Abstract

Optical freeform surfaces are of great advantage in excellent optical performance and integrated alignment features. It has wide applications in illumination, imaging and non-imaging, etc. Machining freeform surfaces on infrared (IR) materials with ultra-precision finish is difficult due to its brittle nature. Fast tool servo (FTS) assisted diamond turning is a powerful technique for the realization of freeform optics on brittle materials due to its features of high spindle speed and high cutting speed. However it has difficulties with large slope angles and large rise-and-falls in the sagittal direction. In order to overcome this defect, the balance of the machining quality on the freeform surface and the brittle nature in IR materials should be realized. This paper presents the design of a near-rotational freeform surface (NRFS) with a low non-rotational degree (NRD) to constraint the variation of traditional freeform optics to solve this issue. In NRFS, the separation of the surface results in a rotational part and a residual part denoted as a non-rotational surface (NRS). Machining NRFS on germanium is operated by FTS diamond turning. Characteristics of the surface indicate that the optical finish of the freeform surface has been achieved. The modulation transfer function (MTF) of the freeform optics shows a good agreement to the design expectation. Images of the final optical system confirm that the fabricating strategy is of high efficiency and high quality. Challenges and prospects are discussed to provide guidance of future work.

© 2017 Optical Society of America

1. Introduction

Machining freeform surfaces on infrared (IR) optical materials with a high finish is a hot topic for several aspects of requirements in recent decades [1]. However, it is still a difficult task due to brittle nature of IR materials. It is known that germanium has excellent electrical and optical property and gains attention as an IR material for important applications in the fields of aerospace, telecommunications and high-speed transistors [2–4]. It was claimed that germanium can be removed in a ductile mode when the material is under high hydrostatic pressure, thus optical surface quality can be achieved [5, 6]. When the undeformed chip thickness is below a critical value dependent upon material properties and tool edge radius, the material is removed by extrusion [7, 8]. Molecular dynamics simulations [9, 10] also predicted ductile behavior of brittle materials based on its properties and inspired confidence of machining such kind of materials. Consequently, it is probable to obtain freeform surface on brittle materials with optical finish by mechanical process.

Freeform optics are complex surfaces including smooth surfaces (e.g. polynomial surface and toroidal) and micro-structured surfaces (e.g. microlens array and compound eye), which are increasingly applied to both imagining and optical illumination systems [11–13]. Among these applications, f-theta lens, which is a typical kind of smooth surfaces, manages to play a role of freeform imaging lens with a wide linear field-of-view (FOV) [14]. Compound eye usually has extremely small volume and very short processing time, so it was expected to be applied in high-speed object tracking and depth perception [15]. As for machining process, previous study showed that diamond milling is an alternative approach to obtain freeform surface on brittle semi-conductor materials. Machining of silicon carbide using polycrystalline diamond end mills has been realized [16], results provided a guidance for milling hard and brittle materials. Single crystal germanium refractive Alvarez lens was fabricated by diamond milling for use in the mid-wavelength infrared [17], nanometric surface roughness values were achieved. This fabrication process worked well overall. Moreover, efforts have been made to enhance the surface uniformity by eliminating effects of tool side feed during turning process by introducing an extra degree of freedom tool servo [18, 19]. On the basis of above, coordinated-axis diamond turning configured with FTS is a powerful technique to machine freeform optics. As for ultra-precision turning operation, a special custom kinematic mounting equipment in five-axis free-form micromachining was utilized to machine optic array on germanium with nano-scale RMS and center-to-center positioning accuracy under a micron [20]. Moreover, silicon, zinc selenide and zinc sulfide were all machined by diamond turning in a ductile mode [21–24]. In fast tool servo (FTS) machining process, complex surfaces with small sag can be realized with a high efficiency and high quality [25].

Application of FTS was introduced to the compensation of motion errors of machine tools [26], it is also well-known that this technique was applied for freeform surface generation in recent decades [27]. It can be inferred from above that FTS machining has been recently focused on machining micro-structured surfaces but rarely on smooth surface, while most smooth surfaces are machined by diamond milling. Because it has been widely known that the approach of FTS could have difficulties with large slope angles and large rise-and-falls in sagittal direction on the surface [28, 29]. However, FTS has its own advantages for it has a much higher efficiency due to its excellent stroke and reciprocal frequency, so it is alternative to machine freeform optics on IR materials. As for other turning process (e.g. slow slide servo, SSS) which has also been adopted to machine micro-structured surfaces [30] but it cannot achieve high efficiency remove of brittle material so the fracture may occur in a large tool feed rate. On the contrary, when the spindle speed and cutting speed are both very high, it leads to a small material removal amount in one single cut. Therefore it can realize high speed machining on freeform optics. In addition, comparing to diamond milling, FTS can also realize smooth surface and it is an alternative approach from time-saving and surface generation quality perspective [29].

In this paper, we presented FTS-assist diamond turning on a germanium freeform surface. In order to take the advantage that freeform with gentle fluctuation can be perfectly realized by FTS, rise-and-falls in sagittal direction should be reduced. A near-rotational freeform surface (NRFS) should therefore be proposed to avoid difficulties in machining. Moreover, such NRFS should satisfy the functional requirement in the optical system, which means it only constraint the variation amplitude of the surface shape but it still plays a role of traditional freeform optics. Herein, a surface whose shape equation is the sum of aspheric and x-y polynomial is selected to design as a NRFS. Designing of a NRFS and machining procedure are analyzed to demonstrate the high quality FTS turning technique. Results and some issues are discussed to explain the surface generation. Finally, conclusions and challenges are given to point out the future development on machining freeform optics on brittle materials.

2. Basic principle of designing and ultra-precision turning of NRFS

2.1 Set up of machining freeform optics on IR materials

The strategy of machining freeform optics on IR materials via ultra-precision turning is shown in Fig. 1. For good turning quality and function, the design of NRFS should come first. Then the component analysis on NRFS is operated to separate the original surface into a rotational surface and a non-rotational surface (NRS). The height range of NRS is defined as non-rotational degree (NRD), which describes the upper limit and lower limit characterizing the distribution of height in NRS. This index is derived to evaluate the flexibility of FTS. Suitable machining parameters are selected carefully, spindle speed, feed rate, and tool geometry are taken into account in order that the material is removed in a ductile mode. Then the corresponding tool path is generated and compensated, which is supposed to be decomposed into motion trajectories for machine axes and FTS reciprocating motion. Machined freeform surface is characterized by microscopic instrument to detect the surface damage. Measurement data and details are discussed to verify the machining quality and efficiency under the selected machining parameter. Finally, the optical performance is evaluated by comparing the measured Modulation Transfer Function (MTF) with the design one. Sample IR images of the optical system are provided to demonstrate the functional effect. Above all a systematic research on ultra-precision turning freeform optics on IR materials is presented to propose and investigate the application of this technique.

 figure: Fig. 1

Fig. 1 Strategy of machining freeform optics on IR materials.

Download Full Size | PPT Slide | PDF

2.2 Design strategy of NRFS

In the surface design of NRFS, the initial surface is set as aspheric surface, which is of rotational symmetry. Thus the surface is described as

z(r)=cr21+1(1+k)c2r2+i=1Airi,
where c is the curvature, k is the conic coefficient, r is the radial variable in polar coordinate which equals to (x2 + y2)1/2 in Cartesian coordinate. While acting as a freeform optics, the second terms of Eq. (1) should translate into the term of x-y polynomial, which is expressed as
z(x,y)=cr21+1(1+k)c2r2+m=1n=1Amnxmyn,
where m=1n=1Amnxmyn is the polynomial part of the surface which is expressed as

m=1n=1Amnxmyn=A10x+A01y+A20x2+A11xy+A02y2+.

The result of Eq. (2) can be obtain by adding a small fluctuation to the initial aspherical surface, namely

z(x,y)=z(r)+δ(x,y).

Since the object is to obtain a NRFS, the difference δ(x, y) between NRFS and the initial should be minimized. In other words, in order to make the freeform optics closed to aspherical surface, the coefficients of Cartesian variables in Eq. (3) are in a certain relation. It can be inferred that the amplitude of independent items xmyn should be minimized after transforming the Cartesian variables into polar ones, so the coefficients Amn should satisfy some certain conditions according to coordinate transformation approach. Since x and y coordinate and the radius coordinate r satisfied the coordinate transformation, namely x2 + y2 = r2. So the coefficients should yield to (A2n,0x2n + A0,2ny2)n≈anr2n (n = 1, 2, 3...), otherwise the surface will be so non-rotational that the freeform surface is not NRFS since the angle variable φ is introduced. Thus it can be derived that the coefficients Amn should satisfy

A2m,2n=Cm+nman,m,n=1,2,3,
where C is the combination operator, an is the coefficient. In this way, for instance, the ratio of A20 and A02 shall be about 1:1, and that of A40, A22, and A04 is limited to approximately 1:2:1. On the analogy of this, that of A60, A42, A24, and A06 is limited to approximately 1:3:3:1, the rest can be done in the same manner. Based on the constraint in coefficient mentioned above, other coefficients are input to introduce minor changes to the surface in order to realize the surface function.

Based on the constraint in coefficient mentioned above, other coefficients are input to introduce minor changes to the surface in order to realize the surface function. It has been used in the design of IR imaging system consisting of three lenses for the uncooled infrared detector. It possesses the following parameters, including F number of 1.0, focal length of 17.6mm and field of view (FOV) of 40° × 30°. As shown in Fig. 2(a), the spherical surfaces are adopted in the whole system except the rear surface of the third lens using freeform optical surface, which is NRFS optimized according to the proposed method. NRFS significantly decreases the system distortion as well as reduces the number of optical lens. After optimization design, the RMS of spot size for all of FOV is less than the pixel size of 20μm, and the distortion value is less than 2.0%. Figure 2(b) shows the MTF curve, providing that the system is designed to approach the diffraction limit. Hence, introduction of the designed NRFS improves the structure and performance, and it reduces the lens number of system to improve the transmission rate of infrared signal. Moreover, NRFS does not only play a role of traditional freeform surface but also achieve a relatively low degree of complexity.

 figure: Fig. 2

Fig. 2 Design of (a) structure of IR optical system, (b) spot diagram and (c) MTF of system.

Download Full Size | PPT Slide | PDF

2.3 Component analysis on NRFS

It is demonstrates in Fig. 3(a) that the final surface can be designed and the freeform surface closed to a rotational surface after a series of optimum designing process discussed above. Based on the discussion on the Section 2.2, the designed freeform surface should be separated in to a rotational surface and a NRS (whose height range is NRD as described above), namely

z(x,y)=zr(r)+zn(x,y),
where the first term zr(r) is the rotational component and zn(x, y) is NRD. Since the rotational components only relate to radial coordinate r, it can be expressed as a 2D surface curve. In this case it is written as
zr(r)=cr21+1(1+k)c2r2+a1r2+a2r4+a3r6+,
where the coefficient an equals to the average of coefficients of x2n and y2n in Eq. (3), namely

 figure: Fig. 3

Fig. 3 (a) Design of NRFS and its (b) Rotational 2D surface curve and (c) NRS.

Download Full Size | PPT Slide | PDF

an=A2n,0+A0,2n2,n=1,2,3.

Thus the rotational part is presents as a 2D surface curve in Fig. 3(b). It can be seen that the curve is much closed to any radial profile of the designed surface. After extracting it, NRD can be furtherly obtained. It is indicated in Fig. 3(c) that NRS has very small sag inside the aperture, NRD of [−26.443, 29.638]μm is derived. Moreover, NRD is of symmetrical distribution at both ends of zero point, which is in favor of the motion performance of FTS. It can be also noticed that two peak-and-valleys appear in each circle, indicating that the reciprocal frequency of FTS is two times of the spindle speed. These indexes are completely within the motive capacity of FTS at a high spindle speed. Therefore, FTS can guarantee the cutting range of the surface. Thus it can be confirmed that the design of a NRFS, which is suitable for ultra-precision turning by FTS, has been achieved.

As for practical machining process, NRFS is separated into the rotational component and NRD, the former one is described by a 2D surface curve which is traversed by x-slide and z-slide of the machine, and it generates the rotational part of the surface coupling with a rotating spindle; while NRD with little sag which is traversed by FTS. Hence the resultant motion trajectory of cutting tool is a space curve on the NRFS. The trajectory is azimuth sampling whose projection is spiral curve on x-y plane.

3. FTS-assisted ultra-precision machining of NRFS

3.1 Machining parameters

The ultra-precision turning of NRFS on single crystal germanium was performed on a Moore Nanotech 250 ultra-precision diamond turning machine with a FTS system mounted on W-axis. The spindle speed is set 2500rpm according to the common turning spindle speed for germanium with the feed rate of 1mm/min. Diamond tool with 0.5mm nose radius is selected as cutting tool and the rake angle is −12°. According to the process of FTS diamond turning with an arc cutting tool, the undeformed chip thickness is relative to cutting parameters and tool geometry. As shown in Fig. 4, assume that the feed direction is towards the center of the surface, then the distance of the adjacent tool position is

L=f2+Δz02,
where f is the tool feed per revolution(mm/rev), Δz0 is the difference along z-direction between two positions. The calculation of undeformed chip thickness in FTS diamond turning can be derived from the geometric relations. Point P-1 and P0 represent the previous cut and current cut, respectively, Rt is the tool nose radius and a0 is the depth of cut for the current pass. The undeformed chip thickness is the segment locating on the radius line of the current cutting tool nose radius (e.g. Segment FG represents a certain one.) which varies from zero at the tip (point A) of the tool to a maximum at the top of the previous uncut region (point F) and to zero again at the top of the current uncut region (point D). It can be seen that segment CE is the maximum undeformed chip thickness, which can be expressed as
dm=Rt[(Rt2(Rta0Δz0)2f)2+(Rta0)2]1/2,
and the position of the maximum undeformed chip thickness is

 figure: Fig. 4

Fig. 4 Schematic of FTS-assisted ultra-precision turning.

Download Full Size | PPT Slide | PDF

αm=arccos(Rta0Rtdm).

Under the above cutting condition, it can be inferred that the calculation of undeformed chip thickness is similar to the situation in end face turning when Δz0 is very small. It can therefore guarantee that undeformed chip thickness is below the brittle-ductile depth when the depth of cut is a few microns [6]. Owing to FTS turning, such high spindle speed can be adopted to machine freeform optics on brittle materials since FTS can realize high frequency so that a very small material removal can be achieved in two adjacent revolution during a high spindle speed turning process. Thus the machining parameters mentioned above including spindle speed, feed rate and tool geometry are selected for the machining process.

3.2 Tool path generation and compensation

Tool path is generated via the azimuth sampling on the surface, the height values on the path are derived by calculating rotational 2D surface curve data and 3D values of NRD, where the former data controls the motion of x-slide and z-slide and the residual controls the motion of FTS. The azimuth sampling on the surface is shown in Fig. 5(a). In this way, the projection of the relative tool-workpiece movement to the x-y plane is a spiral curve (ρ0, φ0), which can be determined by the feed rate fS (mm/min) and spindle speed S (rpm), the coordinate of the motion in z-slide (denoted as z0) and FTS (denoted as w0) can be expressed as

{ρ0=RwfStiφ0=2πStiz0=zr(r)|r=ρw0=zn(x,y)|x=ρ0cosφ0,y=ρ0sinφ0,
and
ti=60iSNt,i=0,1,,RwSNtfS,
where Rw is the aperture radius of the workpiece, Nt is the number of dotted spiral curve. It can be seen that very coordinate variable can be equivalently expressed by the time parameter ti which contains a certain dot number Nt. Thus, tool path can be characterized by the motion of the x and z-slides Fig. 5(b), it can be inferred that x-slide and z-slide motions are the sampling of rotational 2D surface curve, while the motion of FTS can be described by Fig. 5(c), which only consists one degree of freedom, namely the reciprocal movement in W-axis. Thus the resultant ideal tool path (x(ti), y(ti), z(ti)) is obtained.

 figure: Fig. 5

Fig. 5 Schematic of (a) tool path on NRFS, (b) x-slide and z-slide motions and (c) FTS motion.

Download Full Size | PPT Slide | PDF

After tool path generation, compensation should be committed in order to improve the machining accuracy and quality, where geometries of the diamond tool [31]. Herein, tool nose radius difference is modified by a compensation process via a z-direction compensation method to make it content to a stable cutting condition [32]. Finally the machining tool path after compensation is downloaded into the controller of the machine and decomposed into the motion trajectories for each axis.

4. Results and discussions

4.1 Measurement and metrology of NRFS

To demonstrate the flexibility of the machining approach, the NRFS on germanium with the aperture of 30mm is successfully machined. Illustration of FTS-assisted ultra-precision turning is shown in Fig. 6(a), it consists of four axes, X-axis, Z-axis, C-axis and W-axis. The X-axis and Z-axis translational slides provide feed in radial and cutting depth direction respectively while C-axis is driven by spindle speed. The germanium workpiece is adhered to a steel plank which is vacuum-chucked to the spindle. While the spindle is rotating at a high speed, FTS is reciprocating with a corresponding high frequency. X-slide and Z-slide are traversing the rotational surface curve coupling with the movements above. All these kinematical states generate the final NRFS. As described above, the spindle speed was 2500rpm with the feed rate of 1mm/min in machining. Machining result is shown in Fig. 6(b), it can be seen that mirror surface has been achieved by FTS-assisted ultra-precision turning. Since there does not exist mottled image or white streak on the surface, it can be inferred that few macro brittle factures occurred during cutting process. Further measurement should be operated on a high accuracy measurement device.

 figure: Fig. 6

Fig. 6 Illustration of (a) FTS-assisted ultra-precision turning and (b) machined germanium surface.

Download Full Size | PPT Slide | PDF

In order to investigate the machine quality, three-dimensional data of NRFS are measured by Sensofar white light interferometer. With the accurate movement of the device, different areas of the surface are measured, as shown in Fig. 7(a). After removal of flatness error in terms of a high degree polynomial, surface roughness is obtained. The data are captured with the objective amplitude of 20 × and the profile 1360 × 1024 pixel. Results of surface roughness on different areas are shown in Fig. 7(b). It can be seen that a relatively uniform distribution of the roughness are achieved which agree well with the expectation, and the profile of cutting tool contributes towards data distribution since the data has the same form of waviness with the turning profile. This indicates that a small tool feed benefits to final cutting quality. Moreover, Ra values are between 2nm and 4nm which indicates that the machine quality is nanometric level. Therefore it is validated that under the selected parameter, the cutting mode is ductile which agrees to the theoretical analysis. It is also found that Ra value at the outer position in the aperture is larger than that at the center, because the slope at the outer area is larger which cause a larger scallop height during the turning procedure. This demonstrates that ultra-precision turning by FTS can achieve excellent surface finish on machining smooth freeform optics.

 figure: Fig. 7

Fig. 7 Illustration of (a) measurement by white light interferometer and (b) results of surface roughness on different area.

Download Full Size | PPT Slide | PDF

To demonstrate more details of the machining results, details of the machined NRFS are presented. Since the depth of cut is set 2μm, it guarantees the ductile removal of material under ultra-precision cutting. The number of control points is 216,000, which indicates that a relatively small amount of points is used to characterize the NRFS comparing to a traditional freeform surface. Furthermore, the RMS is 19.1nm. As well as surface roughness, the value of RMS indicates that high quality surface can be achieved by ultra-precision turning. Based on high quality, a relatively high spindle speed are achieved and the tool feed rate is raised, thus the advantage of high frequency feature possessed by FTS are taken. As a result, the machining time is 14min 33s, which proves that this strategy also improve the machining efficiency.

4.2 Optical performance verification

The NRFS is finally assembled into an IR imaging lens, whose design is shown in Fig. 5(a). In order to show the optical performance of the final IR optical system, MTF has been measured by Image Master Universal MTF measuring equipment. Results are shown in Fig. 8, which are in a good agreement to the design values in Fig. 5(b). It can be seen that the testing of optical system produces MTF curves approaching the diffraction-limited curve, indicating that a highly effect has been achieved in designing and manufacturing NRFS. Above all, it guarantees a good function of the final optical system. At this stage in the alignment, the optical system well is within the diffraction limit and would highly perform at long wavelength infrared (LWIR).

 figure: Fig. 8

Fig. 8 (a) MTF measuring system and (b) results in FOV of 0°, 10° and 20°.

Download Full Size | PPT Slide | PDF

As shown in Fig. 9(a), the final optical system was established including assembly of lens and an uncooled infrared imaging detector. Figure 5 shows sample images captured by the assembled optical system. It can be seen that images are clear with excellent quality for the capturing distances of 10m, 30m and 500m, which indicates that the device perform highly well. Furthermore, noise equivalent temperature difference (NETD) and minimum resolvable temperature difference (MRTD) are measured to evaluate the comprehensive performance of such infrared imaging system. Among these two indexes, NETD is irrelevant with the working distance of the system, it only reflects on the noise amplitude against temperature, while MRTD is the temperature sensitivity under detecting objects with different spatial frequency. As for this designed system, NETD is 0.065K, and MRTD is 0.3K at the characteristic frequency of 0.44cycles/mrad. It can be concluded from above that the infrared imaging system possesses fine over-all properties.

 figure: Fig. 9

Fig. 9 Illustration of (a) optical system structure and sample images at the distance of (b) 10m, (c) 30m, and (d) 500m.

Download Full Size | PPT Slide | PDF

5. Conclusions

In this paper, a novel machining approach of freeform optics on IR materials has been proposed. Ultra-precision turning based on FTS has been proposed based on FTS generates smooth freeform optics, and this strategy synchronously overcome the difficulties in machining freeform optics with high quality and in cutting brittle materials with low defects. Furthermore, machining efficiency has been improved simultaneously. Good optical performance has been achieved to confirm the broad prospects of this technique. The main conclusions can be drawn as follows:

  • (1) A newly developed machining approach has been proposed. Both designing and machining freeform optics on IR materials have been studied. Via a small removal of the material between adjacent cut during ultra-precision turning, a small material removal can be realized which leads to ductile cutting of brittle material. This strategy provides a guidance to machine freeform optics on IR materials with high quality and optical performance.
  • (2) By constraining the coefficients of x-y polynomial, NRFS has been designed to cater for the feature of ultra-precision turning. It not only lower the variation amplitude of freeform surfaces, but also plays a role of traditional freeform surface as usual. A small NRD can be obtained from NRFS, indicating that NRFS is suitable for FTS-assisted ultra-precision turning.
  • (3) FTS-assisted ultra-precision turning has realized ductile cutting of IR material. Nanometric surface finish has been achieved in germanium NRFS machining. Results indicate that both high quality and high efficiency can be achieved. Good optical performance has been obtained and the final optical system functions well.

Broad prospects are in store to be investigating since ultra-precision cutting techniques have a promising effects on freeform optics in brittle materials. Great challenges emerge that a full understanding of mechanical process is necessary, and effects on tool-workpiece contact remain to be investigated to optimize the manufacture of freeform optics on brittle materials. Newly developed ultra-precision cutting techniques will be a hot research field in the future.

Funding

National Natural Science Foundation (Grant No. 51375337, 61635008); Tianjin Research Program of Application Foundation and Advanced Technology (No. 14JCQNJC05200).

Acknowledgments

The authors would like to express their sincere thanks to Y.B. Lu and Y.X. Xiang for the preparation of experiments.

References and links

1. F. Z. Fang, X. D. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” Cirp. Ann-manuf. Techn. 62(2), 823–846 (2013).

2. L. Colace, G. Masini, S. Cozza, G. Assanto, F. DeNotaristefani, and V. Cencelli, “Near-infrared camera in polycrystalline germanium integrated on complementary-metal-oxide semiconductor electronics,” Appl. Phys. Lett. 90(1), 011103 (2007). [CrossRef]  

3. X. Hu, Y. Li, F. Fang, X. Li, J. Li, Y. Chen, X. Zhang, L. Chai, C. Wang, A. B. Fedotov, and A. M. Zheltikov, “Enhancement of terahertz radiation from GaP emitters by subwavelength antireflective micropyramid structures,” Opt. Lett. 38(12), 2053–2055 (2013). [CrossRef]   [PubMed]  

4. Y.-C. Yeo, X. Gong, M. J. van Dal, G. Vellianitis, and M. Passlack, “Germanium-based transistors for future high performance and low power logic applications,” in 2015 IEEE International Electron Devices Meeting (IEDM) (IEEE, 2015), pp. 2.4. [CrossRef]  

5. P. N. Blake and R. O. Scattergood, “Ductile-regime machining of germanium and silicon,” J. Am. Ceram. Soc. 73(4), 949–957 (1990). [CrossRef]  

6. J. Yan, K. Maekawa, J. i. Tamaki, and A. Kubo, “Experimental study on the ultraprecision ductile machinability of single-crystal germanium,” JSME Int. J., Ser. C 47(1), 29–36 (2004).

7. F. Z. Fang, H. Wu, and Y. Liu, “Modelling and experimental investigation on nanometric cutting of monocrystalline silicon,” Int. J. Mach. Tools Manuf. 45(15), 1681–1686 (2005). [CrossRef]  

8. F. Z. Fang, H. Wu, W. Zhou, and X. Hu, “A study on mechanism of nano-cutting single crystal silicon,” J. Mater. Process. Technol. 184(1), 407–410 (2007). [CrossRef]  

9. H. Tanaka, S. Shimada, and L. Anthony, “Requirements for ductile-mode machining based on deformation analysis of mono-crystalline silicon by molecular dynamics simulation,” Cirp. Ann-manuf. Techn. 56(1), 53–56 (2007).

10. M. Lai, X. Zhang, and F. Fang, “Nanoindentation-induced phase transformation and structural deformation of monocrystalline germanium: a molecular dynamics simulation investigation,” Nanoscale Res. Lett. 8(1), 353 (2013). [CrossRef]   [PubMed]  

11. X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterisation-a fresh strategy,” Cirp. Ann-manuf. Techn. 56(1), 553–556 (2007).

12. L. Li and A. Y. Yi, “Design and fabrication of a freeform microlens array for a compact large-field-of-view compound-eye camera,” Appl. Opt. 51(12), 1843–1852 (2012). [CrossRef]   [PubMed]  

13. A. Bauer and J. P. Rolland, “Visual space assessment of two all-reflective, freeform, optical see-through head-worn displays,” Opt. Express 22(11), 13155–13163 (2014). [CrossRef]   [PubMed]  

14. J. Zhu, T. Yang, and G. Jin, “Design method of surface contour for a freeform lens with wide linear field-of-view,” Opt. Express 21(22), 26080–26092 (2013). [CrossRef]   [PubMed]  

15. L. Li and A. Y. Yi, “Development of a 3D artificial compound eye,” Opt. Express 18(17), 18125–18137 (2010). [CrossRef]   [PubMed]  

16. M. Arif, M. Rahman, and W. Y. San, “Analytical model to determine the critical conditions for the modes of material removal in the milling process of brittle material,” J. Mater. Process. Technol. 212(9), 1925–1933 (2012). [CrossRef]  

17. B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014). [CrossRef]  

18. Z. Zhu, X. Zhou, D. Luo, and Q. Liu, “Development of pseudo-random diamond turning method for fabricating freeform optics with scattering homogenization,” Opt. Express 21(23), 28469–28482 (2013). [CrossRef]   [PubMed]  

19. Z. Zhu and S. To, “Adaptive tool servo diamond turning for enhancing machining efficiency and surface quality of freeform optics,” Opt. Express 23(16), 20234–20248 (2015). [CrossRef]   [PubMed]  

20. M. L. Barkman, B. S. Dutterer, M. A. Davies, and T. J. Suleski, “Free-form machining for micro-imaging systems,” in MOEMS-MEMS 2008 Micro and Nanofabrication (ISOP, 2008), paper 68830G.

21. T. Shibata, S. Fujii, E. Makino, and M. Ikeda, “Ductile-regime turning mechanism of single-crystal silicon,” Precis. Eng. 18(2-3), 129–137 (1996). [CrossRef]  

22. F. Z. Fang, V. Venkatesh, and G. Zhang, “Diamond turning of soft semiconductors to obtain nanometric mirror surfaces,” Int. J. Adv. Manuf. Technol. 19(9), 637–641 (2002). [CrossRef]  

23. S. Shojaee, T. Harriman, Y. Qi, D. Lucca, B. Dutterer, M. Davies, and T. Suleski, “Spatial variations in stress and crystal quality in diamond turned ZnSe surfaces measured by Raman spectroscopy,” Manu. Lett. 2(2), 35–39 (2014).

24. Y. Su, X. Chen, X. Guo, G. Rui, X. Liu, F. Zhang, C. Yang, Y. Xiao, and Z. Xu, “New fabrication technology in single point diamond turning for IR aspheric optical parts,” in 7th International Symposium on Advanced Optical Manufacturing and Testing Technologies (AOMATT 2014) (ISOP, 2014), paper 92811L.

25. T. A. Dow, M. H. Miller, and P. J. Falter, “Application of a fast tool servo for diamond turning of nonrotationally symmetric surfaces,” Precis. Eng. 13(4), 243–250 (1991). [CrossRef]  

26. J.-D. Kim and D.-S. Kim, “Waviness compensation of precision machining by piezo-electric micro cutting device,” Int. J. Mach. Tools Manuf. 38(10-11), 1305–1322 (1998). [CrossRef]  

27. K. Garrard, T. Bruegge, J. Hoffman, T. A. Dow, and A. Sohn, “Design tools for freeform optics,” in Optics & Photonics 2005 (ISOP, 2005), pp. 58740A–58711.

28. J. F. Cuttino, A. C. Miller, and D. E. Schinstock, “Performance optimization of a fast tool servo for single-point diamond turning machines,” IEEE/ASME Trans. Mechatron. 4(2), 169–179 (1999). [CrossRef]  

29. G. E. Davis, G. L. Herrit, and A. R. Hedges, “Diamond turning considerations in the manufacture of beam shaping optics,” in SPIE Optical Engineering + Applications (ISOP, 2011), paper 81300K.

30. M. Mukaida and J. Yan, “Ductile machining of single-crystal silicon for microlens arrays by ultraprecision diamond turning using a slow tool servo,” Int. J. Mach. Tools Manuf.in press.

31. F. Z. Fang, X. D. Zhang, and X. T. Hu, “Cylindrical coordinate machining of optical freeform surfaces,” Opt. Express 16(10), 7323–7329 (2008). [CrossRef]   [PubMed]  

32. D. P. Yu, S. W. Gan, Y. San Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9–12), 1137–1152 (2012). [CrossRef]  

References

  • View by:

  1. F. Z. Fang, X. D. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” Cirp. Ann-manuf. Techn. 62(2), 823–846 (2013).
  2. L. Colace, G. Masini, S. Cozza, G. Assanto, F. DeNotaristefani, and V. Cencelli, “Near-infrared camera in polycrystalline germanium integrated on complementary-metal-oxide semiconductor electronics,” Appl. Phys. Lett. 90(1), 011103 (2007).
    [Crossref]
  3. X. Hu, Y. Li, F. Fang, X. Li, J. Li, Y. Chen, X. Zhang, L. Chai, C. Wang, A. B. Fedotov, and A. M. Zheltikov, “Enhancement of terahertz radiation from GaP emitters by subwavelength antireflective micropyramid structures,” Opt. Lett. 38(12), 2053–2055 (2013).
    [Crossref] [PubMed]
  4. Y.-C. Yeo, X. Gong, M. J. van Dal, G. Vellianitis, and M. Passlack, “Germanium-based transistors for future high performance and low power logic applications,” in 2015 IEEE International Electron Devices Meeting (IEDM) (IEEE, 2015), pp. 2.4.
    [Crossref]
  5. P. N. Blake and R. O. Scattergood, “Ductile-regime machining of germanium and silicon,” J. Am. Ceram. Soc. 73(4), 949–957 (1990).
    [Crossref]
  6. J. Yan, K. Maekawa, J. i. Tamaki, and A. Kubo, “Experimental study on the ultraprecision ductile machinability of single-crystal germanium,” JSME Int. J., Ser. C 47(1), 29–36 (2004).
  7. F. Z. Fang, H. Wu, and Y. Liu, “Modelling and experimental investigation on nanometric cutting of monocrystalline silicon,” Int. J. Mach. Tools Manuf. 45(15), 1681–1686 (2005).
    [Crossref]
  8. F. Z. Fang, H. Wu, W. Zhou, and X. Hu, “A study on mechanism of nano-cutting single crystal silicon,” J. Mater. Process. Technol. 184(1), 407–410 (2007).
    [Crossref]
  9. H. Tanaka, S. Shimada, and L. Anthony, “Requirements for ductile-mode machining based on deformation analysis of mono-crystalline silicon by molecular dynamics simulation,” Cirp. Ann-manuf. Techn. 56(1), 53–56 (2007).
  10. M. Lai, X. Zhang, and F. Fang, “Nanoindentation-induced phase transformation and structural deformation of monocrystalline germanium: a molecular dynamics simulation investigation,” Nanoscale Res. Lett. 8(1), 353 (2013).
    [Crossref] [PubMed]
  11. X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterisation-a fresh strategy,” Cirp. Ann-manuf. Techn. 56(1), 553–556 (2007).
  12. L. Li and A. Y. Yi, “Design and fabrication of a freeform microlens array for a compact large-field-of-view compound-eye camera,” Appl. Opt. 51(12), 1843–1852 (2012).
    [Crossref] [PubMed]
  13. A. Bauer and J. P. Rolland, “Visual space assessment of two all-reflective, freeform, optical see-through head-worn displays,” Opt. Express 22(11), 13155–13163 (2014).
    [Crossref] [PubMed]
  14. J. Zhu, T. Yang, and G. Jin, “Design method of surface contour for a freeform lens with wide linear field-of-view,” Opt. Express 21(22), 26080–26092 (2013).
    [Crossref] [PubMed]
  15. L. Li and A. Y. Yi, “Development of a 3D artificial compound eye,” Opt. Express 18(17), 18125–18137 (2010).
    [Crossref] [PubMed]
  16. M. Arif, M. Rahman, and W. Y. San, “Analytical model to determine the critical conditions for the modes of material removal in the milling process of brittle material,” J. Mater. Process. Technol. 212(9), 1925–1933 (2012).
    [Crossref]
  17. B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014).
    [Crossref]
  18. Z. Zhu, X. Zhou, D. Luo, and Q. Liu, “Development of pseudo-random diamond turning method for fabricating freeform optics with scattering homogenization,” Opt. Express 21(23), 28469–28482 (2013).
    [Crossref] [PubMed]
  19. Z. Zhu and S. To, “Adaptive tool servo diamond turning for enhancing machining efficiency and surface quality of freeform optics,” Opt. Express 23(16), 20234–20248 (2015).
    [Crossref] [PubMed]
  20. M. L. Barkman, B. S. Dutterer, M. A. Davies, and T. J. Suleski, “Free-form machining for micro-imaging systems,” in MOEMS-MEMS 2008 Micro and Nanofabrication (ISOP, 2008), paper 68830G.
  21. T. Shibata, S. Fujii, E. Makino, and M. Ikeda, “Ductile-regime turning mechanism of single-crystal silicon,” Precis. Eng. 18(2-3), 129–137 (1996).
    [Crossref]
  22. F. Z. Fang, V. Venkatesh, and G. Zhang, “Diamond turning of soft semiconductors to obtain nanometric mirror surfaces,” Int. J. Adv. Manuf. Technol. 19(9), 637–641 (2002).
    [Crossref]
  23. S. Shojaee, T. Harriman, Y. Qi, D. Lucca, B. Dutterer, M. Davies, and T. Suleski, “Spatial variations in stress and crystal quality in diamond turned ZnSe surfaces measured by Raman spectroscopy,” Manu. Lett. 2(2), 35–39 (2014).
  24. Y. Su, X. Chen, X. Guo, G. Rui, X. Liu, F. Zhang, C. Yang, Y. Xiao, and Z. Xu, “New fabrication technology in single point diamond turning for IR aspheric optical parts,” in 7th International Symposium on Advanced Optical Manufacturing and Testing Technologies (AOMATT 2014) (ISOP, 2014), paper 92811L.
  25. T. A. Dow, M. H. Miller, and P. J. Falter, “Application of a fast tool servo for diamond turning of nonrotationally symmetric surfaces,” Precis. Eng. 13(4), 243–250 (1991).
    [Crossref]
  26. J.-D. Kim and D.-S. Kim, “Waviness compensation of precision machining by piezo-electric micro cutting device,” Int. J. Mach. Tools Manuf. 38(10-11), 1305–1322 (1998).
    [Crossref]
  27. K. Garrard, T. Bruegge, J. Hoffman, T. A. Dow, and A. Sohn, “Design tools for freeform optics,” in Optics & Photonics 2005 (ISOP, 2005), pp. 58740A–58711.
  28. J. F. Cuttino, A. C. Miller, and D. E. Schinstock, “Performance optimization of a fast tool servo for single-point diamond turning machines,” IEEE/ASME Trans. Mechatron. 4(2), 169–179 (1999).
    [Crossref]
  29. G. E. Davis, G. L. Herrit, and A. R. Hedges, “Diamond turning considerations in the manufacture of beam shaping optics,” in SPIE Optical Engineering + Applications (ISOP, 2011), paper 81300K.
  30. M. Mukaida and J. Yan, “Ductile machining of single-crystal silicon for microlens arrays by ultraprecision diamond turning using a slow tool servo,” Int. J. Mach. Tools Manuf.in press.
  31. F. Z. Fang, X. D. Zhang, and X. T. Hu, “Cylindrical coordinate machining of optical freeform surfaces,” Opt. Express 16(10), 7323–7329 (2008).
    [Crossref] [PubMed]
  32. D. P. Yu, S. W. Gan, Y. San Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9–12), 1137–1152 (2012).
    [Crossref]

2015 (1)

2014 (3)

S. Shojaee, T. Harriman, Y. Qi, D. Lucca, B. Dutterer, M. Davies, and T. Suleski, “Spatial variations in stress and crystal quality in diamond turned ZnSe surfaces measured by Raman spectroscopy,” Manu. Lett. 2(2), 35–39 (2014).

A. Bauer and J. P. Rolland, “Visual space assessment of two all-reflective, freeform, optical see-through head-worn displays,” Opt. Express 22(11), 13155–13163 (2014).
[Crossref] [PubMed]

B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014).
[Crossref]

2013 (5)

2012 (3)

L. Li and A. Y. Yi, “Design and fabrication of a freeform microlens array for a compact large-field-of-view compound-eye camera,” Appl. Opt. 51(12), 1843–1852 (2012).
[Crossref] [PubMed]

M. Arif, M. Rahman, and W. Y. San, “Analytical model to determine the critical conditions for the modes of material removal in the milling process of brittle material,” J. Mater. Process. Technol. 212(9), 1925–1933 (2012).
[Crossref]

D. P. Yu, S. W. Gan, Y. San Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9–12), 1137–1152 (2012).
[Crossref]

2010 (1)

2008 (1)

2007 (4)

X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterisation-a fresh strategy,” Cirp. Ann-manuf. Techn. 56(1), 553–556 (2007).

L. Colace, G. Masini, S. Cozza, G. Assanto, F. DeNotaristefani, and V. Cencelli, “Near-infrared camera in polycrystalline germanium integrated on complementary-metal-oxide semiconductor electronics,” Appl. Phys. Lett. 90(1), 011103 (2007).
[Crossref]

F. Z. Fang, H. Wu, W. Zhou, and X. Hu, “A study on mechanism of nano-cutting single crystal silicon,” J. Mater. Process. Technol. 184(1), 407–410 (2007).
[Crossref]

H. Tanaka, S. Shimada, and L. Anthony, “Requirements for ductile-mode machining based on deformation analysis of mono-crystalline silicon by molecular dynamics simulation,” Cirp. Ann-manuf. Techn. 56(1), 53–56 (2007).

2005 (1)

F. Z. Fang, H. Wu, and Y. Liu, “Modelling and experimental investigation on nanometric cutting of monocrystalline silicon,” Int. J. Mach. Tools Manuf. 45(15), 1681–1686 (2005).
[Crossref]

2004 (1)

J. Yan, K. Maekawa, J. i. Tamaki, and A. Kubo, “Experimental study on the ultraprecision ductile machinability of single-crystal germanium,” JSME Int. J., Ser. C 47(1), 29–36 (2004).

2002 (1)

F. Z. Fang, V. Venkatesh, and G. Zhang, “Diamond turning of soft semiconductors to obtain nanometric mirror surfaces,” Int. J. Adv. Manuf. Technol. 19(9), 637–641 (2002).
[Crossref]

1999 (1)

J. F. Cuttino, A. C. Miller, and D. E. Schinstock, “Performance optimization of a fast tool servo for single-point diamond turning machines,” IEEE/ASME Trans. Mechatron. 4(2), 169–179 (1999).
[Crossref]

1998 (1)

J.-D. Kim and D.-S. Kim, “Waviness compensation of precision machining by piezo-electric micro cutting device,” Int. J. Mach. Tools Manuf. 38(10-11), 1305–1322 (1998).
[Crossref]

1996 (1)

T. Shibata, S. Fujii, E. Makino, and M. Ikeda, “Ductile-regime turning mechanism of single-crystal silicon,” Precis. Eng. 18(2-3), 129–137 (1996).
[Crossref]

1991 (1)

T. A. Dow, M. H. Miller, and P. J. Falter, “Application of a fast tool servo for diamond turning of nonrotationally symmetric surfaces,” Precis. Eng. 13(4), 243–250 (1991).
[Crossref]

1990 (1)

P. N. Blake and R. O. Scattergood, “Ductile-regime machining of germanium and silicon,” J. Am. Ceram. Soc. 73(4), 949–957 (1990).
[Crossref]

Anthony, L.

H. Tanaka, S. Shimada, and L. Anthony, “Requirements for ductile-mode machining based on deformation analysis of mono-crystalline silicon by molecular dynamics simulation,” Cirp. Ann-manuf. Techn. 56(1), 53–56 (2007).

Arif, M.

M. Arif, M. Rahman, and W. Y. San, “Analytical model to determine the critical conditions for the modes of material removal in the milling process of brittle material,” J. Mater. Process. Technol. 212(9), 1925–1933 (2012).
[Crossref]

Assanto, G.

L. Colace, G. Masini, S. Cozza, G. Assanto, F. DeNotaristefani, and V. Cencelli, “Near-infrared camera in polycrystalline germanium integrated on complementary-metal-oxide semiconductor electronics,” Appl. Phys. Lett. 90(1), 011103 (2007).
[Crossref]

Bauer, A.

Blake, P. N.

P. N. Blake and R. O. Scattergood, “Ductile-regime machining of germanium and silicon,” J. Am. Ceram. Soc. 73(4), 949–957 (1990).
[Crossref]

Cencelli, V.

L. Colace, G. Masini, S. Cozza, G. Assanto, F. DeNotaristefani, and V. Cencelli, “Near-infrared camera in polycrystalline germanium integrated on complementary-metal-oxide semiconductor electronics,” Appl. Phys. Lett. 90(1), 011103 (2007).
[Crossref]

Chai, L.

Chen, Y.

Colace, L.

L. Colace, G. Masini, S. Cozza, G. Assanto, F. DeNotaristefani, and V. Cencelli, “Near-infrared camera in polycrystalline germanium integrated on complementary-metal-oxide semiconductor electronics,” Appl. Phys. Lett. 90(1), 011103 (2007).
[Crossref]

Cozza, S.

L. Colace, G. Masini, S. Cozza, G. Assanto, F. DeNotaristefani, and V. Cencelli, “Near-infrared camera in polycrystalline germanium integrated on complementary-metal-oxide semiconductor electronics,” Appl. Phys. Lett. 90(1), 011103 (2007).
[Crossref]

Cuttino, J. F.

J. F. Cuttino, A. C. Miller, and D. E. Schinstock, “Performance optimization of a fast tool servo for single-point diamond turning machines,” IEEE/ASME Trans. Mechatron. 4(2), 169–179 (1999).
[Crossref]

Davies, M.

S. Shojaee, T. Harriman, Y. Qi, D. Lucca, B. Dutterer, M. Davies, and T. Suleski, “Spatial variations in stress and crystal quality in diamond turned ZnSe surfaces measured by Raman spectroscopy,” Manu. Lett. 2(2), 35–39 (2014).

Davies, M. A.

B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014).
[Crossref]

DeNotaristefani, F.

L. Colace, G. Masini, S. Cozza, G. Assanto, F. DeNotaristefani, and V. Cencelli, “Near-infrared camera in polycrystalline germanium integrated on complementary-metal-oxide semiconductor electronics,” Appl. Phys. Lett. 90(1), 011103 (2007).
[Crossref]

Dow, T. A.

T. A. Dow, M. H. Miller, and P. J. Falter, “Application of a fast tool servo for diamond turning of nonrotationally symmetric surfaces,” Precis. Eng. 13(4), 243–250 (1991).
[Crossref]

Dutterer, B.

S. Shojaee, T. Harriman, Y. Qi, D. Lucca, B. Dutterer, M. Davies, and T. Suleski, “Spatial variations in stress and crystal quality in diamond turned ZnSe surfaces measured by Raman spectroscopy,” Manu. Lett. 2(2), 35–39 (2014).

Dutterer, B. S.

B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014).
[Crossref]

Evans, C.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” Cirp. Ann-manuf. Techn. 62(2), 823–846 (2013).

Falter, P. J.

T. A. Dow, M. H. Miller, and P. J. Falter, “Application of a fast tool servo for diamond turning of nonrotationally symmetric surfaces,” Precis. Eng. 13(4), 243–250 (1991).
[Crossref]

Fang, F.

X. Hu, Y. Li, F. Fang, X. Li, J. Li, Y. Chen, X. Zhang, L. Chai, C. Wang, A. B. Fedotov, and A. M. Zheltikov, “Enhancement of terahertz radiation from GaP emitters by subwavelength antireflective micropyramid structures,” Opt. Lett. 38(12), 2053–2055 (2013).
[Crossref] [PubMed]

M. Lai, X. Zhang, and F. Fang, “Nanoindentation-induced phase transformation and structural deformation of monocrystalline germanium: a molecular dynamics simulation investigation,” Nanoscale Res. Lett. 8(1), 353 (2013).
[Crossref] [PubMed]

Fang, F. Z.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” Cirp. Ann-manuf. Techn. 62(2), 823–846 (2013).

F. Z. Fang, X. D. Zhang, and X. T. Hu, “Cylindrical coordinate machining of optical freeform surfaces,” Opt. Express 16(10), 7323–7329 (2008).
[Crossref] [PubMed]

F. Z. Fang, H. Wu, W. Zhou, and X. Hu, “A study on mechanism of nano-cutting single crystal silicon,” J. Mater. Process. Technol. 184(1), 407–410 (2007).
[Crossref]

F. Z. Fang, H. Wu, and Y. Liu, “Modelling and experimental investigation on nanometric cutting of monocrystalline silicon,” Int. J. Mach. Tools Manuf. 45(15), 1681–1686 (2005).
[Crossref]

F. Z. Fang, V. Venkatesh, and G. Zhang, “Diamond turning of soft semiconductors to obtain nanometric mirror surfaces,” Int. J. Adv. Manuf. Technol. 19(9), 637–641 (2002).
[Crossref]

Fedotov, A. B.

Fujii, S.

T. Shibata, S. Fujii, E. Makino, and M. Ikeda, “Ductile-regime turning mechanism of single-crystal silicon,” Precis. Eng. 18(2-3), 129–137 (1996).
[Crossref]

Gan, S. W.

D. P. Yu, S. W. Gan, Y. San Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9–12), 1137–1152 (2012).
[Crossref]

Gong, X.

Y.-C. Yeo, X. Gong, M. J. van Dal, G. Vellianitis, and M. Passlack, “Germanium-based transistors for future high performance and low power logic applications,” in 2015 IEEE International Electron Devices Meeting (IEDM) (IEEE, 2015), pp. 2.4.
[Crossref]

Harriman, T.

S. Shojaee, T. Harriman, Y. Qi, D. Lucca, B. Dutterer, M. Davies, and T. Suleski, “Spatial variations in stress and crystal quality in diamond turned ZnSe surfaces measured by Raman spectroscopy,” Manu. Lett. 2(2), 35–39 (2014).

Harriman, T. A.

B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014).
[Crossref]

Hildebrand, D. S.

B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014).
[Crossref]

Hong, G. S.

D. P. Yu, S. W. Gan, Y. San Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9–12), 1137–1152 (2012).
[Crossref]

Hu, X.

Hu, X. T.

Ikeda, M.

T. Shibata, S. Fujii, E. Makino, and M. Ikeda, “Ductile-regime turning mechanism of single-crystal silicon,” Precis. Eng. 18(2-3), 129–137 (1996).
[Crossref]

Jiang, X.

X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterisation-a fresh strategy,” Cirp. Ann-manuf. Techn. 56(1), 553–556 (2007).

Jin, G.

Kim, D.-S.

J.-D. Kim and D.-S. Kim, “Waviness compensation of precision machining by piezo-electric micro cutting device,” Int. J. Mach. Tools Manuf. 38(10-11), 1305–1322 (1998).
[Crossref]

Kim, J.-D.

J.-D. Kim and D.-S. Kim, “Waviness compensation of precision machining by piezo-electric micro cutting device,” Int. J. Mach. Tools Manuf. 38(10-11), 1305–1322 (1998).
[Crossref]

Kubo, A.

J. Yan, K. Maekawa, J. i. Tamaki, and A. Kubo, “Experimental study on the ultraprecision ductile machinability of single-crystal germanium,” JSME Int. J., Ser. C 47(1), 29–36 (2004).

Lai, M.

M. Lai, X. Zhang, and F. Fang, “Nanoindentation-induced phase transformation and structural deformation of monocrystalline germanium: a molecular dynamics simulation investigation,” Nanoscale Res. Lett. 8(1), 353 (2013).
[Crossref] [PubMed]

Li, J.

Li, L.

Li, X.

Li, Y.

Lineberger, J. L.

B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014).
[Crossref]

Liu, Q.

Liu, Y.

F. Z. Fang, H. Wu, and Y. Liu, “Modelling and experimental investigation on nanometric cutting of monocrystalline silicon,” Int. J. Mach. Tools Manuf. 45(15), 1681–1686 (2005).
[Crossref]

Lucca, D.

S. Shojaee, T. Harriman, Y. Qi, D. Lucca, B. Dutterer, M. Davies, and T. Suleski, “Spatial variations in stress and crystal quality in diamond turned ZnSe surfaces measured by Raman spectroscopy,” Manu. Lett. 2(2), 35–39 (2014).

Lucca, D. A.

B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014).
[Crossref]

Luo, D.

Maekawa, K.

J. Yan, K. Maekawa, J. i. Tamaki, and A. Kubo, “Experimental study on the ultraprecision ductile machinability of single-crystal germanium,” JSME Int. J., Ser. C 47(1), 29–36 (2004).

Makino, E.

T. Shibata, S. Fujii, E. Makino, and M. Ikeda, “Ductile-regime turning mechanism of single-crystal silicon,” Precis. Eng. 18(2-3), 129–137 (1996).
[Crossref]

Masini, G.

L. Colace, G. Masini, S. Cozza, G. Assanto, F. DeNotaristefani, and V. Cencelli, “Near-infrared camera in polycrystalline germanium integrated on complementary-metal-oxide semiconductor electronics,” Appl. Phys. Lett. 90(1), 011103 (2007).
[Crossref]

Miller, A. C.

J. F. Cuttino, A. C. Miller, and D. E. Schinstock, “Performance optimization of a fast tool servo for single-point diamond turning machines,” IEEE/ASME Trans. Mechatron. 4(2), 169–179 (1999).
[Crossref]

Miller, M. H.

T. A. Dow, M. H. Miller, and P. J. Falter, “Application of a fast tool servo for diamond turning of nonrotationally symmetric surfaces,” Precis. Eng. 13(4), 243–250 (1991).
[Crossref]

Mukaida, M.

M. Mukaida and J. Yan, “Ductile machining of single-crystal silicon for microlens arrays by ultraprecision diamond turning using a slow tool servo,” Int. J. Mach. Tools Manuf.in press.

Passlack, M.

Y.-C. Yeo, X. Gong, M. J. van Dal, G. Vellianitis, and M. Passlack, “Germanium-based transistors for future high performance and low power logic applications,” in 2015 IEEE International Electron Devices Meeting (IEDM) (IEEE, 2015), pp. 2.4.
[Crossref]

Qi, Y.

S. Shojaee, T. Harriman, Y. Qi, D. Lucca, B. Dutterer, M. Davies, and T. Suleski, “Spatial variations in stress and crystal quality in diamond turned ZnSe surfaces measured by Raman spectroscopy,” Manu. Lett. 2(2), 35–39 (2014).

Rahman, M.

D. P. Yu, S. W. Gan, Y. San Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9–12), 1137–1152 (2012).
[Crossref]

M. Arif, M. Rahman, and W. Y. San, “Analytical model to determine the critical conditions for the modes of material removal in the milling process of brittle material,” J. Mater. Process. Technol. 212(9), 1925–1933 (2012).
[Crossref]

Rolland, J. P.

San, W. Y.

M. Arif, M. Rahman, and W. Y. San, “Analytical model to determine the critical conditions for the modes of material removal in the milling process of brittle material,” J. Mater. Process. Technol. 212(9), 1925–1933 (2012).
[Crossref]

San Wong, Y.

D. P. Yu, S. W. Gan, Y. San Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9–12), 1137–1152 (2012).
[Crossref]

Scattergood, R. O.

P. N. Blake and R. O. Scattergood, “Ductile-regime machining of germanium and silicon,” J. Am. Ceram. Soc. 73(4), 949–957 (1990).
[Crossref]

Schinstock, D. E.

J. F. Cuttino, A. C. Miller, and D. E. Schinstock, “Performance optimization of a fast tool servo for single-point diamond turning machines,” IEEE/ASME Trans. Mechatron. 4(2), 169–179 (1999).
[Crossref]

Scott, P.

X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterisation-a fresh strategy,” Cirp. Ann-manuf. Techn. 56(1), 553–556 (2007).

Shibata, T.

T. Shibata, S. Fujii, E. Makino, and M. Ikeda, “Ductile-regime turning mechanism of single-crystal silicon,” Precis. Eng. 18(2-3), 129–137 (1996).
[Crossref]

Shimada, S.

H. Tanaka, S. Shimada, and L. Anthony, “Requirements for ductile-mode machining based on deformation analysis of mono-crystalline silicon by molecular dynamics simulation,” Cirp. Ann-manuf. Techn. 56(1), 53–56 (2007).

Shojaee, S.

S. Shojaee, T. Harriman, Y. Qi, D. Lucca, B. Dutterer, M. Davies, and T. Suleski, “Spatial variations in stress and crystal quality in diamond turned ZnSe surfaces measured by Raman spectroscopy,” Manu. Lett. 2(2), 35–39 (2014).

Smilie, P. J.

B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014).
[Crossref]

Suleski, T.

S. Shojaee, T. Harriman, Y. Qi, D. Lucca, B. Dutterer, M. Davies, and T. Suleski, “Spatial variations in stress and crystal quality in diamond turned ZnSe surfaces measured by Raman spectroscopy,” Manu. Lett. 2(2), 35–39 (2014).

Suleski, T. J.

B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014).
[Crossref]

Tamaki, J. i.

J. Yan, K. Maekawa, J. i. Tamaki, and A. Kubo, “Experimental study on the ultraprecision ductile machinability of single-crystal germanium,” JSME Int. J., Ser. C 47(1), 29–36 (2004).

Tanaka, H.

H. Tanaka, S. Shimada, and L. Anthony, “Requirements for ductile-mode machining based on deformation analysis of mono-crystalline silicon by molecular dynamics simulation,” Cirp. Ann-manuf. Techn. 56(1), 53–56 (2007).

To, S.

van Dal, M. J.

Y.-C. Yeo, X. Gong, M. J. van Dal, G. Vellianitis, and M. Passlack, “Germanium-based transistors for future high performance and low power logic applications,” in 2015 IEEE International Electron Devices Meeting (IEDM) (IEEE, 2015), pp. 2.4.
[Crossref]

Vellianitis, G.

Y.-C. Yeo, X. Gong, M. J. van Dal, G. Vellianitis, and M. Passlack, “Germanium-based transistors for future high performance and low power logic applications,” in 2015 IEEE International Electron Devices Meeting (IEDM) (IEEE, 2015), pp. 2.4.
[Crossref]

Venkatesh, V.

F. Z. Fang, V. Venkatesh, and G. Zhang, “Diamond turning of soft semiconductors to obtain nanometric mirror surfaces,” Int. J. Adv. Manuf. Technol. 19(9), 637–641 (2002).
[Crossref]

Wang, C.

Weckenmann, A.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” Cirp. Ann-manuf. Techn. 62(2), 823–846 (2013).

Whitehouse, D.

X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterisation-a fresh strategy,” Cirp. Ann-manuf. Techn. 56(1), 553–556 (2007).

Wu, H.

F. Z. Fang, H. Wu, W. Zhou, and X. Hu, “A study on mechanism of nano-cutting single crystal silicon,” J. Mater. Process. Technol. 184(1), 407–410 (2007).
[Crossref]

F. Z. Fang, H. Wu, and Y. Liu, “Modelling and experimental investigation on nanometric cutting of monocrystalline silicon,” Int. J. Mach. Tools Manuf. 45(15), 1681–1686 (2005).
[Crossref]

Yan, J.

J. Yan, K. Maekawa, J. i. Tamaki, and A. Kubo, “Experimental study on the ultraprecision ductile machinability of single-crystal germanium,” JSME Int. J., Ser. C 47(1), 29–36 (2004).

M. Mukaida and J. Yan, “Ductile machining of single-crystal silicon for microlens arrays by ultraprecision diamond turning using a slow tool servo,” Int. J. Mach. Tools Manuf.in press.

Yang, T.

Yao, J.

D. P. Yu, S. W. Gan, Y. San Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9–12), 1137–1152 (2012).
[Crossref]

Yeo, Y.-C.

Y.-C. Yeo, X. Gong, M. J. van Dal, G. Vellianitis, and M. Passlack, “Germanium-based transistors for future high performance and low power logic applications,” in 2015 IEEE International Electron Devices Meeting (IEDM) (IEEE, 2015), pp. 2.4.
[Crossref]

Yi, A. Y.

Yu, D. P.

D. P. Yu, S. W. Gan, Y. San Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9–12), 1137–1152 (2012).
[Crossref]

Zhang, G.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” Cirp. Ann-manuf. Techn. 62(2), 823–846 (2013).

F. Z. Fang, V. Venkatesh, and G. Zhang, “Diamond turning of soft semiconductors to obtain nanometric mirror surfaces,” Int. J. Adv. Manuf. Technol. 19(9), 637–641 (2002).
[Crossref]

Zhang, X.

X. Hu, Y. Li, F. Fang, X. Li, J. Li, Y. Chen, X. Zhang, L. Chai, C. Wang, A. B. Fedotov, and A. M. Zheltikov, “Enhancement of terahertz radiation from GaP emitters by subwavelength antireflective micropyramid structures,” Opt. Lett. 38(12), 2053–2055 (2013).
[Crossref] [PubMed]

M. Lai, X. Zhang, and F. Fang, “Nanoindentation-induced phase transformation and structural deformation of monocrystalline germanium: a molecular dynamics simulation investigation,” Nanoscale Res. Lett. 8(1), 353 (2013).
[Crossref] [PubMed]

Zhang, X. D.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” Cirp. Ann-manuf. Techn. 62(2), 823–846 (2013).

F. Z. Fang, X. D. Zhang, and X. T. Hu, “Cylindrical coordinate machining of optical freeform surfaces,” Opt. Express 16(10), 7323–7329 (2008).
[Crossref] [PubMed]

Zheltikov, A. M.

Zhou, W.

F. Z. Fang, H. Wu, W. Zhou, and X. Hu, “A study on mechanism of nano-cutting single crystal silicon,” J. Mater. Process. Technol. 184(1), 407–410 (2007).
[Crossref]

Zhou, X.

Zhu, J.

Zhu, Z.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

L. Colace, G. Masini, S. Cozza, G. Assanto, F. DeNotaristefani, and V. Cencelli, “Near-infrared camera in polycrystalline germanium integrated on complementary-metal-oxide semiconductor electronics,” Appl. Phys. Lett. 90(1), 011103 (2007).
[Crossref]

Cirp. Ann-manuf. Techn. (3)

H. Tanaka, S. Shimada, and L. Anthony, “Requirements for ductile-mode machining based on deformation analysis of mono-crystalline silicon by molecular dynamics simulation,” Cirp. Ann-manuf. Techn. 56(1), 53–56 (2007).

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” Cirp. Ann-manuf. Techn. 62(2), 823–846 (2013).

X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterisation-a fresh strategy,” Cirp. Ann-manuf. Techn. 56(1), 553–556 (2007).

IEEE/ASME Trans. Mechatron. (1)

J. F. Cuttino, A. C. Miller, and D. E. Schinstock, “Performance optimization of a fast tool servo for single-point diamond turning machines,” IEEE/ASME Trans. Mechatron. 4(2), 169–179 (1999).
[Crossref]

Int. J. Adv. Manuf. Technol. (2)

D. P. Yu, S. W. Gan, Y. San Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9–12), 1137–1152 (2012).
[Crossref]

F. Z. Fang, V. Venkatesh, and G. Zhang, “Diamond turning of soft semiconductors to obtain nanometric mirror surfaces,” Int. J. Adv. Manuf. Technol. 19(9), 637–641 (2002).
[Crossref]

Int. J. Mach. Tools Manuf. (2)

J.-D. Kim and D.-S. Kim, “Waviness compensation of precision machining by piezo-electric micro cutting device,” Int. J. Mach. Tools Manuf. 38(10-11), 1305–1322 (1998).
[Crossref]

F. Z. Fang, H. Wu, and Y. Liu, “Modelling and experimental investigation on nanometric cutting of monocrystalline silicon,” Int. J. Mach. Tools Manuf. 45(15), 1681–1686 (2005).
[Crossref]

J. Am. Ceram. Soc. (1)

P. N. Blake and R. O. Scattergood, “Ductile-regime machining of germanium and silicon,” J. Am. Ceram. Soc. 73(4), 949–957 (1990).
[Crossref]

J. Mater. Process. Technol. (2)

F. Z. Fang, H. Wu, W. Zhou, and X. Hu, “A study on mechanism of nano-cutting single crystal silicon,” J. Mater. Process. Technol. 184(1), 407–410 (2007).
[Crossref]

M. Arif, M. Rahman, and W. Y. San, “Analytical model to determine the critical conditions for the modes of material removal in the milling process of brittle material,” J. Mater. Process. Technol. 212(9), 1925–1933 (2012).
[Crossref]

JSME Int. J., Ser. C (1)

J. Yan, K. Maekawa, J. i. Tamaki, and A. Kubo, “Experimental study on the ultraprecision ductile machinability of single-crystal germanium,” JSME Int. J., Ser. C 47(1), 29–36 (2004).

Manu. Lett. (1)

S. Shojaee, T. Harriman, Y. Qi, D. Lucca, B. Dutterer, M. Davies, and T. Suleski, “Spatial variations in stress and crystal quality in diamond turned ZnSe surfaces measured by Raman spectroscopy,” Manu. Lett. 2(2), 35–39 (2014).

Nanoscale Res. Lett. (1)

M. Lai, X. Zhang, and F. Fang, “Nanoindentation-induced phase transformation and structural deformation of monocrystalline germanium: a molecular dynamics simulation investigation,” Nanoscale Res. Lett. 8(1), 353 (2013).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (1)

Precis. Eng. (3)

B. S. Dutterer, J. L. Lineberger, P. J. Smilie, D. S. Hildebrand, T. A. Harriman, M. A. Davies, T. J. Suleski, and D. A. Lucca, “Diamond milling of an Alvarez lens in germanium,” Precis. Eng. 38(2), 398–408 (2014).
[Crossref]

T. Shibata, S. Fujii, E. Makino, and M. Ikeda, “Ductile-regime turning mechanism of single-crystal silicon,” Precis. Eng. 18(2-3), 129–137 (1996).
[Crossref]

T. A. Dow, M. H. Miller, and P. J. Falter, “Application of a fast tool servo for diamond turning of nonrotationally symmetric surfaces,” Precis. Eng. 13(4), 243–250 (1991).
[Crossref]

Other (6)

K. Garrard, T. Bruegge, J. Hoffman, T. A. Dow, and A. Sohn, “Design tools for freeform optics,” in Optics & Photonics 2005 (ISOP, 2005), pp. 58740A–58711.

M. L. Barkman, B. S. Dutterer, M. A. Davies, and T. J. Suleski, “Free-form machining for micro-imaging systems,” in MOEMS-MEMS 2008 Micro and Nanofabrication (ISOP, 2008), paper 68830G.

Y. Su, X. Chen, X. Guo, G. Rui, X. Liu, F. Zhang, C. Yang, Y. Xiao, and Z. Xu, “New fabrication technology in single point diamond turning for IR aspheric optical parts,” in 7th International Symposium on Advanced Optical Manufacturing and Testing Technologies (AOMATT 2014) (ISOP, 2014), paper 92811L.

G. E. Davis, G. L. Herrit, and A. R. Hedges, “Diamond turning considerations in the manufacture of beam shaping optics,” in SPIE Optical Engineering + Applications (ISOP, 2011), paper 81300K.

M. Mukaida and J. Yan, “Ductile machining of single-crystal silicon for microlens arrays by ultraprecision diamond turning using a slow tool servo,” Int. J. Mach. Tools Manuf.in press.

Y.-C. Yeo, X. Gong, M. J. van Dal, G. Vellianitis, and M. Passlack, “Germanium-based transistors for future high performance and low power logic applications,” in 2015 IEEE International Electron Devices Meeting (IEDM) (IEEE, 2015), pp. 2.4.
[Crossref]

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 Strategy of machining freeform optics on IR materials.
Fig. 2
Fig. 2 Design of (a) structure of IR optical system, (b) spot diagram and (c) MTF of system.
Fig. 3
Fig. 3 (a) Design of NRFS and its (b) Rotational 2D surface curve and (c) NRS.
Fig. 4
Fig. 4 Schematic of FTS-assisted ultra-precision turning.
Fig. 5
Fig. 5 Schematic of (a) tool path on NRFS, (b) x-slide and z-slide motions and (c) FTS motion.
Fig. 6
Fig. 6 Illustration of (a) FTS-assisted ultra-precision turning and (b) machined germanium surface.
Fig. 7
Fig. 7 Illustration of (a) measurement by white light interferometer and (b) results of surface roughness on different area.
Fig. 8
Fig. 8 (a) MTF measuring system and (b) results in FOV of 0°, 10° and 20°.
Fig. 9
Fig. 9 Illustration of (a) optical system structure and sample images at the distance of (b) 10m, (c) 30m, and (d) 500m.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

z ( r ) = c r 2 1 + 1 ( 1 + k ) c 2 r 2 + i = 1 A i r i ,
z ( x , y ) = c r 2 1 + 1 ( 1 + k ) c 2 r 2 + m = 1 n = 1 A m n x m y n ,
m = 1 n = 1 A m n x m y n = A 10 x + A 01 y + A 20 x 2 + A 11 x y + A 02 y 2 + .
z ( x , y ) = z ( r ) + δ ( x , y ) .
A 2 m , 2 n = C m + n m a n , m , n = 1 , 2 , 3 ,
z ( x , y ) = z r ( r ) + z n ( x , y ) ,
z r ( r ) = c r 2 1 + 1 ( 1 + k ) c 2 r 2 + a 1 r 2 + a 2 r 4 + a 3 r 6 + ,
a n = A 2 n , 0 + A 0 , 2 n 2 , n = 1 , 2 , 3 .
L = f 2 + Δ z 0 2 ,
d m = R t [ ( R t 2 ( R t a 0 Δ z 0 ) 2 f ) 2 + ( R t a 0 ) 2 ] 1 / 2 ,
α m = arc cos ( R t a 0 R t d m ) .
{ ρ 0 = R w f S t i φ 0 = 2 π S t i z 0 = z r ( r ) | r = ρ w 0 = z n ( x , y ) | x = ρ 0 cos φ 0 , y = ρ 0 sin φ 0 ,
t i = 60 i S N t , i = 0 , 1 , , R w S N t f S ,

Metrics