Surface Plasmon Polaritons
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Surface plasmon polaritons (SPPs) are
electromagnetic waves In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength, ran ...
that travel along a
metal A metal () is a material that, when polished or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, electricity and thermal conductivity, heat relatively well. These properties are all associated wit ...
dielectric In electromagnetism, a dielectric (or dielectric medium) is an Insulator (electricity), electrical insulator that can be Polarisability, polarised by an applied electric field. When a dielectric material is placed in an electric field, electric ...
or metal–air interface, practically in the
infrared Infrared (IR; sometimes called infrared light) is electromagnetic radiation (EMR) with wavelengths longer than that of visible light but shorter than microwaves. The infrared spectral band begins with the waves that are just longer than those ...
or visible-frequency. The term "surface plasmon polariton" explains that the wave involves both charge motion in the metal ("
surface plasmon Surface plasmons (SPs) are coherent delocalized electron oscillations that exist at the interface between any two materials where the real part of the dielectric function changes sign across the interface (e.g. a metal-dielectric interface, such ...
") and electromagnetic waves in the air or dielectric ("
polariton In physics, polaritons are bosonic quasiparticles resulting from strong coupling of electromagnetic waves (photon) with an electric or magnetic dipole-carrying excitation (state) of solid or liquid matter (such as a phonon, plasmon, or an exc ...
"). They are a type of
surface wave In physics, a surface wave is a mechanical wave that propagates along the Interface (chemistry), interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occu ...
, 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). Hence, SPPs can have a higher momentum and local field intensity. 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
photolithography Photolithography (also known as optical lithography) is a process used in the manufacturing of integrated circuits. It involves using light to transfer a pattern onto a substrate, typically a silicon wafer. The process begins with a photosensiti ...
beyond the
diffraction limit In optics, any optical instrument or systema microscope, telescope, or camerahas a principal limit to its resolution due to the physics of diffraction. An optical instrument is said to be diffraction-limited if it has reached this limit of res ...
. 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 Photonics is a branch of optics that involves the application of generation, detection, and manipulation of light in the form of photons through emission, transmission, modulation, signal processing, switching, amplification, and sensing. Eve ...
data storage, light generation, and bio-photonics.


Excitation

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. 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
dispersion relation In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the ...
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 In physics, total internal reflection (TIR) is the phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely refl ...
. Nevertheless, coupling of photons into SPPs can be achieved using a coupling medium such as a
prism PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...
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 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
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
. We use a coordinate system where the metal–dielectric interface is the z=0 plane, with the metal at z<0 and dielectric at z>0. The
electric Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
and
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
s as a function of position (x,y,z) and time ''t'' are as follows: :E_(x,y,z,t) = E_0 e^ :E_(x,y,z,t) = \pm E_0 \frac e^ :H_(x,y,z,t) = H_0 e^ where * ''n'' indicates the material (1 for the metal at z<0 or 2 for the dielectric at z>0); * ''ω'' is the
angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
of the waves; * the \pm is + for the metal, − for the dielectric. * E_x,E_z are the ''x''- and ''z''-components of the electric field vector, H_y is the ''y''-component of the magnetic field vector, and the other components (E_y,H_x,H_z) are zero. In other words, SPPs are always TM (transverse magnetic) waves. * ''k'' is the
wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength) ...
; 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. k_x is always the same for both materials, but k_ is generally different from k_ * \frac = -\frac, where \varepsilon_1 is the
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter (epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more ...
of material 1 (the metal), and ''c'' is the
speed of light in vacuum The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
. As discussed below, this can also be written\frac = \frac. A wave of this form satisfies Maxwell's equations ''only'' on condition that the following equations also hold: :\frac + \frac = 0 and :k_^2+k_^2=\varepsilon_n \left(\frac\right)^2 \qquad n=1,2 Solving these two equations, the dispersion relation for a wave propagating on the surface is :k_=\frac \left(\frac\right)^. In the free electron model of an electron gas, which neglects attenuation, the metallic dielectric function is :\varepsilon(\omega)=1-\frac, where the bulk plasma frequency in SI units is :\omega_=\sqrt where ''n'' is the electron density, ''e'' is the
charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aqu ...
of the electron, ''m'' is the effective mass of the electron and is the permittivity of free-space. The
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns * Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variat ...
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". 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 :\omega_=\omega_/\sqrt. In the case of air, this result simplifies to :\omega_=\omega_/\sqrt. If we assume that ''ε''2 is real and ''ε''2 > 0, then it must be true that ''ε''1 < 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 In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter (epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more ...
. The dielectric function of a metal is expressed ''ε''1 = ''ε''1′ + ''i''⋅''ε''1″ where ''ε''1′ and ''ε''1″ are the real and imaginary parts of the dielectric function, respectively. Generally >> ''ε''1″ so the wavenumber can be expressed in terms of its real and imaginary components as :k_=k_'+i k_''=\left frac \left( \frac\right)^\right+ i \left frac \left( \frac\right)^ \frac\right 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
electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
, so at a distance ''x'', the intensity has decreased by a factor of \exp\. 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 :L=\frac. 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 In electromagnetism, skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with gre ...
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 :z_=\frac \left(\frac \right)^ 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

File:SPP silver-air interface 370nm.gif, The
electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
(E-field) of an SPP at the silver-air interface, at the frequency where the free-space wavelength is 370 nm. The animation shows how the E-field varies over an optical cycle. The
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter (epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more ...
of silver at this frequency is . The picture is 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 In electrostatics, a perfect conductor is an idealized model for real conducting materials. The defining property of a perfect conductor is that static electric field and the charge density both vanish in its interior. If the conductor has excess ...
, and the SPP is called a Sommerfeld–Zenneck wave, with almost the same wavelength as the free-space wavelength. The permittivity of silver at this frequency is . The picture is 6 μm across horizontally.


Experimental applications

Nanofabricated systems that exploit SPPs demonstrate potential for designing and controlling the propagation of
light Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
in matter. In particular, SPPs can be used to channel light efficiently into
nanometer 330px, Different lengths as in respect to the Molecule">molecular scale. The nanometre (international spelling as used by the International Bureau of Weights and Measures; SI symbol: nm), or nanometer (American spelling Despite the va ...
scale volumes, leading to direct modification of 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 A biosensor is an analytical device, used for the detection of a chemical substance, that combines a biological component with a physicochemical detector. The ''sensitive biological element'', e.g. tissue, microorganisms, organelles, cell recep ...
, 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 A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is a type of material engineered to have a property, typically rarely observed in naturally occur ...
''. Artificial SPP modes can be realized in
microwave Microwave is a form of electromagnetic radiation with wavelengths shorter than other radio waves but longer than infrared waves. Its wavelength ranges from about one meter to one millimeter, corresponding to frequency, frequencies between 300&n ...
and terahertz frequencies by metamaterials; these are known as spoof surface plasmons. The excitation of SPPs is frequently used in an experimental technique known as
surface plasmon resonance Surface plasmon resonance (SPR) is a phenomenon that occurs where electrons in a thin metal sheet become excited by light that is directed to the sheet with a particular angle of incidence (optics), angle of incidence, and then travel parallel to ...
(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,
wavelength In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
or
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
.
Surface plasmon Surface plasmons (SPs) are coherent delocalized electron oscillations that exist at the interface between any two materials where the real part of the dielectric function changes sign across the interface (e.g. a metal-dielectric interface, such ...
-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. 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. 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. In
surface second harmonic generation Surface second harmonic generation is a method for probing interfaces in atomic and molecular systems. In second harmonic generation (SHG), the light frequency is doubled, essentially converting two photons of the original beam of energy ''E'' int ...
, 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 non-linear optical effect. This larger signal is often exploited to produce a stronger second harmonic signal. 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.


Materials used

Surface plasmon polaritons can only exist at the interface between a positive-
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter (epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more ...
material and a negative-permittivity material. 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, 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, which leads to a high
plasma frequency Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability ...
. (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 and optimizing the deposition conditions of existing materials. Both the loss and polarizability of a material affect its optical performance. The quality factor Q_ for a SPP is defined as \frac. 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. The quality factors and SPP propagation lengths were calculated using the optical data from th
AlAgAu
an
Cu
films. Silver exhibits the lowest losses of current materials in both the visible, near-infrared (NIR) and telecom wavelengths. 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. However, an interband transition at ~470 nm greatly increases the losses in gold at wavelengths below 600 nm. Aluminum is the best plasmonic material in the ultraviolet regime (< 330 nm) and is also CMOS compatible along with copper.


Other materials

The fewer electrons a material has, the lower (i.e. longer-wavelength) its
plasma frequency Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability ...
becomes. Therefore, at infrared and longer wavelengths, various other plasmonic materials also exist besides metals. These include
transparent conducting oxide Transparent conducting films (TCFs) are thin films of optically transparent and electrically conductive material. They are an important component in a number of electronic devices including liquid-crystal displays, OLEDs, touchscreens and photo ...
s, which have typical plasma frequency in the NIR- SWIR infrared range. At longer wavelengths, semiconductors may also be plasmonic. Some materials have negative permittivity at certain infrared wavelengths related to
phonon A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized vibration mode can be defined a ...
s rather than plasmons (so-called ''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
grating A grating is any regularly spaced collection of essentially identical, parallel, elongated elements. Gratings usually consist of a single set of elongated elements, but can consist of two sets, in which case the second set is usually perpendicu ...
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 \Delta k = k_- k_. The grating
harmonics In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st harm ...
of a periodic grating provide additional momentum parallel to the supporting interface to match the terms. :k_=k_ \pm n\ k_\text=\frac \sin \pm n \frac, where k_\text is the wave vector of the grating, \theta_0 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 In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ...
of many gratings of different periodicities. Kretschmann proposed that a statistical
correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables ...
be defined for a rough surface :G(x,y)=\frac\int_A z(x',y')\ z(x'-x,y'-y)\, dx'\, dy', where z(x,y) is the height above the mean surface height at the position (x,y), and A is the area of integration. Assuming that the statistical correlation function is
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
of the form :G(x,y)=\delta^2\exp\left(-\frac\right) where \delta is the
root mean square In mathematics, the root mean square (abbrev. RMS, or rms) of a set of values is the square root of the set's mean square. Given a set x_i, its RMS is denoted as either x_\mathrm or \mathrm_x. The RMS is also known as the quadratic mean (denote ...
height, r is the distance from the point (x,y), and \sigma is the correlation length, then the
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of the correlation function is :, s(k_\text), ^2=\frac \sigma^2 \delta^2 \exp \left( - \frac\right) where s is a measure of the amount of each
spatial frequency In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components (as determined by the Fourier tra ...
k_\text 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 s is discrete and exists only at k=\frac, 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, s 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 dI per
solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poin ...
d \Omega per incident intensity I_ is :\frac=\frac\frac, t_^p, ^2 \ , W, ^2 , s(k_\text), ^2 where , W, ^2 is the radiation pattern from a single
dipole In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: * An electric dipole moment, electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple ...
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 :, W, ^2=A(\theta,, \varepsilon_, )\ \sin^2 \ \varepsilon_1, )^ - \sin2 with : A(\theta,, \varepsilon_1, ) = \frac \frac where \psi is the polarization angle and \theta is the angle from the ''z''-axis in the ''xz''-plane. Two important consequences come out of these equations. The first is that if \psi=0 (s-polarization), then , W, ^2=0 and the scattered light \frac=0. Secondly, the scattered light has a measurable profile which is readily correlated to the roughness. This topic is treated in greater detail in reference.


See also

* Dyakonov surface waves * Graphene plasmonics *
Localized surface plasmon A localized surface plasmon (LSP) is the result of the confinement of a surface plasmon in a nanoparticle of size comparable to or smaller than the wavelength of light used to excite the plasmon. When a small spherical metallic nanoparticle is irr ...
*
Plasmonic lens In nano-optics, a plasmonic lens generally refers to a lens for surface plasmon polaritons (SPPs), i.e. a device that redirects SPPs to converge towards a single focal point. Because SPPs can have very small wavelength, they can converge into a ver ...
*
Superlens A superlens, or super lens, is a lens which uses metamaterials to go beyond the diffraction limit. The diffraction limit is a feature of conventional lenses and microscopes that limits the fineness of their resolution depending on the illumination ...
*
Surface plasmon Surface plasmons (SPs) are coherent delocalized electron oscillations that exist at the interface between any two materials where the real part of the dielectric function changes sign across the interface (e.g. a metal-dielectric interface, such ...
*
Surface plasmon resonance Surface plasmon resonance (SPR) is a phenomenon that occurs where electrons in a thin metal sheet become excited by light that is directed to the sheet with a particular angle of incidence (optics), angle of incidence, and then travel parallel to ...
*
Surface wave In physics, a surface wave is a mechanical wave that propagates along the Interface (chemistry), interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occu ...


Notes


References


Further reading

* * Free PDF download. * Free PDF download. * Free PDF download.


External links

* "''Submitted as coursework for AP272. Winter 2007''". {{DEFAULTSORT:Surface Plasmon Polaritons Quasiparticles Metamaterials Plasmonics Surface waves