Augustin Fresnel

Michell asked whether rays coming from the stars are refracted differently from rays from terrestrial sources.

Robison and Wilson 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 speed of light.

Arago 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 provided 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.
• He made it more precise by assuming that the aethereal density in any body is proportional to the square of the refractive index.

Let:

• c be the speed of light in vacuo
• c1 is its speed in a material body at rest

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 moving, 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.

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 gave it a slightly different form.

He supposed that the whole of the aether within the body was moving together.

The aether entered the body in front, and being immediately condensed, and issuing from it behind, where it is immediately expanded.

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.

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 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.

He divided a beam of light into 2 portions:

1. One passed 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

2. 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 2 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.

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 is at right angles to the direction of motion

…

• light from a star is incident normally on this face

The rays experience no refraction at incidence. 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

…

The telescope by which the emergent wave-front B1D is received is itself being carried forward by the earth’s motion.

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 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
• experiments in refraction and interference are not influenced by any motion which is common to the source, apparatus, and observer
• light travels between given points of a moving material system by the path of least time

These predictions were confirmed by Respighi in 1861, and Hoek in 1868 who used a telescope filled with water and a terrestrial source of light.

He 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 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 was now to reconcile:

• polarization and
• the wave-theory

Young was baffled by it.

In 1816, Arago visited him to show a new experimental result that 2 pencils of light, polarized in planes at right angles, do not interfere with each other under circumstances in which ordinary light shows interference-phenomena.

• Insted, they always give by their reunion the same intensity of light, whatever be their difference of path.

Young then 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 wrote his ideas to Arago in a letter dated Jan. 12, 1817.

In an article on “Chromatics” in September 1817:

Young wrote a letter to Arago April 29, 1818 explaining transverse vibrations, comparing light to the undulations of a cord agitated by one of its extremities.

Arago showed this to Fresnel who at once saw the true explanation of the non-interference of beams polarized in perpendicular planes.

The latter effect could even be made the basis of a proof of the correctness, of Young’s hypothesis.

The vibration of each beam has 3 components:

1. One along the ray 2-3. The other 2 at right angles to it

The Arago-Fresnel experiment showed 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 seen as like the vibrations of sound in air.

• Its direction of vibration is the same as that of propagation.

Fresnel gave 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:

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.