Abstract

The gyrator transform is a useful tool for optical information processing applications. In this work we propose a multi-stage phase retrieval approach based on this operation as well as on the well-known Gerchberg-Saxton algorithm. It results in an iterative algorithm able to retrieve the phase information using several measurements of the gyrator transform power spectrum. The viability and performance of the proposed algorithm is demonstrated by means of several numerical simulations and experimental results.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
    [CrossRef] [PubMed]
  2. E. N. Leith and J. Upatnieks, "Reconstructed wavefronts and communication theory," J. Opt. Soc. Am. 52, 1123-1128 (1962).
    [CrossRef]
  3. E. N. Leith and J. Upatnieks, "Wavefront reconstruction with continuous-tone objects," J. Opt. Soc. Am. 53, 1377-1381 (1963).
    [CrossRef]
  4. E. N. Leith and J. Upatnieks, "Wavefront reconstruction with diffused illumination and three-dimensional objects," J. Opt. Soc. Am. 54, 1295-1301 (1964).
    [CrossRef]
  5. J. P. Guigay, "Fourier-transrorm analysis of Fresnel diffraction patterns and in-line holograms," Optik 49, 121-125 (1977).
  6. M. R. Teague, "Deterministic phase retrieval: a Green’s function solution," J. Opt. Soc. Am. 73, 1434-1441 (1983).
    [CrossRef]
  7. T. E. Gureyev and K. A. Nugent, "Rapid quantitative phase imaging using the transport of intensity equation," Opt. Commun. 133, 339 - 346 (1997).
    [CrossRef]
  8. 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).
  9. 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]
  10. J. H. Seldin and J. R. Fienup, "Iterative blind deconvolution algorithm applied to phase retrieval," J. Opt. Soc. Am. A 7, 428-433 (1990).
    [CrossRef]
  11. J. R. Fienup, "Phase-retrieval algorithms for a complicated optical system," Appl. Opt. 32, 1737-1746 (1993).
    [CrossRef] [PubMed]
  12. D. L. Misell, "A method for the solution of the phase problem in electron microscopy," J. Phys. D: Applied Physics 6, L6-L9 (1973).
    [CrossRef]
  13. D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
    [CrossRef]
  14. D. S. Acton, P. D. Atcheson, M. Cermak, L. K. Kingsbury, F. Shi, and D. C. Redding, "James Webb Space Telescope wavefront sensing and control algorithms," Proc. SPIE 5487, 887 (2004).
    [CrossRef]
  15. G. R. Brady and J. R. Fienup, "Nonlinear optimization algorithm for retrieving the full complex pupil function," Opt. Express 14, 474-486 (2006).
    [CrossRef] [PubMed]
  16. B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, "Phase retrieval algorithm for JWST Flight and Testbed Telescope," Proc. SPIE 6265, 626511 (2006).
    [CrossRef]
  17. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, "Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain," Opt. Lett. 21, 842 (1996).
    [CrossRef] [PubMed]
  18. J. A. Rodrigo, T. Alieva, and M. L. Calvo, "Experimental implementation of the gyrator transform," J. Opt. Soc. Am. A 24, 3135-3139 (2007).
    [CrossRef]
  19. J. A. Rodrigo, T. Alieva, and M. L. Calvo, "Optical system design for orthosymplectic transformations in phase space," J. Opt. Soc. Am. A 23, 2494-2500 (2006).
    [CrossRef]

2007

2006

2004

D. S. Acton, P. D. Atcheson, M. Cermak, L. K. Kingsbury, F. Shi, and D. C. Redding, "James Webb Space Telescope wavefront sensing and control algorithms," Proc. SPIE 5487, 887 (2004).
[CrossRef]

1998

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

1997

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

1993

1990

1988

1983

1977

J. P. Guigay, "Fourier-transrorm analysis of Fresnel diffraction patterns and in-line holograms," Optik 49, 121-125 (1977).

1973

D. L. Misell, "A method for the solution of the phase problem in electron microscopy," J. Phys. D: Applied Physics 6, L6-L9 (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 35, 237-246 (1972).

1964

1963

1962

1948

D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Acton, D. S.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, "Phase retrieval algorithm for JWST Flight and Testbed Telescope," Proc. SPIE 6265, 626511 (2006).
[CrossRef]

D. S. Acton, P. D. Atcheson, M. Cermak, L. K. Kingsbury, F. Shi, and D. C. Redding, "James Webb Space Telescope wavefront sensing and control algorithms," Proc. SPIE 5487, 887 (2004).
[CrossRef]

Alieva, T.

Aronstein, D. L.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, "Phase retrieval algorithm for JWST Flight and Testbed Telescope," Proc. SPIE 6265, 626511 (2006).
[CrossRef]

Atcheson, P. D.

D. S. Acton, P. D. Atcheson, M. Cermak, L. K. Kingsbury, F. Shi, and D. C. Redding, "James Webb Space Telescope wavefront sensing and control algorithms," Proc. SPIE 5487, 887 (2004).
[CrossRef]

Basinger, S.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Bely, P.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Brady, G. R.

Burg, R.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Calvo, M. L.

Cermak, M.

D. S. Acton, P. D. Atcheson, M. Cermak, L. K. Kingsbury, F. Shi, and D. C. Redding, "James Webb Space Telescope wavefront sensing and control algorithms," Proc. SPIE 5487, 887 (2004).
[CrossRef]

Craig, L.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Dean, B. H.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, "Phase retrieval algorithm for JWST Flight and Testbed Telescope," Proc. SPIE 6265, 626511 (2006).
[CrossRef]

Dorsch, R. G.

Femiano, M.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Fienup, J. R.

Fuentes, F. J.

Gabor, D.

D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
[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 35, 237-246 (1972).

Guigay, J. P.

J. P. Guigay, "Fourier-transrorm 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]

Hadaway, J.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Jacobson, D.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Kingsbury, L. K.

D. S. Acton, P. D. Atcheson, M. Cermak, L. K. Kingsbury, F. Shi, and D. C. Redding, "James Webb Space Telescope wavefront sensing and control algorithms," Proc. SPIE 5487, 887 (2004).
[CrossRef]

Kissil, A.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Leith, E. N.

Lowman, A.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Lyon, R.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Mendlovic, D.

Misell, D. L.

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

Mosier, G.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Navarro, R.

Nieto-Vesperinas, M.

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]

Rakoczy, J.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Redding, D. C.

D. S. Acton, P. D. Atcheson, M. Cermak, L. K. Kingsbury, F. Shi, and D. C. Redding, "James Webb Space Telescope wavefront sensing and control algorithms," Proc. SPIE 5487, 887 (2004).
[CrossRef]

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Rodrigo, J. A.

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).

Schunk, R. G.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Seldin, J. H.

Shi, F.

D. S. Acton, P. D. Atcheson, M. Cermak, L. K. Kingsbury, F. Shi, and D. C. Redding, "James Webb Space Telescope wavefront sensing and control algorithms," Proc. SPIE 5487, 887 (2004).
[CrossRef]

Shiri, R.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, "Phase retrieval algorithm for JWST Flight and Testbed Telescope," Proc. SPIE 6265, 626511 (2006).
[CrossRef]

Smith, J. S.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, "Phase retrieval algorithm for JWST Flight and Testbed Telescope," Proc. SPIE 6265, 626511 (2006).
[CrossRef]

Teague, M. R.

Upatnieks, J.

Wilson, M.

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

Zalevsky, Z.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D: Applied Physics

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

Nature

D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
[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-transrorm analysis of Fresnel diffraction patterns and in-line holograms," Optik 49, 121-125 (1977).

Proc. SPIE

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, "Phase retrieval algorithm for JWST Flight and Testbed Telescope," Proc. SPIE 6265, 626511 (2006).
[CrossRef]

D. C. Redding, S. Basinger, A. Lowman, A. Kissil, P. Bely, R. Burg, R. Lyon, G. Mosier,M. Femiano, M. Wilson, R. G. Schunk, L. Craig, D. Jacobson, J. Rakoczy, and J. Hadaway, "Wavefront Sensing and Control for a Next-Generation Space Telescope," Proc. SPIE 3356, 758 (1998).
[CrossRef]

D. S. Acton, P. D. Atcheson, M. Cermak, L. K. Kingsbury, F. Shi, and D. C. Redding, "James Webb Space Telescope wavefront sensing and control algorithms," Proc. SPIE 5487, 887 (2004).
[CrossRef]

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 (11)

Fig. 1.
Fig. 1.

Flow chart corresponding to the proposed algorithm.

Fig. 2.
Fig. 2.

HG4,3 (a) and LG+ 3,1 (b) modes with beam waist w = 1 mm. Intensity and phase distributions are displayed at first and second row, respectively.

Fig. 3.
Fig. 3.

Retrieved phase distribution after N × M = 30 iterations for the case of HG4,3 (a). The phase retrieval is calculated with M = 1, 2 and 3 constraint planes corresponding to angles: α j=1 = 45 o , α j=2 = 65 o and α j=3 = 80 o . The evolution of RMS error concerning these phase retrieval examples is displayed in (b).

Fig. 4.
Fig. 4.

Retrieved phase distribution after N × M = 23 iterations for the case of LG+ 3,1 mode, top panel. The phase retrieval is calculated by using constraint planes involving angles: α= 45 o , 55 o , 65 o , 70 o and 80 o . The evolution of RMS error is also displayed as a function of the number of constraint planes as well as for the iterations utilized.

Fig. 5.
Fig. 5.

(a) Intensity distribution (constraint plane) corresponding to the LG+ 3,1 mode transformed under GT operation with angle α = 80 o . This image as well as the rest of constraint planes are disturbed by a random additive noise with a SNR value of 20 dB and 15 dB, as displayed in (b) and (c) respectively. It permits to study the proposed phase retrieval algorithm from noisy images. To help visualization these images have been saturated.

Fig. 6.
Fig. 6.

Retrieved phase distribution after N × M = 23 iterations for the case of LG+ 3,1 mode, top panel. The phase retrieval is now calculated form measurements of noisy intensity distributions (with SNR of 20 dB) corresponding to constraint planes: α = 45 o , 55 o , 65 o , 70 o and 80 o . The evolution of RMS error is also displayed as a function of the number of constraint planes as well as for the iterations utilized.

Fig. 7.
Fig. 7.

(a) GT optical setup. Each generalized lens L1,2 is an assembled set of two cylindrical lenses (b). The transformation is only reached by the proper rotation (with angles ϕ 1 and ϕ 2) of the lenses.

Fig. 8.
Fig. 8.

(a) Experimental GT setup. The intensity distribution of the output signal is registered by a CCD camera with pixel size of 4.6 μm. (b) Generalized lens constructed by assembling two plano-convex cylindrical lenses, which are rotated at angle ϕ 1 and ϕ 2 respectively.

Fig. 9.
Fig. 9.

Numerical simulation for GT setup: first and second rows correspond to intensity and phase distributions, respectively. Experimental results (a)-(d) are displayed at the third row. Input signal is the LG+ 3,1 mode (a), which phase distribution has to be retrieved.

Fig. 10.
Fig. 10.

Retrieved phase distribution after N × M = 45 iterations for the case of LG+ 3,1 mode, top panel. These results are obtained from the measured GT intensity distributions displayed in Fig. 9(b)-(d) at angles: α = 135 o , 115 o and 100 o . The evolution of the RMS error is also displayed.

Fig. 11.
Fig. 11.

Phase distribution W(x,y) associated to the wavefront distortion arising from the optical setup. It is mainly caused by the SLM structure used for the generation of the input signal LG+ 3,1 mode, Eq. (4).

Tables (1)

Tables Icon

Table 1. RMS error values after N × M = 36 iterations. M is the number of constraint planes.

Equations (5)

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

F α ( r o ) = 1 sin α f i ( x i , y i ) exp ( i 2 π ( x o y o + x i y i ) cos α ( x i y o + x o y i ) sin α ) d x i d y i ,
F α ( r o ) = 1 2 λz sin α f i ( x i , y i ) exp ( i 2 π ( x o y o + x i y i ) cos α ( x i y o + x o y i ) 2 λz sin α ) d x i d y i ,
H G m , n ( r ; w ) = 2 1 / 2 H m ( 2 π x w ) H n ( 2 π y w ) 2 m m ! w 2 n n ! w exp ( π w 2 r 2 ) ,
LG p , l ± ( r ; w ) = w 1 ( p ! ( p + l ) ! ) 1 / 2 [ 2 π ( x w ± i y w ) ] l L p l ( 2 π w 2 r 2 ) exp ( π w 2 r 2 ) ,
RMS = ( Σ E in e E in e i ϕ rt 2 ) 1 2 × ( Σ E in e 2 ) 1 2 ,

Metrics