Photothermal and photoacoustic spectroscopies are sensitive analysis methods applicable to low absorbance analyte measurements.¹ The analytical signals result from changes in the physical state of the sample upon absorption of light from an excitation source. The fact that signal magnitudes are proportional to excitation irradiance has led to the use of powerful laser excitation sources with small focused beams. However, the higher irradiance produced by even relatively weak laser sources often results in nonlinear sample absorption.

On one hand, nonlinear absorption results in measurement errors that are difficult to detect and account for. Detection requires apparatuses that simultaneously measure photothermal signals and excitation laser irradiance or power. The data are recorded as a time series and subsequently displayed, empirically modeled, and/or analyzed. Data are most useful when the excitation laser irradiance is varied over a substation range. Because photothermal spectrometry is sensitive to errors the overlap of excitation and probe laser beams, attenuation schemes that do not produce beam offset or deflection are required. In addition, laser mode variations that occur over time or pulse-to-pulse can mask underlying nonlinear effects. Laser mode control is often a prerequisite to obtaining data useful for nonlinear diagnostics. Knowledge of the nonlinear response can help in design of analytical procedures that are accurate. In addition, semi-empirical models of nonlinear sample response can be used as a tool for increasing analytical information. For example, individual components of gas mixtures have been identified and quantitatively measured using semi-empirical modeling of nonlinear absorption at a single wavelength.²

On the other hand, measurement and modeling of the nonlinear absorption can result in physical insight and important quantitative information regarding the excited states of the species under study. Bleaching is a common nonlinear effect that occurs in measuring condensed phase solutions of organic dyes.³ Bleaching occurs when the analyte has long-lived excited metastable states, i.e., a triplet state. Excited state populations affect the absorption, primarily by depleting the ground state, and secondarily by optical absorption. The absorption coefficient of the excited state may be larger or smaller than the ground state. Comparison of modeled absorption and relaxation processes to experimental observation has resulted in the determination of excited state absorption cross section and relaxation rate constant information on a number of organic dyes.⁴

There are three main steps involved in modeling nonlinear absorption of molecular systems:

1. Determine number density or relative concentration of all states of the species during excitation.
2. Integrate the energy or sum the power absorbed by each state during irradiation.
3. Calculate signal strength from the irradiance-dependent energy or power absorbed.

From these, an irradiance- or integrated irradiance- dependent signal model can be formulated and used to determine the effects of varying rate parameters on the resulting nonlinear behavior.

The first step requires a model of the excited states and how these excited states connect to one another. Kinetic rate models are often found in the literature. For example, generalized kinetic rate models for organic dye species are common in the optical engineering literature since excited state dynamics are important to the understanding of dye laser and optical limiter technology. The excited state model is a set of coupled differential rate expressions that must be solved under the experimental irradiance conditions. Since most relaxation process are first or pseudo-first order, analytical solutions for the number densities can usually be obtained for constant-irradiance conditions. Symbolic language processors may be used to obtain analytical expressions from the set of differential equations describing the number density of ground and excited states. Analytical expressions may also be obtained if the time-scales for excitation and relaxation are very different. For example, in cases of pulsed laser excitation where the pulse duration is either much shorter or much longer than relaxation times, the rate expressions can be simplified by using steady-state approximations or by removing slower relaxation rate processes.

Numerical integration must be used when analytical solutions cannot be obtained. Integration requires a model for the time-dependent irradiance. We have found that the irradiance-dependent signal changes with pulse shape.⁵ So one must have a good measurement of the time-dependence of the excitation pulse. Laser pulses are generally exponential, for gas lasers, or Gaussian, for flash lamp or Q-switched lasers.

Approximations may be used to simplify the kinetic models and give physical insight regarding relaxation processes. For short-pulsed laser excitation, excited state relaxation is often slow compared to the rate for optical excitation. In this case the excited state concentration initially populated by the excitation laser is calculated based on an exponential law using the ratio of the integrated optical irradiance, H(r) (J m^-2), assumed to be cylindrically symmetric with r being radius, to the integrated irradiance for ground state bleaching, H_S

    (1)

The excited population, N^*(H) (m^-3), subsequently relaxes, perhaps via metastable intermediate states. H_S depends on the particular model, but is generally

    (2)

s (m²) is the absorption cross section and hn (J) the photon energy. The absorbed energy is directly proportional to the excited state number density. Since each transition requires the absorption of one photon, the energy density is, U(H)=hn N*(H) (J m- ²). This is the simplest case of optical bleaching. The excited state energy density is a function of integrated irradiance, which, in turn, has the spatial dependence of the excitation laser beam. For the Gaussian laser beam

    (3)

Q (J) is pulse energy and w (m) is the electric field radius. The initial excited state energy density is no longer a Gaussian, but is flattened due to optical bleaching.

To obtain the thermal response, the excited state decay rate distribution is convoluted with the impulse-response for hydrodynamic relaxation. For first order or consecutive first order decay, the spatial distribution of energy production, or power, is that of the initial excited state. For higher order kinetics, the spatial power distribution must be obtained from the number densities. This typically involves numerical rate expression integration. The resulting temperature change distribution is subsequently used to determine the relative amplitude of the optical element or the acoustic wave using the impulse-response for Gaussian beams.

Modeling the initial excited state distribution as a superposition of Gaussian functions with different beam waist radii facilitates this process. For the simple bleaching example, the initial excited state energy density can be expanded as the series

    (4)

Here, H=2Q/p w² is the integrated irradiance. Each term in the series is Gaussian, with a beam waist radius w(m)²=w²/m. The resulting signal is a superposition of terms resulting from each Gaussian element. The relative photothermal signal strength can be calculated based on this distribution. For an excited state relaxation rate that is much faster than hydrodynamic relaxation, signal strengths are simply the sum of strengths, S(Q,w), for each term. If the signal strength is S(Q,w) for a normal Gaussian beam excitation, the resulting signal is

(5)

For continuous and chopped-continuous laser excitation, steady-state populations in the optically- coupled ground and excited states are used to determine the excitation laser irradiance-dependent absorption coefficient, a (E), based on the bleaching irradiance, E_S, through

(6)

Bleaching irradiance depends on the relaxation rate model, but is generally

    (7)

t (s) is the excited state relaxation time constant. The sample-heating rate is then the absorption coefficient-excitation irradiance product, Ea (E). Sample heating by excited state absorption is incorporated as an additive term. The heating rate is the product of the space-dependent concentration, the ground or excited state absorption coefficient, and the space-dependent irradiance. Photothermal signals are again calculated based on the theoretical response of a linear absorption system using a Gaussian laser beam. The sample heating rate is first modeled by a superposition of Gaussian sources, the appropriate optical element or photoacoustic wave amplitude is found for each source. The resulting signal is the superposition of the responses to each Gaussian source term.

Finally, the intermediate case, wherein excited state relaxation occurs during the pulsed excitation, can be modeled. To obtain a signal solution, rate equations describing ground and excited state populations are numerically integrated for a series of H and for a time-dependent laser pulse profile. The absorbed energy is accumulated during integration by summing ground and excited state absorbance-irradiance products. The resulting U(H) vector is then used to calculate the instantaneous signal by first, finding the space-dependent energy density produced by the Gaussian beam, second, modeling the spatial profile as a series of Gaussian functions, and finally, calculating the signal, as in Eq. 5. Final excited state populations may be used to calculate signal evolution through relaxation kinetics. For certain experiments, like photothermal lens and deflection, signal estimates can be obtained directly from derivatives of the U(H) vector.

A schematic diagram of a typical apparatus used to study excitation power- or energy-dependent nonlinear absorption effects in our laboratory is shown in Figure 1. Many elements of this apparatus are arranged or specifically used to control the polarization and mode of the excitation and probe laser beams. If possible, TEM mode control is accomplished using both intracavity apertures and external spatial filters. The energy or power of the excitation laser must be accurately measured. We often use certified detectors to calibrate less expensive measurement devices used during the experiments. The laser monitors are placed in the same plane as the sample, and if needed, pinhole apertures are used to insure that the measured power or energy is that at the center of the beam focused into the sample.

Excitation laser beam attenuation is accomplished using methods developed in our laboratory. We have found two methods that work well. First is to use a venetian blind attenuator followed by spatial filtering in the conjugate plane of a lens.⁶ This method is shown in the figure. High order diffraction is thus filtered and the resulting beam is Gaussian. The overall transmission is variable from 0 to nearly 1 by changing the angle of the venetian blind attenuator. The other method is to use 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 in this case.

The sample cell is usually of a short optical pathlength so that the excitation beam waist radius is relatively constant. Dilute solutions are used. The sample transmission should be 95% or better is insure that irradiance does not change substantially along the beam direction. Where possible, flowing samples are used. Flowing removes photolyzed material from the cell between laser pulses. The focused beam waist radius must be carefully measured. We usually use a razor blade attached to a micrometer-driven translation stage to position the blade in the beam. The beam waist is calculated from plots of the blade translation versus transmission.

The photothermal signal is digitized with a 12 to 16 bit analog-to-digital converter in a PC data collection computer. Alternatively, signals are digitized using an 8 bit transient waveform recorder. Each transient is analyzed using a matched filter.⁷ The filter function is determined from the experimental impulse-response. It is important that the signal and excitation laser power or energy are simultaneously recorded.

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 some numerical integration. Analytical results can be written as FORTRAN source code for easy incorporation into nonlinear regression routines written in the same language.

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