## Abstract

Freeform optics has been regarded as the next generation of the optical components, especially those with non-circular apertures are playing an increasingly significant role in scanning field and specialized optical system. However, there still exist challenges to machine non-circular optical freeform surface. This paper is focused on highly efficiently generating freeform surfaces with optical surface quality by ultra-precision turning using a fast tool servo (FTS). A systematic strategy of machining smooth freeform surfaces with rectangular aperture is proposed in this paper. The contour of freeform optics is decomposed and assigned to the motions of slide and FTS back-and-forth. An optimized model is established for deriving the profile of the rotational component to cater for the capacity of FTS. Tool path reconstruction is carried out to generate a smooth tool trajectory and modified the contour to cater for the stroke of FTS. Simulation is adopted to analyze the machining property of a typical rectangular freeform surface. A rectangular freeform surface is efficiently machined via the proposed method, where a micron level profile error and nanometric finish in Ra are realized. Characteristics of reflection are analyzed via experiment and simulation. Prospects of such machining approach are discussed to provide guidance to future study.

© 2017 Optical Society of America

## 1. Introduction

With the rapid development of optics and manufacturing techniques, the geometry of optical elements has become more and more complex. Optical freeform surfaces, including smooth surfaces and micro-structured surfaces, are usually without rotational symmetry [1]. They are important devices in modern optical systems. Machining of freeform surfaces with a high finish has significant meanings in several aspects of requirements in recent decades [2, 3]. The shape of freeform surface are continually upgraded and modernized which demands increasingly efficient and accurate manufacture of surfaces with all kinds of apertures.

#### 1.1 Applications of non-circular freeform surfaces in optical fields

In traditional imaging applications, optical components are usually of circular aperture because it simplify the design and analysis of optical systems. The circular optical system decreases design issues where the object and image are rotationally symmetric with respect to the optical axis. Freeform optics are only introduced to reduce the volume and suppress the distortion. However, in modern optical fields, freeform surfaces are evolving so rapidly that the majority of the freeform components in optical system are of noncircular aperture, e.g. off-axis mirror [4–6], F-theta lens [7, 8], and freeform prisms [9, 10], etc.

It has been proved that for imaging applications with high aspect ratio, freeform optics assist to providing solutions with clearly better overall imaging performance both in on-axis systems and off-axis systems [11]. Thus the apertures of the components should be rectangular or even strip. Such form of edge also makes the alignment feasible since configures of the fixture can be easy to design and realized. F-theta lens is a typical high aspect ratio freeform optics in on-axis optical systems, and it is the core optical part in optical system of Laser Scanning Unit (LSU). It is responsible for realizing the uniform linear spot of the laser corresponding to the scanning angle, which determines the performance of the laser printer [8, 12, 13].

Another example is a rectangular-view off-axis three-mirror system which consists of three freeform optics based x-y polynomial surface [14], the non-circular freeform surfaces not only provide diffraction limited performance but also extend the usable field by an order of magnitude in area compared to traditional design. Moreover, it is able to meet the alignment issue and fit the space of the whole device.

Freeform prisms are increasingly frequently serving as off-axis display systems, e.g. optical see-through head-mounted display optical see-through head-mounted display [10], Virtual reality (VR) lens [15] and OLED display system [16]. Such optical components usually magnify the image in a microdisplay, the freeform surfaces are to the prism to eliminate distorted see-through view of a real-world scene. The prisms are shaped in rectangular aperture to provide a corresponding image of display, each surface maintains the strict shape accuracy and spatial position and orientation, which requires a complex design strategy and a superior difficulty in manufacturing process.

These applications clearly emphasizing the trends of non-circular freeform optics and their benefits in optical engineering system. The manufacturing approach is in urge demand to keep up with the trend of development so as allow the practical use of freeform optics with varieties of apertures.

#### 1.2 Difficulties and potentials in efficient machining of non-circular freeform surfaces

Freeform optics with non-circular aperture are usually machined via grid line diamond machining [16], diamond milling [17], raster milling [18] and fly-cutting [19]. Unfortunately, these machining techniques are often time-consuming. Nowadays, both high accuracy and high efficiency of optical freeform surfaces machining is in urge demand. Since grid line diamond machining is of high accuracy but time consuming, Fast tool servo (FTS) assisted ultra-precision has been regarded as an increasingly promising machining technique to manufacture ultra-precision freeform surfaces in electrical and optical fields [20, 21]. However, ultra-precision turning of non-circular freeform surfaces has been rarely discussed, as the tool trajectory of single point diamond turning (SPDT) is a spiral in three-dimensional space. There are difficulties in taking advantages of ultra-precision turning, especially when the freeform surface has a great sag variation within the aperture.

As for the freeform surfaces machined by FTS diamond turning, it is inferred from previous research were laid on micro arrays on planar, or surfaces with cycle structure [22–24]. Gao et al. presented the machining of a large area sinusoidal grid surface, whose profile is a superposition of sinusoidal waves in the X- and the Y-direction with spatial wavelengths of 100 μm and amplitudes of 100 nm [25]. Kong et al. enabled FTS machining the efficient technology for fabricating high quality microlens arrays with sub-micrometric form accuracy and nanometric surface finish and established a theoretical model for the prediction of surface generation in FTS assisted machining of microlens arrays in SPDT [26]. However, in ultra-precision machining with FTS, the resultant motion of machine tool motion is the composition of z-slide of machine and high frequency reciprocating movement of FTS. There is a lack of comprehensive research so that the constraint of surface separation and tool path generation still remain ambiguous and even less than systematic. Understanding of surface generation principle by FTS is essential to efficient and high quality machining of smooth optical freeform surfaces, and it makes an increasing varieties of surfaces with special characters practical to be machined by ultra-precision turning. In a previous study [27], we presented the design of a near-rotational freeform surface (NRFS) with low non-rotational degree (NRD) to constraint the variation of traditional freeform optics to utilize the feature of FTS. It is emphasized that FTS assisted ultra-precision turning can possess an advantage when the spindle speed is relatively high, and it benefits in improving the machining efficiency.

This study will be focused on FTS-assisted diamond turning of smooth curve surfaces with rectangular aperture. Principles of machining freeform surfaces by FTS is investigated, surface decomposing methods are proposed to obtain rotational and non-rotational components of freeform surfaces, tool path construction of non-rotational component is proposed to guide the process of surface data. Then, a typical rectangular smooth curve surface, i.e. F-theta lens is taken as an example to be analyzed, and construct the optimized tool path. FTS motion performance is discussed to predict the machining effect. Finally the surface is machined by FTS assisted ultra-precision machining, both topographic and optical characterization are carried out to verify the feasibility and effect of the machining approach.

## 2. Principles of machining freeform surface using FTS assisted ultra-precision turning

The arrangement of FTS-assisted diamond turning freeform surfaces with non-circular aperture is illustrated in Fig. 1. The non-circular aperture freeform surfaces must be extended to circular aperture firstly, because the machine tool transverses the whole circular aperture in the turning process. The contour of freeform surfaces should be decomposed into z-slide motion and FTS back-and-forth motion, wherein component analysis is conducted. Tool path reconstruction is formulated in order that the tool motion can be limited within the capacity of FTS. Machine tool path is then generated and compensated due to surface decomposition to suppress form error. Kinematics characterization and frequency domain analysis are carried out in order to predict the effect of reconstructed tool path. Machining parameters, including tool geometry, feedrate and spindle speed, are determined based on surface feature and work material. An F-theta surface is machined according to the above analysis. Surface topography and surface quality are measured and evaluated to validate the proposed methods. Optical performance are characterized with relevant simulation to give guidance to optimizing the processing parameters and machining strategy. Hence, a systematic research on optimizing FTS turning freeform surfaces is then researched to present the approach of manufacturing non-circular smooth curve surfaces and even other complex surfaces.

#### 2.1 Surface component analysis of extended circular freeform surface

In practical FTS machining, machining tool path should be generated to realize the manufacturing. It is usually converted from the original freeform surfaces, which is defined in a Cartesian system. Thereby, it needs to be transformed to the machine system, consisting of the C-axis—the spindle, X-axis—the x-slide, Z-axis—the z-slide, and W-axis—the FTS. According to the feature of kinematic controller in FTS system, an extra 3D compensation data is produced and assigned to FTS motion. Thus the surface component analysis is critical, including surface decomposition and the evaluation of non-rotational surface (NRS). The surface should be separated in order to obtain the non-rotational component, whose sag range in the whole circular aperture is defined as NRD. Current decomposing methods are the searching for an appropriate contour which can fit the waviness on every sagittal direction. Herein, a decomposing method based on the equation of the surface shape is proposed to find out the equation of the rotational surface generatrix, namely *z _{r}*. In order to demonstrate such method, a typical example of freeform surface is shown in Fig. 2. The original freeform surface is in rectangular aperture [see Fig. 2(a)], it should be extended to circular aperture in order that it can be machined by ultra-precision turning. The extended freeform surface is shown in Fig. 2(b). Herein, a certain profile along radial direction can be selected as the rotational surface generatrix, namely

*θ*is a certain arc coordinate on the projection of cutting plane. Hence, the rotational component can be obtained whose expression only contains the radial variable

*r*[see Fig. 2(c)], so the rotational component

*z*can be expressed as

_{r}Based on above analysis, the object of surface component analysis is to finish surface compartment, which is the optimization of the profile of rotational surface. It requires the searching of the angle *θ* which leads to: (1) minimum range of NRD and (2) symmetric distribution of NRS contour [see Fig. 2(d)]. In this way such issue can yields to a convex optimization below:

*w*

_{1}and

*w*

_{2}is the normalized weight of each index, namely

*w*

_{1},

*w*

_{2}

*>*0 and

*w*

_{1}

*+ w*

_{2}

*=*1, moreover, variables

*r*and

*φ*are within the range of the surface aperture denoted as Ω. It is now translated into a one-element optimization problem with extreme value in objective function. Denote

*θ*

_{0}as the optimizing result, thus the final angle can be searched to obtain the profile of rotational surface. Thus the rotational component is

*z*(

*r*cos

*θ*

_{0},

*r*sin

*θ*

_{0}), and the NRS can be derived by subtracting the rotational component from the original surface.

#### 2.2 Interpolation strategy of the surface contour

It can be found that when dealing with surfaces which are completely non-rotational (e.g. toroidal surface), sag of NRD remains the same whatever profile is selected as the rotational surface profile. When the surface aperture is rectangular instead of circle, the surface contour outside the aperture can be modified into appropriate value. Hence, in this case, interpolation transition is an alternative way to solve the issue. Herein, Hermite interpolation is introduced to deliver the surface contour, because it can keep arbitrary rank derivatives invariant on the whole surface. The interpolation equation between n knots is expressed as

*ξ*is the parameter of interpolating node,

*z*

_{j}^{(}

^{i}^{)}is the

*i*th derivative of the (

*j*+ 1)th point on the undelivered surface,

*φ*is the (2

_{ij}*n*+ 1)th order polynomials in the Hermite interpolation. In order to guarantee the smoothing interpolation transition, the first order Hermite interpolation is applied, which is

*α*and

_{j}*β*are the (2

_{j}*n*+ 1)th order polynomial. The interpolation of the surface contour is shown in Fig. 3. Variation range of surface contour in sagittal direction is transversed by FTS. After interpolation, such value is greatly reduced, and it becomes more symmetrically distributed around zero to meet the requirements of FTS. NRD can be therefore minimized to obtain stroke of FTS. Thus, interpolation transition enables such kind of surface machined by FTS diamond turning.

#### 2.3 Tool path reconstruction

Based on the analysis in Section 2.1, the tool motion can be decomposed according to surface component analysis. On one hand, the rotational component *z _{r}* is expressed as a two dimensional curve profile which only contains the radial variable

*r*. Thus it can be realized by conventional diamond turning with the x- and z-slide of the machine. On the other hand, as shown in Fig. 4, the shape of NRS consists of a series of points which are the sampled data of contour value. Herein, according to the controller in our FTS system, machining tool path is generated via azimuth sampling on freeform surface. These points are arranged in concentric circles from the outside to the inside with a constant angle interval on each concentric circle. It can be inferred from the subplot on x-y plane that the concentric circles generate a polar grid to characterize the NRS, where

*N*is the number of concentric circles and

_{r}*N*is the number of cutting points on every single circle. Since it is unfavorable that freeform surfaces these have some common defects (such as pits, streaks, and waviness), which are primarily due to insufficient constraints of NRS, it is necessary to propose instructions to the reconstructing tool path in order to obtain these data.

_{t}The characteristics of NRS directly affects the feasibility and quality of FTS machining. The sag of the NRS requires to be smaller than the FTS stroke. As for the frequency of the separated surface, it depends on the parameters in the machining process, assuming that the spindle speed is *S*, and the tool feedrate is *f*, thus the motion of every revolution trajectory is a function of time *t* (which is conducted as the parametric variable), and it is expressed as

Surface frequency domain can be treated as the Fourier transformation of *z _{n}* in a revolution which has the most peaks or valleys. Since the amplitude-frequency characteristic meets the feature FTS, the feature of the surface should satisfy some further constraints. Considering the kinematical feature and capacity of FTS, the constraints of non-rotational component

*z*can be summarized as following mathematical description:

_{n}- (1) The range of non-rotational component
*z*should be within the stroke of FTS and the frequency domain should be within the bandwidth of FTS;_{n} - (2) The first order derivative of non-rotational component
*z*_{n}_{1}is continuous differentiable, and oscillates around zero. Its absolute value should not be larger than the velocity threshold of FTS; - (3) The second order derivative of non-rotational component
*z*_{n}_{2}is continuous and without mutation, and the absolute value should not be larger than the acceleration threshold of FTS.

Furthermore, tool path compensation should be adopted to guarantee the machine accuracy. Herein, a compensation of z-direction is introduced to realize a stable tool nose radius compensation [28].

The impact levels of difference processes on surface feature are summarized in Table 1. In comparison with some previous research [29–31], it can be seen that distribution of NRD is mainly determined by surface decomposing, while frequency domain and kinematics of FTS are strongly affected by reconstruction of tool path and cutting parameters, respectively. These methods on pre-processing the machining surface all have a great impact on machine efficient and accuracy. Hence they are key techniques on FTS-assisted diamond turning freeform surface.

## 3. Simulation of the machining approach

Based on the machining approach established in the above section, the machining property of freeform surfaces with rectangular aperture can be studied. An F-theta lens is analyzed to simulate the machining approach of FTS diamond turning. The surface parameters of F-theta lens are listed in Table 2. A diamond tool with a nose radius of 0.5 mm is selected as the machine tool. The spindle speed is 500 rpm which is suitable for ultra-precision turning and the feed rate is 1 mm per minute. The cutting point distribution is selected as 846 × 260 in *N _{r}* ×

*N*to guarantee the form accuracy of the machined surface.

_{t}The surface edge should be firstly transformed into a circular aperture, thus the surface aperture is extended to 15.5 mm in radial dimension [see Fig. 5(a)]. Figure 5(b) is the NRD distribution of the surface within the rectangular aperture when choosing different angle *θ* of the surface profile as the rotational surface profile. It can be inferred that there exists large variation in the sag of NRD with different angle. Therefore, it requires that the objective function coefficient *w*_{1} in Eq. (3) should be larger than *w*_{2} so that the sag of NRD is small enough and the symmetric distribution of NRD is taken into account as well. Figure 5(c) shows the value of the objective function when (*w*_{1}, *w*_{2}) = (0.6, 0.4). It makes the objective function minimum, where *S* (*θ*_{0}) = 1.315 × 10 ^{– 4}, the corresponding *θ*_{0} equals to 88.65°.

Figure 6(a) is the NRS after surface decomposition when *z _{r}* (

*r*) =

*z*(

*r*cos88.65°,

*r*sin88.65°). The corresponding NRD is [–479.227, 26.133] μm, which is considered too large for FTS to track at a relatively high frequency. Hermite polynomial interpolation is employed here to deliver the contour outside the rectangular aperture in order to control the range of NRD, because it can keep the derivative of the surface edge invariant and change the surface contour outside the aperture. Figure 6(b) is the NRS after being delivered, it can be seen that the sag of NRS has been rapidly decreased. Due to the fact the direction of Hermite polynomial interpolation is along the circumferential direction, it is somewhat inevitable that the delivered NRS is not continuous in radial direction. These discontinuous areas locates outside the aperture of the freeform surface, so it does not affects the final surface quality. The FTS motion should be smooth and fluent as long as the points in each concentric circle are smooth. The delivered NRD has been reduced to [–51.542, 26.133] μm, which is greatly improved by the above interpolation strategy. Hence the reconstructive tool path can be generate according to the NRS by azimuth sampling on each concentric circle in the extended aperture.

Frequency domain is derived by Fourier transform of the NRS, it reflects the feature of the FTS reciprocal motion. The edge of the surface is usually sampled to be analyzed so that the frequency domain can be evaluated because the sag value is the largest at the edge of the NRS.

Figure 7 is the amplitude-frequency characteristics of the NRS before and after delivered. Such characteristics represent the property of FTS during the cutting process. A small amplitude and bandwidth in frequency domain benefits in a high accurate motion and a good tracking effects in the manufacture of surface with optical finish. It can be inferred that the characteristic amplitude-frequency of the original NRS is 25.36 μm @16.67 Hz whilst that of the delivered NRS is 4.70 μm @16.67 Hz. Results show that the amplitude has been dropped about ten fold in the delivering process except for extending the range of the frequency to 1.65 μm @83.34 Hz, which can be neglected considering the ability of FTS. These changes contribute to improving the FTS kinetic characteristic and enable such technique to machine non-circular freeform surface.

## 4. Experimental results and discussions

The machining experiment of the analyzed F-theta lens has been carried out to validate the effectiveness of the proposed theory and methods. Figure 8 is the illustration of the machining process. The x- and z-slides feed in radial and cutting depth direction, respectively, while the spindle provides the feed in circumferential direction. The F-theta lens workpiece is clamped on the spindle via a special fixture. Diamond tool is driven by FTS is reciprocating with a corresponding high frequency to mesh with the spindle motion. The experimental parameters are provided in Table 3. The machining cycle is less than half an hour, which is far more efficient than other machining approach. A commercial optical tracking profiler is adopted to analyze the form errors so as to evaluate the machining accuracy whilst the surface quality is obtained by white light interferometer. Optical response is analyzed to evaluate the machining approach so that the prospect of such efficient method is discussed.

#### 4.1 Topographical information

Figure 9(a) is the topographic result of the F-theta lens, where 9 profiles in X direction and 5 profiles in Y direction is measured by an optical tracking profiler. The profile pitch is 3.875 mm in X direction and 1.5 mm in Y direction. The interval of measuring data is about 0.645 μm, which depends on the slope of the surface. The measured sag of the surface is 873.426 μm which is fairly agreed with the predicted value, namely 873.553 μm.

In order to present a further evaluation of the form accuracy, central profile is selected to analysis the profile error. Figure 9(b) illustrates the data of the profile at the center and the corresponding profile error. It can be seen that the profile error is from a few hundreds of nanometers to about 1.45 μm, which decreases from the center to the both side. It is probably due to the system error of the optical tracking in measuring process, because the tracking system cannot perform well in dealing with the minimal change of sag at the center. On the other hand, the coordinate distortions in the X and Z direction caused by the machining errors are also among the influence factors of profile error. The compensation of these identified machining errors can be regarded as the future work to improve the machining accuracy of these typical smooth freeform surface.

Three-dimensional surface metrological detail of F-theta lens are obtained by a Sensofar white light interferometer in order to better validate the machining approach. Figure 10 is the results of surface roughness at different areas, including the center, the midfield and the edge. Based on the removal of flatness error in terms of a high degree polynomial, the data are captured with a 20 × objective amplitude. It can be inferred that the surface roughness Ra value remains nanometric level and it increases from center to edge, where the surface roughness at the center is 5.5 nm in Ra and those at the edge are no more than 15.0 nm in Ra. Surface quality corresponds the texture of ultra-precision turning, which is much smaller than that in milling or fly-cutting. Hence, ultra-precision turning has a distinct advantage in suppressing the machining texture during generating freeform surface. Such results confirm that the cutting strategy guarantees an optical manufacturing effect.

#### 4.2 Optical performance

The surface diffraction behavior is regarded as the tool mark in radial direction, which is similar to side-feeding feature in fly-cutting [32]. It can be seen from Fig. 11(a) that the reflected spot from the surface center is more energy concentrated than that in fly-cutting, indicating the feasibility and benefit in ultra-precision turning. Texture surrounded the spot is recognized as the image of tool marks, due to which there exist a week diffraction ring (because the tool marks have a similar feature with Fresnel lens but not so obvious). Figure 11(b) is the reflection characteristics at the edge of the surface aperture. The tool mark residual is large due to the corresponding surface slope, resulting in reflection spot with several high order diffraction. Moreover, it is found that the high order diffraction spot spread along circumferential direction, because the control dot distance of the tool path is large at the edge, where the tool trajectory is linear interpolated.

In order to investigate the behavior of diffraction effect, simulations are carried out using classic field tracing [33]. The tool marks are replicated on the surface to calculate the detail of optical field. Results are shown in Fig. 12, where amplitudes of the electric field components distribution fairly agrees with that of the optical intensity in the experiment (*R _{t}* = 0.5 mm,

*f*= 2 μm). It is further noticed that the reflective light is more energy concentrated when a small tool mark pitch is adopted. Large tool feed pitch results in series diffractive phenomena because these side-feed tool marks generate a periodic structure which plays a role of optical grating. On the other hand, a moderately large tool nose radius can assisted improving the reflection characteristics of the freeform surface, but perfect cutting tools with large nose radius are hard to fabricate and it may cause surface defect. Hence, future study of ultra-precision turning non-circular freeform surface should be focused on the improvement of cutting parameter and cutting strategy and eliminating the tool mark induced diffraction effect.

## 5. Conclusions

This paper provides an approach to machining smooth freeform surfaces with rectangular aperture via ultra-precision turning. Principles and constraints has been given to enhance the performance during machining process. Optimization of surface decomposition has been introduced and its advantages have been taken to improve cutting stability. A typical rectangular freeform surface has been successfully machined by such technique at a relatively high spindle speed to validate the machining strategy. The main conclusions can be drawn as follows:

- (1) Ultra-precision turning using FTS is regarded as an efficient approach to machining freeform surfaces with non-circular aperture when the edge of the aperture can be extend to circular edge. It is distinguished from fly-cutting and milling process because it has much higher efficiency and a fast-traveled tool trajectory. The configuration of FTS upgrades the applicability in ultra-precision turning freeform surfaces.
- (2) An optimization model has been given to search for the best rotational surface profile to be decomposed and obtain NRS with a minimized and symmetrically distributed NRD, aiding in manufacturing freeform optics with high accuracy and high quality. Reconstruction of tool path can be made enhance the performance of FTS via modifying the z-direction motion of the tool trajectory which locate outside the aperture of the surface.
- (3) Surface topographic details show that form error of a micron level and surface roughness of 5.41 nm has been achieved. The analysis of optical performance show that the effects of tool marks in other cutting process can be suppressed by FTS assisted ultra-precision turning to some expense. Machining time are greatly reduced which confirm the feasibility and efficiency.

## Funding

National Natural Science Foundation (Grant No. 61635008 & 51320105009), National key Research and Development Program of China (Grant No. 2016YFB1102203), and the ‘111’ project by the State Administration of Foreign Experts Affairs and the Ministry of Education of China (Grant No. B07014).

## Acknowledgments

The authors would like to express their sincere thanks to Y.B. Lu and Y.X. Xiang for the preparation of experiments and thanks to N. Yan for the valuable advice on simulation of field tracing.

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