Pulse-Induced Thermal Lensing in Kerr Media

VALERY P. KOZICH,* FLORENCIO E. HERNANDEZ, t and ARISTIDES MARCANO O. Centro de Ffsica, Instituto Venezolano de Investigaciones Cient~cas, Apartado 21 827, Caracas 1020A, Venezuela

The thermal lens effect induced by nanosecond laser pulses in a media having large Kerr nonlinearity is studied. It is shown that thermal lensing is affected by pump-beam self-focusing due to the Kerr nonlinearity, which increases the sensitivity of the thermal lens technique to detect small linear or nonlinear absorption. Index Headings: Thermal lensing; Self-focusing; Z-scan.

INTRODUCTION

The thermal lens technique was found to be a sensitive method to study weak absorption. ~-4 When intense pulsed radiation is used, either linear or nonlinear absorption can lead to local heating of the medium and local changes in the refractive index. Thermal refractive index changes as small as 10 -6, caused by the absorbance of ca. 3 × 10 -4 , c a n be measured without difficulty using pulsed excitation. 4,5 For the cw laser-based technique the best absorbance detection limits for liquids fall near 10-7. 2

It is known that thermal effects in liquids produce a negative diverging lens, which makes it difficult to main- tain a well-collimated beam along the entire length of a long sample. As the thermal lens signal has build-up time, determined by the acoustic propagation time across the excitation laser spot size (a range of tens of nanoseconds to microseconds), other fast nonlinear contributions to the refractive index can dominate during the short-pulse excitation. The Kerr contribution can induce a positive converging lens during the pulsewidth and can affect the propagation of the pump beam in the medium and, as a consequence, the distribution of the deposited heat. Be- cause of the self-focusing effect in Kerr medium, the in- tensity of the pump beam can be rather high along the entire interaction path in a sample. This intensity can increase the sensitivity of the thermal lensing technique when measuring small absorptions, especially when deal- ing with nonlinear absorption.

In this paper we present the results of a study of the thermal effect in solution of solvents characterized by high Kerr nonlinearity dominant during the pump pulsewidth.

Z-SCAN MEASUREMENTS OF KERR AND THERMAL NONLINEARITIES

There are a number of solvents, such as CS2 or benzene, known to be good Kerr media. We have developed single- and dual-beam z-scan techniques 4-6 to measure Kerr and thermal maximum changes in the refractive indices of these solvents. We used 7-ns pulses at a wavelength of

Received 23 May 1995; accepted 21 August 1995. * Author to whom correspondence should be sent. On leave from In-

stitute of Physics, Academy of Sciences ofBelarus, F. Skaryna Avenue, 70, Minsk, Belarus.

t Permanent address: Universidad Central de Venezuela, Caracas, Ven- ezuela.

532 nm from a Nd:YAG laser as the excitation light source. The cw He-Ne laser radiation (k0 = 632.8 nm) served as the probe beam in dual-beam z-scan.

The cw probe radiation enables one to observe the temporal shape of the signal, to separate fast and slow nonlinear changes in the refractive index, and, as a result, to measure induced changes due only to heating. 4,5 The thermal signal is caused mainly by changes in density, following changes in temperature, and rises gradually, reaching the maximum long after the few-nanosecond moment of the heat delivery. The index changes due to the Kerr effect follow the changes of the pump pulse in- tensity and take place only during the pulsewidth. In order to detect the Kerr-induced signal in a dual-beam scheme, one has to have a detection system with high time res- olution and very good filtering of the scattered pump light. Therefore we opted to use single-beam z-scan to measure local variations in the refractive index caused by the Kerr effect.

The experimental schemes of single- and dual-beam z-scan have been described and discussed elsewhere. 4,6 We used a previously developed Gaussian decomposition method 4 to model z-scan dependencies and fit experi- mental and calculated curves. The probe field is assumed to be Gaussian and to propagate in the +z direction

E(z, r, t )= Eo(t) exp XoS(Z) X o ~ i ] ' (1)

where so is the confocal parameter, s(z) = So(1 + zVs~); R(z) = z(1 + soVz 2) is the radius of curvature of the wave- front; and Eo(t) denotes the amplitude of the radiation electric field at the focus. The phase-distorted field at the sample position z is given by

Ed(z, r) = E(z, r)exp[-iA~(z, r)]. (2)

In the case of cubic nonlinearity and Gaussian intensity distribution in a short pump pulse relative to the thermal diffusion time, the phase-distorted field can be expressed at the plane placed distance d from the beam waists as

Ea(d, z, r, t) o o

= Eo(t) ~_~ {([--iA4}o] m) m = 0

x exp [

• ( ' +

-" [m!(1 - zZ/pS)m(1 + z2/s~) '/2

.(gZ + G2),/,]}

7rr2G ?~0(d- z)(g 2 + G 2)

G + i tan- ~ , (3)

1804 Volume 49, Number 12, 1995 0003-7028/95/4912-180452.00/0 APPLIED SPECTROSCOPY © 1995 Society for Applied Spectroscopy

where

Z) [---~-- 1 Xo m

a = ( d - + 2 ] Ls(z) x, po(1 -7-z /p i '

d x z + so 2

g = z2 + S2o ,

and P0 and Xp are confocal parameter and wavelength of the pump beam, respectively; A,I,o is the axis-induced phase shift at the focus.

With the use o f Eqs. 1-3, one can calculate the depen- dence o f the probe beam energy S(z, A~) transmitt ing through the diaphragm as a function of the sample po- sition z for both single- and dual-beam z-scan:

S(z, A,bo) = F(z, ACbo) - F(z, AGo = 0) (4) F(z, AGo = O)

where F(z, AGo) = f;' n dr LEa(d, z, r, ~xO0) l 2, and rl is the aperture radius.

In Fig. 1, the measured single-beam z-scan of a 2-mm- thick benzene cell and the fitting curve are shown. They indicate positive (self-focusing) induced change in the re- fractive index. The index change (An), t ime-averaged over the pulsewidth, was found to be equal to (1.5 ± 0.2) x 10 -5 on the axis of the beam at the focus. The differ- ence, which manifests itself more by sharper wings in the experimental curve, is observed between experimental and calculated dependencies. We attribute this difference to the deviat ion of the pump beam from being perfectly Gauss±an. 7

The Kerr nonlinearity can be treated as instantaneous relative to the light pulsewidth. One can then obtain, for a temporal ly Gaussian pulse, that peak-on-axis index change at the focus An given by

An = k/2(An) . (5)

Accounting for the pump pulse energy of 200 uJ and beam waist o f 45 um, we calculated the peak intensity to be 1.1 G W / c m 2 and the nonlinear coefficient r/2 = (1.8 ± 0 .4) × 10 -14 cmVW.

A two-beam z-scan for thermally induced changes in benzene is shown in Fig. 2. The probe beam has been focused to the 20-um beam waist. As we detect the max- imum of the thermal signal, which is reached in > 40 ns after the pump pulse leaves the sample, we could select the pure thermal contr ibution and measure thermal non- linearity. We have found that the m ax im u m on-axis ther- mal change in the refractive index at the focus is (An)max = - ( 4 . 3 +_ 0.3) x 10 -6 and an absorption coefficient = 1.4 x 10 -3 cm -~.

Similar z-scan measurements of CS2 at a pump energy of 40 uJ give (An) = (2.2 ± 0.2) x 10 -5, which corre- sponds to n2 = (2.5 ± 0.4) x 10 -14 cmVW for the Kerr nonlinearity, (An)m,~ = - ( 4 . 3 _+ 0.3) × 10 -5 for the ther- mal nonlinearity, and absorpt ion coefficient a = 1.4 × 10 -2 cm -1. Taking into account that the thermal signal reaches its m ax im u m more than 40 ns after the heat delivery and approximating the signal growth as next to linear, 8 one may estimate that no more that 20% of its m a x i m u m value could be in effect during the pulsewidth. In a single-beam z-scan technique, one does not separate different contributions; one measures the Kerr nonlin-

0 . 1

m

o 0.0

-0.1

In n l nu • u~~m~~m~ ' I ' I ' I '

- 5 . 0 - 2 . 5 0 . 0 2 . 5 5 . 0

Z (cm) FIG. 1. Measured (points) and calculated (solid curve) single-beam z-scan of a 2-mm-thick cuvette with benzene using 7-ns pulses at X = 532 nm. It depicts positive change in the refractive index with a ADo = 0.35.

o

0 . 0 5

0 . 0 3

0 . 0 0

- 0 . 0 3

- 0 . 0 5

m

R

' I ' I i I t

- 6 - 3 0 3 6

Z ( c m )

Fxo. 2. Measured (points) and calculated (solid curve) two-beam z-scan for maximum thermal changes in the refractive index induced by 7-ns pulses at X = 532 nm and probed by cw He-Ne laser light in a 2-mm- thick cuvette with benzene. It depicts negative change in the refractive index with a ~ o = -0.085.

APPLIED SPECTROSCOPY 1805

6 m

=m

1.5

1.0

0.5

0.0

0.00

[ ]

[ ]

[ ] I

m

[ ]

[]

# m

[] e## # m m

,,m ,S*** mg ~#

• • • A • A • •

0.03 0.06 0.09

Pump Energy (m J) FIe. 3. Measured pump energy dependence of a thermo-optical signal in a 2-cm-thick cuvette with benzene for linearly (squares) and circularly (rombs) polarized pump pulses, and for linearly polarized pulses in the same cuvette filled with ethanol (triangles). The thermal signal is de- tected as a change in the intensity of the probe light of cw He-Ne laser passing through a 2-ram aperture located 80 cm from the sample.

earity diminished partially by the thermal contribution. This means that to get refractive index change (An) de- scribed by pure Kerr nonlinearity, one has to increase the calculated values by approximately 15% of the maximum ( A n ) m a x for the thermal nonlinearity. Therefore one should perhaps increase the measured values of the Kerr non- linearity coefficient n2 by approximately 4% for benzene and 25% for CS2.

From the single-beam z-scan measurements, we con- clude that the Kerr nonlinearity dominates substantially during the pulsewidth, because the Kerr and thermal non- linearities have different time evolutions. Therefore pos- itive refractive index changes leading to self-focusing af- fect the propagation of the pump beam in such media and, as a consequence, influence the distribution of the deposited heat and the focal length &the induced thermal lens.

T H E R M A L LENSING DUE TO LINEAR AND NONLINEAR ABSORPTION

If the sample is thin or if nonlinear effects have a neg- ligible influence on the propagation &the beam, then the dependence of the thermal signal amplitude against the intensity of the pump beam is linear for linear absorption and squared for two-photon absorption? At high inten- sities, the saturation of the signal should occur due to optical aberrations 4 or absorption saturation effects. 9,1°

We have studied thermal lensing caused by vibrational overtone absorption of second-harmonic pulses from the Nd:YAG laser in a 2-cm-length cell filled with benzene or ethanol. The pump beam was focused shortly behind the entrance window of the cell to the beam waist of 30 #m. To get higher sensitivity of thermal lens measure- ments, the probe beam was focused more tightly and its focus has been displaced relative to the focus of the pump beam. 11

In Fig. 3, squares display the energy dependence of the thermal signal in benzene when the pump light is linearly polarized. At low energy the dependence is linear, as it should be for one-photon absorption. At higher pulse energies the dependence becomes nonlinear. To prove that quick nonlinear growth of the signal after 30 #J pump energy was created by self-focusing the pump beam, we have measured the same dependence for the circular po- larized pump light (rombs) and for linear polarized pump light in ethanol (triangles). The use of circular polarized light in benzene diminishes orientational contributions to the Kerr nonlinearity fourfold, and in Fig. 3 one cannot find such nonlinear growth, in ethanol as well, where this nonlinearity is small. Therefore, we attribute the nonlin- ear growth of the thermal signal to self-focusing caused by the orientational Kerr nonlinearity.

It is difficult to correctly model thermal lensing under conditions of self-focusing of the pump beam, but one can get a quali tat ive picture using aberrationless approximation 12,13 or nonlinear ABCD matrix formalism, derived from the aberrationless theory.14 If the pump beam power does not reach the value of catastrophic collapse inside the sample cell, its radius w(z) can be written as 12-14

( w2(z)=P P 1 + 1 - - - 2-5 , (6) P=]PoJ

where P is the pump power and P , is the critical power that depends on the nonlinearity coefficient n2 of the me- dium. For simplicity, the incident beam is supposed to be parallel, i.e., focused on the entrance of the cell.

In calculations, we assume the phase shift due to a thermal lens is very small. In this case the thermal effect for an infinitesimal propagation of the probe wave does not influence the beam size, which suffers the same vari- ations as a linear propagation, whereas the wave-front curvatures are affected by an additional nonlinear change proportional to the pump beam power. We consider the sample as many thin slices and make use of Eqs. 3 and 4 to calculate the energy of a distorted pump field passing through the diaphragm. This field is given as a super- position of the field distorted by the induced phase shift at a number of discrete positions z of thin slices inside the cell. Calculating the probe beam parameter modifi- cations at a plane inside a sample, we do not account for the beam distortion before this plane. Therefore this ap- proximation resembles thin lens approximation when the induced phase shift along a thick sample is integrated and considered as a thin lens placed at a specific plane.

If there is self-focusing, the radius of the pump beam self-focused behind the sample is given by

= -- A ~ c r / / p 2 l , (7)

1806 Volume 49, Number 12, 1995

where A,I,o and A~= are induced phase shifts correspond- ing to P and P, . One should use Eq. 7 instead of the usual equation for a Gaussian beam, which does not have the term (1 - A ~ o / ~ , ) , and take it into account in Eq. 3.

The calculated dependence of the thermal signal as a function of the phase shift Aft0, which is proportional to the pump power, and the sample length L = P0 is shown in Fig. 4. The Z~kCI~cr is equal to 0.01, which means that P~ P , ranges between 0 and 2. One can see that the thermal signal grows nonlinearly versus pump power, but not as much as was observed in the experiment. This means that the main growth corresponds to the pump power P > P=(1 + p~/L2), i.e., when the focal point moves into the cell. It was found that in this case the model of non- linear saturation should be used to avoid infinite intensity at focal points and to evaluate the electric field in the focal region.~2,t3 In a medium with saturable nonlinearity, the beam periodically focuses and defocuses, and the dis- tance between periodic foci decreases as the pump power of the beam is increased. ~2,~3 Perhaps self-focusing under this regime maintains the high intensity of the pump beam throughout the entire sample length and gives rise to the quick nonlinear growth of the thermal signal versus the pump pulse energy.

We have studied the influence of self-focusing on ther- mal signal due to two-photon absorption in diphenyl- butadiene dissolved in benzene or CS2. As we have seen, nanosecond pulses induce both thermal and Kerr changes in the refractive indices of these solvents, but Kerr changes dominate during the pulsewidth. To subtract the contri- bution due to linear absorption in solvent, we use the two-channel differential scheme shown in Fig. 5. Both

1.O

U v

o.s c

0.0 ' I

0 . 0 0 0 . 0 1 0 . 0 2

A D o FIG. 4. Calculated dependence of thermal signal versus on-axis in- duced phase shift A~o at the pump-beam waist with (solid line) and without (dashed line) self-focusing. The confocal parameter of the pump beam Po was chosen to be equal to a sample length, and phase shift After to be 0.01.

BSI

gs i L I L2

D He-Ne L A S E R

N d : Y A G L A S E R

DETI

C E L L I

C E L L 2

FIG. 5. The two-channel experimental apparatus for differential ther- mal lensing. The difference of thermal signals induced in cell 1 and cell 2 is measured.

channels are identical and have equal pump light power. Cells filled with solution and pure solvent are placed in the first and the second channels, respectively.

In Fig. 6, the dependencies of a thermal signal on the energy of the pump beam are shown. One can see the advantage of detecting a differential signal under self- focusing conditions. We succeeded in detecting diphenyl- butadiene in benzene at concentrations as small as 10 -s M.

We have found that in CS2 the detection limit is higher. The thermal signal due to nonlinear absorption in di- phenylbutadiene has been detected at the concentration c >- 4.8 × 10 -s M. The nonlinearity in CS2, despite the

U v

U

0 . 9

0 . 7

0 . 5

0 . 3

0.1 []

0

-0 .1

0 . 0 0

[]

LJ (3

A

N

t %

' I '

0 . 0 2

[J

LJ

A

A A

<9 ~

I

0 . 0 4

P u m p E n e r g y ( m J) FIG. 6. Pump energy dependence of a thermal signal in a 4-cm-thick cuvette with 2.4 × 10 -8 M solutions of diphenylbutadiene in benzene (open squares) and pure benzene (triangles). Rombs display the depen- dence of the difference signal when cuvettes with benzene are located in both channels; solid squares display the difference when one of the cuvettes is filled with diphenylbutadiene solution.

APPLIED SPECTROSCOPY 1807

critical power of self-focusing being lower than for ben- zene, perhaps saturates more quickly because of other nonlinear effects (such as backward-stimulated Raman and Bfillouin scattering). Intensity of the pump beam does not maintain a high level along the entire cell length. Moreover, the linear absorption-producing background signal in CS2 is substantially higher than in benzene.

CONCLUSION

It was observed experimentally that the sensitivity of the thermal lens technique to detect small absorption is increased if the medium under investigation has large Kerr nonlinearity. It occurs because of self-focusing of the exciting beam and the maintenance of high intensity along the sample length of a few centimeters. We found that the amplitude of the thermal signal due to linear absorption, for example, in benzene could be up to three times higher when compared with a case where self-fo- cusing did not take place.

Whereas nonlinear absorption depends on the pump beam intensity, the discussed technique has an additional advantage if applied to the detection of thermal changes in the refractive index due to two-photon absorption. Using nonlinear absorption, we have detected diphe- nylbutadiene dissolved in benzene at a concentration c = 10 -s M, whereas it could not be detected at this concen- tration in other solvents, such as, for example, ethanol, in spite of lower background signal due to linear absorp- tion.

A C K N O WLED G M EN TS

The authors gratefully acknowledge partial financial support from the Consejo Nacional de Investigaciones Cientificas y Tecnologicas (CON- ICIT), Caracas, Venezuela, Fundamental Research Foundation of Be- larus, Minsk, Belarus, and International Centre for Theoretical Physics, Trieste, Italy.

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