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 K2/S2 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 Optical Society of America

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References

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  1. G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).
    [Crossref]
  2. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [Crossref] [PubMed]
  3. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part II: Nonhomogeneous layers,” J. Opt. Soc. Am. 44, 330–335 (1954).
    [Crossref]
  4. K. Klier, “Absorption and scattering in plane parallel turbid media,” J. Opt. Soc. Am. 62, 882–885 (1972).
    [Crossref]
  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. Giovanelli, “Radiative transfer in non-uniform media,” Aust. J. Phys. 12, 164–170 (1959).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  14. W. Schiesser, LEANS-III Introductory Programming Manual (Lehigh U. P., Bethlehem, Pa., 1971).

1972 (1)

1968 (1)

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).
[Crossref]

1957 (1)

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. Giovanelli, “Radiative transfer in non-uniform media,” Aust. J. Phys. 12, 164–170 (1959).
[Crossref]

1956 (1)

(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 (1)

1948 (1)

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. Giovanelli, “Radiative transfer in non-uniform media,” Aust. J. Phys. 12, 164–170 (1959).
[Crossref]

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).
[Crossref]

Klier, K.

Kortüm, G.

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

Kubelka, P.

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).
[Crossref]

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).
[Crossref]

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).

Aust. J. Phys. (1)

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. Giovanelli, “Radiative transfer in non-uniform media,” Aust. J. Phys. 12, 164–170 (1959).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Theory Appl. (1)

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).
[Crossref]

Proc. Nat. Acad. Sci. USA (1)

(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 (8)

(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.

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

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

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).
[Crossref]

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

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Figures (7)

FIG. 1
FIG. 1

Illustration of the convergence of the generalized Newton-Raphson operator for the particular case of r(1) = 0.2601 in a slab of unitary scattering power with black or vacuous backing, r(0) = 0. The trial solution r0 corresponds to the homogeneous KM solution with αKM = 1.0 and λ0 = −0.02. The distances ρi, i = 1, 2, …, 6, for the illustrated iterations are respectively, 3.12 × 102, 4.63 × 10−1, 2.21 × 10−1, 9.28 × 10−3, 4.52 × 10−6, and 3.84 × 10−11.

FIG. 2
FIG. 2

Perspective view of the variation with scattering depth of the reflectance in unitary scattering power slabs for: (a) the optimal inhomogeneous solutions; (b) the corresponding homogeneous KM solutions. The grid separations in logαKM and p are 0.1 and 0.02, respectively.

FIG. 3
FIG. 3

Perspective view of the optimal inhomogeneous distributions associated with the reflectance solutions of Fig. 2(a). The grid separations are the same as in Fig. 2.

FIG. 4
FIG. 4

Selected reflectances and their corresponding optimal distributions for various front boundary reflectances, r(1), which are listed to the right of the curves. These cases, as in the previous figures, correspond to unitary scattering power slabs with r(0) = 0.

FIG. 5
FIG. 5

Comparison of the unitary scattering power results, with r(0) = 0, for the optimal inhomogeneous and KM homogeneous distributions in terms of the quantities of (a) Eq. (1.7) and (b) Eq. (2.7). Over the range of boundary values investigated, which may be denoted by αKM, the optimal distributions α result in enhancements of transmission that range from being almost nil to over three orders of magnitude, while the fractional reduction in I3 relative to the homogeneous case ranges from 30% to 80%.

FIG. 6
FIG. 6

Effects of scattering depth and frontal boundary reflectance, for the r(0) = 0 case, on the reflectances and optimal distributions. The three boundary conditions r(P) = 0.25, 0.50, and 0.75 are utilized, where for r(P) = 0.25, P = 1, 2, 3, 4, and 5; for r(P) = 0.50, P = 1.5, 2, 3, 4, and 5; for r(P) = 0.75, P = 3.2, 4, and 5. The correspondence between associated r and α curves is made by considering the terminal values at p = P and noting that αi(P) > αj(P) when ri(P) < rj(P), where i and j correspond to the values 0.25, 0.50, and 0.75 for r(P).

FIG. 7
FIG. 7

Comparison of the effects of the rear boundary condition, r(0) = 0, 0.25, 0.50, 0.75, and 1, on the behavior of r and α with r(P) = 0.25 for slabs with scattering powers of 5 and 1 (inserts).

Tables (2)

Tables Icon

TABLE I Efficiency and transmittance enhancement effects of optimal distributions as a function of the scattering power, P, and the reflectance, r(P), for slabs with r(0) = 0.

Tables Icon

TABLE II Effects of the backing, r(0), on the efficiency and transmittance enhancement of optimal distributions in slabs with P = 5 for various r(5) values.

Equations (31)

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p ( x ) = 0 x d t S ( t ) .
d I d p = ( 1 + α ) I - J ,
d J d p = I - ( 1 + α ) J ,
d r d p = 1 - 2 r - 2 α r + r 2 ,
T ( p ) = exp { - p P d t [ 1 + α ( t ) - r ( t ) ] } .
r ( p ) = [ 1 - r ( 0 ) ( 1 + α KM ) ] sinh ( β p ) + r ( 0 ) β cosh ( β p ) [ 1 + α KM - r ( 0 ) ] sinh ( β p ) + β cosh ( β p ) ,
T ( 0 ) = [ 1 - 2 r ( P ) ( 1 + α KM ) + r 2 ( P ) ] 1 / 2
T ( 0 ) = [ [ r ( P ) - r ˆ ] ( 1 r ( 0 ) - r ˆ ) ] 1 / 2 ,
I 1 = 0 P d p α 2 ( p ) ,
I 2 = 0 P d p [ α 2 + λ ( d r d p - 1 + 2 r + 2 α r - r 2 ) ] ,
δ I 2 = 0.
α = - λ r ,
d r d p = 1 - 2 r + 2 λ r 2 + r 2 ,             and
d λ d p = 2 λ ( 1 - λ r - r ) ,
I 3 = 0 P d p α ( p )
λ ( p ) λ ( P ) = exp ( 2 P p d t [ 1 + α ( t ) - r ( t ) ] ) ,
( λ ( p ) λ ( P ) ) 1 / 2 = T ( p ) .
d α d p = - λ ( 1 - r 2 ) 0.
d d p [ r n λ n ] = [ f ( r n , λ n ) g ( r n , λ n ) ] .
d d p [ r n λ n ] = C ( r n - 1 , λ n - 1 ) [ r n λ n ] + D ( r n - 1 , λ n - 1 ) ,
C ( r n - 1 , λ n - 1 ) = [ f / r f / λ g / r g / λ ] ( r n - 1 , λ n - 1 ) = [ - 2 ( 1 - 2 λ n - 1 r n - 1 - r n - 1 ) 2 r n - 1 2 - 2 λ n - 1 ( λ n - 1 + 1 ) 2 ( 1 - 2 λ n - 1 r n - 1 - r n - 1 ) ] ,
D ( r n - 1 , λ n - 1 ) = [ f g ] ( r n - 1 , λ n - 1 ) - C ( r n - 1 , λ n - 1 ) [ r n - 1 λ n - 1 ] = [ 1 - r n - 1 2 ( 1 + 4 λ n - 1 ) 2 λ n - 1 r n - 1 ( 1 + 2 λ n - 1 ) ] .
ρ n = max p [ 0 , P ] ( | 1 - r n r n - 1 | , | 1 - λ n λ n - 1 | ) ,
[ r n λ n ] = k 1 [ r n 1 λ n 1 ] + k 2 [ r n 2 λ n 2 ] ,
k 1 ( d d p [ r n 1 λ n 1 ] - C [ r n 1 λ n 1 ] ) + k 2 ( d d p [ r n 2 λ n 2 ] - C [ r n 2 λ n 2 ] ) = D ,
( k 1 + k 2 ) D = D ,
k 1 + k 2 = 1.
r n 1 ( P ) k 1 + r n 2 ( P ) k 2 = r ( P ) .
[ k 1 k 2 ] = 1 r n 1 ( P ) - r n 2 ( P ) [ 1 - r n 2 ( P ) - 1 r n 2 ( P ) ] [ r ( P ) 1 ] ;
α ( p ) = α ( 1 ) ( 1 - 10 - k p ) ,
η = max p [ 0 , 1 ] r α - r α KM ,