The setup for measuring the SHG is described in the supporting online material (22) . We expect that the SHG strongly depends on the resonance that is excited. Obviously, the incident polarization and the detuning of the laser wavelength from the resonance are of particular interest. One possibility for controlling the detuning is to change the laser wavelength for a given sample, which is difficult because of the extremely broad tuning range required. Thus, we follow an alternative route, lithographic tuning (in which the incident laser wavelength of 1.5 mm, as well as the detection system, remains fixed), and tune the resonance positions by changing the SRR size. In this manner, we can also guarantee that the optical properties of the SRR constituent materials are identical for all configurations. The blue bars in Fig. 1 summarize the measured SHG signals. For excitation of the LC resonance in Fig.  1A (horizontal incident polarization), we find an SHG signal that is 500 times above the noise level. As expected for SHG, this signal closely scales with the square of the incident power ( Fig. 2A) . The polarization of the SHG emission is nearly vertical (Fig. 2B) . The small angle with respect to the vertical is due to deviations from perfect mirror symmetry of the SRRs (see electron micrographs in Fig. 1 ). Small detuning of the LC resonance toward smaller wavelength (i.e., to 1.3-mm wavelength) reduces the SHG signal strength from 100% to 20%. For excitation of the Mie resonance with vertical incident polarization in Fig. 1D , we find a small signal just above the noise level. For excitation of the Mie resonance with horizontal incident polarization in Fig. 1C , a small but significant SHG emission is found, which is again po- Although these results are compatible with the known selection rules of surface SHG from usual nonlinear optics (23), these selection rules do not explain the mechanism of SHG. Following our above argumentation on the magnetic component of the Lorentz force, we numerically calculate first the linear electric and magnetic field distributions (22); from these fields, we compute the electron velocities and the Lorentz-force field ( fig. S1 ). In the spirit of a metamaterial, the transverse component of the Lorentz-force field can be spatially averaged over the volume of the unit cell of size a by a by t. This procedure delivers the driving force for the transverse SHG polarization. As usual, the SHG intensity is proportional to the square modulus of the nonlinear electron displacement. Thus, the SHG strength is expected to be proportional to the square modulus of the driving force, and the SHG polarization is directed along the driving-force vector. Corresponding results are summarized in Fig. 3 in the same arrangement as Fig. 1 to allow for a direct comparison between experiment and theory. The agreement is generally good, both for linear optics and for SHG. In particular, we find a much larger SHG signal for excitation of those two resonances (Fig. 3, A and C) , which are related to a finite magnetic-dipole moment (perpendicular to the SRR plane) as compared with the purely electric Mie resonance (Figs.  1D and 3D ), despite the fact that its oscillator strength in the linear spectrum is comparable. The SHG polarization in the theory is strictly vertical for all resonances. Quantitative deviations between the SHG signal strengths of experiment and theory, respectively, are probably due to the simplified SRR shape assumed in our calculations and/or due to the simplicity of our modeling. A systematic microscopic theory of the nonlinear optical properties of metallic metamaterials would be highly desirable but is currently not available.
UBP is comparable with the latter-half of an autoencoder REF .
Reducing the Dimensionality of Data with Neural Networks
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