A Centered Second-Order Finite Volume Scheme for the ...Unité de recherche INRIA Sophia Antipolis...

Preview:

Citation preview

A Centered Second-Order Finite Volume Scheme for the

Heterogeneous Maxwell Equations in Three Dimensions

on Arbitrary Unstructured Meshes

Serge Piperno, Malika Remaki, Loula Fezoui

To cite this version:

Serge Piperno, Malika Remaki, Loula Fezoui. A Centered Second-Order Finite Volume Schemefor the Heterogeneous Maxwell Equations in Three Dimensions on Arbitrary UnstructuredMeshes. RR-4161, INRIA. 2001. <inria-00072461>

HAL Id: inria-00072461

https://hal.inria.fr/inria-00072461

Submitted on 24 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--41

61--

FR

+E

NG

ap por t de r ech er ch e

THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A centered second-order finite volume scheme for theheterogeneous Maxwell equations in three

dimensions on arbitrary unstructured meshes

Serge Piperno — Malika Remaki — Loula Fezoui

N° 4161

Avril 2001

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : 04 92 38 77 77 - International : +33 4 92 38 77 77 — Fax : 04 92 38 77 65 - International : +33 4 92 38 77 65

! "# %$"#&'($$)* ,+.-0/213!452 -06 7689$6": ; 3*

-<-=8 >9 ? @"# $

ABDC!EFBHGIKJ?BDCMLNPORQTSVUXWYIKZU\[BD]^UXZ>I_QT`NFa9WYUcb%BDdDNXa9I

egfih6j2k0lnmporqsjtiuwv'xyqz|k;xz~xyqj2qwvxqsz| krxh!j2k!;zj2~iuk6k!

%yzk6xvqsj2v|

v~i~Xzx k_kMfik6fik|X9l!mpryqump*~vkM

!6 '¡i¢£¥¤ k¥~iyz'kv*t¦6qk!|x§9¨P©ªuqs«k6z| q¬xyqz|®­¯z°xfik¥±?²xv³iqsuqx2z­Pxyfikk!6z| ©ªz k!v66tiyvxk´|iqxkzutij2kyfik6j2k~iyz~Xzk ³r2k6j®v«rq­¯zTxyfikxqsj2k;© zj®vq|zustxqsz|2z­>µ¶v'·°k!uukM¸Dtv'xyqz|nq|\fik;xyk6yzk!|iztj2k qsv¹?·qxfºjnk6xyvuusqsv| v³z³q|iR³zti| vc6z| q¬xyqz|6» ¤ krqsk6u vk!0k!|ik6vuDti¦®6qk!|Dx96z| qxqsz|¹'vusq ­¯z9v|r0´|iqxkzutij2k~vxqxqsz|q|*x·°z¥v| xfiyk6k~v;k qsj2k6|qz|!»¼tij2k6yqw6vuxkMxfiz'·½xfik~Xzxyk6|Dxqwvuz­%xyfiqszyqq|vuP´|q¬xykzutjnkyfik!jnkqs|¶z|ik¹x·°zv| xfk!k*~v6k qj2k!|qsz|­¯zxfik*|rtij2k6yqs!vuzustxyqz|¾z­?µ¾v'·k6usukM¸tv'xqsz|qs|xyfik0xqsj2k zj®vq|>»¿¾ÀDÁFÂyÃÄ 'Å£ k!ukMÆxyzj2v|ik6xqwj¹°´|iqxk¾zustij2kj2k6xfiz !¹;k6|Dxyk6yk \Ç tk!!¹ukMv~©È­¯zHxyqj2kyfik!jnk¹i±?²0xyv³qusq¬x¹ik6|k6y®j2k;xyfiz 6¹ti|xytÆxytik j2kMfikM6¹iv³zy³iq|*³Xzti| vy;z| qxqsz|

ÉÊ>ËÌÍ0ÎKÊ>ÏÐMίÑÌ>ÎÓÒ°ÐÔPÕÖÆ×MÐÙØÚ¯ÛÆÜÆÖÆÛÆÝÏ!ÞÆßMàÙáãâyÚÓÒä;åYáæßÞÆçæáæèÊFéêÙéìëÐÏ!éíYîÆéÆï ÕáæßéíKäÙÞ6ðèÈÞÆßÙàMáãâMï áæäÙíYáãâMï ñí

%$":- 6 "# Fi¾6 (® P9 T 7° 7 °*R ;-# ! ®;8 5® -0;* (+.-0/2136R6!"# ;6; ":!6; 8$)

£ ¼zt j2z|xyz|¥ti|ik6z| qxqsz| kxyv³iqusqx ±?² k2xr~Xk§¨ ~Xzt_ti| f j®vk!|4zustij2k!¥´|iqs t kM;z| ©Èz k*~iz~zD ~vk6j®v«Dq%~Xztusv zustxqsz| kM ¸Dtvxqsz| kµ¶v'·°k!uu¥k6| zj2vq|kHxyk6j2~zk!uY¹k6|½j2qusqk!tVf x zh!|ikHk6x¾vÙk! k!;z| qxqsz|vt uqsj2q¬xyk!j xyvususqs¸DtikM9zt*v³zy³v|DxkM6»¼zt z|i|z|ti|ik6z| q¬xyqz|nt¦v|Dxk kxv³iqsuqx k°­¯zj2k°xyh! | yvuk¹'vusq k~Xzti k!?~vxqxqsz|?k6|®zustij2k!%´|iqsvy³iqxvqskM6¹'k!| k!t®zt2xyzqw qsj2k6|qsz| kM~v;k»k!qsjtiuwv'xyqz||rtij yqs¸DtikMgjnz|Dxyk6|Dxuk_~Xzxyk6|Dxqsk6u k6k;xxk_|iztik!uusk0j xfiz k» Ä M  ¢ _£ usk!;xyzj®v| xyqsj2k¹ÙzutjnkM´|iqw6¹ Ç ti*;k!|xy 6¹'yf j®vvtxk6©Èj2ztixz|¹xv³iqsuqx ±°²¹j xyfiz k |ik!qk¹j®vqsuuwvk|iz|¾xyt;xti ¹i6z| q¬xyqz|usqj2qxkv³z³v|Dxk

"!$#%'&(*)+,!.-0/132456578/:9'

; < ® ?;*

egfkjnz k!uqs|i_z­Fxyk6j®9q|rzusDqs|i¥k6usk!;xyzj®v|ik;xyqs·gvÙk!fv9«r|iz'·|®v«rqs| z­>=Æyk6qs|rk6|Dxyqz|?=@ 'lA°xyfiztifcxfikkMzutixqsz| z­xyfikxqsjnk6© zj®vqs|\µ¾vr·k6usu?kM¸Dtv'xyqz|z|^~v6kyq 6» µ¾v|r qCBFk6yk6|Dxxr~Xk!z­9j2k6xfiz fvÙk0³Xk6k6|Htk »g§qs|iqxk qDBXk!k!|;kegqsj2k;© zj®vqs|FEY§e HGj2k6xf©z IE¯³vk z|KJTk!k yfik6j2k @ LA°z_z| qj2~iusqw;qxxqsj2kfk6j2k! @ AMGƹ9vyk2k;¦;qsk6|Dxj2zxus¶z|xyt;xtiyk yk6tiuwvq !¹%·fik6yk!v´|iq¬xykk!uk!j2k6|Dxnj2k6xfiz !¹?³vk z|\ti|xytÆxytiyk j2k!fikM6¹!v| k!vu·q¬xyf\|v'xtyvuus^;zj2~iuk6ck!zj2k;xyqsk! @ N Aª¹?³itxqs| t;k¾fikMvÙDc6zj2~itxv'xyqz|2z­¥j2vj®vxyqs6k!!»O0v'xfk6yq|ij®v|r¶v Ùv|Dxyvk!!¹P§qs|iqxkQP%zustij2k2egqj2k;© zj®vqs|REY§?P¥e SGj2k;xfz 6v|vusz³k®³vk z|Rt|xytÆxtk j2k!fikM_v| k6x_q z­ qDBFk6yk6|Dxqwvu%z~k!yvxzQEKv| ´|iq¬xyk®k6usk;©j2k!|x_j®vyj2vxyqs6k!Gtqs|iTO¥k!k6| ­¯zyjtiuwv­¯z¥xfik2qs|xyk6yvxqsz|¶z'k6´|iq¬xyk2zustij2k!QEYk!k @ U A­¯zvnk!Dqsk6·Vz­9|rtij2k6yqs!vuPj2k;xyfiz tk q|H°zjn~txyvxqsz|vuWV%ukMÆxyzj2v|ik6xqw6v| @ XLA>­¯zv|v66tiyvxk¥k!rqk!· z­9§e 3v| §7Pe j2k6xfiz GÆ»

¤ k¥vkq|Dxk!kMxyk fk6ykqs|§7Pe ½j2k;xyfiz 6¹v?fvÙk³Xk6k6| k6k6usz~Xk qs|xyfik~vx°kMv6¹|izx|k!;kMyvyqsunz|³Xz D©Y´xxk 6zDz q|vxkM @ r¹iLAX³txz|ti|xyt;xtiyk ´|iqxk¥k6usk6j2k6|Dxxyqsv|itusv©xyqz| @ i¹Yr¹ N ¹MAzz|0xzxyvuus k!xytÆxytik jnkMfk! @ LAª»µzk%~kM;qwk!u¹!·k?;z|q k!>vxyv| v ´|iq¬xyk2zustij2k2v~i~iyzÙqj®vxqsz|>¹PqÈ» k»*v~iqsk!6k6·qwk2;z|xv|Dx!¹ qsy;z|Dxqs|Dtzt!¹ZO0vusk6y«rq|©Èxr~k´|iqxkk!uk!j2k6|DxTv~i~iyzÙqj®vxqsz| @ 6lAÈ»9xyfikµ¾v'·k6usuxk6j q|nxv|qsk6|Dx%xv'xkqw9frr~k!³Xzusqsgv| j®vÙ³Xkyk6·yqxxk!|q|H6z|k6y'v'xyqk­¯zj¹Xq¬x¥qs|v'xytivuPxyztkv®|rtij2k6yqw6vuv~i~iyzÙqj®vxqsz|³vk z|6z|k6y'v'xyqkfk6j2k!!¹qs|v´xxk6~ qskMÆxus2q|~iqsyk ³r2~iyk6rqszt°·°z«?q|xfik¥´k!u z­9°zj*©~txyvxqsz|vu§ustiq r|vj2qw6!» egfik¾;z|rk6yk6|;kz­´x©ªz k62;z|k!'vxqskt~D·qs| yfik6j2k!fv³Xk6k!|¶k!xyv³iuqwfik ­¯z qDBXk!k!|x¥frD~Xk6y³Xzusqs0kM¸tv'xqsz|q|Rv|r qsj2k6|qsz| @ Aª¹Xv| ±\[0k6yzkMxyqj®vxk!*z­^][`_²KE¯·fik!ka]cqw2v¶fvyvÆxk!qwxqw®j2k!f^qcb6kLG_fvÙk³Xk6k6|\~iyz'k yk!6k6|Dxus ­¯z*vk6|ik!yvuFfDr~Xk6y³zuqw¥k!¸Dtvxqsz| @ dY(AÈ»

egfikxyv³iqusqxz­´|iqxkzutjnkgyfik6j2k!9fv³k!k6|2qs|Dk!xqsvxk qs|;kgj®v|rk!vy!¹M­¯z%yk6tusvq tq|ixfk\P%z|*¼k6tij®v||*v|vusrqw @ Azxyfik°j2z q´k k!¸Dtvxqsz|v|vusqw @ U AÈ»fezxyf*v|vus©k! z|izx kMvuk!q¬xyfik6¥·qxf¶³Xzti| vy6z| qxqsz|z·q¬xyf¶|iz|¾k!tiuwvq !»egfik*6z|6k6~xz­ezxvuPTvyqwv'xqsz|qsj2q|iqwfiq|gEYe P HGgfik!j2k¹r~iyz~Xzk ³rihvxyk6| @ Ù,Aȹusk!v °xz2±\j©ìxv³iqsusq¬xyk!tiuxy°­¯z´|q¬xyk¥zustij2k¥yfik!jnkMz||z|yk6tiuwvyq °z|uqs|z|ik_~v;k qsjnk!|qz|>¹iv| 6v|³Xkyk6uwv'xyk xz_xyfikq k!v0z­>¨>z6vuV9rxyk6j*tij qsjnqs|iqwfq|ikEK¨ZV SG?fik!j2k¹~z~zDk ³rQlvjnkMz|@ !AȹD·fiqwf®k;rxk!| ?zj2k~iyz~Xk6xqsk!%xzk6k6vu qsj2k6|qsz|!»?mì|v~ik!rqzt%~v~k! @ Y(Aª¹·kfvÙkqs|rkMxyqDv'xk xyfik±°²©ªxyv³iqsuqxHz­°´x©ªz k6_ti~r·qs| ´|iqxkzustij2kfk6j2k!_qs| x·°z¶v| xfiyk6k~v;k qsj2k6|qsz|*z|ct|xytÆxtk j2k!fik!!¹9­¯z*xfik|rtij2k6yqs!vugzutxyqz| z­xyfikxyqj2k6© zj®vqs|µ¶v'·°k!uuXk!¸Dtvxqsz|!»4ntvq©Èz~xqsj®vuFt¦6qk!|xxyv³qusq¬x6z| q¬xyqz|·°k!k¥z³xvqs|ik z|vy³iq¬xyyv´|iq¬xyk;©ªzutij2k2~vxyq¬xyqz|6¹P·fqsf4·k6yk*j2k6|Dxyqz|ik v| v;xtvuus¶tk @ Aª¹v| ·fiqwf k;rxk!| vk6|ik!yvuP;z| q¬xyqz| k6yqk ­¯z§iyqsk yqsf xyk6j®gq|k!|ik6vu @ N AÈ»

hz'·°k!k6M¹'ti~r·qs| yfik!jnkM%vyk°k6y qwv~i~Xzqs|xyq|·fik!|2tk ­¯z9xfik|rtij2k!qw6vuiqsjtiuwv'xyqz|z­Pk6usk!Æxyzj®v|k;xqwg·gvÙk!T~iz~vDv'xyqz|>¹q|6kxfk|rtij2k6yqs!vu qCBFtqz|q| t;k ³r*t~D·qs| qs|iHo|k!;kMyvyqs|*°zj2~itxv'xyqz|vuD§ustiq D|vj2qs!Pxyzuqsj2q¬xti|i~ifrqw6vuzD6qususvxqsz|oj2v«kMPuz|i©ªyti|6zj2~itxv'xyqz|pEYv'x¥ukMvx¥z'k6k6k6vu~Xk6yqsz `Gk!q|v66tiv'xk»^mì|¶xfiqw¥~v~Xk6M¹F·°k*q|rkMxyqDv'xykv|ik6· ´|iqxk0zutij2kjnk6xfiz ~iyz~Xzk ³rk6j®v«Dq @ U AF­¯z?xyfik|rtij2k!qw6vuFzustxyqz|z­µ¶v'·°k!uukM¸Dtv'xyqz|¥q| fik;xyk6yzk6|ik!ztjnk qwvi»mì|Rxfv'x_fik;xyk6yzk6|ik!zt­¯yvj2k6·zy«X¹PvkMtu¬x_z­°k;qsxk!|;kv| t|iqs¸Dtik!|ik!y¥z­Txyfik®zutixqsz|¶qw¥|iz'· vÙ'vqsusv³iusk @ N Aª»k!j2v«rq 0§?P¥e ,j2k;xfz qs0³vk z|

ÌÌ äqrLsÜ,s

l 9 "*#%/L9 ?(9

vkM;z| ©ªz k6usk!v~©È­¯z2xqsjnk2yfik6j2k*v| z|HkM;z| ©ªz k66k6|Dxk!k |rtij2k6yqs!vu Ç tk!!»¥egfqsyfik!jnkrqsk6u ®v4zyqq|vu;z|k!'vxqsk´|iq¬xyk¶zutjnkj2k;xyfiz ¹·q¬xyf |iz4|Dtjnk!qw6vu qDBXtqsz|>»§qs|iqxkzustij2k!z­vy³iqxvyfv~Xkvk;z|q k!k ¹v·°k!uuv>x·zxr~Xk!z­³Xzt| vy;z| qxqsz|EYv³zy³iqs|iv| j2k;xvusuqw*³zti| v¾;z| qxqsz|GÆ»_ z|ikn~iyk6rqztu­¯z¥ti~r·qs| yfik6j2kMz|v³q¬xyyv_j2k!fikM6¹Dv|2k6|ik!D©Yxr~Xkj2k6xfiz ¹ vÙ·|­¯yzj zjnkg´|iq¬xykk6usk6j2k!|xT~zrz­K @ X,Aª¹Dqs%tk xyz2~iz'kxyfik±?²;©ªxyv³iqsuqxz­xfikyfik!jnk»

egfiqw~v~Xk6qszyv|qDb!k fv>­¯zusuz'·!»Wmì|k!;xqsz|*i¹!·kTqs|xyz t;k?v~k!t z©ª6z|k6y'v'xqsk­¯zyjz­µ¾vr·k6usuFk!¸Dtvxqsz|qs|xfk!k qj2k6|qsz|°­¯zfk;xk!zk!|ik6ztTj2k qsv¹ ¤ k_vuwzqs|xyz t;k_zj2k|zxyvxqsz|!»*mì| kMÆxyqz|Hi¹X·°k*kM6vuuxfikn;k!|xyk6yk usk!v~©È­¯zkM;z| ©ªz k6´|iq¬xyk;©ªzutij2kyfik!jnk~z~zDk ³r¶k6j®v«Dq9­¯z¥xfik2xyqj2k zj®vq|4zutixqsz|Hz­gµ¾v·°k!uu9k!¸Dtvxqsz|!»mì|4k!Æxyqz|Rl¹P·°k~z'kxfik0t¦;qsk6|Dxgxyv³iqsuqx;z| qxqsz|­¯z?xyfiqwyfik!jnk»%egfiqs~iyzrz­>qw³vk z|®xyfik k;´|q¬xyqz|z­Fv qwy;yk;xkk!ukMÆxyzj2v|ik6xqw?k!|ik6y» ¤ kvuwz¥~z'k°qs|nxfiqw?k!;xqsz|*xfvxTxfiqw%k!|ik6yqw%k6vÆxyu6z|k6yk q­|iz v³z³iqs|iR³zti| vqsk!2vyk~ikMk!|Dx!»\§qs|vusus¹·°k~iyk!k6|Dxnqs| kMÆxyqz| X zj2k|rtij2k!qw6vu?yk!tiu¬xqs| x·zRv| xfk!k qsj2k6|qz|!¹%³zxfc­¯zfizj2zk!|ik6ztv| fik6xk!zk6|k6ztj2k qsv»

$ $) +.-0/213645® -0;*

¤ k6z|q k62µ¾v'·k6usu?k!¸Dtvxqsz|*qs| xfiyk6k~v;k qj2k!|qsz|kEKfik;xyk6yzk!|ik6zt_usq|k!v*qszxyz~iqwj2k qtj ·qxf|izzti;k¹·q¬xyf¶~v;k0'vyrq|nk!ukMÆxyqw0~k!j2qxxyqrqx°v| j®v|ik6xqw~Xk6yj2k;©v³iqsuqx¹xfik_usz6vuPusqsfDx~Xk6k T³Xk6qs|i®qk6|³r ! Ȳ"$#LGÆ»Tegfik_k!ukMÆxyqw´k6u %&"(' %*),+-%*./+0%213?v| xfik_j®v|ik6xqw´k6u 4 "5'6 4 )7+ 4 .8+ 4 13Tk6yq¬­¯

9:; :< 8= %=?> "A@B 4 +

= 4=?> "&C2@DBE%GF

egfk!k¶k!¸Dtvxqsz|vykHk6xqs| v ³Xzti| k ~Xzusrfik yvu zj2vq|AH z­JILK»V%k!r·fik!k¾z| xfik zj®vqs|®³zti| v = H¥¹k6ivÆxyunz|kz­Pxyfikx·z~Xzyqs³iusk³Xzti| vyn6z| qxqsz|Tqw°k;xM9vjnk6xyvuusqs³Xzt| vyH6z| qxqsz| E¯z| = HON*¹vzti| vjnk6xyvuusqs2z³k!;x_z_qs|q kv6vÙrqx¾­¯z_k6ivj2~iusk,G¥zv|Rv³z³q|i³zti| v¶6z| q¬xyqz|RE¯z| = HQPD¹>~Xzyq³ius¾z|Rxfknztxk!0³zti| v¾z­Txfik zj®vqs|= H j ¹ik6k§qstiyk2,GÆ»mì|®xyfiyk6k~v;k qsj2k6|qz|!¹xyfikj2k;xvusuqw³Xzt| vy2;z| q¬xyqz|®·qxkMJRS BT%U" RV ¹r·q¬xyfWRS xfikt|iq¬xvy*ztx·gv %|izj®vuÈ»7mì|®xfik­¯zusuz'·qs|i¹v_´yx©ªz k6?oqusk!©µEXiusuk!°v³z³q|i6z| q¬xyqz|®qstk z|xyfikv³z³iqs|i*³zti| v = HQPD¹i·fqsf·qxkM

RS BE%&"$C YRS BZRS B 4 ,F E,G

egfikµ¶v'·°k!uu>xyk6j!v|³k0xyyv|­¯zyjnk qs|Dxzvn~k6t z©ª6z|k6y'v'xyqk­¯zyjEM9:; :< /= %

=[>Z\^] )= 4= \W] .[= 4

=`_&\^] 1`= 4=ba " V +

*= 4=[> C ] ) = %

= C ] . = %=`_ C ] 1 = %

=ba " V + EYG

ίÑÌ>ÎÓÒ

"!$#%'&(*)+,!.-0/132456578/:9' X

ΩmδΩ

8δΩ δΩa⇔

Ω

δΩm

RCS of a wing Cavity resonance

§qstk23M zj®vq| H v| zj®vqs|³zti| v

·fk6ykxyfikv|Dxyq¬©ìrj2j2k;xyqwj®vxyqs6k! ] )i¹ ] .*v| ] 10vk0qk6|³D

] ) " V V VV V #V C# V + ] . " V V C#V V V# V V

+ ] 1 " V # VC# V VV V V F

§zv|rkMÆxyzRS " ' S ) + S . + S 1 ¹xfikj2vxyq¬ ] " S ) ] ) \ S . ] . \ S 1 ] 1 qwv|Dxq©ìrj2j2k;xyqwv| z~Xk6v'xkMv ] "UCGRS B » $ 6-®&') "#9$®

!#"$¤ kvtij2k·k qw~zDkz­v|^v³q¬xyyvR~vxqxqsz| z­xfik zj®vqs| H qs|DxzHv¶´|iqxk|rtij*³k!*z­6z|i|k!Æxyk ~zurfik vuF´|iqxkzustij2k!EKk!vfz|ik_·q¬xyf¾vn´|iqxk_|rtij³Xk6z­­Kv;kM`G;»T§izk;ivj2~iusk¹xyfiqw°vtij2~xqsz|;z'k!y%xfikx·°zn6vk!?z­>k6xk;r©ì;k!|xyk6yk v| k6usk6j2k6|Dx©ª6k6|Dxk!k ´|iqxk¥zustij2k!@ Y(Aª»T§iz?k!vf2´|iqxk¥zutij2kzH=6;k!uu =%#&¹'(& k6|izxkMTqxyzutij2k¹Dv| )&v| *&vykkM~Xk!;xqsk!uxyfikusz6vuk6usk!Æxyqw~Xk6yj2q¬xxqsDqx2v| j®v|ik;xyqs~k!j2k!v³iqsuqxz­Xxyfikj2k qtij¹r·fiqwfvykvytjnk 6z|xyv|Dxgqs|xfik6k6usu*% & »

¤ k6vuuq|Dxk!­Kv;k0³Xk;x·k6k6|x·zn´|iqxkzustij2k!xfk6qsq|Dxyk6kMÆxqsz|¹i·fik6|k6k!q¬xqsvn~Xzusrfik;© yvu>ti­Kv6k»T§zk!vfqs|Dxk!|vu>qs|xyk6­Kv6k,+ &.- "/% &0 % - ¹i·k k!|izxyk_³D^RS &1- "('6 S &.- ),+ S &1- . + S &1- 1 xyfik¾qs|xyk6yvuz'k6*xfik¾qs|Dxk!­Kv;kz­¥xfik¾ti|q¬xvy |izyj®vuȹ?zyqsk6|Dxk ­¯yzj2% & xyz'·v 3% - » egfikyvj2k k6´|iqxqsz|vyk¶k;rxyk6| k xzc³Xzt| vy^qs|Dxk!­Kv;k!0E¯qs| xfikRq|Dxyk6kMÆxqsz| z­¥xyfik zj®vqs|³Xzt| vy = HON54 = HQP·qxf¾vn³zti| v2´|iq¬xyk_zutij2kLGƹxyfik_q| k;76;zyk!~z| qs|ixyzv*´Æxyq¬©xyqzt;k6usuztxyq kxyfik zj®vqs|>» ¤ k k6|izxk³D R8S &.- " ' 8S &.- ),+ 8S &1- .7+ 8S &.- 1 xfk|izj®vusqcb6k |zyj2vusR8S &.- "RS &.-:9; RS &1-; »T§zgxyfikv«k¥z­%qsj2~iusqs6q¬x¹·°k0tk0xfik_|zxyvxqsz| ] &1- " ] =<.> " ; RS &.-; ] 8=<.> »§qs|vusu¹·°k k!|izxyk0³D@? & xfikk;xz­q| qs6k!gz­xfk¥|k6qsfr³zqs|i´|iqxk_zutjnkMz­xyfik¥´|iqxkzustij2k3% & E¯fvÙDqs|iv|Hqs|Dxk!­Kv;knq| ;zj2j2z|"GÆ» ¤ k k6|izxkn³DA & xfik qw6k6xknj2k!vtiykz­Txfik

ÌÌ äqrLsÜ,s

9 "*#%/L9 ?(9

³Xzt| vyz­v*´|iqxk_zustij2k06k6usuY¹·fiqsfqw k;´|ik ³rA & "- < ; RS &.-; F EK3G

5 * #" " egfk06z|k6y'v'xyqk´|q¬xyk_zustij2k_fk6j2k0~iz~zDk ³rk6j®v«rq @ U Aqwg·yq¬xxk6|M

9::::; ::::< & ' & % [& C % &! > \ - <#" $&% ' (&1- " V + & ' & 4 $)*' (& C 4 $&% ' (&! > C - < + [&1- " V + E¯ldG

·fk6ykxyfik¾qs| k;-,qw2uqs|i«k xz4xyfik¶;k!uu% & ¹ ! > qw2xfikxyqj2k¶xk!~>¹Q% & EKkM~» 4 $&% ' (& G®qs®v|v~i~iyzÙqj®v'xyk­¯z_xfkvÙk!yvkz­°xyfikk!ukMÆxyqwTE¯yk!~>»j®v|ik;xyqs,G´k6u z'k!xyfik6k6usu % & v'x_xyqj2k> " S ! > EKkM~» > $&% ' ( " S \ [² ! > GÆ»egfik¶·z vÙk!yvkfv2xyzc³Xk¶ti| k!yxzrz vv~v;k;©ªj2k!v|'vustik¹·fqsfqw k;´|k ­¯zv|D´k6u v| kMvf;k!uu*%#&9³r

& " #' &.&/ < 103320 F§qs|vusus¹ " $% ' (&.- v| + [&1- vyk;k6|Dxyk6yk |rtij2k6yqs!vu Ç tk!!»egfikkMv k!?6v|fik!«xfvx?xfk´|iqxkzustij2kfk6j2k0E¯l G_qw·yq¬xxk!|^vv¶ukMv~©È­¯z¶fk6j2k3Mn|rtij2k6yqs!vuT'vustikMJ% & v| 4 $&% ' (& vyktk qs||Dtjnk!qw6vu Ç tk!xyz®z³xvqs|vn|ik!·½k;xz­|rtij2k6yqw6vuF'vustikMQ% [& v| E4 $) ' (& »

egfik_|rtij2k!qw6vu Ç tk!vk k;´|ik v­¯zuusz'·M4 xfik Ç tk! " $&% ' (&1- v| + [&1- vyk0qsk!|­¯zq|Dxk!|vuPqs|xyk6­Kv6k!³r

" $&% ' (&1- " ] &1-4 $&% ' (& \ 4 $&% ' (-5 + + [&1- " ] &.-

% [& \ % [-5 F E:XG

4 ­¯zvnj2k;xvusuqw0³zti| vq|Dxk!­Kv;k¹·°k0tk¥xfik0­¯zuusz'·q|n|rtij2k!qw6vu Ç tk!M" $% ' (&.- " ] &.-4 $% ' (& + + [&.- " V + EK3G

·fiqwf ;ztu ³Xk¶z³ixyvqs|ik tqs|i^v ´;xqxqszt;k!uu·q¬xyf 4 $&% ' (- " 4 $% ' (& v| % [- "C*% [& »Tegfik0fizqw;k­¯z + [&1- qsg6ukMvyu®vk!6z| ©Èz k!°v~i~iyzÙrqsj®v'xyqz|®z­>xyfik¥6z| q¬xyqz|RS BE% " V v'xxyfikqs|Dxk6­Kv6k¹·fik6yk!vxfiknfizqw;k0­¯z " $&% ' (&.- qw6usvqw6vusustk qs|¶§¥e j2k;xfz 6»§zµ¾vr·k6usu%k!¸Dtvxqsz|0qs| z|ik6© qsj2k6|qsz|>¹Pxyfiqw_·°kMv«¾xykMv'xyjnk!|Dx0usk!v ¥xz¾v|v!;tiv'xyk¥yk Ç k!Æxyqz|z­qs|;zj2q|i2·gvÙkM6»

4 ´|vuus¹r­¯zgv|qs|Dxk6­Kv6k +&.-z|xfk¥v³z³q|i*³zti| v H P ¹rxyfik¥|Dtjnk!qw6vu Ç tk! " $&% ' (&1-v| + [&.- ·qusu>³Xk k;xvqsuk qs|xfikk!¸Dtik!uY»

ίÑÌ>ÎÓÒ

"!$#%'&(*)+,!.-0/132456578/:9' Y

" " # *# " egfk­¯zuusz'·qs|i2k6usk6j2k6|Dxyv®kM¸Dtvusq¬xyqkMgfizu M

RS - & "&CGRS &1- + ] - &"UC ] &.-3F EYG

xxyfikyvj2k¥xqsjnk¹ik!zj2k;xyqw6vuF~iz~k!xyqkMfizu ­¯zk!vf6k6usu*% & M

áæä;åÈéí¯ñwâéèÞñ RS &1- " RV + áæä6åYéíKñsâéèÞñ ] &.- " V F E N G

:g; >-<66Æ=p°6

* #" ¤ kvqj v'xqrqs|i¶v| ~iyz'rq|v¶|ik!6k!yv¶v| zt¦;qsk6|Dx;z| qxqsz|4­¯zxfik±°²Æ©ìxv³iqsusq¬xz­%xfik*´|iqxknzutjnknfk6j2kkEKl© XÙ©ì3Gg·qxf v³zy³iqs|i³Xzti| vy Ç tk!xz³k k;xyk6yj2q|ik » ¤ ktk¥xfkvjnk0«rqs| z­k!|ik6yv~i~iyzvfvgqs| @ Y(Aª¹i·fik6yk_v2¸tv v'xyqs­¯zyj~iuwvÙ°xfik_yzusk0z­v¨v~iti|iz'®­¯t|Æxyqz|¾z­xfik·fzuskk6xz­%|rtij2k6yqs!vuPti|i«r|iz'·|6» ¤ k_´yx~iyz~Xzkxyfik_­¯zuusz'·q| qsy;yk;xyk¥k!|ik6y¹ qskMÆxus k6yqk ­¯zj7xfk_k;~iyk!yqsz|z­xfik_xyzxvuP|rtij2k6yqs!vuPk!ukMÆxyzj®v|ik6xqwk!|ik6y?M

" & ' & & ' % & % & \ & ' 4 % ' (& 4 $% ' (& F E U G

¤ k_yk!6vuufik6yk¥xfvxxfik_k!ukMÆxyzj2v|ik6xqw¥k6|ik!q|xyfik;z|Dxqs|Dtzt!vk0k6yq´k!zj2k;z|k6©'vxqsz|nk!¸Dtv'xyqz|kEYz'r|Dxqs|i xfk6zyk6jiG­¯zxyfikµ¾v'·k6usuixk6j·qxf2|iz0;tyk6|Dx!»9egfiqwxfik!zyk6jxyvxkMxfvx

. = =?> 23 \ . RA,RS 2&0 " V +­¯zv|r®;uszk zustij2k,' ·qxfvyk6tiusv°³Xzt| vy = '2¹·fik!k pxfk¥k!ukMÆxyzj2v|ik6xqwk!|ik6yE¯­¯zv|qwzxyz~iqw¥j2k qstijiGgv| z'r|Dxqs|i k!;xz RA vyk¥kM~Xk!;xqsk6usqk6|³D

" #5"! % ² \#5 4 ² + RA$"A% B 4 F

§znv¶qsk!| j2k;xyvuusqw6vÙrqx¹%q|6k % B S " V v'xxfik³zti| v¹%z'r|Dxqs|i xfik!zyk6j Dqsk6u xyfv'xxfiknk6usk!;xyzj®v|ik;xyqs0k!|ik6yqwk6vÆxyu¾;z|k6yk qs|¶xyfik*!vÙDqx»¥§izuusz'·q|2xfiqw¥q kMvi¹X·°k|v'xtyvuus2xyxyzntk¥xfk_~iz~zDk qsy;yk;xyk¥k!|ik6y E U Ggvvn¨v~iti|iz'2­¯ti|Æxqsz|»

mªx®qwnv³zustxk!u |izx®z³rrqzt·fr4xyfik qsy;yk;xykk!|ik6y E U G*fiztiu ³XkvR~Xzq¬xyqk k6´|iqxk¸Dtv v'xqw*­¯zyj z­gvuuT|rtij2k!qw6vu%ti|i«r|iz'·|iEKuk6xtyvÙ¹xfik% & v| xfik 4 % ' (& GÆ» ¤ k|zxqw;kfk6ykxyfv'xxyfikq¬xytv'xyqz|qw¸Dtiqxk qDBXk!k!|x­¯yzj7xfik_~iyzrz­z­xfik±°²Æ©ìxv³iqsuqx®z­xfk0´yx©ªz k!t~D·qs| ´|q¬xyk;©ªzustij2k¶yfik6j2k¶z­ @ LY,Aȹ·fik!kxyfik¶k!|ik6y\·v®z³rrqztu^vc~zDqxqsk k6´|iqxk¸Dtv v'xqw­¯zj z­vuuti|i«r|iz'·|!»3xxfkRyvj2k¶xyqj2k¹xfikRk6|ik!\~z~zDk fik6yk k!~k!| k6~iuqw;qxus¾z|Hxfikn|rtij2k6yqs!vu9fk6j2k¹>q|;kqx6v|¶³k2z|iu¾·yqxxk!|Rvv¸Dtv v'xyqs­¯zyj z­°vuut|i«r|iz'·| % & + 4 % ' (& xyfiyztifHxyfik®tk®z­xfikk!6z| ~vx_z­xfikyfik!jnk EKldG;»§qs|vuus¹xfik

ÌÌ äqrLsÜ,s

N 9 "*#%/L9 ?(9

'vqwv'xyqz|z­9xfiqw qwy;yk;xk_k!|ik6y tiyq|vnxqsj2k*xk6~¾j2qsfDxusk!v xyz®xk qszt;zj2~itixyv'xyqz|!¹v­¯zgxfik0´yx©ªz k6ti~r·qs| fik!j2k»

mì| xyfik¾­¯zususz'·q|i¹°·k¶fvusu~iyz'kxfv'xxyfik¶~iyz~Xzk k6|k6y¹·qxf v q¬xyqz|vu³zti| v6zykMÆxyqz|4xk6yj®6¹%qw Ä Ä Â ¢ À ¡i tiyqs|iRvHxqsj2kxk!~ ¡Å(¡ Ä XÀ Å À À ¡iÅ '¡ ¢ Ä Ä ¡ Ärà t| k6v§¨>©Èusqs«k;z| qxqsz|z|xyfik_xqsj2k;©ìxyk6~ ! > »egfqs·qsusuDqsk6u xfik~zrz­xfvx%xfikyfik6j2kgqw%±?²Æ©ìxyv³ukHE¯·qxf2xyfikg~iz~zDk k!|ik6yGti| k!Tv0xv³iqsuqx6z| q¬xyqz|z| ! > »

¤ k_~iyz~Xzk­¯zgxfik_k!|ik6y®'vqwv'xyqz| ! " [ C xfik! 96#"R'&K& #%%$&('*)+I2>96'& #% /-,.,96",`"/0/21\1 " 34 576&.- /"

+ [&.- $ 8,`394"+424& /;:Lp:& )565D2>96"T96#%/9'p) ! =<! " ! >

>@?BABCD(?(E7F.GE7FIHKJ(DMLB>ONPJ(A

Q C 'KR % & \ % [& S R " 34 576&1- C ] &1-4 3-4 576& S \ ' 4 3-4 576& + &.- \ + [&.- UT F ( ) <¤ k0fvÙk3M! " & ' R % & \ % [& SWV & '&- % [& C % & X \ *& '& ' 4 $% ' (& V 4 $)*' (& C 4 $&% ' (& \ 4 $% ' (& C 4 % ' (& X

" ! > & - <Q C ' R % & \ % [& S " $% ' (&.- \ ' 4 $&% ' (& R + &.- \ + [&.-YS T F

uugxyk6yj®2q|\xfik zti³iusk¾tij2j®v'xyqz|ºv³z'k6zykM~Xz| xzR´|iqxk¾zustij2kqs|Dxk6­Kv6k!!»\egfikMkxyk6yj®6v|xfk6|³k qsxyq³txk z|zustij2k0q|Dxk!­Kv;kM6» ¤ k_fvÙk M! " ! > F RUZ áæä;åYéíKä'âç \ Z\[ éìåªâçæçæá \ Z â]ÙèÈÞÆí]ÙáæäÙî S + ·qxf

Z áæä6åYéíYäÙâç " áæä;åÈéíKä'âçáæä6åYéíKñsâéè

Q C 'KR % & \ % [& S " $&% ' (&.- C 'KR % - \ % [- S " $% ' (- &\ ' 4 $&% ' (& + &.- \ + [&.- \ ' 4 $% ' (- + - & \ + [- & T +

Z\[ éìåªâçæçæá " [ éìåªâçæçæá áæä6åYéíKñsâéè

Q C ' R % & \ % [& S " $&% ' (&.- \ ' 4 $&% ' (& + &.- \ + [&.- T +Z â]ÙèYÞÆíU]ÙáæäMî " â]ÙèÈÞÆí]ÙáæäÙî

áæä6åYéíKñsâéèQ C ' R % & \ % [& S " $% ' (&.- \ ' 4 $&% ' (& + &.- \ + [&.- T F

§z°xfik¥qs|xyk6y|vuq|Dxyk6­Kv;kxk!j¹r·°k¥6v|yk6~iuwv6kxfik0;k!|Dxk6yk |Dtjnk!qw6vu Ç tirkM³Dnxyfik­¯zj*tiuwvEXG;»_^q|inxyfik_q k6|Dxqxqsk! ' ] &1- "UC ] &1- " ] - &y¹·°k0´| M

Z áæä;åÈéíKä'âç " áæä6åYéíYä'âçáæä6åYéíKñsâéè

Q ' 4 $% ' (& ] &.- R % & \ % [& S \ ' 4 $% ' (- ] - & R % - \ % [- S T F

ίÑÌ>ÎÓÒ

"!$#%'&(*)+,!.-0/132456578/:9' U

§zxfiknj2k;xvusuqwqs|Dxk!­Kv;k_xyk6yj¹P·k*!v|¶yk6~usv;kxyfiknjnk6xyvuusqs Ç tkM³rxfk­¯zj*tiusv0EK3G;» ¤ kkMvqus2´| M

Z\[ éìåÈâçæçæá " [ éìåªâçæçæá áæä6åYéíKñsâéè

Q ' 4 $% ' (& ] &1- R % & \ % [& S T F§qs|vusus¹·k0´| xfvxj2v|r®xk!j®'v|qsf>¹iq|6kâ]ÙèÈÞÆí]ÙáæäÙîáæä6åYéíKñsâéè

Q ' 4 $&% ' (& ] &.- R % & \ % [& S T \ Z [ éåÈâçæçæá \ Z áæä;åÈéíKä'âç "áæä6åYéíYä'âçáæä6åYéíKñsâéè

Q ' 4 $% ' (& ] &.- R % & \ % [& S \ ' 4 $% ' (- ] - & R % - \ % [- S T \[ éìåÈâçæçæá áæä6åYéíKñsâéè

Q ' 4 $% ' (& ] &.- R % & \ % [& S T \â]ÙèYÞÆíU]MáæäÙîáæä;åYéí¯ñsâéè

Q ' 4 $% ' (& ] &1- R % & \ % [& S T " &

- < ' 4 $&% ' (& ] &.- R % & \ % [& S " & ' 4 $&% ' (& - < ] &.-

R % & \ % [& S " V F¤ k0xfk6|fvÙk

! " ! >â]ÙèYÞÆíU]ÙáæäMîáæä;åÈéí¯ñwâéè

Q C ' R % & \ % [& S R " $% ' (&.- C ] &1-4 $% ' (& S \ ' 4 $&% ' (& + &1- \ + [&.- TQ+·fqsfqsk;iv;xus®xfik_yk!tiuxz­xyfik_usk6j2j®vi»

egfkuk!j2j2vv³Xz'kfiz'·9xfv'xTxyfikk6|k6y'vqwv'xyqz|nqwTz|ius tkxzv³zy³iq|_³zti| vqsk!!»|ik6z|k!¸Dtik!|;kqsxyfv'x·qxfz|ius®j2k;xvusuqw³Xzt| vyqsk!!¹xyfik qw6k6xk¥k6|ik!nqwgk;ivÆxus;z|k6yk »¤ k*!v|¶|zxqw;kxfvx·kyk!;z'k6xyfikk6ivÆx0;z|k!'vxqsz|z­%xyfikk!ukMÆxyzj®v|ik6xqw_k6|ik!z­%xfik6z|Dxqs|rtizt¥!vkn­¯zvj2k;xvusuqw®6vÙrqx»Qmª­gzj2kv³z³q|i³Xzti| vyH;z| qxqsz|_vk2~iyk!k6|Dx!¹qs| rqk!·z­¥xfk¾kMtiux®z­¥xfik¾~kM;k q|i4usk6j2j®vi¹·°k¾!v|º|iz'·~iyz~Xzk¾zj2kfzqw;k!*­¯z®xfikv³zy³iq|*³Xzti| vy Ç tk! " $% ' (&.- v| + [&.- M9::; ::< " $&% ' (&1- " ] &.-5 4 $&% ' (& \ )&*& ; RS &1- ;5 % & <.> +·q¬xyfE% & <.> "A% & C ' R8S &1- F % & R8S &.- +

+ [&.- " ] &.-5 % [& C *& & ; RS &.- ;5 4 & $&% ' ( <.> +g·q¬xyf 47& $&% ' ( <.> " 4 $&% ' (& C ' R8S &1- F 4 $&% ' (& R8S &1- F E!3G

mì|Rxfk!k k;´|q¬xyqz|6¹ % & < > v| 4 & $&% ' ( <.> vyk2kM~Xk!;xqsk!uxyfik2xyv|ik!|xyqsvu E¯zxfzz|vususxyzxfik|zyj2vu2RS &.-,G~vxz­%xfik*k!;xzy*% & v| 4 $% ' (& »¥egfikyk!v k6!v|¶vusz|izxqw;kxyfv'xxfik´k6u % & v| 4 $&% ' (& vykvÙ'vqsuwv³iusk·fik!|xfk³Xzti| vy Ç t " $&% ' (&1- qw°tk ¹v| xyfik6|xfv'xxfik´k6u % [& v| 4 $% ' (& vyk0vusz2vÙÙvquwv³uk0·fik!|xyfik_³zti| v Ç t + [&.- qsgtk »

ÌÌ äqrLsÜ,s

M 9 "*#%/L9 ?(9

egfikzyqq|4z­xfikMk Ç tk!j2qsfDxnk6k!j|zx*yk!vuusHz³rDqszt6»amì| ­KvÆx!¹9xyfik64;zyk!~Xz| xzt~D·qs| 4Ç tk!2v'x*xfikv³zy³iqs|i¶³Xzti| vy»4µzyk~iyk!6qsk6us¹xyfikyk!v k6n6v|^fik!« xyfv'xnxfikÇ t " $&% ' (&.- ~iz~zDk v³Xz'k;zyk!~z| xzRxfikxfiyk6kk6usk!;xyqs´k6u 6zj2~Xz|ik!|xz­xfkqD©6zj2~Xz|ik!|xti~r·q| Ç t &.- ' R % & + 4 $&% ' (& S ¹·fik6yk &.- " ) S &.- ) \ . S &.- . \ 1 S &.- 1 ¹xfikj®vxyqs6k! ) ¹ . ¹ 1 ;zyk!~z| xyzxyfik2z~Xk6v'xzyqs|Hxyfik®;z|k!'v'xyqk­¯zj z­?xfk®µ¶v'·°k!uukM¸Dtv'xyqz|0q| ­¯ti|Æxqsz| z­°xyfik;zj2~iuk6xk®k!ukMÆxyzj®v|ik6xqw´k6u ' %G+ 4 ¹9v| xyfikti~k!yy;yqs~x 6zykM~Xz| XxzxyfikT~Xzq¬xyqkT~vxrqwv qwvz|vusqcb!v'xyqz|z­vgj®v'xyqQEYk!k @ Y(A­¯z6zj2~iusk;xk k;xvqsusGÆ»oqj2qsusvus¹ + [&.- ;zyyk!~z| Pxyz¥xfikxfk!kj®v|ik6xqw°6zj2~z|ik6|Dxz­xyfikqr©ª6zj2~z|ik6|Dx9ti~r·q| Ç t &.- ' R % [& + 4 $% ' (& S »

egfiknyk!v k6!v|¶|izxqw;kxfvxxyfik!k Ç tirkMusk!v xzvk6|rtiqs|ik6us´yx©ªz k6xyk!v'xyj2k6|Dxz­Txfikv³zy³iq|¶;z| q¬xyqz|>¹Tq|6kxyfikv~i~iyzÙrqsj®v'xyqz| qw´yx*z k6³Xzxyf^qs|^~v;kKE¯ti~r·qs| Ç tirkM`Gv| q| xyqj2k0E " $% ' (&.- qw_³vk z| % & v| + [&1- z| 4 $&% ' (& GÆ» ¤ qxfczj2k2xyqj2k;©ªqs|xyk6y~zusvxqsz|>¹xyfik!k Ç tk!¥6v|H³k2;zyk!;xk xyzyk!vf¶kM;z| ©Èz k6v6;tyv;­¯z_µ¾v'·k6usuk!¸Dtvxqsz|qs|¶z|ik qj2k6|qsz|>» ¤ k~iusv|xyz;zjn~vyk?xyfik!kxyk!v'xyj2k6|Dxy9z­xyfikv³zy³iqs|i0;z| q¬xyqz|*·qxf%ek6yk6|k6 k!­¯kMÆxus µ¾v'xfik ¨vÙk6 @ lAȹ·fiqwfVvyk¶uwvyk!uºtk qs|V6zj2~itxv'xyqz|vu¥k6usk!;xyzj®v|ik;xyqs!qsjtiuwv'xyqz| @ r¹M¹M¹LX,Aª» hz'·°k!k!!¹ztij®vqs|¾zDvuPfik!kqsxz®k6xvxyv³iqusqxyk!tiu¬xM¹Xv| ·°kvknv;xtvuus¾v³iuskxyzyk!vf¾xfv'x0zvuz|ius­¯z¥xfikMk*´x©Èz k!v6;tyvxk*v³zy³iq| Ç tk!!¹>v·qsusu>³kfiz'·|q|xfkkM¸tk6uÈ»

¤ qxfRxfk2fzqw;kkE!dG­¯zxyfik2³zti| v Ç tk!!¹>v'vustik*­¯z¥xfkn'vqwv'xyqz| ! z­?xfk qwy;yk;xkk!|ik6y E U G k!qsrq|*­¯yzj ¨>k6j2j®vnl»s0qsgqk6|q|xfk

96#"%:& &#%8$ &('*) +424& /;:Lp:& )565D2>96"T:96#%/9' ) ! =<! "$C ! >

>O?BABCD(?(E7F.GE7FIHKJ(DMLB>ONPJ(A

; RS &1- ;5 )& & ' % & < > % & <.> \ % & [ <.> \ *& & ' 4 & 34 576 < > 4 & 4 576 <.> \ 4 & 3-4 576 < > ( ) < ¨>k!jnj®vnl»s0rqk!u ! " ! >

â]ÙèÈÞÆí]ÙáæäÙîáæä;åÈéí¯ñwâéè

Q C ' R % & \ % [& S R " $% ' (&.- C ] &.- 4 $&% ' (& S \ ' 4 $&% ' (& + &.- \ + [&.- UT" ! >

â]ÙèÈÞÆí]ÙáæäÙîáæä;åÈéí¯ñwâéè

' R % & \ % [& S ] &.-5 4 $&% ' (& C ' R % & \ % [& S & & ; RS &.-;5 % & <.>\ ' 4 $&% ' (& ] &.-5 R % & \ % [& S C ' 4 $&% ' (& &)& ; RS &1-;5 4 & % ' ( <.> \ 47& $&% ' ( <.>

" C ! >â]ÙèÈÞÆí]ÙáæäÙîáæä;åÈéí¯ñwâéè

; RS &.-;5 & & ' % & <.> % & <.> \ % & [ < > \ & & ' 4 & $% ' ( < > 4 & % ' ( < > \ 4 & $% ' ( < > ·fqsfqsxfik_yk!tiuxgz­xyfik_uk!j2j2v»

ίÑÌ>ÎÓÒ

"!$#%'&(*)+,!.-0/132456578/:9'

# !& " * egfk qw6k6xkk6|ik! ~iz~zDk q| E U G k!~k!| *k;~iusqs6q¬xyu z| xfikF=Æti~ vxqs|iRyfik6j2k =n­¯zxyfik®j®v|ik;xyqsn´k!u ¹v| xyfik6yk;­¯zk®z|Hxyfik~vxyqs6tiuwvfizqs6k*­¯z + [&1- »egfik!| qxqw_|izxkMvusutiy~iyqsq|xyfv'x_xyfiqsk6|ik!¾·qusuTfvÙkxyz¾³k®;zyk!;xk xyzfk6us~ t0q|4zt_~izrz­»2¨k;xt¥xyfik6|qs|Dxyz t;k_v2;zyk!;xk qw6k6xk0k6|k6y ¹qsk!|³r

" C ! >â]ÙèYÞÆíU]ÙáæäMîáæä;åÈéí¯ñwâéè

; RS &1-;

&*& ; % & <.> ; ² C &)& ;64 & % ' ( < > ; ² F E,G

egfikg~ifrqs!vuj2k!v|iq|z­xfqsT6zykMÆxyk qwy;yk;xkgk!|ik6y_qw9|izxTxvqsfDx­¯z·gv »°zyk!;xqsz|xyk6yj®*vkz|ius yk6uwv'xyk xyzRv³z³iqs|i¶³Xzt| vyqsk!IEK·fiqwfcj2k!v|xyfv'x " q¬­xfik!kvyk|z|ikLGÆ» egfik¾v i q¬xyqz|vugxyk6yj22~z³v³ius4´| xyfik6qs®zyqq|\q|ºxyfikxk!jn~Xzvugqs|;z|qsxk!|;cz­³Xzt| vy|rtij2k6yqw6vu Ç tirkM6»

¤ k6v||iz'·c~z'k?xyfv'x9xfk qsy;yk;xyk°k!|ik6y qw9|iz|i©Èqs|;yk!vqs|i¹M·fk6|nzti%;k!|Dxk6yk ´|iqxkzustij2k_fik!j2k¥qwgtk »Tegfqsqwtij2jnk ti~q|xfk­¯zuusz'·qs|i

96#" :& i&#%$&('*) 0+H:& g9 ` aU" / "9:%,O/ $0+ 9 %" 96``/96" -02 9 5 /4:& "U";/ :/9'/:9' ! " [ C 9 "9:L ,O/! "$C ! >

>@?BABCD(?(E7F.GE7FIHKJ(DMLB>ONPJ(A

; RS &1- ; & & % & < > \ % & [ <.>5

²

\ & & 4 & 4 576 <.> \ 47& 34 576 <.>5

² V F

( ) < egfk~iyzrz­>qw?k!uk!j2k6|Dxyv» ¤ k¥qsjn~u®v xk!j® k6yqrqs|i­¯yzj xyfik6zykMÆxqsz|®qs| xzxyfik_yk!tiu¬xz­9¨k6j2j®v*l» ­¯z ! » ¤ k_fvÙk! "DC ! >

â]ÙèYÞÆíU]ÙáæäMîáæä;åÈéí¯ñwâéè

; RS &1-;5 &*& ' % & <.> % & <.> \ % & [ < > \ &)& ' 4 & $% ' ( < > 4 & % ' ( < > \ 4 & $&% ' ( <.> C ! >

â]ÙèYÞÆíU]ÙáæäMîáæä;åÈéí¯ñwâéè

; RS &1-;

&*& ; % & [ <.> ; ² C &)& ; 4 & $&% ' ( <.> ; ²

\ ! >â]ÙèYÞÆíU]ÙáæäMîáæä;åÈéí¯ñwâéè

; RS &1-;

& & ; % & <.> ; ² C & & ;64 & % ' ( < > ; ² " C ! >

â]ÙèYÞÆíU]MáæäÙîáæä;åYéí¯ñsâéè

; RS &1-; & & % & <.> \ % & [ < >5

²

\ *& & 4 & % ' ( < > \ 4 & $&% ' ( <.>5

² F¤ k¶fvÙk~iz'k xfvxxfk¶fik!j2k E¯l© XÙ©ì'©!dGqwtf xfv'x®xfk qw6k6xk¾k6|k6y qs®|z|©qs|6kMvq|i¹Pfik6|;k®qxqs_³Xzt| k ³r¶qxy_qs|iqxqwvuT'vutik» ¤ k6v| |izxqw;k2fik6yk2xfv'x_xfqs_yk!tiux_qs'vuq qs| k!~k!| k!|xyuz­xfik_'vutik0z­xyfik0xqsj2kxyk6~ ! > »

ÌÌ äqrLsÜ,s

Ù 9 "*#%/L9 ?(9 " " " "$ " mì|4z k!0xz¾~iyz'knxyfv'xztiyfik6j2k®qwxv³iusk¹·k2´|vususRfiz'· xyfv'xxyfik qwy;yk;xk®k!|ik6y ¹t| k6v4xyv³iqusqxc;z| q¬xyqz|\z| ! > ¹qs®v ~Xzq¬xyqk k;´|q¬xyk¸Dtv v'xqw­¯zyj z­xfik¾|rtij2k6yqw6vu'vutikM 4 % ' (& v| % & »%egfiqwTusk!v %xyzxyfikyk!tiux?z­Pxyfiqs°~v~k!!¹xyv«rqs|i_xyfik­¯zj,z­Fxyfik­¯zuusz'·q|xyv³iqsuqx

U 96" :& &#%$&('*) 0+R /.,96/0/ 5 /@"d"/5 9496 :5 #% /K9U,``T96 '&d9 p:9'7 :& I"U" / U9 `T96 $ +a9 i/a" 96"/L96#"L969:LgU9 96 8//9'S)# )g/565 7L"2> % & + 4 4 576& /"0:& ')g:& i& #%I9 ±°² /-,5D6S9 )i:& 96#% ! > 9 H&g'&/

96 M)/3` + &.- + ! > ² # ' & ' -A & A*- ! - & +- & - -+$ 2>96:&k:& p`: :9' :& / 6 `&5D2,` 5c/8,O/ , 96 :& /-,`;:L )# 5D/ ) ,`"3/0/%96 )/3 + &1- + ( ) <¤ k0k;x³v«2xz2xfk k;´|q¬xyqz|z­xyfik qw6k6xk0k6|ik! » ¤ k_fvÙk " & ' & & ' % & % & \ & ' 4 % ' (& 4 $% ' (& C ! >

â]ÙèÈÞÆí]ÙáæäÙîáæä6åYéíKñsâéè

; RS &.-;

&*& ; % & < > ; ² C &)& ; 4 & % ' ( <.> ; ² egfkj®v|ik;xyqs´k!u 4 $&% ' (& qw°qk6|³D EKldG°v| k!~k!| uqs|ikMvyu®z| 4 % ' (& v| xfik Ç tkM + &.- »egfk®usvxxk! k!~k!| uqs|ikMvyu¶qs|cvusu?!vk!0z| xyfik % & v| ­¯z_xfkv³zy³iqs|i³zti| vRz| xfik4 % ' (& xfiyztf EM3G;»Regfik!|^qx*qw*6ukMv*xfvx v| v*·°k!uugvyk¸Dtv yvxqw2­¯zyj®z­xfikt|i«r|iz'·| % & + 4 % ' (& y» ¤ k_|z'· |ik!k v*usz'·k6g³zti| ­¯z » ¤ k_fvÙk " & '(& )& ' % & % & \ *& ' 4 % ' (& 4 $&% ' (&

" & ' & & ; % & ; ² \ & ' & ; 4 % ' (& ; ² \ ! > ' 4 % ' (& - < + &.-

" & ' & & ; % & ; ² \ & ' & ; 4 % ' (& ; ² \ ! > ' 4 % ' (& - <

+ &.- C ] &1-5 % &

" & - < ' & &A & ; RS &.-;=; % & ; ² \

& ' &A & ; RS &.-;=;64 % ' (& ; ² \ ! > ' 4 % ' (& + &.- C ] &1-5 % & F

mì|*xfkgv³z'k°k6~ikMqsz|>¹Ùxyfik zti³uktij2j2vxqsz|nz'k!6k6usus9v| 6k6usu|ik6qsfr³Xz!v|n³kgyk qwxyq³©tixk z'k66k6usuqs|Dxk6­Kv6k!EKq|Dxyk6y|vu>qs|Dxk!­Kv;k!·qusuDv'xfk6x·z;z|Dxyq³itixqsz|g­¯yzj |ik6qsfr³Xzyq|

ίÑÌ>ÎÓÒ

"!$#%'&(*)+,!.-0/132456578/:9' M

6k6usus!¹·fk6yk!vT³zti| v*q|Dxk!­Kv;kM%·qusuDv'xyfik6%z|iunz|kq|r·gv %;z|xyqs³itxyqz| GÆ» ¤ kxfk6|®fvÙk

" áæä6åYéíYäÙâçáæä;åÈéí¯ñwâéè

' & &A & ; RS &.-; ; % & ; ² \ & ' &A & ; RS &1-; ;64 % ' (& ; ² \

! >5 ' 4 % ' (& ] &1- % -\' - -A - ; RS &.- ;=; % - ; ² \

- ' -A - ; RS &1- ;=; 4 % ' (- ; ² C ! >5 ' 4 % ' (- ] &.-% & \[ éìåÈâçæçæá áæä;åÈéí¯ñwâéè

'(&! &A & ; RS &.-; ; % & ; ² \*& '(&A & ; RS &.- ; ;64 % ' (& ; ² C ! >5 ' 4 % ' (& ] &1- % & \

â]ÙèYÞÆíU]MáæäÙîáæä;åYéí¯ñsâéè

'(& )&A & ; RS &1-; ; % & ; ² \*& '&A & ; RS &.-; ; 4 % ' (& ; ² C ! >5 ; RS &.-; *& & 4 & % ' ( <.> ² F

^qs|i¥k6usk6j2k6|Dxvy¥j2qs|izv'xyqz|>z­y6vuwv~iyz tÆx6¹Ùxyfik k;´|q¬xyqz|z­xfikgj®v'xyq ] &.- v| v i q|i6zykMÆxyqz|xk!j®gqs| ¹·k0´| xfv'x

áæä6åYéíYäÙâçáæä;åÈéí¯ñwâéè

; RS &.- ; ' - -A - ; % - ; ² \ & ' &A & ; 4 % ' (& ; ² C ! >5 ;64 % ' (& ; ; % - ;

\ '(& )&A & ; % & ; ² \

!- ' -A - ;64 % ' (- ; ² C ! >5 ; 4 % ' (- ;=; % & ; \[ éìåÈâçæçæá áæä;åÈéí¯ñwâéè

; RS &1-; ' & &A & ; % & ; ² \ & ' &A & ;64 % ' (& ; ² C ! >5 ; % & ; ; 4 % ' (& ; \

â]ÙèYÞÆíU]ÙáæäMîáæä;åYéí¯ñsâéè

; RS &.-; ' & &A & ; % & ; ² \ & ' &A & ;64 % ' (& ; ² C ! > &)& 47& % ' ( <.> ² C ! > &*& % & <.> ²

egfrt!¹q­9vuu qwy;yqj2qs|v|Dxyvk_xyqs;xus|ik!v'xyqk¹rqÈ» k»9xfik0xyqj2kxk6~ ! > qstfxfvx9:; :<,0 q|Dxk!|vuPqs|xyk6­Kv6k,+&.-3+ ! > ² # < >

< > - *&-+-)& !- 0+, ,0 j2k;xvusuqw0q|Dxyk6­Kv;k0|ikMv6k6usu & + ! > < < < +, , ,0 v³z³iqs|inq|Dxyk6­Kv;k0|ikMv6k6usu & + ! > < < < +

qxqw¥;usk!vxfvx V v| xfvx " V ,+ % & " V + 4 % ' (& " V y»egfiqw6z|6ut k!xfik~zrz­xfvxti| k6gxyfik_;z| q¬xyqz|z­egfik6zk!j(l»s¹Dxfk qsy;yk;xykk!|ik6y qwv*~Xzq¬xyqk k6´|iqxk¸Dtv v'xqw­¯zyjz­9vuu>|rtij2k!qw6vu>ti|i«r|iz'·|!» mªxqwvusz2|iz|©ªqs|;yk!vqs|i¹rxyfik6|¾³Xzt| k v| xfikyfik!jnk0qw±?²;©ªxyv³iuk»

2 # /9 2>96'& H& #Qegfk06zj2~vqwz|z­xfkt¦6qk!|xxyv³iqsuqx;z| qxqsz|qsk!|qs|egfk6zyk6jl»s·qxfxfkxv³iqsuqxusqsjnqx0z³xvqs|ik ³r¶vIP%z|R¼k6tij®v|i|Rv|vusqwz­°§iztiqsk6k6qsk!|ij2z k!­¯z0vxytÆxytiyk zxyfiz©z|vuk!tiuwvyq rqsk6u ¥v|¶kMxyqj®v'xyqz|¶z­%xyfiknz~xyqj®vusqxz­%xfiqw0ti¦®6qk!|Dx¥;z| qxqsz|>»0§izv|

ÌÌ äqrLsÜ,s

!l 9 "*#%/L9 ?(9

zxyfizz|vuvxk!qwv|nyq j®v kz­P¸Dtvkgfik6vfik yv ! TB ! _ B ! a ¹'k!j2v«rq @ U AfvT~z'k6|xyfv'xxfikn;k6|Dxyk6yk ´|iqxkzustij2kyfik6j2kIE¯­¯z¥v|¾fizjnzk!|ik6ztgj2k qstijiGqwxyv³iusk_q¬­?v| z|iusq­

! > #! ² \#! _ ² \

#! a ² 5 +·fqsf*j2v«kM~Xzyq³iusk°xfikgtk°z­Xvxqsj2kgxk!~x·qw;k³iqsk6xfv|*·qxfpJTk6k 9yfik6j2k @ LAWEKk6k @ LA­¯zv;zyyk!Æx~zrz­GÆ»§zxfik2vj2knvxkMqwv|yq ¹ztik6|ik!yvut¦;qsk6|Dx¥xyv³iqsuqx6z| q¬xyqz|qw ! > #! \

#! _ \#! a 5 +

·fqsfqw°jnzkkMxyqwÆxyqk¹D³itxg|izxykxyfv'xgq¬xqsz|iu®t¦;qsk6|Dxv| |izx|k!;kMyvy»uxfiztifxfik|k6· §?P¥e yfik6j2k·qxfv¥x·qw;kg´|k6Tq kM;z'k!yxfkvj2kxv³iqsuqxusqsjnqx°v| 2 qs~Xk6qsz|®vJTk!k yfik6j2k¹ixfkj®vqs|¶~iyz~Xk6xz­9xyfiqsfk6j2kqwqxyv³iqsuqxxyzfv| usk;zjn~uk6k!zj2k;xyqsk!tqs|iti|xyt;xtiyk q !» mì|¾xfik*kM¸tk6uȹXzj2k|rtij2k6yqw6vukMtiuxyfz'· xyfik³vqw_~iyz~Xk6xz­xyfikyfik6j2k¹qs|6ut q|i®z|t|xytÆxtk q vyzt| 6zj2~iusk;k6zj2k6xyqkM6»

"#9;- Æ

! " " " # ¤ k6z|q k6v¾6tiyk6|Dx61®z'k60xyfik6zD_k!;xqsz|4z­vyk!Æxv|itiuwv_·gvÙk6tiq k®v'xJ " V EKk6k§qstiykn3GƹFqsk!|¶³r 6 1 _ " 5 _ 9 2, 5 > ¹F·fik!k 2"#6j qsxyfik*·gvÙk!tiq k_·q xfv| $" V F 3O¥fbxyfik¶q|vu­¯yk!¸Dtik6|;» egfik¾q tk fik!k¾;zyk!~Xz| xz5# 5 ~Xzqs|Dxy2~Xk6·gvÙk!uk!|ixyf>»§qtikv| §qstiyklyk6~kMk!|xxyfikk!ukMÆxyqwv| xfik*j2v|ik6xqw0´k!u ~iyz t;k ³rxyfik®;tiyk!|Dx6 1 » ¤ k26v|Rk6knz| §qstk2xyfik2zrz ¸Dtvusqx¾z­Txyfik®v~i~iyzÙqj®vxk*zustxyqz|6¹·fqsf;zjn~vykk6y·°k!uuP·qxfk6vÆxzustxqsz|!»

dy

x

jz

metallic boundary

metallic boundary

§qstiyk MT;tyk6|Dxfik!k;xqs|¾v*yk!;xyv|itiuwv·gvÙk6tiq k

ίÑÌ>ÎÓÒ

"!$#%'&(*)+,!.-0/132456578/:9' LX

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-6 -4 -2 0 2 4 6

Ez

field

X axis

EXACT SOLUTIONAPPROXIMATE SOLUTION

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-6 -4 -2 0 2 4 6

Hy

field

X axis

EXACT SOLUTIONAPPROXIMATE SOLUTION

§qtik_,M%ok!Æxyqz|¾v'x² ¹`%21%E¯usk;­ÓxGv| 4 .gE¯yqfxGÆ»%¼tjnk!qw6vu>v| k6vÆxzustxqsz|!»

§qstiyk0l,MT¼tij2k6yqs!vu%21%EYv³Xz'kLG°v| 4 .gE z'·| GÆ» * #& " !(" 3 "$) " ¤ k*qsjtiuwv'xyk!v'xxk6yqs|i~iyz³iusk6j®v;yzyvqs­¯zqsusq|¶fk;xk!zk!|ik6zt6vkM6» ¤ k´xqsuustij2q|v'xkv¾6zvxk ¼Mvqs­¯zqu?~iz´uskaE qsk6usk!Æxyqw®uwvÙk!¥z­gxfiqw«r|ik!yE" V F #¹·qxf ! " v| *LG³Dnv¥jnz|izfiyzj®v'xyqs°·vÙkguz6vxk yqfDx·v EK·q¬xyf*­¯yk!¸Dtik!|; "&#3F 5 O¥fb,GÆ»%egfik~v6k

ÌÌ äqrLsÜ,s

M 9 "*#%/L9 ?(9

qsy;yk;xyqDbMv'xqsz|Hqsti|xytÆxytiyk v| 6zykM~Xz| xzZ#~zq|Dx~Xk60·gvÙk6usk6|xf EYk!kn§qstiyk X3GÆ»§qstiyk¶4yk6~iyk!k6|Dxy®xfikH6z|Dxzti2z­0xfikR!v'xxyk6yk k!ukMÆxyqw´k6u » ¤ kR6v| |zxqw;kxyfv'xxfik|rtij2k!qw6vuFzustxyqz|fvg|iz~itiyqztzy;qsuuwv'xyqz|vxxyfik_qs|xyk6­Kv6k¥³Xk;x·k6k!|xyfik0x·znj®vxk6yqwvuw6»

* #& 3 "* ¤ k¾~iyk!k6|Dxfik!k¾vv| k;ivj2~iuskxfikR;zj2~itixk yk!tiux®­¯z2xfk¶j®v|k;xqw´k!u y6v'xxk!k ³rv^j2z k6u_v|Dxk!|i|v E¯fik!k4v|Vfik;xyk6yzk!|ik6zt6q;tusv qsk6usk!;xyqs ;rusq| k6!v'xxyk6yk6Gƹ·fqsfVqs~qs;xtiyk z|¶§qtikQYr» egfik_­¯yk!¸Dtik6|;z­9xfk*qs|;q k!|x¥·vÙk;zjnqs|i®­¯yzj xfkyqfx¥q kz­%xfikz³k!;xqw " V F #O¥fb»_egfik*v qtz­Txyfiknusvkj2k;xyvuusqw*6Dusqs| k6¥qw " V F j¹P·fiqsuk*xfikv qtz­Xj®vusurj2k;xyvuusqw°6Dusqs| yqs!vuDq|6utqsz|qw " V F V5 jE¯xfik6k6|Dxk!yvyk?yk6tiusvus_~v;k ·qxf\v| qs|xyk6y'vu?z­ V F #6j%G®¹xfikxfiqw«r|ik!yz­gxfik qk!ukMÆxyqw®uwvÙk6qs " V F #!j»¶egfk~Xk6­¯k!Æx qk!ukMÆxyqwj®v'xyk6yqsvuqw k;´|ik ³r ! " 5 F v|

*»mì|¶xyfiqw¥xk!x_6vk¹Fxyfik2j2k!fHqwv'xyfik6

6zvyk¹q|6k0·°k0fvÙkz|iu # ~zq|Dx~k!·vÙk6usk6|ixf>» ! # #" " " " ¤ k2;z|q k60v| vqs­¯zqu ·qs| z'· ·qxf4vusq¬x_v¥qs|R§qstiyk N »egfiknk!usvxqsk~vvj2k;xyk6z­°xfik qk!ukMÆxyqw9usvÙk6>vyk ! " 5 v|

"U#» ¤ k?;z|q k! qDBXk!k!|Dx>qj2~Xk6­¯k!Æxyu_6z| tÆxqs|ij®v'xyk6yqsvus

E qDBXk!k!|xWGÆ» ¤ kk6| v2j2z|izfizj®v'xqw¥·vÙk0·q¬xyf¾q|6q k6|6kv|iukT" V ¹Xv| ­¯yk!¸Dtik6|;(" V F + ] a » ¤ k;zj2~itxkxfik·vÙk®~iyz~vv'xyqz| v;yzy0xyfikj®v'xyk6yqsvu^EY§qstkM U v| M3GÆ»egfk!k¥qj*tiuwv'xqsz|°·k6yk¥~iz t;k ·q¬xyfv|k6~z|ik6|DxqwvuXxyqj2k_fk6j2k­¯zxfik¥xyk!vxj2k6|Dxz­xfikv i qxqsz|vuiuszyxk6yj®6»k6~Xk6| qs|iz|nxfik6z| tÆxyqrqxz­Xxfik|iz|2~Xk6­¯k!Æx;z| tÆxyzM¹·kk;x qCBFk6yk6|Dxxzxvuk!ukMÆxyqwT´k!u ·qxf2vj2zDzxfxv|q¬xyqz|­¯yzj v|*v~i~vyk6|Dx~k!­¯kMÆx qsk6usk!ÆxyqwHE "V G9xyz*v|vusjnzDxgrj2j2k;xyqwzustxqsz|KEÓ­¯z "U# V3V ¹·°kfvÙk¥vusj2zxTx·°z*~k!­¯kMÆx;z| t;xzGÆ»

§qstiyk%X M_egfik2ti|xytÆxytiyk qsy;yk;xyqDbMv'xyqz|¶­¯z0xyfik®;zDv'xk ¼iMvqs­¯zqu E¯xfik qsk6usk!;xyqsj®vxk6yqwvuP;zyk!~z| °xznxyqwv|iukMgq|q k¥xfik_j®v«k uqs|ik,G;»

ίÑÌ>ÎÓÒ

"!$#%'&(*)+,!.-0/132456578/:9' Y

§qstiyk_7MTegfik_y6v'xxk!k k!ukMÆxyqw´k6u " 3 " :" " " "$ ¤ k~iyk!k6|Dxfik!k®~k!uqsj2q|vHkMtiuxy_­¯z_xyfik|rtij2k6yqs!vu?qsjtiuwv'xyqz| z­°xyfik®xyqj2kk!zustxyqz| z­v|4k6qsk6|jnz k®qs| v¾;ti³qs®j2k6xyvususqs®!vÙDqx¹ ¤ k;z|q k60xyfik0E¹s¹s,G¥j2z k»egfik;t³iqs®!vÙrq¬xqw qwy;yk;xqcb6k ³Dv®xytÆxytik zxfizz|vuFk!tiuwv6ti³Xk!!¹·qxfx·k6|Dx6k6ususqs|¾k!vf qyk!;xqsz|>»¤ kfvÙk®fizk6|Rxz¾~iyk!k6|Dxz|4§qstiyk¾nxyfikk!xyqs!vuT6zj2~z|ik6|Dx 4 1®z­°xyfikj®v|ik;xyqsn´k6u vx*xyfik¾;k!|xyk6nz­xfik6vÙrqx qs|\­¯ti|Æxyqz|\z­xfkxyqj2k»^¼tij2k!qw6vuyk!tiuxynz³ixyvqs|ik ³rFJTk6k yfik!jnkv| ztin´|iqxk¾zutjnkyfik6j2k E¯·qxf ;t³iqsk!uk!jnk!|Dxyvy zutjnkM`G*vyk;zj2~vyv³iusk M|z¶|rtij2k6yqs!vu qwyqs~v'xyqz| qs*vxq´;qwvususR~iyz t6k v| xfik±°²k!yz_·qxf xyfikk6ivÆxnzutxyqz|EK~zq|Dx·qwk qDBXk!k!|;kM­¯z*³Xzxyf^yq `Gqs*k!4j2vuuÈ»4§iznk!vfc|rtij2k6yqs!vufik!j2k¹%·kfvÙk~uzxxk xfikk;iv;x?v| v~i~iyzÙrqsj®v'xyk'vutik;zyyk!~z| qs|ixyz_xfk;z|q k!k n k6yk6kz­­¯yk6k zj»oq|6kQJTk6k yfik!jnk®qs|rzuk!_vxvk6yk yq ¹³Xzxyf =;k;iv;xL=n;tk! qDBXk!!» ¤ k6v|4z³k!kxyfv'xM¹z|xfiqwg~vxyqs6tiusv!vkz­v*k!tiuwvzxfzz|vuXyq ¹Dxfik0|rtij2k6yqs!vu qw~Xk6qz|gv| xfik6zj2~itxv'xyqz|vu>6zxyz­9³zxf¾§7Pe v| §e j2k6xfiz vk_6zj2~vv³uk»

! ;

mì|Hxfiqw¥~v~Xk6M¹F·°k*fvÙk*~iyz~Xzk vt¦6qk!|x0;z| qxqsz|¶­¯zxyfik2xv³iqsuqxz­Txyfik2kM;z| z k6´|iq¬xyk°zustij2kyfik!jnk~iyz~Xzk ³rk!j®v«rq­¯zxfik°xqsj2k zj®vqs|*zustxqsz|z­Fµ¶v'·°k!uuDk!¸Dtvxqsz|qs|®x·zv| xfiyk6k qj2k6|qsz|?q|fik;xyk6yzk!|izt9jnk qwvi»>V%|ik!*kMxyqj®v'xyk!?ukMv tTxz*vt¦6qk!|x§9¨>©Èusq«k?xyv³qusq¬x0;z| qxqsz|z|v³iqxvy´|iq¬xyk?zustij2k!>·qxfjnk6xyvuusqsTzv³zy³iq|³zti| v6z| q¬xyqz|6¹F­¯z_·fiqwfH·k!v«xykMv'xyjnk!|Dxy¥·q¬xyf v'x_ukMvx06z|qwxyk6|Dx¥|rtij2k!qw6vu Ç tk!0fvÙk*³Xk6k!|

ÌÌ äqrLsÜ,s

N 9 "*#%/L9 ?(9

Dielectric

Metal

Free space

§qstiykiY/MO¥k6zj2k6xyEYv³Xz'k¹Xusk;­ÓxGƹWb!zrzj z­°xfik2j2k!f EKv³Xz'k¹PqsfDxGv| y6vxxyk6yk j®v|ik;xyqs´k6u E z'·| G;»

ίÑÌ>ÎÓÒ

"!$#%'&(*)+,!.-0/132456578/:9' U

wave incident

a perfect conductor a conductor (σ >> 1)

a dielectric layer (a slit) ( ε = 2)

§qtiyk N MTv*·qs| z'· uqx

§qstiyk U MTezxvuPk6usk!;xyqs´k!u © " V EKuk6­ÓxGv| "&# V V EKqsfDxG

§qstiyk2!,M9ezxvuPj®v|ik;xyqs´k!u © " V EKuk6­ÓxGv| E"&# V V EKqsfDxG

ÌÌ äqrLsÜ,s

9 "*#%/L9 ?(9

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10 12 14

Hz FVTDHz FDTD

exact (fvtd)exact(fdtd)

§qstiyk3M\P%k6xqw6vuj®v|k;xqw0´k!u v'xxyfik*6k6|Dxk!z­?vjnk6xyvuusqs*;ti³qs!vÙrq¬x­¯z*JTk6k yfik!jnkv| ztig´|q¬xyk_zustij2k_fk6j2k

ίÑÌ>ÎÓÒ

"!$#%'&(*)+,!.-0/132456578/:9' i

tk »Tegfik_xv³iqsusq¬x6z| q¬xyqz|fv~i~Xk6|gxyz2³k_j2zyk¥kMxyqwÆxyqk o³itix·q¬xyfv2;zj2~vv³iusk¥uqsj2q¬xxyqj2k?xk!~^oz|_yk6tusvXq Fxyfv|¥xyfv'xz³xyvq|k ·q¬xyf§iztiqsk6v|vuqs!»Whz'·k6k6M¹Æxfiqw6z| q¬xyqz|qwk6|k6vuÈ»0orzj2k*|rtij2k6yqw6vuyk!tiu¬xqs|¶x·zv| xyfiyk6k qj2k6|qsz|~iyz'k xfiknuwvyk~Xzxyk6|Dxqwvuz­gxfikj2k6xfiz ¹­¯z*·fiqwfck!k6vu%·zy«vykxqsuuvfikMv M*~izrz­z­;z|rk6yk6|;k2v| k!6z| ©z k6v6;tyv;z|¾ti|xytÆxytik yq 6¹X;zt~iuqs|i®·qxfH°µ¾¨cj2k qtij¹Xjtiuxq©ì!vusk;zt~iuqs|iz­j*tiuxqs~iukti³ zj®vq|g·qxf qDBFk6yk6|Dxj2k!fqDb!k!v| xqsjnkxk!~!»

&'

@ A *» »| k6z|>¹ l»ã»Dev|i|k6fiqsuuȹv| _» h»%uk6xyfik!!» #2" /:9'"/5;1\9' #%&/9'H/"&/\:/M) » hk6j2qw~fik6yk»µ¶(O¥vÙ·g© hqusuY¹¼k6· JTzy«X¹P UN l»

@ Alk!v|© vÙrq ek6|vjnzt^v| eti|z k!~ikM6» zj®vq| k!6zj2~zDqxqsz|^jnk6xfiz ­¯znxfikfk6usjnfzuxbgk!¸Dtvxqsz|*v| k!usvxk z~xyqj®vur6z|Dxyzur~iyz³uk!j2!» #2" &/L ¹ME,G M N oN ¹P UU Y»

@ A_§»"e°zti k6uȹi» ©ì»Fµ¾vb!k;x!¹v| hk!uusti»» ^5C:9' )'& /9'/0/kH& /#% 9'-0/(124565 8/:9' ,O/k/a9 96 9 96 5c#% #Q'&( ?O3 5C9'`/:9' g/ D- $ 5D#Q/."9' 96# 5 B+H/L ¹'~vk!PlD3X,orlD»°zj2~itxyq|i¥µk;xyfiz qs|~~iuqsk o;qsk6|6k!v| V%|iq|ik!k6yq|»i¼z''v®o6qk!|;k_%ti³uqwfk66¹mì|»s¹i¼k6·g©JTzy«F¹P U3U »

@ lAl» »e yk6|k6M»Fegfik!k;© qsjnk!|qz|vui~Xk6­¯k!;xus2j2vxyfik usvÙk6%­¯zTxyfikv³z~ixqsz|®z­Pk!ukMÆxyz©j®v|ik6xqw·gvÙkM6» # &-/L ¹M3Y M ordY U ¹ UU »

@ XA_»fOn»°qsvusk;xv| li» ©ª¨?»>¨>qsz|!¹>k q¬xyz6» /-,`(T )#%9'`/5?^"/5 /L9 ¹zutjnk »¼zxfThzuuwv| © VTusk6rqsk6o;qsk6|;k%ti³uqwfk66¹ij2xk! vj¹i¼k!· JTz«F¹ r­¯z ¹P U3U »

@ Al» ©ª»°qsz|iqȹi¨T»i§kb6ztiqȹiv| h»orxk6k»9~i~zÙqsj2vxqsz| k! ¸tv'xqsz| k_µ¶v'·°k!uuF~v k!yf j2v 6k6|Dx k6| u j2k!|x´|iqw6»iek!fi|qs!vuk6~XzxTg©!i¹Lm¼ m¹§k6³iytvy® U3U i»

@ YLAl» ©ª»>°qz|iqv| µ4»Fk!j2v«rqY»¥°zj2~vvqwz| k k!t¾j xfz k! kzustij2k!g´|qsk6|¶k!ukMÆ©xyzj®v| xqwjnk»%ekMfi|iqw6vu>k6~Xzxg©ª!i¹ m¼ m¹µ¾vÙ¶ UU Yr»

@ N Alk!v|©ì%qk!yk°qsz|iqȹ?¨?»§kb6ztiqȹ?¨T»%||ik¹?v| §»%zti~vt » ~vvusuk!u°§7Pe µ¶v'·°k!uuzusk!tqs|iº ti|xyt;xtiyk j2kMfikM6».mì| +:& / /5S@:(9'2 )" "096 / 5 9'#2" /:9'"/5f5c`#%/@"":9' ¹Xµz|Dxk6yk6¹vusq¬­¯z|iqwvi¹F UU Y»

@ U AOn»°zfk6| v| »lzus¹k q¬xyz6» H!: ' #:& , #9'8 5D8/9' 3 -0/(124565 ¹n°zuusk!;xqsz| q v;xqw¸Dtik m¼ m¹*m¼ m z6¸Dtik!|;ztix!¹§iv|;k¹µ¶vf'©3Y2 UU3N »

@ !A0§yv|;qw%°zusuqs|izv| k;xyk6Tµz|i«F»egfik~k!­¯kMÆxyu_j®v'xfik usvÙk6qs|6tirqsuqs|ikMv6zrz q|v'xkM6» H- 9 # ¹> U EKdG6Mæioi U i¹ UU3N »

ÌÌ äqrLsÜ,s

9 "*#%/L9 ?(9@ A0»dV%rj®v ¹re0»dO0vuuszt6x!¹rv| »dhk6y³iq|» & 9 96 :5 #%S#%:& , »Pxyzv~~kMv?qs|ihv| ©

³Xzrz«­¯z¥¼tij2k!qw6vu|vusqw6»X¼zxyf0hzuuwv| © V%uwk6rqk!¥o6qk!|;kn%ti³iusqwfik!y!¹Fj®xyk6 vj¹¼k6· JTzy«F¹ D­¯z ¹P UU3U »

@ MA0»Zhvxyk6|>»Qhqsf kMzutixqsz| oifik6j2k!¥­¯zhr~Xk6y³zuqw°z|k6y'v'xqsz| ¨vÙ·!» # &-/L ¹l U M 3XdY(or U ¹i U3N i»@ !A0»WlvjnkMz|>»xyq¬´X;qwvu qCBFtqz|>¹Pti~r·qs| ³iqwvq|¹Fuqsj2q¬xyk6v| xyfik6qs0kBFk!Æx0z|Rv66tiv;

v| jtiuxqsyq 6z|rk!k6|6kqs|Hxyyv|z|iqwnv| fDr~Xk6z|iqw Ç z'·!»Qmì| :&T * #2" /9'/5 45 9' ! /"/#p9' ()"6\5c/"3H45D9'/ ¹ ltius^'© U UU i»c m ~v~Xk6U '©ìdX U »

@ 6lA0»r¨>kMvq|Dx!» 5D/p 5 9' *0/L#% &/.,`5 9'8J # 9'H /S3^#'&d,p 5 #Q:_9 9 »TTf xfk!qs!¹^|qk6qx _ k9vyqw P^mƹ> U YXi»

@ ,XA_µ¾qsfvk6u§»?9vqs«F¹ ¥vÙDq e¥»°ork6q k!uY¹gv| vÙrj2z| ^¤ »?¨k6j2«k» j2z q¬´k ~k!­¯kMÆxusj®vxyfik uwvÙk!qsj2~iusk6j2k6|Dxyvxqsz|­¯ztk0q|k!ukMÆxyzj®v|ik6xqw~iqw_;z kM6» #2" &-/L ¹!l N E,G MÙXLoX!D¹i UU3U »

@ !A0k6xk6%On»9k6xyz~Xztiusz!» |4xyfikxyk6yj2q|vxqsz| z­xfik~k!­¯kMÆxus4j®v'xyfk uwvÙk!·q¬xyf^usz6vuv³zy³iq|*³Xzti| vy;z| qxqsz|!» #2" &-/L ¹!lEÈG M 3X,odYi¹i UUN »

@ LYLA_oP»i%q~Xk6y|iz»±°²Æ©ìxv³iqsuqx®z­xfik0ti~r·qs| ´xz k!?´|iq¬xyk0zustij2k0fik!j2k­¯z°xyfik0j®v'·°k!uukM¸Dtv'xyqz|qs|x·zv| xfiyk6k qsj2k6|qz|z|*v³q¬xyyvti|xytÆxytik j2k!fikM6» * - -0( 5 -0/'& Z^"/5 # ¹l E,G MM U oF,X N ¹»

@ N A0§»Ùzti~vt v| µ4»Mk6j®v«rqÈ»V9qsxk!|;k°v| ti|qs¸Dtik!|ik!y>z­xfikTj®vr·k6usu xk6j zustxqsz|qs|¶fik6xk!zk6|k6ztv| qsyk6tiuwvj2k qwvi» Z*/3 9 /9 -0/'& ¹xM»X,M U3U oM¹i»

@ U A_µ »?k6j®v«DqÈ» |ik6·´|iqxk¾zustij2kyfik!jnk­¯z2zusrq| j®vr·k6usu 2rxk!j» \- >W& /9'/5 /5) #2" /:9'g/H-0/:& #%/9' 96 5D9'^/" \5D9' "96"96#" ¹ U EK3G M U !o U ¹i»

@ A_µ »Tk6j®v«rqȹ?¨T»%§kb6ztiqȹ°v| §»%zti~vt » ^|^|iztikMvt^yf j®v kxD~Xkzutij2kM´|qsv~i~iusqs¸Dt vti ¸Dtv'xyqz| k®µ¾vr·k6usuk6|¶j2qsuqsk6tRf x yzh!|ik»0ek!fi|iqw6vuk6~Xzx¥g©ìdX¹m¼ m¹lv|rtvy UU3N »

@ Al»æoF»orfv|»7 fvv;xk6yqwxyqs;©È³vk vuszyqxfij ­¯z¾zusrq|iº ¹xqsj2k;© zj®vqs|½µ¶v'·°k!uukM¸Dtv'xyqz|6»mì| + :&^L / 9' -0:96#"%/" >1&d9U,96 ¹k6|iz¹D¼k6'v vi¹dlv|rtv*'© U UU »9 m ~v~Xk6 U Ù©ìldX»

@ Al»æoF»orfv|iv| » µ4»P§qxfik!|>»*,;zj2~vv'xyqkxyt ¶z­°fvyvÆxk!qwxqwÆ©ª³vk vuszqxfj2­¯zgxfikµ¶v'·°k!uuPkM¸Dtv'xyqz|6» # &-/L ¹MX7M Y N or U l¹i UU i»

@ ASP*»orfv|i«'vM¹ ¤ » §» hvusuY¹iv| » h»µzfvj2j2v qwv|>»>Vxqsj2k;© zj®vq| qCBFk6yk6|DxqwvuFzusk6?­¯zk!ukMÆxyzj2v|ik6xqwy6vxxyk6yq|i2~iyz³uk!j2!» , ¹ YYE:XG M Y U oYi¹i UN3U »

ίÑÌ>ÎÓÒ

"!$#%'&(*)+,!.-0/132456578/:9'

@ 'lA0»ev Ç z'k»ik6©Èqs|rk!|xyq|¥k!ukMÆxyzj2v|ik6xqw6Mti~k!y6zj2~itxyq|zutxyqz|*z­Pµ¶v'·°k!uu 9k!¸Dtv'©xyqz|rqwv qyk!;xxyqj2k_qs|Dxk6yvxqsz|z|¾~v;k¥yq 6»%*m~v~k! U '©ª¹ UU i»

@ XA0»iev Ç z'k» # /9'/5?5D:,I//# 9' < :& _9496 9 *H:96#% #%/96K#Q:& , »xyk!ffiztk¹P U3U X»

@ A0»Dev Ç z'kv| V»dez ·qs|>»F¼tij2k6yqw6vuXzutixqsz|2z­Pxyk!v D©ªxyvxkk6usk!;xyzj®v|ik;xyqsgy6vxxk!©qs|i~iyz³iusk6j®tqs|ixyfik*xqsjnk6© k6~Xk6| k!|Dx0j®vr·k6usu k!¸Dtv'xyqz|!» `/ -T9'24/I:W&`0/F & ¹µ¾ee©7M Lo¹i U YXi»

@ 3YLAl» ©ª»"Pqsusv»0°z|rk6yk6|;k_v| k6yzk!xqsj®v'xkMqs|¶´|q¬xykzutjnk*fik!j2k!­¯zk6|ik!yvu>jtu¬©xyq qj2k6|qsz|vu?y6vus;z|k6yÙvxqsz|RusvÙ·!» mÆ»>k;~iusqw;qxk®j2z|izxz|ik®fik!j2k!!» * - -0( 5 -0/'& Z^"/5 # ¹X N EY3G M dY(oi U X¹i UU l»

@ N Al» ©ª»\Pqsusvcv| »4Pqusuk qsk6t>»3°z|Dk6yk!|;k k¶uwv j xyfiz k kMzustij2k!2´|qs~ztiusk!xyh6j2k! k§yqsk yqsf6» b W^`/3 9 /9 H-0/:& ¹XEYX3G6M dYor Y'¹i U3U Yr»

@ U A0» §» ¤ vyj2q|cv| §»4hk;xx!»Vegfk¾jnz q´k k!¸Dtv'xyqz|ºv~i~iyzvf^xyz4xfik¶xyv³iqusqx\v| v66tiyv;v|vusqwz­´|iqxk;© qDBXk!k!|;k0j2k;xyfiz !» #2" &/L ¹!l EY3G6Ms,X U oXY U ¹i U YÙl»

@ A2»PoF» JTk6k»¼tjnk!qw6vuzustxyqz|¾z­Tqs|iq¬xyqsvu³Xzt| vy'vustik~iyz³iusk6j®q|rzusDqs|iµ¾v·°k!uu kM¸Dtv'xyqz|_qs| qwzxyz~qsnj2k qwvi» / ^ /Li/ 6/."3/:9' ¹ EY9©!dG6M LodY¹P U i»

ÌÌ äqrLsÜ,s

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles - B.P. 93 - 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Lorraine : Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - B.P. 101 - 54602 Villers lès Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot St Martin (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - B.P. 105 - 78153 Le Chesnay Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, B.P. 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399

Recommended