# Recent Advancements in Mode Division Multiplexing for Communication and
Computation in Silicon Photonics

Kaveh Rahbardar Mojaver, Seyed Mohammad Reza Safaee, Sunami Sajjanam Morrison,
and Odile Liboiron-Ladouceur All the authors are with the Department of
Electrical and Computer Engineering, McGill University, Quebec, Canada, H3A
0E9, e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>received
1 March 2024.

###### Abstract

Mode Division Multiplexing (MDM) is a technique used over the past decade in
Silicon Photonics (SiPh) to incorporate more data into communication links by
employing higher-order transverse electric or transverse magnetic modes. MDM
was primarily used in optical communication; however, in recent years, there
have been several applications of MDM in optical computing, including both
classical and quantum computing. Although MDM has shown great promise for
increasing the throughput of optical communication and the accuracy and
fidelity of optical computation, there are a few challenges towards expanding
its applications. One major challenge is the lack of process design kits
(PDKs) and building block libraries compatible with standard SiPh foundries.
Here, we present a comprehensive library of MDM components developed using
classical and inverse design, compatible with standard 220 nm SiPh foundries.
The library includes thermo-optic phase shifters, mode multiplexers and
demultiplexers, mode converters, mode exchangers, and multi-mode interference
couplers. We also discuss our recent achievements in MDM for datacom,
classical and quantum optical computing, including a mode-selective switch for
mode-reconfigurable optical add-drop multiplexer (ROADM), multimode multiply-
accumulate operation, and multimode photonic quantum processors.

###### Index Terms:

Integrated optics, optical computing, programmable optical processors, silicon
photonics.

##  I Introduction

Mode Division Multiplexing (MDM) is a technique that incorporates multiple
spatial optical modes inside an optical waveguide or fiber to increase the
capacity of communication channels. Introduced over a decade ago in Silicon
Photonics (SiPh), MDM was primarily aimed at enhancing the throughput of on-
chip communications [2, 3] but has gradually expanded to chip-to-chip
communication through fibers as well [4, 5, 6, 7]. Transverse electric (TE)
and transverse magnetic (TM) modes are solutions to the Maxwell’s equations
for the energy distribution of electromagnetic waves propagating inside a
waveguide or fiber. These modes are orthogonal; hence, they do not interfere
with each other and can be used as separate communication channels.

The key advantage of MDM over wavelength division multiplexing (WDM) lies in
its ability to use a single laser for all channels, significantly saving
power, as lasers constitute the most power-hungry component in communication
links [8]. Additionally, in MDM, mode conversion can be easily achieved
through energy redistribution inside a waveguide [9, 10], while WDM links
require nonlinear effects to alter the wavelength of photons. Importantly, MDM
should not be viewed as a competitor to WDM but rather as a complement,
offering increased degrees of freedom for scaling optical systems [11, 12].

Despite more than a decade since its inception, several challenges hinder the
widespread adoption of MDM. The primary and most significant challenge is
modal crosstalk, where any factor affecting energy distribution inside a
waveguide, such as sidewall roughness or fabrication variations, can lead to
modal crosstalk [13]. Efficient coupling in and out of the chip also remains
challenging in MDM [14]. Additionally, since higher-order spatial modes are
less confined in the waveguide, they require larger radius bends, resulting in
a larger footprint [15]. Modal crosstalk is exacerbated by waveguide
crossings, making complex designs with multiple crossings challenging in MDM
[16]. Another challenge is that the process design kits (PDK) of most standard
SiPh foundries include only single-mode components, necessitating the
development of the required building blocks for MDM system design—an
especially daunting task for research-level and small-scale industry
designers. Even most commercial system-level optical simulators do not support
MDM, as standard PDKs lack MDM components.

Despite these challenges, recent advancements in MDM have showcased its
potential in on-chip and chip-to-chip communication. The coupling of three TE
and one TM modes from a SiPh multimode waveguide to a few-mode fiber achieved
an average crosstalk of -7 dBm, enabling data transmission at a rate of 1.92
Tb/s in the L-band via MDM [6]. Furthermore, a successful demonstration of
1.12 Tb/s data transmission via a multimode rectangular-shaped fiber using MDM
and WDM was achieved with inverse design fiber-to-chip couplers [7]. Beyond
multiplexing data, it has been shown that higher-order modes can be used to
carry the clock or local oscillators in source synchronous communication
systems [17]. Based on a similar approach, a 7.2-Tb/s self-homodyne coherent
transmission over a weakly coupled few-mode fiber using 10 spatial modes has
recently been demonstrated, where nine modes were used to carry the data, and
one mode is used to carry the local oscillator [18].

Beyond optical communication, MDM has garnered interest in optical
computation. Higher-order modes prove useful in enhancing programming speed
and accuracy in programmable optical processors for energy-efficient vector
matrix multiplication widely used in machine learning (ML) and artificial
intelligence (AI) [19]. The orthogonal modes allow for non-coherent summation
of optical energy, finding applications in multimode Multiply-Accumulate (MAC)
operations to increase the number of bits and overcome coherent summation
challenges. Moreover, different spatial modes can be employed to encode
quantum states of light in quantum photonics, facilitating the scalability of
optical quantum systems [20].

In this paper, we start with a brief review of the fundamentals of MDM in
SiPh. Next, we provide a library of MDM building blocks compatible with
standard 220 nm SiPh that can be used by photonic designers to develop various
MDM designs. Then we discuss recent advancements in MDM in datacom, and
optical classical and quantum computing.

##  II MDM: Fundamentals

Figure 1 (a) displays the computed effective refractive index,
$n_{\text{eff}}$, of the first four TE and TM modes in a 220 nm thick strip
waveguide at a wavelength of 1550 nm, plotted against the waveguide width. For
all modes, $n_{\text{eff}}$ increases with the waveguide width. Modes with
$n_{\text{eff}}$ surpassing the refractive index of the SiO2 cladding remain
confined and guided within the waveguide. For instance, with a 430 nm
waveguide width, only the fundamental TE mode is guided, classifying it as a
single mode. However, the number of TE modes increases with the waveguide
width. For a 1.45 µm waveguide, the first three TE modes (TE0, TE1, and TE2)
are guided. Figure 1 (b) displays the cross section of a 1.45 µm by 220 nm
SiPh waveguide and the mode profile of the first three TE modes inside this
waveguide. Figure 1 (c) shows a cross-section of a typical 220 nm thick SiPh
process, including single-mode and multimode waveguides, as well as thermo-
optic phase shifters (TOPS).

The TE and TM modes, serving as orthogonal solutions to the Maxwell equations
within the waveguide, manifest as separate channels suitable for multiplexing
different signals in an MDM system. Optical modes depict the distribution of
electromagnetic wave energy within the waveguide, enabling a relatively
straightforward conversion between them, a double-edged sword. On one hand,
the advantage lies in the ease of converting between two MDM channels by
redistributing the energy within the waveguide, without relying on
nonlinearity effects as required for WDM technology. Section III will showcase
mode converters designed using an inverse design approach. On the other hand,
however, a significant drawback arises in the presence of waveguide sidewall
roughness, where energy redistribution can lead to unwanted coupling between
modes, resulting in modal crosstalk. This issue stands as a key challenge to
be addressed in MDM systems.

Figure 1: (a) Calculated effective refractive index of a 220 nm thick strip
SiPh waveguide versus its width for the first four TE and TM modes. (b)
Profiles (energy density) of the first three TE modes for a 1.45 µm width
case. (c) Cross section view of 220 nm SiPh process.

##  III MDM building blocks

The PDK from most standard SiPh foundries, such as Interuniversity
Microelectronics Centre (IMEC), Advanced Micro Foundry (AMF), and Global
Foundries (GF), typically includes only single-mode components. Consequently,
designing multimode photonic systems poses a challenge, as designers often
need to create all components, including fundamental ones like splitters and
combiners. To address this issue, we present an open-access SiPh library
dedicated to multimode components [21, 22]. This comprehensive library
comprises essential building blocks such as Multimode Interferometers (MMIs),
mode (de)multiplexers, mode-sensitive TOPS, and mode-insensitive TOPS,
catering to both classical and quantum optical computing in multimode systems.
Notably, this library seamlessly integrates with the standard 220 nm thick
SiPh technology platform available through commercial microfabrication
foundries.

###  III-A MDM Multimode Interferometers

MMIs are widely employed in integrated optics for splitting and combining
optical signals. In an MDM system, an MMI capable of simultaneously
splitting/combining all the desired modes is necessary.

Figure 2:  Library of MDM building blocks available through the online library
[21]: (a) MMI that splits/combines the first three TE modes (TE0, TE1, TE2)
using conventional design [23, 24]. (b) SWG-based MMI with a compact footprint
for the first two TE modes [25]. (c) Mode-insensitive thermo-optic phase
shifter (MI-TOPS) that applies the same phase shift to the first three TE
modes [23]. (d) SWG-based mode-selective thermo-optic phase shifter (MS-TOPS)
that applies a different phase shift to TE0 and TE1. The mode profile shows
TE0 propagating in the core of the phase shifter made by Silicon, while TE1
has two side lobes propagating in the SWG region with different temperature
coefficients [26]. (e) Compact mode converters enabled using inverse design
for converting between various TE modes [9]. (f) Inverse-designed mode
exchanger for swapping between TE0 and TE1 [10]. (g) Inverse-designed mode
(de)Mux for the first three TE modes. The design is optimized to compensate
for fabrication variations using PreFab [27, 28].

Figure 2 (a) illustrates a mode-insensitive MMI designed for the first three
TE modes (TE0, TE1, TE2). This MMI performs a 50:50 split/combine of all these
modes simultaneously. To achieve the desired two-fold image of the three input
modes, the multimode MMI width is selected to be 10 µm to support a sufficient
number of modes in the multimode region. The beat length ($L_{\pi}$) is
initially calculated using:

$\displaystyle
L_{\pi}=\frac{\pi}{{\beta_{0}-\beta_{1}}}=\frac{{4n_{\text{eff}}W_{m}^{2}}}{{3\lambda_{0}}}$
(1)

where, $\beta_{0}$ and $\beta_{1}$ represent the propagation constants of the
first two TE modes within the MMI. The effective index, denoted as
$n_{\text{eff}}$, the width of the MMI, labeled as $W_{m}$, and the desired
wavelength is $\lambda_{0}$. To optimize the MMI dimensions, we leverage
commercial numerical simulation-based software like Ansys/Lumerical. Following
this optimization process, the determined length of the MMI is 362 µm,
tailored for efficient operation at a wavelength of 1550 nm. Details of the
design are presented in [23, 24]. While highly efficient, the conventional MMI
exhibits a substantial footprint. Figure 2 (b) illustrates the layout of a
broadband compact dual-mode MMI utilizing subwavelength grating (SWG)
structures. The SWG-based MMI is nearly ten times more compact than its
conventional counterpart, showcasing a flat response with low loss and
crosstalk over a bandwidth of 100 nm. Excess loss measurements reveal values
below 0.1 dB and 0.65 dB for TE0 and TE1, respectively, with modal crosstalk
measuring less than -17 dB for both input modes. Comprehensive design details
and experimental validation for this SWG-based MMI can be found in [25].

In the microfabrication process of SWG structures, the SiO2 cladding
deposition may not completely fill the gaps between the fine features of the
SWG, resulting in residual air gaps. To further enhance the MMI’s
specifications, particularly in addressing modal crosstalk, and to mitigate
the impact of fabrication variations, an investigation into the influence of
buried air gaps in the cladding of SiPh structures is conducted and the design
is optimized for the best performance in the presence of air gaps [29].

###  III-B MDM Thermo-Optic Phase Shifters

Applying phase shift to the light is essential in several applications
including switching, sensing, and computation. Employing TOPS’s is the most
common method for applying phase shift to optical waves in integrated optics.
A TOPS typically comprises a waveguide positioned underneath a metal heater.
The heater applies specific heat corresponding to its bias voltage (or bias
current) onto its underneath waveguide. The refractive index of the waveguide
varies with its temperature, and this alteration in the refractive index
results in a change in the phase of light as it travels through the waveguide.
The temperature-dependent phase change in the optical waveguide is [30]:

$\displaystyle\Delta\phi=\frac{2\pi L}{\lambda_{0}}\frac{dn}{dT}\Delta T$ (2)

where $L$ is the TOPS length, $\lambda_{0}$ is the free-space wavelength of
light, $\Delta T$ is the temperature change, and dn/dT is the thermo-optic
coefficient which is approximately $1.86\times 10^{-4}K^{-1}$ for Silicon
[31]. In MDM systems different modes have different propagation constants and
effective refractive indices, hence, for an specific TEx (or TMx) modes with
an effective refractive index of $n_{\text{eff}}$, we rewrite equation eq. 2
as:

$\displaystyle\Delta\phi_{TE_{x}}=\frac{2\pi
L}{\lambda_{0}}\frac{dn_{eff}}{dT}\Delta T.$ (3)

Simulation results using a commercial CAD tool from Ansys Lumerical shows the
change in the rate of the effective index with respect to the temperature
($dn_{\text{eff}}/dT$) converges toward almost the same value for different TE
modes as the TOPS width increases [23]. According to this simulation for a
TOPS with the waveguide width higher that 4 µm the difference between the
$dn_{\text{eff}}/dT$ for the first three TE modes is smaller than $2\%$.
Therefore, a TOPS with the waveguide width higher that 4 µm is considered
mode-insensitive for the first three TE modes (Figure 2 (c)). The application
of mode-insensitive TOPS (MI-TOPS) is mainly where the same phase shift to all
the modes are desired. For example, in a multimode communication system, when
different modes carry different multiplexed signals, if routing all
multiplexed channels in a same direction is desired, space switches with mode
insensitive phase shifters are used [23].

In contrast, several applications including multi-transverse-mode quantum
gates, multi-transverse-mode optical processor (MTMOP) [19], and mode
selective optical switches need mode-selective TOPS (MS-TOPS) offering widely
different thermo-optic coefficients for different modes. Specifically, where
routing a specific mode to a desired path is needed, a MS-TOPS is beneficial.
We showed mode-selective operation of a narrow TOPS with a width smaller than
1 µm [23], [19]. However, the mode selectivity of this TOPS is relatively poor
resulting in less than 1.1 times difference in the thermo-optic coefficient
between TE0 and for TE1 (1.8 and 1.96 for TE0 and TE1, respectively.)

We have developed a MS-TOPS with a distinct separation between the thermo-
optic coefficients of TE0 and TE1, achieved through the incorporation of SWG
structures [26]. Figure 2 (d) displays the scanning electron microscope (SEM)
image of the SWG-based MS-TOPS, providing insight into its design parameters.
The proposed MS-TOPS design relies on the fact that TE0 primarily propagates
within the 500 nm waveguide center, while the TE1 field pattern exhibits two
lobes closer to the waveguide sidewalls. Silicon’s thermo-optic coefficient is
approximately an order of magnitude larger than that of oxide, specifically
$1.86\times 10^{-4}K^{-1}$ for silicon and $0.95\times 10^{-5}K^{-1}$ for SiO2
[31]. By engineering the thermo-optic coefficient of the regions near the
waveguide sidewalls, while maintaining the waveguide center as silicon, a
substantial difference in thermo-optic coefficients between TE0 and TE1 can be
achieved.

The SWG structures feature a duty ratio defined as the length of the Silicon
over the pitch length. For the MS-TOPS shown in fig. 2 (d), the duty ratio is
40% since the width of Silicon in the periodic structure is 88 nm and the
pitch is 220 nm. The Finite-Difference Eigenmode (FDE) Lumerical mode solver
is employed to calculate the effective refractive index of the MS-TOPS for TE0
and TE1 [32, 33]. The SWG section of the MS-TOPS is modeled as a homogeneous
material with an equivalent refractive index. Simulation results presented in
fig. 2 (d) illustrate the TE0 main lobe predominantly in the silicon core and
the two lobes of TE1 in the SWG side structures. Experimental results
demonstrate a mode-selectivity of 1.44 for a duty ratio of 40%, signifying
that the thermo-optic coefficient of TE0 is 44% larger than that of TE1 [26].
A cascaded configuration of the proposed MS-TOPS and a MI-TOPS provides two
degrees of freedom to manipulate the relative phase of each mode independently
as it can be seen in the next sections in developing mode selective switches.

###  III-C Mode converters and mode exchangers

Mode converters and mode exchangers are essential in MDM communications when
changing or swapping channels is required. In multi-mode computations, they
can be used to convert optical data encoded on a specific transverse mode to
another mode. Unlike WDM, which requires nonlinearity to convert the
wavelength, MDM allows for simple designs with small footprints to convert or
exchange transverse modes. Computational inverse design techniques have shown
potential for efficiently designing mode converters and exchangers with
minimal footprints. TE mode converters on the silicon-on-insulator platform
have been experimentally demonstrated using the computationally efficient
shape optimization method [9]. These mode converters, shown in fig. 2 (e),
exhibit conversion efficiencies above 95%, with an insertion loss ranging from
0.3 dB to 1 dB over a wavelength of 1.5 µm to 1.58 µm. They have been
experimentally validated in the time domain with a 28 Gbps non-return-to-zero
(NRZ) and 20 GBaud 4-level pulse-amplitude modulation (PAM-4) payload
transmissions and have been added to the MDM library.

Mode exchangers can exchange between multiple modes; for example, they can
swap TE0 and TE1 or even higher-order modes. A compact 4 µm $\times$ 4 µm
footprint mode exchanger that can swap TE0 and TE1 is designed using inverse
design and shown in Fig. 2 (f). This mode exchanger is demonstrating 0.3 dB
insertion loss and -29.5 dB crosstalk at 1550 nm [10]. Mode exchangers for
higher-order modes, which exchange TE0 mode with TE1 mode as well as TE2 mode
with TE3 mode, are also presented in [10]. As we will discuss in the next
sections, these mode exchangers can be used in Transverse-Mode Encoded Quantum
Processors where it is desired to transfer quantum information encoded on one
mode to another. They can also be used in optical switches to eliminate the
mode dependency of the phase shifters by cascading phase shifters and
switching the modes in between [10].

###  III-D Mode (de)Mux

Conventional mode (de)Mux can be effectively designed by employing adiabatic-
based asymmetrical directional couplers for TE and TM modes, as outlined in
[2]. In a specific example, an adiabatic directional coupler-based mode
multiplexer, featuring insertion losses varying between -0.3 dB to -2.9 dB for
the TE0 mode, -0.9 dB to -3.7 dB for the TE1 mode, and -0.3 dB to -2.2 dB for
the TE2 modes within the 1520 nm to 1600 nm wavelength range, is demonstrated
in [34] and included in the MDM library [21]. Notably, the worst crosstalks
for this (de)Mux arrangement are reported as $-18.3$ dB, $-17.0$ dB, and
$-19.6$ dB for TE0, TE1, and TE2 modes, respectively.

While adiabatic-based (de)Mux configurations are known for their efficiency,
they often come with relatively large footprints. To address this, inverse
design methodologies have been employed to create mode (de)Muxs with smaller
footprints. For instance, a topologically optimized (de)Mux designed for TE0,
TE1, and TE2 modes showcases a compact 4.5 µm $\times$ 4.5 µm footprint as
shown in fig. 2 (g). This (de)Mux has 0.13 dB insertion loss and worst-case
channel crosstalk is below -18.5 dB across a wavelength bandwidth of 1500
nm-1600 nm [35]. A significant challenge associated with inverse-designed
devices lies in their small footprint, including numerous features with small
dimensions. Fabrication variations and imperfections can often lead to
performance degradation, particularly in terms of modal crosstalk for MDM
components. One viable solution involves employing a deep learning model for
predicting fabrication variations in planar integrated silicon photonic
devices. A convolutional neural network (CNN) has been demonstrated to learn
the relationship between a design and its fabricated outcome, allowing
predictions for new, unseen photonic devices based on a small collection of
SEM images [28]. This prediction can then be re-simulated to estimate the
expected variation in the experimental performance of a given device. By
utilizing this technique and incorporating PreFab fabrication prediction tools
[36], a 3-channel (de)Mux with enhanced performance has been experimentally
demonstrated [27].

##  IV MDM in DataCom

MDM technology has found application in intra-chip communication to enhance
data transmission throughput [23]. Additionally, it has proven effective in
chip-to-chip transmission through the utilization of multi-mode or a few-mode
fibers. A recent demonstration showcased the coupling of three TE and one TM
modes from a SiPh multimode waveguide to a few-mode fiber, achieving an
average crosstalk of -7 dBm. This setup enabled data transmission at a rate of
1.92 Tb/s in the L-band via MDM, complemented by WDM [6]. Given the mode
mismatch between waveguide and few-mode fibers, a 3D inverse-tapered polymer
mode converter was employed at the coupling interface to facilitate mode
conversion.

Furthermore, an inverse-designed chip-to-fiber vertical coupler has been
developed for the first four TE modes. This coupler facilitates the coupling
of modes from a multimode waveguide in SiPh to a multimode fiber with a
rectangular core. The shape and energy distribution of the modes within the
waveguide and the rectangular core fiber remain largely unchanged, while the
mode size is adjusted and optimized to enhance coupling efficiency. Using this
technique, a 1.12 Tb/s data transmission via a multimode fiber was
successfully demonstrated based on MDM and WDM [7]. This section highlights
our recent progress in MDM for datacom.

###  IV-A Multimode spatial switches

Multimode spatial switch is similar to the conventional spatial switch with
this difference that each input/output channel carries multiple multiplexed
modes instead of a single mode. In fact, the switch can route from different
inputs to different outputs while all the multiplexed modes go in the same
direction. In other words, the switch features routing multimode signal but
does not have the option to selectievly route speceifc modes. Figure 3 (a)
shows the schematic of a 2 $\times$ 2 MZI-based multimode space switch. The
switch includes balanced MMIs (or multimode directional couplers) as multimode
splitters and combiners. It also includes a MI-TOPS to apply simmilar phase
shift to all the modes, hence, all the multiplexed modes will be routed to the
same direction. All the componenets used in this switch are already available
in the MDM library [21]. This switch can be scaled to higher number of ports
employing conventional optical space switch architectures such as Banyan,
Benes̆, Spanke-Benes̆, etc. [37].

A three-mode MDM space switch is experimental demonstrated for the TE0, TE1,
and TE2 showing approximately -2, -3.7, and -5.2 dB insertion loss for the
TE0, E1, and TE2 modes at 1550 nm, respectively. The corresponding crosstalk
is less than -8.6 (-9), -8(-10.3), and -10 dB (-10.3 dB) within the wavelength
range of 40 nm (1535–1575 nm) for the cross (bar) states, respectively. Error
free 10 Gbps PRBS-31 payload transmission over this switch has been
demosntrated [23]. Using the similar building block, 3 $\times$ 3 and 4
$\times$ 4 MDM switches utilizing TE0 and TE1, shown in fig. 3 (b), were
experimentally demonstrated in [24]. As multimode waveguide crossing often
exhibit large insertion loss and intermodal crosstalk [16], Spanke-Benes̆
topology is used in the design of this switch to avoid the use of multimode
waveguide crossings. The 3-port switch exhibits approximately 2.6 dB and 3.3
dB insertion loss for the longest path with a crosstalk of at most 10 dB and 8
dB over a bandwidth of 40 nm (1530 nm to 1570 nm) for TE0 and TE1 modes,
respectively. The insertion loss measured for the 4-port switch is
approximately 2.7 dB and 3.6 dB at 1550 nm with a corresponding crosstalk less
than 8 dB for the two TE modes, respectively. The suscessful payload
transmission using 10 Gb/s NRZ and 14.0625 Gbaud PAM-4 pseudorandom binary
sequence (PRBS)-31 data signal has been demosntrated on these switches [24]. A
broadband two-mode switch with 60 nm wavelength range from 1530 to 1590 nm has
been also demonstrated by optimizing the MMI and phase shifter design over a
wider range of wavelength [38].

Figure 3: (a) Schematic view of a 2 × 2 multimode space switch, including Mux
and deMux, and an MZI consisting of two MMIs and one MI-TOPS. (b) Scaling the
switch to 3 × 3 and 4 × 4 using Spanke-Benes̆ topology.

###  IV-B Reconfigurable Mode Selective Switch

A mode-selective switch selectively routes a specific mode without undesirable
effects on the others, directing different modes to distinct switch outputs.
This switch also enables selective addition or removal of desired channels in
a multimode waveguide. Despite considerable advancements in MDM components
within SiPh, selectively accessing specific modes without introducing unwanted
interference to the others has remained challenging. This challenge becomes
more pronounced when attempting to extract the fundamental mode from a
multimode bus waveguide while keeping higher-order modes unchanged. Recent
approaches have addressed this issue by using modal field redistribution,
effectively confining modes to specific regions of the bus waveguide [39, 40].
Although efficient, this technique involves redistributing energy among
transverse modes within the waveguide.

We recently introduced a mode-selective switch designed to route various modes
to distinct ports without redistributing the modal fields. By cascading a MS-
TOPS and a MI-TOPS within a MZI, we successfully demonstrate the selective
addition and removal of TE0 or TE1 from a multimode bus waveguide in SiPh.
Figure 4 (a) illustrates the configuration of this mode-selective switch. This
switch takes the form of an MZI comprising two multimode interferometers
serving as 3-dB splitter/combiners. Within one arm, two cascaded TOPS are
employed. MI-TOPS imparts an identical phase shift to both TE0 and TE1, as
detailed in section III. For the SWG duty cycle ratio of 0.4, the MS-TOPS
applies a phase shift to TE0 that is approximately 1.44 times larger than that
applied to TE1 [26]. The phase shift applied to TE1 in the upper arm of MZI is
the sum of the phase shifts applied by the two cascaded TOPS i.e.,
$\phi+\delta$. For TE0, the phase shift applied by the MS-TOPS is 1.44 times
larger than that of TE1 leading to a total phase shift of
$\phi+1.44\times\delta$.

By applying distinct biases to MI-TOPS and MS-TOPS, leading to different
values of $\phi$ and $\delta$, we can achieve arbitrary and distinct phase
shift values for TE0 and TE1. As depicted in fig. 4 (a), the mode-selective
switch can be configured to pass or drop both modes, or selectively drop one
of them. In a reversed configuration, it can also combine two modes applied to
the two inputs of the MZIs. The SWG region in the MS-TOPS is designed to
preserve the modal field distribution of the modes without converting them
into supermodes, all while introducing unique temperature coefficients for TE0
and TE1. The design is fabricated at the Applied Nanotool (ANT) foundry using
electron beam lithography, with a silicon layer thickness of 220 nm. The TOPSs
utilize a 200 nm thick Titanium Tungsten (TiW) metal heater deposited on top
of the 2.2 µm oxide cladding.

Fig. 4 (b) shows a potential application of the mode-selective switch in a
Reconfigurable Optical Add/Drop Multiplexer (ROADM) system. ROADMs, introduced
in the 2000s for WDM networks, responded to evolving network traffic demands
and provided network flexibility [41]. They enable remote path modification of
optical wavelength channels through wavelength-selective switches, allowing
the addition and removal of specific wavelengths at a location in response to
changes in traffic patterns. As a successor to WDM, the WDM/MDM network would
benefit from a switch capable of adding/dropping individual modes. A mode-
selective switch can perform this function in a WDM/MDM system.

In Figure 5(a), eye diagrams for TE0 and TE1 transmission are displayed using
PRBS31 NRZ and PRBS31Q PAM-4 for the two modes when TE0 is in the bar state
and TE1 is in the cross state (dropped). Payload transmission is carried out
with both modes simultaneously transmitted through the switch, resulting in
degradation due to modal crosstalk. Corresponding bit error rate (BER) curves
are presented in Figure 5(b), demonstrating error-free NRZ transmission at 16
Gbps and a BER below $10^{-6}$ for 40 Gbps. For PAM-4 transmission, the BER
remains below the KP4 forward error correction (FEC) limit of $2.2\times
10^{-4}$ for TE0 and even below the HD FEC limit of $3.8\times 10^{-3}$ for
TE1 at 20 GBaud.

This work serves as a proof of concept for a mode-selective switch but was
constrained by the single-etched SOI fabrication, resulting in grating
couplers with 8 dB of coupling loss. This limitation led to challenging fiber-
to-fiber insertion loss, subsequently impacting the BER measurement due to
amplified spontaneous emission (ASE) noise from the required erbium doped
fiber amplifier (EDFA) optical gain in the testbed. Additionally, coupling in
and out of the chip occurs at the fundamental mode, with adiabatic mode
(de)multiplexers converting higher-order modes to fundamental modes. Lower
modal crosstalk in the mode-selective switch can be achieved through multimode
couplers and few-mode fibers at the interface, eliminating the effects of mode
multiplexers/demultiplexers.

Figure 4: (a) Schematic of a mode-selective switch used in a Mode-selective
ROADM capable of adding/dropping TE0 or TE1. (b) Mode-selective ROADM in a
WDM/MDM flexible system dropping the xth TE mode channel and the yth mode.
Figure 5: (a) NRZ and PAM4 BER eye diagrams with a 100 mV/div y-axis for TE0
(Bar state output port) and TE1 (crossbar state output port). (b) The
corresponding BER curves.

##  V MDM in photonic computation

While MDM has a decade-long history in optical communication, its relatively
recent but burgeoning application in optical computing is gaining significant
attention. This section delves into three primary applications of MDM.
Firstly, MTMOP leverage the distinct propagation constants of various
transverse modes to monitor phase shifts, facilitating precise phase
adjustments essential for analog optical processors. Secondly, the Multi-
Transverse-Mode Multiply-Accumulate Unit (MTM-MAC) capitalizes on the
capability of adding different spatial modes through "non-coherent" summation.
This approach addresses challenges in the coherent summation of conventional
MACs, leading to a higher number of resolved output power. Lastly, we showcase
the utilization of different transverse electric modes for encoding quantum
information in Transverse-Mode Encoded Quantum Processors.

###  V-A Multi-Transverse-Mode MZI-based Optical Processors

Programmable optical processors are promising structures for fast and energy-
efficient analog optical computing. These processors can efficiently perform
vector-matrix multiplication [42, 43], which currently consumes the largest
portion of computation resources in machine learning and AI [44]. One main
challenge related to the programmable optical processor is that, as analog
devices, computation accuracy highly depends on programming accuracy and the
phase setting of phase shifters. On the other hand, emerging applications for
programmable optical processors ask for faster programming time so that the
processor can be reconfigured to process multiple tasks. Calibration and
programming of optical processors require sensing optical power and optical
phase. Although sensing the optical power is easily feasible in photonic
integrated circuits using on-chip photodetectors, sensing the optical phase
requires more complex and elaborate hardware. One possible method for
sensing/monitoring the phase shift applied by a phase shifter is to take
advantage of different transverse electric modes as shown in MTMOPs [19]. As
discussed in section III, a MS-TOPS applies different phase shift to different
modes but the ratio of phase shift values remains constant and it depends on
the geometries of the MS-TOPS. For example for an SWG-based MS-TOPS shown in
[26], for the SWG duty ratio of 40%, the thermo-optic coefficient of TE0 is
44% larger than that of TE1. We use this difference in thermo-optic
coefficient in MTMOP to monitor the phase shift applied by the phase shifters.

Figure 6 (a) presents the proposed 2 × 2 building block schematic of the
MTMOP. It uses two orthogonal TE optical modes: TE0 for carrying the main
optical signal and TE1 for performing phase calibration. The internal phase
shifter $\theta$ is a conventional MI-TOPS applying the same phase shift to
the TE0 and TE1 modes. In the MTMOP, we replace the external phase shifter by
an MZI composed of I) an MMI as a 50:50 beam splitter, II) two phase shifters
($\phi$ and $\delta$), III) a second MMI as the beam combiner. The phase
shifter $\delta$ is MS-TOPS with different thermo-optic coefficient
($dn_{eff}/dT$) for TE0 and TE1. The phase shifter $\phi$ is MI-TOPS. The
principle of operation is: we start with setting a bias to $\phi$ as an
initial point for the desired TE0 phase shift. We then sweep the phase shifter
$\delta$ bias voltage until the TE0 signal power at $O_{0}$ is maximized
meaning the TE0 signal passing through the MZI including phase shifters $\phi$
and $\delta$ constructively interferes. This would not be the case for TE1
owing to the fact that the phase shift implied by $\delta$ is different for
TE0 and TE1. Knowing $dn_{eff}/dT$ for TE0 and TE1, we calculate the phase
shift applied to TE0 by measuring the transmission of TE1. In this way we can
monitor the phase shift applied to TE0 by measuring the TE1 transmission.
Knowing the phase shift applied to the TE0, we iterate the process until
achieving the desired phase shift to TE0. Using this process, we can monitor
the phase shift applied by the external phase shifters. Monitoring the phase
shift is helpful in both calibration and programming phase of optical
processors. We can calibrate the MTMOP completely on-chip using photodetectors
and without need for coherent detection.

Figure 6 (b) shows a 4 × 4 MTMOP in the Reck architecture [45] based on the
aforementioned 2 × 2 building block. The idea presented in fig. 6 (b) can be
extended to other programmable optical processor architectures such as Clement
mesh [46], diamond mesh [47], or Bokun mesh [48]. Details of programming
techniques as well as discussion on scalability are presented in [19].

Figure 6: (a) Schematic view of a 2 × 2 building block of MTMOP. TE0 is used
as the main mode for carrying the data. TE1 is for monitoring the phase shifts
in the calibration and programming phase. (b) 4 × 4 MTMOP based on the Reck
architecture along with the phase calibration unit.

###  V-B Multi-Transverse-Mode Multiply-Accumulate Unit

The terminology of the multiply-accumulate operation refers to the calculation
of the product of two numbers and the addition of the result to an
accumulator. A variety of linear mathematical operations, including matrix
multiplications, convolutions, and dot products, could be represented by a
series of MAC operations. That explains why modern high-performance computing
applications require a cornerstone of highly efficient MAC unit hardware in
terms of energy and compute density. Integrated photonics offers a promising
solution for efficient MAC operations, demonstrating improvements over the
electronic counterparts, toward realizing aj/MAC and PMACs/s/mm2 performance
from both energy and compute density perspectives [49], [50]. A photonic MAC
unit architecture based on microring weight banks drives microring resonators
(MRRs) in and out of resonance to modulate the intensity of an input WDM
signal, while a photodetector performs accumulation (weighted summation)
through electro-optic conversion [51]. System analysis suggests improved
energy efficiency with an increase in the size of the MRR weight bank.
However, network size scalability is predominantly limited by the number of
WDM channels and crosstalk arising from the free spectral range (FSR)
limitations [52]. While deploying two-point or contra-directional coupling
schemes can suppress FSR constraints, they come at the expense of an
additional heater, which deteriorates energy efficiency [53], [54]. On the
other hand, increasing the bit resolution not only demands more power-hungry
drivers but is also limited by the finite amount of resonance extinction
ratio.

Recent advancements in MDM components can potentially alleviate the above
challenges by enabling signal modulation on fundamental and higher order
transverse modes, while simultaneously deploying WDM technique [3]. A mode-
selective modulation design for interconnect applications utilized microring
modulators to demultiplex modes from the multimode bus and multiplex the
selected modes back to their originals [55]. Such architecture for a multi-
transverse-mode MAC unit application suffers from an increased insertion loss
and modal crosstalk due to additional mode down and up conversions. In [56],
an MDM-based matrix-vector multiplier is proposed, where mode up and down
conversion is implemented solely for efficient power splitting within the
system. Weighting, on the other hand, is achieved exclusively in the
fundamental mode, requiring coherent summation at the output.

Figure 7: A schematic representation of the proposed MTM-MAC unit for a two-
channel weighted WDM input with $b_{1}$ and $b_{2}$ amplitudes. Fundamental
(TE0) and first-order (TE1) transverse-mode MRRs are responsible for realizing
the least (LSB) and the most (MSB) significant bits of each weight element
($a_{1}$ and $a_{2}$). The MDM building blocks are described in the text.

We recently introduced an MTM-MAC unit architecture that enables direct MRR-
based weighting of the fundamental and first-order transverse-electric modes
(TE0, TE1) as depicted in Figure 7. For simplicity, only two weighted WDM
channels are considered with amplitudes of $b_{1}$ and $b_{2}$. The first-
order mode (TE1) is excited on-chip using a mode converter. Each element
within the weight matrix (e.g., $a_{1}$ and $a_{2}$) is implemented using a
pair of two MRRs coupled to the TE0 and TE1 signal paths, carrying an input
optical power in a ratio of 0.33:0.66. The weighted TE0 and TE1 signals are
directed to a 2×1 multimode interferometer. The orthogonality of the TE0 and
TE1 optical signals allows for their non-coherent superposition in the MMI
output path, without the need for any preceding phase-matching conditions. The
MMI output leads directly to the photodetector, which is inherently
insensitive to modes. As a result, it accumulates both the weighted TE0 and
TE1 signals, thereby completing an MTM-MAC operation. Experimental validation
of a single channel MTM-MAC unit in the SOI platform confirms that a binary
weight control of each MRR-TE0 and MRR-TE1 resonating at the same wavelength
produces a four-level resolved output power. A worst-case modal crosstalk of
-12 dB is observed in the wavelength range of [1530:1550] nm. This WDM-
compatible design is scalable by further deployment of higher-order modes
(TE2, TE3) leading to a more accurate MAC unit hardware. The drawback of this
design lies in the relatively large footprint of the deployed MRR-TE1 [57],
which could be reduced by utilizing subwavelength-grating structures to
minimize the coupling length [58].

###  V-C Transverse Mode Encoded Programmable Quantum Processor

SiPh has been primarily utilized in integrated quantum computing to manipulate
the quantum state of light and realize quantum gates [59]. Quantum information
can be encoded in various degrees of freedom of single photons, including
polarization, path, or wavelength. In recent years, path-encoded programmable
quantum gates have been successfully demonstrated in SiPh, marking a
significant step towards a large-scale integrated photonic quantum system
[60]. However, encoding information in other degrees of freedom of single
photons, such as polarization and transverse mode, is crucial for the
scalability of the system. Recently, attempts have been made to encode
information in transverse electric modes and manipulate it using transverse
mode integrated quantum gates [61]. Although innovative and efficient, the
design presented in [61] is not programmable. Consequently, it can only
perform a specific operation, in that case, Controlled Not (CNOT).
Additionally, since the design is not programmable, fabrication variations
cannot be compensated for through fine-tuning the phase shifters.

Figure 8 depicts a schematic view of a 4 × 4 building block of a programmable
transverse-mode encoded processor that we propose, leveraging TE0 and TE1 to
encode quantum information. The primary advantage of the proposed transverse
mode encoded programmable quantum processor is its ability to independently
manipulate different transverse modes without converting them to the
fundamental mode. All the components used in this block, as shown in fig. 8
(a), are available in the [21] library. The key structure in this block is an
MZI with two multimode input waveguides, each carrying two modes, TE0 and TE1.
In the input of the MUX, we have a generic path-encoded qubit represented as
$a|0\rangle+b|1\rangle$, where $|0\rangle$ and $|1\rangle$ are two basis
states corresponding to the presence of a single photon on the first and
second single-mode waveguides, respectively. The single-mode waveguides are
0.5 µm wide, allowing only the propagation of the fundamental TE0 mode. The
MUX converts the path-encoded state to a coherent superposition of TE0 and TE1
transverse electric mode states, $a|TE0\rangle+b|TE1\rangle$, of a single
photon propagating in a multi-mode waveguide where $a|TE0\rangle$ and
$b|TE1\rangle$ are the basis states corresponding to the propagation of the
single photon on TE0 and TE1 modes, respectively. The multimode waveguide is
0.96 µm wide, allowing the propagation of TE0 and TE1. We use mode-insensitive
MMIs as splitters and combiners of the MZI. The processor utilizes three
multimode TOPSs: $\delta$, $\phi$, and $\theta$. Each multimode TOPS comprises
cascaded MI-TOPS and MS-TOPS, providing two degrees of freedom to
independently control the phase shifts applied to TE0 and TE1, similar to the
approach we discussed for the mode ROADM in Fig. 4. The lower arm includes a
mode exchanger to swap TE0 and TE1, providing a path for transmission between
TE0 to TE1 and vice versa.

Applying the method discussed in [62], we define the transfer matrix of the
structure by multiplying the corresponding transfer matrices of the cascaded
blocks. For a 50:50 splitting ratio of combiners and splitters, the linear
transformation matrix of the input optical field to the output optical field
is:

$\displaystyle\frac{1}{2}\times\begin{bmatrix}e^{j\phi_{0}+j\theta_{0}}&-e^{j\phi_{0}+j\delta_{0}}&je^{j\phi_{0}+j\theta_{0}}&je^{j\phi_{0}+j\delta_{0}}\\\
-e^{j\phi_{1}+\delta_{1}}&e^{j\phi_{1}+\theta_{1}}&je^{j\phi_{1}+\delta_{1}}&je^{j\phi_{1}+\delta_{1}}\\\
je^{j\theta_{0}}&je^{j\delta_{0}}&-e^{j\theta_{0}}&e^{j\delta_{0}}\\\
je^{j\delta_{1}}&je^{j\theta_{1}}&e^{j\delta_{1}}&e^{j\theta_{1}}\\\
\end{bmatrix}$ (4)

Modifying the values of the three TOPS based on the transformation matrix in
eq. (4), we can perform a unitary transformation on the optical fields of the
two paths and two orthogonal modes. The presented design performs a 4 × 4
unitary transformation employing only a single MZI. We can use the building
block presented in Fig. 8 (a) on a larger mesh as shown in Fig. 8 (b) to
realize a linear transformation on larger dimensions. In the presented multi-
transverse-mode processor, the number of MZIs required for an N × N
transformation on Clements mesh is (N(N-2))/8, which is asymptotically one-
fourth of the MZIs required in a single-mode path-encoded system (i.e.,
(N(N-1))/2) [46]. The mesh depth (number of consecutive MZIs in the longest
path) is also half that of a single-mode path-encoded system.

Figure 8:  (a) A schematic illustration of a 4 × 4 building block for the
Transverse Mode Encoded Programmable Quantum Processor, including three
multimode TOPS. Each multimode TOPS is comprised of a MI-TOPS and MS-TOPS. (b)
Scaling the processor to 8 × 8 using Clements architecture.

##  VI Conclusion

In this work, we have presented a comprehensive library of MDM components,
including MI-TOPS, MS-TOPS, MMIs, mode (de)Mux, mode converters, and mode
exchangers. These components are developed using classical and inverse design
and are all compatible with standard 220 nm SiPh foundries. We have also
discussed how MDM can contribute to on-chip and chip-to-chip communication,
providing increased throughput by exploiting higher-order modes. The mode
ROADM presented lays the groundwork for flexible MDM architectures in the
future. Additionally, we have explored MDM applications in computing,
including MTMOP for faster and more accurate programming of optical
processors. The discussion on MTM-MAC highlights its capability for non-
coherent summation of multiple TE modes, offering scalability in the number of
bits for photonic MACs. Furthermore, the transverse mode encoded programmable
quantum processor has been introduced, providing an extra degree of freedom
for encoding quantum information in SiPh.

## Acknowledgment

Authors acknowledge the help from Canadian Microelectronic Corporation (CMC)
for the subsidized multiproject wafer fabrication through Applied Nanotools
(ANT) as well as financial support from the Natural Sciences and Engineering
Research Council of Canada (NSERC) and Fonds de recherche du Québec (FRQNT).

## References

  * [1]
  * [2] D. Dai, J. Wang, and Y. Shi, “Silicon mode (de)multiplexer enabling high capacity photonic networks-on-chip with a single-wavelength-carrier light,” _Opt. Lett._ , vol. 38, no. 9, pp. 1422–1424, May 2013. [Online]. Available: https://opg.optica.org/ol/abstract.cfm?URI=ol-38-9-1422
  * [3] L.-W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergman, and M. Lipson, “Wdm-compatible mode-division multiplexing on a silicon chip,” _Nature communications_ , vol. 5, no. 1, p. 3069, 2014.
  * [4] S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R.-J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle, “6$\times$ 56-gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6$\times$ 6 mimo equalization,” _Optics Express_ , vol. 19, no. 17, pp. 16 697–16 707, 2011.
  * [5] C. R. Doerr and T. F. Taunay, “Silicon photonics core-, wavelength-, and polarization-diversity receiver,” _IEEE Photonics Technology Letters_ , vol. 23, no. 9, pp. 597–599, 2011.
  * [6] O. A. Jimenez Gordillo, A. Novick, O. L. Wang, A. J. Rizzo, U. D. Dave, K. Bergman, and M. Lipson, “Fiber-chip link via mode division multiplexing,” _IEEE Photonics Technology Letters_ , vol. 35, no. 19, pp. 1071–1074, 2023.
  * [7] K. Y. Yang, C. Shirpurkar, A. J. White, J. Zang, L. Chang, F. Ashtiani, M. A. Guidry, D. Lukin, S. Pericherla, J. Yang, H. Kwon, J. Lu, G. H. Ahn, K. V. Gasse, Y. Jin, S.-P. Yu, T. C. Briles, J. M. Stone, D. R. Carlson, H. Song, K. Zou, H. Zhou, K. Pang, H. Hao, L. R. Trask, M. Li, A. M. Netherton, L. Rechtman, J. R. Stone, J. Skarda, L. Su, D. Vercruysse, J.-P. W. MacLean, S. Aghaeimeibodi, M.-J. Li, D. , D. M. Marom, A. E. Willner, J. E. Bowers, S. A. Diddams, P. J. Delfyett, F. Aflatouni, and J. Vuckovic, “Multi-dimensional data transmission using inverse-designed silicon photonics and microcombs,” _Nature Communications_ , vol. 13, 12 2022.
  * [8] C. P. Chen, J. B. Driscoll, R. R. Grote, B. Souhan, R. M. Osgood, and K. Bergman, “Mode and polarization multiplexing in a si photonic chip at 40gb/s aggregate data bandwidth,” _IEEE Photonics Technology Letters_ , vol. 27, no. 1, pp. 22–25, 2015.
  * [9] M. M. Masnad, G. Zhang, D.-X. Xu, Y. Grinberg, and O. Liboiron-Ladouceur, “Fabrication error tolerant broadband mode converters and their working principles,” _Opt. Express_ , vol. 30, no. 14, pp. 25 817–25 829, Jul 2022. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-30-14-25817
  * [10] G. Zhang, D.-X. Xu, Y. Grinberg, and O. Liboiron-Ladouceur, “Efficient mode exchanger-based silicon photonic switch enabled by inverse design,” _Opt. Express_ , vol. 30, no. 12, pp. 20 543–20 553, Jun 2022. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-30-12-20543
  * [11] R. B. Priti and O. Liboiron-Ladouceur, “Reconfigurable and scalable multimode silicon photonics switch for energy-efficient mode-division-multiplexing systems,” _Journal of Lightwave Technology_ , vol. 37, no. 15, pp. 3851–3860, 2019.
  * [12] ——, “A reconfigurable multimode demultiplexer/switch for mode-multiplexed silicon photonics interconnects,” _IEEE Journal of Selected Topics in Quantum Electronics_ , vol. 24, no. 6, pp. 1–10, 2018.
  * [13] C. Williams, G. Zhang, R. Priti, G. Cowan, and O. Liboiron-Ladouceur, “Modal crosstalk in silicon photonic multimode interconnects,” _Opt. Express_ , vol. 27, no. 20, pp. 27 712–27 725, Sep 2019. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-27-20-27712
  * [14] D. Dai and M. Mao, “Mode converter based on an inverse taper for multimode silicon nanophotonic integrated circuits,” _Opt. Express_ , vol. 23, no. 22, pp. 28 376–28 388, Nov 2015. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-23-22-28376
  * [15] Y. Wang and D. Dai, “Ultra-sharp multimode waveguide bends with dual polarizations,” _Journal of Lightwave Technology_ , vol. 38, no. 15, pp. 3994–3999, 2020.
  * [16] S. H. Badri, H. R. Saghai, and H. Soofi, “Multimode waveguide crossing based on a square maxwell’s fisheye lens,” _Appl. Opt._ , vol. 58, no. 17, pp. 4647–4653, Jun 2019. [Online]. Available: https://opg.optica.org/ao/abstract.cfm?URI=ao-58-17-4647
  * [17] C. Williams, B. Banan, G. Cowan, and O. Liboiron-Ladouceur, “A source-synchronous architecture using mode-division multiplexing for on-chip silicon photonic interconnects,” _IEEE Journal of Selected Topics in Quantum Electronics_ , vol. 22, no. 6, pp. 473–481, 2016.
  * [18] G. Qiao, C. Long, Y. Yang, Z. Xu, M. Zuo, Q. Wu, S. Wang, L. Liu, L. Shen, J. Luo, Z. He, H. Ji, W. Hu, Z. Chen, and J. Li, “Mode-division-multiplexing self-homodyne coherent transmission over a weakly coupled few-mode fiber for optical interconnections,” _Opt. Lett._ , vol. 49, no. 4, pp. 981–984, Feb 2024. [Online]. Available: https://opg.optica.org/ol/abstract.cfm?URI=ol-49-4-981
  * [19] K. R. Mojaver and O. Liboiron-Ladouceur, “On-chip optical phase monitoring in multi-transverse-mode integrated silicon-based optical processors,” _IEEE Journal of Selected Topics in Quantum Electronics_ , vol. 28, no. 6: High Density Integr. Multipurpose Photon. Circ., pp. 1–7, 2022.
  * [20] L.-T. Feng, M. Zhang, X. Xiong, D. Liu, Y.-J. Cheng, F.-M. Jing, X.-Z. Qi, Y. Chen, D.-Y. He, G.-P. Guo, G.-C. Guo, D.-X. Dai, and X.-F. Ren, “Transverse mode-encoded quantum gate on a silicon photonic chip,” _Phys. Rev. Lett._ , vol. 128, p. 060501, Feb 2022. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.128.060501
  * [21] “SiPh MDM Library,” https://github.com/KavehMojaver/SiPh_MDM_PDK, accessed: 2024-02-06.
  * [22] K. H. Rahbardar Mojaver and O. Liboiron-Ladouceur, “Multi-transverse-mode silicon photonics for quantum computing,” in _Proceedings of the Great Lakes Symposium on VLSI 2023_ , ser. GLSVLSI ’23. New York, NY, USA: Association for Computing Machinery, 2023, p. 551–556. [Online]. Available: https://doi.org/10.1145/3583781.3590274
  * [23] G. Zhang, H. R. Mojaver, A. Das, and O. Liboiron-Ladouceur, “Mode insensitive switch for on-chip interconnect mode division multiplexing systems,” _Opt. Lett._ , vol. 45, no. 4, pp. 811–814, Feb 2020. [Online]. Available: https://opg.optica.org/ol/abstract.cfm?URI=ol-45-4-811
  * [24] A. Das, G. Zhang, H. R. Mojaver, and O. Liboiron-Ladouceur, “Scalable two-mode 3-port and 4-port mode insensitive silicon photonic switches,” _IEEE Photonics Technology Letters_ , vol. 33, no. 11, pp. 557–560, 2021.
  * [25] H. Shiran, G. zhang, and O. Liboiron-Ladouceur, “Dual-mode broadband compact 2 × 2 optical power splitter using sub-wavelength metamaterial structures,” _Opt. Express_ , vol. 29, no. 15, pp. 23 864–23 876, Jul 2021. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-29-15-23864
  * [26] S. M. R. Safaee, K. R. Mojaver, G. Zhang, and O. Liboiron-Ladouceur, “Silicon photonics mode-selective phase shifter,” _Journal of Lightwave Technology_ , pp. 1–11, 2023.
  * [27] D. Gostimirovic, M. M. Masnad, D.-X. Xu, Y. Grinberg, and O. Liboiron-Ladouceur, “Pre-fabrication performance verification of a topologically optimized mode demultiplexer using deep neural networks,” in _2022 European Conference on Optical Communication (ECOC)_ , 2022, pp. 1–4.
  * [28] D. Gostimirovic, D.-X. Xu, O. Liboiron-Ladouceur, and Y. Grinberg, “Deep learning-based prediction of fabrication-process-induced structural variations in nanophotonic devices,” _ACS Photonics_ , vol. 9, no. 8, pp. 2623–2633, 2022. [Online]. Available: https://doi.org/10.1021/acsphotonics.1c01973
  * [29] H. Shiran, H. Rahbardar Mojaver, J. Bachman, C. Jin, and O. Liboiron-Ladouceur, “Impact of sio2 cladding voids in siph building blocks,” in _2020 IEEE Photonics Conference (IPC)_ , 2020, pp. 1–2.
  * [30] 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,” _Opt. Express_ , vol. 27, no. 8, pp. 10 456–10 471, Apr 2019. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-27-8-10456
  * [31] A. Arbabi and L. L. Goddard, “Measurements of the refractive indices and thermo-optic coefficients of si3n4 and siox using microring resonances,” _Opt. Lett._ , vol. 38, no. 19, pp. 3878–3881, Oct 2013. [Online]. Available: https://opg.optica.org/ol/abstract.cfm?URI=ol-38-19-3878
  * [32] K. H. R. Mojaver, G. Zhang, and O. Liboiron-Ladouceur, “Design of silicon photonic mode-sensitive thermo-optic phase shifter based on subwavelength grating structures,” in _2022 IEEE Photonics Conference (IPC)_ , 2022, pp. 1–2.
  * [33] K. H. R. Mojaver, S. M. R. Safaee, G. Zhang, and O. Liboiron-Ladouceur, “Mode sensitive swg-based phase shifter for optical processors,” in _2023 IEEE Silicon Photonics Conference (SiPhotonics)_ , 2023, pp. 1–2.
  * [34] R. B. Priti, G. Zhang, and O. Liboiron-Ladouceur, “3×10 gb/s silicon three-mode switch with 120° hybrid based unbalanced mach-zehnder interferometer,” _Opt. Express_ , vol. 27, no. 10, pp. 14 199–14 212, May 2019. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-27-10-14199
  * [35] M. M. Masnad, D. Gostimirovic, Y. Grinberg, D.-X. Xu, and O. Liboiron-Ladouceur, “Feature correction of a topologically optimized mode demultiplexer using deep neural networks,” in _2022 IEEE Photonics Conference (IPC)_ , 2022, pp. 1–2.
  * [36] “PreFab,” https://www.prefabphotonics.com/, accessed: 2024-02-06.
  * [37] G. Papadimitriou, C. Papazoglou, and A. Pomportsis, “Optical switching: switch fabrics, techniques, and architectures,” _Journal of Lightwave Technology_ , vol. 21, no. 2, pp. 384–405, 2003.
  * [38] G. Zhang, A. Das, and O. Liboiron-Ladouceur, “A broadband mzi-based thermal-optic mode insensitive switch in silicon-on-insulator,” in _2020 IEEE Photonics Conference (IPC)_ , 2020, pp. 1–2.
  * [39] X. Yi, W. Zhao, C. Li, L. Zhang, Y. Xiang, C. Liu, Y. Shi, L. Liu, and D. Dai, “Reconfigurable optical add-drop multiplexers for hybrid mode-/wavelength-division-multiplexing systems,” _Advanced Photonics Nexus_ , vol. 2, no. 6, p. 066004, 2023. [Online]. Available: https://doi.org/10.1117/1.APN.2.6.066004
  * [40] X. Yi, C. Li, W. Zhao, L. Zhang, Y. Shi, and D. Dai, “On-chip mode-selective manipulation based on the modal-field redistribution assisted with subwavelength grating structures,” _Nanophotonics_ , vol. 12, no. 9, pp. 1809–1821, 2023. [Online]. Available: https://doi.org/10.1515/nanoph-2023-0111
  * [41] T. A. Strasser and J. Taylor, “Roadms unlock the edge of the network,” _IEEE Communications Magazine_ , vol. 46, no. 7, pp. 146–149, 2008.
  * [42] F. Shokraneh, S. Geoffroy-Gagnon, M. S. Nezami, and O. Liboiron-Ladouceur, “A single layer neural network implemented by a $4\times 4$ mzi-based optical processor,” _IEEE Photonics Journal_ , vol. 11, no. 6, pp. 1–12, 2019.
  * [43] F. Shokraneh, M. S. Nezami, and O. Liboiron-Ladouceur, “Theoretical and experimental analysis of a 4 × 4 reconfigurable mzi-based linear optical processor,” _Journal of Lightwave Technology_ , vol. 38, no. 6, pp. 1258–1267, 2020.
  * [44] R. Desislavov, F. Martínez-Plumed, and J. Hernández-Orallo, “Compute and energy consumption trends in deep learning inference,” _arXiv preprint arXiv:2109.05472_ , 2021.
  * [45] M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” _Physical review letters_ , vol. 73, no. 1, p. 58, 1994.
  * [46] W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “Optimal design for universal multiport interferometers,” _Optica_ , vol. 3, no. 12, pp. 1460–1465, 2016.
  * [47] F. Shokraneh, S. Geoffroy-gagnon, and O. Liboiron-Ladouceur, “The diamond mesh, a phase-error- and loss-tolerant field-programmable mzi-based optical processor for optical neural networks,” _Opt. Express_ , vol. 28, no. 16, pp. 23 495–23 508, Aug 2020. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-28-16-23495
  * [48] K. H. R. Mojaver, B. Zhao, E. Leung, S. M. R. Safaee, and O. Liboiron-Ladouceur, “Addressing the programming challenges of practical interferometric mesh based optical processors,” _Optics Express_ , vol. 31, no. 15, pp. 23 851–23 866, 2023.
  * [49] M. A. Nahmias, T. F. de Lima, A. N. Tait, H.-T. Peng, B. J. Shastri, and P. R. Prucnal, “Photonic multiply-accumulate operations for neural networks,” _IEEE Journal of Selected Topics in Quantum Electronics_ , vol. 26, no. 1, pp. 1–18, 2020.
  * [50] A. R. Totović, G. Dabos, N. Passalis, A. Tefas, and N. Pleros, “Femtojoule per mac neuromorphic photonics: An energy and technology roadmap,” _IEEE Journal of Selected Topics in Quantum Electronics_ , vol. 26, no. 5, pp. 1–15, 2020.
  * [51] A. N. Tait, A. X. Wu, T. F. de Lima, E. Zhou, B. J. Shastri, M. A. Nahmias, and P. R. Prucnal, “Microring weight banks,” _IEEE Journal of Selected Topics in Quantum Electronics_ , vol. 22, no. 6, pp. 312–325, 2016.
  * [52] M. A. Al-Qadasi, L. Chrostowski, B. J. Shastri, and S. Shekhar, “Scaling up silicon photonic-based accelerators: Challenges and opportunities,” _APL Photonics_ , vol. 7, no. 2, p. 020902, 02 2022. [Online]. Available: https://doi.org/10.1063/5.0070992
  * [53] H. Shoman, H. Jayatilleka, A. H. K. Park, A. Mistry, N. A. F. Jaeger, S. Shekhar, and L. Chrostowski, “Compact wavelength- and bandwidth-tunable microring modulator,” _Opt. Express_ , vol. 27, no. 19, pp. 26 661–26 675, Sep 2019. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-27-19-26661
  * [54] N. Eid, R. Boeck, H. Jayatilleka, L. Chrostowski, W. Shi, and N. A. F. Jaeger, “Fsr-free silicon-on-insulator microring resonator based filter with bent contra-directional couplers,” _Opt. Express_ , vol. 24, no. 25, pp. 29 009–29 021, Dec 2016. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-24-25-29009
  * [55] H. Jia, X. Fu, T. Zhou, L. Zhang, S. Yang, and L. Yang, “Mode-selective modulation by silicon microring resonators and mode multiplexers for on-chip optical interconnect,” _Opt. Express_ , vol. 27, no. 3, pp. 2915–2925, Feb 2019. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-27-3-2915
  * [56] Q. Ling, P. Dong, Y. Chu, X. Dong, J. Chen, D. Dai, and Y. Shi, “On-chip optical matrix-vector multiplier based on mode division multiplexing,” _Chip_ , vol. 2, no. 4, p. 100061, 2023. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S2709472323000242
  * [57] B. A. Dorin and W. N. Ye, “Two-mode division multiplexing in a silicon-on-insulator ring resonator,” _Opt. Express_ , vol. 22, no. 4, pp. 4547–4558, Feb 2014. [Online]. Available: https://opg.optica.org/oe/abstract.cfm?URI=oe-22-4-4547
  * [58] M. Ye, Y. Li, Z. Chen, L. Zhang, L. Liu, and Y. Yu, “Dual-mode silicon ring resonator based on subwavelength grating directional couplers,” _IEEE Photonics Journal_ , vol. 14, no. 5, pp. 1–4, 2022.
  * [59] X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph _et al._ , “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” _Nature photonics_ , vol. 12, no. 9, pp. 534–539, 2018.
  * [60] C. Taballione, R. van der Meer, H. J. Snijders, P. Hooijschuur, J. P. Epping, M. de Goede, B. Kassenberg, P. Venderbosch, C. Toebes, H. van den Vlekkert _et al._ , “A universal fully reconfigurable 12-mode quantum photonic processor,” _Materials for Quantum Technology_ , vol. 1, no. 3, p. 035002, 2021.
  * [61] L.-T. Feng, M. Zhang, X. Xiong, D. Liu, Y.-J. Cheng, F.-M. Jing, X.-Z. Qi, Y. Chen, D.-Y. He, G.-P. Guo _et al._ , “Transverse mode-encoded quantum gate on a silicon photonic chip,” _Physical Review Letters_ , vol. 128, no. 6, p. 060501, 2022.
  * [62] F. Shokraneh, S. Geoffroy-Gagnon, and O. Liboiron-Laouceur, “High-performance programmable mzi-based optical processors,” in _Silicon Photonics for High-Performance Computing and Beyond_. CRC Press, 2021, pp. 335–365.

|  Kaveh (Hassan) Rahbardar Mojaver (M’17) received the B.S. and M.S. degrees
from Amirkabir University of Technology (Tehran Polytechnic) in 2009 and 2011,
respectively, and the Ph.D. degree from Concordia University in 2018, all in
electrical engineering. From 2018 to 2023, he served as a postdoctoral
researcher at the Photonic DataCom team at McGill University. In 2023, he
joined Colorado State University as an assistant professor. His research
focuses on photonic integration for optical computing, quantum computing, and
data communications. From 2014 to 2018, he was a part of the Reliable Electron
Device Laboratory at Concordia University, where he conducted research on
micro-fabrication and physics-based modeling of III-nitride heterojunction
field-effect transistors. From 2018 to 2023, as a postdoctoral researcher, Dr.
Mojaver conducted research on photonic integrated circuits, mainly focusing on
silicon photonics for optical communications and computations. Dr. Mojaver’s
research works have been published in several journal and conference
publications. He has received the Fonds de Recherche du Québec – Nature et
technologies (FRQNT) postdoctoral scholarship, Concordia Merit scholarship,
Concordia Accelerator Award, and STARaCom scholarship.
---|---
|  S. Mohammad Reza Safaee (M’22) received both his B.S. and M.S. degrees from
the Department of Electrical Engineering at Iran University of Science and
Technology, Tehran, Iran, in 2013 and 2015, respectively, with a major in
electronics. He is currently a Ph.D. candidate at McGill University,
affiliated with the Photonics Systems Group within the Department of
Electrical and Computer Engineering, located in Montreal, Quebec, Canada. His
ongoing research primarily focuses on developing building blocks that enable
on-chip optical computing applications, programmable photonics, and photonic
digital to analog converters.
---|---
|  Sunami Sajjanam Morrison (M’19) received her B.Eng in Electronics and
Communication Engineering at Anna University, Chennai, India, in 2014, she
then went on to obtain her M.S in Electrical and Computer Engineering with
Engineering Practice from Boston University in 2016 focusing on photonics. She
is currently a Ph.D. Candidate in the Electrical and Computer Engineering
Department at McGill University. She is a part of the Photonics Systems Group,
and her speciality lies in the development and optimisation of high-speed
silicon photonic based transmitters for short-reach communication.
---|---
|  Odile Liboiron-Ladouceur (M’95, SM’14) received the B.Eng. degree in
electrical engineering from McGill University, Montreal, QC, Canada, in 1999,
and the M.S. and Ph.D. degrees in electrical engineering from Columbia
University, New York, NY, USA, in 2003 and 2007, respectively. She is
currently an Associate Professor with the Department of Electrical and
Computer Engineering, McGill University. She was the graduate program director
of the department (2017-2023). She was an associate editor of the IEEE
Photonics Letter (2009–2016) and was on the IEEE Photonics Society Board of
Governance (2016–2018). She holds seven granted U.S. patents, and two
provisional patents, and coauthored over 100 peer-reviewed journal papers and
150 papers in conference proceedings. She gave over 30 invited talks on the
topic of photonics for computing, photonic computational design methodologies,
and photonic-electronic co-design. She is the 2018 recipient of McGill
Principal’s Prize for Outstanding Emerging Researcher and the 2023 William and
Rhea Seath award for engineering innovation.
---|---


* [372] S. Schochet.  Symmetric hyperbolic systems with a large parameter.  Comm. Partial Differential Equations, 11(15):1627–1651, 1986.
  * [373] S. Schochet.  The mathematical theory of low Mach number flows.  M2AN Math. Model. Numer. Anal., 39(3):441–458, 2005.
  * [374] M. E. Schonbek.  Existence of solutions for the Boussinesq system of equations.  J. Differential Equations, 42(3):325–352, 1981.
  * [375] L. W. Schwartz.  Computer extension and analytic continuation of Stokes’ expansion for gravity waves.  J. Fluid Mech., 62(3):553–578, 1974.
  * [376] F. J. Seabra-Santos, D. P. Renouard, and A. M. Temperville.  Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle.  J. Fluid Mech., 176:117–134, 3 1987.
  * [377] F. Serre.  Contribution à l’étude des écoulements permanents et variables dans les canaux.  La Houille Blanche, (6):830–872, 1953.
  * [378] J. Shatah and C. Zeng.  A priori estimates for fluid interface problems.  Comm. Pure Appl. Math., 61(6):848–876, 2008.
  * [379] J. Shatah and C. Zeng.  Local well-posedness for fluid interface problems.  Arch. Ration. Mech. Anal., 199(2):653–705, 2011.
  * [380] T. G. Shepherd.  Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics.  Advances in Geophysics, 32:287–338, 1990.
  * [381] S. Shkoller and T. C. Sideris.  Global existence of near-affine solutions to the compressible Euler equations.  Arch. Ration. Mech. Anal., 234(1):115–180, 2019.
  * [382] B. Simon.  Sturm oscillation and comparison theorems.  In Sturm-Liouville theory, pages 29–43. Birkhäuser, Basel, 2005.
  * [383] A. Stefanov and J. D. Wright.  Small amplitude traveling waves in the full-dispersion Whitham equation.  J. Dynam. Differential Equations, 32(1):85–99, 2020.
  * [384] E. M. Stein.  Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 1993.  With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.
  * [385] A. L. Stewart and P. J. Dellar.  Multilayer shallow water equations with complete coriolis force. part 3\. hyperbolicity and stability under shear.  J. Fluid Mech., 723:289–317, 5 2013.
  * [386] J. J. Stoker.  The formation of breakers and bores. The theory of nonlinear wave propagation in shallow water and open channels.  Communications on Appl. Math., 1:1–87, 1948.
  * [387] G. G. Stokes.  On the theory of oscillatory waves.  Trans. Cambridge Philos. Soc., 8:441–455, 1847.
  * [388] D. J. Struik.  Détermination rigoureuse des ondes irrotationelles périodiques dans un canal à profondeur finie.  Math. Ann., 95(1):595–634, 1926.
  * [389] C. Sturm.  Mémoire sur les équations différentielles linéaires du second ordre.  J. Math. Pures Appl. (1), 1:106–186, 1834.
  * [390] C. H. Su and C. S. Gardner.  Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation.  J. Mathematical Phys., 10:536–539, 1969.
  * [391] T. Tao.  Nonlinear dispersive equations, volume 106 of CBMS Regional Conference Series in Mathematics.  Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.  Local and global analysis.
  * [392] T. Tao.  Incompressible fluid equations, 2018.  Blog posts available at https://terrytao.wordpress.com/category/teaching/254a-incompressible-fluid-equations/.
  * [393] M. E. Taylor.  Partial differential equations. III Nonlinear equations, volume 117 of Applied Mathematical Sciences.  Springer-Verlag, New York, 1997.
  * [394] J. F. Toland.  On the existence of a wave of greatest height and Stokes’s conjecture.  Proc. Roy. Soc. London Ser. A, 363(1715):469–485, 1978.
  * [395] L. N. Trefethen.  Spectral methods in MATLAB, volume 10 of Software, Environments, and Tools.  Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
  * [396] T. Truong, E. Wahlén, and M. H. Wheeler.  Global bifurcation of solitary waves for the Whitham equation.  Math. Ann., 2021.
  * [397] S. Ukai.  The incompressible limit and the initial layer of the compressible Euler equation.  J. Math. Kyoto Univ., 26(2):323–331, 1986.
  * [398] R. M. Vargas-Magaña and P. Panayotaros.  A Whitham-Boussinesq long-wave model for variable topography.  Wave Motion, 65:156–174, 2016.
  * [399] R. M. Vargas-Magaña, P. Panayotaros, and A. A. Minzoni.  Linear modes for channels of constant cross-section and approximate Dirichlet–Neumann operators.  Water Waves, 1:343–370, 2019.
  * [400] C. Viotti, D. Dutykh, and F. Dias.  The conformal-mapping method for surface gravity waves in the presence of variable bathymetry and mean current.  Procedia IUTAM, 11:110–118, 2014.
  * [401] F. d. M. Viríssimo and P. A. Milewski.  Three-layer flows in the shallow water limit.  Stud. Appl. Math., 142(4):487–512, 2019.
  * [402] F. d. M. Viríssimo and P. A. Milewski.  Nonlinear stability of two-layer shallow water flows with a free surface.  Proc. A., 476(2236):20190594, 20, 2020.
  * [403] Y. Wang.  Well-posedness to the cauchy problem of a fully dispersive boussinesq system.  J. Dynam. Differential Equations, pages 1–12, 2020.
  * [404] G. Wei, J. T. Kirby, S. T. Grilli, and R. Subramanya.  A fully nonlinear Boussinesq model for surface waves. I. Highly nonlinear unsteady waves.  J. Fluid Mech., 294:71–92, 1995.
  * [405] B. J. West, K. A. Brueckner, R. S. Janda, D. M. Milder, and R. L. Milton.  A new numerical method for surface hydrodynamics.  J. Geophys. Res., 92:11803–11824, 1987.
  * [406] G. B. Whitham.  Variational methods and applications to water waves.  Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 299, 06 1967.
  * [407] G. B. Whitham.  Linear and nonlinear waves.  Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1999.  Reprint of the 1974 original, A Wiley-Interscience Publication.
  * [408] J. Wilkening and V. Vasan.  Comparison of five methods of computing the Dirichlet-Neumann operator for the water wave problem.  In Nonlinear wave equations: analytic and computational techniques, volume 635 of Contemp. Math., pages 175–210. Amer. Math. Soc., Providence, RI, 2015.
  * [409] S. Wu.  The quartic integrability and long time existence of steep water waves in 2D.  arXiv preprint:2010.09117.
  * [410] S. Wu.  Well-posedness in Sobolev spaces of the full water wave problem in $2$-D.  Invent. Math., 130(1):39–72, 1997.
  * [411] S. Wu.  Mathematical analysis of vortex sheets.  Comm. Pure Appl. Math., 59(8):1065–1206, 2006.
  * [412] T. Y. Wu.  A unified theory for modeling water waves.  Adv. in Appl. Mech., 37:1–88, 2001.
  * [413] Z. Xu.  Asymptotic analysis and numerical analysis of the Benjamin-Ono equation.  PhD thesis, University of Michigan, 2010.
  * [414] Y. J. Yang, Y. C. Fang, T. Y. Tang, and S. R. Ramp.  Convex and concave types of second baroclinic mode internal solitary waves.  Nonlin. Processes Geophys., 17(6):605–614, 2010.
  * [415] M. L. Yates and M. Benoit.  Accuracy and efficiency of two numerical methods of solving the potential flow problem for highly nonlinear and dispersive water waves.  Internat. J. Numer. Methods Fluids, 77(10):616–640, 2015.
  * [416] H. Yosihara.  Gravity waves on the free surface of an incompressible perfect fluid of finite depth.  Publ. Res. Inst. Math. Sci., 18(1):49–96, 1982.
  * [417] J. Yu and L. N. Howard.  Exact Floquet theory for waves over arbitrary periodic topographies.  J. Fluid Mech., 712:451–470, 2012.
  * [418] V. E. Zakharov.  Stability of periodic waves of finite amplitude on the surface of a deep fluid.  J. Appl. Mech. Tech. Phys., 9:190–194, 1968.
  * [419] V. E. Zakharov.  Benney equations and quasiclassical approximation in the inverse problem method.  Funktsional. Anal. i Prilozhen., 14(2):15–24, 1980.
  * [420] V. E. Zakharov, A. I. Dyachenko, and O. A. Vasilyev.  New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface.  Eur. J. Mech. B Fluids, 21(3):283–291, 2002.
  * [421] Z. L. Zou, K. Z. Fang, and Z. B. Liu.  Inter-comparisons of different forms of higher-order Boussinesq equations.  In Q. Ma, editor, Advances in Numerical Simulation of Nonlinear Water Waves, chapter 8, pages 287–323. World Scientific, 2010.