page_title: Surface plasmon polariton
section_title: Fields and dispersion relation
hierarchical_section_title: Surface plasmon polariton / Fields and dispersion relation
caption_attribution_description: English: Dispersion curve for surface plasmons. At low k, the surface plasmon curve (red) approaches the photon curve (blue)
context_page_description: Surface plasmon polaritons 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 and electromagnetic waves in the air or dielectric.
  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. 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.
  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.
context_section_description: The properties of an SPP can be derived from Maxwell's equations. We use a coordinate system where the metal–dielectric interface is the  plane, with the metal at  and dielectric at . The electric and magnetic fields as a function of position  and time t are as follows:
  where
  n indicates the material (1 for the metal at  or 2 for the dielectric at );
  ω is the angular frequency of the waves;
  the  is + for the metal, − for the dielectric.
   are the x- and z-components of the electric field vector,  is the y-component of the magnetic field vector, and the other components () 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.  is always the same for both materials, but  is generally different from 
  , where  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.
  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  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 ε₂ is real and ε₂ > 0, then it must be true that ε₁ < 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 ε₁ = ε₁′ + i⋅ε₁″ where ε₁′ and ε₁″ are the real and imaginary parts of the dielectric function, respectively. Generally |ε₁′| >> ε₁″ 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.
caption: Figure 3: Lossless dispersion curve for surface plasmon polaritons.[a] At low k, the surface plasmon curve (red) approaches the photon curve (blue)
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