simulations in 3D, we have shown how the OEi approach can be used to optimise light-matter scattering interactions in the case of transmission through sub-wavelength apertures and optical forces on micro-particles. The generic nature of our approach means that it can be applied to other cases where the measure has a quadratic form and the propagation is linear. In the present paper we have verified the rigor of the method by demonstrating the experimental spot size operator and intensity operator optimization using Laguerre-Gaussian and Bessel light modes using a dual SLM to implement the technique. Future work will aim to extend this method to optimise the size and contrast of optical dark vortices, the linear Raman scattering or the fluorescence of different samples, the optical trapping force, and the angular momentum transfer in optical manipulation.

Acknowledgements {#acknowledgements .unnumbered}
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We thank the UK EPSRC for funding through the Nanoscope Basic Technology grant. Mark Dennis is acknowledged for the introduction to super-oscillations. KD is a Royal Society-Wolfson Merit Award Holder.

Annex: Dimensionality study {#annex-dimensionality-study .unnumbered}
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In this appendix, we verify the convergence properties of our optical eigenmode method as a function of the dimensionality of the probing base used. Here, 