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Light reflection plays a crucial role in a number of modern technologies. Inthis paper, analytical expressions for maximal reflected power in any directionand for any polarization are given for generic planar structures made of asingle material represented by a complex scalar susceptibility. The problem ofoptimal light-matter interactions to maximize reflections is formulated as thesolution of an optimization problem in terms of the induced currents, subjectto energy conservation and passivity, which admits a global upper bound byusing Lagrangian duality. The derived upper bounds apply to a broad range ofplanar structures, including metasurfaces, gratings, homogenized films,photonic crystal slabs, and more generally, any inhomogeneous planar structureirrespective of its geometrical details. These bounds also set the limit on theminimum possible thickness, for a given lossy material, to achieve a desiredreflectance. Moreover, our results allow discovering parameter regions wherelarge improvements in the efficiency of a reflective structure are possiblecompared to existing designs. Examples are given of the implications of thesefindings for the design of superior and compact reflective components made ofreal, imperfect (i.e., lossy) materials, such as ultra-thin and efficientgratings, polarization converters, and light-weight mirrors for solar/lasersails.

How Thin and Efficient Can a Metasurface Reflector Be? Universal Bounds on Reflection for Any Direction and Polarization