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+
+%\frontmatter
+  %detuning
+  \title{Effetto ottico Kerr (OKE) per lo studio dei modi vibrazionali in liquidi molecolari.}
+  \author{Chilleri Gabriele \\ Zolfanelli Lorenzo}
+  \date{08/06/2016}
+  
+  \begin{document}
+\maketitle
+
+  \cleardoublepage
+  \begin{flushright}
+  \null\vspace{\stretch {1}}
+  \emph{Physics would be dull and life most unfulfilling if all physical phenomena around us were linear. Fortunately, we are living in a nonlinear world.\\(Shen)\break}
+  \vspace{\stretch{2}}\null
+  \end{flushright}
+  \cleardoublepage
+%\tableofcontents  
+
+%\mainmatter
+\chapter*{Introduzione}
+\chaptermark{Introduzione}
+Considerando la densità di corrente $\textbf{J}$ in approsimazione di dipolo, si può ricavare l'espressione della polarizzazione di un mezzo dall'equazione di continuità e dalle equazioni di Maxwell. Essa si esprime come
+\begin{equation}
+\left[ \nabla\times \nabla\times + \dfrac{1}{c^2} \dfrac{\partial ^2}{\partial t^2} \right] \textbf{E}(\textbf{r},t) = -\dfrac{4\pi}{c^2} \dfrac{\partial ^2}{\partial t^2} \textbf{P}(\textbf{r},t).
+\end{equation}
+Può essere utile sviluppare la soluzione dell'equazione differenziale in potenze del campo elettrico,
+$$ \textbf{P}(\textbf{r},t)=\textbf{P}^{(1)}(\textbf{r},t) +\textbf{P}^{(2)}(\textbf{r},t) + \textbf{P}^{(3)}(\textbf{r},t)+...\, ,$$ e quindi scrivere i contributi tensoriali come 
+\begin{eqnarray}
+P_i^{(1)}(\textbf{r},t) &=& \int \chi_{ij}^{(1)}(\textbf{r}-\textbf{r}', t - t') \,\textbf{E}_j(\textbf{r}',t')  \nonumber \\
+P_i^{(2)}(\textbf{r},t)&=& \int \chi_{ijk}^{(2)}(\textbf{r}-\textbf{r}',\textbf{r}-\textbf{r}'',t - t',t-t'') \,\textbf{E}_j(\textbf{r}',t')\textbf{E}_k(\textbf{r}'',t'') \nonumber \\
+P_i^{(3)}(\textbf{r},t)&=& \int \chi_{ijkl}^{(3)}(\textbf{r}-\textbf{r}',\textbf{r}-\textbf{r}'',\textbf{r}-\textbf{r}''', t - t',t-t'',t-t''') \,\textbf{E}_j(\textbf{r}',t')\textbf{E}_k(\textbf{r}'',t'')\textbf{E}_l(\textbf{r}''',t''') \nonumber.
+\end{eqnarray}
+Un campo elettrico è capace di modificare le proprietà dielettriche di un mezzo. Infatti il tensore suscettività $\chi_{ij..n}^{(n)}$ è la variabile che descrive propriamente l'interazione radiazione-materia. Inoltre le equazioni di Maxwell forniscono la connessione tra il tensore dielettrico $\epsilon_{ij..n}
+$ e la suscettività nel sistema C.G.S.,$\epsilon_{ij..n} = 1 + 4\pi\chi_{ij..n}$, quindi direttamente anche con l'indice di rifrazione $\eta$, dalla relazione $\eta= \sqrt{\epsilon}$, cioè con le proprietà ottiche di un mezzo.
+I termini del primo termino descrivono l'ottica lineare, mentre i termini successivi quella non lineare. Essi sono messi in evidenza per campi molto intensi come quelli generati dai laser impulsati. I tensori suscettività dipendono fortemente dalle proprietà di simmetria. Il contributo del secondo ordine è nullo per mezzi isotropi e centro-simmetrici, come per alcuni liquidi e il vetro, mentre quello del terzo ordine è sempre presente per ogni simmetria.
+Uno degli effetti legati al termine del secondo ordine è la \emph{generazione di seconda armonica}, SHG. Questo effetto è dovuto all'interazione, ad esempio in un cristallo BBO, tra due pacchetti con diverso vettore d'onda $\textbf{k}_1 \neq \textbf{k}_2$, che si combinano generando un terzo campo con vettore d'onda $\textbf{k}=\textbf{k}_1+\textbf{k}_2$ e con frequenza $\omega=\omega_1+\omega_2$.
+Quando i due campi sono forniti dallo stesso laser, considerando solo il caso in cui la frequenza del laser non è risonante con le transizioni elettroniche del mezzo, e trascurando gli effetti non locali, è possibile ottenere la duplicazione della frequenza della radiazione incidente.
+
+Il termine del terzo ordine della suscettività è invece responsabile dell'\emph{effetto Kerr ottico}, OKE. L'applicazione di un campo elettrostatico produce su un mezzo isotropo, come un liquido o un gas, un'anisotropia ottica transiente. Considerando effetti locali e non risonanti si parla di \emph{birifrangenza indotta}, altrimenti si parla di \emph{dicroismo indotto}. Per mezzo di un secondo impulso, opportunamente polarizzato, può essere misurata l'evoluzione temporale dell'eccitazione ottica dello stato fuori equilibrio del materiale. Da questa misura si osserva la dinamica di rilassamento della birifranenza indotta, dalla quale si può ottenere informazioni sui modi vibrazionali intermolecolari.
+
+\chapter{Apparato strumentale}
+I processi dinamici riguardanti i liquidi molecolari lavorano su intervalli temporali molto veloci, dai femtosecondi fino ai nanosecondi. Per investigare su tali processi è indispensabile lavorare con laser capaci di produrre impulsi molto corti. Tali laser sono definiti \emph{impulsati}. Al contrario del regime continuo, dove il campo elettromagnetico generato deve essere monocromatico, quindi di durata molto lunga e coerente, nel regime pulsato il campo è rappresentabile come somma dei campi dei modi longitudinali, i quali vanno a costituire un inviluppo gaussiano. 
+$$
+\textbf{E}(t)= \sum_{n=-k}^{+k} A_n\, e^{i\left[(\omega_L+n\delta \omega )t+ \varphi_n   \right]}
+$$
+$A_n$ rappresenta la distribuzione gaussiana, $ \omega_L$ il valore della pulsazione in cui il campo assume il massimo valore dell'intensità dell'inviluppo, $n$ il numero dei modi, $\delta\omega$ la distanza tra due modi e $\varphi_n$ la fase dell'$n$-esimo modo. Per il nostro esperimento abbiamo utilizzato un laser con mezzo attivo Ti:sapphire che ci assicura il principio di \emph{mode-locking}, per cui tutti i modi hanno una relazione di fase costante e che le ampiezze dei modi seguono una funzione gasussiana. La durata temporale minima degli impulsi è data dalla condizione di \emph{transform limited}, la quale definisce costante il prodotto $ \Delta \tau_L \Delta \omega_L= k = cost$, che nel caso gaussiano è $ k = 0.441.$ Data la lunghezza d'onda spettrale, questa relazione ci permette di stimare la durata minima del segnale laser. Il laser con mezzo attivo Ti:sapphire è pompato dalla seconda armonica di un laser in continua   $\mathrm{Nd:YVO_4}$, il quale consente l'inversione di popolazione nel secondo mezzo attivo e la generazione di impulsi. Tale sistema laser produce una dispersione positiva dovuta al mezzo attivo, quindi risulta necessario aggiungere un sistema di compensazione composto da due \emph{chirped mirror} ed un \emph{compressore a prismi} che consentirà di ridurre la dispersione anche per gli altri sistemi ottici dell'apparato strumentale. In figura \ref{spettrolaser} si osserva lo spettro dell'impulso laser confrontato con una funzione gausssiano predeterminato. Confrontando l'impulso con la gaussiana si può calcolare da essa la durata minima nelle condizioni di transform limited che risulta di 14 fs. 
+\begin{figure}[h]
+\includegraphics[scale=0.5]{SpettroLaser}
+\caption{Impulso laser in ingresso nell'apparato sperimentale (nero) confrontato con una funzione gaussiana (rosso).}
+\label{spettrolaser}
+\end{figure}
+Una lamina $\dfrac{\lambda}{2}$ è inserita nel setup sperimentale per assicurare che l'impulso laser sia ben polarizzato verticalmente o orizzontalmente. Un beam-splitter permette di dividere il fascio in due: un fascio \emph{pump} (pompa) e un fascio \emph{probe} (sonda). Il fascio pompa permette di generare la birifrangenza nel campione, mentre il fascio sonda serve a misurare tali modifiche delle proprietà ottiche.  
+\begin{figure}[h]
+\includegraphics[width = \textwidth]{setup}
+\caption{Apparato sperimentale.}
+\label{setup}
+\end{figure}
+In figura \ref{setup} si osserva che la procedura di misura pump-probe si manifesta ritardando il fascio di sonda tramite un sistema di specchi montato su una slitta mobile. Variando la separazione temporale tra pump e probe è possibile indagare l'evoluzione nel tempo della birifrangenza indotta dal pump. Quest'ultimo attraversa un chopper, cioè un disco in rotazione che introduce una modulazione dell'intensità del fascio. Esso è usato in combinazione con un amplificatore lock-in, il quale demodulando il segnale, riduce i contributi di rumore. Si parla di tecnica \emph{omodina} quando si misura direttamente il campo segnale con un rivelatore. Nel nostro caso si compie una misura in \emph{eterodina}. Tale tecnica necessita l'utilizzo di un campo locale che viene generato per mezzo di una lamina ${\lambda}/{4}$, cioè con polarizzazione circolare. Un sistema di lenti permette di focalizzare il fascio di pompa e di sonda nella stessa regione del campione. Definiamo campo locale la porzione di fascio la cui polarizzazione non viene alterata dall'interazione con il campione. Mentre la porzione che interagisce contribuirà alla generazione del campo segnale. Le componenti x e y  del campo locale sono interessate da uno sfasamento di ${\pi}/{2}$, mentre quelle del campo segnale avranno un ulteriore sfasamento dovuto alla birifrangenza. Il percorso delle componenti ortogonali viene separato da un cubo polarizzatore e inviato a un rivelatore a due fotodiodi, il quale lavora in amplificazione differenziale. L'intensità data dal modulo quadro della somma dei due campi determina la presenza di un termine, detto \emph{eterodino}, che ha segno opposto per le due polarizzazioni; l'amplificatore differenziale consente di eliminare gli altri contributi.
+$$ S \propto |E_S|^2 + |E_{loc}|^2 + 2\, Re \, E_S \cdot E_{loc}^* $$
+Quindi in uscita dal sistema di rivelazione avremo il segnale 
+$$ S_{ete} = \dfrac{S_+ - S_-}{2} $$
+La tecnica di misura in eterodina favorisce l'eliminazione dei contributi di fase  ed un aumento in intensità del segnale, in quanto presenta un andamento lineare rispetto al campo segnale, al contrario del caso omodino che è quadratico. Tale aspetto facilita  il rivelamento di campi deboli come quello prodotto dall'effetto Kerr.
+
+\chapter{Misura della durata dell'impulso tramite generazione di seconda armonica}
+Un metodo per ricavare la durata dell'impulso laser è sfruttare il fenomeno della generazione di seconda armonica. Come scritto in precedenza, tale fenomeno si ottiene dall'interazione di due impulsi in un mezzo anisotropo, quindi con il termine del secondo ordine della polarizzazione diverso da zero. Per procedere a tale tecnica abbiamo sostituito il campione liquido con un cristallo BBO?? in una disposizione ottica come in figura \ref{shg}.
+\begin{figure}
+\includegraphics[width = \textwidth]{shg}
+\caption{Generazione seconda armonica per la misura temporale dell'impulso laser.}
+\label{shg}
+\end{figure} La generazione di seconda armonica avviene quando i due impulsi giungono sul cristallo nello stesso istante. Con un diaframma posto dopo il campione si interrompono i fasci con frequenza $\omega$ e si lascia propagare il fascio di seconda armonica $2\omega$. Si misura l'intensità del fascio in funzione del ritardo $\tau$ tra i due impulsi, ottenuto tramite la slitta mobile. L'andamento del segnale in funzione di $\tau$ è dato dalla funzione di autocorrelazione 
+$$ S(\tau) = \int dt\,\left[\,| E(t)+ E(t-\tau)|^2\,\right]^2 .$$
+\begin{figure}
+\includegraphics[width = \textwidth]{SHGcorr}
+\caption{Andamento della funzione di autocorrelazione priva di filtro (in nero),in cui si osservano le frange di interferenza e l'andamento con filtro (in rosso) per eliminare i termini costanti. Si può osservare che la funzione di autocorrelazione segue un andamento gaussiano.}
+\label{SHGcorr}
+\end{figure}
+Calcolando numericamente la funzione di autocorrelazione si ottiene l'andamento del segnale. Trascurando i contributi costanti del segnale, l'andamento calcolato risolto tramite una simulazione MATLAB segue una funzione gaussiana. Per mezzo della relazione $\Delta \tau_L= \dfrac{\Delta \tau_S}{\sqrt{2}}$ si ricava la durata dell'impulso laser simulato
+$$\Delta \tau_{L_{sim}}=14.15\,\mathrm{fs},$$
+invece, per quanto riguarda il dato sperimentale si ricava una durata temporale dell'impulso pari a 
+$$\Delta \tau_{L_{exp}}=15.14\,\mathrm{fs}.$$ 
+\begin{figure}
+\centering
+\subfigure[]{\includegraphics[width = 0.48\textwidth]{FitSHG_sim}}
+\subfigure[]{\includegraphics[width = 0.48\textwidth]{FitSHG_exp}}
+\caption{Fit gaussiani dell'impulso laser simulato (a) e dell'impulso laser sperimentale (b).}
+\label{impulsi}
+\end{figure}
+
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+\end{document}
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