Principles and Metrics of Extreme Learning Machines Using a Highly Nonlinear Fiber

TL;DR

The paper investigates an optical Extreme Learning Machine implemented in a highly nonlinear fiber to enable ultrafast computation via Kerr-induced nonlinearities. It defines task-independent metrics PC$^{95}$ and consistency to quantify dimensionality and reliability, and demonstrates how dispersion and fiber length shape the spectral feature space; MNIST classification serves as a task-dependent benchmark. Key findings show that longer fibers and anomalous dispersion increase the effective dimensionality to about PC$^{95}$ ≈ 70–100 under suitable power, with a notable spectral localization near the pump within ~40 nm. Importantly, the study reveals that optimal performance often occurs at moderate input power and dimensionality, achieving up to ~0.88 test accuracy for d = 40 (P_in ≈ 1.72 mW), surpassing the linear baseline, while excessive nonlinearity can degrade consistency and performance. Together, these results establish a framework linking NLSE-driven optical dynamics to computational capacity and point toward ultrafast, low-latency optical co-processing and metrology applications, with future prospects for in-situ learning and real-time operation.

Abstract

Optical computing offers potential for ultra high-speed and low latency computation by leveraging the intrinsic properties of light. Here, we explore the use of highly nonlinear optical fibers (HNLFs) as platforms for optical computing based on the concept of Extreme Learning Machines. Task-independent evaluations are introduced to the field for the first time and focus on the fundamental metrics of effective dimensionality and consistency, which we experimentally characterize for different nonlinear and dispersive conditions. We show that input power and fiber characteristics significantly influence the dimensionality of the computational system, with longer fibers and higher dispersion producing up to 100 principal components (PCs) at input power levels of 30 mW, where the PC correspond to the linearly independent dimensions of the system. The spectral distribution of the PC's eigenvectors reveals that the high-dimensional dynamics facilitating computing through dimensionality expansion are located within 40~nm of the pump wavelength at 1560~nm, providing general insight for computing with nonlinear Schrödinger equation systems. Task-dependent results demonstrate the effectiveness of HNLFs in classifying MNIST dataset images. Using input data compression through PC analysis, we inject MNIST images of various input dimensionality into the system and study the impact of input power upon classification accuracy. At optimized power levels we achieve a classification test accuracy of 88\%, significantly surpassing the baseline of 83.7\% from linear systems. Noteworthy, we find that best performance is not obtained at maximal input power, i.e. maximal system dimensionality, but at more than one order of magnitude lower. The same is confirmed regarding the MNIST image's compression, where accuracy is substantially improved when strongly compressing the image to less than 50 PCs.

Figures (9)

• Figure 1: (a) conceptual diagram of the ELM (b) schematic of the experimental setup for the Extreme Learning Machine implementation composed of a waveshaper (WS), highly nonlinear fiber (HNLF), and an optical spectrum analyzer (OSA) to collect the output fiber spectra. Typical encoding patterns are depicted for both cases of task-dependent and task-independent. (c) Typical fiber output spectra in logarithmic and linear scale for an input dimension $d$ = 22 and an input power $P_{in}$ = 35 mW.
• Figure 2: (a) conceptual diagram of the waveshaper used for data encoding, (b) schematic of the experimental SLM configurations. In Configuration 1, the SLM is tilted and a grating is applied to separate the orders spatially, allowing only modulated light to be coupled forward. In Configuration 2, the SLM is aligned straight, and no grating is applied, resulting in both modulated and unmodulated light propagating together.
• Figure 3: (a) Principal Component Analysis (PCA) results showing the effect of varying input power and channel size on the number of principal components (PCs. The color gradient indicates the number of PCs, with colder colors representing fewer PCs and cooler colors representing a higher number of PCs. This highlights the regions of channel size and input power combinations that maximize or minimize the complexity of the system response. (b) Number of PCs as a function of channel size for a fixed input power of 40 mW, illustrating the trend and identifying the optimal channel size for this input power level. The insets display example Spatial Light Modulator (SLM) patterns and their respective number of channels for selected channel sizes (50, 200, and 500), providing a visual representation of the corresponding SLM configuration at these sizes. (c) Integral over power spectral density measured by the OSA, normalized by the injected power. Optical losses are essentially independent of the injected power.
• Figure 4: (a) Dispersion as a function of wavelength for three fibers (Fiber 1, Fiber 2, and Fiber 3). Fiber 1 exhibits normal dispersion with a relatively flat profile, while Fibers 2 and 3 demonstrate anomalous dispersion above the pump wavelength. (b) Number of principal components (PCs) versus input power for the three fibers. At low power, the number of PCs is similar across all fibers, and at higher power levels, Fibers 2 and 3 generate more PCs, indicating increased spectral complexity due to higher dispersion. c Three example spectra for the highest injection power and the three different fibers, revealing the systematic difference in spectral complexity and broadening between the normal (Fiber 1) and anomalous (fibers 2 and 3) dispersion.
• Figure 5: (a) Spectral distribution of the PCs shows that the ELM's feature space relies mostly on dynamics close to the pump wavelength.(b) Spectral loadings of the first 100 PCs as a function of wavelength for the case of 35 mW, Fiber 3. (c) Number of PCs as a function of input power for three different fiber lengths (1, 2, and 5 meters). The 5-meter fiber consistently exhibits a higher number of PCs compared to shorter fibers, indicating that longer fibers enhance the dimensionality of the output spectra.