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

HKUST IAS, Jan. 11, 2016 Collaborators:  Dr. Wei Tan, Dr. Yong Sun, Tongji Uni.  Prof. Shun-Qing Shen , The University of Hong Kong

同舟共济

• utline
• 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

• M. Z. Hasan et al., Rev. Mod. Phys. 82, 3045 (2010)

topological description for electronic insulators

• 1. Introduction

electronic medium

Metallic, unidirectional Edge state between insulators with different topological orders

Quantum Hall Effect Chiral Edge State with H Broken T-symmetry Quantum Spin Hall Insulator Edge states without H with T-symmetry

• M. Z. Hasan et al., Rev. Mod. Phys. (2010)

From ENGHETA and.ZIOLKOWSKI 2006

Materials Responds To EM Waves: Permittivity ε and Permeabilty µ εµ

± = n

double-positive (DPS) forward-wave propagation single-negative µ-negative (MNG) evanescent wave “barrier” single-negative ε-negative (ENG) evanescent wave “barrier” double-negative (DNG) backward-wave propagation zero-index materials

, > < ε µ

photonic medium: metamaterials with designed ε and μ

Materials Responds To EM Waves: Permittivity ε and Permeabilty µ εµ

± = n

Photonic Conductor Right-Handed Photonic Insulator II MNG Mirror Photonic Insulator I ENG Mirror Photonic Conductor Left-Handed Dirac-Point Related Medium Photonic Graphene?

, > < ε µ

analog to electrons Q1：different topological orders between EMG and MNG mirrors ??

• M. Z. Hasan et al., Rev. Mod. Phys. (2010)

semimetal for x < .07 semiconductor for .07 < x < .22 semimetal for x > .18 bands Ls;a invert at x ~ .04

Manipulating topological order: Electronic band inversion transition from normal insulator (NI) to TI [d=3]

Photonic analog

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

Chirality

Nature materials 2012

Photonic analog

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

For electronic and photonic NI, we have mapping: Schroedinger Eq. ↔ Maxwellʹs Eq. + photonic crytals

PBG as Normal Photonic Insulator

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 !

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, )

at sub-wavelength condition and normal incidence, a PBG structure: with the edge ωε and ω µ :

( )

( )

) ( ) ( , ) ( ) (

2 1 2 2 1 1 2 1 2 2 1 1

= + + = = + + = d d d d d d d d

µ µ µ ε ε ε

ω µ ω µ ω µ ω ε ω ε ω ε

2 10 1 10 1

) ( , ω α µ ω µ ε ε − = =

20 2 2 20 2

, ) ( µ µ ω β ε ω ε = − = A structure made of opaque or "dark" metamaterials!!

ENG for ω < β1/2 MNG for ω < α1/2

Earlier studies:

• 2. Topological Description: theoretical study

T

ω

, < < µ ε , > > µ ε , < = µ ε , = > µ ε , < > µ ε µ-negative gap

DNG band MNG gap DPS band

, = < µ ε , > = µ ε

, > < µ ε

ENG gap

ε-negative gap

ωε ω µ ω µ ωε

band edges 𝝏𝜻,𝝂 inverted

Band inversion transition: metematerial analogs

tailoring ε and µ

Earlier studies:

Evidence: photonic band inversion in metamaterials

recent studies:

2013

Answers to the two questions

massive Dirac Eq.: Maxwell’s Eq.:

( )

x z r y

E i x H ωµ µ −∂ = ( )

x y r z

H i x E ωε ε ∂ = −

[ ]

1 1 2 2

( ) ( )

x x z

i m x V x E ϕ ϕ ϕ ϕ     − ∂ + + =         σ σ metamaterials

1 z

E ϕ ε =

2 y

H ϕ µ =

( )

( ) 2

r r

m x c ω ε µ = − 2

r r

E c ω ε µ = − + ( )

( ) 2

r r r r

V x c ω ε µ ε µ   = + − +  

• 1. Mapping 1D Maxwell’s equations to 1D Dirac equation

<…..> : Average on space

• 2. EMG and MNG mirror as mass inversion in Dirac Eq.

For SNG mirror, if : 𝜗𝑠 ~ − 𝜈𝑠 , 𝐹 = 𝜕 2𝑑 𝜗𝑠 + 𝜈𝑠 ~ 0 Then at low energy 𝐹~0: the behavior of Dirac Eq. ONLY depends on the sign of the mass 𝑛 =

𝜕 2𝑑 𝜗𝑠 − 𝜈𝑠 ~ 𝜕 𝑑 𝜁𝑠 ~ − 𝜕 𝑑 𝜈𝑠

MNG mirror: 𝜁𝑠 > 0. 𝜈𝑠 < 0 positive mass: m > 0 ENG mirror: 𝜁𝑠 < 0. 𝜈𝑠 > 0 negative mass: m < 0 So, the sign of the mass is inverted from MNG to ENG : Different topological orders for MNG and ENG: The first evidence

m > 0 m = 0 m < 0

Su-Schrieffer-Heeger Model for Polyacetylene (Rev. Mod. Phys. 1988)

Mapping the Dirac Eq. to the SSH model (S.Q Shen 2013)

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

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!

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 The gap divided into two parts: EMG and MNG For asymmetry unit cell:

m

AB) (

For symmetry unit cell:

m

ABA) (

The gap described by: EMG or MNG Depending on symmetry of the unit cell.

Topological description: Extend to mirrors made of dielectric multilayers

• Retrieval theory : Smith et al., PRB 2002
• Bloch-wave-expansion theory: Kan et al., PRA 2009

Determination of effective parameters

Effective parameters in gap regime The gap is divided into two parts: EMG and MNG

Dependence on periodic number

20 , 15 , 10 : ) ( = m AB m

non-local effective parameters !! For asymmetry unit cell:

m

AB) (

MNG ENG

Effective parameters in gap regime

It can be shown: Ε and μ are independent of the periodic number local effective parameters

10

) (ABA

First gap Second gap

( )

( ) 2

r r

m x c ω ε µ = −

2

1 1

e r s

C i p Ld γ ε ε ω ω   = − +    

2

1

m r s

p L i Cd γ µ µ ω ω   = − +    

choosing different circuit parameters

• ne gets DNG, ENG, MEG materials

1D Metamaterials Realized By Transmission Line

Eleftheriade et al., 2002; and by Itol et al., 2002

• 3. Topological Description: experimental study

1 ( )

g

dk g d D τ ω π ω π = ∝

DOS:

g

τ

: group delay D : sample length

Band Inversion in photonic chains

Simulations & Experiments

Edge modes in heterostructures made of ENG and MNG: Theory prediction

m < 0 ENG mirror m > 0 MNG mirror

dEdge mode

x

Edge mode:

Jackiw-Rebbi Solution (Phys. Rev. D 1976)

for

Edge mode at the interface between two photonic mirrors with m>0 and m<0

Edge modes in heterostructures made of ENG and MNG: Microwave experiments

Edge modes in photonic chains: Microwave experiments

Poster presented by Jun Jiang

Poster presented by Kejia Zhu

Edge mode in heterostructure: (AB)6M

1D PC Metal Incident light

A: SiO2 B: TiO2 M: Ag

nm d nm d nm d n n

M B A B A

2 . 60 5 . 55 . 89 327 . 2 443 . 1 = = = = = ， ，

• ---- S1 , ----- S2

theoretical results with different losses θ = 0o ….. S3 experimental results θ = 15o

For λ = 589 nm, dM = 60.2 nm T < 1% without edge mode T = 33% with edge mode Enhancement: 30 ∼ 40

MNG mirror m >0

Edge mode in heterostructures made of ENG and MNG: Visible-light experiments

Edge mode in sandwich structure: (AB)6M(BA)6

Metal

Incident light

    

(AB)5M(BA)5 S; A: SiO2 ; B: TiO2 ; M: Ag ; S: glass

nm d nm d nm d n n

M B A B A

1 . 83 5 . 55 , . 89 327 . 2 , 443 . 1 = = = = =

    

• ---- S1 , ----- S2

theoretical results with different losses θ = 0o ….. S3 experimental results θ = 15o

For λ = 589 nm , dM = 83.1 nm T = 0.15% without edge mode T = 38% with edge mode Enhancement: 255

 OPTICAL THICK metal film  FAR FIELD excitation

MNG m >0 MNG m >0

Possible applications: plasmonics

• M. Z. Hasan et al., Rev. Mod. Phys. (2010)

d < dc NI TI

comparing to sandwich structures of electronic TI

Edge state → Resisdence 10-2 !

Extend to 2D structures

Plannar metamaterials made of transmission lines

Band gap inversion transition

, < = µ ε , = < µ ε , = < µ ε , < = µ ε

m > 0 m < 0 m = 0

1 3 2 4

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

1 3 4 2

1 3 4 2

clockwise circularly polarized source counterclockwise circularly polarized source

1 3 4 2

Thank You

• Mapping Maxwell’s equation to Dirac

Eq., it is shown 𝜗-negative and 𝜈-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

• 4. Summary