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OPTICS LETTERS / Vol. 33, No. 13 / July 1, 2008
Top-hat cw-laser-induced time-resolved modemismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials Nelson G. C. Astrath,1 Francine B. G. Astrath,1 Jun Shen,1,* Jianqin Zhou,1 Paulo R. B. Pedreira,2 Luis C. Malacarne,2 Antonio C. Bento,2 and Mauro L. Baesso2 1
National Research Council Canada, Institute for Fuel Cell Innovation, 4250 Wesbrook Mall, Vancouver, British Columbia V6 T 1W5, Canada 2 Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900, Maringá, Paraná, Brazil *Corresponding author:
[email protected] Received April 15, 2008; accepted May 17, 2008; posted May 27, 2008 (Doc. ID 94968); published June 24, 2008
Thermal lens spectroscopy is a highly sensitive and versatile photothermal technique for material analysis, providing optical and thermal properties. To use less expensive multimode non-Gaussian lasers for quantitative analysis of low-absorption materials, this Letter presents a theoretical model for time-resolved modemismatched thermal lens spectroscopy induced by a cw laser with a top-hat profile. The temperature profile in a sample was calculated, and the intensity of the probe beam center at the detector plane was also derived using the Fresnel diffraction theory. Experimental validation was performed with glass samples, and the results were found well consistent with literature values of the thermo-optical properties of the samples. © 2008 Optical Society of America OCIS codes: 120.6810, 300.6430, 350.5340, 350.6830.
First reported by Gordon and co-workers [1], the thermal lens (TL) spectrometry has been explored as a highly sensitive photothermal technique with attractive characteristics such as its remote and nondestructive nature. The TL effect is based on the heat deposition in a sample by a nonradiative decay process following optical energy absorption by the sample. The sample is usually heated by the absorbed optical energy from a TEM00 Gaussian profile laser, resulting in a transverse temperature gradient, which induces a refractive index gradient behaving like an optical lens. The propagation of the TEM00 excitation laser beam, or another TEM00 laser beam (the probe beam) through the TL is affected, resulting in a change in its intensity profile. By measuring the intensity variation, the information on physical properties of the sample can be obtained. The TL technique has been used widely in chemical analysis as well as in the determination of the optical and thermal properties of gas, liquid, and solid samples [2–7] owing to its high sensitivity and versatility. In most of the TL experiments and their associated theoretical models, the TEM00 Gaussian profile for both the excitation and the probe laser beams has been used and considered to be essential. Owing to the high cost of the TEM00 Gaussian excitation laser, it is desirable to use less expensive non-Gaussian excitation laser to replace the Gaussian one in order to expand the applicability of the TL instruments for quantitative analysis [8,9]. Recently, Li and coworkers introduced the top-hat beam excitation into the pulsed TL technique [8,9], theoretically showing that with the optimum geometry its sensitivity was twice higher than the Gaussian excitation TL scheme. However, an analytical theoretical model, in which absolute physical property measurements can 0146-9592/08/131464-3/$15.00
be experimentally performed, needs to be developed for the practical application of top-hat TL spectroscopy. In this Letter, we report a theoretical model for the time-resolved mode-mismatched thermal lens spectroscopy induced by a cw laser with a top-hat intensity profile. The solution of the heat conduct equation permitted us to calculate the profiles of the temperature and then the refractive index in a sample. The propagation of the distorted probe beam after traveling through the sample, owing to an additional phase shift induced by the TL, was derived using the Fresnel diffraction theory, and a time-resolved expression of the probe beam center intensity at a detector was achieved. Experimental validation was also performed with three glass samples, and their optical and thermal properties were quantitatively determined by fitting experimental data to the theoretical model and found to be in a good agreement with the literature values, indicating that the model can be used in practical TL experiments. In a mode-mismatched TL experiment [3], a cw top-hat beam (excitation laser) excites a weakly absorbing sample of thickness l, causing a TL. A weak TEM00 Gaussian beam, collinear with the excitation beam, travels through and probes the TL. The radii of the excitation and probe beams in the sample are 0e and 1P, respectively. The probe beam propagates in the z direction, and the sample is located at z = 0. The distance between the sample and the detector’s plane is Z2, and the distance between the sample and the probe beam waist of a radius 0P is Z1. In this configuration, it is assumed that (i) the sample dimensions are large compared with the excitation beam radius to avoid edge effects, and (ii) the absorbed excitation laser energy by the sample is low so © 2008 Optical Society of America
July 1, 2008 / Vol. 33, No. 13 / OPTICS LETTERS
that the excitation laser can be considered to be uniform along the z direction. The temperature rise distribution within an isotropic sample is given by the solution of the heat conduction differential equation c共 / t 兲T共r , z , t兲 − kⵜ2T共r , z , t兲 = Q共r兲 [10], with the initial condition T共r , z , 0兲 = 0 and boundary conditions T共⬁ , z , t兲 = 0 and 兩T共r , z , t兲 / z兩z=0 = 0. c, , and k are specific heat, mass density, and thermal conductivity of the sample, respectively. The top-hat source profile follows a unit2 兲U共0e − r兲, step function U共x兲 as Q共r兲 = 共PeAe / c0e in which Pe is the excitation beam power, Ae is the optical absorption coefficient of the sample at the excitation beam wavelength, = 1 − e / 具em典, where e is the excitation beam wavelength, 具em典 is the average wavelength of the fluorescence emission, and is the fluorescence quantum efficiency, which competes for a share of absorbed excitation energy. Using the integral transform methods, Laplace, Fourier Cosine, and Hankel, the solution of the heat conduction equation is
冕冉 ⬁
T共r,t兲 = 4T0
0
1−e
2 −t␣20e/4tc
␣ 0e 2
冊
J0共r␣兲J1共0e␣兲d␣ , 共1兲
in which Jn共z兲 is the n-order Bessel function of the first kind. The characteristic thermal time constant 2 / 4D. D = k / c is the thermal diffusivity of the tc = 0e sample, and T0 = PeAe / 4k. The temporal and radial distributions of the temperature rise inside the sample induce a refractive index gradient, acting as an optical element, causing a phase shift ⌽ to the probe beam ⌽ = 共2 / P兲l共ds / dT兲关T共r , t兲 − T共0 , t兲兴 [3]. Here, ds / dT is the temperature coefficient of the optical path length at the probe beam wavelength p. ⌽ describes the distortion of the probe beam caused by the temperature change in the medium. Using the Fresnel diffraction theory, the propagation of a TEM00 Gaussian probe beam from the sample to the detector plane can be obtained as described in [3]. In this Letter only the center point of the probe beam at the detector’s plane is considered.
Fig. 1. Schematic diagram of the time-resolved experimental apparatus. The excitation and probe beams were provided by a diode-pumped solid-state laser 共532 nm兲 and a He–Ne laser 共632.8 nm兲, respectively. Mi, Li, and Pi stand for mirrors, lenses, and photodiodes, respectively.
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Then, the complex amplitude of the probe beam at the center, using cylindrical coordinates, is given by [3] U共Z1 + Z2,t兲 = C
冕
⬁
exp关− 共1 + iV兲g − i⌽共g,t兲兴dg,
0
共2兲 in which g = 共r / 1P兲 , V = Z1 / ZC (when Z2 Ⰷ ZC), ZC is the confocal distance of the probe beam, C 2 = B exp共−i2Z2 / P兲i1P / PZ2 , B is a constant [3], and ⌽共g , t兲 is given by 2
⌽共g,t兲 =
2
冕
⬁
0
4
冉
1 − e−t␣
22 /4t 0e c
␣20e
冊
⫻关1 − J0共冑mg␣兲J1共0e␣兲兴d␣ ,
共3兲
with
=−
PeAel共ds/dT兲 Pk
.
共4兲
2 2 In Eq. (3), m = 1P / 0e indicates the degree of the mode mismatching of the probe beam and excitation beam. Substituting Eq. (3) into Eq. (2) and carrying out numerical integration over g, the intensity I共t兲 at the detector’s plane can be calculated as I共t兲 = 兩U共Z1 + Z2 , t兲兩2. Figure 1 shows a schematic diagram of the experimental apparatus used for the validation. A diodepumped multimode solid-state laser (Melles Griot, Model 85 GLS 309, 532.0 nm) provides the top-hat beam excitation. The excitation beam was expanded using a set of lenses, and a nearly homogeneous area of the excitation beam profile was taken by a pupil and a lens to produce a top-hat intensity profile in the sample. The radius of the excitation laser beam at the sample position was determined by measuring its transverse intensity using a photodiode plus a
Fig. 2. Normalized TL signals I共t兲 / I共0兲 for the LSCAS-2 (circles) and ZBLAN (squares) glass samples; solid lines, best curve fitting to the model. The inset shows fitted as a function of pump power, and the solid lines in the inset represent the best linear fit.
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OPTICS LETTERS / Vol. 33, No. 13 / July 1, 2008 Table 1. Results of the Experimental Measurements
Samples
D (Measured) 共10−3 cm2 / s兲
D (Literature) 共10−3 cm2 / s兲
l (Measured) (mm)
Ae (Measured) 共cm−1兲
⌰ = / 共PeAel兲 (Measured) W−1
⌰ = −共ds / dT兲 / k (Literature) 共W−1兲
LSCAS-2 ZBLAN Soda-lime
5.7± 0.2 3.0± 0.2 5.0± 0.1
5.8 [11] 2.9 [12] 5.0 [5]
1.50± 0.01 1.17± 0.01 2.75± 0.01
1.70± 0.03 0.35± 0.01 1.00± 0.03
−7.1± 0.2 12.2± 0.2 −6.7± 0.2
−7.0 [11] 12.1 [12] −6.2 [5]
pinhole of 5 m diameter in front of the photodiode. Exposure of the sample to the excitation beam was controlled by means of a shutter (ThorLabs, Model SH05), and the signal from a photodiode P1 was used to trigger a digital oscilloscope (Tektronics, Model TDS 3052) to record the TL signal. A weak TEM00 Gaussian He–Ne laser at 632.8 nm (Melles Griot, Model 05LHP151), almost collinear to the excitation beam, probed the TL. Here the absorbed power of the probe beam by the sample is assumed to be negligible compared with that of the excitation beam. The probe beam was focused by lens L4 共f = 20 cm兲, and the sample was positioned near its confocal plane. There was a small angle ␣ ⬍ 1.5° between the excitation and probe beams, and after passing through the TL the probe beam propagated to a photodiode P2 positioned in a far field 共Z2 ⬇ 5 m兲. A pinhole was put in front of the photodiode P2, and only the central part of the probe beam was detected by the photodiode P2 and then recorded by the digital oscilloscope. The values of 1P, ZC, and Z1 were measured, respectively, as described in [3]. In this work, 0e = 275 m, 1P = 920 m, Z1 = 275 mm and probe beam ZC = 18 mm. Then m = 11.15 and V = 15.2. To validate the theoretical model, we performed measurements with three glasses previously analyzed by a thermal lens using Gaussian excitation. The testing samples were: 2 wt. % Nd2O3-doped lowsilica calcium aluminosilicate glass (LSCAS-2) [11], 0.1 wt. % CoF2-doped ZBLAN glass [12], and 2 wt. % Fe2O3-doped soda-lime glass [5]. Figure 2 shows two examples of normalized TL signal for LSCAS-2 and ZBLAN glasses with excitation power of 66 and 104 mW, respectively. One can see the different shapes of TL transients of these two glasses, which are governed by the sign of the ds / dT: LSCAS-2 glass has positive ds / dT, and ZBLAN glass has negative ds / dT. The numerical curve fittings of I共t兲 to the theoretical model, corresponding to the solid curves in Fig. 2, result in the values of the TL characteristic thermal time response tc, and with different excitation powers, as shown in inset of Fig. 2 for both glasses. Table 1 shows the fitted values of thermal diffusivity and the parameter ⌰ = / PeAel of the three glasses. The absorption coefficients were measured as described elsewhere [11] at the excitation wavelength. It is important to mention that is correlated to the thermal and optical properties of the sample, which are fundamental for material characterization. For instance, it is related with fluorescence quantum
efficiency 共兲, which is one of the most important parameters of optical materials [11,12]. For comparison, in Table 1 we show the literature values for the parameters D and ⌰ = −共ds / dT兲 / 共Pk兲, which are in good agreement with the measured ones in this work. The largest standard deviation of the measured D is less than 7% of the mean value, and the largest ratio of the standard deviation to the mean value of the ⌰ is less than 3%, indicating that the optical and thermal properties of the samples can be precisely measured with the top-hat TL model developed in this work. In summary, we have presented a theoretical model of the time-resolved mode-mismatched TL under top-hat cw laser excitation with an expression of the TL signal at the detector plane. The precise experimental results of three glass samples were in good agreement with literature values and validated the model. It can be concluded that with the theoretical model developed in this work, TL measurements can be experimentally performed providing quantitative information for physical properties using less expensive multimode non-Gaussian lasers, such as high-power diode and optical parametric oscillator based lasers with a wide spectral range. References 1. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965). 2. R. D. Snook and R. D. Lowe, Analyst (Cambridge, U.K.) 120, 2051 (1995). 3. J. Shen, R. D. Lowe, and R. D. Snook, Chem. Phys. 165, 385 (1992). 4. S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis (Wiley, 1996). 5. M. L. Baesso, J. Shen, and R. D. Snook, J. Appl. Phys. 75, 3732 (1994). 6. M. Franko and C. D. Tran, Rev. Sci. Instrum. 67, 1 (1996). 7. J. Shen, M. L. Baesso, and R. D. Snook, J. Appl. Phys. 75, 3738 (1994). 8. B. C. Li and E. Welsch, Appl. Opt. 38, 5241 (1999). 9. B. C. Li, S. Xiong, and Y. Zhang, Appl. Phys. B 80, 527 (2005). 10. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, 1959). 11. E. Pelicon, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, D. F. de Souza, S. L. Oliveira, J. A. Sampaio, S. M. Lima, L. A. O. Nunes, and T. Catunda, J. NonCryst. Solids 304, 244 (2002). 12. S. M. Lima, T. Catunda, R. Lebullenger, A. C. Hernandes, M. L. Baesso, A. C. Bento, and L. C. M. Miranda, Phys. Rev. B 60, 15173 (1999).
