## Abstract

Multi-Zonal computer generated holograms (MZ-CGHs) in combination with interferometric wavefront measurements are perfectly suited as optical adjustment tools - especially if the demands on the alignment accuracy are very high. After reviewing the basic idea for alignment with MZ-CGHs, we derive the analytic relation between the interferometrically observed tilt and power values and the associated lens placement errors, including estimates of the applied approximations. This analysis yields the parameters determining the principle sensitivity of the method. Subsequently, the achievable accuracy of large 6″ MZ-CGHs in practical application is tested with a series of different optical measurements which confirm the technical feasibility. The productive use of the technique will be presented in part II of the paper for different examples in the framework of the Euclid space telescope.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the field of precision metrology for aspheric surfaces, the use of *Computer-Generated Holograms* (CGHs) [1–6] has established itself as a state of the art [7,8]. Using CGHs, one can transform spherical into aspherical wavefronts, which resemble the nominal shape of an aspherical surface under examination. The corresponding rays hit the surface at normal incidence and are reflected backwards into themselves. This is the ‘autocollimation condition’, used for interferometrical methods, which are thus also available for asphere metrology.

In doing so, one is typically interested in detecting the surface quality or irregularity, which can be expanded in terms of Zernike polynomials like astigmatism, coma, trefoil, spherical aberration and higher terms. The remaining low order Zernike polynomials piston, tip, tilt and defocus are normally regarded as adjustment artefacts and therefore numerically subtracted.

However, tip, tilt and defocus yield powerful and extremely sensitive alignment signals, which can fruitfully be used in combination with *Multi-Zonal-CGHs* (MZ-CGHs). This is an approach which we developed extensively in the framework of the *Euclid space mission* [9,10], where particularly critical optical adjustment tolerances well below 10 µm for the centring error are required.

This is the first of two papers on MZ-CGHs for high-precision optical adjustment purposes. After an introduction to the basic idea of the alignment procedure with MZ-CGHs and to the type of CGHs we are using, we summarise the analytic relation between the interferometrically observed tilt and power values and the associated lens placement errors. Subsequently, we examine the actually achieved accuracy of MZ-CGHs with optical performance measurements. This is the prerequisite for the use as highly accurate adjustment tool. In the second paper, we will discuss practical applications and examples in the framework of the Euclid space telescope.

## 2. Adjustment procedure with multi-zonal-CGHs: the basic idea

‘Usual’ CGHs for asphere tests generate wavefronts with *continuously* changing radius of curvature as function of the radial distance from the centre to the outer margin. MZ-CGHs follow a similar approach. However, the difference is that the radius of curvature of the generated wavefront experiences *discontinuous jumps* at the zone edges. The wavefront thus separates into a series of annular subareas. In the schematic sketch of Fig. 1, the CGH is illuminated with a plane wave from the left. All subareas build spherical wavefronts, each with a different focal length. Accordingly, a *series of spots along a straight line* arises, see Fig. 1(a), spot 1-5. The thus defined *optical axis* serves as reference for an interferometric adjustment procedure of the lenses in combination with a Fizeau interferometer. The *straightness* of this axis is the *key feature* of the presented method. It can be realized with extremely high precision due to an uninterruptible circular writing process of the structures on the substrate of the CGH [7,11,12]. This property will be proven with optical tests in section 5.

The MZ-CGH is equipped with a bayonet lock, which is directly mounted on a Fizeau interferometer (Fig. 2). The plane wave (Fig. 1, green rays) defines the direction of the optical axis. At first, we have to ensure that the optical axis of the lenses are adjusted parallel to this reference. This can be done in different ways. In the Euclid project, the lenses are mounted in ‘adaption rings’, which are equipped with flat optical measurement interfaces for this purpose [13,14]. The lenses in turn have a plane reference phase, which is adjusted parallel to these surfaces with the help of a coordinate measuring machine. However, these experimental details do not concern our basic idea and will be presented in part II of this paper. For now, it is just important to assume that the lenses are *not tilt* and the adjustment reduces to the lens centring and the spacing.

The lens alignment is then performed according to the following procedure: The lenses are inserted, starting with L3, see Fig. 1(b). Its spherical surface fits to the corresponding wavefront from zone 5, provided that it is located exactly at its nominal position. A lens positioning error along the optical axis is detected as Zernike *defocus* term in the wavefront. A centring error in the lateral plane is observed as Zernike *tip/tilt* term. The correct position is reached, if the fringes of the interference pattern are perfectly nulled. In this case, the centre of curvature of the lens surface coincides with the spot position of the corresponding zone. The same procedure is repeated with L2 and then with L1, which is inserted at the end.

The layout according to Fig. 1 shows two more zones: Zone 1 and 2. They provide signals from the *cat-eye* positions at the vertices of L3 and L1. As they have a larger numerical aperture (*NA*) than the zones that sample the same surfaces in autocollimation position (zone 3 and 5), the Zernike defocus term is more sensitive to the position in z. And unlike the autocollimation positions, the cat-eyes do not depend on possible errors in the radius of curvature of the lenses. Furthermore, the cat-eye zone 2 has a *rectangular* shape that crosses all annular zones. The large radial range of this zone is optimized for *defocus* measurements. It improves the initial z-positioning of the interferometer. The cat-eye zone 1 allows a final distance monitoring after the integration of L1 - L3. In our setup, the z position of L2 is determined mechanically. Cat-eye positions carry however no information on the centring of lenses.

It is worth mentioning that configurations like those of L1 and L3 with simultaneous cat-eye and autocollimation position on a single surface yield a nice side effect: It is possible to measure the radius of curvature of the lenses at a single fixed position and without referring to an additional scale.

The CGH layout of Fig. 1 and Fig. 2 which we treated as an example to illustrate the basic idea of our adjustment procedures is one of several CGHs used in the framework of Euclid’s near infrared spectrometer and photometer instrument [15] alignment tasks [13]. Although we demonstrated the method with spherical wavefronts on spherical surfaces, it is also applicable and actually in use with moderately aspherical surfaces and the corresponding wavefronts. Problems may however occur with extreme aspheres.

## 3. Types of CGHs and interferometric fringe visibility

We continue with a short overview on the type of CGHs we use within the scope of this paper: Imagine an (‘ideal’) plano-convex lens, laterally sliced into planar plates of thickness λ/(*n-1*) with vacuum wavelength λ and refractive index *n*. Now suppose that only the segments bordered by the convex surface are kept and shifted to a common plain; the rest is removed. The result looks like Fig. 3(a). It is very similar to a *Fresnel lens*, however with the additional benefit, that the phase jumps are exactly equal to 2π. It is called a ‘blazed diffractive lens’, and it still behaves basically like its plano-convex counterpart: It focuses parallel rays to a single spot.

Blazed structures are realizable, however costly and hard to control as multiple successive lithographic exposure and etch steps are required [7,11,16]. The probably crudest approximation to the blazed diffractive lens that still keeps the required properties is shown in Fig. 3(b). All phases between 0 and π are approximated by 0, while all phases between π and 2π are approximated by π. This is called a ‘binary phase CGH’. More specifically, this particular diffraction structure is a ‘Fresnel phase plate’. It is remarkable that it still produces the same focal point. However, a qualitative change arises in terms of *additional diffraction orders*. Even orders (including the zero) are supressed, while the transmitted energy is spread into all positive and negative odd orders. The efficiency of the desired first order is accordingly reduced.

The annular diffractive microstructures with alternating phase are the so-called ‘Fresnel zones’. In case of spherical waves, these patterns can easily be described analytically, see Ref [16]. A rough estimation of the *efficiency* of the diffraction orders can be made with the so called ‘scalar thin phase approximation’. Here we consider the diffractive structure as an infinitely thin surface and disregard the vector properties of light. This approach is usually acceptable if the grating period is large compared to its depth. With these assumptions, and for an exact phase shift of π, a Fourier analysis yields an efficiency of the ± 1st orders of *η* _{± 1} = 4*/π*^{2} ~40.5% [16], which is not fare from our actually measured values in the range of 35 – 40%. A corresponding binary *amplitude* CGH only yields a first order efficiency in the order of 10%, which is often not sufficient, however in special cases (e.g. mirrored surfaces) desired.

The diffractive pattern of our CGHs is located on the surface facing the lenses, see Fig. 1(b). It is uncoated and serves simultaneously as flat *Fizeau-reference surface* [17]. This choice yields the advantage that the substrate of the CGH is in the *common path* of reference and signal, and therefore largely excluded from contributing to the measurement errors. The Fizeau- surface is essential for the accuracy of the interferometric measurement. Our substrate is specified with a surface P-V quality of λ/20. An additional error contribution results from slight spatial inhomogeneities of the etching depth. However, this contribution is not dominant in our production process.

The physical etch depth *h* in the substrate corresponding to the π step in transmission equals *h* = ½ λ (*n*-1)^{−1}. In reflection this results in an optical path difference for the *zero order* of 2 *n h = n* (*n*-1)^{−1} λ. For *n*_{suprasil} = 1.451 this is ≈3.2 λ equal to 6.4 π. A fine-adjustment of the etch depth can be used to maximize the *fringe visibility* by adapting reference reflex and test beam to similar amplitudes.

The second, unstructured surface of the fused silica substrate is slightly wedged to remove the corresponding reflex from the optical path to the camera.

The more expensive blazed phase CGHs can be beneficial in certain cases, e.g. when the additional diffraction orders lead to disturbing reflexes. This easily occurs with increasing number of surfaces involved.

## 4. Analytical sensitivity analysis

In this chapter, we summarize the mathematical relation between the interferometrically observed tilt and power signals and the associated lens placement errors.

#### 4.1 Shift along the optical axis: defocusing

The schematic representation of Fig. 4(a) shows a spherical wavefront (red), which is reflected at a spherical lens surface with radius of curvature R (blue). If the focus of the wave meets exactly the centre of curvature of the lens (*ΔR* = 0), all rays are reflected back into themselves, reversing their propagation direction. The converging spherical wave is transformed into a diverging spherical wave without any additional optical path differences.

However, shifting the lens along the optical axis z by a distance Δ*R* (Fig. 4(a)) leads to different centres of curvature between wavefront and lens. For this reason, the reflected rays pick up an additional optical path of 2δ, which increases along the aperture from *x* = 0 to *x*_{max} (see the enlarged details in Fig. 4(b)). This additional path length corresponds to a defocus wavefront- error, which can be observed with a wavefront measuring device like a Fizeau interferometer. In the following, we derive the analytic relation between this wavefront-error and the Zernike defocus polynomial.

The spherical lens surface (Fig. 4(a), blue line) can be described by the formula of a circle with radius R, centred at *z*_{1} = *R:*

*z*

_{1}this yields:

*R*) almost right-angled. The additional optical path length δ(x) is thus in very good approximation given by:

*first approximation*by expanding the square roots in Eqs. (2), (3) and (5) as a Taylor series around

*x*/

*R*= 0. We keep only the quadratic order in x. This yields after some straightforward calculationsThis equation is valid for

*x*/

*R*≪ 1, corresponding to a

*paraxial*(or small-angle) approximation. The remaining parabolic x-dependence agrees with the Zernike defocus polynomial. Higher orders of Zernike terms are neglected. The expression in the brackets of Eq. (6) may be interpreted as the difference in

*optical power*between incoming wavefront and reflecting surface. A

*second approximation*concerns the Taylor expansion of 1/(

*R*+ Δ

*R*) around Δ

*R*= 0. This leads to a linear dependence between optical path length δ and Δ

*R*, valid for Δ

*R*≪ R:

*x*/

*x*

_{max}| ∈ [0, 1] which is normalized to the aperture size, secondly the numerical aperture

*NA*=

*x*

_{max}/

*R*and thirdly the wavefront error Δ

*W*= 2 δ(

*x*), which equals two times the optical path length δ due to the double path setup in reflection. With this nomenclature, Eq. (7) takes the following form (see also [19]):It is remarkable, that the dependence on the radius of curvature

*R*of the lens in Eq. (6)

*vanishes*in the approach of Eq. (8), which merely depends on

*NA*and Δ

*R*.

All Zernike polynomials, except the *piston,* which is a constant (ρ- independent) offset, have a vanishing mean value over a circular aperture. This property follows directly from the orthogonality of the polynomials [18]. However, the mean value of Δ*W*(ρ) according to Eq. (8) is not vanishing. Therefore, we subtract the ρ-independent piston term ½ *NA ^{2}*Δ

*R*(and keep the notation unchanged for brevity):

*W*(

*ρ*) according to Eq. (9) equals zero. All

*physical*properties like peak-to-valley value

*PV*= Δ

*R NA*or the standard deviation σ = ½ 3

^{2}^{-1/2}Δ

*R NA*are however unchanged. With Eq. (9) we finally arrive at the form of the Zernike defocus polynomial

^{2}*Z*

_{4}=

*A*(2

*ρ*

^{2}- 1) with the normalization constant

*A*. The corresponding defocus wavefront-error is hence Δ

*W*=

*c*

_{4}Z

_{4}with the

*Zernike defocus coefficient c*

_{4}. Different sequential indexing conventions are in use. We refer to the

*Noll-indexing*[20] with the defocus polynomial index equal to 4 (1, 2, 3 = piston, tip, tilt). Two definitions of normalizations are widespread: The ‘fringe definition’ (also referred to as ‘University of Arizona’ notation) with

*A*= 1 is normalized to unity magnitude at the edge of the pupil and the ‘standard definition’ (also called ‘Born & Wolf’ or ‘Noll’ notation [19]) with

*A*= 3

^{1/2}, which builds an

*orthonormal*system. A comparison with Eq. (9) delivers the following result for the Zernike defocus coefficient:

*R*of the lens in z and the corresponding Zernike defocus coefficient c

_{4}, which can be determined from a wavefront measurement. Plotting c

_{4}versus Δ

*R*results in a straight line with a slope (which is the

*sensitivity*of the method) equal to ½

*NA*

^{2}or ½ 3

^{-1/2}

*NA*

^{2}, respectively. These equations are valid for

*NA*≪ 1 and Δ

*R*≪

*R*. The sensitivity is inversely proportional to the

*depth of focus*DOF = ½ λ /

*NA*

^{2}.

Note that Eq. (10) refers to the Zernike defocus coefficient of the *wavefront,* which follows the double path setup of Fig. 4 and Fig. 1(b). Wavefront sensors are however often configured to evaluate *surfaces*, not *wavefronts*. In such cases, Δ*W* is already divided by 2 during the evaluation, and therefore all extracted Zernike coefficients are also halved. This must be taken into account.

Now we consider the error of the approximations that we introduced as we went from Eq. (5) to Eq. (7), where we assumed *NA* and Δ*R/R* ≪ 1. We refer to the (accurate) wavefront-error in Eq. (5) as δ^{0} and calculate the relative difference to the approximation δ^{Appr} according to Eq. (7) in percent, assuming x = x_{max}, which corresponds to ρ = 1:

The *relative* deviation *Err* according to Eq. (11) is plotted in Fig. 5(a) as function of *NA* for three different relations of Δ*R/R.* The paraxial approximation in Eq. (6) leads to the increase of *Err* with *NA*, while the approximation in Eq. (7) has the opposite sign and depends on Δ*R/R.* There are zero-crossings, where both effects compensate each other. Note that the limit of *Err* for NA→ 0 equals Δ*R/R*, while the *absolute* deviation reaches 0.

#### 4.2 Lateral shift: Tilt

Now, we consider a lateral shift Δs between lens and wavefront as shown in Fig. 6.

The calculation of the corresponding wavefront-error is quite similar to the case in section 4.1. Again, we determine the path difference of two circles, the blue one (lens) centred at the origin of the coordinate system and the red one (wavefront) shifted by Δ*s* in the x-direction. A Taylor expansion of the square roots (similar to section 4.1) leads until the linear order in x to

*W*is the wavefront error in the double path setup due to the lens shift Δ

*s.*We introduced polar coordinates (ρ, θ) with

*x*/

*x*

_{max}= ρ sin(θ) and again the numerical aperture

*NA*=

*x*

_{max}/

*R.*

Δ*W* consists of two terms. The first one, quadratic in Δ*s,* but independent of ρ, is the Zernike *piston* term, which is linearly independent of other Zernike polynomials. The second one has the form of the Zernike *tilt* polynomial Z_{3}. Again, we distinguish between the *fringe* definition${Z}_{3}^{Frg}\text{}=\rho \mathrm{sin}(\theta )$with the wavefront error $\Delta W\text{\hspace{0.17em}}={c}_{3}^{Frg}\text{\hspace{0.05em}}{Z}_{3}^{Frg}$and the *standard* definition ${Z}_{3}^{Std}\text{}=2\rho \mathrm{sin}(\theta )$ with $\Delta W\text{\hspace{0.17em}}={c}_{3}^{Std}\text{\hspace{0.05em}}{Z}_{3}^{Std}$. This yields the *Zernike tilt coefficient* as:

*c*

_{3}versus the lateral lens shift Δ

*s*leads to a straight line with a slope equal to 2

*NA*or

*NA*, depending on the choice of normalization. Again, the slope is the

*sensitivity*of the method. These equations are valid for

*NA*≪ 1 (and, strictly speaking, additionally (

*x*- Δ

*s*) /R ≪ 1). The percentage relative error - analogous to Eq. (11) - of Eq. (12) due to the approximation is plotted in Fig. 5(b).

Finally, we point out that a shift of an (ideal) sphere in the lateral x-y plane as shown in Fig. 6 is identical to a rotation e.g. around the lens vertex together with a shift along the optical axis z. Hence, a sole tilt measurement cannot distinguish between lateral shift and rotation, because there is no difference in the position of the spherical surface. Fortunately, there are ways to overcome this issue, for instance by sampling both surfaces of a lens (details in paper II).

#### 4.3 Non-circular aperture shapes

Zernike polynomials build a complete set of *orthogonal* functions on a *circular* disk. Therefore, each coefficient of a Zernike expansion is *unique* and thus independent of the number or the choice of included terms. Adding or removing single terms has no impact to other Zernike coefficients. (However, the numerical determination of Zernike coefficients on a computation grid with a least squares fit, as commonly done, leads to a deviation from strict orthogonality and thus to a weak dependence of fit coefficients on the number of terms.)

The multi-zonal-CGHs however typically consist of multiple *annular,* or even more complexly shaped zones. In this case, the orthogonality of the Zernike polynomials disappears. An expansion in Zernike polynomials - in the sense of a least squares fit – is nevertheless still possible. But the uniqueness and physical meaning of the single coefficients is lost [18]. Adding or removing single terms to an existing Zernike expansion requires a new least squares fit, which will in general impact *all* fit coefficients. Even tilt and defocus can become ambiguous depending on higher orders.

Fortunately, there is an easy solution to this issue: The Zernike fit must be *restricted* to the four terms *piston, tip, tilt* and de*focus*. In this case, the initial meaning of the terms remains and the Eqs. (10) and (13) are also applicable to non-circular apertures [18].

The *sensitivity*, which is the constant of proportionality between the Zernike coefficients (tilt and defocus) and the spatial location, depends only on *NA* and not on the shape of the CGH zones. Nevertheless, a reduced aperture area (e.g. annular instead of circular) results in a reduced accuracy in the determination of the Zernike coefficients from a wavefront measurement due to the reduced information. For example, fitting a defocus term to an annular aperture, narrower than a period of the concentric interference-ring-pattern, will become inaccurate. Therefore, CGH designs should be carefully optimized to the special requirements of a setup.

## 5. Optical performance test of MZ-CGHs

In chapter 2, we discussed that the series of spots arising from the MZ-CGH in Fig. 1 defines an optical axis that serves as reference for the interferometric adjustment procedure. We emphasized, that the straightness of this axis is the prerequisite of our method and pointed out that high-precision polar coordinate laser writing systems [7,11,12] are supposed to yield sufficient accuracy even on large 6″ substrates. Now we examine to what extent this expectation proves to be true.

#### 5.1 Experimental setup

A sketch of the verification setup is shown in Fig. 7. The series of foci from the MZ-CGH is aligned parallel to a high precision air bearing hard stone translation stage with an accuracy in the sub-micron range. This task is facilitated using a hexapod with definable pivot point.

The light spots are optically magnified with a micro objective – tube lens combination by a factor of 15.15 and detected with a Peltier cooled low noise CCD camera with 4.54 x 4.54 µm^{2} pixel size and 14 bit A/D converter. This yields an effective pixel size of 0.3 µm. The centre of mass of a single camera frame can be determined with a precision of below 1/10 of the pixel size, which is 30 nm.

#### 5.2 Spot shapes

The series of 5 light spots arising from the first diffraction orders of the CGH (Fig. 1) is presented in Fig. 8. Here, we use the interferometer according to Fig. 7 merely as a plane-wave light source. The upper row shows the camera measurement with dark-frame subtraction, averaged over 50 images. The best focus position is determined as maximum of the *modulation transfer function* (MTF), sampled along the z-distance. The lower row shows a simulation for the ideal system. The spatial range of the plots is 80 µm. The intensity scale is logarithmic, covering 4 orders of magnitude. Only zone 1 generates the well known Airy-distribution of a circular aperture. The complex intrinsic structures of the other spots reflect the special shapes of the corresponding zones on the CGH.

The agreement between simulation and measurement is very convincing for the inner zones 1 and 3 on the CGH, while there are obvious deviations for the outer zones. Especially in zone 4 and 5, asymmetric distortions of the spots are visible. We found that these distortions stay fixed in space when we rotate the CGH by 180° in the bayonet mount. This proves that they are not caused by the MZ-CGH. Furthermore, we used the property that the interferometric fringe pattern with a mirror in the ‘*cat eye*’ position uncovers the wavefront error of the interferometer light source itself. This way we found that aberrations (mainly a trefoil) are already imprinted in the illuminating wave front. This is the origin of the observed perturbations in the point spread functions of zone 4 and 5.

#### 5.3 Camera based straightness measurements

Straightness measurements require an appropriate reference. The straightness of the translation stage (Fig. 7) is determined by the vendor with a maximum deviation of 2.5 µm over the full traverse range of 800 mm. Additional internal interferometric evaluations show a maximum tilt variation over the full range of less than 3 arc seconds. These values are unusually good for a mechanical stage, but still put a limit on the straightness measurements of the MZ-CGH axis.

In order to overcome this limitation, the lateral spot positions are determined via the *difference* between two measurements with the MZ-CGH rotated by 180° in the bayonet mount between the first and the second. In doing so, the deviations due to the CGH change the sign, while the deviations due to the stage stay unchanged. The impact of the stage is thus eliminated by calculating the half difference of the lateral spot positions.

Additional errors occur due to air turbulences and remaining mechanical vibrations on the optical bench. These errors are largely eliminated by averaging over a series of 50 frames for each spot, which takes about 9 seconds, limited by the camera frame rate. An example for the emerging position ‘clouds’ is shown in Fig. 9. Note that the ranges (displayed at the top of the plots) are tiny compared to the diffraction limited spot sizes!

Finally, thermal fluctuations in the lab of < 2 K lead to mechanical drifts. We monitored drifts over several days and performed critical measurements quickly during stable periods.

The thus determined 5 lateral spot positions are evaluated in Fig. 10, upper row, separated into x- and y-coordinates. Each position (indicated by a blue circle) is determined as the mean value of 50 center of mass coordinates (Fig. 9) from 50 corresponding camera frames. The impact of the stage is eliminated as described above. The red line is a linear fit to these 5 positions, and therefore defines *the mean optical axis of the CGH*. The slope of the line indicates the tiny angle between this axis and the direction of the stage. It is an adjustment artifact of no further importance. However, the residuals between spot position and linear fit, shown in the lower row of Fig. 10 are the deviations of the individual spots from the mean optical axis and thus the relevant quantities. They are smaller or in the order of 0.1 µm in both dimensions of the lateral plain. This outcome is well reproduced by several repeated measurements. We never observed deviations beyond a maximum of 0.2 µm for one lateral dimension. This is almost a factor of 50 smaller than our typical adjustment requirements of below 10 µm. Therefore, in the framework of our specifications, we can simply treat the axis as an *ideal straight line*. However, the tiny deviations are partly reproducible and thus assumed to actually originate from the CGH to a certain extent.

#### 5.4 Interferometric measurement of spot distances

The determination of the distances between the spots in z (optical axis) requires an accurate length measurement. The position of the carriage along the direction of motion is optically detected with a contactless measuring tape with a graduation period of 20 µm and a spatial resolution of 0.1 µm. The linearity is specified as ± 3 μm over a distance of 1000 mm and 0.75 μm over 60 mm. (We found that it is actually even more accurate.) The measuring tape is bonded to the granite core of the stage and behaves thermally like this substrate (linear expansion coefficient α ≈3 μm m^{−1} K^{−1}).

Furthermore, we need a definition of the best focus position. With the camera setup according to Fig. 7, a possible valid definition is the maximum of the MTF, sampled along the z-distance. This procedure leads to accurate results, although we observed slight ambiguities as the maximum of the MTF sometimes shows a week dependence on the spatial frequency.

For the sake of brevity, we restrict this report to the interferometric method, which is based on the principle described in chapter 2, and thus of special interest in the current context. The camera in Fig. 7 is replaced by a reflecting reference sphere with high optical surface quality as already shown in Fig. 2. The Zernike defocus coefficient *c*_{4} is the evident best-focus measure. This method is unambiguous, straightforward and highly accurate – we think slightly more accurate than a best focus determination with the camera setup.

Figure 11 shows *c*_{4}(z), extracted from the wavefront, for all 5 spots as function of the z-coordinate in a small range around the best focus, defined as c_{4} = 0. For each position, 3 measurements are taken. Instead of the absolute z-position, we refer to the *difference* to the *nominal* spot locations. Therefore, the intercepts of the linear fits with the z-axis indicate the deviation of the focus from the ideal positions. They are evaluated and summarised in Table 1. Again, the measurements were repeated several times. We referred to the ‘autocollimation’- as well as to the ‘cat-eye’ positions, which is an alternative option with tiny differences in the presence of aberrated wavefronts. We also shifted the stage in z to test a different part of the measuring tape. We found a reproducibility within a range of ± 2 µm. Therefore, the positions of spot 1 to 4 can be assumed to be exact within our precision of measurements. Spot 5 with the largest focal length (f = 379.395 mm) turns out to be reproducibly (6.1 ± 2) µm too close. This is only 0.02 ‰. Measurement errors are expected due to the accuracy of the measuring tape as well as mechanical and thermal drifts and air turbulences.

The slope of the linear fits, which is, according to Eq. (10), ½ *NA*^{2} is used to compute the numerical aperture in Table 1. The agreement between nominal and measured values is very satisfying, although the measurement yields systematically slightly smaller results. This is due to the fact that the annular zones are marked with a software mask for the numeric evaluation, which was chosen slightly smaller than the margin of the zones to avoid edge disturbances.

#### 5.5 Interferometric measurement of centring

The straightness measurement according to section 5.3 can alternatively be carried out interferometrically with the reference sphere. Figure 12 shows the tilt Zernike coefficient *c*_{3}, determined from the wavefront, for all 5 spots as function of the lateral y-shift of the reference sphere in a small range around *c*_{3} = 0. For practical reasons, the shift is actually performed with the hexapod (Fig. 7), while the sphere remains at rest. For each y-position, 5 subsequent measurements are taken. The reproducibility is so high, that the corresponding 5 markers in the plot overlap almost perfectly. However, an alternating deviation of the measurements from the linear fit to both sides can be observed. This is due to the finite positioning accuracy of the hexapod for these tiny spatial steps of 2 µm, which are obviously sometimes slightly smaller and sometimes slightly larger.

The slope of *c*_{3}(y) is according to Eq. (13) simply two times the numerical aperture. We hence actually plot the halved value ½ *c*_{3}(y) in Fig. 12 such that the slope is directly *NA*, as indicated to the right of the figure. The extracted values turn out to be up to 3% larger than nominally expected. We assume that this deviation is due to an inaccuracy of the hexapod for these extremely tiny movements of single microns.

The measurements in Fig. 12 are not corrected for the error of the translation stage as in section 5.3 and therefore not directly suitable for a determination of the straightness of the optical axis of the MZ-CGH. However, we used this approach for other MZ-CGHs where we found very consistent results.

## 6. Conclusions and outlook

We presented an approach for high-precision optical adjustments based on binary phase CGHs with multiple zones (MZ-CGHs) and integrated Fizeau reference surface. The procedure is based on an evaluation of the interferometrically observed tip, tilt and defocus Zernike polynomials from the different zones. We summarized the analytic relations between these ‘displacement aberrations’ and the associated lens placement errors.

The different zones of the MZ-CGH generate an optical axis, which is the key feature of the adjustment strategy. The *straightness* of this axis determines the accuracy of the presented method. It was optically tested with an appropriate experimental setup. The deviations of five spots representing this axis over a distance of almost 300 mm were determined. We demonstrated that all spots are, within a tolerance of 0.2 µm, located on a *straight line*. This outcome is well reproduced by repeated measurements.

The spatial distance between the five spots along the optical axis yielded the following findings: No significant deviations from the nominal positions could be found for spot 1 to 4 within a measurement accuracy of about ± 2 µm. Spot 5, from the outermost annular zone with the largest focal length (f = 379.395 mm) turns out to be reproducibly (6.1 ± 2) µm too close, which is only 0.02 ‰. Measurement errors are expected mainly due to the accuracy of the measuring tape as well as mechanical and thermal drift and air turbulence.

We presented the results for one particular CGH. However, measurements for several similar CGHs with different zone designs support these findings and make the presented method an ideal workhorse for our precision alignment requirements [21]. In Ref [22]. we prove that the optical imaging quality [23] of our recent space optics is consistent with centring errors of the individual lenses in cold environment of less than 5 µm.

A second publication is scheduled to discuss practical applications and examples in the framework of the *Euclid* space telescope, where we applied our approach successfully with extremely critical optical adjustment tolerances.

For future developments, it is conceivable to go from CGHs with multiple zones to designs with one single zone over the full aperture that simultaneously generates the required multiple wave fronts. This approach could improve the adjustment accuracy even further, as the full aperture is available for every single measurement. However, the feasibility is still to be demonstrated.

## Funding

The MPE Euclid participation is supported by DLR under grant 50 QE 1101.

## Acknowledgments

Many thanks to David Wilman for carefully reading the manuscript and to Ralf Bender for the inspiring spirit in our group. We are in deep sorrow about the death of our college and friend Andreas Bode. He died during the preparation of this paper.

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