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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 25 — Dec. 7, 2009
  • pp: 23316–23322
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Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized Gaussian beam

Yiqiong Zhao, David Shapiro, David Mcgloin, Daniel T. Chiu, and Stefano Marchesini  »View Author Affiliations


Optics Express, Vol. 17, Issue 25, pp. 23316-23322 (2009)
http://dx.doi.org/10.1364/OE.17.023316


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Abstract

It is well known that a circularly polarized Gaussian beam carries spin angular momentum, but not orbital angular momentum. This paper demonstrates that focusing a beam carrying spin angular momentum can induce an orbital angular momentum which we used to drive the orbital motion of a micron-sized metal particle that is trapped off the beam axis. The direction of the orbital motion is controlled by the handedness of the circular polarization. The orbiting dynamics of the trapped particle, which acted as an optical micro-detector, were quantitatively measured and found to be in excellent agreement with the theoretical predictions.

© 2009 OSA

1. Introduction

It is well known that circularly polarized laser beams carry spin angular momentum. Each photon of circularly polarized light is assigned an angular momentum of ±ħ. As shown by Beth in 1936 [1

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

], when circularly polarized light passes through a birefringent wave plate a torque can be observed. This is due to the transfer of angular momentum between the light and the plate as the light alters it polarization state. There is another type of angular momentum carried by light, orbital angular momentum, which is usually associated with beams with helical phase structure exp(ilϕ), where l is the winding number of the helix and φ is the azimuthal coordinate. The use of optical tweezers in which microscopic particles can be trapped in tightly focused laser beams has been used extensively to study the angular momentum of light. Spinning or orbiting motion of a trapped particle has been observed when spin or orbital angular momentum is transferred to the particle, respectively. In 2002, O’Neil et al. [2

2. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. 88(5), 053601 ( 2002). [CrossRef] [PubMed]

] experimentally distinguished orbital angular momentum from spin angular momentum when light with both types of momentum interacted with matter. Spin angular momentum is always intrinsic while orbital angular momentum may be either extrinsic or intrinsic. When particles are trapped away from the beam axis, the orbital angular momentum is associated with their orbital motion and is extrinsic, while the spin angular momentum is intrinsic and is associated with the particles’ spinning motion.

Recently, several theoretical calculations have shown that strongly focusing a circularly polarized laser beam would induce an additional helical phase structure into the longitudinal component of the electric field near the focus [3

3. T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A, Pure Appl. Opt. 10(11), 115005 ( 2008). [CrossRef]

5

5. Y. Zhao, J. S. Edgar, G. D. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-Orbital Angular Momentum Conversion in a Strongly Focused Optical Beam,” Phys. Rev. Lett. 99(7), 073901 ( 2007). [CrossRef] [PubMed]

]. Experimentally, we also observed that when scattering particles were trapped by a strongly focused circularly polarized Laguerre-Gaussian (LG) beam with a helical phase wave front, the orbital rotation rates of trapped particles are altered when changing the handedness of the polarization [5

5. Y. Zhao, J. S. Edgar, G. D. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-Orbital Angular Momentum Conversion in a Strongly Focused Optical Beam,” Phys. Rev. Lett. 99(7), 073901 ( 2007). [CrossRef] [PubMed]

]. Though the theoretical calculations showed that difference should also appear in a focused Gaussian beam trap with input plane wavefronts [3

3. T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A, Pure Appl. Opt. 10(11), 115005 ( 2008). [CrossRef]

,5

5. Y. Zhao, J. S. Edgar, G. D. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-Orbital Angular Momentum Conversion in a Strongly Focused Optical Beam,” Phys. Rev. Lett. 99(7), 073901 ( 2007). [CrossRef] [PubMed]

], directly studying how the handedness of the polarization alters the orbital motion of the trapped particle is difficult for two primary reasons. First, the longitudinal component of a focused Gaussian beam is much weaker than the transverse component even in the strongly focusing situation. Therefore, initiating and maintaining the orbital motion is difficult because of inertia, viscous forces and Brownian motion. Second, most particles are trapped on the beam axis in a Gaussian trap, which does not allow the separation of the extrinsic from the intrinsic nature of the angular momentum. In [6

6. V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the Transfer of the Local Angular Momentum Density of a Multiringed Light Beam to an Optically Trapped Particle,” Phys. Rev. Lett. 91(9), 093602 ( 2003). [CrossRef] [PubMed]

] this was circumvented by using a (multi-ringed) high order Bessel beam carrying both spin and orbital angular momentum. Both effects were shown simultaneously with a orbitally rotating particle also rotating on its own axis due to spin angular momentum. By creating an interference ring pattern of the focused circularly polarized Gaussian beam to change the local electric field distribution, the orbital motion of sub-micron sized spherical polystyrene beads was observed when particles were two-dimensionally trapped off the beam axis [7

7. H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75(6), 063409 ( 2007). [CrossRef]

]. However, to our knowledge, using a single Gaussian beam to study the orbital motion of a three-dimensionally trapped single particle has not been shown.

Transfer of local orbital angular momentum to an absorptive ceramic particle in an optical trap was first observed with a focused linearly polarized LG beam with a helical phase by He et al. [8

8. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct Observation of Transfer of Angular Momentum to Absorptive Particles from a Laser Beam with a Phase Singularity,” Phys. Rev. Lett. 75(5), 826–829 ( 1995). [CrossRef] [PubMed]

]. Subsequently, a number of groups investigated the orbital motion of trapped particles driven by the orbital angular momentum carried by beams [2

2. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. 88(5), 053601 ( 2002). [CrossRef] [PubMed]

, 6

6. V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the Transfer of the Local Angular Momentum Density of a Multiringed Light Beam to an Optically Trapped Particle,” Phys. Rev. Lett. 91(9), 093602 ( 2003). [CrossRef] [PubMed]

, 9

9. S. H. Tao, X.-C. Yuan, J. Lin, and Y. Y. Sun, “Influence of geometric shape of optically trapped particles on the optical rotation induced by vortex beams,” J. Appl. Phys. 100(4), 043105 ( 2006). [CrossRef]

, 10

10. A. T. O’Neil and M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185(1-3), 139–143 ( 2000). [CrossRef]

]. Metallic particles, particularly ones with sizes of the order of the wavelength of light or larger, are harder to trap primarily due to their much higher scattering properties than transparent or even absorptive particles. In 2000, O’Neil and Padgett [10

10. A. T. O’Neil and M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185(1-3), 139–143 ( 2000). [CrossRef]

] showed the three-dimensional optical confinement and orbital rotation of micron-sized silver particles off the beam axis using a strongly focused LG beam with a high numerical aperture (NA) microscope objective, as is conventional in optical tweezers research. However, trap stability with Gaussian beam in this case was poor as the particle kept being pushed into the focal region of the beam by the impulse scattering force and then being pushed out a few seconds later. Here, we present the observation of a stable trap for metallic spherical particles by a more weakly focused Gaussian beam. Successful confinement of the metallic particle in this way for considerable time periods enables the detection of the orbital motion of the particle and thus a detailed study of the angular momentum transfer that takes place.

In this paper, we employed suitably sized gold particles as a sensitive probe to detect the absorption of the orbital angular momentum from the focused light. We report the first direct observation of orbital motion of a single micron-sized particle that is three dimensionally trapped away from the beam axis driven by purely focusing a single circularly polarized Gaussian beam with a plane wavefront in the homogeneous solution. The direction of orbital motion is reversed by changing the handedness of circular polarization. The experimental results clearly demonstrate that the focused circularly polarized Gaussian beam transfers the extrinsic orbital angular momentum to the trapped particle to drive its orbital motion, which is in complete agreement with the theoretical calculations. We also quantitatively measured and analyzed the orbital rotational radius and frequency as a function of the laser beam power.

2. Theoretical predictions

The focusing properties of a circularly polarized Gaussian beam were calculated using the vector Debye integral [11

11. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 ( 1959). [CrossRef]

,12

12. A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138(6B), B1561–B1565 ( 1965). [CrossRef]

]. The simulation results showed that a helical term, exp(±iϕ), was induced in the longitudinal component of the beam corresponding to the handedness of the circular polarization of the incident Gaussian beam when focused through an aplanatic lens [3

3. T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A, Pure Appl. Opt. 10(11), 115005 ( 2008). [CrossRef]

,5

5. Y. Zhao, J. S. Edgar, G. D. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-Orbital Angular Momentum Conversion in a Strongly Focused Optical Beam,” Phys. Rev. Lett. 99(7), 073901 ( 2007). [CrossRef] [PubMed]

]. Here, the aplanatic lens plays the role of a partial optical intrinsic-to-extrinsic angular momentum convertor. The conversion effect itself is not a function of the NA of the lens, but the amplitude of the longitudinal component decreases sharply as the NA decreases.

Our theoretical simulation [5

5. Y. Zhao, J. S. Edgar, G. D. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-Orbital Angular Momentum Conversion in a Strongly Focused Optical Beam,” Phys. Rev. Lett. 99(7), 073901 ( 2007). [CrossRef] [PubMed]

] predicts that focusing a circularly polarized Gaussian beam may result in the possibility of driving orbital motion of a highly scattering particle. To separate the influence of intrinsic spin angular momentum, the particle should be held away from the beam axis allowing orbital motion to be observed due to absorbing optical orbital angular momentum. The orbital rotational direction should agree with the sign of circular polarization. Beyond the orbital direction, all the other properties of the orbital motion (orbital frequency, diameter, and trajectories) should be identical for different circular polarizations with the same trapping power, same trapped particle and surrounding environment.

3. Experiments

In our experiment, the fundamental Gaussian beam is generated by an IPG 5 Watt Ytterbium fiber laser at 1064 nm wavelength (Model number: YLD-5-1064-LP). The beam is expanded and re-collimated to fill the back aperture of a dry objective lens (Olympus UPLFLN60X, NA=0.9) in an inverted microscope geometry. Before the objective, a linear polarizer is placed to obtain a pure linearly polarized Gaussian beam, and then a rotatable quarter wave plate is placed immediately after the polarizer to control the polarization. The circularly polarized Gaussian beam passes through the objective and coverslip, and then focused into the water-based sample. The effective NA of the system is
NAeffective=NAobjnairnwater=0.9/1.33=0.68.
In this focal system, the longitudinal component of the optical intensity is around 10% of the total of the focusing Gaussian beam. The sample contained water with pre-injected spherical gold particles ranging from 1.5 to 3.0 µm in diameters (ALFA AESAR Co. Item No. 39818), which was placed into a home-made well on the coverslip. Because the particles were denser than water, prior to trapping, particles sank to the bottom of the well, sitting on the top of the coverslip. The dilution was chosen to balance the needs of reliably catching one particle for study and of avoiding multiple-particle interactions.

In the inverted optical tweezers, the scattering force from the beam to gold particles opposes the effect of gravity on the particles, thus initially pushing the spheres off the coverslip surface and sending them towards and then away from the beam axis, owing to the direction of the scattering force in the transverse direction. If the particle is moving too fast and passes through the equilibrium trapping position, they would fall back to the coverslip. Therefore, to form a stable trap, we carefully selected the trapping power (ranging from 10 to 100 mW after the objective) and particle size (several microns) so as to balance the gravitational and scattering force and to reduce the particle’s acceleration, thereby restricting its trajectory. Micro-fluidic motion was minimized by the use of a polystyrene well with 1 cm diameter, which was mounted in a transparent chamber to avoid air flow. Particles of about 2-3 microns in diameter would reliably trap off-axis several microns below the focal point. The orbital motion of the trapped particle was recorded using a CCD camera through a custom built microscope. The analysis of three-dimensional confinement of micron-sized metal particles can be found in detail in Reference [9

9. S. H. Tao, X.-C. Yuan, J. Lin, and Y. Y. Sun, “Influence of geometric shape of optically trapped particles on the optical rotation induced by vortex beams,” J. Appl. Phys. 100(4), 043105 ( 2006). [CrossRef]

]. Figure 1
Fig. 1 Successive frames of a video recording that show the orbital motion of a ~3µm spherical gold particle three-dimensionally confined by a focused Gaussian beam with right (right column) and left (left column) circular polarization. The laser power measured after the objective for both circular polarizations is 19mW. The scale bar represents 5µm, and the circular arrow denotes the orbital rotational direction.
shows a series of video frames from two typical experiments. The right column shows the clockwise orbital motion of a ~3 micron gold particle driven by focusing right circularly polarized light while the left column shows the counter-clockwise orbital motion driven by focusing left circularly polarized light. The direction of orbital motion is in accordance with the handedness of the circular polarization.

To further study the orbital motion driven by different circular polarizations, the trajectories of trapped particle were recorded for a period of time and then digitally tracked using custom particle tracking software. Figure 2
Fig. 2 (color online). Trajectories of the trapped gold particle as a function of time in the x (a) and y (b) directions. (c) The trajectories of the particle by overlaying 60 seconds of a video recording. Here, red circles and blue squares show the trajectories of the rotating particle driven by right () and left () circular polarization respectively. RCP: right circular polarization; LCP: left circular polarization. The positions are determined as the center of mass of the trapped particle in each frame using custom particle tracking software.
shows the positions of the center of mass of a trapped particle versus time. The origin of the coordinates is at the center of the beam. The particle was trapped first using right circular polarization (red circles) and then with left circular polarization (blue squares). The x and y components of the positions vary sinusoidally with time and have similar amplitudes and frequencies, which indicate a periodic circular trajectory. Furthermore, since the orbital rotational direction switches with polarization, one component of the motion [such as x component shown in Figure 2(a)] is in phase while the other [y component shown in Figure 2(b)] is 180° out of phase. By overlaying the trajectories of right and left circular polarization [Figure 2(c)], it becomes clear that the particle’s orbital center and diameters are the same. These results are in good agreement with the previously discussed theoretical prediction.

Normally laser tweezers make use of the three-dimensional gradient force to trap particles close to the beam focus. The trapping force in this case is proportional to the laser power. However, because of the much higher density of metallic particles, the principle underlying the stable confinement of the particle is quite different because of the need to balance a much larger downward force [10

10. A. T. O’Neil and M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185(1-3), 139–143 ( 2000). [CrossRef]

], that is, it is more akin to optical levitation in the longitudinal direction. In solution, the downward force is constant and equal to the gravitational force minus the buoyancy force, and the scattering force is pointing up and toward the focal point when particle is below the focal plane. Therefore, a stable three-dimensional confinement is formed when the scattering force component in the longitudinal direction balances the net downward force at an equilibrium point near the focus of a given laser intensity and when the integration of the optical force in the transverse direction contributes to the centripetal force to maintain the orbital circular motion. A change in laser power will change the magnitude of the optical force, and hence will drive the trapped particle to a new position where the forces again balance. Since the beam is highly divergent, changing the laser power changes both the vertical position and the distance of the particle from the optical axis. Therefore, the trajectory of a trapped particle will vary with laser power.

Figure 3(a)
Fig. 3 (color online). Orbital rotational radius (a) and frequency (b) versus trapping laser power driven by right (○) and left (□) circular polarization, respectively. The dash lines in (a) (and “+” in (c)) and (b) (and “×” in (c)) is the average value of clockwise and counter-clockwise orbital motions, which is combined in (c) to show that the orbital rotational radius increased and the frequency decreased as the laser power is increased.
and 3(b) show the orbital rotating radius and frequencies for a trapped particle versus laser power as measured after the microscope objective. The results using different circular polarizations are in good agreement with each other, indicating that only the rotating direction changes with the handedness of the polarization. In addition, as the laser power increases the orbital radius increases accompanied by a corresponding decrease in orbital rotational frequency [Fig. 3(c)]. The velocities, calculated as the product of orbital radius and frequency (v=ωr), as shown in Table 1

Table 1. The velocity of a trapped particle rotating around the beam axis as a function of laser powers

table-icon
View This Table
, remained constant at 5.07±0.15μm/s . The standard deviation is small when compared to the value for laser powers ranging from 19mW to 68mW.

We also studied the trapping of these gold particles using an oil immersion objective lens (NAmax=1.3) with variable NA (from 1.0 to 0.7 in oil). Due to the reduction of the longitudinal component of the focusing beam with decreasing NA, the efficiency of the transfer of orbital angular momentum decreased, which caused a reduction in the orbital rotational rate of the gold particle as anticipated.

4. Conclusions

In conclusion, we experimentally demonstrated that a micron sized metallic particle can be trapped using a Gaussian laser beam and that a purely focused circularly polarized Gaussian beam can drive the orbital motion of such gold particles trapped off beam axis in homogenous medium. The direction of the orbital motion is in accordance with the handedness of the circular polarization. Because a Gaussian beam does not carry helical phase associated with orbital angular momentum, this orbital motion driven by the beam must result from the transfer through focusing the laser beam. The micron-sized gold particle is therefore an optical micro-detector for the study of this angular momentum transfer. Since the three-dimensional confinement of the heavy gold particle requires an intensity dependent balancing with gravity, the orbital radius and frequency could be controlled with appropriate adjustments of the laser power, as theoretically predicted. This non-contact tunable dynamic orbital motion of the trapped particle may offer a simple motor to drive micro-fluidic flow, especially when a very precise slow flow rate is required. Furthermore, the extension of this technique to arbitrary particles would allow for non-contact rotational control of materials of interest to higher resolution imaging techniques such as x-ray or electron microscopy.

Acknowledgment

This work was supported under the Seaborg Fellowship of Lawrence Berkeley National Laboratory and by a Laboratory Directed Research and Development grant. The Advanced Light Source at Lawrence Berkeley National Laboratory is supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences Division, of the U.S. Department of Energy. D. Mcgloin is a Royal Society University Research Fellow. D.T. Chiu gratefully acknowledges NSF (CHE0924320) for support.

*Yiqiong Zhao’s current address is Department of Physics and Electro-Optics Program, University of Dayton, OH, 45469

References and links

1.

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

2.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. 88(5), 053601 ( 2002). [CrossRef] [PubMed]

3.

T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A, Pure Appl. Opt. 10(11), 115005 ( 2008). [CrossRef]

4.

Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 ( 2006). [CrossRef] [PubMed]

5.

Y. Zhao, J. S. Edgar, G. D. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-Orbital Angular Momentum Conversion in a Strongly Focused Optical Beam,” Phys. Rev. Lett. 99(7), 073901 ( 2007). [CrossRef] [PubMed]

6.

V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the Transfer of the Local Angular Momentum Density of a Multiringed Light Beam to an Optically Trapped Particle,” Phys. Rev. Lett. 91(9), 093602 ( 2003). [CrossRef] [PubMed]

7.

H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75(6), 063409 ( 2007). [CrossRef]

8.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct Observation of Transfer of Angular Momentum to Absorptive Particles from a Laser Beam with a Phase Singularity,” Phys. Rev. Lett. 75(5), 826–829 ( 1995). [CrossRef] [PubMed]

9.

S. H. Tao, X.-C. Yuan, J. Lin, and Y. Y. Sun, “Influence of geometric shape of optically trapped particles on the optical rotation induced by vortex beams,” J. Appl. Phys. 100(4), 043105 ( 2006). [CrossRef]

10.

A. T. O’Neil and M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185(1-3), 139–143 ( 2000). [CrossRef]

11.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 ( 1959). [CrossRef]

12.

A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138(6B), B1561–B1565 ( 1965). [CrossRef]

OCIS Codes
(140.7010) Lasers and laser optics : Laser trapping
(260.0260) Physical optics : Physical optics
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: September 28, 2009
Revised Manuscript: November 7, 2009
Manuscript Accepted: November 11, 2009
Published: December 4, 2009

Citation
Yiqiong Zhao, David Shapiro, David Mcgloin, Daniel T. Chiu, and Stefano Marchesini, "Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized Gaussian beam," Opt. Express 17, 23316-23322 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23316


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References

  1. R. E. Beth, “Mechanical Detection and Measurement of the Angular Momentum of Light,” Phys. Rev. 50(2), 115–125 (1936). [CrossRef]
  2. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. 88(5), 053601 (2002). [CrossRef] [PubMed]
  3. T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused Gaussian beam,” J. Opt. A, Pure Appl. Opt. 10(11), 115005 (2008). [CrossRef]
  4. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006). [CrossRef] [PubMed]
  5. Y. Zhao, J. S. Edgar, G. D. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-Orbital Angular Momentum Conversion in a Strongly Focused Optical Beam,” Phys. Rev. Lett. 99(7), 073901 (2007). [CrossRef] [PubMed]
  6. V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the Transfer of the Local Angular Momentum Density of a Multiringed Light Beam to an Optically Trapped Particle,” Phys. Rev. Lett. 91(9), 093602 (2003). [CrossRef] [PubMed]
  7. H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75(6), 063409 (2007). [CrossRef]
  8. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct Observation of Transfer of Angular Momentum to Absorptive Particles from a Laser Beam with a Phase Singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995). [CrossRef] [PubMed]
  9. S. H. Tao, X.-C. Yuan, J. Lin, and Y. Y. Sun, “Influence of geometric shape of optically trapped particles on the optical rotation induced by vortex beams,” J. Appl. Phys. 100(4), 043105 (2006). [CrossRef]
  10. A. T. O’Neil and M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185(1-3), 139–143 (2000). [CrossRef]
  11. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]
  12. A. Boivin and E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138(6B), B1561–B1565 (1965). [CrossRef]

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