Fundamentals of Quantum Fourier Optics

All-quantum signal processing techniques are at the core of the successful advancement of most information-based quantum technologies. This article develops coherent and comprehensive methodologies and mathematical models to describe Fourier optical signal processing in full quantum terms for any input quantum state of light. We begin this article by introducing a spatially 2-D quantum state of a photon, associated with its wavefront and expressible as a 2-D creation operator. Then, by breaking down the Fourier optical processing apparatus into its key components, we strive to acquire the quantum unitary transformation or the input/output quantum relation of the 2-D creation operators. Subsequently, we take advantage of the above results to develop and obtain the quantum analogous of a few essential Fourier optical apparatuses, such as quantum convolution via a 4f-processing system and a quantum 4f-processing system with periodic pupils. Moreover, due to the importance and widespread use of optical pulse shaping in various optical communications and optical sciences fields, we also present an analogous system in full quantum terms, namely quantum pulse shaping with an 8f-processing system. Finally, we apply our results to two extreme examples of the quantum state of light. One is based on a coherent (Glauber) state and the other on a single-photon number (Fock) state for each of the above optical systems. We believe the schemes and mathematical models developed in this article can impact many areas of quantum optical signal processing, quantum holography, quantum communications, quantum radars and multiple-input/multiple-output antennas, and many more applications in quantum imaging, quantum computations, and quantum machine learning algorithms.