Cristiaan Huygens Explains the Refraction of Light Waves
This is my fifth post on the Traité de la lumière [Treatise on Light1], published in 1690 by Cristiaan Huygens (1629-1695). The previous posts were
Christiaan Huygens Proposes an Infinite Series of Levels to the Universe
Cristiaan Huygens Presents Ole Rømer's Proof that the Speed of Light is Finite
In this post, I will present Huygens’s demonstration that the wave theory of light explains refraction, in which the direction of a light wave changes direction as it moves from one medium, say air, into another medium, say water.
The argumentation Huygens uses for refraction is very similar to what he used for reflection. However, because there are two media involved, which may correspond to different states of matter (gas, liquid, solid), all capable of transmitting light, he must present how he thinks the aether and ponderable matter interact.
Because of their fluidity, Huygens considers it unproblematic that aether particles would freely intermix with particles of gases or liquids. More difficult is what happens with solids such as glass. Huygens argues that aether particles can freely pass through solids as well:
One can then in this way conceive of transparency in a solid without any necessity that the ethereal matter which serves for light should pass through it, or that it should find pores in which to insinuate itself. But the truth is that this matter not only passes through solids, but does so even with great facility; of which the experiment of Torricelli, above cited, is already a proof. Because on the quicksilver [mercury] and the water quitting the upper part of the glass tube, it appears that it is immediately filled with ethereal matter, since light passes across it. But here is another argument which proves this ready penetrability, not only in transparent bodies but also in all others. [p.30]
Torricelli’s experiment consists of filling a glass tube with mercury (or water), then closing off the open end with a finger, inverting the tube and leaving the open end in a bowl also filled with mercury. The result is that the tube only partially empties itself of the mercury, leaving a space devoid of mercury at the top of the tube. Torricelli argues that two phenomena are taking place. First, air pressure on the mercury in the bowl is preventing the tube from emptying itself. Second, a vacuum has been created above the mercury inside the tube. The second point was controversial, as many scientists, including Huygens, argued that there was still aethereal matter inside the tube above the mercury, as light passes through the upper part of the tube.
When light passes across a hollow sphere of glass, closed on all sides, it is certain that it is full of ethereal matter, as much as the spaces outside the sphere. And this ethereal matter, as has been shown above, consists of particles which just touch one another. If then it were enclosed in the sphere in such a way that it could not get out through the pores of the glass, it would be obliged to follow the movement of the sphere when one changes its place: and it would require consequently almost the same force to impress a certain velocity on this sphere, when placed on a horizontal plane, as if it were full of water or perhaps of quicksilver: because every body resists the velocity of the motion which one would give to it, in proportion to the quantity of matter which it contains, and which is obliged to follow this motion. But on the contrary one finds that the sphere resists the impress of movement only in proportion to the quantity of matter of the glass of which it is made. Then it must be that the ethereal matter which is inside is not shut up, but flows through it with very great freedom. [pp.30-31]
This last paragraph is crucial, because this debate would reappear in the middle of the nineteenth century, when George Gabriel Stokes (1819-1903) attacked Augustin-Jean Fresnel (1788-1827), who was working with Huygens’s aether model. Stokes replaced the aether that flowed freely through the earth with a sort of sticky-jam aether that stayed immobile inside the earth, even calling Fresnel’s and Huygens’s ideas a “violent supposition”:
The phænomenon of aberration may be reconciled with the undulatory theory of light, as I have already shown… without making the violent supposition that the æther passes freely through the earth in its motion round the sun, but supposing, on the contrary, that the æther close to the surface of the earth is at rest relatively to the earth.2
For more on this topic, see my post How Two Productive Research Programs Were Consciously Thrown Out. I will return to this subject in detail in future posts.
Huygens is also trying to explain why different media have different refractive indices. He proposes that there is some kind of interaction between the particles of the aether and of ponderable matter. He first proposes that the waves pass through the aethereal matter, but that the particles of ponderable matter sort of hinder this passage:
The rarity of transparent bodies being then such as we have said, one easily conceives that the waves might be carried on in the ethereal matter which fills the interstices of the particles. And, moreover, one may believe that the progression of these waves ought to be a little slower in the interior of bodies, by reason of the small detours which the same particles cause. In which different velocity of light I shall show the cause of refraction to consist. [p.33]
He then proposes that in fact the light waves might pass through the particles of both the aether and of ponderable matter:
I will indicate the … last mode in which transparency may be conceived; which is by supposing that the movement of the waves of light is transmitted indifferently both in the particles of the ethereal matter which occupy the interstices of bodies, and in the particles which compose them, so that the movement passes from one to the other. And it will be seen hereafter that this hypothesis serves excellently to explain the double refraction of certain transparent bodies. [p.33]
And then he explains that the particles of ponderable matter may themselves be made up of smaller particles, the latter being the ones participating in the transmission of light waves:
Should it be objected that if the particles of the ether are smaller than those of transparent bodies (since they pass through their intervals), it would follow that they can communicate to them but little of their movement, it may be replied that the particles of these bodies are in turn composed of still smaller particles, and so it will be these secondary particles which will receive the movement from those of the ether. [pp.32-33]
Remember, this is 179 years before the appearance of Mendeleev’s Periodic Table of the Elements, and 223 years before the Rutherford-Bohr model of atom.
Once Huygens has presented his extended understanding of the aether, he moves on to explaining refraction, using the following diagram.
Huygens writes:
To explain then the reasons of these phenomena according to our principles, let AB be the straight line which represents a plane surface bounding the transparent substances which lie towards C and towards N. When I say plane, that does not signify a perfect evenness, but such as has been understood in treating of reflexion, and for the same reason. Let the line AC represent a portion of a wave of light, the centre of which is supposed so distant that this portion may be considered as a straight line. The piece C, then, of the wave AC, will in a certain space of time have advanced as far as the plane AB following the straight line CB, which may be imagined as coming from the luminous centre, and which consequently will cut AC at right angles. Now in the same time the piece A would have come to G along the straight line AG, equal and parallel to CB; and all the portion of wave AC would be at GB if the matter of the transparent body transmitted the movement of the wave as quickly as the matter of the Ether. But let us suppose that it transmits this movement less quickly, by one-third, for instance. Movement will then be spread from the point A, in the matter of the transparent body through a distance equal to two-thirds of CB, making its own particular spherical wave according to what has been said before. This wave is then represented by the circumference SNR, the centre of which is A, and its semi-diameter equal to two-thirds of CB. Then if one considers in order the other pieces H of the wave AC, it appears that in the same time that the piece C reaches B they will not only have arrived at the surface AB along the straight lines HK parallel to CB, but that, in addition, they will have generated in the diaphanous substance from the centres K, partial waves, represented here by circumferences the semi-diameters of which are equal to two-thirds of the lines KM, that is to say, to two-thirds of the prolongations of HK down to the straight line BG; for these semi-diameters would have been equal to entire lengths of KM if the two transparent substances had been of the same penetrability. [pp.35-37]
The diagram appears similar to that for reflection, presented in my previous post. However, this time, the light wave AC will not be reflected by the plane AB, but, rather, will pass through the plane AB and change direction. As for the case for reflection, the light source for AC is considered to be so distant that AC can be considered to be a straight line.
In this particular diagram, it is assumed that the transmission of light waves is one-third slower in the medium below AB than in the medium above AB. When the wave piece C reaches point B, the wave piece A will have generated a circular wave SNR with center A whose semidiameter [radius] is 2/3 that of the length CB. Similarly, when each wave piece H has reached its corresponding point K on the plane AB, a new circular wave with center H is generated; when wave piece C reaches point B, that new circular wave will have semidiameter 2/3 that of the length of the corresponding KM.
When wave piece C reaches point B, then all of the intermediate points H will have generated their corresponding circular waves, all of whom have as common tangent the line NB, as is explained by Huygens:
Now all these circumferences have for a common tangent the straight line BN; namely the same line which is drawn as a tangent from the point B to the circumference SNR which we considered first. For it is easy to see that all the other circumferences will touch the same BN, from B up to the point of contact N, which is the same point where AN falls perpendicularly on BN.
It is then BN, which is formed by small arcs of these circumferences, which terminates the movement that the wave AC has communicated within the transparent body, and where this movement occurs in much greater amount than anywhere else. And for that reason this line, in accordance with what has been said more than once, is the propagation of the wave AC at the moment when its piece C has reached B. For there is no other line below the plane AB which is, like BN, a common tangent to all these partial waves. And if one would know how the wave AC has come progressively to BN, it is necessary only to draw in the same figure the straight lines KO parallel to BN, and all the lines KL parallel to AC. Thus one will see that the wave CA, from being a straight line, has become broken in all the positions LKO successively, and that it has again become a straight line at BN. This being evident by what has already been demonstrated, there is no need to explain it further. [p.37]
Huygens then moves on to a similar diagram, this time for which the light wave AB passes from a slower medium to a faster medium.
The second diagram differs from the first in that the wave NB has a higher angle of refraction in the second diagram, while it has a lower angle of refraction in the first.
Since the angle of refraction is greater in the second medium, should the angle of incidence be too great, then there will be no refraction into the second medium: the circular waves that would be generated at the successive intermediate points K would not all end up with a common tangent NB, as in the diagram.
Huygens’s explanation of refraction provides a direct explanation of Snell’s Law, named after Willebrord Snellius [Willebrord Snel van Royen] (1580-1626), although it has been known as early as Ibn Sahl (940-1000): the sines of the angles of incidence and refraction are in inverse relation with the refractive indices of the two media and in direct relation with the speeds of light in the two media. If θ₁ and θ₂ are respectively the angle of incidence and the angle of refraction, n₁ and n₂ are respectively the refractive indices of the two media and v₁ and v₂ are respectively the speeds of light in the two media, then the following holds:
Finally, Huygens concludes that his model supports the position of Pierre de Fermat (1607-1665) over that of René Descartes (1596-1650):
I will finish this theory of refraction by demonstrating a remarkable proposition which depends on it; namely, that a ray of light in order to go from one point to another, when these points are in different media, is refracted in such wise at the plane surface which joins these two media that it employs the least possible time: and exactly the same happens in the case of reflexion against a plane surface. Mr. Fermat was the first to propound this property of refraction, holding with us, and directly counter to the opinion of Mr. Des Cartes, that light passes more slowly through glass and water than through air. [pp.42-43, my emphasis]
I must say that reading Huygens’s Traité de la lumière is a real joy. It is a remarkably clear book: in the first 40 pages or so, he is capable of proposing an aether model for the transmission of light at a finite speed, and explain the workings of reflection and refraction, and demonstrating that this model is compatible with the known properties of these phenomena.
Christiaan Huygens. Treatise on Light. In which are explained the causes of that which occurs in Reflexion, & in Refraction. And particularly in the strange Refraction of Iceland Crystal. Rendered into English By Silvanus P. Thompson. New York: Dover, 1962. First published by Macmillan and Company, Limited, in 1912.
G.G. Stokes M.A. XLVII. On the constitution of the luminiferous æther. Philosophical Magazine Series 3, 32(216):343-349, 1848.