  Nanoscale fabrication techniques, computational inverse design, and fields
from silicon photonics to metasurface optics are enabling transformative use of
an unprecedented number of structural degrees of freedom in nanophotonics. A
critical need is to understand the extreme limits to what is possible by
engineering nanophotonic structures. This thesis establishes the first general
theoretical framework identifying fundamental limits to light--matter
interactions. It derives bounds for applications across nanophotonics,
including far-field scattering, optimal wavefront shaping, optical beam
switching, and wave communication, as well as the miniaturization of optical
components, including perfect absorbers, linear optical analog computing units,
resonant optical sensors, multilayered thin films, and high-NA metalenses. The
bounds emerge from an infinite set of physical constraints that have to be
satisfied by polarization fields in response to an excitation. The constraints
encode power conservation in single-scenario scattering and requisite field
correlations in multi-scenario scattering. The framework developed in this
thesis, encompassing general linear wave scattering dynamics, offers a new way
to understand optimal designs and their fundamental limits, in nanophotonics
and beyond.
