Rapid fabrication of miniature lens arrays by four-axis single point diamond machining

A novel method for fabricating lens arrays and other nonrotationally symmetric free-form optics is presented. This is a diamond machining technique using 4 controlled axes of motion – X, Y, Z, and C. As in 3-axis diamond micro-milling, a diamond ball endmill is mounted to the work spindle of a 4-axis ultra-precision computer numerical control (CNC) machine. Unlike 3-axis micro-milling, the C-axis is used to hold the cutting edge of the tool in contact with the lens surface for the entire cut. This allows the feed rates to be doubled compared to the current state of the art of micro-milling while producing an optically smooth surface with very low surface form error and exceptionally low radius error. © 2013 Optical Society of America OCIS Codes: (220.0220) Optical design and fabrication; (220.1920) Diamond machining; (220.3630) Lenses; (220.4610) Optical fabrication. References and links 1. R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. 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Chou, P. J. Wang, and D. P. Tsai, “Slow tool servo diamond turning of optical freeform surface for astigmatic contact lens,” Proc. SPIE 8126, 812617, 812617-9 (2011). 11. S. Scheiding, A. Y. Yi, A. Gebhardt, L. Li, S. Risse, R. Eberhardt, and A. Tünnermann, “Freeform manufacturing of a microoptical lens array on a steep curved substrate by use of a voice coil fast tool servo,” Opt. Express 19(24), 23938–23951 (2011). 12. Contour Fine Tooling and Technodiamont, “Reaching new heights in clearance and sweep,” http://contourdiamonds.com/GB/publications/CuttingEdge/Contour%20Cutting%20Edge%20April%202012/Contour%20Cutt ing%20Edge%20Apr%202012.pdf. 13. J. Rogers, A. Kärkkäinen, T. Tkaczyk, J. Rantala, and M. Descour, “Realization of refractive microoptics through grayscale lithographic patterning of photosensitive hybrid glass,” Opt. Express 12(7), 1294–1303 (2004). #179436 $15.00 USD Received 8 Nov 2012; revised 19 Jan 2013; accepted 22 Jan 2013; published 5 Feb 2013 (C) 2013 OSA 11 February 2013 / Vol. 21, No. 3 / OPTICS EXPRESS 3557 14. T. D. Milster and T. S. Tkaczyk, “Miniature and Micro-Optics,” in Handbook of Optics 1, M. Bass, ed. (McGraw-Hill Professional, 2010), 22.1–22.50. 15. Z. D. Popovic, R. A. Sprague, and G. A. Connell, “Technique for monolithic fabrication of microlens arrays,” Appl. Opt. 27(7), 1281–1284 (1988). 16. W. Cox, T. Chen, and D. Hayes, “Micro-optics fabrication by ink-jet printers,” Opt. Photon. News 12(6), 32–35 (2001). 17. F. T. O'Neill, C. R. Walsh, and J. T. Sheridan, “Photoresist reflow method of microlens production: modeling and fabrication techniques,” Proc. SPIE 5456, 197–208 (2004). 18. S. Audran, B. Faure, B. Mortini, J. Regolini, G. Schlatter, and G. Hadziioannou, “Study of mechanisms involved in photoresist microlens formation,” Microelectron. Eng. 83(4-9), 1087–1090 (2006). 19. W. H. Hsieh and J. H. Chen, “Lens-profile control by electrowetting fabrication technique,” IEEE Photon. Technol. Lett. 17(3), 606–608 (2005). 20. H. Ottevaere, R. Cox, H. P. Herzig, T. Miyashita, K. Naessens, M. Taghizadeh, R. Völkel, H. J. Woo, and H. Thienpont, “Comparing glass and plastic refractive microlenses fabricated with different technologies,” J. Opt. A, Pure Appl. Opt. 8(7), S407–S429 (2006). 21. N. C. R. Holme, T. W. Berg, and P. G. Dinesen, “Diamond micro-milling for array mastering,” Proc. SPIE 7062, 70620J, 70620J-8 (2008). 22. S. Scheiding, R. Steinkopf, A. Kolbmüller, S. Risse, R. Eberhardt, and A. Tünnerman, “Lens array manufacturing using a driven diamond tool on an ultra precision lathe,” in Vol 2 of 9th International Conference of the European Society for Precision Engineering and Nanotechnology, H. van Brussel ed. (EUSPN 2009), 423–426. 23. S. Scheiding, A. Y. Yi, A. Gebhardt, R. Loose, L. Li, S. Risse, R. Eberhardt, and A. 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Introduction
Novel high performance lens array based optical systems have been emerging over the last decade that are pushing the limits of current lens fabrication techniques.These lens array technologies include high throughput array microscopes, array flow cytometers, and snapshot hyperspectral imaging systems [1][2][3][4] which either require or may be enhanced by arrays of high quality, miniature multi-element objectives with full aberration correction.Several microlens array fabrication techniques have been well established and are in widespread use for imaging applications such as plenoptic light field sensors [5] and three dimensional imaging systems [6], which do not have full aberration correction, as well as non-imaging application such as light emitting diode (LED) illumination [7].All of these techniques are capable of producing lenses with state of the art form and surface finish, but each of these techniques has a limitation that makes it unsuitable or undesirable to meet the demands of emerging miniature lens array imaging systems (Table 1).Single point diamond turning (SPDT), a very fast 2-axis lathe technique, is capable of producing high quality individual lenses in a wide range of focal lengths and diameters.Lens array fabrication with SPDT takes much longer and cannot be automated because the work piece must be manually repositioned for each lens in the array.Other similar diamond turning techniques such as slow tool servo (STS) and fast tool servo (FTS) synchronize the motion of the X and Z axes of SPDT with the work spindle using a spindle encoder, also called a C-axis.This extra degree of freedom enables 3-D fabrication of non-rotationally symmetric objects like lens arrays.Although fast and fully automated, the edge slopes of lenses fabricated by STS and FTS are limited by the diamond tool's relief angle [8][9][10][11].STS/FTS tools with a 40° relief angle have recently become available [12], but it is impossible to achieve a 90° relief angle.Non-machining techniques can provide both automation and steep slopes, but come with other difficulties.Grayscale lithography can produce high quality lenses of arbitrary shapes and with nearly vertical edge slopes, but sags greater than 60 μm are considered extreme for this technique [13,14].Droplets of optically clear material form low f/# spherical lenses naturally due to surface tension.They can be melted and cooled as in thermal reflow or printed using inkjet technology and cured with ultra-violet (UV) irradiation [15,16].Larger lenses can take on aspheric shapes due to the effects of gravity [17,18].Application of an electrostatic field gives greater control over the shape of the lens [19].The overall size of the lens is limited, however, and arbitrary shapes are not currently possible.Other non-cutting techniques such as laser direct writing and deep lithography with protons have been developed, but do not produce lenses with the same quality as thermal reflow and microjet printing [20].Three-axis micromilling is another diamond machining technique which overcomes the limitations on lens geometries found in other lens array fabrication techniques.This process can produce a wide range of lens shapes including steep, low f/# lenses, aspheres, asymmetric lenses, and other freeform optics [9,[21][22][23][24].The drawback of 3-axis micro-milling is that it requires long fabrication times compared to most other lens fabrication techniques.With a feed rate of 100 mm/min and radial feed rate (stepover) of 4 µm per revolution (0.4 mm 2 /min), a 5 mm clear aperture diameter lens requires approximately one hour of machining time.This is the fastest fabrication rate published for 3-axis micro-milling of optically smooth surfaces so far [9,23].A 3x3 array of 5 mm lenses fabricated on a 19.1 mm diameter disk would require a 9 hour finish cut with micro-milling, compared to 3.2 hours with STS, or 1.3 hours with FTS.A system such as an array of miniature microscopes such as the one proposed in [2] requires aspheric lenses with low f/#, deep sag and tight tolerances.Altogether, 240 lenses need to be machined, 192 of which have a diameter larger than 5 mm with sags and edge slopes that fabrication techniques other than 3-axis micro-milling cannot possibly fabricate in an automated process.With micro-milling, each 5 mm diameter lens would require approximately one hour for the finish cut at 0.4 mm 2 /min, totaling 8 days of uninterrupted machining time for the all the lenses in the arrays [9,23].This much fabrication time adds up to a prohibitive expense for prototyping novel, complex, arrayed imaging devices.Fabrication rates for micro-milling have been increasing as the process and equipment have improved, reaching feed rates up to 250 mm/min and fabrication rates up to 1.25 mm 2 /min [26].Despite the improvements, 3-axis micro-milling is still slower than the other automated diamond machining techniques for most lens arrays.A new technique is needed that can match the ability of 3-axis micro-milling to produce a range of lens types of excellent optical quality in an array or freeform pattern in a fraction of the time currently needed.Four-axis single point diamond machining (4-axis SPDM) is the technique that is proposed here, and it meets all the same requirements as 3-axis micro-milling while reaching feed rates of 500 mm/min, 5x faster than published feed rates for micro-milling optical surfaces [9] and double the feed rates possible today.This paper will describe the principles and implementation of the proposed technique and show results demonstrating the reduction in cutting time while achieving equal or better tolerances (roughness, form error, and radius error) as 3-axis micro-milling.Implementation includes the process of converting a 3-axis micro-milling numerical control (NC) program into a 4-axis SPDM NC program and new methods to compensate for tool misalignment in the spiral tool path.To the best of our knowledge, this is the first time that the 4-axis SPDM technique, the methods presented here to correct the tool misalignments in the tool path, and results obtained using these methods have been published.In addition, lenses fabricated using a 3-axis (X,Y,Z) micro-milling process are presented for comparison with lenses fabricated with this new technique.

Principles
Four-axis SPDM shares similarities with both 3-axis micro-milling and conventional SPDT.The path of the tool is a spiral in XYZ, the same as in 3-axis micro-milling.The difference is that the spindle turns much slower, making only one revolution for every loop in the spiral tool path, and it turns in the opposite direction.The 4th axis is the C-axis, which is used to keep the tool angle and position tightly synchronized, as in the video shown in Fig. 1 (Media 1).The tool angle always orients the face of the tool so that it is coplanar with the lens axis of symmetry (Fig. 2).From the point of view of the diamond tool, the motion is exactly the same as SPDT.

Four-axis SPDM vs. other diamond machining techniques
In diamond turning (SPDT, STS, and FTS), the surface finish is affected by a number of factors which include tool quality, stability of the spindle, work-piece fixture and stages, and the radial feed rate or feed per revolution.When the radial feed rate is too high, the tool imprint (i.e.-the grooves left by the tool) becomes the main contributor to surface roughness.The width and depth of these grooves depends on the tool radius and the feed per revolution.In 3-axis micro-milling, the tool imprint is not a groove, but a divot (Fig. 3).These divots are a result of a cutting process in which the tool is not in constant contact with the work piece.
Instead the tool removes a small amount of material once per spindle rotation, leaving behind a divot.The width and depth of these divots depends on the tool radius and radial feed rate, just as the groove width does in diamond turning.The length of these divots depends on the chip per tooth.Chip per tooth is the distance the tool moves in the time it takes the spindle to complete one rotation and is equal to feed rate divided by spindle speed.In order to achieve a surface finish comparable to diamond turning without changing tools, the chip per tooth and feed per revolution in 3-axis micro-milling need to be low enough so that the width, length and depth of the divot are negligible.A decrease in chip per tooth can be accomplished either by decreasing the feed rate or by increasing the spindle speed.In order to increase the feed rate without the tool imprint size, the spindle speed must be increased proportionally as well.
High speed spindles with top speeds of 60,000 rotations per minute (RPM) have been used to micro-mill lenses at feed rates of up to 250 mm/min at a roughness of 5 nm [26].
In 4-axis SPDM, the diamond tool is always in contact with the part, so it is constantly removing material.Thus no divots are left in the surface, the tool imprint is a groove, and surface roughness does not depend on the feed rate.This allows the tool to move at feed rates 2-5x faster than micro-milling while still producing the same or better roughness.Fig. 3. Micro-milling tool imprint on a plastic lens when chip per tooth is too large.The height and spacing of the divots depends on spindle speed, feed rate, and tool size.This surface was obtained using a 0.53 mm radius tool at a 14 µm chip per tooth.

Equipment
The 4-axis SPDM technique was developed on a 250 Ultra-Precision Lathe (UPL) from Moore Nanotechnology Systems, Inc. (Nanotech®).The 250 UPL comes standard with X and Z axes.This 250 UPL is also equipped with the optional C and Y axes.In work spindle mode, the spindle reaches a top speed of 10,000 RPM, so it is not a high-speed spindle.Finished surfaces were cut with a 0.539 mm diamond ball endmill from Contour Fine Tooling, Inc. 3axis micro-milling NC programs were generated by Nanotech's NanoCAM TM 1.0 software.These NC programs have the necessary G-code headers and a spiral tool path that allows the milling tool to cut out the shape of the lens.The conversion of 3-axis micro-milling NC programs to 4-axis SPDM NC programs was done in Matlab® (R2010b).The Nanotech Work piece Error Correction (WEC) system was used to make 2-D profile measurements of test parts.This system includes a linear variable differential transformer (LVDT) and accompanying data acquisition and analysis software.3-D scans of finished surfaces were taken using two Zygo® interferometers, the PTI250 TM Fizeau interferometer and the NewView TM 5032 optical profiler.The Zygo interferometer measurements were analyzed in MetroPro TM 8.2.0.

Tool path conversion
Because 4-axis SPDM and 3-axis micro-milling both require a spiral tool path, the easiest way to create a 4-axis SPDM NC program is to convert it from an existing 3-axis micromilling NC program.The most important change that needs to be made in the header section is the removal of M03 or M04 work spindle commands that activate the spindle asynchronously.The 250 UPL's controller will halt execution if an M03 or M04 is encountered in an NC program while the C-axis is engaged.Other changes may be needed to ensure that the tool safely approaches the part and starts cutting from the correct position in the correct orientation.Next, the C-axis coordinate must be added to each of the XYZ-axis coordinates.It is best to do this using an automated script or executable program which reads in and copies out the XYZ coordinates with the C-axis coordinate appended (Fig. 4).
In order to mimic the SPDT process, the face of the tool must be coplanar with the surface's axis of symmetry.In the spiral tool path generated by NanoCAM 1.0, XY coordinates are defined at regular angular intervals.This means that the C-axis coordinate needs to be incremented by the same angular step.As shown in Fig. 4, the tool starts a cut oriented at 180° and turns clockwise 0.5° every time it moves from one point in the tool path to the next.The direction that the tool turns depends on the tool, but will always be in the opposite direction used for micro-milling.A tool that spins clockwise when micro-milling will turn counter-clockwise during 4-axis SPDM and vice versa.

Tool misalignment and tool path correction
The machine configuration for 4-axis SPDM is the same as in 3-axis micro-milling [24].The diamond tool is mounted to the work spindle by a collet chuck.The collet chuck and tool must be carefully aligned with the spindle centerline for best results.Despite best efforts, though, there will inevitably be some misalignment between the tool and the spindle centerline which can be several microns.The types of tool misalignment errors that can occur in 3-axis micro-milling also occur in 4-axis SPDM, and their effects on form error are the same.These alignment errors are called tool height error (t h ) and tool past center or not to center (t c ) (Fig. 5).The names tool height error and tool past or not to center are used because these misalignments produce the same exact form errors as their well-known counterparts in SPDT.Tool orientation is an additional type of misalignment that is unique to 4-axis SPDM.t h t c t θ Fig. 5. Four-axis SPDM tool misalignment errors when the C-axis coordinate is 0°.Tool height error, t h , and tool past center error (or tool not to center error), t c are the vertical and horizontal distances from the spindle centerline to the tool apex.Tool orientation error, t θ , is the angle between the face of the tool and the plane containing the spindle centerline and the lens axis of symmetry.
Because t h and t c are already well-known in SPDT, there are a number of techniques already used to evaluate the form error of a lens or a spherical test surface and determine their values.Tool height error can be estimated by examining the nub or cone at the center of the lens using a microscope and an eyepiece reticule.MetroPro and WEC both include analysis features that estimate t c based on a test surface measurement.Any technique for measuring t h and t c for SPDT surfaces is equally valid and useful in 3-axis micro-milling and 4-axis SPDM.In these experiments, the WEC measurements were used to calculate t c , and an Edmund Industrial Optics 3x zoom lens and a Sony® SSC-DC193 Color Video Camera were used to determine t h .In the sign convention used here, t h < 0 if the tool is below the spindle centerline when the tool is at the 0° C-axis position.t c < 0 if the tool is not to center.
In SPDT, t h and t c errors can be eliminated by adjusting the physical height of the tool and shifting the XZ coordinate system in the ± X direction.Physical adjustments to the position of a milling tool are more difficult, and a coordinate system shift will not correct either of these errors.Instead, each coordinate in the tool path must be adjusted independently.For each pair of XY coordinates generated by NanoCAM 1.0, which assumes zero misalignment, the adjusted coordinates, X*Y*, can be calculated using Eq.(1). Figure 6 shows a corrected 4axis SPDM NC program segment.The tool is oriented roughly by rotating the tool so that it faces straight up and then setting the Nanotech 250 UPL's program C-axis coordinate to 0. By eye, this can be done with a precision of about 15°.With the help of a camera t θ can be reduced to approximately ± 3°.A more precise estimate of the tool orientation error requires cutting and measuring a spherical test part.
When the tool is oriented incorrectly, it will have an elliptical cutting profile.The cutting profile of a tool in coordinates (S,T) are the radial and axial offsets, respectively, from the commanded tool position to the point of contact with the surface (Fig. 7).For a perfectly aligned tool with zero radius error and waviness, the tool's shape, cutting profile, and programmed tool radius compensation offsets should be identical.In the case of a misaligned tool, the cutting profile differs from the tool shape and programmed tool compensation offsets.The tool cutting profile can be calculated using the WEC measurement of a test surface profile.For nominally spherical test surfaces, the tool profile can be calculated from the measured test surface profile, (X, Z), using Eqs.( 2) and (3).For aspherical surfaces, numerical methods must be used to find the cutting profile.In these equations, F(X) = Z is the commanded tool position in Z as a function of X, and G(X) = F'(X).R t is the programmed tool radius and R s is the programmed test part radius.
The calculated cutting profile of a misaligned tool and its elliptical fit are shown in Fig. 8.The major axis of this ellipse is the tool diameter, and the minor axis depends on the tool orientation error.Tool orientation error can be calculated from the best fit ellipse using Eq. ( 4).The sign of the tool orientation error is ambiguous because a misalignment in either direction will produce the same elliptical cutting profile.The larger the sweep of the tool used to cut the test part, the more accurate the calculation of tool orientation error will be.( ) ( )  3).An off-center ellipse is fit to the calculated cutting profile.Equation (4) gives a value of 14.244° for t θ . 1 minor diameter cos major diameter For low waviness tools, most of the form error can be corrected by removing or compensating tool misalignments as described in this section.Figure 9 shows a measurement of a 2 mm test part using the WEC system over a 3 mm scan range using a 1 mm radius probe tip.Correcting t h , t c , and t θ leads to a surface profile measurement with 0.04% radius error and 21 nm root-mean-square (RMS) form error, both of which are very tight tolerances for plastic lenses.

Results
Two 3x3 convex lens arrays were cut into polystyrene (Rexolite 1422).The outer diameter of each lens is 3 mm, and the radii are 2 mm (f/1.12) and 20 mm (f/11.2) (Fig. 10).The f/11.2 lenses could also be fabricated STS and FTS, but the f/1.12 lenses would require a tool with a 50° relief, where most tools have 40° of relief or less.The roughing passes were done using 3-axis micro-milling.Roughing using 4-axis SPDM would have required a tool with both positive and negative sweep, similar to tools used in SPDT.The ball endmills used in micro-milling have a strictly positive tool sweep.During roughing, a 3-axis micro-milling program can be run at full speed because roughness is not a concern.The finishing passes were done using the new 4-axis SPDM technique.These finishing passes did not include tangential lead-ins or fillets.The tool misalignment compensation techniques described in the previous section were used to eliminate form error as best as possible.The same compensations were applied for every lens in the array.Aside from compensating for tool misalignment, no additional tool path compensation was applied.The top spindle speed reached in C-axis mode was approximately 126 RPM.As the spindle speed approached this limit, acceleration limits reduced the actual feed rate almost linearly.Near the end of the NC program, acceleration limits further reduced both spindle speed and feed rate to nearly zero.The radial feed rate was 5.5 µm/rev.This should theoretically yield the same surface finish as a SPDT lens cut at 2000 RPM and 11 mm/min.A video of a complete finishing pass for a single lens is shown in Fig. 11 (Media 2).The finish pass for each 2 mm radius lens requires 5 minutes and 34 seconds.The finish pass for each 20 mm radius lens lasted 4 minutes and 4 seconds.Form and radius errors for 20 mm lenses were found to be reduced by running 3 warm-up dry runs prior to cutting the lenses.Each of the arrays in Fig. 10 were cut in 50 minutes.A 0.33 mm diameter area at the apex of the 2 mm lens was measured with the NewView optical profiler to assess the surface roughness.Fringe contrast drops off outside this area as the slope of the lens increases, which leads to measurement errors.The best fit sphere and 4th order Zernike polynomials were removed from the measured profile, and a 40 µm high-pass Gauss Spline filter was applied to the residue to obtain the roughness profile.A 12 µm diameter mask was used to exclude the nub left in the middle of the lens from the roughness calculations (Fig. 12).The 20 mm lenses are much shallower, and can therefore be measured over a larger area.A 0.4 mm × 1.2 mm area was scanned from the apex of each 20 mm lens out towards the edge.The PTI250 was used to measure surface form error and lens radii.A 1.27 mm diameter area of the surface was measured using an f/1.5 transmission sphere.This transmission sphere is not fast enough to measure the entire lens surface, but the PTI250 f/0.58 transmission sphere has a negative working distance and cannot be used.An f/4.8 transmission sphere was used to measure a 2.75 mm diameter area of the 20 mm radius lenses (Fig. 13).Focusing error was removed from the form error measurements and a 40 µm low-pass Guass spline filter was applied to the residue to eliminate surface roughness from the surface form error calculation.
Visible in the surface plots of all lenses is a ridge running almost vertically across the face of the lens.This is an artifact caused by following errors in the Y-axis.These artifacts cover a small area, however.Judging from the RMS form errors of these lenses, none of which is larger than 30 nm, the influence of these artifacts on lens performance will be negligible.The RMS surface roughness (R q ), RMS form error, and radius error of both lens arrays are shown in Tables 2 and 3.In order to compare cutting times between 4-axis SPDM and 3-axis micro-milling, 3 additional lenses were micro-milled using the same tool path used to cut the 20 mm lenses shown above.This tool path includes the compensations for tool misalignments t h and t c .A 5 µm chip per tooth was chosen in an effort to match the surface roughness of the 4-axis SPDM lenses.The spindle was run at 7,000 RPM with a feed rate of 35 mm/min.The cutting blocks portion of this NC program, which excludes tangential lead-in, requires 45 min to execute.The surface roughness, form error and radius measurements for these lenses are shown in Table 4.

Surface quality
All of the surface errors measured for both lens arrays are very low and meet low cost tolerances for surface roughness, tight tolerances for surface form error, and state of the art tolerances for radius error (Table 5).Two of the 20 mm radius lenses would meet state of the art tolerances for surface form error.These tolerances are either better than or not significantly different from tolerances previously reported for micro-milling plastic lenses using the same Nanotech UPL 250 [24], and for micro-milling array masters on a Nanoform 250 in cylindrical coordinates [9,23].The worst case tolerances expected for much larger lens arrays is determined by the 3sigma value, whichever of µ ± 3σ is further from 0. Based on this criterion, the roughness of large numbers of lenses is still expected to be similar to the roughness of the lenses shown here.With large numbers of lenses, there may be one or two that fail to meet tight tolerances for form error, but still meet commercial tolerances. 2 mm lenses can be expected to hold state of the art radius tolerances in large numbers, while a small percentage of 20 mm lenses will exceed this tolerance by a few tenths of a percent.Put another way, these results predict an 80% yield for 5x5 lens arrays of 2 mm radii that must meet tight tolerances on form error and state of the art tolerances on radius error for every lens in the array.Similarly, the predicted yield for 5x5 arrays of 20 mm lenses is 54%.The predicted yields for 10x10 arrays are 41% and 11% respectively.For higher yields on larger lens arrays, a 60 nm tolerance on form error must be accepted for any lens array, and a 0.5% tolerance on radius error must be accepted for longer radii.These tolerances are still very suitable for emerging applications of lens arrays.  2 and 3 and the 3sigma, arg max(|x|,x = µ ± 3σ), expected worst case tolerances for larger numbers of lenses.Also included is the classification of each of these tolerances (low cost, commercial, tight, or state of the art).Here, the naming conventions are as follows: state of the art is best tolerances possible, next is tight, then commercial (standard), and then low cost.What is referred to as "state of the art" in [27] for roughness and radius error is called "tight" here, and what is referred to as "extremely tight" in [27] for roughness and radius error is called "state of the art" here.Form error here is called wavefront error in [27] and "standard" is used in [27] where "commercial" is used here.The breaks between state of the art, tight, commercial, and low cost are the same here as they are in [27].

Comparing fabrication rates
At low feed rates, micro-milling fabrication times are proportional to the area of the tool path diameter.At higher feed rates and smaller diameters, acceleration limits have a noticeable effect, and the fabrication rate of micro-milling slows.The same effect is seen in 4-axis SPDM.For lenses with millimeter scale diameter, the effect of acceleration limits can be approximated as a fixed time acceleration penalty added to the total cutting time.In addition to acceleration limits, 4-axis SPDM is also limited by the maximum C-axis spindle speed.The effect of the maximum spindle speed in 4-axis SPDM is to cause the relationship between fabrication time and part diameter to be approximately linear for diameters smaller than F/(πS max ), where F is the feed rate and S max is the maximum spindle speed, and then to transition to a quadratic dependence.Equation ( 5) estimates cutting time for a single micromilled lens in terms of the tool path diameter (D), feed rate (F), radial feed rate (F rad ), and acceleration penalty (P acc ).Equation ( 6) estimates the cutting time of a single lens 4-axis SPDM process.
In the 4-axis SPDM process presented here, F is 500 mm/min, F rad is 0.0055 mm, and S max is 126 RPM.The tool path semi-diameter of the 2 mm radius lens was 2.000 mm and this lens required 5.57 minutes of cutting time.The tool path semi-diameter of the 20 mm radius lens was 1.668 mm and required 4.07 minutes of cutting time.Using Eq. ( 6) to back-calculate P acc yields 32 and 26 seconds respectively.To estimate cutting times for a wider range of part diameters, P acc is assumed to be 29 seconds.
Schieding et al and Davis et al both report fabrication rates of 0.4 mm 2 /min with roughnesses of 7 nm and 9 nm respectively [9,23].Both groups used a Precitech Nanoform 250 with a high speed spindle from Professional Instruments.Lenses with tool path diameters larger than 0.7 mm are expected to take longer to fabricate by micro-milling at 0.4 mm 2 /min than using the 4-axis SPDM process presented here (Fig. 14).A 5 nm surface finish is possible using a Nanotech 350 freeform generator (FG), a 0.5 mm radius tool, 250 mm/min feed rate, and 0.005 mm radial feed for a fabrication rate of 1.25 mm 2 /min, although the feed rate is typically programmed to decrease linearly with each pass in the spiral path [26].The fabrication rate of the lenses micro-milled for comparison with the 4-axis SPDM lenses was 0.1925 mm 2 /min with a roughness of 5.7 nm.Using the same feed rate and chip per tooth, but with a spindle speed of 60,000 RPM instead of 7,000 RPM would yield a 300 mm/min feed rate and a 1.65 mm 2 /min micro-milling fabrication rate.It is unlikely that a 5-6 nm surface roughness would be maintained at these rates, however [26].
Acceleration limits do have an effect on micro-milling time at these feed rates, and they do depend on the machine configuration.Whether the acceleration penalty for 3-axis micromilling is assumed to be zero or 29 seconds as in 4-axis SPDM, the fabrication rate of 4-axis SPDM does not become significantly higher than 3-axis micro-milling until the tool path diameter is approximately 2 mm (Fig. 14).The maximum possible fabrication rate of 4-axis SPDM using the parameters presented here is 2.75 mm 2 /min, but only at diameters too large for most lens arrays.Typical fabrication rates to be expected for millimeter scale lenses are 0.5-2.5 mm 2 /min.The average fabrication rate achieved with the 20 mm lens was 2.12 mm 2 /min.While this is not the fastest rate projected for the process, it is still 5.3x faster than fabrication rates published using cylindrical 3-axis micro-milling on a Nanoform 250 [9,23].The fabrication rates possible today using a 0.5 mm radius tool on a Nanotech 350 FG come closer to the fabrication rates of 4-axis SPDM, but even at 250 mm/min and 0.005 mm stepover [26], the fabrication rate of the 20 mm 4-axis SPDM lenses is still 70% (2.87 minutes) faster.As the process is developed and improved, fabrication rates are expected to get faster still.
With machining processes such as SPDT, the finish cut makes a small amount of the total effort required to prototype an optical system.Setup, alignment, calibration, and fixturing and programming generally take more time than making the finish cut if the configuration needs to be changed.With micro-milling, the fabrication time is a more significant part of the prototyping process.Since fabrication services for prototyping typically cost $100 -$200 per hour, a reduction in cutting time associated with a 70% increase in fabrication rates can lower prototyping costs noticeably.Further cost and time savings can be found in the setup time, due to the fact that spindle only needs to be balanced for operation at 126 RPM.This is much easier and requires less specialized equipment than balancing a high speed spindle at 40,000 -60,000 RPM.

Conclusions
A novel diamond machining technique known as 4-axis single point diamond machining has been presented.Like slow tool servo, fast tool servo, and 3-axis micro-milling, this technique is capable of producing arrays of lenses in a fully automated process.A simple method for converting an existing 3-axis micro-milling NC program to a 4-axis SPDM NC program has been described.NC compensation of 3-axis micro-milling and 4-axis SPDM tool alignments has been proposed.The 4-axis SPDM technique has been shown to produce plastic lens arrays of excellent optical quality.Lens arrays can be expected to easily meet tolerances of 10 nm roughness, 60 nm form error, and 0.5% radius error for every lens in the array.For 3x3 and 5x5 lens arrays, tolerances of 30 nm form error and 0.3% radius error are achievable.4axis SPDM can produce the same range of lens radii, sags, and slopes that 3-axis micromilling currently offers, which is a wider range than STS or FTS.Thus, this technique can be used to produce both the high f/# and low f/# lenses needed in systems such as an array microscope without changing machine configuration.3-axis micro-milling is still as fast if not faster for fabricating micro-lenses.Miniature lenses, on the other hand, with tool paths larger than 2 mm in diameter can be fabricated using 4-axis SPDM 1.2-2x faster than 3-axis micromilling, without the use of a high speed spindle.This takes hours off the total cutting time of a complex lens array.As a result, this technique is expected to reduce the cost of prototyping novel optical systems based on arrays of miniature lenses by anywhere from hundreds to thousands of dollars.

Fig. 1 .Fig. 2 .
Fig.1.Demonstration of 4-axis SPDM (Media 1).The tool moves around the part in a spiral path, rotating once for every loop the tool makes around the work piece.In the tool's frame of reference, the motion is the same as in single point diamond turning.
Fig. 4. (a) Segment of a 3-axis micro-milling NC program generated by NanoCAM 1.0.Highlighted in yellow are changes that were made to the program manually after it was generated in NanoCAM.(b) 4-axis SPDM NC program segment generated from the 3-axis micro-milling NC program on the left.Green highlights indicate changes made to the code in (a) by an automated Matlab script, which added C-axis coordinates to all XYZ coordinates.Yellow highlights indicate changes made in a text editor afterwards.

Fig. 6 .
Fig. 6.(a) An uncorrected 4-axis SPDM NC program segment.(b) The same 4-axis SPDM NC program with numerical correction for tool height error (t h ) and tool not to center error (t c ). t h = −0.0067and t c = 0.00296096.

Fig. 7 .
Fig. 7. Relationship between the surface profile, the profile of the commanded tool position, and the tool profile.The points along the cutting profile can be found by moving the ST coordinate system along the profile of the commanded tool position and finding the intersection between the tool and surface profiles.

Fig. 8 .
Fig. 8. Cutting profile of a misaligned tool in (S,T) coordinates.S and T are what the program radial and axial tool compensation offsets should be.The nominal tool profile is what the cutting profile would be if the tool had no orientation error (t θ = 0).The actual tool cutting profile is calculated using the measured test surface profile and Eqs.(2) and (3).An off-center ellipse is fit to the calculated cutting profile.Equation (4) gives a value of 14.244° for t θ .

Fig. 9 .
Fig. 9. (a) WEC error measurement of a 2 mm radius test part before applying corrections.The errors t h and t c contribute to a nub in the middle of the lens.The 1 mm radius WEC probe has difficulty measuring features of this size, so the nub appears as wide spike instead.(b) WEC error measurement of a 2 mm radius test part after t h , t c , and t θ have been compensated and/or corrected.The radius error of this test part as measured by WEC is 0.869 μm (0.04%), and the RMS form error as measured by WEC is 21 nm.

Fig. 10 .
Fig.10.Two 3x3 polystyrene lens arrays with radii of 20 mm (left) and 2 mm (right).The finish cut for both of these lens arrays was done using the 4-axis SPDM technique.

Fig. 11 .
Fig.11.A complete 4-axis SPDM finishing operation on a 2 mm lens (Media 2).The original video was cropped and compressed to fit the allowed media file size limits, but it is still real time.The playback speed has not been altered.The original, uncompressed, full size video is available upon request.

Fig. 12 .
Fig. 12. Surface profile measurements of a 2 mm lens (left) and a 20 mm lens (right) fabricated using 4-axis SPDM measured with the Zygo® NewView TM 5032 Optical Profiler.RMS roughness and R q are the same.

Fig. 13 .
Fig. 13.Surface form error of a 2 mm lens (top) and a 20 mm lens (bottom) fabricated using 4axis SPDM measured with the Zygo® PTI250.

Fig. 14 .
Fig.14.Fabrication times and fabrication rates of 4-axis SPDM vs. 3-axis micro-milling .The fastest published rate of fabrication for micro-milling is 0.4 mm 2 /min using a Precitech Nanoform 250[9,23].Peak feed rates of 250 mm/min and fabrication rates of 1.25 mm 2 /min are possible using a Nanotech 350 FG while maintaining a surface roughness of 5 nm[26].The acceleration penalty is unknown at these feed rates.4-axis SPDM at 500 mm/min has an acceleration penalty of approximately 29 s.For the sake of comparison, curves for expected micro-milling times at 1.25 mm 2 /min with acceleration penalties of 0 and 29 seconds are shown.

Table 1 . Ultra-precision lens array fabrication techniques and their capabilities
a -The height of a spherical reflow or printed lens does not exceed its diameter.Additional limitations may apply.b -Unless otherwise specified, fabrication rate refers to unit of part semi-diameter fabricated per unit time (mm/min), unit of part area in XY plane fabricated per unit time (mm 2 /min) or unit of part volume fabricated per unit time (mm 3 /min).Only "finishing" steps, those that determine the final shape and surface quality of the lens, are included.* -Indicates that all other fabrication techniques outperform this technique in the given category.These limitations may lead one to consider choosing an alternative fabrication technique.