From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids

We consider propagation of surface plasmon polaritons (SPPs) in linear periodic chains (LPCs) of prolate and oblate metallic spheroids. We show that the SPP group velocity can be efficiently controlled by varying the aspect ratio of the spheroids. For sufficiently small aspect ratios, a gap appears in the first Brillouin zone of the chain lattice in which propagating modes do not exist. Depending on the SPP polarization, the gap extends to certain intervals of the Bloch wave number $q$. Thus, for transverse polarization, no propagating SPPs exist with wave numbers $q$ such that $q_c^{\perp }<\left|q\right|<\pi /h$, $h$ being the chain period. For longitudinally polarized SPPs, the gap spans the interval $\left|q\right|<q_c^{\parallel }$. Here $q_c^{\perp }$ and $q_c^{\parallel }$ are different constants, which depend on the chain parameters, spheroid aspect ratio, and its orientation with respect to the chain axis. The dependence of the dispersion curves on the spheroid aspect ratio leads to a number of interesting effects. In particular, bandwidth of SPPs that can propagate in an LPC can be substantially increased by utilizing prolate or oblate spheroids. When $q$ is close to a critical value, so that $\left|q-q_c^{\perp }\right|⪡\pi /h$ or $\left|q-q_c^{\parallel }\right|⪡\pi /h$, the decay length of the SPPs is dramatically increased. In addition, the dispersion curves acquire a very large positive or negative slope. This can be used to achieve superluminal group velocity for realistic chain parameters. We demonstrate superluminal propagation of Gaussian wave packets in numerical simulations. Both theory and simulations are based on Maxwell equations with account of retardation and, therefore, are fully relativistic.

Figure 1

(Color online) Contour plot of the real part of the dipole sum, $\text{Re}\left[S\left(kh,qh\right)\right]$ for the transverse (top) and the longitudinal (bottom) polarizations. Due to the symmetry $S\left(\xi ,\eta \right)=S\left(\xi ,-\eta \right)$, only the positive half of the first Brillouin zone is shown. Note that, for the transverse polarization, $\text{Re}\left[S\left(kh,qh\right)\right]$ diverges logarithmically (approaches positive infinity) on the light line $q=k$. Near this line, the function changes so fast that it is not feasible to depict it quantitatively using the contour plot; the region of divergence is schematically shown as a diagonal white line in the top panel. There is no divergence for the longitudinal polarization.

Figure 2

(Color online) (a) Dispersion curves and (b) group velocity for transversely polarized SPPs in chains built from prolate spheroids whose axis of symmetry is perpendicular to the chain, for different spheroid aspect ratios $b/a$ and for fixed ratios $h/b=4$ and ${\lambda }_p/h=3.4$. Since the dispersion curves are symmetric with respect to the $k$ axis, only the positive half of the first Brillouin zone $\left(q>0\right)$ is shown. Only data points that were found numerically are plotted. Theoretically, however, all dispersion curves in infinite chains start at the point $k=q=0$ and follow the light line (labeled as $k=q$ in the top panel) for some range of $q$’s. Horizontal arrows indicate central frequencies of the Gaussian wave packets whose simulated propagation is illustrated in Fig. 8 below.

Figure 3

(Color online) Same as in Fig. 2 but for a chain made of oblate spheroids whose axis of symmetry is parallel to the chain and for a different set of aspect ratios. SPP polarization is orthogonal to the chain.

Figure 4

(Color online) Same as in Fig. 2 but for longitudinal SPP polarization and a different set of aspect ratios. Note that, unlike in the case of transverse polarization, the dispersion curves do not have linear small-$q$ segments. As stated in the text, only data points that satisfy $q\ge k$ are plotted, since there are no real-valued solutions in the region $q<k$.

Figure 5

(Color online) Same as in Fig. 3 but for longitudinal SPP polarization and a different set of aspect ratios. Note that, unlike in the case of transverse polarization, the dispersion curves do not have linear small-$q$ segments. As stated in the text, only data points that satisfy $q\ge k$ are plotted, since there are no real-valued solutions in the region $q<k$. Horizontal arrows indicate central frequencies of the Gaussian wave packets whose simulated propagation is illustrated in Fig. 9 below.

Figure 6

Dispersion curves plotted parametrically in the complex $q$ plane for (a) transversely and (b) longitudinally polarized SPPs in an LPC of prolate spheroids. Parameters: $\gamma /{\omega }_p=0.002$, $b/a=0.15$, and other parameters same as in Fig. 2. Dots label the values of the dimensionless parameter $kh/\pi =\omega h/\pi c_h$. In panel (a), only the points in the region $v_p{v}_g<0$ are shown, which corresponds, approximately, to $\left|\text{Re}\left(q\right)\right|h/\pi \gtrsim 0.17$.

Figure 7

Same as in Fig. 6 for an LPC of oblate spheroids whose aspect ratio is $b/a=0.25$

Figure 8

(Color online) Envelopes of transversely polarized wave packets in a chain of $N=5000$ prolate nanospheroids with the aspect ratio $b/a=0.15$ at different moments of time $t$. Spheroids are oriented so that their axes of symmetry are perpendicular to the chain axis. Time is measured in the units of $\tau =h/c_h$. Column (a): ${\omega }_0=0.1{\omega }_p$ and $v_g\approx 0.57c_h$. Column (b): ${\omega }_0=0.05{\omega }_p$ and $v_g\approx 1.16c_h$. Arrows indicate that the wave packet propagates from right to left after being reflected from the far end of the chain.

Figure 9

(Color online) Envelopes of longitudinally polarized wave packets in a chain of $N=5000$ oblate nanospheroids with the aspect ratio $b/a=0.25$ at different moments of time $t$. Spheroids are oriented so that their axes of symmetry coincide with the chain axis. Time is measured in the units of $\tau =h/c_h$. Note that the largest time shown in this figure is two times smaller than the respective quantity in Fig. 8. Column (a): ${\omega }_0=0.25{\omega }_p$ and $v_g\approx 0.77c_h$. Column (b): ${\omega }_0=0.10{\omega }_p$ and $v_g\approx 2.57c_h$. Arrows indicate that the wave packet propagates from right to left after being reflected from the far end of the chain.