(3.5), the intensity of the pulse obtained by SFG or DFG depends on the product of the two incoming intensities (3.2.2.2 Noncollinear Optical Parametric Amplifier
A noncollinear optical parametric amplifier (NOPA) is an optical device capable of producing tunable femtosecond radiation in the visible or near‐infrared region. The output of a NOPA is obtained by the interaction of two beams in a nonlinear crystal: a strong pump (ω p) and a weak and broadband seed (ω s < ω p). The pump is used to amplify the seed intensity, producing a strong output beam, named the signal, while creating another beam named idler at ω i, where ω p = ω s + ω i for energy conservation. In practice, the nonlinear process involved can be seen as difference frequency generation (DFG) between the pump and the seed. If the pump is the second harmonic at 400 nm of the Ti:sapphire beam, ω i falls in the infrared region, and the output signal wavelength can be typically tuned from 490 to 760 nm. Considering that the seed is a broadband pulse, changing the orientation of the nonlinear crystal allows to amplify different wavelengths based on the particular phase matching condition fulfilled in a given orientation [25].
Optical parametric amplification can be obtained either in a collinear or noncollinear geometry. The NOPA configuration uses the latter, allowing to compensate for the group velocity mismatch (GVM) between the two pulses (Figure 3.2a), which limits the interaction length and, therefore, the amplification of the seed into the signal.
Figure 3.2 Top: (a) Wavevectors of pump (
Source: Reprinted from [25], with the permission of AIP Publishing.
Bottom: Generation of supercontinuum pulse (white light) from a red pulse propagating in a centrosymmetric medium and supercontinuum spectrum generated by 800 nm beam pulse focused in a 2 mm D2O quartz cell filtered by a short‐pass filter at 750 nm.
The advantages of a noncollinear configuration can be also explained in different terms. In a noncollinear configuration, the crystal orientation angle at which phase matching occurs for amplification of a given wavelength becomes dependent on the angle α between the pump and signal wavevectors propagating inside the crystal, as in Figure 3.2d. In particular, as demonstrated in Figure 3.2e, the phase matching condition is simultaneously achieved over a very broad wavelength range when α ∼ 3.7∘. This allows to produce very broadband output pulses, which can be then compressed to very short temporal durations even below 10 fs. Usually, the output pulse is not broad as the near‐vertical line in Figure 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
