Photorefractive reservoir computing

4334 Vol. 50, No. 13 / 1 July 2025 / Optics Letters Letter Photorefractive reservoir computing Sebastian Alveteg,1,2,∗Marc Sciamanna,1Alex Fuerbach,2AND Delphine Wolfersberger1 1Chaire Photonique, LMOPS, CentraleSupelec, Metz 57070, France 2School of Mathematical and Physical Sciences, Macquarie University, New South Wales 2109, Australia *sebastian.alv[email protected] Received 28 April 2025; revised 3 June 2025; accepted 5 June 2025; posted 6 June 2025; published 24 June 2025 Reservoir computing (RC) is a machine learning (ML) framework that has gained attention in recent years as the interest in alternative computing paradigms has grown. RC allows the utilization of physical systems to solve ML tasks. We demonstrate the use of the nonlinear photorefractive reservoir computer and perform tasks requiring both nonlinearity and memory, such as chaotic time series prediction. Changing the photorefractive response by adjusting the applied field and laser power controls the characteristics of the reservoir. Optimizing the characteristics of the reservoir for performing a 10-step Mackey–Glass (MG) time series prediction, we achieve a mean square error (MSE) of 5x10−4.© 2025 Optica Publishing Group. All rights, including for text and data mining (TDM), Artificial Intelligence (AI) training, and similar technologies, are reserved. https://doi.org/10.1364/OL.564645 Interest in alternative computing paradigms has increased in recent decades following the observed saturation of rere’s Law. These alternative paradigms tend to be adapted to and excel at specific tasks. Among these is physical reservoir computing (RC), a framework particularly useful for tasks involving temporal dynamics such as chaotic time series prediction, control of system dynamics, partial differential equation computations, and signal processing [1]. RC is implemented in software using an interconnected set of recurrent neurons with fixed weights, called the reservoir, and only a single trained output layer, making the training much easier than standard recurrent neural nets (RNN) [2]. The reservoir comprises mainly of fixed nodes, which makes it possible to replace it with a physical system. Many implementations of physical RC exist in different fields of physics, to name a few within optics: lasers with feedback [3,4], silicon micro rings [5], and multi-scattering systems [6–8]. In this study, we specifically investigate using a photorefractive crystal as a reservoir for RC. When a photorefractive crystal gets exposed to a non-uniform light field, a refractive index grating is formed; this grating forms the basis of the reservoir. Prior work has been done using photorefractive crystals for machine learning and even a proof of concept using a photorefractive crystal for RC [9–11]. However, this study aims to systematically investigate how modifying the crystal’s nonlinear photorefractive response affects the reservoir computer’s performance. The performance is evaluated using three different benchmark tasks: a 10-step Mackey–Glass (MG) chaotic time series prediction, a nonlinear transformation task, and an XOR task. Conventional passive RC utilizes an interconnected reservoir, xk, updated as the input, uk, is changed, xk=f(Winuk+Wresxk−1), with a trained readout mechanism acting as the output layer yk=Woutxk.Wres and Win are the matrices mapping the reservoir and input to the next reservoir state, while Wout maps the reservoir state xkto the output yk[6]. The photorefractive reservoir can be described similarly, except that the reservoir interacts nonlinearly with the input and now evolves in time xk(t)=f∆n(xk−1,uk,in,t);tk<t<tk+td, where tdis the duration of the input into the reservoir, f∆nis the nonlinear time evolution of the reservoir, and xk−1acts as an initial condition. Furthermore, as the readout is not instantaneous, and the reservoir is continuously changing, the readout needs to take this into account, yk=Wout ∫tk+tr tkfc(x(t)).fcis the nonlinear contribution of the camera, and tris the readout duration. Generally, tdand trare independent, tr<td.Wout is the weights mapping the integrated reservoir contribution to the output ykand is determined through ridge regression, with a hyperparameter Γwhich size determines how much to punish overfitting. The photorefractive effect governs the reservoir’s, x(t), time evolution and response to the input. The photorefractive effect results from a combination of phenomena. First, photocarrier generation occurs as the crystal is exposed to light, followed by transport and subsequent recombination of the carriers, creating a space charge field. The space charge field and the electro-optic effect create a refractive index modulation [12]. This modulation, the key part of the reservoir acting as a memory and its buildup, creates the nonlinear response. Given that the input phase is encoded using a spatial light modulator (SLM) (see Fig. 1(a)), the average intensity of the laser over time will only fluctuate slightly, and we can assume that the stationary case can approximate the average refractive index modulation. In the stationary regime, the strength of the index modulation, ∆n, is governed by the following equation: ∆n(I,Eext)=1 2r33n03EextI/(Id+I), where Iis the intensity of the laser, Idthe dark and background intensity, Eext the electrical field applied to the crystal, n0the ordinary refractive index, and r33 the dominant entry from the electro-optic tensor [13]. The equation assumes that the modulation due to diffusion is negligible, which is true in our case other than when Eext =0. We directly measure the index modulation using the method first described in [13], as shown in Figs. 1(b) and 1(c). Here, the 0146-9592/25/134334-04 Journal © 2025 Optica Publishing Group Letter Vol. 50, No. 13 / 1 July 2025 / Optics Letters 4335 Fig. 1. (a) Schematic of the setup. The grayscale image shows an example of a phase mask with light gray representing a πphase shift. (b) Measured |∆n|and response times at different laser intensities. For the response time measurements, the applied field was kept at 1800 V/cm, establishing a lower bound on τr, while the |∆n|measurements were done at 1200 V/cm as the fringes become unstable at higher applied fields. (c) Measured |∆n|and response times at different applied fields. The intensity used for response time measurement was 400 mW/cm2, establishing a lower bound, while |∆n|was measured at 150 mW/cm2as the center part of the beam became hard to discern at higher intensities. (d) Images captured of the crystal output when spatial light modulator (SLM) displays the phase mask in (a). Both with and without an external electric field, the magnification is twice as large in the 0 V/cm case. SLM was replaced by a mirror to obtain an even phase profile; details are provided in Supplement 1. Depending on the direction of the external electric field, Eext, with respect to the c-axis of the crystal, the difference in the index will either be positive or negative and result in either defocusing or focusing of the incoming laser beam; see Fig. 1(d). The transient behavior will also be important when discussing the performance of the reservoir. However, the exact transient response is complicated and outside the scope of this study, as it requires the solution of the Khuktarev equations, a set of nonlinear coupled equations [14], which the beam being encoded by SLM further complicates. We instead directly measure the response time, τr, by exposing the crystal to the laser and measuring the time it takes for the output intensity to decay to e−1of the maximum output; the decay is caused by fanning. The trend for the intensity matches well what has been observed [15]; see Fig. 1(b). However, the decrease in τrwith Eext is somewhat slower than expected; see Fig. 1(c). τrdoes never go much below td, which we fix to 100 ms, meaning that the system will always remain in the transient state as we send data through the reservoir. Finally, we do not observe any major oscillations, with only a hint of some oscillations at higher applied fields and lower laser intensities, indicating that we have a high charge density in the system [16]. We use a 20 ×5×5 mm strontium niobate (SBN:61) crystal doped with 1% Ce, which is exposed to background illumination and put under an applied electric field along the c-axis, parallel to the laser’s polarization. The ambient temperature varies between 21 and 22◦C. The background illumination used is a Thorlabs OSL2 halogen lamp with an adjustable output power with a maximum of 1.4 W. The resulting dark intensity derived from fitting the function in Fig. 1(b) is 41 mW/cm2. We use a 532 nm Coherent Genesis MX 532 laser. The input is phase encoded on the laser using an HOLOEYE PLUTO SLM, and the center of the SLM encodes the laser with a uniform phaseshift ϕk=πuk,in/max(uin), as we are not using any input mask, while the rest is used as bias; see Fig. 1(a). The SLM gets updated with data every td=100 ms. The radius of the laser beam is approximately 220 µm when it enters the crystal. The far field of the laser at the output gets imaged by a camera, with the exposure time, tr, set to 5 ms for all measurements. We also use an ND filter in front of the camera to ensure that the image is not overexposed and a bandpass filter, reducing the white light captured from the background illumination. Before passing through the output layer, Wout, the image is subsampled to 100 ×100 and subsequently normalized. We configure the camera to trigger after the SLM completes the transition to the following input. As tris small compared to td, the camera only captures a small part of the response. In general, all images are first recorded, and the predictions and determination of Wout are done afterward, in an open-loop fashion [17]. We use three different tasks to investigate the properties of our reservoir. The first task is a 2-step XOR task, where the reservoir computer needs to recreate the XOR array from the input array, which was initially created by taking the XOR operation between every second bit in the input array. This task primarily tests the reservoir’s memory capacity. The second task is a chaotic time series prediction of the Mackey–Glass (MG) time series 10 steps ahead; see Fig. 2and Supplement 1 for series generation. Given the Mackey–Glass task’s nonlinear evolution and delayed nature, the task tests both memory and a nonlinear response to the previous entries. The final task is a nonlinear transformation (NL) task; the reservoir needs to transform the input, a sin(x) function, into another nonlinear function. The nonlinear function we use is a sawtooth function similar to in [18]. The reservoir should not need memory for this task, only a nonlinear response of high complexity to the current input [18]. The data for the NL and XOR tasks have 1000 entries, with 75%used for training and 4336 Vol. 50, No. 13 / 1 July 2025 / Optics Letters Letter Fig. 2. Example of the MG 10-step prediction; the applied field here was 800 V/cm and the intensity was 400 mW/cm2with an MSE of 5x10−4. the rest for testing and validation other than a buffer of 50 entries between the sets. We perform the two tasks for each physical configuration thrice and average the performance numbers. The MG data consists of 3000 time steps, with 2000 for training a buffer of 50 steps and 950 steps for validation and testing, and we run the task two times. We use accuracy to evaluate the performance of the 2-step XOR and mean square error (MSE) for the MG and NL tasks. The different fields and laser powers alter the reservoir, meaning that hyperparameter tuning of Γ needs to be performed once for each task and configuration. We erase any index modulation between each measurement through white light illumination of the crystal to ensure similar initial conditions. We perform the tasks at different configurations of applied fields and laser intensities. For each configuration, we adjust the magnification by moving the camera to compensate for differences in defocusing; see Fig. 1(a). The reservoir is trained and evaluated using two different methods. The first is the conventional one, where the first part of the data set is used for training, and the latter is used for testing and validation. The second method is that after the input has been sent through and images have been recorded, the data are randomly assigned to the training and testing data set. We use k-fold cross-validation to vary the assignment as well, with k =4 for the NL and XOR tasks and k=3 for the MG task. The random assignment should remove most of the temporal bias that may be caused inherently by the crystal and should more clearly reveal the relation between the material parameters and the performance. The removal of the bias also requires the hyperparameter Γto be re-tuned. Random sampling should not reduce the difficulty of the XOR task as the data set is random. However, for tasks with temporal dependence, such as the MG task, the difficulty could be lowered as the reservoir now has access to the part of the dynamics of the whole training set. For this reason, the latter method is only used to evaluate the trends in nonlinearity and memory in the system and is not stated as actual performance. Comparing the two methods discussed (see Fig. 3), we see a significant difference in performance between the two, primarily due to the temporal bias removed; see Supplement 1 for further detail. The difference can, in part, be due to the advantage discussed before for the MG task. The existence of temporal bias means that the reservoir is not consistent over time, and the optimal output weights at one point in time will not be the same at a later point in time. Before introducing background illumination, the difference was more significant. The added carrier concentration may reduce the effect of the thermal noise from the dark intensity; furthermore, the white light ensures a more even erasure of the gratings in the crystal, creating a more stable system [19]. In addition to the short-term temporal bias, we also observe a long-term drift in performance, which is why we see a slight performance difference when comparing Figs. 3(a) and 3(b) at the point where the applied field and laser intensity are identical. We observe an increase in performance as we increase the laser intensity for all tasks (see Fig. 3(a)), enhancing both the memory capacity and nonlinearity of the system. The trends for the MG and XOR tasks, especially when the temporal bias is eliminated, fit well with the behavior observed for the strength of the ∆n modulation and inverse of τr; see Fig. 1(b). In contrast, the performance of the NL task is poor. As the NL task relies much more on the nonlinear response of high complexity to each individual input, it is not unexpected that the behavior differs from the other tasks. The poor performance can be attributed to some extent to the early camera trigger and trbeing small compared to τr, meaning that the captured image does not contain the complete nonlinear response. However, it is possible that the reservoir does not provide an output of high enough complexity to perform the task adequately. As we increase the applied field, we observe an initial increase in performance for the NL and MG tasks (see Fig. 3(b)), before it starts saturating at higher applied fields. The saturation can largely be attributed to an increase in temporal drift in the behavior of the reservoir at higher voltages; see Supplement 1 for more details. When the temporal bias is eliminated, we recover a linear increase in performance of the MG task with a stronger correlation to the linear trend of the material parameters. However, observing the trend of the XOR tasks, it is clear that the performance degrades as the field increases and is not purely related to the temporal bias, as this remains true for higher fields when the temporal bias is eliminated. The difference in behavior between the more nonlinear tasks and the XOR task indicates that increasing the applied field increases the nonlinearity of the reservoir. The increase in nonlinearity can also explain the drop in the performance of the XOR task, as it is known that increased nonlinearity can decrease the linear memory of the reservoir [20]. By optimizing these parameters for the MG task, we are able to reach a low MSE of 5x10−4. So far, we have only discussed modifying the physical parameters. However, the parameters of the setup also have an impact. For example, by increasing tr, we generally increased the performance, and similarly, we observed that doubling the data rate increased the performance Letter Vol. 50, No. 13 / 1 July 2025 / Optics Letters 4337 Fig. 3. (a) Shows the performance trends for the different tasks at different intensity levels; the applied field used was 800 V/cm for all. (b) Performance for different applied fields; the intensity used was 400 mW/cm2for the MG and NL task, while 350 mW/cm2for the XOR task. The insets show the performance trend when we eliminate the temporal bias, and the error bars show the variance. The rectangles show the measurements that have the same applied field and laser intensity. further for the MG and XOR tasks. Nevertheless, we kept these parameters constant to have comparable results between the different physical configurations, but they could be of interest in future work. In conclusion, we have experimentally demonstrated a photorefractive reservoir computer in which memory and nonlinear properties can be modified. A link between ∆n, τr, and the performance is revealed. However, the performance is being held back by a temporal bias, which increases with the strength of the applied field. Despite this, the reservoir can perform the MG task at MSE of 5 ×10−4or normalized MSE (NMSE) of 1.6 ×10−2, which is comparable to other state-of-the-art reservoir computers [18,21]. By more deliberately designing [22] or storing part of the reservoir in the memory of the computer [7], one can reach higher performance. The current reservoir using the SBN crystal is limited to lower data rates due to the response time of the crystal, assuming one wants to keep the laser intensity lower; therefore, investigating materials with faster response times, such as InP [23], could be a potential future step assuming the stability issues can be resolved. Funding. HORIZON EUROPE Marie Sklodowska-Curie Actions (101081465). Acknowledgment. Co-funded by the European Union under the Marie Sklodowska-Curie Grant Agreement No. 101081465 (AUFRANDE). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the Research Executive Agency. Neither the European Union nor the Research Executive Agency can be held responsible for them. 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. REFERENCES 1. M. Yan, C. Huang, P. Bienstman, et al.,Nat. Commun. 15, 2056 (2024). 2. H. Jaeger and H. Haas, Science 304, 78 (2004). 3. L. Appeltant, M. Soriano, G. Van Der Sande, et al.,Nat. Commun. 2, 468 (2011). 4. J. Vatin, D. Rontani, and M. Sciamanna, Opt. 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