# Nozaki-Bekki optical solitons

Nikola Opa$\mathrm{\check{c}}$ak<EMAIL_ADDRESS>Institute of
Solid State Electronics, TU Wien, 1040 Vienna, Austria  Harvard John A.
Paulson School of Engineering and Applied Sciences, Harvard University,
Cambridge, MA 02138, USA  Dmitry Kazakov  Harvard John A. Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Lorenzo L. Columbo  Dipartimento di Elettronica e Telecomunicazioni,
Politecnico di Torino, 10129 Torino, Italy  Maximilian Beiser  Institute of
Solid State Electronics, TU Wien, 1040 Vienna, Austria  Theodore P. Letsou
Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, MA 02138, USA  Department of Electrical Engineering and
Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02142,
USA  Florian Pilat  Institute of Solid State Electronics, TU Wien, 1040
Vienna, Austria  Massimo Brambilla  Dipartimento di Fisica Interateneo and
CNR-IFN, Università e Politecnico di Bari, 70125 Bari, Italy  Franco Prati
Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, 22100
Como, Italy  Marco Piccardo  Harvard John A. Paulson School of Engineering and
Applied Sciences, Harvard University, Cambridge, MA 02138, USA  Department of
Physics, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon,
Portugal  Instituto de Engenharia de Sistemas e Computadores – Microsistemas e
Nanotecnologias (INESC MN), 1000-029 Lisbon, Portugal  Federico Capasso
Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, MA 02138, USA  Benedikt Schwarz  Institute of Solid
State Electronics, TU Wien, 1040 Vienna, Austria  Harvard John A. Paulson
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA
02138, USA

###### Abstract

Recent years witnessed rapid progress of chip-scale integrated optical
frequency comb sources. Among them, two classes are particularly significant –
semiconductor Fabry-Perót lasers and passive ring Kerr microresonators. Here,
we merge the two technologies in a ring semiconductor laser and demonstrate a
new paradigm for free-running soliton formation, called Nozaki-Bekki soliton.
These dissipative waveforms emerge in a family of traveling localized dark
pulses, known within the famed complex Ginzburg-Landau equation. We show that
Nozaki-Bekki solitons are structurally-stable in a ring laser and form
spontaneously with tuning of the laser bias – eliminating the need for an
external optical pump. By combining conclusive experimental findings and a
complementary elaborate theoretical model, we reveal the salient
characteristics of these solitons and provide a guideline for their
generation. Beyond the fundamental soliton circulating inside the ring laser,
we demonstrate multisoliton states as well, verifying their localized nature
and offering an insight into formation of soliton crystals. Our results
consolidate a monolithic electrically-driven platform for direct soliton
generation and open a door for a new research field at the junction of laser
multimode dynamics and Kerr parametric processes.

Dissipative temporal solitons – stable solitary localized pulses – emerge
universally in extended nonlinear media, sustained by a dual balance between
the nonlinearity and dispersion/diffusion, as well as between the gain and
dissipation of the system [1]. Their prime examples in optics came from
passively mode-locked lasers [2], optical fibers [3, 4], and passive high-Q
microresonators [5, 6, 7]. Being of special interest for integrated photonics
due to their compact size, microresonators have taken the community by storm.
Fine-tuning of the requisite continuous-wave (CW) optical pump, combined with
a bulk Kerr nonlinearity, is necessary to provide the parametric gain that
leads to bright or dark microresonator solitons [8, 9, 10, 11]. When
outcoupled, the circulating soliton results in a periodic train of short
pulses, generating a broad frequency comb in the spectral domain [12]. These
miniature Kerr combs have ever since been the vanguard of microresonator
technology, finding use in telecommunication [13, 14], ultrafast ranging [15],
high-precision spectroscopy [16, 17], and frequency synthesis [18].

In this work, we demonstrate a new type of optical dissipative solitons –
named Nozaki-Bekki (NB) solitons – in an electrically-driven mid-infrared
(mid-IR) ring semiconductor laser. They arise as localized waveforms in a
family of traveling dark pulses that satisfy ring periodic boundary
conditions. The active laser gain material in our devices simultaneously
provides a giant Kerr nonlinearity and eliminates the external optical pump
and its challenging frequency tuning, which is a vital ingredient for
microresonator combs. The NB soliton regime – first of its kind in a compact
optical system – emerges spontaneously and is directly accessed solely by
tuning the laser driving current, which we validate by using a phase-sensitive
measurement. The experimental findings are corroborated by a complementary
theoretical Maxwell-Bloch formalism with numerical simulations, allowing us to
identify favourable dispersive and nonlinear conditions for NB soliton
formation and their coherent control. Our initial prediction of their
existence originated from the cubic complex Ginzburg-Landau equation (CGLE) –
one of the most celebrated equations in physics that describes spatially
extended systems close to bifurcations [19]. While their stability was long
discussed within the CGLE framework, NB solitons have so far been scarcely
observed in experiments. Unequivocal classification of NB solitons is made
possible by their striking salient characteristics – anti-phase
synchronization of the soliton with the primary mode and a $2\pi$ temporal
phase ramp across the soliton. The localized nature of NB solitons is
conclusively demonstrated by further observing multisoliton states, both in
theory and experiments. Our findings herald a new generation of
monolithically-integrated self-starting soliton generators that lie at the
intersection of semiconductor lasers and Kerr microresonators.

Figure 1:  Parameter space of the CGLE with corresponding laser regimes. The
unidirectional intracavity intensities and the spectra are obtained from
numerical simulations of the CGLE for different points indicated in the
parameter space. The left column shows two states triggered by turbulence
inside the CW linearly-unstable yellow region of the parameter space (defined
by $1+c_{\mathrm{D}}c_{\mathrm{NL}}<0$), where a single-mode field cannot
exist as a steady state. Defect turbulence occurs deep inside the unstable
region (point $P_{1}$) and is aperiodic, exhibiting chaotic temporal
evolution. Close to the border of CW stability (point $P_{2}$), phase
turbulence leads to narrowband homoclon frequency combs [20]. The right column
displays the coexistence (multistability) of a single-mode regime and an NB
soliton, both set in the CW linearly-stable part of the parameter space
($1+c_{\mathrm{D}}c_{\mathrm{NL}}>0$). NB solitons emerge as coherent,
unidirectional propagating dark pulses, characterized with a broad and smooth
spectral envelope. The visible pulse shoulder represents a shock, stabilizing
the waveform in ring periodic boundary conditions.

To study NB solitons, we use quantum cascade lasers – compact and efficient
devices that emit in the mid-IR and THz regions [21, 22, 23] – embedded in a
ring cavity. Ultrafast intersubband transitions of QCLs provide not only the
optical gain, but also a giant Kerr nonlinearity [24, 25], which is several
orders of magnitude larger than in bulk III-V compounds [26]. This is already
exploited for frequency comb formation in Fabry-Pérot (FP) QCLs [27, 24, 28].
There, the multimode operation originates from the spatial hole burning (SHB)
[28], which describes inhomogeneous gain saturation due to a standing wave
inside the cavity, caused by the counter-propagating components of the
electric field. Conversely, a ring cavity has no reflection points and
supports unidirectional field propagation – preventing SHB. Nevertheless,
recent work demonstrated multimode unidirectional ring QCLs due to phase
turbulence at low pumping levels, making SHB nonessential [20]. Since then,
substantial efforts were put towards developing Kerr combs in ring QCLs [29,
30, 31, 32], leading even to bright pulses after spectral filtering [33] and
theoretically-predicted optically-driven solitons [34, 35, 36] – already
anticipating the potential of these devices as an on-chip soliton platform.

We can interpret ring QCL multimode dynamics on the grounds of the CGLE with
periodic boundary conditions [20]. Starting from the more general laser master
equation [28, 37], we derive the cubic CGLE

$\displaystyle\partial_{t}E=E+(1+ic_{\mathrm{D}})\partial^{2}_{z}E-(1+ic_{\mathrm{NL}})|E|^{2}E,$
(1)

where $E$ is the unidirectional electric field, $t$ is time, and $z$ is the
spatial coordinate along the ring cavity (see derivation in the Supplementary
material). The entire parameter space is elegantly constricted to just two
dimensions, which refer to the dispersive ($c_{\mathrm{D}}$) and nonlinear
effects ($c_{\mathrm{NL}}$). In lasers, $c_{\mathrm{D}}$ is partly determined
by the cavity group velocity dispersion (GVD) $k^{{}^{\prime\prime}}$. Another
contribution to the total GVD originates from the gain lineshape, defined with
the linewidth enhancement factor (LEF) – a crucial semiconductor laser
parameter which describes light amplitude-phase coupling [38, 39]. Moreover,
the LEF also defines $c_{\mathrm{NL}}$ and phenomenologically describes the
giant resonant Kerr nonlinearity of QCLs [28, 24]. A linear stability analysis
of the CGLE divides the $c_{\mathrm{D}}-c_{\mathrm{NL}}$ space in two regions
depending on the stability of the single-mode CW solution under small
perturbations (Fig. 1) [19]. Deep within the unstable region, defect
turbulence occurs, characterized with a broad, unlocked spectrum and
chaotically-evolving intracavity intensity (point $\mathrm{P}_{1}$). Closer to
the stable region, the laser undergoes phase turbulence represented by a
narrower spectrum and shallow intensity variations (point $\mathrm{P}_{2}$).
Phase turbulence can eventually lead to frequency combs in the form of
localized coherent pulse-like structures named homoclons [19, 20].

Figure 2:  Experimental and theoretical characterization of fundamental NB
solitons in a monolithic ring laser. a) Microscope image of the QCL ring and
waveguide coupler, with separate electrical contacts. SEM images depict the
ring-waveguide coupling region. b) Output power of the device as a function of
the ring and waveguide currents, $J_{\mathrm{R}}$ and $J_{\mathrm{WG}}$
respectively. c) A narrow optical beatnote of the laser comb at the central
frequency of 13.59$\,\mathrm{GHz}$, equal to the repetition frequency. d)
Experimentally-obtained intensity spectrum and the intermode phases of an NB
soliton at bias currents of $J_{\mathrm{R}}=$1.33$\,\mathrm{kA/cm}^{2}$ and
$J_{\mathrm{WG}}=$0.79$\,\mathrm{kA/cm}^{2}$. Intermode phases between weaker
sidemodes are synchronized in-phase, while the phase of the primary mode is
$\pi$-shifted. Detailed SWIFTS measured data is presented in the Supplementary
material. e) Temporal profiles of the intensity and phase of the emitted
light. Within the width of the NB soliton, the phase changes its value by
$2\pi$ and remains linear during the remainder of the roundtrip, confirming
that the NB soliton is surrounded by a single-frequency constant CW field. f)
Intensity spectrum obtained from numerical simulations of the master equation,
and g) zoomed-in top-portion within the same range of 35$\,\mathrm{dB}$ as the
experimental spectrum in d). The $\pi$ jumps around the primary mode – salient
characteristic of NB solitons – are visible in the intermode phases. h) The
simulated temporal waveforms of the intensity and the phase over two
roundtrips. Larger amplitude contrast compared to e) is attributed to the
limited dynamic range of detection in experiments, which results in a finite
number of spectral modes used for the temporal waveform reconstruction. The
simulated NB soliton is obtained for LEF of 1.25, in agreement with typical
values in QCLs [39]. The cavity dispersion $k^{{}^{\prime\prime}}$ was set to
800$\,\mathrm{fs}^{2}/\mathrm{mm}$, which together with the gain dispersion
brings the total GVD value to about -700$\,\mathrm{fs}^{2}/\mathrm{mm}$ (see
Supplementary material for a discussion on different contributions to the
effective GVD and nonlinearity in a laser).

The stable region of the parameter space sustains single-mode operation.
However, we show that multimode emission can exist even here if we allow for
large perturbations that are beyond the scope of linear stability analysis.
The resulting frequency combs – known as Nozaki-Bekki holes in the CGLE
framework [40, 19, 41] – have a smooth and broad spectral envelope. In the
temporal domain, they correspond to a family of traveling, localized dark
pulses that preserve their shape and connect two stable CW fields, giving a
constant background. These waveforms exist in a narrow region of the parameter
space, and have been so far related to dark solitons [42] and experimentally
observed in chemical systems [43], fluids [44], and long-cavity ring lasers
[45]. In numerical simulations imposing ring periodic boundary conditions,
states comprising an arranged hole and shock pair are structurally-stable even
with inclusion of higher-order nonlinear terms that are not accounted for in
the cubic CGLE and that play a role in real physical systems e.g. lasers [40,
46, 47]. The coexistence of a stable CW field and a hole-shock pair (point
$\mathrm{P}_{3}$) is a prime example of the phenomenon of multistability –
stochastic dependence of the laser state on the starting conditions – thus
corroborating the solitonic nature of these dissipative localized structures
[1, 6], referred to hereafter as NB solitons in our lasers.

A microscope image of the ring QCL is seen in Fig. 2a), along with an
integrated waveguide coupler. The waveguide core is made from the same QCL
material and has separate contacts for independent electrical driving. This
allows to tune the mode indices, the Q-factor, the power of the extracted
light, and the coupling between the waveguide and the ring – making this
configuration an ideal testbed for a plethora of resonant electromagnetic
phenomena [48]. The light outcoupling in previous experiments relied on the
minuscule ring bending losses, thus limiting the extracted power to
submilliwatt levels at room temperature [20]. Using the waveguide coupler to
efficiently extract the light, our devices reach power levels above
10$\,\mathrm{mW}$ (Fig. 2b)), bringing them on par with FP QCLs of a similar
ridge length and width, fabricated from the same wafer [48]. While the
coupling waveguide is biased below the lasing threshold, the ring QCL is
driven above it, where it operates in a unidirectional regime after the
symmetry-breaking point [33, 48]. We find that at currents partially higher
than the lasing threshold $J_{\mathrm{th}}$ (around 1.2$J_{\mathrm{th}}$), the
ring laser emits a multimode field with a narrow beatnote (Fig. 2c)). This
implies a high degree of coherence, typical of a frequency comb.

Figure 3:  Multisoliton states. a) Spectral behavior of an experimental
multisoliton state, obtained for $J_{\mathrm{R}}=$1.28$\,\mathrm{kA/cm}^{2}$
and $J_{\mathrm{WG}}=$0.74$\,\mathrm{kA/cm}^{2}$. The interference between two
NB solitons causes spectral modulation, resulting in smooth lobes separated by
a spectral hole. The intermode phases indicate an additional $\pi$ jump
between the lobes. b) Two NB solitons appear in the intensity waveform,
surrounded by a CW background. The temporal phase sweeps $4\pi$ within the
two-soliton region. c) Zoomed top portion of the simulated spectrum and the
intermode phases, in agreement with a). d) The intensity spectrum with the
indicated zoomed section, shown in c). The spectral modulation, caused by the
interference of the two solitons, cascades through the entire spectrum. e)
Simulated temporal waveforms of the intensity and the phase. The larger
dynamical range compared to the experiment allowed us to distinguish two
individual $2\pi$ ramps in the phase profile, one for each soliton. The LEF
was set to 1.35 and $k^{{}^{\prime\prime}}$ to
600$\,\mathrm{fs}^{2}/\mathrm{mm}$ (total GVD around
-900$\,\mathrm{fs}^{2}/\mathrm{mm}$). f) Intracavity intensity profiles
obtained for the same laser parameters when starting from different random
noise conditions, labeled with seed 1,2, and 3. The coexistence of a CW
single-mode field, a fundamental NB soliton, and a multisoliton state verifies
multistability and the fact that the laser operates in a linearly-stable
parameter region. The distance between the NB solitons in a multisoliton state
is time-invariant, but it changes for different laser parameters or starting
conditions, as is seen from two states in the bottom row. The state in the
bottom left is taken from e). Experimental evidence of a soliton crystal,
which is a special case of multisoliton states where all of the solitons are
equidistant [49], is found in the Supplementary material.

To benchmark comb operation, we employ SWIFTS – an experimental technique that
extracts both the amplitude and phase of spectral modes [50]. The measured
intensity spectrum (Fig. 2d)) consists of a strong primary mode surrounded by
weaker sidemodes that form a smooth envelope, strikingly reminiscent of
soliton spectra in microresonators [5, 6, 9, 10, 11, 7]. The intermode phases
– the phase differences between adjacent modes – are shown below. They are all
synchronized in-phase, except around the primary mode, where $\pi$ jumps
indicate destructive interference due to anti-phase synchronization. This is
evident from the reconstructed intensity profile (Fig. 2e)), where a single
dark pulse circulates around the cavity in an otherwise quasi-constant CW
background – consistent with the predicted fundamental NB soliton. Residual
intensity oscillations are due to the limited detection bandwidth. Within the
width of the NB soliton, the temporal phase exhibits a steep ramp covering
$2\pi$ and remains linear everywhere else – proving that the CW background
around the soliton is constant and contains a single optical frequency equal
to that of the primary mode.

Figure 4:  Coherent control of NB soliton regimes. a) Spectral shift of the
soliton relative to the primary mode as the ring current is swept. Similar
behavior is exhibited if the waveguide current is tuned instead. b) Sketch of
the coupled ring-waveguide system. The bias alters the optical indices of the
ring and the waveguide through carrier- and thermal-induced changes, resulting
in a mismatch (detailed analysis in the Supplementary material). c) Group
delay dispersion, calculated with a coupled mode theory analysis, as a
function of the index mismatch between the ring and the waveguide. Compared to
a single ring cavity, our devices provide superior control of the dispersion.
d) Simulated soliton spectra as the cavity dispersion $k^{{}^{\prime\prime}}$
is swept, demonstrating a spectral shift of the soliton similar to a). The
gainshape contribution to the total GVD is around
-1500$\,\mathrm{fs}^{2}/\mathrm{mm}$. e) Experimentally-obtained intensity
spectrum of a NB soliton at bias currents of
$J_{\mathrm{R}}=$1.39$\,\mathrm{kA/cm}^{2}$ and
$J_{\mathrm{WG}}=$0.85$\,\mathrm{kA/cm}^{2}$ (red), which corresponds to a
bright pulse in the intensity waveform. The soliton sidemodes are stronger
compared with the dark-pulse soliton from Fig. 2d) (replotted in blue). f)
Illustration of the theoretical dependence of the intensity waveform on the
soliton spectrum. We assume ideal NB soliotn intermode phase distribution with
$\pi$ jumps around the primary mode.

To corroborate the experimental findings, we employ numerical simulations of
the master equation derived from the Maxwell-Bloch system (Eq. (S2) in the
Supplementary material) [28, 24]. The obtained comb spectrum, shown in Fig.
2f), has a smooth spectral envelope engulfing a strong primary mode and
spanning over more than 100 cavity modes. Numerical simulations are not
constrained by a small dynamical range as the experiment (around
35$\,\mathrm{dB}$ in Fig. 2d)). Hence, by concentrating on the top part of the
simulated spectrum within the same range (Fig. 2g)), we show a clear agreement
with the experiment. The simulated intermode phases confirm the $\pi$ shift
between the primary mode and the sidemodes, indicating that this is a hallmark
of NB solitons. The simulated temporal intensity in Fig. 2h) matches the
structures predicted by the CGLE theory in Fig. 1. Higher-order dispersion due
to the laser gainshape is the likely cause of the residual small ringing that
trails the NB soliton, as is well-known from microresonator solitons [51, 52,
6, 53]. The combined simulated and experimental temporal waveforms confirm
that NB solitons are surrounded by a CW background – providing a compelling
proof of their dissipative soliton nature.

Localization is another striking soliton feature, clearly noticed in
multisoliton states – spontaneously ordered ensembles of several co-
propagating solitons [5, 9]. Multisoliton states can be indubitably identified
by their ‘fingerprint’ optical spectra, which have a modulated envelope due to
the interference between individual solitons. Fig. 3a) depicts one such
spectrum, consisted of smooth lobes with spectral holes in-between. The
intermode phases indicate that, besides the usual $\pi$ jumps around the
primary mode, an additional $\pi$ jump occurs at the position of the spectral
hole – providing another telltale sign of a multisoliton state. The
reconstructed intensity (Fig. 3b)) displays two distinct dark pulses, during
which the phase changes by $4\pi$ – twice as much as for a fundamental NB
soliton. Master equation simulations verify multisoliton states as well. The
spectral behavior, including the additional $\pi$ intermode phase jump between
the lobes, is seen in Fig. 3c). Other than enabling large amplitude contrast,
the dynamic range of simulations allows to resolve the $4\pi$ change of the
temporal phase in two steps – one for each soliton (Fig. 3e)). Multistability
is yet another crucial dissipative soliton trait predicted in Fig. 1. To
emulate laser starting from spontaneous emission, weak noise from a random
number generator with a defined seed is fed as a starting condition into the
master equation. Changing the seed, while keeping other laser parameters
fixed, led to three states with different number of solitons (Fig. 3f)). The
coexistence of solitons and a single-mode field proves that NB solitons are
found in the CW linearly-stable region of the parameter space, as predicted by
the CGLE.

Separate electrical contacts of the waveguide and the ring provide two
invaluable knobs for soliton control. Demonstrating this, we shift the soliton
spectral lobe from red to the blue side of the primary mode purely by tuning
the ring current in Fig. 4a). Besides changing the gain and the LEF [24, 20],
altering the bias strongly impacts the GVD, as confirmed by a coupled mode
theory analysis of the ring-waveguide configuration (Figs. 4b) & c)). A small
current-induced index mismatch between the ring and the waveguide induces
large changes of the GVD, covering both normal and anomalous values. The key
role of GVD is also obvious from numerical simulations in Fig. 4d), where we
swept the cavity dispersion. Relative to the primary mode, the soliton is
spectrally shifted similarly to Fig. 4a), and in agreement with recent
observations in microresonators [11]. More anomalous GVD sets the laser
operating point in the linearly-unstable parameter region in analogy to Fig.
1, where neither single-mode nor NB soliton regime can exist. Instead,
homoclons form via phase turbulence [20], which we observe both in experiments
and simulations (Supplementary material).

Having the possibility to form bright pulses with high peak power would open
many doors for NB solitons to be used in nonlinear processes, such as
supercontinuum generation [54]. One way of achieving this relies on modifying
the phase of the primary mode to remove the destructive interference and
induce an intense bright pulse. This could be realized by a second active ring
resonator coupled with the waveguide, acting as a notch filter [55]. Another
possibility in this direction is illustrated in Fig. 4e), where we show an NB
soliton spectrum (in red), obtained at a larger bias compared to the soliton
from Fig. 2 (replotted in blue). Astonishingly, temporal intensity of the
former unveils a low-contrast bright pulse. A comparison between the two
soliton spectra reveals that the bright one has stronger sidemodes. To gain an
intuitive understanding, Fig. 4f) conceptually studies the dependence of the
temporal intensity on the corresponding spectrum, while assuming the
characteristic intermode phase distribution of NB solitons (see Supplementary
material for the full analysis). Whereas the primary mode is fixed, the
soliton lobe is gradually increased, with the color coding representing the
two states from Fig. 4e). The initial increase of the soliton sidemodes
enhances the amplitude contrast of a starting dark pulse, until an intensity
equilibrium between the primary mode and the sidemodes is reached. At this
point, destructive interference between the background and the soliton is
complete and the dark pulse reaches zero at its minimum. Further enhancement
of the sidemodes causes a decrease of the pulse amplitude contrast, eventually
reaching a quasi-constant waveform despite a broad spectrum, as is also
experimentally observed (Supplementary material). Finally, additional sidemode
amplification causes the pulse to ’flip over’ – resulting in a bright coherent
pulse. Both the CGLE and the master equation predict the emergence of NB
solitons as dark pulses, however neither of the two treatments takes into
account the delayed carrier population response to amplitude modulations [56].
A better understanding of the link between the carrier dynamics and the
parametric gain, that is necessary for multimode emission, could allow us to
optimize active ring resonators for the emission of high-contrast bright
pulses.

In this work, we have demonstrated a new way of direct spontaneous soliton
generation by utilizing a mid-IR semiconductor laser active material
implemented in an on-chip integrated ring cavity with a coupler waveguide. The
waveguide coupler has an independent bias, ensuring not only higher output
powers, but also providing a powerful knob to control the total dispersion of
the system. Paired with the giant resonant Kerr nonlinearity of the active
material, this solidifies the coupled waveguide-ring configuration as a very
fruitful playground for nonlinear phenomena – including the direct generation
of electrically-driven NB solitons. The soliton regime is demonstrated by
combining both experimental and theoretical results. The number of solitons
within one roundtrip varies stochastically with the initial conditions of the
lasers. This is indicative of a multistability phenomenon, typical for
dissipative solitons in extended systems, and paves the way for independent
soliton addressing [1]. Moreover, we predict that NB solitons are not
platform-dependent and anticipate their demonstration in other semiconductor
laser types, such as the interband cascade- or quantum dot lasers. The
spontaneous formation of NB solitons with current tuning, without the need of
an external optical pump, makes our QCL rings with coupled waveguides ideal
candidates for monolithic soliton generators specifically targeting mid-IR
applications.

## Methods

Device fabrication and operation. The lasers emit at around
8.2$\,\mathrm{\upmu m}$ and consist of GaInAs/AlInAs layers on an InP
substrate, with the band structure design based on a standard single-phonon
continuum depopulation scheme. The waveguides and the narrow gap in the
coupling region are etched using the standard fabrication recipe employing
optical lithography. The waveguide width is 10$\,\mathrm{mm}$, the curved
section of the racetrack is a semicircle with a radius of 500$\,\mathrm{\upmu
m}$, and the length of the straight section is 1.5$\,\mathrm{mm}$. The ring
circumference defines the cavity repetition rate of around
13.6$\,\mathrm{GHz}$. The heat sink temperature was kept at 14∘C in all
experiments.

Experimental setup and characterisation of the soliton states. The measurement
of the spectral amplitudes and phases, that are used for the reconstruction of
the temporal waveform, is done by utilizing the SWIFTS technique [50]. An
overview of the measurement procedure is in the Supplementary material. In
order to record the laser output modulation at the repetition frequency, we
employ a fast quantum well infrared photodetector in a cryostat, cooled with
liquid nitrogen. The setup uses a custom-built high-resolution Fourier
transform infrared (FTIR) spectrometer ($\sim$500$\,\mathrm{MHz}$). A Zurich
Instruments HF2LI lock-in amplifier is used for the acquisition of both the
intensity and SWIFTS interferograms. In order to stabilize the repetition
frequency of the comb, we rely on electrical injection locking via a radio-
frequency (RF) source. However, caution is required since using a too large
injection power will perturb the soliton spectrum and its intermode phases, as
can be seen in the Supplementary material.

Numerical simulations. Simulations of the CGLE are implemented using a pseudo-
spectral algorithm coupled with an exponential time-differencing scheme. The
master equation of a unidirectional field is discretized with a first-order
forward finite difference method. This method hugely benefits from parallel
implementation on a graphics processing unit (GPU), cutting down the execution
time by 3 orders of magnitude compared to a standard implementation on the
central processing unit. As an example, we simulate tens of millions of time
steps within a minute using a NVIDIA GeForce RTX 3070 GPU (tens of thousands
of simulated cavity roundtrips are necessary to ensure that a steady state is
reached). To emulate a laser starting from spontaneous emission, the numerical
simulations are run with random weak noise as a starting condition. The latter
is obtained from a random number generator by defining an integer seed value.

## Author contributions

N. Opačak, D. Kazakov, F. Pilat, T. P. Letsou, and B. Schwarz carried out the
experiments and analyzed the data. M. Beiser fabricated the device. N. Opačak
performed the master equation simulations. L. L. Columbo, M. Brambilla, and F.
Prati did the CGLE simulations and contributed to analysis of the experimental
results. N. Opačak prepared the manuscript with input from all coauthors. N.
Opačak and T. P. Letsou wrote sections of the Supplementary material. B.
Schwarz, M. Piccardo, and F. Capasso supervised the project. All authors
contributed to the discussion of the results.

## Acknowledgements

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(Grant agreement No. 853014), and from the National Science Foundation under
Grant No. ECCS-2221715. T. P. L. would like to thank the support of the
Department of Defense (DoD) through the National Defense Science and
Engineering Graduate (NDSEG) Fellowship Program.

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MIR

    mid-infrared
NB

    Nozaki-Bekki
CW

    continuous-wave
QCL

    quantum cascade laser
FP

    Fabry-Pérot
SHB

    spatial hole burning
CGLE

    complex Ginzburg-Landau equation
GVD

    group velocity dispersion
LEF

    linewidth enhancement factor

##  Supplementary material A Master equation for lasers with a fast gain
medium

Modeling of semiconductor laser multimode dynamics is typically done by
employing the Maxwell-Bloch equations. This method, obtained by coupling the
density matrix formalism with Maxwell’s equations, is fully capable of
describing the coherence and the spatio-temporal evolution of the laser light
inside the cavity. Within the slowly varying envelope approximation and the
rotating wave approximation, which are two commonly-made approximations, the
Maxwell-Bloch equations for an open two-level laser system read

$\displaystyle\begin{split}&\frac{\partial n_{l}}{\partial
t}={J_{t}}+\frac{n_{u}}{T_{lu}}-\frac{n_{l}}{T_{gl}}-\frac{\mu}{\hbar}\operatorname{Im}\\{E\sigma^{\ast}\\},\\\
&\frac{\partial n_{u}}{\partial
t}=J-\Big{(}\frac{1}{T_{lu}}+\frac{1}{T_{gu}}\Big{)}n_{u}+\frac{\mu}{\hbar}\operatorname{Im}\\{E\sigma^{\ast}\\},\\\
&\frac{\partial\sigma}{\partial
t}=-\frac{1+i\alpha}{T_{2}}\sigma+i\frac{\mu}{2\hbar}(1+i\alpha)^{2}E(n_{u}-n_{l}),\\\
&\Big{(}\frac{n_{r}}{c}\frac{\partial}{\partial t}+\frac{\partial}{\partial
z}\Big{)}E=-i\frac{\Gamma\mu\omega_{0}}{n_{r}\varepsilon_{0}cL_{p}}\sigma-\frac{\alpha_{w}}{2}E.\end{split}$
(S2)

In the system above, $n_{u}$ and $n_{l}$ are the surface carrier densities of
the upper and lower laser levels, and $\sigma$ and $E$ stand for the complex
slowly-varying envelopes of the off-diagonal density matrix element and the
electric field, respectively. Temporal and spatial coordinates are given with
$t$ and $z$, and lifetime $T_{ji}$ represents nonradiative transitions from
level $i$ to level $j$ (between the ground, the lower, and the upper level).
The coherence lifetime is labeled as $T_{2}$, $\Gamma$ is the light
confinement factor of the laser cavity, $\mu$ is the dipole matrix element,
$\omega_{0}$ is the optical frequency of the resonant transition, $n_{r}$ is
the real refractive index, $L_{p}$ is the thickness of the laser active
region, $\alpha_{w}$ gives the waveguide power losses, the current density $J$
represents the pumping rate to the upper laser level and $J_{t}$ models the
thermal excitation of carriers to the lower lasing level, where both $J$ and
$J_{t}$ are normalized to the elementary charge. Parameter $\alpha$ is the LEF
at the gain peak and it describes the coupling between the amplitude and the
phase of the light, which is crucial for semiconductor laser dynamics [38,
28].

The set of equations (S2) quantitatively describes the spatio-temporal
evolution of the laser field. However, following the formalism from [28], it
is possible to replace the entire system of Maxwell-Bloch equations with a
single master equation, thus greatly simplifying the theoretical model an
allowing to draw analogies with the CGLE. The master equation is valid for
lasers with a fast gain medium, such as the QCL, where the gain recovery time
is much shorter than the cavity roundtrip time. The master equation, written
for a unidirectional field inside a ring cavity, reads

$\displaystyle\frac{n_{r}}{c}\frac{\partial E}{\partial t}+\frac{\partial
E}{\partial z}$
$\displaystyle=\frac{g(I)}{2}(1+i{\alpha})\Big{(}E-\tilde{T}_{2}\frac{\partial
E}{\partial t}+\tilde{T}_{2}^{2}\frac{\partial^{2}E}{\partial
t^{2}}\Big{)}-\frac{\alpha_{w}}{2}E+i\beta{|{E}|}^{2}E+i\frac{k^{{}^{\prime\prime}}}{2}\frac{\partial^{2}E}{\partial
t^{2}},$ (S3)

where $\tilde{T}_{2}=T_{2}/(1+i\alpha)$, $k^{{}^{\prime\prime}}$ is the cavity
GVD that originates from the material and waveguide geometry, and $\beta$ is
the bulk Kerr nonlinearity coefficient, which we set to zero in this work (a
much larger effective Kerr nonlinearity originates from the gain medium of the
laser itself). Further, $I=|E|^{2}$ is the intensity, and $g(I)$ is the
saturated gain defined by

$\displaystyle g(I)=\frac{g_{0}}{1+\frac{I}{I_{s}}},$ (S4)

where the unsaturated gain is $g_{0}=\Gamma\mu^{2}\omega_{0}T_{1}T_{2}J/(\hbar
n_{r}c\varepsilon_{0}L_{p})$, the saturation intensity is
$I_{s}=2\hbar^{2}/(\mu^{2}T_{1}T_{2})$, and the carrier recombination time is
$T_{1}=(T_{lu}^{-1}+T_{gu}^{-1})^{-1}$ (approximately equal to the gain
recovery time). The complete derivation, starting from the Maxwell-Bloch
system, can be found in detail in [37].

###  A.1 The total GVD and Kerr nonlinearity in the laser

It is noteworthy to underline that the total GVD in the laser consists of two
major contributions. The first, proportional to the coefficient
$k^{{}^{\prime\prime}}$ in the model, describes the dispersion that comes from
the bulk material of choice and also the mode dispersion due to the resonator
geometry. We refer to it as the cavity dispersion and, as an approximation, it
is considered to be a constant pure second-order dispersion. The second major
contribution to the dispersion originates from the the resonant optical
transition itself, and we refer to it as the gain dispersion. It is inherently
included in the Maxwell-Bloch system of equations (S2) in the equation for the
off-diagonal matrix $\sigma$. This equation describes the response of the
material to the electric field, captured by the complex susceptibility in the
spectral domain

$\displaystyle\chi(\omega)=\frac{\mu^{2}(n_{u}-n_{l})}{\varepsilon\hbar
L_{p}}\frac{(1+i\alpha)^{2}}{\omega-\omega_{0}-\frac{i}{T_{2}}},$ (S5)

where $\operatorname{Im}\\{\chi\\}$ models the modified Lorentzian spectral
lineshape of the gain (ideal Lorentzian shape for vanishing LEF $\alpha=0$).
Moreover, the complex susceptibility also describes the spectral dependence of
the refractive index
$n_{r}(\omega)\approx\sqrt{1+\chi_{\mathrm{bulk}}+\operatorname{Re}\\{\chi(\omega)\\}}$.
Keeping in mind the definition $\mathrm{GVD}(\omega)=\frac{2}{c}\frac{\partial
n_{r}}{\partial\omega}+\frac{\omega}{c}\frac{\partial^{2}n_{r}}{\partial\omega^{2}}$,
it becomes obvious how the resonant optical transition induces an additional
gain dispersion. This contribution to the total GVD includes also higher-order
terms besides the second-order dispersion, however, its approximate value at
position of the gain peak is given in relation (S11).

The LEF physically models the coupling between the amplitude and phase of the
light, and it is mathematically defined as the ratio of the carrier-induced
changes of the refractive index and the optical gain [38]. As such, the LEF
describes how the asymmetric spectral shape of $\operatorname{Im}\\{\chi\\}$
alters the value of $\operatorname{Re}\\{\chi\\}$ at the gain peak and shifts
it away from zero ($\operatorname{Re}\\{\chi\\}$ has a zero-crossing at the
gain peak for an ideal symmetric Lorentzian shape of
$\operatorname{Im}\\{\chi\\}$). This physically means that whenever the
carrier population changes, a corresponding variation of not only the gain,
but also of the refractive index is induced. In a laser system, the carriers
are modulated by the propagating light inside the cavity, which results in
population pulsations due to dynamic gain saturation [56]. As a result, any
change of the light intensity indirectly modifies the refractive index of the
medium through these population pulsations. This is tightly related to the
appearance of a Kerr nonlinearity [28, 24]. Its origin is resonant, as it
comes from the gain medium itself and is relevant for optical frequencies
within the gainwidth of the laser. This contrasts with the broadband bulk
nonlinearity, which is proportional to the coefficient $\beta$ in the model.
The above-presented arguments are valid assuming that the population
pulsations can follow the speed of the intensity modulations, which occur at
the cavity repetition frequency. In other words, in media with fast gain
recovery times, a finite value of the LEF phenomenologically induces a giant
resonant Kerr nonlinearity, whose value is approximately given by equation
(S12).

Symbol |  Description |  Value
---|---|---
$T_{lu}$ |  Upper-lower level transition lifetime |  $1.15\>\mathrm{ps}$
$T_{gu}$ |  Upper-ground level transition lifetime |  $6\>\mathrm{ps}$
$T_{gl}$ |  Lower-ground level transition lifetime |  $80\>\mathrm{fs}$
$T_{2}$ |  Dephasing time |  $125\>\mathrm{fs}$
$n_{r}$ |  Refractive index |  $3.37$
$\alpha_{w}$ |  Waveguide power losses |  $4\>\mathrm{cm}^{-1}$
$\mu$ |  Dipole matrix element |  $1.8\>\mathrm{nm}\times\mathrm{e}$
Table S1:  Parameter values used in the master equation.

##  Supplementary material B Complex Ginzburg-Landau equation for ring
semiconductor lasers

The dynamics of ring semiconductor lasers with a fast gain medium close to the
lasing threshold are well captured by the the CGLE [19, 20]. Here we will
start from the more general master equation (S3) and derive the CGLE,
similarly to the procedure in Refs. [20, 34]. In the first step, the term
proportional to $\tilde{T}_{2}\partial E/\partial t$ on the right hand side of
the master equation (S3) is neglected. This is justified, as its main effect
is to introduce a constant shift of the value of the refractive index $n_{r}$.
Additionally, substituting the second time derivative $\partial^{2}/\partial
t^{2}$ with the second spatial derivative $(c/n_{r})^{2}\partial^{2}/\partial
z^{2}$ is an excellent approximation, as the equation deals with slowly
varying envelopes in the rotating wave approximation. Moreover, the system is
switched to a frame of reference that moves together with the propagating
field by applying coordinate transformations $z\rightarrow z-\frac{c}{n_{r}}t$
and $t\rightarrow t$. Equation (S3) then transforms to

$\displaystyle\frac{n_{r}}{c}\frac{\partial E}{\partial t}$
$\displaystyle=\frac{g(I)}{2}(1+i{\alpha})\Big{(}E+\tilde{T}_{2}^{2}\frac{c^{2}}{n_{r}^{2}}\frac{\partial^{2}E}{\partial
z^{2}}\Big{)}-\frac{\alpha_{w}}{2}E+i\beta{|{E}|}^{2}E+i\frac{k^{{}^{\prime\prime}}}{2}\frac{c^{2}}{n_{r}^{2}}\frac{\partial^{2}E}{\partial
z^{2}}.$ (S6)

We can use the Taylor expansion to approximate the saturated gain $g(I)$ in
the vicinity of the stationary intensity
$I_{0}=I_{s}(\frac{g_{0}}{\alpha_{w}}-1)$

$\displaystyle g(I)=\frac{g_{0}}{1+\frac{I}{I_{s}}}\approx
g(I_{0})+\frac{\partial g(I)}{\partial I}\big{|}_{I=I_{0}}(I-I_{0})\approx
g_{1}-g_{2}\frac{I}{I_{s}},$ (S7)

where $g_{1}=\frac{\alpha_{w}}{g_{0}}(2g_{0}-\alpha_{w})$ and
$g_{2}=\frac{\alpha_{w}^{2}}{g_{0}}$. Inserting relation (S7) into equation
(S6) and neglecting the term proportional to
$\sim|E|^{2}\partial^{2}E/\partial z^{2}$ results in

$\displaystyle\begin{split}\frac{\partial E}{\partial
t}=(\eta+i\omega_{s})E+(d_{R}+id_{I})\frac{\partial^{2}E}{\partial
z^{2}}-(n_{R}+in_{I})|E|^{2}E,\end{split}$ (S8)

where the following functions have been introduced

$\displaystyle\begin{split}&\eta=\frac{g_{1}-\alpha_{w}}{2}\frac{c}{n_{r}},\>\>\>\>\>\>\>\>\>\>\>\omega_{s}=\frac{g_{1}{\alpha}}{2}\frac{c}{n_{r}},\\\
&d_{R}=\frac{g_{1}T_{2}^{2}}{2(1+{\alpha}^{2})}\frac{c^{3}}{n_{r}^{3}},\>\>\>\>\>\>d_{I}=\Big{(}\frac{k^{{}^{\prime\prime}}}{2}-\frac{{\alpha}g_{1}T_{2}^{2}}{2(1+{\alpha}^{2})}\Big{)}\frac{c^{3}}{n_{r}^{3}}\\\
&n_{R}=\frac{g_{2}}{2I_{s}}\frac{c}{n_{r}},\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>n_{I}=-\Big{(}\beta-\frac{{\alpha}g_{2}}{2I_{s}}\Big{)}\frac{c}{n_{r}}.\end{split}$
(S9)

Here, the laser net gain is described with the coefficient $\eta$, and
$\omega_{s}$ represents the frequency shift due to the gain asymmetry
quantified by ${\alpha}$ (value of LEF at the gain peak). The complex
diffusion coefficient, given with $d_{R}+id_{I}$, dampens variations of the
field $E$. Its complex value can be easily understood as it will work towards
smoothing any spatial gradient of both the amplitude and phase. Physically, it
emerges from the curvature of the gain (due to its finite bandwidth),
asymmetric gainshape, and the GVD. Lastly, the nonlinearity is described with
$n_{R}+in_{I}$. The real part $n_{R}$ arises from the gain saturation in the
laser and dampens amplitude fluctuations. The imaginary part $n_{I}$ describes
the phase modulation due to the bulk Kerr nonlinearity and a finite LEF.

Equation (S9) can be written in a more elegant way. To do so, the temporal,
spatial variable, and the electric field need to be rescaled
$t\rightarrow\frac{t}{\eta}$,
$z\rightarrow\Big{(}\frac{d_{R}}{\eta}\Big{)}^{1/2}z$, and
$E\rightarrow\Big{(}\frac{\eta}{n_{R}}\Big{)}^{1/2}e^{i\omega_{s}t/\eta}E$ .
Moreover, the dispersive and nonlinear parameters are introduced as
$c_{\mathrm{D}}$ and $c_{\mathrm{NL}}$ in the following way

$\displaystyle\begin{split}&c_{\mathrm{D}}=-{\alpha}+k^{{}^{\prime\prime}}\frac{g_{0}(1+{\alpha}^{2})}{\alpha_{w}(2g_{0}-\alpha_{w})T_{2}^{2}},\\\
&c_{\mathrm{NL}}={\alpha}-\beta\frac{2I_{s}g_{0}}{\alpha_{w}^{2}}.\end{split}$
(S10)

From the previous relation, it is clear how the cavity dispersion
$k^{{}^{\prime\prime}}$ influences $c_{\mathrm{D}}$ and the bulk Kerr
nonlinearity coefficient $\beta$ influences $c_{\mathrm{NL}}$, while the LEF
influences both of them, providing a physical contribution to both the ive and
nonlinear effects. Equation (S10) provides also a straightforward way to
quantify gain dispersion (see subsection A.1) as

$\displaystyle\mathrm{GVD}_{\mathrm{gain}}=-\frac{\alpha}{1+\alpha^{2}}\frac{(2g_{0}-\alpha_{w})\alpha_{w}T_{2}^{2}}{g_{0}},$
(S11)

and also a giant resonant Kerr nonlinearity that arises from the gain,
phenomenologically described with LEF as

$\displaystyle{\beta}_{\mathrm{gain}}=\alpha\frac{\alpha_{w}^{2}}{2g_{0}I_{s}}.$
(S12)

Implementing relation (S10) into equation (S9) allows us to obtain the
conventional form of the CGLE often found in literature [19]

$\displaystyle\frac{\partial E}{\partial
t}=E+(1+ic_{\mathrm{D}})\frac{\partial^{2}E}{\partial
z^{2}}-(1+ic_{\mathrm{NL}})|E|^{2}E.$ (S13)

In comparison with the laser master equation (S3), the entire parameter space
of CGLE has been reduced to just two dimensions defined by the $c_{D}-c_{NL}$
plane. The CGLE represents one of the most known nonlinear equations in
physics. It generally describes the dynamics of oscillating, spatially
extended systems going through phase transitions – covering a vast amount of
nonlinear phenomena on a qualitative, and often on a quantitative level.

##  Supplementary material C SWIFTS measurement and experimental
characterization of the NB soliton

Here we explain briefly the working principle of the Shifted-Wave Interference
Fourier Transform Spectroscopy (SWIFTS) [50], which is used to experimentally
characterize NB solitons. SWIFTS uses a linear optical intensity detector with
a large electrical bandwidth to detect the phase and amplitude of the
intermode beatings of the frequency comb equidistant modes. The optical
frequency and phase of a comb mode $n$ is given by
$f_{n}=nf_{\mathrm{rep}}+f_{\mathrm{ceo}}$ and $\phi_{n}$, respectively
($f_{\mathrm{ceo}}$ is the carrier envelope offset frequency, and
$f_{\mathrm{rep}}$ is the repetition frequency). Each pair of adjacent modes
beats at their difference frequency, which is equal to $f_{\mathrm{rep}}$. In
order to stabilize $f_{\mathrm{rep}}$ against fluctuations or drifting, we
employ electrical injection locking to an external radio-frequency (RF)
source. The phase of the intermode beating is equal to the phase difference of
the adjacent modes $\phi_{n}-\phi_{n-1}$. The temporal waveform of the
frequency comb can be reconstructed by extracting its intensity spectrum and
the phases of all intermode beatings. In order to measure the phases of the
individual intermode beatings, a method to isolate pairs of adjacent comb
lines, i.e. a frequency discriminator, is required. SWIFTS implements this
function in the following way: the light emitted by the frequency comb passes
through a Fourier transform infrared (FTIR) spectrometer as the delay arm is
swept and is subsequently detected by a fast quantum well infrared
photodetector (QWIP). Here, the FTIR spectrometer acts as a frequency
discriminator in the spectral domain. The beatnote signal at the repetition
frequency, detected by the QWIP after the FTIR spectrometer, is downmixed to
about 10$\,\mathrm{MHz}$ with a local oscillator (LO). The quadratures of the
downmixed beatnote signal, $X$ and $Y$, are recorded by a lock-in amplifier as
function of the FTIR interferometer path difference. The reference signal for
the lock-in amplifier is provided by the RF source, to which the comb
repetition rate is locked. This allows to record the so-called SWIFTS
interferograms for both quadratures. The DC signal of the QWIP yields the
intensity interferogram from which the intensity spectrum of the laser output
is obtained via Fourier transform. A Fourier transform of the SWIFTS
interferograms yields the complex SWIFTS spectrum, where the amplitude of the
mode $S_{n}$ is equal to the geometric average of the amplitudes of adjacent
modes in the intensity spectrum $\sqrt{I_{n}I_{n-1}}$, provided that the
characterized state is a coherent frequency comb. If phase noise is present in
the part of the spectrum, or the modes are not equidistant, then that part of
the SWIFTS spectrum will be substantially weaker. As a conclusion, comparing
the envelopes of the SWIFTS and intensity spectra allows to assess the phase
coherence of the multimode field and benchmark frequency comb operation.

Figure S5:  Experimental characterization of a NB soliton. a) Intensity and
quadrature $X$ and $Y$ SWIFTS interferograms. b) Intensity spectrum (top),
SWIFTS spectrum (middle), and the corresponding intermode phases (bottom). The
agreement between the SWIFTS modal amplitudes and the geometric average of
neighboring intensity modes (black markers) indicates that the characterized
state is coherent and that the laser emits a frequency comb. Intermodal phases
show that all of the intermode beatings are synchronized in-phase, except for
the two beatings between the primary mode and its neighboring sidemodes,
meaning that there is a $\pi$ shift in the phase of the primary mode. c) The
reconstructed temporal profiles of the intensity (top), instantaneous
wavenumber (middle), and the phase (bottom). The instantaneous wavenumber
changes only within the region of the dark pulse, where the phase exhibits a
steep ramp covering the value of $2\pi$. The phase is perfectly linear
elsewhere, which corresponds to a constant instantaneous wavevector of around
$1233.5\,\mathrm{cm}^{-1}$.

Fig. S5 displays in detail the measured SWIFTS data and the characterization
of the NB soliton from Fig. 2 of the main manuscript (obtained at bias
currents of $J_{\mathrm{R}}=$1.33$\,\mathrm{kA/cm}^{2}$ and
$J_{\mathrm{WG}}=$0.79$\,\mathrm{kA/cm}^{2}$). The intensity interferogram and
both of SWIFTS quadrature interferograms can be seen in Fig. S5a) as the delay
arm of the FTIR spectrometer is swept. The intensity interferogram has a
unique shape characteristic to pulsed laser emission where a strong CW
background is present. This can be recognized from the interferogram envelope
that exhibits pulse-like shapes superimposed on a constant background. The
pulses occur approximately every 2.25$\,\mathrm{cm}$ in the path delay, which
corresponds to the distance that the light traverses in air during one laser
cavity roundtrip. The information about the spectral intermode phases is
contained in the $X$ and $Y$ SWIFTS interferograms. Both of them have minima
at the zero path delay (where the intensity interferogram exhibits pulses),
strongly indicating that not all of the intermode beatings are synchronized
in-phase. We show the spectral behavior of the NB soliton in Fig. S5b). The
intensity spectrum (top) displays a strong primary mode surrounded by a
multitutde of weaker sidemodes that form a smooth bell-shaped envelope, akin
to typical dissipative soliton spectra found in passive microresonators or
fibers [5, 6, 8, 9, 10, 11, 7, 4, 3]. The modes of the SWIFTS spectrum $S_{n}$
(middle) correspond to the intermode beatings between adjacent optical modes
$n$ and $n-1$. The markers indicate the geometric average of the intensity of
these neighboring modes, mathematically written as $\sqrt{I_{n}I_{n-1}}$. From
the figure, we can see that the SWIFTS modal amplitudes match the values given
by the markers, thus proving coherent frequency comb operation. The phases of
the complex modes in the SWIFTS spectrum correspond to the intermode phases
between adjacent modes of the intensity spectrum ($\phi_{n}-\phi_{n-1}$). The
intermode phase distribution, shown in the bottom of Fig. S5b), indicates that
all of the intermode beatings are synchronized in-phase, except for the two
beatings between the strong primary mode and its neighbors, which are
$\pi$-shifted. If translated to the modal phases $\phi_{n}$, the phases of the
sidemodes are approximately identical (in the general case, an arbitrary
linear distribution), with the exception of the primary mode, which is $\pi$
out of phase. Intuitively, this means that the sidemodes interfere
constructively in a localized portion of the cavity roundtrip, while the
strong primary mode destructively interferes with them. The knowledge of the
intermode phases allows us to extract the phases of the individual modes
$\phi_{n}$ via a cumulative sum. Together with the modal amplitudes, this
enables the temporal reconstruction of the electric field as

$\displaystyle E(t)=\sum_{n}\sqrt{I_{n}}\mathrm{cos}(2\pi f_{n}t+\phi_{n}).$
(S14)

We show the temporal intensity of the NB soliton in the top of Fig. S5c). It
demonstrates a clear dark pulse, which is a consequence of the above-mentioned
destructive interference. The pulse is surrounded by a quasi-constant
intensity which exhibits residual smaller oscillations. These arise due to a
finite number of elements in the Fourier sum in equation (S14), which is a
consequence of a limited dynamic range of detection of the QWIP. Examining the
plotted instantaneous frequency of the output (proportional to the
instantaneous wavenumber) is beneficial for understanding the salient
properties of these waveforms. It is observed that the frequency is swept only
within the region of the dark pulse, indicating that all of the optical modes
contribute only within this portion of the roundtrip. The instantaneous
frequency is constant everywhere else and equal to the optical frequency of
the primary mode (wavenumber $\sim 1233.5\,\mathrm{cm}^{-1}$). This means that
the constant portion of the intensity is indeed comprised of a single-
frequency CW field – indicating the localized soliton nature of the pulse.
From the point of view of the one-dimensional CGLE, the dark pulses can be
viewed as sources that connect two plane waves with identical wavenumbers,
consistent with periodic ring boundary conditions [19, 40, 41]. These
structures are found to be dynamically stable solutions of the CGLE in a
finite, but narrow region of the $c_{\mathrm{D}}$–$c_{\mathrm{NL}}$ parameter
space. In the case when higher-order terms are included, e.g. in a quintic
CGLE with a fifth-order nonlinearity, a co-propagating hole-shock solutions
still persist as stable structures in the waveform under ring boundary
conditions [46, 47].

The temporal phase profile in Fig. S5c) exhibits a steep ramp that covers the
value of $2\pi$ within the width of the pulse, as discussed in the main
manuscript. Together with the $\pi$ jumps of the intermode phases around the
primary mode, this represents a ’fingerprint’ trait of NB solitons that can be
used for their unequivocal classification.

###  C.1 Spectrally shifted NB solitons

Here we present experimental states with the NB soliton spectral envelope
shifted relative to the position of the primary mode, as reported in Fig. 4a)
from the main manuscript.

Figure S6:  Shifting of the soliton spectral envelope relative to the primary
mode. Experimental characterization of two NB solitons where the tuning of the
bias current results in a shift of the spectral soliton envelope from the red
to the blue side of the primary mode (a) and b) respectively). The intermode
phases display the expected two $\pi$ jumps around the primary mode. The
temporal phase profile sweeps over $2\pi$ during the soliton width. However,
the direction of the sweep depends where the soliton envelope is positioned
relative to the primary mode.

The shift of the soliton spectral envelope in Fig. S6 happens as the currents
of the ring and the waveguide are changed. The main reason for this likely
lies in the large change of the total GVD, as discussed in the main
manuscript, and recently observed experimentally in passive microresonators
[11]. Although the soliton envelope may be positioned differently relative to
the primary mode, the expected two $\pi$ jumps of the intermode phases around
the primary mode are still present – indicating that this is indeed a salient
feature of NB solitons.

The temporal profile of the phase exhibits the familiar $2\pi$ ramp within the
width of the soliton. We can observe that the direction of the ramp depends on
whether the soliton spectral envelope is on the red or on the blue side
relative to the primary mode. In a hypothetical state where the soliton
envelope would be perfectly symmetric relative to the primary mode (if the
soliton spectral center of mass coincides with the position of the primary
mode), the $2\pi$ phase ramp would comprise two separate $\pi$ ramps with an
opposite direction. However, this state does not represent a stable fixed
point, and is likely never to occur experimentally.

##  Supplementary material D Experimental evidence of an NB soliton crystal

In the Fig. 3 of the main manuscript, we have shown an experimental and
theoretical characterization of a multisoliton state comprised of two co-
propagating NB solitons in a single roundtrip. A special case of
multisolitonic states, where all of the solitons within one roundtrip are
equidistant, is called a soliton crystal [49]. In the frequency domain, these
waveforms correspond to a harmonic frequency comb whose spacing between
adjacent comb modes is equal to an integer multiple of the free spectral range
(FSR): $\mathrm{N}\>\times\>\mathrm{FSR}$, where $\mathrm{N}$ is the number of
solitons in the soliton crystal.

Figure S7:  Experimental evidence suggesting an NB soliton crystal. Intensity
spectrum of a probable fifth harmonic frequency comb, where the intermode
spacing equals 5 FSRs. The soliton crystal regime is suggested by the smooth
bell-shaped envelope of the spectrum.

A ring resonator employing a laser active region can provide a platform for
studying soliton crystal dynamics as well. In Fig. S7, we report a state with
an intensity spectrum whose intermode spacing is equal to 5 FSRs, potentially
corresponding to five equidistant co-propagating NB solitons inside the ring
cavity. The coherence of the state is suggested by the high suppression ratio
of the fundamental modes that fall beneath the noise floor, leaving only the
harmonic equidistant modes. Furthermore, the modes form a smooth bell-shaped
spectral envelope that indicates the soliton nature of the state. The high
frequency of the intermode beatnote (around 68$\,\mathrm{GHz}$) lies well
above the cutoff frequency of our optical detector, thus prohibiting SWIFTS
characterization to truly assess the coherence of the state. This begs for the
future use of another coherent technique to study soliton crystal dynamics in
active ring resonators.

##  Supplementary material E Impact of strong radio-frequency injection

In order to stabilize the fundamental frequency comb repetition frequency
(around 13.6$\,\mathrm{GHz}$) and perform the SWIFTS measurement, we employ
electrical injection locking via an RF source. To do so, injected power below
0$\,\mathrm{dBm}$ is sufficient for our devices. Going beyond that level and
injecting stronger power, although beneficial for locking the comb repetition
frequency to the external source, perturbs the salient characteristics of
free-running NB solitons. For that reason, we avoid using large RF injection
powers.

Figure S8:  Influence of strong RF injection on the NB soliton. Measured a)
intensity and SWIFTS interferograms, and b) the intensity spectrum, SWIFTS
spectrum, and the intermode phases. The intensity interferogram exhibits a
modulation of the envelope due to the strong RF injection. The intermode
phases are affected and the two $\pi$ jumps around the primary mode cannot be
seen.

The influence of strong RF injection on an NB soliton can be seen in Fig. S8.
The first indication that the electrical injection perturbs the free-running
soliton can be observed from the intensity interferogram. Unlike the flat
interferogram envelope in Fig. S5, we can see that strong RF injection causes
approximately sinusoidal modulation of the interferogram envelope due to the
fact that the strong primary mode develops modulation sidebands. Furthermore,
the intermode phase distribution is affected around the primary mode, and the
’fingerprint’ two $\pi$ jumps are not present anymore.

##  Supplementary material F Homoclons in the theory and experiment

In this section, we present theoretical and experimental results of homoclons
in our lasers, with the aim of distinguishing these states from NB solitons.
Homoclons represent stable frequency combs, characterized by the formation of
localized patterns in the waveform [19]. They arise due to phase turbulence
and have been observed in Ref. [20], which triggered the subsequent research
of ring QCL comb dynamics. They have been observed in ring QCLs, where they
arise due to phase turbulence [20].

Figure S9:  Simulated homoclon states. a) \- d) Exemplary homoclons obtained
with numerical simulations of the master equation (S3). Homoclons are
characterized as localized states that comprise several pulse-like features in
the temporal intensity. The number of these features depends on the
parameters, but also on the starting conditions of the laser. The intermode
phases are distributed in two clusters, separated by $\pi$. The case of one
pulse-like feature in column d), cannot be mistaken with an NB soliton, as the
latter always has two $\pi$ jumps around the primary mode.

Fig. S9 displays four different homoclon states obtained from numerical
simulations of the master equation. The temporal intensity profile exhibits
localized patterns that comprise different number of pulse-like structures.
Their number depends on the laser parameters, but also on the laser starting
condition (phenomenon of multistability). Typical values of LEF that are used
to obtain homoclons are in the range of 1.1-1.3. The cavity dispersion
$k^{{}^{\prime\prime}}$ falls between -1000 and
-1500$\,\mathrm{fs}^{2}/\mathrm{mm}$, which puts the value of the total GVD
between -2500 and -3000$\,\mathrm{fs}^{2}/\mathrm{mm}$, once the gain
dispersion is included (defined by LEF). In sharp contrast to this, NB
solitons are typically obtained for $k^{{}^{\prime\prime}}$ values between 500
and 1200$\,\mathrm{fs}^{2}/\mathrm{mm}$ – meaning that homoclons and NB
solitons are found in different regions of the laser parameter space. In fact,
NB solitons coexist with a CW single-mode operation inside the CW linearly-
stable region of the parameter space, while homoclons can appear only inside
the CW linearly-unstable region where turbulence destabilizes the single-mode
field.

Homoclons can be classified easily by observing their intermode phase
distribution. As is seen from Fig. S9, the intermode phases form two clusters
separated by $\pi$. In the extreme case where the homoclon exhibits only one
pulse-like feature per roundtrip (Fig S9d)), there is a single $\pi$ jump
located at the position of the strongest mode. NB solitons, on the other hand,
always exhibit two $\pi$ jumps in the intermode phases around the primary
mode. Furthermore, the spectral bandwidth of homoclons is narrower compared to
NB soliotns, which is why the amplitude variations in their temporal waveform
are shallow.

Figure S10:  Experimental homoclon state. The intensity spectrum is shown on
top. The corresponding intermode phases are grouped in two clusters separated
by $\pi$.

Experimental evidence of a homoclon is shown in Fig. S10. The state is
obtained for bias currents of $J_{\mathrm{R}}=$1.39$\,\mathrm{kA/cm}^{2}$ and
$J_{\mathrm{WG}}=$0.86$\,\mathrm{kA/cm}^{2}$, from the same device which emits
the NB solitons shown in the main manuscript. This indicates that bias tuning
of our devices provides a powerful knob that controls the GVD over a large
range of values, enabled by the separate electrical contacts of the ring and
the waveguide coupler.

##  Supplementary material G Coupled mode theory for laser dispersion analysis

The experimental verification of homoclon states (section F) and also the
spectral shift of NB solitons (section (C.1) provide two persuasive arguments
that we are able to tune the GVD over a vast range of values purely by
changing the ring and/or waveguide current. In this section, we will
corroborate this hypothesis with theoretical evidence as well.

Mode interactions between the ring and waveguide are well described by the
coupled mode theory (CMT). Using CMT, we can estimate how much the electric
field leaks from one resonator (the ring) to the other resonator (the
waveguide) when the two are in close proximity to one another. For devices
reported in this work, the ring and waveguide are separated by a narrow air
gap ($\sim 0.8\,\upmu$m) across an interaction length $L_{\text{int}}$ ($\sim
1500\,\upmu$m). Bearing in mind that all of the light is generated in the
ring, we write mathematical expressions for the electric field in the ring
($E_{\text{R}}$), and the electric field localized in the waveguide
($E_{\text{WG}}$), after a propagation length $z$ [48]

$\displaystyle E_{\text{R}}(z)$ $\displaystyle=E_{0}\left[\cos{\left(\psi
z\right)}+\frac{i\delta}{\psi}\sin{\left(\psi
z\right)}\right]\exp{\left(-i\delta z\right)}$ (S15) $\displaystyle
E_{\text{WG}}(z)$ $\displaystyle=-E_{0}\frac{i\kappa}{\psi}\sin{\left(\psi
z\right)}\exp{\left(i\delta z\right)},$ (S16)

where $E_{0}$ is the initial electric field amplitude, $\delta$ is the
difference of the propagation constants in the ring and the waveguide,
$\kappa$ is the difference in propagation constants of the two even/odd super-
modes formed in the coupled resonators, and
$\psi=\sqrt{\kappa^{2}+\delta^{2}}$. $E_{\text{R}}(L_{\text{int}})$ is then
used to calculate the group delay dispersion (GDD) of the ring as function of
frequency $\omega$, which is given by

$\displaystyle\text{GDD}_{\text{R}}(\omega)=\frac{\partial^{2}\phi_{\text{R}}}{\partial\omega^{2}},$
(S17)

where $\phi_{\text{R}}$ is phase angle of $E_{\text{R}}(L_{\text{int}})$.

The propagation constants of the ring and waveguide drift from one another
when the two sections are biased at different pumping levels. By measuring the
transmission of a weak probe laser through the ring-waveguide system while
varying their pumping below threshold and fitting the resultant signal to a
transfer matrix model, we extract how the refractive indices (and therefore,
propagation constants) change as a function of injection current. We estimate
the change in the refractive index as a function of pumping to be of the order
$\sim 0.01\ \text{cm}^{2}\ \text{kA}^{-1}$ [48], corresponding to $\delta
n\sim 0.001-0.010$ for practical pumping levels and typical device geometries.
Furthermore, we use a commercial eigenmode solver to calculate the even and
odd super-mode indices for a typical laser cross-section, estimating their
difference to be $\sim 0.007$ for an air gap of $0.8$ $\mu$m.

Fig. S11(a) shows a calculated two-dimensional heat-map for
$\text{GDD}_{\text{R}}$ as a function of both frequency and refractive index
difference, spanning from 1200 $\text{cm}^{-1}$ to 1250 $\text{cm}^{-1}$ in
frequency, and 0.000 to 0.020 in index difference. Fig.S11(b) shows a cut of
the heat-map at 1235 $\text{cm}^{-1}$ – the center of the laser gain
bandwidth, for different values of super-mode index differences (0.006, 0.007,
and 0.008). Note that regardless of the super-mode index difference, the
dispersion changes from normal (negative GDD) to anomalous (positive GDD) as
$\delta n$ is swept. Therefore, slight pumping differences between the ring
and waveguide can have drastic effects on device dispersion. As a conclusion,
our cavity geometry, comprising a coupled ring-waveguide system with separate
electrical contacts, provides a powerful knob to control the total mode
dispersion – far superior compared to a single ring or waveguide cavity.

Figure S11:  Coupled mode theory analysis of the dispersion. a) A two-
dimensional heat-map of the GDD in the ring as a function of frequency
$\omega$ and refractive index difference $\delta n$
($n_{\text{e}}-n_{\text{o}}=0.007$). b) A cut of the heat-map at $\omega=1235\
\text{cm}^{-1}$ for different values of $n_{\text{e}}-n_{\text{o}}$ (0.006,
0.007, and 0.008).

##  Supplementary material H Dark- and bright-pulse NB solitons

Figure S12:  Experimental evidence of a coherent bright pulse. Detailed
experimental characterization of a state from Fig. 4e) in the main manuscript,
obtained at bias currents of $J_{\mathrm{R}}=$1.39$\,\mathrm{kA/cm}^{2}$ and
$J_{\mathrm{WG}}=$0.85$\,\mathrm{kA/cm}^{2}$.

The detailed experimental characterization of the state from Fig. 4e) in the
main manuscript can be seen in Fig. S12. As was explained in the main
manuscript, the obtained state looks similar to the dark NB soliton state from
Fig. S5 (Fig. 2 of the main manuscript), with the temporal intensity being the
exception as it strikingly shows a formation of a bright coherent pulse. It
was argued that this is due to the different intensity of the soliton
sidemodes relative to the primary mode, where this ratio is higher in the case
of a bright pulse than in the case of the dark pulse. Here we will elaborate
more on this.

Figure S13:  Theoretical analysis of the dependence of the intensity waveform
on the evolution of the soliton spectrum. The primary mode is positioned a) at
the center of the soliton spectral envelope, b) away from the center of the
soliton spectral envelope. We define the soliton sidemodes using a
$\text{sech}^{2}$ distribution, whose choice is purely arbitrary and only
serves to illustrate a trend. The intermode phases are assumed to have $\pi$
jumps around the primary mode and be identical everywhere else. Blue line
represents the intensity waveform reconstructed from the modes that are
visible in the corresponding displayed spectrum, while the grey line is
reconstructed using the weaker modes as well. The blue line exhibits residual
oscillations, just like experimentally-recorded intensity waveforms, which is
a consequence of the limited number of terms in the Fourier series used for
the temporal waveform reconstruction in equation (S14). The behavior in both
cases is similar, except that for a symmetric spectrum, the intensity reaches
zero value at certain point where the dark pulse effectively ’flips over’ and
becomes bright.

The theoretical analysis from Fig. 4f) in the main manuscript is displayed in
detail in Fig. S13. Two cases are investigated, when the primary mode is in
the center of the spectral soliton envelope, and shifted from it, which
corresponds more to the experimental spectra. Both cases agree with the
analysis from the main manuscript. Initial weak soliton sidemodes correspond
to a shallow dark pulse in the intensity waveform, whose contrast grows if we
increase the sidemode amplitudes relative to the primary mode. At one point
(second row in Fig. S13), the destructive interference between the primary
mode, which defines the CW background, and the sidemodes is complete,
resulting in zero intensity at the pulse minima. Further increase of the
sidemode amplitudes results in the spectral soliton envelope having a larger
intensity compared to the primary mode, at which point the dark pulse would
effectively ’flip over’. This is especially evident from the evolution in Fig.
S13a), where the intensity reaches zero before the pulse ’flips over’. Further
increase of the sidemode amplitudes decreases the amplitude contrast of the
dark pulse (third row), until a quasi-constant intenisty is obtained (fourth
row). Additionally amplifying the sidemodes above this point results in the
potential emission of high-contrast bright pulses.

mid-IR

    mid-infrared
QWIP

    quantum well infrared photodetector
FSR

    free spectral range
LO

    local oscillator
FTIR

    Fourier transform infrared
SWIFTS

    Shifted-Wave Interference Fourier Transform Spectroscopy
RF

    radio-frequency
NB

    Nozaki-Bekki
CW

    continuous-wave
QCL

    quantum cascade laser
FP

    Fabry-Pérot
SHB

    spatial hole burning
CGLE

    complex Ginzburg-Landau equation
GVD

    group velocity dispersion
LEF

    linewidth enhancement factor
R

    ring
WG

    waveguide
GDD

    group delay dispersion
CMT

    coupled mode theory
  *[CW]: continuous-wave
  *[NB]: Nozaki-Bekki
  *[mid-IR]: mid-infrared
  *[CGLE]: complex Ginzburg-Landau equation
  *[QCLs]: quantum cascade laser
  *[FP]: Fabry-Pérot
  *[SHB]: spatial hole burning
  *[GVD]: group velocity dispersion
  *[LEF]: linewidth enhancement factor
  *[QCL]: quantum cascade laser
  *[SWIFTS]: Shifted-Wave Interference Fourier Transform
Spectroscopy

  *[RF]: radio-frequency
  *[FTIR]: Fourier transform infrared
  *[QWIP]: quantum well infrared photodetector
  *[LO]: local oscillator
  *[FSR]: free spectral range
  *[FSRs]: free spectral range
  *[CMT]: coupled mode theory
  *[GDD]: group delay dispersion