The path RST is in glass of refrangibility μ. Hence the optical length of the path LRS TM is

For the ray LTS'R' M we notice that regular refraction takes place at R', so that we may easily show, in a similar manner to the above, that—

And the optical length of the path LTS' R' M is—

The phase-difference of the rays arriving at M will thus depend upon v, the distance of M from O. In monochromatic light we should thus get a series of bright and dark circles round O; and in white light, coloured circles. Where v = u, that is, for all points on a circle with L L' as diameter, there is brightness for all colours; for there is no path-difference in the rays. Hence this is a white circle. At the centre, v = o, there will be a coloured spot, the colour depending on the value of eu2 Hence the colour will change with the distance of the

μR2

aperture from the centre.

Colours of Mixed Plates.-These were discovered by Young, and are seen when a luminous source is viewed through two glass plates containing between them a very thin layer of two transparent substances intimately mixed up with each other. Brewster obtained them by using a little soap lather between his plates. The explanation of them is that they are due to interferences of two neighbouring rays which traverse, in the space between the glass plates, the two different substances; a difference of phase is thus introduced

R

between them, and they are in a condition to interfere. If e is the distance between the glasses, and μ, μ are the refrangibilities of the two substances, the difference of optical distances traversed by the two rays is, for normal incidence, e(-). Thus there will be interference for light of wave-length whenever

Describe and explain the phenomena presented by the prismatic spectrum of the light reflected from a thin transparent film, not thin enough to show the "colours of thin plates." How and why does the spectrum change as the film gets thinner and thinner until it shows colours? (Lond. B.Sc. Hons., 1884.)

CHAPTER XV.

DOUBLE REFRACTION. POLARIZATION.

WHEN light is refracted through certain crystalline substances peculiar effects are observed, which we shall now consider. About the best substance for producing these effects, and

D

B

that in which they were first discovered, is Iceland spar. This is a crystallized form of carbonate of lime. The form in which it occurs is that of a rhomb or parallelopiped, that is, a six-sided figure, all whose sides are parallelograms. This form is represented in the figure. The angles of any one of the bounding parallelograms are 101° 55' and 78° 5'; and the dihedral angles between the faces of the rhomb are 105° 5' and 74° 55'. The solid angles at A and A' are, each of them, contained by three equal obtuse angles. If the rhomb has all its edges equal, then the line A A' is called the crystallographic axis of the crystal: A A' would then be equally

FIG. 181.

inclined to the three faces meeting in A or A'. In general the axis of the crystal is a straight line given only in direction; and it is any straight line equally inclined to the three faces meeting in a solid angle of the rhomb contained by three obtuse angles. Any plane through the axis is called a principal plane of the crystal; and a principal plane perpendicular to any face is called the principal plane of that face.

If a ray of light falls on such a rhomb, it is not refracted into it along one direction, as in the case of a piece of glass, but along two, giving rise to two refracted rays. Thus, if a black mark is made on a sheet of white paper, and then covered by the rhomb, two images of the mark will be seen on looking through the rhomb. Or, again, if a narrow beam of light is falling on a screen, and giving a bright spot, on putting the rhomb in the way of the beam, with a face normal to it, this is broken up into two, so that two spots are observed. The crystal is said to be doubly refracting.

In the next place, we have to observe that the light in either of these refracted beams possesses remarkable differences from common light, or the light in the beam before it entered the rhomb. In breaking up the beam of common light by means of the rhomb, the same effect is produced however the rhomb is rotated about the path of the beam-the two emergent beams are of equal intensity. Now, if one of these beams be similarly examined by a second rhomb, this will not be the

case.

To show this we may take two similar rhombs, place one of them on the mark on the paper, thus producing with it two images, and place the other on the first. The second will, in general, form two images of each image produced by the first, thus giving rise to four. But if the upper rhomb be rotated over the lower one, this is what is observed: when the faces of the upper rhomb are all parallel to faces of the lower one, so that the principal planes of the contiguous faces are parallel, only two images are seen. As the upper rhomb is rotated from this position, two more images begin to appear, at first very faint, but gradually increasing in distinctness; and the other images diminish in distinctness. And when the upper rhomb has been rotated through a right angle, these images have completely disappeared, and now only the other two are seen. As the rotation continues, the same series of appearances and disappearances occurs in every quarter of

a turn.

The same thing may be shown by means of the beam of light passing through the two rhombs to a screen. Or, if we cut off one of the two beams which emerge from the first rhomb, and examine the other alone by means of the second rhomb, the spot of light will appear brightest when all the faces of the rhombs are parallel, and will gradually become fainter, and completely disappear when the second rhomb has been turned through a right angle, appearing again when the rhomb is turned past this position, and growing to maximum brightness when another right angle has been accomplished; and so on.

From these experiments it is clear that the light in either of the two beams into which a given one is broken up by the crystal of Iceland spar, differs from common light in possessing properties having reference to direction which common light has not. Thus a beam of common light, on entering a rhomb of spar normally to a face, is always broken up into two beams of equal intensity, but a beam which has come from another rhomb is for certain positions of the second rhomb not broken up, and, in general, the intensities of the parts into which it is broken are unequal, and depend upon the position of the second rhomb relatively to the first. The light in either beam coming from a rhomb then possesses some property with regard to direction which must be referred to the manner in which the rhomb is held across its path.

These beams of light are said to be plane-polarized, or each is polarized in a certain plane. Their planes of polarization are two planes containing the directions of the beams, and respectively parallel and at right angles to the principal section of the face which is normal to the incident beam.

As we shall see, the peculiar property of plane polarized light is that the vibrations of the ether particles constituting it all take place along straight lines in a definite direction. And the plane which is called the plane of polarization of a given beam is, in accordance with the views of Fresnel, the plane at right angles to the direction of vibration of the particles.

We see, then, that by means of a rhomb of Iceland spar we can produce a beam of polarized light, and by means of another rhomb we can determine whether a beam is polarized The rhombs used in this way may be called polarizer and analyzer respectively.

or not.

Huyghens investigated by experiment the laws of refraction

of the two rays in Iceland spar into which a single one is broken up. He found that one of these rays is refracted according to the ordinary laws of refraction, the crystal having for this ray a definite index of refraction, no matter how it is cut or how the incident ray falls on it. This refracted ray is called, in consequence, the ordinary ray. The other ray is refracted according to a different law, and not the ordinary law of refraction. The sines of the angles of incidence and refraction do not bear a constant ratio to each other. It is called the extraordinary ray. Huyghens accounts for the position of this ray, and shows how to find it in any given case, as follows: We have seen how to find the refracted ray, on the principles of the wave theory, in the case of a ray falling on the surface of glass, by supposing the points of the surface to become centres of secondary disturbances as soon as the wave-front reaches them, these disturbances spreading out with equal velocities in all directions. Thus the disturbance from a given point has at any instant reached all the points on the surface of a sphere traced in the medium. And the refracted wave-front is the plane which at any instant touches all the spheres which the disturbances have reached; and the refracted ray is normal to this plane. Now, in the case of Iceland spar the disturbances which constitute the extraordinary ray travel with different velocities in different directions through the crystal. The surface which a disturbance at a single point in the crystal would reach all the

points of in a given time, is
not a sphere, but an oblate
spheroid, or ellipsoid of
revolution, whose axis of
revolution is the axis of the
crystal. This surface is the
wave-surface for the extra-
ordinary rays.
The wave-
surface for the ordinary rays,
which is, of course, a sphere,
has its diameter equal to the
axis of revolution, A A', of
the ellipsoid. Thus, if a
disturbance proceeds from

A'

FIG. 182.

the single point O, giving rise to both ordinary and extraordinary waves, at the end of a given time these will have reached the two surfaces of the sphere and spheroid; and