Abstract

We experimentally demonstrated the generation of a dynamic nonlinear vortex beam array by utilizing a fundamental wave with a modulated phase structure, which was incident into a homogeneous nonlinear medium. In our experiment, one-dimensional and two-dimensional second harmonic vortex beam arrays were investigated, and the topological charge of second harmonic vortex beam of different order was measured. This study presents a method of dynamic control of the nonlinear vortex beam array, which may have applications in multiple-particles optical trapping, optical communication, and so on.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

It is well known that optical angular momentum is composed of spin angular momentum and orbital angular momentum (OAM) [1]. The OAM of light was firstly observed in 1930s [2]. In 1992, Allen et al. verified that OAM has an azimuthal phase termeilφ, where l and φ represent the topological charge and azimuth angle, respectively [3]. In the last two decades, optical OAM has attracted great interest due to its potential applications in various research fields, such as optical trapping [4–8], quantum computation [9], optical communications [10]. In the field of optical trapping, the beam carrying OAM can be used to rotate particles with different angular velocities [4–8]. Quantum computation and quantum imaging have been realized by utilizing the entanglement of optical OAM [9]. In addition, such optical OAM could also be used to increase the capacity of optical communications [10]. The nonlinear generation process enables vortex beams to be obtained at new wavelengths, which opens up new possibilities for all-optical switching and manipulation of vortex beams. Hence, second harmonic generation (SHG) [11–14], sum-frequency generation [11–13], difference frequency generation [12] and even high-harmonic [15–17] of the vortex beam has been investigated. To get SHG of the vortex beam, there are two main methods. One is to manipulate the structure of nonlinear photonic crystals (NPCs) [11,13]. The other is to pattern the output facet of a nonlinear crystal [18,19], which combines both the nonlinear process and the beam spatial shaping. However, these two methods possess the same drawbacks, including complex fabrication and unchangeable nonlinear wave patterns. To solve these problems, some researchers turn their attention to pre-shaping the fundamental wave (FW) before its incidence into a nonlinear crystal [14,20–23]. In our previous researches, we have already introduced the structured FW into nonlinear frequency conversion processes. For example, the nonlinear Raman-Nath SHG of different types of FW was achieved [24].

In this paper, we experimentally demonstrated the nonlinear vortex beam array dynamically generated by utilizing the FW with a modulated phase structure via a spatial light modulator (SLM), which was incident into a homogeneous nonlinear medium. In the one-dimensional (1D) case, a series of second harmonic (SH) vortex beam was achieved in 1D direction. In the two-dimensional (2D) case, SH vortex beam array in 2D direction was generated. Besides, the topological charge of SH vortex beam of different order located in such SH vortex beam array was also experimentally measured.

2. Theoretical analysis

The phase structure of the FW was periodically and sharply modulated from 0 to ϕ into fork-shaped structure. The diffraction of FW is negligible in case of long-period modulation of the FW in a sufficient short medium [24]. And supposing the light propagates along the y-axis (longitudinal direction) of the crystal, in 1D case, the FW is given in the form of the Fourier series by:

E1=A1exp[i(k1yωt)]Cmexp[i2πmf(r,φ)iml1φ],
where A1 and k1 are the amplitude and wave vector of the FW, respectively. Cm is the Fourier coefficients (C0=(1+eiϕ)/2, Cm0=i[cos(mπ)1]2mπ(1eiϕ)). r=xx+yy,φ=tan1(z/x) is the azimuthal angle, f(r,φ)=|r|cos(φ)/Λ is the spiral structure function, Λ is spatial period of the plane wave, l1 is the topological charge of the FW. The SH can be expressed asE2=A2exp[i(k2tr+k2yy2ωt)], whereA2, k2t and k2yrepresent the amplitude, transverse and longitudinal wavevector, respectively. Under the non-depletion assumption, the evolution of the SH wave can be written as [15]:
dA2dy=ε0χ(2)A12buexp(ik2yyi2k1y)exp[ik2tri2πuf(r,φ)]exp(iul1φ),
where bu=m,nm+n=uCmCn is the Fourier coefficient of the nonlinear polarization. In Eq. (2), the first exponential term denotes the longitudinal phase mismatch, the middle exponential term is the nonlinear Raman-Nath diffraction, and the last exponential term represents the topological charge of the generated SH vortex beam. As can be seen, the topological charge of the generated SH vortex beam obeys the law of lSH=ul1.

When the phase is modulated in 2D, the FW can be expressed as

E1=A1ei(k1yωt)m=m=+Cmeim[2πf1(r,φ)+lxφ]p=p=+Cpeip[2πf2(r,φ)+lzφ],
where lx and lz are the topological charges of the FW along x and z direction, respectively. Then the coupled wave equation of SH can be written as:
dA2dy=ε0χ(2)A12ei(k2y2k1)yu=u=+buv=v=+bvei[k2tr2πuf1(r,φ)2πvf2(r,φ)]ei(ulx+vlz)φ,
where bu=m,nm+n=uCmCn and bv=p,qp+q=vCpCq. In this case, the topological charge of the SH vortex beam islSH=ulx+vlz.

3. Experimental results and discussion

In our experiments, a Nd:YAG nanosecond laser with a wavelength and a repetition rate of 1064 nm and 20 Hz, respectively, was applied. The sample was a homogenous 5-mol% MgO:LiNbO3 crystal (10 × 0.5 × 10 mm3 in x × y × z dimensions). The FW was an ordinary wave and incident into the crystal along the y direction. The nonlinear process was Type-I (oo-e) phase matching. The FW was modulated by a SLM, whose resolution is 512 × 512 pixels and each with a square area of 19.5 × 19.5μm2. A 4-f system was used to imprint the phase structure on the nonlinear crystal and the beam waist was 2 mm. A shortpass filter was placed after the crystal to obstruct the FW. At last, the generated SH vortex beam array was projected on a screen and then recorded by a camera.

In order to illuminate the relationship between the generated SH vortex beam array and the structure of the FW, we firstly imprinted different fork-shaped structured FW in 1D case onto the nonlinear crystal. The structure of the FW is shown in the left column of Fig. 1, where the fork-shaped phase structure is 0-π/2 sharply modulated. The duty cycle is 0.5. The corresponding experimental results of the SH vortex beams are shown in the right column of Fig. 1, where u represents the Raman-Nath order. The phase singularity (dark core) always exists as long as lSH is nonzero. The radius of the SH vortex ring gets larger as the topological charge increases. According to Eq. (2), the topological charge of the SH vortex beam obeys the law of lSH=ul1 for the uth order SH. In order to determine the topological charge of the vortex beam in different orders, we measured it by utilizing a cylindrical lens [25,26]. The results are displayed in Fig. 2, which corresponds to the lSH=±1,±2,±3 orders of the SH vortex beam in the case of l1=1. The number of dark stripes equals to the topological charge of the SH, which agrees well with the theoretical prediction.

 figure: Fig. 1

Fig. 1 Left column: The different phase structures of the FW with 0-π/2 phase modulation. Right column: The generated SH vortex beams in different orders of nonlinear Raman-Nath diffraction. lSH=ul1 for the uth order SH.

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 figure: Fig. 2

Fig. 2 The measurement of the topological charge by a cylindrical lens when lSH=±1,±2,±3. The number of the dark stripes represents the topological charge.

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When the FW phase was sharply modulated from 0 to π, the odd orders of SH vortex beam vanishes, as shown in Fig. 3. Because the Fourier coefficients Cm are zero when m is even, as a result bu=0 (bu=m,nm+n=uCmCn) if u is odd.

 figure: Fig. 3

Fig. 3 Left column: different phase structures of the FW with 0-π phase modulation. Right column: The generated SH vortex beam in different orders of nonlinear Raman-Nath diffraction. lSH=ul1 for the uth order SH.

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In addition to the generation of SH vortex beam with integer topological charges, the case of fraction topological charge was also investigated. The FW structure is shown in the left column of Fig. 4 and the corresponding experimental results are shown in the right column. It is interesting that the SH vortex beam was not a closed ring when lSH was not an integer. In this case, a dark stripe exists on the SH vortex beam according to Berry’s theory in 2004 [27]. As the topological charge of the SH vortex beam was further tuned away from the integer, the gap becomes larger. For example, the gap is the largest when lSH is a half-integer (lSH=2.5). When lSH is close to an integer (such as lSH=2.9 shown the + 1 order in the right column of the last row of Fig. 4), the SH vortex beam returned to a ring. Similar to the integer cases of SH vortex beam (lSH0), the phase singularity still exists and the radius of the ring is determined by the topological charge lSH.

 figure: Fig. 4

Fig. 4 Left column: Different phase structures of the FW with 0-π/2 phase modulation with fractional topological charge l1=2.5,2.6,2.7,2.9, respectively. Right column: The generated SH vortex beams in different orders of nonlinear Raman-Nath diffraction. lSH=ul1 for the uth order SH.

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Figure 5 shows the process after the vortex beams passing through the cylindrical lens with the topological charge changing from 2 to 3. The original two dark stripes correspond tolSH=2. With the fractional topological charge of SH vortex beams increasing, another dark stripe comes into being and becomes more and more obvious. At last, the right bright stripe splits into two stripes completely when lSH is close to 3.

 figure: Fig. 5

Fig. 5 Evolution of dark stripes with the topological charge changing from 2 to 3.

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To further reveal the flexibility of our method, the 2D SH vortex beam array generation was studied. The structure of the FW is modulated along the x and z direction, as shown in the top row of Fig. 6, where lx and lz denote the topological charge of the fork-shaped FW along the x-axis and z-axis, respectively. The 2D SH vortex beam array was achieved, as shown in the bottom row of Fig. 6. The topological charge of the SH vortex beam located in the SH vortex beam array can be written as lSH=ulx+vlz, where ulx and vlz are the topological charge along x and z-axis, respectively. The topological charges of SH vortex beams of different orders are measured to be identical to the analysis (lSH=ulx+vlz).

 figure: Fig. 6

Fig. 6 Top row: The different phase structures of the FW with 0-π/2 phase modulation. lx and lz represent the topological charge along x and z axis, respectively. Bottom row: The generated SH vortex beams in different orders of nonlinear Raman-Nath diffraction. lSH=ulx+vlz for the (uth,vth) order SH.

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Compared with the method of modulating NPCs, our method is much more convenient, since it is dynamically controlled. Different from [14] where the authors conducted the experiment making use of diffraction of FW, we paid much attention to the nonlinear diffraction process. In ref [14], two FWs with different topological charge were incident into the nonlinear crystal. In our experiment, the whole phase structure of the FW loaded on the SLM was printed onto the nonlinear crystal. Hence, the completely different experimental phenomena were obtained. SH vortex array in 1D and 2D cases are also systematically studied in our work.

4. Conclusion

In summary, we experimentally demonstrated the nonlinear vortex beam array dynamically generated by utilizing the FW, in a homogeneous nonlinear medium. In 1D cases, the topological charge obeys the law of lSH=ul1 in different orders. In 2D case, SH vortex beam array in 2D direction can be generated and the topological charge obeys the law of lSH=ulx+vlz. Besides, the topological charge of SH vortex beam of different orders located in such SH vortex beam array was also experimentally measured. This study presents a method of dynamic control of the nonlinear vortex beam array, which may have applications in multiple-particles optical trapping, optical communication, and so on.

Funding

National Natural Science Foundation of China (NSFC) (11734011, 61235009); National Key R&D Program of China (2017YFA0303700).

References and links

1. 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). [PubMed]  

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

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). [PubMed]  

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

5. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006). [PubMed]  

6. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).

7. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003). [PubMed]  

8. Z. Shen, L. Su, X. C. Yuan, and Y. C. Shen, “Trapping and rotating of a metallic particle trimer with optical vortex,” Appl. Phys. Lett. 109(24), 241901 (2016).

9. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [PubMed]  

10. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

11. N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012). [PubMed]  

12. G. H. Shao, Z. J. Wu, J. H. Chen, F. Xu, and Y. Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching,” Phys. Rev. A 88(6), 063827 (2013).

13. K. Shemer, N. Voloch-Bloch, A. Shapira, A. Libster, I. Juwiler, and A. Arie, “Azimuthal and radial shaping of vortex beams generated in twisted nonlinear photonic crystals,” Opt. Lett. 38(24), 5470–5473 (2013). [PubMed]  

14. S. M. Li, L. J. Kong, Z. C. Ren, Y. Nan Li, C. H. Tu, and H. T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).

15. F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017). [PubMed]  

16. G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014). [PubMed]  

17. D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017). [PubMed]  

18. A. Shapira, A. Libster, Y. Lilach, and A. Arie, “Functional facets for nonlinear crystals,” Opt. Commun. 300, 244–248 (2013).

19. S. Lightman, R. Gvishi, G. Hurvitz, and A. Arie, “Shaping of light beams by 3D direct laser writing on facets of nonlinear crystals,” Opt. Lett. 40(19), 4460–4463 (2015). [PubMed]  

20. A. Libster-Hershko, S. Trajtenberg-Mills, and A. Arie, “Dynamic control of light beams in second harmonic generation,” Opt. Lett. 40(9), 1944–1947 (2015). [PubMed]  

21. B. K. Singh, R. Remez, Y. Tsur, and A. Arie, “Super-Airy beam: self-accelerating beam with intensified main lobe,” Opt. Lett. 40(20), 4703–4706 (2015). [PubMed]  

22. R. Ni, Y. F. Niu, L. Du, X. P. Hu, Y. Zhang, and S. N. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).

23. B. Yang, X. H. Hong, R. E. Lu, Y. Y. Yue, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “2D wave-front shaping in optical superlattices using nonlinear volume holography,” Opt. Lett. 41(13), 2927–2929 (2016). [PubMed]  

24. H. Liu, J. Li, X. Zhao, Y. Zheng, and X. Chen, “Nonlinear Raman-Nath second harmonic generation with structured fundamental wave,” Opt. Express 24(14), 15666–15671 (2016). [PubMed]  

25. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17(26), 23374–23379 (2009). [PubMed]  

26. V. G. Shvedov, C. Hnatovsky, W. Krolikowski, and A. V. Rode, “Efficient beam converter for the generation of high-power femtosecond vortices,” Opt. Lett. 35(15), 2660–2662 (2010). [PubMed]  

27. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).

References

  • View by:

  1. 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).
    [PubMed]
  2. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936).
  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).
    [PubMed]
  4. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [PubMed]
  5. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
    [PubMed]
  6. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
  7. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
    [PubMed]
  8. Z. Shen, L. Su, X. C. Yuan, and Y. C. Shen, “Trapping and rotating of a metallic particle trimer with optical vortex,” Appl. Phys. Lett. 109(24), 241901 (2016).
  9. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
    [PubMed]
  10. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
  11. N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
    [PubMed]
  12. G. H. Shao, Z. J. Wu, J. H. Chen, F. Xu, and Y. Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching,” Phys. Rev. A 88(6), 063827 (2013).
  13. K. Shemer, N. Voloch-Bloch, A. Shapira, A. Libster, I. Juwiler, and A. Arie, “Azimuthal and radial shaping of vortex beams generated in twisted nonlinear photonic crystals,” Opt. Lett. 38(24), 5470–5473 (2013).
    [PubMed]
  14. S. M. Li, L. J. Kong, Z. C. Ren, Y. Nan Li, C. H. Tu, and H. T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).
  15. F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
    [PubMed]
  16. G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
    [PubMed]
  17. D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
    [PubMed]
  18. A. Shapira, A. Libster, Y. Lilach, and A. Arie, “Functional facets for nonlinear crystals,” Opt. Commun. 300, 244–248 (2013).
  19. S. Lightman, R. Gvishi, G. Hurvitz, and A. Arie, “Shaping of light beams by 3D direct laser writing on facets of nonlinear crystals,” Opt. Lett. 40(19), 4460–4463 (2015).
    [PubMed]
  20. A. Libster-Hershko, S. Trajtenberg-Mills, and A. Arie, “Dynamic control of light beams in second harmonic generation,” Opt. Lett. 40(9), 1944–1947 (2015).
    [PubMed]
  21. B. K. Singh, R. Remez, Y. Tsur, and A. Arie, “Super-Airy beam: self-accelerating beam with intensified main lobe,” Opt. Lett. 40(20), 4703–4706 (2015).
    [PubMed]
  22. R. Ni, Y. F. Niu, L. Du, X. P. Hu, Y. Zhang, and S. N. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).
  23. B. Yang, X. H. Hong, R. E. Lu, Y. Y. Yue, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “2D wave-front shaping in optical superlattices using nonlinear volume holography,” Opt. Lett. 41(13), 2927–2929 (2016).
    [PubMed]
  24. H. Liu, J. Li, X. Zhao, Y. Zheng, and X. Chen, “Nonlinear Raman-Nath second harmonic generation with structured fundamental wave,” Opt. Express 24(14), 15666–15671 (2016).
    [PubMed]
  25. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17(26), 23374–23379 (2009).
    [PubMed]
  26. V. G. Shvedov, C. Hnatovsky, W. Krolikowski, and A. V. Rode, “Efficient beam converter for the generation of high-power femtosecond vortices,” Opt. Lett. 35(15), 2660–2662 (2010).
    [PubMed]
  27. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).

2017 (2)

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

2016 (4)

Z. Shen, L. Su, X. C. Yuan, and Y. C. Shen, “Trapping and rotating of a metallic particle trimer with optical vortex,” Appl. Phys. Lett. 109(24), 241901 (2016).

R. Ni, Y. F. Niu, L. Du, X. P. Hu, Y. Zhang, and S. N. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).

B. Yang, X. H. Hong, R. E. Lu, Y. Y. Yue, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “2D wave-front shaping in optical superlattices using nonlinear volume holography,” Opt. Lett. 41(13), 2927–2929 (2016).
[PubMed]

H. Liu, J. Li, X. Zhao, Y. Zheng, and X. Chen, “Nonlinear Raman-Nath second harmonic generation with structured fundamental wave,” Opt. Express 24(14), 15666–15671 (2016).
[PubMed]

2015 (3)

2014 (1)

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[PubMed]

2013 (4)

A. Shapira, A. Libster, Y. Lilach, and A. Arie, “Functional facets for nonlinear crystals,” Opt. Commun. 300, 244–248 (2013).

G. H. Shao, Z. J. Wu, J. H. Chen, F. Xu, and Y. Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching,” Phys. Rev. A 88(6), 063827 (2013).

K. Shemer, N. Voloch-Bloch, A. Shapira, A. Libster, I. Juwiler, and A. Arie, “Azimuthal and radial shaping of vortex beams generated in twisted nonlinear photonic crystals,” Opt. Lett. 38(24), 5470–5473 (2013).
[PubMed]

S. M. Li, L. J. Kong, Z. C. Ren, Y. Nan Li, C. H. Tu, and H. T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).

2012 (2)

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[PubMed]

2011 (1)

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).

2010 (1)

2009 (1)

2006 (1)

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[PubMed]

2004 (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).

2003 (2)

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
[PubMed]

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

2002 (1)

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).
[PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[PubMed]

1992 (1)

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).
[PubMed]

1936 (1)

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

Adhikary, G.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Ahmed, N.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

Allen, L.

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).
[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(11), 8185–8189 (1992).
[PubMed]

Andersen, M. F.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[PubMed]

Arie, A.

Arissian, L.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

Beijersbergen, M. W.

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).
[PubMed]

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).

Beth, R. A.

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

Bloch, N. V.

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[PubMed]

Bouchard, F.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

Bowman, R.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).

Boyd, R. W.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[PubMed]

Brown, G. G.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

Camper, A.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Chappuis, C.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Chen, J. H.

G. H. Shao, Z. J. Wu, J. H. Chen, F. Xu, and Y. Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching,” Phys. Rev. A 88(6), 063827 (2013).

Chen, X.

Cladé, P.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[PubMed]

Corkum, P. B.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[PubMed]

Cucini, R.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Curtis, J. E.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
[PubMed]

De Ninno, G.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Denisenko, V.

Desyatnikov, A. S.

DiMauro, L. F.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Dolinar, S.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

Dovillaire, G.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Du, L.

R. Ni, Y. F. Niu, L. Du, X. P. Hu, Y. Zhang, and S. N. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).

Fazal, I. M.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

Frassetto, F.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Frumker, E.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[PubMed]

Gariepy, G.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[PubMed]

Gauthier, D.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Géneaux, R.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Grier, D. G.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
[PubMed]

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

Gvishi, R.

Hammond, T. J.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[PubMed]

Helmerson, K.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[PubMed]

Hnatovsky, C.

Hong, X. H.

Hu, X. P.

R. Ni, Y. F. Niu, L. Du, X. P. Hu, Y. Zhang, and S. N. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).

Huang, H.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

Hurvitz, G.

Juwiler, I.

K. Shemer, N. Voloch-Bloch, A. Shapira, A. Libster, I. Juwiler, and A. Arie, “Azimuthal and radial shaping of vortex beams generated in twisted nonlinear photonic crystals,” Opt. Lett. 38(24), 5470–5473 (2013).
[PubMed]

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[PubMed]

Karimi, E.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

Kim, K. T.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[PubMed]

Kivshar, Y. S.

Ko, D. H.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

Kong, F.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

Kong, L. J.

S. M. Li, L. J. Kong, Z. C. Ren, Y. Nan Li, C. H. Tu, and H. T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).

Krolikowski, W.

Leach, J.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[PubMed]

Li, J.

Li, S. M.

S. M. Li, L. J. Kong, Z. C. Ren, Y. Nan Li, C. H. Tu, and H. T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).

Li, Z.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

Libster, A.

Libster-Hershko, A.

Lightman, S.

Lilach, Y.

A. Shapira, A. Libster, Y. Lilach, and A. Arie, “Functional facets for nonlinear crystals,” Opt. Commun. 300, 244–248 (2013).

Liu, H.

Lu, R. E.

Lu, Y. Q.

G. H. Shao, Z. J. Wu, J. H. Chen, F. Xu, and Y. Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching,” Phys. Rev. A 88(6), 063827 (2013).

MacVicar, I.

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).
[PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[PubMed]

Miotti, P.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Nan Li, Y.

S. M. Li, L. J. Kong, Z. C. Ren, Y. Nan Li, C. H. Tu, and H. T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).

Natarajan, V.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[PubMed]

Neshev, D. N.

Ni, R.

R. Ni, Y. F. Niu, L. Du, X. P. Hu, Y. Zhang, and S. N. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).

Niu, Y. F.

R. Ni, Y. F. Niu, L. Du, X. P. Hu, Y. Zhang, and S. N. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).

O’Neil, A. T.

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).
[PubMed]

Padgett, M.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).

Padgett, M. J.

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).
[PubMed]

Phillips, W. D.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[PubMed]

Poletto, L.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Qin, Y. Q.

Remez, R.

Ren, Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

Ren, Z. C.

S. M. Li, L. J. Kong, Z. C. Ren, Y. Nan Li, C. H. Tu, and H. T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).

Ressel, B.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Ribic, P. R.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Rode, A. V.

Ruchon, T.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Ryu, C.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[PubMed]

Shao, G. H.

G. H. Shao, Z. J. Wu, J. H. Chen, F. Xu, and Y. Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching,” Phys. Rev. A 88(6), 063827 (2013).

Shapira, A.

K. Shemer, N. Voloch-Bloch, A. Shapira, A. Libster, I. Juwiler, and A. Arie, “Azimuthal and radial shaping of vortex beams generated in twisted nonlinear photonic crystals,” Opt. Lett. 38(24), 5470–5473 (2013).
[PubMed]

A. Shapira, A. Libster, Y. Lilach, and A. Arie, “Functional facets for nonlinear crystals,” Opt. Commun. 300, 244–248 (2013).

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[PubMed]

Shemer, K.

K. Shemer, N. Voloch-Bloch, A. Shapira, A. Libster, I. Juwiler, and A. Arie, “Azimuthal and radial shaping of vortex beams generated in twisted nonlinear photonic crystals,” Opt. Lett. 38(24), 5470–5473 (2013).
[PubMed]

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[PubMed]

Shen, Y. C.

Z. Shen, L. Su, X. C. Yuan, and Y. C. Shen, “Trapping and rotating of a metallic particle trimer with optical vortex,” Appl. Phys. Lett. 109(24), 241901 (2016).

Shen, Z.

Z. Shen, L. Su, X. C. Yuan, and Y. C. Shen, “Trapping and rotating of a metallic particle trimer with optical vortex,” Appl. Phys. Lett. 109(24), 241901 (2016).

Shiloh, R.

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[PubMed]

Shvedov, V.

Shvedov, V. G.

Singh, B. K.

Soskin, M.

Spezzani, C.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Spreeuw, R. J. C.

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).
[PubMed]

Stupar, M.

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

Su, L.

Z. Shen, L. Su, X. C. Yuan, and Y. C. Shen, “Trapping and rotating of a metallic particle trimer with optical vortex,” Appl. Phys. Lett. 109(24), 241901 (2016).

Trajtenberg-Mills, S.

Tsur, Y.

Tu, C. H.

S. M. Li, L. J. Kong, Z. C. Ren, Y. Nan Li, C. H. Tu, and H. T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).

Tur, M.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

Vaziri, A.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[PubMed]

Voloch-Bloch, N.

Volyar, A.

Wang, H. T.

S. M. Li, L. J. Kong, Z. C. Ren, Y. Nan Li, C. H. Tu, and H. T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).

Wang, J.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[PubMed]

Willner, A. E.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

Woerdman, J. P.

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).
[PubMed]

Wu, Z. J.

G. H. Shao, Z. J. Wu, J. H. Chen, F. Xu, and Y. Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching,” Phys. Rev. A 88(6), 063827 (2013).

Xu, F.

G. H. Shao, Z. J. Wu, J. H. Chen, F. Xu, and Y. Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching,” Phys. Rev. A 88(6), 063827 (2013).

Yan, Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

Yang, B.

Yang, J. Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

Yuan, X. C.

Z. Shen, L. Su, X. C. Yuan, and Y. C. Shen, “Trapping and rotating of a metallic particle trimer with optical vortex,” Appl. Phys. Lett. 109(24), 241901 (2016).

Yue, Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

Yue, Y. Y.

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[PubMed]

Zhang, C.

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

B. Yang, X. H. Hong, R. E. Lu, Y. Y. Yue, C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “2D wave-front shaping in optical superlattices using nonlinear volume holography,” Opt. Lett. 41(13), 2927–2929 (2016).
[PubMed]

Zhang, Y.

R. Ni, Y. F. Niu, L. Du, X. P. Hu, Y. Zhang, and S. N. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).

Zhao, X.

Zheng, Y.

Zhu, S. N.

R. Ni, Y. F. Niu, L. Du, X. P. Hu, Y. Zhang, and S. N. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).

Zhu, Y. Y.

Appl. Phys. Lett. (2)

Z. Shen, L. Su, X. C. Yuan, and Y. C. Shen, “Trapping and rotating of a metallic particle trimer with optical vortex,” Appl. Phys. Lett. 109(24), 241901 (2016).

R. Ni, Y. F. Niu, L. Du, X. P. Hu, Y. Zhang, and S. N. Zhu, “Topological charge transfer in frequency doubling of fractional orbital angular momentum state,” Appl. Phys. Lett. 109(15), 151103 (2016).

J. Opt. A, Pure Appl. Opt. (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).

Nat. Commun. (2)

D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. De Ninno, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(14971), 14971 (2017).
[PubMed]

F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, “Controlling the orbital angular momentum of high harmonic vortices,” Nat. Commun. 8(14970), 14970 (2017).
[PubMed]

Nat. Photonics (2)

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).

Nature (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[PubMed]

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

Opt. Commun. (1)

A. Shapira, A. Libster, Y. Lilach, and A. Arie, “Functional facets for nonlinear crystals,” Opt. Commun. 300, 244–248 (2013).

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. (1)

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

Phys. Rev. A (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).
[PubMed]

S. M. Li, L. J. Kong, Z. C. Ren, Y. Nan Li, C. H. Tu, and H. T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).

G. H. Shao, Z. J. Wu, J. H. Chen, F. Xu, and Y. Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching,” Phys. Rev. A 88(6), 063827 (2013).

Phys. Rev. Lett. (5)

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).
[PubMed]

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[PubMed]

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[PubMed]

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[PubMed]

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
[PubMed]

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Figures (6)

Fig. 1
Fig. 1 Left column: The different phase structures of the FW with 0-π/2 phase modulation. Right column: The generated SH vortex beams in different orders of nonlinear Raman-Nath diffraction. l S H = u l 1 for the u t h order SH.
Fig. 2
Fig. 2 The measurement of the topological charge by a cylindrical lens when l S H = ± 1 , ± 2 , ± 3 . The number of the dark stripes represents the topological charge.
Fig. 3
Fig. 3 Left column: different phase structures of the FW with 0-π phase modulation. Right column: The generated SH vortex beam in different orders of nonlinear Raman-Nath diffraction. l S H = u l 1 for the u t h order SH.
Fig. 4
Fig. 4 Left column: Different phase structures of the FW with 0-π/2 phase modulation with fractional topological charge l 1 = 2.5 , 2.6 , 2.7 , 2.9 , respectively. Right column: The generated SH vortex beams in different orders of nonlinear Raman-Nath diffraction. l S H = u l 1 for the u t h order SH.
Fig. 5
Fig. 5 Evolution of dark stripes with the topological charge changing from 2 to 3.
Fig. 6
Fig. 6 Top row: The different phase structures of the FW with 0-π/2 phase modulation. l x and l z represent the topological charge along x and z axis, respectively. Bottom row: The generated SH vortex beams in different orders of nonlinear Raman-Nath diffraction. l S H = u l x + v l z for the ( u t h , v t h ) order SH.

Equations (4)

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E 1 = A 1 exp [ i ( k 1 y ω t ) ] C m exp [ i 2 π m f ( r , φ ) i m l 1 φ ] ,
d A 2 d y = ε 0 χ ( 2 ) A 1 2 b u exp ( i k 2 y y i 2 k 1 y ) exp [ i k 2 t r i 2 π u f ( r , φ ) ] exp ( i u l 1 φ ) ,
E 1 = A 1 e i ( k 1 y ω t ) m = m = + C m e i m [ 2 π f 1 ( r , φ ) + l x φ ] p = p = + C p e i p [ 2 π f 2 ( r , φ ) + l z φ ] ,
d A 2 d y = ε 0 χ ( 2 ) A 1 2 e i ( k 2 y 2 k 1 ) y u = u = + b u v = v = + b v e i [ k 2 t r 2 π u f 1 ( r , φ ) 2 π v f 2 ( r , φ ) ] e i ( u l x + v l z ) φ ,

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