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{{Short description|Concept in optics}}
[[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''' is the [[plane (geometry)|plane]] spanned by the direction of propagation 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.
The term '''''plane of polarization''''' refers to the direction of [[polarization (waves)|polarization]] of [[linear polarization|linearly-polarized]] light or other [[electromagnetic radiation]]. Unfortunately the term is used with two contradictory meanings. 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>▼
▲
In a [[birefringence|birefringent]] (doubly-refractive) crystal, under the old definition, one must also specify whether the direction of propagation means the ray direction 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. Hobson, p.{{nnbsp}}318.</ref>▼
▲
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. Young, part 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!▼
▲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. Young, part 7, [https://books.google.com/books?id=N69MAAAAYAAJ&pg=PA406 p.{{nnbsp}}406].</ref> That plane, which became known as the
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| }}Vertically
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''-
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| }}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
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. 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:Etienne-Louis Malus.jpg|thumb|left|<div style="text-align: center;">Étienne-Louis Malus (1775–1812).</div>]]
The term ''polarization'' was coined by [[Étienne-Louis Malus]] in 1811.<ref name=buch54/> In 1808, in the midst of confirming Huygens' geometric description of double refraction (while disputing his physical explanation), Malus had discovered that when a ray of light is reflected off a non-metallic surface at the appropriate angle, it behaves like ''one'' of the two rays emerging from a calcite crystal.<ref>Buchwald, 1989, pp.{{nnbsp}}31–43; Darrigol, 2012, pp.{{nnbsp}}191–2.</ref><ref group=Note>The angle of reflection at which this modification occurs became known as ''[[Brewster's angle]]'', after its dependence on the refractive index was determined experimentally by [[David Brewster]] in 1815.</ref> As this behavior had previously been known only in connection with double refraction, Malus described it in that context. In particular, he defined the ''plane of polarization'' of a polarized ray as the plane, containing the ray, in which a principal section of a calcite crystal must lie in order to cause only ''ordinary'' refraction.<ref>Buchwald, 1989, p.{{hsp}}45.</ref> This definition was all the more reasonable because it meant that when a ray was polarized by reflection (off an isotopic medium), the plane of polarization was the [[plane of incidence]] and reflection — that is, the plane containing the incident ray, the normal to the reflective surface, and the polarized reflected ray. But, as we now know, this plane happens to contain the ''magnetic'' vectors of the polarized ray, not the electric vectors.<ref>Born & Wolf, 1970, pp.{{nnbsp}}43,{{tsp}}681.</ref>
The plane of the ray and the magnetic vectors is the one numbered [[#plane2b|(2b)]] above. The implication that the plane of polarization contains the ''magnetic'' vectors is still found in the definition given in the online Merriam-Webster dictionary.{{r|merriamW}} Even [[Julius Adams Stratton]], having said that "It is customary to define the polarization in terms of '''E'''", promptly adds: "In optics, however, the orientation of the vectors is specified traditionally by the 'plane of polarization,' by which is meant the plane normal to '''E''' containing both '''H''' and the axis of propagation."<ref
=== Fresnel's choice ===
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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
Fresnel himself found this implication inconvenient; later that year he wrote:
::Adopting this hypothesis, it would have been more natural to have called the plane of polarisation that in which the oscillations are supposed to be made: but I wished to avoid making any change in the received appellations.<ref name=fy406/><ref group=Note>The actual writing of this treatise (Fresnel, 1822) was apparently completed by mid 1821; see I. Grattan-Guinness, ''Convolutions in French Mathematics, 1800–1840'', Basel: Birkhäuser, 1990, vol.{{tsp}}2, p.{{nnbsp}}884.</ref>
But he soon felt obliged to make a less radical change. In his successful model of double refraction, the displacement of the medium was constrained to be tangential to the wavefront, while the force was allowed to deviate from the displacement and from the wavefront.<ref>Aldis, 1879, pp.{{nnbsp}}8–9.</ref> Hence, if the vibrations were perpendicular to the plane of polarization, then the plane of polarization contained the wave-normal but not necessarily the ray.<ref>Aldis, 1879, pp.{{nnbsp}}9,{{hsp}}20.</ref> In his "Second Memoir" on double refraction, Fresnel formally adopted this new definition, acknowledging that it agreed with the old definition in an isotropic medium such as air, but not in a birefringent crystal.<ref name=fh318/>
The vibrations normal to Malus's plane of polarization are electric, and the electric vibration tangential to the wavefront is '''D''' (Fig.{{nnbsp}}1). Thus, in terms of the above numbering, Fresnel changed the "plane of polarization" from [[#plane2b|(2b)]] to [[#plane2a|(2a)]]. Fresnel's definition remains compatible with the Merriam-Webster definition,{{r|merriamW}} which fails to specify the propagation direction. And it remains compatible with Stratton's definition,<ref
What Fresnel called the "more natural" choice was a plane containing '''D''' and a direction of propagation. In Fig.{{nnbsp}}1, the only plane meeting that specification is the one labeled "Plane of vibration" and later numbered [[#plane1|(1)]] — that is, the one that ''modern'' authors tend to identify with the "plane of polarization". We might therefore wish that Fresnel had been less deferential to his predecessors. That scenario, however, is less realistic than it may seem, because even after Fresnel's transverse-wave theory was generally accepted, the direction of the vibrations was the subject of continuing debate.
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=== "Plane of vibration" ===
The principle that refractive index depended on the density of the aether was essential to Fresnel's [[aether drag hypothesis]].<ref>Darrigol, 2012, pp.{{nnbsp}}258–60.</ref> But it could not be extended to birefringent crystals — in which at least one refractive index varies with direction — because density is not directional. Hence his explanation of refraction required a directional variation in [[stiffness]] of the aether ''within'' a birefringent medium, plus a variation in density ''between'' media.<ref>Whittaker, 1910, pp.{{nnbsp}}127,{{tsp}}132–5.</ref>
[[James MacCullagh]] and [[Franz Ernst Neumann]] avoided this complication by supposing that a higher refractive index corresponded always to the same density but a greater elastic ''compliance'' (lower stiffness). To obtain results that agreed with observations on partial reflection, they had to suppose, contrary to Fresnel, that the vibrations were ''within'' the plane of polarization.<ref>Powell, 1856, pp.{{nnbsp}}4–5; Whittaker, 1910, p.{{nnbsp}}149.</ref>
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[[File:Portrait of George Gabriel Stokes (1819-1903), Physicist and Mathematician (2551115803).jpg|thumb|left|<div style="text-align: center;">George Gabriel Stokes (1819–1903).</div>]]
The question called for
In 1852, Stokes noted a much simpler experiment that leads to the same conclusion. Sunlight scattered from a patch of blue sky 90° from the sun is found, by the methods of Malus, to be polarized in the plane containing the line of sight and the sun. But it is obvious from the geometry that the vibrations of that light can only be perpendicular to that plane.<ref>Whittaker, 1910, pp.{{nnbsp}}169–70.</ref>
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The electromagnetic theory of light further emphasized the ''electric'' vibrations because of their interactions with matter,<ref name=bw28/> whereas the old "plane of polarization" contained the ''magnetic'' vectors. Hence the electromagnetic theory would have reinforced the convention that the vibrations were normal to the plane of polarization — provided, of course, that one was familiar with the historical definition of the plane of polarization. But if one was influenced by physical considerations ''alone'', then, as Feynman{{r|feynman-1963}} and the ''Britannica''{{r|luntz}} illustrate, one would pay attention to the electric vectors and assume that the "plane" of polarization (if one needed such a concept) contained those vectors.
However, it is not clear that a "plane of polarization" is needed at all: knowing what field vectors are involved, one can specify the polarization by specifying the orientation of a particular vector, or, as [[Max Born|Born]] and [[Emil Wolf|Wolf]] suggest, by specifying the "plane of vibration" of that vector.<ref
== Remaining uses ==
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In most contexts, however, the concept of a "plane of polarization" distinct from a plane containing the electric "vibrations" has arguably become redundant, and has certainly become a source of confusion. In the words of Born & Wolf, "it is… better not to use this term."<ref>Born & Wolf, 1970, p.{{hsp}}43.</ref>
==See also==
*[[E-plane and H-plane]]
*[[Plane of incidence]]
==Notes==
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<ref name=carcione-cavallini-1995>J.M. Carcione and F. Cavallini, [http://www.lucabaradello.it/carcione/CC95b.pdf "On the acoustic-electromagnetic analogy"], ''Wave Motion'', vol.{{nnbsp}}21 (1995), pp.{{nnbsp}}149–62. (Note that the authors' analogy is only two-dimensional.)</ref>
<ref name=feynman-1963>R.P. Feynman, R.B. Leighton, and M. Sands, ''The Feynman Lectures on Physics'', California Institute of Technology, 1963–2013, Volume {{serif|I}}, [
<ref name=jenkins-white-1976>Cf. F.A. Jenkins and H.E. White, ''Fundamentals of Optics'', 4th Ed., New York: McGraw-Hill, 1976, {{ISBN|0-07-032330-5}}, pp.{{nnbsp}}553–4, including Fig.{{nnbsp}}26{{serif|I}}.</ref>
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* W.S. Aldis, 1879, [https://archive.org/details/chapteronfresnel00aldirich ''A Chapter on Fresnel's Theory of Double Refraction''], 2nd Ed., Cambridge: Deighton, Bell, & Co. / London: George Bell & Sons.
* M. Born and E. Wolf, 1970, ''[[Principles of Optics]]'', 4th Ed., Oxford: Pergamon Press.
* J.Z. Buchwald, 1989, ''The Rise of the Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century'', University of Chicago Press, {{ISBN|0-226-07886-8}}.
* O. Darrigol, 2012, ''A History of Optics: From Greek Antiquity to the Nineteenth Century'', Oxford, {{ISBN|978-0-19-964437-7}}.
* A. Fresnel, 1822, ''De la Lumière'' (''On Light''), in J. Riffault (ed.), [https://books.google.com/books?id=DOoTAAAAQAAJ ''Supplément à la traduction française de la cinquième édition du "Système de Chimie" par Th.{{tsp}}Thomson''], Paris: Chez Méquignon-Marvis, 1822, pp.{{nnbsp}}1–137,{{tsp}}535–9; reprinted in Fresnel, 1866–70, vol.{{nnbsp}}2, [https://books.google.com/books?id=g6tzUG7JmoQC&pg=PA3 pp.{{nnbsp}}3–146]; translated by T. Young as "Elementary view of the undulatory theory of light", ''Quarterly Journal of Science, Literature, and Art'', vol.{{nnbsp}}22 (Jan.–{{hsp}}Jun.{{tsp}}1827), pp.{{nnbsp}}[https://books.google.com/books?id=C7JMAAAAYAAJ&pg=PA127 127–41], [https://books.google.com/books?id=C7JMAAAAYAAJ&pg=PA441 441–54]; vol.{{nnbsp}}23 (Jul.–{{hsp}}Dec.{{tsp}}1827), pp.{{nnbsp}}[https://books.google.com/books?id=r7VMAAAAYAAJ&pg=PA113 113–35], [https://books.google.com/books?id=r7VMAAAAYAAJ&pg=PA431 431–48]; vol.{{nnbsp}}24 (Jan.–{{hsp}}Jun.{{tsp}}1828), pp.{{nnbsp}}[https://books.google.com/books?id=SbJMAAAAYAAJ&pg=PA198 198–215]; vol.{{nnbsp}}25 (Jul.–{{hsp}}Dec.{{tsp}}1828), pp.{{nnbsp}}[https://books.google.com/books?id=N69MAAAAYAAJ&pg=PA168 168–91], [https://books.google.com/books?id=N69MAAAAYAAJ&pg=PA389 389–407]; vol.{{nnbsp}}26 (Jan.–{{hsp}}Jun.{{tsp}}1829), pp.{{nnbsp}}[https://books.google.com/books?id=OyMFAAAAQAAJ&pg=PA159 159–65].<!-- Volume numbers extrapolated from biodiversitylibrary.org/bibliography/2259. -->
* A. Fresnel, 1827, "Mémoire sur la double réfraction", ''Mémoires de l'Académie Royale des Sciences de l'Institut de France'', vol.{{nnbsp}}{{serif|VII}} (for 1824, printed 1827), [https://archive.org/details/mmoiresdelacad07memo/page/44 pp.{{nnbsp}}45–176]; reprinted as "Second mémoire…" in Fresnel, 1866–70, vol.{{nnbsp}}2, [https://books.google.com/books?id=g6tzUG7JmoQC&pg=PA479 pp.{{nnbsp}}479–596]; translated by A.W. Hobson as [https://archive.org/details/scientificmemoir05memo/page/238 "Memoir on double refraction"], in R.{{nnbsp}}Taylor (ed.), ''Scientific Memoirs'', vol.{{nnbsp}}{{serif|V}} (London: Taylor & Francis, 1852), pp.{{nnbsp}}238–333
* A. Fresnel (ed.{{tsp}} H. de Senarmont, E. Verdet, and L. Fresnel), 1866–70, ''Oeuvres complètes d'Augustin Fresnel'' (3 volumes), Paris: Imprimerie Impériale; [https://books.google.com/books?id=1l0_AAAAcAAJ vol.{{nnbsp}}1 (1866)], [https://books.google.com/books?id=g6tzUG7JmoQC vol.{{nnbsp}}2 (1868)], [https://books.google.com/books?id=Zm0VHAcxjV4C vol.{{nnbsp}}3 (1870)].
* E. Hecht, 2017, ''Optics'', 5th Ed., Pearson Education, {{ISBN|978-1-292-09693-3}}.
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* B. Powell (July 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 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, [
{{vpad|1=1ex}}
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[[Category:Antennas (radio)]]
[[Category:History of physics]]
[[Category:Planes (geometry)]]
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