Importance of diffraction in determining the dispersion of designer surface plasmons

By employing a modified modal matching approach, we obtain explicit analytical expressions relating frequency to in-plane wave vector for the surface electromagnetic mode confined at the interface between vacuum and a perfect conductor patterned with a two-dimensional square array of square holes. Our complete analytical formalism takes into account both multiple order waveguide modes and diffracted evanescent waves, hence overcoming the a priori assumptions intrinsic to previous descriptions of the dispersion of these surface waves. We validate our derived dispersion relation through comparison with that recently recorded at microwave frequencies using prism coupling. Finally, we show that diffracted evanescent waves play an important role in determining the dispersion, so that the electric field associated with “designer” surface modes is much more weakly confined to the interface than the field associated with surface plasmons on real metal surfaces.

Figure 1

Interface between vacuum and a two-dimensional, periodic array of square holes in a perfect conductor. The holes considered here are of depth $h$, side $a$, separated by distance $d$ and filled by dielectric ${\epsilon }_h$. Unless otherwise stated, we take ${\epsilon }_h=10$.

Figure 2

(Color online) (a) Real component of the dispersion relation for a sample defined by $d=1.4/k_a^{\infty }$, $h=\infty$, and ${\epsilon }_h=10$, including only specular reflection [green dot-dash line, given by Eq. (13)], and including first- and, subsequently, second-order diffractions [Eq. (19)]. The evanescent diffracted orders are important even well away from the Brillouin-zone boundary: for a surface defined by $d=1.4/k_a^{\infty }$ and $h=\infty$, the Brillouin-zone boundary occurs at $k_x/k_a^{\infty }=\pi /d{k}_a^{\infty }=2.24$. (b) The dispersion relations for a PEC surface perforated by infinitely deep holes obtained by Pendry and co-workers (Refs. 3, 4) (gray solid line) for $d=1.5/k_a^{\infty }$ and ${\epsilon }_h=10$. Also plotted are the dispersion relations obtained from Eq. (21) in this work (i.e., including first-order diffraction) for hole periods defined by $d=1.1/k_a^{\infty }$, $1.2/k_a^{\infty }$, and $1.4/k_a^{\infty }$ (i.e., a Brillouin-zone boundary occurring at $k_x/k_a^{\infty }=2.86$, 2.62, and 2.24, respectively) and ${\epsilon }_h=10$. (c) Note that in (a) and (b) the dispersion bending is not due to the proximity of the Brillouin-zone boundary—the dashed line shows the dispersion relation obtained from Eq. (21) in this work for a structure defined by $d=3.0/k_a^{\infty }$, corresponding to a Brillouin-zone boundary occurring at $k_x/k_a^{\infty }=1.05$ (marked BZ in the figure). As expected, the dispersion is symmetric around the Brillouin-zone boundary. In (a), (b), and (c), the frequency and wave vector are normalized using Eqs. (15, 16), respectively. Higher order surface modes, above the asymptotic frequency, have been left out for clarity.

Figure 3

(Color online) (a) Schematic of the Otto type prism-coupling experiment. (b) Real component of the dispersion relation in the diagonal direction i.e., $k_y=k_x$) for a sample defined by $a=6.96\text{mm}$, $d=9.53\text{mm}$, $h=15\text{mm}$, and ${\epsilon }_h=2.29$. The circles represent the dispersion measured in a prism-coupling experiment. The lines represent the dispersion predicted by Pendry et al. (Ref. 3), García-Vidal et al. (Ref. 4), García de Abajo and Sáenz (Ref. 5) and by Eq. (21) in this work.

Figure 4

(Color online) (a) The dispersion obtained from Eq. (21) for a hole period defined by $d=1.1/k_a^{\infty }$ (i.e., a Brillouin-zone boundary occurring at $k_x/k_a^{\infty }=2.86$) and ${\epsilon }_h=10$, varying the depth of the hole depth $h$. The frequency and wave vector are normalized using ${\omega }_a^{\infty }$ [Eq. (15)] and $k_a^{\infty }$ [Eq. (16)], respectively. (b) The same dispersion relations as in (a) normalized by the asymptotic frequency for finite depth holes, ${\omega }_a$ and corresponding wave vector, $k_a$, allowing us to compare the curvature of dispersion relations with different asymptotic frequencies. For comparison, the normalized SP dispersion at the interface between vacuum and an unstructured metal surface is shown (thin gray line). In both (a) and (b), higher order surface modes, above the asymptotic frequency, have been left out for clarity. (c) Scaled evanescent decay length normal to the vacuum-surface interface, $\mathsf{L}/{\lambda }_0$, for the modes in (a) and (b).