aip advances 2011

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AIP ADVANCES 1, 022154 (2011)

Thermal lens spectrometry: Optimizing amplitude andshortening the transient time

Rubens Silva,1,2 Marcos A. C. de Araujo,1 Pedro Jali,1 Sanclayton G. C.

Moreira,2 Petrus Alcantara Jr.,2 and Paulo C. de Oliveira1,a

1Departamento de F sica, Centro de Ciencias Exatas e da Natureza, Universidade Federal daParaba, Jo ao Pessoa, 58051-970, Paraba, Brazil2Faculdade de F sica, Instituto de Ciencias Exatas e Naturais, Universidade Federal do

Par a, Belem, 66093-020, Brazil

(Received 28 April 2011; accepted 19 June 2011; published online 29 June 2011)

Based on a model introduced by Shen et al. for cw laser induced mode-mismatcheddual-beam thermal lens spectrometry (TLS), we explore the parameters related withthe geometry of the laser beams and the experimental apparatus that influence theamplitude and time evolution of the transient thermal lens (TL) signal. By keepingthe sample cell at the minimum waist of the excitation beam, our results showthat high amplitude TL signals, very close to the optimized value, combined withshort transient times may be obtained by reducing the curvature radius of the probebeam and the distance between the sample cell and the detector. We also derive anexpression for the thermal diffusivity which is independent of the excitation laserbeam waist, considerably improving the accuracy of the measurements. The sampleused in the experiments was oleic acid, which is present in most of the vegetable oilsand is very transparent in the visible spectral range. Copyright 2011 Author(s). Thisarticle is distributed under a Creative Commons Attribution 3.0 Unported License.[doi:10.1063/1.3609966]

I. INTRODUCTION

Since the first reports on the thermal lens effect in 1964 and 1965,13 a great number of ap-

plications exploring this effect was developed, including the measurement of very low absorptioncoefficients of transparent liquids, such as water, ethanol, etc..36 Today this technique is known asThermal Lens Spectrometry (TLS). Another very important application is in the determination ofthermal diffusivities.7, 8 Other studies such as the measurement of quantum efficiency and dimeriza-tion equilibria of laser dyes were also reported.9, 10 In TLS experiments, a Gaussian TEM00 excitationbeam is partially absorbed by the sample, creating a refractive index gradient in the radial direction.When a probe beam travels along the heated path, phase differences appear between points of thewavefront with different radial distances. These phase differences distort the wavefront, causing alenslike effect.

The first systematic study of the influence of beam geometric positions in dual-beam thermallens experiment was done by Berthoud et al..11 They showed that TL signal is maximum when theexcitation beam is focused in the sample cell. Further, when the distance between the focus of theprobe beam and the cell is changed, the signal amplitude pass through a maximum at a distanceapproximately 3 Zc, where Zc is the Rayleigh parameter of the probe beam. Recently, Marcanoet al. optimized the amplitude of TL signals, approaching the theoretical limit value, by expandingand collimating the probe beam.5 With this configuration, the TL amplitude was enhanced by 40%but, on the other hand, the TL transient time increased almost two orders of magnitude. However, it is

aElectronic mail: [email protected]

2158-3226/2011/1(2)/022154/12 C Author(s) 20111, 022154-1
http://dx.doi.org/10.1063/1.3609966http://dx.doi.org/10.1063/1.3609966http://dx.doi.org/10.1063/1.3609966mailto:%[email protected]:%[email protected]://dx.doi.org/10.1063/1.3609966http://dx.doi.org/10.1063/1.3609966http://dx.doi.org/10.1063/1.3609966


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022154-2 Silva et al. AIP Advances 1, 022154 (2011)

worth noting that all the above mentioned works investigated only the optimization of the amplitudeof the TL signals.

In this Letter, we present theoretical and experimental results showing that it is possible, forcertain geometrical configurations of the laser beams and experimental arrangement, to obtain highamplitude TL signals with short transient times. The sample used in the experiments was oleic acid,

which is present in most of the vegetable oils and is very transparent in the visible spectral range.

II. THEORY

The first models describing the thermal lens effect considered the temperature rise in the sampleas a parabolic function ofr, and the thermal lens could be treated as perfect thin lens.2, 12 Taking theaberrantnatureoftheTLintoaccountSheldon et al.13 proposedanaberrantthermallensmodelforthesingle beam configuration. About ten years later, a model for cw laser induced mode-mismatcheddual-beam thermal lens spectrometry (TLS) was introduced by Shen et al..14, 15 These modelstake into account the Fresnel diffraction theory to the imaging of the probe beam at the detectorplane. A model with a different approach was reported by Bialkowski and Chartier,16 where theycalculated cumulative electric-field phase shifts produced by a series of Gaussian photothermal-induced refractive-index perturbations. In this work we will base our discussions on the results

obtained by Shen et al..The scheme of the laser beams is shown in Fig. 1(a) and 1(b). Fig. 1(a) shows the traditional

scheme, with the sample positioned at the minimum waist of the excitation beam, and a probe beamwhich minimum waist is at a distance Z1 from the sample cell. Fig. 1(b) shows a modified versionof the traditional scheme, recently introduced by Marcano et al..5 The modification introduced inRef. 5 is that the probe beam is collimated (R1p = ) and the beam radius at the sample cell (1p)is increased.

According to the model introduced by Shen et al., the intensity of the TL signal at the center ofthe probe beam, in the plane of the detector, is given by

I(t) = I(0)

1 2

arctan

2m V

V2 + (1+ 2m)2

tc/2t+ V2 + 1+ 2m

2(1)

where,

m =21p

2e, (2)

= PeLpk

dn

d T

, (3)

and,

V = 21p

p

1

R1p+ 1

Z2

. (4)

All parameters appearing in Eqs. (1)(4) are described in Table I. A few other parameters will bedescribed as they appear in the text.

The Eq. (1) is widely used to fit the experimental data of the transient TL signals, giving thevalues of the both, and tc parameters. The latter is being used to find the thermal diffusivity,obtained by

D = 2e

4tc(5)

A. The amplitude of the TL signal

In our analysis, instead of working with the beam parameters e, 1p, 0p, and the confocalparameters, we prefer to work withthe parameters m and V described by Eq.(2) and (4). After findingthe m and V that maximize the TL signal, we analyze how they can be adjusted experimentally.


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FIG. 1. Schema of the geometric position of the beams in a mode-mismatched TL experiment.(a) Traditional scheme,introduced by J. Shen et al., and (b) scheme introduced by A. Marcano et al..

With the intensity of the light at the detector given by Eq. (1), the steady-state TL fractionalsignal amplitude may be defined as14

S = I(0) I()I(0)

. (6)

After substitution of Eq. (1) into the above equation we get

S = 1

1 2

tan1

2m V

1+ 2m + V22

. (7)


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TABLE I. Description of the symbols used in Eqs. (1)(4) and in Fig. 1.

Symbol Description

I(t) Intensity of the center of the probe laser at the detectore minimum excitation beam waist

1p probe beam waist at the sample cell positionR1p curvature radius of the probe beam at the cell position

Pe excitation laser powerp probe beam wavelengthk thermal conductivity absorption coefficient at the excitation beam wavelengthL sample thickness

dn /dT temperature coefficient of the refractive indextc characteristic thermal lens time constant

D thermal diffusivityZ1 distance from the probe beam waist to the sample cellZ2 distance from the sample cell to the detector

From this equation, we may see that for a fixed excitation power (fixed ), the maximumamplitude of the TL signal will be obtained when the argument of the arc tangent function ismaximum. I(t) is a function of the two geometrical parameters m and V, and both depend on1p. For the sake of simplicity, we will analyze them as independent parameters. By analyzingthe dependence of S on V, we verified that the amplitude of the TL signal has a maximum whenV = Vopt, with Vopt given by

Vopt =

1+ 2m, (8)where the negative solution corresponds to the situation where the sample is positioned before theprobe beam focus (R1p < 0).

An analysis of the dependence of S on m, keeping V at the maximum value, reveals that S

grows as m goes to infinity, approaching a limit, given by

Smax = 1

1 4

2(9)

Experimentally m may reach very large values by increasing the relation between the probe andexcitation beams waists at the sample position. Marcano et al.5, 19 optimized the TL signal, approach-ing the theoretical limit value, by expanding and collimating the probe beam. In their experimentsthe diameter of the probe beam at the sample position was about 6 mm, which corresponds to a valueofm of the order of 10000. From their data, V is estimated to be only about 30, while the optimizedvalue of V is around 120. This will not affect too much the amplitude of the TL signal. However,as we will see in the next section, it will greatly affect the time necessary to reach the steady-stateregime.

In Fig. 2 we show the amplitude of the TL signals for various values of the parameter m asa function of V. It is noticed that larger values of m gives rise to larger amplitudes of TL signals,but this behavior saturates for m around a few hundreds. For example, the maximum TL signal form = 300 is very close to the TL signal for m = 10000, differing by only 3% from each other atV= 30.

B. The time dependence of the TL signal

Another very important parameter in an experiment is how fast a physical phenomenon is, andhow long a measurement will take. In a single beam TLS, tc is the characteristic time of transientsignal. But in the dual-beam mode-mismatched TLS, it is well known that the larger the probe


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FIG. 2. Amplitude of the TL signal divided by , for = 0.1, for various values of the parameter m (1, 5, 10, 100, 300,10000) as a function of V.

beam diameter is, the longer the signal takes to reach the steady-state regime, and consequently, thelonger will be the measurement time. In order to optimize the TL signal, one must to increase theamplitude without affecting too much the measurement time. To make a quantitative analysis of thetime expended in a dual-beam mode-mismatched TL experiment we define a quantity that we callthe half-amplitude time (th), obtained by

I(th) = I(0)+ I()2

. (10)

Expanding Eq. (1), and dismissing terms proportional to 2 , we get

th =1

2

V2 + (1 + 2m)2

2 + 4m2V2

4m2V2

2 + 4m2V2

tc, (11)

where,

= V2 + 1+ 2m. (12)

Since we have dismissed the terms proportional to 2 in the derivation of Eq. (11), a specialcare must be taken when comparing th given by this equation with the experimental results in caseswhere is large ( 0.1). From numerical simulations of Eq. (1), we observed that a correctionfactor approximately given by (1 /3) in the value of th must be applied. For instance, in ourexperiments an average = 0.132 was obtained, which gives a correction factor equal to 0.956 toth .

For V = Vopt one may show that

th =

1/2 +mVopt

tc (13)


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FIG. 3. Half-amplitude time for the TL signal for various values of the parameter m (1, 10, 100, 1000, 10000) as a functionof V.

By plotting th as a function ofV (see Fig. 3) we verified that, when m >> 1, we may substituteVopt by V in a wide interval around Vopt, and Eq. (13) may be written as

th m

V

tc (14)

This result shows that, by keeping m fixed, we may decrease the half-amplitude time of thetransient TL signals by increasing the parameter V. In practice, this may be performed by reducingthe curvature radius of the probe beam (R1p), and also by reducing the distance (Z2) from thedetectorto the sample cell.

The above results may be explained as follows: An increment in the mode-mismatching param-eter m means that the probe beam will probe a larger area of the sample and the heat will take alonger time to propagate to the border and to reach the steady-state regime. Consequently the TLtransient time will be longer. On the other hand, the amplitude of the TL signal will be larger dueto the increment in the phase difference. The parameter V is related to the initial phase differencebetween the wave passing at the center and the wave passing at the probe beam waist. The larger isthe initial phase difference the faster will be the transient regime.

If instead oftc we use th to estimate the thermal diffusivity, we may verify that a very importantsimplification occurs. By substituting m and V given by Eqs. (2) and (4), into Eq. (14), and then into

Eq. (5), the result is

D =

4

p

1

R1p+ 1

Z2

th

1(15)

which is independent of the excitation beam waist e. Since the precise measurement of the excitationbeam waist usually takes a long time to be done and there is an error associated with its estimation,the use of Eq. (15) will considerably reduce the measurement time, giving more reliable results thanEq. (5). Another important issue is that tc is obtained by a fitting process involving Eq. (1), thatdepends on e. On the other hand th may be obtained by a direct inspection of the experimentalcurve.


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FIG. 4. Experimental setup for thermal lens spectroscopy. M - mirror, P - polarizer prism, BS - nonpolarizer beam-splitterprism, S - sample, PH - pinhole, PD - photodetector, L1 ( f1 = 20 cm), and L2 ( f2 = 10 cm) lenses.

III. THE EXPERIMENTS

We performed a set of experiments in order to demonstrate the above results. The experimentalsetup is shown in Fig. 4. The excitation laser was an Ar+ laser operating at a wavelength of 514 nm,and focused by the lens L1, which has focal length f1 = 20 cm. The excitation power was about5 mW. The probe laser was a He:Ne laser emitting at 632.8 nm, and aligned counter-propagatingwith the excitation laser. With this arrangement, we guarantee that the laser beams are completelysuperposed throughout the sample extension, which thickness was 2 mm. Two polarizers withorthogonal orientations were used at the output of each laser in order to eliminate light from onelaser to enter the cavity of the other. The sample material was oleic acid, which is transparent to theprobe laser but has a small absorption at the excitation laser wavelength.

The experiment was divided into two sets of measurements. The first set of measurements wasdone with the scheme of Fig. 1(a), where the probe beam was focused by a 10 cm focal lengthlens (L2) and the sample cell was positioned at the minimum waist of the excitation laser (e =36 m), that was about 20 cm from this lens. With this configuration the probe laser beam diameterat the sample position was about 1.2 mm (1p = 613 m), which corresponds to the same diameterof the original He:Ne laser beam, and R1p was 10.7 cm for this configuration. The second set ofmeasurements was done with the scheme of Fig. 1(b), where the lens L2 was replaced by a pair ofcollimating lenses, that were adjusted to maintain the same diameter of the first set of measurements.R1p was 1605 cm for this configuration.

The beam radii e, 1p, and 0p, and the Rayleigh parameters of the excitation and probebeams, Zce and Zcp, respectively, were measured by the knife-edge technique.17, 18 For the first

set of measurements R1p was calculated from R1p = (Z21 + Z

2cp)/Z1. For the second set, it wascalculated by a linear regression of the data of the beam radius taken at several positions around the

sample position. For both sets of measurements, the distance Z2, from the detector to the samplecell was varied, and we have studied the thermal lens signal as a function of the parameter V thatis related with Z2 through Eq. (4). Tables II and III show the equivalence between Z2 and V foreach measurement. The TL signal was detected by a silicon photodiode, in front of which there wasa pinhole with a diameter of 200 m. The data acquisition was made by a Tektronix DPO 3012digital oscilloscope, synchronized with the switching-on of the excitation laser, and an average of128 measurements was made.

Fig. 5 shows transient TL signals obtained with the detector placed at different distances fromthe sample cell. In this experiment, we have used the scheme of Fig. 1(b), where the probe beam


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TABLE II. Parameter V as a function ofZ2, obtained from Eq. (4) for the geometry of Fig. 1(a), with e = 36 m, 1p =613 m and R1p = 10.7 cm.

Z2(cm) 145.7 28.7 21.7 15.5 11.2

V 15.6 19.9 21.7 24.5 28.4

TABLE III. Parameter V as a function of Z2, obtained from Eq. (4) for the geometry of Fig. 1(b), with e = 36 m, 1p =613 m and R1p = 1605 cm.

Z2(cm) 207.5 140.5 105.0 70.0 50.5 29.0 20.0 15.0

V 1.02 1.44 1.89 2.78 3.81 6.55 9.44 12.6

FIG. 5. Normalized transient TL signals for m = 290, and various values of the parameter V.

is collimated. The upper curve was taken with the detector at a distance of 207.5 cm from thesample cell, which corresponds to V = 1.02, while the lower curve was taken at a distance of only15 cm, which corresponds to V = 12.6 (see Tables II and III). From Fig. 5 one may notice that anincreasing of the amplitude and shortening of the transient time occurs simultaneously as we reducethe distance of the detector from the sample cell. Since the probe beam diameter is of the order of

1 mm, and the shortest distance of the sample to the detector of about 15 cm, we believe that the farfield approximation is still valid for our experiments.The procedure for obtained the half-amplitude time from the thermal lens experimental data

consists in measuring both the intensity I(0) immediately before the opening of the shutter thatmodulates the excitation laser and the intensity I() after stabilization of the transient regime, asshown in Fig. 6. We then calculate the average between the intensities I(0) and I() and lookat which time it occurs. This is what we call the half-amplitude time (th). This procedure is notautomated but it is very simple and accurate and do not require any fitting. A photodiode is used todetect the switching on of the excitation beam and define the origin of the time axis.

The experimental results, showing the increase of the amplitude of the TL signal, as well as theshortening of the transient time with V are shown in Figs. 7 and 8, respectively. Due to geometrical


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FIG. 6. Measurement of the half-amplitude time of an experimental thermal lens signal.

limitations, the data points for V between 1 and 15 were obtained using the scheme of Fig. 1(b),while the data points for V > 15 were taken using the scheme of Fig. 1(a). The probe beam radiusat the sample cell 1p was set with the same value for both schemes. The experiments confirmed thetheoretical predictions for the amplitude and transient times.

In order to verify the applicability of Eq. (15) for the measurement of thermal diffusivities, wemade another set of measurements where we fixed all the geometrical parameters, with exception ofthe excitation laser beam diameter which increases with wavelength, as shown in Table IV. The Ar

+laser was tuned at four different lines (476 nm, 488 nm, 496 nm, and 514 nm), while its waist variedfrom 33 m to 38 m. For these measurements 1p = 719 m, R1p = 13.2 cm and Z2 = 31.7 cm.With this configuration, we have m > 400, V = 27.6 and, during the experiments the exciting laserpower was adjusted such that < 0.1, which guarantees that errors introduced by our approximationsmay be neglected in comparison with the experimental ones. These measurements showed thatthe values of thermal diffusivities, calculated with Eq. (15), stayed practically constant around10.2 104cm2/s, a value that agrees with that obtained by Dadarlat et al. (10.3 104 cm2/s)using a photopyroelectric method.20 The accuracy of the thermal diffusivity obtained by the use ofEq. (15) is affected only by the accuracy of the measurement of R1p, Z2, and th , which are estimatedto within1%. On the other hand, an accurate estimation of the excitation beam waist e demands along time and special equipments. In this work no special care was taken to the measurement of theexcitation beam waist and an estimated error of its measurement was about 5%. It is worth noting

that the parameter m, appearing in Eq. (1), depends on the square ofe, making it very sensitive touncertainties in e. These uncertainties will affect the accuracy of the characteristic time tc obtainedby the fitting process and, consequently, the accuracy of the thermal diffusivity. An estimation ofthe experimental error, using our method, gives a value around 1%, where errors based on thetraditional fitting of the experimental thermal lens curve with Eq. (1) typically gives an error of about5%.

IV. CONCLUSION

We have shown, theoretically and experimentally, that high amplitude TL signals, very closeto the optimized value, combined with short transient times, may be obtained by reducing the


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FIG. 7. Amplitude of the thermal lens signalas a function of the parameter V. Data points for V up to about 15 were obtainedin the collimated probe beam configuration, and above 15 were obtained in the focused probe beam configuration. The solidline is the result given by Eq. (7) with m = 290 and = 0.132.

FIG. 8. Half-amplitude time of thethermal lens signalas a function of theparameter V. Data pointsfor V upto about 15wereobtained in the collimated probe beam configuration, and above 15 were obtained in the focused probe beam configuration.The dashed line is the result given by Eq. (11), and the solid line is the approximated result given by Eq. (13), both withm = 290 and tc = 3.08 ms.


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TABLE IV. Thermal diffusivities obtained from Eq. (15) for several pumping laser lines. Thermal diffusivities areindependent of pumping beam waist when calculated from Eq. (15).

e(nm) e(m) th (ms) D(104cm2/s)

476 33 47.4 10.2

488 35 46.7 10.3496 37 47.1 10.2514 38 47.9 10.1

curvature radius of the probe beam and the distance between the sample cell and the detector. Wenoticed that, by expanding the probe beam, the amplitude of the TL signal increases together withthe corresponding transient time. We also noticed that the focused probe beam configuration givesrise to shorter transient times than the collimated probe beam configuration, for the same beamdiameter. The definition of a new parameter, that we called half-amplitude time or th , was essentialfor the model analysis presented here. We also showed that the measurement ofth is more accurate,precise and easier to obtain than the characteristic time tc, and that it may be used to estimate the

thermal diffusivity of the samples. The major advantage of this method is that thermal diffusivitywill not depend on the excitation beam diameter for large m. We have successfully demonstratedthis applicability by estimating the thermal diffusivity of oleic acid, which agreed with the literaturevalue.

ACKNOWLEDGMENTS

The authors would like to thank Dr. M. L. Lyra for helpful discussions. This work was supportedby FAPESPA, FINEP, CNPq and CAPES, Brazilian agencies.

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