Beating the diffraction limit
Beating the diffraction limit using optical eigenmodes is easy. The only challenge is to define a quadratic field dependent operator that measures the spot size. There are many possibilities and we had good results using the second order momentum as a measure. Once the operator defined we can simply calculate its eigenvectors and the one having the smallest eigenvalue will correspond to the smallest spot size achievable in this operators context. In the figure on the left we have looked to minimise the spot size in a region of interest defined by the yellow dotted circle.
Experimentally, this is more challenging as any optical system will include aberrations. To counteract this problem we have opted in experimentally determine the intensity optical eigenmodes pf the system. Practically these correspond to the accessible optical degrees of freedom of the system.This experimental calibration is a two step procedure. In a first step, we probe the optical system with any beam shape we can create. For each of these probes we need to measure the phase and amplitude in the region of interest. From these, we can determine all the intensity optical eigenmodes for the given probes and optical system. These modes fully characterise any field that can be created in this context. These experimentally determined modes are then used with the spot size operator to determine the smallest spot achievable. The beauty of this approach is that it automatically corrects and takes into account any aberration encountered in the system.