Download Transversal ultrasonic probe

1
ACOUSTO-OPTICAL CHARACTERIZATION OF A CONTACT
BETWEEN A TIP AND A PLATE
J.P. Nikolovski
CEA LIST : Laboratoire des Interfaces Sensorielles
18, route du Panorama, 92265 Fontenay aux Roses, France.
Email : [email protected]
tip
Abstract
The local strain perturbation induced by the contact
between a tip and a plate is investigated with an optical
heterodyne interferometer as symmetric S0 and
antisymmetric A0 Lamb waves are propagated in the bulk of
the plate. Elasto-optic disturbance analysis is carried out for
each mode and comparison between both modes is made to
best characterize the contact. Two and three wave optical
interference analysis is used to explain the presence of
fringes of equal thickness disturbances. The S0 Lamb mode
turns out to magnify optical effects of the mechanical
contact.
1. Introduction
Ultrasonic testing of solid materials may require
probing very small areas of a material. This can be done by
coupling flat ultrasonic shear wave transducers to conical
tips1. In many cases implying simultaneous detection of
shear and longitudinal components, it can be very difficult
to detect selectively one component from the other
particularly when sub-wavelength contacts interactions are
concerned. At the same time, the nature of the contact
between the probed material and the tip should be
characterized. The purpose of this work is to use the
selectivity of an optical interferometer which has local
access to the normal displacement of the matter to describe
both experimentally and analytically the optical effects of a
sub-wavelength mechanical contact between a free surface
and a tip.
sz
z
Cr-Au
glass plate
sz
x
AIR
+sz
-sz
optical probe beam
A0 Lamb mode
S0 Lamb mode
Figure 1 - Local perturbation caused by the tip in contact
with a sample. The sample is here a glass plate which
propagates S0 and A0 Lamb modes.
Fig. 2 shows in schematics the optical probe. A He-Ne laser
beam is modulated by an acousto-optic modulator of
frequency f0. The laser beam has a circular polarization as it
goes through the plate. The reflected beam is modulated at
the frequency f of the Lamb waves. It is then received by a
photodiode and demodulated.
Dove's prism
fL
PZT
/4
LASER
beam
fL + f 0
AOM
f0
S 0,A
fL
fL + f 0 mod f
R
S
The tip is a conically or parabolically tapered rod made
with Duralumin. The length of the tip, typically 100 mm
long, is large compared to the wavelength. Local
perturbation induced by the tip can be visualized with an
optical heterodyne interferometer2 trough a transparent
sample as shown in fig.1. In the following case the sample
is a glass plate of thickness 1 mm coated on one face with a
Cr-Au layer. The plate propagates the symmetrical Lamb
mode S0 and the antisymmetrical Lamb mode A0. Going
through the plate to see the local disturbance caused by the
tip is not a very convenient way to proceed because in that
case, the optical path of the probe beam is modified both by
surface displacement of the Cr-Au layer and by the
variation of the refraction index n of the glass plate as
Lamb waves goes by. We nevertheless proceedeed that way
because a first C-scan made on the side opposite to the side
of the contact area revealed no signal variations.
f
ANALYSER
f0 mod f
PHOTODIODE
t
Low pass s (t)
filter
Filter
2. Local perturbation induced by the tip
Tip
(f 0)
90°
f0
90°
Figure 2 -The optical heterodyne interferometer: a laser
beam is modulated by an acousto optic modulator (AOM)
of frequency f0. The optical path of the modulated beam is
modified by the propagation of the Lamb modes at the
frequency f. The light beam passing through the plate has a
circular polarization.
Fig. 3 represents a C-scan over a surface of 70 µm x 70 µm
around the contact area in the configuration of fig. 1 for the
S0 Lamb mode. A 2 cm focal length was used giving a spot
of approximately 7 µm of diameter. The pictures shows
regions where the signal is much smaller in amplitude, by
30 dB, than the signal of undisturbed regions as well as
regions where the signal is stronger than the signal of
undisturbed regions
2
Figure 3 - C-Scan on S0 over a surface of 70 µm x 70 µm.
Multiple contacts can be seen.
Phase inversions
 min  2 air  2s z0  2nc ec with n c  n0  n 


e


2


1
 n   ndz

e e





2


 e c  e0  2s z0

 max  2 air  2s z0  n d ed 


 n d  n0  n



0
e d  e0  2s z

The circular polarization has an x and y component. The
variation of the refraction index due to the travelling wave
are along theses axes:


n3
n x    p11S xx  p12 S zz  and n x  n0  n x 
2




n3
n y  
p21  S xx  S zz  and n y  n0  n y 
2


where p11 and p12 are components of the elasto-optic tensor,
and Sxx and Szz are components of the strain tensor.
The displacement components of the S0 Lamb mode at
1 MHz are approximately 3:
Figure 4 - B-scan on the line y = 25 µm of fig.20. Signal
phase inversions are clearly visible.
Regions of strong signals are very close to regions of weak
signals. It can also be noticed that around these regions
phase inversion of the signal occurs as it is showed more
clearly on a B-scan at the position y = 25 µm of fig. 4.
These effects are obviously due to the presence of the tip
and reveal multiple contacts. The following theoretical
analysis will help understand why inversion of the signal
occurs in regions of strong interactions.
2




0
x( mm)   
 sx ( t , x , z )  5sz cos t 






0


2s
2

 sz ( t , x , z )   z z sin t 
x( mm)    



e

nx and ny components are independent from the z
component. Therefore the average variation of the
refraction index in the thickness of the plate is equal to the
local variation at a particular position in the thickness.
e
2
3. Two wave interferences
In the experimental configuration of fig. 1 and 2, we
assume that there is only one reflected beam from the CrAu layer. Let us calculate the minimum and maximum
optical path length. We call air the optical path length in air
0
when the plate is at rest; s z is the z component amplitude
of the S0 Lamb wave mechanical displacement at the
surface of the plate, nc and nd are the average refraction
indexes of the glass plate when its thickness (e) is
respectively minimum (ec), and maximum (ed), whereas n0
is the refraction index of the plate at rest.
e
2
1
1
n i   n i dz  n i  dz  ni
e e
e e

2

2
0
Numerical estimations: at 1 MHz and for s z =15 Å we
have:
S xx (x, t )  5ks z0 sin(t  kx   )  108
. 10 -6 sin(t  kx   )
S zz (x, t )  
2s z0
sin(t  kx   )  81
. 10 -6 sin(t  kx   )
e
n  15
.
p11  0121
.
p12  0.270
n x ( x , t )  
n03
 p11S xx  p12 S zz   0.45 10 6 sin(t  kx   )
2
n y ( x , t )  
n03
 p21S xx  p12 S zz   2.68 10 6 sin(t  kx   )
2
We have now to calculate the electrical field, reflected back
from the Cr-Au layer: The x and y components of the field
are:





2 0
E S  x E0 cos  0 t 
s ( 2  4n0 )  2e0n x sin(t  kx   )   S  
0 z






2 0
y E0 sin  0 t 
s z ( 2  4n0 )  2e0n y sin(t  kx   )   S 

0


3
reference beam. At the output of the photodiode, the
alternative part of the photocurrent is proportional to the
following expression:
ny
Emsint
nz
Emcost
circular polarization
nx
Figure 5 - Projection of the circular polarization vector
entering the glass plate on the x and y unity vectors
The beam has then to go back again through the /4 plate
that has its eigen vectors at 45° from the x and y axis. At
the exit, the electrical field is approximately linear and can
be approximated by the expression:
 0 t 



nx  n y
ES   2x E0 sin  2  0
  sz (2  4n0 )  2e0

2
0
 








 1 cos  0 t  k 2s z0 sin(t  kx   )   S1   R 




0
0

 



n


n
x
y 

 0


2
i( t )  2 E0  1 2 cos  0 t  k  s z 4n0  2e0
 sin(t   )  2kn0 e   S 2   S 2   
2



 








0 n x  0 n y 

  cos  t  k  s 0 ( 2  4n )  2e
sin(

t


)

2
kn
e




0
0
0
S2
R 
 z

 0
 2
2












Cr-Au
S0 Lamb mode




 sin(t  kx   )    
S


4


S1 S2
We see that the total phase fluctuations of the laser beam
due to variations of the refraction index along x and y is the
average of the x and y refraction indexes variations.
Figure 6 - Three wave interferences can be obtained
experimentally by changing the position of the glass plate
according to the focusing lens.

If the total phase fluctuations are much smaller than 2, the
signal spectrum is composed of a central carrier frequency
f0, and lateral frequency rays, the amplitude of which are
given by Bessel functions. The alternative part of the
photocurrent can be developed as follows:

n x0  n 0y

( x, t )   sz0 ( 2  4n0 )  2e0
2


 sin(t  kx   )


Numerical application: If we take:
n0 = 1.5
s z0 = 15 Å
n x ( x , t )  n y ( x , t )
 1.57 10 6 sin ( t  kx   )
2
e0 = 1 mm
(x, t) = (-6. 10 -9 - 3.1 10 -9 )sin ( t - kx +  ) = -3.1* 2 s z sin ( t - kx +  )
As a first conclusion, when the laser beam goes through the
plate the signal is 3.1 times larger and opposite in sign to
the signal that would be obtained if the laser beam was
simply reflected at the surface of the plate.
The expression of the optical signal path shows also that by
strongly blocking the back surface of the plate with the tip,
it is possible to diminish or even change the sign of the
optical length path. This explains the phase inversions of
the signal in the regions of strong interactions.
4. Three wave interferences
We consider now the case where, the laser beam is partially
reflected at the front and back faces of the plate. This
corresponds to a case of interferences with three beams.
Fig. 6 illustrates the experimental situation. After front and
back reflections as well as going through the /4 plate, the
electrical fields attached to the optical beams are:



ES1  1E0 exp i  0t  k 2sz0 sin(t   )   S1
 0t 
 0
  sz ( 2  4n0 ) 
ES 2   2 E0 exp i  

0 nx  0 n y
k 
  2e0
2
 
E R  E0 exp i R 








 sin(t   )  2kn0e   S 
2




where  1 and  2 are constants characterizing the intensity
of the reflected beam compared to the intensity of the





1 cos  0t   S   R  1 2 cos  0t   S   S  2kn0e  

2
2
2

i( t )  

 2 cos  0t   S 2   R  2kn0e









 0


s
k
cos



t







 0 
S2
R
 1z



0
0




 nx   n y 
 0
1 2 k  sz 2n0  e0
 cos  0   t     S 2   S 2  2kn0e  
2







0
0




 nx   n y 
 k  s0 (1  2n )  e

cos



t







2
kn
e


2
z
0
0
0
S
R
0


2


2







 


 





 0

1sz k cos  0   t     S 2   R 



0
0




 nx   n y 
 0
 1 2 k  sz 2n0  e0
 cos  0   t     S 2   S 2  2kn0e  
2







0
0



 nx   n y 
 k  s0 (1  2n )  e
cos  0   t     S 2   R  2kn0e 
0
0

 2  z

2








 


 

The sum of the three terms at frequency f0 has an amplitude
A and a phase  given by the following expression:





1 cos  0t   S   R  1 2 cos  0t   S   S  2kn0e  

2
2
2


  A cos( 0t   )
 2 cos  0t   S 2   R  2kn0e

with






1 cos  R   S  2kn0e  cos  S   S  2kn0e  

2
2
1

A2  12  12 22   22  21 2 

 cos 2 S 2  4kn0e   S1   R

 2




2 sin R   S  2kn0e
tan  
1 cos R   S   12 cos2kn0e   S
2 cos R   S  2kn0e

1 sin  R   S2  12 sin 2kn0e   S2   S2 
2
2
2

  S2 
2
We do the same treatment for the three terms at lateral
frequency f0 + f:
4




 0

1sz k cos  0   t     S 2   R 



0
0




 nx   n y 
 0
1 2 k  sz 2n0  e0
 cos  0   t     S 2   S 2  2kn0e  
2







0
0



 nx   n y 
 k  s0 (1  2n )  e

cos



t







2
kn
e


0
0
0
S2
R
0 

 2  z
2







 


 

 A cos( 0t    )
with


tan   


2


 







the fringe
is moved up
fringes of equal
thickness

A2  12 s2 k 2  12 22 k 2 2ns  en   22 k 2 2ns  en  s 


1s2ns  en cos  R   S 2  2kne 



21 2 k 2 s s1  2n  en cos  S 2   S1  2kne 



 2 s2ns  en  2ns  en2 cos 2 S  4kn0e   S   R 
2
1


2
This leads to the same periodicity between minima. Fig. 8
presents two C-scans with a glass plate having a 1.1 mrd
wedge. The perturbation caused by the tip is clear. Fringes
are moved toward the crest.
2n
i
optical probe beam
Figure 7 - Three wave interferences; fringes of equal
thickness; glass plate with a 1.1 mrd wedge.

1sk sin  R   S 2  1 2 k 2ns  en sin 2kn0e   S 2   S 2 
1sk cos  R   S 2  1 2 k 2ns  en cos 2kn0e   S 2   S 2 


 2 k  s1  2n  en sin  R   S 2  2kn0e


 2 k  s1  2n  en cos  R   S 2  2kn0e
From the ratio of the central and lateral frequency
amplitudes discloses some information on the acoustic
signal.
r2 
A2
A2






1 cos  R   S  2 kn0 e  cos  S   S  2 kn0 e  

2
2
1

12   22  12 22  21 2 

 2 cos 2 S 2  4kn0 e   S1   R




12 s 2 k 2   22 k 2 2ns  en  s 2  12 22 k 2 2ns  en 2 



1s2ns  en cos  R   S 2  2kne 

21 2 k 2 s s1  2 n  en cos  S 2   S1  2 kne 

 2 s2ns  en  2ns  en 2 cos 2 S  4kn0 e   S   R
2
1



 









Numerical application: let us take  1 =  2 = 0.1, and
s = 15 Å:
2. 01  0. 2 cos  R   S2  2 kne  2 cos  S2   S1  2 kne
k 2r 2  k 2
A2
A2

0. 2 cos 2  S2  4 kne   S1   R
2286  180 cos  R   S2  2 kne  1350 cos  S2   S1  2 kne
540 cos 2  S2  4 kn0 e   S1   R
The signal at the output of the interferometer is inversely
proportional to this ratio: The signal is strong when the
following condition is verified:
cos  S2   S1  2 kne  1
which leads to :
2kne  2
i.e.
e =

2n
Consequently, every time the thickness of the plate is
increased by a value of /2n, the signal increases in
amplitude. If the faces of the glass plate are not really
parallel but have a wedge angle =1.1 mrd as illustrated
with fig.7, fringes of equal thickness appear with the
following periodicity between two maxima:
0. 632 m

i

 185 m
2 n 2  1.1 mrd  1.5
On the other way, the signal becomes very small when the
following condition is satisfied.
cos  S2   S1  2 kne  1
Figure 8 - C-scan on the S0 Lamb mode. Disturbance
caused by the tip to the fringes of equal thickness.
The lower picture of fig. 8 shows that the parabolic tip has
little effect on the position of two neighbouring fringes.
This gives an idea of the disturbed area: a region of
approximately 100 µm around the contact area. The
interesting point is to see how far (about 80 µm) a fringe
can be displaced by the presence of the tip. As a matter of
fact, we know that the free surface vertical displacement is
approximately 15Å (i.e. 0.7% of /2n); without any effect
on the refraction indexes, the lateral displacement of the
fringe should be of the order of 0.7% of the fringe period
i.e. close to 1.3 µm instead of 80 µm. We see here how
5
refraction index variations
mechanical disturbances.
help
investigate
small
8. Conclusion
The strain disturbance caused by a metallic tip in contact
with a glass plate can experimentally be seen with an
optical interferometric probe by propagating a S0 Lamb
wave in the region of the contact. The refraction index
variations reveals the regions stressed by the tip.
Acknowledgments
Author is thankful to D. Royer and D. Fournier (ESCPI,
Paris) for helpful discussions.
1
Nikolovski J.P., Royer D., « Local and selective detection of acoustic
waves at the surface of a material ». IEEE ultrasonics symposium
proceedings. (1997) 699-703.
2
Royer D., Dieulesaint, E.,"Mesures optiques de déplacements
d'amplitude 10-4 à 10-2 Angström. Application aux ondes élastiques".
Revue Phys. Appl. 24 (1989) 833-846.
3
Nikolovski J.P. "Détecteur à ondes de Lamb de la Position d'un stylet",
thèse de doctorat, Université Paris VI, 2 Feb. 1995.