Abstract

We present a Jones matrix method useful to analyze coherent optical Fourier processors employing structured polarization. The proposed method is a generalization of the standard classical optical Fourier transform processor, but considering vectorial spatial functions with two complex components corresponding to two orthogonal linear polarizations. As a result we derive a Jones matrix that describes the polarization output in terms of two vectorial functions defining respectively the structured polarization input and the generalized polarization impulse response. We apply the method to show and analyze an experiment in which a regular scalar diffraction grating is converted into equivalent polarization diffraction gratings by means of an appropriate polarization filtering. The technique is further demonstrated to generate arbitrary structured polarizations. Excellent experimental results are presented.

© 2011 OSA

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References

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  1. J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. 39(10), 1549–1554 (2000).
    [CrossRef]
  2. Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
    [CrossRef]
  3. V. Ramírez-Sánchez and G. Piquero, “Global beam shaping with nonuniformly polarized beams using amplitude transmitances,” Opt.Pura Apl. 40, 87–93 (2007).
  4. A. Volke and G. Heine, “Bringing order into light with structured polarizers,” Photonik Int. 2, 6–9 (2008).
  5. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996).
    [CrossRef] [PubMed]
  6. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [CrossRef] [PubMed]
  7. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009).
    [CrossRef]
  8. J. A. Davis, G. H. Evans, and I. Moreno, “Polarization-multiplexed diffractive optical elements with liquid-crystal displays,” Appl. Opt. 44(19), 4049–4052 (2005).
    [CrossRef] [PubMed]
  9. M. Fratz, D. M. Giel, and P. Fischer, “Digital polarization holograms with defined magnitude and orientation of each pixel’s birefringence,” Opt. Lett. 34(8), 1270–1272 (2009).
    [CrossRef] [PubMed]
  10. G. Cincotti, “Polarization gratings: Design and applications,” IEEE J. Quantum Electron. 39(12), 1645–1652 (2003).
    [CrossRef]
  11. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24(9), 584–586 (1999).
    [CrossRef]
  12. J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26(9), 587–589 (2001).
    [CrossRef]
  13. C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33(20), 2287–2289 (2008).
    [CrossRef] [PubMed]
  14. J. L. Martínez, I. Moreno, and F. Mateos, “Hiding binary optical data with orthogonal circular polarizations,” Opt. Eng. 47(3), 030504 (2008).
    [CrossRef]
  15. B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39(9), 2439–2443 (2000).
    [CrossRef]
  16. H.-Y. Tu, C.-J. Cheng, and M.-L. Chen, “Optical image encryption based on polarization encoding by liquid crystal spatial light modulators,” J. Opt. A, Pure Appl. Opt. 6(6), 524–528 (2004).
    [CrossRef]
  17. M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27(21), 1929–1931 (2002).
    [CrossRef]
  18. K. C. Toussaint, S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. 30(21), 2846–2848 (2005).
    [CrossRef] [PubMed]
  19. 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(24), 3549–3551 (2007).
    [CrossRef] [PubMed]
  20. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
    [CrossRef]
  21. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
    [CrossRef]
  22. F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25(17), 1291–1293 (2000).
    [CrossRef]
  23. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge 2007).
  24. I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51(14), 2031–2038 (2004).
    [CrossRef]
  25. I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Binary polarization pupil filter: Theoretical analysis and experimental realization with a liquid crystal display,” Opt. Commun. 264(1), 63–69 (2006).
    [CrossRef]
  26. I. Moreno, C. Iemmi, J. Campos, M. J. Yzuel, and A. Vargas, “Polarization vortices generation by diffraction from a four quadrant polarization mask,” Opt. Commun. 276(2), 222–230 (2007).
    [CrossRef]
  27. A. Martínez-García, I. Moreno, M. M. Sánchez-López, and P. García-Martínez, “Operational modes of a ferroelectric LCoS modulator for displaying binary polarization, amplitude, and phase diffraction gratings,” Appl. Opt. 48(15), 2903–2914 (2009).
    [CrossRef] [PubMed]
  28. I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
    [CrossRef]

2009 (3)

2008 (3)

C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33(20), 2287–2289 (2008).
[CrossRef] [PubMed]

A. Volke and G. Heine, “Bringing order into light with structured polarizers,” Photonik Int. 2, 6–9 (2008).

J. L. Martínez, I. Moreno, and F. Mateos, “Hiding binary optical data with orthogonal circular polarizations,” Opt. Eng. 47(3), 030504 (2008).
[CrossRef]

2007 (4)

V. Ramírez-Sánchez and G. Piquero, “Global beam shaping with nonuniformly polarized beams using amplitude transmitances,” Opt.Pura Apl. 40, 87–93 (2007).

I. Moreno, C. Iemmi, J. Campos, M. J. Yzuel, and A. Vargas, “Polarization vortices generation by diffraction from a four quadrant polarization mask,” Opt. Commun. 276(2), 222–230 (2007).
[CrossRef]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 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(24), 3549–3551 (2007).
[CrossRef] [PubMed]

2006 (1)

I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Binary polarization pupil filter: Theoretical analysis and experimental realization with a liquid crystal display,” Opt. Commun. 264(1), 63–69 (2006).
[CrossRef]

2005 (2)

2004 (2)

H.-Y. Tu, C.-J. Cheng, and M.-L. Chen, “Optical image encryption based on polarization encoding by liquid crystal spatial light modulators,” J. Opt. A, Pure Appl. Opt. 6(6), 524–528 (2004).
[CrossRef]

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51(14), 2031–2038 (2004).
[CrossRef]

2003 (2)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

G. Cincotti, “Polarization gratings: Design and applications,” IEEE J. Quantum Electron. 39(12), 1645–1652 (2003).
[CrossRef]

2002 (1)

2001 (2)

J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26(9), 587–589 (2001).
[CrossRef]

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
[CrossRef]

2000 (3)

1999 (1)

1998 (1)

1996 (1)

1995 (1)

I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
[CrossRef]

Adachi, J.

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Bomzon, Z.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
[CrossRef]

Borghi, R.

Campos, J.

I. Moreno, C. Iemmi, J. Campos, M. J. Yzuel, and A. Vargas, “Polarization vortices generation by diffraction from a four quadrant polarization mask,” Opt. Commun. 276(2), 222–230 (2007).
[CrossRef]

I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Binary polarization pupil filter: Theoretical analysis and experimental realization with a liquid crystal display,” Opt. Commun. 264(1), 63–69 (2006).
[CrossRef]

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51(14), 2031–2038 (2004).
[CrossRef]

I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
[CrossRef]

Chen, M.-L.

H.-Y. Tu, C.-J. Cheng, and M.-L. Chen, “Optical image encryption based on polarization encoding by liquid crystal spatial light modulators,” J. Opt. A, Pure Appl. Opt. 6(6), 524–528 (2004).
[CrossRef]

Cheng, C.-J.

H.-Y. Tu, C.-J. Cheng, and M.-L. Chen, “Optical image encryption based on polarization encoding by liquid crystal spatial light modulators,” J. Opt. A, Pure Appl. Opt. 6(6), 524–528 (2004).
[CrossRef]

Cincotti, G.

G. Cincotti, “Polarization gratings: Design and applications,” IEEE J. Quantum Electron. 39(12), 1645–1652 (2003).
[CrossRef]

Cottrell, D. M.

Davis, J. A.

Ding, J.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Escuti, M. J.

Evans, G. H.

Fernández-Pousa, C. R.

Fischer, P.

Fratz, M.

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

García-Martínez, P.

Giel, D. M.

Gorecki, C.

I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
[CrossRef]

Gori, F.

Guo, C.-S.

Hasman, E.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
[CrossRef]

Heine, G.

A. Volke and G. Heine, “Bringing order into light with structured polarizers,” Photonik Int. 2, 6–9 (2008).

Iemmi, C.

I. Moreno, C. Iemmi, J. Campos, M. J. Yzuel, and A. Vargas, “Polarization vortices generation by diffraction from a four quadrant polarization mask,” Opt. Commun. 276(2), 222–230 (2007).
[CrossRef]

I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Binary polarization pupil filter: Theoretical analysis and experimental realization with a liquid crystal display,” Opt. Commun. 264(1), 63–69 (2006).
[CrossRef]

Javidi, B.

B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39(9), 2439–2443 (2000).
[CrossRef]

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Jureller, J. E.

Juškaitis, R.

Kleiner, V.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Martínez, J. L.

J. L. Martínez, I. Moreno, and F. Mateos, “Hiding binary optical data with orthogonal circular polarizations,” Opt. Eng. 47(3), 030504 (2008).
[CrossRef]

Martínez-García, A.

Massoumian, F.

Mateos, F.

J. L. Martínez, I. Moreno, and F. Mateos, “Hiding binary optical data with orthogonal circular polarizations,” Opt. Eng. 47(3), 030504 (2008).
[CrossRef]

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

McNamara, D. E.

Moreno, I.

A. Martínez-García, I. Moreno, M. M. Sánchez-López, and P. García-Martínez, “Operational modes of a ferroelectric LCoS modulator for displaying binary polarization, amplitude, and phase diffraction gratings,” Appl. Opt. 48(15), 2903–2914 (2009).
[CrossRef] [PubMed]

J. L. Martínez, I. Moreno, and F. Mateos, “Hiding binary optical data with orthogonal circular polarizations,” Opt. Eng. 47(3), 030504 (2008).
[CrossRef]

I. Moreno, C. Iemmi, J. Campos, M. J. Yzuel, and A. Vargas, “Polarization vortices generation by diffraction from a four quadrant polarization mask,” Opt. Commun. 276(2), 222–230 (2007).
[CrossRef]

I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Binary polarization pupil filter: Theoretical analysis and experimental realization with a liquid crystal display,” Opt. Commun. 264(1), 63–69 (2006).
[CrossRef]

J. A. Davis, G. H. Evans, and I. Moreno, “Polarization-multiplexed diffractive optical elements with liquid-crystal displays,” Appl. Opt. 44(19), 4049–4052 (2005).
[CrossRef] [PubMed]

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51(14), 2031–2038 (2004).
[CrossRef]

J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26(9), 587–589 (2001).
[CrossRef]

I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
[CrossRef]

Neil, M. A. A.

Ni, W.-J.

Nomura, T.

B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39(9), 2439–2443 (2000).
[CrossRef]

Oh, C.

Park, S.

Piquero, G.

V. Ramírez-Sánchez and G. Piquero, “Global beam shaping with nonuniformly polarized beams using amplitude transmitances,” Opt.Pura Apl. 40, 87–93 (2007).

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25(17), 1291–1293 (2000).
[CrossRef]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Ramírez-Sánchez, V.

V. Ramírez-Sánchez and G. Piquero, “Global beam shaping with nonuniformly polarized beams using amplitude transmitances,” Opt.Pura Apl. 40, 87–93 (2007).

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Sánchez-López, M. M.

Santarsiero, M.

Schadt, M.

Scherer, N. F.

Sonehara, T.

Stalder, M.

Toussaint, K. C.

Tu, H.-Y.

H.-Y. Tu, C.-J. Cheng, and M.-L. Chen, “Optical image encryption based on polarization encoding by liquid crystal spatial light modulators,” J. Opt. A, Pure Appl. Opt. 6(6), 524–528 (2004).
[CrossRef]

Vargas, A.

I. Moreno, C. Iemmi, J. Campos, M. J. Yzuel, and A. Vargas, “Polarization vortices generation by diffraction from a four quadrant polarization mask,” Opt. Commun. 276(2), 222–230 (2007).
[CrossRef]

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51(14), 2031–2038 (2004).
[CrossRef]

Volke, A.

A. Volke and G. Heine, “Bringing order into light with structured polarizers,” Photonik Int. 2, 6–9 (2008).

Wang, H.-T.

Wang, X.-L.

Wilson, T.

Yzuel, M. J.

I. Moreno, C. Iemmi, J. Campos, M. J. Yzuel, and A. Vargas, “Polarization vortices generation by diffraction from a four quadrant polarization mask,” Opt. Commun. 276(2), 222–230 (2007).
[CrossRef]

I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Binary polarization pupil filter: Theoretical analysis and experimental realization with a liquid crystal display,” Opt. Commun. 264(1), 63–69 (2006).
[CrossRef]

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51(14), 2031–2038 (2004).
[CrossRef]

I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
[CrossRef]

Zhan, Q.

Adv. Opt. Photon. (1)

Appl. Opt. (3)

Appl. Phys. Lett. (1)

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Cincotti, “Polarization gratings: Design and applications,” IEEE J. Quantum Electron. 39(12), 1645–1652 (2003).
[CrossRef]

J. Mod. Opt. (1)

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51(14), 2031–2038 (2004).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

H.-Y. Tu, C.-J. Cheng, and M.-L. Chen, “Optical image encryption based on polarization encoding by liquid crystal spatial light modulators,” J. Opt. A, Pure Appl. Opt. 6(6), 524–528 (2004).
[CrossRef]

Jpn. J. Appl. Phys. (1)

I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995).
[CrossRef]

N. J. Phys. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007).
[CrossRef]

Opt. Commun. (2)

I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Binary polarization pupil filter: Theoretical analysis and experimental realization with a liquid crystal display,” Opt. Commun. 264(1), 63–69 (2006).
[CrossRef]

I. Moreno, C. Iemmi, J. Campos, M. J. Yzuel, and A. Vargas, “Polarization vortices generation by diffraction from a four quadrant polarization mask,” Opt. Commun. 276(2), 222–230 (2007).
[CrossRef]

Opt. Eng. (2)

J. L. Martínez, I. Moreno, and F. Mateos, “Hiding binary optical data with orthogonal circular polarizations,” Opt. Eng. 47(3), 030504 (2008).
[CrossRef]

B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39(9), 2439–2443 (2000).
[CrossRef]

Opt. Lett. (10)

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
[CrossRef]

F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24(9), 584–586 (1999).
[CrossRef]

M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25(17), 1291–1293 (2000).
[CrossRef]

J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26(9), 587–589 (2001).
[CrossRef]

M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27(21), 1929–1931 (2002).
[CrossRef]

M. Fratz, D. M. Giel, and P. Fischer, “Digital polarization holograms with defined magnitude and orientation of each pixel’s birefringence,” Opt. Lett. 34(8), 1270–1272 (2009).
[CrossRef] [PubMed]

K. C. Toussaint, S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. 30(21), 2846–2848 (2005).
[CrossRef] [PubMed]

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(24), 3549–3551 (2007).
[CrossRef] [PubMed]

C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33(20), 2287–2289 (2008).
[CrossRef] [PubMed]

Opt.Pura Apl. (1)

V. Ramírez-Sánchez and G. Piquero, “Global beam shaping with nonuniformly polarized beams using amplitude transmitances,” Opt.Pura Apl. 40, 87–93 (2007).

Photonik Int. (1)

A. Volke and G. Heine, “Bringing order into light with structured polarizers,” Photonik Int. 2, 6–9 (2008).

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Other (1)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge 2007).

Supplementary Material (2)

» Media 1: AVI (4871 KB)     
» Media 2: AVI (4871 KB)     

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Figures (5)

Fig. 1
Fig. 1

Scheme of the optical Fourier transform polarization processor.

Fig. 2
Fig. 2

Scheme of the optical Fourier transform polarization processor to generate polarization diffraction gratings. (a) Illumination with linearly polarized light oriented at 45° produces an output with a periodic map of linear polarizations with variable orientation. (b) Illumination with circularly polarized light produces an output with a periodic map of elliptical polarizations with fixed azimuth and variable ellipticity.

Fig. 3
Fig. 3

Experimental interference fringes at the output plane for (a)(Media 1) Illumination with linearly polarized light oriented at 45° and analyzer at 0°. (c) Illumination with right handed circularly polarized light and analyzer at 45°. (b)-(d) Evolution of the fringe pattern as a function of the orientation (θ) of the final analyzer corresponding to illumination with: (b)(Media 2) Linear polarization at 45° and (d) Circular right polarization.

Fig. 4
Fig. 4

Experimental arrangement for the production of arbitrary polarization map by means of a polarization Fourier processor. PBS is a polarizing beam splitter, QWP is a quarter wave plate oriented at 45°, L1 and L2 are converging lenses, and M a mirror.

Fig. 5
Fig. 5

(a) Desired polarization map. Experimental results obtained at the output plane for (b) absence of the analyzer and analyzer oriented at (c) 0°, (d) 45°, (e) 90° and (f) 135°.

Equations (30)

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f ( x , y ) = ( f x x ( x , y ) f x y ( x , y ) f y x ( x , y ) f y y ( x , y ) ) .
F ( u , v ) = ( F x x ( u , v ) F x y ( u , v ) F y x ( u , v ) F y y ( u , v ) ) ,
u = x λ f , v = y λ f ,
H ( u , v ) = ( H x x ( u , v ) H x y ( u , v ) H y x ( u , v ) H y y ( u , v ) ) .
h ( x , y ) = ( h x x ( x , y ) h x y ( x , y ) h y x ( x , y ) h y y ( x , y ) ) ,
h i j ( x , y ) = FT - 1 { H i j ( u , v ) } ,
H ( u , v ) F ( u , v ) = ( H x x ( u , v ) H x y ( u , v ) H y x ( u , v ) H y y ( u , v ) ) ( F x x ( u , v ) F x y ( u , v ) F y x ( u , v ) F y y ( u , v ) ) = = ( H x x F x x + H x y F y x H x x F x y + H x y F y y H y x F x x + H y y F y x H y x F x y + H y y F y y ) ,
m ( x , y ) = ( h x x f x x + h x y f y x h x x f x y + h x y f y y h y x f x x + h y y f y x h y x f x y + h y y f y y ) ,
m ( x , y ) = f ( x , y ) ¯ h ( x , y ) ,
V 1 ( u , v ) = F ( u , v ) V 0 , V 2 ( x , y ) = m ( x , y ) V 0 .
F ( u , v ) = 1 2 ( δ ( u a , v ) + δ ( u + a , v ) ) 1 .
H ( u , v ) = P ( u a , v ) Q W P θ = 90 º + P ( u + a , v ) Q W P θ = 0 º = = P ( u , v ) { δ ( u a , v ) ( + i 0 0 1 ) + δ ( u + a , v )     ( 1 0 0 + i ) } = = P ( u , v ) ( i δ ( u a , v ) + δ ( u + a , v ) 0 0 δ ( u a , v ) + i δ ( u + a , v ) ) ,
Q W P θ = R ( θ ) ( 1 0 0 + i ) R ( + θ ) ,
h ( x , y ) = p ( x , y ) ( i e + i π a x + e i π a x 0 0 e + i π a x + i e i π a x ) = = p ( x , y ) 2 e i π 4 ( cos ( π a x + π 4 ) 0 0 sin ( π a x + π 4 ) ) ,
H ( u , v ) F ( u , v ) = 1 2 ( i δ ( u a , v ) + δ ( u + a , v ) 0 0 δ ( u a , v ) + i δ ( u + a , v ) ) .
m ( x , y ) = e i π 4 ( cos ( π a x + π 4 ) 0 0 sin ( π a x + π 4 ) ) .
V 0 = 1 2 ( 1 1 ) .
V 2 ( x , y ) = m ( x , y ) V 0 = e i π 4 2 ( cos ( π a x + π 4 ) sin ( π a x + π 4 ) ) .
V 0 = 1 2 ( 1 + i ) ,
V 2 ( x , y ) = m ( x , y ) V 0 = e i π 4 2 ( cos ( π a x + π 4 ) + i sin ( π a x + π 4 ) ) .
V 2 ( x , y ) = ( cos ( α ( x , y ) ) sin ( α ( x , y ) ) ) ,
f ( x , y ) = exp { i [ α ( x , y ) + π a x ] } .
f ( x , y ) = f B I N ( x , y ) 1 .
F ( u , v ) { A ( u a , v ) + A * ( u a , v ) } 1 .
H ( u , v ) F ( u , v ) = A ( u a , v ) Q P W 45 P 0 + A * ( u a , v ) Q P W 45 P 90 .
P 0 = ( 1 0 0 0 ) , P 90 = ( 0 0 0 1 ) ,
Q W P 45 = R ( 45 º ) ( 1 0 0 + i ) R ( + 45 º ) = e i π 4 2 ( 1 i i 1 ) .
H ( u , v ) F ( u , v ) = e i π 4 2 ( A ( u a , v ) i A * ( u a , v ) i A ( u a , v ) A * ( u a , v ) ) .
m ( x , y ) = e i π 4 2 ( e i [ α ( x , y ) + π a x ] i e i [ α ( x , y ) + π a x ] i e i [ α ( x , y ) + π a x ] e i [ α ( x , y ) + π a x ] ) .
V 2 ( x , y ) = ( cos { α ( x , y ) + π a x + π 4 } sin { α ( x , y ) + π a x + π 4 } ) ,

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