Anal. Chem. 1999, 71, 4663-4668
Depth-Resolved Determination of the Absorption Coefficient by Photoacoustic Spectroscopy within a Hydrogel Christian Kopp and Reinhard Niessner*
Institute of Hydrochemistry, Technical University of Munich, Marchioninistrasse 17, D-81377 Munich, Germany
A semiempirical method for the inversion of photoacoustic signal profiles is presented, which allows the depthresolved determination of the absorption coefficient of nonscattering samples. Experimentally determined reference profiles were used to reconstruct photoacoustic signal profiles measured with an indirect photoacoustic sensor system. The system was calibrated with homogeneous hydrogel samples. The developed inversion method was applied to layered hydrogel samples (two or three layers) and to a sample with continuously varying absorption properties. Particularly the investigation of the latter sample showed that the depth resolution of the used photoacoustic detection system is about 10 µm. Photoacoustic spectroscopy (PAS) has been used as a highly sensitive method for nondestructive investigations of inhomogeneous samples.1 This method has been extensively used in chemical, biological, and environmental studies.2 One of the most valuable applications of PAS lies in the field of depth profiling. The absorption of light in a sample causes a thermal and an acoustic wave. Both waves can be used for the depth-resolved analysis of the sample. Depending on the light source (pulsed or intensity modulated excitation) and the detection method, one of these waves dominates. To investigate the thermal wave, a beam, modulated at a certain frequency f, is focused onto the sample surface. The resulting periodic heat flow in the material is a diffusive process, which produces the corresponding thermal wave. The thermal wave propagates toward the sample surface and causes a periodic change of pressure in an adjacent gas volume, which can be detected by a microphone. By varying the modulation frequency f or the phase shift between light and pressure modulation different depths can be analyzed.3 Within a distance of one thermal diffusion length κ (κ ∼ f -1/2), the amplitude of the thermal wave decreases by a factor of e-1. Therefore, the depth range, which contributes to the photoacoustic pressure change in the gas phase, is limited to 1-2κ.3,4 * To whom correspondence should be addressed: (tel) +49-89/7095-7981; (fax) +49-89/7095-7999; (e-mail)
[email protected]. (1) Tam, A. C. Rev. Mod. Phys. 1986, 58, 381-431. (2) Rosencwaig, A. Photoacoustics and Photoacoustic Spectroscopy; Wiley: New York, 1980. (3) Dittmar, R. M.; Chao, J. L.; Palmer, R. A. Appl. Spectrosc. 1991, 45, 11041110. (4) Ma, T.-C.; Munidasa, M.; Mandelis, A. J. Appl. Phys. 1992, 71, 6029-6035. 10.1021/ac990392w CCC: $18.00 Published on Web 09/09/1999
© 1999 American Chemical Society
In the case of aqueous samples, the 2-fold thermal diffusion length corresponds to a maximal depth range of 140 µm (for a practically applicable minimum frequency of 10 Hz), which is insufficient for the envisioned investigation of biofilms (water content 85-95%) in sewage treatment plants. Biofilms, having thicknesses between 100 and 300 µm, are involved in the removal of organic and inorganic pollutants from wastewater.5 This depth limitation can be overcome with the analysis of the acoustic wave, which can propagate over long distances in condensed matter. To investigate acoustic waves in a sample, sensitively pulsed excitation with piezoelectric detection is the method of choice.6 However, due to the lack of an effective inversion method for a real detector geometry, the conversion of photoacoustic signal profiles to profiles of the absorption coefficient was not possible up to now. Karabutov et al.6 presented an inversion procedure restricted to planar acoustic waves, which is not applicable to most experimental conditions because diffraction distorts initially planar wave fronts. The aim of this article is to present a proper inversion method, which considers the diffraction of the acoustic wave front. Photoacoustic Spectroscopy. When a short laser pulse interacts with condensed matter, the absorbed energy is converted into heat by fast nonradiative relaxation processes. Subsequently, the thermal expansion of the instantaneously heated medium causes an acoustic pulse. Therefore, the time-resolved detection of photoacoustic (laser-excited acoustic) pulses allows one to investigate the light absorption during the course of the laser pulse.6,7 The time-resolved photoacoustic signal P(t) can be converted by eq 1 into the corresponding depth-dependent signal
z ) ct
(1)
P(z) for an acoustically homogeneous sample (sound velocity c ) const). For short laser pulses, the spatial depth resolution of photoacoustic depth profiling is determined by the temporal resolution of the acoustic signal acquisition system and by the sound velocity in the medium under study.6 (5) Headley, J. V.; Gandrass, J.; Kuballa, J.; Peru, K. M.; Gong, Y. Environ. Sci. Technol. 1998, 32, 3968-3973. (6) Karabutov, A. A.; Podymova, N. B.; Letokhov, V. S. Appl. Phys. B 1996, 63, 545-563. (7) Karabutov, A. A.; Podymova, N. B.; Letokhov, V. S. Appl. Opt. 1995, 34, 1484-1487.
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Figure 1. Experimental setup for photoacoustic depth profiling.
One considerable advantage of PAS is the great variety of possible detection geometries. The envisioned in situ analysis of a biofilm requires a special indirect detection geometry; i.e., excitation and detection of acoustic waves are performed from the same side of the sample. Photoacoustic Signal Profile. The photoacoustic signal profile P(τ) for indirect photoacoustic detection can be generally described by the following expression:6-8
P(τ) )
βc2 µ(c[τ - τo])φo exp(Cp
∫
c[τ-τo]
τo
µ(ξ) dξ)Λθ(τ - τo) (2)
where τ ) t + (z/c) stands for the time in the coordinate frame, which moves with the velocity of the corresponding waves (τo corresponds to the sample surface), µ is the depth-dependent absorption coefficient of the sample, φo is the energy density (J/ cm2) at the sample surface, β is the thermal expansion coefficient, Cp is the heat capacity of the sample, θ(τ) is the Heaviside unit function, and Λ is a distortion function.8 The latter function is caused by phenomena such as diffraction, attenuation, stress relaxation, and transmittance through interfaces, which occur due to the propagation of an acoustic wave in a medium. The calculation of the photoacoustic signal profile is also complex for plane acoustic waves. Assuming that the acoustic waves of finite dimension are initially almost planar, the profile of the photoacoustic signal will change during propagation (in contrast to spherical acoustic wave fronts). While propagating for a distance z of the order of LD ) πao2/λ (ao is the beam radius and λ the length of the acoustic wave), the cross section of plane waves is approximately twice due to diffraction, and the phase front becomes almost spherical at distances z g 3LD.7 The general diffraction transformation of the photoacoustic signal profile is (8) Oraevsky, A. A.; Jacques, S. L.; Tittel, F. K. Appl. Opt. 1997, 36, 402-415.
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rather complicated. Due to the wavelength dependency of diffraction, the signal has to be corrected in the frequency spectrum. Only in a special case, where uniform absorption and Gaussian transverse light intensity distribution are present, can a paraxial approximation of the photoacoustic signal profile P(τ) be expressed with elementary functions.6 For samples with absorption gradients, as expected here, the forward problem can only be solved numerically. Therefore, it seems almost impossible to precisely solve the inverse problem for real applications, that is, the calculation of the absorption profile from the pressure profile. A semiempirical solution for this problem is presented in a later section. EXPERIMENTAL SECTION Apparatus. Our experimental setup for time-resolved PAS is presented in Figure 1.9,10 A dye laser (FL 3002, Lambda Physik, Go¨ttingen, Germany) pumped by a XeCl excimer laser (EMG 201 MSC, Lambda Physik) is used for excitation. With the laser dye Coumarin 153, the wavelength can be tuned between 520 and 600 nm (a wavelength of 550 nm was selected for all experiments) while the intensity of the laser light, which was measured with a pyroelectric detector (Rj-7100, Laser Precision Corp.), was adjusted by means of wire screens attenuating the pump beam of the dye laser (the pulse energies were in the range of 0.3-2 mJ depending on the wavelength). The laser pulses have a Gaussian temporal light intensity distribution with a full width at half-maximum (fwhm) of 4.5 ( 0.1 ns and are guided via an optical fiber to the sensor head. The signal originating from the detector is fed directly into a preamplifier (Femto Messtechnik, Berlin, Germany). The amplified output signal is recorded with a digital storage oscilloscope (Tektronix TDS 620) for a period of 1 µs, which corresponds to a depth range of 1.5 mm for aqueous samples (c ) 1.5 µm/ns) according to eq 1. Due to the used (9) Kopp, C.; Niessner, R. Appl. Phys. B 1999, 68, 719-725. (10) Kopp, C.; Niessner, R. Analyst 1998, 123, 547-550.
sample rate of 2 × 109 s-1, one pressure transient consists of 2000 data points. The synchronization of the digital storage oscilloscope (DSO) with the laser pulse is performed with a master trigger. To improve the signal-to-noise-ratio, the time-resolved PA signal is averaged by the DSO over 100 pulses. The averaged PA signal is transferred to a personal computer for further data analysis. Photoacoustic Sensor Head. Similar to Karabutov et al.,6,7 a sensor head for the indirect detection of acoustic waves was developed.9 The surface of the medium under study is irradiated through the base and one side of a transparent prism. Due to the oblique trace of the laser pulse across the prism, the illuminated sample area is elliptical (semiaxis: a ) 0.35 cm; b ) 0.28 cm). A wide-band piezoelectric poly(vinylidene fluoride) (PVDF) film is coupled to the opposite side of the base. Both the preamplifier, which is positioned directly behind the piezoelectric film, and the PVDF film are shielded from the electrical background present in the laboratory surroundings. This transducer design allows the detection of acoustic waves with temporal resolutions in the order of nanoseconds, which corresponds to a depth resolution of 10-20 µm for aqueous samples. At an energy density φo ) 1 mJ/cm2, the detection limit for the absorption coefficient is 0.02 cm-1. The detection limit decreases linearly with increasing energy density (an energy density φo ) 200 mJ/cm2 is possible for aqueous samples resulting in a detection limit of 1 × 10-4 cm-1). A detailed description of the sensor head can be found in ref 9. Sample Preparation. To prepare nonscattering models of biofilms, 1.5% (w/w) bacto-agar (Difco Laboratories, Detroit, MI) was dissolved in water, heated between 80 and 90 °C, poured into molds, and then allowed to cool. The hydrogel was colored with different concentrations of a textile dye (Javana, Ku¨nstler-FarbenFabrik, Hallendorf, Germany) that is stable photochemically and thermally. Since the dye does not fluoresce, the total absorbed energy is converted into heat. A concentration of 2.6 g of dye/L of ultrapure water yields an absorption coefficient of approximately 100 cm-1 at a wavelength of 550 nm. Therefore, the concentration of dye controls the light penetration depth. To obtain absorption gradients, sandwiches of hydrogel layers, having different absorption coefficients, were prepared. To guarantee almost perfect evenness and parallel alignment, each layer was created between two glass plates spaced by well-defined adhesive tapes. The thickness of the adhesive tape was about 100-105 µm. Therefore, the thickness of each layer can be varied easily in steps of the tape thickness by sticking one tape on top of the other. The two adjacent layers were separated by a 10-µm-thick polyethylene film to prevent diffusion of the dye between the layers. The absorption coefficient of each sample or layer was determined with a UVvisible spectrophotometer (Beckman, Fullerton, CA). To avoid distortions of the pressure profile by reflections of the pressure waves at the sample boundaries, the thickness of the whole hydrogel sample should be larger than the interesting depth range ( 1: µi ) µi-1 + Afi/[0.144 (kPa cm3/mJ) φ]), while the parameter τf1 determines the boundary of layer 1 facing the sensor head. To continue with the next layer, the profile pm1(τ) ) pm0(τ) Af1pf1ref(τ - τf1) is calculated and the superposition is repeated for pm1(τ). After the following run, the thickness t1 ) c[τs2 - τs1] (c ) 1.50 µm/ns for hydrogels) of the first layer (see Figure 3) and the laser energy density φ1 ) φo exp[-µ1ct1] incident on the second layer (Lambert-Beer’s law) can be calculated. Like this, the complete pressure profile can be analyzed step by step. Finally, the pressure profile can be reconstructed by composing the individual components. Application of the Inversion Method. The developed inversion method was applied to layered hydrogel samples. First, a two-layered hydrogel sample with a decrease in the absorption coefficient was prepared. The pressure profile of this sample was determined experimentally and reconstructed with the developed inversion method. Figure 4 shows the measured and reconstructed pressure profiles of this two-layered sample, which match almost perfectly. Also, the real and reconstructed absorption profiles agree very well (Figure 4). To combine a decrease and an increase of the absorption coefficient, a three-layered hydrogel sample was investigated. The real and reconstructed absorption and pressure profiles obtained for this sample are displayed in Figure 5. The increase and the decrease in the absorption coefficient were accurately reproduced. Despite the good agreement between the measured and reconstructed signal profile, the inverted absorption profile shows an artificial rise of the absorption coefficient in the third layer. According to the reconstruction, a rise corresponds to an offset between reconstructed and real signal profiles. This rise may be caused by noise. An additional reason for the deviation between the real and reconstructed absorption profiles may be the presence
Figure 5. Reconstructed and real absorption profiles for a threelayered sample. The corresponding measured (line) and reconstructed (×) pressure profiles are added (the amplitudes were reduced to 15%).
Figure 6. Reconstructed and calculated absorption profile for the diffusion of a dye into a 100-µm-thick hydrogel layer.
of the polyethylene film, which was used to separate adjacent layers. It had been assumed that the film would not interfere with the signal profile. As a result, the three-layer system reveals that the deviation between real and inverted absorption profiles increases with the complexity of the sample. Finally, the diffusion of the textile dye from a dye solution into a hydrogel was investigated. A 100 µm-thick hydrogel with an absorption coefficient µGel ) 0.4 cm-1 was prepared on the sensor head. Subsequently, a dye solution with an absorption coefficient µSolution ) 6.1 cm-1 was put on the hydrogel layer. About 20-30 s after the dye solution was added, the pressure profile of this sample was measured. The inverted absorption profile of the measured pressure profile is plotted in Figure 6; it is a stepwise approximation to the continuously varying absorption profile. A comparison between the initial absorption profile and the reconstructed absorption profile in Figure 6 shows that the absorption coefficient inside the hydrogel layer (depth range 0-100 µm) Analytical Chemistry, Vol. 71, No. 20, October 15, 1999
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increased. This increase indicated that dye molecules from the solution diffused into the hydrogel layer. Due to the lack of suitable reference techniques, a solution of the diffusion equation was fitted to the experimentally determined absorption profile.13 Resulting fit parameters correspond to an absorption coefficient of 5.98 cm-1 for the dye solution and a lower limit for the thickness of the hydrogel layer of 91 µm. Both values are in good agreement with the sample parameters, which demonstrates that the developed inversion method is reliable even in the case of continuously varying absorption properties. Additionally, this investigation showed that the depth resolution of the used photoacoustic detection system is on the order of 10 µm. Similar to the threelayered sample, an artificial increase of the absorption coefficient occurred at a depth greater than 250 µm. This increase of the absorption coefficient can be related to the reconstruction because no film between solution and hydrogel was present. An explanation for this effect could be that small deviations accumulate with an increasing number of iterations and cause an offset of the reconstructed pressure profile, which results in an artificial increase of the absorption coefficient. (13) Bronstein, J. N.; Semendjajew, K. A. Taschenbuch der Mathematik; Harri Deutsch Verlag: Frankfurt, 1989.
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CONCLUSIONS The presented semiempirical method for the inversion of photoacoustic signal profiles allows the determination of depthresolved absorption profiles for indirect detection geometries. It can be further improved by increasing the number of reference profiles. Additionally, the sensitivity and precision of the sensor system can be enhanced by suppressing the electronic noise, which interferes with the acquired signal profiles. The developed inversion method is applicable to a wide variety of samples. The required reference profiles can be determined experimentally from homogeneous test samples with the same acoustic properties. The inversion procedure was developed for nonscattering samples, but it also should be possible to extend the presented method to scattering samples. ACKNOWLEDGMENT The authors gratefully thank the DFG Special Research Range 411 for financial support.
Received for review April 15, 1999. Accepted August 3, 1999. AC990392W
