High-Contrast Gratings based Spoof Surface Plasmons

Transcription

1 Suppleentary Inforaton Hgh-Contrast Gratngs base Spoof Surface Plasons Zhuo L 123*+ Langlang Lu 1+ ngzheng Xu 1 Pngpng Nng 1 Chen Chen 1 Ja Xu 1 Xnle Chen 1 Changqng Gu 1 & Quan Qng 3 1 Key Laboratory of Raar Iagng an Mcrowave Photoncs Mnstry of Eucaton College of Electronc an Inforaton Engneerng Nanjng Unversty of eronautcs an stronautcs Nanjng 2116 Chna 2 State Key Laboratory of Mlleter Waves Southeast Unversty Nanjng 2196 Chna 3 Departent of Physcs College of Lberal rts an Scences rzona State Unversty US. *Corresponng author: lzhuo@nuaa.eu.cn + These two authors contrbute equally to ths work S1

2 1. Dervaton of the sperson relaton of 2D HCGs-base SSPs wth ultoe network theory Three steps shoul be taken to obtan the analytcal sperson relaton of the HCGs-base SSPs. Frstly the egenfunctons are eterne for Regons I (ar) an (HCGs). Seconly bounary contons are pose at each nterface to obtan the transfer relaton of the attance atrx between each regon. Fnally the generalze transverse resonance technque s use. a y z k x β x k x β ε μ ε μ b I I I V k x Y V Y k x V Suppleentary Fgure 1. (a) The unboune peroc HCGs. (b) The equvalent transsson lne of one unt cell n (a). S2

3 To eterne the egenfunctons of the HCGs regon n Fg. 1(a) n the an text the structure shown n Suppleentary Fgure S1(a) shoul be frst analyze whch s an unboune peroc array. Two electrc blocks n each unt cell are of wths relatve perttvtes an pereabltes an. The equvalent crcut network for the -th (= 1 2 ) surface oe n one unt cell s shown n Suppleentary Fgure S1(b). Thus the egenfunctons for the -th TM oe can be expresse by the oe voltage an oe current as j z H x z I x e y j z Ex x z I xe j z E x z V x e z (S1.1) (S1.2) (S1.3) n whch s the perttvty of ar s the relatve perttvty of lock ( = ) s the angular frequency. Substtutng equatons (S1.1)-(S1.3) nto Maxwell s Equatons we can fn that the -th surface oe voltage V x an current I x n lock satsfy the followng transsson lne equatons where V I x x x x jk Z I x (S1.4) x x jk Y V x (S1.5) Z 1 kx (S1.6) Y kx k (S1.7). S3

4 ccorng to the Floquet theore we have V V. (S1.8) I I The nput-output relaton for the voltage an current colun vector n one unt cell can be enote by V V T I I V V T I I (S1.9) n whch the transsson atrx T for lock can be expresse as T cos kx jz sn kx cos jy sn k k x x. (S1.1) Then the total transforaton relaton n one unt cell can be enote as V V V V I I I I T T T T (S1.11) an the total transsson atrx T for one unt cell s of the for t T t t t. (S1.12) Cobnng equatons. (S1.8) an (S1.11) we can obtan the followng equaton V T 1 I (S1.13) n whch 1 s the unt atrx. ccorng to the lnear algebra theory the exstence of a nontrval soluton for equaton (S1.13) requres that the eternant of the coeffcent atrx vanshes naely Thus S4 et T 1. (S1.14)

5 2 t t 1. (S1.15) The two roots of equaton (S1.15) ust satsfy the followng relaton 1 t t. (S1.16) Wth the assupton that one root s jkx the other root ust be e n whch x e jk k x s the propagaton factor of the Floquet oe. n fnally the sperson equaton of the unboune HCGs can be wrtten as t t cos x (S1.17) 2 n whch x s the loch wave vector. Substtutng equatons (S1.1)-(S1.12) nto equaton (S1.17) we can obtan equaton (1) n the an text. It s known that for each egenvalue there s a corresponng egenvector U satsfyng the followng equaton TU U. (S1.18) ccorng to the transsson lne network theory the voltage an current vectors can be represente by a lnear cobnaton of the two egenvectors. For each egen exctaton V U I (S1.19) V V. I I (S1.2) Then the voltage an current colun vector wthn the -th ( = ) block can be expresse as kxx jz sn kx x cos cos V V ; x I jy sn k I xx kx x (S1.21) S5

6 kxx jz sn kxx cos cos V V. x I jy sn k I x x kxx (S1.22) Thus the voltage an current n one arbtrary unt cell can be wrtten as V I x x V x x V x x I x x. I x x (S1.23) (S1.24) Conserng that V x an I x are peroc functons wth pero they can be expane n Fourer seres as n (S1.25) n xn jk x V x V e x n (S1.26) n xn jk x I x I e x n whch V n an I n are the apltues of the n-th space haronc for the -th oe voltage V x an oe current I x an can be erve n the followng for for the TM oe 1 jk 1 xnx In I xe x Qn Qn (S1.27) V 1 V xe x 1 Q Q jkxnx n n n (S1.28) n whch Q jkxn kx jkxn kx Y f e 1 Y g e 1 j k k j k k n xn x xn x (S1.29) S6

7 Q jk xn j kxn kx jkxn j k xn kx Y 1 e f e Y e g e 1 j k k j k k n xn x xn x (S1.3) f V I Y (S1.31) 2 g f V I Y (S1.32) 2 V I Y (S1.33) 2 g V I Y. (S1.34) 2 Fro above analyss t can be seen that the egensolutons of the peroc structure satsfy the Floquet conton an can be represente as the suaton of nfnte space haroncs. Due to the spatal perocty of the HCGs n the x recton all space haroncs are generally excte everywhere n the whole structure. The egenfunctons n Regon I can be sply represente by an nfnte nuber of transsson lnes each of whch stans for one space haronc. Thus the whole structure can be oele by an equvalent ultoe network shown n Fg. 1(b) n the an text. The key step for the bounary value proble s the constructon of a general soluton for the peroc regons. Snce the egenoes n the unboune HCGs have been eterne the transfer relatonshp of attance atrx for each regon can be obtane by usng the bounary conton that the tangental coponents of the electroagnetc fels are contnuous at the nterface z. Supposng that n Y n whch s an eleent of the output atrx Y n of the HCGs layer s known the electroagnetc fels n Regon can be expresse as S7

8 n whch E tn an z Y z H n tn n z E tn (S1.35) H tn represent the n-th space haronc electrc an agnetc fels n the peroc HCGs (Regon ) respectvely. The rectons of the electroagnetc fels for the TM oes can be enote by Etn xetn an Htn yhtn ( x y an z are the unt vectors along the x y an z rectons respectvely). The transverse fels n the HCGs regon z h for the TM oes can be expresse as oal suaton wth suppressng the constant vectors j l zh j l zh Et x z Vl x ale bl e l j l zh j l zh Ht x z Il x ale bl e l (S1.36) (S1.37) the n -th Fourer coponent s j z h j z h l l Etn x z Vnl x ale bl e l j z h j z h l l Htn x z Inl x ale bl e l (S1.38) (S1.39) n whch a l an b l are the forwar an backwar travelng wave apltues of l -th Floquet oe n the HCGs respectvely an l s the wave vector along the z recton. V nl an I nl are the apltues of the n -th space haronc for Snce the nterface at z T a= a a a an b b b V x an I x respectvely. h s a PEC plane the coeffcent colun vectors b= T satsfyng b Γa (S1.4) S8

9 where Γ=-1 (1 s the unt atrx) s the reflecton coeffcent atrx ue to the PEC plane. Thus accorng to equatons (S1.35) (S1.38)-(S1.4) the nput attance atrx lookng own nto Regon at the nterface z can be erve as 1 h h h h 1 Y n Yn I 1 e Γe 1 e Γe V (S1.41) eanwhle t s easy to eterne the nput attance atrx the nterface z whch s a agonal atrx as Y up lookng up nto Regon I at Y Y (S1.42) up nl n n whch Yn k zn for the TM oe k k k zn an k k 2n n xn xn x. Therefore the coplex egenvalue of the propose structure can be fnally solve by the generalze transverse resonance conton at z nterface S9 et Y Y. (S1.43) up n Thus we can fnally obtan equaton (5) n the an text by solvng equaton (S1.43) base on the assuptons that an kx k. Only the funaental surface oe ( =) s kept wth all hgh-orer ffracton effects neglecte. 2. Dervaton of the sperson relaton of 2D HCGs-base SSPs wth effectve eu approxaton In ths secton we wll show how the sae sperson relaton can be obtane by usng the effectve eu approxatons. Supposng a plane wave s ncent on the surface of the three layere structure wth ar on the top PEC at the botto an a hoogeneous but ansotropc eu layer of h n between wth the electroagnetc paraeters gven by an (enote n equatons (6) an (7) n the an text) shown n Fg. 1(c) n the an text an the

10 only consere oe s the funaental surface oe (=) whch s characterze by the perpencular wave vector. The reflecton an transsson coeffcents for the TM oe fro layer to j are gven by [1] r j k k k z j z j k z j z j (S2.1) t j k 2k z j k z j z j (S2.2) n whch s the wave peance s the x-coponent of the perttvty an y x x s the y-coponent of the effectve pereablty n eu respectvely. y k s the z- z j coponent of the wave vector n eu j an can be wrtten as k k 2 2 k z j x j y j x. The value of r s always -1 when eu j j s a PEC ncatng coplete reflecton of energy wth a phase shft of the electrc fel. To obtan the reflecton coeffcent R we apply the transfer atrx forals [2] n whch the 2 2 atrx D escrbes the j j nterface an l escrbes the propagaton through eu l. D j 1 t j 1 r j r j 1 (S2.3) e k z l h l k z l h e. (S2.4) The total transfer atrx to escrbe the three layer structures n Fg. 1(c) n the an text can then be wrtten as M D12 2D23 where the subscrpts 1 2 an 3 represent the ar the S1

11 hoogeneous layer an the contnuous PEC layer respectvely. The specular reflecton coeffcent R can be obtane fro M by R M 21 M11 where M s the coponent of M j entfe by row an colun j R x z x z k k e k k e x z x z 2h 2h (S2.5) n whch k z an are the wave vector coponent n the ar an hoogeneous eu respectvely (.e. kz kz1 kz2 an kz3 ). 2 2 Thus when kx k ( kz kx k ) we can fnally obtan equaton (9) n the an text by settng the enonator of equaton (S2.5) to zero. Suppleentary References 1. Inan U. S. an Inan. S. Electroagnetc Waves (Prentce Hall 2). 2. Yeh P. Optcal Waves n Layere Mea (Wley 25). S11