## Pure angular momentum generator using a ring resonator |

Optics Express, Vol. 18, Issue 21, pp. 21651-21662 (2010)

http://dx.doi.org/10.1364/OE.18.021651

Acrobat PDF (2406 KB)

### Abstract

This paper reports a pure angular momentum generator using a ring resonator surrounded by a group of nano-rods. The evanescent waves of the circulating light in the ring are scattered by the nano-rods and generate a rotating electromagnetic field, which has only angular momentum but no linear momentum along the axis of rotation. The angular order is determined by the difference between the order of Whispering Gallery mode and the number of the rods, the rotating frequency is equal to the light frequency divided by the angular order. The maximum amplitude of the rotating electromagnetic fields can be 10 times higher than the amplitude of the input field when there are 36 rods (*R*_{rod} = 120 nm, *n _{r}
* = 1.6). The pure angular momentum generator provides a new platform for trapping and rotation of small particles.

© 2010 OSA

## 1. Introduction

*λ*/2

*π*[1

1. J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character **82**(557), 560–567 (1909). [CrossRef]

*λ*is the wavelength. The first experimental observation of the angular momentum in a polarized light was reported in 1936 [2

2. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. **50**(2), 115–125 (1936). [CrossRef]

3. L. Allen, M. W. Beijersbergen, R. J. C. 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]

*l*-order LGM the ratio of angular momentum to linear momentum is increased to

*lλ*/2

*π*. After this study, a number of applications using the angular momentum of the high-order LGM have been demonstrated, including the trapping of low refractive index spheres [4

4. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. **21**(11), 827–829 (1996). [CrossRef] [PubMed]

5. Y. Torii, N. Shiokawa, T. Hirano, T. Kuga, Y. Shimizu, and H. Sasada, “Pulsed polarization gradient cooling in an optical dipole trap with a Laguerre-Gaussian laser beam,” Eur. Phys. J. D **1**(3), 239–242 (1998). [CrossRef]

6. X. P. Zhang, W. Wang, Y. J. Xie, P. X. Wang, Q. Kong, and Y. K. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. **281**(15-16), 4103–4108 (2008). [CrossRef]

7. S. J. Parkin, G. Knöner, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Picoliter viscometry using optically rotated particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **76**(4), 041507 (2007). [CrossRef] [PubMed]

8. J. Leach, H. Mushfique, R. di Leonardo, M. Padgett, and J. Cooper, “An optically driven pump for microfluidics,” Lab Chip **6**(6), 735–739 (2006). [CrossRef] [PubMed]

*spin angular momentum*and the latter is

*orbital angular momentum*. The generation of the orbital angular momentum is a key issue for many applications. For this purpose, the high-order LGMs are used extensively. They are usually generated by changing the phase structure of laser beam using binary-phase diffractive elements [10

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

11. K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, “Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses,” Opt. Express **12**(15), 3548–3553 (2004). [CrossRef] [PubMed]

12. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express **15**(14), 8619–8625 (2007). [CrossRef] [PubMed]

13. C. Y. Chao, W. Fung, and L. J. Guo, “Polymer microring resonators for biochemical sensing applications,” IEEE J. Sel. Top. Quantum Electron. **12**(1), 134–142 (2006). [CrossRef]

14. J. M. Choi, R. K. Lee, and A. Yariv, “Ring fiber resonators based on fused-fiber grating add-drop filters:application to resonator coupling,” Opt. Lett. **27**(18), 1598–1600 (2002). [CrossRef]

15. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. **10**(4), 549–551 (1998). [CrossRef]

16. Y. H. Ja, “Single-mode optical fiber ring and loop resonators using degenerate two-wave mixing,” Appl. Opt. **30**(18), 2424–2426 (1991). [CrossRef] [PubMed]

13. C. Y. Chao, W. Fung, and L. J. Guo, “Polymer microring resonators for biochemical sensing applications,” IEEE J. Sel. Top. Quantum Electron. **12**(1), 134–142 (2006). [CrossRef]

17. L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature **426**(6968), 816–819 (2003). [CrossRef] [PubMed]

15. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. **10**(4), 549–551 (1998). [CrossRef]

## 2. Theoretical analysis

*R*

_{ring}_{,}the ring width

*W*, the number of the nano-rods

_{r}*N*, the rod radius

*R*, the waveguide width

_{rod}*W*, the refractive index

_{w}*n*, the input wavelength

_{r}*λ*, the gap between the ring and the rods

*G*, the gap between the waveguide and the rod

_{rr}*G*. With these parameters, the external radius of the ring is

_{wr}*R*=

_{e}*R*+

_{ring}*W*/2 and the internal radius of the ring is

_{r}*R*=

_{in}*R*-

_{ring}*W*/2. The centers of rods are positioned at (

_{r}*R*

_{s},

*jθ*

_{0}), where

*R*=

_{s}*R*+

_{e}*G*+

_{rr}*R*,

_{rod}*θ*

_{0}= 2π/

*N*, and

*j*= 1, 2, …,

*N*. The gap between the waveguide and the ring is

*G*= 2

*R*+

_{rod}*G*+

_{rr}*G*as shown in Fig. 1(b).

_{wr}18. C. Manolatou, M. J. Khan, S. H. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. **35**(9), 1322–1331 (1999). [CrossRef]

### 2.1 Light coupling effect

20. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser-beam by a spherical-particle,” J. Opt. Soc. Am. A **10**(4), 693–706 (1993). [CrossRef]

*γ*to

*γ′*and the coupling coefficient

*K*to

*K′*. Based on Eq. (1), the relative circulating intensity on the ring is reduced when

*γ*goes higher in the range of (0, 1). An increase of the rod radius would cause higher intensity loss

*γ*and thus lower relative circulating intensity

*I*/

_{c}*I*.

_{i}### 2.2 Generation of pure angular momentum

*r*and angle

*θ*in the polar coordinate system as shown in Fig. 3(a) . The amplitude of the scattered light

*u*(

*r*,

*θ*) from one rod can be expressed as [21]where

*θ*is the scattering angle,

*s*(

*θ*) is the amplitude scattering factor,

*k*is the wave vector, and

*u*

_{o}is the amplitude of the incident light (the evanescent wave). It is assumed that when viewed from any rod, the scattered lights from the neighboring rods are negligible as compared to the evanescent wave of the circulating light. Thus, the multiple scattering effects can be neglected. Based on the superposition principle and Eq. (2), the electromagnetic field at an arbitrary point A can be written aswhere subscript

*j*corresponds to the

*j*

^{th}rod, and

*j*

^{th}rod as the origin.

*N*is equal to the WGM order

*n*(i.e.,

*l*= 0), the electromagnetic field distribution inside the ring resonator becomes a series of concentric circles. Such distribution is close to the pattern of the 0th-order LGM in the plane perpendicular to the beam's propagation direction. When

*l*= 1, the field distribution is same as the pattern of the 1st-order LGM in the plane perpendicular to the beam's propagation direction, and its rotation frequency

*f*is equal to the light frequency. Detailed analyses and discussions of these two special cases are given in the appendix.

_{r}*n*and

*N*is equal to

*l*(

*l*= |

*n*–

*N*|), the electromagnetic field distribution inside the ring resonator has an

*l*-fold rotational symmetry. This distribution is same as the pattern of the

*l*-order (angular order) LGM. The corresponding rotating electromagnetic field has the rotation frequency

*f*, which is equal to the light frequency

_{r}*f*divided by the order

*l*(that is,

*f*=

_{r}*f*/

*l*).

**and angular momentum**

*P***of the rotating electromagnetic field can be calculated as [3**

*J*3. L. Allen, M. W. Beijersbergen, R. J. C. 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]

*ε*

_{0}is the vacuum permittivity. Since the light propagates only in the

*xy*-plane, the linear momentum on the

*z*-direction is zero (i.e., based on Eq. (4),

*P*

_{z}= 0). Consider the rotation around the

*z*-axis (at

*r*

_{z}= 0), the angular momentum carried by the rotating electromagnetic field only exists in the

*z*direction (i.e., based on Eq. (5),

*J*

_{z}≠ 0,

*J*

_{x}= 0,

*J*

_{y}= 0). Thus, along the

*z*-axis there is no linear momentum but only angular momentum. This is why the term ‘pure angular momentum’ is used in this paper.

## 3. Numerical results and discussions

*R*

_{ring}_{,}

*W*,

_{r}*N*,

*R*,

_{rod}*G*,

_{rr}*G*,

_{wr}*W*,

_{w}*n*and

_{r}*λ*. In order to simplify the simulation, the light coupling effect and the pure angular momentum generation will be discussed separately.

### 3.1 Light coupling effect

*R*is set as a constant. Simulation shows that under the conditions |

_{ring}*K*<

*K*′), which is concluded from the calculated results |

*G*is optimized for the highest relative circulating field (|

*E*

_{4}/

*E*

_{1}|), and the optimized value varies with the wavelength

*λ*as shown in Fig. 5(a) . When the rod radius

*R*is increased, the relative circulating field (|

_{rod}*G*/

_{rr}*G*). The other parameters are

_{wr}*G*= 0.2 μm,

*n*= 1.6,

_{r}*λ*= 619.1 nm and

*W*=

_{r}*W*= 0.3 μm. The trend is changed from decrease to increase after the critical value at

_{w}*R*= 0.1 μm, which is due to the overlap of the rod with the ring/waveguide. Further studies show that these conclusions are still correct when the refractive index is increased from 1.6 to 3.476.

_{rod}### 3.2 Generation of pure angular momentum

*N*= 32,

*R*= 2.2 µm,

_{in}*R*= 2.5 µm,

_{out}*R*= 0.1 µm,

_{rod}*G*= 0.1 µm,

_{rr}*n*= 1.6,

_{r}*W*=

_{r}*W*= 0.3 μm, the numerical results show that a series of WGMs have the wavelengths of 578.50 nm, 592.13 nm, 606.81 nm, 621.83 nm, 636.93 nm, 654.89 nm, 672.90 nm, 691.93 nm and 712.07 nm, which correspond to the order

_{w}*n*from 36 to 28 as shown in Fig. 6 . When the wavelength is 654.89 nm (corresponding to

*n*= 33), the electromagnetic field distribution inside the ring resonator has 1-fold symmetric structure as shown in Figs. 7(a) and 7(b), which agrees with the prediction in section 2.2. The corresponding circulating intensity in the ring is 75.4, which indicates that the field intensity decrease per circle is 1/75.4 (input energy/ total energy, ~1.3%) and it is negligible for the theoretical model. The field amplitude on the ring in Fig. 7(a) is set to 1 for the convenience of observation, and the black line in Fig. 7(b) is the intensity distribution along the white line.

*N*. The amplitude distribution of the electromagnetic field is a series of concentric circles as shown in Fig. 8(a) . It matches with the analysis in section 2.2. Figures 8(b) – 8(e) show the 1st – 4th fold electromagnetic field distributions corresponding to the shorter wavelengths and Figs. 8(f) – 8(i) show the electromagnetic field distributions corresponding to the longer wavelengths. Comparing Figs. 8(e) with 8(i), the radial orders (number of maxima along the radius) of the field distributions are different due to the wavelength difference even though the electromagnetic fields have same angular orders.

### 3.3 Discussions

*τ*

_{±}

*of the*

_{l}*l*-order rotating electromagnetic field is defined aswhere

*l*-order electromagnetic field, “+

*l*” corresponds to the condition

*n*>

*N*(

*l*=

*n*–

*N*), and “-

*l*” refers to

*n*<

*N*(

*l*=

*N*–

*n*). The rotating electromagnetic field is a combination of the scattering from the nano-rods, the evanescent field at the rods’ locations as well as the coupling effect of ring-waveguide coupler. Similar to the two steps mentioned above, the Eq. (6) can be rewritten aswhere

*n*-order WGM. The first part on the left side of Eq. (7) is the same as the relative circulating intensity in Eq. (1). The second part presents the scattering efficiency of the group of nano-rods. Since

*E*is set as 1, the generation efficiency of the

_{in}*l*-order rotating electromagnetic field is the same as the maximum intensity of the

*l*-order electromagnetic field. In the second step of the simulation, the intensity of the coupled light from the waveguide is set as 1, which means the intensity at the output of the waveguide is kept as zero. The gap between the waveguide and the ring is adjusted accordingly to maintain the coupling intensity independently of the circulating intensity on the ring.

*n*. When

*n*is increased, the corresponding wavelength

*λ*is decreased for the same design, and the amplitude of the evanescent field around the ring resonator is reduced. Once the wavelength

_{n}*λ*is smaller than 550 nm, the evanescent fields around the ring become very weak, causing a low electromagnetic field inside the ring (

_{n}*n*is decreased,

*λ*goes up, and the radiation loss is enhanced. The solid line in Fig. 6 shows that when

_{n}*λ*is longer than 750 nm, the radiation loss is dominated and the circulating field

_{n}*N*and the refractive index

*n*. For instance, when the

_{r}*n*is increased to 3.476 (

_{r}*N*= 28), the range is narrowed down to only one wavelength at 1350 nm as shown in Fig. 9(a) .

*l*correspond to different maximum amplitudes of the electromagnetic fields inside the ring. When the wavelength is increased from 519.2 nm to 541.6 nm (

*n*= 1.6,

_{r}*N*= 36), the maximum amplitude of the electromagnetic field is decreased from 10 to 3.6, and most energy is scattered away rather than be confined inside the ring as shown in Fig. 9(e) – 9(f).

*N*, the same

*l*correspond different

*n*(i. e.

*n*=

*N*±

*l*) and a wide range of

*λ*. Since the wavelength affects the circulating intensity and the evanescent field, the amplitudes of the generated fields with same order numbers

_{n}*l*cannot present the generation efficiency. To investigate the variations of the maximum value of all the maximum amplitudes

_{max}with

*N*, the rod

*R*as well as the refractive index

_{rod}*n*should be considered. The relationships are shown in Figs. 9(b) and 9(c). With

_{r}*n*= 1.6 and

_{r}*R*= 0.08 µm,

_{rod}_{max}tends to increase when

*N*goes from 24 to 37, and then decreases when

*N*> 37 (see Fig. 9 (b)). For different rod radii, the peak positions are different. The maximum value is 10 times higher than the amplitude of the input field when

*N*= 36,

*R*= 120 nm,

_{rod}*n*= 1.6,

_{r}*λ*= 519.2 nm,

*l*= 5, circulating field

*E*= 14.7. The field distribution is shown in Fig. 9(e). When

_{c}*n*= 1.6, 2.6 or 3.476, the peak position appear at

_{r}*N*= 37, 28 or 24, respectively (see Fig. 9(c)). After the peak position, the increase of

*N*causes a reduction of

_{max}. For example, when

*N*is increased from 24 to 32 (

*n*= 3.476,

_{r}*l =*1),

_{max}is reduced from 1.9 to 0.2 as shown in Figs. 9(g) and 9(h).

*G*also affects

_{rr}_{max}. An increase of

*G*causes a reduction of the scattering from each rod, whereas a decrease of

_{rr}*G*results in higher field loss

_{rr}*α*, which in turn decreases the circulating field on the ring. For

*R*= 100 nm and

_{rod}*n*= 1.6, the optimized value is 80 nm as shown in Fig. 9(d).

_{r}*xy*-plane (

*z*= 0), the electromagnetic field is also symmetric to this plane. In this case, the field has maximum amplitude in the same plane as shown in Fig. 10(a) . In the

*xy*-plane (

*z*= 0), the electromagnetic field distribution is same as previous presentation as shown in Fig. 10(b) and has no linear momentum in z direction. Assuming a particle has a small displacement from this plane, the gradient force will draw it back, like what happens in optical trapping. This means the generated electromagnetic field can stably trap small particles. The scattering light has a divergent angle in the vertical (

*z*) direction due to the small height (400 nm), and the divergent angle can be reduced by increasing the height.

## 4. Conclusions

*l*-order rotating electromagnetic fields are generated when different resonant wavelengths are coupled into the generator. These fields have pure angular momenta but no linear momenta along the axis of rotation. Detailed studies on the influences of different structural parameters on the electromagnetic field have been conducted. The maximum amplitude of generated fields could be 10 times higher than that of the input field. The pure angular momentum generator overcomes the complication of linear momentum to the rotation of the small particles and provides a compact and flexible platform to study the rotational behaviors in micro-scale.

## Appendix

*l*= 0 and

*l*= 1 are studied here so as to exemplify the features of the electromagnetic field.

*n*=

*N*= 32 (correspondingly

*l*= 0), the circumference of the ring is 32 times of the effective wavelength

*λ*, and thus the evanescent waves scattered by different rods (

_{eff}*l*is 1 (

*l*=

*n*–

*N*= 1, i. e.,

*n*= 33, and

*N*= 32), there is a phase difference of π/16 between the two evanescent waves scattered by the

*j*

^{th}rod and the (

*j*+ 1)

^{th}rod. If the phase of the evanescent wave scattered by the 32nd rod is set as 0, the other phases of the evanescent waves scattered by the 1st to 31st rods are shown in Fig 3(b). Point A and point B are on the same concentric circle of the ring, and they are symmetrical to the origin. Based on Eq. (3), the electromagnetic fields at the points A and B can be expressed asConsidering the contribution of the scattering from the 6th and 10th rods to point A, it hasThe phase difference between the evanescent waves scattered by the

*j*

^{th}rod and the 32nd rod is

*π*phase difference. Considering the rotational symmetry of the generator, the total electromagnetic fields at the points A and B should have a

*π*phase difference (

*λ*/32, the 0-phase rod and ±π phase rod are shifted to the next rods, and the

_{eff}*x*-axis and

*y*-axis have a π/16 rotation around the

*z*-axis. As a result, the electromagnetic field distribution inside the ring has a rotation of π/16. Therefore, the rotation frequency

*f*is equal to the light frequency (that is,

_{r}*f*=

_{r}*f*).

## Acknowledgements

## References and links

1. | J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character |

2. | R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. |

3. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

4. | K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. |

5. | Y. Torii, N. Shiokawa, T. Hirano, T. Kuga, Y. Shimizu, and H. Sasada, “Pulsed polarization gradient cooling in an optical dipole trap with a Laguerre-Gaussian laser beam,” Eur. Phys. J. D |

6. | X. P. Zhang, W. Wang, Y. J. Xie, P. X. Wang, Q. Kong, and Y. K. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. |

7. | S. J. Parkin, G. Knöner, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Picoliter viscometry using optically rotated particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

8. | J. Leach, H. Mushfique, R. di Leonardo, M. Padgett, and J. Cooper, “An optically driven pump for microfluidics,” Lab Chip |

9. | S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, and J. Turunen, “Generation of rotating Gauss-Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. |

10. | D. G. Grier, “A revolution in optical manipulation,” Nature |

11. | K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, “Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses,” Opt. Express |

12. | S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express |

13. | C. Y. Chao, W. Fung, and L. J. Guo, “Polymer microring resonators for biochemical sensing applications,” IEEE J. Sel. Top. Quantum Electron. |

14. | J. M. Choi, R. K. Lee, and A. Yariv, “Ring fiber resonators based on fused-fiber grating add-drop filters:application to resonator coupling,” Opt. Lett. |

15. | B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. |

16. | Y. H. Ja, “Single-mode optical fiber ring and loop resonators using degenerate two-wave mixing,” Appl. Opt. |

17. | L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature |

18. | C. Manolatou, M. J. Khan, S. H. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. |

19. | Y. H. Ja, “Generalized theory of optical fiber loop and ring resonators with multiple couplers. 1: Circulating and output fields,” Appl. Opt. |

20. | J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser-beam by a spherical-particle,” J. Opt. Soc. Am. A |

21. | H. C. d. Hulst, Light scattering by small particles (Dover Publications, Inc., New York, 1981). |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(130.3120) Integrated optics : Integrated optics devices

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: August 19, 2010

Revised Manuscript: September 15, 2010

Manuscript Accepted: September 21, 2010

Published: September 29, 2010

**Virtual Issues**

Vol. 5, Iss. 14 *Virtual Journal for Biomedical Optics*

**Citation**

Y. F. Yu, Y. H. Fu, X. M. Zhang, A. Q. Liu, T. Bourouina, T. Mei, Z. X. Shen, and D. P. Tsai, "Pure angular momentum generator using a ring resonator," Opt. Express **18**, 21651-21662 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-21-21651

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### References

- J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 82(557), 560–567 (1909). [CrossRef]
- R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936). [CrossRef]
- L. Allen, M. W. Beijersbergen, R. J. C. 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]
- K. T. Gahagan and G. A. Swartzlander., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]
- Y. Torii, N. Shiokawa, T. Hirano, T. Kuga, Y. Shimizu, and H. Sasada, “Pulsed polarization gradient cooling in an optical dipole trap with a Laguerre-Gaussian laser beam,” Eur. Phys. J. D 1(3), 239–242 (1998). [CrossRef]
- X. P. Zhang, W. Wang, Y. J. Xie, P. X. Wang, Q. Kong, and Y. K. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 (2008). [CrossRef]
- S. J. Parkin, G. Knöner, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Picoliter viscometry using optically rotated particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(4), 041507 (2007). [CrossRef] [PubMed]
- J. Leach, H. Mushfique, R. di Leonardo, M. Padgett, and J. Cooper, “An optically driven pump for microfluidics,” Lab Chip 6(6), 735–739 (2006). [CrossRef] [PubMed]
- S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, and J. Turunen, “Generation of rotating Gauss-Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).
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