# Zero-dispersion Kerr solitons in optical microresonators

Miles H. Anderson  Institute of Physics, Swiss Federal Institute of Technology
(EPFL), Lausanne, Switzerland  Grigory Lihachev  Institute of Physics, Swiss
Federal Institute of Technology (EPFL), Lausanne, Switzerland  Wenle Weng
Institute of Physics, Swiss Federal Institute of Technology (EPFL), Lausanne,
Switzerland  Junqiu Liu  Institute of Physics, Swiss Federal Institute of
Technology (EPFL), Lausanne, Switzerland  Tobias J. Kippenberg
<EMAIL_ADDRESS>Institute of Physics, Swiss Federal Institute of
Technology (EPFL), Lausanne, Switzerland

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED Figure 1:  Localized Dissipative Structures in the
second-/third-order dispersion ($\mathbf{d_{2}/d_{3}}$) plane.  Clockwise from
left: conventional dissipative solitons, dissipative solitons with dispersive-
wave tails, zero-dispersion solitons (orange area), switching waves with
dispersive wave tails, conventional switching wave. Dashed gray line in
figures represent the CW high-state solution. Video available as supplementary
info. Figure 2:  Excitation of switching-waves inside pulse-drive envelope.
(a) Principle of pulse-driven Kerr cavity dissipative structure formation,
with image of resonator with coupling section. (b-e) SW formation in pure
normal dispersion. (b) Contour of the bistable intracavity CW solutions,
plotted for increasing local value of the pump $F(\tau)$, with stable
(unstable) solution in solid (dashed) line. (c) Development of intracavity
field (red-yellow) within the pulse envelope (blue dashed) with increasing
detuning $\zeta_{0}$. Red-dotted lines connect one field slice with the CW
solution distribution in (b). Maxwell points on the pulse envelope marked with
circles. (d) Expanding (top) and contracting (bottom) high-state (dashed)
under CW driving. (e) Spectra of the fields from (c), and integrated
dispersion operator. (f-g) SW formation with strong third-order dispersion.
(f) Intracavity field within the pulse envelope with increasing detuning
$\zeta_{0}$. (g) Spectra of the fields from (f).

Solitons are shape preserving waveforms that are ubiquitous across many
nonlinear dynamical systems Akhmediev and Ankiewicz (2008), and fall into two
separate classes, that of bright solitons, formed in the anomalous group
velocity dispersion regime, and ‘dark solitons’Kivshar and Luther-Davies
(1998); Godey  _et al._ (2014) in the normal dispersion regime. Both types of
soliton have been observed in BEC Becker  _et al._ (2008),
hydrodynamicsChabchoub  _et al._ (2016), polaritonsAmo  _et al._ (2011), and
mode locked lasersGrelu and Akhmediev (2012); Liu  _et al._ (2015), but have
been particularly relevant to the generation of microresonator-based frequency
combs, where they have unlocked chipscale microcombs used in numerous system
level applications in timing, spectroscopy, and communicationsKippenberg  _et
al._ (2018). For microcombs, both bright dissipative solitons, and
alternatively dark pulses based on interlocking switching waves under normal
dispersion, have been studied. Yet, the existence of localized dissipative
structures that fit between this dichotomy Wai  _et al._ (1987) has been
theoretically predicted Parra-Rivas  _et al._ (2017), but proven
experimentally elusive. Here we report the discovery of dissipative structures
that embody a hybrid between switching waves and dissipative solitons,
existing in the regime of (nearly) vanishing group velocity dispersion where
third-order dispersion is dominant, hence termed as ‘zero-dispersion
solitons’. These zero dispersion solitons are formed through collapsing
perturbed switching wave fronts Lobanov  _et al._ (2019), forming clusters of
quantized solitonic sub-structures, which we synthesize in discrete numbers.
The switching waves are formed directly via synchronous pulse-drivingObrzud
_et al._ (2017) of a photonic chip-based Si3N4 microresonator with vanishing
normal dispersion. The resulting frequency comb spectra is extremely broad in
both the switching wave and zero-dispersion soliton regime, reaching 136 THz
or 97% of an octave. Fourth-order dispersion engineering results in dual-
dispersive formation, and a novel quasi-phase matched dispersive wave related
to Faraday instabilityCopie  _et al._ (2016). This exotic, unanticipated
dissipative structure expands the domain of Kerr cavity physics to the regime
near to zero-dispersion, and could present a superior alternative to
conventional bright solitons for broadband comb generation, but equally may
find observation in other fields.

##  I Introduction

Currently, the field of research in optically driven Kerr nonlinear resonators
and dissipative structure formation has been largely focused on the paradigm
of the bright dissipative soliton Kippenberg  _et al._ (2018); Herr  _et al._
(2014); Leo  _et al._ (2010). Bright dissipative solitons (DS) can be thought
of as a particular variety of localized dissipative structure (LDS), solitary
pulses that retain their shape due to the counter-balance between anomalous
dispersion and nonlinearity, and who have a fixed amplitude determined by the
driven-dissipative parameters of the Kerr cavity environment Akhmediev and
Ankiewicz (2008). Bright DS have been widely studied experimentally in
multiple material platforms Kippenberg  _et al._ (2018), and have been
demonstrated as a desirable candidate for numerous integrated frequency comb-
based applications such as massively parallel telecommunications Marin-Palomo
_et al._ (2017) and LiDAR Riemensberger  _et al._ (2019), astro-spectrometer
calibration Ewelina Obrzud  _et al._ (2019); Suh  _et al._ (2019), dual-comb
spectroscopy Suh  _et al._ (2016), and also for metrology enabled by self-
referencing such as absolute frequency synthesis Spencer  _et al._ (2018) and
towards optical atomic clocks Newman  _et al._ (2019). Across optical physics,
DS have been observed in nonlinear systems such as mode-locked lasers, and
transverse laser cavities Lugiato and Lefever (1987); Ackemann and Firth
(2005); Akhmediev and Ankiewicz (2008), and more widely as basic structures in
nonlinear dynamical systems as diverse as plasma physics, neuron propagation,
and chemical reaction systems Nozaki and Bekki (1985); Gonzalez-Perez  _et
al._ (2016); Liehr (2013).

In opposition to bright DS have been dark dissipative structures, commonly
called ‘dark pulses’, which conversely exist in Kerr cavities possessing
_normal_ dispersion Godey  _et al._ (2014); Liang  _et al._ (2014); Xue  _et
al._ (2015); Huang  _et al._ (2015). These dark pulses (alternatively termed
as ‘platicons’ Lobanov  _et al._ (2015)) are in fact formed by the
interlocking of two separate _switching waves_ (SW), connecting the high and
low stable states of the bistable Kerr cavity Parra-Rivas  _et al._ (2016).
Compared to bright dissipative solitons, they have been found to possess an
intrinsically higher optical conversion efficiency between the input pump and
the generated light as normal dispersion allows more comb lines far from the
pump to be on resonance Xue  _et al._ (2017). As such, they have been proposed
as a superior alternative to bright DS for applications which require the
generation of strong comb lines near to the pump center, and have been
demonstrated as a source for massively parallel telecommunications Fülöp  _et
al._ (2018). Switching waves have been classed more generally as ‘domain
walls’ connecting two stable homogeneous states in driven dissipative systems,
seen in $\upchi^{(2)}$ optical parametric oscillators Rozanov  _et al._
(1982); Trillo  _et al._ (1997), semiconductor lasers Ganne  _et al._ (2001),
birefringent optical fibers Malomed (1994); Garbin  _et al._ (2020), and more
widely in hydrodynamic systems and more Pomeau (1986).

In this work, we report the experimental discovery of a third LDS, a _zero-
dispersion_ dissipative soliton (ZDS), which exists at the crossing point
between conventional dissipative solitons and switching waves in the presence
of vanishing second-order dispersion (SOD) giving way to pure- or dominant-
third order dispersion (TOD). It has remained an open question experimentally
whether such a physical link exists. In the simplified case of pure SOD, these
two cases of LDS are diametrically opposed. However, in realistic Kerr
resonators, in the limit of vanishing SOD, there always exists a TOD component
that begins to dominate. DS existing with a TOD component have been well
demonstrated, but only in the sense where TOD acts as an additional
perturbation to a regular DS, forming a dispersive wave tail that is far-
removed from the center of the soliton spectrum Jang  _et al._ (2014); Brasch
_et al._ (2016). On the SW side, only previous theoretical work has
investigated the case of strong TOD, and has predicted that in this regime,
bright solitary pulses, amounting to a self-interlocked SW, may exist Milián
and Skryabin (2014); Parra-Rivas  _et al._ (2017); Bao  _et al._ (2017);
Lobanov  _et al._ (2019). In this sense, as we depict in Fig. 1, it can be
seen that there exists a continuous physical link between conventional DS and
SW structures as a circular path of dispersion is followed in the SOD/TOD
plane. In this depiction (based on numerical simulations), conventional DS
become increasingly dominated by their dispersive wave tail, until when
crossing into the region of normal dispersion, where individual zero-
dispersion solitons are able to form packed clusters. When normal SOD begins
to dominate once more, this cluster becomes unlinked into two separate
switching waves perturbed by TOD, before reaching conventional pure-SOD
switching waves.

In this work, we focus on the formation of ZDS on the normal SOD side of this
plane. We find that this can be accomplished through first generating
switching waves efficiently via _wave-breaking_ using synchronous pulse-
driving of the Kerr cavityObrzud  _et al._ (2017) (Fig. 2(a)), initially
demonstrated in fiber cavities Coen  _et al._ (1999); Luo  _et al._ (2015),
and proposed for microresonators Lobanov  _et al._ (2019), and observing how
these SWs coalesce into newly identified single ZDS states at high pump-cavity
detuning.

##  II Theory

To analyze the optical structures introduced in this work in a simple and
universal fashion, we first consider an optical system described by the
dimensionless Lugiato-Lefever Equation (LLE) Coen and Erkintalo (2013);
Haelterman  _et al._ (1992), now with a non-CW driving term $f(\tau)$:

$\displaystyle\frac{\partial\psi}{\partial
t^{\prime}}=\left(-d_{1}\frac{\partial}{\partial\tau}-id_{2}\frac{\partial^{2}}{\partial\tau^{2}}+d_{3}\frac{\partial^{3}}{\partial\tau^{3}}\right)\psi$
$\displaystyle+(i|\psi|^{2}-i\zeta_{0}-1)\psi+\sqrt{F_{0}}f(\tau)$ (1)

Here, the form taken by the field solutions are determined solely by the
driving strength $F_{0}$ and detuning $\zeta_{0}$, as well as three parameters
$d_{l}$ describing the relative contributions of the first three orders of
dispersion Milián and Skryabin (2014). For simplicity, we set the SOD
parameter $d_{2}=1$ throughout this work, which corresponds to normal
dispersion. Thus, $d_{3}$ describes the contribution of TOD relative to
$d_{2}$, and the first-order dispersion $d_{1}$ corresponds to the offset in
group-velocity between the cavity field $\psi(\tau)$ and the static frame of
the pulse-driving term $f(\tau)$.

Firstly, it is necessary to investigate direct SW formation by pulse-driving
in the simplified case of pure SOD ($d_{1}=d_{3}=0$). We choose a value of
$F_{0}=10$ which is a typical operating point for practical dissipative
structure formation in experiment, and we set a Gaussian pulse as the driving
function $f(\tau)=\exp(-\tau^{2}/\tau_{p}^{2})$, with pulse duration
$\tau_{p}=100$ so as to ensure any SW is significantly shorter in duration
than the background driving function (which is true also in our experiment).
The detuning is swept linearly from some value $\zeta_{0}<0$ up to
$\zeta_{0}=10$. This range covers the region of Kerr cavity bistability, the
CW solutions of which we graph in Fig. 2(b) for different temporal samples of
the pulse-drive amplitude $F(\tau)=F_{0}f^{2}(\tau)$, over $-120\leq\tau\leq
0$. This effectively gives the _local_ CW solution of the intracavity field
over the length of the pulse-drive envelope, divided into the high-state and
low-state solutions $\psi_{H}$ and $\psi_{L}$, which can be solved
analytically (see Methods).

In Fig. 2(c) (with spectra in 2(e)) we show the intracavity field solutions
$\Psi=|\psi|^{2}$ at different values of $\zeta_{0}$ found using the split-
step method Coen  _et al._ (2013) (see Methods). For this direction in
$\zeta_{0}$, the field initially follows the high-state solution
$\psi_{H}(\tau)$ of the Kerr hysteresis. As $\zeta_{0}$ crosses 0, there begin
to exist parts of the intracavity field where the local Kerr resonance-shift
at the edges of the pulse-drive $F(\tau)$ is insufficient to sustain the high-
state $\psi_{H}(\tau)$. Here, the field outside this point falls to the low-
state $\psi_{L}(\tau)$ while the field further inside the pulse background
stays on $\psi_{H}(\tau)$ creating the SW that connects the two states Rozanov
_et al._ (1982).

Figure 3: Simulation: zero-dispersion soliton formation via desynchronized
pulse-driving. $F_{0}=10,d_{2}=1,d_{3}=1$ (a) Intracavity field as detuning
$\zeta_{0}$ is increased. $d_{1}=-1.34$ (b) Individual time-domain and (c)
frequency-domain of the fields in (a) (red arrows). (d) Alternative field
formation with (i) $d_{1}=-1.20$ (ii) $d_{1}=-1.29$ (iii) $d_{1}=-1.36$ (iv)
$d_{1}=-1.37$. (e) (left) Total intracavity energy from (a) (right) Zoom of
the ‘step’ feature, plotted for different values of desynchronization $d_{1}$.
The black line corresponds to the field given in (a).

From here, the two SW locations $\tau_{\mathrm{SW}}$ follow a location within
the pulse-drive envelope $F(\tau)=F_{m}$, which previous theoretical works on
SW stability have termed as the ‘Maxwell Point’ Ganne  _et al._ (2001); Parra-
Rivas  _et al._ (2016), until at $\zeta_{0}\approx 7$ where there exists no
$F(\tau)>F_{m}$ causing the SWs to meet each other and annihilate, failing to
reach their theoretical maximum detuning at $\zeta_{0}=F_{0}=10$. The
stability of the SW fronts within the pulse envelope after formation is due to
the effective ‘outward pressure’ manifesting on $\psi_{H}$. SWs possess an
innate group velocity offset depending on the value of $F$ and $\zeta_{0}$
Coen  _et al._ (1999), where the $\psi_{H}$ tends to undergo expansion, with
the SWs moving outward, when the driving term is larger than a certain value
$F(\tau)>F_{m}$ for a fixed detuning $\zeta_{0}$ (see Fig. 2(d)). When
$F(\tau)<F_{m}$, the $\psi_{H}$ contracts and the SWs move inward. Accepting
this, it becomes clear that if any high-state $\psi_{H}$ existed within a
pulse-drive envelope, whose peak $F_{0}>F_{m}$, it would undergo expansion
until its SW fronts reached a point where $F(\tau_{\mathrm{SW}})=F_{m}$ and
stop.

Considering now a Kerr cavity possessing strong TOD, we choose $d_{3}=1$. Fig.
2(f,g) presents an identical scenario to Fig. 2(b-e), now with TOD enabled.
Here, time-reversible symmetry has been broken, with a strong difference
emerging between the up-SW (here left in time), and the down-SW (on the
right). The down-SW, which corresponds with the dispersive wave-like feature
on the positive side of the spectrum (Fig. 2(g)), follows similar behavior to
that of the SWs under pure SOD in Fig. 2(c,e) including the Maxwell point as
detuning $\zeta_{0}$ is increased. Conversely, the up-SW exists primarily on
the negative side of the spectrum, where the local gradient of the integrated
dispersion operator $d_{\mathrm{int}}=d_{2}\Omega^{2}+d_{3}\Omega^{3}$
($\Omega$ being the dimensionless angular frequency) has become inverted.
Consequently, the dispersive wave element now possesses a positive group-delay
(negative group-velocity shift) and so has changed orientation to face inward
to the high-state $\psi_{H}$, and the up-SW itself has had its Maxwell point
changed so that it retreats inside the pulse-drive envelope much sooner, thus
eliminating the SW fronts at a much earlier detuning at $\zeta_{0}\approx
4.5$.

Overall, the flatness of the dispersion profile on negative frequencies has
heavily skewed the generated spectrum to the one side resulting in a negative
shift to the group velocity for the SW structure as a whole inside $F(\tau)$.
Naturally, introducing a counter-acting group velocity shift in the form of a
negative $d_{1}$ term should help contain the structure within the center of
$F(\tau)$ as the detuning increases. This scenario is presented in Fig. 3. By
now setting $d_{1}=-1.34$, the time-frame of the cavity field continually
moves forward in fast time $\tau$ (here to the left), keeping both SWs near to
the center of the pulse envelope $F(\tau)$ preventing early collapse. The SW
fronts meet together now at $\zeta_{0}=5.1$, where a significant event occurs.
Instead of eliminating each other as in the case of pure-SOD, the SWs become
locked to each other based on the bonding of the down-SW to the modulated wave
of the up-SW, in an event not at all dissimilar to the much reported formation
of ‘dark pulses’ Xue  _et al._ (2015). Whereas dark pulses lock on the
modulations of the low-state $\psi_{L}$, these ‘bright’ structures necessarily
require strong TOD so that a sufficiently powerful modulation exists on the
high-state $\psi_{H}$.

These bright LDS, whose existence was predicted in recent theoretical works
Parra-Rivas  _et al._ (2017); Lobanov  _et al._ (2019) have been termed as
highly modulated ‘platicons’. However, like dissipative solitons, these
structures are self-stable and can freely exist across the broad background of
the pulse-drive. While the SW states shown here for $\zeta_{0}<5.1$ are bound
to the pulse-drive envelope $F(\tau)$ as a whole through the left and right
Maxwell points, these bright structures are self-stable and eventually find a
single _trapping position_ on one edge of $F(\tau)$ just as with pulse-driven
conventional dissipative solitons Obrzud  _et al._ (2017); Hendry  _et al._
(2019); Anderson  _et al._ (2019). In this example, the trapping position is
on the left edge. As the detuning here increases past $\zeta_{0}>5.1$, the
structure (plotted specifically on levels 3-6 of Fig. 3(b,c)) undergoes
progressive collapses reducing in periodicity initially from 5, down to 2. In
this way, these structures can be thought of as clusters of individual
dissipative solitons packed extremely close together. As their existence is
defined by vanishing SOD in favor of TOD, we term them in this work as zero-
dispersion solitons (ZDS(n)) consisting of $n$ bound sub-solitons. The value
$n$ can be identified by the spectral periodicity between the pump and what
was initially the SW dispersive wave, here on the left end of the spectrum.

Fig. 3(d) shows how varying the group velocity shift $d_{1}$ (or
_desynchronization_ in terms of pulse-driving) gives rise to a varying maximum
detuning for ZDS(n) existence, and with different preferred $n$. Here (with
particular attention to Fig. 3(d-iii)), the ZDS(3) follows its trapping
position on the left-hand slope until it crosses the center line where a
trapping position no longer exists and decays, following an asymmetrical
trajectory reminiscent of recent studies on conventional dissipative solitons
Hendry  _et al._ (2019). The cavity energy trace, plotted in Fig. 3(e) for all
values of $d_{1}$ in the vicinity, shows the asymmetrical unfolding of the
characteristic ‘step’ feature we should expect to see in experiment.

Figure 4: Experimental pulse-driven switching wave and zero-dispersion soliton
formation, MR1 (a) Setup, featuring the EO-comb as a pulsed-source. MZM: Mach-
Zehnder modulator, EOM: electro-optic modulator, EDFA: erbium-doped fiber
amplifier, ESA: electronic spectrum analyzer, OSA: optical spectrum analyzer,
OSC: oscilloscope. The input pulse train is coupled into and out of the
microresonator chip via lensed fibers. Left-inset: Spectrum of the 14 GHz EO-
comb before amplification. (b) Spectrogram of the step feature for different
desynchronization about 27.888880 GHz. (c) Microresonator transmission (with
DC value subtracted), with detunings from (f) marked with dashed lines. (d)
Spectrogram of the repetition-rate beatnote during the laser scan in (c). (e)
Long-term beatnote measurement of the final comb state. (f) Stages of
comb/spectrum formation in descending order of detuning (40 dB vertical
offsets). Red block marks spectral filter for beatnote measurement.  Figure 5:
Octave spanning zero-dispersion soliton spectrum, MR1(a) Experimental
measurement of the resonator integrated dispersion $D_{\mathrm{int}}$, along
with the spectrally extended fitted solution, and solution shifted by
$+D_{1}$. Pump frequency (and pump detuning) marked by the vertical red line
(horizontal dashed line). Phase-matched locations of dispersive waves marked
by circles, with momentum mismatch in dotted red arrow. (b) Measured spectrum
of ZDS(4), with dispersive waves marked with arrows corresponding to the
circles in (a). Insets show individual comb lines. The left-most trace is in
gray to indicate it is the second-order diffraction spectrum of DW2, and not
genuine. (c) Frequency-domain and (d) time-domain simulation of ZDS(4) with
dispersive wave tails marked. Figure 6: Experimental zero-dispersion soliton
formation, MR2 (a) Experimental measurement of the resonator integrated
dispersion $D_{\mathrm{int}}$, and spectrally extended fitted solution. (b)
ZDS(3) comb. (c) ZDS(2) combs, measured at maximum detuning using three pump
powers, noted as effective average power coupled to ring (total power towards
chip).

##  III Experimental Results

The pulse-drive source (as shown in Fig. 4(a)) is provided in the form of an
electro-optic comb (EO-comb) Kobayashi  _et al._ (1988); Obrzud  _et al._
(2017), providing pulses with a minimum duration of 1 ps (see Methods for
details), and whose repetition rate is finely controlled by an RF-synthesized
signal $f_{\mathrm{eo}}$. The cavity platform of choice for the experiment is
the chip-based Si3N4 microresonator, in this case having a native FSR of 27.88
GHz. The EO-comb repetition rate is set to exactly half this at
$f_{\mathrm{eo}}=13.944$ GHz due to RF transmission limitations. Other than a
factor-2 reduction on conversion efficiency due to only half the lines being
coupled to the cavity, the experiment behaves the same as one that is fully
synchronous and we can disregard the excess comb lines. Two microresonators
(referred to as MR1 and MR2) in particular are used to generate ZDS, having
two slightly different dispersion profiles causing the formation of ZDS(n) of
different $n$. Their measured dispersion parameters are given in result
figures further below.

Starting with MR1, in order to ensure that a full range of formation behavior
is observed, and spectral extent maximized, the average pulse-power coupled to
the resonator is set to $P_{\mathrm{av}}=350$ mW (12 pJ pulse energy at 28
GHz), approximately 20 times higher than the observed minimum comb generation
threshold. The experiment proceeds the same way as in the theory section,
typical for LDS generation in Kerr cavities, and particularly in pulse-driven
soliton generation Obrzud  _et al._ (2017); Lilienfein  _et al._ (2019);
Anderson  _et al._ (2019). The exact native FSR of the microresonator is first
found by varying the input repetition rate $f_{\mathrm{eo}}$ until the
expected unfolding of the ZDS ‘step’ is observed (Fig. 4(b)). Here, we see an
asymmetrical extension of the step vs. the relative desynchronization
$\delta\\!f_{\mathrm{eo}}=f_{\mathrm{eo}}-D_{1}/2\pi$, as expected based on
Fig. 3(e), although with slightly different precise form due to unaccounted
for higher order effects (the absence of a step at
$\delta\\!f_{\mathrm{eo}}=0$ is a coincidence based on shot-to-shot
statistical variation of formation probability). Based on this measurement, we
find an optimum $f_{\mathrm{eo}}=27.88888$ GHz, with a locking range for ZDS
on the order of $\pm 10$ kHz.

For this value of $f_{\mathrm{eo}}$, the EO-comb seed laser frequency
$\omega_{p}$ is tuned slowly across a resonance frequency $\omega_{0}$ from
the blue-detuned side to the red-detuned side (such that
$\delta\omega=\omega_{0}-\omega_{p}>0$ by conventionHerr  _et al._ (2014))
towards the region of Kerr bistability. Fig. 4(c) plots the output light from
the microresonator during this scan, and at the same time the RF repetition-
rate beatnote of the ZDS is recorded (Fig. 4(d)). The measured comb spectra
are plotted in Fig. 4(f) at four detunings in descending order, after the SW
is formed. Qualitatively, the results behave as the simulations in Fig. 3(c)
predict. Firstly, over the first two rows, we see the spectrum grow wider and
with a sharper dispersive-wave (DW1) located from 182 to 179 THz. Importantly,
we see the spectral fringes either side of the pump (spaced by $\approx 1$ THz
on the first spectrum) increase their period as detuning is increased,
indicating the two SW fronts are moving together within the pulse-drive
envelope. In the last two rows, we see the SWs have coalesced into the ZDS(5),
a 5-period structure, then reducing to a ZDS(4), each time moving the location
of DW1 further to low frequencies. Throughout this scan over resonance, the RF
repetition rate beatnote shows low noise. In fact, the final beatnote, plotted
in Fig. 4(e) is highly stable, inheriting the low-offset phase noise of the
$f_{\mathrm{eo}}$ as supplied by the RF synthesizer. This confirms that the
ZDS has temporally locked to the driving pulse, just as a conventional bright
dissipative soliton would Obrzud  _et al._ (2017); Anderson  _et al._ (2019).

Fig. 5 analyses this final structure in greater detail. We characterize the
broadband dispersion profile of the MR1 using a cascaded three-laser swept
spectroscopy technique Liu  _et al._ (2016). In Fig. 5(a) we plot the measured
integrated dispersion profile $D_{\mathrm{int}}=\omega_{\mu}-\omega_{0}-\mu
D_{1}$, representing the frequency deviation of each resonator mode $\mu$ from
the uniform FSR grid spaced by $D_{1}$ (where the pump mode corresponds to
$\mu=0$). This data is fitted to a fourth-order polynomial centered at
$\omega_{0}=2\pi\times 192.3$ THz where
$D_{\mathrm{int}}\approx\mu^{2}D_{2}/2+\mu^{3}D_{3}/6+\mu^{4}D_{4}/24$
($D_{2}/2\pi=-3.17$ kHz, $D_{3}/2\pi=13.8$ Hz, $D_{4}/2\pi=-15.9$ mHz, all
$\pm 5\%$). In dimensionless parameters, for a fixed $d_{2}=1$ we have a value
$d_{3}=0.38$. The pump frequency detuning $-\delta\omega/2\pi=-1.2$ GHz, which
we obtain from live cavity phase-response measurements Guo  _et al._ (2017),
is also marked. In Fig. 5(b) the entire spectrum of the ZDS(4) is plotted and
features several dispersive waves (DW), the spectral locations of which can be
predicted based on where $D_{\mathrm{int}}(\mu)=-\delta\omega$. The predicted
DW locations do not match perfectly with experiment however, but this can be
explained by the bandwidth-limited dispersion measurement with unknown higher-
order values for $D_{5},D_{6}$ and so forth.

The first, DW1 at 176 THz, will always occur for ZDS formation due to the
requirement for powerful TOD. The second, DW2 at 280 THz, has occurred due to
the overall normal fourth-order dispersion of the waveguide Pfeiffer  _et al._
(2017), but is not required for ZDS formation. The additional dispersive wave,
termed DWq, occurs where the optical comb modes have wrapped by $D_{1}$ so
that $D_{\mathrm{int}}(\mu)+D_{1}=D_{\mathrm{int}}(\mu-1)=-\delta\omega$ or
that, by shifting by one FSR, the linear wave at DWq has accrued a $2\pi$
phase shift relative to the pump wave. This phase-wrapping is commonly known
to allow the formation of so called ‘Kelly’ sidebands in soliton fiber lasers
Kelly (1992). Due to the longitudinal momentum mismatch $\delta\mu=+1$ between
the coupled linear wave and the ZDS comb lines, quasi-phase matching is
required to bridge this gap Hickstein  _et al._ (2018). This particular
microresonator features a brief ‘mode-stripping’ section where the waveguide
width rapidly tapers down to a narrow width in order to stop any higher-order
spatial modes from propagating Kordts  _et al._ (2016), where the waveguide
dispersion changes sharply. This intra-roundtrip disturbance provides phase
modulation to the linear wave at DWq, and is more than sufficient to enable
quasi-phase matching to stimulate resonant radiation Copie  _et al._ (2016).
This effect has been observed in microresonators with a similar intra-
roundtrip modulation of the waveguide width where it was identified as Faraday
instability Huang  _et al._ (2017), and has long been observed in fiber-based
Kerr resonators with longitudinally varying dispersion Luo  _et al._ (2015);
Copie  _et al._ (2016); Nielsen  _et al._ (2018). Fig. 5(c,d) shows numerical
LLE simulations using the real experimental parameters of MR1 (see Methods),
demonstrating close agreement with the form taken by the spectrum
corresponding to a ZDS(4) as shown in Fig. 5(d). In Fig. 5(c), both simulation
results taking into account either a constant or an oscillating intra-
roundtrip dispersion $D_{\mathrm{int}}(z)$, with DWq appearing only in the
latter case. Further simulations and analysis DWq is presented in the
supplementary info.

In order to observe ZDS(n) of lower $n$ we move to MR2, which has its zero-
dispersion wavelength closer to the pump wavelength at 1560 nm. Here, the
dispersion parameters (Fig. 6(a)) are fitted to be $D_{2}/2\pi=-848$ Hz,
$D_{3}/2\pi=12.8$ Hz, $D_{4}/2\pi=-15.9$ mHz, $\pm 5\%$, corresponding to
dimensionless parameters $d_{2}=1$ and $d_{3}=2.11$, further into the zero-
dispersion regime (Fig. 1). In this microresonator, with the same generation
method as above in MR1, we generate ZDS(3) and ZDS(2) in Fig. 6(a) and (b)
respectively. In this microresonator, we do not observe the same DW2 or DWq as
in microresonator 1. For ZDS(2), we present spectral measurements taken using
three increasing input pump powers, each enabling an increased maximum
detuning $\delta\omega$. As shown, as available power is increased, the
overall spectral profile expands, with both DW1 on the left and the anti-
dispersive wave on the right moving outwards, in a similar manor as for
conventional dissipative solitons Lucas  _et al._ (2017).

##  IV Discussion and Summary

In this work, we have experimentally synthesized a novel class of localized
dissipative structure, the zero-dispersion soliton. In terms of figures of
merit, the generated ZDS(2-5)-based combs presented here are extremely
substantial in terms of the product of their total bandwidth and their total
line-count which, as far as we are aware, is a record for a single-structure
in a microresonator. The central body of the ZDS(4) comb in MR1 spans over 76
THz (1830 and 1260 nm), accounting for more than 2,700 comb teeth, spaced by a
detectable 28 GHz repetition rate. When including the DW features, the final
bandwidth becomes 136 THz or 97% of an octave. As the repetition rate is
directly detectable on photodiode, a future work with fine-tuning of the
microresonator dispersion may enable $f$-$2f$ self-referencing with a single
microcomb Spencer  _et al._ (2018). We have further demonstrated the _direct_
generation of switching waves via pulse-driving, creating a highly smooth
ultra-broadband microcomb under normal dispersion conditions. Such normal
dispersion-based microcombs have thus far only been formed in Si3N4 via
modulation instability enabled by spatial mode-couplingXue  _et al._ (2015),
necessitating an extra coupled-microresonator ring with integrated heaters in
order to be deterministic Kim  _et al._ (2019).

The formation of ZDS-based microcombs more generally has expanded the domain
of microcomb generation towards the region of both normal-dispersion and zero-
dispersion, previously not often considered ideal. This lifting of strict
requirements for anomalous dispersion may give greater flexibility in the
Si3N4 fabrication process going forward. The result also demonstrates not only
that microcomb generation can be achieved in a straight-forward fashion in
such waveguide resonators with normal-to-zero dispersion, but that it may be
most _preferable_ for highly broadband comb generation due to the superior
flatness of the comb in the SW regime, as well as the lack of a high-noise
chaotic phase and multi-soliton formation as compared to its anomalous
dispersion-based counterpart. In terms of physics, the experimental
observations of an entirely novel class of LDS – a bright pulse-like structure
which constitutes a link between SW-based and soliton-based LDS – were
presented. To our knowledge, such an entity has not been observed before in
microresonators as of the time of writing. Its discovery in experiment may
open a new area of fundamental research on the nature of dissipative Kerr
solitons and switching waves under one umbrella.

Note: we would like to acknowledge a parallel work by Li _et al._ completed
during preparation for this manuscript observing a similar phenomenon in
fiber-based Kerr cavities Li  _et al._ (2020).

### Acknowledgments

This work was supported by Contract No. D18AC00032 (DRINQS) from the Defense
Advanced Research Projects Agency (DARPA), and funding from the Swiss National
Science Foundation under grant agreements No. 192293. This material is based
upon work supported by the Air Force Office of Scientific Research, Air Force
Materiel Command, USAF under Award No. FA9550-19-1-0250.

### Author Contributions

M.H.A. performed the experiment with the assistance of G. L.. The sample
microresonator was fabricated by J.L.. M.H.A. conducted the simulations and
prepared the manuscript with the assistance W.W., G.L., and T.J.K.. T.J.K.
supervised the project.

### Data Availability Statement

All data and analysis files will be made available via zenodo.org upon
publication.

### Methods

Theory and Simulation The homogeneous solutions to the intracavity field
$\Psi_{\mathrm{CW}}=|\psi_{\mathrm{CW}}|^{2}$, graphed in Fig. 2(b) for
different local pump strength $F(\tau)=F_{0}f^{2}(\tau)$, is first obtained
from the real roots of the cubic polynomial derived from equation II at
equilibrium and with all dispersion $d_{l}=0$ Haelterman  _et al._ (1992), and
subsequently the complex field solution from the resonance condition,

$\Psi^{3}-2\zeta_{0}\Psi^{2}+(\zeta_{0}^{2}+1)\Psi-F(\tau)=0$ (2)

$\psi_{\mathrm{CW}}=\frac{i\sqrt{F(\tau)}}{\Psi_{\mathrm{CW}}-\zeta_{0}+i}$
(3)

with $\Psi_{H}$ and $\Psi_{L}$ being the top and bottom solution respectively.
The simulation presented in Fig. 2 was calculated via the split step method
with a change in detuning rate $d\zeta_{0}/dt^{\prime}=0.01$ to ensure SWs
reached equilibrium at each stage. In Fig. 3,
$d\zeta_{0}/dt^{\prime}=0.000625$ to allow the ZDS to remain at their
trapping/equilibrium position during he detuning increase. The pulse-drive
width $\tau_{p}=50$.

Experiment The EO-comb is comprised of a CW laser, followed by an intensity
modulator and three phase modulators, driven by an RF signal generator (Rhode
& Schwarz SMB100A), generating 50 spectral lines spaced by $f_{\mathrm{eo}}=$
13.944 GHz. The waveform is compressed in time through linear dispersion made
from 300 m of standard SMF-28 and 5 m of dispersion-compensating fiber,
yielding pulses of minimum duration 1 ps as confirmed by frequency-resolved
optical gating (FROG). The Si3N4 microresonators MR1 and MR2 used in this
experiment have been fabricated with the _photonic Damascene process_ Pfeiffer
_et al._ (2018) with a 2350$\times$770 nm2 cross-section, and possess a peak
probable cavity linewidth of
$\kappa/2\pi=(\kappa_{0}+\kappa_{\mathrm{ex}})/2\pi=$ 208 MHz and 150 MHz
(with external coupling rate $\kappa_{\mathrm{ex}}/2\pi=$ 155 MHz and 120 MHz)
for MR1 and MR2 respectively. Their measured dispersion $D_{\mathrm{int}}$ is
expanded in the main text. Effective power coupled to resonator as quoted
above exclude chip insertion loss of 1.6 dB and half of the 14 GHz comb lines
not coupled to the resonator modes at 28 GHz. The RF beatnote measurement in
Fig. 4(d,e) derives from approximately 11 filtered comb lines outside of the
EO-comb spectrum.

Full system model The experimental microresonator results are described by the
full LLE with real parameters

$\displaystyle\frac{\partial A(t,T)}{\partial t}=\quad$
$\displaystyle\mathcal{F}\Big{[}i\big{(}\delta\omega+\mu\cdot
2\pi\delta\\!f_{\mathrm{eo}}+D_{\mathrm{int}}(z,\mu)\big{)}\tilde{A}_{\mu}\Big{]}$
(4) $\displaystyle-$
$\displaystyle\frac{\kappa}{2}A+ig|A|^{2}A+\sqrt{\frac{\kappa_{\mathrm{ex}}P_{0}}{\hbar\omega_{0}}}f_{p}(T)$

acting on photon field $A(t,T)$ over slow/laboratory time $t$ and fast time
$T$ in the co-moving frame of the intracavity field circulating at $D_{1}$,
with frequency domain counterpart $\tilde{A}_{\mu}$ at discrete comb line
indices $\mu$. Included with the linear phase operators $D_{\mathrm{int}}$ and
$\delta\omega$ is the input pulse desynchronization
$\delta\\!f_{\mathrm{eo}}$. The nonlinear coupling parameter
$g_{0}/2\pi=0.054$ Hz (see supp. info for further on this).

For the simulation presented in Fig. 5(c,d), we set $\delta\omega/2\pi=$ 900
MHz, and the input pulse profile $f_{p}(T)=\exp(-T^{2}/T_{p}^{2})$ with
$T_{p}=$ 1.0 ps, $P_{0}=1.8$ W, and static desynchronization
$\delta\\!f_{\mathrm{eo}}=150$ kHz. In order to stimulate the quasi phase-
matched wave at DWq, we set
$D_{\mathrm{int}}(z,\mu)=D_{\mathrm{int0}}(\mu)(1+0.2\cos(\delta\mu\cdot 2\pi
z/L))$, with $\delta\mu=1$ representing a single longitudinal-mode modulation
in the dispersion operator for the resonator of length $L$. All of the real
parameters are related to the dimensionless parameters by the following:
$t^{\prime}=\frac{\kappa}{2}t$, $\tau=D_{1}\sqrt{\frac{\kappa}{D_{2}}}T$,
$\psi=\sqrt{\frac{2g_{0}}{\kappa}}A$,
$d_{l}=\frac{2}{\kappa}\frac{D_{l}}{l!}\big{(}\frac{\kappa}{D_{2}}\big{)}^{\frac{l}{2}}$
for $l=$1-4, $\zeta_{0}=\frac{2\delta\omega}{\kappa}$,
$F_{0}=\frac{8\kappa_{\mathrm{ex}}g_{0}}{\kappa^{3}\hbar\omega_{0}}P_{0}$.

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