This practice defines the methodology for calculating a set of
commonly used statistical parameters
and functions of surface roughness from a set of measured surface
profile data. Its purposes are to
provide fundamental procedures and notation for processing and
presenting data, to alert the reader
to related issues that may arise in user-specific applications, and to
provide literature
references where further details can be found.

The present practice is limited to the analysis of one-dimensional or
profile data taken at uniform
intervals along straight lines across the surface under test, although
reference is made to the
more general case of two-dimensional measurements made over a
rectangular array of data points.

The data analysis procedures described in this practice are generic
and are not limited to specific
surfaces, surface-generation techniques, degrees of roughness, or
measuring techniques. Examples of
measuring techniques that can be used to generate profile data for
analysis are mechanical
profiling instruments using a rigid contacting probe, optical
profiling instruments that sample
over a line or an array over an area of the surface, optical
interferometry, and
scanning-microscopy techniques such as atomic-force microscopy. The
distinctions between different
measuring techniques enter the present practice through various
parameters and functions that are
defined in Sections 3 and 5, such as their sampling intervals,
bandwidths, and measurement transfer
functions.

The primary interest here is the characterization of random or
periodic aspects of surface finish
rather than isolated surface defects such as pits, protrusions,
scratches or ridges. Although the
methods of data analysis described here can be equally well applied to
profile data of isolated
surface features, the parameters and functions that are derived using
the procedures described in
this practice may have a different physical significance than those
derived from random or periodic
surfaces.

The statistical parameters and functions that are discussed in this
practice are, in fact,
mathematical abstractions that are generally defined in terms of an
infinitely-long linear profile
across the surface, or the "ensemble" average of an infinite number of
finite-length profiles. In
contrast, real profile data are available in the form of one or more
sets of digitized height data
measured at a finite number of discrete positions on the surface under
test. This practice gives
both the abstract definitions of the statistical quantities of
interest, and numerical procedures
for determining values of these abstract quantities from sets of
measured data. In the notation of
this practice these numerical procedures are called "estimators" and
the results that they produce
are called "estimates".

This practice gives "periodogram" estimators for determining the
root-mean-square (rms) roughness,
rms slope, and power spectral density (PSD) of the surface directly
from profile height or slope
measurements. The statistical literature uses a circumflex to
distinguish an estimator or estimate
from its abstract or ensemble-average value. For example, Â
denotes an estimate of the
quality A. However, some word-processors cannot place a circumflex
over consonants in text. Any
symbolic or verbal device may be used instead.

The quality of estimators of surface statistics are, in turn,
characterized by higher-order
statistical properties that describe their "bias" and "fluctuation"
properties with respect to
their abstract or ensemble-average versions. This practice does not
discuss the higher-order
statistical properties of the estimators given here since their
practical significance and use are
application-specific and beyond the scope of this document. Details of
these and related subjects
can be found in References (1-10) at the end of this practice.

Raw measured profile data generally contain trending components that
are independent of the
microtopography of the surface being measured. These components must
be subtracted before the
difference or residual errors are subjected to the
statistical-estimation routines given here.
These trending components originate from both extrinsic and intrinsic
sources. Extrinsic trends
arise from the rigid-body positioning of the part under test in the
measuring apparatus. In optics
these displacement and rotation contributions are called "piston" and
"tilt" errors. In contrast,
intrinsic trends arise from deliberate or accidental shape errors
inherent in the surface under
test, such as a circular or parabolic curvature. In the absence of
a-priori information about the
true surface shape, the intrinsic shape error is frequently limited to
a quadratic (parabolic)
curvature of the surface. Detrending of intrinsic and extrinsic trends
is generally accomplished
simultaneously by subtracting a detrending polynomial from the raw
measured data, where the
polynomial coefficients are determined by least-squares fitting to the
measured data.

Although surfaces and surface measuring instruments exist in real or
configuration space, they are
most easily understood in frequency space, also known as Fourier
transform, reciprocal or
spatial-frequency space. This is because any practical measurement
process can be considered to be
a "linear system", meaning that the measured profile is the
convolution of the true surface profile
and the impulse response of the measuring system; and equivalently,
the Fourier-amplitude spectrum
of the measured profile is the product of that of the true profile and
the frequency-dependent
"transfer function" of the measurement system. This is expressed
symbolically by the following
equation:

where:

A = the Fourier amplitudes,

T (fx) = instrument response function or the measurement transfer
function, and

fx = surface spatial frequency.

This factorization permits the surface and the measuring system to be
discussed independently of
each other in frequency space, and is an essential feature of any
discussion of measurement
systems.

Figure 1 sketches different forms of the measurement transfer
function, T(fx):

Case (a) is a perfect measuring system, which has T (fx) = 1 for all
spatial frequencies, 0 ≤
fx≤ ∞ . This is unrealistic since no real measuring
instrument is equally sensitive to all
spatial frequencies. Case (b) is an ideal measuring system, which has
T (fx) = 1 for LFL ≤ f
x≤ HFL and T (f x) = 0 otherwise, where LFL and HFL denote the
low-frequency and high-frequency
limits of the measurement. The range LFL ≤ f x≤ HFL is called
the bandpass or bandwidth of
the measurement, and ratio HFL/LFL is called the dynamic range of the
measurement. Case (c)
represents a realistic measuring system, since it includes the fact
that T (fx) need not be unity
within the measurement bandpass or strictly zero outside the bandpass.

If the measurement transfer function is known to deviate significantly
from unity within the
measurement bandpass, the measured power spectral density (PSD) can be
transformed into the form
that would have been measured by an instrument with the ideal
rectangular form through the process
of digital "restoration." In its simplest form restoration involves
dividing the measured PSD by
the known form of \T(fx)\² over the measurement
bandpass. Restoration is
particularly relevant to measuring instruments that involve optical
microscopes since the transfer
functions of microscope systems are not unity over their bandpass but
tend to fall linearly between
unity at T (0) = 1 and T(HFL) = 0. The need for, and methodology of
digital restoration is
instrument specific and this practice places no requirements on its
use.

This practice requires that any data on surface finish parameters or
functions generated by the
procedures described herein be accompanied by an identifying
description of measuring instrument
used, estimates of its low- and high-frequency limits, LFL and HFL,
and a statement of whether or
not restoration techniques were used.

In order to make a quantitative comparison between profile data
obtained from different measurement
techniques, the statistical parameters and functions of interest must
be compared over the same or
comparable spatial-frequency regions. The most common quantities used
to compare surfaces are their
root-mean-square (rms) roughness values, which are the square roots of
the areas under the PSD
between specified surface-frequency limits. Surface statistics derived
from measurements involving
different spatial-frequency ranges cannot be compared quantitatively
except in an approximate way.
In some cases measurements with partially or even nonoverlapping
bandwidths can be compared by
using analytic models of the PSDs to extrapolate the PSDs outside
their measurement bandwidth.

Examples of specific band-width limits can be drawn from the optical
and semiconductor industries.
In optics the so-called total integrated scatter or TIS measurement
technique leads to rms
roughness values involving an annulus in two-dimensional spatial
frequencies space from 0.069 to
1.48 µm^{−1}; that is, a dynamic range of 1.48/0.069 =
21/1. In contrast, the range
of spatial frequencies involved in optical and mechanical scanning
techniques are generally much
larger than this, frequently having a dynamic ranges of 512/1 or more.
In the latter case the
subrange of 0.0125 to 1 µm^{−1} has been used to
discuss the rms surface roughness in
the semiconductor industry. These numbers are provided to illustrate
the magnitudes and ranges of
HFL and LFL encountered in practice but do not constitute a
recommendation of particular limits for
the specification of surface finish parameters. Such selections are
application dependent, and are
to be made at the users' discretion.

The limits of integration involved in the determination of rms
roughness and slope values from
measured profile data are introduced by multiplying the measured PSD
by a factor equal to zero for
spatial frequencies outside the desired bandpass and unity within the
desired bandpass, as shown in
Case (b) in Fig. 1. This is called a top-hat or binary filter
function. Before the ready
availability of digital frequency-domain processing as employed in
this practice, bandwidth limits
were imposed by passing the profile data through analog or digital
filters without explicitly
transforming them into the frequency domain and multiplying by a
top-hat function. The two
processes are mathematically equivalent, providing the data filter has
the desired frequency
response. Real data filters, however, frequently have Gaussian or RC
forms that only approximate
the desired top-hat form that introduces some ambiguity in their
interpretation. This practice
recommends the determination of rms roughness and slope values using
top-hat windowing of the
measured PSD in the frequency domain.

The PSD and rms roughness are surface statistics of particular
interest to the optics and
semiconductor industries because of their direct relationship to the
functional properties of such
surfaces. In the case of rougher surfaces these are still valid and
useful statistics, although the
functional properties of such surfaces may depend on additional
statistics as well. The ASME
Standard on Surface Texture, B46.1, discusses additional surface
statistics, terms, and measurement
methods applicable to machined surfaces.

The units used in this practice are a self-consistent set of SI units
that are appropriate for many
measurements in the semiconductor and optics industry. This practice
does not mandate the use of
these units, but does require that results expressed in other units be
referenced to SI units for
ease of comparison.

This standard does not purport to address all of the safety concerns,
if any, associated with its
use. It is the responsibility of the user of this standard to
establish appropriate safety and
health practices and determine the applicability of regulatory
limitations prior to use.