Abstract

Lucky imaging, used with some success in astronomical and even horizontal-path imaging, relies on fleeting conditions of the atmosphere that allow momentary improvements in image quality, at least in portions of an image. Aperture synthesis allows a larger aperture and, thus, a higher-resolution imaging system to be synthesized through the superposition of image spatial-frequency components gathered by cooperative combinations of smaller subapertures. A combination of lucky imaging and aperture synthesis strengthens both methods for obtaining improved images through the turbulent atmosphere. We realize the lucky imaging condition appropriate for aperture synthesis imaging for a pair of rectangular subapertures and demonstrate that this condition occurs when the signal energy associated with bandpass spatial-frequency components achieves its maximum value.

© 2008 Optical Society of America

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References

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  1. “Woods Hole summer study 1968: synthetic-aperture optics,” in Proceedings of the August 1967 Woods Hole Summer Study (U.S. National Academy of Sciences, 1968), Vols. I and II.
  2. G. W. Stroke, “Optical aperture synthesis using successive exposure of a single photograph and spatial filtering “low-frequency redundancy” suppression,” Phys. Lett. A 30, 485-486 (1969).
    [CrossRef]
  3. M. Ryle, A. Hewish, and J. R. Shakeshaft, “The synthesis of large radio telescopes by the use of radio interferometers,” IRE Trans. Antennas Propag. 7, 120-124(1959).
    [CrossRef]
  4. M. Ryle, “A new radio interferometer and its application to the observation of weak radio stars,” Proc. R. Soc. London. Ser. A 211, 351-375 (1952).
  5. A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University Press, 2006), Chaps. 5 and 6.
    [CrossRef]
  6. P. G. Tuthill, J. D. Monnier, W. C. Danchi, E. H. Wishnow, and C. A Haniff, “Michelson interferometry with the Keck I telescope,” Publ. Astron. Soc. Pac. 112 (770), 555-565 (2000).
    [CrossRef]
  7. D. H. Rogstad, “A technique for measuring visibility phase with an optical interferometer in the presence of atmospheric seeing,” Appl. Opt. 7, 585-588 (1968).
    [CrossRef] [PubMed]
  8. W. T. Rhodes and J. W. Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,” J. Opt. Soc. Am. 63, 647-657 (1973).
    [CrossRef]
  9. W. T. Rhodes, “Digital processing of synthetic aperture optical imagery,” Opt. Eng. 13, 267-274 (1974).
  10. D. L. Fried, “Probability of getting a lucky short-exposure image through turbulence,” J. Opt. Soc. Am. 68, 1651-1658 (1978).
    [CrossRef]
  11. N. M. Law, C. D. Mackay, and J. E. Baldwin, “Lucky imaging: high angular resolution imaing in the visible from the ground,” Astron. Astrophys. 446, 739-745 (2006).
    [CrossRef]
  12. C. Mackay, J. Baldwin, N. Law, and P. Warner, “High resolution imaging in the visible from the ground without adaptive optics: new techniques and results,” Proc. SPIE 5492, 128-135 (2004).
    [CrossRef]
  13. G. W. Carhart and M. A. Vorontsov, “Synthetic imaging: nonadaptive anisoplanatic image correction in atmospheric turbulence,” Opt. Lett. 23, 745-747 (1998).
    [CrossRef]
  14. M. A. Vorontsov and Gary W. Carhart, “Anisoplanatic imaging through turbulent media: image recovery by local information fusion from a set of short-exposure images,” J. Opt. Soc. Am. A 18, 1312-1324 (2001).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005), Section 6.3.
  16. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174-1180 (1990).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005), Section 6.4.3.
  18. Mathematica 6.0, from the test images file: Boat.
  19. W. T. Rhodes, “Phase closure and lucky imaging,” Appl. Opt. 48, (this issue).

2006 (1)

N. M. Law, C. D. Mackay, and J. E. Baldwin, “Lucky imaging: high angular resolution imaing in the visible from the ground,” Astron. Astrophys. 446, 739-745 (2006).
[CrossRef]

2004 (1)

C. Mackay, J. Baldwin, N. Law, and P. Warner, “High resolution imaging in the visible from the ground without adaptive optics: new techniques and results,” Proc. SPIE 5492, 128-135 (2004).
[CrossRef]

2001 (1)

2000 (1)

P. G. Tuthill, J. D. Monnier, W. C. Danchi, E. H. Wishnow, and C. A Haniff, “Michelson interferometry with the Keck I telescope,” Publ. Astron. Soc. Pac. 112 (770), 555-565 (2000).
[CrossRef]

1998 (1)

1990 (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174-1180 (1990).
[CrossRef]

1978 (1)

1974 (1)

W. T. Rhodes, “Digital processing of synthetic aperture optical imagery,” Opt. Eng. 13, 267-274 (1974).

1973 (1)

1969 (1)

G. W. Stroke, “Optical aperture synthesis using successive exposure of a single photograph and spatial filtering “low-frequency redundancy” suppression,” Phys. Lett. A 30, 485-486 (1969).
[CrossRef]

1968 (1)

1959 (1)

M. Ryle, A. Hewish, and J. R. Shakeshaft, “The synthesis of large radio telescopes by the use of radio interferometers,” IRE Trans. Antennas Propag. 7, 120-124(1959).
[CrossRef]

1952 (1)

M. Ryle, “A new radio interferometer and its application to the observation of weak radio stars,” Proc. R. Soc. London. Ser. A 211, 351-375 (1952).

Baldwin, J.

C. Mackay, J. Baldwin, N. Law, and P. Warner, “High resolution imaging in the visible from the ground without adaptive optics: new techniques and results,” Proc. SPIE 5492, 128-135 (2004).
[CrossRef]

Baldwin, J. E.

N. M. Law, C. D. Mackay, and J. E. Baldwin, “Lucky imaging: high angular resolution imaing in the visible from the ground,” Astron. Astrophys. 446, 739-745 (2006).
[CrossRef]

Carhart, G. W.

Carhart, Gary W.

Danchi, W. C.

P. G. Tuthill, J. D. Monnier, W. C. Danchi, E. H. Wishnow, and C. A Haniff, “Michelson interferometry with the Keck I telescope,” Publ. Astron. Soc. Pac. 112 (770), 555-565 (2000).
[CrossRef]

Fried, D. L.

Goodman, J. W.

W. T. Rhodes and J. W. Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,” J. Opt. Soc. Am. 63, 647-657 (1973).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005), Section 6.3.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005), Section 6.4.3.

Haniff, C. A

P. G. Tuthill, J. D. Monnier, W. C. Danchi, E. H. Wishnow, and C. A Haniff, “Michelson interferometry with the Keck I telescope,” Publ. Astron. Soc. Pac. 112 (770), 555-565 (2000).
[CrossRef]

Hewish, A.

M. Ryle, A. Hewish, and J. R. Shakeshaft, “The synthesis of large radio telescopes by the use of radio interferometers,” IRE Trans. Antennas Propag. 7, 120-124(1959).
[CrossRef]

Labeyrie, A.

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University Press, 2006), Chaps. 5 and 6.
[CrossRef]

Law, N.

C. Mackay, J. Baldwin, N. Law, and P. Warner, “High resolution imaging in the visible from the ground without adaptive optics: new techniques and results,” Proc. SPIE 5492, 128-135 (2004).
[CrossRef]

Law, N. M.

N. M. Law, C. D. Mackay, and J. E. Baldwin, “Lucky imaging: high angular resolution imaing in the visible from the ground,” Astron. Astrophys. 446, 739-745 (2006).
[CrossRef]

Lipson, S. G.

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University Press, 2006), Chaps. 5 and 6.
[CrossRef]

Mackay, C.

C. Mackay, J. Baldwin, N. Law, and P. Warner, “High resolution imaging in the visible from the ground without adaptive optics: new techniques and results,” Proc. SPIE 5492, 128-135 (2004).
[CrossRef]

Mackay, C. D.

N. M. Law, C. D. Mackay, and J. E. Baldwin, “Lucky imaging: high angular resolution imaing in the visible from the ground,” Astron. Astrophys. 446, 739-745 (2006).
[CrossRef]

Monnier, J. D.

P. G. Tuthill, J. D. Monnier, W. C. Danchi, E. H. Wishnow, and C. A Haniff, “Michelson interferometry with the Keck I telescope,” Publ. Astron. Soc. Pac. 112 (770), 555-565 (2000).
[CrossRef]

Nisenson, P.

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University Press, 2006), Chaps. 5 and 6.
[CrossRef]

Rhodes, W. T.

W. T. Rhodes, “Digital processing of synthetic aperture optical imagery,” Opt. Eng. 13, 267-274 (1974).

W. T. Rhodes and J. W. Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,” J. Opt. Soc. Am. 63, 647-657 (1973).
[CrossRef]

W. T. Rhodes, “Phase closure and lucky imaging,” Appl. Opt. 48, (this issue).

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174-1180 (1990).
[CrossRef]

Rogstad, D. H.

Ryle, M.

M. Ryle, A. Hewish, and J. R. Shakeshaft, “The synthesis of large radio telescopes by the use of radio interferometers,” IRE Trans. Antennas Propag. 7, 120-124(1959).
[CrossRef]

M. Ryle, “A new radio interferometer and its application to the observation of weak radio stars,” Proc. R. Soc. London. Ser. A 211, 351-375 (1952).

Shakeshaft, J. R.

M. Ryle, A. Hewish, and J. R. Shakeshaft, “The synthesis of large radio telescopes by the use of radio interferometers,” IRE Trans. Antennas Propag. 7, 120-124(1959).
[CrossRef]

Stroke, G. W.

G. W. Stroke, “Optical aperture synthesis using successive exposure of a single photograph and spatial filtering “low-frequency redundancy” suppression,” Phys. Lett. A 30, 485-486 (1969).
[CrossRef]

Tuthill, P. G.

P. G. Tuthill, J. D. Monnier, W. C. Danchi, E. H. Wishnow, and C. A Haniff, “Michelson interferometry with the Keck I telescope,” Publ. Astron. Soc. Pac. 112 (770), 555-565 (2000).
[CrossRef]

Vorontsov, M. A.

Warner, P.

C. Mackay, J. Baldwin, N. Law, and P. Warner, “High resolution imaging in the visible from the ground without adaptive optics: new techniques and results,” Proc. SPIE 5492, 128-135 (2004).
[CrossRef]

Wishnow, E. H.

P. G. Tuthill, J. D. Monnier, W. C. Danchi, E. H. Wishnow, and C. A Haniff, “Michelson interferometry with the Keck I telescope,” Publ. Astron. Soc. Pac. 112 (770), 555-565 (2000).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

N. M. Law, C. D. Mackay, and J. E. Baldwin, “Lucky imaging: high angular resolution imaing in the visible from the ground,” Astron. Astrophys. 446, 739-745 (2006).
[CrossRef]

IRE Trans. Antennas Propag. (1)

M. Ryle, A. Hewish, and J. R. Shakeshaft, “The synthesis of large radio telescopes by the use of radio interferometers,” IRE Trans. Antennas Propag. 7, 120-124(1959).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Eng. (2)

W. T. Rhodes, “Digital processing of synthetic aperture optical imagery,” Opt. Eng. 13, 267-274 (1974).

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174-1180 (1990).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

G. W. Stroke, “Optical aperture synthesis using successive exposure of a single photograph and spatial filtering “low-frequency redundancy” suppression,” Phys. Lett. A 30, 485-486 (1969).
[CrossRef]

Proc. SPIE (1)

C. Mackay, J. Baldwin, N. Law, and P. Warner, “High resolution imaging in the visible from the ground without adaptive optics: new techniques and results,” Proc. SPIE 5492, 128-135 (2004).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

P. G. Tuthill, J. D. Monnier, W. C. Danchi, E. H. Wishnow, and C. A Haniff, “Michelson interferometry with the Keck I telescope,” Publ. Astron. Soc. Pac. 112 (770), 555-565 (2000).
[CrossRef]

Other (7)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005), Section 6.3.

“Woods Hole summer study 1968: synthetic-aperture optics,” in Proceedings of the August 1967 Woods Hole Summer Study (U.S. National Academy of Sciences, 1968), Vols. I and II.

M. Ryle, “A new radio interferometer and its application to the observation of weak radio stars,” Proc. R. Soc. London. Ser. A 211, 351-375 (1952).

A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University Press, 2006), Chaps. 5 and 6.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005), Section 6.4.3.

Mathematica 6.0, from the test images file: Boat.

W. T. Rhodes, “Phase closure and lucky imaging,” Appl. Opt. 48, (this issue).

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Figures (3)

Fig. 1
Fig. 1

Illustration of aperture synthesis concept: (a) pupil-plane mask containing two square openings, (b) associated OTF, (c) u axis cross section of large-bandwidth OTF H ( u , v ) synthesized by an equal-weighting sum of individual OTF spatial-frequency passbands, (d) same but with weighting appropriate for a conventional OTF. In (c) and (d) the synthesized OTF is denoted by the dashed lines.

Fig. 2
Fig. 2

Cross sections of the modulus of the OTF, | H ( u , 0 ) | , for different values of the subaperture differential wavefront tilt parameter Δ α 12 , assuming that Δ β 12 = 0 : (a)  Δ α 12 = 0 , (b) approximately 1/3 wave differential tilt across the subapertures, (c) one full wave of differential tilt, (d) several waves of differential tilt. Note that the dimples appearing in the passbands in (b) have become full nulls in (c).

Fig. 3
Fig. 3

Numerically processed images showing effects of pupil-plane mask and turbulence: (a) original image, (b) image obtained with two-square-opening aperture, no turbulence, (c) same but with turbulence, (d) “lucky” two-subaperture image. Note in (c) the doubling effect on low-frequency components brought about by significant differential aberration phase tilt.

Equations (21)

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ϕ ( x , y ) = ϕ o + α 1 x + β 1 y + α 2 x 2 + β 2 y 2 + γ 2 x y + .
E = v l v u u l u u | I ^ ( u , v ) | 2 d u d v ,
H ( u , v ) = p ( x , y ) p ( x , y ) NF | x = λ z u y = λ z v ,
p ( x , y ) = p 1 ( x , y ) * * δ ( x + s / 2 , y ) + p 2 ( x , y ) * * δ ( x s / 2 , y ) ,
p i ( x , y ) = rect ( x / w , y / w ) e j ϕ i e j 2 π ( α i x + β i y ) ,
p p = ( p 1 ** δ + 1 + p 2 ** δ - 1 ) ( p 1 ** δ + 1 + p 2 ** δ - 1 ) = p 1 p 1 + p 2 p 2 + ( p 1 p 2 ) ** δ + 2 + ( p 2 p 1 ) ** δ 2 ,
p m ( x , y ) p n ( x , y ) = tri ( x w , y w ) × sinc [ Δ α n m ( w | x | ) , Δ β n m ( w | y | ) ] × e j Δ ϕ m n e j 2 π ( α ¯ x + β ¯ y ) ,
p m ( x , y ) p m ( x , y ) = tri ( x w , y w ) e j 2 π ( α m x + β m y ) ,
p 1 p 1 + p 2 p 2 = tri ( x w , y w ) × [ e j 2 π ( α 1 x + β 1 y ) + e j 2 π ( α 1 x + β 1 y ) ] = tri ( x w , y w ) × cos [ 2 π ( Δ α 12 x + Δ β 12 y ) ] e j 2 π ( α ¯ x + β ¯ y ) .
OTF   passband  :   tri ( x w , y w ) e j Δ ϕ m n e j 2 π ( α x + β y ) ,
OTF     low pass :   tri ( x w , y w ) e j 2 π ( α x + β y ) .
p m ( x ) p n ( x ) = [ rect ( x w ) e j ϕ m 2 e j 2 π α m x ] [ rect ( y w ) e j ϕ n 2 e j 2 π α n x ] .
p m ( x ) p n ( x ) = e j Δ ϕ m n 2 x w 2 w 2 e j 2 π α m ξ e j 2 π α n ( ξ x ) d ξ = e j Δ ϕ m n 2 e j 2 π α n x x w 2 w 2 e j 2 π ( α n α m ) ξ d ξ = e j Δ ϕ m n 2 e j 2 π α n x rect ( ξ x 2 w x ) e j 2 π ( α n α m ) ξ d ξ ,
p m ( x ) p n ( x ) = e j Δ ϕ m n 2 e j 2 π α n x ( w x ) sinc [ ( w x ) ( α n α m ) ] e - j 2 π ( x / 2 ) ( α n α m ) .
p m ( x ) p n ( x ) = w tri ( x w ) sinc [ ( w | x | ) Δ α n m ] e j Δ ϕ m n 2 e j 2 π α ¯ x ,
p m ( x ) p m ( x ) = w tri ( x w ) e j 2 π α m x .
| p m p n | 2 = | + [ rect ( ξ w ) exp ( j ϕ m 2 ) exp ( j 2 π Σ i = 1 N α m ( i ) ξ i ) ] × [ rect ( ( ξ x ) w ) exp ( j ϕ n 2 ) exp ( j 2 π Σ i = 1 N α n ( i ) ( ξ x ) i ) ] d ξ | 2 .
| p m p n | 2 - + | [ rect ( ξ w ) exp ( j ϕ m 2 ) exp ( j 2 π i = 1 N α m ( i ) ξ i ) ] | 2 d ξ × + | [ rect ( ( ξ x ) w ) exp ( j ϕ n 2 ) exp ( j 2 π i = 1 N α n ( i ) ( ξ x ) i ) ] | 2 d ξ .
| p m p n | 2 + | rect ( ξ w ) | 2 d ξ × + | rect ( ( ξ x ) w ) | 2 d ξ = w 2 .
α m ( 1 ) α n ( 1 ) = 0 α m ( 1 ) = α n ( 1 ) .
α m ( i ) ξ i α n ( i ) ( ξ x ) i = 0.

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