Surface plasmon polariton

Surface plasmon polaritons (SPPs) are electromagnetic waves that travel along a metal–dielectric or metal–air interface, practically in the infrared or visible-frequency. 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.
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 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.
Nevertheless, coupling of photons into SPPs can be achieved using a coupling medium such as a 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 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. We use a coordinate system where the metal–dielectric interface is the formula_1 plane, with the metal at formula_2 and dielectric at formula_3. The electric and magnetic fields as a function of position formula_4 and time "t" are as follows:
where
A wave of this form satisfies Maxwell's equations "only" on condition that the following equations also hold:
and
Solving these two equations, the dispersion relation for a wave propagating on the surface is
In the free electron model of an electron gas, which neglects attenuation, the metallic dielectric function is
where the bulk plasma frequency in SI units is
where "n" is the electron density, "e" is the charge of the electron, "m"∗ is the effective mass of the electron and formula_25 is the permittivity of free-space. The 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". 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
In the case of air, this result simplifies to
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. 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
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, so at a distance "x", the intensity has decreased by a factor of formula_29. 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
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
where formula_32 is the polarization angle and formula_33 is the angle from the "z"-axis in the "xz"-plane. Two important consequences come out of these equations. The first is that if formula_34 (s-polarization), then formula_35 and the scattered light formula_36. Secondly, the scattered light has a measurable profile which is readily correlated to the roughness. This topic is treated in greater detail in reference.