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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 22 — Oct. 22, 2012
  • pp: 24835–24842
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Controllable motion of optical vortex arrays using electromagnetically induced transparency

David Shwa, Evgeny Shtranvasser, Yoni Shalibo, and Nadav Katz  »View Author Affiliations


Optics Express, Vol. 20, Issue 22, pp. 24835-24842 (2012)
http://dx.doi.org/10.1364/OE.20.024835


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Abstract

We demonstrate control of the collective motion of an optical vortex array using an electromagnetically induced transparency media. Scanning the frequency detuning between the pump and probe fields changes the susceptibility of the media, producing a unique effective diffraction of the vortex array for each detuning. We measure several experimental configurations and compare them to numerical simulations.

© 2012 OSA

1. Introduction

The optical susceptibility of atomic media due to a weak resonant probe field can be changed dramatically in the presence of a strong pump field. The imaginary part of the susceptibility vanishes, thus leading to a transparency in the otherwise opaque media, while the real part of the susceptibility becomes very steep, resulting a great reduction in the group velocity as compared with the speed of light. This effect is termed electromagnetically induced transparency (EIT) [1

1. K. Boller, A. Imamoglu, and S. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991). [CrossRef] [PubMed]

, 2

2. M. Fleischhauer, A. Imamoglu, and J. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

]. An extremely rich variety of spatially dependent optical phenomena that is derived from the complex susceptibility of EIT media was demonstrated recently, such as waveguiding [3

3. P. K. Vudyasetu, D. Starling, and J. C. Howell, “All optical waveguiding in a coherent atomic rubidium vapor,” Phys. Rev. Lett. 102, 123602 (2009). [CrossRef] [PubMed]

], frequency dependent diffraction, diffractionless propagation, Negative diffraction and induced deflection [4

4. O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, “Elimination, reversal and directional bias of optical diffraction,” Nat. Phys. 5, 665–668 (2009). [CrossRef]

, 5

5. O. Firstenberg, M. Shuker, N. Davidson, and A. Ron, “Elimination of the diffraction of arbitrary images imprinted on slow light,” Phys. Rev. Lett. 102, 043601 (2009). [CrossRef] [PubMed]

]. Transmission of phase dependent patterns were also achieved and highlighted as an effective way to reduce diffusion effects in storage processes [6

6. M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, “Storing images in warm atomic vapor,” Phys. Rev. Lett. 100, 223601 (2008). [CrossRef] [PubMed]

, 7

7. L. Zhao, T. Wang, Y. Xiao, and S. F. Yelin, “Image storage in hot vapors,” Phys. Rev. A 77, 041802 (2008). [CrossRef]

].

The concept behind the manipulation of vortex arrays is the controllable diffraction of the probe image due to the tunable susceptibility. For an undepleted plane wave pump beam, the susceptibility of a spatially varying probe in EIT can be written as [18

18. O. Firstenberg, M. Shuker, R. Pugatch, D. Fredkin, N. Davidson, and A. Ron, “Theory of thermal motion in electromagnetically induced transparency: effects of diffusion, Doppler broadening, and Dicke and Ramsey narrowing,” Phys. Rev. A 77, 043830 (2008). [CrossRef]

]:
χ31(k,ω)=igcK(ω)n0[1+L˜(k,ω)]
(1)
where k is the transverse wavevector of the probe profile, g is the dipole interaction strength, n0 is the density of the vapor, K(ω) is the dimensionless susceptibility in the absence of the pump field and is the following spatio-temporal spectral function :
L˜(k,ω)=K(ω)|Ωc|2i(Δ+ω)ΓhomDk2
(2)

Here Ωc is the Rabi frequency of the coupling field, Δ is the two photon detuning, Γhom is the decoherence rate and D=vth2γ is the Dicke-Doppler diffusion coefficient where γ is the relaxation rate due to collisions with the buffer gas and vth=kBTm is the thermal velocity of the atoms. The dynamics of the probe pattern due to this complex susceptibility is given by E/z=i(χ31λ|k|24π)E, where λ is the optical wavelength. This dynamics can be described as a diffusion equation with a complex diffusion constant, meaning that it undergoes a frequency dependent optical diffraction together with a diffusion due to the atomic motion [18

18. O. Firstenberg, M. Shuker, R. Pugatch, D. Fredkin, N. Davidson, and A. Ron, “Theory of thermal motion in electromagnetically induced transparency: effects of diffusion, Doppler broadening, and Dicke and Ramsey narrowing,” Phys. Rev. A 77, 043830 (2008). [CrossRef]

]. The diffraction component creates a behavior similar to the vortex array motion in free-space, but with a frequency controlled effective diffraction length. Hence, as the two photon detuning is scanned we expect to see the effects that were described before in free-space, such as a rotation of vortex arrays with L≠0 and a contraction in the case of arrays with L=0. The diffusion component leads to a motion also in the direction of the intensity gradient and not only perpendicular to it. This outcome is unique to the propagation in diffusive media and has no counterpart in free-space propagation.

A major advantage of this method for applications such as optical trapping and optical wrenches is the ability to move vortices without a physical change of the input probe field, which is usually a slow process, but a frequency control which is potentially much faster and more robust.

2. Experimental setup

Our experimental setup is described in Fig. 1. As the EIT medium we use the D1 transition of 87Rb, where the Λ configuration is comprised of the hyperfine levels. We use a distributed feedback laser as our source of 795 nm beam. This beam is separated into pump and probe beams using a beam splitter. Both beams pass through an acousto-optic modulator, the probe for amplitude control and the pump for amplitude and frequency control using a double pass configuration. A hyperfine splitting of 6.8 GHz is created using an electro-optic modulator on the probe beam followed by an etalon used to filter out the unwanted 0 and −1 orders. The EOM is used also to compensate for the frequency mismatch of the probe due to the double pass of the pump. The spatial phase dependence of the beam is created using a liquid crystal spatial light modulator (BNS, 512 × 512 pixels). Finally, both the pump and the probe beams are combined using a Glan-Taylor polarizer and pass through a 7.5 cm quartz cell heated to ∼ 40°C. The cell contains isotopically pure 87Rb and 20 Torr Neon as buffer gas. After the cell, the pump and probe beams are separated using another Glan-Taylor polarizer and the probe beam is measured using a CCD camera (PCO Pixelfly qe) with short exposure times of 5–10μs in order to avoid distortions due to phase fluctuations. For the results described here we use a probe beam with a power of 5μW and a pump beam with a power of 150μW and a waist of 2.5 mm.

Fig. 1 Our experimental setup. Orange lines - pump beam, purple lines - probe beam. DFB - distributed feedback laser, PBS - polarizing beam splitter, λ/2, λ/4 - waveplates, AP -aperture, GT - Glan-Taylor polarizer, SL - solenoid, SLM - spatial light modulator, AOM -acousto-optic modulator, EOM - electro-optic modulator, BD - beam dump, μMSμ-metal shield. In the inset - the relevant 87Rb hyperfine level diagram.

3. Results and discussion

Figure 2 shows the phase patterns imprinted on the spatial light modulator for a 2 × 2 vortex array in two configurations. The vortex array in Fig. 2(a) comprises two vortices having OAM of +1 and two vortices having OAM of −1, so the total OAM of the pattern is L=0, while in Fig. 2(b) all the vorices have OAM of +1, so the total OAM of the pattern is L=4. We chose these two configurations in order to demonstrate the effect of EIT on the two generic possibilities of vortex array, namely, L=0 and L≠0.

Fig. 2 Phase patterns of a four vortex array. The phase for each point of the pattern is the multiplication of each vortex phase, Πj exp(imjϕj). ϕj is defined as ϕj = arctan[(yyj)/(xxj)], where (xj, yj) is the location of the jth vortex core. Patterns are imprinted on the spatial light modulator with the gaussian probe beam directed to the center of the pattern. (a) Two m=1 and two m=−1 vortices. (b) Four m=1 vortices.

Figures 3(a) and 4(a) present typical results for a vortex array control by changing the two photon detuning using EIT setup. The spatial profile of the probe is a 0.5 mm waist gaussian beam with a 2 × 2 vortex array embedded in it. Figure 3(a) is produced using the phase pattern depicted in Fig. 2(a), while Fig. 4(a) is produced using the phase pattern presented in Fig. 2(b).

Fig. 3 Experimental results for the collective rotation of four vortices, two m=+1 and two m=−1 with total angular momentum L=0. (a) Pictures for different frequencies with log scaled normalized intensity. The numbers relate to the vertical lines in (c). (b) Pictures of simulations for the same frequencies. (c) The average size of the array measured as the distance between two opposite vortices. The array size at resonance is about half of the original array size. (d) 2D plot of the vortex centers at different detunings. Although each vortex moves in a circular motion as the detuning is scanned, the full array does not rotate but only contracts. Near the resonance (picture (a3)) the four vortices almost combine to one point with zero amplitude and L=0.
Fig. 4 Experimental results for the collective rotation of four m=+1 vortices with total angular momentum L=4. (a) Pictures for different frequencies with log scaled normalized intensity. The numbers relate to the vertical lines in (c). (b) Pictures of simulations for the same frequencies. (c) The average size of the array measured as the distance between two opposite vortices. (d) 2D plot of the vortex centers at different detunings. In this case, as with L=0, each vortex moves in a circular manner as the detuning is scanned, but here due to the fact that all the vortices rotate in the same direction, there is a global rotation of the entire array.

Figures 3(b) and 4(b) show simulations based on Eq. (1) of both the situations discussed with good agreement to the experimental results. The simulation uses the measured temperature and pump intensity, while γ is calculated ab initio using a hard sphere model. g and K(ω) are the only fitting parameters (minor adjustments from calculated/measured values due to experimental uncertainties) used in order to achieve the measured EIT absorption spectrum. Once the EIT spectrum is matched, the spatial propagation is calculated for all patterns without any further fitting.

Figures 3(d) and 4(d) show the dynamics of each vortex center as the two photon detuning is scanned. For L=4 a global rotation accompanied by vortex repulsion is observed (Fig. 4(c)). The rotation is a signature of the non-zero angular momentum of the array while the repulsion has to do with the movement of the vortices due to the gaussian envelope of the beam. This repulsion is higher when the rotation is higher thus leading to a more distorted images for higher total L patterns. The circular trajectory of each vortex as described in Fig. 4(d) is very different from free-space propagation where a linear trajectory was observed [14

14. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993). [CrossRef]

, 15

15. D. Rozas, Z. S. Sacks, and G. A. Swartzlander Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997). [CrossRef]

]. The circular trajectory is due to the combined effects of diffusion and diffraction. L=0 has a different signature with a global contraction (Fig. 3(c)) and no global rotation. This effect happens since each two adjacent vortices have inverse OAM thus leading to rotation in an opposite direction. Moreover, as the pattern with total OAM of L=0 contracts, the vortices start to join together leading to one large dark spot in the middle of the beam. Simulation shows that for strong enough interaction the global contraction for patterns may result in even a total annihilation of the vortices, in agreement with published free-space results of vortex dipoles [19

19. M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam.” J. Opt. Soc. Am. A 25, 1279–1286 (2008). [CrossRef]

].

As a quantitative measurement of the global rotation that arrays with L≠0 experience, we measure the angle of the connecting line between two vortices with respect to the x-axis. A total rotation span of more than 0.3 radians is visible (Fig. 5(a)).

Fig. 5 (a) Rotation angle of two vortices. m=1(red circles), m=−1(blue squares), m=1 and m=−1 (green diamonds). Simulation results for these configurations are plotted by solid lines. (b) The distance between two m=+1 vortex centers (Black circles) and the EIT transmission (Red squares) as a function of the two photon detuning. Both curves follow the same trend.

The rotation has a dispersive signature, similar to the real part of the susceptibility, where it is possible to see an extra rotation in one side of the resonance and a contra free-space rotation on the other side. The EIT media in this case enables a manipulation of the effective optical path, thus allows us to change the optical pattern as if it was taken at different positions along the beam traveling axis. Actually, free space propagation like this would have taken about twice the Rayleigh length which is 1 meter in our case. In this view it is possible to think of the medium as a very high numerical aperture lens [12

12. Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A 375, 2958–2963 (2011). [CrossRef]

]. Moreover, contra free-space rotation is analogous to a backward movement of the beam from the origin. This effect is a signature of an index of refraction which is smaller than one. Figure 5(a) also demonstrates that positive or negative total angular momentum creates an opposite rotation (blue squares and red circles curves), while zero OAM produces no rotation at all (green diamonds). Due to the imaginary part of the complex susceptibility an actual diffusion of the beam occurs. This effect was observed and measured before by Firstenberg et al. [20

20. O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, “Self-similar modes of coherent diffusion,” Phys. Rev. Lett. 105, 183602 (2010). [CrossRef]

]. Here we show an easy way to measure it by looking at the average distance between vortices with the same optical angular momentum as depicted in Fig. 5(b). This increase in the distance between the vortices agrees well with the EIT line presented also in Fig. 5(b).

Larger arrays of vortices such as 3 × 3 and 4 × 4 were also created with similar effects as shown for the 2 × 2 array case. Due to the fact that the vortices cover a large part of the Gaussian beam, the EIT peak diminishes, thus leading to slightly less pronounced effects. As the dynamics of each vortex is affected by the local field in the point of the vortex, excluding the vortex itself, using a Gaussian field offers a way to manipulate even a single off-centered vortex alone [21

21. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003). [CrossRef]

]. Moreover, here we exploit the linear regime of the EIT media, but higher intensity will probe the nonlinear Kerr medium with extended rotation spans, not possible in the linear regime [16

16. D. Rozas, C. T. Law, and G. A. Swartzlander Jr., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997). [CrossRef]

].

Fig. 6 Storage dynamics of two vortices with total L=0. The storage is done under the condition of zero two-photon detuning. (a) Pictures of the array for different storage times. (b) A cross section of the pictures in (a) along the axis connecting the two vortices. The vortices dark spot vanishes for long storage times.

One advantage stored patterns have over slow light patterns is an enhanced purity of the measured vortex array state. The reason for this enhancement is the fact that only the EIT controlled light is stored. In contrast, in the slow light regime, various spectral contaminations can occur in the beam profile due to minor optical pumping effects and incoherent light.

4. Conclusion

We spectrally control the collective motion of optical vortex arrays by the use of EIT media. This control is done by changing the two photon detuning between the pump and probe beams. We describe how to use this method for various patterns of vortex arrays and explain the phenomenological differences between arrays with L=0 and L≠0, namely rotation and expansion of the array. This control may be used for fast and accurate displacement of optical vortices in applications such as optical wrenches and the transfer of optical angular momentum to particles [8

8. H. He, M. Friese, N. 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, 826–829 (1995). [CrossRef] [PubMed]

]. The ability to detect small changes in the displacement of the vortices may also be used in magnetic sensing [23

23. E. E. Mikhailov, I. Novikova, M. D. Havey, and F. A. Narducci, “Magnetic field imaging with atomic Rb vapor,” Opt. Lett. 34, 3529–3531 (2009). [CrossRef] [PubMed]

] and quantum measurements of multiphoton states with nontrivial spatial profiles.

Acknowledgments

We thank J. Howell and H. Eisenberg for fruitful discussions and comments. This work was supported by ISF grant No. 1248/10.

References and links

1.

K. Boller, A. Imamoglu, and S. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991). [CrossRef] [PubMed]

2.

M. Fleischhauer, A. Imamoglu, and J. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

3.

P. K. Vudyasetu, D. Starling, and J. C. Howell, “All optical waveguiding in a coherent atomic rubidium vapor,” Phys. Rev. Lett. 102, 123602 (2009). [CrossRef] [PubMed]

4.

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, “Elimination, reversal and directional bias of optical diffraction,” Nat. Phys. 5, 665–668 (2009). [CrossRef]

5.

O. Firstenberg, M. Shuker, N. Davidson, and A. Ron, “Elimination of the diffraction of arbitrary images imprinted on slow light,” Phys. Rev. Lett. 102, 043601 (2009). [CrossRef] [PubMed]

6.

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, “Storing images in warm atomic vapor,” Phys. Rev. Lett. 100, 223601 (2008). [CrossRef] [PubMed]

7.

L. Zhao, T. Wang, Y. Xiao, and S. F. Yelin, “Image storage in hot vapors,” Phys. Rev. A 77, 041802 (2008). [CrossRef]

8.

H. He, M. Friese, N. 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, 826–829 (1995). [CrossRef] [PubMed]

9.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef] [PubMed]

10.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008). [CrossRef]

11.

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007). [CrossRef] [PubMed]

12.

Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A 375, 2958–2963 (2011). [CrossRef]

13.

D. Neshev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413–421 (1998). [CrossRef]

14.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993). [CrossRef]

15.

D. Rozas, Z. S. Sacks, and G. A. Swartzlander Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997). [CrossRef]

16.

D. Rozas, C. T. Law, and G. A. Swartzlander Jr., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997). [CrossRef]

17.

X. Gan, J. Zhao, S. Liu, and L. Fang, “Generation and motion control of optical multi-vortex,” Chin. Opt. Lett. 7, 1142–1145 (2009). [CrossRef]

18.

O. Firstenberg, M. Shuker, R. Pugatch, D. Fredkin, N. Davidson, and A. Ron, “Theory of thermal motion in electromagnetically induced transparency: effects of diffusion, Doppler broadening, and Dicke and Ramsey narrowing,” Phys. Rev. A 77, 043830 (2008). [CrossRef]

19.

M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam.” J. Opt. Soc. Am. A 25, 1279–1286 (2008). [CrossRef]

20.

O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, “Self-similar modes of coherent diffusion,” Phys. Rev. Lett. 105, 183602 (2010). [CrossRef]

21.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003). [CrossRef]

22.

P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic rubidium vapor,” Phys. Rev. Lett. 100, 123903 (2008). [CrossRef] [PubMed]

23.

E. E. Mikhailov, I. Novikova, M. D. Havey, and F. A. Narducci, “Magnetic field imaging with atomic Rb vapor,” Opt. Lett. 34, 3529–3531 (2009). [CrossRef] [PubMed]

OCIS Codes
(020.1670) Atomic and molecular physics : Coherent optical effects
(070.4690) Fourier optics and signal processing : Morphological transformations
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: March 12, 2012
Revised Manuscript: April 22, 2012
Manuscript Accepted: April 24, 2012
Published: October 16, 2012

Citation
David Shwa, Evgeny Shtranvasser, Yoni Shalibo, and Nadav Katz, "Controllable motion of optical vortex arrays using electromagnetically induced transparency," Opt. Express 20, 24835-24842 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24835


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References

  1. K. Boller, A. Imamoglu, and S. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66, 2593–2596 (1991). [CrossRef] [PubMed]
  2. M. Fleischhauer, A. Imamoglu, and J. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005). [CrossRef]
  3. P. K. Vudyasetu, D. Starling, and J. C. Howell, “All optical waveguiding in a coherent atomic rubidium vapor,” Phys. Rev. Lett.102, 123602 (2009). [CrossRef] [PubMed]
  4. O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, “Elimination, reversal and directional bias of optical diffraction,” Nat. Phys.5, 665–668 (2009). [CrossRef]
  5. O. Firstenberg, M. Shuker, N. Davidson, and A. Ron, “Elimination of the diffraction of arbitrary images imprinted on slow light,” Phys. Rev. Lett.102, 043601 (2009). [CrossRef] [PubMed]
  6. M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, “Storing images in warm atomic vapor,” Phys. Rev. Lett.100, 223601 (2008). [CrossRef] [PubMed]
  7. L. Zhao, T. Wang, Y. Xiao, and S. F. Yelin, “Image storage in hot vapors,” Phys. Rev. A77, 041802 (2008). [CrossRef]
  8. H. He, M. Friese, N. 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, 826–829 (1995). [CrossRef] [PubMed]
  9. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316 (2001). [CrossRef] [PubMed]
  10. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev.2, 299–313 (2008). [CrossRef]
  11. R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett.98, 203601 (2007). [CrossRef] [PubMed]
  12. Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A375, 2958–2963 (2011). [CrossRef]
  13. D. Neshev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun.151, 413–421 (1998). [CrossRef]
  14. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt.40, 73–87 (1993). [CrossRef]
  15. D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett.79, 3399–3402 (1997). [CrossRef]
  16. D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B14, 3054–3065 (1997). [CrossRef]
  17. X. Gan, J. Zhao, S. Liu, and L. Fang, “Generation and motion control of optical multi-vortex,” Chin. Opt. Lett.7, 1142–1145 (2009). [CrossRef]
  18. O. Firstenberg, M. Shuker, R. Pugatch, D. Fredkin, N. Davidson, and A. Ron, “Theory of thermal motion in electromagnetically induced transparency: effects of diffusion, Doppler broadening, and Dicke and Ramsey narrowing,” Phys. Rev. A77, 043830 (2008). [CrossRef]
  19. M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam.” J. Opt. Soc. Am. A25, 1279–1286 (2008). [CrossRef]
  20. O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, “Self-similar modes of coherent diffusion,” Phys. Rev. Lett.105, 183602 (2010). [CrossRef]
  21. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A20, 1635–1643 (2003). [CrossRef]
  22. P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic rubidium vapor,” Phys. Rev. Lett.100, 123903 (2008). [CrossRef] [PubMed]
  23. E. E. Mikhailov, I. Novikova, M. D. Havey, and F. A. Narducci, “Magnetic field imaging with atomic Rb vapor,” Opt. Lett.34, 3529–3531 (2009). [CrossRef] [PubMed]

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