Abstract

Direct detection of exoplanets is possible by use of a technique called nulling interferometry, which is based on destructive interference of light of the bright object and constructive interference of the faint object. In the infrared wavelength region, this implies that light of a star must be attenuated by a certain factor, the so-called rejection ratio, which typically equals 106. This can be achieved by use of phase shifters, which apply a phase shift of π rad with an average error no greater than 2 mrad over a predefined wavelength region. For a 6–18-μm wavelength interval, this is a tough constraint. We show that the 2-mrad constraint can be relaxed if more than two beams participate in the beam recombination. We focus our attention on dispersive phase shifters and show that rejection ratios beyond 106 can be reached easily by use of a system of four or more apertures and simple dispersive phase shifters that consist of only one material.

© 2003 Optical Society of America

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References

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  1. R. N. Bracewell, “Detecting nonsolar planets by spinning infrared interferometer,” Nature 274, 780–781 (1978).
    [CrossRef]
  2. R. N. Bracewell, R. H. MacPhie, “Searching for nonsolar planets,” Icarus 38, 136–147 (1979).
    [CrossRef]
  3. J. R. P. Angel, N. J. Woolf, “An imaging interferometer to study extrasolar planets,” Astrophys. J. 475, 373–379 (1997).
    [CrossRef]
  4. B. Mennesson, J. M. Mariotti, “Array configurations for a space infrared nulling interferometer dedicated to the search for Earthlike extrasolar planets,” Icarus 128, 202–212 (1997).
    [CrossRef]
  5. B. Mennesson, A. Léger, “Direct detection and characterization of extrasolar planets: the Mariotti interferometer,” Workshop Handouts, Orsay-Ville, France, 13–14 April 1999.
  6. A. Karlsson, B. Mennesson, “The Robin Laurance interferometers,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 871–880 (2000).
    [CrossRef]
  7. A. L. Mieremet, J. J. M. Braat, H. Bokhove, K. Ravel, “Achromatic phase shifting using adjustable dispersive elements,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 1035–1041 (2000).
    [CrossRef]
  8. R. M. Morgan, J. Burge, N. Woolf, “Nulling interferometric beam combiner utilizing dielectric plates: experimental results in the visible broadband,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 340–348 (2000).
    [CrossRef]
  9. A. Léger, J. M. Mariotti, B. Mennesson, M. Olivier, J. L. Puget, D. Rouan, J. Schneider, “Could we search for primitive life on extrasolar planets in the near future? The DARWIN project,” Icarus 123, 249–255 (1996).
    [CrossRef]
  10. A. L. Mieremet, J. J. M. Braat, “Nulling interferometry without achromatic phase shifters,” Appl. Opt. 41, 4697–4703 (2002).
    [CrossRef] [PubMed]
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 5.
  12. M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, Handbook of Optics II, 2nd ed. (McGraw-Hill, New York, 1995), Chap. 33.
  13. W. S. Rodney, R. J. Spindler, “Refractive index of cesium bromide for ultraviolet, visible and infrared wavelengths,” J. Res. Natl. Bur. Stand. 51, 123–126 (1953).
    [CrossRef]
  14. Harshaw Optical Crystals Catalogue (Harshaw Chemical Co., Cleveland, Ohio, 1967).

2002

A. L. Mieremet, J. J. M. Braat, “Nulling interferometry without achromatic phase shifters,” Appl. Opt. 41, 4697–4703 (2002).
[CrossRef] [PubMed]

1997

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

B. Mennesson, J. M. Mariotti, “Array configurations for a space infrared nulling interferometer dedicated to the search for Earthlike extrasolar planets,” Icarus 128, 202–212 (1997).
[CrossRef]

1996

A. Léger, J. M. Mariotti, B. Mennesson, M. Olivier, J. L. Puget, D. Rouan, J. Schneider, “Could we search for primitive life on extrasolar planets in the near future? The DARWIN project,” Icarus 123, 249–255 (1996).
[CrossRef]

1979

R. N. Bracewell, R. H. MacPhie, “Searching for nonsolar planets,” Icarus 38, 136–147 (1979).
[CrossRef]

1978

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

1953

W. S. Rodney, R. J. Spindler, “Refractive index of cesium bromide for ultraviolet, visible and infrared wavelengths,” J. Res. Natl. Bur. Stand. 51, 123–126 (1953).
[CrossRef]

Angel, J. R. P.

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

Bass, M.

M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, Handbook of Optics II, 2nd ed. (McGraw-Hill, New York, 1995), Chap. 33.

Bokhove, H.

A. L. Mieremet, J. J. M. Braat, H. Bokhove, K. Ravel, “Achromatic phase shifting using adjustable dispersive elements,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 1035–1041 (2000).
[CrossRef]

Braat, J. J. M.

A. L. Mieremet, J. J. M. Braat, “Nulling interferometry without achromatic phase shifters,” Appl. Opt. 41, 4697–4703 (2002).
[CrossRef] [PubMed]

A. L. Mieremet, J. J. M. Braat, H. Bokhove, K. Ravel, “Achromatic phase shifting using adjustable dispersive elements,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 1035–1041 (2000).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, R. H. MacPhie, “Searching for nonsolar planets,” Icarus 38, 136–147 (1979).
[CrossRef]

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

Burge, J.

R. M. Morgan, J. Burge, N. Woolf, “Nulling interferometric beam combiner utilizing dielectric plates: experimental results in the visible broadband,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 340–348 (2000).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 5.

Karlsson, A.

A. Karlsson, B. Mennesson, “The Robin Laurance interferometers,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 871–880 (2000).
[CrossRef]

Léger, A.

A. Léger, J. M. Mariotti, B. Mennesson, M. Olivier, J. L. Puget, D. Rouan, J. Schneider, “Could we search for primitive life on extrasolar planets in the near future? The DARWIN project,” Icarus 123, 249–255 (1996).
[CrossRef]

B. Mennesson, A. Léger, “Direct detection and characterization of extrasolar planets: the Mariotti interferometer,” Workshop Handouts, Orsay-Ville, France, 13–14 April 1999.

MacPhie, R. H.

R. N. Bracewell, R. H. MacPhie, “Searching for nonsolar planets,” Icarus 38, 136–147 (1979).
[CrossRef]

Mariotti, J. M.

B. Mennesson, J. M. Mariotti, “Array configurations for a space infrared nulling interferometer dedicated to the search for Earthlike extrasolar planets,” Icarus 128, 202–212 (1997).
[CrossRef]

A. Léger, J. M. Mariotti, B. Mennesson, M. Olivier, J. L. Puget, D. Rouan, J. Schneider, “Could we search for primitive life on extrasolar planets in the near future? The DARWIN project,” Icarus 123, 249–255 (1996).
[CrossRef]

Mennesson, B.

B. Mennesson, J. M. Mariotti, “Array configurations for a space infrared nulling interferometer dedicated to the search for Earthlike extrasolar planets,” Icarus 128, 202–212 (1997).
[CrossRef]

A. Léger, J. M. Mariotti, B. Mennesson, M. Olivier, J. L. Puget, D. Rouan, J. Schneider, “Could we search for primitive life on extrasolar planets in the near future? The DARWIN project,” Icarus 123, 249–255 (1996).
[CrossRef]

B. Mennesson, A. Léger, “Direct detection and characterization of extrasolar planets: the Mariotti interferometer,” Workshop Handouts, Orsay-Ville, France, 13–14 April 1999.

A. Karlsson, B. Mennesson, “The Robin Laurance interferometers,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 871–880 (2000).
[CrossRef]

Mieremet, A. L.

A. L. Mieremet, J. J. M. Braat, “Nulling interferometry without achromatic phase shifters,” Appl. Opt. 41, 4697–4703 (2002).
[CrossRef] [PubMed]

A. L. Mieremet, J. J. M. Braat, H. Bokhove, K. Ravel, “Achromatic phase shifting using adjustable dispersive elements,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 1035–1041 (2000).
[CrossRef]

Morgan, R. M.

R. M. Morgan, J. Burge, N. Woolf, “Nulling interferometric beam combiner utilizing dielectric plates: experimental results in the visible broadband,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 340–348 (2000).
[CrossRef]

Olivier, M.

A. Léger, J. M. Mariotti, B. Mennesson, M. Olivier, J. L. Puget, D. Rouan, J. Schneider, “Could we search for primitive life on extrasolar planets in the near future? The DARWIN project,” Icarus 123, 249–255 (1996).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 5.

Puget, J. L.

A. Léger, J. M. Mariotti, B. Mennesson, M. Olivier, J. L. Puget, D. Rouan, J. Schneider, “Could we search for primitive life on extrasolar planets in the near future? The DARWIN project,” Icarus 123, 249–255 (1996).
[CrossRef]

Ravel, K.

A. L. Mieremet, J. J. M. Braat, H. Bokhove, K. Ravel, “Achromatic phase shifting using adjustable dispersive elements,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 1035–1041 (2000).
[CrossRef]

Rodney, W. S.

W. S. Rodney, R. J. Spindler, “Refractive index of cesium bromide for ultraviolet, visible and infrared wavelengths,” J. Res. Natl. Bur. Stand. 51, 123–126 (1953).
[CrossRef]

Rouan, D.

A. Léger, J. M. Mariotti, B. Mennesson, M. Olivier, J. L. Puget, D. Rouan, J. Schneider, “Could we search for primitive life on extrasolar planets in the near future? The DARWIN project,” Icarus 123, 249–255 (1996).
[CrossRef]

Schneider, J.

A. Léger, J. M. Mariotti, B. Mennesson, M. Olivier, J. L. Puget, D. Rouan, J. Schneider, “Could we search for primitive life on extrasolar planets in the near future? The DARWIN project,” Icarus 123, 249–255 (1996).
[CrossRef]

Spindler, R. J.

W. S. Rodney, R. J. Spindler, “Refractive index of cesium bromide for ultraviolet, visible and infrared wavelengths,” J. Res. Natl. Bur. Stand. 51, 123–126 (1953).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 5.

Van Stryland, E. W.

M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, Handbook of Optics II, 2nd ed. (McGraw-Hill, New York, 1995), Chap. 33.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 5.

Williams, D. R.

M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, Handbook of Optics II, 2nd ed. (McGraw-Hill, New York, 1995), Chap. 33.

Wolfe, W. L.

M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, Handbook of Optics II, 2nd ed. (McGraw-Hill, New York, 1995), Chap. 33.

Woolf, N.

R. M. Morgan, J. Burge, N. Woolf, “Nulling interferometric beam combiner utilizing dielectric plates: experimental results in the visible broadband,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 340–348 (2000).
[CrossRef]

Woolf, N. J.

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

Appl. Opt.

A. L. Mieremet, J. J. M. Braat, “Nulling interferometry without achromatic phase shifters,” Appl. Opt. 41, 4697–4703 (2002).
[CrossRef] [PubMed]

Astrophys. J.

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

Icarus

B. Mennesson, J. M. Mariotti, “Array configurations for a space infrared nulling interferometer dedicated to the search for Earthlike extrasolar planets,” Icarus 128, 202–212 (1997).
[CrossRef]

R. N. Bracewell, R. H. MacPhie, “Searching for nonsolar planets,” Icarus 38, 136–147 (1979).
[CrossRef]

A. Léger, J. M. Mariotti, B. Mennesson, M. Olivier, J. L. Puget, D. Rouan, J. Schneider, “Could we search for primitive life on extrasolar planets in the near future? The DARWIN project,” Icarus 123, 249–255 (1996).
[CrossRef]

J. Res. Natl. Bur. Stand.

W. S. Rodney, R. J. Spindler, “Refractive index of cesium bromide for ultraviolet, visible and infrared wavelengths,” J. Res. Natl. Bur. Stand. 51, 123–126 (1953).
[CrossRef]

Nature

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

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 5.

M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, Handbook of Optics II, 2nd ed. (McGraw-Hill, New York, 1995), Chap. 33.

B. Mennesson, A. Léger, “Direct detection and characterization of extrasolar planets: the Mariotti interferometer,” Workshop Handouts, Orsay-Ville, France, 13–14 April 1999.

A. Karlsson, B. Mennesson, “The Robin Laurance interferometers,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 871–880 (2000).
[CrossRef]

A. L. Mieremet, J. J. M. Braat, H. Bokhove, K. Ravel, “Achromatic phase shifting using adjustable dispersive elements,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 1035–1041 (2000).
[CrossRef]

R. M. Morgan, J. Burge, N. Woolf, “Nulling interferometric beam combiner utilizing dielectric plates: experimental results in the visible broadband,” in Interferometry in Optical Astronomy, P. Léna, A. Quirrenbach, eds., Proc. SPIE4006, 340–348 (2000).
[CrossRef]

Harshaw Optical Crystals Catalogue (Harshaw Chemical Co., Cleveland, Ohio, 1967).

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Figures (7)

Fig. 1
Fig. 1

Schematic drawing of an array of N apertures. Each aperture, positioned at B i , provides a beam with amplitude A 0 and undergoes a phase shift ϕ i . Because the optical path from the source to each telescope is different, a geometrical delay B i · r is present in each beam. During recombination, only a fraction f i of each beam is used.

Fig. 2
Fig. 2

Vector representation of the beams for different values of N, when f i = 1 and ϕ i = 2π(i - 1)/N. The superposition of all vectors equals the null vector.

Fig. 3
Fig. 3

Top panels show the DAC interferometer (left) and the interferometer from our analytical solution (right). The bottom panels show the complex representation of each beam at a single wavelength λ0. The advantage of the 2π + 2∊(λ) phase shifter can be seen from these plots. A superposition of the vectors of plots 1, 2, and 3 results in a vector in which only the horizontal component is close to zero. A superposition of the vectors in plots 4, 5, and 6 results in a vector with both the horizontal and the vertical components close to zero.

Fig. 4
Fig. 4

R 21, λ2) (top), R 31, λ2) (middle), and R 41, λ2) (bottom) as a function of λ21. In each panel the lower curve represents cases 1 (top), 3 (middle), and 5 (bottom), and the upper three curves represent cases 2 (top), 4 (middle), and 6 (bottom), each for a different value of λ1. The relative maxima in the curves of λ1 = 2.5 μm and λ1 = 4.0 μm (top) and the curve of λ1 = 2.5 μm (middle and bottom) are caused by the inflection point of CsBr at λ = 4.63 μm. This point is indicated by a vertical bar, which separates the wavelength interval that does not contain the inflection point (left side of the bar) from the interval that does (right side of the bar).

Fig. 5
Fig. 5

Applied phase shifts ϕ i (λ) that are obtained with the downhill simplex method for a three-aperture nulling interferometer over a bandwidth from 6 to 18 μm. As a reference, dashed lines are drawn at the levels π and 2π. The phase shift applied by the delay line and the CsBr plate of the second aperture can be written as a ϕ2(λ) = π + ∊(λ). A similar relation holds true for the third arm, i.e., ϕ3(λ) = 2π + 2∊(λ).

Fig. 6
Fig. 6

Rejection ratio as a function of the error ∊(λ0). Curve A represents the DAC interferometer (ϕ1 = 0, ϕ2 = π + ∊(λ0), and ϕ3 = 0). Curve B is obtained when the 0 phase shifter of the third arm of the DAC interferometer is replaced by a 2π + 2∊(λ0) phase shifter. The rejection ratio is much less sensitive to errors. Curve C is obtained with the solution given in Table 4.

Fig. 7
Fig. 7

R 3 as a function of an error. In each panel, the lower curve (B) is before error compensation, while the upper curve (A) is after corrections are performed with the delay lines. Top panels: R 31, λ2) as a function of an absolute error in f 2 (left) and f 3 (right). Bottom panels: R 31, λ2) as a function of an absolute error in b 2 (left) and b 3 (right). Note the logarithmic scale.

Tables (4)

Tables Icon

Table 1 Proposals for Different Nulling Interferometers That Use from Two to Six Apertures

Tables Icon

Table 2 Analytical Solutions of the Weighting Factors fi for N = 2, …, 6a

Tables Icon

Table 3 Six Situations for Which Numerical Optimization Are Performed

Tables Icon

Table 4 Solution Found for a Three-Aperture Nulling Interferometer over a Wavelength Band of 6–18 μma

Equations (11)

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I N λ ,   r = A 0 2 λ ,   r i = 1 N   f i 2 + 2   i = 1 N j > i N   f i f j × cos ϕ i λ - ϕ j λ + 2 π λ B i - B j   ·   r .
T N λ ,   r = 1 λ 2 - λ 1 λ 1 λ 2 I N λ ,   r A 0 2 λ ,   r i = 1 N   f i 2 d λ ,
R N λ 1 ,   λ 2 = T N λ ,   r p T N λ ,   0 = λ 2 - λ 1 i = 1 N   f i 2 + 2   i = 1 N j > i N   f i f j λ 1 λ 2 cos ϕ ij λ + 2 π λ B ij   ·   r p d λ λ 2 - λ 1 i = 1 N   f i 2 + 2   i = 1 N j > i N   f i f j λ 1 λ 2 cos ϕ ij λ d λ ,
R N λ 1 ,   λ 2 = λ 2 - λ 1 i = 1 N   f i 2 λ 2 - λ 1 i = 1 N   f i 2 + 2   i = 1 N j > i N   f i f j λ 1 λ 2 cos ϕ ij λ d λ .
ϕ i λ = 2 π λ d i + k = 1 M i n i , k λ - 1 b i , k ,
R N = i = 1 N   f i 2 i = 1 N   f i 2 + 2   i = 1 N j > i N   f i f j   cos ϕ i - ϕ j .
ϕ i λ = 2 π λ d i + n λ - 1 b i .
ϕ i = i - 1 π + λ ,   i = 1 , ,   N .
R DAC λ 1 ,   λ 2 = 4 3 + 4 3 1 λ 2 - λ 1 λ 1 λ 2 cos π + λ d λ - 1 2 3 1 λ 2 - λ 1 λ 1 λ 2   2 λ d λ - 1 = 3 2   2 λ - 1 .
R 3 λ 1 ,   λ 2 = 1 + 1 λ 2 - λ 1 λ 1 λ 2 4 3 cos π + λ + 1 3 cos 2 π + 2 λ d λ - 1   1 6 λ 2 - λ 1 λ 1 λ 2   4 λ d λ - 1 = 6 4 λ - 1 .
R 3 λ 1 ,   λ 2 = 1 + 1 λ 2 - λ 1 λ 1 λ 2 1.338   cos π + λ + 0.338   cos 2 π + 2 λ d λ - 1     6.335 × 10 - 5 - 6.541 × 10 - 3 2 λ + 0.1694 4 λ - 1 .

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