Plane of polarization

But Augustin-Jean Fresnel, in his successful effort to explain double refraction under the 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.

[6] In an isotropic medium such as air, the ray and wave-normal directions are the same, and Fresnel's modification makes no difference.

It has been argued that the term plane of polarization, because of its historical ambiguity, should be avoided in original writing.

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.

1, there are three such planes, to which we may assign numbers for ease of reference: In an isotropic medium, E and D have the same direction,[Note 1] so that the ray and wave-normal directions merge, and the planes (2a) and (2b) become one: Polarization was discovered — but not named or understood — by Christiaan Huygens, as he investigated the double refraction of "Iceland crystal" (transparent calcite, now called Iceland spar).

[12] 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.

[13] This axis was parallel to the axes of the spheroidal secondary waves by which he (correctly) explained the directions of the extraordinary refraction.

[2] 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.

The implication that the plane of polarization contains the magnetic vectors is still found in the definition given in the online Merriam-Webster dictionary.

[18][19] Supposing that light waves were analogous to shear waves in 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.

Fresnel himself found this implication inconvenient; later that year he wrote: But he soon felt obliged to make a less radical change.

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.

[24] 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.

[25] The question called for an experimental determination of the direction of vibration, and the challenge was answered by George Gabriel Stokes.

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).

But Stokes's experiments were bound to detect the electric vibrations, because those have the greater propensity to interact with matter.

[Note 4] The electromagnetic theory of light further emphasized the electric vibrations because of their interactions with matter,[5] whereas the old "plane of polarization" contained the magnetic vectors.

**Fig. 1** : Field vectors ( E , D , B , H ) and propagation directions (ray and wave-normal) for linearly-polarized plane electromagnetic waves in a non-magnetic birefringent crystal. ^{[

1

]} 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, $θ = 0$ and Malus's plane merges with Fresnel's.)

**Fig. 2** : Linearly-polarized (plane-polarized) sinusoidal electromagnetic wave in an *isotropic* medium, propagating in the x direction (the ray direction and wave-normal direction), with the electric field vectors E and D in the y direction, and the magnetic field vectors B and H in the z direction. (The situation in a *non* -isotropic medium is more complicated; cf. Fig. 1.)

**Fig. 4** : Printed label seen through a doubly-refracting calcite crystal and a modern polarizing filter (rotated to show the different polarizations of the two images).