## Digital approaches

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### Optical diffraction tomography

There is considerable interest in developing optical microscopes presenting a lateral resolution below the usual Rayleigh criterion, λ/(2NA), where λ is the wavelength of the illumination and NA is the numerical aperture of the imaging system, while retaining the convenience of far-field illumination and collection.

Among the various ways to ameliorate the resolution, it has been proposed to illuminate the sample with many structured illuminations, namely standing waves, and to mix the different images through simple arithmetics.This technique is very close to optical diffraction tomography (ODT) in which the sample is illuminated under various angles of incidence, the phase and intensity of the diffracted far-field is detected along several directions of observation, and a numerical procedure is used to retrieve the map of the permittivity distribution of the object from the far-field data. Experimental and theoretical studies have shown that using several illuminations permits one to exceed the classical diffraction limit by a factor of two.

In all microscopy techniques using several successive illuminations, one needs a numerical procedure to combine the different images and extract the map of the relative permittivity distribution of the object from the scattered far-field. In general, one assumes that the object is a weak scatterer so that there is a linear relationship between the scattered field and the relative permittivity of the object, namely one assumes that the Born approximation is valid. In this case, the transverse resolution limit can be inferred from simple considerations on the portion of the Ewald sphere that is covered by the experiment. It is limited by λ/2(n_{i}+n_{d}) for configurations in which the incident waves propagate in a medium of refractive index n_{i}, while the diffracted waves propagate in a medium of refraction index n_{d}.

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