# Integrated Buried Heaters for Efficient Spectral Control of Air-Clad
Microresonator Frequency Combs

Grégory Moille<EMAIL_ADDRESS>Joint Quantum Institute, NIST/University of
Maryland, College Park, USA  Microsystems and Nanotechnology Division,
National Institute of Standards and Technology, Gaithersburg, USA  Daron
Westly  Microsystems and Nanotechnology Division, National Institute of
Standards and Technology, Gaithersburg, USA  Edgar F. Perez  Joint Quantum
Institute, NIST/University of Maryland, College Park, USA  Microsystems and
Nanotechnology Division, National Institute of Standards and Technology,
Gaithersburg, USA  Meredith Metzler  Center for Nanoscale Science and
Technology, National Institute of Standards and Technology, Gaithersburg, USA
Gregory Simelgor  Microsystems and Nanotechnology Division, National Institute
of Standards and Technology, Gaithersburg, USA  Kartik Srinivasan  Joint
Quantum Institute, NIST/University of Maryland, College Park, USA
Microsystems and Nanotechnology Division, National Institute of Standards and
Technology, Gaithersburg, USA

(September 3, 2024)

###### Abstract

Integrated heaters are a basic ingredient within the photonics toolbox, in
particular for microresonator frequency tuning through the thermo-refractive
effect. Resonators that are fully embedded in a solid cladding (typically
SiO2) allow for straightforward lossless integration of heater elements.
However, air-clad resonators, which are of great interest for short wavelength
dispersion engineering and direct interfacing with atomic/molecular systems,
do not usually have similarly low loss and efficient integrated heater
integration through standard fabrication. Here, we develop a new approach in
which the integrated heater is embedded in SiO2 below the waveguiding layer,
enabling more efficient heating and more arbitrary routing of the heater
traces than possible in a lateral configuration. We incorporate these buried
heaters within a stoichiometric Si3N4 process flow that includes high-
temperature ($>$1000 ∘C) annealing. Microring resonators with a 1 THz free
spectral range and quality factors near 106 are demonstrated, and the resonant
modes are tuned by nearly 1.5 THz, a 5$\times$ improvement compared to
equivalent devices with lateral heaters. Finally, we demonstrate broadband
dissipative Kerr soliton generation in this platform, and show how the heaters
can be utilized to aid in bringing relevant lock frequencies within a
detectable range.

Since their discovery, frequency combs have yielded a plethora of applications
– spectroscopy Dutt  _et al._ (2018), optical clocks Newman  _et al._ (2019),
frequency synthesis Spencer  _et al._ (2018), and distance ranging Suh and
Vahala (2018). Their integration on chip through the use of dissipative Kerr
soliton (DKS) states in different platforms Leo  _et al._ (2010); Herr  _et
al._ (2014); Xue  _et al._ (2015); Li  _et al._ (2017); Guo  _et al._ (2017);
Moille  _et al._ (2020) has led to a focus on low power Stern  _et al._ (2018)
and low footprint Ye  _et al._ (2022) devices for deployable metrology outside
of the laboratory Diddams, Vahala, and Udem (2020). The realization of octave-
spanning microcombs has been made possible by harnessing unprecedented control
of integrated microresonators dispersion Diddams, Vahala, and Udem (2020),
that additionally overlap with the atomic optical transition frequency Yu  _et
al._ (2019) has helped spur interest in their use in portable optical atomic
clocks Drake  _et al._ (2019); Newman  _et al._ (2019). In such applications,
the frequency comb acts as a gear box Diddams, Vahala, and Udem (2020)
translating the optical frequency stability of a comb tooth locked to an
atomic transition to a microwave frequency through the DKS repetition rate
[Fig. 1(a)]. However, the comb needs to be fully stabilized when realizing
such a frequency division scheme. In the clock case, the carrier envelope
offset $f_{\mathrm{ceo}}$ (i.e. the shift from the zero frequency) [Fig. 1(b)]
needs to be locked along with a comb tooth close enough from the optical
atomic transition frequency, here called $f_{\mathrm{lock}}$ [Fig. 1(c)].Their
locking makes the system in Fig. 1(a) stiff, in the sense that only a single
set of geometric parameters (ring width and thickness combination) will
provide for enough power enhancement at the frequencies of interest (often at
the location of dispersive waves (DWs)) while bringing a comb tooth
sufficiently close to the atomic transition frequency. Yet, each of these
goals are essentially driven by two different parameters: the DW spectral
position is defined by the cavity dispersion while $f_{\mathrm{lock}}$ (the
beat note between a comb tooth and the optical atomic transition) and
$f_{\mathrm{ceo}}$ can be controlled by a simple uniform spectral shift of the
comb. Therefore, integrated heaters appear as a suitable solution, leveraging
spectral tuning via the thermo-refractive effect Bosman and Havinga (1963) and
compatibility with $\chi^{(3)}$ microcomb platforms. Frequency tuning ranging
up to hundreds of gigahertz has been demonstrated Xue  _et al._ (2016); Joshi
_et al._ (2016); Helgason  _et al._ (2021); Tikan  _et al._ (2021), though
mostly in resonators fully embedded in a silica cladding.

Figure 1:  Optical atomic clock principle and stiffness of the frequency comb
locking frequency –  (a) Concept of an optical frequency comb acting as a
frequency divider, depicted as a gear box and inspired by Ref. Diddams,
Vahala, and Udem (2020) where the stability of a comb tooth in the optical
domain is divided down to the microwave domain with the clock output being the
microcomb repetition rate. For the division to be as low noise as possible,
the frequency comb needs to be locked. This means that the carrier envelope
offset ($f_{\mathrm{ceo}}$) and a single comb tooth frequency need to be
locked. (b) Concept of $f_{\mathrm{ceo}}$ detection and locking. A tooth from
an octave-spanning frequency comb and the second harmonic of a low frequency
tooth from the same comb are interfered with each other to retrieve
$f_{\mathrm{ceo}}$, which can in principle range from 0 to
$f_{\mathrm{rep}}/2$ and thus potentially be outside the photodetector
bandwidth when $f_{\mathrm{rep}}$ is large (typical for microcombs). The
octave span of the microcomb is obtained thanks to dispersive waves (DWs),
where the cavity dispersion allows for comb tooth resonance enhancement at the
targeted frequencies. (c) The clock relies on the locking of one comb tooth to
the optical transition of an atomic reference, for instance 85Sr at 698 nm.
Therefore, one comb tooth must be close enough to this optical transition to
be able to be locked. As for $f_{\mathrm{ceo}}$, this locking frequency
$f_{\mathrm{lock}}$ can be between 0 and $f_{\mathrm{rep}}/2$. (d) As
dispersion mostly impacts the position of a comb tooth that is amplified
through DW enhancement, a control knob to shift the whole frequency comb is
needed. Integrated heaters that leverage the thermo-refractive effect can
provide this functionality. However, an air-clad ring is needed to support the
dispersion required to reach the 698 nm wavelength of the 85Sr reference.
Ergo, two possibilities exist: lateral heating (i) and the new buried heater
technology introduced here. (ii). Finite element method simulations show that
the heat build-up is much more efficient in the case of the buried heater,
with a temperature inside the ring reaching 420 K against only 365 K for the
same heating power.

The use of air-clad devices is, however, desirable for dispersion engineering
to reach atomic transition frequencies in the short near-infrared and near-
visible Moille  _et al._ (2022) and allows for post-fabrication processing to
tune the geometrical dispersion if needed Moille  _et al._ (2021a). Direct
integration of the heater to the side of the ring results in poor heat build-
up at the ring core [Fig. 1(d.i)], caused by the air trenches that act as an
insulator but are essential to create the ring during the lithography and etch
fabrication steps. The alternative approach we propose in this paper is to
bury the integrated heater below the ring resonator [Fig. 1(d,ii)], where a
few micrometer gap of silica separates the metal from the optical layer,
resulting in no change in the ring losses. Due to the continuous path for heat
transfer between the heater and ring layers, the efficiency of such a buried
heater is much higher than for the lateral one, confirmed by thermal
simulation [Fig. 1(d)]. We demonstrate the fabrication of such unique
integrated heaters, which are wire-bonding ready for integration in optical
clock systems. In addition, fully embedding the heater in silica and
subsequently planarizing the silica layer makes the remainder of the
fabrication process compatible with a typical Si photonics process flow
Thomson  _et al._ (2016). We experimentally show the high tuning efficiency of
the system, where a resonator mode shift of 1.5 THz (equal to 1.5$\times$ the
resonator free spectral range (FSR)) is achieved without degradation of the
resonator quality factor and with limited cross-talk. While tuning of air-clad
resonators has numerous potential applications, including in sensing and
coupling to quantum emitters, we showcase the utility of these heaters in the
context of broadband DKS microcombs. We experimentally demonstrate tunability
of the microcomb with an estimated shift of $f_{\mathrm{ceo}}$ of 15 % of the
FSR and an estimated shift of the locked optical clock frequency
$f_{\mathrm{lock}}$ of close to 2 FSR, while little modification to dispersion
– and hence the DW position – occurs.

Figure 2:  Buried heater fabrication – (a) Process flow starting with a
commercial silicon wafer with thermal silicon dioxide (SiO2). (i) Deposition
of the platinum heater layer, lifted-off from previously patterned resist.
(ii) Burying the heater in another layer of (HDPCVD) SiO2. This layer needs to
be planarized through CMP because of the metal underneath. An annealing step
at 1000 °C for 3 hours is then performed. (iii) Resumption of the standard
fabrication process with 430 nm Si3N4 film deposition through LPCVD followed
by (iv) e-beam patterning and RIE etching to create the photonics layer.
(v-vii) Oxide lift-off using low temperature HDPCVD for air-clad rings with
oxide-clad facets, followed by annealing. (viii) A last etch step through
Si3N4 and SiO2 to enable electrical access to the heater layer. (ix) Dicing
and polishing of the chip for testing. (b) Optical image of the buried heater
to demonstrate the alignment of the metal layer with the photonics one. (c)
Optical image of the chip, showing how the buried heater fabrication enables
arbitrary routing of the metal layer with respect to the waveguide layer.
Figure 3:  Linear characterization of the buried heater resonator system – (a)
Resonant frequency shift probed around 309 THz (970 nm) for the buried heater
compared to the lateral heater for the same nominal resonator-waveguide
system. Cross-talk is characterized by measuring the frequency shift of
neighboring rings, resulting in a cross-talk of $\Xi\approx
13\leavevmode\nobreak\ \%$ for a neighboring device separated by 5 $\mu$m from
the target device. Devices with a larger separation show only a weak reduction
in $\Xi$. (b) Intrinsic ($Q_{\mathrm{0}}$, blue) and coupled
($Q_{\mathrm{c}}$, red) quality factor for different electrical power, i.e.
temperature. The absence of variation in $Q_{\mathrm{c}}$ shows that no
noticeable thermal expansion happens in the system, and the geometry remains
overall unchanged. The error bar is within the marker size. The three insets
are representative transmission spectra at different power (temperature). (c)
Frequency response of the buried heater under small-signal modulation, using a
vector network analyzer to apply the modulation and coherently photodetect the
transmitted signal. The value of $\approx$ 38 µs obtained at the 3 dB cut-off
is consistent with the typical bandwidths of state-of-the-art thermal
frequency shifters.

The fabrication of the system [Fig. 2(a)] starts with a standard commercial
silicon substrate with a 3 µm thick thermal oxide. We then define the heater
pattern using a direct-write lithography maskless aligner (MLA) NIS  [Fig.
2(a-i)]. The platinum is then deposited (electorn beam evaporation) and lifted
off. Another layer of 2.8 µm of SiO2 is deposited above the heater layer using
a 180 °C low temperature High Density Plasma Chemical Vapor Deposition
(HDPCVD) process [Fig. 2(a-ii)]. To allow for the best material and the lowest
absorption, we proceed to anneal the wafer at 1000 °C for three hours. Thanks
to the low diffusivity of metal into silica McBrayer, Swanson, and Sigmon
(1986), the buried heater remains geometrically intact and the silica layer
exhibits low absoprtion after the annealing. A chemical-mechanical polishing
(CMP) step [Fig. 2(a-ii)] is done to create a flat surface before low-pressure
chemical vapor deposition (LPCVD) of a 430 nm thick film of silicon nitride
Moille  _et al._ (2021b) [Fig. 2(a-iii)]. From this point on, the process is
the same as our regular nano-patterning and fabrication of air-clad ring
resonators Moille  _et al._ (2021a). The ring resonator patterns are created
through electron-beam lithography, with the patterns aligned to the underlying
buried heaters using metal alignment marks that are created within the heater
layer. The electron-beam resist is used as a mask for a CHF3 reactive ion etch
(RIE) of the silicon nitride [Fig. 2(a-iv)]. In order to maintain an air-clad
ring resonator for dispersion purposes while allowing for low insertion
losses, a lift-off of low temperature HDPCVD SiO2 using nLOF resist is
performed [Fig. 2(a-v,vi)]. A last annealing step is performed to anneal the
silicon nitride film, which has been shown to reduce absorption in the C-band
through reducing N-H bonds Stein, Peercy, and Sokel (1983) [Fig. 2(a-vii)].
Finally, an inductively coupled plasma (ICP) etch through the silicon nitride
and the substrate silica, combined with a one minute 6:1 buffer oxide etch
dip, reveals the metal layer for electrical contact [Fig. 2(a-viii)]. The chip
is then diced and polished for optical testing [Fig. 2(a-ix)].

Figure 4:  Tuning and control of the DKS microcomb with the buried heater —
(a) Single DKS microcomb obtained at different currents (and temperatures)
through the buried heater, when pumped around 284 THz (1056 nm). The spectra
is obtained through active “cooling” of the resonator using a 309 THz to 312
THz cross-polarized and counter-propagating secondary pump. The cooler
frequency is changed to obtained the soliton yet its frequency does not impact
nonlinear dynamics. The DW at short frequency (long wavelength) is only
slightly shifted by 2 comb teeth, showcasing the small, yet not negligible,
dispersion of the effective (i.e., mode-weighted) thermo-refractive index. (b)
Simulated integrated dispersion for the different current values in (a),
accounting for the dispersion in thermo-refractive index (inset) resulting
from the variation in modal overlap with the Si3N4, SiO2, and air regions. (c)
Shift in dispersive wave mode number with increase in current (electrical
power), including both the measured (circles) and simulated (dashed line)
values, where the simualtions are based on the integrated dispersion
calculation in (b). (d) Frequency shift of the pump with increase in current
(electrical power). (e) Extracted shift of the repetition rate. The repetition
rate is obtained through wavemeter calibration of the pump wavelength and
measuring with an optical spectral analyzer the comb tooth at $\mu=-76$. The
OSA resolution is 20 pm ((i.e.) 2.5GHz resolution), resulting in a measurement
error of about 32 MHz (error bars are within the data point size). (f)
Extracted $f_{\mathrm{ceo}}$ shift with electrical power from the extracted
$f_{\mathrm{rep}}$ and the wavemeter calibrated $f_{\mathrm{pmp}}$. (g)
Extracted $f_{\mathrm{lock}}$ (blue circles) assuming a comb tooth near the
85Sr transition at 698 nm. Here, we extrapolate the closest comb tooth
frequency based on $f_{\mathrm{rep}}$ and $f_{\mathrm{ceo}}$ from (e) and (f).
The buried heater allows for tuning of two consecutive comb teeth across the
reference frequency (green circles), allowing a full tuning of
$f_{\mathrm{lock}}$ between [-$f_{\mathrm{rep}}/2$,$f_{\mathrm{rep}}/2$].

We proceed to characterize the linear performance – spectral tuning, cross
talk and speed – of the buried integrated heater. The experimental setup
consists of a continuous tunable laser (CTL) around 970 nm probing a single
resonant mode of the cavity. The chip is placed on an aluminum sample holder
not thermally connected to the optical table (see supplementary material).
Heaters are addressed from the electrical pads with DC probes connected to a
current source, which can produce output up to 1.5W of electrical power. We
measure the electrical resistance of the integrated heater to be $R\approx
260$ $\Omega$, of the same order of magnitude as typical integrated heaters
Xue  _et al._ (2016); Joshi  _et al._ (2016). Injecting 1.2 W of electrical
power (i.e. a current of 67 mA), we measure a frequency shift of the resonance
of more than 1.5 THz, corresponding to an $\approx 1.5\leavevmode\nobreak\
$FSR red-shift [Fig. 3(a)]; amongst other things, this guarantees the ability
to tune one of the microresonator’s modes onto resonance with any fixed pump
laser wavelength. From simulations (see supplementary materials) this
frequency shift is consistent with an increase in temperature at the ring
above 900 K. We note that this frequency shift is quite large compared to
state-of-the art integrated heaters in the silicon nitride nonlinear photonics
platform that can be obtained through foundry fabrication Rao  _et al._ (2021)
where a shift of only $1/2$ of an FSR was achieved for the same electrical
power. The efficiency of the buried heater is $\eta_{0}=(-1.02\pm 0.012)$
THz/W, an improvement by a factor 5 compared to the lateral heater efficiency
of $\eta_{\mathrm{side}}=(-220\pm 1)$ GHz/W. The uncertainty estimation arises
from the standard deviation in the confidence interval for the linear fit of
the frequency shift with electrical power. In addition, the cross talk of the
integrated heater is limited. The next ring resonator, with its closest point
from the neighboring active heater of about 5 µm – a relatively tight
separation considering the waveguide access needed to optically address the
resonator – only show a thermal shift of $\eta_{+1}=(-136\pm 0.7)$ GHz/W.
Therefore, the cross-talk of our system can simply be defined by
$\Xi=\eta_{+1}/\eta_{0}\approx 13\leavevmode\nobreak\ $%. Interestingly, the
cross talk is not highly dependent with the distance from the heater. Indeed,
the second neighbor from the active heater shows a spectral shift of
$\eta_{+2}=(-111\pm 0.6)$ GHz/W, showing that a significant portion of the
chip is heating. Therefore, cross-talk could be further improved through an
appropriate thermal shunt through the backside of the chip.

We proceed to consider the impact of the buried heater on the optical quality
factor of the ring resonator [Fig. 3(b)]. We found that the intrinsic quality
factor, in addition to being similar to that of our regular fabrication
without the metal layer of $Q_{0}\approx 0.75\times 10^{6}$ Moille  _et al._
(2021a), remains unchanged for any temperature and electrical power applied to
the system. Moreover, we measure that the coupling quality factor
$Q_{\mathrm{c}}\leavevmode\nobreak\ \approx 1\times 10^{6}$ stays fixed at any
temperature, highlighting the little thermal expansion of the system,
resulting in a fixed geometry with temperature. Our coupling is based on a
straight waveguide, but in the future, using pulley-like coupling, which is
more sensitive to geometrical variations Moille  _et al._ (2019a), would give
a better estimate to what extent the thermal expansion is truly negligible. We
also performed dynamical measurements, in the small signal regime using a
vector network analyzer, to measure the recovery time of the buried heater and
the frequency up to which the system can function [Fig. 3(c)]. It exhibits a 3
dB cut-off at about 26 kHz, which corresponds to a recovery time of the system
of $\approx$38 µs. It is worth noting that this value is similar to that of
integrated heaters on top of silica-clad rings Xue  _et al._ (2016), and is
essentially limited by the thermal lifetimes of silicon nitride and silica Li
_et al._ (2017).

The integrated buried heater presented here shows a high efficiency with a
large thermo-refractive mediated spectral shift of the resonant mode. It
remains to characterize how the thermal tuning of a ring resonator impact the
metric associated with the optical atomic clock frequency comb , namely
$f_{\mathrm{rep}}$, $f_{\mathrm{ceo}}$ and $f_{\mathrm{lock}}$. In previous
work, it has been demonstrated that thermal tuning of the ring resonator
allows for small tuning of the the microcomb repetition rate Xue  _et al._
(2016). Here, we also proceed to characterize others metrics, to assess the
impact of the thermal tuning on the cavity dispersion, the carrier envelop
offset and the locking frequency, which represent three other important
quantities for efficient microcomb application in optical atomic clocks. To do
so, we create dissipative Kerr soliton microcombs at different electric powers
[Fig. 4(a)]. Interestingly, the overall shape of the frequency comb remains
mostly the same, with only a slight shift of the DW position, despite the
introduction of another dispersion contribution. Indeed, the variation of the
effective refractive index with temperature is wavelength-dependent given the
discrepancy in the thermo-refractive coefficient between Si3N4 ($\partial
n/\partial T=2.5\times 10^{-5}$ K-1) Elshaari  _et al._ (2016); Moille  _et
al._ (2019b), SiO2 ($\partial n/\partial T=1\times 10^{-5}$ K-1) Elshaari  _et
al._ (2016) and air ($\partial n/\partial T\equiv 0$ K-1) and the variation of
the effective mode area, which decreases with higher frequency (or mode
number) [Fig. 4(b)]. The integrated dispersion, as obtained from finite
element method simulations taking into account Joule heating, temperature
distribution, and the thermo-refractive coefficient (see supplementary
materials), shows only a slight modification across the thermal tuning range.
Therefore, the dispersive waves – essential for achieving octave spans and
self-referencing of a microcomb – would remain mostly unchanged despite
variation of the above parameters. As reported in Fig. 4(c), the experimental
DW shift is only 2 modes, from $\mu=-77$ to $\mu=-79$, and shows a slight
discrepancy from finite element method simulations ($\mu=-74$ to $\mu=-78$).
Although the overall shape of the DKS microcomb remains unchanged, its
fundamental characteristics of relevance for clock applications – repetition
rate and carrier-envelope offset frequency – are tuned with the electrical
power injected. As seen previously, the resonances are red-shifted with the
electrical power, ergo the pump frequency is also red-shifted [Fig. 4(d)]. As
the refractive index of the material increases with the temperature, the group
index should also increase, resulting in a lower repetition rate of the DKS
circulating in the resonator, as measured experimentally [Fig. 4(e)]. Given
the repetition rate shift and the shift of the pump resonance, it is expected
that $f_{\mathrm{ceo}}$ will also vary. We extract it from the measurements of
$f_{\mathrm{pmp}}$ and $f_{\mathrm{rep}}$ as
$f_{\mathrm{ceo}}=f_{\mathrm{pmp}}-Nf_{\mathrm{rep}}$, where $N$ is the
integer that brings $f_{\mathrm{ceo}}$ within the range
[-$f_{\mathrm{rep}}/2$,+$f_{\mathrm{rep}}/2$], and find that
$f_{\mathrm{ceo}}$ tuning of more than 100 GHz (i.e about 10% of
$f_{\mathrm{rep}}$) is achievable [Fig. 4(f)]. This tuning can be crucial to
allow for $f_{\mathrm{ceo}}$ to be shifted into a detection bandwidth of a
photodetector. Finally, we demonstrate the possibility of tuning
$f_{\mathrm{lock}}$ to a stabilized optical reference [Fig. 4(g)]. Here, we
have assumed a comb tooth is present near the 85Sr transition at 698 nm. In
practice, our demonstrated combs in this work do not extend to this wavelength
due to poor comb extraction resulting from the straight waveguide coupling
scheme employed Moille  _et al._ (2019a); however, we have demonstrated
similar combs (without heaters) extending to 698 nm using optimized pulley
couplers Moille  _et al._ (2022). Finally, we note that because the
$f_{\mathrm{lock}}$ comb tooth mode order is large, it provides a large
multiplicative factor on the temperature-induced shift in $f_{\mathrm{rep}}$.
Thus, with only 1 W of electrical power, $f_{\mathrm{lock}}$ shifts across 3
different comb teeth, constituting a span $>2.5f_{\mathrm{rep}}$.

We have demonstrated a new technique to integrate efficient micro-heaters on
chip, with an air-cladding resonator, by burying the heater below the optical
device layer. Once the buried heater is patterned, encased in SiO2, and
planarized, the subsequent process flow is the same as for standard Si3N4
photonic devices. We show that such buried heater is 5$\times$ more efficient
than lateral heaters and exhibits low cross-talk which could be even further
improved by proper thermal shunt of the chip. We then demonstrate use of the
heaters with broadband microcombs, and measure/extract tuning of critical comb
parameters for use in optical atomic clocks. While the microcomb measurements
are indicative of the use of the buried heater technology for nonlinear
integrated photonics applications, we envision many other uses in scenarios
for which air cladding are required. This in particular would include
microresonator-based sensors Yu  _et al._ (2021) and cavity QED experiments
with gas-phase atoms O’Brien, Furusawa, and Vučković (2009); Chang  _et al._
(2020), where both the wide tuning range and negligible impact on resonator
quality factor are important factors.

Data availability
The data that supports the plots within this paper and other findings of this
study are available from the corresponding authors upon reasonable request.

Acknowledgements
The author thanks Nikolai Klimov and Feng Zhou for valuable inputs. The
authors acknowledge partial funding support from the DARPA APHI, DARPA SAVaNT,
and NIST-on-a-chip programs.

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##  Experimental Setup

Figure S1: Experimental Setup: (a) Schematic of the experimental setup with a
tunable laser around 1060 nm (1060 CTL) for either spectroscopy of cavity
modes or to go through a YDFA amplifier for comb generation. In the latter
case, an amplified 980 nm tunable laser (980 CTL) is used to actively-cool the
resonator allowing for adiabatic DKS access. WDM = wavelength division
multiplexer, OSA = optical spectrum analyzer, PC = polarization controller.
(b) Close-up picture of the coupling portion of the setup with the aluminum
sample holder used in Fig. 3, which acts as a thermal insulator. (c)-(d) Close
up of the copper sample mount used in Fig. 4, which is thermally shunted to
the optical table and acts as a heat sink at room temperature. This results in
a much more stable coupling setup allowing for DKS operation. Note that the
coupling was stable enough that the fiber did not need to be adjusted to keep
an optimal coupling for any current used in Fig. 4.  Figure S2: Simulation
workflow to simulate spectral shifts of the resonant mode of the ring
resonator:  (1) we start with a Joule effect simulation in which an input
current applied to the buried integrated heater results in an electrical
potential and the creation of a heat source. (2) Then, we proceed to solve the
Fourier heat equation to find the steady-state of the temperature profile. We
apply a fixed boundary condition on the temperature at the bottom of the
substrate, and an open boundary elsewhere. We map this temperature profile to
the variation of the refractive index profile $\Delta T\partial n/\partial
T|_{\mathrm{(r,z)}}$. (3) This allows for the Maxwell wave equation to account
for the temperature profile which will be reflected in the frequency of
resonance (i.e. respecting the periodic boundary condition in $\phi$, which is
out of the plane of the image). We note that the figures presented are zoomed-
in from the full simulation domain.

The setup used in this work is presented schematically in Fig. S1(a). The
continuously tunable laser (CTL) provides tuning of the wavelength from 1010
nm to 1070 nm to perform spectroscopy, and we adjust the input light
polarization such that we excite the the fundamental transverse electric (TE)
modes (i.e., TE0) of the resonator, by measuring the transmission as a
function of wavelength. In this linear case, the sample mount is made of
aluminum (Fig. S1(b)), through which we noticed experimentally a limited heat
diffusion and an overal increase of the chip temperature, resulting in
coupling instability for creation of DKS. In contrast, we use a copper sample
mount linked to the optical table(Fig. S1(c)-(d)) to perform nonlinear
measurements, where an ytterbium-doped fiber amplifier (YDFA) is used to
produce adequate optical power around 1060 nm, while a (counterpropagating and
orthogonally polarized) 980 nm laser is used to actively cool the resonator to
reach states adiabatically. With its improved thermal conductivity and heat
sinking to the optical table so that the impact of the buried heaters is more
local and the overall chip temperature is not increased as much as in the case
of the Al sample mount. Importantly, the overall frequency tuning range
achieved by the buried heater is nearly the same regardless of the mount
(i.e., comparing Fig. 3(a) and Fig. 4(d)). Interestingly, we believe that use
of the copper mount also helps mitigate convective air flows, making the setup
much more stable. This has allowed us to measure the DKS states at different
electrical powers in Fig. 4 without the need to adjust the fiber position for
optical coupling.

##  Thermal Simulations

In order to accurately account for the integrated buried heater impact on the
optical mode of the ring resonator, we performed so called fully-coupled
simulations, where the Joule heating, heat diffusion and wave equations are
solved as a coupled equations system [Fig. S2]. First, we leverage the
symmetry of the system, where in a cylindrical coordinate system
$\\{r,\phi,z\\}$, periodic boundary conditions are assumed in $\phi$,
simplifying the system to an effective two-dimensional one along $\\{r,z\\}$.
The optical field is given as $E(r,\phi,z)=E(r,z)e^{im\phi}$, with $m$ the
azimuthal mode number of the ring resonance. Although the implemented heater
design breaks this symmetry because it was not fully wrapped around the ring
to avoid heating of the resonator-waveguide coupling region, we will assume
the heat diffusion is strong enough to allow for uniform temperature in
$\phi$. Here, we describe the essential steps taken by the solver, although
these equations are effectively solved simultaneously. We start by simulating
Joule heating, where a current density is injected in the heater assuming the
measured resistance of 260 $\Omega$ [Fig. S2(a)], which we use as a heat power
source for the heat diffusion equation [Fig. S2(b)], allowing us to retrieve
the temperature profile along $\\{r,z\\}$. The boundary conditions are crucial
for accurate simulation, and following the improvement of the setup described
in the previous section, we assume the bottom of the substrate to be at room
temperature and the others as open boundaries. Simulations where the substrate
thickness has been increased did not significantly change the simulations
results. Although the bottom of the substrate may present a higher temperature
than the one assumed here, our assumption of room temperature best reproduces
the experimental data presented in Fig. 4. Finally, using a Seillmeier model
to determine the refractive index of the silicon nitride and silicon dioxide
layers according to ellipsometry measurements previously performed Moille  _et
al._ (2021b), and accounting for the thermo-refractive index coefficient of
each material, we use the simulated profile to determine the refractive index
shift $\Delta n(r,z)$ due to the heating. This allows to model the frequency
shift of a given resonator mode along with complete dispersion of the
resonator modes with electrical power.


# The Combinatorics of Salva Veritate Principles

Norman E. Trushaev 111The University of Vermont, Department of Mathematics

(January 13, 2022)

###### Abstract

Various concepts of grammatical compositionality arise in many theories of
both natural and artificial languages, and often play a key role in accounts
of the syntax-semantics interface. We propose that many instances of
compositionality should entail non-trivial combinatorial claims about the
expressive power of languages which satisfy these compositional properties. As
an example, we present a formal analysis demonstrating that a particular class
of languages which admit salva vertitate substitutions - a property which we
claim to be a particularly strong example of compositional principle - must
also satisfy a very natural combinatorial constraint identified in this paper.

## Introduction

This essay will present a formal framework and some preliminary results
concerning the combinatorial properties of meaning-preserving substitutions of
sentences. Following Quine (and ultimately Leibniz), we refer to such
operations as salva veritate substitutions. 222It should be mentioned that
Quine was primarily concerned with truth-value preserving substitutions (hence
the veritate bit), whereas we will be more broadly concerned with meaning
preserving substitutions, in general, of which the truth-value preserving
substitutions are a proper subset.

The relevant properties of salva veritate substitutions are formalized in an
abstract property that we introduce in the essay, and which we later call
(SST). This property is itself a particular instance (and a very strong one)
of a more general class of linguistic constraints that might be referred to as
compositionality principles, where, by ”compositionality principle”, we simply
mean any property that might be possessed by a language in virtue of the fact
that it exhibits some form of grammatical compositionality.

Our discussion will be divided into three parts. In Part 1, we briefly present
some historical and conceptual background information which will be used to
motivate our subsequent investigations. Part 2 is devoted to constructing our
formal theory and presenting the main result. Finally, in Part 3, we conclude
with a discussion of the preceding findings, and some remarks on directions
for future work.

##  1 Historical and Conceptual Background

By a ”substitution” we mean any operation on strings of a language which
replaces some sub-string with another. The simplest example would be any
operation which replaces some word occurring in a sentence with some other
word, but one can also consider substitutions of arbitrary syntactic
constituents. Such substitution operations have played an important role in
many debates in linguistics and philosophy, and have been particularly
important in the analysis of intensional phenomenon, synonymy, and
analyticity.

The literature on these matters is extensive, and a full review is beyond the
scope of this article. A good starting point, however, is Quine’s classic ”Two
Dogmas of Empiricism” ([11]). Quine asks us to consider a typical analytic
statement

$(1)\indent\text{No bachelor is married.}$

Quine suggests that a core feature of such statements is that they can be
converted into logical truths by ”putting synonyms for synonyms” (Quine, p.
23). In this particular case, if we substitute the phrase ”unmarried man” for
”bachelor”, we obtain the logically tautological sentence

$(2)\indent\text{No unmarried man is married.}$

In effect, analyticity of $(1)$ then rests on synonymy of the linguistic forms
”unmarried man” and ”bachelor”, and the fact that $(2)$ is a logical truth.
The explanatory burden now rests on finding a satisfactory account of
linguistic synonymy. Regarding this matter, Quine points out (p. 27) that ”A
natural suggestion, deserving close examination, is that the synonymy of two
linguistic forms consists simply in their interchangeability in all contexts
without change of truth value - interchangeability, in Leibniz’s phrase, salva
veritate.”

This suggestion forms the starting point of our investigation. We, however,
will not be concerned with examining the philosophical, semantic, or logical
properties of salva veritate substitutions, but rather their combinatorial
properties, a line of investigation which remains largely unexplored.

Although, perhaps not the conventional view, salve veritate substitutions can
be viewed as a specific instance of compositionality. Compositionality is a
property exhibited by certain linguistic structures, whereby the meaning of
any complex expression is determined by the meaning of it’s component parts,
and the manner in which these component parts are combined ([10]).
Compositionality has been an important subject of investigation in linguistic
and logical scholarship since at least the work of Frege in the 19th century
([1], [3], [10]), and continues to be an active area of inquiry today (see,
e.g. [2], [4])

##  2 Formal Machinery

For our purposes, we consider an interpreted language, by which we mean a set
of strings over some alphabet, and a rule for assigning ”meanings” to these
strings. Formally, we define an interpreted language $\mathcal{L}$ to be an
ordered tuple

$\mathcal{L}=(X,S,h,\mathcal{M}),$

where $X$ denotes the alphabet, $S\subset X^{*}$ denotes the set of well-
formed strings of the language (i.e. the things we can assign interpretations
to), $h:S\rightarrow\mathcal{M}$ denotes the interpretation function, which
assigns meanings to strings, and $\mathcal{M}$ is our set of meanings. In
order to provide a fully general account, we make no assumptions about
$\mathcal{M}$. Our ”meanings”, may therefore consist of anything, including
truth-values, propositions, concepts, sentences of a meta-language, etc. For
our purposes, we further assume that the alphabet $X$ is finite.

We also introduce an additional constraint on $\mathcal{L}$. We call this
constraint ”substitutability of synonymous terms” (SST). Formally we have

######  Definition (Substitutability of Synonymous Terms).

Let $u,v\in S$ be well-formed strings satisfying the relation $h(u)=h(v)$.
Then for any well-formed string $\alpha u\beta\in S$, the string $\alpha
v\beta$ is also well-formed, and $h(\alpha u\beta)=h(\alpha v\beta)$.

Informally, this condition states that if strings $u,v\in S$ have the same
meaning, then the operation of substituting $v$ for $u$ is well-defined for
all strings of our language $\mathcal{L}$, and the result of performing such
an operation has no effect on meaning. It is important to note that such a
condition places both a syntactic and a semantic constraint on $\mathcal{L}$.
The syntactic component guarantees that replacing a substring $u$ that occurs
in any well-formed string $w$, with a synonymous substring $v$, always
produces another well-formed string $w^{\prime}$. The semantic component
guarantees that such substitutions have no impact on meaning, i.e. that
synonymous constituents make the same semantic contribution to any linguistic
context in which they occur.

In addition to (SST), we will require another constraint, which we call
”inductive constructibility” (IC). Formally we have

######  Definition (Inductive Constructibility).

Let $\mathcal{L}=(X,S,h,\mathcal{M})$ be an interpreted language. Then we say
that $\mathcal{L}$ satisfies inductive constructibility iff any well-formed
string $w\in S$ of length $n>1$ is equal to the concatenation $w=uv$ of two
strings $u,v\in S$ which satisfy the relation $|w|=n=|u|+|v|$.

Informally, this condition states that any non-trivial string $w$ (i.e. a
string of length $>1$) must be composed of smaller well-formed substrings.
This prevents the existence of strings $w\in S$ that have non-trivial
substrings which themselves are not well-formed. Hence, any non-trivial well-
formed strings can be constructed by concatenating smaller well-formed
strings. In particular, an inductive argument will immediately show that for
$n>1$, any string $w\in S$ of length $|w|=n+1$ is equal to some string
concatenation of the form $w=ua,$ where $|u|=n,$ and $x\in X$.

Before proceeding, we introduce some additional terminology that will be
needed. Given subsets of well-formed strings $S_{1},S_{2}\subseteq S$, we say
than $S_{1}$ is (strictly) more expressive that $S_{2}$ iff $h(S_{2})$ is a
(proper) subset of $h(S_{1})$. Furthermore, for any subset
$S^{\prime}\subseteq S$ we define the expressive power of $S^{\prime}$ to be
equal to $|h(S_{1})|$. We define the expressive power of an interpreted
language $\mathcal{L}=(X,S,h,\mathcal{M})$ to be equal to the expressive power
of $S$. Given a natural number $n\in\mathbb{N}$, we define the $n$-th
generation of $S$ to be the set $\text{gen}(n):=\\{w\in S:|w|\leq n\\}$ of all
well-formed strings of length $\leq n$.

We are now ready for the main result:

######  Theorem.

An interpreted language $\mathcal{L}$ which satisfies (SST) and (IC) has
infinite expressive power iff gen$(n+1)$ is strictly more expressive than
gen$(n)$ for all $n\in\mathbb{N}$.

###### Proof.

Let $\mathcal{L}=(X,S,h,\mathcal{M})$ be an interpreted language which
satisfies (SST) and (IC). We then have two directions to prove.
($\Rightarrow$) Let $\mathcal{L}$ have infinite expressive power, and suppose
for contradiction that gen$(n+1)$ is not strictly more expressive gen$(n)$ for
some $n\in\mathbb{N}$. Then $h(\text{gen}(n))=h(\text{gen}(n+1))$, i.e.
$\text{gen}(n)$ and $\text{gen}(n+1)$ have equal expressive power. In
particular, we see that the expressive power of $\text{gen}(n+1)$ is finite
and equal to

$|h(\text{gen}(n+1))|=|h(\text{gen}(n))|\leq|X|^{n}.$

If $\mathcal{L}$ has infinite expressive power, then there exist meanings
$m\in h(S)$ such that $m\notin h(\text{gen}(n+1))$. Now fix any such meaning
$m$, and let $w\in S$ be a string of minimal length $|w|=k>n+1$ that is
assigned meaning $h(w)=m$. By (IC), $w$ can be expressed in the form $w=uv$
for $u,v\in S$ of lengths $|u|=n+1$ and $|v|=k^{\prime}=k-(n+1)$. Given that
$u$ has length $n+1$, there exists a string $u^{\prime}\in\text{gen}(n)$ such
that $h(u)=h(u^{\prime})$. By (SST), it follows that

$h(u^{\prime}v)=h(uv)=m.$

Now, since $|u^{\prime}v|<|uv|$, this contradicts our assumption that
$|w|=|uv|$ is the shortest possible length of any string that expresses
meaning $h(w)=m$. Hence, the proper inclusion

$h(\text{gen}(n))\subset h(\text{gen}(n+1))$

holds for all $n\in\mathbb{N}$.
($\Leftarrow$) Suppose that $\text{gen}(n+1)$ is strictly more expressive that
$\text{gen}(n)$. Then for all $k\in\mathbb{N}$, there exists a meaning
$m_{k}\in h(\text{gen}(k+1)-\text{gen}(k))$ that is expressible by a string of
length $k+1$, but not by any string of length $\leq k$. Taking the union

$A:=\bigcup_{k=1}^{\infty}\\{m_{k}\\},$

we then obtain an infinite collection of meanings $A\subseteq\mathcal{M}$ that
is contained in $h(S),$ i.e. $A\subseteq h(S)$. Hence $\mathcal{L}$ has
infinite expressive power.

∎

##  3 Conclusion

The theorem of section 2 demonstrates a close connection between the
admissible substitutions of a language, and its expressive power. In effect,
this provides a practical demonstration of the fact that at least some non-
trivial semantic information about a given language can be obtained purely by
examining the meaning-preserving syntactic operations which are permitted by
the language under consideration. Furthermore, this suggests that questions
concerning the syntax-semantics interface, and the relationship between form
and meaning, may have a non-trivial combinatorial component that is entirely
independent of any particular semantic interpretation.

The constraints placed on $\mathcal{L}$ in this paper, however, require
additional refinement in order to be suitable for the description of most
interesting examples of interpreted languages. Substitutability of synonymous
terms (SST) and inductive constructibility (IC) are rather strong assumptions
to make about a language. Such assumptions may in fact be perfectly innocuous
and realistic assumptions for the analysis of many formal languages, but for
many languages of interest, especially in the case natural languages, it seems
plausible that (SST) will need to replaced - substituted, if you will - by a
weaker constraint on substitutability. Well known intensional phenomenon, such
as those identified and investigated by Russell, Kripke, Montague, Partee,
([9], [5], [6], [7]) and others, demonstrate conditions under which meaning is
not preserved under substitutions of synonymous terms. In some cases, this is
likely to require weakening our assumption regarding the preservation of well-
formedness under substitution. In other cases, we will likely have to weaken
our assumption that meaning remains identical under such substitutions. Many
applications will likely require some combination of both.

(IC) also appears to be quite problematic for many natural languages, for a
variety of related reasons. Sentences of natural language are not, in general,
composed of smaller sub-sentences. In the language of section 2, this means
that the non-trivial well-formed strings of a natural language may not be
composed of smaller well-formed substrings - which was a crucial property in
our proof of the theorem presented in section 2. This seems to be closely
related to the fact that most grammars of natural languages construct
sentences not on the basis of concatenation of simpler formula, but rather on
the basis of grammatical relations of either dependency (in the case of
dependency grammars) or constituency (in the case of phrase structure
grammars).

Although significantly more complicated, such grammars do nevertheless impose
relations of grammatical hierarchy and interdependence on the various
syntactic forms of their respective languages. Under such relations, certain
syntactic forms may be seen as more or less primitive in relation to others.
This opens the door to extending our proof strategy used in our proof of the
main result of section 2, to languages with more complicated grammatical
relations. For our purposes, the crucial feature of (IC) is that it allowed us
to relate certain properties of proper substrings to the properties of the
well-formed strings in which they occur. In a similar fashion, we may hope to
identify methods of relating the expressive properties of more primitive
syntactic forms to the expressive properties of the more complex syntactic
forms in which they occur.

Complications aside, there is at least one area in which future investigations
are likely to be considerably simpler. In an attempt to construct the most
general possible theory, we have made no assumptions about the nature of
$\mathcal{M}$. However, the semantics of most languages, whether they be
formal or natural, generally allows for some additional structure on
$\mathcal{M}$. In most cases, this will be some sort of logical, set
theoretic, or algebraic structure. Whatever the case may be, this structure
will furnish us with additional relations (i.e. constraints) between the
various meanings and syntactic forms countenanced by the language. In general,
the stronger these constraints, the more we can say about the combinatorial
relations between the syntax and semantics of the language.

Regarding (IC) and the structure of $\mathcal{M}$, several suggestions of
Pietroski ([8]) appear to be a promising starting point for extending our
methods to the analysis of natural language. In particular, Pietroski’s
analysis of how meanings compose in natural language suggests that many
instances of compositionality may be reducible to logical conjunction.

Future works on these topics, therefore has several areas to explore. First we
can of course seek to prove more theorems about both (SST) and (IC), as we
have done here. More generally, there are a variety of suggestions in the
literature on compositionality, which make explicit claims about how the
meanings of complex expressions of a language are related to the meanings
their component parts. Many of these claims about compositionality are likely
to entail specific combinatorial properties about their languages, which may
then be identified using formal methods similar to those which we have
employed in this paper. Finally, one might hope to obtain a deeper
understanding of the relations between synonymy, intension, and
compositionality by using these methods. These are rather distinct concepts,
and yet they all seem to exhibit specific combinatorial properties. By
identifying and relating the various combinatorial properties associated with
these linguistic phenomenon, we may hope to thereby identify important
relationships between synonymy, intension, and compositionality, and other
related linguistic concepts.

## References

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