Abstract

A design for a single-plane multiple speckle pattern phase retrieval technique using a deformable mirror (DM) is analyzed within the formalism of complex ABCD-matrices, facilitating its use in conjunction with dynamic wavefronts. The variable focal length DM positioned at a Fourier plane of a lens comprises the adaptive optical (AO) system that replaces the time-consuming axial displacements in the conventional free-space multiple plane setup. Compared with a spatial light modulator, a DM has a smooth continuous surface which avoids pixelation, pixel cross-talk and non-planarity issues. The calculated distances for the proposed AO-system are evaluated experimentally using the conventional free-space phase retrieval setup. Two distance ranges are investigated depending on whether the measurement planes satisfy the Nyquist detector sampling condition or not. It is shown numerically and experimentally that speckle patterns measured at the non-Nyquist range still yield good reconstructions. A DM with a surface height of 25 microns and an aperture diameter of 5.2 mm may be used to reconstruct spherical phase patterns with 50-micron fringe spacing.

© 2010 OSA

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References

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  26. http://en.wikipedia.org/wiki/Spread_spectrum

2010

2009

2008

2007

K. A. Nugent, “X-ray noninterferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A 24(2), 536–547 (2007).
[CrossRef]

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75(4), 043805 (2007).
[CrossRef]

2005

2004

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004).
[CrossRef]

S. Yang and H. Takajo, “Quantization error reduction in the measurement of Fourier intensity for phase retrieval,” Jpn. J. Appl. Phys. 43(No. 8B), 5747–5751 (2004).
[CrossRef]

2003

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, “Three-dimensional imaging of microstructure in gold nanocrystals,” Phys. Rev. Lett. 90(17), 175501 (2003).
[CrossRef] [PubMed]

1993

1987

1984

E. Kirkland, “Improved high resolution image processing of bright field electron micrographs: I. Theory,” Ultramicroscopy 15(3), 151–172 (1984).
[CrossRef]

1980

W. O. Saxton, “Correction of artefacts in linear and nonlinear high resolution electron micrographs,” J. Microsc. Spectrosc. Electron. 5, 661–670 (1980).

1973

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D Appl. Phys. 6(18), 2200–2216 (1973).
[CrossRef]

1972

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Agour, M.

Almoro, P. F.

P. F. Almoro, G. Pedrini, P. N. Gundu, W. Osten, and S. G. Hanson, “Phase microscopy of technical and biological samples through random phase modulation with a diffuser,” Opt. Lett. 35(7), 1028–1030 (2010).
[CrossRef] [PubMed]

P. F. Almoro, P. N. Gundu, and S. G. Hanson, “Numerical correction of aberrations via phase retrieval with speckle illumination,” Opt. Lett. 34(4), 521–523 (2009).
[CrossRef] [PubMed]

P. F. Almoro, G. Pedrini, A. Anand, W. Osten, and S. G. Hanson, “Interferometric evaluation of angular displacements using phase retrieval,” Opt. Lett. 33(18), 2041–2043 (2008).
[CrossRef] [PubMed]

P. F. Almoro, G. Pedrini, F. Zhang, A. M. S. Maallo, A. Anand, P. N. Gundu, W. Wang, A. Asundi, W. Osten, and S. G. Hanson, “Fault-tolerant characterization of phase objects using a speckle-based phase retrieval technique,” Int. J. Optomech. (to be published).

A. M. S. Maallo, P. F. Almoro, and S. G. Hanson, “Quantization analysis of speckle intensity measurements for phase retrieval,” Appl. Opt. (to be published).
[PubMed]

Anand, A.

P. F. Almoro, G. Pedrini, A. Anand, W. Osten, and S. G. Hanson, “Interferometric evaluation of angular displacements using phase retrieval,” Opt. Lett. 33(18), 2041–2043 (2008).
[CrossRef] [PubMed]

P. F. Almoro, G. Pedrini, F. Zhang, A. M. S. Maallo, A. Anand, P. N. Gundu, W. Wang, A. Asundi, W. Osten, and S. G. Hanson, “Fault-tolerant characterization of phase objects using a speckle-based phase retrieval technique,” Int. J. Optomech. (to be published).

Andilla, J.

Asundi, A.

P. F. Almoro, G. Pedrini, F. Zhang, A. M. S. Maallo, A. Anand, P. N. Gundu, W. Wang, A. Asundi, W. Osten, and S. G. Hanson, “Fault-tolerant characterization of phase objects using a speckle-based phase retrieval technique,” Int. J. Optomech. (to be published).

Bergmann, R. B.

Brady, G. R.

Camacho, L.

Falldorf, C.

Faulkner, H. M. L.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004).
[CrossRef]

Fienup, J. R.

García, J.

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 (Stuttg.) 35, 237–246 (1972).

Guizar-Sicairos, M.

Gundu, P. N.

Hanson, S. G.

Ichihashi, Y.

Ito, T.

Kirkland, E.

E. Kirkland, “Improved high resolution image processing of bright field electron micrographs: I. Theory,” Ultramicroscopy 15(3), 151–172 (1984).
[CrossRef]

Kohler, C.

López-Quesada, C.

Maallo, A. M. S.

A. M. S. Maallo, P. F. Almoro, and S. G. Hanson, “Quantization analysis of speckle intensity measurements for phase retrieval,” Appl. Opt. (to be published).
[PubMed]

P. F. Almoro, G. Pedrini, F. Zhang, A. M. S. Maallo, A. Anand, P. N. Gundu, W. Wang, A. Asundi, W. Osten, and S. G. Hanson, “Fault-tolerant characterization of phase objects using a speckle-based phase retrieval technique,” Int. J. Optomech. (to be published).

Marron, J. C.

Martín-Badosa, E.

Masuda, N.

Micó, V.

Misell, D. L.

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D Appl. Phys. 6(18), 2200–2216 (1973).
[CrossRef]

Nakayama, H.

Nugent, K. A.

Osten, W.

Pedrini, G.

P. F. Almoro, G. Pedrini, P. N. Gundu, W. Osten, and S. G. Hanson, “Phase microscopy of technical and biological samples through random phase modulation with a diffuser,” Opt. Lett. 35(7), 1028–1030 (2010).
[CrossRef] [PubMed]

P. F. Almoro, G. Pedrini, A. Anand, W. Osten, and S. G. Hanson, “Interferometric evaluation of angular displacements using phase retrieval,” Opt. Lett. 33(18), 2041–2043 (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(4), 043805 (2007).
[CrossRef]

G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30(8), 833–835 (2005).
[CrossRef] [PubMed]

P. F. Almoro, G. Pedrini, F. Zhang, A. M. S. Maallo, A. Anand, P. N. Gundu, W. Wang, A. Asundi, W. Osten, and S. G. Hanson, “Fault-tolerant characterization of phase objects using a speckle-based phase retrieval technique,” Int. J. Optomech. (to be published).

Pfeifer, M. A.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, “Three-dimensional imaging of microstructure in gold nanocrystals,” Phys. Rev. Lett. 90(17), 175501 (2003).
[CrossRef] [PubMed]

Robinson, I. K.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, “Three-dimensional imaging of microstructure in gold nanocrystals,” Phys. Rev. Lett. 90(17), 175501 (2003).
[CrossRef] [PubMed]

Rodenburg, J. M.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004).
[CrossRef]

Saxton, W. O.

W. O. Saxton, “Correction of artefacts in linear and nonlinear high resolution electron micrographs,” J. Microsc. Spectrosc. Electron. 5, 661–670 (1980).

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Schulz, T. J.

Seldin, J. H.

Shimobaba, T.

Shiraki, A.

Sugie, T.

Takajo, H.

S. Yang and H. Takajo, “Quantization error reduction in the measurement of Fourier intensity for phase retrieval,” Jpn. J. Appl. Phys. 43(No. 8B), 5747–5751 (2004).
[CrossRef]

v. Kopylow, C.

Vartanyants, I. A.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, “Three-dimensional imaging of microstructure in gold nanocrystals,” Phys. Rev. Lett. 90(17), 175501 (2003).
[CrossRef] [PubMed]

Wang, W.

P. F. Almoro, G. Pedrini, F. Zhang, A. M. S. Maallo, A. Anand, P. N. Gundu, W. Wang, A. Asundi, W. Osten, and S. G. Hanson, “Fault-tolerant characterization of phase objects using a speckle-based phase retrieval technique,” Int. J. Optomech. (to be published).

Williams, G. J.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, “Three-dimensional imaging of microstructure in gold nanocrystals,” Phys. Rev. Lett. 90(17), 175501 (2003).
[CrossRef] [PubMed]

Yang, S.

S. Yang and H. Takajo, “Quantization error reduction in the measurement of Fourier intensity for phase retrieval,” Jpn. J. Appl. Phys. 43(No. 8B), 5747–5751 (2004).
[CrossRef]

Yura, H. T.

Zalevsky, Z.

Zhang, F.

C. Kohler, F. Zhang, and W. Osten, “Characterization of a spatial light modulator and its application in phase retrieval,” Appl. Opt. 48(20), 4003–4008 (2009).
[CrossRef] [PubMed]

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75(4), 043805 (2007).
[CrossRef]

P. F. Almoro, G. Pedrini, F. Zhang, A. M. S. Maallo, A. Anand, P. N. Gundu, W. Wang, A. Asundi, W. Osten, and S. G. Hanson, “Fault-tolerant characterization of phase objects using a speckle-based phase retrieval technique,” Int. J. Optomech. (to be published).

Zhang, Y.

Appl. Opt.

Appl. Phys. Lett.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004).
[CrossRef]

Int. J. Optomech.

P. F. Almoro, G. Pedrini, F. Zhang, A. M. S. Maallo, A. Anand, P. N. Gundu, W. Wang, A. Asundi, W. Osten, and S. G. Hanson, “Fault-tolerant characterization of phase objects using a speckle-based phase retrieval technique,” Int. J. Optomech. (to be published).

J. Microsc. Spectrosc. Electron.

W. O. Saxton, “Correction of artefacts in linear and nonlinear high resolution electron micrographs,” J. Microsc. Spectrosc. Electron. 5, 661–670 (1980).

J. Opt. Soc. Am. A

J. Phys. D Appl. Phys.

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D Appl. Phys. 6(18), 2200–2216 (1973).
[CrossRef]

Jpn. J. Appl. Phys.

S. Yang and H. Takajo, “Quantization error reduction in the measurement of Fourier intensity for phase retrieval,” Jpn. J. Appl. Phys. 43(No. 8B), 5747–5751 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Stuttg.)

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Phys. Rev. A

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75(4), 043805 (2007).
[CrossRef]

Phys. Rev. Lett.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, “Three-dimensional imaging of microstructure in gold nanocrystals,” Phys. Rev. Lett. 90(17), 175501 (2003).
[CrossRef] [PubMed]

Ultramicroscopy

E. Kirkland, “Improved high resolution image processing of bright field electron micrographs: I. Theory,” Ultramicroscopy 15(3), 151–172 (1984).
[CrossRef]

Other

N. Loomis, L. Waller, and G. Barbastathis, “High-Speed Phase Recovery Using Chromatic Transport of Intensity Computation in Graphics Processing Units,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2010), paper JMA7.

J. Glückstad, and D. Palima, “Generalised Phase Contrast: Applications in Optics and Photonics,” Springer Series in Optical Sciences, Vol. 146, 310 pp (2009).

http://en.wikipedia.org/wiki/Spread_spectrum

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Figures (4)

Fig. 1
Fig. 1

(a) Setup for the conventional free-space multiple plane phase retrieval system. (b) Proposed setup based on an Adaptive Optical (AO) system using a deformable mirror.

Fig. 2
Fig. 2

Effects of recording the speckle field patterns at two measurement regions relative to the Nyquist sampling distance.

Fig. 3
Fig. 3

Numerical simulations. (a) Initial phase at object plane. (b) and (c) are the reconstructed phase maps using intensity patterns recorded at Nyquist and non-Nyquist distances, respectively.

Fig. 4
Fig. 4

Experimental results using conventional free-space setup. (a) and (b) are the reconstructed phase maps using speckle patterns recorded at Nyquist and non-Nyquist distances, respectively. The result in (b) demonstrates successful phase reconstruction of the test wavefront despite violations of the Nyquist condition during the speckle recording. This gives flexibility to the requirement of the surface height in the DM’s in the proposed AO-system.

Tables (1)

Tables Icon

Table 1 Parameters in the Green’s function

Equations (9)

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

U o u t , x , y ( r ) = d r ' U i n , x , y ( r ' ) G ( r ' , r ) ,
G ( r ' , r ) = i k 2 π B exp [ i k 2 B ( A r ' 2 2 r ' r + D r 2 ) ] .
M f r e e s p a c e = { 1 2 i z k s 0 2 z 2 i z k s 0 2 1 }
M A O = ( f 2 f 1 + 4 f 1 f 2 2 d f + 2 i f 1 ( d f d z + f 2 2 ) k s 1 2 f 2 k 2 s 0 2 s 1 2 d f d z f 1 f 2 + f 1 f 2 ( 1 d f 2 i k s 1 2 ) 2 i f f 2 k s 0 2 f 1 f 2 )
G ( r ' , r ) exp [ i k Δ f 2 ( Δ f Δ z + f 2 2 ) ( f 2 f 1 r ' r ) 2 ] .
G ( r ' , r ) exp [ i k 2 L ( r ' r ) 2 ] .
L e f f = Δ f Δ z + f 2 2 Δ f .
Δ L e f f Δ f Δ z + f 2 2 Δ f ( Δ f Δ z + f 2 2 Δ f ) for  Δ f = f 2 2 Δ f .
L e f f = 2 h f 2 / ( D / 2 ) 2

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