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.

Figures (14)

• Figure 1: Overview of the control--probe add-drop cascade $N$-MRR exponential block. (a) Electronic preprocessing maps an arbitrary input sequence $\{x_n\}$ to a nonnegative control signal via $m=\max_n x_n$, $u_n=x_n-m$, and $I_n=u_n+L$ with $L=m-\min_n x_n$. (b) The control signal $I_n$ induces resonance shifts in a cascade of $N$ rings, while a weak fixed-frequency probe propagates through the serial add-drop cascade (the drop output of each ring feeds the next stage), experiencing multiplicative transmission. (c) After photodetection, the block implements $y(I_n)\approx \exp(I_n-L)\approx \exp(x_n-m)$, i.e., the normalized exponential used in softmax.
• Figure 2: Minimax cascade fits at $L = 8$. (a) Linear scale: shallow cascades ($N = 1, 3, 5, 7$) over $I \in [0, 20]$. The target $e^{I-L}$ (black) is progressively better matched as $N$ increases. (b) Log scale: depth comparison ($N = 5, 10, 20, 30$) on $I \in [0, 8]$. Inset zooms into $I \in [6, 8]$ showing convergence. (c) Pointwise relative error showing the Chebyshev equioscillation pattern characteristic of minimax optimality.
• Figure 3: Cross-section of the X-cut TFLN rib waveguide on a SiO2 substrate. The 600nm LiNbO3 film is etched 500nm to form a 1.4µm-wide single-mode rib waveguide. Lateral signal (S) and ground (G) electrode positions are indicated; electrode design details are discussed in Sec. \ref{['sec:electrode']}.
• Figure 4: Top view of the single add-drop micro-ring resonator used in the 3D FDTD simulation. The ring waveguide ($R = 20µm$, $w = 1.4µm$) is evanescently coupled to input and drop bus waveguides through 100nm gaps at coupling points CP1 and CP2.
• Figure 5: Simulated through-port (blue) and drop-port (red) transmission spectra of the single add-drop micro-ring resonator from 3D FDTD. Top: logarithmic scale; bottom: linear scale. Five resonances are visible with $\mathrm{FSR} \approx 8.29nm$.

Theorems & Definitions (12)

• Lemma 1: Slope bound --- rigorous
• proof
• Remark 1: Necessary condition for approximation
• Proposition 1: Log-domain Taylor expansion at flank center
• proof
• Theorem 1: Heuristic depth-scaling law
• proof : Derivation (heuristic)
• Remark 2: Status of Theorem \ref{['thm:scaling']}
• Proposition 2: Conservative log-error bound
• proof : Derivation sketch