Electronic Metamaterials with Tunable Second-order Optical Nonlinearities

The ability to engineer metamaterials with tunable nonlinear optical properties is crucial for nonlinear optics. Traditionally, metals have been employed to enhance nonlinear optical interactions through field localization. Here, inspired by the electronic properties of materials, we introduce and demonstrate experimentally an asymmetric metal-semiconductor-metal (MSM) metamaterial that exhibits a large and electronically tunable effective second-order optical susceptibility (χ(2)). The induced χ(2) originates from the interaction between the third-order optical susceptibility of the semiconductor (χ(3)) with the engineered internal electric field resulting from the two metals possessing dissimilar work function at its interfaces. We demonstrate a five times larger second-harmonic intensity from the MSM metamaterial, compared to contributions from its constituents with electrically tunable nonlinear coefficient ranging from 2.8 to 15.6 pm/V. Spatial patterning of one of the metals on the semiconductor demonstrates tunable nonlinear diffraction, paving the way for all-optical spatial signal processing with space-invariant and -variant nonlinear impulse response.


Optical characterization setup
The optical characterization is carried out via the Maker fringe setup shown in Supplementary Fig.1. The pump beam is generated using a Ti:Sapphire laser emitting 150 fs pulses with a 80 MHz repetition rate at a center wavelength of 800 nm. The polarization state of the pump is defined by a half-wave plate and a long pass filter with cut-off wavelength at 780 nm is set to filter out any signals from other sources in the range of interest. The sample is tilted and fixed at an angle of 45 degrees normal to the incident beam, which is focused onto the sample surface using an 10x objective lens, resulting in a beam size with a radius of 20 µm. At the output, two short wavelength pass filters and one band pass filter with a total optical density of 12 are inserted to filter out the pump light at ω (i.e., 800nm wavelength), ensuring that all photons collected by the photomultiplier (PMT) are at 2ω (i.e., 400 nm wavelength) and consequently generated from the SHG process. The collimated SHG signal from the sample is separated into p-polarized and s-polarized by a polarizer for calculating the different components of the χ (2) tensor. The detected signal in the PMT is then read with an oscilloscope. A commercial 500 µm thick X-cut quartz wafer, exhibiting a nonlinear coefficient χ (2) xxx of 0.64 ± 8% pm/V is used to calibrate the system, 1,2 and the absolute values of χ (2) tensor components from our samples are determined by comparing the generated SHG signals with those from the quartz sample under the same experimental conditions. Bare fused silica substrates are also characterized under the same conditions as samples with grown metamaterials to ensure that substrates do not contribute to any SHG signal. The second-harmonic response from a single layer of a-Si is found to be negligible compared to the large detected signals from the metal films and MSM metamaterials which is expected due to its amorphous nature. The measurement errors in our setup originate mainly from the fluctuation of laser power due to the varying humidity in the environment (±5%), background noises (±20%), and the non-uniformity in the thickness of the deposited thin films (±10%).
In addition, the possibility of counting error (±10%) of photons in the PMT (Hamamatsu Inc., H11461-03) due to pulse-overlapping, as described in the handbook, is also taken into account. In order to minimize these errors, the generated SHG intensities from quartz, metal films and MSM metamaterials are determined by taking the average of those measured from five different spots on each sample. Following Herman's equation, 3 three tensor components, χ (2) zzz , χ (2) xxz and χ (2) zxx can be extracted by fitting the generated s-and p-polarized second-harmonic signals under various polarization angles of the incident pump beam.

Maker fringes analysis
Since a-Si is an amorphous material, we assume that its third-order susceptibility tensor χ (3) components are same as those of an isotropic material with C ",$ space symmetry and thus has 21 nonzero elements, of which only 3 are independent: The existence of a static electric field in z direction is expected to introduce effective χ (2) tensor components: χ (2) xxz , χ (2) xzx , and χ (2) zxx and χ (2) zzz through the EFISH effect.
We assume that the MSM structures are isotropic in the transverse (i.e., in-plane) direction and that multiple reflections within the thin films can be neglected due to the thin nature of constituent films (i.e., 5 and 25 nm) compared to the wavelength of pump light (i.e., 800nm). 4 With the assistance of Maker fringes analysis, 3 these non-zero χ (2) tensor components of the MSM metamaterial: χ (2) xxz , χ (2) zxx and χ (2) zzz can be determined from fitting the generated s-and p-polarized second-harmonic signal intensities at frequency 2ω as a function of the polarization of the fundamental pump beam at frequency ω, measured at a fixed angle of incidence, θ. Also, since the thicknesses of MSM metamaterials are much smaller than the wavelength of the pump beam, it is justified to use the effective medium theory for determining the three non-zero components of χ (2)  Here, following Herman's notations, >' and &' denote the power at fundamental and second harmonic frequency, respectively, and the superscript s/p represents the state of polarization; 3 is the angle of incidence for fundamental frequency beam  The relation between effective susceptibility C33,(/* and three χ (2) tensors is expressed as:

Analysis of the effect of photocurrent on the built-in electric field within a-Si layer
The induced effective χ (2) in our MSM structures comes from the engineered non-zero static electric field within the semiconductor (i.e., a-Si) layer. Since the SHG measurements were carried out using optical fields with photon energies (i.e., 1.55 e.V.) larger than bandgap of a-Si (i.e., 1.1 e.V.), we need to consider the effect of light induced free carrier generation, which may affect the magnitude of the built-in electric field in a-Si, and, consequently, the induced effective χ (2) . In order to quantify the influence of the generated photocurrent on the built-in electric field, we performed I-V measurements (see Supplementary Fig. 2