Fresnel's third assumption was, that the product of the velocity into the square root of the density is constant for all media; which gives

Accordingly (7) becomes

1

V/V1 = (p1/p)1.

These formulæ give the ratios of the intensities of the reflected and refracted light to that of the incident light.

173. When the second medium is more highly refracting than the first, as is the case when light proceeding from air is reflected at the surface of glass, r is always real; but in the converse case, r is imaginary when the angle of incidence exceeds the critical angle. For if u be the index of refraction from air to glass, and light is internally reflected and refracted at the surface of glass in contact with air,

μι.

Since 1, it follows that cos r is imaginary when i > sin1μ? Under these circumstances, the expressions for the amplitudes of

CHANGE OF PHASE.

181

the reflected and refracted waves become complex, and their interpretation in former times was supposed to be a matter of considerable difficulty. The true explanation is this. The incident, reflected and refracted waves are the real parts of the right-hand sides of (3); if therefore A' and A, are real, the reflected and refracted waves are

but if A' and A, are complex, we must write A'= a + iß, A1a1+ẞ1, and the reflected wave is

=

w'a cos x (- la+my - Vt) - ẞ sin x (- la+my — Vt)

= (a2 + B2)1 cos {x (− lx + my − Vt) + tan-1 B/a}

which shows that there is a change of phase.

To find a, ß, we have from the first of (9)

Also (a2 + B2) = A, so that the reflected wave becomes

which shows that the reflection is total, and is accompanied by a change of phase whose value is given by (12).

(9)

174. To find what the refracted wave becomes, we have from

Since a is negative in the second medium, it follows that the wave penetrates only a very short distance, and becomes insensible at a distance of a few wave-lengths1.

175. The preceding theory is rigorous from a dynamical point of view; but when we consider the corresponding problem in which the incident light is polarized perpendicularly to the plane of incidence, we shall find that a difficulty arises, which will be considered in § 180.

The displacements in the three waves are given by (3), and they lie in the plane of xy and are perpendicular to the direction of propagation of the waves.

The condition that the displacements parallel to y should be · continuous gives

The first of these formulæ shows, that the intensity of the reflected light vanishes when

1 Another explanation, differing only in form, is as follows. The hypothesis that the reflected wave, corresponding to the incident wave

A cos 2π-(-x cos i + y sin i - Vt), is A' cos 2πλ ̄1 (x cos i + y sin i - Vt), tacitly involves the assumption, that reflection is unaccompanied by a change of phase. The fact that the amplitude becomes complex when the angle of incidence exceeds the critical angle shows, that this assumption is erroneous in this particular case. We ought therefore to assume that a change of phase takes place, both in the reflected and refracted wave; and we shall find, that the changes of phase are zero, when the angle of incidence is less than the critical angle, and have the above values when it exceeds it.

POLARIZING ANGLE.

183

176. When light of any kind is incident upon a transparent reflecting surface, the vibrations may be resolved into two components respectively in and perpendicular to the plane of incidence; and the first of (16) shows, that the component of the reflected vibration in the plane of incidence vanishes, when the angle of incidence is equal to tan-1μ. It therefore follows, that if common light be incident at this angle, the reflected light will be polarized in the plane of incidence. This is the law which was established experimentally by Brewster.

μ

177. Airy observed, that certain highly refracting substances, such as diamond, never completely polarize common light at any angle of incidence, but the proportion of polarized light is a maximum at the polarizing angle. The subject has been further investigated experimentally by Jamin', who found that for most transparent substances, Brewster's law is true as a first approximation only. It is therefore not possible to completely polarize light by a single reflection, but this may be accomplished by successive reflections from a pile of plates. Jamin also found, that when light which is plane polarized in any azimuth, is reflected from a transparent substance, the reflected light frequently exhibits slight traces of elliptic polarization; this shows, that reflection produces a difference of phase in one or both of the components of the reflected light. The reflection and refraction of light incident perpendicularly upon a glass plate have been experimentally investigated by Rood, and his results show that Fresnel's formulæ are very approximately correct.

178. When light proceeding from glass, is reflected at the surface of a rarer medium such as air, at an angle greater than the critical angle, it will be found that the values of A', A, given by (15) become complex; and it can be shown in the same manner

2

1 Ann. de Chimie et de Phys. (3), xxix. pp. 31 and 263; Ibid. (3), xxx. p. 257. Owing to the extreme smallness of the wave-length of light, compared with the ordinary standards of measurement, it is probable that if the surface of a polished reflector were magnified to such an extent, that the wave-length of light were represented by one inch, the surface of the reflector would appear to be exceedingly rough and uneven. It is therefore by no means improbable, that the secondary effects observed by Jamin, may be due to the fact, that our mathematical machinery is too coarse-grained to take into account inequalities of the reflecting surface, which though excessively minute, are not small compared with the wavelength of light.

3 Amer. Jour. of Science, vol. 1. July 1870.

that the reflection is total, and is accompanied by a change of phase. In fact the reflected wave is

179. The change of phase which accompanies total reflection, was experimentally verified by Fresnel in the following manner.

D

P

Let ABCD be a rhomb of glass, of which the angles at B and D are greater than the critical angle; and let light polarized in a plane which makes an angle of 45° with the plane of incidence (that is the plane of the paper), be incident normally upon the face AB, and after undergoing two reflections emerge at the face DC. The vibrations in and perpendicular to the plane of incidence after emergence, will be represented by

the emergent light will be circularly polarized. Now if