Accurate and robust characterization of metasurfaces and metamaterials in terms of effective parameters is critical to the design of novel metadevices. We compute the Cramér-Rao lower bounds on the variance of any estimator for both the electric and magnetic surface susceptibilities of metasurfaces. We show that retrieval of such effective properties is inherently difficult around resonances, most notably for low-loss metasurfaces. We also put forth a least-squares estimator to mitigate this difficulty for the normal components of susceptibility tensors, which are observed to be the most ill-behaved. The present work is relevant to the development of loss-compensated metasurfaces for which noise has to be closely considered for accurate and robust device characterization.
© 2015 Optical Society of America
Over the past fifteen years, a lot of attention has been devoted to electromagnetic metamaterials and their two-dimensional counterparts, metasurfaces [1–3]. Since the beginning, efforts have been focused on understanding their effective properties to rapidly and efficiently design new and advanced devices, e.g., unconventional imaging systems . In essence, their unusual effective properties stem from the fact that their constituent meta-atoms are both sub-wavelength and resonant, i.e., they are sub-wavelength resonators. This double character presents a substantial challenge for homogenization that has yet to be fully resolved .
Amongst the many homogenization techniques put forward over the years, the ‘retrieval method’ has had success in the literature due to its compatibility with experiments . It was originally derived from a microwave material characterization technique [7,8], and it was later expanded to negative indices for left-handed materials [6,9]. This homogenization technique has been successfully used to design transformation optics devices such as cloaks and antennas [10–13]. Despite its success, the technique suffers from several shortcomings. Most notably, it fails around resonances where the retrieved effective parameters tend to violate both causality and passivity . In the last few years, a flurry of homogenization techniques have been proposed but, due to the topic’s intrinsic complexity, none have yet to be widely accepted.
In this paper, we consider a specific homogenization technique that is dedicated to metasurfaces, namely Generalized Sheet Transition Conditions (GSTC, ). We investigate its particular statistical properties and we provide a blueprint for estimating the sensitivity of any retrieval algorithm to noise. In the second section, we recall the underlying principles of GSTC and the estimator put forward to estimate effective susceptibilities. In the third section, we introduce and compute the Cramér-Rao lower bound on the variance of any effective susceptibilities’ estimator. In the last section, we propose several alternative estimators and select the best one based on simulations of their respective variances.
2. Generalized sheet transition conditions
In the following, we consider the technique developed by Kuester et al. since 2003 [15–17]. It takes into account the fields discontinuity at a metasurface interface via surface impedance and admittance tensors or, alternatively, electric and magnetic surface susceptibility tensors 18], effective conductivities , and expanded the GSTC formalism to bianisotropic metasurfaces [20,21]. From these new boundary conditions, S-parameters can be defined and, after inversion, surface susceptibilities can be obtained [16,17]. For example, considering a medium characterized by diagonal tensors (biaxial medium), the following three components can be retrieved from a TE-polarized plane wave at normal and oblique incidence16,17]. These expressions constitute estimators for the effective susceptibilities with specific statistical properties.
To illustrate this homogenization technique, we consider as an example a 2D square array of cubes (width = 1.4 μm, periodicity = 1.9 μm) made of Lithium Tantalate (LiTaO3) in the frequency range between 19 and 27 THz, (11.1-15.8μm). LiTaO3’s permittivity in the vicinity of its phonon resonance is expressed as22].
We plot on Fig. 1, the x and y components of the electric and magnetic susceptibility tensors. Since these are equal due to the cubic symmetry of the array, we only refer to them as ‘tangential’. Likewise, we will refer to the component along z as ‘normal’. As seen in Fig. 1, there are a great number of resonances many of which are heavily damped. LiTaO3’s permittivity becomes very large right before its transverse phonon frequency (ωT = 26.7 THz, black vertical dashed line on Fig. 1) turning the cubic particles into high-index dielectric resonators [23,24]. As the Reststrahlen band is approached from below the density of resonances increases but most are damped by the increasing material loss. From extensive numerical simulations, it appears that Eqs. (5-7) lead to inconsistent results for different angles of incidence, especially for low-loss metasurfaces and normal components.
To illustrate this point, we plot in Fig. 2 the normal component of the magnetic susceptibility tensor for three different collision frequencies (γ = 0.94, 1.94, 2.94 THz). In Fig. 2(a) and 2(b), we plot the mean susceptibility retrieved from four different oblique angles of incidence . In Fig. 2(c), we plot the log-variance of the susceptibility. As can be seen from Fig. 2, as the collision frequency is diminished the variance of the normal susceptibility increases substantially, most notably around the resonances.
The rapid oscillations of the variance below 26.7 THz, which are especially visible for a collision frequency of γ = 0.94 THz, are due to the increased density of undamped resonances as previously mentioned. It can also be seen that, besides the usual resonances, the normal susceptibility presents several “anti-resonances” where the real part decreases first and subsequently increases while the imaginary part becomes negative (see Fig. 2(a-b)). In the literature, all these surprising features have most often been attributed to a failure of the homogenization technique, namely spatial dispersion . However, to rule out any other possibility, we investigate the statistical properties of the estimators that lead to the increased variance (Eqs. (5-7)).
3. Cramér-Rao lower bound
We define a statistical model of measurement, including noise, for the case of a TE-polarized plane wave (vectors are denoted by bold lowercase and matrices by bold uppercase letters). Similar reasoning can be applied to a TM-polarized plane wave.Eq. (10)) and is a superposition of the deterministic scattering vector s, which depends on the susceptibility vector (see Eq. (11)), and a random noise vector n. Furthermore, the first-order (mean) and second-order (covariance) noise statistics are assumed to be known.
Estimators defined by Eqs. (5-7) are not the only way to retrieve the above effective susceptibilities. Indeed, from four complex S-parameters (reflection and transmission at normal and oblique incidences), there is more than one way to compute three complex susceptibilities. To gauge the performance of different estimators, we can consider three of their fundamental characteristics: bias, consistency, and efficiency . An estimator is said to be unbiased if its expected value is the true value, consistent if its variance tends to zero as the number of samples increases, and efficient if its variance is minimum, i.e., if it reaches the Cramér-Rao lower bound [26,27]. By definition, the Cramér-Rao lower bound limits the lowest possible variance of any unbiased estimator. Its importance lies in the fact that it depends on the data itself but is independent of any specific estimator .
For a multivariate problem such as the one presented above, the covariance matrix of any estimator of the susceptibilities is always superior to the Cramér-Rao lower bound (i.e., the difference of the two matrices is positive-semidefinite). The Cramér-Rao lower bound is given by the inverse of Fisher’s information matrix .28].29]. Circularity is a relatively frequent property of Gaussian random signals .30], with the additional assumption that the noise covariance is independent of , Fisher’s information matrix can be simplified to Eq. (19).Eq. (21) are provided in Appendix A. The Cramér-Rao lower bound for each component of the susceptibility tensors is then given by the corresponding diagonal element of the inverse Fisher’s information matrix.
As a last step, to make sure our results do not depend on the wavelength range considered (susceptibilities are not unitless), we normalize unknowns and modify Fisher’s information matrix accordingly
We plot on Fig. 3, the Cramér-Rao lower bounds (CRLB) for all susceptibilities retrieved from a TE-polarized plane wave (see Eqs. (5-7)). In Fig. 3(a), we plot the tangential components for three different collision frequencies (γ = 0.94, 1.94, 2.94 THz) and two angles of incidence (). In Fig. 3(b), we plot the normal component for four different oblique angles of incidence ) and one collision frequency (γ = 0.94 THz). As can be seen from Fig. 3, all Cramér-Rao lower bounds increase substantially near resonances. To give a sense of the Signal-to-Noise Ratio (SNR) needed for a meaningful retrieval, let us consider an example. To limit the retrieval error to at most 0.05, or −26 dB, the noise level at 19.65 THz must be kept below −46 dB . Such SNRs are achievable in microwaves with vector network analyzers after careful calibration but it might not be so in every frequency range and for every metasurface . Overall, it can be seen that small losses make the retrieval more difficult, similar to the retrieval of classical material parameters , and that, in addition, small angles of incidence also make retrieval of the normal component more difficult. This last statement can be intuitively understood because for such angles the projection of the field on the z-axis is quite small, i.e., this component is probed less effectively. For low-frequency metasurfaces, it can be an issue because most setups are limited to a small angular range around normal incidence . In summary, the variance of any estimator of the susceptibilities is bound to increase because it has to be greater than the Cramér-Rao lower bound. Therefore, the uncertainty witnessed on Fig. 2 around resonances, due to spatial dispersion, is expected to be compounded by this increased variance. This is the coincidence of these two issues that make it extremely difficult to retrieve consistent and meaningful susceptibilities in some cases. It is particularly a concern for experiments on low-loss or loss-compensated metasurfaces.
4. Alternative estimators
Now that the difficulty of retrieving susceptibilities has been established, especially normal components, the question of which estimator can perform best, or put another way, which estimator has a variance closest to the Cramér-Rao lower bound, still remains. We define three alternative estimators and compare them with the one defined by Eq. (7). First, we define two estimators based solely on either the reflection or transmission coefficient. For TE-polarization, they are expressed asEq. (7), only the second terms in Eqs. (24) and (25) have changed and there is a noticeable symmetry to them. Note that there exist an infinite number of estimators that can be obtained by taking a weighted average of the above two.
Finally, thanks to the linearity of the equations, we can define an estimator based on both reflection and transmission coefficients that is closest to the data in a least-squares senseEq. (26) and similar ones obtained for a TM-polarized plane wave are provided in Appendix C. For the normal component of the magnetic susceptibility tensor, Eqs. (7) and (26) perform similarly while Eqs. (24) and (25) have a slightly greater variance (not shown). To make a more meaningful comparison of these four estimators, we turn to a case where the retrieval proves much more difficult. We now consider the normal component of the electric surface susceptibility tensor.
We plot on Fig. 4, the normal component of the electric susceptibility tensor for all four estimators. In Fig. 4(a) and 4(b), we plot the mean susceptibility retrieved from four different oblique angles of incidence . In Fig. 4(c), we plot the log-variance of the susceptibility. As can be seen from Fig. 4, the four estimators yield sensibly different estimates and none predicts a clear-cut resonance. The least-squares estimator, Eq. (26), achieves the lowest variance while the original estimator, Eq. (7), has a variance that prohibits any meaningful retrieval.
To compute the Cramér-Rao lower bound, the true susceptibility is required. Previously, defining the normal component of the magnetic susceptibility tensor was straightforward since all four estimators offered similar estimations. However, this is not the case for the electric susceptibility. Since the electric susceptibility retrieved with the least-squares estimator at 60° is likely the most accurate, we assume it to be the true one. Note that this choice marginally impacts the magnitude of the Cramér-Rao lower bound. We plot on Fig. 5, the Cramér-Rao lower bound of the normal component of the electric susceptibility tensor as well as the variances for all four estimators obtained from 105 simulations with added Gaussian white noise with a noise power corresponding to a SNR of 0 dB. This is done for two angles of incidence () and one collision frequency (γ = 0.94 THz).
As can be seen from Fig. 5, none of the estimators reach the Cramér-Rao lower bound at all frequencies, though they are all fairly close to it near the main resonance at 19.65 THz. It is therefore possible that a better estimator exists although the existence of the bound does not necessarily imply that it can be reached . Globally, the least-squares estimator performs better, a fact that we have also checked on other cases, and should therefore be favored over all other three estimators. In the present case, a high SNR is required to retrieve meaningful susceptibilities (SNR>75 dB) and that might experimentally prove to be extremely challenging. In the last few years, it has been proposed to include gain in metasurfaces in order to achieve loss-compensated operation. However, as one tends to the lossless regime, the practical issue of retrieving meaningful susceptibilities becomes increasingly difficult if not altogether impossible.
We note that, from extensive simulations, the presence of anti-resonances in retrieved normal components is a robust finding. It can happen for either electric or magnetic components depending on the structure. The question remains open as to why this feature only appears for normal components. Several authors have derived Eqs. (5-7) by different means [32,33], which might shed light on this issue. Finally, all the algorithms discussed so far use the frequency data point by point and treat them independently. As causality implies that the real and imaginary parts of the susceptibilities must follow certain broadband dispersion relations (Kramers-Krönig relations ,), this does not appear to be the most efficient retrieval technique. For homogeneous materials, it was proposed some twenty-five years ago to take advantage of causality to decompose material parameters in poles and zeros and then solve a nonlinear least-squares fitting problem . A similar method could be applied to susceptibilities but at present, unfortunately, it remains unclear whether normal components should always follow causality.
In this paper, we have shown that characterization of metasurfaces in terms of effective susceptibilities can prove difficult around resonances not just because of unaccounted spatial dispersion but also because of intrinsic difficulties in the retrieval algorithms. We have shown that every single algorithm, independent of its specific statistical properties, is bound to fail for low-loss metasurfaces if the signal to noise ratio is not extremely high. This is fundamentally linked to a lack of information in the data at these frequencies. Until now, inversion algorithms for effective parameters have been devised by proceeding to a direct inversion of reflection and transmission coefficients as obtained by a given homogenization technique. This does not ensure the robustness of these algorithms. Therefore, we advocate that all effective parameters estimators should be designed from the beginning with noise statistics in mind.
Appendix A: Fisher’s information matrix
The generic expression of the four Fisher’s information matrices appearing in Equation (21), for both TE and TM polarizations, are given by
Appendix B: Reflection and transmission estimators
For a TM-polarized the two estimators based solely on reflection or transmission coefficients are
Appendix C: Least-squares estimator
For a TE-polarized plane wave, the four matrices of the linear least-squares estimators are expressed as
The authors thank Qualcomm Institute (QI) for its support.
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