  Precise engineering of materials and surfaces has been at the heart of some
of the recent advances in optics and photonics. These advances around the
engineering of materials with new functionalities have also opened up exciting
avenues for designing trainable surfaces that can perform computation and
machine learning tasks through light-matter interaction and diffraction. Here,
we analyze the information processing capacity of coherent optical networks
formed by diffractive surfaces that are trained to perform an all-optical
computational task between a given input and output field-of-view. We show that
the dimensionality of the all-optical solution space covering the
complex-valued transformations between the input and output fields-of-view is
linearly proportional to the number of diffractive surfaces within the optical
network, up to a limit that is dictated by the extent of the input and output
fields-of-view. Deeper diffractive networks that are composed of larger numbers
of trainable surfaces can cover a higher dimensional subspace of the
complex-valued linear transformations between a larger input field-of-view and
a larger output field-of-view, and exhibit depth advantages in terms of their
statistical inference, learning and generalization capabilities for different
image classification tasks, when compared with a single trainable diffractive
surface. These analyses and conclusions are broadly applicable to various forms
of diffractive surfaces, including e.g., plasmonic and/or dielectric-based
metasurfaces and flat optics that can be used to form all-optical processors.
