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Analysis of thin conducting layers for plasmonic and photonic devices

Analysis of the dispersion of electromagnetic waves propagating through finite-thickness charge sheets determines when they can be approximated by zero-thickness conducting interfaces.
12 December 2006, SPIE Newsroom. DOI: 10.1117/2.1200612.0503

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 σs. 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 nb, 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

Figure 1. Reflection and transmission from induced space charge layers at interfaces give rise to reflected and transmitted light beams.

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.

Figure 2. A modified Kretschmann-Raether configuration was used as an optical filter. Here, the conducting layer, of conductivity σ, is assumed to have a nonzero thickness d.

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.

Bizhan Rashidian
Department of Electrical Engineering, Sharif University of Technology
Tehran, Iran
Institute for Nanoscience and Technology, Centre of Excellence for Nanostructures
Tehran, Iran

Bizhan Rashidian received his BSc and MSc (with highest honors) from Tehran University, Tehran, Iran, in 1987 and 1989, respectively, and his PhD from Georgia Institute of Technology, Atlanta, in 1993, all in electrical engineering. He has been with the Department of Electrical Engineering, Sharif University of Technology, Tehran, since 1994, and is now a professor. He is also the founding director of the Nanoelectronics Laboratory and the Photonics Laboratory.

Khashayar Mehrany
Department of Electrical Engineering, Sharif University of Technology
Tehran, Iran

Khashayar Mehrany received his BSc, MSc, and PhD (magna cum laude) degrees from Sharif University of Technology, Tehran, Iran, in 1999, 2001, and 2005, respectively, all in electrical engineering. Since then, he has been an assistant professor with the Department of Electrical Engineering, Sharif University of Technology. His research interests include photonics, semiconductor physics, nanoelectronics, and numerical treatment of electromagnetic problems.