Abstract

The paper described is the third part of a trilogy dealing with the principles, performance, and limitations of what the author named “telescope–interferometers” (TIs). The basic idea consists in transforming one telescope into a wavefront error (WFE) sensing device. This can be achieved in two different ways, namely, off-axis and phase-shifting TIs. In both cases the point-spread function measured in the focal plane of the telescope carries information about the transmitted WFE, which is retrieved by fast and simple algorithms suitable to an adaptive optics (AO) regime. The uncertainties of both types of TIs are evaluated in terms of noise and systematic errors. Numerical models are developed to establish the dependence of driving parameters such as useful spectral range, angular size of the observed star, or detector noise on the total WFE measurement error. The latter is found particularly sensitive to photon noise, which rapidly governs the achieved accuracy for telescope diameters higher than 10m. A few practical examples are studied, showing that the TI method is applicable to AO systems for telescope diameters ranging from 10to50m, depending on seeing conditions and magnitude of the observed stars. Also discussed is the case of a space-borne coronagraph, where the TI technique provides high sampling of the input WFE map.

© 2008 Optical Society of America

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References

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  16. F. Gonte, N. Yaitskova, P. Dierickx, R. Karban, A. Courteville, A. Schumacher, N. Devaney, S. Esposito, K. Dohlen, M. Ferrari, and L. Montoya, "APE: a breadboard to evaluate new phasing technologies for a future European Giant Optical Telescope," Proc. SPIE 5489, 1184-1191 (2004).
    [CrossRef]
  17. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
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    [CrossRef]
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2006 (3)

F. Hénault, "Conceptual design of a phase shifting telescope-interferometer," Opt. Commun. 261, 34-42 (2006).
[CrossRef]

A. Schumacher and N. Devaney, "Phasing segmented mirrors using defocused images at visible wavelengths," Mon. Not. R. Astron. Soc. 366, 537-546 (2006).
[CrossRef]

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," Proc. SPIE 6267, 626-726 (2006).

2005 (4)

2004 (2)

F. Gonte, N. Yaitskova, P. Dierickx, R. Karban, A. Courteville, A. Schumacher, N. Devaney, S. Esposito, K. Dohlen, M. Ferrari, and L. Montoya, "APE: a breadboard to evaluate new phasing technologies for a future European Giant Optical Telescope," Proc. SPIE 5489, 1184-1191 (2004).
[CrossRef]

L. M. Stepp and S. E. Strom, "The Thirty-Meter Telescope project design and development phase," Proc. SPIE 5382, 67-75 (2004).
[CrossRef]

2001 (1)

M. Löfdahl and H. Eriksson, "An algorithm for resolving 2π ambiguities in interferometric measurements by use of multiple wavelengths," Opt. Eng. 40, 984-990 (2001).
[CrossRef]

1999 (1)

1998 (1)

1996 (2)

R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

A. Labeyrie, "Resolved imaging of extra-solar planets with future 10-100 km optical interferometric arrays," Astron. Astrophys. Suppl. Ser. 118, 517-524 (1996).
[CrossRef]

1992 (1)

1989 (1)

S. B. Howell, "Two-dimensional aperture photometry: signal-to-noise ratio of point-source observations and optimal data-extraction techniques," Publ. Astron. Soc. Pac. 101, 616-622 (1989).
[CrossRef]

1988 (1)

1982 (3)

J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
[CrossRef] [PubMed]

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

M. Takeda, H. Ina, and S. Koyabashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. A 72, 156-160 (1982).
[CrossRef]

1953 (1)

H. W. Babcock, "The possibility of compensating astronomical seeing," Publ. Astron. Soc. Pac. 65, 229-236 (1953).
[CrossRef]

Appl. Opt. (6)

Astron. Astrophys. Suppl. Ser. (1)

A. Labeyrie, "Resolved imaging of extra-solar planets with future 10-100 km optical interferometric arrays," Astron. Astrophys. Suppl. Ser. 118, 517-524 (1996).
[CrossRef]

J. Mod. Opt. (2)

R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

F. Hénault, "Wavefront sensor based on varying transmission filters: theory and expected performance," J. Mod. Opt. 52, 1917-1931 (2005).
[CrossRef]

J. Opt. Soc. Am. A (2)

N. Yaitskova, K. Dohlen, P. Dierickx, and L. Montoya, "Mach-Zehnder interferometer for piston and tip-tilt sensing in segmented telescopes: theory and analytical treatment," J. Opt. Soc. Am. A 22, 1093-1105 (2005).
[CrossRef]

M. Takeda, H. Ina, and S. Koyabashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. A 72, 156-160 (1982).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

A. Schumacher and N. Devaney, "Phasing segmented mirrors using defocused images at visible wavelengths," Mon. Not. R. Astron. Soc. 366, 537-546 (2006).
[CrossRef]

Opt. Commun. (1)

F. Hénault, "Conceptual design of a phase shifting telescope-interferometer," Opt. Commun. 261, 34-42 (2006).
[CrossRef]

Opt. Eng. (2)

M. Löfdahl and H. Eriksson, "An algorithm for resolving 2π ambiguities in interferometric measurements by use of multiple wavelengths," Opt. Eng. 40, 984-990 (2001).
[CrossRef]

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

Opt. Lett. (1)

Proc. SPIE (3)

F. Gonte, N. Yaitskova, P. Dierickx, R. Karban, A. Courteville, A. Schumacher, N. Devaney, S. Esposito, K. Dohlen, M. Ferrari, and L. Montoya, "APE: a breadboard to evaluate new phasing technologies for a future European Giant Optical Telescope," Proc. SPIE 5489, 1184-1191 (2004).
[CrossRef]

L. M. Stepp and S. E. Strom, "The Thirty-Meter Telescope project design and development phase," Proc. SPIE 5382, 67-75 (2004).
[CrossRef]

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," Proc. SPIE 6267, 626-726 (2006).

Publ. Astron. Soc. Pac. (2)

S. B. Howell, "Two-dimensional aperture photometry: signal-to-noise ratio of point-source observations and optimal data-extraction techniques," Publ. Astron. Soc. Pac. 101, 616-622 (1989).
[CrossRef]

H. W. Babcock, "The possibility of compensating astronomical seeing," Publ. Astron. Soc. Pac. 65, 229-236 (1953).
[CrossRef]

Other (4)

R. Angel, "Imaging extrasolar planets from the ground," in Scientific Frontiers in Research on Extrasolar Planets, D.Deming and S.Seager, eds., Vol. 294 of ASP Conference Series (Astronomical Society of the Pacific, 2003), pp. 543-556.

A. Labeyrie, "Removal of coronagraphy residues with an adaptive hologram, for imaging exo-Earths," in Astronomy with High Contrast Imaging II, C.Aime and R.Soummer, eds., Vol. 12 of EAS Publications Series (European Astronomical Society, 2004), pp. 3-10.

A. Maréchal and M. Françon, Diffraction, structure des images (Dunod, 1970).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes, (Oxford U. Press, 1998).

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Figures (15)

Fig. 1
Fig. 1

Schematic drawing of an off-axis TI.

Fig. 2
Fig. 2

Schematic drawing of a phase-shifting TI.

Fig. 3
Fig. 3

Signal and noise addition in complex plane.

Fig. 4
Fig. 4

Block diagram of TI computer model.

Fig. 5
Fig. 5

WFE reference map used to evaluate TI measurement accuracies ( PTV = 0.996 λ , RMS = 0.353 λ ).

Fig. 6
Fig. 6

Intrinsic measurement error of an off-axis TI ( PTV = 0.009 λ , RMS = 0.001 λ ).

Fig. 7
Fig. 7

Measurement accuracy as function of spectral bandwidth Δ λ (black curves, phase-shifting TI; gray curves, off-axis TI; solid curves, PTV values; dashed curves, RMS values).

Fig. 8
Fig. 8

Typical error map for enlarged spectral bandwidth (off-axis TI with Δ λ = 2 nm , PTV = 0.104 λ , and RMS = 0.010 λ ).

Fig. 9
Fig. 9

Measurement accuracy as function of angular radius of the observed sky object (black curves, phase-shifting TI; gray curves, off-axis TI; solid curves, RMS bias error; dashed curves, photon noise contribution).

Fig. 10
Fig. 10

Typical error map in the presence of photon noise (angular radius = 0.9 mas, PTV = 0.079 λ , RMS = 0.009 λ ).

Fig. 11
Fig. 11

Measurement accuracy as function of main telescope diameter (black lines, phase-shifting TI; gray lines, off-axis TI; dotted lines, r = 0.1 m ; dashed lines, r = 0.25 m ; solid lines, r = 0.5 m ).

Fig. 12
Fig. 12

Measurement accuracy as function of reference pupil diameter (black curves, phase-shifting TI; gray curves, off-axis TI; dotted curves, D = 10 m ; dashed curves, D = 30 m ; solid curves, D = 50 m )

Fig. 13
Fig. 13

Measurement accuracy as function of star magnitude (black curves, phase-shifting TI; gray curves, off-axis TI; dashed curves, r = 0.25 m ; solid curves, r = 0.5 m ).

Fig. 14
Fig. 14

Simulation of WFE retrieval for long exposure times. Top, instantaneous WFE acquisition with Fried’s radius r 0 = 500 mm , PTV = 6.124 λ , RMS = 1.230 λ (left and right, before and after phase unwrapping). Bottom, piston retrieval errors with r 0 = 100 mm (left and right, averaging on PSFs— PTV = 1.000 λ , RMS = 0.265 λ —and on unwrapped OTF phases— PTV = 0.968 λ , RMS = 0.114 λ ). The latter two maps show that initial cophasing errors are not retrieved

Fig. 15
Fig. 15

Measurement accuracy as function of integration time (black curves, phase-shifting TI; gray curves, off-axis TI; dotted curves, r = 10 mm ; dashed curves, r = 25 mm ; solid curves, r = 50 mm ).

Tables (1)

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Table 1 Typical Parameters Used for 5 m Class TIs

Equations (23)

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A P ( x , y ) = A R B R ( x , y ) exp [ i k Δ ( x , y ) ] + A r B r ( x B , y ) ,
A P ( x , y ) = FT [ A P ( x , y ) ] = x , y A P ( x , y ) exp [ i 2 π ( u x + v y ) ] d x d y = A R FT { B R ( x , y ) exp [ i k Δ ( x , y ) ] } + A r FT [ B r ( x B , y ) ] ,
PSF ( x , y ) = A P ( x , y ) 2 .
OTF ( x , y ) = FT 1 [ PSF ( x , y ) ] .
OTF ( x , y ) = A R 2 S R 2 ( 1 S R O R ( x , y ) + C 2 1 S r O r ( x , y ) + C 1 S R { B R ( x , y ) exp [ i k Δ ( x , y ) ] } B r ( x + B , y ) S r + C 1 S R { B R ( x , y ) exp [ i k Δ ( x , y ) ] } B r ( x B , y ) S r ) ,
C = A r A R S r S R .
B 3 R + r
A R 2 S R C ( B R ( x , y ) exp [ i k Δ ( x , y ) ] ) B r ( x , y ) S r = OTF ( x B , y ) B R + r ( x , y ) .
A R 2 S R C B R ( x , y ) exp [ i k Δ ( x , y ) ] OTF ( x B , y ) B R + r ( x , y ) ,
A P ( x , y ) = A R B R ( x , y ) exp [ i k Δ ( x , y ) ] + A r B r ( x , y ) exp [ i ϕ ] ,
A P ( x , y ) = A R FT { B R ( x , y ) exp [ i k Δ ( x , y ) ] } + A r exp [ i ϕ ] FT [ B r ( x , y ) ] .
A R 2 S R C B R ( x , y ) exp [ i k Δ ( x , y ) ] [ OTF 0 ( x , y ) + i OTF π 2 ( x , y ) OTF π ( x , y ) i OTF 3 π 2 ( x , y ) ] 4 .
B R ( x , y ) = η [ 1 + δ B R ( x , y ) ] ,
η A R 2 S R C B R ( x , y ) exp [ i k Δ ( x , y ) ] + σ Δ exp [ i ψ Δ ] = η A R 2 S R C [ 1 + δ B R ( x , y ) ] exp [ i k [ Δ ( x , y ) + δ Δ ( x , y ) ] ] .
σ Δ exp [ i ψ Δ ] = η A R 2 S R C [ δ B R ( x , y ) + i k δ Δ ( x , y ) ] exp [ i k Δ ( x , y ) ] .
σ Δ = η A R 2 S R C [ δ B R 2 ( x , y ) + k 2 δ Δ 2 ( x , y ) ] 1 2 .
k δ Δ ( x , y ) σ Δ η A R 2 S R C .
k δ Δ ( x , y ) 1 ( SNR × C ) .
SNR = η P τ [ η P τ + n Pix ( d τ + r 2 ) ] 1 2 ,
P = S R π ϵ 2 Δ λ B P ( λ ) d λ ,
k δ Δ ( x , y ) = i ( { exp [ i k Δ ( x , y ) ] B r ( x , y ) S r } exp [ i k Δ ( x , y ) ] ) exp [ i k Δ ( x , y ) ] .
OTF ( x , y ) = FT 1 [ Δ λ PSF λ ( x , y ) d λ ] ,
OTF ( x , y ) = FT 1 [ PSF ( x , y ) B F ϵ ( x , y ) ] .

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