Optical kernel machine with programmable nonlinearity

TL;DR

The paper addresses the challenge of implementing nonlinear optical kernels with low power by embedding a programmable nonlinear kernel in a linear scattering cavity.It leverages structural nonlinearity arising from multiple scattering, tunable via wall reflectivity and DMD modulation area, and characterizes this with the Born-series expression $E_{ m out} = \big[V + V(G_0V) + V(G_0V)^2 + \big] E_{ m in}$ and measurements of output correlations.The authors show that kernel expressivity and information capacity increase with nonlinearity and demonstrate parity-function regression up to fifth order, with performance improving as input and output dimensions grow.The work points to scalable, low-power photonic kernels applicable across platforms, with potential enhancements via broadband/pulsed light and real-time optimization to achieve target-adaptive nonlinear responses.

Abstract

Optical kernel machines offer high throughput and low latency. A nonlinear optical kernel can handle complex nonlinear data, but power consumption is typically high with the conventional nonlinear optical approach. To overcome this issue, we present an optical kernel with structural nonlinearity that can be continuously tuned at low power. It is implemented in a linear optical scattering cavity with a reconfigurable micro-mirror array. By tuning the degree of nonlinearity with multiple scattering, we vary the kernel sensitivity and information capacity. We further optimize the kernel nonlinearity to best approximate the parity functions from first order to fifth order for binary inputs. Our scheme offers potential applicability across photonic platforms, providing programmable kernels with high performance and low power consumption.

Figures (6)

• Figure 1: Programmable nonlinear optical kernel. (a) Schematic of experimental implementation of the kernel in a linear scattering cavity. Input light at $\lambda$ = 1550 nm is scattered by the rough cavity wall and a reconfigurable micro-mirror array (DMD). Part of output light is recorded by a CCD camera. (b) Four binary masks written to the DMD. The micro-mirrors (pixels) tilt at angles of $+12^\circ$ and $-12^\circ$, redirecting incident light in 2 directions, each representing 1 and -1 in the input pattern. A representative base pattern with $10 \times 10$ macropixels is repeated to occupy $100\%, 50\%, 25\%, 12.5\%$ area of the DMD. (c) Measured speckle intensity pattern of output light from the scattering cavity. (d) Schematic of numerically-simulated 2D scattering cavity with varying reflectivity of the wall. The cavity dimension is $130 \lambda \times 60 \lambda$. One side of the cavity wall is covered by 12 mirrors, each can be flipped to $+15^\circ$ or $+15^\circ$ angle.
• Figure 2: Tunable structural nonlinearity in a linear multiple-scattering cavity. Linear regression of cavity input-output mapping gives the coefficient of determination $R^2$. (a) In the numerically-simulated 2D cavity $R^2$ decreases with increasing cavity wall reflectivity. (b) In the experimental 3D cavity, $R^2$ drops as the DMD modulated area increases. The decrease of $R^2$ is a result of enhanced structural nonlinearity.
• Figure 3: Output sensitivity to input change. (a) Real part of the correlation between output field patterns for different mirror configurations increases with the correlation between input patterns (mirror configurations) in the simulated 2D cavity with wall reflectivity of 0.92 (blue) and 0.51 (purple). The black dashed curve is for linear input-output mapping. For higher wall reflectivity, the curve deviates more from the linear mapping due to stronger structural nonlinearity. (b) Correlation of output intensity patterns from the experimental 3D cavity increases with the correlation of input patterns on the DMD with the modulation area of 100% (blue), 50% (orange), and 12.5% (purple). More deviations from the linear mapping (black dashed line) result from stronger structural nonlinearity at larger DMD modulation area.
• Figure 4: Information capacity of the multiple-scattering cavity. (a) Information capacity increases with the wall reflectivity of the simulated 2D cavity. SNR = 100 (blue) leads to a larger information capacity than SNR = 10 (orange). (b) Information capacity increases with the DMD modulation area in the experimental 3D cavity with $\mathrm{SNR}=95.2$.
• Figure 5: Kernel regression to parity functions of varying order. The experimental 3D cavity serves a kernel by providing a high-dimensional feature map that maps input patterns on the DMD to optical intensity patterns at cavity output. Binary patterns $(x_1, x_2, ..., x_9)$ are uploaded to the DMD, and repeated multiple times to occupy a certain percentage of the entire modulation area. Two-dimensional output intensity pattern is linearly regressed to parity function functions of first order $f(x) = x_1$ in (a), second-order $f(x) = x_1 x_2$ in (b), and third-order $f(x) = x_1 x_2 x_3$ in (c). The regression error is plotted as a function of the percentage of DMD modulation area, which controls the kernel nonlinearity. The measured intensity pattern consists of approximately 55 speckle grains. (d-f) Magnitude sum of Boolean function coefficients of first order $S_1$ in (d), second-order $S_2$ in (e), and third-order $S_3$ in (f) are computed for varying percentage of DMD modulation area. The regression error for the parity function of order $i$ reaches the minimum at the DMD modulation area where $S_i$ is maximum. Kernel regression has the best performance when its nonlinearity, tuned by the DMD modulation area, matches the order of the parity function.