Floquet-Bloch analysis of refracting Huygens metasurfaces

We investigate the response of passive lossless Huygens metasurfaces (HMS), designed to implement a prescribed refraction, to arbitrary plane-wave excitations. Applying a rigorous Floquet-Bloch (FB) analysis of the cotangent electric and magnetic surface reactance modulations, we discover that the scattered FB modes follow an extraordinary ray-optical model. According to this model, the HMS can be replaced by a virtual region of constant wave impedance; incident power is incoupled into this virtual Fabry-Pérot etalon, gradually leaking to successively higher-order FB modes. Analytical closed-form expressions are derived for the FB coefficients, indicating that the first FB mode in transmission is always dominant; this clarifies previous experimental findings which implied that the HMS retained its functionality over a broad angular range. Furthermore, we point out the implementation problems inherent in the cotangent function and offer a simple, realizable form of the reactive modulation which does not significantly deteriorate the HMS performance. These results provide fundamental physical insight into the working mechanism of passive lossless Huygens metasurfaces in general, revealing a new degree of freedom which enables control of the transmitted field amplitude: the virtual region wave impedance. Importantly, the implied broad acceptance angle of the HMS may be harnessed for synthesis of low-profile multifunctional devices, designed to respond to several excitations with reduced reflections.

Figure 1

Physical configuration of a refracting, passive lossless Huygens metasurface excited by a plane-wave incident (a) at the designated angle ${\theta }_{\mathrm{in}}$ (as in Ref. [21]) or (b) at an arbitrary angle ${\psi }_{\mathrm{in}}={\psi }_0$. In general, if ${\psi }_{\mathrm{in}}\ne {\theta }_{\mathrm{in}}$, the scattered field consists of an infinite number of transmitted and reflected Floquet-Bloch modes.

Figure 2

Ray-optical interpretation of the Floquet-Bloch analysis. To obtain the contribution to the FB transmission (purple) or reflection (green) coefficients [Eq. (7)], one should trace the rays from the incident ray (brown) to the respective mode, multiplying the Fresnel reflection (red) or transmission (blue) coefficients encountered along the trajectory.

Figure 3

Power refraction efficiency ${\eta }_1^T$ as a function of the angle of incidence ${\psi }_0$ and refraction ${\psi }_1$ for nondesignated HMSs (thick annotated contours), designed to be excited at (a) ${\theta }_{\mathrm{in}}=0$ or (b) ${\theta }_{\mathrm{in}}=\pi /4$, and for designated HMSs (thin contours), designed to refract ${\theta }_{\mathrm{in}}={\psi }_0$ to ${\theta }_{\mathrm{out}}={\psi }_1$. For a given ${\theta }_{\mathrm{in}}$, the black dash-dotted lines mark angular boundaries beyond which the first mode becomes evanescent and no coupling from ${\psi }_0$ to ${\psi }_1$ is possible. The black circles denote the working point considered in Fig. 4.

Figure 4

Scattering of an incident plane wave at ${\psi }_{\mathrm{in}}={\psi }_0={49}^{\circ }$ (red open arrow) off HMSs designed with ${\theta }_{\mathrm{out}}={13}^{\circ }$ (dark green), ${\theta }_{\mathrm{out}}={\theta }_{\mathrm{out},\mathrm{opt}}={69}^{\circ }$ (light green), or ${\theta }_{\mathrm{out}}={\psi }_1={79}^{\circ }$ (yellow) to refract the incident plane wave to ${\psi }_1={79}^{\circ }$. (a) Radiation patterns (dB scale), normalized to incident power; the arrow lengths correspond to the relative power carried by the diffraction modes, which are $\delta$ functions in the angular domain. (b) Transmission coefficients $\left|T_n\right|$ and (c) reflection coefficients $\left|R_n\right|$ for the first FB modes $\left|n\right|\le 10$. The propagating FB mode numbers (contributing to the radiation pattern) are circumscribed by a red rectangle in (b) and (c). Black circles in Fig. 3 denote the working point $\left({\psi }_0,{\psi }_1\right)$ considered herein.

Figure 5

Comparison between the spatial variation of the ideal cotangent (thick pink line) and the well-behaved approximated ${\mathrm{ctg}}_{\alpha }\left(ky{\Delta }_{sin}/2\right)$ with three values of the nonideality parameter: $\alpha =0.05$ (green dashed line), $\alpha =0.1$ (red solid line), and $\alpha =0.2$ (blue dash-dotted line).

Figure 6

Power refraction efficiency ${\eta }_1^T$ as a function of the angle of incidence ${\psi }_0$ and refraction ${\psi }_1$ for realizable HMSs designed to be excited at ${\theta }_{\mathrm{in}}=0$, with a nonideality parameter of (a) $\alpha =0.05$ or (b) $\alpha =0.1$.