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

Unidirectional Phenomena in Optic/Acoustic PT Materials 1 2 3 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

Engineering Photonics using Lithography  Engineering photonics in a real ε -µ plane 500nm Photonic waveguides µ ~ λ (~ 500 nm for Si at 1550 nm) Metals Conventional (gold, silver) dielectrics (silicon) Photonic crystals ~ λ /2 ε<0 µ>0 ε>0 µ>0 (~ 300-500 nm for Si at 1550 nm) ε Magnets Metamaterials Plasmonics ~ λ /10 ε>0 µ<0 (<50 nm for visible light) ε<0 µ<0 (not in optical frequencies) Metamaterials ~ λ /5 (<100 nm for visible light) Demand nanofabrication to engineer photonics!

Photonics is More Than Real   Photonics in a real ε -µ plane What is missing? µ ε '' Metals Conventional Gain Medium (gold, silver) dielectrics (silicon) ε '<0 ε ''>0 ε'>0 ε ''>0 ε<0 µ>0 ε>0 µ>0 ε ε ' Magnets Metamaterials Lossy Metals Lossy Dielectrics ε>0 µ<0 ε<0 µ<0 ε'<0 ε ''<0 ε'>0 ε ''<0 (not in optical frequencies) Engineering nanophotonics in a complex dielectric permittivity plane!

Parity-time (PT) Symmetry in Optics Electron Photon Time-dependent Schrodinger equation Paraxial equation of diffraction ∂ 2 −  − ∂ ∂ 2 2 k E Ψ + ∇ Ψ − Ψ = 2  + + + = ( , ) ( , ) ( , ) ( , ) 0 i r t r t V r t r t 1 0 [ ( ) ( )] 0 ik E n x in x E ∂ 2 ∂ ∂ t m 0 2 R I 2( / ) z k k x 0 Hamiltonian in quantum mechanics ˆ = + + 2 ˆ ˆ ˆ / 2 ( ) ( ) H p m V x iV x R I Parity-time symmetry in quantum mechanics ˆ ˆ → = symmetry PT PTH HPT Parity-time optical potentials Parity-time potentials in quantum mechanics → = − symmetry ( ) ( ) PT n x n x R R → = − ˆ ˆ symmetry ( ) ( ) PT V x V x = − − ( ) ( ) n x n x R R I I = − − ˆ ˆ ( ) ( ) e.g. n =const.+exp( i β x ) V x V x I I → − → − Parity and Time Operators ˆ ˆ ˆ ˆ : , P p p x x → − → → − ˆ ˆ ˆ ˆ : , , T p p x x i i 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 non-Hermitian Hermitian from real to complex ε = 2 n PT-optical modulation along light propagation ε ε " ' x ε = ε + ξ β + δξ β ( ) cos( ) sin( ) x x i x 0 Unidirectional invisibility ( δ =1) δ = = = 1: 1, 0 T R L Z. Lin et al., Phys. Rev. Lett. 106, 213901 (2011)

Unidirectional Reflection at δ =1  The PT optical modulation becomes ∆ ε = ξ exp( i β x ) Fourier space: Introducing a unidirectional wave vector of β  Unidirectional Bragg reflection k k 1 1 k β β 2 k 1 δ k k k 2 1 2 Phase mismatch: δ = β + k 1 - k 2 ≠ 0 Phase match: δ = β + k 1 - k 2 =0 (Reflection) (Reflectionless)

Design of One-way Invisible Cloak To Combine Transformation Optics with Non-Hermitian Optics: Coordinate mapping in TO Add PT-optical modulation into virtual space ε =1+ ξ exp( i β x ), µ = 1 r = f ( r' ) = b ( r'-a )/( b-a ) θ = θ ' ε =1, µ = 1 ε ' = A ε A ε ' = f ( r' ) f' ( r' ) ε / r' T /detA For TE light µ ' r' = f ( r' ) µ /[ r'f' ( r' )] µ ' = A µ A T /detA µ ' θ ' = f' ( r' ) r' µ / f ( r' ) A ij = h i ' ∂ x i ' / h i ∂x i X. Zhu et al., Opt. Lett., 38, 15, 2821(2013) X. Zhu et al., Phys. Rev. Lett. 106, 014301 (2011)

One-way Cloak for Plane Waves Strong reflection No reflection PT cloak PT cloak object object For plane waves from the left Unidirectional light reflection (backward) Phase match: δ = β + k 1 - k 2 =0 ω ξ 2 dE dE = ， in =0 sc i E 2 in 2 dx dx k c 2 (reflecting) For plane waves from the right Phase mismatch: δ = β + k 1 - k 2 ≠ 0 dE dE dx = ， in sc =0 0 dx (no reflecting)

One-way Cloak with Forward Reflection Unidirectional light reflection (forward) For plane waves from the left Phase match: δ = β + k 1 - k 2 =0 (reflecting) For plane waves from the right Phase mismatch: δ = β + k 1 - k 2 ≠ 0 (no reflecting) ≠ k k due to reciprocity Note: 1 2

PT Symmetry in Acoustics The analogy between acoustic equations and Maxwell equations ∂ − ∂ 1 ( ) 1 E p − ωµ − = − − ωρ υ = − z ( ) , , i H i θ θ ∂ θ ∂ θ r r r r ∂ − ∂ ( ) , E p − ωµ = − − ωρ υ = − ∂ z , i H i θ θ ∂ r r r r ∂ ∂ − ∂ υ ∂ υ 1 ( ) 1 ( ) 1 ( ) 1 rH H r − ωε − = − − − ωκ − = − − θ θ 1 r r ( ) i E i p ∂ ∂ θ ∂ ∂ θ z z r r r r r r κ − − υ − υ ρ ρ 1 [ ] p θ θ r r − − µ µ ε − 1 [ ] E H H θ θ z r r z H. Chen and C. T. Chan, J. Phys. D: Appl. Phys. 43, 113001 (2010)

PT Symmetry in Acoustics Unidirectional sound reflection Unidirectional acoustic carpet cloaking Exceptional Point: λ = λ = 2 or 1 T − = * 1 1 ( ) ( ) S k S k PT Symmetry   t r λ = ± − [1 (1 ) / ] t i T T =  R ( )  S k 1,2   r t L 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

Bidirectional Transparency in PT Materials ∆ << ( ) n n 0 + ∆ − ∆ − ∆ − ∆ + ∆ + ∆ − ∆ + ∆ n n i n n n i n n n i n n n i n 0 0 0 0 λ = At the Bragg’s condition 2 n L 0 ∗ ∗ + − + + − + ( ) ( ) , ( ) ( ) , M M n n M M n M M n n M M n = = 11 12 0 0 21 22 0 11 12 0 0 21 22 0 r r + + + + ∗ + ∗ + L R ( ) ( ) ( ) ( ) M M n n M M n M M n n M M n 11 12 0 0 21 22 0 11 12 0 0 21 22 0 2 n = 0 . t + + + ( ) ( ) M M n n M M n 11 12 0 0 21 22 0 X. Zhu et al., Opt. Express, 22, 18401 (2014)

For the coefficients ( ) ( ) ( ) ( )( ) − π − − π sin 1 / sin 1 1 / N P P N P P = − M m 11 11 π π / / P P ( ) ( ) ( ) ( ) − π − π sin 1 / N P P sin 1 / N P P = = M m M m π 12 12 π / 21 21 P / P ( ) ( ) ( ) ( )( ) − π − − π sin 1 / sin 1 1 / N P p N P p = − M m π π 22 22 / / p p Transfer matrix of one unit-cell   i −   cos( ) sin( ) k n L k n L   4 m m ∏ 0 0 j j j j = 11 12 n     j   m m   = − 1 21 22 j sin( ) cos( ) in k n L k n L   0 0 j j j j j 2   11 n = 0   P The coefficient ∆   9 n = = = → = = , 1, , 0 0, 1 M M M M r t N mP When We obtain 11 22 12 21 ( ) L R m = ⋅⋅⋅⋅⋅⋅ = = 1,2,3, or 0, 1 R T ( ) L R

One Numerical Example n 0 =3, ∆ n =0.2, λ =1200 nm, L = λ /(2 n 0 )=200 nm 2 2     11 11 3 n = = = 0     275 P ∆     9 9 0.2 n

One-way Light Enhancement N = 275

Bifurcated Fabry-Pèrot Resonances = = = = → = 1, 0 0, =1 M M M M r t 11 22 12 21 ( ) L R FP Resonances = = = = − = = → = − 0, 1 R T 1, 0 0, = 1 M M M M r t 11 22 12 21 ( ) L R ( ) L R

One-way Light Enhancement

Summary   t r =  R ( )  PT Optic/Acoustic System : S k   r t L Z. Lin et al., Phys. Rev. Lett. 106, 213901 (2011) L. Feng et al., Nat. Mater. 12, 108(2013) …… = ≠  0, 0 R R X. Zhu et al., Opt. Lett., 38, 15, 2821(2013) L R  X. Zhu et al., Phys. Rev. X 4, 031042 (2014) X. Zhu et al., Opt. Express, 23, 022274 (2015)   = =  0, 0 R R X. Zhu et al., Opt. Express, 22, 18401 (2014) = − = → 1 L R T | 1| T R R L R   = = ∞ H. Ramezani et al., Phys. Rev. Lett. 113, 263905 0, R R  (2014) L R  