# Dual-mode microresonators as straightforward access to
octave-spanning dissipative Kerr solitons

Haizhong Weng,1 Adnan Ali Afridi,1 Jing Li,1 Michael McDermott,1 Huilan Tu,2
Liam P. Barry,3 Qiaoyin Lu,2 Weihua Guo,2,∗ and John F. Donegan1,
<EMAIL_ADDRESS><EMAIL_ADDRESS>1School of Physics, CRANN and AMBER, Trinity
College Dublin, Dublin 2, Ireland
2Wuhan National Laboratory for Optoelectronics and School of Optical and
Electronic Information, Huazhong University of Science and Technology, Wuhan
430074, China
3Radio and Optical Communications Lab., School of Electronic Engineering,
Dublin City University, Dublin 9, Ireland

###### Abstract

The Kerr soliton frequency comb is a revolutionary compact ruler of coherent
light that allows applications, from precision metrology to quantum
information technology. The universal, reliable, and low-cost soliton
microcomb source is key to these applications. In this work, we thoroughly
present an innovative design strategy for realizing optical microresonators
with two adjacent modes, separated by approximately 10 GHz, which stabilizes
soliton formation without using additional auxiliary laser or RF components.
We demonstrate the deterministic generation of the single-solitons that span
1.5-octaves, i.e., near 200 THz, via adiabatic pump wavelength tuning. The
ultra-wide soliton existence ranges up to 17 GHz not only suggests the
robustness of the system but will also extend the applications of soliton
combs. Moreover, the proposed scheme is found to easily give rise to multi-
solitons as well as the soliton crystals featuring enhanced repetition rate (2
and 3 THz) and conversion efficiency greater than 10$\%$. We also show the
effective thermal tuning of mode separation for stably accessing single-
soliton. Our results are crucial for the chip-scale self-referenced frequency
combs with a simplified configuration.

##  I Introduction

Dissipative Kerr soliton (DKS), as a self-reinforcing wave packet that
maintains its shape while circulating around a microresonator, has been
demonstrated under a double balance between nonlinearity and dispersion, as
well as parametric gain and cavity loss Brasch et al. (2016); Kippenberg et
al. (2018). Due to the unprecedented compactness, low-noise, high power-
efficiency, and broad spectral bandwidth, soliton Kerr combs (microcombs) have
attracted considerable research interest and been extensively studied for
spectroscopy Suh et al. (2016), communications Marin-Palomo et al. (2017),
frequency synthesizer Spencer et al. (2018), optical clock Drake et al.
(2019), microwave photonics Liu et al. (2020) and sensor applications Tan et
al. (2021). Over the past several years, through the substantial exploration
of the fundamental physics and microresonator fabrication, researchers have
realized Kerr solitons in a growing number of platforms, including ultra-high
Q MgF2 Herr et al. (2014), silica Yi et al. (2015), and monolithic integrated
platforms such as Si3N4 Brasch et al. (2016); Joshi et al. (2016); Wang et al.
(2016); Ye et al. (2022), LiNbO3 He et al. (2019), AlGaAs Moille et al. (2020)
and Ta2O5 Jung et al. (2021), as well as the wide-bandgap semiconductors AlN
Weng et al. (2021a); Liu et al. (2021a), SiC Guidry et al. (2022) and GaNZheng
et al. (2022).

Photonic integration of laser pump and passive resonators offers the
possibility of achieving chip-scale operation, but there are significant
challenges to overcome before the widespread deployment of these soliton comb
systems. One key challenge comes from the thermo-optic instability in the
microresonator when the pump enters into the red-detuned regime for soliton
formation. To stably access the soliton state, a number of experimental
techniques were developed including rapid laser frequency scanning Herr et al.
(2014); Liu et al. (2021a); Jung et al. (2021), careful pump power
manipulation Brasch et al. (2016); Yi et al. (2015), or microheater thermal
tuning Joshi et al. (2016); Ji et al. (2021a). With the extra radio-frequency
(RF) generator, modulator, or microheaters, these schemes can bring the short-
lived soliton to a steady state. Self-injection locking (SIL) Stern et al.
(2018); Raja et al. (2019) has been proposed and exploited for turnkey soliton
generation Shen et al. (2020) or even a remarkable octave-spanning soliton
microcomb Briles et al. (2021), by directly coupling a laser chip to a passive
microresonator. This approach enables a miniaturized frequency comb source but
demands challenging photonic integration and great caution to control the
back-reflected Rayleigh scattering. Also, the likely accessible detuning
reduction in SIL will limit the spectral bandwidth and dispersive waves (DWs)
intensity Briles et al. (2021), as well as the SER and total comb power
Voloshin et al. (2021), which are serious issues for the precision metrology
and timing. Recently, dual-pumping for two resonances (2P2R) has been applied
to mitigate the thermal effects thus leading to the deterministic generation
and switching of the DKS Zheng et al. (2022); Zhang et al. (2019); Zhou et al.
(2019); Lu et al. (2019). However, it drastically increases the system
complexity and cost due to the use of another set of laser, amplifier,
polarization controller, and fiber circulator Zheng et al. (2022); Zhou et al.
(2019); Lu et al. (2019). Another dual-pumping scheme but activating a single
resonance (2P1R) was also proposed, where the auxiliary pump is one modulation
sideband away from the main pump Wildi et al. (2019); Nishimoto et al. (2022).

Figure 1: (a) Schematic of different thermal compensation schemes for Kerr
soliton generation. FPC: fiber polarization controller; EDFA: erbium-doped
fiber amplifier; Cir: circulator. (b) Intracavity power versus the pump
detuning for 1P2R system. The power coupled into the auxiliary mode rises
suddenly due to the blue shift of the resonances during soliton formation,
which can, in turn, remedy the intracavity power change thus preventing the
soliton collapse. Blue and red shaded indicate the pump detuning position
relative to the cavity resonance.

A simple and cost-effective octave-spanning Kerr soliton generator needs to be
developed urgently to bring the microcombs towards practical applications.
Here, we present the straightforward access to octave-spanning DKSs by
injecting a single pump to two close resonances (dual-mode) with the same
polarization, which we called the 1P2R-1P scheme. The modes on the blue and
red sides are used for parametric processes and thermal compensation,
respectively. Figure 1 compares different thermal accessible soliton systems
and sketches the 1P2R mechanism. In contrast to dual-pumping, our method
leverages a much simplified setup once the front-end design is properly
managed. The idea used in this paper was proposed by our group and
successfully applied for octave-spanning DKS generation in an AlN microring
resonator (MRR) Weng et al. (2021a). Dual-mode but with mixed polarization
(1P2R-2P) was also found to help the soliton stabilization process Li et al.
(2017), which requires careful adjustment of the polarization. However, the
real potential of the 1P2R scheme needs investigation due to the rigorous
requirements in design and fabrication for ensuring the pump and auxiliary
modes are in close proximity.

This paper presents the production of octave-spanning Kerr solitons with
improved performance through careful design of the microresonators while we
also discuss the feasibility of fabrication with high yield. The advanced
single-soliton features a 17-GHz-wide soliton existence range (SER) and a
200-THz-wide spectral bandwidth. SER means an effective detuning range where a
soliton state is maintained during the pump wavelength tuning Herr et al.
(2014). Moreover, using the same resonance, octave-spanning soliton crystals
at the telecommunication C-band are also demonstrated. Similar soliton
behavior are also observed in multiple chips, and thereby illustrate its
universal nature. The presented results provide a solid strategy for broadband
DKS generation, which is transferable to alternative materials with a tailored
repetition rate (frep). From an application perspective, the 1P2R-1P scheme
paves the way towards making reliable, dynamic, low-cost, and easy-to-operate
soliton microcomb sources.

Figure 2: (a) Simulated resonant wavelengths for the microrings with a radius
of 23.25 $\mu$m, the thickness of 800 nm, and various RWs. (b) Measured
transmission spectra of seven adjacent MRRs with various RWs. RW change step
is 20 nm. The circle denotes the target dual-mode enabling soliton generation
with the 1P2R-1P scheme. (c) Simulated integrated dispersion profiles. Inset:
zoom-in view of the dual-mode. (d) Measured MI comb spectra at a 400 mW on-
chip power for varying RW. Dashed lines indicate the mode interaction
positions. Figure 3: Experimental soliton results of MRR1. (a) Collected
powers of all output (green) and the pump alone (yellow), as well as their
difference (red) at Pin=150 mW. (b) Microcomb evolution map. (c) Optical
spectra of (i) MI comb, (ii) single-soliton, (iii) 2-SC and (iv) 3-SC by
connecting output fiber with the OSA directly. Top inset: RF noise of MI comb
and single-soliton. The photodiode (PD) noise floor is overlapped by that of
the single-soliton. Lower insets describe the solitons distribution. (d)
Single-soliton spectrum at low-frequency. (e) Measured autocorrelation (AC)
traces of various soliton states.

##  II Design of dispersion-engineered dual-mode microresonators

Figure 2 shows the design and resonance characteristics of the dual-mode MRRs.
The 800-nm-thick Si3N4 MRRs that we are using were fabricated by Ligentec
through photonic Damascene processes Liu et al. (2021b). To build a reliable
layout design, we first conducted the simulations with finite element method.
For the MRRs with a fixed radius of 23.25 $\mu$m and varied ring widths (RWs),
the simulated resonant wavelengths are plotted in Fig. 2(a). As the RW
increases, an obvious mode redshift is observed, while the slope
d$\lambda$/dRW is 0.012 and 0.094 for the TE00 and TE10 modes. Consequently, a
1 nm RW variation will lead to a $\sim$0.08 nm adjustment in the mode
separation $\Delta\lambda$ ($\lambda_{10}$-$\lambda_{00}$), i.e., a 100 nm RW
variation will lead to a $\sim$8 nm $\Delta\lambda$ change, which is almost
one free spectra range (FSR). Specifically, the TE00 mode with an angular
number (m) of 164 and TE10 mode (m=151) have a minimum $\Delta\lambda$ of 0.09
nm at $\sim$1563 nm when RW=1.68 $\mu$m. Then the two modes coincide again at
$\sim$1572 nm with a separation of 0.12 nm when RW=1.78 $\mu$m.

Figure 2(b) displays the measured transmission spectra of seven neighbouring
MRRs with RWs discretely increasing from 1.68 to 1.80 $\mu$m. As the RW
increases, both modes show the redshift (denoted by arrows) with the speeds
almost the same as simulations, illustrating that a 20 nm variation in the MRR
dimension is achievable even using the 248 nm DUV lithography. The target
dual-mode in MRR1 (denoted by a circle) has a $\Delta\lambda$ of 0.084 nm,
i.e., $\sim$10.5 GHz, and is enlarged as an inset of Fig. 2(c). The TE00 and
TE10 modes have an FSR of $\sim$1012 and $\sim$979 GHz, respectively. Their
statistical Q factors are 1.1×106 and 4.2×105, respectively. The dual-mode
position shifts to $\sim$1566.3 nm when RW=1.78 $\mu$m, while the microcombs
cannot be tuned to the soliton regime due to a relatively large
$\Delta\lambda$ of 0.262 nm. Overall, the experimental results are consistent
with simulations, suggesting that dual-mode resonators in the C-band can be
reliably attained by tailoring the MRR dimensions.

Near-zero anomalous dispersion is crucial for broadband microcomb generation.
Figure 2(c) presents the calculated integrated dispersion (Dint) Brasch et al.
(2016) of the TE00 mode family. All the MRRs can support dual-DW, where the
Dint equals zero. With the increase of RW, the DW position at low-frequency
has a small blue shift, while the high-frequency DW dramatically drops from
311 to 254 THz. The simulated second-order dispersion D2/2$\pi$ is 37.2 and 19
MHz when RW is 1.68 and 1.78 $\mu$m, respectively. Figure 2(d) summarizes the
measured modulation instability (MI) microcombs when pumping the MRRs with an
on-chip power of Pin=400 mW. All spectra exceed an octave span thanks to the
dual-DW. The groups of dense lines below 130 THz result from the 2nd-order
diffraction of the optical spectrum analyzer (OSA), and are thus artifacts.
The peculiar comb lines with enhanced or reduced power (denoted by dotted
lines) are caused by mode interactions Ramelow et al. (2014). The wider MRRs
tend to have flatter spectral envelopes and stronger comb lines near the DWs
because ofer the low 2nd-order dispersion.

Figure 4: (a)-(c) Soliton results of MRR1. (a) Transmitted power excluding
pump when Pin= 200 mW. (b) The soliton operation wavelength range and
corresponding soliton numbers at different Pin. (c) Typical TSM spectra and
the fitting. TSM spectra obtained from (d) MRR2 and (e) MRR3 when Pin is
around 220 and 320 mW, respectively.

##  III Experimental results

Octave-spanning single-soliton and soliton crystals. In this section, we
present the octave-spanning soliton results using the first 1.68-$\mu$m-wide
device MRR1 as discussed in Fig. 2. All transmission curves and optical
spectra presented in this work are obtained with a forward pump tuning speed
of 1 nm/s, which allows for adiabatic tuning within the cavity. The
experimental setup is similar to the one reported in Weng et al. (2021a) and
can be found in supplementary material. Figure 3(a) shows the simultaneously
measured transmission of all output and pump alone, as well as their
difference when Pin=150 mW. The striking steps related to the single-soliton
formation are observed, with a width of 0.08 nm ($\sim$10 GHz), which is close
to $\Delta\lambda$. By stopping the pump at different wavelengths, we map the
spectra evolution of the microcomb in Fig. 3(b), which confirms the
aforementioned wide SER conclusively. Two dash lines (i) and (ii) indicate the
state of MI comb and single-soliton, whose spectra are plotted in Fig.
3(c)-(i) and -(ii). The spectrum ranges from 136 to 240 THz at MI state but is
conspicuously widened to 125-322 THz for the single-soliton. The soliton
microcomb covers 1.5-octaves and is state of the art Li et al. (2017);
Pfeiffer et al. (2017); Briles et al. (2018). The dual-DW at the frequency of
132 and 311 THz agree with the simulation results very well. The transition
from MI to soliton state can also be verified by the drastic reduction of the
RF intensity noise, as shown in the inset of Fig. 3(c). For comparison, we
also pump the TE00 mode at 1549.1 nm which is far from the auxiliary mode.
Only an MI comb ranging from 136 to 240 THz appears as a final state when
Pin=150 mW (see supplementary material). These results illustrate that the
dual-mode scheme could also decrease the pump power required to reach the
soliton state.

The carrier-envelope offset frequency (fceo) is an important parameter for
microcomb in the applications of metrology and timekeeping, which can be
detected via the f-2f self-referencing technique Del’Haye et al. (2016). A
near-zero fceo is ideal for electronic detection and phase-locking Briles et
al. (2018). Figure 3(d) shows the single-soliton spectrum at low-frequency,
where the circles and crosses indicate the 1st-order (i.e., realistic) and
2nd-order comb lines, respectively. The latter has a spacing of half FSR and
an intensity increasing trend with the frequency decrease. The fceo can be
calculated via

$\displaystyle{f_{ceo}}=2$ $\displaystyle\times(f_{n}-f_{2n}/2)$ (1)

where fn and f2n/2 are frequencies of the adjacent 1st-order and 2nd-order
comb lines. Consequently, an fceo of $\sim$200 GHz is extracted. Different
from the conventional f-2f beatnote detection via frequency doubling, our
calculation accuracy is only at the level of 1 GHz limited by the OSA
resolution. Nevertheless, this simple measurement could help to achieve a low
fceo by optimizing the microring dimension further.

The soliton crystal (SC) is an extraordinary state with regularly distributed
soliton pulses and enhanced comb line power spaced by multiples of the cavity
FSR Cole et al. (2017); Wang et al. (2018). For example, N-SC exhibits comb
lines separated by N×FSR. Such SCs are typically formed in the presence of
avoided mode crossing Karpov et al. (2019); Weng et al. (2021b). In our
scheme, a weak mode coupling occurs between the two neighboring resonances
(see supplementary material) and result in the observation of 2-SC and 3-SC
when adjusting the power slightly, as shown in Fig. 3(c)-(iii) and -(iv),
respectively. Both spectra exceed an octave-spanning range (127-270 THz) and
exhibit stronger comb lines near the pump. To our knowledge, this is the first
report of octave-spanning dissipative SCs centered on the C-band. As regards
3-SC, there are 8 and 15 comb lines with powers greater than 1 mW and 100
$\mu$W, respectively. The in-waveguide comb powers of the single-soliton,
2-SC, and 3-SC are estimated to be 11.5, 22.8, and 31 mW, corresponding to a
conversion efficiency (CE) of 7.7$\%$, 11.4$\%$, and 13.5$\%$. Compared with
conventional single DKSs (a few percent CE) Bao et al. (2014), the CE of SC is
greatly enhanced and we believe it can be further improved by refining the
external coupling rate Jang et al. (2021). Figure 3(e) shows the experimental
pulse traces carried out by an autocorrelator, where the periods of single-
soliton, 2-SC and 3-SC are $\sim$1, $\sim$0.5 and $\sim$0.33 ps, inversely
proportional to the frep of $\sim$1, $\sim$2 and $\sim$3 THz. With sech-
squared fitting, the pulse width of single-soliton is deduced to be 35 fs from
the autocorrelator trace, while an 18 fs width will be resulted from the
spectrum by assuming a transform-limited pulse. This discrepancy can be mainly
attributed to the phase variation across the pulse spectrum, which can be more
precisely determined through a characterization technique like frequency
resolved optical gating (FROG). In the supplementary material, we also present
the 3-SC and 4-SC from other dual-mode devices.

Figure 5: (a)-(d) Measurement results with MRR2. (a) Dial-mode transmission
spectra at different temperatures. (b) The soliton behavior dependence on the
pump power and temperature. (c) Transmission spectra at high pump power with
available soliton steps. (d) Single-soliton spectra obtained at 310, 320, and
330 K. (e) Single-soliton spectra achieved with MRR3 when T=330 K and Pin=290
mW. Dual-mode transmission (left inset) and two adjacent lines at low-
frequency (right inset).

Versatile two-soliton microcombs with 1P2R-1P. By repeatedly scanning the
laser over the dual-mode in MRR1 at 200 mW, four typical transmission spectra
(excluding pump) are observed and recorded, as shown in Fig. 4(a). Besides the
single-soliton, we also obtain the multi-solitons with soliton number (N) of 2
and 3 as well as the switching from N=3 to N=2. Particularly, the two-soliton
microcomb (TSM) has an SER of 0.12 nm ($\sim$15 GHz). We clarify that these
curves are not real comb power considering that some of pump power, even if
lower than the parametric threshold, will be absorbed by the auxiliary TE10
mode. Despite this, the soliton number is easily identified through the OSA
and power meter measurements. Figure 4(b) is a diagram showing the soliton
number and effective detuning range dependence on Pin, which indicates that
the single-soliton can be definitely generated at 140 and 150 mW. As Pin
increases, both the access probability and SER of single-soliton are
decreasing, while the multi-solitons appear with higher possibility. The TSM
attained at 180 mW has a 0.15-nm-wide ($\sim$18.8 GHz) SER, near twice that of
$\Delta\lambda$. The 1P2R-1P scheme is therefore shown to allow the creation
of two-soliton states with ease.

Figure 4(c) shows several spectra of the TSMs when Pin is around 200 mW. These
soliton combs can be reproduced easily and sustained for long periods without
noise. The relatively azimuthal angles of 37.6°, 60.5°, 120°, 172.4°, 175.1°,
and 178.2° are retrieved by fitting the spectral envelope with

$\displaystyle S^{(2)}(\mu)=S^{(1)}(\mu)\times(2+2\times cos(\mu\psi))$ (2)

where $\psi$ is the relative azimuthal angle between the two pulses, $\mu$ is
the comb mode index relative to the pump position, and S(1)($\mu$) is the
spectral of a single-soliton following a sech-squared shape fitted from the
experimental data Brasch et al. (2016). The TSMs with $\psi$ of 37.6°, 175.1°
and 178.2° are reproducible in the other two 1.68-$\mu$m-wide devices (i.e.,
MRR2 and MRR3), as shown in Figs. 4(d) and 4(e), respectively. All resonators
are over-coupled at the pump wavelength, while the MRR1, MRR2, and MRR3 have a
coupling gap of 650, 650, and 550 nm, respectively. Thus for MRR3, the pump
power to access solitons is higher and the powers of individual lines are
stronger at both high and low frequencies. These TSMs have a generally higher
CE compared with the single-soliton, especially for the $\psi$=178.2° case,
which has a CE beyond 10$\%$. Such diverse soliton states with improved CE are
of interest in applications such as optical arbitrary waveform generation
Jiang et al. (2007) and microwave photonic filters (with larger resonators) Xu
et al. (2019). The results of three-soliton microcombs are shown in the
supplementary material.
Thermal control for separation tuning and single-soliton generation. All above
results are obtained by maintaining the substrate temperature at 290 K (17 °C)
with the aid of a thermoelectric cooler (TEC). However, only multi-solitons
can be triggered in MRR2 and MRR3, which possess a relatively large
$\Delta\lambda$ of 0.125 and 0.127 nm. Next, we will show the effective
control of the mode separation and soliton state by changing TEC temperature
T. Figure 5(a) draws the dual-mode transmission spectra in MRR2 at different
temperatures. As the temperature increases, a thermally induced redshift of
the resonant wavelengths are observed with a d$\lambda$/dT of $\sim$0.02 nm/K,
corresponding to a thermo-optic coefficient of $\sim$ 2.3$\times$10-5/K, which
is consistent with the result reported in Arbabi and Goddard (2013). The
$\Delta\lambda$ declines from 0.125 to 0.086 nm when T increases from 290 to
350 K, indicating that the resonant wavelength of the TE00 mode is more
sensitive to the temperature variation. We also note that the coupling between
the two modes is strengthened when they approach each other, leading to the
increase in the extinction ratio of the TE10 mode. Figure 5(b) depicts the
relation between on-chip power and soliton number at various temperatures. It
can be seen that only TSMs can be reached at the temperature of 290 and 300 K,
while the single-solitons arise when T$\geq$310 K. When 310$\leq$T$\leq$330 K,
low power can trigger single-soliton only, while conversely, TSMs tend to be
formed at high power, which is similar to the trend shown in Fig. 4(b).

Table 1: Comparison of octave-spanning DKSs. | Material
---
|  Q-factor
---
(million)
| On-chip
---
power (mW)
| Spectral range
---
(THz)
| frep
---
(THz)
| SER
---
(GHz)
Accessing method
Si3N4Li et al. (2017) | 2 (Qint) | 120$\pm$15 | 129-275 |  $\sim$1 |  $\sim$1.5 |  | Adiabatic pump sweeping (-100 GHz/s)
---
with 1P2R-2P scheme
Si3N4Pfeiffer et al. (2017) |  $\sim$1(Qload) | 455 | 130-280 |  $\sim$1 | - |  | Forward sweeping and backward tuning
---
Si3N4Briles et al. (2018) | - | 200 | 130-310 |  $\sim$1 | - | Fast pump sweeping with frequency shifter
Si3N4Briles et al. (2021) | 2.7(Qint) | 40 | 140-280 |  $\sim$1 | - |  | Self-injection locking
---
AlNWeng et al. (2021a) | 1.4(Qint) |  $\sim$335 | 130-273 |  $\sim$0.37 |  $\sim$10.4 |  | Adiabatic pump sweeping (-125 GHz/s)
---
with 1P2R-1P scheme
AlNLiu et al. (2021a) | 1.6(Qint) |  $\sim$390 | 130-295 |  $\sim$0.43 | - |  | Fast pump sweeping with
---
single-sideband modulator
LiNbO3He et al. (2021) | 1.15(Qload) |  $\sim$600 | 125-268 |  $\sim$0.2 |  $\sim$0.2 |  | Self-start (photorefractive effect)
---
| Si3N4
---
(this work)
$\sim$1.1(Qint) |  | 140
---
(200, 230)
| 125-320
---
(127-270)
|  $\sim$1
---
($\sim$2, $\sim$3)
$\sim$17 |  | Adiabatic pump sweeping (-125 GHz/s)
---
with 1P2R-1P scheme

Figure 5(c) shows examples of transmission spectra measured at high power,
where the relevant soliton number is labelled at the top. Specifically, the
TSM with a notable SER of $\sim$23 GHz is observed when T=300 K and Pin=190
mW. At 320 K, a soliton switching from N=2 to N=1 is observed when Pin=170 mW.
The deterministic access to single-soliton state is also realized at 320 K,
accompanied by a maximum SER of $\sim$17 GHz, slightly wider than the
0.105-nm-wide $\Delta\lambda$ ($\sim$13 GHz). The single-soliton spectra
acquired at 310, 320, and 330 K are plotted in Fig. 5(d) with pump wavelengths
of 1566.550, 1566.750, and 1566.950 nm, respectively. The spectra have similar
profiles and range from 125 to 320 THz, well beyond an octave span. As with
the results originating from MRR1, dual-DW at the frequency of 130 and 312 THz
are observed. The inset exhibits the two adjacent 1st-order and 2nd-order comb
lines near 125.5 THz. At 310, 320, and 330 K, the fceo is calculated to be
$\sim$105, $\sim$108 and $\sim$109 GHz, respectively, which is about half of
the fceo shown in Fig. 3(d) from MRR1.

By setting the temperature of MRR3 at 330 K and tuning the pump wavelength to
1566.885 nm, the single-soliton can be stably accessed [see Fig. 5(e)], which
has a similar profile as Fig. 3(c)-(ii) (MRR1) and Fig. 5(d) (MRR2). In this
case, the $\Delta\lambda$, SER and fceo are $\sim$0.1 nm ($\sim$12.5 GHz),
$\sim$11 GHz and $\sim$100 GHz, respectively. It should be mentioned that the
MRR2 and MRR3, with almost identical mode separation and fceo, are in the same
chip, which strongly suggests the fabrication uniformity. These results
indicate that temperature control will be crucial for the deterministic
creation of single-solitons.

##  IV Discussion

The results demonstrate that the 1P2R-1P scheme is applicable to our
23-$\mu$m-radius Si3N4 MRRs with the proper mode separation (e.g., 10-13 GHz).
The SER of single-soliton is generally equivalent to $\Delta\lambda$, while
the window of multi-soliton is up to almost twice $\Delta\lambda$. Table 1
compares the reported octave-spanning DKSs realized with various platforms.
Clearly, the SER is much expanded with an auxiliary resonance in our 1P2R-1P
scheme. More importantly, the proposed strategy enables the access to soliton
state via straightforward pump frequency control instead of rapid frequency
scan or complicated control. The present octave-spanning single-solitons are
generated by the Si3N4 MRRs with Q of $\sim$1.1 million, but the ongoing
experiments suggest that a 2.7 million Q will reduce the required on-chip
power to <40 mW Briles et al. (2021), which paves a way towards a miniaturized
soliton system that is integrated with a laser diode. We also demonstrate the
octave-spanning soliton crystals generation.

We believe that the proposed 1P2R-1P approach for deterministic access to DKS
could have profound significance on the microcomb field if a design could be
reproduced in fabrication with a reasonable yield. Some solutions can be
adopted to further control the mode separation and improve the yield. First,
according to Fig. 2, a fine (a few nm) ring dimension scan is necessary when
designing the layout to ensure the accuracy of relative variation in
fabrication. Considering the reliability, uniformity, and ability to fabricate
a high density of MRRs in the commercial foundry, a reasonable variation in
MRR dimensions is possible to provide more samples featuring both the desired
dual-mode and low fceo. Second, post-fabrication such as etching Moille et al.
(2021) can be used to tune the resonance characteristics and the mode spacing.
Finally, as investigated with MRR2 and MRR3, temperature control can
effectively modify the mode separation and change the soliton state. The
control can be implemented by a substrate TEC or surface microheaters, which
has been demonstrated for the thermal tuning of frep and fceo Xue et al.
(2016). In practice, we can also tune pump wavelength or change the laser
source if the real dual-mode region deviates from the designed position.

##  V Conclusions

In summary, we demonstrate the accessing of octave-spanning single-soliton,
soliton crystals, and multi-solitons in the dual-mode microresonators via
simply slow pump tuning. In addition to rich soliton states, the conventional
inaccessible soliton step is stabilized now accompanied by an expanded
detuning range. Compared with the results achieved by the 2P2R method using an
independent auxiliary laser Zhou et al. (2019), the demonstrated SER of 17 GHz
here has been significantly enhanced by two orders of magnitude. Such a broad
soliton existence window will greatly enhance the potential for microcomb use
in applications such as parallel FMCW LiDAR Riemensberger et al. (2020). The
soliton behavior dependence on the pump power and temperature is also
explored.

The proposed straightforward and low-cost soliton generation system 1P2R-1P is
universally feasible for appropriately designed microresonators. We have
demonstrated this first in an AlN MRR and expanded on this in the present
study. It should also be possible across other platforms for soliton microcomb
generation in LiNbO3 and SiC as examples. It is foreseeable that, by using a
passive dual-mode microresonator with an upgraded Q such as $\sim$5×106 Ji et
al. (2021b), the photonic integrated octave-spanning coherent microcomb source
will be delivered soon, which is driven by a laser diode with a power of about
100 mW but without amplifier or optical feedback. We also note that a recent
work Lei et al. (2022) shows the positive effects of 1P2R-2P in improving the
timing jitter and effective linewidth of the soliton microcomb lines. We
believe such a simple system and the versatile soliton states will not only
accelerate the achievement of commercial, portable and affordable soliton
microcomb sources but also contribute to the extension of their applications.

Supplementary material See the supplementary material for additional
information.

Acknowledgements This project is supported by the Science Foundation Ireland
(Grant No. 17/NSFC/4918) and the National Natural Science Foundation of China
(Grant No. 61861136001).

Competing interests The authors have no conflicts of interest to disclose.

Data availability The data in my manuscript can be obtained from the
corresponding author.

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Supplementary Material: Dual-mode microresonators as straightforward access to
octave-spanning dissipative Kerr solitons

##  I Experiment setup

Figure S1 shows the experimental setup for soliton microcomb generation and
characterization. The light source used in the experiment is a tunable laser
ranging from 1480 to 1640 nm (Santec TSL-710). An erbium-doped fiber amplifier
(EDFA) is adopted to amplify the laser power for pumping the resonator through
a lensed fiber with a spot size of 2.5 $\mu$m. A fiber polarization controller
(FPC) is utilized to adjust the polarization. The generated soliton microcombs
are collected with another lensed fiber for characterization. The optical
spectra are recorded by two optical spectrum analyzers (1200-2400 nm and
600-1700 nm range). By injecting the output into an electrical spectrum
analyzer (ESA) after a photodiode (PD), the radio-frequency (RF) intensity
noise of the combs can be detected. The autocorrelation measurement shown in
Fig. 3(e) is carried out by the autocorrelator (AC). For the transmission
measurement, the output is divided into two parts, one is for all-transmission
and the other one is for the pump-transmission only after a band-pass filter.
By calculating the difference between the two branches, we can easily identify
the soliton steps in the transmitted curves, as shown in Figs. 3(a) and 4(a)
in the main text.

Figure S1: Experimental setup used for dissipative Kerr solitons (DKSs)
generation. FPC, fiber polarization controller; TEC, thermoelectric cooler;
EDFA, erbium-doped fiber amplifier; OSA, optical spectrum analyzer; OSA1,
600–1700 nm; OSA2, 1200–2400 nm; FBG, fiber Bragg grating; PD, photodiode;
ESA, electrical spectrum analyzer; AC, autocorrelator.

##  II Mode coupling in MRR1 and the generated MI comb

The transmission spectrum of MRR1 measured in a wide range is shown in Fig.
S2(a), where the TE00 and TE10 modes are marked by circles and crosses,
respectively. The dual-mode near 1565 nm used for the soliton generation is
labelled with a relative mode number $\mu$=0. Figure S2(b) plots the
experimental loaded Q factors, where the values of TE00 modes have an abrupt
change between $\mu$=0 and $\mu$=-1 due to the mode coupling. The extinction
ratio of TE00 mode also changes significantly at $\mu$=-1. Figure S2(c) shows
the free spectra ranges (FSRs) of the two mode families. Clearly, the
fundamental and first-order modes have a statistical FSR of $\sim$1012 and
$\sim$979 GHz, respectively. However, because of the mode coupling, the
obvious FSR changes are observed for the TE00 mode at $\mu$=0 as well as TE10
modes at $\mu$=-1 and -2. Such doublet resonances with avoided mode crossing
are pretty general especially in the resonators with relatively small FSR
[1-3]. It has also been exploited for dispersive wave formation [4] and
soliton crystal (SC) generation [5].

Figure S2: (a) Transmission spectrum of MRR1, where the TE00 and TE10 modes
are marked by circles and crosses, respectively. The resonances at relative
mode number $\mu$ = 0 have a minimum separation of 0.084 nm. Experimental (b)
Q factors and (c) FSRs of the TE modes.  Figure S3: (a) The collected output
power of all-transmission, pump-transmission, and their difference when
pumping the TE00 mode at $\mu$=2. (b) Measured spectrum of MI comb.

For comparison, we also pump the TE00 mode at $\mu$=-2, which has a separation
of 0.62 nm with the adjacent TE10 mode. Figure S3(a) shows the measured
transmission spectra at an on-chip pump power (Pin) of 150 mW. There is no
overlap between the two resonances, indicating that the modes will be pumped
independently. By sweeping the pump wavelength to 1549.11 nm, just before
dropping out of the TE00 resonance, we can successfully access the modulation
instability (MI) comb. As shown in Fig. S3(b), the MI comb ranges from 135 to
242 THz, below an octave span. It has a similar envelope like the one obtained
from the fundamental mode at $\mu$=0 [see Fig.3(c)-(i)]. Thus the dual-mode
scheme could decrease the required power to reach the soliton state.

##  III Three-soliton state and soliton crystals from various MRRs

In this section, we will introduce the other two 1.68-$\mu$m-wide dual-mode
MRRs firstly and then present the versatile soliton states. Figures S4(a) and
S4(c) show the transmission spectra of the dual-mode in MRR4 and MRR5, which
have a separation of 0.067 and 0.019 nm, respectively. The transmitted power
curves at 200 mW excluding the pump of MRR4 are plotted in Fig. S4(b), where
the visible steps correspond to the formation of two-soliton (0.02-nm-wide)
and three-soliton (0.11-nm-wide) states. For MRR5, the extinction ratio of the
TE10 resonance is even higher than the TE00 resonance, suggesting a strong
mode coupling between the two modes. 3- and 4-SC are realized with MRR5 using
the slow pump sweeping technique. There is no single-soliton observation for
these two resonators when increasing the TEC temperature, which we attribute
to the relatively small mode separations. Nevertheless, deterministic
generation of the multi-solitons with pronounced soliton existence window
manifests the reliability and practicality of our approach.

Figure S4(d) shows the octave-spanning three-soliton combs generated with
MRR1, MRR3, and MRR4 under the pump power of 210, 320, and 200 mW,
respectively. Similar three-soliton states are reproducible among different
chips. In addition, the three-solitons obtained from MRR3 have stronger
intensities due to the narrower coupling gap between resonator and waveguide.
Figure S4(e) summarizes the spectra of SCs. The 3-SC spectra originating from
MRR4 exceeds an octave-spanning (130-270 THz) with an estimated conversion
efficiency of 12.8$\%$. However, because of the smaller mode separation and
enhanced mode coupling in MRR5, the 3-SC and 4-SC have a relatively narrow
bandwidth (140-230 THz).

Overall, it is easier to access the single-soliton state in the dual-mode
resonators (MRR1, MRR2, and MRR3) with a relatively large mode separation
(e.g., 10-13 GHz). However, we perceive that the mode separation is a critical
but not the only factor that affects the soliton behavior. More intensive work
needs to be implemented in the future to sort out the influence of mode
coupling on thermal compensation and soliton behavior.

Figure S4: Transmission spectrum for the dual-mode in (a) MRR4 and (c) MRR5.
(b) Measured transmitted power curves excluding the pump in MRR4 when Pin=200
mW. Experimental spectra of (d) three-solitons and (e) soliton crystals from
different microresonators.

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