\documentclass{article}
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						 mark=at position .62 with \arrow{stealth}}}}
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\geometry{tmargin=.7in, bmargin=.75in, lmargin=.75in, rmargin = .75in}

\begin{document}
\setlength{\parindent}{0mm}

\title{Optique - Michelson}
\author{Aubin SIONVILLE}
\date{MPI Clemenceau - 2021-2023}
\maketitle

\vspace{0.5cm}
\part*{Interféromètre de Michelson}

A faire : Schéma de l'interféromètre de Michelson
\newpage
\part*{Montage en lame d'air}

\section*{Montage}
\setlength{\columnsep}{80pt}
\begin{multicols}{2}

    \begin{tikzpicture}
        % Axes
        \draw [loosely dashed] (-5, 0) -- (5, 0);
        \draw [loosely dashed] (0, -5) -- (0, 12);

        % Miroirs
        \draw[thick] (3.5,-2.1)--(3.5,2.1); %M1
        \foreach \z in {-1.9,-1.7, ...,2.1}
        {
        \draw (3.5,\z)--++(0.25,-0.15);
        }
        \draw[thick] (-2.1,3)--(2.1,3); %M2
        \foreach \z in {-2.1,-1.9, ...,2.1}
        {
        \draw (\z,3)-- ++(0.15,0.25);
        }
        
        \draw[dashed] (-2.1,4)--(2.1,4);%M1'
        \foreach \z in {-2.1,-1.9, ...,2.1}
        {
        \draw [dashed] (\z,4)-- ++(0.15,0.25);
        }

        % Lentille semi-réfléchissante
        \draw [thick] (-2, -2) -- (2, 2);

        % Lentille convergente
        \draw [<->] (-3.5, -3.5) -- (3.5, -3.5);
        
        % Ecran
        \draw [thick] (-4, -5) -- (4, -5);

        % Rayons
        \draw [red-ray] (-5, 0) -- (0.8, 0.8); %incident

        \draw [blue-ray] (1.46,1.46) -- (2.55,-3.5); %rayon 1
        \draw [blue-ray] (2.55,-3.5) -- (0.33,-5);
	    \draw [blue-ray] (0.8,0.8) -- (1.12,3);
        \draw [blue-ray] (1.12,3) -- (1.46,1.46);
	
        \draw [densely dashed] (1.12,3) -- (1.27,4); %faux rayon 2
        \draw [densely dashed] (1.27,4) -- (1.76,1.76);

        \draw [green-ray] (0.8,0.8) -- (3.5,1.17); %rayon 2
	    \draw [green-ray] (3.5,1.17) -- (1.76,1.76);
        \draw [green-ray] (1.76,1.76) -- (2.92,-3.5);
        \draw [green-ray] (2.92, -3.5) -- (0.33, -5);

        \draw [densely dashed] (0,-3.5) -- (0.33,-5); %aide au dessin
        
        % Fausses sources
        \draw [blue-ray-d] (0,8.1) -- (1.12,3);
        \draw [green-ray-d] (0,9.77) -- (1.27,4);

        % Annotations
        \node at (-5,0.)[left] {$S$};
        \node at (4, -5)[below] {$Ecran$};
        \node at (3.5, 2.1)[above] {$M_1$};
        \node at (2.1, 3)[right] {$M_2$};
        \node at (2.1, 4)[right] {$M_1'$};
        \node at (0,8.1)[left] {$S_1$};
        \node at (0,9.77)[left] {$S_2$};
        \draw (0,4.7) arc (-90:-84:6.8);
        \node at (0.37,4.72)[above] {$i$};
        \node at (0.33, -5)[below] {$M$};

    \end{tikzpicture}

    \section*{Explication}
    Les ondes interfèrent à l'infini : les rayons sont parallèles \newline
    
    \subparagraph*{Différence de marche}
    \[
        \begin{split}
            \delta(x) &= (SM)_{v2} - (SM)_{v1} \\
            &=\cancel{(SS_2)} + (S_2M) - \cancel{(SS_1)} - (S_1M) \\
            &=(S_2M) - (S_1M)
        \end{split}
    \]
    Principe du retour inverse + théorème de Malus $\implies (M_S2) = (MH)$ et $(S_2M)=(HM)$ \newline
    \[
        \begin{split}
            \delta(x) &= \cancel{(S_2M)} - (S_1H)-\cancel{(HM)} \\
            \delta(x) &= \boxed{2en_{\text{air}}\cos(i)}
        \end{split}
    \]
            
    \subparagraph*{Intensité sur l'écran}
    D'après la formule de Fresnel :
    \[\boxed{I(M) = 2I_0 \bigg(1 + \cos\Big( \frac{2\pi}{\lambda_0} 2n_{air}e \cos(i) \Big) \bigg)}\]
    $I = \text{cst} \implies i = \text{cst}$ \newline
    $\boxed{\text{Les franges d'interférence sont des cercles}}$
\end{multicols}
\newpage
\part*{Montage en coin d'air}


\section*{Montage}
A faire : montage en coin d'air

\section*{Explication}
Les franges sont localisées sur le miroir virtuel \newline
\textbf{Les franges d'interférence sont des lignes}

\subparagraph*{Différence de marche}
$$\delta(x) = 2n_{air}eax$$

\subparagraph*{Intensité sur l'écran}
$$\boxed{I(M) = 2I_0 \bigg(1 + \cos\Big( \frac{2\pi}{\lambda_0} 2 n_{air}eax \Big) \bigg)}$$


\end{document}