Surface roughness determination using the acousto-optic technique:
Theory and experiment
R. Briers and O. Leroy
Interdisciplinary Research Center, Katholieke Universiteit Leuven, Campus Kortrijk,
B-8500 Kortrijk, Belgium
S. Devolder, M. Wevers, and P. De Meester
Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven,
B-3001 Leuven, Belgium
Received 14 February 1997; accepted for publication 3 June 1997
With the acousto-optic nondestructive testing technique, phase shifts between the incident and the
reflected ultrasound are measured at critical angles of the investigated material. This letter is
focusing theoretically as well as experimentally on the influence of a small surface roughness some
micrometers of a thick material on the phase information. © 1997 American Institute of Physics.
S0003-69519704631-7
In most of the theoretical and practical works on surface
roughness only the amplitude of the diffracted ultrasonic
field is used for analysis. A detailed overview of all methods
developed to study diffraction gratings and a nearly complete
list of the most important references on this subject are given
in the book of Maystre.
1
A new nondestructive testing tech-
nique based on phase measurements in a critical angle of the
substrate material was developed at the Department of Met-
als and Material Engineering, K. U. Leuven, in close col-
laboration with the Interdisciplinary Research Center at the
K. U. Leuven, Campus Kortrijk. In the experimental setup
this phase information is obtained using two
2
or only one
3
laser beam depending on the kind of information one wants
to obtain.
A theoretical model
4
was elaborated to determine the
influence of a corrugated material on the phase information.
Let the coordinate system be chosen as in Fig. 1 such that the
periodically rough liquid–solid boundary is described by
x , z = f x -z =0 with f x + = f x , 1
in which is the profile periodicity. The peak-to-peak am-
plitude of the roughness is denoted by h . The time-harmonic
incident ultrasonic wave is a Gaussian beam of half-width
w with a displacement u
i
( x , z ) represented by the Fourier
integral
5
u
i
x , z = 1/2
-
+
V k
x
exp i k
x
x +k
z
z dk
x
, 2
with
V k
x
=
w
0
exp - k
i
-k
x
w
0
/2
2
, 3
and whereby
k
z
= k
2
-k
x
2
, 4
k
i
=k sin
i
, 5
w
0
=k /cos w , 6
k being the longitudinal wave number and
i
the angle
of incidence. The displacement u
t
r
( x , z ) of the total reflected
field can be calculated as
u
t
r
x , z =
m=-
m=+
1/2
-
+
R
m
k
x
V k
x
exp i k
m
x -k
zm
z dk
x
=
m=-
m=+
| u
m
r
x , z | exp i
m
r
, 7
or
u
t
r
x , z =| u
t
r
x , z | exp i
t
r
, 8
with | u
t
r
| ,
t
r
, respectively, the amplitude and the phase of
the total reflected field and R
m
( k
x
) the reflection coefficient
of the m -th diffraction order. The value of k
m
can be deduced
from the diffraction equation
6
k
m
=k
x
+m 2 / , 9
and
k
zm
=
k
2
-k
m
2
. 10
To determine the reflection coefficient R
m
( k
x
) in Eq. 7 the
total reflected ultrasonic field u
1
( x , z ) in the liquid medium
1 and the total transmitted ultrasonic field u
2
( x , z ) in the
solid medium 2 are decomposed in an infinite sum of plane
homogeneous and inhomogeneous plane waves
7,8
represent-
ing the different diffraction orders. The amplitudes of these
waves are found implementing the generalized boundary
conditions
9
along the corrugated surface, being the continu-
ity of the normal displacements
u
1
• =u
2
• , 11
and the continuity of stresses
j =1
3
T
ij
1
j
=
j =1
3
T
ij
2
j
; i =1,3. 12
Since these boundary conditions are periodic functions
in x , they can be expanded in Fourier series. A sufficient
condition for the final solution is to demand that the Fourier
coefficients are equal over one period . Equating these Fou-
rier coefficients results in an infinite system of coupled linear
599 Appl. Phys. Lett. 71 (5), 4 August 1997 0003-6951/97/71(5)/599/3/$10.00 © 1997 American Institute of Physics
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