# Process Variation-Aware Compact Model of Strip Waveguides for Photonic
Circuit Simulation

Aneek James, Anthony Rizzo, Yuyang Wang, Asher Novick, Songli Wang, Robert
Parsons, Kaylx Jang, Maarten Hattink, and Keren Bergman This work was
supported in part by the U.S. Advanced Research Projects Agency–Energy under
ENLITENED Grant DE-AR000843 and in part by the U.S. Defense Advanced Research
Projects Agency under PIPES Grant HR00111920014.A. James, Y. Wang, A. Novick,
S. Wang, R. Parsons, K. Jang, M. Hattink, and K. Bergman are with the
Department of Electrical Engineering, Columbia University, New York, NY 10027,
USA. (Corresponding author: Aneek James, e-mail: aej2149@columbia.edu).A.
Rizzo is with the Air Force Research Laboratory Information Directorate, Rome,
NY 13441, USA.© 2023 IEEE. Personal use of this material is permitted.
Permission from IEEE must be obtained for all other uses, in any current or
future media, including reprinting/republishing this material for advertising
or promotional purposes, creating new collective works, for resale or
redistribution to servers or lists, or reuse of any copyrighted component of
this work in other works.

###### Abstract

We report a novel process variation-aware compact model of strip waveguides
that is suitable for circuit-level simulation of waveguide-based process
design kit (PDK) elements. The model is shown to describe both loss and—using
a novel expression for the thermo-optic effect in high index contrast
materials—the thermo-optic behavior of strip waveguides. A novel group
extraction method enables modeling the effective index’s ($n_{\mathrm{eff}}$)
sensitivity to local process variations without the presumption of variation
source. Use of Euler-bend Mach-Zehnder interferometers (MZIs) fabricated in a
300 mm wafer run allow model parameter extraction at widths up to 2.5 $\mu$m
(highly multi-mode) with strong suppression of higher-order mode excitation.
Experimental results prove the reported model can self-consistently describe
waveguide phase, loss, and thermo-optic behavior across all measured devices
over an unprecedented range of optical bandwidth, waveguide widths, and
temperatures.

###### Index Terms:

Silicon photonics, compact modeling, process variation.

##  I Introduction

Silicon photonics (SiPh) has seen explosive growth in demand as a technology
platform, driven by its adoption in data centers (DC), high performance
computing (HPC) [1, 2, 3], quantum computing [4, 5, 6, 7, 8], and radio-
frequency communication systems [9, 10, 11]. SiPh’s rapid rise and maturation
has been enabled by its ability to leverage decades of research in the
complementary metal–oxide–semiconductor (CMOS) industry, drastically reducing
the typical research and development (R&D) costs associated with new
semiconductor technologies [12, 13, 14]. SiPh, however, has not yet been able
to mimic CMOS yield prediction tools for evaluating photonic integrated
circuits (PICs). Yield is a ubiquitous metric used across semiconductor
manufacturing, with improvements in yield being strongly correlated with
reductions in the time and costs associated with PIC design cycles [15, 16,
17]. The need for predictive yield models can be mitigated to some degree by
designing variation-robust devices [18] or PICs such that performance
variations can be tolerated or corrected for post fabrication [19, 20]. In
each of these cases, however, quantitative yield data cannot be determined
prior to fabrication—an obstacle that will be exacerbated as the number of
components per PIC in silicon is projected to scale well into the millions
within the next decade [21]. Circuit designers also need tools to optimize
system-level performance through device-level design choices [22]. To meet
rising circuit design complexity, commercial foundries must develop process
design kits (PDKs) that include compact models that are both parameterized
over a wide range of relevant design and environmental variables and describe
all important device figures of merit [23, 24]. It is essential that strip
waveguides in particular—a critical component of most SiPh circuits—are
accurately modeled according to their expected fabricated performance.

TABLE I: Features for modeling strip waveguide performance. The model in this work describes phase, loss and thermal behavior effects over a broad range of wavelengths and waveguide geometries. Model Features | [25] | [26] | This Work
---|---|---|---
Wavelength [nm] | 1550 | 1520–1570 | 1450–1650
Nominal Width Range [nm] | 480 | 480–500 | 400–2500
Considered Variation Sources |  $w$,$t$ |  $w$,$t$ | Arbitrary
Statistical Parameter Variations | ✓ | ✓ | ✓
Waveguide Scattering Losses | ✗ | ✗ | ✓
Thermo-optic Effect | ✗ | ✗ | ✓
$w$ \- Waveguide Width Variations
$t$ \- Waveguide Thickness Variations

Broadly speaking, there are three ways to construct compact models: (i) look
up table-based models, obtained directly from measurements or device
simulations, (ii) models based on empirical fit functions, and (iii) physics-
based models [23]. Most previously reported work falls under the look-up
table-based category [27, 26, 25, 28, 29]. These models can be parameterized
using look-up tables (LUTs), where interpolation is used to predict the
performance of designs not explicitly defined in the table. Ensuring that LUT
models are accurate over a wide range of input parameters, however, requires
measuring all waveguide figures of merit for every combination of input
parameters; a task that scales exponentially with the number of modeled
independent variables. Prior demonstrations methods also require the explicit
connection of the measured effective and group index variations to a
predefined number of process variation sources, introducing the possibility of
error if any systemic deviations exist between the simulation configuration
and the realities of the fabrication process.

In this paper, we report to the best of our knowledge, the first geometry-
parameterized compact model of strip waveguides that can capture device
performance over a wide range of wavelengths and waveguide geometries (see
Table I). Using a novel derivation of the thermo-optic effect that is accurate
for high-index contrast waveguides, we demonstrate our model’s ability to
describe both scattering loss and the thermo-optic effect as a function of
both design and statistical parameters. A novel group-extraction-based method
allows the characterization of process variations without presumption of a
source or its associated sensitivity. This extraction methodology is used to
construct a model from dozens of geometric variations of Mach-Zehnder
Interferometers (MZIs) fabricated in a 300 mm commercial foundry. These use of
Euler bends in these MZIs permits the characterization of wide waveguide
performance with minimal higher-order mode excitation. Experimental results
validate the model’s accuracy in describing the phase, loss, and thermo-optic
performance across the entire wafer. The model is also implemented in
Verilog-A to demonstrate compatibility with electronic-photonic co-simulation
environments [30, 31, 32]. This work represents a key step toward the modeling
of waveguide-based PDK components, enabling true-to-measurement circuit
simulation at massive integration densities.

##  II Physics-Aware Model Development

Figure 1: a, Example electric field profile taken from Lumerical MODE. b,
Simulated (scatter) and modeled (dashed) effective index vs wavelength for
several waveguide widths. Each waveguide was simulated with a thickness of 220
nm.

Because the mode condition of an optical waveguide is described via a
transcendental equation, completely generalized analytical solutions for the
effective index $\left(n_{\text{eff}}\right)$ are impossible to derive [33].
We therefore propose, as discussed in [34], finding a behavioral model that
accurately captures its dependence on all design parameters over the relevant
ranges of interest. In this section, we develop dependency models for the
design parameters available. The semi-physical nature of the model is then
leveraged to describe both the scattering loss and the thermo-optic
coefficient. Process variations, whether of a design parameter or not, will be
covered in Section IV.

###  II-A Wavelength Dependence

The wavelength dependence of the waveguide $n_{\text{eff}}$ is first
considered. The $n_{\text{eff}}$ of several silicon-on-insulator (SOI)
waveguide geometries were simulated in Lumerical MODE (Fig. 1a). From the
results, it is shown that the wavelength dependence over the S-, C-, and
L-bands for all geometries is well-approximated by a second-order Taylor
expansion for a wide range of waveguide widths sufficiently above the cutoff
condition (Fig. 1b):

$\color[rgb]{0,0,0}n_{\text{eff,
model}}(\lambda)=\sum_{i=0}^{2}\frac{1}{i!}\left.\frac{\partial^{i}n_{\text{eff}}}{\partial\lambda^{i}}\right|_{\lambda=\lambda_{0}}(\lambda-\lambda_{0})^{i}.$
(1)

###  II-B Geometric Dependence

As the Taylor expansion only captures the wavelength-dependence, it is clear
that the fitting parameters $\partial^{2}n_{\text{eff}}/\partial\lambda^{2}$,
$\partial n_{\text{eff}}/\partial\lambda$ and
$\partial^{0}n_{\text{eff}}/\partial\lambda^{0}$ (hereafter referred to as
$n_{\text{\text{eff}},0}$) are responsible for capturing the dependence on
waveguide geometry. With respect to width, all three fitting parameters were
previously found in [35] to be well described by the following behavioral
model:

Figure 2: a-c, Plot of simulated (scatter) and modeled (dashed)
$n_{\text{eff}}$ parameters
$\left[\partial^{2}n_{\text{eff}}/\partial\lambda^{2},\partial
n_{\text{eff}}/\partial\lambda,n_{\text{\text{eff}},0}\right]$ vs waveguide
width (respectively). These values were for a waveguide with a thickness of
220 nm at a wavelength of 1550 nm. d, Comparison of the model (dashed) and
simulated (scatter) $n_{\text{eff}}$ vs waveguide width for different
thicknesses. Simulated at 1550 nm.

$\color[rgb]{0,0,0}\frac{\partial^{i}n_{\text{eff}}}{\partial\lambda^{i}}(w)=p_{i0}\cdot\frac{w^{2}+p_{i1}w+p_{i2}}{w^{2}+p_{i3}w+p_{i4}},$
(2)

for a total of fifteen model parameters. To verify correctness of the model,
all three parameters were fitted to the simulation data with (1)-(2) using
ordinary least squares (OLS) regression. The model was able to match all three
parameters over the entire range of the width sweep (Fig. 2a-c). The close
matching of the modeled and extracted Taylor parameters means that our
modification of (2) still preserves its ability to match the behavior of
effective index as a function of wavelength. By extension, these three Taylor
parameters allow for a robust description of $n_{\text{eff}}$ as a function of
waveguide width (Fig. 2d). The data also demonstrates this agreement is not
unique to any particular waveguide thickness, with different thicknesses
producing different sub-parameter fits.

Figure 3: Comparison between modeled (dashed) and simulated (scatter)
$n_{\mathrm{eff}}$ for higher order modes and the fundamental TM mode. All
waveguides were simulated with a thickness of 220 nm.

Finally, it should be noted that both the numerator and denominator in (2) are
polynomials of equal order. Our model consequently predicts that, for a given
wavelength, the effective index will asymptotically approach a constant value
as $w$ approaches infinity. The value that the model approaches as $w$ tends
towards infinity can be interpreted as the equivalent $n_{\text{eff}}$ of an
infinite slab of the same thickness:

$\lim_{w\to\infty}n_{\text{eff}}(\lambda,w)=n_{\text{slab}}(\lambda).$ (3)

In this way, our behavioral model can elegantly capture all significant
features of effective index for the design parameters of interest. The model’s
accuracy holds true for higher order modes as well, provided that they are
sufficiently far away from their respective waveguide cutoff condition (Fig.
3).

###  II-C Scattering Loss

Scattering loss due to sidewall roughness (SWR) can be a significant source of
loss in most reported waveguide designs, making it critical for designers to
accurately model [36]. In this section, we demonstrate our model’s ability to
capture SWR loss as a function of waveguide geometry. It was first noted in
[37] that the traditional Payne and Lacey model of SWR-induced loss [38, 39]
was found to be identical in behavior to the derivative of the effective index
with respect to waveguide width:

$\alpha_{\text{SWR}}(\lambda,w)=R\frac{\partial}{\partial
w}\left[n_{\text{eff}}(\lambda,w)\right],$ (4)

where $R$ is a proportionality constant. As our model can describe
$n_{\text{eff}}$ as a function of width, a closed-form representation of
$\partial n_{\text{eff}}/\partial w$ can be exactly derived. This equation can
then be fitted to measured waveguide loss data to extract the proportionality
constant. We validate this by fitting (4) to the scattering loss of a 7 $\mu$m
long SOI waveguide with some SWR wall roughness in Lumerical 3D-FDTD (Fig.
4a). The roughness Root Mean Square (RMS) and correlation length were
arbitrarily chosen to be $\sigma_{\text{rms}}=5~{}\mathrm{nm}$ and
$L_{\text{corr}}=1~{}\mathrm{\mu m}$ respectively. These parameters were then
used to generate a random, anisotropic SWR on the waveguide walls [40].
Propagation losses were simulated for waveguide widths ranging from 450 nm to
850 nm. The results of the fitting are shown in Fig. 4b, with our model
closely matching trend of the scattering loss behavior extracted from FDTD
simulations.

Figure 4: a, Graphical representation of a waveguide simulated with some
sidewall roughness. The inset is a magnified view of the waveguide to clarify
the definition of $\sigma_{\mathrm{rms}}$. b, Scattering losses estimated from
FDTD compared to the fit using our model based on Lumerical MODE data.

###  II-D Thermo-Optic Effect

Our model can also completely describe the thermo-optic coefficient of an
arbitrary waveguide geometry without the need for any thermal measurements.
The thermo-optic coefficient of a waveguide mode most importantly requires
knowledge of the confinement factor, which is the fraction of a mode’s power
confined within each constituent waveguide material. Kawakami showed in [41]
that for a waveguide made up of N materials, each with with an index $n_{k}$
and a confinement factor $\Gamma_{k}$:

$\displaystyle\sum_{k}^{N}\Gamma_{k}n^{2}_{k}=n_{\mathrm{g}}n_{\mathrm{eff}}$
(5a) $\displaystyle\sum_{k}\Gamma_{k}=1,$ (5b)

where (5b) is derived from noting that the sum of all confinement factors must
equal unity due to power conservation. A closed-form of the confinement factor
for a two-material waveguide (e.g. SOI wires) can then be derived:

$\displaystyle\Gamma_{\text{core}}=\frac{\displaystyle
n_{\mathrm{g}}n_{\mathrm{eff}}-n_{\text{clad}}^{2}}{\displaystyle
n_{\text{core}}^{2}-n_{\text{clad}}^{2}}$ (6a)
$\displaystyle\Gamma_{\text{clad}}=\frac{\displaystyle
n_{\text{core}}^{2}-n_{\mathrm{g}}n_{\mathrm{eff}}}{\displaystyle
n_{\text{core}}^{2}-n_{\text{clad}}^{2}},$ (6b)

where $\Gamma_{\text{core}}$ is the power contained in the waveguide core and
$\Gamma_{\text{clad}}$ is the power contained in the cladding.

Next, we must obtain an expression that describes the thermo-optic effect on
$n_{\text{eff}}$ in terms of the confinement factor. A common approximation of
the thermo-optic coefficient of $n_{\text{eff}}$ is

$\frac{\partial n_{\text{eff}}}{\partial T}\approx\Gamma_{1}\frac{\partial
n_{\text{1}}}{\partial T}+\Gamma_{2}\frac{\partial n_{\text{2}}}{\partial
T}+\dots,$ (7)

where $\delta$ represents a small perturbation in the values, $\Gamma_{n}$ is
the confinement of the mode within material $n$ and $\partial
n_{\text{n}}/\partial T$ is the thermo-optic coefficient of material $n$ [42].
Though this equation is widely used [43, 44, 45] and may be accurate in
certain scenarios, to the authors’ knowledge it has never been demonstrated to
be a generally accurate approximation. We therefore start from first
principles and consider a general perturbation of the wave equation [46]:

$\delta\left[\beta^{2}_{\mathrm{eff}}\right]=\Gamma_{\text{core}}\frac{\omega^{2}}{c^{2}}\delta\left[n_{\text{core}}^{2}\right]+\Gamma_{\text{clad}}\frac{\omega^{2}}{c^{2}}\delta\left[n_{\text{clad}}^{2}\right],$
(8)

where $\beta_{\mathrm{eff}}$ is the effective wavenumber,
$\Gamma_{\text{core}}$ is the confinement in the waveguide core,
$\Gamma_{\text{clad}}$ is the confinement in the waveguide cladding, and
$n_{\text{core}}$ and $n_{\text{clad}}$ are the core and cladding indices
respectively. Carrying this operation through and combining with (1) (see
Appendix -A for details) yields:

$\displaystyle n_{\text{eff}}(\lambda,w,T)\approx
n_{\text{eff},T_{0}}(\lambda,w)+\frac{\partial n_{\text{eff}}}{\partial
T}\left(T-T_{0}\right)$ (9a) $\displaystyle\frac{\partial
n_{\text{eff}}}{\partial
T}=\Gamma_{\text{core}}\frac{n_{\text{core}}}{n_{\text{eff,
T${}_{0}$}}}\frac{\partial n_{\text{core}}}{\partial
T}+\Gamma_{\text{clad}}\frac{n_{\text{clad}}}{n_{\text{eff,
T${}_{0}$}}}\frac{\partial n_{\text{clad}}}{\partial T},$ (9b)

where $n_{\text{eff},T_{0}}$ is the $n_{\text{eff}}$ at some reference
temperature $T_{0}$. The key addition to (9) compared to prior literature is
the scaling of each thermo-optic term by ratio between the material and
effective indices. As the index contrast between the core and cladding
decreases, our model will approach the (7). Thus it is clear that our model
will outperform (7) in accuracy when describing high index contrast materials,
such as the SOI waveguide geometries prevalent in SiPh.

Figure 5: a Modeled (dashed) and simulated (scatter) confinement factor vs
waveguide width for different thicknesses. b, Comparison between simulated
(scatter), previously reported model (dotted, Eq. (7)) and our work (dashed
line, Eq. (9)) describing $n_{\text{eff}}$ vs Temperature of a 480 x 220 nm
waveguide.

With these expressions, our confinement factor and the thermo-optic
coefficient models can be validated. The simulated confinement factor is
compared to our model prediction at 1550 nm in Fig. 5a. The optical properties
of silicon and silicon dioxide used in our model were taken directly from
[47]. There was a near perfect agreement between the modeled and simulated
confinement factor, showing that the general behavior of confinement factor is
captured by our model (Fig. 5a). The modeled thermo-optic coefficient is
validated by simulating how the $n_{\text{eff}}$ of a SOI waveguide varies
with temperature using Lumerical MODE (Fig. 5b). Silicon was assumed to have a
thermo-optic coefficient of $1.9\text{\times}{10}^{-4}$ K-1 [48] and SiO2 was
assumed to have a thermo-optic coefficient of $1\text{\times}{10}^{-5}$ K-1
[49]. The model and simulations show exceptional agreement from 300 - 1200 K,
despite the fact that our model does not require any data from thermal
simulations or measurements. As predicted, the previously reported model of
the thermo-optic effect (7) significantly under-predicts the expected change
in $n_{\mathrm{eff}}$. It should be noted that in real devices, waveguide
geometry itself is a function of $T$ due to thermal expansion. This can be
accounted for by modeling $w$ as a function of $T$. Experimental results in
Section V-C, however, show that assuming a constant width geometry provides
sufficient accuracy.

Having a model of the thermo-optic effect that is accurate over a wide range
of conditions like this one holds a great deal of potential to enable more
robust design exploration, such as evaluating photonic waveguide heater
designs [50, 51], characterizing self-heating in micro-resonators [52], or
studying the effect of ambient temperature fluctuations in a system.

###  II-E Parameter Extraction

The practical utility of a compact model is greatly determined by the
associated parameter extraction procedure to connect the model to a given
foundry process. This is particularly important when developing statistical
models, as accurate parameter extraction is the only way to guarantee that
process variations are accurately reflected in the model. A popular solution
is to leverage the phase-sensitivity of interferometric optical filters—such
as Mach-Zehnder interferometers (MZIs), microresonators, or arrayed waveguide
gratings (AWGs)—to monitor process variations across a wafer. Regardless of
the chosen device, a shared difficulty lies in accurately guessing what
particular interference fringe position corresponds to a particular fringe
order [53, 26, 25]. Our method is based on the curve-fitting method presented
in [25] and [54], with some additional steps described to include waveguide
dispersion as an extracted parameter.

The first step in parameter extraction is to characterize the group index
($n_{g}$) of a fabricated interferometer from a wavelength sweep of the
device. To enable this, (1) is rearranged into a more suitable form:

$\displaystyle
n_{\text{eff}}(\lambda)=\frac{1}{2}\frac{\partial^{2}n_{\text{eff}}}{\partial\lambda^{2}}\lambda^{2}+B\lambda+C$
(10a) $\displaystyle B=\frac{\partial
n_{\text{eff}}}{\partial\lambda}-\frac{\partial^{2}n_{\text{eff}}}{\partial\lambda^{2}}\lambda_{0}$
(10b) $\displaystyle
C=\frac{1}{2}\frac{\partial^{2}n_{\text{eff}}}{\partial\lambda^{2}}\lambda^{2}_{0}-\frac{\partial
n_{\text{eff}}}{\partial\lambda}\lambda_{0}+n_{\text{eff,0}},$ (10c)

where $B$ and $C$ are fitting parameters that aggregate the 1${}^{\text{st}}$
and 0${}^{\text{th}}$ order terms from (1) respectively. Following the
procedure described in [54], it is first noted that the fringe condition of an
inteferometric device is described by

$\phi=\frac{2\pi}{\lambda}n_{\text{eff}}(\lambda)L=2\pi m,$ (11)

where $\phi$ is the phase difference between the interferometry arms, $L$ is
the path length of the interferometer, $\lambda$ is a particular fringe
wavelength, and $m$ is an integer corresponding to the particular fringe
order. To extract our model parameters, a wavelength sweep of the
interferometric device is required. Once this is performed, a peak finding
algorithm can be used to detect the wavelength of all detected fringes. A
function that relates the relative fringe locations to the $n_{g}$ of the
waveguide is now required. This can be done by defining a continuous function
that will yield an integer value at each of the detected fringe locations. Let
$m_{0}$ represent the particular fringe order corresponding to an arbitrarily
chosen reference fringe located at $\lambda_{0}$. The fringe order $m$ of any
other fringe can be defined relative to this reference as

$m=m_{0}+\int_{\lambda_{0}}^{\lambda}\frac{dm}{d\lambda}\text{d}\lambda=m_{0}+n_{\text{g}}L\cdot\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right).$
(12)

This continuous function now allows us to redefine the measured fringes into a
form suitable for parameter extraction. A reference fringe variable $n$ is now
defined by letting $m=(m_{0}+n)$. Inserting this back into (12) produces:

$n=n_{g}L\cdot\left(\frac{1}{\lambda_{n}}-\frac{1}{\lambda_{0}}\right),$ (13)

where each relative fringe $n$ is located at an associated wavelength
$\lambda_{n}$. Using (13), the $n_{g}$ of the measured device is now directly
related to the measured fringe locations. This fitting equation must now be
extended to our specific model parameters. The $n_{g}$ of a waveguide is
defined to be

$n_{g}=n_{\text{eff}}-\lambda\frac{\partial n_{\text{eff}}}{\partial\lambda}.$
(14)

Combining with (10a) yields an expression for $n_{g}$ in terms of our compact
model:

$n_{\text{g}}=C-\frac{1}{2}\frac{\partial^{2}n_{\text{eff}}}{\partial\lambda^{2}}\lambda^{2}.$
(15)

By inserting (15) back into (13), we can derive an OLS regression-compatible
expression:

$\displaystyle
n=C\Lambda_{C}-\frac{\partial^{2}n_{\text{eff}}}{\partial\lambda^{2}}\Lambda_{S}$
(16a)
$\displaystyle\Lambda_{C}=L\cdot\left(\frac{1}{\lambda_{n}}-\frac{1}{\lambda_{0}}\right)$
(16b)
$\displaystyle\Lambda_{S}=\frac{L}{2}\cdot\left(\lambda_{n}-\frac{\lambda_{n}^{2}}{\lambda_{0}}\right),$
(16c)

where $[\Lambda_{C},\Lambda_{S}]$ are explanatory variables. Performing an OLS
regression between $n$ and $[\Lambda_{C},\Lambda_{S}]$ gives us two of our
three fitting parameters in (10). Finally, $B$ can be calculated by combining
equations (11) and (10a):

$B=\frac{m}{L}-\frac{1}{2}\frac{\partial^{2}n_{\text{eff}}}{\partial\lambda^{2}}\lambda_{m}-\frac{C}{\lambda_{m}},$
(17)

where the only uncertainty is what fringe order $m$ corresponds to each
measured fringe $\lambda_{m}$. Once $B$ is determined from (17), (10b) and
(10c) can be used to determine the original fitting parameters in (1). It
should be noted that each detected fringe ($m$, $\lambda_{m}$) location will
yield very small variations in the B value due to resolution-based uncertainty
in the exact value for $\lambda_{m}$. For a best guess, all values $B_{m}$
taken from each measured fringe $\lambda_{m}$ should be averaged together.

Figure 6: a, Captured spectrum of simulated MZI used for parameter extraction.
The waveguide mode was simulated in Lumerical MODE, and then exported to a MZI
waveguide simulation block in Lumerical INTERCONNECT. b, Linear Regression of
fringe wavelengths to extract the $n_{g}$ performed on the detected fringes
from a. c, Possible $n_{\text{eff}}$ solutions (black, dashed), along with the
actual solution (red), determined by the $n_{g}$ extracted in b.

To validate this method under ideal conditions, an MZI constructed using 480
nm x 220 nm waveguides is simulated in Lumerical INTERCONNECT. To ensure
accuracy, the waveguide’s $n_{\text{eff}}$ was first simulated in MODE and
then exported to a MODE Waveguide element in INTERCONNECT. As the full-width
half-maximum (FWHM) of the MZI does not affect the extracted $n_{\text{eff}}$,
the waveguides were arbitrarily assumed to have a 2.5 dB/cm loss and the
coupling coefficient was chosen to ensure critical coupling. The spectrum of
the simulated MZI is shown in Fig. 6a. Fringe locations were extracted using a
peak finding algorithm. The fringe located closest to the center of the sweep
was arbitrarily chosen as $n=0$. Using (16), OLS regression found
$\partial^{2}n_{\text{eff}}/\partial\lambda^{2}=-0.136$ $\mu$m-2 and
$C=3.9215$ (Fig. 6b). From here, the family of solutions for $n_{\text{eff}}$
is plotted in Fig. 6c. Each particular solution corresponds to a different
guess on the fringe orders detected, e.g. $m_{0}=52$ vs. $m_{0}=53$. The
separation between each $n_{\text{eff}}$ solution plotted in 6c is determined
by the free-spectral range (FSR) of the interferometer, with a larger FSR
corresponding more widely separated solutions.

To determine the correct fringe order of the reference we use the fact that,
from the simulations performed in Section II, we know the waveguide geometry
has an $n_{\text{eff}}$ of 2.411 at the reference fringe location. In Section
III we explain how to increase the accuracy of this estimation to avoid errors
introduced by this simulation. From this, the reference fringe order is found
to be $m_{0}\approx 114.03$. Since fringe orders must be integer numbers, the
result is rounded to the nearest integer 114. By combining (10a)-(10c), the
original fitting coefficients are found to be $\partial
n_{\text{eff}}/\partial\lambda=-1.078$ $\mu$m-1 and $n_{\text{eff,0}}=2.411$.
To evaluate accuracy of our extraction, we define the relative error between
the extracted and simulated $n_{\text{eff}}$’s $\sigma_{\text{error}}$ by:

$\sigma_{\text{error}}=\sqrt{\frac{\int\left(n_{\text{eff,
model}}-n_{\text{eff, sim}}\right)^{2}\text{d}\lambda}{\int n_{\text{eff,
sim}}^{2}\text{d}\lambda}},$ (18)

where $n_{\text{eff, sim}}$ is the effective index from the MODE simulation,
used as a reference to quantify our method’s accuracy, and $n_{\text{eff,
model}}$ is the result from applying our extraction method to the simulated
MZI. Upon evaluation, the total relative error was found to be 0.017%. Since
the order of the reference fringe is correct, the remaining model error is
attributed to inaccuracies in the initial regression fit using (16a)-(16c).

##  III More Robust $n_{\mathrm{eff}}$ Extraction Under Process Variability

Figure 7: a, Plot of the $n_{g}$ error function for one sample. The error
function shows a minimum at roughly 491.5 nm, which closely agrees with the
actual waveguide width of 490 nm. b, Convergence of the etch bias estimate for
different numbers of samples averaged.

The reliability of the extraction is highly sensitive to the guessed value of
the reference fringe order. For the example in Section II-E, we used a priori
knowledge of the $n_{\text{eff}}$ at the reference fringe to estimate its
order. Therefore, any deviation between the assumed and actual waveguide
dimensions risks introducing error. By noting that the initial order estimate
rounded to the nearest integer, we can use (11) to define a boundary beyond
which our fringe order guess will be incorrect [25]:

$|\Delta m|=|n_{\text{eff, actual}}-n_{\text{eff,
guess}}|\leq\frac{\lambda_{m_{0}}}{2L}.$ (19)

We can see that, to raise confidence in the guessed fringe order, either the
accuracy of our $n_{\text{eff}}$ guess must be increased or the
interferometric path length must be decreased. As explained in Section II-E,
our extraction method begins by directly extracting the $n_{g}$ of a given
interferometer via optical sweep. Process variations will therefore appear as
variations in the extracted values for
$\partial^{2}n_{\text{eff}}/\partial\lambda^{2}$ and $C$. By measuring several
devices of the same drawn width across the all measured dies, wafers, and
lots, the influence of the random width and thickness variations can be
eliminated by averaging their extracted fitting parameters. As the sample size
becomes sufficiently large—with the necessary sample size being a function of
the severity of the process variations—any remaining deviation between the
nominal and averaged parameters will be the result of a systemic etch biases
on the waveguide width. We therefore propose estimating this etch bias by
creating a preliminary $n_{\text{eff}}$ model based on the results of a
photonic mode solver, such as Lumerical MODE. Using this model, an equivalent
waveguide width can be found by minimizing the error function

$\min_{w}\sqrt{\frac{\int\left[n_{\text{g, model}}(w,\lambda)-n_{\text{g,
meas}}(\lambda)\right]^{2}\text{d}\lambda}{\int n_{\text{g,
meas}}^{2}(\lambda)\text{d}\lambda}},$ (20)

where $n_{\text{g, meas}}$ is the extracted model of $n_{g}$ using the
averaged extracted parameters and $n_{\text{g, model}}$ is the simulation-
based, width-dependent a priori model of $n_{\text{eff}}$. The
$n_{\text{eff}}$ of our equivalent waveguide width can then be plugged into
the a priori model to provide a more accurate fringe order estimate. In this
way, we can increase the accuracy of our guessed effective index, regardless
of whether the modeled waveguide composition is accurate to the virtual device
composition.

We now discuss the robustness of this optimization routine in the presence of
other systemic non-idealities and its ability to perform etch bias correction.
To do this, we need a ’ground truth’ value for $n_{\mathrm{eff}}$, which we
obtain by simulating all the non-idealities in Lumerical MODE. Subsequently we
perform the parameter extraction using Lumerical INTERCONNECT. By comparing
the extracted $n_{\mathrm{eff}}$ to the known simulated value for
$n_{\mathrm{eff}}$, we can directly evaluate the robustness of our
methodology.

###  III-A Statistical Geometric Variation

To test the extraction procedure’s accuracy under process variations, a
simulation of 100 random variations on the waveguide geometry was run. The
nominal waveguide dimensions were assumed to be 480 x 220 nm. To simulate
systemic variations, each waveguide was arbitrarily assumed to have an etch
bias of +10 nm. Random fluctuations were simulated by subjecting each device
to a normally distributed variation of $3\sigma=5~{}\mathrm{nm}$ on both the
waveguide width and thickness, as this value is consistent with the worst-case
reported values for geometric variations [26, 25, 27]. Each mode profile was
then exported to INTERCONNECT and simulated with interferometer FSRs ranging
from 4 - 40 nm to investigate the effect this had on the extraction error. The
resulting error function for one of these samples, with a ground truth width
of 490nm, is shown in Fig. 7a. We see the convergence behavior of the etch
bias estimate evolves as a function of device sample size increases for
several FSR designs in Fig. 7b. It can be seen that all FSR designs can yield
at least an estimated etch bias within 2 nm of the actual value, indicating
the utility of our etch bias correction.

Figure 8: Plot of mean error in $n_{\text{eff}}$ over the simulation bandwidth
per simulated device. Each FSR was simulated with 100 random deviations from
the target waveguide geometries. Both width and thickness were assumed to have
a $3\sigma=5~{}\mathrm{nm}$.

Fig. 8 shows the relationship between the average, per sample error and the
interferometer FSR. The error is measured in three scenarios: i.) a ‘naïve’
case, where the fringe order is estimated assuming no etch bias; ii.) where
the fringe order is estimated through our etch bias prediction methodology,
based on 30 measured samples; and iii.) where the exact $n_{\text{eff}}$ from
simulations is used to determine the actual fringe orders. The last scenario,
that produced an average per sample error of roughly 0.017% represents an
error floor for the first two. This error floor is completely determined by
errors in the initial $n_{g}$ regression, as well as any fundamental
limitations in our chosen behavioral model. As the FSR is increased, the
average per sample error in both cases improves steadily until it reaches the
aforementioned floor. This is consistent with (19), indicating that a larger
FSR corresponds to a wider margin of error for the fringe order estimate. For
both the naïve and bias compensation methods, there is a critical FSR value
beyond which the fringe order is correctly estimated for all samples. It is
clear, however, that estimating the presence of any etch biases drastically
improves the fringe order accuracy, reaching the error floor for a much
smaller FSR than when using the naïve method.

Figure 9: a, $n_{g}$ relative error vs simulated sidewall angle. b, Comparison
between the simulated (scatter) and estimated (dashed) $n_{\text{eff}}$ for
different sidewall angles. Figure 10: a, Illustration of measured reticles on
a custom 300 mm wafer, with a blown-up microscopic image of a die with 135
MZIs. b, Nominal $n_{\mathrm{eff}}$ and $n_{g}$ model extracted from device
measurements. c, Width-based model extraction for each die tested. d, Total
model parameter $\mu$m variance $\sigma$ explained vs number of principal
components included.e, Plot of the width-independent subparameters for
$n_{\mathrm{eff,0}}$, $\partial n_{\mathrm{eff}}/\partial\lambda_{0}$,and
$\partial^{2}n_{\mathrm{eff}}/\partial\lambda^{2}_{0}$ vs $V$.

###  III-B Sidewall Angle

We now consider how the parameter extraction behaves when used for waveguides
with some sidewall angle. Up to this point, our simulations assumed the
waveguides to have no sidewall angle. Real waveguides, however, typically
deviate from this ideal [55]. To study how our bias correction behaves under
these conditions, a SOI waveguide with the same nominal (480 x 220 nm) design
as before was simulated with a series of sidewall angles from 85 to 90 degrees
as this is a range typical of foundries [56, 25]. As only the aggregate
behavior is being studied, width and thickness variations were not included.
As seen in Fig. 9a, the minimum of the error function optimized in the etch
bias estimation step remain roughly constant for all considered sidewall
angles. This results in very accurate predictions of the effective index from
our model, even though the fundamental geometry is different. We interpret
this as our optimization routine is picking an ‘equivalent’ waveguide width
that matches the extracted $n_{g}$ profile. This equivalent width always seems
to result in a waveguide design with a similar confinement factor and
effective index—and therefore behavior—as seen in Fig. 9b.

###  III-C Material Variation

This method for increasing the accuracy of the guessed $n_{\mathrm{eff}}$
relies on the assumption that the material properties of the fabricated
waveguides generally match the assumed material properties used in the
simulation data used to construct the model. In practice, however, there can
be a great deal of deviation between the assumed and actual optical properties
of the waveguide materials. As a workaround, the authors suggest extracting
and building a model based around the dispersion of the waveguide
$\partial^{2}n_{\mathrm{eff}}/\partial\lambda^{2}$, as this waveguide
parameter can be extracted exactly from measurements. The nominal model of
$\partial^{2}n_{\mathrm{eff}}/\partial\lambda^{2}$ can then replace $n_{g}$ in
(20) to estimate the width of the measured device. This width can then be used
in conjunction with simulation data to assign it an $n_{\mathrm{eff}}$ guess.
Though the limits of such a technique are unclear to the authors, experimental
results in Section V demonstrate to be effective enough for describing the
$n_{\mathrm{eff}}$, loss, and thermo-optic effect for all measured device
performance.

##  IV Extracting Local Parameter Variations

Figure 11: a, Measured vs modeled optical spectrum of a 480 nm waveguide MZI
with $\Delta L=100~{}\mu$m. b, (i) Histogram of extracted $V$ data along with
its associated Gaussian distribution (red, dashed) overlaid on top. (ii)
Spatial map of the average value for $V$ per measured die across the wafer. c,
Mean and standard deviation of $\Delta w$, $n_{\mathrm{eff}}$, and $n_{g}$.

Process variations (e.g. thickness variation, cladding and core index
variations) will appear in our model as variations in the fifteen model
parameters that comprise Eq. (1). Capturing these variations requires the
ability to extract their value locally, which cannot be done just by looking
at the performance of any individual device. It is commonly assumed in prior
literature that most process parameters slowly vary across the entire wafer
[25]. This assumption implies that the values of the parameters comprising our
model also vary slowly across the wafer. The authors therefore propose
analyzing the performance of several waveguide width designs in close
proximity to each other to locally extract all of the fifteen model
parameters. Each local extraction serves as the observations of each model
parameter that are tracked across the entire wafer.

The simplest way to create a statistical model is to treat each of fifteen
sub-parameters as independent statistical variables. This is not ideal,
however, as each additional variable drastically increases the number of
required iterations for accurate Monte Carlo simulations. To minimize model
complexity, we would like to represent each sub-parameter as a linear function
of an ensemble of variables:

$p_{ni}=p_{ni,\mathrm{avg.}}+\vec{s}\cdot\vec{V}.$ (21)

$\vec{V}$ is the vector of variables that represent the process variations.
Minimizing model complexity would be the equivalent of minimizing the size of
$\vec{V}$. $\vec{s}$ describes the corresponding sensitivities of a given
parameter to each element in $\vec{V}$. To minimize the size of $\vec{V}$, we
leverage the fact that each extracted model parameter will be strongly
correlated to one another. This is because the variations in each model
parameter share common origins such as wafer thickness, annealing time, etc.
We therefore propose using principal component analysis (PCA), a technique for
transforming a number of possibly correlated variables into a smaller number
of uncorrelated variables (i.e principal components) [57], to minimize model
complexity. The chosen principal components are then the variables that make
up $\vec{V}$. The chosen principal components are then the variables that make
up $\vec{V}$. The number of components in $\vec{V}$ is flexible (see Appendix
-B for details). Since our waveguide geometry is primarily a function of two
process variables—waveguide width and thickness—we use only the first
principal component to preserve its physical interpretation. The result is a
model of effective index as a function of width and our process
variations–$\Delta w$, representing width variations and an additional
variable we will call $V$, representing an aggregate of other process
variations, including thickness variation:

$n_{\mathrm{eff,model}}(w+\Delta w,V).$ (22)

This full model of $n_{\mathrm{eff}}$ is then used in the local optimization
and re-extraction of each measured device. The cost function is defined as the
sum of the relative $n_{\mathrm{eff}}$ and $n_{g}$ errors to match both the
measured fringe locations and FSR respectively.

Thus, we can employ a two-stage direct statistical compact model extraction
procedure [24]. In the first stage, we use group extraction to obtain the
complete set of fifteen parameters for a uniform device. In a second step, a
subset of model parameters are re-extracted for each member of a large
ensemble of devices measurements. This approach will be the most accurate
representation of how device performance varies across the wafer without any
presumption of variation source, statistical distribution, correlation, and
the resulting model sensitivity to the variation. An inherent strength of this
approach over others is that it is potentially useful for modeling other
waveguide geometries as well. This potential is due to the model designer
having the option of picking the number of principal components based on a
physical assumption on the key process variables or optimize the percentage of
explained parameter variance (see Appendix -B). While further investigation
would be required to confirm this, the methodology’s flexibility holds a great
deal of promise.

##  V Experimental Demonstration

We measured 7 reticles, each with 135 MZIs consisting of 27 different
waveguide widths (w) from 400 nm to 2500 nm and 5 different arm length delays
($\Delta$L) from 100 nm to 500 nm, fabricated on a custom 300 mm full wafer
through AIM Photonics (Fig. 10a). All 135 MZI were measured on reticle 2 while
a smaller subset of 30 MZIs were measured on each of the remaining reticles,
totaling 315 measured devices. Devices with the same waveguide width are
placed adjacently to minimize the impact of local process variations on device
performance. All MZIs have a nominal waveguide height of 220 nm, and grating
couplers designed for quasi-TE polarization are utilized for optical I/O. The
two arms of each MZIs consist of symmetric waveguide bends to mitigate the
impact of bending on the $n_{g}$. For devices with waveguides beyond the
single-mode cutoff width, Euler bends are used to maintain single mode
operation and high mode isolation [18]. A tunable laser was swept from
1450–1610 nm at a 10 pm resolution to characterize the transmission spectrum
of each MZI.

###  V-A Nominal Extraction

A nominal model $n_{\mathrm{eff}}$ is created by averaging the extracted
parameters for all measured devices as shown in Fig. 10b. We apply the
extraction method described in Section II-E to every collected transmission
spectrum. A preliminary model is built using the simulation data described in
Section II to estimate the expected device FSR for each waveguide width
variation. This estimated FSR is then fed into a peak finding algorithm to
extract the $n_{g}$ parameters, and then estimate fringe orders—and, therefore
the $n_{\text{eff}}$—of each measured device. As the measured $n_{g}$ deviated
a great deal from the simulated values, the technique described in Section
III-C is employed where a preliminary model based on
$\partial^{2}n_{\mathrm{eff}}/\partial\lambda^{2}$ is created and used to
estimate waveguide geometry to estimate the fringe order. All three Taylor-
expansion parameters are then derived using (10a)-(10c), and then averaged
across for each width variation across the entire wafer to create a nominal
experimental model. The extraction is then repeated locally for devices that
are close in proximity to one another to extract local values for the model’s
sub-parameters (Fig. 10c).

Fig. 10d shows that using (31) this first principal component can explain
62.7% of all variance in the sub-parameter values across the wafer. The
authors determined that due to the clear connection between the $V$ and the
three model parameters that determine device behavior as $w\rightarrow\infty$,
this principal component was likely capturing width-independent sources of
variance such as thickness variations (Fig. 10e). The authors will now show
that this provides a model robust enough for capturing statistical behavior
while preserving the goal for clear physical interpretation.

###  V-B Statistical Extraction

Figure 12: a, Measured thermo-optic response of a measured MZI device. b,
Extracted value of $\partial T_{\mathrm{chip}}/\partial T_{\mathrm{heater}}$
vs waveguide width using (9).

The sum of the relative $n_{\mathrm{eff}}$ and $n_{g}$ errors is optimized
using the Nelder-Mead algorithm [58]. The drawn waveguide width and the
extracted $V$ from the local extraction performed in Fig. 10c are used as the
initial guesses for $w$ and $V$. Despite the limited sample size of collected
data, we can already see several notable preliminary statistical trends in
$\Delta w$ as a result of our model. The model of $n_{\mathrm{eff}}$ extracted
from each local optimization was found to result in a close agreement between
the measured and modeled MZI performance. The extracted values for $V$
exhibits the intended physical behavior of process parameter that varies
slowly across the wafer (Fig. 11b). Local optimization yielded a total,
average intra-die, and average local device standard deviation $\sigma_{V}$ of
0.603, 0.386, and 0.167 respectively, showing a correlation between device
proximity and their extracted $V$ values. The mean values of $V$ for die both
(i) in close proximity to each other and (ii) equidistant from the center of
the wafer tend to be similar in value, as shown in the inset of Fig. 11b.

Decoupling the process variations of $V$ from the width variations $\Delta w$
enables extraction of width-dependent systemic effects, as shown in Fig.
11c(i). Our method estimates that waveguide widths with smaller mean errors
also tend to have smaller $\sigma_{\Delta w}$ (Fig. 11c(ii)). This carries
over as an explanation for why for $w=2\mu$m, $n_{\mathrm{eff}}$ varies more
than for $w=1.2\mu$m, allowing insight on what waveguide geometry best
minimizes both $\sigma_{n_{\mathrm{eff}}}$ and $\sigma_{n_{\mathrm{g}}}$. This
sort of process insight for circuit designers is only possible due to the
group benchmarking of all device performance within a localized area.

###  V-C Thermo-optic Effect Model Validation

To validate the thermo-optic effect model developed in Section II-D, we re-
characterized the MZI transmission spectra from a single die of the chip shown
in Fig. 10a. The thermal characterization was performed by adhering a Thorlabs
TLK-H polyimide heater to the side of the chip stage. The heater was
controlled by a Thorlabs TC200 Temperature Controller to set the heater
temperature. Thermal paste was applied between the chip and the chip stage to
minimize thermal resistance between the chip and the heater. The thermal
response of one of the tested MZI is shown in Fig. 12. The fringe closest to
1550 nm is tracked at each temperature step and plotted against temperature to
extract $\partial\lambda/\partial T$. This value is then compared to our
predicted value for $\partial\lambda/\partial T$ gained by taking the
derivative of $\lambda$ in (11) with respect to temperature

Figure 13: a, Microscopic image of a die with waveguide spiral test structures
for measuring width-dependent loss. Inset: magnified image showing three of
the test structures. b, Propagation loss measurement and fit data for a 440 nm
waveguide.

$\frac{\partial\lambda}{\partial
T_{\mathrm{chip}}}=\frac{\displaystyle\frac{\Delta L}{m}\frac{\partial
n_{\mathrm{eff}}}{\partial T_{\mathrm{chip}}}\frac{\partial
T_{\mathrm{chip}}}{T_{\mathrm{heater}}}}{\displaystyle 1-\frac{\Delta
L}{m}\frac{\partial n_{\mathrm{eff,model}}}{\partial\lambda}},$ (23)

where $m$ is the order of the tracked fringe, $\Delta L$ is the path length
difference between the two arms, and $\partial T_{\mathrm{chip}}/\partial
T_{\mathrm{heater}}$ represents the heat transfer efficiency from the heater
to the chip itself. This last term is included as the authors only know the
temperature of the resistive heater rather than the chip temperature itself.
We know $\partial T_{\mathrm{chip}}/\partial T_{\mathrm{heater}}\leq 1$ as
heater cannot raise the temperature of the chip to a value higher than its
own. The extracted parameters for $\Delta w$ and $V$ are used in calculating
(23).

On average, the measured thermo-optic effect was found to be 0.91$\times$ our
model’s (9) prediction. This value was found to be independent of waveguide
geometry with the exception of 400 nm (Fig. 12b). This error for 400 nm is
assumed to be because this width is close enough to the cutoff condition for
our model to lose accuracy. In contrast, the measured thermo-optic effect was
1.21$\times$ the previously reported model’s prediction. This implies that
either the chip’s change in temperature is greater than the heater’s or the
previously reported model is incorrect. The change in temperature of the PIC
can only ever be smaller than the heater’s temperature delta, making the old
model’s prediction clearly nonphysical.

###  V-D Scattering Loss Model Validation

Figure 14: a, Plot of measured (scatter) and modeled (line) propagation loss
vs $\partial n_{\mathrm{eff}}/\partial w$. Slope of fit represents $R$ in (4),
while the intercept represents non-SWR loss. b, Plot of measured (scatter) and
modeled (line) propagation loss vs waveguide width.

To validate the scattering loss model, we measured a die with 25 spiral loss
structures consisting of 5 different waveguide widths (w) from 400 nm to 500
nm and 5 different spiral lengths ($\Delta$L) from 1 cm to 5 cm, fabricated on
a custom 300 mm full wafer through AIM Photonics (Fig. 13). Again, all spiral
structures have a nominal waveguide height of 220 nm, and grating couplers
designed for quasi-TE polarization are utilized for optical I/O. The losses of
each spiral length were recorded, and then fit to a linear equation. The slope
of the this fit was taken to be the propagation loss associated with each
waveguide width. The results of our model fit are shown in Fig. 14. The model
built in Section V-B was used to build a model of $\partial
n_{\mathrm{eff}}/\partial w$. Fitting our modeled $\partial
n_{\mathrm{eff}}/\partial w$ to the measured propagation loss yields
proportionality constant of $R=$6.206\text{\times}{10}^{-8}$$ cm and (Fig.
14a). The intercept of the loss fit is interpreted as the aggregate non-SWR
loss, with a value of 0.901 dB. Fig. 14b shows the excellent agreement between
our model and the data, predicting the similar propagation losses of both the
440 nm and 460 nm waveguides. As mentioned in Section V-B, both 400 and 420 nm
waveguide widths are likely near the cutoff condition. Since (4) is only valid
sufficiently far away from this condition, those data points are not included
in the plot.

###  V-E Verilog-A Implementation

To demonstrate its compatibility with electronic-photonic co-simulation, the
circuit model was implemented in Verilog-A within Cadence Virtuoso (Fig. 15a).
As Verilog-A does not inherently support optical signals, some compatibility
code as well as a small library of photonic device models were built based
upon on previously reported demonstrations [59, 32, 60].

Figure 15: a, Cadence Virtuoso schematic of the MZI test circuit. All circuit
models were written in Verilog-A. The optical stimulus is provided by a
continuous-wave (CW) Laser and detected with a photodetector (PD). b,
Comparison of measured and simulated performance for an MZI with $w=2~{}\mu$m
and $\Delta L=100~{}\mu$m.

##  VI Conclusion

In summary, we have demonstrated a novel compact model that can greatly expand
the accuracy of circuit-level simulation capabilities of silicon PICs. In
contrast to prior work that focused on providing metrology information that
could be use to fabrication engineers [61, 26, 62], we present this PDK model
as a tool suitable for true-to-measurement circuit simulation and
optimization. By leveraging this underlying physical behavior and locally
extracting process variations by performing group extraction, we have
demonstrated a framework for building a model of $n_{\mathrm{eff}}$ that is
entirely driven by measurement data. This model was shown to accurately
describing the phase, loss, and thermo-optic behavior of the measured
integrated waveguides over 4$\times$ the optical bandwidth and over 80$\times$
the range of waveguides widths reported in prior work. We envision that the
advancement over prior demonstrations this work represents can support the
development of waveguide-based PDK components and enable the robust
optimization of next generation PICs.

##  VII Acknowledgements

This work was supported in part by the U.S. Advanced Research Projects
Agency–Energy under ENLITENED Grant DE-AR000843 and in part by the U.S.
Defense Advanced Research Projects Agency under PIPES Grant HR00111920014. The
authors thank AIM Photonics for chip fabrication.

###  -A Derivation of Thermo-Optic Model

Starting from (8) from [46], the effect of a thermal perturbation on the
effective index is investigated. Carrying out this perturbation and following
the chain rule yields:

$2\beta\frac{\partial\beta}{\partial
T}=2\Gamma_{\text{core}}\frac{\omega^{2}}{c^{2}}n_{\text{core}}\frac{\partial
n_{\text{core}}}{\partial
T}+2\Gamma_{\text{clad}}\frac{\omega^{2}}{c^{2}}n_{\text{clad}}\frac{\partial
n_{\text{clad}}}{\partial T}.$ (24)

Noting that $\beta=\omega n_{\text{eff}}/c$ and inserting above, the
relationship simplifies to (25). Combining with (1) yields:

$\displaystyle n_{\text{eff}}=n_{\text{eff, T${}_{0}$}}+\frac{\partial
n_{\text{eff}}}{\partial T}(T-T_{0})$ (25a) $\displaystyle\frac{\partial
n_{\text{eff}}}{\partial
T}=\Gamma_{\text{core}}\frac{n_{\text{core}}}{n_{\text{eff}}}\frac{\partial
n_{\text{core}}}{\partial
T}+\Gamma_{\text{clad}}\frac{n_{\text{clad}}}{n_{\text{eff}}}\frac{\partial
n_{\text{clad}}}{\partial T}.$ (25b)

It is noted that $n_{\text{eff}}$ appears on both sides of the equation.
Multiplying both sides by effective index yields a quadratic equation whose
solution is:

$\displaystyle n_{\text{eff}}=\frac{n_{\text{eff,
T${}_{0}$}}}{2}+\frac{1}{2}\sqrt{n_{\text{eff,
T${}_{0}$}}^{2}+4n^{{}^{\prime}}(T-T_{0})}$ (26a) $\displaystyle
n^{{}^{\prime}}=\Gamma_{\text{core}}n_{\text{core}}\frac{\partial
n_{\text{core}}}{\partial T}+\Gamma_{\text{clad}}n_{\text{clad}}\frac{\partial
n_{\text{clad}}}{\partial T}.$ (26b)

The expression can be simplified by noting that $n_{\text{eff,
T${}_{0}$}}^{2}\gg 4n^{{}^{\prime}}$ for typical values for the thermo-optic
coefficients. Understanding this, it is clear that the behavior of the square
root term is approximately linear. The 1st order Taylor expansion of the
square root term is:

$n_{\text{eff,
T${}_{0}$}}+\frac{1}{2}\frac{4n^{{}^{\prime}}}{\sqrt{n_{\text{eff,
T${}_{0}$}}^{2}+4n^{{}^{\prime}}(T-T_{0})}}(T-T_{0}).$ (27)

Noting again that $n_{\text{eff, T${}_{0}$}}^{2}\gg 4n^{{}^{\prime}}$, (27)
simplifies to:

$n_{\text{eff, T${}_{0}$}}+\frac{2n^{{}^{\prime}}}{n_{\text{eff,
T${}_{0}$}}}(T-T_{0}).$ (28)

Replacing the square root term in (26) with this expression and simplifying
will then yield (9).

###  -B Principal Component Analysis

To start, we form a matrix X our of our list of local sub-parameter
extractions, where each column represents a model parameter and each row is an
observation of said parameter:

$\textbf{X}=\left[{\begin{array}[]{cccc}\frac{\partial^{0}n_{\mathrm{eff}}}{\partial\lambda^{0}}_{0,1}&\frac{\partial^{0}n_{\mathrm{eff}}}{\partial\lambda^{0}}_{1,1}&\cdots&\frac{\partial^{2}n_{\mathrm{eff}}}{\partial\lambda^{2}}_{4,1}\\\
\frac{\partial^{0}n_{\mathrm{eff}}}{\partial\lambda^{0}}_{0,2}&\frac{\partial^{0}n_{\mathrm{eff}}}{\partial\lambda^{0}}_{1,2}&\cdots&\frac{\partial^{2}n_{\mathrm{eff}}}{\partial\lambda^{2}}_{4,2}\\\
\vdots&\vdots&\ddots&\vdots\\\
\frac{\partial^{0}n_{\mathrm{eff}}}{\partial\lambda^{0}}_{0,n}&\frac{\partial^{0}n_{\mathrm{eff}}}{\partial\lambda^{0}}_{1,n}&\cdots&\frac{\partial^{2}n_{\mathrm{eff}}}{\partial\lambda^{2}}_{4,n}\\\
\end{array}}\right].$ (29)

A covariance matrix S is then created from X and find its eigenvectors:

$\textbf{S}=\left[{\begin{array}[]{c}\vec{v}_{0}\\\ \vec{v}_{1}\\\ \vdots\\\
\vec{v}_{n}\\\
\end{array}}\right]\left[{\begin{array}[]{cccc}\lambda_{0}&0&\cdots&0\\\
0&\lambda_{1}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\lambda_{n}\\\
\end{array}}\right]\left[{\begin{array}[]{c}\vec{v}_{0}\\\ \vec{v}_{1}\\\
\vdots\\\ \vec{v}_{n}\\\ \end{array}}\right]^{-1},$ (30)

where $\left[v_{0},v_{1},\cdots,v_{n}\right]$ lists the eigenvectors and
$\left[\lambda_{0},\lambda_{1},\cdots,\lambda_{n}\right]$ are their associated
eigenvalues. The eigenvectors of the correlation matrix represent the
directions of the axes where there is the most variance (i.e. the most
information). Each eigenvalue $\lambda_{i}$ is proportional to how much
variance is captured by its associated principal component $v_{i}$. Picking
the eigenvectors with the largest eigenvalues allows us to reduce data
dimensionality at the expense of some accuracy. The percentage of variability
explained by a principal component is calculated as

$\frac{\sum_{i=0}^{M}\lambda_{i}}{\sum_{i=0}^{N}\lambda_{i}},$ (31)

where $\lambda_{i}$ is the eigenvalue for each eigenvector, $M$ is the number
of principal components the designer has chosen to include, and $N$ is the
maximum number of principal components.

## References

  * [1] A. Rizzo, S. Daudlin, A. Novick, A. James, V. Gopal, V. Murthy, Q. Cheng, B. Y. Kim, X. Ji, Y. Okawachi, M. van Niekerk, V. Deenadayalan, G. Leake, M. Fanto, S. Preble, M. Lipson, A. Gaeta, and K. Bergman, “Petabit-scale silicon photonic interconnects with integrated kerr frequency combs,” _IEEE Journal of Selected Topics in Quantum Electronics_ , vol. 29, no. 1: Nonlinear Integrated Photonics, pp. 1–20, 2023.
  * [2] Q. Cheng, M. Bahadori, M. Glick, S. Rumley, and K. Bergman, “Recent advances in optical technologies for data centers: a review,” _Optica_ , vol. 5, no. 11, pp. 1354–1370, Nov 2018.
  * [3] M. Wade, E. Anderson, S. Ardalan, P. Bhargava, S. Buchbinder, M. L. Davenport, J. Fini, H. Lu, C. Li, R. Meade _et al._ , “Teraphy: a chiplet technology for low-power, high-bandwidth in-package optical i/o,” _IEEE Micro_ , vol. 40, no. 2, pp. 63–71, 2020.
  * [4] G. Moody, V. J. Sorger, D. J. Blumenthal, P. W. Juodawlkis, W. Loh, C. Sorace-Agaskar, A. E. Jones, K. C. Balram, J. C. Matthews, A. Laing _et al._ , “2022 roadmap on integrated quantum photonics,” _Journal of Physics: Photonics_ , vol. 4, no. 1, p. 012501, 2022.
  * [5] G. R. Steinbrecher, J. P. Olson, D. Englund, and J. Carolan, “Quantum optical neural networks,” _npj Quantum Information_ , vol. 5, no. 1, pp. 1–9, 2019\.
  * [6] S. Takeda and A. Furusawa, “Toward large-scale fault-tolerant universal photonic quantum computing,” _APL Photonics_ , vol. 4, no. 6, p. 060902, 2019\.
  * [7] N. C. Harris, D. Bunandar, M. Pant, G. R. Steinbrecher, J. Mower, M. Prabhu, T. Baehr-Jones, M. Hochberg, and D. Englund, “Large-scale quantum photonic circuits in silicon,” _Nanophotonics_ , vol. 5, no. 3, pp. 456–468, 2016\.
  * [8] J. E. Bourassa, R. N. Alexander, M. Vasmer, A. Patil, I. Tzitrin, T. Matsuura, D. Su, B. Q. Baragiola, S. Guha, G. Dauphinais _et al._ , “Blueprint for a scalable photonic fault-tolerant quantum computer,” _Quantum_ , vol. 5, p. 392, 2021.
  * [9] D. Marpaung, J. Yao, and J. Capmany, “Integrated microwave photonics,” _Nature photonics_ , vol. 13, no. 2, pp. 80–90, 2019.
  * [10] Y. Yang, Y. Yamagami, X. Yu, P. Pitchappa, J. Webber, B. Zhang, M. Fujita, T. Nagatsuma, and R. Singh, “Terahertz topological photonics for on-chip communication,” _Nature Photonics_ , vol. 14, no. 7, pp. 446–451, 2020.
  * [11] B. Zong, C. Fan, X. Wang, X. Duan, B. Wang, and J. Wang, “6g technologies: Key drivers, core requirements, system architectures, and enabling technologies,” _IEEE Vehicular Technology Magazine_ , vol. 14, no. 3, pp. 18–27, 2019.
  * [12] W. Shi, Y. Tian, and A. Gervais, “Scaling capacity of fiber-optic transmission systems via silicon photonics,” _Nanophotonics_ , vol. 9, no. 16, pp. 4629–4663, 2020.
  * [13] Z. Zhou, R. Chen, X. Li, and T. Li, “Development trends in silicon photonics for data centers,” _Optical Fiber Technology_ , vol. 44, pp. 13–23, 2018\.
  * [14] R. Sabella, “Silicon photonics for 5g and future networks,” _IEEE Journal of Selected Topics in Quantum Electronics_ , vol. 26, no. 2, pp. 1–11, 2019.
  * [15] R. Gardner, J. Bieker, and S. Elwell, “Solving tough semiconductor manufacturing problems using data mining,” in _2000 IEEE/SEMI Advanced Semiconductor Manufacturing Conference and Workshop. ASMC 2000 (Cat. No. 00CH37072)_. IEEE, 2000, pp. 46–55.
  * [16] N. Kumar, K. Kennedy, K. Gildersleeve, R. Abelson, C. Mastrangelo, and D. Montgomery, “A review of yield modelling techniques for semiconductor manufacturing,” _International Journal of Production Research_ , vol. 44, no. 23, pp. 5019–5036, 2006.
  * [17] W. Bogaerts and L. Chrostowski, “Silicon photonics circuit design: methods, tools and challenges,” _Laser & Photonics Reviews_, vol. 12, no. 4, p. 1700237, 2018.
  * [18] A. Rizzo, U. Dave, A. Novick, A. Freitas, S. P. Roberts, A. James, M. Lipson, and K. Bergman, “Fabrication-robust silicon photonic devices in standard sub-micron silicon-on-insulator processes,” _Opt. Lett._ , vol. 48, no. 2, pp. 215–218, Jan 2023. [Online]. Available: https://opg.optica.org/ol/abstract.cfm?URI=ol-48-2-215
  * [19] Y. Wang, P. Sun, J. Hulme, M. A. Seyedi, M. Fiorentino, R. G. Beausoleil, and K.-T. Cheng, “Energy efficiency and yield optimization for optical interconnects via transceiver grouping,” _J. Lightwave Technol._ , vol. 39, no. 6, pp. 1567–1578, Mar 2021. [Online]. Available: http://opg.optica.org/jlt/abstract.cfm?URI=jlt-39-6-1567
  * [20] A. V. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj _et al._ , “Exploiting cmos manufacturing to reduce tuning requirements for resonant optical devices,” _IEEE Photonics Journal_ , vol. 3, no. 3, pp. 567–579, 2011.
  * [21] N. Margalit, C. Xiang, S. M. Bowers, A. Bjorlin, R. Blum, and J. E. Bowers, “Perspective on the future of silicon photonics and electronics,” _Applied Physics Letters_ , vol. 118, no. 22, p. 220501, 2021.
  * [22] Y. Wang, S. Wang, A. Novick, A. James, R. Parsons, A. Rizzo, and K. Bergman, “Dispersion-engineered and fabrication-robust soi waveguides for ultra-broadband dwdm,” in _2023 Optical Fiber Communications Conference and Exhibition (OFC) [To appear]_ , 2023, pp. 1–3.
  * [23] R. Woltjer, L. Tiemeijer, and D. Klaassen, “An industrial view on compact modeling,” in _2006 European Solid-State Device Research Conference_. IEEE, 2006, pp. 41–48.
  * [24] N. Moezi, “Statistical compact model strategies for nano cmos transistors subject of atomic scale variability,” Ph.D. dissertation, University of Glasgow, 2012.
  * [25] Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,” _Photonics Research_ , vol. 6, no. 11, pp. 1008–1020, 2018.
  * [26] Z. Lu, J. Jhoja, J. Klein, X. Wang, A. Liu, J. Flueckiger, J. Pond, and L. Chrostowski, “Performance prediction for silicon photonics integrated circuits with layout-dependent correlated manufacturing variability,” _Optics express_ , vol. 25, no. 9, pp. 9712–9733, 2017.
  * [27] W. A. Zortman, D. C. Trotter, and M. R. Watts, “Silicon photonics manufacturing,” _Optics express_ , vol. 18, no. 23, pp. 23 598–23 607, 2010.
  * [28] Z. Zhang, S. I. El-Henawy, C. Ríos, and D. S. Boning, “Inference of process variations in silicon photonics from characterization measurements,” in _CLEO: Science and Innovations_. Optica Publishing Group, 2022, pp. SF3O–5.
  * [29] T. H. Stievater, N. F. Tyndall, M. W. Pruessner, D. A. Kozak, and W. S. Rabinovich, “Optical and geometric parameter extraction for photonic integrated circuits,” _Optics Express_ , vol. 30, no. 9, pp. 14 453–14 460, 2022.
  * [30] M. J. Shawon and V. Saxena, “Rapid simulation of photonic integrated circuits using verilog-a compact models,” _IEEE Transactions on Circuits and Systems I: Regular Papers_ , vol. 67, no. 10, pp. 3331–3341, 2020.
  * [31] Z. Zhang, R. Wu, Y. Wang, C. Zhang, E. J. Stanton, C. L. Schow, K.-T. Cheng, and J. E. Bowers, “Compact modeling for silicon photonic heterogeneously integrated circuits,” _Journal of Lightwave Technology_ , vol. 35, no. 14, pp. 2973–2980, 2017.
  * [32] C. Sorace-Agaskar, J. Leu, M. R. Watts, and V. Stojanovic, “Electro-optical co-simulation for integrated cmos photonic circuits with veriloga,” _Optics express_ , vol. 23, no. 21, pp. 27 180–27 203, 2015.
  * [33] B. E. Saleh and M. C. Teich, _Fundamentals of photonics_. John Wiley & Sons, 2019.
  * [34] A. James, Y. Wang, A. Rizzo, and K. Bergman, “Flexible, process-aware compact model of effective index in silicon waveguides for commercial foundries,” in _2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD)_. IEEE, 2022, pp. 173–174.
  * [35] A. E. James, A. Wang, S. Wang, and K. Bergman, “Evaluating regression-based techniques for modelling fabrication variations in silicon photonic waveguides,” in _Applications of Machine Learning 2021_ , vol. 11843\. International Society for Optics and Photonics, 2021, p. 1184305.
  * [36] M. J. Heck, J. F. Bauters, M. L. Davenport, D. T. Spencer, and J. E. Bowers, “Ultra-low loss waveguide platform and its integration with silicon photonics,” _Laser & Photonics Reviews_, vol. 8, no. 5, pp. 667–686, 2014\.
  * [37] D. Melati, A. Melloni, and F. Morichetti, “Real photonic waveguides: guiding light through imperfections,” _Advances in Optics and Photonics_ , vol. 6, no. 2, pp. 156–224, 2014.
  * [38] J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” _IEE Proceedings J-Optoelectronics_ , vol. 137, no. 4, pp. 282–288, 1990.
  * [39] F. Payne and J. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” _Optical and Quantum Electronics_ , vol. 26, no. 10, pp. 977–986, 1994.
  * [40] E. Jaberansary, T. M. B. Masaud, M. Milosevic, M. Nedeljkovic, G. Z. Mashanovich, and H. M. Chong, “Scattering loss estimation using 2-d fourier analysis and modeling of sidewall roughness on optical waveguides,” _IEEE Photonics Journal_ , vol. 5, no. 3, pp. 6 601 010–6 601 010, 2013\.
  * [41] S. Kawakami, “Relation between dispersion and power-flow distribution in a dielectric waveguide,” _JOSA_ , vol. 65, no. 1, pp. 41–45, 1975.
  * [42] N. Y. Winnie, J. Michel, L. C. Kimerling, and L. Eldada, “Polymer-cladded athermal high-index-contrast waveguides,” in _Optoelectronic Integrated Circuits X_ , vol. 6897. International Society for Optics and Photonics, 2008, p. 68970S.
  * [43] P. Jean, A. Douaud, T. Thibault, S. LaRochelle, Y. Messaddeq, and W. Shi, “Sulfur-rich chalcogenide claddings for athermal and high-q silicon microring resonators,” _Optical Materials Express_ , vol. 11, no. 3, pp. 913–925, 2021.
  * [44] H. Zhang, J. Chen, J. Jin, J. Lin, L. Zhao, Z. Bi, A. Huang, and Z. Xiao, “On-chip modulation for rotating sensing of gyroscope based on ring resonator coupled with mach-zehnder interferometer,” _Scientific reports_ , vol. 6, no. 1, pp. 1–9, 2016.
  * [45] Z. Zhou, B. Yin, Q. Deng, X. Li, and J. Cui, “Lowering the energy consumption in silicon photonic devices and systems,” _Photonics Research_ , vol. 3, no. 5, pp. B28–B46, 2015.
  * [46] A. Yariv and P. Yeh, _Photonics: optical electronics in modern communications_. Oxford university press, 2007.
  * [47] E. D. Palik, _Handbook of optical constants of solids_. Academic press, 1998, vol. 3.
  * [48] B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” in _SPIE Proceedings_ , E. Atad-Ettedgui, J. Antebi, and D. Lemke, Eds. SPIE, jun 2006. [Online]. Available: https://doi.org/10.1117%2F12.672850
  * [49] A. W. Elshaari, I. E. Zadeh, K. D. Jöns, and V. Zwiller, “Thermo-optic characterization of silicon nitride resonators for cryogenic photonic circuits,” _IEEE Photonics Journal_ , vol. 8, no. 3, pp. 1–9, 2016.
  * [50] D. Coenen, H. Oprins, Y. Ban, F. Ferraro, M. Pantouvaki, J. Van Campenhout, and I. De Wolf, “Thermal modelling of silicon photonic ring modulator with substrate undercut,” _Journal of Lightwave Technology_ , 2022.
  * [51] M. Jacques, A. Samani, E. El-Fiky, D. Patel, Z. Xing, and D. V. Plant, “Optimization of thermo-optic phase-shifter design and mitigation of thermal crosstalk on the soi platform,” _Optics express_ , vol. 27, no. 8, pp. 10 456–10 471, 2019.
  * [52] M. de Cea, A. H. Atabaki, and R. J. Ram, “Power handling of silicon microring modulators,” _Opt. Express_ , vol. 27, no. 17, pp. 24 274–24 285, Aug 2019\. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-27-17-24274
  * [53] C. Oton, C. Manganelli, F. Bontempi, M. Fournier, D. Fowler, and C. Kopp, “Silicon photonic waveguide metrology using mach-zehnder interferometers,” _Optics express_ , vol. 24, no. 6, pp. 6265–6270, 2016.
  * [54] Q. Deng, L. Liu, X. Li, J. Michel, and Z. Zhou, “Linear-regression-based approach for loss extraction from ring resonators,” _Opt. Lett._ , vol. 41, no. 20, pp. 4747–4750, Oct 2016. [Online]. Available: http://opg.optica.org/ol/abstract.cfm?URI=ol-41-20-4747
  * [55] W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” _Laser & Photonics Reviews_, vol. 6, no. 1, pp. 47–73, 2012.
  * [56] W. N. Ye, D.-X. Xu, S. Janz, P. Cheben, M.-J. Picard, B. Lamontagne, and N. G. Tarr, “Birefringence control using stress engineering in silicon-on-insulator (soi) waveguides,” _Journal of Lightwave Technology_ , vol. 23, no. 3, pp. 1308–1318, 2005.
  * [57] H. Abdi and L. J. Williams, “Principal component analysis,” _Wiley interdisciplinary reviews: computational statistics_ , vol. 2, no. 4, pp. 433–459, 2010.
  * [58] S. Singer and J. Nelder, “Nelder-mead algorithm,” _Scholarpedia_ , vol. 4, no. 7, p. 2928, 2009.
  * [59] E. Kononov, “Modeling photonic links in verilog-a,” Ph.D. dissertation, Massachusetts Institute of Technology, 2013.
  * [60] J. C. Leu, “Integrated silicon photonic circuit simulation,” Ph.D. dissertation, Massachusetts Institute of Technology, 2018.
  * [61] Y. Xing, M. Wang, A. Ruocco, J. Geessels, U. Khan, and W. Bogaerts, “Compact silicon photonics circuit to extract multiple parameters for process control monitoring,” _OSA Continuum_ , vol. 3, no. 2, pp. 379–390, Feb 2020. [Online]. Available: https://opg.optica.org/osac/abstract.cfm?URI=osac-3-2-379
  * [62] D. S. Boning, S. I. El-Henawy, and Z. Zhang, “Variation-aware methods and models for silicon photonic design-for-manufacturability,” _Journal of Lightwave Technology_ , vol. 40, no. 6, pp. 1776–1783, 2022.

Aneek James received his B.S. in Electrical and Electronics Engineering from
the University of Georgia, Athens, GA in 2017 and his M.S., and M.Phil., in
Electrical Engineering from Columbia University, New York, NY in 2019 and
2021, respectively. He is working as a Ph.D. candidate in Electrical
Engineering in the Lightwave Research Laboratory under Professor Keren
Bergman. His research interests include modeling fabrication variations in
silicon photonic devices, as well as the testing and automated control of
silicon photonic systems for high-throughput optical interconnects.
---
Anthony Rizzo received his B.S. in Physics from Haverford College, Haverford,
PA in 2017 and his M.S., M.Phil., and Ph.D., all in Electrical Engineering,
from Columbia University, New York, NY in 2019, 2021, and 2022, respectively.
He completed his doctoral research in the Lightwave Research Laboratory at
Columbia University under Professor Keren Bergman, where he led the first
demonstration of data transmission using an integrated Kerr frequency comb
source and silicon photonic transmitter. He is currently a Research Scientist
at the Air Force Research Laboratory (AFRL) Information Directorate in Rome,
NY, with a focus in large-scale silicon photonic systems for quantum
information processing and artificial intelligence.
---
Yuyang Wang received the B.Eng. degree in electronic engineering from Tsinghua
University, Beijing, China in 2015, and the M.S. and PhD degrees in computer
engineering from the University of California, Santa Barbara (UCSB), CA, USA,
in 2018 and 2021 respectively. He is currently a post-doctoral researcher in
the Lightwave Research Laboratory under Professor Keren Bergman. He was a
Design Engineering Intern at Cadence Design Systems in 2018 and a Visiting
Intern at the Hong Kong University of Science and Technology in 2019. His
research interests include variation-aware modeling, design, and optimization
of silicon photonic interconnects and systems.
---
Asher Novick received his M.Eng. and B.S. degrees in Electrical and Computer
Engineering from Cornell University, Ithaca, NY, USA, in 2016 and
2015,respectively. Between 2016 and 2019, he was at Panduit’s Fiber Research
Lab, where he researched and developed new patentable technologies for optical
fiber-based communication in data center and enterprise applications. He is
currently working toward his Ph.D. degree in Electrical Engineering in the
Lightwave Research Laboratory at Columbia University in the City of New York.
His current research interest is in the modeling, design, and testing of
silicon photonic systems and devices for scalable and efficient link
architectures.
---
Songli Wang received his B.Eng. in Optoelectronic Information Science and
Engineering from Harbin Institute of Technology, Harbin, China, in 2019 and
his M.S. in Electrical Engineering from Columbia University, New York, NY in
2020. He is currently working towards the Ph.D. degree in Electrical
Engineering in the Lightwave Research Laboratory at Columbia University. His
current research interests include modeling, design and testing of silicon
photonic devices and systems.
---
Robert Parsons received the B.S. degree in biomedical engineering from George
Washington University, Washington, D.C., USA, and the M.S. degree in
electrical engineering from Columbia University, New York, NY, USA, in 2020
and 2022, respectively. He is currently working toward the Ph.D. degree in
electrical engineering with the Lightwave Research Laboratory, Columbia
University under Professor Keren Bergman. His research interests include the
modeling, testing, and co-optimization of link architectures and constituent
silicon photonic devices for high-bandwidth, energy-efficient optical
interconnects.
---
Kaylx Jang received his B.S. in Electrical Engineering from the University of
California, Irvine, CA in 2020 and his M.S. in Electrical Engineering from
Columbia University in 2022. In 2019 and 2020, he interned in the testing
department at Ayar Labs and in 2021 did a co-op in the Silicon Photonics
Design team at Nokia (former Elenion). He is working as a Ph.D. student in
Electrical Engineering in the Lightwave Research Lab under Professor Keren
Bergman. His research interests include modeling, design, and testing of high-
performance silicon photonic devices for energy efficient and scalable link
architectures.
---
Maarten Hattink is a graduate student with Columbia University, New York, NY,
USA. He received the B.S. and M.S. degrees from the Eindhoven University of
Technology, The Netherlands, in 2015 and 2017, respectively. While pursuing
these degrees, he worked at Prodrive Technologies B.V. as a Software and FPGA
Engineer. He is currently working toward the Ph.D. degree and his research
interest lies in photonic device integration and thermal control.
---
Keren Bergman (S’87–M’93–SM’07–F’09) received the B.S. degree from Bucknell
University, Lewisburg, PA, in 1988, and the M.S. and Ph.D. degrees from the
Massachusetts Institute of Technology, Cambridge, in 1991 and 1994,
respectively, all in electrical engineering. Dr. Bergman is currently a
Charles Batchelor Professor at Columbia University, New York, NY, where she
also directs the Lightwave Research Laboratory. She leads multiple research
programs on optical interconnection networks for advanced computing systems,
data centers, optical packet switched routers, and chip multiprocessor
nanophotonic networks-on-chip. Dr. Bergman is a Fellow of the IEEE and Optica.
---