=Paper=
{{Paper
|id=Vol-3271/Paper13_CVCS2022
|storemode=property
|title=Preliminary Results of Angle-Resolved BTDF Characterization of Optical Transmissive Diffusers
|pdfUrl=https://ceur-ws.org/Vol-3271/Paper13_CVCS2022.pdf
|volume=Vol-3271
|authors=Jinglin Fu,Jeppe Revall Frisvad,Michael Esslinger,Tatjana Quast,Alfred Schirmacher
|dblpUrl=https://dblp.org/rec/conf/cvcs/FuFEQS22
}}
==Preliminary Results of Angle-Resolved BTDF Characterization of Optical Transmissive Diffusers==
https://ceur-ws.org/Vol-3271/Paper13_CVCS2022.pdf
Preliminary Results of Angle-Resolved BTDF Characterization of
Optical Transmissive Diffusers
Jinglin Fu1, Jeppe R. Frisvad2, Michael Esslinger1, Tatjana Quast1 and Alfred Schirmacher1
1
Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany
2
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Richard
Petersens Plads, Building 324, 2800 Kgs. Lyngby, Denmark
Abstract
Optical diffusers are used in many application areas related to illumination. With the purpose
of characterizing the transmission properties of optical diffusers, we present preliminary results
of BTDF (Bidirectional Transmittance Distribution Function) measurement with high angular
resolution at two different wavelengths. The expanded total uncertainty of the measurement is
estimated to be 0.5% in the best case. In this way, the scattering distribution of optical diffusers
could be quantitatively described by their BTDF values with high precision, good repeatability
and reproducibility. The diffusers’ angle-dependent spectral transmittances in the VIS (from
380 nm to 780 nm in the electromagnetic spectrum) are also measured using a commercially
available spectrophotometer. Results are compared to simulation calculated from a simple
model and a first-step explanation for the sample spectral dependency is proposed.
Keywords 1
Optical Diffuser, Diffuse Transmission, BTDF, Angle-resolved Detection, Wavelength
Dependency
1. Introduction
When light impinges on a rough surface or propagates through a thin layer with non-uniformities in
the material, it will deviate from its original trajectory and be scattered. Depending on the type of
interaction, in its extremes the reflection and transmission of light are distinguished and described as
either specular or diffuse. In most cases however, the situation is a mixture of these two components.
Optical diffuse transmission happens almost everywhere in our daily life and people have been
constantly making great use of it. In the field of satellite-based Earth remote sensing, for example,
optical transmissive diffusers are used for the calibration of the on-orbit spectral radiance [1, 2]. In
illumination design, diffusers are integrated in the optical system to control the distribution of
transmitted light [3, 4]. In computer graphics rendering, “diffusers” are also used in different scenarios
to simulate the transfer of light through thin objects in a realistic way [5, 6, 7, 8].
Characterization of diffuse transmission is of great importance, because it provides important
information on the visual appearance of an object. A quantitative characterization of optical diffusers
would be beneficial to both industrial and metrological aspects. Within the JRP EMPIR project BxDiff
(New quantities for the measurement of appearance) [9], different types of measurands (BRDF, BTDF,
BSSRDF) are determined to lay a foundation for better understanding the characterization of the visual
appearance of objects. In this paper, preliminary results of the angle-resolved measurement on the
BTDF of five different types of diffusers are presented. Measurements were carried out at PTB
(Physikalisch-Technische Bundesanstalt), the national metrology institute in Germany. The combined
The 11th Colour and Visual Computing Symposium 2022, September 08–09, 2022, Gjøvik, Norway
EMAIL: jinglin.fu@ptb.de (A. 1); jerf@dtu.dk (A. 2); michael.esslinger@ptb.de (A. 3); tatjana.quast@ptb.de (A. 4);
alfred.schirmacher@ptb.de (A. 5)
ORCID: 0000-0001-8260-8017 (A. 1); 0000-0002-0603-3669 (A. 2); 0000-0002-2228-7853 (A. 5)
©️ 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
total uncertainty depends on the scattering angle as well as the characteristics of the sample and is
estimated to be around 0.5% in the best case. The spectral transmittance at different scattering angles is
also measured on another setup and compared to the first-step simulation result. A possible explanation
for the wavelength behavior is also proposed in the paper.
2. Method
2.1. Bidirectional Transmittance Distribution Function (BTDF)
To fully characterize diffusely scattered light, the BRDF (Bidirectional Reflectance Distribution
Function) was first introduced by Nicodemus [10, 11]. Bartell et al. [12] then extended this concept into
the field of transmission by introducing an analogous term called BTDF (Bidirectional Transmittance
Distribution Function). As illustrated in Figure 1(a), a thin sample with flat surface element d𝐴𝑖
possesses uniform and isotropic scattering properties. Its top surface is uniformly irradiated by a
differential element of irradiance d𝐸𝑖 from a spatial direction described by (𝜃𝑖 , 𝜙𝑖 ). The BTDF is then
defined as the ratio of the scattered transmitted radiance d𝐿𝑡 in another specific direction (𝜃𝑡 , 𝜙𝑡 ), which
is in the transmittance hemisphere of the sample, to the incident irradiance within d𝐴𝑖 :
d𝐿𝑡 (𝜃𝑡 , 𝜙𝑡 ) (1)
BTDF(𝜃𝑖 , 𝜙𝑖 ; 𝜃𝑡 , 𝜙𝑡 ) = ,
d𝐸𝑖 (𝜃𝑖 , 𝜙𝑖 )
where 𝜃 is the polar angle with respect to the normal of the top surface of the sample (positive direction
of the z-axis in Figure 1) and 𝜙 is the azimuth angle within the sample surface plane. The subscripts “i”
and “t” designate incident and transmitted radiation, respectively.
By using a differential expression, the BTDF purely describes sample properties as a geometric
distribution. It characterizes the transmitting properties of one point on the sample in one specific
direction, with contributions from the entire incident radiant flux confined within a certain solid angle.
It takes positive values with the unit per steradian (sr-1) [11]. Besides its geometrical sense, the BTDF
is also related to parameters such as wavelength and polarization of the incident radiation. Thus, the
BTDF is typically expressed as unpolarized values at a certain wavelength.
Z Z
θi
dLi Pi Abeam θ
i
Φi Φi
dAi Ai A M
dEi
Y Y
Φt Φt R
AD
θt
X θt dLt X
Pt
(a) (b)
Figure 1: Geometry of BTDF: (a) general definition and (b) measurement
In practice, however, measurements cannot be realized with infinitesimal quantities. Stover [13]
suggested the measurement conditions to be modified so that measurements are performed with finite
resolution, which is defined by the aperture size of the detection system. It is usually larger than the
irradiation spot of the incident beam and large enough for a sufficiently good signal-to-noise level, but
in the meantime as small as possible to minimize the influence from the instrument itself on the
measured values, which is the convolution of the real BTDF of the sample and the instrument signature.
Since a real sample cannot be perfectly isotropic and uniform, measurements of BTDF should be
considered as an average over a surface area which is much larger than the size of the statistical features
of the sample [12]. As depicted in Figure 1(b), a sample is irradiated by a collimated beam with finite
beam size 𝐴beam and its projection on the sample surface is then
𝐴𝑖 = 𝐴beam / cos 𝜃𝑖 . (2)
The average incident irradiance 𝐸𝑖 within 𝐴𝑖 on the sample surface takes
𝑃𝑖 (3)
𝐸𝑖 = ,
𝐴𝑖
with 𝑃𝑖 the incident optical power measured when no sample is placed in the beam path. The average
transmitted radiance 𝐿𝑡 is the scattered optical power 𝑃𝑡 detected on the other side of the sample within
𝐴𝑖 through the detector solid angle Ω𝑡 :
𝑃𝑡 1 (4)
𝐿𝑡 = ∙ ,
𝐴𝑖 Ω𝑡 cos 𝜃𝑡
where Ω𝑡 is determined by the detector aperture size 𝐴𝐷 and the distance 𝑅 between the detector aperture
and the sample back surface (front surface is the irradiated side):
𝐴𝐷 (5)
Ω𝑡 = 2 ,
𝑅
and the product of Ω𝑡 and cos 𝜃𝑡 is the projected solid angle at scattering angle 𝜃𝑡 . Combining equations
(2) to (5), we obtain the form for evaluating the BTDF from measured signals, when 𝐴𝑖 is relatively
small and the detector’s field of view is constantly underfilled [13, 14]:
𝐿𝑡 𝑃𝑡 𝑅2 (6)
BTDF = = ∙ .
𝐸𝑖 𝑃𝑖 𝐴𝐷 cos 𝜃𝑡
2.2. Experiment Setup
We measure absolute values of BTDFs of the different scattering transmissive samples in BxDiff
using the NaNoRef setup at PTB, which is originally the national reference setup for measurements of
specular reflection [15]. With proper modifications to hardware and the control program, measurements
of diffuse transmission could also be performed with modest uncertainty.
rotating
beam weak achromatic lens
expander irises
diffuser chopper &
detector aperture
laser aperture sample
sources polarizer ND-filter field
monitor aperture
Si-photodiode
on integrating
sphere
Figure 2: Sketch of BTDF setup (modified NaNoRef) at PTB, top view
As illustrated in Figure 2, the central part of the goniometric setup is an octagonal chamber, in which
the sample and the detection system are enclosed to eliminate stray light from the light source and
ambient light. The sample is placed at the center of this chamber and is irradiated by a collimated beam
whose size is restricted by an interchangeable aperture in front of it. The sample holder is a vertical
motorized rotary stage which could rotate around its azimuth (optical axis) freely, so that any desired
sample azimuth could be achieved. The rotary stage is mounted on a motorized three-axes displacement
unit to control the position of the irradiation spot on the sample and to adjust the longitudinal position
of the sample back surface on the optical axis. The displacement unit is fixed on a horizontal rotary
table, which rotates around the central (vertical) axis of the octagon. This rotation axis is perpendicular
to the optical axis, and the intersection point of these two axes lies exactly on the sample back surface.
In this way the polar angle of the incident radiation onto the sample could be controlled. The detection
system is mounted on another horizontal rotary arm, which is concentric to the sample rotation table.
Therefore, both the table and the arm rotate within the same plane around the same axis, which is
referred to as the detection axis.
In the detection system, the detector aperture is interchangeable to different diameters to realize
detection with different angular resolutions and detection solid angles. Directly behind the detector
aperture, an achromatic lens determines the size of the measurement area 𝐴M (Figure 1(b)) by means of
a field aperture, which is placed at the entrance port of an integrating sphere. A calibrated Si-photodiode
is mounted on the detection port of it. The use of an integrating sphere homogenizes the radiation for
detection and reduces speckle effects as well as polarization-related sensitivity of the detector. All these
components of the detection system are fixed on the detection rotary arm and move altogether during
detection. The optical path is folded with an angle as small as possible by a planar mirror due to limited
space inside the chamber. By alternating the size of the field aperture, the size of the measurement area
on the sample back surface could be adjusted. The parameters regarding measurement conditions are
listed in Table 1.
Table 1
General conditions of BTDF measurement
Parameter Notation Value
Wavelength 𝜆 642 nm, 445 nm
Azimuth angle of incidence 𝜙𝑖 0° to 360°
Azimuth angle of detection 𝜙𝑡 0° to 360°
Polar angle of incidence 𝜃𝑖 0°
Polar angle of detection 𝜃𝑡 -25° to 80°
Irradiation beam size 𝐴beam 9.6 / 29.2 / 44.2 / 80.1 mm2
Detector aperture area 𝐴𝐷 19.6 / 81.7 / 156.1 mm2
Distance sample to detector 𝑅 494.5 mm
Measurement area on sample (at 𝜃𝑡 = 0°) 𝐴𝑀 50.3 / 75.4 / 265.9 mm2
Outside of the chamber are the light sources and some beam controlling optics. Two laser sources
are used for the measurement, one is a fiber-coupled diode laser at 642 nm and another one is a diode-
pumped solid-state laser at 445 nm. To reduce speckle effects, a rotating weak diffuser is placed in the
vicinity of the intermediary focus of a beam expander. In this way, the laser beam is expanded to be
able to supply varying diameters of the irradiation spot on the sample. Moreover, the coherence of the
laser radiation could in this way be reduced and the beam profile is homogeneous after collimation.
Polarization of the radiation is controlled by using a half-wave plate and a linear polarizer.
Measurements are performed in s- and p-polarized radiation respectively, with unpolarized detection
and then the unpolarized BTDF value was calculated by the mean value out of the two measurement
results. The measurement is performed using the lock-in technique to suppress any unwanted optical
signals that are not at the chopping frequency. As the BTDF values of different samples vary largely,
neutral-density filters are used to bring the detected power to an appropriate level, reducing the
influence of electronic noise and improving SNR. The two irises prevent stray light from entering the
measurement chamber and filter out the undesired outer parts of the expanded beam. The monitor
detector is used to correct for fluctuation in the laser output power.
For an absolute measurement, the sample was firstly moved out of the optical path. The ND-filter
was set in and the incident power 𝑃𝑖 was measured at 𝜃𝑡 = 0°. Then, the sample was moved into the
optical path with its geometric center irradiated. The sample was aligned to have 𝜃𝑖 = 0° and its azimuth
𝜙𝑖 (or 𝜙𝑡 since measurements were all in-plane) was either predefined for rotationally symmetrical
samples or predetermined for azimuthally sensitive samples by evaluating the intensity distribution on
a screen using a high-resolution camera, which will be introduced in more detail in the next section. A
well-defined azimuthal orientation of the sample is of great importance when measured BTDF values
are compared for samples with dedicated azimuthal dependence. The angle-resolved detection of the
sample scattering distribution was then carried out by rotating the detector arm to each scatter angle 𝜃𝑡 .
The laser beam was not attenuated by the ND-filter when the transmitted scatter signal was measured.
2.3. Samples under study
Five types of samples are characterized. Two of them are quasi-Lambertian diffusers with relatively
constant BTDF in a large range of scatter angles 𝜃𝑡 . One of them, denoted as HOD-500, is a diffuser
made of high purity fused silica (HOD®) manufactured by Heraeus Quarzglas GmbH [16]. It has 50
mm in diameter and 2 mm in thickness and there are micro air bubbles with diameter less than 20 µm
uniformly distributed in the bulk volume. Another quasi-Lambertian sample, denoted as Zenith-250, is
a 0.25 mm thick foil made of Zenith Polymer®, a high-reflecting material using sintered PTFE powder
manufactured by SphereOptics GmbH [17]. It is clamped between two plates with a clear aperture of
45 mm in the center. Types of Mie scattering volume diffuser made of synthetic fused silica, which are
similar to HOD-500 [18, 19, 20] and Zenith-250 [20], have been proven to be good near-to-Lambertian
diffusers.
The scattering characteristics of the other three types of samples are far away from a Lambertian
characteristic. The first one, denoted as DG20-220, is a rotationally symmetrical ground glass diffuser
distributed by Thorlabs, Inc [21]. It has 2 inches in diameter and 2 mm in thickness with one side treated
using 220 grit polish. Thus, this side of the sample is regarded as the active surface and is irradiated in
the measurement. Light is scattered mainly by the surface of this type of sample, leading to a Gaussian-
shaped distribution. The second type of sample, denoted as E28-14, is a holographic diffuser provided
by Temicon GmbH [22]. It consists of a 125 µm thick imprinted PET foil on standard glass substrate
with 2 mm thickness and it has 50 mm edge length. This type of sample has a two-dimensional
Gaussian-shaped scattering characteristic, meaning that the FWHM of the scattering distribution is
different when varying sample azimuth. The last type among these three, denoted as ED1-S20, is also
distributed by Thorlabs, Inc. It is called an Engineered Diffuser™ with 1 inch in diameter and 1.5 mm
in thickness. It has a square-shaped scattering distribution and is thus also azimuthally sensitive. The
scattering characteristic can be attributed to the individually specified microlens-units on the surface
produced by direct laser writing. This type of diffuser is normally used in applications related to beam
shaping [23].
For both holographic and engineered diffusers, determination of their azimuthal orientation prior to
the measurement must be performed. An air-cooled CCD-camera with 1600 pixels × 1200 pixels
resolution was placed towards the back surface of the sample and was horizontally aligned. A half-
transparent diffuse screen was placed between the camera and the sample and adjusted to be
perpendicular to the beam direction. The sample’s scattering distribution falling on the screen could
then be photographed by the camera from behind the screen. A series of photos was taken when the
azimuth was constantly varied with an interval of 0.1° by the rotary stage holding the sample. The
transmitted scattered patterns were then evaluated by different methods and comparable results with an
uncertainty of 0.2° regarding the sample azimuthal orientation could be achieved. For the holographic
diffuser, its cross section imaged on the screen showed an elliptical form and its 𝜙𝑖 = 0° orientation
was defined when the major axis of the ellipse was parallel to the detection plane. For the engineered
diffuser, the 𝜙𝑖 = 0° orientation was given when one edge of the square-shaped pattern was parallel to
the detection plane. The so found orientations were marked by a small dot on the edge of the diffusers
for later performed measurements.
3. Results
3.1. Quasi-Lambertian Diffusers
The measurement results of HOD-500 and Zenith-250 are illustrated in Figure 3(a). Due to the
physical limitation on the detection range, results in the negative angle part smaller than -25° are
extended by measuring the sample at 𝜙𝑖 = 180° in the positive angle range greater than 25°, where the
same value ought to be obtained with an ideal measurement setup. This method was also used in the
measurement of other samples, when the desired detection range was beyond the limitation.
(a) (b)
Figure 3: (a) BTDF of HOD-500 at 642 nm and 445 nm, 𝜃𝑖 = 0°, 𝜃𝑡 varies from -80° to 80°; (b)
Normalized BTDF of HOD-500 and Zenith-250, 𝜃𝑖 = 0°, 𝜃𝑡 varies from -35° to 35°
For the sake of brevity only the unpolarized BTDF values of HOD-500 are shown here.
Measurement showed similar quasi-Lambertian scattering distributions at both wavelengths, with peak
BTDF values of approx. 0.057 sr-1 and 0.061 sr-1, respectively. The smaller amplitude at 445 nm could
be explained by a decrease in the transmittance due to increasing scattering by a larger refractive index
at shorter wavelength and a higher absorption as well. The right axis of the plot corresponds to the curve
of 642 nm and shows the angle-dependent deviation from a perfect Lambertian diffuser in percentage.
At 60° the BTDF drops less than 10% and at 80° approx. 25%, indicating that light is scattered
uniformly by HOD-500 within a fairly large range of angles.
The normalized BTDF values of both quasi-Lambertian diffusers are plotted together in Figure 3(b)
for an intuitive sample-type comparison. Within a small range of angles from -35° to 35° the
discrepancy between the two diffusers already becomes significant. The BTDF of Zenith-250 decreases
faster than that of HOD-500 and is thus less “Lambertian”. This could be attributed to the different
types and sizes of scattering centers in the bulk volume of sample material.
3.2. Ground Glass Diffusers and Holographic Diffuser
The measured BTDF values for DG20-220 with fitted curves are shown in Figure 4(a) for 642 nm
and 445 nm. Data can be fitted with a Gauss-Lorentz model. Compared to the quasi-Lambertian
diffusers this ground glass diffuser has a sharp peak and the distribution is restricted in a small angle
area. The FWHM (Full-Width at Half Maximum) are 15.9° at 642 nm and 16.5° at 445 nm. The slightly
broader distribution at shorter wavelength might be attributed to enhanced multiple in-surface
scattering, which will be discussed in more detail in Section 4.1.
(a) (b)
Figure 4: BTDF of (a) DG20-220 at 642 nm and 445 nm, 𝜃𝑖 = 0°, 𝜃𝑡 varies from -35° to 35°; (b) E28-14
at 642 nm, 𝜙𝑖 at 0° and 90°, 𝜃𝑖 = 0°, 𝜃𝑡 varies from -40° to 40°
The holographic diffuser E28-14 also has a Gaussian-like scattering characteristic, but the
distribution is dependent on its azimuthal orientation. In Figure 4(b), the measured BTDF values with
fitted curves at 642 nm at 0° and 90° azimuths are demonstrated. The fit uses the regular Gaussian
model. The FWHM values are 26.7° at 𝜙𝑖 = 0° and 12.9° at 𝜙𝑖 = 90°, indicating that by changing the
sample azimuthal orientation of 90° the width of the scatter decreases rapidly to about half of its starting
value. Near the peak of the curve at 0° azimuth strong fluctuation in the BTDF values is observed,
which might be the effect of laser speckle. The wavelength dependency of E28-14 is similar to that of
DG20-220. The peak value is smaller and the FWHMs at both azimuths are slightly larger at 445 nm
(not shown).
3.3. Engineered Diffusers
BTDF values of the engineered diffuser ED1-S20 at both wavelengths and at both azimuths are
illustrated in Figure 5. Its scattering distribution is much different from the other diffusers. From about
-7° to 7° the scatter remains almost unchanged, as indicated by the flat top in the curves and then drops
rapidly within approx. 3° from peak value to close to zero. Although the scatter pattern is square-shaped,
there are still slight differences which could be observed from the angle-resolved measurement at
different azimuthal orientations. The wavelength dependency is on one hand similar to that of the
surface scattering diffusers. On the other hand, at the two “horns” of the curve the scatter behaves quite
distinctively at different wavelengths. This might be attributed to the special microlens structure on the
sample surface, whose interaction with light changes with varying wavelength.
Figure 5: BTDF of ED1-S20 at 642 nm and 445 nm, 𝜙𝑖 at 0° and 90°, 𝜃𝑖 = 0°, 𝜃𝑡 varies from -20° to 20°
4. Discussion
4.1. Wavelength Dependency
For a good comparison between different participants within BxDiff, results are to be compared at
a certain wavelength. A quasi-standard wavelength is that of the red HeNe-laser: 632.8 nm. For results
that are not measured at the prescribed wavelength, a proper correction considering the wavelength-
dependent sample properties should be performed. For this purpose, the angle-dependent spectral
transmittance of a set of samples was measured on an Agilent Cary 6000i, a commercially available
spectrophotometer with Universal Measurement Accessory (UMA) mini-goniometer.
(a) (b)
Figure 6: Spectral transmittance of (a) Zenith-250 and (b) E28-14 (𝜙𝑖 = 0°) at different 𝜃𝑡
In Figure 6(a), the diffuse transmittance into a solid angle of 0.0089 sr of Zenith-250 at different
scattering angles is plotted as a function of wavelength. For bulk diffusers, the transmittance is in
general fairly proportional to the wavelength. The variation rate or slope of the curve varies with
different 𝜃𝑡 . At 𝜃𝑡 = 0°, corresponding to the central peak in the angle-dependent scattering
distribution, the BTDF value changes the most with increasing wavelength, whereas at grazing angle
𝜃𝑡 = 80° the BTDF becomes almost constant and show little spectral dependency. The wavelength
dependency is different for surface scattering types. Spectral transmittance of E28-14 is given as an
example in Figure 6(b). The spectral transmittance in the central peak of the angular scattering
distribution rises with wavelength but flattens at around 600 nm and starts to decrease at even higher
wavelengths. The wavelength of the maximum of spectral transmittance 𝜆max also varies with different
𝜃𝑡 . This is observed more clearly by the differential spectral transmittance in Figure 7(a), where the
derivatives of transmittance with respect to wavelength are plotted. Interestingly, the wavelength
dependency becomes completely “reversed” at larger 𝜃𝑡 . Instead of a positive slope, transmittance
decreases constantly with increasing wavelength in the whole VIS. As a result of this wavelength
dependency, not only the BTDF value but also the shape of the angle-resolved scattering distribution
would be modified by the wavelength correction. For example, correction from a larger wavelength to
a smaller one would cause the peak of the distribution to drop while at the falling edges the BTDF
would be lifted, resulting in a larger FWHM. This also agrees with the comparison between measured
curves of the surface scattering samples at different wavelengths.
(a) (b)
Figure 7: Differential spectral transmittance of (a) E28-14 (𝜙𝑖 = 0°) measured at different 𝜃𝑡 and (b)
simulation using a single scattering model
Scattering effects can likely explain this behavior. We performed a preliminary comparison to
simulation results using a single scattering model where a surface scattering sample consists of an active
top surface with microfacets and small scattering particles on a smooth transparent substrate. The
microfacets obey a normal distribution determined by the surface roughness (standard deviation in
height). The calculation is based on Rayleigh scattering where the wavelength dependence is modified
according to particle size by an exponent in [0.0925, 1] (smaller exponent for larger particles). This
method has been found to fit reasonably well with the Lorenz-Mie theory [24]. Even without
considering multiple scattering, a similar tendency in the simulated spectral transmittance could be
observed. This is indicated most clearly by the derivatives of the simulated spectral transmittance
demonstrated in Figure 7(b). Compared to the measured one, the magnitude of this simulated effect is
smaller, but it is expected to be enhanced by multiple scattering in the sample active surface. Further
calculations are in progress and we expect that use of multiple in-surface scattering and scalar
diffraction theory can bring the model closer to the observation [25, 26].
4.2. Preliminary Uncertainty Analysis
Explicit uncertainty analysis is still in progress. An estimation of the combined measurement
uncertainty is around 0.5% (k = 2) in the best case (𝜃𝑡 = 0°) for all sample types. Contributions to the
estimated uncertainty budget of HOD-500 at 445 nm and 𝜃𝑡 = 0°, as an example, are listed in Table 2.
Determination of the BTDF measurement uncertainty ΔBTDF is performed according to Stover [13].
The uncertainty mainly consists of the statistical errors from the two measured signals, ∆𝑃𝑖 and ∆𝑃𝑡 ,
the errors from the measurement of detection solid angle, ∆Ω𝑡 , and from the receiver’s angular position,
∆𝜃𝑡 . The combined uncertainty ΔBTDF could be expressed by equation (7):
(7)
∆BTDF ∆𝑃𝑖 2 ∆𝑃𝑡 2 ∆Ω𝑡 2 Δ𝜃𝑡 ∙ sin 𝜃𝑡 2
= √( ) + ( ) + ( ) +( ) .
BTDF 𝑃𝑖 𝑃𝑡 Ω𝑡 cos2 𝜃𝑡
At other angles the measurement uncertainty varies mainly with the measured signal amplitude and
the cosine value of the detection angle. For both quasi-Lambertian samples, ΔBTDF remains unchanged
in a large angle area and eventually increases to over 2% at grazing angles. For the other three types of
samples, the signal amplitude drops quickly at the edges of the scattering distribution, leading to a sharp
increase in the total uncertainty, from 0.5% to more than 3%. This estimate might hold only for slowly
varying BTDF characteristics. At least for parts of the scattering distributions depicting a sudden
ΔBTDF
variation, a term proportional to must be added to the final uncertainty budget.
Δ𝜃𝑡
Table 2
Preliminary uncertainty budget of HOD-500, at 𝜆 = 445 nm, 𝜃𝑡 = 0°
Source Notation Relative uncertainty
Irradiation power ∆𝑃𝑖 0.00031
Scattered transmitted power ∆𝑃𝑡 0.00014
Detection solid angle ∆Ω𝑡 0.00153
Distance sample to detector Δ𝑅 0.00054
Detector aperture area Δ𝐴𝐷 0.00134
ND-filter Δ𝜏𝐹 0.00268
Receiver angle ∆𝜃𝑡 0.00005
Estimated total uncertainty (k = 2) ΔBTDF 0.005 / 0.5%
5. Conclusion
Preliminary results of the BTDF measurement involved in the EMPIR project BxDiff are presented
in this paper. Five types of transmissive diffusers, two bulk scattering, one pure surface scattering, one
holographic and one with engineered surface, were measured at the modified NaNoRef setup of PTB
with a preliminary total uncertainty of approx. 0.5% (k = 2) at 0° detection angle (corresponds to the
angle of specular transmission). The angle-dependent spectral transmittance of all samples was
measured using a commercially available spectrophotometer to provide supporting information for a
possible wavelength-dependent correction for the benefit of the comparison at a prescribed wavelength.
An interesting phenomenon of the interaction between angle- and wavelength-dependency of the
spectral transmittance was observed for the holographic diffuser and possible explanations are proposed
from first-step simulation results. Further study on the wavelength dependence of different sample types
will be carried out to provide a more detailed quantitative explanation. Our aim is to acquire thorough
information on the relation between sample microscopic structures and their scattering properties as
well as a better knowledge of BTDF measurement on optical translucent materials with high precision.
6. Acknowledgements and Disclaimer
The authors gratefully thanks Eva Velke for the calibration of ND-filters used in the measurement
and Marius Bagusch for providing measurement data of the area of different apertures used in the
measurement. We also thank the members of the mechanical workshop Morten John, Jörn Hauffe,
Marcel Janik and Henry Ganz for the construction of several mechanical parts of the setup.
We gratefully acknowledge the support of the Braunschweig International Graduate School of
Metrology B-IGSM.
The results reported in this paper are derived from activities within the frame of the EMPIR project
18SIB03 BxDiff (New quantities for the measurement of appearance), that has received funding from
the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon
2020 research and innovation programme.
The mentioned diffusers dealt with in this paper are examples of commercially available products
only and using these specific diffusers in investigations does not constitute an endorsement of these
products by the authors or their institutions.
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