Abstract

Expressions relating the bispectral reflectance of a stack of n fluorescing layers to each individual layer’s reflectance and transmittance are derived. This theoretical framework is used together with recently proposed extensions of the Kubelka–Munk model to study the fluorescence from layered turbid media. For one layer over a reflecting background, the model is shown to give the same results as a previous model. The extension to n layers with different optical properties allows simulating the bispectral reflectance from a pad of layered turbid media. The applicability of the model is exemplified with an optimization of fluorophore distribution in layered turbid media.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. J. C. Zwinkels and F. Gauthier, “Instrumentation, standards, and procedures used at the National Research Council of Canada for high-accuracy fluorescence measurements,” Anal. Chim. Acta 380, 193–209 (1999).
    [CrossRef]

2011

2010

2009

2008

2003

J. Swartling, A. Pifferi, A. M. K. Enejder, and S. Andersson-Engels, “Accelerated Monte Carlo models to simulate fluorescence spectra from layered tissues,” J. Opt. Soc. Am. A 20, 714–727 (2003).
[CrossRef]

D. Y. Churmakov, I. V. Meglinski, S. A. Piletsky, and D. A. Greenhalgh, “Analysis of skin tissues spatial fluorescence distribution by the Monte Carlo simulation,” J. Phys. D: Appl. Phys. 36, 1722–1728 (2003).
[CrossRef]

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka-Munk model,” Color Res. Appl. 28, 4–14(2003).
[CrossRef]

2001

1999

J. C. Zwinkels and F. Gauthier, “Instrumentation, standards, and procedures used at the National Research Council of Canada for high-accuracy fluorescence measurements,” Anal. Chim. Acta 380, 193–209 (1999).
[CrossRef]

1993

S. D. Howison and R. J. Lawrence, “Fluorescent transfer of light in dyed materials,” SIAM J. Appl. Math. 53, 447–458(1993).
[CrossRef]

1990

1988

1954

1948

1860

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. London 11, 545–556 (1860).
[CrossRef]

Andersson, M.

Andersson-Engels, S.

Churmakov, D. Y.

D. Y. Churmakov, I. V. Meglinski, S. A. Piletsky, and D. A. Greenhalgh, “Analysis of skin tissues spatial fluorescence distribution by the Monte Carlo simulation,” J. Phys. D: Appl. Phys. 36, 1722–1728 (2003).
[CrossRef]

Coppel, L. G.

L. G. Coppel, M. Andersson, and P. Edström, “Determination of quantum efficiency in fluorescing turbid media,” Appl. Opt. 50, 2784–2792 (2011).
[CrossRef]

L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Paper Making Research SymposiumE. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio University, 2009), http://miun.diva-portal.org/smash/get/diva2:222641/FULLTEXT01.

Culver, J. P.

Delaney, J.

Edström, P.

L. G. Coppel, M. Andersson, and P. Edström, “Determination of quantum efficiency in fluorescing turbid media,” Appl. Opt. 50, 2784–2792 (2011).
[CrossRef]

L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Paper Making Research SymposiumE. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio University, 2009), http://miun.diva-portal.org/smash/get/diva2:222641/FULLTEXT01.

Enejder, A. M. K.

Gauthier, F.

J. C. Zwinkels and F. Gauthier, “Instrumentation, standards, and procedures used at the National Research Council of Canada for high-accuracy fluorescence measurements,” Anal. Chim. Acta 380, 193–209 (1999).
[CrossRef]

Greenhalgh, D. A.

D. Y. Churmakov, I. V. Meglinski, S. A. Piletsky, and D. A. Greenhalgh, “Analysis of skin tissues spatial fluorescence distribution by the Monte Carlo simulation,” J. Phys. D: Appl. Phys. 36, 1722–1728 (2003).
[CrossRef]

Howison, S. D.

S. D. Howison and R. J. Lawrence, “Fluorescent transfer of light in dyed materials,” SIAM J. Appl. Math. 53, 447–458(1993).
[CrossRef]

Keijzner, M.

Kokhanovsky, A. A.

Kubelka, P.

Lawrence, R. J.

S. D. Howison and R. J. Lawrence, “Fluorescent transfer of light in dyed materials,” SIAM J. Appl. Math. 53, 447–458(1993).
[CrossRef]

Liebert, A.

Lindquister, M.

L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Paper Making Research SymposiumE. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio University, 2009), http://miun.diva-portal.org/smash/get/diva2:222641/FULLTEXT01.

Macdonald, R.

Meglinski, I. V.

D. Y. Churmakov, I. V. Meglinski, S. A. Piletsky, and D. A. Greenhalgh, “Analysis of skin tissues spatial fluorescence distribution by the Monte Carlo simulation,” J. Phys. D: Appl. Phys. 36, 1722–1728 (2003).
[CrossRef]

Nieto-Vesperinas, M.

Ntziachristos, V.

Pattanayak, D. N.

Pifferi, A.

Piletsky, S. A.

D. Y. Churmakov, I. V. Meglinski, S. A. Piletsky, and D. A. Greenhalgh, “Analysis of skin tissues spatial fluorescence distribution by the Monte Carlo simulation,” J. Phys. D: Appl. Phys. 36, 1722–1728 (2003).
[CrossRef]

Pilon, L.

Ripoll, J.

Schmitt, J. M.

Shakespeare, J.

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka-Munk model,” Color Res. Appl. 28, 4–14(2003).
[CrossRef]

Shakespeare, T.

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka-Munk model,” Color Res. Appl. 28, 4–14(2003).
[CrossRef]

Simonot, L.

Star, W. M.

Stokes, G. G.

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. London 11, 545–556 (1860).
[CrossRef]

Storchi, P. R. M.

Swartling, J.

Thoury, M.

Wabnitz, H.

Walker, E. C.

Yodh, A. G.

Yudovsky, D.

Zhou, G. X.

Zolek, N.

Zwinkels, J. C.

J. C. Zwinkels and F. Gauthier, “Instrumentation, standards, and procedures used at the National Research Council of Canada for high-accuracy fluorescence measurements,” Anal. Chim. Acta 380, 193–209 (1999).
[CrossRef]

Anal. Chim. Acta

J. C. Zwinkels and F. Gauthier, “Instrumentation, standards, and procedures used at the National Research Council of Canada for high-accuracy fluorescence measurements,” Anal. Chim. Acta 380, 193–209 (1999).
[CrossRef]

Appl. Opt.

Color Res. Appl.

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka-Munk model,” Color Res. Appl. 28, 4–14(2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D: Appl. Phys.

D. Y. Churmakov, I. V. Meglinski, S. A. Piletsky, and D. A. Greenhalgh, “Analysis of skin tissues spatial fluorescence distribution by the Monte Carlo simulation,” J. Phys. D: Appl. Phys. 36, 1722–1728 (2003).
[CrossRef]

Opt. Express

Proc. R. Soc. London

G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. London 11, 545–556 (1860).
[CrossRef]

SIAM J. Appl. Math.

S. D. Howison and R. J. Lawrence, “Fluorescent transfer of light in dyed materials,” SIAM J. Appl. Math. 53, 447–458(1993).
[CrossRef]

Other

L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Paper Making Research SymposiumE. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio University, 2009), http://miun.diva-portal.org/smash/get/diva2:222641/FULLTEXT01.

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Figures (3)

Fig. 1.
Fig. 1.

Schematic description of a structure with n layers. Each layer is described by the bispectral reflectance and transmittance matrices that can be calculated from the spectral scattering (S) and absorption (K) coefficients and the quantum efficiency (Q) of the layer.

Fig. 2.
Fig. 2.

Normalized reflectance R12tot/Q1 versus top layer thickness. The proposed model (diamonds) gives the same results as the Simonot et al. [15] model (curves) for two-layer media.

Fig. 3.
Fig. 3.

Normalized reflectance R12tot/Q1 versus top and bottom layer thickness t1 for (a) a medium made of a middle layer and two identical outer layers and (b) an opaque pad of the same medium. Three different scattering coefficients in the outer layers [30μm1 (solid curve), 70μm1 (dash-dotted), and 110μm1 (dashed)] and two different pairs of absorption coefficients at λ1 (black and gray) are shown. Increased light scattering in the outer layer reduces the fluorescence efficiency. At t1=15μm, fluorophores are more efficient in the outer layer when K1(λ1)>K2(λ1).

Tables (1)

Tables Icon

Table 1. Scattering (S) and Absorption (K) Coefficients of the Simulated Layer at Excitation (λ1) and Emission Wavelength (λ2)

Equations (34)

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R12tot=R121+T121R222T221+T121R222R221R222T221++T111R122T221+T111R122R221R222T221++T111R112T121+T111R112R111R112T121++T111R112R121R222T221+T111R112R121R222R221R222T221++T111R112R111R122T221+T111R112R111R122R221R222T221++T111R112R111R112R121R222T222+T111R112R111R112R121R222R221R222T22++,
R12tot=R121+T121R222T221X22+T111R122T221X22+T111R112T121X11+T111R112R121R222T221X11X22+T111R112R111R122T221X11X22,
R12tot=R121+T121R222T221+T111R122T2211R221R222+T111R112R121R222T221+T111R112R111R122T221(1R111R112)(1R221R222).
T12tot=T121T222+T121R222R221T222+T121R222R221R222R221T222++T111T122+T112R112R111T122++T111R122R221T222++T111R122R221R222R221T222++T111R112R121T222++T111R112R121R222R221T222++,
T12tot=T121T2221R221R222+T111T1221R111R112+T111R122R221T222+T111R112R121T222(1R111R112)(1R221R222).
R11tot=R111+(T111)2R1121R111R112,
T11tot=T111T1121R111R112.
β(λ2|E)=Rλ2λ2tot+1E(λ2)λ1<λ2Rλ1λ2totE(λ1),
R(λ2)=c1+c2+A+Bα(λ1)2α(λ2)2,
T(λ2)=D1eα(λ2)t+D2eα(λ2)t+D3eα(λ1)t+D4eα(λ1)t,
A=a2K(λ1)Q[ϵ(λ2)+S(λ2)α(λ1)],
B=b2K(λ1)Q[ϵ(λ2)+S(λ2)+α(λ1)],
D1=[ϵ(λ2)+α(λ2)]c1S(λ2),D2=[ϵ(λ2)α(λ2)]c2S(λ2),
D3=[ϵ(λ2)+α(λ1)]A[α(λ1)2α(λ2)2]S(λ2)K(λ1)Qa2S(λ2),
D4=[ϵ(λ2)α(λ1)]B[α(λ1)2α(λ2)2]S(λ2)K(λ1)Qb2S(λ2),
c1=q1+[ϵ(λ2)α(λ2)]c2ϵ(λ2)+α(λ2),
c2=q1[ϵ(λ2)+α(λ2)]1e2α(λ2)tq2eα(λ2)t1[ϵ(λ2)α(λ2)][ϵ(λ2)+α(λ2)]1e2α(λ2)t,
q1=A[ϵ(λ2)+α(λ1)]+B[ϵ(λ2)α(λ1)]α(λ1)2α(λ2)212K(λ1)Q(a+b)S(λ2),
q2=Aeα(λ1)t+Beα(λ1)tα(λ1)2α(λ2)2,
a=1+r(λ1)1r(λ1)2e2α(λ1)t,b=[1+r(λ1)]r(λ1)e2α(λ1)t1r(λ1)2e2α(λ1)t,
r=Sϵ+α.
R12layer=R(λ2)R22layer.
T12layer=T(λ2)T22layer.
R12tot=C1α(λ1)2α(λ2)2+C2α(λ1)2α(λ2)2+C3[1+β(λ2)]+C4[1β(λ2)],
C1=QK(λ1)A1[α(λ2)/β(λ2)+α(λ1)],
C2=QK(λ1)A2[α(λ2)/β(λ2)α(λ1)],
C3=1α(λ1)2α(λ2)21[1+β(λ2)]B1(λ2)eα(λ2)t[1β(λ2)]B2(λ2)eα(λ2)t×{B2(λ2)[D1+D2]eα(λ2)t[1+β(λ2)]([C1Rg(λ2)D1]eα(λ1)t+[C2Rg(λ2)D2]eα(λ1)t)},
C4=11+β(λ2)(D1+D2α(λ1)2α(λ2)2+C3[1β(λ2)]),
A1=B2(λ1)eα(λ1)t[1+β(λ1)]B1(λ1)eα(λ1)t[1β(λ1)]B2(λ1)eα(λ1)t,
A2=B1(λ1)eα(λ1)t[1+β(λ1)]B1(λ1)eα(λ1))t[1β(λ1)]B2(λ1)eα(λ1)t,
D1=QK(λ1)A1[α(λ2)/β(λ2)α(λ1)],
D2=QK(λ1)A2[α(λ2)/β(λ2)+α(λ1)],
B1(λ)=1+β(λ)Rg(λ)[1β(λ)],
B2(λ)=1β(λ)Rg(λ)[1+β(λ)],

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