Abstract

We consider the Neumann boundary-value problem for curvature adaptive optics systems. We show that, because curvature sensors average over extended regions of the wave front, inconsistent data for the solution of the Neumann problem result when the measurements are treated as local. Because this inconsistency is generally resolved passively in the adaptive mirror itself, it can be interpreted as an uncontrolled degree of freedom of the system. We offer several procedures for treating the data in a more consistent fashion.

© 2001 Optical Society of America

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References

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  1. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  2. C. Roddier, F. Roddier, “Combined approach to the Hubble Space Telescope wave-front distortion analysis,” Appl. Opt. 32, 2992–3008 (1993).
    [CrossRef] [PubMed]
  3. C. Roddier, F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277–2287 (1993).
    [CrossRef]
  4. F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
    [CrossRef]
  5. F. Roddier, “Maximum gain and efficiency of adaptive optics systems,” Publ. Astron. Soc. Pac. 110, 837–840 (1998).
    [CrossRef]
  6. F. Rigaut, B. L. Ellerbroek, M. J. Northcott, “Comparison of curvature-based and Shack–Hartmann-based adaptive optics for the Gemini telescope,” Appl. Opt. 36, 2856–2868 (1997).
    [CrossRef] [PubMed]
  7. C. Schwartz, E. Ribak, S. G. Lipson, “Bimorph adaptive mirrors and curvature sensing,” J. Opt. Soc. Am. A 11, 895–902 (1994).
    [CrossRef]
  8. F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
    [CrossRef]
  9. This is of course trivially true because, in general, the slopes sk are discontinuous, having been averaged over finite regions. In this case it is easy to see that this is resolved by the intrinsic stiffness of the mirror that prevents the formation of a slope or surface discontinuity, resulting in a smooth slope distribution. This illustrates the interplay of mechanical properties and boundary conditions in the determination of the final shape of curvature systems.
  10. G. Barton, “Peculiarities of the Neumann problem,” in Elements of Green’s Functions and Propagation, Oxford Science Publications (Clarendon, Oxford, UK, 1989), Sect. 6.1, p. 142.
  11. This can be seen most easily when Eq. (1) is integrated over R and by use of Gauss’s law.
  12. L. Salas, “Variable separation in curvature sensing: fast method for solving the irradiance transport equation in the context of optical telescopes,” Appl. Opt. 35, 1593–1596 (1996).
    [CrossRef] [PubMed]
  13. I. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34, 1232–1237 (1995).
    [CrossRef]
  14. F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
    [CrossRef] [PubMed]

1998 (2)

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

F. Roddier, “Maximum gain and efficiency of adaptive optics systems,” Publ. Astron. Soc. Pac. 110, 837–840 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

I. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34, 1232–1237 (1995).
[CrossRef]

1994 (1)

1993 (2)

1991 (2)

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
[CrossRef] [PubMed]

1988 (1)

Arsenault, R.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Barton, G.

G. Barton, “Peculiarities of the Neumann problem,” in Elements of Green’s Functions and Propagation, Oxford Science Publications (Clarendon, Oxford, UK, 1989), Sect. 6.1, p. 142.

Boyer, C.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Crampton, D.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Ellerbroek, B. L.

Fletcher, J. M.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Gigan, P.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Graves, J. E.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Han, I.

I. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34, 1232–1237 (1995).
[CrossRef]

Jagourel, P.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Lai, O.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Lipson, S. G.

Northcott, M.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Northcott, M. J.

Ribak, E.

Rigaut, F.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

F. Rigaut, B. L. Ellerbroek, M. J. Northcott, “Comparison of curvature-based and Shack–Hartmann-based adaptive optics for the Gemini telescope,” Appl. Opt. 36, 2856–2868 (1997).
[CrossRef] [PubMed]

Roddier, C.

Roddier, F.

Rouan, D.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Salas, L.

Salmon, D.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Schwartz, C.

Stilburn, J.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Thomas, J.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Veran, J. P.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Appl. Opt. (5)

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

Opt. Eng. (1)

I. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34, 1232–1237 (1995).
[CrossRef]

Publ. Astron. Soc. Pac. (3)

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J. P. Veran, P. Gigan, D. Crampton, J. M. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada-France-Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

F. Roddier, “Maximum gain and efficiency of adaptive optics systems,” Publ. Astron. Soc. Pac. 110, 837–840 (1998).
[CrossRef]

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Other (3)

This is of course trivially true because, in general, the slopes sk are discontinuous, having been averaged over finite regions. In this case it is easy to see that this is resolved by the intrinsic stiffness of the mirror that prevents the formation of a slope or surface discontinuity, resulting in a smooth slope distribution. This illustrates the interplay of mechanical properties and boundary conditions in the determination of the final shape of curvature systems.

G. Barton, “Peculiarities of the Neumann problem,” in Elements of Green’s Functions and Propagation, Oxford Science Publications (Clarendon, Oxford, UK, 1989), Sect. 6.1, p. 142.

This can be seen most easily when Eq. (1) is integrated over R and by use of Gauss’s law.

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Equations (13)

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

2ux, y=ψx, y,  rrf,
ψx, y=i=1ncciΘix, y,
Sx, y=k=1nsskΦkx, y,
urr=rf=k=1nsskΦkx, y,
Rψx, ydA=Rnux, ydl.
i=1ncciAi=k=1nsskLk,
πrf2c=2πrfs,
c=i=1ncciAii=1nsAi=1πrf2i=1ncciAi,
s=k=1nsskLkk=1nsLk=12πrfk=1nsskLk.
s=43πα1+rf-rf1+rf.
frf=43παrfrf+2rf-1rf+1.
λ=c¯-2rfs¯.
τ=σλσc¯=1-2rfσs¯σc¯,

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