Development and comparison of primary facilities for traceable BSSRDF measurements

The BSSRDF measuring system of the CSIC was well described in [15], although some modifications have been carried out since then. The incandescent light source has been replaced by a laser driven light source (LDLS) (EQ-99X, Hamamatsu), the optical system has been optimized to obtain the maximum amount of light in the minimum feasible irradiated area, and the detection system, the charged-coupled device (CCD) camera, has been replaced by a high sensitivity CMOS camera (Orca-Flash4.0 V3, Hamamatsu). A scheme of the setup used for these measurements is shown in figure 3.

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The emitting plasma of the light source (S), which has a full width at half maximum (FWHM) of around 200 µm, is imaged by the lens L3 on the entrance slit of the monochromator (Mc), thus also obtaining the image at the exit slit of the Mc. Next to the Mc, a pinhole wheel (PW) has been placed. This PW contains 16 different pinhole sizes, with diameters from 20 µm to 2 mm, which allows the irradiated area on the sample surface to be modified, although in this case only one specific position of the PW has been selected for all the measurements, the smallest one (20 µm-diameter). Lenses L1 and L2 form an optical imaging system between the PW plane (P) and the sample surface plane (P') with a magnification factor $\beta^{^{\prime}}\simeq-4$. This means that the irradiated area on the sample is approximately four times larger than the selected pinhole, without taking into account aberrations and a possible small soft focus of the image which lead to an actual irradiated area of around 140 µm in diameter. There is also a filter wheel (FW) between L1 and L2 with five different positions: clear position (no-filter), dark position (opaque filter) and three neutral density filters with nominal transmittances of 10%, 1% and 0.01%. For positioning the sample, a six-axis robot arm (R6) (Stáubli TX-40) has been used, with a mechanical holder that has a black background to minimize reflections of the light going through the samples. Lastly, for the detection, a high sensitivity CMOS camera provides a spatial resolution on the sample surface of 20 µm × 20 µm per pixel, approximately, and it is placed on a platform that can move around the sample on a cog-wheel ring.

The BSSRDF is defined theoretically in equation (1). As mentioned in the previous section, it is not possible to measure differential quantities, so differential elements are replaced by the radiance, $L_{\mathrm{ssr}}(\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}})$, and the incident radiant flux, $\Phi_{\mathrm{i}}(\mathbf{x_\mathrm{i}},\mathbf{r_\mathrm{i}})$. The radiance can be expressed in terms of the reflected radiant flux, $\Phi_{\mathrm{r}}(\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}})$, as it follows [16]:

$$\begin{align} L_{\mathrm{ssr}}\left(\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}}\right) = \frac{\Phi_{\mathrm{r}}\left(\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}}\right)}{A_{\mathrm{p}}\omega_{\mathrm{r}}} = \frac{\Phi_{\mathrm{r}}\left(\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}}\right)}{A_{\mathrm{r}}\cos{\theta_{\mathrm{r}}}\omega_{\mathrm{r}}}, \end{align} \tag{ 2 }$$

where $A_{\mathrm{p}}$ is the apparent area of the surface element under evaluation, $A_{\mathrm{r}}$ is the actual area of this surface element, $\omega_{\mathrm{r}}$ is the collection solid angle of the detection system and $\theta_{\mathrm{r}}$ is the collection polar angle.

When the collection is made at $\theta_{\mathrm{r}} = 0^{\circ}$, $A_{\mathrm{r}}$ corresponds to area of the field of view (FOV) of a pixel of the camera, $A_{\mathrm{fov}}$. Nevertheless, when the collection is made at an arbitrary $\theta_{\mathrm{r}}$, $A_{\mathrm{r}}$ is defined as follows:

$$\begin{align} A_{\mathrm{r}} = A_{\mathrm{fov}}/\cos{\theta_{\mathrm{r}}}. \end{align} \tag{ 3 }$$

Thus, the radiance $L_{\mathrm{ssr}}(\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}})$ is expressed in terms of the geometrical parameters of the detection system:

$$\begin{align} L_{\mathrm{ssr}}\left(\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}}\right) = \frac{\Phi_{\mathrm{r}}\left(\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}}\right)}{A_{\mathrm{fov}}\omega_{\mathrm{r}}}. \end{align} \tag{ 4 }$$

The BSSRDF can then be expressed as a radiant flux quotient multiplied by a geometrical factor:

$$\begin{align} f_{\mathrm{ssr}}\left(\mathbf{x_\mathrm{i}},\mathbf{r_\mathrm{i}};\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}}\right) = \frac{1}{A_{\mathrm{fov}}\omega_{\mathrm{r}}}\frac{\Phi_{\mathrm{r}}\left(\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}}\right)}{\Phi_{\mathrm{i}}\left(\mathbf{x_\mathrm{i}},\mathbf{r_\mathrm{i}}\right)}. \end{align} \tag{ 5 }$$

The radiant flux is directly related with the dark-subtracted response of a pixel of the CMOS camera, N (in units of counts), as follows:

$$\begin{align} N = \frac{\lambda}{hc}K\tau\eta_{\mathrm{e}}t_{\mathrm{exp}}\Phi, \end{align} \tag{ 6 }$$

where the factor $\lambda/hc$ (with h the Planck constant and c the speed of light in vacuum) represents the inverse of the energy of a photon with wavelength λ, K is the conversion factor between photoelectrons and counts (K = 2.1 counts per photoelectron for this camera), τ is the transmittance of the camera lens, $\eta_{\mathrm{e}}$ is the external quantum efficiency of the pixels of the CMOS sensor and $t_{\mathrm{exp}}$ is the exposure time. The acquisitions taken with the CMOS camera are high-dynamic-range (HDR) acquisitions. This means that the exposure time of each pixel has been adjusted to avoid low signal-to-noise ratios (SNR). Additionally, to reduce the spatial noise, a $10\times10$ pixels binning was done. This means that each pixel of the binned images represents the signal of 100 pixels of the original sensor. Therefore, $A_{\mathrm{fov}}$ represents the area of the FOV of 100 pixels and the spatial resolution of the detection system is reduced, which is taken into account in the calculation of the BSSRDF values and its uncertainty. The obtained binned images are then projected to the SSRS through the corresponding collection polar angle, $\theta_{\mathrm{r}}$, in each case, as it is explained in [15].

The transmittance of the camera lens and the external quantum efficiency of the pixels may vary when measuring the incident radiant flux and the reflected radiant flux, because of the way how the light reaches the camera. In the first case, the measuring solid angle is determined by the irradiation solid angle, $\omega_{\mathrm{i}}$, while in the second case, it is determined by the collection solid angle of the camera, $\omega_{\mathrm{r}}$. Thus, it is necessary to introduce a responsivity correction factor, $C_{\mathrm{rr}}$, which is defined as [15]:

$$\begin{align} C_{\mathrm{rr}} = \frac{\tau_{\mathrm{r}}\eta_{\mathrm{e,r}}}{\tau_{\mathrm{i}}\eta_{\mathrm{e,i}}}, \end{align} \tag{ 7 }$$

where indexes 'r' and 'i' refer to reflected and incident radiant flux measurements, respectively. Furthermore, for the measurement of the incident radiant flux, the neutral density filter of the FW with a nominal transmittance of 0.01% had to be used to avoid the saturation of the pixels of the CMOS sensor. Therefore, the measurement equation of the BSSRDF in the goniospectrophotometer at CSIC becomes:

$$\begin{align} f_{\mathrm{ssr},k} = \frac{\tau_\mathrm{nd}}{A_{\mathrm{fov}}\omega_{\mathrm{r}}}\left(\frac{N_{\mathrm{r},k}}{\sum_{k}N_{\mathrm{i},k}}\right)\left(\frac{t_{\mathrm{exp,i},k}}{t_{\mathrm{exp,r},k}}\right)\frac{1}{C_{\mathrm{rr}}}\frac{1}{C_{\mathrm{nl},k}}, \end{align} \tag{ 8 }$$

where $f_{\mathrm{ssr},k}$ represents the BSSRDF value of the binned pixel 'k', $\tau_\mathrm{nd}$ is the transmittance of the neutral density filter used for the incident radiant flux acquisition, $N_{\mathrm{r},k}$ and $N_{\mathrm{i},k}$ are the response in counts of the binned pixel 'k' for the reflected radiant flux acquisition and the incident radiant flux acquisition, respectively, and $t_{\mathrm{exp,r},k}$ and $t_{\mathrm{exp,i},k}$ are the exposure times used by the binned pixel 'k' in the reflected radiant flux acquisition and in the incident radiant flux acquisition, respectively. $C_{\mathrm{nl},k}$ is a correction factor accounting for the non-linearity of the response of the pixel 'k'.

The square of the relative standard uncertainty of the BSSRDF can be expressed as the quadrature sum of the relative standard uncertainties of all variables involved in equation (8):

$$\begin{align} u_\mathrm{r}^2\left(f_{\mathrm{ssr},k}\right) & = u_\mathrm{r}^2\left(A_\mathrm{fov}\right) + u_\mathrm{r}^2\left(N_{\mathrm{r},k}\right) + u_\mathrm{r}^2\left(\sum_{k}N_{\mathrm{i},k}\right) \notag\\ & \quad + u_\mathrm{r}^2\left(\tau_\mathrm{nd}\right)\;+ u_\mathrm{r}^2\left(\omega_\mathrm{r}\right)\notag\\ & \quad + u_\mathrm{r}^2\left(\frac{t_{\mathrm{exp,i},k}}{t_{\mathrm{exp,r},k}}\right) + u_\mathrm{r}^2\left(C_\mathrm{rr}\right) + u_\mathrm{r}^2\left(C_{\mathrm{nl},k}\right). \end{align} \tag{ 9 }$$

This equation is fulfilled since the errors on the values of the variables involved in the measurement equation are not correlated, except for the term of the sum of the response of the binned pixels in the incident radiant flux acquisition, which is fully correlated. Nevertheless, as the response of the reflected radiant flux acquisition is divided by the incidence radiant flux acquisition with the same camera, this correlation is canceled. On the other hand, it is true that the errors of the BSSRDF values at different geometries are correlated, which should be taken into account when integrating these values to obtain another magnitude from the BSSRDF, but it does not affect the uncertainty of each BSSRDF value to be compared. The estimation of the uncertainty of integrated magnitudes is out of the scope of our current research, but it should be address in the future, likely through the Monte Carlo method [17] due to the complexity of the measurement.

Strictly speaking, the uncertainties introduced by the angular and spatial resolutions of the measuring system should also be accounted in equation (9). Nevertheless, they depend on the specific characteristics of the sample being measured, making necessary to have a prior characterization of the BSSRDF of the sample. The samples that have been measured in this work present a very smooth angular distribution far from the specular direction and a smooth spatial distribution outside the irradiated area, and both conditions are fulfilled for the compared BSSRDF values. Furthermore, the robot arm and the detection platform are very precise, so the error in the angle is supposed to be very small, as the error in the $\mathbf{x}_{\mathrm{r}}$ position on the sample surface, since the detection system has a very high spatial resolution. These effects can be neglected because their contribution is supposed to be much smaller than other main uncertainty sources present in the current measurements of the BSSRDF as the noise due to the response of the pixels.

To measure $A_{\mathrm{fov}}$ a calibrated 10 mm stage micrometer with 50 µm divisions was used. It was placed on the sample plane and it was irradiated with a broadband uniform beam. Then, an horizontal and a vertical image of the micrometer were taken with the CMOS camera in transmittance, using the same acquisition configuration as in the measurements. The horizontal and vertical dimensions of the pixels in mm on the sample surface were obtained from the number of pixels between the first and the last division of the micrometer, resulting the multiplication of both dimensions in $A_{\mathrm{fov}}$. Thus, the relative standard uncertainty of $A_{\mathrm{fov}}$ is determined by the standard deviation of the number of pixels of four repetitions and the uncertainty on the calibration of the stage micrometer, and it was estimated at 0.1% by applying the uncertainty propagation law.

The analysis of the uncertainty due to the response of the pixels of the CMOS camera is essentially the same as explained in [15], but now the noise parameters of the camera are different, since the CMOS camera has replaced the previous CCD camera. The readout noise, $\sigma_{\mathrm{r}}$, of the new detection system was estimated at 0.86 e⁻ rms, the conversion factor, K, at 2.1 counts per e⁻, and the mean thermal electrons rate of the CMOS sensor, $s_{\mathrm{t}}$, at 0.011 e⁻ s⁻¹ per pixel. The method used for the estimation of the correction factor is described in the literature [18]. The standard uncertainty of the dark-subtracted response of the pixel 'j', $u(N_{j})$, becomes:

$$\begin{align} u\left(N_{j}\right) = K\sqrt{2\sigma_\mathrm{r}^2 + t_{\mathrm{exp},j}\left[2s_\mathrm{t} + 2s_\mathrm{s} + K^{-1}N_{j}\right]}, \end{align} \tag{ 10 }$$

where $s_{\mathrm{s}}$ is the rate of the electrons generated per pixel by the stray light in the laboratory in the moment of the measurement and the factor 2 comes from the two acquisitions needed for subtracting the dark signal.

The dark-subtracted binned pixel response is the result of summing the dark-subtracted response of 100 pixels, thus its uncertainty is the square root of the quadrature sum of the uncertainty of each single pixel. The relative standard uncertainty of the dark-subtracted binned pixel response is therefore:

$$\begin{align} u_{\mathrm{r}}\left(N_k\right) = \frac{\sqrt{\sum_{j_{k}}u^{2}\left(N_{j_{k}}\right)}}{\sum_{j_{k}}N_{j_{k}}}, \end{align} \tag{ 11 }$$

where $u(N_{j_{k}})$ is calculated from equation (10) and the additions include the 100 pixels that are being accounted for the binned pixel 'k'. Since $u(N_{j_{k}})$ and $N_{j_{k}}$ are almost constant for a given bin 'k', the impact of the noise in the relative uncertainty is reduced approximately by a factor of 10.

The transmittance of neutral density filters was measured with a calibrated spectrophotometer in the past (see [12]). A relative standard uncertainty of 0.48% was obtained for the filter with a nominal transmittance of 0.01%, the one used for measuring the incident radiant flux.

The calculation of the collection solid angle and its uncertainty analysis was carried out following the same procedure as explained in [15] but now with the CMOS camera. In this case, the angular steps were reduced from 1° to 0.5°, and the resulting relative standard uncertainty became 0.23%.

The uncertainty of the exposure time used by each pixel of the CMOS sensor is very small compared to the minimum integration time used in the acquisitions. So, the relative standard uncertainty of the exposure time ratio can be considered negligible.

The responsivity correction factor is obtained from the measurement data of the collection solid angle, and its calculation is also explained in [15]. Its relative standard uncertainty for the CMOS camera is 0.19%.

The non-linearity of the CMOS sensor was corrected using the methods described in [19]. Nevertheless, the applied correction factors have a relative standard uncertainty estimated at 0.6%.

The goniospectrophotometer setup for BSSRDF measurements at CNAM, shown in figure 4, has been designed to perform an absolute measurement of BSSRDF and is described in [20]. The sample, held on a robot arm, is illuminated by a narrow and directional light beam. The reflected radiance is measured over a small area in the collection direction using a luminance meter as a detection system. Contrary to a camera-based detection, the luminance meter provides a single value of luminance for the area within the FOV. It is therefore combined with a spatial scanning system (i.e. translation stages) to measure luminance at any location on the sample.

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The source of this setup has been designed to illuminate the sample with a very bright and thin beam. It comprises an LDLS (EQ-99X, Hamamatsu) and an optical system optimized to obtain a light spot with a diameter of about 50 µm on the sample. This source is shared with CNAM's 'µBRDF' setup and is described in detail in [14]. The source is located on a breadboard and can be rotated around the sample in the horizontal plane using the rotation ring. The incident flux on the sample is measured by placing the source directly in front of the detection system. The source aperture is adjusted to 1.5° to ensure no vignetting on the detection system for the source direct measurement while maximizing optical power. The detection system is a luminance meter (Pritchard) equipped with a photomultiplier tube. This allows to measure luminance as low as $1\times10^{-3}$ cd m⁻². The diameter of the detection FOV on the sample plane, which limits the spatial resolution of the measurements, is about 1 mm. The sample is held at the center of the goniospectrophotometer by a six-axis robot arm (RV-12S, Mitsubishi). For BSSRDF measurements, the robot has its wrist oriented upward and the sample is held vertically by a clamp. In this way, light can be freely transmitted through the sample with no risk of back reflection. Using the robot rotations and the rotation ring, sample and source are oriented to meet the chosen geometry of measurement for the incident beam and collection directions ($\mathbf{r}_\mathrm{i}$ and $\mathbf{r}_\mathrm{r}$). The location of the incident beam ($\mathbf{x}_\mathrm{i}$) can be controlled by using the translation tools of the robot arm.

The X and Y translations to be applied to the detection system for observing the reflected radiance from the position $\mathbf{x}_\mathrm{r}$ on the sample surface are determined by the geometry of measurement. The location of the collection point in the coordinate system of the detection can be obtained by applying to the vector ($x_\mathrm{r}$, $y_\mathrm{r}$, 0) three consecutive rotations of angle β around the x-axis, α around the y-axis and γ around the z-axis, with ($\alpha, \beta, \gamma$) angles that describe the orientation of the sample and that are linked to the angular geometry of measurement ($\theta_\mathrm{i}$, $\phi_\mathrm{i}$, $\theta_\mathrm{r}$, $\phi_\mathrm{r}$) using formulas given in [21]. In the study presented in this paper, only in-plane measurements were performed, which simplifies the calculation of the detection position:

$$\begin{align} &X = x_{\mathrm{r}}\cos\left(\theta_{\mathrm{r}}\right), \end{align} \tag{ 12 }$$

$$\begin{align} &Y = y_{\mathrm{r}}. \end{align} \tag{ 13 }$$

The BSSRDF, $f_{\mathrm{ssr}}(\mathbf{x_\mathrm{i}},\mathbf{r_\mathrm{i}};\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}})$, is defined as the ratio of the reflected radiance $L_{\mathrm{ssr}}(\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}})$ to the incident radiant flux $\Phi_{\mathrm{i}}(\mathbf{x_\mathrm{i}},\mathbf{r_\mathrm{i}})$. The incident radiant flux is measured by placing the source directly in front of the detection system. In this configuration, the detection system measures the luminance $L_{\mathrm{source}}$ and the associated dark signal $L_{\mathrm{source,dark}}$. The incident radiance flux is then calculated using the following equation:

$$\begin{align} \Phi_{\mathrm{i}} = \left(L_{\mathrm{source}}-L_{\mathrm{source,dark}}\right)A_{\mathrm{fov}}\omega_{\mathrm{r}}, \end{align} \tag{ 14 }$$

with $A_{\mathrm{fov}}$ being the area of the detection FOV on the sample plane and $\omega_{\mathrm{r}}$ the collection solid angle.

The reflected radiance is measured after placing the source, the detection system and the sample in the chosen geometry of measurement. The luminance meter directly measures the luminance $L_{\mathrm{r}}(\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}})$ and the associated dark signal $L_{\mathrm{r,dark}}$. Finally, the measurement equation for BSSRDF is:

$$\begin{align} f_{\mathrm{ssr}}\left(\mathbf{x_\mathrm{i}},\mathbf{r_\mathrm{i}};\mathbf{x_\mathrm{r}},\mathbf{r_\mathrm{r}}\right) = \frac{1}{A_{\mathrm{fov}}\omega_{\mathrm{r}}}\left(\frac{L_{\mathrm{r}}-L_{\mathrm{r,dark}}}{L_{\mathrm{source}}-L_{\mathrm{source,dark}}}\right). \end{align} \tag{ 15 }$$

The relative standard uncertainty of the BSSRDF, $u_{\mathrm{r}}(f_{\mathrm{ssr}})$, can be related to the uncertainties on each variable of equation (15) as follows:

$$\begin{align} u_{\mathrm{r}}^2\left(f_{\mathrm{ssr}}\right) & = \frac{u^2\left(L_{\mathrm{r}}\right) + u^2\left(L_{\mathrm{r,dark}}\right)}{\left(L_{\mathrm{r}}-L_{\mathrm{r,dark}}\right)^2} + \frac{u^2\left(L_{\mathrm{source}}\right) + u^2\left(L_{\mathrm{source,dark}}\right)}{\left(L_{\mathrm{source}}-L_{\mathrm{source,dark}}\right)^2} \notag\\ & \quad + u_{\mathrm{r}}^2\left(A_{\mathrm{fov}}\right) + u_{\mathrm{r}}^2\left(\omega_{\mathrm{r}}\right). \end{align} \tag{ 16 }$$

In this first study, the uncertainties induced by angular and spatial positioning errors are only partially accounted for from the repeatability measurement. As light becomes increasingly diffuse with subsurface scattering, negligible uncertainties are expected to be induced by errors on the angular positioning of the sample. Errors on the spatial $\mathbf{x}_{\mathrm{r}}$ position of the sample have a higher impact, especially close to the center, and will be thoroughly evaluated in future works.

The luminance $L_{\mathrm{r}}$ was independently measured three times for each sample. The uncertainty on $L_{\mathrm{r}}$ is given by the standard deviation on the measurement divided by $\sqrt{3}$. The value of the uncertainty $u(L_{\mathrm{r}})$ is strongly impacted by sensor noise and gets high when the signal is very low: it is inferior to 5% (and as low as 1%) when the measured luminance is superior to 0.2 cd $\mathrm{m}^{-2}$, and as high as 50% when the measured luminance is lower than 0.001 cd $\mathrm{m}^{-2}$.

The uncertainty on $L_{\mathrm{source}}$ has been evaluated at 2.4%. This uncertainty accounts for the reproducibility of the measurement, evaluated by asking the operator to realign the instrument in front of the source for 10 independent measurements. This uncertainty also accounts for the instrument resolution and the non-linearity of the luminance meter, which is used in a high-dynamic mode for measurements on the sample, but in a low-dynamic mode for the direct measurement of the source.

The uncertainty on $L_{\mathrm{dark}}$ and $L_{\mathrm{source,dark}}$ has also been statistically evaluated at 1.4%. This value accounts for stray light, sensor noise and instrument resolution.

The area $A_{\mathrm{fov}}$ was measured by taking a photograph of a ruler placed on the sample plane with the viewfinder of the luminance meter. The relative standard uncertainty for this measurement is 6.7%, limited by the poor accuracy that can be achieved in measuring the FOV diameter. When the measured luminance signal is not too noisy, this component constitutes the largest contribution to the total uncertainty.

The luminance meter comprises a lens that creates some vignetting and yields a non-uniform luminance measurement over the collection solid angle. Consequently, the solid angle can be expressed as:

$$\begin{align} \omega_{\mathrm{r}} = \frac{A^{\prime}_{\mathrm{diaph}}}{d^2}, \end{align} \tag{ 17 }$$

where $A^\prime_{\mathrm{diaph}}$ is the equivalent aperture area that accounts for the luminance distribution within the solid angle, and d is the distance between the sample and the detection's entrance aperture. The equivalent aperture area is $1.41\times10^{-3}\,\mathrm{m}^2$ (43.2 mm for the equivalent diameter) and the sample to aperture distance is 1.931 m.

The luminance distribution over the collection solid angle can be estimated by placing an iris diaphragm just in front of the lens (which corresponds to the position of the detection's entrance aperture), and by measuring the luminance of a uniformly illuminated sample for various apertures of the iris diaphragm. To compute the equivalent aperture area $A^\prime_{\mathrm{diaph}}$, two luminance measurements were performed on an uniformly illuminated Spectralon sample. First, the iris diaphragm diameter was chosen small enough (24 mm) to ensure that the light going through the luminance meter lens was not vignetted and the luminance $L_{\mathrm{center}}$ was measured with a diaphragm of area $A_{\mathrm{center}}$. Then, the iris diaphragm was removed, and the luminance $L_{\mathrm{total}}$ was measured. Assuming that the luminance distribution is uniform at the center of the FOV of the luminance meter, the equivalent aperture area is given by:

$$\begin{align} A^\prime_{\mathrm{diaph}} = {L_{\mathrm{total}}}\times\frac{A_{\mathrm{center}}}{L_{\mathrm{center}}}. \end{align} \tag{ 18 }$$

The relative uncertainty on the equivalent aperture area (2.6%) and on the sample to aperture distance (0.5%) yields the relative uncertainty on the collection solid angle, which is 2.8%.

The BSSRDF measurement facility at KUL (Light & Lighting Laboratory) is a modified commercial NFG, type Rigo 801-300 (TechnoTeam GmbH). This is the system to which the BSSRDF scale was transferred. Typically, an NFG is used to measure ray files and radiation patterns from light sources. However, as the measurement device is equipped with a calibrated imaging luminance measurement device (ILMD), it can be modified to also measure BSSRDF. The modifications to the NFG include a customized laser illumination setup and sample holder.

The illumination part, shown in figure 5, consists of a laser diode (532 nm, type CPS532 - 4.5 mW) followed by an aperture of 1 mm in diameter. This aperture is imaged toward the center of the NFG using a lens with a focal length of 33 cm. The resulting illumination spot at the center of the NFG has a diameter of 2.6 mm. In order to minimize laser speckle, a rotating front surface diffuser, which has a bidirectional transmittance distribution function with FWHM of 8°, is positioned between the laser and the aperture. The collimated laser beam is enclosed by a hollow metal tube onto which the sample holder at the center of the NFG is attached.

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During normal NFG operation, this hollow metal tube is used to provide electrical connections to the light source to be measured. During BSSRDF measurements, a sample holder is positioned at the center of the NFG, as is shown in figure 6. The sample holder, which was custom designed and 3D printed, allows for various angles of incidence within a single plane (i.e. only the polar incident angle can be manually altered, while the azimuthal incident direction is fixed).

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Alignment of the sample is done using the same alignment procedure as when measuring light sources with the NFG. During this procedure a live view of the ILMD is shown with an overlay of the internal coordinate system of the NFG. This allows the front surface of the sample to be positioned exactly at the center. Next, the incident angle is set manually by rotating the sample holder, reading the spatial offsets of the edges of the sample from the ILMD live view, and calculating the corresponding angle of incidence.

Once the sample is aligned, the rotating arm onto which the ILMD is mounted is rotated to capture multiple HDR luminance images from different viewing directions. The FOV of the ILMD can be adjusted by changing the ILMDs lens. For the BSSRDF measurements in this article, a lens with an FOV of 80 mm was chosen.

Although this is not a primary facility developed for traceable BSSRDF measurements, it provides BSSRDF measurements with SI units through the comparison with a Spectralon calibration tile whose spectral radiance factor [22] for a geometry of 0°:45°, $\beta_{\mathrm{e}}$, is known. The BSSRDF measurement equation for this measuring setup is the following:

$$\begin{align} f_{\mathrm{ssr},k} = \frac{L_{\mathrm{s},k}}{L_{\mathrm{t},k}}\frac{\beta_{\mathrm{e}}\cos{\theta_{\mathrm{r}}}}{\pi A_{\mathrm{fov}}}, \end{align} \tag{ 19 }$$

where $L_{\mathrm{s},k}$ and $L_{\mathrm{t},k}$ denote the luminance of the pixel 'k' of the HDR luminance image of the measured sample and the calibration tile, respectively, $\theta_{\mathrm{r}}$ is the collection polar angle and $A_{\mathrm{fov}}$ the area of the FOV of a pixel of the ILMD.