Lez #15+16 Propagation in linear and non-linear regime, solitons

  We discussed some scripts for numerical simulations, based on the symmetrized split-step Fourier method. The following cases have been presented: (1) propagation in dispersive materials; (2) propagation in nonlinear material; (3) propagation in presence of both dispersion and nonlinearity; (4) nonlinearity in the special regime of low (but not vanishing) dispersion; (5) soliton formation.

Lez #13+14 Discrete Fourier Transform and split-step methods for non linear propagation

  We introduced the split-step Fourier method as a tool to solve the pulse-propagation problem in nonlinear dispersive media. Building on the Baker-Hausdorff formula, we have discussed the accuracy of both the standard and the symmetrized split-step Fourier methods that will be used for the numerical simulations. Then we introduced the Fast Fourier Transform (FFT) as an algorithm to compure the Discrete Fourier transform (DFT) in a computer simulation, stressing the role of the sampling frequency and of the maximum time window in terms of frequency resolution and maximum resolved frequency in the conjugate domain.Some examples of FFTs have been discussed 

Lez #11+12 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.

Lez #9+10 Helmotz solution in linear dispersive (beta2) regime II: 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: UNCHIRPED 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).

Lez #3+4 From Maxwell to Helmotz equations

 We recalled Maxwell's equations in the most general form (presence of medium, external charges and currents). We then retrieved the solution in vacuum without sources, recalling the concept of plane wave, Poynting vector, Fourier transform (derivative and convolution theorem). We then moved to the case of a dielectric medium (non magnetic), introducing within the linear approximation the time-dependent (non local) relation between polarization and electric field in the homogeneous and isotropic case. This implies a convolution in time domain. For example under an impulsive electric field (Dirac-delta(t)) the material response occurs even at finite times as P(t)=eps0*chi(t). Moving to the frequency domain, we derived the Helmotz equation in which the material properties enter via the (generally complex) frequency dependent dielectric constant. Please install MATLAB on your laptops (freely distributed by Sapienza: https://www.uniroma1.it/it/pagina/software-gratuito), which we will soon use to simulate pulse propagation.