femtosecond two photon absorption measurements based on the accumulative photo-thermal effect and...

1

Femtosecond two-photon absorption measurements based on the accumulative

photo-thermal effect and the Rayleigh interferometer

1Department of Chemistry, 2CREOL, The College of Optics and Photonics, University of Central Florida, P.O. Box 162366, Orlando, Florida 32816-2366, USA

*[email protected]

Luis Rodriguez,1, 2.* Hyo-Yang Ahn,1

and Kevin D. Belfield1, 2

Rodriguez et al. Optics Express, 17, 19617 (2009).

2

Background

1) A single laser pulse passes through the sample and part of the absorbed energy is converted into heat, a localized temperature rise is induced in the sample.2) This temperature distribution produces a change in the spatial distribution of the refractive index.3) TL distorts the wave front of the laser beam at the exit of the sample. By measuring this distortion, the nonlinear absorbance of the sample can be estimated.

Lens

Sample

Pinhole Detector

Laser beam

Laser pulse

τ

T

w

Dwtc 42=

Lens

Sample

Pinhole Detector

Laser beam

Laser pulse

τ

T

τ

T

w

Dwtc 42= Dwtc 42=

3

Background

They observed:1) T>>tc: the Thermal lens (TL) vanishes after the laser pulse, allowing the sample to return to the initial temperature.2) T<<tc: the sample does not return to its initial temperature and an accumulative heating is generated in the sample.

TL effect overcomes the optical Kerr effect.

4

Theoretical considerations

2( )( ) ( ) (1)

I rQ r I r

Lβ∆= =

Sample

Lens

Focal plane

<1>

<2>

<3>

α

Sample

Lens

Focal plane

<1>

<2>

<3>

α

1) Beams <1>, <2> and <3> are delivered by a HRR laser system. <1> and <3> are placed in the same optical axis.2) <1> and <2> have identical phases, intensities, and diameters. 3) <3> induces a localized TL yielding a refractive index to be compared.4) Sample is a transparent optical medium, being characterized by nonlinear absorption coefficient β and negligible linear absorbance αL.5) <1> and <2> are focused together by a positive lens, producing a parallel interference pattern.

Because αL<<1 we assume that the heat source, radial energy flow into the sample in a unit of volume and a unit of time, is only generated by absorption of two photons:

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( )2 22

2( ) exp 2 (1.a)

PI r r w

wπ= −

2( , ) 1( , ) ( ) (2)

T r tD T r t Q r

t Cρ∂∆ − ∇ ∆ =

∂Sample

Lens

Focal plane

<1>

<2>

<3>

α

Sample

Lens

Focal plane

<1>

<2>

<3>

α

Where:

We solved the heat conduction equation:

The following expression for the temperature distribution is obtained:

(2.a)DC

κρ

=

2 2 2

2 2 0

1 1 4( , ) exp ' (3)

1 4 ' 1 4 '

t

c c c

P r wT r t dt

w t t t t t

βπ κ

−∆ = + + ∫

2 4 (3.a)ct w D=


6

Sample

Lens

Focal plane

<1>

<2>

<3>

α

Sample

Lens

Focal plane

<1>

<2>

<3>

α

The temperature distribution induces an additional phase in the wavefront of the probe beam <1>:

Where the photo-thermal phase shift is:

2( , ) ( , ) (4.a)

L dnr t T r t

dT

πφλ

= ∆

( ) ( )2 22

21

4 1( , ) ln 1 4 1 (5)

2 ! 1 4

i i

ci c

r wLP dnr t t t

w dT ii t t

βφπ λκ

∞

=

− = + + − +

∑

The exponential integral in Eq. (3) is writing as a power series and the photo-thermal phase shift is described by:

[ ]1 1( , ) ( , )exp ( , ) (4)e iE r t E r t r tφ=


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Sample

Lens

Focal plane

<1>

<2>

<3>

α

Sample

Lens

Focal plane

<1>

<2>

<3>

α

In order to evaluate the magnitude of the induced phase shift we consider the phase variation around the optical axis of the beam <3>. This is done by calculating first:

Followed by the evaluation of Eq. (6) when the steady state is reached tc/t<<1:

( )2

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( ) ( , ) (0, )

4 1 1 (6)

2 ! 1 4

ii

i c

t w t t

LP dn

w dT ii t t

φ φ φ

βπ λκ

∞

=

∆ = −

− = − +

∑

2

2( ) (0) 1.32 (7)

2

LP dn

w dT

βφ φ φπ λκ

∆ = ∆ ∞ − ∆ ≈

This expression is obtained with the assumption that the stationary TL is a perfect lens with no aberration, its accuracy is suitable for a small phase shift (∆φ<1) measured on-axis pump beam. This is because experimental variables such as optimal position of the sample and experimental setup geometry are not required.


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Sample

Lens

Focal plane

<1>

<2>

<3>

α

Sample

Lens

Focal plane

<1>

<2>

<3>

α

If the sample is fluorescent, Eq. (7) must consider the following quantity:

Because the two-photon up-converted fluorescence occurs in addition to the heat source.

( )1 2 (7.a)eλ λ− Φ

[ ]( ) ( ) / (7.b)e E d E dλ λ λ λ λ λ= ∫ ∫is the average wavelength of the emitted light, where E(λ) is the two-photon up-

converted fluorescence spectrum of the sample. The TL phase shift is written as:

2

21.32 1 (8)

2 2 e

LP dn

w dT

β λφπ λκ λ

Φ ∆ = −


9

Sample

Lens

Focal plane

<1>

<2>

<3>

α

Sample

Lens

Focal plane

<1>

<2>

<3>

α

1) The dimension of TL is smaller than the separation between beams <1> and <2>.2) Only beam <1> is locally altered by TL.3) The phase of <2> remains constant.4) TL produces a small phase shift in the interference pattern while the fringe spacing remains constant.5) At the focal plane of the lens, the intensity distribution of the interference pattern in the x-direction is given by:

The phase is extracted by taking the Fourier transform in two dimensions [1]:

2( , ) ( , ) ( , ) cos( ( , )) (9)

xi x y a x y b x y x y

x

π φ= + +∆

( ){ }( ){ }

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

exp ( , ) ( , ) ( , ) (10)

I X Y A X Y FT j x y B X Y X X Y

FT j x y B X Y X X Y

φ δ

φ δ

= + − ⊗ ⊗ − ∆

+ ⊗ ⊗ + ∆

[1] Rodriguez et al. Opt. Commun. 247, 453–460 (2005).


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Sample

Lens

Focal plane

<1>

<2>

<3>

α

Sample

Lens

Focal plane

<1>

<2>

<3>

α

By using a band pass filter we extracted one of these Dirac’s delta:

( ){ }exp ( , ) ( , )

( , ) (10.a)

G FT j x y B X Y

X X Y

φ

δ

= ⊗

⊗ + ∆

Taking the inverse 2D-FT of G:

( )( , ) ( , )exp ( , ) (11)g x y b x y j x yφ= −

To eliminate the function b(x,y), an interferogram in absence of TL is evaluated making the same Fourier treatment:

( )( , )( , ) exp ( , ) (12)

( , )

g x yh x y j x y

b x yφ= −

The photo-thermal phase shift at any point can be determined by:

{ }{ }

Im ( , )( , ) arctan (13)

Re ( , )

h x yx y

h x yφ

=


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Beam Splitter

Beam ExpanderPin Holes

Sample

Lens 1

Lens 2

CCD Camera

Mirror

fs Laser

Neutral Density Filter

Optical Attenuator

<1>

<2>

<3>

α

ShutterBeam Splitter


Sample

Lens 1

Lens 2

CCD Camera

Mirror

fs Laser


Optical Attenuator

<1>

<2>

<3>

α

ShutterShutter

Mira 900

Experimental setup

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Beam Splitter


Sample

Lens 1

Lens 2

CCD Camera

Mirror

fs Laser


Optical Attenuator

<1>

<2>

<3>

α

ShutterBeam Splitter


Sample

Lens 1

Lens 2

CCD Camera

Mirror

fs Laser


Optical Attenuator

<1>

<2>

<3>

α

ShutterShutter

Experimental setup

Linear interferogram nonlinear interferogram

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Signal processing [2]

From the experimental interference pattern (a), only the high frequency components and noise are removed by using a low pass Fourier filter (LPF), the carrier frequency and the intensity levels of the interferogram remain constants (b).

I1 = imread(“I1.jpg”);I1a= filter2LR(h,I1);

LPF

(a) (b)

[2] C. Gorecki, Pure Appl. Opt. 1, 103-110 (1992)

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1) The 2D Fast Fourier Transform (FFT) is applied to the filtered pattern (c) and its nonlinear spatial spectrum can be observed in (d):

Snl = fft2(I1a);imagesc(fftshift(log(1+abs(Snl))));

2) (d) shows the three terms of Eq. (10), represented in this image as three distinct Dirac’s spots.

( ){ }( ){ }

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

exp ( , ) ( , ) ( , ) (10)

I X Y A X Y FT j x y B X Y X X Y

FT j x y B X Y X X Y

φ δ

φ δ

= + − ⊗ ⊗ − ∆

+ ⊗ ⊗ + ∆

3) The red square frame in (d) indicates the Dirac’s distribution selected by the band pass filter. This filter basically consisted in cropping the selected spot from the image.

[sAnl, rect] = imcrop;

FFT

(c)

(d)

Signal processing

15

The selected Dirac’s distribution (e) is again transformed by using an inverse 2-D FFT algorithm to obtain the complex function shown in (f):

gnl = ifft2(Gnli1);

The same Fourier treatment was performed on the linear interferogram in order to obtain the function b(x,y), and the function h is calculated:

h=gnl./gn0;

i-FFT

Beam deformation induced by thermal lens

(e)

(f)

Signal processing

16

The photo-thermal phase shift map is obtained by:

Phase=-angle(h);mesh(Phase);

The red square centered in the middle of the map indicates the area where the photo-thermal phase shift is evaluated:

rect2=[100 100 5 5];Phase1=imcrop(Phase,rect2);

In this area an average value of the magnitude of the phase difference is calculated:

Averg=mean(mean(Phase1));

∆φ=1.8 Rad

{ }{ }

Im ( , )( , ) arctan (13)

Re ( , )

h x yx y

h x yφ

=

Signal processing

17

Experimental results

1) Figures (a) and (b) show typical experimental interferograms obtained from 1 mm of path length of CS2 and 240 mW. (b) represents the nonlinear interferogram.

2) The two pinhole interferometer system provided a high spatial quality of the reference and probe beam. As a consequence, the statistical uncertainty of the measurements is reduced.

3) (c) shows the intensity profiles determined from the middle of each image in the horizontal direction (x-direction). The phase is weakly shifted to the right side (red line) when the stationary thermal lens is induced, only the phase is affected.

18The photo-thermal phase shift map obtained from CS2 sample.


19

By using this expression and the parameters given in the table for CS2, we obtain for the 2PA coefficient of CS2 [5]:

The nonlinear absorption coefficient is estimated using the previously figure and Eq. (8):

2

21.32 1 (8)

2 2 e

LP dn

w dT

β λφπ λκ λ

Φ ∆ = −

It is in good agreement with that previously reported by Ganeev et al. [6]:

( ) 135.6 0.2 10 m Wβ −= ± ×

134.2 10 m Wβ −= ×

[5] Rodriguez et al. Optics Express, 17, 19617 (2009). [6] Ganeev et al. Applied Physics B 78, 433–438 (2004).

Material

κ

(Wm-1K-1)

dn dT1 4( )10K − −

w

(µm)

P

(W)

Φ

-

eλ

(nm)

[ ]C

4( )10M −

CS2 0.161 5.00 6 0.24 0.00 - -

RhB 0.200 3.94 37 0.15 0.70 583 1.0

Rh6G 0.200 3.94 37 0.16 0.95 572 2.5

Fluorescein 0.598 0.91 6 0.35 0.90 545 1.9


20

(b)

(d)

200 250 300 350 400 450 500

0.1

Fluorescein

R2= 0.987Slope = 1.8 ± 0.1

Pho

to-t

herm

al p

hase

Flu

ores

cein

(R

ad)

Incident power (mW)

100 150 200 250 300 350 400

0.1

1

RhB

R2=0.987Slope=2.1± 0.1

Pho

to-t

herm

al P

hase

RhB

(R

ad)

Incident power (mW)

(a)

(c)

100 150 200 250 300

0.1

1

Rh6G

R2=0.995Slope=2.0± 0.1

Pho

to-t

herm

al p

hase

Rh6

G (

Rad

)

Incident power (mW)

100 200 300 400 5000.01

0.1

1

CS2

R2=0.997Slope=2.02 ± 0.03

Pho

to-t

herm

al p

hase

CS

2 (R

ad)

Incident power (mW)

(b)

(d)

200 250 300 350 400 450 500

0.1

Fluorescein

R2= 0.987Slope = 1.8 ± 0.1

Pho

to-t

herm

al p

hase

Flu

ores

cein

(R

ad)

Incident power (mW)

100 150 200 250 300 350 400

0.1

1

RhB

R2=0.987Slope=2.1± 0.1

Pho

to-t

herm

al P

hase

RhB

(R

ad)

Incident power (mW)

(b)

(d)

200 250 300 350 400 450 500

0.1

Fluorescein

R2= 0.987Slope = 1.8 ± 0.1

Pho

to-t

herm

al p

hase

Flu

ores

cein

(R

ad)

Incident power (mW)

100 150 200 250 300 350 400

0.1

1

RhB

R2=0.987Slope=2.1± 0.1

Pho

to-t

herm

al P

hase

RhB

(R

ad)

Incident power (mW)

(a)

(c)

100 150 200 250 300

0.1

1

Rh6G

R2=0.995Slope=2.0± 0.1

Pho

to-t

herm

al p

hase

Rh6

G (

Rad

)

Incident power (mW)

100 200 300 400 5000.01

0.1

1

CS2

R2=0.997Slope=2.02 ± 0.03

Pho

to-t

herm

al p

hase

CS

2 (R

ad)

Incident power (mW)

(a)

(c)

100 150 200 250 300

0.1

1

Rh6G

R2=0.995Slope=2.0± 0.1

Pho

to-t

herm

al p

hase

Rh6

G (

Rad

)

Incident power (mW)

100 200 300 400 5000.01

0.1

1

CS2

R2=0.997Slope=2.02 ± 0.03

Pho

to-t

herm

al p

hase

CS

2 (R

ad)

Incident power (mW)

Log-log plot of the experimental measurements, the calculated slope indicates a quadratic dependence between the phase shift and the incident power: ∆φ∝P2.


21

To estimate the resolution of our results, Eq. (8) is rewritten in the following simplified form:

( )dn dT P qφ λκ ∆ =

0q LIβ=

This q value is difficult to detect using the classical nonlinear transmittance technique.

Where q is the nonlinear absorbance. We define the accumulative enhancement factor (terms enclosed in the squared brackets) with respect to a nonlinear transmission technique.

( )( )

4 1 1 1

4 6

0.03 , 90 , 5 10 , 0.161 , 800

10 10

Rad P mW dn dT K Wm K nm

dn dT P q

φ κ λ

λκ

− − − −

−

∆ = = = × = =

= ⇒ ≈

We also suppose that the nonlinear phase shift::

is much smaller than the photo-thermal phase when low incident intensity is used.

2 02 Ln Iπ λ


22

Absolute 2PA cross section spectra measured at different excitation wavelengths for the Fluorescein (a),

Two-photon absorption coefficients (β) can be also expressed in term of the 2PA cross section (δ) by means of:

2[ ] AC N hβ δ ν=

[7] Makarov et al. Opt. Express 16, 4029-4047 (2008).

[7]720 740 760 780 800 820 840 860

10

20

30

40

50

Fluorescein, H2O pH=11

2PA

cro

ss s

ectio

n (G

M)

Excitation wavelength (nm)

(b)

720 740 760 780 800 820 840 860

10

20

30

40

50


2PA

cro

ss s

ectio

n (G

M)


720 740 760 780 800 820 840 860

10

20

30

40

50


2PA

cro

ss s

ectio

n (G

M)


(b)(a)


23

And Rhodamine B, Rhodamine 6G, (b). Good agreement was obtained with reference [7].

[7] Makarov et al. Opt. Express 16, 4029-4047 (2008).

720 740 760 780 800 820 840 860 880 90040

60

80

100

120

140

160

180

200

220

240

RhB, CH3OH

Rh6G, CH3OH


2PA

cro

ss s

ectio

n of

RhB

(G

M)

0

10

20

30

40

50

60

70

80

2PA

cross section of Rh6G

(GM

)

(a)

720 740 760 780 800 820 840 860 880 90040

60

80

100

120

140

160

180

200

220

240

RhB, CH3OH

Rh6G, CH3OH


2PA

cro

ss s

ectio

n of

RhB

(G

M)

0

10

20

30

40

50

60

70

80

2PA


(GM

)

720 740 760 780 800 820 840 860 880 90040

60

80

100

120

140

160

180

200

220

240

RhB, CH3OH

Rh6G, CH3OH


2PA

cro

ss s

ectio

n of

RhB

(G

M)

0

10

20

30

40

50

60

70

80

2PA


(GM

)

(a)(b)

720 740 760 780 800 820 840 860 880 90040

60

80

100

120

140

160

180

200

220

240

RhB, CH3OH

Rh6G, CH3OH


2PA

cro

ss s

ectio

n of

RhB

(G

M)

0

10

20

30

40

50

60

70

80

2PA


(GM

)

(a)

720 740 760 780 800 820 840 860 880 90040

60

80

100

120

140

160

180

200

220

240

RhB, CH3OH

Rh6G, CH3OH


2PA

cro

ss s

ectio

n of

RhB

(G

M)

0

10

20

30

40

50

60

70

80

2PA


(GM

)

720 740 760 780 800 820 840 860 880 90040

60

80

100

120

140

160

180

200

220

240

RhB, CH3OH

Rh6G, CH3OH


2PA

cro

ss s

ectio

n of

RhB

(G

M)

0

10

20

30

40

50

60

70

80

2PA


(GM

)

(a)(b)

[7]


24

Conclusions

A method was developed that allows fast and accurate measurements of the 2PA cross sections in liquid samples using two readily obtained images. This method relies on determination of the phase shift of interference pattern induced by an accumulative thermal lens effect in a liquid sample.

The technique was validated by measuring 2PA coefficients of standard nonlinear media CS2 (neat) and the 2PA cross section of widely used organic molecule standards (Rhodamine B, Rhodamine 6G and Fluorescein), providing results that were in good agreement with those previously reported in the literature.

Since the experimental data can be acquired rapidly, this technique may be useful to determine 2PA coefficients for samples susceptible to photodegradation, such as photoinitiators, reagent for photouncaging, and photosensitizers.

The results obtained using the accumulative photo-thermal effect and Rayleigh interferometery show that this new technique is robust, sensitive, and has the accuracy to measure 2PA cross sections in dilute solutions.

25

Thanks!

AcknowledgmentsWe wish to acknowledge the National Science Foundation (ECS-0621715 and CHE-0832622) for support of this work.

femtosecond two photon absorption measurements based on the accumulative photo-thermal effect and...

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