ARTICLE IN PRESS. Optik 118 (2007)

Transcription

1 ARTICLE IN PRESS Optk 118 (007) Optk Optcs Wavs and rays n unaxal brfrngnt crystals Marı a C. Smon,1, Karn V. Gottschalk Laboratoro d Óptca, Dpartamnto d Físca, Facultad d Cncas Exactas y Naturals, Unvrsdad d Bunos Ars, 148 Bunos Ars, Argntna Rcvd 6 January 006; accptd 3 March 006 Abstract In ths papr, w prsnt a mor laborat and complt way of th formalsm that w hav dvlopd n succssv stps for ray tracng through unaxal brfrngnt mda n vw of th optcal dsgn. Wth th obtand formulas w analyz n dtal what w s through a clavag calct crystal. r 006 Elsvr GmbH. All rghts rsrvd. Kywords: Brfrngnc; Ansotropc mda; Unaxal crstal 1. Introducton Th doubl rfracton phnomnon was dscovrd by Erasmus Bartholnus [1] n 1669 bcaus somon, whos nam and surnam w gnor, took a calct crystal from Icland to Kopnhagn. In Fg. 1, w show th brfrngnc n a calct crystal or Icland spar that s not from Icland but from a lm pt of Co rdoba- Argntn. Fg. 1(a) s a photograph of th doubl rfracton as can b obsrvd lookng through a calct crystal and n Fg. 1(b) w show how a lasr bam ncdnt on th crystal s dvdd nto two bams. Chrstan Huygns (1678) studd th doubl rfracton phnomnon xprmntally usng th Sun as lght sourc and xpland xtraordnary rfracton wth surprsng succss [] by mans of hs wav thory. Durng hs nvstgatons h also dscovrd that ach of th rays that orgnatd n th rfracton n calct can b xtngushd whn passng through a scond calct Corrspondng author. E-mal addrss: hohn@ntzn.com.ar (M.C. Smon). 1 CONICET Carrra dl Invstgador. crystal f ths s rotatd around th drcton n whch lght travls. In 1809, Etn Los Malus [3] dscovrd lght polarzaton by rflcton,.., h found that whn lght s rflctd on th surfac of a transparnt mdum at a partcular angl, t acqurs th sam proprty as ach of th bams gnratd n th doubl rfracton phnomnon. Latr nvstgatons showd othr amazng phnomna that tak plac n brfrngnt crystals and ld to th constructon of mportant optcal lmnts. Arnold Sommrfld [4] ntroducs us to th optcs of crystal statng: Up to hr w hav assumd that optcal mda ar sotropc. Howvr all th plntud of th optcal subtlnss s frst manfstd n ansotropc mda. It s not strang, thrfor, that th Optcs of brfrngnt crystals s an appalng topc to many rsarchrs. Th frst problm that appars s th prsnc of two rfractd rays, on satsfyng Snll s law and trmd ordnary ray, and th othr not satsfyng ths law and hnc calld xtraordnary ray. Th dvlopmnt of formulas that, basd on th knowldg of th ncdnc drcton and th paramtrs /$ - s front mattr r 006 Elsvr GmbH. All rghts rsrvd. do: /j.jlo

2 458 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) non-magntc unaxal brfrngnt crystals, whch ar non-conductng and non-absorbng. Thn, th consttutv rlatons ar ~D ¼ ~E, (1) ~B ¼ m v ~H, () Fg. 1. Brfrngnc n a calct crystal: (a) doubl mag obsrvd wth a nakd y and (b) th lasr bam ncdnt prpndcular to th crystal surfac s dvdd n two bams. dfnng th ntrfac, calculat th drcton of th xtraordnary rfractd ray, has gvn rs to numrous publcatons. Rvwng all ths publcatons, whch ar vald ways to solv th problm vn though n som cass thy do not dffr much, s nthr possbl nor ncssary hr. Our am n ths papr s to prsnt a mor laborat and complt way of th formalsm w prvously dvlopd n succssv stps and whch s publshd n dffrnt artcls to addrss dffrnt applcatons and optcal dsgns [5 9]. W choos as startng pont Maxwll s quatons and th consttutv rlatons of th mda and at frst w gv a dtald dscrpton of th structur of th plan wavs that can propagat through th unaxal crystal. Aftrwards w show a way of solvng rfracton n ntrfacs sotropc mdum crystal and crystal-sotropc mdum. Fnally, w apply th dvlopd formalsm to xplan doubl rfracton as can b sn n a clavag calct crystal.. Plan wavs n unaxal crystals Th structur of plan wavs propagatng through a matral mdum s dtrmnd by Maxwll s quatons and th corrspondng consttutv rlatons [4,10]. Hr, w shall tak nto account th homognous whr ~E s th lctrc fld vctor, ~D s th lctrc dsplacmnt, ~H s th magntc fld vctor, ~B s th magntc nducton, m v s th magntc prmablty of vacuum and s th dlctrc tnsor. For th mda that w ar consdrng, thr s a coordnat systm, calld th prncpal axs systm, n whch th dlctrc tnsor s dagonal: 0 1 ¼ C A (3) whr 1,, 3 th prncpal dlctrc constants. In th cas of unaxal crystals, two of ths constants ar th sam,..: 1 ¼ ¼ o, 3 ¼. ð4þ Dfnng, as t s usual, th prncpal phas vlocts, w hav u o ¼ 1 ; u m v ¼ 1 (5) o m v and dnotng by z 1 ; z and z 3 th unt vctors along th prncpal axs, th consttutv rlaton (1) may b wrttn n th form ~E z 1 ¼ m v u ~ o D z 1 ~E z ¼ m v u ~ o D z (6) ~E z 3 ¼ m v u ~D z 3 : Takng nto account ths proprts of th brfrngnt mda, Maxwll s quatons hav th followng form n th MKS raconalzd unts systm: r ~ H ¼ q~ D qt ; r~ E ¼m v q ~ H qt ; (7) r~d ¼ 0; r ~H ¼ 0: In ths cas a wav quaton for th vctors ~E, ~D and ~H s not obtand, as t occurs for sotropc mda. But w can postulat a plan wav soluton: ~E ¼ ~E j ; ~D ¼ ~D j ; ~H ¼ ~H j (8) wth j ¼ p l ½ N ~r utšþj 0 (9) whr N s th unt vctor normal to th wav front, u s th phas vlocty, l th wavlngth and j o, th ntal

3 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) phas. Rplacng th rlatonshps gvn n Eqs. (8) nto quaton s systm (7), w obtan N ~H ¼u~D, (10a) N ~E ¼ m v u ~H, N ~ D ¼ 0, N ~H ¼ 0. (10b) (10c) (10d) By substtutng Eq. (10b) nto Eq. (10a) and by dvlopng th doubl vctoral product, w gt m v u ~D ¼ ~E ðn ~EÞ N. (11) Ths vctoral quaton can b wrttn for th prncpal componnts by combnng ths wth rlatons (6) and thn th followng systm of quatons for th prncpal componnts of ~D s obtand: n h u þ u o o N z 1 1 D ~ z 1 þ u o N z N z 1 D ~ z þ u N z 3 N z 1 ~D z3 ¼ 0, u o N z 1 N z D ~ z 1 n h þ u þ u o o N z 1 D ~ z þ u N z 3 N z D ~ z 3 ¼ 0, u o N z 1 N z 3 ~D z1 þ u o N z N z 3 D ~ z n h þ u þ u o N z 3 1 D ~ z 3 ¼ 0. ð1þ Ths systm has a no trval soluton whn th assocatd dtrmnant of th coffcnts vanshs. Ths condton gvs rs to two possbl valus for th phas vlocty for a crtan propagaton drcton of th plan wav. Ths two vlocts ar.1. Th ordnary wav Th ordnary wav propagats wth a phas vlocty u o. Ths s th sam for any N-drcton. Th rlatonshps btwn th prncpal componnts of th flds ar obtand from Eqs. (1) combnng wth th consttutv rlatons (6) and Eq. (10b): ~D 0 z 1 ¼ N z D ~ N 0 z ; D ~ 0 z 3 ¼ 0, (15) z 1 ~E 0 z 1 ~H z 1 ¼ N z N z 1 E ~ 0 z ; E ~ 0 z 3 ¼ 0, (16) N ¼ z 1 N z ~H z ; h N z 3 1 ~H z 3 ¼ ~H z N z N, ð17þ z 3 ~B 0 z 1 ~B 0 z 3 ¼ (a) N ¼ z 1 ~B 0 z ; N z h N z B ~ z N z N z 3. ð18þ u 0 ¼ u o, (13) qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u 00 ¼ u þ u o u N z 3, (14) whr w s that both phas vlocts ar qual to u o whn th normal wav front concds wth z 3 ( N ¼ z 3 ). For ths rason th prncpal axs z 3 s calld th optc axs of th crystal [4,10]. For any N-drcton dffrnt from th z 3 -drcton, th two-phas vlocts ar dffrnt and for ach on a dffrnt soluton from Eq. (1) wll b obtand. Th two rsultng wavs hav vry dffrnt charactrstcs and thy ar known as ordnary- and xtraordnary-wav. W shall dscrb ths two wavs sparatly. (b) Fg.. Structur of th wavs n a unaxal crystal: (a) ordnary wav and (b) xtraordnary wav.

4 460 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) From ths quatons w s that th ordnary wav s lnarly polarzd wth ~D 0 and E ~ 0 prpndcular to th plan dfnd by th optc axs (z 3 ) and th unt vctor N normal to th wav front (Fg. (a)). In gomtrcal optcs, th concpt of lght ray whos drcton s that of nrgy flux s usd,.., whn th lght ray s wll dfnd, t concds wth Poyntng vctor. For th ordnary ray w hav R o ¼ N (19) as t may b sn on th vctoral schm n Fg. (a). Summarzng, w may say that th charactrstcs of th ordnary wav ar th sam as plan wavs propagatng through sotropc mda, xcptng polarzaton. In fact, whras n sotropc mda all polarzatons ar prmssbl (ncludng no-polarzd lght), n brfrngnt mda th ordnary wav has a crtan lnar polarzaton for ach drcton of propagaton and othr polarzatons ar not prmssbl. (a).. Ttraordnary wav To ttraordnary wav corrsponds th phas vlocty u 00 gvn n Eq. (14). Ths dpnds on th drcton of propagaton N n such a way that u 00 ¼ u o whn N ¼ z 3,andu 00 ¼ u whn N? z 3. For any othr drcton, u 00 has a valu btwn u o and u. Fg. 3 rprsnts u 00 and u o n a plan contanng th optc axs for postv and ngatv crystals. Th rlatons btwn th prncpal componnts of th flds ~D 00, ~E 00, ~H 00 and ~B 00 rsultng from Eqs. (6), (10b), (1) and (14) ar ~D 00 N z 1 ¼ z 1 ~D 00 z ; N z h N ~D 00 z 3 1 z 3 ¼ ~D 00 z N z N, ð0þ z 3 ~E 00 N z 1 ¼ z 1 ~E 00 z ; N z h ~E 00 u N z z 3 ¼ ~ E z, ð1þ N z N z 3 ~H 00 z 1 ~B 00 z 1 u o ¼ N z N z H ~ z ; H ~ z3 ¼ 0, ¼ N z N z 1 () ~B 00 z ; ~B 00 z 3 ¼ 0. (3) Thus, polarzaton of ttraordnary wav propagatng n drcton of N s prpndcular to th ordnary on propagatng n th sam drcton. Thrfor, ~D 00 s (b) Fg. 3. Ordnary phas vlocty (crcumfrnc) and xtraordnary phas vlocty (ovalod). Postv unaxal crystal. Ngatv unaxal crystal. coplanar wth ~E 00 and N, and all of thm ar prpndcular to ~E 0, as t can b apprcatd on Fg.. Furthr ~D 00 s not paralll to ~E 00, as ~D 00? NU Th Poyntng vctor that rprsnts th ray R s prpndcular to ~ E 00 and H ~ 00, as t appars n Fg. (b). Snc th ray drcton s dffrnt from th normal wav front drcton ( N), a ray vlocty may b dfnd. Ths s th vlocty of advanc of phas n th nrgy flux drcton. Dnotng ths vlocty wth v 00,whav v 00 ¼ u00 cos Z, (4) whr Z s th angl btwn N and R or btwn ~D 00 and ~E 00, as t s showd n Fgs. b and 4. Th ray drcton can b calculatd from th vctoral dagram of Fg. 4 and from Maxwll quatons (Eqs. (10)) [5 7] or drctly from th calculus of Poyntng vctor from Eqs. (1) and (). Th rsultng vcoral rlaton s R ¼ 1 u N f þ u o u N z 3 z3 (5) n wth th normalzaton factor f n ¼ u4 þ u4 o. u4 N z 3 (6)

5 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) th optcal path OPL ¼ c v 00 OP, (9) whr v 00 th phas vlocty along th ray drcton gvn n Eq. (8)..4. Th dffrnc btwn ray and normal wav front Fg. 4. Dagram that dscrbs th plan wav n th plan normal to ~H vctor. Snc cos Z ¼ R N, from Eqs. (14), (4) and (5) v 00 may b xprssd n th form v 00 ¼ u4 þ u4 o u4 N z 3 u þ u o, (7) u N z 3 whch may also b wrttn n trms of R as follows: v 00 u u o ¼ q ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u o þ u. (8) u o R z 3 Summarzng, th structur of ttraordnary wav s notably dffrnt from th on of wavs propagatng n sotropc mda. Lk ordnary wavs, ttraordnary on has a lnar polarzaton dtrmnd by th unt vctor N and th optc axs, and othr polarzatons ar not possbl. Furthr, th phas vlocty vars wth th drcton of wav propagaton. But, th mor notabl charactrstc s th dffrnc btwn th normal wav front and th lght ray. Ths aspct of xtraordnary wavs dsrvs a mor xtnsv and t has gvn rss to many problms and a grat numbr of publcatons..3. Th optcal path On of th mor dlcat problms s th dfnton of optcal path. In a prcdng papr [11] w hav dfnd th optcal path as th gomtrcal path along th ray multpld by th rato c=v 00 (c s th lght vlocty n vacuum). It s mplct n th rlaton btwn u 00 and v 00 that th optcal path s qual to th gomtrcal path along th normal wav front multpld by c=u 00 (4). From ths two possbl forms to calculat th optcal path th frst on s prfrabl bcaus P (Fg. 4) s th pont rachd by th lght and t s always wll dfnd whras P 0 must b calculatd from P. Thn W now show by how and how much th normal wav front ( N ) dffrs from th lght ray ( R ). In ordr to facltat th followng analyss, lt us rwrt Eqs. (5) and (6) n trms of th prncpal rfractv ndcs n o and n (n o ¼ c/u o yn ¼ c/u ): R ¼ 1 h n on f þ ~V 3 (30) wth ~V 3 ¼ n n o N z 3 z3 (31) and th normalzaton factor f ¼ n4 o þ n4 n4 o N z 3 (3) Ths mans that whn w know th drcton of th normal wav front ( N ), th ray drcton s obtand addng a vctor along th optc axs drcton. Whn th ray drcton s known, th normal wav front drcton s obtand by mans of th nvrs rlaton: N ¼ 1 h n R g þ ~V 0 3 (33) wth ~V 0 3 ¼ n o n R z 3 z3 (34) and th normalzaton factor g ¼ n4 þ n4 o. n4 R z 3 (35) Th sgn of ~V 3 (or ~V 0 3 ) dpnds on th rlatonshp btwn th prncpal rfractv ndcs and on th sgn of th scalar product ( N z 3 ) (or ( R z 3 )). Th possbl cass for postv crystals (n 4n o ) and for ngatv crystals (n on o ) ar rprsntd n Fg. 5. In th sam fgur t may b sn that th ray coms nar th optc axs for postv crystals and t movs away from th optc axs for ngatv crystals. Dfnng th angls S and Z as n Fg. 5, w gt cos S ¼ N z 3 ; cos Z ¼ N R (36) and usng Eqs. (30) (3), w obtan cos Z ¼ n o þ n n o cos q S ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff n 4 o þ n4. (37) n4 o cos S W consdr both angls S and Z always postv, bcaus t s nough for our analyss hr. In Fg. 6, w

6 46 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) Fg. 6. Th angl btwn th normal wav front and th ray as a functon of th angl btwn th normal wav front and th optc axs, for calct (n ¼ 1:49, n o ¼ 1:66) and vatrt (n ¼ 1:65, n o ¼ 1:55). valu s obtand for Fg. 5. Vctoral schm of th rlaton btwn th normal wav front and th ray drcton for (a) postv crystals (n 4n o ) and (b) ngatv crystals (n on o ). rprsnt Z as a functon of S for vatrt (n o ¼ 1:55; n ¼ 1:65) and calct (n o ¼ 1:66; n ¼ 1:49). For S ¼ p=, ths s N? z 3, Z s zro snc N z 3 ¼ 0, and for S ¼ 0, ths s N ==z 3, Z rsults zro bcaus ~V 3 == N. W can also s n Fg. 6 that an angl S m, for whch th angl btwn N and R s maxmum xsts. Calculatng th drvatv, w fnd tan S m ¼ n (38) n o and th maxmum valu for Z rsultng from Eq. (37) S ¼ S m s tan Z max ¼ n o n. (39) n o n Th asymmtry of th curvs that rprsnt Z as a functon of S (Fg. 6) s du to th lctd rfrnc angl. Instad of th angl btwn N and z 3 (S), f w lct th angl btwn R and z 3 (O), th maxmum tan O m ¼ n o (40) n as t may b obtand from Eqs. (33) (35). From rlatons (38) and (40) t follows that th maxmum valu for th angl btwn th normal wav front and th ray appars whn th bsctrx of th angl btwn N and R forms an angl of 451 wth th optc axs. In a rcnt papr [1], Avndan o-aljo obtand th sam rlatons (38) and (39) for th maxmum angl btwn th ordnary ray and ttraordnary on, n a plan-paralll plat whn th ncdnt ray s normal to th plat. W shall rfr latr to ths concdnc. W hav mad a dtald dscrpton of th plan wavs whch can propagat n a brfrngnt crystal bcaus th structur of th wavs s fundamntal to analyz and xplan th phnomna of rflcton and rfracton of lght n ntrfacs wth ansotropc mda. 3. Rflcton and rfracton n ntrfacs btwn an sotropc mdum and a unaxal crystal 3.1. Lght s ncdnt from sotropc mdum Lt us consdr an ntrfac btwn an sotropc mdum wth rfracton ndx n and a unaxal crystal whos prncpal rfractv ndcs ar n o and n. Lght s ncdnt from th sotropc mdum and s rflctd and rfractd on th surfac of th crystal that w consdr plan (Fg. 7). Th prncpal axs of th crystal (z 1,z, z 3 ) ar orntd n such a way that th z -axs s contand n th ntrfac and th optc axs z 3 forms an

7 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) w obtan S s ¼ 0, (45a) S t ¼ S t, (45b) θ 1 u o N o s ¼ 0, (46a) 1 u o 1 N o t ¼ S t, (46b) u Fg. 7. Rflcton and rfracton n an sotropc mdum unaxal crystal ntrfac. angl y wth th ntrfac. Th projcton of th optc axs on th ntrfac s n th z-axs drcton (Fg. 7) and x s th unt vctor normal to th ntrfac. Furthrmor, w dnot by S th unt vctor n th drcton of th ncdnt ray. For ntrfacs btwn sotropc mda, th ncdnc plan s th plan dfnd by th ncdnt ray and th normal to th ntrfac. W can us ths concpt for sotropc mdum-brfrngnt crystal ntrfacs, snc th ncdnt ray propagatng through th sotropc mdum concds wth th normal wav front. W us thn th followng coordnat systm: x: unt vctor normal to th ntrfac, s: unt vctor contand n th ntrfac and normal to th ncdnc plan, t: unt vctor along th ntrscton of th ntrfac wth th ncdnc plan. In ths systm, th unt vctor n th optc axs drcton s z 3 ¼ x ðz 3 x Þþ s ðz 3 s Þþ t z 3 t (41) and th unt vctor normal to th wav front s S ¼ x S x þ t S t. (4) In trms of th angls dfnd n Fgs. 6 and 7, w hav z 3 ¼x sn y þ s cos y sn d þ t cos y cos d, (43) S ¼ x cos a þ t sn a. (44) W dnot by S th unt vctor normal to th rflctd wav front. Th unt vctors normal to th rfractd wav fronts ar: N o, for th ordnary wav and N, for ttraordnary on. From th boundary condtons that must b obyd on th ntrfac, 1 N u 00 s ¼ 0, (47a) 1 u 00 1 N t ¼ u S t. (47b) In Eqs. (45) w rcognz th rflcton laws and n Eqs. (46), th rfracton laws for sotropc mda. Ths mans that th rflcton s symmtrc. For th ordnary rfractd ray, Snll s law s vald and both rays, rflctd and ordnary rfractd, ar contand n th ncdnc plan. For ttraordnary rfractd wav front, Eqs. (47) hav bn obtand. From Eq. (47a), t may b sn that th unt vctor N s also contand n th ncdnc plan. Th qualty gvn n Eq. (47b) may b ntrprtd as a gnralzd Snll s law that wrttn n trms of th angls and th rfractv ndcs s n sn a ¼ n 00 sn b, (48) whr n 00 ¼ c=u 00 dpnds on th rfractd ray drcton. From Eq. (14) wrttn n trms of th ndcs n and n o, w obtan n 00 n n o ¼ q ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff n o þ n, (49) n o N z 3 whr N s our unknown. By wrtng ( N z 3 ) as functon of n 00 and th ncdnc drcton by mans of Eqs. (47), from Eq. (49) th followng bquadratc quaton for n 00 s obtand: a 4 ðn 00 Þ 4 þ a ðn 00 Þ þ a 0 ¼ 0, (50) a 4 ¼ h x, h a ¼ n S t ð ht Þn n o 4h, xt n S t h a 0 ¼ n S t ð ht Þn n o þ 4h 4. xt n4 S t ð51þ Th soluton of ths quaton taks th form n 00 ¼ 1 h pffffff h n o D hxt n S t þ h x n S t (5) x wth D ¼ n h s n, S t (53)

8 464 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) whr w hav dfnd th followng auxlary paramtrs: ¼ n o þ n n o ð z3 x Þ, (54) h t ¼ n o þ n, n o z3 t (55) h s ¼ n n n o ð z3 s Þ, (56) t ¼ n n o ð z3 x Þ z 3 t, (57) v σ δ v z whr, h t and h s ar always postv. Th sgn of t dpnds on th rlatonshp btwn th ndcs and th optc axs orntaton. Th componnts of th rfractd wav front ar n S t hx N t ¼ n 00, (58) pffffff n o D n S t hxt N x ¼ n 00 (59) and th componnt along th optc axs drcton s gvn by n h o pffffff N z 3 ¼ z n 00 ð 3 x Þ D þ no z 3 t n S t. (60) Snc th unt vctor normal to th wav front has bn obtand, th ray drcton can b calculatd by mans of Eqs. (30) (3). Th componnts of th rsultng unt vctor R may b wrttn n th form: pffffff n o D R x ¼ n 00, (61) f R s ¼ n o n pffffff n o ð z3 s Þ ðz 3 x Þ D þ no z 3 t n S t n 00, ð6þ f pffffff n o t D þ no h s n S t R t ¼ n 00, (63) f whr R x s always postv. Ths dtrmns th sgn of th root n Eqs. (5), (59), (60), (6) and (63). From Eq. (59) t may b sn that N x can b ngatv bcaus rfracton angl b (Fg. 8(a)) can b gratr than p=, as w showd n dtal n many arlr paprs [8,9,13,14]. Fnally, w calculat th phas vlocty along th ray drcton. By mans of Eq. (7) wrttn n trms of th rfractv ndcs w obtan v 00 ¼ c f n 00 n, (64) n o whr Eqs. (3) and (49) hav bn usd. Comparng th obtand formula wth that of ntrfacs btwn sotropc mda, w s that xtraordnary rfracton s dffrnt from ordnary rfracton n th followng aspcts: For ttraordnary rfractd wav front a gnralzd Snll s law s vald wth a rfractv ndx that dpnds on th ncdnc drcton. Ttraordnary rfractd wav front s contand n th ncdnc plan, but th corrspondng xtraordnary ray s not (sav som spcal cass of symmtry). Hr, w mak a commnt about th wavs polarzaton. For ntrfacs btwn sotropc mda, th polarzatons of rflctd and rfractd wavs dpnd on th polarzaton of th ncdnt wav. In th cas of an sotropc mdum-brfrngnt crystal ntrfac, th polarzaton of th rfractd wavs s dtrmnd by th propagaton drcton and th optc axs orntaton (s Eqs. (15) (18) and (0) (3)) and t do not dpnd on th polarzaton of th ncdnt wav. δ v t v z OPTICAL AXIS PROJECTION Fg. 8. Incdnt plan and ntrfac: (a) th normal wav fronts ar contand n th ncdnt plan and (b) th ntrfac: z-axs s th projcton of optc axs on th ntrfac.

9 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) Lght s ncdnt from th crystal Whn lght s ncdnt from brfrngnt mdum on an ntrfac btwn ths mdum and an sotropc on wth rfractv ndx n, thr ar two possbl ncdnt wavs: th ordnary ncdnt wav and ttraordnary ncdnt wav. In both cass, ach ncdnt wav wll gv rs to two rflctd wavs and on rfractd wav, as t s showd n Fg. 9. InFg. 9(a) th ncdnt ray s ordnary and th ncdnc plan s dfnd as for sotropc mda, as th plan dfnd by th ncdnt ray and th drcton normal to th ntrfac. As n arlr cas, from th boundary condtons that must b obyd on th ntrfac, t s sn that th drctons normal to th ordnary rflctd wav and th rfractd on ar contand n th ncdnc plan. Whn th ncdnt wav s ttraordnary on, th ray s not concdnt wth th normal wav front. Ths gvs rs to th problm of th dfnton of th ncdnc plan. Howvr, th fact that ttraordnary rfractd or rflctd normal wav fronts ar n th ncdnc plan suggsts us to dfn th ncdnc plan as th plan dfnd by th ncdnt normal wav front and th unt vctor normal to th ntrfac. Thn, all th normal wav fronts th ncdnt ( N ), th ordnary rflctd ( N 0 ), ttraordnary rflctd ( N 00 ) and th rfractd on ( S ) ar contand n that plan, shown n Fg. 9(b). Nthr th ncdnt xtraordnary ray nor ttraordnary rflctd on s contand n th ncdnc plan. Whn th ncdnt ray drcton s known, th drcton of th normal wav front may b calculatd by mans of Eqs. (33) (35). Th drctons of th rflctd and rfractd wavs ar obtand from th boundary condtons. For th rflctd wav t s found that th rflcton angl dos not concd wth th ncdnt on nthr for th wav fronts drctons nor for th corrspondng rays drctons. (Ths concdnc appars n th cas of both wavs, ncdnt and rflctd on, bng ordnary.) A dtald dscrpton of ths asymmtrc rflcton can b found n th papr Intrnal rflcton n unaxal crystals I. Gomtrcal dscrpton [9]. Hr w shall confn our attnton n th rfracton whch was not xpland n dtal but t was calculatd whn th ray tracng was mad [5,6,15 17] Whn th ncdnt ray s th ordnary on, Snll s law s vald: n o sn a o ¼ n sn b o. (65) (a) Whn th ncdnt ray s ttraordnary on, w must frst calculat N from Eqs. (33) to (35). In ths way, th ncdnc angl a s obtand: N t ¼ sn a ¼ 1 n R t n g n o R z 3 z3 t ð66þ wth g gvn n Eq. (35). Nxt w apply Snll s law wth a rfractv ndx that dpnds on th ncdnc drcton accordng to Eq. (49): n 00 sn a ¼ n sn a, (67) n 00 n n o ¼ q ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff n o þ n. (68) n o N z 3 (b) Fg. 9. Incdnt plan for a unaxal crystal sotropc mdum ntrfac: (a) ordnary ncdnt ray and (b) xtraordnary ncdnt ray. Whn a ray tracng through a plan paralll plat, prsm, wdg or lns s mad, ttraordnary ray rfractd on th frst ntrfac (sotropc mdumcrystal) s obtand n a form w hav alrady shown: by calculatng n frst plac th normal wav front N and thn th ray R. Hnc for th followng ntrfac (crystal-sotropc mdum) both drctons ar known. From N th rfractd ray drcton s obtand by

10 466 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) mans of formula (67) (68). Th R drcton dtrmns th pont of ncdnc on th scond ntrfac. By mans of th dvlopd formula, th ray tracng can b mad to obtan th mags formd through brfrngnt lmnts, as sn n many xampls publshd n arlr paprs [5 7,15]. Th formula wrttn hr ar gnral,.., thy ar applcabl to ntrfacs btwn an sotropc mdum and a unaxal crystal wth any optc axs drcton. But just bcaus thy ar gnral, thy ar dffcult to rad. Thrfor, w apply t to a smpl but vry llustratv cas. θ 4. What do w s through a calct crystal? If w ar fortunat nough to fnd a calct crystal whch s suffcntly transparnt and larg and w look through t, w wll s a doubl mag such as th on of Fg. 1(a). Ths phnomnon whch surprsd Erasmus Bartholnus n 1669 [1] and worrd Chrstan Huygns [], stll surprss and worrs us, as can b sn from th numbr of publcatons tratng ths subjct n rcnt yars. Chrstan Huygns, who had just dvlopd hs thory on lght wavs and had bn abl to xplan rflcton and rfracton n transparnt mda, also had to study n dtal th strang rfracton n calct bcaus t smd to dmolsh hs xplanatons of rfracton basd on wav thory. But hs thory rmand wondrfully vald and s stll mployd. Erasmus Bartholnus dscrbd brfrngnc prfrably n trms of mags but Chrstan Huygns usd th languag of lght rays. Concrnng calct crystal, h sad: a ray of Sun whch s ncdnt on a surfac of th crystal, s dvdd n two and travls through th crystal n that way. And thn h adds It s stll a gnral law n vry transparnt objct, that a ray whch s ncdnt normally to th surfac, passs wthout undrgong rfracton whl an oblqu ray s always rfractd. Howvr n ths crystal, th normal ray undrgos rfracton and thr ar som oblqu rays that go through straght []. Nowadays, w do not rsort to sun rays for our xprncs but to artfcal lght sourcs and w can show brfrngnc wth a lasr bam. In what follows w s how to xplan xtraordnary rfracton usng th formalsm that w hav dvlopd startng from Maxwll s quatons Th ncdnt ray s prpndcular to th crystal In Fg. 10, w show ray tracng for ncdnc normal to th ntrfac. In ths cas a ¼ 01 and thrfor b o ¼ 01 and b ¼ 01 for any valu of n, n o,orn 00. Ths mans that n th crystal th unt vctor normal to th xtraordnary wav front concds wth th ordnary Fg. 10. Ray tracng for prpndcular ncdnc. ray ( N ¼ R o ). In spt of ths, brfrngnc s vsbl bcaus of th fact that ttraordnary ray dos not concd wth th normal to th wav front ( N a R ) and th ncdnc pont on th scond ntrfac (P) sata dstanc D from th ncdnc pont (Q) of th ordnary ray. If Z s th angl btwn R and N and L th thcknss of th crystal, dstanc D s gvn by D ¼ L tan Z, (69) whr tan Z can b obtand from Eq. (37). Furthrmor, f w tak nto account that S ¼ p= þ y, w hav D ¼ Ln n o ð z3 x Þðz 3 z Þ n o þ n n o ð z3 x Þ. (70) Thus, th dstanc btwn th two rays mrgng from th crystal dpnds on th crystal s thcknss, th orntaton of optc axs and th prncpal rfractv ndcs of th crstal, but t dos not dpnd on th ndx of sotropc mdum (n ). W hav arlr mntond that M. Avndan o Aljo [1] showd that th dstanc btwn ordnary ray and xtraordnary ray s a maxmum whn th optc axs forms an angl S m ¼ atan n (71) n o wth th normal to th ntrfac. Th maxmum angl btwn both rays rsultng n ths cas s Z max ¼ atan n o n. (7) n o n Ths formulas ar n agrmnt wth Eqs. (38) and (39) that w hav obtand for th maxmum angl btwn th normal wav front ( N ) and th ray ( R ).

11 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) Ths s owng to th fact that for prpndcular ncdnc th ordnary ray ð R o Þ concds wth ttraordnary normal wav front ( N )(Fg. 10). W hav obtand for calct th maxmum valu Z max ¼ 6:181 for S m ¼ 41:911. In a clavag calct crystal th angl btwn th crystal s surfac and th optc axs s clos to 451. By calculatng for y ¼ 451, a slghtly smallr valu than th maxmum s obtand (Z ¼ 6:141). Dscovry of brfrngnc wth a calct crystal s part way to ts natural clavag. In fact, th orntaton of optc axs n a natural clavag calct crystal s so that, for normal ncdnc, th dffrnc btwn normal wav front and ray has a valu clos to th maxmum possbl on. 4.. Th ncdnc plan contans th optcal axs For oblqu ncdnc, th dstanc btwn th ordnary and xtraordnary rays dos not concd wth th shft btwn th normal to th wav front and th ray. Lt us consdr th ncdnc plan contanng th optcal axs (Fg. 11). From Eqs. (58) to (63), wthðz 3 s Þ ¼ 0 and z 3 t ¼ ð z3 z Þ, th followng formulas for angls b and r ar obtand [1]: tan b ¼ h xzn qffffffffffffffffffffffffffffffffffffffffffffffffffffffffff S t þ no n n S t n S t n n o h zn, S t (73) θ Fg. 11. Th unt vctors normal to th wav fronts and th rays n th ncdnc plan that contans th optc axs. tan r ¼ n on n S t q ffffffffffffffffffffffffffffffffffffffffffffffffffffffffff þ hxz n S t q ffffffffffffffffffffffffffffffffffffffffffffffffffffffffff (74) n S t wth gvn n Eq. (54) and h z ¼ n o þ n n o ð z3 z Þ, (75) z ¼ n n o ð z3 x Þðz 3 z Þ. (76) For th ordnary ray, w hav n S t tan b o ¼ q ffffffffffffffffffffffffffffffffffffffffffffffffffffffffff n o. (77) n S t Fg. 1(a) rprsnts (r b o ) and (r b ) as functons of th ncdnc angl a for n ¼ 1 and y ¼ 451. It may b sn that th maxmum dffrnc btwn ray (r ) and normal wav front (b ) s obtand whn a ¼4:881. For ths ncdnc angl b ¼3:091 and r ¼þ3:091,.., th normal to th ntrfac s th bsctrx of ðr b Þ¼Z. That ths angl Z s maxmum whn th angl btwn th bsctrx and th optc axs s 451, hav bn shown n Scton. But, ths angl s not vsbl bcaus w do not s th normal wav front. W s th rays, ths s (r b o ). Ths dffrnc ncrass a lttl for postv ncdnc angls and rachs a maxmum of for a ¼ 16:11. If w rprsnt (r a) as a functon of a, w obsrv that for a ¼ 15:651 ttraordnary ray s not dvatd. In rlaton ths, Chrstan Huygns has sad: thr ar oblqu rays that go through straght. Now, lt us s anothr proprty of xtraordnary rfracton found mprcally by Chrstan Huygns. Consdrng a prncpal scton of th crystal and th rfractons of rays ncdnt from oppost sds of th normal to th ntrfac, th rfractd rays fnd th scond ntrfac at th ponts PðþaÞ and PðaÞ. Ths ponts ar at th sam dstanc from pont P, th pont of ncdnc th ray that s prpndcularly ncdnt on th crystal (Fg. 13). Ths may b analytcally dmonstratd from our formulas. Th dstancs btwn PðþaÞ and PðaÞ to th orgn Q ar, rspctvly: QP ðþaþ ¼ L tan r ðþaþ, (78) QP ðaþ ¼ L tan r ðaþ (79) and wth Eq. (74), w obtan 3 QP ðþa QP ða 6 Þ ¼ L4 6 Þ ¼ L4 n o n n sn a qffffffffffffffffffffffffffffffffffffffffffffffffffffff þ z7 5, (80) n sn h a x n o n n snðaþ qffffffffffffffffffffffffffffffffffffffffffffffffffffff þ z7 5, (81) n sn h a x 3

12 468 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) θ Fg. 13. Ray tracng for normal and oblqu ncdnc n th ncdnc plan that contans th optc axs. Th ncdnc ponts PðþaÞ and PðaÞ ar at th sam dstanc of pont Pða ¼ 0). Fg. 1. Dvatons for th rays and th normal wav fronts for calct (n ¼ 1:49, n o ¼ 1:66, y ¼ 451, d ¼ 01). (a) Th angl btwn th normal wav front and th ray and th angl btwn both rays, as functons of th ncdnc angl. (b) Dvaton of ttraordnary rfractd ray wth rspct to th ncdnc drcton. For a ¼ 15:651 th dvaton s zro. For p=pap01 and for 15:651papp=, jaj4jr j. Thus, th rfractd ray movs away from th ntrfac. For 01pap15:651, jajojr j. Thus, th rfractd ray approachs to th ntrfac. whr QP ¼ Lz =. Now, w hav PP ðaþ ¼PP ðþaþ (8) and ths what Chrstan Huygns mprcally obtand Th ncdnc plan dos not contan th optcal axs Accordng to Chrstan Huygns th abov-mntond symmtry s not lmtd to th prncpal scton contanng th optcal axs but also for any othr scton of th crystal. Hnc, lt us consdr an ncdnc plan not contanng th optcal axs. Th coordnats l s and l t of th ncdnc ponts on th scond surfac of th crystal ar (Fg. 14) " # " R t l t ¼ L ; l s ¼ L # R s, (83) R x R x whr R x, R s and R t ar gvn n Eqs. (61), (6) and (63). Usng ths quatons, w obtan " l t ¼ L t þ n # oh s n S t pffffff (84) D " l s ¼ L s þ n # oh st n S t pffffff (85) D wth s ¼ n n o ð z3 x Þðz 3 s Þ, (86) h st ¼ n n o ð z3 s Þ z 3 t. (87) Thn, for normal ncdnc, th coordnats ar l t0 ¼ L h xt ; l s0 ¼ L h xs. (88) Thus, for 7a w hav th followng solutons: l t ðaþl t0 ¼ L n oh s n sn ð aþ pffffff, (89) D l s ðaþl s0 ¼ L n oh st n sn ð aþ pffffff. (90) D Thrfor, th ncdnc ponts PðaÞ, Pða ¼ 0Þ and PðþaÞ on th scond crystal s surfac rsultng from th

13 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) Fg. 14. Th ncdnc ponts on th scond surfac of th crystal for any ncdnc plan. rays ncdnt on th frst surfac at a dtrmnd ncdnc plan, ar along a straght ln (Fg. 14). From Eqs. (89) and (90) w also hav that th dstanc btwn PðþaÞ and Pða ¼ 0Þ s qual to th dstanc btwn PðaÞ to Pða ¼ 0Þ, and ths s also what Huygns obsrvd. That straght ln forms an angl x wth th ncdnc plan and passs across Pða ¼ 0Þ, whr tan x ¼ l sðaþl s0 ¼ h st. (91) l t ðaþl t0 h s Th quaton of that straght ln may b wrttn as l s ¼ n n o ð z3 sþ Lðz 3 x Þþ z 3 t lt. (9) h s In Fg. 14 w obsrv that pont P * s on th ncdnc plan. Th coordnats of ths pont ar l t ¼ L ð z 3 x Þ ; l s ¼ 0. (93) z 3 t Ths ncdnc pont corrsponds to th rfractd ray that s prpndcular to th optc axs and, thrfor, t concds wth th normal wav front. Howvr, ths pont dos not appar at an ntrfac btwn a clavag calct crystal and ar bcaus th corrspondng angl r * s n ths cas gratr than th crtcal angl. Fnally, lt us analyz what happns whn th crystal s rotatd. It s wll known that f w draw a pont on a sht of papr and w plac a calct crystal on t and w rotat t, thn on of th mags rotats around th othr. If w look at th mag prpndcularly, th xtraordnary mag dscrbs a crcumfrnc around th ordnary on. Ths s dpctd n Fg. 15(a). Th coordnats of pont P whch dfn th poston of th mag (usd to obtan Fg. 15(a)) ar gvn by Eqs. (88a) and (88b) whch can b wrttn as l t0 ¼ L h xz cos d, (94) Fg. 15. Curv that dscrbs ttraordnary mag whn th crystal rotats: (a) prpndcular ncdnc, (b) a ¼ 11 and (c) a ¼ 01. l s0 ¼ L h xz sn d. (95) If w look n an oblqu way, th curv dscrbd by th mag s mor complcatd, as can b sn n Fgs. 15(b) and (c). Ths fgurs wr obtand calculatng th coordnats of th ncdnc pont on th scond surfac of th crystal. In a prvous papr [7], by mans of ray tracng through plan paralll unaxal plats for convrgng bams, w hav shown that ths coordnats concd up to frst ordr wth th mag poston. 5. Conclusons Startng from Maxwll 0 s quatons, w hav dvlopd a formalsm to trac rays through unaxal

14 470 ARTICLE IN PRESS M.C. Smon, K.V. Gottschalk / Optk 118 (007) brfrngnt crystals ncludng a dtald dscrpton of th structur of th plan wavs whch can propagat through th crystal. Th quatons for ttraordnary rfracton wr st nlargng and gnralzng th concpts and laws known for ntrfacs btwn sotropc mda. Thus w hav rdfnd th ncdnc plan as th plan dtrmnd by th unt vctor prpndcular to th ntrfac and, nstad of th ray, th normal to th wav front. W hav sn that wth ths dfnton, all normals to th ncdnt, rfractd and rflctd wav fronts l n th ncdnc plan. Th drcton of ttraordnary rfractd normal wav front rsults from Snll s law usng a rfracton ndx whch dpnds on th ncdnc drcton and th ray drcton s obtand addng to th normal to th wav front, a vctor paralll to th optcal axs. Th rfracton at th ntrfac crystal-sotropc mdum s calculatd usng Snll s law and w obtan th drctons of th normal wav front and, consquntly, of th ray n th sotropc mdum. Th ncdnc pont n th scond ntrfac s obtand from th drcton of th ray calculatd n th frst rfracton. Thn w analyz th phnomna that taks plac whn lookng through a clavag calct crystal that hav bn alrady obsrvd and dscrbd by Erasmus Bartholnus and Chrstan Huygns. In th frst plac w show that for normal ncdnc, brfrngnc s obsrvd bcaus th drcton of th xtraordnary ray dos not concd wth th normal wav front. In ths partcular cas, th normal to th xtraordnary wav front and th ordnary ray concd and ths s why th maxmum angl btwn th normal to th wav front and th ray s, for normal ncdnc, qual to th maxmum angl btwn th ordnary and xtraordnary rays. W also showd that for clavag calct, w ar vry clos to ths maxmum. For oblqu ncdnc along th ncdnc plan contanng th crystal optcal axs, usng our formulas, w showd that thr s an ncdnc angl for whch th xtraordnary ray dos not dvat. W also obtand formulas for th coordnats of th ncdnc ponts on th scond crystal surfac and w showd that all th ncdnc ponts of th rays orgnatd n th sam ncdnc plan ar algnd along a straght ln. Morovr w dmonstratd analytcally th symmtrs obsrvd xprmntally by Chrstan Huygns. Thn w showd how ttraordnary mag rotats about th ordnary mag whn th crystal s rotatd about th drcton n whch lght advancs. W can thn fnsh th dscrpton of what w s through a clavag calct crystal, although w do not dar affrm that that dscrpton s complt. W would rathr say that all th abundanc of optcal subtlts [4] that can b dscovrd n ansotropc mda sm to b nxhaustbl. Acknowldgmnts Th pc of calct of th photographs was rcovrd from many dfctv calct from a lm pt n Co rdoba (Argntn). W ar gratful to Dr. Juan M. Smon for th clavag of th crystal. W ar gratful to Dr. Rodolfo M. Echarr for th ngnous assstanc to obtan th photographs. Ths work was carrd out wth a grant from th Unvrsty of Bunos Ars. Rfrncs [1] E. Bartholnus, Vrsuch mt dm doppltbrchndn sla ndschn Krstall, d zur Entdckung nr wundrbarn un außrgwo hnlchn Brchung fu hrtn, Ostwald 0 s klasskr dr xaktn Wssnschaftn, Akadmsch Vrlagsgsllschaft, M.B.H., Lpzg, 19 (pp. 1 35). [] C.H. von Zulchn, Abhanlung u br das Lcht, Ostwald 0 s klasskr dr xaktn Wssnschaftn, Akadmsch Vrlagsgsllschaft, Nr. 0, M.B.H., Lpzg, 1913, pp [3] E`.L. Malus, Sur un propt d la lum r rflch, Mm. Phys. Chm Soc.dA` rcul (1809) [4] A. Sommrfld, Optk Vorlsungn u br thortsch Physk, Band IV, 115, Akadmsch Vrlagsgsllschaft Gst & Portg, G. Lpzg, [5] M.C. Smon, Ray tracng formulas for monoaxal optcal componnts, Appl. Opt. (1983) [6] M.C. Smon, R.M. Echarr, Ray tracng formulas for monoaxal optcal componnts: vctoral formulaton, Appl. Opt. 5 (1986) [7] M.C. Smon, Imag formaton through monoaxal planparalll plats, Appl. Opt. 7 (1988) [8] M.C. Smon, R.M. Echarr, Intrnal total rflcton n monoaxal crystals, Appl. Opt. 6 (1987) [9] M.C. Smon, R.M. Echarr, Intrnal rflcton n unaxal crystals I. Gomtrcal dscrpton, J. Mod. Opt. 37 (1990) [10] M. Born, E. Wolf, Prncpls of Optcs, ffth d., Prgamon Prss, Oxford, Nw York, Toronto Sdny, Braunschwg, 1975 (pp ). [11] M.C. Smon, K.V. Gottschalk, Optcal path n brfrngnt mda and Frmat 0 s prncpl, Pur Appl. Opt. 7 (1988) [1] M. Avndan o-aljo, Analyss of th rfracton of th xtraordnary ray n a plan-paralll unaxal plat wth an arbtrary orntaton of th optcal axs, Opt. Exprss 13 (005) [13] M.C. Smon, L.I. Prz, Total rflcton n unaxal crystals, Optk 8 (1989) [14] M.C. Smon, L.I. Prz, Evanscnt wavs n total rflcton n unaxal crystals, Optk 86 (1990) 18. [15] M.T. Gara, M.C. Smon, Imag formaton through brfrngnt unaxal wdgs, Pur Appl. Opt. 5 (1996) [16] S. Mc Clan, R. Chpman, L. Hllman, Abrratons of a horzontal-vrtcal dpolarzr, Appl. Opt. 31 (199) [17] J.P. Lsso, A.J. Duncan, W. Sbbtt, M.J. Padgtt, Abrratons ntroducd by a lns mad from a brfrngnt matral, Appl. Opt. 39 (000)