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| {{Condensed matter physics}}
| | Entirely those pessimistic all but the futurity of traditional books shouldn't lather scarce so far. A new�Church bench field constitute that Americans get been meter reading more ebooks, merely they haven't entirely replaced impress books. The per centum of American adults World Health Organization scan ebooks has adult to 28 percent, up from 23 percentage at the closing of 2012. It's a amazingly humbled number, considering�some other Church bench study launch that 46 pct of hoi polloi in real time ain a pill or e-reader of about genial. |
| '''Surface plasmon polaritons''' (SPPs), are [[infrared]] or [[visible spectrum|visible]]-frequency [[electromagnetic waves]], which travel along a [[metal]]-[[dielectric]] or metal-air interface. The term "surface plasmon polariton" explains that the wave involves both charge motion in the metal ("[[surface plasmon]]") and electromagnetic waves in the air or dielectric ("[[polariton]]") | |
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| They are a type of [[surface wave]], guided along the interface in much the same way that light can be guided by an optical fiber. SPPs are shorter in wavelength than the incident light (photons).<ref name=nist-plasmo-mm/> Hence, SPPs can have tighter [[laser|spatial confinement]] and higher [[field intensity|local field intensity]].<ref name=nist-plasmo-mm/> Perpendicular to the interface, they have subwavelength-scale confinement. An SPP will propagate along the interface until its energy is lost either to either absorption in the metal or scattering into other directions (such as into free space).
| | What's even out to a greater extent noteworthy is that ebooks appear to be supplementing fixture books. Near seven in ten adults reported that they study black and white books and ebooks at the same time. Entirely 4 per centum of the great unwashed reported to be "ebook only" readers. VII in decade adults learn print and ebooks in concert This is a planetary house that only because the great unwashed are haste to bribe tablets and e-readers doesn't intend they wish alternate to meter reading lone on those physical science devices. |
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| Application of SPPs enables [[subwavelength optics]] in microscopy and [[lithography]] beyond the [[diffraction limit]]. It also enables the first steady-state micro-mechanical measurement of a fundamental property of light itself: the momentum of a photon in a dielectric medium. Other applications are [[photonic]] data storage, light generation, and bio-photonics.<ref name=nist-plasmo-mm>
| | Photographic print remains the cornerstone of Americans' reading material habits, and the canvas besides showed a gain for interpretation in world-wide.�Some 76 percent of adults take a al-Qur'an in approximately formatting in the shoemaker's last 12 months (issue that, spew videos and [http://Search.Huffingtonpost.com/search?q=Cyberspace&s_it=header_form_v1 Cyberspace] memes). Americans hush bang to understand ?? in fact, they've upright expanded how they do so to let in the scoop of both the print and extremity worlds.<br><br>If you loved this article therefore you would like to obtain more info pertaining to [http://batuiti.com/shop/iphone-5-5s-5c/spigen-sgp-slim-armor-case-for-iphone-5-5s-5g-cover/ slim armor] nicely visit our internet site. |
| {{cite web
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| | last =NIST researchers
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| | first =Nanofabrication Research Group
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| | title =Three-Dimensional Plasmonic Metamaterials
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| | publisher =National Institute of Science and Technology
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| | date =
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| | url =http://www.nist.gov/cnst/nrg/3d_plasmonic_metamaterials.cfm
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| | accessdate =2011-02-15}}
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| *{{cite web
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| | last =NIST researchers
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| | first =Nanofabrication Research Group
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| | title =Opto-mechanical Devices for Measuring Nanoplasmonic Metamaterials
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| | publisher =National Institute of Science and Technology
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| | date =
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| | url =http://www.nist.gov/cnst/nrg/nanoplasmonic_lab.cfm
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| | accessdate =2011-02-15}}
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| *{{NIST-PD|article=Three-Dimensional Plasmonic Metamaterials |url=http://www.nist.gov/cnst/nrg/3d_plasmonic_metamaterials.cfm}}</ref><ref name=berkely-grin>
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| {{cite web
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| | last =Yarris
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| | first =Lynn
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| | title =GRIN Plasmonics...
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| | publisher =U.S. Department of Energy National Laboratory Operated by the University of California
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| | date =
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| | url =http://www.nist.gov/cnst/nrg/3d_plasmonic_metamaterials.cfm
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| | format =Online news release
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| | accessdate =2011-02-15}}</ref><ref name=w-barnes>
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| {{cite journal
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| |last1=Barnes|first1=William L.|last2=Dereux|first2=Alain|last3=Ebbesen|first3=Thomas W.|title=Surface plasmon subwavelength optics|doi=10.1038/nature01937|pmid=12917696|year=2003|page=824|issue=6950|volume=424
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| |journal=Nature
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| |url=http://www.nature.com/nature/journal/v424/n6950/pdf/nature01937.pdf
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| |bibcode = 2003Natur.424..824B }}
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| *{{cite journal |last1=Huidobro |first1=Paloma A. |last2=Nesterov |first2=Maxim L. |last3=Martín-Moreno |first3=Luis |last4=García-Vidal |first4=Francisco J. |doi=10.1021/nl100800c |title=Transformation Optics for Plasmonics |pmid=20465271 |year=2010 |page=1985 |issue=6 |volume=10 |journal=Nano Letters|url=http://www.uam.es/gruposinv/nanophot/Web_grupo_archivos/papers/huidobro_june2010.pdf|bibcode = 2010NanoL..10.1985H |arxiv = 1003.1154 }} Free PDF download for these peer reviewed articles.
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| *PDF from arxiv.org - [http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.1154v2.pdf Transformation Optics for Plasmonics]. 15 pages.</ref><ref name=nanoplasmonics/>
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| ==Excitation==
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| {{Multiple image|direction=vertical|align=right|image1=Prism_Coupler.png|image2=Grating_Coupler.png|width=300|caption1=Figure 1: (a) Kretschmann and (b) Otto configuration of an Attenuated Total Reflection setup for coupling surface plasmons. In both cases, the surface plasmon propagates along the metal/dielectric interface {{anchor|fig2anchor}}|caption2=Figure 2: Grating Coupler for Surface Plasmons. The wave vector is increased by the spatial frequency }}
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| SPPs can be excited by both electrons and photons. Excitation by electrons is created by firing electrons into the bulk of a metal. As the electrons scatter, energy is transferred into the bulk plasma. The component of the scattering vector parallel to the surface results in the formation of a surface plasmon polariton.<ref>{{cite journal|author=S.Zeng ''et al.''|title=Size dependence of Au NP-enhanced surface plasmon resonance based on differential phase measurement |journal=Sensors and Actuators B: Chemical |year=2012|doi=10.1016/j.snb.2012.09.073|last2=Yu|first2=Xia|last3=Law|first3=Wing-Cheung|last4=Zhang|first4=Yating|last5=Hu|first5=Rui|last6=Dinh|first6=Xuan-Quyen|last7=Ho|first7=Ho-Pui|last8=Yong|first8=Ken-Tye|volume=176|pages=1128 }}</ref>
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| If a free-space photon comes from air towards a smooth metal surface, it ''cannot'' excite an SPP at the metal-air interface. The reason is that if the photon and SPP have the same frequency, then they necessarily have different in-plane wavevectors. This incompatibility is analogous to the lack of transmission that occurs during [[total internal reflection]]. Analogously, an SPP on a smooth metal surface ''cannot'' lose energy to radiation into the dielectric (if the dielectric is uniform).
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| Nevertheless, coupling of photons into SPPs can be achieved using a coupling medium such as a [[Prism (optics)|prism]] or grating to match the photon and surface plasmon wave vectors. A prism can be positioned against a thin metal film in the Kretschmann configuration or very close to a metal surface in the Otto configuration (Figure 1). A grating coupler matches the wave vectors by increasing the parallel wave vector component by an amount related to the grating period (Figure 2). This method, while less frequently utilized, is critical to the theoretical understanding of the effect of surface [[surface roughness|roughness]]. Moreover, simple isolated surface defects such as a groove, a slit or a corrugation on an otherwise planar surface provides a mechanism by which free-space radiation and SPs can exchange energy and hence couple.
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| ==Dispersion relation==
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| [[File:Coordinates.png|thumb|Figure 3: Coordinate system for 2 material interface]]
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| The electric field of a propagating electromagnetic wave can be expressed
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| : <math>E= E_{0}\exp[i(k_{x} x + k_{z} z -\omega t)]\,</math>
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| where ''k'' is the [[wavenumber|wave number]] and ω is the frequency of the wave. By solving [[Maxwell's equations]] for the [[electromagnetic wave]] at an interface between two materials with relative [[Permitivity|dielectric functions]] ''ε<sub>1</sub>'' and ''ε<sub>2</sub>'' (see figure 3) with the appropriate continuity relation the boundary conditions are<ref name="Raether">{{cite book |last=Raether |first=Heinz |year=1988 |title=Surface Plasmons on Smooth and Rough Surfaces and on Gratings |location=New York |publisher=Springer-Verlag |isbn=3540173633| series=Springer Tracts in Modern Physics '''111'''}}</ref><ref name="M.G. Cottam">{{cite book |last=Cottam |first=Michael G. |year=1989 |title=Introduction to Surface and Superlattice Excitations |location=New York |publisher=Cambridge University Press |isbn=0750305886}}</ref>
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| :<math>\frac{k_{z1}}{\varepsilon_1} + \frac{k_{z2}}{\varepsilon_2} = 0 </math>
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| and
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| :<math>k_{x}^2+k_{zi}^2=\varepsilon_i \left(\frac{\omega}{c}\right)^2 \qquad i=1,2</math>
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| where ''c'' is the [[speed of light]] in a vacuum, and ''k<sub>x</sub>'' is same for both media at the interface for a surface wave. Solving these two equations, the dispersion relation for a wave propagating on the surface is
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| :<math>k_{x}=\frac{\omega}{c} \left(\frac{\varepsilon_1\varepsilon_2}{ \varepsilon_1+\varepsilon_2}\right)^{1/2}.</math>
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| [[File:Dispersion Relationship.png|thumb|Figure 4: Dispersion curve for surface plasmon polaritons. At low ''k'', the surface plasmon curve (red) approaches the photon curve (blue)]]
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| In the [[Free_electron_model#Dielectric_function_of_the_electron_gas|free electron model of an electron gas]], which neglects attenuation, the metallic dielectric function is<ref>{{cite book |last=Kittel |first=Charles |authorlink=Charles Kittel |year=1996 |title=Introduction to Solid State Physics |edition= 8th |location= Hoboken, NJ |publisher=John Wiley & Sons |isbn=0-471-41526-X}}</ref>
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| :<math>\varepsilon(\omega)=1-\frac{\omega_{P}^2}{\omega^2},</math>
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| where the bulk plasma frequency in SI units is
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| :<math>\omega_{P}=\sqrt{\frac{n e^2}{{\varepsilon_0}m^*}}</math>
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| where ''n'' is the electron density, ''e'' is the [[Electron charge|charge]] of the electron, ''m''<sup>*</sup> is the [[Effective mass (solid-state physics)|effective mass]] of the electron and <math>{\varepsilon_0}</math> is the permittivity of free-space. The [[Dispersion (optics)|dispersion]] relation is plotted in Figure 4. At low ''k'', the SPP behaves like a photon, but as ''k'' increases, the dispersion relation bends over and reaches an [[asymptotic limit]] called the "surface plasma frequency". Since the dispersion curve lies to the right of the light line, ''ω = k·c'', the SPP has a shorter wavelength than free-space radiation such that the out-of-plane component of the SPP wavevector is purely imaginary and exhibits evanescent decay. The surface plasma frequency is the asymptote of this curve, and is given by
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| :<math>\omega_{SP}=\omega_P/\sqrt{1+\varepsilon_2}.</math>
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| In the case of air, this result simplifies to
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| :<math>\omega_{SP}=\omega_P/\sqrt{2}.</math>
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| If we assume that ''ε<sub>2</sub>'' is real and ''ε<sub>2</sub>'' > 0, then it must be true that ''ε<sub>1</sub>'' < 0, a condition which is satisfied in metals. Electromagnetic waves passing through a metal experience damping due to Ohmic losses and electron-core interactions. These effects show up in as an imaginary component of the [[dielectric function]]. The dielectric function of a metal is expressed ε<sub>1</sub> = ε<sub>1</sub>' + i·ε<sub>1</sub>" where ε<sub>1</sub>' and ε<sub>1</sub>" are the real and imaginary parts of the dielectric function, respectively. Generally |ε<sub>1</sub>'| >> ε<sub>1</sub>" so the wavenumber can be expressed in terms of its real and imaginary components as<ref name="Raether"/>
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| :<math>k_{x}=k_{x}'+i k_{x}''=\left[\frac{\omega}{c} \left( \frac{\varepsilon_1' \varepsilon_2}{\varepsilon_1' + \varepsilon_2}\right)^{1/2}\right] + i \left[\frac{\omega}{c} \left( \frac{\varepsilon_1' \varepsilon_2}{\varepsilon_1' + \varepsilon_2}\right)^{3/2} \frac{\varepsilon_1''}{2(\varepsilon_1')^2}\right].</math>
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| The wave vector gives us insight into physically meaningful properties of the electromagnetic wave such as its spatial extent and coupling requirements for wave vector matching.
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| ==Propagation length and skin depth==
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| As an SPP propagates along the surface, it loses energy to the metal due to absorption. The intensity of the surface plasmon decays with the square of the [[electric field]], so at a distance ''x'', the intensity has decreased by a factor of exp[-2k<sub>x</sub>"x]. The propagation length is defined as the distance for the SPP intensity to decay by a factor of ''1/e''. This condition is satisfied at a length<ref name="Homola">{{cite book |last=Homola |first=Jirí |year=2006 |title=Surface Plasmon Resonance Based Sensors. Springer Series on Chemical Sensors and Biosensors, '''4'''|location=Berlin |publisher=Springer-Verlag|isbn=3-540-33918-3}}</ref>
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| :<math>L=\frac{1}{2 k_{x}''}.</math>
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| Likewise, the electric field falls off evanescently perpendicular to the metal surface. At low frequencies, the SPP penetration depth into the metal is commonly approximated using the [[skin depth]] formula. In the dielectric, the field will fall off far more slowly. The decay lengths in the metal and dielectric medium can be expressed as<ref name="Homola"/>
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| :<math>z_{i}=\frac{\lambda}{2 \pi} \left(\frac{|\varepsilon_1'|+\varepsilon_2}{\varepsilon_i^2} \right)^{1/2}</math>
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| where ''i'' indicates the medium of propagation. SPPs are very sensitive to slight perturbations within the skin depth and because of this, SPPs are often used to probe inhomogeneities of a surface.
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| ==Animations==
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| <gallery heights="220px" widths="400px" perrow="2">
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| File:SPP silver-air interface 370nm.gif|The [[electric field]] (E-field) of an SPP at the silver-air interface, at the frequency where the free-space wavelength is 370nm. The animation shows how the E-field varies over an optical cycle. The [[permittivity]] of silver at this frequency is {{nowrap|(-2.6 + 0.6i)}}. The picture is {{nowrap|(0.3 × 370nm)}} across horizontally; the SPP wavelength is much smaller than the free-space wavelength.
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| File:SPP silver-air interface 10um.gif|The E-field of an SPP at the silver-air interface, at a much lower frequency corresponding to a free-space wavelength of 10μm. At this frequency, the silver behaves approximately as a [[perfect electric conductor]], and the SPP is called a [[surface wave|Sommerfeld Zenneck wave]], with almost the same wavelength as the free-space wavelength. The permittivity of silver at this frequency is {{nowrap|(-2700 + 1400i)}}. The picture is {{nowrap|(0.6 × 10μm)}} across horizontally.
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| </gallery>
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| ==Experimental applications==
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| Nanofabricated systems that exploit SPPs demonstrate potential for designing and controlling the propagation of [[light]] in matter. In particular, SPPs can be used to channel light efficiently into [[nanometer]] scale volumes, leading to direct modification of [[Normal mode|resonate frequency dispersion]] properties (substantially shrinking the wavelength of light and the speed of light pulses for example), as well as field enhancements suitable for enabling strong interactions with [[nonlinear metamaterials|nonlinear materials]]. The resulting enhanced sensitivity of light to external parameters (for example, an applied electric field or the dielectric constant of an adsorbed molecular layer) shows great promise for applications in sensing and switching.
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| Current research is focused on the design, fabrication, and experimental characterization of novel components for measurement and communications based on nanoscale plasmonic effects. These devices include ultra-compact plasmonic interferometers for applications such as [[biosensing]], optical positioning and optical switching, as well as the individual building blocks (plasmon source, waveguide and detector) needed to integrate a high-bandwidth, infrared-frequency plasmonic communications link on a silicon chip.
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| In addition to building functional devices based on SPPs, it appears feasible to exploit the dispersion characteristics of SPPs traveling in confined metallo-dielectric spaces to create photonic materials with artificially tailored bulk optical characteristics, otherwise known as ''[[photonic metamaterials|metamaterials]]''.<ref name=nanoplasmonics>
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| {{cite web
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| | last =NIST researchers
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| | first =Nanofabrication Research Group
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| | title =Nanoplasmonics
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| | publisher =National Institute of Science and Technology
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| | date =
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| | url =http://www.nist.gov/cnst/nrg/nanoplasmonics.cfm
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| | format =Online
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| | accessdate =2011-02-15}}
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| *{{NIST-PD|article=Nanoplasmonics|url=http://www.nist.gov/cnst/nrg/nanoplasmonics.cfm}}</ref>
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| The excitation of SPPs is frequently used in an experimental technique known as [[surface plasmon resonance]] (SPR). In SPR, the maximum excitation of surface plasmons are detected by monitoring the reflected power from a prism coupler as a function of incident angle or [[wavelength]]. This technique can be used to observe [[nanometer]] changes in thickness, density fluctuations, or molecular absorption.
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| [[Surface plasmon]]-based circuits, including both SPPs and [[localized plasmon resonance]]s, have been proposed as a means of overcoming the size limitations of photonic circuits for use in high performance data processing nano devices.<ref>{{cite journal|doi=10.1126/science.1114849|title=Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions|year=2006|last1=Ozbay|first1=E.|journal=Science|volume=311|issue=5758|pages=189–93|pmid=16410515|bibcode = 2006Sci...311..189O }}</ref>
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| The ability to dynamically control the plasmonic properties of materials in these nano-devices is key to their development. A new approach that uses plasmon-plasmon interactions has been demonstrated recently. Here the bulk plasmon resonance is induced or suppressed to manipulate the propagation of light.<ref>{{cite journal|doi=10.1088/0957-4484/23/44/444004|title=Plasmon–plasmon interaction: Controlling light at nanoscale|year=2012|last1=Akimov|first1=Yu A|last2=Chu|first2=H S|journal=Nanotechnology|volume=23|issue=44|pages=444004|pmid=23080049}}</ref> This approach has been shown to have a high potential for nanoscale light manipulation and the development of a fully CMOS- compatible electro-optical plasmonic modulator.
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| CMOS compatible electro-optic plasmonic modulators will be key components in chip-scale photonic circuits.<ref>{{cite journal|author=Wenshan Cai, Justin S. White, and Mark L. Brongersma|doi=10.1021/nl902701b|title=Compact, High-Speed and Power-Efficient Electrooptic Plasmonic Modulators|year=2009|journal=Nano Letters|volume=9|issue=12|pages=4403–11|pmid=19827771|bibcode = 2009NanoL...9.4403C }}</ref>
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| In [[surface second harmonic generation]], the second harmonic signal is proportional to the square of the electric field. The electric field is stronger at the interface because of the surface plasmon resulting in a [[Nonlinear optics|non-linear optical effect]]. This larger signal is often exploited to produce a stronger second harmonic signal.<ref>{{cite journal|author=V. K. Valev|doi=10.1021/la302485c|title=Characterization of Nanostructured Plasmonic Surfaces with Second Harmonic Generation|year=2012|journal=Langmuir|volume=28|issue=44|pages=15454–15471}}</ref>
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| The wavelength and intensity of the plasmon-related absorption and emission peaks are affected by molecular adsorption that can be used in molecular sensors. For example, a fully operational prototype device detecting casein in milk has been fabricated. The device is based on monitoring changes in plasmon-related absorption of light by a gold layer.<ref>{{cite journal|doi=10.1016/j.stam.2006.12.010|format=free download pdf|title=A localized surface plasmon resonance based immunosensor for the detection of casein in milk|year=2007|author=Minh Hiep, Ha|journal=Science and Technology of Advanced Materials|volume=8|pages=331|last2=Endo|first2=Tatsuro|last3=Kerman|first3=Kagan|last4=Chikae|first4=Miyuki|last5=Kim|first5=Do-Kyun|last6=Yamamura|first6=Shohei|last7=Takamura|first7=Yuzuru|last8=Tamiya|first8=Eiichi|bibcode = 2007STAdM...8..331M|issue=4 }}</ref>
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| ==Effects of roughness==
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| In order to understand the effect of roughness on SPPs, it is beneficial to first understand how a SPP is coupled by a [[Diffraction grating|grating]] [[#fig2anchor|Figure2]]. When a photon is incident on a surface, the wave vector of the photon in the dielectric material is smaller than that of the SPP. In order for the photon to couple into a SPP, the wave vector must increase by <math>\Delta k = k_{SP}- k_{x, \text{photon}}</math>. The grating [[harmonics]] of a periodic grating provide additional momentum parallel to the supporting interface to match the terms.
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| :<math>k_{SPP}=k_{x, \text{photon}} \pm n\ k_\text{grating}=\frac{\omega}{c} \sin{\theta_0} \pm n \frac{2\pi}{a}</math> | |
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| where <math>k_\text{grating}</math> is the wave vector of the grating, <math>\theta_0</math> is the angle of incidence of the incoming photon, ''a'' is the grating period, and ''n'' is an integer.
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| Rough surfaces can be thought of as the [[Superposition principle|superposition]] of many gratings of different periodicities. Kretschmann proposed<ref name="Kretschmann1">{{de icon}} {{cite journal |last=Kretschmann |first=E. |date=April 1974 |title=Die Bestimmung der Oberflächenrauhigkeit dünner Schichten durch Messung der Winkelabhängigkeit der Streustrahlung von Oberflächenplasmaschwingungen |journal=[[Optics Communications]] |volume=10 |issue=4 |pages= 353–356 |doi=10.1016/0030-4018(74)90362-9|bibcode = 1974OptCo..10..353K }}</ref> that a statistical [[correlation function]] be defined for a rough surface
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| :<math>G(x,y)=\frac{1}{A}\int_A z(x',y')\ z(x'-x,y'-y)\, dx'\, dy'</math>
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| where <math>z(x,y)</math> is the height above the mean surface height at the position <math>(x,y)</math>, and <math>A</math> is the area of integration. Assuming that the statistical correlation function is [[Gaussian function|Gaussian]] of the form
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| :<math>G(x,y)=\delta^2\exp\left(-\frac{r^2}{\sigma^2}\right)</math>
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| where <math>\delta</math> is the [[root mean square]] height, <math>r</math> is the distance from the point <math>(x,y)</math>, and <math>\sigma</math> is the correlation length, then the [[Fourier transform]] of the correlation function is
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| :<math>|s(k_\text{surf})|^2=\frac{1}{4 \pi} \sigma^2 \delta^2 \exp \left( - \frac{\sigma^2 k_\text{surf}^2}{4}\right)</math>
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| where <math>s</math> is a measure of the amount of each [[spatial frequency]] <math> k_\text{surf}</math> which help couple photons into a surface plasmon.
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| If the surface only has one Fourier component of roughness (i.e. the surface profile is sinusoidal), then the <math>s</math> is discrete and exists only at <math>k=\frac{2\pi}{a}</math>, resulting in a single narrow set of angles for coupling. If the surface contains many Fourier components, then coupling becomes possible at multiple angles. For a random surface, <math>s</math> becomes continuous and the range of coupling angles broadens.
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| As stated earlier, SPPs are non-radiative. When a SPP travels along a rough surface, it usually becomes radiative due to scattering. The Surface Scattering Theory of light suggests that the scattered intensity <math>dI</math> per [[solid angle]] <math>d \Omega</math> per incident intensity <math>I_{0}</math> is<ref name="Kretschmann2">{{cite journal |last=Kretschmann |first=E. |year=1972 |title=The angular dependence and the polarisation of light emitted by surface plasmons on metals due to roughness |journal=Optics Communications |volume=5 |issue=5 |pages= 331–336 |doi=10.1016/0030-4018(72)90026-0 |bibcode = 1972OptCo...5..331K }}</ref>
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| :<math>\frac{dI}{ d \Omega\ I_{0}}=\frac{4 \sqrt{\varepsilon_{0}}}{\cos{\theta_0}}\frac{\pi^4}{\lambda^4}|t_{012}^p|^2 \ |W|^2 |s(k_\text{surf})|^2</math>
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| where <math>|W|^2</math> is the radiation pattern from a single [[dipole]] at the metal/dielectric interface. If surface plasmons are excited in the Kretschmann geometry and the scattered light is observed in the plane of incidence (Fig. 4), then the dipole function becomes
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| :<math>|W|^2=A(\theta,|\varepsilon_{1}|)\ \sin^2{\psi} \ [(1+\sin^2 \theta /|\varepsilon_1|)^{1/2} - \sin{\theta}]^2</math>
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| with
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| :<math> A(\theta,|\varepsilon_1|) = \frac{|\varepsilon_1|+1}{|\varepsilon_1|-1} \frac{4}{1+\tan{\theta}/| \varepsilon_1|}</math>
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| where <math>\psi</math> is the polarization angle and <math>\theta</math> is the angle from the ''z''-axis in the ''xz''-plane. Two important consequences come out of these equations. The first is that if <math>\psi=0</math> (s-polarization), then <math>|W|^2=0</math> and the scattered light <math>\frac{dI}{ d \Omega\ I_{0}}=0</math>. Secondly, the scattered light has a measurable profile which is readily correlated to the roughness. This topic is treated in greater detail in reference.<ref name="Kretschmann2"/>
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| ==See also==
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| * [[Surface plasmon]]
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| * [[Surface plasmon resonance]]
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| * [[Plasmonic lens]]
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| * [[Superlens]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| *{{cite journal|last1=Ebbesen|doi=10.1038/35570|first1=T. W.|last2=Lezec|first2=H. J.|last3=Ghaemi|first3=H. F.|last4=Thio|first4=T.|last5=Wolff|first5=P. A.|title=Extraordinary optical transmission through sub-wavelength hole arrays|year=1998|page=667|volume=391|journal=Nature|url=http://arep.med.harvard.edu/pdf/Ebbsen98.pdf|bibcode = 1998Natur.391..667E|issue=6668}}
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| *{{cite journal|last1=Hendry|doi=10.1103/PhysRevLett.100.123901|first1=E.|last2=Garcia-Vidal|first2=F.|last3=Martin-Moreno|first3=L.|last4=Rivas|first4=J.|last5=Bonn|first5=M.|last6=Hibbins|first6=A.|last7=Lockyear|first7=M.|title=Optical Control over Surface-Plasmon-Polariton-Assisted THz Transmission through a Slit Aperture|pmid=18517865|year=2008|page=123901|issue=12|volume=100|journal=Physical Review Letters |url=http://shannon.ex.ac.uk/research/emag/pubs/pdf/Hendry_PRL_2008.pdf|bibcode=2008PhRvL.100l3901H}} Free PDF download.
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| *{{cite journal|last1=Barnes|first1=William L.|last2=Dereux|first2=Alain|last3=Ebbesen|first3=Thomas W.|title=Surface plasmon subwavelength optics|doi=10.1038/nature01937|pmid=12917696|year=2003|pages=824|issue=6950|volume=424|journal=Nature
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| |url=http://access.ex.ac.uk/research/emag/wlb/nature_2003_424_824.pdf|bibcode = 2003Natur.424..824B }} Free PDF download.
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| *{{cite journal|last1 =Pitarke|first1 =J M|last2 =Silkin|first2 =V M|last3 =Chulkov|first3 =E V|last4 =Echenique|first4 =P M|doi =10.1088/0034-4885/70/1/R01|title =Theory of surface plasmons and surface-plasmon polaritons|year =2007|pages =1|volume =70|journal =Reports on Progress in Physics
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| |url =http://dipc.ehu.es/etxenike/admin/documentos/archivos/publicaciones/307RPP2007.pdf|arxiv = cond-mat/0611257 |bibcode = 2007RPPh...70....1P }} Free PDF download.
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| ==External links==
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| *{{Cite web
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| | last =White
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| | first =Justin
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| | authorlink =
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| | coauthors =
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| | title =Surface Plasmon Polaritons
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| | work =
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| | publisher =[[Stanford University]]. Physics department
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| | date =March 19, 2007
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| | url =http://large.stanford.edu/courses/2007/ap272/white1/
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| | format =Online
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| | doi =
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| | accessdate =}} "''Submitted as coursework for AP272. Winter 2007''".
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| | |
| {{DEFAULTSORT:Surface Plasmon Polaritons}}
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| [[Category:Optics]]
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| [[Category:Engineering]]
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| [[Category:Metamaterials]]
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| [[Category:Condensed matter physics]]
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