Abstract

Techniques for retrieving the phase of an optical field typically depend on assumptions about the amplitude of the field in a desired plane, usually a pupil plane. We describe an approach that makes no such assumptions and is capable of retrieving both the amplitude and phase in the desired plane. Intensity measurements in two or more planes are used by a nonlinear optimization algorithm to retrieve the phase in the measurement planes. The complex field (amplitude and phase) in the desired plane is then computed by simple propagation. We show simulation results and examine the convergence of the algorithm.

© 2006 Optical Society of America

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References

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  1. J. R. Fienup, "Phase Retrieval Algorithms: a Comparison," Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  2. J.N. Cederquist, J.R. Fienup, C.C. Wackerman, S.R. Robinson and D. Kryskowski, "Wave-Front Phase Estimation from Fourier Intensity Measurements," J. Opt. Soc. Am. A 6, 1020-1026 (1989).
    [CrossRef]
  3. J. R. Fienup, "Phase-retrieval Algorithms for a Complicated Optical System," Appl. Opt. 32, 1737-1746 (1993).
    [CrossRef] [PubMed]
  4. J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, "Hubble Space Telescope Characterized by Using Phase-retrieval Algorithms," Appl. Opt. 32, 1747-1767 (1993).
    [CrossRef] [PubMed]
  5. 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]
  6. C. Roddier and F. Roddier, "Combined Approach to Hubble Space Telescope Wave-Front Distortion Analysis," Appl. Opt. 32, 2992-3008 (1993).
    [CrossRef] [PubMed]
  7. S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, and J. Georges, "Sensing wave-front amplitude and phase with phase diversity," Appl. Opt. 41, 2095-2102 (2002).
    [CrossRef] [PubMed]
  8. J. H. Seldin and R. G. Paxman, "Joint Estimation of Amplitude and Phase from Phase-Diversity Data," in "Signal Recovery and Synthesis" Topical Meeting of the Optical Society of America (June, 2005), paper JTuB4.
  9. Joseph W. Goodman, Introduction to Fourier Optics 2nd Ed. (McGraw-Hill, 1996), Chap. 3.
    [PubMed]
  10. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1986), Chap. 10.
  11. D. Malacara and S. L. DeVore, "Interferogram Evaluation and Wavefront Fitting" in Optical Shop Testing, D. Malacara, Ed. (Wiley, 1992), Chap. 13.
  12. G. R. Brady and J. R. Fienup, "Retrieval of Complex Field Using Nonlinear Optimization", in "Signal Recovery and Synthesis" Topical Meeting of the Optical Society of America (June, 2005), postdeadline paper JTuC3.

Appl. Opt.

J. R. Fienup, "Phase Retrieval Algorithms: a Comparison," Appl. Opt. 21, 2758-2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, "Phase-retrieval Algorithms for a Complicated Optical System," Appl. Opt. 32, 1737-1746 (1993).
[CrossRef] [PubMed]

J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, "Hubble Space Telescope Characterized by Using Phase-retrieval Algorithms," Appl. Opt. 32, 1747-1767 (1993).
[CrossRef] [PubMed]

C. Roddier and F. Roddier, "Combined Approach to Hubble Space Telescope Wave-Front Distortion Analysis," Appl. Opt. 32, 2992-3008 (1993).
[CrossRef] [PubMed]

S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, and J. Georges, "Sensing wave-front amplitude and phase with phase diversity," Appl. Opt. 41, 2095-2102 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J.N. Cederquist, J.R. Fienup, C.C. Wackerman, S.R. Robinson and D. Kryskowski, "Wave-Front Phase Estimation from Fourier Intensity Measurements," J. Opt. Soc. Am. A 6, 1020-1026 (1989).
[CrossRef]

J. Phys. D: Appl. Phys.

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]

Optical Shop Testing

D. Malacara and S. L. DeVore, "Interferogram Evaluation and Wavefront Fitting" in Optical Shop Testing, D. Malacara, Ed. (Wiley, 1992), Chap. 13.

Topical Meeting of the Optical Society

G. R. Brady and J. R. Fienup, "Retrieval of Complex Field Using Nonlinear Optimization", in "Signal Recovery and Synthesis" Topical Meeting of the Optical Society of America (June, 2005), postdeadline paper JTuC3.

J. H. Seldin and R. G. Paxman, "Joint Estimation of Amplitude and Phase from Phase-Diversity Data," in "Signal Recovery and Synthesis" Topical Meeting of the Optical Society of America (June, 2005), paper JTuB4.

Other

Joseph W. Goodman, Introduction to Fourier Optics 2nd Ed. (McGraw-Hill, 1996), Chap. 3.
[PubMed]

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1986), Chap. 10.

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Figures (11)

Fig. 1.
Fig. 1.

(a) Steps used in two-step propagation from pupil to measurement plane. (b) Example of propagations taking place when calculating the objective function, corresponding to step B of the algorithm.

Fig. 2.
Fig. 2.

(a) Amplitude and (b) phase functions used to generate intensity patterns and that will be retrieved by our algorithm. The phase is shown modulo 2π.

Fig. 3.
Fig. 3.

Schematic of system modeled in simulations.

Fig. 4.
Fig. 4.

Simulated measured intensity patterns in the three measurement planes. The dynamic range was compressed to bring out the noise in the background.

Fig. 5.
Fig. 5.

Master plane (-60 mm plane) and reconstructed intensities (-40 mm and -25 mm planes) formed by propagating field with magnitudes given by master plane and our retrieved phases.

Fig. 6.
Fig. 6.

Initial guess for (a) amplitude and (b) phase in pupil and resulting reconstructed (c) amplitude and (d) phase for the case where two intensity measurements in the -60 mm and -25 mm planes were used. The error in the phase from the actual, shown in Fig. 2 (b), is 0.029 waves RMS.

Fig. 7.
Fig. 7.

Initial guess for (a) amplitude and (b) phase in pupil and resulting reconstructed (c) amplitude and (d) phase for the case where all three intensity measurements were used. The error in the phase from the actual, shown in Fig. 2 (b), is 0.017 waves RMS.

Fig. 8.
Fig. 8.

Initial guess for (a) amplitude and (b) phase in pupil, the resulting reconstructed (c) amplitude and (d) phase, and actual (e) amplitude and (f) phase for a more complicated aperture. The error in the phase from the actual is 0.020 waves RMS.

Fig. 9.
Fig. 9.

Reconstruction error versus number of iterations for cases where two planes and threes planes have been used.

Fig. 10.
Fig. 10.

Schematic diagram of a method to improve retrieved phase results.

Fig. 11.
Fig. 11.

(a) Amplitude and (b) phase result from our algorithm. (c) Thresholded amplitude and (d) Zernike fit phase. (e) Final phase estimate after running through a phase-only retrieval algorithm.

Tables (1)

Tables Icon

Table 1. RMS error of pupil plane phase formed by inverse propagating from each of the measurement planes and by averaging the pupil plane field with two different methods.

Equations (12)

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F j ( r , s ) = I j ( r , s )
G j ( r , s ) = F j ( r , s ) exp [ i θ j ( r , s ) ] .
G kj ( r , s ) = IDFT { DFT { G j ( r , s ) } exp ( i 2 π Δ z kj { 1 λ 2 [ ( t M d r ) 2 + ( u N d s ) 2 ] } 1 2 ) } .
G kj ( r , s ) = P kj [ G j ( r , s ) ] .
E j = k j r , s W k ( r , s ) [ F k ( r , s ) G kj ( r , s ) ] 2
E j θ j ( r , s ) = 2 Im [ G j ( r , s ) k j G jk w * ( r , s ) ] ,
G jk w ( r , s ) = P kj [ G kj w ( r , s ) ]
G kj w ( r , s ) = W k ( r , s ) [ F k ( r , s ) G kj ( r , s ) G kj ( r , s ) G kj ( r , s ) ] .
G f ( p , q ) = exp { i π λ f [ ( p d p ) 2 + ( q d q ) 2 ] } DFT { g 0 ( m , n ) }
d p = λ f M d m , d q = λ f N d n .
G j ( r , s ) = P 0 j [ g 0 ( m , n ) ] .
g ( m , n ) = P j 0 [ G j ( r , s ) ] ,

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