On the use of the Reciprocity-Gap functional in inverse scattering from planar cracks
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On the use of the Reciprocity-Gap functional in inverse scattering from planar cracks Amel Ben Abda — Fabrice Delbary — Houssem Haddar. N° 5290 Juin 2004. N 0249-6399. ISRN INRIA/RR--5290--FR+ENG. Thème NUM. apport de recherche.
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(72) $ ' 1 N1 H L$W4 ( I 1 H 1( W # 1N Wu˜ +. ^0_ovp\¯_ac.uTrt_ac+¥uac Π ypiÅoua~(¯u rz_acC*x Å σ .uT¤Mc"ac*rzcWxzbdyuacWT§±xz~b& ÅAu~(¯¥¨cPqc~§ « ¢ c p³_ ¥¥3u~(¯¦qy|pz.vpzpirt_acdvp³cC~§1rz_acdxtcW.y¨axt~q*y¨rsn8°§±vau*rzy~u¥ RG rt~xtcW.~(£cWx Π u [u] « [u] ¢ cp³_ ¥¥Íqc.ua~rzc9¤on rt_a"c 5a~vaxty¨cWx1rtxtup³§±~xtb~u Π qc ucW§±~x v ∈ L (Π) ¤on F Z §±~x¥¥ ξ ∈ R , ξ · Nˆ = 0. F(v)(ξ) = v(x) exp(iξ · x) dx 5~x θ ∈ C a¯c9qc uc ∈R . lqy¨bda¥c.¥.va¥rzy~up0pz_a~(¯¶rt_(r§±v(x; ~x θ θ)p³v= _exp(iθ rz_ (r |θ|· x)= kx v(·; u θ) ∈ H(Ω) Π Π. 1. 3. Π. 3. 3. ˆ) RG(v(·; θ)) = i(θ · N. Z. [u]Π exp(iθ · x)ds(x). () Z½ Á *M½ +-,/. « ¢ c0ptpzvabdcIrz_r ypÅoua~(¯uu 9yp3qc uacW¤Anrt_ac)cP{v (rzy~u ¯_c.xtc Nˆ ypCTvauyµrCua[u]~xtb ¥Lrt~ Π u γ y|pCΠTy£cWu *~u psr uArW«35xz~bua~(¯ ~u √ qcWuax·~rtNcWˆp+=rz_aγc .~bda¥c*ºpt{Avxtcxz~o~r¯y¨rz_8ua~uuacWrzy£c2y¨b yuxtn xzrW«UQ c*r ξ ∈ R ξ · Nˆ = 0 au
(73) qc uc σ. Π. 3. θ(ξ) = ξ −. ÕÕÑZ\[]^_. p. ˆ k 2 − |ξ|2 N..
(74) W V $1
(75) BW C. i¤o£oy~vpz¥¨n ^0_c.xtc*§±~xtc. θ(ξ) · θ(ξ) = k 2. uCrz_acWxzc.§±~xtc. v(·; θ(ξ)) ∈ H(Ω). «Ifip³yuaL:R <)A~uaccPpzy¨¥nC£c.xty cWp rt_(r. Z p p RG(v(·; θ(ξ))) = −i k 2 − |ξ|2 exp(−iγ k 2 − |ξ|2 ) [u]Π exp(iξ · x) dx. Π. §±~x¥¥ ξ ∈ R , ξ · Nˆ = 0, : < ¯ _y _ c*ºqa¥y|*y¨rz¥n¹qc.rzcWxzbdyuac [u] ¤onrtÅAyua8rt_ac yuA£c.x p³c5~vaxty¨cWx9rzx u ps§±~xzb ~§0rz_ac xty_Ar_u pzy|qc« ) Z½ Á *M½ +-,/. ½ , Á $ Nˆ « Q c*r S ¤ crt_ac9vauay¨ripza_ac.xtc9u
(76) qc ua!c a§±~x θˆ ∈ S F([u]Π )(ξ) =. i exp(iγ. p. k 2 − |ξ|2 ) RG(v(·; θ(ξ))) p k 2 − |ξ|2. 3. Π. . ˆ = Λ(θ). 1½ D
(77) \P
(78) \ 1 W1. sup. ϕ∈kS; ϕ⊥θˆ. W1
(79) W " $W 2k B M $ η
(80) \ ±Nˆ
(81) \B 1\ H = I Λ E
(82) I ¢ c xtp³r2ua~rty.c"rt_(r Λ(±N) ˆ = 0 p³yu*c. |RG(v(·; ϕ))|.. \
(83) &
(84) 1 18W
(85) 6 H E \4 N 1 W \ H = . f 0 ≤ η < 2k F([u]Π ). " §±~x2¥¥ ϕ ⊥ Nˆ «+eg~(¯¦ptpzvabdc ϕ)) = 0 ¢ « i c a W c q v.cgrt_(r RG(v(·; ϕ)) = 0 §±~x rt_acgc.ºqyp³rzcWu*ci~§ θˆ ∈ S p³v _ rz_r θˆ 6= ±Nˆ u Λ(RG(v(·; ˆ =0 θ) ¥¨¥ ϕ ∈ k S pzv _Zrt_(r ϕ · θˆ = 0 « y|pu cW¥¨¥y¨ p³c .c.uArzcWxzcP¹r9rz_ac ~xzyyu¹~§0bds~xºoy|p E = {ϕ − (ϕ · N u8bdy¨u~xgºoy|p ˆ2|)θˆNˆ·;Nˆϕ| «∈lqkS; y¨u.c ϕθˆ ⊥6= ±θ}ˆ Nˆ 2|θˆ · N|ˆ < 2k «!c.u *c F([u] ) y|py|qcWurtyW¥¥¨n8#.cWxz~ 2k~u ¯_acWxzc u xtc0rz_cgc.u q ~y¨uAr pL~§$rt_acb (s~x)(ºqy|pL~§ «U@nd*~uArty¨uovay¨rsn F([u] ) E\{S , S } y|p #.cWxz~d~u E ¯_ay _8S *~urtxtSqy|rtprt_ac9_ono ~rz_acPp³y|p0~§rz_ac9¥c.bdb a« E ½ q
(86) \ 1 R
(87) UN = * ' M 0 "
(88) \ 1 W1 ,1 H\ f \
(89) Π. 1. 2. 1. 2. Π.
(90) 1 F([u] )
(91) MN 1 E
(92) &L EN4\
(93) R H L
(94) V W 4
(95) V 0B IM 1 \4$ L
(96) \B 11 . k. Π. ) Z½ Á *M½ +-,/.¸ ½ $ , ½ « ¢ c ptpzvabdcC_c.xtc+rt_(r yp2Åoua~(¯u¹u¹pz_¥¥Lqc*rtc.xtbdy¨uac rt_ac+M~pzy¨rzy~u´~§Lrz_ac"a¥|uac Π «9^0_ay|ΠpibC~vauArtprz~rz_ac"c.£(¥v(rty¨N~ˆu´~§Irt_acd*~u psr uAr γ = x · Nˆ §±~x ¥¨¥ x ∈ Π « 1½
(97) Nˆ U6 D
(98) 1
(99) \ $ 1 4 f \
(100) 3
(101) RG 6= 0
(102) \ γ " \ \ H W B
(103) I loyu*c u yp0¤M~vau qcW ~uac9.u~¤qrty¨u8Caxty¨~xzy\¥~(¯1cWx0uvaaMc.x¤M~vau ap u γ ~§ γ «Zσlq⊂y¨u.Ωc RG 6=Ω 0 [u] 6= 0 u F([u] ) 6= 0 « hT~xzcW~(£cWx Mrz_cZp³va ~x³r9~§ [u] γ yp .~bdr apz~ F([u] ) y|pu¥nArzy|i~u R « !cWu*c2rt_ac.xtc2c*ºqyp³rtp θ ∈ k S pzv _rz_r :R < ˆ | < π u RG(v(·, θ)) 6= 0. 0 < (γ − γ ) |θ · N .
(104).
(105). . −. +. Π. 2. Π. Π. Π. +. −. ʱË3ÕÍÊÏÇ.
(106)
(107) . ¢ cpzc*r. ˆ )N ˆ ξ = θ − (θ · N. G. u. ˆ N ˆ θ0 = ξ − (θ · N). u RG(v(·, θ )) = −c exp(−iγ(θ · Nˆ )) «)^0_c*~u psr uAr γ yp0rt_ac.xtc*§±~xzc2vauy{AvacW¥¨n
(108) qc*rtc.xtbCyuacPyu. ˆ )) RG(v(·, θ)) = c exp(iγ(θ · N Z ˆ) [u]Π exp(is · ξ) ds c = i(θ · N. ¯_c.xtc. [γ − , γ + ]. ¤on rz_ac9x (rty¨~. «Lfgpzy¨ua3:R <1~uc9c*r 0. Π. RG(v(·, θ))/RG(v(·, θ 0 )).. " 7 k 7 %93;d
(109) %2Í0 *
(110) 4² 8,*7 ¢ c".~upzy|qc.xy¨u´rz_ay|p2p³cPrzy~u´rz_acCp³McW.y¥3Wpzc+~§)pt.r³rtc.xty¨uaZaxz~¤a¥c.b p MyU« c«i¯_acWu y|pp³y¨rzvrzcP yu §±xtc.cgpz*c« ¢ ci¯~va¥|d¥y¨Åcrz~" ~y¨uAr1~vqr)rt_acg¥yuaÅd¤ c.rs¯1cWc.u RG udrt_ac§ x cW¥ σ(rzrzc.xtu rt_(r y|p*~bdbd~u¥¨nCvpzcW p1art9yuZyuo£cWxtpzcpzW(rzrzc.xtyuac*ºqMc.xty¨bdcWur p)¥yÅcyu '0). +»« ¢ cgrz_acWuqcPpz.xzy¤Mcgyu rtc.xtb p0~§ y¨u.yac.uArga¥|uac2¯06£cPp)rt_ac+pzvay¨rt¤¥¨c _a~y.c2~§Ia(r +§±~xrz_c9y¨uo£cWxtpzcaxz~¤a¥c.b8« 71~u p³y|qc.x §±~xy¨u psr u.c u ∈ H (R \ σ) pt(rzy|p³§±nAyua y¨u R : C < ∆u + k u = 0 ~ u ∂ u=0 σ, ¯_c.xtc ¯y¨rz_ y|pu
(111) y¨u *y|qc.uAra¥|uac2¯06£c2u¯_acWxzc u u=u +u y|prz_cpzW(r³rtc.xtcW¯6£uc(x)pt(rty=p³§±noexp(ikx y¨u"rt_ac+·l d)ˆ-bCbdcWx³§±|cWd|ˆ¥=xt1qy|(rzy~u
(112) *~uqy¨rzy~u . 1 loc. 3. 2. 3. n. i. s. i. vuayµ§±~xzbd¥n°¯y¨rz_ xtcWpz cPrrt~ p³nobdqrt~rzy|c*ºqup³y~u. s. lim. xˆ = x/r. r(∂r us − ikus ) = 0. «8¬»r+yp¯1cW¥¨¥1ÅAu~(¯u '0)P,+)rt_(r+rz_acZpt.r³rtc.xtcW c.¥|¹_prz_ac. r=|x|→∞. eikr (u∞ (ˆ x) + O(1/r)) r. vuayµ§±~xzbd¥nT¯yµrt_xtcWpzMcWrrz~ ¯_ac.xtc yp2.¥¨¥cW´rz_acC§ x c.¥|°r³rtc.xtuÍ«diu°rz_acd~rt_ac.x2_u §±xt~b&rt_ac xtc.axtcWpzc.uArtrzy~uTrt_axˆc.~xzcWb ~§0ux ayrzyua
(113) pz~¥vqrty¨~u´rz~
(114) rt_ac!c.¥bd_a~¥¨r # cP{Av(rty¨~u '0)W,+ Íy¨§ qcWua~rtcWp0rt_ac2§±vaubdc.uArt¥ pz~¥vqrzy~u8y£cWuyuTqybCcWupzy¨~urt_axtc.c9¤on Φ(x, y) = exp(ik|x − Φ(x, y) §±~x x 6= y qrt_ac.u
(115) rz_cpzW(r³rtc.xtcW c.¥|8.u
(116) ¤Mc9xzcWaxzcPp³cWuArzcW~vartpzyqc σ ¤on y|)/(4π|x − y|) us (x) =. ∞. 5xz~b. us (x) =. rz_cpznAbdqrt~rtyc*ºqupzy¨~u ∇y Φ(x, y) =. ÕÕÑZ\[]^_. Z. σ. ˆ ) [u] (y) dy. (∇y Φ(x, y) · N Π. exp(ik|x|) (−ikˆ x exp(−ikˆ x · y) + O(1/|x|)) 4π|x|.
(117) W V $1
(118) BW J. vuayµ§±~xzbd¥n §±~x¥¨¥ y ∈ σ ¯1c9qcWxzy£c ik ˆ (ˆ x · N) 4π. Z. :G1<. ^0_c.xtc*§±~xtc!q.~bdxtyua+rt_ay|pc*ºqaxtcWptp³y~u¯yµrt_>: <~uac2c.rtp u∞ (ˆ x) = −. u∞ (ˆ x) = −. σ. [u]Π (y) exp(−ikˆ x · y)dy.. : J <. RG(v(·, −kˆ x)) 4π. ¯_c.xtc θ) = exp(iθ · x) «
(119) iuac pzc.cPprz_acWxzc.§±~xtcCrz_ac .¥¨~Ap³cd¥y¨uaŤ c.rs¯1cWc.urz_ac xtcW.y¨xz~q*y¨rsn´A §±vurty¨~v(x, u¥gu§ x cW¥ªr³rtc.xtuÍ«E^0_acTy|qc.uArtyµrsn(: J < ¥|p³~¹y¨u qyW(rtcWpdrt_(r rt_acT§±vau*rzy~u¥ .~uArty¨u p bd~xtcTy¨uq§±~xzb rzy~upCrt_u¸rt_acT§ x c.¥|¸r³rzcWxzu uªrz_rZy|p ¯_on?y¨r¥¥~(¯p c*ºqa¥RG y.yµr xtcW.~up³rzxtv*rzy~upyuZrt_ac.p³c2~§ a¥|uac9.xt ÅopW« hT~xzcW~(£cWx 1y¨u¸rt_ac°.p³cT~§pzW(r³rtc.xty¨u axz~¤a¥c.b 0~uac´Wu b Åc´*¥cWx rz_ac°pzpzvabdqrty¨~up yu Q c.bdb pq0« )gu a« a« ^0_ci*~uqy¨rzy~u RG 6= 0 _a~¥ap1y¨§ ud~ua¥¨ndy¨§$rz_aciy¨u *y|qc.uAr1¯06£c u yp1p³v _ rt_(r dˆ· Nˆ 6= 0 «Ii¤o£oy¨~vpz¥¨n yµ§ dˆ· Nˆ = 0 rz_c.u RG = 0 «3eg~(¯ ptpzvabdc RG = 0 (rz_acWu u = 0 u rt_ac.xtc*§±~xzc u = 0 ¤AnT cW¥¨¥y| _ pi¥c.bdb Zu Trz_c+vauay|{Avacd*~uArty¨uovrzy~uTaxty¨u *ya¥¨c«9^0_ac".~uqy¨rzy~u ~u
(120) rz_acpt.r³rzcWxzcWxrz_acWu
(121) ybda¥y¨cPp ~u σ urz_acWxzc.§±~xtc dˆ· Nˆ = 0 « ∂ u¢ =ci0¯c.xtcua~r¤a¥¨cgrz~"¥¨yuaÅdrz_ciptpzvabd∂qrzyu~u =~§Í0¥¨cWbdbd Cq«Ô2rt~"pz~bdcg*~uqy¨rzy~u ~u rz_ciyu*y|qcWur ¥uac2¯6£c« ½ q ' 8 H V4 L
(122) \$V
(123) \
(124) \V V I"
(125) B N = D . i. ∞. s. n. n. i. .
(126). 4 $N H $$ 1( \ ( $W H ; N N *L4. 1 $ 3 (D ; $ N 0; ID N8$ . /
(127) ± 8
(128) 8 W1N ± ± ± σ. IBW
(129)
(130) \V
(131) \V
(132) \$ 1∂ n u U±ikλ DE u = 0 σ 1 1&λ I"
(133) B $ W . %¦,*, ³ d %9*
(134) ¢ cdrzxtcWr9y¨u¹rz_yppzcW*rzy~urz_ac Wpzcd~§h°(ºq¯1cW¥¨¥ p2cW{Av(rty¨~up2¯_acWxzc xzcWaxzcPp³cWuArtp28pz.xzcWc.uÍ«2¥¥ rt_ac9xtcWpzva¥µr p0~§ p³cPrzy~uTd.u¤McqcWxzy£cP yu8C£c.xtnZp³ybdy¨¥|xb uauac.xP« σ¢ cac.ua~rzc . H(curl , Ω) := {E ∈ L2 (Ω)3 ; curl E ∈ L2 (Ω)3 }. uyµ§ qcWua~rtcWprz_c2pzvaxz§ .c2ay¨£c.xtc.u *c~Mc.x (rz~x1~u rt~Crz_ac9div c.ºArtc.xty¨~x0~§ Ω orz_acWuT¯1c9qc uac s. 1. 1. ∂Ω. u n rt_ac2ua~xtb ¥\rz~. ∂Ω. qyxtcWrtcW. 1. H − 2 (div, ∂Ω) := {E ∈ H − 2 (∂Ω)3 ; E · n = 0 ; divs E ∈ H − 2 (∂Ω)}. ^0_ac9cW¥¨cPrtxzy| c.¥| 1. ¯_ay| _
(135) ypC£cWrt~x0yu pt(rtyp'cWp E ∈ H(curl , Ω) u yu Ω \ σ ~u σ ~u ∂Ω 1. 1. H − 2 (curl , ∂Ω) := {E ∈ H − 2 (∂Ω)3 ; E · n = 0 ; divs (E × n) ∈ H − 2 (∂Ω)} E(x) R3 (i) curl curl E − k 2 E = 0 ˆ =0 (ii) E × N (iii) E × n = F. : <. ʱË3ÕÍÊÏÇ.
(136) .
(137) . ¯_c.xtc y|prt_acC¤ ~vauaxznTart
(138) u ypirz_c"ua~xzb ¥ rt~ qyxzcPrzcPTrt~Zrt_ac c.ºorzc.xty~Fx ~∈§ ΩH«3^0_(div, cqy¨xtcW∂Ω) *rLaxt~¤a¥c.b_ pI¤Mc.cWudp³rzv qy¨cPCy¨uRn'HG=+¯_acWudyµr).~xtxzcPp³M~∂Ωup rt~2pzW(r³rtc.xty¨u xz~¤a¥¨cWb (yR« c«3rz_c xtp³rIcW{Av(rty¨~u+yu;:R <y|pIpzrzy|p cP+yu R \σ ¯yµrt_ E × Nˆ = 0 ~u σ E = E +E Ä yp9ucWurty¨xtc p³~¥¨varzy~urz~Th´(ºq¯c.¥¥ pcW{Avrzy~up2u E y|p2rz_ac pt.r³rzcWxzcP c.¥|´rt_(r+pt(rtyp'cWp E rt_ac+loy¥¨£c.xz·»3h Aa¥¨¥c.x0x qy|(rty¨~u.~uqy¨rzy~u Y − 12. 3. i. i. s. s. lim (curl E s × x − ik|x|E s ) = 0. v uayµ§±~xzbd¥n y¨uª¥¥0ay¨xtcW*rzy~up xˆ « ¬»rdypdpz_a~(¯u yu ' J,+0rz_r"rt_ac.xtc
(139) c*ºqyp³rtpCvauay|{vc
(140) pz~¥vqrzy~u E yu «L^0_ac9¯c.¥¥µ·» ~Ap³cPquacPpzp~§3axt~¤a¥c.b :R <y¨§ y|pua~ru
(141) c.yc.uo£(¥vac2~§rz_ach°(ºq¯1cW¥¨¥ H (curl , R ) ~ cWxtrz~xgyu Ω \ σ .u°¤Mc qc.xty¨£cWTyu Z£c.xtn8pzybCy¥|x2¯k6n« ¢ cCvpzc paqy¨rzy~u ¥Ia(r §±~xrz_ac yuo£cWxtpzcixz~¤a¥¨cWb rz_c9¤ ~vauaxzn £(¥¨vacPp ~u ∂Ω, G := (curl E) ¯_c.xtc qcWua~rtcWp+rz_acrtuacWurty¥1 xzr"~§°pzvax³§ *c£cW*rz~x cW¥ V « ^0_acxzcP*yaxz~q.yµrsn §±vurty¨~(Vu¥ ) RG y|p0ua~(¯ qc uacP~u
(142) rz_acpz.c y¨u H (Ω) := {V ∈ H(curl , Ω) / curl curl V − k V = 0 Ω}, ¤on Z Z : )P < RG (V ) = − F · (curl V ) ds + (V × n) · G ds ¯_c.xtc8rz_acy¨uArzcWx ¥|p pz_a~va¥|¸¤Mc´vauqcWxtp³rz~o~q pZqv¥y¨rsny¨xtyua ¤ c.rs¯1cWc.u H (div, ∂Ω) u :Ïrz_ypC*~uA£c.uArzy~u ¯y¥¥1¤McÅc.qrCy¨u?rz_acxzcWb yuay¨u°~§rt_ay|p"pzcW*rzy~Eu <*? « 5xz~brt_ac H (curl , ∂Ω) lorz~ÅcWp1§±~xtb+v¥du rt_ac9¤ ~vauaxzn*~u qyµrty¨~?u :R(·»y¨y <)~uac9qcWxzy£cPp Z : ) )=< ˆ ds §±~x¥¨¥ V ∈ H (Ω). RG (V ) = [(curl E) ] · (V × N) ¢ c p³_ ¥¥Iaxz~(£c x p³rrt_(r9y¨§ yp2u~r9yac.uArzy|.¥¨¥n 0 rz_acWu σ¯ *~yu.yqcPp¯yµrt_rt_ac p³va ~x³r9~§ rt_ac0svabd8~§ (curl E) *xt~ptp RGΠ « 1½ /
(143) 1 Π U3 V
(144) \U
(145) W1 4 F \
(146) D
(147) 1 RG 6= 0
(148) \8lovaa [(curl E) ] = σ¯ $
(149) I ^0_ac9axt~o~§ ~§ rz_ay|pxtcWpzva¥¨r§±~¥¥¨~(¯prt_ac+ptbdc9¥y¨uacPpgp0rz_ac+axz~o~§ Q c.bdb q0« )« ¢ y¨rz_~vqr ¥~ptpi~§)c. uacWxt¥¨y¨rsn~uac+b 6n´pzpzvabdcrt_(r Π = {x ∈ R ; x = 0} "« Q c*r E , E , E ¤McCrz_ac WxzrzcPp³y|uT*~bd ~uac.uArtpg~§ « QÍc.r ¤Mc"u´~Mc.u´.~uauacPrtcW´pzc*ri.~uArty¨uy¨ua σ MpznobCbdc.rzxty ¯y¨rz_8xtcWpz cPr0rz~ Π «L©gc uac W ⊂ Ω |x|→∞. 3. loc. 2. T. T. M. 2. M. T. M. ∂Ω. ∂Ω. − 12. − 21. T. M. σ. M. Π. M. T. M. T. Π. 3. ˜+ = E. ÕÕÑZ\[]^_. . 3. E + (x1 , x2 , x3 ) (−E1+ (x1 , x2 , −x3 ), −E2+ (x1 , x2 , −x3 ), E3+ (x1 , x2 , −x3 )). 1. 2. 3. x ∈ W +, x ∈ W −..
(150) )P. W V $1
(151) BW. i ¤o£oy~vpz¥¨n ˜ = 0 yu W . curl curl E˜ − k E eg~(¯ pzpzvabdcdrz_c c*ºqy|psrtc.u.c ~§8xzr σ ⊂ σ pzv _¹rz_rrz_acp³vaxz§ .c bdcWpzvaxtc ~§ σ yp+ua~r 0 u
(152) p³v _rz_ (r [(curl E) ] = 0 ~u σ « Q c*r W ¤ c9u8~ cWu
(153) .~uauacPrtcW
(154) p³va¤ p³c.r~§ W pzv _Zrt_(r u (W ∩ σ) ⊂ σ «Tloyu*c [(curl E) ] = 0 u [(E) ] = 0 ~u σ 3~uc _p W ∩ σ 6= ∅ y¨u u ¥|p³~ [(curl E ) ] = 0 u [(E ) ] = 0 ~u σ « ^0_c.xtc*§±~xtc! curl curl E − k E = 0 W y¨u «"lqy¨u.c yp9.~uaucWrtcW \rt_acCvauay|{vcd.~uArzyuov(rty¨~u´axty¨u *ya¥¨c curl curl E cWupzvaxzcPp9rt˜_ac.u?−rz_kEr ˜ E == 0E˜ Wyu W u Wrt_ac
(155) ptbdcaxzyu.y¨a¥c
(156) c.upzvaxtcWp9rt_(r E˜ = E y¨u W « 71~upzcW{Avac.uArt¥¨n ~u u §±xt~b :I)W)
(157) <+~uac
(158) c.rtp+rz_r RG = 0 L¯_ay| _?y|p"y|p .~uArzx ay*rzy~uZ¯[(curl y¨rz_
(159) rt_aE)c9_onA]M~=rt_acW0pzy|p0~§ σrt_ac9¥c.bdb a« 2ip2y¨urz_ac Wpzc+~§1rz_ac !cW¥¨bd_a~¥µ$r #dcW{Av(rty¨~u $vpzy¨u
(160) pz~bdcdp³McW.y¥rtcWp³r2§±vaurty¨~up ~uacd.u c.ºqa¥¨~yµr :I)W)
(161) <"rt~ xzcP*~(£c.xCrz_acT¥uac Π u [(curl E) ] « 5a~x (θ, pˆ) ∈ C × R Vqy¨xtcW*rzy~uq· M~¥|xt0y #W(rty¨~u.~vaa¥!c q¯c9qc uac +. 2. +. ±. 0. T. 0. 0. 2. +. 0. 0. Π. 0. 0. +. 0. 2. +. 0. +. T. T. T. Π. +. Π. T. T. 0. Π. 0. Π. 0. −. +. 0. M. Π. T. 3. 3. Π. u pt(rtyp'cPph°(ºq¯1cW¥¨¥ p)cP{v (rzy~u pu rt_ac.xtc*§±~xzc V (·; θ, pˆ) ∈ V (·; θ, pˆ) « ¢ c¥|p³~d_p6ˆ£·cgθ rt=_(0r curl « () Z½ Á * ½ +, . [(curl E) ] V« (·;¢ θ,cpˆ)ptp³∈vabdHc (Ω) _acWxzc rz_r Π y|p9Åoua~(¯u u y|pqc uacW ¤onrz_ac cP{Av(rty¨~u Y x · Nˆ = γ ¯_c.xtc Nˆ ypidvauay¨riua~xzb ¥$rz~ Π u γ y|p y¨£c.uT*~up³rtuArP« Q c*r ξ ∈ R ˆ = 0 u
(162) qc uac ξ 6= 0 ξ · N p ˆ ˆ × ξ, ˆ u pˆ(ξ) = N θ(ξ) = ξ − k − |ξ| N ¯_c.xtc ξˆ = ξ/|ξ| « ^0_acWu¸rz_ac°.~va¥¨c (θ(ξ), pˆ(ξ)) pt(rty'p cWp θ · θ = k u pˆ · θ = 0 « loy¨u *c ˆ = ξˆa~uac9qcWav*cPp0§±xz~b : ) )
(163) <1rz_ (r pˆ(ξ) × N V (x; θ, pˆ) := pˆ exp(iθ · x) x ∈ R3 .. 5~x. θ · θ = k2 HM (Ω). M. T. Π. 3. 2. 2. 2. RGM (V (·; θ(ξ), pˆ(ξ))) = exp(−iγ. iurz_c9~rz_c.x_u pzy¨u *c. !. !gc.u.c. Z Π. p k 2 − |ξ|2 ). [(curl E)T ]Π · ξˆ exp(iξ · s) ds.. :I)PW<. ˆ =0 ξ·N. p ˆ ˆ + k 2 − |ξ|2 ξ. θ(ξ) × pˆ(ξ) = |ξ|N p ˆ = k 2 − |ξ|2 ξˆ × N ˆ (θ(ξ) × pˆ(ξ)) × N p p RGM (curl V (·; θ(ξ), pˆ(ξ))) = i k 2 − |ξ|2 exp(−iγ k 2 − |ξ|2 ). u. Z Π. . :I)W <. ˆ ) exp(iξ · s) ds. [(curl E)T ]Π · (ξˆ × N. ʱË3ÕÍÊÏÇ.
(164)
(165) 5~x2¥¥. ^0_c.xtc*§±~ξxtc!∈oy¨R§3~uac9ξ p³6=c.rtp0 3. u . )) ˆ =0 ξ·N. A(ξ) = RGM (V (·; θ(ξ), pˆ(ξ))). y¨r§±~¥¥~(¯p)§±xt~b : )61<0u?: )PW<1rz_ (r . F [(curl E)T ]Π (ξ) = exp(iγ. p. Mrz_acdy¨x (ξ,ˆ ξˆ × Nˆ ) §±~xtb piu°~xzrz_~ua~xzb ¥¤pzy|pg~§ Π « u B(ξ) = RG (curl V (·; θ(ξ), pˆ(ξ))) :I). <. k2. M. −. |ξ|2 ). :I)PW<. ! B(ξ) ˆ ˆ ˆ (ξ × N) . A(ξ) ξ − i p k 2 − |ξ|2. ^0_yp y|qcWurtyµrsnªua~(¯axt~(£Ay|qcPpCrz_acWu¶u c*ºqa¥y|*y¨r xzcP*~u psrtxzv*rzy~u¸~§ E) ] ¤on?r Åoyua rz_ac yuo£cWxtpzc5a~vxzyc.x9rtxtup³§±~xtbk~§0rt_ac xty¨_r"_u pzy|qc«.2..~x qyua
(166) rz~T[(curl rt_ac axtc.£oy~vp"Q c.bdb 3rz_yp ¥¨¥~(¯pvprz~ xtcW.~up³rzxtv*r σ yµ§3rz_c9¤ ~vauaxzn £(¥¨vac F ypgp³v _rz_ (r RG 6= 0 « ) Z½ Á *M½ +-,/. ½ , Á $ Nˆ eg~(¯E¥c*r S ¤Mc2rz_ac9vauyµrgpza_acWxzc9u qc uac2§±~x pˆ ∈ S T. . Π. M. Λ(ˆ p) =. 1½ / "
(167) \ W
(168). sup θ∈k S; θ⊥pˆ. |RG(V (·; θ, pˆ))|.. H $ \4 D
(169) \B 11 U
(170) 1B $ 1 . VN 1D $ η 0 ≤ η < 2k F([(curl E) ] )E· w V E 1 H = $ 3 E /
(171) \$ w2k
(172) 1 & M
(173) V $ E
(174) ±Nˆ 1
(175) \ E H = Λ T. Π.
(176) I ¢ c xtp³rua~rzy|*crz_r pzy¨u.c §±~x8¥¥ Nˆ « eg~(¯ (·; θ, pˆ)) « =5axt0~b:rz_ay|p¯θc+⊥ pzpzvabd c2rz_c+c*ºqy|psrtc.u.c~§ pˆ ∈ SΛ(±pzv N)ˆ_8rz_=r 0pˆ 6= ±NˆRG(V u Λ(ˆ qcPqv*c+rz_r p) = 0 ± § ~ x ¥ ¥ z p v Z _ t r _ ( r « RG(V (·; θ, pˆ)) = 0 pˆ = 0 ^0_c.xtc*§±~xtc!A§±~x¥¥ θ ∈ k S θp³v∈ _krzS_ (r θ · pˆ = 0θ ·RG(V « !gc.u.c!avpzyua.: )W)
(177) <¯c9_6£c (·; θ, pˆ)) = 0 ˆ )(x0 · N ˆ )) exp(i(θ · N. Z. : )=C <. . ˆ =0 [(curl E)T ]Π exp(ix · ξ)dx · (ˆ p × N). ¯_c.xtc x ypgp³~bCc2M~yuAr0yu Π « ˆ N ˆ ; θ ∈ kS; θ ⊥ pˆ} y|p8u¶cW¥¨¥y¨ p³c *cWurtc.xtcW¶rrt_ac¹~xty¨y¨u¶~§b (s~x8(ºqy|p E = {θ − (θ · N) ˆ u ~§bdy¨u~x(ºqy|p~§¥c.urz_ pˆ × N ˆ |k < 2k :Rp³yu.c pˆ 6= ±N ˆ <*«LhT~xtc.~(£c.x E:I)
(178) C < w = 2k 2|ˆ p·N ˆ kˆ p × Nk noyc.¥|ap F([(curl ~u a¯_ay| _8*~uArtxtqy*rtp1rz_c9_AnoM~rt_acWpzyp0~§rt_ac9¥c.bdb a« ) Z½ Á *M½ E) +-,/.¸] ) ·½ w =$ 0, ½ ΠE« ¢ c ptpzvabdcC_c.xtc+rt_(r Nˆ yp2Åoua~(¯u¹u¹pz_¥¥Lqc*rtc.xtbdy¨uac rt_ac+M~pzy¨rzy~u´~§Lrz_ac"a¥|uac Π «9^0_ay|pibC~vauArtprz~rz_ac"c.£(¥v(rty¨~u´~§Irt_acd*~u psr uAr γ = x · Nˆ §±~x ¥¨¥ x ∈ Π « 1½ /
(179) Nˆ U.
(180) 1
(181) \ 1 W1 4 F \
(182) D
(183) \ RG 6= 0
(184) \ γ " \ \ H W B
(185) I loyu*c u yp0¤M~vau qcW ~uac9.u~¤qrty¨u8Caxty¨~xzy\¥~(¯1cWx0uvaaMc.x¤M~vau ap γ u γ ~§ γ « σ ⊂ Ω Ω Π. 0. T.
(186).
(187). . . Π. M. −. +. ÕÕÑZ\[]^_.
(188) )6. W V $1
(189) BW. ort_ac.xtc2c*ºqyp³rtp u pˆ ∈ S pzv _Zrt_(r RG(V (·; θ, pˆ)) 6= 0 «I^0_acWxzc.§±~xtc! u «3hT~xzcW~(£cWx6rt_acgpzvaaM~xzr)~§ E) ] y|pL.~bd*r pz~ y|pu¥¨nArty~u « gc.u.c2rz_acWxzc2c.ºoy|p³rtp θ ∈ k S pzv [(curl _Zrt_(r ˆ | < π u F([(curl E) ] )(ξ) 6= 0 0 < (γ − γ ) |θ · N ¯_c.xtc ξ = θ − (θ · Nˆ )Nˆ « lqy¨u.c F([(curl E) ] )(ξ) 6= 0 rz_acWxzc9c.ºqyp³rtp w ∈ R w · Nˆ = 0 p³v _
(190) rt_(r F([(curl E) ] )(ξ) · « ¢ c1rz_acWu"pzc*r pˆ = θ × w «^0_ovp3¯c)_6£c RG(V (·; θ, pˆ)) 6= 0 « ¢ cpzc*r θ = ξ−(θ·Nˆ )Nˆ « w 6= 0 fip³yua. : ) )
(191) <~ucc*r p kθ × wk ˆ )) u RG(V (·; θ , pˆ)) = c exp(−iγ(θ · N ˆ )) RG(V (·; θ, pˆ)) = c exp(iγ(θ · N Z ¯_c.xtc c = ( [(curl E) ] exp(is · ξ) ds) · (ˆp × Nˆ ) « ^0_ac
(192) *~u psr uAr γ y|p+rz_acWxzc.§±~xtc vauay|{Avac.¥n ac*rzcWxzbdyuacWyu [γ , γ ] ¤on rz_c9xtrzy~ loyu.c. RGM 6= 0 θ ∈ kS [(curl E)T ]Π 6= 0 F([(curl E)T ]Π ) 6= 0 ! F([(curl E)T ]Π ) R2 +. T. −. T. T. Π. Π. 3. T. Π. Π. 0. 0. Π. T. −. Π. +. RG(V (·; θ, pˆ))/RG(V (·; θ 0 , pˆ)).. 8 7 0 *1%+, .7 8,*7 "%2*+-%
(193) , iu¥¨n rt_ac.~vp³rzy|2.p³c2¯y¥¨¥$¤Mc+*~upzyqcWxzcP yu
(194) rz_ay|ppzcWrty¨~uÍ« ¢ c9vp³c9pznourt_ac*rty9(rt"rt_(rg.~xtxzc.· pzM~uapgrz~rz_cdpt.r³rtc.xty¨uaZ~§)yu*y|qcWur2a¥|uacC¯06£cWpW«2^0_ac+§±~xz¯0x Tpz~¥£c.xy|pg¤p³cP´~u´xzc.§±~xtb+vq· ¥|(rty¨u8rz_acZpt.r³rtc.xty¨ua8axt~¤¥¨cWbp+u y¨uArtc.x ¥)cW{Av(rty¨~u vp³yua´Tq~va¤a¥c*·»¥6nc.x2M~rzcWuArzy|¥U«
(195) ^0_ac xtcWpzva¥¨rzyuadcW{Avrzy~u
(196) ypqy|pt*xtc*rz0y #.cPZv p³yua P cWqc2c.¥c.bdc.uArtpW« . 1. .
(197)
(198) !#"%$& '(
(199) )*,+-
(200) !./
(201) 0.'
(202) !
(203) ^0_c.xtc+xtc+pz~bdcuovabdc.xtyW¥3qy21 .va¥µrty¨cPp¥¨yuaÅcP8¯yµrt_´rt_ac"aa¥yW(rzy~uT~§Irz_ac"y¨uo£c.x p³c+pt _acWbCc"qc*·. pt*xty¤ cP y¨u
(204) p³cPrty¨~u
(205) o«I^0_ac x psr0~uaciyp0pzpz~q*y|(rzcPd¯yµrt_ rt_acac*rzcWxzbdyu(rty¨~u ~§ rz_acua~xtb ¥ rz~"rz_ac _ ~p³rLa¥uac Π « ^0_cgp³cP*~udqy21 .va¥µrsndxty|p³cPpI¯yµrt_drz_acqc*rtc.xtbdy¨urzy~ud~§\rt_acua~xzb ¥arz~9rt_aca¥|uac .xt Å « ^0_ac.xtc1y|p ua~2axzy~xtyop³cW¥¨cPrzy~uC~§rz_ac0bdcWp³vaxtc.bdcWur p :±yU« c«¤M~vau axtna(r < rz_ (rIy¨u p³vaxtcWp rt_ac ptΠp³vbCarzy~u°~u F([u] ) xzcP{Avay¨xtcW°¤onT¥¨cWbCb Tq«Ôo«V!~(¯c.£cWxgrt_ac p³v 1 .y¨cWuAr2.~uqy¨rzy~u¹y£cWu yuZxtc.b xzÅ q«Ô9y|p)yu § r0cWp³yc.x)rz~d _acW Åduovabdc.xtyW¥¥¨n« lqy¨u.cgrt_ac.xtcgy|p opz~§ x oua~+rz_acW~xtc*rtyW¥M¥¨yuaÅ ¤Mc*rs¯c.cWu rz_yp0*~uqy¨rzy~u
(206) u rz_ac2 _a~y|*ci~§$rt_aci¤M~vuaxtn a(r Arz_ci~ua¥¨n pznqpsrtc.b (rtypsrtxtrzcWnCyp rtxzy|¥qu+cWxzxt~x3¤on£(xtnoy¨uairz_cuovab"¤ cWx k ~x3rz_ac0¤M~vauxtna(r f «¬uCpt.r³rzcWxzyuaaxt~¤a¥c.b p (rz_ay|p rtxzy|¥u
(207) c.xtxt~x0¯1~va¥|u~r¤Mc*~p³rz¥npzyu*c9uovabdc.xty|.¥$c.ºq cWxzybdc.uArtppz_a~(¯cWZrz_rrz_yp.~uayµrty¨~u _~¥|ap2§±~,x %³¥¨+¥ ("§±xtcW{AvacWu*ycWp k u ¥¨¥1y¨u.yac.uAr"qyxzcPrty¨~up dˆ u~u ~xzrz_~~u¥ rt~ Nˆ .« !~(¯c.£c.x ´¤ c.r³rzcWx _a~y|*c ¯~v¥ ¤Mc rt_acZvpzcZ~§¥¨~(¯ §±xtcW{AvacWu*n :ϧ±~x"c.ºqbda¥¨c ¯_ac.xtc y|p rt_acqy|bdc*rtc.x~§ rz_c*x Å <« ^0_ay|p _a~y|*caxtc.£c.uArtp0_ay_
(208) k ~pt*y¥¥rzy~up~§ k[u]< :R2π/D p³cWc 53yvaxtc )
(209) <0Du rt_ac.xtc*§±~xzc2¥¥~(¯pu
(210) cWpzyc.xqc*rtc.xtbdy¨urzy~u
(211) ~§3rz_aMc #Wc.xt~p~§rz_ac2§±vurty¨~u¥ Λ « Π. Π. ʱË3ÕÍÊÏÇ.
(212) )P.
(213) . ^ ~?xtcW.~(£c.x rz_ac ~Ap³y¨rzy~u¶~§9rz_ac¹a¥|uac~uac_p rz~ªp³cWc.Å θ pt(rzy|p³§±nAyua(:RW<*« ¬u pzW(r³rtc.xty¨u c.ºq cWxzybdc.uArtpi~uac".u°cWp³rt¤¥¨y|p³_Zpznop³rzcWb (rzy|+ _~y|*c~§ θ §±~xuon
(214) §±xzcP{Avac.u.n8yµ§Irt_acCay¨xtcW*rzy~u dˆ ~§ rt_ac9y¨u *y|qc.uAra¥|uac2¯06£cyp¯1cW¥¨¥Í _a~Ap³cWuÍ« ¬u8xzrzy|*va¥|x ArtÅcrz_ay|pqyxtcWrty¨~u
(215) pz~Crz_r pt(rtyp'cPp3rz_ac x p³r xtcW{Avayxtc.bdc.uArI~ud*~uqy¨rzy~u.:R1<*«U5axt~b : J <3¯c_ 6£c RG(v(·, θ)) = « axt~b rt_ac9~qrtyW¥\rt_ac.~xzcWb~uac2_p ˆ | < π/k(γ + − γ − ). 0 < |dˆ · N. ^0_c.u. θ = k dˆ ˆ 5 4πu∞ (−d). ˆ = k ku∞ k2 2 =(u∞ (−d)) L 4π. ¯_y _´pz_a~(¯p0rt_(r RG(v(·, θ)) 6= 0 pzy¨u.c u 6= 0 : dˆ · Nˆ 6= 0<«^0_acWxzc.§±~xtc θ = kdˆ pt(rtyp'cWprz_ac .~uayµrty¨~?u :UW <« Q c*rvp0ua~rzy|*crz_ry¨u
(216) ax rty.ciy¨ry|p1bd~xzc2*~uA£c.uayc.uArrz~ _uacgrz_ac2§±xtcW{AvacWu*n uZu~r0rz_ac yu.yqcWuAr qyxtcWrty¨~uÍ« 2gp3bdc.uArzy~uacP¤Mc*§±~xzc (uAvbCcWxzy|.¥oc*ºq cWxzybdc.uArtp pz_a~(¯cW2rt_(r RG(v(·, θ)) 6= 0 y¨§ yp+ua~u ~x³rt_a~~u¥3rz~ «Teg~(¯ y¨§¯1c §±vx³rt_ac.xcWupzvaxzc k < π/(γ − γ ) rt_ac.u uon θ ua~u ~x³rtθ_a~~u¥rt~ Nˆ ¯~v¥¤ cNˆ.~uo£cWuayc.uArW« ^0_ac xtp³rLq2y 1 .va¥µrsn"yp ¥yuaÅcP+rt~rz_accW£6¥¨v (rzy~uC~§ ) ~vqr p³y|qcrz_acaypt0~§Mxtqy¨v p k «3^0_ay|p a2y 1 *v¥µrsn9qcWxzy£cPp\§±xz~b rz_ac1y¨¥¥µ·»M~pzcWquacPpzp ~§qrt_ac)yuo£c.x pzF([u] cIaxz~¤a¥c.b8«hT~xzc)axzcP*y|p³cW¥¨n .rt_acc.£(¥v(rty¨~u ~§ F([u] ) y¨uo£~¥£cWp9rzcWp³r c.¥|ap v(·, θ(ξ)) ¯yµrt_*~bda¥¨c.º¹£(¥¨vacP £cPrt~x p θ(ξ) «^0_acuAvbCcWxzy|.¥ cW£(¥v(rty¨~u~§ RG(v(·, θ(ξ))) ypvaup³rt¤¥¨c ¤McWWvpzcd~§0rz_ac c.ºq ~uac.uArzy|¥L§ rt~x e √ « ¬uZrz_ac§±~¥¨¥~(¯y¨u+uovabdc.xtyW¥ rzxty¥p1¯civpzc2+xt~va_ xzcWva¥|xty|pzrzy~u axt~q*cWptp1¯_ay _
(217) *~up³y|p³rtp1y¨urz_ac rtxzvu.rzy~u ~§rt_a8c 5~vaxtyc.xrtxtup³§±~xtbk¤on c.£(¥v(rty¨u F([u] ) §±~x |ξ| < k u yuo£cWx³rty¨ua°yµr"rz~ ~¤qrty¨u8u8axz~6ºqyb (rzy~u~§ [u] « ∞. +. −. Π. Π. −. ˆ |ξ|2 −k2 (x·N). Π. Π. .
(218) . . ¬ uEy¨bda¥c.bdc.uArty¨uaªu y|qc.uArty+ W(rty¨~uE¥~xzy¨rz_ab ¤pzcW¸~u¶bdcWp³vaxtc.bdcWur p ~uc°_p rz~?Åc.cW¶yu bdyuTrt_(r9bdcWp³vxzcP´a(r xzcCp³v¤qscW*rirt~ua~ypzc!Mrt_acdc *\cWr p~§)¯_ay| _°_ 6£crt~¤Mc psrtvqycW$«+¬u rt_ac"§±~¥¥¨~(¯yua
(219) uAvbCcWxzy|.¥ rtxzy|¥|pirz_acdart
(220) xtc!\
(221) $yU« c«~¤artyuacP´¤on´Z¤M~vuaxtnTc.¥c.bdcWur .~bdavqr (rty¨~u p³va¤ascWr"rz~´cWxzxt~x p 3u¹rz_acWn xt!c 3_ac.u.!c L¥¨xtcWqn°ua~ypzn«^ ~Trt_a?c & 1 1N ¯c_6£c0aqcP W & icWuac.x (rtcWC¤onCM 5~xzrzx udxz~vqrzyuac«3eg~rzy|*crz_rLrz_acip³_ c xtcW.~up³rzxtv*rzy~u rt_(rdypCrt_ac8p³cP*~u ?p³rzc.~§~vaxCyac.uArzy .rzy~uaxz~q.cWptp Ly|pCua~rCrz_ (r p³cWupzyµrty¨£c rt~Trz_cbda¥y¨rzvqc~§rt_acZua~ypzc : va rz~¹ V<*«T¬uqc.cP rz_cZx uq~bu~y|p³c y|p ¥µrtc.xtcW ¯_acWu ¯1c cW£(¥v(rtc [u] p³yu*crt_aypIsvabdTypW¥|*va¥|(rtcW£Ay|du
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