Plane of polarization: Difference between revisions

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[[File:Field-vectors-and-propagation-directions.svg|thumb|300px|'''Fig.{{nnbsp}}1''':{{big| }}Field vectors ('''E''',{{hsp}}'''D''',{{hsp}}'''B''',{{hsp}}'''H''') and propagation directions (ray and wave-normal) for linearly-polarized plane electromagnetic waves in a non-magnetic birefringent crystal.{{r|lunney-weaire-2006}} The plane of vibration, containing both electric vectors ('''E''' & '''D''') and both propagation vectors, is sometimes called the "plane of polarization" by modern authors. Fresnel's "plane of polarization", traditionally used in optics, is the plane containing the magnetic vectors ('''B''' & '''H''') and the ''wave-normal''. Malus's original "plane of polarization" was the plane containing the magnetic vectors and the ''ray''.  (In an isotropic medium,  {{math|''θ'' {{=}} 0}}  and Malus's plane merges with Fresnel's.)]]

For [[light]] and other [[electromagnetic radiation]], the '''plane of polarization''', also known asis the '''[[plane of vibration''', is(geometry)|plane]] spanned by the direction of propagation ([[Poynting vector]]) and either the [[electric vector]] or the [[magnetic vector]], depending on the convention. It can be defined for [[polarization (physics)|polarized]] light, remains fixed in space for ''[[linear polarization|linearly-polarized]]'' light, and undergoes [[axial rotation]] for ''[[circular polarization|circularly-polarized]]'' light.

Unfortunately the two [[plane (geometry)|plane]] conventions are contradictory. As originally defined by [[Étienne-Louis Malus]] in 1811,<ref name=buch54>Buchwald, 1989, p.{{hsp}}54.</ref> the plane of polarization coincided (although this was not known at the time) with the plane containing the direction of propagation and the ''magnetic'' vector.<ref>Stratton, 1941, p.{{hsp}}280; Born & Wolf, 1970, pp.{{nnbsp}}43,{{tsp}}681.</ref> In modern literature, the term ''plane of polarization'', if it is used at all, is likely to mean the plane containing the direction of propagation and the ''electric'' vector,<ref name=luntz/> because the electric field has the greater propensity to interact with matter.<ref name="bw28">Born & Wolf, 1970, p.{{hsp}}28.</ref>

For waves in a [[birefringence|birefringent]] (doubly-refractive) crystal, under the old definition, one must also specify whether the direction of propagation means the ray direction ([[Poynting vector]]) or the wave-[[normal (geometry)|normal]] direction, because these directions generally differ and are both perpendicular to the magnetic vector (Fig.{{nnbsp}}1). Malus, as an adherent of the [[corpuscular theory of light]], could only choose the ray direction. But [[Augustin-Jean Fresnel]], in his successful effort to explain double refraction under the [[wave theory of light|wave theory]] (1822 onward), found it more useful to choose the wave-normal direction, with the result that the supposed vibrations of the medium were then consistently perpendicular to the plane of polarization.<ref name=fh318>Fresnel, 1827, tr.&nbsp;Hobson, p.{{nnbsp}}318.</ref> In an [[isotropy|isotropic]] medium such as air, the ray and wave-normal directions are the same, and Fresnel's modification makes no difference.

Fresnel also admitted that, had he not felt constrained by the received terminology, it would have been more natural to define the plane of polarization as the plane containing the vibrations and the direction of propagation.<ref name=fy406>Fresnel, 1822, tr.&nbsp;Young, part&nbsp;7, [https://books.google.com/books?id=N69MAAAAYAAJ&pg=PA406 p.{{nnbsp}}406].</ref> That plane, which became known as the '''plane of ''vibration''', is perpendicular to Fresnel's "plane of polarization" but identical with the plane that modern writers tend to call by that name!

It has been argued that the term ''plane of polarization'', because of its historical ambiguity, should be avoided in original writing. One can easily specify the orientation of a particular field vector; and even the term ''plane of vibration'' carries less risk of confusion than ''plane of polarization''.<ref>Born & Wolf, 1970, pp.{{nnbsp}}28,{{hsp}}43.</ref>

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Because innumerable materials are [[dielectric]]s or [[electrical conductor|conductors]] while comparatively few are [[ferromagnetism|ferromagnets]], the [[reflection (physics)|reflection]] or [[refraction]] of EM waves (including [[light]]) is more often due to differences in the ''electric'' properties of media than to differences in their magnetic properties. That circumstance tends to draw attention to the ''electric'' vectors, so that we tend to think of the direction of polarization as the direction of the electric vectors, and the "plane of polarization" as the plane containing the electric vectors and the direction of propagation.

[[File:Screen dish antenna.jpg|thumb|left|'''Fig.{{nnbsp}}3''':{{big|&nbsp;}}Vertically- polarized parabolic-grid [[microwave]] antenna. In this case the stated polarization refers to the alignment of the electric ('''E''') field, hence the alignment of the closely spaced metal ribs in the reflector.]]

Indeed, that is the convention used in the online ''Encyclopædia Britannica'',{{r|luntz}} and in [[Richard Feynman|Feynman]]'s lecture on polarization.{{r|feynman-1963}} In the latter case one must infer the convention from the context: Feynman keeps emphasizing the direction of the ''electric'' ('''E''') vector and leaves the reader to presume that the "plane of polarization" contains that vector — and this interpretation indeed fits the examples he gives. The same vector is used to describe the polarization of radio signals and [[antenna (radio)#Polarization|antennas]] (Fig.{{nnbsp}}3).<ref name="auto">Stratton, 1941, p.{{hsp}}280.</ref>

If the medium is magnetically isotropic but electrically ''non''-isotopicisotropic (like a [[birefringence|doubly-refracting]] crystal), the magnetic vectors '''B''' and '''H''' are still parallel, and the electric vectors '''E''' and '''D''' are still perpendicular to both, and the ray direction is still perpendicular to '''E''' and the magnetic vectors, and the wave-normal direction is still perpendicular to '''D''' and the magnetic vectors; but there is generally a small angle between the electric vectors '''E''' and '''D''', hence the same angle between the ray direction and the wave-normal direction (Fig.{{nnbsp}}1).{{r|lunney-weaire-2006}}<ref>Born & Wolf, 1970, p.{{hsp}}668.</ref>{{tsp}} Hence '''D''', '''E''', the wave-normal direction, and the ray direction are all in the same plane, and it is all the more natural to define that plane as the "plane of polarization".

This "natural" definition, however, depends on the theory of EM waves developed by [[James Clerk Maxwell]] in the 1860s — whereas the word ''polarization'' was coined about 50 years earlier, and the associated mystery dates back even further.

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[[File:Calcite and polarizing filter.gif|frame|'''Fig.{{nnbsp}}4''':{{big|&nbsp;}}Printed label seen through a doubly-refracting calcite crystal{{hsp}} and a modern polarizing filter (rotated to show the different polarizations of the two images).]]

Polarization was discovered — but not named or understood — by [[Christiaan Huygens]], as he investigated the [[birefringence|double refraction]] of "Iceland crystal" (transparent [[calcite]], now called [[Iceland spar]]). The essence of his discovery, published in his ''Treatise on Light'' (1690), was as follows. When a ray (meaning a narrow beam of light) passes through two similarly oriented calcite crystals at normal incidence, the ordinary ray emerging from the first crystal suffers only the ordinary refraction in the second, while the extraordinary ray emerging from the first suffers only the extraordinary refraction in the second. But when the second crystal is rotated 90° about the incident rays, the roles are interchanged, so that the ordinary ray emerging from the first crystal suffers only the extraordinary refraction in the second, and vice versa. At intermediate positions of the second crystal, each ray emerging from the first is doubly refracted by the second, giving four rays in total; and as the crystal is rotated from the initial orientation to the perpendicular one, the brightnesses of the rays vary, giving a smooth transition between the extreme cases in which there are only two final rays.<ref>Huygens, 1690, tr.&nbsp;Thompson, pp.{{nnbsp}}92–4.</ref>

Huygens defined a ''principal section'' of a calcite crystal as a plane normal to a natural surface and parallel to the axis of the obtuse solid angle.<ref>Huygens, 1690, tr.&nbsp;Thompson, pp.{{nnbsp}}55–6.</ref> This axis was parallel to the axes of the [[spheroid]]al [[Huygens–Fresnel principle|secondary waves]] by which he (correctly) explained the directions of the extraordinary refraction.

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[[File:Augustin Fresnel.jpg|thumb|<div style="text-align: center;">Augustin-Jean Fresnel (1788–1827).</div>]]

In 1821, [[Augustin-Jean Fresnel]] announced his hypothesis that light waves are exclusively ''[[transverse wave|transverse]]'' and therefore ''always'' polarized in the sense of having a particular transverse orientation, and that what we call ''[[unpolarized'' light]]'' is in fact light whose orientation is rapidly and randomly changing.<ref>Buchwald, 1989, pp.{{nnbsp}}227–9.</ref><ref name=fresnel-1821a>A. Fresnel, "Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées" et&nbsp;seq., ''Annales de Chimie et de Physique'', Ser.&nbsp;2, vol.&nbsp;17, pp.&nbsp;102–11 (May&nbsp;1821), 167–96 (June&nbsp;1821), 312–15 ("Postscript", July&nbsp;1821); reprinted (with added section nos.) in H.&nbsp;de Sénarmont, E.&nbsp;Verdet, and L.&nbsp;Fresnel (eds.), ''Oeuvres complètes d'Augustin Fresnel'', vol.&nbsp;1 (1866), pp.&nbsp;609–48; translated as "On the calculation of the tints that polarization develops in crystalline plates, &amp;&nbsp;postscript", {{Zenodo|4058004}} (Creative Commons), 2021.</ref> Supposing that light waves were analogous to [[s-wave|shear waves]] in [[elasticity (physics)|elastic solids]], and that a higher [[refractive index]] corresponded to a higher [[density]] of the [[luminiferous aether]], he found that he could account for the partial reflection (including polarization by reflection) at the interface between two transparent isotropic media, provided that the vibrations of the aether were perpendicular to the plane of polarization.<ref>Darrigol, 2012, p.{{nnbsp}}212.</ref> Thus the polarization, according to the received definition, was "in" a certain plane if the vibrations were ''perpendicular'' to that plane!

Fresnel himself found this implication inconvenient; later that year he wrote:

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==See also==

*[[E-plane and H-plane]]

*[[Plane of incidence]]

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* B. Powell (July&nbsp;1856), [https://archive.org/stream/s4philosophicalmag12londuoft#page/n13/mode/2up "On the demonstration of Fresnel's formulas for reflected and refracted light; and their applications"], ''Philosophical Magazine and Journal of Science'', Series&nbsp;4, vol.{{nnbsp}}12, no.{{hsp}}76, pp.{{nnbsp}}1–20.

* J.A. Stratton, 1941, ''Electromagnetic Theory'', New York: McGraw-Hill.

* [[E. T. Whittaker]], 1910, [[A History of the Theories of Aether and Electricity|''A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century'']], London: Longmans, Green, &amp; Co.

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