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

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

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

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

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

\begin{strip}
 \begin{author_descr}

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

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

   % \textsuperscript{(1)} Centre for Quantum Technologies,


   % \textsuperscript{(1)()} Centre for Quantum Technologies, National University of Singapore Department of Physics, 
   % \textcolor{blue}{\uline{christian.kurtsiefer@gmail.com}}

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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.
    % Identifying the onset of coherence of a laser diode is of interest when characterizing laser diodes.  
    We present a technique to characterize the onset of coherence in a
    semiconductor laser diode using interferometric photon correlation
    measurements, and we observe with increasing injection current a
    transition of light emitted by the diode from chaotic, to a
    chaotic-coherent light mixture, to coherent.
    % \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 characteristic of a semiconductor laser diode is its threshold current. 
The laser diode emits laser light when operated above this current, 
and behaves as a light-emitting diode (LED)
when operating below.
% 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 threshold current is 
% by measuring the output power of the laser against the injection current, 
to identify the injection current where the emitted optical power 
changes sharply\cite{Siegman1986}. 
However, this method cannot determine whether the light emitted has changed from being incoherent (LED) to coherent laser radiation,
as the intensity measurement does not probe the nature of the light.  

To do this,
 % to distinguish between LEDs and lasers is by
one can measure its second-order photon correlation $g^{(2)}$,
where LEDs exhibit a ``bunching'' signature $g^{(2)}(0)>1$ associated with chaotic light,
while lasers exhibit $g^{(2)}=1$\cite{choi2007evolution,hostein2010demonstration}.
% 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 the lasing regime\cite{van2000laser,folkes2010correlated}
may result in second-order photon correlation measurements exhibiting $g^{(2)}(0)>1$,
when $g^{(2)}=1$ is expected. 
Thus, another method to investigate the transition of the nature of the
emitted light is required to identify the onset of coherence. 

We present such a method that is based on 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
are observed for different laser diode currents.
Interferometric photon correlation measurements were initially used to study spectral diffusion in molecules\cite{brokmann2006photon,PhysRevA.76.033824}, 
and were also applied to differentiate between chaotic light and coherent light with amplitude fluctuations\cite{Lebreton2013Theory,lebreton2013exp}. 
Both types of light sources exhibit a ``bunching'' $g^{(2)}$ signature, 
and hence are indistinguishable by standard second-order photon correlation
measurements. In contrast, interferometric photon correlation measurements
reveal the fraction of coherent light 
in the chaotic-coherent mixture of light emitted by the laser diode
as it transits from the LED regime to the lasing regime,
allowing to more quantitatively characterise the onset of coherence.

\section{Interferometric Photon Correlations}
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

\begin{equation}
    E_{A,B}(t) = \frac{E(t) \pm E(t+\Delta)}{\sqrt{2}}
    \label{eqn:fields}
\end{equation}
at the output ports of the interferometer. 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}

\begin{table*}
\begin{center}
\begin{tabular}{c c c c}
  \hline\hline
  Type of Light & $E(t)$ & $g^{(2)}(\tau=t_{2}-t_{1})$ & $g^{(2X)}_\text{int}(\tau=t_{2}-t_{1})$  \\[0.5ex]
  \hline
  Chaotic &$E_{0}\frac{e^{i\omega t}}{\sqrt{N}}\sum_{j}^{N}e^{i\phi_{j}(t)}$&$1+e^{-2\|\tau\|/\tau_{c}}$ & $-e^{-2\Delta/\tau_{c}}-e^{-2\|\tau\|/\tau_{c}}$\\[1ex]
  Coherent &$E_{0}e^{i\omega t}e^{i\phi_{j}(t)}$& 1 & $-e^{-2m/\tau_{\phi}}$\\
  \hline\hline
\end{tabular}
\end{center}
\caption{Expressions of $g^{(2)}$ and $g^{(2X)}_\text{int}$ for chaotic and
  coherent light. Here, $N$ 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
  $m=\text{min}(\Delta,\|\tau\|)$.\label{tab:type_of_light}}
\end{table*}

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
 \begin{equation}
    \begin{split}
        &g^{(2X)}(t_{2}-t_{1})\\
      \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}
% where remaining terms not shown in the final expression
% where the 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 $g^{(2)}$ terms in Eqn.~\ref{eqn:expansion} have the form of conventional second-order photon correlation functions, 
whereas the second group of terms are simply versions of $g^{(2)}$ that are
time-shifted in their argument by a time $\Delta$ and $-\Delta$, respectively.
The terms grouped as $g^{(2X)}_\text{int}$ occur as a result of light interfering 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 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 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}.
% Expected theory plots of interferometric correlations of chaotic light 

\begin{figure}[h]
    \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}
    \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{figure}

\section{Interferometric Photon Correlations near the Lasing Threshold}
The observed optical power emitted by a laser diode (Photodigm PH760DBR) is shown in
Fig.~\ref{fig:sample_plots}(a) as a function of the injection current, where a
sharp change in intensity is observed around 52.80\,mA.
We now measure $g^{(2X)}$ near this transition current.
% lasing threshold current to be around 52.8\,mA, as shown in Figure 1d. 

To measure $g^{(2X)}$,
we spectrally filter  
the light emitted by the laser diode% is first spectrally filtered
using a grating monochromator and an 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 propagation delay between the two paths of the
interferometer. This is accomplished with a 10\,m long single mode optical fibre (Fig.~\ref{fig:setup}).
Photoevents at each output port of the interferometer are detected by single photon avalanche detectors
and time-stamped. 
The correlations are then extracted numerically by
histogramming the time differences of all pairs of the timestamped events.
The histogram is correctly normalised by multiplying the total measurement time
and dividing by the product between the number of events at each channel and the size of the timing bins.  
% the numbe 
% The correlation are then extracted numerically from the 
% enteries of timestamp 
% 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}(b)-(d).
% 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}[h]
    \begin{center}
   % \includegraphics[width=\columnwidth]{../../figures/experimental_setup/g2X_setup.png}
   \includegraphics[width=\columnwidth]{./g2x_setup.png}
    % \includegraphics[scale=1]{figure1.pdf}
    \end{center}
    \caption{Experimental setup for measuring interferometric photon correlations. The spectrally filtered light from the laser diode 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}


\begin{figure}
    \begin{center}
    % \includegraphics[width=\linewidth]{../../figures/sample_g2x_plots/cross_sample.png}    
    \includegraphics[width=\linewidth]{./cross_sample.png}    
    \end{center}
    % \includegraphics[width=\columnwidth]{../figures_and_plots/combined_data/comparison.png} \hspace*{0.5cm}
  \caption{(a) sampled power emitted by the laser diode as a function of injection current. (b)-(d) $g^{(2X)}$ measurements showing signatures of chaotic light at 52.55\,mA (red), mixture at 52.80\,mA (blue) and coherent light at 53.20\,mA (green).} \label{fig:sample_plots}
\end{figure}

\section{Extracting percentage of coherent light in mixture}
The measurements of $g^{(2X)}$ were fitted with the weighted sum of $g^{(2X)}$ expressions for chaotic and coherent light as shown in Tab.~\ref{tab:type_of_light}. 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 from the expression
\begin{equation}
    \frac{1-g^{(2X)}(0)}{0.5}\,.
    \label{eqn:extract}
\end{equation}

The extracted fraction of coherent light for different injection currents is shown in Fig.~\ref{fig:fraction_vs_current}. It was found to increase with the injection current, and saturates at 0.8, 
not reaching the expected value of 1 even at the largest current (53.4\,mA)
where the laser was clearly operating above threshold. 
We attribute this observation to the limited visibility of the interferometer, 
which results in $g^{(2X)}(0)>0.5$ even for coherent light, 
leading to a smaller fraction of coherent light by simply using Eqn.~\ref{eqn:extract}.
The extracted fraction of coherent light takes even an unphysical negative value at 52.50\,mA and 52.55\,mA
due to a small peak appearing at $g^{(2X)}(0)$, which we again attribute to
the limited visibility of the interferometer. Increasing the visibility of the
interferometer should hopefully resolve this issue.

% \begin{figure}[h]
%     \begin{center}
%     % \includegraphics[width=\linewidth]{../../figures/coherent_percentage_and_intensity_vs_current/coherent_percentage_and_intensity_vs_current.png}
%     \includegraphics[width=\linewidth]{./coherent_percentage_and_intensity_vs_current.png}    
%     \end{center}
%     % \includegraphics[width=\columnwidth]{../figures_and_plots/combined_data/comparison.png} \hspace*{0.5cm}
%   \caption{Sampled intensity of the laser diode against injection current (black), and percentage of coherent light against injection current (red) extracted from the fit of $g^{(2X)}$ measurements.} \label{fig:fraction_vs_current}
% \end{figure}
\begin{figure}[h]
    \begin{center}
    % \includegraphics[width=\linewidth]{../../figures/coherent_percentage_and_intensity_vs_current/coherent_percentage_and_intensity_vs_current.png}
    \includegraphics[width=\linewidth]{./coherent_percentage_vs_current.png}    
    \end{center}
    % \includegraphics[width=\columnwidth]{../figures_and_plots/combined_data/comparison.png} \hspace*{0.5cm}
  \caption{Fraction of coherent light as a function of injection current extracted from the fit of $g^{(2X)}$ measurements.} \label{fig:fraction_vs_current}
\end{figure}

\section{Conclusion}
We presented a method to extract the fraction of coherent light emitted by a laser diode at different injection currents from interferometric photon correlation measurements. The fraction of coherent light was found to increase with injection current, which agrees with the expectation that the light emitted by the laser diode changes from an incoherent LED to coherent laser radiation. 

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