Comparison of optical surface roughness measured by stylus profiler,
AFM and white light interferometer using power spectral density
CHEN Jianchao, SUN Tao*, WANG Jinghe
Center for Precision Engineering, Harbin Institute of Technology, 92 West DaZhi street, Harbin,
P.R. China 150001
Surface roughness measurements were performed on fused-silica, silicon wafer, and glass-ceramic (Zerodur) by
mechanical stylus profiler, atomic force microscope (AFM), and white light interferometer (WLI). Because of the
differences of spatial frequency bandwidth of the measurement instruments surface roughnesses are actually not directly
comparable. In this study, a novel method has been developed to directly compare the roughnesses measured with
different instruments using power spectral density (PSD) function which can be calculated from the measurement data.
The RMS roughnesses were obtained by integrating areas in the overlapping regions of two or more instruments so that
roughnesses measured with different instruments could be directly compared. The agreement among RMS roughnesses
measured with the different techniques improved considerably, and the remaining differences could be explained as
being caused by surface features to which the instruments responded differently. This fruitful work also provides a great
guidance for the selection of surface roughness measurement instruments.
Keywords: surface roughness, AFM, stylus, white light interferometer, power spectral density (PSD)
There is an increasing necessity to use smooth surfaces with a roughness in the sub-nanometer range for X-ray optics,
semiconductor application, and mirrors with high reflectivity, which requires the ability to specify the roughness of
surfaces more explicitly. Currently, to improve the measurement accuracy, roughness measurements made on the same
samples are carried out by using different types of instruments with different bandwidth limitation[3, 4], and the surface
characteristic is customarily described by the statistic parameter root-mean-square (RMS) roughness. Numerous attempts
have been made to compare the measurement results made on smooth optical surfaces with different instruments and to
compare the performances of various types of instruments by investigating the discrepancies of the measured RMS
roughnesses[4, 5]. However, these RMS roughnesses are actually not directly comparable because the measurement
instruments and techniques have different surface spatial frequency bandwidth. It is now possible to use power spectral
density (PSD), a powerful and compact expression of surface characteristics, to correct for the spatial frequency
bandwidth limit differences of the individual instruments. The one dimensional PSD represents the surface height
squared (roughness power) per spatial frequency. The integration over a frequency interval delivers the average
roughness power respectively the square of RMS. And the two dimensional PSD gives information about the relative
contribution of all the possible surface spatial frequencies. For a one-dimensionally rough (isotropic) surface, for which
the height is independent of displacements in a particular direction, we can describe the surface by means of a one
dimensional PSD which also can be calculated from the two dimensional PSD.
In this paper, surface topography (profiles) were first measured on fused-silica, silicon wafer, and glass-ceramic
(Zerodur) using white light interferometer (WLI), atomic force microscope (AFM), and mechanical stylus profiler
respectively. One dimensional PSDs for the stylus profiler and two dimensional PSDs for the AFM and the WLI of each
sample were calculated from the digitized measurement data. And then we calculated one dimensional PSDs for the
AFM and the WLI from the two dimensional PSDs. Finally, RMS roughnesses were calculated from the areas under the
PSD curves for the overlapping frequency range to directly compare measurement results.
Well-polished fused-silica, standard chemical-mechanical polished silicon wafer, and polished glass-ceramic (Zerodur)
were used in this study. For these samples, the surface finish was found to be essentially isotropic (Section 4), allowing
straightforward comparison of the one dimensional (1D) PSD spectra (Section 3) obtained by convolution of the two
dimensional (2D) PSD distributions (Section 3) measured with the white light interferometer (WLI) and the atomic force
microscope (AFM) and the 1D PSD spectra measured with mechanical stylus profiler.
A white light interferometer (WLI) with 10x and 50x objectives, the CCI (Coherence Correlation Interferometer) 2000
developed by Taylor Hobson, was initially used to measure the samples. CCI 2000 is a non-contact areal measurement
system based upon white light interferometry. In this system, a light beam with very short coherence length is split into
two beams, one directed to the sample and the other directed to a smooth reference mirror, and then the light beams
reflected by the sample and by the reference mirror are recombined and transmitted to an array camera. Due to the low
coherence of the light used, interference fringes can be produced only in the case that the optical path length to the
sample and the reference is almost identical. Therefore, when a scanning actuator is used to move the microscope
vertically with regard to the sample, this equal path condition can be detected for each local area of the surface
corresponding to each pixel of the camera, which will then result in a topographic image.
An atomic force microscope (AFM) with an silicon nitride cantilever having a tip radius of about 20 to 50 nm was then
used in the contact mode to obtain up to seven images (512 × 512 pixels) covering areas of 5 × 5 μm2, 10 × 10 μm2, 20 ×
20 μm2, 40 × 40 μm2, 50 × 50 μm2, 60 × 60 μm2, and 80 × 80 μm2 of the samples. In the contact mode, the sharp tip is
maintained in contact with the surface under very small loads (typically in the order of 1nN, obviously, the tip
indentation in the samples studied here is not significant). As the tip is moved across the sample surface, the normal
force on the tip is kept constant by moving the tip away or closer to the scanned surface by a control feedback system.
The movement of tip represents the surface height variation. The instrument used here was a NanoScope Dimension
3100 AFM with NanoScope software, from Digital Instruments, Inc. (DI), now Veeco Metrology Group.
Linear profiles of the samples were finally measured by a mechanical stylus profiler (model Form Talysurf PGI 1240
manufactured by Taylor Hobson) using a 2 μm standard conical diamond stylus tip. Five different positions were
measured along the same direction on each sample, and the length of each profile is 4960 μm (data point spacing, 0.125
μm; stylus speed, 0.5 mm/s; number of data points, 39680).
Characteristics of the measurement results obtained with these three instruments are summarized in Table 1. The column
headed “Size” contains the size of the image (WLI, AFM) or profile length (stylus profiler). The one headed “Number”
contains the resolution of topographic images (WLI, AFM) or the number of traces and the number of points in each
trace (stylus profiler). Another one headed “Spatial frequency range” contains the computed frequency range as
explained in Section 3. And the last column gives a descriptive marker for each measurement.
Table 1: Characteristics of the measurement results.
Instrument Size Number Spatial frequency range (μm-1) Label
WLI 10x 1761 μm × 1761 μm 512 × 512 5.7 × 10 to 0.15 CCI 10x
WLI 50x 360 μm × 360 μm 256 × 256 2.8 × 10 to 0.35 CCI 50x
AFM 5 μm× 5 μm 512 × 512 0.2 to 51.2 AFM 5μm× 5μm
AFM 10 μm × 10 μm 512 × 512 0.1 to 25.6 AFM 10μm× 10μm
AFM 20 μm × 20 μm 512 × 512 5.0 × 10-2 to 12.8 AFM 20μm× 20μm
AFM 40 μm× 40 μm 512 × 512 2.5 × 10 to 6.4 AFM 40μm× 40μm
AFM 50 μm × 50 μm 512 × 512 2.0 × 10 to 5.12 AFM 50μm× 50μm
AFM 60 μm× 60 μm 512 × 512 1.7 × 10 to 4.27 AFM 60μm× 60μm
AFM 80 μm× 80 μm 512 × 512 1.25 × 10 to 3.2 AFM 80μm× 80μm
Stylus Profiler 4960 μm 5 × 39680 -4
2.0 × 10 to 4.0 PGI
This measurement was not performed on silicon wafer.
All topographic images of surfaces in this paper are in the form of digitized data of surface heights, either as an one
dimensional (1D) profile in the form of z(x) where x is the coordinate on the surface (0 < x < L), L is the profile length,
and z is the surface height measured relative to the mean surface level; or as a two dimensional (2D) array in the form of
z(x, y) where x and y are the coordinates on the surface. The area was in the form of a square that was restricted to the
interval 0 < x, y < L, where L is the maximum dimension and z(x, y) is measured relative to the mean surface plane. The
height data z(x) and z(x,y) generally contain trending components such as offset, tilt, and curvature that are independent
of the microtopography of the surface being measured. These components must be subtracted before calculation of the
PSD and RMS roughness. Detrending of the 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. All of the instruments used in this study are equipped with data analysis softwares having the
detrending capability. Therefore, the height data z(x) and z(x,y) used for PSD calculation here were exported from the
instruments after being corrected from offset, tilt, and curvature by their own softwares. For example, each AFM image
(z(x,y) height data) was put through two correction steps: (1) a Plane-fit program within the NanoScope software was
used to remove any tilt, bow, or S-shapes from the overall image, and (2) a Flatten routine within the NanoScope
software was used to remove unwanted features from the individual scan lines.
The PSD of the 1D profile is customarily defined in its limiting integral form for continuous data sets as 
1 1/ 2 L
PSD1D ( f ) lim
L L 1/ 2 L
z ( x ) exp( 2 ifx )dx , (1)
Where z(x) is the surface profile data and the PSD variable f is the spatial frequency of the surface roughness and is
related to the lateral dimension of the surface features. According to Eq. (1), the RMS roughness of a 1D profile (also
defined in terms of z(x)) can be expressed in terms of the PSD as
1 1/ 2 L
z ( x) dx 20 PSD1D ( f )df ,
L 1/ 2 L
This relationship is very useful as the basis of the comparison method developed in this study.
In practice, a finite number of values of z are measured as noted above. Therefore, it can be assumed that the surface
roughness data set consists of N values for z(x) that are measured at equally spaced intervals ∆x over a total length
L=N∆x. Then the 1D PSD for digitized data can be written as
PSD1D ( f j ) z k exp ( 2 i ) jk N , (3)
N k 1
Where j = 1,2,...,N/2. Eq. (3) gives an expression for the PSD calculated from the profile at the spatial frequency fj =j∆f =
j/(N∆x) = j/L. Thus the frequency range of a computed PSD starts at the lower end with 1/L and goes up to N/2 times of
this frequency, 1/(2∆x). The unit of the 1D PSD is length to the third power.
The 2D PSD of a surface represented by a topographic image z(x,y) is defined by
1 1/ 2 L 1/ 2 L
PSD2 D ( f x , f y ) lim
L L2 1/ 2 L
1/ 2 L
z ( x, y ) exp 2 i( f x x f y y ) dy ,
Where the PSD variables fx and fy are the spatial frequencies of the surface roughness. This 2D PSD has a dimension of
(length)4. In the case of discreet measurements with pixel dimensions ∆x and ∆y, M and N pixels in the x and y
directions, respectively, the 2D PSD distribution can be evaluated from the height distribution zm,n via equation
xy M 1 N 1
PSD2 D ( f l , f k ) z m,n exp 2 i(ml M nk N ) , (5)
MN m0 n0
Where 0 ≤ l ≤ M-1 and 0 ≤ k ≤ N-1. As noted above, the area of topographic images studied here was in the form of a
square with the same data points in the both x and y directions. So in this study, there are M = N, ∆x = ∆y, and also fl =
l/(N∆x) = l/L, fk = k/(N∆x) = k/L, where L is measurement length in either x or y direction.
All samples used in this study were isotropic, and consequently the PSD2D(fx, fy) had polar symmetry according to Eq.
(4). For these surface, we can get the 1D PSD(fx) by integrating the 2D PSD over fy (vice versa, PSD(fy) obtained by
integrating over fx). Thus, the relationship between the 1D and 2D PSD can be written
PSD1D ( f x ) PSD2 D ( f x , f y )df y , (6)
According to Eq. (5), the digital equivalent of Eq. (6) yields the discreet integrated 1D PSD as
PSD1D ( fl ) PSD2 D ( f l , f k )f k , (7)
It can be seen From Eq. (6) and Eq. (7) that PSD1D(fx) is essentially an average of the 1D PSD computed from traces at
fixed values of y. This is the basic idea of computing the averaged PSD as discussed below.
For profile measurements made with the stylus profiler, a 1D PSD can be obtained by applying Eq. (3). In order to be
able to compare this PSD with those obtained with the 3D measurement systems (AFM, WLI) for a given surface, we
used Eq. (7) to get a conversion from the 2D isotropic form to the corresponding 1D PSD. However, the 1D PSD
calculated from single profile obtained with stylus profiler is sensitive to the noise in the measurement and thus cannot
be used in our study. To solve this problem, a simple conversion base on the Eq. (7), calculating 1D PSD results from
each profile and then forming an averaged PSD, is employed. Although this process is time-consuming, as it is necessary
to take many independent profile measurements, it has been done here for cases of stylus profiler. We operated the stylus
profiler to get P (in our cases, P=5) surface profiles of length L that contains N data and calculated the PSD1D(k) for each
profile according to Eq. (3), then the averaged 1D PSD can be written
PSD1Dave = PSD1D(k) ( f j ) ,
P k 1
Where the PSD1Dave is the form of averaged 1D PSD.
4. RESULTS AND DISCUSSION
The RMS roughnesses measured with the three different profiling and imaging instruments are listed in Table 2, and the
unit of each roughness is nm (nanometer). With the exception of the stylus profiler that measured profiles along single
line, all the other two instruments gave topographic images with data points in a square array.
Table 2: RMS roughnesses of samples.
WLI AFM Stylus
Sample 5 μm 10 μm 20 μm 40 μm 50 μm 60 μm 80 μm
10x 50x Profilera
× 5 μm × 10 μm × 20 μm × 40 μm × 50 μm × 60 μm × 80 μm
Fused-silica 3.57 6.24 1.32 1.36 1.87 2.45 2.42 2.53 4.35 6.93
Glass-ceramic 3.56 5.59 1.46 1.56 2.18 3.72 3.47 3.06 3.64 5.42
Silicon wafer 4.46 2.46 -- -- 1.43 1.74 2.13 2.86 3.49 7.86
Stylus profiler values are the average of five readings at different positions on the samples.
It is difficult to compare roughness values taken using different instruments because of the different lateral resolutions,
different data point spacings, and thus the different size measurement areas, especially when one considers that the range
of frequencies that any one instrument can resolve is necessarily limited and unique to that instrument. For these reasons,
the PSDs of the samples studied here were calculated and were used to give RMS roughnesses for common bandwidths.
However, in order to be able to apply Eq. (6) to obtain 1D PSD by convolution of the 2D PSD distributions measured
with the WLI and the AFM, we had first to investigate the property of samples surfaces—whether they were isotropic or
not before introducing the PSDs. Figure 1 presents a AFM image of each sample surface with 20 μm × 20 μm scan area
and 20 nm vertical scale. They demonstrate the good isotropic properties of the sample surfaces.
a) b) c)
Figure 1: 20 μm × 20 μm AFM images with 20 nm data scale of the a) fused-silica, b) glass-ceramic, and c) silicon wafer surfaces.
The isotropic properties of surfaces allow us to use Eq. (7) to calculate the 1D PSDs from the 2D PSDs which were
calculated from the topographic images measured with the AFM and the WLI according to Eq. (5). The 1D PSDs for
different instruments cover different regions of the spatial frequency, as seen in Table 1. We would here show the 1D
PSDs obtained for each sample in Figure 2-4.
One-dimensional PSD (nm ·μm)
10 AFM 10μmx10μm
-4 -3 -2 -1 0 1 2
10 10 10 10 10 10 10
Spatial Frequency (μm-1)
Figure 2: PSDs for fused-silica from data obtained using all the instruments.
One-dimensional PSD (nm ·μm)
0 AFM 10μmx10μm
Band D AFM 5μmx5μm
-4 -3 -2 -1 0 1 2 3
10 10 10 10 10 10 10 10
Spatial Frequency (μm )
Figure 3: PSDs for glass-ceramic from data obtained using all the instruments.
10 AFM 80μmx80μm
One-dimensional PSD (nm ·μm)
2 AFM 40μmx40μm
0 Band D
10 Band A
10 -4 -3 -2 -1 0 1 2
10 10 10 10 10 10 10
Spatial Frequency (μm )
Figure 4: PSDs for silicon wafer from data obtained using all the instruments.
The PSDs make it possible for us to compare RMS roughnesses that were made on different instruments by choosing
common bandwidth regions. We selected five different bandwidth regions to accommodate the various instruments:
Band A covering range 5.675×10-4 --0.1453 µm-1 (stylus profiler (PGI), WLI with 10x objective (CCI 10x) ), Band B
covering the range 2.776×10-3--0.3525 µm-1 (PGI, WLI with 50x objective (CCI 50x)), Band C covering the range 2.776
×10-3--0.1453µm-1 (PGI, CCI 10x, CCI 50x), Band D covering the range 0.0125--0.1453µm-1 (PGI, CCI 10x, CCI 50x,
AFM), and Band E covering the range 0.0125--3.2µm-1 (PGI, AFM). The RMS roughness calculated for the five
different bands according to Eq. (2) are given in Table 3, and the unit of each roughness is nm (nanometer).
Table 3: RMS roughnesses
Band A Band B Band C
(5.675×10-4 --0.1453 µm-1) (2.776×10-3--0.3525 µm-1) (2.776 ×10-3--0.1453µm-1)
Sample PGI CCI 10x PGI CCI 50x PGI CCI 10x CCI 50x
Fused-silica 2.220 3.570 2.024 6.238 1.920 3.154 4.529
Glass-ceramic 3.414 3.562 2.931 5.595 2.737 2.968 4.060
Silicon wafer 4.626 4.463 3.784 2.457 3.544 2.109 2.009
Band D Band E
Sample PGI CCI 10x CCI 50x AFM PGI AFM
Fused-silica 1.751 2.728 3.956 2.092 2.273 2.862
Glass-ceramic 2.546 2.557 3.659 1.833 3.732 2.445
Silicon wafer 3.209 1.203 1.715 3.660 5.211 4.932
When the PSD curves for the different instruments are close together, as for glass-ceramic (Figure 3), silicon wafer
(Figure 4) in the frequency range 5.675×10-4 --0.1453 µm-1 and glass-ceramic (Figure 4) in the frequency range
0.0125--3.2µm-1, the RMS roughness values will be in closer agreement than ones when the curves for the different
instruments are further apart. To better understand the reason for the discrepancy when the PSDs are far apart, we
discussed each instrument in turn. In Figure 2, for fused-silica, the PSDs for the CCI 10x and CCI 50x diverge from the
PGI PSD and AFM PSD at the low-frequency end, and the CCI curve is much too high. There may be an error in the
integral order of interference at discontinuities. From the AFM image (Figure 1, a)), it can be seen that some large peaks
scatters throughout the surface. These peaks may cause the algorithms that determined the zero path difference for the
individual pixels mixed up and assigned an integral half-wave jump where was none, and thus the roughness would be
much higher. The CCI PSDs are also slightly higher than others in the case of glass-ceramic, whose surface also has
some small peaks as shown in Figure 1 b). Furthermore, as an optical method, WLI is sensitive to a number of surface
qualities besides the surface height. These include optical constants, surface slopes, fine surface features that cause
diffraction, and deep valleys in which multiple scattering may occur. In addition, scattering from surfaces within the
optical system produces stray light in the system that can affect the accuracy of an optical profiling method. In this study,
the fused-silica and the glass-ceramic are transparent samples with low reflectivity and high scattering which will cause
higher roughnesses, whereas the silicon wafer with high reflectivity has the PSDs that are aligned better with those of the
stylus profiler as can be seen from Figure 4.
In Figure 2-4, two fundamental peaks (at two frequency points: 6.0×10-2 µm-1 and 9.0×10-2 µm-1 or so) can be found in
the PGI (stylus profiler) PSD for each sample. They have considerably higher PSD values than do the rest of the curves,
presumably due to the fact that the mechanical measurement involve a dynamic measurement system, which is subject to
vibration and drive-rate fluctuations. And the stylus profiler PSDs for fused-silica and the glass-ceramic curl down at the
highest spatial frequencies. The shape can be generally attributed to the inability of the stylus profiler with limited-sized
stylus to resolve the surface structures within highest spatial frequency range. Whereas, as a ultramicroscopic scale of
surface measurement technique, AFM can resolve the true surface structures in that frequency range. As can be seen
from Figure 2-4, the agreement between the seven different PSDs corresponding to the seven AFM scan sizes within the
frequency range from approximately 1 µm-1 to 20 µm-1 is remarkable, suggesting that the measurements performed with
AFM are of high degree of consistency and stablility. At lower frequencies, between about 1.25 × 10-2 µm-1 and 1µm-1,
there is a significant spread for the PSD magnitude obtained in the scans over different areas, and these lower frequency
roll-off systematically seen for all AFM measurement shown in Figure 2-4 is due to the detrending procedure discussed
in Section 3.
Surface roughness measurements were performed on fused-silica, silicon wafer and glass-ceramic (Zerodur) by
mechanical stylus profiler, atomic force microscope (AFM) and white light interferometer (WLI). The RMS roughnesses
determined from the different measurements for each sample varied widely. The discrepancies are mainly due to the
differences of spatial frequency bandwidth of the measurement instruments. In order to better compare the
measurements, power spectral density (PSD) were calculated for each sample and measuring instrument. And RMS
roughnesses were calculated from the areas under the PSD curves for the same upper and lower spatial frequency limits.
In this way, the problem of each instrument having different spatial frequency bandwidths has been removed, and thus
the RMS roughnesses measured with different instruments could be directly compared.
The agreement between RMS roughnesses measured with the three instruments in some spatial frequency ranges is
gratifying, and some significant discrepancies were also revealed in the RMS roughnesses and in the PSDs, which may
be due to characteristics of the different instruments. Therefore, the characteristics of surface measurement instruments
have to be kept in mind when examing the PSD or the RMS roughness of a surface, and thus the spatial frequency
reponse characteristics of each instrument will be the subject of further investigations.
This study was supported by the 111 Project, contract No. B07018.
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