Laser-based linear and nonlinear guided elastic waves at ... · Laser-based linear and nonlinear...

Ultrasonics 54 (2014) 39–55

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Ultrasonics

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Laser-based linear and nonlinear guided elastic waves at surfaces (2D)and wedges (1D)

0041-624X/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ultras.2013.05.013

⇑ Corresponding author. Tel.: +49 6221 545205; fax: +49 6221 544255.E-mail address: [email protected] (P. Hess).

Peter Hess a,⇑, Alexey M. Lomonosov a,b, Andreas P. Mayer c

a Institute of Physical Chemistry, University of Heidelberg, D-69120 Heidelberg, Germanyb General Physics Institute, Russian Academy of Sciences, 119991 Moscow, Russiac Faculty B+W, HS Offenburg, University of Applied Sciences, 77723 Gengenbach, Germany

a r t i c l e i n f o

Article history:Received 18 January 2013Received in revised form 23 May 2013Accepted 24 May 2013Available online 11 June 2013

Keywords:Laser ultrasonicsOne- and two-dimensional waveguidesSurface acoustic waves (SAWs)Wedge waves (WWs)

a b s t r a c t

The characteristic features and applications of linear and nonlinear guided elastic waves propagatingalong surfaces (2D) and wedges (1D) are discussed. Laser-based excitation, detection, or contact-freeanalysis of these guided waves with pump–probe methods are reviewed. Determination of materialparameters by broadband surface acoustic waves (SAWs) and other applications in nondestructive eval-uation (NDE) are considered. The realization of nonlinear SAWs in the form of solitary waves and as shockwaves, used for the determination of the fracture strength, is described. The unique properties of disper-sion-free wedge waves (WWs) propagating along homogeneous wedges and of dispersive wedge wavesobserved in the presence of wedge modifications such as tip truncation or coatings are outlined. Theoret-ical and experimental results on nonlinear wedge waves in isotropic and anisotropic solids are presented.

� 2013 Elsevier B.V. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402. Surface acoustic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.1. Linear surface acoustic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.1.2. Excitation and detection of SAWs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.1.3. Determination of elastic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.1.4. Nondestructive evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2. Nonlinear surface acoustic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2.2. Experimental. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2.3. Solitary surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.2.4. Shock waves and determination of fracture strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.2.5. Nonlinear SAWs by wave interaction with scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3. Wedge waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1. Linear wedge waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1.1. Theoretical approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1.2. Excitation and detection of wedge modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1.3. Properties of perfect wedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1.4. Dispersive wedge waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2. Nonlinear wedge waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.2. Experimental. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3. Applications of wedge waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
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40 P. Hess et al. / Ultrasonics 54 (2014) 39–55

1. Introduction

The interest in ultrasonic guided waves covers a wide field,from nondestructive evaluation (NDE) and structural health mon-itoring (SHM) of industrial systems with dispersive three-dimen-sional (3D) ultrasonic waves to basic research on high-qualitylow-dimensional waveguides, for example, one-dimensional (1D)ultrasonic waves without dispersion and diffraction. Acousticwaves can be guided by different types of boundaries, for example,free boundaries, solid–liquid, solid–gas interfaces, or interfaces be-tween two different solids. Normally such waves propagate alongthe boundary over substantially long distances.

Even in well-established conventional applications of ultrason-ics to macroscopic objects such as rails there is a tendency to useguided waves for inspection, which is connected with reduced cost,less inspection time, and greater coverage [1]. These ultrasonicwaves are constrained by the boundaries of the particular geomet-ric structure and include, for example, torsional waves in a tube orLamb waves in a plate but also other modes. Permanent sensorinstallation for long distance diagnostics by conventional transduc-ers, e.g., in rails, is a new approach in SHM. A serious challenge ofguided modes in complex geometries is their dispersive and mul-timode nature. In NDE applications, substantial computations areneeded for data analysis under these conditions.

Currently, practical applications of guided waves for sensingsurface damage and cracks in rails, for example, are dominatedby conventional transducers and sensors, while the use of lasersis still under development (for a review see [2]). However, acontact-free fast inspection system has been described, using apulsed laser for excitation of guided ultrasonic waves and multi-ple air-coupled sensors for detection of transverse-type smallcracks with 1 mm depth in the rail head [3]. Recently forSHM purposes also noncontact laser excitation and sensing, aswell as laser-based piezo-ceramic transducer excitation andsensing systems have been developed [4]. These detection sys-tems can be used for laser-based wireless power and data trans-mission schemes in remote guided waves and impedancemeasurements. Envisaged aircraft, wind turbine, bridge, highspeed rail, and nuclear power plant applications have been dis-cussed [4].

Laser ultrasonics yields wavefield images with a much higherspatial resolution than conventional transducers. Since for laserdiagnostics no baseline data is needed, laser scanning techniquesare less sensitive to environmental and operational changes. A sim-ilar situation is encountered in current studies of material nonlin-earity using nonlinear Lamb waves. While most of theinvestigations have employed conventional transducers only fewstudies use a laser interferometer for detection that provides clearadvantages over contact finite-size wedge receivers [5]. The goal ofthese measurements is to identify plasticity damage, e.g., due todislocation pile-ups, by monitoring generation of the second har-monic of the sound wave prior to initiation of microcracks. Thenonlinear acoustic behavior of cracks has also been studied byall-optical probing, since actuators may mask the effect to be mea-sured owing to nonlinear acoustic transformations at their inter-face with the surface to be probed [6]. As nonlinear signature forthe characterization of imperfect contacts between interfaces orcrack faces frequency mixing has been selected in theseexperiments.

In the following we focus on two particular types of guidedwaves, propagating along a solid half-space, namely surface acous-tic waves (SAWs), or along the edge of a solid wedge bounded bytwo planes, which are called wedge waves (WWs). These wavesare localized at the guiding plane or the apex line, respectively,hence they do not probe the whole sample.

In geometries like planar surfaces (2D) of homogeneous mediaand ideal wedges (1D), guided acoustic waves exist that are notdispersive. In the 1D case, diffraction is absent, too, whereas acous-tic mode conversion may happen under certain conditions.

This review concentrates on the contact-free laser excitationand/or detection of 2D SAWs, propagating along plane surfaces,and 1D WWs, travelling along the apex of a wedge. The uniqueproperties of laser radiation such as directivity, power density,and the possibility of focusing the beam allow unique experimentsto be performed that are not possible by conventional ultrasonics.Pulsed lasers give access to linear but also strongly nonlinear ultra-sonic pulses. The acoustic bandwidth can be easily controlled,which is of great practical interest, since industrial materials mayshow strong frequency-dependent attenuation, limiting the obser-vation distance. Therefore, it can be anticipated that laser-basedwaveguide methods will find their niches, beside the usually muchcheaper conventional transducer techniques, for quality control ofadvanced high-quality materials. Especially, this can be expectedwhen spatial resolution, contact-free performance, limited sampleaccess, or sensitivity are of concern.

Surface acoustic waves were discovered in 1885 by Lord Ray-leigh [7]. This wave concept was first applied to seismic waves(‘‘infrasound’’). Only much later, with the advent of piezoelectrictransducers and interdigital structures on piezoelectric materials,was the field of guided-wave ultrasonics born. Laser ultrasonicsdeveloped even later, after the discovery of lasers in 1960, and gaveaccess to narrowband and broadband ultrasonic waves, as well asto linear and nonlinear elastic pulses [8]. A characteristic feature ofSAWs is that they penetrate only about one wavelength deep intothe solid with the displacement decaying exponentially. Note thatthe penetration depth is wavelength dependent and therefore dif-ferent depths can be probed simultaneously with a broadbandSAW pulse. We can say that the surface acts as a waveguide, keep-ing the elastic energy at the free surface. In fact, these waves prop-agate even around corners and edges. In SAWs particle motion iselliptically polarized with in-plane and out-of-plane displacementcomponents that can be measured optically. Furthermore, planewaves with defined propagation direction and minimized diffrac-tion losses can be launched by focusing the exciting laser pulseto a narrow line at the surface.

When a length scale is introduced into the system, for exampleby a layer, the dispersion-free SAWs become dispersive. The result-ing dispersion effect opens up the possibility of determining themechanical properties of thin films and coatings. This is one ofthe well-established applications of laser-based linear SAWs. An-other one is the localization and sizing of surface-breaking cracks.The excitation of solitary waves in dispersive layered systems ispossible with nonlinear SAW pulses, when dispersion and nonlin-earity is balanced. For the generation of cracks and the measure-ment of the fracture strength of anisotropic crystals in definedcrystallographic configurations, strongly nonlinear SAW pulseswith steep shock fronts are needed [8].

Besides surfaces also solid wedges can function as waveguides.Here the edge of a wedge guides the elastic waves and the energystays near the wedge tip during propagation. Therefore, thesewaves are called wedge waves (WWs), or sometimes edge waves,or line waves. Since most authors use now the term ‘‘wedge wave’’this is employed throughout this review. The 1D acoustic WWswere first discovered by numerical calculations in 1972, indepen-dently by Lagasse [9] and by Maradudin and coworkers [10], as anew type of fundamental acoustic waves propagating dispersion-free along the perfect edge of a homogeneous wedge. This is trueonly, as long as no length scale is introduced into the wedge sys-tem. Any distortions of the ideal wedge geometry by modificationssuch as inhomogeneity, truncation, coatings lead to dispersion of

0 10 20 30 404.6

4.8

5.0

5.2

5.4

5.6

5.8

Velo

city

c, k

m/s

Angle from <100>, degrees

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

D(c) at 40o

SAWp-SAW

f-SW

s-SW

Fig. 1. Dependence of the determinant D(c) on the phase velocity calculated for thesilicon (001) plane in the direction 40� relative to h100i. The lowest local minimumcorresponds to the SAW and the slowest shear wave (s-SW), the mid one to theleaky pseudo-surface wave (p-SAW), and the third one to the fast shear wave (f-SW).

P. Hess et al. / Ultrasonics 54 (2014) 39–55 41

the phase velocity that can be used, of course, to evaluate the devi-ations from the ideal geometry of the particular wedgeconfiguration.

Currently, the field of WWs is still in fast scientific and technicaldevelopment and no industrial applications of these 1D guidedwaves are presently known. However, several feasibility studieshave been performed in recent years pointing to interesting poten-tial applications of these low-dimensional elastic waves in NDE,sensors, and actuators, e.g., in fluidic dynamics or ultrasonicmotors.

2. Surface acoustic waves

2.1. Linear surface acoustic waves

2.1.1. TheorySurface acoustic waves propagate along the interface between

two elastic media, at least one of which is a solid. The long-wave-length elastic waves can be treated within the framework of classi-cal continuum elasticity theory [8,11]. SAWs are solutions of the setof wave equations for the displacement components that usuallysatisfy the boundary conditions of a mechanically stress-free sur-face. The solutions can generally be presented as a linear combina-tion of inhomogeneous plane waves with wavevectors parallel tothe surface that propagate in a defined direction with the samephase velocity and that decay exponentially in depth. Typicallymore than 90% of the energy of the surface wave remains concen-trated at a range of the order of one wavelength below the surface.

Following a standard approach, we start with the linear equa-tion of motion for the displacement field ui(x1,x2,x3; t), i = 1, 2, 3in Lagrangian coordinates xi, i = 1, 2, 3, which denote the positionof a mass element in the undeformed state of the solid [12]. Inthe deformed state of the material, this mass element is displacedto its new position x + u. The solid occupies the half-space x3 P 0.In this notation the equation of motion is

q@2

@t2 ui ¼@

@xjrji; ð2:1Þ

where q is the mass density and the stress tensor r depends on thedisplacement gradients ukl ¼ @uk=@xl linearly: rij ¼ cijklukl.

In Eq. (2.1) and in the following equations, summation over re-peated Cartesian indices is implied. Substituting the stress tensorin this form in Eq. (2.1) provides the wave equation

q@2

@t2 ui ¼ cijkl@2

@xj@xluk ð2:2Þ

in the case of a homogeneous elastic medium.Solutions to Eq. (2.2) are sought in the form of inhomogeneous

plane waves

ui ¼ ai expðikðl � x� ctÞÞ orui ¼ ai expðikðl1x1 þ l2x2 þ l3x3 � ctÞÞ; ð2:3Þ

where the first two components of the vector l = (l1, l2, l3) are nor-mally assumed to be real with l2

1 þ l22 ¼ 1 and determine the propa-

gation direction parallel to the surface x3 = 0.The complex component l3, having a positive imaginary part in

the case of surface waves, describes the decay of the wave field indepth. Quantities ai denote the polarization of the solution and itsamplitude, c is the phase velocity.

Substitution of Eq. (2.3) into Eq. (2.2) yields

qc2ai ¼ cijklljllak: ð2:4Þ

Eq. (2.4) represents a system of three (i = 1,2,3) homogeneous lin-ear equations for the quantities ai. It possesses a nontrivial solutiononly if its determinant equals zero:

detðcijklljll � qc2dikÞ ¼ 0: ð2:5Þ

For given l1, l2, and c, the latter is an algebraic equation of sixth or-der in l3 with real coefficients. It possesses either real roots or com-plex-conjugated pairs of roots. Real l3 correspond to the bulk waves,which do not decay with depth. Thus Eqs. (2.4) and (2.5) can beused to find the bulk waves in anisotropic media. In fact, whenspecifying l = (l1, l2, l3) as a real unit vector in the propagation direc-tion of the bulk wave, Eq. (2.4) is the Christoffel equation, havingthe form of an eigenvalue problem. The eigenvalues c2 of the Chris-toffel matrix, i.e., the three roots of Eq. (2.5), are the squares of thephase velocities of one (quasi-)longitudinal wave and two (quasi-)shear waves. The corresponding polarization vectors a follow fromEq. (2.4).

In order to find surface wave solutions, we treat c as a parame-ter in Eq. (2.5). For c below a certain threshold, none of the roots l3of Eq. (2.5) is real, and we select the three roots lðnÞ3 , n = 1, 2, 3, withpositive imaginary parts to satisfy the condition of the wave fieldvanishing at x3 ?1. Thus the general solution to Eq. (2.2) that de-cays in depth has the form of a linear combination of three partialwaves:

ui ¼X3

n¼1

CnaðnÞi exp ik lðnÞ3 x3 þ l1x1 þ l2x2 � ct� ��

: ð2:6Þ

Vectors a(n) may be normalized in an arbitrary way, but it is prefer-able to normalize them to the a1 component, for example, set a1 = 1(or u1 = 1), because the u2 and u3 components may be zero for someconfigurations in crystals.

Now we make use of the fact that the surface of the solid is trac-tion-free, i.e. r3j = c3jklukl = 0 at x3 = 0 for j = 1, 2, 3. In order to sat-isfy this boundary condition we should find those coefficients Cn

which make the wave field u in the form of Eq. (2.6) satisfy thetraction-free boundary conditions.

Using the expression for ui in the form of Eq. (2.6), the boundarycondition is reduced to

c3jkl

X3

n¼1

CnaðnÞk lðnÞl ¼ 0: ð2:7Þ

Here, we have defined lðnÞ ¼ ðl1; l2; lðnÞ3 Þ and Eq. (2.7) represents a

homogeneous system of three linear equations in Cn (j = 1, 2, 3). This

0.25 0.50 0.75 1.00 1.25 1.50

0.0

0.2

0.4

u3

u2

Dis

plac

emen

t com

pone

nts u i

, a.u

.

Depth, d/λ

u1

0

Fig. 2. Displacement components calculated for the p-SAW branch and thewavevector direction of 40� (see Fig. 1).

Fig. 3. Experimental setup for pulsed laser excitation of linear SAWs and interfer-ometric detection of surface displacements or laser-probe-beam deflection tomeasure velocities.


system possesses a nontrivial solution if the determinant D(c) of the3 � 3 matrix (Mjn) with components Mjn ¼ c3jkla

ðnÞk lðnÞl vanishes,

DðcÞ ¼ detðMjnÞ ¼ 0: ð2:8Þ

Note, that the quantities aðnÞk and lðnÞl are functions of the unknownvalue of the phase velocity c of the surface wave. A conventionalnumerical procedure to calculate c searches for the value that min-imizes the absolute value of D(c). It starts from some guess value forc, which is usually chosen somewhat smaller than the lowest bulkvelocity, and varies it to find the absolute minimum (or zero) ofthe modulus of the determinant. The value of c that makes thedeterminant vanish is the velocity of the wave that satisfies theequation of motion Eq. (2.2), the traction-free boundary conditionEq. (2.7), and decays in depth. With this value of c, aðnÞk and lðnÞl arealso known, and Eq. (2.7) gives access to the three weighting coef-ficients Cn, and finally the displacement field of the surface wavecan be obtained in the form of Eq. (2.6).

A typical dependence of the determinant D(c) calculated for thesilicon (001) plane is shown in Fig. 1 for the direction 40� relativeto the h100i direction. The vectors a(n) were normalized such thataðnÞ1 ¼ 1. Three local minima can be seen: the one with the lowest ccorresponds to the SAW and the slowest shear wave (s-SW), themid one is the leaky pseudo-surface acoustic wave (p-SAW), andthe third one is associated with the fast shear wave (f-SW). Onlythe first minimum reaches zero, indicating that the boundary con-ditions can be satisfied. The one correspondent to the p-SAW re-tains a small nonzero value, so this wave can exist only insuperposition with a slower bulk mode, which is the s-SW in thiscase. Physically, the p-SAW emits the s-SW as it propagates, andtherefore loses its energy continuously. Since this energy loss isrelatively small, the p-SAW can be observed experimentally.

In anisotropic media velocities of acoustic modes are in generaldependent on the propagation direction. To find this dependenceone must either rotate the coordinate system and convert the ten-sor (cijkl) correspondingly, or retain the coordinate system andspecify the propagation direction by varying l1, l2. Then the minimaof D(c) vary and deliver the phase velocity as a function of thepropagation direction, as shown in Fig. 1.

We consider here the displacement field of SAW pulses that de-pends on two spatial coordinates, namely the direction of SAWpropagation and the depth coordinate, and on time. The depen-dence on the third spatial coordinate can be neglected for a SAWpulse that is excited by a homogenous line source, which can berealized by focusing the excitation laser pulse with a cylindricallens to a line on the surface. For a sufficient length of the linesource, effects such as diffraction and beam steering (groupvelocity and wavevector are not parallel) are negligible. The

displacement components calculated by Eq. (2.6) for the p-SAWbranch and the wavevector direction of 40� are shown in Fig. 2.Due to lack of symmetry, the transversal component u2 is nonzero,in other words the polarization plane is tilted away from the sag-ittal plane. The particle motion in the wave field is elliptic, but u1

becomes zero at the depth about kx3 = 1, so that the ellipse degen-erates into a vertical line. At greater depths the elliptic motion re-sumes, but with the opposite rotation direction. The surface wavein this direction is strongly coupled with the s-SW, polarized par-allel to the surface, as can be seen in Fig. 1. Correspondingly, polar-ization of the SAW in this direction is almost parallel to the surface,which makes it difficult to detect.

Besides Rayleigh waves in isotropic solids and generalized Ray-leigh waves in anisotropic solids, SAWs in semi-infinite isotropicand anisotropic solids covered with a layer with different mechan-ical properties will be investigated. Usually it is assumed that thefilm consists of a homogeneous solid of uniform thickness in inti-mate contact with the substrate. The resulting dispersion of thephase velocity in such a system can be used to extract informationon the mechanical properties of the layered system, especially onunknown mechanical properties of the layer itself.

2.1.2. Excitation and detection of SAWsPurely laser-based experiments of the pump–probe type have

important advantages. Such an all-optical noncontact techniqueprovides a wide variety of excitation and detection conditions[8]. These include broadband excitation with pulsed laser radiationby a point or line source, narrowband excitation of wave trainswith a mask or the interference of two short coherent laser pulses.The excitation bandwidth is controlled by the width of the line orpoint source. In linear experiments the pulse energy of the excita-tion laser is selected for thermoelastic excitation of SAW pulses,which have amplitudes in the few-nanometer range. Since the cor-responding strain is in the 10�5�10�4 region, damage of the sur-face of the solid can be avoided under these conditions.

Detection is performed with a continuous wave (cw) laser beamthat allows the absolute interferometric measurement of the sur-face displacement or the determination of the transient surfaceslope or velocity by laser-probe-beam deflection [8]. To recordthe surface displacement an actively stabilized Michelson or het-erodyne interferometer is employed, while the surface velocity isregistered by a knife edge or position-sensitive detector. The detec-tion bandwidth is controlled by the size of the point focus of the cwprobe laser. In Fig. 3 such a laser setup with line source and inter-ferometric detection or laser-probe-beam deflection is shown.

In measurements of the elastic coefficients of films a two-pointprobe setup is used to determine the frequency dependence of thephase velocity. By fitting the normal or anomalous dispersion ofone or several acoustic modes the Young’s modulus or the wholeset of elastic coefficients can be obtained, respectively. The most


advanced laser scanning NDE method of surface-breaking cracksemploys a coupled excitation-and-detection laser arrangement,and the crack is moved step-wise from one side to the other ofthe fixed pump–probe setup. This procedure yields the completeacoustic field around the crack and information on the stiffnessreduction in the crack.

2.1.3. Determination of elastic coefficientsSAWs provide a versatile tool for the determination of the elas-

tic coefficients of, for example, advanced materials and coatings.This has been demonstrated for synthetic single-, micro-, andnanocrystalline diamond, deposited by chemical vapor deposition(CVD) [13]. For anisotropic single-crystal diamond the linear stiff-ness tensor has three independent components, for example, C11,C12, and C44, instead of only two for an isotropic material. In moststudies of micro- and nanocrystalline diamond, for example, theassumption is made that the material is elastically isotropic, reduc-ing the number of independent elastic coefficients to only two, e.g.,the Young’s modulus and the Poisson ratio.

Thin films and coatings are widely applied in industry to modifysurface properties, such as antireflection coatings used in optics,protective films of hard materials such as diamond, and metalliccoatings. A versatile tool for the measurement of the Young’s mod-ulus of thin films is the dispersion of SAWs [13]. More than onefilm property, such as the Young’s modulus, film thickness, or den-sity, may be extracted if the thickness is in the micrometer rangeand the bandwidth of the broadband SAW pulse approaches sev-eral hundred megahertz in a system with large acoustic contrast,as is usually the case for diamond layers. On the other hand, anessentially linear dispersion effect is observed for thinner films ora lower frequency bandwidth, providing only one property, e.g.,the Young’s modulus. Since the dispersion method is not very sen-sitive to the Poisson ratio, values for this property may not be veryaccurate and it may be permissible to take the Poisson ratio fromthe literature.

As mentioned before, introducing a layer on a half-space resultsin a dispersion effect of the SAW velocity, which can be effectivelyused for the evaluation of mechanical film parameters. The physi-cal description of the wave in a coated half-space is similar to thecase of the homogeneous half-space, but with additional condi-tions for the layer-substrate interface [14]. Let us consider a filmwhich occupies the domain x3 2 (�h,0) deposited onto the half-space x3 > 0. Then the boundary condition for the free surface inthe form of Eq. (2.7) must be modified now for x3 = �h

r3j ¼ ~c3jkl

X6

n¼1

eC n~aðnÞk~lðnÞl expð�ik~lðnÞ3 hÞ ¼ 0: ð2:9Þ

Here, a tilde denotes the quantities corresponding to the film, and~lðnÞ1 ¼ l1;

~lðnÞ2 ¼ l2. We note that in the film, all six roots ~lðnÞ3 ,

5.0

5.1

5.2

5.3

5.4

Phas

e ve

loci

ty, k

m/s

Frequency, MHz0 250 500 750 1000

(a)

Fig. 4. Dispersion curves for (a) the stiffening case (diamond film on s

n = 1, . . . ,6, have to be taken into account, because in general thesolutions in the film need not decay with depth. In many practicalcases however, when the film is relatively thin, it can be sufficientto consider only the solutions in the film, which decay with thedepth x3, so the summation is reduced to 3 roots. At the interfacex3 = 0 displacements are continuous:

X3

m¼1

CmaðmÞi ¼X6

n¼1

eCn~aðnÞi ð2:10Þ

and the components r3j of stress are continuous as well:

c3jkl

X3

m¼1

CmaðmÞk lðmÞl ¼ ~c3jkl

X6

n¼1

eCn~aðnÞk~lðnÞl : ð2:11Þ

Eqs. (2.9), (2.10), and (2.11) form a system of 9 linear equations inCm and eCm. Its determinant must be zero to admit a nontrivial solu-tion. The phase velocity of the SAW is sought in the same way as forthe uncoated half-space, but now it is dependent on the wavenum-ber via kh (see Eq. (2.9)). It is important to discriminate two types ofcoatings, acoustically ‘‘faster’’ or stiffening film on a ‘‘slower’’ sub-strate (‘‘anomalous dispersion’’), and the opposite case, when thefilm is ‘‘slower’’ as compared to the substrate (‘‘normal dispersion’’).As examples for these configurations we present dispersion curvesfor a diamond film on silicon (see Fig. 4a) and a silicon oxide layeron silicon (see Fig. 4b), respectively.

The coatings were assumed to be isotropic and the thickness ofthe diamond film was 700 nm and that of the silica film 1.4 lm. Atthe low frequency limit both curves tend to the phase velocity ofpure silicon, which is 4.92 km/s. The high frequency limit is theRayleigh velocity of the film. The principle difference betweenthe two structures is that the transition between the two limitingcases is always monotonous for the loading case, whereas for thestiffening layer it can take a more complicated shape as shown inFig. 4a. The slope of the dispersion curve at zero frequency can con-veniently be obtained via an effective boundary condition at thesurface of the substrate, correct to first order in kh. Dispersioncurves can be obtained experimentally by comparing Fourier trans-forms of the waveforms detected at several distances. Fitting thetheoretical dispersion curve to the experimental one allows theevaluation of unknown material parameters of the film, such aselastic moduli, density or thickness if the substrate properties areknown.

A complete set of elastic moduli has been determined for a110 lm thick anisotropic heteroepitaxial quasi-monocrystallinediamond layer [15]. In the special multimode photoacoustic tech-nique employed for analysis, acoustic surface and interface modesand SAW propagation in different crystallographic directions havebeen taken into account in the analysis. In Fig. 5a and b the exper-imentally studied dispersive and nondispersive acoustic modes aredisplayed together with the corresponding theoretical analysis of

Phas

e ve

loci

ty, k

m/s

0 250 500 750 1000

3.8

4.0

4.2

4.4

4.6

4.8

Frequency, MHz

(b)

ilicon) and (b) silicon-dioxide film on silicon (normal dispersion).

6

7

8

11

Phas

e ve

loci

ty, k

m/s

Frequency, MHz

Substrate direction: <100> <110>

0 50 100 150 200 0 50 100 150 200

6

8

10

12

14

2

Frequency (MHz)

Phas

e ve

loci

ty, k

m/s

Shear cut-off

Longitudinal cut-off

1

(a) (b)

Fig. 5. (a) Measurement of two surface modes in the two crystallographic directions, [001] and [011], for a diamond layer on silicon. (b) Joint Green’s function calculationsfor a diamond layer on silicon in the [001] direction. Mode 1 is the Rayleigh wave propagating in diamond and mode 2 is the pseudo-surface mode propagating partly insilicon.


the mode structure. An advantage of this photoacoustic method isthat the diamond film does not necessarily have to be removedfrom the Ir/YSZ (yttria-stabilized zirconia) Si(001) substrate, ifthe properties of the nucleation layer and substrate can be takeninto consideration with reasonable accuracy. Analysis of theexperimental data yielded the following stiffness coefficients:C11 = 1146 ± 4.8 GPa, C12 = 178 ± 46 GPa, and C44 = 562 ± 3.7 GPa[15]. Similar to homoepitaxial diamond deposition, C12 shows thelargest deviations, but C11 and C44 also deviate, within experimen-tal error, from the generally accepted set of elastic coefficients ofC11 = 1080.4 ± 0.5 GPa, C12 = 127.0 ± 1.0 GPa, and C44 = 576.6 ±0.5 GPa. Possible systematic errors involved in the evaluation ofthe effects of the nucleation layer could not be excluded in thiswork.

The laser-based SAW dispersion technique is extensively used.An example is a study of the dependence of the Young’s modulusof microcrystalline CVD diamond films on the deposition condi-tions. By varying the methane concentration in the source gas be-tween 0.5% and 2.0%, it was shown that the Young’s modulus,extrapolated to zero methane pressure (zero growth rate), ap-proaches the often used value of E = 1143 GPa calculated for ran-dom aggregates of polycrystalline diamond with ideal grainboundaries. For the lowest measured content of 0.5% CH4 the stiff-ness was E = 1080 GPa [16]. However, we have to bear in mind thatthe lower the methane content, the lower the growth rate and, ofcourse, no growth occurs at a methane pressure of zero.

Another example is the observation of SAW dispersiongenerated by gradients in the mechanical and elastic propertiesof millimeter-thick microcrystalline diamond plates [17]. These

Samplescan

Laser pulseLasersource

Laserprobe

Crack

Fig. 6. Experimental arrangement of crack, probe spot, and SAW source (zone I).The configurations of zones II and III were realized by moving the sample withrespect to the fixed optical pump–probe setup until the crack passed the source.

measurements of the Rayleigh velocity clearly demonstrated thevariation of the mechanical properties between growth and nucle-ation sides. A further application is the demonstration of the posi-tive role played by oxygen on the growth rate and on the stiffnessas a function of the oxygen content in the source gas [18].

Nanocrystalline diamond with a crystallite size <100 nm canreach a Young’s modulus of 1100 GPa for low methane concentra-tion (0.5%) and high power densities in the range of about 25 W/cm3, even when the film thickness is only 140 nm [19]. The impor-tance of the nucleation density on the mechanical film quality hasbeen demonstrated for columnar-structured diamond films withcolumn diameters of less than 100 nm. The Young’s modulus ob-tained for low nucleation density (�1010 cm�2) of 517 GPa wasroughly half that of a film or plate grown with a much highernucleation density (>1011 cm�2) of 1120 GPa [20]. The finding ofa measurable reduction of the film stiffness at a grain size belowapproximately 100 nm is in good agreement with SAW dispersionmeasurements on nanocrystalline samples with a grain size be-tween 60 nm and 9 nm, where the Young’s modulus ranged from1050 to 700 GPa [21]. However, it is important to note that quitedifferent stiffnesses of 609 GPa [22] and 1160 GPa [23] have beenreported for comparable grain sizes of approximately 150 nm.

2.1.4. Nondestructive evaluationAn important application of linear SAWs is NDE of surface-

breaking cracks. Since the surface is often subjected to the higheststress levels, the degradation of materials usually starts at the sur-face. These irreversible degradation processes may lead to fatigueor stress corrosion cracking, generating partially closed cracks with

SAW penetration depth / crack depth

0 1 2 3 4 5 6

crackcrackSAW

SAW

Fig. 7. Illustration of the ratios of SAW penetration depths and crack depth for thelowest and highest wavelength of a typical wideband SAW pulse and a crack depthof 50 lm.


a finite penetration depth. Linear SAWs or Rayleigh waves withsuitable wavelengths are ideally suited to detecting such surfacediscontinuities because the penetration depth of these guided sur-face waves can be selected for optimum detection sensitivity.

The first purely laser-based NDE experiments were performedwith the goal of localizing artificial slots and extracting their size[24,25]. Since a laser beam can be moved easily along the surface,the extracted information can be extended by shifting the lasersource, the laser probe, or both with respect to a surface crack.For example, in the first investigation using the scanning lasermethod only the source was swept across the crack; the SAWwas registered at a fixed surface position using a laser or conven-tional transducer [26]. In the most advanced scanning techniquethe pulsed excitation laser and the cw probe laser are coupled atfixed distance and the sample with the microcrack is scanned be-low the fixed pump–probe arrangement, as illustrated in Fig. 6[27]. The information content increases dramatically if the acousticfield around the crack is studied for the following three configura-tions: the source and probe to the left of the crack (zone I), thecrack between pump and probe (zone II), and the pump and probeto the right of the crack (zone III). This double scanning approach islimited to all-optical setups.

Laser-based SAW methods have been most frequently appliedto artificial rectangular slots and v-shaped notches, whereas inves-tigations on real cracks are still rare. The latter include stress cor-rosion cracks [28], cracks induced by fatigue processes [26], andfinite-size cracks originating from impulsive elastic loading bynonlinear SAW pulses with steep shock fronts [29]. Usually realis-tic cracks are not completely open but have partially touchingcrack faces near the tip. These partially closed cracks are of enor-mous practical importance.

While most laser-based NDE experiments have been conductedbetween 1 and 20 MHz, it proved possible to extend laser ultrason-ics into the frequency range of 10–200 MHz [27]. These two fre-quency regions correspond roughly to well-detectable crackdepths of about 1 mm and 10–100 lm, respectively. It is importantto note that with nanosecond laser pulses SAW pulses with a band-width extending over 1–2 orders of magnitude can be launched,probing simultaneously different depths in the micrometer-to-mil-limeter range with a single broadband SAW pulse. Fig. 7 comparesthe penetration depths of such a pulse with the typical depth of�50 lm of a crack generated by impulsive load. The highest sensi-tivity can be achieved for a crack depth comparable to the meanwavelength of the interrogating SAW pulse.

0 200 400 600 800 10000

20

40

60

80

100

120

140

160

180

200

IIIIII

TR

L->S

L->R

LRL

R

R->S

R

RR

Distance x1 (µm)

Tim

e (n

s)

RR

RL L->S

Fig. 8. Time�distance plot of a laser scanning experiment. The 3D picturerepresents the complete scattered acoustic field in the vicinity of the crack,including the incoming, transmitted, and reflected Rayleigh waves. The gray scalerepresents the measured velocity profiles.

Some unique ways to extract information about cracks from la-ser scanning experiments include analysis of the reflection andtransmission coefficients of the incoming SAW, determination ofthe reduced interfacial stress of partially closed cracks by the finitedifference method (FDM), and signal enhancement effects occur-ring when the pump or probe laser hits the crack, as described inmore detail in the following [27].

In Fig. 8 a time�distance plot of a laser scanning experiment isdisplayed. The 3D picture represents the complete scattered acous-tic field in the vicinity of the crack, including the incoming, trans-mitted, and reflected Rayleigh waves. The gray scale shows themeasured velocity profiles. The reflected SAW velocity profile ob-served in zone I consists of two positive peaks with a trough. Sucha tripolar waveform is expected for partially closed cracks. Obvi-ously, the motion of the rear face of the crack affects the wave pro-file to a comparable extent as the front face, indicating asubstantial transient interaction of the two crack faces duringreflection. The physical basis for the evaluation of the crack sizeis the comparison of the reflected pulse shape with the initialundisturbed Rayleigh pulse, which is easily accessible in zone I.Let us consider the reflection coefficient, which depends on the ac-tual shape of the defect. In the simplest case of a semicircular(‘‘half-penny shaped’’) crack, extending normal to the surface, thedepth also describes the size. The reflection coefficient was calcu-lated for both the simulated waveforms and the correspondingmeasured ones as the ratio of the spectral amplitudes of the re-flected and incident waves. The depth of the crack was evaluatedby fitting the FDM-simulated reflection coefficient to the experi-mentally measured one by a least squares procedure. From thisfit a value of 34 lm was extracted for a semicircular microcrack.This crack depth is in reasonable agreement with the direct opticalmeasurement of the crack radius of 49 lm. Note that the lowerpart of the partially closed crack may not be seen acoustically,whereas optically it still may scatter light, owing to the smallerwavelength of light involved in optical detection.

Another approach for crack sizing, which is independent of theSAW amplitude, is the measurement of the time needed by theSAW pulse to travel along the complete crack faces. For crackdepths larger than the SAW wavelength the depth is proportionalto the time delay. The bipolar SAW pulses transmitted throughan impulsive shock-induced crack show a clear time delay or phaseshift, as can be seen in Fig. 7 as small upward shift of the horizontalRayleigh wave trace in zone II. In fact, the time lag of a transmittedbroadband SAW pulse depends on the depth/wavelength ratio.This offers an alternative to evaluating crack sizes from the fre-quency dependence of the phase lag. In the present analysis thetransmitted bipolar SAW was first simulated using the FDM, andthen the phase of the transmission coefficient was estimated as afunction of the depth/wavelength ratio. This was fitted to theexperimentally measured frequency dependence of the phase shiftwith the crack size as the only free fitting parameter. The crackdepth evaluated by the transmission method was 40 lm, in rea-sonable agreement with the reflection method.

To simulate the interaction at the crack interface a simple one-parameter model has been developed, which allows the interfacialinteraction to be modified continuously between completely openand completely closed. This stress parameter e reduces all stress-tensor components by the same scalar factor (0 6 e 6 1). Withinthe framework of a mass–spring representation, this correspondsto a weakening of the spring’s stiffness by a factor of e betweenthe masses situated on each side of the crack opening. Conse-quently, this reduced-stress model describes the softening of thematerial’s stiffness by the crack defect. While FD calculations foran open crack (e = 0) result in a bipolar shape of the reflectedSAW, a value of e = 0.08 reproduced the measured tripolar profile(see Fig. 9). The reduction of the interfacial stress-tensor

Transmission

Surfa

ce s

lope

U31

50 75 125 150 125 150

Reflection

Time (ns)

(a) (b) (c)

Fig. 9. Comparison of FD simulations with experiment: (a) thermoelastic excitation, (b) reflection from the crack, and (c) transmission through the crack. Experimentalwaveform: red, simulation with e = 0: green, simulation with e = 0.08: black.

Fig. 10. Dependence of kernel F in evolution Eq. (2.12) (a) on its argument g and (b) on the wavevector direction.


components to about 8% of the value in the intact lattice of theideal solid indicates a relatively open crack. Of course, such a singleparameter cannot describe the real behavior of a partially closedcrack. It must be considered as a mean interfacial stress or stiffnessaveraged over the whole length of the crack. Consistent with thisfinding a two-parameter model yields the correct tripolar shapefor the reflected wave only when the tensile and shear stress com-ponents have the same value as fitted for the one-parameter case.

A quantitative description of the signal enhancement effectsobserved near the crack in laser scanning experiments is a difficulttask, since microscopic properties of the crack and of the interactionof the excitation or probe laser with the crack faces are involved,and exact information on the crack opening and laser beam radiusmay not be available. Therefore, only a qualitative comparisonbetween experiments and model calculations can be performed.The present FD simulations, using a step width of 1 lm, yield abouta factor of two for the enhancement of the motion at the front edge.Values in this range have been reported by several authors forcracks extending perpendicular to the surface. When the spatialresolution of the probe beam is decreased from about 2 lm to about5 lm the enhancement effect disappears. In agreement with thesesimulations only a moderate enhancement of about 20% has beenfound under comparable experimental conditions.

It is important to note that in real systems cracks also grow atangles different from 90�. For example, in the case of rolling con-tact fatigue in rails, initial crack growth occurs at angles of about20–25� to the surface. A number of in depth theoretical investiga-tions were carried out in the past on the diffraction of ultrasonicwaves by an elastic edge or wedge [30–33]. In principle, a wedgewith a stress-free boundary may be considered as an open crackwith infinite length. Tilted defects have been inspected by bulkwaves using a conventional transducer for measuring the ampli-tude and phase of the wedge diffraction coefficient [34], and goodagreement has been found with results of diffraction theory.Detecting and sizing tilted cracks with a piezo-electric transduceris a difficult task since longitudinal, shear, and Rayleigh wavesare involved. A simpler process, namely scattering of Rayleighwaves from a surface-breaking crack, several wave lengths long,has been treated to extract the Rayleigh reflection and transmis-sion coefficients [35]. A detailed experimental study of the interac-tion of Rayleigh waves with cracks has been performed byall-optical laser excitation and detection [36]. The observed varia-tion of the reflection and transmission coefficient has been com-pared with 3D finite element calculations for a wide range ofangles and depths, which are in good agreement with experiments.These simulations allow new insight into the displacements within

Position sensitive detectorsLaser pulse

cw laserprobes

Absorptionlayer

Fig. 11. Setup for strong laser excitation by the absorption layer method and two-point-probe detection of the propagating SAW.


the sample and the wave-crack interaction processes such asmode-conversion at the bottom of the crack [36].

The term ‘‘edge waves’’ or ‘‘edge resonance‘‘ is frequently usedfor waves guided by the edge of a plate with parallel surfaces[37,38] or a resonance in the reflection of Lamb waves from theedge of a plate [39]. We would like to state here that the use ofthe term ‘‘edge wave’’ that sometimes is used instead of ‘‘wedgewaves’’ does not comprise these highly dispersive guided waves,which are an interesting topic by itself.

2.2. Nonlinear surface acoustic waves

2.2.1. TheoryWith the absence of dispersion and reduced diffraction losses

for plane SAWs, these waves are of particular interest for studyingnonlinear acoustic waves and their effects [40]. An essential prere-quisite for nonlinear behavior is that the attenuation length due tointrinsic damping is longer than the length scale on which wave-form evolution due to nonlinearity takes place (‘‘shock formationdistance’’). By pulsed laser irradiation, plane SAW pulses with finiteamplitudes can be launched, which develop steep shock fronts byfrequency-up conversion (simultaneously frequency-down con-version takes place) during wave propagation along an elasticallynonlinear medium. A nonlinear evolution equation has been de-rived [41–45] describing the corresponding changes of the SAWpulse profile during propagation in a nonlinear homogeneoushalf-space. It has the form

icR@

@x1Bðx; x1Þ ¼ x

Z x

0Fðx0=xÞBðx0; x1ÞBðx�x0; x1Þdx0=2p

�

þ 2Z 1

x

xx0

F�ðx=x0ÞBðx0; x1ÞB�ðx0 �x; x1Þdx0=2p�ð2:12Þ

Here, cR is the phase velocity of the Rayleigh waves and the wave-vector direction is the x1-direction (possibly after a rotation of thecoordinate system). In Eq. (2.12) B(x) is the Fourier transform ofa displacement gradient um1, m = 1, 2, 3 at the surface,

um1ðx1; x2;0; tÞ ¼ 2ReZ 1

0BðxÞ exp i

xcRðx1 � cRtÞ

� �dx2p

� �; ð2:13Þ

or a linear combination of such displacement gradients that areassumed to be approximately independent of x2. In Eq. (2.13) Redenotes the real part. B(x) is allowed to vary gradually with x1.The kernel function F(g) depends on ratios of second-order andthird-order elastic moduli of the substrate. It may be expressed asan overlap integral over the three components of displacement gra-dients associated with linear Rayleigh waves and the sixth-ranktensor (Sijklmn). Its components are the coefficients of the third-orderterms in an expansion of the density of potential energy U inpowers of the displacement gradients,

U ¼ 12

cijkluijukl þ16

Sijklmnuijuklumn þ Oðu4Þ; ð2:14Þ

and are linear combinations of second-order and third-order elasticmoduli,

Sijklmn ¼ cijklmn þ dikcjlmn þ dimcjnkl þ dkmcijnl; ð2:15Þ

where cijklmn are third-order elastic moduli and dmn is the Kroneckersymbol.The kernel F(g) has a simple functional form,

FðgÞ ¼X

n;n0 ;n00

Mðn;n0;n00ÞlðnÞ

�

3 þ glðn0Þ

3 þ ð1� gÞlðn00 Þ3

: ð2:16Þ

The coefficients Mðn; n0;n00Þ can be expressed in terms of the quan-tities lðnÞj ; aðnÞj , Cn, n = 1, 2, 3, and qc2

R introduced in Section 2.1.1., andthe tensor (Sijklmn). Obviously, they strongly depend on the choice ofthe displacement gradient component in Eq. (2.13). In isotropicmedia, F(g) is purely real if B(x) is the Fourier transform of u11

and purely imaginary if B(x) is the Fourier transform of u31. Inanisotropic solids and propagation geometries of low symmetry,F(g) may possess both imaginary and real parts.

The dependence of the function F(g) on the wavevector direc-tion on the (111) surface of silicon is shown in Fig. 10, correspond-ing to the Fourier transform of u31. The relative size of real andimaginary parts depends on g and on the wavevector direction.This has strong consequences for pulse shape evolution, especiallyon the shape of solitary pulses.

Taking into account weak dispersion results in an additionalterm in the evolution equation [40]. By controlling the dispersioneffect, for example by the film thickness, it is possible to balancethe nonlinear and dispersive effects, if they occur on the samelength scale. In the case of dispersion of SAWs resulting from a thinfilm on the surface, the additional term on the right-hand side ofthe evolution Eq. (2.12) has the form DxnB(x). The coefficient Ddepends on the linear acoustic mismatch of film and substrateand may be negative or positive, corresponding to normal oranomalous dispersion, respectively. Normally, the exponent n isequal to 2 and D is proportional to the film thickness. Conse-quently, the dispersion term in Eq. (2.12) is the same as the onein the Benjamin–Ono equation. In fact, the extended evolutionequation provides solitary solutions that represent the typicalpulse shapes observed under such dispersive conditions. The shapeof the solitary pulses is strongly influenced by the kernel F(g) in thenonlinear evolution equation. It turns out that it is not so much thedependence of F on its argument but the average phase angle of thecomplex quantity F that is relevant here and that strongly dependson the anisotropy of the substrate material. With the exponentn = 2 in the dispersion term, the solitary pulses give rise to the fol-lowing forms of the displacement gradients at the surface:

um1ðx1; x2;0; tÞ ¼ jSmðjðð1� jÞx1 � cRt þ x0ÞÞ; ð2:17Þ

where j > 0 and x0 are parameters. The upper sign in Eq. (2.17) isfound in the case of normal dispersion, the lower sign in the caseof anomalous dispersion. For isotropic substrates, S1 is an even func-tion and S3 is an odd (bipolar) function of its argument. In aniso-tropic substrates, this is no longer the case, and S3 may have analmost even shape of the Mexican-hat type. The numerical simula-tions reveal that these are solitary pulses and not real solitons, sincethey do not survive collisions with each other. Only under excep-tional conditions (n = 3) could almost elastic collisions be seen inthese simulations.

2.2.2. ExperimentalAs mentioned above, SAW pulses with finite amplitude must be

launched at the source to realize nonlinear surface pulses. Strongerexcitation with nanosecond laser pulses can be achieved by addinga highly absorbing layer at the source region that explosively

0

0

-100 -50 0 50 100 -50 0 50

Retarded time (ns)

Shea

r dis

plac

emen

t gra

dien

t (ar

b. u

nits

)

(a) (b)

(c) (d)

Fig. 12. Theoretical bipolar (a) and Mexican-hat-shaped (b) solitary SAW pulses calculated using a nonlinear evolution equation with linear dispersion term. Measuredsolitary pulses for the systems silica/NiCr film (c) and silicon/SiO2 film (d).


evaporates upon pulsed laser irradiation (see Fig. 11) [46]. The re-coil momentum exerted on the surface leads to an efficient excita-tion of bulk waves but also of surface waves. By sharply focusingthe pump laser with a cylindrical lens into a narrow line at the sur-face, a plane surface wave propagating in a well-defined directionis obtained. Its bandwidth may extend to several hundred mega-hertz. Vertical particle displacements in the range of 100–200 nmcan be achieved and the power flux through unit square at the sur-face is 10–100 times that observed for thermally-generated linearSAW pulses.

With the ‘‘absorption layer’’ method, strains in the range of 0.01have been realized. Since the critical strain of many brittle materi-als is in this region, mechanical fracture can be achieved, e.g., in sil-ica [47] and silicon [29,47]. Such fracture processes can occurrepetitively because after fracture frequency-up conversion takesplace in the advancing pulse until the critical strain is reachedagain. Therefore, a whole more-or-less regular field of spatiallylimited cracks may be generated along the SAW propagation direc-tion with this impulsive failure method. It is important to point outthat frequency-up conversion concentrates the elastic energy ofthe guided surface wave in an even smaller depth from the surfaceof the solid.

Fig. 11 shows the complete pump–probe setup, which is slightlymodified for nonlinear excitation by adding the absorption layer.Furthermore, the two laser probe positions allow the evolution ofthe SAW pulse shape to be monitored directly [48]. Usually thepulse profile measured at the first probe spot is inserted into thenonlinear evolution equation as an initial condition and the profilecalculated for the second probe spot is compared with experiment.This arrangement has been employed in all experiments on solitarywaves and the measurement of the fracture strength of silicon pre-sented here.

2.2.3. Solitary surface wavesSince SAWs show no dispersion in a homogeneous half-space, a

film was deposited on the substrate to generate solitary pulses. In alayered system the film thickness introduces a new length scaleresponsible for a geometric film-based dispersion effect. Since thislength scale is much larger than the interatomic spacing, intrinsiclattice-based dispersion can be neglected. During propagation in

the nonlinear and dispersive medium, stable bipolar or Mexican-hat-shaped solitary pulses are formed, which travel faster thanthe linear Rayleigh velocity, whereas the small oscillatory tail (so-called ‘‘radiation’’) moves with lower velocity for a loaded system,or vice versa for the stiffening case [49–51].

Fig. 12a and b shows two stationary periodic solutions of thenonlinear SAW evolution equation with a linear dispersion term,namely a bipolar and a Mexican-hat profile. A bipolar pulse wasobserved for a NiCr film loading the silica substrate, leading to nor-mal dispersion as depicted in Fig. 12c. On the other hand, an addedTiN film stiffens silica, leading to anomalous dispersion thatchanges the polarity of the solitary pulse. In this latter case the sol-itary pulse travels with a speed lower than the linear Rayleighvelocity [49]. A Mexican-hat-shaped solitary pulse was found bycovering the Si(111) surface with a thermally grown 110 nm thicksilicon dioxide layer, responsible for normal dispersion (seeFig. 12d) [49]. In this system the theoretically predicted Mexican-hat-shaped profile was observed for the first time [50]. The ob-served bipolar and Mexican-hat-shaped pulses agree well withnumerical solutions of the nonlinear evolution equation with anessentially nonlocal character of nonlinearity and a linear disper-sion term of the Benjamin�Ono type. In this respect the nonlinearevolution of surface waves in solids differs from that in shallow flu-ids, which can be described by the Korteweg–de Vries (KdV) equa-tion [51]. The realization of solitary waves at surfaces and solitonsin crystals achieved by nanosecond and picosecond laser ultrason-ics, respectively, has been reviewed in [52].

2.2.4. Shock waves and determination of fracture strengthAs discussed before, nonlinear SAWs can develop stress peaks

which grow with the propagation distance and thus generatecracks when the critical strain of the material is reached[29,53,54]. The crack size is in the region of 50–100 lm, deter-mined by the length of the SAW pulse (tens of nanoseconds) andthe crack speed that is on the order of the Rayleigh velocity of 3–5 km/s. In order to evaluate the stress of the SAW pulse, it wasmeasured at two different distances from the source by laser-probe-beam deflection, first at 1–2 mm from the source and theother at a distance of about 15–20 mm. A stress calibration proce-dure was applied, based on numerical integration of the nonlinear

Si(111)

{111}

Si(111)<112>SAW

Fig. 13. (a) Image of crack penetration into the bulk from the Si(111) surface. The SAW pulse propagated from left to right in the h�1 � 12i direction. The angle betweensurface and image plane is 45�. (b) Atomic scheme of crack formation for the easy cracking geometry Si(111) h�1 � 1 2i with biaxial crack nucleation and propagation alongthe {11–1} cleavage plane, tilted by 19.5�.


evolution equation Eq. (2.12). The waveform detected at the short-er distance was fed into Eq. (2.12) as an initial condition with thecalibration factor a, which relates the detected waveform U(t)and the gradient u31 so that u31 = aU(t). The calibration factor awas found by fitting the result of integration to the waveform de-tected at the remote location. Then all the strain and stress compo-nents can be determined everywhere in the sample, in particular atthe location, where a crack was located. This provides an estimateof the material strength.

The spectrum of the initial laser-excited transient was limitedto about 200 MHz, mainly owing to the laser pulse duration of8 ns. Since in a nonlinear medium, such as a silicon single crystal,both frequency-up conversion and frequency-down conversionprocesses are taking place, a lengthening of the pulse profile occurssimultaneously with shock formation. This effect is proportional toits magnitude, and therefore the pulse length can be used as a sen-sitive measure of the nonlinear increase of strain. In particular,when the shock fronts become steeper and steeper this quantitycan be determined quite accurately [55].

For Si(111) we made the observation that counterpropagatingnonlinear SAW pulses, i.e., those moving in opposite directions,for example in the h�1 � 12i and h11 � 2i directions on theSi(111) plane, develop completely different nonlinear pulse shapes[29]. In the easy-cracking geometry Si(111) h�1 � 12i the tensilefracture strength was about 4 GPa. The surface-nucleated crackspropagated into the bulk along the {11–1} cleavage plane, whichis inclined by 19.5� to the normal of the free surface (seeFig. 13a). According to the boundary conditions for SAWs, onlythe tensile opening stress r11 is nonzero directly at the surface inthe initial coordinate system. This tensile stress of r11 = 4 GPacan be represented by a set of orthogonal components in the coor-dinate system associated with the tilted {11–1} cleavage plane.The calculated value of rT

11 is equal to 3.6 GPa and can be consid-ered as an estimate of the opening strength of silicon in this specialgeometry. In fact, a combination of mode I (tensile opening) andmode II (in-plane shear sliding) processes is expected to controlthis fracture geometry. The resulting stress components for a biax-ial fracture mechanism in the tilted coordinate system arerT

11 ¼ 3:6 GPa and rT31 ¼ �1:3 GPa. Fig. 13b illustrates the biaxial

fracture components with respect to the {11–1} cleavage planefor this geometry. Extensive results for the formation of surface-breaking cracks in different geometries (crystal planes and direc-tions) have been obtained for single-crystal silicon. The criticalfracture stresses determined for various selected geometries variedbetween 3 and 7 GPa [55].

With the extension of crystalline silicon devices to smaller andsmaller sizes the dependence of the mechanical strength on thesystem size becomes an important issue. In micro-electro-mechan-ical-system (MEMS) and nano-electro-mechanical-system (NEMS)applications, the mechanical stability is essential for their manipu-lation, functionalization, and integration into more complex sys-tems. In general it is expected that the strength increases withdecreasing size of the system owing to the smaller number of crys-tal defects such as voids, microcracks, or dislocations in the smallervolume. In early work on the fracture strength of silicon whiskerswith diameters on the micrometer scale (�1–20 lm) tensile frac-ture strengths of 2–8 GPa were found. These results are in the samerange as the values measured for well-defined geometries. This canbe interpreted by the assumption that similar fracture geometriesand failure mechanisms were involved in these failure processes[56].

In recent years, several techniques, such as the chemical-vapor-deposition (CVD), vapor–liquid–solid (VLS), and CVD-VLS methods,have been developed to grow silicon nanowires with diametersdown to the few-nanometer range. Currently, however, it is verydifficult to extract general conclusions from this pioneering work,since contradictory results have been reported for the size effects[56]. It seems that the strength of silicon can increase to about12 GPa, as the nanowire diameter decreases to 100–200 nm [57]and 15–60 nm [58] in wires grown along the [111] direction. Thisvalue is already near to the theoretical strength for tensile cleavageof silicon along the {111} plane of 22 GPa obtained by ab initio cal-culations for an ideal defectfree silicon lattice [59].

2.2.5. Nonlinear SAWs by wave interaction with scatterersThe source of nonlinear elasticity can not only originate from

the intrinsic nonlinearity of the perfect bulk material, as used inshock wave formation, but also from mechanically damaged sam-ples or local defects. This includes, for example, dislocations orcracks. SAWs have several advantages to image scatterers locatedat the surface in comparison to Lamb waves. Up to now such scat-tering experiments have been performed mostly with conventionaltransducers, e.g., piezo-electric wedge transducers, whereas lasershave been used especially for detecting the second order harmon-ics generated by the material’s damage [60]. In this work, anincrease in the second harmonic amplitude was detected thatwas induced during low cycle fatigue tests and by monotonic ten-sile load of the material above the yield stress of the nickel-basesuperalloy. A goal of these experiments is the development of moresensitive methods, allowing NDE in the early stages of fatigue prior

Fig. 14. ASF wedge modes calculated for a 45� wedge of silica: (a) fundamental orfirst mode (slowest); and (b) second mode (faster). The colors display thedisplacements along the edge.


to crack formation, by observing the nonlinear features of the non-linear interaction process.

Another nonlinear acoustic signature of a crack is frequencymixing. Here a pair of laser beams is intensity modulated at twodifferent frequencies and is used to induce parametric wave inter-action, which proves the elastic nonlinearity of the crack. Opticalinterferometry is applied to detect the acoustic waves at mixed fre-quencies [61]. The highest contrast in crack imaging is found forpartially closed cracks. Alternatively, dual frequency mixing canbe achieved with a transducer used as low frequency pump SAWand a laser employed as high frequency probe SAW [62]. The non-linear interaction has been measured as a phase modulation of theprobe SAW and equated to a velocity change. The velocity-stressrelationship has been considered as a measure of the materialnonlinearity.

3. Wedge waves

3.1. Linear wedge waves

3.1.1. Theoretical approachesAs stated above, linear WWs were first discovered by numerical

calculations in 1972, independently by Lagasse [9] and by Marad-udin and coworkers [10]. The semi-analytical finite element meth-od (FEM), used originally by Lagasse, is still being used [9,63,64].He assumed translational invariance along the apex and reducedthe three-dimensional (3D) problem to a two-dimensional (2D)one by introducing a 1D wavevector. He found an empirical for-mula for the dependence of the phase velocity of the anti-symmet-rical flexural (ASF) modes on the wedge angle H

CW ¼ cR sinðnHÞ; ð3:1Þ

where cR is the Rayleigh velocity of the isotropic material, and n isthe mode number [65]. The main advantage of this method, besidesits fast convergence, is its flexibility concerning shape modifica-tions. It leads directly to the dispersion relations for modifiedwedge cross sections such as truncation or rounding of the wedgetip.

An alternative approach, introduced by Maradudin et al. is basedon an expansion of the displacement field in Laguerre functions[10,66]. Since the Laguerre functions are products of an exponentialand a polynomial function rapid convergence is guaranteed. Thismethod has been extended to anisotropic wedges [67] includingnonlinear problems [68]. It provides approximate analytical expres-sions for the displacement field and does not introduce any artificiallength scale, as is the case in the FEM method.

Both methods are based on Hamilton’s principle with theLagrangian density for the displacement field ui(x1,x2,x3; t), i = 1,2, 3

‘ ¼ 12

q _um _um � uklcklmnumnð Þ: ð3:2Þ

Here _um denotes the partial derivative of um with respect to time.The displacement field is represented in the form

umðx1; x2; x3; tÞ ¼ expðiðkx1 �xtÞÞUmðx2; x3Þ: ð3:3Þ

The x1 axis is chosen along the tip of the wedge. Inserting this in theLagrangian density, integrating over the cross section A of thewedge in the x2�x3 plane, and averaging over a time period 2p/xleads to the functional

J ¼Z Z

Aqx2U�mUm � DlðkÞUk½ ��cklmn½DnðkÞUm��

dx2dx3; ð3:4Þ

where we have introduced the operator Dm(k) with componentsD1(k) = ik, D2(k) = @/@x2, and D3(k) = @/@x3.When applying the finiteelement method, the area A is divided into (usually triangular) ele-ments and Um(x2,x3) is expanded in shape functions N(j)(x2,x3),

Umðx2; x3Þ ¼X

j

NðjÞðx2; x3ÞUðjÞm ð3:5Þ

with the node variables �UðjÞm as expansion coefficients. As shapefunctions, polynomials of at least second order were employed. Inthe case of second-order polynomials, there are six shape functionsand six nodes for each element. Insertion of Eq. (3.5) in the func-tional J and variation with respect to the node variables leads to ageneralized eigenvalue problem of the form

x2Xm0 ;j0

Mðjj0 Þmm0U

ðj0 Þm0 ¼

Xj0 ;m0

Kðjj0 Þ

mm0 ðkÞUðj0 Þm0 ð3:6Þ

with the symmetric mass matrix (Mðjj0 Þmm0 ) and Hermitian (general-

ized) stiffness matrix (Kðjj0 Þ

mm0 ). The latter depends on the wavenumberk.For the Laguerre function approach, a linear transformation (con-formal mapping) is performed from the original coordinates x2, x3 tonew dimensionless variables g, f:

gf

�¼ k R

$ x2

x3

�: ð3:7Þ

The 2 � 2 matrix R$

maps the wedge of given wedge angle to a rect-angular one and its two faces to the two planes g = 0 and f = 0. Thefunctions

Umðx1; x2Þ ¼ eUmðg; fÞ ð3:8Þ

are subsequently expanded in products of two Laguerre functions,

eUmðg; fÞ ¼X

k;l

ukðgÞulðfÞaðmÞkl ð3:9Þ

with (in general complex) expansion coefficients aðmÞkl . The Laguerrefunctions are defined in the usual way,

ulðnÞ ¼ expð�n=2ÞLlðnÞ; ð3:10Þ

where Ll(n) is the lth Laguerre polynomial.Insertion of Eqs. (3.8) and(3.9) in the functional J, truncation of the sums over k and l, and var-iation with respect to the expansion coefficients yields a Hermitianeigenvalue problem,

xk

� �2aðmÞij ¼

Xn;k;l

HðmnÞijkl aðnÞkl : ð3:11Þ

For sharp-angle wedges, convergence can be improved by introduc-ing a scale factor in the arguments of the Laguerre functions [69].

In both approaches, the displacements and phase velocities areextracted from the smallest eigenvalues and the correspondingeigenvectors that solve the (generalized) eigenvalue problem. Bothmethods deliver the frequencies corresponding to elastic wavesthat are totally localized at the wedge tip.

In addition to these numerical approaches, analytical resultswere obtained for the wave speed and displacement field of wedgewaves in the limit of very small wedge angles on the basis ofthin-plate theory or, equivalently, in an expansion of the velocityand displacement field in powers of the wedge angle [70,71].

Pumplaser

Probe laser

Variable angle transducer

Fig. 15. Pump–probe setup with angle-tunable wedge transducer for the excitationand probe-beam deflection for detection of p-WWs near the wedge tip.


Furthermore, a large number of analytical results referring tosharp-angle as well as obtuse-angle wedges have been developedwithin the approximation of geometrical acoustics. These ap-proaches are described in more detailed reviews [72,73].

First we consider theoretical results on the number of wedgemodes, their phase velocity, and their symmetry as a function ofthe wedge angle. Generally, the number of modes increases withdecreasing wedge angle and the higher modes successively ap-proach the Rayleigh velocity. In Fig. 14 two ASF wedge modesare shown. The large deviation of the velocity of fundamentalmodes with low wedge angle, with respect to SAWs, is sufficientto ensure strong guiding. On the other hand, the higher modes oflarger wedge angles tend to couple effectively with SAWs. For thisreason only one fundamental mode is expected to be detectable forwedge angles >45�. Similar to the waves guided by traction-freesurfaces, ‘‘leaky’’ or pseudo-wedge waves (p-WWs) can exist, cou-pled to SAWs or even bulk acoustic waves. A rigorous proof of theexistence of localized waveguide modes for isotropic wedges withangles <90� has been presented [74]. However, a fundamental anti-symmetric ASF mode may exist even for aperture angles up to101.25� [66,75]. A symmetric mode may propagate for someintervals of angles greater than 90� [75]. There is also an overlap-ping region, where both anti-symmetric and symmetric modescoexist. However, the domain of existence of symmetric wedgemodes refers to Poisson ratios smaller than 0.14 and hence coversmainly non-standard isotropic materials [74]. Results reported in[66,76] indicate that symmetric modes existing at larger Poissonratios have phase velocities very close to the Rayleigh wave veloc-ity of the isotropic medium. Currently, essentially all experimentshave been performed and interpreted in terms of ASF wedge

2.0 2.1 2.2 2.3 2.4 2.5 2.6

-6

-4

-2

0

2

5.4 km/s p-WW

4.2 km/s WW

4.54 km/s SAW

u 31, (

a.u.

)

Time, μs

Fig. 16. Fundamental WW, SAW, and supersonic p-WW, detected with the all-optical pump–probe setup.

modes. It is important to note that classification of wedges by sym-metry is quite often used, however, this may not be applicable inthe case of anisotropic materials, since the mid-apex plane wouldhave to be a symmetry plane.

3.1.2. Excitation and detection of wedge modesIn early work, interdigital structures placed across the edge of

the piezoelectric wedge [77] or a pair of broadband shear transduc-ers bonded to the ends of the wedge [78] were employed for thegeneration and detection of flexural WWs. Compared with thesemethods the application of a pulsed laser for excitation and a cwlaser to monitor the surface displacement with an interferometer[79] or the surface velocity with a laser-based probe-beam-deflec-tion setup [80,81] has several advantages. Therefore the laser tech-niques are mainly used in the more recent work on WWs (seeFig. 15). Most important is the high spatial resolution, which al-lows the pump and probe to be placed very near the tip, but alsothe accurate measurement of the phase velocity along the edge.Furthermore, to monitor the decay of the displacement perpendic-ular to the edge tip a high spatial resolution is needed. The highpower achievable with laser radiation makes this approach mostsuitable for future studies on nonlinear WWs.

Recently, a novel device for the selective excitation of guidedwaves by a pulsed laser-operated angle-tunable wedge transducerhas been introduced that allows, for example, the separate efficientexcitation of WWs, SAWs, and pseudo-wedge waves (p-WWs) [81].The separate detection of these modes can be seen in Fig. 16. Theadvantage of this laser-based transducer is that the desired wedgeor surface mode can be launched with maximal efficiency, whileexcitation of all other guided modes is more or less suppressed.This compensates for a possible loss in simplicity and spatial reso-lution using this particular laser-based contact transducer.

3.1.3. Properties of perfect wedgesThe requirements on the quality of wedges and especially the

wedge tip vary with the wavelength employed but are generallyhigh. It seems to be difficult to reach a sufficient wedge qualityin most materials by conventional cutting and polishing of thematerial. Anisotropic single crystals with pronounced cleavageplanes provide a unique possibility to prepare high quality samplesby cleavage along the weakest cleavage plane [82,83]. In fact, up tonow cleaved wedges have reached the highest perfection andessentially dispersion-free propagation of WWs could be observed.In general, even wedge modifications in the micrometer range maybe critical and generate measurable dispersion effects, when thegigahertz frequency range is approached. On the other hand, sucha sensitivity of the phase velocity to wedge modifications can beemployed to assess the quality of edges in cutting tools and blades.The effort needed for the preparation of suitable samples is onereason why the field of dispersion-free wedges is not yet furtheradvanced. The lack of versatile easy-to-apply excitation and detec-tion methods is another reason.

As an example of a nearly perfect wedge we consider experi-ments with a cleaved silicon wedge consisting of two (111) planesforming a wedge angle of 70.5� [81]. For selective wave excitationthe optical variable-angle transducer and for detection the probe-beam-deflection setup were employed. With this arrangementthe fundamental WW, the SAW, and a supersonic pseudo-wedgewave (p-WW) were excited separately. By measuring the propaga-tion time of these waves between the source and various detectionpoints along the edge the phase velocity was measured, clearlyshowing that the p-WW is faster than the SAW with its wavevectoralong the edge. The wavevector along one of the wedge faces iscontrolled by varying the angle u of the wedge transducer [81].The slowest mode was the localized WW propagating at4.21 km/s for u 43�, then the SAW propagating along the tip

Fig. 17. (a) Decay of displacement perpendicular to wedge tip for the fundamental WW; and (b) Same decay for the p-WW.


was generated at u 39� with a velocity of 4.54 km/s. Furthervelocity scanning revealed a new wave at u 32�, propagatingwith a velocity of 5.4 km/s, which is substantially above the SAWvelocity of silicon in this direction. By shifting the probe laser per-pendicular to the edge the strong signal decay with separationfrom the tip could be verified directly, as displayed in Fig. 17a. Thisdecay is weaker for the p-WW and its coupling with the SAW isclearly visible at the surface, as can be seen in Fig. 17b. This obser-vation clearly indicates that the plane wave front is lost withdepth, contrary to the behavior of the localized WW. A more de-tailed analysis shows that the supersonic wave couples not onlywith the SAW but also with the slower bulk shear waves, whichcannot be seen as easily.

3.1.4. Dispersive wedge wavesCurrently, the vast majority of published papers deals with dis-

persive WWs. Reasons for dispersion are, for example, truncationof the wedge tip or coating of one or both wedge faces. Still, ofcourse, the sample must be prepared with great care so that theregistered dispersion effect originates only from the consideredsource and not from additional wedge deficiencies.

Let us consider a rounded wedge and continuous flaws alongthe ridge of a rectangular waveguide. An ideal sharp wedge haszero thickness at the tip and no characteristic dimension and con-sequently no dispersion. Truncation by rounding of the wedge tipincreases the apex angle close to the tip, causing higher frequencycomponents to travel with a higher speed (closer to the Rayleighvelocity), whereas low frequencies propagate essentially with thevelocity of the rectangular wedge [84]. Conversely, a wedge withsharp protruding defects acts as a wedge with an effective apex an-gle smaller than 90� close to the tip. Thus, high-frequency wavesconcentrated near the tip have a slower velocity than the low-fre-quency components travelling with the 90�-wedge velocity. Thesequalitative explanations agree with the frequency dependence ob-served experimentally, where the normal and anomalous disper-sions disappear with increasing acuteness and transition to theundistorted shape of the wedge tip, as expected [84].

As an example of the effect of a coated layer on one of thewedge faces we discuss the cases of an aluminum wedge coatedwith a copper film and a brass wedge coated with an aluminumfilm [85]. Acoustically the first system corresponds to the loadingcase on a plane surface with a slow coating and the second tothe stiffening case with a fast coating. The corresponding normaland anomalous dispersion behaviors were studied for several

wedge angles [85]. Interestingly, there is an important differencebetween SAW propagation on a coated plane surface and WWpropagation on a wedge concerning sensitivity. While the influ-ence of a micrometer-thick layer can be enormous at the verytip, it may become immeasurably small at larger distances fromthe wedge tip, where the wedge is broadened considerably. Fur-thermore, the distance that still has an influence on the dispersionbehavior is, of course, greater in the long-wavelength regime thanin the small-wavelength regime.

Wedge modes have also been used to detect the adsorption ofmoisture at the wedge faces in a humidity sensor. The applicabilityof the corresponding velocity effect has been demonstrated bycoating the wedge tip with a hygroscopic film that adsorbs watereffectively [86]. It has also been demonstrated that the velocityof the fundamental ASF mode decreases upon water loading forwedge angles of 20�90� [87]. This effect is more pronounced fora larger mass density ratio of liquid/solid. A reduction of the phasevelocity by fluid loading was also found for wedges with very smallwedge angles [88] and has been confirmed theoretically [89,90]and experimentally [91,92].

A first report on the visualization of an isolated artificial rectan-gular notch placed on the wedge tip has been published recently[93]. Here a movable pulsed laser was scanned over the area ofone wedge face. The signals of the propagating waves were regis-tered with a shear transducer fixed at the end of the wedge. Thedetected signals were piled up into a data cube (x,y, t) and pictureswere generated by time-gating at various elapsed times. With thisvisualization system the reflection and transmission of the ASFmode through the defect and mode conversion of the ASF modeinto a SAW at the notch could be observed. Despite the fact thatthe notch was relatively large, with a size in the millimeter range,this technique opens up the way to dynamic visualization of iso-lated defects on edges using WWs.

3.2. Nonlinear wedge waves

3.2.1. TheoryOne important impetus for the investigation of WWs is the

expectation that strong nonlinearities and large strains, owing tothe extreme localization of elastic energy in the apex of the wedge,can be realized experimentally. Early measurements, using acleaved LiNbO3 crystal, were rather disappointing, since harmonicgeneration and parametric conversion were even less effectivethan estimated for nonlinear SAWs. In fact, third-order generation


outperformed second-order generation above a certain excitationpower, because generation of even harmonics and second-ordernonlinear effects cancelled out for symmetry reasons [82]. Unfor-tunately, the role played by nonlinearity in isotropic and aniso-tropic wedges presently is not well understood, especially whatquantitative aspects are concerned.

Nonlinear evolution equations for symmetric and for anti-sym-metric wedge waves in isotropic media have been derived [94]. Theequation for symmetric waves contains a nonlinearity of second-order and looks very similar to Eq. (2.12) valid for SAWs. In fact,when defining B(x) as the Fourier transform of a nonzero displace-ment gradient component um1 at the tip of the wedge,

um1ðx1;0;0; tÞ ¼ 2ReZ 1

0BðxÞ exp i

xcWðx1 � cW tÞ

� �dx2p

� �; ð3:12Þ

where cW is the phase velocity of a linear wedge wave, the followingevolution equation is obtained:

icW@

@x1Bðx; x1Þ ¼ x

Z x

0Gðx0=xÞBðx0; x1ÞBðx�x0; x1Þdx0=2p

�

þ 2Z 1

x

xx0� �2

G�ðx=x0ÞBðx0; x1ÞB�ðx0 �x; x1Þdx0=2p�: ð3:13Þ

Like the function F(g) in the evolution Eq. (2.12) for nonlinear SAWs,the kernel function G(g) depends on ratios of second-order andthird-order elastic moduli of the elastic medium the guided waveis propagating in.

This type of evolution equation also holds for anisotropic mediain wedge geometries of sufficiently low symmetry such that a clas-sification of the wedge modes into symmetric and anti-symmetricis not possible. The kernel function G(g) was computed for variouswedge geometries of silicon crystals [68,95]. Apart from quantita-tive differences of the kernel functions F and G, the main differencebetween the evolution Eq. (2.12) for SAWs and (3.13) for symmet-ric WWs is the power of the factor x/x0 in the second integral onthe right-hand sides. The exponent 2 for WWs in comparison to 1for SAWs indicates that frequency down-conversion is even lesseffective for wedge waves than it is for surface waves. Therefore,wave steepening and shock formation of wedge waves may be fa-vored as compared to surface waves.

The nonlinear evolution equation for ASF waves in isotropicmedia contains an effective nonlinearity of third order. Little isknown about its size apart from earlier work for wedges with verysmall wedge angle [96]. Here, the nonlinearity is of purely geomet-ric character and therefore governed by the linear Lamé constantsonly, while the third-order and fourth-order elastic moduli areirrelevant in this case.

3.2.2. ExperimentalFor isotropic wedges experimental results on dispersion-free

wedges are rare. It may be expected that the influence of anisot-ropy on the nonlinear behavior of 1D WWs can be more pro-nounced than for 2D SAWs, at least for selected geometries orcrystallographic configurations. For rectangular edges of silicon itwas found theoretically that no edge mode may exist for the caseof one surface normal being along the h001i direction and theother along the h110i direction, in agreement with preliminaryexperiments [68]. For a second geometry with the two surfacenormals along cubic axes, a well-localized edge mode was foundtheoretically and confirmed experimentally. The resonant sec-ond-order nonlinearity vanishes for this geometry and nonlinearwaveform evolution is governed by an effective nonlinearity ofthird order. In a third rectangular geometry with the surface nor-mals h011i and h111i, an edge mode was found by calculations,which is predicted to exhibit appreciable second-order nonlineareffects, including the formation of solitary pulses in the presenceof weak dispersion [68]. Spiking and shock formation predicted

on the basis of the evolution Eq. (3.13) [97] could be confirmedexperimentally for this particular geometry. Note that these non-linear features are strongly geometry-dependent and reflect thestrong effects of anisotropy.

Despite the fact that strong nonlinearity of certain edge-local-ized modes is expected from these theoretical considerations, 1Dsolitary waves, feasible after introducing weak dispersion by tipmodification or coating of one or both wedge faces, have not yetbeen realized, whereas nonlinear wedge waves developing steepshock profiles during propagation have been seen.

3.3. Applications of wedge waves

A number of possible applications have been discussed for lin-ear and nonlinear edge waves and selected feasibility studies havebeen carried out already for linear WWs, however, no devices areon the market, as in the case of the well-developed SAWs. Suchenvisaged future applications include NDE of defects in cuttingtools or turbine blades, sensor applications measuring the velocitychange upon wedge face and tip modifications, ultrasonic motors,stirring and acoustic streaming in fluidics, and aquatic propulsion.Nonlinear WWs would open the door to 1D solitary waves in NDEof absorbing media and the investigation of the mechanical frac-ture strength and failure behavior of wedges. With improved prep-aration techniques for high-quality samples and strongly improvedcomputing power for simulations, 1D wedge waves may soon leavethe shadow of the well-established 2D surface waves. Especially inthe intricate situation of anisotropic wedges, with respect to exper-iments, the simultaneous support of the selection of sample config-urations by theoretical simulations is indispensable, owing to therather complex behavior of these systems and the large varietyof geometrical possibilities.

Acknowledgements

Financial support of this work by the Deutsche Forschungs-gemeinschaft (DFG Grant MA 1074/10,11) and the RussianFoundation for Basic Research (10-02-00863-a) is gratefullyacknowledged.

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