Objective Goos and H?nchen discovered in 1947 that when an incident beam of finite size undergoes total internal reflection on the interface of two media, the actual reflection point shifts laterally along the incident plane relative to the incident point, and the shift is known as the Goos-H?nchen (G-H) shift. A finite-width incident beam can be compared to a series of plane waves travelling in different directions. These plane waves have different reflections at the interface of the two media. After the superposition of the differences in intensity and phase of all reflected light, the incident beam shifts to a certain extent in the transverse direction. G-H shift has great potential applications in optical isolation, optical sensing, and integrated optics. However, in general, the G-H shift at the material interface is very small, which is only a few times the wavelength. Therefore, it is not conducive to observation, measurement, and practical application. Hyperbolic metamaterials (HMMs) are a type of highly anisotropic uniaxial material named after their hyperbolic dispersion relations. HMMs have a wide range of applications, including light field localization, enhanced spontaneous emission, and subwavelength imaging. In this paper, we present the enhancement, direction transformation, and critical wavelength modulation of the G-H shift on the surface of the subwavelength HMM slab.

Methods Researchers proposed models including steady-state phase, energy transfer, and plane wave linear expansion functions to calculate the G-H shift of reflected light. In the laboratory, position-sensitive detectors, weak measurement, and interferometry are usually used to observe the properties of G-H shift. For nonmagnetic media, the dielectric coefficient of HMM is in the form of a second-order tensor. When the vertical component of the dielectric coefficient is positive and the parallel component is negative, its dispersion surface is a hyperboloid of bilobate type, which is called Ⅰ-type HMM. When the vertical component of a dielectric coefficient is negative and the parallel component is positive, the dispersion surface is a hyperboloid of univalent type, which is called Ⅱ-type HMM. Currently, the most common artificial HMMs include a multilayer structure of metal and dielectric stacked with subwavelength thickness and a metal nanowire array embedded in a dielectric. The multilayer HMM design is determined by the target spectral range, loss, and impedance matching. We investigate the effects of incident wavelength, filling factor, and background permittivities on the properties of G-H shift using the effective medium theory and the stationary-phase method.

Results and Discussions First, we calculated the real and imaginary parts of the equivalent permittivity of the material slab with the incident wavelength (Fig. 2) and determine the types of HMM materials at different incident wavelengths. Then, the variation of the G-H shift of different types of the subwavelength HMM slab with incident angle is calculated (Fig. 3). Figs. 3(a)--(d) present the bulk material, Ⅰ-type HMM, elliptical HMM, and Ⅱ-type HMM, respectively. Fig. 3 shows that the G-H shift of the HMM slab is 100 times that of bulk material under the same incident parameters. In addition, under the same structural parameters, the G-H shift of Ⅰ-type HMM is more significant than that of Ⅱ-type and elliptical HMM slabs.

Second, we calculated the G-H shift with the increasing incident angle when the incident wavelengths are 325 and 350 nm, respectively [Fig. 4(a)], which shows that the G-H shift in the opposite direction can be obtained only by changing the incident wavelength. Fig. 4(b) shows the variation of the reflection phase of the surface of the HMM slab with the incident angle. The phase mutation point corresponds to the G-H shift peak, and when the phase suddenly decreases, the G-H shift is positive, and otherwise,it is negative. Moreover, the relationship between the maximum G-H shift and incident wavelength is calculated (Fig. 5). We discovered that a critical wavelength exists between the positive and negative G-H shifts. The G-H shift is positive when the incident wavelength is less than the critical wavelength, and otherwise, it is negative. The greater the G-H shift is, the closer the incident wavelength is to the critical wavelength, indicating symmetry.

Finally, we studied the effects of the silver filling factor and background dielectric constant on the G-H shift characteristics (Fig. 6). We found that increasing the filling factor and the dielectric constant of the background medium is equivalent to the blue shift of the critical wavelength. When the incident wavelength is less than the critical wavelength, the direction of the G-H shift will change. When the incident wavelength is greater than the critical wavelength, the direction of the G-H shift remain unchanged.

Conclusions In the present study, the intensity and direction characteristics of G-H shift of the subwavelength HMM slab have been revealed. It shows that except enhancing the value of the G-H shift, the HMM slab can achieve the direct transformation of the G-H shift under different incident wavelengths. A critical wavelength exists between the positive and negative G-H shifts. The G-H shift is positive (negative) when the incident wavelength is less (greater) than the critical wavelength. At the same time, the closer the incident wavelength is to the critical wavelength, the larger the G-H shift is. We also discovered that the filling factor and background permittivities can be used to tune the critical wavelength. The critical wavelength presents blue-shift as the filling factor or background permittivities increases (decreases) (red-shifts). We believe that the G-H shift at the surface of the subwavelength HMM slab is very promising for potential applications in novel all-optical isolators, optical sensing, and integrated optoelectronic devices considering these intriguing discoveries.