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Spectral analysis of the finish of polished optical surfaces
Wear, 83 (1982)
189
189 - 201
SPECTRAL ANALYSIS OF THE FINISH OF POLISHED OPTICAL SURFACES*
E. L. CHURCH U.S. Army
Armament
Research
and Development
Command,
Dover,
NJ07801
(U.S.A.)
H. C. BERRY Applied (U.S.A.)
Mathematics
Department,
Brookhaven
National
Laboratory,
Upton,
NY 11973
(Received May 27,1982)
Summary Surface finish is a critical factor in many high performance optical components with major limitations arising from scattering due to topographic surface roughness. Scattering theory shows that the relevant statistic of the residual surface height fluctuations is their power spectral density. In this paper, methods of estimating such spectra from surface profile measurements are described and are illustrated by the analysis of mechanical stylus measurements of several polished surfaces which have nanometer height fluctuations over the surface wavelength range of about 10’ - lo3 pm.
1. Introduction Optical surfaces form an important class of engineering surfaces. There has been an increased interest in the metrology of such surfaces in recent years due to new and more stringent requirements in the fields of lasers, X-ray astronomy and microscopy, and synchrotron radiation. At the present state of the art the principal effect of imperfect surface finish is scattering due to the residual topographic height fluctuations. Theory shows that the statistic of these fluctuations that determines the scattering is their twodimensional power spectral density as a function of surface frequency [l]. In fact, when the magnitude of the surface height projected along the direction of propagation is much less than the radiation wavelength (a version of the Rayleigh smoothness criterion), the scattered intensity is a simple mapping of this power spectrum. Scattering from rougher surfaces depends on higher order as well as the second-order correla*Paper presented at the Second International Conference on Metrology and Properties of Engineering Surfaces, Leicester Polytechnic, Leicester, Gt. Britain, April 14 - 16, 1982. 0043-1648/82/0000-0000/$02.75
0 Elsevier Sequoia/Printed
in The Netherlands
190
tions, although even in this case the scattering can be described in terms of the power spectrum provided that the height fluctuations are described as a gaussian process. At the First International Conference on Metrology and Properties of Engineering Surfaces, several papers were presented on the determination of surface finish by light-scattering techniques [l - 31. In this paper the measurement or, more precisely, the estimation of the power spectral densities of surface roughness from digitized profile data is discussed and these techniques are illustrated by the analysis of mechanical stylus measurements of several polished optical quality surfaces. Although these analysis techniques are discussed in the context of mechanical measurements of polished optical surfaces, they are also directly applicable to optical stylus measurements, to rougher surfaces and to surfaces generated by other finishing processes such as diamond turning.
2. Power spectral densities The power spectral density of a linear surface profile Z(X) is defined as 2
+L
(& I/
K(f) = ,1;m_
-L
dx
exp(-i2lrfX)Z(3t)
I>
(1)
where f is the surface frequency, 2L is the record length and ( >denotes the ensemble average. The subscript 1 denotes this as the one-dimensional or profile spectrum as distinct from the twodimensional or area spectrum that enters into the discussion of surface scattering:
where f, and fY are the rectangular components of the surface frequency, A is the surface area and Z(x,y) is the surface height in the X-Y plane. The relationship between these two spectral densities is discussed in Section 8; at this point we concentrate on the profile spectrum. These definitions are statistical abstractions. In reality we must deal with finite rather than infinite record lengths, sampled rather than continuous data and a few members of the ensemble rather than an infinite set. The problem of estimating the power spectrum from real data is a highly developed field of mathematical and engineering statistics and is treated in many excellent books [4, 51 and reviews [6,7]. In this paper we emphasize the classic technique of periodogram analysis and mention one additional technique (maximum entropy autoregressive analysis) in passing. Figure 1 is a flow chart of the routine we have set up at the Brookhaven National Laboratory under the aegis of the National Synchrotron Light Source (NSLS) for the evaluation of optical surfaces using stylus measurements.
191 Taiystep
Measurement
I
Pilter
1 Sample, Quantize I ReCOrd ---.--w-m-____
-mm--I
----_-
Red
View /
r
I I
Edit
Detrend I
I
Select
Order
Autoregressive
I
Fit
P'(f)
Fig. 1. Flow chart for the spectral estimation routine discussed in the text.
In the measurements to be described the stylus data were obtained from Talystep measurements made at the National Bureau of Standards in Gaithersburg, MD. In those measurements the analogue signal generated by the stylus transducer was filtered, sampled and quantized, and the digitized profile data were recorded on magnetic tape for subsequent analysis. Details of the process are given in an earlier publication in collaboration with Vorburger and coworkers [8] . The subsequent analysis consisted of reading the tape, viewing and editing the profile records and detrending the data by removing a least-
192
squares quadratic profile. The detrended data were then fed into two special spectral estimation routines: the left branch using the fast Fourier transform to generate periodogram-based estimates and the right branch using the Burg maximum entropy routine to generate the spectrum based on an autoregressive model. The output of the routine shown consists of four spectral estimates as a function of surface frequency: the periodogram P’, the smoothed periodogram S, the average periodogram P and the maximum entropy estimate E.
3. Limitations There are four major limitations involved in spectral estimation from real data: bandwidth limits, aliasing, trending and statistical instability. We review these briefly below. Real data are taken over a finite record length 2L and are sampled at a finite interval AX = D. As a result reliable spectral estimates can only be made over a limited range of surface frequencies or wavelengths, namely 2L > ;
> 20
The left-hand limit says that the maximum wavelength is limited by the record length and the right-hand limit that the minimum wavelength is limited by twice the sampling interval. Surface wavelengths shorter or longer than these limits are not lost in the measurement process, however, but reappear as false spectral components within the interval in eqn. (3) through the phenomena of aliasing and trending. To avoid aliasing effects, surface frequencies greater than the Nyquist frequency f = l/20 must be reduced to a negligible level by filtering the analogue signal before sampling. In practice this can be done by the introduction of an electronic filter, by the mechanical-electrical filtering action of the stylus transducer, by the smoothing effect of the stylus “tip” or, more unreliably, by the natural high frequency fall-off observed in the spectra of many surfaces. This last possibility is to be avoided if there is interest in the derivatives of the profile or, equivalently, in the higher moments of the power spectral density, as discussed below. Data trends can be real in the sense that they are due to the presence of surface components with wavelengths longer than the record length (as in time series) or unreal in that they are artifacts of the measurement process. For example, the stylus of the Talystep is scanned in a circular arc over the surface being measured, which is mounted on an adjustable tilt stage. Within limits, adjustment or maladjustment of the stage can remove or introduce an arbitrary quadratic trend into the measured data. In practice this feature is used to remove real trends which might otherwise cause the profile signal to exceed the operating range of the instrument. However, it can introduce a spurious quadratic trend. For this reason we detrend the data by removing
193
a least-squares quadratic polynomial from each sampled record before further analysis. Antialiasing and detrending are imperfect processes and there is distortion of the spectral estimates near the upper and lower limits given by eqn. (3). The practical limits (which can be quantified) are generally taken to lie within those limits by moderate factors; factors of 3 in the examples discussed below. The final limitation of real data is their statistical instability. Spectral estimates, and finish parameters derived from them, are subject to finite statistical fluctuations which reflect the random nature of the surface roughness being measured. A measure of the degree of these fluctuations is their coefficient y of variation: 2 l/2 root-mean-square fluctuation = (4) -Y= mean ( k1 where k is the effective number of degrees of freedom included in the estimate. The weak dependence of y on k forewarns us that considerable effort may be required to produce an estimate with a small coefficient of variation. The amount of effort is to be justified by the application for which the estimates are made.
4. Periodogram
spectral estimates
The periodogram of a profile is the square magnitude of its Fourier coefficients as a function of surface frequency. If the profile contains no frequency higher than the Nyquist frequency, its spectrum is uniquely determined by its sampled values and the periodogram P’(f) can be expressed directly in terms of those values: i2nmn exp(---_
(5)
where D is the sampling interval, N is the total number of sampled points (2L = ND) and Z(nD) is the sampled profile height. The quantity F(m) = 1 except for F(N/2) = i, 6 is a delta function and f, is the discrete surface frequency :
(6) The expression within the vertical rules in eqn. (5) is the discrete Fourier transform of Z(nD) which is usually evaluated using the fast Fourier transform algorithm, the most common form of which requires N to be a power of 2. For this reason eqn. (5) is given in the form appropriate for even N. The periodogram is a poor estimator of the power spectral density of the ensemble since it represents the Fourier amplitudes of a particular
194
realization and therefore fluctuates wildly from frequency to frequency and from realization to realization. In short, it has very poor statistical stability. Each point in the periodogram corresponds to two degrees of freedom (real and imaginary parts) and exhibits a coefficient of variation of about unity. In light scattering these fluctuations appear as “speckle”. There are two classical ways of stabilizing the periodogram: windowing and averaging. Windowing involves the convolution of the periodogram with a window function G(f) to generate a smoothed spectral estimate S(f): (7) of statisThe second method involves the averaging of the periodograms tically equivalent profiles to generate the average periodogram P(f): (9) c C&-m) ” where M is the total number of profiles included in the average. The coefficient of variation of these stabilized estimates is given by eqn. (4) where k is determined by the effective window width in the first case (for a rectangular window, h is twice the number of included points) and Iz = 2M in the second case, for uncorrelated profiles. These estimates are frequently taken to be x2 distributed with 2k degrees of freedom, which permits a simple discussion of their confidence limits [ 91.
P(f) = f
5. Autoregressive
spectral estimates
The second class of spectral estimators considered here is based on an autoregressive model of the discrete random process Z(nD). The problem is to estimate the real autoregressive coefficients a, in the stochastic difference equation i
a,Z{(n
- v)D} =
(9)
f-2
where t is the order of the autoregressive process and e is a unit variance white noise process. The power spectral density of the autoregressive process of order t is then -2
exp(-i2rfV)
(19) V One popular procedure which we have implemented uses the maximum entropy of Burg algorithm for determining the coefficients where the order is either preselected or determined within the routine using the Akaike information criterion [ 5,7] . Low order autoregressive spectral estimates appear as highly smoothed versions of the corresponding periodogram estimates. An advantage of such estimates is that they provide a model-based mechanism for extrapolating
E(f)
=
2
a,
195
the spectra outside the range of measurement given by eqn. (3). Such extrapolations are required for the derivation of the two-dimensional power spectrum from its one-dimensional form and other applications.
6. Finish parameters It is convenient to summarize the information about the surface roughness contained in the spectral density function in terms of a set of finish parameters that are directly related to properties of the profile. This can be done by making use of the fact that the power spectrum of the qth derivative of a profile equals the spectrum of its height multiplied by (27rf)q , plus the fact that the variance of a quantity equals the integral of its power spectrum. In this way the following estimators can be generated for the height variance u2, the slope variance ?n2 and the “curvature” variance c2:
c2 =
d22 ’
WC)
0-1)dx2 where the symbol P’ on the right-hand side is one of the spectral estimates of the profile. These expressions also apply to bandwidth-limited (filtered) spectra, in which case the quantities on the left-hand side are also bandwidth limited. Examples of this are given in Section 7. Estimates of these parameters are subject to statistical fluctuations. However, since they represent integrals (moments) of the corresponding spectral estimates they generally involve a large number of degrees of freedom and therefore tend to be relatively stable. However, exceptions may occur when the spectrum is dominated by a relatively narrow band of frequencies [9] .
7. Illustrations We illustrate the procedures described above by quoting results obtained for several highly polished optical surfaces. Figure 2 shows the profile of a platinumcoated Zerodur flat measured over a record length of 1500 pm (1.5 mm) with a sampling interval of 0.375 pm, corresponding to 4000 sampling points. The vertical scale is in &ngstrijms (1 A = 0.1 nm) and the horizontal scale is in microns.
196
Figure 3 is the averaged periodogram estimate of the spectrum of this surface taken over five profiles. The vertical scale is the power spectral density in cubic microns and the horizontal scale is the surface frequency in reciprocal microns (both on logarithmic scales). The estimates cover the frequency range from 1/2L = 6.7 X 1Om-4 pm-- 1 to l/20 = 1.3prn~-‘, although the usable range is taken to lie between about l/500 = 2 X 10 3 (urn- ’ and l/2 = 5 X 10-l pm- i. The significant feature to note in these figures is that, although this surface is relatively smooth at high frequencies, it is much rougher at low frequencies. Figure 4 is the profile of a gold-coated fused silica flat and Fig. 5 is its corresponding averaged periodogram spectral estimate, again taken over five profiles. In this case the high frequency part of the spectrum is more intense than that of the first surface but its low frequency part is much less intense. This is obvious in both the configuration-space and frequency-space illustrations. Obviously, the first surface would be better in an application where high frequency behavior (large-angle scattering) was more important, while
III i I’\\(‘t:
Fig. 2. Raw profile
I\
\I l(‘l3oll:‘ll:li-‘
of surface
1”
5
EK-111-2s.
Fig. 3. Averaged
periodogram
spectral
estimate
of the profile
of surface
EK-111-2s.
Fig. 5. Averaged
periodogram
spectral
estimate
of the profile
of surface
EK-1-3s.
197
the second surface would be much better suited to applications where low frequency behavior (small-angle scattering) was critical. Both of these surfaces are early test surfaces for the X-ray telescope to be included in the Advanced X-ray Astrophysical Facility and were provided by Dr. M. Zombeck of the Harvard-Smithsonian Center for Astrophysics. Further details concerning these surfaces are given elsewhere [lo] . Figure 6 is the profile of a silicon carbide surface, chemically vapor deposited onto a sintered silicon carbide substrate, which was polished at the National Physical Laboratory under the direction of Dr. A. Franks. (The sample was provided by M. R. Howells and P. Z. Takacs, NSLS, Brookhaven National Laboratory .) Figure 7 is its averaged periodogram. This surface is seen to be remarkably smooth over the entire range of measurement, significantly smoother than either of the preceding surfaces. The high frequency part of the spectrum is, in fact, comparable with the background noise which is approximately constant (white) with a value of about 5 X lo-* E.tm3. The tufts at f = 0.6 - 1.0 pm-’ fall outside the usable range of measurement and are believed to be due to vibration and pick-up rather than to surface features. Table 1 summarizes the values of finish parameters derived from the averaged periodogram estimates of the surfaces shown plus the instrumental background for a stationary stylus, which have been subtracted (as the variances) from the preceding surface profile data. Values of the root-mean-square profile roughness u, slope m and curvature c are given for various ranges of surface wavelength. Table 1, third column, includes wavelengths between 2 and 12 pm (corresponding to the high frequency behavior of the surfaces), the fourth column is for 12 - 512 pm (corresponding tc low frequencies), the fifth column is the total range from 2 to 512 pm and the last column is for the full range from the Nyquist wavelength of 0.75 pm to the record length of 1500 E.tm.The significance of this last column is limited by the fact that it includes parts of the spectral estimate which involve extraneous features and are distorted by aliasing, filtering and detrending effects.
Fig. 6. Raw profile of surface Sic 4. Fig. 7. Averaged periodogram spectral estimate of the profile of surface Sic 4.
198 TABLE
1
Root-mean-square values of the profile heights CJ, slopes m and curvatures c determined by integrating the averaged periodogram spectral estimates in Figs. 3, 5 and 7 over the indicated ranges of surface wavelengths Surface
Parameter
Value of parameter for following ranges of surface wavelength 12.512j.lm
2-12pm
2-512pm
0.75-1500pm
3.13
16.6
16.9
17.6
0.487
0.133
0.505
1.06 x 1O-3
3.84
1.06 x 1O-3
1.07 5.75
x
)
6.33 1.15 2.67 x 1O-3
5.35 0.128 4.93 x lo-’
8.29 1.15 2.67 x 1O-3
9.52 2.66 1.47
x lo-*
0 (A) m (mrad) c (pm-’ )
2.02 0.336 7.45 x 10-4
2.09 0.0416 1.64 x lo-’
2.91 0.338 7.45 x 1O-4
3.46 0.836 4.37 x 10-j
u (A) m (mrad) c (Mm-‘)
1.45 0.278 6.68
0.676 0.0205 8.29 x 1O-6
1.60 0.279 6.68 x 1O-4
2.21 0.903 5.54 x 1O-3
EK-111-2s Pt on Zerodur (Figs. 2, 3)
u (A) m (mrad) c (pm-’ )
EK-1-3s Au on SiOr (Figs. 4, 5)
u (A) c (Ctm-’
Sic 4 Chemical vapor deposition (Figs. 6, 7) Stationary stylus (no illustration)
m (mrad)
x
10m4
x
1O-5
1O-3
These three surfaces are similar in that they have root-mean-square profile roughnesses of the order of a nanometer, slopes of the order of 10e3 rad and radii of curvature of the order of lo2 I.tm over the range of surface wavelengths of about 10’ - lo3 pm. However, they are dissimilar in that their spectral shapes are distinctly different. The fact that they are different shows that there is no universal spectral shape and that measurements in a narrow spectral region do not necessarily determine the behavior in other regions.
8. Relationship
between
one-dimensional
The general relationship sional power spectral densities w,V,
between is
and twodimensional the one-dimensional
spectra and twodimen-
1= j-
dfy w,VJ,l (12) -m If IV2 is given, it is a simple matter to calculate W,, i.e. to deduce the onedimensional spectrum from the twodimensional spectrum. However, the inverse is not possible in general, although there are two particular cases for which it is possible: an extreme anisotropic surface and an isotropic surface. An extreme anisotropic surface lying parallel to the y axis has ~2(fx*f,)
=
~,(fxW,)
which has a trivial inverse.
(13)
199 An isotropic surface is one for which the two-dimensional spectrum is a function only of f where f 2 = f,* + fY2. In that case eqn. (12) can be rewritten as
J
W,(f) = 2 f
f’ df’ (f’Z_f2)1/2
(14)
Wf’)
which is a form of an Abel or half-integral transform this is the inverse Abel or half-derivative expression W*(f)
df’
= -11 n
f
(f’*-f*)l’*
!Jdf’
K(f’)
[ 111. The inverse of
(15)
which may be written in a variety of mathematically equivalent forms. These expressions show that to determine W,%,(f) we must know Wls(f ‘) for all f’ > f. However, since any experimental estimate of the profile spectrum W1 is bandwidth limited, it is strictly impossible to determine W2. To accomplish this inversion, we must invoke independent information concerning the properties of the surface spectrum in addition to isotropy: some model-dependent form which will smooth and extrapolate the profile spectrum to high frequencies. One method is to use the autoregressive model discussed earlier; another is to use some mathematically convenient form for the covariance function or power spectrum. A popular family of functions of this type has covariance functions of the form r”KV(27rur) where r is the lag, u is a constant and K, is a modified Bessel function [ 121. The corresponding form of the ddimensional power spectrum is
(16) where d = 1, 2 and I’ is the gamma function. This family has the virtues of reducing to the firstorder autoregressive form for v = f and giving inverse power law spectra at high frequencies f $ a. It also provides a convenient means of comparing onedimensional and two-dimensional finish parameters and exploring their dependences on surface bandwidth limits. Finish parameters derived from the central moments of one-dimensional and twodimensional spectra of a given surface are generally different even when the integration is carried out over surface frequencies from 0 to 0~: the height variances of the profile and area are equal, the variance of the profile slope is half that of the surface gradient, the variance of the profile curvature is 3/8 that of the surface and so on [ 1 l] . These relationships are independent of the shapes of the power spectra. In contrast, the relationships between bandwidth-limited parameters generally depend on the spectral shapes and the bandwidth limits. An interesting exception occurs for inverse power law spectra; in this case the ratios of profile to area parameters depend only on the profile power law and are independent of the type of parameter (i.e. CJ, m or c) and the bandwith limits [ 8,121. These and related results are readily derived from eqn. (16).
200
Although we have not yet made an extensive study of the reeonstruction of the twodimensional spectra from the onedimensional profile measurements, pre~in~y analysis indicates that the results are in reasonable agreement with independent measurements of surface scatter [8] .
9. Conclusions We have described procedures for estimating the power spectral densities of surface topography from sampled profile data and have illustrated the method by analyzing mechanical stylus data for several highly polished optical surfaces. The issues of surface bandwidth limits, statistical stability, finish parameters and the determination of area spectra and parameters from profile me~uremen~ have been addressed. We conclude that stylus techniques offer a sensitive if indirect method of measuring the finish of polished optical surfaces, particularly at long surface wavelengths, which may be especially important for practical applications [8,13].
We thank Drs. M. R. Howells and P. Z. Takacs of the Brookhaven National Laboratory for encouragement and assistance with this work, Dr. T. V. Vorburger of the National Bureau of Standards for valuable discussions of stylus techniques and for making the profile measurements described and Dr. M. V. Zombeck of the He’d-Smithsoni~ Center for Astrophysics for permission to use his unpublished data. H. C. Berry acknowledges the support of the U.S. Department of Energy. Certain items of commercial equipment are identified in this paper in order to specify the experimental procedure. In no case does such an identification imply recommendation or endorsement by the U.S. Department of Defense or the U.S. Department of Energy.
Nomenclature a %
A :
D ; F G
k
correlation parameter autoregression coefficient surface area root-mean-square profile curvature spatial dimension profite sampling interval maximum entropy spectral estimate surface frequency weighting factor frequency window (--1)“2 effective number of degrees of freedom
201 modified Bessel function half-record length root-mean~quare profile slope number of profiles spatial index number of sample points per profile averaged periodogram spectral estimate periodogram degree of derivative smoothed periodogram spectral estimate autoregressive order power spectral density coordinates in surface plane residual surface height coefficient of variation gamma function delta function unit variance white noise process dummy index root-mean-square profile height lag parameter
References 1 E. L. Church, The measurement 2 3 4 5 6 7 8 9 10 11 12
13
of surface texture and topography by differential light scattering, Wear, 57 (1979) 93 - 105. E. G. Thwaite, The direct measurement of the power spectrum of rough surfaces by optical Fourier transformation, Wear, 57 (1979) 71 - 80. L. H. Tanner, A comparison between Talysurf 10 and optical measurements of roughness and surface slope, Wear, 57 (1979) 81 - 91. G. M. Jenkins and D. G. Watts, Spectral Analysis and its Applications, Holden Day, San Francisco, CA, 1968. M. B. Priestly, Spectral Analysis and Time Series, Vol. 1, Univariafe Series, Academic Press, New York, 1981. R. B. Blackman and J, W. Tukey, The measurement of Power Spectra from the Point of View of Communications Engineering, Dover Publications, New York, 1958. S. M. Kay and S. L. Marple, Jr., Spectrum analysis - a modern perspective, Proc. IEEE, 69 (1981) 1380 - 1419. E. L. Church, M. R. Howells and T. V. Vorburger, Spectral analysis of the finish of diamond-turned mirror surfaces, Proc. Sot. Photo-Opt. Znstrum. Eng., 315 (1982). E. L. Church, Satistical fluctuations of total integrated scatter measurements, J. Opt. Sot. Am., 71 (1981) 1602A. M. V. Zombeck, C. C. Wyman and M. C. Weisskopf, High-resolution X-ray scattering measurements for advanced X-ray astrophysical facility, Opt. Eng., 21 (1982) 63 - 72. E. L. Church, H. A. Jenkinson and J. M. Zavada, Relationship between surface scattering and microtopographic features, Opt. Eng., 18 (1979) 125 - 136. E. 1;. Church, The role of spatial bandwidth limits in the measurement and interpretation of second-order statistical properties, Proc. 26th Conf. on the Design of Experiments in Army Reseoreh, ~e~Q~oprnent and Testing, in AR0 Rep. 81-2. M. Stedman, The metrological evaluation of grazing-incidence mirrors, Proc. SOC. Photo-Opt. instrum. Eng., 316 (1982) 2 - 8.
189
189 - 201
SPECTRAL ANALYSIS OF THE FINISH OF POLISHED OPTICAL SURFACES*
E. L. CHURCH U.S. Army
Armament
Research
and Development
Command,
Dover,
NJ07801
(U.S.A.)
H. C. BERRY Applied (U.S.A.)
Mathematics
Department,
Brookhaven
National
Laboratory,
Upton,
NY 11973
(Received May 27,1982)
Summary Surface finish is a critical factor in many high performance optical components with major limitations arising from scattering due to topographic surface roughness. Scattering theory shows that the relevant statistic of the residual surface height fluctuations is their power spectral density. In this paper, methods of estimating such spectra from surface profile measurements are described and are illustrated by the analysis of mechanical stylus measurements of several polished surfaces which have nanometer height fluctuations over the surface wavelength range of about 10’ - lo3 pm.
1. Introduction Optical surfaces form an important class of engineering surfaces. There has been an increased interest in the metrology of such surfaces in recent years due to new and more stringent requirements in the fields of lasers, X-ray astronomy and microscopy, and synchrotron radiation. At the present state of the art the principal effect of imperfect surface finish is scattering due to the residual topographic height fluctuations. Theory shows that the statistic of these fluctuations that determines the scattering is their twodimensional power spectral density as a function of surface frequency [l]. In fact, when the magnitude of the surface height projected along the direction of propagation is much less than the radiation wavelength (a version of the Rayleigh smoothness criterion), the scattered intensity is a simple mapping of this power spectrum. Scattering from rougher surfaces depends on higher order as well as the second-order correla*Paper presented at the Second International Conference on Metrology and Properties of Engineering Surfaces, Leicester Polytechnic, Leicester, Gt. Britain, April 14 - 16, 1982. 0043-1648/82/0000-0000/$02.75
0 Elsevier Sequoia/Printed
in The Netherlands
190
tions, although even in this case the scattering can be described in terms of the power spectrum provided that the height fluctuations are described as a gaussian process. At the First International Conference on Metrology and Properties of Engineering Surfaces, several papers were presented on the determination of surface finish by light-scattering techniques [l - 31. In this paper the measurement or, more precisely, the estimation of the power spectral densities of surface roughness from digitized profile data is discussed and these techniques are illustrated by the analysis of mechanical stylus measurements of several polished optical quality surfaces. Although these analysis techniques are discussed in the context of mechanical measurements of polished optical surfaces, they are also directly applicable to optical stylus measurements, to rougher surfaces and to surfaces generated by other finishing processes such as diamond turning.
2. Power spectral densities The power spectral density of a linear surface profile Z(X) is defined as 2
+L
(& I/
K(f) = ,1;m_
-L
dx
exp(-i2lrfX)Z(3t)
I>
(1)
where f is the surface frequency, 2L is the record length and ( >denotes the ensemble average. The subscript 1 denotes this as the one-dimensional or profile spectrum as distinct from the twodimensional or area spectrum that enters into the discussion of surface scattering:
where f, and fY are the rectangular components of the surface frequency, A is the surface area and Z(x,y) is the surface height in the X-Y plane. The relationship between these two spectral densities is discussed in Section 8; at this point we concentrate on the profile spectrum. These definitions are statistical abstractions. In reality we must deal with finite rather than infinite record lengths, sampled rather than continuous data and a few members of the ensemble rather than an infinite set. The problem of estimating the power spectrum from real data is a highly developed field of mathematical and engineering statistics and is treated in many excellent books [4, 51 and reviews [6,7]. In this paper we emphasize the classic technique of periodogram analysis and mention one additional technique (maximum entropy autoregressive analysis) in passing. Figure 1 is a flow chart of the routine we have set up at the Brookhaven National Laboratory under the aegis of the National Synchrotron Light Source (NSLS) for the evaluation of optical surfaces using stylus measurements.
191 Taiystep
Measurement
I
Pilter
1 Sample, Quantize I ReCOrd ---.--w-m-____
-mm--I
----_-
Red
View /
r
I I
Edit
Detrend I
I
Select
Order
Autoregressive
I
Fit
P'(f)
Fig. 1. Flow chart for the spectral estimation routine discussed in the text.
In the measurements to be described the stylus data were obtained from Talystep measurements made at the National Bureau of Standards in Gaithersburg, MD. In those measurements the analogue signal generated by the stylus transducer was filtered, sampled and quantized, and the digitized profile data were recorded on magnetic tape for subsequent analysis. Details of the process are given in an earlier publication in collaboration with Vorburger and coworkers [8] . The subsequent analysis consisted of reading the tape, viewing and editing the profile records and detrending the data by removing a least-
192
squares quadratic profile. The detrended data were then fed into two special spectral estimation routines: the left branch using the fast Fourier transform to generate periodogram-based estimates and the right branch using the Burg maximum entropy routine to generate the spectrum based on an autoregressive model. The output of the routine shown consists of four spectral estimates as a function of surface frequency: the periodogram P’, the smoothed periodogram S, the average periodogram P and the maximum entropy estimate E.
3. Limitations There are four major limitations involved in spectral estimation from real data: bandwidth limits, aliasing, trending and statistical instability. We review these briefly below. Real data are taken over a finite record length 2L and are sampled at a finite interval AX = D. As a result reliable spectral estimates can only be made over a limited range of surface frequencies or wavelengths, namely 2L > ;
> 20
The left-hand limit says that the maximum wavelength is limited by the record length and the right-hand limit that the minimum wavelength is limited by twice the sampling interval. Surface wavelengths shorter or longer than these limits are not lost in the measurement process, however, but reappear as false spectral components within the interval in eqn. (3) through the phenomena of aliasing and trending. To avoid aliasing effects, surface frequencies greater than the Nyquist frequency f = l/20 must be reduced to a negligible level by filtering the analogue signal before sampling. In practice this can be done by the introduction of an electronic filter, by the mechanical-electrical filtering action of the stylus transducer, by the smoothing effect of the stylus “tip” or, more unreliably, by the natural high frequency fall-off observed in the spectra of many surfaces. This last possibility is to be avoided if there is interest in the derivatives of the profile or, equivalently, in the higher moments of the power spectral density, as discussed below. Data trends can be real in the sense that they are due to the presence of surface components with wavelengths longer than the record length (as in time series) or unreal in that they are artifacts of the measurement process. For example, the stylus of the Talystep is scanned in a circular arc over the surface being measured, which is mounted on an adjustable tilt stage. Within limits, adjustment or maladjustment of the stage can remove or introduce an arbitrary quadratic trend into the measured data. In practice this feature is used to remove real trends which might otherwise cause the profile signal to exceed the operating range of the instrument. However, it can introduce a spurious quadratic trend. For this reason we detrend the data by removing
193
a least-squares quadratic polynomial from each sampled record before further analysis. Antialiasing and detrending are imperfect processes and there is distortion of the spectral estimates near the upper and lower limits given by eqn. (3). The practical limits (which can be quantified) are generally taken to lie within those limits by moderate factors; factors of 3 in the examples discussed below. The final limitation of real data is their statistical instability. Spectral estimates, and finish parameters derived from them, are subject to finite statistical fluctuations which reflect the random nature of the surface roughness being measured. A measure of the degree of these fluctuations is their coefficient y of variation: 2 l/2 root-mean-square fluctuation = (4) -Y= mean ( k1 where k is the effective number of degrees of freedom included in the estimate. The weak dependence of y on k forewarns us that considerable effort may be required to produce an estimate with a small coefficient of variation. The amount of effort is to be justified by the application for which the estimates are made.
4. Periodogram
spectral estimates
The periodogram of a profile is the square magnitude of its Fourier coefficients as a function of surface frequency. If the profile contains no frequency higher than the Nyquist frequency, its spectrum is uniquely determined by its sampled values and the periodogram P’(f) can be expressed directly in terms of those values: i2nmn exp(---_
(5)
where D is the sampling interval, N is the total number of sampled points (2L = ND) and Z(nD) is the sampled profile height. The quantity F(m) = 1 except for F(N/2) = i, 6 is a delta function and f, is the discrete surface frequency :
(6) The expression within the vertical rules in eqn. (5) is the discrete Fourier transform of Z(nD) which is usually evaluated using the fast Fourier transform algorithm, the most common form of which requires N to be a power of 2. For this reason eqn. (5) is given in the form appropriate for even N. The periodogram is a poor estimator of the power spectral density of the ensemble since it represents the Fourier amplitudes of a particular
194
realization and therefore fluctuates wildly from frequency to frequency and from realization to realization. In short, it has very poor statistical stability. Each point in the periodogram corresponds to two degrees of freedom (real and imaginary parts) and exhibits a coefficient of variation of about unity. In light scattering these fluctuations appear as “speckle”. There are two classical ways of stabilizing the periodogram: windowing and averaging. Windowing involves the convolution of the periodogram with a window function G(f) to generate a smoothed spectral estimate S(f): (7) of statisThe second method involves the averaging of the periodograms tically equivalent profiles to generate the average periodogram P(f): (9) c C&-m) ” where M is the total number of profiles included in the average. The coefficient of variation of these stabilized estimates is given by eqn. (4) where k is determined by the effective window width in the first case (for a rectangular window, h is twice the number of included points) and Iz = 2M in the second case, for uncorrelated profiles. These estimates are frequently taken to be x2 distributed with 2k degrees of freedom, which permits a simple discussion of their confidence limits [ 91.
P(f) = f
5. Autoregressive
spectral estimates
The second class of spectral estimators considered here is based on an autoregressive model of the discrete random process Z(nD). The problem is to estimate the real autoregressive coefficients a, in the stochastic difference equation i
a,Z{(n
- v)D} =
(9)
f-2
where t is the order of the autoregressive process and e is a unit variance white noise process. The power spectral density of the autoregressive process of order t is then -2
exp(-i2rfV)
(19) V One popular procedure which we have implemented uses the maximum entropy of Burg algorithm for determining the coefficients where the order is either preselected or determined within the routine using the Akaike information criterion [ 5,7] . Low order autoregressive spectral estimates appear as highly smoothed versions of the corresponding periodogram estimates. An advantage of such estimates is that they provide a model-based mechanism for extrapolating
E(f)
=
2
a,
195
the spectra outside the range of measurement given by eqn. (3). Such extrapolations are required for the derivation of the two-dimensional power spectrum from its one-dimensional form and other applications.
6. Finish parameters It is convenient to summarize the information about the surface roughness contained in the spectral density function in terms of a set of finish parameters that are directly related to properties of the profile. This can be done by making use of the fact that the power spectrum of the qth derivative of a profile equals the spectrum of its height multiplied by (27rf)q , plus the fact that the variance of a quantity equals the integral of its power spectrum. In this way the following estimators can be generated for the height variance u2, the slope variance ?n2 and the “curvature” variance c2:
c2 =
d22 ’
WC)
0-1)dx2 where the symbol P’ on the right-hand side is one of the spectral estimates of the profile. These expressions also apply to bandwidth-limited (filtered) spectra, in which case the quantities on the left-hand side are also bandwidth limited. Examples of this are given in Section 7. Estimates of these parameters are subject to statistical fluctuations. However, since they represent integrals (moments) of the corresponding spectral estimates they generally involve a large number of degrees of freedom and therefore tend to be relatively stable. However, exceptions may occur when the spectrum is dominated by a relatively narrow band of frequencies [9] .
7. Illustrations We illustrate the procedures described above by quoting results obtained for several highly polished optical surfaces. Figure 2 shows the profile of a platinumcoated Zerodur flat measured over a record length of 1500 pm (1.5 mm) with a sampling interval of 0.375 pm, corresponding to 4000 sampling points. The vertical scale is in &ngstrijms (1 A = 0.1 nm) and the horizontal scale is in microns.
196
Figure 3 is the averaged periodogram estimate of the spectrum of this surface taken over five profiles. The vertical scale is the power spectral density in cubic microns and the horizontal scale is the surface frequency in reciprocal microns (both on logarithmic scales). The estimates cover the frequency range from 1/2L = 6.7 X 1Om-4 pm-- 1 to l/20 = 1.3prn~-‘, although the usable range is taken to lie between about l/500 = 2 X 10 3 (urn- ’ and l/2 = 5 X 10-l pm- i. The significant feature to note in these figures is that, although this surface is relatively smooth at high frequencies, it is much rougher at low frequencies. Figure 4 is the profile of a gold-coated fused silica flat and Fig. 5 is its corresponding averaged periodogram spectral estimate, again taken over five profiles. In this case the high frequency part of the spectrum is more intense than that of the first surface but its low frequency part is much less intense. This is obvious in both the configuration-space and frequency-space illustrations. Obviously, the first surface would be better in an application where high frequency behavior (large-angle scattering) was more important, while
III i I’\\(‘t:
Fig. 2. Raw profile
I\
\I l(‘l3oll:‘ll:li-‘
of surface
1”
5
EK-111-2s.
Fig. 3. Averaged
periodogram
spectral
estimate
of the profile
of surface
EK-111-2s.
Fig. 5. Averaged
periodogram
spectral
estimate
of the profile
of surface
EK-1-3s.
197
the second surface would be much better suited to applications where low frequency behavior (small-angle scattering) was critical. Both of these surfaces are early test surfaces for the X-ray telescope to be included in the Advanced X-ray Astrophysical Facility and were provided by Dr. M. Zombeck of the Harvard-Smithsonian Center for Astrophysics. Further details concerning these surfaces are given elsewhere [lo] . Figure 6 is the profile of a silicon carbide surface, chemically vapor deposited onto a sintered silicon carbide substrate, which was polished at the National Physical Laboratory under the direction of Dr. A. Franks. (The sample was provided by M. R. Howells and P. Z. Takacs, NSLS, Brookhaven National Laboratory .) Figure 7 is its averaged periodogram. This surface is seen to be remarkably smooth over the entire range of measurement, significantly smoother than either of the preceding surfaces. The high frequency part of the spectrum is, in fact, comparable with the background noise which is approximately constant (white) with a value of about 5 X lo-* E.tm3. The tufts at f = 0.6 - 1.0 pm-’ fall outside the usable range of measurement and are believed to be due to vibration and pick-up rather than to surface features. Table 1 summarizes the values of finish parameters derived from the averaged periodogram estimates of the surfaces shown plus the instrumental background for a stationary stylus, which have been subtracted (as the variances) from the preceding surface profile data. Values of the root-mean-square profile roughness u, slope m and curvature c are given for various ranges of surface wavelength. Table 1, third column, includes wavelengths between 2 and 12 pm (corresponding to the high frequency behavior of the surfaces), the fourth column is for 12 - 512 pm (corresponding tc low frequencies), the fifth column is the total range from 2 to 512 pm and the last column is for the full range from the Nyquist wavelength of 0.75 pm to the record length of 1500 E.tm.The significance of this last column is limited by the fact that it includes parts of the spectral estimate which involve extraneous features and are distorted by aliasing, filtering and detrending effects.
Fig. 6. Raw profile of surface Sic 4. Fig. 7. Averaged periodogram spectral estimate of the profile of surface Sic 4.
198 TABLE
1
Root-mean-square values of the profile heights CJ, slopes m and curvatures c determined by integrating the averaged periodogram spectral estimates in Figs. 3, 5 and 7 over the indicated ranges of surface wavelengths Surface
Parameter
Value of parameter for following ranges of surface wavelength 12.512j.lm
2-12pm
2-512pm
0.75-1500pm
3.13
16.6
16.9
17.6
0.487
0.133
0.505
1.06 x 1O-3
3.84
1.06 x 1O-3
1.07 5.75
x
)
6.33 1.15 2.67 x 1O-3
5.35 0.128 4.93 x lo-’
8.29 1.15 2.67 x 1O-3
9.52 2.66 1.47
x lo-*
0 (A) m (mrad) c (pm-’ )
2.02 0.336 7.45 x 10-4
2.09 0.0416 1.64 x lo-’
2.91 0.338 7.45 x 1O-4
3.46 0.836 4.37 x 10-j
u (A) m (mrad) c (Mm-‘)
1.45 0.278 6.68
0.676 0.0205 8.29 x 1O-6
1.60 0.279 6.68 x 1O-4
2.21 0.903 5.54 x 1O-3
EK-111-2s Pt on Zerodur (Figs. 2, 3)
u (A) m (mrad) c (pm-’ )
EK-1-3s Au on SiOr (Figs. 4, 5)
u (A) c (Ctm-’
Sic 4 Chemical vapor deposition (Figs. 6, 7) Stationary stylus (no illustration)
m (mrad)
x
10m4
x
1O-5
1O-3
These three surfaces are similar in that they have root-mean-square profile roughnesses of the order of a nanometer, slopes of the order of 10e3 rad and radii of curvature of the order of lo2 I.tm over the range of surface wavelengths of about 10’ - lo3 pm. However, they are dissimilar in that their spectral shapes are distinctly different. The fact that they are different shows that there is no universal spectral shape and that measurements in a narrow spectral region do not necessarily determine the behavior in other regions.
8. Relationship
between
one-dimensional
The general relationship sional power spectral densities w,V,
between is
and twodimensional the one-dimensional
spectra and twodimen-
1= j-
dfy w,VJ,l (12) -m If IV2 is given, it is a simple matter to calculate W,, i.e. to deduce the onedimensional spectrum from the twodimensional spectrum. However, the inverse is not possible in general, although there are two particular cases for which it is possible: an extreme anisotropic surface and an isotropic surface. An extreme anisotropic surface lying parallel to the y axis has ~2(fx*f,)
=
~,(fxW,)
which has a trivial inverse.
(13)
199 An isotropic surface is one for which the two-dimensional spectrum is a function only of f where f 2 = f,* + fY2. In that case eqn. (12) can be rewritten as
J
W,(f) = 2 f
f’ df’ (f’Z_f2)1/2
(14)
Wf’)
which is a form of an Abel or half-integral transform this is the inverse Abel or half-derivative expression W*(f)
df’
= -11 n
f
(f’*-f*)l’*
!Jdf’
K(f’)
[ 111. The inverse of
(15)
which may be written in a variety of mathematically equivalent forms. These expressions show that to determine W,%,(f) we must know Wls(f ‘) for all f’ > f. However, since any experimental estimate of the profile spectrum W1 is bandwidth limited, it is strictly impossible to determine W2. To accomplish this inversion, we must invoke independent information concerning the properties of the surface spectrum in addition to isotropy: some model-dependent form which will smooth and extrapolate the profile spectrum to high frequencies. One method is to use the autoregressive model discussed earlier; another is to use some mathematically convenient form for the covariance function or power spectrum. A popular family of functions of this type has covariance functions of the form r”KV(27rur) where r is the lag, u is a constant and K, is a modified Bessel function [ 121. The corresponding form of the ddimensional power spectrum is
(16) where d = 1, 2 and I’ is the gamma function. This family has the virtues of reducing to the firstorder autoregressive form for v = f and giving inverse power law spectra at high frequencies f $ a. It also provides a convenient means of comparing onedimensional and two-dimensional finish parameters and exploring their dependences on surface bandwidth limits. Finish parameters derived from the central moments of one-dimensional and twodimensional spectra of a given surface are generally different even when the integration is carried out over surface frequencies from 0 to 0~: the height variances of the profile and area are equal, the variance of the profile slope is half that of the surface gradient, the variance of the profile curvature is 3/8 that of the surface and so on [ 1 l] . These relationships are independent of the shapes of the power spectra. In contrast, the relationships between bandwidth-limited parameters generally depend on the spectral shapes and the bandwidth limits. An interesting exception occurs for inverse power law spectra; in this case the ratios of profile to area parameters depend only on the profile power law and are independent of the type of parameter (i.e. CJ, m or c) and the bandwith limits [ 8,121. These and related results are readily derived from eqn. (16).
200
Although we have not yet made an extensive study of the reeonstruction of the twodimensional spectra from the onedimensional profile measurements, pre~in~y analysis indicates that the results are in reasonable agreement with independent measurements of surface scatter [8] .
9. Conclusions We have described procedures for estimating the power spectral densities of surface topography from sampled profile data and have illustrated the method by analyzing mechanical stylus data for several highly polished optical surfaces. The issues of surface bandwidth limits, statistical stability, finish parameters and the determination of area spectra and parameters from profile me~uremen~ have been addressed. We conclude that stylus techniques offer a sensitive if indirect method of measuring the finish of polished optical surfaces, particularly at long surface wavelengths, which may be especially important for practical applications [8,13].
We thank Drs. M. R. Howells and P. Z. Takacs of the Brookhaven National Laboratory for encouragement and assistance with this work, Dr. T. V. Vorburger of the National Bureau of Standards for valuable discussions of stylus techniques and for making the profile measurements described and Dr. M. V. Zombeck of the He’d-Smithsoni~ Center for Astrophysics for permission to use his unpublished data. H. C. Berry acknowledges the support of the U.S. Department of Energy. Certain items of commercial equipment are identified in this paper in order to specify the experimental procedure. In no case does such an identification imply recommendation or endorsement by the U.S. Department of Defense or the U.S. Department of Energy.
Nomenclature a %
A :
D ; F G
k
correlation parameter autoregression coefficient surface area root-mean-square profile curvature spatial dimension profite sampling interval maximum entropy spectral estimate surface frequency weighting factor frequency window (--1)“2 effective number of degrees of freedom
201 modified Bessel function half-record length root-mean~quare profile slope number of profiles spatial index number of sample points per profile averaged periodogram spectral estimate periodogram degree of derivative smoothed periodogram spectral estimate autoregressive order power spectral density coordinates in surface plane residual surface height coefficient of variation gamma function delta function unit variance white noise process dummy index root-mean-square profile height lag parameter
References 1 E. L. Church, The measurement 2 3 4 5 6 7 8 9 10 11 12
13
of surface texture and topography by differential light scattering, Wear, 57 (1979) 93 - 105. E. G. Thwaite, The direct measurement of the power spectrum of rough surfaces by optical Fourier transformation, Wear, 57 (1979) 71 - 80. L. H. Tanner, A comparison between Talysurf 10 and optical measurements of roughness and surface slope, Wear, 57 (1979) 81 - 91. G. M. Jenkins and D. G. Watts, Spectral Analysis and its Applications, Holden Day, San Francisco, CA, 1968. M. B. Priestly, Spectral Analysis and Time Series, Vol. 1, Univariafe Series, Academic Press, New York, 1981. R. B. Blackman and J, W. Tukey, The measurement of Power Spectra from the Point of View of Communications Engineering, Dover Publications, New York, 1958. S. M. Kay and S. L. Marple, Jr., Spectrum analysis - a modern perspective, Proc. IEEE, 69 (1981) 1380 - 1419. E. L. Church, M. R. Howells and T. V. Vorburger, Spectral analysis of the finish of diamond-turned mirror surfaces, Proc. Sot. Photo-Opt. Znstrum. Eng., 315 (1982). E. L. Church, Satistical fluctuations of total integrated scatter measurements, J. Opt. Sot. Am., 71 (1981) 1602A. M. V. Zombeck, C. C. Wyman and M. C. Weisskopf, High-resolution X-ray scattering measurements for advanced X-ray astrophysical facility, Opt. Eng., 21 (1982) 63 - 72. E. L. Church, H. A. Jenkinson and J. M. Zavada, Relationship between surface scattering and microtopographic features, Opt. Eng., 18 (1979) 125 - 136. E. 1;. Church, The role of spatial bandwidth limits in the measurement and interpretation of second-order statistical properties, Proc. 26th Conf. on the Design of Experiments in Army Reseoreh, ~e~Q~oprnent and Testing, in AR0 Rep. 81-2. M. Stedman, The metrological evaluation of grazing-incidence mirrors, Proc. SOC. Photo-Opt. instrum. Eng., 316 (1982) 2 - 8.