## Abstract

We demonstrate the precise figure measurement of a one-inch (25.4 mm) diamond-turned 90° off-axis commercial-quality parabolic mirror. The test is carried out with a phase-shifting Fizeau interferometer fitted with a spherical reference surface, auxiliary components and a flat return mirror. We present a detailed and systematic appraisal of the necessary steps for alignment and calibration of the instrument and the alignment of the parabolic mirror. Alignment errors and interferometric sensitivity variations are characterised and corrected, and the results give some insight into the diamond-turning process.

©2009 Optical Society of America

## 1. Introduction

Parabolic mirrors are probably the earliest-used [1] and easiest-to-test [2, 3] aspheric optics. A simple technique to make them is single-point diamond turning [4]. Parabolic mirrors can focus a plane wavefront, or, when used in reverse, collimate a spherical wavefront. Utilising this property, we can place the parabolic mirror between a spherical and a plane surface, where either end can be the interferometer, and set up a convenient null test for wavefront errors. A null test is conceptually very simple in that we will detect no fringes in the interferometer when the mirror is a perfect paraboloid.

In this first of the two papers exploring two possible null testing configurations, we discuss the configuration with a spherical wave emanating from the interferometer, the parabolic mirror collimating this spherical wave, and a flat mirror returning this collimated wavefront to the interferometer. We can name this the “SR” (spherical reference) method. For on-axis parabolae, the flat mirror must have a hole in the centre in which the focal point of the parabolic mirror is located, but from a certain off-axis angle upward (depending on the numerical aperture of the mirror), the test becomes easier in that a plane mirror without a hole can be used. In theory, the test for the off-axis mirror is then 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.

The mirror we have used for this study is a small, 90° off-axis commercial-quality parabolic mirror (Edmund Optics NT47-097), 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 an off-axis offset and therefore effective focal length of 50.8 mm. The material is 6061 aluminium, the reflective surface is diamond-turned and protected with an SiO_{2} coating, and the figure error specification is ¼ wave rms (*λ* = 632.8 nm). Application examples for off-axis parabolic mirrors can be found in Refs. [5] and [6].

This type of mirror lends itself well to a general appraisal of OAP testing because (i) a 90° OAP has large changes in curvature over its surface and is hence not readily amenable to substituting a null test by spherical waves [7, 8]; (ii) the apparent surface distortions between the two viewing directions (“collimated” and “spherical” side) are largest; and (iii) its numerical aperture (NA) of 0.24 is relatively large, so that variations of the interferometric sensitivity (to be discussed further in Section 3.4) across the mirror surface will not be negligible. These parameters will allow us to study the influence of geometry on the measurement results, and difficulties found in the test described here should appear to a diminished extent in tests of less extreme OAPs. 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 as the front end of the Fizeau interferometer. Therefore, we have zoomed the image, i.e. adapted the magnification of the optical system so that the interferogram approximately fills the screen and we get good spatial resolution.

For highest precision, we also require a calibration of the other surfaces involved, in this case the transmission sphere and the return mirror. We will briefly describe this calibration [9, 10], and then proceed to the practical problems of setting up the OAP tests. During our experiments we have found that the alignment is not as easy as the simple geometrical description suggests. Some work on this general problem has been done in the past [11–14], and a generalised method and algorithm have been described [14]. We build on this work by simplifying the approach taken there, present a systematic alignment technique, and investigate the performance of the method and the optical quality of the OAP.

Fig. 1 shows a sketch of the SR test and its practical set-up. Besides the OAP, we used a WYKO 6000 interferometer [15] and a custom-made glass flat, polished in our own Australian Centre for Precision Optics (ACPO) workshop to *λ*/100 peak-to-valley (PV) surface flatness.

## 2. Calibrations

#### 2.1 Instrument alignment

To set up the measurement system, first the transmission sphere must be aligned correctly with respect to the optical axis of the interferometer after it is installed in the instrument. The WYKO 6000 [15], and many other interferometers, have an alignment or “spot” mode that switches to the optical Fourier plane and allows tilt removal by bringing spots to an overlap. However, we have determined that this is not accurate enough, possibly due to residual errors in the lens array that constitutes the transmission sphere, and have found it necessary to finish the alignment in interferometric “fringe” mode. This is best done by placing an uncoated glass flat in the focus of the converging wavefront (cat’s-eye position), and then removing all tilt from the resulting interference pattern, as shown in Fig. 2.

Using a high-reflectivity mirror will lead to confusing multiple-beam interference due to multiple reflections and make the alignment more difficult, but can be useful at the end of the process. If highest accuracy is required, one can then work towards getting all multiple-beam patterns to overlap exactly.

Since this cat’s-eye test involves a point reflection, the interferometer is not operated in Fizeau mode, where the common path between object and reference error cancels instrument flaws. On the contrary, the reflection swaps the halves of the wavefront diametrically and doubles all antisymmetric errors [16], such as coma, which is very clearly visible in Fig. 2.

#### 2.2 Transmission sphere

In our calibration approach we follow Griesmann et al. [10], 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 ensure that its remaining uncertainty will approach, or fall below, the uncertainty for a single measurement (i.e. the measurement repeatability). Based on the statistical work on roughness testing in Ref. [17], we can express the compound rms error in a measurement of a calibration ball against a reference transmission sphere by

where *σ _{TS}* denotes the rms wavefront error of the transmission sphere and

*σ*that of the calibration ball. If we can then average

_{ball}*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

where RDF stands for “reference data file”. Depending on *σ _{ball}*, a sufficiently large

*N*will ensure that

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

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 nanometre.

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. [10] that overlapping surface segments do not necessarily invalidate the assumption of statistical independence for our purposes. Still, the statistics here are slightly different from Ref. [17] 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 localised, 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 “TS” reference surface adequately 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). Since 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 evaluated data for 100 random measurements at NA = 0.24, masked with the actual interferogram region of the OAP, and found an improvement in the uncertainty with *N*
^{-0.45}, in good approximation of the expected *N*
^{-0.5} relation. The final calibration data are shown in Fig. 3. The centre of the transmission sphere does not coincide with the centre of the aperture, for two subtle reasons: (i) the optical axes of transmission sphere and OAP only need to intersect at the common focus, but need not be collinear – when they are not lined up, a shift between the respective centres will occur; (ii) even if the axes were perfectly aligned, the horizontal offset would remain, because the ray incidence angles in the aperture of the OAP are not symmetrical about 45°. Indeed, they span a range from 37° to 51°, as explained in more detail in Section 3.4, and therefore the centres cannot coincide; on comparing Fig. 3(b) with Fig. 14(b), the horizontal offsets match quite well.

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 power term, which in spherical metrology is usually treated as an alignment error and removed.

#### 2.3 Return Flat

The return flat was measured with a high-precision Fizeau interferometer [18] that was calibrated with an advanced three-flat test scheme also relying on averaging [19]. The diameter of the flat is 50 mm, and a figure map of the central 45 mm aperture is shown in Fig. 4.

As the diameter of the tested OAP 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. Strictly speaking, however, a null-test mirror should be perfectly flat, since otherwise the OAP cannot be used exactly as designed (i.e. producing or receiving a flat wavefront) on the path to, or from, the flat mirror.

## 3. Testing of the parabolic mirror

Let us define the co-ordinate system as shown in Fig. 5. In contrast to Ref. 14, where the alignment has been described in terms of pitch, yaw, and focus error, we do not use tilts here because we believe that the method is easier to understand and apply if the original parameters are expressed in orthogonal coordinates of *t*, *r*, and *z*, as sketched in Fig. 5. These coordinates are vaguely based on cylindrical coordinates, where *t* is tangential to the shell that forms the surface of the rotation paraboloid, and has no *z* component. The *r* direction is “radial”, and *z* is the direction of the optical axis. The coordinate system is attached to the laboratory (i.e. a 3-axis translation stage), not the OAP, so that the geometry shown in Fig. 5 is only applicable after the OAP has initially been rotated into this system by using the tilt controls and the rotary stage on which the OAP is mounted.

#### 3.1 Preliminary alignment

During the alignment of the OAP, each change made to the position or rotation of the OAP also affects the relative alignment between the OAP and the flat return mirror, and results in large numbers of tilt fringes appearing in response to small adjustments. The resulting complications in understanding the fringe patterns and correcting the tilts of the return mirror simultaneously, while adjusting the OAP, make this adjustment strategy protracted and impractical. Thus we need a component that will simply return the incoming beam without adding tilt. In the case of plane waves, this is accomplished by a retroreflector, in its simplest version being a cube corner. This has also been described in Ref. [14] as a significant time saver in alignment and we have set it up as shown in Fig. 6.

The alignment starts by ensuring that the beam leaving the OAP is roughly collimated; this may require some larger-scale 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 cube corner in its centre. While a cube corner will also retro-reflect the test beam when it is off centre, 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.

Once we have a spot of object light appearing on the interferometer’s camera monitor, we can iteratively optimise the alignment. However, in alignment mode, effectively the power spectrum of the interferogram is observed. This is a small spot on the screen only when there are essentially no fringes in the interferogram; but a strong defocus, creating a continuum of spatial fringe frequencies, can smear out this spot enough to make it very faint or invisible on the monitor. Therefore it is important that the interferometer be set to fringe mode from the start.

The iterative process of aligning the OAP with a cube corner is shown in Fig. 7. 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. We are not yet in a position to go by fringes, but it is possible to adjust ∆*t* almost perfectly in the first iteration. While ∆*r* and ∆*z* influence each other, and ∆*t* influences both of them, ∆*t* is not altered by adjustments of either ∆*r* or ∆*z*. This is true of all off-axis paraboloids, and thus we can state the general rule that ∆*t* should always be optimised first.

The off-axis angle of the optical surface introduces a coupling of ∆*r* and ∆*z*, which is strongest for 90° OAPs, where the slant of the surface is 45° 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. 5) 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. We now also start to see the six-ray pattern coming from the cube corner. After adjusting ∆*r* to remove the strong focus error, we obtain the image shown in Fig. 7-2A. Due to the coupling between ∆*r* and ∆*z*, removing the focus error has brought some of the astigmatism back, so we start the next iteration.

A residual amount of t error is removed for 2B, and for the removal of ∆*z*, we can already go by a fringe pattern that we adjust to become circular, as in 2C. At this stage, if the transmission sphere is not aligned properly as described in 2.1, the fringe pattern will not be concentric with the vertex of the cube corner [14]. Removing ∆*r* again starts the last iteration, in the third row of Fig. 7. Again, a small amount of error in *t* has become visible in 3A, which we have removed in 3B. Finally we need to minimise 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, as demonstrated in Fig. 8 and Media 1.

For the fringe pattern to fill the aperture completely, the cube corner must be centred properly; this is unlikely to happen at the first attempt. However, since the use of the cube corner is only an interim step, it is sufficient to monitor most of the fringe pattern.

Likewise, whether the instrument is zoomed in or not, it is not assured that the interferogram will appear in the centre of the screen. This is because the rotation of the OAP about the focus of the spherical wavefront is in principle arbitrary, which shifts the interferogram about in the interferometer aperture (cf. Fig. 3(b)). The larger the difference between the numerical apertures of parabola and transmission sphere, the more freedom exists in this respect. To ensure the alignment will work as described and the three translation directions have small coupling, the setup should adhere closely to the conditions depicted in Fig. 5.

As can be seen in Fig. 7, the 6-ray pattern from the cube corner looks decentred and distorted in the aperture. This is not due to instrument or alignment errors, but is an intrinsic feature of parabolic mirrors, whose reflection angle is a function of the off-axis distance. As shown in Fig. 9, the image of the 6-ray pattern, imparted to the collimated beam at its exact centre, is projected onto a curved surface before it returns to the instrument, and therefore becomes distorted. The higher the NA of the mirror, the more pronounced the effect gets, as the range of angles within the aperture will increase.

A simulated interferogram, displayed in Fig. 10, shows the same distortion as observed.

The curvature [20] is the inverse of the radius of curvature (ROC) as a function of position, and for the radial direction of a paraboloid is given by

where *p* = 1/4*f* = 1/(101.6 mm), and its radial rate of change, given by

is largest for 60° off-axis mirrors where ∂*z*/∂*r*=0.5, and simpler tests with best-fit reference wavefronts [7, 8] would suffer from the largest problems with fringe densities. As shown in Fig. 10, this problem is alleviated at 90° off-axis angle, but still the rate of change in curvature is quite large. We have previously stated in Ref. [21] that the curvature change is largest for 90° off-axis mirrors; however that is incorrect, as Fig. 10 clearly shows.

It is also important to note that a perfect OAP alignment, and assessment of the OAP errors, are not yet possible. The “outer” part of the OAP (surface angle steeper than 45°) always looks larger than the “inner” part as seen from the interferometer through the transmission sphere. We also observe a re-distribution of irradiance that causes a brightness gradient in the interferogram. Due to the “ray-swapping” properties of the cube corner, the irradiance reaching the “outer” part will be concentrated in the “inner” part upon return from the cube corner, since the centre of the cube corner is exactly above the 45° point (where by “point” we mean the one in the OAP’s central plane of symmetry). Conversely, the irradiance reaching the “inner” part will be diluted upon return to the interferometer. Fig. 11(a) shows the actual brightness distribution.

While the brightness distribution becomes distinctly asymmetric, the opposite is true of the phase map. We cannot eliminate point-symmetrical errors, since each beam going into a cube corner comes back on the opposite side of its corner vertex, so that each pair of diametrically opposite points will pick up the same amount of error. An example of a point-symmetrical phase map is shown in Fig. 11(b) where the ripple structures in the interferogram, which come from diamond-turning, are each overlaid onto the respective other side from the cube corner’s vertex.

#### 3.2 Final alignment

Therefore, after most of the fringes are removed, the retroreflector is no longer useful and is replaced by the precision flat. 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 minimise measurement errors by air movement, but more importantly to minimise the offset of the returning beam on the OAP that an alignment error of either part (OAP or mirror) will create from now on. Depending on the quality of the cube corner used, and the changed appearance of the fringe pattern with the flat return mirror 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, and it is possible to lose the alignment again if it is not done in very small steps.

#### 3.3 Corrections of alignment errors

To carry out a high-precision measurement, we also must address the effect of alignment errors on the measured result. Inevitably, there will be slight errors in the final alignment, both from residual tilts of the return mirror, and from deviations of the linear alignments in *r*, *z*, and *t* from their perfect settings. Since the surface is not perfect, there will always be remaining fringes and it may be hard to judge what the best setting is for taking the measurements. In tests of plane surfaces, we usually subtract only tilt; in tests of spherical surfaces, we subtract tilt and power; but can we simply subtract astigmatism and coma in a test of an OAP and consider them to be alignment errors? In general, this is not the case. It has been found that for each degree of freedom of (mis)alignment, some coupling between different Zernike or Seidel terms exists, which can be modelled by error functionals [22, 23]. A system of equations or a least-squares fit can then be constructed which interprets the surface errors correctly by subtracting from the measurement the best-fitting alignment error functionals. As long as the misalignment is small, the functionals consist of linear combinations of low-order errors such as power, astigmatism, coma, spherical aberration, and trefoil. In other words, a perfect alignment is generated a posteriori, the residual surface errors appearing are minimised, and the remark “shape errors could be interpreted as adjustment errors” [23] applies only insofar as one can then effectively be turned into the other as desired, depending on what parameter in the optical system needs to be optimised.

In cases where the test wavefront is not normal to the surface being tested, the theoretical modelling of the misalignment gradients becomes quite laborious and it is much easier to resort to the empirical strategy described in Ref. [24], which relies on recording deliberate small misalignments for each degree of freedom and using the wavefront errors thus generated to arrive at an estimate of the misalignment gradients. It is important to compare the errors of each displacement direction with those of the “neutral” position, rather than using equal and opposite displacements [21] which will fail to reveal errors of even symmetry.

We have therefore used the procedure of Ref. [24] to generate the error matrix as follows:

where each row of the matrix lists the *n* Zernike error terms created by misalignment in the *k*
^{th} degree of freedom. We let *n* run from 3 to *N* =24, since we want to capture trefoil (Z_{9} and Z_{10}) and quatrefoil (Z_{16} and Z_{17}). We start with the power term and avoid including Z_{1} and Z_{2} (tilt), because tilt can also be associated with the return mirror and therefore eludes unambiguous compensation. We can then visualise the experimentally-determined measurement responses to deliberate displacements as error maps, as in Fig. 12. In the experiments, the actual misalignments for each degree of freedom (*r*, *z* and *t*) were about half a wave; however, for the display in Fig. 12, we have normalised the principal Zernike errors to one wave, but removed them from the phase map so that the coupled second-order effects can be seen.

Now, to minimise the errors of the measurement result, we are only allowed to subtract linear combinations of the wavefront maps of Fig. 12 from our measured wavefront, and need to do so in a way that minimises the squared errors in the corrected result:

where subscript *m* is for “measured”. This gives us a solution for (∆*r*, ∆*z*, ∆*t*) and a corrected wavefront map for the OAP. A well-aligned interferogram and the final optimised wavefront map are shown in Fig. 13.

The least-squares fit gives us (∆*r*, ∆*z*, ∆*t*) = (0.0685, −0.0805, −0.0903) waves, so the final alignment has been within 1/10 fringe, which is quite good, but still not perfect. No Zernike terms other than tilt have been subtracted from Fig. 13(b); and after the above considerations it should be clear that removing isolated terms such as focus or astigmatism is now neither necessary nor allowed. The fitted displacement vector changes only in the third digit when *N* = 15 Zernike coefficients are used, and in the second digit when only 8 coefficients are used; however, as we have seen, it is important to capture at least trefoil.

#### 3.4 Correction for sensitivity errors

Thus far we have only displayed the results as wavefront maps - this is sufficient if the quantity of interest is the distortion of wavefronts traversing the parabolic mirror. However, if we are interested in the spatial deviations of the mirror surface itself, e.g. to optimise a diamond-turning or polishing process, we will have to correct the wavefront map for the variation in interferometric sensitivity. The OAP is 90° off-axis, which means that reflection at 45° folds the optical axis by 90°, and we find an interferometric sensitivity of 0.354 wavelengths per fringe (double-pass testing at normal incidence would yield 0.25 wavelengths per fringe, and the 45° incidence causes a slight de-sensitisation). However, with an OAP like the one we are measuring here, the sensitivity varies significantly over the surface. The light cone from the transmission sphere impinges on the mirror surface at angles spanning a range from 37° to 51° from the surface normal, with concomitant sensitivity changes from 0.313 to 0.397 wavelengths per fringe. Clearly, a precision measurement requires that this variation be taken into account, which can be done according to

where *h*(*x*,*y*) is the true surface height, *w*(*x*,*y*) is the error map in waves (cf. Fig. 11–Fig. 13), and *λ* is the wavelength. Since we are dealing with a paraboloid, the slope ∂*z*/∂*x* is linear in *x*; the correction, however, is not, because it is determined by the cosine of the slope. For conceptual clarity, let us note that we are using the angle between the *x* axis and the OAP’s surface tangent as a proxy for the angle between the *z* axis (collimated beam direction) and the normal to the OAP’s surface here. It is this latter angle that determines the interferometric sensitivity.

In practice, the fringe analysis program allows only to enter a single constant for the sensitivity, so that the correction must be applied to the data after we calculate the wavefront map. This is done by a re-normalisation of results. When evaluating ∂*z*/∂*x*, the difficulty arises that we are viewing the OAP surface along the *x* axis (which we can take to be the *r* direction in Fig. 5(a) in a polar co-ordinate system. This means that we do not have a linear correspondence of an image co-ordinate (here: horizontal) with the *x* axis of the OAP. However, the correction to apply is still linear because for lower numerical apertures, the image co-ordinates in a spherical Fizeau interferogram are proportional to the ray angles, and the ray angles are equal to arctan(∂*z*/∂*x*), so that all we need to do is replace arctan(∂*z*/∂*x*) in Eq. (9) by the horizontal image co-ordinate, with proper scaling and offset. The fact that we are observing the OAP from its centre of symmetry also accounts for the fact that the surface slope is independent of the vertical image co-ordinate, which expresses only a rotation of the parabola. The correction map is shown in Fig. 14(a).

The “neutral” line for the re-scaling is not in the centre of the aperture, since the mirror is asymmetric; this can be seen very clearly in Fig. 14(b), where a co-ordinate grid is overlaid on a wavefront map - and this is also what we have already seen during the calibration in Fig. 3(b). It is important to realise that this overlaid grid is different from that in Fig. 9(b) in that it represents angles here, and spatial co-ordinates in Fig. 9(b), where the mentioned non-linearity is directly visible. Moreover, the mapping of spatial co-ordinates undergoes a point reflection upon traversing the wavefront’s focus, so that we now see the “inner” part of the OAP (*r* <50.8 mm, ∂*z*/∂*x*<1, angle of incidence <45° and hence more sensitive measurement, corrected by values <1) on the left, and the “outer” part (r >50.8 mm, ∂*z*/∂*x*>1, angle of incidence >45° and hence less sensitive measurement, corrected by values >1) on the right side.

There are interesting implications to this sensitivity gradient; essentially the re-scaling will lead to parts of symmetrical errors “leaking” into antisymmetrical errors and vice versa. This means that an error ∆*r* will create focus error and a smaller amount of coma; and errors ∆*z* and ∆*t* will create astigmatism (0° and 45° respectively), and a smaller amount of trefoil error (of odd-even and even-odd symmetry, respectively). This effect provides a good explanation for the aberrations summarised in Fig. 12, as can be seen in simulated results in Fig. 15. Here, one wave of focus (Z_{3}), 0° astigmatism (Z_{4}), or 45° astigmatism (Z_{5}), respectively, have been added, then the errors due to the non-constant sensitivity were simulated (taking the map of Fig. 14 (a) for division, not multiplication) and the main misalignment subtracted again to inspect the secondary effects.

On comparing Fig. 15 with Fig. 12, the coma error from the simulation is not found in the experiment; the reason for this is not clear but could have to do with the non-perfect surface of the OAP itself. This discrepancy shows that the changing sensitivities are not sufficient for a complete description of the errors. The trefoil errors found in the simulation are smaller than the ones in the experiment; part of the reason could be that in the simulation, only isolated Zernike terms were added, whereas in the experiment displacements were introduced to cause errors of prescribed magnitude in the respective Zernike terms, but as discussed above, these do not occur in isolation.

Before considering the final result, we need to address the question at which step in the evaluation procedure to apply the correction. As we can see from Eq. (9), the correction is of the form

which means that it is a linear operation and can in principle be applied either before or after carrying out the least-squares fit of Eq. (8). We have tried this and have arrived at similar, but not equal results, with the better result being obtained from using uncorrected image data for the fit and correcting only the final result. This is not only simpler, but also desirable from a practical point of view, as the uncorrected data are perfectly valid as wavefront maps, and applying the correction at the very last step avoids building up computational errors. The final result after applying all corrections is presented in Fig. 16.

We can now proceed to an interpretation of the OAP’s diamond-turned surface.

## 4. Conclusion

The vertically oriented corrugations of about 20 nm PV amplitude are in the direction of the tool path during diamond turning, and therefore it is safe to say that they come from the tool cutting through the substrate material. The width of the corrugations is on the mm scale and therefore probably not associated with individual passes of the tool, which is usually advanced in μm steps. However, we must bear in mind that this OAP is a mass-produced and low-cost item and therefore we should not expect the highest possible precision. There are also vertical ripples starting at the top rim and replicating its contour downwards over the surface while dying out. This probably reveals that the OAP was not turned as part of a complete paraboloid and then trepanned, but was cut as a 1-inch cylinder first and then diamond-turned: each time the diamond tip makes contact with the workpiece, a slight radial oscillation is triggered that is damped out by the time the tool has traversed the workpiece. Finally, the distinct trefoil error of the surface could be an effect of clamping the piece in three locations for diamond-turning. The amplitude of the observed distortions is consistent with the distance of the OAP surface to the back plane of the mirror substrate: the left side of the surface map shown corresponds to the “low” side of the OAP, where one clamp may have been located on what then became the central symmetry axis. The right side, corresponding to the “high” side of the OAP, bears weaker traces of two clamps. Even though the magnitude of the errors may not necessitate a high-precision measurement like the one demonstrated, the OAP shows an rms surface error of *λ*/12, reaching the specification of *λ*/4 rms very easily.

In this study we have explored the steps necessary to carry out a high-precision surface figure measurement of an off-axis parabolic mirror with a spherical test wave and a flat return mirror. A small 90° OAP of relatively large numerical aperture is used, and is found suitable to highlight the technical difficulties in implementing this test correctly.

The preparation encompasses proper alignment of the transmission sphere in the Fizeau interferometer, and potentially a calibration of the reference surface. We have described a systematic approach to aligning the OAP which makes setting up the test very quick and easy, but requires a retroreflector as additional hardware.

To account for alignment errors properly, we have adapted from the literature a simple empirical method, and its results are in good agreement with simulations of errors caused by varying interferometric sensitivity over the OAP’s surface. We have demonstrated how these errors can be removed for a final appraisal of the diamond-turning process and the optical quality of the mirror.

## References and links

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