Abstract

The bidirectional reflectance distribution function (BRDF) describes several features of a material surface. A one-shot imaging system is proposed here to obtain an in-plane color mapping of light direction corresponding to surface BRDF distribution. Measurement of a surface inclination angle distribution and detection of microstructure on material surfaces are shown to be attainable by the proposed imaging system.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In many manufacturing processes, a distribution of surface inclination angle defined by an angle between a normal vector of a local (infinitesimal) surface and that of a base plane is inspected for quality control of product surfaces. For example, in metal bonding processes such as welding, soldering, and brazing, a contact angle between a solid plate and a molten metal is an indicator of adhesion and needs to be measured for quality check. There are several methods to measure the contact angle with images captured by cameras [1,2]. When the contact angle is too small, however, it is difficult to measure the angle accurately. Moreover, it is often difficult to obtain an in-plane distribution of the contact angle. In addition to the surface inclination angle, a distribution of micrometer-scale defects on a product surface is also often inspected for quality control in many manufacturing processes. However, it is too small to be captured by conventional cameras.

The bidirectional reflectance distribution function (BRDF) describes several features of a material surface [35]. Actually, the surface inclination angle and the micrometer-scale defects affect the BRDF distribution. Thus, the in-plane BRDF distribution is considered to be available for measuring a surface inclination angle distribution and detecting a micrometer-scale defect distribution.

A one-shot imaging system is proposed here to obtain an in-plane color mapping of light direction corresponding to surface BRDF distribution. The imaging system is here called a one-shot BRDF imaging system or briefly one-shot BRDF. The remainder of this paper is organized as follows. First, a basic structure of the one-shot BRDF imaging system is described. Second, measurement of a surface inclination angle distribution of an aluminum cone with a height of 37 µm and a base circle diameter of 2 mm by means of the one-shot BRDF is demonstrated, which can be considered to be applicable to in-plane contact angle measurement. Third, an in-plane light direction distribution extracted from a surface BRDF of a white plastic plate is measured, which is considered to be applicable to inspection of micrometer-scale defects on the surface. Lastly, discussions and conclusions are described.

2. One-shot light direction color mapping imaging system

Figure 1 shows a schematic cross-sectional view of the one-shot BRDF. The one-shot BRDF consists mainly of an illumination optical system and an imaging optical system. The illumination optical system has an LED and a collimator lens that can convert the diverging light rays emitted from the LED to collimated light rays. The collimated light rays are reflected by a beam splitter and travel toward a material surface. The imaging optical system has an imaging lens and a multicolor filter that consists of concentric regions with different color filters [6,7]. The multicolor filter is placed at the focal plane of the imaging lens at a distance f from a principal plane of the imaging lens. The optical axis of the imaging lens is set to z - axis in a Cartesian coordinate system. The multicolor filter is parallel to the xy plane. The reflected light rays from the material surface pass through the beam splitter and are refracted by the imaging lens. The refracted light rays pass through the multicolor filter and are imaged on an image sensor. In this way, a light ray reflected by an object point in the material surface is imaged on an imaging point in the image sensor with its color selected depending on its direction. An angle θ of the light ray with respect to the optical axis passing through a certain point r in the multicolor filter can be derived on the basis of the geometrical optics under the paraxial approximation as

$$r = f\theta, $$
where the r denotes the radial distance of the point r from the optical axis.

 

Fig. 1. Schematic cross-sectional view of one-shot BRDF imaging system (one-shot BRDF). The one-shot BRDF consists mainly of an illumination optical system and an imaging optical system. The imaging optical system has an imaging lens and a multicolor filter that is placed at the focal plane of the imaging lens. The optical axis of the imaging lens is set to z - axis in a Cartesian coordinate system. The multicolor filter is parallel to the x - y plane.

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An incident light ray on a material surface with its direction parallel to the optical axis is reflected as shown in Fig. 2 in the plane of incidence. A normal vector of n on the local (infinitesimal) surface has an angle α to the optical axis. When the surface is sufficiently smooth, the reflection can be practically considered to be a specular reflection with an angle θ to the optical axis. The angle θ can be written on the basis of the law of regular reflection as

$$\theta = 2\alpha. $$
Using Eqs. (1) and (2), a resolution of the surface inclination angle measurement, Δα, can thus be derived by a radial interval Δr of adjacent regions of the multicolor filter as
$$\Delta \alpha = \frac{{\Delta r}}{{2f}}. $$

 

Fig. 2. Schematic view of plane of incidence. A normal vector of n on the local (infinitesimal) surface has an angle α to the optical axis. When the surface is sufficiently smooth, reflection can be considered to be a specular reflection with an angle θ to the optical axis.

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3. Results

3.1 Measurement of in-plane surface inclination angle distribution

Figure 3 shows an unprocessed top view image of a micrometer-size aluminium cone obtained by means of the one-shot BRDF. The image reveals qualitatively an in-plane surface inclination angle distribution. The multicolor filter of the one-shot BRDF is set to the one that consists of 32 graded hue color regions with outermost diameter of 22.0 mm where the red region is center as shown in Fig. 4. The inclination angle resolution in principle amounts to about 0.09 degrees calculated using Eq. (3) with the focal distance f of 105 mm and Δr of 0.34 mm. The cone fabricated by a machining process has a height of about 40 µm with a base circle diameter of about 2000 µm. Surface roughness of the cone is considered to be sufficiently small. The apex angle of the cone is around 176 degrees, which means that the base angle (surface inclination angle) is around 2 degrees. A color contour map is constructed by one-shot BRDF images of a polished aluminium mirror with various inclination angles to the optical axis at intervals of 0.25 degrees. Note that the interval of the color contour of 0.25 degrees is set to larger than the ideal interval of the multicolor filter of 0.09 degrees.

 

Fig. 3. An unprocessed image of a micrometer-size aluminium cone obtained by the one-shot BRDF imaging system. The cone fabricated by a machining process has a height of about 40 µm with a base circle diameter of about 2.0 mm. The apex angle of the cone is around 176 degrees, which means that the base angle (surface inclination angle) is around 2 degrees. A color contour map is constructed by the one-shot BRDF images of a mirror with various inclination angles at intervals of 0.25 degrees.

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Fig. 4. Multicolor filter used for measurement of surface inclination angle distribution. Outermost diameter is 22.0 mm. The filter consists of graded hue color regions where the red region is center.

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Figure 5 shows a three-dimensional measurement of the micrometer-size aluminium cone by means of white light interferometer system (ZYGO) [8]. The three-dimensional measurement is accurate enough for micrometer-size objects. However, it takes much more time than the one-shot BRDF imaging. Three different views of the cone are shown: (a) perspective view, (b) top view, and (c) side view. The height of the cone is 37.0 µm. At the middle height of the cone, the surface inclination angle is about 2.1 degrees. Both the top and the bottom of the cone has corner R shapes.

 

Fig. 5. Three-dimensional measurement of the micrometer-size aluminium cone. The measurement is performed by means of a white light interferometer system (ZYGO). Three different views of the cone are shown: (a) perspective view, (b) top view, and (c) side view. The height of the cone is 37.0 mm. At the middle height of the cone, the surface inclination angle is about 2.1 degrees. Both the top and the bottom of the cone have corner R shapes.

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In-plane continuous distribution of the surface inclination angle can be obtained by the one-shot BRDF using hue color interpolated mapping [7] as shown on the left-hand side in Fig. 6. On the right-hand side of the figure, a surface inclination angle distribution calculated from the white light interferometer measurement is shown. The inclination angle distributions are contoured with a color scale. A standard deviation of the one-shot BRDF image within a radius of 1000 µm is 0.225 degrees with respect to the white light interferometer measurement. There is good agreement between the two results, which validates that the one-shot BRDF can accurately measure the surface inclination angle distribution. An in-plane distribution of a contact angle should thus also be measurable by the one-shot BRDF imaging system.

 

Fig. 6. In-plane continuous distributions of surface inclination angle measured by (a) the one-shot BRDF imaging system and (b) the white light interferometer system. Inclination angle is contoured with a color scale. A standard deviation of the one-shot BRDF image within a radius of 1000 µm is 0.225 degrees with respect to the measurement by the white light interferometer system.

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3.2 Detection of micrometer-size defects on surface

A micrometer-size defect on a product surface scatters an incident light. A BRDF distribution can thus be used to detect such small defect. For example, a defect of a shallow scratch on a surface of a white plastic with depth of about 3 µm can be detected by the one-shot BRDF as shown in Fig. 7. The multicolor filter of the one-shot BRDF is set to the one that consists of two graded hue color regions (i.e., red and blue regions) with outermost diameter of 22.0 mm where the blue region is center with peripheral red region as shown in Fig. 8. Figure 7 shows (a) a conventional camera image of the shallow scratch with the depth of 3 µm on the white acrylic plate under ambient lighting, (b) the one-shot BRDF unprocessed image of the same object, (c) cross-sectional depth of the scratch measured by the white light interferometer system along p - p’ line that is indicated in (b). The depth of the scratch is plotted with respect to the distance along p - p’ line. It can be shown that the shallow scratch is difficult to be detected by the conventional camera. On the other hand, the one-shot BRDF can detect the shallow scratch. This is because the light incident on the scratch is scattered by the scratch with a slight tilt angle to the optical axis, which makes the scattered light pass through the red color region of the multicolor filter.

 

Fig. 7. (a) a conventional camera image of the shallow scratch with depth of 3 µm on the white acrylic plate under ambient lighting, (b) the one-shot BRDF unprocessed image of the same object, (c) cross-sectional depth of the scratch measured by the white light interferometer system along p - p’ line that is indicated in (b).

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Fig. 8. Multicolor filter used for detection of micrometer-size defects on surface. Outermost diameter is 22.0 mm. The filter consists of two graded hue color regions where the blue region is center.

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4. Discussion

The unprocessed image shown in Fig. 3 should be sufficient to reveal a qualitative distribution of the surface inclination angle. The in-plane continuous distributions of the surface inclination angle shown on the left-hand side in Fig. 6 is a quantitative distribution, which can be used for several numerical treatments.

The reflection is assumed to be specular in the measurement of the in-plane surface inclination angle distribution. The measurement error would be large when the reflection is diffusive. An intensity of the specular reflection, however, is often larger than that of the diffusive reflection. Thus, the error due to the diffusive reflection might be negligible to some extent.

Maximum surface inclination angle that can be measured is limited by diameters of an entrance pupil of the imaging lens and the multicolor filter. The diameter of the multicolor filter, however, can be made smaller when the focal length is set to smaller value on the basis of Eq. (1).

Light scattering reflection caused by the micrometer-size defect is proved to be detected by means of the one-shot BRDF. This means that the one-shot BRDF can identify whether reflection is specular or diffusive.

5. Conclusions

The one-shot BRDF imaging system is proposed here to obtain an in-plane color mapping of light direction extracted from surface BRDF. It is demonstrated that the imaging system is applicable both to the surface inclination angle measurement and to the micrometer-size defect inspection. The in-plane distribution of the surface inclination angle of the aluminium cone can be measured with the error of 0.225 degrees. The shallow scratch on the white plastic plate with 3 µm depth can be detected by the one-shot BRDF imaging system whereas it is difficult to be detected by a conventional camera under ambient lighting. These results prove the potential of the one-shot BRDF imaging system for measurements of surface inclination angle distributions and inspections of microstructures on material surfaces.

Disclosures

The authors declare no conflicts of interest.

References

1. G. Dutra, J. Canning, W. Padden, C. Martelli, and S. Dligatch, “Large area optical mapping of surface contact angle,” Opt. Express 25(18), 21127–21144 (2017). [CrossRef]  

2. D. Luo, L. Qian, L. Dong, P. Shao, Z. Yue, J. Wang, B. Shi, S. Wu, and Y. Qin, “Simultaneous measurement of liquid surface tension and contact angle by light reflection,” Opt. Express 27(12), 16703–16712 (2019). [CrossRef]  

3. B. K. P. Horn and R. W. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 18(11), 1770–1779 (1979). [CrossRef]  

4. S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Experimental analysis of bidirectional reflectance distribution function cross section conversion term in direction cosine space,” Opt. Lett. 40(11), 2445–2448 (2015). [CrossRef]  

5. F.E. Nicodemus, J.C. Richmond, and J.J. Hsia, “Geometrical considerations and nomenclature for reflectance,” Final Report National Bureau of Standards, October (1977).

6. H. Ohno and H. Kano, “Depth reconstruction with coaxial multi-wavelength aperture telecentric optical system,” Opt. Express 26(20), 25880–25891 (2018). [CrossRef]  

7. W. L. Howes, “Rainbow schlieren and its applications,” Appl. Opt. 23(14), 2449–2460 (1984). [CrossRef]  

8. ZYGO Corporation, “NewView 9000,” https://www.zygo.com/?/met/profilers/newview9000/

References

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  1. G. Dutra, J. Canning, W. Padden, C. Martelli, and S. Dligatch, “Large area optical mapping of surface contact angle,” Opt. Express 25(18), 21127–21144 (2017).
    [Crossref]
  2. D. Luo, L. Qian, L. Dong, P. Shao, Z. Yue, J. Wang, B. Shi, S. Wu, and Y. Qin, “Simultaneous measurement of liquid surface tension and contact angle by light reflection,” Opt. Express 27(12), 16703–16712 (2019).
    [Crossref]
  3. B. K. P. Horn and R. W. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 18(11), 1770–1779 (1979).
    [Crossref]
  4. S. D. Butler, S. E. Nauyoks, and M. A. Marciniak, “Experimental analysis of bidirectional reflectance distribution function cross section conversion term in direction cosine space,” Opt. Lett. 40(11), 2445–2448 (2015).
    [Crossref]
  5. F.E. Nicodemus, J.C. Richmond, and J.J. Hsia, “Geometrical considerations and nomenclature for reflectance,” Final Report National Bureau of Standards, October (1977).
  6. H. Ohno and H. Kano, “Depth reconstruction with coaxial multi-wavelength aperture telecentric optical system,” Opt. Express 26(20), 25880–25891 (2018).
    [Crossref]
  7. W. L. Howes, “Rainbow schlieren and its applications,” Appl. Opt. 23(14), 2449–2460 (1984).
    [Crossref]
  8. ZYGO Corporation, “NewView 9000,” https://www.zygo.com/?/met/profilers/newview9000/

2019 (1)

2018 (1)

2017 (1)

2015 (1)

1984 (1)

1979 (1)

Butler, S. D.

Canning, J.

Dligatch, S.

Dong, L.

Dutra, G.

Horn, B. K. P.

Howes, W. L.

Hsia, J.J.

F.E. Nicodemus, J.C. Richmond, and J.J. Hsia, “Geometrical considerations and nomenclature for reflectance,” Final Report National Bureau of Standards, October (1977).

Kano, H.

Luo, D.

Marciniak, M. A.

Martelli, C.

Nauyoks, S. E.

Nicodemus, F.E.

F.E. Nicodemus, J.C. Richmond, and J.J. Hsia, “Geometrical considerations and nomenclature for reflectance,” Final Report National Bureau of Standards, October (1977).

Ohno, H.

Padden, W.

Qian, L.

Qin, Y.

Richmond, J.C.

F.E. Nicodemus, J.C. Richmond, and J.J. Hsia, “Geometrical considerations and nomenclature for reflectance,” Final Report National Bureau of Standards, October (1977).

Shao, P.

Shi, B.

Sjoberg, R. W.

Wang, J.

Wu, S.

Yue, Z.

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Figures (8)

Fig. 1.
Fig. 1. Schematic cross-sectional view of one-shot BRDF imaging system (one-shot BRDF). The one-shot BRDF consists mainly of an illumination optical system and an imaging optical system. The imaging optical system has an imaging lens and a multicolor filter that is placed at the focal plane of the imaging lens. The optical axis of the imaging lens is set to z - axis in a Cartesian coordinate system. The multicolor filter is parallel to the x - y plane.
Fig. 2.
Fig. 2. Schematic view of plane of incidence. A normal vector of n on the local (infinitesimal) surface has an angle α to the optical axis. When the surface is sufficiently smooth, reflection can be considered to be a specular reflection with an angle θ to the optical axis.
Fig. 3.
Fig. 3. An unprocessed image of a micrometer-size aluminium cone obtained by the one-shot BRDF imaging system. The cone fabricated by a machining process has a height of about 40 µm with a base circle diameter of about 2.0 mm. The apex angle of the cone is around 176 degrees, which means that the base angle (surface inclination angle) is around 2 degrees. A color contour map is constructed by the one-shot BRDF images of a mirror with various inclination angles at intervals of 0.25 degrees.
Fig. 4.
Fig. 4. Multicolor filter used for measurement of surface inclination angle distribution. Outermost diameter is 22.0 mm. The filter consists of graded hue color regions where the red region is center.
Fig. 5.
Fig. 5. Three-dimensional measurement of the micrometer-size aluminium cone. The measurement is performed by means of a white light interferometer system (ZYGO). Three different views of the cone are shown: (a) perspective view, (b) top view, and (c) side view. The height of the cone is 37.0 mm. At the middle height of the cone, the surface inclination angle is about 2.1 degrees. Both the top and the bottom of the cone have corner R shapes.
Fig. 6.
Fig. 6. In-plane continuous distributions of surface inclination angle measured by (a) the one-shot BRDF imaging system and (b) the white light interferometer system. Inclination angle is contoured with a color scale. A standard deviation of the one-shot BRDF image within a radius of 1000 µm is 0.225 degrees with respect to the measurement by the white light interferometer system.
Fig. 7.
Fig. 7. (a) a conventional camera image of the shallow scratch with depth of 3 µm on the white acrylic plate under ambient lighting, (b) the one-shot BRDF unprocessed image of the same object, (c) cross-sectional depth of the scratch measured by the white light interferometer system along p - p’ line that is indicated in (b).
Fig. 8.
Fig. 8. Multicolor filter used for detection of micrometer-size defects on surface. Outermost diameter is 22.0 mm. The filter consists of two graded hue color regions where the blue region is center.

Equations (3)

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r = f θ ,
θ = 2 α .
Δ α = Δ r 2 f .

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