I've work on exactly this problem and the answer is extremely controversial!
At the heart of the controversy is a disconnect between tomography (3D imaging) and the surface maps (2.5D, "we all live in a hologram") approaches to image formation. The "we all live in a hologram" approach has recently seen a lot of commercial/experimental success much to the chagrin of the tomography old timers.
Different optical systems have drastically different abilities to re-focus the image. Further, each system operates on different physical assumptions. Lets talk about a few:
Light-field systems such as the Lytro can produce selective focus for macroscopic objects, on the other hand similar affects can be achieved with digital filters.
At low NA (>0.3), coherent (laser) illumination matches well with the linear systems approach to wave propagation in a half-space given by the Fresnel propagator. You can apply the Fresnel propagator and focus and de-focus the system, although you need some kind of measure of the complex field. Popular approaches include Hartmann-Shack style wave front sensors (similar to Lytro), traditional interferometry, and transport of intensity (TIE). The images in these systems look pretty terrible because junk somewhere in the beam path (perhaps a few cm away) contributes to image formation.
Now stuff starts getting wacky when coherence effects are introduced. On the other hand the image looks better under incoherent illumination. The first problem is that even for systems that are linear with the object's scattering potential (not many!), coherence introduces a spatial band limit. 795
That is to say objects outside of the spatio-temporal coherence (you can't actually split these) radius fade away. So, now your system is only sensitive to a narrow range of object sizes, and you start getting optical sectioning. This is typically called the "coherence gating" and was famously used by OCT. In practice, you can't propagate the field beyond whats allowed by the coherence. In most broadband systems this means you can't, digitally, move the field up-and-down by more than a few microns! (and it gets worse at high NAs)
On the other hand you can simply move the sample (or adjust the focus). When you do this a new degree of freedom is introduced, which is the origin of tomographic imaging (diffraction tomography). For people who use these kind of systems their world is made of 3D slices. They would tell you that one cannot simply refocus the light indefinitely.
Another problem is that many systems are not actually linear with the object, for example, the infamous halo in phase contrast microscopy. In those cases you can't meaningfully digitally focus/refocus the sample because your system typically doesn't measure the field due to the sample.
On the theoretical side there is a lot of debate as to if fields are random, but the positions taken by various research groups are closely tied to their experimental setups. 795