transmission subwavelength imaging

Transcription

1 Extee stiffness hypeboic eastic etaateia fo tota tansission subwaveength iaging Hyuk Lee 1, Joo Hwan Oh, Hong Min Seung 1, Seung Hyun Cho 3, and Yoon Young Ki 1,,* 1 WCU Mutiscae Design Division, Schoo of Mechanica and Aeospace Engineeing, Seou Nationa Univesity, 599 Gwanak-o, Gwanak-gu, Seou , Koea Institute of Advanced Machines and Design, Seou Nationa Univesity, 599 Gwanak-o, Gwanak-gu, Seou , Koea 3 Koea Reseach Institute of Standads and Science, 67 Gaeong-o, Yuseong-gu, Daeon , Koea Suppeentay Note Anaytic odeing Ou unit ce syste can be anayticay expained with a ass-sping ode as shown in Fig. S1 (a). As a unit ce, we choose the ass-sping syste encosed in the ed box shown in Fig. S1 (b). The ass M is consideed to be connected with ass. Theefoe, one can ode this syste by an equivaent syste consisting of effective ass and stiffness in x and y diections as suggested at the ight side of Fig. S1 (b). The hoizonta and vetica dispaceents of ass M at the () th unit ce ae denoted by U and V, i, espectivey. The ass M is connected to the ass in the adacent unit ce by sping s, and oca esonatos by couped spings as iustated in the figue. The coefficients (,, ) of the couped sping can be defined as * Coesponding Autho, Pofesso, yyki@snu.ac.k, phone , fax

2 Fig. S1. (a) Mass-sping ode of unit ce configuation. (b) Dispaceent notations and a set of asses to be consideed as a new unit ce fo deiving the paaetes. Its effective ass and stiffness tes ae equivaenty epesented in a sipe ass-sping syste in the ight. Fx u F y v o Fx u F y v. (S1) whee F i is the appied foce in the i (i=x,y) diection and the dispaceent coponents of ass in the x, y diections ae denoted by u and v. Stiffness epesents shea stiffness whie, noa stiffness. The off-diagona te is the couping stiffness appeaing due to the incination of the eastic ediu connecting M and. The dispaceent vaiabes of two esonato asses ae denoted by ( u, v ) and ( u, v ). Basic equation fo one-diensiona sipe ass-sping syste To show biefy how the dispesion eation and effective paaetes ae etieved, we conside ony the x-diectiona coponents in the ight figue of Fig. S1 (b). The equiibiu equation of the () th ass in the x diection can be expessed in tes of the x-diectiona --

3 dispaceent U as U x, eff x, eff ( Ui 1, U ) x, eff ( Ui 1, U ). (S) t whee x, eff is the effective stiffness in the x diection. Assuing tie haonic wave otion at an angua fequency of, the dispaceent U is e-witten as U exp[ i( t- kxx)] whee k is the wavenube. By using U / t U, Ui 1, exp(- ikxd) U, and Ui 1, exp( ikxd) U the dispesion eation can be expessed as,, (exp(- ik d) exp( ik d) ) xeff xeff x x. (S3) This is the basic fo of sipe peiodic ass-sping eation and wi be appied in this wok to deive the effective paaetes. x-diectiona paaetes: esonant ass density Let us fist deive the paaetes in the x diection. Because we conside ony ongitudina wave otion, the vetica dispaceents of M, V, wi be oitted. Aso, fo the syetic condition, v, v, 0. Thus, the equations of otion fo the ass coponents in the () th ce becoe i i U M s U U U u u u u U t ( i1, i1, ) ( ) u U U u t ( 1 ) u U U u t ( 1 ), (S4a), (S4b). (S4c) Fo ongitudina otion in the x diection, the dispaceents can be assued not to vay aong the y diection. In this case, the foowing conditions ae satisfied: -3-

4 U U U, (S5a) 1 1 u u 1, (S5b) u u 1. (S5c) Fo steady state tie-haonic wave otion at an angua fequency of, and conditions (S5a-c), equations (S4a) to (S4c) educe to MU s(exp( ik d) exp( ik d) ) U ( u u U ) x x, (S6a) u ( U u ), (S6b) u ( U u ). (S6c) Equation (S6b) and (S6c) can be e-witten as Substituting equation (S7) into (S6a) yieds u u U. (S7) 4 MU s(exp( ikxd) exp( ikxd) ) U U. (S8) By e-witing this equation in the fo of basic sipe ass-sping dispesion eation, the foowing equation can be obtained 4 ( M ) s(exp( ik d) exp( ik d) ). (S9) x x Copaing this with (S3), the effective paaetes can be defined as 4 M M xeff,, (S10a) s, (S10b) xeff, whee / is the x-diectiona esonant fequency. Theefoe, ony the ass te becoes the function of wheeas the effective stiffness is independent of. -4-

5 y-diectiona paaetes: esonant stiffness As fo the y diection, a siia appoach wi be ade. Hee, the ony diffeence is that the x-diectiona otion of esonatos is aways couped with the y-diectiona otion. Thus, the equations of otion fo ass coponents in the () th ce can be witten as foows: V M v v v v V u u u u t ( ) ( 1 1) v V V v t ( 1 ) v V V v t ( 1 ) u V V u t 1 u V V u t 1, (S11a), (S11b), (S11c), (S11d). (S11e) Assuing tie-haonic wave otion, V V exp[ i( t kyy)] and V 1 exp( ikyy) V, equation (S11) becoes MV 4 V (1 exp( iky)) v (1 exp( iky)) v (1exp( ik )) u (1 exp( ik )) u y y, (S1a) v (1 exp( ik )) V v, (S1b) y v (1 exp( ik )) V v, (S1c) y u (1 exp( ik )) V u, (S1d) y u (1exp( ik )) V u. (S1e) y Re-witing equations (S1b-e) with espect to V yieds -5-

6 (1 exp( ik )) v v V y (S13a) (1 exp( ik )) u u V y (S13b) By substituting equation (S13a-b) into (S1a), the dispesion eation fo the y diection can be obtained as 4 ( ) (exp( ) exp( ) ) M ik y iky (S14) Hee, it ust be noted that in contast to the foe case, not ony the effective ass but aso stiffness tes ae functions of. As equation (S14) shows, the y-diectiona esonance occus ainy due to the te. Fo the seected geoety (d=14, = ), one can identify, M,,, and as those shown in Tabe S1. As shown in Tabe S1, so that the y-diectiona esonance occus at a uch highe fequency than that fo the x-diectiona esonance. Because is eated to shea oduus and, to tension, is usuay uch age than. Howeve, in a specia situation when the opeating fequency is uch owe than the y-diectiona esonant fequency /, one can assue that. (S15) If the assuption (S15) is vaid, equation (S14) educes to ( ) (exp( ) exp( ) ) M ik ik (S16) Consequenty, the effective ass and stiffness fo the y diection becoe M, (S17a) yeff, / yeff,. (S17b) -6-

7 M 1.87 GPa 1.69 GPa 3.41 GPa.53e-4 kg 7.89e-5 kg Tabe. S1. The cacuated vaues fo ass and stiffness pats of the anaytica ass-sping ode fo the continuu unit ce. The cacuations wee pefoed by the finite eeent ethod with COMSOL Mutiphysics 3.5a. -7-

8 Fig. S1. (a) Nueica siuation fo two souces as cose as 1 attice constant (14 o 0.09λ). The specific paaetes ae pesented in the ight side. A the siuation set-up and condition wee the sae as those stated in the anuscipt. (b) Obtained intensity pofies confoing the subwaveength iaging capabiity. -8-

9 Fig. S. Additiona siuations fo diffeent thicknesses (not beonging to the Faby-Peot esonance o finite esonance) of eastic etaateia enses. Thicknesses of 7 ces ( 1 ) and 17 ces (.43 ) ae consideed. The esoved iages ae ceay obseved in a figues. -9-

10 -10-

11 Fig. S3. (a) Tansission cuve denoting the thee cases (ipedance atched case, Faby-Peot esonance case, and neithe one) to confi the subwaveength esoution capabiity by nueica siuation. (b) Ipedance atched case. (c) Faby-Peot esonance case. (d) Neithe case. Hee, it ust be noted that a thee cases beong to the hypeboic dispesion ange as shown in Fig. b in the anuscipt. In othe wods, a thee cases ae capabe of esoving subwaveength souces, athough the iage quaity dops when tansission is not adequate. -11-

12 Fig. S4. Concept dawing fo pactica appications in the utasonic iaging technique. By integating ou etaateia ens into a conventiona non-destuctive iaging syste, we expect to find hidden cacks/faws that ae unde subwaveength scae. Because ou etaateia ens poises good tansission and woks as an endoscope, it coud act as a waveguide that ae usefu unde hash envionents (whee pipes ae buied undewate o deep down the eath o in adioactive suoundings). -1-