Abstract

Spectral-bandwidth constraints to ensure controlled amounts of redundancy are established for a class of two-dimensional partially redundant arrays (PRA’s). In the IR, where speckle statistics are poor, the telescope–atmosphere modulation transfer function is determined solely by the PRA geometry. Signal-to-noise-ratio estimates, an optimum aperture criterion, and a six-element PRA example are presented.

© 1981 Optical Society of America

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References

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  1. 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]
  2. F. Roddier, “Speckle interferometry through small multiple apertures: Michelson stellar interferometer and aperture synthesis in optics,” Opt. Commun. 10, 103–105 (1974).
    [Crossref]
  3. T. M. Brown, “Reconstruction of turbulence—degraded images using nonredundant aperture arrays,” J. Opt. Soc. Am. 68, 883–889 (1978).
    [Crossref]
  4. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  5. D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,” J. Opt. Soc. Am. 63, 971–980 (1973).
    [Crossref]
  6. K. T. Knox, “Image retrieval from astronomical speckle patterns,” J. Opt. Soc. Am. 66, 1236–1239 (1976).
    [Crossref]
  7. G. J. M. Aitken and D. L. Desaulniers, “Restoration of atmospherically degraded images using complex spectral ratios,” Opt. Commun. 28, 26–29 (1979).
    [Crossref]
  8. F. Roddier, “Les effets de la turbulence atmospherique sur la formation des images visibles et infrarouges,” J. Opt. (Paris) 10, 299–303 (1979).
    [Crossref]
  9. F. Sibille, A. Chelli, and P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).
  10. D. W. McCarthy, F. J. Lowe, and R. Howell, “Design and operation of an infrared spatial interferometer,” Opt. Eng. 16, 569–574 (1977).
    [Crossref]
  11. M. J. E. Golay, “Point arrays having compact, nonredundant autocorrelations,” J. Opt. Soc. Am. 61, 272–273 (1971).
    [Crossref]
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 120.
  13. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [Crossref]
  14. J. M. Robillot, “Effets atmospheriques et measures radio-interferometriques,” J. Opt. (Paris) 10, 304–307 (1979).
    [Crossref]
  15. M. G. Miller, “Noise considerations in stellar speckle interferometry,” J. Opt. Soc. Am. 67, 1176–1184 (1977).
    [Crossref]

1979 (4)

G. J. M. Aitken and D. L. Desaulniers, “Restoration of atmospherically degraded images using complex spectral ratios,” Opt. Commun. 28, 26–29 (1979).
[Crossref]

F. Roddier, “Les effets de la turbulence atmospherique sur la formation des images visibles et infrarouges,” J. Opt. (Paris) 10, 299–303 (1979).
[Crossref]

F. Sibille, A. Chelli, and P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).

J. M. Robillot, “Effets atmospheriques et measures radio-interferometriques,” J. Opt. (Paris) 10, 304–307 (1979).
[Crossref]

1978 (1)

1977 (2)

D. W. McCarthy, F. J. Lowe, and R. Howell, “Design and operation of an infrared spatial interferometer,” Opt. Eng. 16, 569–574 (1977).
[Crossref]

M. G. Miller, “Noise considerations in stellar speckle interferometry,” J. Opt. Soc. Am. 67, 1176–1184 (1977).
[Crossref]

1976 (1)

1974 (1)

F. Roddier, “Speckle interferometry through small multiple apertures: Michelson stellar interferometer and aperture synthesis in optics,” Opt. Commun. 10, 103–105 (1974).
[Crossref]

1973 (2)

1971 (1)

1970 (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

1966 (1)

Aitken, G. J. M.

G. J. M. Aitken and D. L. Desaulniers, “Restoration of atmospherically degraded images using complex spectral ratios,” Opt. Commun. 28, 26–29 (1979).
[Crossref]

Brown, T. M.

Chelli, A.

F. Sibille, A. Chelli, and P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).

Desaulniers, D. L.

G. J. M. Aitken and D. L. Desaulniers, “Restoration of atmospherically degraded images using complex spectral ratios,” Opt. Commun. 28, 26–29 (1979).
[Crossref]

Fried, D. L.

Golay, M. J. E.

Goodman, J. W.

Howell, R.

D. W. McCarthy, F. J. Lowe, and R. Howell, “Design and operation of an infrared spatial interferometer,” Opt. Eng. 16, 569–574 (1977).
[Crossref]

Knox, K. T.

Korff, D.

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lena, P.

F. Sibille, A. Chelli, and P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).

Lowe, F. J.

D. W. McCarthy, F. J. Lowe, and R. Howell, “Design and operation of an infrared spatial interferometer,” Opt. Eng. 16, 569–574 (1977).
[Crossref]

McCarthy, D. W.

D. W. McCarthy, F. J. Lowe, and R. Howell, “Design and operation of an infrared spatial interferometer,” Opt. Eng. 16, 569–574 (1977).
[Crossref]

Miller, M. G.

Rhodes, W. T.

Robillot, J. M.

J. M. Robillot, “Effets atmospheriques et measures radio-interferometriques,” J. Opt. (Paris) 10, 304–307 (1979).
[Crossref]

Roddier, F.

F. Roddier, “Les effets de la turbulence atmospherique sur la formation des images visibles et infrarouges,” J. Opt. (Paris) 10, 299–303 (1979).
[Crossref]

F. Roddier, “Speckle interferometry through small multiple apertures: Michelson stellar interferometer and aperture synthesis in optics,” Opt. Commun. 10, 103–105 (1974).
[Crossref]

Sibille, F.

F. Sibille, A. Chelli, and P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).

Astron. Astrophys. (2)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

F. Sibille, A. Chelli, and P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).

J. Opt. (Paris) (2)

J. M. Robillot, “Effets atmospheriques et measures radio-interferometriques,” J. Opt. (Paris) 10, 304–307 (1979).
[Crossref]

F. Roddier, “Les effets de la turbulence atmospherique sur la formation des images visibles et infrarouges,” J. Opt. (Paris) 10, 299–303 (1979).
[Crossref]

J. Opt. Soc. Am. (7)

Opt. Commun. (2)

G. J. M. Aitken and D. L. Desaulniers, “Restoration of atmospherically degraded images using complex spectral ratios,” Opt. Commun. 28, 26–29 (1979).
[Crossref]

F. Roddier, “Speckle interferometry through small multiple apertures: Michelson stellar interferometer and aperture synthesis in optics,” Opt. Commun. 10, 103–105 (1974).
[Crossref]

Opt. Eng. (1)

D. W. McCarthy, F. J. Lowe, and R. Howell, “Design and operation of an infrared spatial interferometer,” Opt. Eng. 16, 569–574 (1977).
[Crossref]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 120.

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Figures (4)

Fig. 1
Fig. 1

Response islands of the four-hole PRA shown in the upper-left-hand corner. Hatched areas are the islands at f = fl; dashed islands are at f = 1.3fl.

Fig. 2
Fig. 2

Islands along the ux axis of a seven-element PRA projected onto the fux plane to illustrate temporal–spatial-frequency behavior.

Fig. 3
Fig. 3

A cut through H(u0i) parallel to the f axis for two values of uD/u0i. The curves are normalized with respect to H(u0i) at f = f0.

Fig. 4
Fig. 4

The six-hole PRA and its islands in the right half of the spatial-frequency plane. The OTF is symmetrical about the uy axis. The outer circles in the two diagrams represent the 150-cm aperture and its corresponding spatial-frequency upper limit.

Equations (24)

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H i ( u ) = D 2 2 [ cos - 1 ( ξ ) - ξ ( 1 - ξ 2 ) 1 / 2 ] ξ 1 = 0 ξ > 1 ,
I ( u ) = - [ N H 0 ( u , f ) + i = - N N H i ( u , f ) ] O ( u , f ) A ( u , f ) G ( f ) d f ,
δ f / f = [ 1 - ( 2 - γ ) β ] / ( N m - 0.5 ) .
σ ϕ 2 = σ Φ 2 f 0 2 = D 0 ( r ) ,
I ( u ) = - P ( Φ ) | f l f u H ( u , f ) A ( u , f ) d f | d Φ ,
σ ϕ ( u D / u ) = D 0 1 / 2 ( r ) ( D / r ) < 1 ,
σ ϕ ( δ f / f 0 ) = D 0 1 / 2 ( r ) { [ 1 - ( 2 - γ ) β ] / ( N m - 0.5 ) } < 1 ,
I ( u ) = T ( u ) O ( u ) + B p ( u ) + B ( u ) ,
S = ( h f 0 ) a M 0 T e Δ λ ,
σ I 2 = I 2 - I 2 = B p 2 + B 2 = ( h f 0 ) 2 ( N p + N B ) ,
N p = N a a M 0 T e Δ λ
N B = [ N a a L s + ( A - N a a ) L a ] Ω T e Δ λ ,
SNR I = ( κ η α ) 1 / 2 a M 0 ( T e Δ λ ) 1 / 2 [ N a a M 0 + A L B Ω ] 1 / 2 .
SNR I = ρ a M 0 ( T e Δ λ ) 1 / 2 / ( A L B Ω ) 1 / 2 ,
I 2 = S M + B p 2 + B 2 .
σ M 2 = I 4 - I 2 2 = σ s 2 + σ B 2 + 2 ( S M + B p 2 ) B 2 ,
σ s 2 = I s 4 - I s 2 2 = B p 2 2 + 2 S M B p 2
σ B 2 = B 4 - B 2 2 = B 2 2 .
SNR M = a 2 M 0 2 T e Δ λ / A Ω L B
SNR s = M 0 2 a s T t Δ λ T e M 0 2 a s T t Δ λ T e + M 0 + Ω L B ,
SNR s = M 0 2 a s T t Δ λ T e / Ω L B .
a Δ λ 1 / 2 = ( k Δ f 1 / 2 ) f a f b H ( u , f ) d f ,
β = D / d = 0.8 / ( 2 - γ )
δ f / f 0 = δ λ / λ c = 0.2 ( N m - 0.5 ) .