Lez #51+52 Principles of advanced microscopies I. RIKES.

 We described the basics of RIKES: a vibrationally resonant combination of pump and stokes can induce optical anisotropy which generates an electric field along a direction orthogonal to the incoming stokes, via a four wave mixing process. This effect can be homodyne detected, or via balanced detection, to measure separately the real and imaginary parts of the non linear susceptibility.

Lez #49+50 Principles of advanced microscopies I. Coherent Vibrational imaging.

We introduced the basic principles of linear and non linear imaging, with or without vibrational sensitivity. We then discussed the basics of CARS imaging, the nor resonant background issues and how to resolve them.

Lez #47+48 FSRS II: molecular photoexcitations and transient states: the example of Tiophene

We then discussed the case of MPT studied by FSRS, where an ultrafast motion outside the Franck Condon region is observed, followed by vibrational cooling and singlet to triplet internal conversion. This example illustrates the synergy among transient absorption, FSRS and DFT calculations.   

Lez #45+46 Time-Domain Impulsive Vibrational Spectroscopy:

 We discussed the Impulsive Vibrational Scattering technique, in which a pair of temporally separated pulses is used to map vibrational excitations in the time domain. The experimental setup follows a pump-probe scheme, where an ultrashort broadband pump pulse generates vibrational coherences via two interactions with spectral components shifted by one vibrational quantum. The coherences modulate the optical properties of the medium, which are then monitored by a delayed probe. We derived the analytical expressions using the diagrammatic formalism, assuming impulsive beams. On classroom some lecture notes and a few slides discussed.

Lez #43+44 IRS signal. RRS under different pulse temporal profile

 We investigated the IRS1 response under various resonance conditions, demonstrating that the resulting lineshapes - positive, negative or dispersive - depend on the Raman pump wavelength relative to the sample's absorption and the probed vibrational mode.

Then, we derived an expression for the RRS signal as a function of arbitrary spectral profiles of the Raman pump and probe pulses. This formula was then used to numerically evaluate the Raman gain under different experimental conditions, specifically varying the pump pulse temporal profile and its relative delay with respect to the probe.

Lez #41+42 DephasingII SRS blue side

 Final considerations on dephasing. Homogeneous and inhomogeneous limits (fast and slow modulation regimes) and motional narrowing. Spectral line shapes: Lorentzian, Gaussian, and Voigt. Derivation of the SRS response in the blu side (IRS diagram) for a monochromatic pump and a generic Stokes beam

Lez #39+40 Dephasing

We discussed from a microscopic point of view how out-of-diagonal elements of the density matrix are damped due to the interaction with the environment. As a reference, you can use chapter 5 of the “Principles of Nonlinear Optical Spectroscopy: A Practical Approach” by P. Hamm

Lez #37+38 FSRS I: Heme proteins and molecular movies

 We introduced the Femtosecond Stimulated Raman Scattering (FSRS) experimental scheme, where SRS is exploited to track the evolution of a system upon the photoexcitation of an Actinic pump pulse, preceding the Raman and Stokes pair. The experimental layout has been introduced and the temporal/spectral resolutions of FSRS have been compared with the time-resolved spontaneous Raman counterpart. Then we started to discuss the paradigmatic case of Myoglobin, where FSRS has been exploited to: i) interpret transient absorption experiments to discriminate between two possible contrasting scenarios related to the existence of transient electronic states vs vibrational cooling. ii) determine how the absorbed photon energy, initially stored in delocalized low frequency heme modes, is very efficiently funneled into specific high frequency modes prior to slow dissipation through the protein 

Lez #35+36 SRS signals

 We evaluated the SRS (RRS1 diagram) response in resonant and non resonant case, for monochromatic Raman pump and probe beams.

Lez #33+34 Diagrams for 3rd order processes: pump-probe and stimulated Raman

We re-discussed the pump probe response (stimulated emission (SE), bleaching and excited state absorption (ESA)) commenting on the differences between bleaching and SE. 

We then derived the double sided Feynman diagrams associated with stimulated Raman processes, i.e. for a three level systems (2 electronic states, with the ground state having 2 vibrational levels) in presence of two different pulses, namely the Raman pulse and the Stokes pulse, detecting the heterodyne response arising on top of the stokes beam.

Lez #31+32 Double sided Feynman diagrams: linear absorption and introduction to 3rd order

 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.

We started to extend the same approach to third order processes. Among them, pump-probe experiments with the first 2 simultaneous interactions with the pump, and the third interaction with a probe.

Lez #29+30 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 #27+28 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 #25+26 Principles of laser II and Wavemixing I

 We first finalized the laser lecture introducing the solid state laser principle. Then 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).

Lez #23+24 First ESONERO

Lez #21+22 Experimental techniques for pulsed laser characterization. Autocorrelator.

  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 convenient approach to characterize an optical pulse.

Lez #19+20 Simulazione di Esonero

We solved an old, slightly modified test. 

Lez #17+18 Principles of Laser I

 Recap on Absorption, Spontaneous and stimulated emission. Laser principles, treshold condition. 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.

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.

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.