Abstract

Phase retrieval employs very simple data collection hardware and iterative algorithms to determine the phase of an optical field. We have derived limitations on phase retrieval, as applied to optical surface and wavefront metrology, in terms of the speed of beam (i.e., f-number or numerical aperture) and amount of aberration using arguments based on sampling theory and geometrical optics. These limitations suggest methodologies for expanding these ranges by increasing the complexity of the measurement arrangement, the phase-retrieval algorithm, or both. We have simulated one of these methods where a surface is measured at unusual conjugates.

© 2009 Optical Society of America

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References

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  1. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).
  2. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27-29 (1978).
    [CrossRef] [PubMed]
  3. J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrieval from experimental far-field data,” Opt. Lett. 13, 619-621 (1988).
    [CrossRef] [PubMed]
  4. 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]
  5. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737-1746 (1993).
    [CrossRef] [PubMed]
  6. 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, 1747-1768 (1993).
    [CrossRef] [PubMed]
  7. 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 (2005).
    [CrossRef]
  8. G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316-28 (2002).
    [CrossRef]
  9. Joseph W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2003), Chap. 9.
  10. For example, the Sony ICX625 CCD Image Sensor.
  11. P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, “Extending the range of interferometry through subaperture stitching,” Proc. SPIE TD02, 134-7 (2003).
  12. M. Bray, “Stitching interferometer for large optics using a standard interferometer: description of an automated system [for ICF optics],” Proc. SPIE 3047, 911-18 (1997).
  13. J. R. Fienup, “Phase retrieval for undersampled broadband images,” J. Opt. Soc. Am. A 16, 1831-1839 (1999).
    [CrossRef]
  14. S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
    [CrossRef]
  15. W. T. Welford, Aberrations of Optical Systems (Taylor & Francis, 1986), Chap. 7.
  16. W. T. Welford, Aberrations of Optical Systems (Taylor & Francis, 1986), Chap. 8.

2005

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 (2005).
[CrossRef]

2003

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, “Extending the range of interferometry through subaperture stitching,” Proc. SPIE TD02, 134-7 (2003).

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

2002

G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316-28 (2002).
[CrossRef]

1999

1997

M. Bray, “Stitching interferometer for large optics using a standard interferometer: description of an automated system [for ICF optics],” Proc. SPIE 3047, 911-18 (1997).

1993

1989

1988

1978

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

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 (2005).
[CrossRef]

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 (2005).
[CrossRef]

Barty, A.

G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316-28 (2002).
[CrossRef]

Bradsher, L. S.

G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316-28 (2002).
[CrossRef]

Bray, M.

M. Bray, “Stitching interferometer for large optics using a standard interferometer: description of an automated system [for ICF optics],” Proc. SPIE 3047, 911-18 (1997).

Cederquist, J. N.

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 (2005).
[CrossRef]

Dillon, D. R.

G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316-28 (2002).
[CrossRef]

Dumas, P.

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, “Extending the range of interferometry through subaperture stitching,” Proc. SPIE TD02, 134-7 (2003).

Fienup, J. R.

Fleig, J.

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, “Extending the range of interferometry through subaperture stitching,” Proc. SPIE TD02, 134-7 (2003).

Forbes, G.

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, “Extending the range of interferometry through subaperture stitching,” Proc. SPIE TD02, 134-7 (2003).

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

Goodman, Joseph W.

Joseph W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2003), Chap. 9.

Johnson, M. A.

G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316-28 (2002).
[CrossRef]

Kang, M. G.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Kryskowski, D.

Marron, J. C.

Murphy, P. E.

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, “Extending the range of interferometry through subaperture stitching,” Proc. SPIE TD02, 134-7 (2003).

Nguyen, N. Q.

G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316-28 (2002).
[CrossRef]

Park, M. K.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Park, S. C.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Paxman, R. G.

Phillion, D. W.

G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316-28 (2002).
[CrossRef]

Robinson, S. R.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

Schulz, T. J.

Seldin, J. H.

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 (2005).
[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 (2005).
[CrossRef]

Snell, F. J.

G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316-28 (2002).
[CrossRef]

Sommargren, G. E.

G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316-28 (2002).
[CrossRef]

Wackerman, C. C.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Taylor & Francis, 1986), Chap. 7.

W. T. Welford, Aberrations of Optical Systems (Taylor & Francis, 1986), Chap. 8.

Appl. Opt.

IEEE Signal Process. Mag.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Optik (Jena)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237-246 (1972).

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 (2005).
[CrossRef]

G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316-28 (2002).
[CrossRef]

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, “Extending the range of interferometry through subaperture stitching,” Proc. SPIE TD02, 134-7 (2003).

M. Bray, “Stitching interferometer for large optics using a standard interferometer: description of an automated system [for ICF optics],” Proc. SPIE 3047, 911-18 (1997).

Other

W. T. Welford, Aberrations of Optical Systems (Taylor & Francis, 1986), Chap. 7.

W. T. Welford, Aberrations of Optical Systems (Taylor & Francis, 1986), Chap. 8.

Joseph W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2003), Chap. 9.

For example, the Sony ICX625 CCD Image Sensor.

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Figures (6)

Fig. 1
Fig. 1

Sketch showing the key components required for phase-retrieval-based optical metrology.

Fig. 2
Fig. 2

(a) Plot of the minimum f-number and (b) maximum numerical aperture that can be measured using phase retrieval with a bare detector array, assuming a sampling parameter Q min = 1 .

Fig. 3
Fig. 3

(a) Plot of the minimum f-number and (b) maximum numerical aperture that can be measured using phase retrieval and an imaging optical system with a magnification of 10 × , assuming a sampling parameter Q min = 1 .

Fig. 4
Fig. 4

Sketches of arrangements where a point source is placed at object point O and is imaged to image point O using the (a) convex and (b) concave spherical surfaces under test to form a converging beam that can be adequately sampled. Point F is the focus of the surface and point C is the center of curvature of the surface. Distances l and l are the object and image distances, and R c is the radius of curvature of the surface.

Fig. 5
Fig. 5

Sketch of arrangement simulated. A point source is placed 162.4 mm in front of a concave surface with a radius of curvature of 341.4 mm and diameter of 100 mm . The point source is imaged to a point 3333 mm in front of the mirror. A detector array is used to measure the intensity pattern in two planes 30 and 60 mm inside the image plane.

Fig. 6
Fig. 6

Simulation of a phase-retrieval result for a wavefront whose nominal shape includes 50 waves of spherical aberration. The wavefronts shown have the spherical aberration term removed and only the residual errors that are of interest are shown. The scales to the right are in units of waves. (a) The true wavefront used in the simulation, (b) the recovered wavefront, and (c) the difference between the true wavefront and the recovered wavefront, having an RMS of 0.0054 waves and a PV of 0.026 waves.

Equations (36)

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d ξ = λ z N d x ,
N d x = Q D ,
Q = λ z d ξ D = λ f / # d ξ
f / # Q min d ξ λ .
NA λ 2 Q min d ξ .
f / # Q min d ξ m λ ,
NA m λ 2 Q min d ξ .
R / # = | R c | D ,
2 R c = 1 l + 1 l ,
f / # = ( 2 D R c + D l ) 1 .
( m λ Q min D d ξ + 2 R c ) 1 l < R c 2 .
W 040 = { D 64 R / # ( 1 f / # 1 R / # ) 2 concave   ( R c < 0 ) D 64 R / # ( 1 f / # + 1 R / # ) 2 convex   ( R c > 0 ) .
N d ξ 2 z D { [ W ( ρ ) ρ ] max [ W ( ρ ) ρ ] min } ,
[ W ( ρ ) ρ ] max [ W ( ρ ) ρ ] min N d ξ 2 f / # .
[ W ( ρ ) ρ ] max [ W ( ρ ) ρ ] min λ N 2 Q min .
[ W ( ρ ) ρ ] max λ N 4 Q min .
I ( ρ ) = 1 2 { 1 + cos [ 2 π λ W ( ρ ) ] } .
2 π λ | W ( ρ + Δ ρ ) W ( ρ ) | max π ,
2 π λ [ | W ( ρ ) ρ | max Δ ρ ] π .
| W ( ρ ) ρ | max λ N 4 .
W ( ρ ) = W 040 ρ 4 ,
W ( ρ ) = W 040 ( ρ 4 3 2 ρ 2 ) .
[ W ( ρ ) ρ ] max = W 040 .
W 040 λ N 4 Q min .
[ W ( ρ ) ρ ] max N d ξ 4 m f / # .
[ W ( ρ ) ρ ] max N λ 4 Q min ,
S 1 = surfaces A 2 y a ( u a n u a n ) ,
A = n i a ,
W 040 = 1 8 S 1 .
u a = D / 2 l .
n u a = n u a y a 2 R c .
u a = D ( 1 2 l 1 R c ) .
i a = u a + y a R c = D 2 ( 1 R c 1 l ) .
W 040 = D 4 64 R c ( 1 R c 1 l ) 2 ,
W 040 = D 4 64 R c ( 1 l + 1 R c ) 2 ,
W 040 = { D 64 R / # ( 1 f / # 1 R / # ) 2 concave   ( R c < 0 ) D 64 R / # ( 1 f / # + 1 R / # ) 2 convex   ( R c > 0 ) .

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