Capítulo 2 Deducci#n de la Fuerza de Casimir 2.1. Sistema Real En

!"# $&%(')% *,+(-.0/21 3 45/2376
:8 9;=<?>@;BADCFEHGI>@JHKMLNLPOH>Q;=9SRHTQ;=U&LP<VJS9SCW;YXZESTQ;(<Q[M\F9]E^C_T@CNK`CbacJS;(TQd=CeRS;fgCF<h[MU&[MT);=9i>@Th; RHLP<
ESK`CFA=CF<jU&;=RH[`CF9i>@;kJS9^Cl9HLnmF;=RSLF<@CoacLPTQUlJSK`CFA([M\P9prqtsuUlJHvxwF;=9S;(T@CFKvxyzJS;{9HLVTQ;=yzJS[I;=Th;{;=K;=U&ESKM;(L
RS;|U,L}RS;=KILP<g;(<QE;=A=G~^A=LF<k<hLPOSTQ;|KILP<€UxCt>@;=Th[`CFKI;=<y}JH;)A(LP9HacLPThUxCF9"KMCF<€ESKMCFADCF<(‚
fjLF9S<Q[IRS;=Th;=U&LP<€JS9"<Q[I<h>Q;=U,C S acLPThUxC_RSLlRS;ƒRHLP<€ESK`C_ADCF<gESCFT@CFKI;=KMCF< a = 1, 2 <Q;(E^CFT@C_R^CF<€JS9^C
RS[M<?>„CF9SA([`C L ELPTjJH9^CoADCDm}[MR^C_R V A=LPU&Lo<Q;…UoJS;=<?>@TQCo;=9"K`C7~^wPJSTQC"†ˆ‡}‚I‰nŠY‚H8:<?>„CF<€ESKMCFADC_<jESJS;(RS;=9
<Q;=T&;=9TQ;DC_KM[MRSCFR<Q[M<?>@;(UxCF<&U,‹F<ŒA=LPU&ESKM;ŽhLP<=:A(LPU&Lem_CFTh[`CF<ŒE;=KMGIA=JSKMCF< †JH9S[MR^C_<,L‘<Q;=ESCFT@CFRSCF<„ŠY
U,;(RS[MLP<“’SLFU,LPwF”=9S;(LP<L7[M9H’SLPU&LPwP”=9H;=LP<(z>QT@CF9H<QE^CFTh;=9i>@;(<jJ,LPE^CFA(LP<=}RS[M<h[ME^Ct>@[ImFLP<:Ll9HLSzKML}ADCFKI;=<“L
A=LP9•RH[M<QE;=Th<Q[I\P9•;=<hE^CFA([`CFK—–E;=ThL˜<Q[IU,EHKM;=U&;=9i>Q;)KMCF<kKIK`CFU,CFTh;=U&LP<€ESK`C_ADCF<(‚u™0Cx[I9z>Q;=T?aCFA=;)RS;lA=CFR^C
JS9^CŒRS;|K`CF<€ESKMCFADC_<€;=<h>@‹&A(LPKML}ADC_R^CV<QLPOSTh;|;=K;šh; z ;=9 z ‚^›€RS;(Ux‹_<=^;=K<h[M<?>@;=U,CV;=<œ[M9H~S9S[I>QLŒC&KIL
K`CFThwPL˜RS;(KESK`C_9SL xy ;(9W;=KA(J^CFKU,CF9i>@[I;=9S;oJH9^C"<Q[MU&;(>QTQGMC&>@TQCF<QKMCFA=[ILP9^CFK—‚žfjLP9S<Q[IRS;=Th;=U&LP<|CF’SLFT@C
JS9acLF>Q\P9 A=LP9EžLPKMCFTQ[IdDCFA([M\P9 α RS;(9z>QTQL,RS;lKMC&A=CŸm}[IR^CFR V yzJS;l[I9SA=[IRS;|<QLPOSTh;)JH9^C,RS;)K`CF<2ESKMCFADC_<
<h;oTQ;¡ ^;šQC&A=LP9bJS9^C˜CFU&ESKI[I>@JHR RH;lESThLPO^CFOS[IKM[IR^CFR r v<h;)>QT@CF9S<hU,[>@;|’^CFA([`C&;=K¢UxC_>Q;=Th[`CFKA(LP9
a
ESTQLPOSCFOS[MKI[MR^C_R T = 1 − |r | zRSLP9SRS;kESJS;(RS;ƒCFOS<hLPTQO;=Th<Q;k;¡XZA=[>„CF9SRHLlwPTQCFRSLP<£RS;œKM[MO;=T?>„CFR&;=KI;=AY¤
>@TQ\F9S[MA(LP<gv"m}[MOSTQCFA=[ILP9^CFKI;=<=HL&>QT@CF9S<hU,[>@[ITQ<Q;…Ux‹F<œCFKIK`‹oRS;7K`CVESKMCFADCVE;=ThRS[M”(9SRSLP<h;)E^C_T@CŒ<Q[M;(U,EHTQ;F‚
fjLPU,LlKMLP<“acLF>@LF9S;=<gEHJS;=RS;(9 CFOS<QLFTQO;=TQ<h;FS;(KuA=CFU,ELl;(KM;(A(>@ThLPU,CFwP9S”¡>@[MA(LlRS;…9}JH;=<h>QTQLŒ<Q[I<h>Q;=U,ClTQ;Y¤
a
α
a
α
a
α 2
a
‰n¥
¦¨§{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
1
‰ŸÀ
2
V
r2α
x
L
z
z1
z2
Á¨[MȚwPСJHÑ{T@ÆYCŒÚg‡ZÓn‚ÄωPÌr͗ÂHÒtÃÅןÔQÄÇÆ¡É0È?ןÆ(Ô:ÉÆYʈԚʈÛtËnÔ?Ìr͗ܟÊÎÌrÆYÝ(ÊˆÑ ÌÏÆ(É :Ô ÌrÄÏÒtɐ͗È?ÊÎСÆYÑ{ÑkɐàáÒtÐYÓ_͈СÔhÑnʐÕ_ÔQÈ?ÉÌrÔQʗÔ?ÉÛtÔ?ÔcÑ âÆ(ןÐ(ÉhÖ ãŸÆYËtɗɗСÔ?ʈÓtËnÆ¡ÌÇʈןÆ(Ð(×nÉ Æ¡ÉžÓF͈ʈСơÊ0ÑtÒnɐÚgÑ_ÆjÌr͗ÈhÌÇןÆ@Ð(ØDÉÌÏ×nÆ(ל؄Æ(ȚÙÇÆ
Ö
Ö
ÞFß
Þ
CFK£acLFTQU,CJS9ä<h[M<h>Q;=U,CbA=JS‹F9i>@[MA(LWCFOS[I;=Th>QLb;(9ä;=KjA(J^CFK;=Kj9zåSU&;=ThLNRS;˜E^C_Th>@GIA=JSKMCF<)v‘K`C ;=9S;(TQwPGMC
9SLB<QLP9eADCF9i>Q[MR^CFRH;=<|A=LP9H<Q;=T?m_CFR^CF<(‚0æ®LPT|KML˜>@CF9i>@LS9HL EHJS;=RS;&<Q;(T|RS;=<hA=TQ[>@LBA=JS‹F9i>@[MA=CFU&;=9i>@;Œ<Q[I9
[M9i>@ThLZRSJHA=[MT&;¡XZESKIGMA([I>„C_U,;(9z>Q;KILP<xwFT@CFRSLF<˜RS;•KI[MO;=Th>@CFRRH;•KMLP<,UxCt>@;=Th[`CFKI;=<&yzJS;NA=LP9ZacLPTQU,CF9çCFK
<Q[M<?>@;(UxCHKML,A=J^C_K¨TQ;(yzJS[M;(TQ;o’^C_A=;=T…U,L}RS;=KILP<œU,[IA=TQLF<QA=\FES[MA(LP<kRH;oKMLP<œU&[M<QU&LP<(‚uè}[M9•;(UoO^CFThwPLS;=9
;=yzJS[MKI[MOSTh[ML|>@;(TQU&LZRS[I9^‹FU&[MA(L7’SCFOSTQGMC7acLF>QLP9S;(<gESTQLŸmF;=9S[I;=9i>@;(<œRH;=Kžm_CFA(GML)Ux‹F<gCFKMKM‹7RS;…K`CF<ESK`CFA=CF<v
LF>@ThLP<2T@CFRS[MCFRSLP<2ELPT2;=K0U,C_>@;(TQ[`C_KU,[I<QU&LS‚^8:<h>QLP<…A(LPU&Ež;(9S<@CFThG`CF9 C&KMLP<€acLF>QLP9S;=<œ;=9 V E;=ThRS[MRSLF<
EžLPT“>@TQCF9S<hU,[I<Q[M\F9&v&ELPTjC_OS<QLPThA=[I\P9‚}8:<h>@LF<jacLF>@LF9S;=<jCFE^CFTh;=A=;(9˜A=LF9˜ESTQLFO^CFOS[IKM[MRSCFR T v,<h[M9,JS9^C
TQ;=KMCFA=[I\P9&RS;…aCF<h;…A=LF9˜KMLP<“acLF>QLP9S;=<Ež;(TQRS[IRSLP<(‚H8:<gELPT£;=<QLVyzJS;…<Q;…EHJS;=RS;ƒA(LP9SA(KMJS[IT£RH[MA=[I;=9SRHLlyzJS;
K`CF<2ESThLPES[M;(R^CFRS;(<kRH;)KMC&TQCFRS[`C_A=[M\F9 RH;=9i>@ThLxRS;7K`CŒADCDm}[MR^C_R V RH;=E;=9SRS;(9•åS9S[IADCFU&;=9i>Q;)RS;7K`CŒwP;Y¤
LPU,;¡>@ThG`C…RS;(KS<Q[I<h>Q;=U,CHPK`CkA=J^CFK^;(<h>@‹|A=CFT@C_A(>@;(TQ[IdDCFR^CƒELPTKMC{<Q;=ESCFT@CFA([M\P9 L ;=9i>@Th;œKMCF<ESKMCFADC_<€é2ê^ëtì
í ëŸîƒï„ëŸðcñgï¡òˆð¡ó^ôšð„îkõ}ðœì@ðcö2ðh÷Pòˆøtó r iKILP<:A=J^C_KM;=<“A=CFT@CFA¡>@;(TQ[Md=CF9ŒA=LPU&ESKI;(>„C_U,;(9z>Q;€K`CƒTQ;(<QESJH;=<h>@C)\PEZ>@[¤
ADC)RH;…K`C_<ESKMCFADC_<=‚}ùgCF<Q‹F9SRSLP<h;…;(9˜;=<?>„CF<€Ct~^TQU,CFA=[ILP9S;(<=i<Q;kESJS;(RS;{RS;(<@CFThTQLPKIK`CFT:JS9SC)acLPTQUlJSK`CFA([M\P9
E^CFT@C,K`C,acJS;(TQd=C˜RS;ŒfgCF<h[MU&[MTœ<Q[M9b’^CFA=;(Tƒ<QJSELP<h[MA=[ILP9S;(<…<QLFOSTQ;VKMC,A=LFU,ELP<Q[IA=[I\P9bRS;lK`CF<kESK`CFA=CF<=žL
<QJ TQ;=<hESJS;(<h>„CZ–Å>QLZRSCŒKMCo[M9HacLFTQU,CFA=[I\P9˜TQ;(KM;¡mFC_9z>Q;|<Q;|’^CFKIK`CoA=LF9z>Q;=9S[IR^CV;=9 r ‚
α = 1, 2
raα
z1
z2
V
α
a
α
a
α
a
¨¦ §{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
$&%Q$&% *,+(-.0/21 3 ú+(û.0+(ûj+¡ü
‰nq
jf LF9S<h>QTQJHvPCFU,LF<JS9,<Q[I<h>@;(UxCƒ~^A¡>@[MA([ML&pþý_s S acLPThUxC_RSL|EžLPT“RSLF<£E;=KIGMA(JSK`CF<:[I9H~^9S[>„CFU&;=9i>Q;€RS;=Ká¤
wiCFR^CF<(žA(LPKML}ADC_R^CF<œ;=9 z CFK®[IwPJ^CFK0yzJS;V;(9•;=K®<Q[M<?>@;(UxCxTh;DCFK S ‚u8:<hA=LPwP;(U,LF<{K`CF<kCFU&ESKM[>@JSRH;=<œRS;
TQ;( S;¡XZ[M\P9 ~^A(>Q[MA([`CF<2;¡XHCFA¡>„CFU&;=9i>Q;)[IwPJ^CFKI;=<2yzJS;)KMCF<2RS;7K`CF<2ESKMCFADCF<2Th;DCFKI;=< r ‚ÿ…;7;=<?>„CŒacLPTQU,C&<h;
wiCFT@C_9z>Q[Md=C…yzJS;2;=KSA=CFU,ELƒRS;€TQCFRS[`C_A=[M\F9oRS;=9i>QTQL|RS;2K`C…A=CDmZ[IR^CFRV~^A¡>@[MA([`C{<h;DCƒ;=KSU&[M<hU,L…RS;=9i>@ThL|RS;
K`CVADCDm}[MR^C_RbTQ;=CFKˆ‚^›k<Q[IU,[I<QU&LSH<hJSEžLF9S;=U&LP<œy}JH;l;=9;=K0<Q[I<h>Q;=U,CŒ~^A¡>@[IA=[ML&KMLF<2acL_>@LP9S;(<kEHJS;=RS;(9•<Q;(T
>@T@C_9S<QU&[I>Q[MRSLP<kRS;&K`C˜ADCDm}[MRSCFR‘’SCFA=[MCB;(K:;=<hE^CFA=[IL"mFC_A=GMLB;YXZ>Q;=Th9SLbA(LP9WJS9^C CFU,EHKM[I>QJSRNRS;V>@TQCF9S<š¤
U,[I<Q[M\F9 t ‚}8:<QA(LPwP;(U,LP< t RS;œUxC_9S;=TQC)yzJS; |t | + |r | = 1 }CF<Q;(wPJSTQCF9SRSL)K`C|A=LP9S<h;=T?mFC_A=[M\F9˜RS;
;=9S;(TQwPGMCH_<h[M9o9H;=A=;(<Q[IR^CFRŒRS;j>QLPUxC_T®;(9VA=JS;(9z>@C{KMLP<®wPT@C_RSLP<RS;jKM[MO;=T?>„CFR)[M9i>@;(TQ9SLF<¨RS;gKMCF<¨E;=KIGMA(JSK`CF<
~^A(>Q[MA=[MCF<=‚ ƒT@C_A=[`C_<jC7;=<h>QLS}EžL}RS;=U&LP<£;=<?>@JSRS[MCFT£KMLP<£;=[IwP;=9S;(<h>@CFRSLP<RS;=KADCFU&EL);(KM;=A¡>@ThLPUxC_wP9S”(>Q[MA(L
;=9 S <Q[M9"U&;=d(A=KMCFTQKILoA=LP9B9S[I9SwPåS9";(<h>@CFRSL&;=KM;(A(>QTQ\P9S[IA=LV9S[žmZ[IOST@C_A=[MLF9^CFKˆ‚}8:KA=CFU,ELVLPO;=RS;(A=;|K`C_<
;=A=JSCFA=[ILP9S;=<kRS; bCtX j;(KMK¢RS;=9i>QTQL,vBacJS;(T@C&RS;)KMC&A=CDmZ[IR^CFR^vBA([M;=T?>„CF<œA=LF9SRS[MA([MLP9H;=<œRS;)A(LP9i>@LPTh9SL
;=9eK`CF<|<QJHEž;(Th~^A([M;=<7[M9i>Q;=TQ9SCF< z = z ‚0æ¨C_T@CEL}RS;=T|RS[I<QA=Th;(>Q[MdDC_T{vWA=LP9i>„C_T7KILP<|U,L}RSLP<(KM;&CF<h[Mw_¤
9^CFU&LP<JH9x>„CFU,C SL7~^9S[>@L W = L + L + L CFK<Q[I<h>Q;=U,C7~^A¡>@[MA([ML)A=LPU&ESKM;¡>@L7;…[IU,ELP9S;(U,LP<
A=LP9SRH[MA=[ILP9S;(<)RS;,A=LP9i>QLPTQ9SLb’SLPU&LPwP”=9H;DCF<(¢yzJS;xESJS;=RH;=9ä<Q;(TlRS;xÿ…[MTh[MA„’SKI;(>= œ;=JSU,CF9‘L•RS;,mFLP9
7C_TQU,CF9H;(9KMCF<€<QJHEž;(Th~^A([M;=<œ;¡X}>Q;=TQ9SCF<
H>@LPU,CF9SRHL,C_Ku~^9^CFKRH;=KAD‹_KMA=JHKMLVKMLF<€KMGMU&[I>Q;=<
z = z ,z
†—‡Z‚‰nŠ
L , L , W → ∞.
™0CW~^wPJSTQC†—‡Z‚LJFŠŒUoJH;=<h>QT@Ce;=K2<Q[I<h>@;(UxCN~^A¡>@[MA([MLS‚:›€yzJSG L v L <QLP9]KMLP<&CF9SA„’HLP<xRS; K`C_<
TQ;=wF[MLP9S;(<lm_CFA=GMCF<lRH; S CF9i>@;=<Vv‘RS;=<hESJS”(<&RH;"K`CF<oEHK`CFA=CF<l[I9H~^9S[>„CFU&;=9i>@;,RS;=KIwiCFR^C_< 1 v 2 ¨TQ;Y¤
<QE;=A(>Q[Im_CFU&;=9i>@;_¨v L = L ‚ …CFOH[M;=9HRSL•~_QCFRSLNKILP<lm_CFKILPTQ;(<oRS; L v L A=CFR^C•acLF>Q\P9 yzJS;
CFO^CF9SRHLP9S;lKMC&A=CDmZ[IR^CFRb~SA(>@[IA=[MCHÅTQ;=wFTQ;=<QCFT@‹,Cx;=KIK`CŒRS;=<hESJS”(<ƒRS;)m}[M<h[I>@CFT2K`CF<œTQ;(wP[MLP9H;=< I v _L III
A=LP9JS9^CVaCF<Q;7;=<hEž;(A=GI~SADCH‚8:<h>@C&aCF<h;7m_CFThG`CVT@‹FEH[MR^CFU&;=9i>Q;7A(LP9bKMCoacTQ;=A(JS;=9HA=[`CŒvBA=LP9;=K>„C_UxC HL
RS;=K<h[M<h>Q;=U,CV<Q[ L v L <hLP9<hJH~^A([M;=9i>Q;=U&;=9i>@;)wPT@CF9HRS;=<(^EžLPT€KILŒy}JH;)ESThLPU&;=RS[MCF9SRSLV9zJS;(<h>@ThLP<
TQ;=<hJSKI>@CFRSLP<£<QLFOSTQ;œJS9˜;(9S<Q;(UoOSKM;œRS;œ<Q[I<h>@;(UxC_<RS;œKMLF9SwP[I>QJSRS;(<£RH[Iac;=Th;=9i>@;(<jELZRHTQG`C_U,LP<:<h[MUoJHK`CFT:KMC
aCF<Q;7CFKI;DC_>QLPTQ[MCl;=<hEž;(T@CFR^CŒ;=9 ;=K<h[M<?>@;=U,CVTQ;=CFKˆ‚
0
a
α
a
α
a
α 2
a
α
a
α 2
a
0
a
I
II
III
0
I
3
III
I
III
0
II
I
III
I
III
¦¨§{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
I
III
II = V 0
r2α
LI
z0
tα
2
LII
z1
‰Ÿý
LIII
z2
z3
?ÈÁ¨Ð¡[MÑtwPɗJHÔ?ʈT@Ø¡C&Ô®‡ZÄÏÆg‚χZÔhÂSÑnÃÅÔ?ÊÄÇ¡Æ¡ÙÇÈhÆjÆ¡ÉɐÌÏÕFÑkȎÔ?͗ÌÇÜnȚȚÌÇÌrơ͈É0ÆYÊ0ÌrџơÕ_ÄiÑnڀÌþ͈ƄÔhɗ͗ÌrÔhڀʗÌÇÆ¡ÆYÄrÄzÔQ×nÉ0Ô:ȚÐ(ÄÏÆ(хɞÄÇÓtÆ¡ÄÏÉÆ(È?Æ¡É Þ Æ( Éu͗ʈÔhÉÒnʈԓÔ(Ô?ÌrÄZÐ(ÑnɐÌÇÔhÉcÉ͈Ô?ɗڀСхÆgØ@ʈÆ(ÔhȚÆYÙÇÄ Æ¡Ö É Þ ÈšÐ(Ñ ÍˆÆ¡Ä ÒnԣɗÔ
æ®CFT@C7<Q[IU,ESKI[I~^A=CFT:;=KÅESThLPOSKI;=U,CHP>@TQC_>„CFTh;=U&LP<“A=LP9,JS9x<h[M<?>@;=U,C)9HLlyzJS[MTQCFKˆ‚P8“<?>@Lo[IU,EHKM[MA=C…yzJS;
K`CF<£LP9SR^C_<g;=KI;=A(>QTQLPU,CFwP9H”(>@[IADCF<£A=JZvPLŒADCFU&ELl<Q;…’^CFKMKI;kELPKMCFTQ[IdDCFRSL7;=9"JH9^CoRS[MTh;=A(A=[M\F9x9SLPThUxC_KuCFK
ESK`CF9HL7RH;œ[I9SA=[IRS;=9HA=[`CZFacLPTQU,CFRSLƒELPT:;(KÅmF;=A¡>@LPT“RS;œLP9SR^C|[M9HA=[MRH;=9i>@;gvŒKMC{9SLPTQU,CFK^C|K`Cƒ<hJSEž;(Th~^A([M;_
;=<:RS;(A=[MT(P;=K^A=CFU&EžL{A(LP9&EžLPKMCFTQ[IdDCFA([M\P9 s P9SL|<Q;2U&;=d(A=K`C{A(LP9ŒK`CF<¨LP9SR^CF<:A(JHvPL7ADC_U,ELƒUxC_wP9S”(>Q[MA(L
;=<…9HLPTQU,CFK0CFK0ESKMCF9SLxRH;)[I9SA=[IRS;=9SA([`CH^;(<…RS;(A=[IT=A=LP9EžLPKMCFTQ[IdDCFA([M\P9 p ‚ž›€RH;=U,‹F<=;=K0<Q[I<h>Q;=U,C,<Q;(T@‹
[M9im_CFTQ[MCF9i>@;…acTh;=9i>@;)CVTQLF>@CFA=[ILP9S;=<€;(9 xy ‚
›k’SLPTQC|A(LP9S<h[MRS;(TQ;=U&LP<:JS9SC|LF9SR^CƒA=LP9ŒELPK`C_TQ[Md=CFA=[I\P9 s RS;gacTQ;(A=JS;(9SA=[MC ω A(LP9VmP;(A(>@LFT£RH;œLF9SR^C
~ CVKMLVK`C_TQwPLVRS;|KMCo[M9i>@;(ThaCFA(;F‚Å8:KADCFU&EL&;(KM”(A(>@Th[MA(LV;=<h>@‹&RSCFRSL&EžLPT€KMCoE^CFT?>@;|TQ;=CFKRS;
Q
†—‡Z‚χPŠ
~ ~r) = E~ e
E(t,
φ(z),
RSLP9SRS;…<Q[M9xEž”(TQRS[IR^ClRS;kwP;=9S;(T@CFKI[MR^C_Rx;=<hA=Th[MOS[IU,LP<K`C)CFU,EHKM[I>QJSR&A=LPU&L E~ = (0, E , 0) z>@LFUxCF9HRSL
C Q~ ;(9¹K`CNRS[ITQ;=A(A=[I\P9]RH; x ‚“›€RS;=U,‹F<(£KMCWacJH9SA=[I\P9]RH;bLP9SRSCe9SLFTQU,CFKM[IdDCFR^C φ(z) LPO;=RS;(A=;K`C
;=A=JSCFA=[I\P9BRS; k;=KIU,’HLPKI>Qd{;=9 ADCFRSC&Th;=wP[I\P9˜m_CFA=GMCHHv";(<h>„‹ŒR^CFRSCŒELPT
†—‡Z‚ PŠ
φ(z) = α e + β e
,
RSLP9SRS; r <h;|TQ;(~S;=TQ;)CVADCFRSC&JS9SCŒRH;)KMCF<j>@Th;=<œTQ;(wP[MLP9H;=<€RS;|9zJS;=<?>@ThLx<Q[I<h>Q;=U,CH‚S›…<h[MU&[M<hU,LS
†—‡Z"‚ !iŠ
ω
q = k +Q =
raα
o
tα
a
i(Qx−ωt)
o
ikz
r
q
2
r
−ikz
2
c
o
¨¦ §{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
‰$#
;=<|;=K9zåSU&;=ThLBRS;VLP9SR^CZuvNKMLF<{A=L};(~^A([M;(9z>Q;=< α v β RS;VK`CxacJH9SA=[I\P9NRS;ŒLP9SR^C"E^C_T@C˜ADC_R^C"JS9^C
RS;BK`CF<o>QTQ;=<ŒTh;=wP[ILP9S;=<Œ<h; LFOH>@[I;=9S;(9]RS;BKMCF<VA=LP9HRS[MA([MLP9S;(<ŒRS;BA=LP9i>@LFTQ9SLWväRS;BK`C•A(LP9SRS[IA=[I\P9RS;
9SLPTQU,CFKI[MdDC_A=[M\F9
†—‡Z‚Ï¥PŠ
sen kL %œ;
= 1,
L |α | + |β | + 2
αβ e
dz |φ(z)| =
kL
RSLP9SRS; z RS;=9SL_>„CäCFKkESJS9i>@LäU&;=RS[IL RS;NKMC TQ;(wP[M\P9 r ‚j8“9ç;(K{KMGIU,[>@;e†—‡Z‚I‰ŸŠ¡jKMC A=LF9SRS[MA([M\P9RS;
9SLPTQU,CFKI[MdDC_A=[M\F9xESJS;(RS;7<Q[MU&ESKI[I~^A=CFTQ<h;{C
†—‡Z‚rÀiŠ
L (|α | + |β | ) + L (|α | + |β | ) = 1.
æ®CFT@C"A=CFKMA(JSK`CFTkK`CF<…A=LPU&EžLF9S;=9i>@;(<ƒRS;=K¨ADCFU&EžL"U,CFwP9H”(>@[IA=L B~ = (B , 0, B ) JZ>@[MKI[Md=CFU,LF<kKMC
™0;¡v"RS'; &Ž9SRSJHA=A=[I\P9 ;=KM;(A(>QTQLPU,CFwP9S”¡>@[IADCoRS;7ÁÅCFT@C_R^CDvW†?‰F‚I‰ŸÀPŠ¡SELPT€K`CoA(J^CFK
†—‡Z‚ÇqFŠ
iω ~
−∂ E ı̂ + ∂ E k̂ = B.
c
ÿ…;7CFyzJSGyzJS;
†—‡Z‚ÏýPŠ
∂φ(z)
E
B (t, ~r) = i e
r
Z
X
2
S0
r
r
2
r
r
r
2
∗ 2ikzr
r r
r
r
!
r
I
I
2
I
2
III
III
2
III
2
x
z
v
y
x
z
y
o i(Qx−ωt)
x
q
∂z
—† ‡Z‚"#iŠ
ÿ…;bC_A=JS;(TQRSL‘A=LP9]K`CN;=KI;=A(>QTQL}RS[M9S‹FU,[IADCbA(K`‹F<h[MA=C]p _s—:K`CNRS;=9H<Q[MRSCFRRS; ;=9H;=TQwFG`CWESTQLPU&;=RH[MLW;=<?>„‹
R^CFR^C&EžLPT u = (|E| + |B| )/(16π) ‚žèZ[¢K`C,[I9i>@;=wFT@CFU&LP<2<QLPOHTQ;)>QLZRHLx;=K¢mPLPKIJSU&;=9bRH;=K¢<Q[M<?>@;(UxC
~^A(>Q[MA=[IL S ^LPOH>Q;=9S;(U,LF<€K`CV;=9S;(TQwPGMCV;=KM;(A(>QTQLPU,CFwP9S”¡>@[IADC
†ˆ‡}‚I$‰ )PŠ
A
Q sen kL
U=
L |α | + |β | + 2
Reα β e
|E | ,
8π
q
kL
RSLP9SRS; A ;=<£;=Kŋ_TQ;DC|RS;œK`C{<hJSEž;(Th~^A([M;_}KMCƒA=J^CFK^A(LP9S<h[MRS;(T@CFTh;=U&LP<“[M9H~^9H[I>„CƒC_KS~^9^CFKI[Md=CFT;=KÅA=‹FKMA(JSKMLS‚
8“9 ;=KKIGMU&[I>Q;o†—‡Z‚I‰ŸŠjK`Co;=9H;=TQwFG`CV;=KI;=A(>QTQLPU,CFwP9H”(>@[IADCo;(<h>„‹ŒR^CFRSCŒELPT
†ˆ‡}‚I‰P‰ŸŠ
A
A
|E | .
U=
L (|α | + |β | ) + L (|α | + |β | ) |E | =
Bz (t, ~r) =
2
Q i(Qx−ωt)
Eo e
φ(z).
q
(#
2
0
X
r
r
2
r
2
2
2
r
8π
h
I
I
2
I
2
III
III
2
r
∗ 2ikzr
r r
r
III
2
i
o
!
2
8π
o
o
2
2
¦¨§{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
‡)
›kKg[M9H[MA=[ILRH;"K`Cb<h;=A=A([M\P9 †š‰P‚"!iŠ)’^CFOSKMCFU&LP<)RH;"[MU&EžLP9H;=TlA([M;(Th>„C_<oA=LP9HRS[MA([MLP9S;(<oRS;˜acThLP9i>@;=TQC
>„CFKI;=<jyzJS;|;(KuADC_U,ELo;=KI”=A¡>@TQ[IA=LoRH;=9i>@ThL&RH;ƒK`ClA=CŸm}[IR^CFR"ESJSRH[M;=<h;ƒ;=<?>„CFTgA(J^CF9i>@[IdDCFRHLVRS;ƒK`ClU&[M<hUxC
acLPTQU,C"y}JH;xKML ;=<?>„‹bJH9^CA(JS;=ThR^CRS;,wPJS[>„CFThT@C"<QJth;¡>„Cb;(9 <hJS<l;YXZ>QTQ;(U,LP<(‚¨™0CF<7A=LF9SRS[MA([MLP9H;=<7RS;
A=LP9i>@LFTQ9SL&JS<QJSCFKM;(<k<hLP9KMCVA=LP9i>@[I9zJS[MR^C_RRS;7A=[I;=T?>„CF<2A=LFU,ELP9S;(9z>Q;=<2RS;7KMLP<2A=CFU,ELP<€;(KM”=A¡>@Th[MA=LF<2v
UxCFwF9S”(>Q[MA=LF<;=9)K`Cg<QJHEž;(Th~^A([M;£RS;=Ki<h[M<h>Q;=U,C€TQ;DC_Kˆ=v|”(<h>„C_<¢A(LP9SRSJHA=;=9oC2KILP<m_CFKMLFTQ;=<0RS; r ‚Ÿ›…KIwPJS9^C_<
RS;lKMCF<…A(LPU,ELP9S;(9i>@;=<kRS;=K®ADCFU&ELx;=KI;=A¡>@TQLFUxCFwF9S”(>Q[MA=L&<QLF9NRS[M<hA=LP9i>@[I9zJ^CF<…;(9NK`CŒ<QJSE;=T?~^A=[I;oTQ;=CFKˆ
CFKuU&;=9SLF<g;=9"U&LZRH;=KMLF<j<Q[IU,EHKM;=<(ZRH;=OS[IRSLVCoyzJS;ƒ<Q;ƒ[I9SRSJSA(;=9"A=CFTQwiC_<jv˜A(LPTQTh[M;=9i>Q;=<g<hJSEž;(Th~^A([`CFKI;=<(
K`CF<“A=JSCFKM;(<jCFESCF9i>„CFKIK`CF9ŒCFKÅA=CFU,EL|;=9˜;(KÅUxC_>Q;=Th[`CFKpá$‰ )nsˆ‚}ÿ…;(OS[MRSLlC7;(<h>@ClCFA(JSUoJSKMCFA=[I\P9ŒRS;kA=CFTQwPCF<
vlA(LPTQTh[M;=9i>Q;=<:;=9VKMC…<QJHEž;(Th~^A([M;€RS;(KHU&;(>„C_Kˆt;=KHA=‹FKMA(JSKMLkRS[MTh;=A¡>@LƒRS;gKMCkacJS;(TQdDC{RH;2™0LFTQ;=9i>Qd€<QLPOSTh;gK`C_<
ADCFThwiCF<£;=9xKMC7<hJSE;=Th~SA=[M;œRS;œK`CF<“ESKMCFADC_<A(LP9SRSJSA(;{C)CFUoOS[I+w *S;=RSCFRS;=<(‚z8:<h>„C_<j<QLF9xOS[I;=9xA(LP9SL}A=[IR^CF<=
vçCFE^C_TQ;=A(;=9jEžLPTx;šh;=U&ESKILSj;(9 ;(K{AD‹_KMA=JHKML RS;NKMCF<,acJH;=TQd=CF<B;=9i>@Th;NK`CF<xESK`CFA=CF<˜RS;NJH9 <h[MU&ESKM;
A=LP9SRH;=9S<QCFRSLPT(‚i›…KS[IwPJ^CFKZy}JH;œ;(9,RH[MA„’SL|;šh;(U,EHKMLS_EžL}RS;=U&LP<;¡m}[I>„C_TQK`C_<<Q[^A=CFKMA(JSK`CFU&LP<®K`CF<acJS;(TQd=CF<
;=9B>@”=ThU,[I9SLP<gRH;=K>@;(9S<QLFTkRS;|;(<hacJS;(TQd=LF<k;(9KIJSwiCFTgRH;7Th;=A(JSTQTh[MT2CVK`CoacJS;(TQdDCŒRS;7™LPTQ;(9i>@dF‚
›A=LF9z>Q[M9zJ^CFA([M\P9 ADC_KMA=JHK`CFU&LP<€;=K>Q;=9S<hLPT…RS;7;=<?acJS;=Thd=LP<k;=9NC_KMwPJS9SCŒELP<h[MA=[I\P9 z RS;(9z>QTQL,RS;lKMC
ADCDm}[MR^CFRe(p #_s—
†ˆ‡}‚I‰n‡FŠ
1 %2;
δ
T =
[E E + B B − (|E| + |B| ) ],
2
y}JH;7;(99zJS;(<h>QTQLxA=CF<QL&<Q;|Th;=RSJS8πA(;lC
†ˆ‡}‚I$‰ PŠ
1 %2;
1
T =
[E E + B B − (|E| + |B| )].
8π
2
,œ>@[IKM[IdDCF9SRHLoK`CF<€;(A=J^C_A=[MLF9S;=<)†ˆ‡}‚LJPŠY`†—‡Z‚rýiŠšvN†ˆ‡Z"‚ #iŠjLPOH>Q;=9S;(U,LF<
†ˆ‡}‚I-‰ !iŠ
k
E
1
−T =
|φ(z)| + |∂ φ(z)| .
16π q
q
Á¨[M9^C_KMU&;=9i>@;_HJH>Q[MKM[IdDCF9HRSLoK`Co;(A=J^C_A=[M\F9 †—‡Z‚I‰F‰nŠRS;|K`CV;(9S;=ThwPG`CV;=KI;=A¡>@TQLFUxCFwF9S”(>Q[MADC)LPOH>@;(9S;=U&LP<
k
†ˆ‡}‚I‰n¥FŠ
1
U
−T =
.
|φ(z)| + |∂ φ(z)|
q
q
2A
™0CŒ;=A_‚¨†—‡Z‚‰n¥PŠjacJS;7LPOH>@;(9S[MRSCŒESCFT@CŒJS9b<hLPKML&U&LZRSLŒ;(KM;=A¡>@ThLPUxC_wP9S”(>Q[MA(LS‚S™0CF<œA=LP9HRS[MA([MLP9S;(<2RS;
A=LP9i>@LFTQ9SLœ;=9 z v z KIKM;¡mFC_9)C2JS9^C2ESTh[MU&;=T@C€A(J^CF9i>@[IdDCFA([M\P9|RS;£KMLP<0U&LZRHLP<9SLPThUxCFKI;=<RS;(Kz<Q[M<?>@;(UxCH–
α
a
ij
i
zz
"
2
o
2
zz
3
2
z
2
2
∗
j
i
∗
z
z
zz
0
∗
j
2
∗
z
2
2
2
2
2
ij
2
z
2
2
z
2
!
#
¨¦ §{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
‡Z‰
;=<h>QL]yzJS[I;=Th;NRS;=A([MT"yzJS;eKMLP<˜U&L}RSLP<"RS;(Kƒ<Q[M<?>@;(UxC ;(<h>„‹_9 Th;=<h>QTQ[I9SwP[MRHLP<"CA=[M;(Th>@CF<˜acTQ;(A=JS;(9SA=[MCF<
v‘A=LPU&EžLF9S;=9i>@;(<o9SLPThUxC_KM;=< k RH;=KmF;=A¡>@LPToRH;˜LP9SR^CRS[M<hA=TQ;¡>„CF<(‚/.¢LPU,CF9SRSL•;(9äA=JS;(9i>„C•K`C_<
ω
A=LP9i>@Th[MOSJHA=[MLF9S;=<£RS;2>@L}RSLP<£KMLP<£U,L}RSLP<(i;=<hA=TQ[IOS[MU&LP<£CFK^>@;(9S<QLFTjRS;œ;=<?acJS;=Thd=LP<gA(LPU&L)JS9SC)<QJHUxC7RS;
A=LP9i>@Th[MOSJHA=[MLF9S;=<€RS;(K>@[IEžL†ˆ‡}‚I‰n¥FŠ¡
†ˆ‡}‚I‰ŸÀPŠ
1
k
1
1
h̄ω ,
−T =
N +
|φ (z)| + |∂ φ (z)|
2A
q
q
2
RSLP9SRS;7K`CV;(9S;=ThwPG`C U ;(<h>@‹,R^CFRSC&;(9b>@”(TQU&[M9SLP<€RH;=K09zåSU,;(TQL,ESTQLPU&;=RH[MLŒRS;|acLF>@LP9H;=< N yzJS;l<h;
;=9SA(JS;=9i>@TQCF9b;(9;(K;=<?>„CFRSL n ‚
æ¢LZRH;=U&LP<BTQ;=;(<QA(TQ[MOH[MT";=<?>@;N>@;(9S<QLFT RH;e;(<hacJS;(TQd(LP<[I9i>@TQL}RSJSA([M;(9SRSLJS9^C õ}ð¡óHî¡ò1õ 0iõ í ëŸ2ï 0 í õ}ð
ð„î¡3ô 0iõ}ëŸîjRS;|UxC_9S;=TQCoy}JH;);(<QA=LF9SR^CFU&LP<2K`Co<hJSUxCV<hLPOSTQ; n ‚Åæ:Th[MU&;=TQLVRH;(~^9S[IU,LF<
&ŽU
†ˆ‡}‚I‰nqFŠ
1 &ŽU
1
ρ = ρ (z) = −
G (z, z ) + ∂ ∂
G (z, z )
,
2π
k
RSLP9SRS;
†ˆ‡}‚I‰ŸýPŠ
φ (z)φ (z )
G (z, z ) ≡
;=<2K`ClacJS9HA=[M\F9RS4; ƒTQ;(;=9BE^CFTQC&KMCo;=A=JSCFA=[I\P9BRS; k;=KIU,k̃’HLP−KI>Qkd{;=9 JS9^CŒRS[MU&;=9H<Q[M\F9HLPO;=RH;=A=[I;=9SRHL
n
n
2
n
2
n
X
zz
n
2
n
2
z n
2
n
!
n
n
k2
k2
n
n
0
0
2 z z
k̃ 2
X
0
k̃ 2
0
k̃ 2
2
n
z 0 →z
0
∗
n
n
2
n
ˆ† ‡}‚I‰$#PŠ
v"K`CF<€A(LP9SRS[IA=[ILP9S;=<€RH;|acThLP9i>@;(T@CŒCFRS;(A=J^CFRSCF<=‚ÿ…;¡~^9S[MU&LP<€;(K9}åHU,;(TQL&RS;|LP9SR^CVA=LFU,ESKI;šhLVA(LPU,L
†ˆ‡}‚LJ )PŠ
k̃ = k + iη
(η > 0),
vBCFKu~^9SCFKRS;=KA=‹FKMA(JSKMLo>QLPU,CFTQ;(U,LP<j;=KKMGIU,[>@;œA=J^C_9SRSL η → 0 ‚5kLF>@CFU&LP<€yzJS;|;=9 ;=<h>Q;)KIGMU&[I>Q;
&ŽU
†ˆ‡}‚LJZ‰ŸŠ
φ (z)φ (z )δ(k − k ),
G(z, z ) = −2π
EžLPT7KML A=JSCFKˆJH>Q[MKI[MdDC_9SRSLBK`CF<7;=A(J^CFA([MLP9S;(<"†—‡Z‚I‰ŸqPŠ…v¹†—‡Z‚‰ŸýiŠ¡0ELZRH;=U&LP<|TQ;=;(<QA(TQ[MOH[MT7;=K“>@;=9H<QLPT)RS;
;=<hacJH;=TQd(LP<œA=LPU&L
†ˆ‡}‚LJP‡FŠ
h̄c
k
−T =
dk f ρ .
(k̃ 2 + ∂z2 )Gk̃2 (z, z 0 ) = δ(z − z 0 ),
+
X
0
∗
n
n
0
n
zz
A
Z
2
2
q
k2
2
2
n
¨¦ §{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
›€y}JHG[M9i>@ThLZRSJnh[MU&LP<g;=K9zåSU&;=ThLŒRS;|LZA(JSE^CFA([M\P9
‡P‡
†ˆ‡}‚LJPŠ
;=9 ;(yzJS[MKI[MOSTh[ML"RS;&JS9 ;(<h>@CFRSLbRH;&acTh;=A=JH;=9SA([`C ω C KMCB>Q;=U&Ež;(T@C_>QJST@C T = 1/(k β) 0A(LP9 k KMC
A=LP9S<?>„CF9i>Q;jRS;€ùLPKI>Qd=U,CF9S9ŸRS;£acLPTQU,C2y}JH; f (ω ) = N +1/2 ‚n8“9l9}JH;=<h>QTQLƒA=‹FKMA(JSKML2>@LPU,CFTh;=U&LP<
SEžLFT€KMLŒyzJS; f = 1/2 ‚
T =0
ÿ…CFRSL7y}JH; φ RS;(<QA=Th[MO;œCFK^ADCFU&EL7;(KM”(A(>@Th[MA(Lx†c;=A_‚†ˆ‡Z‚χPŠQŠ¨ESCFT@C|;=K^A=CF<QL7RS;œEžLPKMCFTQ[IdDCFA([M\P9 s yzJS;
;=<h>@CFU&LP<¢>QT@C_>@CF9SRSLHt;(KZEHTQ[MU&;=T>Q”=TQU&[M9HLkRS;(KZKMCFRSLœRS;=Th;=A„’SL…RS;jKMCk;=A_‚S†—‡Z‚‰nqPŠESJS;(RS;g[M9i>Q;=TQEHTQ;(>@CFTQ<h;
A=LPU&LNK`CbA(LP9i>@Th[MOSJSA([M\P9 ð 7í 6 ï¡ôìYòˆ2ï 0 C•KMC•RS;=9H<Q[MRSCFRKML}ADCFK£RS;B;=<?>„CFRSLF<V;=KM;(A(>QTQLPU,CFwP9S”¡>@A(LP<=‚:8:<?>@L
<QJSwP[I;=Th;b;=KkyzJS;N[MRS;(9i>@[I~Sy}JH;=U&LP<˜CFKk<Q;(wPJS9SRSL >Q”=ThU,[I9SL A(LPU,L K`C‘A(LP9i>@Th[MOSJSA([M\P9UxCFwF9S”(>Q[MADC‘C
RS[MA„’^C,RS;=9S<h[MR^C_RNRS;o;=<?>„CFRHLP<{v•C,K`C,<hJSU,C ρ A=LFU,L,K`CxRS;(9S<Q[IR^CFR•>QLF>„CFK—‚8“<?>„C˜[IRS;=9i>@[~^ADC_A=[M\F9
Ež;(TQU&[I>@;…JS9^CŒ[M9i>@;(TQESTh;(>„C_A=[M\F9 <h;=9SA([MKIK`ClRS;(K0TQ;=<hJSKI>@CFRSL†ˆ‡}‚LJP‡FŠ¡‚
ÿk;=U&LP<h>QTQ;=U&LP<|CF’HLPT@C"yzJS;ŒK`CxacJS9HA=[M\F9NRS8; |Th;=;=9]†—‡Z‚I‰DýiŠ…RS;=9i>QTQL RS;VK`C˜A=CŸm}[IR^CFRW<Q;ŒESJS;(RS;
;=<QA(TQ[IOS[MT€RS;|JS9SCVacLFTQU,CoUx‹F<€<h;=9SA([MKIK`CH‚Hÿ…;˜†—‡Z‚$‰ #iŠELZRH;=U&LP<€A=LP9HA=KMJH[MT€yzJS;|E^CFTQC z 6= z 
†—‡Z‚Ç9‡ !zŠ
(k̃ + ∂ )G (z, z ) = 0.
æ®LPT7KML">„C_9z>QLbELZRH;=U&LP<|ESK`CF9i>Q;DCFT|RSLP<7<QLFKMJSA([MLP9S;(<|<Q;=ESCFT@CFRSCF<lRS;&K`C˜acJS9HA=[M\F9 RS:; |Th;=;(9 E^CFTQC
ADCFR^Com_CFKILPTgRS; z 
<Q[ (z − z ) > 0,
A(z )φ (z),
†ˆ‡}‚LJP¥FŠ
G(z, z ) =
<Q[ (z − z ) < 0,
B(z )φ (z),
RSLP9SRS; φ (z) v φ (z) <hLP9<hLPKMJHA=[MLF9S;=<NRS;KMCF<W;=A=JSCFA=[ILP9S;=<WRS;ADC_U,EL ’SLPU&LPwP”=9H;DCF<NyzJS;
A=JSU&ESKM;(9 A=LP9 >@L}R^CF<BK`CF<"A(LP9SRS[IA=[ILP9S;=<BRS;WacTQLF9z>Q;=TQCE^C_T@C z > z v E^CFTQC z < z Th;=<QE;=AY¤
>@[Im_CFU&;=9i>Q;F“U,[I;=9i>@TQCF<&y}JH; A(z ) v B(z ) <QLF9¸ï„ëtóHî¡3ô 0Fó^ôŽð„î&C RH;(>@;(TQU&[M9^C_T,C_JS9SyzJS;•ELZRHTQG`C_9
RS;=E;=9SRH;=TœRS;=Km_CFKILPTg;=KI;=wP[IRSLVE^CFTQC z ‚
,2>@[IKM[Md=CF9SRSL"KMCBacJH9SA=[I\P9‘;(<QA=CFKM\P9‘EžL}RS;(U,LP<|;(<QA=Th[MOS[IT|K`C"<QLPKIJSA=[I\P9eRS;,KMCBacJH9SA=[I\P9‘RH;
; ƒTQ;(;=9
A=LPU&L
†—‡Z‚LJ_ÀiŠ
G(z, z ) = A(z )φ (z)θ(z − z ) + B(z )φ (z)θ(z − z).
βh̄ω
1
f = coth
2
2
!
B
n
B
n
k2
0
2
0




0
>
2
z
0



<
0
0
k̃ 2
0
>
0
<
0
0
0
0
0
0
0
>
0
0
<
0
¨¦ §{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
‡
,k9^C mF;=dN>@;(9S[M;(9SRSLK`C <hLPKMJSA([M\P9RS;NK`C ;(A=J^CFA([M\P9ç’HLPU,LFwP”=9S;=CHjKMC <QJS<?>@[>@JS[IU,LP<,;=9 K`C [I9S’SL_¤
U,LPwF”=9S;=CH‚æ®CFT@C";=<?>@LBADC_KMA=JHK`CFU&LP<…KMCF<{RS;(TQ[mFC_R^CF<ƒE^CFThA=[MCFKM;(<{RS;VK`CxacJH9SA=[I\P9NRS;<ƒTQ;=;(9‘A(LP9WTQ;Y¤
<QE;=A(>QLxC z ‚
∂z G(z, z 0 ) = A(z 0 )φ0> (z)θ(z − z 0 ) + B(z 0 )φ0< (z)θ(z 0 − z)
†ˆ‡}‚LJPqFŠ
+ (A(z 0 )φ> (z) − B(z 0 )φ< (z)) δ(z − z 0 ),
v"JH>@[IKM[IdDCF9SRHLoK`CF<€EHTQLPES[I;=R^C_RS;=<2RS;|K`CVÿk;=KI>@CVRS;7ÿ…[MTQCFAF
∂z G(z, z 0 ) = A(z 0 )φ0> (z)θ(z − z 0 ) + B(z 0 )φ0< (z)θ(z 0 − z)
†ˆ‡}‚LJFýPŠ
+ (A(z 0 )φ> (z 0 ) − B(z 0 )φ< (z 0 )) δ(z − z 0 ).
ÿ…;=Th[Im_CF9SRSLŒ9zJS;¡mFC_U,;(9z>Q;);(<h>@C&;YXHEHTQ;=<h[M\P9 A=LF9Th;=<QE;=A¡>@L,C
z
∂z2 G(z, z 0 ) = A(z 0 )φ00> (z)θ(z − z 0 ) + B(z 0 )φ00< (z)θ(z 0 − z)
+ (A(z 0 )φ0> (z 0 ) − B(z 0 )φ0< (z 0 )) δ(z − z 0 )
†ˆ‡}‚LJ#PŠ
+ (A(z 0 )φ> (z 0 ) − B(z 0 )φ< (z 0 )) δ 0 (z − z 0 ).
Zè JS<h>Q[I>QJHvP;(9SRSL G(z, z ) v ∂ G(z, z ) ;=9oKMCk;(A=J^CFA([M\P9˜†ˆ‡Z‚‰$#iŠYŸvlTh;=A=LFTQR^CF9HRSL{yzJS; φ (z) v
<@C_>Q[M<haC_A=;=9BKMCŒ;(A=J^C_A=[M\F9 RH;7LF9SR^CH^KIKM;=wPCFU,LF<gCFK<h[MwPJH[M;=9i>Q;ƒ<Q[M<?>@;(UxCVRS;|;=A(J^CFA([MLP9S;(<
0
2
z
0
>
A(z 0 )φ0> (z 0 ) − B(z 0 )φ0< (z 0 ) = 1
A(z 0 )φ> (z 0 ) − B(z 0 )φ< (z 0 ) = 0
RS;=KA(J^CFKLPOH>Q;=9S;(U,LF<œELPT€;(KU,”¡>@L}RSLVRS;|RS;(>Q;=ThU,[I9^CF9i>@;(<œKILP<€A=L};(~SA=[M;(9i>@;=<
A(z 0 ) =
φ< (z 0 )
W
B(z 0 ) =
φ> (z 0 )
,
W
v
φ< (z)
†—‡Z‚ )iŠ
†—‡Z‚ Z‰nŠ
†ˆ‡}‚ i‡FŠ
†ˆ‡}‚ +PŠ
¨¦ §{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
RSLP9SRS;
‡=!
ˆ† ‡}‚ !iŠ
;=<";=K?> ThLP9S<2@z[MCF9SLS‚jÁ¨[M9^C_KMU&;=9i>@;<QJS<?>@[I>QJS[MU&LP<•†ˆ‡Z‚"i‡PŠ&v±†ˆ‡Z‚"+iŠ&;=9¸†ˆ‡Z‚χFÀiŠŒE^CFTQC LPOH>@;(9S;=TBKMC
<QLPKIJSA=[I\P9˜RS;{KMCl;=A(J^CFA([M\P9e†—‡Z‚I‰-#iŠ£RS;(9i>@TQLVRH;ƒK`ClA=CŸm}[IR^CFR‚H™¢C7acJS9SA=[I\P9"RS;'ƒTQ;(;=9B;=<?>„‹oR^C_R^CoEžLFT
†ˆ‡}‚ i¥FŠ
φ (z )φ (z)θ(z − z ) + φ (z )φ (z)θ(z − z)
G(z, z ) =
,
φ (z )φ (z ) − φ (z )φ (z )
RSLP9SRS;
†ˆ‡}‚ PÀPŠ
φ (z) = e
+r e
W = φ< (z 0 )φ0> (z 0 ) − φ> (z 0 )φ0< (z 0 )
0
<
0
0
>
v
<
0
0
>
0
−ik̃(z−z1 )
<
0
>
<
0
<
0
>
0
0
s ik̃(z−z1 )
1
ˆ† ‡}‚ iqFŠ
<QLP9]K`CF<Œ<QLFKMJSA([MLP9S;(<,RH;bK`CN;(A=J^CFA([M\P9]RS;bLF9SR^C‘;(9JS9SC‘RS[IU,;(9S<Q[I\P9]A(LP99zåSU&;=ThL RS;LP9SR^C k̃
LPOž;(RS;=A([M;(9SRSL K`CF<ƒA(LP9SRS[IA=[MLF9S;=<|RS;ŒA=LF9z>QLPTQ9HLbCFEHTQLPES[MCFR^CF<…;=9 z = z v z †CBKMC"RS;=Th;=A„’^C RS;
K`C,EHK`CFA=C 1 vK`C&[Md(y}JH[M;=ThR^CxRS;lK`C,EHK`CFA=C 2 TQ;(<QE;=A(>Q[Im_CFU&;=9i>@;DŠ¡‚u8“9N9zJS;(<h>QTQL"AD‹_KMA=JHKML,>QLPU,CFU,LF<
;=KA=CF<QL&;=9 yzJS; L 1/η –H;=<2RS;(A=[IT=^’S[IA=[MU&LP<JS<QL&RS;=KKIGMU&[I>Q;o†—‡Z‚I‰ŸŠjUxCF9HR^CF9SRSLV;(K>„CFU,C SLVRH;=K
<Q[M<?>@;(UxCC[M9Z~^9S[I>QL•CF9i>Q;=<lRH;˜UxC_9SR^CFT η CbA=;(TQLS‚®ÿ…;,;=<h>@C•UxC_9S;=TQCH09SLP<lRH;=<Q’SCFA=;(U,LP<lRS;xKMCF<
<QJSE;=T?~^A=[I;=<œ;=9 z v z ‚
è}JS<h>Q[I>@JZvP;=9HRSLN†—‡Z‚ FÀiŠve†ˆ‡}‚ iqFŠ;=9 †ˆ‡Z"‚ i¥PŠjKIKM;(wiCFU&LP<gC
†ˆ‡}‚ PýPŠ
1
G(z, z ) =
[(e
+r r e
φ> (z) = eik̃(z−z2 ) + r2s e−ik̃(z−z2 )
+
1
0
−
2
3
ik̃((z−z 0 )−L)
0
W
s s −ik̃((z−z 0 )−L)
1 2
0
0
+r1s eik̃((z+z )−(z1 +z2 )) + r2s e−ik̃((z+z )−(z1 +z2 )) )θ(z − z 0 )
0
0
+(e−ik̃((z−z )+L) + r1s r2s eik̃((z−z )+L)
0
RSLP9SRS;
0
+r1s eik̃((z+z )−(z1 +z2 )) + r2s e−ik̃((z+z )−(z1 +z2 )) )θ(z 0 − z)],
L = z2 − z1
;=<œ;=K0C_9SA„’SL&RS;|K`CoADCDm}[MRSCFRSv";(KA> ThLP9S<2@z[MCF9SL
W = 2ik̃ e−ik̃L − r1s r2s eik̃L .
W
;(<
†ˆ‡}‚ +#PŠ
¦¨§{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
‡P¥
æ®CFT@C,’^CFKMKMCFT€K`CŒRS;(9S<Q[IR^CFR•KILZA=CFK0RS;l;(<h>„C_RSLP<V†—‡Z‚‰nqPŠ€A=LP9S<h[MRS;(TQ;(U,LP<œESTh[MU&;=TQL&K`CŒacJS9SA([M\P9RS;
|Th;=;=9 yzJS;|TQ;(<QJS;(KImP;7K`Co;(A=J^C_A=[M\F9RS; œ;=KMU&’SLPK>@d{RH;=9i>@ThL,RS;|KMCoADCDm}[MR^CFR E^CFTQC z > z 
†ˆ‡}"‚ !1)PŠ
φ (z )φ (z)θ(z − z )
,
G(z, z ) =
0
0
<
>
φ< (z 0 )φ0> (z 0 )
0
0
− φ> (z 0 )φ0< (z 0 )
†ˆ‡}‚"!S‰ŸŠ
1 ik̃((z−z 0 )−L)
0
[e
+ r1s r2s e−ik̃((z−z )−L)
W
G(z, z 0 ) =
0
0
+r1s eik̃((z+z )−(z1 +z2 )) + r2s e−ik̃((z+z )−(z1 +z2 )) ],
v"<QJ <Q;(wPJS9SR^CŒRS;(TQ[Im_CFRSCŒTh;=<hEž;(A(>@LxC z v z 
0
†ˆ‡}‚"!z‡FŠ
k̃ 2 ik̃((z−z 0 )−L)
0
[e
+ r1s r2s e−ik̃((z−z )−L)
W
∂z ∂z 0 G(z, z 0 ) =
0
0
−r1s eik̃((z+z )−(z1 +z2 )) − r2s e−ik̃((z+z )−(z1 +z2 )) ],
ESJS;=<)’S;=U&LP<ƒ>@LFUxCFRHL ;(9 A(JS;=9i>„Cy}JH; ∂ ∂ &ŽU G(z, z ) = &ŽU
CF’SLPTQCŒ;(KKMGIU,[>@;œA=J^CF9HRSL z → z EžLPT€KMCV[Md=yzJS[I;=ThR^CH
0
z z0
[∂z ∂z 0 G(z, z 0 )]
‚¢fjLP9H<Q[MRH;=T@C_9SRSL
0
†ˆ‡}‚"!1PŠ
1 −ik̃L
[e
+ r1s r2s eik̃L
W
G(z, z) =
+ r1s eik̃(2z−(z1 +z2 )) + r2s e−ik̃(2z−(z1 +z2 )) ],
†ˆ‡}‚"!+!iŠ
k̃ 2 −ik̃L
[e
+ r1s r2s eik̃L
W
∂z ∂z 0 G(z, z) =
− r1s eik̃(2z−(z1 +z2 )) − r2s e−ik̃(2z−(z1 +z2 )) ].
èZJS<h>Q[I>QJS[MU&LP<g;(<h>@LF<œTh;=<hJSKI>@CFRSLP<2;=9 K`Co;(AF‚®†ˆ‡}‚I‰nqFŠ¡
k + k̃
1 &ŽU
ρ
= −
e
k2
"
2π
+
2
2
k2 W
!
2πA
+
−ik̃L
+ r1s r2s eik̃L
†ˆ‡}‚"!z¥FŠ
#
k 2 − k̃ 2 s ik̃(2z−(z1 +z2 ))
s −ik̃(2z−(z1 +z2 ))
r1 e
+ r2 e
,
k2 W
v"KMJS;(wPLV;=9 ;=K>Q;=9S<hLPTœRS;|;=<hacJH;=TQd(LP<l†ˆ‡Z‚χP‡PŠY
h̄c
k &ŽU
T
=
dk f
zz
!
Z
2
2
q
"
!
k 2 + k̃ 2 −ik̃L
s s ik̃L
e
+
r
r
e
1 2
k2 W
#
k 2 − k̃ 2 s ik̃(2z−(z1 +z2 ))
r1 e
+ r2s e−ik̃(2z−(z1 +z2 )) ,
k2 W
!
†ˆ‡}‚"!iÀPŠ
¦¨§{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
ŽU
‡FÀ
†ˆ‡}‚"!zqFŠ
Z
h̄c
f&
1 2
dk 2
k + k̃ 2 e−ik̃L + r1s r2s eik̃L
2πA
q
W
1 2
2
s ik̃(2z−(z1 +z2 ))
s −ik̃(2z−(z1 +z2 ))
.
+
k − k̃
r1 e
+ r2 e
W
#
A>
B@
W
Tzz =
Ž9SA=KIJS[MU&LP<gKMCo;¡XZESTQ;(<Q[I\P9䆗‡Z‚ iŠE^CFTQCV;=K ‘TQLP9S< z[`CF9SL 
h̄c
f &ŽU
k + k̃
e
T
=
dk
[
&
zz
4πA
+
Z
2
2
q
2
ik̃
!
!
†ˆ‡}‚"!iýPŠ

−ik̃L
+ r1s r2s eik̃L 

e−ik̃L − r1s r2s eik̃L

k 2 − k̃ 2  r1s eik̃(2z−(z1 +z2 )) + r2s e−ik̃(2z−(z1 +z2 )) 
].
e−ik̃L − r1s r2s eik̃L
ik̃
v"JH>@[IKM[IdDCFU&LP<£K`CV;=A_‚¢†ˆ‡Z‚χ)iŠYH>@LPU,CF9SRHLV;=KKMGIU,[>@; η → 0 
†ˆ‡}‚"!1#PŠ
f &ŽU
e
+r r e
2k
h̄c
dk
.
T =
4πA
q
e
−r r e
ik̃
æ®L}RS;=U&LP<€ADC_UoOS[`C_T€K`Co[M9i>Q;=wPTQCFK<QLFOSTQ; dk EžLPT€JH9^CV<QLPOSTh; dk
†ˆ‡}‚Ç¥)PŠ
h̄c
k &ŽU
i e
+r r e
−T
=
dkf
.
πA
q
−r r e
k̃ e
Á¨[M9^C_KMU&;=9i>@;_H;=U&ESKI;DCFU&LP<g;(KyzJS4; &ŽU [i(a + ib)] = %œ; [a + ib] ^E^CFTQCŒLFOH>@;(9S;=T
†ˆ‡}‚Ç¥Z‰ŸŠ
k %œ;
h̄c
1+r r e
dkf
−T
=
.
πA
q
k̃(1 − r r e
)
èZ[Å<Q;=wFJS[MU&LP<“;=KÅU&[M<QU&L|ESTQL}A=;(RS[MU&[M;(9i>@LSiC_’SLPT@C|E^CFTQC7’SCFKMKMCFT:;=K^>@;(9S<QLFTRS;œ;=<?acJS;=Thd=LP<jJH>@[IKM[Md=CF9SRSL
K`CxacJS9SA([M\P9WRS8; ƒTQ;(;=9WyzJS;,Th;=<hJS;=KmP;&K`C˜;=A(J^CFA([M\P9WRS<; k;(KMU&’SLPK>@dlRH;=9i>@ThLRS;ŒKMC˜ADCDm}[MR^CFReE^CFTQC
RSLP9SRS; z <Q;,~th\bCF9i>Q;=TQ[ILPTQU&;=9i>Q;FvWKMJH;=wPLB>@LPU,CFU&LP<ƒ;=K“KIGMU&[I>@;lA=J^CF9HRSL z → z ELPT
z <z
K`CoRS;(TQ;=A„’SC,;=9 †ˆ‡Z‚‰nqPŠYSA=LF9SRSJSA(;lCV;¡XHCFA¡>„CFU&;=9i>Q;);(KU,[I<QU&LVTQ;=<hJSKI>@CFRSLS‚S8“<œRS;(A=[MT(S;=K>Q;=9S<hLPT2RS;
;=<hacJH;=TQd(LP<V;=<V;(KjU&[M<hU,L<Q[M9ä[IU,ELPT?>„CFT|A=\FU,Lb>QLPU&;=U&LP<l;(KjKMGIU,[>@; z → z ‚›€RH;=U,‹F<= T 9SL
RS;=E;=9SRH;˜RS;=K£m_CFKMLPT|RS; z ;=9 KMCADCDm}[MRSCFR m_CFA=GMCA(LPU,LEžL}RSThG`CFU&LP<|;=<hEž;(T@CFToESCFT@C JS9^CbTh;=wP[I\P9
mFC_A=G`Co;(9JH9^C&<h[I>QJ^CFA=[I\P9";=<?>„CFA([MLP9SCFTQ[MCH‚
æ®CFT@C7;=9SA(LP9i>@TQCFTj;(K>Q;=9S<hLPTRH;…;=<?acJS;=Thd=LP<ƒ†ˆ‡Z‚χP‡PŠRS;(9z>QTQLoRH;kKMC7A=CŸm}[IR^CFRxESCFT@C7JS9^C)LF9SR^C)A(LP9
EžLPKI[Md=CFA=[I\P9 p ELZRS;(U,LF<o<Q[IU,ESKI;=U&;=Ul>Q;,Th;=;(U,ESKMCFdDC_T E~ → B~  B~ → −E~ v r → r ;=9KMC
RS;=RSJHA=A=[I\P9CF9i>@;=Th[MLPT(SLPOH>Q;=9S[I;=9SRHL&JH9Th;=<QJHKI>„C_RSLŒ<Q[MU&[MKMCFTCb†—‡Z‚Ç¥}‰nŠ¡‚
+
Z
zz

2
2

2
Z
zz
zz
0
!


3

Z
3
−ik̃L
−ik̃L




s
1
s
1
−ik̃L

s ik̃L
2

s ik̃L
2
s
1
s
1
−ik̃L
s
1
s
1

s ik̃L
2

s ik̃L
2

s 2ik̃L
2
s 2ik̃L
2

0
0
0
zz
s
a
p
a
¦¨§{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
‡Pq
.¢L}RSLP<¨KILP<¢TQ;=<hJSKI>@CFRSLP<¨CF9i>@;(TQ[MLFTQ;=<0acJS;(TQLP9oLFOH>@;(9S[MRSLF<¨E^CFTQCkJS9)mFC_KMLPT~_hLkRS;=K}mP;=A¡>@LPT®RS;gLP9HR^C
E^CFT@C_KM;=KIL Q~ ‚0èZJSU,CF9SRHL KMC˜;=A(J^CFA=[I\P9]†ˆ‡Z‚Ï¥Z‰nŠœ<QLPOSTh; Q~ vWC ^C_RS[M;(9SRSL"K`C˜A(LP9i>@Th[MOSJSA([M\P9WC_9^‹FKMLFwiC
E^CFT@CVELPK`C_TQ[Md=CFA=[I\P9 p ^LPOH>Q;=9S;(U,LP<€~^9SCFKMU&;=9i>@;ƒ;(K>@;(9S<QLPT2RS;|;(<hacJS;(TQd=LF<œ>QLF>„CFK—
−Tzz
k3
h̄c Z 2 Z
d
Q
dkf
=
4π 3
q
2; 1 + r r e + 1 + r r e ,
†ˆ‡}‚Ç¥P‡FŠ
×
RSLP9SRS;b’S;=U&LP<xRH;šQCFRSL [MU&ESKMG>@A=[>„C•KMk̃(1
C‘RS;(−Ež;(r9SrRS;=e9HA=[`C‘)RS;=Kkk̃(19}åHU,−;(rTQL rRSe;bLZA()JSE^CFA([M\P9 f v]RS;bKMCF<
CFU,EHKM[I>QJSRS;(<gRS;|TQ;¡ ^;¡XZ[M\F9 >@CF9i>@LŒ;=9 Q~ A=LFU,LV;(9 k ^LŒRS;|U,CF9S;(T@CV;=yzJS[mFC_KM;=9i>Q;F^;=9 Q v q ‚
8:9 KMC ;(AF‚j†ˆ‡Z‚Ï¥P‡PŠ{;(<QA=Th[MOS[IU,LF<7KMC <hJSU,C<QLFOSTQ; Q~ A=LPU&LbJS9SC[M9i>Q;=wPTQCFK:;=U&ESKM;=CF9SRSLKMLP<7CFTš¤
wPJSU&;=9i>@LP<)JS<QJSCFKM;(<B†hpá‰P‰=s`Š¡‚®æ¨CFTQCbELZRS;(Tl9zJSU&;=TQCFToKMLF<)ELP<Q[IOSKM;(<)m_CFKILPTQ;(<lRS; Q~ A(LP9S<Q[IRS;=TQCFU&LP<
ESTQ[IU,;(TQL7JS9˜<Q[I<h>Q;=U,C)A=LF9˜JS9^Cl<hJSE;=Th~SA=[M;k~^9S[I>@C A ;=9˜A(JHvPLP<jOžLPThRS;=<j[MU&EžLP9H;=U&LP<“A=LP9HRS[MA([MLP9S;(<
RS;bacTQLF9z>Q;=TQCH‚j™0C‘acLFTQU,C‘U,‹F<x<h[MU&ESKM; RS;•ESThLZA(;=RS;(T˜;=<"EHTQLPELP9S;(TxyzJS;NK`Ce<QJSE;=T?~^A=[I;•<Q;=C JS9
A=J^CFRHT@CFRSLVRH;{K`CFRHL H vxEž;(RS[MT;=9"<QJH<€OžLPThRS;=<jA=LP9SRH[MA=[ILP9S;(<jE;=TQ[I\ZRH[MADC_<jRS;ƒùLPTQ9Z¤ˆmPLPC9 |CFThUxC_9
E^CFT@CVKILP<€ADCFU&ELP<=S[—‚r;_‚I
†—‡Z‚Ï¥ iŠ
~
~ + H, y, z) = E(x,
~
E(x,
y, z) = E(x
y + H, z)
v
%

s
1
s
1

p p 2ik̃L
1 2
p p 2ik̃L
1 2
s 2ik̃L
2
s 2ik̃L
2


—† ‡Z‚Ï¥=!zŠ
8“<?>„CF<lA=LP9SRH[MA=[ILP9S;(<)[IU,ESKI[MA=CF9Wy}JH;˜KMLP<|mP;(A(>QLPTQ;(<oRS;˜LP9HR^CE^C_T@CFKI;=KMLF< Q~ RS;(Ož;(9 Ež;(Th>Q;=9S;(A=;=TŒC_K
A=LP9thJH9z>QL {Q~ } ^RSLP9SRS;
†—‡Z‚Ç¥F¥PŠ
~ = 2π (h, k).
Q
8“<?>@LP<®mFC_KMLPTh;=<¢Ež;(TQU&[I>@[IRSLP<acLPThUxCF9)JS9^C…Th;(>QGMA=JHK`HCkA(J^CFRSTQCFR^C…A(JHviC{A(;=KIR^C…JS9S[>„CFTh[`C€>@[I;=9S;JS9ŒCFTQ;=C
†—‡Z‚Ç¥_ÀiŠ
∆ Q = (∆Q) = (2π/H) = 4π /A,
v";=9 ;=KKIGMU&[I>@; A → ∞  ∆ Q → d Q → 0 SK`Co<hJSU,C&<hLPOSTQ; Q~ <h;)C_ESTQLDXZ[MU,CVCVK`CV[M9i>@;(wPT@C_K
†—‡Z‚Ç¥FqPŠ
A
A
∆ Q... →
d Q...
... =
~
~ + H, y, z) = B(x,
~
B(x,
y, z) = B(x
y + H, z).
hk
hk
2
2
X
~
Q
2
2
2
2
4π 2
X
hk
2
4π 2
Z
2
¦¨§{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
Placa 1
‡Fý
Placa 2
L
D/E7F+GHIKJL"MNO-P?QRPBSUTWV3X'PBY[Z9R]\_^a`$P?b Qdc=PTWR;efRP?g]X4`$P?YX4cY XihX X'YX
`$PkY X XlY[m n$o9n9m(T^9prq5XsV3X
2 c=PTWR}Q~lSXsY Y~`$P
PBn9ht^infTWV3XsV[Y XuvRPBVWw2XSW^lxVP/Y XycYXlhBX PBSAn9PBhtP2SWXlVWm ^zVPBSUTXlVAY Xdht^infTWVm"xR=1htm {ln?`P2Y+Z9j R]\_^`$P|b Qd
2
YXlSAc9Y XihXlSBp
ˆf‘I“’-I-”E7†5I=†–•5’iŽ2E7’iE‘I
ˆf—:ˆfk‹<E7—B‹8…K˜™Grˆšˆiƒ‚5G„B…‡ˆf7I
−Tzz
V0
ŒrGrˆi—f›Œœ…+H'’f…+r—ŽHBGr’f’iE7+A›ˆi’-I‹<Œœ…žˆf7ˆi’iŽ2H2…+‹8IF+rŸlŽE ’f…žˆf¡I‹a¢rI—£’f…+E r’fE †rˆL
€ƒ‚5G9„B…‡†rˆŠ‰7‹<ŒˆlŽG
’-I-”E7†5I†KHBˆ-I
V
¤/IH2Iƒ…+¢¥ŽˆfˆfH¦‘Id§_GrˆiH2¨-IŽ…=Ž]I1—B…+¢rH2ˆ7I©Œ‘I’fI
‘I}Œr7I’-I
Ž2ˆfrˆi‹8…+—¦˜™GrˆzH2ˆi—BŽ]I=Hªˆi1‰7‹<ŒˆlŽG'˜™Grˆ‚5G«+ˆz†rˆi—2†rˆ
2
I’fE7I}ˆf5E7¥•5rE Ž2…rL=­ªI}•5FGrHI8®J™L M™¯¦‹aGrˆf—ŽH2I'G<ˆi—2˜™Grˆi‹:I'†rˆ©‘I—/’f…+1ŽHBE7¢rG’fE7…rˆf—†rˆi
2¬
‚5G9„B…<†rˆ'‰7‹<ŒˆlŽGC—2…+¢H2ˆ'‘I°Œr7I’-I
I’fE7I°ˆfE7¥•5rE Ž2…
L±­¦I²’f…+1Ž2H2E7¢Gr’fE +³IA‚5G9„B…<†rˆ'‘I°Œr7I’-I
2
2¬
—2ˆ;…+¢¥ŽE7ˆirˆ:—BE7F+GE7ˆf†r…´ …+—‹<E7—B‹8…³Œ5I—B…+—a†rˆžI=H2H2E ¢5I›¦ŒœˆfH2…´H2ˆfˆi‹8Œ‘I¨fIr†r…µ7IŠŒr7I’-I
Œœ…+Hˆi
1
—2E7—Žˆi‹:I}’i…+‹8ŒGrˆf—Ž…£Œœ…+HI‹a¢rI—|Œr7I’-I— « ›=«HBˆfˆf‹<Œr7I¨-I†r…£‘IkŒ‘I’fI Œ…H|ˆf—BŒ5I’fE …}”=I’f‰ …rL
1 2
2
­¦GˆfF+…r›ˆfœ‚5G9„B…a†ˆ}‰7‹<ŒˆlŽG;ˆi;ˆfˆf—BŒ5I’fE …a”=I’f‰ …‹8¶—dI ‘¶'†rˆ£I‹¢5I—yŒ‘I’fI—yˆi—BŽ]¶a†rI†r…°Œœ…+Hy7I
‹8E —2‹8Iˆf’iG5I’fE +‡®JL·+J+¯]›rŒˆiH2…²—2G—BŽE¸ŽG«ˆfr†…
Z
r2α → 0 ¹
ˆi—BŽ2…8ˆi—
Z
h̄c
k 3 ºƒˆ 2
®J™L»·=¼1¯
.
−Tzz = 3 d2 Q dkf
4π
q
k̃
º©ˆf—Ž]I†r…aˆf—Žˆ}HBˆf—BGr ŽI†r…†ˆk‘I'ˆf’iG5I’iE7+“®WJL·+J+¯|ˆkE71Ž2ˆfF+H2Ir†r…4—2…+¢rHBˆkˆi[¶HBˆ-I›™…+¢¥Žˆfˆf‹<…+—
‘I§_GrˆiH2¨-I°Ž…=Ž]IA—B…+¢rHBˆ7I°Œr‘I’fI
Fz
h̄c
= 3
A
2π
†r…+r†rˆ
Z
2
›
2
dQ
Z
k 3 º©ˆ
dkf
q
"
#
1
1
,
+
k̃(ξ s − 1) k̃(ξ p − 1)
ξ α ≡ r1α r2α e2ik̃L
−1
.
®J™L»·=½1¯
®J™L ¾+¿1¯
À …+‹8…8ŒrGrˆi†rˆˆi—2ŒœˆfHI=H2—2ˆaI:Œ5IHŽE7H©†rˆ4‘I²ÁˆiH2’iˆfHIž­ˆl«Š†ˆaÂ?ˆl΅+[›—BˆaŒrGrˆi†rˆa‹<…+—BŽ2HIH©˜™Grˆa7I
§_GrˆfHB¨-I²—2…+¢rHBˆ'‘IaŒr7I’-I
1
ˆf—©†rˆ'E F+G5I[ŽI‹8IÄr…a«ž†rˆ'—BE7F+r…²…+ŒGrˆf—Ž…rL
¦¨§{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
‡#
8:9o;(KZEHTQLPOSKI;=U,C2acGI<Q[IA=LœK`C€acTQ;=A(JS;=9HA=[`C ω ;=<¨TQ;DC_Kzv7ELP<Q[>@[mFCZ‚næ®LPT¢KMLœ>„CF9i>@LH q = ω/c >@CFUoOS[I”=9
KMLV;=<(‚Å8:K9}åHU,;(TQLŒRS;|LP9SR^C k TQ;(ESTQ;(<Q;(9z>@C,LP9HR^CF<€yzJS;|<Q;7ESTQLFE^CFwiCF9";(9KMCoRS[MTh;=A(A=[M\F9 z A=JSCF9SRSL
;=<oELP<Q[>@[mPL k ∈ (0, ∞) ‚ kLF>@CFU,LF<lyzJS;"KMLF<lELP<Q[IOSKM;(<)m_CFKMLFTQ;=<lRS; z ;(<h>„‹_9]C_A=LF>@CFRSLP<oRH;=9i>@ThL
RS;,KMC A=CDmZ[IR^CFR0KILA(J^CFK“Ež;(TQU&[I>Q;&KMC ;YXZ[M<h>Q;=9SA([`C RS;xU&LZRHLP<|ESTQLPEH[MLP<|;(m_CF9H;=<QA(;=9i>@;(<=‚:æ¢LZRH;=U&LP<
[M9SA(LPTQELPT@C_Tj”=<?>@LP<2C_KAD‹FKIA=JSKILo<Q[uEž;(TQU&[I>@[IU,LF<£yzJS; k >@CFUoOS[I”=9x>QLPU,;kmFC_KMLPTh;=<g[IUxCFwF[M9^CFTh[MLP<“RS;ƒKMC
acLPTQU,C k = iκ ‚æ®CFT@CxwPCFT@CF9i>Q[MdDC_TkyzJS; q <h;DC˜TQ;=CFKˆÅRS;(Ož;(U,LF<{;¡XZ[MwF[MTœ;=9N;=<?>@;oA=CF<QLxy}JH; κ < Q ‚
8“<|RS;(A=[IT=K`C"[I9z>Q;=wPTQCFK<hLPOSTQ; k ;=9eK`C";=A_‚†ˆ‡Z‚Ï¥ #iŠ…<h;,Th;DCFKI[MdDC˜<hLPOSTh;&KMC˜>@T@CDvF;=A(>QLPTQ[MC k = iQ →
‚gæ®LPT LF>@ThLK`CFRSLH Q ;=<B9S;(A=;=<QCFTQ[MCFU&;=9i>@;eTQ;=CFKƒESJS;(<bK`C_<˜mFC_TQ[`C_OSKM;(< x v y ESJH;=RS;(9
0 → ∞
>@LPU,CFTm_CFKMLFTQ;=<£CFThOS[I>QT@CFTh[`CFU&;=9i>Q;gwPT@C_9SRS;=<(–zJS9^C)LF9SR^C|y}JH;kRS;(ADCF[IwiC|;¡XZEžLF9S;=9SA([`CFKIU,;(9i>@;2;=9,JS9^C
RS[MTh;=A=A([M\P9BA(TQ;=A(;=ThG`CV;¡XZELP9S;=9HA=[`C_KMU&;=9i>@;ƒRS;7U,CF9S;=TQCo9SL,CFA=LF>@CFR^CV;=9 K`CoRH[MTQ;(A=A([M\P9BLPESJH;=<h>@CH‚
8:<oRH[IacGMA([MK“;(m_CFKIJ^CFT7RS[MTh;=A¡>„CFU&;=9i>@;xK`C;(AF‚g†ˆ‡Z‚Ï¥ #iŠ|ESJS;=<l;=Kj[I9i>@;=wFT@CF9SRHL•A=LP9i>Q[M;=9H;xacJS9SA([MLP9S;(<
y}JH;o<Q;lm}JS;(KImF;=9NT@‹_ES[MR^C_U,;(9z>Q;lLP<hA=[IK`CF9i>@;(<…A(LP9HacLPThU,; k CFJSU&;=9i>@CH‚è};=TQGMC,A(LP9imP;=9H[M;=9i>Q;V;(m_CFKMJSCFT
K`Ck;(AF‚S†ˆ‡Z‚Ï¥ #iŠ0A=LP9o>Q”=A=9H[MADC_<RS;jm_CFTh[`CFOSKI;A(LPU&ESKM;ŽQCH‚_èZ[M9o;(UoO^CFThwPLStK`CœE^CFTh>Q;gTQ;=CFK}RS;gJS9^C…acJH9SA=[I\P9
CF9^CFKIGI>@[IADC{9SL7;=<jCF9^CFKIGI>Q[MADCZ‚Pæ¨C_T@C)A(LP9imP;(Th>@[ITjC_K[I9i>@;=wFT@CF9SRHL);(9xJS9^C|acJS9SA([M\P9,CF9^CFKIGI>Q[MADC{RH;=O;=U&LP<
<@CFA=CFyT %2;2RS;œK`C|[M9i>@;(wPT@CFK—1‚ kL_>„CFU&LP<:yzJS;…;(9x;(K[I9i>@;=T?m_CFKML k ∈ (0, ∞) ;=K^aCFA¡>@LPT dk k /q ;(<Th;DCFK—‚
›…<Q[IU,[I<QU&LSz;=9";(K[M9i>@;(Thm_CFKIL k ∈ (iQ, 0) ;(KuaCFA¡>@LPT dk k /q >„CFUlOS[M”(9˜;=<€Th;DCFK—‚Hæ¢LPT€KMLl>@CF9i>@LSZ;=9
CFUoO^CF<€Th;=wP[ILP9S;(<€EžL}RS;=U&LP<€Th;=;=U&ESKMCFdDCFT€KMCo;=A_‚¢†ˆ‡Z‚Ï¥ #iŠjELPT
†ˆ‡}‚ÏÀH‰ŸŠ
F
h̄c %2;
k
1
1
.
dQ
=
dkf
+
A
2π
q (ξ − 1) (ξ − 1)
›…’SLPTQCl<QGžEžL}RS;(U,LP<j;=U&ESKM;=CFT£>Q”=A(9S[MA=CF<gRS;{m_CFTh[`CFOSKI;2A=LPU&ESKM;ŽQClE^CFTQClADC_UoOS[`C_T9zJS;(<h>@TQCV[M9i>@;(wPT@CFK
CŒJH9^C&;(yzJS[Im_CFKI;=9i>@;ƒyzJS;7<Q;=C&U,‹F<€<Q;(9SA=[IKMKMCoRS;|;(m_CFKMJSCFT=‚
æ®CFT@CoA=CFUoOS[MCFTj9zJS;(<h>QTQL&A=LP9i>@LFTQ9SLVRS;ƒ[I9i>@;=wFT@CFA([M\P9˜;(9BK`Cl;(AF‚0†—‡Z‚ÏÀZ‰nŠ£9SL_>„CFU&LP<gESTh[MU&;=ThLlyzJS;
;=KSA(LP9i>@LPTh9SLƒA=;(TQT@C_RSL,†U&LP<h>QT@CFRSL…;=9ŒK`Ck~SwPJST@C|‡}"‚ !zŠ¢;=9Œ;=KSyzJS; k m_C{RS©; )ƒC ∞ RS;(<QESKMCFdDC_RSL|KM[MwF;=T?¤
CFU,;(9i>@;g’^C_A=[`C…CFTQTh[MO^CHnKIJS;=wPLkmFCƒC i∞ C{KML…K`CFThwPL…RS;jJS9ŒA=[MThA=JSKIL…RS;€TQCFRS[IL…[M9Z~^9S[I>QL…v7~^9^CFKIU,;(9z>Q;
TQ;=wFTQ;=<QC˜CFK0LPTh[MwP;(9•CxKIL,KMCFTQwPL&RS;(K®;Žh;l[MU,CFwP[I9^CFTh[MLV9SL,;=9SA([M;=ThT@Cx9H[M9SwPåH9bELPKMLS^A(LPU&LxELZRS;(U,LF<
3
3
z
3
Z
2
Z
2
iQ→0→∞
"
s
p
#
¦¨§{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
+)
k 00
iQ
k0
D/E7F+GHI:JL"ÅNœÆªmXsÇiVXlQdXdPBn£PYeiR9PSWPm"Y R9SUTWV3XdRnhBXsQdm"n9^ƒ`$Pm nfTWPÇiVXihtm {ln}SU^ixVPPY¥cY Xln^ƒh^lQdcY P_\_^ƒefRPn9^
PBn9htm PVV3Xzc^lY ^iSBp
2
(r1α r2α ) ei4kL
Espejo 1
r1α r2α ei2kL
Espejo 2
r2α ei2kL
L
D/E7F+GHIJL»·™N™ÈœSUcP_\_^fSASUPBc9XsV3Xl`$^fS5Rn9X`$mS_T3Xsn9hm X pfɪn©uv^lTW{lnƒPBSVWPZ9P_\UXi`$^m n$o9nm"TXlQyPBnfTWP/Pn©YXzhBXgfm`Xi`eiR9P
P|Ì9X Xsn?c=^iY"^fSAPn}Y XyPÍ-c9VWP2SUm {lnk`$PY XuvRPBVWw2Xp
Ê _S T^iS[uv^iVWQXsnp-Ë'^iSUTWV3XsQd^iSAefRPn^dPBSAc^iSWm xY"PefRL
j
‹8…+—ŽH2IHd’f…‹8…²—2E F+Grˆ=L5­…+—©Œ…+ …+—†rˆ'‘Iaˆi’fG5I=’fE7Î®WJL ¾¥Ï9¯zˆf—Ž]¶³†5I†…+—©Œ…+H7I°ˆlÐ¥ŒrH2ˆi—2E +
1 − r1α r2α e2ikL = 0.
®WJL ¾1J+¯
Ñ E78ˆf‹a¢5I=H2F+…r›™Ÿi—BŽ]I4ˆf—y7I’i…+r†rE ’fE +:˜™Grˆ?†rˆf¢œˆfž’iGr‹8Œ7E7H …+—z‹8…†r…+—zr…+HB‹:I ˆf—†rˆ?‘I'’-I-”E7†5I†[›
’f…+‹<…°‹8…—BŽH2I‹<…+—©I°’f…+1ŽE ™G5I’fE +C’f…+ŠI-«Gr†5I²†rˆ'‘I•5FrL®WJL·+¯sL
À …r—2E †rˆfHBˆ´†r…+—8—2GrŒœˆfH•5’fE ˆf—8H2ˆi‚rˆf’iŽ2…+HI=—f›zGr5I¡ˆi§_H2ˆi1Žˆµ†rˆ³‘IK…Ž2HI›’f…–’f…ˆi•r’fE7ˆi1Žˆf—:†rˆ
H2ˆi‚rˆlÐ¥E7+
«
HBˆf—2Œœˆf’lŽE¸”I=‹8ˆi™Ž2ˆL҃5Iµ…+r†rI´’f…+ÓI=‹8Œr E Ž2Gr†
E r’fE †rˆ:—B…+¢rHBˆ;‘I´ŒH2E7‹<ˆfH2I
r1α r2α
1
Œr‘I’fI›ª†r…+r†rˆ:—2ˆžH2ˆl‚5ˆt„2Iµ’i…+ÓI‹<Œr7E¸ŽGr†¡†rˆ;HBˆi‚5ˆsÐE + α L|€Î—2ˆiF+GrE †5IµE7r’iE7†rˆ8—2…+¢H2ˆ;7I´…ŽH2I
r1
Œr‘I’fI›fHBˆi‚5ˆ„2¶r†r…—2ˆzˆfˆf77II …+H2Id’f…+4I‹<Œr7E¸ŽGr† α «}’i…+4G5I©§UI=—2ˆ ikL ›-«'I—B‰+—2G’fˆf—BE ”=I‹<ˆf1Žˆ=L
r2
e
¬
À I†5Id”+ˆi¨z˜™GrˆzI=Ž2HI-”E7ˆi—Id‘Id’fI-”¥E †5I†I†r˜™GrE ˆfHBˆGr5Id§UI—2ˆ ikL «}—BGI‹8Œ7E Ž2Gr†}—Bˆ|”+ˆ‹aGr¸ŽE7Œ7E7’fI†5I
e
¨¦ §{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
H‰
CFKI>Q;=TQ9SCFR^CFU&;=9i>@;€ELPT r L7EžLFT r ‚ZèZ[^<hJSUxC_U,LP<:>QLZR^C_<£KMCF<:A=LF9z>QTQ[IOSJSA=[ILP9S;(<C|K`C|TQ;¡ ^;=A¡>@[mZ[IR^CFR
9SLP<2TQ;(<QJSK>„CFTQ‹ŒJH9^CV<QJSU,CVwP;=LPU&”(>QTQ[IADCoyzJS;7<Q;|ESJS;(RS;)Th;=;(<QA=Th[MOS[IT€A=LPU&L
†ˆ‡}‚Ï+À PŠ
r
r r e
=
.
r
1−r r e
kLF>@CFU,LF<lyzJS;BKMLP<)EžLFKMLP<oRH;˜;=<?>„CN;¡XZESTh;=<Q[I\P9<QLP9KILP<oU&[M<QU&LP<)y}JH;"KMLP<oRH;=<QA(TQ[>@LP<VELPTlKMC•;=A_‚
†ˆ‡Z‚rÀi‡PŠY‚^8:9;(<h>„CŒ<Q[>@J^C_A=[M\F9BacGI<Q[IADCHZ;=KyzJS;7K`CoTQ;¡ ^;=A¡>@[mZ[IR^CFRB>@;(9SwiCŒEžLPKILP<g[IU,ESKI[MA=ClyzJS;|A=LP9 <Q\PKIL
’^CFA=;(T:[M9SA([MRS[IT®JH9VEžL}A=LƒRS;€KIJSd€C{JS9SC{RS;€K`CF<¨ESK`CFA=CF<=_”=<?>„C{TQ;¡ ^;šQCƒJS9SC{ADCF9i>@[IR^CFRV[I9H~^9S[>„C…RS;€KIJSdF
Lƒy}JH;œ9H[S<Q[Iy}JH[M;=TQC{;=<“9S;(A=;=<QCFTQ[IL|[M9SA([MRS[IT¨KMJHdgE^CFTQC…mP;=ThK`CƒTQ;¡ ^;šQC_R^CH‚iÿ…[IA„’SL|RS;€LF>QTQL|U,L}RSLStESJS;(RS;
’^CFO;=T¢KMJSd£RS;=9i>QTQL{RS;K`C2ADCDm}[MR^C_Ro<Q[I9l9S;(A=;=<h[MR^C_RoRS;jJH9^Ck;YXHA([I>@CFA=[I\P9l;YXZ>Q;=Th9^C…<Q[I;=U&ESTQ;“v7A=J^C_9SRSL
<Q;&A=JSU&ESK`C"KMC A(LP9SRS[IA=[M\F9†—‡Z‚rÀi‡PŠ¡‚0ÿk;=O;,<h;=T7A=K`C_TQL y}JH;,;(<h>„C <Q[>@J^CFA([M\P9WELZRHTQG`C"<hLP<h>Q;=9S;(TQ<Q;x;=9
;=K¨>Q[M;=U&EL"<QLPKMCFU,;(9i>@;o<h[:9SLB’}JHOS[M;(T@CBCFOS<hLPKMJH>@CFU&;=9i>@;V9^C_R^C"RS;ŒRS[M<h[ME^CFA([M\P9užLBELZRSThG`C˜RS;(ADCF;(T
;=9B;=Ku>Q[M;(U,ELo<Q[u’}JHOS[M;(T@CVCFOS<hLPTQA([M\P9uHE;=TQLŒ;(9 ;(yzJS[MKI[MOSTh[ML)>Q;=ThU,L}RS[M9S‹FU,[IA=L7;=KA=CFU&EžLŒCF<hLZA([`CFRHL
CB;(<h>QLP<)U&L}RSLP<{9SL ELZRSThG`C˜A(TQ;=A(;=T|;=9e;=K>Q[M;(U,ELSuELPTƒKIL A(J^CFK:KILP<ƒA=;(TQLP<|RS;ŒK`C˜;(AF‚†ˆ‡}‚ÏÀi‡FŠ…9SL
ESJS;=RH;=9>Q;=9S;(T€acTQ;=A(JS;=9HA=[`C_<œA(LP9 E^CFTh>Q;=<2[MU,CFwP[I9^CFTh[`CF<£EžLF<Q[I>Q[Im_CF<(H[ˆ‚þ;F‚r &ŽU q ≤ 0 ‚
8:9ŒA=LF9SA=KIJS<Q[I\P9FKMC…>@TQCŸvF;=A¡>@LPTh[`C{A=;(TQTQCFR^CƒRS;=<hA=Th[I>„C|CFThTQ[IO^C…9SLƒ;(9SA=[I;=TQTQCƒ9S[M9SwFåS9VEžLFKMLS‚_™JS;=wPLH
9SLF>„C_U,LP<…y}JH;VK`C˜A=LF9z>QTQ[IOSJSA=[I\P9WC˜K`C˜[I9i>@;=wFT@CFK¨CF9i>@;=Th[MLPT…ESTQLŸmP;(9S[M;(9i>@;ŒRS;=K¨<Q;=U&[MA(GMThA=JSKIL˜[M9H~S9S[I>QL
;=<29zJSK`CVESJS;(<kKMCo;¡XZELP9S;=9HA=[`C_K
†ˆ‡}‚ÏÀ !iŠ
e
=e
e
;=9N;=K¨RS;=9SLFU,[I9^CFRSLPTkRS;VK`C,;=AF‚†ˆ‡Z‚rÀH‰nŠœ<Q;ŒCF9zJSKMCxE^CFTQCxmFC_KMLPTh;=<{RS; k A=LPU&ESKI;šhLP<{[I9H~^9S[>„CFU&;=9i>Q;
wPT@CF9HRS;=<(‚0æ®LPTƒKILB>@CF9i>@LSuK`Cx>@TQCDvP;=A¡>@LPTh[`C"<QLFOSTQ;ŒK`C˜yzJS;&;(m_CFKIJ^CFU&LP<{KMC˜[M9i>@;(wPT@CFK®;=9 k ;=9eK`C˜;=A_‚
†ˆ‡Z‚rÀH‰nŠESJS;(RS;jRH;(acLPThUxCFTh<Q;“A=LF9zm}[ITh>@[I;=9SRHLP<Q;;=9lJH9^Ck[I9z>Q;=wPTQCFKi<QLFOSTQ;£;=K};šh;£[MU,CFwP[I9^CFTQ[ILgRS;(<QRS; k =
’SCF<h>@C i∞ ‚P8:9Œ;=<?>@;€A=CF<QL k <h\PKML{<h;=TQ‹|[IUxC_wP[M9^C_TQ[`C€ELPTKIL{A=J^C_KSK`CF<®;¡XZELP9S;=9HA=[`C_KM;=<:LF<QA=[IK`CF9i>Q;=<
iQ
y}JH;)C_E^CFTQ;(A=GMCF9B;=9 K`CV;=A_‚¢†ˆ‡Z‚rÀH‰nŠj<h;7A(LP9im}[M;=T?>@;(9bCF’HLPT@CV;=9 ;¡XZELP9S;(9SA=[MCFKM;(<œRH;=A=Th;=A([M;=9i>Q;=<=‚
æ¢LPT2LF>@ThLŒK`CFRSLHSCF’SLFT@C k = iκ A=LP9 κ > Q ‚^æ¢LPT2A=LP9S<h[MwPJH[M;=9i>Q;F
†—‡Z‚ÏÀP¥PŠ
Q +k =Q −κ =q <0
α
1
α
2
α
1
X
n
α α 2ikL n
1 2
2ikL
2
2
α
1
α α 2ikL
1 2
2ik 0 L −2k 00 L
2
2
2
¦¨§{©:ª=«…¬®­£¯±°H² ³{´“³"¬j¦€¦¨µ¡¶€·¸³{´¹­§»ºƒ¬®´“¼k½u§³{´±¦¨§ƒ¾}µ?¿µš¼
v q ≡ iu CF’SLPTQCV;=<2[MU,CFwP[I9^CFTQ[MCHiA=LF9
u=
q
i‡
†ˆ‡}‚ÏÀPÀPŠ
κ2 − Q 2
TQ;DC_Kˆ‚^fjLF9;(<h>@LF<kA=CFUoOS[ILP<=SKMCo;=A_‚¢†ˆ‡Z‚rÀH‰nŠj<h;7;YXZESTQ;(<@CxC_’SLPT@CVA(LPU,L
F
†ˆ‡}‚ÏÀiqFŠ
κ
1
h̄c %œ;
1
dκf
=−
+
dQ
.
A
2π
u (ξ − 1) (ξ − 1)
Á¨[M9^C_KMU&;=9i>@;_9SLF>@CFU,LF<)yzJS;˜<hLPOSTh;x;=<?>„C>QT@CDvP;(A(>QLPTQ[MCRS;,[M9i>@;(wPT@CFA([M\P9 κ  u veA=LPU&LmF;=Th;=U&LP<
RS;=<hESJS”=<Œ>„C_UoOS[M”(9äKMLP<lA=L};(~^A([M;=9i>Q;=<ŒRS;BTQ;¡ ^;¡XZ[M\F9 †cESJS;(<&;(<h>@CFT@‹F9R^C_RSLP<V;=9>@”(TQU&[M9SLF<oRS;"KMLF<
9}åHU,;(TQLP<2RS;|LP9SRSCŒRH;(~^9S[IRSLP<2ELP<Q[>@[ImFLP<„Šœ^<QLP9 TQ;=CFKM;(<=‚^æ¢LPT2KMLo>„C_9z>QLSS;(K[M9i>@;(wPT@CF9HRSLŒ;=<2TQ;=CFKˆHv
K`C{acJS;=ThdDC7EžLPT“JS9H[MR^CFR&RS;k‹FTQ;=C7<hLPOSTh;kA=CFR^C)JH9^C)RS;œK`C_<:ESK`CFA=CF<£C7>Q;=U&Ež;(T@C_>QJST@CƒA=;(TQL˜† f = 1/2Š
;=<h>@‹&RH;=<QA(TQ[>„CVEžLFT
†ˆ‡}‚ÏÀPýPŠ
1
h̄c
1
F
κ
=−
+
dQ
.
dκ √
Z
z
3
z
A
4π 3
Z
2
2
Z
∞
Q
Z
2
∞
Q
2
"
s
"
κ2 − Q2 (ξ s − 1)
p
#
(ξ p − 1)
#