Abstract

We show that three-dimensional incoherent primary sources can be reconstructed from finite-aperture Fresnel-zone mutual intensity measurements by means of coordinate and Fourier transformation. The spatial bandpass and impulse response for three-dimensional imaging that result from use of this approach are derived. The transverse and longitudinal resolutions are evaluated as functions of aperture size and source distance. The longitudinal resolution of three-dimensional coherence imaging falls inversely with the square of the source distance in both the Fresnel and Fraunhofer zones. We experimentally measure the three-dimensional point-spread function by using a rotational shear interferometer.

© 1999 Optical Society of America

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    [CrossRef]
  15. M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Ardent, G. R. Gindi, “Three-dimensional radiographic imaging with a restricted view angle,” J. Opt. Soc. Am. 69, 1323–1333 (1979).
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  16. D. H. DeVorkin, “Michelson and the problem of stellar diameters,” J. Hist. Astron. 6, 1–18 (1975).
  17. J. D. Armitage, A. Lohmann, “Rotary shearing interferometry,” Opt. Acta 12, 185–192 (1965).
    [CrossRef]
  18. C. Roddier, F. Roddier, “Imaging with a coherence interferometer in optical astronomy,” in Image Formation from Coherence Functions in Astronomy, C. V. Schooneveld, ed., Vol. 76 of International Astronomical Union Colloquium 49 (Reidel, Dordrecht, The Netherlands, 1979), pp. 175–179.
    [CrossRef]
  19. K. Itoh, Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A 3, 94–100 (1986).
    [CrossRef]
  20. K. Itoh, T. Inoue, T. Yoshida, Y. Ichioka, “Interferometric supermultispectral imaging,” Appl. Opt. 29, 1625–1630 (1990).
    [CrossRef] [PubMed]
  21. K. Itoh, T. Inoue, Y. Ichioka, “Interferometric spectral imaging and optical three-dimensional Fourier transformation,” J. J. Appl. Phys. 29, L1561–L1564 (1990).
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    [CrossRef]
  23. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  24. J. J. Knab, “Interpolation of band-limited functions using the approximate prolate series,” IEEE Trans. Inf. Theory IT-25, 717–720 (1979).
    [CrossRef]

1996

1995

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

1993

A. M. Zarubin, “Three-dimensional generalization of the van Cittert–Zernike theorem to wave and particle scattering,” Opt. Commun. 100, 491–507 (1993).
[CrossRef]

1990

K. Itoh, T. Inoue, Y. Ichioka, “Interferometric spectral imaging and optical three-dimensional Fourier transformation,” J. J. Appl. Phys. 29, L1561–L1564 (1990).

K. Itoh, T. Inoue, T. Yoshida, Y. Ichioka, “Interferometric supermultispectral imaging,” Appl. Opt. 29, 1625–1630 (1990).
[CrossRef] [PubMed]

1988

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 97–166 (1988).
[CrossRef]

1986

1985

1981

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

1979

A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979).
[CrossRef]

J. J. Knab, “Interpolation of band-limited functions using the approximate prolate series,” IEEE Trans. Inf. Theory IT-25, 717–720 (1979).
[CrossRef]

M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Ardent, G. R. Gindi, “Three-dimensional radiographic imaging with a restricted view angle,” J. Opt. Soc. Am. 69, 1323–1333 (1979).
[CrossRef]

1978

A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).

1975

D. H. DeVorkin, “Michelson and the problem of stellar diameters,” J. Hist. Astron. 6, 1–18 (1975).

1967

1965

J. D. Armitage, A. Lohmann, “Rotary shearing interferometry,” Opt. Acta 12, 185–192 (1965).
[CrossRef]

Ardent, J. W.

Armitage, J. D.

J. D. Armitage, A. Lohmann, “Rotary shearing interferometry,” Opt. Acta 12, 185–192 (1965).
[CrossRef]

Armstrong, J. T.

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

Barrett, H. H.

Carter, W. H.

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

Chiu, M. Y.

Chou, C.

Devaney, A. J.

A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979).
[CrossRef]

DeVorkin, D. H.

D. H. DeVorkin, “Michelson and the problem of stellar diameters,” J. Hist. Astron. 6, 1–18 (1975).

Frieden, B. R.

Gindi, G. R.

Hutter, D. J.

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

Ichioka, Y.

K. Itoh, T. Inoue, Y. Ichioka, “Interferometric spectral imaging and optical three-dimensional Fourier transformation,” J. J. Appl. Phys. 29, L1561–L1564 (1990).

K. Itoh, T. Inoue, T. Yoshida, Y. Ichioka, “Interferometric supermultispectral imaging,” Appl. Opt. 29, 1625–1630 (1990).
[CrossRef] [PubMed]

Inoue, T.

K. Itoh, T. Inoue, T. Yoshida, Y. Ichioka, “Interferometric supermultispectral imaging,” Appl. Opt. 29, 1625–1630 (1990).
[CrossRef] [PubMed]

K. Itoh, T. Inoue, Y. Ichioka, “Interferometric spectral imaging and optical three-dimensional Fourier transformation,” J. J. Appl. Phys. 29, L1561–L1564 (1990).

Itoh, K.

K. Itoh, T. Inoue, Y. Ichioka, “Interferometric spectral imaging and optical three-dimensional Fourier transformation,” J. J. Appl. Phys. 29, L1561–L1564 (1990).

K. Itoh, T. Inoue, T. Yoshida, Y. Ichioka, “Interferometric supermultispectral imaging,” Appl. Opt. 29, 1625–1630 (1990).
[CrossRef] [PubMed]

K. Itoh, Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A 3, 94–100 (1986).
[CrossRef]

K. Itoh, “Interferometric multispectral imaging,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1996), Vol. 35, pp. 145–196.
[CrossRef]

Johnston, K. J.

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

Knab, J. J.

J. J. Knab, “Interpolation of band-limited functions using the approximate prolate series,” IEEE Trans. Inf. Theory IT-25, 717–720 (1979).
[CrossRef]

LaHaie, I. J.

Lohmann, A.

J. D. Armitage, A. Lohmann, “Rotary shearing interferometry,” Opt. Acta 12, 185–192 (1965).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

Mertz, L.

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

Mozurkewich, D.

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

Ohtsuka, Y.

Roddier, C.

C. Roddier, F. Roddier, “Imaging with a coherence interferometer in optical astronomy,” in Image Formation from Coherence Functions in Astronomy, C. V. Schooneveld, ed., Vol. 76 of International Astronomical Union Colloquium 49 (Reidel, Dordrecht, The Netherlands, 1979), pp. 175–179.
[CrossRef]

Roddier, F.

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 97–166 (1988).
[CrossRef]

C. Roddier, F. Roddier, “Imaging with a coherence interferometer in optical astronomy,” in Image Formation from Coherence Functions in Astronomy, C. V. Schooneveld, ed., Vol. 76 of International Astronomical Union Colloquium 49 (Reidel, Dordrecht, The Netherlands, 1979), pp. 175–179.
[CrossRef]

Rosen, J.

Simpson, R. G.

Swenson, G. W.

Wolf, E.

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

Yariv, A.

Yoshida, T.

Zarubin, A. M.

A. M. Zarubin, “Three-dimensional generalization of the van Cittert–Zernike theorem to wave and particle scattering,” Opt. Commun. 100, 491–507 (1993).
[CrossRef]

Appl. Opt.

IEEE Trans. Inf. Theory

J. J. Knab, “Interpolation of band-limited functions using the approximate prolate series,” IEEE Trans. Inf. Theory IT-25, 717–720 (1979).
[CrossRef]

J. Hist. Astron.

D. H. DeVorkin, “Michelson and the problem of stellar diameters,” J. Hist. Astron. 6, 1–18 (1975).

J. J. Appl. Phys.

K. Itoh, T. Inoue, Y. Ichioka, “Interferometric spectral imaging and optical three-dimensional Fourier transformation,” J. J. Appl. Phys. 29, L1561–L1564 (1990).

J. Math. Phys.

A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

J. D. Armitage, A. Lohmann, “Rotary shearing interferometry,” Opt. Acta 12, 185–192 (1965).
[CrossRef]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

Opt. Commun.

A. M. Zarubin, “Three-dimensional generalization of the van Cittert–Zernike theorem to wave and particle scattering,” Opt. Commun. 100, 491–507 (1993).
[CrossRef]

Opt. Lett.

Optik

A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).

Phys. Rep.

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 97–166 (1988).
[CrossRef]

Phys. Today

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

Other

C. V. Schooneveld, ed., Image Formation from Coherence Functions in Astronomy, Vol. 76 of International Astronomical Union Colloquium 49 (Reidel, Dordrecht, The Netherlands, 1978).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

C. Roddier, F. Roddier, “Imaging with a coherence interferometer in optical astronomy,” in Image Formation from Coherence Functions in Astronomy, C. V. Schooneveld, ed., Vol. 76 of International Astronomical Union Colloquium 49 (Reidel, Dordrecht, The Netherlands, 1979), pp. 175–179.
[CrossRef]

K. Itoh, “Interferometric multispectral imaging,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1996), Vol. 35, pp. 145–196.
[CrossRef]

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

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Figures (13)

Fig. 1
Fig. 1

Measurement geometry for coherence imaging. An incoherent primary source distribution in the source volume is imaged by use of two-point correlation measurements drawn from the correlation plane. The separation between the volumes is greater than the extent of either volume. The correlation point coordinates are (x 1, y 1) for the first point and (x 2, y 2) for the second point. The correlation points are scanned throughout the correlation plane to yield the mutual intensity. r1, r2, vectors from the source volume origin to (x 1, y 1) and (x 2, y 2); ŝ 1, ŝ 2, unit vectors parallel to r1 and r2; rs, position vector in the source volume.

Fig. 2
Fig. 2

Coordinate geometry in the sampling plane. Source inversion is simplified by use of the transformed coordinates (Δx, Δy) = (x 1 - x 2, y 1 - y 2), (, ŷ) = [(x 1 + x 2)/2, (y 1 + y 2)/2], q = Δxx̂ + Δ.

Fig. 3
Fig. 3

Relationship between the Cartesian source coordinates and the projective coordinates. The origin of longitudinal coordinate z sp is in the correlation plane. Longitudinal projective coordinate z′ = 1/z sp has an origin at z sp = ∞ and is equal to 1/R at the center of the source volume. In the small-angle approximation, the transverse projective coordinates x′ = zx s and y′ = zy s correspond to the angles θ x and θ y between the y = 0 and x = 0 planes and the ray from the correlation plane origin to the real-space source point.

Fig. 4
Fig. 4

Geometry of the MSI correlation plane: r max, radius of the system aperture; d, sampling-point separation; , distance between the midpoint of the sampling points and the origin; ϕ, angle between the interferometer beam and the x axis.

Fig. 5
Fig. 5

Band volume for MSI sampling. The band volume is plotted in the real-space Fourier space of the source density for the Fraunhofer zone and in the projective-space Fourier space for the Fresnel zone. The coordinate axes correspond to the Fourier coordinates (u x , u y , u z ). The transverse coordinates are normalized with respect to r max0 R. The longitudinal coordinate is normalized with respect to r max 20 R 2. Because r max/R ≪ 1, the normalization frequency for the longitudinal axis is less than it is for the transverse axes. The missing cone in the Fourier space along the u z axis is characteristic of limited-angle tomographic systems.

Fig. 6
Fig. 6

Basic structure of a rotational shear interferometer. The RSI is a Michelson interferometer in which the plane retroreflection mirrors have been replaced with folding mirrors. The folding axes of the mirrors lie in the transverse plane at angles ϕ and -ϕ with respect to the x axis. The output port interferes differentially rotated wave fronts from the two mirrors.

Fig. 7
Fig. 7

Band volume for linear translation RSI sampling. The situation is identical to that of Fig. 4, except that the normalization of the u z axis is now x g max r max0 R 2. x g max is the linear displacement range for the RSI. For the RSI of this figure, ϕ = π/4 and the fold axes of the two mirrors are perpendicular. In this geometry, the RSI is also called a wave-front folding interferometer.

Fig. 8
Fig. 8

Band volume for imaging with a circularly translated RSI. The situation is identical to that of Fig. 6, except that x g max now represents the radius of the circle about which the optical axis of the RSI is translated.

Fig. 9
Fig. 9

Surface plot of the 3D MSI impulse response in the x′–z′ plane. The vertical axis is normalized to the maximum response. The spatial axes are in projective coordinates, with units of inverse meters for the longitudinal axis and radians for the transverse axis. The impulse response is approximately shift invariant in the projective space; it is not shift invariant in real space. To obtain the real-space impulse response one adds 1/R to the longitudinal range and takes the inverse. For an impulse at 1 m, a point at z′ = 0.1 is at z = 1/(1 + 0.1) = 0.91. A point at z′ = -0.1 is at z = 1/(1 - 0.1) = 1.11.

Fig. 10
Fig. 10

Cross section of the RSI impulse response in the x′–z′ plane under the same constraints as for Fig. 9.

Fig. 11
Fig. 11

RSI used to measure the mutual coherence of the four-LED test object and RSI impulse response. The RSI consisted of (a) a 5 cm × 5 cm × 5 cm cube beam splitter with (b) two 5 cm × 5 cm folding mirrors. Each folding mirror was constructed from two separate mirrors affixed to each other at a 90-deg angle, giving a full 5 cm × 5 cm square aperture. A Princeton Instruments 512 × 512 backilluminated CCD was used as the focal-plane array. For longitudinal delay, one of the folding mirrors was placed upon a piezoelectric-driven flexure stage in conjunction with an inductive positioning sensor. The RSI was mounted upon two linear bearings and was translated over a 5-cm length by an Aerotech translation stage.

Fig. 12
Fig. 12

Experimental cross section of the RSI impulse response in the x′–z′ plane. The four corners of the plane and the peak are labeled with their Cartesian coordinates in real space, in meters, relative to the origin of the focal-plane array. This impulse response was sampled by a linearly translated RSI by use of the procedure and the experimental parameters described in the text.

Fig. 13
Fig. 13

Experimental reconstruction of a four-light-emitting-diode test source, as sampled by the RSI. The 50% power density isosurface is shown. The LED’s appear to be of different sizes because they were in fact of different intensities. The source is shown in projective coordinates, but the corners are labeled in Cartesian coordinates, in meters, relative to the origin of the focal-plane array. These data were taken by a RSI by use of the procedure and the experimental parameters described in the text.

Equations (19)

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

Jr1, r2=k02π2σ Irs×expjk0|r1-rs|-|r2-rs||r1-rsr2-rs|d3rs,
Jr1sˆ1, r2sˆ2=I˜sˆ1-sˆ2λ0expjk0r1-r2λ02r1r2,
sˆ1x1Riˆx+y1Riˆy+1-x12+y122R2iˆz,
u=sˆ1-sˆ2λ0Δxλ0Riˆx+Δyλ0Riˆy-xˆΔx+yˆΔyλ0R2iˆz,
J3DΔx, Δy, q=1λ02R2 I˜ux=Δxλ0R, uy=Δyλ0R, uz=-qλ0R2.
Irs * Pρrs=λR2 ρ J3DΔx, Δy, q×expj2πxsΔxλ0R+j2πysΔyλ0R-j2πzsqλ0R2dΔxdΔydq,
|r1-rs|zsp+x1-xs22zsp+y1-ys22zsp+.
Jr1, r2=σIrspλ2zsp2 exp-j2πλ0zspxsΔx+ysΔy+j2πλ0zspxˆΔx+yˆΔyd3rsp,
x=xszsp,  y=yszsp,  z=1zsp
dx=zdx-xdzz2,  dy=zdy-ydzz2,  dz=-dzz2.
dx=zdx,  dy=zdy,  dz=z2dz.
J3DΔx, Δy, q=σIx, y, zλ02z2×exp-j2πλ0xΔx+yΔy+j2πλ0 zqd3r,
J3DΔx, Δy, q=1λ2 I˜pΔxλ0, Δyλ0, qλ0,
Ix, y, zz2 * Pρr=λρ J3DΔx, Δy, q×expj2πλ0xΔx+yΔy-j2πλ0 zqdΔxdΔydq.
Δx=d cosϕ,  Δy=d sinϕ,  q=dˆd.
Δxxf, yf=2yf sin2θ,  Δyxf, yf=2xf sin2θ,
xˆxf, yf, xg=2xf cos2θ+xg,  yˆxf, yf, yg=-2yf cos2θ+yg,
q=Δxxf, yfxˆxf, yf, xg+Δyxf, yfyˆxf, yf, xg=yfxg-xfygsin2θ.
xres=Cxλd,  zres=Czλd2,

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