\documentclass[10pt,a4paper,twocolumn]{article} %% two column, final layout

\usepackage{ol2}
\usepackage[draft]{hyperref}
\usepackage{amsmath}

\begin{document}

\twocolumn[ %% activate for two-column option


\title{Fourier Optics}
\date{11th of January 2013} 
\author{Vincent Maioli}
\begin{abstract}


All OL manuscripts should be
submitted in two-column format to assist length estimate. OSA no
longer requires submission of single-column double-spaced text.
This template, along with associated style files, should be used
to prepare an article for submission to \textit{Optics Letters}
(OL). Examples are given of how to account for some of the factors
that affect the accuracy of the length estimate: figures, tables,
equations, and author affiliations. Note that this template is
only a guideline, and not a guarantee, of final page
length.

\end{abstract}

 ] %% activate for two-column option

\noindent Authors should prepare all OL manuscripts with the
\texttt{[twocolumn]} option. This will assist both the author and
OSA staff in estimating final page count. Preparing a length-check
manuscript also involves setting tables and figures within the
body of the paper with appropriate sizing commands and placing the
\verb+\twocolumn[...]+ command around the title\-page elements (as
explained \hyperlink{frontmatter}{below}). There are also
instructions at the end of the template for setting up author
affiliations properly. Note that it is no longer necessary (or
desirable) to set articles in single column or to set figures and
tables at the back of the manuscript.

\bigskip

Sample code for the preamble is as follows:

\small
\begin{verbatim}
\documentclass[10pt,twocolumn]{article}
\usepackage{ol}
%% Figures should be placed in body
%% of manuscript and
%% sized appropriately.
\end{verbatim}
\normalsize



The command \verb+\twocolumn[...]+ must be placed around the titlepage elements in the two-column option. Note that proper figure, table, and caption environments should be used (see samples below).

\begin{figure}[htb]
\centerline{\includegraphics[width=7.5cm]{richardson-f1.eps}}
\caption{One-column figure set with the figure environment.}
\end{figure}

\begin{figure*}[htb]
\centerline{\includegraphics[width=10cm]{richardson-f1.eps}}
\caption{Two-column figure set with the figure* environment.}
\end{figure*}



\textbf{Displayed equations} may be the most problematic for
two-column display. \emph{Optics Letters} equations are usually
set in one column; breaks and alignment should bring out the
structure of the math:
%% LaTeX
%\begin{eqnarray}
%{\dot{E}_{x,y}} &=&\frac{1}{2}\left( 1+j\alpha \right) \left( G_{x,y}-\gamma \right) %E_{x,y}  \label{Eq1} \nonumber \\
%&&+\kappa E_{x,y}\left( t-\tau \right) \exp \left( -j\Omega
%_{x,y}\tau \right) \nonumber \\
%&& + (\beta _{sp}N)^{1/2} \xi _{x,y}.
%\end{eqnarray}

%% amsmath
\begin{align}
{\dot{E}_{x,y}} &=\frac{1}{2}\left( 1+j\alpha \right) \left( G_{x,y}-\gamma \right) E_{x,y}  \label{Eq1} \nonumber \\
&\quad+\kappa E_{x,y}\left( t-\tau \right) \exp \left( -j\Omega
_{x,y}\tau \right) \nonumber \\
&\quad + (\beta _{sp}N)^{1/2} \xi _{x,y}.
\end{align}


Use standard LaTeX or AMSTeX environments. For equations that \textit{must} span two columns, it is possible to use a float environment, e.g., \verb+\begin{figure*}...\end{figure*}+.
Such an environment will not interfere with figure or table numbering (which is controlled by the caption), but it \textit{will} cause equations to float, often with unwanted consequences.

\textbf{Figures} should  be set to one-column size \mbox{($\sim$8.3 cm)} whenever possible; \textbf{tables} should also be set to one column whenever possible,
but tables with more than five columns will probably need to be set to two columns. For two-column layout, figures and tables can be set across both columns with the alternate figure and table environment commands
\verb+\begin{figure*}...\end{figure*}+ instead of \verb+\begin{figure}...\end{figure}+. Note that tables are typeset and cannot be reduced in size like art, which may require more space than in the submitted paper.

\subsection*{Sample figure environment:}
\small
\begin{verbatim}
\begin{figure}[htb]
\centerline{
\includegraphics[width=8.3cm]{richardson-f1.eps}}
 \caption{Sample figure.}
\end{figure}
\end{verbatim}
\normalsize

\textbf{References} callouts are now formatted with the \texttt{cite} package, which produces bracketed reference style (e.g., [1]).
For online callouts, e.g., see  [1], the words ``Ref.'' and
``Refs.'' are not required.

Before submitting, authors who use BibTeX should first run BibTeX,
then paste the contents of the output file \texttt{*.bbl} into the
\texttt{*.tex} manuscript file. Our electronic submissions system
cannot process BibTeX directly.

\paragraph{The following files are included in this distribution:}
\begin{itemize}\itemsep-2pt
\item\texttt{OLpagelength.tex} \ Template and instructions
\item\texttt{ol2.sty} \ Style file
\item\texttt{ol.bst} \ BibTeX style file for short-form citations
(as published)
\item\texttt{osajnl.bst} \ BibTeX style file for full-length
citations to include on fourth page to assist editors and
reviewers
\item\texttt{richardson-f1.eps} \ Sample .eps figure.
\end{itemize}

\bigskip

\begin{table}
  \centering
  \caption{Sample Table}\begin{tabular}{ccccc} \\ \hline
    % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
    TEST & TEST & TEST & TEST & TEST \\ \hline
    TEST & TEST & TEST & TEST & TEST \\
    TEST & TEST & TEST & TEST & TEST \\
    TEST & TEST & TEST & TEST & TEST \\ \hline
  \end{tabular}
\end{table}

\begin{table*}[htb]
  \centering
  \caption{Sample Table}\begin{tabular}{ccccccccc} \\ \hline
    % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
    TEST & TEST & TEST & TEST & TEST & TEST & TEST & TEST & TEST\\ \hline
    TEST & TEST & TEST & TEST & TEST & TEST & TEST & TEST & TEST\\
    TEST & TEST & TEST & TEST & TEST & TEST & TEST & TEST & TEST\\
    TEST & TEST & TEST & TEST & TEST & TEST & TEST & TEST & TEST\\ \hline
  \end{tabular}
\end{table*}

Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. Dummy text. [use \verb+\pagebreak+ to balance final column]


:):):):):):):):):):):):):) ohohohohoh :):):):):):)

COUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUCOUC

I/ coherent imaging : 
II/ incoherent imaging : 
Incoherent transfer function : in order to have an accurate measure of the effect of the system over the frequency components of an image, we used the grating of highest frequency at our disposal

%// 1 : first frequency -> we were using slide number 2 which had 11 periods visible on screen, 
%// for a screen size of 348² pixels
%// -> period of 348/11 pixels
%// one pixel : 3.75µm ; period of 348*3.75/11 µm -> frequency of 11/1.305 = 8.43 mm^-1


in all cases : image of a point : no coherence  ? anyway, in all cases image of a point is the point spread function of the system


semi-coherent

aliasing

hard or soft cutoff

edge ringing

incoherent transfer function plot and coherent transfer function

effect of defocus on image (we effectively image the field present in front or behind the object...)
folding of the cosines because the variation is more important than the DC term

not able to completely cut the fundamental frequencies (iris can't be closed enough)

references : goodman
? (born \& wolf ?)

--------------------
The purpose of Fourier optics is to describe a given optical system with the tools of the Fourier theory, and more precisely in terms of its effects over the frequency contents of an image.

The behaviour of the system is wholly different between conditions of coherent and incoherent imaging, and the first two parts of this reports will present those behaviours. The last part will describe the effects due to the use of a digital camera and a discussion of the behaviour of the system under conditions of mixed coherence.

[add results]


\section*{experimental set-up}



The optical system under study consisted in two identical lenses, sharing a common focal point where was placed  

 is .... It is represented in the figure \ref{setup}

In the case were the aperture iris is completely open, and when the system is perfectly focused, the image is exactly the intensity distribution at the first focal plane of the second lens.


This 4-f system provides considerable insight in the theory of Fourier optics, as it links to significant results : one being that a lens realises the Fourier transform of its front focal plane in its back focal plane, and the other being that a general imaging system realises the multiplication of the Fourier spectrum of the object by its transfer function. Here it is higligthted as the transfer function comes mostly from the aperture stop, plced precisely at the comon focal plane of the two lenses, realizing directly the frequency filtering process.

\section{Coherent illumination}

two amplitudes at one point are summing up => intensity is not the sum of intensity : interference cross-term appears => system linear in term of amplitudes

\section{Incoherent illumination}

no interference => cross-term vanishes => two *intensities* at one point are summing up =>  system linear in intensities

\section{Mixed coherence}

\section{Digital camera}

\pagebreak
\begin{thebibliography}{99}


\bibitem{Gu} Y. and G. Gbur, ``Scintillation of Airy beam arrays in atmospheric
turbulence,'' Opt. Lett. \textbf{35,} 3456 (2010).

\end{thebibliography}

\pagebreak

\section*{Informational Fourth Page}
In this section, please provide full versions of citations to
assist reviewers and editors (OL publishes a short form of
citations) or any other information that would aid the peer-review
process.


\begin{thebibliography}{99}

\bibitem{Gu} Y. and G. Gbur, ``Scintillation of Airy beam arrays in atmospheric
turbulence,'' Opt. Lett. \textbf{35,} 3456--3458 (2010).

\end{thebibliography}


-------------------
\end{document}
