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 Massless Dirac Equation     H v p v p F x x F y y Dirac Cones at the Brillouin Zone Boundary K Graphene [from wiki] and K’ K.S. Novoselov, etc., Science 306 , 666 (2004) Integer quantum Hall effect Klein tunneling Y. B. Zhang, etc., Nature 438 , 201 (2005) M. I. Katsnelson, etc., Nat. Phys. 2 , 620 (2006)

Pseudospin-1 system with artificial crystals in ultracold atom systems Dice or T3 lattice D. F. Urban, etc., Phys. Rev. B 84 , 115136(2011) Line-centered-square optical lattice R. Shen, etc., Phys. Rev. B 81 , 041410 (2010) Problems: 1. Extremely low temperature, around 1 microkelvin 2. Very narrow working band width

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

Conical dispersion at k=0 for certain 2D photonic crystals 2D Photonic cystals Effective medium crossing Dirac-like cones, 𝜁 = 𝜈 = 0 at the Dirac-like Accidental degeneracy point frequency of monopole and dipole excitations Problems with Dirac equation (pseudospin ½) description: 1. Omission of the flat band 2. Berry phase is 𝜌

Spin-orbit interaction with pseudospin 1 from Maxwell’s equations TE case: Hx, Hy and Ez are nonzero components. Matrix equation from Maxwell’s equations     iH H      x y  (Transverse)      0 0  i x E     z y         0 0 iH H           x y           0  0 2 0 i i      H iH     x y x y      x y  (Longitudinal)   0 0       E    𝜁 = 𝜈 = 0 z     0 0   i x     H i H     y x y Matrix equation in k space:       0 0 0 0 k ik  x y             0 ( ) 0 2 0  k ik k ik    x y x y D       0 0  0 0  k ik   x y 𝑒𝜁 𝑒𝜈 Here 𝜁 = 𝜕 𝐸 and 𝜈 from the first order approximation of 𝜕𝜁 and 𝜕𝜈 𝑒𝜕 = 𝜕 𝐸 𝑒𝜕 𝜕=𝜕 𝐸 𝜕=𝜕 𝐸 near 𝜕 = 𝜕 𝐸 :   d d | |                       ( ) ( ) ( ) ( )         D D D D D D d D d D

Spin- orbit interaction with pseudo spin 1 from Maxwell’s equations    0 0   , where , and left multiply 𝑉 −1 on both sides, Let 𝜔 = 𝑉𝜔     0 2 0 U      0 0      0 0 k ik  x y  1 c 1           0 ( ) v v  k ik k ik   g g x y x y D dn 2     p  n 0 0 k ik    p x y d Let 𝜀𝜕 = 𝜕 − 𝜕 𝐸 , and 𝑇 as the spin vector for spin 1. Spin matrices for spin 1      0 1 0 0 0 i     1 1    1 0 1 S 0 S i i     x y 2   2   0 1 0    0 0  i Spin-orbit Hamiltonian:       H v S k g

Spin-orbit interaction with pseudospin 1 from Maxwell’s equations The eigen vectors for the Hamiltonian,         i i e se k k     1 1          ( ) k   ( , 1) 0 ( , 0) sv k s   sv k s ( )   k 2 g s g 2 s   2     i    e i  k  se k   (𝑙) is set to be unity. In the two eigenvectors with 𝑡 = ±1 , electric field 𝐹 𝑨 (𝑙) , we have .     For arbitrary 𝐹 𝑨 ( ) k ( ) k ( ) k E 1 T z Eigen vectors in terms of electromagnetic fields:          ( ) k ( ) k ( ) k ( ) k H iH / ( ) iH H    x y  x y 1    ( ) k 0 ( 0)       s   ( ) k ( ) k 2 (s 1) E L T z     2   ( ) k ( ) k H iH        ( ) k ( ) k / ( ) iH H x y   x y Berry phase:         0 i d k k k k

Photonic analog of gate voltage Length scale of photonic crystal changes: Frequency and wave vector change    a a a 1         r r r  1   k k k  1 ( 1           ) v v  g k g k k  Hamitonian with 1D variation    ( )I H v g S k V x of potential V(x) I is the 3X3 identity matrix No backscattering Impedance match for normal incidence near the Dirac-like point frequency 𝜕 𝐸

Boundary conditions from spin-1 Hamiltonian Assuming the wave function , we can have three      T ( 2 ) 1 2 differential equations from the pseudospin-1 Hamiltonian,       v [ ]       g 2 2   ( ) i V x   1   2 x y           v [ ]         g 1 1 3 3   ( ) i i V x     2   2 x y x y       v [ ]       g 2 2   ( ) i V x   3   2 x y Integrate the first equation from 𝑦 0 − 𝜗 to 𝑦 0 + 𝜗 ( 𝑦 = 𝑦 0 is the interface), and take the limit as 𝜗 → 0 ,     v v     x  x  x [ ]       0 0 0 g g 2 2 ( ) i dx dx V x dx   1    2 x 2 y x x x 0 0 0 Assume 𝑊(𝑦) and the wave functions are finite,       ( ) ( ), i.e., 2 is continuous at the bounda ry. x x 2 0 2 0

Boundary conditions from spin-1 Hamiltonian Similarly,            ( ) ( ) ( ) ( ), x x x x 1 0 3 0 1 0 3 0 i.e., 𝜔 1 + 𝜔 3 is continuous at the boundary, and [ ]      ( ) ( ) is continuous. V x 3 1 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. E z    1 + are continuous H is continuous. 3 y [ ]       ( ) ( ) is contin uous B is continuous. V x 3 1 x

Klein tunneling for a square potential barrier   2 2 cos cos  k q T ph 2 sin q D       2 2 2 2 2 2 cos cos cos x (cos cos ) q D x k q q k 4 𝜄 𝑙 and 𝜄 𝑟 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 r' 15 /14 a a r

Klein tunneling for a square potential barrier The photon potential , frequency and group velocity: v      0.2962 c 2 /15 0 / 2 V V g 0 D The electron potential and energy:   v  / / / / E v v V V v 0 F g F g

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

Band structure of 1D superlattice Dispersion relation from TMM method: 2 cos 𝑙 𝑦 𝑀 − 2 cos 𝑟 2𝑦 𝑒 cos 𝑟 1𝑦 𝑀 − 𝑒 +𝑡𝑡 ′ sin 𝑟 2𝑦 𝑒 sin 𝑟 1𝑦 (𝑀 − 𝑒) 𝜒 𝜄 𝑟 1 , 𝜄 𝑟 2 = 0 2 2 = (𝜀𝜕 + 𝑊 2 + 𝑙 𝑧 0 𝑟 1𝑦 2 )/𝑤 𝑕 2 2 + 𝑙 𝑧 2 = 𝑟 2𝑦 𝜀𝜕 − 𝑊 0 /2 /𝑤 𝑕 𝜒 𝜄 𝑟 1 , 𝜄 𝑟 2 = cos 𝜄 𝑟 1 + cos 𝜄 𝑟 2 cos 𝜄 𝑟 2 cos 𝜄 𝑟 1 𝑙 𝑦 is the Bloch wave vector and 𝑙 𝑧 is the y component of the wave vector. When 𝑀 = 2𝑒 and 𝜀𝜕 = 0 , we have 𝑙 𝑦 = 0 independent of 𝑙 𝑧 and 𝑊 0 ( 𝑊 0 ≠ 0). Near 𝜀𝜕 = 0 and for small 𝑙 𝑧 , the band dispersion can be expanded as     sgn ( ) sv k s g x

Super collimation in 1D superlattice of photonic crystals Superlattice realized by stacking layers of PCs, PC1 and PC2 with equal thickness.    15 , 2 30 d a L d a    0.06 / v L g a is the lattice constant of PC2.

Super collimation in 1D randomized media Reduced enter frequency: 𝜀𝜕 𝑑 = 0.015𝜌𝑤 𝑕 /𝑒 Half width of the initial Gaussian wave packet: 𝑠 0 = 40𝑒 1D disordered photonic potential Uniform distribution: 𝑊 ∈ [−𝑋, 𝑋] 𝑊 ≡ 𝑊/𝑤 𝑕 𝑋: randomness strength