Lez #13+14 Principles of Laser

 Recap on Absorption, Spontaneous and stimulated emission. Laser principles. Passive resonator. Is a laser superior to a conventional sources? Trading spatial and temporal coherence with flux. Statistical properties of light. How to make a laser: 2/3/4 level systems. Different type of lasers, He-Ne, ruby, excimers, p-n junctions and semiconductor based photonic devices.

NumSim #3+4 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 #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. 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 and qualitatively explained the wave breaking phenomenon.

NumSim# 1+2 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 #9+10 Helmotz solution in linear dispersive (beta2) regime I: 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. 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 and a variation of the compex z-dependent phase. 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. 

Lecture #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 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, 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).