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Ellipsometry by specular reflection has been reworked as a precise surface normal vector detection method for the geometrical shape study of a glossy object. When the object is illuminated by circularly polarized light, the surface normal vector defines the shape of the reflection polarization ellipse; the azimuth and ellipticity are determined by the angle of the incident plane and the angle of incidence, respectively. The tilt-ellipsometry principle of tilt detection is demonstrated experimentally with a metallic polygon and a cube sample.
©2013 Optical Society of America
1. Introduction
Optical methods for three-dimensional (3D) surface shape measurement and 3D display technology are being put to practical use in various fields owing to progress in image measurement and computer technologies. In general, when an object is observed by a microscope or telescope, the 2D projection image has lost depth information. Therefore, conventional triangulation methods, interferometry, fringe projection moiré topography, and confocal microscopy have been developed and widely applied for optical 3D shape measurement. These optical methods reconstruct the 3D shape using the principle of height contour mapping of a landscape based on height measurements.
In robotics applications, shape recognition by diffuse reflection polarization has been reported by many groups and utilizes polarimetry techniques to measure the degree of polarization [1–6]. In these reports, the illumination of unpolarized light is used, except in Koshikawa’s works [7, 8], and depolarization by rough surfaces is assumed, which results in tilt detection with a relatively low accuracy. Although robotics applications are limited to shape recognition, they show that the facet information of the object can be directly measured by using polarization, and the 3D shape can be successfully reconstructed in real-time. The polarization image detection necessary for real-time acquisition is available in commercial polarization cameras [9], and various applications being developed to extract various surface information by polarization [10].
In this study, we extended the polarimetric method used in robotics applications to ellipsometry under irradiation by totally polarized light. The slope and azimuthal angles of the reflection surface are determined by reflected polarization states measured precisely by ellipsometry. As illumination, circularly polarized light that is independent of the direction of the surface normal is used. The 3D shape can be reconstructed using the normal vector of the tangent plane at the reflection point. The readout method of the surface azimuth and slope angle information by ellipsometry for 3D shape measurement is called “tilt-ellipsometry” in this manuscript.
2. Principle of tilt-ellipsometry
Ellipsometry is widely utilized for investigating surfaces and optical properties of thin films [11]. Generally, an ellipsometer precisely measures reflected polarization at a known and fixed angle of incidence to determine the refractive index and layer thickness of the sample. Meanwhile, the tilt-ellipsometer derives shape information about azimuthal and slope angles at reflection points from measuring the reflected ellipse of the sample with known optical properties. Coordinate systems used in this manuscript are shown in Fig. 1.
Fig. 1 Coordinate systems of (a) reflection ellipse and (b) surface normal vector used in this manuscript. θ and ε indicate azimuth and ellipticity angles of reflection polarization, respectively. α and β show azimuth and slope angles of surface normal vector, respectively. Right circular polarization light is illuminated at incident angle ϕi and the polarized light is reflected at the reflection angle ϕr.
Figure 2 shows a geometrical diagram of specular reflection from a sphere sample as observed from the z-direction. Each ray of circular polarization illumination arrives from a different direction, satisfying the law of reflection with the reflected ray direction fixed to z. In addition, the surface normal should be included in the plane of incidence by the law of reflection. Thus, the two angles defining the surface normal are the angle ϕ to the z-direction and the azimuth α from the x-direction, which appear as the ellipticity and azimuth of the ellipse of each reflection polarization. In Fig. 2, the plane of incidence at reflection points A, B, and C is consistent with the xz-plane, while only the angles of incidence are different from each other. At point D, the incident angle is the same as point A, although the plane of incidence is coincident with the xz-plane, which is rotated on the z-axis at an angle of α. As the plane of incidence can be defined by the incident and reflected light, the plane of incidence inevitably contains the normal vector of the tangent plane at the reflection point. It is clear that the surface normal can be determined from the incident angle ϕ and the azimuth α of the incident plane. Therefore, the 3D shape can be obtained by integral operations of surface normal vectors.
Fig. 2 Tilt-ellipsometry concept of specular reflection shown by a spherical object observed from the z-direction. ϕn and α indicate incident angles at point n and the azimuthal angle of the incident plane, respectively.
In conventional ellipsometry measurements, linearly polarized light is used as illumination. The origin of the azimuthal angle of linear polarization is defined by the p-direction of the plane of incidence as the reference frame. On the other hand, in the case of tilt-ellipsometry, an object is illuminated by right or left circular polarization, which does not depend on the reference frame of the coordinate. Circular polarization ensures the rotational symmetry in any reference frame at the reflection.
Figure 3 shows angle of incidence variations of relative complex amplitude reflectances under right circular polarization (RCP) illumination plotted in a complex plane. Complex refractive indices n-ik of 1.5, 2.90-3.07i, 1.36-7.59i, and 0.385-3.41i were used for calculations assuming glass, Fe, Al, and Au surfaces at a wavelength of 632.8 nm. All materials reflect left and right circular ellipses at incident angles of 0° and 90°, respectively. In a transparent material such as glass, the phase difference between the p- and s-components of polarization by reflection changes 180° to 0° around the Brewster angle, and hence, the azimuthal angle of the reflected polarization ellipse is perpendicular to the plane of incidence. The handedness of the reflection ellipse changes at the Brewster angle at which the linearly polarized light is reflected. In absorbing materials such as metal, although variations in the handedness of reflected polarization are as same as those of transparent materials, the azimuth of reflection ellipses lean from the incident plane. This reflection characteristic is predicted by calculating the relative complex amplitude reflectance.
Fig. 3 Angle of incidence variations of relative complex amplitude reflectance under right circular polarization illumination plotted in a complex plane. Complex refractive indices n-ik of 1.5, 2.90-3.07i, 1.36-7.59i, and 0.385-3.41i were used for the calculation assuming glass, Fe, Al, and Au surfaces for a wavelength of 632.8 nm.
As mentioned above, the ellipticity angle of the reflection polarization varies with the angle of incidence, and hence, the incident angle of an object with known optical properties can be derived using the ellipticity angle. Figure 4 shows angle of incidence variations for ellipticity angles under RCP illumination for glass, water, Fe, Al, Au, and Si surfaces at a wavelength of 632.8 nm. The angle of incidence and ellipticity angle show a one-to-one correspondence with all materials. Therefore, the slope angle of a surface normal vector can be obtained from the measured ellipticity angle.
Fig. 4 Angle of incidence variations of ellipticity angle ε under right circular polarization for glass, water, Fe, Al, Au, and Si surfaces for a wavelength of 632.8 nm.
At point D shown in Fig. 2, the ellipticity angle of the reflection polarization is coincident with that at point A where the incident angle is the same as point D. However, the azimuth of the reflection ellipse rotated on the z-axis at angle α is observed. As the azimuthal angle of the reflection ellipse can be derived by calculations from optical properties at the reflection point, the azimuthal angle of an incident plane coincident with the azimuth of the surface normal vector can be determined by the azimuth of the reflection polarization. Especially in transparent materials, the azimuth of the surface normal vector can be directly determined by the azimuth of the reflection ellipse because the azimuth of reflection polarization is perpendicular to the incident plane. Figure 5 shows the reflection polarization map, observed from the z-direction, of expected specular surface reflections of a metallic sphere under RCP illumination. The shading in the middle shows the region of left-handed polarization.
Fig. 5 Reflection polarization map (observed from z-direction) of expected specular surface reflection of a metallic sphere under right circular polarization illumination. Shading in the middle shows the region of left-handed polarization.
3. Mueller calculation of tilt-ellipsometry
In this section, the difference between 3D shape measurements under RCP illumination and unpolarized illumination used in robotics applications is shown using Mueller matrices.
When the Stokes vector of incident light Ein is reflected at a specular surface, the Stokes vector of reflection light Eout can be calculated as follows:
T and S show Mueller matrices for a rotator at angle α and specular object, respectively. T and S can be described as where Ψ and Δ in Eq. (3) are ellipsometric parameters at the reflection surface. Therefore, the reflection polarization Eout under RCP illumination of Ein = (1 0 0 1)T can be expressed by Eq. (1) asMeanwhile, the reflection polarization Eout under unpolarized illumination of Ein = (1 0 0 0)T can be obtained byUnder unpolarized illumination, information of the relative phase Δ of p- and s-components is lost by reflection, as expressed in Eq. (5). Therefore, the slope angle of the surface normal vector must be determined only by the angle of incidence variation in the relative amplitude attenuation Ψ of the p- and s-components. In the case of absorbing materials, the variation in the relative amplitude attenuation is small, as shown in Fig. 3, whereas a large change in Ψ is observed in transparent materials. This is one of the reasons that the measurement object is limited to transparent materials under unpolarized illumination. In tilt-ellipsometry, under circularly polarized illumination, slope and azimuth angles can be determined using angle of incidence variations in both the relative amplitude attenuation Ψ and the relative phase Δ of p- and s-components shown in Eq. (4).4. Experiments for the proof-of-concept
For the proof of this concept, tilt-ellipsometry measurements were carried out. Figure 6 shows the experimental setup schematically. To illuminate the sample from all around, a dome-type illuminator was used. The light is guided by an optical fiber from a Halogen lamp and is emitted at the bottom of the dome. The light reflected at the inner dome passes through the rolled RCP film inserted into the dome to illuminate sample uniformly. The polarization image of the sample for a wavelength of 632.8 nm is measured by a rotating analyzer method with a CCD camera. Photographs of a thick Au-coated polygon and stainless steel cube sample used for measurements are shown in Fig. 7.
Fig. 6 Experimental setup for proof-of-concept experiments. Circular polarization illumination was made by a commercially available circular polarizing film rolled and inserted in the dome-type illuminator. A rotating analyzer method was used for reflection polarization measurements with a Polaroid sheet. An interference filter for a wavelength of 632.8 nm was used.
Fig. 7 Photographs of Au-coated polygon and stainless steel cube for tilt-ellipsometry measurements.
Figures 8(a) and 8(b) shows measured polarization maps of azimuth of the Au-coated polygon and stainless steel cube, respectively, and Figs. 8(c) and 8(d) show ellipticity angles of the Au-coated polygon and stainless steel cube, respectively. Imperfections in the RCP illumination in the left middle area in Figs. 8(a) and 8(c) were caused by the connection of the rolled polarizing film shown in Fig. 5. Although handedness of the ellipticity angle cannot be measured by the rotating analyzer method, we determined the handedness by calculating the angle of incidence variation of ellipticity angles referring to the complex refractive indices of 0.385-3.41i for the Au polygon and 2.90-3.07i for the cube. As each facet of both the polygon and the cube has the same azimuthal angle, the distribution of azimuth shows one color in each facet. Furthermore, the ellipse with the same ellipticity angle was observed in each facet because each facet has the same slope angle.
Fig. 8 (a) and (b) Measured polarization maps of azimuth of Au-coated polygon and stainless steel cube, respectively; (c) and (d) Ellipticity angle of Au-coated polygon and stainless steel cube, respectively. Imperfections in the right circular polarization illumination in the left middle area in (a) and (c) were caused by the connection of the rolled polarizing film shown in Fig. 5.
Using the measured azimuthal and ellipticity angle distributions, 3D shapes were successfully reconstructed as shown in Fig. 9. 3D reconstruction methods such as shape-from-shading and/or photometric stereo methods [12–15] are applicable in our method. The azimuth of the surface normal vector can be directly obtained from the measured azimuthal angle of the reflection ellipse. On the other hand, the incident angle is derived by angle of incidence variations in the ellipticity angle shown in Fig. 4 to extract the slope angle of the surface normal vector. In the reconstruction, convex shapes were assumed to identify the azimuth of α and α + π. By our proposed method, 3D shapes with steep edges and apexes, which are difficult to measure by interferometry, were accurately restored.
Fig. 9 (a) 3D shapes of a polygon and (b) cube reconstructed using the measured reflection ellipse shown in Fig. 8.
5. Summary
We proposed and proved a new concept for 3D shape measurement of specular objects by tilt-ellipsometry. Under circularly polarized illumination, the azimuth and slope angles of the facet could be derived by determining the parameters of the reflected polarization ellipse. To demonstrate our concept, 3D shapes of a Au-coated polygon and a stainless steel cube were successfully reconstructed by determining ellipsometric parameters under right circular polarization illumination. Compared with the measurement of the degree of polarization used in robotics applications, tilt-ellipsometry can obtain the surface normal vector from two measured ellipsometric parameters. Therefore, a high precision achieved by ellipsometry in various applications is expected for 3D shape measurement. Our method utilizing a recently developed, real-time polarimetric technique is promising and has practical potential.
Acknowledgment
This research was partially supported by a Grant-in-Aid for Young Scientists (B), No. 23760042 from the Japan Society for the Promotion of Science (JSPS).
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