# Low intensity saturation of an ISB transition by a mid-IR quantum cascade laser

Mathieu Jeannin, Eduardo Cosentino, Stefano Pirotta, Mario Malerba,
Giorgio Biasiol, Jean-Michel Manceau, Raffaele Colombelli

► To cite this version:

Mathieu Jeannin, Eduardo Cosentino, Stefano Pirotta, Mario Malerba, Giorgio Biasiol, et al. Low intensity saturation of an ISB transition by a mid-IR quantum cascade laser. Applied Physics Letters, 2023, 122 (24), 10.1063/5.0153891. hal-03910837v3

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Low intensity saturation of an ISB transition by a mid-IR quantum cascade laser

Mathieu Jeannin,<fnref n="1" /> Eduardo Cosentino,<fnref n="1" /> Stefano Pirotta,<fnref n="1" /> Mario Malerba,<fnref n="1" /> Giorgio Biasiol,² Jean-Michel Manceau,<fnref n="1" /> and Raffaele Colombelli<fnref n="1" />

¹) Centre de Nanosciences et de Nanotechnologies, CNRS UMR 9001, Université Paris Saclay,
10 Boulevard Thomas Gobert, 91120 Palaiseau, France

²) Laboratorio TASC, CNR-IOM, Area Science Park, S.S. 14 km 163.5, Basovizza I-34149 Trieste,
Italy

(*raffaele.colombelli@u-psud.fr)

(*mathieu.jeannin@universite-paris-saclay.fr)

(Dated: 25 May 2023)

We demonstrate that absorption saturation of a mid-infrared intersubband transition can be engineered to occur at moderate light intensities of the order of 10-20 kW.cm⁻² and at room temperature. The structure consists of an array of metal-semiconductor-metal patches hosting a judiciously designed 253 nm thick GaAs/AlGaAs semiconductor heterostructure. At low incident intensity the structure operates in the strong light-matter coupling regime and exhibits two absorption peaks at wavelengths close to 8.9 µm. Saturation appears as a transition to the weak coupling regime - and therefore to a single-peaked absorption - when increasing the incident intensity. Comparison with a coupled mode theory model explains the data and permits to infer the relevant system parameters. When the pump laser is tuned at the cavity frequency, the reflectivity decreases with increasing incident intensity. When instead the laser is tuned at the polariton frequencies, the reflectivity non-linearly increases with increasing incident intensity. At those wavelengths the system therefore mimics the behavior of a saturable absorption mirror (SESAM) in the mid-IR range, a technology that is currently missing.

Absorption saturation is one of the simplest non-linear optical process. In the case of interband transitions in semiconductors it relies on Pauli blocking of photoexcited carriers, reducing the ability of the material to absorb light. This saturation process is in general governed by the lifetimes and oscillator strength of the involved transition, and the frequency of the exciting light. Absorption saturation of interband transitions in semiconductors quantum wells (QWs) can be triggered with lasers at moderate intensities due to the long exciton lifetimes, which fill up the final states in the conduction band. Conversely, intersubband (ISB) transitions in doped QWs present very short (ps-scale) lifetimes and typical low photon energy of tens or hundreds of meV. Contrary to interband transitions, absorption saturation of ISB transitions does not stem from a filling of the upper state of the transition, but from a depletion of the available carriers in the ground state whose density is fixed by the doping introduced in the structure. These characteristics leads to high saturation intensities, in the sub-MW.cm⁻² range that are only attained with very high power lasers like CO₂ laser<fnref n="1" />, OPOs²⁻⁵ or even free electron lasers⁶. These high intensity values are *in principle* not compatible with conventional semiconductor lasers

output powers, which has so far limited the application of ISB-based saturable absorbers. Some recent work mentioned saturation-like behavior of ISB transitions triggered using a simple QCLs, but in that case saturation was considered a drawback for non-linear light generation⁷,⁸.

ISB transitions are recognized for being a technological cornerstone in mid-infrared (mid-IR) optoelectronics, with the invention of the quantum cascade laser⁹ that led to the development of practical mid-IR semiconductor lasers¹⁰. Since then, ISB transitions have contributed to the extension of semiconductor optoelectronic devices to the mid-IR and THz ranges of the electromagnetic spectrum. Examples are the demonstration and commercialization of mid-IR photodetectors¹¹⁻¹⁵, free-space amplitude modulators¹⁶,¹⁷. As far as mid-IR saturable absorbers are concerned, initial results on narrow gap semiconductors¹⁸ lead to recent efforts focused on finding new material systems relying e.g. on type II superlattices¹⁹ or two dimensional materials like graphene²⁰ and transition metal dichalcogenides²¹ (see Ref. ²²,²³ for recent reviews). Hence, reducing the absorption saturation intensity in ISB devices is technologically appealing as it would allow to exploit the design flexibility of ISB transitions to implement mid-IR saturable absorbers. Demonstrating low-intensity saturation of an ISB-based devices is therefore an important first step.

In a recent theoretical work, we have developed a unified formalism to describe absorption saturation of ISB transitions coupled to microcavities²⁴. In particular, we

<footnote n="1">¹ Authors to whom correspondence should be addressed: Mathieu Jeannin (mathieu.jeannin@universite-paris-saclay.fr) and Raffaele Colombelli (raffaele.colombelli@u-psud.fr)</footnote>
have shown that the nature of absorption saturation radi-
cally changes in cavity-coupled systems depending on the
light-matter interaction regime, quantified by the vac-
uum Rabi frequency $\Omega_R$:

$$
\Omega_R^2 = f_w f_{12} \frac{\Delta n e^2}{4\epsilon\epsilon_0 m^* L_{qw}} \equiv a\Delta n \quad (1)
$$

where $f_w$ is the filling fraction of the QWs inside the active region, $f_{12}$ is the transition oscillator strength hereafter approximated to the infinite square QW limit ($f_{12} = 0.96$), $e$ and $m^*$ are the electron charge and effective mass, $\epsilon_0$ and $\epsilon$ are the vacuum permittivity and the semiconductor relative permittivity, $L_{qw}$ is the QW width, and $\Delta n = n_1 - n_2$ is the population difference between the first ($n_1$) and the second ($n_2$) electronic levels of the QW. The absorption saturation manifests itself with the collapse of the light-matter coupling as both $\Delta n$, and thus $\Omega_R$, tend to zero.

In this work, we experimentally demonstrate that the
absorption saturation of mid-IR intersubband transitions
can be drastically reduced down to the kW.cm⁻² range.
Such low values are compatible with the output power
of commercially available quantum cascade lasers (QCL),
interband cascade lasers (ICL) and emerging mid-IR fiber
lasers²⁵,²⁶. The saturation intensity $I_{sat}$ is defined as
the intensity for which the population difference $\Delta n$ is
reduced to half the value of the introduced doping $n_s$,
or - equivalently - the population in the excited level is
$n_s/4$:

$$
\Delta n_{I_{sat}} = \frac{n_s}{2} \iff n_{2,sat} = \frac{n_s}{4}. \quad (2)
$$

To achieve saturation with a low incident intensity, a
careful balance has to be found. Following Ref.²⁴, struc-
tures with low doping can exhibit low saturation intensi-
ties when introduced in a properly designed cavity. How-
ever, if the doping is so small that the polariton split-
ting cannot be resolved, the spectral signature of the
saturation is masked by the cavity absorption. Increas-
ing the doping allows to increase the frequency splitting,
but leads to a linear increase in saturation intensity as
$I_{sat} \propto \Omega_R^2$. The best choice is therefore to operate at the
onset of the strong light-matter coupling regime, where
a moderate $\Omega_R$ allows to spectrally resolve the polari-
ton states while retaining a low $I_{sat}$. This constitutes a
first steps towards a unique semiconductor saturable ab-
sorber mirror²⁷⁻²⁹ (SESAM) technology that could have
great impact for mid-IR lasers.

We have designed a saturable absorber active region
composed of seven repetitions of GaAs/Al₀.₃₃Ga₀.₆₇As
QWs (6.5/14 nm, total thickness 253 nm) delta-doped
with Si atoms with surface density of 2 10¹¹ cm⁻² (sam-
ple HM4448). Additional delta-doping layers on each
side of the multi-QW structure are introduced to com-
pensate for Fermi level pinning at the semiconductor-
metal interfaces. A reference sample (HM4445) with the
same, but undoped, active region has also been fabri-
cated to measure the optical response of the bare cav-

<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Colorized SEM image of a typical patch antenna array defining the dimensions *s* and *p*. (b) Self-consistent Schrödinger-Poisson simulation of a period of the structure showing the two confined electronic states. (c) Sketch of a single patch antenna with the electric field amplitude of the fundamental TM₀₀ mode.</figcaption></figure>

ity. The active region is processed using standard Au-
Au thermo-compressive wafer bonding³⁰, selective sub-
strate removal in citric acid and electron beam lithogra-
phy to fabricate arrays of metal-metal patch cavities as
shown in Fig. 1(a). The semiconductor between adja-
cent Au patches is removed by ICP-RIE using SiCl₄. A
self-consistent Schrödinger-Poisson simulation (Fig. 1(b))
shows the conduction band profile and confined electronic
states of the QWs, with an energy separation of 138 meV
corresponding to a transition wavenumber of 1110 cm⁻¹
(λ ≈ 9 µm). The patch cavity size *s* ranges from 1.25 µm
to 1.35 µm to tune the frequency of the fundamental
TM₀₀ (Fig. 1(c)) mode across the ISB transition. The
distance *p* between adjacent cavities is fixed at *p* = 1 µm
to operate close to the critical coupling condition, ensur-
ing maximal energy funneling in the system.

The samples are initially characterized in a Fourier-
transform infrared (FTIR) spectrometer equipped with
a Cassegrain microscope objective, to measure the re-
lectivity spectrum of each cavity array. The results are
shown in Fig. 2(a) for both the undoped QWs sample
(purple dashed lines) and the doped QWs sample (blue
solid line). The undoped sample exhibits a single ab-
sorption dip, corresponding to ohmic dissipation in the
cavity. The absorption frequency $\omega_c$ decreases with in-
creasing patch size *s* following the usual relation:

$$
\omega_c = \frac{\pi c}{n_{eff}s} \quad (3)
$$

where $n_{eff}$ is an effective refractive index. On the con-
trary, the reflectivity spectra of the doped QWs sample
exhibit two absorption dips, on each side of the bare cav-
ity absorption, as highlighted in the inset of Fig. 2. We fit
the datasets with Lorentzian absorption lines and report
the extracted peak frequencies as a function of patch size
$s$ in Fig. 2(b) (open symbols). The two absorption dips in
the doped QWs sample exhibit an anti-crossing behavior,
characteristic of the strong light-matter coupling regime.
The low (resp. high) frequency branch corresponds to
the lower (resp. upper) polariton. First, the dispersion
relation of the (undoped) cavity is fitted (purple solid
<figure><img src="image_3.png" /><figcaption>FIG. 2. (a) Reflectivity of the bare cavity arrays (purple dashed line) and of the doped cavity arrays (blue solid lines) as a function of the cavity size. The spectra are offset by 0.5. Inset: zoom on the s = 1.35 µm spectrum around the polariton splitting. (b) Cavity and polaritons dispersion relation as a function of the patch size. The symbols are the experimental data (circles: undoped cavities, squares: doped cavities) extracted from Lorentzian fits of the spectra. The solid lines are fit to eq. (3) and eq. (4) respectively.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 3. (a) Reflectivity spectra of the doped s = 1.35 µm cavity array under low (purple circles) and high (blue circles) intensity excitation. The shaded areas indicate the tuning ranges of the QCL chips. (b) Nonlinear reflectivity as a function of intensity for three different wavelengths (arrows in (a)). Open symbols correspond to experimental data, and solid lines to the CMT prediction. The vertical dashed lines indicate the saturation condition. The horizontal error bars represent uncertainty on the evaluation of the spot size.</figcaption></figure>

line) according to eq. (3), providing $n_{eff} = 3.5$, slightly larger than $n_{GaAs} \approx 3.3$ at these frequencies. This stems from the strong confinement of the electric field between the two metallic plates, together with field leakage and reflection phase at the edges of the cavity. Then, the dispersion relation of the polariton branches is fitted using the following secular equation$^{31}$:

$$ (\omega^2 - \omega_{isb}^2)(\omega^2 - \omega_c^2) = 4\Omega_R^2\omega_c^2 \quad (4) $$

where $\omega_{isb}$ is the intersubband transition frequency and $\omega_c$ is the cavity frequency. In the limit of very low excitation, we can replace $\Delta n \approx n_s$ in the expression of the Rabi frequency.

The slight discrepancy between the secular equation (4) and the data can have several causes, e.g. differences in the fabrication of the two samples resulting in actual patch sizes s slightly different between the two samples. To correct for this effect, we assign the cavities sizes by comparing the frequency of their higher order modes (unperturbed by the ISB transition). From the fit we extract a Rabi splitting of $2\Omega_R = 70 \text{ cm}^{-1}$, which is slightly below the minimal splitting of $78 \text{ cm}^{-1}$ measured in the s = 1.35 µm cavity case. These values should be compared with the resonance frequency and the cavity/ISB transition linewidths (resp. $\gamma_c^{tot}$ and $\gamma_{isb}^{tot}$), yielding $\frac{\Omega_R}{\omega_{isb}} = 0.04$ and $\frac{2\Omega_R}{\gamma_{c, isb}^{tot}} = 0.65$. The Rabi frequency is only a fraction of the transition frequency, but the Rabi splitting is of the order of the dissipation rates of the system. This places the system de facto at the onset of the strong light-matter coupling regime, as we can still resolve the polariton splitting.

We then measure the non-linear reflectivity of the doped QWs array closest to resonance (s = 1.35 µm) as a function of the excitation intensity. The reflectivity is probed with a home-built microscope setup using a commercial tunable QCL (Daylight MIRCAT) as a source and a liquid-nitrogen cooled mercury cadmium telluride detector$^{16}$. The QCL emission is amplitude modulated at 10 kHz with pulse widths of 500 ns (0.5% duty cycle), and lock-in detection is employed. The incident intensity can be tuned through a half-wave plate and a polarizer, and with the laser injection current (See the Supplementary Material for a complete description). The reflectivity of the patch cavity array is normalized against that of a smooth Au surface. Fig. 3(a) reports the reflectivity spectra of the sample at low intensity (blue symbols) as well as at the maximum intensity available from the QCL (purple symbols). For low impinging intensity, we retrieve the two absorption dips (inset of Fig. 2(a)) with two polariton states. On the contrary, the high intensity spectrum displays only one absorption dip, centered between the two polariton states.

We then select three frequencies from the spectrum in Fig. 3(a): The cavity central frequency (1080 cm$^{-1}$, purple arrow), the upper polariton frequency (1112 cm$^{-1}$, light blue arrow) and the frequency at which the reflectivity change is the largest (1140 cm$^{-1}$, black arrow). Note that the latter frequency accidentally lies at the crossover between two QCL chips, hence the small jump in the high intensity reflectivity spectrum. With the laser tuned at these three frequencies we continuously scan the incident intensity. The value of the exciting power (in
mW) is measured using a thermoelectric detector, and the spot size (20 µm radius at 1/e²) is determined with a knife-edge measurement to infer the excitation intensity (in W.cm⁻²). The results are presented in Fig. 3(b) (open symbols). Pumping within the polariton gap at the cavity frequency leads to a decrease in reflectivity as the transition saturates, and the response of the system converges to the one of the bare cavity. On the contrary, pumping at the upper polariton frequency or on the high frequency edge of the spectrum leads to a gradual increase in reflectivity.

We use temporal coupled mode theory (CMT) to obtain an analytical expression relating the sample reflectivity (and the population difference Δn) to the incident laser intensity. The ISB transition and the cavity mode are modeled as oscillators with parameters (ωᵢ, γᵢ, Γᵢ) representing respectively their oscillation frequency, non-radiative, and radiative dampings. In the present case, the radiative coupling of the ISB transition to free-space

radiation (Γᵢₗb = 0) can be neglected³²,³³. The three equations describing the evolution of the amplitude aᵢ of each oscillator, as well as the in- and out-coupling equation of the exciting field s⁺ and s⁻ are thus²⁴:

$$ \frac{da_{isb}}{dt} = (i\omega_{isb} - \gamma_{isb})a_{isb} + i\Omega_R a_c \quad (5) $$

$$ \frac{da_c}{dt} = (i\omega_c - \gamma_{nr} - \Gamma_r)a_c + i\Omega_R a_{isb} + \sqrt{2\Gamma_r}s^+ \quad (6) $$

$$ s^{-} = -s^{+} + \sqrt{2\Gamma_{r}}a_{c} \quad (7) $$

that can be solved in the harmonic regime (s⁺ = eᵢωt) to express analytically the absorption Aᵢₗb = 2γᵢₗb |aᵢₗb/s⁺|² solely due to the ISB transition, and the reflectivity R = |s⁻/s⁺|². Assuming (without loss of generality) that the cavity is at resonance with the ISB transition, i.e. ω_c = ωᵢₗb = ω₀, we have:

$$ A_{isb}(\Delta n) = \frac{4\gamma_{isb}\Gamma_r\Omega_R^2}{[(\omega - \omega_0)^2 - \Omega_R^2 - \gamma_{isb}(\gamma_{nr} + \Gamma_r)]^2 + [(\gamma_{nr} + \Gamma_r + \gamma_{isb})(\omega - \omega_0)]^2} \quad (8) $$

$$ R(\Delta n) = \frac{[(\omega - \omega_0)^2 - \Omega_R^2 + \gamma_{isb}(\Gamma_r - \gamma_{nr})]^2 + [(\Gamma_r - \gamma_{nr} - \gamma_{isb})(\omega - \omega_0)]^2}{[(\omega - \omega_0)^2 - \Omega_R^2 - \gamma_{isb}(\gamma_{nr} + \Gamma_r)]^2 + [(\gamma_{nr} + \Gamma_r + \gamma_{isb})(\omega - \omega_0)]^2} \quad (9) $$

These expressions directly depend on the population difference Δn through the Rabi frequency (1). We then link the excited state excited population of the ISB transition n₂ in steady-state condition under an incident intensity I using:

$$ n_2 = \frac{I}{N_{qw}\hbar\omega} \tau_{12} A_{isb}(\Delta n) \quad (10) $$

where τ₁₂ is the upper state lifetime (of the order of the ps) and Nqw is the number of QWs. Inserting eq. (8) in eq. (10) leads - after inversion - to the following intensity-dependent excited state population³⁴:

$$ I = \frac{\hbar\omega N_{qw} n_2 \left[ \left[ \gamma_{isb}(\gamma_{nr} + \Gamma_r) - ((\omega - \omega_0)^2 - a(n_s - 2n_2)) \right]^2 + (\omega - \omega_0)^2 (\gamma_{isb} + \gamma_{nr} + \Gamma_r)^2 \right]}{4\tau_{12} \gamma_{isb} \Gamma_r a(n_s - 2n_2)} \quad (11) $$

where the coefficient a has been defined in eq. (1). Importantly, using (9), this result allows us to establish a correspondence between the incident intensity and the measured non-linear reflectivity variation through the excited state population.

All the experimental results (reflectivity spectra and non-linear reflectivity curves) are fitted by combining eq. (9) and eq. (11) and using a single set of parameters. The results are presented as solid lines in Fig. 3. The total quality factor Qtot = ωc/Γr+γnr = 6 and the electronic doping n_s = 2 · 10¹¹cm⁻² are obtained from the experiment: the former one from the experimental total

cavity linewidth of the undoped sample (Fig. 2(a)), and the latter one from the experimental value of the Rabi frequency using eq. (1). The only free parameters are the ratio between the radiative and non-radiative cavity dissipation rates, and the ISB linewidth γᵢₗb. The fit yields Qᵣ = 9.5 ± 0.3 and Qᵣₙ = 13 ± 0.1, and Qᵢₗb = 13 ± 0.7. The analytical results are presented in solid lines in Fig. 3.

The calculations quantitatively reproduce all features of the experimental data. We observer the collapse of the polariton states from the low intensity to the high intensity reflectivity spectra. We also observe the de-
crease of the non-linear reflectivity as a function of intensity when pumping in the gap, and the increase of the non-linear reflectivity when pumping at the polariton frequency or on the high frequency edge. In particular eq. (11) permits to deduce the excited state population from the non-linear reflectivity curves, an information that cannot be extracted from the sole measurements. This allows us to mark the saturation condition (2), shown in vertical dashed lines in Fig. 3(b). Depending on the pump frequency, saturation is achieved for intensities of 10 kW.cm⁻² to 20 kW.cm⁻².

We now examine the limitations of the model and its agreement to experimental data. A first approximation is made when considering that the ISB transition frequency does not depend on the saturation of the transition. In reality collective effects in confined electron gas renormalize the transition energy through the depolarization shift, according to the following formula:

$$ \omega_{isb}^2 (\Delta n) = \omega_{isb,0}^2 + \frac{e^2}{4\epsilon\epsilon_0 m^* L_{qw}} \Delta n \quad (12) $$

where the square root of last operand on the right-hand side is known as the plasma frequency. In mid-IR doped QWs, and at such moderates dopings, the depolarization shift is small³⁵. The validity of this approximation is gauged a posteriori by the quantitative agreement between theory and experimental results. This effect should be considered at larger doping densities, or in the case of THz ISB transition where the depolarization shift is generally a larger fraction of the transition energy³⁶,³⁷.

On the experimental side, we identify two main limitations. The first one is the non-homogeneous excitation of the patch cavity array by the (Gaussian) laser beam: all the cavities probed by the beam actually experience different excitation intensities and thus exhibit different reflectivity spectra, possibly resulting in a broadening of the response. The second limitation is the precise determination of the pump spot size. Given the moderate output power of the QCL, a strong focusing is needed to reach high enough intensities. Even when performed with great care, knife-edge measurements of the focal spot size are difficult and can lead to variations in the intensity inferred from the power measurement, as shown by the error bars in Fig. 3(b).

Finally, since the sample is excited using relatively long laser pulses, it is important to estimate the importance of thermal effects in the structure and in particular to estimate whether the observed effect could simply be a manifestation of heating. We report in Supplementary Material an estimation of thermal effects, showing that the nonlinear reflectivity change reported here is indeed due to absorption saturation.

In conclusion, we have demonstrated absorption saturation of an ISB transition at a record low intensity (10-20 kW.cm⁻²) by a judicious engineering of the light-matter coupling, making it compatible with mid-IR semiconductor lasers. The measured non-linear reflectivity

properties have validated the analytical model of saturation in weak and strong-coupling regime that we had developed in Ref.²⁴. This result constitutes a first step towards semiconductor saturable absorber mirrors (SESAM) able to cover the mid-IR spectral range. To further improve this system, future work will focus on improving the main figures of merit of the device: saturation intensity, reflectivity contrast, and non-saturable losses. Furthermore, given the fast decoherence times of the order of hundreds of fs³⁸, and the fast population relaxation times of the order of a few ps²,³,⁵,³⁹,⁴⁰ of ISB transitions confirmed in recent demonstration of final state stimulation of ISB polaritons⁴¹, we expect fast saturation and recovery dynamics on the order of few ps⁴². We conclude by emphasizing that this approach is in principle intrinsically scalable to the full mid-IR range. ISB transitions in III-V semiconductors are well mastered, and can cover the entire mid-infrared. And the cavity electrodynamics concepts enabling the observed ultra-low intensity saturation are also fully scalable to other wavelengths. We therefore believe that this experimental demonstration represents the first step towards a robust technology for mid-infrared SESAMs. It could have impact for mid-IR lasers, judging from the success SESAMs had at shorter near-infrared wavelengths²⁷⁻²⁹.

## SUPPLEMENTARY MATERIAL

See supplementary material below for a complete description of the non-linear reflectivity setup and an estimation of thermal effects in the structure.

## DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## ACKNOWLEDGMENTS

This work was partially supported by the European Union Future and Emerging Technologies (FET) Grant No. 737017 (MIR-BOSE), and by the French National Research Agency: project SOLID (ANR-19-CE24-0003) and IRENA (ANR-17-CE24-0016). This work was done within the C2N micro nanotechnologies platforms and partly supported by the RENATECH network and the General Council of Essonne. M.M. acknowledges support from the Marie Skłodowska Curie Action, Grant Agreement No. 748071. We acknowledge the technical help of the C2N cleanroom staff. We thank Iacopo Carusotto, Ammar Hideur and Adel Bousseksou for scientific discussions, Cristiano Ciuti for granting us access to the Laboratoire Matériaux et Phénomènes Quantiques cleanroom (CNRS UMR 7162, Université Paris-Cité), and Pascal
Filloux for assistance in the ICP-RIE etching of the struc-
tures.

# I. SUPPLEMENTARY - EXPERIMENTAL SETUP

<figure><img src="image_5.png" /><figcaption>FIG. 4. Experimental setup used for the measurement of non-linear reflectivity. λ/2: halfwave plate. Pol.: KRS5 wire grid polarizer. B.S.: 3mm thick 90/10 ZnSe beam splitter. MCT: Mercury Cadmium Telluride detector.</figcaption></figure>

The experimental setup dedicated to the non-linear re-
flectivity measurement is presented in Fig. 4. The tun-
able QCL is a MIRCAT from Daylight solutions equiped
with two chips emitting from 1024 cm⁻<fnref n="1" /> to 1140 cm⁻<fnref n="1" />
 chip 1) and from 1040 cm⁻<fnref n="1" /> to 1320 cm⁻<fnref n="1" /> (chip 2), in
an external cavity configuration with a grating as out-
put coupler. The intensity impingin on the sample is
controlled by varying either the driving current of the
QCL, or by using a combination of halfwave plate and
KRS5 wire grid polarizer. Light is focused on the sample
trough an anti-reflection coated ZnSe objective (Innova-
tion Photonics) with 12 mm focal length. The imaging
of the laser spot on the sample is done using a mid-IR
microbolometric camera (FLIR Systems). The MCT is
a J15D19 model from Judsone Teledyne. The QCL is
driven in pulsed mode with a repetition rate of 10 kHz
and pulse widths of 500 ns, and the measurement is per-
formed using a lockin detection technique.

II. SUPPLEMENTARY - ESTIMATION OF THERMAL EFFECTS

Since the structure is pumped with a moderately in-
tense laser beam and relatively long pulses, it is impor-
tant to consider thermal effects. While a comprehensive
and detailed thermal analysis of the system is beyond the
scope of this work, we provide here and estimate showing
that the observed behavior cannot originate from thermal
excitation of the carriers.

Using a commercial self-consistent Schrödinger-
Poisson solver (NextNano++), we simulate the structure
for different temperatures from 300 K to 1000 K. From

<figure><img src="image_6.png" /><figcaption>FIG. 5. (a) Band structure of a single QW with the two relevant conduction band states. (b) Occupation (carrier densities in 10¹<fnref n="1" /> cm⁻<fnref n="2" />) of each level for three selected temperatures. (c) Normalized population difference as a function of the temperature. (d) Reflectivity as a function of the temperature obtained from the CMT model (main text) using the thermal variation of the population in (c) as an input.</figcaption></figure>

the simulation, we can extract the occupation of the first
two energy levels in the QW, and thus the population
difference as a function of the temperature. All these re-
sults are reported in Fig. 5. Using the same parameters
as the main text for the frequencies, dampings and dop-
ing, we plug the simulated variation of Δn(T) (Fig. 5(c))
in the CMT model and obtain the expected reflectivity
change as a function of the temperature (Fig. 5(d), bot-
tom x-axis). The latter are compared to the experimen-
tally observed variation of the reflectivity as a function of
the impiging intensity (top x-axis). One can see that to
observe the same level of saturation, the sample temper-
ature should reach above 800 K, which should result in
alloying of GaAs and Ti/Au layers and destruction of the
sample. We thus estimate that the observed non-linear
reflectivity does not originate from thermal effect, and is
indeed only based on optical absorption saturation.

<footnote n="1">¹F. H. Julien, J.-M. Lourtioz, N. Herschkorn, D. Delacourt, J. P. Pocholle, M. Papuchon, R. Planel, and G. Le Roux, “Optical saturation of intersubband absorption in GaAs-Al x Ga 1-x As quantum wells,” *Applied Physics Letters* **53**, 116–118 (1988).</footnote>

<footnote n="2">²A. Seilmeier, H.-J. Hübner, G. Abstreiter, G. Weimann, and W. Schlapp, “Intersubband relaxation in GaAs- Al x Ga 1-x As</footnote>
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