## Abstract

The diffuse reflectance spectra and the trichromatic coordinates of diffusing stratified paints are modeled. Each layer contains its own pigments, and their optical properties are first determined from experiments. The radiative transfer equation is then solved by the auxiliary function method for modeling the total light scattered by the stratified systems. The results are in good agreement with experimental spectra and validate the modeling. The calculations are then applied on the same stratified systems to study the influence of the observation angle in a bidirectional configuration and to study the influence of the thickness of the layers in a given configuration. In both cases, the reflectance spectra and the trichromatic coordinates are calculated and compared.

© 2007 Optical Society of America

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### Equations (17)

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(1)
$$\frac{d{f}^{\pm}(\stackrel{\u20d7}{u},z)}{dz}=\mp (k+s)\frac{{f}^{\pm}(\stackrel{\u20d7}{u},z)}{\mid \mathrm{cos}\phantom{\rule{0.2em}{0ex}}\theta \mid}\pm \frac{s}{4\pi}{\int}_{4\pi}\frac{f({\stackrel{\u20d7}{u}}_{1},z)}{\mid \mathrm{cos}\phantom{\rule{0.2em}{0ex}}{\theta}_{1}\mid}p(\stackrel{\u20d7}{u},{\stackrel{\u20d7}{u}}_{1})2\pi \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}{\theta}_{1}d{\theta}_{1}\pm \frac{s}{4\pi}{F}^{\pm}\left(z\right)p(\stackrel{\u20d7}{u},{\stackrel{\u20d7}{u}}_{0}),$$
(2)
$$f({\stackrel{\u20d7}{u}}_{1},z)={f}^{+}({\stackrel{\u20d7}{u}}_{1},z)+{f}^{-}({\stackrel{\u20d7}{u}}_{1},z).$$
(3)
$$\frac{d{f}^{\pm}(\mu ,\tau )}{d\tau}=\mp \frac{{f}^{\pm}(\mu ,\tau )}{\mid \mu \mid}\pm \frac{\varpi}{4\pi}\frac{\mu}{\mid \mu \mid}[{\int}_{{\mu}_{1}=0}^{1}\frac{{f}^{+}({\mu}_{1},\tau )+{f}^{-}({\mu}_{1},\tau )}{\mid {\mu}_{1}\mid}2\pi d{\mu}_{1}+{F}^{\pm}\left(\tau \right)].$$
(4)
$$A\left(\tau \right)={\int}_{0}^{1}\frac{{f}^{+}({\mu}_{1},\tau )+{f}^{-}({\mu}_{1},\tau )}{\mid {\mu}_{1}\mid}d{\mu}_{1},$$
(5)
$$A\left(\tau \right)={\int}_{0}^{1}\frac{{f}^{+}({\stackrel{\u20d7}{u}}_{1},\tau )+{f}^{-}({\stackrel{\u20d7}{u}}_{1},\tau )}{\mid {\mu}_{1}\mid}d{\mu}_{1}.$$
(6)
$$\frac{d{f}^{\pm}(\mu ,\tau )}{d\tau}=\mp \frac{{f}^{\pm}(\mu ,\tau )}{\mu}\pm \frac{\varpi}{2}t\left(\tau \right).$$
(7)
$${f}^{+}(\mu ,\tau )={f}^{+}(\mu ,0)\mathrm{exp}(-\frac{\tau}{\mu})+\frac{\varpi}{2}{\int}_{s=0}^{\tau}t\left(s\right)\mathrm{exp}\left(\frac{s-\tau}{\mu}\right)ds,$$
(8)
$${f}^{-}(\mu ,\tau )={f}^{-}(\mu ,{\tau}_{h})\mathrm{exp}\left(\frac{\tau -{\tau}_{h}}{\mu}\right)+\frac{\varpi}{2}{\int}_{s=\tau}^{{\tau}_{h}}t\left(s\right)\mathrm{exp}\left(\frac{\tau -s}{\mu}\right)ds,$$
(9)
$$A\left(\tau \right)=\frac{\varpi}{2}{\int}_{s=0}^{{\tau}_{h}}H(\tau ,s)t\left(s\right)ds+{\int}_{\mu =0}^{1}[{f}^{+}(\mu ,0)\mathrm{exp}(-\frac{\tau}{\mu})+{f}^{-}(\mu ,{\tau}_{h})\mathrm{exp}\left(\frac{\tau -{\tau}_{h}}{\mu}\right)]\frac{d\mu}{\mu},$$
(10)
$$H(\tau ,s)={\int}_{\mu =0}^{1}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(\frac{\mid \tau -s\mid}{\mu}\right)\frac{d\mu}{\mu}$$
(11)
$${f}^{-}(\mu ,{\tau}_{h})={R}_{paint\to background}\left(\mu \right)\lfloor {f}^{+}(\mu ,{\tau}_{h})+{F}^{+}\left({\tau}_{h}\right)\rfloor .$$
(12)
$${f}^{-}(\mu ,0)={R}_{paint\to background}\left(\mu \right)[{f}^{+}(\mu ,{\tau}_{h})+{F}^{+}\left({\tau}_{h}\right)]\mathrm{exp}(-\frac{{\tau}_{h}}{\mu})+\frac{\varpi}{2}{\int}_{s=0}^{{\tau}_{h}}t\left(s\right)\mathrm{exp}(-\frac{s}{\mu})ds$$
(13)
$$\mathrm{or}\phantom{\rule{0.3em}{0ex}}{f}^{-}(\mu ,0)=\frac{{\rho}_{g}}{\pi}B\mu \phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-\frac{{\tau}_{h}}{\mu})+\frac{\varpi}{2}{\int}_{s=0}^{{\tau}_{h}}t\left(s\right)\mathrm{exp}(-\frac{s}{\mu})ds.$$
(14)
$${\rho}_{f}({\mu}_{i},{\mu}_{f})=\frac{\pi}{{F}_{0}}\frac{{f}_{f}\left({\mu}_{f}\right)}{{\mu}_{f}},$$
(15)
$${\rho}_{f}({\mu}_{i},{\mu}_{f})=\frac{\pi}{{F}_{0}}\frac{T\left(\mu \right)}{{n}^{2}\mu}{f}^{-}(\mu ,0).$$
(16)
$$\rho =\frac{1-{\rho}_{g}(a-b\phantom{\rule{0.2em}{0ex}}\mathrm{coth}\left(bSh\right))}{a+b\phantom{\rule{0.2em}{0ex}}\mathrm{coth}\left(bSh\right)-{\rho}_{g}};\phantom{\rule{1em}{0ex}}a=\frac{K+S}{S}$$
(17)
$$b=\sqrt{{a}^{2}-1}.$$