Abstract

Conventional Hartmann sensor processing relies on locating the centroid of the image that is formed behind each element of a lenslet array. These centroid locations are used for computing the local gradient of the incident aberration, from which the phase of the incident wave front is calculated. The largest aberration that can reliably be sensed in a conventional Hartmann sensor must have a local gradient small enough that the spot formed by each lenslet is confined to the area behind the lenslet: If the local gradient is larger, spots form under nearby lenslets, causing a form of cross talk between the wave-front sensor channels. We describe a wave-front reconstruction algorithm that processes the whole image measured by a Hartmann sensor and a conventional image that is formed by use of the incident aberration. We show that this algorithm can accurately estimate aberrations for cases in which the aberration is strong enough to cause many of the images formed by individual lenslets to fall outside the local region of the Hartmann sensor detector plane defined by the edges of a lenslet.

© 1998 Optical Society of America

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References

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  1. M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).
  2. R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winker, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Sindle, R. A. Cleis, “Two generations of laser-guide-star adaptive-optics experiments at the starfire optical range,” J. Opt. Soc. Am. A 11, 310–314 (1994).
    [CrossRef]
  3. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  4. A. Watson, “Hubble successor gathers support,” Science 272, 1735 (1996).
    [CrossRef]
  5. T. Reichhardt, A. Abbott, D. Swinbanks, “What will be the next big thing?” Nature (London) 381, 465 (1996).
    [CrossRef]
  6. J. R. Feinup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
    [CrossRef]
  7. A. Wirth, A. Jankevics, F. Landers, C. Baird, T. Berkopec, “Final report on the testing of the CIRS telescopes using the Hartmann technique,” Tech. Rep. NAS5-31786, Task 013 (Adaptive Optics Associates, Cambridge, Mass., 1993).
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. J. R. Feinup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef]
  10. D. A. Pierre, Optimization Theory with Applications (Dover, New York, 1986).
  11. M. A. Branch, A. Grace, MATLAB Optimization Toolbox (Math Works, Natick, Mass., 1996).
  12. A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Signals and Systems, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1997).
  13. R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993).
  14. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

1996 (2)

A. Watson, “Hubble successor gathers support,” Science 272, 1735 (1996).
[CrossRef]

T. Reichhardt, A. Abbott, D. Swinbanks, “What will be the next big thing?” Nature (London) 381, 465 (1996).
[CrossRef]

1994 (1)

1993 (1)

1982 (1)

1976 (1)

Abbott, A.

T. Reichhardt, A. Abbott, D. Swinbanks, “What will be the next big thing?” Nature (London) 381, 465 (1996).
[CrossRef]

Baird, C.

A. Wirth, A. Jankevics, F. Landers, C. Baird, T. Berkopec, “Final report on the testing of the CIRS telescopes using the Hartmann technique,” Tech. Rep. NAS5-31786, Task 013 (Adaptive Optics Associates, Cambridge, Mass., 1993).

Berkopec, T.

A. Wirth, A. Jankevics, F. Landers, C. Baird, T. Berkopec, “Final report on the testing of the CIRS telescopes using the Hartmann technique,” Tech. Rep. NAS5-31786, Task 013 (Adaptive Optics Associates, Cambridge, Mass., 1993).

Boeke, B. R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Branch, M. A.

M. A. Branch, A. Grace, MATLAB Optimization Toolbox (Math Works, Natick, Mass., 1996).

Cleis, R. A.

Ellerbroek, B. L.

Feinup, J. R.

Fugate, R. Q.

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grace, A.

M. A. Branch, A. Grace, MATLAB Optimization Toolbox (Math Works, Natick, Mass., 1996).

Higgins, C. H.

Jankevics, A.

A. Wirth, A. Jankevics, F. Landers, C. Baird, T. Berkopec, “Final report on the testing of the CIRS telescopes using the Hartmann technique,” Tech. Rep. NAS5-31786, Task 013 (Adaptive Optics Associates, Cambridge, Mass., 1993).

Jelonek, M. P.

Landers, F.

A. Wirth, A. Jankevics, F. Landers, C. Baird, T. Berkopec, “Final report on the testing of the CIRS telescopes using the Hartmann technique,” Tech. Rep. NAS5-31786, Task 013 (Adaptive Optics Associates, Cambridge, Mass., 1993).

Lange, W. J.

Marron, J. C.

Moroney, J. F.

Nawab, S. H.

A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Signals and Systems, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1997).

Noll, R. J.

Oliker, M. D.

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Signals and Systems, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1997).

Pierre, D. A.

D. A. Pierre, Optimization Theory with Applications (Dover, New York, 1986).

Reichhardt, T.

T. Reichhardt, A. Abbott, D. Swinbanks, “What will be the next big thing?” Nature (London) 381, 465 (1996).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).

Ruane, R. E.

Schulz, T. J.

Seldin, J. H.

Sindle, D. W.

Slavin, A. C.

Spinhirne, J. M.

Swinbanks, D.

T. Reichhardt, A. Abbott, D. Swinbanks, “What will be the next big thing?” Nature (London) 381, 465 (1996).
[CrossRef]

Watson, A.

A. Watson, “Hubble successor gathers support,” Science 272, 1735 (1996).
[CrossRef]

Welsh, B. M.

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).

Wild, W. J.

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Signals and Systems, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1997).

Winker, D. M.

Wirth, A.

A. Wirth, A. Jankevics, F. Landers, C. Baird, T. Berkopec, “Final report on the testing of the CIRS telescopes using the Hartmann technique,” Tech. Rep. NAS5-31786, Task 013 (Adaptive Optics Associates, Cambridge, Mass., 1993).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993).

Wynia, J. M.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nature (London) (1)

T. Reichhardt, A. Abbott, D. Swinbanks, “What will be the next big thing?” Nature (London) 381, 465 (1996).
[CrossRef]

Science (1)

A. Watson, “Hubble successor gathers support,” Science 272, 1735 (1996).
[CrossRef]

Other (8)

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).

D. A. Pierre, Optimization Theory with Applications (Dover, New York, 1986).

M. A. Branch, A. Grace, MATLAB Optimization Toolbox (Math Works, Natick, Mass., 1996).

A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Signals and Systems, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1997).

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

A. Wirth, A. Jankevics, F. Landers, C. Baird, T. Berkopec, “Final report on the testing of the CIRS telescopes using the Hartmann technique,” Tech. Rep. NAS5-31786, Task 013 (Adaptive Optics Associates, Cambridge, Mass., 1993).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (4)

Fig. 1
Fig. 1

Diagram of the Hartmann sensor concept.

Fig. 2
Fig. 2

Wave-front sensor image with an aberrated input wave field. Simulated image formed in the focal plane of a Hartmann sensor for the case of a strongly aberrated input wave front. The grid of squares superimposed upon the image indicates the location of the subapertures. In this case a circular pupil was also imposed, as indicated by truncation of the subapertures near the edges of the pupil. A negative image is shown for clarity.

Fig. 3
Fig. 3

Block diagram of the aberration-sensing concept.

Fig. 4
Fig. 4

Examples of inputs to the wave-front sensing simulation: (a) The input aberration. (b) Simulated conventional image. (c) Simulated wave-front sensor image with the edges of the subapertures superimposed.

Tables (3)

Tables Icon

Table 1 Aberration-Sensing Results for Sensing a Linear Combination of Zernike Polynomials 2–11a

Tables Icon

Table 2 Aberration-Sensing Results for Sensing Only Spherical Aberration, Zernike Polynomial 11a

Tables Icon

Table 3 Aberration-Sensing Results for Sensing Only Defocus Aberration, Zernike Polynomial 4a

Equations (37)

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ϕ ab x = ϕ ab R x n = 11.5 Z 11 x n ,
t H x = S   exp j   k w 2 f l | x - x c s | 2 rect x - x c s d ,
U ab x = P x exp j ϕ ab x ,
U L x = U ab x t H x .
H A f = exp j 2 π   f l λ 1 - λ | f | 2 1 / 2 ,
U D x D = - 1 U L x H A f ,
I D x D = | U D x D | 2 .
U I x i = κ U ab x ,
I I x i = | U I x i | 2 .
ϕ ˜ ab R x n = k = 2 K   α k Z k x n ,
J α ¯ = x D I D x D 1 / 2 - Ĩ D x D ,   α ¯ 1 / 2 2 + x i I I x i 1 / 2 - Ĩ I x i ,   α ¯ 1 / 2 2 ,
k J = J α ¯ α k .
H kl = 2 J α ¯ α k α l .
J D α ¯ = x D   Ĩ D x D ,   α ¯ + x D   I D x D - 2   x D Ĩ D x D ,   α ¯ I D x D 1 / 2 ,
J D α ¯ α k = α k - 2   x D Ĩ D x D ,   α ¯ I D x D 1 / 2 = - x D I D x D Ĩ D x D ,   α ¯ 1 / 2 Ĩ D x D ,   α ¯ α k .
Ĩ D x D ,   α ¯ α k = 2 Ũ D x D ,   α ¯ α k   Ũ D * x D ,   α ¯ ,
Ũ D x D ,   α ¯ = x D   t H x exp j   k   α k Z k x h W x ,   x D ,
Ĩ D x D ,   α ¯ α k = - 2 x   t H x Z k x exp j   k   α k Z k x × h W x ,   x D Ũ D * x D ,   α ¯ ,
J D α ¯ α k = 2 x D I D x D Ĩ D x D ,   α ¯ 1 / 2 Ũ D * x D ,   α ¯ × x   t H x Z k x exp j   k   α k Z k x   h W x ,   x D   = 2 x   t H x Z k x exp j   k   α k Z k x × x D I D x D Ĩ D x D ,   α ¯ 1 / 2 Ũ D * x D ,   α ¯ h W x ,   x D .
I D x D Ĩ D x D ,   α ¯ 1 / 2 Ũ D * x D ,   α ¯
J I α ¯ α k = 2 x   P x Z k x exp j   k   α k Z k x × x i I I x i Ĩ I x i ,   α ¯ 1 / 2 Ũ I * x i ,   α ¯ h I x ,   x i ,
I I x i Ĩ I x i ,   α ¯ 1 / 2 Ũ I * x i ,   α ¯ .
ϕ ab x = k = 2 K   α k Z k x .
ϕ l x = k w 2 f l   x 2 ,
1 2 π d ϕ l x d x = x λ f l .
1 2 π d ϕ l x d x max = d 2 λ f l .
T min = 2 λ f l d .
T min Δ x 3
Δ x 2 λ f l 3 d .
ϕ H f = 2 π f l λ 1 - λ f 2 .
1 2 π d ϕ H f d f 2 λ f l f .
1 2 π d ϕ H f d f max 2 λ f l 1 2 Δ x = λ f l Δ x .
P = Δ x λ f l .
P Δ f 3 .
Δ f = 1 N Δ x ,
N 3 λ f l Δ x 2 .
2 = k = 2 K α ˜ k - α 2 .

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