Abstract

The basic principle and theoretical relationships of an original method are presented that allow the wave-front errors of a ground or spaceborne telescope to be retrieved when its main pupil is combined with a second, decentered reference optical arm. The measurement accuracy of such a telescope–interferometer is then estimated by means of various numerical simulations, and good performance is demonstrated, except in limited areas near the telescope pupil’s rim. In particular, it permits direct phase evaluation (thus avoiding the use of first- or second-order derivatives), which will be of special interest for the cophasing of segmented mirrors in future giant-telescope projects. Finally, the useful practical domain of the method is defined, which seems to be better suited for periodic diagnostics of space- or ground-based telescopes or to real-time scientific observations in some specific cases (e.g., the central star in instruments that search for extrasolar planets).

© 2005 Optical Society of America

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References

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  1. R. V. Shack, B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  6. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
    [CrossRef]
  7. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  8. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [CrossRef]
  9. R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
    [CrossRef]
  10. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik (Stuttgart) 35, 237–246 (1972).
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    [CrossRef]
  12. R. Angel, “Imaging extrasolar planets from the ground,” in Scientific Frontiers in Research on Extrasolar Planets, D. Deming, S. Seager, eds., ASP Conf. Ser.294, 543–556 (2003).
  13. J. L. Codona, R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J. 604, L117–L120 (2004).
    [CrossRef]
  14. A. Labeyrie, “Removal of coronagraphy residues with an adaptive hologram, for imaging exo-Earths,” in Astronomy with High Contrast Imaging II, C. Aime, R. Soummer, eds., Vol. 12 of EAS Publications Series (European Astronomical Society, 2004), pp. 3–10.
  15. A. A. Michelson, F. G. Pease, “Measurement of the diameter of alpha Orionis with the interferometer,” Astrophys. J. 53, 249–259 (1921).
    [CrossRef]
  16. M. Takeda, H. Ina, 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]
  17. M. Born, E. Wolf, Principles of Optics, 6th ed. (London, 1980).
  18. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [CrossRef] [PubMed]
  19. V. Coude, Du Foresto, G. Perrin, C. Ruilier, B. P. Mennesson, W. A. Traub, M. G. Lacasse, “FLUOR fibered instrument at the IOTA interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 856–863 (1998).
    [CrossRef]
  20. F. Reynaud, J. J. Alleman, P. Connes, “Interferometric control of fiber lengths for a coherent telescope array,” Appl. Opt. 31, 3736–3743 (1992).
    [CrossRef] [PubMed]
  21. S. B. Shaklan, F. Roddier, “Single-mode fiber optics in a long-baseline interferometer,” Appl. Opt. 26, 2159–2163 (1987).
    [CrossRef] [PubMed]
  22. F. Hénault is preparing a manuscript entitled “Signal to noise ratio of phase sensing telescope interferometers.”

2004 (1)

J. L. Codona, R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J. 604, L117–L120 (2004).
[CrossRef]

1996 (1)

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

1994 (1)

R. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368, 203–207 (1994).
[CrossRef]

1993 (1)

1992 (2)

1988 (2)

1987 (2)

1982 (3)

M. Takeda, H. Ina, 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]

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).
[CrossRef]

1977 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik (Stuttgart) 35, 237–246 (1972).

1971 (1)

R. V. Shack, B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

1921 (1)

A. A. Michelson, F. G. Pease, “Measurement of the diameter of alpha Orionis with the interferometer,” Astrophys. J. 53, 249–259 (1921).
[CrossRef]

Alleman, J. J.

Angel, R.

J. L. Codona, R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J. 604, L117–L120 (2004).
[CrossRef]

R. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368, 203–207 (1994).
[CrossRef]

R. Angel, “Imaging extrasolar planets from the ground,” in Scientific Frontiers in Research on Extrasolar Planets, D. Deming, S. Seager, eds., ASP Conf. Ser.294, 543–556 (2003).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (London, 1980).

Codona, J. L.

J. L. Codona, R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J. 604, L117–L120 (2004).
[CrossRef]

Connes, P.

Coude, V.

V. Coude, Du Foresto, G. Perrin, C. Ruilier, B. P. Mennesson, W. A. Traub, M. G. Lacasse, “FLUOR fibered instrument at the IOTA interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 856–863 (1998).
[CrossRef]

Fienup, J. R.

Foresto, Du

V. Coude, Du Foresto, G. Perrin, C. Ruilier, B. P. Mennesson, W. A. Traub, M. G. Lacasse, “FLUOR fibered instrument at the IOTA interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 856–863 (1998).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik (Stuttgart) 35, 237–246 (1972).

Gonsalves, R. A.

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

Hardy, J. W.

Hénault, F.

F. Hénault is preparing a manuscript entitled “Signal to noise ratio of phase sensing telescope interferometers.”

Ina, H.

M. Takeda, H. Ina, 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]

Koliopoulos, C. L.

Koyabashi, S.

M. Takeda, H. Ina, 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]

Labeyrie, A.

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

Lacasse, M. G.

V. Coude, Du Foresto, G. Perrin, C. Ruilier, B. P. Mennesson, W. A. Traub, M. G. Lacasse, “FLUOR fibered instrument at the IOTA interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 856–863 (1998).
[CrossRef]

Lane, R. G.

Lefebvre, J. E.

Mennesson, B. P.

V. Coude, Du Foresto, G. Perrin, C. Ruilier, B. P. Mennesson, W. A. Traub, M. G. Lacasse, “FLUOR fibered instrument at the IOTA interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 856–863 (1998).
[CrossRef]

Michelson, A. A.

A. A. Michelson, F. G. Pease, “Measurement of the diameter of alpha Orionis with the interferometer,” Astrophys. J. 53, 249–259 (1921).
[CrossRef]

Paxman, R. G.

Pease, F. G.

A. A. Michelson, F. G. Pease, “Measurement of the diameter of alpha Orionis with the interferometer,” Astrophys. J. 53, 249–259 (1921).
[CrossRef]

Perrin, G.

V. Coude, Du Foresto, G. Perrin, C. Ruilier, B. P. Mennesson, W. A. Traub, M. G. Lacasse, “FLUOR fibered instrument at the IOTA interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 856–863 (1998).
[CrossRef]

Platt, B. C.

R. V. Shack, B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Ragazzoni, R.

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

Reynaud, F.

Roddier, C.

Roddier, F.

Ruilier, C.

V. Coude, Du Foresto, G. Perrin, C. Ruilier, B. P. Mennesson, W. A. Traub, M. G. Lacasse, “FLUOR fibered instrument at the IOTA interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 856–863 (1998).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik (Stuttgart) 35, 237–246 (1972).

Shack, R. V.

R. V. Shack, B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Shaklan, S. B.

Takeda, M.

M. Takeda, H. Ina, 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]

Tallon, M.

Traub, W. A.

V. Coude, Du Foresto, G. Perrin, C. Ruilier, B. P. Mennesson, W. A. Traub, M. G. Lacasse, “FLUOR fibered instrument at the IOTA interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 856–863 (1998).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (London, 1980).

Appl. Opt. (6)

Astrophys. J. (2)

J. L. Codona, R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J. 604, L117–L120 (2004).
[CrossRef]

A. A. Michelson, F. G. Pease, “Measurement of the diameter of alpha Orionis with the interferometer,” Astrophys. J. 53, 249–259 (1921).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. (2)

R. V. Shack, B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nature (1)

R. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368, 203–207 (1994).
[CrossRef]

Opt. Eng. (1)

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

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik (Stuttgart) 35, 237–246 (1972).

Other (5)

R. Angel, “Imaging extrasolar planets from the ground,” in Scientific Frontiers in Research on Extrasolar Planets, D. Deming, S. Seager, eds., ASP Conf. Ser.294, 543–556 (2003).

M. Born, E. Wolf, Principles of Optics, 6th ed. (London, 1980).

V. Coude, Du Foresto, G. Perrin, C. Ruilier, B. P. Mennesson, W. A. Traub, M. G. Lacasse, “FLUOR fibered instrument at the IOTA interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 856–863 (1998).
[CrossRef]

F. Hénault is preparing a manuscript entitled “Signal to noise ratio of phase sensing telescope interferometers.”

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

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Figures (12)

Fig. 1
Fig. 1

Coordinate systems.

Fig. 2
Fig. 2

Example of a telescope PSF modulated by the reference pupil arm (left, linear scale; right, logarithmic scale).

Fig. 3
Fig. 3

MTF of the global system (schematic).

Fig. 4
Fig. 4

Typical example of a global MTF (logarithmic scale).

Fig. 5
Fig. 5

(a) Reference and (c) reconstructed WFEs of a pure defocus defect and (d) their two-dimensional difference map. (b) Modulus of the crossed OTF term. Gray levels are scaled to PTV values listed in Table 1.

Fig. 6
Fig. 6

Cross sections of the reference, reconstructed, and difference WFEs for a pure defocus defect.

Fig. 7
Fig. 7

(a) Reference and (c) reconstructed WFEs of low spatial frequency defects and (d) their two-dimensional difference map. (b) Modulus of the crossed OTF term. Gray levels are scaled to PTV values listed in Table 1.

Fig. 8
Fig. 8

(a) Reference and (c) reconstructed WFEs of segmented mirror defects and (d) their two-dimensional difference map. (b) Modulus of the crossed OTF term. Gray levels are scaled to PTV values listed in Table 1.

Fig. 9
Fig. 9

(a) Reference and (c) reconstructed WFEs of random defects and (d) their two-dimensional difference map. (b) Modulus of the crossed OTF term. Gray levels are scaled to PTV values listed in Table 1.

Fig. 10
Fig. 10

Implementation of the wave-front sensing method in a real telescope: the Fizeau configuration.

Fig. 11
Fig. 11

Implementation of the wave-front sensing method in a real telescope: the intermediate configuration.

Fig. 12
Fig. 12

Implementation of the wave-front sensing method in a real telescope: the Michelson configuration.

Tables (1)

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Table 1 Synthesis of Simulation Results

Equations (29)

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A P ( x , y ) = A R B R ( x , y ) exp [ i 2 π Δ ( x , y ) / λ ] + 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 P ( x , y ) = A R FT { B R ( x , y ) exp [ i 2 π Δ ( x , y ) / λ ] } + A r FT [ B r ( x - B , y ) ] ,
FT { B R ( x , y ) exp [ i 2 π Δ ( x , y ) / λ ] } = π R 2 M ( u , v ) exp [ i Ψ ( u , v ) ] .
FT [ B r ( x - B , y ) ] = π r 2 m ( u , v ) exp ( - i 2 π u B ) ,
A P ( x , y ) = A { M ( u , v ) exp [ i Ψ ( u , v ) ] + C A C m ( u , v ) exp ( - i 2 π u B ) } ,
A = A R π R 2
C A = A r / A R ,
C = r 2 / R 2 .
I P ( x , y ) = A 2 ( M 2 ( u , v ) + C A 2 C 2 m 2 ( u , v ) + C A C M ( u , v ) m ( u , v ) { exp [ i Ψ ( u , v ) + i 2 π u B ] + exp [ - i Ψ ( u , v ) - i 2 π u B ] } ) ,
I P ( x , y ) = A 2 { M 2 ( u , v ) + C A 2 C 2 m 2 ( u , v ) + 2 C A C M ( u , v ) m ( u , v ) cos [ Ψ ( u , v ) + 2 π u B ] } .
C P ( x , y ) = FT - 1 [ I p ( x , y ) ] = A 2 [ C P 1 ( x , y ) + C A 2 C 2 C P 2 ( x , y ) + C A C C P 3 ( x , y ) + C A C C P 4 ( x , y ) ] .
C P ( x , y ) = A 2 [ 1 π R 2 O R ( x , y ) + C A 2 C 2 1 π r 2 O r ( x , y ) + C A C × 1 π R 2 1 π r 2 ( { B R ( x , y ) exp [ i 2 π λ Δ ( x , y ) ] } B r ( x + B , y ) ) + C A C 1 π R 2 1 π r 2 × ( { B R ( - x , - y ) exp [ - 2 π λ Δ ( - x , - y ) ] } B r ( x - B , y ) ) ] ,
C P ( x , y ) = 1 1 + C A 2 C ( O R ( x , y ) + C A 2 C O r ( x , y ) + C A C × { B R ( x , y ) exp [ i 2 π λ Δ ( x , y ) ] } B r ( x + B , y ) π r 2 + C A C { B R ( - x , - y ) × exp [ - i 2 π λ Δ ( - x , - y ) ] } B r ( x - B , y ) π r 2 ) .
B > 3 R + r .
1 + C A 2 C C A C B R + r ( x , y ) C P ( x - B , y ) = { B R ( x , y ) exp [ i 2 π λ Δ ( x , y ) ] } B r ( x , y ) π r 2 .
B R ( x , y ) exp [ i 2 π λ Δ ( x , y ) ] 1 + C A 2 C C A C B R + r ( x , y ) C p ( x - B , y ) .
Δ ( x , y ) λ 2 π arctan { Im [ B R + r ( x , y ) C P ( x - B , y ) ] Re [ B R + r ( x , y ) C P ( x - B , y ) ] } mod [ λ ] ,
C P 1 ( x , y ) = FT - 1 [ M 2 ( u , v ) ] = FT - 1 { M ( u , v ) exp [ i Ψ ( u , v ) ] × M ( u , v ) exp [ - i Ψ ( u , v ) ] }
C P 1 ( x , y ) = FT - 1 { M ( u , v ) exp [ i Ψ ( u , v ) ] } FT - { M ( u , v ) exp [ - i Ψ ( u , v ) ] } .
C P 1 ( x , y ) = 1 π R 2 { B R ( x , y ) exp [ i 2 π λ Δ ( x , y ) ] } 1 π R 2 { B R ( - x , - y ) exp × [ - i 2 π λ Δ ( - x , - y ) ] } .
{ B r ( x , y ) exp [ i 2 π λ Δ ( x , y ) ] } { B R ( - x , - y ) exp × [ - i 2 π λ Δ ( - x , - y ) ] } = π R 2 O R ( x , y )
C P 1 ( x , y ) = 1 π R 2 O R ( x , y ) .
C P 2 ( x , y ) = 1 π r 2 O r ( x , y ) .
C P 3 ( x , y ) = FT - 1 { M ( u , v ) m ( u , v ) exp [ i Ψ ( u , v ) + i 2 π u B ] } ,
C P 3 ( x , y ) = FT - 1 { M ( u , v ) exp [ i Ψ ( u , v ) ] } FT - 1 [ m ( u , v ) ] FT - 1 [ exp ( i 2 π u B ) ] .
C P 3 ( x , y ) = 1 π R 2 { B R ( x , y ) exp [ i 2 π λ Δ ( x , y ) ] } 1 π r 2 B r ( x , t ) δ ( x + B ) ,
C P 3 ( x , y ) = 1 π R 2 1 π r 2 ( { B R ( x , y ) exp [ i 2 π λ Δ ( x , y ) ] } B r ( x + B , y ) ) .
C P 4 ( x , y ) = 1 π R 2 1 π r 2 ( { B R ( - x , - y ) exp [ - i 2 π λ Δ ( - x , - y ) ] } B r ( x - B , y ) ) .

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