Abstract

We report on optical rotational Doppler frequency shift experiments in the context of a counter-intuitive optomechanical phenomenon that is the angular analog of so-called negative optical radiation forces, which involves spin-orbit scattering of light. In practice, spin-orbit opto-mechanical effects arising from the interaction between polarized light and azimuthally varying birefringent optical elements are retrieved from mechano-optical experiments that involve spatial of the medium. Two kinds of experiments (single-beam and two-beam geometries) are performed and both approaches are discussed in the framework of previous dynamical geometric phase and rotational Doppler shift experiments based on spin and/or orbital angular momentum of light.

© 2015 Optical Society of America

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  1. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
    [Crossref]
  2. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1–3 (1998).
    [Crossref]
  3. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
    [Crossref]
  4. S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in gaussian beams,” J. Opt. Soc. Am. A 24, 430–443 (2007).
    [Crossref]
  5. J. Chen, J. Ng, K. Ding, K. H. Fung, Z. Lin, and C. T. Chan, “Negative optical torque,” Sci. Rep. 42, 6386 (2014).
    [Crossref]
  6. D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nature Photon. 8, 610–614 (2014).
    [Crossref]
  7. A. Dogariu, S. Sukhov, and J. J. Saenz, “Optically induced ’negative forces,”’ Nature Photon. 7, 24–27 (2013).
    [Crossref]
  8. O. Brzobohaty, V. Karasek, M. Siler, L. Chvatal, T. Cizmar, and P. Zemanek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ’tractor beam,”’ Nature Photon. 7, 123–127 (2013).
    [Crossref]
  9. V. Kajorndejnukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nature Photon. 7, 787–790 (2013).
    [Crossref]
  10. O. V. Angelsky, A. Y. Bekshaev, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Y. Zenkova, “Orbital rotation without orbital angular momentum: mechanical action of the spin part of the internal energy flow in light beams,” Opt. Express 20, 3563–3571 (2012).
    [Crossref] [PubMed]
  11. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nature Commun. 5, 3300 (2014).
    [Crossref]
  12. S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
    [PubMed]
  13. A. Canaguier-Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” Phys. Rev. A 89, 033841 (2014).
    [Crossref]
  14. A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).
  15. K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
    [Crossref]
  16. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
    [Crossref] [PubMed]
  17. Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultra-short light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
    [Crossref]
  18. M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
    [Crossref]
  19. G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
    [Crossref]
  20. M. Born and E. Wolf, Principles of Optics (Pergamon, 2005).
  21. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New. J. Phys. 9, 78 (2007).
    [Crossref]
  22. G. E. Somargren, “Up/down frequency shifter for optical heterodyne interferometry,” J. Opt. Soc. Am. 65, 960–961 (1975).
    [Crossref]
  23. B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979).
    [Crossref]
  24. R. Simon, H. J. kimble, and E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
    [Crossref] [PubMed]
  25. F. Bretenaker and A. L. Floch, “Energy exchange between a rotating retardation plate and a laser beam,” Phys. Rev. Lett. 65, 2316 (1990).
    [Crossref]
  26. K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
    [Crossref] [PubMed]
  27. N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105, 136402 (2010).
    [Crossref]
  28. P. J. Allen, “A radiation torque experiment,” Am. J. Phys. 34, 1185–1192 (1966).
    [Crossref]
  29. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
    [Crossref]
  30. L. Allen, M. Babiker, and W. L. Power, “Azimuthal doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
    [Crossref]
  31. G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. 132, 8–14 (1996).
    [Crossref]
  32. I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational frequency shift,” Phys. Rev. Lett. 78, 2539–2542 (1997).
    [Crossref]
  33. G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
    [Crossref] [PubMed]
  34. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 3217–3219 (1998).
    [Crossref]
  35. L. Chen and W. She, “Sorting photons of different rotational doppler shifts (rds) by orbital angular momentum of single-photon with spin-orbit-rds entanglement,” Opt. Express 16, 14629–14634 (2008).
    [Crossref]

2015 (2)

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

2014 (5)

J. Chen, J. Ng, K. Ding, K. H. Fung, Z. Lin, and C. T. Chan, “Negative optical torque,” Sci. Rep. 42, 6386 (2014).
[Crossref]

D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nature Photon. 8, 610–614 (2014).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nature Commun. 5, 3300 (2014).
[Crossref]

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
[PubMed]

A. Canaguier-Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” Phys. Rev. A 89, 033841 (2014).
[Crossref]

2013 (3)

A. Dogariu, S. Sukhov, and J. J. Saenz, “Optically induced ’negative forces,”’ Nature Photon. 7, 24–27 (2013).
[Crossref]

O. Brzobohaty, V. Karasek, M. Siler, L. Chvatal, T. Cizmar, and P. Zemanek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ’tractor beam,”’ Nature Photon. 7, 123–127 (2013).
[Crossref]

V. Kajorndejnukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nature Photon. 7, 787–790 (2013).
[Crossref]

2012 (2)

2011 (1)

M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

2010 (1)

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105, 136402 (2010).
[Crossref]

2008 (2)

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref] [PubMed]

L. Chen and W. She, “Sorting photons of different rotational doppler shifts (rds) by orbital angular momentum of single-photon with spin-orbit-rds entanglement,” Opt. Express 16, 14629–14634 (2008).
[Crossref]

2007 (2)

S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in gaussian beams,” J. Opt. Soc. Am. A 24, 430–443 (2007).
[Crossref]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New. J. Phys. 9, 78 (2007).
[Crossref]

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

2003 (1)

Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultra-short light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref]

2002 (1)

1998 (4)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1–3 (1998).
[Crossref]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 3217–3219 (1998).
[Crossref]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[Crossref]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

1997 (1)

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational frequency shift,” Phys. Rev. Lett. 78, 2539–2542 (1997).
[Crossref]

1996 (1)

G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. 132, 8–14 (1996).
[Crossref]

1994 (1)

L. Allen, M. Babiker, and W. L. Power, “Azimuthal doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
[Crossref]

1990 (1)

F. Bretenaker and A. L. Floch, “Energy exchange between a rotating retardation plate and a laser beam,” Phys. Rev. Lett. 65, 2316 (1990).
[Crossref]

1988 (1)

R. Simon, H. J. kimble, and E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[Crossref] [PubMed]

1979 (1)

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979).
[Crossref]

1975 (1)

1966 (1)

P. J. Allen, “A radiation torque experiment,” Am. J. Phys. 34, 1185–1192 (1966).
[Crossref]

1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

Alfano, R. R.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref] [PubMed]

Allen, L.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[Crossref]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 3217–3219 (1998).
[Crossref]

L. Allen, M. Babiker, and W. L. Power, “Azimuthal doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
[Crossref]

Allen, P. J.

P. J. Allen, “A radiation torque experiment,” Am. J. Phys. 34, 1185–1192 (1966).
[Crossref]

Angelsky, O. V.

Arnold, S.

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979).
[Crossref]

Babiker, M.

L. Allen, M. Babiker, and W. L. Power, “Azimuthal doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
[Crossref]

Bekshaev, A. Y.

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nature Commun. 5, 3300 (2014).
[Crossref]

O. V. Angelsky, A. Y. Bekshaev, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Y. Zenkova, “Orbital rotation without orbital angular momentum: mechanical action of the spin part of the internal energy flow in light beams,” Opt. Express 20, 3563–3571 (2012).
[Crossref] [PubMed]

Beresna, M.

M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New. J. Phys. 9, 78 (2007).
[Crossref]

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

Bialynicka-Birula, Z.

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational frequency shift,” Phys. Rev. Lett. 78, 2539–2542 (1997).
[Crossref]

Bialynicki-Birula, I.

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational frequency shift,” Phys. Rev. Lett. 78, 2539–2542 (1997).
[Crossref]

Biener, G.

Bliokh, K. Y.

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nature Commun. 5, 3300 (2014).
[Crossref]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 2005).

Brasselet, E.

D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nature Photon. 8, 610–614 (2014).
[Crossref]

Bretenaker, F.

F. Bretenaker and A. L. Floch, “Energy exchange between a rotating retardation plate and a laser beam,” Phys. Rev. Lett. 65, 2316 (1990).
[Crossref]

Brzobohaty, O.

O. Brzobohaty, V. Karasek, M. Siler, L. Chvatal, T. Cizmar, and P. Zemanek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ’tractor beam,”’ Nature Photon. 7, 123–127 (2013).
[Crossref]

Canaguier-Durand, A.

A. Canaguier-Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” Phys. Rev. A 89, 033841 (2014).
[Crossref]

Chan, C. T.

J. Chen, J. Ng, K. Ding, K. H. Fung, Z. Lin, and C. T. Chan, “Negative optical torque,” Sci. Rep. 42, 6386 (2014).
[Crossref]

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
[PubMed]

Chen, J.

J. Chen, J. Ng, K. Ding, K. H. Fung, Z. Lin, and C. T. Chan, “Negative optical torque,” Sci. Rep. 42, 6386 (2014).
[Crossref]

Chen, L.

Chvatal, L.

O. Brzobohaty, V. Karasek, M. Siler, L. Chvatal, T. Cizmar, and P. Zemanek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ’tractor beam,”’ Nature Photon. 7, 123–127 (2013).
[Crossref]

Cizmar, T.

O. Brzobohaty, V. Karasek, M. Siler, L. Chvatal, T. Cizmar, and P. Zemanek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ’tractor beam,”’ Nature Photon. 7, 123–127 (2013).
[Crossref]

Courtial, J.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[Crossref]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 3217–3219 (1998).
[Crossref]

Dahan, N.

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105, 136402 (2010).
[Crossref]

Dholakia, K.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 3217–3219 (1998).
[Crossref]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[Crossref]

Ding, K.

J. Chen, J. Ng, K. Ding, K. H. Fung, Z. Lin, and C. T. Chan, “Negative optical torque,” Sci. Rep. 42, 6386 (2014).
[Crossref]

Ding, W.

V. Kajorndejnukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nature Photon. 7, 787–790 (2013).
[Crossref]

Dogariu, A.

V. Kajorndejnukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nature Photon. 7, 787–790 (2013).
[Crossref]

A. Dogariu, S. Sukhov, and J. J. Saenz, “Optically induced ’negative forces,”’ Nature Photon. 7, 24–27 (2013).
[Crossref]

Evans, S.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref] [PubMed]

Floch, A. L.

F. Bretenaker and A. L. Floch, “Energy exchange between a rotating retardation plate and a laser beam,” Phys. Rev. Lett. 65, 2316 (1990).
[Crossref]

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1–3 (1998).
[Crossref]

Frischwasser, K.

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105, 136402 (2010).
[Crossref]

Fung, K. H.

J. Chen, J. Ng, K. Ding, K. H. Fung, Z. Lin, and C. T. Chan, “Negative optical torque,” Sci. Rep. 42, 6386 (2014).
[Crossref]

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New. J. Phys. 9, 78 (2007).
[Crossref]

Garetz, B. A.

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979).
[Crossref]

Gecevicius, M.

M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

Genet, C.

A. Canaguier-Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” Phys. Rev. A 89, 033841 (2014).
[Crossref]

Gertus, T.

M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

Gorodetski, Y.

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105, 136402 (2010).
[Crossref]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref] [PubMed]

Hakobyan, D.

D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nature Photon. 8, 610–614 (2014).
[Crossref]

Hanna, S.

Hanson, S. G.

Hasman, E.

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105, 136402 (2010).
[Crossref]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref] [PubMed]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
[Crossref]

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1–3 (1998).
[Crossref]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

Hirao, K.

Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultra-short light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref]

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New. J. Phys. 9, 78 (2007).
[Crossref]

Kajorndejnukul, V.

V. Kajorndejnukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nature Photon. 7, 787–790 (2013).
[Crossref]

Karasek, V.

O. Brzobohaty, V. Karasek, M. Siler, L. Chvatal, T. Cizmar, and P. Zemanek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ’tractor beam,”’ Nature Photon. 7, 123–127 (2013).
[Crossref]

Kazansky, P. G.

M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultra-short light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref]

kimble, H. J.

R. Simon, H. J. kimble, and E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[Crossref] [PubMed]

Kleiner, V.

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105, 136402 (2010).
[Crossref]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref] [PubMed]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
[Crossref]

Lin, Z.

J. Chen, J. Ng, K. Ding, K. H. Fung, Z. Lin, and C. T. Chan, “Negative optical torque,” Sci. Rep. 42, 6386 (2014).
[Crossref]

Maksimyak, A. P.

Maksimyak, P. P.

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New. J. Phys. 9, 78 (2007).
[Crossref]

Milione, G.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref] [PubMed]

Ng, J.

J. Chen, J. Ng, K. Ding, K. H. Fung, Z. Lin, and C. T. Chan, “Negative optical torque,” Sci. Rep. 42, 6386 (2014).
[Crossref]

Nieminen, T. A.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1–3 (1998).
[Crossref]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

Nienhuis, G.

G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. 132, 8–14 (1996).
[Crossref]

Niv, A.

Nolan, D. A.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref] [PubMed]

Nori, F.

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nature Commun. 5, 3300 (2014).
[Crossref]

Padgett, M. J.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[Crossref]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 3217–3219 (1998).
[Crossref]

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Power, W. L.

L. Allen, M. Babiker, and W. L. Power, “Azimuthal doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
[Crossref]

Qiu, C.-W.

V. Kajorndejnukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nature Photon. 7, 787–790 (2013).
[Crossref]

Qiu, J.

Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultra-short light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New. J. Phys. 9, 78 (2007).
[Crossref]

Robertson, D. A.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[Crossref]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 3217–3219 (1998).
[Crossref]

Rubinsztein-Dunlop, H.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1–3 (1998).
[Crossref]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

Saenz, J. J.

A. Dogariu, S. Sukhov, and J. J. Saenz, “Optically induced ’negative forces,”’ Nature Photon. 7, 24–27 (2013).
[Crossref]

She, W.

Shimotsuma, Y.

Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultra-short light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref]

Siler, M.

O. Brzobohaty, V. Karasek, M. Siler, L. Chvatal, T. Cizmar, and P. Zemanek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ’tractor beam,”’ Nature Photon. 7, 123–127 (2013).
[Crossref]

Simon, R.

R. Simon, H. J. kimble, and E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[Crossref] [PubMed]

Simpson, S. H.

Somargren, G. E.

Sudarshan, E. C. G.

R. Simon, H. J. kimble, and E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[Crossref] [PubMed]

Sukhov, S.

V. Kajorndejnukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nature Photon. 7, 787–790 (2013).
[Crossref]

A. Dogariu, S. Sukhov, and J. J. Saenz, “Optically induced ’negative forces,”’ Nature Photon. 7, 24–27 (2013).
[Crossref]

Wang, S. B.

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
[PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 2005).

Zemanek, P.

O. Brzobohaty, V. Karasek, M. Siler, L. Chvatal, T. Cizmar, and P. Zemanek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ’tractor beam,”’ Nature Photon. 7, 123–127 (2013).
[Crossref]

Zenkova, C. Y.

Am. J. Phys. (1)

P. J. Allen, “A radiation torque experiment,” Am. J. Phys. 34, 1185–1192 (1966).
[Crossref]

Appl. Phys. Lett. (1)

M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nat. Commun. (1)

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
[PubMed]

Nature (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

Nature Commun. (1)

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nature Commun. 5, 3300 (2014).
[Crossref]

Nature Photon. (4)

D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nature Photon. 8, 610–614 (2014).
[Crossref]

A. Dogariu, S. Sukhov, and J. J. Saenz, “Optically induced ’negative forces,”’ Nature Photon. 7, 24–27 (2013).
[Crossref]

O. Brzobohaty, V. Karasek, M. Siler, L. Chvatal, T. Cizmar, and P. Zemanek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ’tractor beam,”’ Nature Photon. 7, 123–127 (2013).
[Crossref]

V. Kajorndejnukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nature Photon. 7, 787–790 (2013).
[Crossref]

New. J. Phys. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New. J. Phys. 9, 78 (2007).
[Crossref]

Opt. Commun. (3)

L. Allen, M. Babiker, and W. L. Power, “Azimuthal doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
[Crossref]

G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. 132, 8–14 (1996).
[Crossref]

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rep. (1)

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

Phys. Rev. (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

Phys. Rev. A (1)

A. Canaguier-Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” Phys. Rev. A 89, 033841 (2014).
[Crossref]

Phys. Rev. Lett. (10)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultra-short light pulses,” Phys. Rev. Lett. 91, 247405 (2003).
[Crossref]

R. Simon, H. J. kimble, and E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[Crossref] [PubMed]

F. Bretenaker and A. L. Floch, “Energy exchange between a rotating retardation plate and a laser beam,” Phys. Rev. Lett. 65, 2316 (1990).
[Crossref]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref] [PubMed]

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105, 136402 (2010).
[Crossref]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[Crossref]

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational frequency shift,” Phys. Rev. Lett. 78, 2539–2542 (1997).
[Crossref]

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref] [PubMed]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 3217–3219 (1998).
[Crossref]

Phys. Rev. X (1)

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse Spin and Momentum in Two-Wave Interference,” Phys. Rev. X 5, 011039 (2015).

Sci. Rep. (1)

J. Chen, J. Ng, K. Ding, K. H. Fung, Z. Lin, and C. T. Chan, “Negative optical torque,” Sci. Rep. 42, 6386 (2014).
[Crossref]

Other (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, 2005).

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Figures (6)

Fig. 1
Fig. 1

Illustration of intuitive and counter-intuitive mechanical manifestations of an optical torque per photon τz exerted on matter by an incident Gaussian beam carrying (spin) optical angular momentum sz per photon along its propagation direction z. (a) Right-handed situation, szτz > 0 whatever sz. (b) Left-handed situation, szτz < 0 whatever sz.

Fig. 2
Fig. 2

Design and characterization of azimuthally varying half-wave plate retarders of azimuthal order m = 1/2 (upper row), 1 (middle row) and 3/2 (bottom row) used in this work, with 532 nm operating wavelength. For each m: the left panel shows slow-axis spatial distribution in the (x, y) plane, where segments refer to the local slow-axis local orientation; the middle panel is the image of the inhomogeneous retarder observed between crossed linear polarizers whose direction are indicated as white cross in panel (d); the right panel displays output intensity profile for circularly polarized incident Gaussian beam, which corresponds to an optical vortex beam of topological charge ±2m where the sign depends on the incident polarization state handedness.

Fig. 3
Fig. 3

Sketch of the two-beam rotational Doppler experimental set-up used to videorecord the interference pattern between (i) the frequency shifted output beam emerging from the sample, labeled c σ ( ω + 2 σ Ω ( m 1 ) ), illuminated by collimated circularly polarized Gaussian beam with waist radius w ≈ 1 mm and helicity σ, c σ ( ω ), and (ii) a collinear reference beam with helicity −σ, c σ ( ω ). Optical elements: P is linear polarizer, HWP is half-wave plate, QWP is quarter-wave plate, QP is q-plate, M is mirror, PBS is polarizing beam splitter, NPBS is non-polarizing beam splitter, CCD is imaging device.

Fig. 4
Fig. 4

Two-beam rotational Doppler experimental results for m = 1/2 (upper row), m = 1 (middle row) and m = 3/2 (bottom row) by using the set-up depicted in Fig. 3. For each m: left panel displays a snapshot of the interference intensity pattern observed in the (x, y) plane at t = 0; middle panel shows the time dependence of the correlation coefficient C between intensity patterns at t = 0 and t where the solid line (for m = 1/2 and 3/2) refers to fit by sinusoidal function, see text for details; right panel is a volumetric representation of the recorded interferogram in the (x, y, t) spatiotemporal frame during T = 120 s.

Fig. 5
Fig. 5

Single-beam rotational Doppler experimental sketch that aims at videorecording the interference pattern between the two frequency shifted circular components c σ ( ω + 2 σ Ω ( 1 m ) ), owing to the output polarizer oriented along the y axis for linearly polarized incident Gaussian beam along the x axis with waist radius w ≈ 1 mm. Optical elements: Pin/out are input/output linear polarizers, QP is q-plate, and CCD is imaging device.

Fig. 6
Fig. 6

Single-beam rotational Doppler experiment for m = 1/2 (left column), m = 1 (middle column) and m = 3/2 (right column) following the set-up shown in Fig. 5. Snapshot of the intensity profile in the (x, y) plane of the linearly polarized output component that is orthogonal to that of a linearly polarized incident Gaussian beam is shown in the inset of each panel whereas the spatiotemporal behavior are represented with volumetric rendering over a duration that corresponds to two full rotations of the sample, T ≈ 30 s. Upper row: experiment. Bottom row: model.

Equations (11)

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J static ( m ) = ( 0 e + 2 i m ϕ e 2 i m ϕ 0 ) .
τ z ( σ ) = τ z , spin ( σ ) + τ z , orbital ( σ ) ,
τ z ( σ ) = 2 σ ( 1 m ) .
τ z ( χ ) = 1 2 [ ( 1 + sin 2 χ ) τ z ( + ) + ( 1 sin 2 χ ) τ z ( ) ] ,
τ z ( χ ) = 2 sin 2 χ ( 1 m ) .
δ ω = 2 σ Ω ( m 1 ) .
J dynamic ( m ) = ( 0 e + 2 i [ m ϕ + ( 1 m ) Ω t ] e 2 i [ m ϕ + ( 1 m ) Ω t ] 0 ) .
E after QP ( m , Ω = 0 ) = J static ( m ) ( 1 / 2 , 1 / 2 ) T = cos ( 2 m ϕ ) x + sin ( 2 m ϕ ) y .
E after QP ( m , Ω ) = J dynamic ( m ) ( 1 / 2 , 1 / 2 ) T , = cos ( 2 [ m ϕ + ( 1 m ) Ω t ) ] x + sin ( 2 [ m ϕ + ( 1 m ) Ω t ) ] y ,
E after P out ( m , Ω ) = sin ( 2 [ m ϕ + ( 1 m ) Ω t ) ] y .
I ( r , t ) = ( r / w ) 4 | m | exp ( 2 r 2 / w 2 ) sin 2 ( 2 [ m ϕ + ( 1 m ) Ω t ] ) ,

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