PHOTOTHERMAL DEFLECTION TECHNIQUEsabotin.ung.si/~isschool/2018Erice/RLV.pdfGas Vact sg Photoacoustic signal Effective cavity length …you can hear the light (Bell,1880) PHOTOTHERMAL

Dipartimento SBAI, Università di ROMA “ La Sapienza”, Via A. Scarpa 16, 00161 ROMA - ITALY

PHOTOTHERMAL DEFLECTION TECHNIQUE

Theory and applications: the experience at “La Sapienza” in Rome

Roberto Li Voti

• PHOTOTHERMAL TECHNIQUES

• PRINCIPLE OF PHOTOTHERMAL DEFLECTION

•THE HEAT DIFFUSION

• MEASUREMENT OF THERMAL DIFFUSIVITY

• OTHER APPLICATIONS

Thanks to Photothermal at Roma La Sapienza

Grigore L. LEAHU, Stefano PAOLONI, Concita SIBILIA, Mario BERTOLOTTI

PHOTOTHERMAL EFFECTS

PHOTOTHERMAL TECHNIQUESPhotothermal reflection scheme

Probe incidente

Probe deflesso

Probe non deflesso

Piano di rivelazione

Campione

hx

x

2 h cos( )θx

2 g Lx

θ

S

( ) ( ) ( )LxgxhS 2cos2 +θ=

Photothermal reflection signal

S displacementL distance between sample and detectorh distortion height g distortion gradient

PHOTOTHERMAL TECHNIQUESPhotoreflectance scheme

PHOTOTHERMAL TECHNIQUESPhotoreflectance microscope

Schematic set up

Magnification

PHOTOTHERMAL TECHNIQUESRadiometric technique

Radiation detector

Sample

Chopper

Pump laser

Mirror

Lens

Infrared lensθ

( ) ( ) ( ) ( )

( )

−λπ=λ

λδ∆∂

λ∂λλθε=

λ 1

125

2

2

Tkhcd

d

ddd

Bde

hcW

TT

WRTrAsinS( ) TAsinTS o δθεσ= 234

TTW o δεσ=δ⇔ 34Stephan-Boltzman law

W = ε σ T4

σ = 5.67x10-12 Wcm-2K-4

Lambert lawBroadband filter

Lambert lawSelective filter at λd

Tr transmission of the optics R detector sensitivity

S, PTR signal

PHOTOTHERMAL TECHNIQUESPhotoacoustic technique

( )dTVlRT

sdP

rg2

g2

π+

γ=

Pr ( )fDls ggg π= ,min

P acoustic pressureT temperatureγ specific heat ratioR radius of the celllg length of the cellVr residual volumesg effective lengthD gas thermal diffusivityf modulation frequency

Backing

Segnale PA Microfono

Cella fotoacustica

2r 2R

lg

Campione

Finestra trasparente

Fascio incidentemodulato

Gas

Vact

sg

Photoacoustic signal Effective cavity length

…you can hear the light (Bell,1880)

PHOTOTHERMAL TECHNIQUESPhotopyroelectric technique

He-Ne Laser

Sample

PVF2

Backing

Pre-Amp

Lock-inChopper

( )∫−=

d

dxtxTdt

dpAi

dp

l

l0

,

p pyroelectrica constant of the material A area of the detectorip currentT temperature rise

Photopyroelectric signal

PHOTOTHERMAL TECHNIQUESOther optical techniques

raggio deflesso

MIRAGGIO

oggettoimmagine

∫∇=Φpath

tTdsdT

dn

n

1r

laser di pompa

Φ

fascio deflesso

fascio indeflessoCampione

Aria

Trace gas analysis device

Mirage effect

Schematic set up

Deflection angle

PHOTOTHERMAL TECHNIQUESPhotothermal deflection technique

Photothermal deflectionM

irage

Optical Beam Deflection

PRINCIPLE OF PHOTOTHERMAL DEFLECTION

0rVrnkrV 222 =+∇ )()()(

)()()( rS k ierArV ⋅=

=∇⋅∇+∇

=∇−⋅+∇

0SA2SA

0SnAkA

2

2222 )(

σ⋅=∇⇒=∇ rnS nS 22

nds

dS =

( ) νρ

σσσσ rrr

rr

n

ds

dn

ds

dn

ds

dn

ds

ndS

ds

d

ds

dSn +=+==∇=

∇=∇

νρ

=σ=∇r

rn

ds

dnnt

HELMOLTZ Equation RAY OPTICS Equation

RAY OPTICS Solution Refractive indexRay unit vector

σr

vr

σr

RAY

s

0s =

ρ

BENDING EFFECT

Light bends only due to the transverse gradient

n

( ) ( ) ( ) ( ) ( )0000s σ+ν⋅Φ≅σ⋅Φ+ν⋅Φ=σ rrrvrcossen

∫ν⋅∇=Φ

path

t ds n

n r

h2

h1

Φ

Φh1

h2

σ

σ

o

∫∫

∫∫

∂∂=⋅∇=Φ

∂∂=⋅∇=Φ

path 2path

2th

path 1path

1th

ds h

n

n

1ds

n

hn

ds h

n

n

1 ds

n

hn

2

1

r

r

WEAK DEFLECTION

ds

dnnt

σ=∇r

Integration over s

σrUndeflected ray

( ) ( ) dsn

n0s

s

0

t∫∇=σ−σ rr

s

0s =

( )sσr

( )0σr

( )0νr

Final ray

2-D WEAK DEFLECTION

Deflection formula

( )0σr

( )0νr

∇⋅+

∇⋅=Φ

Φ++Φ+Φ=Φ

∑ ∫∫i path

t

ipath

t

f

n

T

nds

n

f ds

n

T

...

i

∂∂

∂∂r

rrrr

n1 ffT

)(...)()( 0nnn

0111

0 fff

nff

f

nTT

T

ndn −⋅

∂∂++−⋅

∂∂+−⋅

∂∂=

NATURE OF THE DEFLECTION

Taylor expansion of the refractive index

Photothermal deflection

∫∇=Φpath

tdT

dn

ndsT

1r

( )RT

ApTpn

2

31, +=

Tipo di composto A [m3 mol-1] *10-6

Aria 4.6

Ossigeno 4.05

HCl 6.68

Vapore acqueo 3.72 CS2 22

C3H6O 16.2

( )2o

65

T

p103431p108567

dT

dn −− ⋅+⋅−= .

.

AIR OPTOTHERMAL COEFFICIENT

Best approximation

In gases

=

−=

32

2

2

3

2

3

oTo

oTo

RT

Ap

dT

nd

RT

Ap

dT

dn

50 150 250 350 450

-5

-6

-8

p=1013 mBar

p=200 mBar

p=50 mBar

Temperature (K)

Air opththermal coefficient

10

10

-710

10

( )2o

65

T

p103431p108567

dT

dn −− ⋅+⋅−= .

.

AIR OPTOTHERMAL COEFFICIENT

0 0.2 0.4

0.0090.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0

Offset (mm)

Pressure dependance

750mBar

650mBar

500mBar

350mBar

200mBar

0 200 400 600 800

0.008

0.006

0.004

0.002

0

Pressure (mBar)

Maximum deflection signal

De

flect

ion

Sig

nal

Pressure of Nitrogen at 77K

EXPERIMENTAL RESULTS ON THEAIR OPTOTHERMAL COEFFICIENT

SIGNAL DETECTION

Φ=

Φ=

dz

dy

z0

y0

20

2

20

0w

d

nw

d1wdw

πλ≅

πλ+⋅=)(

)(

)()(

)(),( dw

zzyy2

2

2

20

20

edw

P2zyI

−+−−⋅

π=

FilterPositionSensor

Lock-in PC

Deflected / Undeflected beam

POSITION SENSOR EQUATIONS

BEAM CENTER

SPOT-SIZE BROADENING

BEAM INTENSITY ON THE DETECTOR

Zone A Zone B

Zone CZone D

Deflected beam

Beam center

Undeflected beamzoo y

z

y

( )yz000

A2

Pw

dw

yz8

4

PP Φ−Φ⋅

λπ=

−⋅π

⋅=)(

( )yz000

B2

Pw

dw

yz8

4

PP Φ+Φ⋅

λπ=

+⋅π

⋅=)(

( ) Azy000

C P2

Pw

dw

zy8

4

PP −=Φ−Φ⋅

λπ=

−⋅π

⋅=)(

( ) Byz000

D P2

Pw

dw

yz8

4

PP −=Φ+Φ⋅

λπ−=

+⋅π

⋅−=)(

+++−++−=∆

+++−−+=∆

DCBA

DCBA

o

l

DCBA

DCBA

o

v

PPPP

PPPP

V

V

PPPP

PPPP

V

V

Φλ

π=∆

Φλ

π=∆

y0

l

z0

v

w8V

w8V

z2

zerfπ

≅)(

POWER ON THE DETECTOR

Weak deflection approximation

DEFLECTION SIGNAL

Zone A Zone B

Zone CZone D

y

z

Zone A Zone B

Zone CZone D

y

z

Lateral signalVertical signal

•Thermal diffusivity and effusivity measurements

•Absorption spectroscopy

•Effusivity and optical absorption depth profiling

•Measurement of the attenuation in optical waveguides

•Evaluation of the thickness of thin layers

•Trace gas analysis

•Characterization of metallic surfaces

Main applications

Experimental Setup

monocromator

chopperPump lamp

Probe laser

0

0.2

0.4

0.6

0.8

1.0

395 405

1000 nm700 nm400 nm

-5 0nm

+5

Monochromatorbandwidths

He- Ne laser

Mirrors

CCl4

300 W Xe Lamp housing

Lens

Lens

Deflectionangle

ChopperGrating

Sample

Cell

x,y,z stage

Monochromator

PC

Refsignal

PreAmpl

Motordriver

Lock-inAmplifier

Photothermal Deflection Spectroscopy Set-up

0

20

40

60

80

100

400 500 600 700 8000

20

40

60

80

400 5

s-polarized wavep-polarized wave

wavelength, nm wavelength, nm

Op

tical

ref

lect

ance

Op

tical

ref

lect

ance

45°

75°

55°

65°

85°

85°

75°

55° 6545°

Optical Reflectance Spectroscopy

θ

SiAlN

Courtesy of A.Passaseo

Photothermal Deflection Spectroscopy on AlN

0

0.2

0.4

0.6

0.8

1.0

400 500 600 700Wavelength, nm

Nor

mal

ized

Sig

nal,

a.u.

PDS

Optical reflectance 0°

calculated optical absorbance 0°θ

Si

AlN

probe

Device under test

4 MLsGaAs 35 nm

GaAs4 MLs

GaAs 35 nm

GaAs

GaAs 35 nm

GaAs 4 MLs35 nm

4 MLs

GaAs 5 nmGaAs 200 nm

Si doped GaAs substrate

QD

QD

QD

QD

QD

QD

QD

roughness

3500 nm

section

Courtesy of A.Passaseo

Position sensor

Pump beam

Probe beam

InGaAs QD

GaAs :Si

“Back” configuration

InGaAs QD

GaAs :Si

“Front” configuration

Photothermal Deflection Spectroscopy

CCl4 liquid

-40

-30

-20

-10

0

10

800 900 1000 1100 1200

QD fronteQD retroGaAs fronte

Wavelength, nm

Pha

se, d

eg.

PTD, QD-InGaAs/GaA-SI

0.05

0.1

0.2

0.5

1

800 900 1000 1100 1200

QD fronteQD retroGaAs-SI wafer

Wavelength, nm

Def

lect

ion

sign

al, a

.u.

PDS, QD-InGaAs/GaA-SI

Experimental Results on Quantum Dots

3D Photonic crystal: SiO2 synthetic opal

Characterization of VO2/SiO2 invese opals

500 550 600 650 700 750 800 850 900 950 1000 1050 11000

10

20

30

40

50

60

70

80

90

100100

1.313

Ri 0,

Ri 1,

Expi 11⋅

Exphi 11⋅

1.099 103×521.135 λi

wavelength, nm

wavelength, nm

hot

coldcold

•Thermal diffusivity measurements






•Characterization of metallic surfaces

Main applications

THE HEAT DIFFUSION

=∂∂ρ+⋅∇

∇−=

wt

TcF

TkFr

rHeat flux

Energy conservation law

k

w

t

T

D

1T2 −=

∂∂−∇ Fourier diffusion equation

F heat fluxT temperature risew power densityk thermal conductivityD thermal diffusivityρ densityc specific heat

( )[ ]

( ) Dt4zyx231

222

eDtc8

1tzyxG ++−

πρ=,,,

Green function in pulsed regime Temperature solution for unitary pulse placed in the origin at t=0

PULSED REGIMEunitary pulse

“thermal wave”

( ) ( )

( ) ( )4cos,

,~ 1

πωω β

−−===

−

+−−

ll

l

xtAetxT

AeAexTx

xjx

fD π=l

Thermal diffusion lengthk

wT

dx

Td ~~~

22

2

−=− β

Pump beam

k

we

t

T

Dx

T tjω

−=∂∂−

∂∂ 1

2

2

x

Thermal waves

THEORY OF THERMAL WAVES

Heating period Cooling period

Plane “thermal wave”( ) ( )

( ) ( )[ ] ( )ϕωωω

ω

ββ

+−==

===−

+−−⋅−

ll

lrr

ztAeezTtzT

AeAeAezTztj

zjzr

cos,~

Re,

,~ 1

Plane source

Amplitude of temperature

Phase of the temperature

lzAe−

lz−

Complex quantity method

0 0.4 0.8 1.2 1.6 2 2.4 2.8

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 0.4 0.8 1.2 1.6 2 2.4 2.8

0

-20

-40

-60

-80

-100

-120

-140

-160

-180

Normalized path z/l

Phase (degree)

Amplitude normalized

Normalized path z/l

fD π=l

Thermal diffusion length

PLANE THERMAL WAVE

PLANE THERMAL WAVE

0 2 3

A

A/2

0

-A

-A/2

Distance from the source

Negative envelope

Positive envelope

ω t=π/2

ω t=π/3

ω t=0

ω t= −π

l l l

THE HEAT DIFFUSION

THERMAL WAVE GENERATION

Pump beam

lens

Surface

coldhot

hot period

cold period

laser on

laser off Thermal wavelength

surfaceSAMPLE

1D - PLANE THERMAL WAVE SPHERICAL THERMAL WAVE

THE HEAT DIFFUSION

Plane “thermal wave”

( ) ( )l

zjz

D

jz eAeAeAzT

+−⋅−− ⋅=⋅=⋅=

1,

~ω

βω fD π=l


0~

~

2

2

=

− TD

j

dz

Td ω

IThdz

Tdk

z

=+−=

~~

0

( ) IAhk =⋅+⋅ βhk

IA

+=

β

( ) ( ) lzj

g

zD

j

g

zD

jz e

je

Ie

je

Ie

hD

jk

Ie

hk

IzT +−

⋅−⋅−− ⋅=⋅≈⋅

+=⋅

+= 1,

~

ωωωβω

ωωβ

( ) ( )[ ]

−−== −

4cos,

~Re,

πωω

ω ω

l

l zte

e

IezTtzT z

g

tj

PLANE THERMAL WAVE GENERATIONon the soil due to the seasonal light oscillations

Boundary condition on heat flux at the surface

Heat diffusion equation in harmonic regime

2β

z=depth01

2

2

=−t

T

Ddz

Td

∂∂

Days (from Jan 1st)

Tem

pera

ture

, °C

Max temp

Min temp

Ave

rage

da

ily in

tens

ity, W

/m2dx 365:2:

4=ππ

dayx 45=

x

Measured temperature oscillations during the year in depth

( )

−−⋅= −

4cos,

πωω l

l zte

e

IAtzT z

s

rad

Tyear 86400*365

22 ππω ==

L

r

φ

Pumpbeambeam

Probe

Campione

ab

Diffusione

Assorbimento

( )( ) ( )[ ]ll

rjKk

jeP

dT

dn L

++−

−=−

12

111π

α

Collinear configuration

0 0.4 0.8 1.2 1.6 2 2.4 2.8

0

-1

-2

Fase (radianti) della deflessione collineare

Distanza normalizzata

0.2

1

r/µ0 0.4 0.8 1.2 1.6 2 2.4 2.8

1

0

-1

Distanza normalizzata

Logaritmo dell'ampiezza deflessione collineare

0.20.4

0.6

0.1

1

0.8

r/µ

φµ

∆∆= r

Phase method

( ) µϕϕ ry o −=

Z

Pump beam

Probe beam

n

t

x

Y

y

φ

φ

Vertical offset z

Sample

Vertical deflection

Lateral deflection

Lateral offset x

Transverse configuration

Pump beam

lens

Surface

( ) ( )T x t Ae t xx, cos= −− llω

Surface thermal wavel = D fπ

Phase φAmplitude


Thermal Diffusivity

0 0.1 0.2 0.3 0.4

18016014012010080604020

0-20-40-60-80-100

Phase (degree)

Horizontal offset (mm)

1600 Hz

900 Hz

625 Hz

400 Hz

225 Hz

ϕ∆∆−= x

l∆x

∆φ

x

Transverse configuration

Schema Skimming

Campione spesso

Fascio di prova

Fascio di riscaldamento modulato

b

a

xy

z

s

L

zmin

scansione

assorbimento conducibilitàdiffusività

αk1D1

conducibilitàdiffusività

koDo

Aria

r

( )( ) δδδαδ

δδα

π

δδω

dDDjkkjj

yeDDj

ek

P

dT

dn

oo

DDjza

oD

bj

oz

o

o ∫∞

+−

−

+++++

+

=Φ0 1

21

22

28

12

8

1 222

cos21

12

2

2

l

l

ll

( )( ) δδδαδ

δδα

π

δδω

dDDjkkjj

ye

ek

P

dT

dn

oo

DDjza

D

bj

oy

o

o ∫∞

+−

−

+++++

−=Φ0 1

21

22

28

8

1 222

sin1

12

2

2

l

l

ll

( )( )

( )

( ) ( )

++−

−

∞→

→

+++

−=Φ

+−

∞

∫

ll

l

l

l

l

l

yjK

k

Pj

dT

dn

e

k

P

dT

dn

jj

dy

k

P

dT

dn

o

yj

o

oy

11

20

22

sin1

11

1

1

022

1

π

α

α

α

δαδ

δδδα

π

φµ

∆∆= y( ) µϕϕ yy o −=

zbf

DC ⋅+

π=l

air

sample

z

pure mirage

bouncing in vacuumbouncing

1a 1b

Sample

lensposition sensor

Probe laser

pumpbeam

2

prism 60° prism 60°glass glass

PROBE BEAM

WW W

HEAT EXCHANGER

SUPPORTER

SAMPLE HOLDER

GASEOUS NITROGEN

POSITION SENSOR

W WINDOWS

xy

PUMP BEAM

SAMPLE

xz

CAPILLARY

VACUUMLIQUID NITROGEN

ZOOM

GLUE

PROBE BEAM

y

SAMPLE HOLDER

SCAN

PUSH DOWN

WSAMPLE

CollaborationCSM, Alenia, IRTEC CNR, Univ. Trieste

80 90 100

0.07

0.06

0.05

f = 16 Hz

Temperature, K

Thermal Diffusivity (cm /s)2

SUPERCONDUCTION NORMAL STATE

YBCO

h

W

L

d

a

COPPER HEAT SINK

BULKACTIVEREGION

PROBE BEAM

zy

x

medium 2

medium 1

medium 3AIR

scan

0 40 80 120 160 200 240 280

TEMPERATURE DIODE N°2 (K)

offset from the mirror (µm)

60

50

40

30

20

10

0

10 KHz

2.5 KHz

225 Hz

COPPERHEAT SINK

SEMICONDUCTORMONOCRYSTAL

Drive Current 70 mAThreshold Current 50 mA

h = 0.120 mmW= 0.420 mmL= 0.200 mmd= 0.008 mma= 0.010 mm

Es. LASER DIODE DOUBLE HETEROSTRUCTUREAlGaAs/GaAs, λ=800 nm

current, mA

Characterization of SiO2 /GaN opal sampleP

hase

, de

g.

Ln (A), a.u.

3 Hz

6 Hz3 Hz

6 Hz

Offset x, mm

0

100

200

300

400

-1.0 -0.5 0 0.5 1.0-9.0

-7.5

-6.0

-4.5

-3.0

0

100

200

300

400

0 0.2 0.4 0.6 0.8

Cha

ract

eris

tic le

ngth

, µm

Inverse square root frequency (1/Hz2)

silica-GaN 25%silica-GaN 70%

GaN 25% GaN 70% GaN invertedSilica

Unfilled

THERMAL WAVE REFLECTION AND REFRACTIONfor plane waves

( ) ( ) ( )[ ] ( ) ( )[ ]

( ) ( ) ( )[ ]

=

+=

θ+θβ−

θ−θβ−θ+θβ−

zxsin2

zxsinzxsin1

222

111111

tAezxT

rAeAezxT

cos

'cos'cos

,~

,~

Medium 2Medium 1

z

x

∂∂=

∂∂

=

z

Tk

z

Tk

TT

22

11

21

~~

~~Temperature must be conserved at z=0

Normal heat flux must be conserved at z=0

( ) ( )

θ=θ

θ=θ

22

11

11

sinD

1sin

D

1

'

( ) ( )( ) ( )

( )( ) ( )2211

11

2211

2211

ee

e2t

ee

eer

θ+θθ=

θ+θθ−θ=

coscos

cos

coscos

coscos

( )211 DDarcsin=θ≤θ lim

ρ

2 2l

ζ

A

air

invar

− l /224 2l

heating line-2 2 l

“Thermal Snell law”

“Thermal Fresnel coefficients”

Numerical simulation of the temperature field at the Invar-Air interface. The diffusivities are DInvar=0.05 cm2/s,Dair=0.2 cm2/s. The incidence angle isθ1=20°<θlim =30°and the refracted angle is consequentlyθ2=43°.

What the thermal effusivity is?

ckDke ρ==

THERMAL WAVE REFLECTIONexperimental evidence

Absorbing layer

Pump beam

Incident wave

Reflected

Thermal mirrorO

wave

x

z

Probe2θ

y

d

L

Experimental setup

( ) ( ) ( )[ ]x2sinz2zair

airair rAeAezxT θ−θββ− += cos,~

∫

∫

∂∂

=∂

∂

=Φ

∂∂

=∂

∂

=Φ

y

aireff

airz

y

aireff

airx

z

TL

dT

dn

n

1dy

z

T

dT

dn

n

1

x

TL

dT

dn

n

1dy

x

T

dT

dn

n

1

ALdT

dn

n

1C aireffβ

−=

( ) ( ) ( )[ ][ ] ( )[ ]( ) ( ) ( )[ ][ ] ( )[ ]θ−≅θ−=Φ

θ≅θ=Φθ−θββ−

θ−θβ

2r1Ce2reC

2rsinCe2rsinC

x2sinz2zz

x2sinz2x

airair

air

coscos~

~

cos

cos

Temperature in air

Deflection in air

Factor independent on the angle

Final expression for the deflection

0 2 4 6 8 10

2

1

0

Frequency square root, Hz1/2

Sign

al a

mpl

i tude

, m

V


( ) ( ) ( )[ ][ ] ( )[ ]( ) ( ) ( )[ ][ ] ( )[ ]θ−≅θ−=Φ

θ≅θ=Φθ−θββ−

θ−θβ

2r1Ce2reC

2rsinCe2rsinC

x2sinz2zz

x2sinz2x

airair

air

coscos~

~

cos

cos

zΦ~

xΦ~


Am

pli t

ude

rati

o

Inci

den

ce a

ng

le,

deg

0 2 4 6 8

0.3

0.2

0.1

0

15°

11°10°

5°

0°


( )( )

( )( ) ( )θ−=

θ+θ−≅

θ−θ=

ΦΦ= tg

21

2sin

2r1

2rsinR

z

x

coscos~

~Deflection ratio

THERMAL WAVE REFRACTIONexperimental evidence

Incident wave

x

z

Probe

θ

O

Pump beam spot

Refracted waveθ

Rotationstage

2

1

medium 2

medium 1

n

θ1

θ2

y

β2

d

Experimental setup

( ) ( ) ( )[ ]zxsinr2

121222 tAetAezxT θ−θ+θ−θβ−β− == cos,~ rr

Refracted thermal wave (in air)

( ) ( ) ( )[ ][ ]( ) ( ) ( )[ ][ ]zxsin

12z

zxsin12x

12122

12122

etC

esintC

θ−θ+θ−θβ−

θ−θ+θ−θβ−

θ−θ=Φ

θ−θ=Φcos

cos

cos~

~

Deflection ratio

( )12z

x tgR θ−θ=ΦΦ= ~

~

Deflection in the second medium (air)


0 20 40 60

90

80

70

60

50

40

30

20

10

010 30 50

Incidence angle, deg

Sig

nal a

mpl

itud

e, µ

VzΦ~

xΦ~

ANOMALOUS THERMAL REFRACTED FIELD

( )211 DDarcsin=θ>θ lim

( )( ) ( ) ( ) ( ) 1sin

D

Dj1sin

j1

2

12

1

2

21

1BeT−θζ−±ρθ+−

± =ζρ ll,~

What happens when the Snell law is not valid?

ρ

2 2l

ζ

air

invar

−l /224 2l

heating line-2 2 l

Temperature field at the Invar- Air interface. The incidence angle isθ1=70°>θlim .

ρl212

6 2l


( )[ ]112 R θ+θ=θ arctg( )12z

x tgR θ−θ=ΦΦ= ~

~

0 0.2 0.4 0.6 0.8

1

0.5

0

10°

Incidence angle, sin ( )θ

20° 30° 40° 50° 60°

1R

efra

ctio

n a

ngle

, si

n (θ 2)

Linear scale in θ Linear scale in sin(θ )

airInP DD

0 20 40

50

40

30

20

10

0

-1010 30 50

Incidence angle, deg

1

1.7

2.2

5

Re

fra

ctio

n a

ngle

, de

g

DInP=0.44 cm2/s

Dinvar=0.04 cm2/s

( ) ( )22

11

sinD

1sin

D

1 θ=θ

Snell law

THERMAL WAVE INTERFEROMETRY

BASIC PRINCIPLE

To generateplane thermal wavesof a given frequency at the front surface of the sample byheating it periodically with a pump laser beam.

The wavespropagate inside the structure and, if they approach a buried layer withdifferent thermal properties, they are partiallyreflectedgiving rise, together with theincident waves, to an interference effect at the front surface.

DETECTION

APPLICATIONS

Photoacoustic

Radiometry

Photothermal Deflection techniques

Nondestructive evaluation of the thermophysical properties and

the thickness of layered samples

INTERNAL TEMPERATURE RISE

MATERIAL/BULK INTERFACE

BOUNDARY CONDITIONS AT THE SURFACE (z=0)

I; Power intensity

Temperature

heat flux

Thermal Resonator

Forward thermal wave

Backward thermal wave

Pump laser beam1 D illumination

L (Cavity Length)

Buried layer

at Front surfaceTemperature rise

1° Layer


zz BeAezT ββ− +=)(

bulkm

bulkm

ee

eeR

+−=

L

L

Ae

Be

waveincidence

wavereflectedR β−

β==

( )( )BAk

dz

dTkI

BA0T

−β=−=

+=

LeARB ⋅−⋅⋅= β2

( )LeAkdz

dTkI ⋅−−=−= ββ 21

( )Lk

IA ⋅−−

= ββ 2Re1

1

-0.9

+0.9

TEMPERATURE RISEAT THE SURFACE

PHASE SIGNAL ( ) ( )

−

π−=ϕ

π−

π−

DfL42

DfL2

eR1

eDfL2Rsin2arcf tan

( )( )

( )

−

+=

+−

+−

DfLj

DfLj

air

eR

k

IoT

/12

/12

Re1

1π

π

β


( )2/ DfL π

bulkm

bulkm

ee

eeR

+−=

-0.9

+0.9

phase

-0.9

+0.9

( )2/ DfL π

amplitude

0 0.4 0.8 1.2 1.6 2 2.4

40

30

20

10

0

-10

-20

-30

-40

Depth normalized to the thermal diffusion length

R=0.9

R= -0.9

R=0.6

R= -0.6

R= -0.3

R=0.3

R=0

Phase shift (degree)

TEMPERATURE RISEAT THE SURFACE

PHASE SIGNAL

Note that the interference effect(oscillation) is seen when the cavitylength is of the same order of thethermal diffusion length of theresonator and when the coefficient Ris large enough.

The maximum oscillation is of 45°degrees and is obtained in case ofperfect reflection from the buriedlayer with R=±1

( ) ( )

−

π−=ϕ

π−

π−

DfL42

DfL2

eR1

eDfL2Rsin2arcf tan

( )( )

( )

−

+=

+−

+−

DfLj

DfLj

air

eR

k

IoT

/12

/12

Re1

1π

π

β

DfL /π


Opaque coating

Bulk

Air

Vertical Deflection

Illuminated area

Probe beam

Pump beam

z

L

THERMAL WAVE INTERFEROMETRYDiffusivity measurement by Mirage

( )( )[ ]( )[ ]

+−−+−+⋅

ω+=

c

c

aircsurf LjRR

LjR

jee

IT

l

l

12exp1

12exp1ˆ21

2 11 ≈+−=

airc

airc

ee

eeR

bc

bc

ee

eeR

+−=2

Inox L=200µm

D=0.04, 0.046 or 0.06 cm2/s

( )

⋅−⋅⋅−=ϕ−ϕ=ϕ∆ −

−

c

c

L

Lc

refeR

eLsinRarc

l

ll

422

22

1

22tan

Opaque coating

Bulk

Air

Vertical Deflection

Illuminated area

Probe beam

Pump beam

z

L


( )( )[ ]( )[ ]

+−−+−+⋅

ω+=

c

c

aircsurf LjRR

LjR

jee

IT

l

l

12exp1

12exp1ˆ21

2 11 ≈+−=

airc

airc

ee

eeR

bc

bc

ee

eeR

+−=2

Inox L=200µm

D=0.046 cm2/s

( )

⋅−⋅⋅−=ϕ−ϕ=ϕ∆ −

−

c

c

L

Lc

refeR

eLsinRarc

l

ll

422

22

1

22tan

Ph

ase

con

tra

st,

deg.

Spot size

-40

-30

-20

-10

0

0 2.5 5.0 7.5 10.0 12.5

(1)

(2)

(3)

Spot size = 1mm, 2.3mm, 5mm

Opaque coating

Bulk

Air

Vertical Deflection

Illuminated area

Probe beam

Pump beam

z

L


( )( )[ ]( )[ ]

+−−+−+⋅

ω+=

c

c

aircsurf LjRR

LjR

jee

IT

l

l

12exp1

12exp1ˆ21

2 11 ≈+−=

airc

airc

ee

eeR

bc

bc

ee

eeR

+−=2

Inox L=200µm

D=0.046 cm2/s

( )

⋅−⋅⋅−=ϕ−ϕ=ϕ∆ −

−

c

c

L

Lc

refeR

eLsinRarc

l

ll

422

22

1

22tan

Spot size = 2.3 mm

0

0.5

1.0

0.02 0.03 0.04 0.05 0.06 0.07

Sta

nda

rd d

evia

tion

, deg

.

Thermal diffusivity, cm2/s

Spot size, mm1 1.5 2 2.5 3 3.5

(1)

(2)

Opaque coating

Bulk

Air

Vertical Deflection

Illuminated area

Probe beam

Pump beam

z

L

THERMAL WAVE INTERFEROMETRYDiffusivity measurement by Mirage – Reflectivity

bc

bc

ee

eeR

+−=2

Inox L=200µm

D=0.046 cm2/sSpot size = 2.3 mm

-2.0

-1.5

-1.0

-0.5

0

1 2 3 4 5


Ref

lect

ivity

( )( ) ( )

( ) ( ) ( ) ⇒

π+−⋅=

⋅+⋅

⋅−⋅=Γ

∞→

∞→

csurfp

surf

surfp

surf

surfD

fLjR

pTpfTf

pTpfTf

f 12expˆlimˆ

ˆlimˆ

2

( ) [ ]( ) csurf

csurf

DfL

RDfL

π−=Γ

+π−=Γ

⇒2arg

ln2ln 2

•Thermal diffusivity and effusivity measurements






•Evaluation of the photoelastic constants

Main applications

Università degli Studi di Roma “La Sapienza” - Dipartimento di Energetica - Via A Scarpa 14 - 00161 - RomaPer informazioni Prof. M.Bertolotti Tel. 06.49916542 - Email: [email protected]

IR PDS device for trace gas analysis

Ge, Brewster angle

Gas chamber

He-Ne probe

CO2 pump

Position Sensor

Hot zone

Cold zone

Cold zone

Probe He-Ne

Undeflected beam

Deflected beam

Trace Gas Analysis – Infrared Photothermal Deflection Spectroscopy

( ) ( ) ( ) LAew

y

wc

eP

dT

dn

ny wy

L

απωρ

α

≅⋅

−

−=Φ −−

2

22

112

r

Collinear configuration

Photothermal deflection angle

yPump CO2 w

L

0

1

2

3

4

9.0 9.5 10.0 10.5

Lungezza d'onda, µm

Seg

nale

di d

efle

ssio

ne, a

.u.

Spettro di assorbimento CO2

Experimental results – Test on CO 2

9R

9P

10R 10P

0

0.005

0.010

0.015

0.020

0.025

9.5 10.0

exp.

Lunghezza d'onda, µm

Seg

nale

di d

efle

ssio

ne, V

/W

Spettro di assorbimento di una miscella 50 ppm C2H

4

Experimental Results – Test on C 2H4

10P(14) di C2H4

0

0.005

0.010

0.015

0.020

0.025

9.0 9.5 10.0 10.5

Etilenecarbossimetilcellulosa

Lunghezza d'onda, µm

Seg

nale

di d

efle

ssio

ne, V

/W

Experimental results on carbossimetilcellulosa

Carbossimetilcellulosa

Temperature of the treatment Concentration of the emitted ethylene

450°C 7.07 ppm

480°C 46.8 ppm

PLANE THERMAL WAVE RESONATOR

Forward wave

Backward wave

Pump laser beam

1 D illumination

L (Cavity Length)

2° thermal mirror

Absorbing layer1° thermal mirror (glass)

Probe beam

h

Experimental setup

0.1 0.3 0.5 0.7 0.9 1.1 1.3

Cavity length (mm)

1

0.5

0

-0.5

Photothermal deflection signal

Phase

Amplitude logarithm

frequency 36 Hz

THERMAL WAVE RESONATORExperimental evidence in air

• PHOTOTHERMAL TECHNIQUES

• PRINCIPLE OF PHOTOTHERMAL DEFLECTION

•THE HEAT DIFFUSION

• MEASUREMENT OF THERMAL DIFFUSIVITY

• OTHER APPLICATIONS

CONCLUSIONS

PHOTOTHERMAL DEFLECTION TECHNIQUEsabotin.ung.si/~isschool/2018Erice/RLV.pdfGas Vact sg Photoacoustic signal Effective cavity length …you can hear the light (Bell,1880) PHOTOTHERMAL

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