Abstract

Previous work by Allen, demonstrated that optical beams possess orbital angular momentum. Other work has shown that a random, phase-only disturbance can impart ±1 orbital angular momentum states to propagating waves. However, the field preceding the formation of these ±1 states was unknown. In this paper, we identify the unique field that leads to the formation of a pair of branch points, indicators of orbital angular momentum. This field is then verified in a bench-top optical experiment.

© 2012 OSA

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  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]
  2. 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,” Bull. Am. Phys. Soc. 75, 826–829 (1995).
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    [CrossRef]
  4. A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. J. S. Bary, D. A. Weintraub, and J. H. Kastner, “Detection of molecular hydrogen orbiting a ‘naked’ T Tauri star,” Astrophys. J. Lett. 576, L73–L76 (2002).
    [CrossRef]
  15. N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. 492, 883–922 (2008).
    [CrossRef]
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    [CrossRef]
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  23. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).
  24. D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electr. Eng. 18, 467–483 (1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012 (2)

T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “The aggregate behavior of branch points - persistent pairs,” Opt. Express 2, 1046–1059 (2012).
[CrossRef]

2011 (3)

2010 (3)

A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010).
[CrossRef]

D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points - measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18, 22377–22392 (2010).
[CrossRef] [PubMed]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

2009 (2)

2008 (2)

2005 (1)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

2004 (4)

2003 (2)

S. J. van Enk, “Entanglement of electromagnetic fields,” Phys. Rev. A 67, 022303 (2003).
[CrossRef]

I. D. Maleev and G. A. Swartzlander, “Composite optical vorticies,” J. Opt. Soc. Am. B 20, 1169–1176 (2003).
[CrossRef]

2002 (1)

J. S. Bary, D. A. Weintraub, and J. H. Kastner, “Detection of molecular hydrogen orbiting a ‘naked’ T Tauri star,” Astrophys. J. Lett. 576, L73–L76 (2002).
[CrossRef]

2000 (1)

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

1999 (1)

W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett. 521, L63–L66 (1999).
[CrossRef]

1998 (1)

1995 (2)

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[CrossRef]

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,” Bull. Am. Phys. Soc. 75, 826–829 (1995).

1993 (1)

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

1992 (3)

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]

D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electr. Eng. 18, 467–483 (1992).
[CrossRef]

D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef] [PubMed]

‘t Hooft, G. W.

Allen, L.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[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. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Barnett, S. M.

Bary, J. S.

J. S. Bary, D. A. Weintraub, and J. H. Kastner, “Detection of molecular hydrogen orbiting a ‘naked’ T Tauri star,” Astrophys. J. Lett. 576, L73–L76 (2002).
[CrossRef]

Beijersbergen, M. W.

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]

Benseny, A.

A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010).
[CrossRef]

Bianchini, A.

T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

Blake, G. A.

W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett. 521, L63–L66 (1999).
[CrossRef]

Boyd, R. W.

Calvo, G. F.

A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010).
[CrossRef]

Courtial, J.

Dymale, R. C.

Elias, N. M.

N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. 492, 883–922 (2008).
[CrossRef]

Eliel, E. R.

Engelberg, Y. M.

Franke-Arnold, S.

Fried, D. L.

Friese, M. E. J.

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,” Bull. Am. Phys. Soc. 75, 826–829 (1995).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, 1st ed (John Wiley & Sons, 1978).

Gibson, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Gruneisen, M. T.

Hanna, S.

He, H.

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,” Bull. Am. Phys. Soc. 75, 826–829 (1995).

Heckenberg, N. R.

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,” Bull. Am. Phys. Soc. 75, 826–829 (1995).

Hogerheijde, M. R.

W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett. 521, L63–L66 (1999).
[CrossRef]

Indebetouw, G.

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

Johnston, D. C.

D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electr. Eng. 18, 467–483 (1992).
[CrossRef]

Kastner, J. H.

J. S. Bary, D. A. Weintraub, and J. H. Kastner, “Detection of molecular hydrogen orbiting a ‘naked’ T Tauri star,” Astrophys. J. Lett. 576, L73–L76 (2002).
[CrossRef]

Kelly, P. R.

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

Kloosteboer, J. G.

Maleev, I. D.

Mari, E.

T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

Matson, C. L.

Miller, W. A.

Mompart, J.

A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010).
[CrossRef]

Monken, C. H.

Oemrawsingh, S. S. R.

Oesch, D. W.

Padgett, M. J.

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[CrossRef] [PubMed]

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[CrossRef]

Pas’ko, V.

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

Picon, A.

A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010).
[CrossRef]

Plaja, L.

A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010).
[CrossRef]

Pors, B. J.

Rhoadarmer, T. A.

T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553, 112–126 (2004).
[CrossRef]

Romanato, F.

T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

Roso, L.

A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010).
[CrossRef]

Rubinsztein-Dunlop, H.

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,” Bull. Am. Phys. Soc. 75, 826–829 (1995).

Ruschin, S.

Sanchez, D. J.

Simpson, S. H.

Sponselli, A.

T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

Spreeuw, R. J. C.

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]

Swartzlander, G. A.

Sweiti, A. M.

Tamburini, T.

T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

Tewksbury-Christle, C. M.

D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “The aggregate behavior of branch points - persistent pairs,” Opt. Express 2, 1046–1059 (2012).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

Thi, W. F.

W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett. 521, L63–L66 (1999).
[CrossRef]

Thidé, B.

T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

Tyler, G. A.

van Dishoeck, E. F.

W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett. 521, L63–L66 (1999).
[CrossRef]

van Enk, S. J.

S. J. van Enk, “Entanglement of electromagnetic fields,” Phys. Rev. A 67, 022303 (2003).
[CrossRef]

van Houweilingen, J. A. W.

van Zadelhoff, G. J.

W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett. 521, L63–L66 (1999).
[CrossRef]

Vasnetso, M.

Vaughn, J. L.

Vazquez de Aldana, J. R.

A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010).
[CrossRef]

Verstegen, E. J. K.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed (Cambridge Univesity Press, 1962).

Weintraub, D. A.

J. S. Bary, D. A. Weintraub, and J. H. Kastner, “Detection of molecular hydrogen orbiting a ‘naked’ T Tauri star,” Astrophys. J. Lett. 576, L73–L76 (2002).
[CrossRef]

Welsh, B. M.

D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electr. Eng. 18, 467–483 (1992).
[CrossRef]

Woerdman, J.

Woerdman, J. P.

S. S. R. Oemrawsingh, J. A. W. van Houweilingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosteboer, and G. W. ‘t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. 43, 688–694 (2004).
[CrossRef] [PubMed]

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]

Appl. Opt. (3)

Astron. Astrophys. (1)

N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. 492, 883–922 (2008).
[CrossRef]

Astrophys. J. Lett. (2)

W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett. 521, L63–L66 (1999).
[CrossRef]

J. S. Bary, D. A. Weintraub, and J. H. Kastner, “Detection of molecular hydrogen orbiting a ‘naked’ T Tauri star,” Astrophys. J. Lett. 576, L73–L76 (2002).
[CrossRef]

Bull. Am. Phys. Soc. (1)

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,” Bull. Am. Phys. Soc. 75, 826–829 (1995).

Comput. Electr. Eng. (1)

D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electr. Eng. 18, 467–483 (1992).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

New J. Phys. (2)

A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010).
[CrossRef]

T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

Opt. Commun. (2)

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[CrossRef]

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

Opt. Express (6)

Opt. Lett. (1)

Phys. Rev. A (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]

S. J. van Enk, “Entanglement of electromagnetic fields,” Phys. Rev. A 67, 022303 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

Proc. SPIE (2)

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553, 112–126 (2004).
[CrossRef]

Other (3)

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed (Cambridge Univesity Press, 1962).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, 1st ed (John Wiley & Sons, 1978).

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Figures (6)

Fig. 1
Fig. 1

Cartoon representation of the space of the Fresnel transform between Plane 1 and Plane 2 with indicated parameters and coordinates. The intended branch point pair, positive and negative are indicated by the red and green dots respectively in Plane 2.

Fig. 2
Fig. 2

Phases of the Fresnel transform pair for NF = 31 with a branch point separation, δ = 3. (a) The precursor phase. (b) The created hidden phase.

Fig. 3
Fig. 3

Images of the applied DM commands and hidden phase pair from our experimental set-up. (a) Calculated precursor phase from Eq. (6) for desired branch point pair arrangement. (b) Applied DM commands. (c) Measured hidden phase at the WFS from the applied DM commands. (d) Desired branch point phase calculated from Eq. (3).

Fig. 4
Fig. 4

Back propagated phase. (a) Phase calculated from back-propagating the generated hidden phase for a pair of branch points. (b) Phase calculated from back-propagating a top-hat function; standard Fresnel rings. (c) The Precursor Phase; the difference of the back-propagated hidden phase (a) and the Fresnel rings (b).

Fig. 5
Fig. 5

Empirical fit to the precursor phase for a range of branch point separations, δ for Fresnel transform based on a Fresnel number of 26. (a) Precursor phase associated with the indicated separations. The cross sections for the (b) symmetric and (c) asymmetric axis are shown. The red curve is from the phase shown in (a) and the blue curve is generated by Eq. (4) using the identified variables.

Fig. 6
Fig. 6

Empirical fit to the precursor phase for a range of Fresnel numbers for a hidden phase based on a pair separation of 3. (a) Precursor phase associated with the indicated Fresnel numbers. The cross sections for the (b) symmetric and (c) asymmetric axis are shown. The red curve is from the phase shown in (a) and the blue curve is generated by Eq. (4) using the identified variables.

Equations (7)

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A 2 ( x , y ) e i ϕ 2 ( x , y ) = - - A 1 ( ξ , η ) e i ϕ 1 ( ξ , η ) h ( x - ξ , y - η ) d ξ d η .
h ( x , y ) = e i k z i λ z exp [ i k 2 z ( x 2 + y 2 ) ] ,
ϕ hid = tan 1 ( y y p x x p ) tan 1 ( y y n x x n ) ,
P s = c 1 J 1 [ ω s ( x 2 + y 2 ) ] - c 0 J 0 [ ω s ( x 2 + y 2 ) ] , P a = c a J 1 2 2 [ ω a ( x 2 + y 2 ) ] y ,
c 1 = 4 δ N F 5 R , c 0 = 3 δ N F 5 R , ω s = π N F R 2 , c a = 2 π δ N F R 2 , ω a = π N F 2 R 2 .
ϕ pre = δ N F R [ 4 5 J 1 ( π N F ( r R ) 2 ) - 3 5 J 0 ( π N F ( r R ) 2 ) ] + 2 π δ N F R 2 J 1 2 2 ( 1 2 π N F ( r R ) 2 ) [ ( y - y 0 ) cos θ - ( x - x 0 ) sin θ ] .
A 1 ( x , y ) e i ϕ 1 ( x , y ) = 𝔉 - 1 [ 𝔉 { A 2 ( x , y ) e i ϕ 2 ( x , y ) } H ( z ) ] ,

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