Surface plasmon (polariton)

A surface plasmon polariton (SPP) is a surface wave that propagates along a flat interface between a metal and a dielectric, made of a collective oscillation of the conduction electrons in the metal coupled to an electromagnetic field. The field is largest at the interface and decays exponentially into both media: into the dielectric over a fraction of a wavelength, a few hundred nanometers, into the metal over a few tens of nanometers. The wave exists only for TM (p-polarized) light and only where the real part of the metal's permittivity is negative and larger in magnitude than the dielectric's, which holds for silver across the visible and near infrared and for gold at wavelengths beyond about 550 nm. At 633 nm, an SPP on gold in water propagates about 4 µm (intensity 1/e length); on silver, about 25 µm. The short name "surface plasmon" usually means the same thing; localized surface plasmons are the related resonances of metal nanoparticles.

Dispersion relation and length scales

For a metal of permittivity $\varepsilon_m$ and a dielectric of permittivity $\varepsilon_d$, the SPP propagation constant is

$$k_{sp} = k_0 \sqrt{\frac{\varepsilon_m\,\varepsilon_d}{\varepsilon_m + \varepsilon_d}},$$

with $k_0 = 2\pi/\lambda$. Because $\varepsilon_m$ is negative, the square root exceeds $\sqrt{\varepsilon_d}$: the SPP has a larger wavevector than light in the dielectric and cannot be excited by a beam incident from that side. Taking $\varepsilon_m = -11.75 + 1.26i$, the Johnson and Christy value for gold at 633 nm (other tabulations and real films differ by up to about 20%), the effective index $k_{sp}/k_0$ is 1.045 against air and 1.445 against water. The imaginary part sets the intensity propagation length $L = 1/(2\,\text{Im}\,k_{sp})$: 9.8 µm against air and 3.7 µm against water. For silver ($\varepsilon_m = -18.3 + 0.48i$, from the same data set) the lengths are 65 µm and 25 µm, which is why silver is preferred where loss matters and gold where chemical stability does.

The field decays from the surface with 1/e depths of 1/|Im $k_z$| in each medium. For gold and water at 633 nm these are about 180 nm into the water and 27 nm into the metal. The short depth in the dielectric is the basis of SPR sensing: only the refractive index within a couple of hundred nanometers of the surface affects the wave.

Excitation

To make up the wavevector mismatch, light is brought in through a medium of higher index. In the Kretschmann configuration, a thin metal film, about 50 nm of gold, is deposited on the base of a glass prism, and p-polarized light incident beyond the critical angle produces an evanescent wave that tunnels through the film. When the tangential wavevector $k_0 n_p \sin\theta$ matches $\text{Re}\,k_{sp}$, power transfers to the SPP and the reflectance drops sharply. For a prism with $n_p$ = 1.515 and water on the gold, the resonance lies at 72.5°, beyond the glass–water critical angle of 61.6°. The Otto configuration leaves an air gap between prism and metal instead, in the same way a prism coupler excites a slab waveguide mode. Gratings, rough surfaces and sub-wavelength apertures supply the missing momentum by diffraction.

Sensing and other uses

Surface plasmon resonance (SPR) biosensors track the resonance angle or wavelength as molecules bind to a functionalized gold surface. In the Kretschmann geometry above, the resonance angle shifts by about 160° per refractive index unit of the water, so an angular resolution of 10⁻⁴ degree resolves index changes of about 6 × 10⁻⁷. SPR instruments measure binding kinetics without fluorescent labels.

Localized surface plasmons on metal nanoparticles produce strong absorption and scattering peaks; for a small sphere the resonance falls where $\varepsilon_m \approx -2\varepsilon_d$, near 520 nm for gold in water, which gives colloidal gold its red color. The field enhancement at nanoparticles and gaps underlies surface-enhanced Raman spectroscopy. Plasmonic waveguides confine light below the diffraction limit at the cost of propagation lengths of micrometers or less, and plasmonic modulators and detectors exploit that small volume where loss is tolerable.

Pitfalls

Only p-polarized light couples, so an SPR dip that does not vanish when the polarization is rotated to s is not a plasmon. Film thickness controls the depth of the dip: too thin a film overcouples and too thick a film undercouples. Real films are rougher and lossier than bulk data predict, and thin adhesion layers of chromium or titanium under the gold add damping and broaden the resonance.

Common questions

What is the difference between a surface plasmon and a plasmon polariton?

A plasmon is a quantized oscillation of the electron density; a surface plasmon polariton is that oscillation coupled to light at an interface. In optics the terms are used almost interchangeably.

Why can s-polarized light not excite a surface plasmon?

The wave requires an electric field component normal to the surface to drive surface charge, and an s-polarized field lies entirely parallel to it.

Why gold and silver?

Both have a large negative real permittivity with a relatively small imaginary part in the visible and near infrared. Silver has lower loss; gold resists oxidation and is easy to functionalize.

References: S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007); H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988); E. Kretschmann and H. Raether, Z. Naturforsch. A 23, 2135 (1968); P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370 (1972); J. Homola, S. S. Yee and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3 (1999).