# Coherent Control of Mid-Infrared Frequency Comb by Optical Injection of
Near-Infrared Light

Kenichi N. Komagata<EMAIL_ADDRESS>Laboratoire Temps-Fréquence,
Institut de Physique, Université de Neuchâtel, Avenue de Bellevaux 51, 2000
Neuchâtel, Switzerland  Alexandre Parriaux  Laboratoire Temps-Fréquence,
Institut de Physique, Université de Neuchâtel, Avenue de Bellevaux 51, 2000
Neuchâtel, Switzerland  Mathieu Bertrand  Institute for Quantum Electronics,
ETH Zurich, Auguste-Piccard-Hof 1, 8093 Zurich, Switzerland  Johannes
Hillbrand  Institute for Quantum Electronics, ETH Zurich, Auguste-Piccard-Hof
1, 8093 Zurich, Switzerland  Mattias Beck  Institute for Quantum Electronics,
ETH Zurich, Auguste-Piccard-Hof 1, 8093 Zurich, Switzerland  Valentin J.
Wittwer  Laboratoire Temps-Fréquence, Institut de Physique, Université de
Neuchâtel, Avenue de Bellevaux 51, 2000 Neuchâtel, Switzerland  Jérôme Faist
Institute for Quantum Electronics, ETH Zurich, Auguste-Piccard-Hof 1, 8093
Zurich, Switzerland  Thomas Südmeyer  Laboratoire Temps-Fréquence, Institut de
Physique, Université de Neuchâtel, Avenue de Bellevaux 51, 2000 Neuchâtel,
Switzerland

###### Abstract

We demonstrate the use of a low power near-infrared laser illuminating the
front facet of a quantum cascade laser (QCL) as an optical actuator for the
coherent control of a mid-infrared frequency comb. We show that by appropriate
current control of the QCL comb and intensity modulation of the near-infrared
laser, a tight phase lock of a comb line to a distributed feedback laser is
possible with 2 MHz of locking bandwidth and 200 mrad of residual phase noise.
A characterization of the whole scheme is provided showing the limits of the
electrical actuation which we bypassed using the optical actuation. Both comb
degrees of freedom can be locked by performing electrical injection locking of
the repetition rate in parallel. However, we show that the QCL acts as a fast
near-infrared light detector such that injection locking can also be achieved
through modulation of the near-infrared light. These results on the coherent
control of a quantum cascade laser frequency comb are particularly interesting
for coherent averaging in dual-comb spectroscopy and for mid-infrared
frequency comb applications requiring high spectral purity.

##  I Introduction

Quantum cascade lasers (QCL) emitting frequency combs were first demonstrated
in 2012 [1] and have since then become an established technology for fast and
broadband mid-infrared (MIR) spectroscopic applications, such as time-resolved
studies of microsecond-scale molecular dynamics [2], high-pressure and
temperature thermometry in shock tubes [3], or high-resolution measurements of
molecular spectra [4, 5, 6]. Some of their key assets are their low footprint
and large optical power compared to other types of MIR combs, especially in
the 8–10 µm spectral range that is usually reached via nonlinear frequency
conversion [7, 8, 9]. Indeed, QCLs are electrically-driven devices that
directly emit frequency combs in the MIR with a power that can reach 1 W [10].

In the context of comb spectroscopy, the use of two mutually coherent combs
with slightly different repetition rates, namely dual-comb spectroscopy (DCS),
has shown great potential for ultra fast and high resolution measurements
without the need of complex and expensive instruments [11, 12]. DCS has been
well demonstrated with mode-locked lasers in the near-infrared (NIR), but the
MIR is more interesting as molecules generally have stronger absorption
features, which is advantageous for many applications such as trace gas
detection [13], or isotope ratio measurements [14]. DCS with QCLs is then
highly interesting as it provides a compact, low footprint and high-resolution
spectrometer [4, 5, 6].

In DCS, Comb stabilization is not strictly necessary thanks to the
availability of computational phase correction of free-running lasers [15].
Nevertheless, stabilization of the combs allows accurate frequency referencing
and arguably more flexibility, e.g., smaller repetition rates. Moreover, to
properly establish QCLs as a source of choice for MIR comb applications such
as optical frequency synthesis [16, 17, 18], demonstrating high spectral
purity and coherent control can be considered as important as increasing their
bandwidth and detecting their offset frequency [19].

For QCLs, actuation on the drive-current is the most straightforward way to
phase-lock a comb line [20, 21, 22], while the other degree of freedom, namely
the repetition rate, is locked by electrically injecting a radio-frequency
(RF) signal close to the round-trip frequency [23]. These two handles allow
the coherent control of QCL combs and can be used together [18]. However, as
for distributed feedback (DFB) QCLs [24, 25], we expect the intrinsic
frequency noise of the QCL comb and the stabilization by drive current to
involve the same physical process, i.e., current to temperature to refractive
index change. In that case, the time scale of the stabilization will always be
close to the cutoff of the noise process, thus limiting the performance of the
lock. A solution adopted for other types of lasers is to employ another
actuator bound by time scales much faster than the noise source such as opto-
optical modulation [26].

For QCLs, light illumination at another wavelength also enables its control
and can play as an actuator. The technique has been used on DFB-QCLs emitting
a single wavelength for applications such as fast switching [27], gain
enhancement [28], stabilization [29], and frequency modulation [30]. More
recently, a QCL comb emitting in the THz range was locked via the intensity
modulation of a white LED [31]. We also note that illumination of resonant
light enabled the mutual lock between two MIR QCL combs via injection locking
[32]. However, this only allows mutual locking at the same optical frequency,
whereas modulation by off-resonant light offers more possibilities.

Indeed, light illumination at a different wavelength can influence various
parameters with different strengths, for example to allow pure frequency
modulation of a DFB-QCL [33]. Moreover, multiple locking-scheme could be
possible for combs, such as the locking of the repetition rate via current
actuation [34, 35] combined with locking of the comb line frequency with NIR
light. Finally, the NIR is supported by mature technologies allowing a wide
range of possibilities. Among others, the high modulation bandwidths reaching
a few tens of GHz could enable the injection locking of the QCL comb
repetition frequency. In light of the above, there is a compelling interest to
investigate the potential of NIR light illumination for the coherent control
of MIR QCL combs.

In this work, we characterize the influence of NIR light illuminating the
front facet of a MIR QCL comb emitting in the 8 µm range. We measure its
transfer function on the comb frequencies and compare it to the more
conventional electrical actuation. We demonstrate that the limits of
electrical actuation for phase-locks can be bypassed using the intensity
modulated NIR light by tightly locking a QCL comb to a DFB-QCL. Lastly, we
show that the repetition rate of the QCL can also be injection-locked by
intensity modulation of the NIR light.

##  II Experimental setup

Figure 1: Schematic showing the experimental setup used to phase lock a line
from a QCL comb to a DFB-QCL using an intensity modulated NIR CW laser, and
for characterization. The electrical blue wires represent the path for
characterizing the repetition frequency and a comb line, whereas the orange
wires represent the path for phase locking. CW: continuous wave, EAM: electro-
absorption modulator, DM: dichroic mirror, FD: Frequency discriminator, PD:
Photodetector, FTIR: Fourier transform infrared spectrometer.

The experimental setup considered here is presented in Figure 1 and pivots
around a QCL comb. The laser is controlled in current and temperature with a
custom-made driver that sets the operation point of the laser to 1200 mA (1.45
times the lasing threshold) and 0. A frequency comb centered around 1305 cm-1
is emitted, with approximately 80 lines, a total power of 126 mW, and a
repetition frequency of 11.057 GHz. In QCL combs, the comb modes beating
together in the Fabry-Perot cavity lead to a measurable voltage oscillating at
the repetition frequency [36]. Thus, two wire bonds connect the top of the QCL
waveguide near the front and back facet of the laser to RF waveguides on a PCB
chip to efficiently inject and extract the repetition rate independently of
the drive current.

A custom-made dichroic mirror produced by ion beam sputtering is placed in the
optical comb path to direct NIR light from a continuous wave (CW) laser at
1.55 µm (Optilab, DFB-1550-EAM-12-K) to the front facet of the QCL via the
collimation lens (Thorlabs, C037TME-F). The NIR laser has an output power
around 1 mW and can be modulated up to 12 GHz using an integrated electro-
absorption modulator (EAM). The beam is aligned into the QCL so as to maximize
the frequency response (see Sect. III). The amount of light effectively
reaching the front facet is 55% of the emitted power of the NIR laser and
these losses are mainly due to the transmission of the QCL collimation lens at
1.55 µm. The polarization of the NIR light was fixed, however, it did not seem
to change the experimental results.

After passing through the dichroic mirror, the comb is mixed on a 50/50 beam
splitter with a CW MIR light generated by a DFB-QCL (Alpes Laser). The latter
is driven at a current of 191 mA and a temperature of 0 , to emit at a
frequency $f_{\text{cw}}=$ 1309.79 cm-1, which is within the spectral range of
the frequency comb. A typical optical spectrum of the comb and the DFB
recorded with a Fourier transform infrared spectrometer (Bristol 771A-MIR) is
presented in Figure 2. The beating $f_{b}$ between the comb and the DFB-QCL is
recorded on a fast photodetector (VIGO, PV-4TE-10.6).

Figure 2: Typical optical spectrum generated by the QCL comb and the
continuous wave generated by the DFB-QCL.

In the following section we will start with studying the frequency response of
the QCL comb when NIR light illuminates its front facet

##  III Response characterization

We start with the static response before moving on to the frequency dependent
response.

###  III.1 Static response

First, we slowly vary the NIR power reaching the QCL from 0 mW to 0.6 mW, and
measure on a RF spectrum and phase noise analyzer (Rohde & Schwarz, FSWP26)
the frequency shift of a comb line $f_{n}$ via its beating with the DFB-QCL,
and of the repetition rate $f_{r}$, measured directly via the independent
channel for RF extraction on the QCL comb. For comparison, we also measure the
frequency shifts when the drive current of the QCL comb is varied over 1 mA.
These results are presented in Figure 3(a, b), where the shift in $f_{r}$ is
scaled by the mode number $n=3550$ for better comparability with $f_{n}$.

Figure 3: Static response of the QCL comb frequencies to a change in (a) drive
current or (b) illuminated NIR power. The shift in $f_{r}$ is scaled by $n$.
In (b), the response is reported for two alignment conditions (see main text).
Dashed lines are fitted functions to the experimental data. Error bars are
plotted when larger than the data marker.

For a small variations (1 mA) of the QCL drive currents, the frequencies shift
linearly with a superimposed sinusoidal modulation on $f_{r}$ due to back-
reflections [37]. The fitted functions to the experimental data (dashed lines
in Figure 3) using a linear model supplemented by a sine wave give an average
slope of -210 MHz/mA and -85 kHz/mA for $f_{n}$ and $f_{r}$ respectively,
although the slope for $f_{r}$ is dependent on the drive current (position
within the modulation) due to the back reflections. The response to low power
NIR light suggests a quadratic trend. Moreover, the shift ratio between
$f_{n}$ and $nf_{r}$ depends on the alignment of the NIR light on the active
region of the QCL comb. The main alignment method used in this article
(alignment 1, described in subsection III.2) is to maximize the dynamic
response of $f_{n}$ at modulation frequencies of 100 kHz. Another method would
be to maximize the static shift of $f_{n}$ (alignment 2). In the latter case,
$f_{r}$ shows a sinusoidal modulation that is nearly nonexistent for alignment
1, and varies far less than for alignment 1. For alignment 1, the slope
interpolated at zero power are -315 MHz/mW and -570 kHz/mW for $f_{n}$ and
$f_{r}$ respectively.

###  III.2 Dynamic response

We now take an interest in the frequency-dependent response of the QCL comb. A
lock-in amplifier (LIA, Zurich Instruments, UHFLI) modulates the power of the
NIR light via the EAM or the comb drive current through the laser driver, see
the blue path in Figure 1. We study the response of three comb characteristic
frequencies, namely, the offset frequency $f_{0}$, $f_{r}$, and $f_{n}$.
Naturally, the three frequencies are coupled to each other through the comb
equation:

$f_{n}=f_{0}+nf_{r}\quad,$ (1)

where $n$ is an integer. Due to the modulation of the drive current or the NIR
power set by the LIA at frequency $\omega$, the comb, i.e., its frequencies
respond as:

$f_{i}=f_{i}^{(0)}+\Delta f_{i}\sin{(\omega t+\theta_{i})}\quad,$ (2)

where $i=\\{0;r;n\\}$ indexes the three comb frequencies under study,
$f_{i}^{(0)}$ is the average value of $f_{i}$, $\Delta f_{i}$ the peak
amplitude of $f_{i}$ due to the modulation, and $\theta_{i}$ is the phase of
the response. To measure the amplitudes and phases, we convert the frequency
modulation of $f_{i}$ to a voltage modulation using a frequency discriminator
(FD) and demodulate this voltage on the LIA.

For this purpose, the repetition rate of the comb is extracted electrically as
before, amplified and down-mixed to 60 MHz. As $\Delta f_{r}$ is only on the
order of 20 kHz, we take the 10th harmonic and down-mix it to 21 MHz before
sending it to FD-1 (Miteq, FMDM-21.4/4-2, see Figure 1), whose output is
connected to the LIA. As for the comb line $n$, the amplitude and phase
response of the comb line $f_{n}$ is encoded in the beating signal with the
DFB-QCL as:

$f_{b}=f_{n}^{(0)}+\Delta f_{n}\sin{(\omega
t+\theta_{n})}-f_{\text{cw}}\quad.$ (3)

The signal $f_{b}$ detected on the photodetector is then filtered, amplified,
divided by 15 and 3 (RF bay, FPS-15-8, FPS-3-8), up-mixed to 60 MHz (not
shown), and fed to a FD (Miteq, FMDM-60/16-4BC), whose output is connected to
the LIA, as shown in Figure 1. The division step allows a larger frequency
excursion to be measured than the linear range of the FD. The frequency to
voltage conversion ratios, including all division and multiplication steps for
$\Delta f_{r}$ and $\Delta f_{n}$ are respectively ($8.2\pm 0.4$) V/MHz and
($5.93\pm 0.06$) mV/MHz.

Regarding $f_{0}$, this frequency can not be directly detected in practice
since $f$–to–$2f$ interferometry [19] is currently unavailable to QCL combs
but fluctuations thereof can be detected [35], which allows the measurement of
its transfer function. However, for experimental simplicity, we can compute it
from the transfer function of $f_{n}$ and $f_{r}$. Indeed, according to Eq.
(1), (2) and the elastic tape model [38, 39], we have:

$\begin{aligned} \Delta f_{0}\sin{(\omega t+\theta_{0})}=&\Delta
f_{n}\sin{(\omega t+\theta_{n})}\\\ &-n\Delta f_{r}\sin{(\omega
t+\theta_{r})}\end{aligned}\quad.$ (4)

Therefore, Eq. (4) yields that $\Delta f_{0}$ and $\theta_{0}$ are
respectively the modulus and phase of the complex value $\Delta
f_{n}\exp{(i\theta_{n})}-n\Delta f_{r}\exp{(i\theta_{r})}$. Here, $n=3551$ is
set by the wavenumber of the DFB-QCL at 1309.79 cm-1, and by the repetition
rate.

Figure 4 shows the resulting phase $\theta_{i}$ and amplitude responses
$\Delta f_{i}$ of the frequencies ($f_{n},nf_{r},f_{0}$) to modulation of the
electrical drive current, $\Delta I$ = 200 µA, and the NIR power, $\Delta P$ =
50 µW, where $\Delta$ represents the peak amplitude of the modulation. Note
that $\Delta f_{r}$ is scaled by $n$ for better comparison with $\Delta
f_{n}$. Also, the contributions of various components of the characterization
scheme (i.e. the FDs and the laser driver) were measured and deducted in order
to obtain the laser response as faithfully as possible. Moreover, the
responses were measured with two different settings for the ranges [1 Hz, 100
Hz] and [100 Hz, 10 MHz], leading to negligible mismatches at 100 Hz.

Figure 4: Response of the comb frequencies ($f_{n},nf_{r},f_{0}$) to
modulation of the drive current (a, c) and of the intensity of the NIR light
(b, d). Panels (a, b) present the frequency excursions $\Delta f_{i}$ while
panels (c, d) show the phase response $\theta_{i}$. The dashed line in (c) is
the phase response of $f_{n}$ with the laser driver.

Focusing first on the electrical actuation in Figure 4(a, c), we observe that
the amplitude response $\Delta f_{n}$ decreases steadily with the modulation
frequency setting the 3-dB modulation bandwidth to 30 kHz, before decreasing
sharply after about 200 kHz. The phase response remains flat up to 10 kHz with
a 90° modulation bandwidth at 420 kHz, which is coherent with previous tight-
locking results [20]. Moreover, the response of $f_{n}$ closely mimic that of
DFB-QCLs [25, 40]. $f_{r}$ follows a similar behavior as $f_{n}$ with a 3-dB
and 90° modulation bandwidth of 80 kHz and 840 kHz respectively. The dashed
line in Figure 4(c) is the phase response of $f_{n}$ with the laser driver,
and has an 90° modulation bandwidth of 280 kHz. The difference between $\Delta
f_{n}$ and $n\Delta f_{r}$ gives the response $f_{0}$, which has a flatter
response in amplitude with a 3-dB modulation bandwidth of 160 kHz. The
uncertainty on the measurements of $\Delta f_{n}$ and $n\Delta f_{r}$, in
particular, the slope of the FDs which depend on the input RF power induces a
large absolute uncertainty on $\Delta f_{0}$. As for the phase, its response
is also flatter, reaching $-70$° near 1 MHz, after which the measurement is no
longer accurate due to the lack of sensitivity. The (quasi-)fix point [38, 39]
increases with modulation frequency from mode number 1850 at 1 Hz to mode
number 2200 at 100 kHz.

Although the transfer function of a MIR QCL frequency comb has already been
reported in Ref. [35], our measurements highlight a 90° modulation bandwidth
one order of magnitude above what was previously shown. This is in agreement
with the response of DFB-QCLs [41, 25, 40] and other QCL combs, measured
directly [42] or demonstrated in a phase-lock loop [20, 22]. This discrepancy
with the measured modulation bandwidth could be attributed to the frequency
response of the bias-tee used in Ref. [35]. Furthermore, we observed in Figure
3 that modulations of $f_{r}$ due to back reflections locally change the
slope, such that the ratio between $\Delta f_{0}$ and $\Delta f_{n}$ and the
fix points are expected to be modulated as well.

We now turn to the optical actuation shown in Figure 4(b, d). For this
measurement, the NIR laser was aligned on the QCL to the maximize the dynamic
response of $f_{n}$ at a 100 kHz modulation. We observe a nearly flat phase
response for $f_{r}$ up to 1 MHz, with a small resonance near 33 Hz and a
start of a roll-off near 1 MHz. At the resonance near 33 Hz the amplitude
response decreases by a factor 2, and then remains nearly flat apart from the
onset of the roll-off near 1 MHz. The 33-Hz resonance also marks a change of
regime for $f_{n}$, due to the crossing of $n\Delta f_{r}$ with $\Delta
f_{0}$, the latter having a similar response as $f_{r}$, although a smaller
relative change in amplitude response, and an opposite phase response. Thus,
for modulation frequencies lower (higher) than 33 Hz, $\Delta f_{0}$ is
smaller (higher) than $n\Delta f_{r}$, such that the response of $f_{n}$ flips
sign from 180° to 0° from 1 Hz to 1 kHz. The respective (quasi-)fixed points
are at mode numbers 3250 and 3950. An associated dip in the amplitude response
of $f_{n}$ is also visible. Above 1 kHz, the response is flat up to about 50
kHz, where the response increases in phase and magnitude before showing a
signature of a roll-off near 1 MHz.

The mechanism causing frequency modulation due to the drive current change is,
identically to DFB-QCLs, a change of the refractive index [25]. In the case of
a NIR injection, the strong inter-band absorption in the InGaAs quantum wells
(the absorption coefficient $\alpha=6000$ cm-1 at 1550 nm), that are part of
the active region of the QCL, is responsible for the modulation of the
refractive index through the generation of carriers in the vicinity of the NIR
injection point. Also, we believe that the response below 30 Hz is dominated
by thermal processes (heating of the QCL), as the sign of the responses are
equal to drive current modulation and as the optimization of maximum static
shift of $f_{n}$ causes modulation of $f_{r}$ as with drive current changes.

From the perspective of locking the QCL comb, the actuation via the NIR power
is advantageous as it offers a higher bandwidth than the drive current
actuation for all comb frequencies, even when the driver response has been
accounted for. It is especially well suited for the stabilization of $f_{0}$
if it can be measured. The stabilization of $f_{n}$ will require a
countermeasure against the sign reversal as detailed in the next section. As
for $f_{r}$, the increased actuation bandwidth is not as interesting given
that it can be tightly-locked by actuation on the drive current [35].

##  IV Mutual stabilization

We now seek to implement a mutual lock between one comb line $f_{n}$ of the
QCL comb and a DFB-QCL as a proof-of-principle demonstration. The electrical
part of the setup is adapted according to the orange path in Figure 1. The
phase fluctuations between the two lasers are obtained by mixing the signal
$f_{b}$ after division by 15 with a synthesized reference frequency locked to
the maser. This error signal is injected into two PID controllers (Vescent,
D2-125). The output of one PID controller is fed back to the driver of the QCL
comb to modulate the electrical current. The output of the second PID
controller is fed back, after high-pass filtering with a 2nd-order custom-
designed filter with a cutoff at 1 kHz, to the EAM to modulate the intensity
of the NIR CW laser. We also monitor the beatnote with the RF spectrum and
phase noise analyzer. In this way, slow ($<100$ kHz) corrections of $f_{n}$,
which cannot be handled by the NIR light due to the sign reversal, are done by
the drive current, while the NIR light extends the available bandwidth and
cancels added noise from the drive current, i.e. the servo bump.

Figure 5 (a) and (b) respectively show the results of the stabilization of
$f_{b}$ in terms of phase noise and RF power spectrum. When only the
electrical actuation is considered, we observe two bumps in the phase noise
power spectral density (PSD). The first is due to the limit of the integrator
at about 100 kHz, while the second near 300 kHz is the servo bump, which
nearly coincides with the 90° modulation bandwidth of the combined driver and
laser (see dashed blue line in Figure 4(c)). The white phase noise from 1 Hz
to about 100 Hz is due to the reference oscillator, which due to the division
by 15, has its contribution increased by 23 dB. The resulting RF spectrum
shows a Gaussian shape topped with a coherent peak.

Figure 5: Tight-locking of a QCL comb to a DFB-QCL. (a) Phase noise power
spectral density (PSD) of the mutual beatnote in free-running mode, electrical
actuation, and both electrical and optical actuation. (b) RF power spectrum at
100 Hz RBW of the beatnote between one line of the QCL comb and the DFB-QCL
depending on the stabilization scheme used. Inset: Zoom on the beatnote at 1
Hz RBW.

The addition of the optical actuation reduces the phase noise PSD to a value
in the order of -80 dBc/Hz, below the $\beta$-line [43], for all Fourier
frequencies. As a result, the integrated phase noise is decreased from 2.16
rad to 200 mrad (see inset in Figure 5(a)). The electrical servo bump is
eliminated, while a new servo bump appears at 2 MHz. This fast bandwidth
allows to employ a lower division ratio of 4: the residual phase noise from 1
Hz to 3 kHz is set by the reference oscillator. At higher frequencies, we
believe that the mismatch between the reference voltages of the two different
servo controller added noise to the system. Moreover, the bumps near 3 kHz
coincide with the cutoff frequency of the high-pass filter. Further
optimization of the electronic components could improve the lock and decrease
the integrated phase noise further, including the use of a single dual-output
servo controller. We also believe the stabilization bandwidth could be
increased by shortening all the cables, fibers, and free-space paths. In terms
of spectrum (Figure 5(b)), the power in the coherent peak improves by 20 dB.
The height of the pedestal is decreased by 15 dB, such that the difference
between the coherent peak and the top of the pedestal is 50 dB at a RBW of 100
Hz, or about 25 dB more than in Ref. [20] at 500 Hz RBW. The inset shows a
zoom over the peak at a 1 Hz resolution bandwidth (RBW) with an SNR of 70 dB.

In parallel, the repetition frequency of the comb is stabilized by RF
injection locking using a resonant RF signal with 15 dBm of power delivered by
a signal generator (Rohde & Schwarz, SMF100A) referenced to a maser. This RF
signal is sent to the QCL via the dedicated channel for RF injection. Thus
both degrees of freedom are tightly-locked simultaneously with low residual
phase noise. The electrical injection of the repetition rate is the usual
approach for its stabilization. However, the previous results suggest that the
repetition rate could also be injection-locked by modulating the NIR light.

##  V Repetition rate injection via NIR illumination

We thus modulated the EAM of the NIR laser near the repetition frequency near
11.06 GHz while monitoring the generated voltage modulation of the QCL via the
dedicated RF extraction port on a RF spectrum analyzer. At low NIR power and
close to resonance ($1$ MHz offset), the modulation frequency was picked up by
the QCL, which thus acted as a NIR detector (see Figure 6(a)). By slightly
increasing the average NIR power with an amplifier to 1.5 mW at the QCL and
the RF power on the EAM to 17 dBm (estimated modulation depth close to 100%),
we were able to injection-lock the repetition frequency to the synthesizer and
obtain the resolution-limited signal shown in Figure 6(b) at 50 Hz RBW. We
obtained a lock range of a few kHz when scanning the modulation frequency
across the free running repetition frequency (see Figure 6(c)). The phase
noise PSD was reduced up to a Fourier frequency of about 3 kHz compared to the
free-running case (Figure 6(d)). We expect that larger locking ranges could be
achieved when using more NIR power, which will be one of the study we will
present in another dedicated article.

Figure 6: Injection locking of the repetition rate by the NIR laser. (a, b)
Electrically measured RF spectrum of the QCL with (a) the free running
intermode beat and the detection of a weak off-resonant NIR modulation near
$-1$ MHz, and (b) the injection-locked intermode beat using 1.5 mW of NIR
light with 100% modulation depth. (c) Stacked RF spectra (50 Hz RBW) with the
increasing modulation frequency crossing the natural repetition frequency and
causing injection-locking over a range of a few kHz. The dashed horizontal
line indexes the acquisition shown in (b). (d) Phase noise power spectral
density (PN PSD) of the intermode beat signal in the free-running and
injection locked regime.

##  VI Discussion

In this article, we used a low power NIR CW light illuminating the front facet
of a MIR QCL frequency comb as an optical actuator for the phase stabilization
of a comb line and as a mean to achieve coherent injection locking of the
repetition rate. First, by characterizing the response of the QCL, we showed
that intensity modulation of the NIR light offers a higher modulation
bandwidth compared to conventional drive current modulation. Then, we
implemented a stabilization scheme exploiting the NIR light to extend the
locking-bandwidth from 300 kHz to over 2 MHz, which resulted in an increase of
the SNR by 35 dB and an integrated phase noise as low as 200 mrad. Finally, we
showed that the QCL can act as a detector of NIR light modulated at a
frequency of 11 GHz, and that it can be injection-locked in such a way.

We believe that the tighter mutual stabilization enabled by the high bandwidth
of the NIR light could lead to higher sensitivities through coherent averaging
in DCS [44, 22], which is currently one of the main applications of QCL combs.
In this regard, a comprehensive comparison with the performance of
computational coherent averaging [15] is necessary. The high bandwidth could
also allow locking to enhancement cavities for further improvement of the
sensitivity [45]. Moreover, high spectral purity in the MIR is relevant for a
variety of applications such as quantum control of molecules [46], tests of
fundamental physics [47, 48], and generally for the study of molecules via
high-resolution spectroscopy [49, 50, 5]. Thus, we expect our results to
facilitate the application of QCL frequency combs in the MIR.

As a further outlook, we anticipate that other wavelengths [33] could be more
suitable for orthogonal control without sign reversal of the comb properties
such as the offset and repetition frequencies, while perhaps they could also
be used to dynamically tune laser parameters such as dispersion, gain, or
nonlinearity. Faster modulations from 10 MHz to a few GHz could be
investigated as a way to achieve (pure) frequency modulation of the comb.
Then, full optical control of the QCL comb merely driven by a battery could be
envisaged. To keep the compactness of the device and its low footprint, light
could be delivered by fiber or directly generated and modulated on the same
chip if low power is sufficient. Optical control could also be investigated in
interband cascade lasers [51]. Moreover, the recent work on free-space
communications using QCL combs [42] inspires the adaptation of injection
locking of the QCL by NIR light for direct conversion of NIR telecommunication
streams to MIR signals. We believe that increasing the injected optical power
could increase the locking range of the repetition rate to a sufficient level
for this application. Further work including theoretical and numerical
investigations of the laser dynamics [52, 53] are necessary to understand the
full potential of controlling QCLs via optical means.

###### Acknowledgements.

We thank Stéphane Schilt for support in the early stage of the investigation.
We thank Alpes Laser for providing the DFB-QCL used in this work. We
acknowledge fundings from the Schweizerischer Nationalfonds zur Förderung der
Wissenschaftlichen Forschung (40B2-1_176584). Data underlying the results
presented in this paper will be made available on an open server.

## References

  * Hugi  _et al._ [2012] A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, Mid-infrared frequency comb based on a quantum cascade laser, Nature 492, 229 (2012).
  * Klocke  _et al._ [2018] J. L. Klocke, M. Mangold, P. Allmendinger, A. Hugi, M. Geiser, P. Jouy, J. Faist, and T. Kottke, Single-Shot Sub-microsecond Mid-infrared Spectroscopy on Protein Reactions with Quantum Cascade Laser Frequency Combs, Anal. Chem. 90, 10494 (2018).
  * Pinkowski  _et al._ [2021] N. H. Pinkowski, P. Biswas, J. Shao, C. L. Strand, and R. K. Hanson, Thermometry and speciation for high-temperature and -pressure methane pyrolysis using shock tubes and dual-comb spectroscopy, Meas. Sci. Technol. 32, 125502 (2021).
  * Lepère  _et al._ [2022] M. Lepère, O. Browet, J. Clément, B. Vispoel, P. Allmendinger, J. Hayden, F. Eigenmann, A. Hugi, and M. Mangold, A mid-infrared dual-comb spectrometer in step-sweep mode for high-resolution molecular spectroscopy, J. Quant. Spectrosc. Radiat. Transf. 287, 108239 (2022).
  * Agner  _et al._ [2022] J. A. Agner, S. Albert, P. Allmendinger, U. Hollenstein, A. Hugi, P. Jouy, K. Keppler, M. Mangold, F. Merkt, and M. Quack, High-resolution spectroscopic measurements of cold samples in supersonic beams using a QCL dual-comb spectrometer*, Mol. Phys. 120, e2094297 (2022).
  * Komagata  _et al._ [2023] K. N. Komagata, V. J. Wittwer, T. Südmeyer, L. Emmenegger, and M. Gianella, Absolute frequency referencing for swept dual-comb spectroscopy with midinfrared quantum cascade lasers, Phys. Rev. Res. 5, 013047 (2023).
  * Iwakuni  _et al._ [2018] K. Iwakuni, G. Porat, T. Q. Bui, B. J. Bjork, S. B. Schoun, O. H. Heckl, M. E. Fermann, and J. Ye, Phase-stabilized 100 mW frequency comb near 10 µm, Appl. Phys. B 124, 128 (2018).
  * Krzempek  _et al._ [2019] K. Krzempek, D. Tomaszewska, A. Głuszek, T. Martynkien, P. Mergo, J. Sotor, A. Foltynowicz, and G. Soboń, Stabilized all-fiber source for generation of tunable broadband fCEO-free mid-IR frequency comb in the 7 – 9 µm range, Opt. Express 27, 37435 (2019).
  * Hoghooghi  _et al._ [2022] N. Hoghooghi, S. Xing, P. Chang, D. Lesko, A. Lind, G. Rieker, and S. Diddams, Broadband 1-GHz mid-infrared frequency comb, Light Sci. Appl. 11, 264 (2022).
  * Jouy  _et al._ [2017] P. Jouy, J. M. Wolf, Y. Bidaux, P. Allmendinger, M. Mangold, M. Beck, and J. Faist, Dual comb operation of $\lambda$ $\sim$ 8.2 µm quantum cascade laser frequency comb with 1 W optical power, Appl. Phys. Lett. 111, 141102 (2017).
  * Coddington  _et al._ [2016] I. Coddington, N. Newbury, and W. Swann, Dual-comb spectroscopy, Optica 3, 414 (2016).
  * Picqué and Hänsch [2019] N. Picqué and T. W. Hänsch, Frequency comb spectroscopy, Nat. Photonics 13, 146 (2019).
  * Giorgetta  _et al._ [2021] F. R. Giorgetta, J. Peischl, D. I. Herman, G. Ycas, I. Coddington, N. R. Newbury, and K. C. Cossel, Open-Path Dual-Comb Spectroscopy for Multispecies Trace Gas Detection in the 4.5–5 µm Spectral Region, Laser Photonics Rev. 15, 2000583 (2021).
  * Parriaux  _et al._ [2022] A. Parriaux, K. Hammani, C. Thomazo, O. Musset, and G. Millot, Isotope ratio dual-comb spectrometer, Phys. Rev. Res. 4, 023098 (2022).
  * Sterczewski  _et al._ [2019] L. A. Sterczewski, J. Westberg, and G. Wysocki, Computational coherent averaging for free-running dual-comb spectroscopy, Opt. Express 27, 23875 (2019).
  * Jost  _et al._ [2002] J. D. Jost, J. L. Hall, and J. Ye, Continuously tunable, precise, single frequency optical signal generator, Opt. Express 10, 515 (2002).
  * Argence  _et al._ [2015] B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, M. Abgrall, C. Chardonnet, C. Daussy, B. Darquié, Y. Le Coq, and A. Amy-Klein, Quantum cascade laser frequency stabilization at the sub-Hz level, Nat. Photonics 9, 456 (2015).
  * Komagata  _et al._ [2022] K. N. Komagata, M. Gianella, P. Jouy, F. Kapsalidis, M. Shahmohammadi, M. Beck, R. Matthey, V. J. Wittwer, A. Hugi, J. Faist, L. Emmenegger, T. Südmeyer, and S. Schilt, Absolute frequency referencing in the long wave infrared using a quantum cascade laser frequency comb, Opt. Express 30, 12891 (2022).
  * Telle  _et al._ [1999] H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation, Appl. Phys. B 69, 327 (1999).
  * Cappelli  _et al._ [2016] F. Cappelli, G. Campo, I. Galli, G. Giusfredi, S. Bartalini, D. Mazzotti, P. Cancio, S. Borri, B. Hinkov, J. Faist, and P. De Natale, Frequency stability characterization of a quantum cascade laser frequency comb, Laser Photonics Rev. 10, 623 (2016).
  * Westberg  _et al._ [2017] J. Westberg, L. A. Sterczewski, and G. Wysocki, Mid-infrared multiheterodyne spectroscopy with phase-locked quantum cascade lasers, Appl. Phys. Lett. 110, 141108 (2017).
  * Komagata  _et al._ [2021] K. Komagata, A. Shehzad, G. Terrasanta, P. Brochard, R. Matthey, M. Gianella, P. Jouy, F. Kapsalidis, M. Shahmohammadi, M. Beck, V. J. Wittwer, J. Faist, L. Emmenegger, T. Südmeyer, A. Hugi, and S. Schilt, Coherently-averaged dual comb spectrometer at 7.7 µm with master and follower quantum cascade lasers, Opt. Express 29, 19126 (2021).
  * Hillbrand  _et al._ [2019] J. Hillbrand, A. M. Andrews, H. Detz, G. Strasser, and B. Schwarz, Coherent injection locking of quantum cascade laser frequency combs, Nat. Photonics 13, 101 (2019).
  * Tombez  _et al._ [2012] L. Tombez, S. Schilt, J. Di Francesco, P. Thomann, and D. Hofstetter, Temperature dependence of the frequency noise in a mid-IR DFB quantum cascade laser from cryogenic to room temperature, Opt. Express 20, 6851 (2012).
  * Tombez  _et al._ [2013a] L. Tombez, F. Cappelli, S. Schilt, G. Di Domenico, S. Bartalini, and D. Hofstetter, Wavelength tuning and thermal dynamics of continuous-wave mid-infrared distributed feedback quantum cascade lasers, Appl. Phys. Lett. 103, 031111 (2013a).
  * Hoffmann  _et al._ [2013] M. Hoffmann, S. Schilt, and T. Südmeyer, CEO stabilization of a femtosecond laser using a SESAM as fast opto-optical modulator, Opt. Express 21, 30054 (2013).
  * Zervos  _et al._ [2007] C. Zervos, M. D. Frogley, C. C. Phillips, D. O. Kundys, L. R. Wilson, M. Hopkinson, and M. S. Skolnick, All-optical switching in quantum cascade lasers, Appl. Phys. Lett. 90, 053505 (2007).
  * Chen  _et al._ [2009] G. Chen, C. G. Bethea, and R. Martini, Quantum cascade laser gain enhancement by front facet illumination, Opt. Express 17, 24282 (2009).
  * Tombez  _et al._ [2013b] L. Tombez, S. Schilt, D. Hofstetter, and T. Südmeyer, Active linewidth-narrowing of a mid-infrared quantum cascade laser without optical reference, Opt. Lett. 38, 5079 (2013b).
  * Peng  _et al._ [2016] C. Peng, G. Chen, J. Tang, L. Wang, Z. Wen, H. Zhou, and R. Martini, High-Speed Mid-Infrared Frequency Modulation Spectroscopy Based on Quantum Cascade Laser, IEEE Photonics Technol. Lett. 28, 1727 (2016).
  * Consolino  _et al._ [2021] L. Consolino, A. Campa, M. De Regis, F. Cappelli, G. Scalari, J. Faist, M. Beck, M. Rösch, S. Bartalini, and P. De Natale, Controlling and phase-locking a THz quantum cascade laser frequency comb by small optical frequency tuning, Laser Photonics Rev. 15, 2000417 (2021).
  * Hillbrand  _et al._ [2022] J. Hillbrand, M. Bertrand, V. Wittwer, N. Opačak, F. Kapsalidis, M. Gianella, L. Emmenegger, B. Schwarz, T. Südmeyer, M. Beck, and J. Faist, Synchronization of frequency combs by optical injection, Opt. Express 30, 36087 (2022).
  * Peng  _et al._ [2017] C. Peng, H. Zhou, L. Zhu, T. Chen, Q. Liu, D. Wang, J. Li, Q. Peng, G. Chen, and Z. Li, Purified frequency modulation of a quantum cascade laser with an all-optical approach, Opt. Lett. 42, 4506 (2017).
  * Cappelli  _et al._ [2019] F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. Siciliani de Cumis, P. Cancio Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. De Natale, and S. Bartalini, Retrieval of phase relation and emission profile of quantum cascade laser frequency combs, Nat. Photonics 13, 562 (2019).
  * Shehzad  _et al._ [2020] A. Shehzad, P. Brochard, R. Matthey, F. Kapsalidis, M. Shahmohammadi, M. Beck, A. Hugi, P. Jouy, J. Faist, T. Südmeyer, and S. Schilt, Frequency noise correlation between the offset frequency and the mode spacing in a mid-infrared quantum cascade laser frequency comb, Opt. Express 28, 8200 (2020).
  * Piccardo  _et al._ [2018] M. Piccardo, D. Kazakov, N. A. Rubin, P. Chevalier, Y. Wang, F. Xie, K. Lascola, A. Belyanin, and F. Capasso, Time-dependent population inversion gratings in laser frequency combs, Optica 5, 475 (2018).
  * Teng  _et al._ [2023] C. C. Teng, J. Westberg, and G. Wysocki, Gapless tuning of quantum cascade laser frequency combs with external cavity optical feedback, Opt. Lett. 48, 363 (2023).
  * Telle  _et al._ [2002] H. Telle, B. Lipphardt, and J. Stenger, Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements, Appl. Phys. B 74, 1 (2002).
  * Dolgovskiy  _et al._ [2012] V. Dolgovskiy, N. Bucalovic, P. Thomann, C. Schori, G. Di Domenico, and S. Schilt, Cross-influence between the two servo loops of a fully stabilized Er: Fiber optical frequency comb, J. Opt. Soc. Am. B 29, 2944 (2012).
  * Hangauer  _et al._ [2014] A. Hangauer, G. Spinner, M. Nikodem, and G. Wysocki, High frequency modulation capabilities and quasi single-sideband emission from a quantum cascade laser, Opt. Express 22, 23439 (2014).
  * Borri  _et al._ [2011] S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, Frequency-Noise Dynamics of Mid-Infrared Quantum Cascade Lasers, IEEE J. Quantum. Electron. 47, 984 (2011).
  * Corrias  _et al._ [2022] N. Corrias, T. Gabbrielli, P. De Natale, L. Consolino, and F. Cappelli, Analog FM free-space optical communication based on a mid-infrared quantum cascade laser frequency comb, Opt. Express 30, 10217 (2022).
  * Di Domenico  _et al._ [2010] G. Di Domenico, S. Schilt, and P. Thomann, Simple approach to the relation between laser frequency noise and laser line shape, Appl. Opt. 49, 4801 (2010).
  * Chen  _et al._ [2018] Z. Chen, M. Yan, T. W. Hänsch, and N. Picqué, A phase-stable dual-comb interferometer, Nat. Commun. 9, 3035 (2018).
  * Foltynowicz  _et al._ [2013] A. Foltynowicz, P. Masłowski, A. J. Fleisher, B. J. Bjork, and J. Ye, Cavity-enhanced optical frequency comb spectroscopy in the mid-infrared application to trace detection of hydrogen peroxide, Appl. Phys. B 110, 163 (2013).
  * Sinhal  _et al._ [2020] M. Sinhal, Z. Meir, K. Najafian, G. Hegi, and S. Willitsch, Quantum-nondemolition state detection and spectroscopy of single trapped molecules, Science 367, 1213 (2020).
  * Jansen  _et al._ [2014] P. Jansen, H. L. Bethlem, and W. Ubachs, Perspective: Tipping the scales: Search for drifting constants from molecular spectra, J. Chem. Phys. 140, 010901 (2014).
  * Cournol  _et al._ [2019] A. Cournol, M. Manceau, M. Pierens, L. Lecordier, D. B. A. Tran, R. Santagata, B. Argence, A. Goncharov, O. Lopez, M. Abgrall, Y. Le Coq, R. Le Targat, H. Alvarez Martinez, W. K. Lee, D. Xu, P. E. Pottie, R. J. Hendricks, T. E. Wall, J. M. Bieniewska, B. E. Sauer, M. R. Tarbutt, A. Amy-Klein, S. K. Tokunaga, and B. Darquié, A new experiment to test parity symmetry in cold chiral molecules using vibrational spectroscopy, Quantum Electron. 49, 288 (2019).
  * Changala  _et al._ [2019] P. B. Changala, M. L. Weichman, K. F. Lee, M. E. Fermann, and J. Ye, Rovibrational quantum state resolution of the C60 fullerene, Science 363, 49 (2019).
  * Germann  _et al._ [2022] M. Germann, A. Hjältén, V. Boudon, C. Richard, K. Krzempek, A. Hudzikowski, A. Głuszek, G. Soboń, and A. Foltynowicz, A methane line list with sub-MHz accuracy in the 1250 to 1380 cm-1 range from optical frequency comb Fourier transform spectroscopy, J. Quant. Spectrosc. Radiat. Transf. 288, 108252 (2022).
  * Bagheri  _et al._ [2018] M. Bagheri, C. Frez, L. A. Sterczewski, I. Gruidin, M. Fradet, I. Vurgaftman, C. L. Canedy, W. W. Bewley, C. D. Merritt, C. S. Kim, M. Kim, and J. R. Meyer, Passively mode-locked interband cascade optical frequency combs, Sci. Rep. 8, 3322 (2018).
  * Villares and Faist [2015] G. Villares and J. Faist, Quantum cascade laser combs: Effects of modulation and dispersion, Opt. Express 23, 1651 (2015).
  * Opačak and Schwarz [2019] N. Opačak and B. Schwarz, Theory of Frequency-Modulated Combs in Lasers with Spatial Hole Burning, Dispersion, and Kerr Nonlinearity, Phys. Rev. Lett. 123, 243902 (2019).