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In pursuit of higher inference accuracy, deep neural network (DNN) models have significantly increased in complexity and size. To overcome the consequent computational challenges, scalable chiplet-based accelerators have been proposed. However, data communication using metallic-based interconnects in these chiplet-based DNN accelerators is becoming a primary obstacle to performance, energy efficiency, and scalability. The photonic interconnects can provide adequate data communication support due to some superior properties like low latency, high bandwidth and energy efficiency, and ease of broadcast communication. In this paper, we propose SPACX: a Silicon Photonics-based Chiplet ACcelerator for DNN inference applications. Specifically, SPACX includes a photonic network design that enables seamless single-chiplet and cross-chiplet broadcast communications, and a tailored dataflow that promotes data broadcast and maximizes parallelism. Furthermore, we explore the broadcast granularities of the photonic network and implications on system performance and energy efficiency. A flexible bandwidth allocation scheme is also proposed to dynamically adjust communication bandwidths for different types of data. Simulation results using several DNN models show that SPACX can achieve 78% and 75% reduction in execution time and energy, respectively, as compared to other state-of-the-art chiplet-based DNN accelerators.