The phenomenon of nonlinear optical birefringence in uniaxial crystals

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1 Th phmf liar ptical birfrigc i uiaial crstals Timtr Pascal, Adam Usma, J. C. Odd Dpartmt f Phsics, Fdral Uivrsit f Tchlg, Yla. P.M.B Yla. Adamawa Stat, Nigria. ptimtr@ah.cm; aausma@ah.c.uk (Rcivd 20 March 2011; accptd 28 Ju 2011) Abstract Birfrigc is a phm that prducs dubl valu atur f rfractiv idics i uiaial crstals. This phm givs ris t ffcts such as rdiar () ad trardiar wavs () ad Krr ffct which prducs liar cfficits calld Krr cstats. I this wrk, twlv uiaial crstals ar csidrd. Th miimum lctric fild itsit that is rquird t prduc liar birfrigc is dtrmid. Th magituds f th vaishig dilctric tsr cmpts ad thir crrspdig critical agls hav bvaluatd fr th twlv uiaial crstals. Th rsult shws that i gativ uiaial crstals, lctrmagtic wavs travl fastr i th trardiar ais tha i th rdiar as. Th rvrs is hwvr btaid i psitiv uiaial ais. Th wrk has als prvidd th sris t b mpld wh dtrmiig th Krr cstats fr vrdr liar cfficits. A mtivati fr this wrk is a attmpt t mak liar ptical phma accssibl t phsics udrgraduat. Kwrds: Birfrigc, uiaial crstals, Krr cstats, lctric fild itsit ad rfractiv idics. Rsum Birrfrigcia s u fóm qu prduc l dbl - valr atural d ls ídics d rfracció cristals uiaials. Est fóm da lugar a fcts cm l rdiari () las las trardiarias () l fct Krr, qu prduc cficits lials cstats llamads Krr. Est trabaj, dc cristals uiaials s csidrads. El míim d itsidad d camp léctric qu s rquir para prducir la birrfrigcia lial s dtrmia. Las magituds d las cmpts dl tsr diléctric ula sus águls crrspdits crítics ha sid valuads para dc crsitals uiaials. El rsultad mustra qu cristals uiaials gativs, las das lctrmagéticas viaja más rápid l j trardiari qu ls js cmus. Simbarg, s l ctrari btids uj uiaial psitiv. El trabaj tambié ha prprciad la sri qu s mpla para dtrmiar las cstats d ls cficits d Krr, iclus para lial. Ua mtivació para st trabaj s u itt d hacr ls fóms óptics lials d accs a la licciatura d Física. Palabras clav: Birrfrigcia, cristals uiaials, cstats d Krr, itsidad d camp léctric ls ídics d rfracció. PACS: A; C; Fk; gb ISSN I. INTRODUCTION Crstals cahibit a umbr f itrstig ptical prprtis such as dubl rfracti, ptical rtatir plariatiffcts [1]. Matrial i which tw f th cmpts f dilctric cstats ar qual is trmd uiaial crstal. This class icluds trigal, ttragal ad hagal crstals. Th dilctric tsr ca b writt i matri frm as (1) whr ε, ε ad ε ar th vaishig dilctric tsr cmpts. Birfrigc i aistrpic (uiaial) mdium has ma itrprtatis i liar ptics. I [2] it is dfid as a phm that prducs multi valu atur f rfractiv id, whil [3] dfis it as dubl rfracti. Hwvr, its ffct givs ris t ma phma such as; rdiar ad trardiar wavs, Krr ffct ad liar rfractiv idics. Th phmf birfrigc was first discvrd wh it was bsrvd that crstals f Iclad spar, a frm f calcium, frmd dubl imags f bjcts s thrugh thm [4]. Th rfractiv id f a matrial is th factr b which lctrmagtic radiati is slwd dw (rlativ t vacuum) wh it travls isid th matrial. Ma traspart slids ar pticall istrpic, maig that th Lat. Am. J. Phs. Educ. Vl. 5, N. 2, Ju

2 id f rfracti is qual i all dirctis thrughut th crstalli lattic. All istrpic crstals (.g. th cubic crstals) hav quivalt as that itract with light i a similar mar, rgardlss f th crstal ritati with rspct t icidt light wavs [5]. Aistrpic crstals th thr had hav crstallgraphicall distict as ad itract with light i a mar that is dpdt up th ritatif th crstalli lattic with rspct t th icidt light. Accrdig t [5], wh a aistrpic crstal rfracts light, th rsultig ras ar plarid ad travl at diffrt vlcitis. O f th ras travls with th sam vlcit i vr dircti thrugh th crstal ad is trmd th rdiar ra, whil th thr travls with a vlcit that is dpdt up th prpagati dircti withi th crstal. This is rfrrd t as trardiar ra. Th distac f sparati btw th rdiar ad trardiar ras icrass with icrasig crstal thickss. Th tw idpdt rfractiv idics f aistrpic crstals ar quatifid i trms f thir birfrigc. Thus, th birfrigc B f th crstal as dfid b [5] is giv as: B. (2) high lw Th Phm Of Nliar Optical Birfrigc I Uiaial Crstals ptic ais ptic ais (a) FIGURE 1. (a) rdiar ad (b) trardiar wavs i a uiaial birfrigt crstal. (ii) Thr ar als wavs that ar liarl plarid paralll t th pla frmd b th ptic ais ad th dirctif icidc. Ths ar rfrrd t as trardiar wavs (). Th wav is dflctd at th budaris, ad th ras mrgig frm th it fac ar displacd with rspct t th icidt ras. Th rfractiv id f this tp f wav dpds th dirctif prpagati. This ffct f dubl rfractir birfrigc is furthr dmstratd i Fig. 2 [8]. I Fig.2, subscripts 0 idicats icidt wav, whil, 1 ad 2 idicat th rfractd wavs. (b) whr high is th largst rfractiv id ad, lw is th smallst rfractiv id. Eq. (2) agrs with th dfiiti giv b [6] that birfrigc rfrs t th diffrc btw th multipl rfractiv idics hibitd b a aistrpic matrial such as quart. I this wrk, th sris t b mpld ibtaiig th Krr cstats fr vrdr liar suscptibilit is prpsd. Th miimum valu f th lctric fild itsit t caus birfrigc i sm uiaial crstals is dtrmid. Th dilctric tsr cmpts fr th uiaial crstals ar als prstd; thir valus rval which f th as th spd f light will b mr rtardd fr th tw classs f uiaial crstals. II. THE ORDINARY AND EXTRAORDINARY WAVES I a aistrpic mdium, thr is a dirctif prpagatif th icidt wav i which th rfractiv id is idpdt f th plariati. This dircti is kw as th ptic ais. Fr uiaial crstal, thr is a sigl ptic ais, whil i biaial crstal, thr ar tw ptic as [7, 8]. This ffct is illustratd i Fig.1. Accrdig t [7], tw tps f wavs ca prpagat i a uiaial crstal; (i) Wavs liarl plarid prpdicular t th pla frmd b th ptic ais ad th dirctif icidc. This is calld ardiar ais (). I this cas, th fild simpl passs thrugh th crstal i th pctd r rdiar wa, satisfig Sll s law. Th ra crrspdig t th wav whs rfractiv id is idpdt f th dirctif prpagati is calld th rdiar ra. k 0 si 0 k 0 si 0 Itractif rmal surfacs with pla f icidc. FIGURE 2. Dubl rfracti at a budar f aaistrpic mdium. III. REFRACTIVE INDEX DUE TO BIREFRINGENCE I Fig.2, th rfractiv id fr rdiar wav is dtd b, ad is idpdt f th dirctif prpagati. Th rfractiv id fr trardiar wav is dtd b (θ), ad dpds th dirctif prpagati (θ) rlativ t th ptic ais. This ffct as arlir mtid is kw as birfrigc. Accrdig t [9], th bhaviur f rfractiv id is usuall dscribd i trms f rfractiv id surfac, i.. th id llipsid. I th cas f th rdiar ra, it is a Lat. Am. J. Phs. Educ. Vl. 5, N. 2, Ju k k 1 k 2 ptic ais

3 Timtr Pascal, Adam Usma, J. C. Odd sphr; whil fr trardiar ra, it is allipsid, as plaid i [8]. That is, i trms f llipsid, this ffct bcms a thr dimsial bd with clidrical smmtr. Tw idics f rfracti ar th idtical ( = ), s that th pla itrsctig prpdicular t th ptical ais frms a circl. If - ais is csidrd as th ais f th clidrical smmtr (th ptical ais f a uiaial crstal), th fr uiaial crstal, [10] dfid th pricipal idics f rfracti as θ k ad, (3) 2 2 whr ε is dilctric cstat i fr spac ( ~ F/m); ε, ε ad ε ar th dilctric cstats alg, ad as. Fr uiaial crstals ε = ε. It is als a kw fact that rfractiv id ad critical agl f matrials ar rlatd b 1 1 sic ad si c, (4) whr c ad c ar th critical agls at th rdiar ad trardiar as rspctivl. Th critical agl f a matrial dtrmis whthr a itral ra will b rflctd back it th matrial. As shw i Eq. (4), it is a fuctif th rfractiv id, ad hc, th highr th rfractiv id th lwr th critical agl. (a) (c) FIGURE 3. Id llipsid f (a) gativ uiaial cristal ( < ); (b) psitiv uiaial crstal ( > ) ad (c) uiaial crstal with ais as th rtatial smmtric ais. I Fig. 3 (a), th rfractiv id fr th rdiar ra is gratr tha that fr th trardiar ra. I Fig. 3 (b), th rfractiv id fr th rdiar ra is lss tha that fr th trardiar ra. I Fig. 3 (c), th plariatif th rdiar ad trardiar wavs ca b dtrmid usig th id llipsid [10]. Fr uiaial crstal, th rfractiv idics i th ad dirctis ca b rprstd b. Th llipsid has th - ais as th rtatial smmtric ais, with rfractiv id. If th llipsid is dfrmd ad tiltd thrugh a rtati i spac with rspct t th rigial llipsid, th rigial id llipsid i th pricipal crdiat as is giv as [8, 10]. (5) 1. Fr uiaial crstals, tw f th rfractiv idics ar idtical ( = = ), ad =, ad, that this particular llipsid ca b charactrid b tw valus f rfractiv idics; (th rdiar id), which crrspds t r, ad (trardiar id) crrspdig t [2]. Eq. (5) ca thrfr b writt as 1. (6) Th prjctif th llipsid ca furthr b dpictd th pla as shw i Fig. 4. Th plariatif th rdiar wav pits prpdicular t th - pla, [11]. I Fig.4, th plariatif th trardiar wav is alg th vctr OA, ad th id f rfracti is thus (θ). It fllws thrfr that fr a agl θ (b) Lat. Am. J. Phs. Educ. Vl. 5, N. 2, Ju

4 A ( ) si, 7( a) ( ) cs. 7( b) ( ) O 0 FIGURE 4. Prjctif llipsid alg th pla. Th Phm Of Nliar Optical Birfrigc I Uiaial Crstals itsit), it is calld ptical Krr ffct. Accrdig t [15], Krr ffct is a maifstatif alctric fild a matrial. [7] dfid th Krr ffct as thus th diffrc i th rfractiv idics ( ) fr light plarid paralll ad prpdicular t th ptic ais is prprtial t th squar f th applid fild E. I [10] it is prssd as TABLE I. Idics f rfractif sm uiaial crstals. Crstal Calcit (CaCO 3 ) Lithium ibat (LiNbO 3 ) Lithium tatalit (LiTaO 3 ) Quart (SiO 2 ) Rutil (TiO 2 ) Brl (B 3 Al 2 ) Calml (Hg 2 Cl 2 ) Magsium Flurid (MgF 2 ) Sapphir (Al 2 O 3 ) Sdium Nitrat (NaNO 3 ) Pridt (Mg, F) SiO Ic (H 2 O) KE (10) Th quatif th llips (prjctif th llipsid with = 0) is thus (8) Eq. (8) is th quatif allipsid f rvluti [12]. Frm Eqs. (7) ad (8), btais cs si. (9) ( ) Th id f rfracti varis frm (θ) = fr θ = 0 t (θ) = fr θ = 90 [13]. Eq. (9) shws that th id is idpdt f th dirctif prpagatif th wav vctr. Chi (1989), [10], classifid th uiaial crstals it tw; crstals with > ar rfrrd t as psitiv uiaial crstals, whil ths with < ar calld gativ uiaial crstals. Tabl I shws th rfractiv ídics f sm uiaial crstals. IV. THE KERR EFFECT I [14] th Krr ffct is dfid as ithr light iducd dubl rfracti, r a itsit dpdt rfractiv id. I liar ptics, strg fild f high itsit (such as utput f a lasr) ca caus a mdium s rfractiv id t var as th light passs thrugh it. If th rfractiv id varis quadraticall with th fild (liarl with th whr λ is th vacuum wavlgth f light, ad K is kw as th Krr cstat (K cstat). Th Krr cstat is a umrical factr (a cfficit) dfid b [16] fr dd umbr liar suscptibilitis as 1!2 K. (11) 1 1!! 2 2 Frm Eq. (10) ad Tabl II, th miimum lctric fild E that is rquird t caus birfrigc i ths classs f uiaial crstals caasil b calculatd. This is shw i Tabl III, ad it ivlvs th valus f scd rdr ad th furth rdr Krr cstats. Th valu f λ that was usd cms i as a rsult f csidrig th crrspdig miimum valu f th TH frquc (1TH), ad th spd f light (~ m/s 2 ). It is imprtat t t that f th tw classs f crstals (i.. istrpic ad aistrpic), liar ffcts such as Scd Harmic Grati (SHG), ad thr highr vrdr liar ffcts that aris as a rsult f th itractif TH radiati cal b bsrvd i aistrpic crstals, ut f which uiaial crstals ar ampls [9]. It is thrfr apprpriat t us th miimum valu i th TH rag i th valuati prcss i Tabl III. Usig Eqs. (1) ad (3), th vaishig dilctric tsr cmpts fr th twlv (12) uiaial crstals hav b calculatd (s Tabl III). Lat. Am. J. Phs. Educ. Vl. 5, N. 2, Ju

5 Timtr Pascal, Adam Usma, J. C. Odd TABLE II. Birfrigc ad Critical agl f sm uiaial crstals. Crstal c ( ) c ( ) B Calcit (CaCO 3 ) Lithium ibat (LiNbO 3 ) Lithium tatalit (LiTaO 3 ) Quart (SiO 2 ) Rutil (TiO 2 ) Brl (B 3 Al 2 ) Calml (Hg 2 Cl 2 ) Magsium Flurid (MgF 2 ) Sapphir (Al 2 O 3 ) Sdium Nitrat (NaNO 3 ) Pridt (Mg, F) SiO 4 Ic (H 2 O) Frm Eqs. (12) ad (13), th liar suscptibilit ca b writt as a sris f trms f icrasig rdr f v liarit as 1 (2) ( ; E) ( ;,, ) E 4 3 (4) 3 ( ;,, ;, ) E ( 1)!2 ( ) 1 ( ;,... ; ) E (14) 2!! 2 2 TABLE III. Elctric fild rquird t Prduc Birfrigc i sm uiaial crstals. Crstal Elctric fild Itsit (V/m) Nr dilctric Tsr Cmpts Scd rdr Furth rdr ε = ε (10-11 ) ε (10-11 ) 1 3 K 2 K 16 Calcit (CaCO 3 ) Lithium ibat (LiNbO 3 ) Lithium tatalit (LiTaO 3 ) Quart (SiO 2 ) Rutil (TiO 2 ) Brl (B 3 Al 2 ) Calml (Hg 2 Cl 2 ) Magsium Flurid (MgF 2 ) Sapphir (Al 2 O 3 ) Sdium Nitrat (NaNO 3 ) Pridt (Mg, F) SiO 4 Ic (H 2 O) VI. CONCLUSION V. DISCUSSIONS Frm Tabl II, caasil idtif a psitiv ad a gativ uiaial crstal. Hwvr, th psitiv uiaial crstals td t hav highr critical agls tha th gativ uiaial crstals. Ecpt fr Rutil (TiO 2 ) ad Calml ((Hg 2 Cl 2 ) that hav rlativl highr birfrigc, ad hc lwr valus f th critical agls. This mas that i phsical applicatis, Rutil ad Calml ar capabl f prducig fu picturs (ut f fcus), sic matrials that hibit strg birfrigc prduc fu picturs [6]. Similarl, i gativ uiaial crstals, th valus f th critical agls ar highr i th trardiar ais tha th rdiar as. This shws that lctrmagtic wav will travl fastr i th trardiar ais tha i th rdiar as. Hwvr, btais th rvrs i psitiv uiaial crstals. This culd als b bsrvd i th vaishig dilctric cmpts (s Tabl III). This wrk has brught ut th diffrt ffcts that ca b prducd as a rsult f liar ptical birfrigc i twlv uiaial crstals. Ths iclud th valus f th lctric fild cssar t prduc birfrigc; th critical agls fr th tw diffrt as i uiaial crstals, th dilctric tsr cmpts fr th diffrt as f th crstals, ad th sris that is rquird t giv th cssar Krr cstats fr v liar cfficits. Th wrk has als shw that i gativ uiaial crstals, th spd f light travls fastr i th trardiar ais tha th rdiar as. Th rvrs f this rsult is hwvr btaid i psitiv uiaial crstals. REFERENCES [1] Widlich, A. ad Wilki, A., Ralistic Rdrig f Birfrigc i Uiaial crstals, ACM , Trasactis Graphics, 20TBD (2008). Lat. Am. J. Phs. Educ. Vl. 5, N. 2, Ju

6 [2] Sms, R. ad Cs, J., Optical Guidd Wavs ad Dvic, (McGraw Hill Bk Cmpa, USA, 1993). [3] Baldwi, C. G., A Itrducti t Nliar Optics, (Plum Prss, USA, 1975). [4] Chik Obi, B., Itrducti t th Phsics f Sud ad Light, (Hima Educatial Bks, USA, 1991). [5] Abramwit, M. ad Davids, M., Optical Birfrigc, (Olmpus Amrica Rsarch, U.S.A., 2010). [6] Kllr, B., Rfractiv id ad Critical agl, (Uitd Stats Factrs Guild, USA, 2010). [7] Milli, W. P. ad Ebrl, H. J., Lasrs, (Jh Wil ad Ss, Ic, USA, 1988). [8] Ubachs, W., Nliar Optics, Lctur Nts, Dpt. f Phsics ad Astrm, (Uivrsit f Amstrdam, Th Nthrlads, 2001). [9] Laud, B. B., Lasrs ad Nliar ptics, 2 d Ed. (Jh Wil ad Ss, USA, 1991). Th Phm Of Nliar Optical Birfrigc I Uiaial Crstals [10] Chi, L. S., Fudamtals f Lasr Optlctrics, (Wrld Scitific, USA, 1989). [11] Yariv, A., Optical Elctrics, 4 th Ed. (Saudrs Cllg Publishrs, USA, 1991). [12] Jhs, E. R. ad Kikmistr, L. F., Calculus, (All ad Bac, Ic., USA, 1959). [13] Yariv, A., Quatum Elctrics, 2 d Ed. (Jh Wil ad Ss, Ic., USA, 1975). [14] Ni, W., Optical Nliarit: Phma, Applicatis, ad Matrials, Advacd Matrials 5, (1993). [15] Laarvic, Z., Radsavijric, R., Psic, M. ad Osmkrvic, P., Applicatif Krr Elctr ptic Effct t Elctric Fild Masurmts i Trasfrmr ils, J OPTOELECTRON ADV M. 3, 8 (2006). [16] Butchr, C. N. ad Cttr, D., Th Elmts f Nliar Optics, (Cambridg Uivrsit Prss, Cambridg, 1993). Lat. Am. J. Phs. Educ. Vl. 5, N. 2, Ju