Le déplacement latéral et la distorsion d'un faisceau optique borné à la réflexion sur une interface liquide-solide ont

été étudiés antérieurement lorsque l'incidence est égale à l'angle de Rayleigh . Des résultats analogues à ceux de

l'optique dans le cas d'un faisceau acoustique : il y a déplacement angulaire quand le solide est dissipatif, et il y a

déplacement du foyer quand l'angle d'incidence diffère légèrement de l'angle critique, Nous présentons des expressions

analytiques de ces déplacements dans le cas où la largeur spectrale du faisceau est petite par rapport à la constante

d'atténuation a de l'onde de Rayleigh. Dans le cas de faisceaux plus larges, nous présentons des résultats numériques

qui mettent en évidence des déplacements relativement importants pour des champs réfléchis .The lateral shift and profile distortion of a bounded beam incident at the Rayleigh angle from a liquid onto a

solid has been studied in the past for parallel or divergent beams . In a manner analogous to results obtained in

optics, we show that an acoustic beam can also exhibit an angular shift when loss is present in the solid and a

shift in the focal point when the angle of incidence deviates slightly from the critical angle . The extent of these

shifts is given by analytical expressions for beams whose spectral width is small compared to the attenuation

constant a of the Rayleigh wave . For beams having broader spectral widths, we present numerical results for

the reflected fields which reveal the presence of relatively strong shifting effects 

BERTONI, H.

Traitement du signal

I-Revues

Non-specular reflection of convergent beams
from Iiquid-solid interface
Réflexion non spéculaire de faisceaux convergents par une interface solide-liquide
Henry L . BERTONI
Department of Electrical Engineering and Computer Science, Polytechnic
Institute of New York, 333 Jay St ., Brooklyn, NEW YORK 11201
Professor Bertoni has carried out research and lectured on topics of acoustic and electromagnetic wave propagation
and scattering. One aspect of these studies has been to clarify and apply the critical angle phenomena that occur
when a bounded beam is reflected from layered and periodic structures
. His current research interests include the
application of critical angle phenomena to material characterization using the acoustic microscope
.
C. W. HSUE
Department of Electrical Engineering and Computer Science, Polytechnic
Institute of New York, 333 Jay St ., Brooklyn, NEW YORK 11201
T. TAMIR
Department of Electrical Engineering and Computer Science, Polytechnic
Institute of New York, 333 Jay St ., Brooklyn, NEW YORK 11201
SUMMARY
The lateral shift and profile distortion of a bounded beam incident at the Rayleigh angle from a liquid onto a
solid has been studied in the past for parallel or divergent beams . In a manner analogous to results obtained in
optics, we show that an acoustic beam can also exhibit an angular shift when loss is present in the solid and a
shift in the focal point when the angle of incidence deviates slightly from the critical angle . The extent of these
shifts is given by analytical expressions for beams whose spectral width is small compared to the attenuation
constant a of the Rayleigh wave . For beams having broader spectral widths, we present numerical results for
the reflected fields which reveal the presence of relatively strong shifting effects .
KEY WORDS
Acoustic beams, Rayleigh angle, focal shift, angular shift
.
FAISCEAU BORNÉ, RÉFLEXION ET RÉFRACTION
RÉSUMÉ
Le déplacement latéral et la distorsion d'un faisceau optique borné à la réflexion sur une interface liquide-solide ont
été étudiés antérieurement lorsque l'incidence est égale à l'angle de Rayleigh . Des résultats analogues à ceux de
l'optique dans le cas d'un faisceau acoustique : il y a déplacement angulaire quand le solide est dissipatif, et il y a
déplacement du foyer quand l'angle d'incidence diffère légèrement de l'angle critique, Nous présentons des expressions
analytiques de ces déplacements dans le cas où la largeur spectrale du faisceau est petite par rapport à la constante
d'atténuation a de l'onde de Rayleigh
. Dans le cas de faisceaux plus larges, nous présentons des résultats numériques
qui mettent en évidence des déplacements relativement importants pour des champs réfléchis .
MOTS CLÉS
Faisceaux acoustiques, angle de Rayleigh, déplacement de foyer, déplacement angulaire
.
Traitement du Signal
201
volume 2 - n° 1 sp - 1985
CONTENTS
1. Introduction
2. Reflection of paraxial beams
3. Incidence near the Rayleigh critical
angle oR
4. Finite width beams
5. Conclusion
References
NON-SPECULAR REFLEXION OF CONVERGENT BEAMS FROM A LIQUID-SOLID INTERFACE
1 . Introduction
The reflection of an acoustic beam incident at the
Rayleigh angle from a liquid onto a solid has been
studied for well collimated and divergent beams [1-
3]; however, the reflection of convergent beams, such
as those produced by a concave transducer or a lens,
has not yet been examined explicitly . The reflection
acoustic microscope [4] employs an extreme form of
a convergent beam, while beams having more modest
convergence can be envisioned for NDE applications .
Previous studies of acoustic-beam reflection have
dealt primarily with the lateral shift resulting from
coupling to the leaky Rayleigh wave [1-3] . or with the
strong absorption effect produced by the losses in the
solid [5,6]. These effects have also been studied for
optical reflection . In addition, the study of optical
beams has shown an angular shift in the direction of
propagation, and a shift in the distance along the
beam axis to the waist or focal plane [7,8] . The angu-
lar shift results from the variation of the magnitude
p (0) of the reflection coefficient R (9) with the inci-
dence angle 90 , while the focal shift is due to the
derivative of the phase of R (0) with respect to 9 .
The present study reviews the derivation of the lateral,
focal and angular shifts for the case of a two-dimen-
sional Gaussian beam having small angular diver-
gence. Using the accepted pole-zero approximation
[1-3] for R (0), these shifts are evaluated for incidence
near the Rayleigh critical angle 0, Because this leads
to simple but approximate formulas, which hold only
for beams with limited angular divergence, numerical
calculations are presented that show the effects also
for beams having greater divergence .
2. Reflection of paraxial beams
A convergent acoustic beam incident at the angle 9 0
is depicted in Figure 1 . The beam is assumed to be
two dimensional with no variation along the coordi-
nate perpendicular to the plane of the paper. In the
absence of the solid, the beam would be Gaussian
with waist at a distance l from the origin, as indicated
by the dashed lins in Figure 1 . The beam that would
be reflected geometrically from a rigid surface is also
shown with waist at a distance 1 along the reflected
beam axis. In addition to the x, z coordinate system,
Traitement du Signal
1202
1
volume 2 - n' 1 sp - 1985
x
i
Fig. 1 . - Incident and geometrically reflected beams
with coordinate systems centered at the foci .
we have indicated the systems (x i, zi) and (x„ z,) for
the incident and reflected beams, respectively .
For harmonie time dependence exp ( - i ta t), the pres-
sure of the incident beam is given by :
( 1 )
	
PI (xi, zi) =
wo
exp [- (xi/wi) 2
+ ikz;] .
w i
Here k is the wavenumber in the liquid, wo
is the 1/e
half-width of the beam at its waist z i =O, and wi
is
defined by:
(2) wi = wô + i 2 z i/k .
For a well collimated incident beam, the pressure at
the interface is obtained by replacing z i by -1 in (2),
and setting x i = - x cos 90 and zi = -1 + x sin 0 0
in (1) . The pressure of the reflected beam can be
obtained from the Fourier transform of the incident
beam multiplied by the plane wave reflection coeffi-
cient R (kx), where kx is the wavenumber along x .
Thus:
w e
-ikt
(3) p, (x, z) = cos 90
2
1l "
X co
J
R
(kx)
e-(kx-ki)2 (wi/2
cos 00
)2 +i (kx x+ki
z) dk
x,
where k i =k sin 9 0 and kZ=k 2 -kx .
FAISCEAU BORNÉ, RÉFLEXION ET RÉFRACTION
The integrand in (3) will have significant amplitude
only over a narrow interval along the kx axis centered
at k .. As a result, we may approximate k Z by the
Fresnel expansion :
(4) k Z: kcos0o -tan0o (kx-k i)-
(k
.-k i)2 .
2 k cos' 0 0
Using the identity R (kx) = exp [In R (kx)], we may also
approximate R (kx) as :
(5) R (kx)
=R (k i) e`
(kx-ki) L/cos
eo+i
(kx
-k
i ) 2 F/2 k cos2 00
where L and F represent the followin differential
forms evaluated at kx =k . . Using p=IR (k x) I and
IF =arg R(kx) these forms are given by:
dIF
(6 a)
	
L=cosO 0 ( - -
+
i l dk ),
x p x
(6 b)
F=kcos200~
dk
+iI
( 1 dk )2 1 dk-
x p x p x
With the approximations (4) and (5), the integrand
of (3) has the form of an exponential with first and
second power terras in (kx -k i) so that its solution
can be obtained in closed form . The result, when
expressed in terms of the (x„ z,) coordinates, is :
W'
ikz, -(x,-L)2/w ;
p, (x„ z,)=-R(ki)e e ,
W,
w, =wô+i-(z,-x,tan00 -F) .
If the reflecting surface is that of a solid half space,
and if 0 0 is greater than the shear-wave critical angle,
then p =1 in the absence of dissipation in the solid
for kx near k i, so that its derivatives vanish . As a
result, L and F are real quantities . From (7) it is seen
that the axis of the reflected beam is shifted to x, = L,
while the beam waist is shifted to the plane
z,= Ltan 00+F. The lateral shift of the beam axis
parallel to itself, and the focal shift are depicted in
Figure 2. The boundaries of the geometrically reflec-
ted beam are shown dashed, and those of the shifted
beam are given by the solid curves .
When p varies with kx, both L and F can have
real and imaginary parts, so that F= F' + i F" and
L = L'+ i L". The imaginary part F" causes a widening,
or narrowing of the 1/e half width at the beam waist,
whereas the focal shift is given by the real part F' .
The lateral shift of the beam is given by the real
part L' . To see the significance of L", we seek the
maximum of the beam amplitude by examining the
minima of the real part of the exponent in (7) . Substi-
tuting L' for x, in (8), and then requiring that the
Traitement du Signal
203
derivative of Re[(x,-L)2/w'] with respect to x, vanish
leads to the condition :
(9) x' -L = 2L
=tan(i 0) .
z,-L'-F' k [w0+2 F"/k]
This condition indicates that the direction of propaga-
tion of the beam has undergone an angular shift i\0
from the z, axis .
3. Incidence near the Rayleigh critical angle OR
The variation of the reflection coefficient at a solid
surface for angles near 0R has been shown to be
dominated by the pole at kP and zero at k0 that lie
near kR = k sin 0R . This variation is embodied in the
approximation:
(10) R(kx)=(kx-ko)l(kx-k,) .
Numerical investigations have found that Re (k0) and
Re(k i,) are very close to kR . It is also found that
Im (k.) has a contribution a, due to leakage or
radiation into the liquid, and a contribution a d due
to dissipation in the solid . In the absence of dissipa-
tion, the pole and zero have conjugate locations so
that Im (k0) _ -ai. Dissipation in the solid causes the
zero to move up in the complex plane by the distance
a d [3,5]. Thus the pole and zero locations are given
approximately by :
1
kP =kR +i(ad +a,),
k O =kR +i(a d -a,) .
With the reflection coefficient given by
(10) and (11),
the quantity L is found to be :
(12) L=2a,cos0 0
X
[(k i -kR) 2 +(aj-ad)]+i2(k i -k
R)ad
(11)
[(k
i - kR) 2
+ (ai +
ad)2]
[(k i - kR) 2 + (al - ad) 2]
It is also found that :
(13) F=k L2
[(ki
- kR)
-
i ad]/al-
Fig. 4. - Power flow fines and amplitude profiles showing lateral
shift for 0o =0
R
=45°, w
o
=2.12,, or., =0.08k and a,=0.
volume 2 - n° 1 sp - 1985
NON-SPECULAR REFLEXION OF CONVERGENT BEAMS FROM A LIQUID-SOLID INTERFACE
Xr
Fig. 2. -
Lateral and focal shifts of the reflected beam .
Geornetrically reflected beam is shown dashed .
When no dissipation is present in the substrate,
a,=0
and L and F are real, as discussed previously . The
maximum value for L occurs when k,=k R, in which
case L = 2 cos 0o/a,, which is the well known Schoch
displacement. For the phase-match condition, it is
seen that F = 0. The variation of F with k ; is depicted
in Figure 3 for ad =0. The focal shift can be away
from the surface (k, > kR) or towards the surface
(k, < k R) . The maximum focal shift is seen to occur at
k;-k,= ±a,/~\/13-,
at which points I F I/L=0.86 (k/a,) cos 00 . Since k/a,
is typically ten or more, the maximum focal shift is
at least several times the maximum lateral shift. While
the focal shift is larger than the lateral shift, well
collimated beams have a large depth of focus, thus
making the focal shift more difficult to observe .
When loss is present (a,94) we sec from (12) and
(13) that L and F are complex. For incidence at the
Rayleigh angle, L is real, in which case F'<0, so that
(8) yields
Re (w 2)>W ô .
This condition implies a widening of the beam at its
waist. If the beam is incident well away from the
Rayleigh angle, F" can be positive, corresponding to
a narrowing of the reflected beam at its waist . For
low to moderate loss with a d < ( 1/2) oc,, and assuming
wa » R2 F"/k,
the angular shift is also maximum when k ; - kR : + a,/
3, for which
A0 : 2. 6 a d cos 00/kw0 ai
.
4. Finite width beams
The foregoing analysis applies when (5) is valid over
the spatial frequency range
I
k x - k ; I <--_ 2 cos 00/w0 . For
R (kx) of (10), this requirement implies that
Traitement du Signal
204
F/cost 80
OCi
/J3
Fig
. 3. - Focal shift as a function
of the leaky wave attenuation constant
.
2 cos 0o/w o be substantially less than a, . In the far
field region, the incident beam has an angular diver-
gence width 20. : 4/kw 0 . Thus the results for the
varions shifts apply when 20 . is substantially less
than 2 a,/k cos 00 . As an example, if 00 = 45° and a,/k
has value 0.1, then 2 0. must be less than 17° . Since
a,/k is usually less than 0.1, the angular beam width
is correspondingly reduced.
Fig . 5. - Power flow lines and amplitude profiles
showing focal shift for 0
0 =48
.3" .
Using an approach similar to that of reference [1],
we have calculated the power flow lines and amplitude
profiles of the reflected beam for w 0 =2.1 X, for which
2 0,„=17°, when 0R =45° and a,=0.08 k . The results
are plotted in Figures 4 and 5 for angles of incidence
00 =45° and 48.3°. Since the half-width of the fre-
quency spectrum for this beam is 0 .11 k, which is
greater than a,, these parameters lie outside the range
of validity of the expressions (12) and (13) for the
beam shifts, so that only qualitative agreement can
be expected .
For 00 =45°, the lateral shift L in Figure 4 is about
2.1 X, which is in fair agreement with the value 2 .8 X
obtained from (12) . A slightly smaller lateral shift is
found for 0 0 =48.3°, in agreement with (12) . The
volume 2 - n° 1 sp - 1985
separation bétween the power flow lins is an indica-
tion of beam width and indicates a shift of about 8 ~,
as opposed to the value of 14 X predicted by
(13) .
FAISCEAU BORNÉ, RÉFLEXION ET RÉFRACTION
Fig
. 6. - Amplitude profiles for showing angular shift
a,=0
.04k, 0° =48.3" .
The effect of loss on the reflected beam is shown in
Figure 7 for 00 =48.3° and w o =2.1?. The solid has
the same parameters as used in Figures 4-6, except
that aa =al /2. Besides the lateral and focal shifts, the
propagation of the beam axis is shifted by about 2°
from the specular direction, which is smaller than the
value of 3 .4° given by (9) .
5. Conclusion
We have shown that a convergent beam reflected at
a liquid-solid interface exhibits a lateral shift, a shift
in the location of the focus and an angular shift in
the direction of propagation . Simple expressions for
these shifts were derived when the beam is incident
near the Rayleigh critical angle and the angular diver-
gence of the beam is small . For beams having greater
angular divergence, numerical evaluation of the field
indicates that the shifts still occur, but with reduced
values .
Acknowledgement
This work was supported by the National Science
Foundation and by the Center for Advanced Techno-
logy in Telecommunications, which is funded in part
by the New York State Science and Technology Foun-
dation under its Centers for Advanced Technology
Program .
REFERENCES
H. L. BERTONI and T
. TAMIR, Unified theory of Rayleigh
angle phenomena for acoustic beams at liquid-solid
interfaces, Appl. Phys ., 2, 1973, pp. 157-172 .
[2] M. A. BREAZEALE, L. ADLER and G. W. Scorr, Interac-
tion of ultrasonic waves incident at the Rayleigh angle
onto a liquid-solid interface, J.A.P., 48, 1977, pp.
530-
537 .
T. D. K. NGOc and W, G. MAYER, Numerical integra-
tion method for reflected beam profiles near Rayleigh
angle, J.A.S.A ., 67, 1980, pp. 1149-1152
.
[4] C . F. QUATE, A. ATALAR and H. K. WICKRAMASINGHE,
Acoustic microscopy with mechanical scanning-A
review, Proc. IEEE, 67, 1979, pp. 1092-1113 .
H. L. BERTONI and Y . L. Hou, Effects of damping in a
solid on acoustic beams reflected at the Rayleigh critical
angle, Proc. 10th Symp . on NDE, San Antonio, 1975,
pp .
136-142 .
[6] R. L. RICHARDSON, A new technique for measuring metal
surface properties at the Rayleigh critical angle, Proc.
1983 IEEE Ultrasonics Symp ., Atlanta, pp. 895-897 .
C. K . CARNIGLIA and K. R . BROWNSTEIN, Focal shift and
ray model for total internai reflection, J.O.S.A ., 67,
1977, pp . 121-122.
I. A. WHITE, A.
W. SNYDER and C. PASK, Direction
change of beams undergoing partial reflection, J.O .S.A.,
67, 1977, pp.
703-705 .
Traitement du Signal
1205 1
volume 2 - n° 1 sp - 1985
[1]
[3]
[51
[7]
[8]


Réflexion non spéculaire de faisceaux convergents par une interface solide-liquide

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