Abstract

Multiple-telescope interferometry for high-angular-resolution astronomical imaging in the optical–IR–far-IR bands is currently a topic of great scientific interest. The fundamentals that govern the sensitivity of direct-detection instruments and interferometers are reviewed, and the rigorous sensitivity limits imposed by the Cramér–Rao theorem are discussed. Numerical calculations of the Cramér–Rao limit are carried out for a simple example, and the results are used to support the argument that interferometers that have more compact instantaneous beam patterns are more sensitive, since they extract more spatial information from each detected photon. This argument favors arrays with a larger number of telescopes, and it favors all-on-one beam-combining methods as compared with pairwise combination.

© 2003 Optical Society of America

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2002 (1)

S. K. Saha, “Modern optical astronomy:  technology and impact of interferometry,” Rev. Mod. Phys. 74, 551–600 (2002).
[CrossRef]

2001 (7)

P. Stoica, T. L. Marzetta, “Parameter estimation problems with singular information matrices,” IEEE Trans. Signal Process. 49, 87–90 (2001).
[CrossRef]

A. Quirrenbach, “Optical interferometry,” Annu. Rev. Astron. Astrophys. 39, 353–401 (2001).
[CrossRef]

T. Nakajima, “Sensitivity of a ground-based infrared interferometer for aperture synthesis imaging,” Publ. Astron. Soc. Pac. 113, 1289–1299 (2001).
[CrossRef]

T. Nakajima, H. Matsuhara, “Sensitivity of an imaging space infrared interferometer,” Appl. Opt. 40, 514–526 (2001).
[CrossRef]

E. Serabyn, M. M. Colavita, “Fully symmetric nulling beam combiners,” Appl. Opt. 40, 1668–1671 (2001).
[CrossRef]

J. Berger, P. Haguenauer, P. Kern, K. Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34 (2001).
[CrossRef]

R. Molina, J. Núñez, F. J. Cortijo, J. Mateos, “Image restoration in astronomy—a Bayesian perspective,” IEEE Signal Process. Mag. 18, 11–29 (2001).
[CrossRef]

1999 (3)

C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 46, 587–593 (1999).
[CrossRef]

F. Roddier, S. T. Ridgway, “Filling factor and signal-to-noise ratios in optical interferometric arrays,” Publ. Astron. Soc. Pac. 111, 990–996 (1999).
[CrossRef]

D. Leisawitz, J. C. Mather, S. Harvey Moseley, X. Zhang, “The Submillimeter Probe of the Evolution of Cosmic Structure (SPECS),” Astrophys. Space Sci. 269, 563–567 (1999).
[CrossRef]

1998 (4)

A. Blain, R. Ivison, I. Smail, “Observational limits to source confusion in the millimetre/submillimetre waveband,” Mon. Not. R. Astron. Soc. 296, L29–L33 (1998).
[CrossRef]

D. Hughes, S. Serjeant, J. Dunlop, M. Rowan-Robinson, A. Blain, R. G. Mann, R. Ivison, J. Peacock, A. Efstathiou, W. Gear, S. Oliver, A. Lawrence, M. Longair, P. Goldschmidt, T. Jenness, “High-redshift star formation in the Hubble Deep Field revealed by a submillimetre-wavelength survey,” Nature 394, 241–247 (1998).
[CrossRef]

A. Barger, L. Cowie, D. Sanders, E. Fulton, Y. Taniguchi, Y. Sato, K. Kawara, H. Okuda, “Submillimetre-wavelength detection of dusty star-forming galaxies at high redshift,” Nature 394, 248–251 (1998).
[CrossRef]

S. Withington, J. Murphy, “Modal analysis of partially coherent submillimeter-wave quasi-optical systems,” IEEE Trans. Antennas Propag. 46, 1651–1659 (1998).
[CrossRef]

1997 (1)

I. Smail, R. J. Ivison, A. W. Blain, “A deep submillimeter survey of lensing clusters:  a new window on galaxy formation and evolution,” Astrophys. J. 490, L5–L8 (1997).
[CrossRef]

1996 (2)

L. Mugnier, G. Rousset, F. Cassaing, “Aperture configuration optimality criterion for phased arrays of optical telescopes,” J. Opt. Soc. Am. A 13, 2367–2374 (1996).
[CrossRef]

K. L. Bell, Y. Ephraim, H. L. Van Trees, “Explicit Ziv–Zakai lower bound for bearing estimation,” IEEE Trans. Signal Process. 44, 2810–2824 (1996).
[CrossRef]

1993 (1)

S. W. Wedge, D. B. Rutledge, “Wave computations for microwave education,” IEEE Trans. Educ. 36, 127–131 (1993).
[CrossRef]

1992 (2)

S. W. Wedge, D. B. Rutledge, “Wave techniques for noisemodeling and measurement,” IEEE Trans. Microwave Theory Tech. 40, 2004–2012 (1992).
[CrossRef]

P. Jakobsen, P. Greenfield, R. Jedrzejcwski, “The Cramer–Rao lower bound and stellar photometry with aberrated HST images,” Astron. Astrophys. 253, 329–332 (1992).

1991 (3)

J. G. Cohen, “Tests of the photometric accuracy of image restoration using the maximum entropy algorithm,” Astron. J. 101, 734–737 (1991).
[CrossRef]

S. W. Wedge, D. B. Rutledge, “Noise waves and passive linear multiports,” IEEE Microwave Guid. Wave Lett. 1, 117–119 (1991).
[CrossRef]

S. R. Kulkarni, S. Prasad, T. Nakajima, “Noise in optical synthesis images. II. Sensitivity of an  nC2 interferometer with bispectrum imaging,” J. Opt. Soc. Am. A 8, 499–510 (1991).
[CrossRef]

1989 (1)

1986 (2)

1984 (1)

E. Weinstein, A. J. Weiss, “Fundamental limitations in passive time-delay estimation—Part II: wide-band systems,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 1064–1078 (1984).
[CrossRef]

1983 (1)

A. J. Weiss, E. Weinstein, “Fundamental limitations in passive time-delay estimation—Part I:  narrow-band systems,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 472–486 (1983).
[CrossRef]

1982 (1)

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef] [PubMed]

1968 (1)

A. C. Gately, D. J. R. Stock, B. R.-S. Cheo, “A network description for antenna problems,” Proc. IEEE 56, 1181–1193 (1968).
[CrossRef]

1967 (1)

H. Bosma, “On the theory of linear noisy systems,” Philips Res. Rep. Suppl. 10, 1–190 (1967).

1961 (1)

J. Butler, R. Lowe, “Beam-forming matrix simplifies design of electronically scanned antennas,” Electron. Des. 9, 170–173 (1961).

1945 (1)

C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).

Barger, A.

A. Barger, L. Cowie, D. Sanders, E. Fulton, Y. Taniguchi, Y. Sato, K. Kawara, H. Okuda, “Submillimetre-wavelength detection of dusty star-forming galaxies at high redshift,” Nature 394, 248–251 (1998).
[CrossRef]

Beichman, C. A.

C. A. Beichman, “Terrestrial Planet Finder: the search for life-bearing planets around other stars,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 719–723 (1998), see also http://planetquest.jpl.nasa.gov .
[CrossRef]

Bell, K. L.

K. L. Bell, Y. Ephraim, H. L. Van Trees, “Explicit Ziv–Zakai lower bound for bearing estimation,” IEEE Trans. Signal Process. 44, 2810–2824 (1996).
[CrossRef]

Berger, J.

J. Berger, P. Haguenauer, P. Kern, K. Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34 (2001).
[CrossRef]

Blain, A.

A. Blain, R. Ivison, I. Smail, “Observational limits to source confusion in the millimetre/submillimetre waveband,” Mon. Not. R. Astron. Soc. 296, L29–L33 (1998).
[CrossRef]

D. Hughes, S. Serjeant, J. Dunlop, M. Rowan-Robinson, A. Blain, R. G. Mann, R. Ivison, J. Peacock, A. Efstathiou, W. Gear, S. Oliver, A. Lawrence, M. Longair, P. Goldschmidt, T. Jenness, “High-redshift star formation in the Hubble Deep Field revealed by a submillimetre-wavelength survey,” Nature 394, 241–247 (1998).
[CrossRef]

Blain, A. W.

I. Smail, R. J. Ivison, A. W. Blain, “A deep submillimeter survey of lensing clusters:  a new window on galaxy formation and evolution,” Astrophys. J. 490, L5–L8 (1997).
[CrossRef]

Bosma, H.

H. Bosma, “On the theory of linear noisy systems,” Philips Res. Rep. Suppl. 10, 1–190 (1967).

Butler, J.

J. Butler, R. Lowe, “Beam-forming matrix simplifies design of electronically scanned antennas,” Electron. Des. 9, 170–173 (1961).

Cassaing, F.

Cheo, B. R.-S.

A. C. Gately, D. J. R. Stock, B. R.-S. Cheo, “A network description for antenna problems,” Proc. IEEE 56, 1181–1193 (1968).
[CrossRef]

Clinthorne, N. H.

C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 46, 587–593 (1999).
[CrossRef]

Cohen, J. G.

J. G. Cohen, “Tests of the photometric accuracy of image restoration using the maximum entropy algorithm,” Astron. J. 101, 734–737 (1991).
[CrossRef]

Colavita, M. M.

Cortijo, F. J.

R. Molina, J. Núñez, F. J. Cortijo, J. Mateos, “Image restoration in astronomy—a Bayesian perspective,” IEEE Signal Process. Mag. 18, 11–29 (2001).
[CrossRef]

Cowie, L.

A. Barger, L. Cowie, D. Sanders, E. Fulton, Y. Taniguchi, Y. Sato, K. Kawara, H. Okuda, “Submillimetre-wavelength detection of dusty star-forming galaxies at high redshift,” Nature 394, 248–251 (1998).
[CrossRef]

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N. J., 1946).

Danchi, W.

M. Shao, W. Danchi, M. J. DiPirro, M. Dragovan, L. D. Feinberg, M. Hagopian, W. D. Langer, C. R. Lawrence, P. R. Lawson, D. T. Leisawitz, J. C. Mather, S. H. Moseley, M. R. Swain, H. W. Yorke, X. Zhang, “Space-based interfero-metric telescopes for the far infrared,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 772–781 (2000).
[CrossRef]

Danchi, W. C.

D. T. Leisawitz, W. C. Danchi, M. J. DiPirro, L. D. Feinberg, D. Y. Gezari, M. Hagopian, W. D. Langer, J. C. Mather, S. H. Moseley, M. Shao, R. F. Silverberg, J. Staguhn, M. R. Swain, H. W. Yorke, X. Zhang, “Scientific motivation and technology requirements for the SPIRIT and SPECS far-infrared/submillimeter space interferometers,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 36–46 (2000).
[CrossRef]

Delabre, B.

M. Faucherre, B. Delabre, P. Dierickx, F. Merkle, “Michelson versus Fizeau type beam combination—Is there a difference?” in Amplitude and Intensity Spatial Interferometry, Proc. SPIE1237, 206–217 (1990).
[CrossRef]

Dierickx, P.

M. Faucherre, B. Delabre, P. Dierickx, F. Merkle, “Michelson versus Fizeau type beam combination—Is there a difference?” in Amplitude and Intensity Spatial Interferometry, Proc. SPIE1237, 206–217 (1990).
[CrossRef]

DiPirro, M. J.

D. T. Leisawitz, W. C. Danchi, M. J. DiPirro, L. D. Feinberg, D. Y. Gezari, M. Hagopian, W. D. Langer, J. C. Mather, S. H. Moseley, M. Shao, R. F. Silverberg, J. Staguhn, M. R. Swain, H. W. Yorke, X. Zhang, “Scientific motivation and technology requirements for the SPIRIT and SPECS far-infrared/submillimeter space interferometers,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 36–46 (2000).
[CrossRef]

M. Shao, W. Danchi, M. J. DiPirro, M. Dragovan, L. D. Feinberg, M. Hagopian, W. D. Langer, C. R. Lawrence, P. R. Lawson, D. T. Leisawitz, J. C. Mather, S. H. Moseley, M. R. Swain, H. W. Yorke, X. Zhang, “Space-based interfero-metric telescopes for the far infrared,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 772–781 (2000).
[CrossRef]

Dragovan, M.

M. Shao, W. Danchi, M. J. DiPirro, M. Dragovan, L. D. Feinberg, M. Hagopian, W. D. Langer, C. R. Lawrence, P. R. Lawson, D. T. Leisawitz, J. C. Mather, S. H. Moseley, M. R. Swain, H. W. Yorke, X. Zhang, “Space-based interfero-metric telescopes for the far infrared,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 772–781 (2000).
[CrossRef]

Dunlop, J.

D. Hughes, S. Serjeant, J. Dunlop, M. Rowan-Robinson, A. Blain, R. G. Mann, R. Ivison, J. Peacock, A. Efstathiou, W. Gear, S. Oliver, A. Lawrence, M. Longair, P. Goldschmidt, T. Jenness, “High-redshift star formation in the Hubble Deep Field revealed by a submillimetre-wavelength survey,” Nature 394, 241–247 (1998).
[CrossRef]

Efstathiou, A.

D. Hughes, S. Serjeant, J. Dunlop, M. Rowan-Robinson, A. Blain, R. G. Mann, R. Ivison, J. Peacock, A. Efstathiou, W. Gear, S. Oliver, A. Lawrence, M. Longair, P. Goldschmidt, T. Jenness, “High-redshift star formation in the Hubble Deep Field revealed by a submillimetre-wavelength survey,” Nature 394, 241–247 (1998).
[CrossRef]

Eiroa, C.

A. J. Penny, A. Leger, J. Mariotti, C. Schalinski, C. Eiroa, R. J. Laurance, M. Fridlund, “Darwin interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 666–671 (1998), see also http://sci.esa.int/darwin .
[CrossRef]

Ephraim, Y.

K. L. Bell, Y. Ephraim, H. L. Van Trees, “Explicit Ziv–Zakai lower bound for bearing estimation,” IEEE Trans. Signal Process. 44, 2810–2824 (1996).
[CrossRef]

Faucherre, M.

M. Faucherre, B. Delabre, P. Dierickx, F. Merkle, “Michelson versus Fizeau type beam combination—Is there a difference?” in Amplitude and Intensity Spatial Interferometry, Proc. SPIE1237, 206–217 (1990).
[CrossRef]

Feinberg, L. D.

M. Shao, W. Danchi, M. J. DiPirro, M. Dragovan, L. D. Feinberg, M. Hagopian, W. D. Langer, C. R. Lawrence, P. R. Lawson, D. T. Leisawitz, J. C. Mather, S. H. Moseley, M. R. Swain, H. W. Yorke, X. Zhang, “Space-based interfero-metric telescopes for the far infrared,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 772–781 (2000).
[CrossRef]

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M. Shao, W. Danchi, M. J. DiPirro, M. Dragovan, L. D. Feinberg, M. Hagopian, W. D. Langer, C. R. Lawrence, P. R. Lawson, D. T. Leisawitz, J. C. Mather, S. H. Moseley, M. R. Swain, H. W. Yorke, X. Zhang, “Space-based interfero-metric telescopes for the far infrared,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 772–781 (2000).
[CrossRef]

D. T. Leisawitz, W. C. Danchi, M. J. DiPirro, L. D. Feinberg, D. Y. Gezari, M. Hagopian, W. D. Langer, J. C. Mather, S. H. Moseley, M. Shao, R. F. Silverberg, J. Staguhn, M. R. Swain, H. W. Yorke, X. Zhang, “Scientific motivation and technology requirements for the SPIRIT and SPECS far-infrared/submillimeter space interferometers,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 36–46 (2000).
[CrossRef]

Shepp, L. A.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef] [PubMed]

Silverberg, R. F.

D. T. Leisawitz, W. C. Danchi, M. J. DiPirro, L. D. Feinberg, D. Y. Gezari, M. Hagopian, W. D. Langer, J. C. Mather, S. H. Moseley, M. Shao, R. F. Silverberg, J. Staguhn, M. R. Swain, H. W. Yorke, X. Zhang, “Scientific motivation and technology requirements for the SPIRIT and SPECS far-infrared/submillimeter space interferometers,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 36–46 (2000).
[CrossRef]

Smail, I.

A. Blain, R. Ivison, I. Smail, “Observational limits to source confusion in the millimetre/submillimetre waveband,” Mon. Not. R. Astron. Soc. 296, L29–L33 (1998).
[CrossRef]

I. Smail, R. J. Ivison, A. W. Blain, “A deep submillimeter survey of lensing clusters:  a new window on galaxy formation and evolution,” Astrophys. J. 490, L5–L8 (1997).
[CrossRef]

Staguhn, J.

D. T. Leisawitz, W. C. Danchi, M. J. DiPirro, L. D. Feinberg, D. Y. Gezari, M. Hagopian, W. D. Langer, J. C. Mather, S. H. Moseley, M. Shao, R. F. Silverberg, J. Staguhn, M. R. Swain, H. W. Yorke, X. Zhang, “Scientific motivation and technology requirements for the SPIRIT and SPECS far-infrared/submillimeter space interferometers,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 36–46 (2000).
[CrossRef]

Stock, D. J. R.

A. C. Gately, D. J. R. Stock, B. R.-S. Cheo, “A network description for antenna problems,” Proc. IEEE 56, 1181–1193 (1968).
[CrossRef]

Stoica, P.

P. Stoica, T. L. Marzetta, “Parameter estimation problems with singular information matrices,” IEEE Trans. Signal Process. 49, 87–90 (2001).
[CrossRef]

Swain, M. R.

D. T. Leisawitz, W. C. Danchi, M. J. DiPirro, L. D. Feinberg, D. Y. Gezari, M. Hagopian, W. D. Langer, J. C. Mather, S. H. Moseley, M. Shao, R. F. Silverberg, J. Staguhn, M. R. Swain, H. W. Yorke, X. Zhang, “Scientific motivation and technology requirements for the SPIRIT and SPECS far-infrared/submillimeter space interferometers,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 36–46 (2000).
[CrossRef]

M. Shao, W. Danchi, M. J. DiPirro, M. Dragovan, L. D. Feinberg, M. Hagopian, W. D. Langer, C. R. Lawrence, P. R. Lawson, D. T. Leisawitz, J. C. Mather, S. H. Moseley, M. R. Swain, H. W. Yorke, X. Zhang, “Space-based interfero-metric telescopes for the far infrared,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 772–781 (2000).
[CrossRef]

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[CrossRef]

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J. Berger, P. Haguenauer, P. Kern, K. Perraut, F. Malbet, I. Schanen, M. Severi, R. Millan-Gabet, W. Traub, “Integrated optics for astronomical interferometry. IV. First measurements of stars,” Astron. Astrophys. 376, L31–L34 (2001).
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[CrossRef]

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968).

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef] [PubMed]

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W. K. Kahn, W. Wasylkiwskyj, “Coupling, radiation, and scattering by antennas,” in Proceedings of the Symposium on Generalized Networks, Vol. 16 of Microwave Research Institute Symposia Series (Polytechnic, New York, 1966), pp. 83–114.

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[CrossRef]

S. W. Wedge, D. B. Rutledge, “Wave techniques for noisemodeling and measurement,” IEEE Trans. Microwave Theory Tech. 40, 2004–2012 (1992).
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[CrossRef]

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Weinstein, E.

E. Weinstein, A. J. Weiss, “Fundamental limitations in passive time-delay estimation—Part II: wide-band systems,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 1064–1078 (1984).
[CrossRef]

A. J. Weiss, E. Weinstein, “Fundamental limitations in passive time-delay estimation—Part I:  narrow-band systems,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 472–486 (1983).
[CrossRef]

Weiss, A. J.

E. Weinstein, A. J. Weiss, “Fundamental limitations in passive time-delay estimation—Part II: wide-band systems,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 1064–1078 (1984).
[CrossRef]

A. J. Weiss, E. Weinstein, “Fundamental limitations in passive time-delay estimation—Part I:  narrow-band systems,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 472–486 (1983).
[CrossRef]

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C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 46, 587–593 (1999).
[CrossRef]

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[CrossRef]

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M. Shao, W. Danchi, M. J. DiPirro, M. Dragovan, L. D. Feinberg, M. Hagopian, W. D. Langer, C. R. Lawrence, P. R. Lawson, D. T. Leisawitz, J. C. Mather, S. H. Moseley, M. R. Swain, H. W. Yorke, X. Zhang, “Space-based interfero-metric telescopes for the far infrared,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 772–781 (2000).
[CrossRef]

D. T. Leisawitz, W. C. Danchi, M. J. DiPirro, L. D. Feinberg, D. Y. Gezari, M. Hagopian, W. D. Langer, J. C. Mather, S. H. Moseley, M. Shao, R. F. Silverberg, J. Staguhn, M. R. Swain, H. W. Yorke, X. Zhang, “Scientific motivation and technology requirements for the SPIRIT and SPECS far-infrared/submillimeter space interferometers,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 36–46 (2000).
[CrossRef]

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D. T. Leisawitz, W. C. Danchi, M. J. DiPirro, L. D. Feinberg, D. Y. Gezari, M. Hagopian, W. D. Langer, J. C. Mather, S. H. Moseley, M. Shao, R. F. Silverberg, J. Staguhn, M. R. Swain, H. W. Yorke, X. Zhang, “Scientific motivation and technology requirements for the SPIRIT and SPECS far-infrared/submillimeter space interferometers,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 36–46 (2000).
[CrossRef]

M. Shao, W. Danchi, M. J. DiPirro, M. Dragovan, L. D. Feinberg, M. Hagopian, W. D. Langer, C. R. Lawrence, P. R. Lawson, D. T. Leisawitz, J. C. Mather, S. H. Moseley, M. R. Swain, H. W. Yorke, X. Zhang, “Space-based interfero-metric telescopes for the far infrared,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 772–781 (2000).
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[CrossRef]

E. Weinstein, A. J. Weiss, “Fundamental limitations in passive time-delay estimation—Part II: wide-band systems,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 1064–1078 (1984).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. Withington, J. Murphy, “Modal analysis of partially coherent submillimeter-wave quasi-optical systems,” IEEE Trans. Antennas Propag. 46, 1651–1659 (1998).
[CrossRef]

IEEE Trans. Educ. (1)

S. W. Wedge, D. B. Rutledge, “Wave computations for microwave education,” IEEE Trans. Educ. 36, 127–131 (1993).
[CrossRef]

IEEE Trans. Med. Imaging (1)

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theory Tech. (1)

S. W. Wedge, D. B. Rutledge, “Wave techniques for noisemodeling and measurement,” IEEE Trans. Microwave Theory Tech. 40, 2004–2012 (1992).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 46, 587–593 (1999).
[CrossRef]

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K. L. Bell, Y. Ephraim, H. L. Van Trees, “Explicit Ziv–Zakai lower bound for bearing estimation,” IEEE Trans. Signal Process. 44, 2810–2824 (1996).
[CrossRef]

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[CrossRef]

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A. Blain, R. Ivison, I. Smail, “Observational limits to source confusion in the millimetre/submillimetre waveband,” Mon. Not. R. Astron. Soc. 296, L29–L33 (1998).
[CrossRef]

Nature (2)

D. Hughes, S. Serjeant, J. Dunlop, M. Rowan-Robinson, A. Blain, R. G. Mann, R. Ivison, J. Peacock, A. Efstathiou, W. Gear, S. Oliver, A. Lawrence, M. Longair, P. Goldschmidt, T. Jenness, “High-redshift star formation in the Hubble Deep Field revealed by a submillimetre-wavelength survey,” Nature 394, 241–247 (1998).
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A. Barger, L. Cowie, D. Sanders, E. Fulton, Y. Taniguchi, Y. Sato, K. Kawara, H. Okuda, “Submillimetre-wavelength detection of dusty star-forming galaxies at high redshift,” Nature 394, 248–251 (1998).
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See http://olbin.jpl.nasa.gov .

See http://www.nrao.edu .

M. Shao, “SIM: the space interferometry mission,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 536–540 (1998), see also http://sim.jpl.nasa.gov .
[CrossRef]

C. A. Beichman, “Terrestrial Planet Finder: the search for life-bearing planets around other stars,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 719–723 (1998), see also http://planetquest.jpl.nasa.gov .
[CrossRef]

A. J. Penny, A. Leger, J. Mariotti, C. Schalinski, C. Eiroa, R. J. Laurance, M. Fridlund, “Darwin interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 666–671 (1998), see also http://sci.esa.int/darwin .
[CrossRef]

M. Faucherre, B. Delabre, P. Dierickx, F. Merkle, “Michelson versus Fizeau type beam combination—Is there a difference?” in Amplitude and Intensity Spatial Interferometry, Proc. SPIE1237, 206–217 (1990).
[CrossRef]

S. Prasad, “Sensitivity limits on an image-plane fiber interferometer at low-light levels,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE2200, 51–59 (1994).
[CrossRef]

The author is preparing a manuscript to be called “Thermal noise and correlations in photon detection.”

S. W. Wedge, “Computer-aided design of noise microwave circuits,” Ph.D. thesis (California Institute of Technology, Pasadena, Calif., 1991).

M. Shao, W. Danchi, M. J. DiPirro, M. Dragovan, L. D. Feinberg, M. Hagopian, W. D. Langer, C. R. Lawrence, P. R. Lawson, D. T. Leisawitz, J. C. Mather, S. H. Moseley, M. R. Swain, H. W. Yorke, X. Zhang, “Space-based interfero-metric telescopes for the far infrared,” in Interferometry in Optical Astronomy, P. J. Lena, A. Quirrenbach, eds., Proc. SPIE4006, 772–781 (2000).
[CrossRef]

D. T. Leisawitz, W. C. Danchi, M. J. DiPirro, L. D. Feinberg, D. Y. Gezari, M. Hagopian, W. D. Langer, J. C. Mather, S. H. Moseley, M. Shao, R. F. Silverberg, J. Staguhn, M. R. Swain, H. W. Yorke, X. Zhang, “Scientific motivation and technology requirements for the SPIRIT and SPECS far-infrared/submillimeter space interferometers,” in UV, Optical, and IR Space Telescopes and Instruments, J. B. Breckinridge, P. Jakobsen, eds., Proc. SPIE4013, 36–46 (2000).
[CrossRef]

P. Kern, F. Malbet, eds., Integrated Optics for Astronomical Interferometry (Bastinelli-Guirimand, Grenoble, France, 1996).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the pairwise beam combination scheme for a single baseline between telescopes t and t. The inputs t and t on the left represent the single-mode beams from the two telescopes, after division T-1 ways. The four outputs on the right are sent to photon-counting detectors.

Fig. 2
Fig. 2

Schematic diagram of the Butler matrix all-on-one beam combination scheme. The inputs on the left represent the single-mode beams from all T telescopes; each of the T outputs on the right, which are sent to photon-counting detectors, contain some contribution from all T telescope inputs.

Fig. 3
Fig. 3

Comparison of the angular response functions for pairwise and Butler beam combining. Top, typical response corresponding to two telescopes in a pair-combined array; the separation between the telescopes is 27L1 for this example. Bottom, typical response of a Butler-combined array; in this case, there are 10 uniformly spaced telescopes, with a distance 3L1 between telescopes, so that the array size is 27L1.

Fig. 4
Fig. 4

Variation of the normalized sensitivity with telescope array size for the case of Butler beam combination. The source is assumed to have a uniform spatial distribution. The horizontal axis gives the spatial position θ in units of λ/L1; note that the full width at half-power of the single-element pattern is 0.886λ/L1. The vertical axis gives the Cramér–Rao normalized sensitivity bound (see text for details). For Butler beam combination, increasing the array size improves the normalized sensitivity. The dotted curve shows that the sensitivity degradation toward the edges of the field of view scales as the reciprocal of the single-element beam pattern. The upper sensitivity limit calculated with the LS deconvolution method is indistinguishable from Cramér–Rao bound; thus this plot gives the actual achievable sensitivity.

Fig. 5
Fig. 5

Similar to Fig. 4 but calculated for pairwise beam combination. Normalized sensitivity is essentially independent of array size. Again, the upper limit from the LS method is calculated to be the same as the Cramér–Rao lower bound.

Fig. 6
Fig. 6

Comparison of normalized sensitivities for Butler versus pairwise beam combining for a ten-element array when uniform sources are observed; the sensitivity advantage for Butler combining is more than a factor of 3.

Fig. 7
Fig. 7

Similar to Fig. 4 but calculated for a point source in the center of the field instead of for a uniform source. Note that although the normalized sensitivity to the point source is nearly constant, the sensitivity for off-source pixels improves substantially with array size. The upper limit from the LS method is somewhat worse than the Cramér–Rao lower bound and is shown as the upper solid curve for the ten-element array. This curve lies near the Cramér–Rao lower bound for the six-element array and below the bound for the three-element array.

Fig. 8
Fig. 8

Similar to Fig. 7 but with pairwise beam combination assumed; there is no sensitivity improvement with array size. The sensitivity of the LS method is indistinguishable from the Cramér–Rao bound, so the plot gives the actual sensitivity.

Fig. 9
Fig. 9

Comparison of normalized point-source sensitivities for Butler combination and pairwise combination for a ten-element array. Although the sensitivities to the point source itself are comparable, Butler combination is an order of magnitude more sensitive for off-source pixels. The upper solid curve gives the LS upper sensitivity limit for the Butler-combined array, and it lies well below the Cramér–Rao lower bound for the pairwise-combined array.

Tables (1)

Tables Icon

Table 1 Array Configurationsa

Equations (85)

Equations on this page are rendered with MathJax. Learn more.

bi=jSijaj.
Einc(r)=2η0λdΩa(Ω)exp[+jknˆ(Ω)r].
Eout(r)=2η0λ dΩb(Ω)exp[-jknˆ(Ω)r].
Qinc=dΩ|a(Ω)|2.
Qα=i|biα|2.
a=a(Ω)aiα,
S=S(scat)S(rec)S(trans)S(refl).
b(Ω)=dΩS(scat)(Ω, Ω)a(Ω)+jβSjβ(trans)(Ω)ajβ,
biα=dΩSiα(rec)(Ω)a(Ω)+jβSiαjβ(refl)ajβ.
Siα(trans)(Ω)=Siα(rec)(Ω),
S(scat)(Ω, Ω)=[S(scat)]T(Ω, Ω),
Miα,jβ=δiα,jβ-dΩ[Siα(rec)(Ω)]*Sjβ(rec)(Ω)-kγ[Skγ,iδ(refl)]*Skγ,jβ(refl).
dΩ|Siα(rec)(Ω)|21-jβ|Sjβiα(refl)|21.
dΩ[Siα(rec)(Ω)]*Sjβ(rec)(Ω)=δiα,jβ.
Qα=i|biα|2=idΩSiα(rec)(Ω)a(Ω)2.
aq(Ω, ν)aq*(Ω, ν)=Aqq(Ω, ν)δ(Ω-Ω)×δ(ν-ν).
F(pˆ, sˆ, ν)=1λ2qqdΩnˆ(Ω)sˆ×pˆ*eˆq(Ω)Aqq(Ω, ν)eˆq*(Ω)pˆ|1-nˆ(Ω)pˆ|2.
Ftotal(ν)=1λ2qdΩAqq(Ω, ν).
Aqq(Ω, ν)=hν n(Ω, ν)δqq,
Qα(ν)=qqdΩAqq(Ω, ν)i[eˆqSiα(rec)(Ω, ν)]×[Siα(rec)(Ω, ν)eˆq]*.
Qα(ν)=dΩA(Ω, ν)Rα(Ω, ν),
Rα(Ω, ν)=i|Siα(rec)(Ω, ν)|2.
mα(ν)=dΩRα(Ω, ν).
Qα(ν)=12 F(ν)λ2Rα(Ωp, ν),
αRα(Ωp, ν)2Atelλ2.
mα2AtelΔΩαλ2.
ρα(Ω, ν)=λ22Atel Rα(Ω, ν),
αρα(Ω, ν)1.
Qα(ν)=2Atelλ2dΩA(Ω, ν)ρα(Ω, ν).
A(Ω, ν)=sA¯(Ωs, ν)Us(Ω),
Qα(ν)=s2ΔΩsAtelλ2 A¯(Ωs, ν)ρ¯α(Ωs, ν),
ρ¯α(Ωs, ν)=1ΔΩsΔΩsdΩρα(Ω, ν).
Nα=dνsN(Ωs, ν)ρ¯α(Ωs, ν),
N(Ωs, ν)=τ 2ΔΩsAtelλ2 n¯(Ωs, ν)
Nα=s,fN(Ωs, νf)ρ¯α(Ωs, νf),
N(Ωs, νf)=τΔνf2ΔΩsAtelλ2 n¯(Ωs, νf)
ρ¯α(Ωs, νf)=1ΔΩsΔνfΔνfdνΔΩsdΩρα(Ω, ν).
pαc=ρ¯α(Ωs, νf).
λc=N(Ωs, νf)
μα=Nα=cpαcλc.
αpαc1,
0pαc1.
St(rec)(Ω, ν)=exp[+iknˆ(Ω)rt]S(rec)(Ω, ν).
bt(ν)=dΩSt(rec)(Ω, ν)a(Ω, ν).
dΩ[St(rec)(Ω, ν)]*St(rec)(Ω, ν)=dΩ|S(rec)(Ω, ν)|2exp[-iknˆ(Ω)(rt-rt)]0,
biα(ν)=tSiα,t(comb)bt(ν).
biα(ν)=tSiα,t(comb)(ν)St(rec)(Ω, ν)a(Ω, ν).
Rα(Ω, ν)=itSiα,t(comb)(ν)St(rec)(Ω, ν)2.
ρα(Ω, ν)=λ2AT Rα(Ω, ν),
pβc=pαc;awa,
wa=TaaTa,
σc2=(Δλˆc)2λc,
pαcpαc=0
Mij(θ)= ln fθi ln fθj=dxf(x|θ)  ln f(x|θ)θi ln f(x|θ)θj.
σi2=(θˆi-θi)2(M-1)ii.
σi2(M-1)ii(Mii)-1.
f(N|λ)=α μαNαNα! exp(-μα),
 ln fλc=α-pαc+Nαpαcμα,
Mcc(λ)= ln fλc ln fλc=α pαcpαcμα=α pαcpαccpαcλc,
σc2(M-1)cc.
σc2αpαc2cpαcλc-1.
M=PTΓ-1P,
1σc2Mcc=αDc pαc2μα,
μα=cpαcλcpαcλc0,
1σc2αDcpαc2pαcλc=αDcpαcλc1λc,
ln f=α(Nα ln μˆα-μˆα).
 ln fλˆc=αpαc-1+Nαμˆα=0
vα(a)=-1+Nαμˆα(a),
αμˆα(a)vα(a)=αμˆα(b)vα(b)=0,
αμˆα(a)vα(b)=αμˆα(b)vα(a)=0.
αμˆα(a)=αμˆα(b)=αNα
αNαμˆα(a)μˆα(b)-1=αNαμˆα(b)μˆα(a)-1=0.
αNαg(xα)=0,
μˆα=cpαcλˆc=Nα.
μˆ(LS)=(PTP)-1PTN.
μˆ(LS)=(PTP)-1PTμ=(PTP)-1PTPλ=λ.
C(LS)=ΔμˆLS(ΔμˆLS)T=(PTP)-1PTΓP(PTP)-1,
C(LS)=μ(PTP)-1.
μˆ(LS)=μˆ(ML)=P-1N.
S(rec)(θ)=L1λ1/2sincπL1λ θ.
St(rec)(θ)=expi 2πxtλ θS(rec)(θ).
ρα(θ, ν)=λLT Rα(θ, ν).
b1(t, t)=12T-1 (bt+bt),b2(t, t)=12T-1 (bt-bt),b3(t, t)=12T-1 (bt+ibt),b4(t, t)=12T-1 (ibt+bt).
Rα(Ω, ν)=14(T-1) |St(rec)(θ)+iSt(rec)(θ)|2
bα=1T t=1Tbt expi 2παtT,

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