Analytical relations describing piezooptic effect in tetragonal crystals

Transcription

1 Aalytical relatis describig piezptic effect i tetragal crystals, Mytsyk B., Demyayshy N. ad Kst Ya. Istitute f Physical Optics, Dragmav St., Lviv, Ukraie Karpek Physics ad Mechaics Istitute, 5 Naukva St., 790 Lviv, Ukraie. Itrducti Received:.0.0 Abstract. We have derived aalytical relatis ecessary fr iterpretig piezptic effect ad calculati f all piezptic cefficiets the basis f iterfermetric techique, which take a micr-wedge shape f samples it csiderati. The aalysis is preseted fr the tetragal crystals f pit symmetry classes, ad /m. The relatis valid fr the remaiig symmetry classes, mm, m ad /mmm represet simple particular cases f the geeral relatis. Keywrds: piezptic effect, ptical idicatrix, mechaical stress, elastic strai, iterfermetric techiques. PACS: 78.0.Hp UDC: Experimetal studies f piezptic effect (POE i crystallie materials d t ivlve remarkable prblems wheever e deals with the crystals belgig t the cubic r rthrhmbic systems ad higher-symmetry grups f the tetragal ad hexagal systems (mm,, m, /mmm, mm,, m, ad /mmm [ 8]. The matter is that the tesrs describig POE fr these symmetry grups iclude ly s-called 'pricipal' cmpets ( im, with i, m =,, ad the idices i, m crrespdig t the directis f light plarizati ad uiaxial mechaical stress, respectively ad 'diagal' cmpets, 55 ad. O the ther had, there ca happe mre cmplicated situati whe the POE tesr ctais bth '-pricipal' ad '-diagal' cmpets such as,, 5, etc. Amg the well-kw crystallie materials, the examples are LiNbO ad LiNbO :MgO (the pit symmetry class m, ad La Ga 5 SiO (the class [9, 0]. Netheless, the cefficiets have bee determied fr LiNbO ad α-bab O (the symmetry class m usig a s-called trsi methd [, ]. Fllwig frm the matrices f piezptic cefficiets (POCs, the tetragal crystals may be divided it tw subgrups: ( the first e embraces the symmetry classes, ad /m; they are characterised by cmplicated POC matrices with te idepedet cefficiets, icludig five -pricipal es (a s-called 'rtatial' cefficiet, a 'shiftig' cefficiet, ad 'rtatial-shiftig' cefficiets, 5 ad ; ( the secd subgrup embraces the grups, mm, m ad /mmm, with seve idepedet cefficiets; ly tw f these, ad, are -pricipal. The crystals belgig t the first, lwer-symmetry subgrup f the tetragal system (the Ukr. J. Phys. Opt. 0, Vlume, Issue 0

2 Mytsyk B. et al classes, ad /m reveal the same frm f the POE tesr [8, ]. We are t stress that, up t date, reliable experimetal data have bee reprted fr their POCs. This is due t lack f relevat aalytical relatis, which are ecessary fr iterpretig the experimetal results ad calculatig the -pricipal POCs this basis. The aim f the preset wrk is just t slve this prblem. Belw we will describe the POE fr the first tetragaal subgrup ly, sice the aalytical relatis fr the secd e represet particular cases f the mre cmplicated relatis fr the first subgrup.. Opeig remarks ad frmulati f the prblem I rder t study the POE fr the crystals belgig t the first tetragal subgrup, it is ecessary t prepare five samples with differet crystallgraphic rietatis shw schematically i Fig.. At the same time, samples f ly three differet rietatis are eeded fr the crystals f the secd subgrup (see Fig. a, b, ad c. a b c d e Fig.. Schemes f sample rietatis fr studyig the POE i tetragal crystals: (a direct crystallgraphic cut sample, (b Х/5-cut, (c Z/5-cut, (d Z/.5-cut ( =.5, ad (e В-cut (see the text. Let us remid that the pricipal POCs (і, m =,, ca be btaied the basis f theretical relatis derived i the wrks [9, 0]. The mst cmm experimetal techique is based iterfermetric measuremets ad additial csiderati f experimetal errrs that appear due t micr-wedge shape f a sample: im Skm ( i. ( i im ' im i Here λ is the light wavelegth i vacuum, i the refractive idex f crystal, S km the cmpets f elastic cmpliace tesr, ad i m ad ' i m = i m d k dete the peratig stresses defied with respect t a precedig cycle f measuremets (fr ay details see [9]. I the last frmulae, the parameter im meas the stress iducig a half-wave phase retardati, d k the sample thickess alg the directi f light wave vectr k, ad ' i m the peratig stress fr the wedgelike sample rtated by 80 arud the directi f light beam. Fr cmpletig the POC matrix f the tetragal crystals, e shuld rely the aalytical relatis fr the -pricipal POCs,,,, ad 5, which are aalgues f the relatis derived earlier i the study []. Ufrtuately, the theretical relatis suggested i the wrk [] are characterised by a umber f disadvatages. Belw we will itemise them ad utlie the pssible ways ut.. The relatis metied abve facilitate btaiig ly the tw cefficiets, ad. Mrever, these relatis are writte i a frm that excludes ay pssibility fr csiderati f a 0 Ukr. J. Phys. Opt. 0, Vlume, Issue

3 Aalytical relatis wedge-like shape f real experimetal samples, whe determiig the half-wave stresses. Hwever, it is kw that a eglect f eve very small wedges (a case which we refer t as a 'micrwedge' ca lead t table errrs while iterpretig the experimetal results. Let us tice that the cefficiet appears i the theretical relatis with its sig udefied. It meas that e shuld smehw distiguish betwee the directis ad (see Fig. с. This is a difficult prblem. Ideed, usually the psitive directis f the pricipal axes Х, Х ad Х (r simply, ad f the ptical ellipsid are determied ad the directis, 5 ad are distiguished respectively frm, 5 ad usig a piezelectric effect [, ]. Hwever, sice the crystals f the pit grup /m reveal piezelectric effect, it is ecessary t chse sme ther criteria t defie the directis ad ad the determie the sig f the cefficiet. The simplest sluti is as fllws. The magitude ad the sig f the POC ca be fud usig a relati which des t ctai the idefiite sig, i.e. crrespds t sme ther experimetal cditi []. The the relati that ivlves the ucertaity ca be used ly t determie mre accurately the magitude f the cefficiet. This becmes pssible sice the latter relati ctais the Piss ctributi f a very simple frm ad, therefre, the relevat errr ctributes very little t the ttal experimetal errr f the cefficiet.. The relatis used fr determiati f the cefficiets, ad 5 iclude cumbersme cmbiatis f these POCs. Thus it is impssible t determie these cefficiets separately.. The experimetal cditis eablig determiati f the sums f POCs have t yet bee csidered. Mrever, it wuld be very useful t cmpare these sums fud experimetally with the same sums calculated frm the ivlved cefficiets btaied i idepedet ad separate experimets. It is bvius that ay reliable studies f the POE fr the tetragal crystals f the first subgrup require theretical relatis eablig separate determiati f the cefficiets, ad 5. These relatis shuld be writte fr the tw right-haded crdiate systems, fr which the directis ad, * ad *, ad ad are iterchaged (see Fig. c, d, e. The latter ca, i priciple, help fidig ut the cditis fr uambiguus determiati f these POCs. S-called 'symmetric cditis' f piezptic experimets shuld als be csidered (see Table, with further aalysis f the relatis btaied. I case if the relatis tur ut t be differet fr these experimetal cditis, e shuld frmulate the relevat recmmedatis fr uambiguus determiati f the -pricipal POCs. Table. Aalytical relatis fr determiati f POCs i the crystals f symmetry classes, ad /m usig half-wave stress techique, which accutig fr micr-wedge shape f samples Experimetal cditis Relatis Sample f Х/5 -cut m=( k= ( i=( ( S S S S ( ( ' ( Ukr. J. Phys. Opt. 0, Vlume, Issue 0 Т.

4 Mytsyk B. et al Experimetal cditis m= k=( i= m=( k= ( i= m= k= ( i=( m=( k= Relatis ( S S ' k( ( S S S S ( ' ( ( S S ( ' ( ( S S i= ( ' ( m=( k= ( S S i= ( ' ( m= k=( i= m= k= ( i=( m=( k= ( i=( m=( k= ( i= Sample f Z/5 -cut S ' k ( S ( ' ( (S S S ( ' ( ( S S S ( ' ( Sample f Z/5 -cut m= k= i=( ( ' ( ( ' ( fr the idices i brackets e has m =, k =, ad i = ( m= k= S i=( ( ' ( ( ' ( fr the idices i brackets e has m =, k =, ad i = ( Sample f Z/.5 -cut m= k= ( S ' k (* i= m= k= ( S i= ( ' ( ( Т. Т. Т. Т.5 Т. Т.7 Т.8 Т.9 Т.0 Т. Т. Т. Т. 0 Ukr. J. Phys. Opt. 0, Vlume, Issue

5 Aalytical relatis Experimetal cditis m= ( k= ( i= ( m= ( k= ( i= m= ( k= i= ( m= ( k= i= ( m= k= i = ( m= k= i= ( m= k= i= ( m= k= i= ( m=b ( B k= і= Relatis ( ta ( ta ( (ta ta [( S S ta cs ' ( ( S ( ta S (ta ta ]( cs [( S S ta ( * ' ( ( ta (ta ta ]( S S ( ta ( ta ( (ta ta S cs ' cs ( ( ( ta ( ta ( (ta ta S cs ' cs ( ( ( (ta ta cs ( ( ta ta ' ' ( (ta ta Sample f Z/.5 -cut cs ' ' ( ( ta ta S ' ' S ' ' Sample f B-cut ' B( B B( B ( S S S S S S Т.5 Т. Т.7 Т.8 Т.9 Т.0 Т. Т. Т. Ukr. J. Phys. Opt. 0, Vlume, Issue 05

6 Mytsyk B. et al Experimetal cditis m=b ( B k= і= m=b ( B k= і= m=b ( B k= і= Relatis S B ' B ' B B ( ( 5 ( S S S S S S B( B ' B( B 5 B ' B ' S B B Fttes: ( i cases whe the peratig stresses appearig i a frmula are idicated by idices i brackets (e.g.,, e shuld put ( it the frmula fr the direct experimetal cditis ad fr the symmetric cditis; ( the upper ad lwer sigs f the terms, 5 ad S appearig i Eqs. (Т. ad (Т.5 fr the В-cut sample crrespd t the cditis m B ad m B, respectively. Т. Т.5 Т.. Experimetal maifestatis f POE fr a Х/5 -cut sample Csider the relati used fr determiati f π basig the measuremets f ptical path chage 0 k at i=, k= ad m= []: ( d ( S S S S d(. ( 8 I the frame f half-wave stress techique we have = λ/, with im beig the half-wave stress ad d the peratig stress. Let us isert these relatis it Eq. ( ad take it accut the cditis eeded fr cmpesati f the errr caused by the wedge-like shape f sample (i a maer aalgical t Eq. (, e shuld replace / im with (/ im + / ' im /. The e gets the relati ( S S S S, ( ' which ca be used fr evaluatig the cefficiet the basis f experimetally determied peratig stresses ad '. I case whe s-called symmetric experimetal cditis are dealt with (i=, k= ad m=, e btais the relati aalgical t Eq. (, where the parameters ad ' are replaced by ad ' (see Eq. (Т. i Table. Belw we will csider a umber f experimetal gemetries relyig a Х/5-cut sample (see Fig. b, which are used fr checkig reliability f the POCs values ad sigs. Basig up the example f crystals belgig t the pit symmetry classes ad m, the authrs f the wrks [9, 0] have demstrated fr the first time that the Х/5-cut sample allws fr determiig the pricipal cefficiet i tw differet experimetal gemetries. The cefficiet fr the classes Ukr. J. Phys. Opt. 0, Vlume, Issue

7 Aalytical relatis, ad /m ca als be determied usig the Х/5-cut sample (tice that derivati f the crrespdig theretical relatis is the same as i Refs. [9, 0]. The relevat relatis valid fr the direct ad symmetric experimet gemetries are gathered i Table (see Eq. (Т.. Ntice als that the Х/5 о -cut sample admits experimetal gemetries, fr which the sums f pricipal POCs ca be fud. The a cmparis f these sums with the crrespdig sums arisig frm the cefficiet values determied idepedetly the direct-cut samples (see Fig. a wuld serve as a reliability criteri fr the POCs. Let us csider the experimetal gemetry give by m=, k= ad і=. The relati B cefficiets i, which ca easily be btaied after differetiatig the ptical impermeability B =/, is valid fr the light plarizati directi і=. Let us evaluate the chage i B i the ptical impermeability cefficiet, usig the equati f the POE B i im m. ( The summati i Eq. ( is perfrmed ver the idex m=,,...,, ad m dete the mechaical stress tesr cmpets. Nw it is ecessary t fid the frm f the stress tesr [ ] the basis f the relati [] m 5 a b c bc ac ab [ ] [,,,,, ] [,,,,, ], (5 where a, b, c are the directial csies f the uiaxial ladig frce vectr P, is the value uiaxial pressure. Obviusly, if the frce acts alg the directi m= (see Fig. b, the we have a = b = cs5 = ad c = cs90 = 0. I terms f Eq. (5, the [ ] tesr (fr m= may be give as 0,,,,0,0. ( Hece, the -zer cmpets f the tesr [ ] are as fllws: /. Usig Eq. ( ad the matrix f POCs fr the symmetry classes, ad /m [8, ], e gets B (. (7 Thece the relati fr fllws: (. (8 Oe has t isert this relati it the frmulae describig the chage i the ptical path fr the light prpagatig thugh sample []: d d, (9 where k i k k k i k meas the strai ccurrig alg the directi k f light prpagati. Here the star idicates that the strai by a lie give by x = 0 ad x = x, which crrespds t the ptical beam directi k= (see Fig. : Ukr. J. Phys. Opt. 0, Vlume, Issue 07 k i the k-directi differs frm the cmpets f the strai tesr k. T derive the sample strai alg the directi k= (see [], we are t csider a crss secti f a surface f strai tesr, x x x xx 5x x x x m, (0

8 Mytsyk B. et al x x x, ( with,, beig the cmpets f the strai tesr. Usig Eq. (, e ca fid the crdiate x crrespdig t the crss secti pit f the lie k = ad the surface determied by Eq. (0: x. ( Fig.. Crss secti f characteristic surface f the strai tesr by a lie k = (see the text: x = 0, x = x. The semi-axis f the surface give by Eq. (0 is determied as e (see Fig. ad [, ]. Frm the ther side, e ca fid the semi-axis value as e x x x. After accutig fr Eq. (, e gets e* x ε * ε ε ε * The latter yields i the fllwig sample strai ccurrig alg the directi k= : (. ( The cmpets, ad f the strai tesr are t be determied issuig frm the Hk s law ( k Skm m ad usig the elastic cmpliace matrix S km [8, ] fr the classes, ad /m, ad the stress tesr cmpets [ ] frm Eq. ( ( S S, S S / /, ad S /. Isertig these relatis it Eq. (, e arrives at the strai ccurrig alg the directi k= uder the acti f uiaxial stress alg the directi m=: ( S S S S. ( Basig Eqs. (, (8 ad (9, e ca derive the sum f the pricipal POCs fr the experimetal gemetry f Х/5 о -cut sample (the ptical path chage Piss strai:, k with accutig fr the ( d ( S S S S d (. (5 This may be specified fr the cases f half-wave stress techique ad wedge-like sample shape as. ( ( S S S S ' 08 Ukr. J. Phys. Opt. 0, Vlume, Issue

9 Aalytical relatis The relati fr the symmetric experimetal cditis f determiig the sum (m=, k=, ad і= are aalgical ad ca be btaied after substitutig the peratig stresses, ' by, '. These tw relatis are preseted i Table (see Eq. (Т.. Ntice that the Х/5 о -cut sample als allws accmplishig the additial experimetal gemetries. I particular, the direct cditis give by m=, k= ad і=, ad the symmetrical es (m=, k= ad і= eable fidig the sum ; the pair f direct ad symmetrical (i brackets cditis specified by m=(, k= ad і= ca result i the sum ; ad the cditis m=(, k= ad і= yield i the sum. All the aalytical relatis fr the Х/5 о -cut samples are give i Table by Eqs. (Т. (Т., tw f which are symmetrically idetical t Eq. (Т.. These relatis facilitate determiig the cefficiet fr the bth direct ad symmetrical experimetal cditis. Basig the ther te relatis, e ca fid the cefficiet π ad the ther sums f the pricipal POCs. The direct ad symmetric experimetal cditis differ by the directis f light prpagati ad plarizati ( r, as well as by the directis f mechaical ladig. Table shws that all f the relatis fr the direct ad symmetric cditis are i fact the same, i.e. the results fr the POCs r their sums are idepedet f chice f the directis r. As a csequece, e ca defie these directis, alg with the crrespdig experimetal cditis, as symmetrically idetical. Belw we will shw that, fr the grups f symmetry, ad /m, the directis ad (fr a Z/5 -cut sample ad the directis ad * (fr a Z/.5 -cut sample see Fig. c, d are als symmetrically idetical. Thus, e ca crrectly ad uambiguusly determie the -pricipal POCs irrespective f slvig the prblem f defiiti f these directis. Oe shuld emphasise i this respect that the aalysis [] has prved the directis ad t be t symmetrically idetical fr the trigal symmetry grups,, m ad. Relevat recmmedatis fr uambiguus chice f these directis have als bee frmulated i this wrk.. Experimetal maifestatis f POE fr Z/5º-, Х/.5º- ad В-cut samples Abve we have csidered i detail hw t derive the theretical relatis fr the POE assciated with the Х/5º-cut sample. The fial relatis used fr determiig the -pricipal POCs fr the samples f ther rietatis are preseted i Table. Each lie f Table cmbies the tw relatis referred t the direct ad symmetric experimetal cditis, the latter beig put i brackets. The mai peculiarities f these relatis will be discussed belw ad the apprpriate practical cclusis will be draw... Z/5º-cut sample Takig the cditis i=, m= ad k= (r k= see Fig. i Eq. (, e ca btai the fllwig relati fr determiati f the cefficiet π : S (, (7 ' where the cditi S = S hlds true. A cmparis f Eq. (Т.7 fr the Z/5º-cut sample (see Table with Eq. (7 fr the direct-cut sample testifies that these relatis are the same. Nw let us write ut the relatis fr the cefficiet uder the cditis i=, m=, k= ad i=, m=, k=, fllwig frm Eq. (: Ukr. J. Phys. Opt. 0, Vlume, Issue 09

10 Mytsyk B. et al 0 S (, (8 ' S (. (9 ' After cmparig these equatis with each ther ad with Eqs. (Т.8, ad takig it accut that the equality is satisfied fr the symmetry classes uder test, e arrive at the cclusi that all f these relatis are idetical. The peratig stresses are als the same ( = = =. Hwever, the latter equalities cat be true i case whe the samples have a micr-wedge shape. Netheless, the fllwig equalities remai t be valid fr the sums f the reciprcals i the latter case: im. (0 We shuld als stress that the Z/5º-cut sample allws fr fur differet experimetal gemetries fr measurig the rtatial-shiftig cefficiet. These gemetries are described by Eqs. (Т.9 ad (Т.. Netheless, Eqs. (Т. remai t be preferable. Ideed, the crrespdig experimetal errr fr the POC is the assciated ly with the errr f determiig the peratig stresses, while utilisati f Eqs. (Т.9 will give rise t additial errrs referred t determiati f the Piss-ctributi term ad t the pricipal POCs ad... Z/.5 0 -cut sample Let us write ut the aalytical relati fr determiati f the sum f POCs []: * d* cs d* S S S S [ ( ta ( ta ( (ta ta ] cs [( ta ( ta (ta ta ](, where the half-wave stress techique ad the csiderati f a wedge-like sample shape are iteded, as always. Let us substitute * i Eq. ( with λ/ ad with ** (see Eqs. ( ad ( ad the relevat cmmets. The the fllwig relatis fr the sums expressed i terms f the peratig stresses ** ad ' ** will take place: ( ta ( ta ( (ta ta [( S S ta cs ** ' ** S ( ta S (ta ta ](. This frmula is valid fr the experimetal cditis i=m=* ad k=. The authrs f the wrk [], frm which Eq. ( is take, have t aalysed the symmetric experimetal cditis i=m= ad k=*. Therefre it is t clear whether the ambiguity i determiati f the sum exists fr the direct ad symmetric cditis. Let us csider the relatis fr the symmetric cditis aalgical t Eqs. ( ad (. Fr this aim we shuld ( Ukr. J. Phys. Opt. 0, Vlume, Issue (

11 Aalytical relatis establish the frm f the stress tesr fr the case f m=. Accrdig t Fig., the directial csies a, b, c f the vectr P (m= may be fud as P = P(a, b, c = P(csα, siα, 0. ( Fig.. A scheme explaiig determiati f directial csies fr the vectr P (see the text. Cmbiig Eq. ( ad Eq. (5, e ca derive the cmpets f [ ] tesr: * cs,si,0,0,0, si cs * cs, ta,0,0,0, ta. T determie the refractive idex chage i *, e shuld itersect the ptical idicatrix by the lie i= (the equati f lie x = x taα ad x = 0 see Fig. ad use the equati f the perturbed idicatrix []: ( Fig.. Crss secti f perturbed ptical idicatrix by a lie i = (see the text. The result is r 5 ( B B x ( B B x ( B B x B x x B x x B x x. (5 ( B B x ( B B x ta B x ta, x, B B B ta B ta B B ta B The refractive idex * * is equal t (see Fig. * * x x x ta x cs r, while csiderig Eq. ( ad the equality B =B =/, we have * * cs B B ta B cs B ( ta B. cs ta cs Ukr. J. Phys. Opt. 0, Vlume, Issue B B The same refractive idex uder the cditi f -perturbed idicatrix (see Eq. (7 ad the cditis B = 0 is equal t. The the chage i the refractive idex equals t * *, i.e. we have * * *. ( (7

12 Mytsyk B. et al * * B cs. cs B The relatis fr the parameters (i =,, fllw frm the frm f the POC matrix ad the Bi tesr [ * ], as well as Eqs. (, ( ad (: With Eqs. ( ad (8 this leads t B cs ( ta ta, B cs ( ta ta, B cs ( ta ta. * * cs [ ( ta ( ta ( (ta ta ]. (9 The sample strai * * (k=* uder the acti f the stress tesr [ * ] (see Eq. ( ad Fig. 5 ca be fud after itersectig the characteristic surface by the lie give by the frmulae x = x taα ad x = 0 (see Eq. (0: x x x ta ta, x It is see frm Fig. 5 that the semi-axis * * reads as ta (8. (0 ta r x, cs cs ta ta x x x ta * * * cs ( ta ta. ( Fig. 5. Crss secti f characteristic surface f the strai tesr by a lie k = * (see the text. Fially, the strai tesr cmpets, ad may be btaied frm the Hk s law, the frm f the [ * ] tesr (see Eq. (, ad the elastic cmpliace matrix S km : S S S cs ( S S ta S ta ta cs ( S ta S ta S ta. ta cs ( ta ta ta,, ( Basig Eqs. (9, (, ( ad (9, e gets cs [ ( ta ( ta ( (ta ta ] * d* S S S S d* cs [( ta ( ta (ta ta ] (. Ntice that Eq. ( is idetical t Eq. (. Hece, we have the tw idepedet relatis fr de- ( Ukr. J. Phys. Opt. 0, Vlume, Issue

13 Aalytical relatis termiati f the POC sum i the tw symmetrically idetical experimetal cditis: i=m=*, k= *, i=m= *, k=*. ( Ather pricipled questi is as fllws: d the relatis fr * liked t the Z/.5-cut sample (see Eqs. ( ad ( ad Fig. deped the chice f right-haded crdiate system ad, respectively, the chice f directis * ad *? Fr example, let us rtate the crdiate system arud the axis by 80 (see Fig.. Uder such cditis the directis * ad * are iterchaged. Hwever, the sums f the POCs are give by the idetical relatis i the bth cases, as shw fr the particular cditis ( differig just by the directis * ad *. Oe ca als demstrate a pssibility fr arbitrary chice f the directis * ad * fr the mst geeral case. Fr this aim let us rtate a sample shw i Fig. by 90 arud the axis (see Fig.. This will result i replacig the axis by the axis, ad vice versa. Ideed, i tetragal crystals these axes are idetical with respect t bth the refractive idices ad the POCs. As a result, we itrduce the agle α ad the directis * ad * the same as i Fig.. Fig.. A sample rtated arud the axis by 90º with respect t that shw i Fig. (see the text. Thus we have prved that the sum is uambiguusly determied fr the case f Z/.5-cut samples. The crrespdig Eqs. ( ad ( are w cmbied it Eq. (Т.5 i Table, which ca be used i the frame f half-wave stress techique with takig it accut micrwedge sample shapes. Ntice that the relatis ( ad (Т.5 at α=5 are trasfrmed it Eqs. (Т.9 fr the Z/5cut sample, which d t ctai the sum. Usig the samples f Z/.5º-cut, it is pssible t determie the cefficiets, ad, as well as the sum (see Table ad Eqs. (Т., (Т., (Т., (Т. ad (Т.. Eqs. (Т. ad (Т. fr determiati f the cefficiets ad d t deped the agle α. They are idetical bth t Eqs. (Т.7 ad (Т.8 fr the Z/5º-cut samples ad t the crrespdig relatis btaied fr the direct-cut samples (see Eqs. (7 (9. Eq. (Т. icludes a cmplicated sum f the elastic cmpliace cefficiets S km (ΣS km. The latter ca be derived while isertig the expressi fr the cefficiet it this relati (tice that the π cefficiet is determied usig the direct-cut sample. Isertig the sum ΣS km it Eq. (Т.5 eables fidig the sum with essetially reduced errr. Eqs. (Т.9 ad (Т.0 are als imprtat because they d t ctai the Piss strai ctributi ad thus the experimetal errrs fr the sum ca be small eugh... В-cut sample... Geeral csideratis I rder t describe cmpletely the POE fr the classes, ad /m, we are t derive the relatis used whe determiig the shiftig cefficiet π ad the rtatial-shiftig cefficiet 5. T determie the POC, it is ecessary t prepare a sample, which esures that the light Ukr. J. Phys. Opt. 0, Vlume, Issue

14 Mytsyk B. et al is plarised alg the directi i= ad the cmpet σ f the mechaical stress tesr m remais zer. Determiati f 5 meas utilisati f a sample allwig t prvide the light plarizati i= ad a zer cmpet 5. We have fud ut that these cditis ca be prvided with a sample f s-called В-cut. As see frm the scheme f such a sample give by Fig. 7, e has t make cuts at the agles f 5 with respect t the directis ad, usig the iitial Х/5-cut sample (the plaes f the cuts are shw by dtted lies i Fig. 7; see als Fig. e fr mre details. Fig. 7. A scheme f В-cut sample fr determiig π ad π 5 cefficiets; В ad B faces are idicated by arrws (see the text. Fig. 7 testifies that, with the light prpagatig alg the directi k=, it is pssible t esure the plarizati parallel t the i= ad i= directis. Fr fidig the chages i the refractive idices δ ad δ, as well as the strai * ccurrig alg the directi k=, e shuld btai the stress tesr fr the case whe the uiaxial pressure frce Р is applied alg the directis perpedicular t the faces В ad B (i.e., m В ad m B. It is easy t shw that the directial csies a, b, c f the vectr P uder the cditi f m В are as fllws: P В = P В (a, b, c = P В (cs5, cs0, cs0 = P В (,,, while uder the cditi f m B e has P B P B(cs5, cs0, cs0 P (,,. B Accrdig t Eq. (5, the stress tesrs fr the cases f mв ad m B are give by B,,,,,, (5 B,,,,,. (... Idepedet determiati f the rtatial cefficiet The cditis eeded fr determiig the cefficiet are as fllws: і=, k=, ad m В. After calculatig the refractive idex chage δ ad the sample strai * ad csiderig Eq. (5, e ca fid the ptical path chage the basis f Eq. (9. Whe m В, we btai ( d 8 ( S S S S S S d (. 8 The uiaxial pressure directi parallel t m B wuld lead t the sig reversal fr the cmpets 5 ad (see Eq. (. The the relati allwig determiati f the POC will differ by the sigs f the terms ad S : (7 Ukr. J. Phys. Opt. 0, Vlume, Issue

15 Aalytical relatis ( d 8 ( S S S S S S d (. 8 Eqs. (7 ad (8 that take the micr-wedge shape f samples it csiderati ad ca be used with the half-wave techique are icluded i Table (see Eq. (Т.. Ntice that, apart f, Eq. (Т. ctais the fur pricipal POCs ad a cmplicated sum f the S km cefficiets. Ptetially this ca brig abut icreasig errrs ad, as a result, lw experimetal accuracy fr the cefficiet. O the ther had, the sigs f the cefficiet i the tw versis f Eq. (Т. writte fr the direct ad symmetric experimetal cditis are ppsite. The e ca exclude all the pricipal POCs ad all the elastic cmpliaces, except fr S, whe usig the differece betwee these versis f Eqs. (Т. writte uder the cditis m B ad mв. As a csequece, e derives a simple relati give by Eq. (Т., which is preferable while calculatig the POC.... Determiati f the POC differece 5 Uder the cditis csidered abve (і=, k= ad mв it is ecessary t chage the directi f light plarizati by 90, resultig i і=. Calculatig δ ad with the prcedures described abve ad isertig these parameters it Eq. (9, e ca btai the relati that icludes the POC 5 : [ ( ( ] ( S (. S S S S S d 8 5 d Chagig pressure directi t m B gives rise t a similar relati, which differs by the sigs f the terms ( 5 ad S : [ ( ( 5 ] d (0 ( S S S S S S d (. 8 The bth frmulae are cmbied i Eq. (Т.5, where the upper ad the lwer sigs f the terms (π 5 π ad S crrespd t the cditis mв ad m B, respectively. Fllwig frm -idetity f Eqs. (9 ad (0 at m В ad m B (see als Eq. (Т.5, e ca cclude that arbitrary chice f the directis В ad B ca lead t ambiguus determiati f the POC differece ( 5. The same ccers the cefficiet calculated the basis f Eqs. (7, (8 ad (Т.. Hwever, the differece f the tw versis f Eq. (Т.5 writte fr the cases f m В ad m B (see Eq. (Т. reveals ambiguity with respect t the POC cmbiati ( 5. Besides, mst f the POCs ad the S km cefficiets are the excluded frm the frmula ad, as a result, the 5 value ca be determied with high eugh accuracy. The same als refers t Eq. (Т. used fr calculatig the POC (see the cmmets appearig belw Eq. (8. Ukr. J. Phys. Opt. 0, Vlume, Issue 5 (8 (9

16 Mytsyk B. et al After isertig the π cefficiet it Eq. (Т. ad Eq. (Т.5 (r Eqs. (Т.7 (Т.0, e ca fid the POCs 5 ad, respectively. It is bvius that the POC btaied the basis f Eqs. (Т.7 (Т.0 shuld reveal a relatively lwer errr, sice the Piss ctributi i these frmulae is either abset r assciated with s sigle elastic cmpliace cefficiet S. At the same time, the Piss strai ctributi i Eq. (Т.5 is liked with a cmplicated cmbiati f the cefficiets experimetal errr. S km, ad s the errr f each f them will ievitably ctribute t the ttal... A chice f crdiate system Let us fially csider ather pricipled questi: des the ambiguity f calculati f the POC ad the POC differece 5 deped the chice f right-haded crdiate system? It has bee shw i the wrks [9, 0, ] that the ambiguity i the chice f the directis ad fr the trigal crystals f the symmetry classes, m ad m des lead t ambiguus calculati f the cefficiets, ad. I rder t slve the prblem, let us csider experimetal maifestatis f the POE fr the В-cut samples f crystals with the symmetries, ad /m. Let us have the tw right-haded crdiate systems i which the directis ad are iterchaged (tice that the directis ad i Fig. 8 are iterchaged with respect t the crrespdig directis preseted i Fig. e. The the light plarizati ca be aliged with the directis і= ad і= (see Fig. 8, the light prpagati directi with k=, while the tesrs [ B ] ad [ ] wuld differ frm thse give B by Eqs. (5 ad (. Belw we will derive the latter tesrs. If the cditi PВ is fulfilled, the the agle betwee the vectr P ad the directi is equal t 5 (see Fig. 8. The directial csie a f the vectr P is equal t cs5=. It is see frm Fig. 8 that the prjecti f P the directi is equal t P = Рcs5 = Р. ( Fig. 8. A scheme f rietatis f crystallgraphic directis, ad which differ frm thse shw i Fig. e (see the text. T determie the directial csies b ad c, e shuld fid the prjectis f P up the axes ad. The mutual arragemet f the directis,,,, ad is schematically represeted i Fig. 9. It fllws frm Fig. 9 that Р = P cs5= P, Р = P cs5= P. ( Fig. 9. Arragemet f directis,,,, ad fr fidig prjectis f the P vectr axes ad (see the text. With accutig fr Eq. (, Eqs. ( may be rewritte as Р = Р/, Р = Р/ r b = Р /Р = /, c = Р /Р = /. Puttig the quatity Ukr. J. Phys. Opt. 0, Vlume, Issue

17 Aalytical relatis it Eq. (5, e readily btais the tesr P = P(а, b, c = P(, /, / B : B,,,,,. ( Aalgically fr the case f P B we have P = P ( B B, /, /, ad the tesr [ ] may be preseted as B,,,,, B. ( It is see that the sigs f 5 ad have chaged, whe cmpared t Eq. (. The usig the refractive idex chages δ ad ad the strai arisig whe the stresses defied by Eqs. ( ad ( are applied, e ca calculate the ptical path k. Fr example, at і=, k= ad m В e gets ( d 8 (5 ( S S S S S S d (. 8 This relati is idetical t Eq. (7 used fr calculati f the cefficiet π. Uder the cditi m B e ca btai the relati fr π idetical t Eq. (8. The same ccers the prblem f determiati f the POC differece 5 : at і=, k= ad mв (r m B e btais the relatis idetical t Eq. (9 r Eq. (0. At the same time, Eqs. (Т. ad (Т. remai idetical i the bth crdiate systems (see Fig. e ad Fig. 8. Hece, the directis В ad B are t symmetrically idetical fr the classes, ad /m, ulike the directis ad, ad, ad * ad *. Hwever, the differeces f the tw versis f Eq. (Т. ad Eq. (Т.5 writte fr the cases f mв ad m B wuld result respectively i Eqs. (Т. ad (Т.. The latter relatis reveal ambiguity ccerig the prblem f calculati f the POCs ad the POC cmbiati 5. This cclusi is als true f experimetal determiati f the POCs 5 ad, the relatis fr which iclude the POC. I ther wrds, the right-haded crdiate system fr the В-cut sample ca be chse arbitrarily, usig ly the mst geeral recmmedatis fr chsig crystallgraphic axes fr the tetragal crystals. 5. Cclusis We have derived the aalytical relatis which describe all f the POCs fr the tetragal crystals belgig t the symmetry classes, ad /m. These relatis take a micr-wedge shape f real samples it csiderati. We have shw that a umber experimetal gemetries assciated with idirect cuts f crystal samples eable determiig the pricipal POCs π im (i, m =,, ad their sums f the type,, etc. After calculatig these POCs the basis f ur theretical results, e ca cmpare their values im with thse experimetally btaied usig the direct-cut samples. Such a cmparis seems t be a strg eugh test f reliability f the ex- Ukr. J. Phys. Opt. 0, Vlume, Issue 7

18 Mytsyk B. et al perimetal results ad a effective way t reveal a piezptic idetity f samples btaied frm differet parts f the same crystallie bule r frm differet bules. T determie the POCs fr the secd subgrup f tetragal system (i.e., fr the pit symmetry grups, mm, m, ad /mmm, it is eugh t use the aalytical relatis fr the direct-cut, Х/5- ad Z/5-cut samples. Refereces. Narasimhamurty T S, Veerabhadra R K ad Petterse H B, 97. Phtelastic cstats f ADP. J. Mater. Sci. 8: Narasimhamurty T S, 99. Phtelastic behavir f Rchelle salt. Phys. Rev. 8: Adrushchak A S, Mytsyk B H, Rmashk V А ad Seglish Ya А, 99. Piezptic effect i Ba x Sr -x Nb O crystals. Ukr. Fiz. Zhur. : 8.. Feldma A, Hrwitz D, Waxler R M ad Ddge M J. Optical materials characterizati. Natial Bur. Stad. (USA. Tech. Nte 99 ( Martyyuk-Lttska I, Mys O, Dudk T, Adamiv V, Smirv Y ad Vlkh R, 008. Acust-ptic iteracti i α-bab O ad Li B O 7 crystals. Appl. Opt. 7: 5.. Martyyuk-Lttska I, Dudk T, Mys O, Rmayuk G ad Vlkh R, 009. Acustptic iteracti ad phtelastic prperties f Li B O 7 ad α-bab O crystals at the wavelegth f m. Ukr. J. Phys. Opt. 0: Smushkv I V, Kapla M S ad Sumi V I, 970. The temperature depedece f piezptic cstats f KCl ad KBr. Fiz. Tverd. Tela. : Narasimhamurty T S, Phtelastic ad electrptic prperties f crystals. New Yrk: Pleum Press ( Mytsyk B H, Adrushchak A S ad Gaskevich G I, 007. Cmprehesive studies f piezptical effect i lagasite crystals. Ukr. J. Phys. 5: Mytsyk B G, Adrushchak A S, Demyayshy N M, Kst YaP, Kityk A V, Madracci P, Schraz W, 009. Piez-ptic cefficiets f MgO-dped LiNbO crystals. Appl. Opt. 8: Vasylkiv Yu, Savary V, Smaga I, Skab I ad Vlkh R, 0. O determiati f sig f the piez-ptic cefficiets usig trsi methd. Appl. Opt. 50: Skab I, Smaga I, Savary V, Vasylkiv Yu, Vlkh R, 0. Trsi methd fr measurig piezptic cefficiets. Cryst. Res. Tech. :,. Sirti Yu I ad Shasklskaya M P, Fudametals f crystal physics. Mscw: Nauka (979.. Mytsyk B, 00. Methds fr the studies f the piez-ptical effect i crystals ad the aalysis f experimetal data. Part I. Methdlgy fr the studies f piez-ptical effect. Ukr. J. Phys. Opt. :. Mytsyk B., Demyayshy N. ad Kst Ya., 0. Aalytical relatis describig piezptic effect i tetragal crystals. Ukr.J.Phys.Opt. : 0 8. Анотація. Одержано аналітичні співвідношення для визначення усіх п єзооптичних коефіцієнтів за інтерферометричним методом із урахуванням мікроклиновидності реальних зразків. Представлено результати для тетрагональних класів симетрії, і /m. Співвідношення для точкових груп, mm, m і /mmm є простими частковими випадками загальних співвідношень для п єзооптичного ефекту в тетрагональних кристалах. 8 Ukr. J. Phys. Opt. 0, Vlume, Issue