Advanced Approach for Slopes Measurement by Non - Contact Optical Technique

B. Trentadue. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 4, ( Part -1) April 2017, pp.24-29

www.ijera.com DOI: 10.9790/9622-0704012429 1 | P a g e

Advanced Approach for Slopes Measurement by Non - Contact
Optical Technique
B. Trentadue
Dipartimento di Meccanica, Matematica e Management – Politecnico di Bari Viale Japigia 182 – 70126 Bari,
Italy
ABSTRACT
A numerical computation of a very advanced experimental method to acquire shapes is introduced in this paper.
The basic equations that relate the measurement of slopes to the basic geometric and optical parameters of the
system are derived. The sensitivity and accuracy of the method are discussed. In order to validate the accuracy
and the applicability of this method, the qualitative slope behavior of a loaded metallic layer is given.
Keywords: Slope, curvature, optical technique
I. INTRODUCTION
The moiré reflection method [1] is an optical
technique to obtain the slope of reflecting surfaces
with very small curvatures. Since its introduction
by Lichtenberg, a number of alternate optical
arrangements have been proposed to observe the
loci of constant projected curvature [2], [3],[4],[5].
This paper presents a new version of reflection
moiré. Previous versions have used incoherent
illumination and coarse pitch gratings.
II. OBSERVATION OF SLOPE FRINGES
WITH COHERENT ILLUMINATION
The observation of slope fringes requires the
projection of a grating, called reference grating,
into a control or master grating.
The imaging is done by reflecting the reference
grating on a mirror surface whose slope is to be
measured.
The moiré fringes produced by the two gratings are
the loci of projected constant slope fringes. Since
the curvature of a surface is a second order tensor,
three components of the tensor must be measured.
The alternative method proposed in this paper uses
a well-known phenomenon analyzed in [6].
When a grating is illuminated with coherent
collimated illumination, the grating is reproduced
in the space at distances [6],
(1)
where z is the coordinate perpendicular to the
grating, p is the grating pitch, λ is the wave length
of the illumination light and n is an integer 1,2,3....
Fig. 1 shows one set up that can be used to observe
the fringes. The reference grating is projected on
the mirror surface whose slope is to be measured
by means of a semi-reflecting, semi-transparent
RESEARCH ARTICLE OPEN ACCESS



mirror. The moiré pattern is produced by observing
the reference grating through the master grating.
In the first method, a translucent screen is located
behind the master grating. The observation plane
can be changed by adding an optical system and
projecting the pattern onto a screen. The sensitivity
of the method shows to be dependent on the
distance between the mirror surface and the master
grating. The location of the master grating must
conform with equation (1). In practice, the position
of the reference grating is adjusted until maximum
visibility fringes are obtained. Fig. 2 shows the
optical equivalent of the observation setup.
Two parallel gratings are observed with
illumination perpendicular to the grating plane. The
normal illumination produces symmetrical optical
paths for all the orders that interfere.
The observed fringes depend on the slope of the
surface and not on the gap between the two
gratings. If the observation is made in the zero
order direction, the sequence of orders contributing
to the fringes will be given by:
Considering an order other than zero, the condition
of minimum deviation [7] must be used to obtain
symmetrical paths for all the orders so as to
observe the patterns obtained. Since the resultant
order contains many wave fronts, the resulting
fringes are multiple beam interference fringes [7],
[8]. As mentioned in [7], the slit and bar type of
amplitude modulating gratings under the stated
observation conditions will produce fringes that
have the fundamental period.
Of course, the presence of the harmonics changes
the shape of the fringes. Since the different orders
diverge in space, the distance between the two
gratings is restricted to values that limit the amount
of shearing of the different wave fronts to a
reasonable amount.
Essentially, the system is operating as a shear
interferometer and the values of the obtained slopes
correspond to points whose location is known
within the amount of shearing.
III. DERIVATION OF THE EQUATION OF
THE LINES OF CONSTANT SLOPE
This derivation applies to the optical setup shown
in Fig. 1. The first grating (Fig. 2) is situated at a
distance d0 from the mirror surface; the second
grating is situated at a distance d1.
The sum of d o and d1 must satisfy the condition
given by equation (1) otherwise they are arbitrary
quantities. The phases of wave fronts are referred
to the origin of the coordinate system.
The first grating is illuminated in the direction,
which is perpendicular to its plane by a plane wave
front of wavelength λ and amplitude E0.
As this wave front passes through the first grating,
it is multiplied by the transmission function of
grating 1, thereby producing the different
diffraction orders. The angles corresponding to the
diffraction orders are given by,
The derivation presented in this paper follows the
general lines of the derivation given in [9] for in-
plane displacements. In the present case, we are
dealing with out-of-plane displacements.
In [9] the combination of the different diffraction
orders is discussed in detail and an abbreviated
derivation will be given later in this paper.
We will now concentrate on the nth order shown in
Fig.2.
The coordinates system is shown in the same
figure.
The origin is taken in the plane of the first grating,
z, increasing as one moves towards the mirror
under study.
After the first grating, the wave front is given by:
( ) n
n
nn
n
zi
p
nx
i
i
n
zxsini
i
nn eeeEkeeEkzxE
θ
λ
ππ
φ
θθ
λ
π
φ
cos
22
0
cos
2
0),(
−+−
==
(4)
where φn and kn are constants for the grating and
order considered.



For simplicity, we can consider them relative to the
zero order (φ0=0 and k0=1).
Reaching the surface, the wave front is:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
+−
≅−
n
n
xwd
p
nx
i
i
nn eeEkxwdxE
θ
λ
π
φ
cos
)(
2
0000
000
))(,( (5)
Upon reflection on the surface of the object,
the wave fronts are rotated by twice the local slope
of the object. The reflected wave front is then given
by:
( )
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+−−+
⎟
⎠
⎞
⎜
⎝
⎛
+−−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
+−
≅ x
xw
zxwd
x
xw
sinxx
i
xwd
p
nx
i
i
nn
n
n
n
n
eeeEkzxE ∂
∂
θ
∂
∂
θ
λ
π
θ
λ
π
φ
)(
2cos))((
)(
2)(
2
cos
)(
2
0
0
00
0
0
000
),( (6)
The second grating is located in plane z=d0-d1
therefore, the amplitude incident on the second
grating is:
( ) ( ) ( )
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+−+
⎟
⎠
⎞
⎜
⎝
⎛
+−−−+−
≅− x
xw
xwd
x
xw
sinxxxwdsinx
i
i
nn
n
nnn
n
eeEkddxE ∂
∂
θ
∂
∂
θθθ
λ
π
φ
)(
2cos))((
)(
2)(cos)(
2
010
0
01
0
0000
),(
(7)
In Fig. 1 a diaphragm is added that filters all the
other orders, except the zero order. Therefore, the
only wave fronts that contribute to the pattern have
the final order 0. This means that incident order +n
must be rotated back to order 0=+n-n. The final
expression of the amplitude in the plane of the
second grating for the nth wave front (+n,-n) is
(kn=k-n and φn=φ-n ) for a symmetric grating with
transfer function ,
( ) ( )
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+−+
⎟
⎠
⎞
⎜
⎝
⎛
+−−−+−
=
−=−
x
xw
xwd
x
xw
sinxxxwdsinx
i
ip
nx
i
i
n
nnnfinal
n
nnn
nn
eeeeEk
ddxExTddxE
∂
∂
θ
∂
∂
θθθ
λ
π
φ
π
φ
)(
2cos))((
)(
2)(cos)(
2
2
2
0
2
10
)2(
10
0
01
0
0000
),().(),(
(8)
where:
(9)
Therefore, the phase at point x in the plane of the
second grating depends on the deflection and slope
of the surface at a different point ( ). This point
differs for each order since its position depends on
n. This will be interpreted as shear interferometry.
Angles and are supposed to be small,
and . Therefore, we can approximate the
trigonometric functions:
(10)
This formula gives the wave front coming from the
second grating whose order was +n after the first
grating. It is valid for positive or negative values of
n, and shows that the wave fronts are translated
combinations of the slope and deflection of the
mirror. The wave fronts are sheared.
For the zero order term:
(11)
As previously explained, only the zero order light
coming from the second grating is collected. This is
done by introducing a lens system and filter in the
focal plane of the first lens. Consequently, the light
distribution observed is the result of interference
between all wave fronts with order 0 after the two
gratings: (+n,-n).
Combining all contributions:
∑
−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++
=
N
Nn
p
n
dxfi
i
n
dd
i
T eekeExE n
)(
2
22
)(2
0
1
10
)(
λ
λ
π
φλ
π
(12)
Where N is the highest diffraction order reaching
the second grating.
The intensity measured is:



∑ ∑−= −=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−+
−
=
=
N
Nn
N
Nm
p
m
dxf
p
n
dxfi
i
mn
TTT
eekkE
xExExI
mn
)()(
2
)(2222
0
11
)().()(
λλ
λ
π
φφ
(13)
Each term corresponds to shear interference of
wave fronts whose phase is given by f(x).
We use the first order approximation for f(x):
x
xf
xxfxxf
∂
∂ )(
)()( Δ+≅Δ+
(14)
Each term is a complex number, but considering
the ranges for n and m, each term has its conjugate
and they recombine to give a real intensity:
⎭
⎬
⎫
⎩
⎨
⎧
−+−= )(2
)(
)(
2
cos2)( 1
222
0 mnmn
nm
x
xf
mnd
p
kkExI φφ
∂
∂π
(15)
Note: all terms where n=m are background
intensity terms.
( ) ( )
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−≅
2
2
)(
212 1
x
xw
d
x
xw
x
xf
∂
∂
∂
∂
∂
∂
(16)
Classically, the effect of curvature is neglected,
and reflection Moiré patterns are analyzed
considering the pattern as containing the slope
only:
⎭
⎬
⎫
⎩
⎨
⎧
−−−≅ )(2
)(
)(2
2
cos2)( 1
222
0 mnmn
nm
x
xw
mnd
p
kkExI φφ
∂
∂π
(17)
In order for this approximation to be reasonable,
the curvature of the mirror must be very small,
which makes the correction term negligible. The
exact condition is:
(18)
or, defining R(x) as the local radius of curvature of
the mirror:
(19)
The resulting intensity is the
summation of a number of cosine fringe patterns
encoded with the slope of the surface. Each term
(n,m) gives a pattern with a different sensitivity.
(20)
The fundamental harmonic is given by the lowest
sensitivity, i.e. for . This corresponds to
the interference between successive orders, for
example (1,-1) (or (-1,1)) and (0,0):
(21)
All other patterns are harmonics of this
fundamental pattern. Furthermore, approximations
of the derivative are varying in quality among these
terms, since the greater the shear, the worse the
approximation in (14). For this reason, the
fundamental pattern is the most precise. Moreover,
interference between shifted terms ((n, -n) and
(n+1,-n-1), for example, where n is not small)
produce a shifted slope, so that the resulting pattern
is a composite of shifted patterns.
However, the global pattern is of the form:
(22)
although the shape is not exactly sinusoidal due to
the presence of harmonics [10]. The first step in
analyzing the Moiré pattern must therefore be
filtering, to ensure that the fringes are truly cosine
fringes.
IV. EXPERIMENTAL
The moiré reflection technique was applied to
determine the curvatures of a real dental
impression. The collimation of the light was



adjusted using a λ/20 mirror. The position of the
light source was adjusted so that no residual fringe
can be observed in the field. Fig. 3 shows the steps
to obtain the final shape reconstruction.
The analysis of the reflection moiré fringes is
performed by computer fringe analysis. The
HOLOSTRAIN system [11] analyzes the fringes as
spatially frequency modulated signals and
determines the modulating function, extends the
signal beyond the boundaries of the specimen and
computes its derivative. In the case of in-plane
deformation the modulating function is the
displacement function of the points of the surface.
In the case of Reflection Moiré, the modulating
function is the slope function of the surface. The
derivative of the function is the curvature [12]. The
derivatives can be computed with a very high
accuracy. In the particular examples shown in this
paper, a grating of pitch 0.254 mm was used and
the distance d1 was 300 mm. This means that
sensitivities of 1.2 x10-5 radians in the
measurement of slopes can be achieved.
V. CONCLUSION
The equations developed in this paper allow us to
define a more accurate method for surface slope
measurement. Through a new powerful automatic
image analysis system, the whole shape of any kind
of irregular surface can be reconstructed with very
high precision.
REFERENCES
[1] Lichtenberg, F. K., “The Moiré method, a
new experimental method for the determination of
moments in small slab models,” Proc. Soc. Exp.
Stress Analysis, 2, Vol. 12, pp. 83-98, 1954-1955.
[2] Rieder, G.,Ritter, R.,
“Krummungsmessung an belasteten Platten nach
dem Ligtenberschen Moire Verfarhen,” Forsch.
Ing. -Wes. 2, Vol. 31, pp. 33-44, 1965.
[3] Pedretti, M.,”Nouvelle methode de Moire
pour l’analyse des plaques flechies,” Doctoral
Thesis, Ecole Polytechnique de Lausanne, 1974.
[4] Richter, R., Herbst, M., “Ein Optisches
System zur Aufnahme von Vermungsgrossen
dynamische belasteten Platten,” Forsch. Ing. -Wes.
3, Vol. 42, pp. 82-85, 1974.
[5] Chiang, F. P., Jaisingh, G., “A new optical
system for Moire methods, Exp. Mech., 11, Vol.
14, pp. 459-462, 1974.
[6] Sciammarella, C. A., Davis, D., “Gap
effect in Moire fringes observed with coherent
monochromatic collimated light,” Exp. Mech., 8,
Vol. 10, pp. 459-466,1968.
[7] Guild, J., “The interference system of
crossed diffraction gratings,” Chapter III, Oxford at
Clarendon Press, 1956.
[8] Sciammarella, C. A., “Use of gratings in
strain analysis,» Journal of Physics E. Scientific
instruments, Vol. 5, pp. 833-845, 1972.
[9] Sciammarella, C. A., Chang, T. Y.,
“Optical differentiation of displacement patterns
using shearing interferometry wave front
reconstruction,” Exp. Mechanics, Vol. 11, 3, pp.
97-104,1971.
[10]Sciammarella C. A., Bhat, G., “Two
dimensional Fourier transform methods for fringe
pattern analysis,” Proceedings of the VII
international congress in Exp Mechanics, SEM,
Vol. 2, pp. 1530-1538, 1992.
[11] Sciammarella C,.A. Yoshida S., Lamberti L.,
“Advancement of Optical Methods in Experimental
Mechanics”SEM, Springer 2014.



[12] Sciammarella C. A.,. Sciammarella F. M.,
“Experimental Mechanics of Solids” Wiley 2012.

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