Abstract

A phase-space analysis of a rotational-shear volume interferometer is presented. It is shown that, in a fairly general condition of this interferometer and by using a Fourier transform method to retrieve spectrodirectional images, a defocus region of the source location exists near the interferometer. This indicates that the usual Fourier transform method has its focal point at an infinite distance. By focusing on finite-depth sources located in the defocus region, a new interferometric method to obtain both three-dimensional spatial information and spectral information of a stationary, quasi-homogeneous, polychromatic source distribution is developed. A key element of this method is the use of a filter function that acts on the volume interferogram of the rotational-shear volume interferometer. This filter function realizes, for a particular position of a monochromatic point source located within the defocus region, a diffraction-limited resolution of the interferometer for the spectrodirectional image.

© 2001 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. K. Yoshimori, K. Itoh, “Interferometry and radiometry,” J. Opt. Soc. Am. A 14, 3379–3387 (1997).
    [CrossRef]
  9. For a more common experimental situation in which one of the prisms travels along the optical axis while the other one fixes, the zcomponent of R(ρ) in Eq. (1) is replaced by Rz+ρz/2.This ρz dependence of R(ρ) introduces an additional defocus into the spectrodirectional image [Eq. (3)]. However, this defocus can be corrected by the filter function of the form in Eq. (20), with R(ρ) replaced by the new function.
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    [CrossRef]
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    [CrossRef]
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2000 (2)

H. Arimoto, K. Yoshimori, K. Itoh, “Interferometric three-dimensional imaging based on retrieval of generalized radiance distribution,” Opt. Rev. 7, 25–33 (2000).
[CrossRef]

K. Yoshimori, K. Itoh, “An operator method of the theory of optical coherence and radiometry,” Opt. Rev. 7, 34–43 (2000).
[CrossRef]

1999 (2)

1998 (1)

1997 (1)

1996 (1)

1995 (1)

1990 (1)

1988 (1)

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

1986 (1)

1968 (1)

1891 (1)

A. A. Michelson, “On the application of interference methods to spectroscopic measurements,” Phil. Mag. 31, 338–346 (1891).
[CrossRef]

Arimoto, H.

H. Arimoto, K. Yoshimori, K. Itoh, “Interferometric three-dimensional imaging based on retrieval of generalized radiance distribution,” Opt. Rev. 7, 25–33 (2000).
[CrossRef]

H. Arimoto, K. Yoshimori, K. Itoh, “Retrieval of the cross spectral density propagating in free space,” J. Opt. Soc. Am. A 16, 2447–2452 (1999).
[CrossRef]

Brady, David J.

Ichioka, Y.

Inoue, T.

Itoh, K.

Littlejohn, R. G.

Marks, Daniel L.

Michelson, A. A.

A. A. Michelson, “On the application of interference methods to spectroscopic measurements,” Phil. Mag. 31, 338–346 (1891).
[CrossRef]

Ohtsuka, Y.

Roddier, F.

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

Rosen, J.

Stack, Ronald A.

Walther, A.

Winston, R.

Yariv, A.

Yoshida, T.

Yoshimori, K.

K. Yoshimori, K. Itoh, “An operator method of the theory of optical coherence and radiometry,” Opt. Rev. 7, 34–43 (2000).
[CrossRef]

H. Arimoto, K. Yoshimori, K. Itoh, “Interferometric three-dimensional imaging based on retrieval of generalized radiance distribution,” Opt. Rev. 7, 25–33 (2000).
[CrossRef]

H. Arimoto, K. Yoshimori, K. Itoh, “Retrieval of the cross spectral density propagating in free space,” J. Opt. Soc. Am. A 16, 2447–2452 (1999).
[CrossRef]

K. Yoshimori, “Radiometry and coherence in a nonstationery optical field,” J. Opt. Soc. Am. A 15, 2730–2734 (1998).
[CrossRef]

K. Yoshimori, K. Itoh, “Interferometry and radiometry,” J. Opt. Soc. Am. A 14, 3379–3387 (1997).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Lett. (1)

Opt. Rev. (2)

H. Arimoto, K. Yoshimori, K. Itoh, “Interferometric three-dimensional imaging based on retrieval of generalized radiance distribution,” Opt. Rev. 7, 25–33 (2000).
[CrossRef]

K. Yoshimori, K. Itoh, “An operator method of the theory of optical coherence and radiometry,” Opt. Rev. 7, 34–43 (2000).
[CrossRef]

Phil. Mag. (1)

A. A. Michelson, “On the application of interference methods to spectroscopic measurements,” Phil. Mag. 31, 338–346 (1891).
[CrossRef]

Phys. Rep. (1)

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

Other (1)

For a more common experimental situation in which one of the prisms travels along the optical axis while the other one fixes, the zcomponent of R(ρ) in Eq. (1) is replaced by Rz+ρz/2.This ρz dependence of R(ρ) introduces an additional defocus into the spectrodirectional image [Eq. (3)]. However, this defocus can be corrected by the filter function of the form in Eq. (20), with R(ρ) replaced by the new function.

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Figures (3)

Fig. 1
Fig. 1

Schematic on the rotational-shear volume interferometer. Three-dimensional polychromatic sources are distributed about the origin of the Cartesian coordinate system.

Fig. 2
Fig. 2

The upper part of the figure shows the transverse components of the mean position R=(Rx, Ry) and separation ρ=(ρx, ρy) of a pair of points R±ρ/2, at which the split optical fields are superposed at X=(X, Y) on the observation plane. When the rotation angle θ, which is twice of that of the prisms, is smaller than π/2, R is a function of ρ and is distributed on the observation plane, as illustrated in the lower part of the figure. If θ=π/2, however, R coincides with R0 and there is no dispersion of the viewing point.

Fig. 3
Fig. 3

Impulse-response functions of the spectrodirectional image, computed for a number of source locations: (a) for distant sources, (b) for sources at the critical distance, (c) for sources in the defocus region.

Equations (34)

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R(ρ)=-ρy2 tan θ, ρx2 tan θ, Rz,
X(ρ)=-ρy2 sin θ, ρx2 sin θ, ρz2.
h(k1)=d3ρ exp(-ik1·ρ)A(ρ)×Γ[R(ρ)-ρ/2, R(ρ)+ρ/2],
W1(X)=1,if X lieswithinthevolumeinterferogram0,otherwise.
Bω(r, s)=k2(2π)3c d3ρ exp(-ik·ρ)×Γ(r-ρ/2, r+ρ/2),
Bω(R0, s)s=0.
h(k1)=d3r0dωdΩM(r, ks; k1)Bω(r, s).
M(r, k; k1)=d3ρ exp[-i(k1-k)·ρ]×A(ρ)δ3[r-R(ρ)],
Bω(r, s)=δ(ω-ω0)Δ[s-s0(r)].
h(k1|r0, k0)=d3rM [r, k0s0(r); k1].
h(k1|r0, k0)=d3ρ exp(-i{k1-k0s0[R(ρ)]}·ρ)A(ρ).
M(r, k; k1)=A˜(k1-k)δ3(r-R0),
A˜(k)=d3ρ exp(-ik·ρ)A(ρ).
h(k1|r0, k0)=A˜[k1-k0s0(R0)],
W1 (X)=Circ(X/a)Rect(Z/d).
h(κ, s|γ)=201ρdρJ0[β(1+κ)sρ]×sincβzκ-1+κ2s2+ρ22γ2,
γc=(βz/2π)1/2=(d/λ0)1/2,
h(κ, s|γ)=2J1[β(1+κ)s]β(1+κ)ssinc  βzκ-1+κ2s2ifγγc,
γ<γc.
h(k1; r0, k0)=d3ρ exp(-ik1·ρ)A(ρ)Y(ρ; r0, k0)×Γ(R(ρ)-ρ/2, R(ρ)+ρ/2).
h(k1; r0, k0)=d3r0dωdΩM(r, ks; k1; r0, k0)×Bω(r, s),
M(r, k; k1; r0, k0)=d3ρ exp[-i(k1-k)·ρ]A(ρ)×Y(ρ; r0, k0)δ3[r-R(ρ)].
h(k1; r0, k0|r0, k0)=d3rM [r,k0s0(r); k1; r0, k0]=d3ρ exp(-i{k1-k0s0[R(ρ)]}·ρ)A(ρ)Y(ρ; r0, k0).
Y(ρ; r0, k0)=exp(ik0{s0(R0)-s0[R(ρ)]}·ρ).
r|Γˆ(t)|r=Γ(r, t; r, t)=V*(r, t)V(r, t)¯,
h(k1)=d3ρ exp(-ik1·ρ)A(ρ)R(ρ)+ρ/2|Γˆ|R(ρ)-ρ/2=λd3ρ exp(-ik1·ρ)A(ρ)λ|R(ρ)-ρ/2×R(ρ)+ρ/2|Γˆ|λ=Tr[Mˆ(k1)Γˆ].
Mˆ(k1)=d3ρ exp(-ik1·ρ)×A(ρ)|R(ρ)-ρ/2R(ρ)+ρ/2|.
M(r, k, k1)=d3ρ exp(-ik·ρ)×r+ρ/2|Mˆ(k1)|r-ρ/2=d3q exp(iq·r)×k+q/2|Mˆ(k1)|k-q/2.
M(r, k; k1)=d3ρ exp[-i(k1-k)·ρ]×A(ρ)δ3[r-R(ρ)],
k|r=exp(-ik·r)/[(2π)3]1/2.
K(r, k)=d3ρ exp(-ik·ρ)r+ρ/2|Γˆ|r-ρ/2=d3q exp(iq·r)k+q/2|Γˆ|k-q/2.
h(k1)=  d3r d3k(2π)3M(r, k; k1)K(r, k),
Bω(r, s)=k2(2π)3cK(r, k),
h(k1)=d3r0dωdΩM(r, ks; k1)Bω(r, s).

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