  For over a century diffraction theory has been thought to limit the
resolution of focusing and imaging in the optical domain. The size of the
smallest spot achievable is inversely proportional to the range of spatial
wavevectors available. Here, we show that it is possible to locally beat the
diffraction limit at the expense of efficiency. The method is based on the
linearity of Maxwell's equations and that the interaction between light and its
surroundings may be considered quadratic in nature with respect to the
electromagnetic fields. We represent the intensity and spot size as a quadratic
measure with associated eigenmodes. Using a dynamic diffractive optical
element, we demonstrate optical focussing to an area 4 times smaller than the
diffraction limit. The generic method may be applied to numerous physical
phenomena relating to linear and measurable properties of the electromagnetic
field that can be expressed in a quadratic form.
