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

(1)

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

Sergey 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

(2)

Directivity patterns and pulse profiles of ultrasound emitted by laser action on 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 Gusev

^1,a)

1

LAUM, UMR-CNRS 6613, Universit e du Maine, 72085 Le Mans, France

2

IMMM, UMR-CNRS 6283, Universit e du Maine, 72085 Le Mans, France

3

LSPM, 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. V

^C

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

solid

^13,14

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/$30.00 115, 044902-1

VC

2014 AIP Publishing LLC

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II. GENERAL THEORETICAL DESCRIPTION OF THE LASER 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

~ ~

/

₁

ðx; k

x

; zÞ ¼ i 2a

1

~ ~

r ~

ⁱ_n1

ðx; k

x

; a

1

Þ þ R

¹¹_ll

r ~ ~ ~

ⁱ_n1

ðx; k

x

; a

1

Þ

þ a

1

a

2

T

_ll²¹

r ~ ~ ~

ⁱ_n2

ðx; k

x

; a

2

Þ

e

^ia¹^z

/ ~ ~

₁

ðx; k

x

Þe

^ia¹^z

; (1)

~ ~

w

₁

ðx; k

x

; zÞ ¼ i 2a

1

R

¹¹_lt

r ~ ~ ~

ⁱ_n1

ðx; k

x

; a

1

Þ

þ a

1

a

2

T

²¹_lt

r ~ ~ ~

ⁱ_n2

ðx; k

x

; a

2

Þ

e

^ib¹^z

w ~ ~

₁

ðx; k

x

Þe

^ib¹^z

: (2)

The Fourier transforms for the derivation and manipulation of the solutions in Eqs. (1) and (2) are defined by

~ ~

Fðx; ~ k

x

; k

z

Þ ¼ ð

1

ð

1

ð

1

Fðt; x; zÞe

^iðxtk^x^xk^z^zÞ

dtdxdz;

Fðt; x; zÞ ¼ 1 ð2pÞ

³

ð

1

ð

1

ð

1

~ ~

Fðx; ~ k

x

; k

z

Þe

^iðxtk^x^xk^z^zÞ

dxdk

x

dk

z

: (3) In Eqs. (1) and (2), x is the cyclic frequency of the laser- induced normalized stress fields r

ⁱ_n1;2

and of the acoustic fields, k

x

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 r

ⁱ_kl

is isotropic in iso- tropic and cubic solids, i.e., r

ⁱ_kl

r

ⁱ

d

kl

. Note that x-components of the wave vectors are equal for longitudinal and shear waves and also in both media

^22,24

(see Fig. 1), while the z-components are different. i.e., k

zl1;2

a

1;2

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k

_l1;2²

k

_x²

q

and k

zt1;2

b

_1;2

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k

²_t1;2

k

²_x

q , where sgnðRea

1;2

Þ ¼ sgnðReb

_1;2

Þ

¼ sgnðxÞ for real a

1;2

and b

_1;2

and sgnðIma

1;2

Þ ¼ sgnðImb

_1;2

Þ

¼ 1 for imaginary a

1;2

and b

1;2

. Here, x=t

l1;2

k

l1;2

and x=t

t1;2

k

t1;2

are the wave numbers of the compression/dila- tion and shear acoustic waves in medium “1” and medium “2,”

with t

l1;2

and t

t1;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, r

ⁱ_n1;2

¼ r

ⁱ_1;2

= ðq

_1;2

t

²_l1;2

Þ, where q

_1;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 D

l1,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.

(4)

interaction with the interface. This wave is denoted in the right-hand-side of Fig. 1 by D

l1

. 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 (R

¹¹_ll

is the reflectivity coefficient). This wave is denoted in Fig. 1 by R

¹¹_ll

. The third term describes the waves that are generated in the second medium and transmitted into the first medium across the interface (T

_ll²¹

is the transmission coefficient). This wave is denoted in Fig. 1 by T

_ll²¹

. 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 D

l1,

in reflection from the interface (R

¹¹_lt

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 R

¹¹_lt

. 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 D

l2

in trans- mission across the interface (T

²¹_lt

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 T

²¹_lt

. 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 R

¹¹_lt

, R

¹¹_ll

, T

¹²_ll

and T

_lt¹²

of acoustic potentials was reminded. The resultant system is

b

₁

b

₂

k

x

k

x

k

x

k

x

a

1

a

2

b

₁

l

₂₁

b

₂

c

1

l

₂₁

c

₂

c

₁

l

21

c

₂

a

1

l

₂₁

a

2

0 B B B B

@

1 C C C C A

R

¹¹_lt

T

_lt¹²

R

¹¹_ll

T

_ll¹²

0 B B B B B @

1 C C C C C A

¼ k

x

a

1

c

₁

a

1

0 B B B B

@ 1 C C C C A ; (4)

where the compact notations l

₂₁

l

₂

=l

₁

and c

_1;2

k

x

k

²_t1;2

=ð2k

x

Þ are introduced, l

_1;2

are the second Lame con- stants (shear moduli). The transmission coefficients T

²¹_lt

and T

_ll²¹

necessary for the evaluation of Eq. (2) can be obtained either from T

¹²_lt

and T

_ll¹²

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

²²

or by solving the following system, which in turn can be derived from Eq. (4) using symmetry principles

b

₂

b

₁

k

x

k

x

k

x

k

x

a

2

a

1

b

₂

l

₁₂

b

₁

c

₂

l

₁₂

c

₁

c

₂

l

₁₂

c

₁

a

2

l

₁₂

a

1

0 B B B B

@

1 C C C C A

R

²²_lt

T

_lt²¹

R

²²_ll

T

_ll²¹

0 B B B B B @

1 C C C C C A

¼ k

x

a

2

c

₂

a

2

0 B B B B

@ 1 C C C C A : (5)

Thus, the general solution for the laser-generated acoustic field in medium “1” in the (x; k

x

; 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 k

x

:

/ ~

₁

ðx; x; zÞ w ~

₁

ðx; x ; zÞ

!

¼ 1 2p

ð

þ1

1

~ ~

/

₁

ðx; k

x

Þe

^ia¹^z

~ ~

w

₁

ðx; k

x

Þe

^ib¹^z

0 @

1 A

e

^ik^x^x

dk

x

: (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 k

x

. 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 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x

²

þ z

²

p from their excitation region on the angle of emission h, i.e., the angle between the direction of observation and the z-axis, are

/ ~

₁

ðx; r; hÞ w ~

₁

ðx; r; hÞ

!

¼ cos h ffiffiffiffiffiffi k

l1

p / ~ ~

₁

ðx; k

l1

sin hÞ ffiffiffiffiffiffi

k

t1

p w ~ ~

₁

ðx; k

t1

sin hÞ 0

@

1 A

ffiffiffiffiffiffiffiffi

i 2pr r e

^ik^l1^r

e

^ik^t1^r

!

: (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 / ~ ~

₁

ðx; r; hÞ ¼ / ~ ~

₁

ðx; r; hÞ and w ~ ~

₁

ðx; r; hÞ

¼ w ~ ~

₁

ðx; 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 THE LASER-INDUCED THERMO-ELASTIC STRESSES

The spectrum of the normalized thermo-elastic stress is controlled by the spectrum of the laser-induced temperature rise

^21,23

~ ~

r

ⁱ_n1;2

~ ðx; k

x

; k

z

Þ ¼ a

1;2

½ð1 þ

1;2

Þ=ð1

1;2

Þ T ~ ~ ~

1;2⁰

ðx; k

x

; k

z

Þ;

(8) where a, , and T

⁰

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

(5)

inhomogeneous. The laser-induced heat release Q

1

ðt; 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 Q

1

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 W ðzÞ 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 ~

1;2⁰

ðx; k

x

; k

z

Þ H

1;2

ðx; k

_x

; k

z

Þ f ~ ðxÞ Uðk ~

x

Þ: (9) The general expression of the function Hðx; k

_x

; k

z

Þ is

H

1

ðx; k

_x

; k

z

Þ ¼ 1

v

₁

ðk

_z²

d

²₁

Þ W ~ ðk

z

Þ þ v

₁

k

z

v

₂

d

2

v

₁

d

1

þ v

₂

d

2

W ~ ðd

1

Þ

;

H

2

ðx; k

_x

; k

z

Þ ¼ ð1Þ

ðv

₁

d

1

þ v

₂

d

2

Þðk

z

þ d

₂

Þ W ~ ðd

1

Þ:

(10) Here, d

1;2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ix= D

1;2

k

²_x

p can be associated with the

projection of the thermal wave vector ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ix=D

1;2

p on the

z-axis, D

1;2

are the thermal diffusivities and v

_1;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 Q

1

¼ f ðtÞUðxÞW ðzÞ. 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 exp½ð2t=s

L

Þ

²

; UðxÞ

¼ exp½ð2x=dÞ

²

; WðzÞ ¼ ðI=lÞexpðz=lÞ, respectively, where s

L

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 ð ffiffiffi p p

s

L

=2Þexp½ðs

L

x=4Þ

²

; Uðk ~

x

Þ ¼ ð ffiffiffi

p p

d=2Þexp½ðdk

x

=4Þ

²

;

W ~ ðk

z

Þ ¼ ðI=lÞðl

¹

ik

z

Þ

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 s

L

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 W ~ ðk

z

Þ ﬃ W ~ ðd

1

Þ ﬃ I, holds, and consequently the description in Eq. (10) simplifies to

H

1;2

ðx; k

_x

; k

z

Þ ¼ ð61ÞI

ðv

₁

d

1

þ v

₂

d

2

Þðk

z

7 d

_1;2

Þ : (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

H

1;2

ðx; k

_x

; k

z

Þ ¼

ffiffiffiffiffiffiffiffi D

1;2

p I

ðixÞð ffiffiffiffiffi e

1

p þ ffiffiffiffiffi e

2

p Þ : (13) Here, e

1,2

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:

/ ~

₁

ðx;r;hÞ¼P

1

I 1þR

¹¹_ll

ðhÞþP

2=1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh ðt

l1

=t

l2

Þ

²

sin

²

h

q T

_ll²¹

ðhÞ

8 <

:

9 =

; Uðk ~

l1

sin hÞ f ~ ðxÞ

x

ffiffiffiffiffiffiffiffiffiffiffiffiffi i 2pk

l1

r r

e

^ik^l1^r

; (14)

(6)

w ~

₁

ðx; r; hÞ ¼ P

1

I cos h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt

t1

=t

l1

Þ

²

sin

²

h

q R

¹¹_lt

ðhÞ

8 <

: þ P

2=1

cos h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt

t1

=t

l2

Þ

²

sin

²

h

q T

_lt²¹

ðhÞ

9 >

=

> ; Uðk ~

t1

sin hÞ f ~ ðxÞ

x

ffiffiffiffiffiffiffiffiffiffiffiffiffi i 2pk

t1

r r

e

^ik^t1^r

: (15) Here, the compact notation P

1

is introduced for the following combination of physical parameters P

1

^a¹^ð1þ¹^Þ

ffiffiffiffi

_D

1

p 2ð11Þð^p

ffiffiffi

_e₁_þ^p

ffiffiffi

_e₂_Þ

. The parameter P

2=1

^a²^ð1þ²^Þð1¹^Þ

ffiffiffiffi

_D

2

p a1ð1þ1Þð12Þ

ffiffiffiffi

_D

1

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 Uðk ~

x

Þ 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 1 þ R

¹¹_ll

ðhÞ þ P

2=1

cos h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt

l1

=t

l2

Þ

²

sin

²

h

q T

_ll²¹

ðhÞ;

(16) N

w

ðhÞ cos h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt

t1

=t

l1

Þ

²

sin

²

h

q R

¹¹_lt

ðhÞ

þ P

2=1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh ðt

t1

=t

l2

Þ

²

sin

²

h

q T

_lt²¹

ðhÞ: (17)

The role of the other factors in Eqs. (12) and (13), which are angle-dependent, i.e., of Uðk ~

l;t1

sin hÞ / ð ffiffiffi

p p d=2Þ exp½ðdx sin h=4t

l;t1

Þ

²

ð ffiffiffi

p p

d=2Þexp½ðpd sin h=2k

l;t1

Þ

²

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 k

l;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 k

l;t1

) to the plane compression/dilatation wave emis- sion in the case of defocused laser irradiation (d k

l;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 noticed

¹⁵

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 THE DELTA-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 technique

^2–5

), 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 C

_d

, 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 C

_d

), t

t1

< t

l1

< t

t2

< t

l2

. 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., t

t1

< t

l1

< t

t2

< t

l2

, results in particular ordering of the propagation angles of the interact- ing waves h

t1

< h

l1

< h

t2

< h

l2

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.

(7)

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 h

l1=l2

¼ arcsinðt

l1

=t

l2

Þ 19:1

. At angles larger than

h

l1=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 h

l1=t2

¼ arcsinðt

l1

=t

t2

Þ 27:0

. At angles larger than h

l1=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 h

l1=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 h

l1=t2

. At the same time, the emission of the longitudinal waves in Fe should stop at angles above h

l1=l1

¼ arcsinðt

l1

=t

l1

Þ 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 h

l1=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 (m

²

s

¹

) 7.80 10

⁴

0.23 10

⁴

0.84 10

⁴

P

2/1

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 t

_l1

=t

_l2

t

_l1

=t

_t2

t

_t1

=t

_l2

t

_t1

=t

_t2

t

_t1

=t

_l1

t

_t2

=t

_l2

arcsinðt

l1

=t

_l2

Þ arcsinðt

l1

=t

_t2

Þ arcsinðt

t1

=t

_l2

Þ arcsinðt

t1

=t

_t2

Þ arcsinðt

t1

=t

_l1

Þ arcsinðt

t2

=t

_l2

Þ

(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

(8)

directivities of the reflection coefficient R

¹¹_ll

, 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 C

_d

. 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 h

l1=l2

¼ arcsinðt

l1

=t

l2

Þ 19:1

. At angles larger than

h

l1=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 h

l1=t2

¼ arcsinðt

l1

=t

t2

Þ 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 C

_d

side cannot be transformed into propagating longitudinal waves in Fe. It can be seen that this zero in the transmission coefficient T

_ll²¹

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 C

_d

. 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 C

_d

are comparable (see the estimate of the characteristic parameter P

2=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.

(9)

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 h

t1=l2

¼ arcsin ðt

t1

=t

l2

Þ 10:2

. At angles larger than h

t1=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 h

t1=t2

¼ arcsinðt

t1

=t

t2

Þ 14:2

. At angles larger than h

t1=t2

, the shear acoustic waves in diamond are evanescent and they stop to transport acoustic energy from the interface. Thus, at angles larger than h

t1=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 h

t1=l1

¼ arcsin ðt

t1

=t

l1

Þ 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) R

¹¹_lt

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 C

_d

(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) T

_lt²¹

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 C

_d

. 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 C

_d

are comparable (see the estimate of the characteristic parameter P

2=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

interface between iron and diamond.

(10)

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 h

t2=l2

¼ arcsinðt

t2

=t

l2

Þ 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 (R

²²_lt

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 u

L

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 u

T

, 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

~ e

x

þ @/

@z þ @w

@x

~ e

z

¼ @/

@x @w

@z ; 0; @/

@z þ @w

@x

;

then it is straightforward to evaluate that

~

u

L

ðx; r; hÞ ¼ ð2=t

l

ÞðixÞ / ~

₁

ðx; r; hÞ;

~

u

T

ðx; r; hÞ ¼ ð2=t

t

ÞðixÞ w ~

₁

ðx; 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

2 div~ u ¼ 1

2 ð~ g

_xx

þ ~ g

_zz

Þ ¼ 1 2

@ u ~

_x

@x þ @~ u

_z

@z

¼ ð1=2t

²_l

ÞðixÞ

²

u ~

₁

ðx; r ; hÞ : (19) In the wave propagating at shear acoustic velocity, the strain component which can be probed by laser radiation, is

~

g

_xz

ðx; r; hÞ ¼ 1 2

@~ u

x

@z þ @ u ~

z

@x

¼ ð1=2t

²_t

Þcosð2hÞðixÞ

²

w ~

₁

ðx; 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 P

1

I 2t

³⁼²_l

ffiffiffiffiffiffiffiffi 1 2pr r

jN

/

ðhÞj ffiffiffiffi p x

Uðx ~ sin h=t

l1

Þ

f ~ ðxÞe

^ik^l1^rþip=4þiu^N

; (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

interface between iron and diamond.

(11)

and

~

g

_xz

ðx; r; hÞ ¼ P

1

I 2t

³⁼²t

ffiffiffiffiffiffiffiffi 1 2pr r

jN

w

ðhÞjcosð2hÞ ffiffiffiffi

p x

Uðx ~ sin h=t

t1

Þ f ~ ðxÞe

^ik^l1^rþip=4þiu^N

: (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 jN

u;w

ðhÞj and the phases /

_N

of the functions N

u;w

ðhÞ and factorized the diminishing of the sig- nal amplitude characteristic to 2D geometry, / 1= ﬃﬃ

p r . 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

gðs ¼ t r=tÞ ¼ 1 2p

ð

þ1

1

ffiffiffiffi x

p Uðx ~ sin h=tÞ f ~ ðxÞe

ixsþip=4þiu_N

dx;

(23) where s ¼ t r=t is the retarded time, while t ¼ t

l1;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 pds

L

2s

³⁼²a

@

@ s ffiffiffiffiffi j sj

p e

^s²⁼²

cos p 4 þ u

_N

sgnðsÞI

¹

4

s

²

=2

þsin p 4 þ u

_N

I

¹

4

s

²

=2 pds

L

2s

³⁼²a

@

@ s Wð sÞ

¼ pds

L

2s

³⁼²a

e

^s²⁼²

ffiffiffiffiffi j sj p

(

sgnðsÞ s

²

I

⁵

4

s

²

=2 h

ð s

²

1ÞI

¹

4

s

²

=2 i sin p

4 þ u

_N

þ s

²

I

³

4

s

²

=2 I

¹

4

s

²

=2

cos p 4 þ u

_N

)

pds

L

2s

³⁼²a

Hð sÞ ; (24)

where s 2s=s

a

, s

a

ðh;s

L

;d=tÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s

L2

þ ðdsin h=tÞ

²

q

ðd=tÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðts

L

=dÞ

²

þsin

²

h q

is the angle-dependent characteristic du- ration of the strain pulse and I

6¹₄;³₄;⁵₄

are 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 s

L

, through the non- dimensional parameter s

L

s

L

t=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 u

_N

of 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.

(12)

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, s

L

¼ 0:5 ns, d ¼ 5 lm, and t

l

¼ 5900 m/s, t

t

¼ 3200 m/s, s

L;l

s

L

t

l

=d ¼ 0:59, s

L;t

s

L

t

=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., s

L;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 s

L

ts

L

=d is significantly smaller than 1, while, when s

L

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 s

L;l

. It can be seen that for s

L;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 s

L;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 s

L;l

. It can be seen that for s

L;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 s

L;l

s

L

t

l

=d ¼ 0:59. The modulation functions for shear waves are evaluated for non-dimensional parameter s

_L;t

s

_L

t

_t

=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 s

_L;l

s

_L

t

_l

=d.

FIG. 11. 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 shear wave. The modulation functions are evaluated

for few characteristic values of non-dimensional parameter s

_L;t

s

_L

t

_t

=d.

(13)

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 s

L

, 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 results

^12,14,15

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/C

_d

and the interface Fe/C

_d

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- gation

¹⁹

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., s

L

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 theories

^12,14,15

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 prediction

³⁴

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