(\[Eq:SRRpolarizsimple\]),  (\[Eq:prefactorSRR\]), one needs to explicitly verify that each eigenvalue has positive imaginary part. The eigenvalues of Eq. (\[Eq:SRRpolarizsimple\]) are$A^\mathrm{static}_\pm=\alpha(\omega)\lambda_\pm$ with $\lambda_\pm=\frac{\eta_E+\eta_H\pm\sqrt{(\eta_E-\eta_H)^2+4\eta_C^2}}{2}$. Since $\mathrm{Im}(\alpha(\omega))\geq0$ and $\lambda_{\pm}$ are real, we find that both eigenvalues have positive imaginary part only if both $\lambda_+$ and $\lambda_-$ are positive. Thus, loss sets an additional constraint on the polarizability tensor, and limits the magneto-electric coupling to $$|\eta_{C}|\leq\sqrt{\eta_E\eta_H}. \label{Eq:crosscouplelimit}$$ This result implies a very important limitation on magneto-electric scatterers: it states that a magneto-electric cross coupling ($\eta_C$) can only be generated if there is a sufficiently strong directly electric, and directly magnetic response. We note that this constraint is very similar to the constraint on the magneto-electric cross coupling in constitutive tensors derived for homogeneous bi-anisotropic media in Ref.  that recently attracted attention in the framework of proposals for repulsive Casimir forces [@silveirinha; @soukouliscasimir]. While our derivation was specific for split rings, we note that similar constraints hold for all magneto-electric scatterers. In the presence of material loss, the magneto-electric coupling terms are limited by the fact that all electrostatic eigenpolarizabilities must have positive imaginary part.

Predicted scattering properties of single split rings\[section:singleSRRexp\]
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In the remainder of the paper we discuss some insights that the 