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When a = 0, a' = 0; and when a = 90°, a' = 90°. Accordingly, when the plane of polarization of the incident ray coincides with, or is perpendicular to, the plane of incidence, it is unchanged by reflexion. When 0+0=90°, a' = 0, and the plane of polarization of the reflected ray coincides with the plane of incidence, whatever be the azimuth of the incident ray.

(161) The plane of polarization of a polarized ray is changed by refraction, as well as reflexion, but in an opposite direction, the plane being removed farther from the plane of incidence, instead of approaching it. This movement of the plane of polarization increases with the incidence; being nothing when the ray falls perpendicularly upon the refracting surface, and greatest when the incidence is most oblique. The law of the change is expressed by the simple formula,

cotan a = cotan a cos (0 – 0');

in which a and a' denote (as before) the angles which the planes of polarization form with the plane of incidence, before and after refraction. This law was discovered experimentally by Sir David Brewster : it is a necessary consequence of the theory already given, and is deduced by a process exactly similar to that of the preceding article.

CHAPTER X.

ELLIPTIC POLARIZATION.

(162) WHEN an ethereal molecule is displaced from its position of equilibrium, the forces of the neighbouring molecules are no longer balanced, and their resultant tends to drive the particle back to its position of rest. The displacement being supposed to be very small, in comparison with the intervals between the molecules, the force thus excited will be proportional to the displacement; and from this it follows, according to known mechanical principles, that the trajectory described by the molecule will be an ellipse, whose centre coincides with the position of equilibrium. Hence the vibration of the ethereal molecules is, in general, elliptic, and the nature of the light depends on the direction and relative magnitude of the axes. By the principle of transversal vibrations, these elliptic vibrations are all in the plane of the wave; their axes, however, may either preserve constantly the same direction in that plane, or they may be continually shifting. In the former case, the light is said to be polarized; in the latter, it is unpolarized, or common light.

The relative magnitude of the axes of the ellipse determines the nature of the polarization. When the axes are equal, the ellipse becomes a circle, and the light is said to be circularly polarized. On the other hand, when the lesser axis vanishes, the ellipse becomes a right line, and the light is plane-polarized the vibrations being in this case confined to a single plane passing through the direction of the ray.

* This is not strictly true, except in homogeneous or uncrystallized media.

In intermediate cases, the polarization is called elliptical; and its character may vary indefinitely between the two extremes of plane polarization and circular polarization.

(163) An elliptic vibration may be regarded as the resultant of two rectilinear vibrations, at right angles to one another, which differ in phase.

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For, let x and y denote the distances of the molecule of the ether from its position of rest, in the two rectangular directions; a and b the amplitudes of the component vibrations; and t the time. Then

whence

x = a sin (vt - a), y=b sin (vt -ẞ);

7)

a- B = are (sin =) - are (sin =)

a

Taking the cosines of both sides, and clearing the result of radicals, we obtain

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This is the equation of an elliqse referred to its centre.

When the component vibrations are equal in amplitude, and differ 90° in phase,

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The path described by the molecule is then a circle.

(164) The nature of the elliptic polarization is completely defined, when we know the direction of the axes of the ellipse, and the ratio of their lengths.

These may be determined experimentally. In fact, when the elliptically-polarized ray is transmitted through a doublerefracting prism, whose principal section is parallel to one

of the axes of the ellipse, it is resolved into two plane-polarized rays, one of which has the greatest possible intensity, and the other the least. Accordingly, the direction of the principal section, for which the two pencils are most unequal, is the direction of one of the axes; and the square roots of the intensities are in the ratio of their lengths.

The direction of the axes of the ellipse may be more conveniently determined by turning the prism until the two pencils are of equal intensity: the principal section is then inclined at an angle of 45° to each of the axes.

(165) When a plane-polarized ray undergoes reflexion, the reflected light is, generally, elliptically-polarized. For a plane-polarized ray may be resolved into two, polarized respectively in the plane of incidence, and in the perpendicular plane; and we shall presently see that the effect of reflexion is, in general, to alter the phases of these two portions, and by a different amount. Hence the reflected light is compounded of two plane-polarized rays, whose vibrations are at right angles, and whose phases are no longer coincident; it is therefore elliptically polarized (163).

The first case in which this effect was observed was that of total reflexion.

When the angle of incidence exceeds the angle of total reflexion (the light passing from the denser into the rarer medium), the expressions for the intensity of the reflected light, given in (156), become imaginary. But it is obvious that, in this case, the intensity of the reflected light is simply equal to that of the incident, there being no refracted pencil. How, then, are the imaginary expressions to be interpreted? They signify, according to Fresnel, that the periods of vibration of the incident and reflected waves, which had been assumed to coincide at the reflecting surface, no longer coincide there when the reflexion is total; or, in other words, that the ray undergoes a change of phase at the moment of reflexion. The

amount of this change has been deduced by Fresnel, by a most ingenious train of reasoning, based upon the interpretation of imaginary formulæ. It varies with the incidence; and is different for light polarized in the plane of incidence, and in the perpendicular plane.

In the case of light polarized in any azimuth, we have only to conceive the incident vibration resolved into two, one in the plane of incidence, and the other in the perpendicular plane. The phases of these vibrations being differently altered by reflexion, the reflected vibration will be the resultant of two vibrations at right angles to one another, and differing in phase,—the amount of the difference depending upon the angle of incidence: this vibration, consequently, will be elliptic, and the reflected light elliptically polarized. When the azimuth of the plane of polarization of the incident ray is 45°, the amplitudes of the resolved vibrations will be equal; and if, moreover, their difference of phase is a quarter of an undulation, the ellipse will become a circle, and the light will be circularly polarized.

(166) Reducing his formula to numbers, in the case of St. Gobain's glass, Fresnel found that the difference of phase of the two portions of the reflected light amounted to one-eighth* of an undulation, when the angle of incidence was 54° 37'. Polishing, therefore, a parallelopiped of this glass, whose faces of incidence and emergence were inclined to the other faces at these angles, it followed

that a ray RR'R"R", inci

dent perpendicularly on

one of these sides, and

once reflected at each of

R"

R

R

the others, at R' and R", would emerge perpendicularly at the remaining side, the difference of phase in the two portions of

* In order to produce a difference of phase of a quarter of an undulation by a single reflexion, the refractive index should be

4.142.

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