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.  

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