Optical Eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions

We report a mathematically rigorous technique which facilitates the optimization of various optical properties of electromagnetic fields in free space and including scattering interactions. The technique exploits the linearity of electromagnetic fields along with the quadratic nature of the intensity to define specific Optical Eigenmodes (OEi) that are pertinent to the interaction considered. Key applications include the optimization of the size of a focused spot, the transmission through sub-wavelength apertures, and of the optical force acting on microparticles. We verify experimentally the OEi approach by minimising the size of a focused optical field using a superposition of Bessel beams.