## Structure of optical singularities in coaxial superpositions of Laguerre–Gaussian modes |

JOSA A, Vol. 27, Issue 12, pp. 2602-2612 (2010)

http://dx.doi.org/10.1364/JOSAA.27.002602

Acrobat PDF (1172 KB)

### Abstract

We investigate optical singularities in coaxial superpositions of two Laguerre–Gaussian (LG) modes with a common beam waist from the viewpoints of a general formulation of phase structure, experimental generation of various superposition beams, and evaluation of the generated beams’ fidelity. By applying a holographic phase-amplitude modulation scheme using a phase-modulation-type spatial light modulator, output fidelity beyond 0.960 was observed under several typical conditions. Additionally, an elliptic-type folded singularity, which provides a different class of phase structures from familiar helical singularities, was predicted and observed in a superposition involving two LG modes of both radially and azimuthally higher orders.

© 2010 Optical Society of America

## 1. INTRODUCTION

1. L. Allen, S. M. Barnett, and M. J. Padgett, *Optical Angular Momentum* (Institute of Physics, 2003). [CrossRef]

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

3. M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in **34**, 8877–8888 (2001). [CrossRef]

4. M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A **457**, 2251–2263 (2001). [CrossRef]

5. J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. **7**, 55 (2005). [CrossRef]

6. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. **40**, 73–87 (1993). [CrossRef]

7. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A **56**, 4064–4075 (1997). [CrossRef]

8. F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. **250**, 218–230 (2005). [CrossRef]

9. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express **14**, 8382–8392 (2006). [CrossRef] [PubMed]

10. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. **274**, 8–14 (2007). [CrossRef]

11. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express **17**, 9818–9827 (2009). [CrossRef] [PubMed]

12. Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher-order multiringed Laguerre–Gaussian beams by using a spatial light modulator,” Opt. Lett. **32**, 1411–1413 (2007). [CrossRef] [PubMed]

13. 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 **25**, 1642–1651 (2008). [CrossRef]

11. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express **17**, 9818–9827 (2009). [CrossRef] [PubMed]

5. J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. **7**, 55 (2005). [CrossRef]

14. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. **25**, 191–193 (2000). [CrossRef]

15. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express **15**, 8619–8625 (2007). [CrossRef] [PubMed]

6. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. **40**, 73–87 (1993). [CrossRef]

7. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A **56**, 4064–4075 (1997). [CrossRef]

8. F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. **250**, 218–230 (2005). [CrossRef]

9. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express **14**, 8382–8392 (2006). [CrossRef] [PubMed]

10. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. **274**, 8–14 (2007). [CrossRef]

11. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express **17**, 9818–9827 (2009). [CrossRef] [PubMed]

16. 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]

17. J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A **15**, 1132–1138 (1998). [CrossRef]

18. J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity **20**, 1907–1925 (2007). [CrossRef]

## 2. FORMULATION OF SINGULARITIES IN TWO-LG-MODE SUPERPOSITIONS

*z*as a propagation direction. The complex amplitude of the radially

*z*and is defined aswith

*z*coordinate in expressions. Moreover,

*r*is rescaled as

### 2A. Two-Mode Superposition of Azimuthally Same-Order LG Modes

*ϕ*-dependence of

*r*exceeds the radial zero point, while the signature of light amplitude changes around the radial discontinuities in the pure radially higher-order LG modes. As a result, the phase structure of Eq. (3) is the same as that of

### 2B. General Formula for Two-LG-Mode Superpositions

*δ*works only to rotate the total beam profile, since the

*ϕ*-dependence of the beam profile is determined by the exponential factor in Eq. (6) and

*δ*only gives an offset to

*ϕ*. Hence we can omit

*δ*without loss of generality in the case of two-mode superposition.

*ε*is assumed to be small to remove the influence of other zero points. The phase value

*Ψ*at position

*u*, respectively, and

*π*arbitrariness and discontinuity at

*Ψ*varies from zero to

*ϕ*from zero to

*l*]. Such central phase singularities commonly appear in the coaxial superpositions of more than two LG modes having radial mode indices larger than unity.

*ρ*to divide the circle into

*ρ*’s depending on the definite form of the polynomial

*ε*is an infinitesimal variable [Fig. 1b]. For notational simplicity, we choose the origin of angular variables

*ϕ*and

*φ*as the direction of the zero point (i.e.,

*α*). With these preparations, the following relations between

### 2C. Two-Mode Superposition of Radially Lowest-Order LG Modes

6. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. **40**, 73–87 (1993). [CrossRef]

7. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A **56**, 4064–4075 (1997). [CrossRef]

8. F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. **250**, 218–230 (2005). [CrossRef]

9. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express **14**, 8382–8392 (2006). [CrossRef] [PubMed]

10. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. **274**, 8–14 (2007). [CrossRef]

**17**, 9818–9827 (2009). [CrossRef] [PubMed]

**40**, 73–87 (1993). [CrossRef]

**56**, 4064–4075 (1997). [CrossRef]

**250**, 218–230 (2005). [CrossRef]

**14**, 8382–8392 (2006). [CrossRef] [PubMed]

**274**, 8–14 (2007). [CrossRef]

**17**, 9818–9827 (2009). [CrossRef] [PubMed]

### 2D. Folded Singularity in Two-Mode Superposition of General LG Modes

17. J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A **15**, 1132–1138 (1998). [CrossRef]

18. J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity **20**, 1907–1925 (2007). [CrossRef]

**40**, 73–87 (1993). [CrossRef]

*l*and

*ε*, we finally obtain the following expression in the Cartesian coordinates (

18. J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity **20**, 1907–1925 (2007). [CrossRef]

*ϕ*. This means that

17. J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A **15**, 1132–1138 (1998). [CrossRef]

**20**, 1907–1925 (2007). [CrossRef]

## 3. EXPERIMENTAL RESULTS AND DISCUSSION

### 3A. Experimental Setup

16. 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]

19. J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. **61**, 1023–1028 (1971). [CrossRef]

19. J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. **61**, 1023–1028 (1971). [CrossRef]

20. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**, 156–160 (1982). [CrossRef]

21. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. **103**, 422–428 (1993). [CrossRef]

22. 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]

23. C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. **242**, 163–169 (2004). [CrossRef]

12. Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher-order multiringed Laguerre–Gaussian beams by using a spatial light modulator,” Opt. Lett. **32**, 1411–1413 (2007). [CrossRef] [PubMed]

13. 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 **25**, 1642–1651 (2008). [CrossRef]

### 3B. Two-Mode Superposition of Azimuthally Same-Order LG Beams

*z*-directional propagation of the superposition of

*N*-mode superposition of radially

*ϕ*, meaning that

*ϕ*at

*π*phase change when

*r*passes through

### 3C. Two-Mode Superposition of Radially Lowest-Order LG Beams

15. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express **15**, 8619–8625 (2007). [CrossRef] [PubMed]

*et al.*[15

15. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express **15**, 8619–8625 (2007). [CrossRef] [PubMed]

5. J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. **7**, 55 (2005). [CrossRef]

21. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. **103**, 422–428 (1993). [CrossRef]

23. C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. **242**, 163–169 (2004). [CrossRef]

### 3D. Folded Singularity in Two-Mode Superposition of General LG Modes

### 3E. Superposition of Three or More Radially Lowest-Order LG Beams

*l*of the component modes has the same signature, we can extend the previous discussion (Subsection 2C) to superpositions of more than three

*N*-mode superposition of

*z*:where the final expression is again given from the fundamental theorem of algebra. Here, we note that all

## 4. SUMMARY AND CONCLUSION

## APPENDIX A: REQUIREMENTS FOR THE POSITIONS OF OFF-CENTERED ZERO POINTS IN TWO-MODE SUPERPOSITION

*ϕ*. This means that such zero points compose a circular line (radial singularity) rather than a set of solitary zero points. Conversely, solitary zero points can appear only at

## APPENDIX B: FOLDED SINGULARITIES IN SUPERPOSITIONS OF HERMITE–GAUSSIAN MODES

**15**, 1132–1138 (1998). [CrossRef]

*x*and

*y*around an elliptic-type folded singularity. It is obvious that the asymptotic form of complex amplitude must be more than second order to satisfy the condition of folded singularity in Subsection 2D [

**40**, 73–87 (1993). [CrossRef]

*n*and

*m*indicate the mode indices regarding

*x*and

*y*directions, respectively. The three types of folded singularity are expressed as the following superpositions of HG modes

**20**, 1907–1925 (2007). [CrossRef]

## ACKNOWLEDGMENTS

1. | L. Allen, S. M. Barnett, and M. J. Padgett, |

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

3. | M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in |

4. | M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A |

5. | J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. |

6. | G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. |

7. | M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A |

8. | F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. |

9. | J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express |

10. | V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. |

11. | S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express |

12. | Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher-order multiringed Laguerre–Gaussian beams by using a spatial light modulator,” Opt. Lett. |

13. | 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 |

14. | J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. |

15. | S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express |

16. | 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. |

17. | J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A |

18. | J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity |

19. | J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. |

20. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. |

21. | I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. |

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

23. | C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. |

**OCIS Codes**

(090.1760) Holography : Computer holography

(100.5090) Image processing : Phase-only filters

(120.5060) Instrumentation, measurement, and metrology : Phase modulation

(140.3300) Lasers and laser optics : Laser beam shaping

(260.6042) Physical optics : Singular optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 14, 2010

Manuscript Accepted: October 19, 2010

Published: November 17, 2010

**Citation**

Taro Ando, Naoya Matsumoto, Yoshiyuki Ohtake, Yu Takiguchi, and Takashi Inoue, "Structure of optical singularities in coaxial superpositions of Laguerre–Gaussian modes," J. Opt. Soc. Am. A **27**, 2602-2612 (2010)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-27-12-2602

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### References

- L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003). [CrossRef]
- J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974). [CrossRef]
- M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime,” J. Phys. A 34, 8877–8888 (2001). [CrossRef]
- M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A 457, 2251–2263 (2001). [CrossRef]
- J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). [CrossRef]
- G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993). [CrossRef]
- M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997). [CrossRef]
- F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005). [CrossRef]
- J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express 14, 8382–8392 (2006). [CrossRef] [PubMed]
- V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007). [CrossRef]
- S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009). [CrossRef] [PubMed]
- Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher-order multiringed Laguerre–Gaussian beams by using a spatial light modulator,” Opt. Lett. 32, 1411–1413 (2007). [CrossRef] [PubMed]
- 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 25, 1642–1651 (2008). [CrossRef]
- J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000). [CrossRef]
- S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express 15, 8619–8625 (2007). [CrossRef] [PubMed]
- 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]
- J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A 15, 1132–1138 (1998). [CrossRef]
- J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity 20, 1907–1925 (2007). [CrossRef]
- J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1023–1028 (1971). [CrossRef]
- M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982). [CrossRef]
- I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993). [CrossRef]
- 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]
- C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004). [CrossRef]

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