Fourier Neural Operators (FNOs) have gained increasing attention for their effectiveness in extracting frequencydomain features and efficiently approximating functions, making them wellsuited for classification tasks. However, the absence of specialized photonic hardware has limited the acceleration of FNO inference. In this study, we introduce what we believe is the first photonic hardware framework dedicated to speeding up the Fourier layer of an FNO. Our approach employs a frequency domain convolutional photonic chip and a micro-ring array chip, achieving 5-bit quantization precision in the inference process. On the Radio ML 2016.10b dataset, our Fourier convolutional neural network achieves a peak identification accuracy of 95.50%, outperforming standard convolution-based networks. These findings highlight the transformative potential of co-designing software and hardware, demonstrating how photonic computing can deliver specialized acceleration for critical AI components and substantially improve inference efficiency. Ultimately, this work lays a foundation for integrating photonic technologies into next-generation AI accelerators, pointing to a promising direction for further research and development in optoelectronic hybrid computing.