The fact that the photothermal refraction signal amplitude is proportional to excitation irradiance has prompted researchers to use more powerful lasers in order to decrease detection limits. Although higher irradiance enhances the signal, it may also result in nonlinear effects. Absorption-based nonlinear effects are due mostly to dynamic excited state population changes producing optical saturation, bleaching, and multiple- photon absorption. Most nonlinear effects become more problematic using short-pulsed excitation lasers where instantaneous irradiances are high and the excitation time scales become short compared to complete ground state recovery. Nonlinear effects result in analytical calibration problems that are difficult to solve for all but the most controlled laser sources.

We have observed nonlinear bleaching effects in both pulsed and continuous laser excited experiments. These are observed using even modest (mW) optical powers or (m J) pulsed laser energies. Although this may be thought to limit the utility of photothermal spectroscopy to the analysis of all but the most favorable analytes, the nonlinear absorption also opens up new areas of application. In particular, the irradiance or integrated irradiance-dependent photothermal signals may be analyzed yielding a variety of excited state kinetic and optical absorption cross section data. Due to the enhanced sensitivity of photothermal refraction spectroscopy over transmission experiments, the data are easily obtained using low concentration samples.

Irradiance-dependent photothermal refraction signal data obtained for organic dye substances are used to illustrate the method. Using continuous excitation, the data can be used to determine excited triplet state absorption cross sections and lifetimes of the excited metastable triplet states. An additional parameter is found when using pulsed laser excitation sources. In particular, we have been able to deduce rate constants for T₂ to T₁ relaxation after excited state excitation. These sub-nanosecond lifetimes can be deduced by competitive excitation.

The data is rather easily obtained and does not require complicated excitation-probe arrangements typically employed for excited state studies. The data is more directly related to excited state absorption effects thought to be useful for the design of optical limiters.

Methods for deriving models for the photothermal lens signals are discussed in another paper in the conference. An energy level diagram is shown in Figure 1. Species used in this study are modeled as three- or four-level systems with fast intersystem crossing rates (ISC) from the S₁ state. No evidence for excited single state, S_1® S₂, absorption has been observed and this mechanism is not included in the model. The model nonlinear photothermal lens signal equations are found by; first, solving the differential equations for the individual state number densities; second, integrating the energy absorbed during irradiation; finally, calculating the signal strength from the irradiance-dependent power distribution. The irradiance-dependent signal is formulated and used to determine rate parameters by data regression.

The resulting equations for pulsed laser excitation have previously been published. For continuous laser excitation, the strength of the photothermal lens is given by the relatively simple expression for steady-state (long irradiation times)

(1)

(dn/dT) (K^-1) is the thermal-optical coefficient, k (Wm^-1K^-1) the sample thermal conductivity, l (m) is the sample pathlength, and E (W m- ²) is the continuous excitation laser irradiance. s ₁ and s ₃ (m²) are the cross sections for ground singlet state and excited triplet state absorption at the excitation wavelength and the Y_H are heat yields included to account for radiative energy loss. The bleaching irradiance is

(2)

t =1/k_T-S (s) is the T₁ state relaxation time constant and f _T is the quantum yield for triplet production

(3)

k_IC (s^-1) is the fluorescence and internal conversion rate constant. Heat loss through fluorescence emission is accounted for in the singlet state heat yield

(4)

f _f is the fluorescence quantum yield, and n _f is the frequency of maximum fluorescence emission. It is more facile to view these results by an equation contained an excitation irradiance absorptance coefficient

(5)

The small-signal inverse focal length is

(6)

The irradiance-dependent absorption coefficient is

(7)

There is no evidence for triplet state luminescence and Y_H^(T) is assumed to be unity for data regression purposes.

A schematic diagram of the apparatus used to measure the continuous laser excitation power-dependent nonlinear absorption is shown in Figure 2. A 514.5 nm argon ion laser is used or excitation. The output first passes though a line filter and is spatial filtered. Excitation laser beam attenuation is accomplished using a linear graded neutral density filter mounted to a translation stage. A motorized stage is used to smoothly vary the attenuation throughout the range. The beam is periodically blocked to obtain zero readings. The laser power monitor is placed in the same plane as the sample. An aperture is not needed because the laser beam is spatial filtered.

The probe laser is a 2 mW, 632.8 nm HeNe laser. The collinear excitation and probe laser beams are focused into the sample with a 0.25 numerical aperture, 14.8 mm focal length microscope objective. The focused beam waist radius is measured carefully using a razor blade attached to a micrometer-driven translation stage to position the blade in the beam. A square quartz tube with 1 mm inside path length is used as a sample cell. Sample solutions are continuously flowed through the sample cell to reduce errors due to analyte photodegradation. The short path insures that the excitation beam waist radius is relatively constant through the sample. After exiting the sample cell, the excitation laser beam is blocked by a laser line filter. The probe beam passes through the pinhole aperture, then is sent to the silicon photovoltaic detector. Outputs from the photovoltaic detectors are buffered using transimpedance amplifiers.

The photothermal lens signal is recorded by using a 50ns, 8 bit wave form acquisition board that plugs directly into the bus of the data collection computer. The captured photothermal signal is processed using adaptive matched signal procedures. All photothermal lens data is processed to produce signals proportional to inverse focal length. Reference photodiode channels are recorded with a 16-bit analog-to-digital converter. Excitation laser energy is calibrated with a certified detector.

The photothermal signal and reference channel data are recorded on the PC data collection computer. Data processing is performed using a combination of spreadsheet, symbolic language processor, and curve fitting programs. The symbolic language processor is used to obtain analytical solutions for theoretical signal modeling and can perform numerical integration. Analytical results are exported as FORTRAN source code for incorporation into nonlinear regression routines written in the same language. A Marquardt algorithm is used to perform the regression to the nonlinear functions.

Reagents used in this study are eosinY, erythrosin (B) (Sigma) and 1,1’-diethyl-2,2’-cyanine iodide or pseudoisocyanine iodide (Eastman). Cobalt (II) nitrate (Mallinckrodt) is used as the linear absorption standard. Stock solutions of each sample are prepared by dissolving weighted amount of the solutes in ethanol (McCormick 200). Working solutions are then freshly prepared by dilution to about 10 m M. Solutions are filtered using 0.2 m m filter cartridges. Sample absorbance is measured with a spectrophotometer. The 514.5 nm absorbances of all samples are around 10^-3 AU in the 1 mm path length sample cell. Deoxygenation is performed by bubbling solvent-saturated nitrogen through the solutions for 30 minutes.

The model equation gives the inverse focal lengths as a function of excitation irradiance. In order to obtain the absolute absorption cross sections, the 'apparatus constant', which relates the signal, measured in volts, to the inverse focal length, must be determined. Most simple metal complexes do not exhibit nonlinear, irradiance-dependent absorption effects with the irradiance used in this study. Solutions of cobalt (II) in ethanol are used to determine the 'apparatus constant' and to verify linearity of the apparatus response. Data sets were obtained using cobalt (II) solutions with well-known absorbance prior to all measurements on the nonlinear absorbers. The cobalt (II) samples exhibited a linear irradiance-dependent signal behavior over the range of excitation irradiances and signal voltages used for the nonlinear absorbing dye samples.

A typical continuous laser induced bleaching data set is illustrated in Figure 3. Regression analysis is used to obtain the irradiance dependent absorption coefficient according to Eq. 7. This, in turn, is used to determine triplet state absorption cross sections and triplet-to-singlet relaxation rate constants. These results are tabulated in the table. The absorption cross sections are for 514.5 nm Ar⁺ laser excitation. Singlet (s₁) and triplet (s₃) cross sections are and the triplet-singlet relaxation rate constants, k_T-S, are determined from the regression parameters. The triplet quantum yield is estimated from the literature value for the fluorescence quantum yield, assuming no internal conversion (IC). This value used is verified by the experiment since the dye sample absorbance is known and the apparatus response is calibrated with cobalt (II) solutions.

The difference observed between the relaxation constants obtained in air saturated and in deoxygenated solution can be attributed to the quenching of the erythrosin triplet state by oxygen.

With pulsed excitation laser, bleaching of the triplet state of the erythrosin is observed and the relaxation rate constants k_TT are determined. Due to high irradiance and short pulse duration, the slower kinetics cannot be observed. The continuous laser excited photothermal lens experiments described here bridge the gap between the linear spectrometry, at very low irradiance, and the pulsed-laser excitation studies where the irradiance is exceedingly high due to the short excitation duration. In combination, the continuous and pulsed experiments allow determination of rate processes with time constants ranging from picoseconds to seconds.

1. Bialkowski, S. E. Photothermal Spectroscopy Methods for Chemical Analysis New York: Wiley, 1996
2. Giuliano, C. R., and Hess, L. D. IEEE J. Quant. Electron. QE-3 358-367 (1967)
3. Chartier, A., and Bialkowski, S. E. Anal. Chem. 67 2672-2684 (1995)
4. Bialkowski, S. E. Appl. Opt. 32 3177-3189 (1993)