Bernard J. Giron Castro: E-mail<EMAIL_ADDRESS>
#  Memory capacity analysis of time-delay reservoir computing based on silicon
microring resonator nonlinearities

Bernard J. Giron Castro  DTU Electro, Technical University of Denmark, Ørsteds
Plads, 2800 Kgs. Lyngby, Denmark  Christophe Peucheret  Univ Rennes, CNRS,
UMR6082 - FOTON, 22305 Lannion, France  Francesco Da Ros  DTU Electro,
Technical University of Denmark, Ørsteds Plads, 2800 Kgs. Lyngby, Denmark

###### Abstract

Silicon microring resonators (MRRs) have shown strong potential in acting as
the nonlinear nodes of photonic reservoir computing (RC) schemes. By using
nonlinearities within a silicon MRR, such as the ones caused by free-carrier
dispersion (FCD) and thermo-optic (TO) effects, it is possible to map the
input data of the RC to a higher dimensional space. Furthermore, by adding an
external waveguide between the through and add ports of the MRR, it is
possible to implement a time-delay RC (TDRC) with enhanced memory. The input
from the through port is fed back into the add port of the ring with the delay
applied by the external waveguide effectively adding memory. In a TDRC, the
nodes (virtual) are multiplexed in time, and their respective time evolutions
are detected at the drop port. The performance of MRR-based TDRC is highly
dependent on the amount of nonlinearity in the MRR. The nonlinear effects, in
turn, are dependent on the physical properties of the MRR as they determine
the lifetime of the effects. Another factor to take into account is the
stability of the MRR response, as strong time-domain discontinuities at the
drop port are known to emerge from FCD nonlinearities due to self-pulsing
(high nonlinear behaviour). However, quantifying the right amount of
nonlinearity that RC needs for a certain task in order to achieve optimum
performance is challenging. Therefore, further analysis is required to fully
understand the nonlinear dynamics of this TDRC setup. Here, we quantify the
nonlinear and linear memory capacity of the previously described microring-
based TDRC scheme, as a function of the time constants of the generated
carriers and the thermal of the TO effects. We analyze the properties of the
TDRC dynamics that generate the parameter space, in terms of input signal
power and frequency detuning range, over which conventional RC tasks can be
satisfactorily performed by the TDRC scheme.

###### keywords:

Reservoir computing, microring resonator, photonic neural network, memory
capacity

##  1 Introduction

The current rise of artificial intelligence and neural network algorithms has
demonstrated their potential applications in every scientific discipline and
everyday life. Current neural network algorithms, while powerful, require a
large number of computing resources for their often complex training
procedures[1, 2, 3]. Therefore, more powerful and sophisticated computing
architectures with lower power consumption are currently under research.
Neuromorphic computing promises to push forward the current boundaries of
conventional computing. Some of its potential features are massive parallel
computing, collocation of the memory, and the computing processor which avoids
the so-called Von Neumann bottleneck that limits the achievable throughput of
current computing hardware [1, 2, 3].

Under the described context, we focus on reservoir computing (RC). RC is a
type of neuromorphic computing approach, based on recurrent neural networks,
which is characterized by the capability of solving complex computational
tasks while requiring a relatively simple and fast training process of the
output layer [1, 2, 3]. The data at the input layer is mapped to a higher
dimensional reservoir state through nodes interconnected with random and fixed
weights. The nodes in the reservoir layer provide the nonlinear dynamics and
complexity required to accurately address computational tasks. However, if the
influence of the dynamics is too strong, it might deter any influence of the
input in the reservoir state. The effect of the current and past input on the
reservoir response at a given instant is essential in computing tasks that
require memory. Nevertheless, the buffering of past inputs must be finite, as
otherwise, it would considerably increase the computing load of RC. Hence,
such conditions set two of the main features of a functional RC: First, a
certain degree of nonlinearity so that the input data is projected into a
high-dimensional reservoir state following a nonlinear temporal expansion.
This expansion allows RC to linearly differentiate the RC states to address
them through linear regression at the output layer. Secondly, the fading
memory[4], by which the input signals of the reservoir keep propagating over a
finite time through the nodes of the reservoir. Consequently, a temporary
dependence of the current RC state on previous inputs is formed. Other
factors, such as the number of nodes, the impact of neuron biases, and the
sparsity are also important in RC, although they are not the main focus of
this study [3].

Among the approaches to physical RC, photonics has demonstrated some
interesting implementations through a variety of different photonics devices,
such as Mach-Zehnder modulators in a optoelectronic loop[5], laser sources [6,
7, 8], semiconductor optical amplifiers[9] and microring resonators [10], etc.
The dynamics of those devices have demonstrated the potential to provide the
aforementioned benefits of neuromorphic computing. Some of these works
minimize the number of nonlinear nodes required by the scheme using time
multiplexing of a single physical nonlinear node in a method called time-delay
RC (TDRC). Each virtual node is processed through the nonlinear node, one at a
time, and the RC output is determined after all the virtual nodes are sampled.
Several photonic TDRC implementations are found in the literature[9, 5, 11].
MRR-based TDRC schemes have achieved high performance in diverse time-series
prediction tasks in previous studies [10, 12]. The conventional approach of
the scheme, further detailed in Section 2, uses an external waveguide
connected between the through and add ports of an add-drop MRR to provide
delayed feedback of the input and enhance the memory of the system [10].

The required nonlinear dynamics are triggered in the MRR by two-photon
absorption (TPA). Due to TPA, an excess of free carriers emerges in the MRR,
which in turn produces free-carrier dispersion (FCD). The free carrier
absorption within the cavity generates heat. Both the resulting FCD and the
thermo-optic (TO) effect change the refractive index of the silicon MRR
waveguide. This causes a nonlinear shift of the cold cavity resonance where
FCD causes a blue shift and the TO effect a red shift, leading to opposite
shifts of the resonance and an oscillatory self-pulsing of the cavity optical
modal amplitude. [10, 13]. However, FCD and the TO effect usually differ in
the order of magnitude of their timescales. The timescale of FCD is determined
by the free carrier lifetime, $\tau_{\textrm{FC}}$ while the timescale of the
thermal effects is related to the thermal diffusion time constant
$\tau_{\textrm{th}}$. It is better to work with optical pulses modulated by a
sequence with a symbol rate closer to the lowest of the two timescales, i.e.,
$\tau_{\textrm{FC}}$ to increase the TDRC processing speed [10]. Recently, we
studied the effects of varying $\tau_{\textrm{FC}}$, $\tau_{\textrm{th}}$, as
well as the waveguide losses, on the performance of MRR-based TDRC [12]. The
results showed a high correlation, in terms of the timescales of the nonlinear
processes, between the prediction performance and the transitions from linear
to nonlinear regimes of the MRR. The regime of low levels of nonlinearity and
the regime of self-pulsing in the MRR were both found to be detrimental to the
performance of the RC while the best performance was found in a narrow regime
between them. Nevertheless, such findings were limited to the NARMA-10 task.

Here, we numerically deepen the analysis of the memory and nonlinearity
properties of silicon MRR-based TDRC and relate it to the insights obtained
for the NARMA-10 task [12]. We also focus on the role that both nonlinearity
and memory play in this scheme when addressing a variety of RC conventional
computing tasks with different applications. In this way, a better
understanding of the properties of MRR-based TDRC can be achieved, which
allows further generalization of the insights obtained in the previous
study[12], to other tasks. The manuscript is structured as follows: Section 2
describes the working mechanism of the TDRC scheme analyzed in this work as
well as relevant insights from previous studies. Section 3 outlines the memory
capacity metrics used in this article. Additionally, we analyze the existence
in this scheme of the (potentially) universal trade-off between memory and
nonlinearity as previously investigated in the literature and for the studied
TDRC. The results of the study are presented in Section 4 and discussed in
further detail in Section 5. Section 6 summarizes the main conclusions of this
work.

##  2 Silicon MRR-based TDRC

The system under investigation is depicted in Fig. 1. The input data sequence
$u(n)$, at a 1.0 GBd symbol rate, is multiplied by a masking sequence $m(n)$,
which is generated from a uniform distribution in the range [0.0, 1.0]. In the
masked input signal, $X(n)$, we set the random and fixed weights of each
virtual node. Therefore, the mask sequence length must be equal to the desired
number of nodes, $N$. Throughout this work, $N=50$. An optimized bias $\beta$
is added to the masked input, forming $\hat{X}(n)$. The electric signal is
linearly modulated on a continuous wave signal generated by a laser source.
The optical signal has an average power $\overline{P}_{\textrm{in}}$ and an
angular frequency $\omega_{\textrm{p}}$. The value of $\omega_{\textrm{p}}$ is
detuned by $\Delta\omega$ from the resonance frequency of the MRR,
$\omega_{\textrm{0}}$. A root square function accounts for the optoelectronic
transformation performed in the simulation as the model is based on the
electric field at the input port. Hence, $X_{\textrm{in}}(n)$ denotes the
resulting linearly modulated optical power at the input port. As
aforementioned, the through and add ports of the MRR are linked by an external
waveguide which adds a delay of $\tau_{\textrm{d}}=0.5$ ns to the optical
signal that propagates to the add port. The linear combination of the electric
fields coupling from the cavity and the add port is detected by a photodiode.
The output electrical signal is post-processed to sample the virtual nodes and
train the weights of the output layer by means of ridge regression. The
prediction of the TDRC is averaged over 10 different seeds used to generate
the original input sequences of each addressed computing task. Comprehensive
details regarding the simulation model of the system as well as the optical
parameters used for the simulations are presented in our aforementioned study
[12].

Figure 1: MRR-based TDRC setup scheme under investigation.

##  3 Information Processing Capacity in RC

It is important to introduce some useful memory-related metrics to quantify
the memory capacity of the studied system. On the other hand, memory is not
always a very relevant feature for some computational tasks and may even be
detrimental if it counters the nonlinear features of the reservoir.
Furthermore, the performance of memory-demanding tasks can be compromised if
the nonlinearity of the reservoir is too high. Because of this trade-off
between nonlinearity and memory [14] it is important to also introduce a
relevant metric for the nonlinearity of this type of photonic TDRC. This
allows us to establish a clearer picture of the balance between the memory and
nonlinear features.

###  3.1 Linear, nonlinear and total memory capacity

The fading memory of a reservoir entails that it cannot solve unbounded-time
memory tasks and therefore, the number of previous inputs in the past that it
can accurately learn is limited. Thus, it is possible to quantify the memory
capacity of RC within a finite upper bounded range. The linear memory
capacity, as traditionally defined[4], is a measurement of the ability of RC
schemes to reconstruct a target function defined by previous inputs of the RC.
In other words, the task of the RC is to reconstruct the input (usually
uniformly distributed) $k$ steps in the past, $y_{k}(n)=u(n-k)$. The capacity
for a specific value of $k$ is then[4]:

$C[y_{k}]=1-\textrm{NMSE}[y_{k}],$ (1)

$\textrm{NMSE}=\frac{1}{L_{\textrm{data}}}\frac{\sum_{n}\left(\hat{y}(n)-y(n)\right)^{2}}{\sigma_{y}^{2}},$
(2)

where NMSE is the normalized mean square error, $\hat{y}(n)$ and $y(n)$ are
the estimated and target sequences, respectively, with a length
$L_{\textrm{data}}$. $\sigma_{y}^{2}$ is the standard deviation of $y(n)$. It
has been demonstrated[15] that the maximum number ($k_{\textrm{max}}$) of
significant $k$ steps contributing to the memory capacity, is upper-bounded by
the number of nodes. Hence, for this work $k_{\textrm{max}}=50$. Subsequently,
the total linear memory capacity is defined as the sum of the capacities over
all values of $k\leq k_{\textrm{max}}$:

$C_{\textrm{lin}}=\sum_{k}^{k_{\textrm{max}}}{C[y_{k}]}.$ (3)

A generalization of the memory capacity for higher-order functions is possible
by considering the reconstruction of a set of basis functions in the Hilbert
space of fading memory functions[15, 16]. Sometimes referred to as nonlinear
memory capacity, it is determined by considering a basis out of finite
products of higher-order Legendre polynomials. For example, the second-order
memory capacity, referred to as quadratic memory capacity[9], is reconstructed
using the second-order Legendre polynomial as the target function:
$y_{k}(n)=3u^{2}(n-k)-1$. The basis function of the $i^{\textrm{th}}$-order is
denominated $\mathbb{P}_{i}$, so that $y_{i}(n)=\mathbb{P}_{i}$. Thus, the
nonlinear capacities can be expressed in a general expression as:

$C_{i}=\sum_{k}^{k_{\textrm{max}}}{1-\textrm{NMSE}[\mathbb{P}_{i},\hat{y}_{i}(n)]}.$
(4)

Then, the total memory capacity $MC$, also referred to as information
processing capacity, [4, 16] is the sum of all capacities of the
$i^{\textrm{th}}$ order that are calculated up to the highest order of
function considered, $M$:

$MC=\sum_{i}^{M}{C_{i}}.$ (5)

###  3.2 Nonlinearity metric of the MRR-based TDRC

The coefficient of determination $R^{2}$ was used in the previous study of the
system[12] in an attempt to quantify the nonlinear transformation of the input
data at the drop port of the MRR before being affected by the nonlinearity
given by the photodiode. This metric provided some insight into how the output
states of the RC could be fully accounted as a linear transformation of the
input data (a value close to 1.0), if there is no direct relation between them
(value of 0.0), or somewhere in between (where the best performance was
obtained).

Here, we use a metric more closely related to the physics involved in the
dynamics of the MRR by quantifying the frequency detuning of the resonance
resulting from the nonlinear effects occurring in the MRR, named
$\delta_{\textrm{NL}}(t)$. This term is hence, the sum of the frequency
detuning due to FCD and the one due to the TO effect[12]:

$\delta_{\textrm{NL}}(t)=\frac{1}{2\pi}\left[\Delta\omega_{\Delta
N}(t)+\Delta\omega_{\Delta T}(t)\right].$ (6)

Nevertheless, the value of $\delta_{\textrm{NL}}(t)$ is dependent on the
instant value of the data sequence. Consequently, we calculate the standard
deviation of $\delta_{\textrm{NL}}(t)$, i.e.
$\sigma(\delta_{\textrm{NL}}(t))$, over the total length of the sequence. This
gives a clearer insight into the strength of the nonlinear effects considered
in the MRR for the processed data sequence.

###  3.3 Relation between memory capacity, nonlinearity, and task performance

The values of each order of capacity and $MC$ give us a general idea of the
memory capabilities of a particular RC scheme. Nonetheless, the translation of
memory capacity to task performance is not straightforward. In principle,
memory might not be the main feature required by RC to accurately address a
specific task. Other tasks may be more reliant on the nonlinearity and
dimensionality of the scheme (kernel quality[3]). Previous works[14, 15] have
demonstrated an apparent universal trade-off between nonlinearity and memory
capacity in dynamical systems that require nonlinear transformations of the
input to achieve a dimensionality expansion. Therefore, the memory capability
of RC is ultimately limited by the amount of nonlinearity required in a
specific scheme and task. Additionally, it has been shown that in some RC
implementations, the total memory capacity might be poorly correlated with the
performance of a specific task [16], which is an important insight to take
into account in this analysis. Furthermore, different orders of memory
capacity can have different computing significance when addressing a
particular task, depending on the mathematical definition of each task. All
around, it is evident the many factors to consider when analyzing the memory
capabilities of the system under investigation.

##  4 Results

###  4.1 MRR-based TDRC tasks and benchmarks

First, we introduce in this subsection a set of computing tasks, that have
been previously defined and addressed with high performance in photonic TDRC
implementations, and that are considered in this work [5, 9, 11, 10]. Each of
them has different performance metrics and diverse requirements of memory and
nonlinearity. Thus, we verify if there is a common behaviour between their
performances based on the nonlinear and memory dynamics of the MRR-based TDRC.
It is important to note that each of the performance results could be further
improved by an optimization of the optical parameters of the modulated optical
signal, which is outside the main scope of this work. The reference task for
this work, based on the previous analysis of this system [12], is the
discrete-time tenth-order nonlinear auto-regressive moving average (NARMA-10).
We present the results in terms of performance metric, of varying
$\tau_{\textrm{FC}}$ in each of the implemented computing tasks for a range of
$\overline{P}_{\textrm{in}}=[-20:20]$ dBm and a $\Delta\omega/2\pi$ range of =
$\pm 300$ GHz.

####  4.1.1 NARMA-10 task

NARMA-10 is a one-step ahead time-series prediction where the computing target
is the function expressed as:

$y(n+1)=0.3y(n)+0.05y(n)\left[\sum_{i=0}^{9}y(n-i)\right]+1.5u(n-9)u(n)+0.1.$
(7)

The sequence $u(n)$ is generated from a uniform distribution over the interval
[0.0, 0.5]. The mask $m(n)$ is taken from a uniform distribution in the range
[0.0, +1.0]. The performance is measured by calculating the NMSE between the
predicted and target sequences as defined in Eq. 2. 2000 data points were used
for training and 2000 for testing. Each of the analyzed tasks uses a warm-up
sequence of 200 data points before both the training and testing sets to
mitigate the effects of the initial state of the TDRC.

Figure 2: Regions A, B, and C in terms of $\overline{P}_{\textrm{in}}$ and
$\Delta\omega/2\pi$, when solving the NARMA-10 task.

Three regions of different levels of performance were identified[12] in the
parameter space of $\overline{P}_{\textrm{in}}$ vs $\Delta\omega$ when the
MRR-based TDRC addresses the NARMA-10 task. The extension of each regime
highly depends on $\tau_{\textrm{FC}}$ and $\tau_{\textrm{th}}$. An instance
of the results showing those regions (labeled A, B, C) is shown in Fig. 2 for
$\tau_{\textrm{FC}}$ = 10 ns and ${\tau_{\textrm{th}}=50\ \textrm{ns}}$ with a
waveguide attenuation of 0.8 dB/cm. Region A corresponds to an approximately
linear regime of the MRR cavity in which the NMSE value of the prediction is
upper-bounded by the contribution of nonlinearity given by the photodiode. In
region B, an adequate level of nonlinearity is present in the system and the
best performance is achieved (NMSE $<$ 0.2). Region C is characterized by the
presence of self-pulsing, a nonlinear phenomenon in which the fast
oscillations of the optical signal highly disturb the prediction performance
of the system, making it completely unable to address the task (NMSE $>$ 1.0).
By varying $\tau_{\textrm{FC}}$ to 10 ps and 25 ns, we can observe how the
defined regions are changed. The parameter space of region B is extended while
improving the prediction performance of the system[12] (Fig. 3).

Figure 3: TDRC NMSE when solving the NARMA-10 task as a function of
$\overline{P}_{\textrm{in}}$ and $\Delta\omega/2\pi$ for a
$\tau_{\textrm{FC}}$ value of: a) 10 ps, b) 10 ns, c) 25 ns. Minima and maxima
values are displayed as extreme values of the colorbar legend.

####  4.1.2 IPIX radar task

An experimental backscattered radar signal from the ocean surface is used in
this prediction task. The signal was measured by the McMaster University IPIX
radar [17]. There are two data sets available, which refer to the low (average
wave height of 0.8 meters) and high (average wave height of 1.8 meters) sea
states. We focus on the high sea states dataset in this study. The target of
the task is the measured signal, shifted $k$ steps in the future. Therefore,
we calculate the NMSE between the prediction and the actual data of the
signal. The dataset contains the in-phase and in-quadrature outputs of the
radar demodulator. Hence, the 2-D signal is first flattened and then processed
sequentially by the system. We use 1000 samples for the training and 1000 for
the testing. We simulated the system for $k=1$ and $k=2$. The performance of
the TDRC when addressing the radar task as a function of
$\overline{P}_{\textrm{in}}$ and $\Delta\omega$ is shown in Fig. 4 for
different values of $\tau_{\textrm{FC}}$. Fig. 4 a) - c) correspond to $k=1$
and d) - f) to $k=2$. For visualization purposes, we present the results for
$k=1$ as the base-10 logarithm of the NMSE.

Figure 4: NMSE of the TDRC when solving the radar task as a function of
$\overline{P}_{\textrm{in}}$ and $\Delta\omega/2\pi$ for a
$\tau_{\textrm{FC}}$ value of: 10 ps (a, d), 10 ns (b, e) and 25 ns (c, f),
with $k=1$ (a - c) and $k=2$ (d - f).

####  4.1.3 Signal classification

Classification tasks are useful to test the nonlinear capabilities of the
system as they require a higher dimensional mapping space of the input as well
as some degree of memory. In this task, the TDRC must be able to correctly
classify square and sine waveforms. The input sequence $u(n)$ consists of
randomly ordered sequences of sine and square waves that are discretized over
12 points per period. The masking sequence $m(n)$ is uniformly distributed
within the range [0, +1]. An accurate classification of the input signal must
result in a predicted value of 1.0 if the signal is a square waveform, and a
predicted value of 0.0 if the signal is a sine waveform. We measure the
performance with the classification accuracy of the TDRC, which is calculated
by dividing the number of accurate predictions by the total number. We use a
training set of 2000 samples and a testing set of 1000 samples. The results in
terms of the accuracy of the prediction vs the variations of
$\tau_{\textrm{FC}}$ are shown in Fig. 5.

Figure 5: Accuracy of the TDRC when solving the signal classification task as
a function of $\overline{P}_{\textrm{in}}$ and $\Delta\omega/2\pi$ for a
$\tau_{\textrm{FC}}$ value of: a) 10 ps, b) 10 ns, c) 25 ns.

####  4.1.4 Wireless channel equalization

Another task of interest in RC is the capability of channel equalization in
telecommunications. In the following task, a signal has been propagated
through a wireless channel disturbed by noise. The channel is modeled as a
linear system that is also affected by a combination of second-order and
third-order nonlinear distortions due to multi-path propagation. For this
task, the original signal $d(n)$ is an independent, identically distributed
sequence of values $\\{3,-1,+1,+3\\}$. The output of the linear wireless
channel model $q(n)$ is expressed as follows:

$\begin{split}q(n)=0.08d(n+2)-0.12d(n+1)+d(n)+0.18d(n-1)-0.1d(n-2)\\\
+0.091d(n-3)-0.05d(n-4)+0.04d(n-5)+0.03d(n-6)+0.01d(n-7).\end{split}$ (8)

To generate the input sequence of the system, $u(n)$ we take the output of the
channel $q(n)$ after it has been impacted by pseudo-random additive Gaussian
noise with zero mean, which we note as $v(n)$. $u(n)$ also includes the
higher-order nonlinear distortions and it is defined as:

$u(n)=q(n)+0.036q(n)^{2}-0.011q(n)^{3}+v(n).$ (9)

In the preprocessing of the input, we introduce a bias: $u(n)+5$, before
masking the input sequence. The masking sequence $m(n)$ is generated from a
uniform distribution over the interval $[-1,+1]$. The training stage of the
TDRC minimizes the square error between the reconstructed and original
signals. At the post-processing stage, the output of the RC is approximated to
the closest symbol in $\\{3,-1,+1,+3\\}$, which generates the reconstructed
signal $\hat{y}(n)$. As performance metric, we calculate the symbol error
ratio (SER) between the original and reconstructed signals at a signal-to-
noise ratio (SNR) of 32 dB. The obtained results as a function of
$\overline{P}_{\textrm{in}}$ and $\Delta\omega$ are shown in Fig. 6 for
increasing values of $\tau_{\textrm{FC}}$ in base-10 logarithm scale. 10000
data samples are used for training and 10 different subsets of 10000 samples
for testing.

Figure 6: SER of the channel equalization task as a function of
$\overline{P}_{\textrm{in}}$ and $\Delta\omega/2\pi$ for a
$\tau_{\textrm{FC}}$ value of: a) 10 ps, ${\textrm{b})\ 10\ \textrm{ns}}$, c)
25 ns. Minimum SER = 1.7$\times 10^{-4}$ at $\tau_{\textrm{FC}}$ = 10 ps,
$\overline{P}_{\textrm{in}}$ = 20 dBm, $\Delta\omega/2\pi$ = -55 GHz.

###  4.2 Memory capacity vs free-carrier lifetime

Now, we focus on obtaining the variation of the memory capacity over the
parameter space defined in terms of $\overline{P}_{\textrm{in}}$ and
$\Delta\omega/2\pi$ as we did with the performance metric of the task while
varying the value of $\tau_{\textrm{FC}}$. It is necessary to set an upper
limit to the order of memory capacity that is going to be calculated.
Therefore, based on the mathematical expressions that define each of the
tasks, we have considered the nonlinear memory capacity up to the
$3^{\textrm{rd}}$ order to have the most significance in this study. We use
the same input sequence as for the NARMA-10 task to calculate $MC$.

Figure 7: Total memory capacity (up to $3^{\textrm{rd}}$ order) of the
simulated MRR-based TDRC as a function of $\overline{P}_{\textrm{in}}$ and
$\Delta\omega/2\pi$ for a $\tau_{\textrm{FC}}$ value of: a) 10 ps, b) 10 ns,
c) 25 ns.

###  4.3 Nonlinear detuning vs free-carrier lifetime

Similarly, we simulate the nonlinear detuning for the same range of power,
frequency detuning, and set of values of $\tau_{\textrm{FC}}$ (Fig. 8). The
results correspond to simulations addressing the NARMA-10 task. However, due
to the bias added to the input sequences, it can be assumed that the
differences in the value of $\sigma(\delta_{\textrm{NL}}(t))$ between the
different tasks are negligible. Hence, we assume
$\sigma(\delta_{\textrm{NL}}(t))$ to have the same value in every task at a
given $\overline{P}_{\textrm{in}}$ and $\Delta\omega/2\pi$. Such bias is a
requirement of the mathematical model of the system to generate a
quasimonochromatic optical input signal (small amplitude variations) [12].
Experimentally, this assumption for $\sigma(\delta_{\textrm{NL}}(t))$ would
not necessarily hold depending on the optical modulation procedure of each
task.

Figure 8: Nonlinear detuning in the silicon MRR cavity of the simulated MRR-
based TDRC as a function of $\overline{P}_{\textrm{in}}$ and
$\Delta\omega/2\pi$ for a $\tau_{\textrm{FC}}$ value of: a) 10 ps, b) 10 ns,
c) 25 ns.

###  4.4 Memory capacity vs performance and nonlinear detuning

In this subsection, we compare the task-independent metric of the memory
capacity with the performance of the NARMA-10 (0 dBm), signal classification
(0 dBm), radar (0 dBm, $k=2$), and channel equalization ($20\ \textrm{dBm}$)
tasks. The analysis is quantified in terms of $\Delta\omega$ for a single
value of $\overline{P}_{\textrm{in}}$ = 0 dBm and different values of
$\tau_{\textrm{FC}}$. This comparison in a 2-D frequency sweep offers a
clearer picture of the relation between each performance metric and the memory
capacity.

Figure 9: NARMA-10 task: Prediction NMSE (a - c), nonlinear detuning (d - f)
and total memory capacity as a function of $\Delta\omega/2\pi$ for a
$\overline{P}_{\textrm{in}}$ = 0 dBm and $\tau_{\textrm{FC}}$: 10 ps (a, d),
10 ns (b, e), and 25 ns (c, f). Figure 10: Signal classification task:
Prediction accuracy (a - c), nonlinear detuning (d - f) and total memory
capacity as a function of $\Delta\omega/2\pi$ for a
$\overline{P}_{\textrm{in}}$ = 0 dBm and $\tau_{\textrm{FC}}$: 10 ps (a, d),
10 ns (b, e), and 25 ns (c, f). Figure 11: Radar task: Prediction NMSE (a -
c), nonlinear detuning (d - f) and total memory capacity as a function of
$\Delta\omega/2\pi$ for$\overline{P}_{\textrm{in}}$ = 0 dBm, $k=2$ and
$\tau_{\textrm{FC}}$: 10 ps (a, d), 10 ns (b, e), and 25 ns (c, f). Figure 12:
Channel equalization task: Prediction SER (a - c), nonlinear detuning (d - f)
and total memory capacity as a function of $\Delta\omega/2\pi$ for
$\overline{P}_{\textrm{in}}$ = 20 dBm and $\tau_{\textrm{FC}}$: 10 ps (a, d),
10 ns (b, e), and 25 ns (c, f).

##  5 Discussion

The results corresponding to the prediction performance of each task (Fig. 3,
4, 5 and 6) demonstrate that the regions (A, B, C) of the parameter space
defined by $\Delta\omega/2\pi$ for $\overline{P}_{\textrm{in}}$, generally
persist, even among tasks of different computing nature and applications.
Region C achieves the highest level of nonlinear detuning as shown in Fig. 8.
This detuning is sufficient to set the resonance of the MRR too far from the
optical signal frequency, pushing the cavity into a self-pulsing behaviour,
out of resonance. As the nonlinear detuning for a specific value of
$\Delta\omega/2\pi$ and $\overline{P}_{\textrm{in}}$ remains constant from
task to task, it is not surprising that the self-pulsing behaviour is located
in the same parameter space for every task. Hence, the large performance
penalty of region C is distributed over the same range of $\Delta\omega/2\pi$
and $\overline{P}_{\textrm{in}}$ for every task and represents the lowest
performance (Fig. 3, 4, 5 and 6) and the lowest memory capacity (Fig. 7)
achieved. Self-pulsing harms heavily the memory due to its characteristic
disrupting behaviour which prevents the TDRC from maintaining consistency at
the training[12]. Likewise, all the tasks converge to the best performance
value in a fully linear regime as $|\Delta\omega/2\pi|$ increases while
$\sigma(\delta_{\textrm{NL}}(t))$ decreases towards the order of tens of kHz
(Fig. 8). In other words, the MRR operates in region A. As aforementioned,
this performance of the TDRC in region A is achievable thanks to the intrinsic
nonlinearity of the system at photodetection. Region A also benefits from a
moderate memory capacity as shown in Fig. 7.

Nevertheless, some important features of this system are highlighted by the
difference between the tasks in the extension of region B (best performance of
the system) over the parameter space. We take the results of the NARMA-10 task
(Fig. 3) as a reference for comparison. For the NARMA-10 task, region B is
characterized by a moderate level of nonlinear detuning, in the order of
hundreds of MHz (Fig. 3, 8 and 10). Region B spans over both negative detuning
(dominance of the TO effect) and positive detuning (dominance of FCD). Fig. 7
shows that region B has the highest amount of memory capacity as long as self-
pulsing is not triggered [12]. However, when less memory is required, like the
radar task, the distinction between the performances of region A and B is not
as significant Fig. 4(a - c) as the memory capacity in the linear regime
appears to be enough to achieve good performance. The performance difference
between different levels of nonlinearity becomes more clear when the
prediction time horizon is increased $(k=2)$ Fig. 4 and 11(d - f), although it
is not comparable to the clear difference of performance in the NARMA-10 task.
On the other hand, the signal classification task appears to require a higher
nonlinear detuning (in the order of GHz as shown in Fig. 10) with respect to
the NARMA-10 task. This difference of one order of magnitude narrows the size
of region B over the parameter space. This requirement of a higher level of
nonlinearity is to be expected due to the classification nature of the task.

The wireless channel equalization task involves the most different
requirements of nonlinearity and memory in comparison to the rest of the
tasks. In fact, it is hard to establish a clear picture of the right
properties of the system to achieve good performance in this task. As shown in
Fig. 6, it definitely benefits from higher levels of power, unlike the other
tasks, which under certain conditions, can achieve good performance at lower
power as well. Nonetheless, the span of the parameter space of low SER is very
narrow for this system as it is limited to a small set of values of
$\Delta\omega/2\pi$ and $\overline{P}_{\textrm{in}}$. As shown in Fig. 12 the
performance of this task seems to be the best in the limited range of values
where very high nonlinear detuning (1$\times 10^{10}$ \- 1$\times 10^{14}$ Hz)
and high memory capacity are able to coexist. However, this often happens when
the MRR is on the verge of self-pulsing, which penalizes the memory capacity
if triggered, making region B considerably narrower than in the rest of the
tasks. The channel equalization and radar tasks, which involve signals
affected by noise and nonlinear distortions, appear to narrow the extension of
region B if compared to that of other tasks. Particularly, within the range
$\Delta\omega/2\pi\approx[-60,60]$ GHz with
$\overline{P}_{\textrm{in}}\approx[-20,0]$ dBm (not taking into account the
parameter space where region C is present). The nonlinear detuning appears to
be insufficient to achieve proper training of the system, even if the memory
capacity is high in that range. Consequently, the performance becomes even
worse than in the linear regime, and comparable to the lowest performance due
to self-pulsing in region C. As previously studied[16], this poor correlation
between memory capacity and performance may occur in TDRC under certain
conditions of the system and tasks.

As in our previous study[12], the variation of $\tau_{\textrm{FC}}$ provides
the potential of improving the performance of each of the tasks, just like it
was demonstrated initially with the NARMA-10 task. As the results also
demonstrate in this work (Fig. 7), the maximum memory capacity can also be
increased by decreasing $\tau_{\textrm{FC}}$, and the size of region B can be
extended by mitigating the self-pulsing, which also extends the parameter
space where high memory capacity is present. Therefore, a reconfigurable
implementation of this system in which control of the cavity free carrier
lifetime is possible e.g., with a PIN junction, in addition to the control of
input power and frequency detuning, would potentially make the photonic TDRC
capable of handling different applications.

##  6 Conclusion

Throughout this work, we have analyzed the performance, memory capacity, and
nonlinearity properties of a silicon MRR-based TDRC. The results provided
insight into the specific differences in the computing nature of each task.
These differences, determine the requirement of input power and frequency
detuning of each task within the region of adequate nonlinearity provided by
the MRR. We also have validated that the impact of the free carrier lifetime
on the defined regions is extrapolated to other tasks different from NARMA-10.
This extrapolation is particularly evident in the results regarding memory
capacity and nonlinear detuning, which are task-independent under the
established simulation conditions. Our work allows us to further generalize
the beneficial or detrimental impact of self-pulsing, nonlinear, and linear
regimes of an MRR on tasks covering a diversity of applications.

###### Acknowledgements.

This work has received funding by Vetenskapsrådet (BRAIN, grant n. 2022-04798)
and Villum Founden (OPTIC-AI VIL29334, and VI-POPCOM 54486).

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