Optical Materials xxx (2006) xxx xxx www.elsevier.cm/lcate/ptmat The idicative surfaces f the phtelastic effect i Cs 2 HgCl 4 biaxial crystals M.V. Kaida a, B.V. Tybika a, A.V. Zadrzha b, A.S. Adrushchak a, W. Schraz c, B. Sahraui d, A.V. Kityk e, * a Lviv Plitechic Natial Uiversity, 12 S. Badery Str., 79013 Lviv, Ukraie b The Lviv Cllege f the Ifrmati Cmmuicatial Techlgies State Uiversity, 12 Vldymyra Velykg Str., 79026 Lviv, Ukraie c Istitut für Experimetalphysik, Uiversität Wie, Strudlhfgasse 4, A-1090 Wie, Austria d Labratire POMA, CNRS UMR 6136, Uiversité d Agers, 2 Bulevard Lavisier, 49045 Agers Cedex, Frace e Istitute fr Cmputer Sciece, Techical Uiversity f Czestchwa, Al. Armii Krajwej 17, 42-200 Czestchwa, Plad Received 2 Jue 2005; accepted 30 September 2005 Abstract We preset a ew methd t calculate the idicative surfaces f the phtelastic effect i biaxial crystals. The crrespdig equatis are derived ad applied t the calculatis f the idicative surfaces f the lgitudial ad trasverse phtelastic effects i rthrhmbic Cs 2 HgCl 4 crystals. By meas f the idicative surfaces the spatial aistrpy f phtelastic iteracti i Cs 2 HgCl 4 crystals is evaluated. Ó 2005 Elsevier B.V. All rights reserved. PACS: 42.70.a; 42.70.Nq; 42.79.Jq; 78.20.e Keywrds: Idicative surface; Biaxial crystals; Phtelastic effect; Cs 2 HgCl 4 crystals 1. Itrducti The last decade is characterized by a grwig iterest the ivestigatis f parametrical liear ptical phemea (electr-ptic, phtelastic r acust-ptic effects) i slid crystal ad amrphus media. Ccerig aistrpic materials the activity have bee much shifted t lw-symmetry crystals (see e.g., [1 6]), amely due t pssibility f their applicatis i differet ptelectric devices, such as electr-ptical ad acust-ptical mdulatrs ad deflectrs. The applicati f lw-symmetry aistrpic materials i such devices (usually as media fr electr-ptical ad acust-ptical cells) requires a kwledge the spatial aistrpy fr physical prperties * Crrespdig authr. Tel.: +48 34 3250838; fax: +48 34 3250823. E-mail addresses: kityk@ap.uivie.ac.at, kityk@el.pcz.czest.pl (A.V. Kityk). f the crystals [1,7,8]. The mai prblems rise here i regard t the ptimizati f the electr-ptic r phtelastic iteracti gemetry which ly makes pssible t use these materials with a maximal efficiecy. Such prblem may be slved i terms f the idicative surfaces (IS), which directly give a gemetrical characterizati f the spatial aistrpy f the parametrical ptical effects represeted by third- r frth-rak tesrs [9]. Preset paper, deals with biaxial Cs 2 HgCl 4 crystals. A Mach Zehder iterfermeter techique have bee used t study the piez-ptical prperties f Cs 2 HgCl 4 crystals at rm temperature givig a cmplete set f piez-ptical p m ad phtelastic p i tesr cmpets [6,10]. Cmplete sets f the elastic tesr C ij ad the elastic mdulus S ij are kw frm the acustical studies [11]. Substatial phtelastic effect ad lw ultrasic velcities i these crystals determie relatively high figure f merit M 2 fr the istrpic acust-ptical diffracti (fr certai 0925-3467/$ - see frt matter Ó 2005 Elsevier B.V. All rights reserved. di:10.1016/j.ptmat.2005.09.079
2 M.V. Kaida et al. / Optical Materials xxx (2006) xxx xxx gemetries f acust-ptical iteractis M 2 is abut f 100 110 10 15 s 3 /kg [6]) thereby such material ca be csidered as rather prmisig cadidate fr applicatis i acust-ptical devices. This is the reas why the calculatis f the IS f phtelastic effect i Cs 2 HgCl 4 crystals are imprtat, i particular fr the ptimizati f the phtelastic iteracti gemetry givig further imprvemet f the acust-ptical efficiecy. 2. The directial csies fr cstructi f the IS I geeral case the cmpets f the s-rder rak tesr T 0 uh... i a arbitrary Cartesia crdiate system (where u,, h,...= 1, 2 r 3 are the idices each crrespdig t e f the rthgal axes X 0 1, X 0 2 r X 0 3 ) are related t the tesr cmpets T pcl... i the pricipal crystallphysic system (where p,c,l,...= 1, 2 r 3 are the idices each crrespdig t e f the pricipal axes X 1, X 2 r X 3 ) by the relati [9]: T 0 uh... ¼ a upa c a hl...t pcl... ð1þ where a up,a c,a hl are the directial csies f the rtated crdiate system X 0 1,X0 2,X 0 3 with respect t the pricipal crdiate system X 1,X 2,X 3 (see Fig. 1). I a case f the phtelastic effect these axes cicide with the directis f applied mechaical stress r light plarizati. I rder t build up the IS e must calculate tesr cmpets T 0 uh as the fucti f directial csies a up,a c,a hl... Fr the ptical effects i uiaxial crystals the axes f the arbitrary rtated crdiate system are related with the directi f light prpagati ad its pssible plarizatis, what ideed decreases the umber f IS [7,8]. Wede- te: k is the directi f light prpagati ad r is the radius vectr f IS, which is parallel t e f the permitted light plarizatis, j cicides with the ther light plarizati cmpet rthgal t the vectrs k ad r. We chse X 0 1 kk,x 0 2 kj ad X 0 3kr as is shw i Fig. 1 fr uiaxial crystals. Thus the directial csies yield a matrix f rthgal trasfrmati [8] frm the pricipal crystallphysical crdiate system t the arbitrary rtated crdiate system. The directial csies f the radius-vectr r Fig. 1. Spatial psiti f the rtatig crdiate system X 0 1,X0 2,X0 3 with respect t the crystallphysical crdiate system X 1,X 2,X 3 fr uiaxial crystals. i spherical crdiates u ad h ca be writte i the fllwig well-kw frm: a r1 ¼ si h cs u; a r2 ¼ si h si u; a r3 ¼ cs h ð2þ whereas the directial csies fr light plarizati cmpets k ad j read as [8,9]: a j1 ¼si u; a j2 ¼ cs u; a j3 ¼ 0; a k1 ¼ cs h cs u; a k2 ¼ cs h si u; a k3 ¼si h ð3þ where j ad r cicide with the plarizatis f the rdiary ad extrardiary waves, respectively. By usig these equatis the idicative surfaces f lgitudial ad trasverse phtelastic effects were built up fr uiaxial crystals f lithium ibate [7], (Ba x Sr 1x )Nb 2 O 7 [12] ad b-barium brate [8]. This priciple wrks fr biaxial crystals t. The depedece f tw pssible directis i f light plarizatis a light prpagati directi k fr biaxial crystals is deduced by Lagrage s udetermied multipliers k,l. A fucti F [13] is defied F ¼ 0:5igi 0:5kðii 1Þþ lðkiþ, where g is impermeability tesr. Defiig the extreme values i a plae perpedicular t the vectr k, e btais [13]: F =i ¼ gi ki þ lk ¼ g qq a iq ka iq þ la kq ¼ 0; ð4þ where q = 1,2,3,a iq are the directial csies crrespdig t the vectr i, k ¼ igi ¼ g qq a 2 iq is the iverse squared refractive idex f the light wave with the plarizati parallel t the i-directi ad l is a deflecti f electric field vectr f light wave frm the wave frt plae [13]. Eq. (4) ca be als derived frm the Maxwell s equatis [13]. I crystallphysical crdiate system the tesr g has ly diagal cmpets g qq, s we will use its shrt tati g q. The directial csies fr k ad j vectrs ca be defied by csiderig ly e f light plarizatis i.e., ikr i Eq. (4), thereby a iq = a rq. Expressig k a iq, e btais the directial csies fr the vectr k: a k1 ¼ l 1 Dg 2 cs 2 h Dg 3 si 2 h si 2 u si h cs u; a k2 ¼ l 1 Dg 3 si 2 h cs 2 u Dg 1 cs 2 h si h si u; ð5þ a k3 ¼ l 1 Dg 1 si 2 u Dg 2 cs 2 u si 2 h cs h; where Dg 1 = g 2 g 3, Dg 2 = g 3 g 1, Dg 3 = g 1 g 2, which fllw frm the cyclic permutati rule f idices: 1 2 3, 3 1 2 r 2 3 1. I rder t determie the directial csies fr j, we take it accut that j =[r,k], thus frm Eqs. (5) ad (2) we btai: a j1 ¼ Dg 1 l 1 si h cs h si u; a j2 ¼ Dg 2 l 1 si h cs h cs u; ð6þ a j3 ¼ Dg 3 l 1 si 2 h si u cs u. Sice the sum f square f the directial csies fr k ad j is equal e, frm Eqs. (5) r (6) we get: l ¼si h ðdg 1 Þ 2 cs 2 h si 2 u þðdg 2 Þ 2 cs 2 h cs 2 u þðdg 3 Þ 2 si 2 h si 2 u cs 2 u 1=2. ð7þ
M.V. Kaida et al. / Optical Materials xxx (2006) xxx xxx 3 T elimiate a sig ambiguity i Eq. (7) we csider the case h = 0 ad u = 0, i.e., whe the rtated crdiate system cicides with crystallphysic e. I such case the directial csies are defied as: a k1 ¼ 1; a k2 ¼ 0; a k3 ¼ 0; a j1 ¼ 0; a j2 ¼ 1; a j3 ¼ 0; ð8þ thus by isertig Eq. (7) it Eqs. (5) ad (6) the directial csies fr the vectrs k ad j get the fllwig frm: qffiffiffiffiffiffiffiffiffiffiffiffiffi a k1 ¼ a j2 ¼Dg 2 ðdg 2 Þ 2 ; a k2 ¼ 0; a k3 ¼ 0; a j1 ¼ 0; a j3 ¼ 0. Fr a cicidece f cditis (9) ad q (8), ffiffiffiffiffiffiffiffiffiffiffiffiffi it is ecessary t multiply Eqs. (5) ad (6) ðdg 2 Þ 2 =Dg 2 ¼ jdg 2 j=dg 2. The the sig f l is well defied ad the directial csies fr the vectrs k ad j have the frm: a k1 ¼ l 0 Dg 2 cs 2 h Dg 3 si 2 h si 2 u cs u; a k2 ¼ l 0 Dg 3 si 2 h cs 2 u Dg 1 cs 2 h si u; a k3 ¼ l 0 Dg 1 si 2 u Dg 2 cs 2 u si h cs h; ð10þ a j1 ¼ l 0 Dg 1 cs h si u; a j2 ¼ l 0 Dg 2 cs h cs u; a j3 ¼ l 0 Dg 3 si h si u cs u; where l 0 = jl 1 Dg 2 sihj/dg 2. I the same way we have elimiated a sig ambiguity fr the directial csies f the vectrs k ad j. The relatis (10) shw a chage f directial csies as a fucti f agles h ad u fr k ad j directis i biaxial crystals. They are geeral ad thus ca be als applied t particular cases, e.g., fr uiaxial crystals (g 1 = g 2 ). The agles h ad u defie the directi f the radius-vectr r, which is parallel t e f the light plarizati vectrs i. Usig these relatis, we ca write the equatis ad cstruct the IS f phtelastic effect i biaxial crystals. 3. Cstructi ad aalysis f IS fr the phtelastic effect i Cs 2 HgCl 4 crystals The geeral equati f the idicative surfaces fr the lgitudial ad trasverse phtelastic effect i Cs 2 HgCl 4 crystals directly fllws frm Eq. (1) by takig it accut all zer cmpets f the phtelastic tesr fr the pit grup f symmetry mmm: p 0 im ¼ a2 i1 a2 m1 p 11 þ a 2 i1 a2 m2 p 12 þ a 2 i1 a2 m3 p 13 þ a 2 i2 a2 m1 p 21 þ a 2 i2 a2 m2 p 22 þ a 2 i2 a2 m3 p 23 þ a 2 i3 a2 m1 p 31 þ a 2 i3 a2 m2 p 32 þ a 2 i3 a2 m3 p 33 þ 4a i2 a i3 a m2 a m3 p 44 þ 4a i1 a i3 a m1 a m3 p 55 þ 4a i1 a i2 a m1 a m2 p 66 ; ð9þ ð11þ where a i1, a i2, a i3 ad a m1, a m2, a m3 are the directial csies f the plarizati vectr i ad the directi f applied mechaical defrmati m, respectively. I Eq. (11) all the cmpets f the fur-rak phtelastic tesr are give i the matrix represetati (see e.g., [13,14]). Isertig Eqs. (2) ad (10) it Eq. (11), we btai a set f three equatis describig the IS f the lgitudial ad trasverse phtelastic effect, which read: (1) IS f the lgitudial phtelastic effect p 0 ii (h,u) [ikmkr]: p 0 ii ðh;uþ¼ p 11 cs 4 u þðp 12 þ p 21 þ 4p 66 Þsi 2 ucs 2 u þp 22 si 4 u si 4 h þ ðp 31 þ p 13 þ 4p 55 Þcs 2 u þðp 23 þ p 32 þ 4p 44 Þsi 2 u si 2 hcs 2 h þ p 33 cs 4 h; ð12þ (2) IS f the trasverse phtelastic effect fr light plarizati p ðiþ imðh; uþ½ikr; mkjš: h p ðiþ imðh; uþ ¼ p 11 ðdg 1 Þ 2 þ p 22 ðdg 2 Þ 2 þ p 33 ðdg 3 Þ 2 þ 4p 44 Dg 3 Dg 2 þ 4p 55 Dg 3 Dg 1 þ 4p 66 Dg 2 Dg 1 si 2 h cs 2 h si 2 u cs 2 u þ p 12 ðdg 2 Þ 2 cs 4 u þ p 21 ðdg 1 Þ 2 si 4 u si 2 h cs 2 h þ p 13 ðdg 3 Þ 2 cs 2 u þ p 23 ðdg 3 Þ 2 si 2 u si 4 h si 2 u cs 2 u i þ p 31 ðdg 1 Þ 2 si 2 u þ p 32 ðdg 2 Þ 2 cs 2 u cs 4 h h ðdg 1 Þ 2 cs 2 h si 2 u þðdg 2 Þ 2 cs 2 h cs 2 u þðdg 3 Þ 2 si 2 h si 2 u cs 2 ui 1; ð13þ (3) IS f the trasverse phtelastic effect fr mechaical defrmati p ðmþ im ðh; uþ½mkr; ikjš: h p ðmþ im ðh; uþ ¼ p 11 ðdg 1 Þ 2 þ p 22 ðdg 2 Þ 2 þ p 33 ðdg 3 Þ 2 þ 4p 44 Dg 3 Dg 2 þ 4p 55 Dg 3 Dg 1 þ 4p 66 Dg 2 Dg 1 si 2 h cs 2 h si 2 u cs 2 u þ p 21 ðdg 2 Þ 2 cs 4 u þ p 12 ðdg 1 Þ 2 si 4 u si 2 h cs 2 h þ p 31 ðdg 3 Þ 2 cs 2 u þ p 32 ðdg 3 Þ 2 si 2 u si 4 h si 2 u cs 2 u i þ p 13 ðdg 1 Þ 2 si 2 u þ p 23 ðdg 2 Þ 2 cs 2 u cs 4 h h ðdg 1 Þ 2 cs 2 h si 2 u þðdg 2 Þ 2 cs 2 h cs 2 u þðdg 3 Þ 2 si 2 h si 2 u cs 2 ui 1; ð14þ where the upper idex f p ðiþ im r p ðmþ im meas which vectr i r m is parallel t the radius-vectr r f the idicative surface. Oe must meti, that Eqs. (13) ad (14) ctai the value l 0, givig a psitive sig (Dg 2 = g 3 g 1 > 0) fr the Cs 2 HgCl 4 crystal. The idicative surfaces ad their steregraphic prjectis btaied by meas f Eqs. (12) (14) are shw i Fig. 2(a) (c). I ur calculatis we iserted the magitudes f the refractive idices 1 = 1.6498,
4 M.V. Kaida et al. / Optical Materials xxx (2006) xxx xxx Fig. 2. The idicative surfaces (a) (c) ad their steregraphic prjectis (d) (f) f the lgitudial p 0 ii (h,u) (a,d) ad trasverse pðiþ imðh; uþ (b,e), p ðmþ im ðh; uþ (c,f) phtelastic effects i Cs 2 HgCl 4 crystals. 2 = 1.669 ad 3 = 1.6491 (k = 632.8 m) [6] ad the magitudes f phtelastic cstats p 11 = 0.40, p 12 = 0.40, p 13 = 0.39, p 21 = 0.26, p 22 = 0.29, p 23 = 0.34, p 31 = 0.17, p 32 = 0.19, p 33 = 0.25, p 44 = 0.034, p 55 = 0.026 ad p 66 = 0.032 [10]. Geeral type cstructi f idicative surfaces was perfrmed by meas f ew prgram Calc3D. This prgram is a ready prduct, wrkig i Widws evirmet. The image is built up i spherical crdiate system by varyig the agels h ad u. I additi, it is pssible t set a clr ad trasparecy fr each pit. Illumiati rmal t the surface were als take it accut. The ivisible parts f surface were cut usig Z-buffer algrithm. At develpig f this prgram the techlgy f virtual machie ad OpeGL library were used, that allws t btai high quality images. The cstructi priciple f steregraphic prjectis is well-kw ad described i [9,13]. They are shw i Fig. 2(d) (f). The IS f the lgitudial ad trasverse phtelastic effects i Cs 2 HgCl 4 crystals are characterized by the fllwig prperties: the IS i Fig. 2(a) (c) d t pssess a rtati symmetry i ay directi that agree with Germa s therem [14]; hwever, they keep all the elemets f mmm pit symmetry, amely three mutually perpedicular twfld axes ad three plaes f symmetry. This physical prperty is i agreemet with the kw priciple f Curie Neuma [9]. the IS f the trasverse effect (Fig. 2(b) ad (c)) have a ctiuus set f values alg the mai axes. Fr istace, the effective magitudes alg X 1, X 2 ad X 3 axes are i the rage f p 12 p 13, p 21 p 23, p 31 p 32 fr the IS p ðiþ imðh; uþ, ad i the rage f p 21 p 31, p 12 p 32, p 13 p 23 fr the IS p ðmþ im ðh; uþ, respectively. Alg the X 1 axis the chage f the radius-vectr f the IS p ðiþ imðh; uþ frm p 12 t p 13 is caused by a fixati f the directi ikrkx 1 with a pssible chage f the directi m i the plae perpedicular t X 1 axis. I ctrast t biaxial crystals, such ctiuus set f values is t bserved alg the mai axes f uiaxial r cubic crystals, because p 31 = p 32, p 13 = p 23 ad a j3 = 0 i Eq. (3). Oly the crystals f cubic symmetry 23 ad m3 are excepti. Their ctiuus set f values is bserved alg the X 3 - axis i accrdace with peculiarity f their phtelastic tesr, i.e., due t the fact that p 31 5 p 32 ad p 13 5 p 23 (see e.g., [15]). the aalytical search fr extreme values at IS i the case f Cs 2 HgCl 4 crystals, as it was de fr lithium ibate
M.V. Kaida et al. / Optical Materials xxx (2006) xxx xxx 5 [7] r b-barium brate [8] crystals by applyig the cditis p/h = 0 ad p/u = 0, ca be btaied ly lim lim ½p 0 iiðh; uþš ¼ p 11 ðlgitudial maximumþ; u!0 h!p=2 fr the lgitudial effect. S Eq. (12) t slve is f the type: lim lim½p ðiþ u!p=2 h!0 imðh; uþš ¼ p 31 ðtrasverse miimumþ; p 0 ii ðu; hþ ¼aðuÞ si4 ðhþþbðuþsi 2 ðhþ cs 2 ðhþþccs 4 ðhþ lim lim ð15þ h!p=2 u!0 ½pðiÞ imðh; uþš ¼ p 12 ðtrasverse maximumþ; Differetiatig with respect t h yields lim lim ½p ðmþ im ðh; uþš ¼ p 31 ðtrasverse miimumþ; u!0 h!p=2 h p0 ii ðu; hþ ¼2 si h cs hð2½aðuþbðuþþcšsi2 h lim lim ½p ðmþ im ðh; uþš ¼ p 12 ðtrasverse maximumþ. h!p=2 u!p=2 þ½bðuþ2cšþ. ð16þ ð20þ Therefre, extreme values ccur fr sih = 0, csh = 0 ad (2[a(u) b(u) +c]si 2 h +[b(u) 2c]) = 0. Thus it is evidet, that the extremal values ccur at h extr. =0,p/2,p,... ad ( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 1 bðuþ2c ~h extr. ¼ arcsi. ð17þ 2 aðuþbðuþþc Accrdig t (17) we ca calculated the ~ h extr. fr differet values f /. Fr / =90 e btais ~ h extr.: ( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 1 p ~h extr. ¼ arcsi 23 þ p 32 þ 4p 44 2p 33 ð18þ 2 p 22 ðp 23 þ p 32 þ 4p 44 Þþp 33 which i the case f Cs 2 HgCl 4 crystals is equals 37. Isertig this, the magitude f extremal lgitudial effect amuts t lim u!p=2 ¼ 1 4 lim h! ~ ½p0 iiðh; uþš h extr. ðp 23 þ p 32 þ 4p 44 Þ 2 4p 22 p 33 ðp 23 þ p 32 þ 4p 44 Þðp 22 þ p 33 Þ ¼ ~p ðlgitudial miimumþ; ð19þ which i ur case is equal 0.23. This ifrmati might be useful, as it cvers the depedece the phtelastic cmpets. Besides such extreme values als the extreme values fr trasverse effect were btaied umerically usig ur sftware ad are preseted i Table 1 (here the agular parameters f directis are give ly fr radius-vectr f IS). Kwig the extreme directis fr the lgitudial ad trasverse effects e may take the limits t btai crrespdig extremal magitudes: Oe must be stressed that the rder f takig the limits is imprtat here, i.e., lim u (lim h ) 5 lim h (lim u ). Table 1 presets the agular parameters fr ly e directi eglectig all ther symmetry equivalet directis. The latter es may by btaied by emplyig a reflectig f the crrespdig radius vectrs i three crystallgraphic plaes f symmetry. All the IS are characterized by almst the same magitudes i their extreme pits because the values f p 11 ad p 12 cstats are early equal fr Cs 2 HgCl 4 crystals. the aistrpy pwer fr all IS is als shw i Table 1. It was btaied usig the relati [8]: c =(V sphere jv + V j)100%/v sphere, where V sphere =4pjf extr. j 3 /3 is the sphere vlume crrespdig t a certai radius: jf extr. j = max(jf mi j,jf max j); f mi ad f max are the miimum ad maximum values f the IS, respectively, V + ad V are the vlumes f the psitive ad egative parts f the surface, respectively. Takig it accut, that all IS i Fig. 2(a) (c) have the psitive parts ly, we have applied a simplified frmula fr the calculati f aistrpy pwer, i.e., c =(V sphere V + )100%/ V sphere. cmparig the IS f phtelastic effect (see Fig. 2(a) (c)) ad f the piez-ptical e [16], e may realize a substatial differece, i particular piez-ptical effect shws a mre cmplicated structure f IS, icludig the iversi f the sig fr bth trasverse IS p ðiþ im(h, u) ad p ðmþ im (h, u). That is why the aistrpy pwer fr piez-ptical effect [16] is abut f 1.5 2 times larger as cmpared t the phtelastic effect. The directis crrespdig t the extreme values f trasverse piez- ad phtelastic effects practically cicide whereas fr the lgitudial effect [p 0 ii (h,u) adp0 ii (h,u)] they clearly differ. It may be explaied by the fact that p 33 > p 11 fr the piez-ptical effect ad p 33 < p 11 fr phtelastic effect. Table 1 The extreme values ad aistrpy pwer fr IS f phtelastic effect i Cs 2 HgCl 4 crystals IS Miimal magitude fr the IS ad its directi Maximum magitude fr the IS ad its directi Aistrpy pwer Magitude h ( ) u ( ) Magitude h ( ) u ( ) V sphere (uit) 3 V + (uit) 3 c (%) p 0 ii (h,u) ~p ¼ 0:23 37 90 p 11 = 0.40 90 0 0.80 0.32 60 p ðiþ im ðh; uþ p 31 = 0.17 0 0 p 12 = 0.40 90 0 0.80 0.36 55 p ðmþ im ðh; uþ p 31 = 0.17 90 0 p 12 = 0.40 90 90 0.80 0.42 48
6 M.V. Kaida et al. / Optical Materials xxx (2006) xxx xxx Therefre the extreme value fr lgitudial IS p 0 ii (h,u) ccurs at h =0, i.e. alg the X 3 axis whereas fr IS p 0 ii (h,u) the extreme magitude is at h =90 ad u =0, i.e., alg the X 1 axis. 4. Cclusis I cclusi we preset here a ew methd t calculate the IS fr the phtelastic effect i biaxial crystals. The crrespdig equatis has bee derived ad applied t the calculatis f the IS related t the lgitudial ad trasverse phtelastic effects i rthrhmbic Cs 2 HgCl 4 crystals. The maximum magitudes at the IS f the lgitudial ad trasverse phtelastic effects are revealed alg the pricipal crystallgraphic directis. This ideed defies a set f the ptimized sample gemetries recmmeded fr desigers that develp acust-ptical mdulatrs ad deflectrs built up usig Cs 2 HgCl 4 crystals as phtelastic media. By meas f the IS the spatial aistrpy f the phtelastic iteracti i Cs 2 HgCl 4 crystals is evaluated. Ackwledgemet Preset wrk was supprted by Sciece ad Techlgy Ceter i Ukraie (prject. #3222). Refereces [1] A. Médez, A. García-Cabañes, E. Diéguez, J.M. Cabrera, J. Appl. Phys. 86 (1999) 2038. [2] X. Yi, X.Q. Wag, M.K. Lu, D. Xu, Phys. Status Slidi A 191 (2002) 267. [3] W. Kamisky, A.M. Glazer, Z. Kristallgr. 212 (1997) 283. [4] A.K. Bai, K. Veerabhadra Ra, P. Chad, T. Yamaguchi, M. Wada, Jp. J. Appl. Phys. 37 (1998) 5618. [5] H. Hellwig, J. Liebertz, L. Bhatý, J. Appl. Phys. 88 (2000) 240. [6] M.V. Kaida, A.V. Zadrzha, A.S. Adrushchak, A.V. Kityk, Opt. Mater. 22 (2003) 263. [7] B.G. Mytsyk, Ya.V. Pryriz, A.S. Adrushchak, Cryst. Res. Techl. 26 (1991) 931. [8] A.S. Adrushchak, V.T. Adamiv, O.M. Krupych, I.Yu. Martyyuk Lttska, Y.V. Burak, R.O. Vlkh, Ferrelectrics 238 (2000) 863. [9] L.A. Shuvalv, A.A. Urusvskaya, I.S. Zheludev, et al.physical prperties f crystals, 4, Spriger-Verlag, Berli, 1988. [10] M.V. Kaida, A.V. Zadrzha, A.S. Adrushchak, A.V. Kityk, Appl. Opt. 41 (2002) 5341. [11] A.V. Kityk, A.V. Zadrzha, Ya.I. Shchur, I.Yu. Martyyuk- Lttska, Ya. Burak, O.G. Vlkh, Aust. J. Phys. 51 (1998) 943. [12] A.S. Adrushchak, B.G. Mytsyk, Ukraie J. Phys. 40 (1995) 1216. [13] M. Br, E. Wlf, Priciples f ptics, Pergam Press, Oxfrd, 1980. [14] J.F. Nye, Physical prperties f crystals, Clared press, Oxfrd, 1992. [15] B.G. Mytsyk, A.S. Adrushchak, N.M. Demyayshy, L.M. Yakvleva, Crystallgr. Rep. 41 (1996) 472. [16] M.V. Kaida, A.S. Adrushchak, M.M. Klymash, A.V. Kityk, Ya.V. Bbitski, Ukr. Fiz. Zh. 48 (2003) 1104.