Spatial Goos-Hänchen shift in photonic graphene

We present a theoretical study of the spatial Goos-Hänchen shift occurring at the interface between two photonic graphene lattices. Numerical simulations of light propagation in such a wave-guiding system are also presented to corroborate our findings. Our results show that the combined action of the honeycomb lattice structure and the discrete nature of wave-guiding systems give rise to a giant and negative Goos-Hänchen shift.

Figure 1

Left: Honeycomb geometry. The waveguides are located at the vertices. Right: A mesh of the honeycomb structure. The coordinates $m$ and $n$ are illustrated. For all $a_{m,n}, m+n$ is chosen to be even (waveguides marked red). For all $b_{m,n}, m+n$ is chosen to be odd (waveguides in blue).

Figure 2

(a) Dispersion relation of photonic graphene. The upper and the lower bands touch each other in the Dirac points. (b) Near these points, the dispersion relation is linear. Here, $\kappa =1$ and $\delta \beta =0$ have been used.

Figure 3

Schematic of the interface used in the waveguide array. The interface is induced by different coupling constants (${\kappa }_1, {\kappa }_2$) and a change of the detuning (0, $\theta , \delta \beta$).

Figure 4

Comparison of the dispersion relation (with $k_m=0$) on both sides of the interface. (a) The dispersion on the right side is scaled due to the choice ${\kappa }_1<{\kappa }_2, \delta \beta =0$. (b) It is only shifted by choosing ${\kappa }_1={\kappa }_2, \delta \beta >0$.

Figure 5

Modulus (a, c) and phase (b, d) of the reflection (solid black line) and transmission (dashed red) coefficients with parameters $\delta \beta =\theta =0, {\kappa }_1=1, {\kappa }_2=2$. (a, b) Behavior of $\rho$ and $\tau$ away from the Dirac point (i.e., $k_n=0.4$). (c, d) Reflection and transmission coefficients in correspondence to the Dirac point (i.e., $k_n=\frac{\pi }{3}$).

Figure 6

Reflection (solid black line) and transmission (dashed red) coefficients of the power with parameters $\delta \beta =\theta =0, {\kappa }_1=1, {\kappa }_2=2$. (a) The case where $k_n$ is chosen away from the Dirac point (i.e., $k_n=0.4$). (b) The case where $k_n$ is chosen in correspondence with the Dirac point (i.e., $k_n=\frac{\pi }{3}$).

Figure 7

Dimensionless Goos-Hänchen shift as obtained with analytical theory given by Eq. (16) (solid black line) and numerical simulations obtained by integrating Eqs. (2) (red circles). The parameters used to obtain both graphs are ${\kappa }_1={\kappa }_2=1$ and $\delta \beta =\theta =3({\kappa }_1+{\kappa }_2)$, in order to ensure total internal reflection. Two cases, $k_n=0.4$ and $k_n=\pi /3$, are shown.

Figure 8

(a) Transverse view of light evolution in photonic graphene. The various snapshots represent different propagation distances along the $z$ direction. (b) Longitudinal view of light evolution in photonic graphene. For the sake of simplicity, only one transverse dimension, corresponding to the direction where the GH shift takes place, is shown. In (a)–(c), the interface is represented by a solid red line. In (a), moreover, the first snapshot corresponds to the input facet of the wave guiding structure, where a broad Gaussian beam is inserted into the waveguide array. In the third snapshot, in particular, the interaction of such a Gaussian beam with the interface between the two different arrays can be seen. Here, no light propagates to the right part. The negative GH shift can then be estimated from (b), by considering the extrapolated light rays of geometric optics which intersect in front of the interface [see (c)].