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Angular Momenta , Geometric Phases, and Spin-Orbit Interactions of Light and Electron Waves

Angular Momenta , Geometric Phases, and Spin-Orbit Interactions of Light and Electron Waves. Konstantin Y. Bliokh Advanced Science Institute, RIKEN, Wako- shi , Saitama, Japan In collaboration with:

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Angular Momenta , Geometric Phases, and Spin-Orbit Interactions of Light and Electron Waves

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  1. Angular Momenta, Geometric Phases,and Spin-Orbit Interactions of Light and Electron Waves Konstantin Y. Bliokh Advanced Science Institute, RIKEN, Wako-shi, Saitama, Japan In collaboration with: Y. Gorodetski, A. Niv, E. Hasman (Israel); M. Alonso (USA), E. Ostrovskaya (Australia), A. Aiello (Germany),M. R. Dennis (UK), F. Nori (Japan)

  2. Basic concepts: Angular momenta, Berry phase, Spin-orbit interactions, Hall effects • Theory of photon AM • Application to Bessel beams • Electron vortex beams • Theory of electron AM

  3. Angularmomentum of light 1. Intrinsicspin AM (polarization) kc 2. Extrinsic orbital AM (trajectory) rc 3. Intrinsic orbital AM (vortex)

  4. Angular momentum: Spin and orbital Different mechanical action of the SAM and OAM on small particles: orbital motion spinning motion O’Neil et al., PRL 2002; Garces-Chavez et al., PRL 2003

  5. Geometric phase 2D: 3D: AM-rotation coupling: Coriolis / angular-Doppler effect

  6. Geometric phase dynamics,S: Φ geometry,E: C - Berry connection, curvature (parallel transport)

  7. Spin-orbit Lagrangian and phase • SOI Lagrangian from Maxwell equations Bliokh, 2008 2F0 F0 Berry phase: Rytov, 1938; Vladimirskiy, 1941; Ross, 1984; Tomita, Chiao, Wu, PRL 1986 • for Dirac equation

  8. Spin-Hall effect of light • ray equations • (equations of motion) 2D D  + Liberman& Zeldovich, PRA 1992; Bliokh& Bliokh, PLA 2004, PRE 2004; Onoda, Murakami, Nagaosa, PRL 2004

  9. Spin-Hall effect of light Berry phase Spin-Hall effect Anisotropy Bliokh et al., Nature Photon. 2008

  10. Spin-Hall effect of light ImbertFedorov transverse shift Fedorov, 1955; Imbert, 1972; …… Onoda et al., PRL 2004; Bliokh & Bliokh, PRL 2006; Hosten & Kwiat, Science 2008

  11. Spin-Hall effect of light geometry, phase: dynamics, AM:

  12. Spin-Hall effect of light Remarkably, the beam shift and accuracy of measurements can be enormously increased using the method of quantum weak measurements: Angstrom accuracy!

  13. Orbit-orbit interaction and phase • OOI Lagrangian 2F0 F0 Φ • Berry phase C Bliokh, PRL 2006; Alexeyev, Yavorsky, JOA 2006; Segev et al., PRL 1992; Kataevskaya, Kundikova, 1995

  14. Orbital-Hall effect of light • vortex-depndent shifts and orbit-orbit interaction –1 0 1 –2 2 Fedoseyev, Opt. Commun. 2001; Dasgupta & Gupta, Opt. Commun. 2006 Bliokh, PRL 2006

  15. Basic concepts: Angular momenta, Berry phase, Spin-orbit interactions, Hall effects • Theory of photon AM • Application to Bessel beams • Electron vortex beams • Theory of electron AM

  16. Angularmomentum of light • total AM = OAM +SAM? • z-components andparaxialeigenmodes: • s - polarization, • l - vortex

  17. Nonparaxial problem 1 • transversality constraint: • 3D  2D “the separation of the total AM into orbital and spin parts has restricted physical meaning. ... States with definite values of OAM and SAM do not satisfy the condition of transversality in the general case.” A. I. Akhiezer, V. B. Berestetskii, “Quantum Electrodynamics” (1965) though

  18. Nonparaxial problem 2 • nonparaxial vortex though “in the general nonparaxial case there is no separation into ℓ-dependent orbital and s-dependent spin part of AM” S. M. Barnett, L. Allen, Opt. Commun. (1994) spin-to-orbital AM conversion?.. Y. Zhao et al., PRL (2007)

  19. Modified AM and position operators compatible with transversality: • projected SAM and OAM • cf. S.J. van Enk, G. Nienhuis, JMO (1994) - spin-dependent position M.H.L. Pryce, PRSLA (1948), etc. Spin-dependent OAM and position signify the spin-orbit interaction (SOI) of light: (i) spin-to-orbital AM conversion, (ii) spin Hall effect.

  20. Modified AM and position operators Modified operators exhibit non-canonical commutation relations: cf. S.J. van Enk, G. Nienhuis (1994); I. Bialynicki-Birula, Z. Bialynicka-Birula (1987), B.-S. Skagerstam (1992), A. Berard, H. Mohrbach (2006) But they correspond to observable valuesand are transformed as vectors:

  21. Helicity representation • all operators become diagonal ! - Berry connection cf. I. Bialynicki-Birula, Z. Bialynicka-Birula (1987), B.-S. Skagerstam (1992), A. Berard, H. Mohrbach (2006) 3D  2D reduction and diagonalization:

  22. Basic concepts: Angular momenta, Berry phase, Spin-orbit interactions, Hall effects • Theory of photon AM • Application to Bessel beams • Electron vortex beams • Theory of electron AM

  23. Bessel beams in free space • mean values • field • SAM and OAM • for Bessel beam • Berry phase The spin-to-orbit AM conversion in nonparaxial fields originates from the Berry phase associated with the azimuthal distribution of partial waves.

  24. Quantization of caustics • spin-dependent intensity • quantized GO caustic: • fine SOI splitting !

  25. Plasmonic experiment - circular plasmonic lens generates Bessel modes Y. Gorodetski et al., PRL (2008)

  26. Spin and orbital Hall effects • azimuthally truncated field (symmetry breaking along x) B. Zel’dovich et al. (1994), K.Y. Bliokh et al. (2008) • orbital and spin Hall effects of light

  27. Plasmonic experiment Plasmonic half-lens produces spin Hall effect: K. Y. Bliokh et al., PRL (2008)

  28. Basic concepts: Angular momenta, Berry phase, Spin-orbit interactions, Hall effects • Theory of photon AM • Application to Bessel beams • Electron vortex beams • Theory of electron AM

  29. Angularmomentum of electron 1. Spin AM k k ‘non-relativistic’ kc 2. Extrinsic orbital AM rc 3. Intrinsic orbital AM? ‘relativistic’

  30. Scalar electron vortex beams

  31. Scalar electron vortex beams

  32. Scalar electron vortex beams

  33. Orbital-Hall effect for electrons • equations • of motion Murakami, Nagaosa, Zhang, Science 2003 Bliokh et al., PRL 2007

  34. Scalar electron vortex beams

  35. Scalar electron vortex beams

  36. Scalar electron vortex beams

  37. Basic concepts: Angular momenta, Berry phase, Spin-orbit interactions, Hall effects • Theory of photon AM • Application to Bessel beams • Electron vortex beams • Theory of electron AM

  38. Bessel beams for Dirac electron • Dirac equation: • plane wave • polarization bi-spinor cross- terms E>0 • spinor in the rest frame cross- terms E<0

  39. Bessel beams for Dirac electron • spectrum • spin-dependent intensity • SOI strength Bliokh, Dennis, Nori, arXiv:1105.0331

  40. Angular momentum • Foldy-Wouthhuysen • transformation • projector onto E>0 subspace cross- terms E>0 cross- terms E<0 • OAM and SAM for • Bessel beams Bliokh, Dennis, Nori, arXiv:1105.0331

  41. Magnetic moment for Dirac electron • magnetic moment from current • operator ! • final result slightly differs from Gosselin et al. (2008); Chuu, Chang, Niu, (2010)

  42. Spin and orbital AM of light, geometric phase, spin- and orbital-Hall effects of light • General theory for nonparaxial optical fields: modified SAM, OAM, and position operators. • Bessel-beams example, spin-orbit splitting of caustics and Hall effects • Electron vortex beams, AM of electron, orbital-Hall effect. • Relativistic nonparaxial electron: Bessel beams from Dirac equation • General theory of SAM, OAM, position, and magnetic moment of Dirac electron.

  43. THANK YOU!

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