Chapter 14 Light in an Optical Medium

Before the nineteenth century, Isaac Newton’s particle model and Christiaan Huygens’s wave model could both explain most optical phenomena, but with opposite predictions for the speed of light in matter.

Both Newton and Huygens explained Snell’s law of refraction as a sudden change in speed at the optical interface, but with one crucial difference: Newton’s light particles should speed up, not slow down, when entering a more optically dense medium.[1] As Newton explained, attractive forces would accelerate a light particle upon entering a medium, and slow it again when exiting. As the force components parallel to the air-glass interface cancel, that component of the velocity would be unaffected, thus bending the trajectory. In this way, Newton derived Snell’s law of refraction, with a clear and understandable mechanism. Huygens, on the other hand, noticed that a wave’s speed also depends on the medium involved, but in the opposite sense—the waves must slow down to follow Snell’s law.

In 1669, Rasmus Bartholin discovered double-refraction, where a crystal, such as calcite, splits a beam of light as it refracts. How a medium could have two different wave speeds was a puzzle. Furthermore, how could a longitudinal wave carry enough information to account for double refraction? A particle, however, could possess mass, velocity, and the ability to spin about an axis, favoring Newton’s particle model of light.

Also in the 1660s, the Jesuit priest Francesco Maria Grimaldi observed the diffraction of light. This would seem to greatly favor Huygens, but Newton explained it as ripples in a local medium, caused by the impact of his light corpuscles. Thus, the two models remained deadlocked, until 1725 when James Bradley observed the aberration of starlight (see pp. 600 and 723), where the position of stars shifts due to the motion of the Earth. Bradley’s observation could be easily explained with Newton’s theory, providing definitive evidence favoring the particle model. However, that all changed with the turn of the nineteenth century.

Thomas Young’s 1804 double slit experiment (p. 650) came down definitively on the side of light being a wave. In the years after he published the results, Young attempted to explain all of optical phenomena in terms of the wave model. Although he had much success, he could still not explain the effect that some crystals have on light rays.

When light passes through two identical crystals, such as tourmaline, the intensity of the emerging light varies as the second crystal is rotated. The transmitted light is brightest when the axes of the two crystals are aligned, but is completely extinguished when the axes are oriented perpendicular to each other.   Astonishingly, when a third crystal is placed between the two perpendicular ones, some light, again, passes through the last crystal.

Newton explained these results as due to an asymmetry in the shape of the particles. Just as a pick axe can only pass through prison bars aligned one way, only particles aligned parallel to the crystal lattice can pass. Light, the logic went, behaves like little dipoles, thus the origin of the confusing term polarization of light.

During Napoleon’s reign, investigations by Étienne-Louis Malus continued to dog Young’s wave theory of light.[2] Unpolarized light passing through a birefringent crystal, such as Iceland spar, separates into two rays, producing two images. While double refraction was known to Newton and Huygens, Malus observed something interesting when observing sunlight reflected from the windows of the Luxembourg Palace. The two images of the reflected setting sun dramatically differed, and, moreover, as he rotated the crystal the images alternately brightened and dimmed.

Augustin-Jean Fresnel saved the wave model by proposing that light is not longitudinal, but rather transverse like the waves on a string.[3] Fresnel, and his mentor François Jean Dominique Arago, showed that beams of perpendicularly polarized light do not interfere, which led, with a hint from Young, to Fresnel’s transverse wave theory of light.[4]

Siméon Denis Poisson, who vigorously opposed the wave theory of light, pointed out what he saw as a damning feature of Fresnel’s work. Poisson showed that Fresnel’s theory predicted a bright spot in the center of the shadow of a circular screen, which, according to Poisson, was certainly an absurd result.[5] Arago performed the experiment and did indeed find the impossible bright spot, which is now known, in a nice bit of historical irony, as the Poisson spot.[6]

By the middle of the nineteenth century the transverse wave theory seemed the only possible way of accounting for such diverse optical phenomena as reflection, refraction, diffraction, and polarization. Light is a wave, nearly everyone agreed. Then, how and through what does it propagate?

Michael Faraday’s experiments on capacitors with dielectrics had suggested a possible propagation mechanism. Since a slight displacement of charge can push the next atom down the line, perhaps this is how light propagates too? If space were filled with matter, then this process would convey the electric force in much the same manner as small elastic deformations convey the normal force through your chair. The was a problem, however, all these sound-like mechanisms suggested compression waves, not the transverse wave of Fresnel’s theory.

On the other hand, both longitudinal, and transverse, waves can travel through an elastic solid. For example, earthquakes produce both longitudinal p-waves and transverse s-waves. Perhaps, thought  George Gabriel Stokes, light simply consists of transverse waves propagating through the aether, analogous to s-waves traveling through the earth. As he put it:

Undoubtedly, it does violence to the ideas that we should have been likely to form a priori of the nature of the aether, to assert that it must be regarded as an elastic solid in treating of the vibrations of light. When, however, we consider the wonderful simplicity of the explanations of the phenomena of polarization when we adopt the theory of transversal vibrations, and the difficulty, which to me at least appears quite insurmountable, of explaining these phenomena by any vibrations due to the condensation and rarefaction of an elastic fluid such as air, it seems reasonable to suspend our judgment, and be content to learn from phenomena the existence of forces which we should not beforehand have expected.[7],[8]

One problem that bedeviled the elastic solid model of the aether was how to explain the existence of a medium that would be rigid enough to support transverse waves, yet fluid enough to allow planets and stars to glide through unhindered. This difficulty stimulated investigations of elastic solids and how they respond to various forces.

Stokes’s model of light propagating through an elastic solid fascinated a precocious teen studying mathematics at the University of Edinburgh. At the age of 18, James Clerk Maxwell was finally deemed old enough to present a paper on his own to the local chapter of the Royal Society. His paper, titled “On the Equilibrium of Elastic Solids,” tackled this question and was reported by the secretary with the following description:[9]

This paper commenced by pointing out the insufficiency of all theories of elastic solids, in which the equations do not contain two independent constants deduced by experiments. One of these constants is common to liquids and solids, and is called the modulus of cubical elasticity [µ]. The other is peculiar to solids, and is here called the modulus of linear elasticity [m]. The equations of Navier, Poisson, and Lamé and Clapeyron, contain only one coefficient; and Professor G. G. Stokes of Cambridge, seems to have formed the first theory of elastic solids which recognized the independence of cubical and linear elasticity, although M. Cauchy seems to have suggested a modification of the old theories, which made the ratio of linear and cubical elasticity the same for all substances. Professor Stokes has deduced the theory of elastic solids from that of the motion of fluids, and his equations are identical with those of this paper, which are deduced from the following assumptions.

In an element of an elastic solid, acted on by three pressures at right angles to on another, as long as the compressions do not pass the limits of perfect elasticity—

1st, The sum of the pressures, in the three rectangular axes, is proportional to the sum of the compressions in those axes.

2nd, The difference of the pressures in two axes at right angles to one another, is proportional to the difference of the compressions in those axes.

Or, in symbols:—
$1.\ \left( {{P}_{1}}+{{P}_{2}}+{{P}_{3}} \right)=3\mu \left( \frac{\delta x}{x}+\frac{\delta y}{y}+\frac{\delta z}{z} \right)\quad \quad 2.\left\{ \begin{matrix} \left( {{P}_{1}}-{{P}_{2}} \right)=m\left( \frac{\delta x}{x}-\frac{\delta y}{y} \right) \\ \left( {{P}_{2}}-{{P}_{3}} \right)=m\left( \frac{\delta y}{y}-\frac{\delta z}{z} \right) \\ \left( {{P}_{3}}-{{P}_{1}} \right)=m\left( \frac{\delta z}{z}-\frac{\delta x}{x} \right) \\ \end{matrix} \right.$
µ being the modulus of cubical, and m that of linear elasticity.

These equations are found to be convenient for the solution of problems, some of which were given in the latter part of the paper.

The paper concluded with a conjecture, that as the quantity , (which expresses the relation of inequality of pressure in a solid to the doubly-refracting force produced) is probably a function of ; the determination of these quantities for different substances might lead to a more complete theory of double refraction, and extend our knowledge of the laws of optics.[10]

After that, Maxwell attended university at Cambridge. At the time, students sat for a grueling eight-day examination in order to graduate, where each question would be more difficult than the one before. A prize was given to the student who scored highest on the final 17 questions, which in 1854 were written by Professor George Gabriel Stokes. Much to Maxwell’s delight, the last question on the exam read:

Plane polarized light is transmitted, in a direction parallel to the axis of the crystal, across a thick plate of quartz cut perpendicular to the axis, and the emergent light, limited by a screen with a slit, is analyzed by a Nicol’s prism combined with an ordinary prism; describe the appearance presented as the Nicol’s prism is turned round, and from the phenomena deduce the nature of the action of quartz on polarized light propagated in the direction of the axis.[11]

Maxwell did not get the top score on the overall exam, but he did tie for the Smith prize for the best score on the final 17 questions.

Maxwell continued being fascinated by the relationship between elasticity and optics, and by 1856 had already landed an academic job. He moved to northern Scotland, where he taught, got married, and worked on problems in optics—and also showed that Saturn’s rings were neither fluid or solid, but rather made of small objects. Then Maxwell caught another lucky break; he lost his job.

Maxwell landed on his feet at King’s College London, where he took advantage of the intellectually stimulating environment and immersed himself in the works of Faraday, Ampère, Gauss, and William Thomson, in order to develop a mathematical theory of electricity and magnetism. In a sense, the fundamental laws that govern electricity and magnetism where understood by the time Maxwell took up his examination of them. Coulomb’s law allowed one to calculate and predict the electrostatic interactions among charges, while Ampère’s force law between currents and the formula credited to Biot and Savart could be used to calculate magnetic fields in the case of steady currents. Finally, Faraday’s discovery of electromagnetic induction demonstrated how a changing magnetic field induces an electric field.

The field concept was still in its infancy, but investigators such as Faraday and William Thomson used it with increasing success, especially in the laboratory. In formulating his own approach to the subject of electricity and magnetism, Maxwell emphasized his understanding of Faraday’s idea of fields conveyed through the mathematics of fluids and solids:

As I proceeded with the study of Faraday, I perceived that his method of conceiving phenomena was also a mathematical one, though not exhibited in the conventional form of mathematical symbols. I also found that these methods were capable of being expressed in the ordinary mathematical forms, and thus compared with those of the professed mathematicians.

For instance, Faraday, in his mind’s eye, saw lines of force traversing all of space, where the mathematicians saw only centers of force attracting at a distance…When I translated what I considered to be Faraday’s ideas into a mathematical form…I found that several of the most fertile methods of research discovered by the mathematicians could be expressed much better in terms of ideas derived from Faraday than in their original form.[12]

His work ultimately vindicated Faraday’s field conception of electromagnetism, and also confirmed Faraday’s intuitive hunch that light was somehow connected with “vibrations of … lines of force.”[13] Maxwell achieved this by adding an insight of his own to the equations which govern electrical and magnetic phenomena.

Maxwell’s insight was the converse of Faraday’s discovery of 1831 that a changing magnetic field induces an electric field. Based on theoretical arguments arising from questions of symmetry and current continuity, Maxwell predicted that a changing electric field induces a magnetic field. With this insight, Maxwell was able to develop a fully electromagnetic theory of light.

Maxwell discovered that self-propagating electromagnetic waves would travel through space at a constant speed, which happened to equal the previously measured speed of light. Like others, Maxwell felt that the electromagnetic waves required a medium through which to travel. In the course of his work, he returned to his early research project and the work of his former professor—now good friend and colleague—George Stokes.

Maxwell applied Stokes’s model of the aether as his basis for light. This allowed him to imagine the electric and magnetic fields in much the same way as shear stress and strain are related in perfectly elastic solids. In his model, the permittivity of a dielectric was analogous to the same inverse linear elasticity modulus m that Maxwell presented when he was eighteen:
$\varepsilon \sim \frac{1}{m}$,
and the permeability of a magnetic material provided the inertia, so:  $\mu \sim {{\rho }_{m}}$.
Then he could calculate the transverse wave speed in this medium as:
$v=\sqrt{\frac{m}{{{\rho }_{m}}}}\sim \sqrt{\frac{\tfrac{1}{\varepsilon }}{\mu }}\sim \frac{1}{\sqrt{\varepsilon \,\mu }}$.
Maxwell stated this result in his 1862 paper “On Physical Lines of Force”:

The velocity of transverse undulations in our hypothetical medium, calculated from the electro-magnetic experiments of M.M. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.[14]

Although the argument that Maxwell presented in his 1862 paper was suggestive, it was far from conclusive. In 1865, Maxwell derived an explicit wave equation for both the electric and magnetic fields. Maxwell’s wave equation predicted that those electric and magnetic fields which satisfy the wave equation are transverse waves, which propagate at a speed in free space that was in complete agreement to the speed of light.

In 1873, he published A Treatise on Electricity and Magnetism containing a full mathematical description of the behavior of electric and magnetic fields in matter, the predictions of which have been fully confirmed by experiment.

The title of Maxwell’s final chapter is “Theories of Action at a Distance.” In it he criticizes the time-dependent theories of Gauss and Weber:

There appears to be, in the minds of these eminent men, some prejudice, or a priori objection, against the hypothesis of a medium in which the phenomena of radiation of light and heat and the electric actions at a distance take place. It is true that at one time those who speculated as to the causes of physical phenomena were in the habit of accounting for each kind of action at a distance by means of a special aethereal fluid, whose function and property it was to produce these actions. They filled all space three or four times over with aethers of different kinds, the properties of which were invented merely to ‘save appearances,’ so that more rational enquirers were willing rather to accept not only Newton’s definite law of attraction at a distance, but even the dogma of Cotes [in the preface to Newton’s Principia, 2nd edition], that action at a distance is one of the primary properties of matter, and that no explanation can be more intelligible than this fact. Hence the undulatory theory of light has met with much opposition, directed not against its failure to explain the phenomena, but against its assumption of the existence of a medium in which light is propagated.[15]

His “constant aim in this treatise” was that “we ought to endeavor to construct a mental representation of all the details of” the aether.[16]

[1] Isaac Newton, Principia, Mathematical Principles of Natural Philosophy: A New Translation, trans. I Bernard Cohen and Anne Whitman (Berkeley: University of California Press, 1999), Book I, Section 14, Proposition 94, p. 622.

[2] E-L. Malus, “Théorie de la Double Réfraction,” Mémoires présentés à l’Institut des sciences par divers savants, 2 (1811), 303-508. See also, Buchwald, Jed Z., The Rise of the Wave Theory of Light, (Chicago: University of Chicago Press, 1989), 54-64.

[3] Augustin Fresnel, “Memoirs on the Diffraction of Light,” The Wave Theory of Light – Memoirs by Huygens, Young and Fresnel, (New York: American Book Company, 1900), 79–145.

[4] Augustin Fresnel, “On the Action of Rays of Polarized Light upon Each Other,” The Wave Theory of Light – Memoirs by Huygens, Young and Fresnel (American Book Company, 1900), 145–156.

[5] A. Fresnel, Complete Works: Theory of Light, 3 vols., (Paris: Imprimiere Impériale, 1866, 1868, 1870), 1: 366-372.

[6] A. Fresnel, Complete Works: Theory of Light, 3 vols., (Paris: Imprimiere Impériale, 1866, 1868, 1870), 1: 369.

[7] G.G. Stokes, “On the constitution of the luminiferous aether,” Philosophical Magazine 29, 3rd series, (1848), 6-10.

[8] David B. Wilson, “George Gabriel Stokes on Stellar Aberration and the Luminiferous Ether,” The British Journal for the History of Science, 6, No. 1 (Jun., 1972), 57-72.

[9] In the quote that follows, Maxwell uses the term “cubical elasticity” [ ] to denote what we now call bulk modulus.

[10] James Clerk Maxwell, Esq. (Communicated by the Secretary), “On the Equilibrium of Elastic Solids,” in Proceedings of The Royal Society of Edinburgh, Vol. 2, December 1844 to April 1850, Neill and Co., Edinburgh, 1851.

[11] George Gabriel Stokes, Esq. M.A., The Smith Prize Exam of 1854, from the website of the James Clerk Maxwell Foundation (http://www.clerkmaxwellfoundation.org).

[12] From the preface to J.C. Maxwell, Treatise on Electricity and Magnetism, vol. 1, 3rd ed. (1873; rpt. New York: Dover, 1954).

[13] Faraday, “Thoughts on Ray Vibrations,” 1844, Phil. Magazine; “On the physical character of the lines of magnetic force,” 1852, Phil. Magazine.   See page 102 of this chapter for more details.

[14] J.C. Maxwell, “On Lines of Physical Force,” Philosophical Magazine, 21 & 23 (1862), Part III.

[15] J.C. Maxwell, A Treatise on Electricity and Magnetism, vol. 2, 3rd ed. (1873; rpt. New York: Dover, 1954), Art. 865, p. 492.

[16] Ibid., Art. 866, p. 493.