Photonic integrated circuits are finding use in a variety of applications including optical transceivers, LIDAR, bio-sensing, photonic quantum computing, and Machine Learning (ML). In particular, with the exponentially increasing sizes of ML models, photonics-based accelerators are getting special attention as a sustainable solution because they can perform ML inferences with multiple orders of magnitude higher energy efficiency than CMOS-based accelerators. However, recent studies have shown that hardware manufacturing and infrastructure contribute significantly to the carbon footprint of computing devices, even surpassing the emissions generated during their use. For example, the manufacturing process accounts for 74% of the total carbon emissions from Apple in 2019. This prompts us to ask -- if we consider both the embodied (manufacturing) and operational carbon cost of photonics, is it indeed a viable avenue for a sustainable future? So, in this paper, we build a carbon footprint model for photonic chips and investigate the sustainability of photonics-based accelerators by conducting a case study on ADEPT, a photonics-based accelerator for deep neural network inference. Our analysis shows that photonics can reduce both operational and embodied carbon footprints with its high energy efficiency and at least 4$\times$ less fabrication carbon cost per unit area than 28 nm CMOS.