Rigorous local-to-global theory for reflectionless wide-angle refracting metasurfaces designed by local impedance matching (generalized S-parameters)

A passive, lossless refracting metasurface can steer a normally incident plane wave to a wide deflection angle with almost no reflection when each of its unit cells is individually impedance-matched using generalized scattering parameters, i.e., two-port S-parameters whose port reference impedances equal the angularly dependent TE/TM wave impedances on the incident and transmitted sides, realized by three cascaded admittance sheets. This was demonstrated in the originating work only by full-wave (HFSS Floquet-port) simulations and later by experiment, resting on the local periodicity assumption that each cell in the transversally inhomogeneous array scatters as it would in an infinite periodic array. The open problem is to develop a rigorous theory of this local-to-global transfer: prove, disprove, or quantitatively bound the total power that the actual periodic inhomogeneous metasurface reflects into backward Floquet modes, in terms of the supercell period, the rate of variation of the sheet parameters across the supercell, the deflection angle, and the layer geometry, thereby giving the empirically observed near-reflectionless behavior a firm theoretical foundation.

Background

A metasurface is an electrically thin array of subwavelength scatterers (meta-atoms) engineered to transform an incident electromagnetic wave, for example refracting it toward an anomalous direction prescribed by a phase gradient (the generalized Snell law). For a periodic structure, scattering decomposes into Floquet modes — the discrete diffraction orders of a periodic boundary-value problem — and a 'reflectionless refractor' is one that couples essentially all incident power into the single desired transmitted diffraction order and none into backward (reflected) or spurious orders. For plane waves at oblique incidence, the transverse wave impedance that a surface 'sees' depends on angle: for TE (electric field transverse to the plane of incidence) polarization it is $\eta\sec\theta$ and for TM (magnetic field transverse) polarization $\eta\cos\theta$, where $\eta$ is the intrinsic impedance of the surrounding medium and $\theta$ the ray angle. Refracting from $\theta_\mathrm{i}$ to $\theta_\mathrm{t}$ therefore confronts each metasurface cell with different wave impedances on its two faces, and ignoring this mismatch produces strong reflections at wide deflection angles. Wong, Epstein, and Eleftheriades (IEEE Antennas and Wireless Propagation Letters, 2016) solved this by using generalized scattering parameters — S-parameters computed with different reference impedances at the two ports, namely the local angularly dependent wave impedances $Z_\mathrm{S}$ (source side) and $Z_\mathrm{L}$ (load side) — so that the per-cell generalized reflection coefficients $G_{11}$ and $G_{22}$ vanish. The required asymmetric two-port impedance matrix was realized by three cascaded electric admittance sheets separated by thin dielectric substrates, a structure that implements an omega-bianisotropic response (a magnetoelectric coupling between electric and magnetic surface response, shown by Asadchy et al. to be necessary for perfectly controlling refraction). This per-cell design rests on the local periodicity assumption — the assumption that each unit cell in the nonuniform array scatters as if it were part of an infinite periodic array of identical cells — an approximation akin to ray optics that underlies most Huygens'-metasurface and bianisotropic-metasurface syntheses — Huygens' metasurfaces realize the required boundary conditions with paired electric and magnetic surface currents, like the classical Huygens sources of diffraction theory — (e.g., the power-conserving local theory of Epstein and Eleftheriades). Its accuracy was validated numerically (HFSS — High Frequency Structure Simulator, a commercial full-wave finite-element electromagnetic solver — Floquet-port analysis showing 98.89% of power in the desired 80° mode with 0.35% total reflected power) and later experimentally (Chen et al., refraction to 71.8° with negligible measured reflection), but no rigorous theory establishes when, and how accurately, per-cell local impedance matching transfers to the globally inhomogeneous periodic structure (whose period is the supercell — the full repeat length of the phase-gradient pattern, containing one ramp of the required phase profile and typically many subwavelength unit cells). The most closely related rigorous result, by Pérez-Arancibia, Pestourie, and Johnson, is an adiabatic theorem for scalar variable-impedance surfaces: the locally uniform approximation converges as the surface varies more and more slowly, with a computable hierarchy of higher-order corrections that are generically nonzero — indicating that exact global reflectionlessness cannot be expected in general, so the honest target is a quantitative theory with conditions and bounds.

Problem Statement

For a periodic, passive, lossless metasurface that refracts a monochromatic plane wave from incidence angle $\theta_\mathrm{i}$ to transmitted angle $\theta_\mathrm{t}$, designed so that every unit cell, treated in the local periodicity approximation as an isolated two-port, is impedance-matched through generalized scattering parameters referenced to the local angularly dependent wave impedances ($Z_\mathrm{S}=\eta\sec\theta_\mathrm{i}$, $Z_\mathrm{L}=\eta\sec\theta_\mathrm{t}$ for TE; $Z_\mathrm{S}=\eta\cos\theta_\mathrm{i}$, $Z_\mathrm{L}=\eta\cos\theta_\mathrm{t}$ for TM, with generalized reflection coefficients $G_{11}=G_{22}=0$) and realized as cascaded admittance sheets (e.g., the three-sheet cell with closed-form sheet admittances), establish a rigorous theory of the exact scattering by the transversally inhomogeneous periodic structure. Specifically: determine, by proof, quantitative bound, or explicit counterexample, the total power coupled into backward Floquet modes (and spurious transmitted modes) as a function of the supercell period relative to the wavelength, the rate of variation of the sheet admittance/impedance profile across the supercell, the deflection angle $\theta_\mathrm{t}-\theta_\mathrm{i}$, and the dielectric layer parameters; in particular, give conditions under which the reflected power is zero or negligibly small, and account for the small residual reflections observed in full-wave simulations and experiments (e.g., about 0.35% reflected power at 80° deflection). The theory to be established must connect the per-cell generalized S-parameter matching condition to the exact Floquet-mode scattering of the entire inhomogeneous metasurface, and its quantitative content is the parameter dependence of the coupled backward- and spurious-mode power, the conditions for zero or negligible reflection, and an account of the observed residuals.

References

1. Joseph P. S. Wong, Ariel Epstein, George V. Eleftheriades, 'Reflectionless Wide-Angle Refracting Metasurfaces', IEEE Antennas and Wireless Propagation Letters, 2016, DOI: 10.1109/LAWP.2015.2505629, https://doi.org/10.1109/lawp.2015.2505629
2. Ariel Epstein, George V. Eleftheriades, 'Arbitrary Power-Conserving Field Transformations With Passive Lossless Omega-Type Bianisotropic Metasurfaces', IEEE Transactions on Antennas and Propagation, 2016, DOI: 10.1109/TAP.2016.2588495, arXiv:1604.01100, https://arxiv.org/abs/1604.01100
3. V. S. Asadchy, M. Albooyeh, S. N. Tcvetkova, A. Díaz-Rubio, Y. Ra'di, S. A. Tretyakov, 'Perfect control of reflection and refraction using spatially dispersive metasurfaces', Physical Review B 94, 075142, 2016, DOI: 10.1103/PhysRevB.94.075142, arXiv:1605.02044, https://arxiv.org/abs/1605.02044
4. Michael Chen, Elena Abdo-Sánchez, Ariel Epstein, George V. Eleftheriades, 'Experimental Verification of Reflectionless Wide-Angle Refraction via a Bianisotropic Huygens' Metasurface', URSI General Assembly and Scientific Symposium, 2017, DOI: 10.23919/URSIGASS.2017.8105103, arXiv:1703.06669, https://arxiv.org/abs/1703.06669
5. Michael Chen, Elena Abdo-Sánchez, Ariel Epstein, George V. Eleftheriades, 'Theory, design, and experimental verification of a reflectionless bianisotropic Huygens' metasurface for wide-angle refraction', Physical Review B 97, 125433, 2018, DOI: 10.1103/PhysRevB.97.125433, arXiv:1812.05084, https://arxiv.org/abs/1812.05084
6. Carlos Pérez-Arancibia, Raphaël Pestourie, Steven G. Johnson, 'Sideways adiabaticity: Beyond ray optics for slowly varying metasurfaces', Optics Express 26, 30202, 2018, DOI: 10.1364/OE.26.030202, arXiv:1806.05330, https://arxiv.org/abs/1806.05330
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