An optical system reversibly transforms a light input to a light output. In the geometrical-optics approximation, the light input (or output) is a function from ℝ⁴ (four-tuples of real numbers) to spectra. If we reduce the spectra down to RGB, which is reasonable for some purposes, this is a function ℝ⁴ → ℝ³; if we reduce it down to monochrome, which is reasonable for other purposes, it’s a function ℝ⁴ → ℝ. We could think of this as a four-dimensional scalar field. One view of the four dimensions is that they are the X and Y coordinates where light enters (or exits) the system boundary and the θ and φ angles at which it enters. That is, the system can do different things with light that enters at the same angle at different points, or with different angles at the same point.
So, in monochrome, the overall system behavior is a function (ℝ⁴ → ℝ) → (ℝ⁴ → ℝ). But this is still an extremely loose description, because there are many such functions that we cannot realize as an optical system, and there are others that we can realize only with great difficulty.
(In wave mechanics, the input and output are very much simpler; at a given wavelength, each point on the boundary of the system only has a single complex phase and amplitude. This fact allows holographic optics to achieve astonishing performance for single-wavelength systems. Unfortunately the situation with multiple wavelengths becomes complicated again, while geometric optics either depends not at all on wavelength or only depends on wavelength in a very simple way, with color filters and whatnot. So here I will focus on geometric optics.)
For reasons of convenience, I’m going to focus on optical systems that don’t dissipate energy, so the amount of light coming out is the same as the amount of light going in. This seems like kind of a stupid focus, since all actual systems do absorb some light, often most of it. Then they convert that light into heat. But it turns out that, if we treat this as a sort of aberration, we can derive some very interesting properties of optical systems that don’t have it, and then we can figure out how real systems behave by adding in light absorption as a sort of correction.
Another limitation, given conservation of energy, is that optical systems must be reversible. That is, if a certain beam of light going into the system produces a certain distribution of light coming back out of the system, if we send this second distribution of light back in, it will come out as the first beam of light, just going the other way. This seems not to be true in our day-to-day experience, and this requires some examination. For example, a laser pointer shining on white paint produces a spot that can be seen from any direction, so we know it’s throwing off light in every direction, in a way that’s called Lambertian reflection, and yet light going into the spot from every direction doesn’t go back into the laser pointer. We explain this by saying that the paint is full of many different microscopic facets, each of which throws off light in a particular direction when the laser hits it, and there are so many of them even in that little spot that the light seems to go in every direction at once! And if we could shoot a very, very thin beam of light at each facet, in just the reverse of the direction that the laser was making it shine before, all of those beams would be redirected perfectly back to where the laser pointer had been.
Now, in reality, these facets are often so tiny that the geometrical optics approximation breaks down, and we have to use wave mechanics to see what will happen. But it turns out that wave mechanics is reversible too; reversibility is not just a consequence of the geometrical-optics approximation, but a property of the wave-mechanical nature of light that survives in the geometrical-optics approximation.
But if our system consists entirely of macroscopic features — mirror-smooth surfaces that are perhaps curved or have edges, everything either polished metallic mirror-bright or transparent — then, indeed, any transformation that the system produces can in fact be time-reversed in this way. And you can do it in practice, not just in theory, because you don’t need microscopically tiny slivers of light the way you do for the white paint.
This imposes some restrictions on the mathematical form of our system. It can’t, for example, transform two different distributions of incoming light into the same distribution of outgoing light, because then if you time-reversed the outgoing light, it wouldn’t know which of these two different distributions it should produce. The function must be bijective, invertible.
But is that the only restriction? Can we realize any arbitrary invertible (ℝ⁴ → ℝ) → (ℝ⁴ → ℝ) function as an optical system? No, not even close.
One of the strongest restrictions is linearity.
Most optical systems are linear, in the sense that different beams of light don’t interact with each other. If you have some beam A and the system transforms it to f(A), and some other beam B and the system transforms it to f(B), then if you shoot both of those beams of light at the system at once, A + B, then the distribution of light that comes out will be exactly f(A) + f(B). You can have a lens, for example, bend one light beam a bit to the left, and the other a bit to the right, but you can’t have it bend the first light beam to the right when the second one is present, or to the left otherwise.
Now, this is just an approximation, but under most circumstances, it’s a very, very good approximation, and it takes very sensitive instruments to detect departures from linearity. It’s actually a much better approximation than geometrical optics is, because you can see the diffraction phenomena produced by wave mechanics very easily in everyday life, if you know where to look; they’re quite strong whenever you have objects on the scale of a few microns involved, such as your eyelashes. They’re rarely more than one or two orders of magnitude away from visibility. Departures from linearity, by contrast, are usually six or more orders of magnitude away from visibility. So nonlinear optical systems are substantially more difficult to build.
There are a few that are common, though. Fluorescence is usually pretty linear, but it often has a substantial time constant, which means that it departs from instantaneous linearity. Optically-pumped lasers, however, are a sort of nonlinear fluorescence phenomenon: you don’t get a laser beam at all until the gain of the lasing medium rises past 1, as limited by the Q of your cavity. And the most common kind of green laser isn’t a green laser at all; it’s an infrared laser with a frequency-doubling crystal on the front of it, and that’s a nonlinear phenomenon — it doesn’t start happening until the light intensity is above a certain level.
Other nonlinear optical phenomena include phase-conjugating mirrors, Kerr cells, the self-focusing of intense laser beams, and soliton transmission, which is a sort of temporal analogue of spatial self-focusing. Any dielectric inevitably behaves nonlinearly to light passing through it, since its overunity refractive index is due to its response to the electric field of light being different from the response of the vacuum, and that’s an effect that inevitably reaches a limit at some field strength. Normally, though, light’s electric field is far too weak for us to notice this nonlinearity.
But, in the geometrical optics approximation, we invariably ignore these nonlinearities, because they are tiny in everyday life. So our transfer function is, in effect, transferring every separate light beam that could enter our apparatus into some distribution of light at the output. So our transformation function can be computed from a sort of point spread function of the form ℝ⁴ → (ℝ⁴ → ℝ).
However, the requirement that the function be reversible means that as the input light beam shrinks toward a perfectly collimated beam entering at a single point†, the output light beam must also shrink toward being such a thing, except perhaps at discontinuities. So it’s actually even simpler, and this is a simple case of a more general principle called “conservation of étendue”.
XXX is this really correct?
† For wave-mechanical reasons you can’t actually make a perfectly collimated beam entering at a single point — there’s a diffraction limit on the divergence — but here we’re talking about properties of the geometric-optics approximation.
Étendue is a quantity that
Electric current passes through a battery electrolyte not as free electrons, as in a metal, but as positive metal ions, and this is true whether you’re charging or discharging the battery. The positive ions are formed from the metal at the surface of the positive electrode, which has electrons running away from it down a wire, through a circuit, and back around to the negative electrode, where they travel to the surface of the metal and neutralize arriving positive ions, thus transmuting them back into insoluble metal.
You can use this process to coat some random conductive thing with a layer of metal, which is called galvanizing or electroplating — or electroforming, if you do it long enough — or to remove a thin layer from the surface of a piece of metal, which is called electropolishing, or cathodic corrosion if you do it by accident, like on a metal ship hull.
Because electric fields are strongest around edges and sharp points, electropolishing tends to remove those, leaving a mirror-like finish on initially rough metal. Also, since it doesn't
At the currents typically used, this process typically deposits around a nanometer per second of metal on one electrode and removes around a nanometer per second from the other. Much lower or higher currents don’t work as well.
https://en.wikipedia.org/wiki/View_factor https://en.wikipedia.org/wiki/Lagrange_invariant https://en.wikipedia.org/wiki/Etendue