Time-varying metamaterials, which rely on rapid changes in linear permittivity, have opened new avenues in wave phenomena and provided fresh perspectives on classical physics. Time-varying photonics, manipulating the linear susceptibility (χ(1)) at optical frequencies, enables ultrafast switching, frequency shifting, spectral modulation, beam steering, and non-reciprocal devices. This field also holds promise for time crystals, coherent wave control, and lasing. Nonlinear effects, traditionally explained through perturbative changes in medium polarization, have been observed in time-varying media. For instance, frequency shifts of harmonic light have been seen in high-index silicon and germanium metasurfaces. In these systems, perturbative changes in high-Q cavities lead to reflectance changes and frequency shifts. However, resonance engineering limits bandwidth and requires precise design. Coherent control of SHG has been achieved in 2D materials, and harmonic modulation via photocarrier excitation and dark excitons has also been reported. Epsilon-near-zero (ENZ) materials, especially transparent conductive oxides, offer an alternative platform for time modulation at near-infrared frequencies. They combine large linear permittivity changes with ultrafast response, without needing strong photonic resonances or nanostructuring. In ITO, ultrafast refractive index changes result from photocarrier excitation, enabling demonstrations of time refraction, single and double-slit diffraction, and single-cycle dynamics. Harmonic-level studies have shown frequency-shifted high harmonics in CdO thin films, and four-wave mixing exhibiting time refraction and diffraction. ITO itself has been used for harmonic generation, with enhancements of third harmonic generation using plasmonic resonances and surface SHG leveraging the normal electric field component for p-polarized light. While recent findings suggest changes in ITO's nonlinear susceptibility, the effects of non-perturbative time modulation on χ(n) and nonlinear polarization P(n) remain largely unexplored. This research focuses on SHG from a time-varying ITO/air interface under non-perturbative time modulation, comparing it to fundamental frequency modulation, to demonstrate the impact of ITO's linear susceptibility modulation on its second-order nonlinear susceptibility.

Previous research has explored harmonic generation in various contexts, primarily focusing on perturbative approaches. Studies using high-index materials like silicon and germanium demonstrated frequency shifts in harmonic light due to changes in the linear susceptibility within high-Q cavities. However, these methods often suffer from limitations in bandwidth and require precise design and fabrication. Other works have focused on coherent control of SHG in 2D materials and modulation of harmonic light using photocarrier excitation or dark excitons. The use of ENZ materials, particularly transparent conductive oxides like ITO, has emerged as a promising alternative due to their ability to exhibit large and ultrafast changes in linear permittivity without requiring strong photonic resonances. Previous studies with ITO have shown time refraction, single and double-slit diffraction, and single-cycle dynamics, primarily in the near-infrared region. While some research has investigated high-harmonic generation in ENZ materials and SHG in ITO, the effects of non-perturbative time modulation on the nonlinear susceptibility remain under-explored.

The experiment involves SHG generated at the surface of a 310 nm ITO film by a p-polarized probe beam at a fundamental frequency (f = 230 THz) incident at 45°. The interface undergoes time modulation via optical pumping with a pump pulse (f = 230 THz, 225 fs FWHM) at 53° incidence. A delay stage controls the relative arrival time of pump and probe pulses. A spectrometer analyzes the reflected beam at fundamental and harmonic frequencies using spectral filters. The ITO film's ENZ frequency (248 THz) is below the probe frequency, ensuring operation in the metallic region for observing a large reflectivity drop during the metallic-to-dielectric transition under pumping. The generated second and third harmonic signals are measured as a function of probe intensity. The power dependence deviates slightly from the expected values (2 and 3 for SHG and THG, respectively), attributed to self-modulation at higher intensities. The focus is on SHG in reflection due to its surface sensitivity and simplified interpretation. Low probe intensity (<5 GW/cm²) ensures perturbative SHG generation without the pump, isolating the time modulation from the pump beam. The nonlinear susceptibility dynamics are evaluated by measuring the fundamental beam's reflectivity change to assess the linear material property modulation. A transfer matrix method (TMM) models the contribution of the fundamental frequency change. A simple Drude model with a time-varying plasma frequency and scattering rate describes ITO's permittivity change during optical pumping. The model reproduces the modulation amplitude at the fundamental frequency. The SHG is treated as a single nonlinear surface polarization source, its time evolution determined by changes in linear and nonlinear susceptibilities. An anharmonic oscillator model predicts the evolution of χ(1)(f,t) and χ(2)(t). The model shows that the modulation depth of χ(2)(t) is larger than that of χ(1)(f,t), even at moderate intensities, due to the cubic dependence of 1/χ(2) on the electron scattering rate. The extracted χ(2) relative change matches the anharmonic oscillator model at low to medium intensities but deviates at higher intensities, likely due to higher-order effects. The Berreman resonance is suppressed during modulation due to a sharp increase in the electron scattering rate, reducing the surface fields responsible for SHG. For double-slit diffraction, a stretched probe pulse (691 fs) interacts with the medium modulated by two subsequent 225 fs pump pulses. The reflected SHG spectrum is then measured. A Fourier model is employed to model the change in SHG and other fundamental signals as a function of delay and intensity.

The study reveals a significant modulation contrast (up to 93%) of the second harmonic signal with a pump intensity of 100 GW/cm². This modulation is significantly larger than the modulation observed at the fundamental frequency. The experimental results show that a substantial portion of this enhancement originates from the temporal modulation of the second-order nonlinear susceptibility, χ(2). A simple anharmonic oscillator model effectively captures the observed modulation of χ(2) at lower pump intensities, highlighting the strong coupling between linear and nonlinear susceptibilities. At higher pump intensities, deviations from the model suggest the influence of higher-order nonlinear effects. The SHG spectrum exhibits considerable broadening and red-shifting at slightly negative delays, when the probe coincides with the pump's leading edge. This spectral modulation is highly sensitive to the delay between the pump and probe pulses, offering potential applications in optical computing and machine learning. A simple reconstruction algorithm successfully reconstructs waveforms, demonstrating the potential for nonlinear waveform transformation. The high contrast between modulated and unmodulated states in the SHG signal, combined with the availability of cost-effective visible-range equipment, enhances the feasibility of implementing these algorithms. Double-slit diffraction experiments show that the time-varying effects contributing to SHG modulation combine coherently. The strong suppression of the carrier frequency peak in the time-modulated SHG signal improves the visibility of spectral oscillations, facilitating the extraction of information regarding the fast dynamics of ITO. This allows for an estimation of the fast excitation and relaxation timescales, which were not readily accessible in previous studies.

The findings demonstrate the significant impact of non-perturbative time modulation on the nonlinear susceptibility in a time-varying ITO interface. The anharmonic oscillator model provides a reasonable explanation of the observed behavior at lower intensities, showing the strong interplay between linear and nonlinear optical properties. The experimental results offer a valuable benchmark for more advanced theoretical models to explain the non-perturbative regime at higher intensities. The amplified time-varying effects in the frequency domain, due to enhanced effective nonlinearities, open up promising avenues for applications in optical computing, machine learning, and ultrafast spectroscopy. The strong sensitivity of the spectral features to the delay time provides the basis for various applications such as image reconstruction and processing.

This study provides experimental evidence for significant modulation of the second-order nonlinear susceptibility in a time-varying ITO interface. A simple anharmonic oscillator model successfully predicts this behavior at lower intensities, highlighting the coupled nature of linear and nonlinear responses. The enhanced nonlinearities lead to amplified time-varying effects in the frequency domain, with potential applications in optical computing, machine learning, and ultrafast spectroscopy. Further theoretical investigation is needed to fully explain the high-intensity behavior and extend this understanding to other nonlinear processes.

The anharmonic oscillator model, while providing a good approximation at lower intensities, deviates from experimental data at higher intensities. This deviation suggests the presence of higher-order nonlinear effects that are not captured by the model. The model also makes simplifying assumptions, such as constant effective mass and scattering rate for electrons at the surface and a constant asymmetric potential term. The study focuses on SHG at the air/ITO interface, neglecting potential contributions from the ITO/substrate interface.