Theory
Electromagnetism
In order for SPR to occur photons and electrons must couple to form a plasmon. This is based upon their electro-magnetic waves coupling, the electromagnetic equation for a plasmon (derived from Maxwell's equations) is
When k is imaginary this equation describes an evanescent wave. k for a wave propagating through a material is given by , where is the dielectric constant, an optical property. For metals the dielectric constant is
, where is the frequency of excitation, is the viscous damping parameter, and . The in the equation for
represents the density of electrons in the material.
Thickness Dependence
To talk about thickness dependence we start with the Fresnel's equations for the reflectivity of light in p polarization, and apply them to the interfaces between the air (0) and the prism glass (1), and the prism glass and the silver foil (2) as shown in equations below, where
When the surface plasmon polariton line intersects with the light line coupling occurs. In the graph (left) the light line is for light in a vacuum, and the SPP line and light line will never intersect. We will be using a light whose wave vector is modified by sending it through a glass prism [1]. This modification (via Snell's law) will shift the light light to the right so there is an intersect between the two lines and coupling will occur.
for each is the complex dielectric function of the material [2].
These equations for reflectivity at the interfaces can be evaluated under the conditions that the magnetic and electric field of the electromagnetic wave must be continuous at the boundaries between the materials for the total reflectivity of the system with a dependence on thickness [2]. In the equation below
is the wavelength of the incident field, and
is the thickness of the silver film [2].
We take the complex conjugate of the above equation to obtain the reflectance values we will measure as it has both real and imaginary terms [2].
This equation represents the reflectance values, and when plotted against theta for various thicknesses will give the expected SPR graph.
