196 



UNDULATOKl' THEORY 01' LIGHT. 



§ VIII.— CIRCULAR AND ELLIPTICAL I'OLAlilZATION I5Y REFLECTION. 



In all that precedes it Las been tacitly assnmed that tlu; initial phase oi" the 

 reflected midulation is a continuation of the final phase of the incident, or of 

 the same; reversed ; and also that the virtual origins of the elementary waves of 

 which we suppose the resultant relic, .ed wave to be composed, are in one inva- 

 riable surface, whatever be the azimuth of the incident molecular movements. 

 If these assumptions are entirely true, the expressions for the molecular velo- 

 city of the reflected wave ought to correspond with observation in all cases. 

 These expressions, nevertheless, fail for the case of total reflection at the second 

 surfaces of denser media, as will be; apjjarent if we substitute; the value of n, 

 which, in the cas« supposed, is less than unity in the formuhc. 



2 / ■^li^ — mi^t — cost \ 2 /2 _/ ^'^'^^ — ^"^^' — w'^costx 2 



Xy/ri^ — sin^2 + cos£/ ' \ Vw'"' — siu^i + w'^cos:/ 



AVhen sin^jr=?i^, v"^ and «'^ each nr. 1 ; or the reflection is total. Wc; know, 

 experimentally, that it continues to be total for all higher values of £ ; but the 

 radicals in tin; foregoing become imaginary. Mr. Fresnel, therefore, concluded 

 that reflection in some manner raodiiies the ^>A«.sy' of the undulation. Experi- 

 ment proves that it does so, and also that the degree of the modiflcation depends 

 upon the azimuth of the molecular moven:ients, and upon the incidence. 



The conversion of plane into circular polarization by reflection in " Fresnel's 

 rhombs " has been described. The manner in which this change taki^s place 

 may now be understood. If the plane ray is incident in either of the principal 

 azimuths 0^ or 90°, its plane of polarization is not afl'ccted by reflection. But 

 if its azimuth be 45°, it emerges from the rhomb after having undergone two 

 total reflections circularly polarizcMl. Now the plane polarized ray in azimuth 

 45°, is equivalent to two plane ])olarized rays of half the int(>nsit.y in azimuths 

 0" and 90." And as these components would singly undergo no sensible; 

 change of plane by reflection, while jointly fhey produce a circularly polarized 

 ray, we infer that one of them has been advanced or retarded upon the other 

 by a quarter of an undulation. If the ray had undergone only one reflection 

 in the rhomb, or if it had urulergone three, it would have; em(>rged neither piano 

 polarized nor cii-cularly polarized. If it had uiulergone four, it would have 

 emerged plane polarizetl again, with a change of 90° from its original azimuth. 

 Now, all th(!se i)henomena are represented by the equation |i] for the resultant 

 of vibrations at right angles to each other, which is as follows : 

 a'2_y2_|_^22.2 — 2aa'x7/cosO=a'^ahui^O. 



If we substitute for 0, Avhich is the interval between the two compounded 

 undulations, the more convenient symbol 2---, in which — expresses the differ- 

 ence of phase as a fraction of an undulation, the equation takes the form, 

 a'^y^+a^x^ — 2aa'xijcos2- =a'^ahm^2-- . 



This (Mjuatfon becomes the (Mpiation of a circle if we make w^a', and 

 -=^, or an odd nufltiple of ;{. It is then x^-\-7/'^::zza^. It is the equation of an 



eflipse for any values of - if a is not = a'. Hence the necessity of the con- 



dition that the original plane of polarization should be in azimuth 45'^, that the 

 components into which the velocity is decomposed in the principal azimuths 



may be ecpial. It is an e([uation of an (ellipse when az~a', if y is not = :|, 

 or an odd niidlijile of ,[. The ellipse becomes a straight line if - =|, or an odd 



