Abstract
Optical metasurfaces (subwavelength-patterned surfaces typically described by variable effective surface impedances) are typically modeled by an approximation akin to ray optics: the reflection or transmission of an incident wave at each point of the surface is computed as if the surface were "locally uniform," and then the total field is obtained by summing all of these local scattered fields via a Huygens principle. (Similar approximations are found in scalar diffraction theory and in ray optics for curved surfaces.) In this paper, we develop a precise theory of such approximations for variable-impedance surfaces. Not only do we obtain a type of adiabatic theorem showing that the "zeroth-order" locally uniform approximation converges in the limit as the surface varies more and more slowly, including a way to quantify the rate of convergence, but we also obtain an infinite series of higher-order corrections. These corrections, which can be computed to any desired order by performing integral operations on the surface fields, allow rapidly varying surfaces to be modeled with arbitrary accuracy, and also allow one to validate designs based on the zeroth-order approximation (which is often surprisingly accurate) without resorting to expensive brute-force Maxwell solvers. We show that our formulation works arbitrarily close to the surface, and can even compute coupling to guided modes, whereas in the far-field limit our zeroth-order result simplifies to an expression similar to what has been used by other authors.
Highlights
Optical metasurfaces, subwavelength structures described by an effective sheet impedance [25, 20, 18, 19, 43, 42, 17, 16, 38, 2], are being designed for large-area optical devices using models in which the far-field reflection/transmission coefficients are computed at each point assuming a uniform surface—as explained below, we refer to these as “ray-optics” models
We use the technical machinery of surface-integral equations (SIEs) [13, 29] and a “locally uniform approximation” of the metasurface to show that the ray-optics approximation is the far-field zero-th order term in a convergent series (Section 4 and Appendix C), that each successive correction can be computed by performing integrals, and that the next-order correction scales as ε2
We demonstrate through numerical experiments that the ray-optics approximation produces far-field errors that vanish as ε2, and more generally as ε2N+2 if we include N th-order corrections (Fig. 5)
Summary
Subwavelength structures described by an effective sheet impedance [25, 20, 18, 19, 43, 42, 17, 16, 38, 2], are being designed for large-area optical devices using models in which the far-field reflection/transmission coefficients are computed at each point assuming a uniform (or periodic) surface—as explained below, we refer to these as “ray-optics” models This is a good approximation for surfaces (or unit cells) that are varying slowly, a fact that is closely connected to the “adiabatic theorem” [23, 21] for waves propagating through slowly varying media. In the presence of guided modes, which correspond to poles that appear in Gp at certain wavevectors [10], we show that this simplifies the integrals in our perturbative expansion (via a steepest-descent approximation) if one is mainly interested in coupling to guided modes (Appendix F and Figs. 6 and 10)
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