Replicate measurements of the SHG phase in air were performed using various laser spot positions on a given glass slide or nickel electrode with 15 in different electrodes so as to account for variations in the measurements that come along with slight variations in how the sample cell is assembled and mounted between replicates/trials (please see Supplementary Information Fig. ). We first obtain the absolute zero phase from the uncoated portion of a glass microscope slide having one half coated with 10 nm nickel and then move the sample cell over by a few millimeters to determine the fitted phase of the metal nanolayer. We mount the slide in the electrochemical cell and contact it with air (no electrolyte present). The uncoated part of the glass slide is fully transparent at the wavelengths employed here, not birefringent, and the surface potential is zero, so that the SHG response is purely real, i.e. 𝐸 !"#,%&'((:'*+ 𝑒 *-,"#$%%:$'( = real, with 𝜑 ,%&'((:'*+ =0°.