This brief note explains why the Michel-Levy colour chart for birefringence looks like this...

Transcription

1 This bief note explains why the Michel-Levy colou chat fo biefingence looks like this...

2 Theoy of Levy Colou Chat fo Biefingent Mateials Between Cossed Polas Biefingence = n n, the diffeence of the efactive indices fo the two ays Fequency of ays is unchanged in mateial, f = ω / π, ω = angula fequency Wavenumbe of ays changes to n k and n k whee in vacuum and is the wavelength in vacuum. So the wavelengths ae shote in the mateial, / n and / n Wave amplitude in vacuum popotional to i{ kx ωt} exp. π k = is the wavenumbe Ray amplitudes in mateial popotional to exp i{ n kx ωt} and exp i{ n kx ωt} The phase of the wave in vacuum would be geate by an amount kx at a distance of x (at the same time) So a phase diffeence can be conveted to an equivalent distance in vacuum by diving by k The two ays at the same time and at the same position in the mateial diffe in phase by ( n n )kx. This is equivalent to a distance in vacuum of ( n n )x. Hence the etadation = ( n n )x is the distance in vacuum which would cause the same phase diffeence as the two ays expeience afte passing a eal distance of x though the mateial. Lines on the Levy chat ae x (thickness) up the y-axis vesus, which theefoe / n n has a slope of ( ) Rationalisation of the Finge Colous (see Figue, next page):- Peaks in colous occu when = ( n + 0.5), fo n = 0,,, (see last page fo why) so these colous ae expected to be found in the following -anges, fom, to, n = 0 n = n = violet blue geen yellow oange ed n = 3 This gives a easonable epesentation of the chat (see diagam on next page) It is white below 90 It blus out due to ovelaps at fouth ode and beyond The eason why thee ae no naow violet/blue/geen bands in the fist ode ange is explained on the last page

3 7 6 5 n = 0, fist ode ed stats at 30 n =, second ode Cleaest ode since all colous well sepaated thid ode Reasonably clea ode but... n= ed ovelaps with n=3 blue so that thee is no clea ed o blue but only a washed out puple, stating 550 fouth ode Ovelaps getting wose. n = Ovelaps eveywhee. Colous almost washed out. So chats do not extend into fifth ode Ovelaps eveywhee. Tending towads unifom white. n = Fouth ode geen ends ~995. Last colou band on some chats n = 6 0 white below 90 violet stats at 570 and again at 950 n = 3 n =

4 Why ae the peaks at = ( n + 0.5) not at = n? This is because of the cossed-polas. In-Phase Case Second polaise This is the pojection of ay onto the plane of the second polaise n axis If the two ays ae in-phase at this point, this is the diection of the pojection of ay onto the plane of the second polaise Fist polaise defines diection of the two ays which is in-phase n axis Hence, if the ays ae in-phase, i.e., if = n fo n = 0,,, then the waves destuctively intefee because thei pojections onto the plane of the second polaise ae in opposite diections. Convesely, if the ays ae 80 o out-of-phase, i.e., if = ( n + 0.5) fo n = 0,,, then the waves constuctively intefee because ay is now pointing in the opposite diection and the pojections of the two ays onto the plane of the second polaise ae now in the same diection, thus... Out-of-Phase Case n axis Second polaise n axis Both ays now poject in this diection onto the plane of the second polaise Fist polaise

5 How does this wok algebaically? The ay amplitudes in the mateial ae popotional to i{ nkx ωt} i{ n kx ωt} exp and exp espectively. The two waves combine with a elative minus sign so the total amplitude is, exp i { n kx ωt} i{ n kx ωt} exp But a wave intensity is popotional to the absolute squae of its amplitude, which is, The peaks in ( expi{ nkx ωt} expi{ nkx ωt} )( exp i{ nkx ωt} exp i{ nkx ωt} ) = cos( n n ) kx ( n n ) = 4sin kx θ sin occu at θ = ( n + 0.5) π fo n = 0,, So the peaks in a given colou, of wavelength, occu at, n n n kx = n π x = ( n + 0.5)π i.e., fo = ( n n ) x = ( n 0.5) How could the colou chat be calculated moe accuately? + Why does the chaacte of the colous change ode-by-ode? The spectum (the intensity of light at wavelength ) is given as deived above by ( n ) ( n n ) n πx π sin kx = sin = sin. This is plotted against below fo the case of the geen lines at odes, 3 and 4, namely at = 773, 87 and 803, all of which coespond to a peak at = 55 (geen), as can be checked by dividing by.5,.5 and 3.5 espectively. Accodingly the spectum shown below eaches a peak at = 55 in all thee cases. Howeve the specta ae othewise vey diffeent. The second ode spectum has a lage contibution fom aound the geen wavelength only coesponding to quite a pue geen line at second ode; The thid ode spectum has a substantial amount of violet and ed as well as geen, coesponding to a less pue geen at thid ode; The fouth ode spectum has as much violet and ed as geen, coesponding to a paticulaly washed-out geen at fouth ode.

6 . Specta at Second, Thid and Fouth Ode Geen Lines Second Ode (=773) Thid Ode (=87) Fouth Ode (803) Relative Intensity violet 450 (geen) ed 750 wavelength, Exactly the same method fo the fist ode whee a geen line might have been expected, i.e., fo = 0.5 * 55 = 57, poduces the almost flat spectum given below. This explains why, in the fist ode, whee geen (o violet o blue) lines might have been expected, at 90-85, thee ae none - but athe just white.. Spectum at Fist Ode Geen Line (n=0) 0.8 Relative Intensity violet ed 750 (geen) wavelength,