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\documentclass[a4paper, oneside, twocolumn, notitlepage, 10pt]{extarticle_ecoc}
\usepackage{ecoc}

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%-------------------------------------------------- Title -----------------------------------------------------%

\title{Determining the Onset of Lasing Using Interferometric Photon Correlations}%

%------------------------------------------------- Authors-----------------------------------------------------%

\author{
    Xi Jie Yeo\textsuperscript{(1)}, 
    Alvin Leow Zhen Wei\textsuperscript{(2)}, 
    Lijiong Shen\textsuperscript{(3)}, 
    Peng Kian Tan\textsuperscript{(3)}, 
    Christian Kurtsiefer\textsuperscript{(4)}
}

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%------------------------------------------ Description of Authors ----------------------------------------------%

\begin{strip}
 \begin{author_descr}

   \textsuperscript{(1)} Centre for Quantum Technologies,
   \textcolor{blue}{\uline{yeo.xi\_jie@u.nus.edu}}

   \textsuperscript{(2)} National University of Singapore Department of Physics,

   \textsuperscript{(3)} Centre for Quantum Technologies,


   \textsuperscript{(4)} Centre for Quantum Technologies, National University of Singapore Department of Physics, 
   \textcolor{blue}{\uline{christian.kurtsiefer@gmail.com}}

 \end{author_descr}
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%-------------------------------------------------- Abstract ---------------------------------------------------------%

\begin{strip}
  \begin{ecoc_abstract}
    % NOTE: Don't use a blank line here but start abstract right away to avoid an extra line break
    % Second-order photon-correlation measurements $g^{(2)}$ have been used as a 
    % method to classify light into bunching, coherent and antibunched~[1].  
    % However, a light source exibiting bunching statistics $g^{(2)}(0)>1$, is not always indiciative of an absense of coherent light. 
    % We observe coherent light in a chaotic-coherent light mixture using interferometric photon-correlation measurements $g^{(2X)}(\tau)$
    % on a laser diode operating near the threshold current. 
    % Our experimental results of $g^{(2X)}$ also show the laser emitting chaotic light when operating at a current below the lasing threshold and coherent light behaviour above the lasing threshold.
    % Your 45 words abstract, which will appear in the conference program, should be an explicit summary of the paper that states the problem, the methods used as well as the major results and conclusions. It should be complementary to rather than a repeat of the title.
    Semi-conductor laser diodes are commonly accepted as operating as light-emitting diodes below the laser threshold current
    and as lasers above the laser threshold current.


    \textcopyright2022 The Author(s)
  \end{ecoc_abstract}
\end{strip}

%-------------------------------------------------- Introduction Section -------------------------------------------------------%
% \section{Sections of a Paper}
% A conference paper has a clear structure. 

% The introduction section begins with a paragraph introducing the research and its relevance to the community. Subsequently, the state of the art is given. A "However" passage comments on what the world is still missing, has never done and hence is still waiting for. Finally, a "In this paper" paragraph concisely outlines how these issues have been addressed and what this paper is about.

% The paper then continues with a section on the operation principle, the methods, the experiments, the simulations, etc.. Add sections with headings as needed.

% The work and results are summarised with the conclusions. This is also the place for a potential outlook. 

% The paper ends with the acknowledgements and references.
\section{Introduction}
A key characterisation of a semiconductor laser diode is its laser threshold current, 
where the laser diode emits laser light when operated above this current, 
and as a light-emitting diode (LEDs)
when operating below this current.
Thus methods to determine the onset of lasing,
where the laser diode transits from a LED to laser,
is of importance to the community.
A common method to determine the laser threshold current is 
by measuring the output intensity of the laser against the injection current, 
and identifying the injection current where the intensity against injection current curve changes from one linear region to another linear region with a steeper gradient\cite{Siegman1986}. 
However, it is inconclusive to use such a method to determine whether the light emitted has changed from being an LED to a laser. 
A method to distinguish between LEDs and lasers is by measuring its second-order photon correlation $g^{(2)}$,
where LEDs exhibit a ``bunching'' signature $g^{(2)}(0)>1$ associated with chaotic light,
and lasers exhibit $g^{(2)}=1$.
This method of measuring $g^{(2)}(0)$ against injection current has been used to infer the transition of the laser diode from the LED regime to lasing regime\cite{choi2007evolution,hostein2010demonstration}.

However, amplitude noise in lasers when operating near the lasing threshold current has been reported, 
which would result in second-order photon correlation measurements exhibting $g^{(2)}(0)>1$,
even when $g^{(2)}=1$ is expected. 
Thus, a method to more clearly track the transition of the light emitted from LED to laser is required
in order to identify the onset of coherence. 

In this paper, we demonstrate a method to identify the onset of coherence of lasers
by measuring interferometric photon correlations.
In this technique, 
light from the laser diode is sent through an asymmetric Michelson interferometer and the timing correlations between the photoevents detected at the output ports of the interferometer were measured as the laser diode was operated at varying currents.
Interferometric photon correlation measurements were initially used to study spectral diffusion in molecules\cite{brokmann2006photon,PhysRevA.76.033824}, 
and were also to use differentiate between chaotic light and coherent light with amplitude fluctautions\cite{Lebreton2013Theory,lebreton2013exp}, 
where both type of light sources exhibits a ``bunching'' $g^{(2)}$ signature, 
and hence indistinguishable by standard second-order photon correlation measurements. 
We measure interferometric photon correlations at varying operating currents of the laser
which allows us to measure the percentage of coherent light 
in the chaotic-coherent mixuture of light emitted by the laser diode
as it transits from the LED regime to the lasing regime,
and identify the onset of coherence more definitively in the process.

\section{Concept of Interferometric Photon Correlations}
With reference to Fig.~\ref{fig:setup}, light with an electric field profile $E(t)$ is through an asymmetric Michelson interferometer with path difference in time $\Delta$ and produces the electric fields
\begin{equation}
    \begin{split}
        &E_{A}(t) = \frac{E(t) - E(t+\Delta)}{\sqrt{2}},\\
        &E_{B}(t) = \frac{E(t) + E(t+\Delta)}{\sqrt{2}},
    \end{split}
\end{equation}
respectively at the output ports of the interferometer. We note that path difference in time $\Delta$ is chosen to be longer than the coherence time of the light.

\begin{figure}[t]
    \begin{center}
   \includegraphics[width=\columnwidth]{../../figures/experimental_setup/g2X_setup.png}
    % \includegraphics[scale=1]{figure1.pdf}
    \end{center}
    \caption{Experimental setup for measuring interferometric photon correlations. The light is split at a non-polarising beam splitter and coupled into a delay fibre before recombining at another non-polarising beam splitter. APD: Avalanche single photon detectors.}\label{fig:setup}
\end{figure}

The intensity correlations between these two ports is 
\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}\\
      = & + \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2}+\Delta)E(t_{2}+\Delta)E(t_{1}+\Delta)} \rangle/4 \\
        & + \langle {E^{*}(t_{1})E^{*}(t_{2})E(t_{2})E(t_{1})} \rangle /4 \\
        & + \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2})E(t_{2})E(t_{1}+\Delta)} \rangle/4 \\ 
        & + \langle {E^{*}(t_{1})E^{*}(t_{2}+\Delta)E(t_{2}+\Delta)E(t_{1})} \rangle/4 \\
        & - \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2})E(t_{2}+\Delta)E(t_{1})} \rangle/4 \\
        & - \langle {E^{*}(t_{1})E^{*}(t_{2}+\Delta)E(t_{2})E(t_{1}+\Delta)} \rangle/4,
    \end{split}
\end{equation}
where remaining terms not shown in the final expression each average out to zero due to dithering and the normalisation of through division of the term $\langle E^{*}_{A}(t_{1})E_{A}(t_{1})\rangle \langle E^{*}_{B}(t_{2})E_{B}(t_{2})\rangle$ is implict. 

The first two terms in the above equation has the form of the conventional second-order photon correlation function $g^{(2)}$, whereas the next two terms are $g^{(2)}$ functions shifted in time by $\Delta$ and $-\Delta$ respectively. 
The last two terms
, where each term is denoted as $g^{(2X)}_\text{int}$, 
occurs as result of the light from each path of the interferometer interfering with each other at the output ports. 

The expressions for $g^{(2)}$ and $g^{(2X)}_\text{int}$ for coherent and chaotic light are shown in Tab.~\ref{tab:type_of_light}, 
using the models of electric fields for coherent and chaotic light found in Ref.~\cite{Loudon2000}. 

\begin{table*}[t]
   \centering
\caption{Expressions of $g^{(2)}$ and $g^{(2X)}_\text{int}$ for chaotic and coherent light. $N$ 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.} \label{tab:type_of_light}
      \begin{tabular}{||c | 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{table*}

% \begin{table*}[t]
%    \centering
% \caption{Expressions of interferometric photon correlations $$g^{(2X)}$ for chaotic and coherent light. $N$: the number of independent emitters of coherent light in the chaotic source, $\tau_{c}$: coherence time of the chaotic source, $\tau_{\phi}$: coherence time of coherent light,  $m=\text{min}(\Delta,\|\tau\|)$}
% \label{tab:type_of_light}%
%       \begin{tabular}{||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{table*}

The plots 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 a predicted plot for a measurement 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}.
% Expected theory plots of interferometric correlations of chaotic light 

\begin{figure}
    \begin{center}
    % \includegraphics[width=\columnwidth]{../figures_and_plots/theoretical_simulation/theoretical_simulation.png} \hspace*{0.5cm}
    % \includegraphics[width=0.7\linewidth]{LogoECOC.png}
    \includegraphics[width=\linewidth]{../../figures/theoretical_simulation/theoretical_simulation.png}

  \caption{Theoretical plots 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), $\tau_{c}$ is the coherence time of the chaotic light, the coherence time of the coherent light is $\tau_{\phi} = 2 \tau_{c}$ and the delay between the interferometric arms $\Delta= 5\tau_{c}$.} \label{fig:theory_plots}
  \end{center}
\end{figure}

\section{Measuring Interferometic Photon Correlations Across Lasing Threshold}
We measure the intensity emitted by the laser diode against the current supplied to it, 
and estimated the current where a sharp change in intensity is observed
to be around 52.80 mA for this laser diode.
% lasing threshold current to be around 52.8\,mA, as shown in Figure 1d. 
We then measure $g^{(2X)}$ at current values across 52.8 mA.  
The light emitted by the laser diode is first spectrally filtered
by a grating monochromator and etalon,
with an effective spectral filter bandwidth of approximately 6 GHz.
The spectrally filtered light is sent through an asymmetric Mach-Zehnder interferometer 
with a 50 ns timing delay between the two paths of the interferometer,
by the use of a 10 m long single mode fibre. 
Photoevents at each port of the interferometer is detected using single photon avalanche detectors 
and a timing correlation each detector's photoevents is performed to measure the interferometric photon correlations. 
% Hanbury Brown and Twiss type second-order photon correlation measurement is done by repeating the measurement while blocking the light going into the delyayed arm of the interferometer (dotted lines).
Sample measurements showing signatures of coherent, mixture and chaotic light are shown in Fig.~\ref{fig:sample_plots}.
% 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{figure}
    \begin{center}
    \includegraphics[width=0.7\linewidth]{LogoECOC.png}    
    % \includegraphics[width=\columnwidth]{../figures_and_plots/combined_data/comparison.png} \hspace*{0.5cm}
  \caption{Measurements for $g^{(2)}$ and $g^{(2X)}$ of the laser below threshold at 52.55\,mA (red), near threshold at 52.80\,mA (blue) and above threshold at 53.20\,mA (green). The delay between the 2 arms of the interferometer $\Delta$ is aboout 50\,ns, which agrees with where the smaller peaks are expected to appear in the $g^{(2X)}$ measurements for the currents 52.55\,mA and 52.8\,mA. } \label{fig:sample_plots}
  \end{center}
\end{figure}


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%-------------------------------------------------- Conclusions Section ———————————————————————————%

\section{Conclusions}

Good luck with your submission!

%-------------------------------------------------- Acknowledgements Section -------------------------------------------------------%

\section{Acknowledgements}

We wish to thank all of you for submitting a paper to ECOC 2022.

%-------------------------------------------------- Bibliography Section -------------------------------------------------------%

\printbibliography

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