Abstract

A ray-tracing calculation that uses reflection and transmission coefficients for layers with refractive-index matched boundaries leads to the corresponding coefficients for refractive-index mismatch. The model is compared with Monte Carlo calculations for a range of layer parameters. The absorption by a mismatched layer is higher than the corresponding layer with matched boundaries and relatively insensitive to the extent of scattering anisotropy. The model should be useful for practical calculations on biological tissues where refractive-index mismatch is usually present.

© 1992 Optical Society of America

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References

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  1. F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989).
    [CrossRef] [PubMed]
  2. R. R. Anderson, H. Beck, U. Bruggemann, W. Farinelli, S. L. Jacques, J. Parrish, “Pulsed photothermal radiometry in turbid media: internal reflection of backscattered radiation strongly influences optical dosimetry,” Appl. Opt. 28, 2256–2262 (1989).
    [CrossRef] [PubMed]
  3. R. A. J. Groenhuis, H. A. Ferwerda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory,” Appl. Opt. 22, 2456–2462 (1983).
    [CrossRef] [PubMed]
  4. S. L. Jacques, S. A. Prahl, “Modeling optical and thermal distributions in tissue during laser irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
    [CrossRef] [PubMed]
  5. S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).
  6. H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980), Vols. 1 and 2.
  7. S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. (SPIE Optical Engineering, Bellingham, Wash., 1989), pp. 102–111.
  8. W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).
  9. J. W. Ryde, “The scattering of light in turbid media—part I,” Proc. R. Soc. London Ser. A 131, 451–464 (1931).
    [CrossRef]
  10. R. G. Giovanelli, “Reflection by semi-infinite diffusers,” Opt. Acta 2, 153–162 (1955).
    [CrossRef]
  11. J. Strong, Concepts of Classical Optics (Freeman, San Francisco, Calif., 1958).
  12. W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]

1990 (1)

W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1989 (2)

1987 (1)

S. L. Jacques, S. A. Prahl, “Modeling optical and thermal distributions in tissue during laser irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

1983 (1)

1955 (1)

R. G. Giovanelli, “Reflection by semi-infinite diffusers,” Opt. Acta 2, 153–162 (1955).
[CrossRef]

1931 (1)

J. W. Ryde, “The scattering of light in turbid media—part I,” Proc. R. Soc. London Ser. A 131, 451–464 (1931).
[CrossRef]

Anderson, R. R.

Beck, H.

Bolin, F. P.

Bruggemann, U.

Cheong, W. C.

W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Egan, W. G.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

Farinelli, W.

Ference, R. J.

Ferwerda, H. A.

Giovanelli, R. G.

R. G. Giovanelli, “Reflection by semi-infinite diffusers,” Opt. Acta 2, 153–162 (1955).
[CrossRef]

Groenhuis, R. A. J.

Hilgeman, T. W.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

Jacques, S. L.

R. R. Anderson, H. Beck, U. Bruggemann, W. Farinelli, S. L. Jacques, J. Parrish, “Pulsed photothermal radiometry in turbid media: internal reflection of backscattered radiation strongly influences optical dosimetry,” Appl. Opt. 28, 2256–2262 (1989).
[CrossRef] [PubMed]

S. L. Jacques, S. A. Prahl, “Modeling optical and thermal distributions in tissue during laser irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. (SPIE Optical Engineering, Bellingham, Wash., 1989), pp. 102–111.

Keijzer, M.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. (SPIE Optical Engineering, Bellingham, Wash., 1989), pp. 102–111.

Parrish, J.

Prahl, S. A.

W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

S. L. Jacques, S. A. Prahl, “Modeling optical and thermal distributions in tissue during laser irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. (SPIE Optical Engineering, Bellingham, Wash., 1989), pp. 102–111.

Preuss, L. E.

Ryde, J. W.

J. W. Ryde, “The scattering of light in turbid media—part I,” Proc. R. Soc. London Ser. A 131, 451–464 (1931).
[CrossRef]

Strong, J.

J. Strong, Concepts of Classical Optics (Freeman, San Francisco, Calif., 1958).

Taylor, R. C.

Ten Bosch, J. J.

Van de Hulst, H. C.

H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980), Vols. 1 and 2.

Welch, A. J.

W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. (SPIE Optical Engineering, Bellingham, Wash., 1989), pp. 102–111.

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Lasers Surg. Med. (1)

S. L. Jacques, S. A. Prahl, “Modeling optical and thermal distributions in tissue during laser irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

Opt. Acta (1)

R. G. Giovanelli, “Reflection by semi-infinite diffusers,” Opt. Acta 2, 153–162 (1955).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. W. Ryde, “The scattering of light in turbid media—part I,” Proc. R. Soc. London Ser. A 131, 451–464 (1931).
[CrossRef]

Other (5)

J. Strong, Concepts of Classical Optics (Freeman, San Francisco, Calif., 1958).

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).

H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, New York, 1980), Vols. 1 and 2.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds. (SPIE Optical Engineering, Bellingham, Wash., 1989), pp. 102–111.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

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Figures (2)

Fig. 1
Fig. 1

Ray tracing in a plane-parallel layer illuminated at normal incidence with refractive-index mismatch at the phase boundaries. The solid lines depict rays generated by normally incident flux. The dashed lines depict rays generated by internal reflections. The dashed line depicts the unscattered incidence flux. The layer transmission coefficients are fn for unscattered flux, T n for normal incidence exluding the scattered flux, and Tu for isotropic incidence. The layer reflection coeffiecients are Rn for normal incidence and Ru for isotropic incidence. The internal reflection coefficients for diffuse incidence are r1 and r2. Specular reflections are not shown. The continuation of ray tracingfor n1 = -n2 leads to Eqs. (1) and (2) with C and D given by Eqs. (4), (5), and (6).

Fig. 2
Fig. 2

The dependence of layer coefficients on refractive-index mismatch. The parameters are b = 4 and a = 0.95. The layer reflection, transmission, and absorption coefficients are R, T, and A, respectively. The solid lines are for g = 0, and the dashed lines are for g = 0.875. The value of A is almost independent of g for mismatched boundaries.

Tables (2)

Tables Icon

Table I Comparison of Present Model with Monte Carlo Simulation

Tables Icon

Table II Comparison of Present Model with Calculations of Giovanellia and Monte Carlo Calculations for a Semi-infinite Layer with Refractive-index Mismatch

Equations (12)

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R = ( 1 r ) ( C R n + D T n ) ,
T = f n + ( 1 r ) ( C T n + D R n ) ,
( C + D ) = j = 0 r j ( R u + T u ) j = 1 / [ 1 r ( R u + T u ) ] .
C = ( 1 r R u ) / M ,
D = r T u / M ,
M = 1 2 r R u + r 2 ( R u 2 T u 2 ) .
P = l , m , n N l m n a l b m c n = m = 0 P m ,
C r = ( 1 r 2 R u ) / M , C t = ( 1 r 1 R u ) / M .
D r = r 2 T u / M , D t = r 1 T u / M .
R = R sp 1 + ( 1 R sp 1 ) ( 1 r 1 ) ( C r R n + D r T n ) ( 1 + R sp 2 f n ) ,
T = ( 1 R sp 1 ) [ ( 1 R sp 2 ) f n + ( 1 r 2 ) ( C t T n + D t R n ) ( 1 + R sp 2 f n ) ] .
R = R sp 1 + ( 1 R sp 1 ) ( 1 r ) R n ( 1 r R u ) .

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