Abstract

A model is presented of a fluorescent ink halftone. Unlike a nonfluorescent ink, which only absorbs, a fluorescent ink absorbs higher-energy photons and emits lower-energy photons. The amount of fluorescent light produced depends on the percent absorption of the incident light. For fluorescent ink printed on paper, both photon scattering within the paper substrate and multiple internal reflections between the ink layer and the paper substrate significantly increase the percent absorption, so a realistic model must include these effects. The model presented here utilizes the generalized Clapper–Yule theory, which accounts for photon diffusion that is due to both scatter and internal reflection. It is shown that while multiple internal reflections alone only marginally increase the percent absorption, when there are both scattering and internal reflection, the percent absorption is increased significantly. The current study is a theoretical model and does not present experimental results.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Emmel, R. D. Hersch, “Spectral prediction model for a transparent fluorescent ink on paper,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Applications (Society for Imaging Science and Technology, Springfield, Va., 1998), pp. 116–122.
  2. P. Emmel, R. D. Hersch, “A ‘one channel’ spectral colour prediction model for transparent fluorescent inks on a transparent support,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Applications (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 70–77.
  3. G. L. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 41, 643–656 (1997).
  4. G. L. Rogers, “The effect of light scatter on halftone color,” J. Opt. Soc. Am. A 15, 1813–1821 (1998).
    [CrossRef]
  5. J. S. Arney, “A probability description of the Yule–Nielsen effect,” J. Imaging Sci. Technol. 41, 633–640 (1997).
  6. F. R. Clapper, J. A. C. Yule, “The effect of multiple internal reflections on the densities of halftone prints on paper,” J. Opt. Soc. Am. 43, 600–603 (1953).
    [CrossRef]
  7. H. R. Kang, Color Technology for Electronic Imaging Devices (SPIE Press, Bellingham, Wash.1997), Chap. 2.
  8. G. L. Rogers, “A generalized Clapper–Yule model of halftone reflectance,” Color Res. Appl. (to be published).
  9. G. Kortum, Reflectance Spectroscopy (Springer, New York, 1969), pp. 107–108.
  10. D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Natl. Bur. Stand. 29, 329–332 (1942).
    [CrossRef]
  11. J. L. Saunderson, “Calculation of the color of pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942);D. B. Judd, G. Wyszecki, Color in Business, Science, and Industry (Wiley, New York, 1975), p. 453.
    [CrossRef]
  12. G. L. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42, 341–345 (1998).

1998

G. L. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42, 341–345 (1998).

G. L. Rogers, “The effect of light scatter on halftone color,” J. Opt. Soc. Am. A 15, 1813–1821 (1998).
[CrossRef]

1997

G. L. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 41, 643–656 (1997).

J. S. Arney, “A probability description of the Yule–Nielsen effect,” J. Imaging Sci. Technol. 41, 633–640 (1997).

1953

1942

Arney, J. S.

J. S. Arney, “A probability description of the Yule–Nielsen effect,” J. Imaging Sci. Technol. 41, 633–640 (1997).

Clapper, F. R.

Emmel, P.

P. Emmel, R. D. Hersch, “Spectral prediction model for a transparent fluorescent ink on paper,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Applications (Society for Imaging Science and Technology, Springfield, Va., 1998), pp. 116–122.

P. Emmel, R. D. Hersch, “A ‘one channel’ spectral colour prediction model for transparent fluorescent inks on a transparent support,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Applications (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 70–77.

Hersch, R. D.

P. Emmel, R. D. Hersch, “A ‘one channel’ spectral colour prediction model for transparent fluorescent inks on a transparent support,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Applications (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 70–77.

P. Emmel, R. D. Hersch, “Spectral prediction model for a transparent fluorescent ink on paper,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Applications (Society for Imaging Science and Technology, Springfield, Va., 1998), pp. 116–122.

Judd, D. B.

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Natl. Bur. Stand. 29, 329–332 (1942).
[CrossRef]

Kang, H. R.

H. R. Kang, Color Technology for Electronic Imaging Devices (SPIE Press, Bellingham, Wash.1997), Chap. 2.

Kortum, G.

G. Kortum, Reflectance Spectroscopy (Springer, New York, 1969), pp. 107–108.

Rogers, G. L.

G. L. Rogers, “The effect of light scatter on halftone color,” J. Opt. Soc. Am. A 15, 1813–1821 (1998).
[CrossRef]

G. L. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42, 341–345 (1998).

G. L. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 41, 643–656 (1997).

G. L. Rogers, “A generalized Clapper–Yule model of halftone reflectance,” Color Res. Appl. (to be published).

Saunderson, J. L.

Yule, J. A. C.

J. Imaging Sci. Technol.

G. L. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 41, 643–656 (1997).

J. S. Arney, “A probability description of the Yule–Nielsen effect,” J. Imaging Sci. Technol. 41, 633–640 (1997).

G. L. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42, 341–345 (1998).

J. Natl. Bur. Stand.

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Natl. Bur. Stand. 29, 329–332 (1942).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

P. Emmel, R. D. Hersch, “Spectral prediction model for a transparent fluorescent ink on paper,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Applications (Society for Imaging Science and Technology, Springfield, Va., 1998), pp. 116–122.

P. Emmel, R. D. Hersch, “A ‘one channel’ spectral colour prediction model for transparent fluorescent inks on a transparent support,” Proceedings of the IS&T/SID Color Imaging Conference C: Color Science, Systems, and Applications (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 70–77.

H. R. Kang, Color Technology for Electronic Imaging Devices (SPIE Press, Bellingham, Wash.1997), Chap. 2.

G. L. Rogers, “A generalized Clapper–Yule model of halftone reflectance,” Color Res. Appl. (to be published).

G. Kortum, Reflectance Spectroscopy (Springer, New York, 1969), pp. 107–108.

Cited By

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

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Ink layer of thickness t is bordered on its lower boundary by the paper substrate and on its upper boundary by air. There is a diffuse downward-moving stream, I-, and a diffuse upward-moving stream, I+.

Fig. 2
Fig. 2

Upper-boundary condition consists of two terms: the incident light I0 and the internally reflected upward stream r^1I+(t).

Fig. 3
Fig. 3

Lower-boundary condition consists of an infinite number of terms corresponding to the different orders of scattering/internal reflection between the paper substrate and the ink-layer lower boundary.

Fig. 4
Fig. 4

Reflectance from the ink layer into the paper substrate is r from regions void of ink and rτ2 from regions with inkτ2 because the light passes through the ink twice.

Fig. 5
Fig. 5

Percent of energy absorbed as a function of scattering length for several different cases. Scattering length is in units of screen period, d. (a) Upper limit of complete scattering with internal reflection (x¯/d1). (b) Varying scattering length with internal reflection. (c) Varying scattering length with no internal reflection (r=0). (d) Lower limit of no scattering with internal reflection (x¯=0). (e) No scattering and no internal reflection (x¯=0 and r=0).

Fig. 6
Fig. 6

Percent of energy absorbed as a function of ink coverage for several different cases. (a) Upper limit of complete scattering with internal reflection. (b) Scattering length x¯/d=1.5 with internal reflection. (c) Scattering length x¯/d=1.5 with no internal reflection (r=0). (d) Lower limit of no scattering with internal reflection (x¯=0). (e) No scattering and no internal reflection (x¯=0 and r=0).

Equations (78)

Equations on this page are rendered with MathJax. Learn more.

ddz I+(z; λ)=-Γ(λ)I+(z; λ)+Q2 f(λ)×Γ(λ)[I+(z; λ)+I-(z; λ)]dλ,
-ddz I-(z; λ)=-Γ(λ)I-(z; λ)+Q2 f(λ)×Γ(λ)[I+(z; λ)+I-(z; λ)]dλ.
Γ(x, y; λ)=2C1(x, y)(λ),
f(λ)(λ)0,
I-(x, y, t)=I0+r^1(x, y)I+(x, y, t),
r^1(x, y)=rC1(x, y),
I+(x, y, 0)=(1-r^0)(h*I-(x, y, 0)+h*[r^0h*I-(x, y, 0)]+h*{r^0h*[r^0h*I-(x, y, 0)]}+),
r^0(x, y)=rC0(x, y).
h(x, y)dxdy=rg,
H(x, y)=(1-r)[h(x, y)+rh*h(x, y)+r2h*h*h(x, y)+],
Rp=rg(1-r)1-rgr.
R=R(0)+R(1)+R(2)+,
R(n)(x, y)=R(n-1)(x, y)*(r^0h),
R(0)(x, y)=(1-r^0)h(x, y).
I+(x, y, 0)=R*I-(x, y, 0).
I(λ)=1areaarea[1-r^1(x, y)]I+(x, y, t; λ)dxdy.
RH(λ)=I(λ)I0(λ).
I±(z)=v±(z)+u±(z),
±ddz v±(z)=-Γv±(z),
±ddz u±(z)=FλΓ[v-(z)+v+(z)],
Fλg=Q2 f(λ)g(λ)dλ.
I=Iv+Iu.
v-(z)=v-(t)exp[-Γ(t-z)],
v+(z)=v+(0)exp[-Γz].
v-(t)=I0+r^1v+(t),
v+(0)=R*v-(0),
v-(t)=I0+r^1v+(0)exp(-Γt),
v+(0)=R*[v-(t)exp(-Γt)].
v+(0)=R*[I0exp(-Γt)+r^1v+(0)exp(-2Γt)].
v+(0)=I0n=0v+(n),
v+(0)=R(0)*exp(-Γt),
v+(1)=R(0)*[r^1v+(0)exp(-2Γt)]+R(1)*exp(-Γt),
v+(2)=R(0)*[r^1v+(1)exp(-2Γt)]+R(1) *[r^1v+(0)exp(-2Γt)]+R(2)*exp(-Γt),
v+(0)=(1-r^0)h*exp(-Γt),
v+(1)=(1-r^0)h*{[r^0+r^1exp(-2Γt)]v+(0)}=(1-r^0)h*{[r^0+r^1exp(-Γt)]h*exp(-Γt)},
v+(2)=(1-r^0)h*{[r^0+r^1exp(-2Γt)]v+(1)}=(1-r^0)h*([r^0+r^1exp(-2Γt)]h *{[r^0+r^1exp(-2Γt)]h*exp(-Γt)}).
v+(0)=I0(1-r^0){h*T+h*(rih*T)+h*[rih*(rih*T)]+},
T(x, y; λ)=exp[-Γ(x, y; λ)t],
T(x, y; λ)=C0(x, y)+τ(λ)C1(x, y),
τ(λ)=exp[-2(λ)t],
ri=r^0+r^1exp(-2Γt),
ri=rC0+rτ2C1.
Iv=(1-r^1)exp(-Γt)v+(0).
Iv=I0(1-r){Th*T+Th*(rih*T)+Th*[rih*(rih*T)]+}.
Iv=I0(P00+P01τ+P10τ+P11τ2),
P=n=0(pr)np,
r=r00rτ2.
pnm=(1-r)Cnh*Cm.
pnm=pnm(1-r)μm,
P10=p10+kp1krkkpk0+jkp1jrjjpjkrkkpk0+ ,
u-(0)=u-(t)+S,
u+(t)=u+(0)+S,
S=Fλ(1-τ)C1[v+(0)+v-(t)],
1-exp(-Γt)=(1-τ)C1.
u-(t)=r^1u+(t),
u+(0)=R*u-(0),
u+(t)=R*[r^1u+(t)+S]+S.
u+(t)=n=0u+(n).
u+(0)=[1+R(0)]S,
u+(1)=R(0)r^1u+(0)+R(1)S,
u+(2)=R(0)r^1u+(1)+R(1)r^1u+(0)+R(2)S,
u+(0)=S+(1-r^0)h*S,
u+(1)=(1-r^0)(h*r^1S+h*r^0h*S+h*r^1h*S),
u+(2)=(1-r^0)(h*r^0h*r^1S+h*r^1h*r^1S+h*r^0h*r^0h*S+h*r^0h*r^1h*S+h*r^1h*r^0h*S+h*r^1h*r^1h*S).
u+(t)=S+(1+r)(1-r^0)×(h+rh*h+r2h*h*h+)*S.
u+(t)=S+(1-r^0)(1+r)1-r H*S.
Iu=(1-r^1)u+(t).
Iu=Rp(1+rg)/rgS,
S=Fλ(1-τ)[I0μ1+(1+rτ)C1v+(0)],
C1v+(0)=I0{C1h*T+C1h*(rih*T)+C1h*[rih*(rih*T)]+}.
C1v+(0)=I01-r (P10+P11τ),
S=FλI0(1-τ)μ1+1+rτ1-r (P10+P11τ),
I=I0(P00+P01τ+P10τ+P11τ2)+ Rp(1+rg)/rgFλI0(1-τ)μ1+1+rτ(1-r)×(P10+P11τ).
Ck2=Ck,
C0C1=0,
C0+C1=1.
μm=1areaCm(x, y)dxdy,
μ0=1-μ1.

Metrics