Upper-limited angular Goos-Hänchen shifts of Laguerre-Gaussian beams

The angular Goos-Hänchen shift of vortex beam is investigated theoretically when a Laguerre-Gaussian (LG) beam is reflected by an air-metamaterial interface. The upper limit of the angular GH shift is found to be half of the divergence angle of the incident beam, i.e., |Θup| = (|ℓ| + 1)^1/2/k₀w₀, with ℓ, k₀, and w₀ being the vortex charge, wavenumber in vacuum, and beam waist, respectively. Interestingly, the upper limited angular GH shift is accompanied by the upper-limited spatial IF shift. A parameter F is introduced to compare the total beam shift with the beam size. F varies with the vortex charge ℓ and the propagation distance zr. The values of F at zr = ∞ plane can approach 0.5, which are always larger than those at zr = 0 plane. These findings provide a deeper insight into optical beam shifts, and they may have potential application in precision metrology.