3-D micro-structure measurement method based
on shadow moiré using Scanning
Yasuhiko Arai and Shunsuke Yokozeki
Kansai University and Jyouko Applied Optics Laboratory
3-3-35, Yamate-cho, Suita, Osaka 564-8680
ABSTRACT
The novel precise three dimensional shape measurement method using SEM and moiré topography has been proposed. The
possibility for measurement of wave length order by this method has been discussed using the results of experiments for
confirming the principle of this idea. In the experiments, the method with high resolution power based on the new
measurement method is also proposed by employing the fringe scanning technology for the shadow moiré. The optical system
is constructed with SEM using backscattering electrons, the grating holder which can shift the position of the grating, and the
grating of which the pitch is 120 micro meter. Measured results using a bearing ball as a sample show that the high resolution
measurement around one macro meter can be performed by introducing the fringe scanning method to the new measurement.
Introduction
When a micro structure which is smaller than the diffraction limit of light is observed, an electron microscope is usually
employed. The electron microscope is an equipment which can observe some micro structures by using electrons [1].
Generally, electron microscopes are roughly classified as TEM (Transmission electron microscopes) and SEM (Scanning
electron microscopes) [1, 2]. In TEM, an electron beam is emitted to a thin test piece, then, the inside structure of the test
piece can be observed by using any electrons which are through the object. On the other hand, in SEM, a focused electron
beam is emitted to the surface of an object, then, the surface of the object can be observed by secondary electrons,
backscattering electrons, or X ray and so on [1].
The spatial resolution power of the SEM is very high, because the focused electron beam is used in the measurement. The
depth of focus of the SEM is also much deeper than a general optical micro scope. So, an object which has some rough
surfaces can be readily observed three-dimensionally. The process of producing a test piece also is easier than that of the
TEM. Some test pieces of metal and semiconductor can be observed directly without process of producing a replica. So, SEM
is widely employed in industrial fields [1].
In this paper, a new precise shape measurement method, which is based on the optical technologies, is proposed by using
SEM as the popular instrument. A SEM image which is taken with secondary electron beam is shown in Fig.1(A). We can
observe the micro cantilever three-dimensionally by the SEM image. Generally, a shape of object is observed by using
secondary electron beam.
The micro cantilever image which is taken using back
scattering electrons is shown in Fig1(B). Here, the same
micro cantilever is observed with back scattering electrons
and secondary electrons in Fig.1(A) and (B). In Fig.1(B), a
shadow of the cantilever exists on the bottom floor. The
shadow can not be observed in the image (Fig.1(A)) which
is taken by secondary electron beam. That is, there is a
shadow of the object on the floor, when we use SEM with
back scattering electron beam. By using this phenomenon
which produces shadows, a new shape measurement
method for micro structure can be proposed based on moiré
topography [3-5] which has been known well as the optical
(A) By secondary electrons (B) By backscattered electrons
shape measurement method. Moiré topography is classified
to the shadow moiré and the projection moiré [5]. In the
Figures1 SEM images
shadow moiré, the measurement is performed by using one
sheet of grating which is set in front of the measured object.
On the other hand, the grating is projected on the surface of the object in the projection moiré, then the projected grating
image is deformed by the shape of the object. The deformed grating image is taken by CCD camera. Then, 3-D shape
measurement can be performed.
In this paper, a novel shape measurement method for micro structures using the concept of the shadow moiré is proposed.
The measurement optical set-up for the new method is constructed with a SEM and a grating. The grating is produced using
micro machining techniques. Then, because the new method is based on the optical technology, the method is discussed on
the optical view point in the same manner as the shadow moiré in this paper, even if SEM is employed as the main component
of this measurement system.
The 3-D shape measurement method based on feature of emitting secondary electrons of SEM has already been proposed [6].
And, products based on this idea have been commercially dealt with in the industrial fields. This method is based on the
technology for reconstructing 3-D shape from 2-D intensity distribution of image. The method is widely employed in industrial
field. In this method, it is reported that resolution power of the spatial measurement is about a few nm. However, the method
requires very expensive and multi high resolution detectors, because 2-D intensity distribution of image has to be detected
precisely and two-dimensionally. As the results, the equipment based on this idea is generally very expensive.
On the other hands, the method for detecting the micro deformation using electric beam has also been reported [7]． In this
method which is called electron beam moiré, micro grating which is produced by electric rethograph is used as a deformable
model grating. Then, the electric beam which is illuminated on an object is used as a master grating. As the results, the
deformation of the object can be detected as moiré fringes． This method realizes the detection of deformation of the object
with a very high resolution. However, the use of this method is limited in the field of the deformation measurement. Though
this method also uses an electron beam for measurement, the method is clearly a different measurement technology from
proposed method, which can measure a three dimensional object shape, in this paper.
Furthermore, the micro three-dimensional shape measurement already has been performed by using ordinary optical shadow
moiré technology [8] as same as like the method proposed in this paper. However, because these methods are based on
ordinary optical technologies, the improvement of the spatial resolution by these methods can not be hoped enough for the
measurement of sub-micro meter order.
The new method proposed in this paper requires only three parts; SEM which can detect backscattering electrons, a grating
and a grating holder for measurement of micro structures. And, the new technique has an important feature, as follows. In the
ordinary optical measurement, for example; interferometry, the spatial resolution power along the optical axis is a few of nm,
however, the lateral resolution power is a few of µm. In such a measurement, the resolution power of each axis of 3-D
directions is unequal in each coordinate. In the new method, the lateral spatial resolution power is a few of 10nm, because the
measurement system in this method is constructed by SEM. At the present time, the resolution power of the proposed method
is around 1µm. However, if high resolution measurement along the optical axis will be improved much more by using fringe
scanning method and a fine grating, the measurement with harmony of spatial resolution power in 3-D can be realized in this
new method on the range of nanometer order.
In this paper, firstly, the novel three dimensional shape measurement method based on the moiré topography method using
SEM with back scattering electrons is proposed and, the validity of the measurement principle is confirmed by experiment
results. Secondly, the high resolution measurement method [9] based on fringe scanning method for shadow moiré [10,11] is
also proposed. Then, the problems for the practical use of this method are discussed in the error analysis.
Measurement system and methods
Introduction of shadow moiré to SEM system
The shadow moiré measurement system is constructed as shown in Fig.2. The fringe depth hN can be calculated by Eq.(1)
[3,10,11].
h
N
=
P 0 Nb
l − NP
(1)
where, P0 is pitch of grating, N is fringe number, b is the length between grating and lens of camera, l is the length between
light source and lens of camera.
Furthermore, the high resolution measurement method can be easily realized by introducing some fringe scanning
technologies to moiré topography. To introduce the new idea to SEM system, the measurement system which has the same
function of ordinary optical shadow moiré has to be set-up in the SEM vacuum chamber. The inside of the vacuum chamber is
shown in Fig.3. In the chamber, there is an electron lens at center. And, detector (scintillatior) is set at side of the lens. Now, as
shown in Fig.4, we think the electron lens and the detector as the camera lens and the light source of the shadow moiré
system shown in Fig.2, respectively. Then, the grating is inclined for corresponding to the grating in shadow moiré system
shown in Fig.2. Then, the grating is set parallel to the connecting line between the lens and the detector. By setting this system,
the measurement system corresponding to that of the shadow moiré can be constructed using a SEM, a grating, and a grating
holder. The grating is produced of SiO2 and by the micro machining technology. The thickness of the grating is 1µm. The moiré
fringe images on plane and sphere surfaces are shown in Fig.5. For taking these images, the grating of which pitch is 200µm
0
E
P0
ℓ
S
b
Grating
N=1
h1
h2
N=2
Figure 3 Position of electron lens and detector in SEM chamber
Figure2 Moiré topography
Figure 5 Moiré fringes on plane and sphere surface
Figure 4 Schematic diagram of measurement system
is used. In each image, the shadow of the grating exists on the surface, then it can be confirmed that the moiré fringes are
occurred with the grating and the shadow of the grating.
High resolution measurement method by employing fringe scanning method[10,11]
In previous section, it is shown that the shadow moiré measurement system based on SEM can be constructed. In this section,
the possibility of the high resolution measurement of the proposed method is discussed by introducing the fringe scanning
method. On the measurement system with SEM and shadow moiré optical system as shown in Fig.2, the high resolution
measurement can be performed by shifting the grating along the optical axis [10,11].
Moiré profile IM(x,z) of moiré image as shown in Fig.5 is given as Eq. (2)[10,11].
I M ( x , z ) = α ( x , z ) + β ( x , z ) cos
2π l h
P 0 (h + b )
(2)
where, α(x,z) is bias component. β(x,z) is amplitude of moire fringes intensity.
Moiré pattern intensity is rewritten to Eq.(3) and Eq.(4) from Eq.(2), when the grating is shifted by ±bshift from original
position[10,11].
I
I
M + b shift
M − b shift
( x , z ) = α ( x , z ) + β ( x , z ) cos
( x , z ) = α ( x , z ) + β ( x , z ) cos
2 π l ( h + b shift )
P 0 (h + b
)
2 π l ( h − b shift )
P 0 (h + b
)
(3)
(4)
For removing the influence of α(x,z) and β(x,z) from Equations (2), (3) , and (4), the function as shown in Eq.(5) is
proposed[10,11].
Electron lens
Electron beam
Detector
Micro meter head
ℓ
b
Precision table
Grating
Shift
Measured object
Figure 6 Grating holder for shifting grating
f (h) =
Figure 7 Grating holder
I M ( x, z ) − I M + b shift ( x, z )
I M −b shift ( x, z ) − I M ( x, z )
2 π l ( h + b shift )
2π l h
− cos
P0 ( h + b )
P0 ( h + b )
=
2 π l ( h − b shift )
2π l h
cos
− cos
P0 ( h + b )
P0 ( h + b )
cos
(5)
The root; h(x,z) of the function; Eq.(5) can be solved using the corresponding value to f(h) which is calculated from the
intensities of real measured results. In this calculation, Newton-Raphson method is employed as shown previous paper[10,11].
Next, the high resolution measurement is performed using above discussion. In order to shift the grating along the optical axis,
the apparatus shown in Fig.6 is prepared in this paper. The apparatus is constructed with grating holder which can move the
grating straight along the optical axis. Then, the movement of the grating is performed by manual operation. The apparatus is
set in the chamber of SEM as shown in Fig.7. For measurement of micro structures, we have to use the grating with a small
pitch. However, because the measurement apparatus is mere manual operation, the precision of positioning is around 1µm.
The present positioning accuracy of the measurement system is not enough for checking precisely the validity of the micro
measurement method. So, the grating of which pitch is comparatively large is prepared in this paper in order to investigate the
validity of the measurement principle.
Calibration of measurement system
We can measure the shape of object by using the system shown above. However, when we try to realize the measurement by
this system, it is very difficult for a SEM's user to define exactly the position of principal points of electronic lens and detector
like the optical shadow moiré system shown in Fig.2. We can not set also the length between lens and grating or the length
between lens and detector as long as we can not define exactly the position of the principal points of electronic lens and the
detector. For solving this problem, a method is also proposed using 3 kinds of bearing balls of which diameters are known.
Three bearing balls with three kinds of known diameters, d1=15/32, d2=13/32, and d3=9/32 inch, are set as the test pieces.
Generally, the accuracy of sphericity of the bearing balls is about 0.5µm according to makers’ catalog data． And the
roughness of that is averagely 0.02µm． So, the bearing ball is the best test pieces in these measurement ranges for
confirming the principle of the measurement.
Here, every ball touches on the grating, when they are measured by new method. They are measured three times. From these
results, the length ( l ) between light source and lens of camera and the length (b) between grating and lens of camera can be
defined by choosing the best condition that difference between the measured results and the original diameter is as small as
possible. Concretely, the ball of which diameter is known is set as a test piece as shown in Fig.8. Then, as shown above the
bearing ball is contacted on the grating. The position of Point-P2 can be detected from the image by the visual observation of
operator. As the results, the position of the center of the moiré fringes pattern can be detected. The radius of moiré fringes
pattern is estimated from moiré image. Finally, fringe depth, h1 can be estimated geometrically because the diameter of the
test ball is known. Now, Eq. (1) is dealt with as a function which includes two unknown variables, when the measured result(h1)
ℓ
b
P2
Ball
o
a
Grating
P1
h1
r
Figure 8 Schematic diagram of definition Figure 9 Moiré fringe image
of l and b using balls
Figure 10 Measurement result
is substituted into Eq. (1). Such calculation is repeated by using balls with three kinds of different diameters. As the results,
lengths, l and b, can be defined experimentally.
Performance evaluation of the method
Measurement results without fringe scanning methods
From calibration method, parameters for measurement system can be defined as l =50.0mm and b=37.6mm. Next, a
measurement based on shadow moiré is performed by using the ball of which diameter is 9/32 inches. The moiré fringes
pattern image is shown in Fig.9. The measured result of the ball on the line(A-A') which is shown in Fig.9 is plotted in Fig.10. In
this measurement, because the ball is also contacted on the grating, three dimensional coordinate of center of ball can be
estimated by using the position of center of moiré fringes pattern and the radius of the ball. The ideal shape of ball on the
line(A-A') shown in Fig.10 can be calculated by using the position of center and the radius of ball. The calculated outline of the
ball is also shown in Fig.10 as a solid line. It is confirmed that the measured results agree with calculated results of the ball
surface well. Standard deviation of the difference between measured and calculated results is 18.8 µm. Furthermore,
estimated diameter of the ball from measured result by least squares method is 7.025mm. It is confirmed that the error of the
measurement can be defined as 0.119mm(1.7%). From the measured results, it is confirmed that the proposed measurement
method using SEM and shadow moiré technology is validity as three dimensional shape measurement method.
Measurement results by using the fringe scanning method
The pitch of the grating is 120µm in this experiment. The ball of which diameter is 13/32 inch is measured as the test piece in
this measurement. Then, the measurement parameters in this measurement system are also defined using the method shown
in section 3, as follows, the shifting value of the grating(:bshift) is ±50µm. And, l =50.0mm, b=31.0mm. Under these conditions,
three moiré fringes images concerning equations (2),(3) and (4) are taken, and shown in Fig.11. In Fig.11, the shadow of the
grating on the surface of the ball can be observed clearly. It can be confirmed that the phase of moiré fringes changes by the
shift of the grating. The Fourier transform result of the moiré fringes pattern image shown in Fig.11 (b) is shown in Fig.12. It is
also confirmed that there are some noise component and the component of grating. To remove such surplus components from
fringe scanning process, a low pass filtering process is used in this paper. The filtered results of moiré images shown in Fig.11
are shown in Fig.13.
The profiles of the moiré fringes image on the line(B-B') shown in Fig.13 are shown in Fig.14. The calculated ratio f(h) on the
line(B-B') using the profiles shown in Fig.14 is shown in Fig.15. However, when the denominator of Eq.(5) is around zero, the
ratio f(h) is infinite. In order to avoid the problem of the convergence condition of Newton- Raphson method in this calculation,
when the value of function(:f(h)) using real intensities is more than one, the numerator and denominator of Eq.(5) are changed
mutually, then g(h) is defined as the reciprocal of Eq. (5)[10,11]. The ratio for solving fringe depth, h, can be prepared within ±1
by using f(h) and g(h).
(A) -50µm
(C) +50µm
(B) Original position
Figure 11 Moiré fringe images for fringe scanning
(A) -50µm
Figure12 Moiré image in frequency domain
(B) Original position
(C) +50µm
Figure 13 Filtered moiré intensities
Figure 14 Profiles of moiré fringes
Figure 15 Ratio functions f(h) and g(h)
Figure 16 Measured result of ball with fringe scanning method
The profile of surface of ball on the line(B-B') shown in Fig.13 as the measurement result is shown in Fig.16. In this case,
because the ball touches the grating at bshift= -50µm, the center of the ball can be also estimated in the fringe image shown
Fig.11(b). According to this calculation, the ideal shape of surface of the ball can be calculated and also plotted in the Fig.16. It
is confirmed that the measured results and the ideal calculating results agree well. The standard deviation of the difference
between them is 1.21µm. From experimental results, the validity of the proposed measurement method using SEM with
backscattering electrons is confirmed. Furthermore, it is also confirmed that the high resolution measurement can be
performed by introducing the fringe scanning method to the proposed method. When the grating of which pitch is 120µm is
used in the measurement, though the grating pitch is comparatively large, it is confirmed that the measurement within around
1µm can be performed.
Error analysis
Effect of set up error of the optical system
The measurement principle of this method is based on the shadow moiré. In the shadow moiré, because the fringe depth; hN,
is calculated by using Eq. (1), the influence of the errors of the pitch of grating; P0, the length between principal point of the
lens and grating; b, and the length between principal points of lens of light source and camera; l can be discussed by using
the propagation of errors. Then, the effects of the propagation of errors can be shown as Equations (6) and (7) [12].
∆h = ∆P0
∂h
∂h
∂h
+ ∆b
+ ∆l
∂P0
∂b
∂l
⎛ ∂h ⎞
2
⎛ ∂h ⎞
2
⎛ ∂h ⎞
(6)
2
⎟⎟ σ p2 + ⎜ ⎟ σ b2 + ⎜ ⎟ σ l2
σ h2 = ⎜⎜
⎝ ∂l ⎠
⎝ ∂b ⎠
⎝ ∂P0 ⎠
0
(7)
By substituting Eq. (1) into Eq. (6), the influence of the errors concerning the pitch of grating; P0, the length between principal
point of the lens and grating; b, and the length between principal points of lens of light source and camera; l can be discussed
under the assumption that P0, b, and l include 10 percent of real value as an error. Under this assumption, each error is
concretely assumed as ∆P0:0.012mm, ∆b:3.76mm, and ∆ l : 5.0mm in this discussion. As shown in the fringes image of
Fig.11(b), the fringe number(:N) of the measurement is dealt with as from 2 to 4 in the practical measurement. Then, when N is
assumed as 2, the total error concerning such these error sources is estimated as 0.055mm. At the same time, when N=3, the
total error can be also estimated as 0.082mm, and when N=4, it can be estimated as 0.110mm. Furthermore, in order to inhibit
an influence of total error of measurement within around 1 µm in the measurement system, it is confirmed that ∆P0, ∆b, and
∆ l have to be less than 140 nm, 45µm, and 60µm, respectively, according to the method of equal effects [12]. From these
calculated results, it is confirmed that the error of the pitch of the grating is quantitatively most sensitive to the total error in the
optical system. In this case that the chamber of the ordinary commercial SEM is employed as the measurement system, it is
also confirmed that the accuracy of the quantitative of the pitch should be about 400 times than that of b and l .
On the other hand, the random error in the measurement can be also discussed using Eq. (7). Assuming that the pitch of
grating; P0, the length between principal point of the lens and grating; b, and the length between principal points of lens of light
source and camera; l include 5 percent error, the standard deviation of the fringe depth can be estimated as 16 µm at N=2. In
the same way, when N=3, it is also estimated as 24 µm, and when N=4, it is 31µm. As shown above, it is confirmed that the
influence of set up error of the measurement system to the measurement result is able to be estimated and discussed by using
the propagation of errors in the proposed method.
Effect of shift error of the fringe scanning method
Because the proposed method is based on shadow moiré technology and fringe scanning method, the shifting error of the
grating would influence to the total error. Now, the total error is discussed when the shifting value of the grating includes an
error. In order to discuss the influence of the shifting error of the grating, firstly, the shape of the sphere is calculated under the
same condition of the measured result shown in Fig.16. This shape of the sphere is the model of evaluation. As the results,
fringe depth hN is given as the coordinate value of the section of the sphere as shown in Fig.17. Consequently, the fringe
profiles are given by substituting the coordinate value of the section of the sphere into Equations (2), (3), and (4). Assuming
that the shifting values of the grating include some errors in this calculating process, the shifting errors of the grating can be
virtually given to the fringe profiles shown in Equations (3), and (4). In the discussion of the influence of the shifting errors of
the grating, the ratio function shown as Eq. (5); f(h) is calculated using these fringe profiles which include the factor of the
shifting error. Then, the fringe depth is calculated using these fringe profiles. By calculating the standard deviation of the
difference between the calculated fringe depth result and the shape of the sphere which has already calculated without any
errors, the influence of the shifting errors of the grating can be investigated. The results are shown in Fig.18. This result shows
that the shifting errors of the grating influence sensitively to the total measurement result. The total error of the measurement
increases suddenly, when the shifting errors are established as, for example, -2 µm and 5µm in Equations (3) and (4),
respectively. In order to discuss this phenomenon, the shape of the detected result is shown in Fig.19.
As shown in Fig. 19, it is confirmed that the existence of the shifting errors in this fringe scanning method brings undulation in
the shape of the measured results. Then, these influences are very sensitive to shifting errors. It is confirmed that the shifting
errors of the grating should be less than ±1.2 µm according to the result shown in Fig.18, in order to inhibit an influence of total
error of measurement within about 1 µm.
h [μm]
100
50
Position [µm]
Figure 17 Simulation model
10
5
Fringe depth
h [µm]
h=-√5.1592-x2 +5.209
5.159 mm
Standard deviation
of error [μm]
20
200
100
0
0
0
Shift error
5
in Eq.(3) [µm]
-5
Shift error
in Eq.(4) [µm] -1000
10
-500
0
Position [µm]
500
1000
Figure19 Cause of error in
fringe scanning method
Figure 18 Simulation result for estimation of error
in fringe scanning method
As shown results, it is confirmed that the high resolution measurement method can be performed by introducing SEM with
back scattering electrons to shadow moiré and by employing fringe scanning method for shadow moiré. Furthermore, even if a
grating of which pitch is comparatively large (for example, 120µm) is employed, it is confirmed that the shape measurement by
proposed method can be performed within around 1µm by using deciding method of parameters of the measurement system
and the error analysis shown in this paper.
Conclusions
In this paper, the novel three dimensional shape measurement method based on the moiré topography using SEM with back
scattering electrons was proposed． The validity of the method was discussed by using results of the experiments. It was
confirmed that the measurement principle was valid in the experiments. Furthermore, by introducing the fringe scanning
method, it was also confirmed that the method can be improved to the high resolution measurement. At the same time, error
analysis of the measurement system was performed. As the results, even if the grating of which pitch is 120µm is employed, it
was confirmed that the measurement within around 1µm can be performed by the proposed method.
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