Analysis of thin conducting layers for plasmonic and photonic devices

The interaction of electromagnetic waves with conducting interfaces has recently been studied, and has inspired several possible applications.^1–3 In these structures, interesting phenomena arise from free two-dimensional interface-charge layers generated at dielectric interfaces. For instance, a new type of photonic crystal, similar to Kronig-Penny electronic crystals, has been implemented, and novel types of optical band structures have been observed.⁴

Previously, these free-charge layers were modeled using the conductive-interface approximation, in which the interface charge is confined to a zero-thickness layer with conductivity σ_s. The effect of finite layer thickness, and the asymptotic approach toward the conducting interface approximation, is thoroughly studied here for the first time.

We rigorously analyze two different regimes for such structures. First, we consider propagation of optical waves through sub-wavelength free-charge layers and its reflection and transmission coefficients for both transverse-electric (TE) and transverse-magnetic (TM) polarization. Second, we investigate optical slow waves localized at the interface of two dielectrics with an interface charge layer between them and their corresponding effective index of propagation. A special case of unusual wave propagation through such structures is briefly discussed. The electromagnetic response of optical filters based on surface wave excitation at a conducting interface is re-examined, including the effect of nonzero conducting layer thickness.

As illustrated in Figure 1, we consider a dielectric host of refractive index n_b, in which a conducting layer with thickness d and normalized surface conductivity α is induced. The equivalent refractive index and the propagation of electromagnetic waves are readily studied using the transfer matrix method.⁵

For TM-polarized waves, the conducting interface approximation is justified only for conducting layers whose thicknesses are smaller than 0.0001 of the free-space wavelength. On the other hand, to avoid absorption loss, the angular frequency of incident wave ω should satisfy ωτ ≫ 1, where τ denotes the mean free time of charge carriers.⁶ Consequently, lossless induced space-charge layers illuminated by TM-polarized waves should be thinner than 1nm to be satisfactorily modeled by two-dimensional surface conductivity.

For TE-polarized waves, the conducting-interface approximation is justified for conducting layers thinner than 0.1 of the free-space wavelength. Accordingly, lossless induced space-charge layers illuminated by TE-polarized waves should be thinner than 1μm to be satisfactorily modeled by two-dimensional surface conductivity.

Electromagnetic-wave propagation through such structures is unremarkable, save for the special case at which the equivalent refractive index takes infinitely small values, resulting in infinitely large sensitivities. This special case, together with the reasons behind this queer phenomenon, can be explained using the total amplitude reflection coefficient formula.⁶

We also explored the possibility of surface-electromagnetic wave propagation through an induced space-charge layer. We used the standard effective-index technique to study the dispersive behavior of TE- and TM-polarized surface waves. A modified Kretschmann-Raether configuration, shown in Figure 2, was used to implement optical filters based on the excitation of such surface waves.

The propagation of electromagnetic waves through the induced interface-charge layer depends on the density of induced charge carriers and on the incidence angle of the illuminating wave. However, our studies show that the critical ratio of wavelength to charge-layer thickness, beyond which the asymptotic conducting-interface approximation is valid, is usually higher for TM-polarized waves than it is for TE-polarized waves, with values of about 1000 and 10, respectively.

A similar situation occurs for optical slow waves localized at the interface of two dielectrics with an induced interface change between them. Compared with TE-polarized waves, TM-polarized waves demand a higher ratio of wavelength to thickness before the effective index of propagation equals that of the conducting interface approximation.