Optical processors, built with "optical neurons", can efficiently perform high-dimensional linear operations at the speed of light. Thus they are a promising avenue to accelerate large-scale linear computations. With the current advances in micro-fabrication, such optical processors can now be 3D fabricated, but with a limited precision. This limitation translates to quantization of learnable parameters in optical neurons, and should be handled during the design of the optical processor in order to avoid a model mismatch. Specifically, optical neurons should be trained or designed within the physical-constraints at a predefined quantized precision level. To address this critical issues we propose a physics-informed quantization-aware training framework. Our approach accounts for physical constraints during the training process, leading to robust designs. We demonstrate that our approach can design state of the art optical processors using diffractive networks for multiple physics based tasks despite quantized learnable parameters. We thus lay the foundation upon which improved optical processors may be 3D fabricated in the future.