Analysis of reflectance curve of turbid media and determination of the non-surface complex refractive index

Jiwen Sun; Jin Wang; Qing Ye; Jianchun Mei; Wenyuan Zhou; Chunping Zhang; Jianguo Tian

doi:10.1364/OE.23.024602

1. Introduction

Measurements of complex refractive index (CRI), namely $n + i k$ , are widely used in many fields, because it is one of the unique properties of the material itself, and can reflect a lot of information, such as concentration and temperature. Among these diverse measurement methods, it is commonly used to fit reflectance curve to experimental data according to Fresnel's Formula (FF), because of convenience and high efficiency. However, for turbid media with a large extinction coefficient $k$ , there are some deviations between the fitting curve and experimental data, especially far away from the vicinity of the critical angle [1–4 ]. Accordingly, there is a deviation between the fitting CRI and the real one. For this phenomenon, several research teams gave some interpretations and proposed different methods to improve the goodness of fit and accuracy of fitting parameters [2–5 ].

In our previous work, based on the non-uniformity at the boundary of turbid media, a gradient complex refractive index multilayered model (GCRIMM) was proposed [6]. With this model, the fitting curve fits experimental data above the critical angle fairly well, and the non-surface CRI is obtained. Nevertheless, the evanescent wave is the basis of our calculation, so the experimental data below the critical angle cannot be used. As a result, half of reflectance data are wasted, and the deviations between the experimental data below the critical angle and the fitting curve with FF cannot be directly explained by GCRIMM.

In this paper, the concept of penetration depth below the critical angle is introduced, according to Goos-Hänchen shift below the critical angle. By making an analogy with the relationship between the penetration depth and Goos-Hänchen shift of the evanescent wave, the value of penetration depth below the critical angle is obtained approximately. Then, the general formula of GCRIMM is derived, and based on it, 20% and 30% Intralipid solutions, the typical polydisperse turbid media, and the monodisperse suspension of rutile $T i O_{2}$ powder, are taken as examples to calculate their reflectance curves below the critical angle, with slightly adjusted fitting parameters determined from the experimental data above the critical angle. Compared with the fitting curve with FF, calculated reflectance results are close to experimental data, which indicates that the deviations between the fitting curve with FF and experimental data below the critical angle can be explained by GCRIMM and the values of fitting parameters can be adjusted precisely. In other words, our model for experimental data both above and below the critical angle is self-consistent. Finally, combined with the variation of penetration depth, it is drawn that reflectance data in the vicinity of the critical angle can better reflect the information of non-surface CRI of turbid media.

2. Theory

For the transparent media, when total internal reflection occurs, the time average of Poynting vector of transmitted field in the normal direction is zero, but the nonzero instantaneous value points that energy entries into optically thinner medium from optically denser medium and comes back. Besides, there is a phase difference, from 0 to $π$ , between the reflected light and the incident light, and there is also a short distance between the practical reflected beam and the geometrically reflected beam, namely well-known Goos-Hänchen shift. That is to say, on the condition that incident angle is larger than the critical angle, light propagates a distance in the form of evanescent wave and reaches a certain penetration depth, after it is incident into optically thinner medium. Evidently, for the evanescent wave, penetration depth $d_{p}$ is closely associated with Goos-Hänchen shift $D$ . In terms of their equations(TE wave), Eqs. (1) and (2) , a triangular geometric relationship can be obtained, as shown in Fig. 1 , where $α$ and $β$ are equal to the incident angle $θ_{i}$ . This can be equivalently understood as the situation that reflection takes place at the depth of $d_{p}$ instead of the physical interface between the two media [7]. For TM wave, $D_{p}$ can be obtained by $D_{p} = D_{s} \cdot n_{2}^{2} / n_{1}^{2}$ , where $n_{1}$ and $n_{2}$ are the refractive index(RI) of the incident medium and the reflecting medium respectively [8]. Although the perfect triangular relationship disappears, if $n_{1}$ is close to $n_{2}$ , the geometric relationship between $D_{p}$ and $d_{p}$ can also be approximated as a triangle.

Fig. 1 Illustration of the triangular relationship between Goos-Hänchen shift $D_{s}$ (TE wave) and penetration depth $d_{p}$

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D_{s} = \frac{λ}{π} \cdot \frac{\sin θ_{i}}{{(n_{1}^{2} \sin^{2} θ_{i} - n_{2}^{2})}^{1 / 2}} = 2 Z \cdot \cos θ_{i}

d_{p} = \frac{λ}{2 π {(n_{1}^{2} \sin^{2} θ_{i} - n_{2}^{2})}^{1 / 2}} = \frac{D_{s}}{2 \cdot \sin θ_{i}} = \frac{Z}{\tan θ_{i}}

According to Eq. (1) derived by the stationary phase method, Goos-Hänchen shift exists only above the critical angle, and diverges to infinity at the critical angle [8]. However, for turbid media, because of nonzero $k$ , it can be given that Goos-Hänchen shift exists not only above the critical angle but also below the critical angle, and is continuous at the critical angle, by Eq. (8) in [6]. In fact, even for the transparent media, if incident beam is regarded as a bounded Gaussian beam, the same result can be obtained [9,10 ]. It should be noted that, above the critical angle, the results directly calculated by Eq. (8) in [6] are consistent with experimental data, while below the critical angle, the calculated results are smaller than experimental data [9,10 ]. Hence, in calculation, the part below the critical angle is adjusted appropriately in terms of experimental data.

It is given that reflected waves both below and above the critical angle undergo the Goos-Hänchen shift from experiment and theory, so this suggests that, interacting with optically thinner medium, reflected wave propagates along the interface in the form of the surface wave which has a finite beam width. It can be inferred that the surface wave below the critical angle should be similar to the evanescent wave, so there is a penetration depth below the critical angle. Besides, from the perspective of the energy flow, Goos-Hänchen shift marks energy exchange at the interface, which implies the corresponding penetration depth necessarily exists [11,12 ]. Since Goos-Hänchen shift exists below and above the critical angle, penetration depth should also accordingly exist and be continuous at the critical angle. According to the above two points, penetration depth below the critical angle is introduced. For the penetration depth of the evanescent wave, it has been proposed and its expression has been derived before, whereas for the penetration depth below the critical angle, it is only presented in metal optics, but is not used to describe other media. Therefore, there is not a proper formula for the latter. It is worth emphasizing that the penetration depth below the critical angle is different from the depth where the amplitude of transmitted field attenuates to $1 / e$ , because it describes the reflected wave instead of transmitted wave. For the evanescent wave, its beam width, namely the penetration depth, is positively correlated with the propagation length, namely Goos-Hänchen shift. Naturally, below the critical angle, there should also be a positive correlation between Goos-Hänchen shift and penetration depth. Thus, approximate triangular relationship is adopted to calculate penetration depth based on Goos-Hänchen shift, as the situation above the critical angle shows.

In previous work, GCRIMM was presented, and under total internal reflection, the reflectance expression was also obtained by energy flow. It is briefly concluded as follows. Due to the effect of the boundary of turbid media, the distributions of particle number density and average particle size are not uniform, but follow the rule that the closer to the boundary, the less the particle number density and the smaller the average particle size. Naturally, CRI reduces with the decrease of depth. Above the critical angle, by considering the evanescent wave, a layered approach was employed to calculate energy loss in each layer, as shown in Fig. 2(a) , where $n$ and $k$ are the real and imaginary parts of CRI respectively, and increase with subscript $M$ , or the ordinal number of layer. Very importantly, layer thickness $d$ should be thin enough to ensure that CRI in each layer can be regarded as a constant. Nevertheless, this ideal model is not suitable for the practical calculation, because too many layers lead to the extreme value problem induced by lots of parameters and consuming much time. As a result, two simplified models, namely two-layer model and three-layer model, were adopted in the fitting process. The former is conductive to calculating non-surface CRI, while the latter can reveal the change of CRI with depth more clearly. In this paper, the simplified three-layer model is employed, as shown in Fig. 2(b), where $d_{1}$ , $d_{2}$ and $d_{p}$ denote the thicknesses of the first and second layer and the penetration depth respectively. Finally, the reflectance expression can be written as,

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}},

where,

l

denotes propagation distance of evanescent wave in the corresponding layer.

Fig. 2 (a)Illustration of the GCRIMM. The boundary part of turbid media is divided into M layers and energy flows in different layers attenuate in terms of CRI in corresponding layer. (b) Illustration of the simplified three-layer model. The boundary part of turbid media is divided into 3 layers.

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Combined with the above analysis, the reflected wave below the critical angle is similar to the evanescent wave. In other words, it also has the Goos-Hänchen shift and the penetration depth, so reflectance below the critical angle can be calculated by the same way. It is noteworthy that there are two differences between Eq. (3) and the general formula. The first one is that, in our data range, the penetration depth above the critical angle is large, that is, always keeping $d_{p} > d_{1} + d_{2}$ , but the penetration depth below the critical angle may be smaller than $d_{1} + d_{2}$ . Therefore, the general formula should be piecewise expressed according to different conditions. The second one is that, if the energy loss is ignored, the reflectance above the critical angle is invariably unity, whereas the reflectance below the critical angle is obtained by FF where the imaginary part of CRI is set to zero. The two points bring about the following general formula,

\begin{array}{l} R = R_{f} \cdot {\begin{cases} \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 \cdot l_{3}}}{1 - e^{- 2}}, d_{p} > d_{1} + d_{2}, \\ \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 e^{- \frac{4 π}{λ} \cdot k_{2} \cdot l_{2}}}{1 - e^{- 2}}, d_{1} < d_{p} < d_{1} + d_{2}, \end{cases} \\ R_{f} = {\begin{cases} 1, θ_{i} \geq θ_{c}, \\ {(\frac{n_{a} \cos θ_{i} - n_{0} \cos θ_{t}}{n_{a} \cos θ_{i} + n_{0} \cos θ_{t}})}^{2}, θ_{i} < θ_{c} . \end{cases} \end{array}

Here,

θ_{c}

is the critical angle,

θ_{t}

is the refraction angle, and

n_{a}

denotes the average refractive index in the region where energy flows.

3. Experiment

In our experiment, traditional internal reflection experimental setup, shown in Fig. 3 , was used to measure reflectance of sample at different angles [6]. Light from a He-Ne Laser is divided into reference beam and measuring beam by a splitter M. The reference beam is directly received by detector D1, while the measuring beam is received by detector D1, after passes through a polarizer P, an aperture diaphragm A and the prism-sample interface successively.

Fig. 3 Schematic diagram of experimental setup

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The typical biological tissue phantom, namely Intralipid solution, which primarily consists of water, soybean oil and phospholipids, was used for measurement, because of its properties of low absorption and high scattering. Before measurement, 20% and 30% Intralipid solutions should be shaken adequately to avoid normal stratification. Here, prism (F4) with a RI of 1.6166 at 632.8 nm was chosen.

Another idealized monodisperse sample, namely suspension of rutile $T i O_{2}$ powder with a volume fraction of 1.8%, was also measured. It was made by $T i O_{2}$ nanospheres with the particle size of 100nm and distilled water, and a sufficient mixing is necessary. In measurement deposition of $T i O_{2}$ nanospheres should be avoided. Besides, because RI of the suspension is small than Intralipid solution, prism (K9) with a RI of 1.5143 at 632.8 nm was chosen.

4. Results and discussion

In terms of the fitting process described in [6], reflectance data of 20% and 30% Intralipid solutions under total reflection are firstly fitted by our three-layer model, and the fitting parameters are shown in Table 1 .

Table 1. Original fitting results of 20% and 30% Intralipid solutions with our three-layer model

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Based on known formula for Goos-Hänchen shift and experimental data in [10], Goos-Hänchen shift near the critical angle can be obtained. Correspondingly, the penetration depth near the critical angle also can be determined, following the rule that above the critical angle, it is directly calculated by Eq. (2) in [6], and below the critical angle, it is approximately calculated by the above triangular relationship. These results are shown in Fig. 4 , where red curves and blue curves denote the Goos-Hänchen shift and the penetration depth respectively, and position of the critical angle $θ_{c}$ is marked.

Fig. 4 Goos-Hänchen shift (red curve) and the penetration depth (blue curve) near the critical angle for (a) 20% Intralipid solution and (b) 30% Intralipid solution

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Then, all parameter values in Table 1 and results of Goos-Hänchen shift and penetration depth in Fig. 4 are taken into Eq. (4) to directly calculate the reflectance below the critical angle. For 20% and 30% Intralipid solutions, obtained results and fitting curves based on FF are respectively shown in Figs. 5 (a) and (c) .

Fig. 5 The comparisons of calculated results with adjusted fitting parameters based on our three-layer model and fitting curves with FF for Intralipid solutions with different concentrations in different incident ranges (a) for 20% Intralipid solution below the critical angle (b) for 20% Intralipid solution above the criticle angle (c) for 30% Intralipid solution below the critical angle (d) for 30% Intralipid solution above the criticle angle

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Apparently, compared with the fitting curves based on FF, our calculated results fit experimental data better. It should be noted that in calculation process, $n'_{1}$ and $n'_{2}$ are adjusted slightly on the basis of original fitting values in Table 1. Specifically, they are adjusted to 1.3590 and 1.3642 for 20% Intralipid solution, as well as to 1.3602 and 1.3799 for 30% Intralipid solution. Although the values of $n'_{1}$ and $n'_{2}$ are changed, there is little influence on the goodness of fit above the critical angle, as shown in Figs. 5 (b) and (d). Generally, goodness of fit is used to estimate the consistency between the measured curve and fitting curve. According to Eq. (11) in [13], the goodness of fit can be calculated, and their values are 0.9974 and 0.9982 respectively for 20% and 30% Intralipid solutions. These values that are close to 1 denote that the fitting curves are reliable. In fact, within a certain range, $n'_{1}$ and $n'_{2}$ cannot significantly affect the reflectance results above the critical angle. In other words, if experimental data only above the critical angle are used, there may be some deviations for the fitting values of $n'_{1}$ and $n'_{2}$ , even though mean squared error of curve is minimum. However, the subtle changes of $n'_{1}$ and $n'_{2}$ can result in a significant impact on calculated reflectance below the critical angle. If we directly use unadjusted $n'_{1}$ and $n'_{2}$ to calculate reflectance below the critical angle, there are some deviations between the calculated values and the measured values. Hence, $n'_{1}$ and $n'_{2}$ are adjusted slightly in a proper range according to the experimental data below the critical angle, and more accurate parameter values reflecting the change trend of refractive index can be given. Compared with the previous fitting method, this is an improvement.

The monodisperse sample, a suspension of rutile $T i O_{2}$ powder whose volume fraction is about 1.8% and solute particle size is about 100nm, is measured and its results are shown in Fig. 6 . Similarly, experimental data above the critical angle is fitted by our three-layer model at first, in order to obtain the original fitting parameters. Then, the reflectance below the critical angle is calculated directly by Eq. (4), and the refractive index of the first two layers, namely $n'_{1}$ and $n'_{2}$ , are adjusted slightly to ensure that calculated results are consistent with the experimental data. Finally, the reflectance near the critical angle is obtained with the adjusted fitting parameters shown in Table 2 , as displayed in Fig. 6. It is easy to see that our model is closer to the experimental data than FF, both below and above the critical angle. Correspondingly, the goodness of fit is about 0.9941.

Fig. 6 The comparison of calculated results with adjusted fitting parameters based on our three-layer model and fitting curves with FF for the suspension of rutile $T i O_{2}$ powder (volume fraction of about 1.8%) (a) below the critical angle (b) below the critical angle

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Table 2. Adjusted fitting results of the suspension of rutile $T i O_{2}$ powder (volume fraction of about 1.8%) with our three-layer model

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In general, on the basis of these fitting parameters determined by the experimental data above the critical angle, the reflectance values below the critical angle are calculated and are consistent with the experimental data, which well verifies the self-consistency of GCRIMM. This suggests the reason that below the critical angle, fitting curves with FF deviate from experimental data can also be explained by our layered model, as the similar reason for deviations above the critical angle is given in [6]. In Fig. 4, it is obvious that the penetration depth is the deepest at the critical angle, and the farther away from the critical angle, the shallower the penetration depth becomes. According to Eq. (3) in [6], it is easy to know that incident energy flow between any two depths depends on the penetration depth. Taking the effect of non-uniformity at the boundary into account, the proportion of incident energy flow in three different layers to total incident energy flow will change with the incident angle. This leads to such a fact that the farther away from the critical angle, the lower the proportion of incident energy flow in the last layer becomes, as shown in Fig. 7 . Resultantly, the contribution of the non-surface CRI decreases. In other words, as incident angle gets farther away from the critical angle, the energy in the layers with smaller CRI becomes more. As a result, the total energy loss coming from extinction coefficient $k$ becomes less and reflectance increases. This is the reason that experimental data are larger than the fitting curve with FF, far away from the vicinity of the critical angle.

Fig. 7 The proportions of incident energy flow in three different layers to total incident energy flow versus incident angle for (a) 20% Intralipid solution and (b) 30% Intralipid solution

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Besides, because the penetration depth in the vicinity of the critical angle is relatively deep, the experimental data in this small region can better reflect the information of practical non-surface CRI of turbid media. All data and partial data are fitted with FF respectively, and the results of both are presented in Table 3 . The range of partial data is decided, depending on whether the penetration depth is much greater than $d_{1} + d_{2}$ . Here, one third of the value of penetration depth at the critical angle is chosen as a lower limit.

Table 3. Comparison of CRIs fitted with all data and partial data for Intralipid solutions and the suspension of rutile $T i O_{2}$ powder

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The relative error between $k_{a l l}$ fitted with all data and $k'_{3}$ in three-layer model can be calculated by $(k'_{3} - k_{all}) / k'_{3}$ . For 20% and 30% Intralipid solutions, the results are 11.10% and 14.80% respectively. Similarly, for 20% and 30% Intralipid solutions, the relative errors between $k_{p a r}$ fitted with partial data and $k'_{3}$ are just 0.54% and −1.46%. Also, for the suspension of rutile $T i O_{2}$ powder, the relative error between $k_{a l l}$ and $k'_{3}$ , and that between $k_{p a r}$ and $k'_{3}$ are 16.17% and 3.68% respectively. Thus it can be seen that, for extinction coefficient, the results fitted with experimental data in the vicinity of the critical angle are close to the non-surface values, but on the contrary, the results fitted with all experimental data deviate from the non-surface values. As for the relative errors of refractive index, all results in both data ranges are much less than 1%. This is because that the rapidly changing part of reflectance curve decides the fitting value of refractive index for FF, and both the global range and the chosen local range include the vicinity of the critical angle, namely the rapidly changing part [14]. Accordingly, their fitting values $n_{a l l}$ and $n_{p a r}$ are almost same and are close to the non-surface value $n'_{3}$ . Collectively, the fitting CRI with experimental data in the vicinity of the critical angle approximately equals to the CRI in last layer, that is, non-surface CRI of turbid media, which confirms the above analysis that the experimental data in the vicinity of the critical angle can better reflect the information of practical non-surface CRI of turbid media. As a result, if the change trend of CRI is not needed, fitting reflectance data in the vicinity of the critical angle with FF is a simple method to directly obtained non-surface CRI of turbid media. However, in [3], extinction coefficient by local fitting method is smaller than that by traditional global fitting method. Maybe, the reason for this situation comes from the relatively large oscillation above the critical angle.

5. Conclusion

Generally speaking, this paper is an extension of the GCRIMM. For turbid media, Goos-Hänchen shift is regarded as a bond to link the penetration depth of the evanescent wave and the reflected wave, and the penetration depth below the critical angle is proposed. Also, the fitting parameters obtained from experimental data only above the critical angle are adjusted slightly, and then are taken into the general formula. The calculated reflectance results below the critical angle are fairly consistent with measured data. This indicates the self-consistency of our model, as well as a method to improve the accuracy of fitting parameters. Finally, based on GCRIMM, the essence of an easy method that reflectance data in the vicinity of the critical angle are fitted with FF to obtain the non-surface CRI of turbid media is revealed. In practice, if the distribution of CRI of turbid media is a concern, our three-layered model can be employed, while if only the non-surface information of turbid media is needed, this simple method is a better choice. Certainly, the simple method, which can avoid measuring plenty of data, is equally applicable to transparent media and absorbing media.

Acknowledgments

The authors thank the Chinese National Key Basic Research Special Fund (grant 2011CB922003), the Natural Science Foundation of China (grant 61475078, 61405097), the Science and Technology Program of Tianjin (Grant 15JCQNJC02300, 15JCQNJC02600), the International Science & Technology Cooperation Program of China (grant 2013DFA51430).

References and Links

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Conce-ntration	$d_{1}$	$n'_{1}$	$k'_{1}$	$d_{2}$	$n'_{2}$	$k'_{2}$	$n'_{3}$	$k'_{3}$
20%	20nm	1.3620	0.204 $\times 10^{- 3}$	100nm	1.3654	0.683 $\times 10^{- 3}$	1.3661	1.503 $\times 10^{- 3}$
30%	25nm	1.3650	0.205 $\times 10^{- 3}$	125nm	1.3800	1.132 $\times 10^{- 3}$	1.3815	2.459 $\times 10^{- 3}$

materials	$n_{all}$	$k_{all}$	$n_{p a r}$	$k_{p a r}$
20% Intralipid solution	1.36602	0.001339	1.36599	0.001495
30% Intralipid solution	1.38085	0.002099	1.38085	0.002494
the suspension of rutile $T i O_{2}$	1.33624	0.002691	1.33603	0.003092

Conce-ntration	$d_{1}$	$n'_{1}$	$k'_{1}$	$d_{2}$	$n'_{2}$	$k'_{2}$	$n'_{3}$	$k'_{3}$
20%	20nm	1.3620	0.204 $\times 10^{- 3}$	100nm	1.3654	0.683 $\times 10^{- 3}$	1.3661	1.503 $\times 10^{- 3}$
30%	25nm	1.3650	0.205 $\times 10^{- 3}$	125nm	1.3800	1.132 $\times 10^{- 3}$	1.3815	2.459 $\times 10^{- 3}$

materials	$n_{all}$	$k_{all}$	$n_{p a r}$	$k_{p a r}$
20% Intralipid solution	1.36602	0.001339	1.36599	0.001495
30% Intralipid solution	1.38085	0.002099	1.38085	0.002494
the suspension of rutile $T i O_{2}$	1.33624	0.002691	1.33603	0.003092

Analysis of reflectance curve of turbid media and determination of the non-surface complex refractive index

Abstract

1. Introduction

2. Theory

3. Experiment

4. Results and discussion

5. Conclusion

Acknowledgments

References and Links

Cited By

Figures (7)

Tables (3)

Equations (4)

Optics Express