Photothermal SpectroscopyLecture 1 - Principles

Aristides Marcano Olaizola (PhD)Research ProfessorDepartment of Physics and EngineeringDelaware State University, US

Outlook

1. Photothermal effects: thermal lens and thermal mirror.

2. Sensitivity of the photothermal methods.

3. Photothermal characterization of materials.

4. Photothermal spectroscopy – a new kind of spectroscopy

Photothermal are those effects that occur in matter due to the generation of heat that follows the absorption of energy from electromagnetic waves.

Photoelastic – changes in density due to temperature

Photorefractive – change of the refraction index due to temperature

TVV T

TTnn

There are two major characteristics of the photothermal effects:

• Universality• Sensitivity

In any interaction of light and matter there is always a release of heat

Consider one absorbing atom contained in 1 L of water

h

Consider also that a beam of light illuminates the sample continuously. The atom will absorb one photon and will release the energy of this photon toward the surrounding water molecules (heating) in 10-10- 10-13 s.

Thermal diffusion will remove the generated heat. However, this effect is slow. It will take between tens of ms to seconds to equilibrate the temperatures. During this time the atom will accumulate the energy of 108-1013 photons. This can raise the temperature an average of 10-3 oC.

Photothermal method has a phase character. The signal is in most of the cases proportional to the change of phase

TTnL

2

S. E. Bialkowski. “Photothermal Spectroscopy Methods forChemical Analysis”. New York: Wiley, 1996.

Photothermal Effects

Thermal lens

Focused beam

Photothermal Mirror Effect

Thermal lens act like a phase plate

E

)r(ieE r

z

)r(nL2)r(

L

)()( rTTnrn

The change in temperature T is proportional to absorption

To calculate the induced phase we calculate first the distribution of temperature generated thanks to the absorption of a Gaussian beam in the sample.

Excitation Gaussian Beam Intensity

Gaussian Beam Amplitude

oo

o

zzartgi

zRrki

zar

azatEtrzE

)(2)(exp

/)()(),,(

2

2

2

22 /1)( oo zzaza

zzzzR o /)( 22

beam spot radius

curvature radius

2/2oo az Rayleigh range

z sample’s position

For a given sample’s position z and for continuous excitation (CW) the intensity of the excitation beam is

2

2

2

2exp2),(ar

aPtrW o

This function has axial symmetry. It is convenient to solve the Laplace equation in cylindrical coordinates.

where Po is the total light power

Thermal diffusivity equation –Laplace Equation

We write the Laplace equation considering axial symmetry

D thermal diffusivity coefficient absorption coefficient

pC heat capacity density

pCtrWTD

tT ),(2

In cylindrical coordinates with axial symmetry

2

22 1

zrr

rr

We will also neglect the dependence on z (thin lens approximation)

The solution of this equation was first obtained by Whinnery in 1973 (add ref here)

ddtrGtWC

trTt

p

0 0

),,,(),(),(

)(4

exp)(2

)(2/),,(22

tDr

tDtDrItrG o

Io is the modified Bessel function of zeroth order

Using the table integral

2

22

0

22

24/expexp)(

ppbdpbIo

We obtain

4/oo PT

and is the thermal conductivity coefficient

where

darTtrTctt

o )/21/(1

1

22 /2exp)/1(),(

Datc 4/2

-20 -10 0 10 200.000

0.001

0.002

0.003

0.004

t=0.1 s

t=1 s

t=10 s

t=100 sTe

mpe

ratu

re c

hang

e (o K

)

r/re

For water using a 30 mW of 532 nm light

Field of Temperatures generated by the absorption of a beam of light

Refraction index depends on temperature

22

2

o TTnT

Tnn)T(n

For most of the solvents first order is enough. For example for ethanol

TC1043.1)T(n 1o4

Thermal lens works as a phase plate(thin lens approximation)

E

))r(iexp(E

)r(

r

)(2)( rTTnLr

p

The solid samples the thermoelastic effects add an additional term

)(2)( rTnTnLr T

p

Where T is the linear thermo-elastic coefficient

CW excitaion

1

))(/21/(1

))(2exp(1),,(ztt

o

c

dgzmtzg

The phase difference with respect to the center of the beam is

)(/)()( 22 zazazm p

peo dTdnLP 2

where

)t,z,0(n)t,z,r(n2)t,z,r( p

Using the results obtained for the temperature

Mode matching coefficient

D

ApertureSample

Focusing lens

Single beam photothermal lens (PTL) experiment

Pump-probe experiment (m>1)

Probe focusing lens

Beamsplitter

sample

Aperture

Detector

Pump filter

Pump focusing lens

ap z, ab d

zop

zob

Advantages of the pump-probe experiment

1.Higher sensitivity2.Time dependence experiments possible3.Spectroscopy possible by using tunable pump sources.4.Detection technology in the visible. No need of UV or IR detectors.5.Different experimental configurations possible.

Probebeam

L2B

S A

DF

L1

ae

Pumpbeam

z d

Pump-probe optimized mode-mismatched experiment (m>>1)

We define the PTL signal as

)0,,()0,,(),,(

),(tzW

tzWztWtzS

p

pp

b

pp rdrrtzEtzW0

22,,,(),,(

where z is the sample position, b is the aperture radius.

For the sake of simplicity we can consider the radius of the aperture small (b→0). Then the signal can be calculated as

2

22

)0,0,,(

)0,0,,(),0,,(),(

tzE

tzEtzEtzS

p

pp

Plane of the sample Detection plane

We suppose r, r’ << (d-z)

R

r

'r

d-z=d’

x

y y’

x’

’

We calculate the probe amplitude at the far field using the Fresnel approximation

Using a Fresnel diffraction approximation we obtain*

0

)),()1(exp(),0,,( dgtgigiVtzEp

1

))(/21/(1

))(2exp(1),,(ztt

o

c

dgzmtzg

dzzzzzV oppopopp /]1)/[(/ 2

zop is the Rayleigh range of the probe field, zp is the probe beam waist position, d is the detector position

* Shen J, LoweRD, Snook RD (1992) Chem Phys165: 385-396. DOI:10.1016/0301-0104(92)87053-C.

For small phases we can also obtain*

c

co ttVmmV

tmVttzS/22121

/4tan),(222

1

)(/)()( 22 zazazm p

peo dTdnlP

Dzaztc 4/)()( 2

* Marcano A, Loper C, Melikechi N (2002) Pump probe mode mismatched Z-scan, J OptSoc Am B 19: 119-124.

-2 -1 0 1 2

-0,004

0,000

0,004T

L si

gnal

1 s

0.01 s0.001 s

Sample position (cm)The TL signals were calculated using the following parameters: o=0.01, p=632 nm, e=532 nm, zop=0.1 cm, ze=0.1 cm, d=200 cm, D=0.891 10-3 cm2/s and different time values as indicated.

m=1

-20 -10 0 10 20

0,0000

0,0015

0,0030

0.01 s

0.1 s

1 sT

L si

gnal

Sample position (cm)The TL signals were calculated using the following parameters: o=0.01, p=632 nm, e=532 nm, zp=100 cm, ze=0.1 cm, L=200 cm, ae=0 and D=0.891 10-3 cm2/s and different time values as indicated

Optimized mode mis-match experiment

Mode-mismatched scheme

Probe transmission through the aperture

0 5 100

20

40

T(z,t)To

Phot

ocur

rent

(nA

)

Time (s)

Transmitted through the aperture probe light (632 nm) using 250 mW of 532 nm excitation beam of light . The sample is a 2-mm column of water

Experimental PTL signal

PTL signal calculated using the data of previous slide. The solid line is the theoretical fitting of the data.

o

o

TTtTtzS

)(),(

(a)

(b)

Time (sec)

TL

sign

al (a

rb. u

nits

)

0 . 0 0 . 5 1 . 0- 5 0 0 0

0

5 0 0 0

0 . 0 0 . 5 1 . 0

- 0 . 5

0 . 0

0 . 5

PTL signal as a function of time from the distilled water sample (a) and BK7optical glass slab (b). The doted line is the pump field time dependence. Thesignal-to-noise ratio for the PTL is 10 and 75000 for BK7 glass and water,respectively.

-1 0 -5 0 5 1 00 .0

1 .0 x 1 0 -7

(a)

(b)

Sample position (cm)

-10 0 10-0 .0010

-0 .0005

0.0000

Z-scan signal for the distilled water sample(a) and the BK7 optical glass slab (b).

Marcano A. , C. Loper and N. Melikechi, Appl. Phys. Lett., 78, 3415-3417 (2001).

-10 0 10

-0.0006

-0.0004

-0.0002

0.0000PT

L si

gnal

4 Hz

7 Hz

Sample position (cm)

PTL signal from water for two different chopping frequencies : 4 y 7 Hz.

0 2 40.0

0.2

0.4

0.6

T=0

T=15 cm-1

PTL

Sig

nal

Time (s)

T=8.6 cm-1

a b

0 2 4

0.0

0.4

0.8

1.2

T=15 cm-1

Nor

mal

ized

PT

L S

igna

l

Time (s)

T=0

PTL signal is nearly scattering free

a- PTL signal as a function of time for samples (water with latex microspheres) with turbidities of 0, 8.6 and 15 cm-1 obtained using 80 mW of 532 nm CW light from the DPSS Nd-YAG laser,b- Normalized PTL signal for T=0 (black) and T=15 cm-1 (light grey) over the stationary values showing the different signal-to-noise ratios.

2 4 6 8 101E-4

1E-3

0.01

0.1

1

Turbidity (cm-1)

PTL signal

Transmittance

PTL signal as a function of turbidity The solid line is the model prediction.The dependence of transmittance is also plotted for comparison

0.0 0.5 1.0-0.6

-0.3

0.0

0.3

0.6

Time (s)

Pump

PTL signalMeat 2 mm

PTL

Sig

nal

1 10

0.01

0.1

PTL

Sig

nal

Turbidity (cm-1)

Whole milk in water

The signal can be detected in highly turbid samples

Marcano A., Isaac Basaldua, Aaron Villette, Raymond Edziah, Jinjie Liu, OmarZiane, and Noureddine Melikechi, “Photothermal lens spectrometrymeasurements in highly turbid media”, Appl. Spectros. 67 (9), 1013-1018, 2013(DOI: 10.1366/12-06970).

Photothermal Mirror MethodThis method involves the detection of the distortion of a probe beam whose reflection profile is affected by the photo-elastic deformation of a polished material surface induced by the absorption of a focused pump field

Pump beam

ReflectedDistorted Beam

Photothermal Micromirror

The theoretical model used to explain the PTM method is based on the simultaneous resolution of the thermo-elastic equation for the surface deformations and the heat conduction equation.

pCtrITDT

t),(2

Tuu T )1(2)21( 2

where is the Poisson ratio, T is the thermo-elastic

0),,( tzT

Boundary conditions

000

zzzzrz Normal stress components

),0,0(),0,(2 tutru zzp

Initial condition 0)0,,( zrT

The phase difference will be

u(z,r,t)

z

r

Sample

Pumpbeam

Scheme of the PTM experiment

•L. Malacarne, F. Sato, P. R. B. Pedreira, A. C. Bello, R. S. Mendes, M. L.Baesso, N. G. C. Astrath and J. Shen, “Nanoscale surface displacementdetection in high absorbing solids by time-resolved thermal mirror”, Appl.Phys. Lett.92(13), 131903/3 (2008). DOI: 10.1063/1.2905261.•F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M.L. Baesso, N. G. Astrath and J. Shen, “Time-resolved thermal mirror method:A theoretical study”, J. Appl. Phys.104(5), 053520/9 (2008). DOI:10.1063/1.2975997.•L. C. Malacarne, N. G. C. Astrath, G. V.B.Lukasievicz, E. K. Lenzi, M. S.Baesso and S. Bialkowski, “Time-resolved thermal lens and thermal mirrorspectroscopy with sample-fluid heat coupling: A complete model for materialcharacterization’, Appl. Spectros.65(1), 99-104 (2011). DOI: 10.1366/10-06096.•V. S. Zanuto, L. S. Herculano, M. S. Baesso, G.V.B. Lukazievicz, C. Jacinto,L. C. Malacarne and N.G.C. Astrath, “Thermal mirror spectrometry: anexperimental investigation of optical glasses”, Opt. Mat.35(5), 1129–1133(2013).DOI: 10.1016/j.optmat.2013.01.003.•N. G. C. Astrath, L. C. Malacarne, V. S. Zanuto, M. P. Belancon, R. S.Mendes, M. L. Baesso and C. Jacinto, “Finite size effect on the surfacedeformation thermal mirror method”, J. Opt. Soc. Am. B28(7), 1735-1739(2011). DOI:10.1364/JOSAB.28.001735.

References for PTM theory

dgmJtftg oo ),(8/exp),( 2

0

22

4exp2

2),(

2

ErfErfcf

wherectt / arg /;

Erf(x) is the error function and Erfc(x) is the complementary error function.

For opaque materials

A. Marcano, G. Gwanmesia, M. King and D. Caballero, Opt. Eng., 53(12), 127101 (2014). DOI:10.1117/1.OE.53.12.127101.

1pTo ))(1(P

is thermal quantum yield is the thermal conductivityP is the pump power

Datc 4

2

PTM time build-up

0

)),()1(exp(),( dgtgigiVtEp

Diffraction theory provides the value of the field amplitude of the center of the probe beam at the detection plane in a similar way it does for the PTL case

2

22

)0,(

)0,(),()(

tE

tEtEtS

p

ppPTM

0 2000 40000.0

0.5

1.0

Nor

mal

ized

PT

M S

igna

l

Time (t/tc)

=-0.1, zp=100

=-0.1, zp=2000

0 2000 40000.0

0.1

0.2

0.3

=-0.1, zp=100

PTM

Sig

nal

Time (t/tc)

=-0.1, zp=2000

PTM signal for different probe beam Rayleigh ranges

0.0 0.5 1.00.0

0.5

1.0PT

M st

atio

nary

val

ue

PTM signal as a function of PTM phase amplitude

Ref

Pump laser

Probe laser

S

Ampl.

Osc.

SG

DL

FL

B1

B2

M1

M2

M3

D

A

PTM mode-mismatch set-up

0 500 1000 15000.0

0.5

1.0

Nor

mal

ized

PT

M S

igna

l

Time (in tc units)

Glass filter532 nm, 15 mW

tc=1.6 10-3 s.Smax=0.56

0 1000 2000 30000.0

0.2

0.4

0.6

0.8

1.0

1.2

Nor

mal

ized

PT

Mt/tc

Glassy carbon445 nm, 146 mW, 3 Hz tc=60 10-6 sSstationary=0.235

0 500 1000 1500 20000.0

0.5

1.0Ni, 532 nm, 50 mW

tc=1.2 10-5 sSstationary=0.091

Nor

mal

ized

PT

M S

igna

l

Time (in tc units)

PTM signal from Ni and glassy carbon samples using 532 nm and 445 nm diode lasers

0.1 1 10 1001

10

100

1000

Sn

Pt

Ti

Cu

Ni

Quartz

t c (s)

D (10-6 m2s-1)

Equation y = a + b*x

Weight No Weighting

Residual Sum of Squares

0.02141

Pearson's r -0.99712Adj. R-Square 0.99282

Value Standard Error10-6 s Intercept 2.76241 0.0527210-6 s Slope -0.98565 0.03746

Dy2TiO5

Glass

Values of tc versus D

0.01 0.1 1 10 1000.01

0.1

1

10

Dy2Ti

2O

7

Dy2TiO5

SnNi

Cu

QuartzTi

Glassy Carbon

S e/P (W

-1)

o/P (W-1)

Glass

Stationary value versus PTM phase

Conclusions

PTL and PTM are versatile and sensitive technique to determine the absorption and photothermal properties of matters.

The use of pump-probe configuration allows the implementation of PTL an PTM spectroscopy as new method of analysis.