Abstract

A method is described for analyzing point source imagery collected with various amounts of defocus to obtain wavefront slope data at the exit pupil of an imaging system. Integration of this slope data yields the system wavefront aberration function. The method is based on a geometric optics interpretation of intensity point spread function measurements in the caustic region near focus. Algorithm performance is examined through Monte Carlo simulations. Application of the method to segmented-aperture systems is also explored. The proposed method is used to generate initial wavefront estimates for an iterative phase retrieval algorithm, significantly improving the capture range over a blind phase retrieval approach when segment tilt errors are large.

© 2011 Optical Society of America

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References

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  1. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  2. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  3. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006(1980).
    [CrossRef]
  4. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225(1988).
    [CrossRef] [PubMed]
  5. A. J. Devaney, R. A. Gonsalves, and R. Chidlaw, “Application of phase retrieval techniques to adaptive imaging systems,” J. Opt. Soc. Am. 67, 1422 (abstract) (1977), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-67-10-1361.
  6. 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]
  7. E. Acosta, S. Bará, M. A. Rama, and S. Rios, “Determination of phase mode components in terms of local wave-front slopes: an analytical approach,” Opt. Lett. 20, 1083–1085(1995).
    [CrossRef] [PubMed]
  8. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  9. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmorgorov phase screen,” Waves Random Media 2, 209–224(1992).
    [CrossRef]
  10. URL:http://keckobservatory.org/.
  11. URL:http://rhea.as.utexas.edu/.
  12. URL:http://www.salt.ac.za/.
  13. URL:http://www.jwst.nasa.gov/.
  14. G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
    [CrossRef]
  15. Performance is given here in terms of the wavefront error seen in reflection, which is a factor of 2 larger than the specifications reported in , which are in terms of surface error.
  16. F. Shi, G. Chanan, C. Ohara, M. Troy, and D. C. Redding, “Experimental verification of dispersed fringe sensing as a segment phasing technique using the Keck telescope,” Appl. Opt. 434474–4481 (2004).
    [CrossRef] [PubMed]
  17. D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
    [CrossRef]
  18. L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
    [CrossRef]
  19. J. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).
  20. “ZEMAX Optical Design Program, User’s Guide” (ZEMAX Development Corporation, 2006).
  21. S. T. Thurman and J. R. Fienup, “Phase retrieval with signal bias,” J. Opt. Soc. Am. A 26, 1008–1014 (2009).
    [CrossRef]
  22. T. Zielinski and J. R. Fienup, “Extending wavefront sensing capture range for segmented systems through tip and tilt estimation,” in Frontiers in Optics, Tech. Dig. (CD) (Optical Society of America, 2006), paper FMF4.

2009 (1)

2007 (2)

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

2004 (1)

1998 (1)

1995 (1)

1992 (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmorgorov phase screen,” Waves Random Media 2, 209–224(1992).
[CrossRef]

1989 (1)

1988 (1)

1980 (1)

1977 (3)

A. J. Devaney, R. A. Gonsalves, and R. Chidlaw, “Application of phase retrieval techniques to adaptive imaging systems,” J. Opt. Soc. Am. 67, 1422 (abstract) (1977), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-67-10-1361.

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
[CrossRef]

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
[CrossRef]

1976 (1)

Acosta, E.

Acton, D. S.

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Aronstein, D. L.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Bará, S.

Bowers, C. W.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Carey, L.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Cederquist, J. N.

Chanan, G.

Chidlaw, R.

A. J. Devaney, R. A. Gonsalves, and R. Chidlaw, “Application of phase retrieval techniques to adaptive imaging systems,” J. Opt. Soc. Am. 67, 1422 (abstract) (1977), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-67-10-1361.

Contos, A.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Contos, A. R.

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmorgorov phase screen,” Waves Random Media 2, 209–224(1992).
[CrossRef]

Dean, B.

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

Dean, B. H.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Dekens, F.

Devaney, A. J.

A. J. Devaney, R. A. Gonsalves, and R. Chidlaw, “Application of phase retrieval techniques to adaptive imaging systems,” J. Opt. Soc. Am. 67, 1422 (abstract) (1977), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-67-10-1361.

Feinberg, L. D.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Fienup, J. R.

S. T. Thurman and J. R. Fienup, “Phase retrieval with signal bias,” J. Opt. Soc. Am. A 26, 1008–1014 (2009).
[CrossRef]

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]

T. Zielinski and J. R. Fienup, “Extending wavefront sensing capture range for segmented systems through tip and tilt estimation,” in Frontiers in Optics, Tech. Dig. (CD) (Optical Society of America, 2006), paper FMF4.

Fried, D. L.

Glindemann, A.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmorgorov phase screen,” Waves Random Media 2, 209–224(1992).
[CrossRef]

Gonsalves, R. A.

A. J. Devaney, R. A. Gonsalves, and R. Chidlaw, “Application of phase retrieval techniques to adaptive imaging systems,” J. Opt. Soc. Am. 67, 1422 (abstract) (1977), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-67-10-1361.

Goodman, J.

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

Hansen, K.

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

Hayden, W.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Hudgin, R. H.

Kirkman, D.

Kryskowski, D.

Lane, R. G.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmorgorov phase screen,” Waves Random Media 2, 209–224(1992).
[CrossRef]

Lyon, R. G.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Mast, T.

Meza, L.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Michaels, S.

Nelson, J.

Noll, R. J.

Ohara, C.

Rama, M. A.

Redding, D. C.

Rios, S.

Robinson, S. R.

Roddier, F.

Sabatke, E.

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Schwenker, J.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

Shi, F.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

F. Shi, G. Chanan, C. Ohara, M. Troy, and D. C. Redding, “Experimental verification of dispersed fringe sensing as a segment phasing technique using the Keck telescope,” Appl. Opt. 434474–4481 (2004).
[CrossRef] [PubMed]

Shields, D.

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Shiri, R.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Smith, J. S.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Smith, S.

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

Southwell, W. H.

Thurman, S. T.

Towell, T.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

Troy, M.

Wackerman, C. C.

Zielinski, T.

T. Zielinski and J. R. Fienup, “Extending wavefront sensing capture range for segmented systems through tip and tilt estimation,” in Frontiers in Optics, Tech. Dig. (CD) (Optical Society of America, 2006), paper FMF4.

Appl. Opt. (3)

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Proc. SPIE (2)

D. S. Acton, T. Towell, J. Schwenker, D. Shields, E. Sabatke, A. R. Contos, K. Hansen, F. Shi, B. Dean, and S. Smith, “End-to-end commissioning demonstration of the James Webb Space Telescope,” Proc. SPIE 6687, 668706 (2007).
[CrossRef]

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
[CrossRef]

Waves Random Media (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmorgorov phase screen,” Waves Random Media 2, 209–224(1992).
[CrossRef]

Other (8)

URL:http://keckobservatory.org/.

URL:http://rhea.as.utexas.edu/.

URL:http://www.salt.ac.za/.

URL:http://www.jwst.nasa.gov/.

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

“ZEMAX Optical Design Program, User’s Guide” (ZEMAX Development Corporation, 2006).

Performance is given here in terms of the wavefront error seen in reflection, which is a factor of 2 larger than the specifications reported in , which are in terms of surface error.

T. Zielinski and J. R. Fienup, “Extending wavefront sensing capture range for segmented systems through tip and tilt estimation,” in Frontiers in Optics, Tech. Dig. (CD) (Optical Society of America, 2006), paper FMF4.

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Figures (9)

Fig. 1
Fig. 1

Plane z = 0 corresponds to the exit pupil of an imaging system. Point P = ( 0 , 0 , R ) is the geometric image of an on-axis point source. The dashed curve in the exit pupil plane represents a reference sphere of radius R centered on P, while the solid curve represents the aberrated wavefront. The system aberrations are characterized by the wavefront aberration function w ( x , y ) . y m indicates the height of ray m in the pupil plane. The dotted line represents the ideal ray trajectory that passes through P, with ray height r y , m ( z ) . The solid line represents the actual ray trajectory, which deviates from the ideal ray height by ε y , m ( z ) . The angular deviation s y , m of the actual ray from the ideal ray is equal to the slope of the wavefront aberration function w ( x , y ) / y .

Fig. 2
Fig. 2

Simulation example: (a) wavefront aberration function w ( x , y ) in units of wavelength, where the small white square indicates the pupil plane coordinates ( x m , y m ) of a particular ray m; (b) the corresponding joint probability distribution p m , joint ( s x , s y ) for the angular deviation of ray m computed using Eqs. (5, 6); (c) wavefront slope data ( s x , m , s y , m ) at M = 235 points obtained through Eq. (7); (d) corresponding wavefront aberration function reconstructed using Eqs. (8, 9, 10) with K = 91 basis functions; (e)–(i) through-focus intensity measurements I n ( x , y ) of the system PSF, where the small white squares indicate the ideal image domain coordinates [ r x , m ( z n ) , r y , m ( z n ) ] for ray m shown in (a); and (j)–(n) the corresponding probability distributions p m , n ( s x , s y ) computed by low-pass filtering each I n ( x , y ) , shifting by the ideal ray coordinates [ r x , m ( z n ) , r y , m ( z n ) ] , and scaling to angular coordinates [see Eq. (5)]. The joint probability distribution p m , joint ( s x , s y ) shown in (b) is the product of the individual distributions p m , n ( s x , s y ) shown in (j)–(n) [see Eq. (6)].

Fig. 3
Fig. 3

Monte Carlo simulation results: plot of the RMS error in the estimated wavefront σ K versus the RMS strength of the actual wavefront aberrations w ( x , y ) . These results correspond to the case where each w ( x , y ) was an atmospheric phase screen with Kolmogorov statistics, and five PSF measurements were available.

Fig. 4
Fig. 4

Same as Fig. 3, but for the case where each w ( x , y ) was an atmospheric phase screen with Kolmogorov statistics projected onto the first K = 91 Zernike polynomials, and five PSF measurements were available.

Fig. 5
Fig. 5

Same as Fig. 3, but for the case where each w ( x , y ) was an atmospheric phase screen with Kolmogorov statistics projected onto the first K = 91 Zernike polynomials, five PSF measurements were available, and a centroiding algorithm was used to determine the optic axis for each PSF measurement.

Fig. 6
Fig. 6

Same as Fig. 3, but for the case where each w ( x , y ) was an atmospheric phase screen with Kolmogorov statistics projected onto the first K = 91 Zernike polynomials, only three PSF measurements were available, and a centroiding algorithm was used to determine the optic axis for each PSF measurement.

Fig. 7
Fig. 7

Segmented-aperture simulation example: (a) actual wavefront aberration function for a system with J = 18 segments (segment tilt errors dominate the wavefront error), (b) corresponding initial wavefront estimate of segment tilt errors obtained using the proposed wavefront slope estimation technique, (c) wavefront difference between (a) and (b), and (d) corresponding wavefront estimate obtained using the blind phase retrieval approach. All wavefronts are displayed in units of wavelength.

Fig. 8
Fig. 8

Results of the capture range study using the proposed method: RMS error of estimated wavefronts versus the actual RMS wavefront errors, where the circles and crosses represent converged and unconverged trials, respectively.

Fig. 9
Fig. 9

Results of the capture range study: fraction of simulation trials that converged to the desired true solution versus the wavefront error scaling parameter κ. The dashed curve represents the blind phase retrieval results, while the solid curve represents results using the proposed method.

Tables (1)

Tables Icon

Table 1 Ordering and Form of Wavefront Basis Functions ϕ j , k ( x , y ) and Statistics of Segment Wavefront Error Coefficients c j , k Used for the Capture Range Study

Equations (11)

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r x , m ( z ) = x m [ 1 ( z / R ) ] r y , m ( z ) = y m [ 1 ( z / R ) ] .
s x , m = w ( x m , y m ) x s y , m = w ( x m , y m ) y .
ε x , m ( z ) = s x , m z ε y , m ( z ) = s y , m z .
I n ( x , y ) = | A ( x , y ) exp [ i 2 π λ w ( x , y ) ] × exp [ i π λ ( 1 z n 1 R ) ( x 2 + y 2 ) ] exp [ i 2 π λ z n ( x x + y y ) ] d x d y | 2 ,
p m , n ( s x , s y ) = c n I n [ z n s x + r x , m ( z n ) , z n s y + r y , m ( z n ) ] * L ( s x , s y ) ,
p m , joint ( s x , s y ) = n = 1 N p m , n ( s x , s y ) .
( s x , m , s y , m ) = max ( s x , s y ) [ p m , joint ( s x , s y ) ] .
w ( x , y ) = k = 1 K c k ψ k ( x , y ) ,
m = 1 M [ ψ j ( x m , y m ) x ψ k ( x m , y m ) x + ψ j ( x m , y m ) y ψ k ( x m , y m ) y ] = δ j , k ,
c k = m = 1 M [ s x , m ψ k ( x m , y m ) x + s y , m ψ k ( x m , y m ) y ] .
P ( x , y ) = j = 1 J A j ( x , y ) exp [ i 2 π k = 1 K c k ϕ k ( x , y ) + c j , k ϕ j , k ( x , y ) ] ,

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