Abstract

A technique for retrieving the cross-spectral density propagating in free space across an arbitrary plane is proposed. Theoretical descriptions are based on the propagation law of the cross-spectral density described in the Fourier domain. Results of an experiment demonstrating retrieval of the cross-spectral density across an arbitrary reference plane are also reported. The cross-spectral density in the observation area is measured by use of a wave-front-folded interferometer, and the retrieved cross-spectral densities and the intensity distribution agree well with the theoretical expectations.

© 1999 Optical Society of America

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References

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  1. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  2. A. A. Michelson, Studies in Optics (University of Chicago Press, Chicago, 1962).
  3. F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 92–166 (1988).
    [CrossRef]
  4. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  5. D. F. V. James, H. C. Kandpal, E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J. 445, 406–410 (1995).
    [CrossRef]
  6. H. C. Kandpal, K. Saxena, D. S. Mehta, J. S. Vaishya, K. C. Joshi, “Angular separation of a pair of sources by spectral interferometric technique,” J. Mod. Opt. 42, 447–454 (1995).
    [CrossRef]
  7. H. Arimoto, Y. Ohtsuka, “Determination of the angular separation and the spectra of two-point sources from spectral coherence measurements,” Opt. Lett. 23, 724–726 (1998).
    [CrossRef]
  8. J. Rosen, A. Yariv, “General theorem of spatial coherence: application to three-dimensional imaging,” J. Opt. Soc. Am. A 13, 2091–2095 (1996).
    [CrossRef]
  9. J. Rosen, A. Yariv, “Reconstruction of longitudinal distributed incoherent sources,” Opt. Lett. 21, 1803–1805 (1996).
    [CrossRef] [PubMed]
  10. R. G. Littlejohn, R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).
    [CrossRef]
  11. K. Yoshimori, K. Itoh, “Interferometry and radiometry,” J. Opt. Soc. Am. A 14, 3379–3387 (1997).
    [CrossRef]
  12. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

1998 (1)

1997 (1)

1996 (2)

1995 (2)

D. F. V. James, H. C. Kandpal, E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J. 445, 406–410 (1995).
[CrossRef]

H. C. Kandpal, K. Saxena, D. S. Mehta, J. S. Vaishya, K. C. Joshi, “Angular separation of a pair of sources by spectral interferometric technique,” J. Mod. Opt. 42, 447–454 (1995).
[CrossRef]

1993 (1)

1988 (1)

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 92–166 (1988).
[CrossRef]

Arimoto, H.

Goodman, J. W.

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

Itoh, K.

James, D. F. V.

D. F. V. James, H. C. Kandpal, E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J. 445, 406–410 (1995).
[CrossRef]

Joshi, K. C.

H. C. Kandpal, K. Saxena, D. S. Mehta, J. S. Vaishya, K. C. Joshi, “Angular separation of a pair of sources by spectral interferometric technique,” J. Mod. Opt. 42, 447–454 (1995).
[CrossRef]

Kandpal, H. C.

D. F. V. James, H. C. Kandpal, E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J. 445, 406–410 (1995).
[CrossRef]

H. C. Kandpal, K. Saxena, D. S. Mehta, J. S. Vaishya, K. C. Joshi, “Angular separation of a pair of sources by spectral interferometric technique,” J. Mod. Opt. 42, 447–454 (1995).
[CrossRef]

Littlejohn, R. G.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

Mehta, D. S.

H. C. Kandpal, K. Saxena, D. S. Mehta, J. S. Vaishya, K. C. Joshi, “Angular separation of a pair of sources by spectral interferometric technique,” J. Mod. Opt. 42, 447–454 (1995).
[CrossRef]

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Michelson, A. A.

A. A. Michelson, Studies in Optics (University of Chicago Press, Chicago, 1962).

Ohtsuka, Y.

Roddier, F.

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 92–166 (1988).
[CrossRef]

Rosen, J.

Saxena, K.

H. C. Kandpal, K. Saxena, D. S. Mehta, J. S. Vaishya, K. C. Joshi, “Angular separation of a pair of sources by spectral interferometric technique,” J. Mod. Opt. 42, 447–454 (1995).
[CrossRef]

Vaishya, J. S.

H. C. Kandpal, K. Saxena, D. S. Mehta, J. S. Vaishya, K. C. Joshi, “Angular separation of a pair of sources by spectral interferometric technique,” J. Mod. Opt. 42, 447–454 (1995).
[CrossRef]

Winston, R.

Wolf, E.

D. F. V. James, H. C. Kandpal, E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J. 445, 406–410 (1995).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

Yariv, A.

Yoshimori, K.

Astrophys. J. (1)

D. F. V. James, H. C. Kandpal, E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J. 445, 406–410 (1995).
[CrossRef]

J. Mod. Opt. (1)

H. C. Kandpal, K. Saxena, D. S. Mehta, J. S. Vaishya, K. C. Joshi, “Angular separation of a pair of sources by spectral interferometric technique,” J. Mod. Opt. 42, 447–454 (1995).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Lett. (2)

Phys. Rep. (1)

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 92–166 (1988).
[CrossRef]

Other (4)

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

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

A. A. Michelson, Studies in Optics (University of Chicago Press, Chicago, 1962).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the optical system for the general theorem.

Fig. 2
Fig. 2

Geometry of the optical system assumed for the numerical simulations.

Fig. 3
Fig. 3

Calculated cross-spectral density across reference planes: (a) real and (b) imaginary parts at Σr, (c) real and (d) imaginary parts at Σo.

Fig. 4
Fig. 4

Cross sections of the real part of the cross-spectral densities. (a) and (b) correspond to Figs. 3(a) and 3(c).

Fig. 5
Fig. 5

Schematic of the wave-front-folded interferometer for measuring the cross-spectral density.

Fig. 6
Fig. 6

(a) Real and (b) imaginary parts of the measured cross-spectral density. (c), (e), and (g) are the real parts and (d), (f), and (h) are the imaginary parts of retrieved W(zr)(x, x; v) for each zr.

Fig. 7
Fig. 7

(a) Cross section of |W(zr)(x, x; v)| across the source plane (a) along Δx=0 and (b) along xo=0.

Equations (20)

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(2+k2)U(r; ν)=0,
U(r,zr; ν)=12π -U˜(k, zr; ν)×exp(ikr)d2k,
U˜(k, zr; ν)=12π -U(r, zr; ν)×exp(-ikr)d2r,
2zr2+kz2U˜(k, zr; ν)=0,
kz=(k2-k2)1/2(kk)i(k2-k2)1/2(kk).
U˜(k, 0; ν)=exp(ikzzr)U˜(k, zr; ν).
W(zr)(r, r; ν)=U*(r, zr; ν)U(r, zr; ν),
W˜(zr)(k, k; ν)
=12π2-W(zr)(r, r; ν)
×exp[-i(kr-kr)]d2rd2r.
W˜(zr)(k, k; ν)=U˜*(k, zr; ν)U˜(k, zr; ν).
W˜(o)(k, k; ν)=exp[i(kz-kz)zr]W˜(zr)(k, k; ν).
W˜(zr)(k,k; ν)=exp[-i(kz-kz)zr]W˜(o)(k,k; ν).
W(zr)(r, r; ν)
=-exp[-i(kz-kz)zr+i(kr-kr)]W˜(o)(k, k; ν)d2kd2k.
S(r, zr; ν)=W(zr)(r, r; ν).
S(zs)(x; ν)=s(zs)(ν)rect(x/D),
W(zs)(x, x; ν)=[S(zs)(x; ν)S(zs)(x; ν)]1/2δ(x-x).
W(zr)(x, x; ν)
=-exp[i(kz-kz)(zs-zr)+i(kxx-kxx)]W˜(zs)(kx, kx; ν)dkxdkx,

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