A General Study of Internal Conical Refraction Phenomenon of Some Hollow Beams in Biaxial Gyrotropic and on-gyrotropic Crystals

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1 Vol. I. pp: -3 Jue 6 A Geeral Study of Iteral Coial Refratio Pheomeo of Some Hollow eam i iaxial Gyrotropi o-gyrotropi Crytal Faroq Saad Abdelmajid elafhal * Laboratory of Nulear Atomi Moleular Phyi Departmet of Phyi Faulty of Siee Chouaïb Doukkali Uiverity P. 4 El Jadida Moroo belafhal@mail.om (Reeived 5 th May 6; Aepted t Jue 6; Publihed: th Jue 6) Abtrat- I thi paper we tudy the effet of iteral oial refratio ui ome hollow beam a hollow Gauia beam (HG) otrollable dark-hollow beam (CDH) throuh biaxial yrotropi o-yrotropi rytal. The aalytial expreio of the amplitude the iteity ditributio of thee beam i the oidered rytal are derived. Some umerial imulatio are alo ive to illutrate the variatio of the iteity i radial diretio veru the parameter of the iidet beam a the beam order the propaatio ditae Z the yrotropi parameter the ratio r /ω betwee the radiu ylidrial of oial refratio r radiu wait beam ω. The reult provide more eeral harateriti propaatio by a oial refratio of HG CDH i biaxial o-yrotropi rytal. Additioally the propaatio of the flat-topped the Gauia beam both i biaxial yrotropi o-yrotropi rytal are dedued a partiular ae of the preet ivetiatio. Key Word: Iteral oial refratio; iaxial yrotropi rytal; iaxial o-yrotropi rytal; Hollow Gauia beam; Cotrollable dark-hollow beam; Flat-topped beam.. ITRODUCTIO The pheomeo of oial refratio wa diovered theoretially by W. Hamilto i 83 [] ofirmed experimetally by H. Lloyd [] i the ame year. The traformatio of iteral oial refratio ha bee deribed i the eometrial opti i Ref. [3]. Thi pheomeo reult from the propaati of a moohromati liht beam alo a opti axi of a biaxial rytal a a ylider. It repreet the mai ubjet that ha bee tudied i everal work [4-9]. The effet of iteral oial refratio i the rytal wa itrodued tudied i theory by elky Khapalyuk [6] erry [7] i baed o two mai equatio. Thi theory wa explored i detail by elky Stepaov [8] elafhal [9] for the ae of o-yrotropi biaxial rytal. They have how that the evolutio of oial refratio deped o the ratio r /ω betwee the ri radiu of ylider the wait radiu of iidet beam. Thee reult were developed eeralized to hiral biaxial rytal by elky Stepaov []. They have demotrated ompared the iteity ditributio reult by a umerial imulatio experimetal tudy oeri the iteral oial refratio of fudametal Gauia beam i biaxial Copyriht IJPOT All Riht Reerved yrotropi of α-iodi aid rytal. O the other h erry et al. [ ] have alo tudied the effet of hiralty for two dimeiole parameter. Reetly there have appeared may tudie devoted to udert the effet of oial refratio both i theory experimet [3-9]. Amo thee Peet Zolotukhi [3] have demotrated the traformatio of irularly liearly polarized Gauia iput beam i biaxial rytal irularly polarized Lauerre Gau iput beam ha alo bee treated [4 5]. Additioally the oial diffratio iteity of liht beam with top-hat iput beam ha bee tudied by R. T. Dary et al. [6]. Alo Turpi et al. [7] have alo ivetiated polarized uper-gauia oial refratio beam i biaxial rytal. More reetly Turpi et al. [8] have tudied the traformatio of liht beam throuh a biaxial rytal by ui a ew formalim able to how the traitio of double refratio alo ued thi formalim for the propaatio of o-homoeeouly polarized beam throuh a biaxial rytal. The traformatio of radialy azimuthally polarized iput liht beam ha alo bee demotrated i Ref. [9]. The theory of the iteity ditributio of iteral oial refratio for a liht beam pai throuh a biaxial yrotropi rytal i more ompliated tha it i i the ae of a oyrotropi rytal. Our aim here i to ivetiate a theoretial tudy of oial refratio pheomeo for a ew kid of hollow beam a: HG CDH repetively. The preet paper i oraized a follow: I etio we preet a theoretial eeral deriptio of the traformatio of liht beam i yrotropi biaxial rytal by iteral oial refratio. I etio 3 a aalytial expreio to deribe the traformatio of HG i a biaxial of yrotropi rytal i tudied ome partiular ae will be treated diued from the preeted theory. I etio 4 the traformatio of CDH flat-topped beam propaati i a biaxial yrotropi o-yrotropi rytal are alo etablihed. I etio 5 everal umerial example are alo ive to illutrate our aalytial reult. Fially we olude our work i lat etio.. ITERAL COICAL REFRACTIO FOR A LIGHT EAM I IAXIAL GYROTROPIC CRYSTALS I yrotropi of biaxial rytal the iteity ditributio of iteral oial refratio i ivetiated. The iidet Pae

2 Vol. I. pp: -3 Jue 6 Fi. : Shemati repreetatio of the experimetal arraemet. beam i propaated perpediularly alo to oe of the opti axe a a ylider. Fi. how the hemati view of the experimetal etup to tudy the oial diffratio pheomeo by ui a biaxial rytal. Now we oider the plae of a oial refratio ri i the polar oordiate ytem a: x-r r oϕ yr iϕ where r i the radial ompoet i ylidrial oordiate ϕ i the azimuth ale r i the radiu of the ylider of liht emeri o a biaxial rytal. The emeri liht of a eletri field of beam by iteral oial refratio i biaxial yrotropi rytal ha two ompoet ive by [] t Ex D[ k ( r) + ( k oϕ+ k iϕ) ( r) (-a) k ]exp( ik z) t E D[ k ( k y + k ]exp( ik z) Copyriht IJPOT All Riht Reerved oϕk iϕ) (-b) where k oγ k iγ with γ i the ale of polarized iidet beam are the field radial futio depedee i the oial refratio ri defii by iterated expreio a follow ( ρ) exp o( ρ ) π A kr + (-a) J krρ ρdρ J i( kr ) ρ + A( ρ) i π ρ + (-b) ( krρ) ρ dρ v A( ρ) π kr ρ + ( ρ ) ρ ρ ρ + J kr d (-) where z Z (3-a) α r l (3-b) ε v ' α (3-) 4 D ( ) 3 ( ) ikl + 8 (3-d) ε (3-e) ε ε ε l εε 3 (3-f) ε α ( ε ε )( ε ε ) (3-) 3 ε ε I Eq. J are the zeroth the firt order eel futio Z i the propaatio ditae whih i meaured from the foal plae r i the radiu of ylider of liht emeri from the rytal i the yrotropi parameter i the idex of refratio ε z > ε y > ε x repreet the priipal value of the dieletri teor πk f J ( krρ) rdr A ρ (4) where ƒ(r) i the eletri field ditributio of the oidered laer beam k i the wave umber. Thi lat equatio ive the Fourier imae of the field ditributio i the iidet beam ro-etio. From Eq. (-a) (-b) (-a) (-b) (-) whih deribe the outoi beam depedee of the azimuth ale ϕ the iteity ditributio of the outoi field i ive by I Y D { + Re( + )i( ϕγ)}. + + Re( )o( ϕγ) (5) Thi lat equatio repreet the iteity ditributio i biaxial yrotropi rytal by oial refratio whe it tur throuh the ale γ aroud the polar oordiate ytem. Now we will diu the iteity ditributio of the iteral oial refratio for ome hollow beam propaati i biaxial yrotropi rytal by oideri a hollow Gauia Cotrollable dark-hollow ditributio i etio 3 etio 4 repetively. 3. HOLLOW GAUSSIA FIELD I IAXIAL GYROTROPIC CRYSTALS I thi etio we ive the iteity ditributio of the iteral oial refratio for the ae of HG i biaxial yrotropi rytal. Aumi that the eletri field ditributio of HG i the ylidrial oordiate ytem at the oure plae z expreed by [] f G ( β r ) exp( βr )... (6) where G i a otat repreeti the amplitude oeffiiet i the order of the HG β/ω with ω i the Gauia wait at the plae z. Alo HG a be expreed a a uperpoitio of a erie of Lauerre-Gauia mode a Pae

3 Vol. I. pp: -3 Jue 6 m f G exp( β r ) L ( αr ) (7) m m m where L m (.) deote Lauerre polyomial with mode order m( m ) i a biomial oeffiiet αβ. y ubtituti Eq. (7) ito Eq. (4) ui the followi iteral formula [] ( αx ) J ( xy) v+ β x ν v v v 4β x e L ν dx β ( β α) y e (8) αy v L 4β( α β) after tediou iteral alulatio Eq. (4) beome πk ω ρ A( ρ) G exp 4 (9) m k ω ρ L m m m y itrodui Eq. (9) ito Eq. (-a) (-b) (-) the outoi field of HG after pai throuh a biaxial yrotropi rytal i oial refratio a be writte a ( krρ) ρdρ m G + m ( m) k ω ρ Lm o J ( krρ) ρ dρ m m G + m ( m) m k ω ρ i L m J G v + k ω ρ i L m m ρ ikzρ 4 Copyriht IJPOT All Riht Reerved ( kr ) ρ + y ρ 4 ρ 4 ( kr ρ + ) ρ + ( kr ρ + ) J ( krρ) ρdρ ρ + m m( m) (-a) (-b) (-) Thee lat iteral overe a be evaluated umerially. They repreet the mai firt reult of thi tudy whih deribe the traformatio of HG i biaxial yrotropi rytal by iteral oial refratio. 3.. Partiular Cae I the previou etio we have demotrated that the iteity ditributio of the iteral oial refratio for a HG a be propaated i biaxial yrotropi rytal. From our fidi etablihed i Eq. (-a) (-b) (-) oe a dedue partiular ae by hooi ome value of v Cae of hollow Gauia field throuh a biaxial oyrotropi rytal: Thi ae i obtaied whe v. Coequetly Eq. (-a) (-b) (-) redue to m ρ r G + m m m 4 (-a) k ω ρ Lm o( krρ) J( krρ) ρdρ m ρ r G + m m m ( ) 4 (-b) k ω ρ L i( kr ρ) J ( krρ) ρdρ m Thee equatio deribe the field ditributio of the iteral oial refratio of HG i biaxial o-yrotropi rytal. Note that for the ae of o-yrotropi rytal vaihe the iteity ditributio doe ot deped o the azimuth ale ϕ. So Eq. (5) a be rewritte a: D { } I +. () Thi equatio repreet the iteity ditributio of the iteral oial refratio for a iidet hollow Gauia beam i biaxial o-yrotropi rytal Cae of Gauia ditributio throuh a biaxial yrotropi rytal: I thi partiular ae whe Eq. (- a) (-b) (-) et the form ρ exp 4 (3-a) o( kr ρ + ) J ( krρ) ρdρ i( kr ) ρ + J ( kr ) d i ρ exp 4 (3-b) ρ + ρ ρ v k ρ kr ρ + J ( krρ ) ρd ρ ρ + ω ρ 4 (3-) whih repreet the eletrial field of the iteral oial refratio for a Gauia beam throuh a biaxial yrotropi rytal. Note that thee equatio are the ame a Eq. (5) of Ref. [] Cae of Gauia field throuh a biaxial o-yrotropi rytal: For thi peial ae whe v Eq. (-a) (-b) (-) implify to ρ exp exp 4 (4-a) o( krρ) J( krρ) ρdρ Pae 3

4 Vol. I. pp: -3 Jue 6 ρ exp exp 4 (4-b) ρ ρ ρ ρ i( kr ) J ( kr ) d. Thee lat equatio ive the aalytial formula of the eletri field of a Gauia beam propaati throuh a biaxial o-yrotropi rytal. Thee relatio are i aordae with reult of Ref. [89]. 4. COTROLLALE DARK-HOLLOW DISTRIUTIO OF CIRCULAR SYMMETRY I IAXIAL GYROTROPIC CRYSTALS I thi etio we tudy the iteity ditributio of the iteral oial refratio for a itereti kid of hollow beam alled Cotrollable dark-hollow beam (CDH) pai throuh a biaxial yrotropi rytal. The field ditributio of thi beam i the ylidrial oordiate ytem at iput plae i defied a follow [] r r f r exp (5) ω pω where( ) i a biomial oeffiiet i the order of CDH ω i the width beam p i a real poitive parameter (p<). Alo the area of the dark reio at the etre a be otrolled by the parameter p. y ierti Eq. (5) ito Eq. (4) realli the iteral formulae [] v v+ / at u / 4a t e J ( ut) dt e (6) v v+ ( a) Copyriht IJPOT All Riht Reerved u with Re α> Re υ> after tediou iteral alulatio Eq. (4) a be writte a πk ω A( ρ) ( ). (7) k ω ρ p ρ p 4 4 y itrodui Eq. (7) ito Eq. (-a) (-b) (-) we a expre the outoi field of a CDH after pai a biaxial yrotropi rytal a k k ω ρ 4 ( ) p ρ p 4 o( + ) ω ρ 4 kr ρ ( ) J krρ ρdρ p ρ p 4 i( kr ρ + ) J ( kr ) d ρ + ρ ρ ρ (8-a) (8-b) v k ω ρ 4 ( ) p ρ p 4 i( kr ρ + ). ρ + J krρ ρdρ (8-) Thee formula ive the eletrial field of the CDH after traveli a biaxial yrotropi rytal i iteral oial refratio they it repreet the eod mai reult of thi paper. 4.. Partiular Cae I the above etio we have ivetiated the iteity ditributio of the iteral oial refratio for a CDH pai throuh biaxial yrotropi rytal. From our reult etablihed i Eq. (8-a) (8-b) (8-) ome partiular ae a be obtaied by hooi ome value of N p Cae of Cotrollable dark hollow ditributio throuh a biaxial o-yrotropi rytal: From Eq. (8-a) (8-b) (8-) by putti oe obtai k k ω ρ 4 ( ) p ρ p 4 o ω ρ 4 kr ρ J ( ) krρ ρdρ p ρ p 4 (9-a) (9-b) i( kr ρ) J ( krρ) ρdρ. Thee equatio deribe the field ditributio of the oial refratio for a CDH after pai throuh a biaxial o-yrotropi rytal Cae of flat-topped ditributio throuh a biaxial yrotropi rytal: For thi ae by taki > p i Eq. (5) the iidet beam obtaii orrepod to the expreio of the eletri field with irular flat-topped beam whih i ive by [3] r f r () ω Eq. (8-a) (8-b) (8-) redue to Pae 4

5 Vol. I. pp: -3 Jue 6 ρ 4 o( + ) kr Copyriht IJPOT All Riht Reerved ρ J krρ ρdρ ρ 4 i( kr ρ + ) J ( krρ) ρ dρ ρ + v 4 i( kr ) ρ + J ( kr ) d ρ + ρ ρ ρ ρ (-a) (-b) (-) where i the order of the flat-topped beam. Thee are the aalytial formula of outoi eletri field of the flat-topped beam throuh a biaxial yrotropi rytal i oial refratio. I additio whe thee lat equatio orrepod to the reult whih have obtaied i the previou ae (3..) for a Gauia beam i biaxial yrotropi rytal Cae of flat-topped ditributio throuh a biaxial oyrotropi rytal: Whe > p Eq. (8-a) (8-b) (8-) et the form ρ r exp 4 (-a) o( kr ρ) J ( krρ) ρdρ ρ r exp 4 (-b) i( kr ρ) J ( krρ) ρdρ Thee equatio repreet the field ditributio of flattopped beam propaati i a biaxial o-yrotropi rytal with (r). Additioally whe thee relatio are oitet with the reult of Gauia beam whih have dedued i etio (3..3). 5. UMERICAL SIMULATIOS AD DISCUSSIO The aim of thi etio i to ivetiate aalyze the aalytial expreio of the iteity ditributio of the iteral oial refratio for HG CDH throuh biaxial yrotropi o-yrotropi rytal where the waveleth of He-Ne laer i fixed at λ63.8 m. We will hooe the other parameter a follow: the rytal leth l.4 m the beam width ω34µm the radiu of the ylider of the liht emeri from the rytal r 4 µm the upper iteratio limit i p.5. I our imulatio the ifluee of the eometrial radiu r the beam radiu ω the beam order the yrotropi parameter the propaatio ditae Z are illutrated. We will perform the reult of umerial imulatio i the followi etio. 5.. Cae of the Hollow Gauia beam I order to tudy the traformatio of HG pai throuh a biaxial yrotropi o-yrotropi rytal alulated expreed by aalytial expreio i etio 3 we will ive ome umerial imulatio for differet beam order ( ) at Z ( 3 mm) whih are preeted i Fi. (-5) obtaied by ui the parameter previouly metioed. I Fi. (-3) we diplay the iteity profile of the iteral oial refratio of HG i biaxial yrotropi rytal whih are alulated by the above aalytial reult elaborated i Eq. (5) (-a) (-b) (-) for two value of the beam order at differet propaatio ditae Z with the polarized iidet beam (γ).45. From Fi. the iteity ditributio of HG for beam order N equal to zero varie with differet propaatio ditae Z (ee Fi. (a-)). For Z it a be how from that the piral phae hape of the ri i oberved take o a lok-wie piral ditributio (ee Fi. (a)). O ireai the propaatio ditae Z the briht pot beome more obervable the piral ri alo ireae. Furthermore the axial pike appear a Z ireae (ee Fi. ). For thi ae the obtaied reult orrepod to the fudametal Gauia beam i biaxial yrotropi rytal. Fi. 3 preet the iteity profile of HG a Fi. but here for the beam order i equal to oe. We a how from thee fiure of differet value of Z that the piral of the ri i oberved ireae. Additioally the briht pot of the iteity ditributio beome larer the axial pike i how (ee Fi. 3 (a-)). Geerally oe a ee that the briht pot the piral ri a be haed whe Z are ireai (ee Fi. -3). Fi. 4 5 ive the reult of umerial imulatio etablihed i the above ae (3..). Thee reult orrepod to iteitie alulated from Eq. (-a) (-b) () whih deribe the traformatio of hollow Gauia beam i a biaxial o-yrotropi rytal for differet value of beam order hooe the parameter whih are ued i the previou fiure. Fi. 4 illutrate the iteity profile of HG with order at differet propaatio ditae Z. Whe Z the iteity of thi beam preet a etral dark pot double briht ri eparated by dark ri (ee Fi. 4 (a)). A Z ireae the eparatio of ri i how the etral dark pot dereae radually. The the briht pot appear at the etre with further ireae of propaatio ditae Z (ee Fi. 4 ). I thi ae the reult obtaied are idetial to the reult for the ae of the fudametal Gauia beam i Ref. [3]. Fi. 5 ive the iteity profile of HG a Fi. 4 but i thi time for the beam order N i equal to oe. From Fi. 5 (a) we a ee that at Z the iteity et everal briht ri with differet iteitie urroudi the etral dark pot. O ireai the propaatio ditae Z the dark pot redue Pae 5

6 Vol. I. pp: -3 Jue 6 a) Fi. : The iteity ditributio of the iteral oial refratio for a HG i yrotropi rytal for beam order at (a) Z Zmm Z3mm. a) Fi. 3: The iteity ditributio of the iteral oial refratio for a HG i yrotropi rytal for beam order at (a) Z Zmm. Copyriht IJPOT All Riht Reerved Pae 6

7 Vol. I. pp: -3 Jue 6 Fi. 3: The iteity ditributio of the iteral oial refratio for a HG i yrotropi rytal for beam order at Z3mm. (a) Fi. 4: Iteity ditributio of HG i biaxial o- yrotropi rytal omputed for beam order at (a) Z Zmm Z3mm. a) Fi. 5a: Iteity ditributio of HG i biaxial o- yrotropi rytal omputed for beam order with r 4µm ω34µm Z. Copyriht IJPOT All Riht Reerved Pae 7

8 Vol. I. pp: -3 Jue 6 Fi. 5: Iteity ditributio of HG i biaxial o- yrotropi rytal omputed for beam order with r 4µm ω34µm Zmm Z3mm. radually the etral iteity with briht pot beome more obervable (ee Fi. 5 ). Geerally whe the beam order i ireai at ome propaatio ditae Z the dark pot redue radually the iteity of the briht ri i formed alo hae. 5.. Cae of the Cotrollable dark-hollow beam The iteity profile of CDH i biaxial yrotropi o yrotropi rytal are etablihed ive by aalytial expreio i the above etio 4. So we will preet ome umerial imulatio example to illutrate our aalytial reult whih are preeted i Fi. (6-). Fi. 6 7 ive the iteity ditributio of the iteral oial refratio for a CDH i biaxial yrotropi rytal alulated from Eq. (5) (8-a) (8-b) (8-) with the ame parameter ued i Fi. p.9 for two value of beam order ( ) propaatio ditae Z ( 3 mm). I Fi. 6 we illutrate the iteity profile of oial refratio ri for a CDH with the order N i equal to oe at differet propaatio ditae Z. A Z the briht pot of the iteity ditributio i oberved the phae hape take o a lok-wie piral ditributio (ee Fi. 6 (a)). A Z ireae the piral of the ri beome faiter (ee Fi. 6 ). Additioally oe a how that thi piral diappear the axial iteity pike i formed a Z ireae (ee Fi. 6 ). Whe the beam order N i ireai at ome propaatio ditae Z the variatio of iteity of oial refratio ri are how (ee Fi. 7). The imulatio reult of thee fiure are imilar to thoe of Fi. 6. Oe a ee from that the briht pot the piral of the ri a be adjuted by hai Z. (a) Fi. 6a: Iteity profile of oial refratio ri of CDH i biaxial yrotropi rytal for beam order at Z. Copyriht IJPOT All Riht Reerved Pae 8

9 Vol. I. pp: -3 Jue 6 Fi. 6: Iteity profile of oial refratio ri of CDH i biaxial yrotropi rytal for beam order at Zmm Z3mm. (a) Fi. 7: Iteity profile of oial refratio ri of CDH i biaxial yrotropi rytal for beam order at (a) Z Zmm Z3mm. Copyriht IJPOT All Riht Reerved Pae 9

10 Vol. I. pp: -3 Jue 6 (a) Fi. 8: Iteity ditributio of iteral oial refratio for a CDH i biaxial o-yrotropi rytal for beam order p.9 (a) Z Zmm Z3mm. (a) Fi. 9: Iteity ditributio of CDH i biaxial o-yrotropi rytal omputed for beam order at (a) Z Zmm. Copyriht IJPOT All Riht Reerved Pae

11 Vol. I. pp: -3 Jue 6 Fi. 9: Iteity ditributio of CDH i biaxial o-yrotropi rytal omputed for beam order at Z3mm. (a) Fi. : Iteity profile of flat-topped beam throuh biaxial yrotropi rytal i oial refratio for beam order N at (a) Z Zmm Z3mm. a) Fi. a: Iteity profile of flat-topped beam throuh biaxial o-yrotropi rytal i oial refratio for beam order Z. Copyriht IJPOT All Riht Reerved Pae

12 Vol. I. pp: -3 Jue 6 Fi. : Iteity profile of flat-topped beam throuh biaxial o-yrotropi rytal i oial refratio for beam order Zmm Z3mm. Fi. 8 9 preet the umerial reult of the iteity ditributio of the oial refratio for a CDH pai throuh biaxial o-yrotropi rytal whih omputed i the above ae (4..) by the ue of Eq. () (9-a) (9-b) repetively for two beam order ( ) at propaatio ditae Z ( 3 mm) ued the ame parameter a thoe for the previou fiure. Fi. 8 ive the iteity profile for beam order at ome propaatio ditae Z. Oe a ee that at Z the etral iteity of thi beam keep the dark pot urrouded by everal briht ri whih are formed with differet iteitie. A Z ireae the dark pot dereae radually. The the briht pot alo appear at the etre. With further ireae of the beam order at propaatio ditae Z the variatio of iteity profile i how (ee Fi. 9). The umerial reult of thee lat fiure are imilar to thoe of Fi. 8 we a ee from that the dark pot the iteity of the briht ri a be haed by ireai Z. Fi. ive the iteity ditributio of the iteral oial refratio for a flat-topped beam propaati i a biaxial yrotropi rytal whih i elaborated i the above ae (4..) by Eq. (5) (-a) (-b) (-) for a beam order equal to two for three value of propaatio ditae Z ( 3 mm). Oe a ee that Fi. i imilar to Fi. 6 7 but i thi ae for p N>. The iteity profile of thi beam et alo it piral hape of the ri. The thi piral diappear the briht pot appear at the etre a Z ireae (ee Fi. ). Fi. ive the radial ditributio of the iteral oial refratio for a flat-topped beam pai throuh a biaxial o-yrotropi rytal alulated i the partiular ae (4..3) by the ue of Eq. () (-a) (-b). Oe a ee that the reult obtaied i thi fiure are imilar to thoe of Fi. 8 9 but thi time for > the Copyriht IJPOT All Riht Reerved parameter p i equal to zero. It lear from that whe Z the etral dark pot with two briht ri are oberved. Additioally the dark pot the iteity of double briht ri are formed they a be haed by ireai Z (ee Fi.). 6. COCLUSIO I ummary we have ivetiated the propaatio of ome hollow beam like: HG CDH i a biaxial yrotropi o-yrotropi rytal by iteral oial refratio. Aalytial expreio of the iteity ditributio of thee beam pai throuh a traparet lab of a biaxial yrotropi rytal ut perpediularly to oe of it wave axe are derived. The imulatio reult of the iteity ditributio of thee beam are draw. The ifluee of ome parameter of thee iidet beam the propaatio ditae Z the yrotropi parameter the ratio betwee the ri radiu of the oial refratio ylider r the wait radiu ω are diued. Our tudy eeralize the reult previouly etablihed i the ivetiatio o the iteral oial refratio of HG CDH throuh a biaxial oyrotropi rytal. Fially the flat-topped the fudametal Gauia beam propaati i biaxial rytal yrotropi or o-yrotropi are alo dedued a partiular ae i our fidi. ACKOWLEDGMET The firt author wa upported by the Miitry of hiher Eduatio Sietifi Reearh of Yeme. REFERECES [] W. R. Hamilto Third upplemet to a eay o the theory of ytem of ray Tra. R. Irih Aad. vol. 7 pp Pae

13 Vol. I. pp: -3 Jue 6 [] H. Lloyd O the pheomea preeted by liht i it paae alo the axe of biaxial rytal Tra. R. Irih Aad. vol. 7 pp [3] M. or E. Wolf Priiple of Opti Peramum Oxford 98. [4] A. J. Shell N. loembere Laer tudie of iteral oial diffratio. II. Iteity patter i a optially ative rytal α-iodi aid J. Opt. So. Am. vol. 68 o. 8 pp [5] A. J. Shell N. loembere Seod harmoi oial refratio Opt. Comm. vol. o. pp [6] A. M. elky A.P. Khapalyuk Iteral oial refratio of bouded liht beam i biaxial rytal Opt. Spetro. (USSR) vol. 44 pp [7] M. V. erry Coial diffratio aymptoti: fie truture of Poedorff ri axial pike J. Opt. A. vol. 6 pp [8] A. M. elky M.A. Stepaov Iteral oial refratio of oheret liht beam Opt. Comm. vol. 67 pp [9] A. elafhal Theoretial iteity ditributio of iteral oial refratio Opt. Comm. Vol. 78 pp [] A.M. elky M.A. Stepaov Iteral oial refratio of liht beam i biaxial yrotropi rytal Opt. Comm. vol. 4 pp. -6. (See alo erratum to thi paper publihed i Opt. Comm. Vol. 8 PP. 7 ). [] M. V. erry M R Jeffrey M. Mauripur Orbital Spi aular mometum i oial diffratio J. Opt. A: Pure Appl. Opt. vol. 7 pp [] M. V. erry M R Jeffrey Chiral oial diffratio J. Opt. A: Pure Appl. Opt. vol. 8 pp [3] V. Peet D. Zolotukhi Free-pae evolutio of foued Gauia beam traformed by oial diffratio i a biaxial rytal Opt. Comm. vol. 83 pp [4] V. Peet Coial refratio formatio of multiri foal imae with Lauerre-Gau liht beam Opt. Lett. vol. 36 o. 5 pp [5] V. Peet Experimetal tudy of iteral oial refratio i a biaxial rytal with Lauerre-Gau liht beam J. Opt. vol. 6 pp [6] R. T. Dary D. MCloky K. E. allatie J. G. Luey P. R. Eatham J. F. Doe Coial diffratio iteity profile eerated ui a top-hat iput beam Opt. So. Am. vol. o. 9 pp [7] A. Turpi Yu. V. Loiko T. K. Kalkjiev H. Tomizawa J. Mompart Super-Gauia oial refratio beam Opt. Lett. vol. 39 pp [8] A. Turpi Yu. V. Loiko T. K. Kalkjiev J. Mompart Liht propaatio i biaxial rytal J. Opt. vol. 7 pp [9] A. Turpi A. Vara A. Lizaa F. A. Torre-Ruiz I. Etévez I. Moreo J. Campo J. Mompart Traformatio of vetor beam with radial azimuthal polarizatio i biaxial rytal J. Opt. So. Am. A. vol. 3 o. 5 pp [] Y. Cai X. Lu Q. Li Hollow Gauia beam their propaatio propertie Opt. Lett. vol. 8 o. 3 pp [] I. S. Gradhtey I. M. Ryzhik Table of Iteral Serie Produt 5 th Editio Aademi Pre New York 994. [] Z. Mei D. Zhao Cotrollable dark hollow beam their propaatio harateriti J. Opt. So. Am. A vol. o. 9 pp [3] Y. Cai Q. Li Liht beam with elliptial flat-topped profile J. Opt. A Pure Appl. Opt. vol. 6 pp Copyriht IJPOT All Riht Reerved Pae 3