laser ultrasonics for material characterization and defect...

Laser ultrasonics for material characterization and defect detection

Christ GlorieuxLaboratory of Acoustics

Soft Matter and Biophysics

Department of Physics and Astronomy

KU Leuven, Belgium

[email protected]

Alexander Graham Bell & Charles Sumner, 1880Godfather of photoacoustics

Photothermal and photoacoustic phenomena⇒ optical, thermal and elastic properties

photothermal effect

T

z

sinusoidal SAW

gaussian SAW burst SAW

photoacoustic effect

µ

⇒ information on transport properties: ⇒ thermal diffusivity/diffusion length& acoustic velocity and damping/wavelength

λ

Photoacoustic tomography: optical information

Overview

• Laser ultrasonics in layered samples: elastic/thermal depth information from dispersion

• Optical detection schemes• Michelson interferometer• Phase mask interferometer• Sagnac interferometer• Laser beam deflection • Speckle knife edge detection (SKED)• Laser Doppler vibrometry• Photorefractive interferometer• Modulated optical reflection• Brillouin oscillations

• Laser ultrasonics for material characterization: case studies• Calculation of guided wave dispersion and photothermal and photoacoustic displacements

• Laser ultrasonics for defect detection and application for non-destructive testing

Laser ultrasonics on layered samples:

Elastic/thermal depth information from dispersion

vR1

vR2

vR3

Rayleigh waves: wavelength dependent penetration depth

Laser ultrasonics on layered samples: elastic information from guided wave velocity dispersion

• Surface acoustic waves

– Penetration depth ~λ

– Multilayers: dispersion

coating

substrate

substrate

coating

Dispersion curve:

Laser ultrasonics on layered samples: elastic information from guided wave velocity dispersion


0.1 1 10 100 1000

2000

3000

4000

5000

6000

7000

8000

9000

cT,Sn = 1586 m.s-1 cR,Sn = 1485 m.s-1

cL,Si = 8430 m.s-1

cR,Si = 5148 m.s-1

frequency (MHz)

velo

city

(m.s

-1)

Sn

Cu

Si

CuxSny

Photoacoustic characterization of elastic properties of (sub-)micron sub-surface layers

Multilayer sample

0

1

2

0

1

2

P (a

.u.)

0 2 4 6 8 100

1

2

3

z (µm)RayleighSezawa

5 MHz

50 MHz

240 MHz

0.1 1 10 100 1000

2000

3000

4000

5000

6000

7000

8000

9000

cT,Sn = 1586 m.s-1 cR,Sn = 1485 m.s-1

cL,Si = 8430 m.s-1

cR,Si = 5148 m.s-1

frequency (MHz)

velo

city

(m.s

-1)

Sn

Cu

Si

CuxSny

Photoacoustic characterization of elastic properties of (sub-)micron sub-surface layers


Photoacoustic characterization of elastic properties of free-standing films and plates: extraction of velocity dispersion

0

100

50

0.5

1000 806040200

1

10 20 30 40 50 60 70 80 90 100

20

40

60

80

100

0 10 20 30 40 50 60 70 80 90 100

-5

0

5

( , ) ( )

( ) exp( )

with =

( )

kc

= ( , ) exp( ) 2

S x t S x ct

S k i t ikx dk

ddk S k i t ikk xc

ω

ω

ω ω ωδ ωπ

+∞

−∞

+∞ +∞

−∞ −∞

= −

= −

−−

∫

∫ ∫

Photoacoustic characterization of elastic properties of free-standing films and plates: extraction of velocity dispersion

x

t

t

prob

e-pu

mp

dist

ance

frequency (Hz) 10

610

710

810

9

2050

2100

2150

2200

phas

e ve

loci

ty (m

/s)

k

ω2D Fourier transform

c=ω/k

( , ) ( ) exp( ) with = k c( )

( ( ( ), ) exp( ) 2

)

S x t S k i t ikx dk k

ddk S k c i t ikxk

ω ω

ω ω ωπ

δ ω ω

+∞

−∞

+∞ +∞

−∞ −∞

= −

= − −

∫

∫ ∫

substrate

coating

Photothermal characterization of thermal properties of free-standing films and plates: extraction of effective thermal diffusivity dispersion

Optical detection schemes

Laser ultrasonics ⇒ detection schemes

detection tecnhiques

optical

laser beam deflection

knife edge detection

speckle knife edge detection

(SKED)

diffraction interferometric

no scattering

Michelson common-path

Sagnac

birefringent crystal

Phase mask

scattering

laser Doppler

speckle

shearography

photorefractive

optical fiber

modulated optical

reflection

Brillouin oscillations

piezoelectric

Deflection angle ∆θ ≅ ξ/w ~ spatial derivative of wave packet ~ high pass response

w = characteristic lateral dimension of wave packet= displacement ξ or acoustic wavelength λacoustic

Deflection displacement: δ=∆θRelative differential intensity:∆I/I=δ/φ2φ2 is the diameter of the reflected beam at lens L2φ2=F2/F1φ1φ1 is the diameter of the incoming beam at lens L1

So that ∆I/I=δ/φ2=∆θ F1/φ1=(ξ/w)(F1/φ1)

E.g.w=10 µm, F1=50mm,φ1=5mmDisplacement detection limit:ξmin=(∆I/I)min (F1/φ1)-1 w=10-10w=10-15 m(W/Hz)1/2

E.g. 300Hz bandwidth, 75µW probe laser power:Typical light intensity change detection limit:10-9(W/Hz)1/2

ξmin,typ=2pm

2

20

222

2

2 exp2

exp

x dxdI OI x dx

ξ

φ δ ξφφ π

φ

+∞

−∞

− = ≅ = −

∫

∫

F2

F1

Z

Detectors

δ

θ

Laser ultrasonic detection schemes: laser beam deflection

wξ

Z

F2

F1

Detectors

t

difference signal

Laser ultrasonic detection schemes: laser beam deflection

Z

F2

F1

Detectors

t

difference signal

Z

F2

F1

Detectors

t

difference signal

Z

F2

F1

Detectors

t

difference signal

Z

F2

F1

Detectors

t

difference signal

F2

F1

X

Z

Detectors

Pixel signal sign assignment based on steady state light pattern32 x 32 photodiode pixel array

Laser ultrasonic detection schemes: speckle knife edge detector (SKED)

1 2 1 2( ) 2 cos( )I t I I I I φ= + + ∆PHOTO

DETECTOR

NON-CONTACT FAST SENSITIVE

2 ξφ πλ

∆ =λ/4

displacement δ(t)optical phase ∆φ

light intensity

Laser ultrasonic detection schemes: Michelson interferometer

http://www.google.be/url?sa=i&rct=j&q=green%20speckle%20pattern&source=images&cd=&cad=rja&docid=zkCBJICvNDn68M&tbnid=uLOwdajLjxguFM:&ved=0CAUQjRw&url=http://en.wikipedia.org/wiki/Speckle_pattern&ei=9ellUr7DMeOsiAeCoIC4Cw&bvm=bv.55123115,d.aGc&psig=AFQjCNE67QnvWmEVkMR2jI4yhkycNj3C5g&ust=1382497073739568

PD

Fig. 2a. Grating Interferometer Transmission Mode

Fig. 2b. Grating Interferometer Reflection Mode

Pm1 L1 L2 Pm2

PC

Probe

Pump

Probe

PD

Pump

Sample

OptionalSample

L/2 L/4

Al-coatedsubstrate

CpL

Pm

Transmission mode

Reflection mode

Laser ultrasonic detection schemes: phase mask interferometer

http://kino-ap.eng.hokudai.ac.jp/interferometer.html

Laser ultrasonic detection schemes: common path interferometer: Sagnac configuration

http://kino-ap.eng.hokudai.ac.jp/interferometer.html

Laser ultrasonic detection schemes: common path interferometer: time delay by birefringent crystal

Laser ultrasonic detection schemes: laser Doppler vibrometry

• 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐸𝐸𝑟𝑟 𝑡𝑡 ² + 𝐸𝐸𝑠𝑠 𝑡𝑡 ² + 2 ∗ 𝐸𝐸𝑟𝑟 𝑡𝑡 ∗ 𝐸𝐸𝑠𝑠 𝑡𝑡 ∗ cos(𝜑𝜑)

• with 𝜑𝜑 = 𝑘𝑘 ∗ ∆𝑥𝑥 = 2𝜋𝜋∆𝑥𝑥/𝜆𝜆

• 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐸𝐸𝑟𝑟 𝑡𝑡 ² + 𝐸𝐸𝑠𝑠 𝑡𝑡 ² + 2 ∗ 𝐸𝐸𝑟𝑟 𝑡𝑡 ∗ 𝐸𝐸𝑠𝑠 𝑡𝑡 ∗ cos 2𝜋𝜋 𝑟𝑟𝑟𝑟−𝑟𝑟𝑠𝑠𝜆𝜆

• 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐼𝐼𝑟𝑟 + 𝐼𝐼𝑠𝑠 + 2 ∗ 𝐼𝐼1𝐼𝐼2cos 2𝜋𝜋 𝑟𝑟𝑟𝑟−𝑟𝑟𝑠𝑠𝜆𝜆

cos 2𝜋𝜋 𝑟𝑟𝑟𝑟−𝑟𝑟𝑠𝑠𝜆𝜆

= cos 𝜔𝜔𝑟𝑟 − 𝜔𝜔𝑠𝑠 ∗ 𝑡𝑡 = cos 𝜔𝜔0 + 𝜔𝜔𝐴𝐴𝐴𝐴𝐴𝐴 − 𝜔𝜔𝑡𝑡 − ∆𝜔𝜔𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∗ 𝑡𝑡

cos 2𝜋𝜋 𝑟𝑟𝑟𝑟−𝑟𝑟𝑠𝑠𝜆𝜆

= cos 𝜔𝜔𝐴𝐴𝐴𝐴𝐴𝐴 − ∆𝜔𝜔𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∗ 𝑡𝑡

𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐼𝐼𝑟𝑟 + 𝐼𝐼𝑠𝑠 + 2 ∗ 𝐼𝐼1𝐼𝐼2 cos 𝜔𝜔𝐴𝐴𝐴𝐴𝐴𝐴 − ∆𝜔𝜔𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∗ 𝑡𝑡 detected byphotodetector

( )0

( )( ) sample

sample AOM

v tt

cω ω ω∆ = +

Doppler effect

Phase locked loopPLL-detector

δ (m)

T

0 21

1 sin ( / 2)transmittedetalon

I IF φ

=+ ∆

0 0 2

2

0 2

11 sin ( / 2)sin ( / 2)

1 sin ( / 2)

reflectedetalon

etalon

etalon

I I IF

FIF

φ

φφ

= −+ ∆

∆=

+ ∆

δ (m)

R

R: mirror reflectivityF=4R/(1-R)2: Finesse∆φetalon=2π∆(ξ/λoptical)∆λoptical/λ0,optical=dξ/dt/c

Cavity resonance time:τcavity ~ Fdcavity/ce.g. dcavity=10mm, F=100:bandwidth ∆f=τcavity

-1=300MHz

cavity width dcavity

Laser ultrasonic detection schemes: Fabry-Perot interferometer

Laser ultrasonic detection schemes: broadband ←→ narrowband excitation

NarrowbandBroadband

λmin

Laser ultrasonic detection schemes: transient grating excitation

Heterodyne diffraction configuration

Laser ultrasonics:

material characterization

case studies

Laser ultrasonic applications: elastic characterization of hardened steel

excitation beam

bulk waves

thermal diffusion

← SAW SAW →

Laser excitation of surface acoustic waves (SAW) bulk waves thermal diffusion field

Optical detection

detection

excitation

Laser ultrasonic applications: elastic characterization of protective layer on fuel cladding

Laser ultrasonic applications: elastic characterization of protective layer on fuel claddingThermoelasticity GESA remelted 20 µm FeCrAlY on T91 steel

GESA sample



SAW velocity at different grating spacings


Extracted coating thickness

from xy-scan from yx-scan


Submicron oxide layer

d (µm) cR (m/s) ρ (kg/m³)

coating 0.6±1.0 (3.2±0.7) 10³ (7±11) 10³

substrate 2826 7990

Laser ultrasonic applications: elastic characterization of rough polymer coated steel sample

phasegrating

sample

L2 (sph.)

f1 f1+f2 f2

pumpprobe

differential photodetector

L1 (cyl.)phasegrating

sample

L2 (sph.)

f1 f1+f2 f2

pumpprobepumpprobe

differential photodetector

L1 (cyl.)

0 0.1 0.2 0.3 0.4 0.5

0

time (µs)

0 50 100 150 2000

0.05

frequency (MHz)

ampl

itude

(a

rb.u

nits

)

λ=29µm

50 100 150 200 300

0.1

1

wave

lengt

h (µ

m)

frequency (MHz)

wave

num

ber (

µm-1

)

93.5

43.529.121.814.6

9.56.6

k(f) polymer on steel

340 nm Al on polymer on steel

daverage,polymer=3µm

nanoindentation:

0 50 100 150 200 2501000

2000

3000

frequency (MHz)

velo

city

(m.s

-1)

(a)50 100 150

1000

2000

3000

frequency (MHz)

velo

city

(m.s

-1)

(b)

0 50 100 150 200 2501000

2000

3000

frequency (MHz)

velo

city

(m.s

-1)

(c)50 100 150

1000

2000

3000

frequency (MHz)

velo

city

(m.s

-1)

(d)

polymer on steel

2.4,3.3.6 GPa

0.15,0.30,0.35 GPa

Laser ultrasonic applications: elastic characterization of rubber layer

zxx

z

Laser ultrasonic applications: elastic characterization of rubber layer

2000

4000

6000

8000

velo

city

(m.s

-1)

longitudinal velocityCuxSny

shear velocityCuxSny

10 100

2000

4000

6000

8000

frequency (MHz)

velo

city

(m.s

-1)

thickness Sn

10 100frequency (MHz)

thickness CuxSny

Laser ultrasonic applications: elastic characterization of sub-micron intermetallic layer

Sn-Cu interdiffusion layer

Fitting parameters:

vL = 5400 m/svT = 2100 m/sd1 = 1.63 µmd2 = 5.10 µmd3 = 0.50 µmρ = 7900 kg/m³

10 100

2000

4000

6000

8000

10000

frequency (MHz)

velo

city

(m.s

-1)

4500 5000 5500 6000 6500 70000

0.5

1

1.5

2

vL CuxSny (m.s-1)

χ2 / χ2 m

in2050 2100 2150 2200

0

0.5

1

1.5

2

vT CuxSn

y (m.s-1)

χ2 / χ2 m

in

1 1.5 20

0.5

1

1.5

2

dSn (µm)

χ2 / χ2 m

in

4.8 5 5.2 5.40

0.5

1

1.5

2

dCu Sn (µm)

χ2 / χ2 m

in

Laser ultrasonic applications: elastic depth profiling of functionally graded materials

Experimental resultparameter study

38

E = 770 ± 26 GPaρ = 3250 ± 50 kg.m-3

L = 1.4 ± 0.04 µm

Laser ultrasonic applications: elastic characterization of sub-micron nanocrystalline diamond layer

39

Laser ultrasonic applications: thermal characterization of sub-micron nanocrystalline diamond layer

Laser ultrasonic applications: elastic characterization of Ni2MnGa layer

Calculation of guided wave dispersion and photothermal and photoacoustic

displacements

Bulk wave propagation in solidsPropagating quantities:• density ρ• displacement vector ui• strain tensor εij= ½(∂ui/∂xj+∂uj/∂xi)≡½(ui,j+uj,i)• stress components σij• velocity vector vi

2,

0 0 ,2ij ji

i ij jj

uut x

σρ ρ σ

∂∂≡ = ≡

∂ ∂

Governing equations:

• Newton

• Hooke

Strain

Stress

( )( )( ) ( )( )

2 2 20

2 2 20 , , ,

2 2

2

ij ijkl kl

ij L T kk ij T ij

L T k k ij T i j j i

c

c c c

c c u c u u

σ ε

σ ρ ε δ ε

ρ δ

=

= − +

= − + +

Bulk wave propagation in solids

2,

0 0 ,2ij ji

i ij jj

uut x

σρ ρ σ

∂∂≡ = ≡

∂ ∂

Combining Newton and Hooke:

( )( )( ) ( )( )

2 2 20

2 2 20 , , ,

2 2

2

ij ijkl kl

ij L T kk ij T ij

L T k k ij T i j j i

c

c c c

c c u c u u

σ ε

σ ρ ε δ ε

ρ δ

=

= − +

= − + +

( ) ( )( )( )

2 2 20 0 , , ,

2 2 2, ,

2

2

i L T k kj ij T i jj j ii

i L T j ji T i jj

u c c u c u u

u c c u c u

ρ ρ δ= − + +

= − +

Bulk wave propagation in solids: harmonic solutions in 2D

Combining Newton and Hooke:

( , , ) ( , ) exp( )( , , ) ( , ) exp( )x z t x z i t ikxx z t x z i t ikx

ϕ ϕ ωψ ψ ω

= −= −

2 2 22

2 2 2

2 2 22

2 2 2

L

T

kz c t

kz c t

ϕ ω ϕϕ

ψ ω ψψ

∂ ∂− + = −

∂ ∂

∂ ∂− + = −

∂ ∂

2 2 22 2

2 2 2

2 2 22 2

2 2 2

LL

TT

k pz c t

k pz c t

ϕ ω ϕ ϕ

ψ ω ψ ψ

∂ ∂= − ≡ ∂ ∂ ∂ ∂

= − ≡ ∂ ∂

( )( )

( , , ) exp( ) exp( ) exp( )

( , , ) exp( ) exp( ) exp( )L L

T T

x z t A p z B p z i t ikx

x z t C p z B p z i t ikx

ϕ ω

ψ ω

= + − −

= + − −

Harmonic proposal solution for plane waves running in the positive x-direction

Harmonic plane waves running in the positive x-direction with a depth profile in the z-direction.

Surface wave propagation in semi-infinite solids

( )( )

( , , ) exp( ) exp( ) exp( )

( , , ) exp( ) exp( ) exp( )L L

T T


x z t C p z D p z i t ikx

ϕ ω

ψ ω

= + − −

= + − −

Harmonic plane wave components running in the positive x-direction with a depth profile in the z-direction.

Surface waves: no energy far away from the surface

( , , ) exp( )exp( )( , , ) exp( )exp( )

L

T

x z t B p z i t ikxx z t D p z i t ikx

ϕ ωψ ω

= − −= − −

Surface waves: no normal and no shear stress at the free surface: σzz=0 and σxz=0

( ) ( )( )2 2 20 , , ,2ij L T k k ij T i j j ic c u c u uσ ρ δ= − + +

=0

=0

2 2

2 2

20

2T T

L T

Bk p ikpDikp k p

+ − = +

Rayleigh determinant of homogeneous set of equations should be zero

Implicit condition relation between wave number k and angular frequency ω:dispersion relation

2 1/2 1/22 2 2 24

2 2 2 2 2 2 2 21 2 1 1 4 1 1 0T T T L

kk c k c k c k cω ω ω ω

+ + + + − + + =

4 2 2 4 22 4 0T T T Lk k p p k p p+ + − =

cR≡ω/k=f(cL,cT,ρ,ω,k)

NO DISPERSION

Guided wave propagation in a free-standing platewith thickness L

( )( )

( , , ) exp( ) exp( ) exp( )

( , , ) exp( ) exp( ) exp( )L L

T T



ϕ ω

ψ ω

= + − −

= + − −

Harmonic plane wave components running in the positive x-direction with a depth profile in the z-direction.

4 unknowns A,B,C,D to be found from 4 boundary condition equations:No normal and shear stress at plate boundaries z=0 and z=L:σzz(z=0)=0σzx(z=0)=0σzz(z=L)=0σzx(z=L)=0

( ) ( )( )2 2 20 , , ,2ij L T k k ij T i j j ic c u c u uσ ρ δ= − + +

0

ACBD

=

f(cL,cT,ρ,ω,k,L)=0

Lamb determinant of homogeneous set of equations should be zero

cLamb(cL,cT,ρ,ω,L)

DISPERSION

MULTIPLE SOLUTIONS PER FREQUENCY

Zero search of determinant of set of equations, looking for root k-values for every value of ω

Guided wave propagation in a free-standing platewith thickness L


S0-mode A0-mode

Aluminium plate L = 2 mm, ρ = 2700 kg/m³, vT = 3040 m/s, vL =6420 m/s, vR = 2846 m/s

zxx

z

• Dispersion• Multiple modes• Both

running (Im(k)=0) andlocal modes (Im(k)≠0)

Guided wave propagation in multilayers

( )( )

1 1

1 1

1 1 1

1 1 1

( , , ) exp( ) exp( ) exp( )

( , , ) exp( ) exp( ) exp( )

L L

T T



ϕ ω

ψ ω

= + − −

= + − −

( )( )

2 2

2 2

2 2 2

2 2 2

( , , ) exp( ) exp( ) exp( )

( , , ) exp( ) exp( ) exp( )

L L

T T



ϕ ω

ψ ω

= + − −

= + − −

( )( )

3 3

3 3

3 3 3

3 3 3

( , , ) exp( ) exp( ) exp( )

( , , ) exp( ) exp( ) exp( )

L L

T T



ϕ ω

ψ ω

= + − −

= + − −

0

1

2

3

layer

σzz(z=0)=0σzx(z=0)=0

σzz+(z=d1)=σzz

- (z=d1)σzx

+ (z=d1)=σzx- (z=d1)

uz+ (z=d1)=uz

- (z=d1)ux

+ (z=d1)=ux- (z=d1)

σzz+(z= d1+d2)=σzz

- (z= d1+d2)σzx

+ (z= d1+d2)=σzx- (z= d1+d2)

uz+ (z= d1+d2)=uz

- (z= d1+d2)ux

+ (z=d1+d2)=ux- (z= d1+d2)

A3=0C3=0

12 unknownsA1,B1,C1,D1A2,B2,C2,D2A3,B3,C3,D3

12 equations

What if there is a delamination?

1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 1,10 1,11 1,12 1

2,1 1

3,1 1

4,1 1

5,1 2

6,1 2

7,1 2

8,1 2

9,1 3

10,1 3

11,1 3

12,1 12,12 3

...

M M M M M M M M M M M M AM BM CM DM AM BM CM DM AM BM CM M D

0=

What if there is a source?


Determinant of homogeneous set of equations should be zero

cLamb(cL,cT,ρ,ω,L)

DISPERSION CURVE SOLUTIONS k(ω), c(ω)

Zero search of determinant of set of equations, looking for root k-values for every value of ω

f(material properties,ω,k,L)=0


0.1 1 10 100 1000

2000

3000

4000

5000

6000

7000

8000

9000

cT,Sn = 1586 m.s-1 cR,Sn = 1485 m.s-1

cL,Si = 8430 m.s-1

cR,Si = 5148 m.s-1

frequency (MHz)

velo

city

(m.s

-1)

Sn

Cu

Si

CuxSny

Multilayer sample

Guided wave propagation in multilayers

50

2

2

T C T Qtx

ρκ κ

∂ ∂− = −

∂∂( )

02

cos( )( , )2

Q kxT xk i C

ωπ κ ω

=+

0( , ) exp( )Q x t Q i t ikxω= + ( ) ( )2

02 2 2 2

sin( )( , )2

L

L

k c Q qxu xC k i k c

γωπρ α ω ω

−=

+ −2 2

2 2 2

1

L

u u Txx c t

γ∂ ∂ ∂− =

∂∂ ∂

0

0

( , ) ( ) cos( )( , ) cos( )

Q x t Q t kxQ x Q kx

δω=

⇒ =

cL: longitudinal speed of soundC: specific heat capacityρ: densityκ: thermal conductivityγ: thermal expansion coefficientα = κ/ρC : thermal diffusivity

log(t)

dens

ity g

ratin

gam

plitu

deSolution of longitudinal wave equation without and with a source

WITH A THERMAL SOURCE

( )2 0

( , ) exp( ) cos( )2

exp( ) cos

(

( )2

( )

12

) dT x t i t kx

di

QS

k i Ct kQ x

ωωπ

ω

π κω

ω

ωπω

+∞

−∞

+∞

−∞

=

=+

∫

∫

determinant!

Residue theorem?

Kramers rule?

51

2

2

T C T Qtx

ρκ κ

∂ ∂− = −

∂∂( )

02

cos( )( , )2

Q kxT xk i C

ωπ κ ω

=+

0( , ) exp( )Q x t Q i t ikxω= + ( )( )2

02 2 2 2

sin( )( , )2

L

L

k c Q kxu xC k i k c

γωπρ α ω ω

−=

+ −2 2

2 2 2

1

L

u u Txx c t

γ∂ ∂ ∂− =

∂∂ ∂

0

0

( , ) ( )sin( )( , ) sin( )


δω=

⇒ =


log(t)

dens

ity g

ratin

gam

plitu



( )( )2

2 2 2 2 0

( , ) exp( ) sin( )2

exp

( )

( ) s2

(

)2

)

in(L

L

du S

k cC k i

x t i t kx

di t kx

Q

k cQ

ωωπ

ωωπ

ω

γπρ α ω

ω

ω

+∞

−∞

+∞

−∞ −

=

=−+

∫

∫

determinant!

Residue theorem?

52

2

2

T C T Qtx

ρκ κ

∂ ∂− = −

∂∂( )

02

cos( )( , )2

Q kxT xk i C

ωπ κ ω

=+

0( , ) exp( )Q x t Q i t ikxω= + ( )( )2

02 2 2 2

sin( )( , )2

L

L

k c Q kxu xC k i k c

γωπρ α ω ω

−=

+ −2 2

2 2 2

1

L

u u Txx c t

γ∂ ∂ ∂− =

∂∂ ∂

0

0

( , ) ( )sin( )( , ) sin( )


δω=

⇒ =

( )202 2

2

sin( )( , ) exp cos( ) sin( ) ( )1

L LL

L

Q kx ku x t k t kc t kc t tckCk

c

γ αα θαρ

= − − + +

( )20 sin( )( , ) exp ( )Q kxT x t k t tC

α θρ

= −


log(t)

dens

ity g

ratin

gam

plitu



determinant!

Thermoelastic excitation

Proposal solutions

Potentials wave equation

with

Displacements at the surface

with

with

with

k,ω domain thermal driving source

Transient grating in reflection mode: displacement response

2 remaining unknowns B and D are determined via 2 boundary conditions

Stresses are determined by Duhamel-Neumann relation

A = 0C = 0 effect of surface tension

effect of gravitation thermal driving source


Unknowns A,C,E

~ source

effect of surface tension quasi-Rayleigh determinant

2x2 set of equations in unknown coefficients B and D

Solution:


Solution for arbitrary source = 2D (k→x,ω→t) Fourier transform of solution for harmonic excitation, with source spectrum I(ω,k) as weighting function

Transient grating in reflection mode: displacement response:arbitrary source

( )

( ) ( )00

00

( , , ) ( , ) exp ( )

( , )( , , ) ( , ) exp( ( , ) ) ( , ) exp( ( , ) ) ( , ) exp( ( , ) ) exp

( , )( , , ) ( , ) exp( ( , ) ) ( , ) exp(

L L

T T

I x z t d dk I k ikx i t z

I kx z t d dk A k p k z B k p k z E k k z ikx i tI

I kx z t d dk C k p k z D k pI

ω ω ω δ

ωφ ω ω ω ω ω ω σ ω ω

ωψ ω ω ω ω

+∞ +∞

−∞ −∞

+∞ +∞

−∞ −∞

= +

= + − + − +

= + −

∫ ∫

∫ ∫

( ) ( )( , ) ) expk z ikx i tω ω+∞ +∞

−∞ −∞

+∫ ∫

( ) ( )

( , , ) ( ) ( ) ( ) ( , ) 1

( , , 0) ( , ) ( , ) ( , ) ( , ) ( , ) exp z L

I x z t x z t I k

u x t z d dk p k B k k E k ikD k ikx i t

δ δ δ ω

ω ω ω σ ω ω ω ω+∞ +∞

−∞ −∞

= ⇒ =

⇓

= = − − + +∫ ∫

( ) ( )

( ) ( )

( )

0 00 0

0 0

0

0

0

( , , 0)

exp( ) exp( ) exp( ) exp

( , )

( ,

z z zz z

z

L L T

z

L

I kI

u k z

z x

p B p z E z ikD p z ikx i t

p B E ikD I k

ω φ ψ

φ ψ

σ σ ωω

ωσ

= =

=

=

= = ∇ + ∇×

∂ ∂ = + ∂ ∂

= − − − − + − +

= − − + ( )0

exp ) ikx i tI

ω+

Poles!Residue theorem

Laser ultrasonics for defect detection and application for non-destructive testing

3 mm

Lamb waves on a thin Cu membrane …Typical result from full field Doppler xy scanning

Laser ultrasonic laser Doppler xy scanning imaging: example of material characterization and non-destructive testing

Related references

Prague

Czech Republic

Hasselt, Belgium

Mexico City, Mexico

Leuven, Belgium

Michal Landa, Pavla Stoklasova, Hanus Seiner, Petr Sedlak

Ken Haenen, Paulius Pobedinskas

Bert Verstraeten, Jan Sermeus, Liwang Liu, Kris Vanstreels

Angel Cifuentes, Salvador Alvarado, Ernesto Marin

Kelly Martinez, Luis Silvestre Vaca Oyola, Jonathan Guarachi

Osamu Matsuda

THANK YOU FOR YOUR ATTENTION

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