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Abstract
When measuring surfaces it is always a challenge to differentiate whether differences to the expected form originate from positioning errors or from surface errors. In interferometry it is common to subtract tilt and power terms from the measurement result to remove misalignment contributions. This is a suitable approximation for spherical surfaces with small NA. For high NAs and increasing deviations from a spherical shape, which applies to aspheres and freeforms, additional terms show increasing magnitudes. A residual error remains after subtraction of tilt and power. Its form depends on the surface’s nominal shape and oftentimes has a non-negligible magnitude, therefore imposing the risk of being misinterpreted as topography error.
© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical elements are the centerpiece of numerous devices, from cameras to measuring instruments or in laser applications. Spherical lenses are the simplest elements and are therefore widely used. Aspherical lenses and freeformed elements with more complex geometries provide more degrees of freedom for the design of optical systems, which makes them attractive to use and the number of applications rises steadily. During the manufacturing process of all those elements, quality control is applied to make sure that the actual parameters of the produced lenses lie within their specifications. A widespread method, especially in high precision manufacturing, is interferometry. Form deviations but also misalignment of the surface under test (SUT) results in fringes in the interferogram and perfect alignment is nearly impossible due to imperfect alignment stages.
In order to remove surface errors correctly during the manufacturing process, it is important to be able to distinguish surface errors from positioning errors, especially with the current possibilities of high precision finishing. Also for the comparison of different measurements performed by various measurement systems as in [1], the consideration of error components caused by misalignment is fundamental. The removal of best fitting spheres is a compromise, due to the lack of more suitable tools. In this paper we will show that the errors introduced by mere subtraction of tilt and power as a compensation for SUT misalignment increase with the deviation of the SUT from a spherical form and also depend on the measurement setup. We introduce the ratio $PV_{res}/PV_{total}$ as a measure of how well such a subtraction approximates the error components caused by misalignment. Here, $PV_{total}$ is the peak-to-valley value of the misalignment error term and $PV_{res}$ the peak-to-valley value of misalignment error after subtraction of tilt and power, i.e. after subtraction of the best fitting plane and parabola. In section 4, we discuss that the errors caused by misalignment are depending not only on the surfaces’ shapes but also on the measurement method and illustrate these issues exemplary in section 5.
2. Ambiguity of surface and positioning errors
Both surface errors and positioning errors lead to optical path differences and therefore to fringes in the interferogram. The interferogram shows the difference between the imperfect misaligned surface and the nominal shape at nominal position. The cause for the deviation (misalignment or surface error) cannot be determined. A clear assignment is in principle not possible. Only additional information about the SUT’s position can help to assess the share of error caused by misalignment. This information can be provided by fiducials or external measurements [2,3]. Without this information the solution of the problem stays ambiguous and cannot be resolved. In many cases the assumption is made that certain shares of the error are more likely caused by misalignment than by deviations of the topography. The probability might be high for tilt and power terms. But as we will show in this paper, tilt and power are only approximations. Especially when measuring aspheres and freeforms higher order terms with non-negligible magnitudes arise from misalignment. With them it is more difficult to judge where they have their origin without careful analysis of the surface’s geometry and the measuring setup. If the misalignment is not taken into account correctly, the deviation to the expected topography will be misinterpreted as a surface error. That will lead to wrong corrections of the surface geometry within the iterative manufacturing process and, at worst, prohibits a good convergence of the process. It is also a problem as long as there are no reference surfaces available and the performance and accuracy of the different measurement setups and techniques for asphere and freeform metrology are assessed by comparing results of round robin tests [4]. In case of stitching, compensators are being estimated [5] to consider positioning errors and wrong assumptions lead to stitching errors, especially low frequency shape errors [6–8].
3. Misalignment of spheres in nulltest interferometers
It is interesting that even for the case of spherical surfaces a subtraction of power and tilt might be critical and insufficient as well. Before we discuss aspheres and freeforms we review the state of the art that is known for spheres.
Spherical elements are widely used and can be easily tested under nulltest conditions in Fizeau- or Michelson-Interferometers. In nulltest condition the sphere’s center and the focus of the spherical testwave coincide. Usually it is assumed that a defocus of the sphere leads to a quadratic power term and a decentering or tilt of the sphere results in a linear tilt term. To compensate for those positioning errors, those terms are subtracted from the evaluated interferogram. Analysis of the exact formulations show, that the power term is only a rough approximation of the error when a sphere is defocused [9–12], whereby the tilt term describes a decentering fairly well [10]. One property of these exact formulations is, that they consider the optical path difference of the incident rays caused by misalignment, since that is the quantity an interferometer can measure. This is shown in Figs. 1 and 2 for a defocused and a decentered sphere.
3.1 Defocus of spheres
For the geometry of a defocused sphere, as shown in Fig. 1, the law of sine states:
Note that $\Delta z < 0$ in Fig. 1. The angles are defined as
This results in
As already discussed by de Groot et. al. [12] the residual error after subtracting a parabolic term (usually referred to as "power term") is proportional to $\Delta z$ and strongly depending on the NA ($=r_{max}/R$). He stated that subtracting power is sufficient for NA < 0.25 or f-numbers > 2, but the answer to the question of what is sufficient can be quite different depending on the context and the requirements for the respective measuring task. Using Eq. (4) inversely, requirements for the positioning accuracy can be derived, if a certain residual error should not be exceeded after power term removal.
Alternatively, a more accurate description of the error must be used [9–12].
3.2 Decentering of spheres
In contrary to the defocussing of spheres, the decentering has not been discussed so far. When a sphere is decentered, the height difference can be derived in analogy to the height difference for a defocused sphere. The geometry is sketched in Fig. 2. Note that $z_1 = \sqrt {R^2-r^2}$. $\Delta h$ is then given by
Figure 3 shows $\Delta h$ for a sphere of $R = 40$ mm with $r_{max}= D/2 = 25$ mm, decentered by $\Delta x = 10 \mathrm {\mu }$m. It can be seen, that it consists of a tilt term and an additional cylindrial term. The peak-to-valley value (PV) of the cylindrical term (0.49 nm) is much smaller than the PV of the total error of 12.50 $\mathrm {\mu }$m ($PV_{res}/PV_{total}=0.004 \%$) and it might be suitable for most applications to subtract the tilt term only. But for some applications, especially when testing spheres with high NA or when highest demands on precision and accuracy are placed, it might be important to be aware that the tilt term poses a very well yet not exact approximation.
Fig. 3. Resulting height difference $\Delta h$ from decentering a sphere (R = 40 mm) by 10 $\mathrm {\mu }$m. After subtraction of a tilt term (left), a residual cylindrical term remains (right).
4. Misalignment of aspheres and freeforms: How to calculate the resulting error
But how is the situation when measuring aspheres and freeforms? How well do tilt and power describe the errors that are introduced by misalignment in their cases? For the formulation of the exact height difference $\Delta h$ caused by misalignment, it is of great importance to consider the change the misalignment causes in the optical path length of the testing rays. So the difference between the nominally positioned and the misaligned surface has to be determined in the direction of the rays, as it was done for spheres in section 3. The first difficulty is, that there are a number of methods to measure aspheres and freeforms: using compensators like computer generated holograms (CGHs) [13,14], applying stitching methods [15,16], using multiple tilted waves such as the Tilted Wave Interferometer [17] and others. Therefore a general equation is not available. All those methods have their own wavefront geometry and that’s why a formulation of the height difference is depending on both the method as well as on the surface’s geometry. Secondly, due to the changed position of the SUT, the local gradient at a certain position $(x,y)$ will change and with it the angle of the reflected rays, which will lead to additional retrace errors. Unfortunately, retrace errors heavily depend on the optical design of the individual setup [18] so a general description cannot be formulated. However, optics design evolves to reduce the problem [19]. Still, retrace errors represent an additional source of error that needs to be characterized and considered.
4.1 Misalignment errors in sag direction
A simple approach to get an idea of the errors induced by misalignment, is to compare the misaligned surface with the surface at its nominal position and get the difference in sag direction. For points $(x|y)$ the sag of the nominal surface $f(x,y)$ and the sag of the misaligned surface $f_2(x,y)=f(x-\Delta x, y- \Delta y) +\Delta z$ are calculated and subtracted from each other. This is also the way the asphericity or deviation from a best fitting sphere is usually specified.
From Fig. 4 it can be seen that the characterization of the error introduced by defocus in that way is impossible, since the difference between $f(x,y)$ and $f_2(x,y)=f(x,y)+\Delta z$ will always be a constant offset of $\Delta z$. This fact already shows how far this approach is from standard interferometer reality. The deviation in sag direction can therefore only be used for an analysis of the impacts of misalignment in the remaining degrees of freedom.
Fig. 4. Defocused (left) and decentered (right) specimen and the difference to the nominally positioned surface in sag direction.
The approach to determine the difference of the sag values of nominally positioned and the misaligned surface is only valid for interferometers with plane wavefronts or cartesian coordinate measuring machines. In interferometers with convergent or divergent test wavefronts the rays are not parallel to the z-axis and the surface difference in sag direction is therefore not the correct point of view.
4.2 Difference in ray direction
For the correct calculation of the optical path difference caused by positioning errors, it is necessary to consider the direction of the incoming rays. If the rays’ pathways are described by vector $\vec {l}$, the intersection point $P_a = (x_a, y_a, f(x_a,y_a))$ with the nominal surface at nominal position ($f(x,y)$) as well as the intersection point $P_b = (x_b, y_b, f_2(x_b,y_b))$ with the displaced surface ($f_2(x,y)=f(x-\Delta x, y-\Delta y)+\Delta z$) can be determined for each ray. Subsequently, the Euclidean distance between points $P_a$ and $P_b$ could be determined, but this approach loses the information whether point $P_a$ lies in front or behind point $P_b$ from the point of view of the incident ray. Therefore one cannot differentiate between a shortening and an elongation of the optical path length by the misalignment. Alternatively, the distance of the points $P_a$ and $P_b$ from the ray’s (virtual) intersection with the z-axis at point $(0|0|z_0)$ can be taken into account. The geometry is sketched for a nulltest setup in Fig. 5.
The height difference $\Delta h$ is then given by
As can be seen from Eq. (6), the height difference caused by misalignment depends on both the surface geometry $f(x,y)$ and the ray direction ($z_0$). In interferometric measurements in nulltest setup the resulting optical path difference (OPD) can be approximated by $OPD = 2\cdot \Delta h$ for small misalignment values, where retrace errors can be neglected. In other configurations an additional component caused by a change of the reflection angle and possibly resulting retrace errors must be considered as well.
In the following subsections the approach of calculating the difference in ray direction is applied for setups with beams impinging perpendicularly on the SUT and interferometers with spherical wavefronts.
4.2.1 Aspheres and freeforms tested with spherical wavefronts
Usually, interferometers provide planar or spherical wavefronts. Aspheres and Freeforms cannot be tested with them just like that in most cases. The deviation between the spherical wavefront and the non-spherical SUTs is too big, leading to non-resolvable fringes in the interferogram. Therefore only aspheres and freeforms with very small asphericities can be tested without further intervention. For measuring aspheres and freeforms with higher asphericities, annular or subaperture stitching can be applied to get resolvable fringes over the whole aperture.
For spherical wavefronts the rays’ pathways can be described by
For defocusing, Zhao et. al. [20] proposed an analytical formulation. They also define the deviation from a Best Fit Sphere in radial direction, in contrary to the common description in sag direction. In most cases, a description of the radial asphericity will be more meaningful than the deviation in sag direction.
4.2.2 Aspheres and freeforms in nulltest
Another method to measure aspheres and freeforms is to adapt the test wavefront to the SUT’s shape, e.g. by the use of computer generated holograms (CGHs) [13,14]. They make the incoming rays hit the surface perpendicularly and therefore provide nulltest conditions.
Dörband et al. [13] proposed a method to compensate for misalignment errors, when measuring aspheres in CGH setups. They utilize a raytracing of the whole measurement system and fit the resulting wavefront at the sensor with Zernike polynomials. Thereby a sensitivity matrix is set up with the derivatives of the Zernike coefficients with respect to decentering and tilt of the aspheric surface and the hologram. After measuring the wavefronts, the misalignment can be reconstructed by solving the system of equations and the errors caused by misalignment can be compensated. The advantage of this method is, that the effects of misalignment within the whole system, including retrace errors, are considered. But this is its disadvantage at the same time: the effects can only be computed when the system is known. Here we want to discuss the effects of misalignment for nulltest setups, independently from the exact system.
Usually, in nulltest setups the rays hit the SUT perpendicularly. Therefore, $\vec {l}$ can be formulated with the help of the surface’s normal vector $\vec {n}$, which points in ray direction:
To find point $P_b$, the intersection of $\vec {l}_{NT}$ with the misaligned surface $f_2(x,y)$ a non-linear system of equations needs to be solved, analogous to Eq. (8). For the calculation of the resulting height difference $\Delta h$ according to Eq. (6), the virtual intersection of $\vec {l}$ with the z-axis is necessary. The test wavefront has no single focus point and every ray intersects with the optical axis at its own z-coordinate. So for each point $P_a$, a corresponding $z_0$ value needs to be calculated by
For aspheres, which are rotationally symmetric ($r = \sqrt {x^2+y^2}$), the above described procedure and equations can be simplified in case of a defocusing, since this happens on the axis of symmetry. Especially Eq. (9) can be rewritten as a linear equation:
To find the intersection point $P_b$, where $f(r)+\Delta z = f_l(r)$, it is then only necessary to solve one nonlinear equation instead of a system of equations.
4.2.3 Experimental validation
In order to verify the presented calculation approach, exemplary measurements were performed on an asphere [14] in nulltest configuration. In the test arm of a Twyman-Green-Interferometer a combination of an interferometer objective and a CGH provide the aspherical test wavefront (Fig. 6 on the left). The asphere was misaligned deliberately by $\Delta x = 5, 10, 20$ and $50 \mathrm {\mu }$m by the use of a micrometer screw. The difference of the OPD maps recorded at the nominal and the misaligned position was analyzed and compared to the prediction calculated according to Eqs. (6), (8) and (9). Note that the OPD maps need to be corrected for the strong distortion introduced by the CGH first. In Fig. 6 (right) the amplitude of the dominant Zernike terms (Orthonormal Zernike circle polynomials [21] in ANSI notation [22]) of the measurements are compared to the predicted coefficients and show very good agreement.
Fig. 6. Left: measurement setup, consisting of a Twyman-Green-Interferometer, a CGH and an asphere. The asphere’s position can be manually altered using micrometer screws. Right: comparison of measured and simulated values of the dominant Zernike terms for different decentering values.
Two different approaches for calculating the error resulting from misalignment have been presented. Both have in common that the result depends on the surface’s geometry. One should have in mind, that in a first analysis the nominal shape will be used, however, the measurement result will reveal the real surface shape. For small surface deviations the influence on the misalignment error analysis might be small but for higher deviations an iterative process could be implemented. The main difference of the two approaches is the consideration of the ray direction. How this influences the resulting errors is shown exemplary and discussed in the following sections.
5. Exemplary surfaces and their errors
So let us have a look at the errors resulting from decentering and defocusing aspheres and freeforms and the presented approaches, in particular the difference in sag direction as well as the height difference in ray direction in a nulltest setup. Since $\Delta h$ depends on the individual form description of the considered surface, a general discussion is difficult. Thus, the effects of defocusing and decentering of nonspheric surfaces are discussed exemplary for three nonspheric surfaces: an asphere (see Table 1) and two freeforms, Freeform1 (see Table 2) and a Two-Radii Specimen (Freeform2, see Table 3 and Fig. 7). They all have a clear aperture of $D = 50$ mm. The radii of curvature of their respective vertices lie between 34.3 mm (Asphere1) and 52 mm (Freeform1), but the radii of their respective Best Fit Spheres are close to each other (43.45 mm for Asphere1, 45.76 mm for Freeform1 and 39.75 mm for the Freeform2). The chosen surfaces mainly differ in their deviation from the Best Fit Sphere, as depicted in Fig. 8. The aspheric deviation of Asphere1 is rotationally symmetric, whereas the deviation of Freefrom1 shows a strong astigmatic component.
Fig. 8. Deviation from their respective Best Fit Spheres of the three exemplary surfaces. Left: Asphere1, middle: Freeform1, right: Two Radii Freeform2.
Table 1. Parameters of Asphere1
Table 2. Parameters of Freeform1
Table 3. Parameters of Two Radii Specimen (Freeform2), see [23]
5.1 Defocusing
As already mentioned in section 4.1, the difference in sag direction between a nominally positioned and a surface axially displaced by $\Delta z$ will always be a constant. Therefore the analysis of defocused aspheres and freeforms will only be executed in ray direction, namely in a nulltest setup. The results for spherical wavefronts are comparable and will not be discussed separately.
Asphere1
Figure 9 shows the resulting $\Delta h$ for a defocus by $\Delta z = -10 \mathrm {\mu }$m of the Asphere1.
Fig. 9. Height difference $\Delta h$ resulting from a defocus of $\Delta z = -10 \mathrm {\mu }$ of the Asphere1 from its nulltest position (top row). The subtraction of a power term leaves a residual error, shown in the second row.
It shows that a parabolic approximation of the height difference is highly inaccurate. The PV of $\Delta h$ is 1.389 $\mathrm {\mu }$m, whereby the PV after subtraction of a power term still remains at 285 nm ($PV_{res}/PV_{\Delta h}=20.5 \%$). This residual error can be fitted by the Zernike polynomials $Z_{12}$ (spherical aberration), $Z_{24}, Z_{40}$ and $Z_{60}$ and is reduced to 0.4 nm after subtraction of these terms. Numerous high order terms must be taken into account until the residual error is negligiby small. A subtraction of all the terms determined by this analysis in full would lead to a wrong topography as well as ignoring them. For a correct compensation of the misalignment error the ratio of the terms must be considered as well.
Freeform1
Looking at the resulting $\Delta h$ for the defocused Freeform1, $\Delta z = -10\,\mathrm {\mu }$m, see Fig. 10, the strong dependency on the surface’s sag function $f(x,y)$ becomes obvious.
Fig. 10. Height difference $\Delta h$ resulting from a defocus of $\Delta z = -10\,\mathrm {\mu }$m of the Freeform1 from its nulltest position (left). The subtraction of a power term leaves a residual error (middle), which consists mainly of astigmatism. After subtraction of power and astigmatism, a residual error of 54.8 nm (PV) is still left over (right).
The PV value of $\Delta h$ is 1.746 $\mathrm {\mu }$m. After subtraction of the power term 374.7 nm remain, which have an astigmatic shape ($PV_{res}/PV_{\Delta h}=21.5 \%$) - correspondig to the nominal shape of this freeform. If astigmatism is subtracted as well, error components of higher order show. They still have a PV of 54.8 nm. Apparently, a mere subtraction of power can only compensate the misalignment partially.
The values of the PV of $\Delta h$ as well as the residual PVs after subtracting higher order terms are proportional to the amount of defocus, as shown in Fig. 11. Therefore the ratio $PV_{res}/PV_{\Delta h}$ remains constant.
Fig. 11. Left: Power Term and PV of residual error for a defocused Freefrom1 in a nulltest setup, calculated in ray direction for a variable defocus. Right: Magnitude of higher order terms of fitted error $\Delta h$, depending on $\Delta z$.
This means that a power term will only compensate a part - always the same part - of the error resulting from misalignment. In the example of Freeform1 only approximately 80% of the error (PV) can be removed from the result by subtracting a power term. Only the absolute residual error will be smaller the better the alignment is. With this knowledge it is possible to analyze the error resulting from misalignment for a given value of $\Delta z$, determine the power term in the measurement result and subtract higher order terms in the right magnitude, corresponding to the analyzed ratio.
The ratio $PV_{res}/PV_{\Delta h}$ is depending on the deviation from a sphere, as illustrated by the following simulations. Freeform1 consists of a spherical part ($k=0$) and an added polynomial description. By multiplying the polynomial part with a factor $b$, the amplitude of the polynomial and the magnitude of the asphericity can be varied.
The results for such a variation of the asphericity, shown in Fig. 12, confirm the assumption, that the more a surface deviates from a sphere, the worse a power term can describe a defocussing error.
Fig. 12. Left: Power Term and PV of residual error for a defocused Freeform1 in a nulltest setup, calculated in ray direction for a variable asphericity. Right: Ratio $PV_{res}/PV_{\Delta h}$, depending on asphericity.
From both examples we can see that a parabolic approximation only compensates part of the error resulting from defocusing in case of aspheres and freeforms. It is worse the higher the asphericity is, but the ratio between residual error and parabolic approximation remains constant for different values of defocusing for each surface. How the residual error looks like mainly depends on the surface’s sag function or geometry.
5.2 Decentering
When a specimen is decentered, the sag function of the misaligned surface is $f_2(x,y) = f(x-\Delta x,y-\Delta y)$.
Asphere1
In case of decentering the Asphere1 by $\Delta x = 10\,\mathrm {\mu }$m, the resulting error both in sag direction and in ray direction in a nulltest setup is mainly tilt - as expected. But after subtraction of the tilt term, a residual error of $1\,\mathrm {\mu }$m PV ($PV_{res}/PV_{\Delta h} = 10 \%$), when calculated in sag direction, and a residual error of $1.265\,\mathrm {\mu }$m ($PV_{res}/PV_{\Delta h} = 13 \%$), in case of nulltest, remain. It is obvious that such a high residual errors are not negligible. Fitting the error with Zernike terms, it shows that apart from tilt the errors mostly consists of primary and secondary coma terms. After the subtraction of primary coma, the residual errors still have residual errors of 55 nm and 115 nm for the two approaches respectively. The difference between nominal and misaligned asphere, calculated in sag direction, is shown in Fig. 13. The height difference of the asphere, calculated in ray direction in a nulltest setup is shown in Fig. 14. Note that the errors resulting from the two approaches are similar but far from being the same.
Fig. 13. Resulting $\Delta h$ from a decentered asphere (Asphere1), when $\Delta h = f(x,y)-f(x-\Delta x,y)$: difference in sag direction (left), as well as residual errors after subtraction of tilt (middle) and tilt and primary coma (right).
Fig. 14. Height difference $\Delta h$ resulting from a decentering of $\Delta x = 10\,\mathrm {\mu }$m of the Asphere1 from its nulltest position (left). The subtraction of a tilt term leaves a residual error, mainly consisting of coma (middle). After additional subtraction of coma, error components of higher order show (left).
Freeform1
When the Freefrom1 is decentered by $\Delta x = 10\,\mathrm {\mu }$m, the difference in sag direction contains a coma term besides the tilt, as shown in Fig. 15.
Fig. 15. Resulting $\Delta h$ from the decentered Freeform1, when $\Delta h = f(x,y)-f(x-\Delta x,y)$: difference in sag direction (left), as well as residual errors after subtraction of tilt (middle) and tilt and coma (right).
After subtraction of this coma, instead of a secondary coma, as it was the case of the decentered asphere, a remaining trefoil shows. When calculating $\Delta h$ in ray direction, the trefoil is much more dominant, as depicted in Fig. 16. But even after subtraction of both coma and trefoil a higher order component with a PV value of 19.1 nm is left.
Fig. 16. Height difference $\Delta h$ resulting from a decentering of $\Delta x = 10\,\mathrm {\mu }$m of the Freeform1 from its nulltest position (left above). The subtraction of a tilt term leaves a residual error, mainly consisting of a trefoil (right above). After additional subtraction of trefoil, a residual coma shows (left below). Subtracting Tilt, Trefoil and Coma leaves a residual error of 19.1 nm (right below).
Analogous to the variation of the defocus $\Delta z$ (Fig. 11), the PV values are proportional to the value of decenter $\Delta x$, at constant ratio $PV_{res}/PV_{\Delta h} = 2.4 \%$.
Freeform2: Two-Radii Specimen
The difference in sag direction of the decentered Two-Radii freeform also consists of mainly tilt and coma. After subtraction of tilt the remaining error shows a ratio of $PV_{res}/PV_{\Delta h} = 8.86 \%$, showing a dominant coma term. Additionally subtracting the coma term leaves a complex topography that resembles the surface’s geometry (Fig. 17).
Fig. 17. Resulting $\Delta h$ from the decentered Freeform2 (Two-Radii), when $\Delta h = f(x,y)-f(x-\Delta x,y)$: difference in sag direction (left), as well as residual errors after subtraction of tilt (middle) and tilt and coma (right).
As a summary, the simple tilt term removal is no sufficient approximation for the compensation of decentering in case of aspheres and freeforms and the additional error components depend strongly on the surfaces’ geometry. Therefore it is necessary to individually analyze the surfaces to be tested to take the misalignment into account correctly when evaluating interferograms. The ratio of the error components’ magnitudes additionally depend on the chosen approach. We will discuss this in more detail in the following section.
6. Comparison of approaches to determine and compensate misalignment Errors
As it was presumed, it could be observed in section 5, that the consideration of the rays’ directions in determining the errors resulting from misalignment, makes a difference. A close look at Figs. 13–16 show that the errors appear differently for the two calculation approaches. To analyse this more, we will have a look at the Zernike coefficients of the residual error after subtracting tilt. They are presented in bar plots in Figs. 18 and 19 for a decentering of the Asphere1 and the Freeform1, once for $\Delta h$ in sag direction, once for the calculation in the rays’ direction in case of nulltest condition.
Fig. 18. Zernike Fit of Residual error of $\Delta h$ that arises for a decentering by $\Delta x = 10\,\mathrm {\mu }$m for the Asphere1, left in sag direction, as described in sec. 4.1, right as described in sec. 4.2.2.
Fig. 19. Zernike Fit of Residual error of $\Delta h$ that arises for a decentering by $\Delta x = 10\,\mathrm {\mu }$m for the Freeform1, left in sag direction, as described in sec. 4.1, right as described in sec. 4.2.2.
In both cases the same coefficients are dominant but show different magnitudes and signs. Also their ratio is different, which is striking in case of coma and trefoil for the freeform surface Freeform1, compare also Figs. 15 and 16. For a fast analysis of the dominant terms, it might be sufficient to calculate the difference in sag direction. However, to compensate the misalignment correctly it is necessary to subtract the terms in the right proportion from the measurement result, since part of the higher terms could also be part of the surface error and should therefore not be removed completely from the measurement result. The equations presented in section 4.2 provide the basis to get the right ratio of the terms’ magnitudes for the applied measurement method and its ray geometry.
7. Describing misalignment of conic surfaces with power and tilt
Most aspherical and freeform surfaces depend on a variety of parameters. It is therefore difficult to derive general dependencies and correlations. But one group of aspheres can be described by only two parameters: conic surfaces are fully defined by the conic constant $k$ and their ratio of $R/D$. We investigated how well power and tilt describe misalignment errors of these surfaces. For this purpose, a nulltest setup was assumed, misalignments of $\Delta z = -10\,\mathrm {\mu }$m and $\Delta x = 10\,\mathrm {\mu }$m were applied and the resulting errors $\Delta h$ fitted by power and tilt. After subtraction of power and tilt, the magnitudes of the residual errors are determined. The ratios $PV_{res}/PV_{\Delta h}$ and $RMS_{res}/RMS_{\Delta h}$ are a measure of how well power and tilt approximate the real errors.
For both defocusing (Fig. 20) and decentering (Fig. 21) it shows, that for smaller $R/D$ ratios (which means higher NA), the residual error is bigger relative to the total error. Therefore the approximation by power and tilt is worse. The same applies for increasing deviation from a sphere or increasing deviation from $k=0$. What could already be seen in Fig. 12 is confirmed: The higher the asphericity, the bigger are $PV_{res}/PV_{\Delta h}$ and $RMS_{res}/RMS_{\Delta h}$, which means that only power and tilt approximate the effects of misalignment worse.
Fig. 20. Ratios of residual error after subtracting power from the total error $\Delta h$ to the total error for defocused conic surfaces in nulltest setups ($\Delta z = -10\,\mathrm {\mu }$m). Left: Ratio of PV values, right: Ratio of RMS values
Fig. 21. Ratios of residual error after subtracting power from the total error $\Delta h$ to the total error for decentered conic surfaces in nulltest setups ($\Delta x = 10\,\mathrm {\mu }$m). Left: Ratio of PV values, right: Ratio of RMS values.
8. Conclusion
Without external references, misalignment errors can in principle not be distinguished from shape errors. Nevertheless there is a practical need to remove error components that are most likely caused by misalignment from the measurement result. For the compensation of misalignment errors it has been shown that power and tilt are no sufficient approximations for both aspheres and freefroms. We showed that the higher the deviation from a sphere is, the higher is the ratio of residual error and total error $PV_{res}/PV_{total}$. This ratio can serve as a tool to assess how good the alignment has to be when subtracting power and tilt has to be sufficient. In our examples, only down to 80 % of the error could be compensated by subtracting power and tilt. The subtraction of all error components that could arise might be a dangerous alternative, since especially higher order terms could easily be part of the surface error as well. For a correct subtraction the right ratio of the error components is important, but this is highly depending on the surface’s geometry and the measurement mode, as shown. An exact analysis is therefore strongly recommended, if errors resulting from misalignment are to be taken into account correctly.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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