Editing Optics (section)

===Physical optics===
{{Main|Physical optics}}
In physical optics, light is considered to propagate as waves. This model predicts phenomena such as interference and diffraction, which are not explained by geometric optics. The [[speed of light]] waves in [[air]] is approximately 3.0×10<sup>8</sup>&nbsp;m/s (exactly 299,792,458&nbsp;m/s in [[vacuum]]). The [[wavelength]] of visible light waves varies between 400 and 700&nbsp;nm, but the term "light" is also often applied to infrared (0.7–300&nbsp;μm) and ultraviolet radiation (10–400&nbsp;nm).

The wave model can be used to make predictions about how an optical system will behave without requiring an explanation of what is "waving" in what medium. Until the middle of the 19th century, most physicists believed in an "ethereal" medium in which the light disturbance propagated.<ref>MV Klein & TE Furtak, 1986, Optics, John Wiley & Sons, New York {{ISBN|0-471-87297-0}}.</ref> The existence of electromagnetic waves was predicted in 1865 by [[Electromagnetic waves#Derivation from electromagnetic theory|Maxwell's equations]]. These waves propagate at the speed of light and have varying electric and magnetic fields which are orthogonal to one another, and also to the direction of propagation of the waves.<ref>{{cite journal |last=Maxwell |first=James Clerk |author-link=James Clerk Maxwell |title=A dynamical theory of the electromagnetic field |url=http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf |doi=10.1098/rstl.1865.0008 |journal=Philosophical Transactions of the Royal Society of London |volume=155 |page=499 |year=1865 |url-status=live |archive-url=https://web.archive.org/web/20110728140123/http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf |archive-date=2011-07-28 |bibcode=1865RSPT..155..459M |s2cid=186207827 }} This article accompanied a December 8, 1864, presentation by Maxwell to the Royal Society. See also [[A dynamical theory of the electromagnetic field]].</ref> Light waves are now generally treated as electromagnetic waves except when [[Optics#Modern optics|quantum mechanical effects]] have to be considered.

====Modelling and design of optical systems using physical optics====

Many simplified approximations are available for analysing and designing optical systems. Most of these use a single [[Scalar (physics)|scalar]] quantity to represent the electric field of the light wave, rather than using a [[Euclidean vector|vector]] model with orthogonal electric and magnetic vectors.<ref name = "Born and Wolf">M. Born and E. Wolf (1999). ''Principle of Optics''. Cambridge: Cambridge University Press. {{ISBN|0-521-64222-1}}.</ref>
The [[Huygens–Fresnel principle|Huygens–Fresnel]] equation is one such model. This was derived empirically by Fresnel in 1815, based on Huygens' hypothesis that each point on a wavefront generates a secondary spherical wavefront, which Fresnel combined with the principle of [[Superposition principle|superposition]] of waves. The [[Kirchhoff's diffraction formula|Kirchhoff diffraction equation]], which is derived using Maxwell's equations, puts the Huygens-Fresnel equation on a firmer physical foundation. Examples of the application of Huygens–Fresnel principle can be found in the articles on diffraction and [[Fraunhofer diffraction]].

More rigorous models, involving the modelling of both electric and magnetic fields of the light wave, are required when dealing with materials whose electric and magnetic properties affect the interaction of light with the material. For instance, the behaviour of a light wave interacting with a metal surface is quite different from what happens when it interacts with a dielectric material. A vector model must also be used to model polarised light.

[[Computer simulation|Numerical modeling]] techniques such as the [[finite element method]], the [[boundary element method]] and the [[transmission-line matrix method]] can be used to model the propagation of light in systems which cannot be solved analytically. Such models are computationally demanding and are normally only used to solve small-scale problems that require accuracy beyond that which can be achieved with analytical solutions.<ref>{{cite book|author=J. Goodman|year=2005|title=Introduction to Fourier Optics|edition=3rd|publisher=Roberts & Co Publishers|isbn=978-0-9747077-2-3|url= https://books.google.com/books?id=ow5xs_Rtt9AC}}</ref>

All of the results from geometrical optics can be recovered using the techniques of [[Fourier optics]] which apply many of the same mathematical and analytical techniques used in [[acoustic engineering]] and [[signal processing]].

[[Gaussian beam|Gaussian beam propagation]] is a simple paraxial physical optics model for the propagation of coherent radiation such as laser beams. This technique partially accounts for diffraction, allowing accurate calculations of the rate at which a laser beam expands with distance, and the minimum size to which the beam can be focused. Gaussian beam propagation thus bridges the gap between geometric and physical optics.<ref>{{cite book|author=A.E. Siegman|year=1986|title=Lasers|url=https://archive.org/details/lasers0000sieg|url-access=registration|publisher=University Science Books|isbn=978-0-935702-11-8}} Chapter 16.</ref>

====Superposition and interference====
{{Main|Superposition principle|Interference (optics)}}

In the absence of [[nonlinear optics|nonlinear]] effects, the superposition principle can be used to predict the shape of interacting waveforms through the simple addition of the disturbances.{{sfnp|Young|Freedman|2020|pp=1187–1188}} This interaction of waves to produce a resulting pattern is generally termed "interference" and can result in a variety of outcomes. If two waves of the same wavelength and frequency are ''in [[phase (waves)|phase]]'', both the wave crests and wave troughs align. This results in [[constructive interference]] and an increase in the amplitude of the wave, which for light is associated with a brightening of the waveform in that location. Alternatively, if the two waves of the same wavelength and frequency are out of phase, then the wave crests will align with wave troughs and vice versa. This results in [[destructive interference]] and a decrease in the amplitude of the wave, which for light is associated with a dimming of the waveform at that location. See below for an illustration of this effect.{{sfnp|Young|Freedman|2020|p=512, 1189}}

{|
|-
|'''combined<br> waveform'''
|colspan="2" rowspan="3"|[[File:Interference of two waves.svg|class=skin-invert-image]]
|-
|'''wave 1'''
|-
|'''wave 2'''
|-
|
|'''Two waves in phase'''
|'''Two waves 180° out <br>of phase'''
|}

[[File:Dieselrainbow.jpg|thumb|right|upright=1.35|When oil or fuel is spilled, colourful patterns are formed by thin-film interference.]]
Since the Huygens–Fresnel principle states that every point of a wavefront is associated with the production of a new disturbance, it is possible for a wavefront to interfere with itself constructively or destructively at different locations producing bright and dark fringes in regular and predictable patterns.{{sfnp|Young|Freedman|2020|pp=1191–1192}} [[Interferometry]] is the science of measuring these patterns, usually as a means of making precise determinations of distances or [[angular resolution]]s.<ref name=interferometry>{{cite book|author=P. Hariharan|title=Optical Interferometry|edition=2nd|publisher=Academic Press|place=San Diego, US|year=2003|url=http://www.astro.lsa.umich.edu/~monnier/Publications/ROP2003_final.pdf|isbn=978-0-12-325220-3|url-status=live|archive-url=https://web.archive.org/web/20080406215913/http://www.astro.lsa.umich.edu/~monnier/Publications/ROP2003_final.pdf|archive-date=2008-04-06}}</ref> The [[Michelson interferometer]] was a famous instrument which used interference effects to accurately measure the speed of light.<ref>{{cite book|author=E.R. Hoover|title=Cradle of Greatness: National and World Achievements of Ohio's Western Reserve|place=Cleveland|publisher=Shaker Savings Association|year=1977}}</ref>

The appearance of [[Thin film optics|thin films and coatings]] is directly affected by interference effects. [[Antireflective coating]]s use destructive interference to reduce the reflectivity of the surfaces they coat, and can be used to minimise glare and unwanted reflections. The simplest case is a single layer with a thickness of one-fourth the wavelength of incident light. The reflected wave from the top of the film and the reflected wave from the film/material interface are then exactly 180° out of phase, causing destructive interference. The waves are only exactly out of phase for one wavelength, which would typically be chosen to be near the centre of the visible spectrum, around 550&nbsp;nm. More complex designs using multiple layers can achieve low reflectivity over a broad band, or extremely low reflectivity at a single wavelength.

Constructive interference in thin films can create a strong reflection of light in a range of wavelengths, which can be narrow or broad depending on the design of the coating. These films are used to make [[dielectric mirror]]s, [[interference filter]]s, [[heat reflector]]s, and filters for colour separation in [[colour television]] cameras. This interference effect is also what causes the colourful rainbow patterns seen in oil slicks.{{sfnp|Young|Freedman|2020|pp=1198–1200}}

====Diffraction and optical resolution====
{{Main|Diffraction|Optical resolution}}
[[File:Double slit diffraction.svg|class=skin-invert-image|upright=1.35|right|thumb|Diffraction on two slits separated by distance {{mvar|d}}. The bright fringes occur along lines where black lines intersect with black lines and white lines intersect with white lines. These fringes are separated by angle {{mvar|θ}} and are numbered as order {{mvar|n}}.]]

Diffraction is the process by which light interference is most commonly observed. The effect was first described in 1665 by [[Francesco Maria Grimaldi]], who also coined the term from the Latin {{lang|la|diffringere}} {{gloss|to break into pieces}}.<ref>{{cite book |last= Aubert |first= J. L. |date= 1760 |title= Memoires pour l'histoire des sciences et des beaux arts |trans-title= Memoirs for the history of science and fine arts |language= fr |publisher= Impr. de S.A.S.; Chez E. Ganeau |place= Paris |page= [https://archive.org/details/memoirespourlhi146aubegoog/page/n151 149] |url= https://archive.org/details/memoirespourlhi146aubegoog}}</ref><ref>{{cite book |last= Brewster |first= D. |date= 1831 |title= A Treatise on Optics |publisher= Longman, Rees, Orme, Brown & Green and John Taylor |place= London |page= [https://archive.org/details/atreatiseonopti00brewgoog/page/n113 95] |url= https://archive.org/details/atreatiseonopti00brewgoog}}</ref> Later that century, Robert Hooke and Isaac Newton also described phenomena now known to be diffraction in [[Newton's rings]]<ref>{{cite book |last= Hooke |first= R. |date= 1665 |title= Micrographia: or, Some physiological descriptions of minute bodies made by magnifying glasses |url= https://archive.org/details/micrographiaorso00hook |place= London |publisher= J. Martyn and J. Allestry |isbn= 978-0-486-49564-4}}</ref> while [[James Gregory (astronomer and mathematician)|James Gregory]] recorded his observations of diffraction patterns from bird feathers.<ref>{{cite journal |last= Turnbull |first= H. W. |date= 1940–1941 |title= Early Scottish Relations with the Royal Society: I. James Gregory, F.R.S. (1638–1675) |journal= Notes and Records of the Royal Society of London |volume= 3 |pages= 22–38 |doi= 10.1098/rsnr.1940.0003|doi-access= free }}</ref>

The first physical optics model of diffraction that relied on the Huygens–Fresnel principle was developed in 1803 by Thomas Young in his interference experiments with the interference patterns of two closely spaced slits. Young showed that his results could only be explained if the two slits acted as two unique sources of waves rather than corpuscles.<ref>{{cite book |last= Rothman |first= T. |date= 2003 |author-link= Tony Rothman |title= Everything's Relative and Other Fables in Science and Technology |publisher= Wiley |place= New Jersey |isbn= 978-0-471-20257-8 |url-access= registration |url= https://archive.org/details/everythingsrelat0000roth}}</ref> In 1815 and 1818, Augustin-Jean Fresnel firmly established the mathematics of how wave interference can account for diffraction.{{sfnp|Hecht|2017|p=5}}

The simplest physical models of diffraction use equations that describe the angular separation of light and dark fringes due to light of a particular wavelength ({{mvar|λ}}). In general, the equation takes the form
<math display="block">m \lambda = d \sin \theta</math> where {{mvar|d}} is the separation between two wavefront sources (in the case of Young's experiments, it was [[Double-slit experiment|two slits]]), {{mvar|θ}} is the angular separation between the central fringe and the {{nowrap|{{mvar|m}}-th}} order fringe, where the central maximum is {{math|1= ''m'' = 0}}.{{sfnmp |1a1=Hecht|1y=2017|1pp=398–399 |2a1=Young|2a2=Freedman|2y=2020|2p=1192}}

This equation is modified slightly to take into account a variety of situations such as diffraction through a single gap, diffraction through multiple slits, or diffraction through a [[diffraction grating]] that contains a large number of slits at equal spacing.{{sfnmp |1a1=Hecht|1y=2017|1pp=488–491 |2a1=Young|2a2=Freedman|2y=2020|2pp=1224–1225}} More complicated models of diffraction require working with the mathematics of [[Fresnel diffraction|Fresnel]] or [[Fraunhofer diffraction]].<ref name=phyoptics>{{cite book |last= Longhurst |first= R. S. |date= 1968 |title= Geometrical and Physical Optics |edition= 2nd |publisher= Longmans |location= London |bibcode= 1967gpo..book.....L}}</ref>

[[X-ray diffraction]] makes use of the fact that atoms in a crystal have regular spacing at distances that are on the order of one [[angstrom]]. To see diffraction patterns, x-rays with similar wavelengths to that spacing are passed through the crystal. Since crystals are three-dimensional objects rather than two-dimensional gratings, the associated diffraction pattern varies in two directions according to [[Bragg reflection]], with the associated bright spots occurring in [[Diffraction topography|unique patterns]] and {{mvar|d}} being twice the spacing between atoms.{{sfnmp |1a1=Hecht|1y=2017|1p=497 |2a1=Young|2a2=Freedman|2y=2020|2pp=1228–1230}}

Diffraction effects limit the ability of an optical detector to [[optical resolution|optically resolve]] separate light sources. In general, light that is passing through an [[aperture]] will experience diffraction and the best images that can be created (as described in [[Diffraction-limited system|diffraction-limited optics]]) appear as a central spot with surrounding bright rings, separated by dark nulls; this pattern is known as an [[Airy pattern]], and the central bright lobe as an [[Airy disk]].{{sfnp|Hecht|2017|p=482}} The size of such a disk is given by <math display="block"> \sin \theta = 1.22 \frac{\lambda}{D}</math> where {{mvar|θ}} is the angular resolution, {{mvar|λ}} is the wavelength of the light, and {{mvar|D}} is the [[diameter]] of the lens aperture. If the angular separation of the two points is significantly less than the Airy disk angular radius, then the two points cannot be resolved in the image, but if their angular separation is much greater than this, distinct images of the two points are formed and they can therefore be resolved. [[John Strutt, 3rd Baron Rayleigh|Rayleigh]] defined the somewhat arbitrary "[[Angular resolution#The Rayleigh criterion|Rayleigh criterion]]" that two points whose angular separation is equal to the Airy disk radius (measured to first null, that is, to the first place where no light is seen) can be considered to be resolved. It can be seen that the greater the diameter of the lens or its aperture, the finer the resolution.{{sfnmp |1a1=Hecht|1y=2017|1p=485 |2a1=Young|2a2=Freedman|2y=2020|2p=1232}} [[Astronomical interferometer|Interferometry]], with its ability to mimic extremely large baseline apertures, allows for the greatest angular resolution possible.<ref name=interferometry />

For astronomical imaging, the atmosphere prevents optimal resolution from being achieved in the visible spectrum due to the atmospheric [[scattering]] and dispersion which cause stars to [[Scintillation (astronomy)|twinkle]]. Astronomers refer to this effect as the quality of [[astronomical seeing]]. Techniques known as [[adaptive optics]] have been used to eliminate the atmospheric disruption of images and achieve results that approach the diffraction limit.<ref>{{cite thesis |last= Tubbs |first= Robert Nigel |date= September 2003 |type= PhD thesis |title= Lucky Exposures: Diffraction limited astronomical imaging through the atmosphere |url= http://www.mrao.cam.ac.uk/telescopes/coast/theses/rnt/ |publisher=Cambridge University |archive-url= https://web.archive.org/web/20081005013157/http://www.mrao.cam.ac.uk/telescopes/coast/theses/rnt/ |archive-date= 2008-10-05}}</ref>

====Dispersion and scattering====
{{Main|Dispersion (optics)|Scattering}}

[[File:Light dispersion conceptual waves.gif|thumb|right|Conceptual animation of light dispersion through a prism. High frequency (blue) light is deflected the most, and low frequency (red) the least.]]

Refractive processes take place in the physical optics limit, where the wavelength of light is similar to other distances, as a kind of scattering. The simplest type of scattering is [[Thomson scattering]] which occurs when electromagnetic waves are deflected by single particles. In the limit of Thomson scattering, in which the wavelike nature of light is evident, light is dispersed independent of the frequency, in contrast to [[Compton scattering]] which is frequency-dependent and strictly a [[quantum mechanical]] process, involving the nature of light as particles. In a statistical sense, elastic scattering of light by numerous particles much smaller than the wavelength of the light is a process known as [[Rayleigh scattering]] while the similar process for scattering by particles that are similar or larger in wavelength is known as [[Mie scattering]] with the [[Tyndall effect]] being a commonly observed result. A small proportion of light scattering from atoms or molecules may undergo [[Raman scattering]], wherein the frequency changes due to excitation of the atoms and molecules. [[Brillouin scattering]] occurs when the frequency of light changes due to local changes with time and movements of a dense material.<ref>{{cite book|author1=C.F. Bohren  |author2=D.R. Huffman |name-list-style=amp |title=Absorption and Scattering of Light by Small Particles|publisher=Wiley|year=1983|isbn=978-0-471-29340-8}}</ref>

Dispersion occurs when different frequencies of light have different [[phase velocity|phase velocities]], due either to material properties (''material dispersion'') or to the geometry of an [[optical waveguide]] (''waveguide dispersion''). The most familiar form of dispersion is a decrease in index of refraction with increasing wavelength, which is seen in most transparent materials. This is called "normal dispersion". It occurs in all [[dielectric|dielectric materials]], in wavelength ranges where the material does not absorb light.<ref name=J286>{{cite book|author=J.D. Jackson|title=Classical Electrodynamics|edition=2nd|publisher=Wiley|year=1975|isbn=978-0-471-43132-9|page=[https://archive.org/details/classicalelectro00jack_0/page/286 286]|url=https://archive.org/details/classicalelectro00jack_0/page/286}}</ref> In wavelength ranges where a medium has significant absorption, the index of refraction can increase with wavelength. This is called "anomalous dispersion".<ref name=J286/>

The separation of colours by a prism is an example of normal dispersion. At the surfaces of the prism, Snell's law predicts that light incident at an angle {{mvar|θ}} to the normal will be refracted at an angle {{math|arcsin(sin (''θ'') / ''n'')}}. Thus, blue light, with its higher refractive index, is bent more strongly than red light, resulting in the well-known [[rainbow]] pattern.{{sfnp|Young|Freedman|2020|p=1116}}

[[File:Wave group.gif|frame|Dispersion: two sinusoids propagating at different speeds make a moving interference pattern. The red dot moves with the [[phase velocity]], and the green dots propagate with the [[group velocity]]. In this case, the phase velocity is twice the group velocity. The red dot overtakes two green dots, when moving from the left to the right of the figure. In effect, the individual waves (which travel with the phase velocity) escape from the wave packet (which travels with the group velocity).]]

Material dispersion is often characterised by the [[Abbe number]], which gives a simple measure of dispersion based on the index of refraction at three specific wavelengths. Waveguide dispersion is dependent on the [[propagation constant]].{{sfnp|Hecht|2017|pp=202–204}} Both kinds of dispersion cause changes in the group characteristics of the wave, the features of the wave packet that change with the same frequency as the amplitude of the electromagnetic wave. "Group velocity dispersion" manifests as a spreading-out of the signal "envelope" of the radiation and can be quantified with a group dispersion delay parameter:

<math display="block">D = \frac{1}{v_\mathrm{g}^2} \frac{dv_\mathrm{g}}{d\lambda}</math>

where {{math|''v''{{sub|g}}}} is the group velocity.<ref name=optnet>{{cite book |author1=R. Ramaswami |author2=K.N. Sivarajan |title=Optical Networks: A Practical Perspective |url=https://books.google.com/books?id=WpByp4Ip0z8C |isbn=978-0-12-374092-2 |publisher=Academic Press |location=London |year=1998 |url-status=live |archive-url=https://web.archive.org/web/20151027164628/https://books.google.com/books?id=WpByp4Ip0z8C&printsec=frontcover |archive-date=2015-10-27 }}</ref> For a uniform medium, the group velocity is

<math display="block">v_\mathrm{g} = c \left( n - \lambda \frac{dn}{d\lambda} \right)^{-1}</math>

where {{mvar|n}} is the index of refraction and {{mvar|c}} is the speed of light in a vacuum.<ref>Brillouin, Léon. ''Wave Propagation and Group Velocity''. Academic Press Inc., New York (1960)</ref> This gives a simpler form for the dispersion delay parameter:

<math display="block">D = - \frac{\lambda}{c} \, \frac{d^2 n}{d \lambda^2}.</math>

If {{mvar|D}} is less than zero, the medium is said to have ''positive dispersion'' or normal dispersion. If {{mvar|D}} is greater than zero, the medium has ''negative dispersion''. If a light pulse is propagated through a normally dispersive medium, the result is the higher frequency components slow down more than the lower frequency components. The pulse therefore becomes ''positively [[chirp]]ed'', or ''up-chirped'', increasing in frequency with time. This causes the spectrum coming out of a prism to appear with red light the least refracted and blue/violet light the most refracted. Conversely, if a pulse travels through an anomalously (negatively) dispersive medium, high-frequency components travel faster than the lower ones, and the pulse becomes ''negatively chirped'', or ''down-chirped'', decreasing in frequency with time.<ref>{{cite book|author1=M. Born  |author2=E. Wolf |name-list-style=amp |author-link = Max Born|title=Principle of Optics|publisher=Cambridge University Press|year=1999|location=Cambridge|pages=14–24|isbn=978-0-521-64222-4}}</ref>

The result of group velocity dispersion, whether negative or positive, is ultimately temporal spreading of the pulse. This makes dispersion management extremely important in optical communications systems based on [[optical fibre]]s, since if dispersion is too high, a group of pulses representing information will each spread in time and merge, making it impossible to extract the signal.<ref name=optnet />

====Polarisation <span class="anchor" id="Polarization"></span>====
{{Main|Polarisation (waves)}}

Polarisation is a general property of waves that describes the orientation of their oscillations. For [[transverse wave]]s such as many electromagnetic waves, it describes the orientation of the oscillations in the plane perpendicular to the wave's direction of travel. The oscillations may be oriented in a single direction ([[linear polarisation]]), or the oscillation direction may rotate as the wave travels ([[circular polarisation|circular]] or [[elliptical polarisation]]). Circularly polarised waves can rotate rightward or leftward in the direction of travel, and which of those two rotations is present in a wave is called the wave's [[polarimetry|chirality]].{{sfnmp |1a1=Hecht|1y=2017|1pp=333–334 |2a1=Young|2a2=Freedman|2y=2020|2pp=1083,1118}}

The typical way to consider polarisation is to keep track of the orientation of the electric field [[vector (geometry)|vector]] as the electromagnetic wave propagates. The electric field vector of a plane wave may be arbitrarily divided into two perpendicular [[vector components|components]] labeled {{mvar|x}} and {{mvar|y}} (with {{math|'''z'''}} indicating the direction of travel). The shape traced out in the x-y plane by the electric field vector is a [[Lissajous curve|Lissajous figure]] that describes the ''polarisation state''.{{sfnp|Hecht|2017|p=336}} The following figures show some examples of the evolution of the electric field vector (blue), with time (the vertical axes), at a particular point in space, along with its {{mvar|x}} and {{mvar|y}} components (red/left and green/right), and the path traced by the vector in the plane (purple): The same evolution would occur when looking at the electric field at a particular time while evolving the point in space, along the direction opposite to propagation.

<div style="float:left;width:170px">
[[File:Polarisation (Linear).svg|class=skin-invert-image|center|Linear polarisation diagram]]
{{center|''Linear''}}
</div>
<div style="float:left;width:170px">
[[File:Polarisation (Circular).svg|class=skin-invert-image|center|Circular polarisation diagram]]
{{center|''Circular''}}
</div>
<div style="float:left;width:170px">
[[File:Polarisation (Elliptical).svg|class=skin-invert-image|center|Elliptical polarisation diagram]]
{{center|''Elliptical polarisation''}}
</div>
{{Clear}}

In the leftmost figure above, the {{mvar|x}} and {{mvar|y}} components of the light wave are in phase. In this case, the ratio of their strengths is constant, so the direction of the electric vector (the vector sum of these two components) is constant. Since the tip of the vector traces out a single line in the plane, this special case is called linear polarisation. The direction of this line depends on the relative amplitudes of the two components.{{sfnmp|1a1=Hecht|1y=2017|1pp=330–332|2a1=Young|2a2=Freedman|2y=2020|2p=1123}}

In the middle figure, the two orthogonal components have the same amplitudes and are 90° out of phase. In this case, one component is zero when the other component is at maximum or minimum amplitude. There are two possible phase relationships that satisfy this requirement: the {{mvar|x}} component can be 90° ahead of the {{mvar|y}} component or it can be 90° behind the {{mvar|y}} component. In this special case, the electric vector traces out a circle in the plane, so this polarisation is called circular polarisation. The rotation direction in the circle depends on which of the two-phase relationships exists and corresponds to ''right-hand circular polarisation'' and ''left-hand circular polarisation''.{{sfnmp|1a1=Hecht|1y=2017|1pp=333–334|2a1=Young|2a2=Freedman|2y=2020|2p=1123}}

In all other cases, where the two components either do not have the same amplitudes and/or their phase difference is neither zero nor a multiple of 90°, the polarisation is called elliptical polarisation because the electric vector traces out an [[ellipse]] in the plane (the ''polarisation ellipse'').{{sfnmp|1a1=Hecht|1y=2017|1pp=334–335|2a1=Young|2a2=Freedman|2y=2020|2p=1124}} This is shown in the above figure on the right. Detailed mathematics of polarisation is done using [[Jones calculus]] and is characterised by the [[Stokes parameters]].{{sfnp|Hecht|2017|pp=379–383}}

=====Changing polarisation=====
Media that have different indexes of refraction for different polarisation modes are called ''[[birefringence|birefringent]]''.{{sfnp|Young|Freedman|2020|p=1124}} Well known manifestations of this effect appear in optical [[wave plate]]s/retarders (linear modes) and in [[Faraday rotation]]/[[optical rotation]] (circular modes).{{sfnp|Hecht|2017|pp=367,373}} If the path length in the birefringent medium is sufficient, plane waves will exit the material with a significantly different propagation direction, due to refraction. For example, this is the case with macroscopic crystals of [[calcite]], which present the viewer with two offset, orthogonally polarised images of whatever is viewed through them. It was this effect that provided the first discovery of polarisation, by [[Erasmus Bartholinus]] in 1669. In addition, the phase shift, and thus the change in polarisation state, is usually frequency dependent, which, in combination with [[dichroism]], often gives rise to bright colours and rainbow-like effects. In [[mineralogy]], such properties, known as [[pleochroism]], are frequently exploited for the purpose of identifying minerals using polarisation microscopes. Additionally, many plastics that are not normally birefringent will become so when subject to [[mechanical stress]], a phenomenon which is the basis of [[photoelasticity]].{{sfnmp |1a1=Hecht|1y=2017|1p=372 |2a1=Young|2a2=Freedman|2y=2020|2pp=1124–1125}} Non-birefringent methods, to rotate the linear polarisation of light beams, include the use of prismatic [[polarisation rotator]]s which use total internal reflection in a prism set designed for efficient collinear transmission.<ref>{{cite book |author=F.J. Duarte |author-link=F. J. Duarte |title=Tunable Laser Optics |edition=2nd |publisher=CRC |year=2015 |location=New York |pages=117–120 |isbn=978-1-4822-4529-5 |url=http://www.tunablelaseroptics.com |url-status=live |archive-url=https://web.archive.org/web/20150402145942/https://www.tunablelaseroptics.com/ |archive-date=2015-04-02 }}</ref>

[[File:Malus law.svg|class=skin-invert-image|right|thumb|upright=1.6|A polariser changing the orientation of linearly polarised light. In this picture, {{math|1= ''θ''{{sub|1}} – ''θ''{{sub|0}} = ''θ''{{sub|i}}}}.]]

Media that reduce the amplitude of certain polarisation modes are called ''dichroic'', with devices that block nearly all of the radiation in one mode known as ''polarising filters'' or simply "[[polariser]]s". Malus' law, which is named after [[Étienne-Louis Malus]], says that when a perfect polariser is placed in a linear polarised beam of light, the intensity, {{mvar|I}}, of the light that passes through is given by

<math display="block"> I = I_0 \cos^2 \theta_\mathrm{i} ,</math>
where {{math|''I''{{sub|0}}}} is the initial intensity, and {{math|''θ''{{sub|i}}}} is the angle between the light's initial polarisation direction and the axis of the polariser.{{sfnmp |1a1=Hecht|1y=2017|1p=338 |2a1=Young|2a2=Freedman|2y=2020|2pp=1119–1121}}

A beam of unpolarised light can be thought of as containing a uniform mixture of linear polarisations at all possible angles. Since the average value of {{math|cos{{sup|2}} ''θ''}} is 1/2, the transmission coefficient becomes

<math display="block"> \frac {I}{I_0} = \frac {1}{2}\,.</math>

In practice, some light is lost in the polariser and the actual transmission of unpolarised light will be somewhat lower than this, around 38% for Polaroid-type polarisers but considerably higher (>49.9%) for some birefringent prism types.{{sfnp|Hecht|2017|pp=339–342}}

In addition to birefringence and dichroism in extended media, polarisation effects can also occur at the (reflective) interface between two materials of different refractive index. These effects are treated by the [[Fresnel equations]]. Part of the wave is transmitted and part is reflected, with the ratio depending on the angle of incidence and the angle of refraction. In this way, physical optics recovers [[Brewster's angle]].{{sfnp|Hecht|2017|pp=355–358}} When light reflects from a [[Thin-film optics|thin film]] on a surface, interference between the reflections from the film's surfaces can produce polarisation in the reflected and transmitted light.

=====Natural light=====
[[File:CircularPolarizer.jpg|right|thumb|upright=1.8|The effects of a [[photographic filter#Polarizer|polarising filter]] on the sky in a photograph. Left picture is taken without polariser. For the right picture, filter was adjusted to eliminate certain polarisations of the scattered blue light from the sky.]]
Most sources of electromagnetic radiation contain a large number of atoms or molecules that emit light. The orientation of the electric fields produced by these emitters may not be [[statistical correlation|correlated]], in which case the light is said to be ''unpolarised''. If there is partial correlation between the emitters, the light is ''partially polarised''. If the polarisation is consistent across the spectrum of the source, partially polarised light can be described as a superposition of a completely unpolarised component, and a completely polarised one. One may then describe the light in terms of the [[degree of polarisation]], and the parameters of the polarisation ellipse.{{sfnp|Hecht|2017|p=336}}

Light reflected by shiny transparent materials is partly or fully polarised, except when the light is normal (perpendicular) to the surface. It was this effect that allowed the mathematician Étienne-Louis Malus to make the measurements that allowed for his development of the first mathematical models for polarised light. Polarisation occurs when light is scattered in the [[earth's atmosphere|atmosphere]]. The scattered light produces the brightness and colour in clear [[sky|skies]]. This partial polarisation of scattered light can be taken advantage of using polarising filters to darken the sky in [[science of photography|photographs]]. Optical polarisation is principally of importance in [[chemistry]] due to [[circular dichroism]] and optical rotation (''circular birefringence'') exhibited by [[optical activity|optically active]] ([[chirality (chemistry)|chiral]]) [[molecules]].{{sfnp|Hecht|2017|pp=353–356}}