Catalogação na fonte Universidade Federal de Alagoas Biblioteca Central Divisão de Tratamento Técnico

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U F I W A F C S G 2011 S W C S G ! O : "# # $ C &O : "# # 2011 % & ' Catalogação na fonte Universidade Federal de Alagoas Biblioteca Central Divisão de Tratamento Técnico Bibliotecária Responsável: Helena Cristina Pimentel do Vale S586c Silva, Willamys Cristiano Soares. Geração e Caracterização de feixes possuindo momento angular orbital / Willamys Cristiano Soares Silva. – 2011. 69 f.: il. Orientador: Jandir Muguel Hickmann. Co-Orientador: Dilson Pereira Caetano. Tese (doutorado em Física da Matéria Condensada) – Universidade Federal de Alagoas. Instituto de Física. Maceió, 2011. Bibliografia. f. 67-69. 1. Difração. 2. Feixes Laguerre-gauss. 3. Feixes Bessel. 4. Momento Angular orbital. I. Título. CDU: 535.4 • • • ! # $ % # + ' & ( # & ,- . / • ) * ( 2 $ 3 4 # 5 • 7 * * # $ • • & $ • • 1 " 0 3* (5 ! 6( $ 7 # $ 8 * * 9 , # # - * *, "; 8 7 , ,# 9 ) : * 8 * 7 8 9 ' , 8 # * : 8 * ( 8 8 # l , # , - "; ( 8 ) 5* 9 4 $ # < < < $ # * < $ $ . $ $ # )+ ) # < $ - = * $ * l < $ < $ # 8 $ = $ $ . $ $ - = "; = $ = $ # # # $ $ "; #+ " +* $ > $ $ ) # .# ?@ * A p = 0 #A p = 1 p Ap=2 ?? l = 1 #A l = 2 Al=3 "; Ap=3 ?B "; z=0 e) HG > a) HG ?F 9 C@ 8 ) A l = 1 #A l = 2 Al=4 C@ C? * - "; C? 9 A l = ±1 #A l = ±2 8 CC c) ?D 8 ) 8 b) HG f ) HG ?E Al=3 A p=0 ?B "; d) HG l =2 Al=4 ?C HG z = 0 - "; A l = ±3 CF Al=1 #A l = 2 * CB CE 8 ) CG a) l = 1 CF H( 8 CD b) l = 4 γ = 0 a) CG ) l=1 b) EI l=4 8 ) CB E@ 8 - . * "; > A l = 1 #A l = 2 CJ Al=3 E? 8 ) " . * > A l = 1 #A l = 2 CD 8 A l = 1 #A l = 0 C @I A l = −1 8 - A l = 2 #A l = 3 Al=4 A l = 3 E? "; EC "; EE C @@ # 4 EF C @? 4 8 #A l = 4 "; N =5 9 > A l = −1 Al=9 C @C 4 8 #A l = 1 "; N =6 9 > A l = −5 Al=7 E@ ( # EG $ EB *, . # * " F@ E? : ( $ 8 # - "; ( EC 7 9 , * $ A l = 1 #A l = 2 .# EC * # : A l = 3 A l = 4 # $ l+1 A'# . F? #A - $ " # 8 # 0 FC EE 9 A $ . : 9 #A A # 0 FF l=3 EF 8 - "; l=3 FG EG ' ( $ 8 - "; # , ( E? 3 EB : ( A l = −1 #A l = −2 ( $ " 8 ) EC . # : A l = −3 A l = −4 , # 3 # " ( E? ( A l = 1 #A l = 2 FB EC . Al=3 Al=4 : FJ EJ : ( # $ 8 . l = 10 #A l = 11 - "; : A l = 12 A A l = 13 . / 0 # * GI ED ! E @I # 8 M # A l = 0 #A l = 1 E @@ 2 G@ 8 - Al=2 8 "; G? # 8 E @? 2 $ M - "; 8 A l = 1 #A l = 2 Al=3 A l = 4 GC # 8 - "; A l = −2 #A l = −3 GC ! " # # $ % & ' ?@ 4 @G ?? ' : 8< , ?C 5 # ?C@ 9 @D # ?C? ( 8 - ?CC ( 8 ) ?@ "; ?F CI ?E C? ( " ! & ! ) * + (, C@ 4 CE C? ; ( 8 - CC ; ( 8 ) CE 9 "; CF CB ( 8 .# CF E@ EB , - ! + . ) * / ,0 E@ 4 ED E? 1 @B # FI E C '8 GI EE GE ! '1 ) 2 '0 / ! " . 9 $ * , 5 & ! , 8 # , * *" XX 8 8< 5* " 8 K " , L@M . 9 9 3 , 5 . 5 $ 3 N N 9 9 "$ 3 " ' 9 # * O 3 N , 9 # , 9 # , 0 * " ' @DID 5 8 + 6 * * L?M 8 , 5 9 9 # @DCG ) $ LCM # 8 9 ' $ 3 9 λ, * * 9 8 8 # @? * 9 λ/2π ) * 9 λ/2 / ! " ( # 9 "$ 3 N P 9 8 # N * 5 ' @DD? # P # 8 # 9 φ, exp(ilφ) 9 ' 5 l, Q $ , 8 9 $ 9 , LEM 8 ! 8 * $ , , 7 LF GM * : LBM 9, , # # LJM 8 $ 9 9 8 LD @IM 6 5 "; # ' : 5* - 8 "; LEM / " 8 . 9 9 8 * 3 " 9 "; R + # 8 - 8 * 9, 8 , ! ' * * 8 L@@M 8 ! 8 * : / ! " , 8 $ L@?M * L@CM * N * * ' @DJB # 8 5 $ ! * $ 3 # 8 8 ) N 6 8 L@EM / * : , ! ) 8 : * * 8 ) , 9 ! 5 0 L@FM , 8 0 , 8 ) ) * 9 θ # 9 8 0 , # 5* 7 * ' 0 8 5 8 # L@GM ' , 5* " : 8 * ) 8 ) 5 ) * L@BS@DM . , 8 , . 9 # *9 8 U T 9 # " 9 # 5* # * 9 R L?IM *, 8 L?@S?CM 9 8 L?EM 5 *, * * / ! " 1 # $ 5 7 9 1 * 8 - "; 8 5 7 5 ) # 7 5 , 2 8 # # 5 $ , 9 #, 8 , 9 5 - $ "; * 3 8 3 ) / " 5 , # 8 9 8 . , 5 8 8 # 3 4 - 0 " "; 8 ) 3 # , # $ 8 8 , * * " # P 8 # 7 5 4 5 9 #/ $ % #/ & ! 7 5 8 9 . 9 * 9 > H # # , * Vl , Q ) 3 8 # # . A # A O exp(ilφ) 93* 8 V 8 5 , l 9 5 # '8 8 8 - "; , 8 8 5 , $ 9 5 4 : , 7 5 #! * 9 # 9 9 8 # : 9 # 8 * - # 8 * l l "; 5 5 5 8 8 exp(ilφ) 6 3 8< , : $ 8 ) L@E @FM @G 8 $ 9 ' #/# 34 #/# 5 $ * 34 + 5 6 7 $ * + 6 7 0 3 : 8< , L@M ' : ! * 4 6 : , : , 8< *3 > ∂ ∂t V? @A = − ∇× = ∇ = ρ ε V? CA ∇ = 0, V? EA ∂ ∂t +ε , V? ?A , ρ * * : , : , , # * ! * * 8 8< : * 6 & # φ * , > =∇× 6 , 9 * , * *3 * * , ε , 7 & ∇× * # , V? @A V? FA . > #/# 34 5 $ * + + ∇× . ∂ ∂t 6 8 V? GA = 0. V? GA > φ ∂ . ∂t = − ∇φ − & # : V? FA 1 6 V? BA ∇×∇× ∇×∇× −∇ # V? ?A +ε ∂ ∂t = ∇(∇ ∇φ+ > ∂ ∂t V? BA )+ε > ∇ ∂φ = ∂t . 8 > V? CA ε ∇ −∇φ − ∂ ∂t V? JA = . # )−∇ ∂ + ∇(∇ ∂t +ε V? BA V? DA = ρ, V? @IA ρ . ε V? @@A > ∇ φ+ 9 ∂ (∇ ∂t : )=− 8< N : V? DA V? @@A " : # - 9 > ∇ c= √ ǫ , * 9 + *3 1 ∂φ = 0, c ∂t V? @?A : V? DA V? @@A #/( $ 9 & 0 ρ 1∂ φ =− c ∂ t ǫ V? @CA > ∇ φ− : V? @CA : #/( 1∂ c ∂t − ∇ =− V? @EA . V? @EA ! V? @?A * N : 8< $ ! *3 9 & 7 , # " : 5 # 7 *3 , , , L@M ! > × =ε V? @FA . 3 m *, * > m= . × = ε ×( × , # 6 V? @GA ). N 9 * × = × (∇ × × × =( )= j * > (Ej ∇Aj ) − ( ) −( ) j ∇) , V? @BA #/( $ x, y 9 * z & & # V? @BA = ε × m = ε j 6 9 = ∇) j : V? @GA (Ej ∇Aj ) − ( j # > ∇) [Ej ( × ∇)Aj ] − ×( * ×( #: V? @JA . ∇) > [∇j (Ej × )] − × ( ∇) = (∇ )+ ×( V? @DA ∇) V? ?IA . 8< * = 0 ∇ V? @DA , 8 m > =ε j ' # [Ej ( × ∇)Aj ] − # [∇j (Ej × j : V? @DA V? ?@A )] + × V? ?IA V? @JA . , # ! > m =ε j [Ej ( × ∇)Aj ] dv − j [∇j (Ej × )] dv+ * # [∇j (Ej × & ( × , )] dv = s Ej ( × . V? ??A )dv > V? ?CA )dsj . r →∞ 5 #/(/ $ # , , ( , > = ε m = m = ε = ε # )dv + ε [Ej ( × ∇)Aj ] dv j V? ?EA + ( × 5 )dv [Ej ( × ∇)Aj ] dv j ! ( × * V? ?EA , 5 * , 3 3 # 9 9 ! , 9 $ ×▽ 0 O 0 3 # 5 9 # , #/(/ ' $ @DD? # LEM 9 8 # exp(ilφ) l, 8 * 9 5 l 9 3 $ 9 ! > V? ?FA ( ,t) = u( ) exp [i(kz − ωt)] x, , * 3 9 8 x k= π λ , * λ, #/(/ $ ## 8 ω, * P # u( ) , 8 8 8 ∂u ∂z 8 , , 8 ∂ u ∂ z k ∂u ∂z 8 ku , > ∇t u + 2ik ∇t = ∂ ∂x + , ∂ ∂y - ∂u = 0, ∂z V? ?GA * , * , 1.2 *3 # 8 #V A , 9N # #, P > ∂u ( ) = 0, ∂x = 0 =⇒ ∇ 9 " V? ?BA $ V? BA , N 8 =− ∂ ∂t V? ?JA W V? @GA W 8 * × ! > Re ( × * ⋆ )= 1 ( 2 ⋆ × V? ?FA + × 8 ⋆ ). V? ?DA : , #/(/ $ #( , > ∂ = iωu exp [i (kz − ωt)] x ∂t ∂u ∂u = ∇× = + iku exp [i (kz − ωt)] y − exp [i (kz − ωt)] z. ∂z ∂y V? CIA = − ⋆ ∂u ∂x = −iωu⋆ × V? C@A ∂u ∂u y+ z + ωk |u| z, ∂y ∂z V? C?A > =0 ⋆ × V? CCA = −iωu⋆ ∇u + ωk |u| z, * × ⋆ V? CEA = iωu∇u⋆ + ωk |u| z. , , , " > = Re [ε ⋆ × 7 ]= ε [ 2 ⋆ × + × ⋆ ]= iωε (u∇u⋆ − u⋆ ∇u) + ε ωk |u| z. 2 V? CFA , 8 * # 8 u ! > V? CGA u(r, φ, z) = u (r, z) exp(ilφ), 8 7 # * V? CGA * 8 8 , 5 V? ?BA 8 9 l, 9 , 8 - 8 : 5 8< L?GM # 8 8 #/(/ $ 6 #, 9 8 8 9 ∂u ∂z , ku , 8 = iωε 2 > * r φ z ∂u⋆ ∂u − u⋆ ∂r ∂r u r+ lε ω |u | φ + ε ωk |u | z, r 3 , 5 * ! + V? CBA 7 , * c " $ r 3 8 φ, 3* # " z z , =− > iωε lε ω z |u | r + r 2 u 4 ∂u⋆ ∂u − u⋆ ∂r ∂r z + 2ikr |u | 8 φ + lε ω |u | z. V? CJA 3 , 8 z 3 & # * 8 , u = cpz = ε ω |u | 9 8 , * L?BM # 8 , dv z U = = u dv > lε ω |u | dv ε ω |u | dv = l . ω V? CDA #/(/# * " #1 # ,> lε ω |u | dv dv z = = z 8 , dv = ε ωk |u | dv λ l =l . k 2π 9 * & 8 # N ,N ω R Nl # ! U 8 ! V? CDA l ω 8 exp(ilφ) 0 # 3 0 8 " * # 8 9 z # exp(ilφ) 8 - #/(/# 6 "; * "; * 8 ' , "; , 8 # 5 3* ρ $ " φ L?JM > upl (r, φ, z) ∝ (−1) ×Llp V? ?GA * , p : # (LG) : - l " ! - , 9 8 U 8 V? EIA √ r 2 w(z) l exp − kr r exp −i exp(ilφ) w (z) 2R(z) z 2r exp −i(2p + l + 1) arctan w (z) zr , V? E@A #/(/# * " #' w(z) = w * , 5 8 9 5 p Q - zr z V? ECA 8 * z w 2, 2 + (2p + l + 1) arctan $ Llp , , z z ; , l *, , * p=0 + . (p + 1) 3 # "; K N = 2p + |l| l , 9 2π , V? E?A R(z) = z 1 + zr = kw - z zr 1+ ( l=0 8 - Q V? @A "; $ N 8 * * l * ( V? ?A z #/(/# ( * * " ? @> p A p = 0 #A p = 1 ( > ( ? ?> l = 1 #A l = 2 ( > #7 Ap=2 Ap=3 "; z = 0 l = 2 V?I@@A Al=3 V?I@@A Al=4 "; z=0 p=0 A #/(/# * . " - #8 "; 9" l=0 $ 9 8 ! ,# 3 5 l, # $ 8 # - * - # "; 9 l LEM " LGlp HGmn L?DM . 5 ; # . 3 9 3 . , 8 5 (m, n) p = min(m, n) HGmn , : 8 8 (p, l) l = |m − n| N = m+n . HGmn LGlp 9 # . "; > √ x √ y x +y Hm Hn exp(ikz) 2 2 w (z) w(z) w(z) z k(x + y ) exp −i(m + n + 1) arctan × exp −i , V? EEA 2R(z) zr Emn (x, y, z) ∝ exp − Hn (ξ) , K "; n . ( V? CA #/(/# * " ( ? C> d) HG e) HG ( > #0 "; "; - : "; 8 : # 8 # # - * K "* ,# 1 LG± = √ (HG 2 " "; 8 V? EFA ± HG , ) . # * L?DM 9 , , # # "; 8 c) HG ! 9 8 K O b) HG V?I@@A . - > a) HG f) HG ! ! 3 * 0 #/(/( * ; (: 2 + *9 5* ' , 0 $ 8 8 * θ = λ πw "; - LCIM 7 "; 8 5 8 $ 8 #/(/( . " * * ; K $ $3 : 9 , L@EM ' 8 , 9 : * : $ ! $ $ l, * # $ 5 5 9 5 # # * 8 ) $ * @DJB $ *9 8 9 9 ) 8 > V? EGA E(ρ, φ, z) = E Jl (αρ) exp(ilφ) exp(iβz), E , Jl , * l α * 9 . ) k= 8 , ρ φ z * $ # # α +β " β 9 8 > I(ρ, φ, z > 0) = I(ρ, φ). V? EBA #/(/( * ; ( 4 8 * * 8 *3 7 8 ) * ' # 9 8 # 7 , 0 ) ) 5* 8 ) * L@F CJM 8 ) 9 8 ) *! ( , 9 VC @A ( ( > . 8 7 8 , , 8 , , ? E> 8 ) 38 8 , (l = 0) ! ( V? FA 9 V?I@@A ) $ , # ) 8 0 8 . 8 5 L@GM l ( V? FA . 8 #/, ! (# ( ? F> Al=3 Al=4 ( > A l = 1 #A l = 2 V?I@@A #/, 6 8 ) ! 9 , * 9 8 # > 9 8 H 9 8 9 # l 9 8 exp(ilφ) # 8 9 : # # 8 9 5 N 5 * $ 3 Q "$ 3 9 * #/, ! 7 (( 5 8 8 # 5 "; # ) * 7 l 8 # 8 9 8 (/ " ! & ! ) * + (/ ! 8 9 # 3 9 , 8 8" , * ' 8 # * 3* LBM : *3 8 , 8 * : : 0 LC@M L@C C?M 8 , 9 # 8 8 * 8 , # # , 8 CE 8 0 (/# " ! (/# 6 * " " ! (1 * , " # - ' "; $ 8 , , $ * " 8 $ 8 5 8 9 .# $ - ( 8 ; "; l . , VC @A ! - "; ! LCCM l# : # l ' : * ( VC ?A $ ( ( 8 ( VC @AV A C @> - > 8 "; V?I@@A - "; * A l = ±1 #A l = ±2 l=1 l=2 9 A l = ±3 * (/# " ! ( C ?> ( > . , * " (' 8 - "; Al=1 #A l = 2 V?I@@A $ 3 # , ,# 6 - * , ! , "; * 9 # 8 $ 8 ,# 8 3 , 8 3 3 8 * - , "; f ; * , $ , $ * * 9 * 9 , 9 5 , 9 LG −l 3 λ/2 ( * 9 2f LG π ' "; 5 , l, " - 9 O 5 * π/2 ' λ/4 9 LCBM & √ 2f * 3 * 5 "; ! 9 3 9 "; 5 $ VC CA * (/( " ! * 5 - ; (7 ' # * "; ! "; . "; - ," "; 9 ( ( > (/( C C> * V?IIJA " ! * ; 8 ) # , Q , 3 , 8 ) 7 ) 3 " 9 8 8 38 V 8 8 : ) L@FM 9 # 5* K A LCDM $ LCJM * , (/( " ! * 8 ; ) (8 *, $ ) 7 8 8 # : 5 $ 9 Q . R, 2π ρ ρ 9 $ φ ! LCJM> t(ρ, φ) = exp i 2πνρ cos(φ) + lφ − VC @A * 0 exp [i (2πνρ cos(φ))] 0 $ 0 $ γ 9 ν = $ ρ , ( π ρ VC EA $ , P 8 8 ) $ z , = ρ R λ γ P λ λ, K exp −i ρπ ρ 0 : 8 ) $ ρ # . 5 8 ) * ) L@FM 8 ) l =1 l =4 6 (/( " ( ) ! * ; (0 C E> 8 a) l = 1 ( > ) H( 8 b) l = 4 V?I@@A * γ = 0 $ $ ( VC FA 3 Q , 5 l 8 ) (/( " ! ( * ; ,: C F> ( > ( γ = 0 a) l=1 l=4 V?I@@A VC GA 8 ) $ ρ = 1mm R = 10mm 8 b) l=1 0 * , 8 19, 4m 9 > ν = 6000m− (/, & ( ( ! * $ C G> > % , 8 ) V?I@@A (/, & ! * $ % W 5 8 8 8 , # 7 , 9 # : #, 6 # * 3 , , $ ' *9 K *" 9 8 8 , 9 8 3 , ' ! ( * O VC BA 8 Q " 8 # : 9 " 5 , ( VC JA (/, & ( ! * $ C B> . * ( ( > 8 "; > A l = 1 #A l = 2 C J> Al=3 8 ) > A l = 1 #A l = 2 . * > Al=3 V?I@@A 7 * ,# V?I@@A " ( % 5 5 9 , # 2 9 9 8 4 &9 2 2 # 8 L?IM 7 8 *, , , , * 5* " 5 8 (/, & ! * $ % ,( T 8 - "; , I(x, y) ∝ cos λ, 8 # * ( ( C D> A l = 1 #A l = 0 > VC DA 0 "; VC ?A 8 - "; A l = −1 V?I@@A O # d, * 8 - , VC ?A , a, ∆φ(y) , . ( πax ∆φ(y) + λd 2 , l = 0 $3 * ( ! VC DA V A ! ( VC DA V A * ! , # , 8 - , y 4 &9 "; # , 2 2 (/, & ! * VC ?A * ( ( C @I> A l = 2 #A l = 3 ( > $ % ,, : l>1 VC @IA 8 - "; Al=4 V?I@@A * 5* # ' 7 , 5 * * # $ ) $ ) ! # # L?@M * 8 K V $ 4A # 8 8 $ 7 $ * : 8 *, 8 - 4 . "; 4 , (/, & ( $ ! * $ N− ∝ exp (−ilαn ) exp i n k = 2π/λ , , * ka (xcosαn + ysinαn ) z 9 VC CA αn = 2πn/N , a, N , 9 Q ( : 4 8 VC @@A V A V#A V A "; 5 ( N =5 l = 0, 1, 2 # ,1 , IlN ( % N VC CA 7 l VC @@A V A V A VA N =6 * * 4 8 $ # 9 , 5* 8 ( ( > C @@> # VC @@A 4 V?I@@A 4 (/, & l ! * p, l + pN $ Q % N , Q N− IlN pN ∝ exp [−i(l + pN)αn ] exp i n 4 ka (xcosαn + ysinαn ) z N− IlN pN ∝ 7 exp (−ilαn ) exp(−ipNαn ) exp i n VC EA exp(−ipNαn ) = 1 ! # 4 4 l = 1 8 - N = 5 4 l = −5, 1, 7 7 ( # l=9 > V?I@@A ( VC @?A VC @CA : 8 - N =6 " * C @?> 8 - " 8 l´ = ... − 11, −5, 7, 13, ... 4 ( IlN = IlN pN N = 6 8 VC EA "; 8 l = l + pN ( $ # ; ; ka (xcosαn + ysinαn ) z 9 8 - l ,' "; 9 > 4 N =5 A l = −1 #A l = 4 A (/1 ! ( ,7 C @C> 8 - "; 9 > l=7 ( > 4 N =6 A l = −5 #A l = 1 A V?I@@A R 3 # * : 4 8 8 , "; , * 9 # P * * L?@M # (/1 5* 9 8 ! 7 5 , 8 - "; , $ # 8 3 # * $ , # ! 3 / 5 - * , 9 * ( #, 9 "; 8 , 5 " # "; , , ) *3 , 8 9 " , 5 8 (/1 ! ,8 T , K 9 4 #5 P # T 4 # , 9 # P 8 9 7 8 , 5 , 5* 5* ,/ - ! + . ) * / ,/ ! 7 5 , # # 8 ( $ 8 8 , 7 5 8 8 $ 7 # # * * N # 8 # , * 3* 8 # ED ,/# - ) ,/# ! + 1: -) ! + 8 - 8 , "; 5 ! z=0 > √ r 2 w U (r, φ, z = 0) = w , , p=0 5 8 r= 9 l exp − r w VE @A exp(ilφ) x +y , φ = tan− l, 8 y x # τ (x , y ) 8 ( $ V # E(x, y, z) = 9 VE ?A kx = kz x * VE ?A ky = kz y kx * VE ?A # VE CA U (x , y , z = 0)τ (x , y ) exp [−i (kx x + ky y )] dx dy . VE CA !3 8 " # z ( τ (x , y ) " LCIM> : * , # "; −i x +y U (x , y , z = 0)τ (x , y ) exp ik z + λz 2z ik × exp − (xx + yy ) dx dy z ky 7 8 - , # E(kx , ky ) = A * # U (x, y, z = 0) 7 " * * 5 O 8 , ,/# - ) ! + 1 VE CA ( * # # LCEM ( VE @A # 0 ( # 3 E @> ( $ *, . ( > # * V?I@@A ' * # $ 2 * VE CA $ # VE @A * V # # 0 3 : ( A : ( l " , $ ( VE ?A ,/# - ) ( ! + E ?> 1# : ( $ ( EC 7 * $ . : A l = 4 .# * # 9 l = 1 #A l = 2 # $ ( > Al=3 # .# #, , - "; , A l+1 V?I@@A 7 l 8 30 : 8# * $ Q 38 $ 3 , # # * ,/# - ) ( ! + 1( VE ?AV A/ . : ( R# VE ?A 8 " $ 8 9 # # * # E C> 9 # A '# # #A - $ 8 # 0 ( 6 , 9 0 ( # > " # V?I@@A & 8 8 # # # 0 / ( . VE CAV A 3 0 #  φ(y) = sin−  y a +y  , VE EA ,/# - ) ! + a, 1, $ /+ 0 '8 , φ(y) 9 √ 2 3y φ(y) ≈ . a 8 # VE FA 0 3" * $ 9 8 √ 3 U ∼δ x − a exp 6 x 8 √ i2 3ly a VE GA # y & # VE GA VE CA √ i2 3ly a √ 3 a exp δ x − 6 E(kx, ky ) = * ( exp [−i (kx x + ky y )] dx dy . 9 # ( * kx ky VE JA , √ 3a exp −i kx , 6 √ 2 3l E(kx, ky ) = δ ky − a ( , . 9 , 9 * Nl , * $ , A # N VE GA ( y ! VE JA * 90 * ky V VE BA 9 " #, 0 $ " ,/# - ) ! + 11 $ # 0 5 ( : # # ! l 3 ( 0 VE EA # 0 $ ( * , * $ 0 " : * : LCFM . , ( ( A VE FA E E> $ 9 # ( > 9 #A l=3 V?I@@A A 0 . : ,/# - ) ( ! + 1' E F> ( 8 - > "; l = 3 V?I@@A 7 # $ # : & 8 ( 3 √ 2 3l E(kx , ky ) = δ ky + a √ 3a exp −i kx , 6 * * # 3 ( VE GA * VE DA 8 ky " 180 9 * ,/# - ) ( ( ! + 17 E G> ' $ 8 - , ( EC . A l = −4 ( > "; ( E? # # 3 A l = −1 #A l = −2 A l = −3 : V?I@@A 8 ) z=0 ! # 8 , > U (ρ, φ, z = 0) = E Jl (αρ) exp(ilφ) E , Jl , ) l α, VE @IA * ,/# - ) ! + 18 * ρ φ & # VE @IA # 9 VE CA 8 * " : : 8 ) # 8 - "; # ' , ( 8 E B> ( $ 8 > V?I@@A * ) , ( VE BA< 8 : 3 A l = 1 #A l = 2 : 5 $ ( * Al=3 ( Al=4 EC . ( E? : # # ,/# - ) ! + 10 6 # Q * : 8 38 * * 8 $ ( 3 l VE BA Q * Q 8 8 ) VE BAV#A Q # l=2 * * 38 0 8 , * N , l = N −1 38 ' 3 , * 38 , ( # * l =3−1=2 * l > 11 ( VE JA * ,N =3 ,/( 3*+ ': ( E J> : ( # . : l = 11 A l = 12 A l = 13 . / 0 ( ,/( > 8 - "; A l = 10 #A # * V?I@@A 3*+ 7 * 8 8 ( $ VE DA # , . 8 , ,/( 3*+ ' ( ( > E D> ! 8 M $ M V?I@@A 6 K λ = 514.5nm 8 # 10mW . 8 9 # $ LCGM . $ 9 8 - "; " *, 0 5 6 3 # # # * f = 30cm # : : CCD V ( A ED@I & *, 4X Y4"@EIJ $ 1, 75mm 6 0 8 7 4 * $ ' * 4 < O - #W4'Z ,/( 3*+ '# 8 6.1 ( , VE @IAV A # 3 E @I> # A l = 0 #A l = 1 ( > , * # 8 - "; Al=2 ( VE @@A 0 VE @@AV A 0 8 # 8 8 7 V?I@@A . ( # # 0 ( # " # 9 * ( VE @?A , # 8 4 4◦ $ # * , " 7 # 8 # ,/( 3*+ ( '( E @@> 2 8 ( ( > E @?> 2 8 ( > - 8 "; # A l = 1 #A l = 2 Al=3 V?I@@A - V?I@@A 8 "; A l = −2 # #A l = −3 Al=4 ,/, ! ', 6 # * 8 # 8 # 7 , # , 3 * 8 # $ 8 * # $ - ( "; # VE @IAV#A l = 1 l = 2 * ,/, ! 7 5 8 8 # # *, . , # * * 8 # # * * * 5* " , * Q Q 0 l ' 8 , 38 8 *, # , 8 , V A , 1/ ! 4 * W 8 9 9 5 8 # * # 8 9 5 - 8 "; 8 8 9 ) : 8 $ 5 8 " - 9 * "; 8 ) *3 ' , ; 8 8 ) 8 " * , 8 L?IM ) $ # , , , L?@M * * *" 9 # P ' &9 * * ) 8 " $ 9 &9 - , $ " # * 8 l , 8 l ± pN 7 # 0 GF 3 8 # .# * 9 $ * l Q l , , .# * $ 3 30 * l 38 * "$ 3 30 l # 3 Q *, , Q * l 38 l = N −1 N 38 * * , # # , . Q 8 38 0 *, # ( # * 8 * ' , * * * * 9 5 8 8 $ 8 LE@M GG * , " # ) 2 L@M ) C * L?M - LCM 2 ) $ LEM - Z + 7 O . 07 ?CD V?II@A L@JM W ; , " $3* 9 ; * Z & ## % $ 7 , 0 @EF V?II?A L@DM ; L?IM 4 &9 L?@M ; ; ) W ; , " $3* 9 2 2 $ * - (, ?D?B V?IIDA L?CM $ "&$ ; ) - "- L?EM ' - . Z ) ! L??M 4 % $ - . #8 GFB V?IICA DDD V?IIGA ( # $+ 2 * - - : < @IIJI@ V?IIJA + - "/ & ( - Z Z . & . & (, CGJG V?IIDA $3* 9" $+ 2 * :1< IFCDIE V?I@IA L?FM ; 2 ( < % + Z ) % * # 4 7

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