Abstract

The radiation that is due to the scattering of an incoming light wave from an object or that is generated by a source distribution is considered. It is assumed that the field propagation can be described by the equation of radiative transport or, in the P1 approximation, by the diffusion equation. It is shown by explicit examples that there is a class of object (source) distributions that generate scattered (radiation) fields that are identically zero outside these distributions. The existence of these invisible object and source distributions leads to serious complications for the inverse, i.e., the reconstruction, problem.

© 1997 Optical Society of America

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References

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  1. An excellent overview of the state of the art is given in Photon Propagation in Tissues, B. Chance, D. T. Delpy, and G. J. Mueller, eds., Proc. SPIE 2626, 296–346 (1995).
    [Crossref]
  2. B. J. Hoenders, Inverse Problems in Optics (Springer-Verlag, Berlin, 1978), pp. 41–82.
  3. If the scatterer is not weak, a similar reasoning shows that it is possible to construct nonscattering distributions for any nth-order perturbation series expansion.
  4. K. Kim and E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
    [Crossref]
  5. Ph. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 2, Eq. (11.3.44), with n=m=0.
  6. E. Amaldi, “The production and slowing down of neutrons,” in Encyclopedia of Physics, S. Flügge, ed., Vol. 38/2 of Springer Encyclopedia of Physics (Springer-Verlag, Berlin, 1959), Sec. 118, comments between Eqs. (118.23) and (118.24).

1995 (1)

An excellent overview of the state of the art is given in Photon Propagation in Tissues, B. Chance, D. T. Delpy, and G. J. Mueller, eds., Proc. SPIE 2626, 296–346 (1995).
[Crossref]

1986 (1)

K. Kim and E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
[Crossref]

Kim, K.

K. Kim and E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
[Crossref]

Wolf, E.

K. Kim and E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
[Crossref]

Opt. Commun. (1)

K. Kim and E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
[Crossref]

Proc. SPIE (1)

An excellent overview of the state of the art is given in Photon Propagation in Tissues, B. Chance, D. T. Delpy, and G. J. Mueller, eds., Proc. SPIE 2626, 296–346 (1995).
[Crossref]

Other (4)

B. J. Hoenders, Inverse Problems in Optics (Springer-Verlag, Berlin, 1978), pp. 41–82.

If the scatterer is not weak, a similar reasoning shows that it is possible to construct nonscattering distributions for any nth-order perturbation series expansion.

Ph. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 2, Eq. (11.3.44), with n=m=0.

E. Amaldi, “The production and slowing down of neutrons,” in Encyclopedia of Physics, S. Flügge, ed., Vol. 38/2 of Springer Encyclopedia of Physics (Springer-Verlag, Berlin, 1959), Sec. 118, comments between Eqs. (118.23) and (118.24).

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Equations (40)

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(Ω·+ρσt)L(x, Ω)
-ρσt4π4π dΩf(Ω, Ω)L(x, Ω)-(x, Ω)=0.
σt=σs+σa,
f(Ω, Ω)=f(Ω, Ω),
14π4π dΩf(Ω, Ω)=σs/σt.
f(Ω, Ω)=σs/σt,
(Ω·-ρσt)G(x, x, Ω, Ω)
+ρσt4π4π dΩf(Ω, Ω)G(x, x, Ω, Ω)
=δ(x-x)δ(Ω-Ω).
L(x, Ω)=4π dΩ v dxΩ·[G(x, x, Ω, Ω)×L(x, Ω)+G(x, x, Ω, Ω)L(x, Ω)]-4π dΩ v dx(x, Ω)×G(x, x,Ω, Ω)ifxv.
L(x, Ω)=4π dΩ σ dxσ(Ω·n)×G(x, x, Ω, Ω)L(x, Ω)-4π dΩ v dx(x, Ω)G(x, x, Ω, Ω).
(x, Ω)=(Ω·+ρσt)H(x, Ω)-ρσt4π×4π dΩf(Ω, Ω)H(x, Ω),xvs.
H(x, Ω)=0,xboundaryσsofvs.
4π dΩ v dx(x, Ω)G(x, x,Ω, Ω)dx=4π dΩ vsdxG(x, x, Ω, Ω)[(Ω·+ρσt)×H(x, Ω)-ρσt4π4π dΩf(Ω, Ω)H(x, Ω)]dx=4π dΩ σs dxσ(Ω·n)×[G(x, x, Ω, Ω)H(xσ, Ω)]+4πdΩ vsH(x, Ω)(-Ω·+ρσt)×G(x, x, Ω, Ω)-ρσt4π4π dΩf(Ω, Ω)×G(x,x, Ω,Ω)dx,xvs.
L(x, Ω)=4π dΩσsdxσ(Ω·n)×G(x, x, Ω, Ω)L(x, Ω),xvs.
1cNt=·(DN)-μaN+S,
D=[3(μs+μa)]-1,
S=·(DH)-μaH,
n·H(x=xσs)=0,H(x=xσs)=0,xσs.
·[DG(x-x)]-μaG(x-x)=δ(x-x).
N(x)=v (N(x){·[DG(x, x)]}-G(x, x)×{·[DN(x)]+S(x)}) dx=σ DG(x, x)nN(x)-N(x)nG(x, x)dxσ-v G(x, x)S(x)dx.
v G(x, x)S(x)dx
=vs G(x, x){·[DH(x)]-μaH(x)}dx.
=σs DG(xσ, x)nH(xσ)-H(xσ)nG(xσ,x)dxσ
+vs H(x){·[DG(x, x)]-μaG(x, x)}dx.
vsH(x){·[DG(x, x)]-μaG(x, x)}dx=0,
xvs.
N(x)=σs DG(x, x)nN(x)-N(x)nG(x, x)dxσ,
xvs.
N=N0+N1,J=J0+J1,D=D0+D1,
μa=μao+μa1,N0N1,D0D1,
μa0μa1.
·(D0N0)-μa0N0+S0=0.
·(D0N1)-μa0N1+Swb=0,
Swb=·(D1N0)-μa1N0.
N(r)=0a G(r, r)S(r)r2dr,
G(r, r)
=ikD j0(kr)h0(1)(kr)rrikD j0(kr)h0(1)(kr)rr,k=-(μ0/D).
0a j0(kr)S(r)r2dr=0.
0a j1(kr)S(r)r2dr=0.

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