Abstract

A segmented-aperture telescope such as the Multiple-Mirror Telescope will suffer from phase errors unless the segments are aligned to within a small fraction of a wavelength. Such a coherent alignment of the segments is difficult to achieve in real time. An alternative is to record the images degraded by phase errors and to restore them after detection by using phase-retrieval techniques. In this paper we describe the use of Gonsalves’s phase-diversity method (which was previously used to combat atmospheric turbulence) to correct imagery blurred by a misaligned segmented-aperture telescope. Two images are recorded simultaneously: the usual degraded image in the focal plane and a second degraded image in an out-of-focus plane. An iterative gradient-search algorithm finds the phase error of the telescope that is consistent with both degraded images. We refer to this technique as the method of multiple-plane measurements with iterative reconstruction. The final image is obtained by a Wiener–Helstrom filtering of the degraded image using the retrieved phase errors. The results of reconstruction experiments performed with simulated data including the effects of noise are shown for the case of random piston phase errors on each of six segments.

© 1988 Optical Society of America

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References

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  1. E. K. Hege, J. M. Beckers, P. A. Strittmatter, D. W. McCarthy, “Multiple Mirror Telescope as a phased array telescope,” Appl. Opt. 24, 2565–2576 (1985).
    [CrossRef] [PubMed]
  2. T. S. Mast, J. E. Nelson, “Status report on the W. M. Keck Observatory and Ten Meter Telescope,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 204–206 (1986).
    [CrossRef]
  3. D. Enard, “The ESO Very Large Telescope project: present status,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 221–226 (1986).
    [CrossRef]
  4. K. L. Shu, S. Eisenberg, “Planning the National New Technology Telescope (NNTT): I. Optical designs,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 66–79 (1986).
    [CrossRef]
  5. R. R. Butts, S. J. Cusumano, J. S. Fender, C. R. DeHainaut, “Phasing concept for an array of mutually coherent laser transmitters,” Opt. Eng. 26, 553–558 (1987).
    [CrossRef]
  6. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
    [CrossRef] [PubMed]
  7. R. A. Gonsalves, R. Childlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.207, 32–39 (1979).
    [CrossRef]
  8. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [CrossRef]
  9. D. G. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).
  10. C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. 57, 297–303 (1967).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

1987

R. R. Butts, S. J. Cusumano, J. S. Fender, C. R. DeHainaut, “Phasing concept for an array of mutually coherent laser transmitters,” Opt. Eng. 26, 553–558 (1987).
[CrossRef]

1985

1982

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

1975

1967

Beckers, J. M.

Butts, R. R.

R. R. Butts, S. J. Cusumano, J. S. Fender, C. R. DeHainaut, “Phasing concept for an array of mutually coherent laser transmitters,” Opt. Eng. 26, 553–558 (1987).
[CrossRef]

Childlaw, R.

R. A. Gonsalves, R. Childlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.207, 32–39 (1979).
[CrossRef]

Cusumano, S. J.

R. R. Butts, S. J. Cusumano, J. S. Fender, C. R. DeHainaut, “Phasing concept for an array of mutually coherent laser transmitters,” Opt. Eng. 26, 553–558 (1987).
[CrossRef]

DeHainaut, C. R.

R. R. Butts, S. J. Cusumano, J. S. Fender, C. R. DeHainaut, “Phasing concept for an array of mutually coherent laser transmitters,” Opt. Eng. 26, 553–558 (1987).
[CrossRef]

Eisenberg, S.

K. L. Shu, S. Eisenberg, “Planning the National New Technology Telescope (NNTT): I. Optical designs,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 66–79 (1986).
[CrossRef]

Enard, D.

D. Enard, “The ESO Very Large Telescope project: present status,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 221–226 (1986).
[CrossRef]

Fender, J. S.

R. R. Butts, S. J. Cusumano, J. S. Fender, C. R. DeHainaut, “Phasing concept for an array of mutually coherent laser transmitters,” Opt. Eng. 26, 553–558 (1987).
[CrossRef]

Gonsalves, R. A.

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

R. A. Gonsalves, R. Childlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.207, 32–39 (1979).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Hege, E. K.

Helstrom, C. W.

Luenberger, D. G.

D. G. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).

Mast, T. S.

T. S. Mast, J. E. Nelson, “Status report on the W. M. Keck Observatory and Ten Meter Telescope,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 204–206 (1986).
[CrossRef]

McCarthy, D. W.

Nelson, J. E.

T. S. Mast, J. E. Nelson, “Status report on the W. M. Keck Observatory and Ten Meter Telescope,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 204–206 (1986).
[CrossRef]

Shu, K. L.

K. L. Shu, S. Eisenberg, “Planning the National New Technology Telescope (NNTT): I. Optical designs,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 66–79 (1986).
[CrossRef]

Strittmatter, P. A.

Wyant, J. C.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Eng.

R. R. Butts, S. J. Cusumano, J. S. Fender, C. R. DeHainaut, “Phasing concept for an array of mutually coherent laser transmitters,” Opt. Eng. 26, 553–558 (1987).
[CrossRef]

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

Other

D. G. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).

R. A. Gonsalves, R. Childlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.207, 32–39 (1979).
[CrossRef]

T. S. Mast, J. E. Nelson, “Status report on the W. M. Keck Observatory and Ten Meter Telescope,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 204–206 (1986).
[CrossRef]

D. Enard, “The ESO Very Large Telescope project: present status,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 221–226 (1986).
[CrossRef]

K. L. Shu, S. Eisenberg, “Planning the National New Technology Telescope (NNTT): I. Optical designs,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 66–79 (1986).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

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Figures (7)

Fig. 1
Fig. 1

Cross section of the MMT adapted for multiple-plane imaging.

Fig. 2
Fig. 2

Simulation of the aperture for a MMT-like system.

Fig. 3
Fig. 3

Piston configuration representing a true solution for simulations.

Fig. 4
Fig. 4

Modulation transfer functions for aligned and misaligned systems with and without phase diversity.

Fig. 5
Fig. 5

Noiseless-data images and reconstructions: (A) original object (Jupiter), (B) image of object through aligned system, (C) diversity image for aligned system (0.5-wave diversity), (D) Wiener–Helstrom restoration of (B), (E) image through misaligned system, (F) diversity image for misaligned system, (G) restoration of (E) obtained by using a Wiener–Helstrom filter for an aligned system, (H) restoration of (E) obtained by using a Wiener–Helstrom filter constructed with MMIR parameter estimates, and (I) restoration of (E) obtained by using a Wiener–Helstrom filter constructed with true misalignment parameters.

Fig. 6
Fig. 6

One-dimensional cuts of normalized objective function created by varying a single piston parameter: (A) variation over one wave with noisy data, (B) structure at a magnification of 10×, (C) structure at a magnification of 100×, (D) objective function resulting from truncated error metric with noisy data, and (E) objective function resulting from truncated error metric with noiseless data.

Fig. 7
Fig. 7

Data images with 1% noise added and reconstructions: (A) original object, (B) image of object through aligned system, (C) diversity image for aligned system (0.5-wave diversity), (D) Wiener–Helstrom restoration of (B), (E) image through misaligned system, (F) diversity image for misaligned system, (G) restoration of (E) obtained by using a Wiener–Helstrom filter for an aligned system, (H) restoration of (E) obtained by using a Wiener–Helstrom filter constructed with MMIR parameter estimates, (I) restoration of (E) obtained by using a Wiener–Helstrom filter constructed with true misalignment parameters.

Tables (2)

Tables Icon

Table 1 Multiple-Initial-Estimate Experiment Results

Tables Icon

Table 2 Reconstruction Parameters for 1 % Noisea

Equations (28)

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g ( x ) = f ( x ) * s ( x ) ,
G ( u ) = F ( u ) S ( u ) .
S ( u ) = C ( u ) C ( u ) ,
C ( u ) = n = 1 N A n ( u ) exp [ i ϕ n ( u ) ] ,
C d ( u ) = n = 1 N A n ( u ) exp { i [ ϕ n ( u ) + Δ ϕ ( u ) ] } ,
S d ( u ) = C d ( u ) C d ( u ) ,
G ( u ) = S ( u ) F ( u ) ,
G d ( u ) = S d ( u ) F ( u ) .
E = u | G ( u ) Ŝ ( u ) F ̂ ( u ) | 2 + u | G d ( u ) Ŝ d ( u ) F ̂ ( u ) | 2 .
F ̂ ( u ) = Ŝ * ( u ) G ( u ) + Ŝ d * ( u ) G d ( u ) | Ŝ ( u ) | 2 + | Ŝ d ( u ) | 2 .
E = u | G ( u ) Ŝ d ( u ) G d ( u ) Ŝ ( u ) | 2 | Ŝ ( u ) | 2 + | Ŝ d ( u ) | 2 .
F ̂ ( u ) = G ( u ) Ŝ * ( u ) | Ŝ ( u ) | 2 + P n ( u ) P f ( u ) ,
Δ ϕ ( u ) = π Δ Z λ Z f 2 | u | 2 ,
Number of waves of defocus = Δ Z 8 λ ( F # ) 2 ,
E = u | G Ŝ d G d Ŝ | 2 | Ŝ | 2 + | Ŝ d | 2 ,
E k m n = u k m n | G Ŝ d G d Ŝ | 2 | Ŝ | 2 + | Ŝ d | 2
= u k m n ( G Ŝ d G d Ŝ ) ( G * Ŝ d * G d * Ŝ * ) Ŝ Ŝ * + Ŝ d Ŝ d * ,
E k m n = u ( Ŝ d Ŝ k m n Ŝ Ŝ d k m n ) ( G Ŝ * + G d Ŝ d * ) ( G d * Ŝ * G * Ŝ d * ) + c . c . ( | Ŝ | 2 + | Ŝ d | 2 ) 2 ,
C ( u ) = n = 1 N Circ ( u u n ) exp [ i m = 1 M k m n θ m ( u u n ) ] ,
Circ ( u ) = { 1 if | u | r 0 otherwise ;
S = C C .
S ( u ) = { n = 1 N Circ ( u u n ) exp [ i m = 1 M k m n θ m ( u u n ) ] } × { n = 1 N Circ ( u u n u ) × exp [ i m = 1 M k m n θ m ( u u n u ) ] } d u .
S k p q = Circ ( u u q ) i θ p ( u u q ) × exp [ i m = 1 M k m q θ m ( u u q ) ] × { n = 1 N Circ ( u u n u ) × exp [ i m = 1 M k m n θ m ( u u n u ) ] } d u Circ ( u u q u ) i θ p ( u u q u ) × exp [ i m = 1 M k m q θ m ( u u q u ) ] × { n = 1 N Circ ( u u n ) × exp [ i m = 1 M k m n θ m ( u u n ) ] } d u .
B p q ( u ) Circ ( u u q ) i θ p ( u u q ) exp [ i m = 1 M k m q θ m ( u u q ) ] .
S k p q = ( B p q C ) + ( C B p q ) .
( C B p q ) ( u ) = [ ( B p q C ) ( u ) ] * .
B q C = i θ n = 1 N exp [ i θ ( k q k n ) ] × Circ ( u u q ) Circ ( u u n ) = i θ n = 1 N exp [ i θ ( k q k n ) ] × ( Circ Circ ) ( u + u n u q ) .
S k q .

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