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
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