Scattering Symmetries in Diffraction Gratings

Abstract

Metasurfaces enable powerful control of electromagnetic waves using subwavelength planar structures, but their deeply subwavelength periodicity typically suppresses propagating diffraction orders, which limits the number of available scattering channels. Diffraction gratings and metagratings overcome this limitation by supporting multiple propagating diffraction orders, thus providing additional degrees of freedom for controlling wave propagation. However, when several diffraction channels are present, it becomes nontrivial to predict how spatial symmetries combined with reciprocity affect the overall scattering response. For this purpose, we develop a formalism to determine the scattering symmetries of diffraction gratings supporting multiple diffraction orders. The approach is based on constructing a global scattering matrix that connects all incident and scattered diffraction channels and on introducing matrix representations of spatial symmetry operations acting on the field amplitudes. From these representations, we derive an invariance condition that directly constrains the sub-scattering matrices associated with each pair of diffraction orders. This provides a rigorous approach for computing the grating scattering coefficients imposed by symmetry and reciprocity. We illustrate the application of this approach via several examples and show how metagratings may be used to achieve, for instance, angle-asymmetric transmission and extrinsic chiral effects.

Figures (6)

• Figure 1: From diffraction orders to the definition of open scattering channels. a) TM diffraction orders for a grating with parameters $n_\text{s}=n_\text{i}=1$, $\Lambda=1$ and $\theta_\text{i}=45^\circ$. b) Convention used to define the open channels in all four quadrants. Since only TM waves are considered, only quadrants 1 to 4 are depicted.
• Figure 2: Naming and polarization conventions for the field amplitudes of the (a) incident and (b) scattered waves propagating along a given diffraction order channel $d$.
• Figure 3: Scattering situation with redundant diffraction channels. (a) A TM polarized wave is incident on a grating with the parameter set $\{1,1,45^\circ,\sqrt{2}/2\}$. (b) The grating diffracts the incident wave into diffraction orders 0, 1 and 2 leading to a redundant definition of the open channels as defined in Fig. \ref{['fig_do2chan']}.
• Figure 4: Application of the symmetry operation $\sigma_{x}$ on the system depicted in Fig. \ref{['fig_schematicdo']}a. (a) Initial situation identical to the one in Fig. \ref{['fig_schematicdo']}a. (b) Situation after the application of $\sigma_{x}$.
• Figure 5: Scattering symmetries for various gratings. The gratings are assumed to either be infinite along the $y$-direction or at least be $\sigma_{y}$ symmetric. Top row: scattering between 0$^{\text{th}}$-order channels. Bottom row: scattering between 0$^{\text{th}}$-order and 1$^{\text{st}}$-order channels. Identical arrow colors indicate identical scattering coefficients.