Long-range surface plasmons in dielectric-metal-dielectric structure with highly anisotropic substrates

We present a theoretical study of long-range surface plasmons propagating in a thin metallic film clad between two identical uniaxial anisotropic dielectric crystals. We show that the proper orientation of the optical axis of the crystal with respect to the metal surface enhances the propagation length of surface plasmons. Since the proper orientation depends on surface plasmon frequency, we give the results for the propagation length in a wide range of frequencies, including the telecommunication region. To increase the role of anisotropy, we consider artificial substrates from photonic crystals, which possess an order of magnitude stronger anisotropy than the natural optical crystals. We propose Kronig-Penney model for plasmonic crystal where the substrate is a periodic sequence of dielectric delta peaks. In this model the dispersion relation for surface plasmon has a band structure where the band width tends to zero when the frequency approaches the resonant frequency.

Figure 1

(Color online) Dispersion curves for different orientations of optical axis of the anisotropic dielectric crystals with metal film of thickness $d=50\text{nm}$. The dotted curve represents dispersion curve for equivalent isotropic dielectric with dielectric constant ${\epsilon }_{is}=\sqrt{{\epsilon }_x{\epsilon }_z}$.

Figure 2

(Color online) The top figure shows propagation length for parallel $\left({\epsilon }_x>{\epsilon }_z\right)$ and perpendicular $\left({\epsilon }_x<{\epsilon }_z\right)$ orientations for metal thickness $d=20\text{nm}$. The bottom figure shows the same curves between $\omega =0.21{\omega }_s$ and $\omega =0.24{\omega }_s$.

Figure 3

(Color online) Dispersion curves and light lines (ll) for different orientations for metal thicknesses $d=5\text{nm}$ and $d=200\text{nm}$. Light lines for equivalent isotropic case $\left({\epsilon }_x={\epsilon }_z\right)$ are not included in the figure. The anomalous dispersion curves correspond to metal thickness $d=5\text{nm}$ (top figure).

Figure 4

(Color online) Propagation length for parallel, perpendicular orientations, and for the equivalent isotropic case with metal film of thickness $d=50\text{nm}$. $\omega =0.36{\omega }_s$ corresponds to $\lambda =0.961\mu \text{m}$ and $\omega =0.45{\omega }_s$ corresponds to $\lambda =0.768\mu \text{m}$.

Figure 5

(Color online) ${\omega }_{cr}/{\omega }_s$ as a function of thickness $d$ of the metal film. The inset shows the difference between propagation lengths (obtained using Eqs. (5, 7)) as a function of frequency for metal thickness $d=200\text{nm}$.

Figure 6

(Color online) Propagation length $L\left(\omega \right)$ for different orientations of dielectric crystal with metal film of thickness $d=50\text{nm}$.

Figure 7

(Color online) Penetration depth of surface plasmons for different orientations of dielectric crystal with metal film of thickness $d=50\text{nm}$.

Figure 8

Kronig-Penney model for plasmonic crystal.

Figure 9

The first four allowed bands for a plasmonic crystal of Si semispace and periodic substrate with ${\epsilon }_0=3$.

Figure 10

The rhs of Eq. (17) vs frequency. The band edges are the intersections of the curve with two horizontal lines at the level $\pm 1$.

Figure 11

(Color online) Dispersion curves for plasmonic crystal with ${\epsilon }_0=3$ in the limit $ka,qa⪡1$ [Eq. (22), solid line] and for plasmonic structure with a homogeneous dielectric substrate with ${\epsilon }_x=1$ and ${\epsilon }_z=1+{\epsilon }_0$ [Eq. (23), dashed line]. The inset shows the region of low frequencies. The frequency is normalized to ${\omega }_s={\omega }_p/\sqrt{1+{\epsilon }_0}={\omega }_p/2$.