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Auditing for fairness often requires relying on a secondary source, e.g., Census data, to inform about protected attributes. To avoid making assumptions about an overarching model that ties such information to the primary data source, a recent line of work has suggested finding the entire range of possible fairness valuations consistent with both sources. Though attractive, the current form of this methodology relies on rigid analytical expressions and lacks the ability to handle continuous decisions, e.g., metrics of urban services. We show that, in such settings, directly adapting these expressions can lead to loose and even vacuous results, particularly on just how fair the audited decisions may be. If used, the audit would be perceived more optimistically than it ought to be. We propose a linear programming formulation to handle continuous decisions, by finding the empirical fairness range when statistical parity is measured through the Kolmogorov-Smirnov distance. The size of this problem is linear in the number of data points and efficiently solvable. We analyze this approach and give finite-sample guarantees to the resulting fairness valuation. We then apply it to synthetic data and to 311 Chicago City Services data, and demonstrate its ability to reveal small but detectable bounds on fairness. 

Abstract Optical metasurfaces with subwavelength thickness hold considerable promise for future advances in fundamental optics and novel optical applications due to their unprecedented ability to control the phase, amplitude, and polarization of transmitted, reflected, and diffracted light. Introducing active functionalities to optical metasurfaces is an essential step to the development of next-generation flat optical components and devices. During the last few years, many attempts have been made to develop tunable optical metasurfaces with dynamic control of optical properties (e.g., amplitude, phase, polarization, spatial/spectral/temporal responses) and early-stage device functions (e.g., beam steering, tunable focusing, tunable color filters/absorber, dynamic hologram, etc) based on a variety of novel active materials and tunable mechanisms. These recently-developed active metasurfaces show significant promise for practical applications, but significant challenges still remain. In this review, a comprehensive overview of recently-reported tunable metasurfaces is provided which focuses on the ten major tunable metasurface mechanisms. For each type of mechanism, the performance metrics on the reported tunable metasurface are outlined, and the capabilities/limitations of each mechanism and its potential for various photonic applications are compared and summarized. This review concludes with discussion of several prospective applications, emerging technologies, and research directions based on the use of tunable optical metasurfaces. We anticipate significant new advances when the tunable mechanisms are further developed in the coming years. 

A gate-tunable plasmonic optical filter incorporating a sub- wavelength patterned metal–insulator–metal metasurface heterostructure is proposed. An additional thin transparent conducting oxide (TCO) layer is embedded in the insulator layer to form a double metal–oxide-semiconductor configu- ration. Heavily n-doped indium tin oxide (ITO) is em- ployed as the TCO material, whose optical property can be electrically tuned by the formation of a thin active ep- silon-near-zero layer at the ITO–oxide interfaces. Full-wave electromagnetic simulations show that amplitude modula- tion and shift of transmission peak are achievable with 3–5 V applied bias, depending on the application. Moreover, the modulation strength and transmission peak shift increase with a thinner ITO layer. This work is an essential step toward a realization of next-generation compact photonic/ plasmonic integrated devices.