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

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

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

0,0 >< εµ

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?

0,0 >< εµ

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 :

( )

( ) 0)()(

,0)()(

21

2211

21

2211

=++

=

=++

=

dddd

dddd

µµµ

εεε

ωµωµωµ

ωεωεωε

2101101 )( ,ωαµωµεε −==

2022202 ,)( µµωβεωε =−=

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

0,0 << µε 0,0 >> µε

0,0 <= µε 0,0 => µε

0,0 <> µε

µ-negative gap

DNG band

MNG gap

DPS band

0,0 =< µε 0,0 >= µε

0,0 >< µε

ENG gap ε-negative gap

ωε ω µ

ω µ ωε

band edges 𝝎𝜺,𝝁 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.: 0 ( )x z r yE i x Hωµ µ−∂ =

0 ( )x y r zH i x Eωε ε∂ = −

[ ] 1 1

2 2( ) ( )x x zi m x V x E

ϕ ϕϕ ϕ

− ∂ + + =

σ σ

metamaterials

1 0 zEϕ ε= 2 0 yHϕ µ=

( )( )2 r rm xc

ω ε µ= −

2 r rEc

ω ε µ= − +

( )( )2 r r r rV xc

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 : 𝜖𝑟 ~ − 𝜇𝑟 ,𝐸 =𝜔2𝑐

𝜖𝑟 + 𝜇𝑟 ~ 0

Then at low energy 𝐸~0: the behavior of Dirac Eq. ONLY depends on the sign of the mass

𝑚 = 𝜔2𝑐

𝜖𝑟 − 𝜇𝑟 ~ 𝜔𝑐

𝜀𝑟 ~ − 𝜔𝑐𝜇𝑟

MNG mirror: 𝜀𝑟 > 0. 𝜇𝑟 < 0 positive mass: m > 0

ENG mirror: 𝜀𝑟 < 0. 𝜇𝑟 > 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 𝑓𝑓𝑓 ∆𝑡 > 0𝜋 𝑓𝑓𝑓 ∆𝑡 < 0

winding number:

𝜈

= �0 𝑓𝑓𝑓 Δ𝑡 > 0 𝑓𝑓 𝑚 > 0 and 𝐌𝐌𝐌 𝐦𝐦𝐦𝐦𝐦𝐦 1 𝑓𝑓𝑓 Δ𝑡 < 0 𝑓𝑓 𝑚 < 0 and 𝐄𝐌𝐌 𝐦𝐦𝐦𝐦𝐦𝐦

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: mAB)(

For symmetry unit cell: mABA)(

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:)( =mAB m

non-local effective parameters !! For asymmetry unit cell: mAB)(

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 rm x

cω ε µ= −

0 21 1 e

rs

C ip Ld

γε

ε ωω = − +

0 21 m

rs

p L iCd

γµ

µ ωω = − +

choosing different circuit parameters one 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( ) gdkgd D

τω

π ω π= ∝DOS:

gτ : 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 0 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

nmdnmdnmd

nn

M

BA

BA

2.605.550.89

327.2443.1

===

==，

，

----- 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

nmdnmdnmd

nn

M

BA

BA

1.835.55,0.89

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

0,0 <= µε

0,0 =< µε

0,0 =< µε

0,0 <= µε

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 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

4. Summary