%\documentclass[landscape,a0b,final,a4resizeable]{a0poster}
%\documentclass[landscape,a0b,final]{a0poster}
%\documentclass[portrait,a0b,final,a4resizeable]{a0poster}
\documentclass[portrait,a0,final]{a0poster}
%%% Option "a4resizeable" makes it possible ot resize the
%   poster by the command: psresize -pa4 poster.ps poster-a4.ps
%   For final printing, please remove option "a4resizeable" !!

\usepackage{epsfig}
\usepackage{multicol}
\usepackage{pstricks,pst-grad}
\usepackage{graphicx}
\usepackage{epstopdf}
\usepackage{color}
\usepackage{tabulary}
\usepackage{amsmath}
\usepackage{mathtools}
\usepackage[utf8]{inputenc}
\usepackage{hyperref}
% \usepackage{lmodern}
% \usepackage{xcolor}
%\usepackage[usenames,dvipsnames,svgnames,table]{xcolor}



\newcommand{\abs}[1]{| #1 |}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Definition of some variables and colors
%\renewcommand{\rho}{\varrho}
%\renewcommand{\phi}{\varphi}
\newrgbcolor{nus_blue}{0 0.129 0.388}
\newrgbcolor{nus_gold}{1 0.259 0}
\newrgbcolor{nus_lite}{1 0.259 0}
\newrgbcolor{lite}{.70 .70 .70}
\newrgbcolor{liter}{.95 .95 .95}
\setlength{\columnsep}{3cm}
% \setlength{\columnseprule}{2mm}
\setlength{\parindent}{0.0cm}


%% Figures and tables
%% The multicols package doesn't support the floats, figure and table.
%% Use tablehere and figurehere in place of table and figure.

\makeatletter
\newenvironment{tablehere}
  {\def\@captype{table}}
  {}

\newenvironment{figurehere}
  {\def\@captype{figure}}
  {}
\makeatother


% \figcaption replaces - replacement for \caption
% necessary, since in the multicols environment \figure and
% therefore \caption won't work

\setcounter{figure}{1}
\newcommand{\figcaption}[1]{
  %\vspace{0.05cm}
  \begin{center}
  \begin{quote}
    {\large {\sc Figure} \arabic{figure}: #1}
  \end{quote}
  \end{center}
 % \vspace{1cm}
  \stepcounter{figure}
}

\setcounter{table}{1}
\newcommand{\tabcaption}[1]{
  %\vspace{0.05cm}
  \begin{center}
  \begin{quote}
    {\normalsize {\sc Table} \arabic{table}: #1}
  \end{quote}
  \end{center}
 % \vspace{1cm}
  \stepcounter{table}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                Poster                        %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newenvironment{poster}{
  \begin{center}
  \begin{minipage}[c]{0.98\textwidth}
}{
  \end{minipage} 
  \end{center}
}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Begin of Document
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

% \vspace*{2cm}


\newrgbcolor{lightblue}{0. 0. 0.80}
\newrgbcolor{white}{1. 1. 1.}
\newrgbcolor{whiteblue}{.80 .80 1.}
\newrgbcolor{darkgreen}{0 .5 0}


\begin{poster}
\large \sf

%%%%%%%%%%%%%%%%%%%%%
%%% Header
%%%%%%%%%%%%%%%%%%%%%
 %%% CQT_Logo
\begin{minipage}[t]{\textwidth}

      \begin{minipage}[c]{0.05\textwidth}
        \begin{center}
          \includegraphics[width=14cm,angle=0]{./figures/logos/CQT_Logo.pdf}
        \end{center}
      \end{minipage}
\hspace{8cm}
      %%% Title
      \begin{minipage}[c]{0.7\textwidth}
        \begin{center}
% {\sc \Huge Measuring chaotic-coherent emission mixture \linebreak across the lasing threshold \\}
{\sc \Huge Characterising the Onset of Lasing \\[11mm] Using Interferometric Photon Correlations}
% {\sc \Huge Interferometric Photon Correlations of \\[11mm] a Distributed Feedback Laser}
        \end{center}
      \end{minipage}
      %%% NUS-Logo
      \begin{minipage}[c]{0.05\textwidth}
        \begin{center}
          \includegraphics[width=9cm,angle=0]{./figures/logos/NUS_Logo.pdf}
        \end{center}
      \end{minipage}
	\begin{center}
	\vspace{15mm}
	{\Large Yeo Xi Jie$^{1,*}$, Alvin Leow$^2$, Hwang Jaesuk$^1$, Shen Lijiong$^1$, Tan Peng Kian$^1$ and Christian Kurtsiefer$^{1,2}$
            \\[4mm]
            $^1$Centre for Quantum Technologies, National University of Singapore, Singapore 117543\\
            $^2$Department of Physics, National University of Singapore, Singapore 117542\\
            ($^*$\href{email:yeo.xi\_jie@u.nus.edu}{email: yeo.xi\_jie@u.nus.edu)}}
	\end{center}
\end{minipage}

\vspace{11mm}

%%%%%%%%%%%%%%%%%%%%%
%%% Content
%%%%%%%%%%%%%%%%%%%%%
\begin{multicols}{2}
\setlength{\parskip}{1ex plus 0.5ex minus 0.2ex}

%%%%%%%%%%%%%%%%%%%%%
\begin{center}{\bf \Large Abstract}\end{center}
%%%%%%%%%%%%%%%%%%%%%
The threshold current is a key characteristic of a semiconductor laser diode. The laser diode emits coherent laser light when operating above this current, 
and behaves as a light-emitting diode (LED) when operating below. 

We present a technique to characterise the onset of coherence in a semiconductor laser diode using interferometric photon correlation measurements, which were initially used to study spectral diffusion in molecules\,[1], 
and were also applied to differentiate between chaotic light and coherent light with amplitude fluctuations\,[2]. 

Using this technique, we observe the onset of coherent light emission by the laser diode at an injection current in agreement with the threshold current determined via optical power measurements.
From this measurement, we were also able to extract the percentage of coherent light in the mixture. 

%%%%%%%%%%%%%%%%%%
\begin{center}{\bf \Large Interferometric photon correlation measurement $g^{(2X)}$}\end{center}
%%%%%%%%%%%%%%%%%%

% Interferometric photon correlation measurements 

For $g^{(2X)}$ measurement
% we spectrally filter  
light emitted by the laser diode
is sent through an asymmetric Mach-Zehnder interferometer 
with a propagation delay between the two paths of the interferometer
larger than the coherence time of the light.
% We accomplish this with a 180\,m long single mode fibre, introducing a propagation delay of 900\,ns.
  % accomplished with a 180\,m long mode optical fibre.
% The propagation delay is accomplished with a 180\,m long single mode optical fibre.
Photoevents at each output port are detected by single photon avalanche detectors and time-stamped. The correlations are numerically extracted from the time-stamped events.


\begin{figurehere}
    \begin{center}
   % \includegraphics[width=\columnwidth]{../../figures/experimental_setup/g2X_setup.png}
   \includegraphics[width=\columnwidth]{./figures/g2x_setup/g2x_setup.png}
    % \includegraphics[scale=1]{figure1.pdf}
    \end{center}
    \figcaption{Experimental setup for measuring interferometric photon correlations.} \label{fig:setup}
    % The light from the laser diode is split at a non-polarising beamsplitter, with light from one of the output ports coupled into a delay fibre. The light from the other output port recombines with the light from delay fibre at another non-polarising beamsplitter. The reflections at the beamsplitter introduces a phase shift to the light. Photodetectors detect photoevents at each output port of this beamsplitter which are time-stamped, and from which correlations are extracted.} \label{fig:setup}
\end{figurehere}

% The number of coincidences in each timing bin are then extracted numerically by
% histogramming the time differences of all pairs of the timestamped events.
% The value of $g^{(2X)}$ at each timing bin is obtained by multiplying the number of coincidences with the the total measurement time,
% and dividing by the product between the number of events at each channel and the size of the timing bins.

Expressing the electric field of the light from the laser diode as $E(t)$, the propagation delay between the two paths of the interferometer as $\Delta$,
and using the fact that dithering averages out some terms to 0, 
the correlation between the two ports expands to
% \begin{equation*}
%     \begin{split}
%         &g^{(2X)}(t_{2}-t_{1})\\
%         % =& \frac{\langle E^{*}_{A}(t_{1})E^{*}_{B}(t_{2})E_{B}(t_{2})E_{A}(t_{1})\rangle}{ \langle E^{*}_{A}(t_{1})E_{A}(t_{1})\rangle \langle E^{*}_{B}(t_{2})E_{B}(t_{2})\rangle} \\
%       \propto& \langle {E^{*}(t_{1})E^{*}(t_{2})E(t_{2})E(t_{1})} \rangle /4 \\
%         & \underbrace{+ \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2}+\Delta)E(t_{2}+\Delta)E(t_{1}+\Delta)} \rangle/4}_{\text{conventional }g^{(2)}\text{ terms}} \\
%         & + \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2})E(t_{2})E(t_{1}+\Delta)} \rangle/4 \\ 
%         & \underbrace{+ \langle {E^{*}(t_{1})E^{*}(t_{2}+\Delta)E(t_{2}+\Delta)E(t_{1})} \rangle/4}_{\text{time-shifted }g^{(2)}\text{ terms}} \\
%         & - \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2})E(t_{2}+\Delta)E(t_{1})} \rangle/4 \\
%         & \underbrace{- \langle {E^{*}(t_{1})E^{*}(t_{2}+\Delta)E(t_{2})E(t_{1}+\Delta)} \rangle/4}_{g^{(2X)}_\text{int}\text{ terms}}.
%     \end{split}
%     \label{eqn:expansion}
% \end{equation*}
\begin{equation*}
% \begin{split}
  g^{(2X)}(t_{2}-t_{1}) \propto  
    \begin{aligned}
    % &g^{(2X)}(t_{2}-t_{1}) \propto  \\
      &\begin{rcases}
        ~~\, \langle {E^{*}(t_{1})E^{*}(t_{2})E(t_{2})E(t_{1})} \rangle \\
        + \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2}+\Delta)E(t_{2}+\Delta)E(t_{1}+\Delta)} \rangle
      \end{rcases}\text{conventional }g^{(2)}\text{ terms} \\
      \\[0.5mm]
      &\begin{rcases}
        + \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2})E(t_{2})E(t_{1}+\Delta)} \rangle\\
        + \langle {E^{*}(t_{1})E^{*}(t_{2}+\Delta)E(t_{2}+\Delta)E(t_{1})} \rangle
      \end{rcases} g^{(2)}\text{ terms time-shifted by }\pm\Delta\\
      \\[2mm]
      &\begin{rcases}
        - \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2})E(t_{2}+\Delta)E(t_{1})} \rangle \\
        - \langle {E^{*}(t_{1})E^{*}(t_{2}+\Delta)E(t_{2})E(t_{1}+\Delta)} \rangle
      \end{rcases} \text{interference terms}\\
    \end{aligned}
% \end{split}
\end{equation*}

By substituting expressions of $E(t)$ for chaotic light and coherent light [3], models of $g^{(2X)}$ for chaotic, coherent and mixture between chaotic and coherent light can be calculated [4].

For chaotic light, $g^{(2X)}$ measurement exhibit a bunching signature peaking at time differences plus and minus the propagation delay $g^{(2X)}(\pm\Delta) = 1.25$, and zero time difference $g^{(2X)}(0) \approx 1$

For coherent light, $g^{(2X)}$ measurement exhibit an anti-bunching signature with at zero time difference $g^{(2X)}(0) = 0.5$. 

For a mixture of chaotic and coherent light, both the bunching and anti-bunching signatures in $g^{(2X)}$ can be measured, with their respective features scaled proportionally to their fraction in the mixture.

\begin{figurehere}
    \begin{center}
    \includegraphics[width=\linewidth]{./figures/theoretical_simulation/theoretical_simulation.png}    
    \end{center}
  \figcaption{Calculated $g^{(2X)}$ plots with different ratios of chaotic-coherent light mixture; percentage of coherent light indicated on the top-left corner of each plot. Blue, dash: bunching signature associated with chaotic light at $t_{2}-t_{1} = \pm\Delta$. Red, solid: anti-bunching signature associated with coherent light at $t_{2}-t_{1} = 0$. The characteristic time-widths of these features are the coherence times of the respective chaotic/coherent light. } \label{fig:theory_plots}
\end{figurehere}

\vfill\null
\columnbreak
%%%%%%%%%%%%%%%%%%
\begin{center}{\bf \Large Coherence around threshold}\end{center}
%%%%%%%%%%%%%%%%%%
A common method to determine the threshold current is to identify the injection current where the emitted optical power and spectral width changes sharply. 
However, optical power and spectrum measurement cannot determine whether the light emitted by the laser diode has changed from being chaotic (LED) to coherent laser radiation.

To determine this change from chaotic to coherent laser emission, the second-order photon correlation $g^{(2)}$ can be measured.
% where coherent light exhibit $g^{(2)}=1$, while LEDs exhibit a bunching signature $g^{(2)}(0)>1$.
% % associated with chaotic light
% % However, amplitude noise in the lasing regime may result in second-order photon correlation measurements exhibiting $g^{(2)}(0)>1$, when $g^{(2)}=1$ is expected. 
% The bunching signature has a characteristic timescale which is inversely related to the bandwidth of the chaotic light.
However without spectrally filtering the light, 
the timescale of the bunching signature associated with chaotic light may be unresolvable by detectors.
% shorter than the timing jitter of existing photodetectors and time-stamping units,
% This limits the ability to resolve the bunching signature,
This may result in second-order photon correlation measurements of chaotic light exhibiting $g^{(2)}=1$ which coherent laser emission also exhibits.
% , when $g^{(2)}(0)>1$ is expected.

Performing $g^{(2X)}$ measurements may complement existing techniques to determine the threshold current and the change from incoherent to coherent laser radiation by identifying the injection current at the onset of $g^{(2X)}(0)<1$. 
We compare the performance between $g^{(2)}$ and $g^{(2X)}$, by using the same photodetectors and time-stamping unit, and without spectral filtering the emitted light from the laser diode.
% We identify this sharp change by finding the sharpest change in gradient of the power-current curve.
% With this method, we determine the threshold current to be at 36.4\,mA for the semiconductor laser used in our experiment. 

% Alternatively, another method to determine the threshold current is to identify the injection current where the spectral width of the emitted light changes sharply. Using spectral measurement of the light

\begin{figurehere}
    \begin{center}
    \includegraphics[width=\linewidth]{./figures/4_in_1_plot/four_in_one_vs_current.png}    
    \end{center}
  \figcaption{(From Top to Bottom) Respective measurements of $g^{(2X)}$, optical power, spectrum and $g^{(2)}$ as a function of injection current. Using the second-derivative of the optical power-current curve, we determine the threshold current to be at 36.4\,mA (dashed line). }
  % To compare the performance between $g^{(2)}$ and $g^{(2X)}$, the same photodetectors and time-stamping unit were used for the measurements, and no spectral filters were used on the emitted light from the laser diode.}
  \label{fig:fraction_vs_current}
\end{figurehere}

%%%%%%%%%%%%%%%%%%
\begin{center}{\bf \Large Extracting fraction of coherent light from $g^{(2X)}(0)$}\end{center}
%%%%%%%%%%%%%%%%%%
% The value of $g^{(2X)}(0)$ ranges from 0.5 (for coherent light) to 1  (for chaotic light). 
We extract the fraction of coherent light in the mixture using $[1-g^{(2X)}(0)]/0.5$. From our set of measurements, we extract the largest percentage of coherent light in the mixture to be 89.4\%.
\begin{figurehere}
    \begin{center}
    \includegraphics[width=\linewidth]{./figures/g2x_0_vs_current/g2x_0_vs_current.png}    
    \end{center}
  \figcaption{Interferometric photon correlations at zero time difference as a function of injection current. We measure an increasing fraction of coherent light after the threshold current of 36.4\,mA, with a deviation from the decreasing trend observed between 46.4\,mA to 48.4\,mA (dotted lines).} \label{fig:fraction_vs_current}
\end{figurehere}

% %%%%%%%%%%%%%%%%%%
% \begin{center}{\bf \Large Comparison to second-order photon correlations $g^{(2)}$}\end{center}
% %%%%%%%%%%%%%%%%%%
% % Provides information on photon statistics, whether the light is bunched (super-Poissonian) or random (Poissonian). Poissonian light can be laser light, but when bunching not resolvable due to limited detector response or a-priori information on the spectrum of the lasing mode.


% We perform second-order photon correlations and with the  

% \begin{figurehere}
%     \begin{center}
%     \includegraphics[width=\linewidth]{./figures/g2_g2x_vs_current/g2_g2x_vs_current.png}    
%     \end{center}
%   \figcaption{Fraction of coherent light as a function of injection current extracted from the fit of $g^{(2X)}$ measurements. The dip amplitude extracted from the fit of the interferometric timing correlation measurement as a function of the injection current. The correlation measurement was separately fitted to a constant function and two-sided decaying exponential function. } \label{fig:fraction_vs_current}
% \end{figurehere}


% % %%%%%%%%%%%%%%%%%
% \begin{center}{\bf \Large Characterising the transition to coherent light}\end{center}
% % %%%%%%%%%%%%%%%%%%
% % We measure the power emitted by the laser against the current supplied to it, and determined the lasing threshold current to be around 52.8\,mA, as shown in Figure 1d. We measure $g^{(2)}$ and $g^{(2X)}$ for the 760\,nm DBR laser at 52.55\,mA (below threshold), 52.80\,mA (near the threshold) and 53.20\,mA (above the threshold): 
% \begin{figurehere}
%     \begin{center}
%     \includegraphics[width=\linewidth]{./figures/g2x_0_vs_current/g2x_0_vs_current_exponential_fit_residuals.png}    
%     \end{center}
%   \caption{The reduced $\chi^{2}$ between the two fits were compared and the fit with a reduced $\chi^{2}$ closer to 1 was chosen. } \label{fig:fraction_vs_current}
% \end{figurehere}
\end{multicols}
% \begin{center}{\bf \Large References}\end{center}
\vspace{-1cm}
{\large
\begin{flushleft}
  {\bf \Large References}\\[5mm]
  \begin{tabular}[]{clcl}
          {[1]}& X. Brokmann et al.\textit{Opt. Express} \textbf{14}, 6333 (2006).&{[3]}&R. Loudon, \textit{The Quantum Theory of light}. Oxford, UK, \textit{(2000).}\\
          % {[1]}&M.~ Fox, \textit{Quantum Optics: An Introduction}, (Oxford Univ. Press, Oxford, UK, 2006) \\
          {[2]}&A.~Lebreton et al., \textit{Phys. Rev. Lett. \textbf{110}, 163603 (2013).} & {[4]}&A.~Lebreton et al., \textit{Phys. Rev. A. \textbf{88}, 013801 (2013).} \\
  % {\bf \Large References}\\[5mm]
  % \begin{tabular}[]{clcl}
  %         {[1]}& X. Brokmann et al.\textit{Opt. Express} \textbf{14}, 6333 (2006).\\
  %         {[2]}&A.~Lebreton et al., \textit{Phys. Rev. Lett. \textbf{110}, 163603 (2013).}\\
  %         {[3]}&R. Loudon, \textit{The Quantum Theory of light}. Oxford, UK, \textit{(2000).}\\
  %         {[4]}&A.~Lebreton et al., \textit{Phys. Rev. A. \textbf{88}, 013801 (2013).} \\          
  \end{tabular}
\end{flushleft}
}

\end{poster}
\end{document}



% % The light under test with an electric field profile $E(t)$ passes through an asymmetric Michelson interferometer with propagation delay $\Delta$ between the two arms, with electric fields at the output ports of the interferometer;
% % \begin{equation}
% %     E_{A,B}(t) = \frac{E(t) \pm E(t+\Delta)}{\sqrt{2}}.
% %     \label{eqn:fields}
% % \end{equation}
% %  % The propagation delay $\Delta$ is chosen to be longer than the coherence time of the light.
% % The correlation between these two ports is given by
% % \begin{equation}
% %     \begin{split}
% %   &g^{(2X)}(t_{2}-t_{1})\\
% %    =& \frac{\langle E^{*}_{A}(t_{1})E^{*}_{B}(t_{2})E_{B}(t_{2})E_{A}(t_{1})\rangle}{ \langle E^{*}_{A}(t_{1})E_{A}(t_{1})\rangle \langle E^{*}_{B}(t_{2})E_{B}(t_{2})\rangle}.
% % \end{split}
% % \end{equation}

% % Using expressions in Eqn.~\ref{eqn:fields}, 
% % and the fact that dithering averages out some terms to 0,
% % the correlation between the two ports expands to


% % The expected correlations of $g^{(2)}$, $g^{(2X)}_\text{int}$ and $g^{(2X)}$ for coherent and chaotic light
% % are shown in Fig.~\ref{fig:theory_plots}.
% % We also show the 
% % expected correlations of a light source with 50:50 mixture of coherent and chaotic light,
% % obtained by an equally weighted sum of $g^{(2)}$ and $g^{(2X)}$ for coherent and chaotic light 
% % as predicted in Ref.~\cite{Lebreton2013Theory}.

% % \begin{center}
% % \begin{tabular}{||p{0.25\linewidth} | c |c| c||} 
% %  \hline
% %   \centering Type of Light & $E(t)$ & $g^{(2)}(\tau=t_{2}-t_{1})$ & $g^{(2X)}_\text{int}(\tau=t_{2}-t_{1})$  \\ [0.5ex] 
% %  \hline\hline
% %  \centering Chaotic &$E_{0}\frac{e^{i\omega t}}{\sqrt{N}}\sum_{j}^{N}e^{i\phi_{j}(t)}$&$1+e^{-2\|\tau\|/\tau_{c}}$ & $e^{-2\Delta/\tau_{c}}+e^{-2\|\tau\|/\tau_{c}}$\\ 
% %  \hline
% %  \centering Coherent &$E_{0}e^{i\omega t}e^{i\phi_{j}(t)}$& 1 & $e^{-2m/\tau_{\phi}}$\\
% %  \hline
% %  \hline
% % \end{tabular}
% % \end{center}
% % where $N$ is the number of independent emitters of coherent light in the chaotic source, $\tau_{c}$ is the coherence time of the chaotic source, $\tau_{\phi}$ is the coherence time of the coherent light and $m=\text{min}(\Delta,\|\tau\|)$; the smaller of these 2 numbers. The $g^{(2)}$ and $g^{(2X)}$ for a mixture of coherent-chaotic light can be found by a weighted sum of the respective components of $g^{(2)}$ and $g^{(2X)}$.

% % \begin{figurehere}
% %     \begin{center}
% %     \includegraphics[width=\linewidth]{./theoretical_simulation.png}

% %   \caption{Expected correlations of $g^{(2)}$, $g^{(2X)}_{\text{int}}$ and $g^{(2X)}$ for chaotic light (red), coherent light (green) and 50:50 chaotic-coherent light mixture (blue), where $\tau_{c}$ is the coherence time of the chaotic light, $\tau_{\phi} = 2 \tau_{c}$ is the coherence time of the coherent light, and $\Delta= 5\tau_{c}$ is the propagation delay between the interferometric arms.}\label{fig:theory_plots}
% %   \end{center}
% % \end{figurehere}