Abstract

Mechanical action caused by the optical forces connected with the canonical momentum density associated with the local wavevector or Belinfante’s spin angular momentum is experimentally verified. The helicity-dependent and the helicity-independent forces determined by spin momenta of different nature open attractive prospects for the use of optical structures for manipulating minute quantities of matter of importance in nanophysics, nanooptics and nanotechnologies, precision chemistry and pharmacology and in numerous other areas. Investigations in this area reveal new, extraordinary manifestations of optical forces, including the helicity-independent force caused by the transverse helicity-independent spin or vertical spin of a diagonally polarized wave, which was not observed and exploited up to recently. The main finding of our study consists in a direct experimental demonstration of the physical existence and mechanical action of this recently discovered extraordinary transverse component of the spin here arising in an evanescent light wave due to the total internal reflection of a linearly polarized probing beam with azimuthal angle 45° at the interface between the birefringent plate and air, which is oriented perpendicularly to the wavevector of an evanescent wave and localized over the boundary of the transparent media with polarization-dependent refraction indices.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
  9. A. Ashkin and J. P. Gordon, “Stability of radiation-pressure particle traps: an optical Earnshaw theorem,” Opt. Lett. 8(10), 511–513 (1983).
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  13. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
    [Crossref]
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  16. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
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    [Crossref]
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    [Crossref]
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2016 (1)

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[Crossref]

2015 (3)

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9(12), 789–795 (2015).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

2014 (1)

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

2013 (3)

S. M. Barnett and M. V. Berry, “Super weak momentum transfer near optical vortices,” J. Opt. 15(12), 125701 (2013).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum, and angular momentum,” New J. Phys. 15(3), 033026 (2013).
[Crossref]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonic well-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. Rap. Publ. 8, 13032 (2013).
[Crossref]

2012 (3)

2009 (2)

M.V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[Crossref] [PubMed]

2005 (1)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

2000 (2)

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000).
[Crossref]

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184(1–4), 67–71 (2000).
[Crossref]

1998 (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(6691), 348–350 (1998).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1986 (1)

H. C. Ohanian, “What is spin?” Am. J. Phys. 54(6), 500–505 (1986).
[Crossref]

1983 (1)

1936 (1)

R. A. Beth, “Mechanical Detection and Measurement of the Angular Momentum of Light,” Phys. Rev. 50(2), 115–125 (1936).
[Crossref]

Aiello, A.

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9(12), 789–795 (2015).
[Crossref]

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonic well-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. Rap. Publ. 8, 13032 (2013).
[Crossref]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[Crossref] [PubMed]

Allen, L.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184(1–4), 67–71 (2000).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Angelsky, O. V.

Antognozzi, M.

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[Crossref]

Ashkin, A.

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000).
[Crossref]

A. Ashkin and J. P. Gordon, “Stability of radiation-pressure particle traps: an optical Earnshaw theorem,” Opt. Lett. 8(10), 511–513 (1983).
[Crossref] [PubMed]

Banzer, P.

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9(12), 789–795 (2015).
[Crossref]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonic well-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. Rap. Publ. 8, 13032 (2013).
[Crossref]

Barnett, S. M.

S. M. Barnett and M. V. Berry, “Super weak momentum transfer near optical vortices,” J. Opt. 15(12), 125701 (2013).
[Crossref]

Bauer, T.

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonic well-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. Rap. Publ. 8, 13032 (2013).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Bekshaev, A. Y.

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[Crossref]

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

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum, and angular momentum,” New J. Phys. 15(3), 033026 (2013).
[Crossref]

Bekshaev, A. Ya.

Bermingham, C. R.

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[Crossref]

Berry, M. V.

S. M. Barnett and M. V. Berry, “Super weak momentum transfer near optical vortices,” J. Opt. 15(12), 125701 (2013).
[Crossref]

Berry, M.V.

M.V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).

Beth, R. A.

R. A. Beth, “Mechanical Detection and Measurement of the Angular Momentum of Light,” Phys. Rev. 50(2), 115–125 (1936).
[Crossref]

Bliokh, K. Y.

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

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

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum, and angular momentum,” New J. Phys. 15(3), 033026 (2013).
[Crossref]

K. Y. Bliokh and F. Nori, “Transverse spin of a surface plasmon,” Phys. Rev. A 85(6), 061801 (2012).
[Crossref]

Dennis, M. R.

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[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(6691), 348–350 (1998).
[Crossref]

Gordon, J. P.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

Hanson, S. G.

Harniman, R. L.

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[Crossref]

Hayward, R.

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[Crossref]

Heckenberg, N. R.

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(6691), 348–350 (1998).
[Crossref]

Herzig, H. P.

Hoerber, H.

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[Crossref]

Leuchs, G.

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9(12), 789–795 (2015).
[Crossref]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonic well-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. Rap. Publ. 8, 13032 (2013).
[Crossref]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[Crossref] [PubMed]

Lindlein, N.

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonic well-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. Rap. Publ. 8, 13032 (2013).
[Crossref]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[Crossref] [PubMed]

Maksimyak, A. P.

Maksimyak, P. P.

Marquardt, C.

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonic well-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. Rap. Publ. 8, 13032 (2013).
[Crossref]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[Crossref] [PubMed]

Mokhun, I. I.

Neugebauer, M.

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9(12), 789–795 (2015).
[Crossref]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonic well-demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. Rap. Publ. 8, 13032 (2013).
[Crossref]

Nieminen, T. A.

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(6691), 348–350 (1998).
[Crossref]

Nori, F.

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

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

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum, and angular momentum,” New J. Phys. 15(3), 033026 (2013).
[Crossref]

K. Y. Bliokh and F. Nori, “Transverse spin of a surface plasmon,” Phys. Rev. A 85(6), 061801 (2012).
[Crossref]

Ohanian, H. C.

H. C. Ohanian, “What is spin?” Am. J. Phys. 54(6), 500–505 (1986).
[Crossref]

Padgett, M. J.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184(1–4), 67–71 (2000).
[Crossref]

Rockstuhl, C.

Rodríguez-Fortuño, F. J.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

Rubinsztein-Dunlop, H.

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(6691), 348–350 (1998).
[Crossref]

Senior, J.

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[Crossref]

Simpson, S.

M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12(8), 731–735 (2016).
[Crossref]

Spreeuw, R. J.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Tyurin, A. V.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Zayats, A. V.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

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Supplementary Material (4)

NameDescription
» Visualization 1: MP4 (601 KB)      Visualization 1
» Visualization 2: MP4 (3784 KB)      Visualization 2
» Visualization 3: MP4 (535 KB)      Visualization 3
» Visualization 4: MP4 (1432 KB)      Visualization 4

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Figures (6)

Fig. 1
Fig. 1

Experimental arrangement: 1 – glass cut rectangular prism (n = 1.52); 2 – 2 mm-height ring cuvette; 3 – water (n = 1.33); 4 – plate with gold nanoparicles at its upper surface; 5 – white-light illuminating source; 6 – ССD-camera.

Fig. 2
Fig. 2

Notations for the analysis of the propagation of a wave in a birefringent plane-parallel microplate with gold nanoparticles at upper surface. Lower part: O O – optical axis of a plate: γ– angle of incidence at surface 2 (interface plate-air), φ is the azimuth of polarization of the probing beam, θ– is the angle between optical axis of a plate and the azimuth of polarization of the wave 45° impinging on a surface 2. Upper part represents distribution of forces in the evanescent wave acting on gold particles deposited at the surface of birefringent plate: F oz – force caused by the canonical orbital momentum, F sz – longitudinal component of a force caused by the vertical spin momentum, F z – longitudinal resulting optical force, F y ( F sy ) – transversal component of a force caused by the vertical spin momentum, F r – resulting optical force in k -direction causing rectilinear motion of the plate. The angle α between directions z and k , in our case is about 15°.

Fig. 3
Fig. 3

Dependence of the torque on the angle of incidenceγof the beam impinging on the second surface of a plate, for which one of the plate axes coincides with the azimuth of polarization of the incident beam 45°: Curves 1 and 2 correspond to the torque of the beam propagating to the boundary plate-air and from this boundary, respectively; curve 3 represents the resulting torque.

Fig. 4
Fig. 4

The resulting force in the z-direction ( F z ), the transverse diagonal polarization-dependent force induced by the vertical spin momentum in the y-direction ( F y ) and the ratio of optical forces in the longitudinal and transversal directions ( F z / F y ) as a function of the incidence angleγ.

Fig. 5
Fig. 5

Motion of a plate illuminated by laser radiation: rotation and rectilinear motion of a plate with 1W-power laser radiation (a); returning the plate into initial position when the laser is turned off (b) (see Visualization 2).

Fig. 6
Fig. 6

Rotation and rectilinear motion of a plate at an angle of 15°, approximately, to the z-direction induced by 1W-power laser beam impinging on a plate at an angle of 58° (see Visualization 4).

Equations (5)

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< T >=[ ε 0 E E * + 1 μ 0 B B * 1 2 ( ε 0 | E | 2 + 1 μ 0 | B | 2 )I]
E inc12 =( y E y (cosΘcosΦ+isinΦcosΘ)exp(ik n o d/cosγ)exp(kκd/cosγ)+ z E z (sinΘcosΦicosΘsinΦ)exp(ik n e d/cosγ)exp(kκd/cosγ) )exp(iωt)
E inc21 =( y E y (cosΘcosΦ+isinΦsinΘ)exp(i2k n o d/cosγ)exp(iδ)exp(kκd/cosγ)+ z E z (sinΘcosΦicosΘsinΦ)exp(i2k n e d/cosγ)exp(kκd/cosγ) )exp(iωt)
E ev =Eexp(iωt)( x 1 1+ | m | 2 + y m 1+ | m | 2 k k z + z (i) 1 1+ | m | 2 κ k z )exp(i k z zκx),
p z = p oz + p sz = A 2 8πω [ ( k z + m 2 k 2 k z + κ 2 k z )2 κ 2 k z ]exp(2κx)

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