Is There a Fundamental Limit to Spatial Resolution in Phase Microscopy? - Biomedical Optical Phase Microscopy and Nanoscopy - page 337

Figure 16.6
A possible way in which evanescent subwavelength periodic illumination can give super-resolution.
(A) Illumination optics and (B) frequency plane coverage using three different illumination
orientations.

spatial frequencies. They showed that an improvement in image resolution could be
obtained by using the analysis of many different images of this type.

16.6 Imaging Three-Dimensional Phase Objects

Now, the question is what techniques might allow super-resolution of phase objects
considerably thicker than the wavelength. This is essentially a question of three-dimensional
imaging of the optical density. Since there is clearly no way of creating extended volume
fringes with subwavelength spacing, structured illumination cannot be extended beyond the
2 k m limit. On the other hand, in parallel with the determination of three-dimensional crystal
structures from diffraction data, we would expect diffraction synthesis to be possible and
the limitations to be similar to those in one and two dimensions. Now, in a single
diffraction pattern, the three-dimensional Fourier transform of the object is sampled only on
the Ewald sphere, which has radius 2 k 0 and passes through the origin of k space. In Fourier
space, the Ewald sphere represents geometrically the requirement for conservation of
photon energy during elastic scattering (see, e.g., Ref. [11] ). As the incident beam (direction
of the vector k 0 ) is rotated with respect to the sample, the total accessible region of Fourier
space is thus a sphere with radius 2 k 0 and volume (32/3) πk 0 3 . A coherent incident beam
size of D allows this in principle to be sampled in k space on a scale of (2 π / D ) 3 , which
means that the maximum number of data points in the three-dimensional transform is about
m 5 (2 D / ) 3 . Applying the results of Donoho et al. [18] to three dimensions, this would

λ