# Time Localized Tilted Beams in Nearly-Degenerate Laser Cavities

A. Bartolo  Université Côte d’Azur, CNRS, Institut de Physique de Nice, 06560
Valbonne, France  Institut d’Electronique et des Systèmes, CNRS UMR5214, 34000
Montpellier, France  N. Vigne  Institut d’Electronique et des Systèmes, CNRS
UMR5214, 34000 Montpellier, France  M. Marconi  Université Côte d’Azur, CNRS,
Institut de Physique de Nice, 06560 Valbonne, France  G. Beaudoin  Centre de
Nanosciences et de Nanotechnologies, CNRS UMR 9001, Université Paris-Saclay,
91120 Palaiseau, France.  K. Pantzas  Centre de Nanosciences et de
Nanotechnologies, CNRS UMR 9001, Université Paris-Saclay, 91120 Palaiseau,
France.  I. Sagnes  Centre de Nanosciences et de Nanotechnologies, CNRS UMR
9001, Université Paris-Saclay, 91120 Palaiseau, France.  A. Garnache  Institut
d’Electronique et des Systèmes, CNRS UMR5214, 34000 Montpellier, France  M.
Giudici  Université Côte d’Azur, CNRS, Institut de Physique de Nice, 06560
Valbonne, France<EMAIL_ADDRESS>
###### Abstract

We show that nearly degenerate Vertical External-Cavity Surface-Emitting
Lasers emit tilted beams of time localized structures, i.e. mode-locked light
pulses which can be individually addressed. These beams feature a Gaussian
profile and they are emitted in pairs with opposite transverse k-vector.
Because they are phase locked, their interference leads to a non homotetic
pattern in the near-field emission of the laser. When a single pair is emitted
this is a stripe pattern. Our analysis discloses the role of spherical
aberrations of the cavity in stabilizing this spatio-temporal mode-locked
regime and in selecting the value of the transverse k-vector.

Please note: Abbreviations should be introduced at the first mention in the
main text – no abbreviations lists. Suggested structure of main text (not
enforced) is provided below.

## Introduction

Multimode photonics is a novel research subject devoted to the generation and
control of complex light states for applications to information processing,
photonic computing, sensing, and imaging [1, 2]. In lasers the control of
spatial profile of the light emitted requires self-imaging resonators (also
called degenerate cavities [3]) and a broad-area pump. These large aspect-
ratio (or large Fresnel number) laser systems have been implemented using
solid state laser [4, 5, 6] and Vertical External-Cavity Surface-Emitting
Lasers [7, 8, 9, 10]. In large Fresnel number resonators the presence of
multistability between different emission profiles lead to the appearance of
Localized Structures (LS) [11, 12, 13, 14] which, in their simplest form, are
individually addressable bright spots of light in the transverse section of
the laser cavity [15, 16]. These dissipative solitons can be used as
fundamental bricks for structuring the laser profile and they have
applications to information processing [17, 18, 19, 20]. Localized structures
have been also observed in the longitudinal dimension of optical resonators
driven by an injected field: these temporal LS (TLS) are pulses circulating
inside the cavity which can be individually addressed [21, 22, 23, 24]. In
semiconductor lasers, TLS have been implemented within the regime of passive
mode-locking induced by a saturable absorber [25, 26]. When the cavity round-
trip time is larger than the gain recovery time and the modulation depth of
the saturable absorber overcomes a critical value [27], a variety of mode-
locked states with a different number of pulses per round trip coexist with
the off solution. In these conditions, mode-locked pulses can be individually
addressed [28].

More recently, multimode lasers in the three spatial dimensions were
demonstrated. Spatio-temporal modelocking has been reported in optical fibers
[29, 30, 31] and temporally localized non homotetic patterns have been
observed in Vertical External-Cavity Semiconductor Lasers (VECSELs) [32].

In this paper we analyze the behavior of a nearly-degenerate VECSEL cavity
operated in the regime of TLS and we explore the laser emission when the laser
cavity is unstable. While for normal diffraction this laser platforms emits
time localized Turing patterns with an hexagonal structure in far-field [32],
here we show that spherical aberrations of the laser cavity stabilize the
emission of pairs of tilted beams with opposite transverse wavevectors whose
interference leads to non homotetic spatial patterns. This experimental result
discloses the leading role of aberrations in nearly-degenerate cavities and it
gives evidence of a novel regime of spatio-temporal mode-locking based on
phase-locked tilted beams circulating in the cavity. Our experimental
observations are in agreement with theoretical results describing nearly-
degenerate VECSEL cavity operated in the regime of TLS [33].

## Results

Figure 1: The VECSEL cavity ABCD matrix parameters are set to $C\gtrapprox 0$
($\Delta z$=-2.74 mm) and $B\gtrapprox 0$ ($\Delta x$ is varied in the range
$-10\mu m<\Delta x<35\mu m$). Near field (a,c,e) and far field (b, d, f)
emission profiles for three positions of the SESAM: (a) and (b) $\Delta
x=-10\mu m$, (c) and (d) $\Delta x=26\mu m$ and (e) and (f) $\Delta x=35\mu
m$. (g) Tilting angle of the emitted beams with respect to the optical axis as
a function of $\Delta x$.

We consider a large aspect-ratio Vertical External-Cavity Surface-Emitting
Laser (VECSEL) operated in the regime of TLS, i.e. featuring i) cavity round-
trip time ($\tau$) larger than the carrier’s recombination time ($\tau_{g}$)
and ii) SESAM’s saturable losses larger than a critical value (typically
$\Delta R>8\%$, [26]). Large aspect-ratio is achieved by using a nearly self-
imaging external cavity and by pumping the gain mirror with a flat-top
elliptical beam having a size of $90\times 50\mu\mathrm{m}$. The details of
the experimental setup are described in Methods.

The spatial profile of the VECSEL emission depends on the $B$ and $C$ values
of the ABCD ray transfer matrix describing the round-trip propagation of the
field in the external cavity. As shown by the Huygens-Fresnel diffraction
integral [34, 33], $B$ controls diffraction of the beam travelling in the
cavity, while $C$ controls its wavefront curvature. Close to the self-imaging
condition (SIC) and for the choice of the optical elements used in our
external cavity (see Methods and Supplemental Document), the value of $B$ can
be controlled by shifting the SA mirror around its SIC position ($x_{0}$)
while the value of $C$ can be controlled by shifting one of the cavity lenses
(cf. $L_{2}$ in Fig. 7) around its SIC position ($z_{0}$). By calling these
displacements $\Delta x=x-x_{0}$ and $\Delta z=z-z_{0}$ respectively,
$B=8\Delta x$ while $C=-2\,\Delta z/64\mathrm{mm}^{2}$ (see Eq. 1) [32, 35].

Close to SIC, for positive diffraction and for focusing wavefront curvature
(($B>0,C<0$), well-defined temporal localized hexagonal patterns appear with a
far-field having a strong component in the axial direction of the VECSEL [32,
33].

When approaching SIC for an overall defocusing wavefront curvature ($C>0$), a
stable cavity is obtained for negative diffraction ($B<0$) and, in the
interval $-50\mu m<\Delta x<0\mu m$, the VECSEL emits an axial fundamental
Gaussian mode whose waist decreases as $\Delta x$ is increased (Fig. 1(a) and
(b)). For $\Delta x>0$ no axial mode can be supported for emission. However,
instead of switching off, the laser emits non homotetic patterns having a far-
field distribution in the form of a ring (Fig. 1(c) and (d)). The diameter of
this ring increases with $\Delta x$ and, correspondingly, the spatial
frequency of the near-field profile increases, as shown in Fig. 1(g). The
system is emitting a set of tilted plane-waves having transverse wavevectors
in different spatial directions but all sharing the same module. The circular
symmetry of the far-field emission is rather fragile and it is broken for
large value of the transverse wave vectors. Then, the circle in the far-field
breaks up in spots 1(f) which are paired: each transverse wavevector
$\overrightarrow{k_{\perp}}$ is linked to the opposite transverse wavevector
$-\overrightarrow{k_{\perp}}$. By blocking the component
$\overrightarrow{k_{\perp}}$ in the cavity, the emission at
$-\overrightarrow{k_{\perp}}$ is also removed. We conclude that the system
emits pairs of tilted plane-waves, the first (the second) leaves the gain
section with a transverse wavevector $\overrightarrow{k_{\perp}}$
($-\overrightarrow{k_{\perp}}$) and comes back after one roundtrip with the
opposite transverse wavevector $-\overrightarrow{k_{\perp}}$
($\overrightarrow{k_{\perp}}$). These two waves are phase locked and their
interference gives birth to fringes onto the surface of the gain section and
of the SESAM (as in Fig. 1(c) and (e).

Figure 2: Near-field (a, c, e) and far-field (b, d, f) profiles of VECSEL
emission in presence of a small anisotropy in the laser cavity. for different
values of $\Delta x$: (a),(b) $\Delta x$ = 0$\mu m$, (c),(d) $\Delta x$ =
30$\mu m$ and (e),(f) $\Delta x$= 35$\mu m$; $\Delta z$= -2.74 $mm$ and
$P_{p}$=175$mW$

Conical emission of tilted waves is broken by inserting an anisotropic element
in the set-up (as a glass window) or simply by slightly tilting an optical
element of the cavity ($L_{2}$ $L_{3}$ or the SESAM). Then, only two points
are left in the far-field (Fig. 2, revealing an emission of two tilted waves
having opposite transverse wavevector $\pm\overrightarrow{k_{\perp}}$.
Interference of the two beams give birth to a stripe pattern onto the surface
of the gain section and of the SESAM. The angular dependence of the tilted
wave emission as a function of $\Delta x$ follows the same curve shown in Fig.
1 (c).

Figure 3: Near-field and far-field profiles of a single tilted beam. Panels
(a) and (b) show respectively the near-field and the far-field intensity
distribution of the VECSEL emission with a pair of counterpropagating tilted
beams. In (c) and (d) one of the two beams ($-\overrightarrow{k_{\perp}}$) has
been filtered out by placing a circular aperture in a fourier plane of the
detection path. Panel (c) show the near field distribution of the remaining
tilted beam and panel (d) the corresponding far-field profile. Panels (e) and
(f) show the near field and far field of the $-\overrightarrow{k_{\perp}}$
tilted beam by filtering $+\overrightarrow{k_{\perp}}$.

For examining the intensity profile of a single tilted wave circulating in the
cavity we may prevent interference between the two opposite wave vectors
outside the cavity. This has been done by analyzing the situation of a single
pair of tilted beams and filtering one of the two spots in a far-field plane
of the detection path. The near field profile of the single beam remaining
turns out to have a gaussian profile, as shown in Fig. 3.

Figure 4: a) Coexisting time resolved output of the VECSEL corresponding to
the profile shown in Fig. 2 c) and d). These mode-locked states with a number
of pulses per roundtrip ranging from one to four correspond to the same
pattern profile upon an intensity scaling factor which depends on the number
of pulses. The width of the pulses emitted is below the time resolution of our
detection system hence their width is smaller than 20 ps.. b) Total output
power emitted by the VECSEL versus the pump power $P_{p}$ showing the
stability of the different pulsating states and of the off solutions, ranging
from no pulse to four pulses per roundtrip. c) Optical spectra for each
pulsing state shown in a)

The VECSEL emission profiles described above are acquired with CCD camera
which cannot resolve the temporal behavior of the laser emission. The analysis
of laser dynamics is obtained by using an array of fast detectors capable of
monitoring different region of the near field emission with a bandwidth of
about 10 GHz or a single detector monitoring the total intensity emitted with
a bandwidth of about 33 GHz, see the Methods section for more details. This
analysis shows that the time behavior of the patterns of Fig. 1 and 2
correspond to mode-locked states with a number of pulses per roundtrip
spanning from one to four, as shown in Fig. 4. These states and the off
solution coexist in a large interval of the pump level, as shown in Fig. 4b).
Their stability ranges $P_{p,a}<P_{p}<P_{p,b}$ share the same upper limit
$P_{p,b}$=178 mW, which corresponds to the VECSEL threshold, while the lower
limit $P_{p,a}$ increases with the number of pulses per round-trip.
Coexistence of the four states is observed for 150 mW<$P_{p}$<178 mW. This
multistability is the signature of the TLS regime, where the pulses can be
individually addressed by shining short pump pulses onto the VECSEL [26, 32].
It is worth to underline that these different states are associated to the
same near and far field profile upon an intensity scaling factor which depends
on the the number of pulses travelling in the cavity. In Fig. 4 c) we show
also the optical spectra for each pulsing state shown in Fig. 4 a) disclosing
a spectral envelope of $\approx 1$ nm, FWHM. It is worth noting that the
resolution of the optical spectrum analyzer (0.06 nm, i.e. 15 GHz) does not
allow to resolve the spectral combs corresponding to the mode-locked states
shown in Fig. 4 a) whose teeth separation is in the range of .25-1.25 GHz.
Finally, we remark that the spatially resolved detection indicates that the
pulsing activity is synchronous throughout the entire near-field section of
the pattern emitted by the VECSEL.

Figure 5: (a) Near-field profile of the stripe pattern when the perturbation
beam targeting one stripe of the pattern is applied (a) and after perturbation
beam removal (b). (c) Spatiotemporal diagram of the writing process of the
temporal localized stripe pattern. The evolution of the TLS roundtrip after
rountrip is represented by a black trace, while the pump evolution is
represented by using a color code. Stationary value of the pump is = 174 mW
and addressing is obtained by sending a 120 ps pump pulse, 10 mW peak power
onto the gain mirror between round-trip number 10800 and round-trip number
15400.

In our optically pumped VECSEL temporal LSs can be individually addressed by
shining short pump pulses onto the gain mirror. The system is set by the CW
optical pump in the multistable parameter region ($150$mW$<P_{p}<178$mW) where
LSs exist. An additional laser, capable of providing optical pulses of about
120 ps (FWHM) at arbitrary rate, is used to generate pump pulses which are
overlapped to the CW pump. These pulses have a Gaussian spatial profile and a
waist of 8 $\mu m$ which matches approximately the size of the stripes of the
pattern (Fig. 5a)). Pulses peak power is chosen to be sufficiently large to
drive the VECSEL beyond the upper limit of the multistable region shown in
Fig. 4b), where only the solution composed by four pulses per roundtrip is
stable. Finally, the addressing pulse is sent to the gain mirror synchronously
with the cavity roundtrip for about five thousand roundtrips. The addressing
process is depicted in Fig. 5 by using a space-time diagram where the pump
value is represented by using a color code, while the trajectory of the LS is
represented by a black trace. In Fig. 5a), we choose an initial condition
where no LS is present inside the cavity before the addressing pulse. The pump
pulse is sufficiently short to switch-on a single LS which persists after the
perturbation is removed.

Other initial conditions can be chosen with similar results, provided that the
addressing pulse is separated in time from the preexisting LS of at least
$\tau_{g}$. This addressing technique can be adapted for erasing LSs [32].
Erasure of LSs can be also obtained by feedbacking an emitted pulse onto the
gain mirror with proper time delay, as described in [36].

The position in the gain mirror section targeted by the addressing beam is
important for a successful addressing. One must target the region where, at
the switching, a bright fringe will appear in the near field, as shown in Fig.
5a) and b). The position of these fringes is fixed by the position of the
optical elements which, as explained above, determines the angle of the
transverse k-vector. If one targets the region in between two fringes no
switching is observed.

## Discussion

Continuous wave tilted beam emission resulting in stationary patterns have
been previously observed in large aspect-ratio laser and explained as a Turing
instability [37, 38, 39]. In these situations a positive detuning between the
gain curve resonance and the closest resonator resonance is at the origin of
this instability. Then, the laser emits tilted beams whose frequency matches
the gain peak and whose longitudinal component of the wavevector fulfills the
resonance condition of the resonator. The matching of the two resonances is
enabled by the tilting of the emitted beam, i.e. by a finite transverse
component of its wavevector. This mechanism does not apply to our system where
the set of longitudinal cavity resonances is very dense (less than 500 MHz
free spectral range) compared to the width of all other relevant spectral
filtering curves, such as microcavities resonances (> 9 nm, i.e. more than 1
THz), gain and saturable absorption curves (more than 10 THz). Moreover, we
have verified that the angle of the tilted waves emitted is not depending on
any parameter of the active or passive media (temperature and pumping). As
shown in Fig. 1, this angle depends instead on the distance between the SESAM
and its collimator L4.

Figure 6: Path of pairs of tilted beams travelling at different angles in a
4-lenses telecentric cavity where only the lens nearby the SESAM ($L_{4}$)
exhibit spherical aberrations with $\alpha$>0\.

In our system, the origin of tilted waves can be understood by considering
spherical aberrations of the optical elements used in the cavity which result
in a global effect of the imaging property of the entire resonator. The most
important contribution to aberrations comes from the short focal length
collimators and, in particular, from $L_{4}$ where, because of the
magnification factor of $L_{2}$ and $L_{3}$ (see Methods), the incident beam
is at the largest distance from the optical axis. The effect of aberrations on
the beam path is schematically shown in Fig. 6: the emitted beam angle versus
the distance between L4 and the SESAM ($d_{5}$) can be understood by assuming
the focal length of $L_{4}$ dependent on the radial distance $R$ between the
optical axis and the incidence point of the incoming beam:
$f(R)=f_{0}(1+\alpha R^{2})$. Accordingly, if $\alpha>0$, for a range of
distances $d_{5}$ larger than $f_{0}$ there is a set of off-axis Gaussian
beams which is stable, as shown schematically in figure 6. The sign of
$\alpha$ depends mainly on $L_{4}$ parameters and, by using other lenses, it
was possible to test the effects of negative values of spherical aberration
$\alpha$. In these cases, we observe the tilted beam angle increasing by
shortening the distance $d_{5}$, thus confirming our interpretation [40]. It
is worthwhile noting, that when the SESAM is replaced by a high reflective
mirror, stationary tilted beams are emitted by the VECSEL [41]. In this
situation, they do not necessarily appear in pairs and a single transverse
wavevector tilted beam can be observed in the cavity [40]. This experimental
evidence illustrates that the emission of pairs of phase locked tilted beams
is induced by the presence of a saturable absorber. Interference between the
paired beams lead to intensity fringes onto the SESAM surface which can
saturate the absorber and enable laser emission. In conclusion, we have
analyzed a nearly-degenerate VECSEL cavities in the regime of TLS and we have
given evidence of a novel spatio-temporal mode-locking regime supported by
spherical aberrations of the laser resonator. Our experimental observations
are corroborated by a recent theoretical model [33]. Possible applications are
related to the ability of addressing individually the pulses circulating in
the cavity which leads to information processing. The emission of several
pairs of TLS tilted beams travelling in the cavity with different azimuthal
angles with respect to the optical axis suggests to use them as multiple
frequency comb beams [42].

## Methods

Figure 7: Experimental set-up showing the L-shape nearly-8f cavity VECSEL.
$d_{1}$: distance between the gain section and lens $L_{1}$, $d_{2}$: distance
between $L_{1}$ and lens $L_{2}$, $d_{3}$: distance between $L_{2}$ and lens
$L_{3}$, $d_{4}$: distance between $L_{3}$ and lens $L_{4}$, $d_{5}$: distance
between $L_{4}$ and the SESAM, HRM= high reflectivity beam splitter (>99.5$\%$
at 1.060 nm).

The VECSEL cavity has an L-shape and it is delimited by a gain mirror (also
called 1/2 VCSEL) and by a semiconductor SA mirror (SESAM). The former is
based on a GaAs substrate with 12 strain-balanced InGaAs/GaAsP quantum wells
(QWs) designed for barrier optical pumping and emitting at 1.06
$\mu\mathrm{m}$. The gain mirror is optically pumped at $808\,$ nm by a flat-
top elliptical profile having an horizontal axis of 90 $\mu$m and a vertical
one of 50 $\mu$m. The SESAM features a single strained InGaAs/GaAs QW located
at $1\sim 2$ nm from the external surface leading to a carrier’s recombination
rate approximately two orders of magnitude faster than the gain medium which
is compatible with a stable passive mode-locking and with the existence of
temporal localized states (TLS) [43]. Light extraction from the cavity occurs
by transmission through a high reflective beam splitter (>99.5 reflectivity at
1.060 nm) which reflects the intracavity radiation beam. The output beam from
the VECSEL is sent to the detection part where the far-field and near-field
profiles are imaged on two CCD cameras. The near-field is also imaged on an
array of optical fibers for spatially resolved detection at 10 GHz bandwidth.
Finally, the total emission is monitored by a 33 GHz bandwidth detection
system and by an optical spectrum analyzer.

Both the gain mirror and the SESAM have been engineered to be operated in a
nearly degenerate cavity. The high level of losses of such a cavity has
required to increase the confinement factor in the gain mirror by enhancing
the microcavity effect. Moreover, for enabling multistable mode-locked states,
it is required the SESAM modulation depth of the saturated/unsaturated
reflectivity (also called saturable losses) to be larger than a critical
amount ($A=8\%$) [25, 44, 43]. By increasing the finesse of the SESAM
microcavity, we achieve a modulation depth $A=23\%$ at the micro-cavity
resonance peak (1.0667 $\mu m$). Fine control of the modulation depth
experienced by the intracavity field is set by controlling the detuning
between the two microcavities cavities
($\delta\lambda=\lambda_{B}-\lambda_{G}$) [26], the smaller the detuning the
higher the contrast. The results described here has been obtained for $5.5$
nm$>\delta\lambda>4\,$nm which enables wide multistable response and a
threshold value accessible to our pump power ($P<350$ mW). More details on the
samples used can be found in [35].

The VECSEL external cavity has been designed to fulfill the requirement
$\tau>\tau_{g}\sim 1$ ns and self-imaging condition (SIC) after one roundtrip.
In addition, the SESAM and gain mirror need to be placed in conjugate planes
with a magnification factor $M$ larger than one for saturating efficiently the
SESAM.

Accordingly we use a four-lenses arrangement where the first lens ($L_{1}$),
the one closest to the gain section) and the last lens ($L_{4})$, the one
closest to the SESAM) are large numerical aperture collimators (Thorlabs
C240TME-1064, $f_{1}=f_{4}=f_{c}=$8 mm) and $L_{2}$ and $L_{3}$ are achromatic
lenses (Thorlabs AC254-XX-C) having $f_{2}=100$mm and $f_{3}=$200 mm. Self-
imaging condition can be achieved through a telecentric arrangement of these
optical elements, i.e. lenses are placed at distances given by the sum of
their focal lengths ($d_{1}$=$f_{1}$, $d_{2}$=$f_{1}$+$f_{2}$,
$d_{3}$=$f_{2}$+$f_{3}$, $d_{4}$=$f_{3}$+$f_{4}$, $d_{5}$=$f_{4})$. thus
making a total cavity length $L$=632 mm (cavity round-trip time $\tau\approx
4.2\,$ns). However, the presence of a pump induced lens onto the gain section
having a focal length $f_{th}$ spanning from 10 to 80 mm depending on the pump
level [45], requires slight shift of the position of $L_{2}$ and of the
position of the SESAM to achieve SIC [35].

By calling $x$ the offset of $d_{5}$ with respect to telecentric position
($x=d_{5}-f_{c}$) and $z$ the offset of $d_{2}$ with respect to telecentric
position ($z=d_{2}$$-(f_{c}+f_{2})$) (cf. Fig. 7), the SIC in presence of the
pump induced lens is obtained at: $z_{0}(f_{th}$)$=-f_{c}^{2}/2f_{th}$ and
$x_{0}(f_{th})=-(f_{c}^{4})/(2M^{2}f_{2}^{2}f_{th})$ [35]. For focal lengths
values in our experiment, one finds that $z_{0}$ is of the order of few
millimeters, while $x_{0}$ will be of few microns since $f_{2}>>f_{c}$ .

It is useful to describe the behavior of the $ABCD$ rondtrip matrix close to
SIC. By calling $\Delta x=x-x_{0}$ and $\Delta z=z-z_{0}$ the deviations of
the SESAM position and of $L_{2}$ position from the SIC condition, this matrix
can be approximated to [32, 35]

$\left(\begin{array}[]{cc}A_{RT}&B_{RT}\\\
C_{RT}&D_{RT}\end{array}\right)=\left(\begin{array}[]{cc}1&\;\;2\,M^{2}\Delta
x\\\ -2\dfrac{\Delta z}{f_{c}^{2}}&1\end{array}\right)$ (1)

Accordingly, we can identify as $\Delta x$ the experimental parameter
controlling the diffraction coefficient and as $\Delta z$ the experimental
parameter controlling the wavefront curvature. Stability of the cavity
requires $\Delta x>0$ when $\Delta z>0$ and $\Delta x<0$ when $\Delta z<0$.

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## Acknowledgements

We acknowledge J. Javaloyes, F. Maucher and S. Gurevich for useful
discussions. We acknowledge funds from Agence National de la Recherche (ANR):
projet BLASON (ANR-18-CE24-0002) and projet KOGIT (ANR-22-CE92-0009) and
funding from Région PACA OPTIMAL.

## Author contributions statement

A.G. and M.G. conceived the experiment(s), A.B., N.V. and M.M. conducted the
experiment(s), A.B., N.V., M.M, A.G. and M.G. analysed the results. G.B., K.P
and I.S. realized the semiconductor samples. M.G. and A.G. reviewed the
manuscript.

## Additional information

## Competing financial interests

The authors declare no competing financial interests.