Unidirectional Phenomena in Optic/Acoustic PT Materials 1 2 3 - - PowerPoint PPT Presentation

Unidirectional Phenomena in Optic/Acoustic PT Materials

Xue-Feng Zhu, Jie Zhu & Xiang Zhang

• 1. Huazhong University of Science and Technology
• 2. The Hong Kong Polytechnic University
• 3. University of California at Berkeley

3 1 2

Engineering Photonics using Lithography

• Engineering photonics in a real ε-µ plane

Conventional dielectrics (silicon)

ε>0 µ>0

Magnets

ε>0 µ<0

(not in optical frequencies)

Metamaterials

ε<0 µ<0

ε µ

Demand nanofabrication to engineer photonics!

Metals (gold, silver)

ε<0 µ>0

Photonic crystals ~ λ/2 (~ 300-500 nm for Si at 1550 nm) Plasmonics ~ λ/10 (<50 nm for visible light) Metamaterials ~ λ/5 (<100 nm for visible light) 500nm Photonic waveguides ~ λ (~ 500 nm for Si at 1550 nm)

Photonics is More Than Real

Conventional dielectrics (silicon)

ε>0 µ>0

Magnets

ε>0 µ<0

(not in optical frequencies)

Metamaterials

ε<0 µ<0

ε µ

• What is missing?

Engineering nanophotonics in a complex dielectric permittivity plane!

Metals (gold, silver)

ε<0 µ>0

ε''

Gain Medium

ε'>0 ε''>0

Lossy Dielectrics

ε'>0 ε''<0

Lossy Metals

ε'<0 ε''<0

ε'

ε'<0 ε''>0

• Photonics in a real ε-µ plane

Parity-time (PT) Symmetry in Optics

Time-dependent Schrodinger equation

2 2

( , ) ( , ) ( , ) ( , ) 2 i t t V t t t m Ψ Ψ Ψ ∂ + ∇ − = ∂ r r r r  

Paraxial equation of diffraction

2 2 1 2

[ ( ) ( )] 2( / )

R I

k E ik E n x in x E z k k x

− − ∂

∂ + + + = ∂ ∂ Parity-time symmetry in quantum mechanics

2

ˆ ˆ ˆ ˆ / 2 ( ) ( )

R I

H p m V x iV x = + +

Hamiltonian in quantum mechanics

ˆ ˆ symmetry PT PTH HPT → =

Parity-time potentials in quantum mechanics

ˆ ˆ symmetry ( ) ( ) ˆ ˆ ( ) ( )

R R I I

PT V x V x V x V x → = − = − −

Parity-time optical potentials

symmetry ( ) ( ) ( ) ( )

R R I I

PT n x n x n x n x → = − = − −

e.g. n=const.+exp(iβx)

Parity and Time Operators

ˆ ˆ ˆ ˆ : , P p p x x → − → − ˆ ˆ ˆ ˆ : , , T p p x x i i → − → → − Electron Photon

• C. E. Ruter et al., Nature Phys. 6, 192 (2010)

Non-Hermitian Optics

Engineering material index from real to complex: PT symmetry in optics

Index modulation from real to complex

Hermitian non-Hermitian

PT-optical modulation along light propagation

( ) cos( ) sin( ) x x i x ε ε ξ β δξ β = + +

' ε " ε x

• Z. Lin et al., Phys. Rev. Lett. 106, 213901 (2011)

1: δ =

1,

L

T R = =

Unidirectional invisibility (δ=1)

2

n ε =

• The PT optical modulation becomes

Fourier space: Introducing a unidirectional wave vector of β

• Unidirectional Bragg reflection

Unidirectional Reflection at δ=1

∆ ε=ξexp(iβx)

β

1

k

2

k

1

k

2

k

1

k

β

1

k

2

k δ

Phase match: δ=β+k1-k2 =0 Phase mismatch: δ=β+k1-k2 ≠ 0

(Reflection) (Reflectionless)

Design of One-way Invisible Cloak

To Combine Transformation Optics with Non-Hermitian Optics:

Add PT-optical modulation into virtual space Coordinate mapping in TO

• X. Zhu et al., Opt. Lett., 38, 15, 2821(2013)
• X. Zhu et al., Phys. Rev. Lett. 106, 014301 (2011)

ε' = AεA

T/detA

Aij = hi'∂xi'/hi∂xi µ' = AµA

T/detA

r = f(r') = b(r'-a)/(b-a) θ =θ '

For TE light

ε' = f(r')f'(r') ε/r' µ'r' = f(r')µ/[r'f'(r')] µ'θ ' = f'(r')r'µ/f(r')

ε=1+ξexp(iβx), µ=1 ε=1, µ=1

One-way Cloak for Plane Waves

2 2 2

=0 2

in sc in

dE dE i E dx dx k c ω ξ = ，

Phase match: δ=β+k1-k2 =0

(reflecting)

Phase mismatch: δ=β+k1-k2 ≠ 0

(no reflecting)

=0

in sc

dE dE dx dx = ，

Unidirectional light reflection (backward)

For plane waves from the left For plane waves from the right

PT cloak

• bject

Strong reflection

PT cloak

• bject

No reflection

One-way Cloak with Forward Reflection

Phase match: δ=β+k1-k2 =0

(reflecting)

Phase mismatch: δ=β+k1-k2 ≠ 0

(no reflecting)

Unidirectional light reflection (forward) For plane waves from the left For plane waves from the right

1 2

≠ k k

due to reciprocity Note:

PT Symmetry in Acoustics

• H. Chen and C. T. Chan, J. Phys. D: Appl. Phys. 43, 113001 (2010)

The analogy between acoustic equations and Maxwell equations

1

1 , , ( ) 1 1

r r r

p i r p i r r i p r r r

θ θ θ

ωρ υ θ ωρ υ υ υ ωκ θ

−

∂ − = − ∂ ∂ − = − ∂ ∂ ∂ − = − − ∂ ∂ ( ) 1 ( ) , ( ) , ( ) ( ) 1 1 ( )

z r r z r z z

E i H r E i H r rH H i E r r r

θ θ θ

ωµ θ ωµ ωε θ ∂ − − − = − ∂ ∂ − − = − ∂ ∂ ∂ − − − = − − ∂ ∂

1

[ ]

r r

p

θ θ

υ υ ρ ρ κ − − −

1

[ ]

z r r z

E H H

θ θ

µ µ ε − − −

PT Symmetry in Acoustics

Unidirectional sound reflection Unidirectional acoustic carpet cloaking

( )

R L

t r S k r t   =    

* 1

( ) ( ) S k S k

−

=

1,2

[1 (1 ) / ] t i T T λ = ± −

PT Symmetry Exceptional Point:

1 2 or

1 T λ λ = =

• X. Zhu et al., Phys. Rev. X 4, 031042 (2014)

Other Unidirectional Phenomena in PT Materials

• 1. Bidirectional transparency and one-way light

enhancement

• 2. Bifurcated Fabry-Pèrot resonances with

unidirectional field enhancement

At the Bragg’s condition 2n L λ =

Bidirectional Transparency in PT Materials

• X. Zhu et al., Opt. Express, 22, 18401 (2014)

n n i n + ∆ + ∆ n n i n − ∆ + ∆ n n i n − ∆ − ∆ n n i n + ∆ − ∆

( ) n n ∆ <<

11 12 21 22 11 12 21 22

( ) ( ) , ( ) ( )

L

M M n n M M n r M M n n M M n + − + = + + +

11 12 21 22 11 12 21 22

( ) ( ) , ( ) ( )

R

M M n n M M n r M M n n M M n

∗ ∗ ∗ ∗

+ − + = + + +

11 12 21 22

2 . ( ) ( ) n t M M n n M M n = + + +

Transfer matrix of one unit-cell

4 11 12 1 21 22

cos( ) sin( ) sin( ) cos( )

j j j j j j j j j j j

i k n L k n L m m n m m in k n L k n L

=

  −     =         −  

∏

2

11 9 n P n   =   ∆  

The coefficient

( )

( )

( )( )

( )

11 11

sin 1 / sin 1 1 / / / N P P N P P M m P P π π π π − − − = −

( )

( )

12 12

sin 1 / / N P P M m P π π − =

( )

( )

21 21

sin 1 / / N P P M m P π π − =

( )

( )

( )( )

( )

22 22

sin 1 / sin 1 1 / / / N P p N P p M m p p π π π π − − − = − For the coefficients

N mP =

11 22 12 21 ( )

, 1, , 0, 1

L R

M M M M r t = = → = =

When We obtain

( )

• r

0, 1

L R

R T = =

1,2,3, m = ⋅⋅⋅⋅⋅⋅

n0=3, ∆n=0.2, λ=1200 nm, L=λ/(2n0)=200 nm

One Numerical Example

2 2

11 11 3 275 9 9 0.2 n P n     = = =     ∆    

275 N =

One-way Light Enhancement

Bifurcated Fabry-Pèrot Resonances

( )

0, 1

L R

R T = =

FP Resonances

11 22 12 21 ( )

1, 0, =1

L R

M M M M r t = = = = → =

11 22 12 21 ( )

1, 0, = 1

L R

M M M M r t = = − = = → = −

One-way Light Enhancement

Summary

( )

R L

t r S k r t   =    

PT Optic/Acoustic System:

1

0, 0, | 1| 0,

=

= ≠     = =  − = →   = = ∞   

L R L R T L R L R

R R R R T R R R R

• Z. Lin et al., Phys. Rev. Lett. 106, 213901 (2011)
• L. Feng et al., Nat. Mater. 12, 108(2013)……
• X. Zhu et al., Opt. Lett., 38, 15, 2821(2013)
• X. Zhu et al., Phys. Rev. X 4, 031042 (2014)
• X. Zhu et al., Opt. Express, 23, 022274 (2015)
• X. Zhu et al., Opt. Express, 22, 18401 (2014)
• H. Ramezani et al., Phys. Rev. Lett. 113, 263905

(2014)