## Position measurement of non-integer OAM beams with structurally invariant propagation |

Optics Express, Vol. 20, Issue 25, pp. 27429-27441 (2012)

http://dx.doi.org/10.1364/OE.20.027429

Acrobat PDF (2329 KB)

### Abstract

We present a design to generate structurally propagation invariant light beams carrying non-integer orbital angular momentum (OAM) using Hermite-Laguerre-Gaussian (HLG) modes. Different from previous techniques, the symmetry axes of our beams are fixed when varying the OAM; this simplifies the calibration technique for beam positional measurement using a quadrant detector. We have also demonstrated analytically and experimentally that both the OAM value and the HLG mode orientation play an important role in the quadrant detector response. The assumption that a quadrant detector is most sensitive at the beam center does not always hold for anisotropic beam profiles, such as HLG beams.

© 2012 OSA

## 1. Introduction

*πN*, with

*N*an integer value that is equivalent to the OAM content of such a beam. After the first investigation of the astigmatic transformation of Hermite-Gaussian (HG) modes into Laguerre-Gaussian (LG) modes [1

1. E. Abramochkin and V. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. **83**, 123–125 (1991). [CrossRef]

*ℓ*of the LG modes [2

2. 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]

3. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. **42**, 217–223 (1995). [CrossRef]

4. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics **5**, 343–348 (2011). [CrossRef]

5. S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’tHooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. **95**, 240501 (2005). [CrossRef] [PubMed]

6. B. J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. **13**, 064008 (2011). [CrossRef]

7. J. Sato, M. Endo, S. Yamaguchi, K. Nanri, and T. Fujioka, “Simple annular-beam generator with a laser-diode-pumped axially off-set power build-up cavity,” Opt. Commun. **277**, 342–348 (2007). [CrossRef]

8. G. C. G. Berkhout and M. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express **18**, 13836–13841 (2010). [CrossRef] [PubMed]

9. F. Tamburini, B. Thidé, G. Molina-Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. **7**, 195–197 (2011). [CrossRef]

10. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A **82**, 023817 (2010). [CrossRef]

*non-integer*values of OAM that potentially broadens the OAM beams applications. By applying non-integer OAM beams to existing OAM beams applications, we introduce a degree of freedom of optical manipulation. Non-integer OAM beams have found uses in high-dimensional quantum information processing [6

6. B. J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. **13**, 064008 (2011). [CrossRef]

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]

*integer*OAM beam generation techniques have been introduced. The first demonstration used the so-called ‘

*π*/2-mode converter’, which belongs to a family of astigmatic mode converters that applies the appropriate Gouy phase to create well defined mode indices of LG beams carrying integer OAM [12

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

13. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. **112**, 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**, 183–188 (1996). [CrossRef]

3. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. **42**, 217–223 (1995). [CrossRef]

15. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. **45**, 1231–1237 (1998). [CrossRef]

*ℓ*but different radial mode index

*p*. Although the OAM content of such a beam is well defined, the spatial field distribution evolves during propagation. This mode impurity problem holds also when employing q plates [16

16. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. **96**, 163905 (2006). [CrossRef] [PubMed]

17. G. Machavariani, N. Davidson, E. Hasman, S. Bilt, A. Ishaaya, and A. A. Friesem, “Efficient conversion of a Gaussian beam to a high purity helical beam,” Opt. Commun **209**, 265–271 (2002). [CrossRef]

21. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett **34**, 34–36 (2009). [CrossRef]

*non-integer*OAM beam generation, only a handful of studies have been carried on. One of the initial ideas was to use off-axis illumination of an SPP; equivalently, one may use a non-integer 2

*π*phase step SPP [5

5. S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’tHooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. **95**, 240501 (2005). [CrossRef] [PubMed]

*ℓ*nor

*p*mode purity [22

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

23. J. B. Götte, K. O’Holleran, D. Preece, F. Flossman, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express **16**, 993–1006 (2008). [CrossRef] [PubMed]

*ℓ*| ≤ 1 [24

24. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction,” Opt. Express **18**, 16480–16485 (2010). [CrossRef]

25. A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or LaguerreGaussian modes and the variable-phase mode converter,” Opt. Commun. **181**, 35–45 (2000). [CrossRef]

26. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A: Pure Appl. Opt. **6**, S157–S161 (2004). [CrossRef]

28. E. G. Abramochkin, E. Razueva, and V. G. Volostnikov, “General astigmatic transform of Hermite-Laguerre-Gaussian beams,” J. Opt. Soc. Am. A **27**, 2506–2513 (2010). [CrossRef]

29. E. J. Lee, Y. Park, C. S. Kim, and T. Kouh, “Detection sensitivity of the optical beam deflection method characterized with the optical spot size on the detector,” Curr. Appl. Phys. **10**, 834–837 (2010). [CrossRef]

30. Y. Panduputra, T. W. Ng, A. Neild, and M. Robinson, “Intensity influence on Gaussian beam laser based measurements using quadrant photodiodes,” Appl. Opt. **49**, 3669–3675 (2010). [CrossRef] [PubMed]

*integer*OAM beams [31

31. N. Hermosa, A. Aiello, and J. P. Woerdman, “Quadrant detector calibration for vortex beams,” Opt. Lett. **36**, 409–411 (2011). [CrossRef] [PubMed]

*non-integer*OAM beams based upon HLG modes.

26. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A: Pure Appl. Opt. **6**, S157–S161 (2004). [CrossRef]

25. A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or LaguerreGaussian modes and the variable-phase mode converter,” Opt. Commun. **181**, 35–45 (2000). [CrossRef]

26. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A: Pure Appl. Opt. **6**, S157–S161 (2004). [CrossRef]

## 2. Generation of structurally propagation invariant light carrying non-integer OAM

### 2.1. Experimental set-up

*π*/2 astigmatic mode converter [12

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

*f*, separated at a distance

*integer*OAM is achieved when the symmetry axes of the HG beam are oriented at an angle

*α*= 45° with respect to the symmetry axes of the cylindrical lenses [12

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

*α*= 45°.

*α*, i.e. the angle between the symmetry axes of cylindrical lenses and the symmetry axes of the input HG beam [28

28. E. G. Abramochkin, E. Razueva, and V. G. Volostnikov, “General astigmatic transform of Hermite-Laguerre-Gaussian beams,” J. Opt. Soc. Am. A **27**, 2506–2513 (2010). [CrossRef]

*ℓ*= (

*n*−

*m*)sin2

*α*, with

*n*and

*m*the mode index of high order HG beams [26

**6**, S157–S161 (2004). [CrossRef]

27. J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A **70**, 013809 (2004). [CrossRef]

*d*and the position of the cylindrical lenses pair with respect to the mode matching lens [32]. However, for aligning purposes, the approach of varying

*α*is more attractive when tuning the non-integer OAM value.

_{01}mode with varying orientation angle

*α*on a ‘

*π*/2-mode converter’. The Gouy phase

*φ*experienced by the projected mode after traversing the cylindrical lenses is 2

*α*. Note that the symmetry axes of the outgoing HLG beam are always aligned to the projection axes of the cylindrical lenses.

*λ*= 632.8 nm, situated at the centre of an open two-mirror cavity allowing for a generation of up to the third order of the HG mode family (i.e. HG

_{3,3}). The laser is forced to operate in a single higher order HG mode by insertion of a 18

*μ*m diameter copper wire normal to and rotatable with respect to the axis of the laser cavity. The strength and location of a mode matching lens and a pair of cylindrical lenses are chosen such that they create integer OAM beams when the wire is orientated at

*α*= 45°. By rotating the wire about the optical axis, we tune the parameter

*α*to generate the HLG modes. This is different from the two previous techniques; where two Dove prisms and two cylindrical lenses are rotated to flip the HG mode before being converted into HLG modes [25

25. A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or LaguerreGaussian modes and the variable-phase mode converter,” Opt. Commun. **181**, 35–45 (2000). [CrossRef]

*α*is tuned by rotating the cylindrical lenses [26

**6**, S157–S161 (2004). [CrossRef]

*always*aligned to the quadrant detector measurement axes which greatly simplifies the quadrant detector operation.

### 2.2. Characterization of non-integer OAM beams

*α*. The open laser cavity is forced to operate at the first higher order HG mode, i.e HG

_{0,1}. The first two rows display the measured intensity profiles of Fig. 3(a) the incoming HG

_{0,1}and Fig. 3(b) the outgoing HLG

_{0,1|[}

_{α}_{:0°,90°]}beams at the far-field after the collimating lens. Our generated HLG beam profiles match with the calculation shown in Fig. 3(c). In the calculated images, we have used a color map to indicate the phase profile of the generated HLG modes. For outgoing HLG profiles being the analytic interpolation between a HG mode and a LG mode, we observe a more flat wavefront inside the high intensity areas (note the even color tone). Inside the dark intensity areas, the phase value increases non-linearly along the azimuthal direction. The phase singularity of zero-OAM beams at

*α*=

*N*× 90° forms a line (most left and most right images of Fig. 3(c)), whereas for integer OAM beams at

*α*= (2

*N*+1)×45° it forms a vortex (the center image of Fig. 3(c)), with

*N*an integer number. Apart from the overall scaling due to diffraction and slight astigmatism due to imperfect alignment, the generated HLG beams are structurally stable during propagation, as shown in Fig. 4.

*ℓ*number of lines at the dark centre of the beam, which is also the case for the centre image of Fig. 5. In the case of beams with non-integer OAM values, the branching gradually dissolves into separated shifted lines as shown by the measurement result (a) and confirmed by the calculation (b).

*ℓ*= (

*n*−

*m*)sin2

*α*[26

**6**, S157–S161 (2004). [CrossRef]

27. J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A **70**, 013809 (2004). [CrossRef]

*ℓ*changes each time

*α*crosses the value of

*N*× 90°. Secondly, the HLG mode profile rotates by 90° each time

*α*crosses the value of (2

*N*+ 1) × 45° with

*N*an integer number. Take the example of Fig. 3(b) and 3(c), where we have tuned the angle 0° ≤

*α*≤ 90° to obtain 0 ≤

*ℓ*≤ 1. Although all the HLG modes shown have positive values of

*ℓ*, the mode profiles for

*α*≤ 45° are rotated 90° with respect to the profiles for

*α*≥ 45°. Therefore, for an identical

*ℓ*or OAM values, there are two possible orthogonal orientations of the HLG beams. The orthogonal orientation of HLG beams also greatly simplifies the calibration procedure when measuring the beam position using a quadrant detector, as will be discussed in the next section.

## 3. Quadrant detector response to HLG beam displacement

*I*

_{A},

*I*

_{B},

*I*

_{C}and

*I*

_{D}which are generated when an optical beam strikes the active area. Its position-current relation can be written as:

*U*(

*x,y*)|

^{2}the intensity of the impinging beam, for shifts along the

*x*– and

*y*–axis of the quadrant detector, respectively. When using our generation technique, the quadrant detector axes coincide with the transverse beam axes. To obtain the nominal beam displacement, the quadrant detector signal

*I*/

_{x,y}*I*

_{Σ}has to be normalized to the slope of this relationship curve, i.e. the calibration constant

*K*.

*x*≪

*w*, and the calibration constant

*K*is derived around the beam center where the slope of positional-current relationship is linear. Previously, the quadrant detector calibration constant for LG beams as a function of

*ℓ*has also been derived for small displacements around the beam center [31

31. N. Hermosa, A. Aiello, and J. P. Woerdman, “Quadrant detector calibration for vortex beams,” Opt. Lett. **36**, 409–411 (2011). [CrossRef] [PubMed]

*K*for the displacement along the

*x*–axis of HLG

_{n,m|α}mode is also valid for the displacement along the

*y*–axis of HLG

*|*

_{m,n}*mode.*

_{α}*x*≪

*w*will not always give the most sensitive position measurement. This is due to the fact that for some cases, the profile cross section |

*U*(

*x,y*)|

^{2}of the HLG modes along the axis of displacement, has near zero values across one displacement axis. As an example, let us observe HLG modes for

*α*> 45° in Fig. 3(b). The low intensity values at the beam centre across the

*y*-axis is certainly the least sensitive area to measure beam displacement along the

*x*–axis. Interchangeably, the quadrant detector is least sensitive for beam displacement along the

*y*–axis around the centre area of HLG modes for

*α*< 45°. Therefore, it is important to find the region where the quadrant detector can operate with the highest sensitivity.

### 3.1. Analytical solutions of quadrant detector calibration

*mode. For didactic purposes, we take the example of a radial mode index*

_{n,m|α}*p*= 0 and an azimuthal mode index

*ℓ*= 1 (i.e. HLG

_{0,1|α}), and investigate the quadrant response for the beam displacement along the

*x*–axis. By applying the distribution function of HLG

_{0,1|α}given in Ref. [26

**6**, S157–S161 (2004). [CrossRef]

*x*-axis displacement relationship normalized to the beam radius

*w*for HLG

_{0,1|α}to be Note that due to the symmetry axes, the same expression is found for the beam displacement of HLG

_{1,0|α}mode along the y-axis, substituting the index

*x*with

*y*.

_{0,1|α}beams. The position-current relationship curves in Fig. 6(b) reveal that there are different linear regions with a constant slope (calibration constant

*K*) for different values of

*α*. The linear region shifts to a higher

*x/w*value for

*α*> 45°, coinciding with the peak intensity cross section along the displacement axis, at around

*x/w*= 0.7. For

*α*< 45° the beam cross-section along the

*x*-axis resembles that of a Gaussian profile and the range of linearity is around the beam center.

_{0,1|α}modes, where we have used a quadrant detector from NewFocus model 2921 with an active area of 10mm × 10mm. Figure 6(b) shows the the match between our data (open circles and dots) and the analytical solution (solid and dashed lines). It is important to realize that there exists two values of

*α*for each HLG

_{n,m|α}mode that give the same

*ℓ*values but with orthogonally oriented spatial distribution. These orthogonally oriented modes have different calibration constants

*K*, as plotted in Fig. 7.

*K*. This calibration procedure is particularly relevant for potential applications using HLG modes as non-integer OAM beams: in beam shifts measurements, high precision metrology, optical manipulation using tweezers or scanning near-field optical microscopy.

## 4. Conclusion

*α*. For example,

*ℓ*= 1 can be constructed from HLG

_{0,2|[}

_{α}_{:15°]}, which actually produces a phase distribution that is different from that of a LG

_{0,1}mode. In applications such as OAM beams shifts or optical manipulation using OAM beams, noticeable differences will occur when addressing an integer OAM value by using either LG modes or HLG modes.

*always*aligned to the axes of quadrant detectors; which simplifies the operation and calibration of the detector.

*K*of a quadrant detector for HLG beams agrees with Ref. [31

31. N. Hermosa, A. Aiello, and J. P. Woerdman, “Quadrant detector calibration for vortex beams,” Opt. Lett. **36**, 409–411 (2011). [CrossRef] [PubMed]

*ℓ*, where the beam profile is isotropic or cylindrically symmetric. The assumption that a quadrant detector is most sensitive at the beam center does not always hold for general astigmatic modes, i.e. HLG modes, that has an anisotropic beam profiles.

*ℓ*values and the HLG mode orientation play a role in the quadrant detector response. Furthermore, the anisotropic nature of HLG beams creates different regions having linear response of a quadrant detector when measuring beam positional shift. The beam positional measurement is most sensitive around the peak of the HLG mode profile. Our result can easily be extended to arbitrarily higher order HLG beams as solutions of light carrying higher order non-integer OAM.

## Acknowledgments

## References and links

1. | E. Abramochkin and V. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. |

2. | 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 |

3. | H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. |

4. | M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics |

5. | S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’tHooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. |

6. | B. J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. |

7. | J. Sato, M. Endo, S. Yamaguchi, K. Nanri, and T. Fujioka, “Simple annular-beam generator with a laser-diode-pumped axially off-set power build-up cavity,” Opt. Commun. |

8. | G. C. G. Berkhout and M. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express |

9. | F. Tamburini, B. Thidé, G. Molina-Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. |

10. | M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A |

11. | S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express |

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

13. | M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. |

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. |

15. | J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. |

16. | L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. |

17. | G. Machavariani, N. Davidson, E. Hasman, S. Bilt, A. Ishaaya, and A. A. Friesem, “Efficient conversion of a Gaussian beam to a high purity helical beam,” Opt. Commun |

18. | S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A |

19. | S.-C. Chu and K. Otsuka, “Doughnut-like beam generation of Laguerre-Gaussian mode with extremely high mode purity,” Opt. Commun. |

20. | N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A |

21. | T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett |

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

23. | J. B. Götte, K. O’Holleran, D. Preece, F. Flossman, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express |

24. | D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction,” Opt. Express |

25. | A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or LaguerreGaussian modes and the variable-phase mode converter,” Opt. Commun. |

26. | E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A: Pure Appl. Opt. |

27. | J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A |

28. | E. G. Abramochkin, E. Razueva, and V. G. Volostnikov, “General astigmatic transform of Hermite-Laguerre-Gaussian beams,” J. Opt. Soc. Am. A |

29. | E. J. Lee, Y. Park, C. S. Kim, and T. Kouh, “Detection sensitivity of the optical beam deflection method characterized with the optical spot size on the detector,” Curr. Appl. Phys. |

30. | Y. Panduputra, T. W. Ng, A. Neild, and M. Robinson, “Intensity influence on Gaussian beam laser based measurements using quadrant photodiodes,” Appl. Opt. |

31. | N. Hermosa, A. Aiello, and J. P. Woerdman, “Quadrant detector calibration for vortex beams,” Opt. Lett. |

32. | S. J. M. Habraken, “Cylindrical lens mode converters and OAM,” Personal Communication (2010). |

**OCIS Codes**

(040.5160) Detectors : Photodetectors

(120.4640) Instrumentation, measurement, and metrology : Optical instruments

(260.6042) Physical optics : Singular optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 19, 2012

Revised Manuscript: November 1, 2012

Manuscript Accepted: November 1, 2012

Published: November 26, 2012

**Citation**

A. M. Nugrowati, W. G. Stam, and J. P. Woerdman, "Position measurement of non-integer OAM beams with structurally invariant propagation," Opt. Express **20**, 27429-27441 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-25-27429

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### References

- E. Abramochkin and V. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun.83, 123–125 (1991). [CrossRef]
- 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. A45, 8185–8189 (1992). [CrossRef] [PubMed]
- H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt.42, 217–223 (1995). [CrossRef]
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