# Surface plasmon (or phonon) polaritons

Another radiative phenomenon that is worthwhile to discuss here is the optical plasmon (or phonon) polariton. Plasmons are quasiparticles associated with oscillations of plasma, which is a collection of charged particles such as electrons in a metal or semiconductor [1]. Plasmons are longitudinal excitations that can occur either in the bulk or at the interface. The field associated with a plasmon is confined near the surface, while the amplitude decays away from the interface. Such a wave propagates along the surface, and it is called a surface electromagnetic wave. Surface plasmons can be excited by electromagnetic waves and are important for the study of optical properties of metallic materials, especially near the plasma frequency, which usually lies in the ultraviolet.

In addition to the requirement of evanescent waves on both sides of the interface, the polariton dispersion relations given below must be satisfied [1,2]:

$\frac{k_{1z} }{\varepsilon _{1} } +\frac{k_{2z} }{\varepsilon _{2} } =0 \;\;\;\; for \;TM \;waves$

(1)

$\frac{k_{1z} }{\mu _{1} } +\frac{k_{2z} }{\mu _{2} } =0 \;\;\;\; for \;TE \;waves$

(2)

This means that the sign of permittivity must be opposite for media 1 and 2 in order to couple a surface polariton with a TM wave. A negative Re(ε) exists in the visible and near infrared for metals like Al, Ag, W, and Au. When Eq. (1) is satisfied, the excitation of surface plasmon polariton (SPP) interacts with the incoming radiation and causes a strong absorption. Lattice vibration in some dielectric materials like SiC and SiO2 can result in a negative Re(ε) in the mid-infrared. The associated surface electromagnetic wave is called a surface phonon polariton (SPhP). On the other hand, magnetic materials having negative permeability are necessary to excite a surface polariton for a TE wave. Some metamaterials can exhibit negative permeability in the optical frequencies, and negative index materials exhibit simultaneously negative permittivity and permeability in the same frequency region. Therefore, both TE and TM waves may excite SPPs with negative index materials [2] or with bilayer materials of alternating negative ε and μ, the so-called single negative materials [3].

The condition for the excitation of surface polaritons is that the denominator of Fresnel’s reflection coefficient be zero. A pole in the reflection coefficient is an indication of a resonance. Taking a TM wave for example, one can solve Eq. (1) to obtain [4]

$k_{x} =\frac{\omega }{c} \sqrt{\frac{\mu _{1} /\varepsilon _{1} -\mu _{2} /\varepsilon _{2} }{1/\varepsilon _{1}^{2} -1/\varepsilon _{2}^{2} } }$

(3)

This equation is called the polariton dispersion relation, which relates the frequency with the parallel component of the wavevector. For nonmagnetic materials, it becomes

$k_{x} =\frac{\omega }{c} \sqrt{\frac{\varepsilon _{1} \varepsilon _{2} }{\varepsilon _{1}^{} +\varepsilon _{2}^{} } }$

(4)

One should bear in mind that the permittivities are in general functions of the frequency. For a metal with a negative real permittivity, the normal component of the wavevector is purely imaginary for any real kx, because $\mu \varepsilon \omega ^{2} /c^{2} <0$. Thus, evanescent waves exist in metals regardless of the angle of incidence.

The requirement of evanescent waves on both sides of the interface prohibits the coupling of propagating waves to the surface polaritons. Figure 1 qualitatively shows a dispersion curve of surface polaritons from Eq. (4) along with the dispersion line of the light propagating in a dielectric having a refractive index nd, suggesting that the propagating light cannot excite the surface polariton. In order to couple propagating light with surface polaritons, we critically need a coupler that can shift the dispersion line of the light to match the parallel (or in-plane) wavevector component of the surface polaritons [1]. Two conventional surface polariton couplers being widely accepted are a metal-coated prism and a metallic grating structure, whose configurations and mechanisms of light-SP(h)P coupling are schematically illustrated in Fig. 1. For a metal-coated prism coupler (or Kretschmann coupler), the light in-plane wavevector becomes kx = npk0sinθ , where np is the refractive index of the prism and k0 = ω / c is the propagating wavevector in vacuum, when the light is incident through the prism on the metallic thin film with the incidence angle θ. The dispersion curve for the surface polaritons propagating on the opposite side of the metallic thin film can thus be satisfied for a given incidence angle as shown in Fig. 1(b). When the light is incident on the grating structure, the Bloch-Floquet condition becomes kx,j = kx + 2πj / Λ, where j is the diffraction order and Λ is the grating period.

Figure 1. Schematic diagrams of surface polariton couplers and dispersion curves. (a) A metal-coated prism coupler can excite surface polartions (b) by making use of the total internal reflection on the prism-metal interface. (c) A metallic grating structures can also excite surface polaritons by (d) light diffraction to several orders.

The light in-plane wavevector can increase by a factor of j / Λ depending on the diffraction order, shifting the light dispersion to couple with the SPP dispersion: see Fig. 1(d). It should be noted that the grating coupler can excite multiple surface polaritons even at the normal incidence. Figure 2 shows the reduced dispersion relation for a binary grating made of Ag with Λ = 1.7μm [4]. The dispersion curves (dash-dotted lines) are folded. The solid lines correspond to an incidence angle of 30° and are also folded. The intersections identify the location where SPPs can be excited for a TM wave incidence, when the magnetic field is parallel to the grooves.

Figure 2. Dispersion relation of SPP as manifested by an Ag grating [4]. Note that kx = (ω / c)sinθ

The excitation of surface polaritons makes a significant effect on near-field thermal radiation because the incident photon energy is resonantly transferred to collectively oscillate charges in materials upon the surface polariton excitation, resulting in a sharp peak in the light absorptance spectrum (or a sharp dip in the light reflectance spectrum). Thus near-field radiative heat transfer can be greatly enhanced with the surface polariton excitation. Moreover, surface polaritons play a crucial role in tailoring the spectral and directional radiative properties of materials. For example, coherent thermal emission can be realized by exciting surface polaritons in grating structures and truncated photonic crystals. Further discussion is deferred to later sections.

## References

[1] Raether, H., 1988, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer, Berlin.

[2] Park, K., Lee, B. J., Fu, C. J., and Zhang, Z. M., 2005, “Study of the Surface and Bulk Polaritons with a Negative Index Metamaterial,” Journal of the Optical Society of America B, 22, pp. 1016–1023.

[3] Fu, C. J., Zhang, Z. M., and Tanner, D. B., 2005, “Planar Heterogeneous Structures for Coherent Emission of Radiation,” Optics Letters, 30, pp. 1873–1875.

[4] Zhang, Z. M., 2007, Nano/Microscale Heat Transfer, McGraw-Hill, New York.