# Binary neutron star mergers as the source of the highest energy cosmic rays

Glennys R. Farrar<EMAIL_ADDRESS>Center for Cosmology and Particle Physics,
Department of Physics
New York University, New York, NY 10003, USA

###### Abstract

We propose that ultrahigh energy cosmic rays are produced in binary neutron
star mergers. Interpreting the highest energy events as r-process nuclei
eliminates the need for exotic sources, while the observed near-universal
maximum rigidity of ultrahigh energy sources can be understood as due to the
uniformity of jets generated by the gravitationally-driven dynamo, given the
narrow range of total binary neutron star masses. We discuss evidence for this
scenario, and its prediction of coincidences between neutrinos above 10 PeV
and gravitational waves.

Introduction
Major advances in ultrahigh energy cosmic ray (UHECR) observations in the past
two decades have lead to a much better understanding of their spectrum and
composition, but the central question remains: What are the systems
responsible for accelerating cosmic rays to the highest observed energies? We
argue here that two particularly puzzling features of the observations
strongly suggest that extragalactic cosmic rays are produced in binary neutron
star (BNS) mergers, with the highest energy events being r-process nuclei such
as tellurium. This BNS-merger scenario explains all current observations,
within present uncertainties, as we show.

Options for UHECR sources are constrained by the observed rate and energetics,
the spectrum and composition, inhomogeneities in arrival directions (dipole
and higher multipoles and hotspots), and constraints from extremely high
energy events such as the recent Amaterasu particle Abbasi  _et al._ (2024a).
Additional crucial facts are that individual accelerators have a nearly
universal maximum rigidity, which – under the conventional assumption that the
heaviest UHECRs are iron nuclei – requires an entirely different source class
to explain the highest energy events Abbasi  _et al._ (2024a); Unger and
Farrar (2024). After giving a brief overview, we review the constraints on
UHECR sources and note why previously proposed source candidates are
problematic in one way or another, motivating the need to seek new
possibilities. We show that BNS-mergers can explain all of the observations as
of now, and close with a discussion of tests and implications of this
scenario.

Overview
The very large exposures of the Pierre Auger Observatory in Argentina and the
Telescope Array in Utah (TA) have resulted in $\approx$ 50,000 events having
been recorded above 10 EeV ($10^{19}$ eV) Abdul Halim  _et al._ (2023a), of
which more than 40 are above 100 EeV Abdul Halim  _et al._ (2023b). At these
energies, Galactic cosmic rays (GCRs) make a negligible contribution. The
cutoff in the spectrum at the highest energies is due to a combination of
limited accelerating-power of the sources and Greisen-Zatsepin-Kuzmin
processing Greisen (1966); Zatsepin and Kuzmin (1966); Halim  _et al._ (2023).
For protons, GZK energy loss is due primarily to inelastic scattering on
cosmic microwave background photons. For nuclei, interactions with the CMB and
other extragalactic background light cause spallation or breakup; these
interactions conserve the energy per nucleon. The rigidity, $\mathcal{R}\equiv
E/Ze$, where $E$ is energy and $Ze$ is charge, is twice the energy per
nucleon, for common nuclei except protons, and thus is approximately conserved
during propagation.

The observed composition above 10 EeV is consistent with UHECRs being nuclei
with a relatively narrow range of rigidities centered on $\mathcal{R}\approx
4$ EV. (See, e.g., Fig. 4 of Bister and Farrar (2023).) Below 10 EeV lies a
transition region where the low end of the extragalactic spectrum overlaps the
highest energy Galactic cosmic rays. The light extragalactic component is
well-explained as resulting from spallation of primary UHE nuclei as they pass
through the environment surrounding the accelerator Unger  _et al._ (2015).
The protons’ origin as spallation products – revealed by their having a common
energy per nucleon rather than rigidity, with respect to other UHECRs – is a
key indicator of the validity of this mechanism.

Constraints on UHECR sources
1\. Universal maximum rigidity: Arguably the most puzzling feature of UHECRs
is that the spread in nuclear masses appearing at any given energy is so small
Aab  _et al._ (2014), such that the rigidity spectrum is very narrow – only a
factor-few in width. It is to be expected that different nuclei accelerated in
an individual source have essentially the same spectrum in rigidity, since
known acceleration mechanisms depend only on rigidity and that is preserved
during propagation as discussed above. However for source classes proposed up
to now, different sources should exhibit different maximum rigidities so that
an ensemble of sources would in general have a broad rigidity spectrum,
contrary to observations Halim  _et al._ (2023). This problem was
systematically investigated in Ref. Ehlert  _et al._ (2023), where it was
shown that none of the conventional source candidates (blazars, long GRBs,
TDEs) have a sufficiently small population variance to yield a distribution of
maximum rigidities consistent with the UHECR composition data. (Seyfert
galaxies – low luminosity AGN – were found to be marginally compatible within
the scope of the analysis but vulnerable to exclusion when all sources of
variance are taken into account Ehlert  _et al._ (2023).)

2\. The Hillas criterion: A candidate UHECR source must be capable of
accelerating UHECRs to the observed rigidity. The characteristic size of the
accelerating region, $R$, must be large enough to confine the UHECR – i.e.,
$R$ cannot be significantly smaller than the Larmor radius of the cosmic ray.
This leads to the Hillas criterion relating the maximum rigidity to which
UHECRs are accelerated to the size and field strength in the source:

$\mathcal{R}_{\rm max,EV}\lesssim 3\times 10^{-11}\,\Gamma_{\rm jet}\,R_{\rm
km}\,B_{\rm G}~{},$ (1)

where $B_{\rm G}$ is the magnetic field strength of the jet in Gauss,
$\Gamma_{\rm jet}$ is its relativistic boost factor and $R_{\rm km}$ is the
characteristic size of the confinement region in km; see, e.g., Farrar and
Gruzinov (2009). Since the Poynting luminosity is proportional to the cross-
sectional area times $\vec{E}\times\vec{B}$, the Hillas condition (1) on $RB$
means that the minimum Poynting luminosity of the jet is determined,
independently of $R$ and $B$ separately. This implies a condition on the
(isotropic equivalent) total power of a jet capable of accelerating CRs to a
rigidity $\mathcal{R}_{\rm max,EV}$; e.g.. from Farrar and Gruzinov (2009):

$\mathcal{L}_{\rm bol}>\frac{c}{6}\,\Gamma_{\rm jet}^{4}B^{2}R^{2}\approx
10^{41}\,\Gamma_{\rm jet}^{2}\,\mathcal{R}_{\rm max,EV}^{2}\,{\rm erg/s}~{}.$
(2)

If UHECRs were protons so $\mathcal{R}=E$, this would be a very stringent
requirement and rule out all but extremely luminous sources.

3\. Total UHECR energy injection: In addition to the conditions (1) and (2), a
source class capable of explaining all UHECRs must account for the observed
total flux of UHECRs. Since UHECRs lose energy during propagation due to GZK
processes, a steady replenishment is required. The volumetric energy-injection
rate for UHECRs above 10 EeV is Muzio  _et al._ (2022)

$\dot{\mathcal{Q}}=6\times 10^{44}\text{ erg\, Mpc}^{-3}\,\text{yr}^{-1}~{}.$
(3)

(The low energy limit chosen to define $\dot{\mathcal{Q}}$ only matters at the
$\mathcal{O}(1)$ level since the spectrum emerging from the source is hard,
$dN/dE\sim E^{-p}$ with $p\lesssim 1$ Halim  _et al._ (2023); the 10 EeV
threshold adopted here avoids contamination by Galactic cosmic rays.) For
continuous sources, Eq. (3) constrains the product of the number density of
sources times their mean luminosity in UHECRs: $n_{S}\,\mathcal{L}_{\rm
UCR}=\dot{\mathcal{Q}}$. For transient sources, the constraint is
$\Gamma_{S}\,\mathcal{E}_{\rm UCR}=\dot{\mathcal{Q}}$, where $\Gamma_{S}$ is
the volumetric rate of the transient sources and $\mathcal{E}_{\rm UCR}$ is
the total energy in UHECRs above the specified threshold, produced in an
average transient event.

4\. Number density of contributing sources: The observations of UHECR arrival
directions are very constraining on candidate source classes, even without
individual sources having been identified. Auger finds that the higher
multipoles beyond dipole are consistent with those of an isotropic
distribution; this upper limit on small and intermediate scale inhomogeneities
was used in Ref. Bister and Farrar (2023) to constrain the density of UHECR
sources to

$n_{S}\gtrsim 10^{-3.5}\,{\rm Mpc}^{-3}.$ (4)

This number density constraint disfavors rare continuous sources such as
blazars, powerful AGN and starburst galaxies, apart from a marginally viable
narrow window for sufficiently high extragalactic magnetic smearing. Figure 1
in the Appendix reproduced from Bister and Farrar (2023) displays the
constraint.

For transients, the constraint (4) is on the effective source density
$n_{S}^{\rm eff}\equiv\Gamma_{S}\,\tau$, where $\Gamma_{S}$ is the volumetric
rate of transients producing UHECRs above the energy threshold and $\tau$ is
the mean arrival time spread of the UHECRs from a transient event within the
GZK horizon. Due to UHECR deflections in the magnetic fields between source
and detector, the UHECRs arriving at the Milky Way are spread over an angle
$\psi$ and arrive with time delay up to $\tau$ Achterberg  _et al._ (1999):

$\displaystyle\psi$ $\displaystyle\approx
30^{\circ}\beta_{\text{EGMF}}\sqrt{D_{\textrm{M}pc}}/\mathcal{R}_{\rm EV}$ (5)
$\displaystyle\tau$ $\displaystyle\approx
0.14\,(D_{\textrm{M}pc}\beta_{\text{EGMF}}/\mathcal{R}_{\rm
EV})^{2}\,{\textrm{Myr}}~{},$ (6)

where the extragalactic magnetic smearing parameter is

$\beta_{\mathrm{EGMF}}\equiv
B_{\mathrm{EGMF}}/\mathrm{nG}\,\sqrt{L_{c}/\mathrm{Mpc}}~{}.$ (7)

Commonly adopted estimates are $B_{\mathrm{EGMF}}\approx 1$ nG, and $L_{c}$ in
the range of 10 kpc to 1 Mpc, giving $\beta_{\mathrm{EGMF}}$ from 0.1 to 1.
For jetted sources, continuous or transient, the contributing source density
is reduced relative to the actual density by the fraction $f_{\rm contrib}$,
which may be crudely estimated as $f_{\rm contrib}\approx{\rm
sin}(\sqrt{\delta_{j}^{2}+\psi^{2}})$ with $\delta_{j}$ being the jet opening
angle. The mean source distance of UHECRs above 32 EeV is 70 Mpc Bister and
Farrar (2023), so for a 2-sided jet whose opening angle is 10∘ the sky-
averaged value of $f_{\rm contrib}$ ranges from 0.2 for small
$\beta_{\text{EGMF}}$ to 1 for $\beta_{\text{EGMF}}\gtrsim 0.6$. Long GRBs and
visibly jetted TDEs – those TDEs whose jets are visible almost immediately –
are disfavored as the exclusive source of UHECRs by this criterion, being
marginally possible only if the mean $\beta_{\text{EGMF}}$ is near its maximum
allowed value Bister and Farrar (2023); see Fig. 1 in the Appendix.

5\. Amaterasu Event: The very energetic cosmic ray Amaterasu detected by TA
Abbasi  _et al._ (2024b) provides direct evidence that at least some UHECRs
are produced by transient events, as demonstrated in Unger and Farrar (2024).
In brief, because of its high energy, GZK energy losses during propagation
severely restrict the distance of Amaterasu’s source. Its nominal energy, in
view of its likely composition, is $E_{\text{nom}}=(212{\pm}25)$ EeV for which
the source must be within $\approx 13$ Mpc Unger and Farrar (2024). Taking
into account the uncertainties in energy and composition, Ref. Unger and
Farrar (2024) backtracked Amaterasu through the Galactic magnetic field using
the recent UF23 suite of GMF models Unger and Farrar (2023), which capture the
uncertainty in GMF deflections. The maximum source volume contains only
ordinary galaxies and a few low-luminosity AGNs, suggesting that Amaterasu was
created by a transient event in an otherwise ordinary galaxy.

Source candidates
Early favorite candidate UHECR sources were the jets found in powerful AGN or
long GRBs Vietri (1995); Waxman (1995). However both are now essentially
excluded, except as the source of a small fraction of events, by (i) the
source density requirement, $n_{S}\gtrsim 10^{-3.5}\,{\rm Mpc}^{-3}$, (ii) the
standard source requirement, and (iii) the absence of those sources in the
Amaterasu source volume. An additional problem for long GRBs is that they
cannot account for the volumetric energy injection rate, $\dot{\mathcal{Q}}$,
unless the UHECR energy in GRBs exceeds the energy in gamma rays by two orders
of magnitude Farrar and Gruzinov (2009).

AGN minimally satisfying the bolometric luminosity condition to accelerate to
5 EV, $\mathcal{L}_{\rm bol}\gtrsim 10^{42.5}$ erg/s, may be marginally
consistent with the number density requirement – c.f., the AGN luminosity
function in Duras, F.  _et al._ (2020). However still lower luminosity AGN are
much more abundant – Zaw et al. Zaw  _et al._ (2019), finds that 40% of the
SDSS galaxies out to $z=0.07$ are either broad-line AGN or satisfy the Kewley
narrow-line AGN criterion – so the narrow rigidity distribution of UHECRs (the
standard reach condition) is implausible for continuous AGN; see also Ehlert
_et al._ (2023).

Another possibility for UHECR acceleration was suggested in Farrar and
Gruzinov (2009): short-lived but powerful jets in giant flares of otherwise
low-luminosity AGNs, produced by a disk instability or accretion event, or the
jets of tidal disruption events (TDEs) when a star passes close enough to a
supermassive black hole to be shredded, forming an accretion disk and
sometimes a jet. TDEs were subsequently discovered van Velzen  _et al._ (2011)
and the rate found to be potentially sufficient van Velzen and Farrar (2014).
Many TDEs have now been detected, and several spatio-temporal correlations
between TDEs and high energy astrophysical neutrinos have been reported Stein
_et al._ (2021); Reusch  _et al._ (2022); Jiang  _et al._ (2023); see Piran
and Beniamini (2023) for a recent discussion. However it remains to be seen
whether jetted TDEs are sufficiently abundant to explain the smoothness of the
UHECR arrival direction distribution (4) and the energy injection requirement
(3). Extreme-jetted TDEs – the rare cases when a powerful jet is almost
immediately apparent – are too rare Bister and Farrar (2023) – but recent
observations and theory suggest that jets may only become visible through
radio emission a year or more after the merger event, so the jury is out on
whether jetted TDEs could by themselves produce all the UHECRs. However even
if capable TDEs are sufficiently abundant, the near-universal maximum rigidity
of UHECR sources is a serious challenge to the TDE scenario because TDEs are a
very diverse population unlikely to have a narrow distribution of acceleration
capability.

A distinctly different but appealing proposal is that UHECRs are accelerated
in accretion shocks of galaxy clusters and filaments Simeon  _et al._ (2023).
However the Amaterasu event’s origination in a region not containing a galaxy
cluster, requires that mild as well as massive shocks accelerate to the
highest energies, which seemingly would imply a very broad distribution of
source maximum rigidities, contrary to the data.

Binary Neutron Star mergers & UHECRs
Binary neutron star mergers provide a natural rationale for the near-universal
maximum rigidity: Gravity and angular momentum govern the dynamo process which
generates the jets, and the range of total binary mass is very narrow with
just a 5% spread around the mean 111The website of P. Freire lists 20
confirmed binary neutron star systems whose masses have been measured; see
Özel and Freire (2016) for a review. The mean value of the total mass in these
20 cases is $(2.642\pm 0.137)M_{\odot}$. It may seem surprising that the range
of total mass of BNS systems is so narrow, given that neutron star masses have
been observed from as low as about $1M_{\odot}$ to above 2 $M_{\odot}$, but
neutron stars with masses significantly different from
$\approx\\!1.33\,M_{\odot}$ are generally paired with white dwarfs, normal
stars or black holes, while the formation of a binary NS without disruption of
the binary appears to occur only for a narrow mass range; see Sgalletta  _et
al._ (2023) for a discussion and population synthesis modeling of binary
neutron stars.. Thus the jets produced in BNS mergers can be expected to be
nearly the same from one merger to another 222The wide variation in the light
curves of short GRBs does not contradict this; simulations constrained to the
kilonova associated with the binary neutron star merger GW170817 show that the
variation in the observed light curves of short GRBs is attributable to
different viewing angles of the jet and different circumbinary medium Xie  _et
al._ (2018).. The very high-resolution GR-MHD simulation of the merger of two
1.35 $M_{\odot}$ neutron stars in Ref. Kiuchi  _et al._ (2023) demonstrates
that dynamo action produces a pair of jets with fields in excess of $10^{15}$
G on multi-km length scales, easily satisfying the Hillas criterion.

Having motivated that BNS mergers can have a common maximum rigidity at the
required factor-few level, we turn to the effective source density and global
UHECR energy injection requirements. Both of these depend on the BNS merger
rate. Estimating the rate depends on assumptions, so it is reassuring that two
different methods give similar results, albeit presently with large
uncertainties. The LIGO-Virgo-Kagra collaboration’s conservative range on the
BNS merger rate is 10-1700 $\rm{Gpc^{-3}\,yr^{-1}}$ Abbott  _et al._ (2023).
This is compatible with the recent detailed study of the afterglows of 29
short GRBs Rouco Escorial  _et al._ (2023) which finds, using events with
well-measured opening angles, a beaming-corrected short GRB rate of
$1786^{+6346}_{-1507}$ $\rm{Gpc^{-3}\,yr^{-1}}$. Below, we take the BNS merger
rate to be $\Gamma_{3}\,\\!\times\\!10^{3}\,{\rm Gpc}^{-3}\,{\rm yr}^{-1}$.

The contributing source density constraint is readily satisfied by BNS
mergers, as follows. Using Eq. (6), with mean source distance of 70 Mpc and
mean rigidity 4.5 EV for the more demanding 32 EeV threshold Bister and Farrar
(2023) requires $n_{\rm eff}\gtrsim 34\,\Gamma_{3}\,\beta_{\rm EGMF}^{2}\,{\rm
Mpc}^{-3}$. The opening angle of the jet is 12∘ Kiuchi  _et al._ (2023) so
$f_{\rm contrib}\approx 0.2-1$ and the effective source density requirement
derived in Bister and Farrar (2023), $n_{\rm eff}\gtrsim 10^{-3.5}\,{\rm
Mpc}^{-3}$, is satisfied as long $\beta_{\rm EGMF}^{2}\,\Gamma_{3}\,f_{\rm
contrib}\gtrsim 10^{-5}$. The conservative lower bound on the BNS merger rate
from gravitational waves gives $\Gamma_{3}>10^{-2}$ Abbott  _et al._ (2023) so
with $f_{\rm contrib}\approx 0.6$, the observed source density requirement is
satisfied as long as $\beta_{\rm EGMF}\gtrsim 0.04$, as is expected.

Accounting for the entire observed UHECR energy injection rate is also
possible within present uncertainties. The simulation of Ref. Kiuchi  _et al._
(2023) finds a Poynting luminosity in excess of $10^{51}$ erg/s and a
magnetically-driven wind with kinetic energy $5\\!\times\\!10^{50}$ erg, when
the simulation ends about 150 ms post-merger. For a typical 1s duration, this
suggests an energy budget for UHECRs of
$1.5\,\epsilon_{1.5}\\!\times\\!10^{51}$ erg, giving the observed
$\dot{\mathcal{Q}}$ if $\epsilon_{1.5}\Gamma_{3}=0.4$, comfortably in the
constrained range. Simulations of particle acceleration in BNS merger jets are
needed, especially through the recently-appreciated mechanism of plasmoid-
mediated magnetic reconnection Comisso and Sironi (2019); Zhang  _et al._
(2021); Li  _et al._ (2023) which is expected to be dominant in Poynting-flux
dominated outflows, because Kiuchi  _et al._ (2023) does not take into account
particle acceleration and feedback. Simulations are also needed to check that
photons do not overly dissociate nuclei; Metzger  _et al._ (2011) showed this
is not a problem for long GRBs and it is likely also the case for BNS merger
jets.

Tests and implications
R-process UHECRs and the highest energy CRs: In the BNS-merger scenario, most
UHECRs are typical nuclei of the ambient medium around the binary, which get
swept up in the jets and accelerated. They may be spallated to some extent
while escaping through the medium surrounding the accelerator and during
extragalactic propagation Unger  _et al._ (2015). In addition, a contribution
of r-process elements can be expected. Half of all r-process nuclei are
produced by nucleosynthesis in binary neutron star mergers which is generally
thought to take place in ejecta in the equatorial plane of the merger Kasen
_et al._ (2017). Occasionally, a heavy nucleus can migrate into a jet and be
accelerated, enhancing the fraction of r-process elements in UHECRs relative
to the Galactic average. The masses of r-process nuclei peak around $A=130$
(Te-Xe; Z = 52-54) and $A=195$ (Pt; Z = 78); see Cowan  _et al._ (2021) for a
recent review. Taking $\mathcal{R}\approx 4.5\pm 0.5$ EV to approximate the
mean universal rigidity Bister and Farrar (2023), the energies of these peak
r-process nuclei after acceleration in the source are $E\approx 240\pm 25$ EeV
and $350\pm 40$ EeV.

The proposal here that UHECRs originate in BNS mergers, and hence naturally
contain some component of r-process elements, provides the first compelling,
non-exotic explanation for the highest energy UHECR ever recorded: the Fly’s
Eye “OMG" event Bird  _et al._ (1995). Its energy was originally estimated to
be 320$\pm 90$ EeV. Simply rescaling that value using the modern air
fluorescence yield Ave  _et al._ (2013) implies a more accurate estimated
energy assignment is 250$\pm 70$ EeV; a more detailed re-analysis updating the
pressure dependence is warranted. As we now show, the Fly’s Eye data is
comfortably explained as an r-process nucleus for energy assignments in this
range. A similar analysis applies to the Amaterasu particle.

Since the rigidity range is very narrow, we can estimate the charge of the OMG
event for any given energy using the mean rigidity of 4.5 EV Bister and Farrar
(2023). This results in an estimated charge of $(56\pm
15)\left(\frac{4.5}{\mathcal{R}}\right)$ for $E=250\pm 70$ EeV, suggesting
this event originated near the Te-Xe r-process peak, since the charge loss of
heavy nuclei in propagation is small 333An example relevant for this purpose
is the detailed study of the Amaterasu particle reported in Unger and Farrar
(2024). Existing codes to calculate energy losses during propagation do not
treat higher masses than Fe but Ref. Metzger  _et al._ (2011) gives an
approximate treatment which suggests that nuclei heavier than iron may have
energy-loss horizons considerably larger than Fe, which would be important in
quantitatively interpreting abundances of extreme energy cosmic rays in future
datasets..

The depth-of-shower-maximum of the OMG event was measured to be $X_{\rm
max}=815^{+60}_{-53}\,{\rm g\,cm}^{-2}$ including both systematic and
statistical uncertainties Bird  _et al._ (1995). We compare this to the
expectations under the illustrative interpretation of $(A,E)=(130,250\,{\rm
EeV})$. The predicted $X_{\rm max}$ for Fe at the nominal energy is $800\,{\rm
g\,cm}^{-2}$, using Sybill23d to simulate the shower development. The depth-
of-shower-maximum shifts with nuclear mass and incident cosmic ray energy as
$d\langle X_{\rm max}\rangle/d\,{\rm log_{10}A}\approx-d\langle X_{\rm
max}\rangle/d\,{\rm log_{10}E}\approx-60\,{\rm g\,cm}^{-2}$ Kampert and Unger
(2012). For E= 250 EeV the tellurium identification predicts $X_{\rm
max}=773\,{\rm g\,cm}^{-2}$, consistent with observation within $1\sigma$.

In sum, the Fly’s Eye data on both the energy deposit and the depth of shower
maximum of the OMG event fit well with its being an ultrahigh energy cosmic
ray nucleus in the tellurium peak of r-process nucleosynthesis. Thus the very
existence of this event can be considered tentative evidence of the binary
neutron star merger origin of UHECRs. (If the maximum UHECR charge is 26,
which appears to have been universally assumed up to now, the Fly’s Eye event
must either have an unknown exotic origin or have a rigidity
$\mathcal{R}\gtrsim 10$ EV: more than twice that of other UHECRs. With such a
high rigidity, by Eq. (3) its source should have at least an order of
magnitude higher luminosity than the average source. For additional
perspectives on this event see Elbert and Sommers (1995); Risse  _et al._
(2004); Gnatyk  _et al._ (2016); Fitoussi  _et al._ (2020).) With a large
enough sample of well-measured events above 150 EeV, as may be possible with
proposed future detectors such as GCOS Hörandel (2021), the contributions of
different r-process nuclei to extreme energy CRs could be mapped out.

EHE neutrinos associated with BNS mergers: As noted in the Introduction, the
light composition of UHECRs below the ankle is nicely explained as due to
interactions of higher-mass UHECRs with photons or gas in the environment of
the accelerator Unger  _et al._ (2015). These interactions produce a
population of neutrinos with peak energy of about 20 PeV Muzio  _et al._
(2021); see Fig. 3 in the Appendix reproduced from Muzio  _et al._ (2021). In
the BNS merger scenario for UHECR production, neutrinos in the $20$ PeV peak
energy range are produced by UHECRs interacting on their way out of the source
environment. Thus every neutrino detected in this energy range should be
accompanied by the gravitational wave (GW) produced by the neutron star
merger. Since the cosmic ray, GW and neutrino travel at essentially the speed
of light, the arrival time delay at observation between the GW and neutrino is
mainly due to the extra path length of the UHECR relative to the GW on account
of its magnetic deflections prior to the interaction in the source environment
which produces the neutrino. This needs to be estimated in order to determine
the relevant window for spatial-temporal coincidences between gravitational
waves and neutrinos in the 20 PeV peak. From Fig. 24 of
https://urldefense.com/v3/__https://icecube-
gen2.wisc.edu/science/publications/tdr/__;!!DZ3fjg!6NZ5v26I9gwHIP05-l4KWTTb-
JIldymaQRXBFOooXDZ16djGbKmZNb4W60kqzabAOZZKCXme8pL-VA$ , the sensitivity of
IceCube-Gen2 to diffuse astrophysical neutrinos after 10 years of operation
should be an order of magnitude below the predicted peak flux Muzio  _et al._
(2021). Moreover according to Ref. Gupta  _et al._ (2023), the efficiency for
detection of a BNS merger within $z=2$ at SNR $\geq 10$ should be 40% with the
“4020ET" GW detection network. So, coincident observations of a BNS merger and
a neutrino in this very high energy population, which would confirm the BNS
origin of UHECRs, looks promising with next-generation detectors.

Summary
We have proposed that binary neutron star mergers are the source of ultrahigh
energy cosmic rays. Due to the very narrow range of total mass of binary
neutron stars and the fact that the magnetic field in the jet is produced by a
dynamo whose energy source is gravitational, it is plausible that UHECR
production is almost universal from one merger event to another. This can
explain the small source-to-source variation in UHECR maximum rigidity, which
is a severe challenge to conventional source types as demonstrated in Ehlert
_et al._ (2023). The observed narrow range of maximum source rigidities
indicates that previously considered source types, which have a wide range of
maximum rigidities, can make at most a subdominant contribution to UHECR
production. The rate of binary neutron star mergers and power of their jets
are compatible with BNS mergers accounting for all UHECRs within present
constraints, but more precise determinations could challenge this scenario as
well.

Since neutron star mergers are the site of r-process nucleosynthesis, our
proposal suggests the existence of UHECRs with a composition significantly
heavier than Fe. An immediate success of this scenario is that the Fly’s Eye
“OMG" event is naturally explained as an r-process nucleus in the tellurium
peak, rather than being an extreme outlier requiring a uniquely powerful
accelerator or exotic origin. The same applies to the Amaterasu particle if
its true energy is close to its nominal value. In the future, UHECR data
should be interpreted allowing for the possibility of a component of r-process
nuclei, and codes to model UHECR energy losses during extragalactic
propagation should be extended to include r-process nuclei.

## Acknowledgments

I am grateful to T. Bister, M. Muzio and M. Unger for fruitful collaborations
and for valuable input which has stimulated and enriched the work presented
here. I also thank B. Dawson, M. Unger and A. A. Watson for discussions of an
updated interpretation of the Fly’s Eye event, and I much appreciate helpful
information from L. Comisso, M. Mapelli, A. MacFadyen and B. Metzger. This
research has been supported by the National Science Foundation PHY-2013199.

##  I Appendix A: Amaterasu and Andromeda

The source locus of TA’s Amaterasu event is devoid of distinctive source
candidates Unger and Farrar (2024), but does contain our neighbor Andromeda
(M31), a galaxy of comparable mass to the Milky Way at a distance of 765 kpc.
In a scenario such as ours where every galaxy hosts UHECR production in
proportion to its BNS merger rate, M31 should be an a priori relatively
powerful source. This raises the question of whether – were M31 the source of
Amaterasu – could that somehow be discovered and relic evidence of the event
investigated.

To gauge the likelihood that a BNS merger in M31 could have produced
Amaterasu, we calculate the expected number of BNS mergers in M31 contributing
to UHECRs detected today. Analysis of Milky Way double-NS systems leads to an
estimated BNS merger rate in the Milky Way of $32^{+19}_{-9}\,{\rm Myr}^{-1}$
Grunthal  _et al._ (2021), which should be applicable to M31 given its
similarities to our Galaxy. Two additional factors enter the probability of
observing a UHECR today from a BNS merger in M31 some time in the past: the
arrival-time spread, $\tau$, and the fraction of UHECRs reaching the Milky
Way, $f_{\rm contrib}$, discussed below Eq. (6); both depend on the
intervening magnetic fields through the smearing factor $\beta$.

The strength and coherence length of the random magnetic field within the
Local Group has not been directly constrained. Using stacked observations of
filaments Ref. Carretti  _et al._ (2023) estimates the average random field
strength at $z=0$ to be $B_{\rm rand}\approx(8-26)\,{\rm
nG}(\delta_{g}/10)^{2/3}$, where $\delta_{g}$ is the gas density relative to
the cosmic mean gas density. But it is not clear what to take for the path-
averaged value of $\delta_{g}$. Amaterasu’s trajectory would likely have
passed through the stellar disks of both M31 and the Milky Way, where the
random field strength reaches $\mu$G levels, but only over a length scale of
10’s kpc. Writing
$\beta_{\mathrm{M}31}=6\,(B_{\mathrm{LG}}/30\,\mathrm{nG})\,\sqrt{L_{c,\mathrm{LG}}/40\,\mathrm{kpc}}$,
gives an estimated arrival time spread
$\tau_{\mathrm{M}31}=184\,\mathrm{kyr}\,(\beta_{\mathrm{M31}}/6)^{2}$ and
$f_{\rm contrib}\approx 1$, implying there is a reasonable chance that a BNS
merger in M31 occurred recently enough to have produced Amaterasu.

Someday it might become possible to identify sites of BNS mergers through
localized over-densities of r-process elements and – very futuristically –
conceivably determine the orientation of the jet from the plane of r-process
enhancement. In a most optimistic world in which magnetic deflections en route
from M31 are small enough for the event to have reached Earth with little
angular dispersion, the UHECR arrival direction to the Galaxy would constrain
the direction of the jet, whose opening angle is $\approx 10^{\circ}$ Kiuchi
_et al._ (2023), thereby potentially implying the geometry of r-process
enhancement. Otherwise, there would be nothing distinctive to argue that a BNS
merger relic was associated with production of Amaterasu. (Note that
deflection in the Milky Way has been accounted for in the discussion of Unger
and Farrar (2024).)

##  II Appendix B: Some figures from cited papers

For the convenience of the reader, we reproduce here a few relevant figures
from the literature.

Figure 1: Figure 11 from Bister and Farrar (2023) showing combined constraints
on the the source number density $n$ and EGMF smearing parameter
$\beta_{\mathrm{EGMF}}\equiv B/\mathrm{nG}\,\sqrt{L_{c}/\mathrm{Mpc}}$; dark
grey regions are favored. The intensity bar shows the number of simulations
out of 1000 total that have both a sufficiently large dipole and sufficiently
small higher moments. The white region is excluded at more than 99% CL.
Characteristic estimates of the number densities of some steady source
candidates are shown with red lines, and indicative locii of transient source
candidates with orange lines; see Ref. Bister and Farrar (2023) for discussion
and references. The line labeled NS-BH mergers corresponds to
$\Gamma_{3}=0.03$. The effect of beaming due to a limited jet opening angle is
not included.  Figure 2: (top row) The azimuthal and radial fields as a
function of distance, from Fig. 3 of Kiuchi  _et al._ (2023); (bottom) the
Poynting flux on a sphere at 500 km, 150 ms after the merger, from Fig. 4 of
Kiuchi  _et al._ (2023).  Figure 3: Figure 1 from Muzio  _et al._ (2022),
extending the analysis of Unger  _et al._ (2015). Left: The CR predictions for
spectrum (top) and composition (bottom) compared to shifted Auger
observations. Right: The neutrino and gamma-ray predictions for this model
(solid and dashed lines, respectively). The total neutrino flux due to UHECRs
(“UHECR $\nu$’s”, solid magenta) is broken down by origin: UHECR interactions
during extragalactic propagation (“Propagation $\nu$’s”, dotted dark magenta),
UHECR photohadronic and hadronic interactions in the source (“Source $\nu$’s”,
dot-dashed and dashed dark magenta, respectively). Neutrinos originating from
a source other than UHECRs (“Non-UHECR $\nu$’s”, dot-dot-dashed dark magenta)
are also shown. The observed and inferred values of the extragalactic gamma-
ray flux, astrophysical neutrino fluxes, flux measurements from the Glashow
event, and upper-bounds on the EHE cosmic neutrino flux from IceCube (black)
and Auger (grey) are shown. See Muzio  _et al._ (2022) for details.

## References

  * Abbasi  _et al._ (2024a) R. Abbasi _et al._ (Telescope Array), Science 382, 903 (2024a).
  * Unger and Farrar (2024) M. Unger and G. R. Farrar, Astrophys. J. Lett. 962, L5 (2024), arXiv:2312.13273 [astro-ph.HE] .
  * Abdul Halim  _et al._ (2023a) A. Abdul Halim _et al._ (Pierre Auger), PoS ICRC2023, 252 (2023a).
  * Abdul Halim  _et al._ (2023b) A. Abdul Halim _et al._ (Pierre Auger), Astrophys. J. Suppl. 264, 50 (2023b), arXiv:2211.16020 [astro-ph.HE] .
  * Greisen (1966) K. Greisen, Phys. Rev. Lett. 16, 748 (1966).
  * Zatsepin and Kuzmin (1966) G. T. Zatsepin and V. A. Kuzmin, JETP Lett. 4, 78 (1966).
  * Halim  _et al._ (2023) A. A. Halim _et al._ (Pierre Auger), JCAP 05, 024, arXiv:2211.02857 [astro-ph.HE] .
  * Bister and Farrar (2023) T. Bister and G. R. Farrar, (2023), arXiv:2312.02645 [astro-ph.HE] .
  * Unger  _et al._ (2015) M. Unger, G. R. Farrar, and L. A. Anchordoqui, Phys.Rev. D92, 123001 (2015), arXiv:1505.02153 [astro-ph.HE] .
  * Aab  _et al._ (2014) A. Aab _et al._ (Pierre Auger), Phys. Rev. D 90, 122006 (2014), arXiv:1409.5083 [astro-ph.HE] .
  * Ehlert  _et al._ (2023) D. Ehlert, F. Oikonomou, and M. Unger, Phys. Rev. D 107, 103045 (2023), arXiv:2207.10691 [astro-ph.HE] .
  * Farrar and Gruzinov (2009) G. R. Farrar and A. Gruzinov, Astrophys. J. 693, 329 (2009), arXiv:0802.1074 .
  * Muzio  _et al._ (2022) M. S. Muzio, G. R. Farrar, and M. Unger, Phys. Rev. D 105, 023022 (2022), arXiv:2108.05512 [astro-ph.HE] .
  * Achterberg  _et al._ (1999) A. Achterberg, Y. A. Gallant, C. A. Norman, and D. B. Melrose,  (1999), arXiv:astro-ph/9907060 .
  * Abbasi  _et al._ (2024b) R. Abbasi _et al._ (Telescope Array), Science 382, 903 (2024b).
  * Unger and Farrar (2023) M. Unger and G. R. Farrar, (2023), arXiv:2311.12120 .
  * Vietri (1995) M. Vietri, Astrophys. J. 453, 883 (1995), arXiv:astro-ph/9506081 [astro-ph] .
  * Waxman (1995) E. Waxman, Phys. Rev. Lett. 75, 386 (1995), astro-ph/9505082 .
  * Duras, F.  _et al._ (2020) Duras, F., Bongiorno, A., Ricci, F., Piconcelli, E., Shankar, F., Lusso, E., Bianchi, S., Fiore, F., Maiolino, R., Marconi, A., Onori, F., Sani, E., Schneider, R., Vignali, C., and La Franca, F., A&A 636, A73 (2020).
  * Zaw  _et al._ (2019) I. Zaw, Y.-P. Chen, and G. R. Farrar, Astrophys. J. 872, 134 (2019), arXiv:1902.03799 [astro-ph.GA] .
  * van Velzen  _et al._ (2011) S. van Velzen, G. R. Farrar, S. Gezari, N. Morrell, D. Zaritsky, L. Ostman, M. Smith, J. Gelfand, and A. J. Drake, Astrophys. J. 741, 73 (2011), arXiv:1009.1627 [astro-ph.CO] .
  * van Velzen and Farrar (2014) S. van Velzen and G. R. Farrar, Astrophys. J. 792, 53 (2014), arXiv:1407.6425 [astro-ph.GA] .
  * Stein  _et al._ (2021) R. Stein _et al._ , Nature Astron. 5, 510 (2021), arXiv:2005.05340 [astro-ph.HE] .
  * Reusch  _et al._ (2022) S. Reusch, R. Stein, M. Kowalski, S. van Velzen, A. Franckowiak, C. Lunardini, K. Murase, W. Winter, J. C. A. Miller-Jones, M. M. Kasliwal, M. Gilfanov, S. Garrappa, V. S. Paliya, T. Ahumada, S. Anand, C. Barbarino, E. C. Bellm, V. Brinnel, S. Buson, S. B. Cenko, M. W. Coughlin, K. De, R. Dekany, S. Frederick, A. Gal-Yam, S. Gezari, M. Giroletti, M. J. Graham, V. Karambelkar, S. S. Kimura, A. K. H. Kong, E. C. Kool, R. R. Laher, P. Medvedev, J. Necker, J. Nordin, D. A. Perley, M. Rigault, B. Rusholme, S. Schulze, T. Schweyer, L. P. Singer, J. Sollerman, N. L. Strotjohann, R. Sunyaev, J. van Santen, R. Walters, B. T. Zhang, and E. Zimmerman, Phys. Rev. Lett. 128, 221101 (2022), arXiv:2111.09390 [astro-ph.HE] .
  * Jiang  _et al._ (2023) N. Jiang, Z. Zhou, J. Zhu, Y. Wang, and T. Wang, Astrophys. J. Lett. 953, L12 (2023), arXiv:2307.16667 [astro-ph.HE] .
  * Piran and Beniamini (2023) T. Piran and P. Beniamini, JCAP 11, 049, arXiv:2309.15644 [astro-ph.HE] .
  * Simeon  _et al._ (2023) P. Simeon, N. Globus, K. S. S. Barrow, and R. Blandford, PoS ICRC2023, 369 (2023).
  * Kiuchi  _et al._ (2023) K. Kiuchi, A. Reboul-Salze, M. Shibata, and Y. Sekiguchi, (2023), arXiv:2306.15721 [astro-ph.HE] .
  * Abbott  _et al._ (2023) R. Abbott _et al._ (KAGRA, VIRGO, LIGO Scientific), Phys. Rev. X 13, 041039 (2023), arXiv:2111.03606 [gr-qc] .
  * Rouco Escorial  _et al._ (2023) A. Rouco Escorial, W. Fong, E. Berger, T. Laskar, R. Margutti, G. Schroeder, J. C. Rastinejad, D. Cornish, S. Popp, M. Lally, A. E. Nugent, K. Paterson, B. D. Metzger, R. Chornock, K. Alexander, Y. Cendes, and T. Eftekhari, Astrophys. J. 959, 13 (2023), arXiv:2210.05695 [astro-ph.HE] .
  * Comisso and Sironi (2019) L. Comisso and L. Sironi, Astrophys. J. 886, 122 (2019), arXiv:1909.01420 [astro-ph.HE] .
  * Zhang  _et al._ (2021) H. Zhang, L. Sironi, and D. Giannios, Astrophys. J. 922, 261 (2021), arXiv:2105.00009 [astro-ph.HE] .
  * Li  _et al._ (2023) X. Li, F. Guo, Y.-H. Liu, and H. Li, The Astrophysical Journal Letters 954, L37 (2023).
  * Metzger  _et al._ (2011) B. D. Metzger, D. Giannios, and S. Horiuchi, MNRAS 415, 2495 (2011), arXiv:1101.4019 [astro-ph.HE] .
  * Kasen  _et al._ (2017) D. Kasen, B. Metzger, J. Barnes, E. Quataert, and E. Ramirez-Ruiz, Nature 551, 80 (2017), arXiv:1710.05463 [astro-ph.HE] .
  * Cowan  _et al._ (2021) J. J. Cowan, C. Sneden, J. E. Lawler, A. Aprahamian, M. Wiescher, K. Langanke, G. Martínez-Pinedo, and F.-K. Thielemann, Rev. Mod. Phys. 93, 015002 (2021).
  * Bird  _et al._ (1995) D. J. Bird _et al._ , Astrophys. J. 441, 144 (1995), arXiv:astro-ph/9410067 [astro-ph] .
  * Ave  _et al._ (2013) M. Ave _et al._ (AIRFLY), Astropart. Phys. 42, 90 (2013), arXiv:1210.6734 [astro-ph.IM] .
  * Kampert and Unger (2012) K.-H. Kampert and M. Unger, Astropart. Phys. 35, 660 (2012), arXiv:1201.0018 [astro-ph.HE] .
  * Elbert and Sommers (1995) J. W. Elbert and P. Sommers, Astrophys. J. 441, 151 (1995).
  * Risse  _et al._ (2004) M. Risse, P. Homola, D. Gora, J. Pekala, B. Wilczynska, and H. Wilczynski, Astropart. Phys. 21, 479 (2004), arXiv:astro-ph/0401629 .
  * Gnatyk  _et al._ (2016) R. B. Gnatyk, Y. N. Kudrya, and V. I. Zhdanov, Advances in Astronomy and Space Physics 6, 41 (2016).
  * Fitoussi  _et al._ (2020) T. Fitoussi, G. Medina-Tanco, and J.-C. D’Olivo, JCAP 01, 042, arXiv:1906.11170 [astro-ph.HE] .
  * Hörandel (2021) J. R. Hörandel (GCOS), PoS ICRC2021, 027 (2021), arXiv:2203.01127 [astro-ph.HE] .
  * Muzio  _et al._ (2021) M. S. Muzio, G. Farrar, and M. R. Unger, Phys. Rev. D to be submitted (2021).
  * (46) https://urldefense.com/v3/__https://icecube-gen2.wisc.edu/science/publications/tdr/__;!!DZ3fjg!6NZ5v26I9gwHIP05-l4KWTTb-JIldymaQRXBFOooXDZ16djGbKmZNb4W60kqzabAOZZKCXme8pL-VA$,  _IceCube-Gen2 Technical Design Report_ , Tech. Rep. (Ice Cube Collaboration).
  * Gupta  _et al._ (2023) I. Gupta _et al._ , (2023), arXiv:2307.10421 [gr-qc] .
  * Grunthal  _et al._ (2021) K. Grunthal, M. Kramer, and G. Desvignes, Monthly Notices of the Royal Astronomical Society 507, 5658 (2021).
  * Carretti  _et al._ (2023) E. Carretti, S. P. O’Sullivan, V. Vacca, F. Vazza, C. Gheller, T. Vernstrom, and A. Bonafede, MNRAS 518, 2273 (2023), arXiv:2210.06220 [astro-ph.CO] .
  * Özel and Freire (2016) F. Özel and P. Freire, ARAA 54, 401 (2016), arXiv:1603.02698 [astro-ph.HE] .
  * Sgalletta  _et al._ (2023) C. Sgalletta, G. Iorio, M. Mapelli, M. C. Artale, L. Boco, D. Chattopadhyay, A. Lapi, A. Possenti, S. Rinaldi, and M. Spera, MNRAS 526, 2210 (2023), arXiv:2305.04955 [astro-ph.HE] .
  * Xie  _et al._ (2018) X. Xie, J. Zrake, and A. MacFadyen, Astrophys. J. 863, 58 (2018), arXiv:1804.09345 [astro-ph.HE] .


# Efficient chip-based optical parametric oscillators from 590 nm to 1150 nm

Jordan R. Stone<EMAIL_ADDRESS>Joint Quantum Institute, NIST/University of
Maryland, College Park, MD 20742  National Institute for Standards and
Technology, Gaithersburg, MD 20899  Xiyuan Lu  Joint Quantum Institute,
NIST/University of Maryland, College Park, MD 20742  National Institute for
Standards and Technology, Gaithersburg, MD 20899  Gregory Moille  Joint
Quantum Institute, NIST/University of Maryland, College Park, MD 20742
National Institute for Standards and Technology, Gaithersburg, MD 20899
Kartik Srinivasan  Joint Quantum Institute, NIST/University of Maryland,
College Park, MD 20742  National Institute for Standards and Technology,
Gaithersburg, MD 20899

###### Abstract

Optical parametric oscillators are widely used to generate coherent light at
frequencies not accessible by conventional laser gain. However, chip-based
parametric oscillators operating in the visible spectrum have suffered from
pump-to-signal conversion efficiencies typically less than $0.1~{}\%$. Here,
we demonstrate efficient optical parametric oscillators based on silicon
nitride photonics that address frequencies between $260$ THz ($1150$ nm) and
$510$ THz ($590$ nm). Pumping silicon nitride microrings near $385$ THz ($780$
nm) yields monochromatic signal and idler waves with unprecedented output
powers in this wavelength range. We estimate on-chip output powers (separately
for the signal and idler) between $1$ mW and $5$ mW and conversion
efficiencies reaching $\approx 15~{}\%$. Underlying this improved performance
is our development of pulley waveguides for broadband near-critical coupling,
which exploits a fundamental connection between the waveguide-resonator
coupling rate and conversion efficiency. Finally, we find that mode
competition reduces conversion efficiency at high pump powers, thereby
constraining the maximum realizable output power. Our work proves that optical
parametric oscillators built with integrated photonics can produce useful
amounts of visible laser light with high efficiency.

##  I Introduction

Lasers operating at visible and near-infrared (NIR) wavelengths are essential
to modern science and technology Bothwell  _et al._ (2022); Awschalom  _et
al._ (2021); Matheus  _et al._ (2019); Weissleder (2001), but affordable
systems typically suffer from poor spectral purity and gaps in spectral
coverage, while higher-performance options are large and expensive. The latter
often rely on bulk nonlinear optics to spectrally translate longer-wavelength
lasers to the targeted frequency, employing either sum-frequency or second-
harmonic generation in $\chi^{(2)}$-nonlinear media Tinsley  _et al._ (2021);
Parrotta  _et al._ (2013); Mildren  _et al._ (2005). Their operational
complexity and substantial power consumption (they often require liquid
cooling systems) renders them impractical in many situations. Hence, it is
desirable to transition the nonlinear wavelength conversion to a more scalable
nonlinear optics platform, e.g., integrated photonics.

One approach is to leverage the wavelength flexibility inherent to optical
parametric oscillators using nonlinear microresonators, which possess large
optical quality factors ($Q$) and small mode volumes to intensify circulating
light and promote efficient nonlinear interactions Vahala (2003); Kippenberg
_et al._ (2004). Recent studies of microresonator-based optical parametric
oscillators ($\mu$OPOs) have demonstrated broad spectral separation between
pump and generated light Sayson  _et al._ (2019); Werner  _et al._ (2012); Jia
_et al._ (2018), low-power operation Lu  _et al._ (2019, 2021), and visible-
wavelength access Lu  _et al._ (2020). While both $\chi^{\rm{(2)}}$ and
$\chi^{\rm{(3)}}$ $\mu$OPOs offer some wavelength flexibility,
$\chi^{\rm{(3)}}$ systems are useful to generate visible light from a NIR
pump; moreover, their natural availability in the popular silicon photonics
platform Moss  _et al._ (2013) can enable their scalable fabrication and
integration with other components, including pump lasers Park  _et al._
(2020). On the other hand, the reported or inferred (e.g., from optical
spectra) conversion efficiencies are $\lesssim 0.1$% Sayson  _et al._ (2019);
Lu  _et al._ (2019, 2020); Fujii  _et al._ (2019); Tang  _et al._ (2020);
Domeneguetti  _et al._ (2021), and the available output power is far too low
for many applications (e.g., $<$10 $\mu$W for previous visible $\mu$OPOs Lu
_et al._ (2020)). Realizing higher conversion efficiencies and output powers
would enable a wide range of on-chip applications and broaden the reach of
silicon photonics in the visible spectrum.

Here, we demonstrate efficient $\mu$OPOs that generate coherent light within
the spectral window between 260 THz and 510 THz (590 nm and 1150 nm). We
measure conversion efficiencies between 3.5 % and 14.5 % with corresponding
on-chip output powers greater than 1 mW (and as high as 5 mW). Our results
spring from efficient broadband waveguide-resonator coupling, which we realize
with pulley-waveguide geometries designed using coupled-mode simulations. In
the rest of this paper, we first introduce the key $\mu$OPO physics and
specify our experimental procedures. Then, we explain our coupled-mode
simulations and present measurements to confirm their accuracy. Next, we
present the optical spectra of 16 different $\mu$OPOs, from which we determine
output powers and conversion efficiencies. Finally, we show how parasitic
nonlinear processes currently constrain the maximum realizable output power.
Our work is an important step forward in the quest for practical, chip-based
sources of visible laser light using nonlinear optics.

Figure 1: Achieving high conversion efficiency ($CE$) across a wide spectral
band with microresonator-based optical parametric oscillators ($\mu$OPOs). (a)
Conceptual depiction of $\chi^{(3)}$ optical parametric oscillation. Two pump
photons (frequency $\omega_{\rm{p}}$) are converted to one signal
($\omega_{\rm{s}}$) photon and one idler ($\omega_{\rm{i}}$) photon. We focus
on generating different $\mu$OPOs (faded arrows) within the spectral window
between $260$ THz and $510$ THz, as marked by the green stripe. (b) Depictions
of microring couplers, as viewed from above, which rely on evanescent coupling
between the microring and either a straight access waveguide (top panel) or a
pulley waveguide (bottom panel). Couplers are defined by three geometric
parameters: the waveguide width, $WW$, the gap (or distance) between the
microring and waveguide, $G$, and the length of waveguide, $L_{\rm{p}}$, that
runs parallel to the microring. (c) Simulated coupling ratios, $K$ (gray), and
measured $\mu$OPO spectra (blue) for identical microrings with different
coupling geometries. The top panel data corresponds to a straight waveguide
($WW=375$ nm; $G=125$ nm) coupler, while bottom panel data corresponds to a
pulley waveguide (same $WW$ and $G$; $L_{\rm{p}}=3$ $\mu$m).

The $\mu$OPOs we consider generate monochromatic signal and idler waves from a
continuous-wave (CW) pump laser through resonantly-enhanced degenerate four
wave mixing (FWM), as depicted in Fig. 1a Boyd (2020). In experiments, we pump
a fundamental transverse-electric (TE1) eigenmode of a silicon nitride
microring near $385$ THz, and FWM transfers energy to TE1 signal and idler
modes. In principle, the range of accessible output frequencies, as
constrained only by energy conservation, is $DC$ to $2\omega_{\rm{p}}$, where
$\omega_{\rm{p}}$ is the pump frequency. However, in practice this range is
dictated by the group velocity dispersion (GVD), which must be engineered such
that FWM to the targeted signal and idler modes is favored (simultaneously
phase- and frequency-matched), but FWM to other modes is suppressed. In
Appendix A, we recount our approach to dispersion engineering that is also
described in Ref. Lu  _et al._ , 2020.

A separate challenge is to ensure that pump power is efficiently converted
into output signal or idler power. Hence, we define the on-chip conversion
efficiency as:

$CE=\frac{P_{\rm{s(i)}}}{P_{\rm{in}}},$ (1)

where $P_{\rm{s(i)}}$ is the signal (idler) power in the waveguide output, and
$P_{\rm{in}}$ is the pump power in the waveguide input. Recent theoretical
work has derived the maximum obtainable $CE$ as:

$CE_{\rm{s(i)}}^{\rm{max}}=\frac{1}{2}\frac{\kappa_{\rm{s(i)}}\kappa_{\rm{p}}}{\Gamma_{s(i)}\Gamma_{\rm{p}}}\frac{\omega_{\rm{s(i)}}}{\omega_{\rm{p}}},$
(2)

where $\kappa_{\rm{s(i)}}$ and $\kappa_{\rm{p}}$ are the waveguide-resonator
coupling rates of the signal (idler) and pump modes, $\Gamma_{\rm{s(i)}}$ and
$\Gamma_{\rm{p}}$ are the total loss rates (i.e., loaded linewidths) of the
signal (idler) and pump modes, and $\omega_{\rm{s(i)}}$ and $\omega_{\rm{p}}$
are the frequencies of the signal (idler) and pump light, respectively Sayson
_et al._ (2019). Clearly, obtaining large $CE$ involves engineering $\kappa$
for both the pump mode and targeted signal/idler modes. Such coupling
engineering is a common problem in the nonlinear optics of Kerr
microresonators; it arises, for example, in the efficient extraction of
octave-spanning Kerr microcombs Moille  _et al._ (2019). The problem is that,
given a straight waveguide evanescently coupled to a microring resonator,
$\kappa$ decreases exponentially with frequency due to decreasing overlap of
the microring mode with the waveguide mode. Hence, when the pump mode is
critically coupled, the signal mode is undercoupled, resulting in low $CE$.
Moreover, when $\omega_{\rm{s}}$ is a visible frequency, the smaller
evanescent decay length compared to NIR frequencies exacerbates the challenge.
One solution is to utilize so-called pulley waveguides, which increase the
physical distance over which the waveguide and microring can exchange energy
Moille  _et al._ (2019); Shah Hosseini  _et al._ (2010). Figure 1b illustrates
the physical differences between straight-waveguide couplers (top panel) and
pulley-waveguide couplers (bottom panel), and it depicts the three geometric
parameters that define such couplers in our study: The waveguide width, $WW$,
the waveguide-resonator gap, $G$, and the pulley length, $L_{\rm{p}}$ (which
approaches zero in the straight-waveguide limit). In Fig. 1c, we present
measurements of $\mu$OPO spectra extracted from nominally identical microrings
using either a straight-waveguide coupler (top panel) or a pulley-waveguide
coupler (bottom panel). These measurements are representative of other
comparisons between the two coupling schemes. While $P_{\rm{i}}$ is roughly
the same in each case, $P_{\rm{s}}$ is approximately $20\times$ greater in the
pulley waveguide. To explain the result, we show in the same panels the
simulated coupling ratio, defined as $K=\kappa/\gamma$, where
$\gamma=\Gamma-\kappa$ is the intrinsic loss rate. Near $\omega_{\rm{s}}$, $K$
is $\approx 8\times$ higher for the pulley-waveguide coupler.

##  II design and test of waveguide couplers

To design couplers for testing, we simulate $\kappa$ spectra for a variety of
coupling geometries with the goal of achieving $K\approx 1$ (i.e., critical
coupling) at frequencies between $260$ THz and $510$ THz. Notably, achieving
high $CE$ only requires that $\kappa$ be optimized at $\omega_{\rm{p}}$ and
either $\omega_{\rm{s}}$ or $\omega_{\rm{i}}$, depending on which output wave
(signal or idler) is to be used. (Moreover, ideally, the unused wave is
undercoupled to reduce threshold power). At the same time, broadband near-
critical coupling is preferable, since then a single coupling geometry is
robust against design imperfections, and it may be used for many different
$\mu$OPOs. Our simulations are based on a coupled mode theory for optical
waveguides Huang (1994), which calculates $\kappa$ according to:

$\kappa=\frac{c}{2\pi Rn_{\rm{g}}}|k_{\rm{t}}|^{2},$ (3)

where $R$ is the microring outer radius, $c$ is the speed of light,
$n_{\rm{g}}$ is the group refractive index, and $k_{\rm{t}}$ is a coupling
coefficient defined as:

$k_{\rm{t}}=\frac{i\omega}{4}\int_{L}\left[\int_{A}(\epsilon_{\rm{WG}}-\epsilon_{\rm{R}})\bm{E_{\rm{R}}^{*}}\cdot\bm{E_{\rm{WG}}}\rm{d}\it{r}\rm{d}\it{z}\right]e^{i\phi}\rm{d}\it{l},$
(4)

where $\epsilon_{\rm{WG(R)}}$ is the dielectric permittivity of the access
waveguide (microring), $\bm{E_{\rm{WG(R)}}}$ is the electric field of the
waveguide (microring) eigenmode, and $\phi$ is an accumulated phase accounting
for the difference in the waveguide and microring propagation constants. The
coordinates $r$ and $z$ are horizontal and vertical coordinates in-plane with
the microring/waveguide cross section, as labeled in the Fig. 2a inset, and
$l$ follows the direction of light propagation. The central integral in Eq.
(4) evaluates the evanescent overlap between the microring and waveguide modes
at the frequency $\omega$. For more details, see Ref. Moille  _et al._ (2019).

Figure 2: Design, simulation, and measurement of coupling rate ($\kappa$). (a)
Simulated $\kappa$ spectra for a silicon nitride (SiN) microring with
waveguide parameters $WW=375$ nm, $G=150$ nm, and $L_{\rm{p}}=0$ $\mu$m (bold
gray curve), $2$ $\mu$m (faded blue), $3$ $\mu$m (bold turquoise), $4$ $\mu$m
(faded green), and $5$ $\mu$m (faded yellow). The circular data points
indicate measured $\kappa$ values for the coupler with $L_{\rm{p}}=0$ $\mu$m.
The inset depicts the waveguide/microring cross section, including labels for
the ring width ($RW$) and height ($H$), and shows axes for the coordinates $r$
and $z$ over which we integrate Eq. 4. (b) Simulated $\kappa$ spectra for a
SiN microring with waveguide parameters $G=150$ nm, $L_{\rm{p}}=3$ $\mu$m, and
different values of $WW$. The circular data points indicate measured $\kappa$
values for the coupler with $WW=375$ nm. The uncertainty in the measured
$\kappa$ values as determined from a nonlinear least squares fit to the
resonator transmission spectrum is smaller than the data point size. Figure 3:
$\mu$OPO spectra, conversion efficiency, and output power. (a) $\mu$OPO
spectra from a series of devices recorded with an optical spectrum analyzer
(OSA). Here, the y axis is normalized such that $0$ dB is equal to $1$ mW. No
correction factors (e.g., from optical losses) are applied to these data. The
transmitted pump light is shown in gray, and the signal/idler light is shown
in color. (b) $CE$ (gray) and $P_{\rm{i}}$ or $P_{\rm{s}}$ (purple; left and
right portions of data, respectively) versus frequency, taking into account
optical losses between the waveguide and OSA (see Appendix B). Error bars are
estimated from the precision uncertainties in our measurements of optical
power.

Figure 2a shows simulated $\kappa$ spectra for a SiN microring with nominal
$R=25$ $\mu$m, ring width $RW=825$ nm, and height $H=500$ nm, which are chosen
to suitably engineer the GVD (see Appendix A). In addition, we choose $WW=375$
nm, $G=150$ nm, and $L_{\rm{p}}$ from $0$ $\mu$m (i.e., a straight-waveguide
coupler) to $5$ $\mu$m. Increasing $L_{\rm{p}}$ results in larger $\kappa$ at
higher frequencies compared to the straight-waveguide coupler (dark grey
line), but it introduces resonances in the $\kappa$ spectra (i.e., regions
where $\kappa\rightarrow 0$) that arise from the coherent energy exchange
between the microring and waveguide. These resonances blueshift when
$L_{\rm{p}}$ is increased. Based on these data, we select $L_{\rm{p}}=3$
$\mu$m for further study because it minimizes variations in $\kappa$ over the
targeted spectral region.

Next, we optimize $WW$ and $G$. The predominant effect of changing $G$ is to
vertically shift the entire $\kappa$ spectrum; i.e., $G$ has little impact on
the spectral location of the coupling resonances. However, the relationship
between $\kappa$ and $WW$ is more complex. Within the range of values
considered, larger $WW$ increases $\kappa$ because the waveguide propagation
constant shifts closer to the microring propagation constant, and the
evanescent overlap between the microring and waveguide modes is not
appreciably changed. At the same time, coupling resonances are redshifted.
Figure 2b shows $\kappa$ spectra for $G=150$ nm, $L_{\rm{p}}=3$ $\mu$m, and
three values of $WW$. Apparently, the flattest $\kappa$ spectra are realized
for $WW$ between $375$ and $400$ nm. After choosing $WW$, $G$ may be chosen to
realize critical coupling near $\omega_{\rm{p}}$.

To assess the accuracy of our simulations, we fabricate an array of SiN
microrings with systematic coupling parameter variations (see Appendix C for
details on our fabrication process), and we measure $\kappa$ for each device.
Specifically, we use either $L_{\rm{p}}=0$ or $L_{\rm{p}}=3$ $\mu$m, $G$
between $110$ nm and $160$ nm, and $WW$ between $350$ nm and $400$ nm. To
measure $\kappa$, we carry out mode spectroscopy using a CW Titanium Sapphire
(TiS) laser, which is tunable from $305$ THz to $415$ THz ($720$ nm to $980$
nm). In addition, we can perform sum frequency generation with the TiS laser
and a $154$ THz ($1950$ nm) laser to generate coherent light from $460$ THz to
$510$ THz ($590$ nm to $650$ nm). We find that simulations predict slightly
larger $\kappa$ values than we measure; to compensate, we reduce $G$ by
approximately $20$ nm in experiments. In Fig. 2a, we present $\kappa$
measurements (gray circles) of a straight-waveguide-coupled device with
$WW=375$ nm and $G=125$ nm. The measured $\kappa$ values are slightly lower
than the corresponding simulation with $G=150$ nm, but both data decrease
exponentially with frequency, which is a known characteristic of straight-
waveguide couplers Moille  _et al._ (2019). Specifically, we measure
$\kappa\approx 800$ MHz near $350$ THz, but it drops sharply to $\kappa\approx
40$ MHz near $500$ THz. In contrast, we observe a more achromatic $\kappa$
spectrum in a pulley-waveguide-coupled device with $WW=375$ nm, $G=135$ nm,
and $L_{\rm{p}}=3$ $\mu$m. Our measurements (green circles) are shown in Fig.
2b, and they agree with the corresponding simulation with $G=150$ nm. Our
measurements indicate that, between $300$ THz and $500$ THz, $\kappa$ takes on
values over the relatively narrow range (compared to the straight-waveguide
coupler) of $180$ MHz to $400$ MHz. We also measure $\gamma\approx 300$ MHz
that is approximately independent of optical frequency. Therefore, according
to Eq. 2, our best pulley-waveguide-coupled devices should support many
different $\mu$OPOs with $CE>1$%.

##  III $\mu$OPO generation and conversion efficiency measurements

To test our prediction, we record $\mu$OPO spectra with a calibrated optical
spectrum analyzer (OSA) and calculate $P_{\rm{s}}$, $P_{\rm{i}}$, and $CE$
values after accounting for optical losses between the waveguide and OSA (for
details, see Appendix B). To generate $\mu$OPOs, we tune $\omega_{\rm{p}}$
into resonance, starting blue detuned and decreasing $\omega_{\rm{p}}$ until
$P_{\rm{s}}$ and $P_{\rm{i}}$ are maximized. We repeat this procedure for
different $P_{\rm{in}}$ values with the goal of maximizing $CE$. To ensure a
variety of $\omega_{\rm{s(i)}}$ values, we engineer the GVD by systematically
varying $RW$ in different devices (see Ref. Lu  _et al._ (2020) and Appendix
A). We utilize pulley-waveguide couplers such as those characterized in Fig.
2, and we find that $CE$ is maximized for $L_{\rm{p}}=3$ $\mu$m, $WW$ between
$375$ nm and $400$ nm, and $G$ between $125$ nm and $135$ nm. In Fig. 3a, we
present a compiled set of $\mu$OPO spectra that we extract from pulley-
waveguide couplers with parameters in the above optimum range. In most cases,
both $P_{\rm{s}}$ and $P_{\rm{i}}$ are greater than $1$ mW, and $P_{\rm{in}}$
is typically between $30$ mW and $45$ mW. However, there are atypical $\mu$OPO
spectra for which either $P_{\rm{s}}$ or $P_{\rm{i}}$ is $\ll 1$ mW. Most
likely, these result from mode interactions that locally alter the microring
GVD and $Q$ Herr  _et al._ (2014); Ramelow  _et al._ (2014). Table II in
Appendix B lists the individual pump, signal, and idler frequencies and powers
for each $\mu$OPO spectrum shown in Fig. 3(a).

To characterize the $\mu$OPO performance, we calculate from Fig. 3a the
largest values of $P_{\rm{s}}$ and $P_{\rm{i}}$ in spectral bins spanning
approximately $20$ THz each, and we plot the results in Fig. 3b along with the
corresponding $CE$ values. We find $P_{\rm{i}}>1$ mW from $264$ THz to $346$
THz and $P_{\rm{i}}>2$ mW from $275$ THz to $346$ THz. In the best case, we
generate $4$ mW of idler power at $315$ THz using $P_{\rm{in}}=34$ mW, which
equates to $CE\approx 12$%. Meanwhile, $P_{\rm{s}}>1.9$ mW from $416$ THz to
$506$ THz, and in the best case, we generate $5$ mW of signal power at $454$
THz using $P_{\rm{in}}=34$ mW, which equates to $CE\approx 14.5$%. Moreover,
as expected from our simulations and measurements of $\kappa$ and Eq. 2, $CE$
decreases in the spectral wings as a result of smaller $\kappa$.

Figure 4: Mode competition constrains the $\mu$OPO output power. The figure
depicts $\mu$OPO spectra at three different pump powers, $P_{\rm{in}}$. For
$P_{\rm{in}}=32$ mW (top panel), only the pump, signal, and idler modes
oscillate. As $P_{\rm{in}}$ is increased (middle and bottom panels), mode
switching occurs, and mode competitions distribute energy to modes other than
the targeted signal/idler pair, leading to reduced $P_{\rm{s(i)}}$. To guide
the eye, we include a red line in each panel that marks the top-panel
$P_{\rm{i}}$ level.

##  IV Limitations on output power

Finally, we discuss the limits to $\mu$OPO output power and analyze an
example. The relationship between $P_{\rm{s(i)}}$, $P_{\rm{in}}$, and other
experimental parameters has been analyzed theoretically Stone  _et al._
(2022). Therein, it was predicted that $P_{\rm{s(i)}}$ increases with
$P_{\rm{in}}$ for $P_{\rm{in}}\gtrsim P_{\rm{th}}$, but further increases in
$P_{\rm{in}}$ lead to saturation or even reduction of $P_{\rm{s(i)}}$. The
reason is that parasitic FWM processes compete with the targeted $\mu$OPO
process. The predominant parasitic FWM process that we observe in experiments
is mode competition between the targeted signal/idler modes and their spectral
neighbors Stone  _et al._ (2022). In Fig. 4, we show the $\mu$OPO spectrum for
a single device as $P_{\rm{in}}$ is increased. For $P_{\rm{in}}=32$ mW, only
the pump, signal, and idler modes oscillate, and $P_{\rm{s(i)}}>1$ mW (top
panel). The idler power in this case is marked by the red line in each panel.
When $P_{\rm{in}}=50$ mW (middle panel), $\omega_{\rm{s}}$ and
$\omega_{\rm{i}}$ shift to higher and lower frequencies, respectively. This
behavior was predicted in Ref. Stone  _et al._ (2022) and termed ‘mode
switching.’ In addition, other modes with frequencies close to
$\omega_{\rm{s(i)}}$ begin to oscillate and steal energy from the $\mu$OPO.
Hence, $P_{\rm{i}}$ decreases to a level below the red line, despite the
increase in $P_{\rm{in}}$. When $P_{\rm{in}}$ is further increased to $70$ mW
(bottom panel), $P_{\rm{s(i)}}$ remains approximately the same, but the power
in the competing modes increases. The behavior demonstrated in this example is
ubiquitous within our $\mu$OPO devices and explains why $P_{\rm{s(i)}}$ cannot
be increased arbitrarily by increasing $P_{\rm{in}}$. Still, increasing $CE$
and $P_{\rm{s(i)}}$ beyond the levels we demonstrate may be possible using
alternate phase- and frequency-matching strategies, such as that reported in
Ref. Zhou  _et al._ (2022). As it stands, the achieved power levels are
relevant for some applications, such as spectroscopy of various coherent near-
infrared and visible systems Xu  _et al._ (2007); Brazhnikov  _et al._ (2020).

##  V Discussion

In conclusion, we have demonstrated energy-efficient $\mu$OPOs with
practically-relevant output powers in a crucial portion of the visible and
near-infrared spectrum. Our results are enabled by broadband pulley-waveguide
couplers that we design using coupled-mode simulations. For the most widely-
separated $\mu$OPOs, we observe relatively lower $CE$ values that are
consistent with undercoupled signal and idler modes. Hence, an important focus
for future work is to extend the spectral bandwidth over which devices are
nearly critically coupled, thus broadening the range of frequencies that can
be efficiently extracted into a single waveguide. Other possible approaches
include using frequency-specific coupling geometries (e.g., one may optimize
$\kappa$ at the $\mu$OPO-specific frequencies $\omega_{\rm{s}}$ and
$\omega_{\rm{p}}$, while neglecting the idler), or to couple multiple
waveguides - each designed for different portions of the $\mu$OPO spectrum -
to one microring. Moreover, new strategies should be devised to avoid
parasitic FWM processes and increase the realizable output power. Nonetheless,
our work is a compelling demonstration that $\mu$OPOs can help satisfy the
demand for compact sources of visible laser light.

###### Acknowledgements.

We thank Khoi Hoang and Tahmid Sami Rahman for carefully reading the
manuscript and giving useful feedback. This project is funded by the DARPA
LUMOS and NIST-on-a-chip programs.

##  Appendix A Engineering GVD to control the $\mu$OPO spectrum

Figure 5: Engineering dispersion in $\mu$OPO systems. (a) Simulated frequency
mismatch parameter, $\delta$, versus frequency. Arrows point to the generated
frequencies, $\omega_{\rm{s}}$ and $\omega_{\rm{i}}$, and the pump frequency,
$\omega_{\rm{p}}$. (b) Predicted $\mu$OPO frequencies for different values of
$RW$ and $\omega_{\rm{p}}$. Different colors/shapes correspond to different
pump modes, with $\omega_{\rm{p}}$ (near $385$ THz) increasing from the blue
circle to the red cross.

The microresonator GVD determines which frequencies, if any, are phase- and
frequency-matched for FWM. When designing $\chi^{\rm{(3)}}$ $\mu$OPOs, a
convenient quantity related to GVD is the frequency mismatch parameter,
$\delta(\nu)=\nu_{\rm{s}}+\nu_{\rm{i}}-2\nu_{\rm{p}}$, where we use a slightly
different notation than in the main text; namely, $\nu_{\rm{s(i)}}$ are the
set of possible signal (idler) mode frequencies, and $\nu_{\rm{p}}$ is the
pump mode frequency. Degenerate FWM is allowed when $\delta(\nu)\gtrsim 0$
(small positive $\delta$ is required to compensate nonlinear shifts). Figure
5a shows a typical $\delta(\nu)$ spectrum as engineered for a
$\chi^{\rm{(3)}}$ $\mu$OPO system. The spectrum exhibits normal dispersion
around $\nu_{\rm{p}}$, which is necessary to suppress comb formation Lu  _et
al._ (2020), and higher-order GVD causes $\delta$ to turn and pass through
zero Stone  _et al._ (2022). In general, oscillation occurs on the mode pair
with smallest positive $\delta$.

The $\delta$ spectrum is controlled through the microring geometry; namely,
the $RW$ parameter. Moreover, a given microring may support several different
$\mu$OPOs, depending on $\omega_{\rm{p}}$. In our devices, a group of four or
five pump modes will yield different $\mu$OPO spectra, with larger
$\omega_{\rm{p}}$ increasing the signal-idler frequency separation. In Fig.
5b, we mark $\omega_{\rm{p}}$, $\omega_{\rm{s}}$, and $\omega_{\rm{i}}$ for
four different $\mu$OPOs at each of three $RW$ values. Importantly, fine-
tuning $RW$ (combined with a strategic choice of $\omega_{\rm{p}}$) can yield
$\mu$OPOs with generated frequencies anywhere in the targeted spectral window.

##  Appendix B Estimating on-chip powers from OSA measurements

Future integrated-photonics systems will route laser light from source to
destination (e.g., from a $\mu$OPO-generating microring to an integrated
atomic vapor cell) via low-loss optical waveguides. Hence, we estimate on-chip
powers and conversion efficiencies, which requires a careful calibration of
optical losses between the waveguide and OSA. In general, such losses are
dispersive, so we measure them for each device at a comprehensive set of
optical frequencies. We use lensed fibers to in- and out-couple light to and
from the waveguide. The lensed-fiber output is split using a fiber-based
$90/10$ coupler that is designed for a wide wavelength span around $780$ nm.
The ‘$90$’ output port is connected to the OSA via optical fiber, while light
from the other port is connected to a photodetector. To estimate losses, we
use a power meter to record the optical power at various points in the
experiment; specifically, we take measurements at the lensed-fiber input,
lensed-fiber output, and OSA input. Finally, we note any discrepancies between
the OSA input and OSA reading. Table 2 shows an example taken at $306$ THz
($980$ nm) for one $\mu$OPO device. Adding the different losses yields the
OSA-to-on-chip power conversion factor, which allows us to convert $\mu$OPO
spectra to on-chip signal/idler powers. Moreover, knowledge of the waveguide-
to-lensed-fiber coupling losses allows us to estimate the on-chip pump power
in experiments, which we use to estimate $CE$ values.

Measurement point | Power ($\mu$W) | Loss (dB)
---|---|---
Input to waveguide | 10 | N/A
Output from waveguide | 2.2 | 3.3 per waveguide facet
Input to OSA (after coupler) | 1.4 | 2
OSA reading | 0.7 | 3
Total |  | 8.3
Table 1: Example of loss calibrations at $306$ THz ($980$ nm) to determine the
on-chip power conversion factor.

Finally, we tabulate the signal and idler frequencies and on-chip powers for
each of the $16$ spectra shown in Fig. 3a. These data are presented in

$\omega_{\rm{p}}/2\pi$ (THz) |  $P_{\rm{in}}$ (mW) |  $\omega_{\rm{s}}/2\pi$ (THz) |  $P_{\rm{s}}$ (mW) |  $\omega_{\rm{i}}/2\pi$ (THz) |  $P_{\rm{i}}$ (mW)
---|---|---|---|---|---
385 | 34 | 423 | 2.2 | 346 | 2.7
378 | 26 | 416 | 1.8 | 339 | 0.2
378 | 20 | 432 | 1.8 | 324 | 2.1
380 | 25 | 437 | 0.02 | 322 | 1.8
385 | 27 | 438 | 0.8 | 332 | 1.4
384 | 30 | 446 | 1.3 | 321 | 0.4
385 | 36 | 454 | 5.0 | 318 | 2.6
385 | 35 | 454 | 4.4 | 316 | 4.0
384 | 28 | 466 | 1.9 | 303 | 1.8
384 | 46 | 467 | 1.7 | 302 | 1.6
384 | 32 | 470 | 3.3 | 298 | 1.9
386 | 33 | 475 | 2.2 | 297 | 0.6
384 | 59 | 479 | 3.4 | 289 | 2.7
385 | 55 | 495 | 2.4 | 275 | 2.3
385 | 34 | 502 | 2.0 | 268 | 1.1
385 | 47 | 506 | 2.5 | 264 | 1.1
Table 2: Pump, signal, and idler frequencies and on-chip powers for the $16$
$\mu$OPO spectra presented in Fig. 3a.

##  Appendix C Fabrication methods

To create device layouts, we use the Nanolithography Toolbox, a free software
package developed by NIST Balram  _et al._ (2016). We deposit stoichiometric
SiN (Si3N4) by low-pressure chemical vapor deposition on top of a $3$
$\mu$m-thick layer of SiO2 on a $100$ mm diameter Si wafer. We fit
ellipsometer measurements of the wavelength-dependent SiN refractive index and
layer thicknesses to an extended Sellmeier model. The device pattern is
created in positive-tone resist by electron-beam lithography and then
transferred to SiN by reactive ion etching using a CF4/CHF3 chemistry. After
cleaning the devices, we anneal them for four hours at $1100$°C in N2. Next,
we perform oxide lift-off to make devices air-clad around their left, right,
and top sides. The facets of the chip are then polished for lensed-fiber
coupling. After polishing, the chip is annealed again.

## References

  * Bothwell  _et al._ (2022) Tobias Bothwell, Colin J Kennedy, Alexander Aeppli, Dhruv Kedar, John M Robinson, Eric Oelker, Alexander Staron, and Jun Ye, “Resolving the gravitational redshift across a millimetre-scale atomic sample,” Nature 602, 420–424 (2022).
  * Awschalom  _et al._ (2021) David Awschalom, Karl K Berggren, Hannes Bernien, Sunil Bhave, Lincoln D Carr, Paul Davids, Sophia E Economou, Dirk Englund, Andrei Faraon, Martin Fejer, _et al._ , “Development of quantum interconnects (quics) for next-generation information technologies,” PRX Quantum 2, 017002 (2021).
  * Matheus  _et al._ (2019) Luiz Eduardo Mendes Matheus, Alex Borges Vieira, Luiz FM Vieira, Marcos AM Vieira, and Omprakash Gnawali, “Visible light communication: concepts, applications and challenges,” IEEE Communications Surveys & Tutorials 21, 3204–3237 (2019).
  * Weissleder (2001) Ralph Weissleder, “A clearer vision for in vivo imaging,” Nature Biotechnology 19, 316–317 (2001).
  * Tinsley  _et al._ (2021) Jonathan N Tinsley, Satvika Bandarupally, Jussi-Pekka Penttinen, Shamaila Manzoor, Sanna Ranta, Leonardo Salvi, Mircea Guina, and Nicola Poli, “Watt-level blue light for precision spectroscopy, laser cooling and trapping of strontium and cadmium atoms,” Optics Express 29, 25462–25476 (2021).
  * Parrotta  _et al._ (2013) Daniele C Parrotta, Alan J Kemp, Martin D Dawson, and Jennifer E Hastie, “Multiwatt, continuous-wave, tunable diamond raman laser with intracavity frequency-doubling to the visible region,” IEEE Journal of Selected Topics in Quantum Electronics 19, 1400108–1400108 (2013).
  * Mildren  _et al._ (2005) Richard P Mildren, Helen M Pask, Hamish Ogilvy, and James A Piper, “Discretely tunable, all-solid-state laser in the green, yellow, and red,” Optics Letters 30, 1500–1502 (2005).
  * Vahala (2003) Kerry J Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
  * Kippenberg  _et al._ (2004) TJ Kippenberg, SM Spillane, and KJ Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity,” Physical Review Letters 93, 083904 (2004).
  * Sayson  _et al._ (2019) Noel Lito B Sayson, Toby Bi, Vincent Ng, Hoan Pham, Luke S Trainor, Harald GL Schwefel, Stéphane Coen, Miro Erkintalo, and Stuart G Murdoch, “Octave-spanning tunable parametric oscillation in crystalline kerr microresonators,” Nature Photonics 13, 701–706 (2019).
  * Werner  _et al._ (2012) Christoph Sebastian Werner, Tobias Beckmann, Karsten Buse, and Ingo Breunig, “Blue-pumped whispering gallery optical parametric oscillator,” Optics Letters 37, 4224–4226 (2012).
  * Jia  _et al._ (2018) Yuechen Jia, Kevin Hanka, Kevin T Zawilski, Peter G Schunemann, Karsten Buse, and Ingo Breunig, “Continuous-wave whispering-gallery optical parametric oscillator based on cdsip2,” Optics Express 26, 10833–10841 (2018).
  * Lu  _et al._ (2019) Xiyuan Lu, Gregory Moille, Anshuman Singh, Qing Li, Daron A Westly, Ashutosh Rao, Su-Peng Yu, Travis C Briles, Scott B Papp, and Kartik Srinivasan, “Milliwatt-threshold visible–telecom optical parametric oscillation using silicon nanophotonics,” Optica 6, 1535–1541 (2019).
  * Lu  _et al._ (2021) Juanjuan Lu, Ayed Al Sayem, Zheng Gong, Joshua B Surya, Chang-Ling Zou, and Hong X Tang, “Ultralow-threshold thin-film lithium niobate optical parametric oscillator,” Optica 8, 539–544 (2021).
  * Lu  _et al._ (2020) Xiyuan Lu, Gregory Moille, Ashutosh Rao, Daron A Westly, and Kartik Srinivasan, “On-chip optical parametric oscillation into the visible: generating red, orange, yellow, and green from a near-infrared pump,” Optica 7, 1417–1425 (2020).
  * Moss  _et al._ (2013) David J. Moss, Roberto Morandotti, Alexander L. Gaeta, and Michal Lipson, “New CMOS-compatible platforms based on silicon nitride and Hydex for nonlinear optics,” Nature Photonics 7, 597–607 (2013).
  * Park  _et al._ (2020) Hyundai Park, Chong Zhang, Minh A. Tran, and Tin Komljenovic, “Heterogeneous silicon nitride photonics,” Optica 7, 336 (2020).
  * Fujii  _et al._ (2019) Shun Fujii, Shuya Tanaka, Mika Fuchida, Hikaru Amano, Yuka Hayama, Ryo Suzuki, Yasuhiro Kakinuma, and Takasumi Tanabe, “Octave-wide phase-matched four-wave mixing in dispersion-engineered crystalline microresonators,” Optics Letters 44, 3146–3149 (2019).
  * Tang  _et al._ (2020) Yulong Tang, Zheng Gong, Xianwen Liu, and Hong X Tang, “Widely separated optical Kerr parametric oscillation in AlN microrings,” Optics Letters 45, 1124–1127 (2020).
  * Domeneguetti  _et al._ (2021) Renato R Domeneguetti, Yun Zhao, Xingchen Ji, Marcelo Martinelli, Michal Lipson, Alexander L Gaeta, and Paulo Nussenzveig, “Parametric sideband generation in cmos-compatible oscillators from visible to telecom wavelengths,” Optica 8, 316–322 (2021).
  * Boyd (2020) Robert W Boyd,  _Nonlinear optics_ (Academic press, 2020).
  * Moille  _et al._ (2019) Gregory Moille, Qing Li, Travis C Briles, Su-Peng Yu, Tara Drake, Xiyuan Lu, Ashutosh Rao, Daron Westly, Scott B Papp, and Kartik Srinivasan, “Broadband resonator-waveguide coupling for efficient extraction of octave-spanning microcombs,” Optics Letters 44, 4737–4740 (2019).
  * Shah Hosseini  _et al._ (2010) Ehsan Shah Hosseini, Siva Yegnanarayanan, Amir Hossein Atabaki, Mohammad Soltani, and Ali Adibi, “Systematic design and fabrication of high-Q single-mode pulley-coupled planar silicon nitride microdisk resonators at visible wavelengths,” Optics Express 18, 2127 (2010).
  * Huang (1994) Wei-Ping Huang, “Coupled-mode theory for optical waveguides: an overview,” Journal of the Optical Society of America A 11, 963–983 (1994).
  * Herr  _et al._ (2014) T Herr, V Brasch, JD Jost, I Mirgorodskiy, G Lihachev, ML Gorodetsky, and TJ Kippenberg, “Mode spectrum and temporal soliton formation in optical microresonators,” Physical Review Letters 113, 123901 (2014).
  * Ramelow  _et al._ (2014) Sven Ramelow, Alessandro Farsi, Stéphane Clemmen, Jacob S Levy, Adrea R Johnson, Yoshitomo Okawachi, Michael RE Lamont, Michal Lipson, and Alexander L Gaeta, “Strong polarization mode coupling in microresonators,” Optics Letters 39, 5134–5137 (2014).
  * Stone  _et al._ (2022) Jordan R Stone, Gregory Moille, Xiyuan Lu, and Kartik Srinivasan, “Conversion efficiency in kerr-microresonator optical parametric oscillators: From three modes to many modes,” Physical Review Applied 17, 024038 (2022).
  * Zhou  _et al._ (2022) Feng Zhou, Xiyuan Lu, Ashutosh Rao, Jordan Stone, Gregory Moille, Edgar Perez, Daron Westly, and Kartik Srinivasan, “Hybrid-mode-family kerr optical parametric oscillation for robust coherent light generation on chip,” Laser & Photonics Reviews , 2100582 (2022).
  * Xu  _et al._ (2007) Xiaodong Xu, Bo Sun, Paul R Berman, Duncan G Steel, Allan S Bracker, Dan Gammon, and Lu J Sham, “Coherent optical spectroscopy of a strongly driven quantum dot,” Science 317, 929–932 (2007).
  * Brazhnikov  _et al._ (2020) Denis Viktorovich Brazhnikov, Stepan Maksimovich Ignatovich, Irina Sergeevna Mesenzova, Anton Mikailovich Mikhailov, Rodolphe Boudot, and Mikhail Nikolaevich Skvortsov, “Two-frequency sub-doppler spectroscopy of the caesium D1 line in various configurations of counterpropagating laser beams,” Quantum Electronics 50, 1015 (2020).
  * Balram  _et al._ (2016) Krishna C Balram, Daron A Westly, Marcelo Davanco, Karen E Grutter, Qing Li, Thomas Michels, Christopher H Ray, Liya Yu, Richard J Kasica, Christopher B Wallin, _et al._ , “The nanolithography toolbox,” Journal of Research of the National Institute of Standards and Technology 121, 464 (2016).