Abstract

The derived laws apply to layers whose scattering coefficient S and absorption coefficient K vary vertically to the surface of the layer. In the general case the differential equations of the preceding paper [ P. Kubelka, J. Opt. Soc. Am. 38, 448 ( 1948)] must be used; the coefficients, however, hitherto constant, now are functions of the distance x from the surface. In the practically important case in which K/S is constant, one may introduce the variable p, such that p0x(x)dx. One reduces thereby the nonhomogeneous to the previously treated homogeneous case.

Transmittance T1,2 and reflectance R1,2 of two nonhomogeneous sheets can be calculated by the following equations:

T1,2=T1T21-R1R2,         R1,2=R1+T12R21-R1R2,

where T1, T2, R1, R2 are the transmittances and reflectances of the single sheets, and R1 represents the reflectance of the first sheet when illuminated in the inverse direction. Analogous formulas for more sheets and formulas relating transmittance, reflectance for specimens upon black, gray or white backing surfaces, and contrast ratio, are derived.

It is shown by theory and experiment that reflectance and absorption of a nonhomogeneous specimen depend on the direction of illumination, whereas transmittance does not.

© 1954 Optical Society of America

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References

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  1. Paul Kubelka and Franz Munk, Z. tech. Phys. 12, 593 (1931). See also abstracts in English given by F. A. Steele, reference 5; Deane B. Judd, reference 6; H. A. Gardner, Physical and Chemical Examination of Paints, Varnishes, Laquers and Colors (Inst. Paints Varnishes Research, Washington, D. C., 1939), ninth edition, p. 10ff.
  2. Paul Kubelka, J. Opt. Soc. Am. 38, 448 (1948).
    [Crossref] [PubMed]
  3. Åke S :son Stenius, Svensk Papperstidn. 56, 607 (1953).
  4. F. A. Steele, Paper Trade J. 100, 37 (1935).
  5. Deane B. Judd, (with collaboration of Harrison, Sweo, Hickson, Eickjoff, Show, and Pfaffenbarger, J. Research Natl. Bur. Standards 39, 287 (1937). Deane B. Judd, Paper Trade J. (1938), January6.
    [Crossref]
  6. Åke S :son Stenius, Svensk Papperstidn. 54, 663 (1951); (reference 6) Svensk Papperstidn. 54, 701 (1951).
  7. Gurevic, Physik. Z. 31, 753 (1930).
  8. Hans Zocher, Kolloidchemisches Taschenbuch, (1948) third edition, p. 83.
  9. Deane B. Judd, J. Research Natl. Bur. Standards 12, 345 (1934) (reference 9); J. Research Natl. Bur. Standards 13, 281 (1934).

1953 (1)

Åke S :son Stenius, Svensk Papperstidn. 56, 607 (1953).

1951 (1)

Åke S :son Stenius, Svensk Papperstidn. 54, 663 (1951); (reference 6) Svensk Papperstidn. 54, 701 (1951).

1948 (1)

1937 (1)

Deane B. Judd, (with collaboration of Harrison, Sweo, Hickson, Eickjoff, Show, and Pfaffenbarger, J. Research Natl. Bur. Standards 39, 287 (1937). Deane B. Judd, Paper Trade J. (1938), January6.
[Crossref]

Deane B. Judd, (with collaboration of Harrison, Sweo, Hickson, Eickjoff, Show, and Pfaffenbarger, J. Research Natl. Bur. Standards 39, 287 (1937). Deane B. Judd, Paper Trade J. (1938), January6.
[Crossref]

1935 (1)

F. A. Steele, Paper Trade J. 100, 37 (1935).

1934 (1)

Deane B. Judd, J. Research Natl. Bur. Standards 12, 345 (1934) (reference 9); J. Research Natl. Bur. Standards 13, 281 (1934).

1931 (1)

Paul Kubelka and Franz Munk, Z. tech. Phys. 12, 593 (1931). See also abstracts in English given by F. A. Steele, reference 5; Deane B. Judd, reference 6; H. A. Gardner, Physical and Chemical Examination of Paints, Varnishes, Laquers and Colors (Inst. Paints Varnishes Research, Washington, D. C., 1939), ninth edition, p. 10ff.

1930 (1)

Gurevic, Physik. Z. 31, 753 (1930).

Eickjoff,

Deane B. Judd, (with collaboration of Harrison, Sweo, Hickson, Eickjoff, Show, and Pfaffenbarger, J. Research Natl. Bur. Standards 39, 287 (1937). Deane B. Judd, Paper Trade J. (1938), January6.
[Crossref]

Gurevic,

Gurevic, Physik. Z. 31, 753 (1930).

Harrison,

Deane B. Judd, (with collaboration of Harrison, Sweo, Hickson, Eickjoff, Show, and Pfaffenbarger, J. Research Natl. Bur. Standards 39, 287 (1937). Deane B. Judd, Paper Trade J. (1938), January6.
[Crossref]

Hickson,

Deane B. Judd, (with collaboration of Harrison, Sweo, Hickson, Eickjoff, Show, and Pfaffenbarger, J. Research Natl. Bur. Standards 39, 287 (1937). Deane B. Judd, Paper Trade J. (1938), January6.
[Crossref]

Judd, Deane B.

Deane B. Judd, (with collaboration of Harrison, Sweo, Hickson, Eickjoff, Show, and Pfaffenbarger, J. Research Natl. Bur. Standards 39, 287 (1937). Deane B. Judd, Paper Trade J. (1938), January6.
[Crossref]

Deane B. Judd, J. Research Natl. Bur. Standards 12, 345 (1934) (reference 9); J. Research Natl. Bur. Standards 13, 281 (1934).

Kubelka, Paul

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

Paul Kubelka and Franz Munk, Z. tech. Phys. 12, 593 (1931). See also abstracts in English given by F. A. Steele, reference 5; Deane B. Judd, reference 6; H. A. Gardner, Physical and Chemical Examination of Paints, Varnishes, Laquers and Colors (Inst. Paints Varnishes Research, Washington, D. C., 1939), ninth edition, p. 10ff.

Munk, Franz

Paul Kubelka and Franz Munk, Z. tech. Phys. 12, 593 (1931). See also abstracts in English given by F. A. Steele, reference 5; Deane B. Judd, reference 6; H. A. Gardner, Physical and Chemical Examination of Paints, Varnishes, Laquers and Colors (Inst. Paints Varnishes Research, Washington, D. C., 1939), ninth edition, p. 10ff.

Pfaffenbarger,

Deane B. Judd, (with collaboration of Harrison, Sweo, Hickson, Eickjoff, Show, and Pfaffenbarger, J. Research Natl. Bur. Standards 39, 287 (1937). Deane B. Judd, Paper Trade J. (1938), January6.
[Crossref]

Show,

Deane B. Judd, (with collaboration of Harrison, Sweo, Hickson, Eickjoff, Show, and Pfaffenbarger, J. Research Natl. Bur. Standards 39, 287 (1937). Deane B. Judd, Paper Trade J. (1938), January6.
[Crossref]

Steele, F. A.

F. A. Steele, Paper Trade J. 100, 37 (1935).

Stenius, Åke S :son

Åke S :son Stenius, Svensk Papperstidn. 56, 607 (1953).

Åke S :son Stenius, Svensk Papperstidn. 54, 663 (1951); (reference 6) Svensk Papperstidn. 54, 701 (1951).

Sweo,

Deane B. Judd, (with collaboration of Harrison, Sweo, Hickson, Eickjoff, Show, and Pfaffenbarger, J. Research Natl. Bur. Standards 39, 287 (1937). Deane B. Judd, Paper Trade J. (1938), January6.
[Crossref]

Zocher, Hans

Hans Zocher, Kolloidchemisches Taschenbuch, (1948) third edition, p. 83.

J. Opt. Soc. Am. (1)

J. Research Natl. Bur. Standards (2)

Deane B. Judd, (with collaboration of Harrison, Sweo, Hickson, Eickjoff, Show, and Pfaffenbarger, J. Research Natl. Bur. Standards 39, 287 (1937). Deane B. Judd, Paper Trade J. (1938), January6.
[Crossref]

Deane B. Judd, J. Research Natl. Bur. Standards 12, 345 (1934) (reference 9); J. Research Natl. Bur. Standards 13, 281 (1934).

Paper Trade J. (1)

F. A. Steele, Paper Trade J. 100, 37 (1935).

Physik. Z. (1)

Gurevic, Physik. Z. 31, 753 (1930).

Svensk Papperstidn. (2)

Åke S :son Stenius, Svensk Papperstidn. 54, 663 (1951); (reference 6) Svensk Papperstidn. 54, 701 (1951).

Åke S :son Stenius, Svensk Papperstidn. 56, 607 (1953).

Z. tech. Phys. (1)

Paul Kubelka and Franz Munk, Z. tech. Phys. 12, 593 (1931). See also abstracts in English given by F. A. Steele, reference 5; Deane B. Judd, reference 6; H. A. Gardner, Physical and Chemical Examination of Paints, Varnishes, Laquers and Colors (Inst. Paints Varnishes Research, Washington, D. C., 1939), ninth edition, p. 10ff.

Other (1)

Hans Zocher, Kolloidchemisches Taschenbuch, (1948) third edition, p. 83.

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Equations (47)

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T 1 , 2 = T 1 T 2 1 - R 1 R 2 ,             R 1 , 2 = R 1 + T 1 2 R 2 1 - R 1 R 2 ,
- d i = - ( S + K ) i d x + S j d x , d j = - ( S + K ) j d x + S i d x
p = 0 x S ( x ) d x ,
S ( x ) d x = d p .
P = 0 X S ( x ) d x .
- d i = - ( S + K / S ) i d p + j d p , d j = - ( S + K / S ) j d p + i d p .
Reflectance :             R = 1 a + b ctgh b P ;
Transmittance :             T = b a sinh b P + b cosh b P ; a ( S + K ) / S ,             b ( a 2 - 1 ) 1 2 .
a ( S + K / S ) = constant ,             S ( x ) = S w D ( x ) ,
P = 0 X S ( x ) d x = S w 0 X D ( x ) d x = S w X w ,
T 1 , 2 = T 1 T 2 ( 1 + R I R 2 + R I 2 R 2 2 + ) = T 1 T 2 1 - R I R 2 .
R 1 , 2 = R 1 + T 1 T I R 2 ( 1 + R I R 2 + R I 2 R 2 2 + ) = R 1 + T 1 T I R 2 1 - R I R 2 .
R 1 , 2 = R 1 + T 1 2 R 2 1 - R I R 2 .
T 1 , 2 = T 1 T 2 1 - R 1 R 2
R 1 , 2 = R 1 + T 1 2 R 2 1 - R 1 R 2 .
T I = T 1 ,
T 1 , 2 = T 1 T 2 / ( 1 - R I R 2 ) .
T I I , I = T I I T I / ( 1 - R 2 R I ) .
T I I , I T 1 , 2 = T I T I I T 1 T 2 .
T I I I , I I , I T 1 , 2 , 3 = T I I , I T I I I T 1 , 2 T 3 = T I T I I T I I I T 1 T 2 T 3 .
T N I I I , I I , I T 1 , 2 , 3 , n = T I T I I T I I I T N T 1 T 2 T 3 T n .
T I = T 1 ;             T I I = T 2 ;             T I I I = T 3 ; T N = T n ;
T N I I I , I I , I = T 1 , 2 , 3 , n .
T 1 , 2 , 3 = T 1 , 2 T 3 1 - R 2 , 1 R 3 ,
T 1 , 2 = T 1 T 2 1 - R 1 R 2 ,
R 2 , 1 = R 2 + T 2 2 R 1 1 - R 2 R 1 ,
T 1 , 2 , 3 = T 1 T 2 T 3 1 - R 1 R 2 - R 2 R 3 + R 1 R 2 2 R 3 - R 1 T 2 2 R 3 .
T 1 , 2 , 3 = T 3 , 2 , 1 ,
T 1 , 2 , 3 T 2 , 3 , 1
T 1 , 2 , 3 T 1 , 3 , 2 .
T 1 , 2 , 3 = T 1 T 2 , 3 / ( 1 - R I R 2 , 3 ) .
T 1 , 2 , 3 , n = T 1 T 2 , 3 , 4 , n 1 - R I R 2 , 3 , 4 , n .
R 1 , 2 , 3 , n = R 1 + T 1 R 2 , 3 , 4 , n 1 - R I R 2 , 3 , 4 , n .
R 1 ( r ) = R 1 + T 2 r 1 - R I r .
T = [ ( R 1 ( r ) - R 1 ) ( 1 / r - R I ) ] 1 2 ,
T = [ ( R I ( r ) - R I ) ( 1 / r - R 1 ) ] 1 2 .
R 1 ( r ) - R 1 1 - r R 1 = R I ( r ) - R I 1 - r R I .
T = [ ( R 1 ( 1 ) - R 1 ) ( 1 - R I ) ] 1 2 ,
R 1 ( 1 ) = R 1 ( r ) - R 1 + r ( R 1 - R 1 ( r ) R I ) r ( 1 - R I )
C 1 ( r ) = R 1 / R 1 ( r )
C 1 ( 1 ) = R 1 / R 1 ( 1 ) ,
C 1 ( 1 ) = r C 1 ( r ) ( 1 - R I ) 1 - r C 1 ( r ) - C 1 ( r ) - r R I .
R = a - ( T 2 + a 2 - 1 ) 1 2 .
a ( x ) = a ( X - x ) .
T 1 , 2 , 3 = T 1 T 2 T 3 1 - ( R 1 R 2 + R 2 R 3 + R 1 R 3 ) + R 1 R 2 R 3 · ( 1 + R 2 2 - T 2 2 ) / R 2 .
( 1 + R 2 2 - T 2 2 ) / R 2 .
( 1 + R 2 2 - T 2 2 ) / R 2 = 2 a 2 = 1 / R 2 + R 2 ,