Abstract

Recently a phase retrieval method using a movable phase plate as modulator has been proposed [Phys. Rev. A 75, 043805 (2007)]. This method is applicable to general complex-valued fields and exhibits rapid convergence and high robustness to noise. In this paper, we demonstrate how to use this technique to characterize the phase shifting properties of a liquid-crystal modulator, and in turn we use the characterized modulator as the modulation device in the presented phase retrieval method. The adoption of a dynamic modulator gives a much more robust and flexible setup.

© 2009 Optical Society of America

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References

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  1. R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Jena) 34, 275-284 (1971).
  2. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 227-246 (1972).
  3. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
    [CrossRef]
  4. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
    [CrossRef] [PubMed]
  5. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  6. D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D: Appl. Phys. 6, L6-L9 (1973).
    [CrossRef]
  7. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795-4797 (2004).
    [CrossRef]
  8. P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval using multiple illumination wavelengths,” Opt. Lett. 33, 309-311 (2008).
    [CrossRef] [PubMed]
  9. K. A. Nugent, G. Peele, H. M. Quiney, and H. N. Chapman, “Diffraction with wavefront curvature a path to unique phase recovery,” Acta Crystallogr. A 61, 373-381 (2005).
    [CrossRef] [PubMed]
  10. F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
    [CrossRef]
  11. R. C. Dixon, Spread Spectrum Systems (Wiley, 1984),http://en.wikipedia.org/wiki/Spread_spectrum.
  12. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668-1670 (2004).
    [CrossRef] [PubMed]
  13. “Phase only LCOS spatial light modulator,” http://www.holoeye.com/.
  14. C. Kohler, X. Schwab, and W. Osten, “Optimally tuned spatial light modulators for digital holography,” Appl. Opt. 45, 960-967 (2006).
    [CrossRef] [PubMed]
  15. G. Bader, R. Bürkle, E. Lueder, N. Fruehauf, and C. Zeile, “Fast and accurate techniques for measuring the complex transmittance of liquid crystal valves,” Proc. SPIE 3015, 93-104 (1997).
    [CrossRef]
  16. P. Hariharan, S. Mujumdar, and H. Ramachandran, “A simple demonstration of the Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).

2008 (1)

2007 (1)

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

2006 (1)

2005 (1)

K. A. Nugent, G. Peele, H. M. Quiney, and H. N. Chapman, “Diffraction with wavefront curvature a path to unique phase recovery,” Acta Crystallogr. A 61, 373-381 (2005).
[CrossRef] [PubMed]

2004 (2)

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668-1670 (2004).
[CrossRef] [PubMed]

2003 (1)

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

1999 (2)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

P. Hariharan, S. Mujumdar, and H. Ramachandran, “A simple demonstration of the Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).

1997 (1)

G. Bader, R. Bürkle, E. Lueder, N. Fruehauf, and C. Zeile, “Fast and accurate techniques for measuring the complex transmittance of liquid crystal valves,” Proc. SPIE 3015, 93-104 (1997).
[CrossRef]

1982 (1)

1973 (1)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D: Appl. Phys. 6, L6-L9 (1973).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 227-246 (1972).

1971 (1)

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Jena) 34, 275-284 (1971).

Bader, G.

G. Bader, R. Bürkle, E. Lueder, N. Fruehauf, and C. Zeile, “Fast and accurate techniques for measuring the complex transmittance of liquid crystal valves,” Proc. SPIE 3015, 93-104 (1997).
[CrossRef]

Bao, P.

Bürkle, R.

G. Bader, R. Bürkle, E. Lueder, N. Fruehauf, and C. Zeile, “Fast and accurate techniques for measuring the complex transmittance of liquid crystal valves,” Proc. SPIE 3015, 93-104 (1997).
[CrossRef]

Chapman, H. N.

K. A. Nugent, G. Peele, H. M. Quiney, and H. N. Chapman, “Diffraction with wavefront curvature a path to unique phase recovery,” Acta Crystallogr. A 61, 373-381 (2005).
[CrossRef] [PubMed]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

Dixon, R. C.

R. C. Dixon, Spread Spectrum Systems (Wiley, 1984),http://en.wikipedia.org/wiki/Spread_spectrum.

Faulkner, H. M. L.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

Fienup, J. R.

Fruehauf, N.

G. Bader, R. Bürkle, E. Lueder, N. Fruehauf, and C. Zeile, “Fast and accurate techniques for measuring the complex transmittance of liquid crystal valves,” Proc. SPIE 3015, 93-104 (1997).
[CrossRef]

Gao, M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

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 (Jena) 35, 227-246 (1972).

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Jena) 34, 275-284 (1971).

Hariharan, P.

P. Hariharan, S. Mujumdar, and H. Ramachandran, “A simple demonstration of the Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

Kohler, C.

Lueder, E.

G. Bader, R. Bürkle, E. Lueder, N. Fruehauf, and C. Zeile, “Fast and accurate techniques for measuring the complex transmittance of liquid crystal valves,” Proc. SPIE 3015, 93-104 (1997).
[CrossRef]

Miao, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

Misell, D. L.

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D: Appl. Phys. 6, L6-L9 (1973).
[CrossRef]

Mujumdar, S.

P. Hariharan, S. Mujumdar, and H. Ramachandran, “A simple demonstration of the Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).

Nagahara, L. A.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Nugent, K. A.

K. A. Nugent, G. Peele, H. M. Quiney, and H. N. Chapman, “Diffraction with wavefront curvature a path to unique phase recovery,” Acta Crystallogr. A 61, 373-381 (2005).
[CrossRef] [PubMed]

Osten, W.

Pedrini, G.

P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval using multiple illumination wavelengths,” Opt. Lett. 33, 309-311 (2008).
[CrossRef] [PubMed]

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Peele, G.

K. A. Nugent, G. Peele, H. M. Quiney, and H. N. Chapman, “Diffraction with wavefront curvature a path to unique phase recovery,” Acta Crystallogr. A 61, 373-381 (2005).
[CrossRef] [PubMed]

Quiney, H. M.

K. A. Nugent, G. Peele, H. M. Quiney, and H. N. Chapman, “Diffraction with wavefront curvature a path to unique phase recovery,” Acta Crystallogr. A 61, 373-381 (2005).
[CrossRef] [PubMed]

Ramachandran, H.

P. Hariharan, S. Mujumdar, and H. Ramachandran, “A simple demonstration of the Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).

Rodenburg, J. M.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

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 (Jena) 35, 227-246 (1972).

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Jena) 34, 275-284 (1971).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

Schwab, X.

Vartanyants, I.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Yamaguchi, I.

Yaroslavsky, L. P.

Zeile, C.

G. Bader, R. Bürkle, E. Lueder, N. Fruehauf, and C. Zeile, “Fast and accurate techniques for measuring the complex transmittance of liquid crystal valves,” Proc. SPIE 3015, 93-104 (1997).
[CrossRef]

Zhang, F.

Zhang, R.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Zuo, J. M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Acta Crystallogr. A (1)

K. A. Nugent, G. Peele, H. M. Quiney, and H. N. Chapman, “Diffraction with wavefront curvature a path to unique phase recovery,” Acta Crystallogr. A 61, 373-381 (2005).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

J. Mod. Opt. (1)

P. Hariharan, S. Mujumdar, and H. Ramachandran, “A simple demonstration of the Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).

J. Phys. D: Appl. Phys. (1)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D: Appl. Phys. 6, L6-L9 (1973).
[CrossRef]

Nature (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342-344 (1999).
[CrossRef]

Opt. Lett. (2)

Optik (Jena) (2)

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Jena) 34, 275-284 (1971).

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 227-246 (1972).

Phys. Rev. A (1)

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Proc. SPIE (1)

G. Bader, R. Bürkle, E. Lueder, N. Fruehauf, and C. Zeile, “Fast and accurate techniques for measuring the complex transmittance of liquid crystal valves,” Proc. SPIE 3015, 93-104 (1997).
[CrossRef]

Science (1)

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Other (2)

R. C. Dixon, Spread Spectrum Systems (Wiley, 1984),http://en.wikipedia.org/wiki/Spread_spectrum.

“Phase only LCOS spatial light modulator,” http://www.holoeye.com/.

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Figures (7)

Fig. 1
Fig. 1

Scheme of the experimental setup used with the SSPR algorithm: z is the distance between the object plane U obj and the modulator plane U, and d is the distance between the modulator plane and the detector plane V.

Fig. 2
Fig. 2

Experimental setup for the characterization of the modulation property of reflective LCDs: BS, beam splitter; Pol, polarizer.

Fig. 3
Fig. 3

Phase shift of the LCoS display measured with the double slit setup described in [14].

Fig. 4
Fig. 4

Retrieved phase of a blazed grating with a period of 12 LCoS pixels obtained by the SSPR method.

Fig. 5
Fig. 5

Retrieved phase of (a) blazed grating and (b) spot array written into the LCoS display obtained by the SSPR method.

Fig. 6
Fig. 6

Schematic experimental setup of the SSPR method implemented with a reflective LCD modulator.

Fig. 7
Fig. 7

Retrieved (a) amplitude and (b) phase of a mount plate illuminated by a plane wave.

Tables (1)

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Table 1 Error Sources in Phase Retrieval and Their Resulting Phase Errors in the Recovered Wave Field

Equations (2)

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d = N Δ X Δ x / λ ,
Δ ξ = 1.22 λ z / S w s 1.22 Δ x z / d ,

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