## Dynamics of a paired optical vortex generated by second-harmonic generation |

Optics Express, Vol. 18, Issue 17, pp. 17796-17804 (2010)

http://dx.doi.org/10.1364/OE.18.017796

Acrobat PDF (842 KB)

### Abstract

We study the dynamics of a paired optical vortex (OV) generated by second-harmonic generation (SHG) using sub-picosecond pulses. By changing the position of a thin nonlinear crystal along the propagation direction, we observe a rotation of two vortices with changing separation distance. The dynamics is well explained by SHG with a beam walk-off, which introduces a contamination of zero-order Laguerre-Gaussian beam (LG_{0}) together with topological charge doubling. The quantitative analysis indicates that the rotation angle of the OVs manifests the Gouy phase while the splitting provides the walk-off angle of the crystal. We also show that the subtraction of LG_{0} is realized by the superposition of LG_{0} with an anti-balanced phase in the pump.

© 2010 Optical Society of America

## 1. Introduction

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

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

3. Y. Tokizane, K. Shimatake, Y. Toda, K. Oka, M. Tsubota, S. Tanda, and R. Morita, “Global evaluation of closed-loop electron dynamics in quasi-one-dimensional conductors using polarization vortices,” Opt. Express **17**, 24198–24207 (2009). [CrossRef]

4. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**, 5448–5456 (2004). [CrossRef] [PubMed]

*of the paraxial wave equation, where*

^{p}_{ℓ}*ℓ*and

*p*are integer. The index

*ℓ*is socalled topological charge or orbital angular momentum (OAM), given by the winding number of phase on the wavefront around the vortex [5

5. 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**, 8185–8189 (1992). [CrossRef] [PubMed]

*p*denotes the number of nodal rings about the beam axis. In the paper, we will focus our attention on singly-ringed LG

^{0}

*modes, and hereafter describe as LG*

_{ℓ}*. Since vortices make an important role in various branches of physics, such as fluid mechanics, condensed matter physics, and astrophysics, the study of OVs provides a universal viewpoint in optical physics and thus has attracted fundamental interest as well [6*

_{ℓ}6. M. Berry, “Making waves in physics,” Nature **403**, 21–21 (2000). [CrossRef] [PubMed]

7. M.J. Paz-Alonso and H. Michinel, “Superfluidlike Motion of Vortices in Light Condensates,” Phys. Rev. Lett. **94** 093901-1–4 (2005). [CrossRef] [PubMed]

*β*-Barium Borate) crystal. By changing the crystal position along the propagation direction, a rotational dynamics of the paired OV including creation and collapse of pairing is observed and quantitatively analyzed. Each dynamics observed in the experiment is well reproduced by the theoretical simulation based on the beam walk-off, which produces the changes in relative amplitude and phase of the composite OVs (LG

_{0}+LG

_{2}) according to the position of the nonlinear crystal. The experiment is simple but provides an accurate evolution of the vortices without changing neither the propagation distance nor crystal length. We also demonstrate the compensation of the walk-off effect by superimposing the LG

_{0}beam in the pump.

## 2. Second-harmonic vortices with a beam walk-off

15. A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. **150**, 372–280 (1998). [CrossRef]

16. F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. **95**, 253901-1–4 (2005). [CrossRef] [PubMed]

17. A. Dreischuh, D. N. Neshev, V. Z. Kolev, S. Saltiel, M. Samoc, W. Krolikowski, and Y. S. Kivshar, “Nonlinear dynamics of two-color optical vortices in lithium niobate crystals,” Opt. Express **16**, 5406–5420 (2008). [CrossRef] [PubMed]

*o*+

*o*) and extraordinary second-harmonic(

*e*) beams. We also assume the crystal thickness is thin enough to satisfy the weak walk-off regime. The comprehensive analysis for various phase-matching conditions can be found elsewhere [15

15. A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. **150**, 372–280 (1998). [CrossRef]

*ℓ*, frequency

*ω*and wavenumber

*k*is given by

*r*,

*φ*, and

*z*are the cylindrical coordinates, and

*E*

_{0}is the amplitude parameter.

*u*is the envelope function of the LG

_{ℓ}*mode denoted as [5*

_{ℓ}5. 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**, 8185–8189 (1992). [CrossRef] [PubMed]

11. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A **56**, 4064–4075 (1997). [CrossRef]

*and beam size*

^{ℓ}_{G}*w*(

*z*) are given by

*w*

_{0}and Rayleigh range

*z*=

_{R}*kw*

^{2}

_{0}/2.

15. A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. **150**, 372–280 (1998). [CrossRef]

*a*in Cartesian coordinates as

_{ℓ}*x*+

*iy*(

*x*−

*iy*) is taken for positive (negative)

*ℓ*. From the coupled wave equations in a quadratic nonlinear crystal, the ordinary polarized fundamental and extraordinary second-harmonic field amplitudes (

*E*and

^{F}*E*) follow the equations

^{S}*α*is the walk-off angle of the extraordinary second-harmonic beam, and

*g*is the coupling coefficients. We note again that the pump depletion is neglected owing to the low conversion efficiency. Since we consider a thin crystal (thickness

*L*), the beam size can be fixed to be constant throughout the crystal. At the center position of the crystal

*z*′, the coupled amplitude equation becomes

*ξ*=

*x*/

*w*(

*z*′),

*η*=

*y*/

*w*(

*z*′),

*t*=

*z*/

*L*and assume

*γ*=

*αL*/

*w*(

*z*′),

*F*= (

*w*

_{0}/

*w*(

*z*′))

^{2}

*E*/(

^{S}*gL*). The second-harmonic envelope at the crystal position

*z*′ in Eq. (7) can be transformed to

*ℓ*= 1. Taking into account the assumption that the crystal is sufficiently thin, Eq. (8) gives

*ξ*|, |

*η*| ≪ 1, which is consistent with the weak walk-off condition.

*F*(

*ξ*,

*η*) = 0 results in two vortices at (

*ξ*

_{0},

*η*

_{0}) = (

*γ*/2,±

*γ*/(2√3)). Here, it should be noted that, while the usual walk-off between fundamental and second-harmonic waves occur in

*x*-direction, another type of walk-off, that is, singular-point splitting arises in

*y*-direction. In this case, the envelope function of SHG amplitude can be described by a product of two vortices,

*s*

_{1}=

*αL*/2 and

*s*

_{2}=

*αL*/(2√3). The splitting of the vortices occurs in the

*y*-direction (crystal axis) with a separation of 2

*s*

_{2}=

*αL*/√3. Equation (10) follows

_{0}and LG

_{2}with different amplitude and phase. The amplitude of the second-harmonic beam consisting of a product of two vortices thus can be transformed to a series of OVs with different charges. It should be emphasized here that a simple SHG process with a walk-off introduces a contamination of

*ℓ*= 0 beam together with topological charge doubling. In addition, both the relative amplitude ∝ 1/

*w*

^{2}and phase difference Φ

^{2}

*− Φ*

_{G}^{0}

*depends on*

_{G}*z*′. As a result, we can evaluate the interaction dynamics of the paired vortex as a function of the crystal position in

*z*-direction.

## 3. Experimental

_{1}pump. Figure 1 (a) shows the schematic of the experimental setup. A mode-locked Ti:sapphire laser with a repetition rate of ~90 MHz and a center wavelength of ~800 nm was used for the light source. The LG

_{1}pulse (fundamental pulse for the SHG) was created from the HG

_{00}pulse from the laser by passing through a spatial phase shifter (SPS) and an astigmatic mode converter (AMC) [18

18. M. W. Beijersbergen, L. Allen, H. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. **96**, 123–132 (1993). [CrossRef]

_{10}with a relative phase shift of

*π*between the left and right halves of the HG

_{00}beam [19

19. Y. Yoshikawa and H. Sasada, “Versatile generation of optical vortices based on paraxial mode expansion,” J. Opt. Soc. Am. A **19**, 2127–2133 (2002). [CrossRef]

_{10}pulse is then fed to an astigmatic lens system consisting of two spherical and cylindrical lens.

_{00}pulse without delay. The fork-like fringe pattern clearly indicates the LG with a topological charge of unity (LG

_{1}) is dominant at this stage. The temporal chirp of the pulse was compensated by a pair of chirp mirrors just after the laser source. The average power of the LG

_{1}was ~10 mW and its pulse duration evaluated by autocorrelation measurements was ~100 fs.

_{1}was focused onto a BBO crystal (CASIX, thickness 0.1 mm) using a lens (

*f*= 100 mm), and the output SHG with type I phase matching was detected by a charge-coupled device (CCD) camera after passing through a UV pass filter. The BBO has a large walk-off angle (~ 3.9° at 800 nm pump) [20

20. K. Kato, “Second-harmonic generation to 2048 Åin *β*-BaB_{2}O_{4},” IEEE J. Quantum Electron. , **QE-22**, 1013–1014 (1986). [CrossRef]

*z*′) was automatically moved along the optical axis by using a motorized linear stage. The optimum focus position of the input pulse is defined as

*z*′ = 0, and we obtained the intensity distributions of the SHG over a range between

*z*′ = ±7.5 mm, which corresponds to

*z*′ = ±2.3

*z*using a Rayleigh length of

_{R}*z*(=

_{R}*πw*

^{2}

_{0}/

*λ*≈ 3.3 mm, where

*w*

_{0}=29

*µ*m at

*λ*=800 nm). Note that

*z*for the fundamental and second-harmonic beams are equal since

_{R}*w*

_{0}of SHG is reduced by a factor of 1/√2.

*z*′. In Fig. 2 (c), there are two distinct vortices, each of which rotates as a pair together with decreasing the separation as the crystal moves away from

*z*′ = 0 (Fig. 2 (b) and (d)), and becomes indistinguishable at |

*z*′| > 2

*z*(Fig. 2 (a) and (e)). These sequential dynamics can be evaluated as trajectories of the vortices in Fig. 3 (a). Note that we detect the SHG at a fixed position, allowing to evaluate the sequential vortices with the same dimension. For simplicity, we plot the positions of the vortices normalized by the beam size of the SHG (

_{R}*w*). Following the same manner as the previous reports on the propagation dynamics [12

_{s}12. D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B **14**, 3054–3065 (1997). [CrossRef]

21. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express **17**, 9818–9827 (2009). [CrossRef] [PubMed]

*ϕ*and distance

_{v}*r*between the two vortices as a function of the crystal position

_{v}*z*′/

*z*in Fig. 3 (b) and (c). Here

_{R}*ϕ*and

_{v}*r*are denoted in Fig. 1 (c), and reflect the relative phase and amplitude of the composite OVs. As shown in Fig. 3 (b),

_{v}*ϕ*changes monotonically between −

_{v}*π*/2 and +

*π*/2, suggesting the Gouy phase rotation [21

21. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express **17**, 9818–9827 (2009). [CrossRef] [PubMed]

22. J. H. Chow, G. de Vine, M. B. Gray, and D. E. McClelland, “Measurement of Gouy phase evolution by use of spatial mode interference,” Opt. Lett. **29**, 2339–2341 (2004). [CrossRef] [PubMed]

23. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express **14**, 8382–8392 (2006). [CrossRef] [PubMed]

*r*decreases symmetrically with respect to

_{v}*z*′ = 0, indicating that the perturbation of LG

_{0}is efficient at

*z*′ = 0 and decreases as increasing |

*z*′|. These

*z*′-dependent splitting suggests the walk-off origin, which will be confirmed below.

*I*= |

*u*

^{S}_{2}|

^{2}where

*u*

^{S}_{2}is given by in Eq. (11). In the calculations, only

*s*

_{2}is a variable parameter to be optimized. We used

*s*

_{2}=4.1

*µ*m, the validity of which will be discussed later. The intensity distributions of the composite OVs are shown in the upper part of Fig. 4, each of which reproduces well the paired OV dynamics in Fig. 2. In the lower part, we also plot the phase distributions in order to clarify the positions of the vortices. Within the focal region, a distinct paired vortex exists owing to the intentional contamination of LG

_{0}∝ 1/

*w*

^{2}. When moving out of focal region, the pair rotates according to the relative Gouy rotation ∆Φ

*= Φ*

_{G}^{2}

*− Φ*

_{G}^{0}

*, and its separation decreases as LG*

_{G}_{0}decreases.

*ϕ*, which reflects the relative phase described by ∆Φ

_{v}*= 2tan*

_{G}^{−1}(

*z*/

*z*), is consistent with the experimental data. Our simple experiment thus provides an evaluation of the the Gouy phase with a fixed detection geometry. On the other hand, the splitting

_{R}*r*in Fig. 5 (c) also shows good agreement with experimental results. We note that

_{v}*r*is determined by the ratio of the amplitude between LG

_{v}_{0}and LG

_{2}, which is given by (

*s*

_{2}/

*w*)

^{2}in Eq. (11). From the experimental data

*r*at

_{v}*z*′ = 0,

*s*

_{2}(=

*r*/2) is evaluated to be 4.1

_{v}*µ*m. Within the assumption that the beam walk-off is weak,

*s*

_{2}can be described by

*α*′

*L*/(2√3), where

*α*′ is the external walk-off angle in the air which satisfies sin (

*α*′/√3) =

*n*(400 nm) sin (

_{e}*α*/√3) from the Snell’s law. In the present case, we obtained the internal walk-off angle

*α*= 5.1°, which is almost consistent with the typical value of BBO for the type-I phase-matching (~ 3.9° at 800 nm pump) [20

20. K. Kato, “Second-harmonic generation to 2048 Åin *β*-BaB_{2}O_{4},” IEEE J. Quantum Electron. , **QE-22**, 1013–1014 (1986). [CrossRef]

_{0}. Therefore we can remove the splitting by introducing LG

_{0}(HG

_{00}) as another fundamental source. Although such cancellation of LG

_{0}perturbation using coherent beam addition has already been reported in [8

8. I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light-beams with screw dislocations,” Opt. Commun. **103**, 422–428 (1993). [CrossRef]

24. A. Bahabad and A. Arie, “Generation of optical vortex beams by nonlinear wave mixing,” Opt. Express **15**, 17619–17624 (2007). [CrossRef] [PubMed]

25. Y. Ueno, Y. Toda, S. Adachi, R. Morita, and T. Tawara, “Coherent transfer of orbital angular momentum to excitons by optical four-wave mixing,” Opt. Express **17**, 20567–20574 (2009). [CrossRef] [PubMed]

_{00}) were splitted into two optical paths, one of which transfers HG

_{00}to LG

_{1}using SPS and AMC. Another path carries LG

_{0}beam through the optical delay line, which consists of a retro-reflector mounted upon a translation stage and a mirror upon a piezoelectric transducer. The former accounts for the difference between two optical paths lengths within the pulse duration while the latter provides a fine position to determine the relative phase

*φ*with respect to LG

_{p}_{1}. The relative amplitude between LG

_{1}and LG

_{0}was controlled by a neutral density filter. The second-harmonic conversion and its detection schemes are the same as in Fig. 1 (a).

*φ*at a fixed crystal position

_{p}*z*′ = 0. The OV pair changes its separation according to

*φ*. A significant reduction of the splitting was observed in the anti-phase condition

_{p}*φ*~

_{p}*π*. For clarity, the SHG images obtained by introducing individual pump beams (LG

_{1}and HG

_{00}) are shown in the lower part. It is important to note that optimization of the relative amplitude is necessary to realize the absence of splitting. In addition, the optimized amplitude varies with

*z*′, indicating that the splitting observed in our experiment does not arise from the fundamental beam but from the SHG.

## 4. Summary

_{0}, the subtraction of which was experimentally realized by the superposition of LG

_{0}with an anti-balanced phase in the fundamental beam.

## Acknowledgments

## References and links

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

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

3. | Y. Tokizane, K. Shimatake, Y. Toda, K. Oka, M. Tsubota, S. Tanda, and R. Morita, “Global evaluation of closed-loop electron dynamics in quasi-one-dimensional conductors using polarization vortices,” Opt. Express |

4. | G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express |

5. | 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 |

6. | M. Berry, “Making waves in physics,” Nature |

7. | M.J. Paz-Alonso and H. Michinel, “Superfluidlike Motion of Vortices in Light Condensates,” Phys. Rev. Lett. |

8. | I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light-beams with screw dislocations,” Opt. Commun. |

9. | K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. |

10. | G. Indebetow, “Optical vortices and their propagation,” J. Mod. Opt. |

11. | M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A |

12. | D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B |

13. | I. Freund, “Optical vortex trajectories,” Opt. Commun. 181, 19–33 (2000). I. D. Maleev, and G. A. Swartzlander, “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169–1176 (2003). [CrossRef] |

14. | K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A |

15. | A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. |

16. | F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. |

17. | A. Dreischuh, D. N. Neshev, V. Z. Kolev, S. Saltiel, M. Samoc, W. Krolikowski, and Y. S. Kivshar, “Nonlinear dynamics of two-color optical vortices in lithium niobate crystals,” Opt. Express |

18. | M. W. Beijersbergen, L. Allen, H. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. |

19. | Y. Yoshikawa and H. Sasada, “Versatile generation of optical vortices based on paraxial mode expansion,” J. Opt. Soc. Am. A |

20. | K. Kato, “Second-harmonic generation to 2048 Åin |

21. | S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express |

22. | J. H. Chow, G. de Vine, M. B. Gray, and D. E. McClelland, “Measurement of Gouy phase evolution by use of spatial mode interference,” Opt. Lett. |

23. | J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express |

24. | A. Bahabad and A. Arie, “Generation of optical vortex beams by nonlinear wave mixing,” Opt. Express |

25. | Y. Ueno, Y. Toda, S. Adachi, R. Morita, and T. Tawara, “Coherent transfer of orbital angular momentum to excitons by optical four-wave mixing,” Opt. Express |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.2620) Nonlinear optics : Harmonic generation and mixing

(050.4865) Diffraction and gratings : Optical vortices

(260.6042) Physical optics : Singular optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 15, 2010

Revised Manuscript: July 21, 2010

Manuscript Accepted: July 28, 2010

Published: August 3, 2010

**Citation**

Y. Toda, S. Honda, and R. Morita, "Dynamics of a paired optical vortex generated by second-harmonic
generation," Opt. Express **18**, 17796-17804 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17796

Sort: Year | Journal | Reset

### References

- K. T. Gahagan and G. A. Swartzlander, Jr., “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996). [CrossRef] [PubMed]
- D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef] [PubMed]
- Y. Tokizane, K. Shimatake, Y. Toda, K. Oka, M. Tsubota, S. Tanda, and R. Morita, “Global evaluation of closed-loop electron dynamics in quasi-one-dimensional conductors using polarization vortices,” Opt. Express 17, 24198–24207 (2009). [CrossRef]
- G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef] [PubMed]
- 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, 8185–8189 (1992). [CrossRef] [PubMed]
- M. Berry, “Making waves in physics,” Nature 403, 21–21 (2000). [CrossRef] [PubMed]
- M. J. Paz-Alonso and H. Michinel, “Superfluidlike Motion of Vortices in Light Condensates,” Phys. Rev. Lett. 94, 093901-1–4 (2005). [CrossRef] [PubMed]
- I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light-beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993). [CrossRef]
- K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123–127 (1992). [CrossRef]
- G. Indebetow, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993). [CrossRef]
- M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997). [CrossRef]
- D. Rozas, C. T. Law, G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997). [CrossRef]
- . I. Freund, “Optical vortex trajectories,” Opt. Commun. 181, 19–33 (2000). I. D. Maleev, and G. A. Swartzlander, “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169–1176 (2003). [CrossRef]
- K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–3745 (1996). [CrossRef] [PubMed]
- A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–280 (1998). [CrossRef]
- F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901-1–4 (2005). [CrossRef] [PubMed]
- A. Dreischuh, D. N. Neshev, V. Z. Kolev, S. Saltiel, M. Samoc, W. Krolikowski, and Y. S. Kivshar, “Nonlinear dynamics of two-color optical vortices in lithium niobate crystals,” Opt. Express 16, 5406–5420 (2008). [CrossRef] [PubMed]
- M. W. Beijersbergen, L. Allen, H. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]
- Y. Yoshikawa and H. Sasada, “Versatile generation of optical vortices based on paraxial mode expansion,” J. Opt. Soc. Am. A 19, 2127–2133 (2002). [CrossRef]
- K. Kato, “Second-harmonic generation to 2048 Å in β -BaB2O4,” IEEE J. Quantum Electron., QE-22, 1013–1014 (1986). [CrossRef]
- S. M. Baumann, D. M. Kalb, L. H. MacMillan, E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009). [CrossRef] [PubMed]
- J. H. Chow, G. de Vine, M. B. Gray, and D. E. McClelland, “Measurement of Gouy phase evolution by use of spatial mode interference,” Opt. Lett. 29, 2339–2341 (2004). [CrossRef] [PubMed]
- J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, "Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express 14, 8382–8392 (2006). [CrossRef] [PubMed]
- A. Bahabad and A. Arie, “Generation of optical vortex beams by nonlinear wave mixing,” Opt. Express 15, 17619–17624 (2007). [CrossRef] [PubMed]
- Y. Ueno, Y. Toda, S. Adachi, R. Morita and T. Tawara, “Coherent transfer of orbital angular momentum to excitons by optical four-wave mixing,” Opt. Express 17, 20567–20574 (2009). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.