# Broad Frequency Shift of Parametric Processes in Epsilon-Near-Zero Time-Varying Media
## 3 Discussion  
The broad frequency shift observed for four-wave mixing around the epsilon-near-zero region only partially arises from the same physical mechanism responsible for the frequency shift of reflected and transmitted beams from a time-varying surface [25]. The fields generated by the nonlinear polarisation experience an adiabatic shift of the refractive index. However, a simple application of that model to our experiment would predict shifts comparable to those reported in [25], which are nearly three times smaller than what is reported here. Another contribution to the frequency shift of all beams originates from the time-varying Fresnel coefficients at the two interfaces air/AZO and AZO/glass. A complex refractive index that varies in time at a faster-than-picosecond time scale, as in our case, leads to significant changes in the wavelength of the impinging fields. We have estimated the magnitude of the effect that such a change would provide assuming a large change in refractive index (\(\Delta n=2\)), delivered in a short, 50 fs time period. Furthermore, even for this extreme case, the time dependence of the complex Fresnel coefficients would only account for 12 nm of shift in NR and 35 nm in PC. Furthermore, despite combining the two effects, we could only account for about 35 % of the observed shift for the forward PC. We therefore speculate that the experimental results indicate possibly non-trivial underlying physical effects that might require a full microscopic model of this experimental study.  
Figure 3: (**a**) Carrier wavelength shift of the PC signal as a function of the pump-probe delay \(\Delta\tau\) (blue-curve) overlapped with the PC generation efficiency (red curve). (**b**) Carrier wavelength shift of the NR signal as a function of the pump-probe delay \(\tau\) (blue curve) overlapped with the NR generation efficiency (red curve). (**c**) Reflection spectrogram as in Figure 2c, with a dashed blue line at the input seed carrier wavelength and a black dotted line at the carrier wavelength of the PC peak. (**d**) Measured wavelength shifts for the PC (blue) and the NR (red) nonlinear products, as a function of the pump intensity.  
Finally, we note that the simple model of an infinitely thin time-varying medium presented in Figure 1a,b would imply a symmetric effect in the forward and backward directions. In contrast, we observe that PC and NR have different spectral dynamics, and each of them matches that of the amplified seed along the reflection (PC) and transmission (NR). Overall, the forward signals generated by the four-wave mixing process show a spectral blue shift with respect to those generated in the backward direction, as seen in Figure 3d. This figure shows the dependence of the wavelength shift for PC (blue) and NR (red) for increasing pump intensities. The two curves follow a qualitatively similar trend, yet with a rigid blue shift of the NR that is 30 nm larger. The asymmetry between forward and backward directions may be attributed to the difference between the Fresnel coefficients experienced by the forward and backward propagating beams. Absorption, which induces an exponential decay of all beams (the attenuation coefficient is \(\alpha=4\pi n_{i}/\lambda=0.0079\) nm\({}^{-1}\)), may also contribute to the observed asymmetries, as well as the cross phase modulation between the pump and the generated radiation, which mainly affects the forward propagating beam. Once again, the simple models outlined above can explain some of the difference between backward and forward, but intriguingly cannot fully account for the observed frequency shift.