Experimental reservoir computing with diffractively coupled VCSELs

TL;DR

This work evaluates the system's memory and solves the 2-bit XOR task and the 3-bit header recognition (HR) task with bit error ratios (BERs) below 1% and the 2-bit digital-to-analog conversion (DAC) task with a root mean square error (RMSE) of 0.067.

Abstract

We present experiments on reservoir computing (RC) using a network of vertical-cavity surface-emitting lasers (VCSELs) that we diffractively couple via an external cavity. Our optical reservoir computer consists of 24 physical VCSEL nodes. We evaluate the system's memory and solve the 2-bit XOR task and the 3-bit header recognition (HR) task with bit error ratios (BERs) below 1\,\% and the 2-bit digital-to-analog conversion (DAC) task with a root-mean-square error (RMSE) of 0.067.

Figures (5)

• Figure 1: a) Scheme of the experimental setup. For full description, see text. Abbreviations not introduced in the text: PD = photodiode, att = attenuator, amp = radio-frequency amplifier, L1/L2/L3/L4 = lenses, PowM = powermeter for alignment, MO = microscope objective. b) Reservoir computing (RC) scheme. The light blue area indicates the external cavity, and the green circles represent the VCSELs. Pseudo-random inputs $r_n$ can be binary (bits) or continuous values between zero and one. The input weights, weights of reservoir-internal connections, and output weights are denoted by $\mathbf{w}^\mathrm{in}$, $\mathbf{w}^\mathrm{res}$, and $\mathbf{w}^\mathrm{out}$, respectively.
• Figure 2: Injection laser time trace, directly measured (blue dashed line) and after reflection at the surface of one VCSEL (red solid line), and response of the same VCSEL (green dotted line). The green dots (red triangles) represent $q_{n,j}$ -- the state of the $j^\mathrm{th}$ node of the reservoir at the $n^\mathrm{th}$ step -- for the responses (reflections). The VCSEL implements the nonlinear transformation required for RC.
• Figure 3: Dynamic responses of four different VCSELs to the injection of uniformly distributed pseudo-random numbers $r_n$ with the injection laser. Every red circle represents the response $q_{n,j}$ to the injection of $r_n$, the blue dots are averages.
• Figure 4: Memory correlation $M_k$ vs. number of time steps back $k$ (a) for $\Delta \lambda = 0{pm}$ with different power ratios $\epsilon$ and for the linear reflections (refl.) at the VCSELs' surfaces and (b) for $\epsilon = 0.20$ with different detuning $\Delta \lambda$ of the injection laser from the average VCSEL emission wavelength.
• Figure 5: a) Bit error ratios (BERs) for the 2-bit, 3-bit and 4-bit header recognition (HR) task. b) Root-mean-square error (RMSE) for 1- to 4-bit digital-to-analog conversion (DAC). Both for different power ratios $\epsilon$, for the reflections (refl.), and for the trivial guess (triv. g.). Lines are guides to the eye.