Considerations on measurement of bidirectional transmittance distribution function of thick samples over a wide range of viewing zenith angles

We delve into theoretical and experimental considerations for determining the spectral bidirectional transmittance distribution function (BTDF) of thick samples across a broad viewing zenith angle range. Nominally, BTDF is defined as the ratio of transmitted radiance to incident irradiance measured from the same plane. However, when employing thick samples for BTDF measurements, the viewing plane of the transmitted beam may shift from the front to the rear surface of the sample, altering the measurement geometry compared to using the sample front surface as the reference plane. Consequently, the viewing zenith angle from the sample rear surface increases relative to the sample front surface, and the sample-to-detector-aperture distance decreases by an amount corresponding to the sample thickness. We introduce a method for determining the BTDF of thick samples, considering the transformation of practical measurement results to a scenario where the measurements are conducted at a very large distance from the sample. To validate the method, we utilize a BTDF facility equipped with two instruments that significantly differ in their sample-to-detector-aperture distances. We evaluate the impact of a 2 mm sample thickness on the BTDF by assessing the ratio of transmitted and incident radiant fluxes as a function of viewing zenith angle relative to the sample rear surface. The evaluation is conducted in the wavelength range from 550 nm to 1450 nm in 300 nm steps, and in the viewing zenith angle range from −70° to 70° in 5° steps. Measurements are performed in-plane at an incident zenith angle of 0°. It is concluded that consistent determination of BTDF of a thick sample is possible by converting the experimental parameters of the real measurements at relatively short distances from the sample to correspond to those that would be obtained from measurements at very large distances from the sample.


Introduction
Bidirectional transmittance distribution function (BTDF) characterizes the visual appearance of an object as perceived from varying incident and viewing angles, in relation to its placement in front of an illumination source.The quantification of appearance of translucent materials is important in many fields of industry, research, and entertainment.For industries involving cosmetics [1], car paint [2], and food [3], BTDF values could play a pivotal role in quality control and the strategic development of products.Earthobserving satellites [4,5] rely on calibrated detectors that have traceability to reference materials calibrated for their BTDF to study Earth and space.Furthermore, the entertainment industry enhances the realism of computer-generated images for movies, virtual reality, and computer games by incorporating real-world BTDF values into their rendering software [6,7].
BTDF was formulated by Bartell et al [8] as the ratio of transmitted radiance to incident irradiance as a continuation of the work on bidirectional reflectance distribution function (BRDF) by Nicodemus et al [9].In practical terms, BTDF can be expressed as the ratio of transmitted power per unit area and solid angle to incident power per unit area.Accordingly, BTDF is a quantity that characterizes the transmitted distribution of incident light in a specific solid angle cone.Characterization of color can be incorporated by measuring irradiance and radiance spectrally, resulting in spectral BTDF, which reveals information on the wavelength dependence of the material's transmittance.However, it is important to note that the parameters for defining BTDF involve differentials, necessitating instruments capable of approximating infinitesimal quantities.Despite this, one prevalent method for measuring BTDF involves goniometers that systematically scan BTDF at discrete points within a hemisphere from the sample's rear surface.
The facility at Aalto University includes two goniometerbased instruments that have been validated for BTDF measurements in our previous work for viewing zenith angles up to 35 • [10].The facility includes an absolute gonioreflectometer [11,12] modified for BTDF measurements [10], referred to as the absolute instrument; and a commercial goniospectrophotometer, Cary 7000, with a goniometer extension, Universal Measurement Accessory (UMA), referred to as the complementary instrument [13].Our facility has actively participated in various intercomparisons between National Metrology Institutes (NMIs), demonstrating proficiency in directionalhemispherical reflectance assessments of reflective diffusers [11,12,14,15] and BTDF evaluations for diverse materials, encompassing transmissive diffusers and structured materials [16].
In preceding studies, the characterization of facilities for measuring BTDF at the NMI level have been explored in a relatively narrow viewing zenith angle range [16][17][18][19].A similar characterization of Aalto's BTDF facility has been presented in [10].However, when extending our investigations to encompass a broad viewing zenith angle range while characterizing thick samples, a novel challenge arises.Notably, the conventional definition of BTDF does not account for the effect of sample thickness on the measurement geometry.Therefore, this work will study BTDF measurements of a 2 mm thick transmissive diffuser in a wide viewing zenith angle range.To evaluate the BTDF of thick samples, the ratio of transmitted and incident radiant fluxes is studied as a function of the viewing zenith angle relative to the sample rear surface.The ratios are recorded using two instruments with a substantial difference in their sample-to-detector-aperture distances.The instruments record significant deviations in their BTDF using the nominal viewing zenith angles relative to the sample front surface but are in reasonably good agreement when using the viewing zenith angles relative to the sample rear surface.An important outcome of this study is that consistent determination of the BTDF of a thick sample is possible by considering the BTDF in the limit of a very large measurement distance and by converting the experimental parameters of the real measurement at short distances to corresponds to those at very large distances.

Measurement setup
To study the effect of sample thickness on BTDF, this work uses two goniometer-based instruments that have been characterized for BTDF measurements in detail in [10].The instruments serve as a tool for validating the method of correcting near-field BTDF measurements to those obtained in the far-field (elaborated in section 3.2).An important feature of these instruments for the present work is their large difference in sample-to-detector-aperture distances.The instruments include an absolute gonioreflectometer, referred to as the absolute instrument; and a commercial spectrophotometer, Cary 7000 UMA, referred to as the complementary instrument [13].
Figure 1 illustrates the key components of the two goniometer-based instruments in Aalto.These instruments consist of a light source and a detection system.Broadband illumination emitted by the light sources is directed towards a double monochromator, which disperses the incident light within a bandwidth of less than 10 nm.An automated polarizer filter-wheel provides linear s-and p-polarized light at the sample, and unpolarized BTDF results can be obtained by averaging the former results.Subsequently, the modulated and collimated beam is projected onto a small spot on the sample's front surface.
BTDF measurements are conducted in the visible and nearinfrared wavelength ranges and the signals are recorded using underfilled Si and InGaAs photodiodes for respective spectra.The detector systems employ solid angles of 0.002 195 sr and 0.0153 sr, corresponding to sample-to-detector-aperture distances of 472.9 mm and 129 mm, and detector aperture diameters of 25.00 mm and 18.00 mm for the absolute and complementary instruments, respectively.The transmitted signals are collected in a viewing zenith angle range from −70 • to 70 • .The instruments' uncertainty in BTDF measurements has been extensively studied in [10].Sources of error affecting the measurement results of each instrument have been identified, and their impact on BTDF has been assessed through the propagation of uncertainty.The expanded uncertainty, assuming a Gaussian probability distribution, is provided with a coverage factor k = 2, corresponding to 95% confidence interval.Key components contributing to the overall uncertainty of each instrument include measurement noise, straylight, and uncertainty in BTDF due to error in setting the viewing zenith angle.Additionally, the complementary instrument is constrained by uncertainties related to the solid angle.

BTDF measurement principles
In the definition of BTDF, the sample is assigned with cylindrical coordinates, where the origin coincides with the front surface of the sample and the z axis is aligned according to the sample normal.BTDF is a function of both the incident and transmitted azimuth φ s and zenith θ s angles, as well as the wavelength of the incident light λ.Subscript s attached to the azimuth and zenith angles denotes whether it corresponds to the incident (s = i), reflected (s = r), or transmitted (s = t) directions of the light.BTDF is defined as [8] where L tλ is the spectral transmitted radiance and E iλ is the spectral incident irradiance.For practical measurements concerning the instruments detailed in section 2, L tλ and E iλ can be derived from the transmitted radiant flux ϕ t (θ i , φ i , θ t , φ t , λ) and the incident radiant flux ϕ i (θ i , φ i , λ) at a small detector aperture.The detector subtends a solid angle Ω in the direction defined by angles θ t and φ t for collecting the transmitted signal within the spectral interval ∆λ.When the detector collects all transmitted light from the irradiated volume of the sample, f tλ simplifies to where only θ t is explicitly marked as the angular argument of ϕ t in following considerations and the factor cos θ t accounts for the narrowing of the beam spot when viewed by the detector at large viewing zenith angles θ t .The solid angle Ω is formed by the cone between a point on the front surface of the sample and the detector aperture of a radius r at the distance R, (3)

BTDF measurements of thick samples
In measurements of BRDF, the apex of the solid angle naturally lies on the front surface of the sample.Consequently, the incident irradiance and reflected radiance are measured on a common plane at z = 0.However, in BTDF measurements the incident irradiance undergoes diffusion through the sample over a distance ∆z.This diffusion has the potential to displace the plane of the viewing area.Displacement of the viewing area from the front surface towards the rear surface would lead to an increase of the viewing zenith angle and a reduction in the effective sample-to-detector-aperture distance.Equation ( 2) does not take into consideration the displacement of the viewing area as a function of the sample thickness.As a result, the viewing zenith angle, referenced from the front sample surface at z = 0, will not necessarily be equal to the viewing zenith angle taken from the rear sample surface at z = ∆z (see figure 2).
relative to equation (3) where Increasing the viewing zenith angle θ * t reduces the projected radiating area, which is accounted for by the cos θ * t factor, and additionally changes the transmitted radiant flux due to potential non-Lambertian property of the sample.Now θ * t , R * z and R y are given by We next consider the case where two different detectors, at distances R and R ′ from the front surface of the sample, measure the transmitted radiant fluxes ϕ t (θ * t ) and where the viewing zenith angle is defined relative to the rear surface of the sample at z = ∆z and the primed symbols refer to the detector at distance R ′ .Noting from figure 2 that R * = R − ∆z cos θ * t within relative corrections of the order of (∆z/R) 2 , we can model the behavior of ϕ t (θ * t ) and ϕ ′ t (θ * t ) as a function of θ * t .Assuming the same incident irradiance and viewing area for the detectors at R and R ′ , the ratio of their signals is given by If the incident irradiances ϕ i and ϕ i ′ are different for measurements at R and R ′ , the right-hand side of equation ( 8) is equal to the ratio τ In equation ( 2), the same reference surface is used for determination of the incident irradiance and transmitted radiance.Thus, evaluation of BTDF of thick samples based on the radiance at the rear surface of the sample is somewhat problematic.A solution to this problem is to determine the BTDF for a thick sample in the limit where ∆z/R ∞ approaches zero, i.e., using virtual measurements at very large distances R ∞ .Then θ * t and the corresponding solid angle Ω * ∞ are equal to the quantities θ t and Ω ∞ , determined using the sample front surface as the reference plane.Based on our earlier considerations, it is beneficial that the transmitted radiant flux ϕ ∞ t (θ t ) = ϕ ∞ t (θ * t ) , in the limit of large distance R ∞ , can be calculated from the radiant flux ϕ t (θ * t ) at a short distance R with the help of equation ( 8), where it is assumed that the hypothetical measurement at large distance R ′ = R ∞ , approaching infinity, is carried out using the solid angle Ω ∞ = Ω of short distance measurements.The radiant flux ϕ ∞ t (θ t ) is then obtained as a function of viewing zenith angle θ t relative to the front surface of the thick sample to be further used in calculation of the BTDF by equation (2).

Results and discussion
In this section, we investigate how measurement geometry affects BTDF measurements of a thick sample over a wide viewing zenith angle range.Notably, we study the BTDF using two different instruments described in section 2, which have a considerable difference in their sample-to-detector-aperture distances.Nominal results of the instruments are expected to show angular dependence in the relative difference of the determined BTDF as a function of viewing zenith angle due to sample thickness.

Experimental data
The measured quantity is the ratio of transmitted and incident fluxes τ (θ t ) = ϕ t (θ t ) /ϕ i as a function of viewing zenith angle relative to the sample front surface.The sample was measured for τ (θ t ) in the viewing zenith angle range from −70 • to 70 • in 5 • steps, and in the spectral range from 550 nm to 1450 nm in 300 nm steps.The measurements were performed in-plane, with incident zenith angle θ i = 0 • and with viewing azimuth angle either φ t = 90 • or φ t = 270 • , corresponding to positive and negative viewing zenith angles θ t , respectively.The reported experimental result is taken as the average of measurements with s-and p-polarized light.
The studied sample is a piece of fused synthetic silica, HOD-500, manufactured by Heraeus Quarzglas GmbH & Co [20].The sample has a diameter of 50 mm and a thickness of ∆z = 2.0 mm.The sample possesses a quasi-Lambertian scattering distribution with some spectral dependence.One advantage of the sample material includes a controllable amount and size of microbubbles in the bulk, which can be optimized for desired optical parameters.The sample has an average bubble density of <2.3% and a bubble size of <20 µm.The microbubbles generate multiple scattering events in the bulk, with a mean free path of 0.056 mm.
Figures 3(a) and (b) show the measured ratio of transmitted and incident radiant fluxes recorded by the complementary (τ ) and absolute instrument (τ ′ ).The ratio is shown as a function of viewing zenith angle θ t determined by the front surface of the HOD-500 sample.The results show that the HOD sample has a smooth increase of τ (θ t ) and τ ′ (θ t ) as a function of wavelength.The difference in τ scales between the instruments can be attributed to the ratio of the solid angles according to equation (8).
Ideally, the ratio τ (θ t ) /τ ′ (θ t ) should be equal to the ratio of solid angles Ω/Ω ′ = 6.97 for a thin sample with ∆z ≈ 0. However, if the results of figure 3 were scaled by their respective solid angles, then their relative difference would be 0.87%, −0.35%, and −5.9% at viewing zenith angles 0 • , 35 • and 70 • , respectively, as an average of results at all wavelengths.The angular dependence is related to the selection of sample front surface as the reference plane for θ t and Ω, as discussed in section 3.

Proper comparison of measured data
In order to properly compare measurement results of two instruments with a considerable difference in their sampleto-detector-aperture distances, they need to be presented as a function of θ * t , i.e. the viewing zenith angle of the detector relative to the sample rear surface.Conversion from angle θ t to angle θ * t is carried out using equations ( 5)-( 7) with distances R and R ′ for the complementary and absolute instrument, respectively.Subsequently, linear interpolation is used to obtain τ (θ * t ) and τ ′ (θ * t ) at the viewing zenith angles θ * t from −70 • to 70 • with 5 • separation.Finally, the results at negative and positive viewing angles are averaged to calculate τ (θ * t )/τ ′ (θ * t ) at positive angles θ * t .Figure 4 shows the adjusted experimental result where the relative angular variation is reduced by more than a factor of two as compared with the case where the front surface of the sample was used as the reference plane.
In addition to the change from angle θ t to angle θ * t , the effect of solid angle ratio Ω * /Ω * ′ as described by equation ( 8) needs to be considered.Figure 4 shows the anticipated angular dependency of τ (θ * t )/τ ′ (θ * t ) using the thick green line.The dependency is derived from detector measurements at varying distances R and R ′ from the sample's front surface.The results show that the model approximates the behavior of measured τ (θ * t )/τ ′ (θ * t ) as a function of θ * t well for angles up to 70 • within the expanded uncertainties of the measurements.
The systematic deviation between the measured data and model in figure 4 can be attributed to the uncertainty in the complementary instrument solid angle [10].The instrument has a relative standard uncertainty of 0.92% in the distance between the sample and detector aperture, which is the main contribution to the systematic deviation of the model and measured ratio.At viewing angles larger than 35 • , the uncertainty increases due to the increased sensitivity to the viewing zenith angle with a standard uncertainty of 0.10 • for both instruments.Those systematic uncertainty contributions are 0.60% and 0.55% at 70 • for the nominator and denominator of the ratio τ (θ * t )/τ ′ (θ * t ), respectively.Thus, the relative systematic deviations of 1% to 3% between the model and data in figure 4 are acceptable.

Correction of short-distance BTDF results
Section 4.1 highlighted the challenges of using the sample front surface as a reference for BTDF measurements of thick samples, mainly due to the apparent shift of the source of the transmitted radiance to the rear surface of the sample.We argue that scattering close to the rear surface of the sample is important for the transmitted radiance because the beam spot at the rear surface is visible at large viewing zenith angles.Due to the total internal reflection of the sample (refractive index n = 1.46 @ 500 nm), internal light with an angle of incidence smaller than 43 • appears to originate at the sample rear surface whereas light at larger angles of incidence remains inside the sample.Combined with the short mean free path of 0.056 mm this means that a majority of the sample volume is not directly visible to the external observer.Therefore, there is a case for the sample rear surface being the virtual source of the transmitted flux leaving the sample.Section 3.2 demonstrated that we can minimize the effect of sample thickness ∆z by considering the sample at a large distance R ∞ where the sample appears thin (i.e.∆z/R ∞ ≈ 0), thereby removing the effect of instrument geometry from the BTDF measurement results of thick samples.Figure 5 shows the BTDF of the HOD-500 sample obtained by the method outlined in section 3.2.The BTDF is shown as an average of negative and positive viewing zenith angle BTDF results.The results are based on the transmitted radiant fluxes  from equation ( 9) scaled by incident irradiance and the factor Ω cos θ t from equation (2). Figure 5 shows that the angle dependent deviations in the relative difference of the instruments are minimized.In appendix, comparison results using the sample front surface as the uncorrected reference for BTDF are shown.The results of figure 5 show that both instruments record identical trends in their BTDF as a function of viewing zenith angle, and that the measurement results agree well within their expanded uncertainties for angles up to 70 • .It is seen that consistent determination of the BTDF of a thick sample is possible by considering the BTDF measurement in the limit of a very large distance R ∞ and by converting the experimental parameters of the real measurement at short distance R to correspond to those at R ∞ .

Conclusions
In this work we presented theoretical and experimental considerations on spectral BTDF measurements of thick samples in a wide viewing zenith angle range.The sample thickness has the potential to shift the viewing area of the transmitted radiance from the front to the rear surface of the sample.Consequently, the viewing zenith angle relative to the sample rear surface increases and the sample-to-detector-aperture distance decreases.
To experimentally evaluate the effect of sample thickness on BTDF, we studied the ratio of the transmitted radiant flux to the incident radiant flux as a function of viewing zenith angle relative to the sample rear surface.Based on experimental results we propose a method for determining the BTDF of thick samples by converting the results of practical measurements to correspond to those that would be obtained from measurements at very large distances from the sample.It is advantageous that the difference of using the sample front or rear surface as the reference plane vanishes at large distances from the sample.
The proposed method was validated by utilizing a BTDF facility employing two instruments that have a significant difference in their sample-to-detector-aperture distance.Using the sample front surface as reference for the solid angle while measuring BTDF, the instruments showed large angle dependent deviations in the relative difference of their measurement results.On the contrary, applying the method proposed in this work to the result of the two instruments successfully minimized the angle dependent deviations.Subsequently, the BTDF values showed good agreement within their expanded uncertainty (k = 2) for viewing zenith angles up to 70 • .

Figure 1 .
Figure 1.Schematic of a goniometer-based instrument in a configuration for BTDF measurements.

Figure 2 .
Figure 2. Sample front surface illuminated by a beam at z = 0 and light diffusion through the sample with a thickness of ∆z.The sample thickness causes a change in the viewing zenith angle θt and in the z-component of the sample-to-detector-aperture distance Rz.

Figure 2
illustrates the impact of longitudinal diffusion on the propagation of incident light through the sample across a distance denoted as ∆z.The case in the figure applies when the final scattering event occurs close to the sample rear surface.The viewing zenith angle θ t then increases to θ * t and the z-component of the sample-to-detector-aperture distance R z reduces to R * z = R z − ∆z.With the reference plane at z = ∆z, the transmitted radiant flux measured by the detector from the sample rear surface also changes because of the increased solid angle,

Figure 3 .
Figure 3. Ratio of transmitted to incident flux τ (θt) = ϕ t (θt) /ϕ i of a HOD-500 sample recorded by (a) the complementary instrument with R = 129 mm and (b) the absolute instrument with R ′ = 472.9mm.The ratio of fluxes is shown at four wavelengths as a function of viewing zenith angle θt determined by the front surface of the sample.The error bars show the expanded uncertainty of measurements (k = 2) according to [10] where the effect of increased relative contribution due to noise and viewing angle uncertainty between θt = 35 • and 70 • are taken into account.

Figure 4 .
Figure 4. Ratio of instrument τ (θ * t ) values for the HOD-500 sample measured at four wavelengths.The ratio is given for the complementary and absolute instrument as a function of the viewing zenith angle θ * t defined by the sample rear surface.Data points are shifted by 0.5 • increments.The error bars show the expanded uncertainty of τ(θ * t )/τ ′ (θ * t ) (k = 2).The model of equation (8) (green stars) shows that some angular dependence is expected when comparing measurement results of detectors at different distances R and R ′ .

Figure 5 .
Figure 5. BTDF of the thick HOD-500 sample using the sample front surface as the reference plane.Experimental data are corrected by equation (9).The red line shows the BTDF determined using the absolute instrument with R ′ = 472.9mm, and the blue dotted line shows the BTDF determined using the complementary instrument with R = 129 mm.BTDF is shown at four wavelengths in the viewing zenith angle range from 0 • to 70 • in 5 • steps.The error bars show the expanded uncertainty of measurements (k = 2).

Figure A1 .
Figure A1.BTDF of the thick HOD-500 sample using two different sample-to-detector aperture distances.The sample front surface serves as the reference plane with no short-distance correction (equation (9)).The red line shows the BTDF measured using the absolute instrument with R ′ = 472.9mm, and the blue dotted line shows the BTDF measured using the complementary instrument with R = 129 mm.BTDF is shown at four wavelengths in the viewing zenith angle range from 0 • to 70 • in 5 • steps.Data points are shifted by 0.5 • (Absolute) and 0.7 • (Complementary) increments for better visibility.The error bars show the expanded uncertainty of measurements (k = 2).