2.1.

Bidirectional Measurements

The five sintered Zenith PTFE samples were purchased from SphereOptics (Concord, New Hampshire). The thicknesses of samples 1 to 5 were measured with a micrometer to be $(0.109\pm 0.003)$ , $(0.259\pm 0.003)$ , $(0.522\pm 0.004)$ , $(1.057\pm 0.008)$ , and $(10.1\pm 0.1)\mathrm{mm}$ , respectively. Table 1 lists the sample parameters as well as some of the measured optical properties. The densities of samples range from $1.5\text{to}1.9\mathrm{g}∕{\mathrm{cm}}^3$ . These samples were cut into $50\times 50\text{-}{\mathrm{mm}}^2$ pieces from a sheet, except for sample 5, which came as a $50\times 50\text{-}{\mathrm{mm}}^2$ piece from the manufacturer.

Table 1

Parameters of the PTFE samples and the directional-hemispherical reflectance and transmittance obtained by integrating the measured BRDF and BTDF at λ=635nm .

A three-axis automated scatterometer (TAAS)²³ with a $635\text{-}\mathrm{nm}$ laser diode was used for the measurement of the BRDF and BTDF. The BRDF is defined as the reflected radiance divided by the incident irradiance,

Figure 1 shows the schematic of light scattering from a semitransparent film. The observation angle is defined separately for BRDF and BTDF as depicted in the figure. For BTDF measurement when ${\theta }_o={\theta }_i$ , both the transmitted collimated light (i.e., unscattered light) and the scattered light may contribute to the detector signal. The measured BTDF should exhibit a noticeable peak at ${\theta }_o={\theta }_i$ when the collimated light transmittance is significant enough.

Fig. 1

Schematic of volume scattering by a slab and the laser scatterometer. The unscattered light is the transmitted collimated beam. The observation angle ${\theta }_o$ is relative to the surface normal of the sample and defined separately for BRDF and BTDF measurements. In the experiments, the direction of the laser is fixed. The sample can be rotated to change the incidence angle ${\theta }_i$ and the detector can be rotated around the sample.

2.2.

Spectral Characterization

The wavelength dependence of $R$ and $T$ was characterized with a monochromator and an integrating sphere at wavelengths from $350\text{to}1050\mathrm{nm}$ .²⁴ The light source was a tungsten-halogen lamp. The rotation of two gratings inside the monochromator and the change of filters at the inlet of the monochromator are capable of achieving a wavelength resolution of $10\mathrm{nm}$ . The rms fluctuation of power from the monochromator was estimated to be 1%. The light exiting the monochromator is directed to the sample mounted either before the sphere entrance port for measuring $T$ or at the back of the sphere wall for measuring $R$ . A chopper is used to obtain a phase-locked optical signal. A silicon detector was located at the bottom port with a baffle to prevent the direct illumination of the detector from the scattered light. The detector output is amplified by a transimpedance preamplifier before being sent to a lock-in amplifier. The integrating sphere has an inner wall of $200\mathrm{mm}$ in diameter. The entrance port is $25\mathrm{mm}$ in diameter. The inner wall is coated with PTFE with a reflectance between 0.98 and 0.99 in the visible spectral range.

2.3.

Theory and Calculations

With the assumptions that the medium does not emit light (i.e., cold medium), the wave-like interactions are negligible, the medium is homogeneous, and the polarization state is neglected, the light scattering in a PTFE film can be modeled using the RTE as

If only the hemispherical properties are of interest, the adding-doubling method is an effective technique that results in high accuracy for obtaining $R$ and $T$ of an absorbing and scattering medium. The adding-doubling method was intro-duced by van de Hulst²⁵ to solve the RTE in a parallel slab composed of multiple layers. In the adding method, if the reflection and transmission functions of each individual layer are known, the reflection and transmission functions of the composite can be calculated. The adding method is termed as the doubling method when the layers are identical in thickness and in all the RTE parameters.²⁰ The reflection and transmission functions for an arbitrarily thick slab can be obtained by repeatedly adding and doubling the layers until the desired thickness is reached. Subsequently, the directional-hemispherical reflectance and transmittance can be calculated by integrating the reflection and transmission functions. Detailed discussion of the adding-doubling method can be found from Refs. 16, 20. The adding-doubling method requires inputs of the scattering albedo, optical thickness ${\tau }_{\lambda }=({\sigma }_{\lambda }+a_{\lambda })d$ with $d$ being the film thickness, and the asymmetric parameter $g$ to predict the directional-hemispherical properties of PTFE films.

A Monte Carlo algorithm was developed to model the BRDF and BTDF by considering volume scattering but neglecting the effect of surface scattering. Furthermore, the Monte Carlo simulation does not consider the dependent multiple scattering and coherent wave-like interactions. After a photon bundle is released from air to the PTFE medium, the propagation step between subsequent scattering events is calculated by $l=-\ln \left(R_1\right)∕({\sigma }_{\lambda }+a_{\lambda })$ , where $R_1$ is a random number between 0 and 1 and is generated based on a quasirandom sequence.²⁶ The previous equation results in the mean free path (i.e., average propagation step) of $1∕({\sigma }_{\lambda }+a_{\lambda })$ . The angle between the direction vectors before and after the scattering is determined by the Henyey–Greenstein scattering phase function given in Eq. 4 following the procedure described by Wang, Jacques, and Zheng.²⁷ After each scattering event, the energy of the photon bundle reduces to ${\omega }_{\lambda }E$ due to the absorption, where $E$ is the energy of the photon bundle before scattering.

In the Monte Carlo simulation, $R$ is calculated as the summation of the reflected photon bundle energy divided by the total energy of the incident photon bundle, which is the product of the number of photon bundles and the energy assigned to each incident photon bundle. Similarly, $T$ is calculated as the summation of the transmitted photon bundle energy divided by the total energy of the incident photon bundles. The calculated $R$ and $T$ using the Monte Carlo simulation with a photon bundle number of $1\times {10}^6$ is in agreement with that using the adding-doubling method within a statistical fluctuation of 0.5%. To determine the BRDF and BTDF, a virtual detector with a solid angle of $1.76\times {10}^{-3}\mathrm{sr}$ is placed in $5\text{-}\mathrm{deg}$ intervals in the corresponding hemisphere. For the calculation of BRDF and BTDF, $2\times {10}^7$ photon bundles are used and each run takes approximately $2.5\mathrm{h}$ of CPU time with a $3.2\text{-}\mathrm{GHz}$ Pentium 4 processor when the thickness of the sample is $0.1\mathrm{mm}$ . The resulting fluctuation is within 1.5% of the BRDF and BTDF values. Additional photon bundles can be used to reduce the statistical fluctuation with longer calculation time.

3.1.

Measured Bidirectional Reflectance Distribution Function and Bidirectional Transmittance Distribution Function

Figure 2 shows the measured BRDF for all five samples and the BTDF for samples 1 to 4 for normal incidence at the wavelength $\lambda =635\mathrm{nm}$ . The transmittance of sample 5 is less than 0.01, and its BTDF cannot be measured due to the limited signal-to-noise ratio. The measured BRDF and BTDF exhibit nearly diffuse characteristics. For a perfectly diffuse reflector, the BRDF should be a constant of $0.318{\mathrm{sr}}^{-1}$ . In a round-robin test of the PTFE material, Early ²⁸ also reported the similar variation of the BRDF of thick PTFE slabs. As the sample thickness decreases, the corresponding BRDF value decreases because of the increased transmission. Interestingly, the hump in the BRDF near ${\theta }_o=0\mathrm{deg}$ becomes more obvious as the thickness decreases. Due to the measurement uncertainty of the laser scatterometer, the measured BRDF does not exhibit perfect symmetry with respect to ${\theta }_o=0\mathrm{deg}$ .

Fig. 2

Measured (a) BRDF and (b) BTDF, averaged over the two polarizations, at normal incidence for $\lambda =635\mathrm{nm}$ .

As expected, the BTDF values increase as the sample thickness decreases. The measurements reveal that the BTDF of samples 2, 3, and 4 are flatter than the corresponding BRDF even though the values are smaller. This can be explained by the multiple scattering of light inside the film. In order for the photon bundles to transmit through the PTFE films, they need to experience numerous scattering events, resulting in random propagation in the medium. Therefore, the BTDF of sample 4 with thickness of $1\mathrm{mm}$ is nearly uniform within $\pm 75\mathrm{deg}$ observation angles.

To obtain the hemispherical properties, the BRDF and BTDF data need to be carefully analyzed and fitted to perform integration. Additional measurements with closely spaced data intervals were performed. As an example, Fig. 3 shows the average of the measured BTDF (circles) and the fitted (solid line) BTDF of sample 1 at normal incidence. Eight measurements were conducted at different locations on the sample. The location on the sample was changed by rotating it by $45\mathrm{deg}$ after each measurement. Because the laser beam was not aligned to the center of rotation of the sample, the location of the laser spot changed when the sample was rotated. Error bars stand for the random uncertainty at the 95% confidence level. The measured results beyond $78\mathrm{deg}$ are not reliable because of the alignment difficulty. Since the measurement exhibits some fluctuations due to the measurement uncertainty and sample inhomogeneity, a polynomial equation was used to fit the BTDF data. For BTDF of sample 1, a quadratic function is sufficient. The standard error of estimate of the fitted equation is $0.0015{\mathrm{sr}}^{-1}$ , which is much smaller than the standard deviation of the data points. Notice that an extrapolation of the fitted equation for ${\theta }_o$ from $78\text{to}90\mathrm{deg}$ is performed before the integration; however, the relative error caused by this extrapolation is estimated to be less than 0.5%. In addition, the bidirectional properties are assumed to be symmetric with respect to ${\theta }_o=0\mathrm{deg}$ .

Fig. 3

Average of eight BTDF measurements for sample 1 and the fitted polynomial curve. The error bars show the measurement uncertainty with 95% confidence interval.

The polynomial fitting functions are integrated to obtain $R$ from the BRDF and $T$ from the BTDF of each sample. When $C_I$ is taken as 1, the summation of $R$ and $T$ is always greater than unity by approximately 6%. While the measurements of TAAS agree well with specular samples (within 2% for a smooth Si), it is about 5% higher for a rough silicon surface and about 5% higher for the $10\text{-}\mathrm{mm}$ -thick PTFE than those obtained from a standard reference instrument at NIST.^{23, 28} This may be caused by the detector collection geometry and the uncertainty of the aperture area. To reduce the bias uncertainty, all TAAS measurement results are reduced by 6%. This may be considered as a self-calibration procedure. With $C_I=0.943$ , the reflectance of sample 5 obtained by integration of BRDF is 0.988, which agrees well with the reported values.⁸ The summation of $R$ and $T$ for each of the thinner samples is about unity, as shown in Table 1. When the sample location is fixed, the variation of repeated BRDF and BTDF measurements is within 2%. The combined uncertainty of TAAS is within 5% with a confidence level of 95%. For BTDF of sample 1, however, the data variation at different locations on the sample is greater than the instrument uncertainty due to sample inhomogeneity.

The effects of incidence angle on the BRDF are shown in Fig. 4 for all samples. The measurements were performed at incidence angle ${\theta }_i=0$ , 30, 50, and $70\mathrm{deg}$ with a linear polarizer. The results are plotted in terms of ${\mathrm{BRDF}}^{*}\cos \left({\theta }_o\right)$ to make the comparison easier. Even at oblique incidence, the diffuse scattering feature is preserved in a large observation angle range except a peak in the forward scattering direction. The peak is much higher for $s$ -polarization than for $p$ -polarization, suggesting that surface scattering becomes important at large angles of incidence. This can be understood by two facts: 1. the surface looks smoother at glazing angles; and 2. the reflectance at the interface between air and PTFE is higher for $s$ -polarization than for $p$ -polarization. The Brewster angle ${\theta }_B={\tan }^{-1}\left(n\right)$ is about $54\mathrm{deg}$ for refractive index $n=1.36$ . At the Brewster angle, the surface reflectance for $p$ -polarization approaches zero. When the data are examined more carefully, the BRDF for ${\theta }_i=50\mathrm{deg}$ is higher than that for ${\theta }_i=30\mathrm{deg}$ at large observation angles. Hence, volume scattering must also play a role in the enhanced forward scattering. It can also be seen that for samples 4 and 5, the peaks at ${\theta }_i=70\mathrm{deg}$ and $60\mathrm{deg}<{\theta }_o<80\mathrm{deg}$ are much lower than for samples 1 to 3. Visual observation reveals that sample 1, 2, and 3 may contain some surface crystallization that has improved the smoothness and specularity of these samples. This may be the reason for the specular peaks in samples 1 to 3 for the $s$ -polarization at ${\theta }_i=30$ and $50\mathrm{deg}$ . At ${\theta }_i=70\mathrm{deg}$ , samples 4 and 5 also exhibit specular peaks for $s$ -polarization but with a smaller peak than that of the thinner samples.

Fig. 4

BRDF of all five samples at various incidence angles for both $p$ - and $s$ -polarizations.

The increased BRDF at oblique incidence results in a reduction in the BTDF as depicted in Fig. 5 , where the ordinate is in terms of ${\mathrm{BTDF}}^{*}\cos \left({\theta }_o\right)$ . It should be noted that the scale of the ordinates is different for each sample. The data points at ${\theta }_o=-5$ and $-10\mathrm{deg}$ were missing for ${\theta }_i=70\mathrm{deg}$ due to blocking of the beam by the rotary stage. As discussed earlier, the BRDF decreases with increasing thickness, and also the polarization has little effect on the BRDF at normal incidence. It can be clearly seen that more reduction of BTDF existed for $s$ -polarization than for $p$ -polarization. On the other hand, because the transmitted light experiences multiple scattering events, they are redirected into random directions. Hence, the BTDF shape appears symmetric about ${\theta }_o=0\mathrm{deg}$ , regardless of the incidence angle.

Fig. 5

BTDF of the four thin-film samples at various incidence angles for both polarizations.

3.2.

Estimation of ${\sigma }_{\lambda }^{\prime }$ and the Ranges of $a_{\lambda }$ , ${\sigma }_{\lambda }$ , and $g$

If the absorption coefficient $a_{\lambda }$ is negligible, the directional-hemispherical properties of a volume scattering medium depend only on the film thickness $d$ and the reduced scattering coefficient ${\sigma }_{\lambda }^{\prime }={\sigma }_{\lambda }(1-g)$ .²⁰ In the present study, ${\sigma }_{\lambda }^{\prime }$ is estimated by comparison of the calculated $R$ -to- $T$ ratio for the given thickness with those obtained by integrating the measured BRDF and BTDF over the corresponding hemisphere. In fact, the ratio $R∕T$ is independent of the bias uncertainty in the scatterometer measurement. Note that the refractive index of PTFE is around 1.36, and the reflectance at the interface between air and PTFE is approximately 2.3% near normal incidence. The reflected light will be distributed hemispherically due to surface roughness and may exhibit a broadened specular peak. In the present study, the adding-doubling method is employed to solve the RTE for the PTFE films, without considering surface scattering due to refractive index mismatch and surface roughness.

To determine ${\sigma }_{\lambda }^{\prime }$ , the ratio $R∕T$ is calculated as a function of ${\sigma }_{\lambda }^{\prime }$ for the thickness $d$ equal to the thicknesses of the four thinner samples. By assuming that all samples have the same ${\sigma }_{\lambda }^{\prime }$ , the standard error of estimate between the calculated and measured $R∕T$ for all four samples is a function of ${\sigma }_{\lambda }^{\prime }$ only. The standard error of estimate between the measured and the calculated $R∕T$ values reaches a minimum at ${\sigma }_{\lambda }^{\prime }=167{\mathrm{cm}}^{-1}$ . Considering the measurement uncertainty and sample-to-sample variation, the reduced scattering coefficient is estimated to be ${\sigma }_{\lambda }^{\prime }=(167\pm 20){\mathrm{cm}}^{-1}$ . Here, the obtained ${\sigma }_{\lambda }^{\prime }$ value is approximately one order of magnitude higher than the reported value in Ref. 22.

Figure 6 shows the measured $R∕T$ ratio and the calculated values using the adding-doubling method with ${\sigma }_{\lambda }^{\prime }=167{\mathrm{cm}}^{-1}$ . The inset shows the results for smaller $d$ values to compare the measurement with the calculation clearly. While in general the reflectance and transmittance are complicated functions of the film thickness, the ratio $R∕T$ exhibits a linear dependence on the film thickness for $d$ from $0.1\text{to}10\mathrm{mm}$ . A linear regression of the adding-doubling calculations yields that $R∕T=0.59{\sigma }_{\lambda }^{\prime }d$ . Using different values of ${\sigma }_{\lambda }^{\prime }$ , it was found that a constant of 0.59 remains the same, regardless of the value of ${\sigma }_{\lambda }^{\prime }$ .

Fig. 6

The ratio of $R∕T$ obtained from the integration of measured BRDF and BTDF (dots) and that calculated using adding-doubling method (triangles), where the inset shows the results for a small $d$ range. The parameters used for the calculation are ${\sigma }^{\prime }=167{\mathrm{cm}}^{-1}$ and $g=0.9$ .

For strongly scattering and nonabsorbing media, the linear dependence of $R∕T$ on ${\sigma }_{\lambda }^{\prime }d$ can be derived from other approximation models such as the Kubelka–Munk model (or two-flux model)²⁹ and the three-flux model.³⁰ Star, Marijnissen, and van Gemert³¹ reported the transformation between the scattering coefficient used in the Kubelka–Munk model and the scattering coefficient ${\sigma }_{\lambda }$ of RTE. By solving the governing equations of these models, it can be shown that the ratio of $R∕T$ is in linear relationship with ${\sigma }_{\lambda }^{\prime }d$ , similar to that calculated from the adding-doubling method. The linear dependence of $R∕T$ on film thickness for strongly scattering and nonabsorbing mediums can be obtained from all these models, including the adding-doubling method, Monte Carlo simulation, Kubelka–Munk model, and the three-flux model. Although there are some variations in the final analytical expressions, depending on different model approximations, the linear relationship provides a very simple approach for the determination of the reduced scattering coefficient of PTFE films by measuring $R$ and $T$ of a sample with known thickness.

Table 2 lists the calculated $R$ and $T$ of the $10\text{-}\mathrm{mm}$ -thick PTFE slab using the adding-doubling method. The parameters are set to be ${\sigma }_{\lambda }^{\prime }=167{\mathrm{cm}}^{-1}$ and $g=0.9$ (i.e., ${\sigma }_{\lambda }=1670{\mathrm{cm}}^{-1}$ ). As mentioned in the previous section, the reduced scattering coefficient is the dominant factor in determining the hemispherical properties. Hence, the calculation shown in Table 2 is also applicable for different $g$ values as long as ${\sigma }_{\lambda }^{\prime }$ is fixed. The absorption coefficient must be less than $0.01{\mathrm{cm}}^{-1}$ for $R$ of the $10\text{-}\mathrm{mm}$ -thick sample to be greater than 0.977. It should be noted that the measured values are usually greater than 0.98.⁸

Table 2

The influence of the absorption coefficient on the directional-hemispherical reflectance and transmittance of the 10-mm -thick PTFE slab when σλ=1670cm−1 and g=0.9 .

As can be seen from Table 2, the adding-doubling calculation suggests that the $10\text{-}\mathrm{mm}$ -thick slab can have a nonzero transmittance as large as 0.01 when the absorption is neglected. To assess the transmittance value, the integrating sphere is used with the $635\text{-}\mathrm{nm}$ laser diode as the light source. The sample is placed at the entrance port. The transmittance of sample 5 ( $10.1\mathrm{mm}$ thick) is estimated to be $0.007\pm 0.002$ , which indicates that the absorption coefficient should indeed be less than $0.01{\mathrm{cm}}^{-1}$ (referring to Table 2). The uncertainty in the integrating sphere measurement for the $10\text{-}\mathrm{mm}$ -thick sample is large due to the low signal-to-noise ratio. The uncertainty associated with the integrating sphere measurements for most samples is greater than that in the scatterometer measurement. Therefore, the integration of BRDF and BTDF allows a more accurate determination of $R$ and $T$ at the laser wavelength. Because the obtained $a_{\lambda }$ is at least four orders of magnitude smaller than the scattering coefficient, the absorption in PTFE films is neglected hereafter. It should be mentioned that the absorption coefficient of PTFE is much smaller than typical biological tissues.

If the collimated light transmittance $T_d$ is significant, the scattering coefficient can easily be determined by using Beer’s law $T_d=\exp (-{\sigma }_{\lambda }d)$ . For highly scattering samples, however, the optical signal received by the detector when ${\theta }_o={\theta }_i$ contains both the collimated light transmitted through the sample without scattering and the scattered light. The correct way to determine $T_d$ is to subtract the power received by the detector at ${\theta }_o=0\mathrm{deg}$ for normal incidence to that at ${\theta }_o=\delta$ , where $\delta$ is a small angle from $1\text{to}3\mathrm{deg}$ that allows the detector to stay away from direct light exposure. Unfortunately, even for the thinnest sample in this study, no peaks in the BTDF can be observed around ${\theta }_o=0\mathrm{deg}$ . The BTDF of sample 1 ( $109\mu \mathrm{m}$ thick) for eight different measurements at normal incidence are plotted in Fig. 7 for ${\theta }_o=0\text{to}6\mathrm{deg}$ with a $1\text{-}\mathrm{deg}$ interval. The fitted curve and the error bounds with 95% confidence interval are also shown. The resulting BTDF is rather flat in this region and no peak can be seen at ${\theta }_o=0\mathrm{deg}$ , suggesting that the transmitted collimated light is overwhelmed by the scattered light in the direction parallel to the incidence. In other words, the contribution of the collimated light transmission to the BTDF must be less than the uncertainty, which is about $\Delta f=0.012{\mathrm{sr}}^{-1}$ at the 95% confidence level. Therefore, the maximum direct transmittance (transmitted power divided by the incident laser power) should be $\Delta f\delta {\Omega }_o=2.2\times {10}^{-6}$ . Setting $T_d=\exp (-{\sigma }_{\lambda }d)=2.2\times {10}^{-6}$ yields ${\sigma }_{\lambda }\approx 1200{\mathrm{cm}}^{-1}$ , which may be regarded as the lower limit of the scattering coefficient. If the scattering coefficient were less than ${\sigma }_{\lambda }\approx 1200{\mathrm{cm}}^{-1}$ , a distinguishable peak, exceeding the upper bound of the uncertainty, would have appeared in the BTDF at ${\theta }_o=0\mathrm{deg}$ .

Fig. 7

BTDF of sample 1 at normal incidence for eight measurements in the observation angle range from $0\text{to}6\mathrm{deg}$ . The solid line represents the quadratic fit of the average BTDF, discussed previously in Fig. 3, whereas the dashed lines denote the uncertainty bounds with 95% confidence interval.

Under the condition that the wave-like interactions and the dependent scattering are negligible, the mean free path should be greater than several wavelengths. Suppose that the minimum mean free path is $2.5\mu \mathrm{m}$ , which is approximately four times the laser wavelength (i.e., $635\mathrm{nm}$ ), then the upper limit of the scattering coefficient is estimated to be $4000{\mathrm{cm}}^{-1}$ . Based on ${\sigma }_{\lambda }(1-g)={\sigma }_{\lambda }^{\prime }=167{\mathrm{cm}}^{-1}$ and $1200{\mathrm{cm}}^{-1}<{\sigma }_{\lambda }<4000{\mathrm{cm}}^{-1}$ , the asymmetric parameter $g$ should be between 0.86 and 0.96, suggesting that PTFE is a strongly forward scattering material. Compared with the parameters of biological tissues, the scattering coefficient of PTFE is approximately ten times larger.¹⁷ Much thinner samples, e.g., $d<30\mu \mathrm{m}$ , would be necessary to precisely determine the scattering coefficient of PTFE.

3.3.

Monte Carlo Simulation

The purpose of using Monte Carlo simulation is to see how volume scattering affects the BRDF and BTDF, without considering surface scattering and polarization effects. In the calculation, the parameters are set to be ${\sigma }_{\lambda }^{\prime }=167{\mathrm{cm}}^{-1}$ and $g=0.9$ , i.e., ${\sigma }_{\lambda }=1670{\mathrm{cm}}^{-1}$ . The simulation results of BRDF and BTDF at normal incidence are shown in Fig. 8 . Compared with the measurements shown in Fig. 2, the Monte Carlo simulation can predict the general trend of the BRDF but with a rapid drop at large observation angles. The Monte Carlo results do not show the hump near ${\theta }_o=0\mathrm{deg}$ , because the hump is associated with the surface scattering effect. Note that the humps are more prominent for samples 1 to 3 than for samples 4 and 5. It should be noted that the Monte Carlo model predicted the same $R$ and $T$ as the adding-doubling method did. For the BTDF, the Monte Carlo simulation also captures the essential features qualitatively; but the resulting BTDF exhibits a much larger angular variation compared with the experimental results. There are two reasons for this discrepancy. First, if surface scattering were included, there will be a total internal reflection for light emerged from PTFE to air at polar angles (inside PTFE) greater than the critical angle ${\theta }_c={\sin }^{-1}(1∕n)\approx 47\mathrm{deg}$ . This critical angle may largely affect the energy distribution at large observation angles. Second, the Henyey–Greenstein scattering phase function is an oversimplified one, and its applicability for predicting the bidirectional scattering properties may be questionable.

Fig. 8

Monte Carlo simulation of (a) BRDF and (b) BTDF at normal incidence.

Figure 9 shows the cosine ${\theta }_o$ modified BRDF and BTDF for sample 2 at oblique incidence. Even without considering polarization dependence and surface scattering, the BRDFs exhibit asymmetric characteristics due to the enhanced forward scattering and reduced backward scattering at increase incidence angles, although the enhancement and reduction are overpredicted in the Monte Carlo simulation. Furthermore, the Monte Carlo simulation cannot capture the specular peaks for $s$ -polarization as observed experimentally. For BTDF, the Monte Carlo simulation predicts the symmetric feature with respect to ${\theta }_o=0\mathrm{deg}$ , but overpredicts the reduction of BTDF at large incidence angles. Future research is needed to develop a more sophisticated Monte Carlo model to include surface reflection and refraction by considering the surface roughness and polarization dependence.

Fig. 9

Monte Carlo simulation of the (a) BRDF and (b) BTDF of sample 2 at different incidence angles.

3.4.

Spectral Reflectance and Transmittance Measurements

The directional-hemispherical reflectance and transmittance of the PTFE films were measured using the system of monochromator and integrating sphere as discussed previously. Care must be taken for the transmittance measurement. The reference signal is obtained by sending the light directly to the back of the sphere. On the other hand, when the sample covers the front entrance port, the transmitted light is diffuse. For the reflectance measurement, sample 5 is used as the reference and it is placed on the back port of the sphere facing the laser. The other samples are interchanged to obtain the ratio of the reflected signal. It is assumed that the reflectance of the $10\text{-}\mathrm{mm}$ -thick PTFE sample is 0.988, as determined by integration of the BRDF. The results of $R$ and $T$ obtained by integrating BRDF and BTDF are shown as square marks. The measured $R$ and $T$ using the diode laser at $635\mathrm{nm}$ with the sphere are shown with the diamond marks. All the measured results agree very well. The relative uncertainty of the reflectance and transmittance measurements was estimated to be 10% with a confidence level of 95%.

Figures 10a and 10b show the measured spectral reflectance and transmittance in the wavelength region from $350\text{to}1050\mathrm{nm}$ of samples 1 to 4. As the thickness decreases, the transmittance increases but the reflectance decreases. As the wavelength increases, the reflectance decreases but the transmittance increases. This suggests that the scattering is stronger toward short wavelengths, as expected for small particle scattering. The reduced scattering coefficient can be calculated using the formula $R∕T=0.59{\sigma }_{\lambda }^{\prime }d$ for each sample at each measurement wavelength. The average ${\sigma }_{\lambda }^{\prime }$ for the four samples and its standard deviation are plotted in Fig. 10c. It can be clearly seen that the reduced scattering coefficient increases as the wavelength is reduced. Furthermore, the value obtained previously with the scatterometer measurement of ${\sigma }_{\lambda }^{\prime }=167{\mathrm{cm}}^{-1}$ at $\lambda =635\mathrm{nm}$ is slightly lower than that obtained with the integrating sphere; but the agreement is within the expanded uncertainty.

Fig. 10

Spectral, directional-hemispherical (a) reflectance and (b) transmittance of the thin-film samples measured with an integrating sphere at normal incidence. The square marks represent data integrated from the scatterometer measurements; diamond marks represent data obtained with the diode laser at $635\text{-}\mathrm{nm}$ wavelength and the integrating sphere. (c) Reduced scattering coefficient calculated from $R∕T$ , as compared with that obtained previously from the scatterometer measurements.