Enhanced control of the Goos–Hänchen shift at graphene-hyperbolic boron nitride multilayer hyper crystal

In this study, we investigate analytically and numerically the Goos–Hänchen Shift (GHS) for reflected and transmitted plane waves in a planar uniaxial anisotropic hexagonal boron nitride (hBN) slab placed in air. An arbitrarily polarized plane wave serves as the excitation source. The Transfer Matrix Method is employed to compute Fresnel coefficients for s- and p-polarization, while the Stationary Phase Method is used to analyze the resulting GHS. For both polarizations, the reflected GHS appears at the left and right boundaries of two well-known reststrahlen bands (RBs) in the far-infrared (FIR) frequency range. Additionally, it is observed within RB1 $<$f $<$ RB2, whereas the transmitted GHS is only present at the right and left boundaries of RB1 and RB2, respectively. Our primary objective is to extend GHS control across a broader frequency range, from FIR to the near-infrared, beyond the conventional RB-limited regime. To achieve this, we integrate a finite number of graphene sheets—modeled as a uniaxial anisotropic finite-thickness medium—with the hBN slab, forming a graphene-hBN multilayer hyper-crystal. We analyze how graphene key parameters, including chemical potential, temperature, and sheet count, influence the GHS. Additionally, the effect of hBN thickness on the reflected GHS spectra is examined across varying incident angles for both polarizations, with and without graphene integration. Our findings offer valuable insights for GHS-based optical devices operating across extended frequency ranges, as well as applications in temperature sensing and hyper-lensing.