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Abstract
In this study, a high-precision and low-cost instrument was developed for the measurement of the wedge angle of wedge plates. The module was designed using two prism-patterned subwavelength gratings, with the −1 order rays extracted using a semiconductor laser beam-splitting device. Subsequently, in reference to Snell’s Law and the diffractive characteristic, the wedge angles of wedge plates were tested and analyzed when the two points on the movable receiving screen were overlapped through adjustment of the distance. This result revealed that the proposed system achieved sub-second ultimate resolution for the wedge angles of the wedge plate under a 1-µm precision requirement for measuring instruments.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Wedge plates (WPs) are widely used in optical fields, such as in image display [1–3], optical measurement [4–6], light source collimation [7–10], material refractive index measurement [11], and wavefront measurement [12]. In these applications, the precision of the wedge angle (WA) is crucial and greatly affects the measurement precision. For example, Ghoorchi-Beygi measured the focal length and proposed that correction of the WA can effectively reduce the absolute error of the system [13]. In the design of a lateral-shearing interferometer, the quantity of phase shift is proportional to the WA. Improvement of the WA error can directly reduce the phase-shift error [14]. Xu analyzed the influence of the WA on the maximum error of ray collimation by using the error measurement of the WA [15].
Interferometry is the current mainstream method for measuring the WA [16–20]. Kumar et al. measured optical windows’ refractive index and the WA simultaneously using Fizeau interferometry and a cyclic path optical configuration [21]. Chatterjee et al. applied multipass optical configuration to measure the residual WA of a transparent parallel plate with high optical quality [16]. Although interference measurement can yield high-precision measurement results, the precision remains insufficient for light sources and experimental instruments. One of the methods is by using centration measurement machine [22–24]. The wedge is calculated by the radius of the spot. The improvement of the measurement accuracy depends on the lens group to enlarge the rotation radius of the spot on the CCD. In this study, subwavelength grating (SWG) is used as the magnifying element to measure the wedge angle, SWG can greatly improve the sensitivity of a system through enhancement of the amplification of the angle and diffraction ray. The angle of −1 order diffraction ray from SWG changes more drastically. Moreover, determining the coincidence of light spots is applied in this study to substitute the related components of the imaging system. Compared with the interferometer, the environmental influence of the design of this study are greatly reduced. The instrument have better stability under micron-order instrument requirements with speckle measurement, and the oil and gas in the production line will hardly affect the measurement results have a serious impact on the interferometer. Therefore, this study can measure the wedge angle with simple optical structure and micron-order instrument requirements.
SWGs are frequently used in different fields of optical measurement [25–27]. In this research, the SWG patterned on a prism was applied for the measurement of the WA. WA measurement was performed with a semiconductor laser light source (λ = 650 nm), two beam splitters, two 45°–45°–90° prisms, and an SWG period of 375 nm. As a result of the high angle amplification, the precision of the measuring instrument used for the WA of WPs was greatly improved.
2. Experimental setup
The schematic of the experimental setup is depicted in Fig. 1. A semiconductor laser with a wavelength of 650 nm was used as the light source. The laser ray passes through the first beam splitter 1 (BS1) to reach the surface of the WP. The ray is reflected from the WP into the measurement system and splits into two after passing through the second beam splitter (BS2). Both rays respectively pass through the SWG surfaces on prism 1 and prism 2. Finally, the two rays are displayed on the receiving screen.
Different angles of the WP result in light coming out of the SWG at different angles, which eventually appear on the receiving screen at different spacing. As depicted in Fig. 2, the light speckles are overlapped by adjusting receiving screen. The WA is then calculated by moving the distance of the receiving screen through application of Snell’s Law and the grating equation.
Figure 3(a) and (b) present the diffraction speckle on the receiving screen during WA measurement. As depicted in Fig. 3(a), the light speckles were not overlapped. The receiving screen was thus adjusted until the light speckles overlapped, as presented in Fig. 3(b). The WA could then be calculated by moving the distance of the receiving screen.
Fig. 3. Light speckle status of the receiving panel. The (a) nonoverlapping speckle and (b) overlapping speckle.
In this research, the measurement of the angle of the WP was facilitated through use of the high angular amplification of the SWG. The SWG is manufactured through use of the ultraprecision machining with single-point diamond tool [28–29], the design of the profiles of SWGs and the angle of the prism must be considered. The optical path in the prism pattern using the SWG is illustrated in Fig. 4.
The relationship between the angles α and β can be derived using Snell’s Law [30], which is expressed as follows:
where nair is the refractive index of air, and np is the refractive index of the prism.The relationship between the angle of diffraction θdiff and the angle of incidence θinc on the SWG was determined through application of the grating equation [31] as follows:
For the convenience of measurement, the relationship between the θinc and θdiff on the SWG surface is first discussed. First, Eq. (3) could be converted as follows:
Figure 5(a) depicts the plot of θdiff against θinc with different λ/Λ based on Eq. (4). To avoid the interference of other-order diffraction light, θinc was set between 41.8182° and 52.7273°. When the ray exits from SWG, except for order 0 and order −2, only order −1 light exists. In addition to the ratio of the wavelength and grating period, the surface morphology of the SWG and incidence angle also greatly affects the diffraction efficiency of the SWG. According to the calculation results presented in Fig. 5(a), different incident angles and grating periods were further simulated.
Fig. 5. Relationship between the θinc and θdiff of (a) different ratios of wavelength and grating periods and (b) different combinations of wavelengths and grating periods.
A more detailed parameters are substituted into Eq. (4). The simulation results are illustrated in Fig. 5(b). The zero-order diffraction ray only followed Snell’s Law, regardless of the changes to the grating period; the θinc remained at 41.8182° when the θdiff was 90°. Therefore, according to the results presented in Fig. 5(b), the θinc was set from 41.8182° to 64.5455°; the grating period was set from 350 to 550 nm, and the influence of other-order diffraction light on the location of the light speckle can be prevented by using a 650 nm laser. Under this precondition, the influence of the incident angle and SWG morphology on diffraction efficiency was further examined.
The impact of the SWG surface topography on the diffraction efficiency was evaluated using CP, which is the ratio of the peak position w to the period Λ, as presented in Fig. 6. On the basis of the fabrication capability and diffraction efficiency noted in a previous study [32]. In this study, CP = 0.5 was selected and applied in this research. A scanning electron microscopy image of the SWG with grating period of 375 ± 10 nm is depicted in Fig. 7.
The calculated λ/Λ, θinc, and θdiff, different wavelengths, incident angles, SWG morphology, and the other parameters listed in Table 1, the influence of diffracted light efficiency was simulated. The simulation results are illustrated in Fig. 8.
Fig. 8. Relationship between the grating depth height and average diffraction efficiency of (a) θinc = 45°; (b) θinc = 55°; and (c) θinc = 65° at CP = 0.5.
Table 1. SWG parameter setting
According to the simulation results presented in Fig. 8, the maximum diffraction efficiency occurred at the region with a period ranging from 350 to 400 nm; grating depth d was at approximately 320 to 380 nm when θinc = 45° and θinc = 55°. Considering the setting of subsequent experiments and universality of the prism, the following settings were used for the following analysis: grating period Λ = 375 nm and θinc = 45°. After the surface shape of the SWG and the incident angle of the diffraction surface were determined, the magnification effects must be verified. According to Eq. (3), Fig. 9 shows the effect of angular amplification at different prism angles θp when the ray reaches the surface of the prism at α=0.
The magnification can increase the incident angle θinc by approximately 1.4 times under a prism design of θp = 45°, and influence from the other-order diffraction light can be prevented. The prism of θp = 45° was applied in the following experiments.
3. System analysis
In this study, a laser with wavelength 650 nm, BS with a fabrication tolerance of 0.01 mm and a linear shift table with an accuracy of 1µm are applied for measurement. Before measurement, the optical setup alignment is necessary. The light source is installed on the kinematic mount with rotatory stage. The standard laser alignment block is applied to align the light source in a long-distance condition. Z-fold parallel configuration is adopted to reduce the eccentricity and tip-tilt error under 0.5” for the rays through the BS1 and BS2. The configuration ensures the alignment of laser ray in front of the prism. All the element can be adjusted by the tip-tilt stage so that the spots move in the same plane.
The accuracy of measurement can be affected in many ways. One possible cause is the perpendicularity error of two beams split by the BS2. The measurement error caused by the perpendicularity error of BS at each incident angle is calculated. According to the statistical data of measurement results, the impact of the perpendicularity error can be reduced. Another possible cause is the coincidence error of the two light spots on the screen. Stray light emitted by SWG and blurred edges affect the coincidence of the light spot. In this study, to obtain clear and sharp light spots, a neutral density filters are set up in front of the light source, and an iris diaphragm is added in front of the receiving screen. When the parallel ray with radius 1 mm enter the system, a light spot with width 5 mm is obtained on the receiving screen. As the receiving screen moves for 1 µm, the width of the spot changes for 1.3 mm. The obvious change effect can significantly reduce the spot coincidence error.
In the actual production process, because the SWG is typically obtained using an ultraprecision processing machine rather than through a semiconductor process, the error of the grating period can therefore be as high as 10 nm. Table 2 lists the moving distances of the receiving screen under different grating periods, indicating that the measurement result is highly sensitive to the ratio of the wavelength and grating period. Moreover, the independent measurement of the wavelength or a part of the SWG period is not accurate to reflect the real condition of the instrument since the SWG is bonded on the prism. The ratio of wavelength to the SWG period is checked on uncertainty. Thus, calibrating the wavelength and grating period after completion of the initial instrument calibration measurement is necessary. This alters the incident angle of the light source to obtain different receiving screen distances.
Table 2. Combination of different wavelengths and grating periods
Finally, the average ratio of the current wavelength to the grating period was revealed to be 1.7422. After correcting the data, the result is consistent with the theoretical design value.
The influence of the receiving screen precision on the WA measurement was further analyzed. With a receiving screen distance error of 1 µm, the WA measurement result was 0.1863”. Under a measuring instrument precision of 1 µm, the precision of the WA measurement can reach the sub-second scale.
The type A evaluation method was employed to evaluate the human error in the measurement process and the accuracy of each component. The eccentricity error of the collimating lens and the perpendicularity error of the BS were measured by Trioptics OptiCentric 300. The WA measurement uncertainties caused by each component are shown in Table 3.
Table 3. Measurement uncertainty caused by each component
4. Results and discussion
In this study, a grating period of 375 ± 10 nm, light source wavelength of 650 nm, and 1-µm precision linear shift table were employed. In the forward simulation, the moving distance of the receiving screen with a WA from 0” to 1” was 0.0052 mm. In the reverse simulation, the limit of the WA with receiving screen movement of 1 µm was 0.1863”. The results demonstrated that the system can measure the WP with sub-second resolution.
A few biplane quartz glasses with WA under 1000” from an existing production line were randomly selected for measurement. Each WP was measured 10 times to verify the stability of the system. The average measurement results of the WP under different WAs are summarized in Table 4. The ranges of results were all less than 2’’, and the standard deviations were less than 1”.
Table 4. System measurement result
Considering that different WA may affect the measurement results of the instrument. In this study, a statistical test (ANOVA) with sample number n = 10 and alpha level = 0.05 are applied to analyze the impact of different WA on the measured results in Table 5. Since the P-Value = 0.43 > 0.05, the variables are not statistically significant. Therefore, the instrument in this study can measure WP of different specifications stably within the design scope.
Table 5. Analysis of variance
Box and whisker plot is used to analyze the measurement results of WA to correct the measurement errors of the instrument, as shown in Fig. 10. The results show good symmetry and normal distribution. Since the distance between inner fence and outer fence is small, the measurement results are highly repeatable. The ANOVA test and Box and whisker plot both verify the sub-second scale measurement of the WA as well as the high stability of the system.
5. Conclusion
In this research, a high-precision WA measurement method for WPs was designed. Compared with the current method of measuring WAs based on the principle of interference, the proposed optical design is simpler. The environmental influence of the design of this study are greatly reduced in contrast to the interferometer. The instrument has better stability under micron-order instrument requirements with speckle measurement. The oil and gas in the production line will hardly affect the measurement results. With the angular amplification of the light deflection ability of the SWG, the proposed system achieved sub-second precision under a 1-µm measuring instrument precision requirement. The precision of the WA measurement for WPs in this study successfully verified the feasibility and applicability of this design. Under the condition of manually adjusting the receiving screen, the single measurement time is about 10-30s. In the future work, the instrument can be further improved into an automated measurement to meet the demand of mass production and detection in the factory. Therefore, the single measurement time can be reduced to within 5s.
Funding
Ministry of Science and Technology, Taiwan (MOST 110-2222-E-007 -009 -MY3, MOST 110-2622-E-007 -002).
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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