Lez #27+28 Double sided Feynman diagrams: Transient absorption

We evaluated the third order non linear response in the pump probe scheme. Two beams are used to accomplish the 3 interacitons, excluding two photon processes and using RWA the number of fiagrams is reduced to 6, corresponding to 3 different processes: excited state absorption, stimulated emission, ground state bleaching. Bleaching and stimulated response give the same response, unless the system is evolving onto the excited state, for example due to vibrational relaxation.

Lez #25+26 Double sided Feynman diagrams: linear absorption

 We introduced the double sided Feynman diagrams to describe the perturbative expansion of the polarization, end explicitly evaluated the linear absorption of a two level system.

Lez #23+24 Radiation - matter, rabi oscillations, perturbative expansion

 Today we recalled the basic concepts of quantum mechanics, introducing the density matrix. We stressed the importance  of density matrix to describe statistical mixtures, and how this is not possible through an eigenstates superposition (for instance a two level system).

We discussed how to solve Bloch equation and Rabi oscillation, introducing the rotating wave approximation to neglect some terms in the integration.

We then introduced Shroedinger, Heisenberg and Interaction schemes (in this latter both eigenfunctions and operators evolve, and the temporal evolution related to the unperturbed Hamiltonian is removed from the wavefunction). Then it makes sense a perturbative expansion of the eigenfunctions and of the density matrix in the Interactions scheme. Finally, we went back to the Shroedinger scheme.

Please refer to the first part of the notes added to classroom.

Lez #21+22 Introduction to 3rd order spontaneous vs stimulated processes, classical approach

We introduced the field-induced polarization, in both the spontaneous and stimulated cases. We discussed the origin of spontaneous Raman, and why it is incoherent, linear and homodyne. As for the selection rules we recalled some basic notions from struttura della materia class. Then we moved to the stimulated Raman, a third order process for which we evaluated different contributions to the polarization. See the material on classroom.

Lez #19+20 wavemixing and OPA II

We derived the solution in the general case of non perfect phase matching, 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). We understood how to realize the phase matching in the collinear and non collinear case, dealing with walk off. In the non collinear case it is possible to amplify a wide bandwith with a wise choide of the incidence geometry and cristal orientation.

Lez #17+18 Wavemixing and OPA

We first finalized the laser lecture introducing mode locking for pulsed laser generation, active and passive methods, gain and losses compensation.  

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.

Primo esonero

Lez #15+16 Experimental techniques for pulsed laser characterisation

 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 convenent approach to characterize an optical pulse.

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).

Lez #3+4 Maxwell to Helmotz

 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.

Lez #1+2 Course overview

Usual approximations adopted for classical radiation matter interaction. Beyond plane and monochromatic waves. Expectations for linear propagation of a wave packet. General features of a non linear process and pulsed light: frequency mixing and temporal resolution. Non (temporal) locality of the susceptibility. From spectroscopy to microscopy.