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The company lytro.com has an Illum and an Immerge [Light Field Camera] which presumably photograph Light Fields.

If I understand correctly the term light field refers to a 5D concept - at each point in 3D space there are rays of light from all directions. Those directions use 2 more dimensions (e.g. $(\theta, \phi)$) and $2+3=5$. This could be called the 5D plenoptic function. I think at each point in this 5D space you have some kind of spectral intensity information. It could be RGB values or a complete spectrum, and would have units of (for example) power per unit area, per unit solid angle, per unit wavelength.

Is this just a name for a thing, or are there real, substantial mathematical things you can do with a light field? For example, if I know the Electric and Magnetic fields in a volume, I suppose could calculate the resulting light field in a straightforward way. But what can I actually DO with a light field besides market it?

Related question and answer, and a random popular article that repeats the term light field over and over.

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The short answer is this: yes, the light field is useful in geometric optics, where you can simulate light as though it were a bunch of classical particles moving on straight lines with definite positions and momenta. This is wrong, since light is excitations in a field, but it works as an approximation. This is the idea behind the framework used in astronomy called radiative transfer, and ray tracing optics in computer graphics.

So, a phase space number density of photons can be written as $\rho(\mathbf{x}, \mathbf{p})$, a density over real space, which is 3D, and momentum space, which is also 3D. The magnitude of $\mathbf{p}$ is the "spectrum" direction, and the direction of $\mathbf{p}$ is the direction the light is traveling. This is related to the spectral radiance, $I_\nu$, by: $$\begin{align}I_\nu(\mathbf{x},\theta, \phi, \nu) &= hcp^3 \rho(\mathbf{x}, \mathbf{p}) \\ &= \frac{h^4 \nu^3}{c^2} \rho(\mathbf{x}, \mathbf{p}), \end{align}$$ where $\mathbf{p} = [ \sin \theta \cos \phi,\ \sin\theta \sin \phi,\ \cos\theta] \times h\nu /c.$

As for what it's useful for in a camera the answer is: it depends. You're engaging in a trade-off where spatial resolution on the chip is sacrificed for improved resolution in other ways. For example, integral field spectrographs use the lost spatial resolution to encode spectral information. In the case of light field cameras, they're using it to encode direction of travel for the light. Roughly, this is equivalent to taking many different images at different focuses at the same time, so you can do fun depth of field tricks with Photoshop by combining them, or just selecting the one with the desired focus.

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