Topological Description for Photonic Mirrors

Transcription

1 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

2 1. Introduction outline 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

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

4 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

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

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

7 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

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

9 Photonic analog Nature materials 2012 Chirality

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

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

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

13 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 ε1 = ε10, µ 1( ω) = µ 10 2 ω β ε 2 ( ω) = ε 20, µ 2 2 = µ 20 ω MNG for ω < α 1/2 ENG for ω < β 1/2 A structure made of opaque or "dark" metamaterials!! at sub-wavelength condition and normal incidence, a PBG structure: with the edge ω ε and ω µ : ε µ ( ω ) ε ( ω ) µ ε1( ωε ) d = d 1 1 µ 1( ωµ ) d = d + ε 2( ωε ) d + d µ ( ω ) d + d 2 2 µ 2 = 0, 2 = 0

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

15 recent studies: 2013 Answers to the two questions 1. Mapping 1D Maxwell s equations to 1D Dirac equation metamaterials Maxwell s Eq.: massive Dirac Eq.: E = iωµ 0 µ ( xh ) x z r y H = ωε i 0 ε ( xe ) x y r z ϕ1 ϕ1 i x x + mx z + V x = E ϕ ϕ [ σ ( ) σ ( )] 2 2 ϕ 1 = ε0ez ϕ 2 = µ 0H y ω mx ( ) = ε r µ r 2c ( ) <..> : Average on space ω E = ε r + µ r 2c ω V( x) = ( ε r + µ r) ε r + µ r 2c

16 2. EMG and MNG mirror as mass inversion in Dirac Eq. For SNG mirror, if : ε r ~ μ r, E = ω 2c ε r + μ r ~ 0 Then at low energy E~0: the behavior of Dirac Eq. ONLY depends on the sign of the mass m = ω 2c ε r μ r ~ ω c ε r ~ ω c μ r So, the sign of the mass is inverted from MNG to ENG : MNG mirror: ε r > 0. μ r < 0 positive mass: m > 0 ENG mirror: ε r < 0. μ r > 0 negative mass: m < 0 Different topological orders for MNG and ENG: The first evidence

17 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 Band inversion in the SSH model Mass inversion in the Dirac Eq.

18 Berry phase: 0 fff t > 0 = π fff t < 0 winding number: ν 0 fff Δt > 0 oo m > 0 and MMM mmmmmm = 1 fff Δt < 0 oo m < 0 and EEE mmmmmm Therefore, MNG and ENG mirrors have Different topological orders!

19 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. For asymmetry unit cell: (AB) m The gap divided into two parts: EMG and MNG For symmetry unit cell: (ABA) m The gap described by: EMG or MNG

20 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

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

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

23 3. Topological Description: experimental study 1D Metamaterials Realized By Transmission Line Eleftheriade et al., 2002; and by Itol et al., 2002 ε r 1 1 γ = C0 i e pε 2 + ω Ld ω s µ r p 1 γ L m = i µ ω Cd ω s choosing different circuit parameters one gets DNG, ENG, MEG materials mx ( ) ω = ε r µ r 2c ( )

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

25 Simulations & Experiments

26 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)

27 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

28 Edge modes in photonic chains: Microwave experiments

29 Poster presented by Jun Jiang

30 Poster presented by Kejia Zhu

31 Edge mode in heterostructures made of ENG and MNG: Visible-light experiments Edge mode in heterostructure: (AB) 6 M MNG mirror m >0 Incident light 1D PC Metal S 1, S 2 theoretical results with different losses θ = 0 o A: SiO 2 B: TiO 2 M: Ag.. S 3 experimental results θ = 15 o n d d A A M = 1.443, n = 89.0 nm, d = 60.2 nm B = B = 55.5 nm For λ = 589 nm, d M = 60.2 nm T < 1% without edge mode T = 33% with edge mode Enhancement: 30 40

32 Edge mode in sandwich structure: (AB) 6 M(BA) 6 Incident light MNG m >0 MNG m >0 Metal n d d (AB) 5 M(BA) 5 S; A: SiO 2 ; B: TiO 2 ; M: Ag ; S: glass A A M S 1, S 2 theoretical results with different losses θ = 0 o = 1.443, n = 89.0 nm, d = 83.1 nm B = B = 55.5 nm.. S 3 experimental results θ = 15 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

33 comparing to sandwich structures of electronic TI M. Z. Hasan et al., Rev. Mod. Phys. (2010) d < dc NI TI Edge state Resisdence 10-2!

34 Extend to 2D structures Plannar metamaterials made of transmission lines Band gap inversion transition ε = 0, µ < 0 ε < 0, µ = 0 m = 0 m > 0 m < 0 ε < 0, µ = ε = 0, µ < 0 0

35 x- direction linearly polarized source y- direction linearly polarized source

36 clockwise circularly polarized source counterclockwise circularly polarized source

37 4. Summary Mapping Maxwell s equation to Dirac Eq., it is shown ε-negative and μ-negative mirrors have different topological orders. Realizing topological modes in structures made of photonic mirrors. Proving new ways of applications based on photonic topological modes. Financial Supports: NSFC, 973 Program of MOST Thank You