Abstract

The dynamic spatial control of light fields is essential to a range of applications, from microscopy to optical micro-manipulation and communications. Here we describe the use of a single digital micro-mirror device (DMD) to generate and rapidly switch vector beams with spatially controllable intensity, phase and polarisation. We demonstrate local spatial control over linear, elliptical and circular polarisation, allowing the generation of radially and azimuthally polarised beams and Poincaré beams. All of these can be switched at rates of up to 4kHz (limited only by our DMD model), a rate ∼2 orders of magnitude faster than the switching speeds of typical phase-only spatial light modulators. The polarisation state of the generated beams is characterised with spatially resolved Stokes measurements. We also describe detail of technical considerations when using a DMD, and quantify the mode capacity and efficiency of the beam generation. The high-speed switching capabilities of this method will be particularly useful for the control of light propagation through complex media such as multimode fibers, where rapid spatial modulation of intensity, phase and polarisation is required.

© 2016 Optical Society of America

Full Article  |  PDF Article
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2016 (2)

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photon. 10, 327–332 (2016).
[Crossref]

N. Radwell, R. Hawley, J. Götte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref] [PubMed]

2015 (6)

2014 (5)

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116, 183105 (2014).
[Crossref]

R. W. Bowman, G. M. Gibson, A. Linnenberger, D. B. Phillips, J. A. Grieve, D. M. Carberry, S. Serati, M. J. Miles, and M. J. Padgett, “‘red tweezers’: Fast, customisable hologram generation for optical tweezers,” Comput. Phys. Commun. 185, 268–273 (2014).
[Crossref]

J. Carpenter, B. J. Eggleton, and J. Schröder, “110×110 optical mode transfer matrix inversion,” Opt. Express 22, 96–101 (2014).
[Crossref] [PubMed]

G. S. Gordon, F. Feng, Q. Kang, Y. Jung, J. Sahu, and T. Wilkinson, “Coherent, focus-corrected imaging of optical fiber facets using a single-pixel detector,” Opt. Lett. 39, 6034–6037 (2014).
[Crossref] [PubMed]

S. A. Goorden, J. Bertolotti, and A. P. Mosk, “Superpixel-based spatial amplitude and phase modulation using a digital micromirror device,” Opt. Express 22, 17999–18009 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (5)

2011 (1)

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[Crossref]

2010 (6)

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
[Crossref]

Y.-X. Ren, M. Li, K. Huang, J.-G. Wu, H.-F. Gao, Z.-Q. Wang, and Y.-M. Li, “Experimental generation of laguerre-gaussian beam using digital micromirror device,” Appl. Opt. 49, 1838–1844 (2010).
[Crossref] [PubMed]

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[Crossref]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

2007 (3)

R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913–1922 (2007).
[Crossref] [PubMed]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett 32, 3549–3551 (2007).
[Crossref] [PubMed]

2006 (1)

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]

2004 (3)

Z. Bouchal and R. Celechovsky, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131 (2004).
[Crossref]

S. Pereira and A. Van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
[Crossref]

Q. Zhan, “Trapping metallic rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
[Crossref] [PubMed]

2003 (2)

E. Galvez, P. Crawford, H. Sztul, M. Pysher, P. Haglin, and R. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

1999 (2)

V. Niziev and A. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D Appl. Phys. 32, 1455 (1999).
[Crossref]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[Crossref]

1998 (2)

M. Friese, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

J. Courtial, “Self-imaging beams and the guoy effect,” Opt. Commun. 151, 1–4 (1998).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

1979 (1)

1967 (1)

J. Dyment, “Hermite-gaussian mode patterns in GaAs junction lasers,” Appl. Phys. Lett. 10, 84–86 (1967).
[Crossref]

1966 (1)

Alfano, R. R.

Allen, L.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Alonso, M. A.

Arita, Y.

Barnett, S. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. G. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic super-sampling,” arXiv preprint arXiv:1607.08236 (2016).

Barter, O.

D. Stuart, O. Barter, and A. Kuhn, “Fast algorithms for generating binary holograms,” arXiv preprint arXiv:1409.1841 (2014).

Beckley, A. M.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Bernet, S.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[Crossref]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Bertolotti, J.

Bianchi, S.

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12, 635–639 (2012).
[Crossref]

Boccara, A.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

Boccara, A. C.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

Bouchal, Z.

Z. Bouchal and R. Celechovsky, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131 (2004).
[Crossref]

Bowman, R. W.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref] [PubMed]

R. W. Bowman, G. M. Gibson, A. Linnenberger, D. B. Phillips, J. A. Grieve, D. M. Carberry, S. Serati, M. J. Miles, and M. J. Padgett, “‘red tweezers’: Fast, customisable hologram generation for optical tweezers,” Comput. Phys. Commun. 185, 268–273 (2014).
[Crossref]

Boyd, R. W.

Brake, J.

Brown, B. R.

Brown, T. G.

Campos, J.

Caravaca-Aguirre, A. M.

Carberry, D. M.

R. W. Bowman, G. M. Gibson, A. Linnenberger, D. B. Phillips, J. A. Grieve, D. M. Carberry, S. Serati, M. J. Miles, and M. J. Padgett, “‘red tweezers’: Fast, customisable hologram generation for optical tweezers,” Comput. Phys. Commun. 185, 268–273 (2014).
[Crossref]

Carminati, R.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

Carpenter, J.

Celechovsky, R.

Z. Bouchal and R. Celechovsky, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131 (2004).
[Crossref]

Chen, C.

Cižmár, T.

M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nat. Photon. 9, 529–535 (2015).
[Crossref]

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[Crossref] [PubMed]

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[Crossref]

Conkey, D. B.

Cottrell, D. M.

Courtial, J.

J. Courtial, “Self-imaging beams and the guoy effect,” Opt. Commun. 151, 1–4 (1998).
[Crossref]

Crawford, P.

E. Galvez, P. Crawford, H. Sztul, M. Pysher, P. Haglin, and R. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

Davis, J. A.

Dholakia, K.

Y. Arita, M. Mazilu, T. Vettenburg, E. M. Write, and K. Dholakia, “Rotation of two trapped micro particles in vacuum: observation of optically mediated parametric resonances,” Opt. Lett. 40, 4751–4754 (2015).
[Crossref] [PubMed]

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[Crossref] [PubMed]

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[Crossref]

Di Leonardo, R.

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12, 635–639 (2012).
[Crossref]

R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913–1922 (2007).
[Crossref] [PubMed]

Ding, J.

X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett 32, 3549–3551 (2007).
[Crossref] [PubMed]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Drori, Y.

Dudley, A.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photon. 10, 327–332 (2016).
[Crossref]

Dyment, J.

J. Dyment, “Hermite-gaussian mode patterns in GaAs junction lasers,” Appl. Phys. Lett. 10, 84–86 (1967).
[Crossref]

Edgar, M. P.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref] [PubMed]

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. G. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic super-sampling,” arXiv preprint arXiv:1607.08236 (2016).

Eggleton, B. J.

Feng, F.

Fink, M.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

Forbes, A.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photon. 10, 327–332 (2016).
[Crossref]

Franke-Arnold, S.

N. Radwell, R. Hawley, J. Götte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref] [PubMed]

Friese, M.

M. Friese, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Galvez, E.

E. Galvez, P. Crawford, H. Sztul, M. Pysher, P. Haglin, and R. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

Galvez, E. J.

Gao, H.-F.

Gibson, G. G.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. G. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic super-sampling,” arXiv preprint arXiv:1607.08236 (2016).

Gibson, G. M.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref] [PubMed]

R. W. Bowman, G. M. Gibson, A. Linnenberger, D. B. Phillips, J. A. Grieve, D. M. Carberry, S. Serati, M. J. Miles, and M. J. Padgett, “‘red tweezers’: Fast, customisable hologram generation for optical tweezers,” Comput. Phys. Commun. 185, 268–273 (2014).
[Crossref]

Gigan, S.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

Gong, L.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116, 183105 (2014).
[Crossref]

Goorden, S. A.

Gordon, G. S.

Götte, J.

N. Radwell, R. Hawley, J. Götte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref] [PubMed]

Grieve, J. A.

R. W. Bowman, G. M. Gibson, A. Linnenberger, D. B. Phillips, J. A. Grieve, D. M. Carberry, S. Serati, M. J. Miles, and M. J. Padgett, “‘red tweezers’: Fast, customisable hologram generation for optical tweezers,” Comput. Phys. Commun. 185, 268–273 (2014).
[Crossref]

Guo, C.-S.

X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett 32, 3549–3551 (2007).
[Crossref] [PubMed]

Haglin, P.

E. Galvez, P. Crawford, H. Sztul, M. Pysher, P. Haglin, and R. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

Hao, X.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
[Crossref]

Hawley, R.

N. Radwell, R. Hawley, J. Götte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref] [PubMed]

Heckenberg, N.

M. Friese, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

Huang, H.

Huang, K.

Ianni, F.

Jang, M.

Jesacher, A.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[Crossref]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Jung, Y.

Kang, Q.

Karimi, E.

Katz, N.

Khadka, S.

Kuang, C.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
[Crossref]

Kuhn, A.

D. Stuart, O. Barter, and A. Kuhn, “Fast algorithms for generating binary holograms,” arXiv preprint arXiv:1409.1841 (2014).

Lavery, M. P.

Lee, W.-H.

Lerner, V.

Lerosey, G.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Li, F.

Li, M.

Li, Y.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116, 183105 (2014).
[Crossref]

Li, Y.-M.

Linnenberger, A.

R. W. Bowman, G. M. Gibson, A. Linnenberger, D. B. Phillips, J. A. Grieve, D. M. Carberry, S. Serati, M. J. Miles, and M. J. Padgett, “‘red tweezers’: Fast, customisable hologram generation for optical tweezers,” Comput. Phys. Commun. 185, 268–273 (2014).
[Crossref]

Litvin, I.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photon. 10, 327–332 (2016).
[Crossref]

Liu, R.

Liu, W.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116, 183105 (2014).
[Crossref]

Liu, X.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
[Crossref]

Lohmann, A. W.

Magana-Loaiza, O. S.

Malik, M.

Manzo, C.

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]

Marrucci, L.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photon. 10, 327–332 (2016).
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G. Milione, M. P. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, and et al., “4× 20 gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de) multiplexer,” Opt. Lett. 40, 1980–1983 (2015).
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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]

Maurer, C.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[Crossref]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Mazilu, M.

Miles, M. J.

R. W. Bowman, G. M. Gibson, A. Linnenberger, D. B. Phillips, J. A. Grieve, D. M. Carberry, S. Serati, M. J. Miles, and M. J. Padgett, “‘red tweezers’: Fast, customisable hologram generation for optical tweezers,” Comput. Phys. Commun. 185, 268–273 (2014).
[Crossref]

Milione, G.

Mirhosseini, M.

Mitchell, K. J.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref] [PubMed]

Moreno, I.

Mosk, A. P.

Naidoo, D.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photon. 10, 327–332 (2016).
[Crossref]

Nesterov, A.

V. Niziev and A. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D Appl. Phys. 32, 1455 (1999).
[Crossref]

Nguyen, T. A.

Ni, W.-J.

X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett 32, 3549–3551 (2007).
[Crossref] [PubMed]

Nieminen, T.

M. Friese, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

Niziev, V.

V. Niziev and A. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D Appl. Phys. 32, 1455 (1999).
[Crossref]

Nolan, D. A.

Nomoto, S.

Padgett, M.

Padgett, M. J.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref] [PubMed]

R. W. Bowman, G. M. Gibson, A. Linnenberger, D. B. Phillips, J. A. Grieve, D. M. Carberry, S. Serati, M. J. Miles, and M. J. Padgett, “‘red tweezers’: Fast, customisable hologram generation for optical tweezers,” Comput. Phys. Commun. 185, 268–273 (2014).
[Crossref]

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. G. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic super-sampling,” arXiv preprint arXiv:1607.08236 (2016).

Paparo, D.

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]

Pereira, S.

S. Pereira and A. Van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
[Crossref]

Phillips, D.

Phillips, D. B.

R. W. Bowman, G. M. Gibson, A. Linnenberger, D. B. Phillips, J. A. Grieve, D. M. Carberry, S. Serati, M. J. Miles, and M. J. Padgett, “‘red tweezers’: Fast, customisable hologram generation for optical tweezers,” Comput. Phys. Commun. 185, 268–273 (2014).
[Crossref]

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. G. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic super-sampling,” arXiv preprint arXiv:1607.08236 (2016).

Piccirillo, B.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photon. 10, 327–332 (2016).
[Crossref]

Piestun, R.

Plöschner, M.

M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nat. Photon. 9, 529–535 (2015).
[Crossref]

Popoff, S.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

Pysher, M.

E. Galvez, P. Crawford, H. Sztul, M. Pysher, P. Haglin, and R. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Radwell, N.

N. Radwell, R. Hawley, J. Götte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref] [PubMed]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref] [PubMed]

Ren, Y.

Ren, Y.-X.

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[Crossref]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Rodenburg, B.

Roux, F. S.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photon. 10, 327–332 (2016).
[Crossref]

Ruan, H.

Rubinsztein-Dunlop, H.

M. Friese, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

Ruocco, G.

Sahu, J.

Schröder, J.

Schubert, W. H.

Serati, S.

R. W. Bowman, G. M. Gibson, A. Linnenberger, D. B. Phillips, J. A. Grieve, D. M. Carberry, S. Serati, M. J. Miles, and M. J. Padgett, “‘red tweezers’: Fast, customisable hologram generation for optical tweezers,” Comput. Phys. Commun. 185, 268–273 (2014).
[Crossref]

Shwa, D.

Spreeuw, R.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Stuart, D.

D. Stuart, O. Barter, and A. Kuhn, “Fast algorithms for generating binary holograms,” arXiv preprint arXiv:1409.1841 (2014).

Sun, B.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref] [PubMed]

Sun, M.-J.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. G. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic super-sampling,” arXiv preprint arXiv:1607.08236 (2016).

Sztul, H.

E. Galvez, P. Crawford, H. Sztul, M. Pysher, P. Haglin, and R. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

Taylor, J. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. G. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic super-sampling,” arXiv preprint arXiv:1607.08236 (2016).

Tyc, T.

M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nat. Photon. 9, 529–535 (2015).
[Crossref]

Van de Nes, A.

S. Pereira and A. Van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
[Crossref]

Vettenburg, T.

Wang, D.

Wang, H.-T.

X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett 32, 3549–3551 (2007).
[Crossref] [PubMed]

Wang, M.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116, 183105 (2014).
[Crossref]

Wang, T.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
[Crossref]

Wang, X.-L.

X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett 32, 3549–3551 (2007).
[Crossref] [PubMed]

Wang, Z.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116, 183105 (2014).
[Crossref]

Wang, Z.-Q.

Welsh, S. S.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref] [PubMed]

Wilkinson, T.

Williams, R.

E. Galvez, P. Crawford, H. Sztul, M. Pysher, P. Haglin, and R. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
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Woerdman, J.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
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Zhan, Q.

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L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116, 183105 (2014).
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Zhou, E. H.

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R. W. Bowman, G. M. Gibson, A. Linnenberger, D. B. Phillips, J. A. Grieve, D. M. Carberry, S. Serati, M. J. Miles, and M. J. Padgett, “‘red tweezers’: Fast, customisable hologram generation for optical tweezers,” Comput. Phys. Commun. 185, 268–273 (2014).
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J. Appl. Phys. (1)

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116, 183105 (2014).
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X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
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J. Phys. D Appl. Phys. (1)

V. Niziev and A. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D Appl. Phys. 32, 1455 (1999).
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S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12, 635–639 (2012).
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Laser Photon. Rev. (1)

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
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Nat. Commun. (3)

N. Radwell, R. Hawley, J. Götte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
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S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
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T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
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D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photon. 10, 327–332 (2016).
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M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nat. Photon. 9, 529–535 (2015).
[Crossref]

Nature (1)

M. Friese, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
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Z. Bouchal and R. Celechovsky, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131 (2004).
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C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
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Opt. Commun. (2)

S. Pereira and A. Van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
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Opt. Express (7)

Opt. Lett (1)

X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett 32, 3549–3551 (2007).
[Crossref] [PubMed]

Opt. Lett. (4)

Optica (2)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

E. Galvez, P. Crawford, H. Sztul, M. Pysher, P. Haglin, and R. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

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]

Sci. Rep. (1)

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref] [PubMed]

Other (2)

D. Stuart, O. Barter, and A. Kuhn, “Fast algorithms for generating binary holograms,” arXiv preprint arXiv:1409.1841 (2014).

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. G. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic super-sampling,” arXiv preprint arXiv:1607.08236 (2016).

Supplementary Material (1)

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» Visualization 1: MP4 (1906 KB)      This video demonstrates the dynamic switching of the polarization state of a vector beam generated using the Digital Micro-mirror Device (DMD) seen in Figure 4

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Figures (6)

Fig. 1
Fig. 1 Experimental set-up: The laser beam is generated by a Zeeman stabilised HeNe laser (Neoark corp Neo-262). The focal length of lens F1 = 40 mm, and F2 = 160 mm. The DMD is a Texas Instruments DLP LightCrafter evaluation model with a 0.3 WVGA chipset. F3 = 300 mm, and APT is a bespoke aperture designed to pass only beams A and B whilst blocking all other orders diffracted from the DMD. HWP1-HWP4 are multi-order half-wave plates. BD1 (ThorLabs BD40) and BD2 (ThorLabs BD27) are calcite beam displacers. M1 and M2 are silvered mirrors. F4 = 200 mm, and F5 = 150 mm. QWP and LP are a quarter-wave plate and a linear polariser that can be inserted before the CMOS camera (we used a Prosilica GC660, or a high-speed Mikrotron EoSens CL MC1362) to measure the spatially resolved polarisation states of the generated beams. The DMD patterns were designed using a LabVIEW program, and two different interface options were used: patterns were either uploaded directly to the DMD via the mini HDMI link for 60 Hz real-time control, or saved as 1-bit bitmaps and preloaded onto the DMD internal memory via USB link, to be played back in high-speed at up to 4 kHz. The inset shows the angles of incident and reflected light on the DMD.
Fig. 2
Fig. 2 DMD hologram design: (a) A radially polarised vector vortex beam with a Laguerre-Gaussian (LG=1,p=0) intensity profile can be formed from the addition of a horizontally polarised Hermite-Gaussian (HGn=1,m=0) spatial mode, and a vertically polarised HG0,1 spatial mode. Here and p are the LG vortex and radial spatial mode indices respectively, and n and m are the HG spatial mode indices. (b) A chirally polarised vector vortex beam (i.e. the polarisation structure is chiral in nature) is formed from the addition of two rotated HG beams of orthogonal linear polarisations. (c) An azimuthally polarised vector vortex beam is formed from the addition of a horizontally polarised HG0,1 spatial mode and a vertically polarised HG1,0 spatial mode. (d) A binary transmittance function displayed on the DMD to generate the radially polarised beam shown in (a). This was designed using Eqns. 15, where �� and �� are the appropriate HG modes shown in (a) each incorporating a different phase tilt to transmit them in different directions. Here we show the negative of the transmission function (i.e. white represents regions where incident light is blocked, and black where it is transmitted). The insets show regions of the pattern in more detail: (i) a region that diffracts light predominantly into beam A, (ii) a region that diffracts a similar intensity of light into beams A and B. (e) A simulation of the field at the Fourier plane of the DMD when displaying the binary transmittance function shown in (d), showing beams A and B spatially separated and with independent intensity and phase profiles. The central beam is the zero diffraction order, and the beams are copied in the lower right quadrant due to the binary nature of the diffraction grating.
Fig. 3
Fig. 3 Experimentally generated vector beams with uniform polarisation. Upper row: (a) Generation of a diffraction limited beam (i.e. generated by diffracting light from over the entire face of the DMD) of diagonal linear polarisation in the Fourier plane of the DMD. This is formed from beams A and B of equal power with a zero radian phase shift between their electric fields. Inset shows the intensity of the beam. Scale bar = 50μm. (b) The same beam as in (a) now viewed in the image plane of the DMD. (c) Generation of a diffraction limited beam of circular polarisation in the Fourier plane of the DMD. This is formed from beams A and B of equal power with a π/2 rad. phase shift between their electric fields. (d) The same beam as (c) now viewed in the image plane of the DMD.
Fig. 4
Fig. 4 Comparison of simulated and experimentally generated vector beams with spatially varying polarisation. Upper row: (a–c) A simulated (a) and experimentally measured (b,c) radially polarised vector vortex beam. (b) is measured in the Fourier plane of the DMD, while (c) is measured in the image plane of the DMD. Insets show the simulated/measured intensity of the field in each case. Middle row: (d–f) Simulated azimuthally polarised (d), Poincaré ‘lemon’ (e), and Poincaré ‘star’ (f) beams. In (e) and (f) the colour signifies the handedness of the local polarisation state. Lower row: (g–i) Experimentally generated and measured polarisation maps of the same beams as the rows above.
Fig. 5
Fig. 5 Higher order polarisation structuring. Generation of a linearly polarised beam possessing a diagonal square lattice of polarisation vortices. The polarisation structure around each vortex alternates between a radial distribution and a 4-pointed star distribution. The expected position of each vortex is marked with a red cross. The inset shows the intensity pattern of the beam, where the vortices can clearly be seen. To create this structure beam A was formed from a HG3,2 spatial mode, and beam B from a HG2,3 spatial mode.
Fig. 6
Fig. 6 High-speed beam switching using a DMD Upper panels (spinning lobe images): images of the generated beam captured through a static linear polariser at ∼7 kHz with a high-speed camera, as the DMD cycles through a series of 10 pre-loaded patterns at a rate of 4 kHz. The orientation of the transmitted polarisation is shown as a white arrow in the lower left hand corner of each panel. We show every second camera frame, and the elapsed time of each frame is also shown in red. The high-speed rotation of the lobes indicates the rotation of the local polarisation within the beam from one DMD pattern to the next. Lower panels (polarisation maps): Experimentally reconstructed polarisation maps of the beam for each DMD pattern. The polarisation map data was not collected at high speed as it required the recording of multiple images of the beam transmitted through different polariser states as described in Section 4. Visualization 1 also shows the camera frames as a slow motion movie.

Equations (10)

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T ( x , y ) = 1 2 + 1 2 sgn [ cos ( p ( x , y ) ) + cos ( q ( x , y ) ) ] ,
p ( x , y ) = ϕ S ( x , y ) + ϕ tilt ( x , y ) ,
q ( x , y ) = arcsin ( S ( x , y ) / S max ) .
E ( x , y ) = [ E A ( x , y ) e i ϕ A E B ( x , y ) e i ϕ B ] ,
𝕊 dual = S dual ( x , y ) e i ϕ S , dual ( x , y ) = W rel e i ϕ global 𝔸 + ( 1 W rel ) 𝔹 ,
P major = [ 1 2 ( I p + | L | ) ] 1 2 , P minor = [ 1 2 ( I p | L | ) ] 1 2 , θ = 1 2 arg ( L ) , h = sgn ( V ) ,
I p = [ ( | E v | 2 | E h | 2 ) 2 + ( | E a | 2 | E d | 2 ) 2 + ( | E | 2 | E + | 2 ) 2 ] 1 2 ,
L = ( | E v | 2 | E h | 2 ) 2 + i ( | E a | 2 | E d | 2 ) 2 .
E trans = | 1 N E T ( x , y ) 𝕊 ( x , y ) * dx dy | 2 ,
N E = [ | T ( x , y ) | 2 dx dy × | 𝕊 ( x , y ) | 2 dx dy ] 1 2 .

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