Abstract

This discussion reconciles differences in literature expressions for the diffusion approximation boundary conditions for the interface between two turbid media with different refractive indices.

© 2002 Optical Society of America

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References

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  1. S. A. Walker, D. A. Boas, E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935–1944 (1998) [see Eqs. (9)–(15)].
    [CrossRef]
  2. J. Ripoll, M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse–diffuse interfaces,” J. Opt. Soc. Am. A 16, 1947–1957 (1999) [see Eqs. (14) and (15)]. This paper adds consideration of transverse flux to the derivation of Aronson.3 The parameters RU(i) and RJ(i) of this reference are equivalent to [1-Rϕ(i)]/2 and 1-RJ(i) when expressed in the parameters used here.
    [CrossRef]
  3. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995) [see Eqs. (34)–(36)].
    [CrossRef]
  4. M. Gerken, G. W. Faris, “Frequency-domain immersion technique for accurate optical property measurements of turbid media,” Opt. Lett. 24, 1726–1728 (1999).
    [CrossRef]
  5. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994). See the paragraph beginning, “We mention in passing …” (p. 2731). This expression is based incorrectly on the effective reflectance coefficient for the interface between a turbid medium and a transparent medium and has a sign error.
    [CrossRef]
  6. S. A. Walker, S. Fantini, E. Gratton, “Effect of index of refraction mismatch on the recovery of optical properties of cylindrical inhomogeneities in an infinite turbid medium,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, B. Chance, R. R. Alfano, eds., Proc. SPIE2979, 219–225 (1997) [see Eq. (6)].
    [CrossRef]
  7. The total flux and surface irradiance approaches are equivalent in the diffusion approximation. For higher-order approximations, the surface irradiance approach must be used. For example, the total flux balance expression in Eq. (7-4) of Ref. 8 is valid only in the diffusion approximation.
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.
  9. M. Keijzer, W. M. Star, P. R. M. Storchi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988) [see Eq. (19)]. Note that Aronson’s subsequent approximations [Ref. 3, Eqs. (38)] are more accurate than this result.
    [CrossRef] [PubMed]
  10. J. Ripoll, M. Nieto-Vesperinas, “Reflection and transmission coefficients for diffuse photon density waves,” Opt. Lett. 24, 796–798 (1999).
    [CrossRef]

1999

1998

1995

1994

1988

Aronson, R.

Boas, D. A.

Fantini, S.

S. A. Walker, S. Fantini, E. Gratton, “Effect of index of refraction mismatch on the recovery of optical properties of cylindrical inhomogeneities in an infinite turbid medium,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, B. Chance, R. R. Alfano, eds., Proc. SPIE2979, 219–225 (1997) [see Eq. (6)].
[CrossRef]

Faris, G. W.

Feng, T.-C.

Gerken, M.

Gratton, E.

S. A. Walker, D. A. Boas, E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935–1944 (1998) [see Eqs. (9)–(15)].
[CrossRef]

S. A. Walker, S. Fantini, E. Gratton, “Effect of index of refraction mismatch on the recovery of optical properties of cylindrical inhomogeneities in an infinite turbid medium,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, B. Chance, R. R. Alfano, eds., Proc. SPIE2979, 219–225 (1997) [see Eq. (6)].
[CrossRef]

Haskell, R. C.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Keijzer, M.

McAdams, M. S.

Nieto-Vesperinas, M.

Ripoll, J.

Star, W. M.

Storchi, P. R. M.

Svaasand, L. O.

Tromberg, B. J.

Tsay, T.-T.

Walker, S. A.

S. A. Walker, D. A. Boas, E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37, 1935–1944 (1998) [see Eqs. (9)–(15)].
[CrossRef]

S. A. Walker, S. Fantini, E. Gratton, “Effect of index of refraction mismatch on the recovery of optical properties of cylindrical inhomogeneities in an infinite turbid medium,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, B. Chance, R. R. Alfano, eds., Proc. SPIE2979, 219–225 (1997) [see Eq. (6)].
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Lett.

Other

S. A. Walker, S. Fantini, E. Gratton, “Effect of index of refraction mismatch on the recovery of optical properties of cylindrical inhomogeneities in an infinite turbid medium,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, B. Chance, R. R. Alfano, eds., Proc. SPIE2979, 219–225 (1997) [see Eq. (6)].
[CrossRef]

The total flux and surface irradiance approaches are equivalent in the diffusion approximation. For higher-order approximations, the surface irradiance approach must be used. For example, the total flux balance expression in Eq. (7-4) of Ref. 8 is valid only in the diffusion approximation.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

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Figures (1)

Fig. 1
Fig. 1

Geometry for boundary condition equations.

Equations (18)

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Eirrad(1)=Ereflected(1)+Etransmitted(2),
Eirrad(2)=Ereflected(2)+Etransmitted(1).
sˆ·zˆ>0L1(sˆ)sˆ·zˆ dΩ
=sˆ·zˆ<0RFresnel(1)[sˆ·(-zˆ)]L1(sˆ)sˆ·(-zˆ)dΩ+sˆ·zˆ>0[1-RFresnel(2)(sˆ·zˆ)]L2(sˆ)sˆ·zˆ dΩ,
sˆ·zˆ<0L2(sˆ)sˆ·(-zˆ) dΩ
=sˆ·zˆ>0RFresnel(2)(sˆ·zˆ)L2(sˆ)sˆ·zˆ dΩ+sˆ·zˆ<0{1-RFresnel(1)[sˆ·(-zˆ)]}
×L1(sˆ)sˆ·(-zˆ)dΩ,
RFresnel(cos θi)
=12ni cos θj-nj cos θini cos θj+nj cos θi2+12ni cos θi-nj cos θjni cos θi+nj cos θj2for0θθc1forθcθπ/2,
L(sˆ)14πϕ+34πJ·sˆ,
J(i)=Jz(i)zˆ+Jt(i)tˆ(i),
ϕ14+Jz12=Rϕ1ϕ14-RJ1Jz12+(1-Rϕ2)ϕ24+(1-RJ2)Jz22,
ϕ24-Jz22=Rϕ2ϕ24+RJ2Jz22+(1-Rϕ1)ϕ14-(1-RJ1)Jz12.
Rϕ(i)=20π/2RFresnel(cos θi)cos θi sin θi dθi,
RJ(i)=30π/2RFresnel(cos θi)cos2 θi sin θi dθi,
Jz1=Jz2=Jz.
(1-Rϕ1)ϕ1+2(RJ1+RJ2)Jz=(1-Rϕ2)ϕ2,
ϕ2-n2n12ϕ1=2RJ1+RJ21-Rϕ2J2=Cn2n1Jz,

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