Photonic reservoir computing based on nonlinear wave dynamics at microscale

High-dimensional nonlinear dynamical systems, including neural networks, can be utilized as computational resources for information processing. In this sense, nonlinear wave systems are good candidates for such computational resources. Here, we propose and numerically demonstrate information processing based on nonlinear wave dynamics in microcavity lasers, i.e., optical spatiotemporal systems at microscale. A remarkable feature is its ability of high-dimensional and nonlinear mapping of input information to the wave states, enabling efficient and fast information processing at microscale. We show that the computational capability for nonlinear/memory tasks is maximized at the edge of dynamical stability. Moreover, we show that computational capability can be enhanced by applying a time-division multiplexing technique to the wave dynamics. Thus, the computational potential of the wave dynamics can sufficiently be extracted even when the number of detectors to monitor the wave states is limited. In addition, we discuss the merging of optical information processing with optical sensing, revealing a novel method for model-free sensing by using a microcavity reservoir as a sensing element. These results pave a way for on-chip photonic computing with high-dimensional dynamics and a model-free sensing method.

lasers, inspired by wave/quantum chaos, have recently been utilized to control the emission properties 21,22 . An interesting feature of such microcavity lasers is the ability to exhibit a variety of spatiotemporal wave dynamics through the interplay of a gain medium and cavity shape 23,24 . Unlike previous work, we utilize such wave dynamics in microcavity lasers driven by an input signal for RC and numerically demonstrate that RC-based information processing can efficiently be achieved at microscale owing to the spatial degrees of freedom based on the high-dimensional dynamics with a long memory effect.
In addition, we discuss the application of microcavity-based processing by using the sensitivity of wave dynamics in a microcavity in an external perturbation. We propose the use of a microcavity as a sensing element as well as a reservoir, resulting in high-dimensional mapping. The merging of optical sensing and the reservoir suggests the possibility of novel sensing without complex post processing and theoretical sensing models. As a proof-of-concept demonstration, we show fast sensing of external reflective index by using the microcavity RC. Figure 1(a) shows a schematic of the proposed system, which consists of a microcavity coupled to an input waveguide and probes (detectors) to form the RC output ŷ. The microcavity includes a nonlinear gain medium, and the cavity shape is designed as the Bunimovich stadium 25 , in which ray orbits are proven to be fully chaotic and the corresponding wave patterns are complex (Fig. 1b). An optical signal encoded with a phase modulation is injected from the input waveguide and is able to reach all parts of the cavity owing to the chaotic multiple reflections at the cavity boundary while nonlinearly amplified by the gain medium. A feature of the stadium cavity is the frequency dependence of its wave pattern, resulting from multiple reflections and the resulting wave mixing (interference) in the cavity; the speckle-like wave pattern is sensitive to the input frequency (Fig. 2a). Actually, as demonstrated in Fig. 2b, the spatial correlation between two wave patterns excited by the input with frequencies ω 0 and ω ω ω = + Δ 0 decreases as Δω increases. This means that the information can be encoded into the wave patterns with instant frequency by phase-modulating the input light. Moreover, the gain medium plays an important role in adding an additional nonlinearity and memory effect for the amplification. The emitted signals from the cavity are detected at the probe points with the sampling time interval τ s . In the simulation, N probes are assumed to be placed around the cavity.

Microcavity-based RC
For RC, we consider the linear readout , 2 , }), and w i is a readout weight. The goal of the processing is to approximate a functional relation between input signal u n ( ) and target signal y n ( ) using output ŷ. To this end, a finite set of training data = u n y n { ( ), ( )} n T 0 is utilized to determine the readout weights, such that the mean square error ∑ | − | T y n y n 1/ ( ) ( ) n 2 is minimized. In the training process, we simply use the least-squares method.

Results and Discussions
To gain an insight into the computational capability of the microcavity-based RC, the numerical simulation was performed using the Maxwell-Bloch (MB) model, in which the gain medium is modeled as a simple two-level system 26 . Although the MB model is a simple model of microcavity lasers, the dynamical lasing phenomena can qualitatively be examined 27,28 . We assumed that the cavity is two-dimensionally extended on a plane, and that the electric field is polarized perpendicular to the plane. For generality, all variables were made dimensionless (see Methods for details), and we discuss the RC capability in reference to the dimensionless variables. (a) Schematic of RC using a microcavity laser. The cavity shape is designed as the Bunimovich's stadium, known as a chaotic cavity, and the cavity is coupled to an input waveguide. The incident light encoded by the signal u is injected to the cavity. The emitted light intensities are detected by the probes, represented by the green triangles, and then used for the output ŷ. The lower figure in (a) shows the refractive index distribution of the stadium cavity coupled to an input waveguide. = . n 3 3 in and = . n 1 0 out denote the refractive indices inside and outside the cavity and waveguide, respectively. (b) An example of the intensity pattern responding to the input light, the frequency of which is tuned to a resonant frequency of the cavity ω 0 . The boundary of the cavity coupled to the waveguide is represented by the white curve. The green triangles represent the probe points to detect the intensity signals. In the simulation, the normalized pumping power ≈ ∞ W W / 1 th was set, where W th is the threshold pumping power.
For the simulations, the refractive index n in inside the cavity was set to be 3.3, and the length L of the major axis of the stadium cavity was ≈1.67λ, where λ is the wavelength of the input light in vacuum. (If λ = 0.85 μm, L would be 1.42 μm). τ s was fixed to be close to the lifetime τ c of the cavity without a gain medium.   www.nature.com/scientificreports www.nature.com/scientificreports/ required nonlinearity and memory, respectively 29 . The input signal u n ( ) is an identically distributed random sequence generated from a uniform distribution between − [ 1,1]. The goal of the task was to reproduce the nonlinearly converted signal y n ( ) with a delay of τ (see the inset of Fig. 4a). To evaluate both the ability to adapt nonlinear tasks and memory capacity of the RC, we introduced the correlation between the target signal y and output ŷ, where 〈·〉 is the mean over time step n, σ z denotes the standard deviation of = z y or ŷ. Next, the nonlinearmemory capacity was defined as the sum of τ ν NM ( ), with τ to infinity: 0 NMC ν corresponds to the linear memory capacity MC in the limit of ν → 0 30 . With the nonlinear-memory capacity ν≠ NMC 0 , one can evaluate both the nonlinearity and memory effects in microcavity lasers at the same time.
We examined the lasing dynamics in the stadium cavity and obtained NMC ν from the intensity signals detected by each probe. In the simulation, = T 1000 samples were used for the training, and the τ ν NM ( ) and NMC ν were evaluated for the T samples. Figure 4 shows the numerical results of    31 , or the so-called echo-state property 2 , which is an important condition for RC 29,32 . In the multimode lasing regime, the spontaneous multi-modal oscillations appear, leading to different results, even from the same input, depending on the initial state of the reservoir. Thus, the appearance of irreproducibility prevents consistent processing of the input information. We measured the consistency, defined as the mean correlation between the output signals starting from two different initial states. (see Method for further detail). As shown in Fig. 4c, the decrease in consistency is linked to the degrade of NMC ν . If consistency can be kept in the multimode lasing regime, better performance may be achieved through nonlinear mixing of different spatial modes.
As shown in the above-mentioned results, the effects of the gain medium, as well as the high-dimensionality of the wave states, play a crucial role in enhancing the computational capability of the RC frameworks. The compensation of the short memory, inherent in compact RC systems, and additional nonlinearity caused by the interaction with the gain medium are an advantage, compared to conventional passive photonic integrated RC 12,14,15 where nonlinearity is introduced only as part of the measurement process.
Effect of cavity shapes. The cavity shapes play a crucial role in the quality of light confinement and the wave dynamics. In the stadium cavity, the chaotic multiple reflections lead to efficient wave mixing dynamics, enabling high-dimensional mapping of the input information into complex wave patterns, as demonstrated in the previous subsection. To gain a further insight into the effect of the wave-chaotic cavity on the RC performance, we also numerically examined the laser dynamics in a non-chaotic cavity where the internal ray orbits do not exhibit chaos. For the non-chaotic cavity, we chose a circular-shaped cavity (Fig. 5a) and compared the RC performance with that obtained in the stadium-shaped lasers with the same area and same pumping power condition in a consistency regime. Figure 5b shows the performance comparison for nonlinear tasks, where τ ν NM ( ) is plotted as a function of the nonlinear parameter ν when the delay is τ = 0. Clearly, τ = ν NM ( 0) of the stadium-shaped laser can outperform that of the circular-shaped laser for all values of ν, which may partly be attributed to a strong wave mixing effect in the stadium cavity. As shown in Fig. 5c, however, for tasks requiring memory with a delay parameter τ  1, the τ ν NM ( ) of the circular-shaped laser is relatively higher than that of the stadium-shaped laser because the circular cavity has a longer cavity-lifetime (lower loss rate) 22 . These results suggest a trade-off between the cavity shapes exhibiting a long-memory effect and nonlinearity. In terms of ray-wave correspondence 33 , the effect of the cavity shape becomes more dramatic for a larger value of a size parameter defined by λ n L/ in ; it is expected to lead to a larger difference in the RC performance. The investigation concerning the aforementioned factors will be an important issue in the RC. the number of probes. In the microcavity-based RC, the number of the signals detected by the probes, N, corresponds to the number of reservoir nodes used to calculate the output ŷ, and it generally affects the capacity to approximate the target signal y t ( ). To obtain a rich variety of responses from the reservoir nodes, the probes should at least be placed at a wavelength-scale distance from each other because shorter distance between the probes results in similar responses, i.e., ≈ x ). This implies that the maximum number of the probes to be  In (b,c), the performance of the stadiumand circular-shaped lasers are compared under the same input condition, same number of probes = N 11, and same pumping condition in a single-mode lasing (consistency) regime. For nonlinear tasks, the NM ν of the stadium-shaped laser is higher than that of the circular-shaped laser, as shown in (b). However, the circularshaped laser outperforms the stadium-shaped laser for tasks requiring long memory τ  1, as shown in (c). (2019) 9:19078 | https://doi.org/10.1038/s41598-019-55247-y www.nature.com/scientificreports www.nature.com/scientificreports/ placed, N max , (effectively corresponding to the maximum number of nodes used to calculate ŷ) is limited by the cavity size and the wavelength. We roughly estimate N max as the ratio of the perimeter of the stadium cavity, π = + P L ( /2 1) , to the characteristic wavelength λ n / in inside the cavity with n in , π λ ∼ + N nL ( /2 1) / max i n , assuming that the spatial autocorrelation of the wave patterns is sufficiently small for a spatial scale larger than λ. For example, ≈ N 1, 600 max nodes can potentially be used in a 0.013 mm 2 footprint when L = 160 μm, λ = 0.85 μm, and = . n 3 3 in . We emphasize that the potential to implement such high-density and large-scale (virtual) nodes is unique to the wave dynamical RC and is not found in conventional photonic integrated RC 12 , comprised of multiple elements.
Using spatiotemporal dynamics for Rc. As mentioned above, the RC performance depends on the number of the probes N used for calculating the output ŷ. In an actual implementation, however, it may be practically difficult to place a large number of probes (or detectors) around a cavity. To overcome the problem, it should be noted that the dynamical information is included in delayed sequences, obtained from a few observables 34 . This suggests the possibility that, even when only a few observables are utilized, the dynamical information can be extracted from the dynamics of a few observables. In addition, the use of mask signals can create a rich variety of the responses from reservoir nodes 17  where w ijk is an optimal weight obtained by using the least-squares method.
An example of the time-multiplexing method for = M 10 is shown in Fig. 6a, where the input information u holds for the period T m , and the colored random signals with the period T m are used as the mask signal because the use of colored noise, or chaotic oscillation, as the mask signals will lead to a good RC performance 35 .  Fig. 6b. We find that with the time-multiplexing method of = M 10 and = K 5, ν NMC for only a single probe = N 1 can be larger than NMC ν without implementing the time-multiplexing method (Fig. 6c). The time-multiplexing method is effective in achieving high RC performance even when the number of the probes is limited. See Supplementary Information for further investigations on the RC performance obtained by the time-multiplexing method.
Sensing applications. Physical RC frameworks generally suggest that physical systems responding to input signals themselves can be utilized as information processing systems. This implies that when physical systems are perturbed by an external stimulus (for example environmental changes), the system itself can be utilized to detect the external stimulus with an appropriate training process, in addition to monitoring internal states or observing unmeasured variables 36 . Here, we consider microcavities for the detection of an environmental physical quantity www.nature.com/scientificreports www.nature.com/scientificreports/ in the RC scheme and demonstrate the identification of a refractive index n out outside the cavity, i.e., refractometric sensing.
As a simple demonstration, we consider the case when a stadium microcavity is surrounded by a medium with a refractive index n out , as shown in Fig. 7a. A randomly phase-modulated light is injected into the stadium cavity and emissions from the cavity are detected with the five probes ( = N 5). When the external refractive index n out changes, the phases of the reflection/transmission and the coupling to the probes changes. Consequently, the detected intensity x i at the probe i also changes (Fig. 7b). We use . Our purpose is to identify the surrounding refractive index n out from the output ŷ after the training of w ij to minimize −n y out . In the training process, we used 100 datasets , where n out was randomly chosen from a region of . ≤ ≤ . n 1 3 15 out . Figure 7c shows the trained output ŷ, where it is assumed that n out randomly changes in time. Clearly, ŷ follows the changes in the index n out with the error of 0.1%, even when n out rapidly changes in a timescale of τ τ ≈ s c . Consequently, n out can be identified with low errors, as shown in Fig. 7d. We remark that the memory of the reservoir (cavity) does not play an essential role in this sensing task. In this sense, the proposed learning-based sensing scheme is related to the extreme learning machine (ELM) 30,37 as well as RC.
We emphasize that the proposed method does not need any precise sensing model, high-quality microcavity, or complex post processing, unlike previous work concerning microcavity sensors 38,39 , where the shift in the resonant frequency in a microcavity due to the change of the refractive index has been measured in accordance with the transmission or reflection spectra.
The results presented in this subsection suggest that in the merging of the optical sensing and learning-based processing, a model-free detection of the external refractive index n out is achieved at a rate of τ 1/ c .

Summary
In this study, we have proposed and demonstrated RC based on nonlinear wave dynamics in a microcavity laser. An advantage of using a microcavity laser as a reservoir is that it can generate high-dimensional, nonlinear dynamics at microscale, enabling high-dimensional mapping of the input information to wave patterns after nonlinear amplification in the gain medium. We emphasize that the high-dimensional mapping to wave patterns is a spontaneous process with low energy loss, achieved by natural multiple reflections, at a short timescale, which can be of the order of the cavity lifetime τ c . This suggests the potential for photonic parallel information processing. www.nature.com/scientificreports www.nature.com/scientificreports/ Furthermore, we proposed the application of a time-multiplexing encoding technique to wave dynamics and demonstrated enhancement in computational performance. This method could be used in situations when only a few detectors are available, owing to a physical constraint and, beyond the example of the microcavity lasers, the method will be applicable to any physical RC systems with spatial degree of freedom but only a few detectors.
Lastly, we discussed the sensing application options of the microcavity-based RC, in which case the microcavity is used as a sensing element as well as a reservoir. The combination of the optical sensing and RC could be used for the model-free identification of physical quantities. These results pave a way to utilize complex wave dynamical systems at microscale for fast photonic information processing and shed light on a potential trend toward model free sensing, using the concept of RC.

Methods
The Maxwell-Bloch model in microcavity lasers. We assume that the thicknesses of microcavities are much smaller than their in-plane dimensions, and that microcavities are treated as two-dimensional objects by applying effective refractive indices n in . To describe the light-matter interaction, we used the Maxwell-Bloch (MB) model where the gain medium inside the cavity is modelled as a two-level system. The Maxwell-Bloch model is a simple model to describe the laser dynamics, but it can qualitatively reproduce lasing phenomena in two-dimensional microcavity lasers 26,27 . The normalized Maxwell-Bloch model is given by where space and time are made dimensionless by the scale transformations ω → x c x / s and ω → t t s , respectively. ω s is a reference frequency close to the transition frequency ω a of the two-level gain medium. In Eqs. (4)(5)(6), E, ρ, W are the (dimensionless) electric field, polarization field, and population inversion component in a two-level gain system, respectively. All of the parameters are also made dimensionless. = n r 2  is the relative permittivity, where the refractive index n r is n in inside the cavity and waveguide, whereas = n n r o ut outside the cavity and waveguide. σ represents the background absorption inside the cavity. Θ x y ( , ) is a step function; Θ is 1 inside the cavity and zero outside the cavity, respectively.
ω ω Δ = / a a s represents the normalized gain center. The two relaxation parameters, γ ⊥ and γ , are the transverse and longitudinal relaxation rates, respectively. ∞ W is the pumping power, representing an excitation power to the gain medium 26,27 . When = ∞ W 0, the gain medium does not essentially affect the wave dynamics.
The Maxwell Eq. (4) was simulated using the finite-difference time-domain (FDTD) method, where a perfect matched layer (PML) was introduced near the boundary of the calculation space to absorb the emission light.
In the stadium cavity, shown in Fig. 1, the radius R of the half circle and major axis length = L R 4 of the stadium were set . n 12 25/( 2 ) in and n 49/ ( 2 ) in in a unit of ω c/ 0 , where ω 0 is the input angular frequency, and c is the light velocity in vacuum. The actual L would be ~1.42 μm if = . n 3 3 in and wavelength λ = 0.85 μm. Although L is shorter than that of a standard microcavity, we restrict ourselves to the cases of short length owing to the lack of computational power. We emphasize that similar results can be essentially obtained in cases of longer length.
⊥ 0 1, γ = − 10 4 . n out was set at 1 for the results shown in Figs. 1-6, whereas in Fig. 7, n out is changed in a range of . ≤ ≤ . n 1 3 15 out . We confirmed that these parameter values do not essentially affect the RC performance. incident wave. The incident light E in is phase-modulated with the input signal u n ( ) (and mask signal when using the time-multiplexing method), and is injected into the cavity via the input waveguide shown in Fig. 1, in 0 where A is the amplitude, ψ is the lowest-order waveguide mode, φ t ( ) is the modulated phase, and ω 0 is the center frequency of the input light. ω 0 is tuned to the resonant frequency of the stadium cavity. The amplitude A is given such that the injection locking to the lasing mode with frequency ω 0 is achieved when R n 12 25/( 2 ) in and input wavelength λ = 0.85 μm, the cavity lifetime τ c is less than 0.1 ps. Thus, sampling at an interval of τ τ ≈ s c is unrealistic. However, we remark that the problem can be moderated in a large cavity because τ c can increase with increasing R 22 .
The mask signal Mask(t) is used only when the time-multiplexing method is applied, and it is given as a colored noise signal with a decay rate of τ 1/ c , which is repeated with a time interval of T m . The use of such colored noise can efficiently excite the reservoir nodes used for the RC 35 . www.nature.com/scientificreports www.nature.com/scientificreports/ estimation of parameter values toward actual experiments. In this study, we have restricted ourselves to a case concerning a small cavity of λ ≈ . L 1 67 because of the limitation of our computational power. We remark that the results in this paper have been presented in the normalized form, in which parameter values available for actual experiments can be estimated. For example, when L = 100 μm, the average mode interval ω π Δ /(2 ) i will be in the order of gigahertz 28 . The photon lifetime τ c in the stadium cavity (without a waveguide) is estimated as 28 ps and can be changed using amplification due to the gain medium. The sampling rate τ s can be set as τ c in an actual experiment. In addition, we also remark that, for a large cavity with L = 100 μm, it is relatively easy to place the probes or single-mode waveguides coupled to detectors around the cavity.
consistency. Consistency is the similarity of the response outputs for a repeated drive signal and is considered one of the important properties of RC. The consistency can be measured in the correlation between the two response outputs, obtained from different initial conditions 31 . We measured the consistency of the wave dynamics in microcavity lasers driven by the input light E in as follows: ,1 ,2 x t ( ) i j ( ) denotes the intensity signal detected by the probe i, which is obtained from an initial state j. 〈·〉 denotes the time average.