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).

Lez #3+4 From 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. Pay attention to the way we treat the real (Taylor expansion) and imaginary (constant absorption at w_0) part of beta(w).

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.

Schedule and tentative program

 The class consists of 60 hours teaching. Lectures will be held in Room 3, Edificio Fermi, on:

Tuesday 12-14

Wednesday 14-16

Thursday 16-18

The tentative program (based on previous editions) is Here. In 2026 there might be minor adjustments, please check on this blog later this year.  

Opening class will be on Tuesday 3 March