Ÿ W W ` ` [ Y ` ` b Y b b Y & X \ V! hz_3v =J_#"XYZ_?LZ[N <Mv ] LZv = G =V_?W = G WZ] G SV]b<ML6[ G LZLZ<;Sus_ I_ G UYR=V_?W WZ]b=J_?[\SV:xN B! hz_ik_

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2 Ÿ W W ` ` [ Y ` ` b Y b b Y & X \ V! hz_3v =J_#"XYZ_?LZ[N <Mv ] LZv = G =V_?W = G WZ] G SV]b<ML6[ G LZLZ<;Sus_ I_ G UYR=V_?W WZ]b=J_?[\SV:xN B! hz_ik_ G JYR=V_?I_?LOSV G =V_ s G J_?W <ML _\]bsvhz_\=%$'& (#) *,+ (-&/ <;=%&809:*;+=<>*;+=*;0'?@* BBA v _D s G J] [KdZ=V<;dE_=JSV] _\+<Mv : ]bamhts G =V_ WZ] U[?YZJJ_?W hz_=v_ ]bl <;=VWZ_\=`SV<v G I]b: ] G =J]FE?_`SVhZ_9=V_ G WZ_\= ]bsvhksvhz_?j_9[?<mlz[?_\dzsj B! hz?:b_?[\sj=v<mi G am_?losv]b[= G WZ] G SJ] <MLK<;sE_\NX SVhZ_HGJILKNMO*QP8PR*;SUTVIL9=&8.0'( hz_\=v_9sjhz_p) *;+rqs*,iut&8p &v9 w6<mv w G [?YZYZI WXZY [J\J]Û_a` WXdY bz\fe_ ` WJh [J\fi Y WJh bz\ji Y ` ` 4 mlo=7 G LZW SVhZ_ ) *;+rq &v9-9=&/ N&/9w6<Mv w G [\YZYZI! hz #"py G SV] <MLZH DB =J_?JYZ:bS`]bL SVhZ_u=V_?: G SJ] <MLZ Û_\fx@yzh@:iQ{U ~} ~ ƒ e_\fˆ lšˆœ x,žh@:i {V ~r} ƒ l hz] [h G =V_u<Mv SVhZ_uv <;=VI [J\fe_-Û_ Y ` W b \fe_-û_` Y <MvlSVhZ_9aM_?LZ_\= G :U G wo_9_d"xy G SV]b<ML [ Y Y 4 ^ \ hz_\=v_ ]b`sjhz_p)švia(#* a*qp/.n?œ&/9wœ=v<mi SVhZ]b v <M:b: <: SVhZ_9=J_?: G SV]b<ML ^ 4 ml Œ7 ž \ e_-û_ hz] [h&] Gz _?W&_ ZG [\Suw G :byz_ B Q S G : J<Kv <M: :b<= uv =V<MI SVhZ_9_?LZ_\=Va;N ª<: G : <MLZaSVhZ_9wO_?[SV<;= Œ Œ x, Ôƒ 4 ml x,7 «Y \ [ X Y b l Y SJhZ #"py G SV] <MLZ DB SVh G S [m Y G LZW SVhZ_ 4

3 Ÿ l ž Ÿ l \ l œ ] a B DB B yz:b_?[\sj=j<mi G amlz_sv] [= G WZ] G SV]b<ML B! hz_9wo_?[\sj<;=v G =J_9JhZ<: L ] L œ ]ba B D3B B QL G q *;$'& TVq SJhZ_ UNXI se<m:b G =V_u[h G LZaM_?W J< SJh G SŽÛ_ ^ Pue_ e G LZW ž B \ Ûe 4 ml Œ7! hz_p+=*qp/il9=&8 a* ) *,+rqs*,iut& P8&v9 w G LZW SVhZ_ +=*QP/IL9=&8 a* œ YR=JSVhZ_\= G LZW SVhZ_ +=*;<>+=Ia?@9=&/ a* ) *,+ qs&v9-9=&8 N&v9 w &80'$V*ŒK \ ^ Ž\ e \ ^ Û_ e e_ ^ -e ž ml ˆŒ7 4 ml Œ7 2 \ Ÿ ^ -e l 4 ml :iœ7 Q S]b`=J_ G WZ] :xn U_?_?L SVh G S`SVhZ_ )P/I0 * MOIQ a* 4 Y 7 \ [ _ { v YZ: :b_#"py G SV] <ML DB q, B F _\=J_9I G amlz]bsjyzwz_ <MvlSVhZ_ MOIQ a* *;? 9:.+! \ y 5 & \ y(' \ &! \ y#" $%! Y ]b?p 4 4 ml =@7

4 { { & X \ V z x _ y œ ] a B DB q B ~z: G LZ_ G wo_ B! hz_d: G LZ_ G wo_wz_?j[=v]bs_?w stn^_#"xy G SJ] <ML DB M [ G LZLZ<;SH_ ] esh] L^=V_ G :V <;=J: W^sE_?[ G YZU_ ]xs <MYZ: W se_ ]bl LZ]bSV_\:bN&:b<MLZa B QL <;=VWZ_\= SJ< WZ_ LZ_ G IK<;=J_3=V_ G : ] esv] [+I<pWZ_?: [?<MLZJ]bWZ_\= G MOIQ a*)'ia? a*œ9 SVh G S`dZ=V<X[\_?_?WZ G :b<mlza SVhZ_ [?<p<;=vwz] L G SJ_ Gr ] ]xsvh G (#) *;? 9:+rIaP$'&8(D9:+ &/tt 9=&/.0 [z4! 7P 4 4 7\ [z4! 7! 7 \! { JhZ<:L ] L œ ]ba B DB#"RB%$ NWZ_?LZ<;SV]bLZaSVhZ_ )švia(#*+stn & \' ] { [z4! 7 {! l SVhZ_u=V_?: G SJ] <MLsE_\S _?_\L SVhZ_ G wo_ G LZW SVhZ_9UdE_?[SJ= G :5WZ] USU=V]bsYRSV]b<ML [ G L se_u_ dz=v_\jj_?w G 4 ml : Œ7 4 ml -x,7 4 & 7 \ [z4! 7 \ y [z4! 7 )(! \+*-,:[z4! 7/. 4 & 7 { )( & \+* {, 4 & 7/. l 4 ml

5 \ \ \ { { { { _ E() E(-) = E() œ ] a B DB#"RB de_\[\sj= G :5WZ_?LZJ]xS N <Mv G G wo_9d G [VtO_\S B d! hz_h G wo_ud G [VtO_\S G LZW SVhZ_uUdE_\[\SJ= G :5WZ] esj=v]bsyrsj] <ML G =V_uIYRSJY G :1œ <MYR=V] _\=`SJ= G LZUv <;=VI B } <MLZJ] WZ_\= G G L _ ZG I d: _ SJhZ_9Ud_?[\SJ= G : WZ_\LZJ]bS N[?<MLZJ]bUSV]bLZa<Mv G se< G JhZ<:L ]bl œ ]ba B DB ZB! hz_ yzyz:b_\= v <;=VIYZ: G UhZ<= SVh G S`SVhZ_HSVhZ_H G wo_u[ G L s_ `=V]bSUSV_?L G 4 & 7\ \ \! hz_h G wo_\d G [VtO_\S [ G L se_ =J]bSJSV_\L G 4 & 7 \ L4! ]bsvhksvhz_9wz_ LZ]bSV]b<ML <Mv SVhZ_H(-& 0V? )(!! &!! &! \ %! &!! & &! & ]! & x & 4! ]! & ]! 7 < T'0'? 9=&/.10 & 4! 4"!7 \$#! &! & 7 ]! 2 4"!7! & 4! 7! & 4! 7 l! 4 ml Œ7 7 4 ml 4 ml :ˆŒ7

6 & X \ V E() 1 œ ] a B DB ZB pg I d: _ Ud_?[\SJ= G : WZ_\LZJ]bS N B ψ( ) 1 2 œ ] a B DB XB i G wo_u=j_?jyz:bsj] LZa+v =V<MI 2π/( 2-1 ) SVhZ_9 G I d: _ ede_?[\su= G :5WZ_?LZJ]bS N B G LZW SVhZ_Hdh G J_ & \' ] B! hz_ G wo_ ]b JhZ<: L ] L œ ] a B D3B XB

7 ! \ \ \ ' \ l ' l m V VR! hz_?@.1š *;+=*Q0V?Œ* P/*Q0a9=š ] SVhZ_ ]bwrsvh <Mv SJhZ_ G wo_9d G [VtO_\S?P O\ y ]! y & ] &! hz_?@.1š *;+=*Q0V?Œ* $'T +ril9=&/.10] SVhZ_ ]bwrsvh <MvlSVhZ_ G wo_ud G [to_s <ML SJhZ_uSV]bI_ Gr ]b?p ] ' 4 ml : Œ7 ] ' 4 mlš iœ7

8 [ [ & X \ V a # V } <MLZJ] WZ_\= S <]bl LZ]bSV_?:xN ]bwz_ 4 \ 7R G wo_9d G [VtO_\SV \ [ \ [ 4! 7 ( 4! 7 ( l 4 mlšq=7! hz_o& 09:*,+r<>*,+=*Q0'?@* <Mv SVhZ_ S < G wo_ d G [VtO_\SV [ G L se_ WZ_?U[\=V]bs_?W^a;= G dhz] [ G : :bn G JhZ<: L ] L œ ] a B DBRB! hz_&j]xsvy G SV] <ML JhZ<= L ] L SJhZ_ amyr=v_]b G J] I d: ] [ G SV]b<ML G ]bs <MLZ:bNjJhZ<= SVhZ_ SVhZ_ ] LOSV_\=Jv _\=V_?LZ[?_ <Mv S <I<MLZ<X[hR=V<MI G SV]x[ G wo_hd G [VtO_\SV?Pl bp 4! 7 \ [ 4! 7 \ [ 4! ]! 7 4! ]! 7! " #$H"% '& ( *)+Lƒ l 4 mlš@@7 Im E E 2 2 E 1 1 œ ] a B DBRB QLTSJ_\=Vv _\=V_\LZ[?_<MvlS < IK<MLZ<p[hR=V<MI G SV] [ G wo_? B Re

9 [ [ [ _ & l l l! hz_ LZ_D [?<MI3s] LZ_?W G wo_ud G [VtO_\S] \! hz_ G I d: ]xsvyzwz_ž[ [ G L se_9<;szs G ] LZ_\W v =V<MI1SVhZ_uv <;=VIYZ: G \ [z4! 7 )( l 4 mlš Œ7 [ \ \ \ 4 <;= v =V<MI œ ] a B D3BRB spn YZJ] LZa3SVhZ_ am_\lz_\= G : ] E?_?W ~ NXSVh G am<;= GR{ SVhZ_?<;=V_\I 7#4 7 4 mlš x,7 [ \ [ [ [! hz_9dh G J_9WZ] E_=V_?LZ[?_ s_\s _?_?L SVhZ_H G wo_? \! 4 G LZW =4! 4 & ] & 7 7 {! hz_ dh G J_! & <MvzSVhZ_ [?<MI s]blz_?w G wo_ G SuSVhZ_I<MI_?LOS DBRB G! & \! O\ ]!! ]b 4 7 l 4 mlš Œ7 [ [ \i!! [ G L&s_<;sZS G ]blz_?w&v =V<MIdœ ] a B 4 hz_ v =V_#"pYZ_?LZ[\N <Mv : ]bamhts] wo_\=jn hz] amh B! hz_\=v_?v <;=J_ <MLZ:bNSVhZ_H* *Q? 9=&/ * q *;Ia0<Mv SVhZ_9<MJ[?] :b: G SV]bLZa _?:b_?[\sj=v<mi G amlz_\sj] [ _?: W$P bp]bsvž& 09:*;0'(-&/9wK] YZJY G : :xn <;sj_=jwo_?w B! hz_]bltsv_\lzj]bs N ]b SVhZ_9dE<: _\= <MvlSJhZ_9= G WZ] G SV] <MLdE_\=YZLZ]bS G =V_ GRB Fƒ [z4! 7 { [z4! 7 { {% 4 mlš ˆŒ7 $ _?[ G YZJ_ { \! 4 mlš Œ7 SVhZ_9_#"XY G SV]b<ML [ G L se_ =J]bSJSV_\L v <;= G I<MLZ<X[hR=V<MI G SV]x[ G wo_ [z4! 7 \f[ 4! ]! 7~ G Fƒ [ :4! H] & 7 \ x@[ Fƒ 4! H] 7 \ [ l! hz] `: _ G WZ`SV< SVhZ_u[?_?LOSJ= G : <;sj_\=uw G SV]b<ML SVh G S 4 [ l 4 ml =7

10 ? r X u V V1! hpyz`svhz #"py G SV] <ML DB q [ G LKs_H`=V]bSJSJ_?L G m\ l 4 ml Œ@7! hz] `] SJhZ_us G J]b[ _#"py G SV] <MLK<Mv SVhZ_u] LOSV_\=Jv _\=V_?LZ[?_ B Q S:b_ G WZ SV<3SVhZ_uv <M: : <: ] LZa<;sU_\=Jw G SV] <MLZ?>! hz_ G wo_\ G LZW G =V_R?Œ.q )P8*Œ9:*;P/w?@.šV*,+r*;09 ]bv =V_?I G ]blzl[?<mlzes G LTS ]bl SV] I_ B QLSVh G S [ G U_SVhZ_+: G US SV_=VI <Mv _#"py G SV] <ML DB#" q, G :bj< =V_\I G ]blzh[?<mlzus G LOS] L SV] I_ B <;SV_;P hz<: _\wo_=p SVh G S`SVhZ_udh G J_ WZ] E_\=V_\LZ[?_ I G N w G =JN G G v YZLZ[SV] <ML <Mv de<mj]xsv] <ML B! hz_ G wo_? G LZW G =V_ & 0'?@.šV*,+r*;09 ] v`svhz_ dh G U_ WZ] _\=V_?LZ[\_ w G =V]b_?= G d] WZ:xN G G v YZLZ[\SJ] <ML <Mv SJ] I_ B Qv SJhZ_9SV]bIK_ G wo_= G am_ <Mv SVhZ_ [?<MU] LZ_HSV_\=VI ] \fi 4 Œ7 SVhZ_m G wo_? G LZW G =V_?Œ.q )P8*Œ9:*;P/wf& 0'?@.šV*,+r*;09 B QeL6SVhZ] u[ G J_ LZ< ]bltsv_=vv _\=V_?LZ[\_+[ G L s_ <;sj_\=uwo_?w B LZ_ G : G NX LZ_?_\WZŽ?@.1š *;+=*Q09 P & 1š9 (#.1T +?@*;( «G LZW «_?I]bSJSV]bLZap G wo_? G LZW ] L <;=JWZ_\= SV<+<;sU_\=JwO_ ]bltsv_=vv _\=V_?LZ[\_ BA L _ G I d: _ <Mv [?<MhZ_=V_?LOS G LZW ] LZ[?<MhZ_=V_?LOSH: ] amhos U<MYR=V[?_? G =J_ am]bwo_?l ] L œ ] a B DB o B ψ 1 a) b) ψ1 S 1 ψ 2 S 1 ψ2 S 2 S 2 œ ] a B DB o B ] amhosj<myr=v[\_? «] LZ[\<MhZ_\=V_?LOS B G LZW «> G 3[?<MhZ_=V_?LOS «]b+svhz_ I]b=J=V<;= ] I G am_k<mv «m s

11 + V? +=IaT'0'šV.a<>*;+ $V& +ria? 9=&/.10 +riat'0'š.<>*,+ $'& w G :b] WK<MLZ:bNv <;=`d: G LZ_H G wo_?\pl bpr] L [ G J_?hZ_=V_9SVhZ_uJ[\=J_?_?L G LZW SVhZ_ : ] amhos U<MYR=V[?_ G =V_ G S ]bl LZ]bSV_ WZ] US G LZ[\_ v =V<MI SVhZ_ WZ] = G [\SV]bLZa <;s _?[\S B! hz_ ]bi G am_\ <MvSVhZ_ :b] amhos J<MYR=V[\_ «_ G LZW SJhZ_9J[=V_?_?LPv <;=VI_?W stn SVhZ_u: _\LZJ_? _ G LZW G =V_u] L LZ]xSV_?:xN v G = G G N B f S 0 L 0 D L V œ ] a B DBRB œ= G YZhZ<Mv _\= WZ] = G [\SV]b<ML B «_;>+:b] amhos3j<myr=v[?_ _r> [?<MLZWZ_?LZU] LZa&:b_?LZ > WZ] = G [\SJ] LZa <;s _?[\S > [?<M:b: _?[SV] LZa : _?LZ > J[\=V_\_?L ' > v <X[ G : :b_?lza;svh1 +=*;(-0V*QP $'& +=Ia? 9=&8.0 +r*;(-0v*;p $'& +ria? 9=&8.0 ]bw G : ]bw v <;= G am_?lz_\= G : :b] amhosse_ G I ]blz[?: YZWZ]bLZa G UdhZ_\=V]b[ G : G wo_ B! hz]b ] I d:b] _? SVh G S SVhZ_:b] amhoshj<myr=v[?_ G LZW SVhZ_J[=V_?_?L6[ G L^sE_ G S LZ]xSV_WZ] US G LZ[?_v =V<MI SVhZ_WZ] = G [\SJ] LZa <;s _?[\S B S 0 D V œ ] a B DB m B œ=v_?jlz_?: WZ] = G [\SV]b<ML B «_;> :b] amhos U<MYR=V[?_ > WZ] = G [SV] LZa <;s _\[\S > J[=V_?_?L1

12 l?u r X u V V1 m & +ria? 9=&8.0 &80 I (-P8&/9]bWZ] J[\YZJJ_?W se_?:b<= ] L&SV_\=JIKu<Mv +=IaT'0'šV.a<>*,+ $'& +=Ia?@9=&/.0zP bp JYZI] LZaKSVh G S SVhZ_ :b] amhosj<myr=v[\_ G LZWSVhZ_ U[\=V_?_?L G =V_ G S ] L LZ]bSJ_ WZ] US G LZ[\_Kv =J<MI SVhZ_ WZ] = G [\SJ] LZa J:b]bS B QL SVhZ]b [ G J_HSVhZ_9WZ] = G [SV] <MLK[ G L s_9wz_\j[\=v]xse_?w stnkyzj] LZad: G LZ_ G wo_? B! hz_hd: G LZ_ G wo_9]b `=V]bSJSJ_?L G G [?<MI d:b_ v YZLZ[\SJ] <ML [J\ hz_\=v_9sjhz_ I G amlz]bsvyzwz_<mvlsvhz_ G wo_hwo_?[\sj<;= ] [ _! \ y 5 {% } G : [?YZ: G SV_uSJhZ_9:b] amhos]bltsv_\lzj]bs N G S d<m] LOS ]bl œ ] a B DB ƒ B z Slit width d 4 ml x,7 4 ml 0 z { dz ρ r θ r ρ P z siθ ρ = r - z siθ œ ] a B DB ƒ B D ] = G [\SV]b<ML ]bl <MLZ_9U: ]bs B! hz_ G wo \LTSV_=V] LZa SJhZ_J:b]bS ] LSJhZ_ ]bltsv_=jw G : h G G L G I d: ]xsvyzwz_r hz] [h+] dz=v<;d<;=jsv] <ML G : SV< B! hz_\=v_\v <;=V_ SVhZ_ G wo_hsvh G S_?LTSJ_\=V_?W SVhR=J<MYZaMh SVhZ_u] LOSV_\=Jw G :l[ G L se_ =V]xSJSV_?L G S d<m] LOS~.<ML SVhZ_9J[\=V_?_\L G [ \ {% 4 ml Œ7

13 \ \ \ # l hz_\=v_ u]b G dz=v<;de<;=usv] <ML G :b]bs N [?<p_\ +[?]b_?los G LZW + V? \ ] 2 l 4 ml hz_ J[=V_?_?L&]b G S G L6] L LZ]xSV_3WZ] US G LZ[\_ v =V<MI SJhZ_ J:b]bSu] I d:xnr] LZa3SVh G S B i hz_?l <;sj_\=jwx] LZa SVhZ]b G LZW [\<MI s] LZ]bLZaSVhZ_uS < _#"XY G SJ] <MLZ`<MLZ_ <;szs G ]blz [J\ { { ) l $ N] LOSV_?a;= G SJ] LZa < wo_\=`svhz_9j: ]xs ] WRSJh <MLZ_H ] :b:$j_?_usvh G S [J\! hz_\=v_?v <;=V_9SVhZ_9:b] amhos]bltsj_?lzj]bs NK] ~ {% { ~ {% ~ { ~ { {% { { ] ]! 4! 7 h! % {% l 4 4 Œ7! hz] `] I d: ]b_? SVh G S O\ [O[ \ -_h -_ \ \! \ y 5 4 ml xœiœ7 m\ #_ \fi l 4 ml xl=7! hz_ ] LOSV_?LZU]bS NK] E?_\=V< hz_?l \ 2Uy <;= G LZW G S`I Gr ]bi YZI w G : YZ_H hz_?l \ji3<;= \ ml x,@7 \ l 4 ml xœ Œ7

14 ? r X u V V1 Amplitude or Itesity 3π 2π π π 2π 3π α 3π 2π π π 2π 3π α œ ] a B DB M B = G dhz] [ G :5J<M:bYRSV] <ML <MvlSVhZ_9WZ] = G [\SV] <MLKd G SJSV_\=JL ] L <MLZ_ J:b]bS B

15 + V? & +ria? 9=&8.0 &80 (-P &v9=( ] WZ]bJ[?YZJJ_\WjsTN YZJ] LZaSVhZ_ œ= G YZLZhZ<Mv _\=Kv <;=VI G : ]bji B! hz] ] I d:b] _? SVh G S SVhZ_ ] LZ[\] WZ_?LOSH: ]bamhts[ G L se_ WZ_\J[\=V]xsE_?W G G d: G LZ_p G wo_ G LZW SVh G S SVhZ_ J[\=J_?_?L^] G S] L LZ]xSV_ WZ] es G LZ[?_ U< SVh G S`SVhZ_ G LZaM:b_ ] SVhZ_9 G I_ v <;= G :b:ese_ G I B z d 0 dz θ ρ z de wave D θ r œ ] a B DB q B D ] = G [\SV]b<ML ]bl SVhZ_ J:b]bS UNRUSJ_?I B! hz_ <;dzsv]b[ G :$WZ] US G LZ[?_ \ & \ l 4 ml x@x,7! hz_h G wo_usjh G S d G JU_? SVhR=J<MYZaMh G L ]bl LZ]bSV_?U] I G : d<;=jsv] <ML <Mv G J: ]xs]b [J\ { { ) l 4 ml

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19 $ ƒ! (1) θ (2) θ i i i θ d si θ d si i F /3& HG JI j% % Š(-/*Œ0 3,'4-/0&3& d

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21 H! $ ƒ! "$#&% /0:-%(')+%('2% 0&Š % Š *Œ0&5&/-2/*Œ0 *Œ)h4 3,'4-/0&3 /< 3Œ/H8:% 0 _ % 4-2/*Œ0 % 0&'% <2<2/*Œ0 6a/-#7'% <ˆ!% Š(-j-2* 8 *Œ0&%;*, &-4,/0&< 4,0&5 -#1 &<a-2#&%!&#?am-?) =! "$#&%!? A -?) ="! /< ] z} ] } ] 8 ] } ] 8 x ] z } G BQ *- 5&/#"!%('% 0:-/4-/0&3 -#&/< > x ƒ Œ²: ƒ v *Œ0-2#&% )+*Š94,ž! ž4,0&% *Œ)h4 Š(*Œžž/Ÿ74-/0&3ž% 0&< *,'$Ÿ /H'2'*,' *Œ)q)+*_Š94,ž!ž% 0&3,-# ] ] 8 > ] } ] 8 ƒ Ž G θ θ i N F /3& HG JI1 - ži4. %(5 5&/#"\'4,Š(-/*Œ0 3,'4-2/0&3& F "$#&% 5&/#"!'4,Š(-/*Œ04-2-2%('0 *Œ) -#&% 9 *,'25&%(' /< <2#&*6a0 /0 /3& G JI! "u#&% /0:-% 0&<ˆ/- /< 3Œ/8:% 0 1 -#&%.% 0&'%(<2<2/*Œ0 > z2{ u T«ª z2{ ƒ 9: "$#&% 4,0&3Œž% #&%('%./<u5&%( 0&% 5 4,< > 2 ] 8 z2{ j} ` z2{ aw ƒ, 1

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