Non-polarized Light

The nature of ordinary non-polarized light was next discussed.

Fresnel wrote[51]

He then proceeded to consider the relation of the direction of vibration to the plane of polarization. “Apply these ideas to double refraction, and regard a uniaxal crystal as an elastic medium in which the accelerating force which results from the displacement of a row of molecules perpendicular to the axis, relative to contiguous rows, is the same all round the axis; while the displacements parallel to the axis produce accelerating forces of a different intensity, stronger if the crystal “repulsive,” and weaker if it is “attractive.”

The distinctive character of the rays which are ordinarily refracted being that of propagating themselves with the same velocity in all directions, we must admit that their oscillatory motions are executed at right angles to the plane drawn through these rays and the axis of the crystal; for then the displacements which they occasion, always taking place along directions perpendicular to the axis, will, by hypothesis, always give rise to the same accelerating forces.

But, with the conventional meaning which is attached to the expression plane of polarization, the plane of polarization of the ordinary rays is the plane through the axis: thus, in a pencil of polarized light, the oscillatory motion is executed at right angles to the plane of polarization.”

This result afforded Fresnel a foothold in dealing with the problem which occupied the rest of his life: henceforth his aim was to base the theory of light on the dynamical properties of the luminiferous medium.

The first topic which he attacked from this point of view was the propagation of light in crystalline bodies. Since Brewster’s discovery that many crystals do not conform to the type to which Huygens’ construction is applicable, the wave theory had to some extent lost credit in this region. Fresnel, now, by what was perhaps the most brilliant of all his efforts,[52] not only reconquered the lost territory, but added a new domain to science.

He had, as he tells us himself, never believed the doctrine that in crystals there are two different luminiferous media, one to transmit the ordinary, and the other the extraordinary waves. The alternative to which he inclined was that the two velocities of propagation were really the two roots of a quadratic equation, derivable in some way from the theory of a single aether. Could this equation be obtained, he was confident of finding the explanation, not only of double refraction, but also of the polarization by which it is always accompanied.

The first step was to take the case of uniaxal crystals, which had been discussed by Huygens, and to see whether Huygens’ sphere and spheroid could be replaced by, or made to depend on, a single surface.[53]

Now a wave propagated in any direction through a uniaxal crystal can be resolved into two plane-polarized components; one of these, the “ordinary ray,” is polarized in the principal section, and has a velocity v1, which may be represented by the radius of Huygens’ sphere—say,

while the other, the “extraordinary ray,” is polarized in a plane at right angles to the principal section, and has a wave-velocity v2, which may be represented by the perpendicular drawn from the centre of Huygens’ spheroid on the tangent-plane parallel to the plane of the wave. If the spheroid be represented by the equation

and if (l, m, n) denote the direction-cosines of the normal to the plane of the wave, we have therefore

But the quantities 1/v1 and 1/v2, as given by these equations, are easily seen to be the lengths of the semi-axes of the ellipse in which the spheroid

is intersected by the plane

and thus the construction in terms of Huygens’ sphere and spheroid can be replaced by one which depends only on a single surface, namely the spheroid

Having achieved this reduction, Fresnel guessed that the case of biaxal crystals could be covered by substituting for the latter spheroid an ellipsoid with three unequal axes—say,

If 1/v1 and 1/v2 denote the lengths of the semi-axes of the ellipse in which this ellipsoid is intersected by the plane

it is well known that v1 and v2 are the roots of the equation in v

and accordingly Fresnel conjectured that the roots of this equation represent the velocities, in a biaxal crystal, of the two plane-polarized waves whose normals are in the direction (l, m, n)

Having thus arrived at his result by reasoning of a purely geometrical character, he now devised a dynamical scheme to suit it.

The vibrating medium within a crystal he supposed to be ultimately constituted of particles subjected to mutual forces; and on this assumption he showed that the elastic force of restitution when the system is disturbed must depend linearly on the displacement.

In this first proposition a difference is apparent between Fresnel’s and a true elastic-solid theory; for in actual elastic solids the forces of restitution depend not on the absolute displacement, but on the strains, i.e., the relative displacements.

In any crystal there will exist three directions at right angles to each other, for which the force of restitution acts in the same line as the displacement: the directions which possess this property are named axes of elasticity. Let these be taken as axes, and suppose that the elastic forces of restitution for unit displacements in these three directions are 1/ε1, 1/ε2, 1/ε3 respectively.

The elasticity should vary with the direction of the molecular displacement seemed to Fresnel to suggest that the molecules of the material body either take part in the luminous vibration, or at any rate influence in some way the elasticity of the aether.

A unit displacement in any arbitrary direction (α, β, γ) can be resolved into component displacements (cos α, cos β, cos γ) parallel to the axes, and each of these produces its own effect independently; so the components of the force of restitution are

This resultant force has not in general the same direction as the displacement which produced it; but it may always be decomposed into two other forces, one parallel and the other perpendicular to the direction of the displacement; and the former of these is evidently

The surface

will therefore have the property that the square of its radius vector in any direction is proportional to the component in that direction of the elastic force due to a unit displacement in that direction: it is called the surface of elasticity.

Consider now a displacement along one of the axes of the section on which the surface of elasticity is intersected by the plane of the wave. It is easily seen that in this case the component of the elastic force at right angles to the displacement acts along the normal to the wave-front.

Fresnel assumes that it will be without influence on the propagation of the vibrations, on the ground of his fundamental hypothesis that the vibrations of light are performed solely in the wave-front.

This step is evidently open to criticism; for in a dynamical theory everything should be deduced from the laws of motion without special assumptions. But granting his contention, it follows that such a displacement will retain its direction, and will be propagated as a plane-polarized wave with a definite velocity.

Now, in order that a stretched cord may vibrate with unchanged period, when its tension is varied, its length must be increased proportionally to the square root of its tension; and similarly the wave-length of a luminous vibration of given period is proportional to the square root of the elastic force (per unit displacement), which urges the molecules of the medium parallel to the wave-front.

Hence the velocity of propagation of a wave, measured at right angles to its front, is proportional to the square root of the component, along the direction of displacement, of the elastic force per unit displacement, and the velocity of propagation of such a plane-polarized wave as we have considered is proportional to the radius vector of the surface of elasticity in the direction of displacement.

Moreover, any displacement in the given wave-front can be resolved into two, which are respectively parallel to the two axes of the diametral section of the surface of elasticity by a plane parallel to this wave-front.

It follows from what has been said that each of these component displacements will be propagated as an independent plane-polarized wave, the velocities of propagation being proportional to the axes of the section,[54] and therefore inversely proportional to the axes of the section of the inverse surface of this with respect to the origin, which is the ellipsoid

But this is precisely the result to which, as we have seen, Fresnel lad been led by purely geometrical considerations; and thus his geometrical conjecture could now be regarded as substantiated by a study of the dynamics of the medium.

It is easy to determine the wave-surface or locus at any instant—say, t = 1—of a disturbance originated at some previous instant—say,t=0—at some particular point—say, the origin. For this wave-surface will evidently be the envelope of plane waves emitted from the origin at the instant t = 0—that is, it will be the envelope of planes

where the constants l, m, n, v are connected by the identical equation

and by the relation previously found—namely,

By the usual procedure for determining envelopes, it may be shown that the locus in question is the surface of the fourth degree

which is called Fresnel’s wave-surface.[55] It is a two-sheeted surface, as must evidently be the case from physical considerations. In uniaxal crystals, for which ε2 and ε3 are equal, it degenerates into the sphere

and the spheroid

It is to these two surfaces that tangent-planes are drawn in the construction given by Huygens for the ordinary and extraordinary refracted rays in Iceland spar. As Fresnel observed, exactly the same construction applies to biaxal crystals, when the two sheets of the wave-surface are substituted for Huygens’ sphere and spheroid.

“The theory which I have adopted,” says Fresnel at the end of this memorable paper, “and the simple constructions which I have deduced from it, have this remarkable character, that all the unknown quantities are determined together by the solution of the problem. We find at the same time the velocities of the ordinary ray and of the extraordinary ray, and their planes of polarization. Physicists who have studied attentively the laws of nature will feel that such simplicity and such close relations between the different elements of the phenomenon are conclusive in favour of the hypothesis on which they are based.”

The question as to the correctness of Fresnel’s construction was discussed for many years afterwards.

A striking consequence of it was pointed out in 1832 by William Rowan Hamilton (b. 1805, d. 1865), Royal Astronomer of Ireland, who remarked[56] that the surface defined by Fresnel’s equation has four conical points, at each of which there is an infinite number of tangent planes; consequently, a single ray, proceeding from a point within the crystal in the direction of one of these points, must be divided on emergence into an infinite number of rays, constituting a conical surface.

Hamilton also showed that there are four planes, each of which touches the wave-surface in an infinite number of points, constituting a circle of contact: so that a corresponding ray incident externally should be divided within the crystal into an infinite number of refracted rays, again constituting a conical surface.

These singular and unexpected consequences of the theory were shortly afterwards verified experimentally by Humphrey Lloyd,[57] and helped greatly to confirm belief in Fresnel’s theory.

However, conical refraction only shows his form of the wave-surface to be correct in its general features, and is no test of its accuracy in all details.

But the following experiments showed that the construction of Huygens and Fresnel is very correct:

• Stokes in 1872[58]
• Glazebrook in 1879[59]
• Hastings in 1887,[60]

Fresnel’s final formulae have since been unassailable.

The dynamical substructure on which he based them is open to objection.

But, as Stokes observed[61]:

“If we reflect on the state of the subject as Fresnel found it, and as he left it, the wonder is, not that he failed to give a rigorous dynamical theory, but that a single mind was capable of effecting so much.”