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SEMI MF1811-0704 © SEMI 2003, 2004 8 or determini stic values. Also known as random roughness. 4.2.31.1 Discussion — For exam ple, the profile ) 2 cos( ) ( φ + π = x f A x Z x (7) is deterministic if φ = const., but rand…
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periodogram. In general, they differ from the
periodogram in that they incorporate different types and
degrees of a-priori physical or mathematical
information about the original data set. The
periodogram, in contrast, includes the maximum
number of degrees of freedom and is always used for
first-cut evaluation and analysis. Details of the
correlation and other spectral estimation methods are
discussed in the literature listed in Section 7. A
mathematical variant of the periodogram estimator is
the correlation method. This is a two-step process that
requires the estimation of an intermediate function, the
autocovariance function, which is then Fourier
transformed to obtain the periodogram estimate of the
power spectrum. This method is not discussed in this
guide since it is indirect, but when properly applied
gives identically the same results as the direct transform
method recommended in this guide.
4.2.29 power spectral density (PSD) — statistical
function that shows how the mean-square (rms)
2
of a
given quantity is distributed among the various surface
spatial frequencies inherent in the profile height. Also
known as power spectrum.
4.2.29.1 Discussion — The two conventional measures
of surface roughness, R
a
and R
q
do not carry any
information about the transverse scale of the surface
roughness. That is, they are independent of how much
the surface profile is squeezed or stretched parallel to
the surface plane. The PSD is the simplest statistic that
carries that important additional information.
4.2.29.2 profile or one-dimensional PSD of the surface
height, [µm
3
] — a function of the spatial frequency, f
x
,
in units of inverse micrometers, µm
–1
, defined as
follows:
0,d)(
2
Lim)(
2
2/
2/
2
1
>
=
∫
+
−
π
∞→ x
L
L
xfi
Lx
fxexZ
L
fS
x
(5)
4.2.29.2.1 Discussion — The subscript “x” on “f
x
”
corresponds to the direction of the profile on the surface
and can be omitted if no confusion is involved. In this
definition the spatial frequency, f
x
, is always positive
and greater than zero. The value at f
x
= 0 corresponds
to the average value of the profile height, which is zero
for detrended profiles. The factor of 2 accounts for the
equal contribution from negative frequencies and en-
sures that the area under the positive-frequency profile
spectrum equals the rms-squared (mean-square) profile
height.
4.2.29.3 area or two-dimensional PSD of the surface
height [µm
4
] — a function of the spatial frequencies in
both the x and y directions on the surface, f
x
and f
y
, in
units of inverse micrometers, µm
–1
, defined as follows:
(){}
,dd2exp),(
1
Lim),(
2
2
+π=
∫∫
∞→
yxyfxfiyxZ
A
ffS
A
yxAyx
+∞<<−∞
yx
ff , (6)
4.2.29.3.1 Discussion — The spatial frequency ranges
included in this definition cover the entire frequency
plane and are not limited to positive frequencies only as
in the case of the profile spectrum. In the case of an
isotropically rough surface the area spectrum is a
function only of the magnitude of the two-dimensional
frequency vector: f = √(f
x
2
+ f
y
2
). The profile spectrum
can be derived from the area spectrum, but the area
spectrum cannot, in general, be derived from the profile
spectrum. Uniaxial and isotropically rough surfaces are
exceptions.
4.2.29.4 Discussion of Units: The surface height
fluctuations of optical surfaces are usually measured in
units of nanometers (1 nm = 10
–3
µm), or the non-SI
units of Ångstroms (1 Å = 10
–4
µm). Values of the
PSDs estimated using height data in these units can be
converted to the recommended units by multiplying by
the following conversion factors:
• To convert S
1
in units of nm
2
µm to units of µm
3
multiply it by 10
–6
,
• To convert S
1
in units of Å
2
µm to units of µm
3
multiply it by 10
–8
,
• To convert S
2
in units of nm
2
µm
2
to units of µm
4
multiply it by 10
–6
, and
• To convert S
2
in units of Å
2
µm
2
to units of µm
4
multiply it by 10
–8
.
If the sample interval is given in millimeters instead of
micrometers, the conversion factors for S
1
should be
multiplied by an additional factor of 10
3
, and those for
S
2
should be multiplied by an additional factor of 10
6
.
4.2.30 radius of curvature,
x
R
)
— radius of a circle
fitted to the measured surface profile.
4.2.30.1 Discussion — When the radius us large
relative to the profile length its magnitude is most
easily determined from the quadratic term in the
detrending polynomial. If the average surface profile is
written as Z(x) = a + bx + cx
2
, the estimate of the radius
of curvature in the x direction is ).2/(1 cR
x
=
)
If Z and x
are expressed in micrometers,
x
R
)
is in micrometers.
Since the radii of curvature of nominally flat surfaces
can be quite large, other reporting units, such as meters
or kilometers, may be more appropriate.
4.2.31 random surface profile — surface height profile
that involves parameters that are distributed according
to statistical distribution laws rather than having fixed
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or deterministic values. Also known as random
roughness.
4.2.31.1 Discussion — For example, the profile
)2cos()( φ+
π
= xfAxZ
x
(7)
is deterministic if φ = const., but random if φ has a
finite-width probability distribution function P(φ). Fin-
ish parameters and functions such as Z(x)
2
, are then the
values of those quantities averaged over P(φ).
4.2.32 restoration — signal-processing procedure in
which measurements are compensated for a non-unit
measurement transfer function by passing them through
a digital filter that restores the effective measurement
function to unity over its bandpass.
4.2.32.1 Discussion — The measured profile can be
restored and the statistics of the restored profile can
then be estimated. The most common spatial- and
frequency-domain filters used for this purpose are
“inverse” and “Wiener” filters. This guide does not
discuss the details of such restoration processes, which
may be found in standard signal-processing texts such
as those listed in Section 7.
4.2.33 RMS profile roughness, R
q
, [nm] — square root
of the mean-square profile roughness.
4.2.34 RMS profile slope, ∆
q
— square root of the
mean-square profile slope.
4.2.34.1 Discussion — The slope is dimensionless,
although the fundamental unit is the radian. In practice
it may be convenient to express the rms slope of highly
polished surfaces in microradians.
4.2.35 sample interval, D, [µm] — distance between
adjacent measurements of the surface height along the x
axis. Also known as sampling interval.
4.2.35.1 Discussion — The sample interval is usually
chosen or recommended by the manufacturer of the
profile instrument being used. The sample interval
defines the Nyquist frequency and hence, the extreme
HFL of the measurement. This guide does not address
measurements with unequal sample intervals or those
made along nonlinear traces over the surface.
4.2.36 sampled profile, Z(x
n
), [nm] — surface height,
Z(x
n
), measured at N equally-spaced points along the x
axis.
4.2.36.1 Discussion — This guide uses the following
indexing convention for the position of the height
samples,
NnDnx
n
,,2,1,)1( L=−= (8)
Therefore the distance between the first and last points
in the profile trace is (N – 1)D.
4.2.37 sampled slope, m(x
n
) — surface slope, m(x
n
),
measured at N equally-spaced points along the x axis
using the same indexing convention as for the sampled
profile.
4.2.37.1 Discussion — Some instruments measure the
surface slope directly, while others, in effect, measure
the surface height at N + 1 points and generate N slope
values using the equation:
[]
NnxZxZ
D
xm
nnn
,,2,1,)()(
1
)(
1
L=−=
+
(9)
4.2.38 slope power spectrum, S
′
(f
x
[µm] — statistical
function that shows how the mean-square profile slope
is distributed over surface spatial frequencies as
follows:
()
)()2(d2exp)(
1
Lim)(
1
2
2
2/
2/
1 xx
L
L
xLx
fSfxxfixm
L
fS π=
π=
′
∫
+
−
∞→
(10)
4.2.38.1 Discussion — This simple connection
between the slope and roughness power spectra permits
one to be determined immediately in terms of the other.
The prime on S
1
, on the left denotes that this is the PSD
of the slope, while the unprimed S
1
on the far right is
the PSD of the height.
4.2.39 spatial frequency, f
x
,[1/µm] — frequency
parameter in the Fourier transform of the surface profile
Z(x).
4.2.39.1 Discussion — The parameter f
x
is related to
the spatial wavelength, d
x
through f
x
= 1/d
x
. Similar
quantities are defined for the y component, and the
magnitude of the two-dimensional spatial-frequency
vector,
22
yx
fff += , that appears in the two-dimen-
sional power spectral density of an isotropically-rough
surface, S
2
(f).
4.2.40 spatial wavelength, d
x
, [µm] — reciprocal of the
spatial frequency, f
x
.
4.2.40.1 Discussion — The mechanical-engineering
community frequently uses the symbol λ for the spatial
wavelength, while the optical community reserves that
symbol for the radiation wavelength.
4.2.41 spectral model — analytic expression for the
power spectral density which contains a number of
adjustable parameters called finish parameters.
4.2.41.1 Discussion — The values of the finish
parameters are obtained by fitting estimates of the PSD
of the surface height or slope fluctuations to the model.
The fitting process performs a number of important
functions: it averages out the fluctuations appearing in
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individual estimates of the power spectrum, it
condenses the data into a few intrinsic surface
parameters, and provides a mechanism for extrapolating
the measured data outside the measurement bandwidth.
4.2.41.2 One example of a spectral model is the ABC
model, which has the following form for the profile
PSD:
2/2
1
])(1[
)(
C
x
x
Bf
A
fS
+
= (11)
and for the two-dimensional spectrum of an isotrop-
ically-rough surface:
2/)1(2
2
])(1[
)(
+
+
′
=
C
Bf
A
fS (12)
where:
.
)2/(
2/)1(
2
1
AB
C
C
A ⋅
Γ
+Γ
⋅
π
=
′
The finish parameters in this model are A, B and C,
which have the dimensions of µm
3
, µm
1
, and µm
0
. This
model is sometimes called the K-correlation model, and
the quantity B/(2π), the correlation length.
4.2.41.3 Another example of a spectral model is the
fractal model, which has the following form for the
profile PSD:
C
x
x
f
K
fS
1
1
)( = (13)
and for the two-dimensional spectrum of an
isotropically-rough surface:
1
2
2
)(
+
=
C
f
K
fS (14)
where:
.
)2/(
2/)1(
2
1
12
K
C
C
K ⋅
Γ
+Γ
⋅
π
=
The finish parameters in this model are K
1
and C, which
have the dimensions of µm
(3–C)
and µm
0
. The
dimensionless number C usually lies between 1 and 3
but need not be an integer. The quantity K
C
is
sometimes referred to as the spectral strength, and the
parameter, C, the spectral index.
4.2.41.4 The fractal model is the limiting case of the
ABC model when the finish parameter B becomes very
large. The value of the intrinsic mean-square profile
and area roughness of the ABC model, obtained by
integrating the ABC spectrum over all frequencies, is as
follows:
2
2
00
1
2
1
2
d)(2d)(
B
A
C
ffSfffSR
xxq
′
−
π
=π==
∫∫
∞∞
(15)
which is finite for C > 1. The intrinsic value of the
mean-square roughness of the fractal model is always
infinite because of its divergence at low spatial
frequencies. In contrast, the measured roughness
values, obtained by integrating only over the measure-
ment bandpass, are finite for the ABC model for any
value of C, and for the fractal model.
4.2.41.5 A third example of a spectral model is the
periodic model, which is for a surface consisting of a
periodic structure and has the following forms for the
profile PSD:
∑
∞=
=
−δ⋅=
k
k
xkx
d
k
fAfS
1
0
2
1
2
1
)( (16)
and for the two-dimensional spectrum:
∑
+∞=
−∞=
δ⋅
−δ⋅=
k
k
yxkyx
f
d
k
fAffS )(
4
1
),(
0
2
||2
(17)
where the k = 0 is excluded. The finish parameters of
this model are the A
k
' s, the Fourier amplitudes of the
periodic profile, and d
0
, the fundamental spatial
wavelength of the periodicity, both expressed in µm.
A
2
/2 is the mean-square roughness of the k
th
harmonic
of the profile (k = 1 is the fundamental), and δ(F) is a
unit-area function that is sharply peaked about the point
F = 0. The value of the intrinsic mean-square profile
and area roughness of the periodic model is as follows:
∑
∫∫∫
∞
=
+∞
∞−
+∞
∞−
∞
===
1
2
2
0
1
2
2
1
d),(d)(
k
k
yyxxxxq
AfffSdfffSR (18)
In contrast, the measured value is the right-hand side
summed over those spectral lines that fall within the
measurement bandpass.
4.2.41.6 Finally, it is possible to develop a composite
model that is made up of a sum of terms involving
different models or models with different parameters, or
both.
4.2.42 trace length, L [µm] — total length of the
surface sampled by a linear profile measurement. Also
known as profile length.
4.2.42.1 Discussion: In the indexing used in this guide
DNxL
N
)1( −
=
=
(19)