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SEMI MF1811-0704 © SEMI 2003, 2004 11 where M 0 (and also M 1 and M 2 ) are evaluated using the general mom e nt expression, ∑ = = N n P P n Z n N M 1 ) ( 1 (23) where: P = 0, 1, or 2. 5.1.3 Piston and tilt detrending is…
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where:
x
N
= the position of the N
th
or last point in the
measurement (with x
1
assumed at zero), and
D = 1 the sampling interval (see Section
4.2.15.1).
The periodogram estimate is based on a Fourier
representation of the surface profile. The basic
periodicity of that expansion is ND rather than the
literal profile length L = (N–1)D. Depending on the
type of FFT used in the practical evaluation of the PSD,
N may be required to be a power of 2, such as 1024,
although in general, there is no restriction on N in this
guide.
4.2.43 transfer function — function of spatial
frequency having a magnitude between zero and one
which describes the sensitivity of a linear measuring
system to the amplitudes of different spatial-frequency
components in the profile being measured. Also known
as measurement transfer function.
4.2.43.1 Discussion — The ideal transfer function is
unity within the measurement bandpass and zero for
frequencies outside the bandpass. Real-world
measurement transfer functions can deviate
significantly from this. The transfer function is the
Fourier transform of the impulse response function of
the measuring apparatus.
4.2.44 uniaxial surface — surface whose roughness is
confined to a particular direction or lay, so that it can be
completely characterized by profile measurements
perpendicular to the lay direction. Surfaces that display
harmonic lines are frequently uniaxial. Also known as
grating-like surface.
4.2.44.1 Discussion — In contrast, an isotropic surface
can also be completely characterized by profile
measurements made in one direction, but there is no
preferred direction as there is for uniaxial surfaces.
Surfaces that are neither uniaxial nor isotropic can be
characterized using the procedures described in this
guide, although profile measurements taken on many
directions across the surface may be needed to generate
a complete statistical description of the surface under
test.
4.2.45 window function, W(x
n
) — bell-shaped or
smooth-edged function that multiplies the detrended
profile data set before it is inserted into the period-
ogram estimation routine. Also known as data window.
4.2.45.1 Discussion — The window function
“smoothes out” possible discontinuities at the ends of
the measured, finite-length data set in order to eliminate
the spurious oscillations that those discontinuities
would otherwise generate in the spectral estimate. As
long as the window function performs its function of
reducing the contributions from the ends of the data
record and has the proper normalization, its shape is of
secondary importance.
4.2.46 zero padding — procedure of adding zero
values to a data set to bring the total number of data
points, N, to a power of two to facilitate the evaluation
of the FFT appearing in the periodogram spectral
estimate.
4.2.46.1 Discussion — The window functions should
be applied to the data set before zero padding. Zero
padding is less important with the ready availability of
arbitrary- N FFT computing packages.
5 Calculations
5.1 Detrending
5.1.1 Introduction — The estimators defined in this
section are based on the analysis of a data set Z(n)
consisting of N discrete values of the surface profile
Z(x
n
= (n–1)D) measured at equally-spaced locations
along a straight line of length L, where n = 1 to N. If
Z(n) is the measured profile, the detrended profile is
given by:
][)()(
2
ncnbanZnZd ⋅+⋅+−= (20)
where the quantity in the square bracket is the quadratic
detrending polynomial. The estimated values of the
polynomial coefficients a, b, and c, denoted by a
)
, b
)
,
and c
)
, respectively, are determined by least-squares
fitting of the polynomial to the measured profile data as
now described. The degree of the detrending
polynomial is chosen by the following considerations:
Removing piston only (zeroth-order polynomial, a) is
useful for instructional purposes but is inadequate in
practice. It affects only the zero-frequency or “dc” term
in the power spectral density. Removing piston and tilt
(first-order polynomial, a + b·n) is sufficient for
removing uncertainties in the rigid-body positioning of
a nominally flat sample in the measurement apparatus.
Removing piston, tilt and curvature (second-order
polynomial, a + b·n + c·n
2
) removes an additional
quadratic term in the profile that may result from
instrumental (extrinsic) effects or true (intrinsic)
curvature in the surface being measured.
5.1.2 Piston detrending is as follows:
anZndZ
)
)
−= )()( (21)
and
0
Ma
+
=
)
(22)
SEMI MF1811-0704 © SEMI 2003, 2004 11
where M
0
(and also M
1
and M
2
) are evaluated using the
general moment expression,
∑
=
=
N
n
P
P
nZn
N
M
1
)(
1
(23)
where:
P = 0, 1, or 2.
5.1.3 Piston and tilt detrending is as follows:
][)()( nbanZndZ
)
)
)
+−= (24)
where:
n = 1, 2, … , N,
a
)
=
]3)12[(
1
2
10
MMN
N
−+⋅
−
+ , and
b
)
=
+
−
−
−
1
2
1
6
1
0
N
M
M
N
.
5.1.4 Piston, tilt, and curvature detrending is as
follows:
][)()(
2
ncnbanZndZ
)
)
)
)
++−= (25)
where:
n = 1, 2, … , N,
a
)
=
],10)12(6)233[(
)2)(1(
3
210
2
MMNMNN
NN
++−++
−−
+
b
)
=
and],)1(30)118)(12(2
)12)(2)(13[(
)4)(1(
6
21
0
22
MNMNN
MNNN
NN
++++−
+++
−−
−
c
)
=
.6)1(6)2)(1[(
)4)(1(
30
210
22
MMNMNN
NN
++−++
−−
+
NOTE 1: The estimated values of the coefficients depend on
the degree of the polynomial being detrended. For example,
the value of the coefficient a
)
for piston detrending and
piston-plus-tilt detrending derived from the same data set are
generally different.
NOTE 2: Despite these apparent differences, the mean values
of each of the detrended profiles given by
∑
=
==
N
n
ndZ
N
1
0)(
1
Mean value
)
(26)
vanishes in all cases. This means that the “dc” value of the
estimated power spectrum of the detrended profile is zero,
which offers a convenient numerical check on the numerical
processing routines used.
NOTE 3: Least-squares fitting routines are available in many
computer packages. Analytic results are given above for
reference and checking.
5.2 RMS Roughness
5.2.1 There are two different estimators for the rms
roughness, R
q
— one expressed in configuration space,
and the other in frequency space, as follows:
∑
=
=
N
n
q
ndZ
N
R
1
22
)(
1
)Config(
))
(27)
and
∑
+
=
=
2
1
1
1
2
)(
1
)Freq(
N
m
q
mS
ND
R
)
)
(28)
1 where )(
1
mS
)
is the periodogram estimate of the PSD
based on )(ndZ
)
(see Section 5.4).
NOTE 4: These two estimates of R
q
2
are mathematically
identical if the periodogram is evaluated using a unit data
window, W(n) = 1. Although a unit window function is not
recommended for general use, the numerical identity of the
Equation 27 and Equation 28 in that case offers a convenient
check on the programming of the periodogram estimator.
NOTE 5: If a non-unit data window is used in the calculation
of the PSD the two estimates of the rms roughness given will
not, in general, be numerically identical for a particular pro-
file measurement. On the other hand, the two estimates are
identical for an ensemble average over a large number of pro-
file measurements. In other words, the two estimates of R
q
2
are statistically the same.
NOTE 6: The first estimator has the advantage of familiarity
and simplicity since it is expressed directly in terms of the
detrended values of the measured profile data. Its disad-
vantage is that it involves, perforce, the transfer function of
the measuring apparatus, and in a nonobvious way. In con-
trast, the frequency-space form may be more complicated to
evaluate but has the advantage that it permits the bandwidth to
be included in the rms value to be varied by selecting the
range of m values included in the frequency sum. In addition,
it permits the effects of a non-unit instrumental transfer
function within that bandpass to be examined directly, and to
be divided out by restoration processes, if its form is known
independently.
NOTE 7: The spectra of real surfaces frequently tend to
diverge at low spatial frequencies so that the values of the rms
roughness obtained using either estimator may depend signi-
ficantly on the value of the LFL of the measurement process,
or chosen as a reference value. In some cases, the presence of
a non-vanishing LFL can give a finite value of the profile
roughness when its intrinsic value is infinite or undefined.
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5.3 RMS Slope
5.3.1 There are two different estimators for the rms
slope, ∆
q
— one expressed in configuration space, and
the other in frequency space, as follows:
∑
−
=
−+=∆
1
1
2
2
)()1([
1
)Config(
N
n
q
ndZndZ
ND
)))
(29)
and
2
)2/(1
1
1
2
)1(2
)(
1
)Freq(
−π
=∆
∑
+
=
ND
m
mS
ND
N
m
q
)
)
(30)
NOTE 8: The magnitudes of these two estimates are
generally different since the first treats the profile as a
collection of straight-line segments connecting the
measurement points, while the second connects them with a
bandwidth-limited interpolation curve and involves a smaller
bias error.
NOTE 9: The notes for the rms roughness estimators just
made generally apply to slope estimates as well. One
difference is that the significant bandwidth effects on the
slope occur principally at the HFL rather than the LFL as is
the case for rms roughness measurements.
5.4 Periodogram Estimators of the Profile Power
Spectral Density
5.4.1 Form for 1-dimensional Power Spectral Density
for height-measuring profilometers is as follows:
.
)()(FFT2
)(
2
1
N
mKmD
mS
⋅⋅
=
)
(31)
The spatial frequency is evaluated at the discrete
valuesof m, with m = 1, 2, … , [1 + (N/2)]:
.
1
ND
m
f
x
−
= (32)
5.4.2 The symbol FFT in Equation 31 stands for
discrete Fourier transform, which is always evaluated
using some version of the fast Fourier transform as
follows:
∑
=
⋅⋅−−π=
N
n
ndZnWNmnim
1
)()(]/)1)(1(2exp[)(FFT
)
(33)
where:
W(n) = window function (see Section 5.6) and
K(m) = book-keeping factor equal to ½ for m = 1 o
r
m = 1 + (N/2) and equal to 1 otherwise.
NOTE 10: Equation 33 applies for the conventional case of
even N. Different forms apply for odd N.
NOTE 11: The case m = 1 corresponds to the zero-frequency
or dc component of the detrended surface profile and m = 2
corresponds to the spatial frequency 1/(ND), which is
essentially the reciprocal of the trace length, ( N–1)D. On the
opposite extreme, the frequency corresponding to m = 1 + N/2
is the Nyquist frequency, 1/(2D). The extreme range of
surface wavelengths included in the measurement is therefore
1/(ND) < f < 1/(2D). In other words, the extreme LFL =
1/(ND), the extreme HFL = 1/(2D), and the dynamic range of
the measurement is N/2.
NOTE 12: A convenient and readable reference to the FFT
and its evaluation is Chapter 12 in Numerical Recipes by
Press, Flannery, Teukolsky, and Vetterling (see Section 7.1).
NOTE 13: The Brookhaven National Laboratory Report (see
Section 7.2) contains further background information on these
procedures along with a computer program and numerical
examples. (The BASIC routines used there involve different
forms for the quantities M
P
appearing in the expressions for
the detrending polynomials than those discussed in this guide,
although the numerical values of the detrending polynomials
are identical in both cases.)
NOTE 14: The periodogram estimator just given is not the
only method of estimating the power spectral density from a
set of profile data, but it is the most direct and common
method. It is sufficient for general use, and is a necessary
first step to be taken before adding embellishments such as
post-processing or considering more complicated estimators.
This guide does not exclude the use of post-processing or
alternative methods of analysis, but does require that the basic
periodogram estimates just described be included in the
discussion for comparative purposes.
5.4.3 Form for 1-dimensional Power Spectral Density
for slope-measuring profilometers is as follows:
N
mKmD
ND
m
mS
)()(FFT2
)1(2
)(
2
2
1
⋅⋅
⋅
−π
=
−
)
(34)
where:
∑
=
−−π
⋅⋅=
′
N
n
N
mni
ndMnWem
1
)1)(1(2
)()()(TFF
)
and )(ndM
)
is the value of the profile slope measure-
ments detrended using either the least-squares piston or
the piston-plus-tilt expressions (see Section 5.1).
NOTE 15: This estimate of the profile power spectrum is the
power spectrum of the profile slope divided by (2πf
m
)
2
. The
prime on the FFT on the left denotes that it involves slope
rather than height data.
NOTE 16: The case m = 1 corresponds to zero spatial fre-
quency and must be excluded in the use of the above
expressions.