Direct imaging of traveling Lamb waves in plates using photorefractive dynamic holography

Abstract

Anisotropic stiffness properties of sheet materials can be determined by measuring the propagation of Lamb waves in different directions, but this typically requires multiple positioning of a suitable transducer at several points or scanning over the area of the sample plate. A laser imaging approach is presented that utilizes the adaptive property of photorefractive materials to produce a real-time measurement of the antisymmetric Lamb traveling wave displacement and phase in all planar directions simultaneously without scanning. Continuous excitation and lock-in methodology is employed, enabling the data to be recorded and displayed by a video camera. Analysis of the image produces a direct quantitative determination of the phase velocity in all directions showing plate stiffness anisotropy in the plane. The method is applicable to materials that scatter light diffusely and provides quantitative imaging of the dynamic surface motion exhibited by traveling elastic waves. A description is given of this imaging process and, for the first time, its ability to perform lock-in measurement of elastic wave displacement amplitude and phase. {copyright} {ital 1999 Acoustical Society of America.}

Full text

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 1
Direct imaging of traveling Lamb waves in plates using
photorefractive dynamic holography
K. L. Telschow, V. A. Deason, R. S. Schley and S. M. Watson
Idaho National Engineering and Environmental Laboratory
Lockheed Martin Idaho Technologies Co.
P.O. Box 1625
Idaho Falls, ID 83415-2209
Received:

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 2
ABSTRACT
Anisotropic stiffness properties of sheet materials can be determined by measuring the propagation
of Lamb waves in different directions, but this typically requires multiple positioning of a suitable
transducer at several points or scanning over the area of the sample plate. A laser imaging approach
is presented that utilizes the adaptive property of photorefractive materials to produce a real-time
measurement of the antisymmetric Lamb traveling wave displacement and phase in all planar
directions simultaneously without scanning. Continuous excitation and lock-in methodology is
employed enabling the data to be recorded and displayed by a video camera. Analysis of the image
produces a direct quantitative determination of the phase velocity in all directions showing plate
stiffness anisotropy in the plane. The method is applicable to materials that scatter light diffusely and
provides quantitative imaging of the dynamic surface motion exhibited by traveling elastic waves. A
description is given of this imaging process and, for the first time, its ability to perform lock-in
measurement of elastic wave displacement amplitude and phase.

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 3
INTRODUCTION
Many optical techniques for measuring ultrasonic motion at surfaces have been developed for use
in applications such as vibration measurement and laser ultrasonics. Most of these methods have
similar sensitivities and are based on time domain processing using homodyne, heterodyne, Fabry-
Perot1, and, more recently, photorefractive interferometry2. Generally, the methods described above
do not allow measurement at more than one surface point simultaneously, requiring multiple beam
movements or scanning in order to produce images of ultrasonic motion over an extended area.
Electronic speckle interferometry and shearography do provide images of vibrational motion over
large surface areas. This method has proven very durable in the field for large displacement
amplitudes and a sensitivity of 1/3000 of the optical wavelength has been demonstrated under
laboratory conditions.3 Full-field imaging of traveling ultrasonic waves using digital shearography
has been recently reported with sensitivity in the nanometer range.4 With this method, optical
interference occurs at the photodetector surface of the camera that records the speckle image from
the sample surface. Multiple image frames are typically recorded and processed in a computer to
produce an output proportional to sample surface displacement. This paper discusses a powerful
alternative method that utilizes the photorefractive effect in optically nonlinear materials to perform
adaptive optical interferometry in an imaging mode.5,6 Optical interference occurs within the
photorefractive material with this technique and the output is an optical image whose intensity
distribution is directly proportional to the sample surface vibration amplitude for small ultrasonic
displacements. Utilizing this approach, no post-processing of the data recorded by a video camera is
required to produce images of the surface vibration amplitude over large areas. The application of
this approach to imaging of standing wave resonant motion in plates has been previously
described.7,8,9 This paper describes results of an investigation into the fundamental operation and
application of this technique to nonstationary waveforms through imaging of traveling Lamb waves

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 4
in plates.10 The ability to measure nonstationary waveforms not only at single points but also as an
image over a surface comes from the inherent lock-in measurement process occurring. This
recording mechanism is developed theoretically as well as experimentally here, for the first time, for
imaging of flexural wave propagation in isotropic plates. Application to imaging flexural wave
propagation in anisotropic plates is also presented. A benefit of the imaging approach is developed
from the Fourier transform of the recorded wavefront that produces a mapping of the propagation
wavevector in all planar directions as a single image. This mapping yields information about the
elastic symmetry of the wave propagation and, therefore, the material microstructure directly.
BACKGROUND
Photorefractivity5refers to that process where optical excitation and transport of electrically
charged carriers within select nonlinear optical materials produces an optical diffraction grating from
the interference pattern developed inside the material. A spatial and temporal charge distribution
results in the photorefractive material that reflects the optical phase information impressed onto the
optical signal beam by the vibrating sample surface. Several optical frequency domain measurement
methods of vibration have been proposed using photorefractive two and four-wave mixing in select
materials.11,12 These provide a time averaged response that is a nonlinear function of the specimen
vibration displacement amplitude. A method using an unconventional photorefractive process has
been reported that provides output linear with the vibration displacement amplitude, but it is limited
to a select group of materials.13 The method reported here utilizes the normal photorefractive effect
to produce an optical grating at a fixed beat frequency between the phase modulated signal and
reference beams. It can be used in a manner that directly measures vibration amplitude and phase
with a response proportional to the Bessel function of order one, providing a linear output for small
amplitudes. The method accommodates rough surfaces, exhibits a flat frequency response above the

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 5
photorefractive cutoff frequency and can be used for detecting both standing and traveling waves. In
this paper, the underlying physics of the optical detection process is developed and application of the
method to full-field imaging of traveling Lamb waves in plates with isotropic and anisotropic
stiffness properties is presented.
EXPERIMENTAL METHOD
The experimental setup for vibration detection is shown in figure 1. A solid state laser source at
532 nm was split into two legs forming the signal and reference beams. The signal beam was
reflected off traveling waves produced at the surface of a nickel plate driven at its center by a
continuously excited piezoelectric transducer. The excited traveling waves occurring on the plate
surface produced a phase modulation sig
δ
of the signal beam. The reference beam was phase
modulated by an electro-optic modulator at a fixed modulation depth ref
δ
. The photorefractive
material was from single crystal Bismuth Silicon Oxide (BSO) of size 10 mm by 10 mm by 2.25 mm
and cut along the <001> and <011> directions. The measured time constant was 0.01 ms. The
modulated beams were combined and interfered inside the BSO photorefractive crystal utilizing an
external angle 2θ = 55 degrees between the beams in order to produce a large response for operation
in the diffusive charge transport regime. The refractive index grating produced within the
photorefractive material can be readout by four-wave or two-wave mixing techniques.
In the four-wave mixing configuration, the reference beam was reflected back into the crystal
along a counter-propagating path that matched the Bragg angle of the photorefractive grating in the
medium. The vibration induced optical phase grating was read out by the resulting diffracted
reference beam, or conjugate signal beam, that propagated backward along the signal beam leg and
was detected by deflecting it with a beamsplitter (not shown) toward a photodetector. Subsequently,

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 6
when only one detection point was being interrogated, the photodetector signal was processed with
conventional electrical lock-in methods to increase the measurement signal to noise ratio through
utilizing a small detection bandwidth. A video charge-coupled device (CCD) camera was employed
to record the demodulated optical phase grating and produce images of the elastic displacement
wavefront over the surface of the plate. Four-wave mixing isolated the signal beam phase
information very effectively from the transmitted signal beam.
A significant drawback of the four-wave mixing approach is the low output intensity of the
readout process. This was improved by employing a two-wave approach that recorded the forward
diffracted beam enhanced by the gain of the two-wave mixing process. However, there was also a
significant component of the directly transmitted signal beam that had to be discriminated against in
order to achieve maximum sensitivity. Optically active photorefractive materials, in this case BSO,
offer a means for providing the needed discrimination through optical activity and anisotropic self-
diffraction, which produce a rotation in the linear polarization of the diffracted reference beam with
respect to that of the transmitted signal beam. By using an appropriate thickness of the
photorefractive crystal, it was possible to achieve a nearly 90° polarization shift between the two
beams.14 This allowed reduction of the directly transmitted signal beam through the use of high
extinction ratio polarizers. The resulting intensity of the diffracted reference beam was dependent on
the vibration displacement and temporally modulated at the frequency difference between the mixing
waves in a manner analogous to the four-wave mixing case. As before, only the output intensity need
be measured to obtain both the vibration amplitude and phase, thereby allowing direct imaging and
ease of interpretation.
FLEXURAL WAVE DISPLACEMENT DISTRIBUTION

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 7
The mechanism that allows recording of traveling wave displacements can be illustrated by
considering a traveling flexural wave in a plate. The classical differential equation of motion for the
out-of-plane displacement of an isotropic homogeneous plate driven by a force per unit area of
),( tf p
ρ
is given by15
),(
2
2
4tf
t
Dp
ρ
ξ
σξ
=
∂
∂
+∇ (1)
where )1(12 2
3
s
Eh
D−
≡ is the bending stiffness of the plate, h
m
ρσ
≡ the mass density per unit area,
m
ρ
= the mass density of the material, s = Poisson’s ratio, E = Young’s modulus, h = the plate
thickness,
ξ
is the displacement normal to the plate surface and
ρ
is the radial spatial coordinate.
Eqn. 1 is valid at low frequencies where the elastic wavelength ( a
λ
) is much larger than the plate
thickness, i.e.
π
λ
λ
π
2
1for,
2a
a
a
ahhkk <<⇒<<= and approximates the lowest antisymmetric
Lamb or flexural wave mode. Consider a traveling flexural wave excited at a single point on the plate
by a piezoelectric transducer undergoing continuous oscillation with frequency
π
ω
2s and phase
s
ϕ
producing a point force per unit area on the plate of 




+−
=)(
2
)(
Re),( 0ss
pti
eFtf
ϕω
πρ
ρδ
ρ
,
where the amplitude of the total force applied to the plate is 0
F and )(x
δ
is the Dirac delta function.
Eqn. 1 can be rewritten utilizing )(
)(),( ssti
et
ϕ
ω
ρξρξ ρ
+−
=, where )(
ρξ
ρ
is the complex
displacement amplitude, as follows

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 8
ρ
ρδ
π
ρξ
ρ
)(
2
)())(( 0
2222
D
F
kk aa =+∇−∇ (2)
where D
ka
2
4
σω
≡. Since Eqn. 2 is a fourth order differential equation, in addition to the two
propagating wave modes, two non-propagating heavily damped modes must also be included in order
to form a complete solution satisfying the boundary conditions. The solution can be found by
adopting the time dependence of the forcing function along with applying the 2-dimensional spatial
Fourier transform, which, for a circularly symmetric function, can be expressed in terms of the
Hankel transform of order zero as16
∫∫ ∞∞ =⇔=
00
00)()(
~
2
1
)()()(2)(
~kdkkJkggdkJgkg
ρ
π
ρρρρρπ
(3)
with
ρρρρ
δ
ρρ
ρ
ρρδ
dkJkJ
k
kk
kdkkJkJ )()(
)(
and,)()(
)( 00
0
00
0
′
=
′
−
′
=
′
−∫∫ ∞∞ . Applying the
transform to Eqn. 2 and using the orthogonality of the Bessel functions yields the radial solution in
the Fourier domain as








+−
=))((
1
)(
~
2222
0
aa kkkk
D
F
k
ρ
ξ
.(4)

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 9
The resultant waveform traveling outward from the excitation point in the spatial domain is obtained
by the inverse transform according to
kdk
kkkk
kJ
D
F
aa 







+−
=∫
∞
))((
)(
2
)( 2222 0
0
0
ρ
π
ρξ
ρ
(5)
which can be integrated using the relation )(
)(
)(
0
022
0
ρ
ρ
kK
kx
xdxxJ =
+
∫
∞,17 where )(
0xK is a Modified
Bessel function of the second kind. The resultant traveling wave solution is given by18





+−
−= )(
)]()([Re),( 1
0
1
00 s
t
s
i
e
a
ikH
a
kHit
ϕω
ρρξρξ
(6)
where s
a
a
a
ac
k
D
F
ω
λ
λ
π
σω
ξ
==≡ ,
2
,
8
0
0 is the antisymmetric traveling mode wavelength with
phase velocity a
c and )()()( 00
1
0xiNxJxH += is the Hankel function of order zero representing a
wave traveling outward from the origin. A central conclusion of this research is that the optical
imaging approach described measures both the elastic wave amplitude and phase over the plate
surface simultaneously in a manner similar to electrical signal lock-in detection. The explicit
separation of the flexural wave displacement into terms depicting the elastic wave amplitude and
phase is not readily apparent from Eqn. 6. However, this separation can be achieved near the origin
and far away by expanding the Hankel functions. Near the origin, the Bessel functions can be
expanded as ...5772.0],)
2
[ln(
2
)(,
4
2
1)( 0
0
0
0=+→−→ →→
γγ
π
x
xN
x
xJ xx to show that

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 10
1)]()([lim 1
0
1
0
0=−
→ixHxH
x, designating 0
ξ
as the displacement amplitude at the source point.
Another way of writing the solution, using the relation )(
2
)( 1
0
0ixH
i
xK
π
=, is





+





+−+= )cos()(
2
)()sin()(),( 0000 ssaassa tkKkNtkJt
ϕωρ
π
ρϕωρξρξ
.(7)
which explicitly shows the traveling wave near and far field behavior as
)
4
sin(
2
),(),sin(),( 00
0
π
ρϕω
ρπ
ξρξϕωξρξ
ρρ
+−+→+→∞→→ ass
a
k
ss
kkt
k
ttt aa .
These expressions show that the displacement at the origin is 0
ξ
and that as the elastic wave travels
away from the origin, its phase increases linearly with radial distance. The normal lock-in method
allows one to detemine the flexural wave amplitude and phase separately and then reconstruct the
displacement completely at any point. The next section shows this procedure explicitly for the
photorefractive detection methodology.
PHOTOREFRACTIVE TRAVELING WAVE DETECTION
The method by which the photorefractive process demodulates the optical phase information can
be illustrated by considering an approximation to the two-wave & four-wave mixing processes. The
integral form of the Hankel function,19
∫=+= ∞−−
−
−
0
)
4
(
0
2
1
)
4
(
1
0)()
2
1(
2
)(
π
π
π
π
xi
u
xi exhdu
x
iu
u
e
e
x
xH , can be used to write Eqn. 6 for the
traveling wave displacement as

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 11









−−−
Φ
=)
4
(
Re),( )(
0ssa
a
kh tki
e
i
eit a
ϕω
π
ρ
ξρξ ρ
( 8)
where )1(
)()()( 00 ik
eikhkh
i
ekh a
aa
a
a+−
−≡
Φ
ρ
ρρρ
. The optical phase shift of the signal
beam, referring to Figure 1 and assuming normal incidence, 0=
ζ
, produced by the traveling wave
can be described using Eqn. 8, with )()( 4aa
kΦ−= +
π
ρρχ
, as
))(sin()(
4
))(,(4
),( 0
ρχϕωρ
λ
πξ
λ
ρ
χ
ϕ
ω
ρ
π
ξ
ρδ
−+=
−+
=ssasig tkh
s
t
s
t. (9)
Using the relation ∑
=∞=
−∞=
n
nnin
exJ
ix
e
θ
θ
)(
)sin( , the electric field amplitude of the optical signal
beam can be represented, referring to figure 1, as
))((
)(
)2(
))(sin(
)2(
)),(2(
),(
0
0
ρχϕω
δ
πν
ρχϕωδ
πν
ρδπν
−+
∑
−⋅
=
−+
−⋅
=
+−⋅
=
∞=
−∞=
ss
n
nsign
ss
s
sssig
ss
s
sigss
ss
tin
eJ
tRki
eI
ti
e
tRki
eI
ttRki
eItrA
r
r
r
r
r
r
(10)
where )(
40
0
ρ
λ
πξ
δ
asig kh=, ss rrR rr
r+= , s
Ithe optical signal beam intensity, s
k is the optical
signal beam wavevector and
ν
is the laser optical frequency. The reference beam is phase modulated

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 12
with magnitude 0ref
δ
, at the frequency
π
ω
2rand phase
r
ϕ
by an electro-optic modulator (EOM)
according to )sin(
0rrrefref t
ϕωδδ
+= , which produces a reference beam amplitude
)(
)(
)2(
)2(
),(
0rr
mrefm
rr
r
refrr
rr
tim
eJ
tRki
eI
tRki
eItrA
ϕω
δ
πν
δπν
+
∑
−⋅
=
+−⋅
=
∞
−∞=
r
r
r
r
(11)
where
r
r
rrR rr
r+= ,
r
Ithe optical reference beam intensity,
r
k is the optical reference beam
wavevector .
Interference inside the crystal produces a spatially and temporally modulated intensity pattern,
assuming the polarizations of the signal and reference beams are the same, as
[]
0
00 2
,,)cos(1 I
II
MIIIrKMII rs
rsrefsig =+=−+Σ+⋅+=
δδ
r
r, (12)
where rs kkK rr
r−= is the grating wavevector and rrss rkrk r
r
r
r⋅−⋅=Σ accounts for path length
differences between the two beams. The interference intensity distribution within the crystal generates
a corresponding space charge electric field distribution. The dynamic behavior of this field is
controlled by the charge carrier mobility and trapping that produces, in the diffusive operation
regime, a single relaxation time response given by20
0
2
I
AA
iE
E
t
Ers
q
scsc ∗
⋅
=+
ττ∂
∂
(13)

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 13
where
τ
is the material response time. The maximum achievable space-charge field,
πε
2
Λ
=A
qqN
E, is
controlled by the concentration of available charge trapping sites A
N, the fringe spacing K
π
2
=Λ ,
the carrier charge qand the permittivity of the medium
ε
. Using Eqn.s 10 through 12, the
interference term in Eqn. 13 becomes
)]([
)()(
2,
,00
0
ρχϕω
δδ
ntrKi
eJJM
I
AA nmnm
mn
mn refmsign
rs −++Σ+⋅
∑
=
⋅∞=
−∞=
∗r
r
(14)
where rsnmrs mnmn
nm
ϕϕϕωωω
−=−= ,.
Using the result of Eqn. 14 and solving Eqn. 13 for the space charge field yields
()












+
+
−+++Σ+⋅
∑
=∞
−∞= ..
)1(2
]
2
[
)()(, 00
,cc
i
ntrKi
e
JJMEtrE nm
nmnm
refmsign
mn
qsc
τω
χϕω
π
δδ
r
r
r(15)
where c.c. stands for complex conjugate. Let
χϕωτωψ π
ntrK nmnmnmnmnm −+++Σ+⋅=Θ= 2
and,)tan( r
r then

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 14
()
.
1
]sin[
)()(
)1(
]sin[]cos[
)()(,
22
00
,
22
00
,








+
−−++Σ+⋅
∑
=








+
Θ+Θ
∑
=
∞
−∞=
∞
−∞=
τω
ψχϕω
δδ
τω
τω
δδ
nm
nmnmnm
refmsign
mn
q
nm
nmnmnm
refmsign
mn
qsc
ntrK
JJME
JJMEtrE
r
r
r
(16)
This can be expanded to
()
()






























+
−−++Σ+⋅
∑
+
Σ+⋅












Ω+
−−Φ+Ω
∑
+
=
∞
≠−∞=
∞
=
22
00
)( ,
222
00
1
0000
1
)](sin[
)()(
sin
1
)][cos(
)()(
)()(
,
τω
ψχϕω
δδ
τ
ψχ
δδ
δδ
nm
nmnmnm
refmsign
mnmn
n
refnsign
n
refsig
qsc
ntrK
JJ
rK
n
tn
JJ
JJ
MEtrE
r
r
r
r
r(17)
where the frequency difference rs
ωω
−=Ω , the phase difference rs
ϕϕ
−=Φ , n
Jis the Bessel
function of the first kind and
τψ
Ω= n
n)tan( . Eqn. 17 represents the electric space charge field
within the photorefractive crystal as a series of terms including a constant term, low frequency terms
at multiples of the difference frequency between the signal and reference beams and higher frequency
terms at multiples of the signal and reference frequencies. In the above configuration, the
photorefractive crystal acts as a mixing and low pass filtering element providing the benefits of lock-
in detection. Therefore the space charge field responds to slowly varying phase modulations
occurring within the material response time constant allowing only the terms around the difference
frequency Ωto be important, assuming that rs,
ω
<<Ω . Employing the low pass filtering, Eqn. 17
for the space charge field becomes

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 15
()
)sin(
1
))(cos(
))(()(2
))(()(
,22
1
0101
0000
Σ+⋅














+
Ω+
−−Φ+Ω
+= rK
t
JJ
JJ
MEtrE sigref
sigref
qsc
r
r
K
r
τ
ψρχ
ρδδ
ρδδ
(18)
which can be more compactly written as
()
)sin());(),((, 0Σ+⋅= rKtfMEtrE sigqsc
r
r
r
ρχρδ
.
The space-charge field modulates the local refractive index through the linear electro-optic effect.
This effect creates a diffraction grating within the crystal that contains the low frequency phase
information desired. Several methods can be used to readout the space charge field and diffraction
grating including (1) four-wave mixing, (2) two-wave mixing with polarization selection, and (3)
electrical measurement through conduction of photoexcited carriers. The magnitude of the index of
refraction grating produced is given by5 2
41
3
0
1sc
Ern
n−= , where 0
n is the average refractive index
of the medium, 41
r is the effective, orientation-dependent electro-optic coefficient in BSO. The
diffracted beam intensity is a direct measure of the grating established and its diffraction efficiency is
determined by the wave coupling constant, according to the scattering theory developed by
Kogelnik,21
));(),((
2
));(),((
cos2cos 00
41
3
0
1tfM
L
tfM
LErn
Ln sigsig
q
ρχρδρχρδ
θλ
π
θλ
π
ζ
Γ
=








=






≡(19)
where Lis the interaction length,
Γ
is the two-wave mixing coupling constant5 and 16.0=
Γ
L for the
BSO crystal used with input beam polarizations along the <001> and perpendicular to the <110>
directions,
λ
is the laser source wavelength, and 2
θ
is the angle between the mixing waves.

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 16
Operation in the four-wave mixing arrangement is described as it provides a simpler analysis for
demonstrating the mechanism whereby lock-in imaging occurs since in this configuration there is no
direct transmitted beam. The reference beam that passes through the crystal is reflected back into the
crystal and diffracts from the photoinduced grating retracing the signal beam path, see figure 1. In the
undepleted pump approximation, the diffracted (conjugate) beam intensity is21
2
34 sin
cos/
ζ
θ
α
L
eII −
=(20)
where 3
I is the back propagated reference beam intensity and
α
is the material absorption
coefficient. The refractive index modulation amplitude generated by the mixing process is generally
small, so that
ζ
ζ
ζ
≈<< )sin(and,1 . The intensity of the diffracted conjugate beam is given by












+−−Φ+Ω








Ω+
+





Γ
=
−
K))(cos()()(
1
)()(4
)()(
2
cos/
0100
22
0100
0
2
00
2
0
2
2
3
4
ψρχδδ
τ
δδ
δδ
θα
tJJ
JJ
JJ
M
L
L
eI
I
sigsig
refref
sigref (21)
where
tan( )
ψ
τ
=Ω . Eqn. 21 shows that the magnitude and phase of the traveling wave have been
placed as arguments of the Bessel functions for the magnitude and as the phase of a low frequency
AC signal. The resultant measured intensity is then proportional to
))(cos()()( 0100
ψρχδδ
−−Φ+Ω∝ tJJI sigsigAC (22)

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 17
which for small traveling wave displacement amplitudes becomes
.
2
2
)(
0
0
with,
))(,(4
))(sin(
4
2
1
))(cos(
2
π
π
ρ
ψϑ
λ
ρχϑρπξ
ψρχ
λ
πξ
ψρχ
δ
+−Φ=
−+Ω
∝
+−−Φ+Ω






∝−−Φ+Ω∝
t
ttI a
kh
sig
AC (23)
Therefore, if Eqn. 23 is compared with Eqn. 9, the optical imaging approach can be seen to provide a
true measure of the traveling wave amplitude and phase, for small amplitudes where .1
),(4 <<
λ
ρ
π
ξ
t
The maximum measured intensity of Eqn. 22 occurs at a phase shift of 1.08 radians, which
corresponds to a traveling wave amplitude of 45.7 nm for a probe wavelength of 532 nm. Comparison
of the AC and DC terms in Eqn. 21, with knowledge of the reference beam modulation amplitude and
the photorefractive crystal time constant, allows absolute calibration of the flexural wave
displacement amplitude even when the maximum signal beam modulation amplitude cannot be
realized.22 Operation with the two-wave mixing method provides similar results whereby the
diffraction process produces an output beam whose AC intensity component is proportional to the
elastic wave displacement. This mode can be more efficient than the four-wave approach in that the
output can be configured to be proportional to the two-wave mixing coupling constant
Γ
, rather than
the square of this quantity. The analysis is complicated by the fact that a large directly transmitted
beam is also present and does not further illustrate the lock-in measurement process under discussion;
therefore, it is not presented here but will be the subject of future work.
SINGLE POINT MEASUREMENTS

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 18
Single point measurements were implemented with the two-wave mixing technique according to
figure 1 by focusing the signal beam to a spot size of about 0.1 mm diameter onto a 0.125 mm thick
pure nickel plate. A piezoelectric transducer with a short length of metal rod ground to a point served
as the source transducer. The tip of the rod was placed in contact with the plate surface by applying a
small amount of pressure and its vibration generated flexural waves in the plate. The plate was
clamped at the outer plate boundaries between two pieces of a viscoelastic damping material23 that
shows damping characteristics superior to natural rubber. Continuous excitation of the transducer
then produced steady traveling waves emanating from the contact point at a prescribed signal
frequency and wavelength according to Eqn. 6. Although this method of excitation was suitable for
the measurements here, difficulties were encountered due to the resonant behavior of the metal
rod/transducer combination that allowed only discrete frequencies to be investigated. Figures 2-4
show the flexural wave amplitudes as a function of distance from the source for frequencies of 8.0,
15.0, and 30.0 kHz. The points are the result of direct measurement by translating the detection beam
along a radius from the source point. The solid lines shown in the figures are the calculated results
from Eqn. 6. Good agreement is seen between the measurements and the calculation using only the
displacement amplitude at the source point and the relative phase between the signal and reference
beam modulations as adjustable parameters. Some deviation is seen in the 8 kHz data at the left of
figure 2 that we attribute to insufficient damping of the wave reflected from the supporting clamp at
the plate edge. Material elastic and physical constants of
3
9.8,31.0,204 cm
g
sGPaE m===
ρ
and mm125.0=h were used for the nickel plate
calculation. In each figure, both the traveling flexural wave displacement amplitude and phase were
recorded by using AC lock-in measurement techniques. Figures 2-4 show the reconstructed
waveforms taking into account the amplitudes and phases measured by the lock-in; the results are

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 19
proportional to the out-of-plane displacement, no absolute calibration was performed. The sensitivity
of the measurement was ultimately limited by optical phase noise produced by the detection laser
and environmental vibrations which had previously been shown to correspond to a minimum
detectability of about 0.002 nm for these point measurements.9 At each frequency, the relative phase
shift between the signal and reference beams was adjusted so that the maximum displacement
amplitude occurred at the source point yielding the results shown. Each figure corresponds to a
snapshot of the traveling wave displacement along the plate at a moment in time. The wavelength of
the flexural wave can be obtained by Fourier transforming the data and employing Eqn. 4.
Measurements up to 1 MHz agreed well with the calculations, as shown in figure 5.
PHOTOREFRACTIVE DYNAMIC HOLOGRAPHIC IMAGING
Since optical interference and the photorefractive effect occur throughout the photorefractive
crystal, the point method described above can be generalized to that of an image of the vibration
over the surface of the plate. The volume character of the photorefractive process creates a grating
distribution that locally records the phase modulation measured from each point of the specimen
surface as long as the surface is accurately represented within the photorefractive crystal. The output
beam intensity can then be measured by an array of detectors, or a highly pixelated device, such as a
CCD camera. Each pixel records the local intensity from a point on the specimen producing an
output proportional to that point’s displacement. Even a diffusely reflecting surface can be measured
if the surface is adequately imaged inside the photorefractive crystal by suitable optics.
The experimental setup for vibration imaging using polarization rotation through anisotropic self-
diffraction is similar to that shown in figure 1 except that additional optical elements are used to
illuminate and image the vibrating surface. A two-wave mixing configuration was used as the rough
surface of the plate diffusely scattered the laser light resulting in insufficient light approaching the

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 20
video camera after diffracting from the grating in the four-wave method. Light from a solid state laser
source (532 nm, 1W), was split into two legs forming the signal and reference beams. The signal
beam was expanded and reflected off the surface of the plate undergoing continuous vibration.
Traveling flexural waves in the plate were excited by a contact piezoelectric transducer in the same
manner as with the point measurements. Light scattered from the plate was imaged in the
photorefractive crystal by a collection lens. The modulated reference beam was also expanded and
projected into the photorefractive crystal to produce the volume holographic grating. An input high
extinction ratio polarizer selected one component of the signal beam from the plate. At the output of
the crystal, the diffracted wavefront was selected through use of another high extinction ratio
polarizer.
Figure 6 shows images of the traveling flexural waves in the plate obtained with the two-wave
mixing method at frequencies of 8,15, and 30 kHz. The expected circular wavefronts due to the
isotropic microstructure of the nickel plate are clearly defined and the relative vibration
displacement phase is readily distinguishable. The figure shows single frame image data at three
different frequencies that have the background subtracted. For qualitative inspection of two
dimensional waveforms from the CCD output, the eye integrates over multiple video frames. If the
difference frequency is held at zero or locked to the camera frame rate of 30 Hz, a stationary
wavefront pattern is observed. This signal averaging makes it possible to easily detect subtle patterns
such as those brought about by the traveling waves. Also the entire pattern can be made to change its
phase continuously at the frequency, Ω, from 1-30 Hz, so that the appearance is that of waves
emanating from the center and traveling outward. This is physically equivalent to the actual
traveling wave motion except that viewing of the wave has been slowed to a much smaller
observation frequency that is held constant and independent of the actual wave frequency. A
sequence of successive frames is shown in figure 7. The frame rate is 30 Hz and the offset frequency

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 21
is about 4 Hz resulting in a continuous change in the relative phases between each image. The result
is a time-lapse image that shows the viewer a wave emanating from the center and traveling outward
and finally attenuated outside the field of view. This quasi-real-time imaging tells the viewer the
wavefront shape from which information about the plate material, such as the elastic constants or the
locations of flaws along the wave path can be determined.
The photorefractive process yields a true picture of the actual wave out-of-plane displacement
motion and does not require any additional processing to generate the images of figure 6. However,
to obtain quantitative measurement of the flexural wave displacement at any point within the image,
the intensity at that point must be compared with the background value. This background value is
obtained from an average of several additional frames recorded at different phase shifts between the
signal and reference images phase locked so as to eliminate the flexural wave displacement when the
average is taken. This procedure was previously illustrated in reference 9 concerning images of
resonant vibrations in plates.
FREQUENCY ANALYSIS OF THE FLEXURAL WAVE IMAGES
The magnitude of the Fourier transform of the traveling wave displacement as a function of the
radial propagation direction, Eqn. (4), shows a real pole at the applied wavevector for the traveling
wave and imaginary poles of the same value that contribute to satisfy the boundary conditions.
Therefore, the Fourier transform image of the traveling wave displacements should be a single ring at
the applied wavevector delineating the propagating mode. Figures 8-10 show calculations and
measurements of the traveling wave displacements as an image over the surface. The calculations
are from Eqn. (6) and the measurements those of figure 6. Beside the images are shown images of
the magnitude of the Fourier transforms. A strong response is seen as a ring at the propagating
wavevector, whose magnitude can be determined to allow calculation of the elastic stiffness of the

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 22
plate, assuming the plate mass density per unit area is known. Both the wavefront and the Fourier
transform images clearly show the isotropic character of the plate. Correspondingly, the figures also
show the magnitude of the Fourier transforms of the measured image data. The large response is
again seen as rings at the propagating wavevector values. This analysis procedure provides
considerable information about the plate in one simple image.
ANISOTROPIC MATERIAL MEASUREMENTS
If the specimen is elastically anisotropic, then the wave speed varies with the propagation
direction. Figure 11 shows this type of behavior for traveling waves in a sheet of carbon fiber
composite. The carbon fiber sheet was approximately 0.18 mm thick with the fibers aligned in
parallel along the vertical direction. The matrix is an isotropic resin material. The highly oblong
wavefront pattern seen in figure 11 shows the anisotropy clearly and immediately. Figure 12 shows
the wavelengths measured for this composite sheet in the directions along (x’s) and perpendicular
(o’s) to the fibers as a function of frequency. Clearly, a great deal of information about the
anisotropic elastic properties of the sheet can be obtained directly from this image measurement
technique.
CONCLUSIONS
An imaging photorefractive optical lock-in traveling wave measurement method has been
described. Detailed operation of the imaging method for recording nonstationary wavefronts through
the lock-in process has been presented. Four-wave and two-wave mixing were described for reading
out the signal producing an output intensity directly proportional to the amplitude of the vibration
being measured at a preset mechanical phase. Point measurements scanned along a propagation
radius produced a spatial snapshot of the amplitude and phase of the traveling waveform. Direct two-

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 23
dimensional surface images of the traveling wave were obtained by expanding the collection optics
and imaging the output beam from the photorefractive material. These images showed the ultrasonic
wavelength and wavefront shape and provided a quantitative method for obtaining the elastic
stiffness symmetry of sheet materials, as illustrated for an isotropic nickel plate and an anisotropic
composite carbon sheet. The method is capable of flat frequency response over a wide range above
the cutoff of the photorefractive effect and is applicable to imaging the ultrasonic motion of surfaces
with rough diffusely reflecting finishes.
ACKNOWLEDGMENTS
This work was sponsored by the U.S. Department of Energy, Office of Science, Office of Basic
Energy Sciences, Engineering Research Program and the INEEL Laboratory Directed Research &
Development program under DOE Idaho Operations Office Contract DE-AC07-94ID13223.

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 24
Figure Captions
Figure 1. Photorefractive two-wave mixing setup for optical vibration detection.
Figure 2. Flexural mode traveling wave surface displacement on a 0.125 mm thick nickel plate
at 8.0 kHz.
Figure 3. Flexural mode traveling wave surface displacement on a 0.125 mm thick nickel plate
at 15.0 kHz.
Figure 4. Flexural mode traveling wave surface displacement on a 0.125 mm thick nickel plate
at 30.0 kHz.
Figure 5. Flexural wave wavelengths as a function of driving frequency for the 0.125mm thick
nickel plate, calculated (line) using parameters described in the text and measured
(circles).
Figure 6. Single frame images of the traveling waves at (top) 8 kHz, (middle) 15 kHz, and
(bottom) 30 kHz.
Figure 7. Time-lapse picture of successive frames of the traveling wave images showing the
emergence of the wavefront from the center of the plate.
Figure 8. Calculated (top) and measured (bottom) traveling wave displacements (left) and
magnitude of the 2-D FFT (right) for the nickel plate at 8.0 kHz.
Figure 9. Calculated (top) and measured (bottom) traveling wave displacements (left) and
magnitude of the 2-D FFT (right) for the nickel plate at 15.0 kHz.
Figure 10. Calculated (top) and measured (bottom) traveling wave displacements (left) and
magnitude of the 2-D FFT (right) for the nickel plate at 30.0 kHz.
Figure 11. Image of a traveling wave in a an anisotropic composite sheet at 37.8 kHz.
Figure 12. Measurements of the wavelength in the vertical and horizontal directions in the
anisotropic composite sheet.

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 25
Figure 1
kr
K
2θ
ks
ρka
ξo(ωs t)
rs
rr
r
Laser
Photorefractive
Material
Signal Beam
Reference Beam
Sheet Specimen
δsig = 4π ξ(ρ, ωs t) cos(ζ)
λ
2ζ
δref
Piezoelectric
Transducer

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 26
−40 −30 −20 −10 0 10 20 30 40
−10
−5
0
5
10
15
20
ξ (arbitrary units)
ρ (mm)
Figure 2

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 27
−40 −30 −20 −10 0 10 20 30 40
−6
−4
−2
0
2
4
6
8
10
12
ξ (arbitrary units)
ρ (mm)
Figure 3

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 28
−40 −30 −20 −10 0 10 20 30 40
−3
−2
−1
0
1
2
3
4
5
6
ξ (arbitrary units)
ρ (mm)
Figure 4

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 29
100101102103
10−1
100
101
102
λ (mm)
f (kHz)
Figure 5

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 30
X (mm)
Y (mm)
−30 −20 −10 0 10 20 30
−30
−20
−10
0
10
20
30
X (mm)
Y (mm)
−30 −20 −10 0 10 20 30
−30
−20
−10
0
10
20
30
X (mm)
Y (mm)
−30 −20 −10 0 10 20 30
−30
−20
−10
0
10
20
30
Figure 6

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 31
Figure 7

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 32
−40 −20 0 20 40
−40
−20
0
20
40
X (mm)
Y (mm)
−0.2 −0.1 0 0.1 0.2
−0.2
−0.1
0
0.1
0.2
ν
x
(mm
−1
)
ν
y
(mm
−1
)
X (mm)
Y (mm)
−40 −20 0 20 40
−40
−20
0
20
40
ν
x
(1/mm)
ν
y
(1/mm)
−0.2 −0.1 0 0.1 0.2
−0.2
−0.1
0
0.1
0.2
Figure 8

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 33
−40 −20 0 20 40
−40
−20
0
20
40
X (mm)
Y (mm)
−0.2 −0.1 0 0.1 0.2
−0.2
−0.1
0
0.1
0.2
ν
x
(mm
−1
)
ν
y
(mm
−1
)
X (mm)
Y (mm)
−40 −20 0 20 40
−40
−20
0
20
40
ν
x
(1/mm)
ν
y
(1/mm)
−0.2 −0.1 0 0.1 0.2
−0.2
−0.1
0
0.1
0.2
Figure 9

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 34
−40 −20 0 20 40
−40
−20
0
20
40
X (mm)
Y (mm)
−0.2 −0.1 0 0.1 0.2
−0.2
−0.1
0
0.1
0.2
ν
x
(mm
−1
)
ν
y
(mm
−1
)
X (mm)
Y (mm)
−40 −20 0 20 40
−40
−20
0
20
40
ν
x
(1/mm)
ν
y
(1/mm)
−0.2 −0.1 0 0.1 0.2
−0.2
−0.1
0
0.1
0.2
Figure 10

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 35
X (mm)
Y (mm)
−100 −50 0 50 100
−100
−80
−60
−40
−20
0
20
40
60
80
100
Figure 11

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 36
30 40 50 60 70 80 90 100
2
3
4
5
6
7
8
9
f (kHz)
λ (mm)
Figure 12

J. Acoust. Soc. Am. 106, 2578-2587 (1999) 37
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