## Nonlinear optics: A surprise in store?

At ultrafast data rates, the ability to use light to control things could speed information processing. However, photons tend not to interact with each other, and so a nonlinear optical material is needed and the response of such materials is typically weak. Alam *et al.* report a surprising finding: that indium tin oxide, a commercially available transparent conducting oxide widely used in microelectronics, exhibits a large nonlinear response. They used a wavelength regime where the permittivity of the material is close to zero and observed a large and fast nonlinear optical response. The finding offers the possibility that other, so far unexplored, materials may be out there for nonlinear optical applications.

*Science*, this issue p. 795

## Abstract

Nonlinear optical phenomena are crucial for a broad range of applications, such as microscopy, all-optical data processing, and quantum information. However, materials usually exhibit a weak optical nonlinearity even under intense coherent illumination. We report that indium tin oxide can acquire an ultrafast and large intensity-dependent refractive index in the region of the spectrum where the real part of its permittivity vanishes. We observe a change in the real part of the refractive index of 0.72 ± 0.025, corresponding to 170% of the linear refractive index. This change in refractive index is reversible with a recovery time of about 360 femtoseconds. Our results offer the possibility of designing material structures with large ultrafast nonlinearity for applications in nanophotonics.

A long-standing goal in nonlinear optics has been the development of materials whose refractive index can be drastically changed using a low-power optical field. Ideally, these materials should possess subpicosecond time response and be compatible with existing complementary metal-oxide semiconductor (CMOS) fabrication technologies (*1*–*2*). Simple calculus shows that, for a given change (∆ε) in the permittivity ε, the resulting change (∆*n*) in the refractive index *n* is given for a lossless material by . We see that this change becomes large as the permittivity becomes small, suggesting that the epsilon-near-zero (ENZ) frequencies of a material system should give rise to strong nonlinear optical properties.

Materials possessing free charges, such as metals and highly doped semiconductors, have zero real permittivity at the bulk plasmon wavelength. A number of authors have reported on the unusual properties of matter under ENZ conditions (*3*–*5*) and on their promise for applications in nonlinear optics (*6*–*10*).

We used commercially available indium tin oxide (ITO), a CMOS-compatible degenerate semiconductor, as the ENZ medium (Fig. 1). The zero-permittivity wavelength of ITO occurs at near-infrared wavelengths and can be tuned by controlling the doping density or by applying a static electric field (*11*, *12*). The *z*-scan technique was used to characterize the intensity-dependent refractive index of ITO for transverse magnetic (TM) polarized light (Fig. 2A) (*13*, *14*). The wavelength-dependent effective nonlinear refractive index coefficient (*I* is the intensity of the laser beam)and effective nonlinear attenuation constant extracted from our measurements are plotted for angles of incidence varying from θ = 0° to θ_{} = 60° (Fig. 2, A and B). The results indicate that ITO exhibits positive and negative , corresponding to self-focusing and saturable absorption, respectively. Our results reveal a substantial wavelength- and angle-dependent enhancement of the material’s nonlinear response at ENZ wavelengths. The measured value of *n*_{2(eff)} (6 × 10^{−5} cm^{2}*/*GW) at the shortest wavelength (970 nm) agrees well with the value reported by Elim *et al*. (*15*). At a wavelength of 1240 nm for normal incidence, and are ~43 and ~53 times larger than the corresponding values at 970 nm, respectively. For TM-polarized light at oblique incident, the nonlinear response is further enhanced. The enhancement factors, defined relative to the values far from the ENZ spectral region (at λ = 970 nm) at normal incidence, are plotted as functions of θ in Fig. 2C. The enhancement tends to increase with θ for 0° *<* θ *<* 60° and decreases sharply for θ *>* 60°. The maximum enhancement factors, measured at θ = 60°, are 1837 and 2377 for and , respectively. Thus, at λ_{0} = 1240 nm, the and values for θ = 60° are ~43 and ~45 times larger than for normal incidence, respectively.

The temporal dynamics of the optical nonlinear response was studied using a degenerate pump-probe transmission measurement. Here, an intense pump pulse and a weak probe pulse at the same wavelength interact with the sample, and the induced change in probe transmittance ∆*T* is measured as a function to the time delay τ between the two pulses. The measured temporal response is proportional to the convolution of the probe’s temporal envelope with the material’s temporal response function. (Fig. 2D). The transient nonlinear response has a rise time no longer than 200 fs (this estimate is limited by our laser pulse duration) and a recovery time of 360 fs. Such ultrafast response would allow all-optical modulation speeds of at least 1.5 THz.

The nonlinear coefficients shown in Fig. 2, A and B, may be slightly overestimated because the *z*-scan method neglects the change in reflectivity caused by ∆*n* (*16*), but this overestimation is no larger than by a factor of ∼1.8. In any case, the measured values are extremely large. In particular, the value of = 0.11 cm^{2}*/*GW measured at θ = 60° is more than two orders of magnitude larger than that of As_{2}Se_{3} chalcogenide glass (*17*) and ~5 times larger in magnitude than that of a recently proposed highly nonlinear metamaterial (*6*). The optical losses of ITO at ENZ wavelengths can be quite large, although in (*16*) we describe some realistic applications that can tolerate this much loss.

We attribute the observed nonlinearity primarily to a modification of the energy distribution of conduction-band electrons as a consequence of the laser-induced electron heating. We describe the nonlinear optical response by means of a phenomenological two-temperature model (*16*, *18**–**20*). Figure 3A shows the calculated temporal evolution of the free-electron temperature (*T*_{e}) and lattice temperature (*T*_{l}) of ITO after irradiation by the laser pulse (denoted by the dashed curve). The free-electron temperature exhibits an ultrafast transient and is limited by the electron-phonon relaxation time of the material (*21*). The normalized transient nonlinear response measured via the degenerate pump-probe technique is well described by the temporal profiles of *T*_{e} convolved with the probe’s intensity envelope, which is plotted as the solid curve in Fig. 2D.

The peak values of *T*_{e} obtained with our model are plotted in Fig. 3B as functions of the wavelength and the angle of incidence. The temperature profile exhibits the main features present in our experimental results, namely a pronounced enhancement of the response at ENZ wavelengths that reaches a maximum for an angle of incidence close to θ = 60°. The general behavior observed in this result can be understood in terms of two contributions. The increasing values of *T*_{e} for longer wavelengths result from the increase in free carrier absorption, and the peak that develops around θ ≈ 60° results from an enhancement of the electric field within the ITO film. This enhancement occurs only for obliquely incident TM polarized light at wavelengths within the ENZ region and follows from the continuity of the normal electric displacement field across an ITO-air interface (*16*, *22*). As discussed in (*16*), Δε results from an effective red shift in the material’s plasma frequency caused by an increase in the free-electron temperature (Δ*T*_{e}). It is important to note that Δε does not scale linearly with Δ*T*_{e} for a large Δ*T*_{e} and that Δ*n* is a nonlinear function of Δε at ENZ. Consequently, a modest field intensity enhancement in the ITO film can lead to a large enhancement of *n*_{2(eff)} at ENZ wavelengths. This is confirmed by our model presented in (*16*), which accounts for such nonlinear relationships between Δ*n*, Δε, and Δ*T*_{e}.

The hot-electron–induced optical nonlinearity of ITO at ENZ wavelengths differs from that of noble metals under infrared irradiation in two ways. First, as argued above, for a given change in permittivity, the nonlinear change in refractive index is always larger in the ENZ region than in non-ENZ regions. Second, the free-electron heat capacity of ITO (4.53 Jm^{−3}K^{−1}) is more than an order of magnitude smaller than that of a noble metal such as gold. Thus, the increase in the free-electron temperature compared with the Fermi temperature and the consequent change in refractive index in ITO is much larger.

For sufficiently large optical intensities, the nonlinear response of ITO at ENZ wavelengths can lead to changes in its refractive index that are larger than the linear refractive index. As a result, the Fresnel reflection and transmission coefficients undergo a large change as a function of the incident optical intensity. To demonstrate this phenomenon, we measured the intensity-dependent transmittance (*T*), reflectance (*R*), and absorptance (*A*) of the sample at 1240 nm for θ = 30° (Fig. 4A). At the lowest intensity, these measurements agree well with the predictions of a simple linear Fresnel analysis. As the intensity is increased, we observe a large monotonic increase (reduction) in transmittance (reflectance). The maximum reduction in absorptance is ~30%, which is consistent with the saturable absorption observed in our *z*-scan measurements. The real part of the refractive index of ITO undergoes a dramatic change from its linear value of 0.42 to a value of 1.14 ± 0.025 for an intensity of 150 GW*/*cm^{2} (Fig. 4B). Similarly, the imaginary part of the index is substantially reduced from its linear value of 0.42 to a value of 0.27 ± 0.015 at this intensity. Both the real and the imaginary parts of the refractive index saturate for even higher input power. We found that these measurements are highly repeatable and that the material does not exhibit a permanent change of its optical properties.

The magnitude of the optically induced ultrafast change of the real part of the refractive index (Δ*n* = 0.72 ± 0.025) and the relative change of 170% in comparison to the linear value are unprecedented. The change in the refractive index corresponds to a change of the permittivity from ε = 0 + 0.352i to ε = 1.22 + 0.61i where i is the square root of −1. This result shows that ITO can exhibit a reversible transition from metallic to a lossy dielectric state with a subpicosecond time response at wavelengths slightly longer than the bulk plasmon wavelength. Moreover, the usual perturbation expansion description of nonlinear optical effects is not applicable for this material at high intensities.

We have shown that a thin ITO film exhibits an extremely large ultrafast third-order nonlinearity at ENZ wavelengths. Moreover, it can acquire an optically induced change in the refractive index that is unprecedentedly large. Our results challenge the notion that the nonlinear optical response is only a perturbation to the linear response. Materials with such a large nonlinear response are expected to enable exotic nonlinear dynamics (*22*) and allow all-optical control of metasurface and active plasmonics devices. Thus, our results introduce a completely new paradigm in nonlinear optics and open new avenues for developing optical nanostructures with large nonlinearity for applications in nanophotonics, plasmonics, and nonlinear nano-optics.

## SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/352/6287/795/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S4