Open Access
Add to BibSonomy
Abstract
In this study, a simple method to determine the reflectance and transmittance of turbid media is proposed. The method is based on a single integrating sphere system. The integrating sphere theory is used to calculate the reflectance and transmittance under diffuse and directional illumination, and an experimental setup is proposed to measure these quantities. This method was used to determine the reflectance and transmittance of poly(vinyl alcohol) (PVA) hydrogels. Monte Carlo calculations were also implemented to validate our results. The comparison shows that the optical properties obtained through this method propagate low values of uncertainty, and a setup simpler than that of the double integrating sphere system can be realized. However, our system is limited to the diffuse–diffuse transmittance. The results suggest that the single sphere system and Monte Carlo method can be a strong combination to calculate the inherent optical properties of PVA samples.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
16 May 2019: Typographical corrections were made to the author listing and author affiliations.
1. Introduction
Integrating spheres are optical instruments usually employed to calculate the optical properties, namely reflectance and transmittance, of a turbid medium [1–3]. These integrating spheres—hereafter referred to as simply “spheres”—are often required in various branches of science, including optics [4–6], radiometry [1,7], medicine [8–10], and biology [11–13].
The integrating sphere method allows us to obtain the optical properties in several configurations of illumination and detection. In addition, this method can be implemented to determine the inherent optical properties of turbid media [8,14–17]. Moreover, they can be used to illuminate a surface with a uniform distribution of diffuse light [6]. However, this versatility comes with some drawbacks. For instance, numerous methods can be followed to obtain the optical properties of turbid media. On one hand, there are single and double integrating sphere systems. On the other hand, one can theoretically express the optical properties in diverse manners depending on the approach chosen. These approaches relate the optical properties with the detected flux (power). We can categorize these approaches into three types. The first type depends on the geometrical properties of the sphere, the second involves only power ratios, and the third is a mixture of the other two types. These approaches have been described in detail in the literature. However, a comparison among them has not yet been reported. This is relevant because measurements using integrating spheres could have large inaccuracies [18–21].
The main purpose of this study is to develop a reliable method to determine the optical properties of turbid media. This will be achieved through a sensitivity analysis of the three approaches mentioned above by using the method of integrating spheres.
The integrating sphere method can be implemented through single or double integrating sphere systems. Here, we will focus on the former, but the advantages and drawbacks of both are briefly mentioned. The single integrating sphere system employs compact relations to obtain the optical properties. Its system calibration is easier and the experimental setup is simpler compared to the double integrating sphere system. In addition, this system is cheaper than the double integrating sphere system. However, it cannot illuminate the sample diffusely in the transmission geometry, and the reflectance and transmittance cannot be measured simultaneously. Nevertheless, as we will see later, these problems can be solved by considering two beams instead of one to illuminate the sphere. The double integrating sphere system is one of the most popular configurations to measure the reflectance and transmittance of a sample [15], mainly because these quantities can be obtained simultaneously. Moreover, with the double integrating sphere system, the sample can be illuminated diffusely in the transmission or reflection geometry. However, this system is difficult to implement and calibrate, and it does not help reduce measurement uncertainties. Furthermore, its experimental setup is more expensive than that of the single integrating sphere system. We believe that a reliable and affordable system for evaluating the optical properties of a turbid medium can be realized through a single integrating sphere system.
In the integrating sphere theory, different approaches can be followed to obtain the optical properties of a system. As mentioned above, we will consider three of the most common ones. In the first approach, the reflectance and transmittance are considered as functions of the geometrical properties of the sphere [22]. In the second approach, these properties are expressed in terms of some parameters known as the constants of the sphere—we will explain them below [16]. In the third approach, the optical properties are expressed as functions of only detected power ratios. Although, we can theoretically use any approach to determine the properties of a medium, it is unclear which yields the best results, for instance, the lowest propagated uncertainty. Therefore, we will apply a sensitivity analysis to compare these approaches and determine the best among them.
Through the comparison among the three approaches, we will be able to propose an inexpensive method to obtain the optical properties of a turbid medium by using a single integrating sphere system. In particular, the reflectance and transmittance of poly(vinyl alcohol) (PVA) will be measured. This biomaterial has been used in a wide range of applications in biomedical science such as dialysis membranes, artificial skin, and surgical repairs [23]. PVA was chosen to evaluate the effectiveness of the proposed method because of its excellent mechanical strength, biocompatibility, and non-toxicity.
2. Methodology
A reliable method is proposed here to obtain the optical properties of turbid media. Our study starts by considering diffuse illumination and diffuse detection, which represents the short diffuse–diffuse configuration. Later, the optical properties under directional illumination and diffuse detection (the directional–diffuse configuration) are investigated. In particular, the reflectance and its propagated uncertainty are analyzed. A Monte Carlo model is used as a reference to compare our experimental results. Finally, we present the preparation of the PVA samples and the experimental setup used to evaluate their optical properties.
2.1 Theoretical considerations for the integrating sphere
Consider a sphere with a total area $A=4\pi r^2$, the interior of which is coated with a highly diffusive material having a reflectance denoted by $m$. It is assumed that the sphere has four ports (see Fig. 1). The areas of the sample’s port, the detector’s port, the beam entrance’s port, and a removable port to exclude the specular component of the beam are denoted by $A_s$, $A_{\delta }$, $A_e$, and $A_h$, respectively. The fractional area of the sphere covered by the diffuse material (remaining surface area of the sphere after removing the ports) is defined as $\alpha = {[A-(A_s+A_{\delta }+A_e+A_h)]}/{A}$. It should be noted that this parameter changes when the removable port is closed ($A_h=0$).
Fig. 1. Schematic diagrams of an integrating sphere used to illuminate a sample with diffuse or directional light. The reflection geometry is shown in (a) and (b), and the transmittance geometry is shown in (c) and (d).
It is also assumed that the sample is located outside the sphere. For generality, it is considered that the sample can produce coherent components (non-scattered diffuse components) through reflection and transmission, namely specular reflection $R_{sp}$ and a transmittivity $T_c$, respectively. However, these coherent components can be eliminated with a removal port, as mentioned above.
The integrating sphere theory is based on the exchange of radiation between the sphere and sample. A usual way to address this issue is through the multiple reflection inside the cavity. However, there are other ways: see Ref [24]. Thus, a relation can be established between the amount of power reflected by the sample and that collected by a detector plugged to the sphere. Here, it is important to determine the type of illumination on the sample because this trait defines the type of optical property obtained: diffuse or directional (see Figs. 1).
2.1.1 Diffuse illumination
First, consider diffuse illumination on the sample. In the reflection geometry, such illumination can be produced by a laser beam incident on the sphere’s wall. Then, according to the integrating sphere theory, the radiation (or power) emitted and received in two arbitrary areas at the surface are related by a constant, which is equal to the ratio between the illuminated surface area and the total area of the sphere [25]. In such a case, the power detected after considering all the reflections in the cavity can be expressed as [22]
where $P_o$ denotes the incident power and $R_d$ represent the diffuse-diffuse reflectance. This relation was obtained under two assumptions. First, the contribution of the detector’s reflectance is negligible compared with that of any of the reflective elements inside the sphere. Second, there is a baffle to prevent the arrival of first reflection from the illuminated spot on the wall at the detector.It should be noted that the diffuse reflectance $R_d$ can be directly extracted from Eq. (1). Hence, it can be expressed as a function of the geometrical properties of the sphere:
Although this apparently is a good way to measure the diffuse reflectance, we should be careful because the propagated uncertainty of this expression can be large, as will be shown later.In practical situations, it is convenient to express the power received by the detector, Eq. (1), in terms of some parameters known as the constants of the sphere [15], instead of their geometrical properties. These parameters help in simplifying Eq. (1) and could reduce the uncertainty in the results. However, we will see that, in some situations, this is not completely true.
The constants of the sphere are defined as [15]
To improve the measurements of the optical parameters of a heterogeneous material, consider Eq. (4) to redefine the constants of the sphere in terms of some power quotients. This is a very useful approach that we will use exhaustively. Two scenarios are considered to evaluate the parameters (see Fig. 2). First, the power detected without a sample is measured, which is denoted by $P_d^{(o)}$. However, the port of the sample remains clear ($R_d=0$ and $A_s\neq 0$) to keep the value of $\alpha$ unchanged: see Eq. (7). Later, a diffuse reflectance standard is placed on the sphere to detect a power $P_d^{(std)}$. Thus, the constants of the sphere can be rewritten as
Fig. 2. Scheme to evaluate the diffuse–diffuse reflectance $R_d$. (a) Sphere without a sample, (b) sphere with a diffuse standard, and (c) with a sample.
Thus, we derived an expression to calculate the diffuse reflectance from the above redefinitions and Eq. (4):
where $\rho _d=P_d^{(o)}/P_d$ and $\rho _{std}=P_d^{(o)}/P_d^{(std)}$. It is emphasized that three independent measurements are needed to retrieve this trait: $P_d^{(o)}$, $P_d^{(std)}$, and $P_d$, which are shown by Fig. 2. Moreover, it should be noted that this expression is a function of power ratios, and it is possible to reduce its uncertainty by increasing the number of measurements.We would like to emphasize that the relation (6) gives us a practical way to evaluate the reflectance under diffuse illumination and detection. We believe that this could be a significant contribution of this work. On one hand, as we will see later, this is an easy way to reduce the uncertainty propagated. On the other hand, the relation is such that it can be implemented with only one integrating sphere. Furthermore, by expressing the optical properties as functions of power ratios, we achieve a practical approach that will be used to obtain the optical properties in other illumination configurations.
2.1.2 Directional illumination
Consider collimated illumination on the sample and the diffuse detection of light (see Fig. 1b). In such a case, the detected power can be expressed as
The asterisk in the above equation emphasizes that a coherent component is included in the integrating sphere. Thus, $\alpha ^*$ is the fractional area of the sphere without the hole of the gloss trap ($A_h = 0$), $R_{ cd}$ is the directional–diffuse reflectance, $R_{sp}$ denotes the specular component of the light, and $P_1$ is the incident power; for generality, $P_1$ is different from $P_o$.Assume, for now, that the specular component is outside the sphere ($R_{sp}=0$): see Fig. 1b. Then, from the quotient of Eq. (7) and (1), a simple relation to calculate the diffuse-directional reflectance is obtained without including the specular component [26]:
This is a very compact expression that allows us to obtain an uncertainty much lower than that in the diffuse–diffuse configuration. Note that there is no asterisk in the above equation, because the specular component is excluded.In some situations, it may be useful to measure the contribution of the coherent component of the beam $R_{sp}$ (see Fig. 1b). Recall that this parameter corresponds to the light that is not scattered diffusely by the sample, as stated above. Then, by substituting Eq. (8) into (7) and solving for $R_{sp}$, we can express the specular reflectance as
It should be noted that the term ${P_r}/{P_d}$ is obtained with the specular component excluded. It can be seen that this expression also propagates a very low uncertainty.Table 1 lists the optical properties for both diffuse and directional configurations as power-ratio functions. The reflectance and transmittance are schematically shown in Fig. 1. As we mentioned, owing to the similarity between these quantities, the transmittance can be calculated through a similar procedure; therefore, its expression is omitted here. However, a detailed derivation of these expressions can be found in Ref. [27]. In particular, it should be noted that a diffuse source of light is needed to measure the transmittance $T_d$. The incident power is assumed to be the same in all configurations in order to simplify the expressions.
Table 1. Optical properties as functions of the detected power obtained through the integrating sphere method.
2.2 Materials and experimental setup
2.2.1 PVA preparation
An aqueous solution of PVA was prepared with a degree of hydrolysis greater than 99% and an average molecular weight (MW) of 85000-124000 pq (Sigma-Aldrich). Four PVA concentrations of 7%, 9%, 12%, and 15% by weight in solution were obtained by gradually heating the appropriate amounts of PVA and demineralized water over a thermal bath from 20 to 85 $^{{\circ} }$C for 90 min. Continuous gentle stirring is required to ensure homogeneity and promote dissolution of the PVA. The solution was allowed to stand for a few hours so that air bubbles, if any, migrate to the surface, from where they can be removed.
A mechanically rigid and optically turbid hydrogel was obtained from the aqueous solution. The solution was poured into containers refrigerated at ${-80}^{\circ}$C and kept for 90 min. Subsequently, the frozen solution was thawed at room temperature $20\pm 3\ ^{{\circ} }$C for 4 h. This constitutes one freezing–thawing cycle. It was observed that, for higher PVA weight concentrations, the hydrogels experienced an improvement in strength and an increase in turbidity with subsequent freezing–thawing cycles.
The hydrogel samples of PVA—hereafter referred to as PVA layers—were obtained by cutting a 100-$\mu$m slide with different concentrations of PVA (7%, 9%, 12%, and 15%). The samples were selected depending on the uniformity of their structure and were placed on a microscope slide with a thickness of 1.1 mm and a refractive index of 1.52 (typical crown glass) to keep its structure unchanged. It was observed that, for a thicker sample (1 mm thickness with a PVA concentration of 7%), no substrate was required.
2.2.2 Experimental measurements
A schematic of the experimental setup used to measure the optical properties of the PVA layers is shown in Fig. 3. The medium is illuminated with an NIR pulsed laser (Brilliant-b; 1064 nm; 10 Hz; 5 ns) with an incident power of $P_o=0.28$ W. An integrating sphere (Ocean Optics, ISP-50-8-R-GT) is used to collect the scattered light. This sphere has a removable port, which is covered with a white coating having a diffuse reflectance of $98\%$, to eliminate the coherent components (specular reflection or transmittivity). This port and the entrance port form an angle of 8$^{{\circ} }$ with respect to the sample surface. The sphere has a radius of 8 cm, and the ports have radii of 1 cm. As per manufacturer specifications, the diffuse reflectances of the sphere walls and the standard (OceanOptics WS-1) are both $98\%$. Furthermore, the constants of the sphere are considered to be $b_1=0.8$ and $b_2=0.25$ according to Ref. [15]. The sphere is attached to an optical detector by an optical fiber (Ocean Optics, P1000-2 UV/VIS). It should be noted that the area of the detector’s port is defined by the diameter of the optical fiber (${\sim} 1000\mu$m). A digital phosphor oscilloscope, 2 GHz, 10 GS/s, and a Si amplified detector in the wavelength range of 200–1100 nm (Thorlabs) are utilized to assess the light inside the sphere. Five mirrors around the sphere were used to illuminate the the sample in different configurations. Two of them were removable in order to change the illumination configuration by modifying the direction of the laser beam. The trajectories of beams 1, 2 and 3 on the figure helped us to illustrate each of the illumination configurations. It is worth noting that these mirrors allow us to use only one source to illuminate the sample in the reflection and transmission geometries. Thus, it can be considered that the incidence power is the same for any beam direction.
Fig. 3. Schematic of the experimental setup used to measure the optical properties of a turbid medium.
The directional reflectance $R_{cd}$ was measured using Eq. (8) presented in Table 1. This relation depends on the powers $P_r$, $P_d$ and sphere wall’s reflectance $m$. Recall that our sphere has a removable aperture to exclude the specular component, and the mirrors allow us to assume that $P_o=P_1$. Thus, the detected powers $P_d$ and $P_r$ can be measured by sequentially illuminating the sphere and sample in two configurations: diffuse and directional. Diffuse illumination on the sample was obtained hitting the wall of the sphere with a directional beam (beam 1). Directional illumination on the sample was obtained hitting the sample with a directional beam (beam 2). Additionally, the directional transmittance $T_{cd}$ could be measured with beams 1 and 3 by following a similar procedure. Finally, each of the measurements was repeated thrice, and the background signal was subtracted in each case.
Our experimental setup can be considered similar to other setups reported in the literature, such as the dual-beam spectrometer reported in Ref. [28] and the experimental setup presented in Ref. [29]. However, we would like to emphasizes that our set up was designed to be able to evaluate all the optical properties in Table 1, namely directional reflectance and transmittance, collimated reflectance and transmittance, and diffuse reflectance, rather than only the directional reflectance, as in the dual-beam spectrometer. In addition, these relations are obtained as a result of the sensitivity analysis of the different approaches discussed previously. Moreover, the previous experimental setups use a diffuse standard to measure the directional–diffuse optical properties; in contrast, our method only uses the standard to calculate the diffuse–diffuse reflectance [Eq. (6)]. Without detracting from the importance of the previous methods, we believe that our system can be another good option to study the reflectance and transmittance of turbid media.
Finally, the uncertainty of the optical properties is calculated through the relation
where $X$ denotes the properties, i.e., $R_d,$ $R_{cd}$, $R_{sp}$, etc., and $\delta x_i$ denotes the uncertainty of the variables of the optical properties.2.3 Monte Carlo calculations
The Monte Carlo method [32] will be used to calculate the reflectance and transmittance of a PVA layer in order to determine its trend.
The layered media is defined in terms of the optical properties reported in Ref. [33]: $\mu _a=0.0342$ mm$^{{-}1}$, $\mu '_s=0.5$ mm$^{{-}1}$, and $n = 1.36$ for $\lambda = 1064$ nm. The scattering coefficient is defined as $\mu _s=\rho _VC_s$ [m$^{{-}1}$], and the absorption coefficient is defined as $\mu _a=\rho _VC_a$ [m$^{{-}1}$], where $\rho _V$ denotes the volumetric density and $C_s$ and $C_a$ denote the scattering and absorption cross sections, respectively. These parameters represent the probability that a photon will be scattered (or absorbed) per unit length. The reduced scattering cross section is defined as $\mu _s'=\mu _s(1-g)$, where $g$ represents the anisotropy of the medium.
We assumed that the medium has a width of $dz=1.8$ mm to ensure that at least one interaction with the medium is obtained [$(\mu _a+\mu _s')dz \approx 1$]. It is also assumed that the substrate is a glass layer with a refractive index of 1.48 and width of 1 mm. The illumination in the medium is considered directional and orthogonal to its surfaces. We saw that an increment in the incident angle to $8^{{\circ} }$ did not produce a significant change in the results.
The Monte Carlo simulation reports the absorption, directional–diffuse reflectance $R_{MC}$, and total transmittance $T_{tot}$, among other optical parameters. The latter is the sum of the collimated $T_c$ and the directional–diffuse transmittance $T_{MC}$. In order to separate these values, it can be proved that the collimated transmittance can be expressed as $T_c=\exp [-( \mu _s+\mu _a)dz ]$. Thus, the directional–diffuse transmittance can be expressed as
In our system, the beam passes through a glass layer before leaving the system, and the transmittance of the collimated beam $T_c$ will be reduced by $T_{mr}$ because of the multiple reflections in the substrate.To simulate the increase in PVA content in the medium, linear increments of the absorption and scattering coefficients are assumed. Recall that the inherent optical properties are defined as the product of the volumetric density $\rho _V$ [m$^{{-}3}$] and the scattering (or absorption) cross section $C_s$ [m$^2$].
The inherent optical properties are equal to the product of the cross-section (absorption or scattering) [m$^2$] and the volumetric density [m$^{{-}3}$]. We will assume that cross sections are constants to reduce variations of $\mu$’s to changes only in density. Therefore, one can obtain an enhancement of the optical properties through an increment in the density by increasing the number of particles of the medium $N_o$. This could be similar to increasing the PVA content. The original density is $\rho _V=N_o/V_{layer}$, where $V_{layer}$ denotes the volume of the layer. Thus, the density can be expressed as a function of the percentage increase of the number of particles, ${\rho (n) = N_o(1+n/100)/V_{layer}=\rho _V(1+n/100)}$, where $n$ is denotes the percentage increase. For example, a zero increment of the density is expressed as $\rho (n=0)=\rho _V$, an increment of 5% is expressed as $\rho (n=5)=\rho _V(1+5\%)$, and so on. Hereafter, these percentage increments will be called %PVA simulated. Thus, the scattering coefficient can be rewritten as
and the absorption coefficient as3. Results and discussion
The uncertainty of the diffuse reflectance $R_d$, considering Eqs. (2) and (10), can be evaluated through the relation
Fig. 4. Comparison of the uncertainties for the three ways of expressing diffuse reflectance. The horizontal axis denotes the relative uncertainty and the vertical axis denotes the total magnitude of uncertainty propagated.The Eq. (10) is represented by solid line. It is considered that $R_d=0.41$ and a power detected ratio of ${\rho _d}=1.122$. (a) Evaluating the uncertainty propagation in Eq. (14). Effect of the change in each variable for a given uncertainty in that variable at $a_s=0.015$, $a_{\delta }=1.5\times 10^{{-}4}$, and $m= 0.98$. The contribution of the wall reflectance $m$ is denoted by diamonds. The contribution of the sample aperture $a_s$ is denoted by dashed line. The contribution of the detector aperture $a_d$ is denoted by circles. (b) Evaluating the uncertainty propagation in Eq. (15). Effect of the change in each variable for a given uncertainty in that variable at $b_1=0.8$ and $b_2=0.25$. Contribution of the power detected $\rho _d$ is represented by diamonds, the constant of the sphere $b_1$ by dashed line and constant of the sphere $b_2$ by circles. (c) Evaluating the uncertainty propagation in Eq. (16). Effect of the change in each variable for a given uncertainty in that variable at $R_d^{(std)}=0.98$, $\rho _D=0.9$, and $\rho _{(std)} =0.75$. Contribution of the power detected ratio $\rho _D$ is represented by dashed line, the reflectance standard $R_d^{(std)}$ by diamonds and power detected ratio with the standard $\rho _{std}$ by circles.
From Fig. 4(a) and Eq. (14), it can be seen that the main contribution to the uncertainty of $R_d$ is the uncertainty of the sphere wall’s reflectance $m$, which is three orders of magnitude higher than the uncertainty of the other variables. Hence, the contributions of the uncertainties of the remaining parameters can be considered negligible, and they are not appreciable in Fig. 4(a). Moreover, the uncertainty propagated by this relation is unacceptable for a quantity that takes values between 0 and 1. Even if the variable $m$ only varies by 1%, the uncertainty propagated to $R_d$ can still be high. Hence, we believe that this is not a good way to determine the diffuse reflectance.
On the other hand, the uncertainty of the diffuse reflectance calculated through Eqs. (4) and (10) is given by
According to Fig. 4(b), the major contribution to the percentage uncertainty of the diffuse reflectance calculated from Eq. (4) is from the constant of the sphere $b_1$ and the power ratio $\rho _d$ equivalently. The latter could be reduced by increasing the number of experiments. In contrast, the parameter $b_1$ is a constant, which could be calculated through Eq. (3) or even through (5). It is not established which of the equations is the best to calculate this value. Nevertheless, Eq. (3) is a function of the ratio $A_{\delta }/A$, which could propagate large values of uncertainty, as was seen before. Therefore, we presume that Eq. (5) could be a better way to determine $b_1$ because it is always possible to increase the number of measurements in order to reduce its uncertainty. This point is clarified in the following analysis.
The uncertainty of the diffuse reflectance obtained from Eq. (6) is shown in Fig. 4(c) and can be written as
The contribution of each term in Eq. (16) is shown in Fig. 4(c). These contributions are lower than those resulting from the other approaches. It can be seen that the largest contribution is given by the power ratio $P_d^{(o)}/P_d$, dotted line. In contrast, the lowest one is from the diffuse reflectance of the standard, $R_{std}$. However, the uncertainty propagated is always less than 1, even when all the variables vary by approximately 10%. This emphasizes the advantage of this approach.
We can see that, among the three approaches to calculate the diffuse reflectance—represented by Eqs. (2), (4), and (6)—the latter propagates the lowest percentage uncertainty. Hence, this result reveals that the most suitable way to calculate the relation from the integrating sphere theory is through power ratios. In particular, these results allow us to propose Eq. (6) as a good candidate to measure the diffuse reflectance. Furthermore, for media with a higher reflectance, the percentage uncertainty in $R_d$ could be smaller, as was pointed out by Pickering et al. [16].
In Fig. 5, the diffuse–diffuse reflectance $R_d$ is presented as a function of PVA content of the sample. This property was achieved through Eq. (6) and the set up shown in Fig. 3. We can observe an enhancement of this reflectance from 7% to 15% of PVA content. This noticeable increment (>20%) is not linearly related with the PVA content. However, it is consistent with the enhancement of scattering in the medium. We can also see that the error bars obtained are less than 5%, we could even reduce them by increasing the number of experiments according to the Eq. (16).
Next, we employed a Monte Carlo simulation [32] as a reference for the directional-diffuse reflectance and transmittance of the light in a medium-layered PVA (see Table 2 and Fig. 6).
Fig. 6. Monte Carlo simulation of a PVA layer as a function of the percentage increment $n$. (a) Directional-diffuse transmittance $T_{MC}$, collimated transmittance $T_c$, and total transmittance $T_{tot}=T_{MC}+T_c$ of the layer. (b) Directional-diffuse reflectance $R_{MC}$ and collimated or specular reflectance $R_{sp}$ of the layer. (c) Absorption of the layer ${\cal A}_{MC}$.
Table 2. Monte Carlo simulation results of a PVA layer for $\mu _a=0.0342$ mm$^{{-}1}$, $\mu '_s=0.5$ mm$^{{-}1}$, and $n = 1.36$ for $\lambda = 1064$ nm over a transparent substrate with a refractive index of 1.48. Note that, owing to energy conservation, ${\cal A}+ (R_{MC}+R_{sp} )+T_{tot}=1$.
Table 2 presents the results of the simulation with the original parameters and without any increment in the density. It can be seen that more than 40% of the light is transmitted. A similar quantity of light is absorbed and diffusely reflected by the medium. An increment of the inherent optical properties, for instance, by increasing the number of particles in the layer, can change these values. It can be useful to evaluate the impact of these changes in the reflectance and transmittance of the system.
Figure 6 show a simulation of this scenario, the effect of enhancement of the inherent optical properties attained through the increase of the number of particles, which will be similar to the increase of concentration of a PVA layer. Here, we assume that the optical cross section remains constant. Figure 6a illustrates the directional–diffuse transmittance $T_{MC}$, the collimated transmittance $T_c$ of the layer, and its sum $T_{tot}$. Fig. 6b shows the directional–diffuse and the specular reflectance, $R_{MC}$ and $R_{sp}$, respectively. Finally, Fig. 6c shows the absorption $\cal A$ of the layer.
The total transmittance $T_{tot}=T_{MC}+T_c$, which is directly obtained from the simulation and the collimated transmittance, remains almost the same between 6 to 18% of PVA content. It should be noted that the collimated transmittance $T_c$ will decrease exponentially as a function of the changes in the inherent optical properties. For an increment of 400% in density ($n= 400$), the collimated transmittance $T_c\approx 0$. This is similar to having a medium with five times the original optical thickness $[(\mu _a+\mu _s) dz]$. In this scenario the transmittance will be ruled by $T_{MC}$
The directional–diffuse reflectance $R_{MC}$ remains almost constant in the range of variation of the inherent optical properties too. The specular component $R_{sp}$ of the medium is constant because of the contrast between the refractive indices of the medium and the environment. It has a value of $R_{sp}= 0.0234$. This could be different for a real PVA layer because of the surface roughness and could produce an enhancement of the diffuse reflectance.
The absorption of the medium $\cal A_{MC}$ can also be considered constant in the range of variation of the inherent optical properties (slope ${\ll} 1$). Its increment of approximately $2 \%$ is proportional to the decrement of the transmittance. Naturally, a medium with a higher optical thickness [$(\mu _a+\mu _s)dz\gg 1$] will have a stronger absorption at the expense of reflectance and transmittance.
These Monte Carlo calculations suggest that a small range of variation of the inherent optical properties could produce a linear change in reflectance and transmittance. This information could be helpful to study the changes in the absorbed and scattered light produced by an increment of the PVA content in a real sample.
Now, the uncertainty of the directional reflectance $R_{cd}$ (see Fig. 1b) is calculated from Eqs. (8) and (10):
The directional reflectance $R_{cd}$ and transmittance $T_{cd}$ of PVA layers on a glass substrate are shown in Fig. 7. The directional transmittance is indicated by the dotted line and directional reflectance by the solid one in Fig. 7a and 7b, respectively. The absorption, shown in Fig. 7c, is obtained from relation ${\cal A}=1-(R_{MC}+T_{MC})$. The transmittance and reflectance were obtained using the experimental setup shown in Fig. 3 and the values in Table 3. To obtain these data, the relations shown in Table 1 for the directional–diffuse configuration (third and fourth rows) were used. It should be noted that the IR source of light utilized to illuminate the sphere allows us to keep the detected (${\sim} 0.135$ W) and incident power (0.25 W) very close, which is noticeable because the spheres can usually reduce the incident power by approximately two orders of magnitude. This shows the advantage of using such light sources in a sphere. Furthermore, this could be very useful to study biological tissue and to characterize PVA samples or other blanks. In particular, this could be implemented in photoacoustic experiments [31] by establishing a robust method to determine the inherent optical properties of the tissue.
Fig. 7. Integrating sphere measurements of a PVA layer with increasing PVA content. (a) Directional–diffuse transmittance $T_{cd}$ and collimated transmittance $T_c$. (b) Directional–diffuse reflectance $R_{cd}$ and (c) absorption ${\cal A}=1- (R_{cd}+T_{cd} )$ of the PVA.
Table 3. Detected power of a set of samples with different PVA contents and a width of 100 $\mu$m.
The directional transmittance decreases as a function of the PVA content. This is consistent with the Monte Carlo results. A linear fit of these data yields a line with a slope of ${-}0.0138$ and intersect of $0.49$ ( $R_{adj}^2=0.29$). On the other hand, these measurements present low values of uncertainty. We can reduce the uncertainties further by increasing the number of measurements according to Eq. (17).
The reduction of transmittance appears to be an effect of light scattering in the system. On one hand, it contributes to the return of light at the entrance face, increasing the reflectance. On the other hand, absorption is enhanced because of the increase in interactions. Recall that this parameter is given by ${\cal A} = 1 - (R_{cd} + T_{cd})$.
The directional diffuse reflectance $R_{cd}$ does not show a dependency on the PVA content. The slope of the linear fit is almost zero (slope = ${-}0.0041$, intersect of $0.24$ with $R_{adj}^2=0.42$). This is expected from the previous Monte Carlo results and is an indirect validation of the results. On other hand, absorption increases as a function of the PVA content. This is also expected from the fact that more interactions with the medium produce an enhanced absorption.
It should be noted that these results correspond to the optical properties of the system composed of PVA layers and a glass substrate. Here, we wish to stress the effect of the glass on the optical properties of the PVA layers. We observed that the substrate could have important effects on our measurements. First, the illumination of the sample will be different in the reflection and transmission geometries because of the different positions of the glass substrate in these geometries. For instance, the substrate can produce spurious reflections inside the sphere and produce a specular reflection for the glass instead of the sample, which can increase the power collected by the detector and result in an overestimation of the directional reflectance. Second, it is assumed that the glass substrate is a flat, non-absorbing homogenous layer. However, we have found that this is not entirely true, because its directional reflectance [see Eq. (8)] is 13%, while the theoretically expected value is zero. Even though this overestimation can be evaluated with the Fresnel theory and the specular reflection be removed with an extra hole in the sphere, one should be careful with this discrepancy. In order to resolve these issues, we could remove the substrate or consider this contribution while conducting measurements.
The experimental results are not perfectly consistent with the Monte Carlo calculations, but they help us confirm the expected trend. The discrepancy between the two can be a consequence of the time the sample was stored for (more than one year). The sample could have dried in this time, resulting in enhanced scattering properties [34,35].
Finally, it is worth noting that the proposed experimental setup can be used to measure all the optical properties of the medium listed in Table 1, with the exception of the diffuse transmittance $T_d$. We believe that this single integrating sphere system could be an affordable alternative to the double integrating sphere system for estimating the optical properties of turbid media.
4. Conclusions
A feasible method to calculate the optical properties of a turbid medium through a single integrating sphere system has been presented. This method has the advantages of compact expressions to obtain the optical properties with a low propagated uncertainty and a simple associated experimental setup. However, it is limited to the diffuse–diffuse transmittance.
An experimental setup to measure the reflectance and transmittance was also proposed as a result of our analysis. This setup utilizes a set of mirrors in order to use only one integrating sphere and one light source to obtain the reflectance and transmittance. This is a very practical approach that helps us increase the number of measurements, precision, and feasibility of measurement.
A Monte Carlo model was utilized to study the amount of scattered and absorbed light in a PVA layer. The model provides reference values to identify the behavior of the reflectance and transmittance of a realistic medium. Moreover, these results validate the use of this method to calculate the inherent optical properties of PVA.
It is suggested that our proposed method is a good candidate for retrieving the inherent optical properties of a turbid material. In particular, the reflectance and transmittance of PVA layers were obtained with a high precision. Additionally, this method can be used as a practical tool to explore the effect of diffuse illumination on homogenous layered media.
Funding
Universidad Autónoma de Sinaloa (PROFAPI2014/211).
Acknowledgements
E. Terán is grateful to the Universidad Autónoma de Sinaloa for financial support through grant PROFAPI2014/211.
References
1. J. M. Davies and W. Zagieboylo, “An integrating sphere system for measuring average reflectance and transmittance,” Appl. Opt. 4(2), 167–174 (1965). [CrossRef]
2. B. J. Hisdal, “Reflectance of perfect diffuse and specular samples in the integrating sphere,” J. Opt. Soc. Am. A 55(9), 1122–1125 (1965). [CrossRef]
3. L. Hanssen, “Integrating-sphere system and method for absolute measurement of transmittance, reflectance, and absorptance of specular samples,” Appl. Opt. 40(19), 3196–3204 (2001). [CrossRef]
4. G. D. Boreman, A. B. Centore, and Y. Sun, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29(4), 339–342 (1990). [CrossRef]
5. S. Nevas, F. Manoocheri, and E. Ikonen, “Gonioreflectometer for measuring spectral diffuse reflectance,” Appl. Opt. 43(35), 6391–6399 (2004). [CrossRef]
6. Z.-m. Zhu, X.-h. Qu, G.-x. Jia, and J.-f. J. Ouyang, “Uniform illumination design by configuration of LED array and diffuse reflection surface for color vision application,” J. Disp. Technol. 7(2), 84–89 (2011). [CrossRef]
7. V. Ahtee, S. W. Brown, T. C. Larason, K. R. Lykke, E. Ikonen, and M. Noorma, “Comparison of absolute spectral irradiance responsivity measurement techniques using wavelength-tunable lasers,” Appl. Opt. 46(20), 4228–4236 (2007). [CrossRef]
8. Y. Pu, W. Wang, M. AL-Rubaiee, S. K. Gayen, and M. Xu, “Determination of optical coefficients and fractal dimensional parameters of cancerous and normal prostate tissues,” Appl. Spectrosc. 66(7), 828–834 (2012). [CrossRef]
9. S. Takano and E. Okada, “Analysis of the diffuse reflectance spectra of skin due to detection system,” in European Conferences on Biomedical Optics, International Society for Optics and Photonics, 80881A (2011).
10. J. Thaung and J. Sjostrand, “Integrated light scattering as a function of wavelength in donor lenses,” J. Opt. Soc. Am. A 19(1), 152–157 (2002). [CrossRef]
11. H. Mamane, J. J. Ducoste, and K. G. Linden, “Effect of particles on ultraviolet light penetration in natural and engineered systems,” Appl. Opt. 45(8), 1844–1856 (2006). [CrossRef]
12. P. Naik and E. J. D’Sa, “Phytoplankton light absorption of cultures and natural samples: comparisons using two spectrophotometers,” Opt. Express 20(5), 4871–4886 (2012). [CrossRef]
13. M. Seyfried, L. Fukshansky, and E. Schafer, “Correcting remission and transmission spectra of plant tissue measured in glass cuvettes: a technique,” Appl. Opt. 22(3), 492–496 (1983). [CrossRef]
14. L. Fukshansky, N. Fukshansky-Kazarinova, and A. M. V. Remisowsky, “Absorption spectra of leaves corrected for scattering and distributional error: a radiative transfer and absorption statistics treatment,” Appl. Opt. 30(22), 3145–3153 (1991). [CrossRef]
15. J. W. Pickering, S. A. Prahl, N. Van Wieringen, J. F. Beek, H. J. Sterenborg, and M. J. Van Gemert, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32(4), 399–410 (1993). [CrossRef]
16. S. A. Prahl, M. J. Van Gemert, and A. J. Welch, “Double-integrating-sphere system for measuring the optical properties of tissue,” Appl. Opt. 32(4), 559–568 (1993). [CrossRef]
17. M. Hammer, A. Roggan, D. Schweitzer, and G. Muller, “Optical properties of ocular fundus tissues-an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40(6), 963–978 (1995). [CrossRef]
18. C. K. Gatebe, J. J. Butler, J. W. Cooper, M. Kowalewski, and M. D. King, “Characterization of errors in the use of integrating-sphere systems in the calibration of scanning radiometers,” Appl. Opt. 46(31), 7640–7651 (2007). [CrossRef]
19. A. C. Hardy, “History of the design of the recording spectrophotometer,” J. Opt. Soc. Am. 28(10), 360–364 (1938). [CrossRef]
20. A. M. Nilsson, A. Jonsson, J. C. Jonsson, and A. Roos, “Method for more accurate transmittance measurements of low-angle scattering samples using an integrating sphere with an entry port beam diffuser,” Appl. Opt. 50(7), 999–1006 (2011). [CrossRef]
21. A. H. Taylor, “Errors in reflectometry,” J. Opt. Soc. Am. 25(2), 51–56 (1935). [CrossRef]
22. J. W. Pickering, C. J. Moes, H. J. C. Sterenborg, S. A. Prahl, and M. J. Van Gemert, “Two integrating spheres with an intervening scattering sample,” J. Opt. Soc. Am. A 9(4), 621–631 (1992). [CrossRef]
23. H. S. Mansur and H. S. Costa, “Nanostructured poly (vinyl alcohol)/bioactive glass and poly (vinyl alcohol)/chitosan/bioactive glass hybrid scaffolds for biomedical applications,” Chem. Eng. J. 137(1), 72–83 (2008). [CrossRef]
24. J. F. Clare, “Comparison of four analytic methods for the calculation of irradiance in integrating spheres,” J. Opt. Soc. Am. A 15(12), 3086–3096 (1998). [CrossRef]
25. K. F. Carr, A Guide to Integrating Sphere Radiometry & Photometry (Labsphere Technical Guide, Labsphere Inc.1997).
26. E. Terán, E. R. Méndez, S. Enríque, and R. Iglesias- Prieto, “Multiple light scattering and absorption in reef-building corals,” Appl. Opt. 49(27), 5032–5042 (2010). [CrossRef]
27. E. Terán-Bobadilla, “Consecuencias del esparcimiento múltiple en la absorción de algunos sistemas biológicos,” PhD thesis, Centro de Investigación en Científica y de Educación Superior de Ensenada, Carretera Ensenada-Tijuana No. 3918 Zona Playitas Código Postal 22860 Apdo. Postal 360 Ensenada, B.C. México, August 2010.
28. S. Prahl, Everything I think you should know about inverse adding-doubling. Oregon Medical Laser Center, Manual of the Inverse Adding-Doubling Program (2011). http://omlc.ogi.edu/software/iad/ (retrieved 01.10.12).
29. T. P. Moffitt, “Compact fiber-optic diffuse reflection probes for medical diagnostics,” Ph.D. dissertation (Oregon Health & Science University, 2007). [CrossRef]
30. G. Kortoum, Reflectance Spectroscopy (Springer-Verlag, 1969).
31. S. L. Jacques, “Coupling 3D Monte Carlo light transport in optically heterogeneous tissues to photoacoustic signal generation,” J. Photoacoust. 2(4), 137–142 (2014). [CrossRef]
32. L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995). [CrossRef]
33. S. Manohar, A. Kharine, J. C. G. van Hespen, W. Steenbergen, and T. G. van Leeuwen, “Photoacoustic mammography laboratory prototype: imaging of breast tissue phantoms,” J. Biomed. Opt. 9(6), 1172–1181 (2004). [CrossRef]
34. T. Nakaoki and H. Yamashita, “Bound states of water in poly (vinyl alcohol) hydrogel prepared by repeated freezing and melting method,” J. Mol. Struct. 875(1-3), 282–287 (2008). [CrossRef]
35. C. M. Hassan and N. A. Peppas, “Structure and applications of poly(vinyl alcohol) hydrogels produced by conventional crosslinking or by freezing/thawing methods,” Adv. Polym. Sci. 153, 37–65 (2000). [CrossRef]
