Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials

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1 UPPLEMENTARY INFORMATION DOI:.38/NMAT33 Drac cones nduced by accdental deeneracy n photonc crystals and zero-refractve-ndex aterals Xueqn Huan #, Yun La #, Zh Hon Han #, Huhuo Zhen, C. T. Chan * Departent of Physcs, Hon Kon Unversty of cence and Technoloy, Clear Water Bay, Kowloon, Hon Kon, Chna # The authors contrbuted equally to ths wor. * Correspondn author: phchan@ust.h. Accdental-deeneracy-nduced Drac pont n a photonc crystal wth a tranular lattce. Here, we show that the accdental-deeneracy-nduced Drac pont (ADIDP can also be realzed usn photonc crystals (PCs wth a tranular lattce. The ε and μ of the cylnders are the sae as those n the square lattce. The radus s chaned to R.84a, but the flln rato reans alost the sae as that n the square lattce case. The dspersons are qualtatvely slar to those of the PC wth a square lattce shown n the text. Two lnear bands and one addtonal flat band ntersect at a trplydeenerate pont at the Γ pont as shown n F. a and a three-densonal dsperson plot shown the Drac cone near the Γ pont s shown n F. b. NATURE MATERIAL Macllan Publshers Lted. All rhts reserved.

2 UPPLEMENTARY INFORMATION DOI:.38/NMAT33 F. a, The band structure of a PC wth a tranular lattce, the radus, the relatve perttvty and pereablty of the cylnder are R.84a, ε.5, μ. b, A three densonal dsperson plot shown frequences as functons of the wave vectors x and y.. Dspersons wth and wthout the accdental deeneracy Here, we show the band structures of several square lattce photonc crystals (PCs wth dfferent rad of the cylnders (and thus dfferent flln ratos. The band structure wth accdental deeneracy and thus ADIDP s shown n F. a, whch has a trply-deenerate pont at the Γ pont at frequency f.54 c/ a. Here, the lattce constant s a, the radus, relatve perttvty and pereablty of the cylnder are set at R.a, ε.5, μ, respectvely. Two lnear branches (forn a Drac cone and a thrd flat branch ntersect at a trply-deenerate pont. If the radus of the cylnder devates fro.a, the trply-deenerate pont wll brea up nto a doublydeenerate pont correspondn to two dpolar odes and a snle onopolar ode. For exaple, when R.9a, as shown n F. b, the trply-deenerate pont s decoupled nto a doubly-deenerate pont ( f.566 c/ a and a snle ode. Whle for the case of R.a, the trply-deenerate pont s also ( f.54 c/ a decoupled, but the doubly-deenerate pont ( f.5 c/ a s lower n frequency than the snle ode ( f.54 c/ a, as shown n F. c. Here, we note that all the dspersons around the doubly-deenerate pont and the snle ode at the Γ pont n Fs. b and c are quadratc. The Drac cone eeres as a consequence of accdental deeneracy at a partcular radus (F. a. NATURE MATERIAL Macllan Publshers Lted. All rhts reserved.

3 DOI:.38/NMAT33 UPPLEMENTARY INFORMATION F. The band structure of the photonc crystals wth square lattce for three dfferent rad of the cylnders, a R.a, b R.9a, c R.a. The lattce constant s a, the relatve perttvty and pereablty of the cylnder are ε.5, μ. 3. Electrc feld aps for the eenstates at the Drac pont In order to understand the physcal nature of the eenstates at the ADIDP, we show the electrc feld patterns of the eenstates near the Drac pont wth a sall alon the Γ X drecton (.476 π / a,. In Fs. 3a and 3b, we show the real x and anary parts of the electrc felds of the lowest frequency state ( f.57 / y c a whch show that the eenode s a cobnaton of a onopole and a transverse anetc dpole. The feld pattern for the hhest frequency state ( f.555 c/ a (not shown here s slar, also a cobnaton of a onopole and a transverse anetc dpole. The real part of the electrc feld of the flat band s plotted n F. 3c and the anary part s alost zero. Fro F. 3c, we can see t s a pure dpolar ode wth anetc feld parallel to the drecton of,.e. a lontudnal anetc dpole. The feld patterns of the odes near the Drac pont tell us that three branches nvolve only a xture of onopolar, dpolar odes. The correspondn anetc felds H are also shown as vector felds n the fures. NATURE MATERIAL 3 Macllan Publshers Lted. All rhts reserved.

4 UPPLEMENTARY INFORMATION DOI:.38/NMAT33 F. 3 The color patterns show the E z and the vector felds show H of the eenstates near the Drac pont wth a sall alon the Γ X drecton (.476 π / a, x. a, The real part of E z and the anary part of H at y f.57 c/ a, b, The anary part of the E z and the real part of H at f.57 c/ a, c, The real part of E z and the anary part of H at f.54 / c a. 4. Multple scattern theory (MT analyss of the ADIDP Here, we apply ultple scattern theory (MT to analyze the dsperson bands near the ADIDP at the Γ pont of the square lattce. Throuh ths analyss, we shall see that the Drac-cone dspersons accopaned by a flat band can eere as a consequence of the accdental deeneracy of the dpolar and onopolar odes at the Γ pont. A. The MT forulaton for the proble The MT equatons can be wrtten as ( ( ( ( b D j b j, ( n,, n j n where b ( and D ( are the Me scattern coeffcent and T-atrx coeffcents of anular oentu nuber for the th scatterer, respectvely. ( n,, j denotes the transforaton atrx that transfors the scattered wave of anular oentu nuber n fro the jth scatterer, nto the ncdent wave of anular oentu nuber 4 NATURE MATERIAL Macllan Publshers Lted. All rhts reserved.

5 DOI:.38/NMAT33 UPPLEMENTARY INFORMATION for the th scatterer. Consder a photonc crystal wth a prtve unt cell. R Applyn Bloch s theore (.e. b ( b e, n whch s the Bloch wave vector and R s the lattce vector of the th scatterer, Eq. ( becoes ( n, ( b D b n n n where n s the lattce su. for n can be obtaned n the recprocal space as n n H ( r + 4 e r nφ ( r ( ( ( n + J n+ π δ n, Ω J ( n+ r J r. (3 Here s the recprocal lattce vector; ω c; r s the nearest nehbor dstance; Ω s the area of the D unt cell, and (, +,.e. + and φ Ar ( + φ are the polar coordnates of the vector. For n <, we have n n. Fro the eenstate feld aps near the ADIDP, we now that they are anly dpolar ( ± and onopolar ( odes. Thus, here we can apply the MT to the subspace spanned by dpolar and onopolar odes, whch results n a 3x3 atrx as: D b D b. (4 D b, In order to obtan the dsperson relatons, we do a sall ( δ ( δ φ sall ω expanson for the eenodes near the Γ pont. In the follown, we consder the case for a square lattce. The case of tranular lattce can be derved slarly. Frst, we do a sall expanson as f δ f + f δ δ + O δ ( ( δ ( ( δ for, and a D ±, ± and ± n Eq. 4. NATURE MATERIAL 5 Macllan Publshers Lted. All rhts reserved.

6 UPPLEMENTARY INFORMATION DOI:.38/NMAT33 F. 4 The recprocal lattce of the square lattce. B. all expanson for In the lt of δ, can be drectly wrtten n the follown for: Y ( ( r J r + π ( ( ( (5 + +, 4 r r + + Ω J( r J r J r where C ( 4 r Y r + ( ( π r ( C s a functon of ω only, and Ω J( r J r J ( r ( ( 4 Ω J r s a functon of both ω and δ. s a suaton of any ters. Each ter s assocated wth a recprocal lattce vector. Here, the superscrpts C and ndcate the parts assocated wth recprocal vectors and, respectvely. Due to the syetry of the recprocal lattce of the square lattce, can be reatly splfed. In the square lattce, for any recprocal lattce vector wth apltude Ar and polar anle ( ψ, we can always fnd three other recprocal lattce vectors, 3 and 4, wth the sae apltude and polar anles ψ ψ + π ψ ψ + π,, 3 ψ ψ + 3 π, respectvely, as shown n 4 6 NATURE MATERIAL Macllan Publshers Lted. All rhts reserved.

7 DOI:.38/NMAT33 F. 4. In the lt of δ, we have f J ( r ( ( J r,,3,4 to,, 3 and 4 n, then we fnd f δ δ + +. If we consder,.e. the suaton of the four ters correspondn δ J + r,,3,4 J( r δ δ + + J( r + δ rj ( r δ δ +,,3,4 J( r J ( r J ( r δ δ + δ r +,,3,4 J( r ( J( r J( r J ( r + δ (.,,3,4 ( ( r ( + + O δ J r J r The zeroth-order part of f s ( r J 4 J r ( ( δ, whch s a functon of ω only. The frst order part s proportonal to. However, snce we have,,3,4 3 and 4, we fnd δ and thus the frst order part,,3,4 J ( r 4 vanshes. Therefore, we obtan f + O ( δ. For other J r ( ( recprocal lattce vectors of, the above analyss also apples. As a result, we J obtan ( r f ( ω + O( δ, where f ( ω J r s only dependent on ω. ( ( C Here, we note that and are both real nubers (Eq. 5. Fro the optcal theore (see e.. Ref., we have ( D UPPLEMENTARY INFORMATION Re n Dn and NATURE MATERIAL 7 Macllan Publshers Lted. All rhts reserved.

8 UPPLEMENTARY INFORMATION DOI:.38/NMAT33 ( Re( D I( D I( D D. The real part of Dn s, whch s n exactly the sae as n n n Dn Dn ( C. Therefore, the ter I( D + + D D s n n n a purely anary nuber. Thus, D and D ± as δ can be approxately wrtten as D A ω + B ω δ, ± ( ( ( ( ω ( ω δ ( D A + B, where A ( ω, A ( ω and ( (6 B ω are all real functons of ω only. Here, we have appled D, because the cylnders are cylndrcally syetrc. D C. all expanson for In the lt of δ, can be wrtten as φ ( r ( ( 4 δ r J e, (7 + φ e + Ω 8J( r J r where C 4 δ r φ e Ω and C 8J( r 4 J φ ( r e ( ( Ω J r are both functons of ω and δ C. It s seen that the frst part s proportonal to δ e φ. The second part When δ, we have φ Ar δ + ψ + ( f s a suaton of any ters le. δ δ + + and δ zˆ φ ( ( ( J r e J r,,3,4, where ψ Ar (. Aan, consder,.e. the suaton of the four ters correspondn to,, 3 and 4 n, then we fnd 8 NATURE MATERIAL Macllan Publshers Lted. All rhts reserved.

9 DOI:.38/NMAT33 UPPLEMENTARY INFORMATION f δ ψ ˆ δ z J + r e +,,3,4 J( r δ δ + + J ( r + δ rj ( r ψ e δ δ δ zˆ + +,,3,4 J( r ψ J( r e J ( r δ δ δ zˆ + δ r + +,,3,4 J ( r ( J ( r ψ J ( r e J ( r zˆ + δ r O ( δ.,,3,4 J( r ( J( r The zeroth-order part of f,.e. e ψ,,3,4 ( ψ J r e J r ( (,,3,4, s proportonal to 3. nce we have e ψ e ψ ψ ψ 4 and e e, thus the zeroth-order part vanshes. The frst order part of f nvolves ters proportonal to ψ or e ψ zˆ δ. Because φ e δ δe and,,3,4,,3,4 ψ e zˆ δ δe φ,,3,4,,3,4 ψ e δ, therefore, the frst order part f δ e φ. lar analyss can be appled for other recprocal lattce vector ters, thus, we have δ e φ C, as ( (. Thus, the lattce su can be wrtten as C ω δe φ, (8 where C ( ω s a real functon of ω only. D. all expanson for In the lt of δ, can be wrtten as φ ( r ( ( 4 δ r J e. (9 + 3 φ 3 e + Ω 48J3( r J3 r C NATURE MATERIAL 9 Macllan Publshers Lted. All rhts reserved.

10 UPPLEMENTARY INFORMATION DOI:.38/NMAT33 where 4 δ r and Ω 3 C φ e 48J3( r 4 J φ ( r e ( ( 3 Ω J3 r are both functons of ω and δ C. s proportonal to δ. s a suaton of any ters le and. Aan, consder f φ ( ( ( J3 r e J r,,3,4 3 ters correspondn to,, 3 and 4 n, then we fnd f,.e. the suaton of the four δ ψ ˆ δ z J3 + r e +,,3,4 J3( r δ δ + + J3( r + δ rj 3( r ψ e δ δ δ zˆ + +,,3,4 J3( r ψ J3( r e J 3( r δ δ δ zˆ + δ r + +,,3,4 J3( r ( J3( r ψ J3 ( r e J 3 ( r zˆ + δ (.,,3,4 J3( r ( r 3( O δ J r The zeroth-order part of f,.e. ( ψ J 3 r e J r ( (,,3,4 3, s proportonal to e ψ ψ ψ ψ ψ 3 4. nce we have e e e e, the zeroth-order part,,3,4 ψ vanshes. Due to e ψ δ and e zˆ δ ψ ψ 3 ( e e 3,,3,4,,3,4 ψ and ψ 4 e e 4, the frst order part of f also vanshes. Therefore, only second order or hher order parts exst, whch leads to f O( δ lar analyss apples for other recprocal lattce vector ters, thus, we have ( δ C O as. Thus, the lattce su can be wrtten as ( ( ω φ δ C, (, where C ( ω, φ s a functon of ω and φ.. NATURE MATERIAL Macllan Publshers Lted. All rhts reserved.

11 DOI:.38/NMAT33 UPPLEMENTARY INFORMATION E. Don a sall ω expanson to obtan the dsperson relatons Besdes a sall δ expanson, we also need to do a sall ω expanson to obtan the dspersons of the eenstates near the Γ pont. At the Γ pont, where δ, we have due to the syetry of ± ± square lattce. The secular equaton of Eq. 4 reduces nto three ndependent equatons,.e. D, D and D. By solvn D, we can obtan the onopolar eenfrequency ω. By solvn D D, we can obtan the dpolar eenfrequency ω d. Near ω and ω d, we can do a sall δω expanson for D and D ± such that the dspersons nearby (δω as a functon of δ can be obtaned. a the eneral case of ω ωd Frst, we consder the eneral case of ω ωd. Near ω, we et D A ( ω( ω ω B( ωδ ( +, where A ( A ( ω ω ω. We also have D A ( ω, n whch the B ( ω δ ter s otted as t s a hh ± order ter. By substtutn the nto the MT equatons n Eq. 4, we obtan φ A Cδe Cδ b φ φ C e ( A( B δ ω ω + δ Cδe b. ( φ C b δ Cδe A Here A, A, B, C and C can all be consdered as nonzero constants n a sall ω and ree. By solvn the secular equaton, we fnd A ( eans that the dsperson of the onopolar band s quadratc. +, whch ω ω Bδ NATURE MATERIAL Macllan Publshers Lted. All rhts reserved.

12 UPPLEMENTARY INFORMATION DOI:.38/NMAT33 A Near ω, we obtan D A ( ω( ω ω + B( ωδ, where d ( ± d ( ω A ( ω ω. We also have D A( ω, n whch the B ( ω δ ter s otted as t s a hh order ter. By substtutn the nto the MT equatons n Eq. 4, we obtan ( ( d φ A ω ω + Bδ Cδe Cδ b φ φ Cδe A Cδe b. ( φ Cδ Cδe ( A( ω ωd Bδ b + Here A, A, B, C and C can be taen as nonzero constants n a sall frequency reon. By solvn the secular equaton, we fnd ( A ω ω + Bδ ± C δ, whch d eans that the dspersons of the dpolar bands are also quadratc. b the accdental deeneracy case of ω ωd ω Now, we consder the specal case of ω ωd ω due to accdental deeneracy. Near ω, we obtan D ( A ( ω( ω ω + B( ωδ, where A ( A ( and D ( A( ω( ω ω ± + B( ωδ, where A ( A ( the nto the MT equatons n Eq. 4, we obtan ( ( ω ω ω, ω ω ω. By substtutn φ A ω ω + Bδ Cδe Cδ b φ φ Cδe ( A ( ω ω Bδ Cδe + b. (3 φ b Cδ Cδe ( A ( ω ω + Bδ Here A, A, B, C and C can be consdered as nonzero constants n a sall frequency ree. The secular equaton of Eq. 3 s a cubc equaton of δω : ( ( ( 4 4 ( A δω Bδ C δ ( Ce φ Cδ Aδω + Bδ Aδω + Bδ + Aδω + Bδ C δ + + I, (4 NATURE MATERIAL Macllan Publshers Lted. All rhts reserved.

13 DOI:.38/NMAT33 UPPLEMENTARY INFORMATION where δω ω ω. By solvn ths secular equaton, we fnd three solutons: one s ( ω ω + O δ, whch corresponds to a band of quadratc dsperson (ben flat near the Γ pont. The other two are ω,3 ω vδ O( δ ± +, where v C, A A whch correspond to two bands wth Drac-le lnear dspersons. By substtutn ω ω nto Eq. 3, and ottn the hh order ters of ( φ φ O δ, we fnd b and b e be. The scattered feld can be wrtten as sc φ φ φ ( ( ( sn ( φ φ E b H r e + bh r e be H r, ndcatn a anetc dpole orented parallel to δ. uch a dpole can be vewed as a lontudnal anetc plasa ode, as t has a anetc feld parallel to δ and s a result of effectve zero pereablty. By substtutn ω,3 ω ± vδ nto Eq. 3 and ottn the hh order ters φ δ, we fnd φ ± v Ab C e b and ± v Ab + Ce b. Ths of O( ndcates that the dpolar and the onopolar odes are coupled toether n the Drac cone. 5. The lontudnal band The lontudnal band s taen nto consderaton n our effectve edu theory and the physcal orn of ths flat band s a anetc lontudnal band nduced by μ eff. Ths flat band ust exst f we have a μ syste and the exstence of such a band n the band structure of the photonc crystal at the Drac Cone s consstent wth the fact that the photonc crystal has an effectvely zero ndex. The band s dspersonless f the μ syste s perfectly hooeneous and as a lontudnal ode, t s a deaf band whch does not couple wth external lht. But μ eff n any real photonc crystal or etaateral wth, whch coprses dscrete NATURE MATERIAL 3 Macllan Publshers Lted. All rhts reserved.

14 UPPLEMENTARY INFORMATION DOI:.38/NMAT33 buldn blocs, there s always soe spatal dsperson, so that the band s not perfectly dspersonless away fro the zone center and the flat band can be excted f lht s ncdent wth non-zero -parallel coponents. In our sulatons and experents, we appled an ncdent plane wave at noral ncdence, whch cannot excte the flat band. At the Drac pont frequency, the flat band can be excted f we use a thtly focused aussan bea ncdence whch carres non-zero -coponents alon the surface. The transtted lht at the ext surface s not plane wave le and the wave pattern s due to the flat band excted nsde the photonc crystal, as shown n F. 5a. But snce ths anetc lontudnal ode has a narrow band wdth, we can avod ths band (f we want to by operatn at slhtly above the Drac frequency, where the n eff s stll very close to zero. Wthout the exctaton of the lontudnal band, the output wave has a plane-wave wavefront aan,, as shown n F. 5b, the sae as a hypothetcal hooeneous ateral wth near zero refractve ndex. We note aan that ths flat band ust be there f the ateral has effectve perttvty and pereablty both equal to zero. F. 5: The sulated E z feld dstrbuton wth a thtly focused aussan bea ncdence, focused at the left nterface of our PC, at the Drac frequency (f.54c/a 4 NATURE MATERIAL Macllan Publshers Lted. All rhts reserved.

15 DOI:.38/NMAT33 UPPLEMENTARY INFORMATION a and slhtly above the Drac frequency (f.56c/a b. Here, a s the lattce constant of the PC, c s the velocty of lht n vacuu. The paraeters of the structure are the sae as F. n the text. 6. The cloan effect for PEC and delectrc objects n the photonc crystal syste wth ADIDP In the text, we deonstrated that wave can pass throuh the photonc crystal syste at the ADIDP wth an ebedded PMC object nsde t (F. 3b. It s treated as a cloan effect n the lterature []. For copleteness, we nuercally llustrate slar effect for ebedded PEC and delectrc objects. As a control calculaton, we replace the PC wth ADIDP by the hooeneous hh delectrc edu ( ε.5, the ncdent waves are alost totally reflected bac by the 9 deree bendn channel wthout any udn effects, as shown n F. 6a. We now put a PEC object or delectrc object ( ε 6 n a 9 deree bendn channel flled wth the PC syste wth ADIDP, the electroanetc waves can stll transt throuh wth lttle dstortons, as are shown n Fs. 6b and 6c, respectvely. Ths s due to the effectve zero perttvty and zero pereablty [Ref. 3]. lar effect can also be observed n snle zero ateral [Ref. 4]. NATURE MATERIAL 5 Macllan Publshers Lted. All rhts reserved.

16 UPPLEMENTARY INFORMATION DOI:.38/NMAT33 F. 6 ulated electrc feld (E z patterns n the 9 deree bendn channel. a, E z dstrbuton f the 9 deree bendn channel s flled wth a hooeneous hh delectrc edu ( ε.5. Wave ncdent fro the lower left hand channel s reflected. E z dstrbuton f the 9 deree bendn channel s flled wth PCs wth an ebedded PEC object b and a delectrc object ( ε 6 c. The ncdent wave s plane wave and the boundary condtons of the channel are PMC. The worn frequency s.54 c/ a. References. hen, P., Introducton To Wave cattern, Localzaton And Mesoscopc Phenoena, prner, Hedelberer (6.. Hao, J., Yan, W., and Qu, M. uper-reflecton and cloan based on zero ndex etaateral. Appl. Phys. Lett. 96, 9 (. 6 NATURE MATERIAL Macllan Publshers Lted. All rhts reserved.

17 DOI:.38/NMAT33 UPPLEMENTARY INFORMATION 3. Nuyen,V. C., Chen, L., and Halteran, K. Total Transsson and Total Reflecton by Zero Index Metaaterals wth Defects. Phys. Rev. Lett. 5, 3398 (. 4. Jn, Y., and He,. Enhancn and suppressn radaton wth soe pereabltynear-zero structures. Opt. Express 8, 6587 (. NATURE MATERIAL 7 Macllan Publshers Lted. All rhts reserved.