PY3101 Optics. Learning objectives. Wave propagation in anisotropic media Poynting walk-off The index ellipsoid Birefringence. The Index Ellipsoid

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1 The Ide Ellpsd M.P. Vaugha Learg bjectves Wave prpagat astrpc meda Ptg walk-ff The de ellpsd Brefrgece 1

2 Wave prpagat astrpc meda The wave equat Relatve permttvt I E. Assumg free charges r currets E. Substtutg fr D ad emplg the usual vectr dett D t E E. E t

3 The wave equat Assumg that we have alged ur Cartesa crdates wth the prcpal aes f the crstal we ma wrte ths ut cmpet frm as E E. E t We shall lk fr sluts f the frm E E e krt. The wave equat Ntg that ad puttg 1, c, the wave equat becmes k k E k E E. c 3

4 4 The wave equat Nw c v. E k k c E s the squared magtude f the phase velct. We ma the use t bta v k The wave equat Ths, the, s a egevalue prblem wth the characterstc equat. ck where we have defed,

5 The wave equat Althugh ths appears t eld a plmal f rder 6, fact the 6 terms drp ut due t the relat a where a s the th Cartesa cmpet f the ut vectr the k drect. Mrever, sce all pwers f are eve, we bta a quadratc., a 4 b c The wave equat I the geeral case ad b a a a a, a 1 a 1 a 1 c. 5

6 The de ellpsd The sluts fr crrespd t tw mdes f vbrat Dfferet cmpets f the plarsat see dfferet refractve dces These mdes f vbrat ma be vsualsed b meas f the de ellpsd The de ellpsd 6

7 The de ellpsd Csder sme arbtrar wavevectr k Takg the tersect f the plae perpedcular t k wth the de ellpsd defes a ellpse The sem-aes f ths ellpse gve refractve dces ad, whch crrespd t the tw mdes f vbrat D ad D Optc aes I geeral, a ellpsd has tw crcular crss-sects I the case f just tw dstct semaes, we have a spherd ad there s just e crcular crss-sect The rmals t these crss-sects are kw as the ptc aes f the crstal 7

8 Optc aes Optc aes ptcal classes Baal crstals tw ptc aes (these are shw as N 1 ad N the prevus Fg.) Uaal crstals l e ptc as (take, cvet, t be alg the -as) Istrpc crstals N ptc as refractve de the same all drects 8

9 Optc aes Fr certa specal k drects, the quadratc wll have repeated rts. I these cases, the ptcal feld wll l see e refractve de (the crss-sect wth the de ellpsd s crcular) These drects are therefre the ptc aes f the crstal ad determed b the cdt b 4ac. Uaal crstals 9

10 Uaal crstals I a uaal crstal, we have (b cvet) = = ad = e. These are kw as the rdar ad etrardar refractve dces respectvel. The ceffcets f the quadratc equat fr are the a a, ad b e a a 1 e 1 c 4 e. Optc aes The dscrmat s b 4ac 4 1 a. e Fr e, the dscrmat s er whe a = 1,.e. whe k s parallel wth the -as. Thus, ths s the ptc as f the crstal 1

11 Uaal crstals The sluts fr are the e 1 e 1a 1 ad. Puttg, cs a Uaal crstals k 11

12 Uaal crstals we ma bta the eplct agular depedece 1 e 1 1 cs. e Specal cases,,., e. Ptg walk-ff 1

13 Ptg walk-ff We te that the D ad D are cstat ver the wavefrt f the prpagat. Each must therefre share the same phase factr k r t. Usg k, D k D. Ths mples k ad D are perpedcular. Ptg walk-ff Farada s Law becmes k E H. Ths meas that H must be perpedcular t bth k ad E. Smlarl, frm the Mawell-Ampere Law k H D. Ths meas that D must be perpedcular t bth k ad H. 13

14 Ptg walk-ff Ths mples that k, E ad D must be cplaar Hwever, althugh k ad D are rmal t e ather, geeral, E wll be at a agle t D gve b cs 1 ED E D. Ptg walk-ff I the meatme, the eerg flw wll stll be gve b the Ptg vectr S EH. Hwever, sce E ad k are t, geeral, perpedcular t e ather, the Ptg vectr s lger the k-drect ths s kw as Ptg walk-ff 14

15 Ptg walk-ff The de ellpsd (secd dervat) 15

16 Ide ellpsd The eerg dest due t the electrc feld s gve b u E 1 DE 1 D E D E D E. Ths ma be re-wrtte u E 1 D D D. Ide ellpsd Makg the chage f varable (asscated wth a scalg) D u E, we the have 1. Ths s the equat f the de ellpsd. 16

17 Uaal crstal k Uaal crstal Fr a uaal crstal, we have e 1. Takg = wth lss f geeralt, frm the fgure,, cs s., 17

18 Uaal crstal Substtutg these epresss t the de ellpsd cs Rearragg ths, we bta as fud earler. e s 1. 1 e 1 1 cs, e Brefrgece 18

19 Duble refract Duble refract eample: calcte 19

20 Brefrgece a uaal crstal The brefrgece s defed as. Fr a wave wth etrardar ad rdar cmpets, E e ad E, prpagatg a drect r, we ma wrte ad E e E E ep t c r ep r E t. c Brefrgece a uaal crstal The secd f these equats ma be re-wrtte E ep r E t ep c c r. Hece, after a dstace r, the rdar wave acqures a retardat r c r. Ths prvdes the bass fr retardat plates (see Plarsat).

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