Abstract

Large segmented telescopes cannot be modeled accurately with fast-Fourier-transform techniques since small features such as gaps between the segments will be inadequately sampled. An analytic Fourier-transform method can be used to model any pupil configuration with straight edges, including tolerance analysis and some types of apodization. We analytically investigated a 32-m segmented primary with 18 hexagonal segments for high-contrast imaging. There are significant regions in the image in which extrasolar planets could be detected. However, the hexagonal profile of the pupil was not as useful as expected. The gaps between the segments, the secondary obscuration, and the secondary spiders must be as small as possible and their edges must be apodized. Apodizing the edges of the individual segments reduced the useful regions in the image since the gaps appeared to be wider.

© 2005 Optical Society of America

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References

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  1. R. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature (London) 368, 203–207 (1994).
    [CrossRef]
  2. P. Nisenson, C. Papaliolios, “Detection of earth-like planets using apodized telescopes,” Astrophys. J. 548, L201–L205 (2001).
    [CrossRef]
  3. J. R. P. Angel, J. H. Burge, J. L. Codona, W. B. Davison, B. Martin, “20- and 30-m telescope designs with potential for subsequent incorporation into a track-mounted pair (20/20 or 30/30),” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 1223–1232 (2002).
  4. J. E. Nelson, “Design concepts for the California Extremely Large Telescope (CELT),” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation, and Commissioning, T. A. Sebring, T. Anderson, eds., Proc. SPIE4004, 282–289 (2000).
    [CrossRef]
  5. J. Hayes, “Fast Fourier transforms and their application,” in Applied Optics and Optical Engineering, 11th ed. (Academic, San Diego, Calif., 1992), Vol. 2, Chap. 2, pp. 55–123.
  6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  7. J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  8. R. Angel, “Imaging planets from the ground,” in Scientific Frontiers in Research on Extrasolar Planets, Vol. 294 of ASP Conference Series, S. Seager, D. Deming, eds. (Astronomical Society of the Pacific, San Francisco, Calif., 2003), pp. 543–555.

2001 (1)

P. Nisenson, C. Papaliolios, “Detection of earth-like planets using apodized telescopes,” Astrophys. J. 548, L201–L205 (2001).
[CrossRef]

1994 (1)

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

Angel, J. R. P.

J. R. P. Angel, J. H. Burge, J. L. Codona, W. B. Davison, B. Martin, “20- and 30-m telescope designs with potential for subsequent incorporation into a track-mounted pair (20/20 or 30/30),” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 1223–1232 (2002).

Angel, R.

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

R. Angel, “Imaging planets from the ground,” in Scientific Frontiers in Research on Extrasolar Planets, Vol. 294 of ASP Conference Series, S. Seager, D. Deming, eds. (Astronomical Society of the Pacific, San Francisco, Calif., 2003), pp. 543–555.

Burge, J. H.

J. R. P. Angel, J. H. Burge, J. L. Codona, W. B. Davison, B. Martin, “20- and 30-m telescope designs with potential for subsequent incorporation into a track-mounted pair (20/20 or 30/30),” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 1223–1232 (2002).

Codona, J. L.

J. R. P. Angel, J. H. Burge, J. L. Codona, W. B. Davison, B. Martin, “20- and 30-m telescope designs with potential for subsequent incorporation into a track-mounted pair (20/20 or 30/30),” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 1223–1232 (2002).

Davison, W. B.

J. R. P. Angel, J. H. Burge, J. L. Codona, W. B. Davison, B. Martin, “20- and 30-m telescope designs with potential for subsequent incorporation into a track-mounted pair (20/20 or 30/30),” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 1223–1232 (2002).

Gaskill, J.

J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Hayes, J.

J. Hayes, “Fast Fourier transforms and their application,” in Applied Optics and Optical Engineering, 11th ed. (Academic, San Diego, Calif., 1992), Vol. 2, Chap. 2, pp. 55–123.

Martin, B.

J. R. P. Angel, J. H. Burge, J. L. Codona, W. B. Davison, B. Martin, “20- and 30-m telescope designs with potential for subsequent incorporation into a track-mounted pair (20/20 or 30/30),” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 1223–1232 (2002).

Nelson, J. E.

J. E. Nelson, “Design concepts for the California Extremely Large Telescope (CELT),” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation, and Commissioning, T. A. Sebring, T. Anderson, eds., Proc. SPIE4004, 282–289 (2000).
[CrossRef]

Nisenson, P.

P. Nisenson, C. Papaliolios, “Detection of earth-like planets using apodized telescopes,” Astrophys. J. 548, L201–L205 (2001).
[CrossRef]

Papaliolios, C.

P. Nisenson, C. Papaliolios, “Detection of earth-like planets using apodized telescopes,” Astrophys. J. 548, L201–L205 (2001).
[CrossRef]

Astrophys. J. (1)

P. Nisenson, C. Papaliolios, “Detection of earth-like planets using apodized telescopes,” Astrophys. J. 548, L201–L205 (2001).
[CrossRef]

Nature (London) (1)

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

Other (6)

J. R. P. Angel, J. H. Burge, J. L. Codona, W. B. Davison, B. Martin, “20- and 30-m telescope designs with potential for subsequent incorporation into a track-mounted pair (20/20 or 30/30),” in Future Giant Telescopes, J. R. P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 1223–1232 (2002).

J. E. Nelson, “Design concepts for the California Extremely Large Telescope (CELT),” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation, and Commissioning, T. A. Sebring, T. Anderson, eds., Proc. SPIE4004, 282–289 (2000).
[CrossRef]

J. Hayes, “Fast Fourier transforms and their application,” in Applied Optics and Optical Engineering, 11th ed. (Academic, San Diego, Calif., 1992), Vol. 2, Chap. 2, pp. 55–123.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

R. Angel, “Imaging planets from the ground,” in Scientific Frontiers in Research on Extrasolar Planets, Vol. 294 of ASP Conference Series, S. Seager, D. Deming, eds. (Astronomical Society of the Pacific, San Francisco, Calif., 2003), pp. 543–555.

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Figures (8)

Fig. 1
Fig. 1

Regular hexagon is the sum of two trapezoids, and each trapezoid is the sum of strips whose length depends on the height of the strip in the trapezoid.

Fig. 2
Fig. 2

Image intensity from a hexagonal pupil can be calculated analytically. As a check, a FFT can then be applied to the image amplitude to recover the original pupil amplitude.

Fig. 3
Fig. 3

Apodized hexagon and its cross section, which falls off as cos2.

Fig. 4
Fig. 4

Analytic PSF irradiance plot for a segmented hexagonal primary 32 m in diameter, with spider supports and secondary.

Fig. 5
Fig. 5

System performance was evaluated with and without the secondary and supports, with varying gap widths, and with two styles of apodization.

Fig. 6
Fig. 6

Regions where planet detection is possible are shown in gray for a perfect hexagonal pupil and for a hexagonal pupil with secondary, supports, and the major edges apodized. The star’s PSF is shown at the center. S–B, signal-to-background.

Fig. 7
Fig. 7

Apodizing the main edges in the primary led to an improvement in the area usable for planet imaging, defined as regions where the signal-to-background ratio is greater than 1. The loss of performance due to the widening of the gaps is clearly visible.

Fig. 8
Fig. 8

Apodization of the individual segments initially leads to a loss in performance because the gaps between segments appear to be wider. Performance recovers somewhat as the apodization width increases.

Tables (1)

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Table 1 Useful Hexagonal Fourier Transforms

Equations (11)

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I ( x , y ) = ( 1 λ f ) 2 { f ( x , y ) exp [ i 2 π ϕ ( x , y ) / λ ] × exp [ i 2 π ( ξ y + η x ) ] δ x δ y } ξ y / λ f , η x / λ f 2 ,
I ( x , y ) = ( 1 λ f ) 2 { k f k ( x , y ) exp [ i 2 π ϕ k ( x , y ) / λ ] × exp [ i 2 π ( ξ y + η x ) ] δ x δ y } ξ y / λ f , η x / λ f 2 ,
F [ f ( x , y ) ] = g ( ξ , η ) ,
{ f ( x - x 0 , y - y 0 , θ ) } = exp [ i 2 π ( x 0 η + y 0 ξ ) ] g ( ξ cos θ + η sin θ , - ξ sin θ + η cos θ ) ,
g ( ξ , η ) = 0 D / 2 { ( y - D ) / 3 ( y - D ) / ( - 3 ) exp [ i 2 π ( ξ y + η x ) ] δ x } δ y = exp [ - i π D ( 2 η 3 + ξ ) ] 4 π 2 ( η 3 - 3 η ξ 2 ) ( ( 3 η - 3 ξ ) × { exp ( i π D 3 η ) - exp [ i π D ( 4 3 η + ξ ) ] } + ( 3 η + 3 ξ ) [ exp ( i π D η / 3 ) - exp ( i π D ξ ) ] ) .
hex ( ξ , η ) = g ( ξ , η ) + g ( - ξ , η ) ,
( 1 λ f ) 2 hex ( y λ f , x λ f ) 2 .
A ( ξ , η ) = 0 ( D - 2 ɛ ) 2 y - ( D - 2 ɛ ) 3 y - ( D - 2 ɛ ) - 3 exp [ i 2 π ( ξ y + η x ) ] δ x δ y ,
B ( ξ , η ) = 0 ɛ ( y - D / 2 ) 3 ( y - D / 2 ) - 3 [ cos π ( ɛ - y ) 2 ɛ ] 2 × exp [ i 2 π ( ξ y + η x ) ] δ x δ y ,
C ( ξ , η ) = exp [ - i 2 π ( D 4 ξ + 3 D 4 η ) B ( ξ cos 2 π 3 - η sin 2 π 3 , ξ sin 2 π 3 + η cos 2 π 3 ) ] .
G ( ξ , η ) = A ( ξ , η ) + A ( - ξ , η ) + exp ( i π D ξ ) B ( ξ , η ) + exp ( - i π D ξ ) B ( - ξ , η ) + C ( ξ , η ) + C ( - ξ , η ) + C ( - ξ , - η ) + C ( ξ , - η ) .

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