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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, the plane of polarization happened to coincide with the plane containing the direction of propagation and the ''magnetic'' vector; but this was not known at the time. In modern literature, the term ''plane of polarization'', if it is used at all, more often refers to the plane containing the direction of propagation and the ''electric'' vector, because the electric field has the greater propensity to interact with matter. That propensity, together with Malus's definition and [[Augustin-Jean Fresnel|Fresnel]]'s speculations on the [[luminiferous aether]], led early investigators to define the "plane of ''vibration''" as perpendicular to the plane of polarization and containing the direction of propagation.

{{afc cleared}}

This history must be taken into account when interpreting the term ''plane of polarization'' in existing literature. Sometimes the meaning can only be inferred from the context. In original writing, confusion can be avoided by specifying the orientation of a particular vector.<br style="margin-bottom: 1ex;" />

== Physics of the term ==

[[File:EM-Wave.gif|thumb|Linearly-polarized (plane-polarized) electromagnetic wave, propagating in the ''x'' direction (the ''ray'' direction), with the electric field '''E''' in the ''y'' direction (vertical) and the magnetic fields '''B''' and '''H''' in the ''z'' direction (horizontal).<br style="margin-bottom: .6ex;" />In an isotropic medium, the '''D''' field is in the same direction as the '''E''' field, and the wave-normal direction is the same as the ray direction, and the ''plane of polarization'' as originally defined by Malus is the ''xz'' plane, which is normal to the '''D''' field.<br style="margin-bottom: .6ex;" />In a double-refracting crystal, the '''D''' direction and the wave-normal direction are still in the ''xy'' plane, and still perpendicular to each other; but there is generally a small angle between '''E''' and '''D''', hence the same angle between the ray and the wave-normal; and the ''plane of polarization'' as originally defined is generally no longer the ''xz'' plane, but is still the plane normal to the '''D''' field.<br style="margin-bottom: .6ex;" />In either case, the plane of ''vibration'' as defined by Fresnel is the ''xy'' plane. But ''modern'' authors tend to identify that plane as the plane of polarization!]]

For [[electromagnetic radiation|electromagnetic (EM) waves]] in an ''isotropic'' medium (that is, a medium whose properties are independent of direction), the [[electric field]] [[Euclidean vector|vectors]] ('''E''' and '''D''') are in one direction, and the [[magnetic field]] vectors ('''H''' and '''B''') are in another direction, perpendicular to the first, and the direction of propagation is perpendicular to both the electric and the magnetic vectors. In this case the direction of propagation is both the ''ray'' direction and the ''wave-normal'' direction (the direction perpendicular to the [[wavefront]]). For a [[linear polarization|''linearly''-polarized]] wave (also called a ''plane''-polarized wave), the orientations of the field vectors are fixed.

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|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]].

If the medium is magnetically isotropic but electrically ''non''-isotopic (like a [[birefringence|double-refracting]] crystal), the magnetic vectors '''H''' and '''B''' 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.{{r|lunney-weaire-2006|p=26–7}} 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 of the plane of polarization depends on the theory of EM waves developed by [[James Clerk Maxwell|Maxwell]] in the 1860s. But the word ''polarization'' was coined about 50 years earlier, and the associated mystery dates back even further.

== History of the term ==

Polarization was discovered — but not named or understood — by [[Christiaan Huygens|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 (perpendicular) 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.{{r|huygens-1690|p=92–4}}

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.{{r|huygens-1690|p=55–6}} 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.

{{clear}}

[[File:Calcite and polarizing filter.gif|thumb|300px|Printed label seen through a double-refracting calcite crystal and a modern polarizing filter rotated to show the different polarizations of the two images.]]

The term ''polarization'' was coined by [[Étienne-Louis Malus]] (pronounced {{respell|ma|LOOSE|'}}) in 1811.{{r|buchwald-1989|p=54}}&nbsp; In&nbsp;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.{{r|buchwald-1989|p=31–43}} As this behavior had previously been known exclusively 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.{{r|buchwald-1989|p=45}} His 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; the component of the electric vector normal to that plane (i.e., parallel to the surface) is reflected to some extent for ''any'' angle of incidence, due to the change in [[permittivity]] at the surface.

The modern implication of Malus's definition — that the plane of polarization contains the ''magnetic'' vectors — survives in the online Merriam-Webster dictionary.{{r|merriamW}}

In 1821, [[Augustin-Jean Fresnel]], having already explained [[diffraction]] in terms of the [[wave theory of light]], announced his hypothesis that light waves are exclusively ''[[transverse wave|transverse]]'' and therefore ''always'' polarized, and that what we call "unpolarized" light is in fact light whose polarization is rapidly and randomly changing.{{r|buchwald-1989|p=227–9}} On that hypothesis, he proceeded to explain nearly all the remaining optical phenomena known at that time.{{r|frankel-1976|p=169}}

In deriving his [[Fresnel equations|eponymous equations]] for the reflection and transmission coefficients at the interface between two transparent media, Fresnel thought in terms of [[s-wave|shear waves]] in [[elasticity (physics)|elastic solids]], and supposed that a higher [[refractive index]] corresponded to a higher [[density]] of the [[luminiferous aether]]. But, as [[James MacCullagh]] later pointed out, that supposition does not work for double-refracting crystals (in which at least one refractive index varies with direction), because density is not directional; under Fresnel's analogy, a complete explanation of refraction would require a directional variation in [[stiffness]] of the aether ''within'' one medium, plus a variation in density ''between'' media. To avoid this complication, MacCullagh supposed that a higher refractive index corresponded to the same density but a greater elastic ''compliance'' (lower stiffness). Fresnel's supposition led to results that agreed with observation (on partial reflection and transmission) if he further supposed that the vibrations were ''normal'' to the plane of polarization. So began the distinction between the "plane of vibration" and the "plane of polarization". MacCullagh, in contrast, had to suppose that the two planes were the same — i.e., that the vibrations were ''within'' the plane of polarization. (This story was recounted in 1856 by [[Baden Powell (mathematician)|Baden Powell]].{{r|powell-1856}})

Thus attention was focused on whether the plane of vibration could be determined experimentally with the technology of the time. Consider a fine [[diffraction grating]] illuminated at normal incidence. At large angles of diffraction, the grating will appear somewhat edge-on, so that the directions of vibration will be crowded towards the direction parallel to the plane of the grating. If the planes of polarization coincide with the planes of vibration (''à&nbsp;la'' MacCullagh), they will be crowded in the same direction; and if the planes of polarization are ''normal'' to the planes of vibration (''à&nbsp;la'' Fresnel), the planes of polarization will be crowded in the normal direction. To measure the crowding, one could vary the polarization of the incident light in equal steps, and determine the planes of polarization of the diffracted light in the usual manner. Such an experiment was devised, and performed in 1849, by [[Sir George Stokes, 1st Baronet|George Gabriel Stokes]], and it found in favor of Fresnel.{{r|powell-1856|p=19–20;{{hsp}}}}{{r|stokes-1849|p=4–5}}

This is surprising in view of MacCullagh's point that density is not directional. If we attempt an analogy between shear waves in a non-isotropic elastic solid and EM waves in a magnetically isotropic but electrically non-isotropic crystal, the density must correspond to the magnetic [[permeability (electromagnetism)|permeability]] (both being non-directional), and the compliance must correspond to the electric [[permittivity]] (both being directional). The result is that the velocity of the solid corresponds to the '''H''' field,{{r|carcione-cavallini-1995}} so that the mechanical vibrations of the shear wave are in the direction of the ''magnetic'' vibrations of the EM wave. But Stokes's experiment was bound to detect the ''electric'' vibrations, because those were the vibrations that interacted with the grating (and with most other objects). In short, MacCullagh's "vibrations" were the ones that had a mechanical analog, but Fresnel's were the ones that were going to be detected in optical experiments. But that clarification raises another difficulty: the analogy between permeability and aether density allows very little variation in aether density among non-magnetic media and is therefore incompatible with Fresnel's hypothesis that aether density is the main criterion of refractive index.{{r|analogies}}

== History meets physics ==

The "vibrations" identified by Fresnel were taken as tangential to the wavefronts.{{r|aldis-1879|p=8,9}} In electromagnetic terms, that identifies the direction of vibration as the '''D''' direction. In a double-refracting crystal, it was conventional to take the ''wave-normal'' direction as the propagation direction to be included in the plane of polarization.{{r|aldis-1879|p=20}} So the plane of polarization was normal to the wavefront. As the vibration was in both the plane of vibration and the plane tangential to the wavefront, both planes being normal to the plane of polarization, it followed that the plane of polarization was simply the plane normal to the vibration.{{r|aldis-1879|p=9}} In electromagnetic terms, that identifies the plane of polarization as simply the plane normal to '''D'''.

Hence the old and new definitions of the plane of polarization may be stated succinctly and precisely as follows:

* Under the old definition, the plane of polarization is the plane normal to the '''D''' field.

* Under the new definition, the plane of polarization is the plane normal to the '''B''' field.

The former definition implies that the plane of polarization contains the magnetic vectors; the latter implies that it contains the electric vectors.

As the new electromagnetic theory further emphasized the ''electric'' vibrations (because of their interactions with matter), whereas the old "plane of polarization" contained the ''magnetic'' vectors, the new theory would have tended to reinforce the convention that the plane of vibration was normal to the plane of polarization — but only if one was familiar with the historical definition of the plane of polarization. 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, assume that the "plane" of polarization (if one needed such a concept) contained those vectors, and not bother to define a separate "plane of vibration". Moreover, it is not clear that a "plane of polarization" is needed at all: knowing what field vectors are involved, we can specify the polarization by specifying the orientation of a particular vector.

== Remaining uses ==

There is at least one context in which the ambiguity of the "plane of polarization" does no harm. In an optically ''[[optical rotation|chiral]]'' medium — that is, one in which the "plane of polarization" gradually rotates as the wave propagates — the choice of definition does not affect the existence or direction ("handedness") of the rotation.

There is also a context in which the original definition might still suggest itself. In a non-magnetic non-chiral crystal of the ''[[optic axis of a crystal|biaxial]]'' class (in which there is no ordinary refraction, but both refractions violate [[Snell's law]]), there are three mutually perpendicular planes for which the speed of light is isotropic within the plane provided that the electric vectors are normal to the plane.{{r|jenkins-white-1976}} That context naturally draws attention to a plane normal to the vibrations as envisaged by Fresnel, and that plane is indeed the plane of polarization as defined by Malus.

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.

== References ==

{{Reflist|refs=

<ref name=aldis-1879>W.S. Aldis, [https://archive.org/details/chapteronfresnel00aldirich ''A Chapter on Fresnel's Theory of Double Refraction''], 2nd&nbsp;Ed., Cambridge: Deighton, Bell, & Co., 1879.</ref>

<ref name=analogies>Concerning the limitations of elastic-electromagnetic analogies, see (e.g.) O.&nbsp;Darrigol, ''A History of Optics: From Greek Antiquity to the Nineteenth Century'', Oxford, 2012, pp.{{nnbsp}}227–32.</ref>

<ref name=buchwald-1989>J.Z. Buchwald, ''The Rise of the Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century'', University of Chicago Press, 1989.</ref>

<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.&nbsp;Sands, ''The Feynman Lectures on Physics'', California Institute of Technology, 1963–2013, Volume&nbsp;{{serif|I}}, [http://feynmanlectures.caltech.edu/I_33.html Lecture&nbsp;33].</ref>

<ref name=frankel-1976>E. Frankel, "Corpuscular optics and the wave theory of light: The science and politics of a revolution in physics", ''Social Studies of Science'', Vol.{{nnbsp}}6, No.{{hsp}}2 (May&nbsp;1976), pp.{{nnbsp}}141–84.</ref>

<ref name=huygens-1690>C. Huygens, ''Traité de la Lumière'' (Leiden: Van der Aa, 1690), translated by S.P. Thompson as ''[http://www.gutenberg.org/ebooks/14725 Treatise on Light]'', University of Chicago Press, 1912.</ref>

<ref name=jenkins-white-1976>Cf.&nbsp; F.A. Jenkins and H.E. White, ''Fundamentals of Optics'', 4th&nbsp;Ed., New&nbsp;York: McGraw-Hill, 1976, Fig.{{nnbsp}}26{{serif|I}} (p.{{hsp}}554).</ref>

<ref name=lunney-weaire-2006> J.G. Lunney and D. Weaire, "The ins and outs of conical refraction", ''Europhysics News'', Vol.{{nnbsp}}37, No.{{hsp}}3 (May–June&nbsp;2006), pp.{{nnbsp}}26–9; [http://dx.doi.org/10.1051/epn:2006305 doi.org/10.1051/epn:2006305].</ref>

<ref name=luntz>M. Luntz (?) et al., [https://www.britannica.com/science/radiation/The-structure-and-properties-of-matter#ref398787 "Double refraction"], ''Encyclopædia Britannica'', accessed 15&nbsp;September 2017.</ref>

<ref name=merriamW>Merriam-Webster, Inc., [https://www.merriam-webster.com/dictionary/plane%20of%20polarization "Plane of polarization"], accessed 15&nbsp;September 2017.</ref>

<ref name=powell-1856>B. Powell, [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 (July&nbsp;1856), pp.{{nnbsp}}1–20.</ref>

<ref name=stokes-1849>G.G. Stokes, [https://archive.org/stream/transactionsofca09camb#page/n15/mode/2up "On the dynamical theory of diffraction"] (read 26&nbsp;November 1849), ''Transactions of the Cambridge Philosophical Society'', Vol.{{nnbsp}}9, Part&nbsp;1 (1851), pp.{{nnbsp}}1–62.</ref>

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== plane of polarization ==

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