**1. Introduction**

Coherent perfect absorption (CPA) was first proposed as a time-reversed version of a laser [1]. Similar to a laser cavity, CPA occurs when light is resonant at specific wavelengths in a high-Q Fabry–Perot optical resonator. However, for CPA, the active gain material is replaced with a moderately lossy medium. Because the system's single-pass losses are typically low, perfect absorption for a given input intensity is extremely sensitive to the Q-factor and resonance wavelength [2,3].

An alternative scheme utilizes deeply subwavelength and highly absorbing materials [4–6]. Here, two counter-propagating coherent beams interfere at the film's surface and create a standing wave. Absorption in the film is then modulated by changing the relative phase of the two beams, or equivalently by scanning the film along the nodes (peak transmittance) and antinodes (peak absorption) of the interference pattern. This approach has been demonstrated in the ultrafast [7]

and quantum regime [8–11], as well as in integrated photonic systems [12–16]. While a resonant cavity is not required, single-pass absorption should be 50% to achieve perfect absorption [17,18]. This is difficult to obtain in conventional dielectrics (too little losses) or metals (too high reflectivity). To circumnavigate this challenge, metasurfaces—nanostructured subwavelength films—with ideal absorptive optical properties have been used to achieve CPA [4,19]. Ideal absorption can be achieved in extremely subwavelength films over a broad range of wavelengths, making metasurface-based CPA advantageous over bulk cavity structures and compatible with integrated photonic platforms [3].

While metasurfaces and other engineered structures can exhibit CPA over large wavelength ranges, the necessary nanofabrication can be a limitation for practical CPA applications. Thin films of epsilon-near-zero (ENZ) materials, such as transparent conductive oxides (TCOs) like aluminum-doped zinc oxide (AZO) or indium tin oxide (ITO), have been proposed as a particularly suitable platform for broadband CPA [20]. ENZ materials exhibit a real part of the dielectric permittivity which crosses zero for wavelengths of practical interest in the near-infrared or visible regions [21,22]. Due to the continuity of the transverse component of the electric field at the interface, the electric field within the ENZ material can be very large and can lead to perfect absorption (PA) when illuminated at a critical angle of incidence [23,24]. In the limit of deeply subwavelength ENZ film, PA is provided by critical coupling the incident light to a fast wave propagating along the ENZ layer [24]. The proposed systems for ENZ PA are multilayer structures where the ENZ thin layer is sandwiched between two dielectrics or a dielectric and a metal structure [25]. At the critical angle where CPA happens (this is often referred to as directional PA), the loss follows a linear relationship with the ENZ film thickness which implies that CPA can occur in an arbitrarily thin ENZ film (with arbitrary small single-pass absorption) [26]. For instance, PA has been demonstrated for films of ITO film thickness as low as 0.02 *λ*<sup>0</sup> (free-space wavelength) and with only 5% single-pass absorption [27]. Electrical tuning of one port directional PA have also been shown in plasmonic strip cavity based on a ENZ thin layer, with a modulation in reflectance of the 15% [28]. Finally, broadband coherent modulation of directional PA in ENZ deeply subwavelength film have been proved by using ITO multylayer structures sandwiched between two ZnSe prisms [20]. The control of nonlinear processes by two port illumination was also theorized for deeply sub-wavelength ENZ slab [29]. Applications of CPA in deeply subwavelength ENZ films could be found in photovoltaic energy conversion or devices such as bolometers which require large absorption with small masses. However, other applications, such as in nonlinear or quantum optics, may benefit from thicker films where the efficiency of the nonlinear process and the parametric gain generally scale with thickness.

Here, we study CPA in films of TCOs near their ENZ wavelength where the film's refractive index exhibits large anomalous dispersion and a near-zero refractive index. Such films can be treated as deeply subwavelength because the effective wavelength will increase drastically for wavelength approaching the ENZ wavelength. We theoretically and experimentally explore the role of this transition region in order to achieve CPA in homogeneous AZO optically thick films and then show how this can be controlled with intense optical pump fields. It was recently shown that the combination of low refractive index and the high damage threshold of these materials allows TCOs to exhibit large and ultrafast Kerr-type optical nonlinearities in the ENZ region [30–36] and behave as efficient time-varying medium [37,38].

We perform CPA experiments in a Sagnac-like interferometer where two counter propagating light pulses are incident normal to the sample. We achieve coherent control of absorption in AZO films with different thicknesses. For all samples the total energy modulation exhibits a maximum value near the ENZ wavelength. We then demonstrate dynamical control of CPA using its strong intensity-dependent refractive index change. Our demonstration of broadband and tunable CPA in homogeneous ENZ films is relevant for practical nanoscale optical-switches and modulators where alternative nano-pattered metasurfaces would suffer from low switching efficiencies and detriments of nanofabrication processes.

#### **2. Theoretical Investigation**

Our optical system consists in two counter-propagating continuous waves (CW) impinging on a homogeneous ENZ film, *E<sup>A</sup> in* and *<sup>E</sup><sup>B</sup> in*, respectively, at normal incidence (Figure 1a). From the transfer matrix method (TMM), we calculate the electric field at the two outputs of our symmetric system, *EC out* and *E<sup>D</sup> out* respectively. By changing the relative phase *φ* between the two counter-propagating beams, we simulate the scenario in which the film is shifted along the propagation direction and calculate the intensity of the two outputs, C and D (Figure 1b). By summing the intensity at the two outputs (*ITot* = *IC* + *ID*), we define the modulation visibility of the total energy as

$$V\_{\text{tot}} = \frac{\left(I\_{\text{Tot}}^{\text{max}} - I\_{\text{Tot}}^{\text{min}}\right)}{\left(I\_{\text{Tot}}^{\text{max}} + I\_{\text{Tot}}^{\text{min}}\right)} \tag{1}$$

where *Imax Tot* and *<sup>I</sup>min Tot* are the maximum and minimum of the total output energy of the system. In principle, for CPA to occur in a thin film, the transmission and reflection coefficients from both sides of the film should be equal (|*r*| = |*t*|) with a phase difference of *ϕrt* = 0 or *π* in order to achieve 100% light absorption. In this situation, the value of the total visibility is 1.

**Figure 1.** (**a**) Bi-directional coherent perfect absorption (CPA) scheme. (**b**) Intensity of the two output beams, C and D, and its sum as we scan the sample position in the propagation direction. This is equivalent to changing the relative phase between the two input fields *φ*.

We consider three different cases by fixing the zero crossing of the real part of the dielectric permittivity at *λENZ* ≈ 1350 nm, but vary the dispersion across the ENZ region as shown in Figure 2a–c where we plot the refractive index profiles (real, *n*, and imaginary, *k*, parts) for three different cases studied. These are calculated from a Drude model

$$
\varepsilon = \epsilon\_{\infty} - \frac{\omega\_p^2}{(\omega^2 + i\gamma\omega)} \tag{2}
$$

where <sup>∞</sup> is the high frequency permittivity, *ω<sup>p</sup>* is the plasma frequency and *γ* is the damping coefficient. It has been shown that this model correctly reproduces the ENZ refractive index for a variety of materials, as ITO and AZO [33,39–41]. In our case, we use <sup>∞</sup> = 3.18 and *<sup>ω</sup><sup>p</sup>* = 2.4745 × <sup>10</sup><sup>15</sup> rad/s. We vary the Drude model damping coefficient, thus increasing losses and reducing the dispersion gradient in *<sup>n</sup>*, from (a) to (c) *<sup>γ</sup>* = 1.0073 × 1013 → 2.4745 × <sup>10</sup><sup>14</sup> rad/s. Figure 2d–f show the visibility *Vtot* of the total energy as a function of the ENZ film's thickness for the three cases shown in Figure 2a–c. In Figure 2d (*γa*), we do not observe coherent modulation of the total energy for thickness below 1000 nm. Due to the high transmission of the thin film, the interference between the reflected and transmitted field is weak. For thicker films, *r* and *t* become more similar and stronger interference is observed. The TMM model predicts visibility with a maximum value close to one that is pinned to a wavelength slightly shorter than *λENZ*. When we increase the optical losses of the ENZ slab, Figure 2e,f, the peak of the visibility becomes broader, exhibiting multiple resonances as the thickness

increases but all with maximum absorption at a wavelength just below *λENZ*. These results show that the system exhibits broadband coherent modulation of the energy with a maximum value close to one just below *λENZ*, independently of the thickness and of the single-pass absorption. We associate this maximum to a Fabry–Perot (FP) like resonance due to interference effects in the Air/AZO/glass system. The fact that the FP resonance is 'locked' before the ENZ wavelength irrespectively of the thickness is due to the ENZ condition [42,43].

**Figure 2.** (**a**–**c**) Real and imaginary part of the refractive index of the three cases with *λENZ* ≈ 1350 nm. (**d**–**f**) Normalized visibility of the total energy as a function of the wavelength for different thicknesses. The dashed red line indicates the *λENZ*. For the dispersion we use <sup>∞</sup> = 3.18 and *<sup>ω</sup><sup>p</sup>* <sup>=</sup> 2.4745 <sup>×</sup> 1015 rad/s. For the damping constant we use *<sup>γ</sup><sup>a</sup>* <sup>=</sup> 1.0073 <sup>×</sup> 1013, *<sup>γ</sup><sup>b</sup>* <sup>=</sup> 0.8053 <sup>×</sup> 1014 and *<sup>γ</sup><sup>c</sup>* <sup>=</sup> 2.3614 <sup>×</sup> <sup>10</sup><sup>14</sup> rad/s.

In the ideal case without losses, the first resonance of an FP cavity is reached when the 2*nd* = *λ*0. Due to the strong gradient of the *n* before the ENZ region, the *λ*<sup>0</sup> at which the first resonance occurs will not scale linearly with *d*, but it will be locked in this spectral range with strong dispersion. Moreover, in a lossy dielectric medium, *r* and *t* become complex and their phases depend on the value of both *n* and *k* of the lossy medium. Here the first resonant order for the FP cavity is reached when 2*nd* = *λ*0(1 − *α*/*π*), where *α* is the phase of the transmission coefficient [18]. Combining the strong dispersion of *n* due to the ENZ condition and the value of *k*, almost perfect modulation of absorption is expected at wavelength just below *λENZ* even for subwavelength thickness (Figure 2f).
