CIRP Annals - Manufacturing Technology 57 (2008) 509–512

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CIRP Annals - Manufacturing Technology

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Uncertainty evaluation for measurements of peak-to-valley surface form errors

Chris J. Evans (1)

Zygo Corporation, Laurel Brook Road, Middlefield, CT, USA

A R T I C L E I N F O

Keywords:

Metrology

Uncertainty

Spatially varying errors

A B S T R A C T

The peak-to-valley deviation of a surface from its specified form is a widely used, conceptually simple

surface specification. Evaluating the uncertainty in PV is not so simple. In the presence of noise, the

measured PV is always too large. This paper shows an approach to correcting that bias. A number of error

sources in interferometric testing of surfaces result in errors that have spatial variations across the

aperture under test. Uncertainty in the correction of these errors, for example, leads to uncertainties that

vary with position. This paper offers a procedure for evaluating the uncertainty in PV in this case.

� 2008 CIRP.

1. Introduction

Increasingly large arrays of data form the foundation forevaluation of geometric form and, in particular, finish (reviewed,for example, in [1]). A plethora of parameters derived from thesedata are defined in the relevant standards, along with importantissues such as filtering, sampling, etc. Expression of uncertaintieshas received less attention, particularly in the case of opticalmeasurements of surface form. This paper will focus on suchinterferometric measurements, although the underlying ideas mayapply more broadly.

The underlying data for form and finish measurements arearrays of length (height) measurements. The most straightforwardapproach to the estimation of uncertainty, therefore, is to generatean array of the same size [2] containing the uncertainties in eachheight. This allows the combination of Type A and Type Buncertainties [3]; for example, combining experimental measure-ments of noise with an estimate of the effect of imperfectalignment which is correlated spatially over the test aperture.When all sources are considered, the result is a matrix of combinedstandard uncertainties.

The uncertainty matrix approach is intellectually satisfying buthas some drawbacks. First, the information density is too high forsome users (who often want a single number). This objection canbe handled by various analyses or parametric characterizations ofthe uncertainty matrix (mean, full width at half maximum, etc.).The second objection to the uncertainty matrix is that it cannot beused easily for decisions of acceptance/rejection [4]; surfaces arenot specified in this form. The relationship between the matrix andthe uncertainty in commonly used parametric descriptions ofsurfaces is not always straightforward.

This paper considers uncertainty in one of the simplestmeasurands, PV, the peak-to-valley deviation of the surface fromits specified form. It is widely recognized that PV has a number ofdrawbacks for characterizing surfaces. Despite that, it remains oneof the most commonly used specifications of optical surfaces [5]and even for some surfaces used in state of the art lithography (e.g.photomask blanks [6]).

0007-8506/$ – see front matter � 2008 CIRP.

doi:10.1016/j.cirp.2008.03.084

2. Bias resulting from measurement noise

In the presence of noise, form parameters are biased;specifically the expectation value is systematically larger thanthe ‘‘true’’ value (Haitjema [7]). This bias must be evaluated andeither the estimate of the measurand compensated (as assumed in[2]) or the uncertainty statement should be adjusted according themethod proposed by Phillips et al. [8].

Evans [9] averaged different sized samples from a population ofrepeated measurements and extrapolated to find the underlyingPV (and rms) of the surface. Davies and Levenson [10] used asimilar population of measurements to estimate analytically boththe bias and the uncertainty in its correction. Both theseapproaches require a large population of experimental data. Mostrecently Haitjema and Morel used Fourier filtering [11] to extract a‘‘corrected’’ profile, which depends on the criterion used todifferentiate noise from the ‘‘true’’ surface. It should be possibleto estimate the bias in PV from a fundamental perspective.

In a modern interferometer, there are two major sources thatresult in both spatially and temporally varying noise: the detector(electron well depth, dark current, amplifier noise, etc.) andturbulence (refractive index variation). Consider the hypotheticalcase of measuring a perfect surface where the noise at each pixel issampled from the same Gaussian distribution. Clearly the expectedvalue of the bias (and the uncertainty in that estimate of the bias)depends only on the sample size and the standard deviation of thedistribution (Fig. 1).

The distributions of Fig. 1, and the increase in the most probablevalue with log of the number of pixels, can be predicted fromExtreme Value Theory (EVT). Perfect surfaces are rarely measured.For imperfect surfaces, the bias in PV will depend on theprobability that the tails of the distribution are sampled close tothe peaks/valleys of the surface. Rather than attempting toconvolve EVT with statistical descriptions of the surface, we useMonte Carlo methods to demonstrate the underlying behaviors.

Simulations where Gaussian noise is added to 1000 � 1000pure sinusoidal surfaces show the sensitivity of the bias to signal-to-noise ratio (Fig. 2). Similar simulations show the bias is largely

Fig. 1. PV bias for unit standard deviation noise for some common CCD array sizes

(simulation).

Fig. 2. PV bias as a function of signal to noise for a simulated surface with 5 cycles/

aperture.

Fig. 3. Simulated bias in PV as a function of kurtosis for fixed skew = 0, 105 samples.

Fig. 4. Mean bias in PV as a function of skew and kurtosis, 105 samples.

C.J. Evans / CIRP Annals - Manufacturing Technology 57 (2008) 509–512510

insensitive to frequency, as expected. The area within some smallheight of a peak (or valley) decreases with increasing frequency,but the number of peaks (or valleys) increases. In principle aFourier description of the surface plus knowledge of the noisestandard deviation could lead to an evaluation of the bias and itsuncertainty.

2.1. Bias prediction from surface statistics

A more straightforward approach, however, comes fromconsidering the higher moments of the surface statistics (skewand kurtosis). Consider again the perfect surface—except with asingle large peak (i.e. a surface with high skewness). Theprobability of that single peak coinciding with peak noise isnegligible, while the remainder of the surface has a very highprobability of sampling the low tail of the noise distribution. Thissuggests that the bias in measured PV will depend in part on theabsolute value of the skew of the surface.

Continuing the gedanken experiment, consider a surfacecomprising a small number of large peaks and valleys (i.e. largekurtosis). The probability of the high tail of the noise distributioncoinciding with a rare peak is low. The same argument applies tothe low tail of the noise coinciding with one of the rare valleys.Hence, a surface with large kurtosis will have a lower bias in itsmeasured PV than one where the surface is mostly at the extremevalues.

This hypothesis that the bias in PV is a function of the surfaceskew and kurtosis can easily be tested using Pearson’s system ofdistributions (calculated using MatLab) to generate height arrays

with known moments. Distributions with fixed PV and variableskew and kurtosis were generated and the effect of Gaussian noiseon the PV was evaluated (e.g. Fig. 3). Note that at high values ofkurtosis, a small number of points are driven to the tails of thedistribution and, for finite sample size, the resulting kurtosis candeviate from the input value. Nevertheless, the simulation clearlyshows that the bias decreases with increasing kurtosis, aspredicted, and that the scatter increases.

Fig. 4 shows the mean bias for a range of different values ofskewness derived from a number of simulations such as thatshown in Fig. 3. The scatter at high kurtosis and low skewness givesincreased uncertainty in the estimate of the bias (for a fixednumber of trials).

2.2. Experimental

By definition it is not possible to evaluate the bias in themeasured PV. However, the variation in the measurement can becompared with predictions.

Fig. 5 is one of a series of 20 sequential measurements from anf/0.75 spherical interferometric test set-up in a well-controlledlaboratory environment. Using Pearson’s method for this samplesize (552,000 pts vs. the 100,000 pts used for the simulationssummarized in Fig. 4) and the measured skew (0.3) and kurtosis(2.7), we estimate a bias of 2.98snoise with a standard deviation ofPV values given by 0.89snoise.

The noise in the experimental data can be evaluated bysubtracting the average of all 20 measurements from theindividual data sets and evaluating the statistics of the residual.For each evaluation we see, in this specific case, slowly movingturbulence plus electronic noise. The turbulence is well fit by a 37

Fig. 5. Single measurement of a spherical cavity. Skew = 0.3, kurtosis = 2.7.

Fig. 7. Standard deviation, nm, as a function of position in the aperture

(mean = 0.49 nm).

Fig. 6. Residual noise after removing low spatial frequency (turbulence) terms.

0.34 nm S.D.

C.J. Evans / CIRP Annals - Manufacturing Technology 57 (2008) 509–512 511

term Zernike fit, leaving a difference map (Fig. 6) with adistribution that is a very good fit to a Gaussian with a standarddeviation of 0.34 nm. Even without removing the apparentturbulence, the noise distribution still appears to be well fit by aGaussian with a standard deviation of 0.6 nm. This suggests thatthe PV of the average of the 20 measurements will be biased by(0.6 � 2.98) = 1.78 nm. The predicted standard deviation of PVvalues is (0.89 � 0.6) = 0.53 nm. The experimentally measuredstandard deviation of the PV for the 20 tests is 0.9 nm.

The model (Section 2.1) using a Gaussian noise distributionwith a standard deviation of 0.6 nm underestimated the experi-mental range of PV values. A better (but computationally intensivefor large data sets) evaluation of the noise is to compute thestandard deviation of the measurements at each pixel (Fig. 7). Thisshows a factor of 4–5 difference in noise between pixels near thecenter of the aperture and those around the edge. The high noisepixels, for this specific case, fall close to the ‘‘valley’’ in the part (atthe edge). Hence the variation in PV values and the bias will both belarger than estimated from the model.

Another potential source for the underestimation is the choiceof the distribution used in the model. Exponential or more ‘‘heavy-tailed’’ distributions will lead to larger bias (and larger experi-mental scatter) in PV. Tools are available for fitting the tails ofdistributions, but require large populations of data.

Practically, the bias always increases PV and specifications arealways of the form PV less than some value. Hence use of aGaussian distribution which may underestimate the bias (andhence the bias correction applied) is conservative from theconsumer’s perspective in terms of part conformance testing.Underestimating the bias will not increase the probability of avendor shipping a part that is out of tolerance.

3. Spatial variation in uncertainty

Noise is not the only source of measurement uncertainty thatleads to uncertainties that have systematic spatial variations.Consider, for example, a case where a Fizeau interferometer andtransmission sphere is used to measure a series of spherical partsurfaces using an industry standard manual 5-axis mount. For anygiven part in this series, the operator may align the test part to theinterferometer leaving residual tilt of say �1/2 fringe visible in the

aperture and with a similar spacing error along the optical axis (i.e.a curved, quadratic fringe). The interferometer software willsubtract the best-fit tilt and power. The systematic errors arisingfrom imperfect alignment may be compensated, leaving anuncertainty term due to the uncertainty in the compensation.More commonly, no compensation is made.

For simplicity consider only the misalignment along the opticalaxis. The set-up error may be considered as a rectangulardistribution with limits at � fringe of power (i.e. a coefficient of�79 nm in the 2r2 � 1 term in a Zernike expansion). Using publishedsensitivity values [8] for a specific f/1.1 system, this results inspurious third order spherical aberration (Zernike coefficient) of�2 nm, with a higher order contribution an order of magnitudesmaller. Since the distribution is assumed to be rectangular, theuncertainty in the Zernike spherical aberration coefficient is 2/H3 nm.Hence the matrix of uncertainties arising from this source is theabsolute value of the height at each point in the aperture for theZernike third order spherical aberration with a coefficient of 2/H3 nm(Figs. 8 and 9).

Figs. 8 and 9 show the significant spatial variation inuncertainty that can arise. A number of other time invariantsources of uncertainty may be treated in this manner—and severalcan be ignored. Uncertainty in the realization of the unit, forexample, is always insignificant in interferometric testing of opticsin a null configuration. Detector non-linearities, source intensityfluctuation, and pzt non-linearity (in a mechanical phase shifter ornon-linearity in wavelength change in a wavelength shiftingsystem) all produce errors that are proportional to the relativephase (height difference) of points on the surface and that dependon the starting phase in the evaluation. Practically, such errors areusually treated as an uncorrected bias since the initial phase isunknown and the sensitivity can be evaluated experimentallyquite easily. The resulting uncertainty can be added (in quad-rature) to the uncertainty maps of Figs. 7 and 8 and other similarmatrices for other sources.

4. Uncertainty in PV

Our purpose here is not to discuss in detail uncertainty ininterferometric measurement of surfaces. Rather, we haveattempted to show so far that PV is biased in a predictable wayand that an uncertainty matrix can be constructed that provides amap showing the spatial variation of uncertainties over the surfaceof the part under test. How does this relate to uncertainty in PV?

One approach is to simply consider the uncertainty in PV astwice the maximum of the uncertainty matrix. This clearlyoverestimates the uncertainty except when two equal maximain the uncertainty matrix coincide with the peak and the valley of

Fig. 9. Radial profile through the data of Fig. 8.

Fig. 8. Uncertainties (nm) arising from positioning error.

C.J. Evans / CIRP Annals - Manufacturing Technology 57 (2008) 509–512512

the surface. Consider Figs. 5 and 8. The peak in data of Fig. 5 fallsvery close to the outer minimum zone of Fig. 8. When uncertaintiesthat vary with position are considered, different points on thesurface may become the peak and the valley.

The obvious approach to evaluating the uncertainty in PV (orany other parameter) is through a Monte Carlo simulation.Assuming that all significant error sources are represented, thenthe simulation is straightforward. A number of estimates of thesurface shape are made combining the best estimate of the surfacewith errors at each pixel computed for each error source andsummed. The PV can be computed for each trial and thedistribution of results provides the estimate of the uncertaintyin PV. The noise can be explicitly included, but the bias still needsto be evaluated and subtracted from the best estimate of the PV.The disadvantage of the Monte Carlo method is that it iscomputationally intensive and must be repeated for everymeasurement made. This requirement is burdensome whenroutine measurements of similar objects are made on the sameinstrument under the same conditions, the only difference beingthe departure of the test surface from nominal.

Another approach is to use the uncertainty matrix and theestimate of the part surface directly to evaluate separately the

uncertainties in the peak and the valley, and then combine them.Specifically, adding and subtracting the uncertainty matrix to thedata and finding the new peaks provide the range of values (uP)within which the peak might reasonably be expect to lie. The rangeof values (uV) within which the valley might reasonably beexpected to lie can be found in exactly the same manner, and thecombined standard uncertainty in the PV is given by their sum inquadrature ðuPV ¼ ðu2

P þ u2VÞ

1=2Þ. This approach assumes that the PVis large compared to the uncertainty and that the plane removed(for flats) is unaffected. The latter assumption is reasonable forleast squares fit planes, but may not be for parameters (e.g. [6])where the plane removed minimizes the PV. In such cases MonteCarlo methods should be used.

5. Concluding remarks

PV is a widely used characterization of surface form, but onewhich is biased in the presence of noise. The bias depends onsurface skew and kurtosis and its expectation can be estimatedusing Pearson’s method.

Typical Type B uncertainties (for interferometric tests ofsurfaces) are spatially correlated. Using the maximum of thematrix of combined standard uncertainties for each height in thedata set to estimate the uncertainty in PV leads to overestimation.Monte Carlo simulations convolving the uncertainty matrix withthe shape of the surface under test provide robust uncertaintyestimates, but may be burdensome in some cases. A simplifiedapproach was proposed in this paper.

In general, better use will be made of the matrix of data wheremore than 2 points are used to characterize the surface. Measure-ment differences between vendors and their customers will beminimized by prior agreement to more robust parameters, such asrms or the newly proposed robust amplitude parameter, PVr [12].

Acknowledgements

W.T. Estler first suggested (to C.E.) the use of an uncertaintymatrix; T.L. Schmitz and A. Davies contributed to its first publisheduse [2]. Numerous colleagues at Zygo have made helpfulcomments, in particular V. Badami.

References

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