Abstract

We present a new approach to processing of interferometric data, which dramatically improves operation on the smallest fraction of the fringe (∼1/1000 of a wavelength and beyond). In particular, this approach, dubbed the variation-invariant subspace tracking approach (VISTA), makes it possible to operate a Michelson interferometer in a highly stable mode in which the estimation of optical path delay becomes invariant (insensitive) to a large class of systematic model errors. This previously unknown invariance property of interferometry follows from the fundamental odd–even symmetry of the channeled spectrum and its derivatives. VISTA offers a powerful algorithmic alternative for alleviating technological challenges in the design of high-precision long-baseline spaceborne interferometers.

© 2001 Optical Society of America

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References

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  1. Space Interferometry Mission: Taking the Measure of the Universe, , 1999. (Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, Calif., 91109).
  2. M. M. Colavita, M. Shao, M. D. Rayman, “Orbiting stellar interferometer for astrometry and imaging,” Appl. Opt. 32, 1789–1797 (1993).
    [CrossRef] [PubMed]
  3. R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
    [CrossRef]
  4. M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).
  5. H. Cramer, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N. J., 1946).
  6. C. E. Padilla, V. I. Karlov, J. Li, H. M. Chun, J. N. Tsitsiklis, R. D. Reasenberg, “A study of fringe tracking for high precision space-based interferometers,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE2477, 63–76 (1995).
    [CrossRef]
  7. C. E. Padilla, V. I. Karlov, Starlight Fringe Detection and Tracking Algorithms for Long-Baseline Space-Based Interferometers, , 1998. (Moldyn, Inc., 955 Massachusetts Avenue, Suite 501, Cambridge, Mass. 02139).
  8. A. Gelb, ed., Applied Optimal Estimation (MIT Press, Cambridge, Mass., 1989).

1993 (1)

1988 (2)

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Babcock, R. W.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Bender, P.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Buffington, A.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Carney, B.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Chandler, J. F.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Chun, H. M.

C. E. Padilla, V. I. Karlov, J. Li, H. M. Chun, J. N. Tsitsiklis, R. D. Reasenberg, “A study of fringe tracking for high precision space-based interferometers,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE2477, 63–76 (1995).
[CrossRef]

Colavita, M. M.

M. M. Colavita, M. Shao, M. D. Rayman, “Orbiting stellar interferometer for astrometry and imaging,” Appl. Opt. 32, 1789–1797 (1993).
[CrossRef] [PubMed]

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Cramer, H.

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

Gorenstein, M. V.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Hershey, J. L.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Hines, B. E.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Huchra, J. P.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Hughes, J. A.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Hutter, D. J.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Johnston, K. J.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Jones, B. F.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Kaplan, G. H.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Karlov, V. I.

C. E. Padilla, V. I. Karlov, J. Li, H. M. Chun, J. N. Tsitsiklis, R. D. Reasenberg, “A study of fringe tracking for high precision space-based interferometers,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE2477, 63–76 (1995).
[CrossRef]

C. E. Padilla, V. I. Karlov, Starlight Fringe Detection and Tracking Algorithms for Long-Baseline Space-Based Interferometers, , 1998. (Moldyn, Inc., 955 Massachusetts Avenue, Suite 501, Cambridge, Mass. 02139).

Li, J.

C. E. Padilla, V. I. Karlov, J. Li, H. M. Chun, J. N. Tsitsiklis, R. D. Reasenberg, “A study of fringe tracking for high precision space-based interferometers,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE2477, 63–76 (1995).
[CrossRef]

Matson, L. E.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Mozurkewich, D.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Padilla, C. E.

C. E. Padilla, V. I. Karlov, J. Li, H. M. Chun, J. N. Tsitsiklis, R. D. Reasenberg, “A study of fringe tracking for high precision space-based interferometers,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE2477, 63–76 (1995).
[CrossRef]

C. E. Padilla, V. I. Karlov, Starlight Fringe Detection and Tracking Algorithms for Long-Baseline Space-Based Interferometers, , 1998. (Moldyn, Inc., 955 Massachusetts Avenue, Suite 501, Cambridge, Mass. 02139).

Pearlman, M. R.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Rayman, M. D.

Reasenberg, R. D.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

C. E. Padilla, V. I. Karlov, J. Li, H. M. Chun, J. N. Tsitsiklis, R. D. Reasenberg, “A study of fringe tracking for high precision space-based interferometers,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE2477, 63–76 (1995).
[CrossRef]

Shao, M.

M. M. Colavita, M. Shao, M. D. Rayman, “Orbiting stellar interferometer for astrometry and imaging,” Appl. Opt. 32, 1789–1797 (1993).
[CrossRef] [PubMed]

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Shapiro, I. I.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Simon, R. S.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Staelin, D. H.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Taylor, R. S.

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Tsitsiklis, J. N.

C. E. Padilla, V. I. Karlov, J. Li, H. M. Chun, J. N. Tsitsiklis, R. D. Reasenberg, “A study of fringe tracking for high precision space-based interferometers,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE2477, 63–76 (1995).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Astron. J. (1)

R. D. Reasenberg, R. W. Babcock, J. F. Chandler, M. V. Gorenstein, J. P. Huchra, M. R. Pearlman, I. I. Shapiro, R. S. Taylor, P. Bender, A. Buffington, B. Carney, J. A. Hughes, K. J. Johnston, B. F. Jones, L. E. Matson, “Microarcsecond optical astrometry: an instrument and its astrophysical applications,” Astron. J. 96, 1731–1745 (1988).
[CrossRef]

Other (5)

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

C. E. Padilla, V. I. Karlov, J. Li, H. M. Chun, J. N. Tsitsiklis, R. D. Reasenberg, “A study of fringe tracking for high precision space-based interferometers,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE2477, 63–76 (1995).
[CrossRef]

C. E. Padilla, V. I. Karlov, Starlight Fringe Detection and Tracking Algorithms for Long-Baseline Space-Based Interferometers, , 1998. (Moldyn, Inc., 955 Massachusetts Avenue, Suite 501, Cambridge, Mass. 02139).

A. Gelb, ed., Applied Optimal Estimation (MIT Press, Cambridge, Mass., 1989).

Space Interferometry Mission: Taking the Measure of the Universe, , 1999. (Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, Calif., 91109).

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Figures (8)

Fig. 1
Fig. 1

Informational errors versus dynamic errors in interferometric observations.

Fig. 2
Fig. 2

Channeled spectra (Iij) as a function of OPD (x) and wavelength (λ).

Fig. 3
Fig. 3

Effect of OPD and wavelength-error coupling with work on a small fraction of a fringe.

Fig. 4
Fig. 4

Odd–even symmetry of channeled spectra and its derivatives as a basis for VISTA. (a) Channeled spectra (CS): Even symmetry with respect to OPD; (b) derivatives of the CS with respect to OPD: odd symmetry with respect to OPD; (c) derivatives of the CS with respect to wavelength: even symmetry with respect to OPD.

Fig. 5
Fig. 5

VISTA expands the view of beam interactions in optics.

Fig. 6
Fig. 6

Degradation of fringe tracking accuracy as a function of the STD of wavelength displacement errors (Case 1: asymmetric OPD control program with respect to zero OPD).

Fig. 7
Fig. 7

VISTA provides invariance to wavelength displacement errors (Case 2: symmetric OPD control program with respect to zero OPD).

Fig. 8
Fig. 8

VISTA versus conventional quarter-wave strategy in the PMM.

Equations (14)

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

Iij=I0ij1+Λj cos2πλ¯jxi+θjχjλχit
χjλ=sincπΔλjλ¯j2xi,χit=sincπΔti2λ¯jx˙i.
yij=Iij(x, ξ)+ηij.
Cx(x, λ)=Iij(x, λ)x,Cλ(x, λ)=Iij(x, λ)λ.
Δyfj=yfj-Ifj(+xr, ξr)=Cx(+xr, ξr)Δx+Cξ(+xr, ξr)Δξ+ηfj,
Δysj=ysj-Isj(-xr, ξr)=Cx(-xr, ξr)Δx+Cξ(-xr, ξr)Δξ+ηsj.
Cx(±xr, ξr)=Iij(x, ξ)xx=±xr,
Cξ(±xr, ξr)=Iij(x, ξ)ξx=±xr.
Iij(-xr, ξr)=Iij(+xr, ξr),
Cx(-xr, ξr)=-Cx(+xr, ξr),
Cξ(-xr, ξr)=Cξ(+xr, ξr),
Δy(f,s)j=Δyfj-Δysj=2Cx(xr, ξr)Δx+Δη(f,s)j,
p(Δη(f,s)j)=Γp(ηf)p(ηs)dηfdηs,
Γ={ηf,ηs : ηf-ηs=Δη(f,s)j}.

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