861 



POLARIZATION OF LIGHT. 



POLARIZATION OF LIGHT. 



optic axis, or the two optic axes, of uniaxal or biaxal crystals. Indeed 

 the subject has been introduced in this place only to enable the reader 

 to form a better idea of the evidence in favour of the view taken in 

 the undulatory theory of the nature of polarization. 



The laws of the phenomena of crystalline plates were most carefully 

 investigated by SI. Biot and Sir David Brewster, and to account for 

 them the former imagined his theory of moveable polarization. Dr. 

 Young first showed that on the undulatory theory the retardation of 

 one of the two oppositely polarized rays which passed through the 

 crystalline plate relatively to the other was precisely that required to 

 produce, by ordinary interference, the tint observed. Huygens's 

 demonstration of the laws of reflection and refraction intimately con- 

 nects refraction with velocity of propagation, and thereby permits of 

 the retardation in question being calculated from the observed pheno- 

 mena of double refraction, without making any assumption as to the 

 nature of polarization. But it remained to be shown why no pheno- 

 mena of interference should be perceived unless the light incident on 

 the crystal were polarized, and the emergent light were subsequently 

 analyzed. 



It occurred to MM. Arago and Fresnel that it would be interesting to 

 examine in what manner the interference of two rays of light, which, 

 as regards length of path, are in a condition to interfere, might be 

 modified by previous polarization of the rays. The memoir contain- 

 ng an account of this important investigation is published in the 1 Oth 

 volume (1819) of the ' Annales de Chimie/ p. 288, and a full account of 

 it is given in Sir Juhn Herschel's Treatise on Light. Tha laws of inter- 

 ference of polarized light in truth might have been obtained at once 

 from the phenomena of the colours of crystalline plates by asiuming 

 thote cdinir* fit IP due to interference; but it was highly important at the 

 time to establish them in a more direct manner, by the observation of 

 what were incontestably fringes of interference. They may be thus 

 briefly enunciated : (1) Two rays of light coming from the same 

 source, and polarized in rectangular planes, are incapable of inter- 

 fering. (2) Two rays coming from the same source, and polarized 

 in the tame direction, interfere exactly like rays of common light. 

 (3) Two rays coming from the same source of common light, 

 and polarized in rectangular planes, may be afterwards analyzed 

 without acquiring thereby the property of mutually influencing each 

 other. (4) But two rays polarized in rectangular planes and after- 

 wards analyzed, do interfere, provided they come from the same source 

 of polarized light. (5) In the phenomena of interference produced 

 by rays which have experienced double refraction, the character 

 of the interference is not determined simply by the difference of path ; 

 it U necessary in certain cases to change this difference by a semi- 

 undulation. The half undulation must be added or not according as 

 the planes of primitive polarization and of analyzation lie in alternate 

 pain, or in the same pan- of opposite quadrants made by the rectan- 

 gular planes of polarization of the two polarized rays transmitted by 

 the crystalline plate. 



These lawg, combined with Malm's law already mentioned, and with 

 the formula which gives the intensity of the light resulting from the 

 interference of two streams of light of known intensities and difference 

 of path, enable us to calculate completely the colours of crystalline 

 plates in polarized light without making any assumption as to what 

 constitutes polarization. 



Let CP be the plane of primitive polarization, CA that of analyza- 

 tion, c o, c E the rectangular planes of polarization of the rays into 



which a ray of any kind is divided by the double refraction of the 

 crystalline plate. Let be the azimuth of c o, and s that of c A, both 

 measured from c P. We may without loss of generality suppose _t_to be 

 between the limits and 90, and between the limits i + 90. 

 Further, let o, e, be the lengths of path in air equivalent in time of 

 being described to the paths of the two rays respectively within the 

 crystal ; let \ be the wave-length in air belonging to any particular 

 kind of light, and take the original intensity of that light as unity. 



By Malus's law the intensities of the two streams, polarized along 

 co, c E, into which the original stream is divided by double refraction, 

 will be cos 1 > sin 5 i, respectively. If each of these be again divided 

 into two, polarized along and perpendicularly to c A, the intensities of 

 the former portions will by the same law be cos 5 i cos* (i >), sin 1 i 

 irin" (it), respectively. The difference of phase of these portions 



n_ 



will be (o e). Now if I, i' be the intensities of two streams of 



common light from the game source, p their difference of phase, the 

 intenmty of the light resulting from their interference will be 1 + 1' + 



2 V (i I 7 ) cos p. But by the second law of interference of polarized 

 light this formula may be applied to the interference of two streams of 

 polarized light which are capable of interfering, as by law 4 the two 

 streams are which are polarized along CA. In the application of 

 the formula we must take y i = cos i cos (i s), V i' = sin i sin (i i) 

 or = sin i sin (s-i), according as i > s or i < s. But by law 5 when 

 i < s we must change the actual difference of path by half an undula- 

 tion, that is, change p by ir, or, which comes to the same, change the 

 sign of one of the radicals Vi or V i'. Hence it will suffice to take 

 V i' = sin i sin (is) in all cases, and omit the addition or subtraction 

 of the half undulation. Hence the expression for the intensity will 

 become 



in 2 i sin 2 (j s) 

 2ir 



+ 2 cos i sin i cos (i s) sin (is) cos (oe) 



A 



(A) 



which may be readily transformed into the more simple expression 



cos 2 i sin 2( sin 2(t ) sin' T (oe) 



(B) 



The discussion of either of these expressions would give account of 

 the observed phenomena of the coloration of crystalline plates in 

 polarized light, but would exceed our limits. Suffice it to remark 

 that in the expression (B) the first term denotes the illumination, 

 alike for all colours, which would exist if the plate were removed, 

 while the second changes materially from colour to colour, in con- 

 sequence of the variation of A, in comparison of which the variation of 

 oe may usually be neglected. 



Now the study of the phenomena of light which are independent of 

 polarization leads us, and that in different ways, to the conclusion that 

 with light of given wave length the square of the amplitude of vibra- 

 tion must be taken as the measure of intensity, and consequently the 

 amplitude of vibration will vary as the square root of the intensity. 

 If now, bearing this in mind, we go over the whole investigation of 

 the colours of crystalline plates, beginning with the first application of 

 Malus's law, and deducing at every step from the intensities which are 

 the objects of direct observation the corresponding amplitudes of 

 vibration, we can hardly fail to be impressed with the idea that in 

 polarized light the vibrations of the ether take place in a rectilinear 

 manner in a direction transverse to that of propagation, and related in 

 some constant manner to the plane of polarization. Unquestionably 

 there must be a transverse something about polarized light which 

 admits of composition and resolution in that way. Now polarized 

 light in all its relations is symmetrical with respect to the plane of 

 polarization and the perpendicular plane, and to no other. For ex- 

 ample, when light is polarized by reflection at the surface of glass, 

 everything must evidently be symmetrical with respect to the plane of 

 reflection. Hence our rectilinear vibrations must also be symmetrical 

 with respect to the plane of polarization, and therefore must either be 

 in or perpendicular to that plane. 



But if such be the nature of polarized light, what notion must we 

 form of the nature of common light ? We have seen that common 

 light, in passing through a rhomb of Iceland spar not limited by a 

 screen, is resolved into two polarized streams, which, mixing on emer- 

 gence, yield a light having all the properties of common light. Nor 

 does the light reflected from the first surface of the rhomb possess any 

 other properties : it may be polarized like common light, or divided 

 by another rhomb into two polarized streams which mix on emergence 

 just as before, so that there is not the slightest ground for supposing 

 that with the light lost by reflection the original beam loses any ele- 

 ment of a different character from that which it retains. It is easy to 

 make out the nature of the vibration which results from the co- 

 existence of two series of rectilinear and transverse vibrations propa- 

 gated in the same direction and taking place in rectangular planes. 

 The vibrations in such a case would take place in planes perpendicular 

 to the direction of propagation, that is, would lie in the fronts of the 

 waves. 



Hence the following important suppositions are adopted in the 

 theory of undulations. 



The vibrations of the ether which constitute light, unlike those of air 

 which constitute sound, tale place, not to and fro in the direction of pro- 

 pagation, but laterally in the tangent planes of the waves. In polariztd 

 liijht the rUtrati'ms are rectilinear, and either parallel or perpendicular 

 to the plane of polarization, a point which for the present may be left 

 undecided. The polarization of common light consists in the resolution 

 of the vibrations into two rectilinear series, the undulations belonging to 

 one of which are obtained apart from those belonging to the other. 



The theory of transversal vibrations was first suggested by Dr. 

 Young, who was led to it by the consideration of certain phenomena 

 of biaxal crystals discovered by Sir David Brewster. It suggested 

 itself independently to the mind of Fresnel, who was led to it by con- 

 sidering the laws of interference of polarized light, made out by Arago 

 and himself. It encountered, at first, much opposition, for it was 

 entirely opposed to preconceived notions. It obliges us in fact to 

 suppose that the vibrations which constitute light are carried on, not 

 by forces by which the ether tends to resist condensation or rarefac- 

 tion, but by forces by which it tends to resist distortion, unaccompanied 



