Gavin R. Putland,  BE PhD

Tuesday, April 04, 2017 (Comment)

What century is this? — the ambiguity of ‘plane of polarization’

Update (1 June 2019):  An improved version of this post has been contributed to Wikipedia as the article “Plane of polarization”, and subsequently revised. The version below does not explain Fresnel's modification of Malus's definition.
Update (26 September 2017):  The text of this page is available for modification and reuse under the terms of the Creative Commons Attribution-Sharealike 3.0 Unported License and the GNU Free Documentation License (unversioned, with no invariant sections, front-cover texts, or back-cover texts).

PLANE OF POLARIZATION is a term to be deciphered in other people's writings but avoided in one's own, so that one's own readers are spared the trouble of deciphering it. The online Merriam-Webster defines it as

the plane in which the magnetic-vibration component of plane-polarized electromagnetic radiation lies

whereas the Britannica, under “The passage of electromagnetic rays”, says:

If the field vector maintains a fixed direction, the wave is said to be plane-polarized, the plane of polarization being the one that contains the propagation direction and the electric vector.

That plane most assuredly does not contain the magnetic vector! (Needless to say, the emphasis in both quotes is mine.) Feynman, in his lecture on polarization, uses the latter definition but doesn't spell it out. He keeps emphasizing the direction of the electric vector, and leaves the reader to infer that the “plane of polarization” is the plane of that vector. That interpretation indeed fits the examples he gives.

So there are two clashing conventions. And to make matters worse, it is hardly ever acknowledged that there are two clashing conventions: the users of each seem to be unaware of the other.

Cutting across this fault-line is another convention, namely that the plane of vibration is normal (perpendicular) to the plane of polarization. Applying that convention to the Britannica definition, one would conclude that the plane of vibration were the one containing the propagation direction and the magnetic vector, which would surprise readers who have worked in radio transmission (as I briefly did). Applying it to the Merriam-Webster definition, one would conclude that the definition shouldn't nominate the same plane for both the vibration and the polarization.

So it seems that the vibration/polarization distinction is not universal either. (Indeed Feynman, as far as I am aware, doesn't use the expression “plane of vibration”.) And why would one bother making such a distinction anyway? How did we get into this mess?

The short answer is that the Britannica/Feynman convention — in which we focus on the electric vector, define the “plane of polarization” as the plane containing that vector and the direction of propagation, and don't bother defining a separate “plane of vibration” — is what we would adopt if we considered the physics alone, whereas the other conventions carry historical baggage.

Let's start with the physics. For electromagnetic (EM) waves in an isotropic medium (that is, a medium whose properties are independent of direction), the magnetic field vectors (H and B) are in one direction, and the electric field vectors (E and D) are in another direction, perpendicular to the first, and the direction of propagation (the ray direction, and the direction normal to the wavefront) is perpendicular to both of the above. Now, 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 the magnetic properties. That circumstance obviously focuses attention on the electric vectors, so that we tend to think of the direction of polarization as the direction of those vectors, and the “plane of polarization” as the plane containing those vectors and the direction of propagation.

If the medium is magnetically isotropic but electrically non-isotopic (like a doubly refractive crystal), the magnetic vectors H and B are still in the same direction, but there is generally an angle between the electric vectors E and D, and the same angle between the ray direction and the direction normal to the wavefront, so that E and the magnetic vectors are still perpendicular to the ray direction, while D and the magnetic vectors are still perpendicular to the wave-normal direction, i.e. tangential to the wavefront. 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 think of that plane as the “plane of polarization”.

(What if the medium is electrically and magnetically non-isotropic? Then H and B are no longer necessarily parallel; and E and H are still perpendicular to the ray direction, but no longer necessarily perpendicular to each other; and D and B are still tangential to the wavefront, but again no longer necessarily perpendicular to each other. Very messy — but also a bit academic, because such materials tend not to be very transparent to EM waves. Accordingly, elsewhere in this article, we assume that the electric fields are perpendicular to the magnetic fields.)

The trouble with the “natural” terminology is that it depends on the theory of EM waves developed by Maxwell in the 1860s, whereas the word polarization was coined about 50 years earlier and the phenomenon was known for more than a century before that.

Polarization was discovered — but not named, much less understood — by Huygens, as he investigated the double refraction of calcite (then known as “Iceland crystal”). 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.

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. This axis was parallel to the axes of the spheroidal secondary waves by which he ingeniously — and correctly — explained the directions of the extraordinary refraction.

The term polarization was coined in 1811 by Étienne-Louis Malus (pronounced “Ma-LOOSE”).  In 1808, in the midst of confirming Huygens' quantitative description of double refraction (while disputing his 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. 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 (Buchwald, 1989, p. 45). His definition was all the more reasonable because it meant that when a ray was polarized by reflection, 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. Unfortunately, as we now know, this happens to be the plane of the magnetic vectors, not the electric vectors!

Malus died young in 1812, depriving the corpuscularists of their best-and-fairest researcher. His discoveries brought dark days for the wave-theorists, who initially had no working hypothesis on the nature of polarization. But in 1821, Augustin-Jean Fresnel, having already explained diffraction in terms of the wave theory, announced his hypothesis that light waves are exclusively transverse and therefore always polarized, and that what we call “unpolarized” light is in fact light whose polarization is rapidly and randomly changing. On that hypothesis, he proceeded to explain nearly all the remaining optical phenomena known at that time. Then he too died young. Although he was later overshadowed by Maxwell's unification of optics and electromagnetics, Fresnel was, and will always remain, the author of the first coherent theory of light.

In deriving his eponymous equations for the reflection and refraction coefficients at the interface between two transparent media (of which Feynman offers his own derivation in the aforesaid lecture), Fresnel thought in terms of shear waves in elastic solids, and supposed that a higher refractive index corresponded to a higher density. But, as James MacCullagh later pointed out, that analogy won't do for doubly refractive crystals (whose refractive index varies with direction), because density is not directional! Accordingly, MacCullagh supposed that a higher refractive index corresponded to the same density but a greater elastic compliance (lower stiffness). Fresnel, in order to produce results that agreed with observation (especially polarization by reflection), had to assume 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 assume that the two planes were the same — i.e., that the vibrations were within the plane of polarization. (The story was told by Baden Powell in 1856.)

So who was right? And, given the technology of the time, how could one tell? 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 (a la MacCullagh), they will be crowded in the same direction; and if the planes of polarization are normal to the planes of vibration (a 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 by George Gabriel Stokes in 1849, and it found in favor of Fresnel.

How was this possible? Didn't MacCullagh have a decisive point — that density is not directional? Indeed he did. And for precisely that reason, if we want to construct 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, and the compliance to the electric permittivity, with the result that the velocity of the solid corresponds to the... ahem... H field, so that the vibration is in the direction of the magnetic vectors! 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 analogy, but Fresnel's were the ones that were going to be observed in optical experiments.

Maxwell's theory of electromagnetic waves did away with the need for mechanical analogies (although the point was slow to sink in). As the new theory further emphasized the electric vibrations, whereas Malus's “plane of polarization” contained the magnetic vectors, the new theory tended to reinforce the convention that the plane of vibration was normal to the plane of polarization — provided, of course, that you knew about Malus's old definition of the plane of polarization. But what if you didn't? Then you would do what Feynman and the Britannica did: pay attention to the electric vectors, assume that the “plane” of polarization (if you need such a concept) contains those vectors, and not bother defining a separate “plane of vibration”. Moreover, it is not clear that a “plane of polarization” is needed at all: now that we know what field vectors are involved, we might as well specify the polarization by specifying the orientation of a particular vector.

Having said that, I should note one context in which the ambiguity of “plane of polarization” does no harm. In a 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. I should also note that in a non-magnetic non-chiral biaxial crystal (one in which there is no ordinary refraction, but both refractions violate “Snell's law”), there are three mutually perpendicular planes in which the speed of light is isotropic within the plane provided that the electric vectors are normal to the plane. 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 the plane containing the electric vibrations is an artifice and an anachronism: if you knew the physics but not the history, you wouldn't think of things that way.

[Last modified April 17, 2017.]

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