Compiled

# Silicon microring synapses enable photonic deep learning beyond 9-bit
precision

Weipeng Zhang  Department of Electrical and Computer Engineering, Princeton
University, Princeton, NJ 08540, USA  Corresponding author:
<EMAIL_ADDRESS>Chaoran Huang  Department of Electrical and Computer
Engineering, Princeton University, Princeton, NJ 08540, USA  The Chinese
University of Hong Kong, Shatin, N.T., Hong Kong SAR, China  Hsuan-Tung Peng
Department of Electrical and Computer Engineering, Princeton University,
Princeton, NJ 08540, USA  Simon Bilodeau  Department of Electrical and
Computer Engineering, Princeton University, Princeton, NJ 08540, USA  Aashu
Jha  Department of Electrical and Computer Engineering, Princeton University,
Princeton, NJ 08540, USA  Eric Blow  Department of Electrical and Computer
Engineering, Princeton University, Princeton, NJ 08540, USA  Thomas Ferreira
de Lima  Department of Electrical and Computer Engineering, Princeton
University, Princeton, NJ 08540, USA  Now at NEC Laboratories America,
Princeton, NJ 08540, USA  Bhavin J. Shastri  Department of Physics,
Engineering Physics and Astronomy, Queen’s University, Kingston, Ontario K7L
3N6, Canada  Vector Institute, Toronto, Ontario M5G 1M1, Canada  Paul Prucnal
Department of Electrical and Computer Engineering, Princeton University,
Princeton, NJ 08540, USA

###### Abstract

Deep neural networks (DNN) consist of layers of neurons interconnected by
synaptic weights. A high bit-precision in weights is generally required to
guarantee high accuracy in many applications. Minimizing error accumulation
between layers is also essential when building large-scale networks. Recent
demonstrations of photonic neural networks are limited in bit-precision due to
crosstalk and the high sensitivity of optical components (e.g., resonators).
Here, we experimentally demonstrate a record-high precision of 9 bits with a
dithering control scheme for photonic synapses. We then numerically simulated
the impact with increased synaptic precision on a wireless signal
classification application. This work could help realize the potential of
photonic neural networks for many practical, real-world tasks. ©2022 Optica
Publishing Group. Users may use, reuse, and build upon the article, or use the
article for text or data mining, so long as such uses are for non-commercial
purposes and appropriate attribution is maintained. All other rights are
reserved.

###### doi:

http://dx.doi.org/xx.xxxx/XX.XX.XXXXXX

††journal: arxiv

##  1 Introduction

Deep neural networks (DNN) have enabled various applications, from fundamental
research in chemistry [1] and biology [2] to examples in daily life such as
financial analysis [3] and cybersecurity [4]. In DNNs, consecutive layers of
neurons (represented as vectors) are interconnected by synaptic weights
(represented as matrices). Practical implementations of DNNs often rely on
certain precision in weighting to maintain usable prediction accuracy and
scale-up network sizes [5, 6]. This is especially true with noisy input
signals [7], whose fluctuations can propagate and accumulate from one layer to
the next. The state-of-the-art DNNs implemented in electronics, like the TPU
[8], have a precision of 8 bits or higher. Emerging technologies such as
photonic or optical neural networks (PNN or ONN) are competitive [9, 10],
given their fundamental benefits in terms of higher interconnection density,
broad bandwidth, and lower energy consumption [11, 12]. Optical synapses are
elements that configure the connection strength, or weights, between two
optical neurons of consecutive layers. In Figure 1(a), which presents an
instanciation of a continuous-time recurrent neural network (CTRNN), these
would loosely correspond to the arrows between different photonic recurrent
neural network blocks (PRNNs) [11]. However, all demonstrated photonic
synapses had been limited to low precision: for example, phase-change
materials (PCM) based synapses have been reported to have 5 bits of precision
[13]; Mach-Zehnder interferometer (MZI) based coherent networks [14, 15] are
susceptible to fabrication error [16] which poses challenges for maintaining
precision when the network scales up; and photonic synapses based on microring
resonators (MRR) have so far been limited to 7 bits [17]. Low precision
undermines the inherent advantages of scalability and restricts the achievable
accuracy of optical neural networks, limiting their practical applications,
such as the classification of wireless signals.

Figure 1: (a) Diagram of a typical photonic neural network (the continuous-
time recurrent neural network, CTRNN [22]), in which, the hidden layer is
built by interconnected photonic recurrent neural network (PRNN) cells. Each
PRNN cell is realized by many microring synapses similar to the one shown in
(b). The input and output layers are convolutional neural networks (CNN) by
electronic components. (b) Schematic of two photonic synapses with dithering
control. Ref. Gen., reference signal generator. MUX, wavelength multiplexer.
BPD, balanced photodetector. LF, low frequency. HF, high frequency. The inlet
displays the build-in bias-tee of the MZM. (c) Zoomed-in micrograph of the MRR
synapse cascading four MRRs in a parallel add/drop configuration. (d) Spectral
response of the MRR synapse measured from both the THRU and DROP ports. (e)
Interposer PCB. The chip, which is wire-bonded on a DIP 24 chip carrier, is
mounted on this PCB. A multi-channel current source is integrated on the broad
and applies currents to the MRRs. The optical input and output are through the
fiber V groove on the top right.

Resonator-based approaches to optical synapses, such as MRRs [18], are
appealing due to their compactness, sensitivity, and innate wavelength-
division multiplexing (WDM) compatibility. However, this resonant nature
results in inter-channel crosstalk (from overlapping spectral filter
responses) and susceptibility to spurious environmental fluctuations (from
high sensitivity). Many methods have been proposed to improve weighting
control, including feedback methods [19] to address the high sensitivity and a
feedforward model [20] to overcome inter-channel crosstalk. For years,
however, the control precision of MRR-based photonic synapse has seemingly
plateaued at around 7 bits [21, 17].

The limited precision, from our perspective, is due to the lack of monitoring
of the entire signal path of the photonic synapse. Previous control methods
only monitor drifts of the MRR resonances and are insensitive to other
fluctuations. We observed that every component in the input signal path could
cause weight drifting and should be considered. For example, the electrical-
to-optical modulator is the component that converts the input electrical
signal onto the photonic path. If its modulation depth fluctuates (usually
caused by polarization drift or temperature drift), the amplitude of the
output signal, namely the weight, will drift even if the MRRs remain perfectly
stable. This scenario, however, is not necessarily captured by the other
feedback methods since the MRR remains on resonance. Thus, the current
performance bottleneck is caused by drifting sources other than the MRR
synapse. To compensate for the thermal drift of a single microring, a
dithering scheme [23, 24, 25, 26] was proposed to lock the microring to its
resonance frequency in a static fashion. Photonic synapses, however, require a
novel implementation of the “dithering” since more than one device needs to be
controlled, and each is dynamically tuned for weighting as opposed to being
statically stabilized. Also, other elements in synapses, besides the MRR
itself, can induce drifts in addition to the thermal drift.

Here, we develop a new dithering scheme to control multiple photonic synapses
simultaneously and experimentally achieve a record high precision of 9 bits.
This scheme introduces a dithering control signal to inputs, allowing
monitoring and stabilizing of the entire optical link comprising the photonic
synapses, from the MZM (where the RF signals enter the system) to the BPD
(where they exit the system), as shown in Figure 1(b). With this, drifts of
weighting caused by environmental changes can be dynamically tracked and
compensated along the entire link. More importantly, our scheme can also track
and compensate for the drift due to inter-channel crosstalk. Thus, accurate
weighting can be guaranteed, especially for high fan-in neurons with multi-
channel frontends (multiple MRRs). We tested this method on a two-MRR synaptic
system (i.e., a neuron with inputs) and achieved about 9 bits of precision,
which, to our knowledge, is the highest recorded bit precision in a photonic
neuromorphic architecture. We also simulated a photonic neural network for a
real-world wireless signal classification problem, and the result shows that
increasing the weighting precision from 5 bits[19] to 9 bits will increase the
classification accuracy from only 50% to over 90% in some cases. Such an
improvement makes photonics competitive against its analog electronic
counterparts, paving the way for practical photonic neural networks for real-
world tasks, including but not limited to intelligent signal processing of RF
signals [27, 28] and optical communications [29].

Figure 2: Procedure of the dithering control. Mat. matrix. The calibration
procedure consists of two steps. Step 1: Set current to zero and record the
dithering waveforms as the reference of weights equal to one. Step 2: Sweep
the current of all the MRRs, record the dithering waveforms and calculate the
weights for each MRR. This step generates a weight-current lookup curve for
each MRR. The weight control procedure performs a multi-channel binary search
to tune the MRRs to the commanded weights. The lookup curves obtained in the
calibration procedure provide the initial searching points for this weight
control procedure.

##  2 Method and Setup

MRRs are WDM-compatible tunable filters where multiple MRRs of different radii
can provide parallel weighting to lightwaves of different wavelengths.
Accordingly, MRR-based photonic synapses can be realized through the
"Broadcast-and-weight" architecture [30]. This architecture establishes
connections between a pair of neurons where input signals from each upstream
neuron are amplitude-modulated at different carrier wavelengths, then filtered
or "weighted" by MRRs, and summed optoelectronically, resulting in an optical
output signal that is sent to downstream neurons [12]. In this way, the
weighting precision of each synapse is key to the overall performance of the
photonic neural network.

A typical MRR-based synaptic system is depicted in Figure 1(b), where the
signal path starts at the Mach-Zehnder modulators (MZM) and ends at the
photodetector. The optical intensity delivered from the laser sources to the
MRR weights is constant in an ideal system. In practice, however, polarization
and temperature drifts, unstable optical alignment, or changing electrical
parasitics can cause uncompensated power fluctuation resulting in an effective
weight error. Fortunately, our dithering control scheme can address all these
non-idealities by monitoring the entire signal path, bringing about
performance improvement of photonic NNs.

As shown in Figure 1(b), we use an MRR weight bank that can process two signal
channels (see detailed description in Ref. [28]). To enable dithering control
in the system, we superimpose a dithering signal, a predefined sinusoidal wave
(generated by 33220A, Keysight) with a frequency usually much lower (< 10MHz)
and an amplitude much smaller (< 100mV) than the input signals, into each
signal path. These dithering signals can be separated with a diplexer
(ZDPLX-2150-S+, Mini-Circuits) and captured synchronously by the oscilloscope
(DPO4032, Tektronix). The oscilloscope is triggered by the sync output of the
reference signal generator to align each captured waveform of the dithering
signals. Since dithering signals share the same path with the signals being
weighted, the variations of the output dithering signals reflect the actual
weight value in real-time. Thus, an accurate weight can be continuously
monitored regardless of the input data statistics, such as sparsity, variance,
etc.

The dithering signal is introduced through the built-in bias tee of the MZMs
(see the inset of Figure 1(b)). Instead of applying pure DC voltages to the
bias ports, we dither the bias voltages at predefined frequencies while the RF
input port remained unchanged. Thus, the only additional components required
are a diplexer and a low-end oscillator. Therefore, this method can be
seamlessly integrated into many demonstrated optical synaptic setups. Scaling
this dithering technique can be done in two ways. For a small system, say less
than four microrings in total, new dithering signals of other harmonic
frequencies must be added, which can be achieved using a frequency doubler or
new generators. For a larger system (> 4 microrings), a more affordable and
efficient way is to use an RF switch. In this way, the dithering signal can be
applied to one microring at a time in a time-multiplexed manner. Both
scenarios require only one diplexer and one scope channel to separate and
record the dithering signals, which do not scale with the number of synapses
and help maintain a low power budget. We demonstrate the first strategy using
a 2 MRR synapse system. This method eliminates the need to directly sense the
MRR resonances locally via a dedicated circuit, as with the feedback methods.
We can replace the source-measure-units (SMU) with voltages/current sources
for driving MRRs, usually of lower cost and smaller footprint, as shown in
Figure 1(e). Removing sensing also eases the electrical overhead required for
large-scale MRR synapse actuation. We include more details of the PCB design
in the supplemental material.

The operational principle of this dithering control method is illustrated in
Figure 2. First, a two-step calibration process is done to obtain a weight-
current ($W\times I$) lookup curve for each channel, providing initial
searching points for the subsequent weight control process. During
calibration, the first step (refer to step 1 in Figure 2) is to tune all the
MRRs to a reference position by applying predefined fixed currents
($i_{\textrm{ref,m}}$). Meanwhile, the MZMs are sequentially dithered, and the
output waveforms (captured by the scope) compose the reference dithered
signals ($r_{m}$, for each channel $m$). Given the sampling length of $N$ and
the total number of channels $M$, the recorded reference signals can be
expressed as $r_{m}(t)$, where $t=0,\Delta T,2\Delta T,3\Delta
T,\ldots,(N-1)\Delta T$ and $m=1,2,\ldots,M$ (index of the channel). Then a
reference matrix A can be constructed using all the $r_{m}(t)$ as its column
(as shown in Eq. 1).

$\begin{split}A&=[r_{1}(t),r_{2}(t),\cdots,r_{M}(t)]^{T}\\\
&=\begin{bmatrix}r_{1}(0)&r_{1}(\Delta T)&\cdots&r_{1}((N-1)\Delta T)\\\
r_{2}(0)&r_{2}(\Delta T)&\cdots&r_{2}((N-1)\Delta T)\\\
\cdots&\cdots&\cdots&\cdots\\\ r_{M}(0)&r_{M}(\Delta
T)&\cdots&r_{M}((N-1)\Delta T)\\\ \end{bmatrix}\end{split}$ (1)

The next step (step 2 in Figure 2) is to turn dithering on for all the
channels and do a full-range multidimensional current sweep of all channels.
The scope records frames of the dithering signals at each intermediate point
in the sweep. For this procedure, each captured waveform, $S(t)=(s(0),s(\Delta
T),\ldots,s((N-1)\Delta T))$ represents the addition of all the dithering
signals. The amplitude of each dithering frequency can be decomposed using Eq.
2,

$W=(w_{1},w_{2},\cdots,w_{M})=\underbrace{A^{-1}}_{M\times
N}\times\underbrace{S}_{N\times 1}$ (2)

where $A^{-1}$ is the pseudo-inverse matrix of matrix $A$. In the vector
$W=(w_{1},w_{2},\ldots,w_{M})$, each element $w_{m}$ represents a correlation
between the weighted sum waveform and the reference dithering signals,
therefore corresponding to the real-time measurement of the weights. Note that
this sweep only needs to be carried out once regardless of the channel count
since each captured waveform can decompose the weights for all the channels.
Then, we can obtain the $W\times I$ (weight-current) lookup curves
corresponding to the applied current and the output weight for each channel,
as step 2 of the calibration procedure shown in Figure 2. In our lab setup,
the frequency of the calibration step needs to be done once a day. Whenever a
new set of weights needs to be commanded, the dithering will be turned on,
allowing the subsequent binary search to precisely tune the microrings to the
target weights. The lookup curves obtained in the previous calibration
procedure provide the starting points. The binary search iteratively adjusts
the current setpoints ($I_{set}$) of the microring according to the measured
weight outputs ($W_{get}$), as the weight control procedure shown in Figure
2). It is worth noting that binary searching is carried out with multi-
dimensional searching, which means the weights of all the channels are
measured and adjusted simultaneously without scaling search time with channel
count. The multi-dimensional searching also minimizes inter-channel crosstalk
since the applied currents account for inter-channel interference. More
details on this weight searching process can be found in section 1 of the
Supplemental Information.

Figure 3: (a) Weight comparison of the actual signal and dithering signal. (b)
Measurement of weighting precision in two MRR synapses. The synapse precision
is evaluated at weights on the grid. Each red dot represents a measured
weight, and three measurements were performed at each point on the grid. (c)
Weighting error for the trials in (b), calculated as follows ($\Delta
w=w_{meas}-w_{target}$) where $w_{meas}$, $w_{target}$ are the measured and
target weights respectively. The dashed and solid circles correspond to 8, and
9 bit-precisions, respectively.

##  3 Result

To show the effectiveness of the dithering control method, we evaluated the
agreement between the dithering signal and the actual signal. We apply a 10
MHz dithering signal atop a 120 MHz sine wave representing the synaptic
signal. After the calibration steps, we swept the current (roughly from 0.5 to
1.6 mA) applied to the MRR heater to vary the weights in equally spaced
increments within a [-1,1] interval as shown in Figure 3(a). Meanwhile, we
recorded the weights derived from the output signal waveforms (high-frequency
output of the diplexer) and the dithering signals (diplexer low-frequency
output) on two oscilloscope channels. Figure 3(a) shows the measured weights
for various applied currents. The dashed and solid curves represent the
weights (left axis) measured by the actual and dithering signals. The error
between the two weight values, calculated as the orange curve in the figure,
is within $0.5\%$ across the whole range. This confirms that this dithering
method can be used as an accurate proxy of the actual weight.

Next, the performance evaluation was extended to use two MRR synapses. Figure
3(b) shows the result on a mesh-plot. This type of mesh-estimation was used in
previous work [20, 17], in which two MRRs are tuned to equidistant grid points
of $(w_{1},w_{2}),w_{1,2}\in[-1,1]$, and the weight accuracy is evaluated at
each grid point. The standard deviation of the measured errors at each point
($\sim$0.0039) gives the equivalent bit-precision of the weight control
performance. We chose a 9x9 mesh, and for each grid point, we repeated the
weight search three times to test the repeatability of the process. Figure
3(c) shows the aggregation of the errors derived from all the tested points.
Compared with previous work, the error is significantly reduced, and a
precision of 9.0 bits is achieved. This reflects a four-fold (or 2-bit)
improvement from the previously best-reported result [21], and over 5-bit
improvement from the first reported MRR synapse [31].

Figure 4: Crosstalk suppression using multi-channel dithering control. Solid
line: weights of channel 2 obtained by multi-channel control. Dashed line:
weights of channel 2 obtained without multi-channel control.

Besides the improved accuracy, the multi-dimensional weight searching also
alleviates inter-channel crosstalk. Such crosstalk is mainly due to the
thermal interference between adjacent photonic devices [32, 33], which causes
small weight errors in adjacent MRRs. To evaluate the improvement in
crosstalk, we did a comparison test whose results are shown in Figure 4. Here,
we first tuned the weights to $(w_{1},w_{2})=(-0.5,-1.0)$, then swept the
weight of channel 1 using the dithering weight while keeping a constant
current applied to channel 2. This method corresponds to a one-dimensional
search in which only channel 1 was considered (refer to the dashed orange
curve). Because of the inter-channel crosstalk, when the current applied to
MRR 1 is changed, the applied current to MRR 2 needs to be slightly adjusted.
Without that adjustment, the weight of MRR 2 drifts by up to 8.8% ($\max
w-\min w=0.088$), as indicated by the orange curve in the figure. Next, we
performed a two-dimensional weight control at the same command weights (solid
line in Fig. 4). In contrast to the one-dimensional case, the multi-
dimensional weight control scheme always considers all channels for every
command set of weights. It thus can compensate for crosstalk as indicated by
the blue curve, reducing the error to 1.5% ($\max w-\min w=0.013$). The nature
of the multi-dimensional search keeps this performance advantage valid and
more significant as the number of MRR channels scales up.

Another benefit of the multi-dimensional search scheme is avoiding long
searching time, especially for large-scale MRR synapses, since it does not
scale linearly with the number of MRRs. For example, we compared the time it
took to find weight(s) for one MRR and two MRRs simultaneously. After
averaging 15 trials, we observed an increase of about $12\%$ for adding a new
MRR ($13s$ to $16s$), rather than the expected $100\%$ increase corresponding
to a linear increase. This advantage can be explained by the frequency
decomposition of the dithering signals: during one search frame, capturing one
waveform can effectively measure the weights of all the MRRs. Accordingly, the
currents applied to all the MRRs can be simultaneously adjusted in a single
step.

Figure 5: Simulation results of prediction accuracy vs bit-precision by a
photonic recurrent neural network on a wireless signal classification task
under SNR of 10dB, 15dB, 20dB, 25dB and 30dB.

To illustrate the impact of the increased bit resolution in photonic neural
networks, we performed a numerical simulation of a photonic NN model under
varying weight precision levels. We used PyTorch [34] (in Python) to simulate
a model of a continuous-time recurrent neural network (CTRNN) [22] (shown in
Figure 1(a)), realized by an MRR-based photonic neural network coupled with an
FPGA. This network was recently successfully employed for wireless signal
classification [35]. The task is to identify 30 identical ZigBee device
transmissions by identifying hidden signatures in their RF waveforms. As shown
in Figure 5, the classification tasks were repeated under different signal-to-
noise ratios (SNRs) and weighing precision conditions. As expected, a higher
weight precision results in higher prediction accuracy. This is particularly
true when inputs have low SNRs since the neural network demands more accurate
weighting to separate subtle signatures easily buried under background noise.
Assuming a minimum threshold of 80% accuracy, increasing precision from 7- to
9-bits can compensate for an SNR degradation from 30dB to 20dB. Given this
result, this improvement of bit-precision by this proposed dithering control
method proves to be practically effective for boosting the performance of
photonic NNs, which is especially useful for low-SNR inputs.

The supplemental material includes another numerical simulation on a
feedforward DNN, a cascading and fully-connected network with two neurons in
each layer. By increasing the network size under the conditions of different
bit-precision, the simulation explores how the weighting error (the deviation
from the expected weight because of a limited bit-precision) accumulates when
expanding the network depth under the conditions of varying bit-precision. We
find that increased bit-precision reduces the error accumulated in the
network, which allows for large networks. Specifically, we find that a 2-bit
improvement in precision (from 7 to 9 bit) enables a network of three times
more layers (from 5 to 18) by maintaining the signal-to-noise ratio (SNR=20).
This result highlights another benefit of our dithering method: the
feasibility of building large networks while maintaining signal fidelity,
fitting the needs for more applications.

##  4 Conclusion

In summary, we proposed and tested a dithering control method for photonic
synapses realized by MRRs, which can boost the accuracy to 9 bits, which is 2
bits higher than the previous best result. The increased bit precision results
in increase of prediction accuracy and enables large networks with a
reasonable cost. As aforementioned, our proof-of-principle setup uses a
benchtop dithering generator, scope, and CPU to complete the control loop.
This limits the speed by a few tens of seconds (for each weight), the
precision by 9-bit, the ease in scalability, and the overall footprint. A
solution is to use an FPGA with high-speed RF I/O coupled with a lock-in
circuit, as shown in [23], to replace these bulky instruments and carry the
dithering control algorithm.

Our dithering approach has several advantages relative to prior work: it is
scalable and can accommodate multi-channel MRR synapses, eliminates the
crosstalk issue, and speeds up the searching time (thanks to the multi-
dimensional searching strategy), requires less frequent calibration, and can
be easily incorporated into other photonic synapses. These advantages have
great potential for current and future resonator and non-resonator-based
techniques, especially in photonic NN applications. The resulting high
accuracy makes it competitive against its electronic counterparts, thus
benefiting various silicon photonics applications and narrowing the
performance gap between electronic and photonic approaches.

FundingDefense Advanced Research Projects Agency (HR-00111990049). National
Science Foundation (ECCS-1642962). The devices were fabricated at the Advanced
Micro Foundry (AMF) in Singapore through the support of CMC Microsystems. B.
J. Shastri acknowledges support from the Natural Sciences and Engineering
Research Council of Canada (NSERC). S. Bilodeau acknowledges funding from the
Fonds de recherche du Québec - Nature et technologies.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

Supplemental document See Supplement 1 for supporting content.

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