Abstract

An analytical technique for extracting phase, visibility, and amplitude information as needed for interferometric astrometry for the Space Interferometry Mission (SIM) is presented. This model accounts for a number of physical and instrumental effects and is valid for the general case of a bandpass filter. I was able to obtain a general solution for polychromatic phasors and to address the properties of unbiased fringe estimators in the presence of noise. For demonstration purposes I studied a rectangular bandpass filter with two different methods of optical path difference (OPD) modulation: stepping and ramping OPD modulation. A number of areas for further studies relevant to instrument design and simulations are outlined and discussed.

© 2003 Optical Society of America

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References

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  1. R. Danner, S. C. Unwin, eds., SIM Interferometry Mission: Taking the Measure of the Universe, NASA Doc. JPL 400-811 (NASA, Washington, D.C., 1999).
  2. S. C. Unwin, S. G. Turyshev, eds., Science with the Space Interferometry Mission, JPL Publ. 02-01 (Jet Propulsion Laboratory, Pasadena, Calif., 2002); see also http://sim.jpl.nasa.gov/ .
  3. A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).
  4. S. G. Turyshev, “Analytic expressions for the while light fringe extraction,” JPL Internal Tech. Memo. 00-0901 (Jet Propulsion Laboratory, Pasadena, Calif., 2000).
  5. M. Milman, S. Basinger, “Error sources and algorithms for white-light fringe estimation at low light levels,” Appl. Opt. 41, 2655–2671 (2002).
    [CrossRef] [PubMed]
  6. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1988), Vol. XXVI, pp. 349–393.
  7. M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
    [CrossRef]
  8. 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).
  9. M. Shao, M. M. Colavita, “Long-baseline optical and infrared stellar interferometry,” Astron. Astrophys. 30, 457–498 (1992).
    [CrossRef]
  10. M. Shao, M. M. Colavita, “Potential of long-baseline infrared interferometry for narrow-angle interferometry,” Astron. Astrophys. 262, 353–358 (1992).
  11. M. M. Colavita, “Fringe visibility estimators for the Palomar Testbed interferometer,” Publ. Astron. Soc. Pac. 111, 111–117 (1999).
    [CrossRef]
  12. A. F. Boden, “SIM astrometric grid simulation development and performance assessment,” JPL Interoffice Memo. 10-005 (Jet Propulsion Laboratory, Pasadena, Calif., 1997).
  13. R. Swartz, “Metrology breaks and the SIM astrometric grid,” JPL Interoffice Memo. 17-063 (Jet Propulsion Laboratory, Pasadena, Calif., 2000).
  14. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  15. W. J. Tango, R. Q. Twiss, “Michelson stellar interferometry,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1980), Vol. XVII, pp. 239–277.
  16. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]
  17. P. R. Lawson, “Phase and group delay estimation,” in Principles of Long Baseline Stellar Interferometry, P. R. Lawson, ed., JPL Publ. 00-009 (Jet Propulsion Laboratory, Pasadena, Calif., 2000).
  18. J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
    [CrossRef]
  19. T. A. ten Brummelaar, “Correlation measurement and group delay tracking in optical stellar interferometry with a noisy detector,” Mon. Notes R. Astron. Soc. 285, 135–150 (1997).
    [CrossRef]
  20. M. Milman, J. Catanzarite, S. G. Turyshev, “The effect of wavenumber error on the computation of path-length delay in white-light interferometry,” Appl. Opt. 41, 4884–4890 (2002).
    [CrossRef] [PubMed]
  21. Note that, by taking the limit v → 0 in Eq. (6) [i.e., sinc(½ kvΔτi) → 1], one recovers the case of stepping OPD modulation with the familiar simple form of an observational equation: ℐi(k) = ℐ0[1 + V sin(ϕ0 + kxi)].

2002 (2)

1999 (2)

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

M. M. Colavita, “Fringe visibility estimators for the Palomar Testbed interferometer,” Publ. Astron. Soc. Pac. 111, 111–117 (1999).
[CrossRef]

1998 (1)

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

1997 (1)

T. A. ten Brummelaar, “Correlation measurement and group delay tracking in optical stellar interferometry with a noisy detector,” Mon. Notes R. Astron. Soc. 285, 135–150 (1997).
[CrossRef]

1994 (1)

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

1992 (2)

M. Shao, M. M. Colavita, “Long-baseline optical and infrared stellar interferometry,” Astron. Astrophys. 30, 457–498 (1992).
[CrossRef]

M. Shao, M. M. Colavita, “Potential of long-baseline infrared interferometry for narrow-angle interferometry,” Astron. Astrophys. 262, 353–358 (1992).

1988 (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).

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Armstrong, J. T.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

Basinger, S.

Benson, J. A.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Boden, A. F.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

A. F. Boden, “SIM astrometric grid simulation development and performance assessment,” JPL Interoffice Memo. 10-005 (Jet Propulsion Laboratory, Pasadena, Calif., 1997).

Bowers, P. F.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Buscher, D. F.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

Catanzarite, J.

Clark, J. H.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Colavita, M. M.

M. M. Colavita, “Fringe visibility estimators for the Palomar Testbed interferometer,” Publ. Astron. Soc. Pac. 111, 111–117 (1999).
[CrossRef]

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

M. Shao, M. M. Colavita, “Potential of long-baseline infrared interferometry for narrow-angle interferometry,” Astron. Astrophys. 262, 353–358 (1992).

M. Shao, M. M. Colavita, “Long-baseline optical and infrared stellar interferometry,” Astron. Astrophys. 30, 457–498 (1992).
[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).

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1988), Vol. XXVI, pp. 349–393.

Dumont, P. J.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Elias, N. M.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Gubler, J.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Gursel, Y.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Ha, L.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[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. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[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).

Hughes, J. A.

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).

Hummel, C. A.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

Hutter, D. J.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[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).

Johnston, K. J.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[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).

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).

Koresko, C. D.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Kulkarni, S. R.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Lane, B. F.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Lawson, P. R.

P. R. Lawson, “Phase and group delay estimation,” in Principles of Long Baseline Stellar Interferometry, P. R. Lawson, ed., JPL Publ. 00-009 (Jet Propulsion Laboratory, Pasadena, Calif., 2000).

Ling, L.-C.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Malbet, F.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Milman, M.

Mobley, D. W.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Mozurkewich, D.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

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).

Palmer, D. L.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Pan, X. P.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Quirrenbach, A.

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

Rickard, L. J.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Shao, M.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

M. Shao, M. M. Colavita, “Long-baseline optical and infrared stellar interferometry,” Astron. Astrophys. 30, 457–498 (1992).
[CrossRef]

M. Shao, M. M. Colavita, “Potential of long-baseline infrared interferometry for narrow-angle interferometry,” Astron. Astrophys. 262, 353–358 (1992).

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).

Simon, R. S.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[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).

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).

Swartz, R.

R. Swartz, “Metrology breaks and the SIM astrometric grid,” JPL Interoffice Memo. 17-063 (Jet Propulsion Laboratory, Pasadena, Calif., 2000).

Tango, W. J.

W. J. Tango, R. Q. Twiss, “Michelson stellar interferometry,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1980), Vol. XVII, pp. 239–277.

ten Brummelaar, T. A.

T. A. ten Brummelaar, “Correlation measurement and group delay tracking in optical stellar interferometry with a noisy detector,” Mon. Notes R. Astron. Soc. 285, 135–150 (1997).
[CrossRef]

Turyshev, S. G.

M. Milman, J. Catanzarite, S. G. Turyshev, “The effect of wavenumber error on the computation of path-length delay in white-light interferometry,” Appl. Opt. 41, 4884–4890 (2002).
[CrossRef] [PubMed]

S. G. Turyshev, “Analytic expressions for the while light fringe extraction,” JPL Internal Tech. Memo. 00-0901 (Jet Propulsion Laboratory, Pasadena, Calif., 2000).

Twiss, R. Q.

W. J. Tango, R. Q. Twiss, “Michelson stellar interferometry,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1980), Vol. XVII, pp. 239–277.

van Belle, G. T.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Wallace, J. K.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

White, N. M.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Yu, J. W.

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Appl. Opt. (2)

Astron. Astrophys. (4)

A. Quirrenbach, D. Mozurkewich, D. F. Buscher, C. A. Hummel, J. T. Armstrong, “Phase-referenced visibility averaging in optical long-baseline interferometry,” Astron. Astrophys. 286, 1019–1027 (1994).

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).

M. Shao, M. M. Colavita, “Long-baseline optical and infrared stellar interferometry,” Astron. Astrophys. 30, 457–498 (1992).
[CrossRef]

M. Shao, M. M. Colavita, “Potential of long-baseline infrared interferometry for narrow-angle interferometry,” Astron. Astrophys. 262, 353–358 (1992).

Astrophys. J. (2)

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

M. M. Colavita, J. K. Wallace, B. E. Hines, Y. Gursel, F. Malbet, D. L. Palmer, X. P. Pan, M. Shao, J. W. Yu, A. F. Boden, P. J. Dumont, J. Gubler, C. D. Koresko, S. R. Kulkarni, B. F. Lane, D. W. Mobley, G. T. van Belle, “The Palomar Testbed Interferometer,” Astrophys. J. 510, 505–521 (1999).
[CrossRef]

Mon. Notes R. Astron. Soc. (1)

T. A. ten Brummelaar, “Correlation measurement and group delay tracking in optical stellar interferometry with a noisy detector,” Mon. Notes R. Astron. Soc. 285, 135–150 (1997).
[CrossRef]

Opt. Eng. (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

M. M. Colavita, “Fringe visibility estimators for the Palomar Testbed interferometer,” Publ. Astron. Soc. Pac. 111, 111–117 (1999).
[CrossRef]

Other (10)

A. F. Boden, “SIM astrometric grid simulation development and performance assessment,” JPL Interoffice Memo. 10-005 (Jet Propulsion Laboratory, Pasadena, Calif., 1997).

R. Swartz, “Metrology breaks and the SIM astrometric grid,” JPL Interoffice Memo. 17-063 (Jet Propulsion Laboratory, Pasadena, Calif., 2000).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

W. J. Tango, R. Q. Twiss, “Michelson stellar interferometry,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1980), Vol. XVII, pp. 239–277.

P. R. Lawson, “Phase and group delay estimation,” in Principles of Long Baseline Stellar Interferometry, P. R. Lawson, ed., JPL Publ. 00-009 (Jet Propulsion Laboratory, Pasadena, Calif., 2000).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1988), Vol. XXVI, pp. 349–393.

S. G. Turyshev, “Analytic expressions for the while light fringe extraction,” JPL Internal Tech. Memo. 00-0901 (Jet Propulsion Laboratory, Pasadena, Calif., 2000).

R. Danner, S. C. Unwin, eds., SIM Interferometry Mission: Taking the Measure of the Universe, NASA Doc. JPL 400-811 (NASA, Washington, D.C., 1999).

S. C. Unwin, S. G. Turyshev, eds., Science with the Space Interferometry Mission, JPL Publ. 02-01 (Jet Propulsion Laboratory, Pasadena, Calif., 2002); see also http://sim.jpl.nasa.gov/ .

Note that, by taking the limit v → 0 in Eq. (6) [i.e., sinc(½ kvΔτi) → 1], one recovers the case of stepping OPD modulation with the familiar simple form of an observational equation: ℐi(k) = ℐ0[1 + V sin(ϕ0 + kxi)].

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Figures (7)

Fig. 1
Fig. 1

Basic geometry of light propagation in stellar interferometry.

Fig. 2
Fig. 2

Monochromatic fringe. Note that the visibility is set to V = 1 and the initial phase offset is chosen to be ϕ0 = π/5.

Fig. 3
Fig. 3

Top, typical OPD modulation stroke with ramping over eight equal temporal bins with durations of 10 ms each. Bottom, monochromatic fringe as a function of OPD modulated by ramping over a wavelength (the initial phase offset is ϕ0 = π/5, the same as in Fig. 2).

Fig. 4
Fig. 4

Top, typical OPD modulation stroke with stepping over eight equal temporal bins with durations of 10 ms. Bottom, monochromatic fringe as a function of OPD modulated in eight equal steps over a wavelength, as shown above. The fringe parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

Left, three independent monochromatic waves presented as functions of the OPD. The phase of the fringes changes as ϕ = ϕ0 + k i x, with external delay corresponding to a phase difference ϕ0 = π/5 and three wave numbers, k 1 = k 0, k 2 = 1.5k 0, and k 3 = 2k 0 (thus the bandwidth is Δk = k 0), where k 0 is a reference wave number. Right, interference pattern of polychromatic light composed from the same three sinusoidal waves shown at the left. Note the drastic change in the character of the interferometric pattern.

Fig. 6
Fig. 6

Left, three monochromatic waves within the narrow finite bandwidth. Right, interference of the polychromatic light that as a superposition of these three waves. Note the envelope correction to the fringe visibility as a function of the OPD.

Fig. 7
Fig. 7

Top, typical behavior of three monochromatic sinusoidal fringes modulated in eight equal steps over a wavelength (as shown at the top of Fig. 4). Parameters for the waves are ϕ = k i x, ϕ0 = 0, and k 1 = 0.75k 0 (thicker dashed lines), k 2 = 0.8k 0 (thinner dashed lines), and k 3 = 0.85k 0 (thin solid lines). Thus the width of the spectral channel is Δk = 0.1k 0. Bottom, polychromatic fringe composed of these three waves.

Equations (128)

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E=E0 expjωt-k · x,
=E · E*time=E012+E022+2E1E2,
k=01+V cos kd0,
V=2121+2ν=2|E01||E02|E012+E022ν.
k, t=01+V sinϕ0+kxt,
ik=01+V sinc½ kvΔτisinϕ0+kxi,
Ek, t=E0k, texpjωt-kx,
k, tE01k, t+E02k, tE01*k, t+E02*k, ttime =E012k+E022k+2|E01k||E02k|γkcosΦk+Δφ12k, t,
ν˜k=νkexpjϕνk,
T˜k=TkexpjϕTk,
γ˜k=ν˜kT˜k=νkTkexpjϕνk+ϕTkγkexpjΦk,
Δφ12k, t=kxt,
k, t1k+2k+21k2k1/2γkcosΦk+kxt.
V˜k=21k2k1/21k+2k γk=VkexpjΦk.
k, t=0k1+VkcosΦk+kxt.
dk, t=0k1+VkcosΦk+kxtdk.
δΦbsk=π2+fk,
fk=fkc+fkck-kc+½fkck-kc2+OΔkc3.
ΦkΦk+δΦbsk=Φk+π2+fk=π2+ϕk.
dk, t=0k1+Vksinϕk+kxtdk,
dk, t=0k0k1+Vksinϕk+kxtdkdt,
dNk, t=αk0k0k1+Vksinϕk+kxtdkdt.
dNk, t=αk0k0k1+Vksinϕk+kxtn · τAdAdkdt,
Nk, t=k0k1+Vksinϕk+kxt,
dNk, t=k0k1+Vksinϕk+kxtdkdt,
NtΔkSIM=kSIM-kSIM+ Nk, tdk, ΔkSIM=kSIM+-kSIM-.
kSIM-kSIM+ Nk, tdk=l=1Lkl-kl+ Nk, tdk=l=1L NltΔkl,
Nlt=1Δklkl-kl+ k0k1+Vksinϕk+kxtdk.
0l=1Δklkl-kl+ k0kdk.
ˆ0lk=k0k0l, 1Δklkl-kl+ ˆ0lkdk=1.
Nlt=0l1+1Δklkl-kl+ 0lkVksinϕk+kxtdk.
V0l=1Δklkl-kl+ ˆ0lkVkdk.
Vˆ0lk=ˆ0lkVkV0lk0kVk0lV0l, 1Δklkl-kl+ Vˆ0lkdk=1.
Nlt=0l1+V0l1Δklkl-kl+ Vˆ0lk×sinϕk+kxtdk.
kl=1Δklkl-kl+ ˆ0lkkdk10lΔklkl-kl+ k0kkdk.
ϕl=1Δklkl-kl+ ˆ0lkϕkdk10lΔklkl-kl+ k0kϕkdk.
ϕkl, ϕklϕl,
Nlt=0l1+V0l sinϕkl+klxt1Δklkl-kl+ Vˆ0lkcosk-klxt+ϕk-ϕkldk+V0l cosϕkl+klxt1Δklkl-kl+ Vˆ0lksink-klxt+ϕk-ϕkldk.
W˜lΔkl, ϕkl, xt=1Δklkl-kl+ Vˆ0lkexpjk-klxt+ϕk-ϕkldk
10lV0lΔklkl-kl+ k0kVk×expjk-klxt+ϕk-ϕkldk.
W˜l=ReW˜l+j ImW˜l,
ReW˜l=1Δklkl-kl+ Vˆ0lkcosk-klxt+ϕk-ϕkldk, ImW˜l=1Δklkl-kl+ Vˆ0lksink-klxt+ϕk-ϕkldk.
Nlt=0l1+V0l sinϕkl+klxtReW˜lxt+V0l cosϕkl+klxtImW˜lxt.
W˜lΔkl, ϕkl, xt=lΔkl, ϕkl, xt×expjΩlΔkl, ϕkl, xt,
lt=Re2W˜l+Im2W˜l1/2, Ωlt=ArctanImW˜lReW˜l.
Nlt=0l1+V0lltsinϕkl+klxt+Ωlt.
Γ˜x=V˜0lW˜lV0ll expjϕkl+Ωl.
Δt=t+-t-=i=1N=8 Δτi, Δτi=ti+-ti-.
NlΔt=t-t+ Nltdt=i=1N=8ti-ti+ Nltdt=i=1N=8 NliΔτi,
Nli=1Δτiti-ti+ Nltdt =1Δτiti-ti+ 0l1+V0l sin ϕklltcosklxt+Ωlt+V0l cos ϕklltsinklxt+Ωltdt.
Nli=0l1+V0l sin ϕklReP˜li+V0l cos ϕklImP˜li,
ReP˜li=1Δτiti-ti+ lΔkl, ϕkl, xtcosklxt+ΩlΔkl, ϕkl, xtdt, ImP˜li=1Δτiti-ti+ lΔkl, ϕkl, xtsinklxt+ΩlΔkl, ϕkl, xtdt.
P˜li=ReP˜li+j ImP˜li.
P˜li=1Δτiti-ti+ lΔkl, ϕkl, xtexpjklxt+ΩlΔkl, ϕkl, xtdt,
P˜li=1Δτiti-ti+expjklxtW˜lΔkl, ϕkl, xtdt,
P˜li=10lV0lΔklΔτiti-ti+kl-kl+ k0kVkexpjkxt+ϕk-ϕkldtdk.
Nl1NlN=1,ImP˜l1,ReP˜l11,ImP˜lN,ReP˜lN 0l0lV0l cos ϕkl0lV0l sin ϕkl,
Ni=N¯i+i, Ei=0, Ei2=σi2.
Cy=σ12000σ220000σN2, Gy=Cy-1=σ1-2000σ2-20000σN-2,
N¯i+i=1, ImP˜li, ReP˜li0l0lV0l cos ϕkl0lV0l sin ϕkl,
N¯i+i=AiαXα,
Xα=iNAiαN¯i, A=ATGyA-1ATGy,
Ai=1, ImP˜i, ReP˜i1, si, ci,
ATGy=1σ121σ221σN2s1σ12s2σ22sNσN2c1σ12c2σ22cNσN2, ATGyA=iN1σi2iNsiσi2iNciσi2iNsiσi2iNsi2σi2iNsiciσi2iNciσi2iNsiciσi2iNci2σi2.
Λ=ATGyA-1=1Δ12ijNsicj-cisj2σi2σj212ijNci-cjsicj-cisjσi2σj2-12ijNsi-sjsicj-cisjσi2σj212ijNci-cjsicj-cisjσi2σj212ijNci-cj2σi2σj2-12ijNsi-sjci-cjσi2σj2-12ijNsi-sjsicj-cisjσi2σj2-12ijNsi-sjci-cjσi2σj212ijNsi-sj2σi2σj2,
Δ=12ijkNsicj-sjciσi2σj2σk2sicj-sjci+sjck-skcj+skci-sick,
A=ATGyA-1ATGy=1DAkkCk,
Ak=ijN1σi2σj2σk2sicj-sjcisicj-sjci+sjck-skcj+skci-sick, k=ijN1σi2σj2σk2ci-cjsicj-sjci+sjck-skcj+skci-sick, Ck=-ijN1σi2σj2σk2si-sjsicj-sjci+sjck-skcj+skci-sick, D=kN Ak=13ijkN1σi2σj2σk2sicj-sjci+sjck-skcj+skci-sick2.
P˜i=ReP˜i+j ImP˜i=pi expjπi,
pi=Re2P˜i+Im2P˜i1/2, πi=arctanImP˜iReP˜i,
Ak=ijNpipj sinπi-πjσi2σj2σk2pipj sinπi-πj+pjpk sinπj-πk+pkpi sinπk-πi, k=ijNpi cos πi-pj cos πjσi2σj2σk2pipj sinπi-πj+pjpk sinπj-πk+pkpi sinπk-πi, Ck=-ijNpi sin πi-pj sin πjσi2σj2σk2pipj sinπi-πj+pjpk sinπj-πk+pkpi sinπk-πi, D=kN Ak=13ijkN1σi2σj2σk2pipj sinπi-πj+pjpk sinπj-πk+pkpi sinπk-πi2.
0=1DkNN¯kAk, 0V0 cosϕ¯=1DkNN¯kk, 0V0 sinϕ¯=1DkNN¯kCk,
V02=kNN¯kk2+ kNN¯kCk2kNN¯kAk2, ϕ¯=arctankNN¯kCkkNN¯kk, 0=kNN¯kAkkN Ak.
lk=lk-kl; Δkl.
k=l=1L lk, lk=0l=const.kkl-, kl+0kkl-, kl+.
0k=const., Vk=const., 0l=const.,
ϕk-ϕkl=d0lk-kl+O2ϕkl2,
W˜lΔkl, ϕkl, xi=10lV0lΔklkl-kl+ k0kVk×expjk-klxt+ϕk-ϕkldk =1Δklkl-kl+expjk-klxt+d0ldk+O2ϕkl2.
W˜lΔkl, ϕkl, xi=1Δkl-Δkl/2+Δkl/2expjκd0l+xtdκ=sin½Δkld0l+xt½Δkld0l+xt.
P˜li=1Δτiti-ti+expjklxtsin½Δkld0l+xt½Δkld0l+xtdt.
P˜li=expjklxtiδP˜li,
δP˜li=1Δτi-Δτi/2+Δτi/2expjklxti+τ-xti×sin½Δkld0l+xti+τ½Δkld0l+xti+τdτ.
Nli=0l1+V0l sinϕkl+klxtiReδP˜li+V0l cosϕkl+klxtiImδP˜li,
xt=i=1N=8 xti, xti=xitti-, ti+0tti-, ti+,
P˜li=expjklxtisin½Δkld0l+xi½Δkld0l+xi.
Nli=0l1+V0l sinc½Δklxi+d0lsinϕkl+klxi.
12 Δklxi+d0l12 Δklxi=πilNΔkSIMk0.
πilnΔkSIMk0πL.
sincΔklz2=n=0-1n2n+1!Δklz22n =1-13!Δklz22+15!Δklz24+O17!Δklz26.
Nli=0l1+V0l1-Δkl2xi+d0l224+Δkl4xi+d0l41920sinϕkl+klxi.
xt=x0+vt,
P˜li=expjklxtiδP˜li,
δP˜li=1Δτi-Δτi/2+Δτi/2expjklvτsin½Δklzτ½Δklzτdτ
Nli=0l1+V0l sinϕkl+klxtiReδP˜li+V0l cosϕkl+klxtiImδP˜li,
Nli=1, sin klxti, cos klxti 1000ReδP˜li-ImδP˜li0ImδP˜liReδP˜Pli 0l0lV0l cos ϕkl0lV0l sin ϕkl.
δP˜li=1Δτi-Δτi/2+Δτi/2expjklvτsin½Δklzτ½Δklzτdτ =1Δτi-Δτi/2+Δτi/2expjklvτ1-Δkl2zτ224+Δkl4zτ41920+OΔkl6z67!26dτ,
δP˜li=sin½klvΔτi½klvΔτi+A˜liΔkl224kl2+˜liΔkl41920kl4+OΔkl67!26kl6,
A˜li=1+1-jklzi2-12 klvΔτi2sin½klvΔτi½klvΔτi-21-jklzicos12 klvΔτi
˜li=1+1-jklzi2-12 klvΔτi22+42-jklzi2+41-12 klvΔτi2×sin½klvΔτi½klvΔτi-45+1-jklzi1-jklzi2-½klvΔτi2cos½klvΔτi.
ReδP˜li=sin½klvΔτi½klvΔτi1+2-kl2zi2-12 klvΔτi2Δkl224kl2-2 cos12 klvΔτiΔkl224kl2+OΔkl45!24kl4,
ImδP˜li=2klzi-sin½klvΔτi½klvΔτi+cos12 klvΔτiΔkl224kl2+OΔkl45!24kl4,
ϕl=1Δklkl-kl+ ˆ0lkϕkdk10lΔklkl-kl+ k0kϕkdk.
ϕk=ϕkl+ϕkklk-kl+122ϕk2klk-kl2+OΔkl3.
ϕl=ϕkl+122ϕk2kl10lΔklkl-kl+ k0kk-kl2dk+OΔkl3.
ϕkl=ϕl-122ϕk2klΔkl2μl2+OΔkl3,
μl2=10lΔkl3kl-kl+ k0kk-kl2dk.
ϕkl=ϕl-p=2P1p!pϕkpklΔklpμlp+OΔklp,
μl0 = 1, μl1 = 0,mx μl2= 10lΔkl3kl-kl+ k0kk-kl2dk,
μlp=10lΔklp+1kl-kl+ k0kk-klpdk, ×0<|μlp|<1,  p.
k-klxt+ϕk-ϕkl=k-klxt+d0l+OΔkl2,
expjk-klxt+d0l+OΔkl2=1+jk-kld0l+OΔkl2expjk-klxt.
W˜lΔkl, ϕl, xt=1+d0lxt+OΔkl2×-+ Vˆ0lkexpjk-klxtdk.
W˜lΔkl, xt=-+Vˆ0lkexpjk-klxtdk.
W˜lΔkl, ϕl, xt=W˜lΔkl, xt+d0lWlΔkl, xt+OΔkl2,
W˜lΔkl, xt=lΔkl, xtexpjΩlΔkl, xtl expjΩl.
W˜lΔkl, ϕl, xt=l expjΩl+d0ll+jlΩlexpjΩl+OΔkl2.
ReW˜lΔkl, ϕl, xi=l cos Ωl+d0ll cos Ωl-lΩl sin Ωl+OΔkl2,
ImW˜lΔkl, ϕl, xi=l sin Ωl+d0ll sin Ωl+lΩl cos Ωl+OΔkl2.
Ak=ijN1σi2σj2σk2sincΔkxi2sincΔkxj2sink¯xi-xjsincΔkxi2sincΔkxj2sink¯xi-xj+sincΔkxj2sincΔkxk2sink¯xj-xk+sincΔkxk2sincΔkxi2sink¯xk-xi,
k=ijN1σi2σj2σk2sincΔkxi2cos k¯xi-sincΔkxj2cos k¯xjsincΔkxi2sincΔkxj2sink¯xi-xj+sincΔkxj2sincΔkxk2sink¯xj-xk+sincΔkxk2sincΔkxi2sink¯xk-xi,
Ck=-ijN1σi2σj2σk2sincΔkxi2sin k¯xi-sincΔkxj2sin k¯xjsincΔkxi2sincΔkxj2sink¯xi-xj+sincΔkxj2sincΔkxk2sink¯xj-xk+sincΔkxk2sincΔkxi2sink¯xk-xi,
D=kN Ak=13ijkN1σi2σj2σk2sincΔkxi2sincΔkxj2sink¯xi-xj+sincΔkxj2sincΔkxk2sink¯xj-xk+sincΔkxk2sincΔkxi2sink¯xk-xi2,
Ak=ijN1σi2σj2σk2sinkxi-xjsinkxi-xj+sinkxj-xk+sinkxk-xi,
k=ijN1σi2σj2σk2cos kxi-cos kxjsinkxi-xj+sinkxj-xk+sinkxk-xi,
Ck=-ijN1σi2σj2σk2sin kxi-sin kxjsinkxi-xj+sinkxj-xk+sinkxk-xi,
D=kN Ak=13ijkN1σi2σj2σk2sinkxi-xj+sinkxj-xk+sinkxk-xi2.
0V cos ϕ0=N¯1σ22+2σ32+σ42-N¯32σ12+σ22+σ42+N¯2+N¯4σ12-σ322σ12+σ22+σ32+σ42, 0V sin ϕ0=N¯4σ12+2σ22+σ32-N¯2σ12+σ32+2σ42+N¯1+N¯3σ42-σ222σ12+σ22+σ32+σ42, 0=N¯1+N¯3σ22+σ42+N¯2+N¯4σ12+σ322σ12+σ22+σ32+σ42.
V2=N¯1σ22+2σ32+σ42-N¯32σ12+σ22+σ42+N¯2+N¯4σ12-σ322+N¯4σ12+2σ22+σ32-N¯2σ12+σ32+2σ42+N¯1+N¯3σ42-σ222N¯1+N¯3σ22+σ42+N¯2+N¯4σ12+σ322, ϕ0=arctanN¯4σ12+2σ22+σ32-N¯2σ12+σ32+2σ42+N¯1+N¯3σ42-σ22N¯1σ22+2σ32+σ42-N¯32σ12+σ22+σ42+N¯2+N¯4σ12-σ32.

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