Abstract

In 1942 Duntley published a paper giving a theory that describes the reflectance and transmittance of light-diffusing mats. This theory avoids the logical objections that apply to the two-constant theory developed by Kubelka and Munk, and other contributors. This paper describes new ways of determining the five coefficients that appear in Duntley’s theory. It discusses the characteristic features of light-scattering mats that follow this theory, emphasizing the points of difference with the two-constant theory. Results of laboratory measurements are given which demonstrate the utility of Duntley’s theory, and show the dangers of using the two-constant theory as a research tool.

© 1965 Optical Society of America

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References

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  1. A. Schuster, Astrophys. J. 21, 1 (1905).
    [Crossref]
  2. P. Kubelka and F. Munk, Z. Tech. Phys. 12, 593 (1931).
  3. S. Q. Duntley, J. Opt. Soc. Am. 32, 61 (1942).
    [Crossref]
  4. H. McNicholas, J. Res. Natl. Bur. Std. 1, 29 (1928).
  5. D. B. Judd and et al., J. Res. Natl. Bur. Std. 19, 287 (1937).
    [Crossref]
  6. Å. S. Stenius, Svensk Papperstid. 54, 663 (1951).
  7. W. J. Foote, Paper Trade J. 109, No. 25, 31 (1939).
  8. L. Nordman (private communication).
  9. E. W. Arnold, Tappi 46, 250 (1963).
  10. J. A. Ryde, Proc. Roy. Soc. (London) 131A, 451 (1931).
  11. J. A. Van den Akker, Tappi 32, 498 (1949).
  12. F. A. Steele, Paper Trade J. 100, No. 12, 37 (1935).
  13. Å. S. Stenius, Svensk Papperstid. 56, 607 (1953).
  14. P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948).
    [Crossref] [PubMed]
  15. K. Shibata, J. Opt. Soc. Am. 47, 172 (1957).
    [Crossref]
  16. F. Kottler, J. Opt. Soc. Am. 50, 483 (1960).
    [Crossref]
  17. A. L. Lathrop, Tappi 47, 789 (1964).
  18. L. A. Jones, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 140 (1922).
    [Crossref]
  19. J. A. Van den Akker, L. R. Dearth, and W. M. Shillcox, J. Opt. Soc. Am. 46, 378 (1956).
  20. L. R. Dearth (private communication).

1964 (1)

A. L. Lathrop, Tappi 47, 789 (1964).

1963 (1)

E. W. Arnold, Tappi 46, 250 (1963).

1960 (1)

1957 (1)

1956 (1)

J. A. Van den Akker, L. R. Dearth, and W. M. Shillcox, J. Opt. Soc. Am. 46, 378 (1956).

1953 (1)

Å. S. Stenius, Svensk Papperstid. 56, 607 (1953).

1951 (1)

Å. S. Stenius, Svensk Papperstid. 54, 663 (1951).

1949 (1)

J. A. Van den Akker, Tappi 32, 498 (1949).

1948 (1)

1942 (1)

1939 (1)

W. J. Foote, Paper Trade J. 109, No. 25, 31 (1939).

1937 (1)

D. B. Judd and et al., J. Res. Natl. Bur. Std. 19, 287 (1937).
[Crossref]

1935 (1)

F. A. Steele, Paper Trade J. 100, No. 12, 37 (1935).

1931 (2)

J. A. Ryde, Proc. Roy. Soc. (London) 131A, 451 (1931).

P. Kubelka and F. Munk, Z. Tech. Phys. 12, 593 (1931).

1928 (1)

H. McNicholas, J. Res. Natl. Bur. Std. 1, 29 (1928).

1922 (1)

L. A. Jones, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 140 (1922).
[Crossref]

1905 (1)

A. Schuster, Astrophys. J. 21, 1 (1905).
[Crossref]

Arnold, E. W.

E. W. Arnold, Tappi 46, 250 (1963).

Dearth, L. R.

J. A. Van den Akker, L. R. Dearth, and W. M. Shillcox, J. Opt. Soc. Am. 46, 378 (1956).

L. R. Dearth (private communication).

Duntley, S. Q.

Foote, W. J.

W. J. Foote, Paper Trade J. 109, No. 25, 31 (1939).

Jones, L. A.

L. A. Jones, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 140 (1922).
[Crossref]

Judd, D. B.

D. B. Judd and et al., J. Res. Natl. Bur. Std. 19, 287 (1937).
[Crossref]

Kottler, F.

Kubelka, P.

P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948).
[Crossref] [PubMed]

P. Kubelka and F. Munk, Z. Tech. Phys. 12, 593 (1931).

Lathrop, A. L.

A. L. Lathrop, Tappi 47, 789 (1964).

McNicholas, H.

H. McNicholas, J. Res. Natl. Bur. Std. 1, 29 (1928).

Munk, F.

P. Kubelka and F. Munk, Z. Tech. Phys. 12, 593 (1931).

Nordman, L.

L. Nordman (private communication).

Ryde, J. A.

J. A. Ryde, Proc. Roy. Soc. (London) 131A, 451 (1931).

Schuster, A.

A. Schuster, Astrophys. J. 21, 1 (1905).
[Crossref]

Shibata, K.

Shillcox, W. M.

J. A. Van den Akker, L. R. Dearth, and W. M. Shillcox, J. Opt. Soc. Am. 46, 378 (1956).

Steele, F. A.

F. A. Steele, Paper Trade J. 100, No. 12, 37 (1935).

Stenius, Å. S.

Å. S. Stenius, Svensk Papperstid. 56, 607 (1953).

Å. S. Stenius, Svensk Papperstid. 54, 663 (1951).

Van den Akker, J. A.

J. A. Van den Akker, L. R. Dearth, and W. M. Shillcox, J. Opt. Soc. Am. 46, 378 (1956).

J. A. Van den Akker, Tappi 32, 498 (1949).

Astrophys. J. (1)

A. Schuster, Astrophys. J. 21, 1 (1905).
[Crossref]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. and Rev. Sci. Instr. (1)

L. A. Jones, J. Opt. Soc. Am. and Rev. Sci. Instr. 6, 140 (1922).
[Crossref]

J. Res. Natl. Bur. Std. (2)

H. McNicholas, J. Res. Natl. Bur. Std. 1, 29 (1928).

D. B. Judd and et al., J. Res. Natl. Bur. Std. 19, 287 (1937).
[Crossref]

Paper Trade J. (2)

W. J. Foote, Paper Trade J. 109, No. 25, 31 (1939).

F. A. Steele, Paper Trade J. 100, No. 12, 37 (1935).

Proc. Roy. Soc. (London) (1)

J. A. Ryde, Proc. Roy. Soc. (London) 131A, 451 (1931).

Svensk Papperstid. (2)

Å. S. Stenius, Svensk Papperstid. 56, 607 (1953).

Å. S. Stenius, Svensk Papperstid. 54, 663 (1951).

Tappi (3)

J. A. Van den Akker, Tappi 32, 498 (1949).

A. L. Lathrop, Tappi 47, 789 (1964).

E. W. Arnold, Tappi 46, 250 (1963).

Z. Tech. Phys. (1)

P. Kubelka and F. Munk, Z. Tech. Phys. 12, 593 (1931).

Other (2)

L. Nordman (private communication).

L. R. Dearth (private communication).

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Figures (5)

F. 1
F. 1

Example that shows the comparison between some of the parameters in the K–M and Duntley theories, and how this comparison changes with mat absorption. (1) B/S(R′,T′) for X=5.0; (2) k(R′,T′)/μ for X=5.0; (3) B/S; (4) k/μ; (5) Q′; and (6) 1+P′.

F. 2
F. 2

Kubelka–Munk scattering and absorption coefficients for a viscose-fiber paper as computed for different mat thicknesses, X. The abscissa is linear with respect to X−1. (1) S(R′,T′,R′) and k(R′,T′,R′); (2) □, S(R′,R′) and k(R′,R′); (3) ○, S(T′,R) and k(T′,R′); (4) △, k(R′,T′).

F. 3
F. 3

Kubelka–Munk scattering and absorption coefficients of a low-reflectance mat for different mat thicknesses, X. The Duntley-theory parameters are B=0.76, B′=0.59, F′=0.75, μ=2.3, μ′=2.0, R′=0.1086, R=0.1262, Q′=2.215, P′=0.171, S=0.6515, and k=2.38. These scattering coefficients and the μ/μ′ ratio are sensibly the same as those given for the pink tissue paper in Table I.

F. 4
F. 4

Difference between the functions R′(X) and T′(X) and their approximate values as computed from the K–M theory. The data correspond to the viscose-fiber paper. The assumed Duntley coefficients are those given in Fig. 2. The K–M coefficients are S=0.8155 and k=0.0204.

F. 5
F. 5

Example that shows the difficulty of determining the Duntley-theory parameters by curve fitting the functions R′(X) and T′(X).

Tables (1)

Tables Icon

Table I Characteristic Duntley coefficients for two paper specimens.

Equations (34)

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R = ( S sinh K X ) / D ,
T = K / D ,
D = ( k + S ) sinh K X + K cosh K X .
T c = exp ( q X ) ,
R = Q R + P T T c P ,
T = Q T + P T c R ( Q 1 ) T c .
P = [ ( μ μ ) B + ( B B ) ( B + F ) ] / C ,
Q = [ ( μ + μ ) F + ( B + F ) ( B + F ) ] / C ,
C = ( q ) 2 K 2 .
Q = ( B P F ) / ( μ + B q ) .
R = Q R P ,
T = Q T .
T a + b = T a T b / ( 1 R a R b ) ,
R a + b = R a + T a T a R b / ( 1 R a R b ) .
R b = ( R a + b R a ) T b / T b + a T a .
q X = ln P T ln ( R + P Q R ) .
q X = ln ( P R Q + 1 ) ln ( T Q T ) .
ln [ T / ( R R ) ] = K X ln R .
S = 2 R K / [ 1 ( R ) 2 ] = b B / b B / ( 1 + P ) when μ is small ;
k = ( 1 R ) K / ( 1 + R ) = ( 1 R ) ( 1 + R ) μ / [ ( 1 R ) ( 1 + R ) ] ( 1 + P ) μ when μ is small .
d S ( R , T ) / d X 1 = ( 1 / b ) × { ln [ 1 ( R ) 2 ] ln Q ln ( 1 R 2 ) } ,
d S ( R , T , R ) / d X 1 = ( 1 / b ) ln ( R / R ) ,
d S ( R , R ) / d X 1 = 1 2 [ d S ( R , T ) / d X 1 + d S ( R , T , R ) / d X 1 ] .
( R R ) / R = [ b exp ( b S X ) ] / ( a sinh b S X b cosh b S X ) .
( R R ) / R T = exp ( b S X ) .
( R R ) / T = ( R R ) / T .
S = ( 1 / b X ) ln [ T R / ( R R ) ] .
S ( T , R , R ) = ( 1 / b X ) { ln R + ln [ T / ( R R ) ] } .
K X = ln R + ln [ T / ( R R ) ] .
S ( T , R , R ) = K / b ( 1 / b X ) ln ( R / R ) .
( R R ) / ( R R 3 ) = exp ( 2 b S X ) ,
S ( R , R ) = ( 1 2 b X ) { ln [ R ( R ) 3 ] ln [ Q ( R R ) ] } .
ln ( R R ) = ln ( R R 3 ) 2 K X ,
S ( R , R ) = K / b + ( 1 2 b X ) × { ln [ R ( R ) 3 ] ln Q ln ( R R 3 ) } .