Abstract

We use the angular Doppler-effect to obtain stable frequency shifts from below one Hertz to hundreds of Hertz in the optical domain, constituting a control of 1 part in 1014. For the first time, we use these very small frequency shifts to create continuous motion in interference patterns including the scanning of linear fringe patterns and the rotation of the interference pattern formed from a Laguerre-Gaussian beam. This enables controlled lateral and rotational movement of trapped particles.

© 2002 Optical Society of America

Full Article  |  PDF Article
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References

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  1. D. Haubrich, M. Dornseifer, and R. Wynands, “Lossless beam combiners for nearly equal laser frequencies,” Rev. Sci. Inst. 71, 338 (2000).
    [Crossref]
  2. K. MacAdam, A. Steinbach, and C. E. Wieman, “A Narrow-Band Tunable Diode-Laser System with Grating Feedback, and a Saturated Absorption Spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098 (1992).
    [Crossref]
  3. S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278 (2001).
    [Crossref] [PubMed]
  4. A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7 (1997).
    [Crossref]
  5. M. MacDonald, L. Paterson, W. Sibbett, P. E. Bryant, and K. Dholakia, “Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,” Opt. Lett. 26, 863 (2001).
    [Crossref]
  6. M. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21 (2002).
    [Crossref]
  7. M. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101 (2002).
    [Crossref] [PubMed]
  8. L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled Rotation of Optically Trapped Microscopic Particles,” Science 292, 912 (2001).
    [Crossref] [PubMed]
  9. B. A. Garetz, “Angular Doppler Effect,” JOSA Lett. 71, 609 (1981).
    [Crossref]
  10. I. Bialynicki-Birula and Z. Bialynicki-Birula, “Rotational Frequency Shift,” Phys. Rev. Lett. 78, 2539 (1997).
    [Crossref]
  11. 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 (1988).
    [Crossref] [PubMed]
  12. D. Schrader, S. Kuhr, W. Alt, M. Müller, V. Gomer, and D. Meschede, “An optical conveyer belt for single neutral atoms,” Appl. Phys. B 73, 819 (2001).
    [Crossref]
  13. P. Hariharan and B. Ward, “Interferometry and the Doppler effect: An experimental verification,” J. Mod. Opt. 44, 221 (1997).
    [Crossref]
  14. C. F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by the electro-optic effect,” Appl. Phys. Lett. 1, 46 (1962).
    [Crossref]

2002 (2)

M. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21 (2002).
[Crossref]

M. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101 (2002).
[Crossref] [PubMed]

2001 (4)

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled Rotation of Optically Trapped Microscopic Particles,” Science 292, 912 (2001).
[Crossref] [PubMed]

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278 (2001).
[Crossref] [PubMed]

D. Schrader, S. Kuhr, W. Alt, M. Müller, V. Gomer, and D. Meschede, “An optical conveyer belt for single neutral atoms,” Appl. Phys. B 73, 819 (2001).
[Crossref]

M. MacDonald, L. Paterson, W. Sibbett, P. E. Bryant, and K. Dholakia, “Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,” Opt. Lett. 26, 863 (2001).
[Crossref]

2000 (1)

D. Haubrich, M. Dornseifer, and R. Wynands, “Lossless beam combiners for nearly equal laser frequencies,” Rev. Sci. Inst. 71, 338 (2000).
[Crossref]

1997 (3)

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7 (1997).
[Crossref]

P. Hariharan and B. Ward, “Interferometry and the Doppler effect: An experimental verification,” J. Mod. Opt. 44, 221 (1997).
[Crossref]

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Rotational Frequency Shift,” Phys. Rev. Lett. 78, 2539 (1997).
[Crossref]

1992 (1)

K. MacAdam, A. Steinbach, and C. E. Wieman, “A Narrow-Band Tunable Diode-Laser System with Grating Feedback, and a Saturated Absorption Spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098 (1992).
[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 (1988).
[Crossref] [PubMed]

1981 (1)

B. A. Garetz, “Angular Doppler Effect,” JOSA Lett. 71, 609 (1981).
[Crossref]

1962 (1)

C. F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by the electro-optic effect,” Appl. Phys. Lett. 1, 46 (1962).
[Crossref]

Alt, W.

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278 (2001).
[Crossref] [PubMed]

D. Schrader, S. Kuhr, W. Alt, M. Müller, V. Gomer, and D. Meschede, “An optical conveyer belt for single neutral atoms,” Appl. Phys. B 73, 819 (2001).
[Crossref]

Arlt, J.

M. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21 (2002).
[Crossref]

M. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101 (2002).
[Crossref] [PubMed]

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled Rotation of Optically Trapped Microscopic Particles,” Science 292, 912 (2001).
[Crossref] [PubMed]

Baird, D.

C. F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by the electro-optic effect,” Appl. Phys. Lett. 1, 46 (1962).
[Crossref]

Berns, M. W.

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7 (1997).
[Crossref]

Bialynicki-Birula, I.

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Rotational Frequency Shift,” Phys. Rev. Lett. 78, 2539 (1997).
[Crossref]

Bialynicki-Birula, Z.

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Rotational Frequency Shift,” Phys. Rev. Lett. 78, 2539 (1997).
[Crossref]

Bryant, P. E.

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled Rotation of Optically Trapped Microscopic Particles,” Science 292, 912 (2001).
[Crossref] [PubMed]

M. MacDonald, L. Paterson, W. Sibbett, P. E. Bryant, and K. Dholakia, “Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,” Opt. Lett. 26, 863 (2001).
[Crossref]

Buhrer, C. F.

C. F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by the electro-optic effect,” Appl. Phys. Lett. 1, 46 (1962).
[Crossref]

Chiou, A. E.

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7 (1997).
[Crossref]

Conwell, E. M.

C. F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by the electro-optic effect,” Appl. Phys. Lett. 1, 46 (1962).
[Crossref]

Dholakia, K.

M. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101 (2002).
[Crossref] [PubMed]

M. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21 (2002).
[Crossref]

M. MacDonald, L. Paterson, W. Sibbett, P. E. Bryant, and K. Dholakia, “Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,” Opt. Lett. 26, 863 (2001).
[Crossref]

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled Rotation of Optically Trapped Microscopic Particles,” Science 292, 912 (2001).
[Crossref] [PubMed]

Dornseifer, M.

D. Haubrich, M. Dornseifer, and R. Wynands, “Lossless beam combiners for nearly equal laser frequencies,” Rev. Sci. Inst. 71, 338 (2000).
[Crossref]

Garetz, B. A.

B. A. Garetz, “Angular Doppler Effect,” JOSA Lett. 71, 609 (1981).
[Crossref]

Gomer, V.

D. Schrader, S. Kuhr, W. Alt, M. Müller, V. Gomer, and D. Meschede, “An optical conveyer belt for single neutral atoms,” Appl. Phys. B 73, 819 (2001).
[Crossref]

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278 (2001).
[Crossref] [PubMed]

Hariharan, P.

P. Hariharan and B. Ward, “Interferometry and the Doppler effect: An experimental verification,” J. Mod. Opt. 44, 221 (1997).
[Crossref]

Haubrich, D.

D. Haubrich, M. Dornseifer, and R. Wynands, “Lossless beam combiners for nearly equal laser frequencies,” Rev. Sci. Inst. 71, 338 (2000).
[Crossref]

Hong, J.

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7 (1997).
[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 (1988).
[Crossref] [PubMed]

Kuhr, S.

D. Schrader, S. Kuhr, W. Alt, M. Müller, V. Gomer, and D. Meschede, “An optical conveyer belt for single neutral atoms,” Appl. Phys. B 73, 819 (2001).
[Crossref]

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278 (2001).
[Crossref] [PubMed]

MacAdam, K.

K. MacAdam, A. Steinbach, and C. E. Wieman, “A Narrow-Band Tunable Diode-Laser System with Grating Feedback, and a Saturated Absorption Spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098 (1992).
[Crossref]

MacDonald, M.

M. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21 (2002).
[Crossref]

M. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101 (2002).
[Crossref] [PubMed]

M. MacDonald, L. Paterson, W. Sibbett, P. E. Bryant, and K. Dholakia, “Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,” Opt. Lett. 26, 863 (2001).
[Crossref]

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled Rotation of Optically Trapped Microscopic Particles,” Science 292, 912 (2001).
[Crossref] [PubMed]

Meschede, D.

D. Schrader, S. Kuhr, W. Alt, M. Müller, V. Gomer, and D. Meschede, “An optical conveyer belt for single neutral atoms,” Appl. Phys. B 73, 819 (2001).
[Crossref]

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278 (2001).
[Crossref] [PubMed]

Müller, M.

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278 (2001).
[Crossref] [PubMed]

D. Schrader, S. Kuhr, W. Alt, M. Müller, V. Gomer, and D. Meschede, “An optical conveyer belt for single neutral atoms,” Appl. Phys. B 73, 819 (2001).
[Crossref]

Paterson, L.

M. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21 (2002).
[Crossref]

M. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101 (2002).
[Crossref] [PubMed]

M. MacDonald, L. Paterson, W. Sibbett, P. E. Bryant, and K. Dholakia, “Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,” Opt. Lett. 26, 863 (2001).
[Crossref]

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled Rotation of Optically Trapped Microscopic Particles,” Science 292, 912 (2001).
[Crossref] [PubMed]

Schrader, D.

D. Schrader, S. Kuhr, W. Alt, M. Müller, V. Gomer, and D. Meschede, “An optical conveyer belt for single neutral atoms,” Appl. Phys. B 73, 819 (2001).
[Crossref]

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278 (2001).
[Crossref] [PubMed]

Sibbett, W.

M. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101 (2002).
[Crossref] [PubMed]

M. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21 (2002).
[Crossref]

M. MacDonald, L. Paterson, W. Sibbett, P. E. Bryant, and K. Dholakia, “Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,” Opt. Lett. 26, 863 (2001).
[Crossref]

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled Rotation of Optically Trapped Microscopic Particles,” Science 292, 912 (2001).
[Crossref] [PubMed]

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 (1988).
[Crossref] [PubMed]

Sonek, G. J.

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7 (1997).
[Crossref]

Steinbach, A.

K. MacAdam, A. Steinbach, and C. E. Wieman, “A Narrow-Band Tunable Diode-Laser System with Grating Feedback, and a Saturated Absorption Spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098 (1992).
[Crossref]

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 (1988).
[Crossref] [PubMed]

Volke-Sepulveda, K.

M. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21 (2002).
[Crossref]

M. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101 (2002).
[Crossref] [PubMed]

Wang, W.

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7 (1997).
[Crossref]

Ward, B.

P. Hariharan and B. Ward, “Interferometry and the Doppler effect: An experimental verification,” J. Mod. Opt. 44, 221 (1997).
[Crossref]

Wieman, C. E.

K. MacAdam, A. Steinbach, and C. E. Wieman, “A Narrow-Band Tunable Diode-Laser System with Grating Feedback, and a Saturated Absorption Spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098 (1992).
[Crossref]

Wynands, R.

D. Haubrich, M. Dornseifer, and R. Wynands, “Lossless beam combiners for nearly equal laser frequencies,” Rev. Sci. Inst. 71, 338 (2000).
[Crossref]

Am. J. Phys. (1)

K. MacAdam, A. Steinbach, and C. E. Wieman, “A Narrow-Band Tunable Diode-Laser System with Grating Feedback, and a Saturated Absorption Spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098 (1992).
[Crossref]

Appl. Phys. B (1)

D. Schrader, S. Kuhr, W. Alt, M. Müller, V. Gomer, and D. Meschede, “An optical conveyer belt for single neutral atoms,” Appl. Phys. B 73, 819 (2001).
[Crossref]

Appl. Phys. Lett. (1)

C. F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by the electro-optic effect,” Appl. Phys. Lett. 1, 46 (1962).
[Crossref]

J. Mod. Opt. (1)

P. Hariharan and B. Ward, “Interferometry and the Doppler effect: An experimental verification,” J. Mod. Opt. 44, 221 (1997).
[Crossref]

JOSA Lett. (1)

B. A. Garetz, “Angular Doppler Effect,” JOSA Lett. 71, 609 (1981).
[Crossref]

Opt. Commun. (2)

M. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21 (2002).
[Crossref]

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7 (1997).
[Crossref]

Opt. Lett. (1)

Phys. Rev. Lett. (2)

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Rotational Frequency Shift,” Phys. Rev. Lett. 78, 2539 (1997).
[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 (1988).
[Crossref] [PubMed]

Rev. Sci. Inst. (1)

D. Haubrich, M. Dornseifer, and R. Wynands, “Lossless beam combiners for nearly equal laser frequencies,” Rev. Sci. Inst. 71, 338 (2000).
[Crossref]

Science (3)

M. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101 (2002).
[Crossref] [PubMed]

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled Rotation of Optically Trapped Microscopic Particles,” Science 292, 912 (2001).
[Crossref] [PubMed]

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278 (2001).
[Crossref] [PubMed]

Supplementary Material (2)

» Media 1: MPEG (1711 KB)     
» Media 2: MPEG (1470 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Adapted Mach-Zehnder interferometer for creating two co-propagating laser beams with a frequency shift between them of 2Ωrot. PBS, polarising beam splitter; M, mirror; λ/2, half wave plate; Rλ/2, rotating λ/2; BS, 50:50 beam splitter; +ħ , right hand circularly polarised light; - ħ , left hand circularly polarised light; Ωrot, rotation frequency of half-wave plate.

Fig. 2.
Fig. 2.

The beat signal produced by interfering two Gaussian beams separated in frequency using the angular Doppler effect.

Fig. 3.
Fig. 3.

1 μm diameter silica spheres moving from left to right as the linear fringes of an interference pattern are scanned using the angular Doppler effect. The particles are trapped (a) along the bright fringes and then move to the right (b)-(d). Continuous motion of the pattern amalgamates all of these spheres on the right hand side of the pattern region (d).

Fig. 4
Fig. 4

Rotating interference patterns: patterns produced in order between l = 1 + l = -1, l = 2 + l = -2 and l = 3 + l = -3 [see video-clip: arlt1.m1v (1,711 KB)].

Fig. 5
Fig. 5

a 5 μm long glass rod trapped in the interference pattern between one Laguerre-Gaussian beam with l = 1 and one with l = -1, is rotated continuously in water using the angular Doppler-effect [see video-clip: arlt2.m1v (1,470 KB)].

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

ħ + I Ω rot = ħ + I Ω rot /
ħ ω 1 + L 1 2 2 I = ħ ω 2 + L 2 2 2 I
Δ ω = ω 1 ω 2 = 2 Ω rot
V = 2 Ω λ sin α
E 1 ( r ̅ ) = E 01 p z exp [ i ( + kz ωt ) ] ;
E 2 ( r ̅ ) = E 02 p z exp [ i ( + kz ( ω + Δ ω ) t ) ]
I ( r ̅ ) = E 1 ( r ̅ ) + E 2 ( r ̅ ) 2
= E 01 2 + E 02 2 + 2 Re { E 01 E 02 * } cos ( 2 l φ + ( Δ ω ) t )
I ( p , φ , z , t ) 2 I 0 ( p , z ) [ 1 + cos ( 2 l φ + ( Δ ω ) t ) ]
= 4 I 0 p z cos 2 ( l φ + ( Δ ω 2 ) t ) .
I ( p , φ , z , t ) = 4 I 0 p z cos 2 [ ψ φ t ]
dt = ( ψ t ) φ ( ψ φ ) t = Ω rot l
E 2 ( r ̅ ) = E 02 p z exp [ i ( kz ( ω + Δ ω ) t ) ] .
I ( r ̅ ) = E 01 2 + E 02 2 + 2 Re { E 01 E 02 * } cos ( + 2 Ω rot t )
dt = 2 Ω rot l

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