Huygen's Principle

Shortly after Roemer’s discovery, the wave-theory of light was greatly improved by Christiaan Huygens (b. 1629, d. 1695).

Back then, Huygens was living in Paris. He communicated his results in 1678 to Cassini, Roemer, De la Hire, and the other physicists of the French Academy.

He prepared a long manuscript on this which he published in 1690 at Leyden,[43] as Traité de la lumière où sont expliquées les causes de ce qui luy arrice dans la réflexion et dans la réfraction. Et parti- culièrement dans l’étrange réfraction du cristal d’Islande, Par C.H.D.Z[44]

Hooke’s hypothesis was that light is essentially a form of motion.

Huygens proved this by the effects observed with burning-glasses. The combustion induced at the focus of the glass dissociated the molecules of bodies. This was a certain sign of motion.

In conformity to the Cartesian philosophy, we seek the cause of all natural phenomena in purely mechanical actions.

Is that motion caused by the medium, as supposed in Hooke’s theory?

Or is that motion like that of a flight of arrows, as in the corpuscular theory?

Huygens decided that Hooke’s theory is the only tenable one. This is because beams of light proceeding in directions inclined to each other do not interfere with each other in any way.

Torricelli previously showed that light is transmitted as readily through a vacuum as through air.

From this, Huygens inferred that the medium or aether in which the propagation takes place must penetrate all matter, and be present even in all vacuums.

The process of wave-propagation he discussed and is now named after him.[45]

Consider a wave-front,[46] or locus of disturbance, as it exists at a definite instant t0.

Then each surface-element of the wave-front may be regarded as the source of a secondary wave, which in a homogeneous isotropic medium will be propagated outwards from the surface-element in the form of a sphere whose radius at any subsequent instant t is proportional to (t-t0).

The wave-front which represents the whole disturbance at the instant t is simply the envelope of the secondary waves which arise from the various surface elements of the original wave-front.[47]

The introduction of this principle enabled Huygens to succeed where Hooke and other contemporary wave-theorists[48] had failed, in achieving the explanation of refraction and reflexion.

His method was to combine his own principle with Hooke’s device of following separately the fortunes of the right-hand and left-hand sides of a wave-front when it reaches the interface between two media. The actual explanation for the case of reflexion is as follows:

Let:

• AB represent the interface at which reflexion takes place
• AHC the incident wave-front at an instant t0
• GMB the position which the wave-front would occupy at a later instant t if the propagation were not interrupted by reflexion.

Then by Huygens’ principle the secondary wave from A is at the instant t a sphere RNS of radius equal to AG.

The disturbance from H, after meeting the interface at K, will generate a secondary wave TV of radius equal to KM.

Similarly, the secondary wave corresponding to any other element of the original wavefront can be found.

The envelope of these secondary waves, which constitutes the final wave-front, will be a plane BN, which will be inclined to AB at the same angle as AC. This gives the law of reflexion.

The law of refraction is established by similar reasoning, on the supposition that the velocity of light depends on the medium in which it is propagated.

Since a ray which passes from air to glass is bent inwards towards the normal, it may be inferred that light travels more slowly in glass than in air.

Huygens explained that:

• the variation of the speed of light from one medium to another was caused by transparent bodies being made up of hard particles which interact with the aethereal matter, modifying its elasticity.
• the opacity of metals was caused by some metals being hard (to account for reflection) and others being soft.
  • The latter destroy the luminous motion by damping it.

The second half of the Théorie de la lumière is concerned with a phenomenon which had been discovered a few years previously by a Danish philosopher, Erasmus Bartholin (b. 1625, d. 1698).

A sailor had brought from Iceland to Copenhagen a number of beautiful crystals which he had collected in the Bay of Röerford.

Bartholin got it and noticed[49] that any small object viewed through one of these crystals appeared double. He found the immediate cause of this in the fact that a ray of light entering the crystal gave rise in general to 2 refracted rays.

• One of these rays was subject to the ordinary law of refraction.
• The other, called the extraordinary ray, obeyed a different law, which Bartholin did not succeed in determining.

The matter had arrived at this stage when it was taken up by Huygens.

Since in his conception each ray of light corresponds to the propagation of a wave-front, the two rays in Iceland spar must correspond to two different wave-fronts propagated simultaneously.

He found this idea easy:

Accordingly, he supposed that a light-disturbance generated at any spot within a crystal of Iceland spar spreads out in the form of a wave-surface, composed of a sphere and a spheroid having the origin of disturbance as centre.

The spherical wave-front corresponds to the ordinary ray, and the spheroid to the extraordinary ray: and the direction in which the extraordinary ray is refracted may be determined by a geometrical construction, in which the spheroid takes the place which in the ordinary construction is taken by the sphere.

Thus, let the plane of the figure be at right angles to the intersection of the wave-front with the surface of the crystal; let AB represent the trace of the incident wave-front; and suppose that in unit time the disturbance from B reaches the interface at T.

In this unit-interval of time the disturbance from A will have spread out within the crystal into a sphere and spheroid: so the wave-front corresponding to the ordinary ray will be the tangent-plane to the sphere through the line whose trace is T, while the wave-front corresponding to the extraordinary ray will be the tangent-plane to the spheroid through the same line. The points of contact N and M will determine the directions AN and AN of the two refracted rays[50] within the crystal.

In the Théorie de la lumière, Huygens did not attempt a detailed explanation of the spheroidal wave, but communicated one later in a letter to Papin,[51] written in December, 1690, “As to the kinds of matter contained in Iceland crystal," he says, “I suppose one composed of small spheroids, and another which occupies the interspaces around these spheroids, and which serves to bind them together.

Besides these, there is the matter of aether permeating all the crystal, both between and within the parcels of the two kinds of matter just mentioned.

I suppose both the little spheroids, and the matter which occupies the intervals around them, to be composed of small fixed particles, amongst which are diffused in perpetual motion the still finer particles of the aether.

There is now no reason why the ordinary ray in the crystal should not be due to waves propagated in this aethereal matter.

To account for the extraordinary refraction, I conceive another kind of waves, which have for vehicle both the aethereal matter and the two other kinds of matter constituting the crystal.

Of these latter, I suppose that the matter of the small spheroids transmits the waves a little more quickly than the aethereal matter, while that around the spheroids transmits these waves a little more slowly than the same aethereal matter.

These same waves, when they travel in the direction of the breadth of the spheroids, meet with more of the matter of the spheroids, or at least pass with less obstruction, and so are propagated a little more quickly in this sense than in the other ; thus the light-disturbance is propagated as a spheroidal sheet.”

Huygens made another discovery[52] of capital importance when experimenting with the Iceland crystal.

He observed that the two rays which are obtained by the double refraction of a single ray afterwards behave in a way different from ordinary light which has not experienced double refraction.

In particular, if one of these rays is incident on a second crystal of Iceland spar, it gives rise in some circumstances to two, and in others to only one, refracted ray.

The behaviour of the ray at this second refraction can be altered by simply rotating the second crystal about the direction of the ray as axis.

The ray undergoing the ordinary or extraordinary refraction according as the principal section of the crystal is in a certain direction or in the direction at right angles to this.

The first stage in the explanation of Huygens’ observation was reached by Newton, who in 1717 showed[53] that a ray obtained by double refraction differs from a ray of ordinary light in the same way that a long rod whose cross-section is a rectangle differs from a long rod whose cross-section is a circle.

In other words, the properties of a ray of ordinary light are the same with respect to all directions at right angles to its direction of propagation. Whereas a ray obtained by double refraction must be supposed to have sides, or properties related to special directions at right angles to its own direction.

The refraction of such a ray at the surface of a crystal depends on the relation of its sides to the principal plane of the crystal.

That a ray of light should possess such properties seemed to Newton[54] an insuperable objection to the hypothesis which regarded waves of light as analogous to waves of sound.

On this point he was in the right. His objections are perfectly valid against the wave-theory as it was understood by his contemporaries,[55] although not against the later theory[56] by Young and Fresnel.