A quantitative phase shifting differential interference contrast (PS-DIC) shearing interferometer is adopted to measure the profile of transparent specimen with inclined surface. The effects of the incline angle on DIC measurement accuracy were studied. The optical model of the test system was constructed and the measurement of surface with various incline angles ranging from 5° to 60° was simulated. The experiments validate the simulation model and show the feasibility of profile reconstruction of inclined structure. It is interested to find that even with an inclined angle of 15°, unwrapping technique is required to make the measurement more accurate. In addition, the measurement can be further improved by taking into account the effects of the change in shear distance on the optical path difference. This study provides useful information that should be considered for complex geometry measurement with quantitative DIC technique.
©2012 Optical Society of America
Transparent components such as prisms, light guide plates, thin-film transistors, and micro-lens array are frequently employed due to the rapid development of optoelectronic industry. As these components are fabricated with complex surface structure to fulfill the need for function and to improve optical performance [1,2], there is an increasing demand for measuring the profile of these transparent components to ensure the product quality. A variety of profilometery techniques may be adopted for the transparent object measurement. These techniques can be categorized into contact type (such as stylus profilometer, and atomic force microscope ) and non-contact type (such as triangulation laser scanner, structure light inspection system, scanning electron microscope, confocal microscope, and interferometer [4–7]). Non-contact type instruments are more suitable for industrial applications because of their higher processing speed and less chance of scratching the examined products.
One such non-contact type technique is differential interference contrast (DIC), which is a beam-shearing interference system in which the reference beam sheared by a minuscule distance. The DIC technique produces interference images that represent the gradients of optical path difference between the two sheared beams. The technique may properly be adopted in transparent specimen measurement with the advantages of high spatial resolution, vibration resistance, and outstanding contrast.
Although the DIC technique was originally used to enhance the image contrast of transparent specimens to its environment only, the capability of quantitative measurement is become crucial in industrial application lately. Lessor et al. [8,9] proposed a quantitative surface topography determination method of reflected DIC. Shimada et al.  developed a reflected DIC microscope system with Nomarski prism movement for phase shifting. Cogswell et al.  proposed a geometric phase shifting technique that can convert conventional transmitted DIC microscope into quantitative mode with the use of de Senarmont compensator and then King et al.  increased the accuracy of measurement by adopting post-calibration process.
When using the quantitative DIC technique, there is few study concentrated their work on the profile measurement of transparent object with inclined surface. Lessor et al.  determined metallic mirror surface topography by using the Nomarksi DIC microscope and developed an optical model for the examination of sloped surfaces. However, the proposed method was only valid for slopes that are small than 3.781° . Ishiwata et al.  proposed a retardation-modulated DIC (RM-DIC) microscope to quantitatively measure the phase difference distribution for microstructures. They using a blazed grating approximation by replacing the light reflected from a gentle slope with diffracted light from a blazed grating. However, no experiment was conducted to verify their approximation.
Theoretically, error is introduced when inclined surface is measured since the light paths will no longer follow the assumed ideal paths and recombined into one interfered light beam any more. In our previous study , transparent objects with sub-micron step height structure were measured with phase shifting DIC (PS-DIC) technique. The experimental results show the feasibility of the technique for step height measurement. It is of interest to further study the feasibility of using the technique for transparent specimens with complex geometry. In this paper, the light propagation mechanisms in transparent specimens with inclined surface are studied and the profile measurement error for such cases is also analyzed. These results can be used as a stepping stone to study the effects of complex geometry on DIC images and profilometery reconstruction.
2. Test system and system simulations
Figure 1 illustrates the schematic diagram of transmitted PS-DIC setup for profile measurement of a prism and the sheared light beams near the conjugate image plane. The tested prism was set with its flat plane facing the objective lens such that prism with a large wedge angle can still be measured without total internal reflection (TIR) of light. In the optical system, DIC prisms (Wollaston prism) were used as the key component to spatially shear and recombine light beams to form a lateral shearing interferometer.
In order to reveal the effects of complex geometry on DIC measurement, the effects of specimen shapes on the prorogation direction of light and DIC images were analyzed. As shown in Fig. 2 , the light will be deflected when passing through a prism structure while it will maintain its original direction when passing through a step height structure.
Considering the sheared light beams passing through a prism, the incident lights deflect twice. As shown in Fig. 3 , the deflection of light can be determined by Snell’s law:
For a prism specimen, the optical path difference, OPD, measured with transmitted PS-DIC system can be written as:Fig. 3. Since the relationships between these parameters can be derived from sine theorem:Fig. 3.
The optical path difference can be expressed as:
As shown in Fig. 3 that the shear distance will change when light passing through a prism, the ratio of change of shear distance can be calculated as following:
The resulting optical path difference becomes:
It should be noticed that the optical path difference is only affected by the incident angle of light for a certain DIC system (constant ∆s) with a test prism (constant n). Figure 4 shows the effects of the incident angle on optical path difference and corresponding phase difference with a shear distance of 3μm and a prism refractive index of 1.519. As shown in Fig. 4, it was interesting to find that the optical path difference is larger than λ/2 even when a small wedge angle of 15° was measured. It was indicated that unwrapping technique will be needed to reconstruct the profile of prism.
It was also observed that the optical path difference is roughly proportional to the wedge angle of prism when both the wedge angle of prism and the incident angle of light are small. However, when the wedge angle of prism is large, the optical path difference is strongly affected by the incident angle of light.
Figure 5 explains how a PS-DIC image is formed with the propagation of light waves. In this system, the monochromatic partially coherence plane wave W is polarized by the polarizer. When the plane wave pass through first DIC prism, the incident wave is sheared into a pair of waves, W1 and W2, that have a minuscule shear distance Δs in space and their polarization states are orthogonal to each other. Then the two waves pass through condenser, specimen, objective lens, and the second DIC prism. The second DIC prism recombines these two waves with an adjustable phase shifting β. The recombined light wave, W2-W1, is interfered by an analyzer. Therefore, the sheared light waves which contain the profile information of specimen can be collected with the DIC system by measuring the intensity of the interference image.
The intensity of the interference image, I, can be written as:
The quantitative phase difference corresponding to the optical path difference of specimen can be estimated by acquisitions series of images with various phase shifting . In this study, four-step phase shifting algorithm was adopted to estimate the phase gradient information of specimen . Four raw DIC images with phase shifting of 0, π/2, π, and 3π/2 in two orthogonal shear directions were collected. The non-iterative Modified Fourier Phase Integration (MFPI)  method was then used to reconstruct the profile of specimen.
The light propagation mechanisms in the transmitted PS-DIC system were simulated with commercial software, Advanced Systems Analysis Program (ASAP®). Figure 6 shows the schematic diagram of the system simulated. The simulation model was consisted of three main sections: illumination subsystem, imaging subsystem and specimen.
The illumination subsystem consisted of a light source, a rotatable polarizer, a quarter-wave plate (QWP), a collection lens, an aperture stop, a field stop, Wollaston prism, and a condenser lens. The wavelength of the light source was assumed to be 550 nm. The rotatable polarizer was used to adjust the phase shifting between the sheared beams. The optical axis of rotatable polarizer was aligned with the fast axis of QWP, in the beginning. The orientation of the fast axis of the QWP with respect to the x axis shown in Fig. 1 was + 45°. A collecting lens was then used to collimate the polarized light. The aperture stop and the field stop were located at conjugate plane of the light source and the detector, respectively. Then Wollaston prism was put in the front focal plane of the condenser. The condenser with a NA of 0.30 illuminated the examined specimen. It should be noticed that through the illumination subsystem, Köhler illumination was reached and the shear distance was 3μm.
The imaging subsystem consisted of an objective lens, a Wollaston prism, an analyzer, an imaging lens, and a detector. The objective lens with a magnification of 20X was used. The second Wollaston prism was located at the back focal plane of the objective lens. The orientation of the optical axis of the analyzer with respect to the x axis was −45°.
3. Simulation results and discussions
Series of simulations were conducted to study the effects of wedge angle on transmitted PS-DIC images and reconstructed profiles. All the parameters used in the simulation are listed in Table 1 .
Figure 7 shows a typical simulation result of the transmitted PS-DIC images of 45° prism. Figure 8 shows the calculated optical path difference with/without unwrapping. Figure 9 shows topography reconstruction of the prism and comparison of profiles reconstructed from the optical path difference with/without unwrapping and compensation, respectively. As shown in Fig. 9, reconstructed profiles of the prism with unwrapping and compensation were much more accurate than the raw data.
Since the reconstructed profile of the prism is an incline surface, it would be much easier to compare the angle estimated based on these profiles with the specified wedge angle to know the accuracy of the DIC approach. Figure 10 shows the comparison of wedge angles estimated for all simulated cases with/without unwrapping technique, and with/without compensation for shear change respectively. After examining these results, the following observations can be made:
3.1 With/without unwrapping for optical path difference
Without unwrapping, the wedge angles estimated were quite different from those specified. When the optical path difference between sheared beams is larger than λ/2, unwrapping is required to reach a more accurate result. It is interested to find that even with an inclined angle of 15°, unwrapping technique is required to make the measurement more accurate. This is so because the corresponding phase difference between two split beams is already larger than π even with a wedge angle of 15°.
In this study, two-dimensional phase unwrapping technique was adopted. The discontinued phase difference values were unwrapped by adding multiple times of 2π . The phase was unwrapped vertically along the leftmost column of the phase difference image first. Each row was then horizontally unwrapped using the respective leftmost unwrapped phase as the initial value for that row. The comparison of phase difference of raw data and the unwrapped phase difference result is shown in Fig. 8. The unwrapped phase difference was then used for phase reconstruction.
As shown in Fig. 10, it was quite obvious that, the wedge angles estimated of all simulated cases with unwrapping technique are much closer to the angle specified than those without unwrapping technique.
3.2 With/without compensation for shear change
Although the accuracy of estimation of wedge angle can be greatly improved by the unwrapping technique, there are still difference between the angle estimated and that specified, especially when a larger wedge angle is examined as shown in Fig. 10. This is so since that the shear distance has changed when light passing through a prism. Thus the measured optical path difference between the two sheared beams should be also compensated. The ratio of change in shear distance and the effects on optical path difference had been derived in Eq. (6) and Eq. (7). By taking into account the effects of change in shear distance on the optical path difference, the accuracy can be further improved as shown in Fig. 10.
3.3 Angle measurement limit
It is also found that the measurement of prisms with wedge angle larger than 45° was impossible. This is so because the light deflected after the prism and could not be collected into imaging subsystem due to large refracted angle, derived by Eq. (2). It is likely that the dominating factors are the numerical aperture (NA) of objective lens in the imaging subsystem. The NA of objective lens limits the maximum cone angle that can be collected into the imaging subsystem. When a 20X (NA = 0.45) objective lens was used in air, the half-angle of the maximum cone of light was about 26.744þ. These limitations might restrict the maximum measurable wedge angle of the system. Based on the model simulated, the maximum measurable wedge angle is about 46þ for the system simulated. Over these ranges, several errors may be introduced into the system, and no compensation technique had so far been proposed.
3.4 Blur area
It was found that measurement errors may occur especially around the edge of the wedge prism. As shown in Fig. 11 , blur area which is corresponding to the edge of the wedge prism (about 6 μm) was observed for each DIC image. Theoretically, when one of the two sheared beams passing through the prism and was deflected while the other one which did not go through the prism and maintain its direction, these sheared beams could not interfere with each other and resulted in the blur area. As a result, miscalculation may occur in this area for profile reconstruction.
In summary, the simulation results indicated that unwrapping and compensation are needed for most cases studied and the NA of objective lens should also be taken into consideration to identify the limit of the current approach. With unwrapping for optical path difference, combining with compensation for shear change, the accuracy can be further improved.
4. Verification experiments
A series of experiments were conducted to verify the simulation results and to test the measurement accuracy. The experimental setup of transmitted PS-DIC measurement system is shown in Fig. 12 . A halogen lamp with a filter (λ = 550nm, FWHM = 8nm) was used as the light source for this system. A de Senarmont compensator  was used to polarize the light and adjust the phase shift between the sheared beams. The de Senarmont compensator composed of a polarizer mounted on the rotation stage, an analyzer, and a quarter-wave plate. A pair of DIC prisms was used to shear and recombine the light beams. The condenser lens and objective lens was used to collect the light for illumination and imaging. A telecentric lens and a CCD detector were used to image the interference image and a personal computer was used for data processing. The specimens were prisms with wedge angles of 30þ and 45þ. The specifications of components used in the experiments are listed in Table 2 and that of the tested specimens are listed in Table 3 .
Comparison between corresponding phase difference derived from simulations and that from experiments was made first. Figure 13 shows typical simulation and experimental results for a 45þ prism measurement. It was shown that the phase difference of raw data and the unwrapped phase difference for both simulation and experimental results are quite similar to each other except the edge area. The edge of tested prism in experiment is not straight so much more irregular results for observed phase difference in this area. Although the experimental phase difference image was relatively noisy around the edge area; the averaging filter  was adopted to smooth noise before unwrapping. Then the discontinued phase difference was unwrapped by two-dimensional phase unwrapping technique.
Figure 14 displays the four transmitted PS-DIC images in horizontal shear direction and Fig. 15 shows the topography reconstruction result. As shown in Fig. 15, the accuracy of reconstructed profile can be greatly improved with phase unwrapping and compensation for shear change.
The experiments for each case were tested 30 times to study the measurement accuracy and repeatability. Table 4 summarizes the experimental measurement results for mean phase difference and reconstructed wedge angle of prisms compared with the simulation results. The average measurement error is about 0.81þ for 30þ prism and 1.12þ for 45þprism. The measurement standard deviation is about 0.467þ for 30þ prism and 0.479þ for 45þprism. As shown in Table 4, the results of reconstructed wedge angle measured by transmitted PS-DIC experiment were very close to those simulated using our optical model. The difference between simulation results and experimental results are about 1.2% for 30þ prism and 3.1% for 45þprism.
In this paper, the feasibility of profile measurement of transparent inclined surface by transmitted DIC was studied. The optical model of transmitted PS-DIC shearing interferometer was constructed and series of simulations for various prisms with wedge angles from 5° to 60° were tested. Experiments were also conducted to verify the simulations.
The simulation study indicated that unwrapping technique is required for profile reconstruction even with a wedge angle as smaller as 15°. This is so because the corresponding phase difference between the two split beams is already larger than π even with a wedge angle of 15°. It is also found that the accuracy of the DIC measurement can be further improved by taking into account the effects of the change in shear distance on the optical path difference.
As a result, when it comes to measurement of transparent specimen with complex structure such as inclined surface, these findings suggest that phase unwrapping of the measured optical path difference should be adopted and the compensation of shear change will further improve the measurement accuracy of the transmitted DIC system.
The authors thank the National Science Council of Taiwan (NSCT), for its financial support and the Center for Measurement Standards, Industrial Technology Research Institute for technical assistance, and financial support, as well.
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