## Visualization of the birth of an optical vortex using diffraction from a triangular aperture |

Optics Express, Vol. 19, Issue 7, pp. 5760-5771 (2011)

http://dx.doi.org/10.1364/OE.19.005760

Acrobat PDF (934 KB)

### Abstract

The study and application of optical vortices have gained significant prominence over the last two decades. An interesting challenge remains the determination of the azimuthal index (topological charge) *ℓ* of an optical vortex beam for a range of applications. We explore the diffraction of such beams from a triangular aperture and observe that the form of the resultant diffraction pattern is dependent upon both the magnitude and sign of the azimuthal index and this is valid for both monochromatic and broadband light fields. For the first time we demonstrate that this behavior is related not only to the azimuthal index but crucially the Gouy phase component of the incident beam. In particular, we explore the far field diffraction pattern for incident fields incident upon a triangular aperture possessing non-integer values of the azimuthal index *ℓ*. Such fields have a complex vortex structure. We are able to infer the birth of a vortex which occurs at half-integer values of *ℓ* and explore its evolution by observations of the diffraction pattern. These results demonstrate the extended versatility of a triangular aperture for the study of optical vortices.

© 2011 Optical Society of America

## 1. Introduction

*et al.*[1

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

*p*with

*p*+ 1 denoting the number of bright high intensity rings around the beam propagation axis. The azimuthal index

*ℓ*characterizes the exp(i

*ℓφ*) phase dependence around the optical axis. It is sometimes denoted as the topological charge or winding number of the light field. The azimuthal index is commonly associated with the orbital angular momentum content of the beam given as

*ℓh̄*per photon. More broadly, we may consider the field of optical vortices which relate to the presence of singular points in the optical field [2

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

3. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszstein-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]

5. V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. **91**, 093602 (2003). [CrossRef] [PubMed]

5. V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. **91**, 093602 (2003). [CrossRef] [PubMed]

6. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. **88**, 053601 (2002). [CrossRef]

7. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. **88**, 013601 (2001). [CrossRef]

*ℓ*values in such a system [8

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

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

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

*ℓ*which characterizes this beam. The azimuthal index

*ℓ*can be determined in a simple and direct way from the form of the pattern. In particular, we demonstrate that the Gouy phase shift causes the 180° rotation of the diffraction pattern as observed by Hickmann

*et al.*[10

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

11. S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. **95**, 240501 (2005). [CrossRef] [PubMed]

*et al.*[12

12. I. V. Basistiy, M. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. **119**, 604–612 (1995). [CrossRef]

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

*pℓ*, where

*ℓ*may now be either integer or non-integer. For the case of the fractional phase step (2

*pℓ*) being non-integer, a key theoretical prediction of Berry is the birth of a vortex within the beam as the fractional phase step reaches and passes a half-integer value. Thus to experimentally investigate such fractional topological charges we need to both generalize the range of fractional phase steps available as well as find a way of studying the evolution. Experimental studies have been performed using interference [14

14. W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. **239**, 129–135 (2004). [CrossRef]

15. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. **6**, 71 (2004). [CrossRef]

## 2. Theoretical background

### 2.1. Basic equations

*λ*and azimuthal index

*ℓ*that is centered on a polygonal aperture located at

*z*= 0,

*z*being the propagation axis. Denoting the transverse coordinates on the aperture plane as (

*x,y*) = (

*ρ*,

*θ*) in either Cartesian or cylindrical polar coordinates, we write the aperture transmission function as

*t*(

*x,y*) =

*t*(

*ρ*,

*θ*). For our specific case

*t*(

*x,y*) describes an equilateral triangular aperture which has unity transmission inside the aperture and zero outside the aperture, and over the spatial extent of the aperture the slowly varying electric field envelope of the incident optical vortex at

*z*= 0 is written as where

*A*measures the field strength, and we hereafter set

*A*= 1. In our experiment we create the far-field diffraction pattern using a standard 2

*f*Fourier transforming optical system based on a lens of focal length

*f*, and we denote the transverse Cartesian coordinates in the focal plane as (

*ξ*,

*η*). The diffracted field at the distance

*z*= 2

*f*past the aperture is then proportional to the Fourier transform of the product of the optical vortex times the aperture transmission function yielding the result [16

16. R. C. Smith and J. S. Marsh, “Diffraction patterns of simple apertures,” J. Opt. Soc. Am. **64**, 798–803 (1974). [CrossRef]

*X*= 2

*πξ*/

*λf*and

*Y*= 2

*πη*/

*λf*are scaled transverse coordinates in the observation plane.

*ℓ*, both integer values and non-integer. We checked our numerical code against the known analytic solution for a uniformly illuminated triangular aperture [16

16. R. C. Smith and J. S. Marsh, “Diffraction patterns of simple apertures,” J. Opt. Soc. Am. **64**, 798–803 (1974). [CrossRef]

*E*(

*X,Y,z*)|

^{2}for a variety of conditions. We note that our main interest here is in the morphology of these intensity profiles, as opposed to the detailed intensity values or the specific spatial sizes. This is relevant in view of the fact that, within the paraxial approximations employed, the intensity profiles generated by Eq. (2) do not depend on the specific triangular aperture size but only on the azimuthal index

*ℓ*. This follows from the fact that the optical vortices in Eq. (1) do not display any particular length scale against which the aperture could be compared.

### 2.2. Integer azimuthal indices

*et. al.*[10

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

*ℓ*= 2, 3, 4 in the upper row, and

*ℓ*= −2,−3, −4 in the lower row, and one observes what Hickmann

*et al.*[10

**105**, 053904 (2010). [CrossRef] [PubMed]

*ℓ*| + 1). We also note that there are (|

*ℓ*| + 1) parallel rows or lines of lobes making up a given truncated optical lattice. Furthermore, the effect of changing the sign of the azimuthal index

*ℓ*→ −

*ℓ*is to rotate the orientation of the truncated optical lattice by 180°, as can be verified by inspecting the upper and lower rows in Fig. 1. Thus the number of lobes in the truncated optical lattice and its orientation can be used as a detector of the azimuthal index of an incident optical vortex.

**105**, 053904 (2010). [CrossRef] [PubMed]

*z*= 0 In writing this we are explicitly assuming that the vortex core of the form

*ρ*

^{|ℓ|}

*e*

^{iℓθ}is isolated by the aperture. To proceed we expand the field beyond the aperture in terms of the complete orthonormal set of LG modes where the real functions

*u*

_{p,|m|}(

*ρ*,

*z*) are given by [17] where

*k*=

*ω*/

*c*,

*w*(

*z*) and

*R*(

*z*) are the Gaussian spot size and radius of curvature of the propagating modes,

*w*(0) =

*w*

_{0}and 1/

*R*(0) = 0. Φ(

*z*) is the Gouy phase-shift of the propagating modes, with Φ(0) = 0 and Φ(

*z*→ ∞) =

*π*/2. Then we may write the field beyond the aperture as an expansion of LG modes where the expansion coefficients

*z*= 0 and are given by We note that based on this definition the following symmetry holds that we shall use later and that this symmetry holds independent of the choice of LG spot size

*w*

_{0}. We now specifically consider the far-field for which the Gouy phase-shift is Φ(

*z*) →

*π*/2 [17], in which case the diffracted field becomes

*ℓ*giving the far-field profile

*m*→ −

*m*, in the third line we used the symmetry (8), and in the fourth line we used

*e*

^{i|m|π}=

*e*

^{imπ}. By comparing with Eq. (11), we therefore see that the far-field intensity pattern is given by Thus the far-field intensity patterns for

*ℓ*= ±|

*ℓ*| have the structure given by |

*F*

^{(ℓ)}(

*ρ*,

*θ*,

*z*)|

^{2}but are rotated by 180° with respect to each other around the vortex center. We note that if we artificially set the Gouy phase-shift to zero Φ(

*z*) = 0 in the far-field in place of the correct value Φ(

*z*) =

*π*/2, then the intensity pattern would not be rotated upon reversing the sign of

*ℓ*, meaning that the rotation has its origin in the Gouy phase-shift. This is the main result of this analysis, and it explains the fact that diffraction of a optical vortex core from a polygonal aperture in general depends upon the sign of the azimuthal index, and furthermore exposes that this dependence stems from the Gouy phase-shift, giving new physical insight into this problem, compared to previous studies [10

**105**, 053904 (2010). [CrossRef] [PubMed]

### 2.3. Fractional azimuthal indices

*ℓ*. Figure 2 shows the intensity patterns obtained for

*ℓ*= 2 → 3 in steps of 0.1. The main feature we wish to point out is that for azimuthal indices in the range 2 <

*ℓ*< 2.3 the intensity pattern remains largely unchanged and close to that for

*ℓ*= 2 in that it has (|

*ℓ*| + 1) = 3 rows in the truncated optical lattice. In contrast for

*ℓ*≥ 2.4 one can see that the intensity pattern starts to distort on the left side as indicated by the circled region in the intensity pattern for

*ℓ*= 2.4 in Fig. 2. This distortion is the beginning of a new row of lobes that develops for

*ℓ*> 2.4, and for

*ℓ*= 3 the new row is fully formed and there are (|

*ℓ*| + 1) = 4 rows in the truncated optical lattice shown in the intensity profile. What the sequence of intensity profiles in Fig. 2 reveals is how the birth of a vortex is manifested in the truncated optical lattice generated by the triangular aperture, and in particular how the optical lattice deforms from having three to four rows as

*ℓ*= 2 → 3 in this example: There is little distortion of the truncated optical lattice from its

*ℓ*= 2 form with 3 rows for 2 <

*ℓ*< 2.3, and the birth of the

*ℓ*= 3 vortex with its 4 rows is clearly evident for

*ℓ*≥ 2.5 until it is completed for

*ℓ*= 3. This sequence of events is perfectly in keeping with the prediction due to Berry [13

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

*ℓ*with the same general result: As the azimuthal index of the incident optical vortex passes (|

*ℓ*| +1/2) the birth of an optical vortex is revealed in the optical lattice produced by diffraction of the optical vortex by the triangular lattice, in keeping with Berry’s prediction [13

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

*ℓ*= 2 → 5, so that several examples can be viewed. To aid viewing the animation pauses at integer values of

*ℓ*so that the reader can view the different examples involving variation between integer values of azimuthal index. The animation paints a fascinating picture in which starting from an integer

*ℓ*azimuthal index the optical lattice has (|

*ℓ*| + 1) rows which persist until the azimuthal index reaches (|

*ℓ*| + 0.3) at which point the optical lattice appears to move to the right as a new row of lobes starts to appear on the left. This process culminates when the azimuthal index reaches (|

*ℓ*| + 1) for which the optical lattice has (|

*ℓ*| + 2) rows, and this scenario is seen in each example displayed in the animation.

## 3. Experiments

*P*

_{max}= 4 mW) served for the monochromatic measurements and a supercontinuum source (Fianium Ltd, 4 ps, 10 Mhz) for the broadband white light measurements. The white light beam was additionally sent through a photonic crystal fiber (Thorlabs LMA-25) in order to obtain a beam featuring a homogeneous Gaussian intensity profile. The laser beam was subsequently expanded with a telescope in order to slightly overfill the chip of a spatial light modulator (SLM) which was a Holoeye LC-R 2500. The SLM operated in the standard first-order configuration and was used to imprint the vortex phase on the incident beam where the LG beam was created in the far-field of the SLM. We used a pair of lenses (lenses L

_{3}and L

_{4}in Fig. 4) in order to filter the first order beam, carrying the vortex, from the unmodulated zero-order beam using an aperture located in the back focal plane of lens L

_{3}. For the broad-band studies a prism (not shown in Fig. 4) was located in the back focal plane of lens L

_{4}. With this, we compensated for the dispersion mediated by the linear phase shift imposed onto the laser beam by the SLM in order to separate the first-order from the zero-order beam. The two lenses are not necessary for the monochromatic studies, but we intended to perform both the monochromatic and the broadband studies on the basis of the same experimental setup. After lens L

_{4}, the created LG beam was incident onto a triangular aperture located at a distance of approximately 3 m from lens L

_{4}. The triangular aperture was realized in a different manner for the monochromatic and broadband studies. We used a second SLM (Hamamatsu X10468-01) for the monochromatic studies. This allowed us to flexibly change the geometry of the triangular aperture using dedicated Labview software. In particular, we were able to take into account the change in size of the LG beam when changing the index

*ℓ*. In contrast, we used static triangular apertures imprinted onto photographic film for the broadband measurements. With this we avoided inducing dispersion onto the broadband LG beam which would require further prism compensation. Finally, lens L

_{5}was used to create the far-field diffraction pattern in the respective back focal plane where a color CCD camera (Basler piA640-210gc, pixel size: 7.4 μm × 7.4 μm) served to record and save images of the diffraction patterns onto the hard drive of a computer. On a final note, the quality of the prism dispersion compensation critically depends on the beam diameter. To account for this, we reduced the size of the broadband LG beam by extracting a central part of the beam using the first SLM. We obtained satisfactory results in terms of dispersion compensation when reducing the beam diameter by a factor of 2.

*ℓ*. The diffraction pattern is blurred and deformed if the size of the triangular aperture is not commensurate with the diameter of the bright ring. Moreover, the pattern intensity is larger for a large thickness while the pattern gets more distinct for a small thickness. We have also adjusted the CCD camera exposure time in order to best highlight the pattern morphology for each recorded pattern. We first show the experimental results for integer azimuthal indices in Fig. 5 where the triangular aperture was the same for

*ℓ*= 2 and

*ℓ*= 3 and adjusted to a larger size for

*ℓ*= 4. The experimental results are in excellent agreement with the theoretical prediction shown in Fig. 1: the number of interference lobes on any one side of the diffraction pattern is equal to (|

*ℓ*| + 1), there are (|

*ℓ*| + 1) parallel lines of lobes in any of the three directions, and the pattern orientation is flipped when the sign of the azimuthal index is changed. We also observe excellent agreement for non-integer azimuthal indices which is best seen through comparison of Figs. 6 and 2 where the former shows diffraction patterns recorded with the CCD camera for

*ℓ*= 2 → 3 in steps of 0.1. The experimental data clearly manifest the expected distortion of the diffraction pattern for

*ℓ*≥ 2.4 as indicated by the circled region. A new row of lobes arises that develops for

*ℓ*> 2.4, and finally the

*ℓ*= 3 diffraction pattern is formed. Our experimental data therefore verify Berry’s prediction [13

**6**, 259–268 (2004). [CrossRef]

*ℓ*= 2 → 3 in steps of 0.1 in Fig. 7. The patterns are less distinct compared to the monochromatic data shown in Figs. 5 and 6 which, apart from the broadband nature of the light, must be attributed to the reduced LG beam size (only half of SLM display used due to dispersion compensation) and the use of static triangular apertures imprinted onto photographic film which did not allow us to adjust the aperture to the LG beam as flexibly as we were able to do in the monochromatic studies. Nevertheless, we can clearly identify the birth of the vortex in the broadband patterns as well and observe all the characteristic effects that is a new row of lobes arises for

*ℓ*> 2.4 as well.

*ℓφ*) exhibits a discontinuity at a definite azimuthal angle that can be physically identified, so that the relative orientation of the triangular aperture and azimuthal position comes into play and indeed dictates the spatial direction in which the optical lattice distorts and the lattice symmetry is broken. Thus, experimentally it should be the case that if the triangular aperture is rotated keeping the azimuthal index and orientation of the optical vortex fixed then the orientation of the optical lattice should also rotate. Moreover, the optical lattice should melt and reassemble at a different spatial location. This is indeed observed as we demonstrate in Fig. 8 on the basis of experimentally recorded monochromatic diffraction patterns. As the triangular aperture is rotated in steps of 90° so does the diffraction pattern. However, the region of the vortex birth does not follow the rotation as indicated by the dashed rectangles indicating the expected regions and the dashed ellipses indicating the actual regions where the birth of the vortex is observed. This observation also explains why the broadband data would not coincide with the monochromatic data if rotated accordingly. The monochromatic measurements were based on apertures created with a second SLM whereas the broadband measurements were performed using a static aperture. As a consequence, these two configurations exhibited different relative orientations between the aperture and the fractional vortex leading to different regions where the birth of the vortex took place.

## 4. Conclusions

## Acknowledgments

## References and links

1. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A |

2. | J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A |

3. | H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszstein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. |

4. | K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Phys. |

5. | V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. |

6. | A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. |

7. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. |

8. | H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. |

9. | C. S. Guo, L. L. Lu, and H. T. Wang, “Characterizing topological charge of optical vortices by using an annular aperture,” Opt. Lett. |

10. | J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. |

11. | S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. |

12. | I. V. Basistiy, M. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. |

13. | M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. |

14. | W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. |

15. | J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. |

16. | R. C. Smith and J. S. Marsh, “Diffraction patterns of simple apertures,” J. Opt. Soc. Am. |

17. | A. E. Siegman, Lasers (University Science, 2004), Chap. 16.4. |

18. | J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. |

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(050.1940) Diffraction and gratings : Diffraction

(140.3300) Lasers and laser optics : Laser beam shaping

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: December 23, 2010

Revised Manuscript: February 14, 2011

Manuscript Accepted: February 15, 2011

Published: March 14, 2011

**Citation**

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, "Visualization of the birth of an optical vortex using diffraction from a triangular aperture," Opt. Express **19**, 5760-5771 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-5760

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### References

- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]
- J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336, 165–190 (1974). [CrossRef]
- H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinszstein-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]
- K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Phys. 56, 261–337 (2008).
- V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003). [CrossRef] [PubMed]
- A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002). [CrossRef]
- G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001). [CrossRef]
- H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. 31, 999–1001 (2006). [CrossRef] [PubMed]
- C. S. Guo, L. L. Lu, and H. T. Wang, “Characterizing topological charge of optical vortices by using an annular aperture,” Opt. Lett. 34, 3686–3688 (2009). [CrossRef] [PubMed]
- J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010). [CrossRef] [PubMed]
- S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005). [CrossRef] [PubMed]
- I. V. Basistiy, M. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995). [CrossRef]
- M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6, 259–268 (2004). [CrossRef]
- W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004). [CrossRef]
- J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” N. J. Phys. 6, 71 (2004). [CrossRef]
- R. C. Smith and J. S. Marsh, “Diffraction patterns of simple apertures,” J. Opt. Soc. Am. 64, 798–803 (1974). [CrossRef]
- See, for example,A. E. Siegman, Lasers (University Science, 2004), Chap. 16.4.
- J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50, 1573–1580 (2003).

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