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Abstract
The measurement of bidirectional scattering distribution function (BSDF) and total transmittance (TT) is a complicated issue. We have proposed a screen image synthesis (SIS) BSDF meter that provides high-speed and precise measurement. However, the measurement of TT using integrating sphere cannot offer a precise measurement. Therefore, we propose a calibration method that uses SIS BSDF meter and responsivity of the integrating sphere. It brings us not only precise BSDF function but precise TT measurement.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Scatterometers are basic tools for many optical applications such as optical modelling [1], building design [2–3], gloss evaluation [4–5], color estimation [6–7], and scattering estimation [8–9]. A high-speed scatterometers can save time and cost. Conventional scatterometers detect data point-by-point, and the process is time consuming [6–7], [10–13]. Whole-field measurements using a scatterometer could take several hours and days, respectively. Therefore, we proposed the screen image synthesis (SIS) technology to apply image-based measurement [14–15]. Like other image-based measurement technologies in the field of the intensity distribution measurement and the bidirectional scattering distribution function (BSDF) measurement [16–26], it shows both high speed and high precision. In the latest version of SIS system, a screen was used to acquire light distribution, a rail-based system was designed to minimize the shadow effect of the automatic measurement system, and an image reconstruction algorithm with an adjustable rotational angle was developed [15]. The SIS system is even more superior than the snapshot technique that uses hemisphere screen [27–28], because it uses flat screen to avoid the cross-talk reflection noise between screen surfaces.
In case the measured BSDF is applied to evaluate the energy efficiency, the total transmittance (TT) is a key parameter, and it is always measured by an integrating sphere. Unfortunately, the measured TT usually has large error, because the response of the integrating sphere is not uniform, and the measured value always varies due to the light distribution of the sample [29–32], i.e., it varies with the BSDF of the sample. As a result, if the sample has asymmetric BSDF, no integrating sphere can produce a precise measurement without rotating the spherical shell. Thus, in this paper, we propose using the bidirectional transmission distribution function (BTDF) data measured by the SIS BSDF meter to calibrate the TT measurement. In the experiment, a 3M tape was used as a sample to show the feasibility of the proposed method, and two standard samples were used to demonstrate the measurement accuracy.
2. SIS system
Figure 1 shows the SIS system which consists of an image-acquiring system and a rotational system. The image-acquiring part includes a CMOS image sensor (IDS GmbH, Germany, UI-2280SE, 2248 × 2048 pixels, pixel size: 3.45µm), a screen and a lens (Edmund Optics, USA, Ultrahigh-resolution, focal length 5 mm). The screen is an antireflection glass plate with a white polymer sheet, its size is 601 mm × 496 mm × 3 mm. The optical property of the polymer sheet is approximated as Lambertian distribution. The sample is 50 cm away from the screen, and the screen is 75 cm away from the CMOS sensor. The sample can be rotated with a three-motor rotating system (Fig. 2). Two motors rotate the sample and the light source together to change the viewing angle of the screen and make 4-π solid angle of scattering light to be captured. θs and φs denote the zenith angle and the azimuth angle of the scattering light, respectively. The other motor controls the incident angle of the laser beam (${\theta _i}$). The camera-controlling process is combined with a high dynamic range (HDR) exposure process. After all the HDR images are captured, we fix intensity distribution with Cosine third law and combine all images to get the BSDF data, i.e., the radiance distribution over the irradiance of the incident light [15]. The cosine-corrected BSDF (CCBSDF) is, for various incident angle, the intensity distribution of the scattering light divided by the irradiance of the incident light. And the cosine-corrected BTDF (CCBTDF) is the transmission part of the CCBSDF.
Fig. 2. θs and φs denote the zenith angle and the azimuth angle of the scattering light, respectively. The incident angle of the laser beam (${\theta _i}$) was changed by rotating the sample.
3. Total transmittance calibration method
The spatial responsivity of photometry $R({{\theta_s},{\varphi_s}} )$ within an integrating sphere is measured in unit of watt and it is not uniform. The measured value depends on the direction of the incident light, and is roughly symmetrical along the normal direction of the detection port. Therefore, the TT value measured by integration sphere always varies with CCBTDF. To solve the problem, a calibration process of the integrating sphere must be included. First, we use a laser beam illuminating the integrating sphere with power L. The incident angle of the laser beam simulates the scattering angle $({{\theta_s},{\varphi_s}} )$ of a sample put in the entrance hole of the integrating sphere. The spatial response distribution function (SRDF) of the integrating sphere is expressed as a unitless function:
where, $R({{\theta_s},{\varphi_s},{t_0}} )$ are the values measured by the integrating sphere, t is the temporal axis, and ${t_0}$ is the calibration time. The absolute value of power $L(t )$ can’t be measured precisely. It changes with time, and the measured $R({{\theta_s},{\varphi_s},t} )\; $ change accordingly. So, we use a power meter PM(t) to calibrate the relative temporal variation caused by $L(t )$, i.e.:The measured ${P_{{\theta _i}{\varphi _i}}}({{t_0}} )$ is affected by the varying laser source, and it should be calibrated by the power meter PM(t), i.e.:
Therefore, Eq. (4) is calibrated by Eq. (2) and Eq. (5).
TT at a specific incident direction $({{\theta_i},{\varphi_i}} )$ is defined as the measurement result of ${I_{{\theta _i}{\varphi _i}}}({{\theta_s},{\varphi_s}} )$ integrated over the transmission hemisphere and divided by the incident power L,
We divide Eq. (7) by Eq. (6) to eliminate A and $L({{t_0}} )$, and then get the calibrated TT.
3. Experiment
Figure 3(a) shows the experimental setup for measuring $R({{\theta_s},{\varphi_s}} )$. The light source is a laser beam with wavelength 532 nm. It illuminates the mirror, and was reflected to the entrance hole of the integrating sphere. We shift and rotate the mirror to change the incident angle of the laser beam. The direction of the incident angle of the laser beam is the same as the zenith angle ${\theta _s}$ of a scattering sample. Because when we put a scattering sample in front of the entrance port, the normal direction of the scattering sample coincides with the normal direction of the entrance port. We put a beam splitter on the light path to direct part of laser beam to the power meter to calibrate the temporal variation of the laser light source. Figure 3(b) shows the inside of the integrating sphere. The diameter of the integrating sphere is 30.5 cm. The diameter of the baffle is 5 cm. The distance between the baffle and spherical shell is 3.5 cm. It caused a high response when the laser hit near the edge of the baffle. Otherwise, the laser hit the baffle or hit out of the baffle would have a similar response. The multiplier of the integrating sphere is about 9.7. We illuminate laser with power L in different angle ${\theta _s}$, and then make the unitless measurement $R({{\theta_s}} )/\textrm{PM}({{\theta_s}} )$, as shown in Fig. 3(c). Since the depolarization property of integrating sphere was proved, the dependence of polarization is not discussed in this paper [33–34]. In small angles, $R({{\theta_s}} )/\textrm{PM}({{\theta_s}} )$ does not have obvious variation, and it is because of the baffle which blocks small-angle-light in front of the detector in the integrating sphere. In around 7°, $R({{\theta_s}} )/\textrm{PM}({{\theta_s}} )$ is the maximum, and the value starts to decay as the angle is getting larger. Because $R({{\theta_s},{\varphi_s}} )$ is symmetrical along the normal axis of the detection port, and the detection port locates on the opposite side of the entrance hole of the integrating sphere. It means $R({{\theta_s},{\varphi_s}} )$ is also symmetrical along the normal direction of the entrance port. So, we only need to measure $R({{\theta_s}} )/\textrm{PM}({{\theta_s}} )$, and then duplicate it to generate $R({{\theta_s},{\varphi_s}} )/\textrm{PM}({{\theta_s}} )$, as shown in Fig. 3(d).
Fig. 3. (a) Experimental setup for the measurement of $R({{\theta_s},{\varphi_s}} )$; (b) the inside of the integrating sphere shows the layout of it; (c) The measured value of R/PM as a function of ${\theta _s}$; (d) $R({{\theta_s}} )/PM({{\theta_s}} )$ is expanded to $R({{\theta_s},{\varphi_s}} )/PM({{\theta_s}} )$ using the axis symmetry.
We test two standard samples to demonstrate the calibration accuracy. Table 1 shows that the measured haze, diffuse transmittance and TT of two standard samples, and they were measured by CMS/ITRI, which is the national metrology institute [35]. The measurement method was according to ISO 14782 [36]. The measured values of TT were 92.1% and 85.5%. We used the SIS-BSDF meter to measure the CCBTDF’ for the two standard samples with normal incident angle. We used the integrating sphere to measure ${P_{{\theta _i}}}$ of the two samples with ${\theta _i}$ equals to 0°. We brought the measured parameters as well as $R({{\theta_s},{\varphi_s}} )$ into Eq. (8), and the calibrated TT turned out to be 91.83% and 83.93%, respectively. Comparing to the standard values, the error of the calibrated values was smaller than 1.5%.
Table 1. The specs of test samples measured according to ISO 14782
Then, we used a 3M tape as the sample to show the feasibility of the proposed method. Because 3M tape is a homogeneous material, the BSDF and TT only change along ${\theta _i}$, and are independent of ${\varphi _i}$. Figure 4 shows the CCBSDF measured by the SIS-BSDF meter. Figure 4(a), 4(b), 4(c), and 4(d) show the CCBSDF with ${\theta _i}$ equals 0°, 30°, 50°, and 70°, respectively. In the TT calibration process, the parameters ${T_{{\theta _i}{\varphi _i}}}$ and ${P_{{\theta _i}{\varphi _i}}}$ in Eq. (8) was simplified to ${T_{{\theta _i}}}$ and ${P_{{\theta _i}}}$ because of the homogeneous of the 3M tape.
Fig. 4. 2D-BTDF and 2D-BTDF of 3M tape in different incident angle θi. (a) θi = 0$^\circ $; (b) θi = 30$^\circ $; (c) θi = 50$^\circ ;$ (d) θi = 70$^\circ $.
We put the 3M tape in front of the entrance pole of the integrating sphere, and change the incident angle of the laser beam to measure the integration transmission power ${P_{{\theta _i}}}$. The measured $R({{\theta_s},{\varphi_s}} )$, the measured $CCBTD{F^{\prime}}({{\theta_i},{\theta_s},{\varphi_s}} )$, and the measured ${P_{{\theta _i}}}$ were brought into Eq. (8) to calculate the calibrated TT (${T_{{\theta _i}}}$). Table 2 shows the calibrated TT. The value decreases as the incident angle increases. It is a reasonable result, because the Fresnel reflection increases with incident angle.
Table 2. The calibrated TT
4. Conclusion
The measurement of BSDF is usually accompanied by the measurement of TT. However, the measurement of TT using integrating sphere induces large errors. Because the SRDF of the integrating sphere is not uniform, the measurement result always varies with the light distribution of the sample. Therefore, we proposed a calibration process. It used the CCBTDF data measured by the SIS BSDF meter, the SRDF of the integrating sphere, and the integration transmission power to make calibration of TT. Due of the symmetry of the integrating sphere, we only need to measure 1D responsivity of the integrating sphere $R({{\theta_s}} )$. In the experiment, and two standard samples were used to demonstrate the measurement accuracy. The TT values were of the two standard samples were measured according to TT standard ISO 14782. In comparison, the error of calibration TT was smaller than 1.5%. A 3M tape was used as sample to show the feasibility of the proposed method. We measured the TT values using the proposed methods. It shows the measured TT decreases as the incident angle increases. It is a reasonable result due to the Fresnel reflection.
Funding
Ministry of Science and Technology, Taiwan (108-2221-E-008 -084 -MY3, 108-2221-E-008 -097 -MY3, 109-2221-E-008-087-MY2, 110-2218-E-008-006 -MBK).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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