For the sample preparation, the coffee and milk solutions have been chosen. The former for their specific scattering feature and the latter for their special absorbing feature. The solutions have been obtained according to the protocol described in [24] and summarized in Table 1 and Table 2 . The photographs of the samples are presented in Figure 1 and Figure 2 .

In this section the SLIPI-1p approach is briefly described in sections. The details can be found in [27] and [28] .

A 450 nm or 638 nm continuous wave (CW) laser beam is used as the illumination source. The incident wave intensity is adjusted by using a density wheel in order to optimize the signal-to-noise ratio while avoiding saturation. Then, the beam is magnified by a factor of 10 by using a telescope device that is made with a pair of positive spherical lenses. An aperture is used to select the central part of the beam where the spatial intensity profile is quite homogeneous. A positive cylindrical lens with a focal length of 200 mm is used to create the sheet of laser light in the basin (L: 40 mm, l: 20 mm and h: 80 mm). To spatially modulate this sheet of laser light, a 2 lines/mm Ronchi grating is directly attached to the basin. This limits bad effects caused by near-field diffraction, known as the Talbot effect. A camera whose field of view direction is 90˚ with the light propagation direction is used to

capture the images. It is an Andor brand Luca (r) CCD camera, with 14-bit electron multiplication. The final image is the result of accumulating more than 100 images to a single frame. The exposure time is between 0.001 and 0.1 seconds to optimize the dynamic range. The use of a Ronchi grating with 2 lp/mm is justified by its ability, in our experimental case, to spatially modulate light by generating a periodic pattern of dark and light bands across the sample with a relatively low spatial frequency (0.5 mm between each line), which is suitable for our study materials, such as milk and also for our camera. Indeed, the cut off frequency of the camera is over the spatial frequency of the Ronchi grating. Figure 3 presents a diagram of the experimental device.

The SLIPI approach used in this study relies on an analysis tool known as synchronous detection [29] . Synchronous detection, or ‘lock-in amplifier,’ is a common instrument used to address signal-to-noise ratio issues in research institutes and laboratories. It is a very powerful tool where signals of interest can be detected even if they are smaller than the accompanying noise signals. The lock-in amplifier is also used to detect and measure very small AC signals down to nano-volts or less, where noise remains a problem [30] . Synchronous detection is most often associated with the analysis of temporally varying signals, but it also works well for spatially modulated signals. To explain the principle of synchronous detection, let’s consider a 1D signal, I(x), with a superimposed periodic variation of amplitude $I_s$ in space:

$I\left(x\right)=I_s\sin \left(2\pi \vartheta x+\phi \right)+I_{MS}$ (1)

where $I_s$ is the modulated light with a ballistic photonic signature; $\vartheta$ is the spatial frequency (mm^-1) of the modulation, and $\phi$ is the spatial phase, which is unknown. The term $I_{MS}$ represents any undesirable non-modulated intensity contribution also detected, such as multiple light scattering or any other surrounding light sources. The goal of synchronous detection analysis is to extract $I_s$ and reject $I_{MS}$. To achieve this, the signal (x) is multiplied by two reference signals $R_1$ and $R_2$, which are computationally generated with a relative phase shift of π/2:

$R_1\left(x\right)=\sin \left(2\pi \vartheta x\right)$ (2)

$R_2\left(x\right)=\cos \left(2\pi \vartheta x\right)$ (3)

By multiplying (x) by these reference signals, $I_1$ and $I_2$ are obtained.

$I_1\left(x\right)=\frac{1}{2}{I}_S\left(\cos \left(\phi \right)-\cos \left(4\pi \vartheta x+\phi \right)\right)+I_{MS}\sin \left(2\pi \vartheta x\right)$ (4)

$I_2\left(x\right)=\frac{1}{2}{I}_S\left(\sin \left(\phi \right)+\sin \left(4\pi \vartheta x+\phi \right)\right)+I_{MS}\cos \left(2\pi \vartheta x\right)$ (5)

The frequency analysis of $I_1$ and $I_2$ reveals three components: (1) a non-modulated component, (2) a component modulated at $2\vartheta$, and (3) another component modulated at $\vartheta$. The last two components can be removed using a low-pass filter in the Fourier domain, with a cutoff frequency below $\vartheta$, resulting in the following expressions

${\tilde{I}}_1\left(x\right)=\frac{1}{2}{\tilde{I}}_S\cos \left(\phi \right)$ (6)

${\tilde{I}}_2\left(x\right)=\frac{1}{2}{\tilde{I}}_S\sin \left(\phi \right)$ (7)

where the tilde indicates that frequency filtering has been applied. From these, the SLIPI-1p image ${\tilde{I}}_S$ can finally be extracted by calculating:

${\tilde{I}}_S=2\sqrt{{\tilde{I}}_1^2+{\tilde{I}}_2^2}$ (8)

An illustration of the SLIPI-1p process is shown in Figure 4 . The example displays the signal from a structured laser light sheet, with effective sections extracted from two different depths, marked as A and B. A decrease in amplitude is observed from column A to column B. The 1D Fourier transforms of curves A and B show a reduction in the strength of the first-order peak (modulation frequency). This frequency is then isolated using frequency filtering (red zone) after applying the synchronous detection algorithm.

Finally, the exponential decay is revealed, as indicated in the SLIPI image and the corresponding $I_s$ curve.

From a modulated image, the modulation amplitude is extracted using Equation (8). The modulated component, $I_s$, corresponds to the simple light diffusion that decreases exponentially with distance. By applying an exponential fit to $I_s$, the extinction coefficient ${\mu }_e$ is directly extracted as indicated by the Beer-Lambert-Bouguer law, which states that, assuming a collimated beam of a single wavelength traverses a uniform medium, the transmittance T is given by:

$T=\frac{I_t}{I_0}={\text{e}}^{-{\mu }_{e}L}={\text{e}}^{-OD}$ (9)

with T corresponding to the transmittance, which is a measure of light attenuation along the distance L. $I_0$ represents the incident light measured using a “blank” reference cuvette containing only the solvent. Then, $I_t$ is the transmitted light, measured considering the solvent-solute mixture. Finally, ${\mu }_e$ is the extinction coefficient and $OD$ is the optical path length.

The extinction coefficient ${\mu }_e$ is the sum of the absorption coefficient ${\mu }_a$ and the scattering coefficient ${\mu }_s$ showed in the Equation (10).

${\mu }_e={\mu }_a+{\mu }_s$ (10)

The absorption ${\mu }_a$ and diffusion ${\mu }_s$ coefficients are mixed in the extinction coefficient by a sum. In this study we would like to separate them, it means calculate both coefficients by only the use of SLIPI measurement.

A simple way to achieve it is to find another equation with the same unknown parameters ${\mu }_a$ and ${\mu }_s$ to have a system of two equation with two unknown parameters. That would permit to obtain ${\mu }_a$ and ${\mu }_s$ with simple calculation. In the following we obtain this second equation from the relations of the Kubeka-Munk theory.

The Kubelka-Munk model is a two-flux radiative transfer model that predicts the reflectance and transmittance of a homogeneous absorbing and scattering layer illuminated by Lambertian radiation. We consider a thin substratum of depth dz and we respectively denote i and j as the downward and upward facing Lambertian fluxes (see Figure 5 ). It is assumed that the variations of these fluxes are due to absorption and scattering.

K and S are the Kubelka-Munk coefficients associated respectively with the absorption and scattering of the medium per length unit. As it gets into the layer of depth $\text{d}z$, the downward flow i decreases by the amount $\left(K+S\right)i\text{d}z$ due to absorption and scattering, and it further increases by the amount $Si\text{d}z$ due to the backscattering loss of the upward flow j. The same reasoning on the upward flow j leads to the following system of differential equations which can be solved in several ways, for example by using the Laplace transformation [31] or the matrix exponential method [32] :

$\frac{\text{d}i}{\text{d}z}=-\left(K+S\right)i+Sj$ (11)

$\frac{\text{d}j}{\text{d}z}=-Si+\left(K+S\right)j$ (12)

Indeed, the relationship between KubelKa-Munk coefficients and transport coefficients has been examined by several researchers previously. They used the radiative transfer equation as a starting point [33] - [35] . In this case the expression for the KubelKa-Munk absorption coefficient K is given by the Equation (13) as a function of absorption coefficient ${\mu }_a$ [34] [36] - [38] .

$K=2{\mu }_a$ (13)

On the other hand, the KubelKa-Munk scattering coefficient relation is obtained by a semi-empirical approach. By using this approach, the relation is derived for the case of incident collimated light. Thus, in isotropic domain, Klier and Gate have shown that the scattering coefficient of KubelKa-Munk noted S can be expressed as a function of the scattering coefficient ${\mu }_s$ by the following expression:

$S=\frac{3}{4}{\mu }_s$ (14)

With the local optical characteristics K and S assumed to be independent of the medium depth, Kubelka-Munk defines the following parameters [34] [36] - [39] :

$\alpha =\sqrt{K\left(K+2S\right)}$ (15)

$\beta =\sqrt{\frac{K}{K+2S}}$ (16)

We note, in addition, that the calculation of the coefficients has been standardized (ISO standard) following the method proposed by Kubelka-Munk [40] . Also it should be said that the Kubelka-Munk theory is one of the theories based on the scattering approximation and is widely used to obtain the optical properties from the measured reflectance spectra. Today this theory is widely used in the fields of multispectral imaging or structured illumination [12] to measure the optical properties of certain media such as turbid media. Moreover, the advantage of this theory is that it provides simple analytical expressions to obtain the optical parameters from the measured total reflectance or transmittance of a diffusive medium [37] . However, the conditions for the application of the Kubelka-Munk theory require a diffuse or collimated, monochromatic and isotropic light source. Also the sample must be homogeneously and isotropically absorbing or scattering. This corresponds perfectly to the SLIPI-1p technique. Thus we can say that the conditions of these equations use are applicable to SLIPI-1p.

Beer Lambert’s law exploited by the SLIPI-1p method allows the determination of the extinction coefficient ${\mu }_e$ from the measurements made by transmittance. However, let us notify here that our medium being isotropic with a detection angle of π/2, we obtain an anisotropy factor $g=0$ which leads to an equality between the reduced scattering coefficient ${{\mu }^{\prime }}_s$ and the scattering coefficient ${\mu }_s$ as suggested by the Equation (17) from [36] .

${{\mu }^{\prime }}_s={\mu }_s\left(1-g\right)$ (17)

The relationship between the extinction coefficient ${\mu }_e$, the absorption coefficient ${\mu }_a$ and the scattering coefficient ${\mu }_s$ given by the Equation (10) does not allow the calculation of the other two coefficients [41] . We end up with an equation with two unknowns. A simple calculation would consist in finding a second equation with the same unknowns. The contribution of KubelKa-Munk’s relations in this study is therefore this second equation that allows us to calculate easily the absorption coefficient ${\mu }_a$ and the scattering coefficient ${\mu }_s$. Therefore, the relationships between the Kubelka-Munk coefficients, ${\mu }_a$ and ${\mu }_s$ are introduced [34] [36] - [38] . In the case of an isotropic medium, these relations are given by the Equations (13) and (14) where K is the fraction of the flux that is absorbed and S is the fraction of the flux that is scattered.

Substituting the Equations (13) and (14) into the Equations (15) and (16), we have the expressions for $\alpha$ and $\beta$ as a function of ${\mu }_a$ and ${\mu }_s$ given by the Equations (18) and (19).

$\alpha =\sqrt{{\mu }_a\left(4{\mu }_a+3{\mu }_s\right)}$ (18)

$\beta =\sqrt{\frac{4{\mu }_a}{4{\mu }_a+3{\mu }_s}}$ (19)

Thus

$\frac{\alpha }{\beta }=\frac{4{\mu }_a+3{\mu }_s}{2}$ (20)

$\alpha \cdot \beta =2{\mu }_a$ (21)

From the Equation (10), we can derive the relation

$4{\mu }_a+3{\mu }_s={\mu }_a+3{\mu }_e$(22)

This relation is replaced in the Equations (19) and (20) to obtain the expression of $\alpha$ as a function of $\beta$, ${\mu }_a$ and ${\mu }_e$ given by the Equation (21).

$\alpha =\frac{\beta \left({\mu }_a+3{\mu }_e\right)}{2}$ (23)

The expression for $\alpha$ thus obtained is replaced in the Equation (20) and an

expression for ${\mu }_a$ as a function of $\beta$ and ${\mu }_e$ given by the Equation (11) is derived.

${\mu }_a=\frac{3{\beta }^2}{4-{\beta }^2}{\mu }_e$ (24)

The Equation (1) and (11) are then used to obtain the Equation (12) which is the expression for ${\mu }_s$ as a function of $\beta$ and ${\mu }_e$.

${\mu }_s=\frac{4\left(1-{\beta }^2\right)}{4-{\beta }^2}{\mu }_e$ (25)

The expressions for ${\mu }_a$ and ${\mu }_s$ are obtained as a function of ${\mu }_e$ which is the measurement that is therefore known and $\beta$ as an unknown parameter. The idea is to find an estimate of the parameter $\beta$. It is easy to notice that we obtain a function $g\left(\beta \right)$ that is given by the Equation (24) when we perform the calculation ${\mu }_s-{\mu }_a$.

$g\left(\beta \right)=\frac{4-7{\beta }^2}{4-{\beta }^2}{\mu }_e$ (26)

The variations of this function combined with $\beta \ge 0$ ; $-{\mu }_e\le {\mu }_s-{\mu }_a\le {\mu }_e$ and impose a value of $\beta$ between 0 and 1 with:

The use of the relations between the KubelKa-Munk coefficients and the radiative transfer coefficients of the light imposes the case of a preponderant diffusion. We can therefore write:

$\frac{{\mu }_a}{{\mu }_s}=\frac{3{\beta }^2}{4\left(1-{\beta }^2\right)}\ll 1\approx \epsilon$ (27)

$\epsilon$ the ratio of ${\mu }_a$ to ${\mu }_s$ is a very small value in terms of which we estimate from (27)

$\beta \le \sqrt{\frac{4\epsilon }{3+4\epsilon }}$ (28)

The calculation of ${\mu }_a$ and ${\mu }_s$ required to know the value of $\beta$. A direct calculation is not possible but $\beta$ can be estimated as suggested by the inequation (28). Therefore, the estimated value of $\beta$ only depend on the accuracy of the estimation of $\epsilon$. In a context of mathematical situations called precision or tolerance, $\epsilon$ must be chosen as a threshold to minimize or compare the two physical parameters obtained [42] - [44] and its current value is 10⁻³ which leads to $\beta \approx 0.036$. This value of $\epsilon$ can be adjusted according to the accuracy requirements and the characteristics of the problem. In the case of this study, we choose to directly ensure of the accuracy of $\beta$. Our approach was to calculate $\beta$ from the Equation (27) using the value of ${\mu }_a$ and ${\mu }_s$ found in the literature for the media used it means milk and coffee. The value found is of the order of 0.03 with respect to milk of a strongly scattering nature [45] - [49] that permit to confirm $\beta \approx 0.036$.

We can overlook the fact that the relationships between the KubelKa-Munk coefficients and the radiative transfer coefficients are largely obtained in the case of predominantly diffusion and rigorously obtain $\beta \approx 0.999$ for the calculation of ${\mu }_a$ and ${\mu }_s$ for highly absorbing solutions such as coffee.