1.

Introduction

Black coatings are widely used for improving the performance of optical instruments, for example, to minimize internal reflections in the optical paths. Accurate characterization of coatings at the transition between the far ultraviolet (FUV) and extreme ultraviolet (EUV) regimes is crucial for a wide range of applications, including lithography¹ and spectroscopy² among others. Space instruments designed for application in solar and heliospheric physics require a highly absorptive coating to suppress the stray light. In particular, collimators that subtend the desired field of view of optical instruments are required to suppress stray light effectively.

The Space Research Center PAS (CBK PAN) is preparing a GLOWS (GLObal Solar Wind Structure) photometer for the NASA’s Interstellar Mapping and Acceleration Probe (IMAP) space mission.³ The GLOWS photometer will measure the intensity of the heliospheric backscatter glow, which is generated by resonant scattering of solar Lyman-$\alpha$ (121.6 nm) photons on interstellar neutral hydrogen (ISN H) atoms penetrating the heliosphere. The GLOWS measurements will enable investigation of the helio-latitudinal structure of the solar wind and radiation pressure acting on ISN H.⁴ To fulfill the mission’s requirements, the instrument must meet stringent stray-light-suppression requirements, which are achieved by a suitable optical design and blackening its optical baffle and collimator surfaces with the Acktar Magic Black™ coating.⁵

The Acktar Magic Black™ is a coating from an entire family of black coatings offered by the Acktar company, which are characterized by a low reflectance in a broad temperature range in a wide range of wavelengths. These properties make the coatings interesting in space-physics and astronomical contexts for applications in spaceborne instruments. Besides the IMAP/GLOWS instrument mentioned above, past or future missions that use the Acktar black coatings include, among others, Solar Orbiter/METIS,⁶ CHEOPS,⁷ Sentinel-4/UVN,⁷ ULTRASAT,⁸ James Webb Space Telescope,⁹ and JUICE.¹⁰

Development of an instrument requires a number of numerical simulations, for which the reflectance properties of the coating need to be known. Reflectance measurements are conventionally converted into the bidirectional reflectance distribution function (BRDF), which provides a rigorous description of reflectance properties as a function of the incidence and scattering/reflection directions. In the literature available to date and in the materials provided by the manufacturer, one can find a relatively complete set of scatterometer measurements of the Acktar Magic Black™ coating for selected wavelengths in the visible range (VIS).^6,7 However, for the UV range, there are available only measurements of the BRDF for 193 nm in the specular plane and of the reflectance for a limited set of the incidence and scattering angles for $\sim 13\mathrm{nm}$.¹¹ Lyman-$\alpha$ diffuse-reflectance characteristics of 10 different materials were discussed in Ref. 12. This study includes the Acktar’s Spectral Black™ adhesive foil, some carbon-nanotube, and carbon-foam coatings, and Laser Black coating manufactured by the Epner Company. The measurements were performed for near-normal incidence and selected scattering angles. Relatively little data on the reflectance of coatings in the UV range, including specifically measurements for $\sim 121.6\mathrm{nm}$, seem to be due to difficulties specific to the FUV and EUV regimes (measurements in vacuum, need for a special type of radiation sources and detectors, calibration issues, etc.). The goal of this paper is to address this gap by measuring the reflectance and developing a phenomenological model of the BRDF for $\sim 121.6\mathrm{nm}$ in the full scattering hemisphere.

The measurements of the reflectance of Acktar Magic Black™ samples discussed in this paper were performed at the Institute of Optoelectronics of the Military University of Technology in Warsaw. Radiation in the EUV and FUV ranges¹³ was produced with a laser plasma radiation source based on a double-stream gas puff target.¹⁴ Due to difficulties specific to the FUV and EUV regimes, the measurements do not cover the entire hemisphere but characterize comprehensively certain parts, including quasi-perpendicular (relative to the sample surface) and quasi-parallel cases. This set of measurements is shown to properly constrain the parameters of the BRDF model proposed in this paper.

The phenomenological BRDF model is based on a series of analytic fits to the measurements discussed above. To obtain the BRDF model for $\sim 121.6\mathrm{nm}$, in the first step, we develop a methodology and pre-validate it using measurement data available in the literature for the VIS, thus a BRDF model for 532 nm is also discussed here as a byproduct of our analysis. The methodology discussed in this paper is more general and can be used also to construct BRDF models for other wavelengths provided that appropriate measurement data are available.

This paper is organized as follows. Section 2 describes the experimental setup and Sec. 3 discusses the obtained raw reflectance measurements. The methodology for the phenomenological model of the BRDF is presented in Sec. 4. The BRDF models for $\sim 532$ and $\sim 121.6\mathrm{nm}$ are derived in Sec. 5. Numerical simulations using our BRDF model are discussed in Sec. 6 to show that they reproduce the original reflectance measurements. Section 7 contains a general discussion of the obtained results and conclusions.

2.

Experimental Setup

The measurements were performed for flat aluminum surfaces covered with the Acktar Black coating. The measurements were taken at various incidence angles ${\theta }_i$ (measured from the normal to the sample) to obtain on-axis—specular reflectance, as well as off-specular reflectance, where the scattering angle ${\theta }_s$ is $\delta =\pm 15\mathrm{deg}$ around the specular direction. Two series of measurements were performed. Series I (specular reflectance) is summarized in Sec. 3.1, where ${\theta }_i=[18,45,75,80]\mathrm{deg}$, and series II (off-specular reflectance) is presented in Sec. 3.2, where ${\theta }_i=[45,75,80]\mathrm{deg}$.

For the reflectance measurements, an experimental setup presented in Fig. 1 was developed, consisting of a laser plasma source, three baffles with diameters of 4.8, 16, and 4.63 mm, and a back-illuminated CCD camera (DX420-BN Andor, Oxford Instruments), cooled down to a temperature of $-20°\mathrm{C}$, with an narrowband VUV/UV optical filter¹⁵ (Acton FN122-XN-1D), hereafter called Lyman-α filter, mounted to transmit the radiation within a narrow wavelength range around the hydrogen line (121.6 nm). The camera’s chip size was $1024\times 256\text{pixels}$, the pixel size was $26\times 26\mu {\mathrm{m}}^2$, and the exposure time 5 s. The distance $z$ between the source and the sample was in the range of 346 to 360 mm, depending on the series of measurements. The distance between the source and the CCD camera was 0.486 to 0.5 m, keeping in mind to always have the source–camera distance constant during the reference and angular measurements. 1.

Fig. 1

A scheme of the experimental setup used in reflectance measurements: (a) with dimensions and visualization of the reference and angular measurements and (b) with a description of each component.

To produce EUV radiation, we used a laser-plasma radiation source based on a double stream Xe/He gas puff target.¹⁶ The source was driven by a Nd:YAG laser system (NL129 EKSPLA, Lithuania) with a pulse energy of 6.27 to 6.43 J and a pulse duration of 6.44 to 6.87 ns. These laser impulses were focused with an $f=100\mathrm{mm}$ lens onto a double-stream gas puff target. Two nozzles are required to produce this type of target. The inner nozzle injects a small amount of working gas (xenon) into the vacuum $200\mu \mathrm{s}$ after receiving a synchronization pulse from the laser power supply. The nozzle remains open for $750\mu \mathrm{s}$. The second ring-shaped nozzle is designed to inject a low-$Z$-number gas, in this case it is helium, to increase the density of the working gas by confining it by the radial flow of the He gas. It opens $400\mu \mathrm{s}$ after receiving a synchronization pulse and remains open for $555\mu \mathrm{s}$. At 1 ms, after the synchronization pulse, a laser pulse is emitted, which irradiates the Xe/He gas puff target to produce a plasma. This plasma emits efficiently the EUV radiation used in the subsequent reflectance measurements. The gas pressures and nozzle opening and closing times were optimized to provide the best radiation intensity in the wavelength range of interest. Energy fluctuations of the EUV source were estimated as $\sim 9\%$. The data were taken for the Xe plasma, which emits a broadband spectrum, spanning from 1 nm to far infrared. A part of the spectrum measured using a grating spectrometer is shown in Fig. 2(a).

Fig. 2

(a) The comparison of the Xe plasma spectrum measured with the LiF filter (pink line), and the spectrum measured with additional Lyman-$\alpha$ filter (blue line). Note that the Xe plasma spectrum with the Lyman-$\alpha$ filter has been multiplied by 10 to better visualize the difference between both spectra. (b) The transmission of the Lyman-$\alpha$ filter used in the reflectance measurements.

For spectral measurements in the FUV/EUV range, an HP Spectroscopy GmbH (formerly known as DR. HOERLEIN + PARTNER GbR) spectrograph was used. The spectrograph is based on a concave flat-field varied-linespace grating, which has 1200 grooves/mm. The spectral range of this instrument is 40 to 280 nm, and the resolution is 0.1 nm. To narrow down the spectrum, an interference Lyman-α filter was used, mounted at the entrance aperture of the CCD camera, to select only radiation in the range 115 to 135 nm with a spectral peak at $\sim 121.1\mathrm{nm}$. The measured peak transmission of the Lyman-$\alpha$ filter was $\sim 5\%$. During the measurements of the Lyman-$\alpha$ filter’s transmission, a lithium fluoride (LiF) filter was used to cut off the strong impact of higher diffraction orders from shorter wavelengths, which were overlapping the radiation from the spectral range near 100 nm. The thickness of LiF filter was 3 mm. The spectrum of Xe plasma radiation transmitted by the LiF filter is presented in Fig. 2(a)—pink line. The spectrum of Xe plasma radiation transmitted by the LiF and Lyman-$\alpha$ filters is presented in Fig. 2(a)—blue line (Xe spectrum with Lyman-α filter was multiplied by 10). The measured transmission of Lyman-$\alpha$ filter is presented in Fig. 2(b). The optimal EUV photon yield was obtained for the gas pressures equal to 9 bars for xenon and 6 bars for helium.

Three baffles/apertures were used to direct the unfiltered EUV beam toward the sample. The center of the sample was located 346 to 360 mm from the plasma. The sample was mounted on a rotation mount so that the incidence angle ${\theta }_i$ of the EUV beam, measured from the normal direction to the sample surface, could be easily changed in the range from 18 deg to 80 deg. The CCD camera with an entrance aperture, defined by the Lyman-α filter placed in front of the CCD detector, was used as a detector. In both measurement series, between the filter surface and the camera chip, a second aperture was mounted. The diameter of the entrance aperture was 7.2 mm in series I and 7.5 mm in series II, which can be seen as a circular intensity pattern in Fig. 3. The distance between the aperture and the camera’s chip was 36.2 mm in series I and 30 mm in series II. The camera’s position was manually adjusted to receive either the specularly reflected beam for each incidence angle (so that the angle $\delta =0\mathrm{deg}$) or the reflected signal for off-specular case ($\delta \ne 0\mathrm{deg}$, see Fig. 1). The distance between the sample’s center and the CCD chip was kept constant during the measurements and was equal to 140 mm. The angle ${\theta }_s$ is the scattering angle and for the specular reflection ${\theta }_i={\theta }_s$. For the off-specular reflection ${\theta }_i\ne {\theta }_s$, and the scattering angle can be described as

Fig. 3

Reflectance raw data taken for xenon at (a) 18 deg, (b) 45 deg, (c) 75 deg, and (d) 80 deg. Note the intensity change indicated by the colorbar.

3.

Experimental Results

3.1.

Specular Reflectance

For the specular reflectance measurements (series I), the reference and reflected signals (hereafter referred to as the angular data) were measured separately. The reference signal was measured without the sample, so the EUV beam illuminated the CCD camera directly, keeping the distance from the plasma to the CCD chip equal to 486 mm, as shown in Fig. 1. Typically, 1 to 10 EUV pulses were integrated in the case of reference measurements, and 30 to 50 EUV pulses were used to measure the reflected signal, depending on the angle of incidence. The variations in the number of pulses are accounted for in the averaged measured reflectances presented in this section.

Measurements were conducted for six flat aluminum samples coated with Acktar Magic Black™. The details of the coating manufacturing process are not known since they are proprietary technology developed by the Acktar company. The samples’ size was $1\times 1{\mathrm{in}}^2$. According to the manufacturer specification, the coating was performed on top of the aluminum sample and the coating was the same for all measured samples. The measurements discussed in this section were performed only for specular reflection for the incidence and reflection angles of ${\theta }_i=[\mathrm{18,45},\mathrm{75,80}]\mathrm{deg}$. Example images from the CCD camera for several angles are presented in Fig. 3. In each image in Fig. 3, there is an imprint that was caused by CCD’s array contamination. The imprint resulted in a decrease in the signal, which was corrected in the post-processing of the measurement data. The angle of 18 deg was the minimal possible angle due to the chamber dimensions and geometry.

Twelve data images were captured as a data series for each angle and each sample. A black square added approximately in the center of each image in Fig. 3 indicates the position of spatial signal integration. The intensity registered by the CCD camera from this region was integrated by summing up all pixel intensities. For each image (including the reference and angular data), the background signal was subtracted. The white-border square area in each image in Fig. 3 shows the region where the background signal was integrated. Both squared regions (with the black and white border) were exactly $100\times 100\text{pixels}$ in size. For each data series, the standard deviation $\sigma$ was calculated, to estimate the error. Dixon’s $Q$-test¹⁷ was used to identify the outliers in a dataset. The test confirmed that with a probability of 99%, there were no outliers during the measurement. The images were averaged to improve the signal-to-noise ratio. The Dixon’s $Q$-test is further described in Sec. 3.3.

For each image, the reflectance ($R$) was calculated as

Eq. (2)

$$R=\frac{n_{\mathrm{ref}}(I_{\mathrm{an},\text{signal}}-I_{\mathrm{an},\text{background}})}{n_{\mathrm{an}}(I_{\mathrm{ref},\text{signal}}-I_{\mathrm{ref},\text{background}})},$$

where $n_{\mathrm{ref}}$ is the number of laser impulses used in the reference measurements and $n_{\mathrm{an}}$ is the angular measurements. For the reference images, $I_{\mathrm{ref},\text{signal}}$ is the signal integrated in the central square, and $I_{\mathrm{ref},\text{background}}$ is the signal integrated inside the square located on the right side of an image. A similar procedure was used for the angular measurements, $I_{\mathrm{an},\text{signal}}$, and $I_{\mathrm{an},\text{background}}$.

Plots with the results are presented in Fig. 4, and the reflectance values are given in Table 1. The reflectance was estimated as the average of the reflectance values for each sample and will be further used to calculate the BRDF value. The standard deviation was calculated from the error propagation formulas, which will be discussed in Sec. 3.4. The results show that the reflectance of the Acktar samples decreases with the decrease of the incidence angle. This kind of behavior (low perpendicular reflectance) is desired for tube-like collimators of photometers (i.e., the GLOWS instrument) because it minimizes the transmission of stray light through the instrument.

Fig. 4

Raw values of the reflectance for each sample and each angle.

Table 1

Results of the angular measurements for six samples coated with Acktar Magic Black™.

Sample	Reflectance @ 18 deg	Reflectance @ 45 deg	Reflectance @ 75 deg	Reflectance @ 80 deg
1	$(5.79\pm 0.33)\times {10}^{-5}$	$(8.46\pm 0.57)\times {10}^{-5}$	$(8.37\pm 0.49)\times {10}^{-4}$	$(1.22\pm 0.10)\times {10}^{-3}$
2	$(5.49\pm 0.30)\times {10}^{-5}$	$(8.35\pm 0.54)\times {10}^{-5}$	$(6.65\pm 0.49)\times {10}^{-4}$	$(1.15\pm 0.09)\times {10}^{-3}$
3	$(5.78\pm 0.31)\times {10}^{-5}$	$(8.64\pm 0.59)\times {10}^{-5}$	$(8.53\pm 0.57)\times {10}^{-4}$	$(1.11\pm 0.09)\times {10}^{-3}$
	$(5.64\pm 0.33)\times {10}^{-5}$	$(8.29\pm 0.59)\times {10}^{-5}$	$(8.29\pm 0.56)\times {10}^{-4}$	$(1.55\pm 0.10)\times {10}^{-3}$
5	$(5.72\pm 0.31)\times {10}^{-5}$	$(9.45\pm 0.53)\times {10}^{-5}$	$(7.16\pm 0.48)\times {10}^{-4}$	$(1.30\pm 0.09)\times {10}^{-3}$
6	$(5.92\pm 0.33)\times {10}^{-5}$	$(9.59\pm 0.60)\times {10}^{-5}$	$(7.29\pm 0.58)\times {10}^{-4}$	$(1.47\pm 0.10)\times {10}^{-3}$
Mean reflectance	$(5.72\pm 1.38)\times {10}^{-5}$	$(8.80\pm 2.45)\times {10}^{-5}$	$(7.73\pm 2.24)\times {10}^{-4}$	$(1.30\pm 0.39)\times {10}^{-3}$

3.2.

Off-Specular Reflectance

The off-specular reflectance measurements (hereafter series II) were performed for a specific incidence angle ${\theta }_i$ while collecting the reflected signal at various angles $\delta \ne 0\mathrm{deg}$ from the specular reflection. The size of the samples used was $40\times 40{\mathrm{mm}}^2$. The off-axis measurements were performed by moving the CCD camera in the range of ($\delta$) from $-15\mathrm{deg}$ to $+15\mathrm{deg}$ from the specular reflection position. Again, at the beginning, the reference data were taken [see Fig. 5(a)]. Next, the data for the angular measurements were obtained. Examples of the data are shown in Figs. 5(b)–5(d).

Fig. 5

Reflectance raw data taken for xenon: (a) reference data and (b) examples of the data taken for the incidence angle ${\theta }_i=80\mathrm{deg}$ for specular reflection, and for off-specular case (c) at ${\theta }_i=80\mathrm{deg}$, $\delta =-5\mathrm{deg}$, and (d) at ${\theta }_i=80\mathrm{deg}$, $\delta =-10\mathrm{deg}$. Note the intensity change indicated by the colorbar.

The results presented in Fig. 6 and Table 2 show that for the incidence angle of 45 deg [Fig. 6(a)], and the reflectance is almost at the same level for each off-specular position. It is more than 10 times smaller in comparison to the results obtained at the position of ${\theta }_i=80\mathrm{deg}$. For the high incidence angles of ${\theta }_i=75\mathrm{deg}$ [Fig. 6(b)] and ${\theta }_i=80\mathrm{deg}$ [Fig. 6(c)], the reflectance increases with the increase of the scattering angle. For the incidence angle of 75 deg, the number of photons captured by the CCD’s array is two times smaller than that for the angle of 80 deg. Moreover, the difference between the highest and smallest results is at a level of $\sim 35\%$. For the incidence angle of 80 deg, the highest reflectance of 0.12% was captured when the camera was moved to $\delta =+5\mathrm{deg}$ from specular reflection. A rotation of the CCD camera in the opposite direction causes a nearly linear decrease of the reflectance by $\sim 50\%$. The measurements marked as N.A. were not possible due to the system geometry.

Fig. 6

Results of the off-specular measurements for the incidence angles of (a) ${\theta }_i=45\mathrm{deg}$, (b) ${\theta }_i=75\mathrm{deg}$, and (c) ${\theta }_i=80\mathrm{deg}$.

Table 2

Results of the off-axis measurements for a sample coated with Acktar Magic Black™.

Reflectance [−]		Incidence angle θi
		45 deg	75 deg	80 deg
Angle δ from the specular-reflection direction	−15 deg	$(7.30\pm 0.38)\times {10}^{-5}$	$(3.28\pm 0.27)\times {10}^{-4}$	$(0.62\pm 0.02)\times {10}^{-3}$
	−10 deg	$(7.50\pm 0.38)\times {10}^{-5}$	$(3.41\pm 0.25)\times {10}^{-4}$	$(0.71\pm 0.02)\times {10}^{-3}$
	−5 deg	$(7.90\pm 0.40)\times {10}^{-5}$	$(5.05\pm 0.37)\times {10}^{-4}$	$(0.85\pm 0.02)\times {10}^{-3}$
	0 deg	$(7.30\pm 0.39)\times {10}^{-5}$	$(5.55\pm 0.38)\times {10}^{-4}$	$(1.03\pm 0.03)\times {10}^{-3}$
	5 deg	$(7.80\pm 0.36)\times {10}^{-5}$	$(5.80\pm 0.41)\times {10}^{-4}$	$(1.25\pm 0.03)\times {10}^{-3}$
	10 deg	$(7.70\pm 0.36)\times {10}^{-5}$	$(5.72\pm 0.40)\times {10}^{-4}$	N.A.
	15 deg	$(6.70\pm 0.28)\times {10}^{-5}$	N.A.	N.A.

4.

Phenomenological Model of the BRDF: Methodology

The measurements described in Secs. 2 and 3 were aimed at the raw reflectance of the Acktar Magic Black™ coating defined as a ratio of the sample-scattered signal intensity registered by the CCD camera to the reference intensity, measured with the beam illuminating the camera directly. The measurements were corrected for the background identified outside of the nominal light spot in the images captured by the camera. In this section, we discuss the process of conversion of the raw-reflectance measurements into the BRDF and the general approach to fitting a phenomenological model of the BRDF.

4.1.

BRDF Function

The raw measurements of the reflectance discussed in Sec. 3 correspond to the ratio $d{P}_s/P_i$ of the scattered power $d{P}_s$ (in photons/s or W units) detected by the camera to the incident power $P_i$, so they depend on the geometry of the experimental setup, in particular on the diameter $d$ of the aperture in front of the CCD camera and the aperture-sample distance $r$. A geometry-independent characterization of the reflectance properties of a surface is possible by using the BRDF defined as

Eq. (6)

$$\mathrm{BRDF}({\theta }_i,{\phi }_i,{\theta }_s,{\phi }_s)=\frac{d{P}_{\mathrm{s}}}{P_{i}d{\mathrm{\Omega }}_s\cos ({\theta }_s)},$$

where $d{\mathrm{\Omega }}_s$ is the solid angle subtended by the detector aperture and $\cos ({\theta }_s)$ accounts for Lambert-cosine’s-law effects as proposed in Ref. 18. It can be shown that $\mathrm{BRDF}=d{L}_s/E_i$ is the ratio of the radiance $d{L}_s$ in a given scattering direction and the incident irradiance $E_i$. The unit of the BRDF is ${\mathrm{sr}}^{-1}$.

In Fig. 7, we show the geometry convention. In the most general case, the BRDF is a function of the incidence direction $({\theta }_i,{\phi }_i)$ and the scattering direction $({\theta }_s,{\phi }_s)$.

Fig. 7

Geometry convention used in this paper.

If the azimuthal angles ${\phi }_i$ and ${\phi }_s$ are measured from the specular plane, we have always ${\phi }_i=0$ (by the definition of our frame), and the dependence on ${\phi }_i$ can be omitted. Hence, only three angles are needed, and $\mathrm{BRDF}=\mathrm{BRDF}({\theta }_i,{\theta }_s,{\phi }_s)$. If only the specular plane is of interest, a convention $\mathrm{BRDF}=\mathrm{BRDF}({\theta }_i,{\theta }_s)$ is used in the literature, where to distinguish between the forward-scattering and backward-scattering parts, ${\theta }_s$ may take both positive and negative values. In our paper, ${\theta }_s>0\mathrm{deg}$ corresponds to the forward-scattering case (${\phi }_s=180\mathrm{deg}$) and ${\theta }_{s }<0\mathrm{deg}$ corresponds to the backward-scattering case (${\phi }_s=0\mathrm{deg}$).

5.

Phenomenological Model of the BRDF—Results

5.1.

In-Specular-Plane Modeling for the VIS Range (532 nm)

There is a substantial amount of information available in the literature describing the BRDF for the Acktar Magic Black™ coating in the VIS of wavelengths, in particular for 532 nm. We use these data to pre-validate our approach and fix values of some fitting parameters that are used later for processing UV measurements.

First, we digitally extracted BRDF measurements from Figs. 2 and 3 in Ref. 7, which are shown as dashed lines in Fig. 8. The data describe the behavior of the BRDF in the specular plane. Then, we selected a subset $S$ of the data containing a relatively small number of data points representing a similar set as the UV-range measurements described in Sec. 3. The data points from the subset $S$ are shown as circles in Fig. 8. The measurements described in Ref. 7 do not cover the case ${\theta }_i=75\mathrm{deg}$, thus, to make the VIS-range set more similar to the UV-range set from Sec. 3, we generated semi-synthetic data (shown as black circles in Fig. 8) for ${\theta }_i=75\mathrm{deg}$ by linear interpolation between the data sets for ${\theta }_i=60\mathrm{deg}$ (cyan dashed line) and ${\theta }_i=80\mathrm{deg}$ (magenta dashed line). The interpolation was done for ln(BRDF), i.e., for the natural logarithm of the BRDF, which corresponds to the interpolation in the logarithmic scale.

Fig. 8

BRDF measurement data for 532 nm (dashed lines), from which a limited subset $S$ of data points (circles) was selected for pre-validation of fitting procedures used later for $\sim 121.6\mathrm{nm}$. The solid lines show the best globally fitted model for 532 nm. The error bar for the mean error $3{\sigma }_E$ of the fit is shown in the middle of the plot. Different colors represent different incidence angles.

A number of fitting functions were tested on the 532-nm data and finally, the following form:

Eq. (7)

$$\begin{gathered} \mathrm{BRDF}({\theta }_i,{\theta }_s)={\mathrm{BRDF}}_0\exp [a_1{\theta }_i({\theta }_s-{\theta }_{s0})+a_2{\theta }_i({\theta }_s-{\theta }_{s0}{)}^{\alpha }+a_3{\theta }_i^2({\theta }_s-{\theta }_{s0}{)}^{\alpha }]\text{for}{\theta }_s\ge {\theta }_{s0}, \\ \mathrm{BRDF}({\theta }_i,{\theta }_s)={\mathrm{BRDF}}_0\text{for}{\theta }_s<{\theta }_{s0}, \end{gathered}$$

was found to provide a reasonable global fit with a relatively small number of parameters ${\mathrm{BRDF}}_0$, $\alpha$, ${\theta }_{s0}$, $a_1$, $a_2$, $a_3$. Note that since in this section, we consider only the variation of the BRDF in the specular plane, it is a function of the incidence angle ${\theta }_i$ and only one scattering angle ${\theta }_s$. Note also that the angle ${\theta }_{s0}$ should not be confused with the specular angle. The angle ${\theta }_{s0}$ is simply defined as an angle measured in the specular plane below which the BRDF is constant in our BRDF model of Eq. (7). For fixed values of parameters $\alpha$ and ${\theta }_{s0}$, the other parameters ${\mathrm{BRDF}}_0$, $a_1$, $a_2$, and $a_3$ can be found for ln(BRDF) using the linear regression method as expansion coefficients in the functional space $[1,{\theta }_i({\theta }_s-{\theta }_{s0}),{\theta }_i({\theta }_s-{\theta }_{s0}{)}^{\alpha },{\theta }_i^2({\theta }_s-{\theta }_{s0}{)}^{\alpha }]$. The linear regression method allows us to obtain a global fit, i.e., to fit measurements for different ${\theta }_i$ [i.e., all the pairs $({\theta }_i,{\theta }_s)$ represented by circles in Fig. 8] at once in one fitting step. Since the fitting procedure operates on ln(BRDF), it minimizes the mean squared error in the logarithmic scale. To verify a posteriori the goodness of fit, we define the mean error

$${\sigma }_E=\frac{1}{N_{\mathrm{S}}}\sum |\ln ({\mathrm{BRDF}}_{\text{fitted}})-\ln ({\mathrm{BRDF}}_{\text{measured}})|,$$

where the sum is computed over $N_S$ data points in the subset $S$ (i.e., over the circles in Fig. 8). Uncertainty propagation rules can be used to show that this error in the logarithmic scale can be considered as an approximation for the relative error

$${\sigma }_{\mathrm{RE}}=\frac{1}{N_S}\sum \frac{|{\mathrm{BRDF}}_{\text{fitted}}-{\mathrm{BRDF}}_{\text{measured}}|}{{\mathrm{BRDF}}_{\text{measured}}}$$

of the BRDF itself in the linear scale.

This fitting step is then repeated for different values of the parameters $\alpha$ and ${\theta }_{s0}$ to find the minimum of the total relative error

$${\sigma }_{\mathrm{TRE}}=\frac{1}{N}\sum \frac{|{\mathrm{BRDF}}_{\text{fitted}}-{\mathrm{BRDF}}_{\text{measured}}|}{{\mathrm{BRDF}}_{\text{measured}}},$$

where the sum is now computed over all of the $N$ available data points for 532 nm (so over the dashed lines in Fig. 8, not only over the circles). In this approach, the parameters $\alpha$ and ${\theta }_{s0}$ can be referred to as hyperparameters (a term used in machine learning for higher-order parameters that are used as fixed in the ongoing step of fitting), and ${\mathrm{BRDF}}_0$, $a_1$, $a_2$, and $a_3$ are regular parameters obtained from the linear regression.

The best globally fitted model for 532 nm is shown by solid lines in Fig. 8. It was obtained for the following values of parameters: $\alpha =4.5$, ${\theta }_{s0}=-12\mathrm{deg}$, ${\mathrm{BRDF}}_0=2.49\times {10}^{-3}{\mathrm{sr}}^{-1}$, $a_1=6.57\times {10}^{-4}{\mathrm{deg}}^{-2}$, $a_2=-2.90\times {10}^{-11}{\mathrm{deg}}^{-1-\alpha }$, and $a_3=9.76\times {10}^{-13}{\mathrm{deg}}^{-2-\alpha }$. The errors of the fit are as follows: ${\sigma }_E=4\%$ and ${\sigma }_{\mathrm{TRE}}=12.9\%$. Using these parameter values in Eq. (7), we obtain a fit that provides an approximation of the behavior of the BRDF for 532 nm in the specular plane as a function of the incidence angle $0\le {\theta }_i\le 90\mathrm{deg}$ and the scattering angle $-90\le {\theta }_s\le 90\mathrm{deg}$ over $\sim 4$ orders of magnitude of the BRDF variability.

5.2.

Modeling the BRDF in the Full Hemisphere for the VIS Range (532 nm)

The description of the BRDF behavior in the specular plane as discussed in Sec. 5.1 does not provide the full information needed for a general case of ray-tracing simulations or other computations aimed at estimating the radiation power scattered in an arbitrary direction. In general, the behavior out of the specular plane also needs to be known. For 532 nm, measurement data of this type are presented in Table 3 and Fig. 7 in Ref. 6. We use the data here to extend our model beyond the specular plane.

Table 3

Comparison of TIS values obtained from the model presented in this paper with actual measurements.

	TIS
	θi=5  deg	θi=30  deg	θi=50  deg	θi=70  deg
Measurements (Ref. 6)	0.8%	0.9%	1.4%	5.2%
Our model [Eqs. (7), (8), and (10)]	0.81%	1.09%	1.68%	4.0%

The BRDF presented in Fig. 8 is characterized by a strongly elevated forward-scattering side (${\theta }_s\ge 0\mathrm{deg}$) and a flat backward-scattering side (${\theta }_s<0\mathrm{deg}$). Figure 7 in Ref. 6 shows that in the full hemisphere, the in-specular-plane profile of the BRDF spreads out of the specular plane, but the width of the spreading is different for different incidence angle ${\theta }_i$. We try to capture this kind of behavior in our analytical model by defining ${\mathrm{BRDF}}^{-}$ and ${\mathrm{BRDF}}^{+}$ as branches of the BRDF corresponding to ${\theta }_s<0\mathrm{deg}$ and ${\theta }_s\ge 0\mathrm{deg}$ in the specular plane, respectively. A full-hemisphere dependence of the BRDF can be then obtained by using a weighted average:

Eq. (8)

$$\mathrm{BRDF}({\theta }_i,\theta ,\phi )=w{\mathrm{BRDF}}^{+}+(1-w){\mathrm{BRDF}}^{-},$$

where the weight

$$\begin{gathered} w=-\frac{|\phi -180|}{{\mathrm{\Delta }}_{\phi }}+1\text{for}|\phi -180|<{\mathrm{\Delta }}_{\phi } \\ w=0\text{for}|\phi -180|\ge {\mathrm{\Delta }}_{\phi } \end{gathered}$$

is a piecewise-linear function of the azimuthal angle $\phi$ in the spherical coordinates. The weight $w$ is conceived to take the value of 1 in the forward-scattering part (${\theta }_s\ge 0\mathrm{deg}$, which corresponds to $\phi =180\mathrm{deg}$) of the specular plane, then to linearly decrease and take the value of 0 when the azimuthal distance $|\phi -180|$ from the forward-scattering part of the specular plane becomes larger than ${\mathrm{\Delta }}_{\phi }$. In this formulation, a given direction in the scattering hemisphere is identified by the azimuthal angle $0\le \phi <360\mathrm{deg}$ and the polar angle $0\le \theta \le 90\mathrm{deg}$. For a given incidence angle ${\theta }_i$ and scattering direction $(\theta ,\phi )$, using Eq. (7) and the best-fit parameters from Sec. 5.1, we can compute ${\mathrm{BRDF}}^{-}=\mathrm{BRDF}({\theta }_i,{\theta }_s=-\theta )$, ${\mathrm{BRDF}}^{+}=\mathrm{BRDF}({\theta }_i,{\theta }_s=\theta )$, and the weight $w(\phi )$. This allows us to finally compute $\mathrm{BRDF}({\theta }_i,\theta ,\phi )$ from Eq. (8). For the weight $w(\phi )$, we need to know the value of the parameter ${\mathrm{\Delta }}_{\phi }$, which determines the width (in the azimuthal angle $\phi$ domain) of the spread of the in-specular-plane profile in the out-of-specular-plane direction. To account for the dependence of the spread on ${\theta }_i$ seen in Fig. 7 in Ref. 6, we propose the following function: ${\mathrm{\Delta }}_{\phi }=c_1{\theta }_i+c_2$ (where $c_1=-0.95$, $c_2=90.5\mathrm{deg}$), which gives isocontours in our Fig. 9 similar to those presented in Fig. 7 in Ref. 6.

Fig. 9

Full-hemisphere dependence of the BRDF model of Eq. (8) for the incidence angles (a) ${\theta }_i=5\mathrm{deg}$, (b) ${\theta }_i=30\mathrm{deg}$, (c) ${\theta }_i=50\mathrm{deg}$, and (d) ${\theta }_i=70\mathrm{deg}$. A polar-plot convention is used (a kind of view from the top), where the radius represents the spherical-coordinates polar angle $\theta$, and the angle around each plot represents the spherical-coordinates azimuthal angle $\phi$. The color encodes the values of the BRDF as shown in the colorbars; additionally, some isocontours are overplotted. The plot was prepared to be directly comparable with Fig. 7 in Ref. 6. The only difference is that in our paper the left-side sector ($90⩽\phi ⩽270\mathrm{deg}$) represents the forward-scattering side, and the right-side sector is the backward-scattering part, whereas in Fig. 7 in Ref. 6 the backward-scattering and forward-scattering sides are defined in the opposite way.

Having a description of the BRDF in the full hemisphere, we can compute the total integrated scatter (TIS) as

Eq. (9)

$$\mathrm{TIS}({\theta }_i)=\iint \mathrm{BRDF}({\theta }_i,\theta ,\phi )\cos (\theta )\sin (\theta )\mathrm{d}\theta \mathrm{d}\phi .$$

The quantity $0\le \mathrm{TIS}({\theta }_i)\le 1$ tells us what fraction of the incident radiation power is scattered irrespective of the direction; in other words, $1-\mathrm{TIS}({\theta }_i)$ is the fraction of the incident radiation power absorbed by the surface.

The BRDF measurements discussed in Refs. 6 and 7 display a folding (cf. Figs. 2 and 3 in Ref. 7 with our Fig. 8) at the highest values of the scattering angle ${\theta }_s$, which has not been taken into account in our considerations so far. This folding is commonly interpreted as a spurious measurement effect, which appears when the forward scattering is measured very close to the surface. However, since the folding is present in real measurements, for direct comparison of TIS resulting from our BRDF model with Table 3 in Ref. 6, we need to include this effect. The effect affects only the forward-scattering branch, thus, we can introduce it as an additional modulation, which effectively works for the highest values of the scattering angle ${\theta }_s$:

Eq. (10)

$${\mathrm{BRDF}}^{+}=\frac{\mathrm{BRDF}({\theta }_i,{\theta }_s=\theta )\{1-\mathrm{erf}[a({\theta }_s-{\theta }_{sf})]\}}{2},$$

where $a=0.35{\mathrm{deg}}^{-1}$ and ${\theta }_{sf}=87\mathrm{deg}$ are the parameters of the folding function, and erf() is the Gauss error function. Figure 10(a) shows a comparison of the folded and unfolded BRDF dependence in the specular plane as included in our approach.

Fig. 10

(a) Illustration of the folding effect that appears in actual BRDF measurements for the highest values of the scattering angle ${\theta }_s$. Solid lines show the unfolded BRDF, and dashed lines the BRDF folded using Eq. (10). (b) Dependence of the maximum BRDF value on the incidence angle. Red circles show the maximum measured values from Figs. 2 and 3 in Ref. 7, green circles show the maximum measured values from the bottom panels of Fig. 7 in Ref. 6, and blue crosses show the maximum values from our model in Eq. (7) with the folding by Eq. (10) included.

In Fig. 10(b), we also compare the maximum measured BRDF values from Refs. 6 and 7 with our model of Eq. (7) with folding by Eq. (10) included. Finally, the values of TIS presented in Table 3 in Ref. 6, i.e., scatterometer measurements in the full hemisphere for Acktar Magic Black™ coating, are compared in Table 3 with the integral of Eq. (9) for our full-hemisphere BRDF model of Eqs. (7), (8), and (10).

5.3.

BRDF Model for the UV Range (∼121.6 nm)

The approach, pre-validated in Secs. 5.1 and 5.2 on the 532-nm measurement data, is now applied to fit the reflectance measurements in the UV range described in Sec. 3. In the first step, Eq. (6) was applied to the raw-reflectance measurements from Tables 1 and 2 to obtain the corresponding BRDF values. For the experimental setup, the solid angle subtended by the camera aperture can be computed as ${\mathrm{\Omega }}_s=2\pi [1-\cos (\rho )]$, where $\rho =d/(2r)$ is the angular radius of the spherical cap related to ${\mathrm{\Omega }}_s$. The diameter $d$ of the aperture in front of the CCD camera and the aperture-sample distance $r$ were slightly different in series I ($d=7.2\mathrm{mm}$, $r=103.8\mathrm{mm}$) of the measurements and series II ($d=7.5\mathrm{mm}$, $r=110\mathrm{mm}$), which is accounted for in the computations presented in this section.

In Fig. 11, we show the BRDF values (circles) computed from the raw-reflectance measurements presented in Tables 1 and 2. In series I, the reflectances for different samples were measured only for the specular angle. For this reason, the measurements are shown in Fig. 11 as a vertical series of circles at ${\theta }_i={\theta }_s$ for ${\theta }_i=[\mathrm{18,45},\mathrm{75,80}]\mathrm{deg}$. Using the data points grouped by the incidence angle ${\theta }_i$, we computed inter-sample spreads (as the standard deviation) and plotted them in the form of $1\sigma$ error bars in Fig. 11, where the arrows connect a given spread with the corresponding group of data points. The measurement uncertainties estimated in Sec. 3 (the relative uncertainty of 6.5% for the reflectance, the absolute uncertainty of 1 deg for the angles ${\theta }_i$ and ${\theta }_s$) are also shown in the form of $3\sigma$ error bars in the middle of the plot.

Fig. 11

Comparison of BRDF measurement data for the UV regime $\sim 121.6\mathrm{nm}$ (circles) with the best globally fitted model (solid lines). The error bars for the mean error of the fit, measurement uncertainties, and inter-sample spreads are shown in the plot. Different colors represent different incidence angles.

The reflectance measurements converted to the BRDF are then used as an input for the fitting procedure. The same functional form and hyperparameter values $\alpha =4.5$ and ${\theta }_{s0}=-12\mathrm{deg}$ are used as those found in Sec. 5.1 (i.e., from the best fit for 532 nm). Similarly to Sec. 5.1, the rest of the parameters of the fit are obtained from the linear regression method, where measurements for different ${\theta }_i$, i.e., all $({\theta }_i,{\theta }_s)$ pairs represented by circles in Fig. 11, can be included in one global fit. The fitted parameter values for the UV regime ($\sim 121.6\mathrm{nm}$) are ${\mathrm{BRDF}}_0=1.14\times {10}^{-2}{\mathrm{sr}}^{-1}$, $a_1=4.02\times {10}^{-4}{\mathrm{deg}}^{-2}$, $a_2=-6.9\times {10}^{-11}{\mathrm{deg}}^{-1-\alpha }$, and $a_3=1.34\times {10}^{-12}{\mathrm{deg}}^{-2-\alpha }$. The error of the fit is ${\sigma }_E=6.4\%$. Using these parameter values in Eq. (7), we obtain a fit that provides an approximation for the behavior of the BRDF for $\sim 121.6\mathrm{nm}$ in the specular plane as a function of the incidence angle $0\le {\theta }_i\le 90\mathrm{deg}$ and scattering angle $-90\le {\theta }_s\le 90\mathrm{deg}$ over $\sim 3$ orders of magnitude of the BRDF variability, as shown in Fig. 11.

One should note that an additional measurement for ${\theta }_i={\theta }_s=60\mathrm{deg}$ (blue circle in Fig. 11) was not included in Tables 1 and 2. It was also not used in the fitting procedure described above. This measurement and the corresponding model curve (blue line) were added to Fig. 11 to verify if unseen-in-fitting measurements agree with the fitted model.

Currently, we have no access to out-of-specular-plane measurement data for $\sim 121.6\mathrm{nm}$, thus the only way of extending the model for the UV regime beyond the specular plane is by using the same formulation as discussed for 532 nm in Sec. 5.2. When computing ${\mathrm{BRDF}}^{+}$ and ${\mathrm{BRDF}}^{-}$, the best-fit-in-the-specular-plane parameter values given above in this section should be used, but for the rest of the procedure one should follow exactly the prescription given in Sec. 5.2. Due to strong similarities between the in-specular-plane BRDF for 532 and $\sim 121.6\mathrm{nm}$, this approach is considered as reasonable by the authors.

6.

Validation by Ray Tracing Simulations

Since the main goal of this paper is to provide a prescription for the BRDF in the full hemisphere for a UV wavelength $\sim 121.6\mathrm{nm}$ that can be used in computations, in this section, we present results of ray tracing simulations to show that the BRDF derived in Sec. 5 indeed gives similar raw reflectance as presented in Sec. 3.

The simulation is based on a simple forward ray tracing for the measurement setup for series II from Sec. 2. Figure 12 schematically shows the simulated setup.

Fig. 12

Schematic view of the simulation setup. The incident beam (shown in red) is formed by three apertures $A_1\text{-}{A}_3$ (green). Two types of rays scattered from the sample (black) are shown. Gray lines represent instances of rays scattered in random directions, and yellow lines show instances of scattered rays that were registered by the detector (blue). Between the sample and the detector, an additional aperture $A_D$ is placed.

A light source with a diameter of 0.5 mm is located 360 mm from the center of a sample of the size $40\times 40{\mathrm{mm}}^2$. Three apertures $A_1\text{-}{A}_3$ of diameters 4.8, 16, and 4.63 mm shaping the beam are placed at 32, 93, and 211 mm from the source, correspondingly. The apertures are centered at the optical axis between the centers of the source and the sample. The planes of the apertures are perpendicular to the axis. The beam from the source is divergent to make sure that the light spot at the last aperture is slightly larger than its diameter. The simulation neglects reflections/scattering from the planes containing the apertures, only rays within the apertures pass through further, and other rays are assumed to be perfectly absorbed. A consequence of this idealization is that effectively only the last aperture is important for the final shape of the beam in the simulations.

For the rays reaching the sample at a given incidence angle (set as a variable parameter in the simulation), the scattering angles are drawn from a three-dimensional probability distribution $p({\theta }_i,\theta ,\phi )$, which is equal to a discretized integrand of TIS from Eq. (9), i.e.,

In practice, in our simulation, a three-dimensional discrete grid of $90\times 90\times 360$ points was used to span the probability dependence on ${\theta }_i$, $\theta$, $\phi$. Drawing scattering directions from such a discrete grid leads to the appearance of some artificial patterns when small regions (in comparison with the probability-grid resolution of $1\times 1\times 1{\mathrm{deg}}^3$) are of interest. To suppress the artificial patterns, we used an additional uniform-distribution-based randomization of the directions of scattered rays within each cell.

At a distance of 110 mm from the center of the sample along a given detector-related direction $({\theta }_D,{\phi }_D)$, an aperture ${\mathrm{A}}_D$ with a diameter of 7.5 mm was placed, which subtended a small solid angle from the entire hemisphere in which the rays are scattered from the sample. Further, at a distance of 30 mm from the aperture, a rectangular detecting plane $25.6\times 6.375{\mathrm{mm}}^2$ was located. Rays traversing the plane were counted as detected.

In Fig. 13, the results of simulations for the incidence angle ${\theta }_i=80\mathrm{deg}$ and different detector angles are juxtaposed with CCD images registered in measurements discussed in Secs. 2 and 3. Simulations are noisier due to a higher Poisson noise caused by smaller count (ray) numbers, but a general agreement between the simulations and measurements is observed.

Fig. 13

Comparison of (a) measured CCD images registered in measurements discussed in Secs. 2 and 3 with (b) ray tracing simulations.

One can see several peculiarities in the measurements that are captured by the simulations. For example, there is a systematic evolution of the light spot from the top to the bottom panel (as the detector angle increases), where the boundaries of the light spot are strongly blurred in the top panel and sharper in the bottom panels. A regular gradient of the brightness of the light spot is seen, the left side being slightly brighter than the right side for the detector angles 65 deg to 80 deg. This gradient in the images is related to the steep gradient of the BRDF in the specular plane for large ${\theta }_i$ and ${\theta }_s$ values, as shown in Fig. 11. However, in the bottom panels (detector angle 85 deg), the gradient becomes reversed (right-hand side of the light spot is now brighter) with respect to the rest of the panels. At this angle, we are so close to the sample surface that the BRDF folding effect (discussed earlier and shown in the left panel of Fig. 10) kicks in. All these peculiarities are seen both in the measurement images and in the simulations.

Figure 14 shows a comparison of the measured and simulated raw reflectance for the entire series II of the measurements. The raw reflectance in the simulations was computed similarly as in the measurements, i.e., in the simulated images, a rectangle on the light spot [see Fig. 13(b)] was defined. The reflectance was computed as the ratio of the averages for the sample-scattered and reference images over the same rectangle. A general agreement between the measurements and simulations is observed in Fig. 14.

Fig. 14

Comparison of the measured (circles) and simulated (crosses) raw reflectance for series II of measurements in the UV regime discussed in Sec. 3. Different colors correspond to different incidence angles.

7.

Discussion and Conclusions

This paper presents the results of reflectance measurements for the Acktar Magic Black™ coating in the UV regime around $\sim 121.6\mathrm{nm}$ and the resulting BRDF function. A laser-plasma X-ray source based on the gas-puff target was used in combination with an Acton narrowband filter to generate radiation in a narrow range of wavelengths at the crossover between the FUV and EUV regimes. Six samples coated with Acktar Magic Black™ were studied in an optical setup with a back-illuminated CCD camera as the photon-flux measuring device. The measurements characterize comprehensively certain parts of the scattering hemisphere, including quasi-perpendicular (relative to the sample surface) and quasi-parallel cases. Tables 1 and 2 along with Figs. 4 and 6 summarize the raw reflectance measurements. Further in the paper, the measurements are shown to properly constrain the parameters of the derived phenomenological model of the BRDF.

Our methodology for the BRDF model was pre-validated in the VIS regime (532 nm), where a substantial body of data on the reflectance properties of the coating is available in the literature. Therefore, a BRDF model for 532 nm is also discussed here as a byproduct of our analyses for the UV regime. Fitted functions are demonstrated in Figs. 8 and 11 to provide a good approximation to the measurements, which is formally confirmed by small errors ${\sigma }_E=4\%$ for the VIS regime and ${\sigma }_E=6.4\%$ for the UV regime. An out-of-specular extension was proposed for the BRDF models based on measurement data for 532 nm available in the literature. Figure 9 and Table 3 in our paper show that the out-of-specular extension agrees with the measurements presented in Ref. 6. Therefore, a full-hemisphere BRDF model is proposed for both 532 and $\sim 121.6\mathrm{nm}$.

In Fig. 11, error bars for inter-sample spreads and measurement uncertainties were added to give the reader a glimpse of how the modeling error compares with other uncertainties. Note that the inter-sample spreads are shown as $1\sigma$, whereas the other errors are shown as $3\sigma$, thus our study suggests that the inter-sample variability of the reflectance is the largest source of uncertainty for the grazing scattering (i.e., ${\theta }_{i }>60\mathrm{deg}$).

In the numerical simulations presented in Sec. 6, the BRDF model for $\sim 121.6\mathrm{nm}$ was used to verify if the raw-reflectance measurements from Sec. 3 were correctly reproduced. This simulation also included images of the light spot created on an imaging detector. The simulations and measurements are in agreement, including certain peculiarities discussed in Sec. 6.

Taking into account the character of the variability of the BRDF for the coating under consideration, the values of the parameter ${\mathrm{BRDF}}_0$ [see Eq. (7)] can be used to get a rough estimate of the TIS for the perpendicular incidence. It is possible because for small incidence angles, the BRDF dependence on the scattering angle is relatively flat (see Figs. 8 and 11) and can be approximated by the Lambertian-scattering model (the same scattering probability in each direction) with $\mathrm{BRDF}=\mathrm{TIS}/\pi =\mathrm{const}$. Therefore, a rough estimation of $\mathrm{TIS}=\pi$ ${\mathrm{BRDF}}_0$ gives $\sim 0.8\%$ for 532 nm and $\sim 3.6\%$ for $\sim 121.6\mathrm{nm}$, which suggests a much higher reflection (or much lower absorption) of the UV radiation as compared to the VIS for the perpendicular incidence. For higher values of the incidence angle, the Lambertian-scattering model does not apply and the full dependence of the BRDF on the scattering angle must be considered. For ${\theta }_i=70\mathrm{deg}$, we obtained a TIS estimation of $\sim 5.8\%$ for $\sim 121.6\mathrm{nm}$, which is comparable with 5.2% obtained in the measurements for 532 nm (see Table 3). Therefore, the Acktar Magic Black™ seems to absorb less UV than VIS radiation, but only for the perpendicular incidence, whereas for higher incidence angles the absorption becomes comparable to that in the VIS range. This can be easily explained if we look at the maximum values in Figs. 2 and 5 and the plateau (corresponding to ${\mathrm{BRDF}}_0$) in the two plots. For the UV regime, the plateau is higher than for the VIS regime, but the maximum value (for the largest incidence and scattering angles) is lower in UV in comparison with VIS. In general, the results obtained in this paper agree with other publications, where the scattering of light from Acktar coatings was studied.^19–22

It is known that the morphology of a given surface is crucial for the absorption and scattering of light. The surface of coatings manufactured by Acktar can be described as cauliflower-surface-like morphology (see, e.g., Fig. 6 in Ref. 19) with irregular curds at many scales. Such structures may be expected to effectively scatter and trap photons in multiple-reflection configurations, which increases the probability of the absorption of light. This type of surface morphology can also strongly reduce the probability of backward scattering of light for small incidence angles. Microscopic mechanisms determining the absorbing properties of coatings are beyond the scope of our paper. However, such theoretical studies explaining the role of the surface morphology can be found elsewhere, e.g., in Ref. 23. The simulations were performed for different idealized models of surface roughness including V-type grooves, sinusoidal structure, and randomly distributed surface irregularities. This theoretical study suggests that for these types of surface morphology, the absorption ratio generally decreases with the incidence angle, which means that the reflectance characterized by the BRDF and TIS can be expected to increase. The results obtained in our paper (low normal-incidence reflection, higher grazing-incidence reflection) are qualitatively consistent with the theoretical results reported in Ref. 23.

The form of the fitting function in Eq. (7) assumes that for ${\theta }_s<{\theta }_{s0}$, the BRDF curves for different ${\theta }_i$ merge, but, apparently, more flexibility for ${\theta }_s<{\theta }_{s0}$ could lead to a better fit between the 532-nm measurements (represented by dashed lines) and the BRDF model (solid lines) in Fig. 8. However, one should bear in mind that the model is fitted to a subset $S$ of the points represented by the dashed lines. Therefore, the model must have significant rigidity, which is ensured by its functional form if a relatively sparse set of measurements is to be fitted.

In summary, we provide a full-hemisphere characterization of the reflectance properties of the Acktar Magic Black™ coating for $\sim 121.6$ and $\sim 532\mathrm{nm}$ as BRDF models fitted to actual measurements. The models are given as closed-form expressions, thus they can be used in manual estimates and incorporated into all kinds of ray-tracing software, including customized ray-tracing computations implemented from scratch or commercial software like, e.g., Zemax or FRED. The model for $\sim 121.6\mathrm{nm}$ is conceived to fill a gap in the literature, regarding the characterization of reflectance of the Acktar Magic Black™ coating in the proximity of the Lyman-α spectral line for hydrogen, which is of interest in satellite-instrument applications as the brightest spectral line in the solar spectrum in the UV range. It is important to emphasize the critical role of the raw-reflectance measurements discussed in Secs. 2 and 3, as constraining the parameters of the BRDF model derived for $\sim 121.6\mathrm{nm}$. As shown in Fig. 14, simulations based on the proposed BRDF model are in full agreement with the raw-reflectance measurements, which cross-validates the measurement and modeling aspects of our work.