Sideways adiabaticity: beyond ray optics for slowly varying metasurfaces

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. * cperezar@mit.edu 1 arXiv:1806.05330v2 [physics.optics]

Inverse design of large-area metasurfaces

“…Another interesting direction to explore would be further development of the theory of nearly periodic structures and locally periodic approximations. In a companion work [46], we develop a rigorous theory of slowly varying (nearly uniform) structures, and show that the analogous "locally uniform" approximation appears as the 0th-order term in a convergent series of integral corrections. A corresponding rigorous theory of higher-order corrections to the locally periodic approximation, analogous to coupled-mode expansions for propagation through nearly periodic media [30], along with efficient numerical methods to obtain corrections, is an important goal for the theory of metasurfaces.…”

“…A corresponding rigorous theory of higher-order corrections to the locally periodic approximation, analogous to coupled-mode expansions for propagation through nearly periodic media [30], along with efficient numerical methods to obtain corrections, is an important goal for the theory of metasurfaces. A closely related problem is coupling radiation to and from guided modes by nearly-periodic surfaces, a version of which is solved in [46]. In another paper [46], we have recently shown that similar local approximations can indeed be used to compute both near fields and coupling to guided waves.…”

“…The key "locally periodic" assumption is that the pillar width (the unit cell) changes sufficiently slowly from one pillar to the next. (This assumption is rigorously quantified in [46], is validated numerically in Sec. 4, and it turns out that we even obtain good accuracy when there are sudden changes in pillar width at a few locations.)…”

Topology optimization of freeform large-area metasurfaces

“…If desired, the optimal design can be further refined by improving the LPA wavefront by minimizing a phase error [15] or wavefront error objective [16]. At the same time, deviations from LPA may be mitigated by considering larger metasurfaces where LPA is more appropriate, by including higher-order corrections to LPA [19], or by incorporating optimization constraints designed to enforce the validity of LPA. For example, the optimization process may be augmented with constraints restricting the structural variations between neighboring cells or specifying a bound on the physical residual:…”

Overlapping domains for topology optimization of large-area metasurfaces

“…In particular, we find that d 1.5a is enough for accurate modeling of many lens designs. (In contrast, there will be other applications such as grating couplers or excitation of surface resonances [23] in which long-range surface propagation is crucial, but such propagation implies a nearly periodic surface in which LPA will be accurate. In this sense, ODA and LPA are complementary.)…”