## Abstract

The system of differential equations of Kubelka-Munk,

$-di=-(S+K)idx+Sjdx, dj=-(S+K)jdx+Sidx$

(i, j⋯ intensities of the light traveling inside a plane-parallel light-scattering specimen towards its unilluminated and its illuminated surface; x⋯ distance from the unilluminated surface S, K⋯ constants), has been derived from a simplified model of traveling of light in the material. Now, without simplifying assumptions the following exact system is derived:

$-di=-12(S+K)uidx+12Svjdx,dj=-12(S+K)vjdx+12Suidx,$

$u≡∫0π/2(∂i/i∂φ)(dφ/cosφ), v≡∫0π/2(∂j/j∂φ)(dφ/cosφ)$, φ≡angle from normal of the light). Both systems become identical when u=v=2, that is, for instance, when the material is perfectly dull and when the light, is perfectly diffused or if it is parallel and hits the specimen under an angle of 60° from normal. Consequently, the different formulas Kubelka-Munk got by integration of their differential equations are exact when these conditions are fulfilled. The Gurevic and Judd formulas, although derived in another way by their authors, may be got from the Kubelka-Munk differential equations too. Consequently, they are exact under the same conditions. The integrated equations may be adapted for practical use by introducing hyperbolic functions and the secondary constants $a=12(1/R∞+R∞)$ and $b=12(1/R∞-R∞)$, (R≡reflectivity). Reflectance R, for instance, is then represented by the formula

$R=1-Rg(a-b ctghbSX)a+b ctghbSX-Rg$

(Rg≡reflectance of the backing, X=thickness of the specimen) and transmittance T by the formula

$T=ba sinhbSX+b coshbSX.$

In many practical cases the exact formulas may be replaced by appropriated approximations.

© 1948 Optical Society of America

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### References

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1. G. G. Stokes, Proc. Roy. Soc. London 11, 545 (1860–62).
[Crossref]
2. H. J. Channon, F. F. Renwick, and B. V. Storr, Proc. Roy. Soc. London 94, 222 (1918).
[Crossref]
3. H. D. Bruce, , Nat. Bur. Stand. (1926).
4. M. Gurevic, Physik. Zeits. 31, 753 (1930).
5. P. Kubelka and F. Munk, Zeits. f. tech. Physik 12, 593 (1931); see also: F. A. Steele, reference 5; Gardner, Handbook Paints, Varnishes, Lacquers and Colors (H. A. Gardner Laboratory, Inc., Bethesda, Maryland, 1935) seventh edition, p. 16; Judd, see reference 6.
6. D. B. Judd, J. Research Nat. Bur. Stand.12, 345and J. Research Nat. Bur. Stand. 13, 281 (1934).
[Crossref]
7. L. Amy, Rev. d’optique 16, 81 (1937).
8. T. Smith, Trans. Opt. Soc. (London) 33, 150 (1931).
[Crossref]
9. Zocher, Kolloidchemisches Taschenbuch.
10. D. B. Judd, J. Research Nat. Bur. Stand. 19, 287 (1937) and Paper Trade J. 106, 5 (1938).
[Crossref]
11. F. A. Steele, Paper Trade J. 100, 37 (1935).
12. D. B. Judd, Symposium on Color, 1931 (A.S.T.M.) (remark in the discussion, following the lecture of R. H. Sawyer).
13. George W. Ingle, , May (1942).

#### 1937 (2)

L. Amy, Rev. d’optique 16, 81 (1937).

D. B. Judd, J. Research Nat. Bur. Stand. 19, 287 (1937) and Paper Trade J. 106, 5 (1938).
[Crossref]

#### 1935 (1)

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

#### 1931 (2)

P. Kubelka and F. Munk, Zeits. f. tech. Physik 12, 593 (1931); see also: F. A. Steele, reference 5; Gardner, Handbook Paints, Varnishes, Lacquers and Colors (H. A. Gardner Laboratory, Inc., Bethesda, Maryland, 1935) seventh edition, p. 16; Judd, see reference 6.

T. Smith, Trans. Opt. Soc. (London) 33, 150 (1931).
[Crossref]

#### 1930 (1)

M. Gurevic, Physik. Zeits. 31, 753 (1930).

#### 1918 (1)

H. J. Channon, F. F. Renwick, and B. V. Storr, Proc. Roy. Soc. London 94, 222 (1918).
[Crossref]

#### Amy, L.

L. Amy, Rev. d’optique 16, 81 (1937).

#### Bruce, H. D.

H. D. Bruce, , Nat. Bur. Stand. (1926).

#### Channon, H. J.

H. J. Channon, F. F. Renwick, and B. V. Storr, Proc. Roy. Soc. London 94, 222 (1918).
[Crossref]

#### Gurevic, M.

M. Gurevic, Physik. Zeits. 31, 753 (1930).

#### Ingle, George W.

George W. Ingle, , May (1942).

#### Judd, D. B.

D. B. Judd, J. Research Nat. Bur. Stand. 19, 287 (1937) and Paper Trade J. 106, 5 (1938).
[Crossref]

D. B. Judd, Symposium on Color, 1931 (A.S.T.M.) (remark in the discussion, following the lecture of R. H. Sawyer).

D. B. Judd, J. Research Nat. Bur. Stand.12, 345and J. Research Nat. Bur. Stand. 13, 281 (1934).
[Crossref]

#### Kubelka, P.

P. Kubelka and F. Munk, Zeits. f. tech. Physik 12, 593 (1931); see also: F. A. Steele, reference 5; Gardner, Handbook Paints, Varnishes, Lacquers and Colors (H. A. Gardner Laboratory, Inc., Bethesda, Maryland, 1935) seventh edition, p. 16; Judd, see reference 6.

#### Munk, F.

P. Kubelka and F. Munk, Zeits. f. tech. Physik 12, 593 (1931); see also: F. A. Steele, reference 5; Gardner, Handbook Paints, Varnishes, Lacquers and Colors (H. A. Gardner Laboratory, Inc., Bethesda, Maryland, 1935) seventh edition, p. 16; Judd, see reference 6.

#### Renwick, F. F.

H. J. Channon, F. F. Renwick, and B. V. Storr, Proc. Roy. Soc. London 94, 222 (1918).
[Crossref]

#### Smith, T.

T. Smith, Trans. Opt. Soc. (London) 33, 150 (1931).
[Crossref]

#### Steele, F. A.

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

#### Stokes, G. G.

G. G. Stokes, Proc. Roy. Soc. London 11, 545 (1860–62).
[Crossref]

#### Storr, B. V.

H. J. Channon, F. F. Renwick, and B. V. Storr, Proc. Roy. Soc. London 94, 222 (1918).
[Crossref]

#### Zocher,

Zocher, Kolloidchemisches Taschenbuch.

#### J. Research Nat. Bur. Stand. (1)

D. B. Judd, J. Research Nat. Bur. Stand. 19, 287 (1937) and Paper Trade J. 106, 5 (1938).
[Crossref]

#### Paper Trade J. (1)

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

#### Physik. Zeits. (1)

M. Gurevic, Physik. Zeits. 31, 753 (1930).

#### Proc. Roy. Soc. London (2)

G. G. Stokes, Proc. Roy. Soc. London 11, 545 (1860–62).
[Crossref]

H. J. Channon, F. F. Renwick, and B. V. Storr, Proc. Roy. Soc. London 94, 222 (1918).
[Crossref]

#### Rev. d’optique (1)

L. Amy, Rev. d’optique 16, 81 (1937).

#### Trans. Opt. Soc. (London) (1)

T. Smith, Trans. Opt. Soc. (London) 33, 150 (1931).
[Crossref]

#### Zeits. f. tech. Physik (1)

P. Kubelka and F. Munk, Zeits. f. tech. Physik 12, 593 (1931); see also: F. A. Steele, reference 5; Gardner, Handbook Paints, Varnishes, Lacquers and Colors (H. A. Gardner Laboratory, Inc., Bethesda, Maryland, 1935) seventh edition, p. 16; Judd, see reference 6.

#### Other (5)

D. B. Judd, J. Research Nat. Bur. Stand.12, 345and J. Research Nat. Bur. Stand. 13, 281 (1934).
[Crossref]

H. D. Bruce, , Nat. Bur. Stand. (1926).

D. B. Judd, Symposium on Color, 1931 (A.S.T.M.) (remark in the discussion, following the lecture of R. H. Sawyer).

George W. Ingle, , May (1942).

Zocher, Kolloidchemisches Taschenbuch.

### Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

### Tables (4)

Table IA Definition of terms used in Section I.

Table IB Definition of terms used in Sections II and III.

Table II Exact formulas for practical use.

Table III Approximations (the formulas compiled in this table may be considered as examples only. When other ones are desired for special purposes they may be got by combining the quoted ones, or developed from the corresponding exact formulas in the ways indicated in Section II).

### Equations (66)

$- d i = - ( S + K ) i d x + S j d x , d j = - ( S + K ) j d x + S i d x$
$- d i = - 1 2 ( S + K ) u i d x + 1 2 S v j d x , d j = - 1 2 ( S + K ) v j d x + 1 2 S u i d x ,$
$R = 1 - R g ( a - b ctgh b S X ) a + b ctgh b S X - R g$
$T = b a sinh b S X + b cosh b S X .$
$d ξ i = d x · ∫ 0 π / 2 ∂ i i ∂ φ · d φ cos φ ≡ u · d x ,$
$d ξ j = d x · ∫ 0 π / 2 ∂ j j ∂ φ · d φ cos φ ≡ v · d x .$
$( σ + ∊ ) i d ξ i = ( σ + ∊ ) u i d x ,$
$∊ i d ξ i = ∊ u i d x$
$σ i d ξ i = σ u i d x$
$( σ + ∊ ) j d ξ j = ( σ + ∊ ) v j d x$
$σ j d ξ j = σ v j d x .$
$- d i = - ( σ + ∊ ) u i d x + σ v j d x ,$
$d j = - ( σ + ∊ ) v j d x + σ u i d x .$
$- d i = - ( S + K ) i d x + S j d x ,$
$d j = - ( S + K ) j d x + S i d x ,$
$v = u = const .$
$∂ i * ∂ φ = i * sin 2 φ ,$
$u * = ∫ 0 π / 2 sin 2 φ d φ cos φ = 2.$
$S = u σ , K = u ∊ .$
$S ≡ u * σ = 2 σ ,$
$K ≡ u * ∊ = 2 ∊ .$
$( S + K ) / S ≡ a ,$
$- d i / S d x = - a i + j ,$
$d j / S d x = - a j + i .$
$d r / S d x = r 2 - 2 a r + 1$
$x = 1 b S A r ctgh 1 - a r 0 b r 0 ,$
$b = ( a 2 - 1 ) 1 2 ,$
$r 0 = 1 a + b ctgh b S x .$
$j = j 0 = r 0 i 0 = i 0 / ( a + b ctgh b S x )$
$i = i 0 = j 0 / r 0 = j 0 ( a + b ctgh b S x )$
$i 0 = const . ( a sinh b S x + b cosh b S x ) = I ( a sinh b S x + b cosh b S x ) / ( a sinh b S X + b cosh b S X )$
$j 0 = const . sinh b S x = R 0 I sinh b S x / sinh b S X .$
$j 0 = I sinh b S x a sinh b S X + b cosh b S X .$
$R ∞ ≡ lim x → ∞ · r , R ∞ = a - ( a 2 - 1 ) 1 2 = a - b .$
$R ∞ = 1 / ( a + b ) ,$
$a = 1 2 ( 1 / R ∞ + R ∞ ) ,$
$b = 1 2 ( 1 / R ∞ - R ∞ ) .$
$b = ( a 2 - 1 ) 1 2$
$a = ( b 2 + 1 ) 1 2$
$S X = 1 b ( A r ctgh a - R b - A r ctgh a - R g b ) ,$
$R = 1 - R g ( a - b ctgh b S X ) a + b ctgh b S X - R g .$
$R = 1 R ∞ ( R g - R ∞ ) - R ∞ ( R g - 1 R ∞ ) exp ( S X ( 1 R ∞ - R ∞ ) ) ( R g - R ∞ ) - ( R g - 1 R ∞ ) exp ( S X ( 1 R ∞ - R ∞ ) )$
$R 0 = 1 a + b ctgh b S X = sinh b S X a sinh b S X + b cosh b S X .$
$T = b / ( a sinh b S X + b cosh b S X ) .$
$S X = [ A R sinh ( b / T ) - A r sinh b ] / b .$
$T = ( 1 - R ∞ ) 2 e - b S X 1 - R ∞ 2 e - 2 b S X ,$
$T = 1 2 ( 1 R ∞ - R ∞ ) 1 2 ( 1 R ∞ + R ∞ ) sinh b S X + 1 2 ( 1 R ∞ - R ∞ ) cosh b S X = 1 - R ∞ 2 ( 1 + R ∞ 2 ) · 1 2 ( e b S X - e - b S X ) + ( 1 - R ∞ 2 ) · 1 2 ( e b S X + e - b S X ) = ( 1 - R ∞ 2 ) e - b S X 1 - R ∞ 2 e - 2 b S X .$
$T 2 + b 2 = ( a - R 0 ) 2 .$
$T = ( ( a - R 0 ) 2 - b 2 ) 1 2 ,$
$R 0 = a - ( T 2 + b 2 ) 1 2 .$
$a = 1 + R 0 2 + T 2 2 R 0 .$
$a = 1 2 ( R + R 0 - R + R g R 0 R g ) .$
$R = R 0 - R g ( 2 a R 0 - 1 ) 1 - R g R 0 .$
$C 1 = R g C R g ( 1 - R 0 ) 1 + R g C R g - C R g - R g R 0$
$a = 1 2 ( R 1 + R 0 - R 1 + 1 R 0 ) ,$
$R 1 = R 0 / C 1 and R = R 0 / C R g .$
$R = R 0 + T 2 R g 1 - R 0 R g .$
$sinh b S X = b S X / 1 ! + b 3 S 3 X 3 / 3 ! + b 5 S 5 X 5 / 5 ! + ⋯ , cosh b S X = 1 + b 2 S 2 X 2 / 2 ! + b 4 S 4 X 4 / 4 ! + ⋯ , ctgh b S X = 1 / b S X + b S X / 3 - b 3 S 3 X 3 / 45 + - ⋯ ,$
$b → 0 a → 1 b = ( a 2 - 1 ) 1 2 ≈ ( 2 ( a - 1 ) ) 1 2 .$
$sinh b S X → 1 2 e b S X ← cosh b S X , ctgh b S X → ( 1 - e - 2 b S X ) → 1.$
$b → 1 2 R ∞ ← a$
$d R 0 = S T 2 d X , d T = - T ( S + E - S R 0 ) d X ,$
$d 2 i / d x 2 = b 2 S 2 i and d 2 j / d x 2 = b 2 S 2 j .$
$i = c 1 sinh b S x + c 2 cosh b S x , j = c 3 sinh b S x + c 4 cosh b S x .$
$ln R ∞ = Arcosh a = - Arsinh b .$
$d ln r / S d x = 2 ( cosh ln r - cosh ln R ∞ ) .$