Abstract

In field retrieval, the amplitude and phase of the generalized pupil function for an optical system are estimated from multiple defocused measurements of the system point-spread function. A baseline field reconstruction algorithm optimizing a data consistency metric is described. Additionally, two metrics specifically designed to incorporate a priori knowledge about pupil amplitude for hard-edged and uniformly illuminated aperture systems are given. Experimental results demonstrate the benefit of using these amplitude metrics in addition to the baseline metric.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  20. D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).
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    [CrossRef]
  22. S. T. Thurman, R. T. DeRosa, and J. R. Fienup, “Amplitude metrics for field retrieval,” presented at the OSA Frontiers in Optics Meeting, Rochester, NY, October 2006.
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    [CrossRef]

2006 (2)

2005 (2)

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

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

2004 (1)

J. J. Green, B. H. Dean, C. M. Ohara, and D. C. Redding, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 994-1003 (2004).

2003 (3)

2002 (1)

1999 (2)

J. R. Fienup, “Phase retrieval for undersampled broadband images,” J. Opt. Soc. Am. A 16, 1831-1839 (1999).
[CrossRef]

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. (Bellingham) 38, 1229-1240 (1999).
[CrossRef]

1997 (1)

1993 (3)

1989 (1)

1985 (1)

D. L. Snyder and M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. 32, 3864-3871 (1985).
[CrossRef]

1982 (2)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829-832 (1982).

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758-2769 (1982).
[CrossRef] [PubMed]

1973 (1)

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

Almoro, P.

Bowers, C. W.

Brady, G. R.

Calef, B.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

Cederquist, J. N.

Dean, B. H.

J. J. Green, B. H. Dean, C. M. Ohara, and D. C. Redding, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 994-1003 (2004).

B. H. Dean and C. W. Bowers, “Diversity selection for phase-diverse phase retrieval,” J. Opt. Soc. Am. A 20, 1490-1504 (2003).
[CrossRef]

DeRosa, R. T.

S. T. Thurman, R. T. DeRosa, and J. R. Fienup, “Amplitude metrics for field retrieval,” presented at the OSA Frontiers in Optics Meeting, Rochester, NY, October 2006.

Fienup, J. R.

Fiete, R. D.

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. (Bellingham) 38, 1229-1240 (1999).
[CrossRef]

Georges, J.

Gerwe, D. R.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829-832 (1982).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

Green, J. J.

J. J. Green, B. H. Dean, C. M. Ohara, and D. C. Redding, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 994-1003 (2004).

Hege, E. K.

Jain, M.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

Jefferies, S. M.

Kryskowski, D.

Lloyd-Hart, M.

Luna, C.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

Marron, J. C.

Miller, J. J.

Miller, M. I.

D. L. Snyder and M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. 32, 3864-3871 (1985).
[CrossRef]

Misell, D. L.

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

Ohara, C. M.

J. J. Green, B. H. Dean, C. M. Ohara, and D. C. Redding, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 994-1003 (2004).

Osten, W.

Paxman, R. G.

J. H. Seldin and R. G. Paxman, “Joint estimation of amplitude and phase from phase-diversity data,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, Technical Digest (Optical Society of America, 2005), paper JTuB4.
[PubMed]

Pedrini, G.

Redding, D. C.

J. J. Green, B. H. Dean, C. M. Ohara, and D. C. Redding, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 994-1003 (2004).

Robinson, S. R.

Roddier, C.

Roddier, F.

Schulz, T. J.

Seldin, J. H.

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]

J. H. Seldin and R. G. Paxman, “Joint estimation of amplitude and phase from phase-diversity data,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, Technical Digest (Optical Society of America, 2005), paper JTuB4.
[PubMed]

Snyder, D. L.

D. L. Snyder and M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. 32, 3864-3871 (1985).
[CrossRef]

Thurman, S. T.

S. T. Thurman, R. T. DeRosa, and J. R. Fienup, “Amplitude metrics for field retrieval,” presented at the OSA Frontiers in Optics Meeting, Rochester, NY, October 2006.

Tiziani, H.

Wackerman, C. C.

Zhang, Y.

Appl. Opt. (7)

IEEE Trans. Nucl. Sci. (1)

D. L. Snyder and M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. 32, 3864-3871 (1985).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Phys. D (1)

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

Opt. Eng. (Bellingham) (2)

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. (Bellingham) 38, 1229-1240 (1999).
[CrossRef]

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829-832 (1982).

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (2)

J. J. Green, B. H. Dean, C. M. Ohara, and D. C. Redding, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 994-1003 (2004).

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

Other (3)

S. T. Thurman, R. T. DeRosa, and J. R. Fienup, “Amplitude metrics for field retrieval,” presented at the OSA Frontiers in Optics Meeting, Rochester, NY, October 2006.

J. H. Seldin and R. G. Paxman, “Joint estimation of amplitude and phase from phase-diversity data,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, Technical Digest (Optical Society of America, 2005), paper JTuB4.
[PubMed]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

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Figures (12)

Fig. 1
Fig. 1

Diagram of 4F system used for experiment.

Fig. 2
Fig. 2

Plot of Γ ( x , κ ) , Γ ( x , κ ) , and Γ ( x , κ ) .

Fig. 3
Fig. 3

Digital scans of the pupil amplitude masks used in the experiment: (a) circle, (b) spiral, (c) nine-aperture triarm, and (d) nine-aperture Golay.

Fig. 4
Fig. 4

Field retrieval results for the circular pupil mask: (a) A ̂ 1 ( m , n ) , (b) A ̂ 2 ( m , n ) , (c) A ̂ 3 ( m , n ) , (d) A ̂ 4 ( m , n ) , (e) A ̂ 5 ( m , n ) , and (f) ϕ ̂ 5 ( m , n ) (in units of radians) with piston tip, tilt, and focus terms removed. Note that ϕ ̂ 5 ( m , n ) is shown only within the aperture at points where A ̂ 5 ( m , n ) > κ 1 .

Fig. 5
Fig. 5

Histograms of the retrieved pupil amplitude for the circular pupil mask: dashed curve, A ̂ 1 ( m , n ) and solid curve A ̂ 3 ( m , n ) . The scale for the vertical axis is logarithmic.

Fig. 6
Fig. 6

Comparison between measured and modeled PSFs using A ̂ 5 ( m , n ) and ϕ ̂ 5 ( m , n ) for the circular pupil mask. Measured PSFs G k ( p , q ) are shown in the left-hand column and modeled PSFs G ̂ k ( p , q ) are shown in the right-hand column. The defocus distances for each PSF are (a), (b) 4 mm ; (c), (d) 0 mm ; and (e), (f) 4 mm .

Fig. 7
Fig. 7

Same as Fig. 4, except for the spiral pupil mask.

Fig. 8
Fig. 8

Same as Fig. 6, except for the spiral pupil mask.

Fig. 9
Fig. 9

Same as Fig. 4, except for the triarm pupil mask.

Fig. 10
Fig. 10

Same as Fig. 6, except for the triarm pupil mask. The defocus distances for each PSF are (a), (b) 4 mm ; (c), (d) 2 mm ; (e), (f) 0 mm ; (g), (h) 2 mm ; and (i), (j) 4 mm .

Fig. 11
Fig. 11

Same as Fig. 4, except for the Golay pupil mask.

Fig. 12
Fig. 12

Same as Fig. 10, except for the Golay pupil mask.

Tables (5)

Tables Icon

Table 1 Details of Various Field Retrieval Estimates

Tables Icon

Table 2 Metric Values for Each Field Retrieval Result with the Circular Pupil Mask a

Tables Icon

Table 3 Metric Values for Each Field Retrieval Result with the Spiral Pupil Mask a

Tables Icon

Table 4 Metric Values for Each Field Retrieval Result with the Triarm Pupil Mask a

Tables Icon

Table 5 Metric Values for Each Field Retrieval Result with the Golay Pupil Mask a

Equations (35)

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E ̂ p ( m , n ) = A ̂ ( m , n ) exp [ i ϕ ̂ ( m , n ) ] ,
A ̂ ( m , n ) = M N B ̂ ( m , n ) ( m , n ) B ̂ ( m , n ) .
E ̂ f ( p , q ) = DFT [ E ̂ p ] = 1 M N ( m , n ) E ̂ p ( m , n ) exp [ i 2 π ( m p M + n q N ) ] ,
U ̂ f ( m , n ) = 1 M N ( p , q ) E ̂ f ( p , q ) exp [ i 2 π ( p m M + q n N ) ] .
U ̂ k ( m , n ) = U ̂ f ( m , n ) exp ( i 2 π z ̂ k 1 λ 2 m 2 Δ m 2 n 2 Δ n 2 ) ,
I ̂ k ( p , q ) = E ̂ k ( p , q ) 2 .
H d ( m , n ) = sinc ( f d N n ) sinc ( f d M m ) ,
H s , k ( m , n ) = exp [ i 2 π ( m p ̂ s , k M + n q ̂ s , k N ) ] ,
g ̂ k ( m , n ) = H s , k ( m , n ) H d ( m , n ) f ̂ k ( m , n ) ,
Φ d = 1 K k = 1 K { ( p , q ) W k ( p , q ) [ α k G ̂ k ( p , q ) G k ( p , q ) ] 2 ( p , q ) W k ( p , q ) G k 2 ( p , q ) } ,
α k = ( p , q ) W k ( p , q ) G ̂ k ( p , q ) G k ( p , q ) ( p , q ) W k ( p , q ) G ̂ k 2 ( p , q ) ,
Φ d = 1 1 K k = 1 K [ ( p , q ) W k ( p , q ) G ̂ k ( p , q ) G k ( p , q ) ] 2 [ ( p , q ) W k ( p , q ) G k 2 ( p , q ) ] [ ( p , q ) W k ( p , q ) G ̂ k 2 ( p , q ) ] .
( m , n ) Φ d B ̂ ( m , n ) s ( m m , n n ) ,
Φ 1 ( κ 1 ) = 1 M N ( m , n ) Γ [ A ̂ ( m , n ) , κ 1 ]
Φ 2 ( κ 2 ) = ( μ , η ) D { 1 M N ( m , n ) Γ [ A ̂ ( m , n ) A ̂ ( m + μ , n + η ) , κ 2 ] } ,
Γ ( x , κ ) = { 2 x 2 3 κ 2 8 x 3 27 κ 3 + x 4 27 κ 4 , x 3 κ 1 , x > 3 κ } ,
Φ d G ̂ k ( p , q )
= 2 K W k ( p , q ) [ ( p , q ) W k ( p , q ) G ̂ k ( p , q ) G k ( p , q ) ] [ ( p , q ) W k ( p , q ) G k 2 ( p , q ) ] [ ( p , q ) W k ( p , q ) G ̂ k 2 ( p , q ) ] 2 { G ̂ k ( p , q ) [ ( p , q ) W k ( p , q ) G ̂ k ( p , q ) G k ( p , q ) ] G k ( p , q ) [ ( p , q ) W k ( p , q ) G ̂ k 2 ( p , q ) ] } .
g ̂ k ( m , n ) = Φ d Re [ g ̂ k ( m , n ) ] + i Φ d Im [ g ̂ k ( m , n ) ] = 1 M N ( p , q ) Φ d G ̂ k ( p , q ) exp [ i 2 π ( m p M + n q N ) ] .
Φ d p ̂ s , k = Im [ ( m , n ) 2 π m M g ̂ k ( m , n ) g ̂ k * ( m , n ) ] ,
Φ d q ̂ s , k = Im [ ( m , n ) 2 π n N g ̂ k ( m , n ) g ̂ k * ( m , n ) ] .
Φ d I ̂ k ( p , q ) = 1 M N ( m , n ) f ̂ k ( m , n ) exp [ i 2 π ( p m M + q n N ) ] ,
f ̂ k ( m , n ) = H ̂ s , k * ( m , n ) H d * ( m , n ) g ̂ k ( m , n ) .
E ̂ k ( p , q ) = 2 E ̂ k ( p , q ) Φ d I ̂ k ( p , q ) ,
U ̂ k ( m , n ) = 1 M N ( p , q ) E ̂ k ( p , q ) exp [ i 2 π ( m p M + n q N ) ] .
Φ d z ̂ k = Im [ ( m , n ) 2 π 1 λ 2 m 2 Δ m 2 n 2 Δ n 2 U ̂ k ( m , n ) U ̂ k * ( m , n ) ] .
U ̂ f ( m , n ) = k U ̂ k ( m , n ) exp ( i 2 π z ̂ k 1 λ 2 m 2 Δ m 2 n 2 Δ n 2 ) ,
E ̂ f ( p , q ) = 1 M N ( m , n ) U ̂ f ( m , n ) exp [ i 2 π ( p m M + q n N ) ] ,
E ̂ p ( m , n ) = 1 M N ( p , q ) E ̂ f ( p , q ) exp [ i 2 π ( m p M + n q N ) ] .
Φ d ϕ ̂ ( m , n ) = Im [ E ̂ p ( m , n ) E ̂ p * ( m , n ) ] ,
Φ d A ̂ ( m , n ) = Re { E ̂ p ( m , n ) exp [ i ϕ ̂ ( m , n ) ] } .
Φ d B ̂ ( m , n ) = sgn [ B ̂ ( m , n ) ] ( m , n ) B ̂ ( m , n ) [ M N Φ d A ̂ ( m , n ) ( m , n ) Φ d A ̂ ( m , n ) A ̂ ( m , n ) ] .
Φ 1 ( κ 1 ) A ̂ ( m , n ) = 1 M N Γ [ A ̂ ( m , n ) , κ 1 ] ,
Φ 2 ( κ 2 ) A ̂ ( m , n ) = ( μ , η ) D 1 M N { Γ [ A ̂ ( m , n ) A ̂ ( m + μ , n + η ) , κ 2 ] Γ [ A ̂ ( m μ , n η ) A ̂ ( m , n ) , κ 2 ] } ,
Γ ( x , κ ) = { sgn ( x ) ( 4 x 3 κ 2 8 x 2 9 κ 3 + 4 x 3 27 κ 4 ) , x 3 κ 0 , x > 3 κ } .

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