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# Surface plasmon

Surface plasmons, also referred to in the literature as surface plasma polaritons, are fluctuations in the electron density at the boundary of two materials. Plasmons are the collective vibrations of an electron gas (or plasma) surrounding the atomic lattice sites of a metal. When plasmons couple with a photon, the resulting particle is called a polariton. This polariton propagates along the surface of the metal until it decays, either by absorption, whereupon the energy is converted into phonons, or by a radiative transition into a photon. Surface Plasmons were first reported in 1957 by R.H. Ritchie[1]. In the following two decades, Surface Plasmons were extensively studied by many scientists, the foremost of whom were Prof. Dr. Heinz Raether, E. Kretschmann, and A. Otto.

## Excitation

Figure 1: (a) Kretschmann and (b) Otto configuration of an Attenuated Total Reflection setup for coupling surface plasmons. In both cases, the surface plasmon propagates along the metal/dielectric interface

Figure 2: Grating Coupler for Surface Plasmons. The wave vector is increased by the spatial frequency

Surface plasmons can be excited by both electrons and photons. Excitation by electrons is realized by shooting electrons into a metal. As the electrons scatter, energy is transferred into the plasma. The component of the scattering vector parallel to the surface results in the formation of a surface plasmon.

Excitation by photons requires the use of a coupling medium such as a prism or grating to match the photon and surface plasmon wave vectors. A prism can be positioned against a thin metal film in the Kretschmann configuration or very close to a metal surface in the Otto configuration (Figure 1). A grating coupler matches the wave vectors by increasing the wave vector by the spatial frequency of the grating (Figure 2). This method, while less frequently utilized, is critical to the theoretical understanding of the impact of surface roughness.

## Dispersion relationship

The electric field of a propagating electromagnetic wave can be expressed

$E= E_{0}\exp[i(k_{x} x + k_{z} z -\omega t)]\,$

where k is the wave number and ω is the frequency of the wave. By solving Maxwell's equations for the electromagnetic wave at an interface between two materials with relative dielectric constants $\varepsilon_1$ and $\varepsilon_2$ (see figure 3) with the appropriate continuity relationships the boundary conditions are found to be[2]

$\frac{k_{z1}}{\varepsilon_1} + \frac{k_{z2}}{\varepsilon_2} = 0$

and

$k_{x}^2+k_{zi}^2=\varepsilon_i \left(\frac{\omega}{c}\right)^2 \qquad i=1,2$

where c is the speed of light in a vacuum. Solving these two equations, the dispersion relationship for a wave propagating on the surface is

$k_{x}=\frac{\omega}{c} \left(\frac{\varepsilon_1\varepsilon_2}{ \varepsilon_1+\varepsilon_2}\right)^{1/2}.$

In the free electron model of an electron gas, the dielectric constant is[3]

$\varepsilon(\omega)=1+\frac{\omega_{P}^2}{\omega^2},$

where the plasma frequency is

$\omega_{P}=\frac{4\pi\times n \times e^2}{m^*}$

where n is the electron density, e is the charge of the electron, and m* is the effective mass of the electron. The dispersion relationship is plotted in Figure 4. At low k, the surface plasmon behaves like a photon, but as k increases, the dispersion relationship bends over and reaches an asymptotic limit. Since the dispersion curve lies to the right of the light curve, ω = k * c, the surface plasmon is non-radiative. Finally solving these equations, the maximum frequency of the surface plasmon is calculated to be

$\omega_{SP}=\omega_P/\sqrt{1+\varepsilon_2}.$

In the case of air, this result simplifies to

$\omega_{SP}=\omega_P/\sqrt{2}.$

If we assume that $\varepsilon_2$ is real and $\varepsilon_2>0$, then it must be true that $\varepsilon_1<0$, a condition which is satisfied in metals. EM waves passing through a metal experience damping due to electrical conductivity and electron-core interactions. These effects show up in as an imaginary component of the dielectric function. The dielectric function of a metal is expressed $\varepsilon_1=\varepsilon_1'+i \varepsilon_1''$ where $\varepsilon_1'$ and $\varepsilon_1''$ are the real and imaginary parts of the dielectric function, respectively. Generally $|\varepsilon_1'| \gg \varepsilon_1''$ so the wavenumber can be expressed in terms of its real and imaginary components[2]

$k_{x}=k_{x}'+i k_{x}''=\left[\frac{\omega}{c} \left( \frac{\varepsilon_1' \varepsilon_2}{\varepsilon_1' + \varepsilon_2}\right)^{1/2}\right] + i \left[\frac{\omega}{c} \left( \frac{\varepsilon_1' \varepsilon_2}{\varepsilon_1' + \varepsilon_2}\right)^{3/2} \frac{\varepsilon_1''}{2(\varepsilon_1')^2}\right].$

The wave vector gives us insight into physically meaningful properties of the electromagnetic wave such as its spatial extent and coupling requirements for wave vector matching.

## Propagation length and skin depth

As a surface plasmon propagates along the surface, it quickly loses its energy to the metal due to absorption. The intensity of the surface plasmon decays with the square of the electric field, so at a distance x, the intensity has decreased by a factor of exp[ − 2kx''x]. The propagation length is defined as the distance for the surface plasmon to decay by a factor of 1 / e. This condition is satisfied at a length[4]

$L=\frac{1}{2 k_{x}''}.$

Likewise, the electric field falls off exponentially normal to the surface. The skin depth is defined as the distance where the electric field falls off by 1 / e. The field will fall of at different rates in the metal and dielectric medium and the skin depth in each medium can be expressed[4]

$z_{i}=\frac{\lambda}{2 \pi} \left(\frac{\varepsilon_1'+\varepsilon_2}{\varepsilon_i^2} \right)^{1/2}$

where i indicates the medium of propagation. Surface plasmons are very sensitive to slight perturbations within the skin depth and because of this, surface plasmons are often used to probe inhomogeneities of a surface.

## Effects of roughness

In order to understand the effect of roughness on surface plasmons, it is beneficial to first understand how a plasmon is coupled by a grating (Figure 2). When a photon is incident on a surface, the wave vector of the photon in the dielectric material is smaller than that of the surface plasmon. In order for the photon to couple into a surface plasmon, the wave vector must increase by Δk = kSPkx,photon. The harmonics of a periodic grating add to the wave vector to match the terms.

$k_{SP}=k_{x, \text{photon}} \pm n\ k_\text{grating}=\frac{\omega}{c} \sin{\theta_0} \pm n \frac{2\pi}{a}$

where kgrating is the wave vector of the grating, θ0 is the angle of incidence of the incoming photon, a is the grating period, and n is an integer.

Rough surfaces can be thought of as the superposition of many gratings of different periodicities. Kretschmann proposed[5] that a statistical correlation function be defined for a rough surface

$G(x,y)=\frac{1}{A}\int_A z(x',y')\ z(x'-x,y'-y)\, dx'\, dy'$

where z(x,y) is the height above the mean surface height at the position (x,y), and A is the area of integration. Assuming that the statistical correlation function is be Gaussian of the form

$G(x,y)=\delta^2\exp\left(-\frac{r^2}{\sigma^2}\right)$

where δ is the root mean square height, r is the distance from the point (x,y), and σ is the correlation length, then the Fourier transform of the correlation function is

$|s(k_\text{surf})|^2=\frac{1}{4 \pi} \sigma^2 \delta^2 \exp \left( - \frac{\sigma^2 k_\text{surf}^2}{4}\right)$

where s is a measure of the amount of each spatial frequency ksurf which help couple photons into a surface plasmon.

If the surface only has one Fourier component of roughness (i.e. the surface profile is sinusoidal), then the s is discrete and exists only at $k=\frac{2\pi}{a}$, resulting in a single narrow set of angles for coupling. If the surface contains many Fourier components, then coupling becomes possible at multiple angles. For a random surface, s becomes continuous and the range of coupling angles broadens.

As stated earlier, surface plasmons are non-radiative. When a surface plasmon travels along a rough surface, it usually becomes radiative due to scattering. The Surface Scattering Theory of light suggests that the scattered intensity dI per solid angle dΩ per incident intensity I0 is[6]

$\frac{dI}{ d \Omega\ I_{0}}=\frac{4 \sqrt{\varepsilon_{0}}}{\cos{\theta_0}}\frac{\pi^4}{\lambda^4}|t_{012}^p|^2 \ |W|^2 |s(k_\text{surf})|^2$

where | W | 2 is the radiation pattern from a single dipole at the metal/dielectric interface. If surface plasmons are excited in the Kretschmann geometry and the scattered light is observed in the plane of incidence (Fig. 4), then the dipole function becomes

$|W|^2=A(\theta,|\varepsilon_{1}|)\ \sin^2{\psi} \ [(1+\sin^2 \theta /|\varepsilon_1|)^{1/2} - \sin{\theta}]^2$

with

$A(\theta,|\varepsilon_1|) = \frac{|\varepsilon_1|+1}{|\varepsilon_1|-1} \frac{4}{1+\tan{\theta}/| \varepsilon_1|}$

where ψ is the polarization angle and θ is the angle from the z-axis in the xz-plane. Two important consequences come out of these equations. The first is that if ψ = 0 (s-polarization), then | W | 2 = 0 and the scattered light $\frac{dI}{ d \Omega\ I_{0}}=0$. Secondly, the scattered light has a measurable profile which is readily correlated to the roughness. This topic is treated in greater detail in reference [6].

## Experimental applications

The excitation of surface plasmons is frequently used in an experimental technique known as surface plasmon resonance (SPR). In SPR, the maximum excitation of surface plasmons are detected by monitoring the reflected power from a prism coupler as a function of incident angle or wavelength. This technique can be used to observe nanometer changes in thickness, density fluctuations, or molecular adsorption.

In surface second harmonic generation, the second harmonic signal is proportional to the square of the electric field. The electric field is stronger at the interface because of the surface plasmon. This larger signal is often exploited to produce a stronger second harmonic signal.

## References

1. ^ Ritchie, R. H. (June 1957). "Plasma Losses by Fast Electrons in Thin Films". Physical Review 106 (5): 874-881. doi:10.1103/PhysRev.106.874.
2. ^ a b Raether, Heinz (1988). Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer Tracts in Modern Physics 111. New York: Springer-Verlag. ISBN 0-387-17363-3.  (Germany: ISBN 3-540-17363-3)
3. ^ Kittel, Charles (1996). Introduction to Solid State Physics, 8th ed., Hoboken, NJ: John Wiley & Sons. ISBN 0-471-41526-X.
4. ^ a b Homola, Jirí (2006). Surface Plasmon Resonance Based Sensors. Berlin: Springer-Verlag. ISBN 3-540-33918-3.
5. ^ (German) Kretschmann, E. (April 1974). "Die bestimmung der oberflächenrauhigkeit dünner schichten durch messung der winkelabhängigkeit der streustrahlung von oberflächenplasmaschwingungen". Optics Communications 10 (4): 353-356. doi:10.1016/0030-4018(74)90362-9.
6. ^ a b (German) Kretschmann, E. (August 1972). "The angular dependence and the polarisation of light emitted by surface plasmons on metals due to roughness". Optics Communications 5 (5): 331-336. doi:10.1016/0030-4018(72)90026-0.