# Broad Frequency Shift of Parametric Processes in Epsilon-Near-Zero Time-Varying Media
## 2 Results  
Phase conjugation and negative refraction result from a four-wave mixing interaction mediated by a Kerr-type nonlinearity and satisfy a parametric amplification energy matching condition: \(\omega_{\text{PC, NR}}=2\omega_{\text{P}}-\omega_{\text{S}}\), where \(\omega_{\text{P}}\) and \(\omega_{\text{S}}\) are the pump (P) and seed (S) frequencies, respectively [26; 27; 28; 29]. The PC and NR are the processes satisfying the additional condition \(\omega_{\text{S}}=\omega_{\text{P}}=\omega_{0}\), leading to \(\omega_{\text{PC, NR}}=\omega_{0}\). We remind the reader that, for each photon of NR and PC generated as _idler_ of the parametric amplification process, a _seed_ photon is also amplified (amplified seed--AS). In the case of a deeply sub-wavelength nonlinear medium, the component of momentum in the direction of the film thickness (\(k_{z}\)) does not need to be conserved, and the only relevant phase matching condition involves the in-plane wavevector components \(\vec{k}_{r}\), according to  
\[\vec{k}_{r}^{\text{(NR,PC)}}+\vec{k}_{r}^{\text{(AS)}}=2\,\vec{k}_{r}^{\text{(P )}}. \tag{1}\]  
Since the pump beam is orthogonal to the nonlinear film, it has a null in-plane wavevector component (\(k_{r}^{\text{(P)}}=0\)), and the above constraint simplifies to \(\vec{k}_{r}^{\text{(NR,PC)}}=-\vec{k}_{r}^{\text{(AS)}}\). The amplification process is coherent and the amplified seed retains the phase of the input seed. Due to the sub-wavelength nature of the nonlinear medium, the longitudinal component of the amplified seed wavevector is undetermined, and the only spatial requirement is that \(\vec{k}_{r}^{\text{(AS)}}=\vec{k}_{r}^{\text{(S)}}\), which has a forward and a backward solution, as shown in Figure 1a,b. In summary, for a seed photon injected into the nonlinear medium, one photon will emerge either as the phase conjugation (PC) or negative refraction (NR) of the input seed. At the same time, another photon (amplified seed) will be generated, either in the backward (AS\({}_{\text{b}}\)) or in the forward (AS\({}_{\text{f}}\)) direction. These last two signals are overlapped to the reflected (RS) and the transmitted (TS) seed, respectively. We assume that pump and seed are co-polarised. In the ideal case described here, the amplified seed can emerge either in the backward or in the forward direction, irrespective of the generation of negative refraction or phase conjugation. We shall see below how the frequency shift of the PC and NR are linked to those of the RS and TS fields. In previous experiments, we showed that PC and NR generated by a sub-wavelength AZO film can achieve larger than unitary internal efficiency, meaning that the PC and NR have, inside the sample, an amplitude larger than the seed [24]. Here we investigate the spectral dynamics of PC and NR, and of the amplified seed, in similar experimental conditions. We performed the degenerate four-wave mixing (FWM) experiment using a pump and probe setup schematically shown in Figure 1c. The output of a mode-locked Ti:sapphire laser with a pulse duration of \(\simeq\)105 fs pumped an optical parametric amplifier (OPA), which generated a pulse train with a wavelength tuneable in the near-IR spectral region. The OPA output was split into two arms. The pump, with high intensity, was injected at normal incidence onto the ENZ film, and was focused to a full width at half maximum beam size (FWHM) of \(\simeq\)460 \(\upmu\)m. The seed, with a lower intensity and smaller beam spot size (\(\simeq\)150 \(\upmu\)m FWHM), was overlapped with the pump coming at a small (\(\simeq\)6\({}^{\circ}\)) angle to the sample normal. Both the pump and the seed were vertically polarised. The 500 nm AZO film was deposited on a glass substrate by laser pulse deposition and exhibits a real part of the relative dielectric permittivity \(\Re[\varepsilon_{\mathrm{r}}]\) crossing zero at \(\lambda_{\mathrm{ENZ}}\simeq 1350\) nm. We choose to excite the AZO film above the crossing point (\(\lambda_{\mathrm{p}}=\lambda_{\mathrm{s}}=1400\) nm).