## Surface plasmon polariton

• Surface plasmon polaritons (SPPs), are infrared or visible-frequency electromagnetic waves that 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").

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). Hence, SPPs can have tighter spatial confinement and higher 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 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.

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 relations (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.

Wavelength Regime Metal ${\displaystyle Q_{SPP}(\times 10^{-3})}$ ${\displaystyle L_{SPP}(\mu m)}$
Ultraviolet (280 nm) Al 0.07 2.5
Visible (650 nm) Ag 1.2 84
Cu 0.42 24
Au 0.4 20
Near-Infrared (1000 nm) Ag 2.2 340
Cu 1.1 190
Au 1.1 190
Telecom (1550 nm) Ag 5 1200
Cu 3.4 820
Au 3.2 730

${\displaystyle E_{x,n}(x,y,z,t)=E_{0}e^{ik_{x}x+ik_{z,n}|z|-i\omega t}}$
${\displaystyle E_{z,n}(x,y,z,t)=\pm E_{0}{\frac {k_{x}}{k_{z,n}}}e^{ik_{x}x+ik_{z,n}|z|-i\omega t}}$
${\displaystyle H_{y,n}(x,y,z,t)=H_{0}e^{ik_{x}x+ik_{z,n}|z|-i\omega t}}$
${\displaystyle {\frac {k_{z1}}{\varepsilon _{1}}}+{\frac {k_{z2}}{\varepsilon _{2}}}=0}$
${\displaystyle k_{x}^{2}+k_{zn}^{2}=\varepsilon _{n}\left({\frac {\omega }{c}}\right)^{2}\qquad n=1,2}$
${\displaystyle k_{x}={\frac {\omega }{c}}\left({\frac {\varepsilon _{1}\varepsilon _{2}}{\varepsilon _{1}+\varepsilon _{2}}}\right)^{1/2}.}$
${\displaystyle \varepsilon (\omega )=1-{\frac {\omega _{P}^{2}}{\omega ^{2}}},}$
${\displaystyle \omega _{P}={\sqrt {\frac {ne^{2}}{{\varepsilon _{0}}m^{*}}}}}$
${\displaystyle \omega _{SP}=\omega _{P}/{\sqrt {1+\varepsilon _{2}}}.}$
${\displaystyle \omega _{SP}=\omega _{P}/{\sqrt {2}}.}$
${\displaystyle k_{x}=k_{x}'+ik_{x}''=\left[{\frac {\omega }{c}}\left({\frac {\varepsilon _{1}'\varepsilon _{2}}{\varepsilon _{1}'+\varepsilon _{2}}}\right)^{1/2}\right]+i\left[{\frac {\omega }{c}}\left({\frac {\varepsilon _{1}'\varepsilon _{2}}{\varepsilon _{1}'+\varepsilon _{2}}}\right)^{3/2}{\frac {\varepsilon _{1}''}{2(\varepsilon _{1}')^{2}}}\right].}$
${\displaystyle L={\frac {1}{2k_{x}''}}.}$
${\displaystyle z_{i}={\frac {\lambda }{2\pi }}\left({\frac {|\varepsilon _{1}'|+\varepsilon _{2}}{\varepsilon _{i}^{2}}}\right)^{1/2}}$
${\displaystyle k_{SPP}=k_{x,{\text{photon}}}\pm n\ k_{\text{grating}}={\frac {\omega }{c}}\sin {\theta _{0}}\pm n{\frac {2\pi }{a}}}$
${\displaystyle G(x,y)={\frac {1}{A}}\int _{A}z(x',y')\ z(x'-x,y'-y)\,dx'\,dy'}$
${\displaystyle G(x,y)=\delta ^{2}\exp \left(-{\frac {r^{2}}{\sigma ^{2}}}\right)}$
${\displaystyle |s(k_{\text{surf}})|^{2}={\frac {1}{4\pi }}\sigma ^{2}\delta ^{2}\exp \left(-{\frac {\sigma ^{2}k_{\text{surf}}^{2}}{4}}\right)}$
${\displaystyle {\frac {dI}{d\Omega \ I_{0}}}={\frac {4{\sqrt {\varepsilon _{0}}}}{\cos {\theta _{0}}}}{\frac {\pi ^{4}}{\lambda ^{4}}}|t_{012}^{p}|^{2}\ |W|^{2}|s(k_{\text{surf}})|^{2}}$
${\displaystyle |W|^{2}=A(\theta ,|\varepsilon _{1}|)\ \sin ^{2}{\psi }\ [(1+\sin ^{2}\theta /|\varepsilon _{1}|)^{1/2}-\sin {\theta }]^{2}}$
${\displaystyle A(\theta ,|\varepsilon _{1}|)={\frac {|\varepsilon _{1}|+1}{|\varepsilon _{1}|-1}}{\frac {4}{1+\tan {\theta }/|\varepsilon _{1}|}}}$
• n indicates the material (1 for the metal at ${\displaystyle z<0}$ or 2 for the dielectric at ${\displaystyle z>0}$);
• ω is the angular frequency of the waves;
• the ${\displaystyle \pm }$ is + for the metal, - for the dielectric.
• ${\displaystyle E_{x},E_{z}}$ are the x- and z-components of the electric field vector, ${\displaystyle H_{y}}$ is the y-component of the magnetic field vector, and the other components (${\displaystyle E_{y},H_{x},H_{z}}$) are zero. In other words, SPPs are always TM (transverse magnetic) waves.
• k is the 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. ${\displaystyle k_{x}}$ is always the same for both materials, but ${\displaystyle k_{z,1}}$ is generally different from ${\displaystyle k_{z,2}}$
• ${\displaystyle {\frac {H_{0}}{E_{0}}}=-{\frac {\varepsilon _{1}\omega }{k_{z,1}c}}}$, where ${\displaystyle \varepsilon _{1}}$ is the permittivity of material 1 (the metal), and c is the speed of light in vacuum. As discussed below, this can also be written${\displaystyle {\frac {H_{0}}{E_{0}}}={\frac {\varepsilon _{2}\omega }{k_{z,2}c}}}$.
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