1 Theory
The theory of Surface Plasmon Resonance draws from three main areas of physics, Solid State physics, Electromagnetism, and Optics.
Overview
Plasmon resonance is a solid-state plasma phenomenon observed when a photon is absorbed by an electron (referred to as a plasmon) and thereby excited into resonance (kittel). Surface plasmon resonance (SPR) refers to resonance which occurs across the boundary between materials with differing dielectric constants, specifically between an insulator and a conductor.
One method of establishing SPR involves shining p-polarized light through a glass prism onto a thin metal foil (connolly). At a specific angle (discussed below), the photons are absorbed by the plasmons, causing resonance; thus, observing an intensity minimum in the reflected light corresponds to the resonance condition being met. This resonance condition is derived from the principles of the conservation of energy and the conservation of momentum, and occurs when the electromagnetic waves of the photon and electron match. (Connolly). This allows the wave functions to couple, permitting the absorption of the photon by the plasmon (kittel).
Electromagnetism
SPR is predicated on the coupling between the photon electromagnetic wave and the electron's electromagnetic wave. Each of these waves are described by the modified wave equation, which is ultimately derived from Maxwell's equations. The solution to the modified wave equation is as follows,
where z is the transverse variable (direction perpendicular to the thin foil), t is time, omega is the angular frequency (which is the same for both the incident light and the plasmon during resonance), and ~k is the complex wave vector.
In order for the two waves to couple, their wave vectors (k) and frequencies (omega) must equal each other. The dispersion relation, which describes the relationship between a wave's frequency and its wave vector, graphically demonstrates the coupling condition. In the dispersion relation image on the left the wave vector of light in a vacuum is represented by the red curve and the wave vector of the plasmons is represented in blue. The frequency of the wave, plotted on the y-axis, corresponds to its energy, and the wave vector, plotted on the x-axis, represents its momentum. Thus, at the intersection of the dispersion curves, both waves have the same frequency and wave vector, satisfying both conservation of energy and conservation of momentum. This is the coupling condition. There are two additional noteworthy aspects to the dispersion relation. First, the plasmon dispersion curve asymptotically approaches the plasmon frequency; in addition, the dispersion curve for light in air (green) never intersects this dispersion curve for plasmons. Thus, for coupling to occur, the slope of the photon dispersion curve must be decreased. Physically this corresponds to attenuating the speed of the photons, which is accomplished by passing the laser beam through a medium with a higher index of refraction, such as glass. This modifies the wave vector according to Snell's law.
The final coupling condition is expressed
Optics
Under this resonance condition, the thickness dependence of SPR can be derived from Fresnel's equations. Fresnel's equations model the reflectance at the boundary between two materials. We are interested in the thickness dependence with respect to the overall reflectance, so we must use Fresnel's equations twice to derive this expression. If we let the subscripts 0, 1, and 2 denote the air, glass, and metal foil respectively, the following two equations model the reflectance between the air-glass and glass-foil interfaces:
where epsilon are the complex-valued dielectric functions for the air, glass, and prism,
,
omega is the frequency of the laser, and theta is the angle of incidence. Deriving the overall reflectance, which includes the thickness dependence, requires evaluating these two equations with the additional constraints that the Electric field and Magnetic field (H) components of the light (electromagnetic) wave are continuous at the boundaries between the air-glass and glass-metal foil interface. Solving this boundary condition problem yields a relation between reflectivity and angle of incidence, with a complex exponential dependence on thickness:
What is observed in the lab is the total real reflectivity, given by
This relation gives the model for the thickness dependence that we tested.
References:
A primary reference for the thickness dependence was the SPR Lab developed by Portland State University, which is attached as a PDF to this webpage.