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UNDULATORY THEORY OF LIGHT. 



UNDULATORY THEORY OF LIGHT. 



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of the wave surface, and join the point of incidence with the points of 

 contact. The two tangent planes will give the directions of the fronts 

 of the two refracted waves, while the two joining-lines will give in 

 direction the courses, and in magnitude the velocities, of propagation, 

 of the two refracted rays ; and if from the point of incidence we let 

 fall perpendiculars on the two tangent planes, they will represent in 

 magnitude and direction the velocities of propagation of the refracted 

 waves, estimated in directions perpendicular to their planes. If the 

 incident waves and the surface of the medium be curved, the same 

 construction will hold good on substituting tangent planes for these 

 curved surfaces, just as in the case of ordinary media, but the front of 

 either refracted wave will of course no longer be plane but curved. A 

 construction based on the same principles gives the courses of inter- 

 nally reflected rays ; and it may be remarked that a single incident 

 ray in general gives rise to-two internally reflected rays. 



The accuracy of Huygens's construction, as has been already 

 observed, was confirmed by the observations of Wollaston and Malus, 

 and it was for some time supposed that other doubly refracting crys- 

 tals resembled Iceland spar, except as to the energy of their double 

 refraction. But Biot discovered that in quartz and several other 

 doubly refracting crystals it is the less refracted instead of the more 

 refracted of the two rays which obeys the ordinary law of refraction, 

 so that in the application of Huygens's construction the oblate must 

 be replaced by a prolate spheroid. This difference does not entail any 

 difference in the state of polarisation : it is still the ordinary ray that 

 is polarised in a principal plane. Crystals in which, as in Iceland spar, 

 the ordinary ray is the more refracted, have been called negative, and 

 those of the other class positive ; terms derived from certain views 

 relating to the corpuscular theory. 



If a plate of Iceland spar cut in a direction perpendicular to the axis 

 be interposed between the polariser and the analyser of a polarising 

 apparatus, a most splendid series of coloured rings make their appear- 

 ance. If the analyser had been turned till the field was dark, the 

 rings are seen to be interrupted by a black cross, the arms of which are 

 parallel and perpendicular to the plane of primitive polarisation, and 

 the order of the tints, beginning from the centre, agrees with that of 

 the reflected system of Newton's rings. If the analyser be turned 

 through 90, the black cross is replaced by a white one in the same 

 position, and the tints now agree with the transmitted system of 

 Newton's rings, but are much more vivid. In intermediate positions 

 of the analyser, the rings are less vivid, and are interrupted by a 

 double cross containing white light of mean intensity, or, in other 

 words, by eight radii, of which every alternate pair are at right angles. 

 The arms of the crosses are parallel and perpendicular to the planes of 

 polarisation of the polariser and the analyser. Similar rings are seen 

 in the case of other doubly refracting crystals of the class now known 

 ad imiaxal ; but sometimes, in consequence of a great chromatic variation 

 in the doubly refracting power of the crystal, the succession of tints 

 deviates materially from that of Newton's rings. 



In quartz, the phenomena manifested by polarised light passing 

 nearly along the axis appeared peculiar, resembling a combination of 

 those belonging to an ordinary uniaxal crystal with those belonging to 

 syrup of sugar ; but the same have recently been discovered in some 

 other uniaxal crystals. 



In investigating the colours exhibited by crystalline plates, Sir David 

 Brewster was led to the discovery that in many in fact, in the greater 

 number of crystals, there exist not one, but two, directions about 

 which coloured rings are seen in polarised light. Such crystals are 

 called triaical, and the two directions in question are called the optic 

 axes. Sometimes, as in sulphate of lime, sugar, &c., the optic axes are 

 widely separated ; and in this case, if a crystal cut perpendicularly to 

 either axis be introduced into the previously dark field of a polarising 

 apparatus, a series of nearly circular coloured rings is seen interrupted 

 by a tingle dark brush, in place of the pair of brushes forming a cross 

 which are seen in uniaxal crystals. If the crystal be turned round an 

 axis coinciding with the optic axis, the black brash turns round in a 

 contrary direction at an equal rate relatively to space, or a double rate 

 relatively to the crystal, whereas in a uniaxal crystal similarly treated 

 the black cross remains stationary. The rings are ordinarily nearly 

 equidistant, whereas in a uniaxal crystal they obey the law of increase 

 of Newton's rings, the squares of their radii increasing in arithmetical 

 progression. In other crystals, like nitre, the optic axes are near each 

 other, and may be seen together, especially if the plate be cut in a 

 direction perpendicular to their middle line. In this case, on intro- 

 ducing the crystal into the dark field a set of coloured curves are seen 

 resembling lemniscates, having the optic axes for poles ; and each 

 optic axis is traversed by a dark hyperbolic brush; and at certain 

 azimuths of the crystal, 90 apart, the two brushes unite and form a 

 cross, one arm of which passes through the optic axes. 



Sir David Brewster also discovered the relation between the optical 

 characters of crystals and their crystallographic forms. It was found 

 that the system of rectangular axes formed by lines bisecting the acute 

 and obtuse angle between the optic axes, and a line perpendicular to 

 their plane, were intimately connected with the crystalline form, so 

 that whenever there existed a plane of crystalline symmetry, two of 

 these axes lay in it. It is found that crystals of the cubic system 

 are singly refracting, those of the rhombohedral and pyramidal systems 

 uniaxal, and those of the prismatic, oblique, and anorthic systems 

 ARTS AND SCI. DIV. VOL. VIII. 



biaxal. No account is here taken of properties like those of syrup of 

 sugar, nor of what Biot has termed lamellar polarisation. 



The explanation of these beautiful coloured rings and curves follows 

 at once from combining the observed laws of double refraction, 

 including therein the polarisation of the refracted rays, with the laws 

 of the interference of polarised light. The latter, as we have seen, 

 admit of a perfectly simple explanation on the hypothesis of transverse 

 vibrations ; it remains to be seen what account that hypothesis can give 

 of the former. 



For some time after the discovery of biaxal crystals, it was supposed 

 that one of the refracted rays followed the law of ordinary refraction, 

 while the other followed some unknown law more complicated than 

 the Huygenian. It was theory which first pointed out to Fresnel, that 

 neither ray followed the ordinary law, an anticipation which he found 

 to be confirmed by experiment. 



Our limits would not permit us to enter into the theory of double > 

 refraction as given by Fresnel ; we shall content ourselves with a brief 

 notice of the principles of the investigation, and a statement of the 

 results to which it conducted him. 



In any theory of double refraction, there are two kinds of laws 

 which have to be accounted for ; those which regulate the velocity of 

 propagation, and those which regulate the state of polarisation. For 

 the two are evidently so bound up together, that any true theory 

 ought to explain both at the same time. 



With regard to the former, if we only knew the form of the wave- 

 surface, all the rest would follow from Huygens's construction. To 

 determine, however, the propagation of a disturbance spreading out 

 on all sides, is a problem presenting many difficulties, some of which 

 may be evaded by the following consideration. Imagine an infinite 

 number of plane waves, the effect of which, severally, is infinitely small, 

 to pass initially through the point from which the disturbance is 

 supposed to emanate. . These will serve to represent initially the 

 disturbance in the neighbourhood of that point, and their effect will 

 elsewhere be insensible. As the time progresses, they will travel 

 along with the velocities belonging to plane waves in their respective 

 directions, and their effect will be insensible except along the surface 

 of their ultimate intersections, which, therefore, will be the wave- 

 surface required. Hence, everything is reduced to the determination 

 of the mode of propagation of a plane wave in an arbitrary direction. 



This problem Fresnel endeavoured to solve by regarding the ether 

 within a crystal as made up of distinct particles acting on one another 

 with forces which are functions of the distances, and considering in 

 the first instance the motion of a single particle supposed to be alone 

 disturbed. The result is, however, meant to be applied to a whole 

 plane of particles constituting a wave, and this application is kept in. 

 view throughout the investigation. The force of restitution called 

 into play by the displacement is accordingly resolved in a direction 

 parallel and perpendicular to the front of the wave, and it is assumed 

 that the latter component produces no effect, because although a single 

 particle would be as free to move in that as in any other direction if 

 impelled, a plane of particles could not so move without compression, 

 whereas vibrations which are strictly transversal take place without 

 compression, to which Fresnel supposes the ether would oppose an 

 immense resistance. Accordingly account is taken only of that com- 

 ponent of the force of restitution which lies in the -plane of the wave, 

 and which therefore the particle, considered as one of a plane of 

 particles, would be free to obey. It is shown that for either of two 

 rectangular displacements parallel to the front of the wave, the com- 

 ponent of the force of restitution which is parallel to the front is also 

 in the direction of displacement, but for a given displacement the 

 force of restitution is different in these two directions. If now the 

 initial displacement be parallel to the front of the wave, but otherwise 

 arbitrary, and if it be resolved in these two directions, the components 

 will be propagated independently of each other, but with different 

 velocities. This accounts both for the double velocity of propagation 

 and for the polarisation, in rectangular planes, of the disturbance pro- 

 pagated with the two velocities respectively. 



These results were mainly deduced from a consideration of the force 

 of restitution called into play by an absolute displacement, whereas it 

 belongs to the fundamental conception of the mechanism of an undu- 

 lation that it is propagated by forces called into play by relative dis- 

 placements. This difficulty by no means escaped Fresnel, who 

 endeavoured to show, by probable reasoning, that the general results 

 would still be the same. 



The actual results which follow from Fresnel's theory may be 

 enunciated in the following laws : 



(1.) In every crystal there exists a system of three rectangular axes 

 (axes of elasticity), with respect to which the optical phenomena are 

 symmetrical. (2.) Let a, b, c, be three parameters belonging to these 

 axes respectively, and representing certain velocities of propagation ; 

 construct the ellipsoid a'-x- + 6 2 J/ ! + c 3 z ; = l, and cut it by a diametral 

 plane parallel to the front of a wave, the reciprocals of the semi-axes of 

 the elliptic section will represent the two normal velocities of the 

 waves which can travel independently in the given direction, and 

 planes perpendicular to the wave front and to the respective semi-axes 

 will be the corresponding planes of polarisation. 



These laws are of a nature to admit of comparison with experiment, 

 either directly or by the consequences which mathematically flow from 



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