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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 3 — Feb. 11, 2013
  • pp: 3379–3387
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Quantitative characterization of the energy circulation in helical beams by means of near-field diffraction

Roland A. Terborg and Karen Volke-Sepúlveda  »View Author Affiliations


Optics Express, Vol. 21, Issue 3, pp. 3379-3387 (2013)
http://dx.doi.org/10.1364/OE.21.003379


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Abstract

We present a method to measure the skew angle of the wave-fronts in an optical vortex, which is directly related with the energy flux. It is based on the analysis of the evolution on propagation of the near-field diffraction pattern generated by a single-slit, consisting of two main lobes that shift in opposite directions depending on the vortex helicity. The transverse displacement of each lobe as a function of the propagation distance allows to quantify the energy circulation. Analytical, numerical and experimental results are compared, showing good agreement. We illustrate the method for the case of Bessel beams, although we also discuss other types of helical beams, such as Laguerre-Gauss and Mathieu beams.

© 2013 OSA

1. Introduction

The study of optical vortices (OVs) has been the focus of great attention in the last two decades due to both, their topological and dynamical properties, which have led to important applications in several areas of physics. Regarding the topology of their wavefronts, OVs exhibit a screw dislocation or phase singularity along their propagation axis [1

1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974). [CrossRef]

], characteristic that may be used, for instance, to store quantum information [2

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

]. In terms of their dynamical properties, OVs are often associated with the presence of orbital angular momentum (OAM) carried by the optical field [3

3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

, 4

4. L. Allen, M. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]

]. Although this is not a property of the vortices per se [5

5. M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009). [CrossRef]

], the most familiar propagation modes with circular symmetry and embedded vortices on-axis, such as Laguerre-Gaussian (LG) and Bessel beams (BBs), do possess OAM, which can be transferred to matter [6

6. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]

8

8. 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, 170406 (2006). [CrossRef] [PubMed]

]. Other beams whose geometry departs from the circular, as the case of elliptical Mathieu beams (MBs) with helical phase [9

9. S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26, 1803–1805 (2001). [CrossRef]

], have also been used to rotate matter [10

10. C. López-Mariscal, J. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express 14, 4182–4187 (2006). [CrossRef] [PubMed]

]. The topological charge or singularity strength of an OV is the integer number l of phase cycles of 2π in a closed contour around the vortex core. It might be positive or negative depending on the handedness of rotation of the wavefronts. In this work, we will deal with three types of optical fields: BBs, MBs and LG beams, which will be referred to, altogether, as helical beams. As a consequence of the rotating phase, helical beams are also characterized by a circulating energy flow in the angular direction or optical current [5

5. M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009). [CrossRef]

], responsible for the torques exerted on small particles by these beams.

Due to the myriad of applications of helical beams, not only for rotating matter but also for microscopy [11

11. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13, 689–694 (2005). [CrossRef] [PubMed]

], quantum information storage and transmission [2

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

, 12

12. G. Molina-Terriza, J. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]

], optical lattices [13

13. K. Volke-Sepúlveda and R. Jáuregui, “All-optical 3D atomic loops generated with Bessel light fields,” J. Phys. B: At. Mol. Opt. 42, 085303 (2009). [CrossRef]

], nonlinear optics [14

14. G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992). [CrossRef] [PubMed]

16

16. A. Ruelas, S. Lopez-Aguayo, and J. C. Gutiérrez-Vega, “Stable solitons in elliptical photonic lattices,” Opt. Lett. 33, 2785–2787 (2008). [CrossRef] [PubMed]

] and astronomy [17

17. G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30, 3308–3310 (2005). [CrossRef]

], to name but a few, there have been big efforts dedicated to their experimental generation [18

18. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989). [CrossRef] [PubMed]

21

21. R. J. Hernández-Hernández, R. A. Terborg, I. Ricardez-Vargas, and K. Volke-Sepúlveda, “Experimental generation of Mathieu-Gauss beams with a phase-only spatial light modulator,” Appl. Opt. 49, 6903–6909 (2010). [CrossRef] [PubMed]

] and characterization. The determination of the topological charge and handedness have been the most common goal. There are techniques based on the interference of a helical beam with a reference plane wave [4

4. L. Allen, M. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]

]; others are based on Young’s experiment with different portions of the same beam [22

22. H. Sztul and R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. 31, 999–1001 (2006). [CrossRef] [PubMed]

]. On the other hand, methods to infer the properties of OVs based on diffraction have also been extensively developed, specially in the last few years. For example, diffraction of a vortex through a half-plane or a single slit allows to determine its handedness [23

23. J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50, 1573–1580 (2003).

25

25. D. Ghai, P. Senthilkumaran, and R. Sirohi, “Single-slit diffraction of an optical beam with phase singularity,” Opt. Laser Eng. 47, 123–126 (2009). [CrossRef]

]. More recently, diffraction by a triangular aperture was shown to give rise to truncated lattice patterns containing information about both, the magnitude and sign of the singularity strength [26

26. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010). [CrossRef] [PubMed]

]. Many other apertures, such as annulus, either circular [27

27. C. Guo, L. Lu, and H. Wang, “Characterizing topological charge of optical vortices by using an annular aperture,” Opt. Lett. 34, 3686–3688 (2009). [CrossRef] [PubMed]

] or elliptical [28

28. H. Tao, Y. Liu, Z. Chen, and J. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B: Lasers Opt. 106, 927–932 (2012). [CrossRef]

], iris diaphragms [29

29. A. Kumar, P. Vaity, and R. Singh, “Diffraction characteristics of optical vortex passing through an aperture-iris diaphragm,” Opt. Commun. 283, 4141–4145 (2010). [CrossRef]

], squares [30

30. P. H. F. Mesquita, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Engineering a square truncated lattice with light’s orbital angular momentum,” Opt. Express 19, 20616–20621 (2011). [CrossRef] [PubMed]

] and refractive elements like axicons [31

31. Y. Han and G. Zhao, “Measuring the topological charge of optical vortices with an axicon,” Opt. Lett. 36, 2017–2019 (2011). [CrossRef] [PubMed]

] have also been used to investigate diffraction of OVs.

Only few experimental works, however, have focused on the energy circulation in helical beams, even when this has been subject of considerable interest from the theoretical viewpoint [32

32. M. V. Berry and M. R. Dennis, “Stream function for optical energy flow,” J. Opt. 13, 064004 (2011). [CrossRef]

, 33

33. A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A: Pure Appl. Opt. 11, 094003 (2009). [CrossRef]

]. For instance, a proposal by Bekshaev is based on the analysis of the distortions produced in an oblique section of a beam due to the energy circulation compared to the geometric-optics expectation, where the shift of the center of gravity of the beam intensity is the characterization parameter [33

33. A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A: Pure Appl. Opt. 11, 094003 (2009). [CrossRef]

]. Although ingenious, this analysis does not seem very useful in practice in the optical domain, due to the high accuracy required for the measurements. In contrast, experimental studies in this direction have relied so far on the use of a Shack-Hartmann sensor to obtain a full map of the transverse energy flow [34

34. J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Opt. Express 14, 11919–11924 (2006). [CrossRef] [PubMed]

, 35

35. F. A. Starikov, G. G. Kochemasov, S. M. Kulikov, A. N. Manachinsky, N. V. Maslov, A. V. Ogorodnikov, S. A. Sukharev, V. P. Aksenov, I. V. Izmailov, F. Y. Kanev, V. V. Atuchin, and I. S. Soldatenkov, “Wavefront reconstruction of an optical vortex by a Hartmann-Shack sensor,” Opt. Lett. 32, 2291–2293 (2007). [CrossRef] [PubMed]

].

In this work, we demonstrate that it is possible to obtain quantitative information about the energy circulation in helical beams by means of diffraction through a single-slit. The pattern consists of two main lobes that shift outwards in opposite directions depending on the vortex helicity. By analysing the evolution of this pattern along the propagation direction in the near-field, we can determine the local skew angle of the Poynting vector. In order to discriminate the effects of simple diffraction from those arising due to the energy circulation, we compare the case of the helical beam with a non-rotating beam of similar characteristics. A comparison among analytical, numerical and experimental results is established. As a case of study we focus our attention on Bessel beams, but we also discuss some results for other helical beams.

2. Energy circulation in helical beams

In the case of paraxial helical light beams with circular cylindrical symmetry, the time-averaged Poynting vector can be written as [3

3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

, 7

7. K. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum S. Opt. 4, S82–S89 (2002). [CrossRef]

]
S=C0|u|2((Sρρ+(lkρσ2k1|u|2|u|2ρ)φ+z).
(1)

Here C0 = 2ε0/2 for free space propagation, (ρ⃗, φ⃗, z⃗) are the unit vectors in circular cylindrical coordinates, l is the topological charge and u = u(ρ, z) represents the complex amplitude of the optical field. The radial component Sρ=ρz/(z2+zR2) for LG beams [3

3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

], with zR denoting the Rayleigh range, and it is zero for BBs [7

7. K. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum S. Opt. 4, S82–S89 (2002). [CrossRef]

]. The parameter σ accounts for the polarization state, taking the values of 1 or −1 for left- or right-handed circular polarization, respectively, while it vanishes for linear polarization. We will only consider here the case of linear polarization. The case of circular polarization deserves a detailed study by itself, since the energy circulation will be modified by the second term in the angular component in Eq. (1), specially in the case of nonparaxial beams. In fact, the experimental method proposed here could be useful for detecting these differences for different polarization states.

Figure 1 shows the calculated transverse energy circulation at a given z plane for three examples of helical light beams: (a) a Bessel beam, (b) an LG mode and (c) a helical Mathieu beam. The analytical expression of the Poynting vector for Mathieu beams looks considerably more complicated than Eq. (1), and depending on the parameters describing the beam (topological charge l and ellipticity), it may exhibit a single vortex of charge l or l single-charged individual vortices along the interfocal line [9

9. S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26, 1803–1805 (2001). [CrossRef]

].

Fig. 1 Transverse components of the Poynting vector for different kinds of helical beams: (a) Bessel beam with l = 9; (b) Single-ringed Laguerre-Gauss beam with l = 6 and (c) helical Mathieu beam of order r = 6 and ellipticity parameter q = 12.

3. Near-field diffraction of an optical vortex by a single-slit

Fig. 2 Experimental setup. Spatial light modulator (SLM) displaying a phase computer generated hologram (CGH); Spatial Filter (SF); Slit (S) of 0.1mm width; CCD camera mounted on a micrometric translation stage, which is free to move along the Z direction.

From each of the obtained images we track the position of the intensity peaks of the two main lobes at different propagation distances. Because the maximum intensity points may shift due to reasons alien to energy circulation (such as diffraction ripples and a slight misalignment of the translation stage) we need a reference beam for comparison, with no energy circulation but with a similar intensity distribution to that of the helical beam in the region of the slit. In our experiments these are a cosine and a sine Bessel modes, for l = 6 and l = 9 respectively. In Fig. 3 we show the experimental images (green) and numerical simulations (gray-scale) of the incident beams (left-most column) and the near field diffraction patterns at different planes z. The numerical simulations were performed by solving the Fresnel diffraction integral at different z-planes with an iterative algorithm. It is easy to notice how the upper and lower lobes move in opposite directions from the slit axis over propagation for the helical beam, whereas the diffraction pattern is symmetric about the slit for the case of non-rotating beams. This is an indication of the energy circulation around the OV.

Fig. 3 Numerical simulation (gray-scale) and experimental images (green) of the impinging beam (left column) and their diffraction patterns at different z-planes for the helical Bessel beam of order l = 6 (bottom rows) and a non-rotating cosine beam as a reference (top rows). The z = 0mm distance would correspond to the nearest possible position of the CCD detector to the slit. The position of the axis of the slit is shown as a dashed red line. The slit has a width of 0.1mm.

Fig. 4 Displacement of the maximum intensity points of a rotating mode relative to the reference beam. The upper and lower lobes are analized separately (left and right plots respectively). The red line is the linear fit for the experimental data and the theoretical displacement is shown in green.

The experimental, numerical and theoretical values of (tan αφ) = 〈Sx〉/〈Sz〉 for the Bessel beams of orders 6 and 9 are summarized in Table 1. We found a fairly good agreement of the experimental and numerical values with respect to the theoretical values; differences are below 6% for the l = 6 mode and below 11% for the l = 9 mode. The difference in the experimental results for the upper and lower lobes might be due a minor misalignment of the slit or to a lack of perfect homogeneity of the intensity around the beam profile.

Table 1. Results for (tan αφ) = 〈Sx〉/〈Sz〉 from the experimental and numerical diffractive analisis for Bessel beams of order 6 and 9. For each mode the first and second rows correspond to the upper and lower lobes respectively. The slit is placed at x = 0mm and has a width of 0.1mm.]

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This technique can also be applied to other kinds of beams, but some further considerations must be taken into account. For circular beams, like LG and BBs, it is clear that in principle, due to the rotational symmetry, the diffraction by a single slit oriented along any arbitrary diameter provides enough information to extrapolate the energy circulation around the whole circumference. In contrast, this method has a more limited application for non-circular beams, like the elliptical MBs for instance. In that case, the orientation and position of the slit would be additional parameters to build up a full map of the energy circulation, which may become a very hard task. On the other hand, for beams that are not propagation invariant, as in the case of LG modes, the plane z at which the slit is placed is an important parameter.

Fig. 5 Simulation of the propagation of single-slit diffraction for a helical MB of order l = 6 and parameter of elipticity q = 12 (top row) and a p = 0, l = 6 LG mode with a Rayleigh range of 59mm (bottom row). The slit is centered at x = 0mm (dashed line) and has a width of 0.1mm.

Table 2. Numerical and theoretical results for (tan αφ) = 〈Sx〉/〈Sz〉 from the analisis of the MB and LG mode shown in Fig. 5.

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4. Discussion and conclusions

We have presented a technique to determine, quantitatively, the energy circulation in an optical vortex by analyzing the evolution on propagation of the near-field diffraction pattern produced by a single slit. Based on a numerical and experimental study, we evaluated the method by comparing our results with theoretical calculations, and found a maximum error of about 10% or less. Among the sources of error in our results are the spatial resolution of the CCD camera in the experiments and the resolution of the numerical grid in the simulations, since for some cases the pixel size might become comparable with the relative displacement of the intensity peaks we are tracking from plane to plane. The width of the slit is an important parameter. Although a narrow slit enables the selection of a portion of beam where the energy flux has uniform direction, so that the trajectory of the lobes will be almost linear, it will produce broader lobes and stronger diffraction effects than a wide slit, making it more difficult to determine the positions of the points of interest. In our experiments the slit has a width of 0.1mm ≈ 188λ and for the case of the BB of order 6 it is 15 the diameter of the innermost ring (for the BB of order 9 it is 17). According to our numerical study, for a given set of parameters defining the beam transverse size, the larger the value of the topological charge l, the better the results, since the measured shifts are also larger, allowing higher accuracy. In the experiments this is not always the case, since there are other factors that can affect the results, such as slight inhomogeneities in the intensity distribution of the generated beams, which are usually larger for higher order beams. However, we want to stress that we successfully applied our method to beams that are smaller than 1mm in diameter, which is a beam size that cannot be analyzed with standard Shack-Hartmann sensors.

This method can be used to compare the energy circulation of different vortices, depending on their topological charge, characteristic size, total optical power or even polarization state. It could also be a straightforward way to analyze the energy flux as a function of the wavelength in the case of polichromatic vortices [37

37. J. Leach and M. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. 5, 154.1–154.7 (2003). [CrossRef]

], or to determine the local energy flux in arrays of vortices. Finally, it is important to point out that, in contrast with refraction-based techniques like the Shack Hartman detector, our method can be applied not only to optical waves, but also to other kinds of waves, such as acoustic vortices [36

36. K. Volke-Sepúlveda, A. O. Santillán, and R. Boullosa, “Transfer of angular momentum to matter from acoustical vortices in free space,” Phys. Rev. Lett. 100, 024302 (2008). [CrossRef] [PubMed]

, 38

38. B. T. Hefner and P. L. Marston, “An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems,” J. Acoust. Soc. Am. 106, 3313–3316 (1999). [CrossRef]

40

40. K. D. Skeldon, C. Wilson, M. Edgar, and M. J. Padgett, “An acoustic spanner and its associated rotational Doppler shift,” New J. Phys. 10, 013018 (2008). [CrossRef]

], electron-wave vortices [41

41. 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, 192–195 (2011). [CrossRef] [PubMed]

], vortices in Bose-Einstein condensates [8

8. 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, 170406 (2006). [CrossRef] [PubMed]

] or X-ray vortices [42

42. A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, and I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002). [CrossRef]

].

Acknowledgments

Authors acknowledge support from DGAPA-UNAM, grant IN100110, and from CONACYT Mexico, grants 132527, 186368 (K. Volke-Sepúlveda) and 323560 (R. A. Terborg). We are also very grateful to Ms. Laura Perez-Garcia for her valuable help in the experiments.

References and links

1.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974). [CrossRef]

2.

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

3.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

4.

L. Allen, M. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]

5.

M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009). [CrossRef]

6.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]

7.

K. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum S. Opt. 4, S82–S89 (2002). [CrossRef]

8.

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, 170406 (2006). [CrossRef] [PubMed]

9.

S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26, 1803–1805 (2001). [CrossRef]

10.

C. López-Mariscal, J. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express 14, 4182–4187 (2006). [CrossRef] [PubMed]

11.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13, 689–694 (2005). [CrossRef] [PubMed]

12.

G. Molina-Terriza, J. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]

13.

K. Volke-Sepúlveda and R. Jáuregui, “All-optical 3D atomic loops generated with Bessel light fields,” J. Phys. B: At. Mol. Opt. 42, 085303 (2009). [CrossRef]

14.

G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992). [CrossRef] [PubMed]

15.

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Shaping soliton properties in Mathieu lattices,” Opt. Lett. 31, 238–240 (2006). [CrossRef] [PubMed]

16.

A. Ruelas, S. Lopez-Aguayo, and J. C. Gutiérrez-Vega, “Stable solitons in elliptical photonic lattices,” Opt. Lett. 33, 2785–2787 (2008). [CrossRef] [PubMed]

17.

G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30, 3308–3310 (2005). [CrossRef]

18.

A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989). [CrossRef] [PubMed]

19.

V. Arrizón, D. Sánchez-de-la Llave, U. Ruiz, and G. Méndez, “Efficient generation of an arbitrary nondiffracting Bessel beam employing its phase modulation,” Opt. Lett. 34, 1456–1458 (2009). [CrossRef] [PubMed]

20.

I. Ricardez-Vargas and K. Volke-Sepúlveda, “Experimental generation and dynamical reconfiguration of different circular optical lattices for applications in atom trapping,” J. Opt. Soc. Am. B 27, 948–955 (2010). [CrossRef]

21.

R. J. Hernández-Hernández, R. A. Terborg, I. Ricardez-Vargas, and K. Volke-Sepúlveda, “Experimental generation of Mathieu-Gauss beams with a phase-only spatial light modulator,” Appl. Opt. 49, 6903–6909 (2010). [CrossRef] [PubMed]

22.

H. Sztul and R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. 31, 999–1001 (2006). [CrossRef] [PubMed]

23.

J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50, 1573–1580 (2003).

24.

J. Masajada, “Half-plane diffraction in the case of Gaussian beams containing an optical vortex,” Opt. Commun. 175, 289–294 (2000). [CrossRef]

25.

D. Ghai, P. Senthilkumaran, and R. Sirohi, “Single-slit diffraction of an optical beam with phase singularity,” Opt. Laser Eng. 47, 123–126 (2009). [CrossRef]

26.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010). [CrossRef] [PubMed]

27.

C. Guo, L. Lu, and H. Wang, “Characterizing topological charge of optical vortices by using an annular aperture,” Opt. Lett. 34, 3686–3688 (2009). [CrossRef] [PubMed]

28.

H. Tao, Y. Liu, Z. Chen, and J. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B: Lasers Opt. 106, 927–932 (2012). [CrossRef]

29.

A. Kumar, P. Vaity, and R. Singh, “Diffraction characteristics of optical vortex passing through an aperture-iris diaphragm,” Opt. Commun. 283, 4141–4145 (2010). [CrossRef]

30.

P. H. F. Mesquita, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Engineering a square truncated lattice with light’s orbital angular momentum,” Opt. Express 19, 20616–20621 (2011). [CrossRef] [PubMed]

31.

Y. Han and G. Zhao, “Measuring the topological charge of optical vortices with an axicon,” Opt. Lett. 36, 2017–2019 (2011). [CrossRef] [PubMed]

32.

M. V. Berry and M. R. Dennis, “Stream function for optical energy flow,” J. Opt. 13, 064004 (2011). [CrossRef]

33.

A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A: Pure Appl. Opt. 11, 094003 (2009). [CrossRef]

34.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Opt. Express 14, 11919–11924 (2006). [CrossRef] [PubMed]

35.

F. A. Starikov, G. G. Kochemasov, S. M. Kulikov, A. N. Manachinsky, N. V. Maslov, A. V. Ogorodnikov, S. A. Sukharev, V. P. Aksenov, I. V. Izmailov, F. Y. Kanev, V. V. Atuchin, and I. S. Soldatenkov, “Wavefront reconstruction of an optical vortex by a Hartmann-Shack sensor,” Opt. Lett. 32, 2291–2293 (2007). [CrossRef] [PubMed]

36.

K. Volke-Sepúlveda, A. O. Santillán, and R. Boullosa, “Transfer of angular momentum to matter from acoustical vortices in free space,” Phys. Rev. Lett. 100, 024302 (2008). [CrossRef] [PubMed]

37.

J. Leach and M. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. 5, 154.1–154.7 (2003). [CrossRef]

38.

B. T. Hefner and P. L. Marston, “An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems,” J. Acoust. Soc. Am. 106, 3313–3316 (1999). [CrossRef]

39.

J. L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. 91, 244302 (2003). [CrossRef] [PubMed]

40.

K. D. Skeldon, C. Wilson, M. Edgar, and M. J. Padgett, “An acoustic spanner and its associated rotational Doppler shift,” New J. Phys. 10, 013018 (2008). [CrossRef]

41.

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, 192–195 (2011). [CrossRef] [PubMed]

42.

A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, and I. McNulty, “Observation of an x-ray vortex,” Opt. Lett. 27, 1752–1754 (2002). [CrossRef]

OCIS Codes
(050.1220) Diffraction and gratings : Apertures
(050.1940) Diffraction and gratings : Diffraction
(050.5080) Diffraction and gratings : Phase shift
(140.3300) Lasers and laser optics : Laser beam shaping
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Diffraction

History
Original Manuscript: November 19, 2012
Revised Manuscript: December 21, 2012
Manuscript Accepted: December 21, 2012
Published: February 4, 2013

Citation
Roland A. Terborg and Karen Volke-Sepúlveda, "Quantitative characterization of the energy circulation in helical beams by means of near-field diffraction," Opt. Express 21, 3379-3387 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3379


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  29. A. Kumar, P. Vaity, and R. Singh, “Diffraction characteristics of optical vortex passing through an aperture-iris diaphragm,” Opt. Commun.283, 4141–4145 (2010). [CrossRef]
  30. P. H. F. Mesquita, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Engineering a square truncated lattice with light’s orbital angular momentum,” Opt. Express19, 20616–20621 (2011). [CrossRef] [PubMed]
  31. Y. Han and G. Zhao, “Measuring the topological charge of optical vortices with an axicon,” Opt. Lett.36, 2017–2019 (2011). [CrossRef] [PubMed]
  32. M. V. Berry and M. R. Dennis, “Stream function for optical energy flow,” J. Opt.13, 064004 (2011). [CrossRef]
  33. A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A: Pure Appl. Opt.11, 094003 (2009). [CrossRef]
  34. J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Opt. Express14, 11919–11924 (2006). [CrossRef] [PubMed]
  35. F. A. Starikov, G. G. Kochemasov, S. M. Kulikov, A. N. Manachinsky, N. V. Maslov, A. V. Ogorodnikov, S. A. Sukharev, V. P. Aksenov, I. V. Izmailov, F. Y. Kanev, V. V. Atuchin, and I. S. Soldatenkov, “Wavefront reconstruction of an optical vortex by a Hartmann-Shack sensor,” Opt. Lett.32, 2291–2293 (2007). [CrossRef] [PubMed]
  36. K. Volke-Sepúlveda, A. O. Santillán, and R. Boullosa, “Transfer of angular momentum to matter from acoustical vortices in free space,” Phys. Rev. Lett.100, 024302 (2008). [CrossRef] [PubMed]
  37. J. Leach and M. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys.5, 154.1–154.7 (2003). [CrossRef]
  38. B. T. Hefner and P. L. Marston, “An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems,” J. Acoust. Soc. Am.106, 3313–3316 (1999). [CrossRef]
  39. J. L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett.91, 244302 (2003). [CrossRef] [PubMed]
  40. K. D. Skeldon, C. Wilson, M. Edgar, and M. J. Padgett, “An acoustic spanner and its associated rotational Doppler shift,” New J. Phys.10, 013018 (2008). [CrossRef]
  41. 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,” Science331, 192–195 (2011). [CrossRef] [PubMed]
  42. A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, and I. McNulty, “Observation of an x-ray vortex,” Opt. Lett.27, 1752–1754 (2002). [CrossRef]

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