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We present a method of measuring the parallelism of two parallel narrow beams by positioning a CCD each before and after the focal plane of the collimation lens. Because of this differential defocused positioning of the two CCDs, the two beam spots cannot overlap on the CCD image plane even if they are nearly parallel, and the distance between the two spots on each CCD can be obtained simultaneously, which eliminates the influence of environmental noise. Our experimental results show that a satisfactory system stability and resolution of 0.1″ and 0.2″, respectively, can be obtained with the proposed method.
© 2016 Optical Society of America
1. Introduction
With the advantages of high measuring accuracy and fast measuring speed, laser interferometers are widely used in many measurement applications [1–4]. Moreover, multi-axis interferometers and differential interferometers are particularly favored in measuring the multiple degrees of freedom of linear displacements or angular displacements [5–7]. The measurement accuracy of laser interferometers is closely related to the parallelism between the parallel narrow laser beams generated by the interferometer in angular displacement measurement [8,9]. In general, the parallelism of the two parallel narrow beams is limited to 20″, and more strictly to 5″ in the case of high-accuracy measurements [10]. Thus, it becomes necessary to measure the parallelism of the narrow beams. According to the autocollimation principle, when two beams radiate away from the collimation lens to its focal plane, they will converge as two beam spots on the focal plane, and the distance between the two spots can be used to calculate their parallelism. However, the beam spots have an actual size of hundreds of microns due to the aberration of the collimation lens and the non-ideal characteristics of the laser beam, which causes the aforementioned method to become invalid in the case wherein the beam parallelism is excellent, i.e., the two beams are nearly parallel. In this case, the two actual beam spots will overlap in the focal plane, and it is difficult to distinguish between them. In order to solve this problem, we previously proposed a time-sharing measurement method [11] in which the overlapping of beam spots can be avoided by means of using a beam-selected structure to allow the two beams converge onto the focal plane in sequence. However, this method is an asynchronous measuring method (the beam spots are measured at different times), and it can suffer from system drift and environmental fluctuations [12]. Consequently, in this study, we present a simultaneous measurement method for beam parallelism based on the differential defocusing principle. In our setup, two CCDs are utilized, with one positioned before and the other after the focal plane of the collimation lens so that the system can be utilized to acquire the distance of the beam spots simultaneously and parallelism can be calculated in real-time. This method is still valid even if the two beams are nearly parallel. In addition, based on this differential defocusing measuring principle, the distance information between the beam spots on each CCD can be subtracted to eliminate the environmental fluctuation influence.
2. Measurement principle
As shown in Fig. 1, when two narrow beams are close to being perfectly parallel, their converged beam spots on the focal plane of the collimation lens are very close to each other. As is well-known, in a practical collimation measurement system, the beam spot has an actual size of hundreds of microns due to the aberration of the collimation lens and non-ideal nature of the laser beam. Thus, the actual spots corresponding to beams 1 and 2 will overlap on the focal plane, and it is difficult to simultaneously determine the beam spot positions.
In order to solve this problem, we position a CCD each before and after of the focal plane, i.e., the differential defocusing arrangement, where the defocused distances from the focal plane are a1 and a2 for CCD 1 and CCD 2, respectively (assuming that the optical axis direction is positive). Though the size of the beam spots undergoes enlargement when the CCD is at the defocusing position, the spots do not overlap when a1 and a2 are sufficiently large. Consequently, we can easily determine the distance between the two beam spots on each CCD, and the parallelism of the two beams can be calculated by means of Eq. (1).
Here, d1 and d2 represent the distances between the two beam spots on CCD 1 and CCD 2, respectively, f denotes the focal length of the collimation lens, and β the parallelism of beams 1 and 2. In this arrangement, it is easy to determine if the two beams are nearly or completely parallel, wherein parallelism will lead to overlapping of the beam spots on the focal plane; however, the two beam spots will not overlap in the imaging planes of CCD 1 and CCD 2 at the differential defocusing positions. This advantage makes it possible to measure the parallelism of the two beams with only one measurement when the parallelism value is very small, i.e., when the two beams are nearly or completely parallel, which cannot be achieved when the CCD is positioned at the focal plane position. In addition, d1 and d2 are obtained simultaneously, which avoids the problem of spot drift in the time-sharing measurement [11].
3. Error analysis
3.1 Influence of CCD position error
In general, it is difficult to align the CCD imaging plane at the absolute defocused position as expected. In actual application, position errors Δa1 and Δa2 are generated, as shown in Fig. 2. In the figure, the blue dotted lines represent the actual CCD image planes. In this case, d1 and d2 are the actual distances between the two beam spots, while it appears that the CCDs are positioned at a1 + Δa1 and a2 + Δa2. The parallelism error Δβ caused by Δa1 and Δa2 can be calculated via Eq. (2).
Assuming β = 5″, f = 500 mm, D = 22 mm. The coordinate origin is the point of intersection of the optical axis of the collimation lens and its focal plane. When a1 = −6 mm and a2 = 6 mm, the calculated parallelism errors are shown in Fig. 3(a). Further, Fig. 3(b) shows the calculated parallelism errors when a1 = −6 mm and a2 = 12 mm. From Fig. 3, we note that the parallelism error is smaller than 0.2″ when the two CCDs are positioned on either side of the focal plane.
Fig. 3 Parallelism errors for CCD deviate position of (a) a1 = −6 mm, a2 = 6 mm, (b) a1 = −6 mm, a2 = 12 mm.
In our study, we also calculated the parallelism errors when the two CCDs were positioned on the same side of the plane; these results are shown in Fig. 4. We note from Fig. 4 that the maximum parallelism error is nearly 3″, which indicates that positioning the CCDs on the same side of the focal plane leads to an obvious increase in the parallelism error with respect to the case with the CCDs positioned on either side of the focal plane. We refer to the latter CCD arrangement as the differential defocusing arrangement in our study.
Fig. 4 Parallelism errors for CCD deviate position of (a) a1 = 12 mm, a2 = 6 mm, (b) a1 = 7 mm, a2 = 6 mm.
In order to further analyze the effect of the CCD position error, we discuss the problem from the viewpoint of the mathematical expectation of the parallelism error. In a practical setup, the errors Δa1 and Δa2 inevitably arise due to errors in alignment and other factors. Let us assume that the position errors Δa1 and Δa2 can be exactly controlled to lie in the range of (−0.01, 0.01) mm, and that the CCD has an equal probability of being located at any position in this range. Subsequently, the mathematical expectation of the parallelism error arising due to the CCD position error can be calculated via Eq. (3). In our study, setting a1 = −6 mm and varying a2, we calculated the expectation of the parallelism error as Fig. 5; Fig. 5(b) shows the magnification of the section marked in red in Fig. 5(a).
It can be observed from Fig. 5 that different positions of CCD 2 lead to different expectations of the parallelism error; the mathematical expectation is larger when the two CCDs are on the same side of the focal plane (the CCD position is “negative” according to our coordinate system) than when on either side of the focal plane (wherein the CCD position is positive). Thus, a larger mathematical expectation indicates a larger parallelism error. A close examination of Fig. 5 reveals why the parallelism error in Fig. 4 is larger than that in Fig. 3. The mathematical expectation is approximately maximum when the two CCDs are located at the same position, i.e., a1 = a2 = −6 mm, and it is approximately minimum when the two CCDs are located at symmetrical positions with respect to the focal plane, i.e., a1 = -a2. According to this analysis, we adopt the equal distance differential defocusing arrangement of CCDs in the design of our measurement system.
As analyzed above, the defocusing position is extremely important in our method. In the practical measurement, we use following steps to determine the defocused distances a1 (a2) and control its change Δa1 (Δa2) within the range of (−0.01mm, 0.01mm): (1) Generating wide parallel beam by a long-focal collimation lens with focal length of 1000 mm; (2) Let the parallel beam go into the collimation lens of the parallelism measurement system, and an auto-focusing method with focusing uncertainty of 1 μm [16] is used to find the focal plane position; (3) Moving the CCD from focal plane position to a designed defocusing position with defocused distances a1 (a2), then fixing the CCD. The moving of CCD is under the monitoring of a double-frequency laser interferometer with displacement measurement resolution less than 10 nm, so a1 (a2) can be precisely determined. Even if the fixing process of CCD cause larger deviation of defocused position, the real-time monitoring of double-frequency laser interferometer can also provide a new value of a1 (a2) to grantee the change Δa1 (Δa2) within the range of (−0.01mm, 0.01mm).
3.2 Influence of CCD being tilted with respect to optical axis
If the CCD image plane is not normal to the optical axis of the collimation lens (assuming that the CCDs tilt by angles of γ1 and γ2 with respect to the optical axis), as shown in Fig. 6, a parallelism error occurs in this case as well. Equation (4) can be used to calculate the parallelism including the CCD tilting error.
Figure 7 shows the parallelism error calculated via Eq. (4), wherein we assume that the ideal parallelism β = 5″, a1 = −6 mm, and a2 = 6 mm. From Fig. 7, we note that the measurement error quickly increases with increase in tilting angle, and thus, the tilting angle of the CCD must be rigorously constrained within the tolerance range of (−3deg, 3deg) to make the parallelism error less than 0.1″.
4. Experiments
Figure 8 shows the experiment setup corresponding to the measurement principle shown in Fig. 1. A collimator was used as a collimation lens with focal length of 500 mm, and two CCDs (POINT GREY Inc. GS-U3-28S4M-C, with 1928 × 1448 pixels and 3.69-μm pixel resolution) were positioned before and after the focal plane of the collimation lens. An HP5517B laser and an optical attenuator were used to obtain the laser beam, and a beam splitter was used to separate and direct the laser beam into the two CCDs. A one-axis differential prism was used to create two narrow parallel beams. The method based on the differential defocusing principle proposed in this paper was compared with the method based on the time-sharing measurement [11].
We use the energy centroid of actual beam spot to represent the spot’s position in this paper. The centroid coordinates along the X-direction for the two beam spots in one CCD image plane are shown in Fig. 9(a). As shown in Fig. 9(a), the drift of the beam spots is approximately 0.3 px along the X-direction.
Fig. 9 Centroid position stability along X-direction as obtained with (a) simultaneous measurement and (b) time-sharing measurement.
As shown in Fig. 9(a), the centroid coordinates of the two beam spots on one CCD exhibit a similar drifting trend with time, and thus, the drift can be offset by subtraction. On the other hand, with the time-sharing measurement based on the autocollimation principle [11], the presence of a measurement interval produces drift, as shown in Fig. 9(b). This drift cannot be offset by subtraction because the centroid coordinates are obtained at different times for the two beams.
Regardless of the method used (differential defocusing principle or time-sharing measurement), the parallelism was calculated by subtracting the centroid coordinates of the beam spots [13–15]. The results of the subtraction are shown in Fig. 10, wherein it can be observed that the differential defocusing principle offers considerably better stability (0.1″) as regards the centroid distance between the beam spots than that obtained with the time-sharing measurement method (0.4″).
Fig. 10 Difference between centroid coordinates along the X-direction of the two beam spots based on (a) differential defocusing principle, (b) time-sharing measurement.
The parallelism of the two beams obtained with a one-axis differential prism was measured by means of both the abovementioned methods. Figure 11 shows the results obtained with both approaches for repeated measurements. It is obvious that the method based on the differential defocusing principle exhibits a significantly better stability (less than 0.1″) with regard to parallelism measurement than the time-sharing measurement (less than 0.6″).
Fig. 11 Parallelism measurement based on (a) differential defocusing principle, (b) time-sharing measurement.
The system resolution of the setup based on the differential defocusing principle was also tested by adding a reflector in the path of one beam. The reflector was adjusted at fine reflection angles by means of a piezoelectric ceramic actuator. The experiment setup for system resolution testing is shown in Fig. 12, and the corresponding results are shown in Fig. 13. We note that the system could not distinguish steps of 0.1″. The results for steps of 0.2″ and 0.4″ can also be converted as sawtooth waveform when the parallelism values at each forward/backward step are averaged, as shown in Fig. 14, which indicates that a resolution of 0.2″ can be achieved by our system.
Fig. 13 Results of system resolution testing for (a) step interval of 0.4″, (b) step interval of 0.2″, (c) step interval of 0.1″.
Fig. 14 Results of system resolution in sawtooth waveform testing for (a) step interval of 0.4″, (b) step interval of 0.2″.
5. Conclusions
In this study, we used the differential defocusing principle to measure the parallelism of two narrow parallel beams. In our setup, two CCDs are positioned at equal distances from the focal plane of the collimation lens on either side of the focal plane. The simultaneous spot measurement enabled by this method greatly reduces the drift and environmental fluctuations. Our experimental results show that a system stability of 0.1″ is achievable along with a system resolution of 0.2″. Thus, we conclude that our system can be used to measure parallelism for two narrow beams with extraordinary precision and stability. Moreover, the application scope of the method can be further expanded to measure parallelism of multi-narrow beams and to measure tiny angles.
Acknowledgments
This research work is funded by the National Natural Science Foundation of China (NSFC) under Grant Nos. 61575075 and 61008031, the Fundamental Research Funds for the Central Universities under Grant No. HIT.NSRIF.2014020, Natural Science Foundation of Heilongjiang Province under Grant No. F2016014, and the Postdoctoral Science-Research Development Foundation of Heilongjiang Province under Grant No. LBH-Q13078.
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