Abstract

In the literature one can encounter at least two different radiative transfer equations for media with spatially varying refractive indices. These are the results of Ferwerda [J. Opt. A Pure Appl. Opt. 1, L1 (1999)] and Tualle and Tinet [Opt. Commun. 228, 33 (2003)]. Accordingly, two different diffusion approximations are derived from these two radiative transfer equations. I reconsider the derivation of the radiative transfer equation in a medium with an inhomogeneous refractive index and confirm the result of Tualle and Tinet. In the diffusion approximation, a simple analytical solution has been found for the steady-state illumination of a nonabsorbing turbid medium with a varying refractive index.

© 2004 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. H. A. Ferwerda, “The radiative transfer equation for scattering media with a spatially varying refractive index,” J. Opt. A, Pure Appl. Opt. 1, L1–L2 (1999).
    [CrossRef]
  3. J.-M. Tualle, E. Tinet, “Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index,” Opt. Commun. 228, 33–38 (2003).
    [CrossRef]
  4. G. C. Pomraning, Radiation Hydrodynamics (Pergamon, Oxford, UK, 1973).
  5. E. G. Harris, “Radiative transfer in dispersive media,” Phys. Rev. 138, B479–B485 (1965).
    [CrossRef]
  6. L. Martı́-López, J. Bounza-Domı́nguez, J. C. Hebden, S. Arridge, R. A. Martı́nez-Celorio, “Validity conditions for the radiative transfer equation,” J. Opt. Soc. Am. A 20, 2046–2056 (2003).
    [CrossRef]
  7. H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2714–2727 (2003).
    [CrossRef]
  8. T. Khan, H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A, Pure Appl. Opt. 5, 137–141 (2003).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970).
  10. T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
    [CrossRef]

2003 (4)

J.-M. Tualle, E. Tinet, “Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index,” Opt. Commun. 228, 33–38 (2003).
[CrossRef]

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2714–2727 (2003).
[CrossRef]

T. Khan, H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A, Pure Appl. Opt. 5, 137–141 (2003).
[CrossRef]

L. Martı́-López, J. Bounza-Domı́nguez, J. C. Hebden, S. Arridge, R. A. Martı́nez-Celorio, “Validity conditions for the radiative transfer equation,” J. Opt. Soc. Am. A 20, 2046–2056 (2003).
[CrossRef]

1999 (1)

H. A. Ferwerda, “The radiative transfer equation for scattering media with a spatially varying refractive index,” J. Opt. A, Pure Appl. Opt. 1, L1–L2 (1999).
[CrossRef]

1997 (1)

1965 (1)

E. G. Harris, “Radiative transfer in dispersive media,” Phys. Rev. 138, B479–B485 (1965).
[CrossRef]

Arridge, S.

Boas, D. A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970).

Bounza-Domi´nguez, J.

Brooksby, B.

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2714–2727 (2003).
[CrossRef]

Chance, B.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Dehghani, H.

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2714–2727 (2003).
[CrossRef]

Durduran, T.

Ferwerda, H. A.

H. A. Ferwerda, “The radiative transfer equation for scattering media with a spatially varying refractive index,” J. Opt. A, Pure Appl. Opt. 1, L1–L2 (1999).
[CrossRef]

Harris, E. G.

E. G. Harris, “Radiative transfer in dispersive media,” Phys. Rev. 138, B479–B485 (1965).
[CrossRef]

Hebden, J. C.

Jiang, H.

T. Khan, H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A, Pure Appl. Opt. 5, 137–141 (2003).
[CrossRef]

Khan, T.

T. Khan, H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A, Pure Appl. Opt. 5, 137–141 (2003).
[CrossRef]

Marti´-López, L.

Marti´nez-Celorio, R. A.

Paulsen, K. D.

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2714–2727 (2003).
[CrossRef]

Pogue, B. W.

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2714–2727 (2003).
[CrossRef]

Pomraning, G. C.

G. C. Pomraning, Radiation Hydrodynamics (Pergamon, Oxford, UK, 1973).

Tinet, E.

J.-M. Tualle, E. Tinet, “Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index,” Opt. Commun. 228, 33–38 (2003).
[CrossRef]

Tualle, J.-M.

J.-M. Tualle, E. Tinet, “Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index,” Opt. Commun. 228, 33–38 (2003).
[CrossRef]

Vishwanath, K.

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2714–2727 (2003).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970).

Yodh, A. G.

J. Opt. A, Pure Appl. Opt. (2)

H. A. Ferwerda, “The radiative transfer equation for scattering media with a spatially varying refractive index,” J. Opt. A, Pure Appl. Opt. 1, L1–L2 (1999).
[CrossRef]

T. Khan, H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A, Pure Appl. Opt. 5, 137–141 (2003).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J.-M. Tualle, E. Tinet, “Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index,” Opt. Commun. 228, 33–38 (2003).
[CrossRef]

Phys. Med. Biol. (1)

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, K. D. Paulsen, “The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach,” Phys. Med. Biol. 48, 2714–2727 (2003).
[CrossRef]

Phys. Rev. (1)

E. G. Harris, “Radiative transfer in dispersive media,” Phys. Rev. 138, B479–B485 (1965).
[CrossRef]

Other (3)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

G. C. Pomraning, Radiation Hydrodynamics (Pergamon, Oxford, UK, 1973).

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970).

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Figures (2)

Fig. 1
Fig. 1

Schematic of the element of the phase space that comprises the element of volume dV in physical space and the element of solid angle dΩ in the direction Ω. Vectors v1Ω and v2Ω show the velocities of the photons entering and leaving the element of volume dV through the surfaces x and x+dx, where v1 and v2 are the speeds of photons at these two surfaces.

Fig. 2
Fig. 2

Distribution of the refractive index in an infinite nonabsorbing turbid medium, n/n0=(1+ae-br2) and the corresponding distribution of the specific intensity, ϕ/ϕ0=(1+ae-br2)2, in the steady state. The parameters a and b are chosen to be a=0.2 and b=0.5 cm-2. Distances on the r axis are in centimeters.

Equations (29)

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dN=f(r, Ω, t)dVdΩ,
δ(dN)1=Ω(vf)dVdΩdt,
dds (nΩ)=n,
Ω˙=vn [n-Ω(nΩ)],
Ω˙|Ω+dΩ=vn {n-(Ω+dΩ)[n(Ω+dΩ)]}=vn [n-Ω(Ωn)]-vn [Ω(ndΩ)+dΩ(Ωn)].
Ω˙θΩθ=-vn (Ωn),
Ω˙φΩφ=-vn (Ωn).
(Ω˙θf)|θ+dθ-(Ω˙θf)|θdΩθdVdΩdt=(Ω˙θf)ΩθdVdΩdt=Ω˙θfΩθ+Ω˙θΩθ fdVdΩdt.
δ(dN)2=(Ω˙θ,φf+fθ,φΩ˙)dVdΩdt,
θ,φ=θeθ+1sin θφeφ,
θ,φΩ˙=-2vn (Ωn).
δ(dN)3=ftdVdΩdt.
δ(dN)=ft+Ω(vf)+Ω˙θ,φf-2vn (nΩ)fdVdΩdt.
δ(dN)=-(μa+μs)vf+vμs4πp(ΩΩ)f(Ω)dΩ+dVdΩdt,
ft+Ω(vf)+vn nθ,φf-2vfn (Ωn)=-(μa+μs)vf+vμs4πp(ΩΩ)f(Ω)dΩ+,
ncLt+ΩL+1n nθ,φL-2n (Ωn)L=-(μa+μs)L+μs4πp(ΩΩ)L(Ω)dΩ+e,
L=14π ϕ+34π (JΩ),
θ,φ=ρ[Ω-Ω(ΩΩ)],
Ω=ρeρ+1ρθeθ+1ρ sin θφeφ,
θ,φ(JΩ)=[Ω-Ω(ΩΩ)](JΩ)=J-Ω(JΩ).
ncϕt+Ωϕ-2n (Ωn)ϕ+3nc(JΩ)t+3Ω(JΩ)+3n (Jn)-9n (Ωn)(JΩ)=-μaϕ-3[μa+μs(1-g)](JΩ)+e,
ncϕt+J+μaϕ=E,
J=-Dϕ+2Dn ϕn-3DncJt,
ncϕt-(Dϕ)+2Dϕn n+μaϕ=E.
ncϕt-Dn2 ϕn2+μaϕ=E.
Dn2 ϕn2=0.
ϕ=Cn2,
n=n0(1+ae-br2),
ϕ=ϕ0(1+ae-br2)2,

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