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\documentclass[a4paper, 12pt,oneside]{article}
%On peut changer "oneside" en "twoside" si on sait que le résultat sera recto-verso.
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%Début du document
\begin{document}
\title{\normalsize{Lab Work Report - Group N$^\circ$\\ XX - Experiment}}
\date{\normalsize{\today}}
\author{\normalsize{Name} 1\and \normalsize{Name 2}}
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%\tableofcontents
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\begin{center}
\large\textbf{\sffamily Experiment N$^\circ$ F10: Fourier optics}\\%
\large\sffamily Group N$^\circ$16: Ancarola Raffaele, Cincotti Armando\\%
\large\sffamily \today\qquad Tarrago Velez Santiago\\%TODO cambiare nome assistente
\end{center}
% Introduction
\section{Introduction}
Based on wave physics, Fourier optics is a much useful tool than geometrical optics
to study light propagation phenomenons, as it takes in account all wave properties of
light. It permits in fact to study perfectly complex phenomenon as diffraction, and it
is at the basis for developping useful technique for image and signal treatments.
In the following report, as an example of direct application of this field in physics,
it is shown that using convergent lenses it is possible to obtain the Fourier transform
of a complex light signal, and that the latter can be easily treated by filtering it directly
on its Fourier transform.
\section{A brief Theroretical basis}
\subsection{Principles of geometrical optic}\label{prince}
Optics of thin convergent lenses can be studied in a first approximation by usig two geometrical principles \cite{ref1}:
\begin{enumerate}
\item Each beam parallel to the main axis of the lens, will refract so that it passes really or virtually through the focal point.
\item Each beam passing through the center of the lens, won't get deflected.
\end{enumerate}
\subsection{Obtaining a Fourier Transform with a lens}
The optical systems that will be studied in the following report can be schematised as :
\begin{minipage}{\textwidth}
\vspace{0.3cm}
\hspace{-0.9cm}
\begin{minipage}{0.45\textwidth}
An object with a certain transmittance function $f(x,y)$, emitting a light signal $U(x,y,z)$
which creates an image $g(x,y) = U(x,y,d)$ on a plane screen positionned at a given distance $z = d$.
One can show that a convergent lens with a given focal plane at distance $f$ permit to obtain the Fourier Transform
$F(\nu_x, \nu_y)$ of the signal emitted in $z=0$, $U(x,y,0)=f(x,y)$, where $\nu_x$ and $\nu_y$ are
the spatial frequencies of the signal in directions $\vec{e}_x$ and $\vec{e}_y$.
\end{minipage}
\hspace{0.005\textwidth}
\begin{minipage}{0.65\textwidth}
\includegraphics[width=\textwidth]{simple_transform.png}
\captionof{figure}{Plane wave composed by one spatial frequency projected in one point of the Fourier plane. \cite{ref2}}
\label{simple_trans}
\end{minipage}
\vspace{0.3cm}
\end{minipage}
In fact, the luminous intensity
of the signal at a distance $f$ from the lens he passed trough, is given by :
\begin{equation}\label{I}
I(x,y) = \frac{1}{(\lambda f)^2}
\abs*{F(\frac{x}{\lambda f}, \frac{y}{\lambda f})} \\
\end{equation}
where $\lambda$ is the wave lenght of the signal. Equation (\ref{I}) proove in fact that
the Fourier Transform can be ``projected'' on the focal plane of the lens (see exemple in Figure \ref{simple_trans}).
One can also show that, considering the inverse Fourier Transform $f(x,y) = F^{-1}(\nu_x,\nu_y)$, if the screen is now
in front of a second lens, on its focal plane, then $g(x,y)$ follows this relation :
\begin{equation}\label{g}
g(-x,-y) \propto F^{-1}(\frac{x}{\lambda f}, \frac{y}{\lambda f}) = f(x,y)
\end{equation}
This means that, appling a second convergent lens to a signal already transformed by one lens,
creates a real inververse image (because of the ``$-$'' sign) of the emitted signal.
{\bf N.B.} the whole mathematical procedure for the computation of equations (\ref{I}) and (\ref{g}) can be found in the
{\it notice} for this report \cite{ref2}.
\paragraph{About the Fourier Plane.} A Fourier Transform qualitatively consist on the projection of a signal (here $f(x,y)$) on each of the vector
from its Fourier basis. In fact, the Fourier Transform decomposes the signal in the frequencies that make it up.
As just explained in this section, a lens consinst in this case in a sort of projection from the real space to the frequencies space. So
at the focal plane, also called {\bf Fourier plane} because it's where the Fourier Transform {\it lays}, can be observed several points consisting
on the set of spatial frequencies that composes the signal $f(x,y)$. Each point on this plane is the projection of a certain frequency
which can be measured using the following relation \cite{ref2} :
\begin{equation}\label{nu}
\nu_x = \frac{x}{\lambda f} = \frac{x^{'}}{\lambda f \gamma}
\end{equation}
where $x$ is the distance in $\vec{e}_x$ direction between the point and
the origin of the Fourier plane where $\nu_x = \nu_y = 0$, $x^{'}$ is the same distance
but mesured on an extended image of the Fourier Plane, and where $\gamma$
is the extension coefficient which will be defined in the next section.
The equation is similar for $\nu_y$.
\subsection{Projection of a real image.} When a real image is projected on a plane in front of the system of lenses and
luminous source, the projection can be smaller or bigger than the image from the source. Let $d$ be the distance between
one lens and a given object, and $d^{'}$ the distance between the same lens and the projected image of this object,
then an
extension coefficient $\gamma$ is geometrically given as :
\begin{equation}\label{gamma}
\gamma = \frac{d^{'}}{d}
\end{equation}
Having $\gamma$, then the following relation is maintained \cite{ref1}:
\begin{equation}\label{d}
\frac{1}{d} + \frac{1}{d^{'}} = \frac{1}{f} \quad \Rightarrow \quad d^{'} = \frac{df}{d - f}
\end{equation}
where $f$ is the distance between the lens and its focal plane.
\section{Experimental setup}
To study an optical image and obtain its Fourier transform, one can start by set up
the {\bf 4f configuration} illustrated in Figure \ref{4f}. All the piecies are mounted on one
long rail and can be moved anytime above it.
%TODO inserisci immagine schema_esperimento.png
\begin{minipage}{\textwidth}
\vspace{0.3cm}
\hspace{-0.8cm}
\begin{minipage}{0.45\textwidth}
It is first needed an image, so a green laser beam with wavelenght $\lambda = 532$ \si{\nano\metre}
followed by a beam expander with focal distance $f_1 = 12$ \si{\milli\metre}
are first mounted on the rail to obtain a large beam of light. Secondly, a lens with focal distance $f_2=100$ \si{\milli\metre}
is positionned at distance $11.5$ \si{\centi\metre} from the laser beam, wich is approximatively the sum $f_1+f_2$
where there would be the combined focals of the two lenses.
\end{minipage}
%\hspace{0.005\textwidth}
\begin{minipage}{0.65\textwidth}
\includegraphics[width=\textwidth]{schema_esperimento.png}
\captionof{figure}{4 focals configuration used to project a Fourier plane and an inverse image. \cite{ref2}}
\label{4f}
\end{minipage}
\vspace{0.3cm}
\end{minipage}
As it follows by the first principle in section \ref{prince}, as the
beam passes through the focal point, it then becames parallel to the rail when it passes through the lens.
\begin{minipage}{\textwidth}
\vspace{0.3cm}
\hspace{-1cm}
\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{foto/house_reverse.png}
\captionof{figure}{Reverse image of Fourier's home.}
\label{rev_img}
\end{minipage}
\hspace{0.01\textwidth}
\begin{minipage}{0.55\textwidth}
Thirdly, a filter with the image of a house is positionned in front of the first lens in orther to get an image.
This filter consist now in the object with transmittance $f(x,y)$ as it is studied in the therotical section
and it will be called {\it Fourier's home} from now on.
Then, following the schema in Figure \ref{4f}, one lens with focal distance $f_3 = 125$ \si{\milli\metre} is positionned at $f_3$ from
the Fourier's home. At a distance of $f_3$ further can be found the Fourier plane. Finally $1.25$ \si{\centi\metre} further another lens
with focal distance $f_3$ is mounted on the rail so that an inverse image of the house is projcted on a screen as it follows by relation
(\ref{g}) (see Figure \ref{rev_img}).
\end{minipage}
\vspace{0.3cm}
\end{minipage}
\section{Results of the experiment and Discussions}
\subsection{Lens law (\ref{d}) verification}
To verify the law that appears on the leftside in (\ref{d}), it is first measured the lenght
of the projected image of Fourier's home when $\gamma = 1$,
which is $l = 0.5$ \si{\centi\metre}. Then for a fixed distance $d$ from a lens with
$f=50$ \si{\milli\metre}, $d^{'}$ is measured by the rightside equation in (\ref{d}) and
the screen is positionned at this distance from the lens. Measuring the lenght $l_{img}$ of the projected
image, the experimental extension coefficient can be measured as $\gamma_{exp} = \frac{l_{img}}{l}$ and compared
to the value of $\gamma_{ref}$ measured by using equation (\ref{gamma}).
The error on all distance measurements is $\Delta x = 0.1$ \si{\centi\metre}.
Table \ref{verification} shows several measurements for this experiment. It can be noticed that
$\gamma_{ref}$ values do not differ a lot from $\gamma_{ref}$ values which fall in the error interval.
This shows that the leftside law in (\ref{d}) holds.
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
$d$ [\si{\centi\metre}] & $d'$ [\si{\centi\metre}] & $l_{img}$ [\si{\centi\metre}] & $\gamma_{ref}$ & $\gamma_{exp}$ \\
\hline
$7.0 $ & $17.5$ & $1.3$ & $2.5 \pm 0.1$ & $2.6 \pm 0.4$ \\
\hline
$8.0 $ & $13.3$ & $0.9$ & $1.7 \pm 0.1$ & $1.8 \pm 0.5$ \\
\hline
$9.0 $ & $11.3$ & $0.7$ & $1.3 \pm 0.1$ & $1.3 \pm 0.7$ \\
\hline
$10.0$ & $10.0$ & $0.6$ & $1.0 \pm 0.1$ & $1.2 \pm 0.4$ \\
\hline
$11.3$ & $9.0 $ & $0.5$ & $0.8 \pm 0.1$ & $1.0 \pm 0.4$ \\
\hline
$13.3$ & $8.0 $ & $0.3$ & $0.6 \pm 0.1$ & $0.6 \pm 0.3$ \\
\hline
$17.5$ & $7.0 $ & $0.2$ & $0.4 \pm 0.1$ & $0.4 \pm 0.3$ \\
\hline
\end{tabular}
\caption{Measurments of the extension coefficient $\gamma$ for different values of $d$}
\label{verification}
\end{table}
\subsection{Fourier plane}
The Fourier's home is an image composed by four quadrants, each one of them composed by parallel
lines following one direction per quadrant (see Figure \ref{fourier_house}).
These lines form a pattern of different spatial frequencies in different direction: $\nu_x$, $\nu_y$ or a composition of both
$\nu_x + \nu_y$. Between the two convergent lenses in configuration 4f, there is the Fourier Plane where
each of these frequencies is ``projected'' in a point on the plane.
The image of the Fourier transform can be extended mounting a convergent lens in front
of it in order to obtain a bigger image as it is shown on Figure \ref{fourier_plane}.
\begin{figure}[h]
\centering
%\hspace{-0.6cm}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{foto/fourier_plane.pdf}
\caption{Extended image of the Fourier plane.} %TODO
\label{fourier_plane}
\end{subfigure}
\hspace{0.08\textwidth}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.9\textwidth]{foto/house_border.pdf}
\caption{Extended image of Fourier's home.} %TODO
\label{fourier_house}
\end{subfigure}
\caption{Fourier plane and Fourier's home sectors, highlighted per direction of frequencies in Fourier plane.}
\end{figure}
On Figure \ref{fourier_plane} can be observed that several points are aligned concentrically on 4 different directions. In fact each
direction derives from one sector of Fourier's home, for exemple the vertical lines on Figure \ref{fourier_house} are separated in frequencies
only on direction $\vec{e}_x$ so this sector ``projects'' the horizontal points on the Fourier plane.
\subsubsection{Verifing relation (\ref{nu}).}
\begin{minipage}{\textwidth}
\vspace{0.3cm}
\hspace{-0.8cm}
\begin{minipage}{0.45\textwidth}
By counting the number of all visible lines for each sector of Fourier's home, the minimum non-zero frequency can be measured
as $\nu_{i} = N/l$, where $N$ is the number of lines and $l$ the lenght of the sector when the image isn't extended.
Then, by extending the image of the Fourier Plane, these frequencies can be measured by relation (\ref{nu}) measuring the distance $x^{'}$
between one point in the first ``circle'' and the center. Table \ref{frequency} shows that these measurement don't coincide, but the order
of the two measurments is the same. This divergence can be explained by the errors in lenght measurements, that occurs even in
the determination of $\gamma$, affected also by the precision of the one that takes the measurments.
\end{minipage}
\hspace{0.05\textwidth}
\begin{minipage}{0.4\textwidth}
\includegraphics[width=\textwidth]{foto/house_elab.pdf}
\captionof{figure}{House image elaborated in order to clearly count lines.}
\label{elab}
\end{minipage}
\vspace{0.3cm}
\end{minipage}
% TODO continua a commentare qui se ti prolunghi
\begin{table}[H]
\centering
\begin{tabular}{cc|c|c|c|c|c|c|}
\cline{2-8}
\multicolumn{1}{ c| }{} & $N$ & $l$ [\si{\milli\metre}] & $x^{'}$ [\si{\centi\metre}] & $\gamma$ & $f$ [\si{\milli\metre}] & $\lambda$ [\si{\nano\metre}] & $\nu_x$ [\si{1/\kilo\metre}] \\
\hline
%
\multicolumn{1}{|c|}{Line counting} & \multirow{2}{*}{$24 \pm 1$} & \multirow{2}{*}{$2.5 \pm 0.5$} & \multirow{2}{*}{$0.5 \pm 0.1$} & \multirow{2}{*}{$7.25$} & \multirow{2}{*}{$50 \pm 1$} & \multirow{2}{*}{$532 \pm 1$} & $9.6 \pm 4.2$ \\
\cline{1-1} \cline{8-8}
\multicolumn{1}{|c|}{Relation \ref{nu}} & & & & & & & $23.3 \pm 3.1$ \\
%\hline
\Xhline{4\arrayrulewidth}
%
\end{tabular}
\caption{Frequency computed with two different methods.}
\label{frequency}
\end{table}
\subsubsection{Filtering the Fourier Plane.}
Once obtained the Fourier Plane, one can try to filter light directly on this one and see
what appens to the real image reconstructed by the second lens in configuration 4f and then
extendend using a convergent lens with $f = 100$ \si{\milli\metre}.
\paragraph{Filtering frequencies in one direction.}
By using two filters composed by a diagonal or horizontal/vertical line, one can filter a whole set of frequencies
on one of the 4 directions illustrated in Figure \ref{fourier_plane}. This results in a real image where all of the spatial frequencies in
that direction have been filtered. As in Fourier's home each sector is composed by lines following only one direction, this
means that this kind of filter hides one of the sectors and let pass light from the three others.
For exemple by filtering the points in $\vec{e}_{x}$ direction where $\nu_y = 0$ (see Figure \ref{filter_dir}),
the sector with horizontal lines in the Fourier's home get hidden (see Figure \ref{filter_res}).
\begin{figure}[h]
\centering
%\hspace{-0.6cm}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{foto/miss_bottom_right.png}
\caption{Result on the house image.}
\label{filter_res}
\end{subfigure}
\hspace{0.08\textwidth}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{foto/fourier_horiz.png}
\caption{Filtering the horizontal direction.}
\label{filter_dir}
\end{subfigure}
\caption{Application of a filter on the fourier plane and visualization of the result.}
\label{filtering}
\end{figure}
\paragraph{Filtering a few or all except one frequencies.}
With a filter consinsting on a flat black surface with one small hole, it is possible to chose only a small set
of frequencies to let pass trough in the Fourier plane. Doing this it is possible to visualize only one spatial
frequency that composes the signal coming from Fourier's Home.
\begin{minipage}{\textwidth}
\vspace{0.3cm}
\hspace{-0.8cm}
\begin{minipage}{0.35\textwidth}
An example can be observed in Figure \ref{top_left_img}, or a more interesting one in Figure \ref{lowpass_img}
where only quite flat surfaces can be observed. This image was in fact obtained
by letting through the central point in the Fourier plane where $\nu_x = \nu_y = 0$, and the filter surface
acts like a low-pass filter.
In a similar way a large black point acts like a high-pass filter if it hides the central points
on Fourier plane. The result of the latter can be observed on Figure
\ref{}%TODO label di altra immagine
where the lines that compose Fourier's home
are much neat than in other pictures.
\end{minipage}
\hspace{0.03\textwidth}
\begin{minipage}{0.3\textwidth}
\includegraphics[width=\textwidth]{foto/top_left.pdf}
\captionof{figure}{What appears of Fourier's home letting pass one frequency only.}
\label{top_left_img}
\end{minipage}
\hspace{0.03\textwidth}
\begin{minipage}{0.3\textwidth}
\includegraphics[width=\textwidth]{foto/house_passebas.png}
\captionof{figure}{House image when a low-pass filter is applied.}
\label{lowpass_img}
\end{minipage}
\vspace{0.3cm}
\end{minipage}
\paragraph{A brief discussion.}
We've observed how it is possible to hide frequencies from a light signal. In fact it has been prooven
that working on the fourier plane it is possible to create every type of frequency filter that is needed.
This kind of manipulations on a signal are really usefull and common in the field of signal treatement.
For example it directly follows from the report that this tecnique can be developped to create light filters
for spatial frequencies in order to hide noise in images, or in radio techniques where superposition
of frequencies and smart filtering permits to transmit audible signal over radio wave \cite{ref3}.
\section{Conclusion}
Throughout the different experiments and measurements, the geometrical relations for convergent
lenses have been verified as the results were quite consistent with theory.
The most interesting part is the projection of the real image on the frequency plane, also called
here {\it Fourier plane}. This kind of manpulation is in fact at the basis of image processing as it
permits to easily treat the light signal by filtering precise frequencies.
As mentionned in the introduction, Fourier optics is in fact an usefull tool for studying light waves
and treating light signals, and much more applications can be found in every kind of field as spectroscopy,
holography, pattern recognition and so on \cite{ref4}.
\section{Annexes}
\subsection{About the error}
\paragraph{Incertitude on distances}
All measured distances taken with the ruler have the incertitude
$\Delta x = 0.1$ \si{\centi\metre}.
\bigskip
\begin{itemize}
\item \textbf{Nota bene}: this incertitude is never shown on values written on tables because it's the same for all of them.
\end{itemize}
\paragraph{Incertitude on $\gamma$}
\bigskip
\begin{itemize}
\setlength\itemsep{0.7em}
\item $\Delta \gamma_{ref} = \frac{\Delta d'}{d} + \frac{d'}{d^2} \Delta d$
\item $\Delta \gamma_{exp} = \frac{\Delta L}{l} + \frac{L}{l^2} \Delta l$
\end{itemize}
\bigskip
where $L$ is the length of the projected image on the screen and $l$ is the real
size of the house.
\paragraph{Incertitude on $\nu_x$}
\bigskip
\begin{itemize}
\item $\Delta \nu_x(N, l) = \frac{\Delta N}{l} + \frac{N}{l^2} \Delta l$
\item $\Delta \nu_x(x^{'}, \lambda, f, \gamma) =
\frac{\Delta x^{'}}{\lambda f \gamma} + \frac{x^{'}}{(\lambda f \gamma)^2} \cdot
(\lambda \gamma \; \Delta f + f \gamma \; \Delta \lambda + f \lambda \; \Delta \gamma)$
\end{itemize}
\subsection{Remaining images}
\begin{figure}[h]
\centering
\begin{subfigure}{0.25\textwidth}
\includegraphics[width=\textwidth]{foto/fourier_diag.png}
\end{subfigure}
\hspace{0.05\textwidth}
\begin{subfigure}{0.25\textwidth}
\includegraphics[width=\textwidth]{foto/fourier_plane_2.jpg}
\end{subfigure}
\hspace{0.05\textwidth}
\begin{subfigure}{0.25\textwidth}
\includegraphics[width=\textwidth]{foto/fourier_vert.jpg}
\end{subfigure}
\\
\begin{subfigure}{0.25\textwidth}
\includegraphics[width=\textwidth]{foto/house_big.jpg}
\end{subfigure}
\hspace{0.05\textwidth}
\begin{subfigure}{0.25\textwidth}
\includegraphics[width=\textwidth]{foto/little_house.jpg}
\end{subfigure}
\hspace{0.05\textwidth}
\begin{subfigure}{0.25\textwidth}
\includegraphics[width=\textwidth]{foto/miss_bottom_left.jpg}
\end{subfigure}
\end{figure}
\section{Literature References}
% Bibliographie
\begin{thebibliography}{99}
\bibitem{ref1}
Dr. D. MARI, Dr. I. TKALCEC, Métrologie Lessons TP I, Optique I : {\it Optique géométrique, Réfraction, Lentilles}
Faculté des sciences de base, Section de physique.
\bibitem{ref2}
Notice des TP de physique F10 : Optique de Fourier.
\bibitem{ref3}
HARALD, Brune, Cours de Physique IV, Faculté des Sciences de Base, Section de Physique, Ecole Polytechnique Fédérale de Lausanne.
\bibitem{ref4}
TYSON, Robert K. Principles and applications of Fourier Optics, Chapter 7: Practical Applications. University of North Carolina at Charlotte, USA, August 2014.
\end{thebibliography}
\end{document}

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