Exploring the bandwidth-limited readout in coherent photonic
reservoir computing
Mohab Abdalla1,2,∗ , Mauricio Gomes1 , Paul Jimenez1 , Raphael Cardoso1 , Clément
Zrounba1 , Guanghui Ren2 , Andreas Boes2,3 , Arnan Mitchell2 , Alberto Bosio1 , Ian
O’Connor1 , Fabio Pavanello1
1. Univ. Lyon, Ecole Centrale de Lyon, INSA Lyon, UCB Lyon, CPE Lyon, CNRS, Lyon Institute of Nanotechnology,
UMR5270, Ecully, 69130, France
2. Integrated Photonics and Applications Centre (InPAC), School of Engineering, RMIT University, Melbourne, VIC
3000, Australia
3. Institute for Photonics and Advanced Sensing (IPAS), University of Adelaide, Adelaide, SA 5005, Australia*mohab.abdalla@ec-lyon.fr

Summary. In this work, we shed light on an essential component in photonic reservoir computing:the photodetector. Through numerical investigations, we showcase the effects of its bandwidth-limited
response on the performance of a coherent time-multiplexed system on two different benchmark tasks:
the bitwise XOR and the Santa Fe time-series. Our results suggest that fast sampling of lower
bandwidth (i.e. slower) photodetectors might improve overall prediction accuracy.

Historically, reservoir computing (RC) emerged in the early 2000s to circumvent the limitations oftraining deep neural networks. With the progress in deep learning during the 2010s, the bulk of interest
in RC has been redirected towards physical implementations. Many photonic implementations have beenexplored in the literature, driven in large part by the promise of high speeds and low energy consumption.
In the first on-chip silicon photonics demonstration of RC [1], it was shown that a largely passive, linear
photonic system can be used for RC as the readout is performed anyway with a photodetector, which
provides a sufficient nonlinearity by producing an output that is proportional to the square of the electricfield. Indeed, this is mainly the case for interference-based coherent systems (i.e. operating on a singlewavelength), as the complex-valued electric fields do add up linearly, but their corresponding intensities
do not. For the simple case of interference between two beams of identical wavelengths and with opticalintensities I1 and I2 and phase difference of ∆ϕ, the resultant intensity Ires is:p(1)Ires = I1 + I2 + 2 I1 I2 cos (∆ϕ)
Considering two independent, non-interfering beams, the last term is dropped and the intensity of the sumof the two beams becomes equivalent to the sum of their individual intensities. What this entails is that,as long as there are paths for optical interference, one can indeed imply that the nonlinearity is being doneat the readout stage, due to the nonlinear dependence of the total intensity on the phase shift between thedifferent participating beams. This is particularly the case for all-optical coherent RC implementations[1, 2, 3], and also in the optical perceptron [4]. On the other hand, in the case of the optoelectronic
approach [5], the photodetector participates purely as a readout mechanism, as there are no paths in the
optical part of the system which lead to interference. Thus, the standard photodetector itself does not’perform’ any nonlinearity (a slight semantic inaccuracy sometimes in the literature), it simply emerges
from the act of probing the corresponding intensity of multiple interfering fields. Complex temporal
dynamics can be achieved by using simple passive elements and providing optical feedback, which can bedone using a variant of our recently proposed architecture in [3], as shown in Fig.1. Varying the phaseshift on one of the waveguides allows the reconfiguration of the rich temporal dynamics of the system.

Figure 1: Conceptual diagram of the system. A modulator and a photodetector (both can be either integrated or off-chip)
perform the electro-optic and optoelectronic conversions, respectively. Passive waveguides in feedback yield rich interference
patterns. The delay line can also be integrated by leveraging (for example) the low-loss lithium niobate on insulator platform.

One could thus utilize this ’free’ nonlinearity as long as:
1. The system is perturbed periodically within the timescale of its own dynamics, governed by theoptical path lengths experienced by the signal before it gets to the readout;
2. There is no requirement for many physical nodes/layers.
In the act of probing the time-averaged optical intensity, the photodetector ’imprints’ associated noiseon the signal as well as a filter-like response due to the imposed RF-bandwidth limitations from thedetector electronics. In most implementations so far, a high-speed detector is assumed beneficial as it
would capture more of the optical dynamics. In this investigation, the output is passed through different
p-i-n photodetectors (modeled with associated noise and a subsequent 4th order butterworth filter), anda fixed sampling rate of 40GSa/s so as to realize 20 time-multiplexed nodes obtained by a high-speed
oscilloscope. Furthermore, the input mask - normally used in time-delay RC - is no longer considered so
as to simplify the study. The output is used to train (using 1000 examples) a least-squares regression
model, which is then applied on a test set of another 1000 samples. The prediction accuracy is thenassessed using the bit error rate (BER) and normalized mean square error (NMSE) metrics.

Figure 2: Results for XOR with (a) two, (b) three, and (c) four bits into the past, and (d) Santa Fe time-series prediction.
The results shown in Fig 2 consistently show better performance for slower photodetectors. For thetemporal bitwise XOR task, the results for u[n] ⊕ u[n − 1] are not shown as BER=0 everywhere. InFig. 2(a) the BER remains below the accepted value using a 2 GHz photodetector. Furthermore, theBER is consistently lower for the slower detectors, as shown in (b),(c). The results for the one-stepahead Santa Fe prediction follow a similar trend. These preliminary findings suggest that engineering
the readout mechanism, which is often neglected, can provide nontrivial gains which can be as importantas engineering the complex system (reservoir) itself. In this case, it is due to the fact that we take into
account the additional timescales introduced by the photodetector’s bandwidth. Moreover, having slower
detectors allows to relax some constraints in terms of responsivity, noise, and overall design.

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