3.2e because the seed is temporally chirped and the interaction with the pump only involves a relatively narrow portion of the seed spectrum. Then, changing the alignment (spatially and temporally) between pump and signal allows to amplify different portions of the spectrum, as shown by the dashed lines in Figure 3.2e.
3.2.2.3 Supercontinuum Generation
The weak seed of a NOPA, usually, is a spectrally broad ultrashort light pulse. These pulses are called white light or supercontinuum pulses [26] and are generated by a third‐order nonlinear phenomenon named self‐phase modulation (SPM), combined to another phenomenon called self‐focusing (SF). Besides the NOPA seed, these white light pulses are also used as probe pulses for transient absorption experiments discussed in the next section. To understand the generation of these supercontinuum pulses, consider a centrosymmetric material such as a liquid or glass (χ (2) = 0). Taking into account third‐order nonlinear effects, the instantaneous polarization can be written as:
(3.7)
which makes it possible to write the square of the refractive index n as:
(3.8)
Therefore, a propagating pulse will produce a change of the refractive index related to its instantaneous intensity. Considering the temporal and spatial distribution of the pulse intensity, and considering that χ (3) is usually very small, the last expression is usually approximated in first order as n ≈ n 0 + n 2 I(r, t), where the nonlinear refractive index n 2 is closely related to χ (3). Considering a Gaussian beam, the intensity depends on position and time. Therefore, one may expect both spatial and temporal changes of the refractive index, which give rise, respectively, to SF and SPM [26].
The SPM is the effect related to the temporal dependence of beam intensity and, in particular, to the variations of the phase of the pulse. The applied field can be written as:
(3.9)
Therefore, the instantaneous phase of the pulse depends on time as:
(3.10)
Considering the instantaneous frequency ω(t):
(3.11)
the last term implies a change (self‐modulation) of the frequency of the propagating light in a way controlled by the intensity envelope of the pulse itself. This effect causes the introduction of new frequencies in the spectrum of the pulse, symmetrically higher and lower than ω 0 according to the alternating signs of
A discussion of SPM, however, cannot be disentangled from the effects of SF. In fact, SF occurs in parallel to SPM because of the spatial dependence of the intensity in the pulse, which, for a Gaussian beam, is much stronger in the center of the beam than the sides. For this reason, the beam is capable of modifying the local refractive index, making it higher at the center of the beam if n 2 > 0. This causes a focusing of the beam along its path, because the region of modified refractive index behaves as a lens.
In laboratory practice, SF is easily observed by prefocusing by an ordinary lens an intense femtosecond laser beam (at least a few μJ per pulse) to a spot of a few tens of μm within a transparent medium with significant χ (3), such as ordinary glass, a water cell, or crystals such as sapphire. Focusing the beam allows to reach a threshold intensity above which the onset of SF occurs. As a consequence, the beam spontaneously shrinks down to a filament with much smaller (a few μm) cross section, within a few millimeters of propagation length. In practice, SF stops when the diameter of the filament is so small (a few μm) that the diffraction is strong enough to balance the effect and prevent further self‐focusing. Obviously, self‐focusing causes a dramatic increase in the local intensity of the electric field. Thereby, SPM is strongly enhanced within the filament, strongly contributing to a dramatic broadening of the pulse spectrum and to the generation of an intense white light. Therefore, the formation of a stable filament is essential to have a stable and intense white light pulse. The final output of these processes is a spectral broad pulse as a consequence of combined SPM and SF, and it is also temporally broad and strongly chirped as a consequence of group velocity dispersion (GVD), as depicted in Figure 3.2. For instance, if the white light is generated from a 800 nm beam passing through a 2 mm cuvette of D2O, the pulse covers a very broad range which is symmetric with respect to 800 nm, from which the visible part can be then selected by a filter (Figure 3.2).
3.3 Transient Absorption Spectroscopy
Ultrafast transient absorption (TA), or pump/probe, spectroscopy is a nonlinear spectroscopic method based on measuring the changes in the absorption spectrum of a system following an external excitation [827–32]. In a TA experiment, the sample is photoexcited by a femtosecond pulse called pump and the variations of the absorption spectrum are measured by another, delayed, ultrafast pulse named probe. The probe is usually spectrally broad (400–700 nm) and this allows to record simultaneously the changes of the absorption spectrum in a wide spectral range. Moreover, the variations in the entire spectrum are recorded at different time delays between the two pulses, yielding kinetic traces of the time‐dependent absorption coefficient at every wavelength.
In these experiments, the pump pulse is normally resonant to one of the electronic transitions of the sample, in order to bring it from the ground state to an upper energy state. Then, the instrument records the intensity of a probe light pulse which has traversed, at certain delay after the excitation, the excited spot on the sample, and compares this with the result of an identical measurement without the pump pulse. As explained hereafter, the TA signal is obtained from the ratio between the probe intensities recorded with and without excitation.
3.3.1 The Experimental Method
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,