# Broad Frequency Shift of Parametric Processes in Epsilon-Near-Zero Time-Varying Media
## 2 Results
This choice was motivated by recent findings showing that the frequency degenerate FWM is more efficient at wavelengths longer than the \(\lambda_{ENZ}\)[18, 24]. For the wavelength of choice, the real and imaginary part of the refractive index of the AZO sample measured by ellipsometry were \(n_{\mathrm{r}}\simeq 0.3\) and \(n_{\mathrm{i}}\simeq 0.89\), respectively. The pump intensity was set to \(\simeq 770\) GW/cm\({}^{2}\). The input seed peak power was kept below 1 GW/cm\({}^{2}\) through all the experiments. We recorded the power spectra of PC (Figure 2a), NR (b), reflection (c), and transmission (d) as a function of the delay between the pump and the seed (\(\tau\), where \(\tau<0\) for the seed impinging on the sample before the pump pulse). From linear measurements, the reflection and transmission coefficients at the seed wavelength are \(R\simeq 0.51\) and \(T\simeq 0.04\). The spectra of the nonlinear signals were collected using the same spectrometer. The normalised spectrograms show that, at the delay corresponding to maximum generation efficiency, both the PC and NR are red-shifted with respect to the input seed, yet with different spectral shifts. Each spectrogram in the four panels is normalised to its maximum. The transmission and reflection spectrograms have a more complex structure that we interpret as the overlap of the signal reflection and transmission with the amplified seed field (AS\({}_{\mathrm{b}}\) and AS\({}_{\mathrm{f}}\), respectively). Figure 1: (**a,b**) describe the two four-wave mixing processes that generate the PC and NR signals, respectively. The pump beam is illustrated with a green arrow while the input seed is shown with a blue arrow. Red arrows show the result of the parametric amplification process, while blue dashed arrows show the linear reflection and transmission. For each of these two four-wave mixing processes, the seed field is amplified and emerges along both the transmitted and reflected directions. (**c**) Sketch of the experimental setup. Pump and seed fields have the same wavelength (1400 nm) and vertical polarisation. The seed is overlapped to the pump with an angle of \(\simeq 6^{\circ}\). The delay between the pump and seed pulses is controlled by a delay line (not shown). A spectrometer is used to collect the radiation emerging along the direction of the seed reflection and transmission (RS \(+\) AS\({}_{\mathrm{b}}\) and TS \(+\) AS\({}_{\mathrm{f}}\)), as well as to characterise the PC and NR. To better understand the spectral evolution, we plot the carrier wavelength shift (blue line) as a function of the pump-probe delay for both the PC and NR signals in Figure 3a,b, respectively. We overlap the wavelength shift with the PC and NR external generation efficiencies (red). The generation efficiency is calculated as the measured signal (PC or NR) energy at a specific pump-probe delay divided by the input seed pulse energy. One can see that the PC is shifted by more than 60 nm with respect to the input seed wavelength, with a maximum shift occurring for the seed overlapped with the leading edge of the pump pulse. Conversely, the NR wavelength shift is larger when the seed is overlapped with either the leading or the trailing edge of the pump pulse, yet with the opposite sign. The reflected seed is expected to drop and slowly regain the original unperturbed wavelength as a function of the pump-probe delay \(\tau\), as shown for an intraband excitation in [15]. This explains, for instance, why the reflection at \(-\)200 fs is stronger than that at 400 fs. Unlike [15], after an initial drop, the reflected signal suddenly increases at a wavelength \(\simeq\)60 nm red-shifted with respect to the input seed, as shown in Figure 3c. We interpret this as the combination of the reflection dynamics already reported in [15] with the signal generated by the parametric process in conjunction with PC and NR. We note that this shift is larger than the probe spectral bandwidth of the input seed (\(\simeq\)40 nm).