## Abstract

This paper complements our previous study on testing a 25.4 mm diameter diamond-turned 90° off-axis commercial-quality parabolic mirror with a spherical test wave in a phase-shifting Fizeau interferometer (Opt. Express **17**, 3196–3210 (2009). In this study I reverse the optical system and use the Fizeau interferometer with a planar reference surface, auxiliary components, and the surface of the transmission sphere as a reflecting spherical return surface. As in the previous paper, I present a description of the necessary steps for alignment and measurement validation. The reversal of the optical system, and associated co-ordinate systems, necessitates some changes of hardware and analysis that provide insight into the underlying symmetries, and may prove useful in a wider context.

© 2009 Optical Society of America

## 1. Introduction

Parabolic mirrors are conceptually the simplest aspheric optics, and have been used in the earliest of telescopes after Newton found out how to make them [1]. Whereas spherical surfaces occur naturally in polishing processes – with flats really being a class of spheres with very long radius of curvature – creating precise paraboloids and other aspheres has remained a challenge to this day [2,3]. Parabolic mirrors can focus a plane wavefront, or, when used in reverse, collimate a spherical wavefront. This allows for convenient null testing; with the parabolic mirror between a spherical and a plane surface, where either end can be the interferometer, there will be no interferometric fringes when the mirror is a perfect paraboloid. Complementing our previous investigation of the null test with a spherical reference wave [4], I will discuss in this paper the configuration with a plane wave emanating from the interferometer, the parabolic mirror focusing it to a spherical wave, and a spherical mirror returning the spherical wavefront to the interferometer via the parabolic mirror. For easy distinction from the “SR” (spherical reference) method, let us name this the “FR” (flat reference) method. For on-axis parabolae, one would place a small and high-quality reflective ball [6] in the focus of the parabola [7,8] so as not to obstruct too much surface area; for off-axis mirrors whose reflected light cone does not intersect the incoming plane wave, it is possible to use a concave spherical mirror to return the wavefront to the interferometer. This is a double-pass test which comes with its own set of caveats [8] mostly related to the concave mirror’s optical quality. Single-pass null tests are possible but technically quite complicated, so that for most practical purposes the double-pass null test is adequate. For a 90° off-axis mirror, the test is then in theory straightforward – set up the parabolic mirror with the “collimated” side pointing at the interferometer, and place a reflective sphere so that its focus coincides with that of the wavefront leaving the parabolic mirror.

The mirror 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 a lateral offset and therefore effective focal length of 50.8 mm. The material is 6061 aluminium, the reflective surface is diamond turned and protected, and the figure error specification is ¼ wave rms.

The diameter of the mirror is much smaller than the 152.4 mm aperture of the WYKO 6000 interferometer used for the test [10], and therefore I have zoomed the image for the measurements, so that the interferogram of the mirror approximately fills the screen resulting in good spatial resolution.

For highest precision, we also require a calibration of the other surfaces involved, here the transmission flat and the return sphere. However, our previous study [4] has shown that after subtraction of all alignment errors, the error of the OAP surface is about 50 nm rms, and therefore more than an order of magnitude greater than the error in the reference surfaces. Therefore I will only give a brief discussion of how a calibration of the surfaces may be achieved, but will not demonstrate the procedure.

Previous work on the alignment of parabolic mirrors [11] has suggested that the SR method is more difficult to use than the FR method; and one of the purposes of the present study is to gain some practical insight on this point.

Fig. 1 shows a sketch of the FR test and its practical set-up. Besides the OAP, we used a WYKO reference flat (uncoated) and a Zygo f/# = 0.75 (NA = 0.68) reference sphere.

## 2. Calibrations

#### 2.1 Instrument alignment

The transmission flat must be aligned correctly with respect to the optical axis of the interferometer after it is installed in the instrument. The WYKO 6000, and many other interferometers, can be switched to alignment or “spot” mode, where the optical Fourier transform of the interferogram plane can be observed and tilt between two plane waves can be removed by bringing the corresponding spots to an overlap. The accuracy of a few fringes that is attainable with this method is not sufficient to arrive at the best alignment for the OAP quickly. Therefore one can use a retroreflector (here, a cube corner) to align the reference flat with the optical axis in fringe mode. For this step, the interferometer is best zoomed out so that the entire fringe pattern produced by the cube corner may be observed, as shown in Fig. 2. I have used a pellicle attenuator, because the high reflectivity of the gold-coated cube corner would otherwise lead to confusing multiple-beam interference and make the alignment more difficult. If highest accuracy is required, the multiple reflections can later be used to enhance sensitivity; when all multiple-beam patterns overlap, the reference flat is aligned exactly.

Since the triple reflection in the cube corner swaps each ray with the one on the opposite side of the vertex, 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 such as coma [12]. During alignment of the transmission sphere [4], a fair amount of coma appeared as a result of point reflection. Since this is not apparent in the present alignment procedure, the coma was most likely related to the transmission sphere itself, not the interferometer; any residual antisymmetric errors are masked by the larger (symmetrical) error of the cube corner itself.

#### 2.2 Transmission flat

In Ref. [4] we have described how the interferometer’s spherical reference surface was calibrated while the return flat was just assumed to be perfect (and known to be very good). The analogous case for the FR set-up would be to calibrate the interferometer’s plane reference surface, and to assume the reflection sphere to be perfect. It is possible in both set-ups to calibrate both surfaces, but this is fraught with additional difficulty for the respective return surface, so that it is much more practical to (i) ascertain acceptable optical quality of the return surface before the OAP test and then neglect its errors, or (ii) re-use a previous calibration file by appropriate symmetry and scaling operations to adapt it to the OAP test.

If larger OAPs are tested in the FR configuration, it may be worthwhile to carry out a three-flat calibration of the reference flat with a modern method [13–15]. A subtlety in the present test is that, even though the test wavefront is planar, we are allowed to subtract focus error, subject to the constraints explained in Refs. [16] and [4]. However, this does not mean that focus error in the reference flat is irrelevant; conceptually, the null test can function properly only when a perfectly collimated wave tests the OAP. If the incident wavefront is not collimated, retrace errors are guaranteed. A surface map as in Fig. 3(a), which is from a generic pitch-polished transmission flat, would therefore not be ideal for a high-precision parabolic mirror test, despite very low flatness error. The medium-scale ripple and the overall focus term will create difficulties in the interpretation of the results. In our case of a 25.4 mm aperture we can still use such a surface without calibration, since the utilised surface portion is very much smaller and of considerably better flatness, as shown in Fig. 3(b). Some of the residual errors may also be absorbed in low-order alignment error terms, but this is acceptable only because in this study the errors of the OAP are much larger than the errors in the reference flat.

#### 2.3 Return Sphere

We have described the calibration of the transmission sphere in Ref. [4]. Whenever such a data set exists, it can be re-used. If the interferogram regions for both tests are of equal size, the transmission sphere reference file can simply be rotated by 180° (accounting for a point reflection as the test wave goes through a focus on its way to the reflection sphere) and used as it is to subtract from the result.

It is important to note here that equal interferogram sizes are the exception rather than the rule, because the zoom setting of the interferometer will generally have to be adapted to the SR or the FR configuration. In our example, the zoom factor for the SR set-up is 3.1 – since the total interferogram region corresponds to NA = 0.68 and the OAP has NA = 0.24, the ratio of the angles will be (arcsin 0.68)/(arcsin 0.24)=3.1. The zoom factor for the FR set-up is 6.0 (magnifying 25.4 mm out of the 152.4 mm to fill the image). The NA on the “spherical” side of the OAP is of course still the same, so that the same region of the spherical surface will be used for reflection as was used previously in transmission, including the axial asymmetry described in Ref. [4].

If the interferogram regions are of different sizes, then either (i) the sphere calibration can be re-scaled, which may impair the spatial resolution of the measurement, or (ii) the reflection sphere is assumed to be perfect, and is neglected, or (iii) a new ball-averaging measurement can be made, as shown in Fig. 4. The calibration ball is placed with its centre in the focus of the converging wave emanating from the OAP, and a calibration sequence is carried out as described in Ref. [17]: an average is built up from measurements of the ball in a number of random rotational positions until the statistical reliability of the average is considered to be a good approximation of an OAP measurement against a perfect sphere. Of course, the additional difficulty arises here that ideally the OAP alignment should not change at all as the average is recorded.

The data set thus obtained can then be used directly as an estimate of the OAP surface, and is therefore not so much a calibration as a fairly laborious measurement. The practical expense of this can be circumvented by using option (ii) above, or if that would cause unacceptable errors, by (iv) using a known “good” surface portion (i.e. of low sphericity error) of the calibration ball as a reflection surface and then neglecting its influence and dispensing with the averaging procedure.

If the averaging with the calibration ball is carried out as described, an additional measurement can be taken after removing the ball; the difference between the two data sets will then represent an estimate of the error in the reflection sphere and should match the transmission sphere calibration file for the SR set-up quite closely. This is based on the assumption that the common-path transmission mode is free from re-trace errors.

For the comparison of SR and FR set-ups presented in this study, the interferograms in both experiments were of equal sizes, and therefore I have modelled the reflection sphere as shown in Fig. 5 (cf. Fig. 3 (b) in Ref. [4]).

A comparison of corrected and uncorrected results for the measurement of the OAP by the FR method revealed that the correction for errors in the reflection sphere improved the detected rms error in the mirror surface from 46.9 to 45. 9 nm, and the PV error from 352.4 nm to 351.5 nm (see Fig. 14). The importance of the absolute improvement is small in this case, but again, this is due to the relatively large error existing on the OAP. Strictly speaking, a null-test sphere should be perfect, since otherwise the OAP cannot be used exactly as designed (i.e. producing or receiving a perfectly spherical wavefront) on the path to, or from, the spherical surface.

## 3. Testing of the parabolic mirror

The definition of the co-ordinate system is shown in Fig. 6. The OAP is adjusted in its rotary stage so that its focus coincides with the focus of the reflection sphere. There are five degrees of freedom for misalignment: the three translations in *t*-, *r*-, and *z*-direction, and rotations *ρ* about the *r* axis and *τ* about the *t* axis. A rotation about the *z* axis does not cause any errors in the focused wavefront, and is equivalent to a *t*-translation. Also, it has been found that the two rotations are coupled with one translation each [18]: in our notation, the coupled pairs are *τ* rotation with *z* translation, and *ρ* rotation with *t* translation. This corresponds to the observation in Ref. [4] that tilt fringes cannot be unambiguously assigned to one or the other degree of freedom; straight fringes in this case come about (and are removed) by moving the reflection sphere in a plane normal to the optical axis (*r* direction), as indicated by the red arrows in Fig. 6. Hence they are not tilt, but displacement fringes in this geometry.

Rotations of the OAP away from the “collimated” optical axis leads to complicated aberrations [19], whereas a translation of the OAP in *r* direction will mostly cause focus error. In the FR configuration, coupling of rotations and translations is difficult to avoid, and the coordinate system is more complicated than that for the SR system. Unless the rotation axes are centred in the OAP’s surface, a rotation in *τ* will also introduce an error in *r*, and rotation in *ρ* will create an error in *t*. Therefore the alignment procedure will be iterative to a larger extent than the one given for the SR set-up.

#### 3.1 Preliminary alignment

The initial alignment of the OAP 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, but generally elliptical. Then a *τ* rotation will make the cross-section of the light cone roughly circular.

If a reflection sphere were used initially, the issues in understanding the complicated fringe pattern would be similar to those described for the SR set-up. Thus, once more it is better to use a retroreflector to start with. In the case of spherical waves, the component returning the incoming beam without adding tilt is simply a plane mirror surface, placed exactly at the focus of the converging wavefront. This is shown in Fig. 7. Assuming the transmission flat has been adjusted properly, this set-up will always indicate the optical axis correctly; however, the way to fill the entire aperture with the retro-reflected wave is to adjust the angle of the plane mirror appropriately, so that the complete light cone leaving the OAP will be reflected back onto it.

Comparing the SR and FR set-ups, we can now see the complementarity and symmetry: the reference surface is aligned with a plane mirror/cube corner, respectively, and the OAP then converts spherical to collimated or vice versa, so that the respective other retroreflecting surface is needed.

Once we have a spot of object light appearing on the interferometer’s camera monitor, we can iteratively optimise the alignment. As before, strong astigmatism or defocus will create a continuum of spatial fringe frequencies and may, in alignment mode, smear out the light spot so much as to make it 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 the aid of a plane mirror is shown in Fig. 8.

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. 8-1 A, 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 Δ*ρ* must be adjusted. There are no fringes visible yet, but it is still possible to adjust Δ*ρ* almost perfectly in the first iteration. As to be seen from Fig. 6(a), Δ*ρ* is not altered by adjustments of either Δ*τ* or Δ*r*. Thus, again in analogy to the SR configuration, we first align the OAP’s central plane of symmetry to include the interferometer’s optical axis.

Continuing from the condition depicted in Fig. 8-1B, a rotation about the t axis will make the reflected wave more circularly symmetric. In the experiments I have found that sometimes a strongly astigmatic pattern can be mistaken for a circular one (bearing in mind that we are observing the fringe pattern, not the focal plane), but one arrives at either one fairly quickly.

After removing Δ*r*, some residual astigmatism becomes visible, so we start the next iteration in Fig. 8-2 A. A residual amount of Δ*ρ* error is removed for 2B, and for the removal of Δ*τ*, we can already go by a fringe pattern that we adjust to become circular, as in 2C. Removing Δ*r* again starts the last iteration, in the third row of Fig. 8. Again, a small amount of error in *ρ* has become visible in 3A, which is removed in 3B. Finally we need to minimise the astigmatism (or Δ*τ*), 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. 9 and (Media 1).

In the FR set-up, the interferogram *will* appear in the centre of the screen, regardless of what rotation about the *z* axis exists; however the interferogram region may be slightly truncated if the retroreflection mirror is not aligned exactly normal to the optical axis. This is likely to happen at the first attempt. However, since the use of the flat mirror is only an interim step, it is sufficient to monitor most of the fringe pattern and the complete aperture is not strictly necessary.

As before in the SR case, we can now neither align the OAP perfectly, nor assess the OAP errors reliably. Light from the “outer” part of the OAP (surface angle steeper than 45°) will be reflected into the “inner” part (i.e. the part closer to the vertex of the parent on-axis parabola) upon reflection by the plane mirror, which concentrates the irradiance in a smaller area upon return to the interferometer. Conversely, the irradiance reaching the “inner” part will be diluted and the intensity will drop. Fig. 10(a) shows the actual brightness distribution.

While the brightness distribution becomes distinctly asymmetric (from left to right in Fig. 10(a)), the opposite is true of the phase map. We cannot eliminate point-symmetrical errors, since each beam being reflected by the mirror comes back on the opposite side of the focus (and hence the optical axis), so that each pair of opposite points will pick up the same amount of error. An example of a point-symmetrical phase map is shown in Fig. 10(b) where the ripple structures in the interferogram, which come from diamond-turning, are each overlaid onto the respective opposite side of the reflection point. The asymmetric distortions in the horizontal direction, and also focusing issues associated with the double-pass set-up [9], wash out the symmetry in the small-scale features somewhat, but the symmetry of errors about the horizontal axis is very distinct.

#### 3.2 Final alignment

As opposed to using the cube corner with its imperfections in Ref. [4], we can select an almost perfect micro-area of the mirror for retroreflection, and hence we can get very close to the best alignment in the preliminary step; nonetheless we will have to make corrections after the mirror is replaced by the reflection sphere, as in Fig. 1(b). Again, a pinhole screen (e.g. a piece of white paper with a small diameter hole) will be very helpful for the initial alignment of the reflection sphere. We cannot minimise the distance between components here, since the foci of OAP and reflection sphere must coincide. The final adjustments must be made very carefully, as now each movement of the OAP will create large amounts of displacement fringes, and it is possible to lose the alignment again if too large changes are made.

Only very small adjustments of the OAP itself should be necessary; if translations must be made, it is best to follow the OAP’ s focus with the translation mechanisms on the auxiliary mount that holds the transmission sphere, so that the OAP does not get displaced in the aperture and we remain able to compare and subtract measurement results without having to correct for image shifts. This is particularly important in our study, where the OAP shows a large amount of fine structure.

#### 3.3 Corrections of alignment errors

To remove remaining alignment errors, I followed the technique of [16] again, recorded deliberate small misalignments for each degree of freedom of the OAP and used the wavefront errors thus generated to arrive at an estimate of the misalignment gradients. I have also tested the coupling between rotations and translations as mentioned above, and have found that an approximate 10:1 ratio exists between rotations and translations, i.e. when Δ*ρ* is adjusted for Z_{5} = 1 wave (0° astigmatism), this causes about 10 times the higher-order aberrations we get when Δ*t* is adjusted for Z_{2} = 1 wave (*y* tilt); Δ*τ* and Δ*z* are coupled in this way as well.

As in Ref. [4], we cannot account for tilts, and instead carry out the displacement estimates over Z_{3} through Z_{24} , again to capture trefoil (Z_{9} and Z_{10}) and quatrefoil (Z_{16} and Z_{17}). The error maps obtained are displayed in Fig. 11. The actual misalignments for each degree of freedom were about half a wave. However, for the representation in Fig. 11, I have normalised the principal Zernike errors for each misalignment to one wave but removed them from the phase map so that the secondary effects can be seen.

Finally I subtract the optimised solution for (Δ* _{r}*, Δ

*τ*, Δ

*ρ*) from the measured wavefront to obtain a corrected map for the OAP. A well-aligned interferogram and the final optimised wavefront map are shown in Fig. 12.

The least-squares fit gives us (Δ*r*, Δ*z*, Δ*t*) = (0.0932, −0.0429, 0.0065) 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. 12(b); 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 15 Zernike coefficients are used, but due to the changed OAP surface perspective in the FR test, it is more important than in the SR test to capture quatrefoil, and 24 Zernike terms or more should be used.

#### 3.4 Correction for sensitivity errors

To convert the distortion of the wavefront traversing the OAP into spatial deviations of the mirror surface, we 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. 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. 10–Fig. 12), and *λ* is the wavelength. In contrast to Ref. [4], the correction is now a function of both spatial coordinates, since we are looking down into the paraboloid from the direction of the rotation axis, which means that the lines of constant slope are now circles. The slope *∂z/∂R* (with *R*
^{2} = *r*
^{2} + *t*
^{2} in the co-ordinates of Fig. 6, with its origin at the origin of the paraboloid) is linear in *R*; the correction, however, is not, because it is determined by the cosine of the slope.

In this case, the correspondence between image co-ordinates and spatial co-ordinates is direct, and therefore easily applied, as shown in Fig. 13(a). Also, the 45° line, where the correction is unity, now goes through the centre of the aperture. This is demonstrated in Fig. 13(b), which, compared to the corresponding figure in Ref. [4], shows that the 45° line is located on exactly the same surface features for the two different measurements.

Again we see the “inner” part of the 0 AP (*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.

As before, the re-scaling causes some “leaking” of errors into higher-order terms. Since the corrections are quite similar, so are the results, and the respective overview in Ref. [4] still applies. Finally we can apply the correction to the wavefront result of Fig. 12(b), and the end result is presented in Fig. 14.

Although the surface maps from the SR and FR tests look very similar, we had previously found 50.3 nm rms and 343.6 nm PV, so there is some discrepancy left to explain. One possible reason is that the relative distortion of the surface between the two versions of the test shrinks the left half of the surface in the FR test; since that part contains two high points, they have smaller area in the phase map from the FR test and therefore the overall rms error is lower. The gap in the phase map at the left edge, caused by insufficient fringe contrast, plays no role for rms or PV. The PV values are fairly close together, so it appears that both measurements are valid to within a few nanometres. For a confirmation that the phase maps are indeed free from residual errors, a coordinate transformation would have to be carried out so as to arrive at the same mapping of physical OAP surface to processed image; but since this study is concerned with the alignment, the provisions necessary for maximal accuracy, and the attainable accuracy in an actual measurement, we do not carry out the transformation.

## 4. Conclusion

I have investigated the steps necessary to carry out a high-precision measurement of an off-axis parabolic mirror with a plane test wave and a spherical return surface. This paper complements our earlier study with a spherical test wave and flat return surface, and the complete study constitutes what we hope to be a useful guide to high-precision off-axis paraboloid alignment and measurement, with some applicability to other aspherics amenable to null testing.

The preparation for the FR test encompasses proper alignment of the transmission flat in the Fizeau interferometer, and potentially a calibration of the reference and reflection surfaces. Based on an alignment algorithm described in the literature [11] and modified and demonstrated in Ref. [4], I have described a systematic approach to aligning the OAP that makes setting up the FR test very quick and easy. The measurement is then carried out in a very similar way to the SR test.

Comparing the practical applicability of the two tests, the SR configuration appears more versatile and easier to use, for the following reasons:

- the spherical reference wave allows adaptation of the test to a large range of OAP sizes and numerical apertures;
- the degrees of freedom for misalignment can be described and implemented as translations, which simplifies the set-up, and the alignment is free of ambiguity;
- calibration of reference surfaces is easier than for the FR technique; for the transmission sphere, the ball-averaging method or a new double-pass technique [20] can be used, and good reflection flats with negligible imperfections are relatively easy to procure;
- the correction for sensitivity errors is theoretically more complex but easier to apply in practice.

To distinguish alignment errors from figure errors properly, it is indispensable for precision measurements to use either a theoretical or experimental description of the secondary misalignment errors. Since a theoretical description seldom does justice to the entire system, and has also been seen to break down in extreme cases [21], it appears preferable to characterise the alignment errors empirically in the actual optical system.

In a final step, the wavefront maps can be converted to surface figure maps via a sensitivity correction that, to the best of my knowledge, has not been described in detail before. If the objective is to improve the figure of an optic, sensitivity corrections are the key to high precision and should be applied in all interferometric tests in which the angle of incidence varies over the surface of the tested optic [22].

To enhance the precision even further, image distortion effects also need to be removed, but this is not usually done interferometrically and I have therefore not dealt with distortion in this study.

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