  A general framework for determining fundamental bounds in nanophotonics is
introduced in this paper. The theory is based on convex optimization of dual
problems constructed from operators generated by electromagnetic integral
equations. The optimized variable is a contrast current defined within a
prescribed region of a given material constitutive relations. Two power
conservation constraints analogous to optical theorem are utilized to tighten
the bounds and to prescribe either losses or material properties. Thanks to the
utilization of matrix rank-1 updates, modal decompositions, and model order
reduction techniques, the optimization procedure is computationally efficient
even for complicated scenarios. No dual gaps are observed. The method is
well-suited to accommodate material anisotropy and inhomogeneity. To
demonstrate the validity of the method, bounds on scattering, absorption, and
extinction cross sections are derived first and evaluated for several canonical
regions. The tightness of the bounds is verified by comparison to optimized
spherical nanoparticles and shells. The next metric investigated is
bi-directional scattering studied closely on a particular example of an
electrically thin slab. Finally, the bounds are established for Purcell's
factor and local field enhancement where a dimer is used as a practical
example.
