Photonic Exponential Approximation via Cascaded TFLN Microring Resonators toward Softmax

Abstract: The rapid growth of large-scale AI models has intensified energy consumption and data-movement challenges in modern datacenters. Photonic accelerators offer a promising path by executing the linear matrix multiplications of transformer inference at high throughput and low energy. However, the softmax attention layer -- which requires element-wise exponentiation followed by normalization -- still relies on electronic post-processing, creating an electro-optic conversion bottleneck that negates much of the potential photonic advantage. We present a cascaded micro-ring resonator (MRR) architecture that synthesizes the per-channel exponential function required by softmax, e^{x_n - max(x)}, over a finite interval with tunable worst-case relative error. A control signal detunes each ring via an electro-optic mechanism; a weak probe at fixed frequency experiences Lorentzian transmission, and cascading N identical stages yields a multiplicative transfer function whose logarithm is approximately linear. We derive mapping rules, depth-scaling estimates, and a minimax fitting formulation, and validate the framework with three-dimensional FDTD simulations of X-cut thin-film lithium niobate (TFLN) add-drop micro-ring resonators. Direct multi-ring FDTD validation extends to a five-ring cascade and confirms agreement with theory primarily over the upper operating range; deeper cascades and higher quality factors are assessed analytically. The cascade implements the per-channel exponential block -- the key missing nonlinearity for photonic softmax; completing a full softmax additionally requires summation and normalization, which we discuss but do not implement here.