# Plane of polarization

The term plane of polarization refers to the direction of polarization of 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 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.

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.

## Physics of the term

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).
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.
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.
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 (EM) waves in an isotropic medium (that is, a medium whose properties are independent of direction), the electric field 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 linearly-polarized wave (also called a plane-polarized wave), the orientations of the field vectors are fixed.

Because innumerable materials are dielectrics or conductors while comparatively few are ferromagnets, the 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.

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,[1] and in Feynman's lecture on polarization.[2] 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 antennas.

If the medium is magnetically isotropic but electrically non-isotopic (like a 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.[3]: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 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 Huygens, as he investigated the 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.[4]: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.[4]:55–6 This axis was parallel to the axes of the spheroidal secondary waves by which he (correctly) explained the directions of the extraordinary refraction.

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 in 1811.[5]:54  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.[5]: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.[5]: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.[6]

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 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.[5]:227–9 On that hypothesis, he proceeded to explain nearly all the remaining optical phenomena known at that time.[7]:169

In deriving his eponymous equations for the reflection and transmission coefficients at the interface between two transparent media, Fresnel thought in terms of shear waves in elastic solids, and supposed that a higher refractive index corresponded to a higher density of the luminiferous aether. This idea could not be extended to double-refracting crystals (in which at least one refractive index varies with direction), because density is not directional. Thus Fresnel's explanation of refraction required a directional variation in stiffness of the aether within one medium, plus a variation in density between media. To avoid this complication, James MacCullagh supposed that a higher refractive index corresponded always to the same density but a greater elastic compliance (lower stiffness). Fresnel's analogies 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 history was recounted by Baden Powell in 1856,[8] and by E.T. Whittaker in 1910.[9]:132–5, 149)

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 la MacCullagh), they will be crowded in the same direction; and if the planes of polarization are normal to the planes of vibration (à 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 George Gabriel Stokes, and it found in favor of Fresnel.[8]:19–20; [10]:4–5

It remains to explain how this result can be reconciled with the non-directionality of density. 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 (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,[11] 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.[12]

## History meets physics

The "vibrations" identified by Fresnel were taken as tangential to the wavefronts.[13]: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.[13]: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.[13]: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[2] and the Britannica[1] 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 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 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.[14] 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

1. ^ a b M. Luntz (?) et al., "Double refraction", Encyclopædia Britannica, accessed 15 September 2017.
2. ^ a b R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, California Institute of Technology, 1963–2013, Volume I, Lecture 33.
3. ^ J.G. Lunney and D. Weaire, "The ins and outs of conical refraction", Europhysics News, Vol. 37, No. 3 (May–June 2006), pp. 26–9; doi.org/10.1051/epn:2006305.
4. ^ a b C. Huygens, Traité de la Lumière (Leiden: Van der Aa, 1690), translated by S.P. Thompson as Treatise on Light, University of Chicago Press, 1912.
5. ^ a b c d 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.
6. ^ Merriam-Webster, Inc., "Plane of polarization", accessed 15 September 2017.
7. ^ E. Frankel, "Corpuscular optics and the wave theory of light: The science and politics of a revolution in physics", Social Studies of Science, Vol. 6, No. 2 (May 1976), pp. 141–84.
8. ^ a b B. Powell, "On the demonstration of Fresnel's formulas for reflected and refracted light; and their applications", Philosophical Magazine and Journal of Science, Series 4, Vol. 12, No. 76 (July 1856), pp. 1–20.
9. ^ E.T. Whittaker, A History of the Theories of Aether and Electricity: From the age of Descartes to the close of the nineteenth century, Longmans, Green, & Co., 1910.
10. ^ G.G. Stokes, "On the dynamical theory of diffraction" (read 26 November 1849), Transactions of the Cambridge Philosophical Society, Vol. 9, Part 1 (1851), pp. 1–62.
11. ^ J.M. Carcione and F. Cavallini, "On the acoustic-electromagnetic analogy", Wave Motion, Vol. 21 (1995), pp. 149–62. (Note that the authors' analogy is only two-dimensional.)
12. ^ Concerning the limitations of elastic-electromagnetic analogies, see (e.g.) O. Darrigol, A History of Optics: From Greek Antiquity to the Nineteenth Century, Oxford, 2012, pp. 227–32.
13. ^ a b c W.S. Aldis, A Chapter on Fresnel's Theory of Double Refraction, 2nd Ed., Cambridge: Deighton, Bell, & Co., 1879.
14. ^ Cf.  F.A. Jenkins and H.E. White, Fundamentals of Optics, 4th Ed., New York: McGraw-Hill, 1976, Fig. 26I (p. 554).