CHAP. V., 3.] 



OPTICS. FKESNEL. 



107 



ircuiar embraced in Fresnel's theory, and equally (though 

 >lariza- unknowingly) confirmed by Sir D. Brewster's labo- 

 rious observations. But on one point Fresnel him- 

 self obtained a signal triumph. Having deduced ex- 

 pressions for the intensity of refracted light, on push- 

 ing them to the limit where refraction out of a denser 

 into a rarer medium becomes impracticable because 

 the light undergoes total internal reflection, the for- 

 mulae became affected by the multiplier ^/^7, and 

 were unsusceptible of arithmetical evaluation. In 

 endeavouring to attach a meaning to these expres- 

 sions, it occurred to him that as the intensity of the 

 totally reflected ray undergoes no change with the 

 angle of incidence, the expression in question might 

 in some way determine the alteration of the phase of 

 the wave (the position and direction of motion of the 

 molecules under consideration) which took place at the 

 instant of reflection. Now, admitting this as likely, 

 it appeared that the phase would vary not only with 

 the angle of internal reflection, but with the plane of 

 polarization of the ray. It had previously been 

 shown by Arago and by himself, that when two oppo- 

 sitely polarized rays meet or interfere, though there 

 is then no destruction of the light, there is usually 

 a remarkable change in its character. There is 

 one position of the interfering wave relative to the 

 primary one in which the combination produces light 

 polarized in a plane exactly intermediate between the 

 planes of previous polarization. If either ray be now 

 accelerated by half a wave-length on the other, the 

 new plane of polarization becomes perpendicular to 

 the former ; but if the shift of either of the primary 

 rays amounts to only one quarter of a wave-length, 

 the motion of the molecules takes place in a circle, 

 and the undulation has a helical form. Now, Fresnel 

 tested his hypothesis concerning totally reflected light 

 by calculating the circumstances of incidence which 

 should produce an effect equivalent to this ; and the 

 result completely verified his bold conjecture. The 

 apparatus employed is called FresneVs Rhomb, which 

 transforms plane-polarized light into light equally 

 reflexible in all azimuths, yet not common light, be- 

 cause it possesses properties which common light does 

 not (such as displaying the rings in crystals) ; this 

 is termed circularly polarized light. 



(492.) Theory of Double Refraction. The difficulty of 



mble re- accounting for double refraction did not consist 

 in showing how a spheroidal wave might be pro- 

 pagated. Young had already shown, in 1809, that 

 it would result from supposing a lamellar arrange- 

 ment of the crystalline molecules so that the ether 

 was differently elastic in a direction parallel to the 

 axis than in a plane or planes perpendicular to 

 that line. 1 Huygens had shown something similar 

 in accounting for terrestrial atmospheric refraction. 

 The difficulty was, to account for two waves travel- 



iction ex- 

 aincd. 



ling at the same time through the same portion of 

 matter with unequal velocities. The moment that 

 the idea of molecular movement transverse to the line 

 of propagation was admitted, it was easy to see that 

 no contradiction was involved in the idea. Two 

 waves might simultaneously travel in the same direc- 

 tion, and through the same medium, provided the 

 molecular displacements were in different planes. So 

 happy a solution could hardly fail to strike such minds 

 as those of Young and Fresnel with the impress of 

 conviction. A closer analysis confirmed the proba- 

 bility. Iceland spar (or rather the ether imprisoned 

 within it) is conceived of as a medium of uniform elas- 

 ticity in all planes perpendicular to the axis, but of a 

 different and greater elasticity in any direction parallel 

 to the axis. It is shown to result from this, that in 

 the direction of the axis alone is the motion of light 

 independent of the plane of the vibrations of which it 

 is composed, and consequently no separation of rays 

 occurs. When a ray moves parallel to what may by 

 an analogy be called the equatoreal plane of the 

 crystal, its undulation will, generally speaking, be 

 resolved into two whose vibrations are parallel and 

 perpendicular to that plane, and which travel with 

 different velocities though in the same direction. If 

 the ray take any other direction through the crystal, 

 both the direction and velocity of the divided rays 

 differ. The form of the extraordinary wave is exactly 

 the spheroid of Huygens. 



But what are we to conclude concerning those crys- (493.) 

 tals (of the discovery of which we shall speak in 5) Theoi 7 

 presenting two axes of double refraction ? Fresnel at^j^ *^ 

 once assumed that the elasticities must in that case axes, 

 vary in three rectangular directions, and he proceeded 

 to calculate the manner of propagation of a wave 

 through a medium thus constituted. I had proposed 

 to attempt some explanation of the steps of his most 

 ingenious and profound argument, but I find it incom- 

 patible with the space at my disposal, and at any 

 rate hardly to be apprehended without the use of 

 symbols or figures. For these reasons I shall merely 

 state the results. When the medium presents un- 

 equal elasticities in three rectangular directions, the 

 surface of the wave consists of two sheets each tra- 

 velling with its peculiar velocity. But neither of 

 these being spherical, the result cannot be expressed 

 by the ordinary law of refraction. In two directions 

 within the crystal, the wave surfaces coincide, or the 

 two rays coalesce. These directions are evidently 

 the optic axes, and the wave surface in their neigh- 

 bourhood has very interesting geometrical and 

 physical properties which have been elucidated 

 by British philosophers, as will be noticed in 

 another section. The true optical axes cannot 

 exceed two, and when two of the three elasticities 

 become equal, they merge into one. This is the 



1 Young's reasoning (Quarterly Review 1809, and Works, vol. i. p. 228) is based on an experiment by Chladni on the differing 

 velocity with which sound is propagated in wood, depending on the direction of the fibres. 



