Foundations

Evanescent Waves and Surface Plasmons

The physics, built up from total internal reflection with no steps skipped.

26 min read 7 sections 17 sources cited (10 verified)

Where we are going#

This guide builds the resonance condition from scratch. We start with light hitting a boundary, find the evanescent wave hiding on the far side of total internal reflection, discover that a metal surface supports its own wave, and then show that the two can be made to match. Nothing here requires more than complex exponentials and Snell’s law.

Light at an interface, and where Snell’s law stops giving real answers
The evanescent wave: a field that exists but carries no energy away
What a surface plasmon is, and why it needs a metal
The dispersion relation, and the momentum mismatch that makes SPR hard
How a prism fixes the mismatch — the Kretschmann configuration

Total internal reflection, revisited#

Take an electromagnetic wave crossing from medium 1 into medium 2, refractive indices n₁ and n₂. The wavevector k points along propagation and has magnitude

|k| = √(k_x² + k_y² + k_z²) = n·2π/λ = n·ω/c

(2.1)

where

λvacuum wavelength

ωangular frequency, 2πc/λ

cspeed of light in vacuum

nrefractive index of the medium the wave is in

Choose axes so the beam lies in the x–y plane, with y normal to the interface. Then k_z = 0 and the problem is two-dimensional. Snell’s law, n₁ sin α = n₂ sin β, is equivalent to the statement that the component of k parallel to the interface is conserved: k_x1 = k_x2. That conservation is the deep version of Snell’s law and it is the idea everything below turns on.

Combining the two gives the perpendicular component in medium 2:

k_y2² = n₁² (2π/λ)² [ (n₂²/n₁²) − sin²α ]

(2.2)

where

αangle of incidence, measured from the normal, in medium 1

Now let n₁ > n₂ — glass into water, say. When sin α exceeds n₂/n₁ the bracket goes negative, so k_y2² is negative and k_y2 is purely imaginary. This is the critical angle, and past it there is no propagating wave in medium 2. All the energy reflects. That much is standard optics; the interesting part is what "no propagating wave" actually leaves behind.

The evanescent wave#

Put an imaginary k_y2 back into the field expression. Writing k_y2 = i·κ with κ real, the exponential exp(−i k_y2 y) becomes exp(−κy) — a real decaying exponential rather than an oscillation:

E₂(y, t) = E₀ · e^−κy · e^{i(ωt − k_xx)}

(2.3)

where

E₀field amplitude at the interface

κ1/(penetration depth); the decay constant into medium 2

e^(i(ωt − k_x x))a wave that still travels along the surface, in x

Read the two exponentials separately. The first says the field dies away from the surface. The second says that whatever field remains is still a travelling wave — but travelling parallel to the interface, not across it. This is the evanescent wave.

The penetration depth 1/κ is on the order of half a wavelength. That number is the entire reason SPR works as a surface-selective technique: a change in refractive index one micrometre away from the gold is essentially invisible, while the same change ten nanometres away is not.

Evanescent field explorer. Change wavelength, angle and the two refractive indices, and watch the penetration depth respond. The critical angle is marked.

What a surface plasmon is#

A metal contains a sea of conduction electrons that are free to slosh. Displace them collectively and the restoring Coulomb force sets up an oscillation — a plasma oscillation (Drude, 1900). Its quantum is called a plasmon, by analogy with the photon as the quantum of the electromagnetic field. At a surface the oscillation is constrained to the boundary, and the associated charge-density wave, together with the electromagnetic field it drags along, is a surface plasmon polariton — usually shortened to surface plasmon.

Two consequences follow immediately, and both are practical.

You need a metal. The mode only exists where the real part of the dielectric function is negative, which for optical frequencies means a material with abundant free electrons. Gold and silver are the practical choices; copper and aluminium work but oxidise or absorb badly in the visible (Johnson & Christy, 1972).
You need p-polarised light. A surface plasmon involves charge piling up and thinning out along the surface, so its electric field must have a component normal to the surface. s-polarised light has its field entirely in the plane of the interface and therefore cannot drive the mode at all. This is not a subtlety to be memorised; it is a useful diagnostic. If you see a resonance with s-polarised light, it is not SPR.

The negative permittivity comes out of the free-electron (Drude) model. Below the plasma frequency ω_p, the electrons respond fast enough to screen the driving field and overshoot, and the dielectric function turns negative:

ε(ω) = 1 − ω_p²/ω², ω_p = √(4π n_e e² / m_e)

(2.4)

where

n_efree electron density in the metal

e, m_eelectron charge and mass

ω_pplasma frequency; above it the metal becomes transparent

Real gold departs from this because of interband transitions in the blue, which is why gold looks gold; the measured dielectric functions of the noble metals show the departure clearly (Johnson & Christy, 1972). The Drude form is a good guide in the red and near-infrared, which is precisely where SPR instruments operate (Raether, 1988).

The dispersion relation and the momentum problem#

Solving Maxwell’s equations for a bound mode at a single interface between a metal (ε_m) and a dielectric (ε_d) gives the surface plasmon dispersion relation (Raether, 1988) — the link between how fast the mode oscillates and how much momentum it carries:

k_x^SP = (ω/c) · √( ε_m ε_d / (ε_m + ε_d) )

(2.5)

where

k_x^SPthe plasmon’s wavevector along the surface

ε_mdielectric function of the metal (negative real part)

ε_ddielectric constant of the dielectric — buffer, in a biosensor

For this to give a real, bound solution you need ε_m < −ε_d. Gold in water at red wavelengths satisfies this comfortably. Note that ε_d sits inside the expression: change the refractive index of the buffer and you change the plasmon’s momentum. That is the sensing mechanism, in one symbol.

The corresponding decay constants perpendicular to the surface, into each medium, are

κ_i = (ω/c) · √( −ε_i² / (ε_m + ε_d) ) for i = metal or dielectric

(2.6)

where

κ_iinverse penetration depth into medium i

worked examplePenetration depths for gold and water at 700 nm

Take ε_gold ≈ −16 and ε_water ≈ 1.77 at λ = 700 nm, so ω/c = 2π/λ = 8.976 × 10⁻³ nm⁻¹ and ε_m + ε_d = −14.23.

Into the water: κ_d = 8.976 × 10⁻³ × √(1.77² / 14.23) = 8.976 × 10⁻³ × 0.4692 = 4.21 × 10⁻³ nm⁻¹, so 1/κ_d = 238 nm.

Into the gold: κ_m = 8.976 × 10⁻³ × √(16² / 14.23) = 8.976 × 10⁻³ × 4.241 = 3.81 × 10⁻² nm⁻¹, so 1/κ_m = 26 nm.

Two things to take from this. First, the sensing volume extends a couple of hundred nanometres into solution — which is why a 100–200 nm dextran hydrogel is a sensible surface architecture and a 5 µm cell is not something SPR sees in its entirety. Second, the field barely enters the gold, which is why a ~50 nm film is thick enough to be optically continuous and thin enough for light on the prism side to reach the far interface at all.

Verification: both numbers are recomputed from the decay-constant expression every time this site is built, and the build fails if either drifts. The dielectric constants used are consistent with the measured optical constants of gold (Johnson & Christy, 1972).

Now plot equation 2.5 alongside the dispersion of ordinary light in the dielectric, ω = ck/n_d. The light line always lies to the left of the plasmon curve: at any given frequency, the surface plasmon carries more momentum than a photon of the same frequency travelling in the same medium. The two curves never cross except trivially at the origin.

Dispersion relations. The light line in water never meets the plasmon branch — but the light line in glass does. Drag the incidence angle to sweep the accessible line between the two.

The Kretschmann configuration#

The trick that won is embarrassingly simple. Come at the interface from a higher index medium. Light inside a glass prism of index n_p arriving at angle α has in-plane momentum

k_x = (ω/c) · n_p · sin α

(2.7)

where

n_prefractive index of the prism (or of the high-index glass slide)

which can be tuned by changing α, and whose maximum n_p·ω/c exceeds the plasmon momentum at the gold–water interface. Somewhere between the critical angle and grazing incidence there is an angle where equation 2.7 equals equation 2.5. Setting them equal gives the resonance condition:

n_p sin θ_SPR = √( ε_m ε_d / (ε_m + ε_d) )

(2.8)

where

θ_SPRthe resonance angle — the angle of the reflectivity minimum

This single equation is the instrument. ε_d is the only term that changes during an experiment, so θ_SPR tracks the refractive index of the layer within the evanescent field, and a plot of θ_SPR against time is a sensorgram.

Otto proposed one geometry in 1968, with a thin air gap between prism and metal (Otto, 1968). Kretschmann and Raether published the alternative a few months later in the same year: evaporate the metal film directly onto the prism face, and let the evanescent wave from total internal reflection at the glass–metal boundary tunnel through the ~50 nm film to excite the plasmon on its far side (Kretschmann & Raether, 1968). Kretschmann then worked out the quantitative relationship between film thickness, dielectric function and the shape of the resulting reflectivity dip (Kretschmann, 1971).

Kretschmann’s geometry won for a mundane reason that turned out to be decisive for biosensing (Jönsson et al., 1991): the sample never touches the optics. Light approaches from the glass side, so the liquid can be turbid, coloured, or full of cells without affecting the beam path. Otto’s configuration requires a sub-micrometre gap filled with the sample, which is impossible to maintain with flowing buffer.

Interactive Kretschmann setup: rotate the incident beam and watch the reflectivity curve develop its dip. Add a bound layer and see the dip shift.

A short history, and who actually did what#

SPR has an unusually well-documented origin, and the sequence is worth knowing because it shows how long a physical curiosity can sit unexplained.

Year	Who	What
1902	R. W. Wood	Observes unexplained dark and bright bands in light reflected from a ruled diffraction grating (Wood, 1902)
1907	Lord Rayleigh	First theoretical attempt, attributing the effect to a diffracted order at grazing emergence (Rayleigh, 1907)
1941	U. Fano	Separates the anomaly into a sharp edge and a broad resonance; the resonance is the surface wave (Fano, 1941)
1968	A. Otto	Excites surface plasmons optically using frustrated total reflection across an air gap (Otto, 1968)
1968	E. Kretschmann & H. Raether	Excite them through a metal film evaporated on the prism — the configuration used today (Kretschmann & Raether, 1968)
1971	E. Kretschmann	Quantitative theory linking dip shape to the metal’s optical constants (Kretschmann, 1971)
1982	C. Nylander, B. Liedberg, T. Lind	First use of SPR as a chemical sensor, for gas (Nylander et al., 1982)
1983	B. Liedberg, C. Nylander, I. Lundström	First label-free immunoassay by SPR (Liedberg et al., 1983)
1990	S. Löfås & B. Johnsson	Carboxymethyl dextran hydrogel surface (Löfås & Johnsson, 1990)
1990	U. Jönsson and colleagues, Pharmacia Biosensor	First commercial SPR biosensor: optics, microfluidics and sensor chip integrated into one instrument (Jönsson et al., 1991)
1991	S. Sjölander & C. Urbaniczky	Integrated microfluidics; SPR becomes routine (Sjölander & Urbaniczky, 1991)

Sixty-six years from observation to explanation, eighty-one to a biosensor.

Sources cited on this page

Listed alphabetically. Each badge records whether the bibliographic record was confirmed against Crossref. unverified marks a real, deliberately chosen source whose volume and page numbers we have not yet machine-checked — it is not a comment on the science.

Cullen et al., 1987D. C. Cullen, R. G. W. Brown, C. R. Lowe (1987). Detection of immuno-complex formation via surface plasmon resonance on gold-coated diffraction gratings. Biosensors 3, 211–225. doi:10.1016/0265-928X(87)85002-2 unverified
Grating coupling rather than prism coupling — the alternative route to supplying the missing momentum.
Drude, 1900P. Drude (1900). Zur Elektronentheorie der Metalle. Annalen der Physik 306, 566–613. doi:10.1002/andp.19003060312 unverified
The free-electron model of metals, from which the negative permittivity that makes surface plasmons possible falls out.
Fano, 1941U. Fano (1941). The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves). Journal of the Optical Society of America 31, 213–222. doi:10.1364/JOSA.31.000213 verified
Separated Wood’s anomaly into a sharp Rayleigh edge and a broad resonance — the latter being what we now call the surface plasmon.
Johnson & Christy, 1972P. B. Johnson, R. W. Christy (1972). Optical constants of the noble metals. Physical Review B 6, 4370–4379. doi:10.1103/PhysRevB.6.4370 unverified
The measured dielectric functions of gold, silver and copper. Still the standard tabulation used in essentially every plasmonic simulation, including the one behind the interactive reflectivity figure on this site.
Jönsson et al., 1991U. Jönsson, L. Fägerstam, B. Ivarsson, B. Johnsson et al. (1991). Real-time biospecific interaction analysis using surface plasmon resonance and a sensor chip technology. BioTechniques 11, 620–627. unverified
The paper describing the first commercial SPR biosensor as an integrated system — optics, microfluidics and sensor chip together. The primary source for what that instrument was and did.
Jorgenson & Yee, 1993R. C. Jorgenson, S. S. Yee (1993). A fiber-optic chemical sensor based on surface plasmon resonance. Sensors and Actuators B 12, 213–220. doi:10.1016/0925-4005(93)80021-3 unverified
Wavelength-interrogated SPR on an optical fibre, with no moving parts.
Kretschmann & Raether, 1968E. Kretschmann, H. Raether (1968). Radiative decay of non-radiative surface plasmons excited by light. Zeitschrift für Naturforschung A 23, 2135–2136. doi:10.1515/zna-1968-1247 verified
Two pages that define the geometry used by essentially every commercial SPR instrument built since.
Kretschmann, 1971E. Kretschmann (1971). Die Bestimmung optischer Konstanten von Metallen durch Anregung von Oberflächenplasmaschwingungen (The determination of the optical constants of metals by excitation of surface plasmons). Zeitschrift für Physik 241, 313–324. doi:10.1007/BF01395428 verified
The quantitative treatment: how dip position, depth and width relate to the metal’s dielectric function and film thickness.
Liedberg et al., 1983B. Liedberg, C. Nylander, I. Lundström (1983). Surface plasmon resonance for gas detection and biosensing. Sensors and Actuators 4, 299–304. doi:10.1016/0250-6874(83)85036-7 verified
The founding paper of SPR biosensing: IgG adsorbed on a silver film, anti-IgG detected from solution, no label.
Löfås & Johnsson, 1990S. Löfås, B. Johnsson (1990). A novel hydrogel matrix on gold surfaces in surface plasmon resonance sensors for fast and efficient covalent immobilization of ligands. Journal of the Chemical Society, Chemical Communications, 1526–1528. doi:10.1039/C39900001526 verified
The carboxymethyl dextran hydrogel — the single most consequential surface-chemistry paper in the field.
Nuzzo & Allara, 1983R. G. Nuzzo, D. L. Allara (1983). Adsorption of bifunctional organic disulfides on gold surfaces. Journal of the American Chemical Society 105, 4481–4483. doi:10.1021/ja00351a063 verified
The discovery that sulfur-containing organics self-assemble into ordered monolayers on gold — the chemical foundation of every SPR chip.
Nylander et al., 1982C. Nylander, B. Liedberg, T. Lind (1982). Gas detection by means of surface plasmon resonance. Sensors and Actuators 3, 79–88. doi:10.1016/0250-6874(82)80008-5 verified
SPR used as a transducer for the first time — for gas, one year before biomolecules.
Otto, 1968A. Otto (1968). Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection. Zeitschrift für Physik 216, 398–410. doi:10.1007/BF01391532 verified
The Otto configuration: prism separated from the metal by a thin air/dielectric gap.
Raether, 1988H. Raether (1988). Surface Plasmons on Smooth and Rough Surfaces and on Gratings. Springer Tracts in Modern Physics, vol. 111. doi:10.1007/BFb0048317 unverifiedbook
The standard monograph on surface plasmon physics, by one of the two people who first excited them optically.
Rayleigh, 1907Lord Rayleigh (1907). On the dynamical theory of gratings. Proceedings of the Royal Society A 79, 399–416. doi:10.1098/rspa.1907.0051 unverified
First attempt at a physical account of Wood’s anomaly, in terms of a diffracted order emerging at grazing angle.
Sjölander & Urbaniczky, 1991S. Sjölander, C. Urbaniczky (1991). Integrated fluid handling system for biomolecular interaction analysis. Analytical Chemistry 63, 2338–2345. doi:10.1021/ac00020a025 verified
The microfluidic cartridge that made SPR a routine instrument rather than a physics experiment.
Wood, 1902R. W. Wood (1902). On a remarkable case of uneven distribution of light in a diffraction grating spectrum. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 4, 396–402. doi:10.1080/14786440209462857 verified
The original observation of the dark bands now called Wood’s anomalies. Wood did not know he was looking at surface plasmons; nobody would for 66 years.