View source for Physics:Surface plasmon polariton

{{Short description|Electromagnetic waves that travel along an interface}}
{{Condensed matter physics}}
'''Surface plasmon polaritons''' ('''SPPs''') are [[Physics:Electromagnetic waves|electromagnetic waves]] that travel along a [[Chemistry:Metal|metal]]–[[Physics:Dielectric|dielectric]] or metal–air interface, practically in the [[Physics:Infrared|infrared]] or [[Physics:Visible spectrum|visible]]-frequency. The term "surface plasmon polariton" explains that the wave involves both charge motion in the metal ("[[Physics:Surface plasmon|surface plasmon]]") and electromagnetic waves in the air or dielectric ("[[Physics:Polariton|polariton]]").<ref name="nano-enhance-spp"/>

They are a type of [[Physics:Surface wave|surface wave]], guided along the interface in much the same way that light can be guided by an optical fiber. SPPs have a shorter wavelength than light in vacuum at the same frequency (photons).<ref name=nist-plasmo-mm/> Hence, SPPs can have a higher [[Physics:Photon#Relativistic energy and momentum|momentum]] and 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 absorption in the metal or scattering into other directions (such as into free space).

Application of SPPs enables subwavelength optics in microscopy and [[Engineering:Photolithography|photolithography]] beyond the [[Physics:Diffraction limit|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/><ref name=berkely-grin/><ref name=w-barnes/><ref name=nanoplasmonics/>

==Excitation==

{{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 }}
SPPs can be excited by both electrons and photons. Excitation by electrons is created by firing electrons into the bulk of a metal.<ref>{{Cite journal| last = Bashevoy | first =M.V.|author2=Jonsson, F. | author3=Krasavin, A.V. |author4=Zheludev, N.I. |author5=Chen Y. |author6=Stockman M.I. | title =Generation of traveling surface plasmon waves by free-electron impact | journal =Nano Letters| volume =6| year =2006| issue =6| pages =1113–5|  doi =10.1021/nl060941v | pmid =16771563| arxiv =physics/0604227| bibcode =2006NanoL...6.1113B| s2cid =9358094}}</ref> 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 name="spp-differential"/>

For a photon to excite an SPP, both must have the same frequency and momentum. However, for a given frequency, a free-space photon has ''less'' momentum than an SPP because the two have different [[Physics:Dispersion relation|dispersion relation]]s (see below). This momentum mismatch is the reason that a free-space photon from air cannot couple directly to an SPP. For the same reason, an SPP on a smooth metal surface ''cannot'' emit energy as a free-space photon into the dielectric (if the dielectric is uniform). This incompatibility is analogous to the lack of transmission that occurs during [[Total internal reflection|total internal reflection]].

Nevertheless, coupling of photons into SPPs can be achieved using a coupling medium such as a [[Physics:Prism (optics)|prism]] or grating to match the photon and SPP wave vectors (and thus match their momenta). 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 [[Physics:Surface roughness|roughness]]. Moreover, simple isolated surface defects such as a groove, a slit or a corrugation on an otherwise planar surface provide a mechanism by which free-space radiation and SPs can exchange energy and hence couple.

==Fields and dispersion relation==
The properties of an SPP can be derived from [[Physics:Maxwell's equations|Maxwell's equations]]. We use a coordinate system where the metal–dielectric interface is the <math>z=0</math> plane, with the metal at <math>z<0</math> and dielectric at <math>z>0</math>. The [[Physics:Electric field|electric]] and [[Physics:Magnetic field|magnetic field]]s as a function of position <math>(x,y,z)</math> and time ''t'' are as follows:<ref name="Raether"/><ref name="M.G. Cottam"/><ref name="ieee"/>

:<math>E_{x,n}(x,y,z,t) = E_0 e^{i k_x x + i k_{z,n} |z| - i \omega t}</math>
:<math>E_{z,n}(x,y,z,t) = \pm E_0 \frac{k_x}{k_{z,n}} e^{i k_x x + i k_{z,n} |z| - i \omega t}</math>
:<math>H_{y,n}(x,y,z,t) = H_0 e^{i k_x x + i k_{z,n} |z| - i \omega t}</math>

where
* ''n'' indicates the material (1 for the metal at <math>z<0</math> or 2 for the dielectric at <math>z>0</math>);
* ''ω'' is the [[Physics:Angular frequency|angular frequency]] of the waves;
* the <math>\pm</math> is + for the metal, − for the dielectric.
* <math>E_x,E_z</math> are the ''x''- and ''z''-components of the electric field vector, <math>H_y</math> is the ''y''-component of the magnetic field vector, and the other components (<math>E_y,H_x,H_z</math>) are zero. In other words, SPPs are always [[Physics:Transverse mode|TM (transverse magnetic)]] waves.
* ''k'' is the [[Physics:Wave vector|wave vector]]; it is a complex vector, and in the case of a lossless SPP, it turns out that the ''x'' components are real and the ''z'' components are imaginary—the wave oscillates along the ''x'' direction and exponentially decays along the ''z'' direction. <math>k_x</math> is always the same for both materials, but <math>k_{z,1}</math> is generally different from <math>k_{z,2}</math>
* <math>\frac{H_0}{E_0} = -\frac{\varepsilon_1 \omega}{k_{z,1}c}</math>, where <math>\varepsilon_1</math> is the [[Physics:Permittivity|permittivity]] of material 1 (the metal), and ''c'' is the [[Physics:Speed of light|speed of light in vacuum]]. As discussed below, this can also be written<math>\frac{H_0}{E_0} = \frac{\varepsilon_2 \omega}{k_{z,2}c}</math>.

A wave of this form satisfies Maxwell's equations ''only'' on condition that the following equations also hold:

:<math>\frac{k_{z1}}{\varepsilon_1} + \frac{k_{z2}}{\varepsilon_2} = 0 </math>

and

:<math>k_{x}^2+k_{zn}^2=\varepsilon_n \left(\frac{\omega}{c}\right)^2 \qquad n=1,2</math>

Solving these two equations, the dispersion relation for a wave propagating on the surface is

:<math>k_{x}=\frac{\omega}{c} \left(\frac{\varepsilon_1\varepsilon_2}{ \varepsilon_1+\varepsilon_2}\right)^{1/2}.</math>

[[File:Dispersion Relationship.png|thumb|Figure 3: Lossless dispersion curve for surface plasmon polaritons.{{efn|name="dispersion"}} At low ''k'', the surface plasmon curve (red) approaches the photon curve (blue)]]
In the [[Physics: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 name="kittel-ssp"/>

:<math>\varepsilon(\omega)=1-\frac{\omega_{\rm P}^2}{\omega^2},</math>

where the bulk plasma frequency in SI units is

:<math>\omega_{\rm P}=\sqrt{\frac{n e^2}{{\varepsilon_0}m^*}}</math>

where ''n'' is the electron density, ''e'' is the [[Physics:Electron charge|charge]] of the electron, ''m''<sup>∗</sup> is the [[Physics:Effective mass (solid-state physics)|effective mass]] of the electron and <math>{\varepsilon_0}</math> is the permittivity of free-space. The [[Physics:Dispersion (optics)|dispersion]] relation is plotted in Figure 3. 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".{{efn|name="dispersion"}} 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

:<math>\omega_{\rm SP}=\omega_{\rm P}/\sqrt{1+\varepsilon_2}.</math>

In the case of air, this result simplifies to

:<math>\omega_{\rm SP}=\omega_{\rm P}/\sqrt{2}.</math>

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 {{abs|''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"/>

:<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>

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.

==Propagation length and skin depth==
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 [[Physics:Electric field|electric field]], so at a distance ''x'', the intensity has decreased by a factor of <math display="inline">\exp\{-2k_{x}'' x\}</math>. 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"/>

:<math>L=\frac{1}{2 k_{x}''}.</math>

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"/>

:<math>z_{i}=\frac{\lambda}{2 \pi} \left(\frac{|\varepsilon_1'|+\varepsilon_2}{\varepsilon_i^2} \right)^{1/2}</math>

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.

==Animations==
<gallery heights="220px" widths="400px" perrow="2">
File:SPP silver-air interface 370nm.gif|The [[Physics:Electric field|electric field]] (E-field) of an SPP at the silver-air interface, at the frequency where the free-space wavelength is 370&nbsp;nm. The animation shows how the E-field varies over an optical cycle. The [[Physics:Permittivity|permittivity]] of silver at this frequency is {{nowrap|(−2.6 + 0.6''i'')}}. The picture is {{nowrap|(0.3 × 370 nm)}} across horizontally; the SPP wavelength is much smaller than the free-space wavelength.

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 [[Physics:Surface wave#Sommerfeld–Zenneck 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 + 1400''i'')}}. The picture is 6&nbsp;μm across horizontally.
</gallery>

==Experimental applications==
Nanofabricated systems that exploit SPPs demonstrate potential for designing and controlling the propagation of [[Physics:Light|light]] in matter. In particular, SPPs can be used to channel light efficiently into nanometer scale volumes, leading to direct modification of [[Physics: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 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.

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.

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 ''metamaterials''.<ref name=nanoplasmonics/> Artificial SPP modes can be realized in [[Physics:Microwave|microwave]] and [[Physics:Terahertz radiation|terahertz]] frequencies by metamaterials; these are known as [[Physics:Spoof surface plasmon|spoof surface plasmon]]s.<ref name="pendry-mimic"/><ref name="pan-metamaterial-particles"/>

The excitation of SPPs is frequently used in an experimental technique known as [[Physics:Surface plasmon resonance|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 [[Physics:Angle of incidence (optics)|incident angle]], [[Wavelength|wavelength]] or [[Physics:Phase (waves)|phase]].<ref name="nanotech-bio"/>

[[Physics:Surface plasmon|Surface plasmon]]-based circuits, including both SPPs and localized plasmon resonances, have been proposed as a means of overcoming the size limitations of photonic circuits for use in high performance data processing nano devices.<ref name="ozbay"/>

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 name="akimov-pp"/> 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.

CMOS compatible electro-optic plasmonic modulators will be key components in chip-scale photonic circuits.<ref name="electro-optic-plasmon"/>

In [[Physics:Surface second harmonic generation|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 [[Physics:Nonlinear optics|non-linear optical effect]]. This larger signal is often exploited to produce a stronger second harmonic signal.<ref name="valev-shg"/>

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 name="immunosensor"/>

==Materials used==
Surface plasmon polaritons can only exist at the interface between a positive-permittivity material and a negative-permittivity material.<ref name="Yeh2005"/> The positive-permittivity material, often called the '''dielectric material''', can be any transparent material such as air or (for visible light) glass. The negative-permittivity material, often called the '''plasmonic material''',<ref name=WestIshii2010/> may be a metal or other material. It is more critical, as it tends to have a large effect on the wavelength, absorption length, and other properties of the SPP. Some plasmonic materials are discussed next.

===Metals===
For visible and near-infrared light, the only plasmonic materials are metals, due to their abundance of free electrons,<ref name="WestIshii2010"/> which leads to a high [[Physics:Plasma frequency|plasma frequency]]. (Materials have negative real permittivity only below their plasma frequency.)

Unfortunately, metals suffer from ohmic losses that can degrade the performance of plasmonic devices. The need for lower loss has fueled research aimed at developing new materials for plasmonics<ref name="WestIshii2010"/><ref name="BoltassevaAtwater2011"/><ref name="BlaberArnold2010"/> and optimizing the deposition conditions of existing materials.<ref name="McPeakJayanti2015"/> Both the loss and polarizability of a material affect its optical performance. The quality factor <math>Q_{SPP}</math> for a SPP is defined as <math>\frac{\varepsilon'^{2}}{\varepsilon''}</math>.<ref name="BlaberArnold2010"/> The table below shows the quality factors and SPP propagation lengths for four common plasmonic metals; Al, Ag, Au and Cu deposited by thermal evaporation under optimized conditions.<ref name="McPeakJayanti2015"/>  The quality factors and SPP propagation lengths were calculated using the optical data from the [http://refractiveindex.info/?shelf=main&book=Al&page=McPeak Al], [http://refractiveindex.info/?shelf=main&book=Ag&page=McPeak Ag], [http://refractiveindex.info/?shelf=main&book=Au&page=McPeak Au] and [http://refractiveindex.info/?shelf=main&book=Cu&page=McPeak Cu] films.<ref name="ieee"/>

{| class="wikitable"
! Wavelength Regime
! Metal
! <math>Q_{SPP} (\times 10^{3})</math>
! <math>L_{SPP} (\mu m)</math>
|-
| Ultraviolet (280&nbsp;nm)
| Al
| 0.07
| 2.5
|-
| Visible (650&nbsp;nm)
| Ag
| 1.2
| 84
|-
| 
| Cu
| 0.42
| 24
|-
| 
| Au
| 0.4
| 20
|-
| Near-Infrared (1000&nbsp;nm)
| Ag
| 2.2
| 340
|-
| 
| Cu
| 1.1
| 190
|-
| 
| Au
| 1.1
| 190
|-
| Telecom (1550&nbsp;nm)
| Ag
| 5
| 1200
|-
| 
| Cu
| 3.4
| 820
|-
| 
| Au
| 3.2
| 730
|}

Silver exhibits the lowest losses of current materials in both the visible, near-infrared (NIR) and telecom wavelengths.<ref name="McPeakJayanti2015"/> Gold and copper perform equally well in the visible and NIR with copper having a slight advantage at telecom wavelengths. Gold has the advantage over both silver and copper of being chemically stable in natural environments making it well suited for plasmonic biosensors.<ref name="Homola2003"/> However, an interband transition at ~470&nbsp;nm greatly increases the losses in gold at wavelengths below 600&nbsp;nm.<ref name="EtchegoinLe Ru2006"/> Aluminum is the best plasmonic material in the ultraviolet regime (< 330&nbsp;nm) and is also CMOS compatible along with copper.

===Other materials===
The fewer electrons a material has, the lower (i.e. longer-wavelength) its [[Physics:Plasma frequency|plasma frequency]] becomes. Therefore, at infrared and longer wavelengths, various other plasmonic materials also exist besides metals.<ref name=WestIshii2010/> These include transparent conducting oxides, which have typical plasma frequency in the NIR-SWIR infrared range.<ref name="spp-doped-film"/> At longer wavelengths, semiconductors may also be plasmonic.

Some materials have negative permittivity at certain infrared wavelengths related to [[Physics:Phonon|phonon]]s rather than plasmons (so-called [[Physics:Reststrahlen effect|''reststrahlen'' bands]]). The resulting waves have the same optical properties as surface plasmon polaritons, but are called by a different term, '''surface phonon polaritons'''.

==Effects of roughness==
In order to understand the effect of roughness on SPPs, it is beneficial to first understand how a SPP is coupled by a [[Physics:Diffraction grating|grating]] 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.

:<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>

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.

Rough surfaces can be thought of as the [[Superposition principle|superposition]] of many gratings of different periodicities. Kretschmann proposed<ref name="Kretschmann1"/> that a statistical [[Correlation function|correlation function]] be defined for a rough surface

:<math>G(x,y)=\frac{1}{A}\int_A z(x',y')\ z(x'-x,y'-y)\, dx'\, dy',</math>

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

:<math>G(x,y)=\delta^2\exp\left(-\frac{r^2}{\sigma^2}\right)</math>

where <math>\delta</math> is the [[Root mean square|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

:<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>

where <math>s</math> is a measure of the amount of each [[Spatial frequency|spatial frequency]] <math> k_\text{surf}</math> which help couple photons into a surface plasmon.

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.

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|solid angle]] <math>d \Omega</math> per incident intensity <math>I_{0}</math> is<ref name="Kretschmann2"/>

:<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>

where <math>|W|^2</math> is the radiation pattern from a single [[Physics:Dipole|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.&nbsp;4), then the dipole function becomes

:<math>|W|^2=A(\theta,|\varepsilon_{1}|)\ \sin^2{\psi} \ [(1+\sin^2 \theta /|\varepsilon_1|)^{1/2} - \sin{\theta}]^2</math>

with

:<math> A(\theta,|\varepsilon_1|) = \frac{|\varepsilon_1|+1}{|\varepsilon_1|-1} \frac{4}{1+\tan{\theta}/| \varepsilon_1|}</math>

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"/>

==See also==
* [[Physics:Dyakonov surface waves|Dyakonov surface waves]]
* [[Physics:Graphene plasmonics|Graphene plasmonics]]
* [[Physics:Localized surface plasmon|Localized surface plasmon]]
* [[Physics:Plasmonic lens|Plasmonic lens]]
* [[Physics:Superlens|Superlens]]
* [[Physics:Surface plasmon|Surface plasmon]]
* [[Physics:Surface plasmon resonance|Surface plasmon resonance]]
* [[Physics:Surface wave|Surface wave]]

==Notes==
{{notelist|refs=
{{efn|name="dispersion"|This lossless dispersion relation neglects the effects of [[Damping ratio|damping]] factors, such as the [[Physics:Attenuation|intrinsic losses]] in metals. For lossy cases, the dispersion curve backbends after the reaching the surface plasmon frequency instead of [[Asymptote|asymptotically increasing]].<ref name="damping-sp"/><ref name="maier-plasmonics-book"/>}}
}}

==References==
{{Reflist|40em|refs=

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<ref name="damping-sp">{{cite journal |first1=E. T. |last1=Arakawa |first2=M. W. |last2=Williams |first3=R. N. |last3=Hamm |first4=R. H. |last4=Ritchie|date= 29 October 1973 |title= Effect of Damping on Surface Plasmon Dispersion |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |volume=31 |issue= 18 |pages=1127–1129 |doi=10.1103/PhysRevLett.31.1127|bibcode=1973PhRvL..31.1127A }}</ref>

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{{cite web
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*{{NIST-PD|article=Nanoplasmonics|url=https://www.nist.gov/cnst/nrg/nanoplasmonics.cfm}}</ref>

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*{{cite web
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*{{NIST-PD|article=Three-Dimensional Plasmonic  Metamaterials |url=https://www.nist.gov/cnst/nrg/3d_plasmonic_metamaterials.cfm}}</ref>

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<ref name=w-barnes>
{{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=824–30|issue=6950|volume=424
|journal=Nature
|bibcode = 2003Natur.424..824B |s2cid=116017}} 
*{{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 |pages=1985–90 |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 |hdl=10044/1/42407 |s2cid=1255444 }} Free PDF download for these peer reviewed articles.
*PDF from arxiv.org – [https://arxiv.org/PS_cache/arxiv/pdf/1003/1003.1154v2.pdf Transformation Optics for Plasmonics]. 15 pages.</ref>

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}}

==Further reading==
*{{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|s2cid=205024396}}
*{{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|hdl=10036/33196|hdl-access=free}} Free PDF download.
*{{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=824–30|issue=6950|volume=424|journal=Nature|url=http://access.ex.ac.uk/research/emag/wlb/nature_2003_424_824.pdf|bibcode=2003Natur.424..824B|s2cid=116017 |url-status=dead|archive-url=https://web.archive.org/web/20110811024623/http://access.ex.ac.uk/research/emag/wlb/nature_2003_424_824.pdf|archive-date=2011-08-11}} Free PDF download.
*{{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|issue =1|journal =Reports on Progress in Physics
|url =http://dipc.ehu.es/etxenike/admin/documentos/archivos/publicaciones/307RPP2007.pdf|arxiv = cond-mat/0611257 |bibcode = 2007RPPh...70....1P |s2cid =46471088}} Free PDF download.

==External links==
*{{Cite web
  | last =White 
  | first =Justin  
  | title =Surface Plasmon Polaritons 
  | publisher =[[Organization:Stanford University|Stanford University]]. Physics department 
  | date =March 19, 2007 
  | url =http://large.stanford.edu/courses/2007/ap272/white1/ 
  | format =Online 
  }} "''Submitted as coursework for AP272. Winter 2007''".

{{DEFAULTSORT:Surface Plasmon Polaritons}}
[[Category:Quasiparticles]]
[[Category:Metamaterials]]
[[Category:Plasmonics]]

{{Sourceattribution|Surface plasmon polariton}}