Cramér-Rao bounds for metasurfaces susceptibilities

Abstract

Over the past fifteen years, a lot of efforts have been focused on understanding the effective properties of metamaterials [1]. In the last few years, metasurfaces in particular have been widely investigated [2]. Several homogenization methods dedicated to them have been proposed but, due to the topic’s complexity, none have yet to be widely accepted. We considered a specific homogenization method dedicated to metasurfaces, namely Generalized Sheet Transition Conditions (GSTC, [3]). This method was chosen because it is compatible with retrieval from reflection and transmission coefficients. In this method, metasurfaces are characterized by electric and magnetic susceptibilities. In the literature, retrieved effective parameters have been shown to violate causality around resonances and this has been attributed to spatial dispersion [4]. In order to determine if spatial dispersion is the only source of this phenomenon, we have investigated the statistical properties of estimators that have been put forward for these susceptibilities. We have thus computed the Cramer-Rao lower bounds on the variance of these estimators. We have shown that this bound increases substantially around resonances making retrieval possible only for very high Signal-to-Noise Ratio (SNR, [5]). Therefore, in experiments, issues arising from spatial dispersion and noise compound and result in non-physical effective parameters. To mitigate this, we have proposed a least-squares estimator for susceptibilities that has a better performance with respect to noise. Sensitivity to noise is particularly acute for low-loss metasurfaces. It often results in required SNRs that are unachievable in practice. The present work is thus relevant to the development of loss-compensated metasurfaces for which the issues posed by retrieval will have to be closely considered for accurate and robust device characterization.