Abstract

The equation relating dot area to half-tone density, when sideways scattering of light takes place in the paper, has been modified to take into account multiple internal reflections. Curves showing the relationship between dot area and density for various ink densities and amounts of first-surface reflectance are given. Callahan’s conclusion that these effects are negligible is not in accordance with the authors’ experience, and the new equations indicate the possibility of densities much higher than the limiting values derived from Callahan’s equations and from our earlier equation which neglected multiple internal reflectance.

© 1953 Optical Society of America

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References

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  1. F. P. Callahan, J. Opt. Soc. Am. 42, 104–105 (1952).
    [Crossref]
  2. J. A. C. Yule and W. J. Nielsen, Proc. Tech. Assoc. Graphic Arts 3, 65–76 (1951).
  3. F. C. Williams and F. R. Clapper, J. Opt. Soc. Am. 40, 796 (1950).
  4. D. B. Judd, J. Research Natl. Bur. Standards 29, 329–332 (1942).
    [Crossref]
  5. A. Murray, J. Franklin Inst. 221, 721–744 (1936).
    [Crossref]
  6. See reference 2, discussion.
  7. J. A. C. Yule and R. Colt, Proc. Tech. Assoc. Graphic Arts 3, 77–82 (1951).

1952 (1)

1951 (2)

J. A. C. Yule and W. J. Nielsen, Proc. Tech. Assoc. Graphic Arts 3, 65–76 (1951).

J. A. C. Yule and R. Colt, Proc. Tech. Assoc. Graphic Arts 3, 77–82 (1951).

1950 (1)

F. C. Williams and F. R. Clapper, J. Opt. Soc. Am. 40, 796 (1950).

1942 (1)

D. B. Judd, J. Research Natl. Bur. Standards 29, 329–332 (1942).
[Crossref]

1936 (1)

A. Murray, J. Franklin Inst. 221, 721–744 (1936).
[Crossref]

Callahan, F. P.

Clapper, F. R.

F. C. Williams and F. R. Clapper, J. Opt. Soc. Am. 40, 796 (1950).

Colt, R.

J. A. C. Yule and R. Colt, Proc. Tech. Assoc. Graphic Arts 3, 77–82 (1951).

Judd, D. B.

D. B. Judd, J. Research Natl. Bur. Standards 29, 329–332 (1942).
[Crossref]

Murray, A.

A. Murray, J. Franklin Inst. 221, 721–744 (1936).
[Crossref]

Nielsen, W. J.

J. A. C. Yule and W. J. Nielsen, Proc. Tech. Assoc. Graphic Arts 3, 65–76 (1951).

Williams, F. C.

F. C. Williams and F. R. Clapper, J. Opt. Soc. Am. 40, 796 (1950).

Yule, J. A. C.

J. A. C. Yule and W. J. Nielsen, Proc. Tech. Assoc. Graphic Arts 3, 65–76 (1951).

J. A. C. Yule and R. Colt, Proc. Tech. Assoc. Graphic Arts 3, 77–82 (1951).

J. Franklin Inst. (1)

A. Murray, J. Franklin Inst. 221, 721–744 (1936).
[Crossref]

J. Opt. Soc. Am. (2)

F. P. Callahan, J. Opt. Soc. Am. 42, 104–105 (1952).
[Crossref]

F. C. Williams and F. R. Clapper, J. Opt. Soc. Am. 40, 796 (1950).

J. Research Natl. Bur. Standards (1)

D. B. Judd, J. Research Natl. Bur. Standards 29, 329–332 (1942).
[Crossref]

Proc. Tech. Assoc. Graphic Arts (2)

J. A. C. Yule and W. J. Nielsen, Proc. Tech. Assoc. Graphic Arts 3, 65–76 (1951).

J. A. C. Yule and R. Colt, Proc. Tech. Assoc. Graphic Arts 3, 77–82 (1951).

Other (1)

See reference 2, discussion.

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Figures (2)

Fig. 1
Fig. 1

Sideways scattering and internal reflections within the paper. Symbols: s=Surface reflectance (factor K used to take account of any specular portion of surface reflection). a=Dot area. t=Ink film transmittance. 1−x=Reflectance of internally incident diffuse light at air surface. r=Fraction of light that is reversed in direction (not absorbed or transmitted by paper) and attempts to emerge from paper surface.

Fig. 2
Fig. 2

Relation of density to dot area calculated from various equations. The following equations were used in plotting the curves: Eq. (1)—considering multiple internal reflections, R = x ( 1 - s ) r ( 1 - a + a t ) 2 1 - r ( 1 - x ) ( 1 - a + a t 2 ) + K s; Eq. (2)—Callahan’s relationship, error in reflectance=ga(1−a)(1−t)2; Eq. (3)—Yule’s relationship, R=1−a(1−Rsn)n; Eq. (4)—Murray-Davies relationship, R=1−a(1−Rs). Curve A—Eq. (1): s=0, t=0, x=0.40; Curve B—Eq. (1): s=.04, t=0, x=0.40; Curve C—Eq. (1): s=0, t=0.306, x=0.40; Curve D—Eq. (1): s=0.04, t=0, x=1.0; Eq. (2): t=0, g=1.0 (surface reflection s=0.04 added); Eq. (3): Rs=0, n=2 (surface reflection s=0.04 added); Curve E—Eq. (1): s=0, t=0.20, x=1.0; Eq. (2): t=0.04, g=0.70; Eq. (3): Rs=0.04, n=2; Curve F—Eq. (4): Rs=0.04; Curve G—Observed reflection densities with 150-line screen.

Equations (11)

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r ( 1 - x ) ( 1 - s ) ( 1 - a + a t ) ( 1 - a + a t 2 ) .
1 st surface reflection = K s , 1 st emergence = r x ( 1 - s ) ( 1 - a + a t ) 2 , 2 nd emergence = r x ( 1 - s ) ( 1 - a + a t ) 2 × [ r ( 1 - x ) ( 1 - a + a t 2 ) ] , n th emergence = r x ( 1 - s ) ( 1 - a + a t ) 2 × [ r n - 1 ( 1 - x ) n - 1 ( 1 - a + a t 2 ) ] n - 1 .
R = K s + x ( 1 - s ) r ( 1 - a + a t ) 2 1 - r ( 1 - x ) ( 1 - a + a t 2 ) ,
R w = K s + [ x ( 1 - s ) r / 1 - r ( 1 - x ) ] .
R = K s 1 - s + r x ( 1 - a + a t ) 2 1 - r ( 1 - x ) ( 1 - a + a t 2 ) ( K s / 1 - s ) + [ r x / 1 - r ( 1 - x ) . ] .
R = K s 1 - s + x ( 1 - a + a t ) 2 1 - ( 1 - x ) ( 1 - a + a t 2 ) ( K s / 1 - s ) + 1 .
R = K S 1 - s + x t 2 1 - ( 1 - x ) t 2 ( K s / 1 - s ) + 1 .
R = 0.4 ( 1 - a ) 2 0.4 + 0.6 a             or             D = - log 0.4 ( 1 - a ) 2 0.4 + 0.6 a .
D = - log 0.4 ( 1 - 0.694 a ) 2 1 - 0.6 ( 1 - 0.907 a ) R = 0.4 [ 1 - a + a ( 0.306 ) ] 2 1 - 0.6 [ 1 - a + a ( 0.306 ) 2 ] .
For curve D : R = K s + ( 1 - s ) ( 1 - a ) 2 . For curve E : R = ( 1 - a + a t ) 2 .
R λ = K s + x ( 1 - s ) r [ 1 - ā ( 1 - t ¯ λ ) ] 2 1 - ( 1 - x ) r [ 1 - ā ( 1 - t ¯ λ 2 ) ] .