Fresnel and Fraunhofer diffraction of a laser Gaussian beam by fork-shaped gratings. Ljiljana Janicijevic and Suzana Topuzoski *

Transcription

1 Fenel an Faunhofe ffacon of a lae Gauan bea by fo-hape gang Ljljana ancjevc an Suana Topuo * nue of phyc Faculy of naual cence an aheac nvey S Cyl an Mehou Sopje epublc of Maceona *Coeponng auho: uana_opuo@yahoo.co Expeon ecbng he voex bea hch ae geneae n a poce of Fenel ffacon of a Gauan bea ncen ou of a on a fo-hape gang of abay nege chage p an voex po n he cae of Faunhofe ffacon by hee gang ae euce. The coon geneal anon funcon of he gang efne an pecale fo he cae of aplue hologa bnay aplue gang an he phae veon. Opcal voex bea o cae of phae ngulay h chage p an p ae he hghe negave an pove ffacon oe bea. The aal pa of he ave aplue ecbe by he pouc of p-h oe Gau-oughnu funcon an a Kue funcon o by he f oe Gau-oughnu funcon an a ffeence of o ofe Beel funcon hoe oe on ach he ngulay chage value. The ave aplue an he neny buon ae cue fo he nea an fa fel n he focal plane of a convegen len a ell a he pecalaon of he eul hen he gang chage p.e. he gang un fo foe no eclnea. The analycal eon fo he voex a ae alo cue. OCS coe: NTODCTON Opcal voce have aace an nceae nee an have becoe vey poan n opcal appng an anpulaon of all pacle. ong he opcal eleen hch geneae opcal voex bea ae phae pal plae [3] helcal axcon [456] foe gang [789] pal one plae []. a ecogne by llen e al. [] fo he f e ha a bea h helcal avefon chaacee by auhal phae lϕ hee ϕ he auhal coonae l an nege poe an obal angula oenu lh pe phoon n popagaon econ.

2 n [3] he auho analye hypegeoec lae bea ha ae geneae by coplex aplue copoe of fou cofaco: he Gauan onen a logahc axcon a pal phae plae an an aplue poe funcon h a poble ngulay a he ogn of he coonae. Foe nefeoga an gang belong o he cla of ffacve opcal eleen hch geneae opcal voce. Bae on he eenal eence he auho n [7] gave he f epo on he poucon of opcal fel h avefon locaon of vaou oe by he ue of hologa poeng fo-hape locaon. n effo fo heoecal ecpon of he Gauan ave fel anfoaon by he foe hologa ha been one n [8]. The auho efne he anon funcon of he foe hologa an ulplyng by he Gauan ncen bea calculae he fel eaely afe he hologa. con of hee coponen-one chagele an o h nuce phae ngulae of oppoe chage. Bu hou ung he eho of olvng he fface ave negal any nfoaon abou he aplue fo an he paal confguaon of he fel coponen n obane. The popoe ecpon of he chage coponen a Laguee-Gauan bea bae only on he eenal fa fel egaon of he ng-le anvee aplue neny pofle an on he ngulay eecon by nefeoec eho. The uaon la h he eul n [9] hee he Faunhofe ffacon of a plane ave by foe gang eae. The auho olve he poble n cylncal coonae pefong he negaon ove he auhal vaable only. By ong o hon ha hee ex a geneaon of bea h phae ngulae bu he aplue neny pofle of uch bea ean unnon. ae poble he calculaon of he voex e hough he aal neny buon o poan n eenal applcaon. Copue geneae hologa have been ue o ceae hollo bea o opcal voce hch ee ue fo gung of col ao [4] appe n he a egon uoune by a epulve pole poenal all nae blue-eune ap. Knong he aal neny buon of he fface fel vey poan becaue he opcal pole poenal paal vaaon epen upon he bea neny pofle [56] a

3 ln [ F] / hee he value of F epen on he auaon neny he naual h of he aoc anon an he feuency eunng fo he aoc eonance. Theefoe he foe gang an hologa a opcal eleen hch geneae opcal voce nee a coplee oluon fo he fface ave an neny buon. n [7] he auho have analycally eve he eon fo ffacon of nonvegng nonffacng Beel bea by fo-hape gang an eclnea gang. The fface ave fel con of a nonvegng aplue euce bea of he ae opologcal chage an oe a he ncen bea a a eo ffacon oe an hghe ffacon oe voex bea hoe chage a u of he ncen bea chage an ha of he foe gang ulple by he ffacon oe. The aplue of he hghe ffacon oe bea ecbe by a u of Gau hypegeoec funcon. n h acle e olve he poble of an ou of a ncen Gauan bea on a gang h a fo-hape ngulay enfe by he nube of nenal eeh Fg. a nclung he cae of abence Fg. b. The ave an he neny buon ae ee hough he Beel funcon hoe oe on ach he ngulay oe of he bea hey ae cue n he nea an fa fel a ell a he pecalaon of he eul fo he cae of eclnea gang. lo he analycal eon fo he Faunhofe ffacon of a Gauan bea ncen on a foe gang h a n he focal plane of a phecal convegen len ae eve an cue. The voex a an he behavo of he bea nea an fa fo he voex cene ae ecbe by analycal foula.. THE TNSMSSON FNCTON OF THE FOK-SHPED GTNG FSG The anon funcon of he fo-hape hologa can be eale by he phoogaphc egaon an eucon of he aance buon of he nefeence beeen a lghly nclne plane ave h aplue an a ave h ngulay of oe p n auhal econ of 3

4 phae ave fon h aplue phooeuce anpaence. coϕ pϕ D. The bnay veon of he gang ae copue geneae an The geneal anon funcon of he fo-hape gang n cylncal coonae ye efne a T ϕ h coϕ pϕ co p D ϕ ϕ D ϕ beng he coonae n gang plane p an nege hong he nube of he nenal eeh of he foe gang. When p he anon funcon un no eclnea gang. The conan D a peo of he eclnea gang an play he ae ole fo he FSG fa fo pole. The pecfcaon of he anon coeffcen epen on he ype of he gang. Fo aplue gang h nuoal anon euaon conan only e h -: /. an ae anon coeffcen of he fl anpaency connece o an n he follong ay: ;. The anon coeffcen of bnay aplue gang ae: / 3... Fo he phae hologa he anon coeffcen ae efne a: β α β α.. h α beng he ba phae eaaon β beng he aplue of he phae oulaon an β χ α co β ; χ α n β enoe Beel funcon of nege oe. Fo bnay phae gang h ecangula pofle he coeffcen ae he follong [8]: ; ; 3.. 4

5 hee χ efne he coeffcen of anon of he phae laye hch eual o one fo eal anon hle α an β have he ae eanng a fo he phae hologa. Fo he cae χ an β follo ha: - α. The pole of he coonae ye locae on he boo of he nenal gang eeh a hon n Fg.. ϕ Π a b Fg.. a Foe gang h p an b eclnea gang hen p. - Fg.. The geoey of he poble. 3. THE FESNEL DFFCTON OF GSSN LSE BEM BY THE FSG The uy of he Fenel ffacon gve an oppouny o cu he paal fo an vegence uale o ffacve peang of he bea geneae by he FSG. To avo he choce of nfne l of he ffacng eleen aea of he negaon convenen he ncen ave fel o be n he fo of a onochoac Gauan bea of avelengh λ. a ance fo a he ncen bea epeene a ϕ hee λ popagaon conan he bea coplex paaee he bea a au. To he bea coplex cuvaue 3 5

6 a eal on-axal au of cuvaue / agne hle / / he bea anvee aplue pofle au. The fo-hape gang uae a ance fo he bea a Fg.. The ncen bea ax pang hough he pole of he gang uae n he plane ϕ play he ole of -ax of he cylncal ye e ae ong h. n he obevaon ceen Π an n he ae e uae a a ance - fo he gang plane he ave fel n he pon can be foun ung he Fenel-Kchoff negal [9] T ϕ ϕ - hee he aea of he gang. co ϕ ϕ 4 The neon of E. an E. n he ffacon negal 4 yel coϕ λ co D coϕ λ co D co ϕ n nϕ pϕ ϕ ϕ n nϕ pϕ ϕ. 5 Fuhe e ue he follong vaable anfoaon ha concen he obevaon plane λ co D n n co 6 h an bu 6

7 D D D / co n g ; co λ λ λ. 7 The pa of he nely nouce vaable an n he obevaon plane Π play he ole of plane pola coonae elae o he pole / D C λ an λ / D C epecvely Fg. 3. C O C pola ax Fg. 3. Geoecal lluaon of he anfoaon 6. The oluon of he negal ove he auhal vaable n E. 5 co ; co ϕ ϕ ϕ ϕ ϕ Φ Φ p p 8 ae ealy foun by he ue of he acoby-nge eny non fo he heoy of Beel funcon [] an ae epeene a [ / ; p p p Φ Φ ]. 9 The neon of h eul n he eon 5 gve. p p p p Theefoe he oal fface fel epeene a a u of eo-h ffacon oe an hghe ffacon oe pove an negave 7

8 h he eo ffacon oe ave aplue an he hghe ffacon oe ave aplue Y p The negal ove he aal vaable n E. an 3 ae enoe a Y Y. 3 4 p Y 5 an hey ae he ell non negal of Beel funcon [] b a l a e > e a l l b b l > ν 4 a ν / Γ µ b µ ν / µ ν b b a M ; ; ν ν µ 4a a Γ ν 4a eplacng he oluon n E. an 3 gve he eo-h ffacon oe bea an he hghe ffacon oe hee Γ p / Γ p e ν > e a > µ. 6 p / p p - p p 7 M p ; M α β; x ae confluen hypegeoec o Kue funcon an he Kue anfoaon fo he hypegeoec funcon [] p p M p ; x ; x x M p 8 8

9 ha been ue. The eo-h ffacon oe bea an onay Gauan chagele bea. Wheea he hghe ffacon oe bea poe phae ngulae h excepon hen p h oppoe chage ncang oppoe helcal avefon chale of he bea fface n he -h an --h ffacon oe epecvely. The aal pa of he ave funcon epeene by he pouc of a oughnu-gauan funcon of oe p an a confluen hypegeoec o Kue funcon of a coplex paaee. p ng he elaon beeen Kue an Beel funcon ee appenx e can ee eon 7 n he fo 9. 4 / p p p Q Q p Q Q Hee e ve ee he aguen of he Beel funcon a 4 Q y noucng a ne coplex cuvaue Q hoe chaacec ae connece o hoe of he ncen Gauan bea n he follong ay ;. f e ue he eny x x ν ν ν he fface coponen 9 can be ee alo a. 4 / / p p p p Q Q p Q Q The exence of he e 9

10 3 Q enue f-oe oughnu-gauan oe of he aplue pofle of he bea. The popagaon axe an o λ / D an λ / D 4 ae axe of eo aplue neny value. They evae fo he axe of he eo-oe bea ax on boh e fo angle δ acan λ D /. The oe of he Beel funcon n 9 an on ach he opologcal chage hch ffe fo he eul n [45] hee he ffacon of a plane ave by copue geneae hologa h anon funcon of a helcal axcon eae an he ffacon coponen ae ecbe by Beel funcon of nege oe hch eual he opologcal chage value of he voex bea coponen. nohe ffeence ha hee he aguen of he Beel funcon epen on he uaac e of he aal coonae nea lnealy. E. 9 an ae la o he cae of ffacon of a plane an Gauan bea by a pal phae plae h ngulay nege oe n peene n eal n [3] hee a a eul of ffacon only one ffacon oe bea obane popagang along he opcal ax efne by he ncen bea ax. n h o a a eul of ffacon of a Gauan bea by foe gang a fan of epaae ffacon oe bea obeve. n he -h ffacon oe 3.. he bea ha opologcal chage p an ecbe by Beel funcon of oe p / an p / epenng on he pouc of he ffacon oe an he gang ngulay oe p. Wheea n [3] he ngulay oe of he fface bea eual o n an he bea ecbe by Beel funcon of oe n / an n / epenng on he phae laye ngulay n. The voex ave fel ecbe by a ffeence of o ofe Beel funcon of faconal oe / accong o D. oa e al. n [] a eul of neacon beeen he voex bea an he bacgoun fel. Fee pace paaxal popagaon of an aay of voce nee n a Gauan bea an each of fo / ha been nvegae n [3] hong ha voce of oppoe chage aac each ohe an can colle an annhlae heea f hey ae of

11 ae opologcal chage he aay ply an o conac n he ho bea an oae gly. The Foue anfo ee a a pouc of a Kue funcon an a onu-gauan funcon pofle. n oe o fn he enon of he voce of he bea 9 e ue he aal neny buon. To fn neceay o ae a epaaon of he eal an he agnay pa n he eon 9 hch no poble becaue of he coplex aguen Q of he Beel funcon. Fo ha pupoe e ue he Neuann aon heoe ee appenx an e can efne he hghe ffacon oe bea hough he Beel funcon of eal aguen a any poon fo he gang plane. The evaon of he neny upon he aal coonae an ang eual o eo n oe o fn he voex a eul n a ancenenal euaon hch a no analycal oluon fo abay chage oe p bu can be olve only nuecally fo pacula value of an p f neee. Soe geneal nfoaon abou he paal behavo of he voex bea pouce by he foe gang an voex a can be obane fo vey nea an fa fel. 4. THE NE-FELD PPOXMTON When an he appoxae value of he Beel funcon aguen Q 4. We oo no accoun ha ; >> [ ] an /. Fo euaon he ave aplue ae obane n he fo p an he neny buon a / p p / - 4 p 4 p 4 5 ˆ 8 p p

12 Fuhe e fn he neny f evave upon he aal coonae ae he eual o eo an ung oe elaon fo Beel funcon e fn ha bee a voex eo he coeponng evave ae eual o eo hen p p Fo he able of Beel funcon e can appoxae he f eo of E. 7 a p /... < 3 / 4 p o he voex a n he nea fel ae / p λ p. 8 a gven all ance value of - he hghe chage oe voce ae e. Fo a ngulay oe p he axu of he lgh ng aoun he voce ae foun on he oaonal paabolo. Coneng he all aguen appoxaon fo he Beel funcon ν x ν x hen x Γ ν foun ha he neny op on o eo nea he voex axe a p ˆ 4 p Γ 8 6 p. 9 ean ha n a gven ffacon oe a a gven ance - he nceang of he voex chage oe ae he all of he voex o becoe oe vecal. Th concluon can be alo een n he Fg. 4 hee he ploe gaph ae bae on he analycal oluon 6 fo he nea-fel neny. n he Fg. 5 he aal neny buon aoun he f ffacon oe voex hon fo ffeen all ance. n he boh gaph he ocllang behavo of he aal neny aoun he value one hen gong aay fo he voex cene clealy een a bgge aal ance fo he voex cene h ngng neny eceang o eo. The paaee ue ae: λ 8 n. The ploe gaph ae bae on E. 6.

13 a.u. Fg. 5. neny buon n he f ffacon oe fo ffacon of a Gauan bea by a foe gang h ngulay p a ance: - o-ahe lne -5 full lne -5 ahe lne a.u. Fg. 4. neny buon n he f ffacon oe fo ffacon of a Gauan bea by a foe gang h ngulay: p o-ahe lne p full lne p3 ahe lne a ance THE F-FELD PPOXMTON When hen e can appoxae Q. ccong o he eon 9 he ave h phae ngulae can be appoxae n he fo 3 p p p p an accongly he nene ae 4 ˆ p p. 3 Expeon 3 ae la o he fa fel neny buon of a Gauan bea fface by pal phae plae of opologcal chage n n [3] bu hey ffe n he oe of he Beel funcon. Hee hey ae eual o epenng on he pouc of he ffacon oe an he gang ngulay oe p nea of n [3] epenng on he phae laye ngulay n. lo he aguen of he Beel funcon hee / p n / / nea of. / 3

14 n he bea paaxal egon e ule he all aguen appoxaon fo he ofe Beel funcon appoxae [ / ] ˆ The voex a ae foun a an fn ha p p p Γ 4 p p. 3 p p. 33 p Fa fo he ngulay ax hen e ue he lage aguen appoxaon fo he ofe Beel funcon an fn ha he neny eceae a ˆ ν 4ν x x x 8x 4 p a.u. a.u Fg. 6. neny buon n he f ffacon oe fo ffacon of a Gauan bea by foe gang h ngulay: p o-ahe lne p full lne p3 ahe lne a ance. Fg. 7. neny buon n he f ffacon oe fo ffacon of a Gauan bea by foe gang h ngulay p a ance: o-ahe lne 3 full lne 5 ahe lne. n he Fg. 6 he aal neny buon aoun he voex n he f ffacon oe a ance an fo ffeen opologcal chage of he gang hon. Wheea n he Fg. 7 he analog 4

15 plo ae ae n oe o ho he neny eceang along he ax fo a gven value of he opologcal chage. The ploe gaph ae bae on E SPECLZTON OF THE ESLTS FO THE CSE p a pevouly agee ha he geneal oluon concen p a a pecal cae.e. hen he gang eclnea nea of beng foe. We ue E. 9 an pecale fo he value p applyng alo he eny x hx x chx [ x x ] [ x x ] / x / an ge he hghe ffacon oe ave aplue n he fo an he coeponng nene 35 ˆ. 36 When p he eo ffacon oe ave aplue an he coeponng neny ˆ Fo he E. 36 an 38 can be conclue ealy ha he axal nene ˆ ˆ ae ffeen fo eo a ece. 7. FNHOFE DFFCTON OF GSSN BEM ON FSG We cone a cala ffacon of a Gauan bea ϕ 39 o 5

16 hee he au of he Gauan bea a a conan aplue value n he bea cene by a foe gang hoe anance gven by E. ncen h a n he gang plane. n he focal plane of a convegen len h focal ance f he neny buon can be foun ung he ffacon negal [9] f f C T ϕ ϕ co ϕ σ ϕ 4 hee C / λf a coplex conan. The negaon n E. 4 ove he aea of he gang bu nce fo > he negaon ove he aal vaable can be pefoe n he neval [ ]. n he poce of negaon e ae ung la anfoaon fo he vaable n he focal plane a hoe gven by E. 6 co λf D co ; n n ;. λf D λf n co ; g D co λf / D The fnal eon fo he fface ave fel n he Foue plane ae f 4 f f fo he eo-h ffacon oe an f p [ p / ] p p f f fo he hghe ffacon oe hee e have enoe f f f 4 λf f. 43 Fo he E. 4 een ha he eo-h ffacon oe fel epeene by a Gauan bgh po h au λf / an neny f o 6

17 ˆ. 44 f f f The ave fel 4 ae la o hoe obane fo he fa-fel appoxaon of Fenel ffacon of a Gauan bea by foe gang h hee ffeence: -The cene of he coponen f n he focal plane hee he voce ex ae he pon C λ f / D an λ f / D C. To nea-by voex bea ae epaae by ance C λf / D C. - n he aguen of he Beel funcon no nea of e have f / gven by E Fo he neny aal buon n he -h ffacon oe obane fo he E. 4 ˆ f p p f f f f f he voex a ae foun n a la ay a ha been one pevouly n econ 5 an ae gven by he eon 45 p p λf p p f. 46 p p n [6] he oluon of a Faunhofe ffacon of a Gauan bea by a helcal axcon euce n a fo of a ee of confluen hypegeoec funcon an n a fo of fne u of he Beel funcon of oe n / fo a phae pal plae of ngulay oe n helcal axcon anfo no phae pal plae hen he axcon angle en o eo. n boh cae he fface bea cae opologcal chage eual o he ngulay oe n of he pal phae plae. 8. CONCLSONS Fo he heoecal eul of h acle e can a ou he follong concluon: -The fo-hape gang h p nenal eeh ffac he Gauan noally ncen bea no a fan of bea hoe nube epen on he ype of he gang. The aplue euce agh- 7

18 hough bea a Gauan fo eal bnay phae gang aben hle he hghe ffacon oe bea evae h epec o he ncen econ ae hollo bea. -The axe of he le bea ae a ue o he phae ngulae of oe p an p ncang he oppoe chale of he avefon-hey ae voex bea. -The aal pa of he ave funcon ecbng he paal fo of he voex bea ae no of he nonffacng nege oe Beel ype no of he Laguee-Gauan ype. They ae ecbe by a oughnu Gauan hape aonally coplcae by e cone of Beel funcon an ofe Beel funcon of eal aguen hoe oe on ach he value of he phae ngulay chage of he voce. They ae of he pal phae plae ype. -The a pace aoun he voex axe ae appe oun by lgh oaonal hypebolo. The peence n he obevaon plane egee by lgh ng of a gven by euaon 8 an 33 fo nea an fa fel epecvely. The ng ae aoun a po h cene C λ / D an λ / D C. -Fo axu value on he ng he neny op on o eo oa he voex axe voex cene n he obevaon plane a n boh nea an fa fel appoxaon. C p -On he ohe e fo he lgh ng n he nea fel appoxaon he neny op en o one 4 / hle n he fa fel op on o eo le p. -The euaon Ψ n Ψ n.. hee Ψ / D co a phae funcon of a plane ave efne n pola coonae ye h C aen a pole epecvely an Ψ ae he phae funcon of he voex bea efne he fnge of he nefeence paen. eale n he plane conan n he fa fel of he voex bea he fnge paen efne by co p n ; n D.. he fo-hape fnge paen h p nenal eeh a a hon n he een epoe by Hecenbeg e al. n [8]. 8

19 PPENDX Le u ue he follong noaon fo he coplex aguen of Kue funcon n E. 7 y. ong he vaey of ecuence elaon afe by he Kue funcon e ue he elaon [] µ ν ν M µ ν ; y ν M µ ν ;y y M µ ν ; y an aapng o he value µ p / ν p ge p p y p M p ; y M p;y M p ; y. 3 p The gh e of h eon elae o he ffeence of o ofe Beel funcon. Tang no accoun ha an p / p y p p y Γ y M p; y 4 p / p y y p p y Γ y M p ; y 5 p e fn ha p M p p / p y ; y Γ p p y y y 6 PPENDX Snce he bea 6 an 9 ae Gauan oulae only a all ance behn he gang hey ovelap each ohe an nefee. bgge ance hey ex epaaely an he neny buon can be calculae epaaely fo each fface coponen. The nene of he bea ae ˆ 7 9

20 [ ]. ˆ B K p p 8 n 8 e have enoe a K 9 hle B p p an ae funcon of eal aguen obane by applcaon of Neuann aon heoe [] hen < u g g u g u ν ν hen < g u u g u g g u ν ν ν ν ν o he Beel funcon p n E.. When < o fo ance < he coeffcen B p p an ae p p p p p p B. 3 Wheea hen > o fo ance > 4

21 hey ae gven a p p p 5 p p p B. 6 The oughnu Gauan hape of he bea 8 ae aonally coplcae by he e [ ] B p p hch ae pa n he eenaon of he voex coe e of he bea. Beng funcon of he ance hey ogehe h he vaable Gauan co econ au clealy ncae ha voex e n a gven ffacon oe epen on he ance an on he chage oe p. Bngng o eo he f evave of eon 8 h epec o he vaable ll gve he euaon fo fnng he neny exee [ ] 7. - ˆ p p p p p p B B B K The oluon gve he voex eo. Fo ou pupoe of poance ae he f axa hch efne he voex e. The ey fo he enfcaon lay n he oluon of he euaon [ ] [ ] B B p p p p. 8 Wh B p p an a ae efne n E euaon 8 ae ahe ancenenal an a no analycal oluon fo abay chage oe p. They can be olve only nuecally fo pacula value of an p. EFEENCES. S. N. Khonna V. V. Kolya M. V. Shnayev V.. Sofe an G. V. plenev The phae oo fle. Mo. Op

22 . M. W. Bejebegen. P. C. Coenel M. Kenen an. P. Woean Helcal-avefon lae bea pouce h a pal phae plae Op. Coun V. V. Kolya.. laov S. N. Khonna V.. Sofe H. Elfo an. Tuunen Geneaon of phae ngulay hough ffacng a plane o Gauan bea by a pal phae plae. Op. Soc Vaaa. Tuunen an. T. Fbeg ealaon of geneal nonffacng bea h copue-geneae hologa. Op. Soc Dav E. Cacole an D. M. Coell neny an phae eaueen of nonffacng bea geneae h a agneo-opc paal lgh oulao ppl. Op V. V. Kolya.. Kovalev. V. Sanov O. Yu. Moeev an V.. Sofe Dffacon of a fne-au plane ave an a Gauan bea by a helcal axcon an a pal phae plae. Op. Soc V. Yu. Bahenov M. V. Vaneov an M. S. Son Lae bea h ce locaon n he avefon P a Zh. Ep. Teo. F N.. Hecenbeg. McDuff C. P. Sh H. ubnen-dunlop an M.. Wegene Lae bea h phae ngulae Op. an Quan. Elecon. 4 S95-S G. F. Ban Phae ngulae n bea.. Phy H. He N.. Hecenbeg an H. ubnen-dunlop Opcal pacle appng h hghe-oe oughnu bea pouce ung hgh effcency copue geneae hologa. Mo. Op N.. Hecenbeg. McDuff. C. P. Sh an. G. Whe Geneaon of opcal phae ngulae by copue-geneae hologa Op. Le L. llen M. W. Bejebegen.. C. Speeu an. P. Woean "Obal angula oenu of lgh an he anfoaon of Laguee-Gauan lae oe" Phy. ev V. V. Kolya an.. Kovalev Faly of hypegeoec lae bea. Op. Soc

23 4. N. Fean. Kaplan an N. Davon Da opcal ap fo col ao n vance n aoc olecula an opcal phyc Eleve Scence pp l T. Ho an K. Dholaa o gung along Laguee-Gauan an Beel lgh bea ppl. Phy. B l an K. Dholaa Geneaon of hgh-oe Beel bea by ue of an axcon Op. Coun Lj. ancjevc an S. Topuo Dffacon of nonvegng Beel bea by fo-hape an eclnea gang n Poceeng of he Sxh nenaonal Confeence of he Balan Phycal non ecan nue of Phyc Vol pp Lj. ancjevc. Moe an M. onoa Dffacon popee of ccula an lnea one plae h apeo pofle of he phae laye Bullen e Socéé e phycen e la ep. Soc. e Maceone n Maceonan. 9. M. Bon an E. Wolf Pncple of Opc Cabge nvey Pe H. Baean an. Eely Hghe Tancenenal Funcon Naua Moco M. bao an.. Segun Hanboo of Maheacal Funcon Dove publ. nc D. oa C. T. La an G.. Salane. Popagaon ynac of opcal voce. Op. Soc.. B G. nebeou Opcal voce an he popagaon. Mo. Op