Figure 1

(a) Geometry of the system: “Sandwichlike” structure containing a graphene layer on a substrate composed of a dielectric spacer (with dielectric constant ${\varepsilon }_1$) and a transparent conductive wafer with a periodic interface relief. (b) Real part of the SPP frequency versus wave vector in the first Brillouin zone, calculated for a single graphene layer at interface of two semi-infinite dielectric media. (c),(d) Edges of the first two lowest gaps (at edge and at center of Brillouin zone) versus modulation depth $a$ (c) or gate voltage $V$ (d). The parameters are the following: ${\varepsilon }_2=1$, ${\varepsilon }_1=3.9$, $h_0=300$nm, $D=20.67\mu$m, $a=0.6$ (b) and (d). In panels (b) and (c) $V=50$V, corresponding to an average chemical potential $\overline{\mu }\approx 0.222$eV. In panel (b) the light lines for the cladding media, $kc/\omega =\sqrt{{\varepsilon }_m}$, are shown dashed ($m=1$) and dash-dotted ($m=2$).Reuse & Permissions

Figure 2

(a) Zero harmonic reflection coefficient $|r_0/H_i{|}^2$ vs frequency and incidence angle for SPPs in periodically modulated graphene with $V=50$V. (b)–(d) Zero harmonic reflection coefficient (red solid lines) and relative square amplitudes, $|r_n/H_i{|}^2$, of $n=1$ (blue dashed lines) and $n=-1$ (black dashed lines) harmonics for the incidence angles $\Theta =0^{\circ }$ (b), $\Theta ={20}^{\circ }$ (c), and $\Theta ={80}^{\circ }$ (d) [the latter two correspond to the vertical dashed lines in panel (a)]. (e),(f) Zero harmonic reflection coefficient vs frequency and gate voltage for $\Theta ={20}^{\circ }$ (e) and $\Theta ={80}^{\circ }$ (f). Other parameters are the same as in Fig. 1. In panels (a), (e), and (f) red (blue) color corresponds to high (low) values of the reflection coefficient. In panels (b) and (d) Fano-type fits are shown by dash-dotted (orange) lines, with $R_0=0.1508$, $\delta R=0.035$, $q=2$, $\hslash {\omega }_0=10$meV (b) and $R_0=0.116$, $\delta R=0.08$, $q=0.08$, ${\omega }_0=10.68$meV (d); $\hslash \gamma =0.1$meV.Reuse & Permissions

Figure 3

Reflection coefficient of zero harmonic $|r_0/H_i{|}^2$ vs frequency $\omega$ and angle of incidence $\Theta$ for ATR excitation of SPPs in graphene with parameters ${\varepsilon }_3=16$, ${\varepsilon }_2=1$, ${\varepsilon }_1=3.9$, $h_0=300$nm, $D=10.34\mu$m, $a=0.6$, $d=10\mu$m, $V=50$ V (a), or 90 V (b).Reuse & Permissions