Editing Diffraction grating (section)

==Theory of operation==
{{main|Diffraction}}
[[Image:Difraction grating reflecting green light.JPG|thumb|A [[Blazed grating|blazed diffraction grating]] reflecting only the green portion of the spectrum from a room's fluorescent lighting]]For a diffraction grating, the relationship between the grating spacing (i.e., the distance between adjacent grating grooves or slits), the angle of the wave (light) incidence to the grating, and the diffracted wave from the grating is known as the grating equation. Like many other optical formulas, the grating equation can be derived by using the [[Huygens–Fresnel principle]],<ref>{{Citation |title=Extended Huygens-Fresnel Principle |url=http://dx.doi.org/10.1117/3.549260.p24 |work=Field Guide to Atmospheric Optics |date=2004 |pages=24 |access-date=2023-09-17 |place=1000 20th Street, Bellingham, WA 98227-0010 USA |publisher=SPIE|doi=10.1117/3.549260.p24 |isbn=978-0-8194-5318-1 |url-access=subscription }}</ref> stating that each point on a wavefront of a propagating wave can be considered to act as a point wave source, and a wavefront at any subsequent point can be found by adding together the contributions from each of these individual point wave sources on the previous wavefront.

Gratings may be of the 'reflective' or 'transmissive' type, analogous to a mirror or lens, respectively. A grating has a 'zero-order mode' (where the integer order of diffraction ''m'' is set to zero), in which a ray of light behaves according to the laws of [[Specular reflection|reflection]] (like a mirror) and [[Snell's law|refraction]] (like a lens), respectively.

[[File:Diffraction Grating Equation.jpg|thumb|400x400px|A diagram showing the path difference between rays of light scattered from adjacent rulings at the same local position on each ruling of a reflective diffraction grating (actually a blazed grating). The choice of + or – in the path difference formula depends on the sign convention for<math>\Theta_\mathrm m</math>: plus if <math>\Theta_\mathrm m = \Theta_\mathrm i</math> describes the case of backscatter, minus if it describes specular reflection.

Note that the pair of the black ray path parts and the pair of the light green ray path parts have no path difference in each pair, while there is a path difference in the red ray path part pair that matters in the diffraction grating equation derivation.]]

An idealized diffraction grating is made up of a set of slits of spacing <math>d</math>, that must be wider than the wavelength of interest to cause diffraction. Assuming a [[plane wave]] of [[Monochromatic radiation|monochromatic light]] of wavelength <math>\lambda</math> at [[Normal (geometry)|normal]] incidence on a grating (i.e., wavefronts of the incident wave are parallel to the grating main plane), each slit in the grating acts as a quasi point wave source from which light propagates in all directions (although this is typically limited to the forward hemisphere from the point source). Of course, every point on every slit to which the incident wave reaches plays as a point wave source for the diffraction wave and all these contributions to the diffraction wave determine the detailed diffraction wave light property distribution, but diffraction angles (at the grating) at which the diffraction wave intensity is highest are determined only by these quasi point sources corresponding the slits in the grating. After the incident light (wave) interacts with the grating, the resulting diffracted light from the grating is composed of the sum of [[Interference (wave propagation)|interfering]]<ref>{{Cite journal |title=Interference of waves |url=http://dx.doi.org/10.1036/1097-8542.348700 |access-date=2023-09-17 |website=AccessScience|doi=10.1036/1097-8542.348700 |url-access=subscription }}</ref> wave components emanating from each slit in the grating; At any given point in space through which the diffracted light may pass, typically called observation point, the path length from each slit in the grating to the given point varies, so the phase of the wave emanating from each of the slits at that point also varies. As a result, the sum of the diffracted waves from the grating slits at the given observation point creates a peak, valley, or some degree between them in light intensity through additive and [[destructive interference]].  When the difference between the light paths from adjacent slits to the observation point is equal to an odd integer-multiple of the half of the wavelength, ''l''<math>l(\lambda/2)</math> with an odd integer <math>l</math>, the waves are out of phase at that point, and thus cancel each other to create the (locally) minimum light intensity. Similarly, when the path difference is a multiple of <math>\lambda</math>, the waves are in phase and the (locally) maximum intensity occurs. For light at the normal incidence to the grating, the intensity maxima occur at diffraction angles <math>\theta_m</math>, which satisfy the relationship <math>d \sin\theta_m = m\lambda</math>, where <math>\theta_m</math> is the angle between the diffracted ray and the grating's [[Normal (geometry)|normal]] vector, <math>d</math> is the distance from the center of one slit to the center of the adjacent slit, and <math>m</math> is an [[integer]] representing the propagation-mode of interest called the diffraction order.

[[File:comparison refraction diffraction spectra.svg|thumb|Comparison of the spectra obtained from a diffraction grating by diffraction (1), and a prism by refraction (2). Longer wavelengths (red) are diffracted more, but refracted less than shorter wavelengths (violet).|390x390px]]
[[File:Mehrfachspalt-Numerisch.png|thumb|Intensity as [[heatmap]] for monochromatic light behind a grating]]

When a [[plane wave|plane light wave]] is normally incident on a grating of uniform period <math>d</math>, the diffracted light has maxima at diffraction angles <math>\theta_m</math> given by a special case of the grating equation as
<math display="block">\sin\theta_m = \frac{m\lambda}{d}.</math>

It can be shown that if the plane wave is incident at angle <math>\theta_i</math> relative to the grating normal, in the plane orthogonal to the grating periodicity, the grating equation becomes
<math display="block">\sin\theta_i + \sin\theta_m = \frac{m\lambda}{d},</math>
which describes in-plane diffraction as a special case of the more general scenario of conical, or off-plane, diffraction described by the generalized grating equation:
<math display="block">\sin\theta_i + \sin\theta_m = \frac{m\lambda}{d\sin\gamma},</math>
where <math>\gamma</math> is the angle between the direction of the plane wave and the direction of the grating grooves, which is orthogonal to both the directions of grating periodicity and grating normal. 
Various sign conventions for <math>\theta_i</math>, <math>\theta_m</math> and <math>m</math> are used; any choice is fine as long as the choice is kept through diffraction-related calculations. When solved for diffracted angle at which the diffracted wave intensity are maximized, the equation becomes
<math display="block">\theta_m = \arcsin\!\left( \sin\theta_i -\frac{m\lambda}{d\sin\gamma}\right ).</math>

The diffracted light that corresponds to direct transmission for a transmissive diffraction grating or [[specular reflection]]<ref>{{Citation |title=specular reflection |url=http://www.springerreference.com/index/doi/10.1007/springerreference_25311 |work=SpringerReference |date=2011 |access-date=2023-09-17 |place=Berlin/Heidelberg |publisher=Springer-Verlag|doi=10.1007/springerreference_25311 |doi-broken-date=1 November 2024 |url-access=subscription }}</ref> for a reflective grating is called the zero order, and is denoted <math>m=0</math>. The other diffracted light intensity maxima occur at angles <math>\theta_m</math> represented by non-zero integer diffraction orders <math>m</math>. Note that <math>m</math> can be positive or negative, corresponding to diffracted orders on both sides of the zero-order diffracted beam.

Even if the grating equation is derived from a specific grating such as the grating in the right diagram (this grating is called a blazed grating), the equation can apply to any regular structure of the same spacing, because the phase relationship between light scattered from adjacent diffracting elements of the grating remains the same. The detailed diffracted light property distribution (e.g., intensity) depends on the detailed structure of the grating elements as well as on the number of elements in the grating, but it always gives maxima in the directions given by the grating equation.

Depending on how a grating modulates incident light on it to cause the diffracted light, there are the following grating types:<ref>{{Cite book|last=Hecht|first=Eugene|title=Optics|publisher=Pearson| year=2017|isbn=978-1-292-09693-3| pages=497| chapter=10.2.8. The Diffraction Grating}}</ref>

* Transmission amplitude diffraction grating, which spatially and periodically modulates the intensity of an incident wave that transmits through the grating (and the diffracted wave is the consequence of this modulation). 
* Reflection amplitude diffraction gratings, which spatially and periodically modulate the intensity of an incident wave that is reflected from the grating.
* Transmission phase diffraction grating, that spatially and periodically modulates the phase of an incident wave passing through the grating.
* Reflection phase diffraction grating, that spatially and periodically modulates the phase of an incident wave reflected from the grating.

An [[optical axis grating|optical axis diffraction grating]], in which the optical axis is spatially and periodically modulated, is also considered either a reflection or transmission phase diffraction grating.

The grating equation applies to all these gratings due to the same phase relationship between the diffracted waves from adjacent diffracting elements of the gratings, even if the detailed distribution of the diffracted wave property depends on the detailed structure of each grating.

===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>