Abstract

The effects of finite telescope pupil sizes on the measurement of fringe visibility in fiber optic stellar interferometry are described. It is shown theoretically that the measured fringe visibility is equal to the cross correlation of the magnitude of the source’s mutual coherence function with the cross correlation of the telescopes’ effective pupil functions. If the telescopes’ effective pupil diameters are not small compared with the width of the source’s mutual coherence function, then the measured fringe visibility will be correspondingly distorted. The theoretical results are verified experimentally in a fiber optic Mach–Zehnder interferometer.

© 1999 Optical Society of America

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References

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  1. J. T. Armstrong, D. J. Hutter, K. J. Johnston, D. Mozurkewich, “Stellar interferometry in the 1990s,” Phys. Today 48(5), 42–49 (1995).
    [CrossRef]
  2. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  3. C. Froehly, “Coherence and interferometry through optical fibers,” in Proceedings of the ESO Conference on Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, M. H. Ulrich, K. Kjär, eds. (European Southern Observatory, Garching, Germany, 1981), pp. 24–27.
  4. S. B. Shaklan, F. Roddier, “Single-mode fiber optics in a long-baseline interferometer,” Appl. Opt. 26, 2159–2163 (1987).
    [CrossRef] [PubMed]
  5. S. B. Shaklan, “Multiple beam correlation using single-mode fiber optics with application to interferometric imaging,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1989).
  6. V. Coudé du Foresto, “Integrated optics in astronomical interferometry,” in Very High Angular Resolution Imaging, J. G. Robertson, W. J. Tango, eds. (Kluwer Academic, Boston, Mass., 1994), pp. 261–271.
  7. J. J. Alleman, F. Reynaud, P. Connes, “Fiber-linked telescope array: description and laboratory tests of a two-channel prototype,” Appl. Opt. 34, 2284–2294 (1995).
    [CrossRef] [PubMed]
  8. R.-R. Rohloff, Ch. Leinert, “Properties of fiber optics for application in astronomical interferometry,” Appl. Opt. 30, 5031–5036 (1991).
    [CrossRef] [PubMed]
  9. S. Shaklan, F. Roddier, “Coupling starlight into single-mode fiber optics,” Appl. Opt. 27, 2334–2338 (1988).
    [CrossRef] [PubMed]
  10. S. T. Ridgway, “The scientific support for a space interferometry mission,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE1947, 2–11 (1993).
    [CrossRef]
  11. J. Yu, S. Shaklan, M. Shao, “High-precision astrometry of crowded fields by interferometry,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE1947, 2–11 (1993).
  12. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
    [CrossRef]
  13. R. E. Wagner, W. J. Tomlinson, “Coupling efficiency of optics in single-mode fiber components,” Appl. Opt. 21, 2671–2688 (1982).
    [CrossRef] [PubMed]
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  15. D. C. O’Shea, W. R. Callen, W. T. Rhodes, Introduction to Lasers and Their Applications (Addison-Wesley, Reading, Mass., 1978).
  16. P. Zhao, J.-M. Mariotti, P. Léna, V. Coudé du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
    [CrossRef]
  17. D. A. Christensen, J. Rotge, A. Klemas, G. Loos, D. Merriman, “Laser diode coherence length variation for balancing fiber optic interferometers,” Opt. Eng. 33, 2034–2038 (1994).
    [CrossRef]

1995 (2)

J. J. Alleman, F. Reynaud, P. Connes, “Fiber-linked telescope array: description and laboratory tests of a two-channel prototype,” Appl. Opt. 34, 2284–2294 (1995).
[CrossRef] [PubMed]

J. T. Armstrong, D. J. Hutter, K. J. Johnston, D. Mozurkewich, “Stellar interferometry in the 1990s,” Phys. Today 48(5), 42–49 (1995).
[CrossRef]

1994 (2)

P. Zhao, J.-M. Mariotti, P. Léna, V. Coudé du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

D. A. Christensen, J. Rotge, A. Klemas, G. Loos, D. Merriman, “Laser diode coherence length variation for balancing fiber optic interferometers,” Opt. Eng. 33, 2034–2038 (1994).
[CrossRef]

1991 (1)

1988 (1)

1987 (1)

1982 (1)

1977 (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Alleman, J. J.

Armstrong, J. T.

J. T. Armstrong, D. J. Hutter, K. J. Johnston, D. Mozurkewich, “Stellar interferometry in the 1990s,” Phys. Today 48(5), 42–49 (1995).
[CrossRef]

Callen, W. R.

D. C. O’Shea, W. R. Callen, W. T. Rhodes, Introduction to Lasers and Their Applications (Addison-Wesley, Reading, Mass., 1978).

Christensen, D. A.

D. A. Christensen, J. Rotge, A. Klemas, G. Loos, D. Merriman, “Laser diode coherence length variation for balancing fiber optic interferometers,” Opt. Eng. 33, 2034–2038 (1994).
[CrossRef]

Connes, P.

Coudé du Foresto, V.

P. Zhao, J.-M. Mariotti, P. Léna, V. Coudé du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

V. Coudé du Foresto, “Integrated optics in astronomical interferometry,” in Very High Angular Resolution Imaging, J. G. Robertson, W. J. Tango, eds. (Kluwer Academic, Boston, Mass., 1994), pp. 261–271.

Froehly, C.

C. Froehly, “Coherence and interferometry through optical fibers,” in Proceedings of the ESO Conference on Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, M. H. Ulrich, K. Kjär, eds. (European Southern Observatory, Garching, Germany, 1981), pp. 24–27.

Goodman, J. W.

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hutter, D. J.

J. T. Armstrong, D. J. Hutter, K. J. Johnston, D. Mozurkewich, “Stellar interferometry in the 1990s,” Phys. Today 48(5), 42–49 (1995).
[CrossRef]

Johnston, K. J.

J. T. Armstrong, D. J. Hutter, K. J. Johnston, D. Mozurkewich, “Stellar interferometry in the 1990s,” Phys. Today 48(5), 42–49 (1995).
[CrossRef]

Klemas, A.

D. A. Christensen, J. Rotge, A. Klemas, G. Loos, D. Merriman, “Laser diode coherence length variation for balancing fiber optic interferometers,” Opt. Eng. 33, 2034–2038 (1994).
[CrossRef]

Leinert, Ch.

Léna, P.

P. Zhao, J.-M. Mariotti, P. Léna, V. Coudé du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

Loos, G.

D. A. Christensen, J. Rotge, A. Klemas, G. Loos, D. Merriman, “Laser diode coherence length variation for balancing fiber optic interferometers,” Opt. Eng. 33, 2034–2038 (1994).
[CrossRef]

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Mariotti, J.-M.

P. Zhao, J.-M. Mariotti, P. Léna, V. Coudé du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

Merriman, D.

D. A. Christensen, J. Rotge, A. Klemas, G. Loos, D. Merriman, “Laser diode coherence length variation for balancing fiber optic interferometers,” Opt. Eng. 33, 2034–2038 (1994).
[CrossRef]

Mozurkewich, D.

J. T. Armstrong, D. J. Hutter, K. J. Johnston, D. Mozurkewich, “Stellar interferometry in the 1990s,” Phys. Today 48(5), 42–49 (1995).
[CrossRef]

O’Shea, D. C.

D. C. O’Shea, W. R. Callen, W. T. Rhodes, Introduction to Lasers and Their Applications (Addison-Wesley, Reading, Mass., 1978).

Reynaud, F.

Rhodes, W. T.

D. C. O’Shea, W. R. Callen, W. T. Rhodes, Introduction to Lasers and Their Applications (Addison-Wesley, Reading, Mass., 1978).

Ridgway, S. T.

S. T. Ridgway, “The scientific support for a space interferometry mission,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE1947, 2–11 (1993).
[CrossRef]

Roddier, F.

Rohloff, R.-R.

Rotge, J.

D. A. Christensen, J. Rotge, A. Klemas, G. Loos, D. Merriman, “Laser diode coherence length variation for balancing fiber optic interferometers,” Opt. Eng. 33, 2034–2038 (1994).
[CrossRef]

Shaklan, S.

S. Shaklan, F. Roddier, “Coupling starlight into single-mode fiber optics,” Appl. Opt. 27, 2334–2338 (1988).
[CrossRef] [PubMed]

J. Yu, S. Shaklan, M. Shao, “High-precision astrometry of crowded fields by interferometry,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE1947, 2–11 (1993).

Shaklan, S. B.

S. B. Shaklan, F. Roddier, “Single-mode fiber optics in a long-baseline interferometer,” Appl. Opt. 26, 2159–2163 (1987).
[CrossRef] [PubMed]

S. B. Shaklan, “Multiple beam correlation using single-mode fiber optics with application to interferometric imaging,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1989).

Shao, M.

J. Yu, S. Shaklan, M. Shao, “High-precision astrometry of crowded fields by interferometry,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE1947, 2–11 (1993).

Tomlinson, W. J.

Wagner, R. E.

Yu, J.

J. Yu, S. Shaklan, M. Shao, “High-precision astrometry of crowded fields by interferometry,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE1947, 2–11 (1993).

Zhao, P.

P. Zhao, J.-M. Mariotti, P. Léna, V. Coudé du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

Zhou, B.

P. Zhao, J.-M. Mariotti, P. Léna, V. Coudé du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

Appl. Opt. (5)

Bell Syst. Tech. J. (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Opt. Commun. (1)

P. Zhao, J.-M. Mariotti, P. Léna, V. Coudé du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

Opt. Eng. (1)

D. A. Christensen, J. Rotge, A. Klemas, G. Loos, D. Merriman, “Laser diode coherence length variation for balancing fiber optic interferometers,” Opt. Eng. 33, 2034–2038 (1994).
[CrossRef]

Phys. Today (1)

J. T. Armstrong, D. J. Hutter, K. J. Johnston, D. Mozurkewich, “Stellar interferometry in the 1990s,” Phys. Today 48(5), 42–49 (1995).
[CrossRef]

Other (8)

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

C. Froehly, “Coherence and interferometry through optical fibers,” in Proceedings of the ESO Conference on Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, M. H. Ulrich, K. Kjär, eds. (European Southern Observatory, Garching, Germany, 1981), pp. 24–27.

S. B. Shaklan, “Multiple beam correlation using single-mode fiber optics with application to interferometric imaging,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1989).

V. Coudé du Foresto, “Integrated optics in astronomical interferometry,” in Very High Angular Resolution Imaging, J. G. Robertson, W. J. Tango, eds. (Kluwer Academic, Boston, Mass., 1994), pp. 261–271.

S. T. Ridgway, “The scientific support for a space interferometry mission,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE1947, 2–11 (1993).
[CrossRef]

J. Yu, S. Shaklan, M. Shao, “High-precision astrometry of crowded fields by interferometry,” in Spaceborne Interferometry, R. D. Reasenberg, ed., Proc. SPIE1947, 2–11 (1993).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

D. C. O’Shea, W. R. Callen, W. T. Rhodes, Introduction to Lasers and Their Applications (Addison-Wesley, Reading, Mass., 1978).

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Figures (5)

Fig. 1
Fig. 1

Fiber coupling geometry. The telescope is represented by a lens with diameter D. The face of the fiber is placed a distance f from the lens, where f is the focal length of the lens. The vector coordinate xp is used at the lens, and xf is used at the fiber face.

Fig. 2
Fig. 2

Fiber optic stellar interferometer. Two apertures (representing the telescopes) are placed a distance z from the source. The light from the apertures is coupled to two SM fibers, and the light output from the two fibers is recombined by a beam splitter and directed onto a detector: BS, beam splitter; L, lens.

Fig. 3
Fig. 3

Diagram of the pupil plane. The vectors x1 and x2 represent the distance from the plane’s origin to the centers of the pupils. The vectors x and x are the distances from the centers of the pupils to given points on the pupils.

Fig. 4
Fig. 4

Mach–Zehnder fiber optic interferometer used to demonstrate the effects of the pupil size on the measured fringe visibility function: BS, beam splitter; D, detector; L, lens; MMF, multimode fiber; SMF, single-mode fiber; T, translation stage; LD, laser diode source.

Fig. 5
Fig. 5

Comparison of theoretical and experimental fringe visibility data for two cases: large input pupil diameter and small input pupil diameter.

Equations (24)

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uc=K-usp(xp)P(xp)u˜f(xp)dxp,
wp=λfπw0,
Peff(xp)=P(xp)u˜f(xp),
uc=K-usp(xp)Peff(xp)dxp.
ID=uD*uD,
uD=K1 exp(-jkfLf1)uc1(x1; t)+K2 exp(-jkfLf2)uc2(x2; t+τ),
ID=K12|uc1(x1; t)|2+K22|uc2(x2; t+τ)|2+2K1K2 Re{uc1(x1; t)uc2*(x2; t+τ)×exp[-jkf(Lf1-Lf2)]}.
ID=I1+I2+2K1K2 Re{Γ(x1, x2; τ)×exp[-jkf(Lf1-Lf2)]},
uc1(x1; t)=K1-usp(ψ; t)Peff 1(x)dx,
uc2(x2; t+τ)=K2-usp(η; t+τ)Peff 2(x)dx.
Γ(x1, x2; τ)=uc1(x1; t)uc2*(x2; t+τ)=--usp(ψ; t)usp*(η; t+τ)×Peff 1(x)×Peff 2*(x)dxdx.
Γ(x1, x2; τ)=--Γsp(χ; τ)Peff 1(x)Peff 2*(x)dxdx.
x=x-χ+(x2-x1).
Γ(x1, x2; τ)=-Γsp(χ; τ)-Peff 2*(x)×Peff 1[x-χ+(x2-x1)]dxdχ.
X12[χ-(x2-x1)]
=Peff 2[χ-(x2-x1)]  Peff 1[χ-(x2-x1)]=-Peff 2(x)Peff 1*[x-χ+(x2-x1)]dx,
Γ(x1, x2; τ)=-Γsp(χ; τ)X12*[χ-(x2-x1)]dχ.
Γ(x1, x2; τ)=Γsp(x2-x1; τ)  X12(x2-x1).
ID=I1+I2+2K1K2 Re{Γsp(x2-x1; τ) X12(x2-x1)exp[-jkf(Lf1-Lf2)]}.
Θ=θ12+θatm+ϕ+kf(Lf2-Lf1),
ID=I1+I2+2K1K2|Γsp(x2-x1; τ)| |X12(x2-x1)|cos(Θ).
Ii=Ki2Γsp(0)  Ai(0),
V=Imax-IminImax+Imin.
V(x2-x1)=2K1K2|Γsp(x2-x1; τ)|  |X12(x2-x1)|K12Γsp(0)  A1(0)+K22Γsp(0)  A2(0).

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