propagation. The resultant light in this case is said to be elliptically polarized.

It therefore follows, that elliptically polarized light may be produced by the composition of two waves of plane polarized light, whose amplitudes and phases are different, and whose planes of polarization are at right angles; and conversely, a wave of elliptically polarized light can always be resolved into two waves of plane polarized light, whose planes of polarization are at right angles.

If A becomes

· ± A', and 8 = (n + 1) π, or e = e' + (n + 1)λ, equation (7)

v2 + w2 = A2.

Here the phases differ by an odd multiple of a quarter of a wave-length; in this case the elements of ether describe circles, and the light is said to be circularly polarized.

A circularly polarized wave may therefore be regarded as compounded of the two plane polarized waves

Let us now suppose, that the observer is looking along the axis of x in the direction in which the light is being propagated, and let us take the upper sign in (8). If P be any element of the ether,

POуy=(2π/λ) (x − Vt − e) =

;

cos &;

y

whence the resultant velocity along OP is zero; and if U be the resultant velocity perpendicular to OP, measured in the direction in which

decreases,

Uv sin - cos = 2πA/T.

POLARIZED LIGHT.

13

Since U is positive, the direction of vibration is from the left-hand to the right-hand of the observer; and a wave of this kind is called a right-handed circularly polarized wave. We see that it compounded of the two plane polarized waves

v = A cos &, w = A sin p.

may be

If the lower sign be taken, the angle POy=π-; whence if U' be the velocity perpendicular to OP', measured in the same direction as before,

U' = v sin 4 + ŵ cos $ = – 2πÁ/t.

Since U' is negative, the direction of vibration is from right to left; and a wave of this kind is called a left-handed circularly polarized wave. It may be compounded of the two plane polarized

waves

v=- · A cos &, w = A sin o.

Conversely, any plane polarized wave may be regarded as being compounded of a right-handed and a left-handed circularly polarized wave, whose amplitudes, phases and velocities of propagation are equal.

If d=nπ, or e=e′ +1⁄2nλ, (7) reduces to v/A = w/A',

which represents a plane polarized wave. From this result, we see that two plane polarized waves cannot compound into another plane polarized wave, unless their phases differ by a multiple of half a wave-length.

The methods by which plane, circularly, and elliptically polarized light can be produced, will be described in a subsequent chapter.

14. We must now consider a proposition, known as the Principle of Huygens.

Let PQ be the front, at time t, of a wave of any form which is travelling outwards; and let P'Q' be the front of the wave at time ť. To fix our ideas we may suppose that the wave is spherical, and the medium isotropic, but the argument will apply to waves of any form, which are propagated in an æolotropic medium.

At time t, the ether in the neighbourhood of P will be in a state of vibration; hence P may be regarded as a centre of disturb

1 According to the definition adopted, the directions of propagation and vibration are the same as those of translation and rotation of a right-handed screw; they are also related in the same manner, as the magnetic force produced by an electric current circulating round the ray,

ance, which propagates a spherical wave. Draw OPP' to meet the front of the wave at time t'; then at the end of an interval tt after the wave has passed over P, the secondary wave

A

produced by the element P will have reached P', and the ether in the neighbourhood of P' will be thrown into vibration. Similarly if Q be any other point on the wave front at time t, and OQQ' be drawn to meet the wave front at time t', it follows that at time t' the ether at Q will be set in motion, owing to the secondary wave propagated by Q. We therefore see, that the wave front at time t' is the envelop of the secondary waves, which may be conceived to diverge from the different points of the wave front at time t.

We are thus led to the following proposition, which was first enunciated by Huygens, and which may be stated as follows:

The effect of a primary wave upon any given point in the region beyond the wave, may be obtained by dividing the primary wave into an indefinite number of small elements, which are to be regarded as centres of disturbance, and finding by integration over the front of the primary wave, the sum of the disturbances produced at the given point by each of the secondary waves.

15. It is a well-known law of Geometrical Optics, that when light is reflected at a given surface, the angle of incidence is equal to the angle of reflection. We shall now prove this law by means of the undulatory theory.

Let AB be any portion of the front of a plane wave, which is reflected at AC, M any point upon it; and draw MP, BC perpendicular to the wave front. Draw AD, CD such that the

REFLECTION AND REFRACTION.

15

angles DAC and DCA are respectively equal to the angles BCA and BAC; and draw PN perpendicular to CD. Let t, T be the times which the wave occupies in travelling to P and C.

also since the triangles ABC and CDA are equal in every respect, and the angle D is therefore a right angle,

Now when the original wave reaches P, this point will become a centre of disturbance, and spherical waves will be propagated; at the end of an interval T-t the wave has reached C, and the secondary wave which diverges from P has reached N, since we have shown that PNV (T-t). Hence all the secondary waves. which diverge from points between A and C will touch AC, which is accordingly the front of the reflected wave at the instant the incident wave has reached C.

Since we have shown that the triangles ABC and CDA are equal in every respect, the angle BCA DAC; whence the angle of incidence is equal to the angle of reflection.

=

16. In order to prove the law of refraction, let V be the velocity of light in the second medium; and with A as centre describe a sphere of radius V'T, and let CE be the tangent drawn from C to this sphere; draw PN' perpendicular to CE. Then PC CB-PM T-t

PN'
AE

which shows when the incident wave has reached C, the secondary wave which diverges from P in the second medium, will have reached N'. Hence CE is the front of the refracted wave.

and

If i, r be the angles of incidence and refraction,

whence

which is the law of sines.

When the second medium is more highly refracting than the first, i>r, whence V> V'; accordingly the velocity of light is greater in a less refracting medium such as air, than in a more highly refracting medium such as glass. It also follows that the index of refraction is equal to the ratio of the velocities of light in the two media.

The direction of the refracted ray may be found by the following geometrical construction, which as we shall hereafter show may be generalized in the case of crystalline media, in which the wave surface is not spherical.

Let AB be the surface of separation of the two media, A the point of incidence. With A as a centre describe two spheres whose radii AB, AC are proportional to the velocities V, V' of light in the two media. Let the incident ray AP be produced to meet the first sphere in P, and let the tangent at P meet the plane AB in T. Then if TQ be the tangent from T to the second sphere, AQ is the refracted ray.