In physics, plasma is an ionized state of matter. The presence of free electrons which is not bound to an atom or molecule makes the plasma electrically conductive. Therefore plasma, the electron density, responds strongly to electromagnetic fields and resonates at a specific frequency defined
where ωp is the plasma frequency, n is the electron density, e is the electric charge of a electron, and m is the mass. Then according to the wave equation,
Eq. (2‐30) indicates that if the frequency of incident light is below the plasma frequency, the light can not penetrate through the metal and the skin depth is
k / 1 .
As photons are quantization of light, plasmon is the quasiparticle resulting from the quantization of plasma oscillations. Thus, plasmons are collective oscillations of the free electron gas density, often at optical frequencies. If those plasmons occur at the interface between a vacuum or material with a positive dielectric constant and that of a negative dielectric constant and are confined to surfaces, they are defined as surface plasmons. Consequently, surface plasmons
are charge density waves propagating along a dielectric‐metal interface.
Furthermore, the quasiparticles resulting from the strong coupling of electromagnetic waves with an electric or magnetic dipole‐carrying excitation are called polaritons. Therefore, a surface plasmon polariton (SPP) is a combined excitation consisting of a surface plasmon and a photon. Since the wave is on the boundary of the metal and the external medium, these oscillations are very sensitive to boundary conditions.
Fig. 2‐7 Schematic illustration of surface plasmons propagating along the dielectric‐metal interface induced by a p‐polarized incident light
To characterize surface plasmon polaritons, consider a p‐polarized electromagnetic wave propagating along a dielectric‐metal interface, as shown in Fig. 2‐7. According to Maxwell’s equations and boundary conditions, the dispersion relation of surface plasmon polaritons can be obtained,
where εd and εm is the dielectric constant of dielectric material and metal, respectively. For most dielectric materials, their dielectric constants are positive;
while dielectric constants of metal are negative and usually their absolute values are greater than that of dielectric materials when the frequency of incident light is below the plasma frequency. Therefore, from Eq. (2‐31a),
wavevector along x direction is real, i.e. surface plasmons propagate along the metallic surface.
The wavevector of surface plasmons along z direction, i.e. perpendicular to the surface, is always imaginary according to Eq. (2‐31b). It implies that magnitude of electromagnetic fields along z direction decays exponentially.
Therefore, we can conclude that surface plasmons are an evanescent field which will not propagate out of the surface. Since the dielectric constants of real metal are also complex, the imaginary part represents exponential decay on magnitude when surface plasmons propagate along the surface.
It means that the magnitude of the wavevector of surface plasmons is greater than that of incident light. Fig. 2‐8 shows the comparison of dispersion curves
for fulfilling the momentum matching condition. There are two common methods: either introducing a periodic structure, such as gratings, corrugations, or a hole array, to provide an additional wavevector or inducing attenuated total reflection (ATR) in the interface to increase the wavevector of incident light,
Fig. 2‐9 Excite surface plasmon polaritons by (a) a grating structure and (b) attenuated total reflection
Due to the sensitive dependence of excitation of surface plasmons on the boundary conditions of a metal‐dielectric interface, the excitation of surface plasmons is used in surface plasmon resonance (SPR). In SPR, the excitation of surface plasmons as a function of incident angle or wavelength is obtained by detecting the reflected power from a prism coupler. Any perturbation in the refraction index, the absorption, or topography will reflect a substantial change in SPR reflectivity. Therefore, this technique can be widely used to observe insignificant variation in thickness, density fluctuations, or molecular adsorption.
Another remarkable application of surface plasmons is to enhance the transmission through subwavelength apertures. T. W. Ebbesen et al.
demonstrated extraordinary transmission through subwavelength hole arrays or a aperture surrounded with corrugations that breaks through the theoretic
ω
limit predicted by Bethe’s theory, as shown in Fig. 2‐10 [35‐36]. According to the surface plasmon theory, with the aid of a periodic structure, such as hole arrays or corrugations, surface plasmon waves that propagate along the interface can be induced. When the period of the structure is half of the effective wavelength of the SPP mode, SPP standing waves which are a combination of electromagnetic waves and surface charges are generated. The electromagnetic field perpendicular to the surface is evanescent, i.e. an exponential decay with the distance away from the surface. It indicates a non‐radiative wave to prevent power propagating from the surface. The enhancement of the transmission can be explained by considering the interaction between the incident light and the surface plasmon modes.
Fig. 2‐10 Transmission spectrum of (a) a circular aperture surrounded with a corrugation structure and (b) a hole array
Consider a hole array in a metal film of dielectric constant εm sandwiched with two lossless dielectric media of dielectric constants ε1 and ε2, respectively, as shown in Fig. 2‐11. In the absence of resonant SPP modes, the energy transfer between the photons on the opposite side is considerably inefficient and thus the transmission is extremely low. In contrast, resonant SPP modes excited by
photons at one of the interfaces enhance the transmission by the diffraction of the evanescent field resulting from SPP modes, indicating the scatter of the field leading to the formation of transmitted light.
In the case of a thin film, interaction between the opposite interfaces makes surface plasmons modes split into two types: symmetric modes of lower frequency and anti‐symmetric modes of higher frequency. Only anti‐symmetric SPP modes induce field enhancement near the metal surface because of longer propagation length, which is inversely proportional to the square of the film thickness. Therefore, the tunneling effect through a thin metal film has to be taken into consideration and dominates the transmitted field.
If the metal film is thick enough, the interaction between SPPs in the opposite interfaces is ignored. Then the corresponding SPP modes induced by incident light propagate independently along the surface with wave vector ksp and decays from the surface exponentially with the attenuation constant kz. The interaction between the incident light and the SPP modes will occurs in three different ways.
Fig. 2-11 Configuration of a hole array in a metal film sandwiched with two dielectric materials
1. In the entrance interface, incident light couples to SPP modes. The ksp2
ksp1
ε1
ε2
entrance plane
exit plane incident light
emitted light
dielectric 1
dielectric 2
εm metal film
hole array
evanescent field is enhanced by SPP modes and the transmission through the holes is increased. In the meantime, the field is also scattered by the periodic array so that some waves are produced accordingly.
2. Incident light cannot couple to SPP modes in the entrance interface. But the incident light that is weakly transmitted through the holes fulfills the boundary conditions of the exit plane so that SPP modes are induced in the exit interface. The electric field enhanced by the SPP modes increases the transmission. Some transmitted light out of the exit plane is also generated as a result of the scattering of the field.
3. If the boundary conditions allow the excitation of SPP modes to take place in both sides of the metal film, the field is influenced by the SPP modes in two interfaces. As a result, the incident field near the interfaces is enhanced by the induced field resulting from the SPP modes.
Furthermore, the interaction between double resonances on both sides delivers energy through the film without exponential decrease and thus results in strong field enhancement. Therefore, SPP modes function like an electromagnetic energy carrier. The incident energy is converted to non‐radiative SPP resonance by coupling light into SPP modes and then transferred to a propagating wave in the form of re‐radiation from the field induced by SPP modes. According to this model, the transmitted field is predicted to be enhanced up to 104 times than that without the SPP modes.