Abstract

Nulling interferometers combine on-axis suppression with high angular resolution, making them ideal instruments for the direct detection of faint planets close to their parent star. The synthesized point-spread function for a rotating nulling interferometer utilizing phase chopping is shown to consist of a main peak, satellite peaks, and their associated sidelobes, and simple analytic expressions are derived for the modulation efficiency and angular resolution. Sufficient angular resolution is vital for the detection and characterization of multiple-planet systems and requires that some configurations be substantially larger than previously thought. The corresponding increase in stellar leakage has a major effect on performance and can be a deciding factor in the choice of array configuration.

© 2005 Optical Society of America

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References

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  1. N. Woolf, J. R. Angel, “Astronomical searches for earth-like planets and signs of life,” Annu. Rev. Astron. Astrophys. 36, 507–537 (1998).
    [CrossRef]
  2. G. W. Marcy, R. P. Butler, “Detection of extrasolar giant planets,” Annu. Rev. Astron. Astrophys. 36, 57–97 (1998).
    [CrossRef]
  3. R. N. Bracewell, “Detecting nonsolar planets by spinning infrared interferometer,” Nature 274, 780–781 (1978).
    [CrossRef]
  4. E. Serabyn, “Nulling interferometry and planet detection,” in Principles of Long Baseline Interferometry, P. Lawson, ed. (Jet Propulsion Laboratory, 2000), pp. 273–292.
  5. G. Vasisht, M. M. Colavita, “Differential phase interferometry with the Keck telescopes,” Proc. SPIE 5491, 567–576 (2004).
    [CrossRef]
  6. V. Ford, P. D. Lisman, S. B. Shaklan, J. T. Trauger, T. Ho, D. Hoppe, A. E. Lowman, “The Terrestrial Planet Finder coronagraph: technology and mission design studies,” Proc. SPIE 5487, 1274–1283 (2004).
    [CrossRef]
  7. D. Coulter, “NASA’s Terrestrial Planet Finder mission: the search for habitable planets,” in Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Planets, M. Fridlund, T. Henning, eds. (European Space Agency, 2003), pp. 47–54.
  8. A. Karlsson, L. Kaltenegger, “The technology of DARWIN,” in Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Planets, M. Fridlund, T. Henning, eds. (European Space Agency, 2003), pp. 41–46.
  9. S. Dubovitsky, O. P. Lay, “Architecture selection and optimization for planet-finding interferometers,” Proc. SPIE 5491, 284–295 (2004).
    [CrossRef]
  10. P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array—design and performance of a modern synthesis radio telescope,” IEEE Proc. 71, 1295–1320 (1983).
    [CrossRef]
  11. O. P. Lay, “Systematic errors in nulling interferometers,” Appl. Opt. 43, 6100–6123 (2004).
    [CrossRef] [PubMed]
  12. O. P. Lay, S. Dubovitsky, “Nulling interferometers: the importance of systematic errors and the X-array,” Proc. SPIE 5491, 874–885 (2004).
    [CrossRef]
  13. T. Velusamy, C. A. Beichman, “Nulling interferometry for extra-solar planet detection: sensitivity and image reconstruction,” in IEEE Aerospace Conference Proceedings (IEEE Press, 2001), Vol. 4, pp. 2013–2025.
  14. J. R. P. Angel, N. J. Woolf, “An imaging nulling interferometer to study extrasolar planets,” Astrophys. J. 475, 373–379 (1997).
    [CrossRef]
  15. B. Mennesson, “Thermal infrared stellar interferometry: observations of circumstellar environments using single-mode guided optics and contributions to the DARWIN space mission,” Ph.D. dissertation (Universite Paris, 1999).
  16. W. C. Danchi, D. Deming, M. J. Kuchner, S. Seager, “Detection of close-in extrasolar giant planets using the Fourier-Kelvin Stellar Interferometer,” Astrophys. J. 597, L57–L60 (2003).
    [CrossRef]
  17. L. Kaltenegger, A. Karlsson, “Requirements on the stellar rejection of the DARWIN mission,” Proc. SPIE 5491, 275–283 (2004).
    [CrossRef]
  18. C. A. Beichman, N. J. Woolf, C. A. Lindensmith, Terrestrial Planet Finder: a NASA Origins Program to Search for Habitable Planets (Jet Propulsion Laboratory, 1999).
  19. A. L. Karlsson, ESTEC, Noordwijk, The Netherlands (personal communication, 2004).
  20. A. L. Karlsson, O. Wallner, J. M. Perdigues Armengol, O. Absil, “Three telescope nuller based on multibeam injection into single-mode waveguide,” Proc. SPIE 5491, 831–841 (2004).
    [CrossRef]

2004 (7)

G. Vasisht, M. M. Colavita, “Differential phase interferometry with the Keck telescopes,” Proc. SPIE 5491, 567–576 (2004).
[CrossRef]

V. Ford, P. D. Lisman, S. B. Shaklan, J. T. Trauger, T. Ho, D. Hoppe, A. E. Lowman, “The Terrestrial Planet Finder coronagraph: technology and mission design studies,” Proc. SPIE 5487, 1274–1283 (2004).
[CrossRef]

S. Dubovitsky, O. P. Lay, “Architecture selection and optimization for planet-finding interferometers,” Proc. SPIE 5491, 284–295 (2004).
[CrossRef]

O. P. Lay, “Systematic errors in nulling interferometers,” Appl. Opt. 43, 6100–6123 (2004).
[CrossRef] [PubMed]

O. P. Lay, S. Dubovitsky, “Nulling interferometers: the importance of systematic errors and the X-array,” Proc. SPIE 5491, 874–885 (2004).
[CrossRef]

L. Kaltenegger, A. Karlsson, “Requirements on the stellar rejection of the DARWIN mission,” Proc. SPIE 5491, 275–283 (2004).
[CrossRef]

A. L. Karlsson, O. Wallner, J. M. Perdigues Armengol, O. Absil, “Three telescope nuller based on multibeam injection into single-mode waveguide,” Proc. SPIE 5491, 831–841 (2004).
[CrossRef]

2003 (1)

W. C. Danchi, D. Deming, M. J. Kuchner, S. Seager, “Detection of close-in extrasolar giant planets using the Fourier-Kelvin Stellar Interferometer,” Astrophys. J. 597, L57–L60 (2003).
[CrossRef]

1998 (2)

N. Woolf, J. R. Angel, “Astronomical searches for earth-like planets and signs of life,” Annu. Rev. Astron. Astrophys. 36, 507–537 (1998).
[CrossRef]

G. W. Marcy, R. P. Butler, “Detection of extrasolar giant planets,” Annu. Rev. Astron. Astrophys. 36, 57–97 (1998).
[CrossRef]

1997 (1)

J. R. P. Angel, N. J. Woolf, “An imaging nulling interferometer to study extrasolar planets,” Astrophys. J. 475, 373–379 (1997).
[CrossRef]

1983 (1)

P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array—design and performance of a modern synthesis radio telescope,” IEEE Proc. 71, 1295–1320 (1983).
[CrossRef]

1978 (1)

R. N. Bracewell, “Detecting nonsolar planets by spinning infrared interferometer,” Nature 274, 780–781 (1978).
[CrossRef]

Absil, O.

A. L. Karlsson, O. Wallner, J. M. Perdigues Armengol, O. Absil, “Three telescope nuller based on multibeam injection into single-mode waveguide,” Proc. SPIE 5491, 831–841 (2004).
[CrossRef]

Angel, J. R.

N. Woolf, J. R. Angel, “Astronomical searches for earth-like planets and signs of life,” Annu. Rev. Astron. Astrophys. 36, 507–537 (1998).
[CrossRef]

Angel, J. R. P.

J. R. P. Angel, N. J. Woolf, “An imaging nulling interferometer to study extrasolar planets,” Astrophys. J. 475, 373–379 (1997).
[CrossRef]

Beichman, C. A.

T. Velusamy, C. A. Beichman, “Nulling interferometry for extra-solar planet detection: sensitivity and image reconstruction,” in IEEE Aerospace Conference Proceedings (IEEE Press, 2001), Vol. 4, pp. 2013–2025.

C. A. Beichman, N. J. Woolf, C. A. Lindensmith, Terrestrial Planet Finder: a NASA Origins Program to Search for Habitable Planets (Jet Propulsion Laboratory, 1999).

Bracewell, R. N.

R. N. Bracewell, “Detecting nonsolar planets by spinning infrared interferometer,” Nature 274, 780–781 (1978).
[CrossRef]

Butler, R. P.

G. W. Marcy, R. P. Butler, “Detection of extrasolar giant planets,” Annu. Rev. Astron. Astrophys. 36, 57–97 (1998).
[CrossRef]

Colavita, M. M.

G. Vasisht, M. M. Colavita, “Differential phase interferometry with the Keck telescopes,” Proc. SPIE 5491, 567–576 (2004).
[CrossRef]

Coulter, D.

D. Coulter, “NASA’s Terrestrial Planet Finder mission: the search for habitable planets,” in Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Planets, M. Fridlund, T. Henning, eds. (European Space Agency, 2003), pp. 47–54.

Danchi, W. C.

W. C. Danchi, D. Deming, M. J. Kuchner, S. Seager, “Detection of close-in extrasolar giant planets using the Fourier-Kelvin Stellar Interferometer,” Astrophys. J. 597, L57–L60 (2003).
[CrossRef]

Deming, D.

W. C. Danchi, D. Deming, M. J. Kuchner, S. Seager, “Detection of close-in extrasolar giant planets using the Fourier-Kelvin Stellar Interferometer,” Astrophys. J. 597, L57–L60 (2003).
[CrossRef]

Dubovitsky, S.

O. P. Lay, S. Dubovitsky, “Nulling interferometers: the importance of systematic errors and the X-array,” Proc. SPIE 5491, 874–885 (2004).
[CrossRef]

S. Dubovitsky, O. P. Lay, “Architecture selection and optimization for planet-finding interferometers,” Proc. SPIE 5491, 284–295 (2004).
[CrossRef]

Ekers, R. D.

P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array—design and performance of a modern synthesis radio telescope,” IEEE Proc. 71, 1295–1320 (1983).
[CrossRef]

Ford, V.

V. Ford, P. D. Lisman, S. B. Shaklan, J. T. Trauger, T. Ho, D. Hoppe, A. E. Lowman, “The Terrestrial Planet Finder coronagraph: technology and mission design studies,” Proc. SPIE 5487, 1274–1283 (2004).
[CrossRef]

Ho, T.

V. Ford, P. D. Lisman, S. B. Shaklan, J. T. Trauger, T. Ho, D. Hoppe, A. E. Lowman, “The Terrestrial Planet Finder coronagraph: technology and mission design studies,” Proc. SPIE 5487, 1274–1283 (2004).
[CrossRef]

Hoppe, D.

V. Ford, P. D. Lisman, S. B. Shaklan, J. T. Trauger, T. Ho, D. Hoppe, A. E. Lowman, “The Terrestrial Planet Finder coronagraph: technology and mission design studies,” Proc. SPIE 5487, 1274–1283 (2004).
[CrossRef]

Kaltenegger, L.

L. Kaltenegger, A. Karlsson, “Requirements on the stellar rejection of the DARWIN mission,” Proc. SPIE 5491, 275–283 (2004).
[CrossRef]

A. Karlsson, L. Kaltenegger, “The technology of DARWIN,” in Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Planets, M. Fridlund, T. Henning, eds. (European Space Agency, 2003), pp. 41–46.

Karlsson, A.

L. Kaltenegger, A. Karlsson, “Requirements on the stellar rejection of the DARWIN mission,” Proc. SPIE 5491, 275–283 (2004).
[CrossRef]

A. Karlsson, L. Kaltenegger, “The technology of DARWIN,” in Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Planets, M. Fridlund, T. Henning, eds. (European Space Agency, 2003), pp. 41–46.

Karlsson, A. L.

A. L. Karlsson, O. Wallner, J. M. Perdigues Armengol, O. Absil, “Three telescope nuller based on multibeam injection into single-mode waveguide,” Proc. SPIE 5491, 831–841 (2004).
[CrossRef]

A. L. Karlsson, ESTEC, Noordwijk, The Netherlands (personal communication, 2004).

Kuchner, M. J.

W. C. Danchi, D. Deming, M. J. Kuchner, S. Seager, “Detection of close-in extrasolar giant planets using the Fourier-Kelvin Stellar Interferometer,” Astrophys. J. 597, L57–L60 (2003).
[CrossRef]

Lay, O. P.

O. P. Lay, “Systematic errors in nulling interferometers,” Appl. Opt. 43, 6100–6123 (2004).
[CrossRef] [PubMed]

O. P. Lay, S. Dubovitsky, “Nulling interferometers: the importance of systematic errors and the X-array,” Proc. SPIE 5491, 874–885 (2004).
[CrossRef]

S. Dubovitsky, O. P. Lay, “Architecture selection and optimization for planet-finding interferometers,” Proc. SPIE 5491, 284–295 (2004).
[CrossRef]

Lindensmith, C. A.

C. A. Beichman, N. J. Woolf, C. A. Lindensmith, Terrestrial Planet Finder: a NASA Origins Program to Search for Habitable Planets (Jet Propulsion Laboratory, 1999).

Lisman, P. D.

V. Ford, P. D. Lisman, S. B. Shaklan, J. T. Trauger, T. Ho, D. Hoppe, A. E. Lowman, “The Terrestrial Planet Finder coronagraph: technology and mission design studies,” Proc. SPIE 5487, 1274–1283 (2004).
[CrossRef]

Lowman, A. E.

V. Ford, P. D. Lisman, S. B. Shaklan, J. T. Trauger, T. Ho, D. Hoppe, A. E. Lowman, “The Terrestrial Planet Finder coronagraph: technology and mission design studies,” Proc. SPIE 5487, 1274–1283 (2004).
[CrossRef]

Marcy, G. W.

G. W. Marcy, R. P. Butler, “Detection of extrasolar giant planets,” Annu. Rev. Astron. Astrophys. 36, 57–97 (1998).
[CrossRef]

Mennesson, B.

B. Mennesson, “Thermal infrared stellar interferometry: observations of circumstellar environments using single-mode guided optics and contributions to the DARWIN space mission,” Ph.D. dissertation (Universite Paris, 1999).

Napier, P. J.

P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array—design and performance of a modern synthesis radio telescope,” IEEE Proc. 71, 1295–1320 (1983).
[CrossRef]

Perdigues Armengol, J. M.

A. L. Karlsson, O. Wallner, J. M. Perdigues Armengol, O. Absil, “Three telescope nuller based on multibeam injection into single-mode waveguide,” Proc. SPIE 5491, 831–841 (2004).
[CrossRef]

Seager, S.

W. C. Danchi, D. Deming, M. J. Kuchner, S. Seager, “Detection of close-in extrasolar giant planets using the Fourier-Kelvin Stellar Interferometer,” Astrophys. J. 597, L57–L60 (2003).
[CrossRef]

Serabyn, E.

E. Serabyn, “Nulling interferometry and planet detection,” in Principles of Long Baseline Interferometry, P. Lawson, ed. (Jet Propulsion Laboratory, 2000), pp. 273–292.

Shaklan, S. B.

V. Ford, P. D. Lisman, S. B. Shaklan, J. T. Trauger, T. Ho, D. Hoppe, A. E. Lowman, “The Terrestrial Planet Finder coronagraph: technology and mission design studies,” Proc. SPIE 5487, 1274–1283 (2004).
[CrossRef]

Thompson, A. R.

P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array—design and performance of a modern synthesis radio telescope,” IEEE Proc. 71, 1295–1320 (1983).
[CrossRef]

Trauger, J. T.

V. Ford, P. D. Lisman, S. B. Shaklan, J. T. Trauger, T. Ho, D. Hoppe, A. E. Lowman, “The Terrestrial Planet Finder coronagraph: technology and mission design studies,” Proc. SPIE 5487, 1274–1283 (2004).
[CrossRef]

Vasisht, G.

G. Vasisht, M. M. Colavita, “Differential phase interferometry with the Keck telescopes,” Proc. SPIE 5491, 567–576 (2004).
[CrossRef]

Velusamy, T.

T. Velusamy, C. A. Beichman, “Nulling interferometry for extra-solar planet detection: sensitivity and image reconstruction,” in IEEE Aerospace Conference Proceedings (IEEE Press, 2001), Vol. 4, pp. 2013–2025.

Wallner, O.

A. L. Karlsson, O. Wallner, J. M. Perdigues Armengol, O. Absil, “Three telescope nuller based on multibeam injection into single-mode waveguide,” Proc. SPIE 5491, 831–841 (2004).
[CrossRef]

Woolf, N.

N. Woolf, J. R. Angel, “Astronomical searches for earth-like planets and signs of life,” Annu. Rev. Astron. Astrophys. 36, 507–537 (1998).
[CrossRef]

Woolf, N. J.

J. R. P. Angel, N. J. Woolf, “An imaging nulling interferometer to study extrasolar planets,” Astrophys. J. 475, 373–379 (1997).
[CrossRef]

C. A. Beichman, N. J. Woolf, C. A. Lindensmith, Terrestrial Planet Finder: a NASA Origins Program to Search for Habitable Planets (Jet Propulsion Laboratory, 1999).

Annu. Rev. Astron. Astrophys. (2)

N. Woolf, J. R. Angel, “Astronomical searches for earth-like planets and signs of life,” Annu. Rev. Astron. Astrophys. 36, 507–537 (1998).
[CrossRef]

G. W. Marcy, R. P. Butler, “Detection of extrasolar giant planets,” Annu. Rev. Astron. Astrophys. 36, 57–97 (1998).
[CrossRef]

Appl. Opt. (1)

Astrophys. J. (2)

W. C. Danchi, D. Deming, M. J. Kuchner, S. Seager, “Detection of close-in extrasolar giant planets using the Fourier-Kelvin Stellar Interferometer,” Astrophys. J. 597, L57–L60 (2003).
[CrossRef]

J. R. P. Angel, N. J. Woolf, “An imaging nulling interferometer to study extrasolar planets,” Astrophys. J. 475, 373–379 (1997).
[CrossRef]

IEEE Proc. (1)

P. J. Napier, A. R. Thompson, R. D. Ekers, “The Very Large Array—design and performance of a modern synthesis radio telescope,” IEEE Proc. 71, 1295–1320 (1983).
[CrossRef]

Nature (1)

R. N. Bracewell, “Detecting nonsolar planets by spinning infrared interferometer,” Nature 274, 780–781 (1978).
[CrossRef]

Proc. SPIE (6)

G. Vasisht, M. M. Colavita, “Differential phase interferometry with the Keck telescopes,” Proc. SPIE 5491, 567–576 (2004).
[CrossRef]

V. Ford, P. D. Lisman, S. B. Shaklan, J. T. Trauger, T. Ho, D. Hoppe, A. E. Lowman, “The Terrestrial Planet Finder coronagraph: technology and mission design studies,” Proc. SPIE 5487, 1274–1283 (2004).
[CrossRef]

S. Dubovitsky, O. P. Lay, “Architecture selection and optimization for planet-finding interferometers,” Proc. SPIE 5491, 284–295 (2004).
[CrossRef]

O. P. Lay, S. Dubovitsky, “Nulling interferometers: the importance of systematic errors and the X-array,” Proc. SPIE 5491, 874–885 (2004).
[CrossRef]

L. Kaltenegger, A. Karlsson, “Requirements on the stellar rejection of the DARWIN mission,” Proc. SPIE 5491, 275–283 (2004).
[CrossRef]

A. L. Karlsson, O. Wallner, J. M. Perdigues Armengol, O. Absil, “Three telescope nuller based on multibeam injection into single-mode waveguide,” Proc. SPIE 5491, 831–841 (2004).
[CrossRef]

Other (7)

B. Mennesson, “Thermal infrared stellar interferometry: observations of circumstellar environments using single-mode guided optics and contributions to the DARWIN space mission,” Ph.D. dissertation (Universite Paris, 1999).

C. A. Beichman, N. J. Woolf, C. A. Lindensmith, Terrestrial Planet Finder: a NASA Origins Program to Search for Habitable Planets (Jet Propulsion Laboratory, 1999).

A. L. Karlsson, ESTEC, Noordwijk, The Netherlands (personal communication, 2004).

T. Velusamy, C. A. Beichman, “Nulling interferometry for extra-solar planet detection: sensitivity and image reconstruction,” in IEEE Aerospace Conference Proceedings (IEEE Press, 2001), Vol. 4, pp. 2013–2025.

D. Coulter, “NASA’s Terrestrial Planet Finder mission: the search for habitable planets,” in Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Planets, M. Fridlund, T. Henning, eds. (European Space Agency, 2003), pp. 47–54.

A. Karlsson, L. Kaltenegger, “The technology of DARWIN,” in Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Planets, M. Fridlund, T. Henning, eds. (European Space Agency, 2003), pp. 41–46.

E. Serabyn, “Nulling interferometry and planet detection,” in Principles of Long Baseline Interferometry, P. Lawson, ed. (Jet Propulsion Laboratory, 2000), pp. 273–292.

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Figures (10)

Fig. 1
Fig. 1

Schematic of a generic nulling interferometer and target system. The target system includes the star and planet(s), located at a large distance from the instrument. The z axis is aligned with the direction to the star, with the x and y axes defined with respect to the target system. The X and Y axes are defined with respect to the array and α is the rotation angle. The instrument is represented as a phased array in which the electric fields are combined with amplitudes Aj and phases θj. Following the combiner, the beams are spatially filtered, dispersed in wavelength, and detected. The signal processing is described in the text.

Fig. 2
Fig. 2

Schematic of the X-Array configuration used in the example calculations.

Fig. 3
Fig. 3

(a) Signal from a unit flux point source for the X-Array in the absence of phase chopping, as a function of rotation angle. (b) The signal from the same point source with phase chopping.

Fig. 4
Fig. 4

Dirty map formed from the cross correlation of the measured signal with templates of the signal expected from a point source at each location on the sky using the X-Array specified in Table 1. The star is located at (0,0); the planet is at (30,0). White and black indicate strong positive and negative correlation, respectively. (a) Measured noise-free signal as a function of rotation angle. (b)–(e) Five examples of the template function.

Fig. 5
Fig. 5

Graphic illustration of the steps needed to calculate the PSF for a rotating nulling interferometer at the location illustrated by the X symbol. Each set of concentric rings represents one or more Bessel functions.

Fig. 6
Fig. 6

Increasing the optical bandwidth effectively smears out the sidelobes of the single-baseline response. The monochromatic case is a Bessel function of zero order. The examples are offset vertically for clarity.

Fig. 7
Fig. 7

(a) Output of the graphic recipe of Fig. 5, showing the locations and relative amplitudes of the main and satellite peaks of the PSF for the X-Array. (b) Numerical computation of the dirty map for a point source at (30,0), showing the same structure. The elongation of the main peak is the result of overlap with the neighboring negative satellite peaks.

Fig. 8
Fig. 8

Schematic illustration of how the PSF shape varies with radial offset from the star. (a) Main and satellite peaks well separated at large radial offsets. (b) Substantial overlap of sidelobes at smaller radial offset. The relative location of the peaks is preserved. (c) Closeup showing the overlap and mutual cancellation of the central lobes of the main and satellite peaks that occurs at small radial offsets. Positive and negative peaks are represented by solid and dashed circles, respectively.

Fig. 9
Fig. 9

Curves showing the sensitivity to planet photons for two phase-chopped nulling arrays as a function of angular offset from the star in dimensionless coordinates. The heavy curve is for the X-Array with 2:1 aspect ratio; the lighter curve is for a Linear DCB configuration (variant B in Table 3). The horizontal dashed lines denote the asymptotic modulation efficiency ηmod∞, as computed from Eq. (15). The IWA is defined as the angular offset at which the curves first reach the value of ηmod∞. For comparison, the horizontal bars denote the full width at half-maximum size of the synthesized PSF [approximation (18)], which is not directly related to the features in the curves.

Fig. 10
Fig. 10

Noise-free dirty maps of a three-planet system from (a) the X-Array example configuration (Table 1) and (b) a 60 m Linear DCB configuration (variant B in Table 3). The planets are point sources of equal brightness at the locations denoted by the circles. The inner planet lies at the IWA of the array in both cases.

Tables (3)

Tables Icon

Table 1 Array Configuration Parameters for the X-Array Example

Tables Icon

Table 2 Redundancy and Baseline Classification for the X-Array

Tables Icon

Table 3 Comparison of Configurations

Equations (19)

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E ( Θ x , Θ y ) j A j exp { i [ ϕ j + 2 π λ ( x j Θ x + y j Θ y ) ] } .
N ( Θ x , Θ y ) = { j A j exp [ i ( ϕ j + 2 π λ [ x j Θ x + y j Θ y ] ) ] } × { k A k exp [ - i ( ϕ k + 2 π λ [ x k Θ x + y k Θ y ] ) ] } = j k A j A k cos [ Δ ϕ j k + 2 π λ ( Δ x j k Θ x + Δ y j k Θ y ) ] ,
Δ x j k ( α ) = Δ X j k cos α - Δ Y j k sin α ,
Δ y j k ( α ) = Δ X j k sin α + Δ Y j k cos α .
N ( Θ x , Θ y , α ) = A 1 2 + A 2 2 + 2 A 1 A 2 cos { Δ ϕ 12 + 2 π λ [ Δ x 12 ( α ) Θ x + Δ y 12 ( α ) Θ y ] } .
N C ( Θ x , Θ y , α ) = 1 2 ( j k A j A k cos { Δ ϕ j k + 2 π λ [ Δ x j k ( α ) Θ x + Δ y j k ( α ) Θ y ] } - j k A j A k cos { - Δ ϕ j k + 2 π λ [ Δ x j k ( α ) Θ x + Δ y j k ( α ) Θ y ] } ) = j k A j A k sin Δ ϕ j k sin { 2 π λ [ Δ x j k ( α ) Θ x + Δ y j k ( α ) Θ y ] }
N C ( Θ x , Θ y , α ) = i C i sin { 2 π λ [ Δ x i ( α ) Θ x + Δ y i ( α ) Θ y ] } ,
C i = 2 j < k k [ A j A k sin Δ ϕ j k × δ ( Δ x j k - Δ x i ) δ ( Δ y j k - Δ y i ) ] .
M ( θ x , θ y , Θ x , Θ y ) = 1 2 π 0 2 π N C ( Θ x , Θ y , α ) × N C ( θ x , θ y , α ) d α .
M ( θ x , θ y , Θ x , Θ y ) = 1 r m s ( N C ) 1 2 π 0 2 π m C m sin { 2 π λ [ Δ x m ( α ) Θ x + Δ y m ( α ) Θ y ] } × n C n sin { 2 π λ [ Δ x n ( α ) θ x + Δ y n ( θ ) θ y ] } d α = 1 2 π rms ( N C ) m n C m C n 0 2 π sin { 2 π λ [ Δ x m ( α ) Θ x + Δ y m ( α ) Θ y ] } sin { 2 π λ [ Δ x n ( α ) θ x + Δ y n ( α ) θ y ] } d α .
M ( θ x , θ y , Θ x , Θ y ) = 1 2 rms ( N C ) m n C m C n [ J 0 ( 2 π B n λ { [ θ x - β x , m n ( Θ x , Θ y ) ] 2 + [ θ y - β y , m n ( Θ x , Θ y ) ] 2 } 1 / 2 ) - J 0 ( 2 π B n λ { [ θ x + β x , m n ( Θ x , Θ y ) ] 2 + [ θ y + β y , m n ( Θ x , Θ y ) ] 2 } 1 / 2 ) ] .
β x , m n ( Θ x , Θ y ) = 1 B n 2 ( Δ X m Δ X n Θ x + Δ Y m Δ X n Θ y + Δ Y m Δ Y n Θ x - Δ X m Δ Y n Θ y ) ,
β y , m n ( Θ x , Θ y ) = 1 B n 2 ( Δ X m Δ Y n Θ x + Δ Y m Δ Y n Θ y - Δ Y m Δ X n Θ x + Δ X m Δ X n Θ y ) .
M max , = 1 2 rms ( N C , ) m C m 2 = rms ( N C , ) = ( 1 2 m C m 2 ) 1 / 2 ,
η mod = M max , j η opt j a j = ( 1 2 i C i 2 ) 1 / 2 j η opt j a j .
η mod = [ 1 2 ( 16 A 4 + 4 A 4 + 4 A 4 ) ] 1 / 2 / 4 ( 10 × 0.1 ) = 0.22.
N * = j k 2 F * A j A k cos ϕ j k J 1 ( 2 π B j k θ * λ ) / ( 2 π B j k θ * λ ) j k 2 F * A j A k cos ϕ j k [ 1 - 0.125 ( 2 π B j k θ * λ ) 2 ] - 9.87 F * θ * 2 λ - 2 j k A j A k cos ϕ j k B j k 2 .
θ FWHM 0.48 λ av B av ,
B av = C i 2 B i C i 2 .

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