Goos-H$\ddot{a}$nchen shifts of bilayer meta-grating with unidirectional guide resonance

TL;DR

This work investigates how Goos-Hänchen shifts behave when light couples to unidirectional guide resonances in bilayer metagratings. It combines numerical mapping of UGRs in structural and synthetic parameter spaces with temporal coupled mode theory to derive how GH shifts relate to resonance phase, group velocity, and the $Q$-factor of the UGRs. The authors identify two anomalous GH behaviors: a resonant peak coexisting with near-constant transmittance due to interference, and a GH magnitude that scales with $rac{ ext{d}oldsymbol{ extomega}_b}{ ext{d}k_x}$ and $Q$, with possible suppression near band dips. Gaussian-beam simulations visualize the shift dynamics and suggest practical opportunities for beam steering, sensing, and integrated photonics using all-dielectric bilayer metastructures.

Abstract

Bilayer meta-gratings with asymmetric structural parameters could host unidirectional guide resonances. The distribution of unidirectional guide resonances in the space of structural parameters and synthetic parameters is identified. As the incident optical beam being resonant with the unidirectional guide resonance, the Goos-H$\ddot{a}$nchen shifts of the scattered beams exhibit two anomalous behaviors: the resonant peak of the Goos-H$\ddot{a}$nchen shift is accompanied by constant transmittance and reflectance; the magnitude of the Goos-H$\ddot{a}$nchen shift is not always proportional to the quality factor of the unidirectional guide resonance. The temporal coupled mode theory analysis reveals that the first anomalous behavior is due to interference between direct scattering and radiation from the unidirectional guide resonance; the Goos-H$\ddot{a}$nchen shifts are proportional to the group velocity as well as the quality factor of the unidirectional guide resonance. Numerical simulations of incidence of Gaussian beam with finite beam width provide intuitive visualization of the Goos-H$\ddot{a}$nchen shift.

Figures (5)

• Figure 1: (a) Schematic of the structure of the bilayer meta-grating in one period. The band structure of the leaky resonant modes as $\Delta g=0.053a$ and $\Delta g=0.068a$ are plotted in (b) and (c), respectively, with $w_1=0.256a$ and $w_2=0.296a$ in both panels. The directionality $\eta$ of the band in panel (b) and (c) with blue and red color are plotted as blue and red curves in panel (d), respectively. The field pattern of the UGRs marked by the green and purple points in panel (b) and (c) are plotted in panel (e) and (f), respectively.
• Figure 2: When $w_2=0.296a$ is fixed, and $w_1$ is scanned, $\Delta g$ of the systems that host UGR with positive and negative $k_x$ versus $w_1$ are plotted as solid black and empty blue dots in panel (a), respectively. The $k_x$, resonant frequency, and $Q$ factor of the corresponding UGR versus $w_1$ are plotted in panel (b), (c) and (d), respectively.
• Figure 3: For the structure in Fig. 1 (b), the angular spectrum of reflectance and transmittance are plotted as black and blue lines in the first row, respectively; the angular spectrum of the phase of the reflected and transmitted field are plotted as black and blue lines in the second row, respectively; the GH shift given by the stationary-phase theory of the reflected and transmitted field are plotted as black and blue lines in the third row, respectively. For the first to fourth column, the incident plane waves are incident from the right-upper, right-lower, left-upper and left-lower background of the meta-grating.
• Figure 4: (a) The maximal GH shift in the angular spectral versus the structural parameter $w_1$ of the branches of UGRs with positive and negative $k_x$ are plotted as solid black and empty blue dots, respectively. (b) The same as panel (a) with the x axis being replaced by the corresponding $Q$ factors of the UGRs. The solid thin line fits the proportional relation between the GH shift and the $Q$ factor.
• Figure 5: Spatial distributions of time-averaged Poynting vectors for incident and transmitted fields under the incident Gaussian beams with incident angles being $\theta_{\text{inc}}=\pm14.9502^o$, are plotted as black and blue arrows, respectively. The field pattern of $E_y$ is plotted as color-scale. The frequency is $af/c=0.445614$, and the structural parameters are the same as the system in Fig. 1 (b). The incident field is from the right-upper, right-lower, left-upper and left-lower background of the meta-grating for the cases in (a), (b) (c) and (d), respectively.