Double transmission-mediums based geometric phase analysis for determining the two surface profiles of transparent object

https://doi.org/10.1016/j.optlaseng.2015.12.008Get rights and content

Highlights

  • A simple and convenient method to measure two-sided surface profiles is introduced.

  • Two transmission-mediums is used to modulated a large variation of phase values.

  • Experimental calculated results with nanometer-level sensitivity are presented.

  • This method allows a fast estimation of the geometry of a transparent specimen.

  • Phase calculation method a strong self-adaption ability to initial rotational errors.

Abstract

The accurate measurement for the surface profiles of transparent object is of significance for quality control in optical devices and precision instruments. Here, a double transmission-mediums based geometric phase analysis method has been developed to evaluate both the upper and lower surface profiles of transparent object. To do this, the tested transparent object is placed above a preprinted lattice pattern. When viewed from above with a CCD camera, any slope variations of the surfaces will lead to distortions of the transmission-lattice patterns. And when changing one side of object’s contact medium, the lattice virtual image with modulated phase is distorted once again. Combined with the derived relationship between phase variations of transmission-lattice patterns and out-of-plane heights of two surfaces, the double-sided surface profiles of transparent object can be reconstructed successfully. With this, the technique, which is verified experimentally, is demonstrated to be a feasible and reliable method. The advantage of this method is that it simplifies the setup and allows a fast estimation of the geometry of a transparent specimen. The double-sided profiles can be decoupled easily according to the big difference of refractive indexes between contact mediums. And the calculation accuracy can be guaranteed by the weighted average from four directions.

Introduction

The three-dimensional (3-D) shape measurement technology of transparent object, as an important means for obtaining surface morphology characteristics, is widely used in quality control, industrial test, engineering design, medical diagnosis and other fields [1], [2]. Therefore, accurate and precise measurement techniques have been of considerable interest, both in research and industry [3], [4]. And much more attention has been paid to the methods that aim to solve such a thorny problem, which can be roughly divided into point and surface measurement, contact and non-contact measurement [5], etc.

Surface measurement methods have the advantages of non-contact, rapid and full-field [6]. One of the methods is digital holography which retrieved three-dimensional information from the relation between phase and height of object [7], [8]. It has very high sensitivity and accuracy, but the measurement range is within the sub-millimeter or micrometer scale [9]. Vedel et al. [10] established the relationship between geometric morphology and polarized parameters and realized the shape reconstruction of optical element based on polarization of specular reflected light. But both direction of illuminant and diffuse reflection on the surface of tested object will influence the accuracy of measurement; Kutulakos et al. [11], [12] presented a theory to reconstruct refractive and specular 3d shape based on the concept of “triangulation of light”, where intersection of three linear light beams is used to locate a given point in three dimensional (3D) space. Shan et al. [13] proposed a method for reconstructing refractive height fields recorded against a known background or set of backgrounds from a single viewpoint. Wetzstein et al. [14] present an approach to reconstructing thin transparent surface by observing the distortion of light field background illumination. Liu et al. [15], [16], [17] successively proposed transmission-carrier techniques for measuring liquid surface deformation by analyzing the virtual images of transmission-carrier. Such methods are applicable for multi-scale deformation of transparent solid or liquid, whereas they are not suitable for measuring the double-side profiles of transparent object.

In many cases, it is necessary to measure the double-sided shape of transparent objects. For instance, many transparent components have different surface curvatures which are heavily used in optical devices and precision instruments. The measurement of optical surfaces can be done by deflectometric means, such as Moire, Ronchi, and Shark–Hartmann methods [18], [19]. Based on this technique, Horneber et al. [20] presented a Phase Measuring Deflectometry (PMD) method to measure the height and slope of lenses. Recent work by Lee [21], the transmission deflectometry technique for measuring surface profiles utilizing two illuminating wavelengths and liquid, is first proposed. It has lower cost and does not require coherent illumination. However, it is usually limited to that there is little difference between the refractive indexes of the two illuminating wavelengths in the same medium. So that the variation of optical path length of the two illuminating wavelengths is not obvious. And a small rotational error or tilt angle inevitably exists between the principal direction of the pre-set fringe and the horizontal (or the vertical), thus calculation errors will be introduced which are difficult to eliminate.

Extending these ideas, a double transmission-mediums technique based geometric phase analysis method is developed for simultaneously measuring the double-sided surface profiles of transparent object. The principle of improved method is mainly based on using two different of object’s contact medium when placing a periodic lattice underneath a tested transparent object. Obviously, the phase of transmission lattice pattern will be modulated twice in the two cases. Combined with the derived relationship between out-of-plane heights of two surfaces and phase variations of transmission-lattice patterns, the double-sided morphology of transparent object can be reconstructed accurately.

Section snippets

Measurement principle for double-sided profiles of transparent object

Suppose that a specifically periodic lattice pattern is placed underneath a transparent object. When viewed from above, the phase of the lattice virtual image will be modulated by the tested object, as shown in Fig. 1(a). And when changing one side of object׳s contact medium from air to other transparent mediums (see Fig. 1(b)), such as water, the lattice virtual image with modulated phase will be distorted again. These distortions of lattice images can be recorded by the CCD camera placed in

Experimental setup and procedure

To be able to measure double-sided profiles simultaneously, an experimental setup was designed as shown in Fig. 3. A positive meniscus lenses was used as tested object with different radius of curvatures. Devices used in this experiment included periodic lattice pattern, positive meniscus lenses, CCD camera, light illuminators and fixed lens mounts. The lattice pattern with a gray level distribution of square wave was generated by using MATLAB language program designed according to the

Conclusions

A simple and feasible method of double transmission-mediums based geometric phase analysis technique has been developed for measuring the surfaces profiles of transparent object. Using the two different contact mediums, the information of double surface profiles were decoupled successfully. The complete quantitative relationship between the profile changes of the double-sided surfaces and the phase variations of transmission-lattice images was systematically derived. An experiment for measuring

Acknowledgments

The authors are grateful to the financial support from the National Natural Science Foundation of China (Nos. 11232008, 11572041 and 11372037), Program for New Century Excellent Talents in University (NCET-12-0036).

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