Abstract

We describe an experimental technique for generating a quasi-monochromatic field with any arbitrary spatial coherence properties that can be described by the cross-spectral density function, W(r1,r2). This is done by using a dynamic binary amplitude grating generated by a digital micromirror device to rapidly alternate between a set of coherent fields, creating an incoherent mix of modes that represent the coherent mode decomposition of the desired W(r1,r2). This method was then demonstrated experimentally by interfering two plane waves and then spatially varying the coherence between them. It is then shown that this creates an interference pattern between the two beams whose fringe visibility varies spatially in an arbitrary and prescribed way.

© 2014 Optical Society of America

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  1. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
    [CrossRef]
  2. B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. OSullivan, M. Mirhosseini, D. J. Robertson, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37, 3735–3737 (2012).
    [CrossRef]
  3. M. Mirhosseini, B. Rodenburg, M. Malik, and R. W. Boyd, “Free-space communication through turbulence: a comparison of plane-wave and orbital-angular-momentum encodings,” J. Mod. Opt. 61, 43–48 (2014).
    [CrossRef]
  4. R. W. Boyd, B. Rodenburg, M. Mirhosseini, and S. M. Barnett, “Influence of atmospheric turbulence on the propagation of quantum states of light using plane-wave encoding,” Opt. Express 19, 18310–18317 (2011).
    [CrossRef]
  5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [CrossRef]
  6. A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
    [CrossRef]
  7. R. W. Boyd, A. Jha, M. Malik, C. O’Sullivan, B. Rodenburg, and D. J. Gauthier, “Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon,” Proc. SPIE 7948, 79480L (2011).
    [CrossRef]
  8. M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
    [CrossRef]
  9. P. S. Considine, “Effects of coherence on imaging systems,” J. Opt. Soc. Am. 56, 1001–1007 (1966).
    [CrossRef]
  10. B. Lin, “Partially coherent imaging in two dimensions and the theoretical limits of projection printing in microfabrication,” IEEE Trans. Electron Devices 27, 931–938 (1980).
    [CrossRef]
  11. J. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
    [CrossRef]
  12. F. Dubois, L. Joannes, and J.-C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. 38, 7085–7094 (1999).
    [CrossRef]
  13. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
    [CrossRef]
  14. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36, 4104–4106 (2011).
    [CrossRef]
  15. Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, “Clustering of solitons in weakly correlated wavefronts,” Proc. Natl. Acad. Sci. USA 99, 5223–5227 (2002).
    [CrossRef]
  16. L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
    [CrossRef]
  17. J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
    [CrossRef]
  18. J. S. Lundeen and C. Bamber, “Procedure for direct measurement of general quantum states using weak measurement,” Phys. Rev. Lett. 108, 070402 (2012).
    [CrossRef]
  19. C. Rickenstorff, E. Flores, M. Olvera-Santamaría, and A. Ostrovsky, “Modulation of coherence and polarization using nematic 90 degree-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).
  20. E. Baleine and A. Dogariu, “Variable-coherence tomography for inverse scattering problems,” J. Opt. Soc. Am. A 21, 1917–1923 (2004).
    [CrossRef]
  21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  22. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
    [CrossRef]
  23. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  24. M. Mirhosseini, O. S. Magaña-Loaiza, C. Chen, B. Rodenburg, M. Malik, and R. W. Boyd, “Rapid generation of light beams carrying orbital angular momentum,” Opt. Express 21, 30196–30203 (2013).
    [CrossRef]
  25. D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
    [CrossRef]
  26. B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
    [CrossRef]
  27. W.-H. Lee, “High efficiency multiple beam gratings,” Appl. Opt. 18, 2152–2158 (1979).
    [CrossRef]
  28. P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D. N. Christodoulides, and Z. Chen, “Trapping and transporting aerosols with a single optical bottle beam generated by moiré techniques,” Opt. Lett. 36, 1491–1493 (2011).
    [CrossRef]
  29. E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
    [CrossRef]

2014

M. Mirhosseini, B. Rodenburg, M. Malik, and R. W. Boyd, “Free-space communication through turbulence: a comparison of plane-wave and orbital-angular-momentum encodings,” J. Mod. Opt. 61, 43–48 (2014).
[CrossRef]

2013

2012

M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
[CrossRef]

B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. OSullivan, M. Mirhosseini, D. J. Robertson, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37, 3735–3737 (2012).
[CrossRef]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

J. S. Lundeen and C. Bamber, “Procedure for direct measurement of general quantum states using weak measurement,” Phys. Rev. Lett. 108, 070402 (2012).
[CrossRef]

C. Rickenstorff, E. Flores, M. Olvera-Santamaría, and A. Ostrovsky, “Modulation of coherence and polarization using nematic 90 degree-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

2011

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[CrossRef]

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

R. W. Boyd, A. Jha, M. Malik, C. O’Sullivan, B. Rodenburg, and D. J. Gauthier, “Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon,” Proc. SPIE 7948, 79480L (2011).
[CrossRef]

P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D. N. Christodoulides, and Z. Chen, “Trapping and transporting aerosols with a single optical bottle beam generated by moiré techniques,” Opt. Lett. 36, 1491–1493 (2011).
[CrossRef]

R. W. Boyd, B. Rodenburg, M. Mirhosseini, and S. M. Barnett, “Influence of atmospheric turbulence on the propagation of quantum states of light using plane-wave encoding,” Opt. Express 19, 18310–18317 (2011).
[CrossRef]

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36, 4104–4106 (2011).
[CrossRef]

2004

2003

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[CrossRef]

2002

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, “Clustering of solitons in weakly correlated wavefronts,” Proc. Natl. Acad. Sci. USA 99, 5223–5227 (2002).
[CrossRef]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[CrossRef]

2001

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef]

1999

1982

1981

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

1980

B. Lin, “Partially coherent imaging in two dimensions and the theoretical limits of projection printing in microfabrication,” IEEE Trans. Electron Devices 27, 931–938 (1980).
[CrossRef]

1979

1970

J. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[CrossRef]

1969

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[CrossRef]

1966

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Andersson, E.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

Baleine, E.

Bamber, C.

J. S. Lundeen and C. Bamber, “Procedure for direct measurement of general quantum states using weak measurement,” Phys. Rev. Lett. 108, 070402 (2012).
[CrossRef]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[CrossRef]

Barnett, S. M.

Bent, N.

Bolduc, E.

Boyd, R. W.

M. Mirhosseini, B. Rodenburg, M. Malik, and R. W. Boyd, “Free-space communication through turbulence: a comparison of plane-wave and orbital-angular-momentum encodings,” J. Mod. Opt. 61, 43–48 (2014).
[CrossRef]

M. Mirhosseini, O. S. Magaña-Loaiza, C. Chen, B. Rodenburg, M. Malik, and R. W. Boyd, “Rapid generation of light beams carrying orbital angular momentum,” Opt. Express 21, 30196–30203 (2013).
[CrossRef]

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
[CrossRef]

B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. OSullivan, M. Mirhosseini, D. J. Robertson, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37, 3735–3737 (2012).
[CrossRef]

M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
[CrossRef]

R. W. Boyd, B. Rodenburg, M. Mirhosseini, and S. M. Barnett, “Influence of atmospheric turbulence on the propagation of quantum states of light using plane-wave encoding,” Opt. Express 19, 18310–18317 (2011).
[CrossRef]

R. W. Boyd, A. Jha, M. Malik, C. O’Sullivan, B. Rodenburg, and D. J. Gauthier, “Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon,” Proc. SPIE 7948, 79480L (2011).
[CrossRef]

Brown, B. R.

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[CrossRef]

Buller, G. S.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

Chen, C.

Chen, Z.

P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D. N. Christodoulides, and Z. Chen, “Trapping and transporting aerosols with a single optical bottle beam generated by moiré techniques,” Opt. Lett. 36, 1491–1493 (2011).
[CrossRef]

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, “Clustering of solitons in weakly correlated wavefronts,” Proc. Natl. Acad. Sci. USA 99, 5223–5227 (2002).
[CrossRef]

Christodoulides, D. N.

P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D. N. Christodoulides, and Z. Chen, “Trapping and transporting aerosols with a single optical bottle beam generated by moiré techniques,” Opt. Lett. 36, 1491–1493 (2011).
[CrossRef]

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, “Clustering of solitons in weakly correlated wavefronts,” Proc. Natl. Acad. Sci. USA 99, 5223–5227 (2002).
[CrossRef]

Considine, P. S.

Dada, A. C.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

Dainty, J.

J. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[CrossRef]

Dogariu, A.

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Dubois, F.

Dudley, D.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[CrossRef]

Duncan, W. M.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[CrossRef]

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Fleischer, J. W.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

Flores, E.

C. Rickenstorff, E. Flores, M. Olvera-Santamaría, and A. Ostrovsky, “Modulation of coherence and polarization using nematic 90 degree-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).

Gauthier, D. J.

R. W. Boyd, A. Jha, M. Malik, C. O’Sullivan, B. Rodenburg, and D. J. Gauthier, “Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon,” Proc. SPIE 7948, 79480L (2011).
[CrossRef]

Gbur, G.

Hernandez, D.

Huang, H.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Huang, S.

Jha, A.

R. W. Boyd, A. Jha, M. Malik, C. O’Sullivan, B. Rodenburg, and D. J. Gauthier, “Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon,” Proc. SPIE 7948, 79480L (2011).
[CrossRef]

Joannes, L.

Karimi, E.

Lajunen, H.

Lavery, M. P. J.

Leach, J.

M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
[CrossRef]

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

Lee, W.-H.

Legros, J.-C.

Lin, B.

B. Lin, “Partially coherent imaging in two dimensions and the theoretical limits of projection printing in microfabrication,” IEEE Trans. Electron Devices 27, 931–938 (1980).
[CrossRef]

Lohmann, A. W.

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[CrossRef]

Lundeen, J. S.

J. S. Lundeen and C. Bamber, “Procedure for direct measurement of general quantum states using weak measurement,” Phys. Rev. Lett. 108, 070402 (2012).
[CrossRef]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[CrossRef]

Magaña-Loaiza, O. S.

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef]

Malik, M.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martin, H.

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, “Clustering of solitons in weakly correlated wavefronts,” Proc. Natl. Acad. Sci. USA 99, 5223–5227 (2002).
[CrossRef]

Mirhosseini, M.

O’Sullivan, C.

R. W. Boyd, A. Jha, M. Malik, C. O’Sullivan, B. Rodenburg, and D. J. Gauthier, “Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon,” Proc. SPIE 7948, 79480L (2011).
[CrossRef]

O’Sullivan, M. N.

Olvera-Santamaría, M.

C. Rickenstorff, E. Flores, M. Olvera-Santamaría, and A. Ostrovsky, “Modulation of coherence and polarization using nematic 90 degree-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).

Ostrovsky, A.

C. Rickenstorff, E. Flores, M. Olvera-Santamaría, and A. Ostrovsky, “Modulation of coherence and polarization using nematic 90 degree-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).

OSullivan, M. N.

Padgett, M. J.

Patel, A.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[CrossRef]

Prakash, J.

Ren, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Rickenstorff, C.

C. Rickenstorff, E. Flores, M. Olvera-Santamaría, and A. Ostrovsky, “Modulation of coherence and polarization using nematic 90 degree-twist liquid-crystal spatial light modulators,” Rev. Mex. Fis. 58, 270–273 (2012).

Robertson, D. J.

Rodenburg, B.

Saastamoinen, T.

Salazar, M.

Santamato, E.

Sears, S. M.

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, “Clustering of solitons in weakly correlated wavefronts,” Proc. Natl. Acad. Sci. USA 99, 5223–5227 (2002).
[CrossRef]

Segev, M.

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, “Clustering of solitons in weakly correlated wavefronts,” Proc. Natl. Acad. Sci. USA 99, 5223–5227 (2002).
[CrossRef]

Situ, G.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

Slaughter, J.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[CrossRef]

Starikov, A.

Stewart, C.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[CrossRef]

Sutherland, B.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[CrossRef]

Tur, M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef]

Waller, L.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

Wang, J.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef]

Wilner, A. E.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Wolf, E.

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[CrossRef]

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Yan, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Wilner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Yue, Y.

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Figures (5)

Fig. 1.
Fig. 1.

Left: a binary amplitude grating composed of a series of rectangular pulses diffracting light into multiple orders. Middle: pulse position modulation where a phase change is induced in the diffracted order as a result of a shift in the pulses. Right: change in the amplitude of the diffracted order by pulse width modulation in which the diffraction efficiency is varied by changing the duty cycle of the binary pulses.

Fig. 2.
Fig. 2.

Experimental setup used to generate any field, W(r1r2). A fully spatially coherent plane wave is prepared by spatial filtering of a He–Ne laser. This collimated beam is reflected off a CGH generated by the DMD, and the desired diffracted order is filtered by a 4f system and imaged onto a CCD.

Fig. 3.
Fig. 3.

Interference fringes formed from the coherent superposition of two plane waves. The left image shows the CGH used to generate the desired mode. The middle image represents the target image, while the right image is an experimental image of the generated mode.

Fig. 4.
Fig. 4.

Interference fringes formed from superposition of two plane waves that are partially coherent with respect to each other. The top figures show the CGHs used to generate the desired modes given by Eq. (16). The bottom left figure represents the target intensity pattern, while the bottom right figure is an experimental image of the generated field.

Fig. 5.
Fig. 5.

Plot of the intensity of the image in Fig. 4 along the 1D slice of r for θ=0, i.e., along the x axis (solid blue line). Also shown as the black-dotted line is the theoretical envelope of the maximum and minimum intensities based on the intended visibility function V(r).

Equations (21)

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W(r1,r2)=U*(r1)U(r2),
W(r1,r2)=nλnψn*(r1)ψn(r2),
W(r1,r2)ψn(r1)d2r1=λnψn(r2).
W(r1,r2)=I(r1)I(r2)μ(r1,r2).
τdet>τs>τcoh.
f(r)=msin(πmq)πmeim(G·r+2πδ),
U1=Uin*sin(πq)πei2πδ,
q(r)=1πarcsin(A(r)),δ(r)=ϕ(r)2π,
cos(G·r+2πδ(r)).
f(r)=H[cos(G·r+2πδ(r))cos(πq(r))],
H(z){0ifz<01ifz0.
U(r)eikx+eikx
G=2π25px(x^+y^),
k=2π100px2π780μm.
ψ1(r)(UA+f(r)UB),ψ2(r)(f(r)UA+UB),
f(r)=V(r)/(1+1V(r)2).
I(r)(1f)2+4fcos2(kx),
V(r)=|sin(κr)|,
2πκ=4k3=2π75px2π580μm.
q(r)=1πarcsin(4fcos2(kx)+(1f)2Imax),
δ1,2(r)=arg(R(ψ1,2)+iI(ψ1,2))=arg((2cos(kx)(1f)cos(kx))i((f1)sin(kx))).

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