Abstract

The usual global reciprocity relations of radiative transfer do not hold for two points located in regions of different index of refraction. Modified reciprocity relations that involve the relative index are derived. The result has computational as well as theoretical consequences.

© 1997 Optical Society of America

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References

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  1. See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120; S. R. Arridge, “The forward and inverse problems in time-resolved infra-red imaging,” pp. 35–64.
  2. J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), pp. 369–373.
  3. J. R. Lamarsh, Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1966), Eq. (5–126), p. 156.
  4. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass.1967), Eq. (9), p. 27.
  5. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [Crossref]
  6. M. Born, E. Wolf, Principles of Optics, 2nd rev. ed. (Pergamon, New York, 1964).
  7. G. Pomraning, The Equations of Radiation Hydrodynamics (Pergamon, Oxford, 1973), Chap. 5, pp. 144–153. The final result is contained in Eq. (5.67).

1995 (1)

Aronson, R.

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[Crossref]

See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120; S. R. Arridge, “The forward and inverse problems in time-resolved infra-red imaging,” pp. 35–64.

Barbour, R. L.

See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120; S. R. Arridge, “The forward and inverse problems in time-resolved infra-red imaging,” pp. 35–64.

Born, M.

M. Born, E. Wolf, Principles of Optics, 2nd rev. ed. (Pergamon, New York, 1964).

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass.1967), Eq. (9), p. 27.

Chang, J.

See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120; S. R. Arridge, “The forward and inverse problems in time-resolved infra-red imaging,” pp. 35–64.

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), pp. 369–373.

Graber, H. L.

See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120; S. R. Arridge, “The forward and inverse problems in time-resolved infra-red imaging,” pp. 35–64.

Lamarsh, J. R.

J. R. Lamarsh, Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1966), Eq. (5–126), p. 156.

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), pp. 369–373.

Pomraning, G.

G. Pomraning, The Equations of Radiation Hydrodynamics (Pergamon, Oxford, 1973), Chap. 5, pp. 144–153. The final result is contained in Eq. (5.67).

Wang, Y.

See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120; S. R. Arridge, “The forward and inverse problems in time-resolved infra-red imaging,” pp. 35–64.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 2nd rev. ed. (Pergamon, New York, 1964).

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass.1967), Eq. (9), p. 27.

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

Other (6)

M. Born, E. Wolf, Principles of Optics, 2nd rev. ed. (Pergamon, New York, 1964).

G. Pomraning, The Equations of Radiation Hydrodynamics (Pergamon, Oxford, 1973), Chap. 5, pp. 144–153. The final result is contained in Eq. (5.67).

See, for instance, R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller et al., eds., Vol. IS11 of Institute Series (SPIE, Bellingham, Wash., 1993), pp. 87–120; S. R. Arridge, “The forward and inverse problems in time-resolved infra-red imaging,” pp. 35–64.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), pp. 369–373.

J. R. Lamarsh, Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1966), Eq. (5–126), p. 156.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass.1967), Eq. (9), p. 27.

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Equations (37)

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G(r, r)=G(r, r),
G(r, Ωˆ; r, Ωˆ)=G(r,-Ω; r,-Ωˆ),
G(r, r)=n2G(r, r)
G(r, Ωˆ; r, Ωˆ)=n2G(r,-Ωˆ; r,-Ωˆ),
·D(r)G(r, r)+Σa(r)G(r, r)=δ(r-r),
SdSeˆ·D(r)[G(r, r)G(r, r)-G(r, r)G(r, r)]
=G(r, r)-G(r, r).
IiSdSeˆi·D(ri)[G(ri, r)G(ri, r)-G(ri, r)G(ri, r)]=Hi(r)G(r, r)-Hi(r)G(r, r),
n2G(r1, r)-G(r2, r)=C(n)eˆ1·J(r, r),
SdSeˆ2·D(r2)G(r2, r)G(r2, r)
=SdSeˆ1·J(r, r)[n2G(r1, r)
-C(n)eˆ1·J(r, r)].
[n2H1(r)+H2(r)]G(r, r)
=[n2H1(r)+H2(r)]G(r, r).
Ωˆ·G(r, Ωˆ; r, Ωˆ)+Σt(r)G(r, Ωˆ; r, Ωˆ)
=dΩ1Σs(r; Ωˆ1Ωˆ)G(r, Ωˆ1; r, Ωˆ)
+δ(r-r)δ(Ωˆ-Ωˆ).
SdrdΩˆeˆ·ΩˆG(r,-Ωˆ)G(r, Ωˆ)
=G(r,-Ωˆ)-G(r, Ωˆ).
G(-Ωˆ)=r(eˆ·Ωˆ)G(Ωˆr),rS,eˆ·Ωˆ>0,
eˆ·Ωˆ>0dΩˆeˆ·Ωˆr(eˆ·Ωˆ)[G(Ωˆr)G(Ωˆ)-G(Ωˆ)G(Ωˆr)].
IiSdrJi(r, r, Ωˆ, r, Ωˆ)=Hi(r)G(r,-Ωˆ)-Hi(r)G(r, Ωˆ),
JidΩˆieˆi·ΩˆiG(-Ωˆi)G(Ωˆi).
dΩˆjeˆj·ΩˆjG(Ωˆj)=-dΩˆieˆi·Ωˆitij(eˆi·Ωˆi)G(Ωˆi),
eˆi·Ωˆi>0.
tij(eˆi·Ωˆi)=tji(-eˆj·Ωˆj).
dΩˆ1eˆ1·Ωˆ1=-n2dΩˆ2eˆ2·Ωˆ2.
J2=eˆ1·Ωˆ1>0dΩˆ2eˆ2·Ωˆ2[G(-Ωˆ2)G(Ωˆ2)-G(Ωˆ2)G(-Ωˆ2)]=-eˆ1·Ωˆ1>0dΩˆ1eˆ1·Ωˆ1t12(eˆ1·Ωˆ1)×[G(-Ωˆ2)G(Ωˆ1)-G(Ωˆ1)G(-Ωˆ2)]=-eˆ1·Ωˆ1>0dΩˆ1eˆ1·Ωˆ1t21(-eˆ2·Ωˆ2)×[G(-Ωˆ2)G(Ωˆ1)-G(Ωˆ1)G(-Ωˆ2)]=eˆ1·Ωˆ1<0dΩˆ1eˆ1·Ωˆ1t21(eˆ2·Ωˆ2)×[G(Ωˆ2)G(-Ωˆ1)-G(-Ωˆ1)G(Ωˆ2)]=-n2eˆ1·Ωˆ1<0dΩˆ2eˆ2·Ωˆ2t21(eˆ2·Ωˆ2)×[G(Ωˆ2)G(-Ωˆ1)-G(-Ωˆ1)G(Ωˆ2)]=n2eˆ1·Ωˆ1<0dΩˆ1eˆ1·Ωˆ1[G(Ωˆ1)G(-Ωˆ1)-G(-Ωˆ1)G(Ωˆ1)]=-n2dΩˆ1eˆ1·Ωˆ1G(-Ωˆ1)G(Ωˆ1)=-n2J1.
[n2H1(r)+H2(r)]G(r,-Ωˆ)
=[n2H1(r)+H2(r)]G(r, Ωˆ),
KSdSeˆ·D(r)[G(r, r)G(r, r)-G(r, r)G(r, r)]/n2(r),
G(r, r)/n2(r)-G(r, r)/n2(r)=0.
K=Sdr/n2(r)dΩˆeˆ·ΩˆG(r,-Ωˆ)G(r, Ωˆ).
G(x, Ωˆ; x, Ωˆ)=(μt/ΔΩˆ)exp(-Σ|x|/μ-Σx/μ).
G(x,-Ωˆ; x,-Ωˆ)
=(μt/ΔΩˆ)exp(-Σ|x|/μ-Σx/μ),
G(x,-Ωˆ; x,-Ωˆ)=n2G(x, Ωˆ; x, Ωˆ),

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