Second harmonic generation at a time-varying interface

The study investigates how non-perturbative, ultrafast temporal modulation of a medium impacts second-harmonic generation (SHG) at an interface. Time‑varying photonics typically modulates the linear susceptibility χ(1) to achieve ultrafast switching, frequency shifting, and nonreciprocity. However, how such strong temporal modulations affect nonlinear processes—particularly the second-order susceptibility χ(2) and the generated SHG—remains unclear. The authors aim to determine whether and how time‑varying changes of the linear response in epsilon‑near‑zero (ENZ) materials, specifically indium tin oxide (ITO), translate into significant and potentially dominant modulation of χ(2), thereby enhancing SHG contrast, frequency broadening, and shifts. The work seeks to extend time‑varying photonics effects into the visible via the harmonic and to evaluate implications for applications such as optical computing, sensing, and spectroscopy.

Prior work established that time‑varying media enable ultrafast modulation, frequency shifting, beam steering, and nonreciprocal behavior by modulating χ(1) at optical frequencies. Nonlinear effects in time‑varying media have been explored mainly using perturbative frameworks or high‑Q resonances in metasurfaces (e.g., silicon and germanium), yielding strong spectral shifts without large material changes, albeit with limited bandwidth and fabrication demands. In ENZ materials (ITO, AZO, CdO), large, ultrafast refractive index changes arise from intraband carrier dynamics, enabling demonstrations of time refraction, time diffraction (single/double slit), and single‑cycle dynamics. ITO has shown enhanced THG via ENZ modes and surface SHG enhanced near Berreman resonances. Recent reports suggest time‑dependent nonlinearities in ENZ platforms. Nonetheless, the explicit role of non‑perturbatively modulated χ(2) and nonlinear polarization P(n) in harmonic generation—and its relative importance versus the modulated fundamental fields—has remained largely unexplored.

Experimental platform: A 310 nm ITO film (Präzisions Glas & Optik GmbH) on SiO2 is interrogated at an air/ITO interface. A p‑polarized probe at f = 230 THz (λ = 1300 nm) impinges at 45° (near the Berreman resonance) to generate reflected SHG at 2f ≈ 460 THz (λ ≈ 650 nm). The interface is optically pumped with a degenerate pump at 230 THz (225 fs FWHM) at 53° incidence. Pump–probe delay is controlled with a delay stage. Reflected spectra at fundamental and harmonic frequencies are acquired with spectrometers using appropriate filters. ITO’s ENZ frequency is 248 THz; pumping shifts the response via intraband transitions. Operating regime: Probe intensity is kept low (<5 GW/cm^2) to ensure perturbative SHG in the absence of the pump and to minimize probe‑induced self‑modulation, while pump intensity is varied up to 100 GW/cm^2 to induce strong time modulation. SHG is measured in reflection to isolate surface contributions and minimize propagation/multiple reflection effects; THG requires higher probe intensities and is not the focus. Linear and nonlinear modeling: Linear permittivity changes under pumping are modeled with a Drude dispersion whose plasma frequency ωp and electron scattering rate γ are time‑dependent due to intraband excitation and nonparabolicity. A transfer matrix method (TMM) computes: (i) the reflectivity modulation at f and 2f, (ii) the surface normal field Ez(f,t) that sources SH polarization, and (iii) the expected SH response from field‑only modulation. Anharmonic oscillator model: To connect χ(1)(t) and χ(2)(t), a simple anharmonic oscillator model (Boyd) is used, linking χ(2)(2ω;ω,ω) ∝ (ea/m)·ωp^2/[D(2ω)D(ω)^2] with D(ω)=ω^2−ω0^2+iγω, and in the Drude limit (ω0=0) giving χ(2) dependence ∝ ωp^2/[(4ω^2+2iγω)(ω^2+iγω)^2]. This predicts a stronger sensitivity of χ(2) to γ (inverse cubic) than χ(1) (linear). Time‑dependent ωp(t) and γ(t) are obtained from rate‑equation‑like fits to the pump intensity I(t) with relaxation Γ. Extraction of χ(2) dynamics: From measured reflectivity changes (fundamental) and TMM‑computed Ez(f,t), the SH field is modeled as ESHG ∝ χ(2)(t)·Ez(f,t)^2 with the Fresnel factor at 2f approximated constant (small measured R(2f) modulation). Comparing measured SHG modulation with field‑only predictions yields the inferred χ(2)(t) relative changes. Spectral/time‑diffraction studies: Time‑diffraction is probed by scanning delay and recording SH spectra. For double‑slit time diffraction, two 225 fs pump pulses create two temporal slits; the probe is stretched to 691 fs with a 4f system, and reflected SH spectra are measured. A phenomenological Fourier model with a fast rise (~4.36 fs 10–90%) and slow decay (~615 fs) of the temporal aperture is used to interpret diffraction order intensities and to estimate fast relaxation components. A non‑degenerate control (probe at 2f from BBO) measures linear changes at 2f.

• Large SHG modulation: Modulation contrast at the second harmonic reaches up to 93% at pump intensity of 100 GW/cm^2, yielding strong spectral broadening and redshift.
• SHG vs fundamental modulation: The SHG modulation depth exceeds that of the fundamental reflectivity R(f) across pump intensities and saturates earlier (above ~50 GW/cm^2). The modulation of reflectivity at 2f, R(2f), is ~13× smaller than the SHG modulation, so χ(1)(2f) variations contribute negligibly to SHG dynamics.
• Power dependences: Measured probe‑intensity scalings yield slopes of 1.72 for SHG and 3.13 for THG (vs ideal 2 and 3), attributed to self‑modulation at higher probe intensities (>20 GW/cm^2).
• Modeling agreement: A Drude model with time‑varying ωp and γ reproduces the fundamental reflectivity modulation. A field‑only SHG model (varying Ez(f,t) at the surface) underestimates the SHG contrast; including a time‑varying χ(2)(t) from the anharmonic oscillator model matches the SHG modulation well at low–medium intensities (<50 GW/cm^2).
• χ(2) sensitivity: The model and extracted data indicate χ(2) is more strongly modulated than χ(1), due to χ(2)’s inverse‑cubic dependence on γ versus χ(1)’s linear dependence. Suppression of the Berreman resonance under pumping reduces Ez at the surface, yet the χ(2) increase dominates the observed SHG modulation.
• Spectral dynamics: Time modulation produces pronounced SH spectral broadening and redshift near the pump leading edge; the overall harmonic spectral shift is approximately twice that observed at the fundamental. Effects saturate at ~70 GW/cm^2.
• Time‑diffraction: Double‑slit time diffraction at SH shows clear spectral oscillations with a visibility of 0.58 ± 0.01 across slit separations, indicating coherent interference of newly generated frequencies. Using the Fourier model, the ratio of blue/red second‑order diffraction intensities (0.16 ± 0.02 measured) is consistent with comparable fast rise and decay times, supporting a short fs relaxation component.
• Computational utility: Using delay‑dependent SH spectra, simple regression reconstructs target waveforms (e.g., sin^2) with mean square error 1.57×10^−4, highlighting potential for optical computing and machine learning in the visible.

The study demonstrates that SHG at a time‑varying ITO interface is driven not only by modulation of the fundamental fields but also by significant, coherent modulation of the second‑order susceptibility χ(2). This resolves a key question: strong temporal changes in linear properties induce concomitant and even stronger relative changes in χ(2), which, despite suppression of surface field enhancement (e.g., Berreman resonance), result in a large SHG modulation depth and rich spectral dynamics. The anharmonic oscillator model captures the main physics and quantitatively matches measurements up to the onset of saturation. These findings extend time‑varying photonics effects into the visible via the harmonic, amplifying time‑diffraction signatures and frequency shifting. The enhanced sensitivity of SH spectra to delay enables practical implementations in ultrafast spectroscopy, sensing, and optical information processing, including regression and potential machine‑learning pipelines. Deviations at high pump intensities point to the need for more complete non‑perturbative and dispersive models, offering a testbed to refine theories of time‑varying nonlinear optics.

The work provides experimental evidence that non‑perturbative temporal modulation in ENZ ITO strongly modulates the second‑order susceptibility, leading to up to 93% SHG contrast, pronounced frequency broadening, and redshifts. A simple anharmonic oscillator model linked to time‑varying Drude parameters explains the SHG modulation at low–medium pump intensities, distinguishing contributions from field modulation and χ(2) dynamics. Time‑diffraction at the harmonic confirms coherent generation of new frequencies with high visibility and improved spectral readability due to suppression of the carrier peak. These insights generalize to other time‑varying platforms and nonlinear processes (e.g., THG, FWM). Future work should develop rigorous non‑perturbative models including dispersion, saturation, and more realistic interfacial physics, extend to different material systems and thicknesses, and leverage the strong SH sensitivity for optical computing, reconfigurable image processing, and broadband sensing.

• Model simplifications: The anharmonic oscillator model neglects higher‑order processes and dispersion, assumes constant interfacial asymmetry parameter a, and equates surface and bulk electron parameters; accuracy degrades at high pump intensities (>50 GW/cm^2) where saturation and higher‑order effects emerge.
• Field‑only and Fourier models: The field‑only SHG model underestimates modulation without χ(2) dynamics. The Fourier time‑diffraction model is phenomenological, omits sample dispersion and phase shifts, and offers qualitative rather than quantitative dynamics.
• Spectral modeling gap: The generation of new frequencies and large shifts at high modulation cannot be fully captured by the present model, indicating a need for comprehensive non‑perturbative, dispersive treatments.
• Experimental constraints: To avoid probe‑induced self‑modulation, probe intensity is limited; THG studies are constrained by required higher probe intensity and complex propagation/multi‑reflection effects.