SURFACE PLASMON RESONANCE AT A SILVER-AIR INTERFACE

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Theory

Surface Plasmon Waves

In a metal with dielectric function ε₁, Maxwell’s wave equation for the electric field is as follows [6]:

where E₁ is the electric field inside material 1, the metal, ω is the frequency of oscillation, c is the speed of light in a vacuum, and r is the position. This means that the magnitude of the wavevector, k, for this wave follows:

which is called the dispersion equation. The wavevector is also shown in the figure to the left, and is necessary to obtain the general solution to the wave equation, which is of the following form [6]:

However, as implied by the figure, the y component can be ignored and the system regarded as 2 dimensional, that is: k_1y= 0. This allows the dispersion equation to be rewritten as:

The dielectric functions of metals are mostly negative, real numbers. Thus fulfilling (4), requires either k_1x or k_1z to be imaginary. Using the boundary conditions for the electric field, as detailed in the following section, it is seen that k_x can be taken to be real and k_z imaginary. Plugging such parameters into (3) results in sustained oscillation of the electric field in the x direction and an exponential decay in the z direction with characteristic length, 1/k_z.

Boundary Conditions and Interface Restraints

Since surface plasmon waves exists on an interface between two materials, the solution to the wave equation is restricted by material 2, the insulator, as well as material 1, the metal. The equations for the electric field in material 2 are the exact same as those for material 1, with the simple substitution of the parameters of material 2, such as its dielectric function, ε₂. Thus, the only additional equations required are the standard electromagnetic boundary conditions on a surface. In the coordinate system established for this experiment, the interface is in the xy plane, with the metal lying above the interfacial plane and the insulator, below. Therefore, the system boundary conditions are [6]:

E_1x= E_2x = E_x

ε₁E_1z= ε2 E_2z

For this coupled wave to exist on each side of the interface, it must satisfy the dispersion equations inside each material and meet the above boundary conditions for continuity. Compiling these restraints gives the dispersion equation of the total surface plasmon wave [1]:

where k_x is the x component of the incident wavevector k. This dispersion relation defines the conditions on k_xand ω that are mapped in this study. It is important to note that s-polarized light, which has k_x= 0, cannot ever satisfy this equation, so this situation was used as a system check during experimentation.

Frequency-Wavenumber Space

The figure to the right shows the dispersion relation of the surface plasmon wave. The case shown is a silver-air interface based on the data obtained by Yang et. al. in 2015 and serves as the literature comparison for this experiment. As discussed earlier, excitation of surface plasmon resonance occurs when there is a coupling of the incident light with the mechanical oscillation of the electrons at the interface. Coupling requires the matching of oscillation frequency and wavevector, and it therefore corresponds with an intersection of two curves in frequency-wavenumber space. Thus, resonance of the surface plasmon with the incident light occurs when their respective frequency and x component of the wavevector are matched at the interface. It is important to note that the solid line in the figure to the right, which represents the dispersion relation of light in a vacuum, never intersects the plasmon curve for the experimental values. This is obvious when considering the slopes of the two, as light will have a slope of c in ω-k space where as the experimental values must have a slope that is strictly less than c. It is for this reason that a glass prism is used; passing light through another material lowers the slope of the curve as [1]:

where n is the refractive index of the prism. This allows the meeting of the requirements for coupling with the surface plasmon wave as noted by the intersection of the dotted and dashed lines in the figure.

Further, having the light be incident at an angle Θ_int allows for light line slope adjustment as well. This is because while not influencing the frequency of the incident wave, it allows for the scaling of the wavevector’s x component by sin(Θ_int). Therefore, at a given frequency, observing a range of incident angles allows a sweep over a range of k_x values. Resonance will be observer at some angle, Θ_int= Θ_plasmon, which can be used for the calculation of resonance conditions knowing the original magnitude of the incident wavevector.

Jesse Grindstaff & Molly Andersen

References:

[1] Pluchery O., Vayron R., and Van K., 2011 “Laboratory experiments for exploring the surface plasmon resonance.” Eur. J. Phys. 32 585–99.

[2] Haes A.J. and Van Duyne R.P., 2002 “A Nanoscale Optical Biosensor: Sensitivity and Selectivity of an Approach Based on the Localized Surface Plasmon Resonance Spectroscopy of Triangular Silver Nanoparticles.” J. ACS 124 (35), 10596-10604.

[3] Barnes W.L., Dereux A. and Ebbesen T.W., 2003 “Surface plasmon subwavelength optics.” Nature 424 824-830.

[4] Drescher, D. G., Ramarkrishnan, N. A., and Drescher, M. J., 2009 “Surface plasmon resonance (SPR) analysis of binding interactions of proteins in inner-ear sensory epithelia.” Meth. Mol. Biol. (493) 323-343.

[5] Yang, H.U., D’Archangel, J., Sundheimer, M.L., Tucker, E., Boreman, G.D., Raschke, M.B., 2015 “Optical dielectric function of silver” Physical Review B 91 234137.

[6] Griffiths, D.J. “Introduction to Electrodynamics.” (Pearson Education, Inc.) 4^th ed. 2013.

[7] Novotny, L., and Hecht, B. “Principles of nano-optics.” (Cambridge University Press, New York, New York). 2006.