Precision of diamond turning sinusoidal structures as measurement standards used to assess topography fidelity

In optical surface metrology, it is crucial to assess the fidelity of the topography measuring signals. One parameter to quantify this is the small-scale fidelity limit T FIL defined in ISO 25 178-600:2019. To determine this parameter, sinusoidal structures are generated, where the wavelengths are modulated according to a discrete chirp series. The objects are produced by means of ultra-precision diamond face turning. Planar areas and regions with slopes below 4° could be produced with form deviations of ≲10 nm. An initial estimate of the cutting tool’s nose radius resulted in a deviation that caused the ridges of the structures to be too narrow by approximately 150 nm, while the trenches were too wide. At the bottom of narrow trenches, deviations are observed in the form of elevations with heights of about 20 to 100 nm. The measurement standard investigated in this study has also been used to characterise optical instruments in a round-robin test within the European project TracOptic, which requires precise knowledge of the geometry of all structures. The geometry of the topography, cosine structures superimposed with form deviations, was measured using the Met. LR-AFM metrological long-range atomic force microscope of the German National Metrology Institute.


Introduction
Conventional methods for inspecting surface texture involve the use of contact stylus instruments.However, these instruments have disadvantages.They only sample a small number of profiles due to the time it takes to measure them.When used on softer surfaces, the contact of the stylus with the surface can cause damage.
Optical topography measuring microscopes, such as confocal microscopes, scanning coherence microscopes (using interferometric principles) and focus variation instruments, are preferred for quickly inspecting flat regions on a surface.However, their interaction with the micro-geometry of surface features and the optical properties of the material can be complex and diverse.Over the last decade, several investigations have been conducted into the limitations of transfer characteristics for steeply inclined surfaces of microstructures, depending on the physical principles of probing and the numerical aperture of the lens system, see for example figure 7 of the article by Lehmann et al [1] and figures 4 and 7 by Mauch et al [2].The bandwidth of signal transfer and signal distortion in surface metrology are limited by several effects.These include optical interaction in the case of optical instruments, and finite stylus size and deformation due to contact on softer materials.The literature has discussed comparisons of the performance of roughness measuring instruments for many years, for example in 2010 by Leach et al [3] and in 2023 by Buchenau et al [4].
The International Standardisation Technical Committee ISO/TC 213 Geometric Product Specification also includes a working group, WG 16, which specifies the metrological characteristics of surface texture and the operating principles of instruments used to measure surface topography.This covers all metrological aspects of characterisation and calibration methods for both surface texture and surface texture measuring instruments.The standard ISO 25 178-600:2019 [5] defines two parameters for quantifying topography fidelity: the width limit for full height transfer W l and the small-scale fidelity limit T FIL .
• The width limit for full height transfer W l is 'the width of the narrowest rectangular groove whose step height is measured within a given tolerance'.
• The small-scale fidelity limit T FIL is 'the shortest spatial wavelength for which both the height and the spatial wavelength of the measured profile or topography deviate by less than a specified amount from the values of the one to which it is being compared' or 'the smallest lateral surface feature for which the reported topography parameters deviate from accepted values by less than a specified fraction'.
To determine these parameters, various types of material measurement standards 1 are defined in ISO 25 178-70:2014 [7], an international standard specifying the characteristics of material measures used for 'the periodic verification and adjustment of areal surface texture measurement instruments'.The following types are regular periodic 1D grating structures with different cross-sectional features: • Type PPS: Periodic sinusoidal shape • Type PPT: Periodic triangular shape • Type PPR: Periodic rectangular shape • Type PPA: Periodic arcuate shape The parameters for topography fidelity require a range of different pitch and height values.To obtain the parameter small-scale fidelity limit using fewer sets of structures on a measurement standard, a modulation of the pitch or wavelength is necessary.The Physikalisch-Technische Bundesanstalt, the German National Metrology Institute, began developing modulated versions of the sinusoidal shape type about sixteen years ago [8,9] and continues to refine them [10] to aid in determining the parameter small-scale fidelity limit T FIL [11].
Material measurement standards are manufactured to represent sequences of cosine waves with constant wavelength for each of the waves and a monotonically changing wavelength from wave to wave according to a chirp sequence as shown in figure 1.The structures of the standard consist of sequences with positive cosines and sequences with negative cosines.This type of standard is called a pseudo-chirp standard.The term chirp is given because the choice of wavelengths follows a discretised chirp series and the term pseudo because the wavelength is constant within a single wave.
Each single wave is a profile element, with the i-th element defined as where s = +1 for the positive cosines and s = -1 for the negative cosines and where The material measure used for interlaboratory comparisons [10] within a large European project of the Horizon 2020 EMPIR programme, TracOptic, has cosine wave sequences with wavelength values listed in table 1.The artefact has several sequences of different amplitudes a or heights h = 2a.
A well-established process for producing high-precision parts such as this type of measuring standard is ultra-precision diamond turning.This technology has been proven effective in manufacturing optical components, including contact lenses, eyeglasses, and lightreflecting surfaces for illumination.The mechanical robustness of the workpiece, the repeatability of the machining process and the commercial availability make ultra-precision turning attractive for the production of material measurement standards with deterministic patterns for surface metrology [12][13][14].
For an artefact that is produced by a cutting process, the finite size of the cutting tool places a limit on the size of the features that can be created.Therefore, Figure 1.Sequence of structures representing positive and negative cosine waves. 1 The two terms material measure and measurement standard are used as synonyms.The International Vocabulary of Metrology VIM [6] uses measurement standard and etalon.However, the term material measure can be found in a number of ISO/TC 213 standardisation documents and is used in the title of the abovementioned ISO 25 178-70:2014.the concave structures are designed to be large enough to fit the size of the tool, i.e. the radii of the concave curvatures must be larger than the radius of the tool nose.The curvature κ at the bottom of the trench, where the slope is zero, is the value of the second derivative of the cosine function The greater symbol with an exclamation mark denotes a 'requirement' and λ tool denotes the wavelength of a cosine structure of amplitude a with a radius of curvature at its extrema that is equal to the radius of curvature of the nose of the tool.To fit the tool nose into a structure trench, the minimum structure wavelength min l must be greater than λ tool , see table 2.
The surfaces of these standards contain deterministic patterns and structures with details on the nanometre scale, spanning several millimetres.Manufacturing scales and wavelengths at sub-micron levels is a challenge for complex mechanical machining processes.
Although ultra-precision machining has been known for almost a century, it remains an area of scientific research.In the 1960s, research into ultra-precision machining focused on large optical surfaces.Taniguchi predicted in the 1980s that ultra-precision manufacturing (UPM) had almost reached its dimensional limits [15].Depending on the overall dimensions of the area of interest, additive or subtractive manufacturing or material forming is capable of producing structures with resolutions and accuracies down to the molecular level [16].Nevertheless, mechanical chipping processes generate roughness values down to R a < 1 nm and form accuracies down to several 10 nm over relatively large surfaces of up to several square metres.These extreme values are due to the demands of space telescopes and the semiconductor industry.
The process is often limited by tool wear with extremely sharp cutting edges with radii below a few tens of nanometres [17].Such sharp edges are necessary for the complex spatial chip generation during an ultra-precise cut of materials.As long as the uncut chip thickness is greater than the radius of the cutting edge, the chipping process is dominated by the energy consumption of the primary shear zone in front of the diamond tool due to the sharp cutting edge.As the uncut chip thickness approaches the dimension of the cutting edge, the chip and surface generation is dominated by a ploughing and sliding effect in front of the tool and in the tool-workpiece contact at the tool flank, which can also lead to considerable wear.
Depending on the process, several metres to hundreds of kilometres can be cut with a single tool.Machining metals is difficult due to the intrinsic chemical and mechanical properties of the crystalline carbon in contact with metals [18,19].Many approaches have been developed to overcome these limitations with ultrasonic actuation of the tool or workpiece, e.g. with integrated laser radiation, with cooling systems or surface modification of the workpiece and/or tools.Nickel-phosphorus (NiP) coatings are often chosen for UPM because of their well-known machinability [20].Here, the phosphorus content, heat treatment and geometry of the coating can affect ultra-precision processes [21].Bao et al underline the fact that the complex chipping process of diamond in NiP is not fully understood today [22].
The type of diamond and the orientation of the cutting planes and edges of the tool are important for machinability and tool life [23,24].They must be correctly selected and precisely operated.For the ultra-precision process, the coolant is important in terms of reducing temperature rise and tribological contact.In addition, the ambient gas can affect the surface degeneration of the diamond, for example, by controlling the oxygen content in the cutting volume.
The cutting process dynamics are affected by various factors, including the relative speed of the workpiece and tool at the point of contact, the feed, the infeed, and the inclination of the cutting edge and rake face.
The generation of the chip involves both plastic and elastic components, such as plastic side flow or the spring-back effect of grains [25].

Manufacturing process
In this study, copper discs of 80 mm diameter are coated with a nickel-phosphorus (NiP) layer of approximately 100 μm thickness.The disc is mounted on the vacuum chuck of the spindle of a Moore 250UPL lathe.The flat face of the disc is textured by the cutting tool, which feeds perpendicular to the axis of rotation of the workpiece, see figure 2. In this way, profiles of deterministic surface structures are generated in the radial directions of the disc.Along the radial directions, the profile is identical within the accuracy of the cutting process and the spiral toolpath of the diamond tool.Circularly symmetrical grooves and ridges are generated.The toolpath must be generated very accurately to include the spatial distance between the calculated tool centre point and the generated surface of the workpiece.The width of the trenches is greater than the size of the cutting tool, which acts as an upper geometric filter for the design of the surface; in the case of sinusoidal structures, the limit is given by equation (4).
The contour of the tool nose is measured perpendicularly to the rake face using a confocal microscope that generates a topography map.The location of the contour is uncertain due to a lack of information about the precise position of the cutting edge in contact with the material to be removed.Figure 3 shows one of the estimated contours.This contour data is then used as structuring element to morphologically dilate the designed cosine structures [26].It contains the geometry of the tool contour, but • it is biased due to the uncertainty of finding the position of the contour in contact with the material, and • it neglects further process influences like elastic and plastic deformations, which are particularly critical in trenches, as illustrated in figure 4, or thermal effects.
Tool wear must be kept within reasonable limits throughout the machining of the workpiece to ensure that the cutting edge geometry is not affected by progressive wear on the rake face, which subsequently affects the chip flow.This is because the cutting edge changes the cutting geometry and process as it wears during chip generation, which in turn affects the   resulting surface geometry.It is recommended to machine at least one workpiece with an acceptable wear rate.
To minimize the curvature of the trenches and ridges within the typical measuring ranges of the instruments to be characterised by the material measurements, the disc is chosen with a sufficiently large diameter and the structures are cut at the outer region.On the other hand, the diameter of the disc must be small enough to minimise the cutting path and therefore tool wear.
After the ultra-precision turning, the circular disc is cut into pieces as schematically shown in figure 5 by means of wire-electro-discharge machining (W-EDM).This provides twelve measurement standards of structures that are almost identical, accounting for any uncertainty caused by the complex manufacturing process.The individual pieces are marked on the front side to adjust their orientation in the measurement device, and an identification number is laser-etched onto the side surface.

Measurement method
To obtain a detailed characterization of the microstructures of measurement standards, a combination of an atomic force microscope (AFM) and a highprecision stage with sub-nanometer resolution and a measurement range of several millimeters is used.At the Physikalisch-Technische Bundesanstalt (PTB), the German National Metrology Institute, such a traceably calibrated instrumental setup has been developed [27][28][29].It is used for research and for calibration services [30].
The home-built AFM operates in amplitude modulation oscillation mode, with intermittent contact, and is used as a zero-point sensor.The high precision stage is a SIOS Nano-Measuring Machine (NMM) [31,32] equipped with a Zerodur ® motion platform as metrology frame, interferometric positioning sensors and autocollimator angle sensors for measuring 6 degrees of freedom (6-DOF) of its motion platform.To achieve rapid scanning over large ranges, a piezo stage is used as an additional vertical axis to quickly track small vertical changes [33].
The pseudo-chirp standard is measured with a Nanosensors Point Probe Plus ® Non-Contact with long cantilever and reflex coating (PPP-NCLR).The manufacturer, Nanosensors, states a radius of curvature r tip of the tip end of r tip < 7 nm (guaranteed r tip < 10 nm).During usage it has a wear increasing the tip radius by a few Nanometres.Estimates on the size of the tip reveal radii of r tip  20 nm.Research is being carried out to characterize the different types of probe tips that are employed on the Met.LR-AFM [34,35].
The approximate value of the height h t of the tip is h tip ≈ 15 μm, so that the structures C9 with trenches as high as h = 2a = 10 μm are almost at the limit of that type of AFM probes.

Evaluation method
The measurement standard is characterised through a three-step process of analysing profile data: 1. determination of reference plateau regions and starting points (the transition points between reference plateaus and largest profile element resp.cosine wave).
2. tilt correction by rotating the profile by the inclination angle of the larger one of the two reference plateaus, either the top or the bottom one, that are marked in figure 6.
3. characterization of each of the profile elements, which are either positive s = + 1 or negative s = − 1 cosine waves according to equation (1).
Steps 2 and 3 assume that the profile begins with the bottom reference plateau, followed by positive waves, and then negative waves, with a flat region in between.The measurement standard is designed so that the distance between the two starting points is the same for any amplitude.Therefore, the size of the flat region between the positive and negative cosine wave elements varies between sequences of specified amplitudes.If the structures are scanned in reverse direction, the data are turned after tilt correction by x = x 0 − x raw for analysis.To maintain the shape of the cosine structures, tilt correction is performed through rotation rather than projection.The à priori knowledge of the nominal wavelength of the design is used to localize each of the waves.This part of analysis procedure sorts the data points such that i n x x and 1, 2 ,..., and where i is the loop index over the profile elements.The characterization in Step 3 is structured as follows 3.1 To find the interval of wave i, the data points that lie within the interval [x start,i , x start,i + 1.1λ design,i ] are used to locate the position x h of the maximum height in case of s = + 1 and lowest depth in case of s = − 1.That interval is ten percent wider than the nominal wavelength to tolerate uncertainties of step 1 and of the wavelength of the manufactured structures with respect to the nominal.

The data points with
are then used to align the nominal cosine to these data points, i.e. to determine the position x E,i , z 0,i of the designed wave: 3.3 Outliers are removed from the set of data points ν = 1,K,n i .The criterion to identify outliers is based on the root mean square of the residuals adopted to the quality of the data.After outlier removal the fit is performed anew.

3.4
To estimate wavelength λ fit,i and amplitude a fit,i of the measurement data, the estimated position ˆx z , i i E, 0, is kept constant to avoid any bias of the wavelength and amplitude estimators due to features that might occure by a limited manufacturing accuracy: The symbols with a hat on top represent the estimated values of the parameters obtained through the optimization process.
For a certain amplitude class of the sequence of profile elements that have both positive and negative waves, N y parallel scan lines are acquired at pairwise distances Δy to test • the influence of the curvature of the concentric trenches and ridges formed by the face turning process and representing the peaks and valleys of the waves and • the homogeneity within the measurement area at the center of the measurement standard (one of the 12 pieces) to be used for optical microscopy that may be disturbed by random effects caused when cutting the material.

Results
The material measure presented in this article is one of 12 pieces of the disc being diamond turned to be used for a round robin within the European project TracOptic of the Horizon 2020 EMPIR programme [10].
For each of the nine different amplitudes N y = 24 parallel profiles with the sequence of waves are scanned with a pairwise distance of Δy = 4 μm approximately covering the region of interest, the measurement area.Within the measurement area where y = 100 μm, the ridges and trenches lying on concentric circles (as shown in figure 5) have a negligible curvature.Therefore, the cosine wave shape is almost identical for all scan lines, except for random uncertainties resulting from manufacturing and measurement processes.
The residuals ε design,ν of the fit according to equation (5) or the residuals ε fit,ν according to equation (6) represent the deviations between the measured height values and the height values of either the design curve or the curve with fitted wavelength The distribution of the residuals indicates the deviation from the cosine structures.One criterion is the root mean square of the residuals rms, where The root mean square roughness R q of the plateau regions is used to define the limits of the root mean square of the residuals for assessing form deviation.As the measurement standard is used under ambient conditions without any specific clean room requirements, there is a possibility of contamination.These contaminations can cause a bias in single profile elements and reference plateau intervals for a few of the scan lines.The values of R q of the reference surface sections (without contaminations) are R q = 4.2 ± 0.7 nm.
The root mean square (rms) values of the residuals were chosen as the criterion for form deviation, with values of 5, 10 and 20 nm.The dimensions of the cutting tool determine the minimum size of the features that can be produced.The width of the trench and the radius of curvature at the bottom of the grooves must be greater than that of the tool.The surface topography features of the workpiece are characterized by • the width of a groove or a valley, • the depth of a groove or a valley, and • the radius of curvature at the bottom of a concave feature.
Regarding trenches and ridges of a sinusoidal cross-sectional profile, the minimum manufacturable feature size is specified as the minimum wavelength λ limit in conjunction with the amplitude a or as the smallest radius of curvature The manufacturing process is limited not only by the depth of the grooves but also by the absolute value of the depth that needs to be cut into the material.The removal of material becomes increasingly difficult with increasing structure heights.There are several factors that influence the cutting process and limit manufacturing precision.
For each of the three requirements on permissible distortions, i.e. the shape deviation criteria rms = 5, 10 and 20 nm, and for each amplitude, we determined the wavelength limit values λ limit .They are listed in table 2. All of the wavelength limits are larger than min l .
For the reader's convenience to avoid jumping through the document, table 3 repeats the information on the minimum wavelengths min l being manufactured as given in table 2.
If the depth of the grooves is 1 μm, then the form can be generated well for sufficiently wide grooves.The wavelength has to be larger than 6 μm for a depth of h = 50 nm and λ > 60 μm for a groove depth of h = 1 μm to maintain a form deviation below rms = 5 nm.Structure grooves of larger depths show significant form deviations of above rms = 5 nm for any of the wavelengths.
The small asperities that fill the bottom of the grooves typically deviate in height from 10 to 40 nm.In the case of very slim trenches, they may reach up to 100 nm.These lead to the limitation of manufacturable feature sizes listed in table 2. All waves, except for those with a height of 10 μm, fall within the acceptable range of form deviation of rms = 20 nm.For example, in case of h = 1 μm the smallest wavelength fulfilling that requirement is λ limit = 15 μm > 11.4 m min l m = .The two subplots of figure 7 illustrate the performance difference of the diamond turning machine tool for wide trenches versus slim trenches using the example of structure C5 with a height of h = 0.7 μm.Subplot A shows the residuals of a negative cosine of nominal wavelength λ design = 42.8 μm with a roughness that complies with that at the plateau regions.The position fit (equation ( 5)) of the designed cosine has a root mean square of the residuals of rms design = 3.2 nm which is consistent with R q .The fit of the wavelength and the amplitude given the previously fitted position (equation ( 6)) has a root mean square of the residuals of rms fit = 3.1 nm with ˆ42.895 m fit l m = and ĥ 700.5 nm fit = . .The residuals indicate a systematic signature that reveals shape distortion, resulting in large values of the root mean square of rms design = 27.8 nm and rms fit = 23.0 nm.There are also structures in which the systematic deviations at the bottom of narrow grooves rise up to 100 nm.
The asperities on the bottom of the groove may have been caused by insufficient space for the chips, the material being cut off, to exit the workpiece's surface, as depicted in figure 4. Furthermore, tool wear may result in form deviation features, especially at the structures located at the inner radii of the disc.Several research groups have extensively studied the diamond turning machining process.For instance, the Chinese groups at Harbin Institute of Technology and Qinhuangdao University collaborated with the Irish University College Dublin [36].The experimental findings of Harbin Institute of Technology, together with two groups from Nanjing University, are supported by simulations, providing a deeper understanding of the process that explains the phenomena [37].The demands on   measurement standards, where the geometric shape is crucial, require a more detailed investigation of the effects that limit the manufacturing process.It has been observed that the sinusoidal shapes are distorted, resulting in slimmer ridges than trenches.The wavelengths of the positive cosine waves are systematically smaller by approximately 0.1...0.2 μm, those of the negative elements are larger respectively.For example, in figure 8, we can see wave sequence C7 with a height of h = 3 μm.Figure 9 shows the height deviations the distortions.
It is therefore assumed that the position of the contour on the tool used to estimate the tool nose radius did not comply with the contour representing the cutting edge that contacts the material for ablation.Figure 10 shows the waves of nominal wavelength λ = 11.4 μm of the 12th of 24 profiles of the sequence of h = 1 μm that lies in the center, at y = 50 μm.It is representative and illustrates well the two above-mentioned effects: 1. the formation of small asperities at the bottom of the grooves, 2. the underestimation of the tool nose radius and its cutting behavior when dilating the geometry to obtain the machining trajectory.
Confusion may arise if the measurement object used to characterise an instrument is not well characterised due to a lack of instrumentation capable of performing reference measurements.Mauch et al observed the small asperities at the bottom of the concave structure of a pseudo-chirp measurement standard, see figure 4 in [2].They interpreted this to be caused by the curved wave front of a Gaussian beam, see figure 7 in [2].However, it is unclear which component of the observed signal belongs intrinsically to the object rather than the optical instrument.
Surface metrology faces a significant challenge in characterising reference standards with sufficient accuracy for assessing topography measuring instruments.It is generally recommended that the instrumentation required to measure a reference standard should have a resolution approximately five times better than the instruments being assessed by the reference standard.PTB's metrological large range AFM, Met.LR-AFM [27][28][29]33], which is now in stable operation for routine calibration services, has made it possible to determine the shape of the artefacts.The shape of each structure on the measurement standard obtained by the Met.LR-AFM can then be used as reference geometry to be subtracted from measurement data acquired by optical instruments, whose response is to be characterized.The reference geometry helps to identify apparent features resulting from optical interaction.
The algorithms developed thus far to determine the small-scale fidelity limit use a model function that is a superposition of a cosine with an offset or a line.Either a given wavelength and amplitude are used to fit the position of the structure element, or all of the geometry parameters are fitted.However, when non-negligible asperities and distortion of the peaks are present, any deviations from the original sinusoidal shape must also be parameterised when calibrating the reference object.The optical measurement data can be corrected for non-ideal shape to obtain an unbiased estimate of the root mean square of the deviations, which can serve as a possible estimate of the small-scale fidelity limit.In future research, we plan to investigate the appropriateness of model functions, such as the use of smoothed cubic splines.

Conclusion
Sinusoidal structures with modulated wavelengths following a discrete chirp series are produced as a reference measurement standard to determine the topography fidelity of topography measuring instruments.The objects consist of sequences of waves with varying wavelengths.Each wave has a constant wavelength and is part of a sequence with different amplitudes, which is referred to as pseudo-chirp standard.The objects are manufactured with ultraprecision diamond face turning.
The manufacturing accuracies are limited to 10 nm for regions with slopes below 4°.Slim trenches exhibit asperities at their base with heights of around 20 up to 100 nm.The cutting tool's nose radius has been estimated with a systematic error, resulting in ridges that are approximately 150 nm too narrow and the trenches too wide respectively.
To determine the geometry of the structures, i.e. the cosines with the form deviations, of the pseudochirp standard PTB's large range metrology atomic force microscope, Met.LR-AFM, was used as reference instrument.
Future tasks involve developing a new approach to determine tool contours with the aim of handling form deviations on the nanometre scale.Additionally, there are plans to provide the international surface metrology community with an analysis procedure that determines the small-scale fidelity limit based on a geometry model of the pseudo-chirp geometry, including manufacturing artifacts caused by the limited precision of the cutting process.

Figure 2 .
Figure2.Face turning: generating profiles of deterministic surface structures in feed direction, which is the radial directions of the disc; for all radials the profile is identical within the accuracy of the chipping process.

Figure 3 .
Figure 3. Tool contour extracted from topography measured by a confocal microscope.

Figure 4 .
Figure 4. Chipped material must flow out when cutting trenches, i.e. concave structures.

Figure 5 .
Figure 5. Schematic drawing showing the cylindrical disc workpiece machined with a surface structure representing the measurement standard symbolically indicated as red concentric circles.The disc is cut into pieces and in this drawing one piece is depicted with the location of the measurement area indicated on its top.

Figure 5
Figure 5 symbolically depicts the concentric trenches and ridges as red circles and the measurement area on top of a single piece.The dashed line depicts the radial of the piece, with the central scan line positioned on it.

Figure 6 .
Figure 6.Diagram of raw data measured with the Metrological LR-AFM system: of and waves of amplitude.The bottom reference region is selected to determine the tilt angle for a subsequent rotation.The evaluation procedure uses the transition point x 0 between bottom plateau and largest positive cosine wave as origin and starting point to localize the positive cosine waves.

Figure 7 .
Figure7.Height deviations of the measurement data of negative cosine waves of C5, h design = 700 nm, with respect to the the designed cosine (residuals of fit equation equation (5)) depicted as gray dash-dotted curve and with respect to the fit of wavelength and amplitude (residuals of fit equation equation (6)) depicted as red dashed curve; top: Subplot A essentially showing the roughness comparable with that at the plateau regions for λ design = 42.8 μm with ˆ42.895 m fit l m = and ĥ 700.5 nm fit =; bottom: Subplot B showing a systematic signature of the residuals revealing the shape deviation for λ design = 9.4 μm with ˆ9.635 m fit l m = and ĥ 675.3 nm fit = .

Figure 8 .
Figure 8. Wavelength deviations Δλ as function of nominal wavelength revealing the effect of underestimating the tool nose radius and its cutting behavior when dilating the geometry to obtain the machining trajectory, example of sequence C7 with height h = 3 μm; top: Subplot A displays Δλ as function of the wavelength λ for the positive cosine waves, bottom: Subplot B displays Δλ for the negative cosine waves.

Figure 9 .
Figure 9. Height deviations Δh as function of nominal height, example of sequence C7 with height h = 3 μm; top: Subplot A displays Δλ as function of the wavelength λ for the positive cosine waves, bottom: Subplot B displays Δh for the negative cosine waves.

Figure 10 .
Figure 10.Measured profile together with fitted curves of a positive and a negative wave of nominal wavelength λ = 11.4 μm of the 12th of 24 profiles of the sequence C6 of h = 1 μm depicted as example to illustrate following effects: (1) the formation of small asperities at the bottom of concave structures, (2) the underestimation of the tool radius and its cutting behavior when dilating the geometry to obtain the machining trajectory.

Table 1 .
Wavelength values λ i /μm with i = 1,K,38 in accordance with a discrete chirp series of the sequences of cosine waves of a measurement standard for the characterization of the topography fidelity of optical topography measuring instruements.

Table 2 .
Overview of the structure heights with the wavelength of their smallest features in comparison with the size of

Table 3 .
Limitation of form accuracies depending on height and wavelength of the negative cosine waves representing grooves.