Photothermal lens aberration effects in two laserthermal lens spectrophotometry

Stephen E. Bialkowski

A comparison of theories describing two laser photothermal lens signals is given. The aberrant nature ofthis lens is accounted for in a theory which treats the propagation of a monitor laser in terms of a phase shiftin this laser beam wave front. The difference between theories are discussed in terms of the predicted signalstrengths and temporal behavior. The aberrant theory results in smaller theoretical signal strengths anddifferent functional relationships between signal and analyte level.

One of the issues that many have addressed throughboth experimental and theoretical investigation is thatof the beam overlap in two laser photothermal lensspectrophotometry (TLS). The problem is difficult toaddress because of the many degrees of freedom in theseexperiments.1 In two laser TLS, the monitor laserbeam is characterized by a beam waist and a radius ofcurvature of the wave front at the photothermal lens.This photothermal lens is formed by absorption of ra-diation from a pump laser beam. Although the pumplaser induced photothermal lens is of a Gaussian profile,its effect on the propagation characteristics of themonitor beam is generally assumed to be due to prop-agation through a parabolic medium. The latter par-abolic lens medium approximation greatly simplifiescalculations. For short optical path lengths, the para-bolic medium is a simple lens defined by only a focallength.

To address the fact that there are several degrees offreedom in these experiments, many researchers haveassumed that a maximum TLS signal will occur whenthe beam waist of the monitor laser is equal to that ofthe pump laser.1-4 Furthermore, since the maximumTLS signal will be obtained when the beam waist of thepump laser is a minimum, i.e., focused into the ab-sorbing medium, this places further constraints on theadjustable experimental parameters, allowing themonitor laser beam characteristics (focus position andminimum spot size) to be varied over a relatively smallrange of values.1 5

The author is with Utah State University, Department of Chem-istry & Biochemistry, Logan, Utah 84322.

Received 3 April 1985.0003-6935/85/172792-05$02.00/0.© 1985 Optical Society of America.

It is the purpose of this paper to address the first as-sumption. The questions that are posed here are: Howaccurate are the parabolic lens approximation equationswhen the beam waists are matched? If this is not agood approximation, what are the optimum beamcharacteristics of the monitor laser in these two laserTLS experiments?

Formation of the photothermal lens by a pump laserpropagating in the fundamental TEMoo mode has beenaddressed by Twarowski and Kliger2 and others. Theresults have been keenly summarized by Fang andSwofford.1 Basically, for pulsed laser excitation, thepump laser produces a time-dependent refractive-indexperturbation given by

n(rt) = no +dn 2 lnlOAEp [ -2r 2 1dTJ rw2(1 + 2t/t,)(PCp [2(1 + 2t/t,)J

where A is the sample absorbance, Ep is the pump laserenergy in joules, cop is the pump laser beam waist elec-tric field radius, tc is the characteristic time constantequal to w2/4K, K being the thermal diffusivity of thesample, and the product 5pCp is the density-dependentheat capacity. In the parabolic lens approximation, thetime-dependent focal length of the thermal lens f(t) isdetermined from the curvature of the refractive indexat zero radius and integrated over the path length of thesample medium6:

1 1 dn 8 nlOAEp 2

f(t) no dT rpC +(2)

The TLS signal is due to the change in monitor laserbeam intensity passing through a pinhole placed somedistance d from the sample. Twarowski and Kligerhave shown that for a monitor laser beam focused at adistance z in front of the sample, the on-axis (r = 0)signal is given by

2792 APPLIED OPTICS / Vol. 24, No. 17 / 1 September 1985

I(t) I(-) =2 d zd + ( + ZR)

I(t) f(t) ZR + (d + z)2

d 2tZ z+ Z2

R (3)(fOt) ZR + (d + z)2 (3

where ZR iS the Rayleigh range of the monitor laser.Several assumptions were made in deriving Eq. (3).

Most important are that the fraction of pump laserenergy absorbed by the sample is small and that thelength of the sample is short. Together, these as-sumptions insure that the amount of energy depositedin the sample is a constant over the length of this sampleand that the radial dependence of the deposited energyis also independent of distance through the sample.

Although the latter theory yields an analytical resultwhich defines the TLS signal as a function of sampleabsorbance, it is intuitively incorrect since it does notaccount for the real profile of the photothermal lens,which is not parabolic, nor does it account for monitorlaser beam waist variations other than through the wavefront curvature implicitly defined in the Rayleigh rangeand focus spot distance. An alternative to the paraboliclens approximation has more recently been describedby Weaire et al. in their studies of nonlinear refractionin InSb.7

In the latter theory of the laser intensity change at thedetector, the effect of the nonlinear refractive index istreated in terms of a radially dependent phase shift inthe light propagation equation. Calculation of thesignal for a two laser TLS experiment can be performedin a similar fashion. The phase shift of a ray of wave-length X propagating through a length of medium thathas undergone a small temperature change of d T is

O-2l (d4)-no dTI

Similarly, for an equivalent on-axis temperature change,the parabolic TLS focal length at t = 0 is

1 41 Idn\~TdT, (5)to noco pdT

and the phase shift is related to the parabolic TLS focallength by

AO = 2 (i), (6)

which can be used to compare directly the result of thetwo theories. A typical TLS focal length is -103 m, andfor cow2 = 7.3 X 10-7 m2 and X = 632.8 nm, which is typ-ical in our experiments,5 the maximum on-axis phaseshift is -0.002 rad. Conversely, for a phase shift of AO= 1, the focal length is -1.8 m, which is about the min-imum focal length we have observed. With such smallphase shifts the theory described by Weaire et al. iscertainly valid.

For simplicity, it is assumed that the monitor laseris focused with a minimum beam waist c 0 in the sampleand that the optical path length of the sample mediumis less than the Rayleigh range. The radially dependentphase shift can be calculated from Eq. (1) as

AO/(r,t) = -7 [no - n(r,t)]. (7)

The electric field of the monitor laser as it emerges fromthe sample at z = 0 is

E(r,o) = E(oo) exp r + iO(rt) (8)

Expanding the Gaussian radially dependent portion ofthis complex phase shift results in the series

E(ro) = E(oo) E_ exp 7m=O m! \wmI

(9)

where C2 = [2mc0 + 2 (t)]/C0w2C(t); cw2(t) = @2(1 +2t/tc) and AOk (o,t) = -41 ln(10)AEp(dn/dT)/W2(t)vCPor can be related to the parabolic focal length throughEqs. (2) and (6). The calculation of the electric field ata distance d from the sample is reduced to the calcula-tion for that of a series of TEMOO beams.

E(r,d) = E(o,o) E [i 00,0)] 1 +d 1/2m=0

X exp r2 (d + ItPm (d)} (10)I [,,-,d 2Rm(d) J IIwhere cm(d) = com(0)(1 + d2/z2 ),1/2 Rm(d) = d(1 +z2/d2), Pm(d) = tan-1 (d/zm), Zm = cw2 (0)/7rX, c02 (0)

- wCop(t)/[ 0 + w)(t)], and k = 27r/X. Finally, the r= 0 electric field is

E(o,d) = E(o,o) [i (1 + d2/z2)-1/2 exp[iPm(d)], (11)m=O m!72

which can be used to calculate the TLS signal since I E *(o,d)E(o,d) at the detector.

Although this result is not analytical and thus lessappealing than that of Eq. 3, the theory does incorporatethe effect of a finite monitor beam waist and the aber-rant nature of the photothermal lens. The calculationsare not as formidable as they first appear. Even withlarge on-axis phase shifts on the order of unity, the se-ries converges rapidly, and calculations of the TLSsignals take well under 1 sec on a DEC VAX computer.It may also be noticed that the first term in the seriesis that of the propagation of the monitor laser in theabsence of any phase shift. This fact shortens the cal-culation time requirements for the TLS signal as de-fined by the intensity dependence in Eq. (3).

The restriction that the monitor laser be focused intothe sample is not necessary but is imposed only tosimplify the calculated signal. The effect of this con-dition on the resulting TLS signals has been tested inaccordance with the theory of Twarowski and Kliger.A program was written which scanned the focus positionof the monitor laser relative to the photothermal lensunder the constraint that the spot size of the pump andmonitor laser be equal at the sample. This constraintlimited the range over which the distance could bevaried. For each monitor laser focus to photothermallens distance, a new detector position was calculated forthe maximum TLS signal.5 It was found that the TLSsignal does not vary with the monitor laser focus posi-tion at the optimum detector distance. Thus the resultspresented below may be taken as being generally validfor all pump-monitor laser optical geometries.

1 September 1985 / Vol. 24, No. 17 / APPLIED OPTICS 2793

To address the questions pertaining to the validityof matching beam waists, a set of conditions similar tothose of our previous experiments was chosen.5 Inparticular, the pump laser beam waist electric field ra-dius was 0.8 mm; a He-Ne laser ( = 632.8 nm) was usedas the monitor; and an optimum detector distance of3.61 m, as determined by the procedure prescribedabove, was used in the calculations. Figure 1 illustratesthe effect of the probe laser beam waist radius on thecalculated TLS signal. The upper line is that for theparabolic lens approximation, and the lower line wascalculated using the phase shift method with matchedbeam waists. Both calculated TLS signals demonstratenonlinearities due to the large phase shift and corre-spondingly small focal length. Although it is not ap-parent from this figure, both lines show similar curva-ture when plotted on a relative scale so that the maxi-mum values of each line are equal. They are, however,not identical. A comparison of the TLS signal theoriesfor a monitor laser beam waist which is much smallerthan that of the pump laser is shown in Fig. 2. In thiscase, the differences between the two theoretical de-scriptions are aparently insignificant.

Another critical point to address in comparing thetwo theories is that of the temporal behavior of the TLSsignal. Figure 3 illustrates the temporal behavior ofboth theories for a small phase shift. Again the beamwaists are matched in the sample. Data of this type,e.g., time resolved, have been used to determine thermalconductivities of the sample medium.810 It is clearfrom this figure that different values of tc would bederived from the two curves. However, both theoreticalcurves result in a straight line when the inverse squareroot is plotted relative to time. Thus the calibrationprocedure used by Bailey et al. 9 is no doubt an impor-tant experimental procedure and should insure accuratethermal conductivity determinations. As may begleaned from Fig. 2, the temporal behavior of the TLSsignal is not dependent on the particular theory whenthe monitor laser beam waist is very small comparedwith that of the pump. Subsequently, thermal con-ductivities may be able to be determined without ap-paratus calibration if this condition is met.

The ratio of the TLS signals calculated using bothmethods is illustrated in Fig. 4 as a function of the beamwaist ratio wo/p. The phase shift TLS signal is equalto that of the parabolic lens in the limit as co approacheszero. This is not surprising given the fact that the focallength of the parabolic lens is calculated from the cur-vature of the refractive-index profile at zero radius. Toaddress the question of the maximum TLS signal, thetheoretical phase shift TLS signal has been calculatedas a function of the beam waist ratio. A typical resultof these calculations is shown in Fig. 5. These data areof a small 0.001 phase shift and again are for conditionsof previous experiments. The fact that the relativeTLS signal increases with the monitor to pump beamwaist ratio is surprising. The curvature of a Gaussianshaped refractive-index profile decreases as the radiusincreases. Since for most materials (dn/dT) is negative,the photothermal lens has a negative on-axis focal

4j

< IZ 3-1-JZ

U)

Z Id Ij

llJ jI 1

II0 . .4 I6 -- 1.0. 0 . 2 . 4 . 6 . 8 1. 0

PHASE ANGLE

Fig. 1. Comparison of the parabolic lens theory A to the phase shift

theory B for wp = 0.8 mm, wo = 0.8 mm, d = 3.61 m.

. 000007 4

.000006 1-J

zc .000005-

cn

U) .000004-z'J-j-j .000003-

C

J .000002-3H

. 0

A-,

.2 .4 .6 .8 1.0

PHASE ANGLE

Fig.2. Comparison of the parabolic lens theory A to that of the phase

shift B for wp = 0.8 mm, wo = 0.025 mm, d = 3.61 m.

1.

-Jz0U)

-J

wa:

0

TIME (T/Tc)

Fig. 3. Relative temporal TLS signals for the parabolic lens theoryA and phase shift theory B for AO = 0.001, cop = 0.8 mm, coo = 0.8 mm,

d = 3.61 m. The magnitude of B is a factor of 6 less than that of Aon an absolute basis.

2794 APPLIED OPTICS / Vol. 24, No. 17 / 1 September 1985

I

0H

-JzW.

U)enzLii

-J

a:

0IW

7.

6

5.

4.

3

2

0 ' I I I0 1 2 3 4 5 6 7 8 9 10

PROBE/PUMP BEAM WAIST RATIO

Fig. 4. Ratio of the parabolic lens TLS signal to that of the phaseshift for conditions as in Fig. 3.

-J/.0-

.8l/

_U)

-H

.6 ^/

/ .4

H.2-

0~0 1 2 4 5 7 8 10

PROBE/PUMP BEAM WAIST RATIO

Fig. 5. Phase shift TLS signal as a function of monitor beam waist.Conditions are those of Fig. 3.

PHASE SHIFT

Fig. 6. Comparison of parabolic lens A to phase shift B for conditionsidentical to that of Fig. 1 but with a different signal definition.

length; that is, the on-axis portion of the monitor laserwill diverge. At larger radii, the photothermal lens hasa positive focal length due to the positive curvature ofthe Gaussian refractive-index profile. In fact,

1 =4r 2

- = K exp(-2r2/W2) -_- f (r) cI WP I

(12)

where the constant K is dependent on the magnitudeof the phase shift. Thus one would expect the TLSsignal to decrease with increasing monitor laser beamwaist, since that portion of the monitor laser beamprofile at radii >op/2 will be focused toward the de-tector, while the on-axis portion will be defocused.

This apparent discrepancy is in part due to the defi-nition of the TLS signal [Eq. (3)] where the change inintensity is divided by the intensity during the time thatthe photothermal lens is present. This signal definitionwas chosen because it is proportional to the on-axistemperature change (neglecting the second-order in-verse focal length term) and subsequently proportionalto the absorbance. In the parabolic lens approximation,this linear dependence is important in analytical ap-plications such as spectroscopy and quantitative spec-trophotometry. Another reason for the TLS signalincrease with increasing monitor laser beam waist radiusis that the strength of the photothermal lens in Eq. (12)decreases in an exponential fashion with squared radius.This Gaussian dependence will result in smaller positivefocal lengths for off-axis rays than the negative on-axisfocal length.

The signal defined in Eq. (3) may result in a maxi-mum linear range in the parabolic lens approximationbut does not necessarily result in a maximum linearrange in the phase shift theory. A comparison of theparabolic lens theory with that of the phase shift for asignal definition of [I(oo) - I(t)]/I(oo) is illustrated inFig. 6. Other than this difference in signal definition,the parameters are the same as those used to calculatethe results in Fig. 1. This signal definition results in alinear signal with respect to phase change for the phaseshift theory. Compared to Fig. 1, this latter datasuggest that this is a better way to define the signal foranalytical applications. However, although the phaseshift theory yields a signal, as defined here, which ex-hibits a linear relationship to the phase shift at thisparticular detector distance, it has been found that atdistances other than this optimum, the signal is notlinear. The reason for this fact is not clear at this pointbut nonetheless suggests an empirical method formaximizing the linear range. That is, (1) the best de-tector distance is calculated from the parabolic lenstheory for the particular optical geometry5 ; (2) the signalis defined as [I(oo) - I(t)]/I(oo). An exact derivationof the optimum distance for signal linearity may beobtained from the phase shift theory intensity equa-tions. To perform this calculation, the zero root of thesecond derivation with respect to detector distance maybe found. It appears that this differential must beevaluated numerically.

To summarize: it has been shown that the paraboliclens approximation is accurate only in the limit of a

1 September 1985 / Vol. 24, No. 17 / APPLIED OPTICS 2795

1

small monitor beam waist relative to that of the pump.However, this should not be taken as an invalidation ofdata interpreted in terms of the latter approximatetheory. Quantitative data on thermal conductivities,analyte concentrations, etc. may still be obtained, butthese data cannot be obtained directly from the TLSsignal. Working curves or calibrations must be ob-tained and the resulting TLS signals interpreted relativeto these results. The monitor beam waist dependenceon the TLS signal has also been more accurately de-scribed using the phase shift theory. The conclusionto be gleaned from Fig. 5 is that the greater the monitorbeam waist the greater the TLS signal. This result mayprove useful for future optimization of experiment de-sign for quantitative measures of trace analytes. 5 Al-though the theory developed here was for a pulsedpump laser created thermal lens, this theory will workequally well for cw laser excitation. Treatment of thelatter TLS signals will vary only in the form of the re-fractive index [Eq. (1)].

This research was supported in part by BRSG S07RR07093-17 awarded by the Biomedical ResearchSupport Grant Program, Division of Research Re-sources, National Institutes of Health.

References1. H. L. Fang and R. L. Swofford "The Thermal Lens in Absorption

Spectroscopy," in Ultrasensitive Laser Spectroscopy, D. S.Kliger, Ed. (Academic, New York, 1983), Chap. 3.

2. A. J. Twarowski and D. S. Kliger, "Multiphoton AbsorptionSpectra Using Thermal Blooming," Chem. Phys. 20, 253(1977).

3. K. Mori, T. Imasaka, and N. Ishibashi, "Thermal Lens Spectro-photometry Based on Pulsed Laser Excitation," Anal. Chem. 54,2034 (1982).

4. C. A. Carter and J. M. Harris, "Comparison of Models Describingthe Thermal Lens Effect," Anal. Chem. 55, 1256 (1983).

5. G. R. Long and S. E. Bialkowski, "Pulsed Infrared Laser ThermalLens Spectrophotometric Determination of Trace-Level Ana-lytes," Anal. Chem. 56, 2806 (1984).

6. W. B. Jackson, N. M. Amer, A. C. Boccara, and D. Fournier,"Photothermal Deflection Spectroscopy," Appl. Opt. 20, 1333(1981).

7. D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith, "Ef-fect of Low-Power Nonlinear Refraction on Laser-Beam Propa-gation in InSb," Opt. Lett. 4, 331 (1979).

8. R. T. Bailey, F. R. Cruickshank, D. Pugh, and W. Johnstone,"Pulsed Source Thermal Lens," J. Chem. Soc. Faraday Trans 2'77, 1387 (1981).

9. R. T. Bailey, F. R. Cruickshank, D. Pugh, A. McLeod, and W.Johnstone, "Gas Phase Thermal Diffusivities by a Thermal LensTechnique," Chem. Phys. 68, 351 (1982).

10. M. C. Gupta, S. D. Hong, A. Gupta, and J. Moacanin, "ThermalDiffusivity Measurements Using a Pulsed Dual-Beam ThermalLens Technique," Appl. Phys. Lett. 37, 505 (1980).

MEASUREMENTS REDUCE LIMIT FOR ANISOTROPY OF SPACEIs there a preferred reference frame in space? According toEinstein's Theory of General Relativity, the answer is "no,'and physicists at NBS have confirmed this with an experimentbased on atomic clocks that is 300 times more sensitive thanany previous measurement. In a test of spatial anisotropy, NBSscientists made comparisons between a hydrogen maser atomicclock and a clock based on laser-cooled beryllium ions storedin a magnetic trap. If there were a preferred direction inspace, the hydrogen maser would be relatively unaffected whilethe rate of the ion-based clock would be dependent on itsorientation in space. The experiments, conducted by JohnPrestage, John Bollinger, Wayne Itano, and Dave Wineland,showed no dependence, within 100 microhertz, of the relativeclock rates on their orientation with respect to the sun orfixed stars. This represents about a 300-fold improvement overthe 1961 Hughes et al./Drever experiments based on carefulnuclear magnetic resonance measurements.

PHYSICS

2796 APPLIED OPTICS / Vol. 24, No. 17 / 1 September 1985