Pseudospin-1 Physics with Photonic Crystals Anan Fang, 2016 - - PowerPoint PPT Presentation

Pseudospin-1 Physics with Photonic Crystals

Anan Fang, 2016 Department of Physics, HKUST

Outline

• Background
• Spin-orbit Hamiltonian with pseudospin-1
• Photonic analog of gate voltage in graphene
• Boundary conditions at the interface
• Super Klein tunneling
• Supercollimation in the superlattice of PCs

and 1D disorder modulation of PCs

• Localization behavior in 1D disordered

Pseudospin-1 system

Background

Dirac Cones at the Brillouin Zone Boundary K and K’ Graphene [from wiki]

F x x F y y

H v p v p    

Massless Dirac Equation

K.S. Novoselov, etc., Science 306, 666 (2004)

Klein tunneling

• M. I. Katsnelson, etc., Nat. Phys. 2, 620 (2006)

Integer quantum Hall effect

• Y. B. Zhang, etc., Nature 438, 201 (2005)

Pseudospin-1 system with artificial crystals in ultracold atom systems

Line-centered-square optical lattice

• R. Shen, etc., Phys. Rev. B 81, 041410 (2010)

Dice or T3 lattice

• D. F. Urban, etc., Phys. Rev. B

84, 115136(2011)

Problems:

• 1. Extremely low temperature, around 1 microkelvin
• 2. Very narrow working band width

Conical dispersion at k=0 for certain 2D photonic crystals ε = μ =0 at the Dirac-like point frequency 𝜕𝐸 Conical dispersion for a square lattice of dielectric cylinders in air r = 0.2a, ε = 12.5

2D Photonic cystals Dirac-like cones, Accidental degeneracy

• f monopole and dipole

excitations Effective medium crossing 𝜁 = 𝜈 = 0 at the Dirac-like point frequency Problems with Dirac equation (pseudospin ½) description:

• 1. Omission of the flat band
• 2. Berry phase is 𝜌

Conical dispersion at k=0 for certain 2D photonic crystals

Spin-orbit interaction with pseudospin 1 from Maxwell’s equations 2 i x y i i x y x y i x y                                                           

x y z x y

H H iH E iH               

(Transverse)

x y z x y

iH i H E H H               

(Longitudinal)

𝜁 = 𝜈 = 0

TE case: Hx, Hy and Ez are nonzero components. Matrix equation from Maxwell’s equations

( ) 0 2

x y x y x y D x y

k ik k ik k ik k ik                                 

Matrix equation in k space:

) ( ( )

|

D

D D D

d d

 

        



    ) ( ( )

|

D

D D D

d d

 

        



   

Here 𝜁 = 𝜕𝐸

𝑒𝜁 𝑒𝜕 𝜕=𝜕𝐸

and 𝜈 = 𝜕𝐸

𝑒𝜈 𝑒𝜕 𝜕=𝜕𝐸

from the first order approximation of 𝜕𝜁 and 𝜕𝜈

near 𝜕 = 𝜕𝐸:

Spin-orbit interaction with pseudo spin 1 from Maxwell’s equations Let 𝜔 = 𝑉𝜔 , where 2 U                 , and left multiply 𝑉−1 on both sides, 1

g p p

c dn n d v       1 ( ) 2

x y g x y x y D x y

k ik v k ik k ik k ik                     Let 𝜀𝜕 = 𝜕 − 𝜕𝐸, and 𝑇 as the spin vector for spin 1. 1 1 1 1 1 2

x

S            Spin matrices for spin 1 1 2

y

i S i i i              Spin-orbit Hamiltonian:

g

H S k v      

Spin-orbit interaction with pseudospin 1 from Maxwell’s equations The eigen vectors for the Hamiltonian,

( )

1 2 2

k k

i k s i

se se

 





             ( , 1)

g

sv k s    

( )

1 2

k k

i k s i

e e

 





            ( , 0)

g

sv k s    In the two eigenvectors with 𝑡 = ±1, electric field 𝐹𝑨

(𝑙) is set to be unity.

For arbitrary 𝐹𝑨

(𝑙), we have .

( ) ( ) ( ) 1 k k k T z

E    

( ) ( ) ( ) ( ) ( ) ( )

/ / ( ) 1 2 (s 1) 2 ( )

k k x y k k T z k k x y

iH H E iH H                       

( ) ( ) ( ) ( ) ( )

( 0)

k k x y k L k k x y

H iH s H iH                 Eigen vectors in terms of electromagnetic fields:

i d       



k k k

k

Berry phase:

Photonic analog of gate voltage Length scale of photonic crystal changes: a a r r a r       Frequency and wave vector change 1      

1 ( 1

)

g k g k k

v v



           1 k k k    ( )I

gS

V H v k x    Impedance match near the Dirac-like point frequency 𝜕𝐸 No backscattering for normal incidence Hamitonian with 1D variation

• f potential V(x)

I is the 3X3 identity matrix

Assuming the wave function , we can have three differential equations from the pseudospin-1 Hamiltonian,

1 2 2)

(     

T

Boundary conditions from spin-1 Hamiltonian

2 2 1

( ) 2

[ ]

g

v i V x x y                  

3 3 1 1 2

( ) 2

[ ]

g

v i i V x x y x y                          

2 3 2

( ) 2

[ ]

g

v i V x x y                  

2 2 1

( ) 2 2

[ ]

x x x g g x x x

v v i dx dx V x dx x y    

     

       

  

Integrate the first equation from 𝑦0 − 𝜗 to 𝑦0 + 𝜗 (𝑦 = 𝑦0 is the interface), and take the limit as 𝜗 → 0, Assume 𝑊(𝑦) and the wave functions are finite,

2 2 2 is continuous at the bounda

( ) ( ), i.e., ry. x x      

1 3 1 3

( ) ( ) ( ) ( ), x x x x            i.e., 𝜔1 + 𝜔3 is continuous at the boundary, and

3 1

( ) ( ) is continuous.

[ ]

V x      Boundary conditions from spin-1 Hamiltonian Similarly, Apply the three boundary conditions from spin=1 Hamiltonian to eigen functions in terms of EM fields in TE case, we have the following correspondence,

2 is continuous

is continuous.  

z

E

3 1+

are continuous H is continuous.   

y 3 1

( ) ( ) is contin B uous is continuous.

[ ]

x

V x      

Klein tunneling for a square potential barrier

2 2 2 2 2 2 2 2 2

cos sin cos cos cos cos (cos cos ) 4

k q ph x x k q q k

T q D q D         

𝜄𝑙and 𝜄𝑟 are angles of wave vector k and q in region I (III) and II, respectively. 𝜀𝜕 = 𝜕 − 𝜕𝐸 . T=1 for all incident angles 𝜄 when 𝜀𝜕 = 𝑊

0/2.

' 15 /14 a a 

r' 15 /14 r 

Klein tunneling for a square potential barrier / /

F g

v v E   / /

F g

v V V v 

2 /15 D

V   0.2962

g

c v  The photon potential , frequency and group velocity:

0 / 2

V   The electron potential and energy:

1D superlattice of photonic crystals Kronig-Penney type photonic potential

Band structure of 1D superlattice 2 cos 𝑙𝑦𝑀 − 2 cos 𝑟2𝑦𝑒 cos 𝑟1𝑦 𝑀 − 𝑒 +𝑡𝑡′ sin 𝑟2𝑦𝑒 sin 𝑟1𝑦(𝑀 − 𝑒) 𝜒 𝜄𝑟1, 𝜄𝑟2 = 0 Dispersion relation from TMM method: When 𝑀 = 2𝑒 and 𝜀𝜕 = 0, we have 𝑙𝑦 = 0 independent of 𝑙𝑧 and 𝑊

0 (𝑊 0 ≠ 0).

𝑟1𝑦

2 + 𝑙𝑧 2 = (𝜀𝜕 + 𝑊

2 )/𝑤𝑕

2

𝑟2𝑦

2 + 𝑙𝑧 2 =

𝜀𝜕 − 𝑊

0/2 /𝑤𝑕 2

𝜒 𝜄𝑟1, 𝜄𝑟2 = cos 𝜄𝑟1 cos 𝜄𝑟2 + cos 𝜄𝑟2 cos 𝜄𝑟1 Near 𝜀𝜕 = 0 and for small 𝑙𝑧, the band dispersion can be expanded as

g x

k sv   sgn ) ( s   𝑙𝑦 is the Bloch wave vector and 𝑙𝑧 is the y component of the wave vector.

Super collimation in 1D superlattice of photonic crystals

0.06 /

g

v L    15 , 2 30 d a L d a   

a is the lattice constant of PC2. Superlattice realized by stacking layers of PCs, PC1 and PC2 with equal thickness.

1D disordered photonic potential Reduced enter frequency: 𝜀𝜕𝑑 = 0.015𝜌𝑤𝑕/𝑒 Half width of the initial Gaussian wave packet: 𝑠0 = 40𝑒 Super collimation in 1D randomized media Uniform distribution: 𝑊 ∈ [−𝑋, 𝑋] 𝑊 ≡ 𝑊/𝑤𝑕 𝑋: randomness strength

Localization length as a function of 𝑙||

d: the thickness of one layer 𝑙||: the wave vector component parallel to the interface

2





 k / 0.053 / / 4

g F

v E v d    

D

    

: the reduced frequency for psedospin-1 photons E: the electron energy in graphene

0 / g

V V v 

0.534 / d W  

2 1 1 2

2( l l n ) n L T L T         Localization length:

𝑈

1 and 𝑈2 are transmission coefficients

for the sample thicknesses 𝑀1 and 𝑀2,

• respectively. <> indicates the ensemble

average for the disordered system.

U /

F

V v 

for photons and for electrons, where 𝑊

0 and 𝑉0 are the

photonic and electronic potentials,

• respectively. 𝑊

∈ −𝑋, 𝑋 .

Localization length as a function of randomness strength 0.20 k k  V / and / [ , ]

g e F

v v V W W   / 0.053 / / 4

g F

v E v d     Minimum localization length exists for a critical value of randomness strength 𝑋

𝑑.

Transmission as a function of normalized potential V / / / /

g e F g F

v V v V v v E E      0.20 k k 

( )

1

• r

for electrons k 0 at the Dirac point 1

k el y

              

( )

1 0 or 0 for photons k 0 at the Dirac-like point 1

k ph y

                      

Plane wave Cylindrical wave

Wave filter

Single PC 1D disordered pseudospin-1 system Half width of beam: 4𝑒

0.0534 / d   

Summary

• 1. We demonstrated that certain dielectric photonic crystals

exhibiting Dirac-like conical dispersion at k=0 can be used to realize photonic pseudospin-1 materials.

• 2. Photonic potential can be varied by a simple change of length

scale of PCs.

• 3. Super Klein tunneling effect for pseudospin-1 photons.
• 4. Super collimation behaviors for 1D PC superlattice and

disordered system.

• 5. Unusual localization behavior for pseudospin-1 photons in 1D

disordered system.