Effect of the gradient of complex refractive index at boundary of turbid media on total internal reflection

Jiwen Sun; Jin Wang; Yuanze Liu; Qing Ye; Huanhuan Zeng; Wenyuan Zhou; Jianchun Mei; Chunping Zhang; Jianguo Tian

doi:10.1364/OE.23.007320

1. Introduction

Measurement of the complex refractive index (CRI), an important optical parameter, has been increasingly applied to biomedicine, environmental science, oceanography and production process [1–4]. Among many measurement technologies, it’s convenient and most common to obtain the CRI by fitting intensity reflectance curve to data according to Fresnel’s Formula. For the transparent media and weakly absorbing media, experimental reflectance data can obey Fresnel’s Formula well [5], while for the scattering turbid media, there are some deviations between the experimental data and the fitting curve in terms of Fresnel’s Formula. Especially above the critical angle, compared with the fitting curve, the experimental reflectance data are relatively large. Moreover, the discrepancies are more significant with the increase of scattering. If fitting method based on Fresnel’s Formula is still employed, there will be some errors on the fitting parameters, particularly the imaginary part of CRI, due to large differences between experimental data and fitting curve above the critical angle.

In order to improve goodness of fit in regression analysis, different explanations and methods have been proposed. An angle-dependent $n_{i}$ , the imaginary part of CRI, used in Fresnel’s Formula was presented, because of the penetration depth depending on the incident angle [6, 7]. Also, its expression is deduced from the penetration depth under total reflection instead of the depth where the intensity of transmitted field attenuated to $e^{- 1}$ at normal incidence in the traditional sense. Since $n_{i}$ is related to the incident angle, a local curve-fitting method was raised to replace conventional global curve-fitting method [8]. Augusto et al. built a coherent-scattering model in the basis of Mie scattering theory and took advantage of multiple reflections formula to calculate the reflectance for low-concentration solution [9, 10]. Just a two-parameter empirical formula was mathematically given to correct reflectance R by I. Niskanen et al. [11].

Intralipid, namely fat emulsion, mainly composed of water, soybean oil and phospholipids, is a typical turbid medium. Because of its properties of low absorption and high scattering, it is often used as the biological tissue phantom in various researches, such as photodynamic therapy [12, 13]. So, the research on optical properties of Intralipid is necessary and useful.

In this paper, taking non-uniformity at the boundary of turbid media into account, a gradient complex refractive index multilayered model (GCRIMM) is raised. Based on the GCRIMM, a simplified three-layer model is used to obtain the expression of reflectance, and in calculation, the energy loss is derived from penetration depth and Goos-Hänchen shift. For convenient fitting process, the simplified three-layer model is approximated as an easier two-layer model. Besides, 20% and 30% Intralipid solutions and rutile $T i O_{2}$ powder suspension are measured, and the reflectance data are fitted according to Fresnel’s Formula and our simplified models respectively. Compared with Fresnel’s Formula, the simplified models can fit data better, and the value of fitting parameter $k$ , the imaginary part of CRI, is clearly larger. Most importantly, our model reveals the physics mechanism of the deviations between the experimental data and the fitting curve with Fresnel’s Formula for turbid medium.

2. Theory

For turbid media, solute particle sizes are not uniform but exit a certain distribution. Some single substance can be regarded as a monodisperse system, as long as particle sizes are approximately equal and particle size distribution is narrow. Nevertheless, most substance must be treated as a polydisperse system. Just as Intralipid, during the emulsification process, fat is wrapped by phospholipid molecules whose one end is lipophilic and the other end is hydrophilic, which forms negatively charged emulsion particles with different particle sizes and a broad particle size distribution [14]. If solute particles are uniformly distributed in the solvent and the volume of solution is infinite, any part of solution can be regarded as an effective medium with the same optical parameters. However, considering the presence of the boundary, there are two reasons to make the ideal situation invalid, shown as Fig. 1(a). Firstly, according to the surface physics, when prism, which mainly consists of $S i O_{2}$ , and emulsion particles are immersed in the water, there is a several-nanometer-thick action zone in the vicinity of the surface, namely solvation layer [15]. If two solvation layers overlap, repulsive hydration force will appear. Moreover, because the isoelectric point of $S i O_{2}$ is around 2, there are negative charges, which are the same as naturally occurring negative surface charges on the emulsion particles, on the surface of $S i O_{2}$ immersed in Intralipid solution [16]. Therefore, even though solution is homogeneous enough, because of the repulsion of solvation force and electrostatic force, emulsion particles can’t tightly cling to the prism surface. This leads to a fact that the particle number density is reduced in the vicinity of the prism surface. Secondly, due to the restriction of the prism surface, emulsion particles can’t pass through the surface. Taking the centroid of every emulsion particle as reference, large particles are far away from the prism surface, while small particles are near to the prism surface. That is, near the prism surface, particle size distribution of emulsion particles isn’t uniform, but there is a gradient. The closer to the prism, the smaller average particle size becomes. In short, the particle number density and the average particle size have a combined effect on CRI of the emulsion sample, and naturally, the closer to the prism, the lower the CRI is.

Fig. 1 Illustrations of (a) particle distribution near the prism surface (b) the GCRIMM (c) the simplified three-layer model.

Download Full Size | PDF

The above-mentioned phenomenon generally exists in most turbid media, yet Fresnel’s Formula is only applicable to uniform media. Therefore, GCRIMM is adopted to calculate reflectance, instead of following Fresnel’s Formula with an assumed constant CRI of test substance [17],

\begin{array}{l} {\begin{cases} R_{s} = \frac{{(n_{0} \cos θ_{1} - u_{2})}^{2} + v_{2}^{2}}{{(n_{0} \cos θ_{1} + u_{2})}^{2} + v_{2}^{2}} \\ R_{p} = \frac{{[{(n_{s}^{2} - k_{s}^{2})}^{2} \cos θ_{1} - n_{0} u_{2}]}^{2} + {(2 n_{s} k_{s} \cos θ_{1} - n_{0} v_{2})}^{2}}{{[{(n_{s}^{2} - k_{s}^{2})}^{2} \cos θ_{1} + n_{0} u_{2}]}^{2} + {(2 n_{s} k_{s} \cos θ_{1} - n_{0} v_{2})}^{2}} \end{cases}, \\ u_{2}^{2} - v_{2}^{2} = n_{s}^{2} - k_{s}^{2} - n_{1}^{2} \sin^{2} θ_{1}, u_{2} v_{2} = n_{s} k_{s}, \end{array}

where,

n_{0}

is the refractive index (RI) of incident medium,

n_{s}

and

k_{s}

are the real part and the imaginary part of the CRI of the test substance respectively, and

θ_{1}

is the incident angle.

In this case that the reflecting medium has a lower RI than the incident medium, when incident angle is greater than critical angle, total internal reflection will occur. In other words, light incidents into the reflecting medium, after a shift, and comes back to the incident medium in the form of the evanescent wave. Thus, in order to analyze the light reflection near the prism surface, GCRIMM is shown in Fig. 1(b). In this model, the reflecting medium is divided into many layers, and $E_{m}$ denotes electric field amplitude in corresponding layer, where $m$ is the ordinal number of layers. $n_{m}$ and $k_{m}$ are the real part and the imaginary part of the CRI in corresponding layer, and they increase gradually as the depth increases. The thickness of each layer $d$ must be very thin so that there are only small changes for $n$ and $k$ between adjacent layers. It is worth noting that interfaces between adjacent layers aren’t specular, due to the solute particles. So, it is obviously invalid to calculate reflectance between adjacent layers by Fresnel’s Formula, except for the prism-sample interface. In the experiments, since reflectance is determined by measuring the intensity of reflected light and incident light, we can consider the total reflection process from the perspective of the energy flow. Energy that enters into the reflecting medium is divided into $M$ parts, and each part accompanied by the loss propagates in a direction parallel to the interface in corresponding layer. Finally, all parts come back to the incident medium together.

The penetration depth of the evanescent wave can be calculated by following formula [18],

\begin{array}{l} d_{p} = \frac{λ}{\sqrt{2} π n_{0}} \cdot \frac{1}{{(\sqrt{υ_{2}^{2} + μ_{2}^{2}} + υ_{2})}^{\frac{1}{2}}}, \\ υ_{2} = \sin^{2} θ_{1} - (n_{s}^{2} - k_{s}^{2}) / n_{0}^{2}, μ_{2} = 2 n_{s} k_{s} / n_{0}^{2}, \end{array}

where

n_{0}

,

n_{s}

,

k_{s}

and

θ_{1}

are the same as the above, and

λ

is the wavelength of incident light in vacuum.

Thus, amplitude in each layer can be obtained in terms of the expression of evanescent wave [18], namely $E_{m} = E \cdot e^{- m \cdot d / d_{p}}$ . Accordingly, intensity in each layer is written as, $I_{m} = I \cdot e^{- 2 m \cdot d / d_{p}}$ . However, it is worth stressing that intensity I of light is defined as the time average of the amount of energy which crosses in unit time a unit area perpendicular to the direction of the energy flow [15]. So, in order to gain the energy flow, we need area integral of intensity on the area which is perpendicular to the direction of the energy flow. Assuming beam width in every layer is a constant $d_{z}$ , energy flow between any two depths can be written as,

w = d_{z} \cdot \int_{d_{1}}^{d_{2}} I \cdot e^{- 2 \cdot \frac{x}{d_{p}}} d x = \frac{d_{z} I d_{p}}{2} \cdot (e^{- 2 \cdot \frac{d_{1}}{d_{p}}} - e^{- 2 \cdot \frac{d_{2}}{d_{p}}}) .

Hence, incident energy flow in the $m th$ layer can be expressed as,

w_{m} = \frac{d_{z} I d_{p}}{2} \cdot [e^{- 2 \cdot \frac{(m - 1) d}{d_{p}}} - e^{- 2 \cdot \frac{m d}{d_{p}}}] .

In view of a loss, intensity can be calculated by the attenuation coefficient $α$ , $I = I_{0} \cdot e^{- α \cdot l}$ , where $l$ is the propagation distance of light in the turbid medium, and α is usually associated with $k$ according to the expression, $α = \frac{4 π}{λ} \cdot k$ . In the $m th$ layer, $d$ is so thin that $k_{m}$ is a constant. Therefore, outgoing intensity in arbitrary depth is the product of incident intensity in the same depth and a factor $e^{- \frac{4 π}{λ} \cdot k_{m} \cdot l_{m}}$ , where $l_{m}$ is the propagation distance of light in the $m th$ layer. Obviously, outgoing energy flow in the $m th$ layer is the product of incident energy flow in this layer and the same factor $e^{- \frac{4 π}{λ} \cdot k_{m} \cdot l_{m}}$ , as follows,

w'_{m} = w_{m} \cdot e^{- \frac{4 π}{λ} \cdot k_{m} \cdot l_{m}} .

Since divergence angle of laser is extremely small, cross-sectional area of the incident beam can be considered equal to the outgoing beam. So, intensity reflectance $R$ can be expressed as the ratio of the total amount of outgoing energy flows to the total amount of incident energy flows. Taking penetration depth $d_{p}$ as a cut-off, we have,

R = \frac{\sum_{m = 1}^{N} w'_{m}}{\sum_{m = 1}^{N} w_{m}} = \frac{\sum_{m = 1}^{n} (e^{- 2 \cdot \frac{(m - 1) d}{d_{p}}} - e^{- 2 \cdot \frac{m d}{d_{p}}}) \cdot e^{- \frac{4 π}{λ} \cdot k_{m} \cdot l_{m}}}{1 - e^{- 2}} .

Because of a small enough $d$ , $l_{m}$ can be regarded as a sum of twice layer thickness and parallel propagation distance of evanescent wave in this layer. In calculation, parallel propagation distance is approximated to the Goos-Hänchen shift $D_{p}$ divided by $\cos θ$ . Accordingly, $l_{m}$ is written as,

l_{m} = \frac{D_{p}}{\cos θ_{1}} + 2 d,

where

D_{p}

is calculated according to CRI in this layer. As p-polarized light from a real RI medium (

n_{0}

) incident into a CRI medium (

n_{s} + i \cdot k_{s}

), under total reflection, Goos-Hänchen shift is represented as [19],

\begin{array}{l} D_{p} = \frac{λ}{2 π} {(C + D) \times [\begin{array}{l} \sin θ_{1} (g_{2} q - g_{1} τ) - \\ \frac{n_{0}^{2}}{\sqrt{A^{2} + B^{2}}} \sin θ_{1} \cos^{2} θ_{1} (g_{2} q + g_{1} τ) \end{array}] + (C - D) \times \frac{2 n_{0}^{2} q τ}{\sqrt{A^{2} + B^{2}}} \sin θ_{1} \cos θ_{1}}, \\ C = \frac{1}{{(g_{1} \cos θ_{1} - q)}^{2} + {(g_{2} \cos θ_{1} - τ)}^{2}}, D = \frac{1}{{(g_{1} \cos θ_{1} + q)}^{2} + {(g_{2} \cos θ_{1} + τ)}^{2}}, \\ g_{1} = \frac{n_{s}^{2} (1 - k_{s}^{2})}{n_{0}}, g_{2} = \frac{2 n_{s}^{2} k_{s}}{n_{0}}, A = n_{s}^{2} (1 - k_{2}^{2}) - n_{s}^{2} \sin^{2} θ_{1} = q^{2} - τ^{2}, B = 2 n_{s}^{2} k_{s} = 2 q τ . \end{array}

So far, reflectance $R$ can be completely obtained by Eqs. (6)-(8). However, in the practical calculation, the more the number of layers, the more the amount of calculation and parameters needed to fit. As a result, the multilayered model is simplified as a three-layer model shown as Fig. 1(c). In the first layer closest to the prism surface with a very thin thickness $d_{1}$ , because emulsion particles can’t tightly cling to the prism and pass through the surface, the number density of emulsion particles is low and the average particle size is small. Naturally, both n1 and k1 are not large. In the second layer slightly far away from the surface of the prism with a relatively large thickness $d_{2}$ , number density of emulsion particles becomes higher and approximately equals to that of the non-surface part of material, while the average particle size is larger than that in the first layer and smaller than that of the non-surface part of material. Thus, $n'_{2}$ and $k'_{2}$ are a bit larger than those in the first layer and lower than those of the non-surface part of material. In the last layer, there is some distance away from the prism surface, so not only the particle number density but also the average particle size approximately equal to that of the non-surface part of material. In other words, $n'_{3}$ and $k'_{3}$ can be treated as the real part and the imaginary part of the CRI of the non-surface part of material, respectively. On the basis of the above analysis, Eq. (6) is simplified as the formula below,

R = \frac{(1 - e^{- 2 \cdot \frac{d_{1}}{d_{p}}}) \cdot e^{- \frac{4 π}{λ} \cdot k_{1} \cdot l_{1}} + (e^{- 2 \cdot \frac{d_{1}}{d_{p}}} - e^{- 2 \cdot \frac{d_{1} + d_{2}}{d_{p}}}) \cdot e^{- \frac{4 π}{λ} \cdot k_{2} \cdot l_{2}} + (e^{- 2 \cdot \frac{d_{1} + d_{2}}{d_{p}}} - e^{- 2}) \cdot e^{- \frac{4 π}{λ} \cdot k_{3} \cdot l_{3}}}{1 - e^{- 2}} .

It should be emphasized that, for the common fitting methods based on Fresnel’s Formula, turbid material is regarded as an effective medium with a constant CRI. Nevertheless, actually, CRI near the prism surface is lower than that of the non-surface part of turbid material. A portion of total incident energy flow propagates in the part with a lower RI, so practical energy loss is less than the result calculated by CRI of the non-surface part of turbid material. Hence, most experimental reflectance data above the critical angle are greater than the fitting value with Fresnel’s Formula.

3. Experiment

In our experimental installation schematically shown as Fig. 2 [20], the incident light powers and the outgoing light powers at different incident angles are detected. Light from a He-Ne laser is divided into two parts through the splitter M. One part is received by detector D1 of a dual-channel power meter (PM320E, Thorlabs, New Jersey, USA) as reference light, in order to eliminate the effects of fluctuations of laser in the experiment. The other part passes through the polarizer P and the aperture diaphragm A, and then becomes collimated p-polarized light. Finally, it is detected by detector D2 after two refractions on air-prism interface and a partial reflection or total internal reflection on prism-sample interface.

Fig. 2 Schematic diagram of experimental installation.

Download Full Size | PDF

In our experiment, the RIs of 20% and 30% intralipid solutions, were expected to be between 1.36 and 1.38, so the triangular prism (F4) with a RI of 1.6166 at 632.8 nm was chosen. Intralipid solution placed for a long time occurs normal stratification, although components and effectiveness of solution don’t change. So, it’s necessary to make it well mixed by sufficient shake before it was poured into the sample pool. Besides, it takes a short time to measure, so Intralipid solution can’t occur stratification during this period. Rutile $T i O_{2}$ powder suspension, an idealized monodisperse sample, whose volume fraction is about 2%, was made by $T i O_{2}$ nanospheres with particle size of 100nm and distilled water. Differently, the triangular prism (K9) with a RI of 1.5143 at 632.8 nm was chosen to adapt RI of the suspension. Because $T i O_{2}$ particles are easy to deposit, we captured the less number of reflectance data for suspension to avoid deposition. According to the geometric relationships and Snell's law, incident angle was derived from the rotation angle, and then the reflectance curve versus angle was obtained. For curve-fitting method with Fresnel’s Formula, there are two fitting parameters, the real part and the imaginary part of CRI, so they can be fitted directly at the same time. However, for curve-fitting method with the simplified three-layer model, there are too many unknown parameters, including three sets of CRI in three different layers as well as two layer thicknesses, so they can’t be fitted at the same time. For this reason, the following steps are adopted to solve this multidimensional unconstrained optimization problem. Firstly, by fitting Eq. (1) to experimental data, the values of the real part and the imaginary part of CRI were obtained and marked as $n_{f}$ and $k_{f}$ . Afterwards, the initial values of $n'_{3}$ and $k'_{3}$ in simplified model were set to $n_{f}$ and $k_{f}$ respectively. According to Eqs. (7)-(11), it can be inferred that the layer thicknesses $d_{1}$ and $d_{2}$ have a more important effect on the curve profile than CRIs in the first and second layer. Hence, $d_{1}$ and $d_{2}$ were fitted preferentially by assuming the proper values of $n'_{1}$ , $k'_{1}$ , $n'_{2}$ and $k'_{2}$ on the basis of previous analysis, $n'_{1} < n'_{2} < n'_{3}$ and $k'_{1} < k'_{2} < k'_{3}$ . After that, the four parameters were fitted accurately with successive approximation method, and then $d_{1}$ and $d_{2}$ were fine tuned to gain other sets of values of $n'_{1}$ , $k'_{1}$ , $n'_{2}$ and $k'_{2}$ . By comparisons, a best-fitting reflectance curve was obtained. Similarly, $n'_{3}$ and $k'_{3}$ were adjusted carefully around $n_{f}$ and $k_{f}$ respectively, and the above steps were repeated many times to gain other best-fitting reflectance curves. Finally, these fitting curves were compared with each other to pick up the one which fits data best, and the set of optimal solutions were ultimately determined. It’s worth emphasizing that the reasonable initial values of parameters and several constraints can avoid multi extreme value problem effectively.

In order to reduce the number of fitting parameters, the first and the second layers can be regarded as a single layer with a CRI, $n_{12} + i \cdot k_{12}$ , so a two-layer model is proposed under this approximation. The reflectance R can be directly calculated by Eq. (6) where N is taken as 2. Corresponding depth of the single layer is denoted as $d_{12}$ . If we are only interested in the $n'_{3}$ and $k'_{3}$ , but not the change trend of CRI near the boundary, the two-layer model is a good choice, because it reduces the amount of calculation greatly. The following results are all obtained by three-layer model, except for the comparisons between the three-layer model and the two-layer model.

4. Results and discussion

The fitting parameters with two different methods are listed in Table 1. By comparing CRIs in different layers in the same concentration solution, for 20% intralipid solution, the ratios of changes on $n$ and $k$ between the second and the last layers to that between the first and the second layers are obtained, $r_{n} = \frac{(n'_{3} - n'_{2})}{(n'_{2} - n'_{1})} = 20.59 %$ and $r_{k} = \frac{(k'_{3} - k'_{2})}{(k'_{2} - k'_{1})} = 171.41 %$ . Similarly, for 30% intralipid solution, $r'_{n} = 10.00 %$ and $r'_{k} = 143.22 %$ are obtained. These results mean that $n$ and $k$ increase as the depth increases, but the change trends of them are inconsistent. In other words, compared with $k$ , change of $n$ occurs mainly in the region closer to boundary. Combining with the above analysis, the reason is that the particle number density and the average particle size increase as the depth increases, and their combined effects on $n$ and $k$ are different. This is qualitatively consistent with the effective medium theory.

Table 1. Fitting results of 20% and 30% Intralipid solutions with Fresnel’s Formula and the simplified three-layer model

View Table | View all tables in this article

Taking solvent water $(n_{w} = 1.3321, k_{w} = 2.5 \times 10^{- 5})$ as reference, comparison between the fitting results with Fresnel’s Formula and our model indicates that, for 20% and 30% intralipid solutions, the relative errors of $n$ calculated by $(n'_{3} - n') / (n' - n_{w})$ , are about 0.236% and 1.361% respectively, while the relative errors of $k$ are 12.251% and 17.160% respectively. The fitting values of $n$ obtained with two different methods are approximately equal. Nevertheless, $k$ fitted with our model is higher than that fitted with Fresnel’s Formula. The reason for relatively large deviation on $k$ is that the changes of the particle number density and the average particle size with the increase of the depth are ignored in the fitting method with Fresnel’s Formula. As for $n$ , its changes concentrate mainly in the first layer, about 20 nanometers thick, and the ratio of the thickness of the first layer to the penetration depth is very low in most data used to fit. Consequently, although the changes of the particle number density and the average particle size with the increase of depth is ignored in the fitting process, there is a negligible effect on the result of $n$ .

The measured reflectance data and fitting curves for undiluted 20% and 30% intralipid solutions are shown in Figs. 3(a) and 3(b) respectively. It is worth noting that, only a portion of data used in fitting process are given in Fig. 3, in order to show the deviations above the critical angle more clearly. Obviously, above the critical angle, there are some deviations between red curves and the experimental data. Especially in the range far away from the critical angle, data are higher than the curve, and the farther away from the critical angle, the larger deviations between experimental data and red curves are. Comparison of Figs. 3(a) and 3(b) indicates that the higher the concentration, the greater the deviations. However, the black curves can correspond to the data very well. In other words, our model works fairly well.

Fig. 3 Experimental data above the critical angle (blue dots) and fitting reflectance curves with Fresnel’s Formula (red solid line) and our simplified three-layer model (black solid line) for (a) 20% intralipid solution (b) 30% intralipid solution.

Download Full Size | PDF

As previously discussed, the discrepancies between experimental data above the critical angle and the fitting curve with Fresnel’s Formula originate from the gradient CRI at the boundary of turbid media. To explain the reason why the discrepancies far away from the critical angle are greater than that near the critical angle, we take 30% Intralipid solution for example. According to Eq. (2), penetration depth decreases as incident angle increases under total reflection, and gradually approaches the depths of the first layer and the second layer, as shown in Fig. 4(a). For this reason, the ratios of incident energy flows in different layers to the total energy flow also change as incident angle increases according to Eq. (3). Evidently, Fig. 4(b) indicates that the ratio of incident energy flow in the last layer decreases with the increase of incident angle, while that in the first layer and the second layer increases with the increase of incident angle. In terms of previous conclusions, namely $n'_{1} < n'_{2} < n'_{3}$ and $k'_{1} < k'_{2} < k'_{3}$ , this means that as incident angle increases, more parts of total incident energy flow propagate in the part with the lower CRI, so the total energy loss becomes smaller. However, if there isn’t a gradient distribution on CRI, or CRIs in first two layers are the same as the last layer, changes of ratios will not affect the energy loss. That is to say, at a larger incident angle, the difference between total amount of outgoing energy flows in the practical situation and fitting value with Fresnel’s Formula is greater, as shown in Fig. 3.

Fig. 4 (a) penetration depth versus incident angle for 30% Intralipid solution (b) ratios of incident energy flows in different layers to the total incident energy flow versus incident angle for 30% Intralipid solution.

Download Full Size | PDF

Indeed, because of simplification of the three-layer model, the CRI within a certain range is artificially assumed to be the same value. This leads to a result that the fitting CRI is the average CRI in the layer. Accordingly, calculated reflectance $R$ is an approximation. However, the resulting error is small enough to be negligible. In calculation, for the more accurate results, the number of layers can be increased.

30% Intralid solution was taken as example to compare the fitting curves based on the three-layer model and the two-layer model. The fitting result is shown in Fig. 5. The blue curve and the black curve almost overlap, and are very closed to data points. It indicates that the high goodness of fit isn’t affected by the number of parameters, which can check our model’s validity indirectly. The fitting values of $n_{12}$ , $k_{12}$ and $d_{12}$ are 1.377, 0.00094 and 151nm respectively, and the values of $n'_{3}$ and $k'_{3}$ are the same as that of three-layer model. Thus, in practical calculations, the two-layer model can obtain the CRI of the non-surface part of material accurately. Certainly, the obtained CRI in the approximation of a single layer is an average CRI in the first and second layers in simplified three-layer model, so there are some differences between two curves more or less.

Fig. 5 The comparison between the three-layer model, the two-layer model and Fresnel’s Formula for 30% Intralipid solution.

Download Full Size | PDF

In fact, GCRIMM is raised to mainly describe a more common polydisperse system. However, there is a gradient distribution of CRI at the boundary of a monodisperse system too. As a result, it is also available for a monodisperse system theoretically. For rutile $T i O_{2}$ powder suspension, a similar result can be obtained in Fig. 6. The blue curve and the black curve almost overlap and fit experimental reflectance better than the red one. This checks the validity of simplified models for a monodisperse system. The fitting parameters are shown in Table 2. Because the particle size is relatively uniform in the monodisperse system, from the previous analysis, it is can be inferred that the range of gradient distribution that is brought from different particle sizes is small. That is to say, it is reasonable that the depth of the second layer is smaller than that in Intralipid solution. As for the depth of the first layer, the larger result than that in Intralipid solution maybe comes from the thicker solvation layer.

Fig. 6 The comparison between the three-layer model, the two-layer model and Fresnel’s Formula for rutile $T i O_{2}$ powder suspension.

Download Full Size | PDF

Table 2. Fitting results of rutile $T i O_{2}$ powder suspension with Fresnel’s Formula, the simplified three-layer model and the simplified two-layer model

View Table | View all tables in this article

In all, the above results show that whichever our model is used is much better than Fresnel’s Formula. The three-layer model can reflects more details at the boundary, while the two-layer model makes fitting process become easy. Importantly, the two simplified models can get almost the same CRI of the non-surface part of material, the information that we most need.

Although some papers have given own fitting methods to make the fitting curves fit the experimental data well, there are some limitations and deficiencies worth exploring. An angle-dependent $n_{i}$ that was proposed in the papers [6, 7] is worth discussing, because $n_{i}$ of material is constant instead of variable. Besides, in the expression of $n_{i}$ , the reciprocal of penetration depth of evanescent wave is regarded as attenuation coefficient $α$ . However, $α$ associated with by $α = \frac{4 π}{λ} \cdot n_{i}$ , is generally regarded as the reciprocal of the depth where the intensity of transmitted field attenuated to $e^{- 1}$ at normal incidence. There are some differences between the two expressions. In the papers [9, 10], interface between the matrix and the composite medium is approximated as a specular surface to use multiple reflections formula. They obtained the reflectance successfully from the perspective of scattering, but there is a limitation that in order to make sure workableness of coherent-scattering model, the concentration of test solution must be low enough. The modified formula of reflectance in the paper [11], is a purely empirical formula without any theoretical explanations.

5. Conclusion

In this paper, GCRIMM is proposed according to the practical state of turbid media at the boundary, in order to explain the phenomenon that the experimental data are greater than fitting results with Fresnel’s Formula above the critical angle. Also, from the perspective of energy flow, p-polarized intensity reflectance is theoretically calculated by using the penetration depth and Goos-Hänchen shift. In experiment, 20% and 30% concentration intralipid solutions and rutile $T i O_{2}$ powder suspension are measured to obtain the values of reflectance at different incident angle. As we expect, above the critical angle, the fitting reflectance curve with simplified models can fit the experimental data fairly well. What's more, compared with the fitting result with Fresnel’s Formula, there is a larger imaginary part of the CRI based on our method.

Compared with the previous papers, this paper more physically reveal the essence of experimental phenomenon, namely the non-uniformity of turbid media at the boundary.

Furthermore, the model and calculation method we present are suitable for not only intralipid solution but also other turbid media. That is to say, by fitting the reflectance curve to experimental data, we can determine internal gradient CRI near the surface of turbid media directly.

Acknowledgments

The authors thank the Chinese National Key Basic Research Special Fund (grant 2011CB922003), the Natural Science Foundation of China (grant 61475078, 61405097), the International Science & Technology Cooperation Program of China (grant 2013DFA51430).

References and Links

1. Y. L. Jin, J. Y. Chen, L. Xu, and P. N. Wang, “Refractive index measurement for biomaterial samples by total internal reflection,” Phys. Med. Biol. 51(20), N371–N379 (2006). [CrossRef] [PubMed]

2. M. Ebert, S. Weinbruch, P. Hoffmann, and H. M. Ortner, “The chemical composition and complex refractive index of rural and urban influenced aerosols determined by individual particle analysis,” Atmos. Environ. 38(38), 6531–6545 (2004). [CrossRef]

3. X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. 34(18), 3477–3480 (1995). [CrossRef] [PubMed]

4. A. J. Jääskeläinen, K. E. Peiponen, and J. A. Räty, “On reflectometric measurement of a refractive index of milk,” J. Dairy Sci. 84(1), 38–43 (2001). [CrossRef] [PubMed]

5. G. H. Meeten and A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6(2), 214–221 (1995). [CrossRef]

6. W. R. Calhoun, H. Maeta, A. Combs, L. M. Bali, and S. Bali, “Measurement of the refractive index of highly turbid media,” Opt. Lett. 35(8), 1224–1226 (2010). [CrossRef] [PubMed]

7. K. G. Goyal, M. L. Dong, V. M. Nguemaha, B. W. Worth, P. T. Judge, W. R. Calhoun, L. M. Bali, and S. Bali, “Empirical model of total internal reflection from highly turbid media,” Opt. Lett. 38(22), 4888–4891 (2013). [CrossRef] [PubMed]

8. W. Guo, M. Xia, W. Li, J. Dai, X. Zhang, and K. Yang, “A local curve-fitting method for the complex refractive index measurement of turbid media,” Meas. Sci. Technol. 23(4), 047001 (2012). [CrossRef]

9. R. G. Barrera and A. García-Valenzuela, “Coherent reflectance in a system of random Mie scatterers and its relation to the effective-medium approach,” J. Opt. Soc. Am. A 20(2), 296–311 (2003). [CrossRef] [PubMed]

10. A. García-Valenzuela, R. Barrera, C. Sánchez-Pérez, A. Reyes-Coronado, and E. Méndez, “Coherent reflection of light from a turbid suspension of particles in an internal-reflection configuration: Theory versus experiment,” Opt. Express 13(18), 6723–6737 (2005). [CrossRef] [PubMed]

11. I. Niskanen, J. Räty, and K.-E. Peiponen, “Complex refractive index of turbid liquids,” Opt. Lett. 32(7), 862–864 (2007). [CrossRef] [PubMed]

12. I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34(12), 1927–1930 (1989). [CrossRef]

13. H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991). [CrossRef] [PubMed]

14. G. Chansiri, R. T. Lyons, M. V. Patel, and S. L. Hem, “Effect of Surface Charge on the Stability of Oil/Water Emulsions during Steam Sterilization,” J. Pharm. Sci. 88(4), 454–458 (1999). [CrossRef] [PubMed]

15. L. He, Y. Hu, M. Wang, and Y. Yin, “Determination of solvation layer thickness by a magnetophotonic approach,” ACS Nano 6(5), 4196–4202 (2012). [CrossRef] [PubMed]

16. G. A. Parks and A. Parks, “The isoelectric points of solid oxides, solid hydroxides, and aqueous hydroxo complex systems,” Chem. Rev. 65(2), 177–198 (1965). [CrossRef]

17. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999)

18. F. M. Mirabella, Internal Reflection Spectroscopy: Theory and Applications (Chemical Rubber Company, 1992)

19. G.-Y. Leng, “Study of Goos-Hänchen shift of reflected beam from absorbing medium,” Acta Opt. Sin. 7(5), 464–467 (1987).

20. Q. Ye, J. Wang, Z.-C. Deng, W.-Y. Zhou, C.-P. Zhang, and J.-G. Tian, “Measurement of the complex refractive index of tissue-mimicking phantoms and biotissue by extended differential total reflection method,” J. Biomed. Opt. 16(9), 097001 (2011). [CrossRef] [PubMed]

Concentration	$n_{f}$	$k_{f}$	$d_{1}$	$n'_{1}$	$k'_{1}$	$d_{2}$	$n'_{2}$	$k'_{2}$	$n'_{3}$	$k'_{3}$
20%	1.36602	1.339 $\times 10^{- 3}$	20 nm	1.3620	0.204 $\times 10^{- 3}$	100 nm	1.3654	0.683 $\times 10^{- 3}$	1.3661	1.503 $\times 10^{- 3}$
30%	1.38085	2.099 $\times 10^{- 3}$	25 nm	1.3650	0.205 $\times 10^{- 3}$	125 nm	1.3800	1.132 $\times 10^{- 3}$	1.3815	2.459 $\times 10^{- 3}$

Model	$n_{f}$	$k_{f}$	$d_{1}$ $(d_{12})$	$n'_{1}$ $(n_{12})$	$k'_{1}$ $(k_{12})$	$d_{2}$	$n'_{2}$	$k'_{2}$	$n'_{3}$	$k'_{3}$
Three- layer	1.34900	5.113 $\times 10^{- 3}$	51 nm	1.3334	0.027 $\times 10^{- 3}$	28 nm	1.3451	3.228 $\times 10^{- 3}$	1.3509	6.039 $\times 10^{- 3}$
Two- layer	1.34900	5.113 $\times 10^{- 3}$	82 nm	1.3390	1.138 $\times 10^{- 3}$				1.3509	6.038 $\times 10^{- 3}$

Concentration	$n_{f}$	$k_{f}$	$d_{1}$	$n'_{1}$	$k'_{1}$	$d_{2}$	$n'_{2}$	$k'_{2}$	$n'_{3}$	$k'_{3}$
20%	1.36602	1.339 $\times 10^{- 3}$	20 nm	1.3620	0.204 $\times 10^{- 3}$	100 nm	1.3654	0.683 $\times 10^{- 3}$	1.3661	1.503 $\times 10^{- 3}$
30%	1.38085	2.099 $\times 10^{- 3}$	25 nm	1.3650	0.205 $\times 10^{- 3}$	125 nm	1.3800	1.132 $\times 10^{- 3}$	1.3815	2.459 $\times 10^{- 3}$

Model	$n_{f}$	$k_{f}$	$d_{1}$ $(d_{12})$	$n'_{1}$ $(n_{12})$	$k'_{1}$ $(k_{12})$	$d_{2}$	$n'_{2}$	$k'_{2}$	$n'_{3}$	$k'_{3}$
Three- layer	1.34900	5.113 $\times 10^{- 3}$	51 nm	1.3334	0.027 $\times 10^{- 3}$	28 nm	1.3451	3.228 $\times 10^{- 3}$	1.3509	6.039 $\times 10^{- 3}$
Two- layer	1.34900	5.113 $\times 10^{- 3}$	82 nm	1.3390	1.138 $\times 10^{- 3}$				1.3509	6.038 $\times 10^{- 3}$

Effect of the gradient of complex refractive index at boundary of turbid media on total internal reflection

Abstract

1. Introduction

2. Theory

3. Experiment

4. Results and discussion

5. Conclusion

Acknowledgments

References and Links

Cited By

Figures (6)

Tables (2)

Equations (9)

Optics Express