L is the propagation length inside the material [19]. Besides broadening, GVD causes a frequency chirp, that is a time dependence of the instantaneous frequency of the pulse, given by Figure 3.1 Panel (a): Simulation of a gaussian pulse centered at 550 nm with FHWM = 5 fs (first curve from the top) after propagation through a SiO2 medium of 1 mm thickness (second curve) and 2 mm (third curve). Panel (b and c): zooms of the tails of the black pulse (squares). Each wavelength is delayed by a different phase, resulting in a longer pulse with a positive chirp (the redder frequencies are faster than the bluer).
Pulse broadening and chirp acquired by femtosecond pulses during their propagation in optical setups need to be put under control in order to preserve good time resolution. One way to do it is to limit the use of transparent optical components, preferring the use of reflective optics only. Some special methods exist to manipulate the chirp, such as what is called a pulse compressor, built by using a pair of prisms or gratings. In a pulse compressor, one can add negative GVD (redder part of a pulse propagates slower than the blue part) which compensates the effect of pulse broadening in a transparent media, recompressing the pulse [19, 20].
Last but not least, dispersion also affects the temporal overlap of two pulses centered at different wavelengths which pass through the same medium, because their group velocities are generally different: this effect is called group velocity mismatch, or GVM. Thereby, if the pulses are initially overlapping in time at a certain point in space, they overlap no more after some propagation distance within a dispersive medium. The GVM effect, for example, can be very important in the generation of pulses through nonlinear effects because it can limit the effective interaction length between the two pulses.
3.2.2 Nonlinear Optics: Basis and Applications
3.2.2.1 Second Harmonic Generation and Sum Frequency Generation
Second harmonic generation (SHG) is a nonlinear optical phenomenon in which two photons of the same frequency, interacting in a nonlinear material, are converted in a single photon with doubled frequency [22]. The polarization
(3.4)
where, in general, χ (n) is a tensor. The first term of the equation describes the phenomena usually encountered in linear optics, while the other describes nonlinear effects at different orders in the electric field. Under certain conditions (χ (2) ≠ 0, as generally occurs in a non‐centrosymmetrical medium), two photons at the same frequency ω 1, passing through an appropriate medium, are combined to generate a new photon with a doubled frequency (2ω 1). The process follows the laws of energy (ω 1 + ω 1 = 2ω 1) and momentum conservation (
where n is the refractive index, L is the optical path within the nonlinear material, Δk = k 2 − 2k 1 is the so‐called phase mismatch, and χ eff is the effective susceptibility which is a certain combination of the components of the χ (2) tensor, which depends on the material and on its orientation. The intensity of the new beam depends on the square of the incident beam intensity, on the length L (with a quadratic dependence if Δk = 0), and on the degree of phase mismatch. Fulfilling the condition Δk = 0, named phase matching, gives maximally efficient SHG, and corresponds to the conservation of momentum in the process. It can be seen as a situation in which first and second harmonic beams propagate in the medium with the same speed. In order to achieve this, the refractive index at ω and 2ω has to be the same. Although this is not generally possible in isotropic media, such a limitation can be overcome by using (uniaxial) birefringent media such as beta‐barium borate (BBO). The latter display two different refractive indexes, ordinary (n o), and extraordinary (n e), for beams with two orthogonal polarizations, where the extraordinary index also depends on the angle θ between the
Because femtosecond pulses are intrinsically broadband, another important parameter in SHG is the extent of spectral bandwidth which is effectively doubled (acceptance bandwidth), not necessarily coincident with the whole pulse bandwidth. In fact, the phase matching condition is exactly fulfilled only at a given wavelength, and therefore it cannot be exactly fulfilled across the entire pulse bandwidth. In practice, assuming that the first harmonic beam at λ 1 propagates as an ordinary beam, and a SHG beam is produced as an extraordinary beam at λ 2 = λ 1/2, the portion dλ 1 of the doubled pulse bandwidth is given by [23]:
(3.6)
Therefore, to increase the bandwidth of the doubled beam, it is necessary to decrease the crystal thickness L, at the cost of SHG efficiency (proportional to L 2). Therefore, according to the experimental requirements, one needs to find the right compromise between the two needs.
SHG is a specific nonlinear process which involves two photons with the same energy. Other second‐order nonlinear processes are possible, such as sum and difference frequency generation (SFG and DFG), where two photons with different energies combine together into a third photon. As for SHG, these processes need to satisfy energy and momentum conservation, which for SFG are expressed by: ω 3 = ω 1 + ω 2 and
