Augustin Fresnel

Michell[34] originially asked whether rays coming from the stars are refracted differently from rays from terrestrial sources.

Robison and Wilson[35] had asserted that the focal length of an achromatic telescope should be increased when it is directed to a star towards which the earth is moving, owing to the change in the relative velocity of light.

Arago[36] tested this and concluded that the light coming from any star behaves in all cases of reflexion and refraction precisely as it would if the star were situated in the place which it appears to occupy in consequence of aberration, and the earth were at rest. The apparent refraction in a moving prism is equal to the absolute refraction in a fixed prism.

Fresnel now set out to provide a theory capable of explaining Arago’s result. He adopted Young’s suggestion, that the refractive powers of transparent bodies depend on the concentration of aether within them; and made it more precise by assuming that the aethereal density in any body is proportional to the square of the refractive index, Thus, if c denote the velocity of light in vacuo, and if c1 denote its velocity in a given material body at rest, so that μ = c/σ1 is the refractive index, then the densities ρ and ρ1 of the aether in interplanetary space and in the body respectively will be connected by the relation

Fresnel further assumed that, when a body is in motion, part of the aether within it is carried along-namely, that part which constitutes the excess of its density over the density of aether in vacuo; while the rest of the aether within the space occupied by the body is stationary. Thus the density of aether carried along is (ρ1-ρ) or (μ2 - 1)ρ, while a quantity of aether of density ρ remains at rest.

The velocity with which the centre of gravity of the aether within the body moves forward in the direction of propagation is therefore

where ω denotes the component of the velocity of the body in this direction. This is to be added to the velocity of propagation of the light-waves within the body, so that in the moving body the absolute velocity of light is

Many years afterwards Stokes[37] put the same supposition in a slightly different form. Suppose the whole of the aether within the body to move together, the aether entering the body in front, and being immediately condensed, and issuing from it behind, where it is immediately rarefied.

On this assumption a mass ρω of aether must pass in unit time across a plane of area unity, drawn anywhere within the body in a direction at right angles to the body’s motion; and therefore the aether within the body has a drift-velocity - ωρ/ρ1, relative to the body: sa the velocity of light relative to the body will be c1 - ωρ/ρ1, and the absolute velocity of light in the moving body will be

or

This formula was experimentally confirmed in 1851 by H. Fizeau[38] who measured the displacement of interference fringes formed by light which had passed through a column of moving water.

The same result may easily be deduced from an experiment performed by Hoek.[39] In this a beam of light was divided into two portions, one of which was made to pass through a tube of water AB and was then reflected at a mirror C, the light being afterwards allowed to return to A without passing through the water: while the other portion of the bifurcated beam was made to describe the same path in the reverse order, i.e. passing through the water on its return journey from C instead of on the outward journey. On causing the two portions of the beam to interfere, Hoek found that no difference of phase was produced between them when the apparatus was oriented in the direction of the terrestrial motion.

Let w denote the velocity of the earth, supposed to be directed from the tube towards the mirror. Let c/μ denote the velocity of light in the water at rest, and c/μ + φ the velocity of light in the water when moving. Let l denote the length of the tube. The magnitude of the distance BC does not affect the experiment, so we may suppose it zero. The time taken by the first portion of the beam to perform its journey is evidently

while the time for the second portion of the beam is

The equality of these expressions gives at once, when terms of higher orders than the first in w/c are neglected,

which is Fresnel’s expression.[40]

On the basis of this formula, Fresnel proceeded to solve the problem of refraction in moving bodies. Suppose that a prism A0 C0 B0, is carried along by the earth’s motion in vacuo, its face A0 C0, being at right angles to the direction of motion; and

A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf that light from a star is incident normally on this face. The rays experience no refraction at incidence; and we have only to consider the effect produced by the second surface A0B0. Suppose that during an interval τ of time the prism travels from the position A0C0B0 to the position A1C1B1, while the luminous disturbance at C0, travels to B1, and the luminous disturbance at A0 travels to D, so that B1D is the emergent wave-front.

Then we have

If we write , and denote the total deviation of the wave-front by δ1, we have

, and therefore (neglecting second-order terms in w/c)

Denoting by δ the value of δ1, when w is zero, we have

Subtracting this equation from the preceding, we have

Now the telescope by which the emergent wave-front B1D is received is itself being carried forward by the earth’s motion; and we must therefore apply the usual correction for aberration in order to find the apparent direction of the emergent ray. But this correction is w sin δ/c, and precisely counteracts the effect which has been calculated as due to the motion of the prison. So finally we see that the motion of the earth has no first-order influence on the refraction of light from the stars.

Fresnel inferred from his formula that if observations were made with a telescope filled with water, the aberration would be unaffected by the presence of the water—a result which was verified by Airy[41] in 1871.

He showed, moreover, that the apparent positions of terrestrial objects, carried along with the observer, are not displaced by the earth’s motion; that experiments in refraction and interference are not influenced by any motion which is common to the source, apparatus, and observer; and that light travels between given points of a moving material system by the path of least time. These predictions have also been confirmed by observation: Respighi[42] in 1861, and Hoek[43] in 1868, experimenting with a telescope filled with water and a terrestrial source of light, found that no effect was produced on the phenomena of reflexion and refraction by altering the orientation of the apparatus relative to the direction of the earth’s motion. E. Mascart[44] in 1872 discussed experimentally the question of the effect of notion of the source or recipient of light in all its bearings, and showed that the light of the sun and that derived from artificial sources are alike incapable of revealing by diffraction-phenomena the translatory motion of the earth.

The greatest problem now confronting the investigators of light was to reconcile the facts of polarization with the principles of the wave-theory. Young had long been pondering over this, but had hitherto been baffled by it. In 1816 he received a visit from Arago, who told him of a new experimental result which he and Fresnel had lately obtained[45]—namely, that two pencils of light, polarized in planes at right angles, do not interfere with each other under circumstances in which ordinary light shows interference-phenomena, but always give by their reunion the same intensity of light, whatever be their difference of path.

Arago had not long left him when Young, reflecting on the new experiment, discovered the long-sought key to the mystery: it consisted in the very alternative which Bernoulli had rejected eighty year’s before, of supposing that the vibrations of light are executed at right angles to the direction of propagation,

Young’s ideas were first embodied in a letter to Arago,[46] dated Jan. 12, 1817. “I have been reflecting,” he wrote, “on the possibility of giving an imperfect explanation of the affection of light which constitutes polarization, without departing from the genuine doctrine of undulations. It is a principle in this theory, that all undulations are simply propagated through homogeneous mediums in concentric spherical surfaces like the undulations of sound, consisting simply in the direct and retrograde motions of the particles in the direction of the radius, with their concomitant condensation and rarefactions.

Yet it is possible to explain in this theory a transverse vibration, propagated also in the direction of the radius, and with equal velocity, the motions of the particles being in a certain constant direction with respect to that radius; and this is a polarization.”

In an article on “Chromatics,” which was written in September of the same year[47] for the supplement to the Encyclopaedia Britannica.

He writes:[48] “If we assume as a mathematical postulate, on the undulating theory, without attempting to demonstrate its physical foundation, that a transverse motion may be propagated in a direct line, we may derive from this assumption a tolerable illustration of the subdivision of polarized light by reflexion in an oblique plane,” by “supposing the polar motion to be resolved” into two constituents, which fare differently at reflexion.

In a further letter to Arago, dated April 29th, 1818, Young recurred to the subject of transverse vibrations, comparing light to the undulations of a cord agitated by one of its extremities. [49] This letter was shown by Arago to Fresnel, who at once saw that it presented the true explanation of the non-interference of beams polarized in perpendicular planes, and that the latter effect could even be made the basis of a proof of the correctness, of Young’s hypothesis: for if the vibration of each beam be supposed resolved into three components, one along the ray and the other two at right angles to it, it is obvious from the Arago-Fresnel experiment that the components in the direction of the ray must vanish: in other words, that the vibrations which constitute light are executed in the wave-front.

The theory of the propagation of waves in an elastic solid was yet unknown. Light was still always interpreted by the analogy with the vibrations of sound in air, for which the direction of vibration is the same as that of propagation.

It was therefore necessary to give some justification for the new departure. With wonderful insight Fresnel indicated[50] the precise direction in which the theory of vibrations in ponderable bodies needed to be extended in order to allow of waves similar to those of light: “the geometers,” he wrote," who have discussed the vibrations of elastic fluids hitherto have taken account of no accelerating forces except those arising from the difference of condensation or dilatation between consecutive layers."

He pointed out that if we also suppose the medium to possess a rigidity, or power of resisting distortion, such as is manifested by all actual solid bodies, it will be capable of transverse vibration. The absence of longitudinal waves in the aether he accounted for by supposing that the forces which oppose condensation are far more powerful than those which oppose distortion, and that the velocity with which condensations are propagated is so great compared with the speed of the oscillations of light, that a practical equilibrium of pressure is maintained perpetually.