Directivity patterns and pulse profiles of ultrasound emitted by laser action on interface between transparent and opaque solids: Analytical theory

Directivity patterns and pulse profiles of ultrasound emitted by laser action on interfacebetween transparent and opaque solids: Analytical theorySergey M. Nikitin, Vincent Tournat, Nikolay Chigarev, Alain Bulou, Bernard Castagnede, Andreas Zerr, and

Vitalyi Gusev

Citation: Journal of Applied Physics 115, 044902 (2014); doi: 10.1063/1.4861882 View online: http://dx.doi.org/10.1063/1.4861882 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Finite element calculations of the time dependent thermal fluxes in the laser-heated diamond anvil cell J. Appl. Phys. 111, 112617 (2012); 10.1063/1.4726231 Generation of inhomogeneous plane shear acoustic modes by laser-induced thermoelastic gratings at theinterface of transparent and opaque solids J. Appl. Phys. 110, 123526 (2011); 10.1063/1.3662921 Effect of laser beam incidence angle on the thermoelastic generation in semi-transparent materials J. Acoust. Soc. Am. 130, 3691 (2011); 10.1121/1.3658384 Two dimensional hydrodynamic simulation of high pressures induced by high power nanosecond laser-matterinteractions under water J. Appl. Phys. 101, 103514 (2007); 10.1063/1.2734538 Nanosecond time-resolved multiprobe imaging of laser damage in transparent solids Appl. Phys. Lett. 81, 3149 (2002); 10.1063/1.1511536

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

http://scitation.aip.org/content/aip/journal/jap?ver=pdfcov

http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/1417871120/x01/AIP-PT/JAP_ArticleDL_091014/AIP-2233_APL_Editor_1640x440.jpg/47344656396c504a5a37344142416b75?x

http://scitation.aip.org/search?value1=Sergey+M.+Nikitin&option1=author

http://scitation.aip.org/search?value1=Vincent+Tournat&option1=author

http://scitation.aip.org/search?value1=Nikolay+Chigarev&option1=author

http://scitation.aip.org/search?value1=Alain+Bulou&option1=author

http://scitation.aip.org/search?value1=Bernard+Castagnede&option1=author

http://scitation.aip.org/search?value1=Andreas+Zerr&option1=author

http://scitation.aip.org/search?value1=Vitalyi+Gusev&option1=author

http://scitation.aip.org/content/aip/journal/jap?ver=pdfcov

http://dx.doi.org/10.1063/1.4861882

http://scitation.aip.org/content/aip/journal/jap/115/4?ver=pdfcov

http://scitation.aip.org/content/aip?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jap/111/11/10.1063/1.4726231?ver=pdfcov



http://scitation.aip.org/content/asa/journal/jasa/130/6/10.1121/1.3658384?ver=pdfcov



http://scitation.aip.org/content/aip/journal/apl/81/17/10.1063/1.1511536?ver=pdfcov

Directivity patterns and pulse profiles of ultrasound emitted by laser actionon interface between transparent and opaque solids: Analytical theory

Sergey M. Nikitin,1,2,3,a) Vincent Tournat,1 Nikolay Chigarev,1 Alain Bulou,2

Bernard Castagnede,1 Andreas Zerr,3 and Vitalyi Gusev1,a)

1LAUM, UMR-CNRS 6613, Universit�e du Maine, 72085 Le Mans, France2IMMM, UMR-CNRS 6283, Universit�e du Maine, 72085 Le Mans, France3LSPM, UPR-CNRS 3407, Universit�e Paris Nord, 93430 Villetaneuse, France

(Received 29 October 2013; accepted 27 December 2013; published online 22 January 2014)

The analytical theory for the directivity patterns of ultrasounds emitted from laser-irradiated

interface between two isotropic solids is developed. It is valid for arbitrary combinations of

transparent and opaque materials. The directivity patterns are derived both in two-dimensional and

in three-dimensional geometries, by accounting for the specific features of the sound generation by

the photo-induced mechanical stresses distributed in the volume, essential in the laser ultrasonics.

In particular, the theory accounts for the contribution to the emitted propagating acoustic fields

from the converted by the interface evanescent photo-generated compression-dilatation waves. The

precise analytical solutions for the profiles of longitudinal and shear acoustic pulses emitted in

different directions are proposed. The developed theory can be applied for dimensional scaling,

optimization, and interpretation of the high-pressure laser ultrasonics experiments in diamond anvil

cell. VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4861882]

I. INTRODUCTION

In the recent years, the application of all-optical non-

contact laser ultrasonics (LU) techniques for the evaluation

of compressed material properties, through high pressure

experiments in a diamond anvil cell (DAC) on samples with

characteristic dimensions between 10 and 100 lm, attracted

an increasing number of researchers.1–6 While picosecond

laser ultrasonics, based on the application of a femtosecond

laser, uses generation and detection of acoustic waves propa-

gating quasi-collinear to the DAC axis,1 the point-source-

point-receiver technique,2,3 and line-source-point-receiver

technique,4,5 based on the application of sharply focused

radiation of sub-nanosecond laser, are monitoring the longi-

tudinal and shear acoustic waves propagating at rather large

angles to this axis. Thus, for the latter two LU-DAC techni-

ques, the knowledge of the directivity patterns of acoustical

waves, emitted from laser-irradiated interface between trans-

parent materials (e.g. diamond or pressure medium such as

KBr or argon) and light-absorbing materials (e.g. metals),

could be useful for scaling and interpretation of the experi-

ments on sound waves propagation in metals and transparent

media compressed in a DAC.

Earlier, both theoretical and experimental studies of

the directivity patterns of laser ultrasound in solids mostly

concentrated on the case when laser irradiates a mechani-

cally free surface of a solid halfspace.6–18 Directivity pat-

terns were evaluated both for the compression/dilatation

and for the shear acoustic waves. The influence on the di-

rectivity patterns of the thermal conductivity of the solid,12

and the laser focusing and laser penetration depth in the

solid13,14 were investigated. The profiles of the emitted

acoustic pulses were the subject of the analysis both in the

case of longitudinal and of shear acoustic waves.10,11,15,16

These research activities had been initiated and were

continuously supported by the applications of laser ultra-

sound for the non-destructive testing of the materials and

structures,7 and the investigations of the directivity pat-

terns of laser ultrasound emitted from a mechanically free

surface are continuing for some specific applications until

now.17,18

The investigations of the directivity patterns of laser

ultrasound emitted from an interface between an optically

transparent and an optically opaque solids have been fueled

by the applications of laser ultrasound in high pressure

research, which started just a few years ago.1–5 Recently, the

directivity patterns of longitudinal and shear acoustic waves

emitted by focusing laser radiation at the interface of dia-

mond with aluminum have been simulated for the first time

numerically.19 Here, we present the analytical descriptions

for the directivity patterns of laser ultrasound, which are

valid for arbitrary combinations of transparent and opaque

materials. The directivity patterns are derived as particular

cases of the known general solution for the acoustic fields

generated by laser radiation in layered media,20,21 by

accounting for the specific features of the sound generation

by thermo-elastic stresses distributed in the volume, which

are essential for laser ultrasonics. We also present the analyt-

ical solutions for the profiles of the longitudinal and shear

acoustic pulses emitted in different directions. The derived

mathematical formulas provide straight opportunity to pre-

dict the acoustic field, which is formed in the diamond anvil

cell after photo-generation and several reflections of bulk

acoustic waves at the interfaces relevant for the experiments

in DAC.

a)Authors to whom correspondence should be addressed. Electronic

addresses: [email protected] and vitali.goussev@univ-

lemans.fr

0021-8979/2014/115(4)/044902/15/$30.00 VC 2014 AIP Publishing LLC115, 044902-1

JOURNAL OF APPLIED PHYSICS 115, 044902 (2014)


149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

http://dx.doi.org/10.1063/1.4861882

http://dx.doi.org/10.1063/1.4861882

http://dx.doi.org/10.1063/1.4861882

mailto:[email protected]



http://crossmark.crossref.org/dialog/?doi=10.1063/1.4861882&domain=pdf&date_stamp=2014-01-22

II. GENERAL THEORETICAL DESCRIPTION OF THELASER ULTRASOUND DIRECTIVITY PATTERN

In order to derive an analytical presentation of the direc-

tivity patterns of the acoustic waves when the laser radiation

is incident from the transparent media “(2)” onto its plane

interface with the light-absorbing media “(1)” (Fig. 1), we use

the general solution obtained for the optoacoustic transforma-

tion at these type of interfaces in Ref. 21. We present mathe-

matical formalism for the case of two-dimensional (2D)

geometry, which is experimentally realized by focusing laser

radiation into a stripe with the length significantly exceeding

its width. The generalization of mathematical approach for the

case of three-dimensional (3D) geometry, which is experi-

mentally realized by focusing laser radiation into a circular

spot, is presented in Appendix A. For definiteness, we analyze

the directivity patterns of acoustic waves in medium “(1),”

because the solutions for the medium “(2)” can be obtained

by symmetry principles. If the origin of the coordinate system

is chosen at the interface of media “(1)” and “(2)” and z-axis

is perpendicular to the interface and directed toward media

“(1),” the solutions for the Fourier spectra of scalar / and vec-

tor w acoustic potentials (Eqs. (13) and (15) from Ref. 21),

which satisfy the conditions of continuity of the mechanical

displacements and stresses at the interface and the conditions

of radiation in the far field,22–24 are

~~/1ðx; kx; zÞ ¼�i

2a1

~~~rin1 ðx; kx;a1Þ þ R11

ll

~~~rin1 ðx; kx;�a1Þ

�

þa1

a2

T21ll

~~~rin2 ðx; kx;a2Þ

�e�ia1z � ~~/1ðx; kxÞe�ia1z;

(1)

~~w1ðx; kx; zÞ ¼�i

2a1

R11lt

~~~rin1ðx; kx;�a1Þ

�

þa1

a2

T21lt

~~~rin2ðx; kx;a2Þ

�e�ib1z � ~~w1ðx; kxÞe�ib1z:

(2)

The Fourier transforms for the derivation and manipulation

of the solutions in Eqs. (1) and (2) are defined by

~~~Fðx; kx; kzÞ ¼ð1�1

ð1�1

ð1�1

Fðt; x; zÞe�iðxt�kxx�kzzÞdtdxdz;

Fðt; x; zÞ ¼ 1

ð2pÞ3ð1�1

ð1�1

ð1�1

~~~Fðx; kx; kzÞeiðxt�kxx�kzzÞdxdkxdkz:

(3)

In Eqs. (1) and (2), x is the cyclic frequency of the laser-

induced normalized stress fields rin1;2 and of the acoustic fields,

kx is the projection of the acoustic wave vectors on the x-axis,

which is in the plane of the interface. We remind here that

photo-induced thermo-elastic stress tensor rikl is isotropic in iso-

tropic and cubic solids, i.e., rikl � ridkl. Note that

x-components of the wave vectors are equal for longitudinal and

shear waves and also in both media22,24 (see Fig. 1), while the

z-components are different. i.e., kzl1;2 � a1;2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

l1;2 � k2x

qand

kzt1;2 � b1;2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

t1;2 � k2x

q, where sgnðRea1;2Þ ¼ sgnðReb1;2Þ

¼ sgnðxÞ for real a1;2 and b1;2 and sgnðIma1;2Þ ¼ sgnðImb1;2Þ¼ �1 for imaginary a1;2 and b1;2. Here, x=tl1;2 � kl1;2 and

x=tt1;2 � kt1;2 are the wave numbers of the compression/dila-

tion and shear acoustic waves in medium “1” and medium “2,”

with tl1;2 and tt1;2 denoting the velocities of the longitudinal

and shear waves, respectively. Although the physical nature of

laser-induced stresses in Eqs. (1) and (2) is not specified yet,

below we analyze the case of thermoelastic stresses caused by

laser heating of the media. They are normalized on the longitu-

dinal moduli of the corresponding media, rin1;2 ¼ ri

1;2=

ðq1;2t2l1;2Þ, where q1;2 are the densities. Although solutions (1)

and (2), borrowed from Ref. 21, are written for the two-

dimensional problem and are well-suited only for the line-

source-point-receiver LU-DAC technique,4,5 the extension of

the general solution to the three-dimensional case is straightfor-

ward (Appendix A).

The solutions in Eqs. (1) and (2) have a clear transparent

physical sense. They account for the fact that the stresses are

induced by laser radiation, in general, in both contacting

media. They also account for the fact that thermo-elastic

stress in isotropic media can generate only compression/dila-

tation acoustic waves.23,25 In the qualitative illustration of

the phenomena under consideration in Fig. 1, the wave vec-

tors of the compression/dilatation waves excited by thermo-

elastic sources are marked by Dl1,l2. Those of these waves,

which propagate from the interface, contribute to the direc-

tivity patterns of laser ultrasound without any additional

transformations. Those of these waves, which are directed

towards the interface are transformed in reflection/transmis-

sion into four waves, both compression/dilatation and shear,

which could contribute to directivity patterns of laser ultra-

sound. In the solution (1) for the compression/dilatation

waves, i.e., for the scalar potential in the first medium, the

first term describes the wave generated in the first medium

and propagating in the bulk of the first medium without any

FIG. 1. Reflection and refraction, with and without mode conversion, of

plane compression/dilatation waves at a plane boundary between isotropic

media.

044902-2 Nikitin et al. J. Appl. Phys. 115, 044902 (2014)


149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

interaction with the interface. This wave is denoted in the

right-hand-side of Fig. 1 by Dl1. The second term describes

the waves also generated in the first medium, but redirected

in the bulk of this medium only after the reflection on the

interface (R11ll is the reflectivity coefficient). This wave is

denoted in Fig. 1 by R11ll . The third term describes the waves

that are generated in the second medium and transmitted into

the first medium across the interface (T21ll is the transmission

coefficient). This wave is denoted in Fig. 1 by T21ll . In the so-

lution (2) for the shear waves, i.e., for the vector potential in

the first medium, the first term describes the wave which is

generated at the interface due to mode conversion of the

compression/dilatation waves, denoted in the left-hand-side

of Fig. 1 by Dl1, in reflection from the interface (R11lt is the

reflection coefficient with mode conversion). The corre-

sponding shear wave, contributing to the directivity pattern

of laser ultrasound is denoted in Fig. 1 by R11lt . The second

term describes the generation of the shear waves at the inter-

face due to mode conversion of the compression/dilatation

waves, denoted in the left-hand-side of Fig. 1 by Dl2 in trans-

mission across the interface (T21lt is the transmission coeffi-

cient with mode conversion). The corresponding shear wave,

contributing to the directivity pattern of laser ultrasound in

the first medium is denoted in Fig. 1 by T21lt . The physical na-

ture of waves contributing to the directivity patterns of laser

ultrasound in the second medium can be understood simi-

larly (see Fig. 1).

The reflection and transmission coefficients, which are

necessary for using the solution in Eq. (1), can be found in

textbooks.22,24,26 In Ref. 21, the classical approach of deriving

the system of algebraic equations for R11lt , R11

ll , T12ll and T12

lt of

acoustic potentials was reminded. The resultant system is

b1 b2 �kx kx

kx �kx a1 a2

b1 l21b2 �c1 l21c2

c1 �l21c2 a1 l21a2

0BBBB@

1CCCCA

R11lt

T12lt

R11ll

T12ll

0BBBBB@

1CCCCCA ¼

kx

a1

c1

a1

0BBBB@

1CCCCA; (4)

where the compact notations l21 � l2=l1 and c1;2 � kx

�k2t1;2=ð2kxÞ are introduced, l1;2 are the second Lame con-

stants (shear moduli). The transmission coefficients T21lt and

T21ll necessary for the evaluation of Eq. (2) can be obtained

either from T12lt and T12

ll in Eq. (4), using symmetry consider-

ations22 or by solving the following system, which in turn

can be derived from Eq. (4) using symmetry principles

b2 b1 kx �kx

�kx kx a2 a1

b2 l12b1 c2 �l12c1

�c2 l12c1 a2 l12a1

0BBBB@

1CCCCA

R22lt

T21lt

R22ll

T21ll

0BBBBB@

1CCCCCA ¼

�kx

a2

�c2

a2

0BBBB@

1CCCCA: (5)

Thus, the general solution for the laser-generated acoustic

field in medium “1” in the (x; kx; z) space is complete.

To get the description of the emitted acoustic field in

(x; x; z) space, it is sufficient to apply to Eqs. (1) and (2) the

following inverse Fourier transformation Eq. (3) over kx:

~/1ðx; x; zÞ~w1ðx; x; zÞ

!¼ 1

2p

ðþ1�1

~~/1ðx; kxÞe�ia1z

~~w1ðx; kxÞe�ib1z

0@

1A

� e�ikxxdkx: (6)

To find the directivity pattern of the laser-induced acoustic

source, it is necessary to evaluate the integral only at large

distances from the source, where the phase of the integrand

is strongly changing even with small variations of kx. This

provides opportunity to approximate the integrals in Eq. (6)

in the general case by using the method of steepest descent

(method of the stationary phase).26 The derived dependences

of the complex amplitudes of the compression/dilatation and

shear cylindrical waves at large distances r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ z2p

from

their excitation region on the angle of emission h, i.e., the

angle between the direction of observation and the z-axis, are

~/1ðx; r; hÞ~w1ðx; r; hÞ

!¼ cos h

ffiffiffiffiffiffikl1

p ~~/1ðx; kl1 sin hÞffiffiffiffiffiffikt1

p ~~w1ðx; kt1sin hÞ

0@

1A

�ffiffiffiffiffiffiffiffi

i

2pr

re�ikl1r

e�ikt1r

!: (7)

The solution in Eq. (7) has been derived by assuming that

x > 0. The solution for negative frequencies x < 0, can be

derived using~~/1ðx; r; hÞ ¼

~~/1�ð�x; r; hÞ and

~~w1ðx; r; hÞ¼ ~~w1

�ð�x; r; hÞ, i.e., with the application of complex conju-

gation. It should be also noted, that neither the interface

waves (Stoneley waves, if they exist) nor the contribution to

far fields from the head waves are included in the solution in

Eq. (6). They could be evaluated through the analysis of the

pole (if it exists) and the branch points of the integrand in

Eq. (6). However, it is worth noting that the head waves, i.e.,

contributions from branch cut integrals, are asymptotically

negligible at large distances, because they diminish in ampli-

tude faster than the bulk waves.26,27 To proceed to the analy-

sis of the particular possible experimental situations, the

distribution of the laser-induced stresses in the contacting

media should be specified.

III. GENERAL THEORETICAL DESCRIPTION OF THELASER-INDUCED THERMO-ELASTIC STRESSES

The spectrum of the normalized thermo-elastic stress is

controlled by the spectrum of the laser-induced temperature

rise21,23

~~~rin1;2 ðx; kx; kzÞ ¼ ��a1;2½ð1þ �1;2Þ=ð1� �1;2Þ�

~~~T 01;2ðx; kx; kzÞ;(8)

where �a, �, and T0

are the linear thermal expansion coeffi-

cient, Poisson ratio, and the temperature rise, respectively.

The laser-induced heating is described in 2D rectangular ge-

ometry, dictated by the line-type spatial structure of the

focused laser radiation,4,5 using the equations of heat

conduction in the contacting materials.20,21,23 The equation

of heat conduction in the light absorbing material is



149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

inhomogeneous. The laser-induced heat release Q1ðt; x; zÞ in

the light-absorbing material, i.e., the increase in the material

thermal energy density per unit time in a unit volume, can be

presented in the form Q1 ¼ f ðtÞUðxÞWðzÞ,23,24 where the

function f ðtÞ describes the laser pulse intensity envelope in

time, the function UðxÞ describes the lateral distribution of

the absorbed laser energy release controlled by laser inten-

sity distribution at the irradiated surface, i.e., by focusing,

and the function WðzÞ describes the distribution of the

absorbed laser energy release in depth controlled by the opti-

cal penetration depth and some other physical parameters of

the light absorbing medium.23,25 The solution of the thermal

conductivity equations, satisfying the conditions of continu-

ity of the temperature and of the heat flux at the interface

between the opaque and the transparent media, can be

derived by the same method of integral transforms, which

has been applied in Ref. 21 to derive the solutions in Eqs. (1)

and (2) of the inhomogeneous wave equations. It is sufficient

to find the solution in the reciprocal Fourier spaces, where it

can be factorized in the following form:

~~~T 01;2ðx; kx; kzÞ � H1;2ðx; kx; kzÞ~f ðxÞ~UðkxÞ: (9)

The general expression of the function Hðx; kx; kzÞ is

H1ðx; kx; kzÞ ¼1

v1ðk2z � d2

1Þ~WðkzÞ þ

v1kz � v2d2

v1d1 þ v2d2

~Wð�d1Þ� �

;

H2ðx; kx; kzÞ ¼ð�1Þ

ðv1d1 þ v2d2Þðkz þ d2Þ~Wð�d1Þ:

(10)

Here, d1;2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ix=D1;2 � k2

x

pcan be associated with the

projection of the thermal wave vectorffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ix=D1;2

pon the

z-axis, D1;2 are the thermal diffusivities and v1;2 are the ther-

mal conductivities of the media in contact. It is worth men-

tioning here that if future experiments in DAC indicate the

importance of thermal boundary resistance at the interface

between the diamond and the sample, then the solution of

the heat conduction problem with modified interface condi-

tions for temperature field and heat flux could be still

obtained by the same mathematical formalism of integral

transforms.

The description of the laser-induced stresses presented

in Eqs. (8)–(10) finalizes the general solution for the ultra-

sound emission by thermoelastic sources. The specific

problems of interest for the practical analysis differ only by

the characteristics of the laser induced heat release

Q1 ¼ f ðtÞUðxÞWðzÞ. The common models for the laser pulse

intensity envelope, distribution of laser intensity across

the focus, and distribution of laser intensity in depth of

light absorbing medium are f ðtÞ ¼ exp½�ð2t=sLÞ2�; UðxÞ¼ exp½�ð2x=dÞ2�; WðzÞ ¼ ðI=lÞexpð�z=lÞ, respectively,

where sL is the laser pulse duration at 1/e level, d is the diam-

eter of the focus at 1/e level, I is the absorbed part of the laser

intensity incident on the interface, and l is the characteristic

heating depth, which in metals can depend not only on the

penetration depth of laser radiation but also on the penetration

depth of the non-equilibrium overheated electrons.28–30 In the

Fourier domain, the description of the heat release is

~f ðxÞ ¼ ðffiffiffipp

sL=2Þexp½�ðsLx=4Þ2�;~UðkxÞ ¼ ð

ffiffiffipp

d=2Þexp½�ðdkx=4Þ2�;~WðkzÞ ¼ ðI=lÞðl�1 � ikzÞ�1: (11)

In some particular experimental situations, depending on

laser focusing, duration of the laser pulses, penetration depth

of the laser radiation and physical properties of the contact-

ing solids, the general theoretical formulas for the tempera-

ture rise, and thermo-elastic stresses can be significantly

simplified.

IV. DIRECTIVITY PATTERNS OF LASER ULTRASOUND

The boundary between laser ultrasonics and laser hyper-

sonics (picosecond laser ultrasonics) is commonly and quali-

tatively considered to pass near 1 GHz frequency. For

example, in accordance with Eq. (11) the spectrum of acous-

tic waves emitted in the LU-DAC type experiments with

sub-nanosecond laser, with sL � 0:5 ns, is limited in

frequencies by f � 1:3 GHz. So these experiments can be

considered as laser ultrasonics experiments. At ultrasonic

frequencies in metals and other good thermal conductors

(in diamond, for example), the depth of the zone heated ei-

ther by penetrating laser radiation or by the transport of the

overheated electrons is thin both thermally and acoustically.

In other words, the initially heated depth is much shorter

than both thermal and acoustic wavelengths, and the heating

can be well approximated by interface-localized heating.

From the mathematics point of view for laser ultrasound, the

relation ~WðkzÞ ffi ~Wð�d1Þ ffi I, holds, and consequently the

description in Eq. (10) simplifies to

H1;2ðx; kx; kzÞ ¼ð61ÞI

ðv1d1 þ v2d2Þðkz 7 d1;2Þ: (12)

Additionally at ultrasonic frequencies, the acoustic wave-

lengths are longer than the thermal wavelengths. In this case,

neglecting the acoustic wave numbers in comparison with

the thermal wave numbers, the solution in Eq. (12) is

reduced to

H1;2ðx; kx; kzÞ ¼ �ffiffiffiffiffiffiffiffiD1;2

pI

ð�ixÞð ffiffiffiffiffie1p þ ffiffiffiffiffi

e2p Þ : (13)

Here, e1,2 denote the thermal effusivities of the contacting

media. The description of the laser-induced stresses at ultra-

sonic frequencies, which is derived above in Eqs. (8), (9), and

(13), provides opportunity to present the directivity patterns of

the laser ultrasound, i.e., Eq. (7), in the following form:

~/1ðx;r;hÞ¼P1I 1þR11ll ðhÞþP2=1

coshffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðtl1=tl2Þ2�sin2h

q T21ll ðhÞ

8<:

9=;

�~Uðkl1sinhÞ~f ðxÞ

x

ffiffiffiffiffiffiffiffiffiffiffiffiffii

2pkl1r

re�ikl1r; (14)



149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

~w1ðx; r; hÞ ¼ P1Icos hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðtt1=tl1Þ2 � sin2 hq R11

lt ðhÞ8<:

þP2=1

cos hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðtt1=tl2Þ2 � sin2 h

q T21lt ðhÞ

9>=>;

�~Uðkt1 sin hÞ~f ðxÞ

x

ffiffiffiffiffiffiffiffiffiffiffiffiffii

2pkt1r

re�ikt1r: (15)

Here, the compact notation P1 is introduced for the following

combination of physical parameters P1 � �a1ð1þ�1ÞffiffiffiffiD1

p

2ð1��1Þðffiffiffie1p þ ffiffiffie2

p Þ.

The parameter P2=1 � �a2ð1þ�2Þð1��1ÞffiffiffiffiD2

p

�a1ð1þ�1Þð1��2ÞffiffiffiffiD1

p characterizes the rela-

tive efficiency of laser light transformation into ultrasound in

the contacting media in the case of interface absorption of

laser radiation.

From the physics point of view, the mathematical com-

binations in the figure brackets in Eqs. (14) and (15) describe

the directivity of the emission of compression/dilatation and

shear acoustic waves, respectively, by laser radiation delta-

localized in 2D geometry, i.e., when laser radiation is

focused on y-axis. Formally, this limiting situation is realized

due to ~UðkxÞ ¼ const / d, when the dependence on angles

rests only in the terms inside the figure brackets. These terms

are the most important parts of the directivity patterns, which

are expected to be sensitive to the relative parameters of the

materials in contact. So we need to evaluate

N/ðhÞ � 1þ R11ll ðhÞ þ P2=1

cos hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðtl1=tl2Þ2 � sin2 h

q T21ll ðhÞ;

(16)

NwðhÞ �cos hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


lt ðhÞ

þ P2=1

coshffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðtt1=tl2Þ2 � sin2 h

q T21lt ðhÞ: (17)

The role of the other factors in Eqs. (12) and (13),

which are angle-dependent, i.e., of ~Uðkl;t1 sin hÞ / ðffiffiffipp

d=2Þexp½�ðdx sin h=4tl;t1Þ2� � ð

ffiffiffipp

d=2Þexp½�ðpd sin h=2kl;t1Þ2�is physically clear. These factors, which depend on laser fo-

cusing, indicate that the higher is the ratio between the laser

beam diameter d and the acoustic wavelength kl;t1—the

smaller is the emission angle. Or, saying differently, these

factors control the transition from a structured directivity

pattern in the case of tight focusing of laser radiation

(d kl;t1) to the plane compression/dilatation wave emis-

sion in the case of defocused laser irradiation (d � kl;t1). At

the same time these factors, being frequency dependent, also

influence the profiles of the emitted ultrasound pulses.

As demonstrated in Appendix A, the solutions (16) and

(17) describe not only the directivity of a delta-localized at

the interface line source in the 2D geometry by also the di-

rectivity of a delta-localized at the interface point source in

the 3D geometry. The similar situation of the absence of

difference in the directivity patterns of delta-localized sour-

ces in 2D and 3D geometries was earlier noticed for the case

of laser action on the mechanically free surface of a solid.15

That is why we start the analysis of the directivity patterns in

Sec. V from the evaluation of the structure of the directivity

patterns of delta-localized sources in Eqs. (16) and (17). The

analysis of the role of the frequency-dependent factors in

Eqs. (14) and (15) will be presented later in Sec. VI. In the

case of the laser action on mechanically free surface of a

solid it was noticed15 that the profiles of the emitted ultra-

sound pulses could be different in 2D and in 3D geometries,

although we are not aware of any, even theoretical, confirma-

tion of this expectation. In Sec. V, we confirm this difference

analytically and we also demonstrate that frequency-

dependent factors in Eqs. (14) and (15) provide in 2D and

3D geometries different modulations of the point-source di-

rectivity patterns. It is worth noting here that our general

theory, presented above, describes as limiting cases the

mechanically free and the infinitely rigid (immobile) interfa-

ces (see Appendix B).

V. DIRECTIVITY PATTERNS OF THEDELTA-LOCALIZED SOURCES

In this section, in view of the perspectives to apply the

developed theory to the laser ultrasonics experiments in

diamond anvil cell (LUDAC technique2–5), we present the

theoretical results for laser ultrasound emitted by a thermo-

elastic source delta-localized on the interface of diamond,

i.e., carbon in the cubic diamond phase denoted as Cd, and

iron (Fe). Important feature of this interface is that both bulk

acoustic velocities in the first of the contacting media (in Fe)

are slower than both acoustic velocities in the second of the

contacting media (in Cd), tt1 < tl1 < tt2 < tl2. This situa-

tion, expected to be typical for practically all the materials in

the diamond anvil cell, is qualitatively illustrated in Fig. 1.

As it has been mentioned above, the translation invariance of

the system along the x-axis requires the conservation in inter-

action of the acoustic waves of the x–components of their

wave vectors (see Fig. 1). Because of this, particular ordering

of the acoustic velocities, i.e., tt1 < tl1 < tt2 < tl2, results in

particular ordering of the propagation angles of the interact-

ing waves ht1 < hl1 < ht2 < hl2 and, most significantly, in

particular ordering and the number of critical angles, which

could play role in the reflection/transmission of the compres-

sion/dilatation acoustic waves by the interface. We define

here as critical angles those angles of observation, at which

one of the acoustic waves interacting at the interface, propa-

gates along the interface.24 For the angles defined as in Fig.

1, this definition means that at angles larger than the critical

one, the x-component of the wave vector for these waves

becomes larger than the total wave vector, while the

z-component becomes imaginary valued. Thus, this wave

becomes evanescent and stops to transfer acoustic energy

away from the surface. The analysis which follows, demon-

strates that critical angles are the characteristic angles in

structuring of the directivity patterns of laser ultrasound,

although they are not the only angles that could be of

importance.



149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

In Fig. 2, we present the results for longitudinal waves

emitted in iron (medium (1)) from its laser-irradiated inter-

face with diamond. The materials parameters used in the

evaluation are listed in Table I. The estimated critical angles

are listed in Table II. We present in Fig. 2 and in some of the

following figures the dependencies on the angle of observa-

tion both of the amplitudes and of the phases of the emitted

potentials. The knowledge of phase directivity patterns is im-

portant for the understanding of some features of the ampli-

tude directivity patterns and is necessary for the evaluation

of the emitted acoustic pulse profiles (Sec. VI).

The results presented in the middle column in Fig. 2

demonstrate the contribution to the total longitudinal laser

ultrasound signal in Fe (Eq. (16)) which is due to the

compression/dilatation waves initially generated in Fe. At

angles smaller than all the critical ones, all five acoustic

waves coupled by the interface, i.e., the incident on the inter-

face longitudinal wave and four acoustic waves created by

the interface, are all propagating, and the latter four transport

the acoustic energy from the interface to the bulk of contact-

ing media. The first characteristic feature in the directivity

patterns of both amplitudes and phases appears at critical

angle hl1=l2 ¼ arcsinðtl1=tl2Þ � 19:1 . At angles larger than

hl1=l2, the longitudinal acoustic waves in diamond are evan-

escent and they stop to transport acoustic energy away from

the interface. The second characteristic feature in the direc-

tivity patterns of both amplitudes and phases appears at

critical angle hl1=t2 ¼ arcsinðtl1=tt2Þ � 27:0 . At angles

larger than hl1=t2, the shear acoustic waves in diamond are

evanescent and they stop to transport acoustic energy away

from the interface. Thus, at angles larger than hl1=t2, the inci-

dent from Fe on the interface acoustic energy does not

induce the emission of the acoustic energy flux into diamond

(medium (2)). From the physics point of view, it would be

reasonable to expect that, under these conditions, the emis-

sion of acoustic waves in Fe would increase with angle

increasing above hl1=t2. At the same time, the emission of the

longitudinal waves in Fe should stop at angles above

hl1=l1 ¼ arcsinðtl1=tl1Þ ¼ 90:0 , when they become evanes-

cent. So the position of the second maximum in the ampli-

tude directivity pattern, which in the middle column of

Fig. 2 is between hl1=t2 and 90.0 , looks reasonable. It is

worth noting that the directivities presented in the middle

column of Fig. 2 are very different from the expected

FIG. 2. Amplitude and phase directivity

patterns of longitudinal ultrasound

emitted in iron by delta-localized sour-

ces, created by laser-irradiation of plane

interface between iron and diamond.

TABLE I. Values of physical parameters of diamond, iron and aluminum,

applied in the manuscript for the estimates and the evaluation of the directiv-

ity patterns.

Physical properties Diamond Iron Aluminum

Longitudinal velocity

of sound (m s�1)

18 000 5900 6420

Transverse velocity

of sound (m s�1)

13 000 3200 3040

Density (kg m�3) 3500 7900 2700

Linear thermal expansion

coefficient (K�1)

1.1� 10�6 11.3� 10�6 23.1� 10�6

Poisson’s ratio 0.20 0.30 0.34

Thermal diffusivities (m2 s�1) 7.80� 10�4 0.23� 10�4 0.84� 10�4

P2/1 � 0.46 0.07

TABLE II. The values of acoustic velocities ratios and of the related critical angles, which manifest themselves in the phenomena of reflection and refraction

of compression/dilatation waves at the interfaces between iron and diamond and between the aluminum and diamond.

Critical angles tl1=tl2 tl1=tt2 tt1=tl2 tt1=tt2 tt1=tl1 tt2=tl2

arcsinðtl1=tl2Þ arcsinðtl1=tt2Þ arcsinðtt1=tl2Þ arcsinðtt1=tt2Þ arcsinðtt1=tl1Þ arcsinðtt2=tl2Þ

(1) Iron 0.328 0.454 0.178 0.246 0.542 0.722

(2) Diamond 19.1 27.0 10.2 14.2 32.8 46.2

(1) Aluminum 0.357 0.494 0.169 0.234 0.473 0.722

(2) Diamond 20.9 29.6 9.7 13.5 28.3 46.2



149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

directivities of the reflection coefficient R11ll , which are pre-

sented for comparison in Fig. 3 and are in accordance with

known theoretical predictions.24,26,31

The results presented in the right column in Fig. 2 dem-

onstrate the contribution to the total longitudinal laser ultra-

sound signal in Fe of the compression/dilatation waves

initially generated in Cd. The compression/dilatation waves

generated in diamond are incident on the interface as propa-

gating waves and carry energy in the bulk of the diamond af-

ter reflection only at the angles smaller than the first critical

angle hl1=l2 ¼ arcsinðtl1=tl2Þ � 19:1 . At angles larger than

hl1=l2, these are the evanescent compression/dilatation waves

thermo-elastically generated in diamond that are transformed

into propagating acoustic waves carrying energy from the

interface. At angles larger than the second critical angle

hl1=t2 ¼ arcsinðtl1=tt2Þ � 27:0 , the shear waves in diamond

also become evanescent and the acoustic energy incident on

the interface from the diamond side could be transported

only into Fe. Both critical angles are clearly manifested in

the right column in Fig. 2 in the directivity pattern of the

phase. However, in the directivity pattern of the phases, an

additional feature in the form of a phase jump of 180 is

clearly seen at an angle of 24.2 between the two critical

angles. This phase shift is a definite fingerprint of the zero in

the amplitude directivity pattern. Thus, our analysis indicates

the existence of an angle at which the evanescent longitudi-

nal waves incident on the interface from the Cd side cannot

be transformed into propagating longitudinal waves in Fe. It

can be seen that this zero in the transmission coefficient T21ll

suppresses the possible feature related to the second critical

angle in the amplitude directivity pattern. The position of the

local maximum in the amplitude directivity pattern between

the second critical angle and 90 can be explained similar to

the case of the middle column in Fig. 2.

The results presented in the left column of Fig. 2 dem-

onstrate the total longitudinal laser ultrasound signal in Fe

which is due to the compression/dilatation waves initially

generated both in Fe and in Cd. The directivities in the left

column are obtained by summation of those in the middle

and right columns and accounting for the relative phase of

different acoustic contributions. Because the efficiencies of

the opto-acoustic conversion in Fe and Cd are comparable

(see the estimate of the characteristic parameter P2=1 in

Table I), both contacting materials importantly contribute

to the resultant directivity pattern. The characteristic fea-

tures related to the above discussed two critical angles and

the local maximum in amplitude directivity between the

second critical angle and 90 are predicted. However, it is

worth noting that if, for different pairs of contacting materi-

als, the zero in the transmission of the longitudinal waves

across the interface without mode conversion takes place at

an angle larger than the second critical angle, then the local

maximum in the directivity pattern could be between this

angle and 90 .In Fig. 4, we present the results of our evaluation of the

transversal waves emitted in iron (medium (1)) from its laser-

irradiated interface with diamond. The results presented in the

middle column in Fig. 4 demonstrate the contribution to the

total shear laser ultrasound signal in Fe (Eq. (17)), which is

FIG. 3. Amplitude and phase of the reflection coefficient for the plane longi-

tudinal acoustic wave incident from iron on its interface with diamond.

FIG. 4. Amplitude and phase directiv-

ity patterns of shear ultrasound emitted

in iron by delta-localized sources, cre-

ated by laser-irradiation of plane inter-

face between iron and diamond.



149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

due to the compression/dilatation waves initially generated in

Fe. At angles smaller than all the critical ones, all five acoustic

waves coupled by the interface, i.e., the incident on the inter-

face longitudinal wave and four acoustic waves created by the

interface, are all propagating, and the latter four waves are

transporting the acoustic energy from the interface to the bulk

of contacting media. The emission of the shear acoustic waves

in the direction normal to the laser-irradiated surface is forbid-

den by symmetry principles.25,32,33 This explains the growth

in amplitude of the emitted shear waves at small angles. The

first characteristic feature in the directivity patterns of both

amplitudes and phases appears at critical angle ht1=l2 ¼ arcsin

ðtt1=tl2Þ � 10:2 . At angles larger than ht1=l2, the longitudinal

acoustic waves in diamond are evanescent and they stop to

transport acoustic energy from the interface. The second char-

acteristic feature in the directivity patterns of both amplitudes

and phases appears at critical angle ht1=t2 ¼ arcsinðtt1=tt2Þ� 14:2 . At angles larger than ht1=t2, the shear acoustic waves

in diamond are evanescent and they stop to transport acoustic

energy from the interface. Thus, at angles larger than ht1=t2,

the incident from Fe on the interface acoustic energy does

not induce the emission of acoustic waves into diamond (me-

dium (2)). The third critical angle ht1=l1 ¼ arcsin ðtt1=tl1Þ �32:8 most clearly manifests itself in the directivity pattern of

the phase. Above this critical angle, the shear acoustic waves

are emitted in Fe due to the mode conversion in reflection

from the interface of the evanescent compression/dilatation

waves generated by laser radiation in Fe. Shear waves become

the only wave transporting the acoustic energy from the inter-

face. The reasoning, similar to the one devoted above to longi-

tudinal waves directivity patterns, leads to the conclusion that

a local maximum in the amplitude directivity pattern,

observed in the middle column in Fig. 4 at angles between the

largest critical angle and 90 , could be expected theoretically.

Additional structuring of the directivity patterns presented in

the middle column in Fig. 4 is due to the existence of an angle

of 18.8 , at which there is no reflection of the compression/di-

latation waves with mode conversion into the shear waves (in

Eq. (17) R11lt ¼ 0).24,26,31 In the directivity pattern of the

phases, this angle manifests itself as �180 phase jump. It is

worth noting here, that the absence of mode conversion of the

compression/dilatation waves in reflection from a mechani-

cally free surface is known to take place at the angle of 45

(Appendix B).22,25

The results presented in the right column in Fig. 4 show

the contribution to the total shear laser ultrasound signal in

Fe (Eq. (17)), which is due to the compression/dilatation

waves initially generated in diamond. They could be inter-

preted similarly to those in the middle column. Three critical

angles clearly manifest themselves in the directivity patterns

both of the amplitudes and of the phases. However, the direc-

tivity patterns for the emission of shear waves in Fe due to

the compression/dilatation waves laser-generated in Cd

(right column in Fig. 4) are less structured in comparison

with those in the middle column, because there is no angle

between 0 and 90 where the transmission of the compres-

sion/dilatation waves with mode conversion is impossible

(in Eq. (17) T21lt 6¼ 0).

The results presented in the left column in Fig. 4 show

the total shear laser ultrasound signal in Fe which is due to

the compression/dilatation waves initially generated both in

Fe and in Cd. The directivities in the left column are obtained

by summation of those in the middle and right columns and

accounting for the relative phases of different acoustic con-

tributions. Both contacting materials significantly contribute

to the resultant directivity pattern, because the efficiencies of

the opto-acoustic conversion in Fe and Cd are comparable

(see the estimate of the characteristic parameter P2=1 in

Table I). The characteristic features related to the above dis-

cussed three critical angles and the local maximum in ampli-

tude directivity between the third critical angle and 90 are

predicted. However, if for different pairs of the contacting

materials, the zero in the transmission of the longitudinal

waves across the interface with mode conversion takes place

at an angle larger than the third critical angle, then the local

maximum in the directivity pattern could be between this

angle and 90 .In Figs. 5 and 6, the directivity patterns of the acoustic

waves emitted in diamond are presented for a qualitative com-

parison with the directivity patterns of the acoustic waves

emitted in iron (Figs. 2 and 4). The longitudinal wave in dia-

mond is the fastest of all acoustic waves in the considered

FIG. 5. Amplitude and phase directivity

patterns of longitudinal ultrasound emit-

ted in diamond by delta-localized sour-

ces, created by laser-irradiation of plane




149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

system. As a consequence, there are no critical angles for the

emission of this wave in diamond and its directivity patterns

are poorly structured (Fig. 5). To the structuring of the direc-

tivity patterns of the shear waves emitted in diamond contrib-

ute a single critical angle ht2=l2 ¼ arcsinðtt2=tl2Þ � 46:2 and

the angle of 36.7 , at which the reflection of the compression/

dilatation waves with mode conversion into shear waves is

absent (R22lt ¼ 0).

The results presented in Figs. 2 and 4–6 provide the

complete description of the directivity patterns of laser ultra-

sound emitted by laser-induced sources delta-localized on

Fe/C interface. The influence on the directivity patterns of

the laser pulse duration and focusing is analyzed in Sec. VI.

VI. PULSE PROFILES OF THE LASER ULTRASOUND

In this section, we first evaluate the profiles of the emit-

ted ultrasound pulses, which could be of interest in the appli-

cations of laser ultrasound to high-pressure research. Then

we analyze the deviations in the directivity patterns of the

laser ultrasound, which are caused by the deviation of the

acoustic sources from the delta-localization.

In the common experimental investigations of the direc-

tivity patterns of laser ultrasound, the acoustic waves are

usually detected either at the free curved surface of a half-

cylinder, when the radiation is focused along the axis of the

cylinder plane cut,8,10,12,15,18 or on the rear surface of the

plate, when the radiation is focused on the front surface.16,17

So the detection is on a mechanically free surface, where the

strain has a minimum, while the displacement has a maxi-

mum. So the detection is done by the optical interferometry

or by beam deflection technique, while in the analytical theo-

ries, it is common to analyze the mechanical displacement

component uL in the compression/dilatation wave, which in

the far field is purely longitudinal, and so this component is

just parallel to the wave propagation direction. The wave,

propagating with shear velocity, is purely transversal in the

far field, and it is common to analyze the displacement com-

ponent uT , which is perpendicular to the wave propagation

direction. As the relations of the mechanical displacement

vector to the acoustic potentials are known,

~u ¼ @/@x� @w@z

� �~ex þ

@/@zþ @w@x

� �~ez

¼ @/@x� @w@z; 0;

@/@zþ @w@x

� �;

then it is straightforward to evaluate that

~uLðx; r; hÞ ¼ �ð2=tlÞðixÞ~/1ðx; r; hÞ;~uTðx; r; hÞ ¼ ð2=ttÞðixÞ~w1ðx; r; hÞ: (18)

In the high pressure laser ultrasonics experiments, the

detection is on the surface strongly loaded by diamond, and

the detection by the reflectometry can be much more efficient

than by the interferometry.2–5 So it would be more useful to

analyze not the displacements, but strains, which are produc-

ing the changes in the optical refractive index of the media. In

the wave propagating at longitudinal sound velocity, the

strain, which can be probed by laser radiation, is equal to one

half of the relative change of the material volume

~gðx; r; hÞ � 1

2div~u ¼ 1

2ð~gxx þ ~gzzÞ ¼

1

2

@~ux

@xþ @~uz

@z

� �

¼ ð1=2t2l ÞðixÞ

2 ~u1ðx; r; hÞ: (19)

In the wave propagating at shear acoustic velocity, the strain

component which can be probed by laser radiation, is

~gxzðx; r; hÞ ¼1

2

@~ux

@zþ @~uz

@x

� �

¼ ð1=2t2t Þcosð2hÞðixÞ2 ~w1ðx; r; hÞ: (20)

The solutions in Eqs. (18)–(20) demonstrate that the directiv-

ity patterns of laser ultrasound could differ depending on the

measured physical parameter. From the solutions in Eqs.

(19) and (20) and in Eqs. (14) and (15) it follows that for the

applications in high pressure experiments, it would be inter-

esting to evaluate theoretically

~gðx; r; hÞ ¼ � P1I

2t3=2l

ffiffiffiffiffiffiffiffi1

2pr

rjN/ðhÞj

ffiffiffiffixp

~Uðx sin h=tl1Þ

� ~f ðxÞe�ikl1rþip=4þiuN ; (21)

FIG. 6. Amplitude and phase directiv-

ity patterns of shear ultrasound emitted

in diamond by delta-localized sources,

created by laser-irradiation of plane




149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

and

~gxzðx; r; hÞ ¼ �P1I

2t3=2t

ffiffiffiffiffiffiffiffi1

2pr

rjNwðhÞjcosð2hÞ

�ffiffiffiffixp

~Uðx sin h=tt1Þ~f ðxÞe�ikl1rþip=4þiuN :

(22)

Note, that, to derive Eqs. (21) and (22), we have just substi-

tuted Eqs. (14) and (15) into Eqs. (19) and (20), separated

explicitly the amplitudes jNu;wðhÞj and the phases /N of the

functions Nu;wðhÞ and factorized the diminishing of the sig-

nal amplitude characteristic to 2D geometry, / 1=ffiffirp

.

From the forms of Eqs. (21) and (22) it is clear that, when

transforming the solutions into the time domain, i.e., by per-

forming the inverse Fourier transform, the dependence of both

strain profiles on time can be factorized by the same integral

gðs¼ t� r=tÞ ¼ 1

2p

ðþ1�1

ffiffiffiffixp

~Uðx sinh=tÞ~f ðxÞeixsþip=4þiuNdx;

(23)

where s ¼ t� r=t is the retarded time, while t ¼ tl1;t1

and N ¼ N/;w, when evaluating normal and shear strains,

respectively. The integral in Eq. (23) for the description of

laser radiation given in Eq. (11) can be calculated analytically.

It should not be however forgotten that, when integrating over

negative frequencies, the integrand should be modified to pro-

vide finally real valued profile. The result of the integration is

gðsÞ ¼ pdsL

2s3=2a

@

@�s

ffiffiffiffiffij�sj

pe��s2=2 cos

p4þ uN

� �sgnðsÞI1

4�s2=2� ��

þsinp4þ uN

� �I�1

4�s2=2� ��

� pdsL

2s3=2a

@

@�sWð�sÞ

¼ pdsL

2s3=2a

e��s2=2ffiffiffiffiffij�sj

p(

sgnðsÞ �s2I�54

�s2=2� �h

�ð�s2 � 1ÞI�14

�s2=2� �i

sinp4þ uN

� �

þ�s2 I�34

�s2=2� �

� I14

�s2=2� ��

cosp4þ uN

� �)

� pdsL

2s3=2a

Hð�sÞ; (24)

where �s�2s=sa, saðh;sL;d=tÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisL

2þðdsinh=tÞ2q

�ðd=tÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðtsL=dÞ2þsin2h

qis the angle-dependent characteristic du-

ration of the strain pulse and I614;�3

4;�5

4are the modified Bessel

functions. The characteristic duration of the strain pulse

depends on the time of sound propagation across the laser

focus ðd=tÞ, on the laser pulse duration sL, through the non-

dimensional parameter �sL� sLt=d, and on the propagation

direction (on the angle of observation h). The pulses propa-

gating at larger angles to the surface normal are longer in du-

ration. In Fig. 7, the profiles of the compression/dilatation

strain pulses in 2D geometry are presented for different

phases /N possible in Eq. (24).

The profiles of the compression/dilatation strain pulses

in 3D geometry, relevant to the earlier reported experiments

in point-source-point-receiver configuration,2,3 are presented

in Fig. 13 in Appendix A. Our analytical results in Eqs. (24)

and (A5) demonstrate that the strain pulse profiles photo-

generated in 2D and in 3D configurations, although being

relatively similar, are in general different. This is illustrated

for two particular values of the phase in Fig. 8. Note that

earlier this possible difference in the strain pulse profiles

generated by line and point sources had been mentioned for

the case of thermoelastic sound generation at the mechani-

cally free surface of the medium.15

The analytical result for the acoustic strain and displace-

ment profiles in Eq. (24) in combination with the phases uNof the delta-localized sources, which have been evaluated in

Sec. V (see Figs. 2 and 4) for different angles of observation,

are used to calculate the peak-to-peak amplitudes, i.e., maxi-

mum minus minimum values in the pulse profiles, as a

function of the emission direction. This “peak-to-peak

directivities” provide additional angle-dependent modulation

FIG. 7. Normalized strain profiles for acoustical pulses emitted by focusing

2D Gaussian beam of Gaussian laser pulses on the interface between opaque

and transparent media. Profiles are presented for various possible phases of

the directivity pattern of delta-localized sources.

FIG. 8. Comparison of acoustic strain profiles emitted by Gaussian laser

beams and Gaussian laser pulses in 2D and 3D experimental geometries at

two particular phases of the directivity pattern of delta-localized sources.



149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

of the directivities patterns evaluated for the delta-localized

sources in Sec. V (see Figs. 2 and 4). It should be also taken

into account that in comparison with the solutions for the

particle displacements in Eq. (18) and the solution for the

compression/dilatation strain in Eqs. (19) and (21), the solu-

tion for the shear strain in Eqs. (20) and (22) contains an

extra angle-dependent factor cosð2hÞ. The evaluated angle-

dependent functions, providing modulation of the directivity

patterns of laser ultrasound, emitted by delta-localized sour-

ces and described by the amplitudes of the solutions in Eqs.

(16) and (17), are presented for the 2D case in Fig. 9. They

are evaluated for the following values of the parameters,

sL ¼ 0:5 ns, d ¼ 5 lm, and tl ¼ 5900 m/s, tt ¼ 3200 m/s,

�sL;l � sLtl=d ¼ 0:59, �sL;t � sLtt=d ¼ 0:32, which are char-

acteristic to the earlier reported LU-DAC experiments in

line-source-point-receiver configuration with Fe.4,5 The

modulation functions for the 3D case are presented in Fig.

14 of Appendix A. It can be concluded that a part of the zero

in modulation function of the shear strain, which is a formal

effect related to the definition of the shear strain in Eq. (20),

the modulation functions in Fig. 9 are smooth and do not pro-

vide additional qualitative structuring of the directivity pat-

terns. However this conclusion, based on the analysis of

Fig. 9, is valid only because the characteristic non-

dimensional parameter, i.e., �sL;l ¼ 0:59, is rather close to 1.

The dependence of the modulation functions on angle could

be much more important if the characteristic non-dimensional

parameter �sL � tsL=d is significantly smaller than 1, while,

when �sL � 1, the dependence of the strain pulse amplitude

on the angle is negligible.

To illustrate this, we present in Fig. 10 the modulation

functions for longitudinal strain and displacement directiv-

ities evaluated at different values of the non-dimensional pa-

rameter �sL;l. It can be seen that for �sL;l ¼ 5:9, the modulation

of the compression/dilatation strain/displacement directivity

due to pulse shape effects is practically negligible and thus,

the directivity of laser ultrasound practically equals the

directivity of delta-localized source. For �sL;l ¼ 0:059, the

modulation functions of compression/dilatation strain/dis-

placement in Fig. 10 significantly support the emission at

angles smaller than 10 –20 and suppress the emission at

larger angles. This results in the expectation of drastic differ-

ences between the experimental directivity patterns of laser

ultrasound and the directivities of delta-localized sources. The

side lobe in longitudinal wave emission, which is predicted

around the angle of �45 in Fig. 2, will be suppressed.

In Fig. 11, we presented the modulation functions for

shear strain/displacement directivities evaluated at different

values of the non-dimensional parameter �sL;l. It can be seen

that for �sL;t ¼ 0:032, the modulations of the compression/dila-

tation strain/displacement directivity due to pulse shape effects

suppress the second lobe in the shear wave emission, which is

predicted around �35 in Fig. 4 for delta-localized sources. As

the first lobe in the directivity of shear waves emission by

FIG. 9. Additional modulations of delta-localized sources directivities, which

are caused by finite duration of the laser pulse and finite width of the laser focus,

for emission of acoustic strain and acoustic displacements in 2D geometry. The

modulation functions for compression/dilatation waves are evaluated for non-

dimensional parameter �sL;l � sLtl=d ¼ 0:59. The modulation functions for

shear waves are evaluated for non-dimensional parameter �sL;t � sLtt=d ¼ 0:32.

FIG. 10. Additional modulations of delta-localized sources directivities,

which are caused by finite duration of the laser pulse and finite width of the

laser focus, for acoustic strain (continuous curves) and acoustic displace-

ment (dashed curves) in compression/dilatation wave. The modulation func-

tions are evaluated for few characteristic values of non-dimensional

parameter �sL;l � sLtl=d.



laser focus, for acoustic strain (continuous curves) and acoustic displace-

ment (dashed curves) in shear wave. The modulation functions are evaluated

for few characteristic values of non-dimensional parameter �sL;t � sLtt=d.



149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

delta-localized sources is also expected at non-zero angle from

symmetry considerations (see Sec. V) and is around �10

angle in Fig. 4, then the pulse shape related modulation func-

tion significantly suppresses this first shear lobe as well.

From the physics point of view, these theoretical predic-

tions correspond to the transition, with diminishing parameter

�sL, from delta-localized sources to plane sources, accompa-

nied by preferential photo-generation of plane compression/

dilatation waves propagating normally to the laser-irradiated

surface, and to suppression of shear waves photo-generation.

The above developed theory provides opportunity to evaluate

the directivity patterns of LU for any intermediate situations

between the emission of strongly divergent acoustic fields and

the emission of the acoustic beams, which are quasi-collinear

with the normal to the interface.

VII. DISCUSSION

The interpretations of the multiple lobes in the ampli-

tude directivity patterns had been proposed in Sec. V on the

basis of the evaluation of the critical angles and the possible

angles for the absence in reflection/transmission of the acous-

tic waves across the interface. These interpretations indicate

that the theoretically predicted in Figs. 2 and 4 directivities of

the delta-localized thermo-elastic stresses can be understood

qualitatively from the physics point of view. In addition,

we demonstrate in Appendix B that the developed theory

perfectly reproduces the well-known results12,14,15 for the di-

rectivity patterns of the laser ultrasound emitted from the

laser-irradiated mechanically free surface of the media.

In the absence of the available experimental measure-

ments for comparison, we have compared the predictions of

our analytical theory with the recently published results of

the numerical evaluation of the directivities patterns in dia-

mond anvil cell by the finite element method.19 Because in

Ref. 19 the interface between the aluminum (Al) and the dia-

mond is considered, we have applied our theory to this com-

bination of the materials. The directivity patterns for the

delta-localized sources, presented in Fig. 12, are evaluated

for the similar material parameters as in Ref. 19. They

should be compared with the directivity patterns presented in

Figs. 5(c) and 5(d) in Ref. 19. By comparing the results in

Fig. 12 with those in Figs. 2 and 4, it can be concluded that

the main theoretical predictions for the interface Al/Cd and

the interface Fe/Cd are qualitatively similar, as it could have

been expected from the absence of important difference

between Al and Fe in values of acoustic velocities, Poisson

ratios and critical angles on the interface with diamond (see

Tables I and II), and consist in the description of structuring

of the directivity patterns. The analytical theory predicts sev-

eral lobes in the directivity patterns both for the longitudinal

and shear waves. These analytical predictions are not sup-

ported by the results of the numerical evaluations presented

in Figs. 5(c) and 5(d) of Ref. 19, where both directivity pat-

terns are poorly structured. In particular, numerical investi-

gation19 does not predict the dominant lobe in the directivity

pattern of the longitudinal waves around �45 , which is ana-

lytically predicted in Fig. 12. It should be mentioned here

that we have evaluated the pulse-shape effects, predicted in

Sec. VI using the same parameters of the laser radiation as

in Ref. 19, i.e., sL ¼ 0:5 ns, d ¼ 5 lm. We have found

that the additional angle-dependent modulation, i.e., due to

pulse-shape effects, cannot cut the side lobe predicted in

Fig. 12 for longitudinal waves around �45 . Thus, our ana-

lytical results are in contradiction with the numerical theory

in Ref. 19. However, we are not considering the predictions

obtained in Ref. 19 as a strong argument against our analyti-

cal theory, because some of the other numerical evaluations

in Ref. 19 are also in contradiction and not only with the

results of the analytical theory but also with the available

experimental results. In fact, the numerically evaluated direc-

tivity patterns for the emission of the shear acoustic waves

from the laser-irradiated mechanically free surface, presented

in Fig. 5(b) of Ref. 19, do not contain the second side lobe at

large angles exceeding 60 (see Fig. 15 of Appendix B),

which is not only well-established by analytical theories12,14,15

but has been recently observed experimentally.18

Finally, one more, although indirect, argument in favor of

validity of the theory, developed by us in this manuscript, is the

known theoretical prediction34 for the modification of the direc-

tivity pattern of the laser ultrasound emitted in liquid in case,

when the surface of the liquid is loaded by a solid plate.

Additional structuring of the directivity pattern, relative to the

pattern of laser ultrasound emitted from a mechanically free sur-

face of laser-irradiated liquid, had been predicted.34 Several

lobes in the directivity pattern are theoretically expected.34

VIII. CONCLUSIONS

We presented the analytical descriptions for the directiv-

ity patterns of ultrasound (LU), emitted from the laser-

irradiated interface between two isotropic solids. The

solutions are valid for arbitrary combinations of transparent

and opaque materials. The directivity patterns are derived by

accounting for the specific features of the sound generation

by the thermo-elastic stresses distributed in the volume,

which are essential for laser ultrasonics. We also presented

FIG. 12. Amplitude directivity patterns of longitudinal and shear ultrasound

emitted in aluminum by delta-localized sources, created by laser-irradiation

of plane interface between aluminum and diamond.



149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

the analytical solutions for the profiles of longitudinal and

shear acoustic pulses emitted in different directions. The

derived mathematical formulas provide straight opportunity

to predict the acoustic field, which is formed in the DAC af-

ter photo-generation and several reflections of bulk acoustic

waves at the interfaces. The developed theory can be

applied in the future for the dimensional scaling the LU-

DAC experiments, optimization of these experiments (in

terms of a choice of the optimal directions/positions for

acoustic waves detection) and the interpretation of the ex-

perimental results.

ACKNOWLEDGMENTS

This research was conducted in the frame of the project

“LUDACISM” supported by the program ANR BLANC

2011.

APPENDIX A: DIRECTIVITY PATTERNS IN THE 3D CASE

The solution for the directivity pattern in the point-

source configuration can be derived by an approach which

is similar to the one shown in the main text for the 2D case.

The spatial Fourier transform along the x-axis in 2D is

replaced in the case of 3D and cylindrical symmetry by the

Hankel (Fourier-Bessel) transform along the transverse ra-

dial coordinate r? in the (x,y) plane. The only modification

in the solutions for the acoustic potentials and the tempera-

ture field in the reciprocal spaces is the replacement of kx

by k?, which is the projection of the wave-vectors on the

(x,y) plane. Equation (5) is replaced by the inverse Hankel

transform over k?, which can be calculated by the method

of steepest descent.26,27 This results in the following form

of Eq. (6) in the 3D case

~/1ðx; r; hÞ~w1ðx; r; hÞ

0@

1A ¼ cosh

kl1~~/1ðx; kl1 sin hÞ

kt1~~w1ðx; kt1 sin hÞ

0B@

1CA

� i

r

e�ikl1r

e�ikt1r

0@

1A: (A1)

So just the multiplier,ffiffiffiffiffiffiffiffiffiikl;t1

q=ð2prÞ, characteristic to

2D geometry, is replaced in 3D cylindrical geometry by

ikl;t1=r. This provides correct diminishing of the acoustic

wave amplitude with distance from the source, / 1=r,

and modifies the profiles of the emitted acoustic pulses.

The description of the normal strain in Eq. (21) is modified

into

~gðx;r;hÞ¼�P1I

2t2l

1

rjN/ðhÞjx~Uðxsinh=tl1Þ~f ðxÞe�ikl1rþip=2þiuN ;

(A2)

and the description of the shear strain in Eq. (22) is modified

into

~gr?zðx; r; hÞ ¼1

2

@~ur?

@zþ @~uz

@r?

� �

¼ �P1I

2t2t

1

rjNwðhÞjcosð2hÞx~Uðx sin h=tt1Þ~f ðxÞ

� e�ikl1rþip=2þiuN : (A3)

The dependence of both strain profiles on time can be factor-

ized by the same integral

gðs ¼ t� r=tÞ ¼ 1

2p

ðþ1�1

x~Uðx sin h=tÞ~f ðxÞeixsþip=2þiuNdx:

(A4)

The integral in Eq. (A4), in case, where the description of

laser radiation is given in Eq. (11), can be calculated analyti-

cally. It should not be forgotten, however, that, when inte-

grating over negative frequencies, the integrand should be

modified to provide finally real valued profile. The result of

the integration is

gðsÞ¼ffiffiffipp

dsL

s2a

@

@�se��s2

cosuNþ ierf i�sð ÞsinuN

�n o

�ffiffiffipp

dsL

s2a

@

@�sWð�sÞ

¼ffiffiffipp

dsL

s2a

� 2ffiffiffipp sinuN�2�se��s2

cosuN�erfið�sÞsinuN½ �

�ffiffiffipp

dsL

s2a

Hð�sÞ; (A5)

where all the notations are the same as in the 2D case analyzed

in Sec. VI and erf is the error function (probability integral).

The evaluated (with the use of Eq. (A5)) profiles of the com-

pression/dilatation strain pulses in 3D geometry, relevant to

the earlier reported experiments in point-source-point-receiver

configuration,2,3 are presented in Fig. 13. The angle-dependent

functions, providing modulation of the directivity patterns of

laser ultrasound, emitted by delta-localized sources and

described by the amplitudes of the solutions in Eqs. (16) and

(17), are presented for the 3D case in Fig. 14. They are eval-

uated for the same parameters of material and laser pulses as

in the 2D case in Sec. VI.

APPENDIX B: DIRECTIVITY PATTERNS OF LASERULTRASOUND EMITTED FROM MECHANICALLY FREEAND RIGID INTERFACES

The directivity patterns of laser ultrasound emitted due

to laser-irradiation of the mechanically free surface of iron,

which are computed as an asymptotic case of the general for-

mulas in Eqs. (16) and (17), are presented in Fig. 15.

On the one hand, the comparison of Fig. 15 with the

well-documented results existing in the literature for

the mechanically free surface,12,14,15 confirms the validity of

our analytical theory in this limiting case. On the other hand,

the results presented in the right column in Fig. 15 clearly



149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

illustrate in this simplest case, the role played in the structur-

ing of the directivity patterns by the critical angles and the

zeros in the transmission/reflection coefficients, which is

revealed in Sec. V. The maximum in the directivity of the

amplitudes of the shear waves is near the critical angle

ht1=l1 ¼ arcsinðtt1=tl1Þ � 32:8 , which manifests itself also

in the directivity pattern of the phase. Above this critical

angle, the shear acoustic waves are emitted in Fe due to the

mode conversion in reflection from the interface of the evan-

escent compression/dilatation waves generated by laser radi-

ation in Fe. Shear waves become the only wave transporting

the acoustic energy from the interface. The splitting of the

directivity pattern for the amplitudes of the shear waves into

two lobes is caused by the absence of mode conversion of

the compression/dilatation waves in reflection from a

mechanically free surface at the angle of 45 (R11lt ¼ 0). This

zero reflectivity manifests itself as 180 phase jump in the di-

rectivity pattern of phase. The directivity patterns of the

shear ultrasound emitted from mechanically free surface

confirm the following general statement formulated in

Sec. V. If the zero in reflection/transmission coefficient for

compression/dilatation wave is achieved at angles larger

than the largest critical angle, then the local maximum in the

amplitude directivity pattern could be achieved between this

zero reflection/transmission angle and 90 , and not between

the largest critical angle and 90 .The directivity patterns of the laser ultrasound emitted

due to laser-irradiation of the surface of iron in contact with

absolutely rigid medium (2) are presented in Fig. 16. They are

computed by significantly increasing the sound velocities and

the shear rigidity in medium (2) relative to the corresponding

values in diamond in Eqs. (16) and (17). We have also derived



laser focus, for emission of acoustic strain and acoustic displacements in 3D

geometry. The modulation functions for compression/dilatation waves are

evaluated for non-dimensional parameter �sL;l � sLtl=d ¼ 0:59. The modula-

tion functions for shear waves are evaluated for non-dimensional parameter

�sL;t � sLtt=d ¼ 0:32.

FIG. 15. Amplitude and phase directivity patterns for longitudinal and shear

ultrasound emitted by delta-localized sources, created by laser-irradiation of

mechanically free surface of iron.

FIG. 13. Normalized strain profiles for acoustical pulses emitted by focusing

3D Gaussian beam of Gaussian laser pulses on the interface between opaque

and transparent media. Profiles are presented for various possible phases of

the directivity pattern of delta-localized sources.

FIG. 16. Amplitude and phase directivity patterns for longitudinal and shear

ultrasound emitted by delta-localized sources, created by laser-irradiation of

interface between an infinitely rigid transparent solid and light-absorbing

elastic solid.



149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

the analytical formulas for the directivity patterns of laser

ultrasound in Fe, assuming the surface of Fe is immobile (sur-

face boundary conditions of zero mechanical displacements),

N/ðhÞ � 1þ R11ll ðhÞ ¼

2 cos hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðtl1=tt1Þ2 � sin2 h

qsin2 hþ cos h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðtl1=tt1Þ2 � sin2 h

q ;

(B1)

NwðhÞ �cos hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


lt ðhÞ

¼ 2 cos h sin h

sin2 hþ cos hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðtt1=tl1Þ2 � sin2 h

q : (B2)

On the one hand, the comparison of results in Fig. 16

with those which can be obtained directly from Eqs. (B1) and

(B2), confirms the validity of our analytical theory in this lim-

iting case. On the other hand, the results presented in the right

column in Fig. B2 clearly illustrate in this simplest case the

role played in the structuring of the directivity patterns by the

critical angles and the zeros in the transmission/reflection

coefficients, revealed in Sec. V. The maximum in the directiv-

ity of the amplitudes of the shear waves is near the critical

angle ht1=l1 ¼ arcsinðtt1=tl1Þ � 32:8 , which manifests itself

also in the directivity pattern of the phase. Above this critical

angle, shear acoustic waves become the only wave transport-

ing the acoustic energy from the interface. The situation is

very similar to the case of mechanically free surface, because

the same single critical angle exists in both limiting cases.

However in the case of absolutely rigid interface, there is no

splitting of the directivity pattern for the amplitudes of the

shear waves into two lobes, because there is always nonzero

mode conversion into shear waves of the compression/dilata-

tion waves in reflection (R11lt 6¼ 0 when 0 < ht1 < 90 ).

1F. Decremps, L. Belliard, B. Perrin, and M. Gauthier, Phys. Rev. Lett.

100, 035502 (2008).2N. Chigarev, P. Zinin, L.-C. Ming, G. Amulele, A. Bulou, and V. Gusev,

Appl. Phys. Lett. 93, 181905 (2008).

3P. V. Zinin, N. Chigarev, D. Mounier, A. Bulou, L. C. Ming, T. Acosta,

and V. Gusev, J. Phys.: Conf. Ser. 215, 012053 (2010).4N. Chigarev, P. Zinin, D. Mounier, A. Bulou, L. C. Ming, T. Acosta, and

V. Gusev, High Pressure Res. 30, 78 (2010).5N. Chigarev, P. Zinin, D. Mounier, A. Bulou, A. Zerr, L. C. Ming, and V.

Gusev, J. Phys.: Conf. Ser. 278, 012017 (2011).6C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, J. Appl.

Phys. 51, 6210 (1980).7C. B. Scruby and L. E. Drain, Laser Ultrasonics Techniques andApplications (Hilger, Bristol, 1990).

8D. A. Hutchins, R. J. Dewhurst, and S. B. Palmer, Ultrasonics 19, 103

(1981); J. Acoust. Soc. Am. 70, 1362 (1981).9L. F. Rose, J. Acoust. Soc. Am. 75, 723 (1984).

10J. D. Aussel, A. Le Brun, and J. C. Baboux, Ultrasonics 26, 245 (1988).11S. Y. Zhang, M. Paul, S. Fassbender, U. Schleichert, and W. Arnold, Res.

Nondestr. Eval. 2, 143 (1990).12P. Zhang, C. F. Ying, and J. Shen, Ultrasonics 35, 233 (1997).13B. Yuhai, P. Liwei, F. Qiping, Y. Yuping, and Y. Chongfu, Ultrasonics 33,

429 (1995).14W. Japing, S. Dufang, and H. Yulong, J. Appl. Phys. 83, 1207 (1998).15J. R. Bernstein and J. B. Spicer, J. Acoust. Soc. Am. 107, 1352 (2000).16S. Raetz, T. Dehoux, and B. Audoin, J. Acoust. Soc. Am. 130, 3691

(2011).17J. M. S. Sakamoto, B. R. Tittman, A. Baba, and G. M. Pacheco, J. Phys.:

Conf. Ser. 278, 012032 (2011).18C. Pei, K. Demachi, H. Zhu, T. Fukuchi, K. Koyama, and M. Uesaka, Opt.

Laser Technol. 44, 860 (2012).19W. Feng, D. Yang, X. Zhu, Y. Guo, and W. Liao, J. Appl. Phys. 111,

013107 (2012).20V. Gusev, C. Desmet, W. Lauriks, C. Glorieux, and J. Thoen, J. Acoust.

Soc. Am. 100, 1514 (1996).21M. Kouyate, T. Pezeril, D. Mounier, and V. Gusev, J. Appl. Phys. 110,

123526 (2011).22L. M. Brekhovskikh and O. A. Godin, Acoustics of Layered Media I:

Plane and Quasi-Plane Waves (Berlin, Springer-Verlag, 1990).23V. E. Gusev and A. Karabutov, Laser Optoacoustics (AIP, New York, 1993).24B. A. Auld, Acoustic Fields and Waves in Solids, 2nd ed. (Krieger,

Malabar, 1990), Vol. 2.25V. E. Gusev, Appl. Phys. Lett. 94, 164105 (2009).26W. M. Ewing, W. S. Jardetzky, and F. Press, Elastic Waves in Layered

Media (McGraw-Hill, New York, 1957).27G. F. Miller and H. Purcey, Proc. R. Soc. London, Ser. A 223, 521 (1954).28V. E. Gusev, Opt. Commun. 94, 76 (1992).29G. Tas and H. Maris, Phys. Rev. B 49, 15046 (1994).30V. E. Gusev and O. B. Wright, Phys. Rev. B 57, 2878–2888 (1998).31http://www.crewes.org/ResearchLinks/ExplorerPrograms/ZE/ZEcrewes.html.32T. Pezeril, N. Chigarev, P. Ruello, S. Gougeon, D. Mounier, J.-M.

Breteau, P. Picart, and V. Gusev, Phys. Rev. B 73, 132301 (2006).33T. Pezeril, P. Ruello, S. Gougeon, N. Chigarev, D. Mounier, J.-M.

Breteau, P. Picart, and V. Gusev, Phys. Rev. B 75, 174307 (2007).34L. M. Lyamshev, Sov. Phys. Acoustics 25, 322 (1979) [Akust. Zh. 25, 566

(1979)].



149.150.51.237 On: Mon, 15 Sep 2014 22:50:52

http://dx.doi.org/10.1103/PhysRevLett.100.035502

http://dx.doi.org/10.1063/1.3013587

http://dx.doi.org/10.1088/1742-6596/215/1/012053

http://dx.doi.org/10.1080/08957950903549519

http://dx.doi.org/10.1088/1742-6596/278/1/012017

http://dx.doi.org/10.1063/1.327601

http://dx.doi.org/10.1063/1.327601

http://dx.doi.org/10.1016/0041-624X(81)90078-0

http://dx.doi.org/10.1121/1.387126

http://dx.doi.org/10.1121/1.390583

http://dx.doi.org/10.1016/0041-624X(88)90013-3

http://dx.doi.org/10.1080/09349849009409494

http://dx.doi.org/10.1080/09349849009409494

http://dx.doi.org/10.1016/S0041-624X(96)00106-0

http://dx.doi.org/10.1016/0041-624X(95)00057-A

http://dx.doi.org/10.1063/1.366817

http://dx.doi.org/10.1121/1.428422

http://dx.doi.org/10.1121/1.3658384

http://dx.doi.org/10.1088/1742-6596/278/1/012032

http://dx.doi.org/10.1088/1742-6596/278/1/012032

http://dx.doi.org/10.1016/j.optlastec.2011.11.018

http://dx.doi.org/10.1016/j.optlastec.2011.11.018

http://dx.doi.org/10.1063/1.3676253

http://dx.doi.org/10.1121/1.416021

http://dx.doi.org/10.1121/1.416021

http://dx.doi.org/10.1063/1.3662921

http://dx.doi.org/10.1063/1.3125243

http://dx.doi.org/10.1098/rspa.1954.0134

http://dx.doi.org/10.1016/0030-4018(92)90409-K

http://dx.doi.org/10.1103/PhysRevB.49.15046


http://www.crewes.org/ResearchLinks/ExplorerPrograms/ZE/ZEcrewes.html



Documents

Directivity patterns and pulse profiles of ultrasound emitted by laser action on interface between transparent and opaque solids: Analytical theory