Abstract

We present an improved phase-retrieval algorithm that is based on the Sziklas and Siegman coordinate transformation (SSCT) and applied to optical surface testing. With the SSCT, a spherical-wave diffraction problem can be transformed into a plane-wave diffraction problem, and the fast Fourier transform can be applied directly in propagation computations. Compared with conventional diffraction propagation methods, the proposed method is simple and relatively fast, and the computation efficiency for the phase-retrieval algorithm can be increased to a certain degree. Analysis and simulation were performed for this method, and simulation results exhibit correct diffraction computation and good phase-retrieval capability. A practical 200mm diameter, f/5 spherical surface was tested; testing results showed good agreement with that of a ZYGO interferometer, which confirmed the feasibility and accuracy of the proposed method.

© 2010 Optical Society of America

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  1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2760 (1982).
    [CrossRef]
  2. G. R. Brady and J. R. Fienup, “Improved optical metrology using phase retrieval,” in Optical Fabrication and Testing 2004, OSA Technical Digest (Optical Society of America, 2004), paper OTuB3.
  3. G. R. Brady and J. R. Fienup, “Phase retrieval as an optical metrology tool,” in Optical Fabrication and Testing 2005, Topical Meeting of the Optical Society of America, SPIE Technical Digest (SPIE, 2005), Vol. TD03, pp, 139–141
  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]
  5. 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]
  6. A. Lowman, D. Redding, S. Basinger, D. Cohen, J. Faust, J. Green, C. Ohara, and F. Shi, “Phase retrieval camera for testing NGST optics,” Proc. SPIE 4850, 329–335 (2003).
    [CrossRef]
  7. P. Atcheson, S. Acton, and P. Lightsey, “Instrument-level phase retrieval wavefront sensing and correction for astronomical telescopes,” Proc. SPIE 4839, 228–239 (2003).
    [CrossRef]
  8. G. R. Brady and J. R. Fienup, “Effect of broadband illumination on reconstruction error of phase retrieval in optical metrology,” Proc. SPIE 6617, 66170I (2007).
    [CrossRef]
  9. G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17, 624–639 (2009).
    [CrossRef]
  10. G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express 14, 474–486 (2006).
    [CrossRef]
  11. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef]
  12. G. R. Brady and J. R. Fienup, “Measurement range of phase retrieval in optical surface and wavefront metrology,” Appl. Opt. 48, 442–449 (2009).
    [CrossRef]
  13. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).
  14. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. J. F. Perkins and R. A. Shatas, “Propagation in unstable and plane-mirror optical resonators,” Appl. Phys. 9, 343–345(1976).
    [CrossRef]
  20. K. E. Oughstun and C. C. Khamnei, “Three-dimensional field structure in open unstable resonators, Part I: passive cavity results,” Opt. Express 4, 388–399 (1999).
    [CrossRef]
  21. X. Zhu, A. Schülzgen, H. Li, H. Wei, J. V. Moloney, and N. Peyghambarian, “Coherent beam transformations using multimode waveguides,” Opt. Express 18, 7506–7520.
    [CrossRef]
  22. J. Azana, “Lensless imaging of an arbitrary object,” Opt. Lett. 28, 501–503 (2003).
    [CrossRef]
  23. V. Yu. Ivanov, V. P. Sivokon, and M. A.. Vorontsov, “Phase retrieval from a set of intensity measurement: theory and experiment,” J. Opt. Soc. Am. A 9, 1515–1524 (1992).
    [CrossRef]
  24. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

2009

2007

G. R. Brady and J. R. Fienup, “Effect of broadband illumination on reconstruction error of phase retrieval in optical metrology,” Proc. SPIE 6617, 66170I (2007).
[CrossRef]

2006

G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express 14, 474–486 (2006).
[CrossRef]

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]

2003

A. Lowman, D. Redding, S. Basinger, D. Cohen, J. Faust, J. Green, C. Ohara, and F. Shi, “Phase retrieval camera for testing NGST optics,” Proc. SPIE 4850, 329–335 (2003).
[CrossRef]

P. Atcheson, S. Acton, and P. Lightsey, “Instrument-level phase retrieval wavefront sensing and correction for astronomical telescopes,” Proc. SPIE 4839, 228–239 (2003).
[CrossRef]

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

J. Azana, “Lensless imaging of an arbitrary object,” Opt. Lett. 28, 501–503 (2003).
[CrossRef]

2000

1999

1993

1992

1982

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

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2760 (1982).
[CrossRef]

1976

J. F. Perkins and R. A. Shatas, “Propagation in unstable and plane-mirror optical resonators,” Appl. Phys. 9, 343–345(1976).
[CrossRef]

1974

E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[CrossRef]

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of 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 (2006).
[CrossRef]

Acton, S.

P. Atcheson, S. Acton, and P. Lightsey, “Instrument-level phase retrieval wavefront sensing and correction for astronomical telescopes,” Proc. SPIE 4839, 228–239 (2003).
[CrossRef]

Allman, B. E.

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.

P. Atcheson, S. Acton, and P. Lightsey, “Instrument-level phase retrieval wavefront sensing and correction for astronomical telescopes,” Proc. SPIE 4839, 228–239 (2003).
[CrossRef]

Azana, J.

Barty, A.

Basinger, S.

A. Lowman, D. Redding, S. Basinger, D. Cohen, J. Faust, J. Green, C. Ohara, and F. Shi, “Phase retrieval camera for testing NGST optics,” Proc. SPIE 4850, 329–335 (2003).
[CrossRef]

Bowers, C.

Brady, G. R.

G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17, 624–639 (2009).
[CrossRef]

G. R. Brady and J. R. Fienup, “Measurement range of phase retrieval in optical surface and wavefront metrology,” Appl. Opt. 48, 442–449 (2009).
[CrossRef]

G. R. Brady and J. R. Fienup, “Effect of broadband illumination on reconstruction error of phase retrieval in optical metrology,” Proc. SPIE 6617, 66170I (2007).
[CrossRef]

G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express 14, 474–486 (2006).
[CrossRef]

G. R. Brady and J. R. Fienup, “Improved optical metrology using phase retrieval,” in Optical Fabrication and Testing 2004, OSA Technical Digest (Optical Society of America, 2004), paper OTuB3.

G. R. Brady and J. R. Fienup, “Phase retrieval as an optical metrology tool,” in Optical Fabrication and Testing 2005, Topical Meeting of the Optical Society of America, SPIE Technical Digest (SPIE, 2005), Vol. TD03, pp, 139–141

Cohen, D.

A. Lowman, D. Redding, S. Basinger, D. Cohen, J. Faust, J. Green, C. Ohara, and F. Shi, “Phase retrieval camera for testing NGST optics,” Proc. SPIE 4850, 329–335 (2003).
[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]

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

Faust, J.

A. Lowman, D. Redding, S. Basinger, D. Cohen, J. Faust, J. Green, C. Ohara, and F. Shi, “Phase retrieval camera for testing NGST optics,” Proc. SPIE 4850, 329–335 (2003).
[CrossRef]

Fienup, J. R.

G. R. Brady and J. R. Fienup, “Measurement range of phase retrieval in optical surface and wavefront metrology,” Appl. Opt. 48, 442–449 (2009).
[CrossRef]

G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17, 624–639 (2009).
[CrossRef]

G. R. Brady and J. R. Fienup, “Effect of broadband illumination on reconstruction error of phase retrieval in optical metrology,” Proc. SPIE 6617, 66170I (2007).
[CrossRef]

G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express 14, 474–486 (2006).
[CrossRef]

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[CrossRef]

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]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2760 (1982).
[CrossRef]

G. R. Brady and J. R. Fienup, “Phase retrieval as an optical metrology tool,” in Optical Fabrication and Testing 2005, Topical Meeting of the Optical Society of America, SPIE Technical Digest (SPIE, 2005), Vol. TD03, pp, 139–141

G. R. Brady and J. R. Fienup, “Improved optical metrology using phase retrieval,” in Optical Fabrication and Testing 2004, OSA Technical Digest (Optical Society of America, 2004), paper OTuB3.

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 (Jena) 35, 237–246 (1972).

Gonsalves, R. A.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Green, J.

A. Lowman, D. Redding, S. Basinger, D. Cohen, J. Faust, J. Green, C. Ohara, and F. Shi, “Phase retrieval camera for testing NGST optics,” Proc. SPIE 4850, 329–335 (2003).
[CrossRef]

Guizar-Sicairos, M.

Ivanov, V. Yu.

Khamnei, C. C.

Li, H.

Lightsey, P.

P. Atcheson, S. Acton, and P. Lightsey, “Instrument-level phase retrieval wavefront sensing and correction for astronomical telescopes,” Proc. SPIE 4839, 228–239 (2003).
[CrossRef]

Lowman, A.

A. Lowman, D. Redding, S. Basinger, D. Cohen, J. Faust, J. Green, C. Ohara, and F. Shi, “Phase retrieval camera for testing NGST optics,” Proc. SPIE 4850, 329–335 (2003).
[CrossRef]

Marron, J. C.

McMahon, P. J.

Moloney, J. V.

Nugen, K. A.

Ohara, C.

A. Lowman, D. Redding, S. Basinger, D. Cohen, J. Faust, J. Green, C. Ohara, and F. Shi, “Phase retrieval camera for testing NGST optics,” Proc. SPIE 4850, 329–335 (2003).
[CrossRef]

Oughstun, K. E.

Paganin, D.

Perkins, J. F.

J. F. Perkins and R. A. Shatas, “Propagation in unstable and plane-mirror optical resonators,” Appl. Phys. 9, 343–345(1976).
[CrossRef]

Peyghambarian, N.

Redding, D.

A. Lowman, D. Redding, S. Basinger, D. Cohen, J. Faust, J. Green, C. Ohara, and F. Shi, “Phase retrieval camera for testing NGST optics,” Proc. SPIE 4850, 329–335 (2003).
[CrossRef]

Roddier, C.

Roddier, F.

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 (Jena) 35, 237–246 (1972).

Schulz, T. J.

Schülzgen, A.

Seldin, J. H.

Shatas, R. A.

J. F. Perkins and R. A. Shatas, “Propagation in unstable and plane-mirror optical resonators,” Appl. Phys. 9, 343–345(1976).
[CrossRef]

Shi, F.

A. Lowman, D. Redding, S. Basinger, D. Cohen, J. Faust, J. Green, C. Ohara, and F. Shi, “Phase retrieval camera for testing NGST optics,” Proc. SPIE 4850, 329–335 (2003).
[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]

Siegman, A. E.

E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[CrossRef]

Sivokon, V. P.

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]

Sziklas, E. A.

E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[CrossRef]

Tiller, J. B.

Vorontsov, M. A.

Wei, H.

Zhu, X.

Appl. Opt.

Appl. Phys.

J. F. Perkins and R. A. Shatas, “Propagation in unstable and plane-mirror optical resonators,” Appl. Phys. 9, 343–345(1976).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

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

Opt. Express

Opt. Lett.

Optik (Jena)

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

Proc. IEEE

E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[CrossRef]

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]

A. Lowman, D. Redding, S. Basinger, D. Cohen, J. Faust, J. Green, C. Ohara, and F. Shi, “Phase retrieval camera for testing NGST optics,” Proc. SPIE 4850, 329–335 (2003).
[CrossRef]

P. Atcheson, S. Acton, and P. Lightsey, “Instrument-level phase retrieval wavefront sensing and correction for astronomical telescopes,” Proc. SPIE 4839, 228–239 (2003).
[CrossRef]

G. R. Brady and J. R. Fienup, “Effect of broadband illumination on reconstruction error of phase retrieval in optical metrology,” Proc. SPIE 6617, 66170I (2007).
[CrossRef]

Other

G. R. Brady and J. R. Fienup, “Improved optical metrology using phase retrieval,” in Optical Fabrication and Testing 2004, OSA Technical Digest (Optical Society of America, 2004), paper OTuB3.

G. R. Brady and J. R. Fienup, “Phase retrieval as an optical metrology tool,” in Optical Fabrication and Testing 2005, Topical Meeting of the Optical Society of America, SPIE Technical Digest (SPIE, 2005), Vol. TD03, pp, 139–141

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (12)

Fig. 1
Fig. 1

Schematic of phase-retrieval technology for optics testing.

Fig. 2
Fig. 2

Schematic of the SSCT. The spherical-wave diffraction can be transformed into plane-wave diffraction.

Fig. 3
Fig. 3

Block diagram of the improved algorithm. This algorithm consists of two parts; the inner part is the basic GS algorithm and the outer part is the SSCT.

Fig. 4
Fig. 4

For a spherical mirror, the size of the diffraction image in the plane z = z 1 is approximately proportional to the distance from the origin to the image plane, | z 1 | .

Fig. 5
Fig. 5

Comparison of defocused images for a 200 mm diameter spherical mirror with different diffraction propagation models.

Fig. 6
Fig. 6

Phase map for the 200 mm diameter spheri cal mirror, PV = 0.5 λ , RMS = 0.0169 λ , where λ is the wavelength of the light source, λ = 0.6328 μm .

Fig. 7
Fig. 7

Simulated defocused images in the planes z = ± 5 mm according to the sphere-to-sphere diffraction propagation model.

Fig. 8
Fig. 8

Simulation results for the phase retrieval of the 200 mm diameter spherical mirror. (a) Retrieved phase map by the improved algorithm after 100 iterations, PV = 0.5056 λ and RMS = 0.0169 λ . (b) Difference between the original phase and the retrieved phase.

Fig. 9
Fig. 9

Spherical mirror and its figure error map. The figure error is measured by a ZYGO interferometer, PV = 0.313 λ , RMS = 0.058 λ .

Fig. 10
Fig. 10

Experimental setup for phase-retrieval measurement of a 200 mm diameter spherical mirror: (a) picture of the experimental setup and (b) schematic diagram of the measurement arrangement.

Fig. 11
Fig. 11

Diffraction images captured by the CCD in the planes z = ± 2 mm .

Fig. 12
Fig. 12

Experimental results of phase retrieval for the 200 mm diameter spherical mirror by the improved algorithm. After 600 iterations (less than 1 min ), the error metric E becomes small enough and the final retrieved figure errors are PV = 0.310 λ and RMS = 0.0575 λ .

Equations (7)

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

E = | I i ( x , y ) | g o ( x , y ) | 2 | .
F m ( v x , v y ) = g m ( x , y ) exp ( j 2 π v x x j 2 π v y y ) d x d y ,
F d ( v x , v y ) = F m ( v x , v y ) exp [ j π λ ( v x 2 + v y 2 ) ( z d z m ) ] ,
g d ( x , y , z ) = F d ( v x , v y ) exp ( j 2 π v x x + j 2 π v y y ) d v x d v y .
{ g m ( x , y , z m ) = [ g m ( x , y , z m ) / z m ] exp [ j k ( x 2 + y 2 ) / 2 z m ] g d ( ξ , η , z d ) = [ g d ( ξ , η , z d ) / z d ] exp [ j k ( ξ 2 + η 2 ) / 2 z d ] ,
{ x = x z m , y = y z m ξ = ξ z d , η = η z d z m z d = z m z d z m z d ,
r 2 = | z 1 | r 1 R .

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