Topological Description for Photonic Mirrors Hong Chen School of - PowerPoint PPT Presentation

Topological Description for Photonic Mirrors Hong Chen School of Physics, Tongji University, Shanghai , China 同舟共济 Collaborators:  Dr. Wei Tan, Dr. Yong Sun, Tongji Uni.  Prof. Shun-Qing Shen , The University of Hong Kong HKUST IAS, Jan. 11, 2016

outline 1. Introduction  topological Insulators, band inversion  photonic analogs, mirrors as photonic insulators 2. Topological description: Theoretical study  mapping 1D Maxwell’s equations to Dirac equation  topological orders for photonic mirrors 3. Topological description: Experimental study  band inversion, microwave experiments  edge modes, microwave and visible light experiments 4. Summary

1. Introduction M. Z. Hasan et al ., Rev. Mod. Phys . 82 , 3045 (2010) topological description for electronic insulators electronic medium

Metallic, unidirectional Edge state between insulators with different topological orders M. Z. Hasan et al ., Rev. Mod. Phys . (2010) Quantum Hall Effect Chiral Edge State with H Broken T-symmetry Quantum Spin Hall Insulator Edge states without H with T-symmetry

photonic medium : metamaterials with designed ε and μ Materials Responds To EM Waves: Permittivity ε and Permeabilty µ = ± εµ n single-negative double-positive ε -negative (DPS) (ENG) forward-wave evanescent wave propagation “barrier” double-negative single-negative µ < ε > (DNG) µ -negative 0 , 0 backward-wave (MNG) propagation evanescent wave “barrier” zero-index materials From ENGHETA and.ZIOLKOWSKI 2006

analog to electrons Materials Responds To EM Waves: Permittivity ε and Permeabilty µ = ± εµ n Dirac-Point Related Medium Photonic Graphene? Photonic Insulator I Photonic Conductor ENG Mirror Right-Handed µ < ε > 0 , 0 Photonic Conductor Photonic Insulator II Left-Handed MNG Mirror Q1 ： different topological orders between EMG and MNG mirrors ??

Manipulating topological order: Electronic band inversion transition from normal insulator (NI) to TI [d=3] M. Z. Hasan et al ., Rev. Mod. Phys . (2010) semimetal for x < .07 semiconductor for .07 < x < .22 semimetal for x > .18 bands L s;a invert at x ~ . 04

Photonic analog Experiments: Wang et al ., Nature 461 , 772 (2009).

Photonic analog Nature materials 2012 Chirality

Mapping between electrons and photons Schroedinger Equation : Maxwell’s Equation : Periodical Structures: electronic band gaps , electron insulators Photonic Crystals photonic band gaps (PBG) , Yablonovitch and John 1987 photonic insulators

PBG as Normal Photonic Insulator For electronic and photonic NI, we have mapping: Schroedinger Eq. ↔ Maxwellʹs Eq. + photonic crytals

Band inversion transition in (2013) electronic systems: theoretical description Dirac Equation (1928) - Dirac matrices, for example: d=1: d=2: Q2 : Dirac Eq. ↔ Maxwell’s Eq. + artificial structures?? we proposed : Metamaterials !

2. Topological Description: theoretical study Earlier studies: Band inversion : metematerial analogs PBG in a 1D stack of ε -negative (ENG)/ µ -negative (MNG) pairs  (Jiang et al., PRE 2004, 2006; Weng et al., PRE 2007; Jiang et al. AIP Adv. 2012, ) α ε = ε µ ω = µ − , ( ) MNG for ω < α 1/2 ω 1 10 1 10 2 β ENG for ω < β 1/2 ε ω = ε − µ = µ ( ) , ω 2 20 2 20 2 A structure made of opaque or "dark" metamaterials!! ε ω + ε ω ( ) ( ) d ( ) d at sub-wavelength condition ε ω = ε ε = 1 1 2 2 0 , ε + d d and normal incidence, 1 2 µ ω + µ ω ( ) ( ) d ( ) d a PBG structure: µ µ µ ω = = 1 1 2 2 0 with the edge ω ε and ω µ : µ + d d 1 2

Earlier studies: Band inversion transition: metematerial analogs T ε = µ < ε > µ = 0 , 0 0 , 0 MNG gap ω ε ω µ ε > µ < 0 , 0 ε > µ > ε < µ < 0 , 0 0 , 0 DPS band DNG band ω µ ω ω ε ε < µ > 0 , 0 ε = µ > ε < µ = 0 , 0 ENG gap 0 , 0 ε -negative gap band edges 𝝏 𝜻 , 𝝂 tailoring ε and µ inverted µ -negative gap Evidence: photonic band inversion in metamaterials

recent studies: 2013 Answers to the two questions 1. Mapping 1D Maxwell’s equations to 1D Dirac equation −∂ = ωµ µ E i ( ) x H x z 0 r y Maxwell’s Eq.: ∂ = − ωε ε H i ( ) x E x y 0 r z metamaterials ϕ ϕ     [ ] − ∂ + + = 1 1 σ σ     massive Dirac Eq.: i m x ( ) V x ( ) E x x z ϕ ϕ     2 2 ω ε = − + µ ϕ = µ ϕ = ε E E H r r 2 0 y 1 0 z 2 c ω ε ω ( ) ( ) =  ε + µ − ε + µ  = − µ V x ( )   m x ( ) r r r r r r 2 c 2 c <…..> : Average on space

2. EMG and MNG mirror as mass inversion in Dirac Eq. 𝜗 𝑠 ~ − 𝜈 𝑠 , 𝐹 = 𝜕 2𝑑 𝜗 𝑠 + 𝜈 𝑠 ~ 0 For SNG mirror, if : Then at low energy 𝐹 ~0: the behavior of Dirac Eq. ONLY depends on the sign of the mass 𝜕 𝜕 𝜕 𝑛 = 2𝑑 𝜗 𝑠 − 𝜈 𝑠 ~ 𝑑 𝜁 𝑠 ~ − 𝑑 𝜈 𝑠 So, the sign of the mass is inverted from MNG to ENG : MNG mirror: 𝜁 𝑠 > 0. 𝜈 𝑠 < 0 positive mass: m > 0 ENG mirror: 𝜁 𝑠 < 0. 𝜈 𝑠 > 0 negative mass: m < 0 Different topological orders for MNG and ENG: The first evidence

Mapping the Dirac Eq. to the SSH model (S.Q Shen 2013) Su-Schrieffer-Heeger Model for Polyacetylene (Rev. Mod. Phys. 1988) m > 0 m = 0 m < 0 Mass inversion in the Dirac Eq. Band inversion in the SSH model

Berry phase: = � 0 𝑔𝑔𝑔 ∆𝑢 > 0 𝜌 𝑔𝑔𝑔 ∆𝑢 < 0 winding number: 𝜉 = � 0 𝑔𝑔𝑔 Δ𝑢 > 0 𝑔𝑔 𝑛 > 0 and 𝐍𝐍𝐍 𝐧𝐧𝐧𝐧𝐧𝐧 1 𝑔𝑔𝑔 Δ𝑢 < 0 𝑔𝑔 𝑛 < 0 and 𝐅𝐍𝐍 𝐧𝐧𝐧𝐧𝐧𝐧 Therefore, MNG and ENG mirrors have Different topological orders!

Topological description: Extend to mirrors made of dielectric multilayers Our study : Guo et al., PRE 2008; Chin. Phys. B 2008 multilayer structures or1D-PC made of dielectrics with ( ε > 1, μ = 1) can act as SNG metamaterials in gap region Depending on symmetry of the unit cell. ( AB ) For asymmetry unit cell: m The gap divided into two parts: EMG and MNG ( ABA ) For symmetry unit cell: m The gap described by: EMG or MNG

Determination of effective parameters  Retrieval theory : Smith et al., PRB 2002  Bloch-wave-expansion theory: Kan et al., PRA 2009 Effective parameters in gap regime The gap is divided into two parts: EMG and MNG

Dependence on periodic number = ( AB m ) : m 10 , 15 , 20 non-local effective parameters !! ( AB ) For asymmetry unit cell: m

Effective parameters in gap regime ( ABA ) 10 MNG ENG Second gap First gap It can be shown: Ε and μ are independent of the periodic number local effective parameters

3. Topological Description: experimental study 1D Metamaterials Realized By Transmission Line Eleftheriade et al., 2002; and by Itol et al., 2002 γ γ     1 1 p 1 µ = − + ε = − + m e    L  i C i r µ 0 ω r ε 0 ω  ω   ω  2 2 p Cd Ld s s ω choosing different circuit parameters ( ) = ε − µ m x ( ) r r one gets DNG, ENG, MEG materials 2 c

Band Inversion in photonic chains τ 1 dk ω = ∝ g DOS: g ( ) π ω π d D τ : group delay D : sample length g

Simulations & Experiments

Edge modes in heterostructures made of ENG and MNG: Theory prediction dEdge mode m < 0 ENG mirror m > 0 MNG mirror 0 x Edge mode: for Jackiw-Rebbi Solution (Phys. Rev. D 1976)

Edge modes in heterostructures made of ENG and MNG: Microwave experiments Edge mode at the interface between two photonic mirrors with m >0 and m <0

Edge modes in photonic chains: Microwave experiments

Poster presented by Jun Jiang

Poster presented by Kejia Zhu

Edge mode in heterostructures made of ENG and MNG: Visible-light experiments Edge mode in heterostructure: (AB) 6 M MNG mirror m >0 ----- S 1 , ----- S 2 1D PC Metal theoretical results with different losses θ = 0 o Incident light ….. S 3 experimental results A: SiO 2 θ = 15 o B: TiO 2 M: Ag = = ， n 1 . 443 n 2 . 327 For λ = 589 nm, d M = 60.2 nm A B = = ， d 89 . 0 nm d 55 . 5 nm T < 1% without edge mode A B = d 60 . 2 nm T = 33% with edge mode M Enhancement: 30 ∼ 40

Edge mode in sandwich structure: (AB) 6 M(BA) 6 MNG m >0 MNG m >0 (AB) 5 M(BA) 5 S;           A: SiO 2 ; B: TiO 2 ; M: Ag ; S: glass Incident = = light n 1 . 443 , n 2 . 327 A B = = d 89 . 0 nm , d 55 . 5 nm A B = d 83 . 1 nm M Meta l ----- S 1 , ----- S 2 ….. S 3 theoretical results experimental results with different losses θ = 15 o θ = 0 o For λ = 589 nm , d M = 83.1 nm T = 0.15% without edge mode T = 38% with edge mode Enhancement: 255  OPTICAL THICK metal film  FAR FIELD excitation Possible applications: plasmonics