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

Lez #9+10 Helmotz solution in linear dispersive (beta2) regime II: initially chirped gaussian pulse

 We discussed as, in linear regime, the presence of a beta2 (GVD) affects a pulse initially chirped in time domain (i.e. adding a phase in time domain). In such case, the pulse can indefinitely broaden upon traveling along z into the material OR, it can get narrower down to a minimum and then broaden again. Hence, providing some chirp it can be useful, for instance, if one aims to get a minimum beam waist at a given point in space (the sample) and NOT in the z position where the pulse is generated (the laser output). Note that starting with a pulse chirped in time domain, the initial bandwidth is also broadened with respect to 1/T0, i.e. the pulse is initially non fourier Transform limited. In principle, one could start with a pulse chirped in frequency domain (only a phase factor in frequency domain) hence modifying the initial time duration. This situation is similar to what is done pre-compensating pulse broadening. See for instance Chirped Pulse Amplification.

Lez #7+8 Helmotz solution in linear dispersive (beta2) regime I: initially Trasform limited gaussian pulse

 We derived the evolution of an initial (z=0) gaussian wavepacket (in time) with phase=0 in presence of beta1 and beta2. We first made a change of reference frame defining a new time variable dependent on the spatial position inside the material (in other words we defined a new time variable to have all the temporal profiles centered at zero at any distance Z within the material). We moved to frequency domain, where the solution in z is easy to obtain (exponential). Then we evaluated the initial condition in w-space and we finally anti-transformed back in time domain. We found a temporal broadening (always a broadening, depending on the squared beta2) and a variation of the complex z-dependent phase (positive or negative, depending on the sign of beta2). This means the pulse acquires some chirp while propagating along z, with different "temporal portions" of the envelope having different frequencies. The bandwidth is unaffected. Hence, the solution at finite Z is no longer Fourier Transform limited. 

Lez #5+6 Helmholtz equation: quasi-monochromatic solution, with beta1 and losses

  We considered as a possible solution a quasi-monochromatic wavepacket, meaning a plane wave with a "carrier" frequency at w0 modulated by a slowing varying envelope. For the z dependence of the slowing varying part we factored out a plane wave-like term with a constant, real phase beta(w_0). Moving to frequency domain we obtained an equation for the envelope depending on beta(w), a complex, frequency dependent quantity. We expanded the real part of beta(w) around w_0 as we expect the spectrum of the solution to be "narrow" with respect to w_0. We considered the first (beta1) and second (beta2) orders of the rel part expansion, while we took the entire imaginary part alpha (responsible for the absorption). We went back to time domain. Retaining only beta1 and the absorption (the imaginary part of beta(w), for which we assume a constant imaginary value) we found for the envelope a solution propagating as a wave with velocity 1/beta1 (group velocity). Losses appear as an exponential decay in real space. As a reference, see the lecture notes, the first Agrawaal chapter 1.2.3 and second: 2.1 e 2.3.1 (Nonlinear Fiber Optics Govind P. Agrawal).