Abstract

The phenomenological model of Kubelka and Munk (KM) for describing the reflection and transmission of diffuse radiation in turbid, plane parallel media is utilized in investigating the nature of inhomogeneous distributions (i.e., the ratio of the absorption and scattering coefficients, K/S, is not necessarily constant) which satisfy certain constraints in finitely thick slabs. The distributions are constrained to be continuous and to minimize the integral of K<sup>2</sup>/S<sup>2</sup> across the slab, while resulting in a specified reflectance when the slab rests upon a backing of specified reflectance. The form of the inhomogeneous distributions is obtained as the solution to the corresponding variational problem, and the associated Lagrange multiplier is found to be algebraically related to the transmittance. The sufficiency of the approach is justified a posteriori by direct comparison with the closed-form solutions of KM for homogeneous distributions. The qualitative nature of such optimal inhomogeneous distributions is discussed with regard to the effects of the boundary conditions and the scattering thickness and is found to be approximately exponential.

© 1977 the Optical Society of America

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  1. G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).
  2. p. Kubelka, "New contributions to the optics of intensely lightscattering materials. Part I, "1 J. Opt. Soc. Am. 38, 448–457 (1948).
  3. P. Kubelka, "New contributions to the optics of intensely lightscattering materials. Part II: Nonhomogeneous layers," J. Opt. Soc. Am. 44, 330–335 (1954).
  4. K. Klier, "Absorption and scattering in plane parallel turbid media," J. Opt. Soc. Am. 62, 882–885 (1972).
  5. S. Chandrasekhar, Radiative Transfer(Dover, New York, 1960).
  6. (a)R. Bellman and R. Kalaba, "On the principle of invariant imbedding and propagation through inhomogeneous media," Proc. Nat. Acad. Sci. USA 42, 629–632 (1956); (b) R. W. Preisendorfer, "A mathematical foundation for radiative transfer theory," J. Math. Mech. 6, 685–730 (1957); (c) R. Bellman, R. Kalaba, and G. M. Wing, "Invariant imbedding and mathematical physics. I. Particle processes," J. Math. Phys. 1, 280–308 (1960); (d) R. Bellman, R. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (Elsevier, New York, 1963); (e) J. O. Mingle, The Invariant Imbedding Theory of Nuclear Transport (Elsevier, New York, 1973).
  7. (a) Reference 1; (b)C. E. Jones and K. Klier, "Optical and spectroscopic methods for the study of surfaces," Ann. Rev. Mater. Sci. 2, 1–32 (1972);(c) W. D. Ross, "Theoretical computation of light scattering power: Comparison between TiO2 and air bubbles," J. Paint Tech. 43, 50–66 (1971); E. Hoffmann, C. J. Lancucki, and J. W. Spencer, "Measurement of the hiding power of paints," J. Oil Col. Chem. Assoc. 55, 292–313 (1972); (e) P. B. Mitton, "Opacity, hiding power, and tinting strength, in Pigment Handbook, Vol. III. Characterization and Physical Relationships, edited by T. C. Patton (Wiley, New York, 1973), pp. 289–339.
  8. The effects of sinusoidal variation (in the plane perpendicular to the surface normal) on the scattering characteristics have been studied:R. G. Giovanelli, "Radiative transfer in discontinuous media," Aust. J. Phys. 10, 227–239 (1957); R. G. Giovanuelli, "Radiative transfer in non-uniform media," Aust. J. Phys. 12, 164–170 (1959).
  9. L. S. Pontryagin, V. G. Boltyansky, R. V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes, translated by D. E. Brown (MacMillan, New York, 1964).
  10. L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory (W. B. Saunders, Philadelphia, 1969).
  11. S. M. Roberts and J. S. Shipman, Two-Point Boundary Value Problems: Shooting Methods (Elsevier, New York, 1972).
  12. R. McGill and P. Kenneth, "A convergence theorem on the iterative solution of nonlinear two-point boundary-value systems," in Proceedings of the XIVth LAF Congress, Paris, 1963, pp. 173–188; R. McGill and P. Kenneth, "Soludion of variational problems by means of a generalized Newton-Raphson operator," AIAA J. 2, 1761–1766 (1964).
  13. A. Miele, "Method of particular solutions for linear, two-point boundary-value problems," J. Opt. Theory Appl. 2, 260–273 (1968); A. Miele, "General technique for solving nonlinear, two-point boundary-value problems via the method of particular solutions," J. Opt. Theory Appl. 5, 382–399 (1970).
  14. W. Schiesser, LEANS-III Introductory Programming Manual (Lehigh U. P., Bethlehem, Pa., 1971).

1972

K. Klier, "Absorption and scattering in plane parallel turbid media," J. Opt. Soc. Am. 62, 882–885 (1972).

(a) Reference 1; (b)C. E. Jones and K. Klier, "Optical and spectroscopic methods for the study of surfaces," Ann. Rev. Mater. Sci. 2, 1–32 (1972);(c) W. D. Ross, "Theoretical computation of light scattering power: Comparison between TiO2 and air bubbles," J. Paint Tech. 43, 50–66 (1971); E. Hoffmann, C. J. Lancucki, and J. W. Spencer, "Measurement of the hiding power of paints," J. Oil Col. Chem. Assoc. 55, 292–313 (1972); (e) P. B. Mitton, "Opacity, hiding power, and tinting strength, in Pigment Handbook, Vol. III. Characterization and Physical Relationships, edited by T. C. Patton (Wiley, New York, 1973), pp. 289–339.

1968

A. Miele, "Method of particular solutions for linear, two-point boundary-value problems," J. Opt. Theory Appl. 2, 260–273 (1968); A. Miele, "General technique for solving nonlinear, two-point boundary-value problems via the method of particular solutions," J. Opt. Theory Appl. 5, 382–399 (1970).

1957

The effects of sinusoidal variation (in the plane perpendicular to the surface normal) on the scattering characteristics have been studied:R. G. Giovanelli, "Radiative transfer in discontinuous media," Aust. J. Phys. 10, 227–239 (1957); R. G. Giovanuelli, "Radiative transfer in non-uniform media," Aust. J. Phys. 12, 164–170 (1959).

1956

(a)R. Bellman and R. Kalaba, "On the principle of invariant imbedding and propagation through inhomogeneous media," Proc. Nat. Acad. Sci. USA 42, 629–632 (1956); (b) R. W. Preisendorfer, "A mathematical foundation for radiative transfer theory," J. Math. Mech. 6, 685–730 (1957); (c) R. Bellman, R. Kalaba, and G. M. Wing, "Invariant imbedding and mathematical physics. I. Particle processes," J. Math. Phys. 1, 280–308 (1960); (d) R. Bellman, R. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (Elsevier, New York, 1963); (e) J. O. Mingle, The Invariant Imbedding Theory of Nuclear Transport (Elsevier, New York, 1973).

1954

1948

p. Kubelka, "New contributions to the optics of intensely lightscattering materials. Part I, "1 J. Opt. Soc. Am. 38, 448–457 (1948).

Bellman, R.

(a)R. Bellman and R. Kalaba, "On the principle of invariant imbedding and propagation through inhomogeneous media," Proc. Nat. Acad. Sci. USA 42, 629–632 (1956); (b) R. W. Preisendorfer, "A mathematical foundation for radiative transfer theory," J. Math. Mech. 6, 685–730 (1957); (c) R. Bellman, R. Kalaba, and G. M. Wing, "Invariant imbedding and mathematical physics. I. Particle processes," J. Math. Phys. 1, 280–308 (1960); (d) R. Bellman, R. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (Elsevier, New York, 1963); (e) J. O. Mingle, The Invariant Imbedding Theory of Nuclear Transport (Elsevier, New York, 1973).

Boltyansky, V. G.

L. S. Pontryagin, V. G. Boltyansky, R. V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes, translated by D. E. Brown (MacMillan, New York, 1964).

Chandrasekhar, S.

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

Gamkrelidze, R. V.

L. S. Pontryagin, V. G. Boltyansky, R. V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes, translated by D. E. Brown (MacMillan, New York, 1964).

Giovanelli, R. G.

The effects of sinusoidal variation (in the plane perpendicular to the surface normal) on the scattering characteristics have been studied:R. G. Giovanelli, "Radiative transfer in discontinuous media," Aust. J. Phys. 10, 227–239 (1957); R. G. Giovanuelli, "Radiative transfer in non-uniform media," Aust. J. Phys. 12, 164–170 (1959).

Jones, C. E.

(a) Reference 1; (b)C. E. Jones and K. Klier, "Optical and spectroscopic methods for the study of surfaces," Ann. Rev. Mater. Sci. 2, 1–32 (1972);(c) W. D. Ross, "Theoretical computation of light scattering power: Comparison between TiO2 and air bubbles," J. Paint Tech. 43, 50–66 (1971); E. Hoffmann, C. J. Lancucki, and J. W. Spencer, "Measurement of the hiding power of paints," J. Oil Col. Chem. Assoc. 55, 292–313 (1972); (e) P. B. Mitton, "Opacity, hiding power, and tinting strength, in Pigment Handbook, Vol. III. Characterization and Physical Relationships, edited by T. C. Patton (Wiley, New York, 1973), pp. 289–339.

Kalaba, R.

(a)R. Bellman and R. Kalaba, "On the principle of invariant imbedding and propagation through inhomogeneous media," Proc. Nat. Acad. Sci. USA 42, 629–632 (1956); (b) R. W. Preisendorfer, "A mathematical foundation for radiative transfer theory," J. Math. Mech. 6, 685–730 (1957); (c) R. Bellman, R. Kalaba, and G. M. Wing, "Invariant imbedding and mathematical physics. I. Particle processes," J. Math. Phys. 1, 280–308 (1960); (d) R. Bellman, R. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (Elsevier, New York, 1963); (e) J. O. Mingle, The Invariant Imbedding Theory of Nuclear Transport (Elsevier, New York, 1973).

Kenneth, P.

R. McGill and P. Kenneth, "A convergence theorem on the iterative solution of nonlinear two-point boundary-value systems," in Proceedings of the XIVth LAF Congress, Paris, 1963, pp. 173–188; R. McGill and P. Kenneth, "Soludion of variational problems by means of a generalized Newton-Raphson operator," AIAA J. 2, 1761–1766 (1964).

Klier, K.

(a) Reference 1; (b)C. E. Jones and K. Klier, "Optical and spectroscopic methods for the study of surfaces," Ann. Rev. Mater. Sci. 2, 1–32 (1972);(c) W. D. Ross, "Theoretical computation of light scattering power: Comparison between TiO2 and air bubbles," J. Paint Tech. 43, 50–66 (1971); E. Hoffmann, C. J. Lancucki, and J. W. Spencer, "Measurement of the hiding power of paints," J. Oil Col. Chem. Assoc. 55, 292–313 (1972); (e) P. B. Mitton, "Opacity, hiding power, and tinting strength, in Pigment Handbook, Vol. III. Characterization and Physical Relationships, edited by T. C. Patton (Wiley, New York, 1973), pp. 289–339.

K. Klier, "Absorption and scattering in plane parallel turbid media," J. Opt. Soc. Am. 62, 882–885 (1972).

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).

Kubelka, P.

P. Kubelka, "New contributions to the optics of intensely lightscattering materials. Part II: Nonhomogeneous layers," J. Opt. Soc. Am. 44, 330–335 (1954).

p. Kubelka, "New contributions to the optics of intensely lightscattering materials. Part I, "1 J. Opt. Soc. Am. 38, 448–457 (1948).

McGill, R.

R. McGill and P. Kenneth, "A convergence theorem on the iterative solution of nonlinear two-point boundary-value systems," in Proceedings of the XIVth LAF Congress, Paris, 1963, pp. 173–188; R. McGill and P. Kenneth, "Soludion of variational problems by means of a generalized Newton-Raphson operator," AIAA J. 2, 1761–1766 (1964).

Miele, A.

A. Miele, "Method of particular solutions for linear, two-point boundary-value problems," J. Opt. Theory Appl. 2, 260–273 (1968); A. Miele, "General technique for solving nonlinear, two-point boundary-value problems via the method of particular solutions," J. Opt. Theory Appl. 5, 382–399 (1970).

Mischenko, E. F.

L. S. Pontryagin, V. G. Boltyansky, R. V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes, translated by D. E. Brown (MacMillan, New York, 1964).

Pontryagin, L. S.

L. S. Pontryagin, V. G. Boltyansky, R. V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes, translated by D. E. Brown (MacMillan, New York, 1964).

Roberts, S. M.

S. M. Roberts and J. S. Shipman, Two-Point Boundary Value Problems: Shooting Methods (Elsevier, New York, 1972).

Schiesser, W.

W. Schiesser, LEANS-III Introductory Programming Manual (Lehigh U. P., Bethlehem, Pa., 1971).

Shipman, J. S.

S. M. Roberts and J. S. Shipman, Two-Point Boundary Value Problems: Shooting Methods (Elsevier, New York, 1972).

Young, L. C.

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory (W. B. Saunders, Philadelphia, 1969).

1 J. Opt. Soc. Am.

p. Kubelka, "New contributions to the optics of intensely lightscattering materials. Part I, "1 J. Opt. Soc. Am. 38, 448–457 (1948).

Ann. Rev. Mater. Sci.

(a) Reference 1; (b)C. E. Jones and K. Klier, "Optical and spectroscopic methods for the study of surfaces," Ann. Rev. Mater. Sci. 2, 1–32 (1972);(c) W. D. Ross, "Theoretical computation of light scattering power: Comparison between TiO2 and air bubbles," J. Paint Tech. 43, 50–66 (1971); E. Hoffmann, C. J. Lancucki, and J. W. Spencer, "Measurement of the hiding power of paints," J. Oil Col. Chem. Assoc. 55, 292–313 (1972); (e) P. B. Mitton, "Opacity, hiding power, and tinting strength, in Pigment Handbook, Vol. III. Characterization and Physical Relationships, edited by T. C. Patton (Wiley, New York, 1973), pp. 289–339.

Aust. J. Phys.

The effects of sinusoidal variation (in the plane perpendicular to the surface normal) on the scattering characteristics have been studied:R. G. Giovanelli, "Radiative transfer in discontinuous media," Aust. J. Phys. 10, 227–239 (1957); R. G. Giovanuelli, "Radiative transfer in non-uniform media," Aust. J. Phys. 12, 164–170 (1959).

J. Opt. Soc. Am.

J. Opt. Theory Appl.

A. Miele, "Method of particular solutions for linear, two-point boundary-value problems," J. Opt. Theory Appl. 2, 260–273 (1968); A. Miele, "General technique for solving nonlinear, two-point boundary-value problems via the method of particular solutions," J. Opt. Theory Appl. 5, 382–399 (1970).

Proc. Nat. Acad. Sci. USA

(a)R. Bellman and R. Kalaba, "On the principle of invariant imbedding and propagation through inhomogeneous media," Proc. Nat. Acad. Sci. USA 42, 629–632 (1956); (b) R. W. Preisendorfer, "A mathematical foundation for radiative transfer theory," J. Math. Mech. 6, 685–730 (1957); (c) R. Bellman, R. Kalaba, and G. M. Wing, "Invariant imbedding and mathematical physics. I. Particle processes," J. Math. Phys. 1, 280–308 (1960); (d) R. Bellman, R. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (Elsevier, New York, 1963); (e) J. O. Mingle, The Invariant Imbedding Theory of Nuclear Transport (Elsevier, New York, 1973).

Other

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).

W. Schiesser, LEANS-III Introductory Programming Manual (Lehigh U. P., Bethlehem, Pa., 1971).

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

L. S. Pontryagin, V. G. Boltyansky, R. V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes, translated by D. E. Brown (MacMillan, New York, 1964).

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory (W. B. Saunders, Philadelphia, 1969).

S. M. Roberts and J. S. Shipman, Two-Point Boundary Value Problems: Shooting Methods (Elsevier, New York, 1972).

R. McGill and P. Kenneth, "A convergence theorem on the iterative solution of nonlinear two-point boundary-value systems," in Proceedings of the XIVth LAF Congress, Paris, 1963, pp. 173–188; R. McGill and P. Kenneth, "Soludion of variational problems by means of a generalized Newton-Raphson operator," AIAA J. 2, 1761–1766 (1964).

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