Editing Diffraction grating (section)

===Quantum electrodynamics===
[[File:Helical fluorescent lamp spectrum by diffraction grating.JPG|thumb|A helical fluorescent lamp photographed in a blazed reflection-diffraction grating, showing the various spectral lines produced by the lamp.]]

[[Quantum electrodynamics]] (QED) offers another derivation of the properties of a diffraction grating in terms of [[photon]]s as particles (at some level). QED can be described intuitively with the [[path integral formulation]] of quantum mechanics. As such it can model photons as potentially following all paths from a source to a final point, each path with a certain [[probability amplitude]]. These probability amplitudes can be represented as a complex number or equivalent vector—or, as [[Richard Feynman]] simply calls them in his book on QED, "arrows".

For the probability that a certain event will happen, one sums the probability amplitudes for all of the possible ways in which the event can occur, and then takes the square of the length of the result. The probability amplitude for a photon from a monochromatic source to arrive at a certain final point at a given time, in this case, can be modeled as an arrow that spins rapidly until it is evaluated when the photon reaches its final point. For example, for the probability that a photon will reflect off a mirror and be observed at a given point a given amount of time later, one sets the photon's probability amplitude spinning as it leaves the source, follows it to the mirror, and then to its final point, even for paths that do not involve bouncing off the mirror at equal angles. One can then evaluate the probability amplitude at the photon's final point; next, one can integrate over all of these arrows (see [[vector sum]]), and square the length of the result to obtain the probability that this photon will reflect off the mirror in the pertinent fashion. The times these paths take are what determines the angle of the probability amplitude arrow, as they can be said to "spin" at a constant rate (which is related to the frequency of the photon).

The times of the paths near the classical reflection site of the mirror are nearly the same, so the probability amplitudes point in nearly the same direction—thus, they have a sizable sum. Examining the paths towards the edges of the mirror reveals that the times of nearby paths are quite different from each other, and thus we wind up summing vectors that cancel out quickly. So, there is a higher probability that light will follow a near-classical reflection path than a path further out. However, a diffraction grating can be made out of this mirror, by scraping away areas near the edge of the mirror that usually cancel nearby amplitudes out—but now, since the photons don't reflect from the scraped-off portions, the probability amplitudes that would all point, for instance, at forty-five degrees, can have a sizable sum. Thus, this lets light of the right frequency sum to a larger probability amplitude, and as such possess a larger probability of reaching the appropriate final point.

This particular description involves many simplifications: a point source, a "surface" that light can reflect off (thus neglecting the interactions with electrons) and so forth. The biggest simplification is perhaps in the fact that the "spinning" of the probability amplitude arrows is actually more accurately explained as a "spinning" of the source, as the probability amplitudes of photons do not "spin" while they are in transit. We obtain the same variation in probability amplitudes by letting the time at which the photon left the source be indeterminate—and the time of the path now tells us when the photon would have left the source, and thus what the angle of its "arrow" would be. However, this model and approximation is a reasonable one to illustrate a diffraction grating conceptually. Light of a different frequency may also reflect off the same diffraction grating, but with a different final point.<ref>{{cite book|last=Feynman|first=Richard|title=QED: The Strange Theory of Light and Matter|year=1985|publisher=Princeton University Press|isbn=978-0691083889 |url=https://books.google.com/books?id=2o2JfTDiA40C}}</ref>