**3. Coherent Absorption and Its Dynamical Control**

We experimentally investigated the behaviour of CPA in ENZ films using AZO films illuminated by two counter-propagating laser beams in a Sagnac-like interferometer configuration. Figure 3a shows a schematic of the set-up. Laser pulses (105 fs FWHM duration, repetition rate 100 Hz) are generated by an Optical Parametric Amplifier (TOPAS) in a tunable range between 1120 nm and 1500 nm. The input power is controlled through a half wave plate and a polarizing beam splitter, which also fixes the input p-polarization (horizontal in the lab frame). The beam is split by a non-polarizing beam splitter into two beams A and B with equal energy and then recombined onto the sample at normal incidence. The AZO film (deposited on a 1 mm thick glass slide) is facing the beam A, whereas the beam B is incident on the substrate side. The two beams are focused down to 50 μm by using a pair of 125 mm lenses. By moving the sample with a piezo-electric stage, interferograms are generated at the output C and D and measured with photodiodes. We used two beam splitters to extract the light from the

interferometer and send it to the photodiodes. A representative example of these interferograms are shown in Figure 3b. In order to calculate the energy visibility in the pulsed case we proceed in the same way as for the CW case, i.e. we evaluate the central portion (where the pulse intensity is maximum) of the interferogram and extrapolate the average values for maximum and minimum of the intensity.

**Figure 3.** (**a**) Schematics of the Sagnac interferometer. (**b**) An example of measurement for *λ*<sup>0</sup> = 1280 nm, assuming energy equal to 1 at the interferometer input. The total modulation of the energy (or absorption) is given by the sum of C and D (green curve). The inset shows a zoom of the interferogram. (**c**) ellipsometer measurement of the index of refraction of AZO 900 nm thick film, (**d**) experimental (dots) and TMM simulation (solid line) of R, T and abs for the same sample.

We investigated two AZO samples of similar optical properties, i.e., *n* and *k*, in the ENZ region, but with thicknesses of 500 and 900 nm (Figure 3c). The real part of the dielectric permittivity crosses zero around 1340 nm for the 900-nm-thick film, which corresponds to where the real and imaginary part of the refractive index are equal (*n*<sup>900</sup> = *k*<sup>900</sup> = 0.34). For the 500-nm-thick sample the *λENZ* is redshifted by 30 nm (*n*<sup>500</sup> = *k*<sup>500</sup> = 0.52 at the *λENZ*) due to small differences in the material deposition. In the spectral range under analysis, *n* of the AZO 900-nm-thick film passes from close to 1 around 1100 nm to less than 0.2 for longer wavelengths. Since *λeff* = *λ*0/*n* inside the medium, the effective length (*Leff* = *L*/*λeff*) of our sample is 0.8 *λ*<sup>0</sup> at 1050 nm, 0.21 *λ*<sup>0</sup> at the (*λENZ* = 1350 nm, and becomes optically deeply subwavelength around 1500 nm (0.1 *λ*<sup>0</sup> ).

We also deposited three 900-nm-thick AZO films on a glass substrate three samples with similar *n* (about 20% difference), but different value of *k* at the crossing point (*k*<sup>1</sup> = 0.34, *k*<sup>2</sup> = 0.30 and *k*<sup>3</sup> = 0.27 for the 900 nm thick film). All the samples exhibit similar optical properties with an absorption close to 60% across the ENZ region (Figure 3d).

We first perform a CPA experiment for the bare glass substrate. In this case the energy modulation is almost zero for all the spectral range of interest. In Figure 4 we report the measured normalized total energy modulation visibility (red circles), together with the values predicted by the TMM (solid lines) for the AZO film. All the samples show the same trend independently from the thickness. In the case of high optical losses, for the different thicknesses the visibility is almost zero in the region where the index of refraction is close to one, whereas it increases and reaches a maximum value up to the 60% just before *λENZ*. As we decrease the value of *k*, the trend of the visibility remains the same for all

the samples, but its maximum value across the transition region decreases. Overall, the experimental results confirm the predictions that CPA can be observed in ENZ films over a broad bandwidth with thicknesses larger than the conventional subwavelength designs. The bandwidth of ∼100 nm is comparable with CPA in deeply subwavelength ENZ single layer (∼150 nm [20]) or white-light cavity (∼100 nm [3]), whereas it is larger than metasurfaces (∼40 nm [4]).

**Figure 4.** Experimental (circles) and transfer matrix method (TMM) simulation (solid line) of normalized visibility of the total energy for aluminum-doped zinc oxide (AZO) 500 nm and 900 nm with different values of *k*. (**a**,**b**) High losses *k*1, (**c**,**d**) middle losses *k*<sup>2</sup> and (**e**,**f**) low losses *k*3. For the TMM simulation we suppose Δ*λ* ∼60 nm.

We finally investigate nonlinear coherent absorption in ENZ films based on modification of the film refractive index through the nonlinear Kerr coefficient. Previously it has been demonstrated that the ENZ condition leads to the enhancement of third order nonlinearities in terms of nonlinear refractive index change for thin film of AZO [32]. This is based on the observation that when the permittivity is close to zero, any nonlinear change Δ*n*, proportional to *χ*(3)/*n*, is enhanced due to the *n* tending to low values. In Ref. [32] a refractive index change of 400% was reported for an AZO film optically pumped with 1.3 TW/cm2 without showing damage of the sample or saturation of the optical Kerr effect. In the same work, at *λ* = 1310 nm a nonlinear susceptibility of *Re*[*χ*(3)] <sup>∼</sup> 4.73 <sup>×</sup> <sup>10</sup>−<sup>20</sup> <sup>V</sup>2/m2 and *Im*[*χ*(3)] <sup>∼</sup> 0.57 <sup>×</sup> <sup>10</sup>−<sup>20</sup> V2/m2 was extrapolated. We therefore illuminated the AZO film in the Sagnac interferometer with two high intensity pulses at normal incidence and same wavelength. The intensities on each side are 0.8 and 0.6 TW/cm2, respectively. By increasing the intensities from the linear regime to these maximum values, we observe that the CPA visibility passes from 68% of the linear case to 35% (Figure 5a,b). The peak of the normalized visibility also redshifts and becomes broader for both the samples, with a nearly 50 nm-shift for

the 500 nm sample. Following the recent works in TCOs, this can be explained by the fact that the dielectric permittivity, and so the optical constants including *λENZ*, exhibit a redshift when it is optically pumped across the ENZ wavelength [31,44]. The redshift of the *λENZ* is also associated to a positive Δ*n* and to a negative Δ*k* [30,32]. Due to the decreasing of *k*, the visibility drops, as we observed for the linear case. While, the shift of the visibility peak is related to the shift of *λENZ* in the same direction, and therefore to the shift of the strong dispersion which the material exhibits at wavelength shorter than the zero-crossing frequency. In Figure 5c,d we plot the experimental results together with TMM simulations. The TMM simulations are obtained considering a ∼60 nm shift of *<sup>ω</sup><sup>p</sup>* and a decreasing of *<sup>γ</sup>* (0.15 × 1015 → 0.09 × <sup>10</sup>15). This correspond to a <sup>Δ</sup>*λENZ* ∼ 60 nm and to a *k*<sup>500</sup> = 0.38 and *k*<sup>900</sup> = 0.24. These results show that enhanced nonlinearities in ENZ materials can be used to add a degree of freedom to tune the efficiency and the bandwidth of coherent absorption.

**Figure 5.** (**a**,**b**) Normalized visibility of the total energy for both samples, 500 nm (**a**) and 900 nm (**b**). The dashed blue curve represents the linear characterization, while the circles is the nonlinear CPA with high beam intensity. (**c**,**d**) Experimental (circles) and TMM simulation (solid line) of normalized visibility of the total energy for AZO 500 nm and 900 nm for the nonlinear CPA.

#### **4. Conclusions**

We theoretically and experimentally demonstrate coherent control of absorption in films of ENZ material. We show that it is possible to achieve a coherent absorption-mediated interferometric effect with a maximum of its effect locked just below the *λENZ* wavelength. Due the strong dispersion at wavelengths below the crossing point, it can be tuned to any wavelength shorter than *λENZ* by varying film thickness and the optical losses. The 60% total visibility achieved in the AZO film could be improved using a CW and collimated beam in order to achieve CPA. By using AZO's strong intensity-dependent nonlinearities, we also showed that it is possible to dynamically tune the visibility of the total energy by simply increasing the intensity of the incoming beam. The possibility to add a degree of freedom for the coherent control of the absorption in ENZ media by using intensity-dependent refractive index proposes a route towards technologies [45] such as optical data processing or devices that require efficient light absorption and dynamical tunability. All the data supporting this manuscript are available at http://researchdata.gla.ac.uk/939/.

**Author Contributions:** Investigation and data curation V.B. and S.V.; Sample fabrication C.D.; code for analysis S.V. and V.B., and code for measurement acquisition T.R.; Formal analysis and writing-original draft V.B., S.V. and D.F.; supervision D.F., V.M.S., A.B. and M.F.; all the author have contributed equally to the writing—review. All authors have read and agreed to the published version of the manuscript.

**Funding:** D. F. acknowledges financial support from EPSRC (UK, Grant No. EP/M009122/1). The Purdue team acknowledges support by the U.S. Office of Naval Research (optical characterization), U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0017717 (sample preparation), and Air Force Office of Scientific Research (AFOSR) award FA9550-18-1-0002.

**Conflicts of Interest:** The authors declare no conflict of interest.
