Lez #23+24 NOPA

 We started by revising the derivation of the phase-matching bandwidth in three-wave mixing, which is inversely proportional to the group velocity mismatch (GVM) between the signal and idler.

We discussed the twofold importance of Optical Parametric Amplification (OPA), which is characterized by frequency tunability and its potential for broadband amplification. We examined the positive feedback loop between the signal and idler that gives rise to the exponential scaling of the OPA gain with the crystal length L, as well as the various factors that limit this gain in the short-pulse regime.

We then discussed the Phase Matching (PM) condition and the necessity of using birefringent nonlinear crystals to satisfy it. Specifically, we explored the use of BBO crystals (which are negative uniaxial) to achieve Type I PM. In this configuration, the signal and idler experience the ordinary refractive index (n_o ), while the pump experiences an effective refractive index -a combination of n_e and n_o- determined by the angle θ. We saw how non-collinear geometry introduces an additional degree of freedom, the pump-signal angle α, to fulfill PM. For a specific value of α (the "magic angle"), this allows us to simultaneously satisfy the PM condition—and hence obtain efficient amplification—over a broad spectral interval of the seed.

Next, we discussed the temporal walk-off between interacting pulses, defining the pulse-splitting length. We studied the optimal crystal length for OPA processes, noting that it varies critically depending on whether the GVM of the signal and idler with respect to the pump have the same or opposite signs.

Finally, we saw the result of the generalization of broadband PM to non-collinear geometries (NOPA). In this case, optimal broadband PM is achieved when the projection of the idler's group velocity onto the signal's propagation direction is exactly equal to the signal's group velocity. Consequently, while collinear broadband amplification requires a degenerate configuration (ω_1 =ω_2), the non-collinear geometry relaxes this constraint.

Lez #21+22 Pulse duration measurements, Autocorrelator

 We discussed different type of autocorrelation techniques for measuring the duration of ultrashort pulses, namely linear autocorrelator, non-collinear SHG autocorrelator and collinear SHG autocorrelator, discussing the different advantages/disadvantages. We also introduced FROG (frequency-resolved optical gating) as a convenient approach to characterize an optical pulse.

Lez #19+20 Esercitazione 1 esonero

Lez #17+18 Wavemixing and OPA

 Starting from Maxwell equation in presence of a dielectric, non magnetic  we re-derived the differential equation for the pulse envelope in presence of linear polarization in a slightly different way.

We then extended to a non linear term, considering the specific case of second order. The most general case is here a three waves mixing, approximating one beam (the pump) much more intense then the others, the signal (initially weak) and the idler (initially empty).

We derived the gain expression in the case of perfect phase matching and in the general case, providing the acceptance bandwidth of the signal for given thickness, non linearity and group velocity mismatch. I commented on the need of having large bandwidth to preserve short pulse durations, but also on the opportunity to use long crystals to obtain efficient spectral filtering for example via second harmonic generation (in which w1=w2 and it acts as a pump, while w3 is the generated signal).

Lez #15-16: Numerical propagation in linear and non-linear regime

After discussing the split-step Fourier method (in both its standard and symmetrized versions) and its numerical convergence, we evaluated scripts for propagating short pulses using the symmetrized approach. The following cases were presented: (1) propagation in a nonlinear material with no dispersion, including the effects of absorption and/or pre-chirp; (2) propagation in the presence of both dispersion and nonlinearity; (3) propagation in the presence of normal dispersion and nonlinearity, with some pre-chirp; (4) nonlinearity in the special regime of low (but non-vanishing) dispersion; (5) soliton formation.

Lez #13+14 Propagation in non-linear media, Kerr effect

 We discussed non linear effects. At first, we observed as in the very common case of centro symmetric materials even susceptibilities must be zero (only odd restoring forces, i.e. even potential energy). Therefore, the most general, lowest order non linear effect is the third order (Boyd 1.5.10 and this link). Such an effect can be introduced in terms of a non linear, intensity dependent refractive index. We solved the equation in time domain, obtaining a z-independent temporal profile and a z dependent phase term. It can be shown (numerically) how the resulting spectrum broadens with z. We then combined the two effects and derived the soliton solution arising when (anomalous) GVD and Kerr compensate for LD=Lnl. We also discussed wave breaking phenomenon, in presence of non linearity with small normal dispersion. 

Lez #11+12 Numerical solution in the linear dispersion regime

 We tested some matlab scripts t perform fourier transform and solve gaussian beam propagation with initial chirp in presence of beta1 and losses