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Abstract
Multi-reflective imaging systems find wide applications in optical imaging and space detection. However, it is faced with difficulties in adjusting the freeform mirrors with high accuracy to guarantee the optical function. Motivated by this, an alignment-free manufacture approach is proposed to machine the optical system. The direct optical performance-guided manufacture route is established without measuring the form error of freeform optics. An analytical model is established to investigate the effects of machine errors to serve the error identification and compensation in machining. Based on the integrated manufactured system, an ingenious self-designed testing configuration is constructed to evaluate the optical performance by directly measuring the wavefront aberration. Experiments are carried out to manufacture a three-mirror anastigmat, surface topographical details and optical performance shows agreement to the designed expectation. The final system works as an off-axis infrared imaging system. Results validate the feasibility of the proposed method to achieve excellent optical application.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Multi-reflective imaging systems (MRIS) have been widely used in imaging optics and space applications since they can provide a long detection distance [1]. The off-axis configuration usually has an offset aperture stop or biased fields to avoid the central obscuration in comparison to the on-axis layout. In this pattern, the optical alignment is practically important in off-axis MRIS. There exist difficulties in aligning the mechanical axis with the optical axis precisely for each off-axis optical component. Such process is tedious and challenging to achieve the optimized performance as designed, especially for freeform optics, which possess no appropriate references [2, 3].
Several approaches have been investigated on the alignment of freeform systems, including MRIS and other complex optical systems. Special mechanical fixtures, including coordinate measuring machines (CMM) [4], auto-collimators and patterns generated by computer-generated hologram (CGH) [5] can provide adjustment references. Computer-aided alignment (CAA) or so called computer-guided alignment (CGA) techniques bring the optical alignments to an optimization problem of reducing the time-consumption [6]. Further studies, such as nodal aberration theory (NAT) [7, 8], take the optical performance, including modulus transfer function (MTF), wavefront [9], or other relative parameters to be the alignment targets.
Based on the fact that the optical alignment efforts is high cost and complicated, some studies are coming up to reduce optical alignment efforts via other optical design and manufacture recent years. Computable initial structure in off-axis three-mirror anastigmat (TMA) was proposed where the primary mirror and the tertiary mirror distribute in the same substrate [10]. Such structure reduce the difficulties in assemble these two mirrors. In a previous study, a unified characterization of freeform optics was used to replace multi-mirror group in a gas cell to realize assembly simplification [11]. On the other hand, we proposed a rotating tool turning method to machine freeform prisms, which is a pioneer work to avoid the complex alignment in the view of in integrated optical manufacture [12]. Imaging snap-together freeform system was also studied using ultra-precision turning techniques, where two optical surfaces was snap-together in a common setup to reduce the alignment effort [13].
On the other hand, the machining accuracy is increasingly instructed by optical performance, in the view of modern optical manufacture. Traditional manufacture of freeform optics are usually focused on the surface topographic details, including form accuracy and surface quality of one freeform surface in a single manufacture period. Researches mainly on the method of compensating the form error of an individual freeform surface via in situ measurement [14, 15]. The measurement always involved dealing data since the machining and measuring coordinates need matching [16, 17]. However, there is still lack of studies on performance-guided manufacture while it has great potential in contributing to the alignment of complex optical systems.
Inspired by the current studies, an alignment-free manufacturing idea and approach is proposed and implemented to machine the freeform off-axis MRIS. The relationship between the optical performance and machine errors is studied and the alignment errors are reduced significantly and even can be neglected. This study intends to recommend an integrated approach for the machining and measurement of complex optical system, which has been proved to obtain the relative good optical performance after only single machining process. The proposed method therefore provides one low-cost and efficient approach to machining the freeform off-axis reflective imaging system.
2. Principles of integrated manufacture
The proposed integrated manufacture method enables the optical system generated only by ultra-precision machining process without extra remounting and alignment. It allows us to directly focus on the final optical performance of the system instead of paying efforts to examine the form accuracy. The machining approach avoids the troubles in measuring the form error since it is hard and tedious to straightly obtain the error value in freeform optics. It is most innovative that it makes a breakthrough in optical manufacture by constructing a routine of “Error identification – Integrated machining – Performance evaluation” instead of repeating the topographic measurement and compensation of each optical component in conventional thinking.
2.1 Manufacture strategy
Figure 1 illustrates the integrated manufacture strategy of alignment-free MRIS. Firstly, the optical design of the MRIS configuration is carried out. Then the appropriate ultra-precision machining configuration is selected according to the distribution of the freeform optics. Before practical machining process, the machine errors should be considered because the cutting process is operated on more than one freeform surfaces, which may amplify the form error in the optical system. Hence the machine error model should be established to analyze the system errors in the machine tool. Based on the error model, a pre-machining is carried out on a typical workpiece in order to identify the machine error, so that the system errors can be identified to minimize the reduction of optical performance. The machining process can be conducted on the MRIS after the above preparation. The optical performance is measured using an interferometer and a detector or reflective equipment since the MRIS is generated without extra alignment step. The optical performance can be directly obtained and evaluate to give the validation with the original design. Hence an alignment-free machining strategy can be constructed and the direct optical performance evaluation can be realized without measuring the form accuracy in single optical component.
2.2 Machining configuration
In order to perform the idea of integrated manufacture, the machining configuration is established to manufacture a MRIS with alignment-free structure. Among these optical reflectors, freeform mirrors are generated one by one via the motions of the cutting tool along the X, Y and Z axes. In the meantime, the spindle is rotating as the synchronization reference of the linear axes. In this study a typical MRIS, i.e. a TMA is taking as an example.
Figure 2(a) is an illustration of this machining approach to generate a TMA. The whole configuration functions based on a four-axis ultra-precision machine using the fly-cutting technology in the raster scanning mode [18]. All the freeform mirrors are mounted on the B axis of the machine according to the system design in advance, and the position relationship between each mirror keeps in agreement with the design values due to the high precision movement without remounting in the ultra-precision machine. The alignment accuracy depends on the machine accuracy, which is generally well above the tolerance requirement in machining process. The alignment errors therefore become insignificant or even can be neglected. It is worth mentioned that the machine errors may induce the alignment errors partially so they should be considered and reduced. It is shown in Fig. 2(b) that the tool marks are generated according to the kinematical feature of the machining configuration. The cutting direction is along X axis, while the feed direction is along Z axis. The material removal is implemented when diamond tool rotates a circle, and it forms one single fish-scale cutting mark according to the cutting path. Hence, the tool feed interval in these two directions should be optimized according to the surface residual height and tool nose radius.
Fig. 2 (a) Alignment-free machining configuration and (b) cutting marks generation for off-axis three-mirror system on ultra-precision turning machine.
Figure 3(a) is the space constraints of the TMA, including the relationship between rotating radius R and the distance between the mirrors. In order to avoid the cutting interference during machining process, the appropriate rotating radius R of fly-cutter is determined by the minimum distance between the primary mirror or tertiary mirror and the second mirror and the surface slope change of the surface in X direction. These constraints are also applicable to other kinds of MRIS, as shown in Fig. 3(b). The rotating radius R is should not exceed half of the minimum distance between the mirrors distributing on the opposite side.
Fig. 3 Rotating radius according to the minimum distance between the corresponding mirrors. (a) TMA and (b) other multi-mirror systems.
3. Analysis on machine errors and performance measurement
According to the above analysis, the machine errors are the significant facts in the final manufacture accuracy. Due to the lack of knowledge about the relationship between its optical performance and errors in manufacturing such TMA, the machine errors should be analysed to identify the dominating errors in machining process. Based on the identification of form errors, the wavefront aberration can be deduced. Hence other items of optical performance can be obtained. Moreover, the optical performance needs to be obtained to give a comprehensive evaluation without giving form error of each freeform surface. A series of practical approaches are proposed to directly measure the wavefront aberration which is directly measured using interferometry techniques.
3.1 Machine errors
The machine errors are categorized into system errors of the machine and tool alignment errors. The system errors contain the motion errors of each axis including the position and angle errors, and the relative position errors between two axes. The tool alignment errors are mainly caused by manual adjustment and measurement errors of cutting tool, including the errors of tool position, nose radius and rotating radius.
The machine error model is established based on the multi-body system theory [19]. This machining system totally has 34 items of machine errors, including 24 position errors of the X, Y, Z and C axes, 5 squareness errors between the axes, 3 tool alignment errors and 2 tool geometric errors. These errors can be simplified into 11 items with regard to almost the same effect on form errors, since the degenerated degrees of freedom reduced. The coordinate distortions of machine error items are listed in Table 1. Herein, the items Δx, Δy and Δz represent the position errors of fly-cutter in three directions, respectively, While ΔαZX, ΔαZY and ΔαYX represent the perpendicular errors between the corresponding two axes. Moreover, Δθmn represents the tilt errors of the linear axes, where the subscript m and n means the angular errors of m-axis rotating along n-axis. In the meantime, ΔR and ΔRt are the errors of rotating radius and tool nose radius, respectively.
Table 1. Coordinate distortion due to machine error items.
The simplified error items can therefore be further categorized into 4 groups, which leads to different effects on the different optical performance. Due to the proposed machining method, some machine errors result in the same error effect on all the mirrors, thus may only transfers the image without reducing optical performance. The specifications of TMA and effects on optical performance are listed in Table 2. It can be inferred that the linear errors and tilt errors result in the position and orientation change of the image, while the nonlinear errors have a negative influence on optical performance.
Table 2. Specifications of freeform off-axis TMA.
Based on the machine errors and ideal surface model, the form errors of three mirrors are calculated, which are then introduced into the variation of optical performance based on optical ray-tracing theory. Wavefront aberration (unit: λ = 632.8 nm) is directly calculated as the basic parameter described optical performance from form errors. Thus other items of optical performance can be deduced. The weight coefficients on optical performance of the 11 error items are computed. The results are shown in Fig. 4. As far as the properties of the ultra-precision machine are concerned, the value of linear errors are set 0.001 mm while the angular errors are set 0.001° as the simulated errors. The results indicate highly agreement with the above effects on the different optical performance.
Table 3 lists the fitting Zernike coefficients of the wavefront aberration caused by the four main machine errors. It can be inferred that each machine errors mainly effects more than one type of aberrations, and the probability of spherical aberration and astigmatism are the largest. Based on the abovementioned analysis, the identification of the machine error can be conducted by a pre-machining approach. A typical workpiece is machined by this configuration and tested, wherein the machine errors are identified. Hence the influence of machine errors on optical performance can be minimized through compensation in advance.
Table 3. Zernike coefficients of the wavefront aberration caused by the main machine errors.
3.2 Direct measurement of optical performance
The proposed integrated manufacture off-axis TMA provides natural convenience to measure the final optical performance directly without any alignment and adjustment between each optical surface. Therefore, a few schemes are built to measure wavefront aberration is using interferometer with an optical sensor or a planar/spherical reference mirror. The optical system can rotate and translate to coordinate the adjustment stage of the detector. In this way, it provides the access to measuring the different angles between central field and horizontal direction, denoted as θ.
Figure 5(a) presents the illustration of using a CCD sensor to capture the point spread function (PSF) image from the parallel rays through the TMA. The experimental setup is similar to that with wavefront sensor. Herein, PSF is used to describe the concentration ability of the optical system. The MTF can be calculated by Fourier transformation according to the PSF. Therefore, optical performance can also be measured by the CCD image method. Figure 5(b) reveals a non-interference measuring method based on the Shack-Hartmann wavefront sensor, which detects the deviation of the planar wavefront. In this method, the planar wavefront is generated by collimator from the objective space, because the interference function of the planar reference wavefront is not needed. The spherical reference mirror is replaced by the wavefront sensor mounted with microscope objective lens. The focus point of microscope objective lens is situated exactly on the image plane, it plays a role of converting the spherical wavefront on the image plane into the planar wavefront. Thus the wavefront sensor can detect the corresponding information. Figure 5(c) shows the measurement configuration based on the concave spherical reference mirror. A planar wavefront enters the optical system from the objective space, which is replaced by an interferometer. The concave spherical mirror is located close to the image plane, where the beam converges at its focus point. The light goes back along the same path and enters into interferometer after reflected by the spherical mirror, and then an interference fringe is formed. Figure 5(d) displays the configuration where the planar reference mirror is located in the objective space. The spherical lens is mounted on the interferometer to provide a spherical wavefront, which is precisely located on the image plane of the optical system. Utilizing the reversibility of optical ray, the measurement of wavefront aberration can be simplified by adjusting the planar mirror and the position of the optical system.
Fig. 5 Optical measurement setup with (a) CCD sensor based on the planar wavefront input from the objective space, (b) wavefront sensor and microscope objective lens or (c) a spherical reference mirror and (d) Optical measurement setup with a reference plane mirror lens based on the spherical wavefront input from image space.
4. Experimental results and discussion
4.1 Optical design
Figure 6(a) presents the optical structure and path of the off-axis TMA. Herein, XY polynomial is adopted to express the freeform mirrors in order to eliminate off-axis aberration. In this way, the system guides light ray effectively and finally achieve the best optical performance. The automated optimization of the freeform mirrors [20] are conducted based on the manufacture constraints, including the curvature and the position of the freeform mirrors. In the meantime, the optical performance is guaranteed by adjusting the surface coefficients. Figure 6(b) demonstrates the ideal wavefront aberration of the central field of view, whereas the scale of the exit pupil is 23.89 mm. It can be seen that such value is less than 0.197λ, whereas λ is 632.8 nm. Thus the TMA satisfies the operational situation of infrared range. The main optical specifications of the designed system are listed in Table 4, including the expected optical performance the structure size of the designed system. It is indicated that the optical design achieve remarkable performance.
Fig. 6 (a) Optical path design and optical performance of and (b) wavefront aberration at central field of view.
Table 4. Specifications of freeform off-axis three-mirror imaging system.
The designed MTF curve is approaching the diffraction limit, confirming the design is effectively practical, as shown in Fig. 7(a), where the central field of view approximate the diffraction limit. In order to investigate the strict requirement of alignment accuracy, The structural tolerance analysis is executed. A statistics method is adopted in the analysis process to investigate the effects of alignment errors (i.e. the self-deviation and relative position deviations of three mirrors) on optical parameters. Figure 7(b) shows an example for the variation of MTF due to the rotating angle deviations along X-axis of the three mirrors. Results indicate that the primary and tertiary mirror should have relatively tight tolerances in angle deviations. The final tolerance is obtained that for the relative position error of 60 μm and the angle error of 0.03° according to the designed requirement of MTF > 0.4 at 100 lp/mm. It is inferred from above analysis that the relative high precision is necessary for alignment.
Fig. 7 (a) MTF at different field of view and (b) variations MTF along according to different angle errors of three mirrors.
4.2 Experimental procedure
Before the integrated machining of the designed TMA, it is necessary to ensure the machining accuracy to satisfy the requirement in optical design. The proper alignment is adjusted carefully and the main machine errors are identified by pre-machining typical reference surfaces. Herein, a spherical surface is machined by this machining configuration with the same cutting parameters as those in the practical machining.
Figure 8(a) shows the form errors of the pre-machined spherical surface with a curvature of 20 mm, the wave aberration can be used to deduct the form error using the transformation relationship. The peak-to-valley (PV) of the form error is 0.414 μm, reflecting that the machine errors should be decomposed. After the compensation of identifying the machine errors, the form error has been reduced by 67%, leading to the form error of 0.136 μm PV, as shown in Fig. 8(b). Moreover, it is illustrated in Fig. 8(c) that the main machine errors can be identified according to the analysis in Section 3.1. It is demonstrated that the source of machine errors are dominate by rotating radius R and tool nose radius Rt, while the squareness errors between axes can be neglected. It confirms that the identification of machine errors is reasonable and capable for the machine configure. Thus the identified errors can be compensated to machine tool to cater for the manufacture of the optical system.
Fig. 8 Form error of a pre-machine spherical surface: (a) before compensation, (b) after compensation and (c) the identified machine errors.
The freeform off-axis TMA is machined by an ultra-precision machine (Moore Nanotech 350FG UPL). Figure 9(a) is the experimental setup for the integrated manufacture. The TMA is fixed on the machine’s Z-slide by a special frame, while the fly-cutter is vacuum-chucked to the spindle. The nose radius of diamond tool is adopted as 6.0 mm so that the feed cycles in Z direction and cutting period can be reduced. The spindle speed is 1500 rpm with 100 mm/min in tool feedrate, leading to a 66.7 μm tool pitch along X direction. The tool feed intervals in Z direction is 15 μm. Key parameters for designing the rotating radius of fly-cutter listed in Table 5. According to the values of curvature radii and minimum distances, the rotating radius of fly-cutter is selected as 50.0 mm. Such value which is smaller than the distance between the second mirror and others, and the curvature radius of all mirrors guarantees interference-free state with tool rotating radius.
Fig. 9 (a) Actual machining setup based on the fly-cutting method and (b) configuration for measuring the optical performance.
Table 5. Key parameters for designing the rotating radius of fly-cutter.
The optical performance is measured immediately after machining using the measurement methods in Section 3.2. Figure 9(b) presents the measuring system. It is established based on the planar reference mirror in objective space, namely the method given in Fig. 5(d). An interferometer (Zygo GPI) is adopted under the work wavelength of λ = 632.8 nm. A spherical lens with the f-number of 3.3 provides the spherical wavefront. The form accuracy of the planar reference mirror is less than λ/10, and the angle θ in Fig. 5(d) is 7.85° at the central field.
4.3 Results and discussion
In order to investigate the machining quality, the surface topography is measured by white light interferometer (Veeko NT9500) with 5 × objective lens and 640 × 480 pixels. Figures 10(a) and (b) illustrate the surface quality and PSD of the machined mirror in the machine’s X- and Z- directions, respectively. Results show that the surface roughness reaches nanometric level (Ra 8.74 nm), proving that optical finish has been achieved in the machined mirrors, while the fish scale cutting marks can be derived from the distribution of PSD. It can be further seen from Fig. 10(c) that the spatial periods of the profile are about 72.6 μm and 29.9 μm in X- and Z- directions. Considering the raster milling path is zigzag along X-direction in the adjacent tool feed, the pitch on the surface topography is doubled in the Z-direction. Such phenomena are in good agreement with the cutting parameters, which proves the proposed machining method possesses good cutting performance. It is pointed out that such fish scale texture would not affect the final use of the TMA since the final imaging system works in the infrared range.
Fig. 10 (a) Surface topography, (b) two dimensional PSD and (c) specification of surface profile in X and Z direction.
Wavefront aberration in the machined TMA is measured by a Zygo GPI interferometer. It demonstrates a 1 × 3 grid of wavefront results at the field angle 0°, + 1° and + 2°, respectively. As shown in Fig. 11(a), the PV of wavefront aberration at the central field is 0.254λ (namely 0.161 μm), which is in agreement to the ideal wave front aberration in optical design (0.197λ). Whereas the edges of the field guarantees the high accuracy under the operation wave range, as shown in Fig. 11(b) and (c). Such value is considerable value for infrared optics, especially when the optical system is generated without any extra alignment. Figure 11(d) demonstrates the Zernike coefficients of wavefront aberration at the central field. It can be inferred that the 5th and 12th items of the Zernike polynomials are the major items, representing that the astigmatism in X direction dominates the wave aberration. Such phenomena corresponds to the combination of design and manufacture process, indicating that the stability and controlling of machine accuracy remains to be optimized to suppress this error component.
Fig. 11 The measurement results of wavefront aberration in the field of (a) 0°, (b) + 1° and (c) + 2° and (d) Zernike polynomials coefficient of the wavefront aberration.
The final TMA works as an infrared imaging system, which is shown in Fig. 12(a) actual sceneries are captured at different working distances. The scenes in distances of 1 m, 10 m and 500 m are snapped in the infrared band in Figs. 12(b)-(d), respectively. The real images satisfy the feature of high resolution and prove excellent performance of such alignment-free system.
Fig. 12 (a) The experimental assembly of infrared imaging system and the captured images in the working distance of (b) 1 m, (c) 10 m and (d) 500 m.
5. Conclusions
An integrated manufacturing method for freeform off-axis multi-mirror system is proposed to overcome the difficulties in alignment process in this paper. The effect of machine errors on optical performance is analyzed to systematically perceive the machining method and guide for achieving manufacturing accuracy. Measurement methods are also established by a self-made experimental equipment to evaluate the optical performance of the integrated system. Experimental results prove the proposed method can achieve the good optical performance only after the individual machining process. Based on this study, the identification and compensation can be implemented to further improve the optical performance. The idea of alignment-free manufacture dedicates to popularizing freeform optics and it demonstrates further feasibility in realizing complex optical system.
Funding
Science Challenge Program (TZ2018006), National Natural Science Foundation of China (Grant No. 61635008 and 51375337) and the Tianjin Research Program of Application Foundation and Advanced Technology (No. 17JCZDJC38600).
Acknowledgments
The authors express their sincere thanks to Yongbin Lu, Hangyu Xu and Yongxu Xiang for the dedicated efforts on the experiments. The authors express their sincere thanks to Jinjin Chen for the kind assistance of capturing real images.
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