Enhanced Nonlinear Refractive Index in $\epsilon$-Near-Zero Materials

New propagation regimes for light arise from the ability to tune the dielectric permittivity to extremely low values. Here, we demonstrate a universal approach based on the low linear permittivity values attained in the $\epsilon$-near-zero (ENZ) regime for enhancing the nonlinear refractive index, which enables remarkable light-induced changes of the material properties. Experiments performed on Al-doped ZnO (AZO) thin films show a sixfold increase of the Kerr nonlinear refractive index ($n_2$) at the ENZ wavelength, located in the 1300 nm region. This in turn leads to ultrafast light-induced refractive index changes of the order of unity, thus representing a new paradigm for nonlinear optics.

Figure 1

(a) Experimental setup. A high intensity, horizontally polarized beam at 785 nm pumps the thin AZO film at normal incidence. The reflection and transmission of a weak probe beam (variable wavelength between 1150 and 1550 nm, vertically polarized, $\sim 10°$ angle of incidence) are simultaneously recorded. For the linear characterization, the pump beam was blocked. (b) Measured reflectivity ($R_0$, blue circles) and transmissivity ($T_0$, red squares) in the linear regime (no pump beam). The error bars have been evaluated as the standard deviation on a sample of 300 measurements. (c) Real (blue circles) and imaginary (red squares) part of the linear permittivity extracted from the data in (a) using an inverse transfer matrix approach. The error bars have been evaluated by extracting $\epsilon$ from the pairs $\{R_0+{\sigma }_{R_0},T_0+{\sigma }_{T_0}\}$ (upper bounds) and $\{R_0-{\sigma }_{R_0},T_0-{\sigma }_{T_0}\}$ (lower bounds), where ${\sigma }_{R_0}$ and ${\sigma }_{T_0}$ are the reflectivity and transmissivity standard deviations, respectively. The large error bars for the real part of the permittivity at longer wavelengths are due to the very low values of transmissivity measured, in turn resulting in a high relative error.

Figure 2

Plots of the theoretically estimated trends for $n_2$ and ${\beta }_2$ from Eqs. (2) and (3). (a) Plot of $(n_r+n_i)/D$ with $D=n_r^{\text{pump}}(n_r^2+n_i^2)$. This term weighs the real part of the nonlinearity, $n_{2r}$. (b) Plot of $(n_r-n_i)/D$. This term weighs the imaginary part of the nonlinearity, ${\beta }_2$. The vertical dashed lines indicate the ENZ wavelength.

Figure 3

(a) Measured reflectivity change $\mathrm{\Delta }R/R_0=(R-R_0)/R_0$, where $R$ is the reflectivity at the probe wavelength with the pump and $R_0$ is the linear value (pump off) for increasing pump intensities: $I_p=435\mathrm{GW}/{\mathrm{cm}}^2$ (the black circles), $I_p=870\mathrm{GW}/{\mathrm{cm}}^2$ (the blue squares), and $I_p=1300\mathrm{GW}/{\mathrm{cm}}^2$ (the red triangles). (b) Same as (a) but for transmissivity change $\mathrm{\Delta }T/T_0=(T-T_0)/T_0$. The error bars have been evaluated by propagating the error of the $R$, $R_0$, $T$, and $T_0$ standard deviations, from a sample of 300 measurements.

Figure 4

(a) Real (blue circles) and imaginary (red squares) part of ${ϵ}_{\text{nl}}$ as a function of the pump intensity for a specific probe wavelength, and corresponding linear fits (the solid lines). (b) Real (blue circles) and imaginary (red squares) part of the ${\chi }^{(3)}$ tensor for different probe wavelengths.

Figure 5

(a) Real part of the nonlinear Kerr index, $n_{2r}$, and (b) nonlinear absorption coefficient, ${\beta }_2=4\pi n_{2i}/\lambda$, obtained from Eq. (2) and the data in Fig. 4 (the blue dots). The red lines represent the expected theoretical result obtained from Eq. (6). (c) Change in the real part of the refractive index, $\delta n_r=n_r(I_p)-n_r(I_p=0)$, for different pump intensities (the left axis). Maximum relative refractive index change, $\delta n_r/n_r(I_p=0)$, at 1390 nm (the right axis). (d) Measured shift of the probe carrier wavelength as a function of the pump intensity (the red curve is the linear fit).