Testing of a diamond-turned off-axis parabolic mirror

Transcription

1 Invited Paper Testing of a diamond-turned off-axis parabolic mirror Jan Burke *a, Kai Wang b, Adam Bramble b a CIRO Materials cience and Engineering Australian Centre for Precision Optics, PO Box 218, Lindfield, NW 2070, Australia b Centre for Lasers and Applications, Department of Physics, Macquarie University, NW 2109, Australia ABTRACT Methods to test on-axis and off-axis parabolic mirrors are standard textbook fare. All we need, we are told, is a spherical wavefront and a plane mirror, or a plane wavefront and a spherical mirror. Contrasting with the implied ease of application, reports on practical experience with these tests are somewhat rare, particularly for off-axis mirrors. We have explored both variations of this testing method with a phase-shifting Fizeau interferometer, auxiliary components, and a one-inch diamond-turned 90º off-axis commercial-quality parabolic test mirror. The testing process is quick and easy only if you know how, and frustrating and time-consuming otherwise. We report on the calibration of the reference surfaces, present a detailed and systematic re-appraisal of the necessary steps for alignment and measurement validation, which have been described previously but in a less straightforward way, and present a brief characterization of the parabolic mirror that gives some insight into the diamond-turning process. Keywords: interferometry; optical testing; sphere calibration; aspheric metrology; off-axis parabolic mirror 1. INTRODUCTION Parabolic mirrors are probably the earliest-used and easiest-to-test aspheric optics. Relying on their ability to focus a plane wavefront, or, when used in reverse, to collimate a spherical wavefront, we can set up the parabolic mirror between a spherical and a plane surface, where either end can be the interferometer, and test for wavefront errors conveniently. Let us name the configuration with a spherical wave emanating from the interferometer the R (spherical reference) method, and the configuration with a flat reference wave the FR method. For on-axis parabolae, the R method requires a testing mirror with a hole in the center, and the FR method needs a small spherical mirror or reflective ball in the beam. Both of these methods result in partial occlusion of the test wavefront. From a certain off-axis angle upward (depending on the numerical aperture of the mirror), the test gets easier in that the occlusions disappear, we do not need specialized components, and can use generic spherical and flat reference surfaces [1]. In theory, the test is straightforward set up the parabolic mirror so that its focus coincides with the focus of the spherical wavefront, and place the flat mirror normal to the direction of the collimated beam. We have explored this test with a small, 90º off-axis commercial-quality parabolic mirror, Edmund Optics NT (subsequently also referred to as OAP for off-axis paraboloid ), with a diameter of 25.4 mm, parent focal length of 25.4 mm and a lateral offset and therefore effective focal length of 50.8 mm. The material is 6061 aluminum, the reflective surface is diamond turned and protected, and the figure error specification is ¼ wave rms. An example application of this type of mirror can be found in Ref. [2]. We have conducted the test in both the R and FR configurations and have found that the alignment is not as easy as the simple geometrical description had us believe. ome work on this general problem has been done in the past [3]-[6], but usually the literature emphasizes the results and mentions the alignment methods only in passing, and a comparison of the two approaches appears to be completely missing. The objective of the test was to gather experience with the testing configurations, to compare their relative performance and ease of use, and to gain an insight into the optical quality of the mirror. The numerical aperture (NA) of the OAP is 0.24, which is much less than the NA of 0.68 of the 4-inch transmission sphere used. Therefore, we have zoomed the * Jan.Burke@csiro.au, ; Fax ; acpo.csiro.au Interferometry XIV: Techniques and Analysis, edited by Joanna chmit, Katherine Creath, Catherine E. Towers, Proc. of PIE Vol. 7063, , (2008) X/08/$18 doi: / PIE Digital Library -- ubscriber Archive Copy Proc. of PIE Vol

2 interferometer image for the R configuration so that the interferogram approximately fills the screen and we get good spatial resolution. For best results, a calibration of the transmission sphere at that setting is required. Conversely, in the FR configuration, the interferogram will fill a diameter of only one inch in the 6-inch aperture, so that here, too, we have adjusted the zoom to give us the best possible resolution. In the following, we will first briefly describe the calibration of the reference sphere[7],[8], and then proceed to the practical problems of setting up the tests. We simplify the previous work on the R test and give step-by step instructions on how the alignment can be completed quickly and easily. We discuss the FR test in comparison and provide results for both methods. Fig. 1 shows a sketch of the R test and its practical set up. Besides the parabolic mirror mentioned above, we used a WYKO 6000 interferometer [9] and a custom-made mirror polished in our own Australian Centre for Precision Optics (ACPO) workshop to λ/100 peak-to-valley (PV) surface flatness, measured with a calibrated highprecision Fizeau interferometer [10]. surface parent FL= 25.4mm focal length=50.25mm O/AP plane mirror Fig. 1. Left: illustration of R parabolic mirror test; right: practical setup of mirror test. The diameter of the mirror is 50 mm, and a figure map of the 45 mm central aperture is shown in Fig. 2. As the diameter of the test mirror is only 25.4 mm, the flatness of the mirror in this test is even better. Therefore, we do not attempt to calibrate the mirror in this setup, and neglect its influence on the result. trictly speaking, however, a null-test mirror should be perfectly flat, since otherwise the OAP cannot be used according to design (i.e. producing or accepting a flat wavefront) on the path to, or from, the flat mirror. Measurement Parameters FiINC -RD F sub-subregion masked 90% Waveleegek em Wedge 0-50 XfYize 111 X 110 Pixel size em Dale 21 eplemker 2001 Time 10:10:26 Ra nm Rms nm 20 Pt. PV nm 2 Pt. PV 6.57 nm Analysis Parameters Terms None Pupil 100 Masks: Fillering None Dala Resonre No Valid Fumes mm I mm I Fig. 2. Figure map of λ/100 mirror. 2. CALIBRATION OF THE REFERENCE PHERE In our calibration approach we follow Griesmann et al.[7], who propose that the reference sphere can be measured against a test ball which is moved in random fashion between measurements. This builds up an average in which the random contribution from the ball approaches zero in the limit. Enough samples should be included in the average to Proc. of PIE Vol

3 ensure that its remaining uncertainty will approach, or fall below, the uncertainty for a single measurement (i.e. the measurement repeatability). The practical test setup is described in more detail in Ref. [11]. Based on the statistical work on roughness testing in Ref. [12], we can express the compound rms error in a measurement of a calibration ball against a reference transmission sphere by 2 2 σ meas = σt + σ ball, (1) where σ T denotes the rms wavefront error of the transmission sphere and σ ball that of the calibration ball. If we can then average N random measurements over uncorrelated portions of the calibration ball (where the transmission sphere remains stationary), the influence of the average on the measurement uncertainty should diminish as 2 2 σ ball σ RDF = σt +, (2) N where RDF stands for reference data file. Depending on σ ball, a sufficiently large N will ensure that σ ball N σ meas 1 meas 2 σ single =, (3) 2 where σ single is a simple measure for the repeatability of a single measurement, which is determined by taking two measurements meas 1 and meas 2 in immediate succession, subtracting them and scaling the resulting rms to represent one measurement. Alternatively, a larger number of measurements can be taken and averaged to construct a standard all, from which the repeatability can be determined by σ single = σ all meas i, (4) i.e. subtracting one of the measurements from the average should result in approximately the same number as the method in Eq. (3). We have found that the results from the two methods agree to within fractions of a nanometer. At NA = 0.24, we sample only 1.6% of the total area of the calibration ball in a single measurement and therefore we have a large number of non-overlapping surface segments at our disposal. However it was also shown in Ref. [7] that overlapping surface segments do not necessarily invalidate the assumption of statistical independence for our purposes. till, the statistics here are slightly different from Ref. [12] insofar as the σ ball that we are calculating does not refer to a spatially uniform random error, but instead to a figure error, where the departures from a best-fit sphere may be highly localized, depending on the actual figure of the calibration and reference spheres. Our initial test for the single-measurement repeatability gave 0.5 nm rms when the calibration ball was measured, and 1.1 nm rms for a measurement of the OAP. Therefore, we consider the T reference surface calibrated when the rms uncertainty of the calibration is below 1 nm. To give us an idea of how many measurements against the calibration ball this will require, we can determine the average rms error between successive measurements of random surface portions of the calibration ball according to Eq. (3), or use Eq. (4). ince the latter method requires a robust average for the reference surface first, we find the simple subtraction of successive measurements to be the faster method by far. We also need to consider what influence the extent of the interferogram region has on the rms. ince we have zoomed the image, the interferogram of a calibration ball will fill the entire rectangular display. The circular outline of the OAP interferogram fills only about half that area, and thus the rms error should decrease when we consider it only for the true interferogram region. For the full video image, we found the rms uncertainty of the calibration ball in a single measurement to be 6.2 nm, and this went down to 4.0 nm when we restricted the region of interest to within the actual OAP interferogram outline, as shown in Fig. 3. Therefore, if the uncertainty decreases as predicted by Eq. (2), we should be able to get a good calibration by averaging over 20 random orientations of the calibration ball. We decided to take more than 20 measurements in order to get a robust estimate of the actual decrease in uncertainty with growing N. A previous calibration run with NA = 0.68 for a different application [11] had shown the uncertainty decreasing as N 0.63, which made an investigation into the behavior for a smaller NA highly interesting and relevant. Here, we evaluated data for 100 random measurements at NA = 0.24, and found an improvement in the uncertainty with N 0.45, in good approximation of the expected N 0.5 relation. The preliminary indication is therefore that either the amount of overlap between the surface portions forming the average, or the second-order spatial statistics of the calibration ball, Proc. of PIE Vol

4 or both, have an influence on the way the errors are averaged out, and the treatment in Ref. [12] probably only approximates the conditions in the random-ball test. The final calibration file is shown in Fig. 3, and is indeed a subset of the full-aperture file presented in Ref. [11]. Measurement Parameters Measurement Parameters Ra Rms 20 Pt. PV 2 Pt. PV Ra 2.820r Rms 3.585r 20 Pt. PV nm 2 Pt. PV nm Analysis Parameters Measurement Parameters Measurement Parameters Ra Rms 20 Pt. PV 2 Pt. PV Ra 1.889r Rms Pt. PV nm 2 Pt. PV nm Fig. 3. Top row: typical error map between individual ball measurement and average of all measurements. Left: full display (scale is 15 nm (black) to +15 nm (white); right: interferogram region only ( 10 to +10 nm scale). Note the different RM error readings. Bottom row: final calibration file after averaging over 100 random ball positions. Left: full display; right: interferogram region only (both scales are 10 to +10 nm). Note that tilt and power terms are removed from these phase maps. With 100 measurements in the average, the uncertainty in the transmission sphere calibration decreases to an extrapolated value of 0.5 nm, which is probably better than we need it for the OAP measurement, and also below the above-mentioned repeatability of 1.1 nm. Nonetheless, Fig. 3 shows that even for small regions of the reference sphere, it can be useful to check for errors that cannot be absorbed in the focus term, which is usually treated as an alignment error in spherical metrology and removed. We do not discuss possible reference flat calibration methods for the FR test here, but just observe that this issue has been solved completely with a method that also relies on averaging [13], and that in this study we simply assume that we can treat a 1-inch region of a 6-inch reference surface as almost perfectly flat. 3.1 R configuration 3. TETING OF THE PARABOLIC MIRROR During the alignment of the system depicted in Fig. 1, our first observation was that each change we made to the position or rotation of the OAP (to make its focus coincide with the focus of the test wavefront) also affected the relative alignment between the OAP and the flat return mirror. The resulting complications in understanding the fringe patterns and correcting the tilts of the return mirror simultaneously while adjusting the OAP, proved intractable within a reasonable time budget. It stands to reason that this problem would be simplified considerably if we could stop thinking about the return mirror while we align the OAP, and could just rely on the wavefront being reflected to where it came from. Of course a component exists that does just that; it is a retroreflector, its simplest incarnation being a corner cube. Consequently we replaced the flat mirror by a gold-coated corner cube, and found (i) that this made the alignment work considerably easier, and (ii) that this had been done before [6]. What follows is largely built on the work in Ref. [6], however we believe that the method is easier to understand and apply if the original parameters of pitch, yaw, and focus error, which are interdependent, are expressed in orthogonal coordinates of t, r, and z, with the axes being defined as in Proc. of PIE Vol

5 Fig. 4. These coordinates are vaguely based on cylindrical coordinates, where t is tangential to the shell that forms the surface of the rotation paraboloid. However, the coordinate system is attached to the laboratory, not the OAP, so that the geometry shown in Fig. 4 is only applicable after the OAP has initially been rotated into this system. z t r Fig. 4. Definition of paraboloid alignment coordinate system in R setup. Dashed line: OAP s central line of symmetry. Right angles between the axes are indicated by solid black or white lines. To set up the measurement system, first the transmission sphere must be aligned correctly with respect to the optical axis of the interferometer. This is best done by placing a mirror in the focus of the converging wavefront (cat s eye position), but with a slight focus error, and centering the resulting fringe pattern in the aperture by tilting the transmission sphere appropriately. The OAP should be on a 3-axis translation stage, as depicted in Fig. 5. The tilt control knobs at the back of the mirror mount, and the rotary stage on which the OAP is mounted, are only used for coarse adjustment so that the geometry of Fig. 4 matches the coordinate system shown in Fig. 5. T tilt controls z t r rotation stage Fig. 5 Left: etting up the OAP on a 3-axis translation mechanism; inserted labels give the relationships to Fig. 4. Right: setup with corner cube for retroreflection of nominally collimated beam. The alignment starts by ensuring that the beam leaving the OAP is roughly collimated; this may require some largerscale rotations and translations of the OAP but can be checked without the need to get any light back into the interferometer. Once the beam is reasonably collimated, we need to ensure that it hits the corner cube in its center. While a corner cube will also retro-reflect the test beam when it is off center, this would not be helpful, as we need the reflected beam to fill the aperture of the mirror on the way back, so that we can see its entire surface and fluff out the fringes more accurately. We will therefore also expect to see the 6-ray pattern characteristic of a corner cube partitioning the interferogram, as shown below. The correct alignment is demonstrated in Fig. 6. Proc. of PIE Vol

6 Fig. 6. Centering the cube corner on the collimated beam. The OAP mount is visible in the foreground, the corner cube is in the center of the picture. White arrow indicates beam location. Once we have a spot of object light appearing on the interferometer s camera monitor, we can iteratively optimize the alignment. In alignment mode of the WYKO 6000 where effectively the power spectrum of the interferogram is observed, and is expected to be a small spot on screen when the fringes are fluffed out a strong defocus, creating a continuum of spatial fringe frequencies, can smear out the spot enough to make it invisible on the monitor, so that it is probably easier and faster to use fringe mode for these alignments. The first two iterations are shown in Fig. 7. 1A 1B 1C 2A 2B 2C Fig. 7. The first two iterations of the OAP alignment. A images: all errors present (for the respective iteration); B images: t removed; C images: z removed. Image 2A results from 1C by removing r, and starts the next iteration. Note that both iterations shown here are carried out in fringe mode. The first time the object light comes into view, the OAP is still likely to be severely misaligned. However, even in a pattern that shows little more than the principal ray, as in Fig. 7-1A, there is enough information to tell us what to do next. The slant of the astigmatic pattern (and also its position in the upper half of the image) tells us that t must be adjusted. Obviously we are not yet in a position to go by fringes, but it is possible to get this adjustment almost exactly right the first time. While r and z influence each other, and t influences both of them, t is not altered by Proc. of PIE Vol

7 adjustments of either r or z. This is true of all off-axis paraboloids. The off-axis angle of the optical surface introduces a coupling of r and z, which is strongest for 90º OAPs, where the slant of the surface is 45º on the optical axis. The circular profile along the parabolic rotation surface leads to a weak coupling between t and the other two parameters, which disappears when t is adjusted properly, as then all tangential planes to the OAP surface in its central line of symmetry (cf. Fig. 4) are parallel to the t direction. In Fig. 7-1B, the t adjustment has been made and the astigmatism is now vertical. We next adjust z until the spot pattern becomes more circular, as in 1C. According to the discussion above, removing a fair amount of focus error brings some of the astigmatism back, so we start the next iteration in Fig. 7-2A. A residual amount of t error is removed for 2B, and for the removal of z, we can already go by a fringe pattern becoming circular, as in 2C. Removing r again starts the last iteration, as depicted in Fig. 8. 3A 3B 3C Fig. 8. The last iteration of the OAP alignment with the retroreflector. The fine vertical lines are a feature of the mirror surface and will be addressed in detail in ection 4 below. Again, a small amount of error in t has become visible in Fig. 8-3A, which we have removed in 3B. Finally we need to minimize the astigmatism (or z), which has been done in 3C. The entire process from 1A to 3C can be finished in less than a minute from almost any initial condition. There are a few more subtleties to mention about the adjustment. Firstly, only when the aperture is filled with the fringe pattern will it become obvious whether the corner cube is accurately centered in the beam; if it is not, the outline of the interferogram will be truncated, as shown in Fig. 9. What counts here is that we get to monitor most of the fringe pattern, but since the retroreflector is only an interim step on the way to a good measurement, the corner cube does not have to be centered perfectly. Fig. 9. Truncated interferogram outlines due to imperfect centering of corner cube in collimated beam. Left: Corner cube shifted left; center: corner cube centered; right: corner cube shifted right. econdly, even with perfect alignment of the reference sphere, it is not assured that the interferogram will appear in the center of the screen, whether the instrument is zoomed in or not. This is because the rotation of the parabola about its focus is in principle arbitrary, which shifts the interferogram about in the interferometer aperture. The larger the difference between the numerical apertures of parabola and transmission sphere, the more freedom we have in this respect. Fig. 10 clarifies this for rotation about the z axis. Proc. of PIE Vol

8 4 Fig. 10. ketch of three possibilities to point the focus of the OAP at the focus of the transmission sphere. In practice however, we will want to have the line from the focus to the center of the OAP normal to the t direction, which centers the interferogram vertically in the aperture (shown in darker shading in Fig. 10) and minimizes the interdependence of alignment errors. Thirdly, the attentive reader will already have noticed that the 6-ray pattern from the corner cube looks decentered and distorted in the aperture. This is not due to instrument or alignment errors, but is an intrinsic feature of parabolic mirrors, whose curvature is a function of the off-axis distance. The higher the NA of the mirror, the more pronounced the effect gets, as the range of curvatures within the aperture will become larger and the image of the 6-ray pattern, imparted to the collimated beam at its exact center, becomes projected onto a curved surface before it returns to the interferometer, which of course leads to distortions. A simulated interferogram, displayed in Fig. 11, shows exactly the same distortion. Moreover, in a parabolic mirror the curvature changes fastest near the 45º slope point (i.e. for a 90º OAP), as shown in Fig. 11, so that for this type of OAP the distortion is largest, and simpler tests with best-fit reference wavefronts [14],[15] would experience the largest problems with fringe densities. Off-axis distasca [a.s.] curvature curvature rate of change slope z/ r Fig. 11. Left: simulated interferogram showing image distortion. Misalignment fringes have been added solely to make the 6-ray pattern more visible. Right: generic plot of OAP parameters. The curvature (solid white line) drops fastest at the 45º point; the derivative of the curvature (dotted white line) has a minimum where the surface slope (black line) equals unity. Fourthly, the ray-swapping properties of a corner cube create interesting phenomena. As seen from the transmission sphere, the outer part of the OAP (surface angle steeper than 45º) is always larger than the inner part, but the light power reaching the outer part will be concentrated in the inner part upon return from the corner cube, since the center of the corner cube is exactly above the 45º point (where by point we mean the one in the OAP s central plane of Proc. of PIE Vol

9 symmetry). Conversely, the light power reaching the inner part will be diluted upon return to the interferometer. The geometry is explained by Fig. 12 a) and b), and Fig. 12 c) shows the actual intensity distribution. a b c I d Fig. 12. a) OAP as seen from interferometer through reference sphere: the 45º point of the OAP is not centered in the aperture. b) OAP as seen from above and through corner cube: the 45º point of the OAP is centered in the aperture. c) Intensity distribution in the object wave; recorded with reference sphere tilted away from optical axis and hence, no fringes. d) Phase map of well-aligned OAP with corner cube. While the intensity distribution becomes distinctly asymmetric, the opposite is true of the phase map. After the fringes are removed, the retroreflector is no longer useful to us. We cannot hope to remove symmetrical errors, since each beam going into a cube corner comes back on the other side of its center, so that each of pair of points will pick up the same amount of error. An example of a point-symmetrical phase map is shown in Fig. 12 d) where the ripple structures in the interferogram, which come from diamond-turning, are each overlaid onto the respective other side of the corner cube s vertex. The asymmetric distortions in the horizontal direction break the horizontal antisymmetry of the small-scale features, but the antisymmetry of errors about the horizontal axis is very good, as the distortions in the vertical direction are symmetrical. We can now remove the corner cube and insert the flat mirror in the beam. For its initial alignment, it is very useful to use a pinhole screen that allows the light from the transmission sphere through, and will catch the reflected wave from the mirror and show us how we need to align the mirror to guide the reflected wave back through the pinhole. Also, the mirror should be as close as possible to the OAP, as in Fig. 1, to minimize measurement errors by air movement, but more importantly to minimize the offset of the returning beam on the OAP that an alignment error of either part (OAP or mirror) will create from now on, for we are not finished yet. Depending on the quality of the corner cube used, and the appearance of the fringe pattern after the bogus point symmetry no longer exists, further fine adjustments may be necessary. This has to be done very carefully, as now each movement of the OAP will create large amounts of tilt fringes on screen and it is possible to lose the alignment again if this is not done in very small steps. Remembering that we have a double-pass arrangement at 45º, the interferometric sensitivity is about 0.35 waves per fringe. With an OAP like the one we are measuring here, the sensitivity varies significantly over the surface. In fact the angle of incidence of the light cone from the transmission sphere on the OAP spans the range from 37º to 51º, with concomitant sensitivity changes (cosine of that angle) from 0.8 to 0.625! Ideally, the measured phase values would need to be re-scaled point-by-point to account for this large range. We have not done that in this comparative study but for high-precision absolute metrology it appears indispensable to us. At least there is now no distortion in the image anymore, as the mirror (if properly aligned) will send each ray back exactly onto itself. Proc. of PIE Vol

10 At this point, we found that fluffing out the fringes completely was not possible which is the expected result, as the OAP has some errors. The question now arises, which errors in the figure map are we allowed to ignore as mere alignment errors, and which errors must we ascribe to the OAP? The relationships between alignment and measurement errors have been investigated before with Zernike polynomials[3]. The analysis in Ref. [3] was carried out for the first eight Zernike polynomials, and it was found that a defocus alignment error creates a small amount of spherical aberration, and an astigmatic alignment error creates a small amount of coma. As a general rule, introducing or removing lower-order alignment errors usually affects higher-order terms [3],[16]. We will now investigate in detail how alignment errors affect the measurement results. Table 1 gives the first 15 Zernike coefficients for alignment errors in each of the axes, where we have introduced approximately equal and opposite amounts of each type of misalignment to get a better estimate of the error terms they cause[16]. Table 1. The first 15 Zernike coefficients (in waves) for various misalignments of the OAP. a3 stands for 3 rd order spherical aberration, o-e means odd symmetry in x and even symmetry in y. Grayed numbers denote coefficients that are neglected in this study; black background denotes the coefficients of interest for the respective alignment error, with the alignment errors introduced deliberately in bold type. Total fit error is not affected by removing terms for display. Zernike coefficient r < 0 r > 0 z < 0 z > 0 t < 0 t > 0 best alignment 1 (tilt x) (tilt y) (focus) (astig. 0º) (astig. 45º) (coma x) (coma y) (a3) (trefoil o e) (trefoil e o) (a5) total fit error The spherical aberration caused by focus error is quite weak. Indeed, by looking at other terms throughout this table, we can see that there are random deviations of about in the Zernike coefficients, so that a deviation of less than 0.01 is not meaningful. We therefore conclude that we can tolerate some focus error (perhaps deliberately introduced to see and remove astigmatism better) because it does not appear to falsify the measurement result. Moving the OAP up and down along the z axis, on the other hand, has considerable effects. As we have seen during the adjustment, astigmatism is introduced, and the question was whether we could afford to ignore this as an alignment error. The strong response of coefficient 9 (o-e trefoil error) to this misalignment, and the appearance of some slight coma as well, show clearly that this is not the case. Note, however, that the largest part of the trefoil error appears to be an actual figure error, since there is a clear positive bias in the numbers. This is corroborated by testing for errors due to misalignments in t direction, where the o-e trefoil error remains constant and is by far the largest of the errors in the table, and the apparent e-o trefoil error can be manipulated by shifting the OAP back and forth along the t direction. Fig. 13 shows the differences between phase measurements taken with opposite misalignments in the three directions. Proc. of PIE Vol

11 Fig. 13. Difference maps of phase measurements with opposite alignment errors. Left: for r (with focus error removed from phase map); center: for z (with focus and astigmatism removed from phase map); right: for ϕ (with focus and astigmatism removed from phase map). Besides the large-scale errors, the translation of the OAP is also revealed by the emergence of the surface fine structure due to subtraction of slightly mismatched locations. The scale is 50 to +50 nm for all results. To exclude the possibility that imperfections in the actual OAP give us misleading interrelations, we have also simulated the errors by ray tracing with a perfect OAP, and the simulated phase maps show the same dependences on misalignment. This careful inspection of the results shows that we cannot allow astigmatism alignment error in a high-precision measurement. However, if the trefoil error were larger, and considered more detrimental in an application than astigmatism, we might consider using the OAP with a deliberately introduced astigmatism to reduce the apparent trefoil error. The bottom line of the table, giving the total fit error, also shows that the higher-order errors are minimized when the OAP is aligned to yield the smallest focus and astigmatism. In other words, we capture more of the errors in the 15- coefficient Zernike fit if we align the OAP as well as possible. We will discuss the actual measurement result at the end of this paper, side-by-side with the result from the FR measurement. 3.2 FR configuration For the sake of consistency, we keep the displacement coordinate system attached to the OAP as we reverse the measurement system, but we must still align the OAP with respect to the axes of the translation stage. Of course, when illuminating the OAP with a collimated beam, translations of the OAP have no effect on the alignment error (assuming we translate the OAP and the spherical return mirror together). Therefore, we need to use rotations in this case, not translations. To denote this, we label the rotations about the t and r axes by Greek letters τ and ρ. Translations along the r axis (varying the distance between the OAP and the mirror) lead to defocus error as before. The geometry is drawn in Fig. 14. z τ r, ρ Fig. 14. Definition of paraboloid alignment coordinate system in FR setup. Dashed line: OAP s central line of symmetry. The reflecting spherical surface is that of the transmission sphere, and is not drawn to its full numerical aperture here. Right angles between the axes are indicated by solid black or white lines. To fluff out straight fringes, the reflection sphere is shifted normal to the axis of the incoming wavefront (indicated by black arrows on the right). It is immediately evident that this coordinate system is more complicated than the one drawn in Fig. 4. If the τ axis is not in the center of the OAP, a rotation in τ will introduce an error in r, so that the procedure will be iterative to a larger extent than the one outlined above. The z axis is drawn only for completeness, but is not used for the alignment of the OAP. Proc. of PIE Vol

12 After mounting the reference flat in the interferometer, it needs to be aligned normal to the optical axis. To accomplish this, we can place the corner cube in the beam, whereupon a fringe pattern will appear immediately. By tilting the reference flat appropriately, we can then exploit the expected point-symmetry and center the fringe pattern on the corner point of the retroreflector, as shown in Fig. 15. Fig. 15. Fringe pattern centered in corner cube, indicating correct alignment of reference flat in interferometer. With a gold-coated corner cube and an uncoated reference flat, the double reflections will create a confusing fringe pattern. For Fig. 15 we have used a pellicle attenuator between reference flat and corner cube, and also temporarily zoomed the imaging unit out from the OAP sub-aperture region, so as to see the entire fringe pattern. The practical implementation of the FR setup is shown in Fig. 16. The collimated direction of the OAP is now pointed at the interferometer, and the OAP is adjusted in its rotary stage so that the focus points in the right direction. The rotation axes are not placed optimally. As pointed out above, a τ rotation will move the OAP surface a fair amount in the r direction. Likewise, a ρ rotation will move the focus up and down in the t direction (cf. Fig. 4). We will keep the description of the alignment brief, since the steps are quite similar to those described in 3.1 but with the system reversed. The initial alignment is made by observing the symmetry in the light cone leaving the OAP. If it is asymmetric with respect to the OAP s central plane of symmetry (i.e. skewed in t direction), a rotation in ρ direction will make it symmetrical. Then a τ rotation will remove astigmatism and we can start to use the transmission sphere. In addition, if we use the 3-axis stage to correct r, we will displace the OAP in the aperture, which will make it harder to compare opposite r with respect to higher-order errors. Moving the OAP in t direction will displace it in the aperture as well, so that it is best to follow the OAP s focus by using the translation mechanisms on the auxiliary mount that holds the transmission sphere, as indicated in Fig. 14. The micrometer screw for r is only just visible in Fig. 16 on the right, the ones for z and t are not. τ p r ρ z p p 5 5 a a a a a P P P r 0 P p p p 0 P Fig. 16. Left: etting up the OAP on a 3-axis translation mechanism; inserted labels give the relationships to Fig. 14. Right: setup with reference sphere used in reflection. Note that r can be adjusted on the reference sphere mount. Proc. of PIE Vol

13 We have found this test considerably harder to optimize than the R configuration. One reason for this is probably the inevitable mixture of rotations and translations for this geometry, but also the high sensitivity of the spherical section of the setup to alignment errors, which makes it more difficult to fluff out the fringes in the interferogram. We therefore disagree with the comment made in Ref. [6] that this version of the test is easier to handle. Of course, practical preferences say nothing about the performance of a test. We conclude this study by investigating the appearance of higher-order errors in the same way as above. Table 2 gives the first 15 Zernike coefficients for alignment errors in each of the three relevant axes, where we have been careful to keep the introduced errors in the same range as above to make the tests as comparable as possible. Table 2. The first 15 Zernike coefficients (in waves) for various misalignments of the OAP. The signs of rotations are taken in the mathematical sense and looking down the respective axis toward the origin as defined in Fig. 14. a3 stands for 3 rd order spherical aberration, o-e means odd symmetry in x and even symmetry in y. Grayed numbers denote coefficients that are neglected in this study; black background denotes the coefficients of interest for the respective alignment error, with the alignment errors introduced deliberately in bold type. Total fit error is not affected by removing terms for display. Zernike coefficient r < 0 r > 0 τ > 0 τ < 0 ρ > 0 ρ < 0 best alignment 1 (tilt x) (tilt y) (focus) (astig. 0º) (astig. 45º) (coma x) (coma y) (a3) (trefoil o e) (trefoil e o) (a5) total fit error In this configuration, the response to focus error is very different from above. A large amount of coma appears (note that we have kept the tilts, known to cause coma in many instances, at very low levels for the r tests), whereas, again, the spherical aberration terms are of little concern. From a practical point of view, this behavior is disadvantageous, because focus error is very hard to remove completely, and likely to drift back into the interferogram over the time it takes to make a few averaging measurements. Rotation in τ causes 0º astigmatism, a small amount of coma, and again as above, o-e trefoil. Rotation in ρ does the same regarding the respective orthogonal (in the Zernike polynomial sense) errors. Comparing the coupling coefficients with those in Table 1, we find that astigmatism goes with equal amounts of coma and trefoil in both setups. In analogy to above, Fig. 17 shows the differences between phase measurements taken with opposite misalignments in the three directions. Proc. of PIE Vol

14 Fig. 17. Difference maps of phase measurements with opposite alignment errors. Left: for r (with focus error removed from phase map); center: for τ (with focus and astigmatism removed from phase map); right: for ρ (with focus and astigmatism removed from phase map). The scale is 50 to +50 nm for all results. ince the bearing for the r adjustment on the transmission sphere mount was a little loose and each adjustment also caused a large amount of tilt which had to be removed again, we simulated this system with a perfect OAP (and perfect misalignments) as well, and found our observations confirmed. This shows that the two systems are not equivalent and that we have practical reasons other than convenience to prefer the R setup. We have not studied OAPs other than 90º, so it is possible that other systems show different sensitivities. If all alignment errors are removed, however, we should still get a valid measurement with this setup which should show the same OAP surface errors as the one from the R measurement. As described above in ection 2, for the FR measurement we have assumed the influence of the reference flat to be negligible. ubtracting the reference sphere error from the FR test is not straightforward. Assuming we use the same region of the reference sphere as in the R test, its error can be approximated by using the Zernike terms from the calibration file created in ection 2. However, we do not take this step here because, as we have seen, the errors found in the transmission sphere do not affect the parameters that we have investigated. 4. REULT AND DICUION After our analyses by Zernike coefficient tables, we can now proceed to comparing the figure error maps from the two measurement methods. Fig. 18 displays the final results. FjI: fl,i,,0 Tflf W.Igtfl Wdg 0.35 XfYi Pj,Ij 0.00 Dt 22 My 2000 Tj,, 21:40:56 Ra nm Rms nm 20 Pt. PV Pt. PV Analysis Parameters T&,! Tilt Pt,tttt Pupil H Ittti tug Nttttu Dttt fltututtttu Nt - Vtlid Ptitttt FjI: W.Igtfl Wdg 0.35 XfYi Pj,Ij 0.00 Dt 22 My 2000 Tj,, 21:42:35 Ra nm Rms nm 20 Pt. PV Pt. PV Analysis Parameters T&,! Tilt Pt,tttt Pupil H Ittti tug Nttttu Dttt fltututtttu Nt Vtlid Ptitttt Fig. 18. Final results of OAP measurements, with overlaid crosshairs to compare perspective errors. Left: from R measurement; right: from FR measurement. The low side of the mirror is on the left. Both scales are 100 to +100 nm. The difference due to the direction from which the mirror is measured is immediately evident. The fine vertical toolmarks appear straight in the R measurement, and curved in the FR measurement. This pattern alone tells us very clearly which measurement was made from the focus of the paraboloid, and which one was made from the collimated side. ince the OAP is a diamond-turned rotation paraboloid, the image from the focus will show the toolmarks as straight lines, whereas the collimated beam looks down upon the circle segments left by the diamond tip. This, of course, also leads to a perspective distortion of the surface. Whereas the low (and darker) vertical strip is distinctly right of the center in the R result, it appears almost in the center of the FR phase map. It is this distortion that causes the discrepancy in trefoil error (0.189 for R vs for FR) and the different rms errors in the phase maps of Fig. 18. Does this mean that the OAP is better when seen from the collimated side? To answer this question, we return to Table 2, check the total fit error, and Proc. of PIE Vol

15 find that in this case the best alignment has not improved the fitting error over the misaligned measurements. Therefore it is likely that a significant error is still hiding in higher-order Zernike coefficients. We have investigated the next set of aberrations with a 24-term Zernike analysis, and have found waves of quatrefoil error ( ) for the FR result (total 24-coefficient fit error waves), against only waves for the R result (total 24-coefficient fit error waves). To check whether this is an effect of the perspective distortion discussed above, we plot only coefficients 9 and 16 (trefoil and quatrefoil) for both results, so that we can compare the appearance of the surface shape to the actual figure maps. This test is shown in Fig. 19. Filb!I.2I Tflft,ThiI a qt,th!i W.Igtfl Wdl 0.35 XfYi Pj,Ij 0.00 Dt 21 My 2000 Tj,, 21:30:55 Ra nm Rms nm 20 Pt. PV Pt. PV Analysis Parameters T&,! Tilt Pt,tttt Pupil H Ittti tug Nttttu Dttt fltututtttu Nt Vtlid Ptitttt FiI: W.Igtfl Wdl 0.35 XfYi Pj,Ij 0.00 Dt 21 My 2000 Tj,, 21:31:53 Ra nm Rms nm 20 Pt. PV Pt. PV Analysis Parameters T&,! Tilt Pt,tttt Pupil H Ittti tug Nttttu Dttt fltututtttu Nt Vtlid Ptitttt Fig. 19. Isolated trefoil and quatrefoil errors for the OAP mirror. Left: from R measurement; right: from FR measurement. By comparison with Fig. 18, it becomes quite plausible that the surface is modeled well in both cases, and that for the FR result it takes a higher quatrefoil term to shift the two high points on the left side of the phase map further to the left. Besides that, the rms and PV error figures for the two phase maps in Fig. 19 are very similar. Our finding is therefore that both methods are equivalent if the initial alignment is carried out carefully, but that the results are not immediately comparable because of perspective distortions. Because of this, it is very important to relate the actual figure map to the Zernike coefficients that model it, to pay attention to the coordinate mapping, and to ensure the error analysis is carried out with sufficiently high-order terms to capture errors that may shift to different Zernike coefficients in response to a coordinate distortion. The figure maps reveal another detail. Besides the toolmarks in the vertical direction, a vertical oscillation is clearly visible, starting from the top rim of the mirror and moving down with decaying amplitude. This may correspond to a radial vibration of the diamond tool that is triggered when the tip first hits the material on each rotation, and then dies down while the tip crosses the mirror. Therefore, most likely the OAP was not trepanned from a larger piece but was individually diamond-turned. Trefoil error in off-axis parabolae has also been observed in a trepanned mirror [14], but it is not clear what role trepanning or standalone machining plays in the generation of trefoil error. 5. CONCLUION We have explored two double-pass testing methods for off-axis parabolic mirrors: an R configuration with spherical reference wave and flat mirror; and the reversed FR setup with flat reference and a spherical mirror. Building on a previously published methodical approach for alignment of the components, we have systematized and simplified the alignment algorithm for the R method. To enable a high-precision measurement, we have used the relatively recent random ball averaging calibration method for the transmission sphere, and have found that for the numerical aperture of 0.24, given by the OAP, the simple statistical 1/ N model for error reduction by averaging works well. The setup procedure for the FR method is not quite so straightforward, and also the fine alignment of the setup turned out to be more difficult in practice than in the R method. For both methods, we have investigated the sensitivity of measurement results to alignment errors, both by measurements and simulations, and have found that they are equivalent, save for a large susceptibility to coma error in the FR setup that does not have a correspondent in the R setup. Therefore, our preference is to test OAPs in the R configuration. It is easier to align, more stable against drift when set up, and systematically superior to the FR setup in that it tolerates focus error quite well. The example OAP that we measured was found to be of relatively good quality and suitable for a measurement precision study, and to meet its λ/4 rms surface error specification very easily. Proc. of PIE Vol

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