# Mathematics of Surface Plasmon Resonance

Surface plasmons are particle waves of the free electron plasma on a metal surface, which can be excited by p-polarized light under the resonance condition (Figure1). A theoretical mathematical description for the resonance condition can be obtained by solving the Maxwell equations for a multilayer optical system , which provides the following mathematical solution for the resonance condition: where ω is the angular frequency of light, c is speed of light in vacuum, and ε0, ε1 and ε2 are the permittivities of the prism, SPR metal layer and the adjacent medium, respectively. Figure 1: Schematic representation of the Kretschmann configuration of the SPR used in the study. A) When the projected wave vector of the incident light do not match the wave vector of surface plasmons no excitation occurs. B) At a certain incident light angle the projected wave vector of the incident light matches the wave vector of surface plasmons, which is dependent on the wavelength of light used and the dielectric properties of the prism (e0), metal layer (e2) and the surrounding medium (e1). See the animation of how the full SPR curve is formed here.

The permittivity and refractive index of materials can be used in their complex forms:  Where ε, ε' and ε" are the complex permittivity, real- and imaginary part of the complex permittivity, respectively. ñ, n and k are the complex refractive index, real- and imaginary part of the complex refractive index, respectively. Permittivity and refractive index have the following relationship: A general answer for a multilayered system linked to measurable or controllable variables can be solved using a transfer matrix formalism of 2´2 matrices. The overall formalism has already been published several times, and it is not in the scope of this article to discuss it in detail again. [2-3]

In practice this matrix formalism is solved by mathematical fitting tools, or by dedicated software tools developed for this, such as the MP-SPR Navi™ LayerSolver™.

## Complex refractive index and extinction coefficients

The refractive is in reality complex number , where the real part n corresponds to the refraction of light, and complex coefficient k to the extinction or absorption of light by the material. The complex refractive index for a sample material can be obtained by using the BioNavis LayerSolver software, designed to calculate thicknesses and optical constants of materials.

## Density and swelling

Density and swelling require MP-SPR measurement with two or more wavelengths. The multiple-wavelength method is able to calculate the real refractive index of a material layer. This layer also typically includes some solvent which lowers the measurement refractive index. The refractive index is a linear product of the different components of a layer: and The refractive index of solvent is either known (from literature) or easily obtained (i.e. product from the SPR fitting of TIR angle for the bulk medium). Similarly the refractive index of the bulk sample material is typically known, or can be obtained by other experiments. When these are obtained, the density of the material can be calculated by solving the volume fraction x.

## Multivariable SPR Experiments

The challenge of SPR in characterizing layer thickness and refractive index is the fact that the SPR spectrum measured in one set of conditions is in practice not sensitive to the unique differences in d and n, and only a continuum solution for the surface plasmon wave vector (ksp) proportional to d and n can be deducted: In practical experiments we can assume that ksp contains all the information and constants that cause the differences in the experimental SPR spectra measured at several wavelengths or in two different media. Hence, we can simplify the relationship for mathematical purposes in the following way. If we measure the SPR spectrum in two different media with a large enough difference in n, then a unique solution for the final sample layer can be relatively easily calculated from the intersection of the two continuum solutions when d1 = d2 = d and n1 = n2 =n, i.e. A similar but slightly more complex approach is to use a multiwavelength approach, as n also has a wavelength dependency, i.e., dn/dλ. However, for relatively small changes, this relationship can to a good approximation be assumed to be linear. Hence, this approximation allows us to find a unique solution for d and n by solving the following equation system for two or three wavelenths: where and It is worth mentioning that the discussion above is only valid for sample layers that do not absorb light at the wavelengths used for SPR spectrum measurement, i.e., for k = 0, which is often the case for organic sample layers. If k ≠ 0, then there is actually a unique solution for the sample layer in the ksp = d(n + ik) space, and the above approach would be unnecessary.

## References

 Granqvist, N., Biomimetic Interfaces for Surface Sensitive Drug Discovery Techniques, http://urn.fi/URN:ISBN:978-952-10-9829-1

 Albers W, Vikholm-Lundin I (2010) Surface Plasmon Resonance on Nanoscale Organic Films. In: Carrara S (ed) Nano-Bio-Sensing. Springer

 Sadowski JW, Korhonen IK, Peltonen JP (1995) Characterization of thin films and their structures in surface plasmon resonance measurements. Optical Engineering 34 (9):2581-2586

How does the SPR peak respond to layer thickness change in thickness in air? Measure of thick samples (waveguide effect). Thickness up to few micrometers (with light nonabsorbing samples) can be measured. See animation.

How does the SPR peak respond to change in thickness in water? Surface measured with two different wavelengths (here 785 nm and 670 nm). The thickness remains the same in both cases. The difference in the curves is due to the change of refractive index in relation to the measured wavelength. See animation

How does the SPR peak respond to change in thickness when the sample absorbs light? Light absorbing samples, such as porphyrins or gold and silver nanoparticles, cause intensity changes to meaured curves. See animation.