of Microelectronics and Information Technology, Royal Institute of Technology, Kista Sweden
Show AbstractIn the present paper, we will first present our recent theoretical and experimental results of negative refraction in 2D photonic crystals. We investigated the coupling efficiency between external plane waves and the Bloch waves, and designed a microsuperlens utilizing the negative refraction in PhCs and discussed the influences of the surface termination on the transmission and the imaging quality. The results of two experiments demonstrating negative refraction in a 2D PhC at the optical communication wavelengths will also presented. We will then present our results on negative refraction in structured metal (perfect electric conductor) surfaces. It is well known that surface waves do not exist at the interface between a perfect electric conductor (PEC) and a dielectric since the electromagnetic wave can not penetrate into the PEC. Even so, it has been shown that the 1D and 2D periodic PEC structures can support surface waves. Nevertheless, since such a structure is a periodic structure, the dispersion relation must be modified by the space periodic modulation. This gives us a chance to explore some new phenomena of surface waves. For example, it is possible to have negative refraction in such a system, and we show that subwavelength imaging of a dipole source by a slab of such a structure is possible. Unlike other negative refraction index material (NIM) for creating metamaterials with plasma like response, the present scheme is readily implemented and in principle lossless. Our experimental attempts at the microwave region will also be also presented.
3:15 PM - **J2.3
Novel Passive and Active Transmission Line Metamaterial Devices.
Christophe Caloz 1
1 , Ecole Polytechnique de Montreal, Montreal, Quebec, Canada
Show AbstractAfter a brief introduction on the fundamentals of transmission line (TL) metamaterials (MTMs), founded on the concept of composite right/left-handedness (CRLH), a number of novel passive and active CRLH TL MTM components, antennas and quasi-optical structures will be presented. The talk will be concluded by a discussion of possible directions for the next generation of MTMs.
3:45 PM - J2.4
A Nanowire-Dielectric Composite Metamaterial
Jerrold Kielbasa 1 , Jiwen Liu 1 , Burak Ucer 1 , David Carroll 1 , Richard Wiiliams 1
1 Physics, The Center for Nanotechnology and Molecular Materials , Wake Forest University, Winston Salem, North Carolina, United States
Show AbstractNegative refractive index materials (NRIM) have been shown to possess fascinating properties which will have technological consequences (e.g. sub-diffraction limit resolution). These materials have not been observed in nature and therefore must be built. Metamaterials with simultaneously negative permittivity ε and permeability μ are designed by imbedding electric and magnetic subunits in a dielectric medium. The metamaterial must be seen as continuous in order for ε and μ to be valid parameters, and therefore some dimensions of the electric and magnetic subunits must be much smaller than the relevant electromagnetic wavelength. This has kept NRIMs in the microwave region. Vis-NIR NRIMs will require nanoscale subunits. It has been predicted that a random array of Ag nanowires will exhibit a broad-band plasmon resonance and may exhibit negative μ at the same frequency.1 We randomly imbedded different amounts of Ag nanowires (l ≈ 5 μm, d ≈ 100 nm) in a TEOS sol-gel host. Ellipsometry and reflectometry were used to measure the optical constants at λ = 1064 nm. Transmission through the material was measured between 300 – 1000 nm in order to observe resonances.
4:30 PM - **J2.5
Photonic Meta Materials, Nano-scale plasmonics and Super Lens
Xiang Zhang 1
1 NSF Center for Nanoscale Science and Engineering, University of California, Berkeley, Berkeley, California, United States
Show AbstractAbstract Recent theory predicted a new class of meta structures made of engineered sub wavelength entities - meta “atoms” and “molecules” which enable the unprecedented electromagnetic properties that do not exist in the nature. For example, artificial plasma and artificial magnetism, and super lens that focuses far below the diffraction limit. If the theory is correct and these unique properties can be realized, it will have profound impact in wide range of applications such as nano-scale imaging, nanolithography, and integrated nano photonics. These photonic “atoms” usually form highly complex structures which present a critical need in developing truly 3D micro and nano-manufacturing techniques which are not available presently.In the first part of this presentation, I’ll discuss a few micro and nano fabrication technologies that we developed for engineering complex meta-structures. In the second part, I’ll discuss sub-lambda photonic “atoms” and “molecules” and the potential applications in nano-scale imaging and lithography. We demonstrated, for the first time, the high frequency magnetic activity at THz generated by artificially structured “molecule resonance”, as well as the artificial plasma. Our experiment also confirmed the key proposition of super lens theory by using surface plasmon. We indeed observed superlensing at near-field. I will discuss our recent experiment on the superlens, as well as its limits. The surface plasmon indeed promises an exciting engineering paradigm of “optical frequency and x-ray wavelength”. This talk will be concluded with a vision of the nano-manufacturing that will enable the new nano plasmonics and other applications.
5:00 PM - J2.6
Fabrication of Multilayer Metal-Dielectric Nanofilms for Coupled Plasmon Resonant Devices.
M. Joseph Roberts 1 , Andrew Guenthner 1 , Geoffrey Lindsay 1 , Merle Elson 1 , Simin Feng 1
1 , NAVAIR NAWCWD, China Lake, California, United States
Show AbstractWe will present a study on the fabrication of multilayer alternating metal-dielectric (M-D-M-...) films. We have developed processes for production of laterally continuous gold or silver layers alternated with glassy functional polymer films in which the thickness is on the order of 40 nm and 100 nm respectively. Such films can be used to study physical phenomena associated with the coupled plasmon resonances [Feng, et al, Optics Express, 13, 4113 (2005)] and the resonant transmission in the forbidden bands [Laroche, et al. Phys. Rev. B 71, 155113 (2005)]. Such films may also find applications in photonic bandgap and other nanoplasmonic devices. Since the surface plasmon and evanescent coupling is a mean to propagate light inside nanocircuits, the investigation of the coupled surface plasmons in the multilayer structures provides us with fundamental knowledge for the future 3D integration of the nanocircuits. Furthermore, the metal is one of the essential components in the construction of negative index medium. When the thickness of each layer is much less than the wavelength, such multilayer structures become metamaterials. Our fabrication technique will be useful for designing negative index medium. The fabrication technique also allows design flexibility, for example, systems with regions of single M-D-M plane together with multilayer structures facilitating tunable multiple plasmon resonance wavelength response from a single system. Multiple plasmon wavelength resonance absorptions may be obtained from such systems. Functional polymer films enable further design flexibility and increase the number of applications of the fabricated devices.
5:15 PM - J2.7
Negative Permeability Due to Hyperfine Transition in Array of Thin Copper Wires.
Jian Qi Shen 1 , Sailing He 1
1 Centre for Optical and Electromagnetic Research,, Zhejiang University, Hangzhou China
Show AbstractIn the literature, negative electric permittivity was obtained by using the ALMWs structure (array of long metallicwires, e.g., Copper wires), which simulates the plasma behavior at microwave frequencies, and the artificial negativemagnetic permeability was built up by using small resonant metallic particles, e.g., the SRRs structure (split ringresonators) with very high magnetic polarizability. A combination of these two structures yields a left-handed medium.Here we suggest an alternative scheme to realize negative permeability in the structure of array of thin Copper wires:specifically, the magnetic-dipole allowed transition between some certain hyperfine-splitting levels (with transitionfrequency in microwave range) in Cooper atom may exhibit strong magnetic response and can lead to negativepermeability under proper conditions (related to the Cooper atomic concentration and operating frequency). Thusthe SRRs structure can be replaced by the above scheme to achieve negative permeability.This work is supported by the National Basic Research Program of China under Project No. 2004CB719800.
5:30 PM - **J2.8
Chemical Route Fabricated Magnetic Structure Exhibiting a Negative Permeability at Infrared Frequencies.
Xiaopeng Zhao 1 , Hui Liu 1
1 Applied Physics, Northwestern Polytechnical University, Xi An City, Shaanxi Province China
Show Abstract
Symposium Organizers
Shih-Yuan (SY) Wang Hewlett-Packard Laboratories
Nicholas Xuanlai Fang University of Illinois, Urbana-Champaign
Lars Thylen Royal Institute of Technology (KTH)
M. Saif Islam University of California-Davis
J3: Artificial Magnetic Materials and Other Novel Properties
Session Chairs
Wednesday AM, April 19, 2006
Room 3018 (Moscone West)
9:45 AM - J3.1
Controllable Helicity Reversal and Photon Polarization Evolution Inside a Photonic Resonant Vapor.
Fei Zhuang 1 , Jian Qi Shen 2
1 Department of Physics and Institute of Condensed Matter Physics, Hangzhou Teacher's College, Hangzhou 310012 China, 2 Centre for Optical and Electromagnetic Research,, Zhejiang University, Hangzhou 310058 China
Show Abstract10:00 AM - J3.2
Artificial Magnetism and Negative Refraction in Periodic and Random Photonic Crystals.
Didier Felbacq 1 , Guy Bouchitte 2
1 physics, university of montpellier , montpellier France, 2 mathematics, university of Toulon, La Garde France
Show AbstractWe present a theory of the artificial magnetic activity in dielectric photonic crystals. We show,by using a rigorous multiple-scale approach, that the presence of Mie resonances at large wavelengths induces a macroscopic magnetic polarization resulting in an effective permeability with anomalous dispersion. This proves that the medium behaves as if it had a negative permeability near the resonances. The results are extended to non-periodic structures and the possibility of having a negative refraction is investigated numerically.ReferencesD. Felbacq, G. Bouchitté, Theory of Mesoscopic Magnetism in Photonic Crystals, Phys. Rev. Lett. 94, 183902 (2005).D. Felbacq, G. Bouchitté, Negative refraction in periodic and random photonic crystals, New J. Phys. 7 (2005) 159.
10:15 AM - **J3.3
Negative-Index Metamaterials: Going Optical
Vladimir Shalaev 1
1 School of Electric and Computer Engineering, Purdue University, West Lafayette, Indiana, United States
Show AbstractIn this presentation we'll consider various approaches for developing optical negative-index materials(ONIMS). New ONIM-based devices and their applications will be discussed.
10:45 AM - J3.4
Negative Refractions in Uniaxially Anisotropic Chiral Media.
Cheng Qiang 1 , Cui Jun 1
1 Radio Department, Center for Computational Electromagnetics and the State Key Laboratory of Millimeter Waves, Southeast University, Nanjing, Jiangsu, China
Show AbstractWe investigate the reflection and refraction properties in uniaxially anisotropic chiral media, where the chirality appears only in one direction and the host medium can be either anisotropic dielectric or anisotropic electric plasma. When a plane wave is incident from free space to the uniaxially chiral media, two eigenwaves are generated with different group and phase velocities, which we call as p+ and p- waves. We have shown that the reflection and refraction properties are closely related to the dispersion relation of the chiral media, and the two eigenwaves behave quite differently with respect to the medium parameters. If the host medium is an uniaxially anisotropic dielectric, we demonstrate that the p+ wave experiences positive phase and group refractions, and the p- wave experiences a positive phase refraction and a negative group refraction at the medium interface when the chirality becomes relatively large. If the host medium is a uniaxially electric plasma whose permittivity component in the chiral direction is negative, we show that the negative group refraction is always supported. If the host medium is a uniaxially electric plasma whose permittivity component perpendicular to the chiral direction is negative, however, several cases with negative refractions may occur. The reflection and refraction properties of a plane wave impinging on the interface of the uniaxially anisotropic chiral have also been investigated in details. We prove that such properties are closely related to the dispersion relation of the chiral medium, and the two eigenpolarizations behave differently with respect to the incident angles. In some cases, only one of the two eigenwaves can be supported and transmitted in the chiral medium. The Brewster angles have also been analyzed and computed numerically.
11:00 AM - J3: ArtMag
BREAK
11:30 AM - **J3.5
Fundamentals of Negative Refraction and Negative Index Materials.
David Smith 1
1 Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina, United States
Show Abstract12:00 PM - **J3.6
Experimental Demonstration of Subwavelength Imaging by Left-handed Composite Materials and Negative Refracting Photonic Crystals.
Ekmel Ozbay 1
1 Nanotechnology Research Center, Bilkent University, Bilkent, Ankara Turkey
Show AbstractWe review the studies conducted in our group concerning electromagnetic properties of metamaterials and photonic crystals with negative effective index of refraction. In particular, we demonstrate the true left handed and subwavelength imaging properties of composite metamaterials by analyzing the electric and magnetic response of the material components systematically. The negative refraction, subwavelength focusing, and flat lens phenomena using 2D dielectric photonic crystals are also presented.
12:30 PM - J3.7
Symmetric and Antisymmetric Modes of Electromagnetic Resonators
Yongmin Liu 1 , Nickolas Fang 2 , Dongmin Wu 1 , Cheng Sun 1 , Xiang Zhang 1
1 , NSF Nanoscale Science and Engineering Center (NSEC), 5130 EtcheVerry Hall, University of California, Berkeley, Berkeley, California, United States, 2 Dept. of Mechanical & Industrial Engineering, University of Illinois,Urbana-Champaign, Urbana, Illinois, United States
Show AbstractSplit-ring resonators (SRRs) are considered as the key building block in designing negative index metamaterials (NIMs), because SRRs can provide negative magnetic permeability which is not obtainable in natural materials. However, the electric resonance associated with SRRs has been overlooked in previous studies. In this paper, we report the detailed investigation of a new resonator structure that exhibits pronounced magnetic as well as electric responses. The unit cell of this new resonator consists of paired split-ring structures. Our numerical simulation indicates the existence of both symmetric and antisymmetric modes at different resonating conditions. The detailed analysis of the simulation results suggests that the symmetric mode is due to the magnetic coupling to resonators, in which the effective permeability could be negative. On the other hand, the antisymmetric mode originating from strong electric coupling is believed to contribute to negative effective permittivity. The results would provide a new pathway to design negative index materials using split-ring resonators.
12:45 PM - J3.8
Negative Permeability of Single-ring Split Ring Resonator in the Visible Light Frequency Region.
Takuo Tanaka 1 , Atsushi Ishikawa 1 2 , Satoshi Kawata 1 2
1 Nanophotonics Lab., RIKEN, Wako, Saitama, Japan, 2 Department of Applied Physics, Osaka University, Suita, Osaka, Japan
Show AbstractNegative permeability of single-ring split ring resonator (s-SRR) is theoretically investigated in the visible light frequency region[1, 2]. To estimate magnetic responses of conductive elements precisely, we determined internal impedance by considering the delay of the current inside the metal structure. The increase of the surface resistivity, which is the real part of the internal impedance, results in the decrease of the resonator’s Q-value. This means the degradation of the tunable range of the permeability. The increase of the internal reactance results in the reduction of the resonant frequency. In our calculation, the surface resistivity saturates at the inherent frequency of each metal as the frequency increases. On the silver case, the saturation value is 0.4 Ω and this value is remarkably smaller than that of gold and copper. On the other hand, the internal reactance is increasing as the frequency increases independently of metal. We concluded that the internal reactance is dominant factor to realize the negative magnetic permeability in the optical frequency region. We also show the frequency dependence of the magnetic permeability of s-SRRs. In the case of s-SRRs made of copper, the minimum value of the magnetic permeability becomes positive value at 550 THz even in the high filling factor condition (11%). In the case of s-SRRs made of gold, only on the filling factor was 11%, the minimum value of the magnetic permeability takes negative value in the entire visible range. On the other hand, the silver SRR exhibits negative magnetic permeability in the visible range even under the low filling factor condition of 3%. Moreover, we concluded that reducing the geometrical capacitance and using silver for SRR are necessary to realize the negative magnetic permeability in the visible light range.[1] A. Ishikawa, T. Tanaka, and S. Kawata, Phys. Rev. Lett. (accepted for publication).[2] A. Ishikawa, T. Tanaka, and S. Kawata, Opt. Commun. (in printing).
J4: Imaging Applications of NIMS
Session Chairs
Wednesday PM, April 19, 2006
Room 3018 (Moscone West)
2:30 PM - J4.1
Realization of Optical Superlens Imaging Below the Diffraction Limit.
Hyesog Lee 1 , Yi Xiong 1 , Nicholas Fang 2 , Werayut Srituravanich 1 , Stephane Durant 1 , Muralidhar Ambati 1 , Cheng Sun 1 , Xiang Zhang 1
1 , University of Californica at Berkeley, Berkeley, California, United States, 2 , University of Illinois at Urbana-Champaign, Urbana, Illinois, United States
Show AbstractRecently, the concept of superlensing has received considerable attention for its unique ability to produce images below the diffraction limit. The theoretical study has predicted a “super-lens” made of materials with negative permittivity and/or permeability, is capable of resolving features much smaller than the working wavelength and a near perfect image can be obtained through the restoration of lost evanescent waves. We have already demonstrated in our earlier work that a 60nm half-pitch object can indeed be resolved with the implementation of a silver superlens with 365nm incident light with λ/6 resolution, which is well below the diffraction limit. In order to further support the imaging ability of our silver superlens, a two-dimensional arbitrary object with 40nm line width was also imaged. In this report, we present experimental and theoretical investigations of optical superlensing through a thin silver slab. In addition, a new superlens imaging result is presented which shows the image of a 50nm half-pitch object at λ/7 resolution.
2:45 PM - **J4.2
Some Recent Studies of Negative Index Meta-materials and Related Structures.
Sailing He 1
1 , Joint Research Center of Photonics of the Royal Institute of Technology (Sweden) and Zhejiang University, Hangzhou China
Show AbstractNegative refraction, first introduced in a left-handed material (LHM) which exhibits a negative index of refraction due to simultaneously negative permittivity and permeability, has recently attracted much attention due to the enormous implication of e.g. perfect or super lens (beating the diffraction limit). Artificial metamaterials with negative refraction index (or left-handed or backward wave behaviors) can give extraordinary physical properties and functionality unattainable with naturally-existing materials. LHMs in the microwave region have been fabricated experimentally, and the impact would be much larger at optical frequencies. Many nano-structured meta-materials with negative refraction have been proposed. A chiral way to the realization of backward waves at optical frequencies has also been suggested and studied recently.Some of our recent results on these metamaterials will be presented. For multi-layered structures composite of right-handed materials (RHMs) and left-handed materials (LHMs), slow propagation (or even trapping) of light is studied as well as some unusual filtering properties. Surface polaritons related to a chiral medium which supports backward waves are also studied, and used to slow down the light effectively. Some complicated structures or devices based on negative index metamaterials will also be presented.
3:15 PM - **J4.3
Electromagnetic "Edge-Modes" at Interfaces: their Role in Pendry's "Superlens" and in "One-Way Waveguides".
F. D. M. Haldane 1
1 Department of Physics, Princeton University, Princeton, New Jersey, United States
Show AbstractBoundaries between media that are "topologically distinct" (i.e., which cannot be continuously transformed into each other without passing though a singularity) typically support inevitably-present degees of freedom that are localized at the interface, and which are related to the essential difference between the media on either side of the interface. Such phenomena are usefully-discussed from a general condensed-matter-physics viewpoint, rather than from a Maxwell-equation-specific viewpoint. A celebrated example in condensed matter physics is the quantum Hall effect (QHE) edge states. In photonics, the surface polariton modes at the boundary between "normal" and NIM media are an example (cond-mat/0206420), as are the undirectional QHE-like modes at boundaries between photonic crystals with non-trivial Chern number derived from a Faraday effect (the "one-way waveguide", cond-mat/0503588). Pendry's "superlens" effect is a formal property of simplified model constitutive relations that allow "fine-tuning" of parameters to produce a completely dispersionless degenerate pair of surface polariton modes with vanishing group velocity at all (surface) wavenumbers. The subwavelength features of the "superlens" image are assembled at the focal point by resonance between the surface modes on opposite faces of the NIM slab, and "perfect lensing" is crucially-dependent on a total absence of dispersion in the fine-tuned model. The completely-local constitutive relations that allow such "fine-tuning" idealize the NIM medium as homogeneous, linear, lossless, isotropic, and structureless. The "superlens" resolution is limited by departures from these idealizations; even if dissipation (lossiness) that inevitably broadens surface modes can be controlled, the microscopic structure of NIM metamaterials produces an absolute limit to resolution. NIM phenomena are inherently interface effects: there is nothing very "exotic" about the NIM medium itself: the new physics is associated with the boundary between media with opposite-sign refractive index (negative relative refractive index). Metamaterials provide the possibility of dramatic engineering of such interfaces. Another exotic possibility for metamaterial engineering is the "one-way waveguide", a direct transcription of quantum-Hall edge states into photonics. In this case, light is confined in the plane of a periodic 2D metamaterial with a Faraday axis normal to the plane, and there is a 1D domain wall separating 2D regions with opposite-direction Faraday axes. It is possible in principle to engineer the metamaterial to have a photonic band gap induced by the Faraday effect; in that case the interface supports modes which flow in one-direction only along the domain wall, with no possibility of back-reflection at bends or imperfections.(This work was supported in part by NSF DMR02-13706 at Princeton Center for Complex Materials).
3:45 PM - J4.4
Strong Effect of Surfaces on Resolution Limit of Negative-index "superlens."
A Bratkovsky 1 , A Cano 2 , A Levanyuk 2
1 , Hewlett-Packard Laboratories, Palo Alto, California, United States, 2 , University Autonoma, Madrid Spain
Show AbstractWe show that a combined effect of surfaces and microscopic structure strongly reduces a resolution limit of an (otherwise perfect) negative-index slab. This effect is illustrated in a model of a sandwich of negative-index materials. As an additional result, we find that the surface electromagnetic modes may have a gap in a spectrum. The gap prevents a resonance between these modes and an incident radiation, and that can make a "subwavelength" imaging less sensitive to losses.
4:30 PM - **J4.5
Phononic Nanophotonics: Sub-wavelength Imaging andExtraordinary Transmission in Mid-infrared Using Silicon Carbide Films.
Gennady Shvets 1 , Dmitry Korobkin 1 , Yaroslav Urzhumov 1
1 Physics, The University of Texas at Austin, Austin, Texas, United States
Show Abstract5:00 PM - **J4.6
Negative Refraction And Focusing Using Transmission-Line Metamaterials.
George Eleftheriades 1
1 Dept of Electrical and Computer Engineering, University of Toronto, Toronto, Ontario, Canada
Show AbstractIn this presentation we will discuss recent developments in transmission-line metamaterials. Transmission-line metamaterials are realized by periodically loading a host transmission-line (TL) medium (e.g. microstrip) with passive elements (inductors and capacitors). For example, a medium exhibiting a negative refractive index (NRI) can be realized by loading a transmission-line network with series capacitors and shunt inductors. The key observation is that there is a correspondence between negative permittivity and a shunt inductor (L), as well as between negative permeability and a series capacitor (C). This allows the synthesis of artificial media (metamaterials) with a negative permittivity and a negative permeability, and hence a negative refractive index. When the unit cell dimension d is much smaller than a guided wavelength, the array can be regarded as a homogeneous effective medium. In these structures, the loading lumped elements could be realized either in chip or in printed form. A major advantage of these NRI-TL media is that they exhibit a broad bandwidth over which the refractive index remains negative. Experimental and simulation results of negative refraction, focusing and sub-diffraction focusing will be presented based on these transmission-line metamaterials. Both planar as well as volumetric NRI-TL media will be presented. Furthermore, implementations using discrete loading elements, as well as fully printed ones, will be shown. Harnessing these metamaterial implementations, several devices and applications will be demonstrated. Finally, possible realizations of the transmission-line metamaterials at infrared and optical frequencies will be discussed.
5:30 PM - J4.7
Molecular Scale Imaging with a Multilayer Superlens.
Pratik Chaturvedi 1 , Nicholas Fang 1
1 Mechanical & Industrial Engineering, Univ. of Illinois, Urbana-Champaign, Urbana, Illinois, United States
Show Abstract Recent theory[1] suggested a thin negative index film should function as a “superlens”, providing image detail with resolution beyond the diffraction limit—a limitation to which all positive index optics are subject. The superlens allows the recovery of evanescent waves in the image via the excitation of surface plasmons. It has been demonstrated experimentally[2] that a silver superlens allows to resolve features well below the working wavelength. Resolution as high as 60 nanometer (λ/6) half-pitch has been achieved. This unique class of superlens will enable parallel imaging and nanofabrication in a single snapshot, a feat that are not yet available with other nanoscale imaging techniques such as atomic force microscope or scanning electron microscope. In this paper, we explore the possibility of further refining the image resolution using a multilayer superlens[3]. Using a stable transfer matrix scheme, our numerical calculations show an ultimate imaging resolution of λ/26. This is made possible using alternating stacks of alumina (Al2O3) and silver (Ag) layers to enhance a broad spectrum of evanescent waves via surface plasmon modes. Furthermore, we present the effect of alterations in number of layers and thickness to the image transfer function. With optimized design of multilayer superlens (working wavelength of 387.5nm), our study indicates the feasibility of resolving features of 15nm and below. However, our tolerance analysis indicates that a 380 nm commercial light source would degrade the imaging resolution to about 19nm, but still better than prevalent imaging techniques. Preliminary experiments are ongoing to demonstrate the molecular scale imaging resolution. The development of potential low-loss and high resolution superlens opens the door to exciting applications in nanoscale optical metrology and nanomanufacturing. REFERENCES1.Pendry, J. B., “Negative refraction makes a perfect lens”, Phys. Rev. Lett., 85(2000)3966.2.Fang, N.; Lee, H.; Sun, C.; Zhang, X., “Sub-diffraction-limited optical imaging with a silver superlens”, Science, 308(2005)534. 3.Ramakrishna, S. A.; Pendry, J. B., “Optical gain removes absorption and increases resolution in a near-field lens”, Phys. Rev., B67(2003)201101.
Symposium Organizers
Shih-Yuan (SY) Wang Hewlett-Packard Laboratories
Nicholas Xuanlai Fang University of Illinois, Urbana-Champaign
Lars Thylen Royal Institute of Technology (KTH)
M. Saif Islam University of California-Davis
J5: Emerging Applications of NIMs
Session Chairs
Thursday AM, April 20, 2006
Room 3018 (Moscone West)
9:45 AM - J5.1
Negative Index Imaging by Si-Based 2D Photonic Crystal Structures
Won Park 1 , Ethan Schonbrun 1 , Qi Wu 1 , Yonghao Cui 2 , Mark Tinker 2 , Jeong-Bong Lee 2 , Tsuyoshi Yamashita 3 , Chris Summers 3
1 Electrical & Computer Engineering, University of Colorado, Boulder, Colorado, United States, 2 Electrical Engineering, University of Texas at Dallas, Richardson, Texas, United States, 3 Materials Science & Engineering, Georgia Institute of Technology, Atlanta, Georgia, United States
Show AbstractWe report theoretical modeling and experimental demonstration of negative index imaging by Si-based photonic crystal (PC) structures in the near-infrared frequency region. PCs provide a unique mechanism for negative index imaging and evanescent wave amplification in which the incident waves are coupled to the slab resonance instead of the surface resonance used in most other negative index materials. Since the slab resonance is not evanescent within the PC slab, it is generally easier to achieve evanescent wave amplification. Furthermore, PCs provide powerful design flexibility due to the linear scalability of Maxwell’s equations. Also, PCs are generally purely dielectric system and thus exhibit lower loss at optical frequencies.First, we designed a PC that exhibits negative index imaging for TM and spatial dispersion for TE polarizations. The effective index for TM light is -1 near λ = 1.5μm so that the PC is index-matched to air, our incident medium. Also we chose the operating wavelength to be slightly below the air light line so that out-of-plane loss is minimized. We then fabricated input waveguide with a specially designed tip so that it would produce incident fields similar to a point source. The output waveguides were 500nm wide photonic wires to directly probe the image plane. The out-of-plane scattering image captured by an infrared camera clearly showed focusing and dispersive behavior for TM and TE polarizations, respectively. Also, TM light illuminated only one output waveguide whereas TM light illuminated 6 waveguides simultaneously. This directly demonstrated the formation of an image with a lateral size less than the vacuum wavelength.We also designed tunable negative index lens based on mechanically tunable PC, which is comprised of a periodic array of high index material in a low index flexible polymer film. Tunability is achieved by applying mechanical force with MEMS actuators. The photonic band structure is extremely sensitive to mechanical stress, consequently producing tunability much greater than that achievable with liquid crystal. We first theoretically investigated the negative index imaging by a PC made of Si nanorods embedded in a polyimide film. The focusing properties were found to be a strong function of both frequency and mechanical stress. With uniaxial strain of up to 10%, we could achieve identical focusing characteristics for a bandwidth of 30%, which corresponds to a width of 500nm at 1.5μm. For experimental demonstration, we fabricated Si nanorods embedded in polyimide thin film. We also designed input waveguides for both plane-wave-like and point-source-like incident fields. The out-of-plane scattering images clearly showed both the negative refraction of a plane wave and focusing of a diverging source at 1.5μm. This new concept of tunable PC allows real-time, dynamic control of photonic band structure, thereby greatly expanding the utility of PCs and enabling novel nanophotonic systems.
10:00 AM - J5.2
Chemical Route Fabricated Magnetic Structure Exhibiting a Negative Permeability at Infrared Frequencies.
Xiaopeng Zhao 1 , Hui Liu 1
1 Applied Physics, Northwestern Polytechnical University, Xi An City, Shaanxi Province China
Show Abstract10:15 AM - **J5.3
DNG, SNG, and ENZ Materials for Optical Nanocircuits and Transparency.
Nader Engheta 1 , Andrea Alu 1 , Mario Silveirinha 1 , Alessandro Salandrino 1
1 Electrical and Systems Engineering Department, University of Pennsylvania, Philadelphia, Pennsylvania, United States
Show AbstractRecent years have witnessed a growing interest and activity in the area of materials with negative or low-valued permittivity and/or permeability, due to the unusual features in their interaction with electromagnetic and optical waves. These materials include double-negative (DNG) media, where both parameters have negative real parts, single-negative materials (SNG) with only one parameter having negative real part, and epsilon-near-zero (ENZ) materials in which the relative permittivity is near zero. Each of these classes of materials exhibits interesting characteristics which can lead to useful potential applications.We have been investigating theoretically various features of these media and have been developing some of the fundamental concepts and theories of wave interaction with a variety of structures and systems involving these material media. These include ultracompact cavity resonators and waveguides, anomalous scattering from sub-wavelength spherical and cylindrical multilayered structures containing such media, transparency and low-observability using plasmonic covers, the concept of nanocircuit lumped elements at optical frequencies, negative magnetic response in IR and visible domains, and higher directivity for compact radiators and sensors, to name a few. In this talk, we will give an overview of two of the areas of our ongoing research, namely, the role of DNG, SNG, and ENZ materials in constructing nano-scale circuit elements at optical frequencies, and the possibility of using these materials as covers for reducing the total scattering cross section of objects of certain sizes and shapes. In the problem of nanocircuits, plasmonic nanoparticles behave as nanoinductors, while the non-plasmonic nanostructures play the role of nanocapacitors. By arranging these nanostructures in a variety of different ways, one can form more complex circuits and systems at optical frequencies. As for the transparency phenomenon, the presence of a cover with ENZ or SNG materials can provide reduction or cancellation of dipolar terms in scattering, resulting in reduction of scattering signature of an object.In this presentation, we will discuss some of our theoretical results along with the physical insights into these findings.
11:00 AM - J5: EmerApp
BREAK
11:30 AM - **J5.5
Fabrication of Optical Negative Index Meta-structure at sub-10 micron using Nanoimprint Lithography.
Wei Wu 1 , Zhaoning Yu 1 , Yongmin Liu 2 , Pratik Chaturvedi 3 , Evgenia Kim 2 , Alex Bratkovski 2 , Ekaterina Ponizovskaya 1 , Nick Fang 3 , Xiang Zhang 2 , S. Y. Wang 1 , R. S. Williams 1
1 Quantum Science Research, Hewlett-Packard Laboratories, Palo Alto, California, United States, 2 , University of California-Berkeley, Berkeley, California, United States, 3 , University of Illinois, Urbana-Champaing, Urbana-Champaign, Illinois, United States
Show AbstractThursday, April 20New Abstract10:30 am *J5.5Fabrication of Optical Negative Index Meta-structure at sub-10 micron using Nanoimprint Lithography. Wei WuWei Wu1*, Zhaoning Yu1, Yongmin Liu2, Pratik Chaturvedi3, Evgenia Kim2, Alex Bratkovski2, Ekaterina Ponizovskaya1, Nick Fang3, Xiang Zhang2 and S.Y. Wang1, R. S. Williams11Quantum Science Research, HP Labs2University of California, Berkeley3University of Illinois, Urbana-ChampaignNegative index meta-materials (NIM) that exhibit unique and unprecedented properties have recently attracted worldwide attention. It has opened doors to new avenues in nano-photonics and optical integration.1-7 However, how to fabricate these meta-structures with high-precision, high-throughput and low-cost remains a challenge especially for short working wavelengths (i.e. infrared). Here, We report the development of an approach to fabricating optical meta-structures operating at sub-10 micron infrared range using nanoimprint lithography (NIL)8. The metal meta-structure consists of a large array (100umX1mm) of L-shaped resonators (LSRs). The smallest dimension is 45 nm with better than 10 nm critical dimension control. Metallic LSRs are chosen because they adapt to a four-fold rotational symmetry and the excitation of the magnetic dipole moment is more isotropic and insensitive to the azimuth angle of the electric field. More details in design, fabrication process flow, theoretical studies and experiment data will be presented.1 V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).2 A. Yariv and P. Yeh, Optical Waves in Crystals (1984).3 J. B. Pendry, Physical Review Letters 85, 3966 (2000).4 D. R. Smith, W. J. Padilla, D. C. Vier, et al., Physical Review Letters 84, 4184 (2000).5 R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).6 N. Fang, Z. Liu, T.-J. Yen, et al., Optics Express 11, 682 (2003).7 N. Fang, Z. Liu, T.-J. Yen, et al., Applied Physics A: Materials Science and Processing 80, 1315 (2005).8 S. Y. Chou, P. R. Krauss, and P. J. Renstrom, Journal of Vacuum Science & Technology B 14, 4129 (1996).
12:00 PM - **J5.6
Sub-wavelength Imaging through Metallic Nanorod Array.
Atsushi Ono 1 , Jun-ichi Kato 1 , Satoshi Kawata 1
1 Nanophotonics Lab., RIKEN, Wako, Saitama, Japan
Show AbstractNegative index material is expected to exhibit interesting optical properties. Especially, superlens effect, which is predicted by John B. Pendry in 2000, is very attractive to overcome the diffraction limit in optical imaging [1]. Although there is no negative index material in nature, Pendry numerically suggested that several metals, only dielectric constant is negative at optical frequencies, behave like a superlens under the electrostatic limit and for the p-polarized fields. X. Zhang experimentally demonstrated this superlens effect by constructing nanolithography system with silver thin film in 2005 [2]. In this presentation, we newly propose a sub-wavelength imaging system at optical frequency regime in an array of metallic nanorods [3]. The near-field components of dipole sources were plasmonically transferred through the rod array to reproduce the image of the dipoles in the other side. We calculated the field distribution at the different planes of imaging process using the finite-difference time-domain (FDTD) algorithm and found that the spatial resolution was 40 nm, which was much beyond the diffraction-limit and was limited by the array pitch. The typical configuration is a hexagonal arrangement with 40 nm periodicity of silver rods of 50 nm height and 20 nm diameter. The image formation highly depends on the coherence and the polarization of the dipole sources, array pitch, and the source-array distance. The principle of our near-field imaging is based on the longitudinal resonance of the localized surface plasmon along a metallic nanorod. The spectral responses of the device are also investigated.References[1] J. B. Pendry, Phys. Rev. Lett. 85, 18 (2000)[2] N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005)[3] A. Ono, J. Kato, and S. Kawata, Phys. Rev. Lett., in press.
12:30 PM - J5.7
Veselago Lens with Gain.
E Ponizovskaya 1 , A Bratkovsky 1
1 , Hewlett-Packard Laboratories, Palo Alto, California, United States
Show Abstract12:45 PM - J5.8
Optical Devices Using Negative Refraction in Photonic Crystals.
Vito Mocella 1 , Principia Dardano 1 2 , Luigi Moretti 1 3 , Ivo Rendina 1
1 IMM - Unita' di Napoli, CNR, Napoli Italy, 2 Dip. Scienze Fisiche, Universita' Federico II, Napoli Italy, 3 DIMET, Universita' Mediterranea, Reggio Calabria Italy
Show AbstractPhotonic crystals (PhCs) represent an alternative way to achieve negative refraction with respect to metamaterials, where the inhomogeneities introduced are much smaller than the wavelength of the incoming radiation. Indeed, PhCs can show a negative refraction effect, that is, the light passing through them, in particular conditions, experience a different output direction with respect to the usual one. On the other hand, introducing a "macroscopic" dielectric function that describes the global behavior of a PhC structure, this last can be seen as a periodical spatial dispersive medium. In such a medium, many monochromatic plane waves of the same frequency can propagate with different phase and group velocities. This is the origin of a number of particular features exhibited by dispersive media, and in particular this is the origin of the Pendellösung phenomenon, due to the phase modulation between coexisting plane wave components. Using such interference phenomenon it is possible to obtain outgoing optical intensities either in positive or in negative directions, by modulating the PhC thickness [1].In this communication, we show how to make use of the interference to control the light propagation at the exit surface of a PhC. The refraction state is a function of the index contrast, of the PhC thickness, of the light incidence angle and polarization. By suitably using these properties it is possible to realize very simple and efficient integrated optical devices. In particular, we will present new results concerning the operation of a polarizing beam splitter [2], where the TM polarization is refracted in positive direction whereas the TE component is negatively refracted. The device, based on an air-holes square-lattice silicon PhC, shows a very high efficiency (90 %) in the broad wavelength range of major interest in the infrared optical communications (all the C-band). With the same original approach described in [2], starting from a design combining equifrequency surfaces and band diagrams, we also propose a new class of wide-angle thermally-controlled optical switches based on the negative refraction effect in PhCs, making use of the large thermo-optic effect present in silicon [3].[1] V. Mocella, "Negative refraction in Photonic Crystals: thickness dependence and Pendellösung phenomenon.", Opt. Express 13, 1361-1367 (2005).[2] V. Mocella, P. Dardano, L. Moretti, and I. Rendina, "A polarizing beam splitter using negative refraction of photonic crystals," Opt. Express 13, 7699-7707 (2005).[3] F. G. Della Corte, M. Esposito Montefusco, L. Moretti, I. Rendina, G. Cocorullo, “Temperature dependence analysis of the thermo-optic effect in silicon by single and double oscillator models”, J. of Applied Phys., 88(12), 7115-7119 (2000).