In this Letter, we theoretically describe photoacoustic signal generation of molecules, for which triplet relaxation can be neglected, by considering the excited state lifetime, the fluorescence quantum yield, and the fast vibrational relaxation. We show that the phase response of the photoacoustic signal can be exploited to determine the excited state lifetime of dark molecules. For fluorescent molecules, the phase response can be used to determine the fluorescence quantum yield directly without the need of reference samples.
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The photoacoustic (PA) effect, also called optoacoustic effect, was discovered in 1880  and describes the generation of acoustic waves following light absorption of a material. In general, if light is absorbed by a molecule, it undergoes a transition from the ground state into an excited state. Thermal relaxation of the excited molecule leads to an increase in local pressure and ultimately to an acoustic wave. Theoretical work on PA signal generation (see, e.g., Refs. [2,3]) serves for a better understanding of the measurement results and allows to design efficient setups. However, the exact events occurring during excitation, that is, electronic excitation and relaxation, are usually neglected, and the heating function is often modeled as a Dirac impulse , that is, light absorption and heat generation are assumed to occur infinitely fast. This approximation is reasonable as long as fast relaxing materials are investigated but, in general, does not hold for molecules relaxing within nanoseconds or even longer. Currently, there is growing interest in the excited state lifetime, as it was found that the lifetime can be used as an alternative contrast mechanism instead of spectral information which requires multiwavelength excitation and spectral unmixing [5,6]. For determination of the excited state lifetimes, pump-probe methods have been employed [5–7].
In this Letter, we theoretically examine the transient behavior of molecules exhibiting just two singlet states (see Fig. 1). We consider the excited state lifetime, the fluorescence quantum yield, and fast vibrational relaxation to derive the frequency dependent amplitude and phase of the PA pressure. A similar formalism, which is restricted to purely nonradiative triplet state relaxation, was found by Keller et al. . We will show that the phase response of the PA signal allows us to assess the excited state lifetime of nonfluorescent molecules, similar to fluorescence lifetime imaging (FLIM) for fluorescent molecules , without the need of pump-probe experiments. For molecules which additionally exhibit radiative relaxation, that is, fluorescence, the proposed method together with knowledge on the absorption and emission spectra allows the direct assessment of the fluorescence quantum yield. Material parameters, like the absorption cross section or the refractive index of the dye solution under investigation, have no influence on the evaluation. Also the method does not require comparison with a well characterized sample with known fluorescence quantum efficiency. This stands in contrast to today’s standard methods in which the overall fluorescence of the investigated sample is compared with a reference dye. In order to avoid the pitfalls of pure optical methods, determination of the fluorescence quantum yield using PA means was demonstrated, for example, in Ref. . However, in the employed underlying theoretical models, the fluorescence lifetime and the vibrational energies were neglected .
Figure 1 shows the Jablonski diagram of a molecule exhibiting two singlet states. A photon (exc, blue) excites an electron from the highest occupied molecular orbital’s (HOMO) ground state into the excited state . From the electron relaxes to , thereby releasing thermal energy (red wavy path in Fig. 1). For dye molecules, this vibrational relaxation usually happens on a timescale of 1 ps [12,13]. From , which is the ground state of the lowest unoccupied molecular orbital (LUMO), the electron can relax into the HOMO via radiative relaxation, that is, fluorescence (green solid path in Fig. 1), or via nonradiative relaxation, that is, thermal relaxation (red solid path in Fig. 1), with a relaxation time constant usually of several nanoseconds. Note that for fluorescence and for thermal relaxation, the electron can relax into any vibrational state within the HOMO. This state is then emptied by fast vibrational relaxation to the ground state , releasing heat. As the latter relaxation happens very fast, in the case of nonradiative relaxation from , one can assume that the electron relaxes from to directly. Also in the case of fluorescence, relaxation from to the ground state contributes to the released heat. The released vibrational energy, , depends on the emitted photon energy , namely, , where denotes the energy difference between LUMO and HOMO (see Fig. 1). For PA measurements, in general, excitation of several thousand molecules is required in order that the pressure exceeds noise level [14,15]. Therefore, for calculation of the released energy, one can use the expectation values of and . The expectation value of the fluorescent photon energy is given by16]. In the following, we assume that excitation occurs only from , that is, no re-excitation from , , or occurs. For an excitation laser pulse that is much shorter than the excited state lifetime , we find the fluorescence , given in numbers of photons, to follow the well-known exponential decay: 2) corresponds to
In Fig. 2(a) the solid green curve represents the fluorescence amplitude over frequency, calculated for an excited state lifetime of . The overall heat dissipation from the chromophores follows a similar tendency [black solid curve in Fig. 2(a)]. However, in contrast to fluorescence, for high frequencies the heat dissipation does not approach 0. The reason is the fast vibrational relaxation from to (blue curve), which is very much faster than the relaxation from to . Therefore, this energy contribution can be approximated by a Dirac delta function in time. We compared the Dirac delta approximation with a model considering also the vibrational relaxation times and found that the approximation holds as long as the to relaxation time is around a factor 100 below the excited state lifetime . Altogether, the heating function can be described by
For the PA signal, not the heating function , that is, the generated heat, but rather the generated pressure is relevant, which is given by the time derivative of the heating function . The PA pressure as a function of the angular frequency therefore is17]. The PA pressure shows a high-pass like behavior [see Fig. 2(b)], with the amplitude being 2(b). The contribution from the vibrational relaxation is . As its timescale lies in the picosecond range, it is linearly proportional to the excitation frequency in the displayed frequency region. is the PA contribution from to . It shows a high-pass behavior. The solid black line in Fig. 2(b) is the overall generated pressure. The curves were calculated according to an Atto 390 dye excited with a wavelength of 405 nm, that is, an excitation photon energy , LUMO lifetime , fluorescence quantum yield , , , from which we derive and . Note that the energy released via fluorescence in Fig. 2(a) is not nine times as high as the thermally released energy, as one could assume from the quantum yield of 0.9, as vibrational relaxation within the HOMO always contributes to the thermal signal, independent of the preceding relaxation.
In Fig. 3(a) the PA amplitude is plotted versus frequency for various values of the fluorescence quantum yield . Lower quantum yields lead to higher signals. In Fig. 3(b) the corresponding PA phase is plotted according to Eq. (7). The phase response has a minimum at a frequency . With increasing fluorescence quantum yield shifts to lower frequencies and additionally the observed phase minimum increases. In Fig. 3(c) the PA amplitude is plotted for different values of the excited state lifetime. The dashed curve symbolizes the contribution of the relaxation. Shorter relaxation times lead to higher PA signals. For long relaxation times, the PA signal is dominated by the vibrational relaxation in the LUMO. In Fig. 3(d) the corresponding PA phase is shown. With increasing lifetime, the frequency shifts to lower frequencies, while the minimum phase stays constant.
From Eq. (7) we can get analytic expressions for the minimum phase and for the angular frequency where the phase becomes minimal, :8) simplifies. For the sum of and simply resembles the energy of the excitation photon which is usually well known in an experiment. By employing the relation for of Eq. (8) the energy level of can be directly calculated by 3(b), which was calculated for , has a of 0.5 rad. According to Eq. (9) the energy difference between and is , which corresponds to 2.86 eV. Note that Eq. (9) can only be applied to dark molecules, that is, .
Now knowing the energy gap , one can directly calculate the relaxation time by making use of the relation for of Eq. (8),
In case the molecule shows fluorescence, that is, , one can measure the excited state lifetime with time-resolved fluorescence measurements (see, e.g., Ref. ). From a spectroscopic measurement, one can additionally assess from the point of intersection between absorption and emission spectrum, which is 2.86 eV for Atto 390. The energy can be calculated from the emission spectrum with Eq. (1). In the expression for of Eq. (8) now all values besides the quantum efficiency are known. By determining the frequency where the phase becomes minimum, , one can calculate the fluorescence quantum efficiency:20–22]. Today, the gold standard is to compare the fluorescence intensity of the investigated sample with a well characterized standard sample with known (see, e.g., Refs. [22,23]). As the measured fluorescence intensity depends on various parameters, such as, absorption coefficient, refractive index of the sample, and so on, the determination of the quantum yield is error-prone .
If triplet relaxation cannot be neglected, the analytic equations given in this Letter cannot be applied. Instead the differential equation system describing the molecule including the triplet state can be solved numerically. Curve fitting can then be applied to find the quantum yield and the singlet and triplet relaxation times. We note that all graphs shown in this Letter were calculated for a point-like PA source at the position of ultrasound generation. In an experiment, attenuation of ultrasound and frequency dependence of the transducer’s sensitivity will significantly change the behavior of measured PA amplitude and phase . For a quantitative evaluation, the measurement system has therefore to be calibrated. Calibration could be done by measuring a sample exhibiting only fast acoustic transients, for instance, a metal film. A similar procedure is used in FLIM, where second harmonic generation is used for calibration. The approximation of a point-like source is valid only for μm thick samples exposed with a tightly focused laser beam, such as in the case of optical-resolution PA microscopy [17,25,26]. If larger volumes are excited, the frequency responses derived in this Letter have to be multiplied with the frequency response of the excited volume .
In summary, we theoretically calculated the PA response of point-like sources by considering the excited state lifetime, the fluorescence quantum yield, and the energy levels of molecules exhibiting only two singlet states. We found that under certain conditions, such as no significant triplet relaxation, for dark molecules it is possible to determine the bandgap energy and the excited state lifetime by measuring the phase of the PA response. Further, we found that for fluorescent molecules for which the absorption and emission spectra are known, the phase of the PA response allows us to determine the fluorescence quantum yield. In the PA experiments, just a single excitation wavelength has to be used. Therefore, no optical parametric oscillators, tunable lasers, or multiple lasers with different wavelengths are required for performing the PA experiments. The obtained results do not depend on the comparison with standard samples for which the absorption cross section or the refractive index may introduce uncertainties. We therefore believe that the proposed method could become an alternative to determine the fluorescence quantum yield in the future.
Austrian Science Fund (FWF) (P27839-N36); European Regional Development Fund (ERDF) (IWB2020:MiCi); Innovative Upper Austria 2020; Federal Government of Upper Austria.
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