Abstract

We propose a new method for determining structures of semitransparent media from measurements of the extinguished power in scattering experiments. The method circumvents the problem of measuring the phase of the scattered field. We illustrate how this technique may be used to reconstruct both deterministic and random scatterers.

© 1999 Optical Society of America

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References

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  1. See, for example, E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.
  2. H. C. van de Hulst, “On the attenuation of plane waves by obstacles of arbitrary size and form,” Physica 15, 740–746 (1949).
    [CrossRef]
  3. P. S. Carney, E. Wolf, G. S. Agarwal, “Statistical generalizations of the optical theorem with applications to inverse scattering,” J. Opt. Soc. Am. A 14, 3366–3371 (1997).
    [CrossRef]
  4. R. G. Newton, “Present status of the generalized Marchenko method for the solution of the inverse scattering problem in three dimensions,” in Inverse Problems in Mathematical Physics, L. Päivärinta, E. Somersalo, eds. (Springer-Verlag, Berlin, 1993).
  5. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  6. Within the accuracy of the first-order Born approximation, D is constant over these values of S. However experimental values of D may require the averaging in Eq. (3.6) owing to noise and multiple scattering.
  7. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  8. R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inf. Theory 3, 182–187 (1957).
    [CrossRef]
  9. G. Gbur, K. Kim, “The quasi-homogeneous approximation for a class of three-dimensional primary sources,” Opt. Commun. 163, 20–23 (1999). This paper is soon to be reprinted in Opt. Commun. owing to a large number of printer’s errors in the original publication.
    [CrossRef]
  10. In order that g˜ satisfy requirements of analyticity in κ′,g˜(|κ|) must be expressible as an analytic function of |κ|2.
  11. See Appendix in G. Gbur, E. Wolf, “Determination of the density correlation function from scattering with polychromatic light,” Opt. Commun. (to be published).
  12. C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

1999 (1)

G. Gbur, K. Kim, “The quasi-homogeneous approximation for a class of three-dimensional primary sources,” Opt. Commun. 163, 20–23 (1999). This paper is soon to be reprinted in Opt. Commun. owing to a large number of printer’s errors in the original publication.
[CrossRef]

1997 (1)

1977 (1)

1957 (1)

R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inf. Theory 3, 182–187 (1957).
[CrossRef]

1949 (1)

H. C. van de Hulst, “On the attenuation of plane waves by obstacles of arbitrary size and form,” Physica 15, 740–746 (1949).
[CrossRef]

Agarwal, G. S.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Carney, P. S.

Carter, W. H.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

Diu, B.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

Gbur, G.

G. Gbur, K. Kim, “The quasi-homogeneous approximation for a class of three-dimensional primary sources,” Opt. Commun. 163, 20–23 (1999). This paper is soon to be reprinted in Opt. Commun. owing to a large number of printer’s errors in the original publication.
[CrossRef]

See Appendix in G. Gbur, E. Wolf, “Determination of the density correlation function from scattering with polychromatic light,” Opt. Commun. (to be published).

Kim, K.

G. Gbur, K. Kim, “The quasi-homogeneous approximation for a class of three-dimensional primary sources,” Opt. Commun. 163, 20–23 (1999). This paper is soon to be reprinted in Opt. Commun. owing to a large number of printer’s errors in the original publication.
[CrossRef]

Laloe, F.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

Newton, R. G.

R. G. Newton, “Present status of the generalized Marchenko method for the solution of the inverse scattering problem in three dimensions,” in Inverse Problems in Mathematical Physics, L. Päivärinta, E. Somersalo, eds. (Springer-Verlag, Berlin, 1993).

Silverman, R. A.

R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inf. Theory 3, 182–187 (1957).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, “On the attenuation of plane waves by obstacles of arbitrary size and form,” Physica 15, 740–746 (1949).
[CrossRef]

Wolf, E.

P. S. Carney, E. Wolf, G. S. Agarwal, “Statistical generalizations of the optical theorem with applications to inverse scattering,” J. Opt. Soc. Am. A 14, 3366–3371 (1997).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[CrossRef]

See Appendix in G. Gbur, E. Wolf, “Determination of the density correlation function from scattering with polychromatic light,” Opt. Commun. (to be published).

See, for example, E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

IRE Trans. Inf. Theory (1)

R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inf. Theory 3, 182–187 (1957).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

G. Gbur, K. Kim, “The quasi-homogeneous approximation for a class of three-dimensional primary sources,” Opt. Commun. 163, 20–23 (1999). This paper is soon to be reprinted in Opt. Commun. owing to a large number of printer’s errors in the original publication.
[CrossRef]

Physica (1)

H. C. van de Hulst, “On the attenuation of plane waves by obstacles of arbitrary size and form,” Physica 15, 740–746 (1949).
[CrossRef]

Other (7)

See, for example, E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.

In order that g˜ satisfy requirements of analyticity in κ′,g˜(|κ|) must be expressible as an analytic function of |κ|2.

See Appendix in G. Gbur, E. Wolf, “Determination of the density correlation function from scattering with polychromatic light,” Opt. Commun. (to be published).

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

R. G. Newton, “Present status of the generalized Marchenko method for the solution of the inverse scattering problem in three dimensions,” in Inverse Problems in Mathematical Physics, L. Päivärinta, E. Somersalo, eds. (Springer-Verlag, Berlin, 1993).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Within the accuracy of the first-order Born approximation, D is constant over these values of S. However experimental values of D may require the averaging in Eq. (3.6) owing to noise and multiple scattering.

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Figures (2)

Fig. 1
Fig. 1

Left, sphere with size parameter ka=3π (radius a=1.5λ); right, sphere of radius ka=10π (a=5λ). In all cases α=0.01/2π; the original profile is shown by long-dashed curves. Short–long-dashed curves, reconstruction of spheres with susceptibility η=0.01i/2π; short-dashed curves, reconstruction of spheres with susceptibility η=(0.01+0.01i)/2π; solid curves, reconstruction of spheres with susceptibility η=(0.0512+0.01i)/2π.

Fig. 2
Fig. 2

Susceptibility intensity function Γ for a medium characterized by an ensemble of independent homogeneous spheres each with susceptibility η=0.21/4π and size parameter ka=4. Dashed curves, original probability distribution, with (a) kσ=100, (b) kσ=10, and (c) kσ=4. Solid curves, reconstructed fluctuation structure function. Both types of curves have been normalized so that the peak values are unity.

Equations (51)

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Ψ(i)(r, t)=ψ(i)(r)exp(-iωt)
ψ(i)(r)=a exp(ikrs0),
Ψ(s)(r, t)=ψ(s)(r)exp(-iωt)
ψ(r)=ψ(i)(r)+ψ(s)(r).
ψ(s)(rs)a exp(ikr)r f(s, s0),
P(e)=|a|2 4πk I f(s0, s0),
f(s1, s2)=k2η(r)exp[-ikr(s2-s1)]d3r,
ψ(i)(r)=a1 exp(ikrs1)+a2 exp(ikrs2),
P(e)(a1, a2)=4πk I [|a1|2f(s1, s1)+a1*a2 f(s1, s2)+a2*a1 f(s2, s1)+|a2|2f(s2, s2)].
P(e)(a1, a2)-P(e)(a1, -a2)
=8πk I[a1*a2f(s1, s2)+a2*a1f(s2, s1)].
D(s1, s2)=k8πa1*a2 {P(e)(a1, ia2)-P(e)(a1, -ia2)+i[P(e)(a1, a2)-P(e)(a1, -a2)]}.
D(s1, s2)=f(s1, s2)-f *(s2, s1).
2ψ(r)+k2ψ(r)=-4πk2η(r)ψ(r).
f(s1, s2)=k2η˜[k(s1-s2)],
η˜(K)=η(r)exp(-iKr)d3r
α(r)Iη(r).
D(s1, s2)=2ik2α˜[k(s1-s2)],
D(s)=12π (2π)dϕD[S(ϕ)+s /2,S(ϕ)-s/2].
αLP(r)=ki16π3 |s|2 exp(ikrs)D(s)d3s.
Iη(r)=0,
η(r1)η(r2)C(r1, r2)Γr1+r22g(r2-r1),
f(s1,s2)=k4Γ˜[k(s1-s2)]G˜-ks1+s22,
G(r)g(r)G(r),
G(r)=exp(ikr)r.
D(s1, s2)=2ik4Γ˜[k(s1-s2)] I G˜-k s1+s22.
G˜(κ)=12π2  g˜(κ-κ)κ2-(k+i)2 d3κ,
G˜(-ks)=1π 01dx-dκκ2×g˜[(k2+κ2+2xkκ)1/2]κ2-(k+i)2.
G˜(-ks)=i22 d ykyg˜(ky),
D(s1,s2)=2ik5Γ˜[k(s1-s2)]22 g˜(ky)ydy.
g(r)=δ(3)(r).
g˜(κ)=1.
D(s1, s2)=2ik5Γ˜[k(s1-s2)],
ΓLP(r)=1i16π3k2 |s|2 exp(ikrs)D(s)d3s,
C(r1, r2)=η02Ba(r1-r0)Ba(r2-r0)×p(r0)d3r0,
Ba(r)=1if|r|a0otherwise,
C(r1, r2)=η02Ba(r+r/2)Ba(r-r/2)×p(r+R)d3r.
C(r1, r2)Γ(R)g(r)+η02Ba(r+r/2)×Ba(r-r/2)rp(R)d3r+,
Γ(R)=4π3 a3η02p(R).
g(r)=1-3r4a+r316a3forr2a0forr>2a.
f(s1, s2; r0)=exp[ikr0(s2-s1)]fa(s1-s2).
f(s1, s2; r0)=fa(s1-s2)p˜[k(s1-s2)],
D(s1, s2)=[fa(s1-s2)-fa*(s2-s1)]×p˜[k(s1-s2)].
D(s)=[fa(s)-fa*(-s)] p˜(ks).
ΓLP(r)=1k3 αLP(r-r)p(r)d3r,
p(r)=(πσ2/2)-3/2 exp(-r2/2σ2).
f(s1, s2)=1k m=0(2m+1) iβmβm-iγm Pm(s1s2),
βm=kjm(Nka)j(ka)-nkjm(Nka)jm(ka),
γm=nkjm(Nka)nm(ka)-kjm(Nka)nm(ka).
D(s)=2ik m=1(2m+1)Rβmβm-iγmPm(1-s2/2).
αLP(r)=m=0 (2m+1)π2kr Rβmβm-iγm×dd(kr) [krjm(kr)nm(kr)].

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