Abstract

In this paper, we designed a binary phase mask (PM) with specific phase modulation characteristic and analyzed the spatial spectrum of the beam passing through the PM. In the case where the difference of phase modulation between two lattices of the binary PM is not equal to π, we found the spatial spectrum has the central spot (direct current component, DC component) except for the central eight strong symmetrical spots and many outer weak symmetrical spots. Based on the multiple-beam interference, the propagation-invariant vortex with a square array can be realized by interference of the eight plane waves with the same wave vectors along the optical axis from the central eight symmetrical spots via the modulated phase values of the central eight symmetric spots. The vortex arrays have two kinds of vortex with an opposite topological charge of l=±1. The helix can be formed with a square array by the interference of the vortex array and the plane wave along the optical axis from the DC component. The helix with outstanding helical intensity distributions have two screw directions, which coincide with the phase distribution of the optical vortex with the square array. The energy efficiency of this method can reach more than 80%. The simulation results demonstrate the feasibility of this method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Vortex beam have the typical orbital angular momentum per photon for its special helical phase distribution [1,2]. It can be applied in the fields of image processing [38], electron acceleration [9,10], optics communication, et. al. The vortex beam can be produced by mode selector in the cavity [11], mode conversion by cylindrical lens [12], spiral phase plate [13,14], liquid crystal spatial light modulator [1518], et. al. The researchers have also proposed many methods for produce vortex arrays, such as, multiple-beam interference [1924], helical phase spatial filtering based on typical 4f optical system [25], fractional Talbot effect [26], diffractive optical elements worked as the beam splitter [2730], et al. The multiple-beam interference is much better than other methods for its high energy efficiency and propagation-invariant characteristic [31]. In addition, by interference the propagation-invariant vortex array with one plane beam along optical axis, it can be formed the helix array. The helix have the special spatial helical intensity distribution. The vortex array and helix array can be applied in the fields of material processing [3235], micro-particle manipulation and sorting [36,37], et.al.

2. Theoretical analysis and simulation results

Based on the multiple-beams’ interference theory, here we proposed an efficient method to generate the vortex and helix with square array by use of one two-dimensional (2D) binary phase mask (PM). Figure 1 is the optical system diagram. L1 is the beam expander and L2 is the beam collimator. We can obtain the collimated wide beam via the laser passed through the beam expander and collimator in turn. L3 and L4 are the Fourier Lens with the same focal length of f. One binary PM was set on the front focal plane of L3 and we can obtain the spatial spectrum of the beam passing through the binary PM on the back focal plane of L3. One filter was set on the back focal plane of L3 and the front focal plane of L4. The filter can not only filter the spatial spectrum, but also modulate the phase distribution of the spatial spectrum. The charge coupled device (CCD) was set behind L4 and it was used to record the intensity pattern of the output optical field.

 

Fig. 1. Optical system diagram.

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Figure 2 show the phase modulation characteristics of the binary PM and the spatial spectrum of the beam passing through the PM. The binary PM shown in Fig. 2(a) has the periodic distribution in four directions, which include the horizontal, vertical and the ± 45 degrees from horizontal to vertical directions [38]. This binary PM can be used for generate the vortex and helix with square array. The binary PM shown in Fig. 2(a) has two kinds of lattices with different gray levels, which represent the two different phase modulation values ±φ on the incident beam. Ideally, when one uniform plane wave with amplitude A passed through the binary PM, the optical field can be simply expressed as

$${u_{PM}}(x,y,0) = A\cos \phi + iA\sin \phi \exp \left\{ {i\pi \left[ {{\textrm{Int}}\left( {\frac{{{k_1}x}}{\pi }} \right) + {\textrm{Int}}\left( {\frac{{{k_1}y}}{\pi }} \right) + {\textrm{Int}}\left( {{k_1}\frac{{x + y}}{\pi }} \right) + {\textrm{Int}}\left( {{k_1}\frac{{x - y}}{\pi }} \right)} \right]} \right\}.$$
where (x, y, z) is the common Cartesian coordinate with horizontal x-axis, vertical y-axis in the transverse plane. Int means taking integers. k1 is the parameter of the binary PM on the transverse plane.

 

Fig. 2. The phase modulation characteristics of the binary phase mask with φ=0.25π and the spatial spectrum of the beam passing through the phase mask.

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Fig. 2(b) shows the spatial spectrum of the binary PM with φ=0.25π shown in Fig. 2(a). From Eq. (1), it is obviously that the amplitude of the DC component correspond to the central spot is Acosφ. In addition to the central spot, there are central eight strong symmetrical spots with the same intensity next to the central spot and many outer weak symmetrical spots in the spatial spectrum. It is evidently there is no central spot for φ=π/2.

By analysis, the components correspond to the central eight strong symmetrical spots shown in Fig. 2(b) have the same amplitude as

$$a = A\sin \phi \sum\limits_{l ={-} \infty }^{ + \infty } {\sum\limits_{m ={-} \infty }^{ + \infty } {{{\left( {\frac{2}{\pi }} \right)}^4}\frac{1}{{({2l - 2m + 1} )(1 - 2l - 2m)({2m + 1} )(2l + 1)}}} } = 0.2702A\sin \phi .$$
The filter shown in Fig. 1 can only permit the central eight symmetric spots and the central spot passing through. Meanwhile, the filter can modulate the phase distribution of the central eight symmetric spots.

At first, we only permit the central eight symmetric spots passing through. Figure 3 shows the phase distribution of the eight spots which can be used to generate the vortex with square array. The phase values of these eight symmetric spots can be realized by modulate the spatial spectrum of the binary PM. L4 can transformed the central eight symmetric spots into eight symmetric plane waves with the same wave vectors along optical axis. For the case of Fig. 3, the complex amplitude of the interference pattern of the eight symmetric plane waves can be deduced as

$$\begin{aligned}{u_{vortex}}(x,y,z) &= 4a\left\{ \sin \left[ {\frac{{2{k_r}}}{{\sqrt {10} }}({x + y} )} \right]\sin \left[ {\frac{{{k_r}}}{{\sqrt {10} }}({x - y} )} \right]\right.\\ & \quad \left.+ i\sin \left[ {\frac{{{k_r}}}{{\sqrt {10} }}({x + y} )} \right]\sin \left[ {\frac{{2{k_r}}}{{\sqrt {10} }}({x - y} )} \right] \right\}\exp (i{k_z}z).\end{aligned}$$
where kr2+kz2=k2, kr=$\sqrt {10} $ k1, k = 2π/λ. λ is the wavelength and k is the wave vector. kr and kz are the wave vectors along transverse plane and the optical axis respectively.

 

Fig. 3. The phase distribution of the eight symmetric spots which can be used for generate the vortex with square array.

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Figure 4 shows the simulation results of the interference pattern expressed by Eq. (3). Figure 4(a) is the normalized intensity distribution of the interference pattern and Fig. 4(b) is the phase distribution of the interference pattern. It is evident that it formed the optical vortex with square array. From Fig. 4(a) and (b), we can see the vortex arrays have two kinds vortex with the opposite topological charge of l=±1. The minimum unit of vortex array is 2×2 lattices.

 

Fig. 4. The simulation results of the interference pattern of the eight symmetric plane wave from the eight symmetric spots shown in Fig. 3.

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Figure 5 shows the contour of the normalized intensity distribution of the single vortex shown in Fig. 4. The difference of values between the adjacent contours is 0.1. Figure 6 shows the phase difference between the single vortex and an ideal optical vortex with l = 1, which has the linear phase gradient along azimuthal direction. From the inner to outer of Fig. 6, the 1st solid line, 2nd solid line, 3rd solid line, …, nth solid line correspond to the 1°, 2°, 3°, …, n° (π/180, 2π/180, 3π/180, …, /180) respectively. From Fig. 6, we can see that the single optical vortex is very close to an ideal optical vortex.

 

Fig. 5. The contour of the intensity distribution of the single vortex shown in Fig. 4.

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Fig. 6. The phase difference between the single vortex shown in Fig. 4 and an ideal optical vortex with l = 1.

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Then we permit the central spot (DC component) passing through. L4 can transform the central spot into the plane wave along optical axis. The total complex amplitude of the interference pattern of the vortex array and DC component can be written as

$$\begin{array}{l} {u_{helix}}(x,y,z) = A\cos \phi \exp (ikz) + \\ 1.0808A\textrm{sin}\phi \left\{ {\sin \left[ {\frac{{2{k_r}}}{{\sqrt {10} }}({x + y} )} \right]\sin \left[ {\frac{{{k_r}}}{{\sqrt {10} }}({x - y} )} \right] + i\sin \left[ {\frac{{{k_r}}}{{\sqrt {10} }}({x + y} )} \right]\sin \left[ {\frac{{2{k_r}}}{{\sqrt {10} }}({x - y} )} \right]} \right\}\exp (i{k_z}z). \end{array}$$
Obviously, the DC component and vortex array have the different wave vectors along the optical axis, it means the interference pattern will changed as the distance z increased. To ensure the interference pattern with the best contrast, the DC component and the vortex array should have the same maximum amplitude. In the paraxial approximation, we can calculated φ≈0.224π.

Figure 7 shows the simulation results of the intensity distribution of the helix with the square array for different z. Here we set φ=0.224π, λ=632.8 nm, kr=0.01k, the size of the images is about 0.5mm×0.5 mm. Figure 7(a)-(f) correspond to z = 110 mm, 112 mm, 114 mm, 116 mm, 118 mm and 120 mm respectively. It is obvious that the helix have the outstanding helical intensity distributions along optical axis. Meanwhile, we found the helix have two type screw directions, rotate clockwise and counterclockwise. This result coincide with the above analysis that the vortex have two kinds vortex with the opposite topological charge of l=±1.

 

Fig. 7. The intensity distribution of the helix with square array at different z. (a) z = 110 mm, (b) z = 112 mm, (c) z = 114 mm, (d) z = 116 mm, (e) z = 118 mm, (f) z = 120 mm.

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Regardless of the energy loss of the optical system, ideally, the energy efficiency of the optical system can be calculated as

$$\eta = {\cos ^2}\phi + 8 \times {({0.2702\sin \phi } )^2} = 82.56\%.$$

3. Conclusion

We proposed a method to generate the vortex and helix with square array. It has the higher energy efficiency than 80%. We obtained the spatial spectrum of the PM with the central spot except the central eight symmetrical spots and many outer weak symmetrical spots by modulate the phase distribution of the binary PM. It can be formed the propagation-invariant vortex with square array by interference of the eight plane wave with the same wave vectors along optical axis from the central eight symmetrical spots with the specific phase distribution. The vortex arrays have two kinds of vortex with the opposite topological charge of l=±1. Then, it can be realized the helix with square array by interference the vortex array with the plane wave along optical axis from the DC component. The helix with the outstanding helical intensity distributions have two type screw directions, rotate clockwise and counterclockwise along the optical axis. The simulation results demonstrate the feasibility of this method. We can also expanded this method to the electron beam [3943], extreme ultraviolet or acoustic wave [4446]. This method can be widely used in many fields, such as material processing [3235], micro-particle manipulation [36], optical sorting [37], telecommunications [47] and et. al.

Funding

National Natural Science Foundation of China (11504096); Natural Science Foundation of Shandong Province (ZR2017MA047); Doctoral Foundation of University of Jinan (XBS1407, XBS1611); School Scientific Research Foundation of University of Jinan (XKY1407, XKY1706).

Disclosures

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

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11. R. Oron, N. Davidson, A. A. Friesem, and F. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000). [CrossRef]  

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13. M. W. Beijersbergen, R. P. C. Coerwinkel, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994). [CrossRef]  

14. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996). [CrossRef]  

15. N. R. Heckenberg, R. Mc Duff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef]  

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References

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  1. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A: At., Mol., Opt. Phys. 45(11), 8185–8189 (1992).
    [Crossref]
  2. S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
    [Crossref]
  3. J. A. Davis, D. E. Mc Namra, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
    [Crossref]
  4. C.-S. Guo, X. Cheng, X.-Y. Ren, J.-P. Ding, and H.-T. Wang, “Optical vortex phase-shifting digital holography,” Opt. Express 12(21), 5166–5171 (2004).
    [Crossref]
  5. G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30(24), 3308–3310 (2005).
    [Crossref]
  6. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005).
    [Crossref]
  7. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow Effects in Spiral Phase Contrast Microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
    [Crossref]
  8. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral interferometry,” Opt. Lett. 30(15), 1953–1955 (2005).
    [Crossref]
  9. J. T. Mendonça and J. Vieira, “Donut wakefields generated by intense laser pulses with orbital angular momentum,” Phys. Plasmas 21(3), 033107 (2014).
    [Crossref]
  10. J. Vieira and J. T. Mendonça, “Nonlinear Laser Driven Donut Wakefields for Positron and Electron Acceleration,” Phys. Rev. Lett. 112(21), 215001 (2014).
    [Crossref]
  11. R. Oron, N. Davidson, A. A. Friesem, and F. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
    [Crossref]
  12. M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic Laser Mode Converters and Transfer of Orbital Angular-Momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
    [Crossref]
  13. M. W. Beijersbergen, R. P. C. Coerwinkel, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
    [Crossref]
  14. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996).
    [Crossref]
  15. N. R. Heckenberg, R. Mc Duff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
    [Crossref]
  16. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
    [Crossref]
  17. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28(11), 872–874 (2003).
    [Crossref]
  18. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic Holographic Optical Tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
    [Crossref]
  19. G. C. G. Berkhout and M. W. Beijersbergen, “Method for Probing the Orbital Angular Momentum of Optical Vortices in Electromagnetic Waves from Astronomical Objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
    [Crossref]
  20. A. Jiménez-Ceniceros, M. Trejo-Durán, E. Alvarado-Méndez, and V. M. Castaño, “Extinction zones and scalability in N-beam interference lattices,” Opt. Commun. 283(3), 362–367 (2010).
    [Crossref]
  21. V. Arrizón, S. Chavez-Cerda, U. Ruiz, and R. Carrada, “Periodic and quasi-periodic non-diffracting wave fields generated by superposition of multiple Bessel beams,” Opt. Express 15(25), 16748–16753 (2007).
    [Crossref]
  22. V. Arrizón, D. Sánchez-de-la-Llave, G. Méndez, and U. Ruiz, “Efficient generation of periodic and quasi-periodic non-diffractive optical fields with phase holograms,” Opt. Express 19(11), 10553–10562 (2011).
    [Crossref]
  23. Y. Han and C. Liu, “Propagation-invariant hollow beams with hexagonal symmetry,” Opt. Commun. 284(9), 2264–2267 (2011).
    [Crossref]
  24. J. Becker, P. Rose, M. Boguslawski, and C. Denz, “Systematic approach to complex periodic vortex and helix lattices,” Opt. Express 19(10), 9848–9862 (2011).
    [Crossref]
  25. C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259(2), 449–454 (2006).
    [Crossref]
  26. G.-X. Wei, L.-L. Lu, and C.-S. Guo, “Generation of optical vortex array based on the fractional Talbot effect,” Opt. Commun. 282(14), 2665–2669 (2009).
    [Crossref]
  27. H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24(4), 505–515 (1977).
    [Crossref]
  28. A. P. Porfirev and S. N. Khonina, “Simple method for efficient reconfigurable optical vortex beam splitting,” Opt. Express 25(16), 18722–18735 (2017).
    [Crossref]
  29. S. Rasouli and D. Hebri, “Theory of diffraction of vortex beams from 2D orthogonal periodic structures and Talbot self-healing under vortex beam illumination,” J. Opt. Soc. Am. A 36(5), 800–808 (2019).
    [Crossref]
  30. A. Porfirev, S. Khonina, Y. Azizian-Kalandaragh, and M. Kirilenko, “Efficient generation of arrays of closed-packed high-quality light rings,” Photonic. Nanostruct. 37, 100736 (2019).
    [Crossref]
  31. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53(7), 537–578 (2003).
    [Crossref]
  32. T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
    [Crossref]
  33. L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. 26(23), 1858–1860 (2001).
    [Crossref]
  34. L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. 27(11), 900–902 (2002).
    [Crossref]
  35. X. L. Yang and L. Z. Cai, “Wave design of the interference of three noncoplanar beams for microfiber fabrication,” Opt. Commun. 208(4-6), 293–297 (2002).
    [Crossref]
  36. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
    [Crossref]
  37. C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283(9), 1889–1893 (2010).
    [Crossref]
  38. D. Hebri and S. Rasouli, “Diffraction from two-dimensional orthogonal nonseparable periodic structures: Talbot distance dependence on the number theoretic properties of the structures,” J. Opt. Soc. Am. A 36(2), 253–263 (2019).
    [Crossref]
  39. J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
    [Crossref]
  40. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
    [Crossref]
  41. T. Niermann, J. Verbeeck, and M. Lehmann, “Creating arrays of electron vortices,” Ultramicroscopy 136, 165–170 (2014).
    [Crossref]
  42. V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of Nondiffracting Electron Bessel Beams,” Phys. Rev. X 4(1), 011013 (2014).
    [Crossref]
  43. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie>, “Generation of electro Airy beams,” Nature 494(7437), 331–335 (2013).
    [Crossref]
  44. Z. Hong, J. Zhang, and B. W. Drinkwater, “Observation of Orbital Angular Momentum Transfer from Bessel-Shaped Acoustic Vortices to Diphasic Liquid-Microparticle Mixtures,” Phys. Rev. Lett. 114(21), 214301 (2015).
    [Crossref]
  45. D. Baresch, J.-L. Thomas, and R. Marchiano, “Observation of a Single-Beam Gradient Force Acoustical Trap for Elastic Particles: Acoustical Tweezers,” Phys. Rev. Lett. 116(2), 024301 (2016).
    [Crossref]
  46. A. Marzo, M. Caleap, and B. W. Drinkwater, “Acoustic Virtual Vortices with Tunable Orbital Angular Momentum for Trapping of Mie Particles,” Phys. Rev. Lett. 120(4), 044301 (2018).
    [Crossref]
  47. M. O. Jensen and M. J. Brett, “Square spiral 3D photonic bandgap crystals at telecommunications frequencies,” Opt. Express 13(9), 3348–3354 (2005).
    [Crossref]

2019 (3)

2018 (1)

A. Marzo, M. Caleap, and B. W. Drinkwater, “Acoustic Virtual Vortices with Tunable Orbital Angular Momentum for Trapping of Mie Particles,” Phys. Rev. Lett. 120(4), 044301 (2018).
[Crossref]

2017 (1)

2016 (1)

D. Baresch, J.-L. Thomas, and R. Marchiano, “Observation of a Single-Beam Gradient Force Acoustical Trap for Elastic Particles: Acoustical Tweezers,” Phys. Rev. Lett. 116(2), 024301 (2016).
[Crossref]

2015 (1)

Z. Hong, J. Zhang, and B. W. Drinkwater, “Observation of Orbital Angular Momentum Transfer from Bessel-Shaped Acoustic Vortices to Diphasic Liquid-Microparticle Mixtures,” Phys. Rev. Lett. 114(21), 214301 (2015).
[Crossref]

2014 (4)

T. Niermann, J. Verbeeck, and M. Lehmann, “Creating arrays of electron vortices,” Ultramicroscopy 136, 165–170 (2014).
[Crossref]

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of Nondiffracting Electron Bessel Beams,” Phys. Rev. X 4(1), 011013 (2014).
[Crossref]

J. T. Mendonça and J. Vieira, “Donut wakefields generated by intense laser pulses with orbital angular momentum,” Phys. Plasmas 21(3), 033107 (2014).
[Crossref]

J. Vieira and J. T. Mendonça, “Nonlinear Laser Driven Donut Wakefields for Positron and Electron Acceleration,” Phys. Rev. Lett. 112(21), 215001 (2014).
[Crossref]

2013 (1)

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie>, “Generation of electro Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

2011 (4)

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

V. Arrizón, D. Sánchez-de-la-Llave, G. Méndez, and U. Ruiz, “Efficient generation of periodic and quasi-periodic non-diffractive optical fields with phase holograms,” Opt. Express 19(11), 10553–10562 (2011).
[Crossref]

Y. Han and C. Liu, “Propagation-invariant hollow beams with hexagonal symmetry,” Opt. Commun. 284(9), 2264–2267 (2011).
[Crossref]

J. Becker, P. Rose, M. Boguslawski, and C. Denz, “Systematic approach to complex periodic vortex and helix lattices,” Opt. Express 19(10), 9848–9862 (2011).
[Crossref]

2010 (3)

A. Jiménez-Ceniceros, M. Trejo-Durán, E. Alvarado-Méndez, and V. M. Castaño, “Extinction zones and scalability in N-beam interference lattices,” Opt. Commun. 283(3), 362–367 (2010).
[Crossref]

C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283(9), 1889–1893 (2010).
[Crossref]

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref]

2009 (1)

G.-X. Wei, L.-L. Lu, and C.-S. Guo, “Generation of optical vortex array based on the fractional Talbot effect,” Opt. Commun. 282(14), 2665–2669 (2009).
[Crossref]

2008 (1)

G. C. G. Berkhout and M. W. Beijersbergen, “Method for Probing the Orbital Angular Momentum of Optical Vortices in Electromagnetic Waves from Astronomical Objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[Crossref]

2007 (1)

2006 (1)

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259(2), 449–454 (2006).
[Crossref]

2005 (5)

2004 (1)

2003 (4)

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

J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28(11), 872–874 (2003).
[Crossref]

Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53(7), 537–578 (2003).
[Crossref]

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[Crossref]

2002 (3)

L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. 27(11), 900–902 (2002).
[Crossref]

X. L. Yang and L. Z. Cai, “Wave design of the interference of three noncoplanar beams for microfiber fabrication,” Opt. Commun. 208(4-6), 293–297 (2002).
[Crossref]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic Holographic Optical Tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[Crossref]

2001 (2)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. 26(23), 1858–1860 (2001).
[Crossref]

2000 (2)

R. Oron, N. Davidson, A. A. Friesem, and F. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
[Crossref]

J. A. Davis, D. E. Mc Namra, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25(2), 99–101 (2000).
[Crossref]

1996 (1)

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996).
[Crossref]

1994 (2)

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[Crossref]

M. W. Beijersbergen, R. P. C. Coerwinkel, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

1993 (1)

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic Laser Mode Converters and Transfer of Orbital Angular-Momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

1992 (2)

N. R. Heckenberg, R. Mc Duff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A: At., Mol., Opt. Phys. 45(11), 8185–8189 (1992).
[Crossref]

1977 (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24(4), 505–515 (1977).
[Crossref]

Agrawal, A.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

Allen, L.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996).
[Crossref]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic Laser Mode Converters and Transfer of Orbital Angular-Momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A: At., Mol., Opt. Phys. 45(11), 8185–8189 (1992).
[Crossref]

Alvarado-Méndez, E.

A. Jiménez-Ceniceros, M. Trejo-Durán, E. Alvarado-Méndez, and V. M. Castaño, “Extinction zones and scalability in N-beam interference lattices,” Opt. Commun. 283(3), 362–367 (2010).
[Crossref]

Anderson, I. M.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

Arie, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie>, “Generation of electro Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Arrizón, V.

Azizian-Kalandaragh, Y.

A. Porfirev, S. Khonina, Y. Azizian-Kalandaragh, and M. Kirilenko, “Efficient generation of arrays of closed-packed high-quality light rings,” Photonic. Nanostruct. 37, 100736 (2019).
[Crossref]

Baresch, D.

D. Baresch, J.-L. Thomas, and R. Marchiano, “Observation of a Single-Beam Gradient Force Acoustical Trap for Elastic Particles: Acoustical Tweezers,” Phys. Rev. Lett. 116(2), 024301 (2016).
[Crossref]

Barnett, S. M.

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[Crossref]

Becker, J.

Beijersbergen, M. W.

G. C. G. Berkhout and M. W. Beijersbergen, “Method for Probing the Orbital Angular Momentum of Optical Vortices in Electromagnetic Waves from Astronomical Objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[Crossref]

M. W. Beijersbergen, R. P. C. Coerwinkel, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic Laser Mode Converters and Transfer of Orbital Angular-Momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A: At., Mol., Opt. Phys. 45(11), 8185–8189 (1992).
[Crossref]

Berkhout, G. C. G.

G. C. G. Berkhout and M. W. Beijersbergen, “Method for Probing the Orbital Angular Momentum of Optical Vortices in Electromagnetic Waves from Astronomical Objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[Crossref]

Bernet, S.

Boguslawski, M.

Bouchal, Z.

Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53(7), 537–578 (2003).
[Crossref]

Boyd, R. W.

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of Nondiffracting Electron Bessel Beams,” Phys. Rev. X 4(1), 011013 (2014).
[Crossref]

Brett, M. J.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Cai, L. Z.

Caleap, M.

A. Marzo, M. Caleap, and B. W. Drinkwater, “Acoustic Virtual Vortices with Tunable Orbital Angular Momentum for Trapping of Mie Particles,” Phys. Rev. Lett. 120(4), 044301 (2018).
[Crossref]

Campos, J.

Carrada, R.

Castaño, V. M.

A. Jiménez-Ceniceros, M. Trejo-Durán, E. Alvarado-Méndez, and V. M. Castaño, “Extinction zones and scalability in N-beam interference lattices,” Opt. Commun. 283(3), 362–367 (2010).
[Crossref]

Chavez-Cerda, S.

Cheng, X.

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

Cottrell, D. M.

Curtis, J. E.

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

J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28(11), 872–874 (2003).
[Crossref]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic Holographic Optical Tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[Crossref]

Dammann, H.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24(4), 505–515 (1977).
[Crossref]

Davidson, N.

R. Oron, N. Davidson, A. A. Friesem, and F. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
[Crossref]

Davis, J. A.

Dennis, M. R.

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of Nondiffracting Electron Bessel Beams,” Phys. Rev. X 4(1), 011013 (2014).
[Crossref]

Denz, C.

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Ding, J. P.

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259(2), 449–454 (2006).
[Crossref]

Ding, J.-P.

Drinkwater, B. W.

A. Marzo, M. Caleap, and B. W. Drinkwater, “Acoustic Virtual Vortices with Tunable Orbital Angular Momentum for Trapping of Mie Particles,” Phys. Rev. Lett. 120(4), 044301 (2018).
[Crossref]

Z. Hong, J. Zhang, and B. W. Drinkwater, “Observation of Orbital Angular Momentum Transfer from Bessel-Shaped Acoustic Vortices to Diphasic Liquid-Microparticle Mixtures,” Phys. Rev. Lett. 114(21), 214301 (2015).
[Crossref]

Foo, G.

Frabboni, S.

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of Nondiffracting Electron Bessel Beams,” Phys. Rev. X 4(1), 011013 (2014).
[Crossref]

Friesem, A. A.

R. Oron, N. Davidson, A. A. Friesem, and F. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
[Crossref]

Fürhapter, S.

Gazzadi, G. C.

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of Nondiffracting Electron Bessel Beams,” Phys. Rev. X 4(1), 011013 (2014).
[Crossref]

Gover, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie>, “Generation of electro Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

Grier, D. G.

J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28(11), 872–874 (2003).
[Crossref]

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

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic Holographic Optical Tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[Crossref]

Grillo, V.

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of Nondiffracting Electron Bessel Beams,” Phys. Rev. X 4(1), 011013 (2014).
[Crossref]

Guo, C. S.

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259(2), 449–454 (2006).
[Crossref]

Guo, C.-S.

C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283(9), 1889–1893 (2010).
[Crossref]

G.-X. Wei, L.-L. Lu, and C.-S. Guo, “Generation of optical vortex array based on the fractional Talbot effect,” Opt. Commun. 282(14), 2665–2669 (2009).
[Crossref]

C.-S. Guo, X. Cheng, X.-Y. Ren, J.-P. Ding, and H.-T. Wang, “Optical vortex phase-shifting digital holography,” Opt. Express 12(21), 5166–5171 (2004).
[Crossref]

Han, Y.

Y. Han and C. Liu, “Propagation-invariant hollow beams with hexagonal symmetry,” Opt. Commun. 284(9), 2264–2267 (2011).
[Crossref]

Han, Y. J.

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259(2), 449–454 (2006).
[Crossref]

Hasman, F.

R. Oron, N. Davidson, A. A. Friesem, and F. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
[Crossref]

Hebri, D.

Heckenberg, N. R.

Herzing, A. A.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

Hong, Z.

Z. Hong, J. Zhang, and B. W. Drinkwater, “Observation of Orbital Angular Momentum Transfer from Bessel-Shaped Acoustic Vortices to Diphasic Liquid-Microparticle Mixtures,” Phys. Rev. Lett. 114(21), 214301 (2015).
[Crossref]

C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283(9), 1889–1893 (2010).
[Crossref]

Jensen, M. O.

Jesacher, A.

Jiménez-Ceniceros, A.

A. Jiménez-Ceniceros, M. Trejo-Durán, E. Alvarado-Méndez, and V. M. Castaño, “Extinction zones and scalability in N-beam interference lattices,” Opt. Commun. 283(3), 362–367 (2010).
[Crossref]

Juodkazis, S.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[Crossref]

Karimi, E.

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of Nondiffracting Electron Bessel Beams,” Phys. Rev. X 4(1), 011013 (2014).
[Crossref]

Khonina, S.

A. Porfirev, S. Khonina, Y. Azizian-Kalandaragh, and M. Kirilenko, “Efficient generation of arrays of closed-packed high-quality light rings,” Photonic. Nanostruct. 37, 100736 (2019).
[Crossref]

Khonina, S. N.

Kirilenko, M.

A. Porfirev, S. Khonina, Y. Azizian-Kalandaragh, and M. Kirilenko, “Efficient generation of arrays of closed-packed high-quality light rings,” Photonic. Nanostruct. 37, 100736 (2019).
[Crossref]

Klotz, E.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24(4), 505–515 (1977).
[Crossref]

Kondo, T.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[Crossref]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic Holographic Optical Tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[Crossref]

Lehmann, M.

T. Niermann, J. Verbeeck, and M. Lehmann, “Creating arrays of electron vortices,” Ultramicroscopy 136, 165–170 (2014).
[Crossref]

Lereah, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie>, “Generation of electro Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

Lezec, H. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

Lilach, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie>, “Generation of electro Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

Liu, C.

Y. Han and C. Liu, “Propagation-invariant hollow beams with hexagonal symmetry,” Opt. Commun. 284(9), 2264–2267 (2011).
[Crossref]

Lu, L.-L.

G.-X. Wei, L.-L. Lu, and C.-S. Guo, “Generation of optical vortex array based on the fractional Talbot effect,” Opt. Commun. 282(14), 2665–2669 (2009).
[Crossref]

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Marchiano, R.

D. Baresch, J.-L. Thomas, and R. Marchiano, “Observation of a Single-Beam Gradient Force Acoustical Trap for Elastic Particles: Acoustical Tweezers,” Phys. Rev. Lett. 116(2), 024301 (2016).
[Crossref]

Marzo, A.

A. Marzo, M. Caleap, and B. W. Drinkwater, “Acoustic Virtual Vortices with Tunable Orbital Angular Momentum for Trapping of Mie Particles,” Phys. Rev. Lett. 120(4), 044301 (2018).
[Crossref]

Matsuo, S.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[Crossref]

Mc Duff, R.

Mc Namra, D. E.

McClelland, J. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

McMorran, B. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

Méndez, G.

Mendonça, J. T.

J. T. Mendonça and J. Vieira, “Donut wakefields generated by intense laser pulses with orbital angular momentum,” Phys. Plasmas 21(3), 033107 (2014).
[Crossref]

J. Vieira and J. T. Mendonça, “Nonlinear Laser Driven Donut Wakefields for Positron and Electron Acceleration,” Phys. Rev. Lett. 112(21), 215001 (2014).
[Crossref]

Misawa, H.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[Crossref]

Mizeikis, V.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[Crossref]

Niermann, T.

T. Niermann, J. Verbeeck, and M. Lehmann, “Creating arrays of electron vortices,” Ultramicroscopy 136, 165–170 (2014).
[Crossref]

Oron, R.

R. Oron, N. Davidson, A. A. Friesem, and F. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
[Crossref]

Padgett, M. J.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996).
[Crossref]

Palacios, D. M.

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Porfirev, A.

A. Porfirev, S. Khonina, Y. Azizian-Kalandaragh, and M. Kirilenko, “Efficient generation of arrays of closed-packed high-quality light rings,” Photonic. Nanostruct. 37, 100736 (2019).
[Crossref]

Porfirev, A. P.

Rasouli, S.

Ren, X.-Y.

Ritsch-Marte, M.

Robertson, D. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996).
[Crossref]

Rose, P.

Ruiz, U.

Sánchez-de-la-Llave, D.

Schattschneider, P.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref]

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Smith, C. P.

Smith, G. M.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996).
[Crossref]

Spreeuw, R. J.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A: At., Mol., Opt. Phys. 45(11), 8185–8189 (1992).
[Crossref]

Swartzlander, G. A.

Thomas, J.-L.

D. Baresch, J.-L. Thomas, and R. Marchiano, “Observation of a Single-Beam Gradient Force Acoustical Trap for Elastic Particles: Acoustical Tweezers,” Phys. Rev. Lett. 116(2), 024301 (2016).
[Crossref]

Tian, H.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref]

Trejo-Durán, M.

A. Jiménez-Ceniceros, M. Trejo-Durán, E. Alvarado-Méndez, and V. M. Castaño, “Extinction zones and scalability in N-beam interference lattices,” Opt. Commun. 283(3), 362–367 (2010).
[Crossref]

Turnbull, G. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996).
[Crossref]

Unguris, J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

Vanderveen, H.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic Laser Mode Converters and Transfer of Orbital Angular-Momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

Verbeeck, J.

T. Niermann, J. Verbeeck, and M. Lehmann, “Creating arrays of electron vortices,” Ultramicroscopy 136, 165–170 (2014).
[Crossref]

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref]

Vieira, J.

J. T. Mendonça and J. Vieira, “Donut wakefields generated by intense laser pulses with orbital angular momentum,” Phys. Plasmas 21(3), 033107 (2014).
[Crossref]

J. Vieira and J. T. Mendonça, “Nonlinear Laser Driven Donut Wakefields for Positron and Electron Acceleration,” Phys. Rev. Lett. 112(21), 215001 (2014).
[Crossref]

Voloch-Bloch, N.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie>, “Generation of electro Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

Wang, H. T.

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259(2), 449–454 (2006).
[Crossref]

Wang, H.-T.

Wang, Y. R.

Wei, G.-X.

G.-X. Wei, L.-L. Lu, and C.-S. Guo, “Generation of optical vortex array based on the fractional Talbot effect,” Opt. Commun. 282(14), 2665–2669 (2009).
[Crossref]

White, A. G.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic Laser Mode Converters and Transfer of Orbital Angular-Momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A: At., Mol., Opt. Phys. 45(11), 8185–8189 (1992).
[Crossref]

Yang, X. L.

Yu, Y.-N.

C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283(9), 1889–1893 (2010).
[Crossref]

Zhang, J.

Z. Hong, J. Zhang, and B. W. Drinkwater, “Observation of Orbital Angular Momentum Transfer from Bessel-Shaped Acoustic Vortices to Diphasic Liquid-Microparticle Mixtures,” Phys. Rev. Lett. 114(21), 214301 (2015).
[Crossref]

Zhang, Y.

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259(2), 449–454 (2006).
[Crossref]

Appl. Phys. Lett. (1)

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[Crossref]

Czech. J. Phys. (1)

Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53(7), 537–578 (2003).
[Crossref]

J. Opt. Soc. Am. A (2)

Nature (2)

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref]

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie>, “Generation of electro Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

Opt. Acta (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24(4), 505–515 (1977).
[Crossref]

Opt. Commun. (12)

C. S. Guo, Y. Zhang, Y. J. Han, J. P. Ding, and H. T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259(2), 449–454 (2006).
[Crossref]

G.-X. Wei, L.-L. Lu, and C.-S. Guo, “Generation of optical vortex array based on the fractional Talbot effect,” Opt. Commun. 282(14), 2665–2669 (2009).
[Crossref]

A. Jiménez-Ceniceros, M. Trejo-Durán, E. Alvarado-Méndez, and V. M. Castaño, “Extinction zones and scalability in N-beam interference lattices,” Opt. Commun. 283(3), 362–367 (2010).
[Crossref]

Y. Han and C. Liu, “Propagation-invariant hollow beams with hexagonal symmetry,” Opt. Commun. 284(9), 2264–2267 (2011).
[Crossref]

C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283(9), 1889–1893 (2010).
[Crossref]

X. L. Yang and L. Z. Cai, “Wave design of the interference of three noncoplanar beams for microfiber fabrication,” Opt. Commun. 208(4-6), 293–297 (2002).
[Crossref]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[Crossref]

R. Oron, N. Davidson, A. A. Friesem, and F. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
[Crossref]

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic Laser Mode Converters and Transfer of Orbital Angular-Momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

M. W. Beijersbergen, R. P. C. Coerwinkel, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996).
[Crossref]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic Holographic Optical Tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[Crossref]

Opt. Express (7)

Opt. Lett. (7)

Photonic. Nanostruct. (1)

A. Porfirev, S. Khonina, Y. Azizian-Kalandaragh, and M. Kirilenko, “Efficient generation of arrays of closed-packed high-quality light rings,” Photonic. Nanostruct. 37, 100736 (2019).
[Crossref]

Phys. Plasmas (1)

J. T. Mendonça and J. Vieira, “Donut wakefields generated by intense laser pulses with orbital angular momentum,” Phys. Plasmas 21(3), 033107 (2014).
[Crossref]

Phys. Rev. A: At., Mol., Opt. Phys. (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A: At., Mol., Opt. Phys. 45(11), 8185–8189 (1992).
[Crossref]

Phys. Rev. Lett. (7)

G. C. G. Berkhout and M. W. Beijersbergen, “Method for Probing the Orbital Angular Momentum of Optical Vortices in Electromagnetic Waves from Astronomical Objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[Crossref]

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

J. Vieira and J. T. Mendonça, “Nonlinear Laser Driven Donut Wakefields for Positron and Electron Acceleration,” Phys. Rev. Lett. 112(21), 215001 (2014).
[Crossref]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow Effects in Spiral Phase Contrast Microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
[Crossref]

Z. Hong, J. Zhang, and B. W. Drinkwater, “Observation of Orbital Angular Momentum Transfer from Bessel-Shaped Acoustic Vortices to Diphasic Liquid-Microparticle Mixtures,” Phys. Rev. Lett. 114(21), 214301 (2015).
[Crossref]

D. Baresch, J.-L. Thomas, and R. Marchiano, “Observation of a Single-Beam Gradient Force Acoustical Trap for Elastic Particles: Acoustical Tweezers,” Phys. Rev. Lett. 116(2), 024301 (2016).
[Crossref]

A. Marzo, M. Caleap, and B. W. Drinkwater, “Acoustic Virtual Vortices with Tunable Orbital Angular Momentum for Trapping of Mie Particles,” Phys. Rev. Lett. 120(4), 044301 (2018).
[Crossref]

Phys. Rev. X (1)

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of Nondiffracting Electron Bessel Beams,” Phys. Rev. X 4(1), 011013 (2014).
[Crossref]

Science (2)

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Ultramicroscopy (1)

T. Niermann, J. Verbeeck, and M. Lehmann, “Creating arrays of electron vortices,” Ultramicroscopy 136, 165–170 (2014).
[Crossref]

Cited By

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Figures (7)

Fig. 1.
Fig. 1. Optical system diagram.
Fig. 2.
Fig. 2. The phase modulation characteristics of the binary phase mask with φ=0.25π and the spatial spectrum of the beam passing through the phase mask.
Fig. 3.
Fig. 3. The phase distribution of the eight symmetric spots which can be used for generate the vortex with square array.
Fig. 4.
Fig. 4. The simulation results of the interference pattern of the eight symmetric plane wave from the eight symmetric spots shown in Fig. 3.
Fig. 5.
Fig. 5. The contour of the intensity distribution of the single vortex shown in Fig. 4.
Fig. 6.
Fig. 6. The phase difference between the single vortex shown in Fig. 4 and an ideal optical vortex with l = 1.
Fig. 7.
Fig. 7. The intensity distribution of the helix with square array at different z. (a) z = 110 mm, (b) z = 112 mm, (c) z = 114 mm, (d) z = 116 mm, (e) z = 118 mm, (f) z = 120 mm.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

u P M ( x , y , 0 ) = A cos ϕ + i A sin ϕ exp { i π [ Int ( k 1 x π ) + Int ( k 1 y π ) + Int ( k 1 x + y π ) + Int ( k 1 x y π ) ] } .
a = A sin ϕ l = + m = + ( 2 π ) 4 1 ( 2 l 2 m + 1 ) ( 1 2 l 2 m ) ( 2 m + 1 ) ( 2 l + 1 ) = 0.2702 A sin ϕ .
u v o r t e x ( x , y , z ) = 4 a { sin [ 2 k r 10 ( x + y ) ] sin [ k r 10 ( x y ) ] + i sin [ k r 10 ( x + y ) ] sin [ 2 k r 10 ( x y ) ] } exp ( i k z z ) .
u h e l i x ( x , y , z ) = A cos ϕ exp ( i k z ) + 1.0808 A sin ϕ { sin [ 2 k r 10 ( x + y ) ] sin [ k r 10 ( x y ) ] + i sin [ k r 10 ( x + y ) ] sin [ 2 k r 10 ( x y ) ] } exp ( i k z z ) .
η = cos 2 ϕ + 8 × ( 0.2702 sin ϕ ) 2 = 82.56 % .

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