Broadband All-Angle Negative Refraction by Phononic Crystals

Transcription

1 Supplmntar Information Broadband All-Angl Ngativ Rfraction b Phononic Crstals Yang Fan Li, Fi Mng, Shiwi Zhou, Ming-Hui Lu and Xiaodong Huang 1 Optimization algorithm and procss Bfor th optimization procss, it is important that th uppr and lowr frqunc limits Ω u and Ω l ar accuratl tractd. Taking a PnC consisting of circular stl rods in air with a filling fraction of 50% for ampl, th variation of radius of th qui-frqunc contour (EFC) curvatur along ΓM is plottd togthr with th first phononic band and th air disprsion lin in Fig. S1a. As discussd in th manuscript, th point whr th sign of radius of EFC curvatur altrs dfins th lowr limit l whil th uppr limit u locats at th intrsction of th first phononic band and th air lin. Fig. S1a shows that AANR dos not ist in th currnt dsign du to. l u Our goal is to find th optimal matrial distribution of th PnC that posssss a broad allangl ngativ rfraction (AANR) frqunc rang at th first band. Th optimization objctiv can b intuitivl st to nlarg th diffrnc btwn th uppr and lowr limit, i.. maimizing ( u l ). Although l and u can b numricall dtrmind, it is quit difficult to mathmaticall formulat ths two valus, which causs th challng in topolog optimization. In this papr, w altrnativl choos to dcras th lowr limit Ω l b maimizing th radius of curvatur at a rfrnc point (its wav vctor l k a / = 1 / 0 ) as shown in Fig. S1a. Th dcras of Ω l is quivalnt to nlarg th AANR frqunc rang sinc Ω u is dtrmind b th first band and constant air disprsion lin. Thus, th optimization problm is dfind as Maimiz: R f f f f f f 3/ f f f (S1) Subjct to: V f N 1 V * (S) whr R is th radius of curvatur at th rfrnc point. f k, f k. f, f and f ar th scond-ordr partial drivativs. is th dsign variabl for lmnt. = 1 dnots lmnt is solid matrial whil = 0 dnots lmnt is air; N is th total

2 numbr of lmnts. V f is th filling fraction of solid matrial and V * is th prscribd volum constraint. (b) FIG. S1 (a) Band structur of th stl-air phononic crstal (solid blu lin) and disprsion rlation in air which is shiftd to M point (rd dashd lin). Frquncis ar normalizd b πc air /a. Th rfrnc point is illustratd b a black circl smbol. Th inst rprsnts th unit cll. (b) Illustration of squar grids within th irrducibl Brillouin zon 1 W considr a squar PnC unit cll with C 4v smmtr. Th band structur and EFCs ar calculatd using finit lmnt mthod. Th first Brillouin zon is dividd into a grid, which is illustratd b a coars grid in Fig. S1b. Tak an arbitrar point 1 on th ΓM boundar for ampl, nighbor vctors as f, f,. f, f and f can b prssd b frquncis of f 4 5 f (S3) k f f and 4 k f (S4) 4 k Thrfor, th objctiv function in Eq. S1 can b furthr simplifid to: Maimiz: R f f f at th rfrnc point (S5) Th snsitivit numbr α for lmnt can b dfind according to th chain rul as R R f i R f i R f i (S6) f f f i4,5 i i1,6,8 i i,3,7 i Th drivativ of Ω i with rgard to dsign variabl can b calculatd b

3 i 1 T K ui i i M u i (S7) whr u i is th ignvctor corrsponding to Ω i. K and M ar lmntal stiffnss and mass matri. A highr positiv valu of th lmntal snsitivit numbr in Eq. S6 indicats that incrasing th dsign variabl of th lmnt (phsicall switching matrial from air to stl) will incras th AANR frqunc rang to a largr tnt. In ordr to maimiz R, it is ncssar to incras th dsign variabls from = 0 to 1 for lmnts with high snsitivit numbrs and dcras th dsign variabls from = 1 to 0 for lmnts with low snsitivit numbrs. Th dsign variabls ar updatd according to th following quation. 1, if th 0, if th (S8) Th thrshold snsitivit numbr th is dtrmind b th volum fraction constraint dfind in Eq. S. Hnc, th nw phononic structur is formd with updatd dsign variabls. This procss is rpatd until an optimum achivs. Mor dtails about BESO can also rfr to Huang and Xi. In th following ampls, BESO starts from th initial guss shown in Fig. S1a. Figur Sa prsnts th volution historis of AANR frqunc rang and th topolog of th phononic unit cll during th optimization procss. Th filling fraction of solid matrial is fid at 50%. It is obsrvd that th AANR frqunc rang constantl grows and finall stabilizs at a maimum valu around 1.19% nar its cntral frqunc whil th gomtr of th unit cll also volvs stabl to an optimal. Th final optimizd structur is surprisingl simpl. Ecpt for th prst linar connctions, th domain occupid b air can b simplifid to four idntical quadrants as shown in Fig.1a in th manuscript. Th simplifid structur in th manuscript, which has an AANR rang of 0.35%, onl causs a vr small discrpanc, indicating th high manufacturing robustnss of this dsign.

4 FIG. S (a) Evolution historis of AANR frqunc rang and topolog during th optimization at a fi filling fraction of 50%. Th whit and gr dnot air and stl, rspctivl. (b) Band structur of th optimizd phononic crstal and corrsponding AANR frqunc rang.

5 Slf-collimation and subwavlngth focusing ffct of a point sourc across an ightlar phononic slab FIG. S3 Schmatic illustration of th wav-bam rfraction law. Whn th radiatd bams from a point sourc S ntr th PnC slab with th incidnt angl α, th will b ngativl rfractd and travl along th grn lins. For a point sourc placd on th lft sid of th flat lns, rfractions of radiatd wavs first mt insid th slab and thn mt again and form an imag I on th right sid. Thus, focusing at frquncis that hav small rfractd angls rquirs a vr wid phononic slab in ordr to form a clar spot I on th right sid. (a) (b) (c) (d) FIG.S4 Normalizd intnsit fild of a point sourc and its imag across an 8-lar phononic slab at a frqunc nar th lowr limit of th AANR rang. (a) Ω=0.194, (b) Ω=0.195, (c) Ω=0.196, (d) Ω= Acoustic wavs at ths frquncis princ slf-collimation within th phononic crstal. (a) (b) (c) (d) FIG.S5 Normalizd intnsit fild of a point sourc and its imag across an 8-lar phononic slab at a frqunc nar th uppr dg of AANR rang Ω=0.15, 0.0, 0.5 and Acoustic wavs at ths frquncis princ subwavlngth focusing on th right sid of th phononic crstal. 3 Validation of mtafluid assumption

6 FIG. S6 Simulation rsults of prssur filds of a point sourc at th normalizd frqunc ω=0.3 across an 8- lar slab with (a) and without (b) mtafluid assumption. Th lft modl (Fig. 1a) ignors th shar modulus of th solid whil th right modl trats th air and stl as fluid and solid sparatl. Th idntical prssur filds dmonstrat th sam focusing ffct at th normalizd frqunc Ω=0.3, which indicats that it is suitabl to trat th solid as mtafluid. Rfrncs 1 Fi Mng, Shuo Li, Han Lin, Baohua Jia, and Xiaodong Huang, Finit Elm. Anal. Ds , 46 (016). X. Huang, Y. M. Xi, Evolutionar topolog optimization of continuum structurs: mthods and applications, John Wil & Sons, Chichstr 010.