ISO/TR 19319:2013

TECHNICAL ISO/TR
REPORT 19319
Second edition
2013-03-15
Surface chemical analysis —
Fundamental approaches to
determination of lateral resolution
and sharpness in beam-based methods
Analyse chimique des surfaces — Approche fondamentale pour
la détermination de la résolution latérale et de la netteté par des
méthodes à base de faisceau
Reference number
©
ISO 2013
© ISO 2013
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ii © ISO 2013 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Terms and definitions . 1
3 Symbols and abbreviated terms . 4
4 Determination of lateral resolution and sharpness by imaging of stripe patterns .7
4.1 Theoretical background . 7
4.2 Determination of the line spread function and the modulation transfer function by
imaging of a narrow stripe .21
4.3 Determination of the edge spread function (ESF) by imaging a straight edge .41
4.4 Determination of lateral resolution by imaging of square-wave gratings .56
5 Physical factors affecting lateral resolution, analysis area and sample area viewed by the
analyser in AES and XPS .96
5.1 General information .96
5.2 Lateral resolution of AES and XPS .97
5.3 Analysis area .104
5.4 Sample area viewed by the analyser .106
6 Measurements of analysis area and sample area viewed by the analyser in AES
and XPS .107
6.1 General information .107
6.2 Analysis area .108
6.3 Sample area viewed by the analyser .109
Annex A (informative) Reduction of image period for 3-stripe gratings .110
Bibliography .113
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
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ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International
Standards adopted by the technical committees are circulated to the member bodies for voting.
Publication as an International Standard requires approval by at least 75 % of the member bodies
casting a vote.
In exceptional circumstances, when a technical committee has collected data of a different kind from
that which is normally published as an International Standard (“state of the art”, for example), it may
decide by a simple majority vote of its participating members to publish a Technical Report. A Technical
Report is entirely informative in nature and does not have to be reviewed until the data it provides are
considered to be no longer valid or useful.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/TR 19319 was prepared by Technical Committee ISO/TC 201, Surface chemical analysis, Subcommittee
SC 2, General procedures.
This second edition cancels and replaces the first edition (ISO/TR 19319:2003), which has been
technically revised.
iv © ISO 2013 – All rights reserved

Introduction
Surface-analytical techniques such as SIMS, AES and XPS enable imaging of surfaces. The most relevant
parameter of element or chemical maps and line scans is the lateral resolution, also called image
1)
resolution. Therefore well defined and accurate procedures for the determination of lateral resolution
are required. Those procedures together with appropriate test specimen are basic preconditions for
comparability of results obtained by imaging surface-analytical methods and performance tests
of instruments as well. This Technical Report is intended to serve as a basis for the development of
International Standards.
Nowadays there is some confusion in the community in the understanding of the term “lateral resolution”.
Definitions originating from different fields of application and different communities of users can be
found in the literature. Unfortunately they are inconsistent in many cases. As a result, values of “lateral
resolution” published by manufacturers and users having been derived by using different definitions
and/or determined by different procedures cannot be compared to each other. It is the intention of this
Technical Report to basically describe different approaches for the characterization of lateral resolution
including their interrelations.
[1]
The term resolution was introduced with respect to the performance of microscopes by Ernst Abbe.
[2]
Later on it was applied to spectroscopy by Lord Rayleigh. It is based on the diffraction theory of light
and the original definition of lateral resolution as “the minimum spacing at which two features of the
image can be recognised as distinct and separate” is in common use in the light and electron microscopy
[3]
communities as documented in the standard ISO 22493:2008.
However, in the surface analysis community a very different approach, the “knife edge method”, is the
most popular one for the determination of lateral resolution. This method is based on evaluation of
an image or of a line scan over a straight edge. Here lateral resolution is characterized by parameters
describing the steepness of the edge spread function ESF. The standard “ISO 18516:2006 Surface Chemical
Analysis – Auger electron spectroscopy and X-ray photoelectron spectroscopy – Determination of lateral
[4]
resolution” is limited to this approach. But the ESF and corresponding rise parameters D are
x-(1-x)
more related to image sharpness than to lateral resolution which refers to two separated features.
The reason why the original meaning of resolution is not commonly implemented in the common
practice in surface analysis is the lack of suitable test specimens having the required features in the
sub-µm range. However, recently a new type of test specimen was developed featuring a series of flat
[5,6]
square-wave gratings characterized by chemical contrast and different periods. Such test specimens
may enable an implementation of the original definition of lateral resolution into practical approaches
in surface chemical analysis.
Having solved the problem of availability of appropriate test specimens another problem has to be
solved: The establishment of a criterion for whether two features are separated or not. The Rayleigh
[2]
criterion was developed for diffraction optics and its application in imaging surface analysis is not
[7]
straightforward. The Sparrow criterion defines a resolution threshold exclusively by the existence of
a minimum between two maxima. Actually, for practical imaging in surface analysis, noise is a relevant
feature especially at the limit of resolution. Therefore the Sparrow criterion will fail to solve the problem.
The solution is to develop a resolution criterion relying on the detection of a minimum between two
features but additionally considering noise effects.
The lateral resolution of imaging systems is strongly related to a number of functions describing the
formation of images:
— the modulation transfer function,
— the contrast transfer function,
— the point spread function,
1) The term “image resolution” is used in the microscopy community whereas in the surface analysis community
the term “lateral resolution” is common practice to distinguish it from “depth resolution”.
— the line spread function and
— and the edge spread function.
Those functions may be utilized to describe the performance of optical instruments and instruments used
for imaging in surface analysis as well. In particular the contrast transfer function has been used successfully
for the benefit of the determination of lateral resolution of imaging instruments in surface analysis.
Section 4 of this report describes the basics of procedures for the analysis of images of stripe patterns,
narrow stripes and step transitions. A comparison of all procedures related to lateral resolution and
sharpness is given in 4.1.7.
Section 5 of the report describes physical factors affecting lateral resolution, analysis area and sample
area viewed by the analyser in Auger electron spectroscopy and X-ray photoelectron spectroscopy.
Section 6 of the report gives guidance on the determination of sample area viewed by the analyser in
applications of Auger electron spectroscopy and X-ray photoelectron spectroscopy.
vi © ISO 2013 – All rights reserved

TECHNICAL REPORT ISO/TR 19319:2013(E)
Surface chemical analysis — Fundamental approaches
to determination of lateral resolution and sharpness in
beam-based methods
1 Scope
This Technical Report describes:
a) Functions and their relevance to lateral resolution:
1) Point spread function (PSF) — see 4.1.1
2) Line spread function (LSF) — see 4.1.2
3) Edge spread function (ESF) — see 4.1.3
4) Modulation transfer function (MTF) — see 4.1.4
5) Contrast transfer function (CTF) — see 4.1.5.
b) Experimental methods for the determination of lateral resolution and parameters related to
lateral resolution:
1) Imaging of a narrow stripe — see 4.2
2) Imaging of a sharp edge — see 4.3
3) Imaging of square-wave gratings — see 4.4.
c) Physical factors affecting lateral resolution, analysis area and sample area viewed by the analyser
in Auger electron spectroscopy and X-ray photoelectron spectroscopy — see Clauses 5 and 6.
2 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
2.1
analysis area
two-dimensional region of a sample surface measured in the plane of that surface from which
the entire analytical signal or a specified percentage of that signal is detected
[SOURCE: ISO 18115:2010, definition 5.8]
2.2
contrast transfer function
CTF
ratio of the image contrast to the object contrast of a square-wave pattern as a function of spatial frequency
Note 1 to entry: In this document the contrast transfer function CTF has been used also with an abscissa expressed
in terms of w /P and is called the generalized contrast transfer function in those cases (cf. 4.4.3.2). w is the
LSF LSF
full width at half maximum of the line spread function LSF
Note 2 to entry: In transmission electron microscopy and other phase sensitive methods the term contrast
transfer function is used with a different meaning considering amplitude as well as phase information.
2.3
cut-off frequency of the contrast transfer function
lowest spatial frequency at which the contrast transfer function CTF equals to zero
Note 1 to entry: In this document the spatial frequency at which the contrast transfer function CTF equals the
threshold of resolution under consideration of noise (cf. 4.4.3.3) is called effective cut-off frequency of the contrast
transfer function.
2.4
edge spread function
ESF
normalized spatial signal distribution in the linearized output of an imaging system resulting from
imaging a theoretical infinitely sharp edge
[SOURCE: ISO 12231:2012, definition 3.43]
2.5
effective cut-off frequency
see cut-off frequency of the contrast transfer function, Note 1 to entry
2.6
effective lateral resolution
minimum spacing of two stripes of a square-wave grating at which the dip of signal intensity between
two maxima of the image is at least 4 times the reduced noise σ
NR
2.7
generalized contrast transfer function
see contrast transfer function, Note 1 to entry
2.8
image contrast
c
i
c = (I –I )/(I +I ) = ΔI/2 I (Michelson contrast), where I , I and I are signal
i max min max min mean max min mean
intensities in the image
Note 1 to entry: Other definitions (not used in this document) include: difference in signal between two arbitrarily
chosen points of interest (P , P ) in the image field, normalized by the maximum possible signal available under
1 2
the particular operating conditions, (ISO 22493:2008, definition 5.3).
cS=−SS
i 21 max
Note 2 to entry: With respect to aperiodic patterns the Weber contrast c = (I – I )/I is used to quantify the
b b
contrast between a feature with the signal intensity I and the background signal intensity I .
b
Note 3 to entry: With respect to periodic object patterns, the terms contrast and modulation often are used
synonymously.
2.9
image resolution
minimum spacing at which two features of the image can be recognised as distinct and separate
[SOURCE: ISO 22493:2008, definition 7.2]
2.10
lateral resolution
minimum distance between two features (in this document the period of a square wave grating) which
can be imaged in that way, that the dip between two maxima is at least 4 times the reduced noiseσ
NR
(cf. 4.4.2.3)
Note 1 to entry: This definition is in accordance with the definition of image resolution given in ISO 22493:2008.
Note 2 to entry: This definition is different from the definition of lateral resolution given in ISO 18115:2010.
2 © ISO 2013 – All rights reserved

2.11
linear system
system whose response is proportional to the level of input signals
[SOURCE: ISO 9334:1995, definition 3.1]
2.12
line spread function
LSF
normalized spatial signal distribution in the linearized output of an imaging system resulting from
imaging a theoretical infinitely thin line
[SOURCE: ISO 12231:2012, definition 3.94]
2.13
modulation
m
measure of degree of variation in a sinusoidal signal
mI=−II +I
() ()
maxmin maxmin
[SOURCE: ISO 9334:1995, definition 3.17]
2.14
modulation transfer function MTF
ratio of the image modulation to the object modulation as a function of spatial frequency
[SOURCE: ISO/IEC 19794-6:2011, definition 4.7]
2.15
noise
time-varying disturbances superimposed on the analytical signal with fluctuations leading to
uncertainty in the signal intensity
Note 1 to entry: An accurate measure of noise can be determined from the standard deviation of the fluctuations.
Visual or other estimates, such as peak to peak noise in a spectrum or in a line scan, may be useful as semiquantitative
measures of noise.
[SOURCE: ISO 18115:2010, definition 5.315]
Note 2 to entry: By averaging over S /4 data points of the line scan over a square-wave grating the standard
PP
1/2 1/2
deviation of noise σ can be reduced by a factor of (S /4) . σ = (4/S ) σ is called reduced noise in this
N PP NR PP N
document (cf. 4.4.2.3). S means number of sampling points per period.
PP
2.16
object pattern
spatial distribution of a sample property seen by the imaging instrument
[SOURCE: ISO 9334:1995, definition 4.1]
2.17
optical transfer function
OTF
frequency response, in terms of spatial frequency, of an imaging system to a sinusoidal object pattern
and Fourier transform of the imaging system’s point spread function
[SOURCE: ISO 9334:1995, definition 3.8]
2.18
point spread function
PSF
normalized distribution of signal intensity in the image of an infinitely small point
[SOURCE: ISO 9334:1995, definition 3.5]
2.19
reduced noise
see noise, Note 2 to entry
2.20
Rose criterion
condition for an average observer to be able to distinguish small features in the presence of noise, which
requires that the change in signal for the feature exceeds the noise by a factor of at least three
[SOURCE: ISO 22493:2008, definition 5.3.7]
2.21
sample area viewed by the analyser
two-dimensional region of a sample surface measured in the plane of that surface from which the
analyser can collect an analytical signal from the sample or a specified percentage of that signal
2.22
sampling points per period
S
PP
grating period divided by sampling step width
Note 1 to entry: For the case of 3-stripe gratings the image of the grating may have a smaller period than the
object grating (cf. 4.4.1.1). In this case it must be explained whether the grating period of the object or the image
is considered.
2.23
signal-to-noise ratio
R
S/N
ratio of the signal intensity to a measure of the total noise in determining that signal
[SOURCE: ISO 18115:2010, definition 5.427]
2.24
spatial frequency
reciprocal of the period of a periodic object pattern (grating)
3 Symbols and abbreviated terms
AES Auger electron spectroscopy
c image contrast
i
c object contrast
o
(c /c ) c /c at the threshold of resolution
i o ThR i o
CTF contrast transfer function
d distance between two narrow stripes
D dip between two maxima
d distance between two consecutive gratings
gr
4 © ISO 2013 – All rights reserved

D data distance of the MTF calculated by Fourier transform
LSF
D dip at the threshold of resolution
ThR
D ESF steepness parameter giving the distance between points of well-defined intensities x
x-(100-x)
and 100-x (e.g. 20 % to 80 %) of the profile over a straight edge
DNR dip-to-noise ratio
erf error function
ESF edge spread function
F fit range
r
FWHM full width at half maximum
G gap between two stripes
G(x) Gaussian function
i(x,y) normalized intensity distribution of measured signals in the image
I incident beam current (in AES)
i
I maximum value of signal intensity in the image of a 3-stripe-grating (A-B-A)
max
I signal intensity of the left maximum in the image of a 3-stripe-grating (A-B-A)
max l
I signal intensity of the right maximum in the image of a 3-stripe-grating (A-B-A)
max r
I signal intensity of the minimum in the image of a 3-stripe-grating (A-B-A)
min
I intensity of the lower plateau of constant concentration
pll
I intensity of the upper plateau of constant concentration
plu
J (r) intensity distribution of detected Auger electrons as a function of the radius r
A
J (r) intensity distribution of detected Auger electrons that were created by backscattered
Ab
J (r) intensity distribution of detected Auger electrons that were created by the incident beam
Ai
k spatial frequency
k steepness parameter of the logistic function
S
L length
L(x) Lorentzian function
L Length of a plateau of constant concentration
pl
LSF line spread function
m modulation
m image modulation
i
m object modulation
o
M length range of measured LSF values
LSF
MTF modulation transfer function
o(x,y) object pattern
OTF optical transfer function
P grating period
P period of the largest non-resolved grating
P period of the first (finest) resolved grating
P period of the second resolved grating
P period at R = 4 determined by interpolation between P and P
int D/RN 0 1
P period at R = 4 determined by extrapolation with P and P
ext D/RN 1 2
PSF point spread function
PSV1 type 1 Pseudo-Voigt function
PSV2 type 2 Pseudo-Voigt function
q grading factor of consecutive grating periods q = P /P
n+1 n
R backscattering factor (in AES)
r radius from the centre of the incident electron beam on the sample surface (in AES)
r effective lateral resolution
e
R ratio of dip-to-reduced-noise
D/RN
R length range where LSF data are used for Fourier transform
LSF
R signal-to-noise ratio
S/N
r upper limit of integration in Formula (65)
max
s mean deviation of w determined by a fitting procedure
LSF
S sampling step width
w
S sampling points per period as a variable
pp
spp dimension unit of the variable S
PP
SIMS Secondary Ion Mass Spectrometry
u uncertainty of a quantity
U expanded uncertainty of a quantity
U combined expanded uncertainty of a quantity
c
w full width at half maximum of a peak function
w full width at half maximum of the Gaussian part of a type 2 Pseudo-Voigt function
G
w full width at half maximum of the upper plateau of constant concentration in an image of
im
6 © ISO 2013 – All rights reserved

w full width at half maximum of the Lorentzian part of a type 2 Pseudo-Voigt function
L
w full width at half maximum of the line spread function
LSF
w width of a stripe in the object pattern
s
x, x´ length variable
XPS X-ray photoelectron spectroscopy
y, y´ length variable
Δr lateral resolution
Δr (50) lateral resolution determined from a 25% to 75% intensity change in a line profile over a
straight edge
η Lorentzian fraction of a Pseudo-Voigt function
σ Gaussian parameter describing the radial distribution of backscattered electrons (in AES)
b
σ Gaussian parameter describing the radial distribution of the incident electron beam (in
i
AES)
σ standard deviation of noise
N
σ standard deviation of reduced noise
NR
4 Determination of lateral resolution and sharpness by imaging of stripe patterns
4.1 Theoretical background
4.1.1 Image formation and the point spread function (PSF)
The imaging process describes the formation of an image as a result of the interaction between an object
and an imaging system. The object may be characterized by the object pattern o(x,y). This is determined
by a distribution of a certain parameter, for instance a concentration of an element, in the object plane (x,
y) and the relation of this parameter to the respective signal intensity seen by the imaging instrument.
The imaging system is represented by its point spread function (PSF). The PSF(x-x´, y-y´) is the normalized
intensity distribution of measured signals in the image i(x´,y´) related to a point at position (x,y) in the
object pattern o(x,y).
For linear systems (cf. terms and definitions) the image is formed by the superposition of all intensity
[8]
distributions produced in the image plane by each individual point of the object pattern o(x,y). This is
mathematically described by the convolution integral
+∞
id(´x´,y )o=−(,xy)PSF(´xx,´yy− ) xyd (1)
∫∫
−∞
This convolution integral can be written as
i(x, yx)o=⊗(),y PSF(x, y) (2)
where ⊗ denotes the convolution operation. Formulae (1) and (2) reveal that the image is a weighted
sum of point spread functions emerging from every point of the object. Figure 1 illustrates the image
formation and the influence of the PSF on the image quality in terms of sharpness.
Figure 1 — Top: Imaging of a square simulated by the convolution of the square with a Gaussian PSF
where ⊗ denotes the convolution operation. Bottom: X-cuts of object, PSF and image, respectively
In the example given in Figure 1 the dimension of the object and the PSF are of the same order of magnitude.
However, for practical applications, two borderline cases of imaging are of particular interest:
1. If the FWHM of the PSF is small compared to the smallest details of the imaged object, then the
convolution yields an image that is very similar to the original object. In that case the imaging
process (Figure 2a) delivers sharp images of the object.
2. If the FWHM of the PSF is large compared to the imaged object, then the convolution yields the PSF
(Figure 2b). The latter case can be exploited to determine the PSF without a deconvolution procedure.
The PSF describes the performance of an imaging instrument with respect to lateral resolution and
the sharpness of images obtained. The smaller the FWHM of the PSF the better is the lateral resolution.
Figure 2 — Two borderline cases of imaging: a) The object is large compared to the FWHM of
the PSF. This case is ideal for imaging. b) The object is small compared to the FWHM of the PSF.
This case is ideal for the determination of the PSF
8 © ISO 2013 – All rights reserved

4.1.2 The line spread function (LSF)
The LSF is the normalized intensity distribution in the image of a narrow line and yields a one-dimensional
description of image quality. According to the model of image formation described above (Figure 1) the
LSF corresponds to the convolution of the PSF with an infinitely narrow line, mathematically described
by the Dirac delta function δ(x):
∞ ∞
LSF(xx)P=−SF(´,)yxδ(´xx)´ddy (3)
∫ ∫
−∞−∞
+∞
= PSF(xy,)dy (4)
∫
−∞
The LSF is generally different from a cross section through the two-dimensional PSF. In Figure 3 this is
demonstrated for the top hat distribution. Only in the case of a PSF represented by a two-dimensional
Gaussian distribution the LSF is identical to the corresponding one-dimensional distribution:
I 22
Gex =−xp ()xx− 0 /2σ (5)
()
()
σπ2
The LSF approach is more often used for the determination of lateral resolution than the PSF approach
and the full width at half maximum (FWHM) of the LSF is often used as a measure of lateral resolution.
However, with the availability of well-defined nanoscaled pointlike objects, the PSF approach may
become relevant in the future, too. When a narrow line is imaged, a considerable number of line scans
can usually be added by appropriate software tools and the LSF information is obtained at reasonable
signal-to-noise ratios.
Finally it should be mentioned that the LSF is not necessarily a Gaussian shaped function. Other shapes
as Lorentzian, Voigt function, etc., are possible (cf. 4.2.1). Therefore two imaging instruments having
LSFs with the same FWHM but with different shapes will differ in the lateral resolution which can be
achieved (this effect will be demonstrated in 4.4.3.1).
4.1.3 The edge spread function (ESF)
The ESF is the intensity distribution in the image of an edge (step transition) measured in the direction
perpendicular to that of the edge. The ESF is the integral of the LSF
x
ESF(xx)(= LSFd´) x´ (6)
∫
−∞
The ESF may be determined by a convolution of the PSF with a step function.
The distance between points of well defined relative intensity (e.g. 12 %–88 %, 16 %–84 %, 20 %–80 %
or 25 %–75 %) in an ESF is often taken as a measure of lateral resolution. For a Gaussian LSF the distance
between the 12 % and 88 % intensity points (indicated in Figure 4) corresponds to its FWHM.
Figure 3 — Determination of the LSF by imaging of a narrow line. Different types of PSF were
convoluted with a narrow line. The z-axis denotes the signal intensity within the images (from [9])
10 © ISO 2013 – All rights reserved

Figure 4 — Determination of the ESF by imaging of an edge. The value D is used as a
12-88
measure of lateral resolution
4.1.4 The modulation transfer function (MTF)
The concept of the optical transfer function (OTF) was developed to characterize the performance of
[8,10]
imaging systems. It was adapted from electronic and communication engineering to optical imaging
and is based on the transfer of sinusoidal signals. “The optical transfer function (OTF) is the frequency
response, in terms of spatial frequency (cf. terms and definitions), of an optical system to sinusoidal
[10]
distributions of light intensity in the object plane” . “The part of OTF describing the reproduction
of contrast is called the modulation transfer function (MTF), while the phase component is called the
[8]
phase transfer function (PTF)” . Both parts of the OTF may be determined by imaging a sine wave
grating. With respect to surface analytical methods, only the MTF is of interest.
The modulation of periodic patterns in objects and images is defined as
mI=−II +I (7)
() ()
maxmin maxmin
where I is the maximum value of a periodic structure and I is the minimum value between two
max min
maxima (cf. Figure 5).
Figure 5 — Definition of modulation m
The MTF describes the transfer of the object modulation m to the image modulation m as a function of
o i
spatial frequency k
MTF (k) = m /m (8)
i o
where k is the reciprocal of the period of a sine wave grating. The object modulation m is based, for
o
instance, on differences in sample composition and it can be determined from the respective image
profile as outlined in 4.4.1.3.
An ideal imaging instrument is characterized by m = m and correspondingly MTF = 1 for all k values.
i o
In reality imaging is always characterized by a decreasing image modulation m vs. increasing spatial
i
frequency (Figure 6). Therefore the MTF can be used to describe the performance of an imaging
instrument. The MTF is directly related to its lateral resolution.
12 © ISO 2013 – All rights reserved

Figure 6 — Imaging of sine wave gratings with different periods and the transfer of modulation
from object to image
Another definition of the optical transfer function OTF is based on the fact, that the OTF is the Fourier
transform (FT) of the point spread function P
∞
OTF(kl,)==FT[PSF(,xy)] PSF(xy,)exp[−+ix2π()kylx]ddy (9)
∫∫
−∞
where k and l are spatial frequency variables associated with the space coordinates (x,y), respectively.
The OTF is a complex function and the MTF is the normalized modulus of the OTF. For the one-dimensional
case (and only this will be treated below) the MTF is given by the Fourier transform of the line spread
function (LSF)
∞ ∞
MTF(kk)O==TF() LSF(xi)exp()− 2πxk ddxx/LSF() x (10)
∫∫
−∞ −∞
A narrow LSF in position space yields a wide MTF in spatial frequency space, and vice versa. The Fourier
transform of a Gaussian distribution is again Gaussian and therefore a Gaussian LSF yields a Gaussian
MTF. Figure 7 demonstrates that both methods of calculating the MTF, Fourier transformation of the LSF
and determination of m from the image of a sine wave grating, yield exactly the same values of the MTF.
i
Figure 7 — Calculation of the modulation transfer function (MTF) by Fourier transform of the
line spread function (LSF). The black dots are m /m values taken from Figure 6
i o
4.1.5 The contrast transfer function (CTF)
Optimal samples for a determination of the lateral resolution of beam-based imaging methods of surface
analysis have a flat surface and a high material contrast. In the sub-100 nm range, this requirement is
fulfilled by square-wave gratings, whereas flat sine-wave gratings are not available. Furthermore, the
sharp contrast at the edges of a square-wave grating enables the determination of the LSF and ESF. For
this reason we describe the determination of lateral resolution (cf. 4.4) using this kind of grating.
In analogy to the modulation m of sine-wave gratings the contrast of square-wave gratings is defined
by c = (I – I )/(I + I ). The variation of contrast with spatial frequency is described by the
max min max min
contrast transfer function
CTF (k) = c /c (11)
i o
where c and c are the contrast of image and object pattern, respectively (cf. terms and definitions). In
i o
Figure 8 simulation results of the imaging of square-wave gratings and sine-wave gratings are displayed
for high and medium resolution and at the limit of resolution as well. The imaging system is represented
here by a Gaussian LSF with 50 nm FWHM. In all cases the contrast c in the image of a square-wave
i
grating is higher than the modulation m in the image of the sine-wave grating. If the grating period
i
is large compared to the FWHM of the imaging system’s LSF (300 nm period grating), the intensity in
the image of the square-wave grating drops to zero between the strips of the grating providing c = 1,
i
whereas this is principally not the case for the sine-wave grating. A plateau of the CTF (c /c = 1) at low
i o
spatial frequencies for square-wave gratings appears accordingly. Imaging of sine-wave gratings yields
for a Gaussian LSF a Gaussian MTF (cf. Figures 7 and 9).
14 © ISO 2013 – All rights reserved

Figure 8 — Imaging of square-wave gratings (black lines) and sine-wave gratings (grey lines) of
different periods. c and m are contrast and modulation in the image of a square-wave and sine-
i i
wave grating, respectively. Note the different length scales
Figure 9 — CTF and MTF determined from images of square-wave gratings and sine-wave
gratings, respectively. The imaging system is characterized by a 50 nm FWHM Gaussian LSF.
The black symbols correspond to values taken from images displayed in Figure 8
4.1.6 Classical resolution criteria
The most commonly used resolution criterion in microscopy is the Rayleigh criterion: “Two point sources
are just resolved if the central maximum of the intensity diffraction pattern produced by one point
[2]
source coincides with the first zero of the intensity diffraction pattern produced by the other” . It is an
empirical estimate of resolution and corresponds to a decrease of intensity (dip) of 19 % (rectangular
aperture) or 26.4 % (radial aperture) from the intensity of the two maxima. The threshold of resolution
defined by the Rayleigh criterion reflects rather the performance of visual inspection than the sensitivity
of modern instruments with sophisticated detectors. Because the Rayleigh criterion needs a rather
clear separation of features (expressed by the depth of the dip), it leads to a resolution which is worse in
comparison to resolutions obtained by more appropriate criteria.
[7]
The Sparrow criterion defines the lowest resolution threshold that is possible in principle: the
appearance of a dip between two maxima of signal intensity. In practical imaging noise prevents
the detection of a very small dip between two maxima. As a consequence the resolution determined
according to the Sparrow criterion is unrealistically high. Three grating profiles are resolved in
Figure 10 according to the Sparrow criterion and only one grating profile is resolved according to the
Rayleigh criterion.
[11]
The Rayleigh criterion, the Sparrow criterion and other so-called classical resolution criteria are
related to the pointspread function of the imaging instrument and do not take into account measurement
conditions such as noise and sampling step width. All classical criteria do not cover object contrast
issues. Therefore they give rather a theoretical limit of resolution and their application in imaging surface
analysis is not straightforward. The application of the Rayleigh criterion and the Sparrow criterion in
surface analysis has been discussed in Ref. [12] but they do not play a role in practical surface analysis.
16 © ISO 2013 – All rights reserved

Figure 10 — Application of the Rayleigh criterion and the Sparrow criterion to simulated image
profiles over square-wave gratings with different periods
4.1.7 Comparison of functions, parameters and methods related to effective lateral resolution
and sharpness
See Tables 1, 2 and 3.
18 © ISO 2013 – All rights reserved
Table 1 — Summary of essential results of the methods related to effective lateral resolution
Feature or Method of determination Relation to effective lateral Advantages Disadvantages Subclause
function resolution
Effective lateral Visual determination of r from an Period of the finest resolved grat- Real time procedure with- Subjective decision whether a grat- 4.4.2
e
resolution, r image of a series of square-wave grat- ing corresponds to r out any data treatment ing is resolved or not
e e
4.4.4
ings
Large uncertainty of r at large
e
values of the grading factor q
Effective lateral Determination of dip-to-reduced-noise Calculated value of r is based on Criterion-based calculation Numerical evaluation of noise and 4.4.2
e
resolution, r ratios R from a linescan over three the resolution criterion (37) of r dips from a linescan over three
e D/RN e
4.4.4
square-wave gratings, calculation of r gratings
e
from the intersection of R (P) with
D/RN
R = 4 according to Formula (39)
D/RN High accuracy of r also at
e
large values of the grading
factor (q ≤ 1.8)
Effective cut-off Determination of the intersection r corresponds to the reciprocal of Criterion based method for Time-consuming evaluation of 4.4.3.3
e
frequency of the between CTF and (c /c ) the effective cut-off frequency of the determination of r noise and dips from a linescan over
i o ThR e
CTF the CTF a series of gratings

Table 2 — Summary of essential results of the methods related to sharpness
Feature or function Method of determination Relation to sharpness Advantages Disadvantages Sublause
CTF (contrast transfer Linescan over a series of square-wave Describes sharpness over Response of an imaging sys- Time-consuming numerical evaluation 4.1.5
function) gratings, image contrast as function the whole range of spatial tem over the whole range of of noise and dips from a linescan over a
4.4.3
of spatial frequency frequencies spatial frequencies series of gratings
LSF (line spread func- Linescan over a narrow stripe LSF describes the sharp- Simple procedure of measure- 4.1.2
tion) ness of an image ment
4.2.1–4.2.5
FWHM of LSF, w Linescan over a narrow stripe, deter- Measure of sharpness of Simple procedure Shape of the LSF is not considered 4.2.1–4.2.5
LSF
mination of FWHM an image
MTF (modulation LSF from linescan over a narrow Describes sharpness over Response of an imaging sys- Tails of LSF have to be measured with 4.1.4
transfer function) stripe the whole range of spatial tem over the whole range of high accuracy
4.2.6
frequencies spatial frequencies

MTF by Fourier transform of LSF Noise affects the spectrum of spatial
frequencies
ESF (edge spread func- Linescan over a straight edge (step ESF describes the sharp- Simple procedure For long-tailed LSFs the 0% and 100% 4.1.3
tion) transition) covering the 0% and ness of an image levels of ESF may be not covered by
4.3
100% levels of signal intensity measurement
D , D , D Linescan over a straight edge (step Measure of sharpness of Simple procedure 0% and 100% levels of ESF may be not 4.1.3
12–88 16-84 20–80
ESF rise parameters transition) covering the 0% and an image covered by measurement, then D
x-(1-x)
4.3
100% levels of signal intensity may be underestimated

20 © ISO 2013 – All rights reserved
Table 3 — Requirements for test samples
Feature Method of determination Type of sample Required sample properties Subclause
Effective lateral resolution r Visual determination of r from an Series of chemical square-wave Grading factor q = P /P of grating periods P: 4.4.1.1
e e n+1 n
image of a series of square-wave grat- gratings A–B–A (materials A and
Optimum: 1.2 ≤ q ≤1.5 4.4.1.2
ings B) with graded grating periods P
Calculation according to Formula (39): q ≤ 1.7 4.4.4
Calculation of r from the intersection
e
of R (P) with R = 4 according to
D/RN D/RN Optimum distance d between two consecutive
gr
Formula (39)
gratings:
Determination of r as the reciprocal of
e  General: 1.5 P ≤ d ≤ 3 P
gr
the effective cut-off frequency of the CTF
Gaussian LSF: P ≤ d ≤ 2 P
gr
Expanded (k=2) uncertainty of certified grating
periods:
U(P) ≤ 10 %
Line spread function, LSF Linescan over a narrow stripe Narrow stripe Stripe width w to ensure Δw < 10 %: 4.2.2
s LSF
FWHM of the LSF, w  Gaussian LSF: w ≤ 0.6 w 4.2.7
LSF s LSF
Modulation transfer function,  Lorentzian LSF: w ≤ 0.4 w
s LSF
MTF
Stripe distance d to ensure Δw < 6 %:
LSF
Gaussian LSF: d ≥ 2.5 w
LSF
Lorentzian LSF: d ≥ 5 w
LSF
Edge spread function ESF Linescan over a straight edge (step tran- Chemical edge: material A–mate- 4.3.3
Plateau lengths L to ensure ΔD < 10 %:
pl x-(1-x)
ESF rise parameters D , sition) covering the 0% and 100% levels rial B with extended lower (A)
...