Figure 1

(a) Schematic of beam reflection from a complex crystal slab of thickness $L$ and lattice period $\Lambda$. The spatial shift $\Delta$ of the barycenter of the reflected beam from the incident beam at the plane $x=0$ is the GH shift. The effective penetration depth of photons in the crystal is $\rho$. (b) Schematic of beam scattering for left and right beam incidence.Reuse & Permissions

Figure 2

Reflectance $R^{\left(l\right)}={\left|r^{\left(l\right)}\right|}^2$ (top plots) and corresponding normalized GH shift $\Delta /\lambda$ (bottom plots) versus incidence angle $\theta$ in a sinusoidal $\mathcal{PT}$-crystal slab [Eq. (15)] for (a) $\sigma =0$ (real-valued potential) and (b) $\sigma =1$ (complex potential). Parameter values are $\Lambda /\lambda =2$, $n_s=1.5$, $L=800\Lambda$, and $A=0.001$. The thin solid lines are the curves obtained by exact transfer matrix calculation using Eq. (1), whereas the dotted curves are the predictions based on coupled-mode equation analysis. In (a) the dotted curves are not plotted because they turn out to be overlapped with the solid ones. The shaded areas in (a) correspond to the transmission bands of the crystal, whereas in (b) they mark the regions where the reflected beam is not appreciably amplified. Note that, owing to the presence of gain, in (b) the reflectance is larger than one in the central region.Reuse & Permissions

Figure 3

Same as Fig. 2, but for $A=0.003$.Reuse & Permissions

Figure 4

Reflection of a Gaussian beam from (a) an ideal metallic mirror (reflection coefficient $r=-1$), (b) the index-modulated crystal of Fig. 2a, and (c) the $\mathcal{PT}$-symmetric sinusoidal crystal of Fig. 2b. In (d) and (e) the angular spectrum of the incident beam is depicted by the dotted lines, whereas the solid curves are the reflectances $|r^{\left(l\right)}{|}^2$ of the two crystals. (f) Intensity profiles of the reflected beams $|E_y^{\left(r\right)}{(x,z)|}^2$, normalized to their peak values, at the plane $x=-3050\lambda$. ${\Delta }_1$ and ${\Delta }_2$ are the GH shifts of the beams reflected by the index-modulated and by the $\mathcal{PT}$-symmetric crystals, respectively.Reuse & Permissions

Figure 5

Behavior of transmittance $T={\left|t\right|}^2$, reflectance $R^{\left(l\right)}={\left|r^{\left(l\right)}\right|}^2$ for left-side incidence, and normalized GH shift $\Delta /\lambda$ in a sinusoidal $\mathcal{P}T$ crystal close to the invisibility regime for (a) $\sigma =0.99$ and (b) $\sigma =1.01$. The other parameter values are the same as in Fig. 2.Reuse & Permissions

Figure 6

Same as Fig. 2, but for a deep sinusoidal grating for (a) $\sigma =0$ and (b) $\sigma =0.9$. Other parameter values are $\Lambda /\lambda =2$, $n_s=1.5$, $L=10\Lambda$, and $A=0.15$. The inset in the top plot of (b) shows the behavior of the crystal transmittance $T={\left|t\right|}^2$ versus the incidence angle. The $\mathcal{PT}$ symmetry-breaking point of the crystal slab with grating amplitude $A=0.15$ is ${\sigma }_c\simeq 1$.Reuse & Permissions