Abstract

We present a new all-digital technique to extract the wavefront of a structured light beam. Our method employs non-homogeneous polarization optics together with dynamic, digital holograms written to a spatial light modulator to measure the phase relationship between orthogonal polarization states in real-time, thereby accessing the wavefront information. Importantly, we show how this can be applied to measuring the wavefront of propagating light fields, over extended distances, without any moving components. We illustrate the versatility of the tool by measuring propagating optical vortices, Bessel, Airy and speckle fields. The comparison of the extracted and programmed wavefronts yields excellent agreement.

© 2014 Optical Society of America

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    [CrossRef]
  4. M. Rueckel, J. A. Mack-Bucher, W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. 103, 17137–17142 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  8. M. Paurisse, M. Hanna, F. Druon, P. Georges, “Wavefront control of a multicore ytterbium-doped pulse fiber amplifier by digital holography,” Opt. Lett. 35, 1428–1430 (2010).
    [CrossRef] [PubMed]
  9. R. Navarro, E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999).
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  13. J. Millerd, N. Brock, J. Hayes, M. North Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
    [CrossRef]
  14. M. North Morris, J. Millerd, N. Brock, J. Hayes, B. Saif, “Dynamic Phase-Shifting Electronic Speckle Pattern Interferometer,” Proc. SPIE 5869, 58691B (2005).
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  17. C. Schulze, A. Dudley, D. Flamm, M. Duparré, A. Forbes, “Reconstruction of laser beam wavefronts based on mode analysis,” App. Opt. 52(21), 5312–5317 (2013).
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  18. G. G. Stokes, ”On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc., 96, 399 (1852).
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    [CrossRef]
  21. V. Denisenko, A. Minovich, A. Desyatnikov, W. Krolikowski, M. Soskin, Y. Kivshar, ”Mapping phases of singular scalar light fields,” Opt. Lett. 33, 89–91 (2008).
    [CrossRef]
  22. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, I. I. Mokhun, ”Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
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    [CrossRef]
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    [CrossRef]
  36. D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode area fibers,” Opt. Lett. 35, 3429–3431 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]

2014 (1)

Q. Cui, M. Li, Z. Yu, “Influence of topological charges on random wandering of optical vortex propagating through turbulent atmosphere,” Opt. Comm. 329, 10–14 (2014).
[CrossRef]

2013 (5)

2012 (3)

2010 (4)

2009 (1)

2008 (1)

2006 (2)

S. R. Chamot, C. Dainty, S. Esposito, “Adaptive optics for ophthalmic applications using a pyramid wavefront sensor,” Opt. Express 14, 518–526 (2006).
[CrossRef] [PubMed]

M. Rueckel, J. A. Mack-Bucher, W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. 103, 17137–17142 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (2)

2002 (3)

2001 (1)

2000 (1)

M. A. A. Neil, R. Juakaitis, M. J. Booth, T. Wilson, T. Tanaka, S. Kawata, “Adaptive aberration correction in a twophoton microscope,” J. Microsc. 200, 105–108 (2000).
[CrossRef]

1999 (1)

1998 (1)

M. Booth, M. Neil, T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. 192, 90–98 (1998).
[CrossRef]

1992 (1)

1975 (1)

1852 (1)

G. G. Stokes, ”On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc., 96, 399 (1852).

Andrews, D.

D. Andrews, Structured Light and Its Applications (Academic, 2011).

Angelsky, O. V.

Artal, P.

Beckers, J.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Beuzit, J.-L.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Booth, M.

M. Booth, M. Neil, T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. 192, 90–98 (1998).
[CrossRef]

Booth, M. J.

M. A. A. Neil, R. Juakaitis, M. J. Booth, T. Wilson, T. Tanaka, S. Kawata, “Adaptive aberration correction in a twophoton microscope,” J. Microsc. 200, 105–108 (2000).
[CrossRef]

Borchardt, J.

Bowman, R. W.

Brock, N.

M. North Morris, J. Millerd, N. Brock, J. Hayes, B. Saif, “Dynamic Phase-Shifting Electronic Speckle Pattern Interferometer,” Proc. SPIE 5869, 58691B (2005).
[CrossRef]

J. Millerd, N. Brock, J. Hayes, M. North Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Campbell, M.

Chamot, S. R.

Cohen, M.

Cui, Q.

Q. Cui, M. Li, Z. Yu, “Influence of topological charges on random wandering of optical vortex propagating through turbulent atmosphere,” Opt. Comm. 329, 10–14 (2014).
[CrossRef]

Dainty, C.

Davidson, F. M.

Denisenko, V.

Denk, W.

M. Rueckel, J. A. Mack-Bucher, W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. 103, 17137–17142 (2006).
[CrossRef] [PubMed]

Desyatnikov, A.

Drexler, W.

Druon, F.

Dudley, A.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, A. Forbes, “Reconstruction of laser beam wavefronts based on mode analysis,” App. Opt. 52(21), 5312–5317 (2013).
[CrossRef]

A. Dudley, Y. Li, T. Mhlanga, M. Escuti, A. Forbes, “Generating and measuring nondiffracting vector Bessel beams,” Opt. Lett. 38(17), 3429–3432 (2013).
[CrossRef] [PubMed]

Duparré, M.

Escuti, M.

Escuti, M. J.

Esposito, S.

Fercher, A. F.

Fernndez, E. J.

Flamm, D.

Forbes, A.

D. Flamm, C. Schulze, D. Naidoo, S. Schröter, A. Forbes, M. Duparré, “All-digital holographic tool for mode excitation and analysis in optical fibers,” J. Lightwave Technol. 31(7), 1023–1032 (2013).
[CrossRef]

A. Dudley, Y. Li, T. Mhlanga, M. Escuti, A. Forbes, “Generating and measuring nondiffracting vector Bessel beams,” Opt. Lett. 38(17), 3429–3432 (2013).
[CrossRef] [PubMed]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, A. Forbes, “Reconstruction of laser beam wavefronts based on mode analysis,” App. Opt. 52(21), 5312–5317 (2013).
[CrossRef]

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
[CrossRef] [PubMed]

C. Schulze, D. Flamm, M. Duparré, A. Forbes, “Beam-quality measurements using a spatial light modulator,” Opt. Lett., 37(22), 4687–4689 (2012).
[CrossRef]

A. Forbes, Laser Beam Propagation: Generation and Propagation of Customized Light (CRC, 2013).

Freund, I.

Georges, P.

Goldstein, D.

D. Goldstein, Polarized Light (Marcel Dekker, 2004).

Guérineau, N.

Hanna, M.

Hayes, J.

M. North Morris, J. Millerd, N. Brock, J. Hayes, B. Saif, “Dynamic Phase-Shifting Electronic Speckle Pattern Interferometer,” Proc. SPIE 5869, 58691B (2005).
[CrossRef]

J. Millerd, N. Brock, J. Hayes, M. North Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Hebert, T.

Hermann, B.

Juakaitis, R.

M. A. A. Neil, R. Juakaitis, M. J. Booth, T. Wilson, T. Tanaka, S. Kawata, “Adaptive aberration correction in a twophoton microscope,” J. Microsc. 200, 105–108 (2000).
[CrossRef]

Kaiser, T.

Kawata, S.

M. A. A. Neil, R. Juakaitis, M. J. Booth, T. Wilson, T. Tanaka, S. Kawata, “Adaptive aberration correction in a twophoton microscope,” J. Microsc. 200, 105–108 (2000).
[CrossRef]

Kim, J.

Kivshar, Y.

Korotkova, O.

Kozawa, Y.

Krolikowski, W.

Lai, O.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Lane, R. G.

Lena, P.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Li, M.

Q. Cui, M. Li, Z. Yu, “Influence of topological charges on random wandering of optical vortex propagating through turbulent atmosphere,” Opt. Comm. 329, 10–14 (2014).
[CrossRef]

Li, Y.

Love, G. D.

Lü, B.

Luo, Y.

Y. Luo, B. Lü, ”Spectral Stokes singularities of partially coherent radially polarized beams focused by a high numerical aperture objective,” J. Opt. 12, 115703(2010).
[CrossRef]

Mack-Bucher, J. A.

M. Rueckel, J. A. Mack-Bucher, W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. 103, 17137–17142 (2006).
[CrossRef] [PubMed]

Madec, P.-Y.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Mhlanga, T.

Millerd, J.

M. North Morris, J. Millerd, N. Brock, J. Hayes, B. Saif, “Dynamic Phase-Shifting Electronic Speckle Pattern Interferometer,” Proc. SPIE 5869, 58691B (2005).
[CrossRef]

J. Millerd, N. Brock, J. Hayes, M. North Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Minovich, A.

Mokhun, A. I.

Mokhun, I. I.

Moreno-Barriuso, E.

Naidoo, D.

Navarro, R.

Neil, M.

M. Booth, M. Neil, T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. 192, 90–98 (1998).
[CrossRef]

Neil, M. A. A.

M. A. A. Neil, R. Juakaitis, M. J. Booth, T. Wilson, T. Tanaka, S. Kawata, “Adaptive aberration correction in a twophoton microscope,” J. Microsc. 200, 105–108 (2000).
[CrossRef]

North Morris, M.

M. North Morris, J. Millerd, N. Brock, J. Hayes, B. Saif, “Dynamic Phase-Shifting Electronic Speckle Pattern Interferometer,” Proc. SPIE 5869, 58691B (2005).
[CrossRef]

J. Millerd, N. Brock, J. Hayes, M. North Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Northcott, M.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Novak, M.

J. Millerd, N. Brock, J. Hayes, M. North Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Padgett, M. J.

Paurisse, M.

Prieto, P. M.

Primot, J.

Queener, H.

Ricklin, J. C.

Rigaut, F.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Rimmer, M. P.

Ritsch-Marte, M.

Roddier, F.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Romero-Borja, F.

Roorda, A.

Rousset, G.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Rueckel, M.

M. Rueckel, J. A. Mack-Bucher, W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. 103, 17137–17142 (2006).
[CrossRef] [PubMed]

Saif, B.

M. North Morris, J. Millerd, N. Brock, J. Hayes, B. Saif, “Dynamic Phase-Shifting Electronic Speckle Pattern Interferometer,” Proc. SPIE 5869, 58691B (2005).
[CrossRef]

Sandler, D.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Sato, S.

Sattmann, H.

Schmidt, O. A.

Schröter, S.

Schulze, C.

Séchaud, M.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Soskin, M.

Soskin, M. S.

Stokes, G. G.

G. G. Stokes, ”On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc., 96, 399 (1852).

G. G. Stokes, Mathematical and Physical Papers (Cambridge University, 1922).

Tallon, M.

Tanaka, T.

M. A. A. Neil, R. Juakaitis, M. J. Booth, T. Wilson, T. Tanaka, S. Kawata, “Adaptive aberration correction in a twophoton microscope,” J. Microsc. 200, 105–108 (2000).
[CrossRef]

Thalhammer, G.

Unterhuber, A.

Velghe, S.

Vyas, S.

Wattellier, B.

William Donnelly, I.

Wilson, T.

M. A. A. Neil, R. Juakaitis, M. J. Booth, T. Wilson, T. Tanaka, S. Kawata, “Adaptive aberration correction in a twophoton microscope,” J. Microsc. 200, 105–108 (2000).
[CrossRef]

M. Booth, M. Neil, T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. 192, 90–98 (1998).
[CrossRef]

Wolf, E.

Wyant, J.

J. Millerd, N. Brock, J. Hayes, M. North Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Wyant, J. C.

Yan, H.

Yu, Z.

Q. Cui, M. Li, Z. Yu, “Influence of topological charges on random wandering of optical vortex propagating through turbulent atmosphere,” Opt. Comm. 329, 10–14 (2014).
[CrossRef]

App. Opt. (1)

C. Schulze, A. Dudley, D. Flamm, M. Duparré, A. Forbes, “Reconstruction of laser beam wavefronts based on mode analysis,” App. Opt. 52(21), 5312–5317 (2013).
[CrossRef]

Appl. Opt. (3)

J. Lightwave Technol. (1)

J. Microsc. (2)

M. A. A. Neil, R. Juakaitis, M. J. Booth, T. Wilson, T. Tanaka, S. Kawata, “Adaptive aberration correction in a twophoton microscope,” J. Microsc. 200, 105–108 (2000).
[CrossRef]

M. Booth, M. Neil, T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. 192, 90–98 (1998).
[CrossRef]

J. Opt. (1)

Y. Luo, B. Lü, ”Spectral Stokes singularities of partially coherent radially polarized beams focused by a high numerical aperture objective,” J. Opt. 12, 115703(2010).
[CrossRef]

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

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

Opt. Comm. (1)

Q. Cui, M. Li, Z. Yu, “Influence of topological charges on random wandering of optical vortex propagating through turbulent atmosphere,” Opt. Comm. 329, 10–14 (2014).
[CrossRef]

Opt. Express (5)

Opt. Lett. (11)

V. Denisenko, A. Minovich, A. Desyatnikov, W. Krolikowski, M. Soskin, Y. Kivshar, ”Mapping phases of singular scalar light fields,” Opt. Lett. 33, 89–91 (2008).
[CrossRef]

H. Yan, B. Lü, ”Spectral Stokes singularities of stochastic electromagnetic beams,” Opt. Lett. 34(13), 1933–1935 (2009).
[CrossRef] [PubMed]

B. Hermann, E. J. Fernndez, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, P. Artal, “Adaptive optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29, 2142–2144 (2004).
[CrossRef] [PubMed]

O. Korotkova, E. Wolf, ”Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
[CrossRef] [PubMed]

S. Velghe, J. Primot, N. Guérineau, M. Cohen, B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. 30, 245–247 (2005).
[CrossRef] [PubMed]

A. Dudley, Y. Li, T. Mhlanga, M. Escuti, A. Forbes, “Generating and measuring nondiffracting vector Bessel beams,” Opt. Lett. 38(17), 3429–3432 (2013).
[CrossRef] [PubMed]

M. Paurisse, M. Hanna, F. Druon, P. Georges, “Wavefront control of a multicore ytterbium-doped pulse fiber amplifier by digital holography,” Opt. Lett. 35, 1428–1430 (2010).
[CrossRef] [PubMed]

D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode area fibers,” Opt. Lett. 35, 3429–3431 (2010).
[CrossRef] [PubMed]

R. Navarro, E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999).
[CrossRef]

I. Freund, ”Poincaré vortices,” Opt. Lett. 26, 1996–1998 (2001).
[CrossRef]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, I. I. Mokhun, ”Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
[CrossRef]

Opt. Lett., (1)

C. Schulze, D. Flamm, M. Duparré, A. Forbes, “Beam-quality measurements using a spatial light modulator,” Opt. Lett., 37(22), 4687–4689 (2012).
[CrossRef]

Proc. Natl. Acad. Sci. (1)

M. Rueckel, J. A. Mack-Bucher, W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. 103, 17137–17142 (2006).
[CrossRef] [PubMed]

Proc. SPIE (2)

J. Millerd, N. Brock, J. Hayes, M. North Morris, M. Novak, J. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

M. North Morris, J. Millerd, N. Brock, J. Hayes, B. Saif, “Dynamic Phase-Shifting Electronic Speckle Pattern Interferometer,” Proc. SPIE 5869, 58691B (2005).
[CrossRef]

Trans. Cambridge Philos. Soc (1)

G. G. Stokes, ”On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc., 96, 399 (1852).

Other (5)

G. G. Stokes, Mathematical and Physical Papers (Cambridge University, 1922).

D. Andrews, Structured Light and Its Applications (Academic, 2011).

A. Forbes, Laser Beam Propagation: Generation and Propagation of Customized Light (CRC, 2013).

D. Goldstein, Polarized Light (Marcel Dekker, 2004).

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Lena, O. Lai, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

Supplementary Material (2)

» Media 1: MOV (2684 KB)     
» Media 2: MOV (2677 KB)     

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Figures (7)

Fig. 1
Fig. 1

A comparison between (a) and (b) the standard, manual Stokes polarimetry and (c) and (d) our corresponding digital method for extracting S2 and S3, respectively. The example illustrated here is an incoherent superposition of two Gaussian fields of orthogonal polarization where the vertical component has an additional azimuthal phase of exp(i3ϕ). P: polarizer; λ/4: quater-waveplate; LCD: liquid crystal display of the SLM; and PG: polarization grating. Accompanying polarizer, quater-waveplate and LCD settings are given.

Fig. 2
Fig. 2

A comparison between (a) the standard, manual movement of a detector along a beam’s propagation and (b) the digital propagation method. The example illustrated here is a Gaussian beam. WS: wavefront-sensing technique consisting of a LCD and PG; D: detector; and L: lens. Accompanying intensity profiles and phase patterns encoded on the LCD for particular propagation distances (z) are given as inserts.

Fig. 3
Fig. 3

(a) Schematic of the experimental wavefront sensing setup. HeNe: helium-neon laser; P: polarizer; L1→5: lenses; M: mirror; LCD: liquid crystal display of the SLM; A: aperture; PG: polarization grating; CCD: CCD detector. Example holograms for the azimuthal phase profile: (b) exp(i3ϕ) and (c) exp(i3ϕ + π/2) for the Stokes measurements, S3 and S2, respectively for extracting (d) the wavefront. (e) Example hologram (with a blazed grating) for extracting (f) the intensity profile.

Fig. 4
Fig. 4

Experimentally measured intensity profiles (a) I45°, I135° and (b) IR, IL (for a vortex beam of = +1) used to calculate the Stokes parameters (c) S2 and (d) S3, respectively needed for the extraction of the (e) [(f)] measured (theoretical) wavefront.

Fig. 5
Fig. 5

Experimentally measured intensity profiles of (a) vortex beams and (d) Bessel beams. Some of the gray-level holograms used are given as inserts. Corresponding (b) [(e)] experimentally measured and (c) [(f)] theoretically calculated wavefronts for near-field vortex (Bessel) beams. The correlation between the measured and theoretical wavefronts varies from a minimum of 0.914 to a maximum of 0.988. Corresponding azimuthal indices are given in the top right corner. The white arrows highlight the handedness of the azimuthal phase. Inserts depict a magnified view of the singularities.

Fig. 6
Fig. 6

Experimentally measured intensity profiles of an (a) Airy beam and (b) speckle field. The gray-level holograms used are given as inserts. Corresponding experimentally measured (Exp.) and theoretically calculated (Th.) wavefronts for a near-field (c) Airy beam and (d) speckle field.

Fig. 7
Fig. 7

Experimentally measured wavefronts for a (a) Gaussian and (b) vortex beam ( = +2) at different propagation planes. Corresponding propagation distances are given in the bottom right corner. The full data (video) can be viewed in (a) ( Media 1) and (b) ( Media 2).

Equations (7)

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U ˜ ( r , ϕ ) = U ˜ ( r , ϕ ) + U ˜ ( r , ϕ ) ,
U ˜ ( r , ϕ ) = | U ( r , ϕ ) | exp ( i δ ( r , ϕ ) ) x ^ and U ˜ ( r , ϕ ) = | U ( r , ϕ ) | exp ( i δ ( r , ϕ ) ) y ^
U ˜ ( r , ϕ ) = | U ( r , ϕ ) | x ^ + | U ( r , ϕ ) | y ^ exp ( i δ ( r , ϕ ) ) ,
U ( r , ϕ ) = exp ( r 2 / ω 0 ) and U ( r , ϕ ) = exp ( r 2 / ω 0 ) exp ( i δ ( r , ϕ ) ) .
S 0 ( r , ϕ ) = | U ( r , ϕ ) | 2 + | U ( r , ϕ ) | 2 = I 0 ° + I 90 ° S 1 ( r , ϕ ) = | U ( r , ϕ ) | 2 | U ( r , ϕ ) | 2 = I 0 ° I 90 ° S 2 ( r , ϕ ) = 2 | U ( r , ϕ ) | | U ( r , ϕ ) | cos ( δ ( r , ϕ ) ) = I 45 ° I 135 ° S 3 ( r , ϕ ) = 2 | U ( r , ϕ ) | | U ( r , ϕ ) | sin ( δ ( r , ϕ ) ) = I R I L
α ( r , ϕ ) = arctan ( S 2 ( r , ϕ ) S 1 ( r , ϕ ) ) and δ ( r , ϕ ) = 1 2 arctan ( S 3 ( r , ϕ ) S 2 ( r , ϕ ) ) .
1 ( [ U ( r ) ] exp ( i k z z ) ) .

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