  All-quantum signal processing techniques are at the core of the successful
advancement of most information-based quantum technologies. This paper 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 paper by introducing a spatially two-dimensional
quantum state of a photon, associated with its wavefront and expressible as a
two-dimensional 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
two-dimensional creation operators. Subsequently, we take advantage of the
above results to develop and obtain the quantum analogous of a few essential
Fourier optical apparatus, 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 paper
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 computations and quantum
machine learning algorithms.
