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If we indicate with I _u and I _p the probe light intensities transmitted through the unexcited (u) and photoexcited (p) sample, and writing the number of absorbers in the system as N ₀ = N _g + N _e, that is the sum of non‐excited (N _g) and excited absorbers (N _e), the Beer–Lambert law can be used to express I _u and I _p in terms of the variation Δσ of the attenuation cross section:

(3.12)

(3.13)

(3.14)

where σ _g and σ _e are the attenuation cross sections in the ground and excited state, respectively, Δσ = σ _e − σ _g, and d is the sample thickness. In practice, the recorded TA signal is most commonly expressed as a differential optical density ΔOD, which is indeed proportional to the change in cross section: ΔOD = ΔσdN _e/2.303. Thus, the TA signal is given by:

(3.15)

This is the quantity that is experimentally obtained from I _u and I _p generally reported in a TA experiment. Considering that the variation of the absorption is normally very small, I _p − I _u << I _u, it is possible to linearize Eq. (3.15) and calculate the TA signal as:

(3.16)

In general, ΔOD(ω) is the superposition of three types of contributions: ground state bleaching (GSB), stimulated emission (SE), and excited state absorption (ESA). Their nature is illustrated in Figure 3.3 using a molecular system with three electronic levels as an example. Similar considerations apply to molecules with more than three levels, or to other types of physical systems, such as bulk solids or nanoparticles.

GSB is the negative TA signal related to the depopulation of the ground state upon pumping. After pumping, photoexcited molecules are no more in the ground state and they cannot contribute anymore to the ground state absorption. Therefore, the probe is less absorbed in the spectral range of the 0 → 1 transition (see Figure 3.3) and this appears as a negative signal in the TA spectrum, resembling one or more of the steady state absorption bands.

Also, SE is a negative signal and it occurs because the probe pulse passing through the excited volume stimulates the emission of a photon from the excited state to the ground state. It produces a negative signal because the detector records an increase in intensity of the probe in the photoexcited system, in the spectral region of the 1 → 0 transition. The spectral shape of SE is very similar to the photoluminescence band except for its negative sign. However, it is worth noting that the SE cross section, and intensity, is proportional to the emission frequency, while the intensity of stationary fluorescence depends on the cube of emission frequency, as it is easy to see from Einstein's coefficients. In a perfect two‐level system, GSB and SE would be perfectly degenerate and be indistinguishable, but in a typical molecular system, they are separated by a Stokes shift.

Figure 3.3 On the left panel, a hypothetical signal decomposed into three contributions. Diagram of the typical signals in a TA experiments: ground state bleaching (GSB), due to the decreasing absorption of the sample; stimulated emission (SE) due to the interaction with the probe which stimulates photon emission and the consequent decay of the excited system back to the ground state; and excited state absorption (ESA) related to the possibility of new transitions toward higher energy states. Upper right: simple scheme of a TA measurement: pump and the probe pulses overlap on the sample, and changing the delay between them allows to measure the changes in the absorption spectrum induced by photoexcitation and subsequent dynamics.

Last but not least, ESA is the only positive signal that is recorded in a TA experiment. It is due to the new electronic transitions, such as 1 → 2, that become possible from the excited state, because probe absorption can bring the system to other higher energy states [33].

Overall, a TA experiment is simultaneously sensitive to several electronic transitions accessible from the ground and excited state(s) of the system. Virtually any photoexcited system is expected to provide several observables that can be used to follow the dynamics, making the technique very versatile. For instance, suppose that, for a given system, the GSB is difficult to observe because it falls out of the probing window. Then, one can still resort to SE or ESA as observables to monitor the evolving system. While a singlet excited state will typically give rise to strong SE, a triplet state is not expected to, due to the very low radiative rate toward the ground state. However, it should still be detectable through an ESA transition toward some higher triplet state, which should remain strongly allowed. Overall, following the spectral position and the intensity of one or more of these signals allows to reconstruct in detail the undergoing relaxations of the system, provided that the time resolution is good enough to resolve all the dynamics of interest (usually <100 fs). The analysis of TA data provides a wealth of information on the relaxation dynamics of the photoexcited system, such as excited state population decay, internal conversion or intersystem crossing, charge and energy transfer, energy relaxation, solvation of the excited state, and more [3, 8, 29, 34].

To achieve femtosecond time resolution, it is necessary to have short pump and probe pulses (50–80 fs) which keep their duration along the setup (for example they should not traverse too much glass) and, moreover, it is necessary that the two pulses are synchronized down to the scale of a few femtoseconds. Synchronization on these timescales cannot be achieved by the use of electronics. The only way to achieve such a precision is to use all‐optical strategies. In fact, a transient absorption setup typically exploits a delay stage in the optical path of one of the beams, which allows to change the relative delay between two pulses increasing or decreasing the path of one of them. It is easy to calculate that an increase in the path of 3 mm corresponds to a delay of 10 ps. In a standard setup, both beams derive from a single laser, the original beam being split in two portions which are used to generate separately the pump and the probe. In general, both pump and probe derive from the same laser source because it would be harder to synchronize with femtosecond accuracy two beams originating from two different lasers. The next section describes in more detail the main technical characteristics of a typical TA setup.