Exploiting duality using metamaterials

Transcription

1 Exploiting duality using metamaterials Jensen Li School of Physics and Astronomy, University of Birmingham, UK Jan 12, 2015

2 Metamaterials with magnetic response Interesting physics by pairing up electric and magnetic responses Cloak a Air Microwave beam (GHz) b Negative index metamaterial Negative Shelby, Smith & Shultz, Science 2001 Negative refractive index Schurig, Smith & Pendry, Science

3 Movitation Electromagnetic duality is symmetry of microscopic Maxwell s equation but broken when materials are present for θ = π/2 Decoupled into TE/TM rather than LCP/RCP Unless ε = μ Possible if we have metamaterials Ivan Fernandez-Corbaton et.al., PRL (2013) 3

4 Outline Investigate phenomena driven by different kinds of dual operations between electric and magnetic fields Enabled by metamaterials by putting electric & magnetic fields in equal footing Restoring duality with metamaterials Helicity-preserving mirror Redefining parity operator using duality Demonstrating Parity-Time-phase transition and unidirectional reflection at exceptional point Gauging dual operation Establishing pseudo magnetic field with metamaterials

5 Restoring duality with metamaterials Helicity/spin angular momentum of photon reversed for a normal mirror at any angle Target: reflecting metasurface preserving helicity at all angles An example of metasurface to have all-angle polarization control LCP LCP 5

6 Spin/polarization state A polarization lives on Poincare sphere North pole: Left handed circular polarization South pole: Right handed circular polarization LCP: xˆ iyˆ / 2 RCP: xˆ iyˆ / 2 6

7 Helicity-preserving mirror Criteria at normal incidence Normal reflection phase: Φ z, Φ x with Φ x φ z = π Sub-λ structures & thickness Effective boundary condition E E z x +ih x ih z z cot 2 cot 2 x 0, 0, Decoupled into circular polarizations ( E ih) ( uˆ sin( 0 z / 2 b ) vˆcos( z / 2 b ))

8 Helicity-preserving mirror Normal mirror flips the spin of light at any angles Spin-preserving mirror has boundary condition: r (, b ) exp(2i tan 1 tan( z / 2 ( cos b ) )) Normal incidence r exp[ i( z 2b )] Geometric phase Gold nano-rods on gold mirror with MgF2 spacer Theory: IEEE Trans. Antennas & Propag. 62, 6274 (2014) Experiment: Adv. Opt. Materials (accepted)

9 Geometric phase in polarization space Flexible phase by rotating the structures Incident wave with LCP transmitted wave of cross-polarization RCP with additional transmitted phase Φ Berry 1 Berry 2 2a v u Enclosed area a x A. Niv et. al. Opt. Lett. (2007) Cross-polar. Conversion: ± e ±i2α

10 Geometric phase in polariztion space Opposite geometric phases for the two spins Material: E tu = t u E u, E tv = t v E v x ± y = exp iα u ± iv E + = 1 1 i E x E 2 1 i E y E x E y = E t+ = 1 1 i cos α sin α t u 0 cos α sin α E t 2 1 i sin α cos α 0 t v sin α cos α (t = u + t v )/2 exp( i2α)(t u t v )/2 E + exp(+i2α)(t u t v )/2 (t u + t v )/2 E v cos α sin α sin α cos α 1 1 i i E + E E u E v u 1 Berry 2 2a a x

11 Transmission Directivity Directivity App: Spin-induced orbital manipulation Spin-dependent refraction Spin coupled to orbital motion B k 0 n t sin θ t n i sin θ i a v x b y (a) b A y u ε ε 2 R z L w y x x w y 2.5 x (c) (a) t t t 0 t 2 L t 12 t 22 Bend 0.8 angle (deg) z x (b) x (c) (d) = Φ Berry x R Phase A 1.0 Spin-dependent refraction 0.5 x t Bending t angle (deg) 2 Incident 0.5V/m Phase B -0.5V/m Incident (b) y x y Φ Berry = ±2α Opt. Exp. 20, (2012) (d) a

12 App: Spin-induced orbital manipulation Inducing orbital angular momentum for angular variation q 1 l = 0, + l = 2, l = 0, l = 2, + (a) Experiment (b) Theory RCP as incident Linear y- polarization as incident (c) (d) Stokes S 3 parameter plotted Red: LCP Blue: RCP Additional rotation of orbitals by around 27 o Nano Lett. 13, 4148 (2013)

13 Optical Spin-Hall Effect (OSHE) Refraction Air-glass interface Hoston, Science (2008). Metasurface X. Yin, Z. Ye, J. Rho, Y. Wang & X. Zhang, Science (2008). Spin can affect orbital motion thru opposite geometric phases: spin Hall effect of light Surface-plasmon-polariton (plasmonic spin-hall effect) K. Y. Bliokh et.al., PRL (2008). The splitting is tiny (comparable to wavelength)

14 App: Holographic interface for SPP A generic manipulation of OSHE Arbitrary specification of target profiles for the two spins possible Orientation of slots = Holographic storage of SPP profiles Experimental setup in generation SPP Collaboration with Hui Liu s group

15 Our scheme: Holographic interface Target profile for a particular spin Geometric Matching Rule Same set of atoms are responsible for both LCP and RCP profiles now Data capacity doubled

16 Plasmonic Spin-Hall effect SIM. EXP. Spin-splitting of orbitals becomes arbitrary Nature Comm (2015) Opposite geometric phases of 2 spins from same set of atoms

17 Coherent control for getting motion picture Based on tunable pixels, Design time frames Static pictures SIM. EXP.

18 Duality can be restored by using metamaterials The geometric phase related can also be used to obtain plasmonic spin-hall effect Next, to investigate the role of dual operation in constructing Parity-Time (PT) symmetric systems Physics enriched by controlling the symmetry breaking 18

19 Parity-time symmetry in QM Quantum Mechanics o Observables are real numbers o General belief: o Real eigenvalues Hermitian operators Parity-time (PT) symmetric Hamiltonian o A weaker condition can still give real eigenvalues o Phase transition (spontaneous symmetry breaking) between the real and complex phases using an external order parameter 2 2 H p x ix * V x V x C. M. Washington Univ., St. Louis

20 Parity-time symmetry in Optics PT-symmetric Hamiltonian can be established in optics Through a complex potential using (x) Pair of modes with complex conjugated Eigenvalues (propagation constants) Real Eigenvalues * V x V x A. Guo et.al., Phys. Rev. Lett. 103, (2009) C. E. Rüter et.al., Nature Physics 6, 192 (2010)

21 Parity time symmetry in optics and metamaterials PT behaviour of a optical system C. E. Rüter et. al. Nature Physics (2010) Coherent Perfect Absorbers & PT phase transition Y. D. Chong et. al. PRL 2010 Y. D. Chong et. al. PRL (2011) PT-symmetric laser absorber S. Longhi, PRA (2010) Parity time synthetic photonic lattices A. Regensburger et. al. Nature (2012) Gain-Driven Discrete Breathers N. Lazarides and G. P. Tsironis PRL (2013) Switching in plasmonic system A. Lupu et.al., OE (2013). Experimental unidirectional reflectionless propagation L. Feng et al. Nature Mater. (2013). PT symmetry related phenomenon can be observed in optics and metamaterials

22 Establishing Hamiltonian with dual operation 1. Polarizing the atoms 2 2 a a 0 a 2. Scattered waves 1. & 2. complete for us to get S-matrix 2i p 2 f ( a a igm), 2i m 2 i f gp, with f, b 0 b a a b a b a b i p b a S b a 2 2 Linearization approx Focus on phenomenon of coherent perfect approximation: b+ = b- = 0 We get Absorption loss Scattering loss E in a+ b- b+ a- (E-field amplitudes)

23 Establishing Hamiltonian with dual operation Coherent Perfect Absorption as an eigenvalue problem S a 0 p p H2 a 0 m m where Hamiltonian is defined by i i 0 a a H2, g f a fb i 0 ib Interpretation: Renormalized (not ) for normal mode consideration: incident waves as effective gain Ideal PT-symmetry condition a a b (gain not needed) Operators: PT, H 0 with 0 i 1 0 i P with P I, T K Phys. Rev. A 87, (2013). CPA: Chong et.al., Phys. Rev. Lett. 105 (2010) 0 1

24 Coherent perfect absorption A way to modulate signal through interference of two beams of different directions Coherent perfect absorber Chong, Ge, Cao & Stone, PRL (2010) Zhang, Macdonald & Zheludev, LSA (2012) Out-of-phase: destructive inteference in-phase: constructive inteference

25 Experimental demonstration Coupling between a bright and a dark atom o Probing PT-symmetry in conventional EIT config o But with intermediate coupling, comparable losses from 2 atoms Bright atom b+ a+ a- b- Metamaterial model configuration Dark atom Transmission-line analog Experiment done by Hong Chen s group in Tongji University

26 Passive system realization Extracting the dipolar model parameters from simulations Fitting t/r spectrum f GHz R, 0.38GHz a R, 0 b 1.2exp s / 0.66mm 1 2 a b

27 Separation s (mm) CPA with ideal PT Broken PT-sym. Spontaneous PT-symmetry breaking Ideal PT-symmetry: Black and blue curve s (or equivalently ) is varied R1=1Ohm, R2=45.5 Ohm satisfying Exceptional point s = 0.8mm R 2 () H a a b 0 i i b 0 i i Separation s (mm) b b, 27

28 Coherent Absorption Spontaneous symmetry breaking PT-Phase transition between eiganvalues behavior 2 absorption peaks (real eigenfrequencies ) for s smaller than 0.8mm Single absorption peak (complex conjugated eigenfrequencies) otherwise Measurements Separation s (mm) Frequency (GHz) Phys. Rev. Lett. 112, (2014). 28

29 Establishing PT-symmetry for uni-directional reflectionless propagation Wave equation z H y E x = ik 0 ε(z) 0 0 μ(z) P-operator redefined E x H y P E H = iπm E z H 0 z I Π = z I 0 Unitary, Hermitian, P 2 = I, symmetry operator of vacuum Pairing mirror and dual operation T-operator T E H = I 0 0 I where E H Π E H is the time-averaged power flow PT-symmetry condition E H ε z = μ z 29

30 Unidirectional reflection Merit = ( R f R b R f +R b ) Cross-potential matching: Unidirectional reflection at subwavelength thickness w/o requiring large gain/loss a/(2c) a/(2c) Standard PT-matching Cross-potential PTmatching Phys. Rev. A 91, (2015)

31 Unidirectional reflection = Exceptional point of Hamiltonian Eigenvalue degeneracy of the S-matrix Equivalently the eigenvalue degeneracy of constitutive matrix Constitutive matrix interpreted as Hamiltonian (Hermitian for lossless) C 1 = 2 S I B ik 0 a S + I B 1 = ε eff iκ eff iκ eff μ eff S = t r b r f t B = H y2 H y1 E x2 E x1 = ik 0a 2 C 1 E x2 + E x1 H y2 + H y1 Inherited PT-symmetry condition ε eff = μ eff and κ eff = κ eff Advantage of cross-potential matching: gain and loss will not be averaged out to get back a Hermitian Hamiltonian

32 Using dual operation to establish PT-symmetric metamaterials A PT-phase transition of coherent perfect absorption can be observed using a totally passive system Unidirectional reflectionless propagation at subwavelength thickness can be observed Next, we will consider a continuous vesion of dual operation to generate pseudo-magnetic field 32

33 Magnetic field and gauge field for electron B = A B unchanged under gauge transformation A A + φ Magnetic field B bends charged particles (Hall effect) Lorentz force qv B Gauge field A modifies the phases (Aharonov-Bohm effect) Geometric phase q/ħ A dr Interference What about for photon? 33

34 Pseudo magnetic field for photon Pseudo magnetic field in momentum space anomalous velocity giving Hall effect of light beam displacement in the order of λ Berry Curvature Berry Curvature M. Onoda, S. Murakami, N. Nagaosa, Phys. Rev. Lett. 93, (2004) Bliokh et.al., Nature Photon. 2, 748 (2008) Hosten et.al., Science 319, 787 (2008) 34

35 Pseudo magnetic field for photon Pseudo magnetic field in real space - lattice of resonators / waveguides in diffraction regime Direction-dependent coupling M. Hafezi et.al., Nature Phys. 7, 907 (2011) dynamic modulation (shifting of dispersion surface) Q. Lin, and S. Fan, Phys. Rev. X 4, (2014) K. Fang, and S. Fan, Phys. Rev. Lett. 111, (2013) K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782 (2012) Additional way (to TO) to bend and guide light by magnetic Lorentz force Strained photonic crystal M. C. Rechtsman et.al., Nature Photon. 7, 153 (2013) 35