Abstract

The optical interferometry community has discussed the possibility of using adaptive optics (AO) on apertures much larger than the atmospheric coherence length in order to increase the sensitivity of an interferometer, although few quantitative models have been investigated. The aim of this paper is to develop an analytic model of an AO-equipped interferometer and to use it to quantify, in relative terms, the gains that may be achieved over an interferometer equipped only with tip–tilt correction. Functional forms are derived for wavefront errors as a function of spatial and temporal coherence scales and flux and applied to the AO and tip–tilt cases. In both cases, the AO and fringe detection systems operate in the same spectral region, with the sharing ratio and subaperture size as adjustable parameters, and with the interferometer beams assumed to be spatially filtered after wavefront correction. It is concluded that the use of AO improves the performance of the interferometer in three ways. First, at the optimal aperture size for a tip–tilt system, the AO system is as much as 50% more sensitive. Second, the sensitivity of the AO system continues to improve with increasing aperture size. And third, the signal-to-noise ratio of low-visibility fringes in the bright-star limit is significantly improved over the tip–tilt case.

© 2007 Optical Society of America

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References

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  1. S. R. Restaino, J. Andrews, J. T. Armstrong, T. Martinez, C. Wilcox, and D. Payne, "Navy prototype optical interferometer upgrade with light-weight telescopes and adaptive optics: A status update," Proceedings of 2006 AMOS Technical Conference (2006), pp. 105-111.
  2. D. F. Buscher, "Getting the most out of C.O.A.S.T.," Ph.D. dissertation (Cambridge University, 1988).
  3. J. E. Baldwin and C. A. Haniff, "The application of interferometry to optical astronomical imaging," Philos. Trans. R. Soc. London Ser. A 360, 969-986 (2002).
    [Crossref]
  4. 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]
  5. D. F. Buscher, J. T. Armstrong, C. A. Hummel, A. Quirrenbach, D. Mozurkewich, K. J. Johnston, C. S. Denison, M. M. Colavita, and M. Shao, "Interferometric seeing measurements on Mount Wilson: Power spectra and outer scales," Appl. Opt. 34, 1081-1096 (1995).
    [Crossref] [PubMed]
  6. J. R. P. Angel, "Ground-based imaging of extrasolar planets using adaptive optics," Nature 368, 203-207 (1994).
    [Crossref]
  7. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
    [Crossref]
  8. C. B. Hogge and R. R. Butts, "Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence," IEEE Trans. Antennas Propag. 24, 144-154 (1976).
    [Crossref]
  9. F. Roddier, "The effects of atmospheric turbulence in optical astronomy," Prog. Opt. XIX, 281-376 (1981).
    [Crossref]

2002 (1)

J. E. Baldwin and C. A. Haniff, "The application of interferometry to optical astronomical imaging," Philos. Trans. R. Soc. London Ser. A 360, 969-986 (2002).
[Crossref]

1995 (1)

1994 (1)

J. R. P. Angel, "Ground-based imaging of extrasolar planets using adaptive optics," Nature 368, 203-207 (1994).
[Crossref]

1981 (1)

F. Roddier, "The effects of atmospheric turbulence in optical astronomy," Prog. Opt. XIX, 281-376 (1981).
[Crossref]

1976 (2)

R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
[Crossref]

C. B. Hogge and R. R. Butts, "Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence," IEEE Trans. Antennas Propag. 24, 144-154 (1976).
[Crossref]

1966 (1)

Andrews, J.

S. R. Restaino, J. Andrews, J. T. Armstrong, T. Martinez, C. Wilcox, and D. Payne, "Navy prototype optical interferometer upgrade with light-weight telescopes and adaptive optics: A status update," Proceedings of 2006 AMOS Technical Conference (2006), pp. 105-111.

Angel, J. R. P.

J. R. P. Angel, "Ground-based imaging of extrasolar planets using adaptive optics," Nature 368, 203-207 (1994).
[Crossref]

Armstrong, J. T.

D. F. Buscher, J. T. Armstrong, C. A. Hummel, A. Quirrenbach, D. Mozurkewich, K. J. Johnston, C. S. Denison, M. M. Colavita, and M. Shao, "Interferometric seeing measurements on Mount Wilson: Power spectra and outer scales," Appl. Opt. 34, 1081-1096 (1995).
[Crossref] [PubMed]

S. R. Restaino, J. Andrews, J. T. Armstrong, T. Martinez, C. Wilcox, and D. Payne, "Navy prototype optical interferometer upgrade with light-weight telescopes and adaptive optics: A status update," Proceedings of 2006 AMOS Technical Conference (2006), pp. 105-111.

Baldwin, J. E.

J. E. Baldwin and C. A. Haniff, "The application of interferometry to optical astronomical imaging," Philos. Trans. R. Soc. London Ser. A 360, 969-986 (2002).
[Crossref]

Buscher, D. F.

Butts, R. R.

C. B. Hogge and R. R. Butts, "Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence," IEEE Trans. Antennas Propag. 24, 144-154 (1976).
[Crossref]

Colavita, M. M.

Denison, C. S.

Fried, D. L.

Haniff, C. A.

J. E. Baldwin and C. A. Haniff, "The application of interferometry to optical astronomical imaging," Philos. Trans. R. Soc. London Ser. A 360, 969-986 (2002).
[Crossref]

Hogge, C. B.

C. B. Hogge and R. R. Butts, "Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence," IEEE Trans. Antennas Propag. 24, 144-154 (1976).
[Crossref]

Hummel, C. A.

Johnston, K. J.

Martinez, T.

S. R. Restaino, J. Andrews, J. T. Armstrong, T. Martinez, C. Wilcox, and D. Payne, "Navy prototype optical interferometer upgrade with light-weight telescopes and adaptive optics: A status update," Proceedings of 2006 AMOS Technical Conference (2006), pp. 105-111.

Mozurkewich, D.

Noll, R. J.

Payne, D.

S. R. Restaino, J. Andrews, J. T. Armstrong, T. Martinez, C. Wilcox, and D. Payne, "Navy prototype optical interferometer upgrade with light-weight telescopes and adaptive optics: A status update," Proceedings of 2006 AMOS Technical Conference (2006), pp. 105-111.

Quirrenbach, A.

Restaino, S. R.

S. R. Restaino, J. Andrews, J. T. Armstrong, T. Martinez, C. Wilcox, and D. Payne, "Navy prototype optical interferometer upgrade with light-weight telescopes and adaptive optics: A status update," Proceedings of 2006 AMOS Technical Conference (2006), pp. 105-111.

Roddier, F.

F. Roddier, "The effects of atmospheric turbulence in optical astronomy," Prog. Opt. XIX, 281-376 (1981).
[Crossref]

Shao, M.

Wilcox, C.

S. R. Restaino, J. Andrews, J. T. Armstrong, T. Martinez, C. Wilcox, and D. Payne, "Navy prototype optical interferometer upgrade with light-weight telescopes and adaptive optics: A status update," Proceedings of 2006 AMOS Technical Conference (2006), pp. 105-111.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

C. B. Hogge and R. R. Butts, "Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence," IEEE Trans. Antennas Propag. 24, 144-154 (1976).
[Crossref]

J. Opt. Soc. Am. (2)

Nature (1)

J. R. P. Angel, "Ground-based imaging of extrasolar planets using adaptive optics," Nature 368, 203-207 (1994).
[Crossref]

Philos. Trans. R. Soc. London Ser. A (1)

J. E. Baldwin and C. A. Haniff, "The application of interferometry to optical astronomical imaging," Philos. Trans. R. Soc. London Ser. A 360, 969-986 (2002).
[Crossref]

Prog. Opt. (1)

F. Roddier, "The effects of atmospheric turbulence in optical astronomy," Prog. Opt. XIX, 281-376 (1981).
[Crossref]

Other (2)

S. R. Restaino, J. Andrews, J. T. Armstrong, T. Martinez, C. Wilcox, and D. Payne, "Navy prototype optical interferometer upgrade with light-weight telescopes and adaptive optics: A status update," Proceedings of 2006 AMOS Technical Conference (2006), pp. 105-111.

D. F. Buscher, "Getting the most out of C.O.A.S.T.," Ph.D. dissertation (Cambridge University, 1988).

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Figures (7)

Fig. 1
Fig. 1

Variation of ρ = τ / a as a function of N, the number of photons delivered to the adaptive optics system. The points are the evaluation of Eq. (18). The curve is the approximation of Eq. (20).

Fig. 2
Fig. 2

Optimum values for a A / r 0 and τ t s / t 0 as a function of the number of photons per coherence volume N delivered to the adaptive optics system.

Fig. 3
Fig. 3

Phase variance σ 2 across a subaperture as a function of the number of photons per coherence volume N delivered to the adaptive optics sensor for the case in which a and τ are optimized.

Fig. 4
Fig. 4

Phase variance across a subaperture normalized to the variance at the optimal values of a and τ. (a) Normalized variance is plotted as a function of a, the subaperture size in units of r 0 , normalized to its optimal value, a opt . The curves, from upper to lower, are calculated for N = 100, 10, and 1. (b) Normalized variance is plotted as a function of τ, the AO servo time constant in units of t 0 , normalized to its optimal value, τ opt . The curves, from upper to lower, are calculated for N = 1, 10, and 100.

Fig. 5
Fig. 5

The fraction of light to send to the adaptive optics system that optimizes the interferometric signal-to-noise ratio as a function of the number of photons per coherence volume presented to the whole system (full curve). For comparison, we plot the same quantity for the tip–tilt case with D / r 0 = 3.9 , the value for maximum sensivity (dashed curve).

Fig. 6
Fig. 6

Photons per coherence volume needed to obtain a fixed signal-to-noise ratio with an optical interferometer, as a function of aperture size, for an adaptive-optics equipped interferometer and for an interferometer with tip–tilt correction only.

Fig. 7
Fig. 7

Signal-to-noise ratio at V = 1 as a function of aperture size in the bright-star limit.

Equations (49)

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σ 2 ( B ) = 6.88 ( B / r 0 ) 5 / 3 .
σ Q 2 = ( λ A ) 2 Q 1 + Q 2 ( A / r 0 ) 2 n ,
σ d 2 = ( θ A ) 2 1 4 π x 2 d x d y = ( θ A 4 ) 2 ,
σ tt 2 = ( 2 π λ ) 2 ( A 4 ) 2 σ Q 2 ,
= ( π 2 ) 2 Q 1 + Q 2 ( A / r 0 ) 2 n .
n = N ( t t 0 ) ( A r 0 ) 2 .
σ tt 2 = ( π 2 4 N ) ( t s t 0 ) 1 [ Q 1 ( A r 0 ) 2 + Q 2 ] .
σ ho 2 = α ( A r 0 ) 5 / 3 ,
Φ 2 ( ν ) = p 0 t 0 5 / 3 ν 8 / 3 for   ν > ν a
= p 0 t 0 5 / 3 ν 0 2 ν 2 / 3 for   ν < ν a ,
ν a = d r 0 A t 0 ,
σ E 2 = β ( t s t 0 ) 5 / 3 γ ( A r 0 ) 1 / 3 ( t s t 0 ) 2 ,
σ 2 = q 1 N a 2 τ + q 2 N τ + α a 5 / 3 + β τ 5 / 3 γ τ 2 a 1 / 3 .
σ 2 τ = q 1 N a 2 τ 2 q 2 N τ 2 + 5 3 β τ 2 / 3 2 γ τ a 1 / 3 ,
σ 2 a = 2 q 1 N a 3 τ + 5 3 α a 2 / 3 + 1 3 γ τ 2 a 4 / 3 .
q 1 N a 2 + q 2 N = 5 3 β τ 8 / 3 2 γ τ 3 a 1 / 3 ,
q 1 N a 2 = 5 6 α a 5 / 3 τ + 1 6 γ τ 3 a 1 / 3 .
N 3 = 6 3 q 2 7 q 1 4 ( 5 α ρ + γ ρ 3 ) 4 ( 10 β ρ 8 / 3 5 α ρ 13 γ ρ 3 ) 7
= 0.499 ρ 3 ( 1 + 0.0341 ρ 2 ) 4 ( 1 + 1.205 ρ 5 / 3 0.443 ρ 2 ) 7 .
ρ = 0.6677 N 0.333 + 1.114 N 0.0078
a = [ N 6 q 1 ( 5 α ρ + γ ρ 3 ) ] 3 / 14 ,
= 2.083 [ N ( ρ + 0.0341 ρ 3 ) ] 3 / 14 .
σ 2 = σ 0 2 + b N c ,
= 0.0449 + 1.261 N 0.450 ,
S 2 = 2 π 2 V 2 N ( 1 f ) ( t I t 0 ) ( π D 2 4 r 0 2 ) e σ 2 ,
σ 2 = σ 0 2 + b ( f N ) c ,
d σ 2 d f = b c N c f c + 1 .
d S 2 d f = N 2 π ( t I t 0 ) ( D r 0 ) 2 e σ 2 N 2 π ( 1 f ) ( t I t 0 ) ( D r 0 ) 2 e σ 2 d σ 2 d f ,
= 0 ,
N c b c = 1 f f c + 1 .
f = 0.448 0.202 log ( N ) + 0.0177 log 2 ( N ) + 0.00341 log 3 ( N )
S 2 = N 2 π ( 1 f ) a 2 e σ 2 .
S 2 τ = N 2 π ( 1 f ) a 2 e σ 2 σ 2 τ ,
= 0 ,
S 2 f = N 2 π a 2 e σ 2 N 2 π ( 1 f ) a 2 e σ 2 σ 2 f ,
= 0 ,
σ 2 τ = q 1 f N a 2 τ 2 q 2 f N τ 2 + 5 3 β τ 2 / 3 2 γ τ a 1 / 3 ,
σ 2 f = q 1 f 2 N a 2 τ q 2 f 2 N τ .
f N = q 1 + q 2 a 2 a 2 τ ( 5 3 β τ 5 / 3 2 γ τ 2 a 1 / 3 ) 1 ,
1 f = 1 + ( 5 3 β τ 5 / 3 2 γ τ 2 a 1 / 3 ) 1 .
S 2 = N V 2 2 π ( 1 f ) ( D r 0 ) 2 e σ 2 .
q 1 N a 2 + q 2 N = 5 3 β τ 8 / 3 2 γ τ 3 a 1 / 3 ,
q 1 N a 2 = 5 6 α a 5 / 3 τ + γ τ 3 6 a 1 / 3 ,
q 2 N + 5 6 α a 5 / 3 τ + 13 6 γ τ 3 a 1 / 3 = 5 3 β τ 8 / 3 .
q 1 N = a 14 / 3 ( 5 6 α ρ + 1 6 γ ρ 3 ) ,
q 2 N = a 8 / 3 ( 5 3 β ρ 8 / 3 5 6 α ρ 13 6 γ ρ 3 ) .
( q 1 N ) 3 / 14 ( q 2 N ) 3 / 8 = ( 5 6 α ρ + 1 6 γ ρ 3 ) 3 / 14 ( 5 3 β ρ 8 / 3 5 6 α ρ 13 6 γ ρ 3 ) 3 / 8 ,
N 9 / 56 = q 2 3 / 8 q 1 3 / 14 ( 5 6 α ρ + 1 6 γ ρ 3 ) 3 / 14 ( 5 3 β ρ 8 / 3 5 6 α ρ 13 6 γ ρ 3 ) 3 / 8
N 3 = 6 3 q 2 7 q 1 4 ( 5 α ρ + γ ρ 3 ) 4 ( 10 β ρ 5 / 3 5 α ρ 13 γ ρ 3 ) 7 .

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