Abstract

In this work, we propose a robust and versatile approach for the characterization of the complex field amplitude of holographically generated coherent-scalar paraxial beams. For this purpose we apply an iterative algorithm that allows recovering the phase of the generated beam from the measurement of its Wigner distribution projections. Its performance is analyzed for beams of different symmetry: Laguerre-Gaussian, Hermite-Gaussian and spiral ones, which are obtained experimentally by a computer generated hologram (CGH) implemented on a programmable spatial light modulator (SLM). Using the same method we also study the quality of their holographic recording on a highly efficient photopolymerizable glass. The proposed approach is useful for the creation of adaptive CGH that takes into account the peculiarities of the SLM, as well as for the quality control of the holographic data storage.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. E. Siegman, Lasers (University Science Books, 1986).
  2. F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser Beam Shaping Applications (CRC Press, 2005).
    [CrossRef]
  3. A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers: A Reprint Volume With Commentaries (World Scientific Publishing Company, 2006).
    [CrossRef] [PubMed]
  4. J. P. Kirk, and A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1023–1028 (1971).
    [CrossRef]
  5. 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]
  6. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).
    [CrossRef]
  7. B. Hennelly, J. Ojeda-Castañeda, and M. Testorf, eds., Phase Space Optics: Fundamentals and Applications (McGraw-Hill, 2009).
  8. T. Alieva, and J. A. Rodrigo, “Iterative phase retrieval from Wigner Distribution Projections,” in “Signal Recovery and Synthesis,” (Optical Society of America, 2009), p. STuD2.
  9. A. Cámara, T. Alieva, J. A. Rodrigo, and M. L. Calvo, “Phase space tomography reconstruction of the Wigner distribution for optical beams separable in Cartesian coordinates,” J. Opt. Soc. Am. A 26, 1301–1306 (2009).
    [CrossRef]
  10. M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).
  11. J. Otón, P. Ambs, M. S. Millán, and E. Pérez-Cabré, “Multipoint phase calibration for improved compensation of inherent wavefront distortion in parallel aligned liquid crystal on silicon displays,” Appl. Opt. 46, 5667–5679 (2007).
    [CrossRef] [PubMed]
  12. J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Aberration production using a high-resolution liquid-crystal spatial light modulator,” Appl. Opt. 46, 2423–2433 (2007).
    [CrossRef] [PubMed]
  13. A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express 15, 5801–5808 (2007).
    [CrossRef] [PubMed]
  14. C. López-Quesada, J. Andilla, and E. Martín-Badosa, “Correction of aberration in holographic optical tweezers using a Shack-Hartmann sensor,” Appl. Opt. 48, 1084–1090 (2009).
    [CrossRef]
  15. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
    [CrossRef] [PubMed]
  16. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [CrossRef] [PubMed]
  17. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
    [CrossRef]
  18. Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
    [CrossRef]
  19. E. G. Abramochkin, and V. G. Volostnikov, “Spiral light beams,” Sov. Phys. Usp. 47, 1177 (2004).
  20. T. Alieva, E. Abramochkin, A. Asenjo-Garcia, and E. Razueva, “Rotating beams in isotropic optical system,” Opt. Express 18, 3568–3573 (2010).
    [CrossRef] [PubMed]
  21. J. P. Guigay, “Fourier-transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).
  22. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  23. T. E. Gureyev, and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
    [CrossRef]
  24. U. Gopinathan, G. Situ, T. J. Naughton, and J. T. Sheridan, “Noninterferometric phase retrieval using a fractional Fourier system,” J. Opt. Soc. Am. A 25, 108–115 (2008).
    [CrossRef]
  25. R. W. Gerchberg, and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  26. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef] [PubMed]
  27. M. Nieto-Vesperinas, R. Navarro, and F. J. Fuentes, “Performance of a simulated-annealing algorithm for phase retrieval,” J. Opt. Soc. Am. A 5, 30–38 (1988).
    [CrossRef]
  28. J. A. Rodrigo, H. Duadi, T. Alieva, and Z. Zalevsky, “Multi-stage phase retrieval algorithm based upon the gyrator transform,” Opt. Express 18, 1510–1520 (2010).
    [CrossRef] [PubMed]
  29. D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407 (1997).
    [CrossRef]
  30. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. 31, 649–651 (2006).
    [CrossRef] [PubMed]
  31. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25, 1642–1651 (2008).
    [CrossRef]
  32. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett. 34, 34–36 (2009).
    [CrossRef]
  33. A. Lizana, A. Márquez, L. Lobato, Y. Rodange, I. Moreno, C. Iemmi, and J. Campos, “The minimum euclidean distance principle applied to improve the modulation diffraction efficiency in digitally controlled spatial light modulators,” Opt. Express 18, 10581–10593 (2010).
    [CrossRef] [PubMed]
  34. I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express 16, 16711–16722 (2008).
    [CrossRef] [PubMed]
  35. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  36. V. Arrizón, G. Méndez, and D. S. de La-Llave, “Accurate encoding of arbitrary complex fields with amplitude-only liquid crystal spatial light modulators,” Opt. Express 13, 7913–7927 (2005).
    [CrossRef] [PubMed]
  37. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 231–1237 (1998).
    [CrossRef]
  38. F. del Monte, O. Martínez, J. Rodrigo, M. Calvo, and P. Cheben, “A volume holographic sol-gel material with large enhancement of dynamic range by incorporation of high refractive index species,” Adv. Mater. 18, 2014–2017 (2006).
    [CrossRef]
  39. O. Martínez-Matos, J. A. Rodrigo, M. P. Hernández-Garay, J. G. Izquierdo, R. Weigand, M. L. Calvo, P. Cheben, P. Vaveliuk, and L. Bañares, “Generation of femtosecond paraxial beams with arbitrary spatial distribution,” Opt. Lett. 35, 652–654 (2010).
    [CrossRef] [PubMed]

2010

2009

2008

2007

2006

J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. 31, 649–651 (2006).
[CrossRef] [PubMed]

F. del Monte, O. Martínez, J. Rodrigo, M. Calvo, and P. Cheben, “A volume holographic sol-gel material with large enhancement of dynamic range by incorporation of high refractive index species,” Adv. Mater. 18, 2014–2017 (2006).
[CrossRef]

2005

2004

E. G. Abramochkin, and V. G. Volostnikov, “Spiral light beams,” Sov. Phys. Usp. 47, 1177 (2004).

2001

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

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

1999

1998

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 231–1237 (1998).
[CrossRef]

1997

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407 (1997).
[CrossRef]

T. E. Gureyev, and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

1996

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

1988

1983

1977

J. P. Guigay, “Fourier-transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

1976

1972

R. W. Gerchberg, and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1971

Abramochkin, E.

Abramochkin, E. G.

E. G. Abramochkin, and V. G. Volostnikov, “Spiral light beams,” Sov. Phys. Usp. 47, 1177 (2004).

Alieva, T.

Allen, L.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 231–1237 (1998).
[CrossRef]

Ambs, P.

Andilla, J.

Ando, T.

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 231–1237 (1998).
[CrossRef]

Arrizón, V.

Asenjo-Garcia, A.

Bañares, L.

Bandres, M. A.

Bentley, J. B.

Bernet, S.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

Calvo, M.

F. del Monte, O. Martínez, J. Rodrigo, M. Calvo, and P. Cheben, “A volume holographic sol-gel material with large enhancement of dynamic range by incorporation of high refractive index species,” Adv. Mater. 18, 2014–2017 (2006).
[CrossRef]

Calvo, M. L.

Cámara, A.

Campos, J.

Carrada, R.

Cheben, P.

O. Martínez-Matos, J. A. Rodrigo, M. P. Hernández-Garay, J. G. Izquierdo, R. Weigand, M. L. Calvo, P. Cheben, P. Vaveliuk, and L. Bañares, “Generation of femtosecond paraxial beams with arbitrary spatial distribution,” Opt. Lett. 35, 652–654 (2010).
[CrossRef] [PubMed]

F. del Monte, O. Martínez, J. Rodrigo, M. Calvo, and P. Cheben, “A volume holographic sol-gel material with large enhancement of dynamic range by incorporation of high refractive index species,” Adv. Mater. 18, 2014–2017 (2006).
[CrossRef]

Cottrell, D. M.

Davis, J. A.

de La-Llave, D. S.

del Monte, F.

F. del Monte, O. Martínez, J. Rodrigo, M. Calvo, and P. Cheben, “A volume holographic sol-gel material with large enhancement of dynamic range by incorporation of high refractive index species,” Adv. Mater. 18, 2014–2017 (2006).
[CrossRef]

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 231–1237 (1998).
[CrossRef]

Dorsch, R. G.

Duadi, H.

Duncan, B. D.

Fernández, E.

Fuentes, F. J.

Fukuchi, N.

Fürhapter, S.

Gerchberg, R. W.

R. W. Gerchberg, and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goda, M. E.

González, L. A.

Gopinathan, U.

Guigay, J. P.

J. P. Guigay, “Fourier-transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Gureyev, T. E.

T. E. Gureyev, and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

Gutiérrez-Vega, J. C.

Hara, T.

Hernández-Garay, M. P.

Iemmi, C.

Inoue, T.

Izquierdo, J. G.

Jesacher, A.

Jones, A. L.

Kirk, J. P.

Konforti, N.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407 (1997).
[CrossRef]

Lizana, A.

Lobato, L.

López-Quesada, C.

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

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] [PubMed]

Márquez, A.

Martín-Badosa, E.

Martínez, O.

F. del Monte, O. Martínez, J. Rodrigo, M. Calvo, and P. Cheben, “A volume holographic sol-gel material with large enhancement of dynamic range by incorporation of high refractive index species,” Adv. Mater. 18, 2014–2017 (2006).
[CrossRef]

Martínez-Matos, O.

Matsumoto, N.

Maurer, C.

Méndez, G.

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407 (1997).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

Millán, M. S.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Moreno, I.

Naughton, T. J.

Navarro, R.

Nieto-Vesperinas, M.

Noll, R. J.

Nugent, K. A.

T. E. Gureyev, and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

Ohtake, Y.

Otón, J.

Padgett, M. J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 231–1237 (1998).
[CrossRef]

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

Pérez-Cabré, E.

Piestun, R.

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[CrossRef]

Razueva, E.

Ritsch-Marte, M.

Rodange, Y.

Rodrigo, J.

F. del Monte, O. Martínez, J. Rodrigo, M. Calvo, and P. Cheben, “A volume holographic sol-gel material with large enhancement of dynamic range by incorporation of high refractive index species,” Adv. Mater. 18, 2014–2017 (2006).
[CrossRef]

Rodrigo, J. A.

Ruiz, U.

Saxton, W. O.

R. W. Gerchberg, and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schechner, Y. Y.

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[CrossRef]

Schmidt, J. D.

Schwaighofer, A.

Shamir, J.

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[CrossRef]

Sheridan, J. T.

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

Situ, G.

Teague, M. R.

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Vaveliuk, P.

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] [PubMed]

Volostnikov, V. G.

E. G. Abramochkin, and V. G. Volostnikov, “Spiral light beams,” Sov. Phys. Usp. 47, 1177 (2004).

Weigand, R.

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] [PubMed]

Yzuel, M. J.

Zalevsky, Z.

Zeilinger, 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] [PubMed]

Adv. Mater.

F. del Monte, O. Martínez, J. Rodrigo, M. Calvo, and P. Cheben, “A volume holographic sol-gel material with large enhancement of dynamic range by incorporation of high refractive index species,” Adv. Mater. 18, 2014–2017 (2006).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407 (1997).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 231–1237 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nat. Phys.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Nature

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

Opt. Commun.

T. E. Gureyev, and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

R. W. Gerchberg, and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

J. P. Guigay, “Fourier-transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Phys. Rev. E

Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996).
[CrossRef]

Science

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp.

E. G. Abramochkin, and V. G. Volostnikov, “Spiral light beams,” Sov. Phys. Usp. 47, 1177 (2004).

Other

M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

A. E. Siegman, Lasers (University Science Books, 1986).

F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser Beam Shaping Applications (CRC Press, 2005).
[CrossRef]

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers: A Reprint Volume With Commentaries (World Scientific Publishing Company, 2006).
[CrossRef] [PubMed]

B. Hennelly, J. Ojeda-Castañeda, and M. Testorf, eds., Phase Space Optics: Fundamentals and Applications (McGraw-Hill, 2009).

T. Alieva, and J. A. Rodrigo, “Iterative phase retrieval from Wigner Distribution Projections,” in “Signal Recovery and Synthesis,” (Optical Society of America, 2009), p. STuD2.

Supplementary Material (1)

» Media 1: MOV (3198 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Flow chart of the proposed phase retrieval algorithm.

Fig. 2
Fig. 2

(a) Intensity and phase distributions of the LG 2 , 4 + beam. (b) Phase distribution retrieved after N × M = 96 iterations using several constraint images M. (c) Evolution of the RMS error as a function of the number of iterations and constraint images. RMS values after N × M = 96 are: 1.23 (M = 1), 0.94 (M = 2), 0.02 (M = 3), and 10−5 (M = 4).

Fig. 3
Fig. 3

(a) Retrieved phase associated to the LG 2 , 4 + and LG 2 , 4 beams, obtained using N × M = 96 iterations and M = 4 constraint images (RMS 10−5). (b)–(e) The four constraint images (Ej) corresponding to each LG beam, displayed at first and second rows respectively.

Fig. 4
Fig. 4

(a) Experimental setup for generating arbitrary paraxial beams. The SLM displays a CGH associated to the complex signal to be generated. BS is a cube beam splitter, L1 and L2 are spherical relay lenses (with a diameter of 5.1 cm) working as telescope 0.4×, focal lengths: f1 = 25 cm and f2 = 10 cm, in our case. Pinhole P is located at Fourier plane of L1 for spatial filtering. (b) SLM’s area for CGH addressing, imaged at plane z0 = 0. Retrieved phase ϕab(x, y) of this region before (c) and after correction (d), see Table 1.

Fig. 5
Fig. 5

Experimental results. Intensity of the LG 2 , 4 + beam generated by means of CGH–type 1 (a) and 2 (b). Fourier spectra of these beams are shown in (c) and (d), correspondingly. The HG6,2 generated by CGH–type 1 (e) and 2 (f) and their power spectra (g) and (h), respectively. The third row displays the power spectra profile, along x axis, corresponding to (c, d) and (g, h): theoretical profile (green line) and the measured ones associated to CGH–type 1 (blue line) and 2 (red line).

Fig. 6
Fig. 6

Intensity and phase distributions of the LG 2 , 4 + (a), LG 2 , 4 + + LG 2 , 4 (b), spiral beam LG 0 , 0 + + LG 0 , 3 + (c), and HG6,2 (d) mode, displayed at the first and second rows correspondingly. (e)–(h) Experimental results: intensity of the generated beam and its phase distribution reconstructed using M = 8 constraint images and N × M = 288 iterations.

Fig. 7
Fig. 7

Experimental results. Measured constraint images Ej (square root of intensity distribution) associated to LG 2 , 4 + mode.

Fig. 8
Fig. 8

Experimental results. (a) Holographically recorded LG 2 , 4 + beam onto a photopolymerizable glass and its power spectrum (b). (c) Retrieved phase of the reconstructed beam and its wavefront distortion (d). (e) Reconstruction of LG 3 , 4 + + LG 3 , 4 beam recorded on a photopolymerizable glass, see Media 1.

Tables (1)

Tables Icon

Table 1 Zernike Coefficients (waves at 532 nm) of Uncorrected and Corrected Wavefront

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

L j ( x , y ) = exp ( i π ( x cos θ + y sin θ ) 2 λ f j ) ,
f out ( r o ) = exp ( i 2 π z / λ ) i λ z f ( r i ) L j ( r i ) exp ( i π ( x o x i ) 2 + ( y o y i ) 2 λ z ) d r i ,
LG p , l ± ( r ; w ) = w 1 ( p ! ( p + l ) ! ) 1 / 2 ( 2 π x ± i y w ) l p l ( 2 π w 2 r 2 ) exp ( π w 2 r 2 ) ,
RMS ( k , j ) = ( | E 0 exp ( i ϕ ) E 0 exp ( i ϕ k , j ) | 2 d r ) 1 / 2 × ( | E 0 exp ( i ϕ ) | 2 d r ) 1 / 2 ,
f ( x , y ) = a ( x , y ) exp [ i ϕ ( x , y ) ] ,
H ( x , y ) = exp [ i ψ ( a , ϕ ) ] .
ψ ( a , ϕ ) = g ( a ) sin ϕ ,
ψ ( a , ϕ ) = g ( a ) ϕ ,
HG m , n ( r ; w ) = m ( 2 π x w ) n ( 2 π y w ) exp ( π w 2 r 2 ) ,
SNR = | f ( r ) | 2 d r × ( | f ( r ) β s e ( r ) | 2 d r ) 1 ,
β = { f ( r ) s e * ( r ) } d r × ( | s e ( r ) | 2 d r ) 1 ,
η = | f * ( r ) β s e ( r ) d r | 2 ,

Metrics