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\begin{document}
\title{Direct measurement of coherent light proportion from a \DIFdelbegin \DIFdel{practical }\DIFdelend laser\DIFdelbegin \DIFdel{source}\DIFdelend }
\author{Xi~Jie~Yeo}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

\author{Eva~Ernst}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}
% \affiliation{Institute fuer Experimentalphysik, Universitaet Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria}

\author{Alvin~Leow}
\affiliation{Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551}

\author{Jaesuk~Hwang}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

\author{Lijiong~Shen}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

\author{Peng~Kian~Tan}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}

\author{Christian~Kurtsiefer}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543}
\affiliation{Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551}
\email[]{christian.kurtsiefer@gmail.com}

\date{\today}
\begin{abstract}
% 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 
estimate the proportion of coherent emission in the light emitted by a \DIFdelbegin \DIFdel{practical laser source }\DIFdelend \DIFaddbegin \DIFadd{semiconductor laser diode }\DIFaddend without spectral filtering,
\DIFdelbegin \DIFdel{using interferometric photon correlation measurements, 
where photon correlations are measured }\DIFdelend \DIFaddbegin \DIFadd{by measuring photon correlations }\DIFaddend between the light emitted from the output ports of an asymmetric Mach-Zehnder interferometer.
Using this technique, 
we determine the proportion of coherent emission \DIFaddbegin \DIFadd{of the brightest mode }\DIFaddend in the light emitted by a laser diode, 
in a \DIFdelbegin \DIFdel{statistical }\DIFdelend mixture of light \DIFdelbegin \DIFdel{from stimulated and spontaneous }\DIFdelend \DIFaddbegin \DIFadd{with coherent and incoherent }\DIFaddend emission,
and a \DIFdelbegin \DIFdel{statistical }\DIFdelend mixture of light \DIFdelbegin \DIFdel{from stimulated }\DIFdelend \DIFaddbegin \DIFadd{with coherent }\DIFaddend emission at two distinguishable spectral bands.


\end{abstract}

% insert suggested PACS numbers in braces on next line
% \pacs{}

\maketitle
\section{Introduction}
%%%%%%%%%%%%
% KEY IDEAS
%%%%%%%%%%%%
% Why are generation of coherent light source important? (in context of CV QKD maybe?)
% Ensuring that these coherent state sources remain emitting coherent states
% Lasers are a source of coherent states
% Conventional means to verify that a source is indeed generating coherent states
% Problem with such a source

% G20 plot against g2x0
% complement existing measurement to see some sort of transition from stimulated to spontaneous

% define coherent light?
\DIFdelbegin \DIFdel{The %DIF < theories that lead to the
invention of lasers can be traced to work
describing the emission process of the light from an atom to be spontaneous or stimulated \mbox{%DIFAUXCMD
\cite{Einstein1916}}\hskip0pt%DIFAUXCMD
.
}\DIFdelend %DIF >  The %theories that lead to the
%DIF >  invention of lasers can be traced to theories
%DIF >  describing the emission process of the light from an atom to be spontaneous or stimulated \cite{Einstein1916}.
%DIF >  An ensemble of light emitters undergoing stimulated emission will emit coherent light that has a well-defined phase,
%DIF >  whereas an ensemble of light emitters undergoing spontaneous emission will emit incoherent light which is randomly phased \cite{Loudon2000}.
\DIFaddbegin 

\DIFadd{In traditional models describing macroscopic lasers,  
the laser undergo a sharp transition 
from undergoing spontaneous emission,
to dominantly stimulated emission~\mbox{%DIFAUXCMD
\cite{Lamb1964,Arecchi1965,Haken1975}}\hskip0pt%DIFAUXCMD
.
%DIF >  in the nature of light emitted by the laser 
%DIF >  occurs at the lasing threshold, 
%DIF >  separating two operating regimes of light emitters in the laser 
}\DIFaddend An ensemble of light emitters undergoing stimulated emission will emit coherent light that has a well-defined phase,
whereas an ensemble of light emitters undergoing spontaneous emission will emit incoherent light which is randomly phased\DIFaddbegin \DIFadd{~}\DIFaddend \cite{Loudon2000}.

\DIFdelbegin \DIFdel{In traditional models of macroscopic lasers \mbox{%DIFAUXCMD
\cite{Lamb1964,Arecchi1965,Haken1975}}\hskip0pt%DIFAUXCMD
, 
the light emitted from a laser is modeled to originate dominantly from stimulated emission. 
}\DIFdelend %DIF >  emitting incoherent light which is randomly phased
%DIF >   or coherent.
%DIF >  the light emitted from a macroscopic laser is modelled to originate dominantly from stimulated emission \cite{Lamb1964,Arecchi1965,Haken1975},
%DIF >  where coherent light that has a well-defined phase is emitted \cite{Loudon2000}. 
%and spontaneous emission is negligible.
%Furthermore,
\DIFdelbegin \DIFdel{These models also predict a phase transition of the nature of emission at the lasing threshold, separating two operating regimes where light emitted is either spontaneous or stimulated.
}\DIFdelend %DIF >  These models also predict 

However, experiments on small lasers \DIFdelbegin \DIFdel{have shown }\DIFdelend \DIFaddbegin \DIFadd{suggest }\DIFaddend that the transition from spontaneous to stimulated emission is not abrupt,
but extends over a range \DIFdelbegin \DIFdel{for the pump power \mbox{%DIFAUXCMD
\cite{Choi2007,Urlich2007,Wiersig2009,Hostein2010,Kreinberg2017}}\hskip0pt%DIFAUXCMD
.
}\DIFdelend \DIFaddbegin \DIFadd{of pump powers~\mbox{%DIFAUXCMD
\cite{RJin1994,Strauf2006,Choi2007,Urlich2007,Wiersig2009,Hostein2010,Pan2016,Kreinberg2017,George2021}}\hskip0pt%DIFAUXCMD
.
%DIF > Mode hop?
}\DIFaddend Across this range\DIFaddbegin \DIFadd{, }\DIFaddend %of input power,
the light emitted from the laser is 
a mixture of spontaneous and stimulated emission.
\DIFdelbegin %DIFDELCMD < 

%DIFDELCMD < %%%
\DIFdel{In these experiments, }\DIFdelend \DIFaddbegin \DIFadd{Thus, observations of }\DIFaddend the transition from spontaneous to stimulated emission
\DIFdelbegin \DIFdel{was characterized by }\DIFdelend \DIFaddbegin \DIFadd{require methods to measure the
%DIF >  components in this mixture.  
proportion of coherent light or incoherent light in this mixture.
In these experiments, the proportion of coherent and incoherent light emitted by the laser 
%DIF >  transition from spontaneous to stimulated emission was characterized 
were estimated by }\DIFaddend measuring
the second-order photon correlation $g^{(2)}$,
which corresponds to the distribution of timing separations between single photon detection events,
using the standard Hanbury-Brown and Twiss scheme\DIFaddbegin \DIFadd{~}\DIFaddend \cite{HBT1956}.
The measurement result can be explained using Glauber's theory of optical coherence\DIFaddbegin \DIFadd{~}\DIFaddend \cite{Glauber1963},
where incoherent light from spontaneous emission would exhibit a ``bunching''
signature with $g^{(2)}(0)>1$,
while coherent light from stimulated emission exhibits a Poissonian
distribution with $g^{(2)}=1$.
\DIFdelbegin %DIFDELCMD < 

%DIFDELCMD < %%%
\DIFdel{The ``bunching'' signature associated with incoherent light has a characteristic timescale inversely related to the }\DIFdelend \DIFaddbegin \DIFadd{The operating conditions
where the laser emits dominantly coherent light 
would then be estimated to be where $g^{(2)} \to 1$.
}

\DIFadd{A limitation of using second-order photon correlation to quantify the proportion of coherent and incoherent light emitted by the laser, is due to situations where incoherent light 
exhibits $g^{(2)}\approx1$, 
indistinguishable from the same measurement of coherent light.
%DIF >  The ``bunching'' signature associated with incoherent light has a characteristic timescale inversely related to the spectral width of the light,
%DIF >  according to the Wiener-Khintchine theorem
%DIF >  % the nature of emission
%DIF >  ~\cite{Wiener1930,Khintchine1934,MandelWolf1995}. 
%DIF >  {(\bf Maybe that needs also an optics reference? Is WK as a mathematical argument enough?)}
For example, 
when the }\DIFaddend spectral width of the \DIFdelbegin \DIFdel{light,
according to the Wiener-Khintchine theorem related through a Fourier transform
\mbox{%DIFAUXCMD
\cite{Wiener1930,Khintchine1934,MandelWolf1995}}\hskip0pt%DIFAUXCMD
. 
%DIF <  {(\bf Maybe that needs also an optics reference? Is WK as a mathematical argument enough?)}
In a practical measurement,
}\DIFdelend \DIFaddbegin \DIFadd{incoherent light is broad,
}\DIFaddend the \DIFaddbegin \DIFadd{characteristic timescale of the ``bunching'' signature is small~\mbox{%DIFAUXCMD
\cite{Wiener1930,Khintchine1934,MandelWolf1995}}\hskip0pt%DIFAUXCMD
,
to the extent that it may be smaller than the detector timing resolution.
As the }\DIFaddend amplitude of the ``bunching'' signature scales with the ratio of characteristic timescale of the light to the timing response of the detectors\DIFdelbegin \DIFdel{\mbox{%DIFAUXCMD
\cite{Scarl1968}}\hskip0pt%DIFAUXCMD
.
Thus,
when the spectral width of the incoherent light is broad, 
to }\DIFdelend \DIFaddbegin \DIFadd{~\mbox{%DIFAUXCMD
\cite{Scarl1968}}\hskip0pt%DIFAUXCMD
,
}\DIFaddend the \DIFdelbegin \DIFdel{extent that the characteristic timescale of the ``bunching'' signature is smaller than the detector timing uncertainty}\DIFdelend \DIFaddbegin \DIFadd{second-order photon correlations measurment of incoherent light exhibits $g^{(2)}\approx1$, 
%DIF >  indistinguishable from the same measurement of coherent light.
and fails to distinguish if the laser is emitting coherent light}\DIFaddend , incoherent light \DIFdelbegin \DIFdel{may exhibit $g^{(2)}\approx1$, 
like coherent light}\DIFdelend \DIFaddbegin \DIFadd{or a mixture of the two}\DIFaddend . 

%DIF >  The ``bunching'' signature associated with incoherent light has a characteristic timescale inversely related to the spectral width of the light,
%DIF >  according to the Wiener-Khintchine theorem 
%DIF >  % the nature of emission
%DIF >  \cite{Wiener1930,Khintchine1934,MandelWolf1995}. 
%DIF >  % {(\bf Maybe that needs also an optics reference? Is WK as a mathematical argument enough?)}
%DIF >  The amplitude of the ``bunching'' signature scales with the ratio of characteristic timescale of the light 
%DIF >  to the timing response of the detectors \cite{Scarl1968}.
%DIF >  Thus, when the spectral width of the incoherent light is broad,
%DIF >  to the extent that the characteristic timescale of the ``bunching'' signature is smaller than the detector timing resolution,
%DIF >  the second-order photon correlations measurment of incoherent light exhibit $g^{(2)}\approx1$, 
%DIF >  indistinguishable from the same measurement of coherent light.
\DIFaddbegin 

\DIFaddend To overcome the \DIFaddbegin \DIFadd{failure to characterise the light emission from  $g^{(2)}$
due to }\DIFaddend limited detector timing \DIFdelbegin \DIFdel{uncertainty}\DIFdelend \DIFaddbegin \DIFadd{resolution}\DIFaddend , 
a narrow band of 
\DIFdelbegin \DIFdel{incoherent }\DIFdelend %DIF > incoherent 
light can be prepared with filters from a wide
\DIFdelbegin \DIFdel{optical spectrum of an incoherent licht source}\DIFdelend \DIFaddbegin \DIFadd{light source with a wide optical spectrum}\DIFaddend ~\cite{PK2017}. 
%DIF >  of an incoherent light source.
The narrow spectral width of a filtered incoherent light has a correspondingly
larger characteristic coherence timescale,
which may be long enough to be resolvable by the detectors.
% need to cite
\DIFdelbegin %DIFDELCMD < 

%DIFDELCMD < %%%
\DIFdel{However}\DIFdelend \DIFaddbegin \DIFadd{Nevertheless}\DIFaddend , for characterising the transition from spontaneous to stimulated
emission of a laser, such spectral filtering presents some shortcomings.
First, as spectral filtering discards light outside the transmission window of
a filter, a result would be inconclusive for the full emission of the source.
Second,
spectral filtering requires \textit{a priori} information or an educated guess of the central frequency and bandwidth of stimulated emission.
Third, it has been shown that spectral filtering below the Schawlow-Townes linewidth of the laser result in $g^{(2)}(0)>1$, similar to light from spontaneous emission\DIFaddbegin \DIFadd{~}\DIFaddend \cite{Neelen1992}. 

\DIFdelbegin \DIFdel{This motivates for methods quantifying }\DIFdelend \DIFaddbegin \DIFadd{These shortcomings motivate for methods that
quantifies }\DIFaddend the proportion of coherent \DIFaddbegin \DIFadd{or incoherent }\DIFaddend light
emitted by a \DIFaddbegin \DIFadd{laser }\DIFaddend source without the need for spectral filtering\DIFdelbegin \DIFdel{.
A method to characterise the stimulated and spontaneous emission from 
a pulsed
laser has been demonstrated before \mbox{%DIFAUXCMD
\cite{George2021}}\hskip0pt%DIFAUXCMD
.
}\DIFdelend \DIFaddbegin \DIFadd{,
in order to properly characterise the transition of laser diode emission from 
spontaneous to stimulated.
%DIF >  and can detect coherent emission into multiple modes,
%DIF >  with these multiple modes incoherent with respect to each other.
}

\DIFaddend %, and we aim to provide such a method for continuous wave lasers.

In this paper, we present a method to quantify \DIFdelbegin \DIFdel{bounds for }\DIFdelend the proportion of
coherent light for a continuous wave laser\DIFaddbegin \DIFadd{, 
without the use of spectral filters}\DIFaddend . 
Specifically, we investigate the
brightest mode of coherent emission from a semiconductor laser diode without
spectral filtering, by 
\DIFdelbegin \DIFdel{using interferometric photon correlations 
.
Earlier methods }\DIFdelend \DIFaddbegin \DIFadd{measuring second-order photon correlations 
between the light emitted from the output ports of an asymmetric Mach-Zehnder interferometer.
%DIF >  interferometric photon correlations.
This measurement, also known as interferometric photon correlation,
was
originally applied to differentiate between incoherent light and coherent light with amplitude fluctuations~\mbox{%DIFAUXCMD
\cite{lebreton2013exp}}\hskip0pt%DIFAUXCMD
.
Earlier variations }\DIFaddend of interferometric photon correlation measurements were used to study
spectral diffusion in organic molecules embedded in solid matrix~\cite{brokmann2006photon,PhysRevA.76.033824}.
\DIFdelbegin \DIFdel{The method of interferometric photon correlation used in this paper was
originally applied to differentiating between incoherent light and coherent light with amplitude fluctuations~\mbox{%DIFAUXCMD
\cite{lebreton2013exp}}\hskip0pt%DIFAUXCMD
.
}\DIFdelend In contrast to second-order photon correlations, 
this method can clearly distinguish between a finite linewidth coherent light and broadband incoherent light\DIFdelbegin \DIFdel{~\mbox{%DIFAUXCMD
\cite{Lebreton2013Theory}}\hskip0pt%DIFAUXCMD
.
We use this }\DIFdelend \DIFaddbegin \DIFadd{,
and has temporal features related to the characteristic timescale of the coherent light
~\mbox{%DIFAUXCMD
\cite{Lebreton2013Theory}}\hskip0pt%DIFAUXCMD
,
which is typically of interest in contrast to the characteristic timescale of the coherent light
in the the temporal features of standard second-order photon correlations.
We extend the use of interferometric photon correlation 
as a }\DIFaddend method to extract the \DIFdelbegin \DIFdel{fraction }\DIFdelend \DIFaddbegin \DIFadd{proportion }\DIFaddend of coherent light
emitted by the laser diode over a range of \DIFdelbegin \DIFdel{pump powers }\DIFdelend \DIFaddbegin \DIFadd{operating currents 
}\DIFaddend across the lasing threshold\DIFdelbegin \DIFdel{,
 }\DIFdelend \DIFaddbegin \DIFadd{.
 }\DIFaddend % as defined by a sharp change in optical power against operating current
% statistical mixture of coherent light with light from spontaneous emission is emitted.
\DIFdelbegin \DIFdel{and in the lasing regime above threshold where coherent light is emitted }\DIFdelend \DIFaddbegin 

\DIFadd{Using this method,
we were also able to identify a
%DIF >  and in the lasing 
regime above the lasing threshold 
where the proportion of coherent light significantly reduced over a range of 
operating currents.
Further investigation
revealed that the laser was emitting }\DIFaddend into
two distinguishable longitudinal modes \DIFdelbegin \DIFdel{.
}\DIFdelend \DIFaddbegin \DIFadd{in this regime.
If conventional second-order correlation was used,
this regime of multimode operation may have been overlooked
as $g^{(2)}=1$ 
even when these two longitiudinal modes 
are incoherent with respect to each other~\mbox{%DIFAUXCMD
\cite{Weber1970,Yin1996}}\hskip0pt%DIFAUXCMD
.
%DIF >  identified  
%DIF >  Furthermore, a laser emitting coherent light in multiple longitiudinal modes 
%DIF >  can exhibit $g^{(2)}=1$ 
%DIF >  even when these two longitiudinal modes 
%DIF >  are incoherent with respect to each other~\cite{Weber1970,Yin1996}.
%DIF >  Thus, laser operation in a single or multiple longitiudinal mode 
%DIF >  cannot be determined by measurements of $g^{(2)}$ alone.  
}\DIFaddend 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{figure}
  \begin{center}
    \includegraphics[width=0.8\columnwidth]{./figures/g2x_setup/g2x_setup_vs2.eps}
  \end{center}
  \caption{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. 
    Light from a laser diode enters an asymmetric Mach-Zehnder Interferometer. % reflections at the beamsplitter introduces a phase shift to the light. 
    Singe photon avalanche photodetectors (APD) 
    \DIFdelbeginFL \DIFdelFL{generate photodetection events
    }\DIFdelendFL at each output port of the interferometer
    \DIFaddbeginFL \DIFaddFL{generate photodetection events}\DIFaddendFL , which are time-stamped to
    extract the the correlations numerically.} \label{fig:setup}
\end{figure}


The setup for an interferometric photon correlation measurement $g^{(2X)}$ is shown in Fig.~\ref{fig:setup}.
Light emitted by the laser diode
is sent through an asymmetric Mach-Zehnder interferometer, with a long
propagation delay $\Delta$ between the two paths of the interferometer that
exceeds the coherence time of the light.

With a light field $E(t)$ at the input, %of the interferometer,
% where each of the sampled light field traverses arms of different lengths.
% The two light fields meet at another beamsplitter,
% with the field in the longer arm of the interferometer is delayed relative to the shorter arm by $\Delta$.
the light fields at the output ports $A,B$ of the interferometer are 
\begin{equation}
    E_{A,B}(t) = \frac{E(t) \pm E(t+\Delta)}{\sqrt{2}}\,,
    \label{eqn:fields}
\end{equation}
with the relative phase shift $\pi$ acquired by one of the output fields from
the beamsplitter. 
%results in the sign difference in one of the terms in Eqn. \ref{eqn:fields}.


% , introducing 900\,ns.The propagation delay $\Delta$ is chosen to be longer
% than the coherence time of the light.
\DIFdelbegin \DIFdel{Using these expressions for the electrical fields, the }\DIFdelend %DIF >  Using these expressions for the electrical fields, 
\DIFaddbegin \DIFadd{The }\DIFaddend temporal correlation of
photodetection events between the two output ports is given by
\begin{equation}
  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}\,\DIFdelbegin \DIFdel{.
}\DIFdelend \DIFaddbegin \DIFadd{,
}\DIFaddend \end{equation}
\DIFdelbegin \DIFdel{Therein}\DIFdelend \DIFaddbegin \DIFadd{therein}\DIFaddend , $\langle \rangle$ indicates an expectation value\DIFdelbegin \DIFdel{and/or an ensemble
average.
 }\DIFdelend \DIFaddbegin \DIFadd{.
 %DIF >  and/or an ensemble average.
}\DIFaddend Using Eqn.~\ref{eqn:fields}, 
% and the fact terms which computes correlation of that ha is 0 at all values of $t_{2}- t_{1}$ 
% averages out some terms to 0,
$g^{(2X)}(t_{2}-t_{1})$ can be grouped in several terms: 
\begin{equation}
  \DIFdelbegin %DIFDELCMD < \begin{aligned}
%DIFDELCMD <    g^{(2X)}(t_{2}-t_{1})=\\
%DIFDELCMD <           = \frac{1}{4}[ &\langle {E^{*}(t_{1})E^{*}(t_{2})E(t_{2})E(t_{1})} \rangle \\
%DIFDELCMD <         & + \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2}+\Delta)E(t_{2}+\Delta)E(t_{1}+\Delta)} \rangle \\[1mm]
%DIFDELCMD <         & + \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2})E(t_{2})E(t_{1}+\Delta)} \rangle\\
%DIFDELCMD <         & + \langle {E^{*}(t_{1})E^{*}(t_{2}+\Delta)E(t_{2}+\Delta)E(t_{1})} \rangle\\[1mm]
%DIFDELCMD <         & - \langle {E^{*}(t_{1}+\Delta)E^{*}(t_{2})E(t_{2}+\Delta)E(t_{1})} \rangle\\
%DIFDELCMD <         & - \langle {E^{*}(t_{1})E^{*}(t_{2}+\Delta)E(t_{2})E(t_{1}+\Delta)} \rangle]\,.\\
%DIFDELCMD <   \end{aligned}%%%
\DIFdelend \DIFaddbegin \begin{aligned}
   g^{(2X)}&(t_{2}-t_{1})=\\
          = \frac{1}{4}[ &\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 \\[1mm]
        & + \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\\[1mm]
        & - \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{aligned}\DIFaddend 
  \label{eqn:expansion}
% \end{split}
\end{equation}
%Terms that were not shown in the expansion were either cancelled out by another term with the same absolute value but opposite sign, or are complex valued and uncorrelated for all values of $t_{2}- t_{1}$, resulting in a correlation of 0 at all times.
% 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.
%
%In this expression, %Eqn.~\ref{eqn:expansion},
The first two terms have the form of conventional second-order photon
correlation functions $g^{(2)}(t_2-t_1)$.
The next two terms are conventional second-order photon correlation functions,
time-shifted forward and backward in their argument by the propagation delay
$\Delta$ in the interferometer.
The last two terms have negative signs and reduce $g^{(2X)}$, leading to a dip
at zero time difference \DIFdelbegin \DIFdel{$t_2-t_1$}\DIFdelend \DIFaddbegin \DIFadd{$t_2-t_1 = 0$}\DIFaddend ,
with a width given by the coherence time of the light.

The expectation values appearing in Eqn.~\ref{eqn:expansion} for $g^{(2X)}$
can be evaluated
by using  \DIFdelbegin \DIFdel{statistical }\DIFdelend % substituting
expressions~\cite{Loudon2000} of $E(t)$ for incoherent light and coherent light \cite{Lebreton2013Theory}.
% We present an expression of $g^{(2X)}$ for a mixture coherent light and a secondary uncorrelated light in Appendix A.
%expected results for %models of
%$g^{(2X)}$ for incoherent, coherent and mixed coherent-incoherent light can
%be calculated

For incoherent light, %a measurement of
$g^{(2X)}$ exhibits a ``bunching'' signature peaking at time differences plus and minus the propagation delay, \DIFdelbegin \DIFdel{$g^{(2X)}(\pm\Delta) = 1+1/4$}\DIFdelend \DIFaddbegin \DIFadd{$g^{(2X)}(\pm\Delta) = 1+(1/4)$}\DIFaddend .
At zero-time difference, 
the expected ``bunching'' signature from conventional second-order photon correlation functions in the first two terms of Eqn.~\ref{eqn:expansion}
and the dip from the last two terms of Eqn.~\ref{eqn:expansion} cancel each
other, resulting in $g^{(2X)}(0)= 1$ \DIFdelbegin \DIFdel{.
}\DIFdelend \DIFaddbegin \DIFadd{(see Fig.~\ref{fig:sample_plots} top).
}\DIFaddend % In the situation where the incoherent light has coherence time shorter than the timing resolution of the detectors,
% the ``bunching'' signatures would be unresolveable

For coherent light, 
since the second-order photon correlation function $g^{(2)}$ has a constant
value of 1, the $g^{(2X)}$ will show the negative contributions from the last two terms of Eqn.~\ref{eqn:expansion},
resulting in $g^{(2X)}(0) = 1/2$ \DIFdelbegin \DIFdel{. 
}\DIFdelend \DIFaddbegin \DIFadd{(see Fig.~\ref{fig:sample_plots} bottom).
}\DIFaddend 

% \begin{figure}
%  \begin{center}
%     \includegraphics[width=\linewidth]{./figures/theoretical_simulation/theoretical_simulation.png}    
%     \end{center}
%   \caption{Calculated $g^{(2X)}$ plots with different ratios of incoherent-coherent light mixture; percentage of coherent light indicated on the top-left corner of each plot. Blue, dash: bunching signature associated with incoherent 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 incoherent/coherent light. } \label{fig:theory_plots}
% \end{figure}


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

% The expected correlations of $g^{(2)}$, $g^{(2X)}_\text{int}$ and $g^{(2X)}$ for coherent and incoherent 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 incoherent light,
% obtained by an equally weighted sum of $g^{(2)}$ and $g^{(2X)}$ for coherent and incoherent light 
% as predicted in Ref.~\cite{Lebreton2013Theory}.
% % Expected theory plots of interferometric correlations of incoherent light 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Extracting fraction of coherent light emitted in a mixture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In order to obtain an interpretation of the nature of the light emitted
beyond just presenting the components of $g^{(2X)}$, we consider a light field
that is neither completely coherent nor incoherent.
We assume that light emitted by the laser is a mixture of coherent
light field $E_\text{coh}$, and \DIFdelbegin \DIFdel{an uncorrelated }\DIFdelend \DIFaddbegin \DIFadd{a }\DIFaddend light field $E_\text{unc}$ \DIFdelbegin \DIFdel{,
which nature }\DIFdelend \DIFaddbegin \DIFadd{uncorrelated to $E_\text{coh}$.
The nature of $E_\text{unc}$ }\DIFaddend can be coherent, incoherent, or a coherent-incoherent mixture.
In the following, we try to extract quantitative information about the
components from the interferometric photon correlations $g^{(2X)}$, namely the
fraction of \DIFdelbegin \DIFdel{coherent }\DIFdelend light intensity of the brightest coherent component in
the light field, and a collective treatment of all the rest.

% we demonstrate a method to extract information on the
% fraction of coherent light intensity of the brightest coherent emission,
% over the mean intensity of the mixture.

We model the light field mixture with an electrical field
\begin{equation}
    E_\text{mix}(t) = \sqrt{\rho}E_\text{coh}(t) + \sqrt{1-\rho}E_\text{unc}(t)\,,
    \label{eqn:mixture_field}
\end{equation}
where $\rho$ is the
fraction of \DIFdelbegin \DIFdel{coherent }\DIFdelend light intensity of the brightest coherent emission, and
%, over the mean intensity of the mixture.
the respective light field terms are normalised 
such that \DIFdelbegin \DIFdel{$\abs{E_\text{res}} = \abs{E_\text{coh}} = \abs{E_\text{unc}}$}\DIFdelend \DIFaddbegin \DIFadd{$\abs{E_\text{mix}} = \abs{E_\text{coh}} = \abs{E_\text{unc}}$}\DIFaddend .


% We consider the scenarios where
 % $g_\text{unc}^{(2)}(0)=1$,
% such as when the uncorrelated light is incoherent with a coherence time unresolvable by the detectors,
% or when the uncorrelated light is another coherent light source,and 
Evaluating photon correlation in Eqn.~\ref{eqn:expansion} with this light model,
and further assuming that \DIFdelbegin \DIFdel{(1) }\DIFdelend \DIFaddbegin \DIFadd{first, }\DIFaddend the propagation delay in the interferometer is
significantly longer than the coherence time scale of the light source,
and \DIFdelbegin \DIFdel{(2) }\DIFdelend \DIFaddbegin \DIFadd{second, }\DIFaddend the interferometer has good visibility yields
%the interferometric photon correlation of the light field mixture at zero time difference is
\begin{equation}
    g_{\text{mix}}^{(2X)}(0) = 2\rho-\frac{3\rho^{2}}{2} + \frac{(1-\rho)^{2}}{2}g_{\text{unc}}^{(2)}(0)\,,
 \label{eqn:g2x_mixture_extractable}
\end{equation}
at zero time difference, with only two remaining parameters, $\rho$ 
and $g_{\text{unc}}^{(2)}(0)$, 
the zero time difference second order photon correlation of the uncorrelated
field (see Appendix~\ref{app:g2x_mixture}). 

% \textbf{(Original Version)}
%This expression, together with the constraint for $rho$ to be a meaningful
%real-valued fraction, $0\le\rho\le$, implies a
\DIFdelbegin \DIFdel{The connection in }\DIFdelend \DIFaddbegin \DIFadd{Using }\DIFaddend Eqn.~\ref{eqn:g2x_mixture_extractable}, together with the
physical \DIFdelbegin \DIFdel{requirement }\DIFdelend \DIFaddbegin \DIFadd{constraint }\DIFaddend $0 \leq \rho \leq 1$ for the coherent light fraction
limits the possible combinations of  $g_{\text{unc}}^{(2)}(0)$ and
$g_{\text{mix}}^{(2X)}(0)$, shown as non-shaded areas in
Fig.~\ref{fig:bounds_plots}; the exact expressions for the boundaries are given
in Appendix~\ref{app:boundaries}.


% We obtain physically possible values of $g_{\text{unc}}^{(2)}(0)$ for a given $g_{\text{mix}}^{(2X)}(0)$,
% by imposing the constraint of $0 \leq \rho \leq 1$ to Eqn.~\ref{eqn:g2x_mixture_extractable}.
% \textbf{Not sure if the expressions that define the solution space here are clunky and should be moved to appendix?}
% This results in the solution space shown in Fig.~\ref{fig:bounds_plots},
% with a lower bound
% \begin{equation}
%         g_{\text{unc}}^{(2)}(0) \geq  \begin{cases}0\,,  &g_{\text{mix}}^{(2X)}(0) \leq \frac{2}{3}\\
%                         3 +\frac{1}{1 - 2 g_{\text{mix}}^{(2X)}(0) }, &g_{\text{mix}}^{(2X)}(0) \in[\frac{2}{3},1]  \\
%                         2g_{\text{mix}}^{(2X)}(0)\, &g_{\text{mix}}^{(2X)}(0) \geq 1 \end{cases},
%  \label{eqn:lowerbound_g2}
% \end{equation}
% and the upper bound for $g_{\text{mix}}^{(2X)}(0) \in [0,\frac{1}{2})$ is 
% \begin{equation}
%     g_{\text{unc}}^{(2)}(0) \leq 2g_{\text{mix}}^{(2X)}(0)\,.
%  \label{eqn:upperbound_g2}
% \end{equation}



% from Eqn.~\ref{eqn:g2x_mixture_extractable} 
% by taking the derivative of $g_{\text{unc}}^{(2)}(0)$ with respect to $\rho$,
% for a given value of $g_{\text{mix}}^{(2X)}(0)$.
% For the solution of Eqn.~\ref{eqn:g2x_mixture_extractable} to give a real-valued $\rho$,
% $g_{\text{unc}}^{(2)}(0)$ is bounded between the hyperbolas


\begin{figure}
    \begin{center}
    \includegraphics[width=\linewidth]{./figures/calculating_bounds/g2_unc_bounds.eps}    
    \end{center}
  \caption{Combinations of $g_{\text{unc}}^{(2)}(0)$ and
    $g_{\text{mix}}^{(2X)}(0)$ that correspond to physical and real-valued
    $\rho$. In shaded areas, no such solution exist. Inset: Zoom into the region $1\leq g_{\text{unc}}^{(2)}(0)\leq 2$, where the uncorrelated light source is assumed to be a mixture of coherent and completely incoherent light, and thermal light.} 
  \label{fig:bounds_plots}
\end{figure}

We can now further assume that the nature of the uncorrelated light source
is some mixture of coherent and completely incoherent light ($g^{(2)}(0)=1$),
and thermal light ($g^{(2)}(0)=2 $). This constrains the second-order
photon correlation of the uncorrelated light:
\begin{equation}
    1\leq g_{\text{unc}}^{(2)}(0)\leq 2.
    \label{eqn:g2_bounds}
\end{equation}
We impose these bounds in Eqn.~\ref{eqn:g2x_mixture_extractable},
and extract the bounds to the fraction of \DIFdelbegin \DIFdel{coherent light of }\DIFdelend \DIFaddbegin \DIFadd{light intensity in }\DIFaddend the brightest coherent emission $\rho$   
with an upper bound,
\begin{equation}
     \rho \leq \sqrt{2 - 2\,g^{(2X)}(0)},
 \label{eqn:fraction_upper_bound}
\end{equation}
and a lower bound,
\begin{equation}
    \rho \geq  \begin{cases}\frac{1}{2}+\frac{1}{2}\sqrt{3-4\,g^{(2X)}(0)}, &\text{for}\, \frac{1}{2} \leq g^{(2X)}(0) \leq \frac{3}{4} \\
    2-2\,g^{(2X)}_{\text{mix}}(0), &\text{for}\, \frac{3}{4} \leq g^{(2X)}(0) \leq 1\end{cases}\,,
 \label{eqn:fraction_lower_bounds}
\end{equation}
with $g^{(2X)}_{\text{mix}}(0)$ ranging from $1/2$ for fully coherent light, 
to 1 for fully incoherent light.

In practice, these two bounds for $\rho$ are
quite tight, and allow to extract a fraction of coherent light in an
experiment with a small uncertainty.

% The conditional expressions of Eqn.~\ref{eqn:fraction_lower_bounds} 
% is due to the minimum allowed value of $g_\text{unc}^{(2)}(0)$ 
% such that the solution of Eqn.~\ref{eqn:g2x_mixture_extractable} 
% gives a real valued $\rho$ 
% for $\frac{3}{4} \leq g^{(2X)}_{\text{mix}}(0) \leq 1$ (see inset Fig.~\ref{fig:bounds_plots})
% The fraction of coherent light in the mixture, $\rho$, can then be extracted
% from $g^{(2X)}(0)$ via
% \begin{equation}
%  \rho = 2\left(1-g^{(2X)}(0)\right)\,.
%  \label{eqn:extract}
% \end{equation}

% \textbf{Version 2}
% Physical boundaries requires $0\leq\rho\leq1$,
% $g^{(2)}_{\text{unc}}\geq0$ and
% $g^{(2X)}_{\text{mix}}\geq0$.
% From Eqn.~\ref{eqn:g2x_mixture_extractable},
% we find the boundaries of $g^{(2)}_{\text{unc}}(0)$,
% and $\rho$ for a given $g^{(2X)}_{\text{mix}}(0)$.
% \begin{equation}
%   \begin{aligned}
%   g^{(2)}_{\text{unc}}(0) \leq 3 +\frac{1}{1-2 g^{(2X)}_{\text{mix}}(0)}, g^{(2X)}_{\text{mix}}(0) < \frac{1}{2}\\
%   g^{(2)}_{\text{unc}}(0) \leq 0 , g^{(2X)}_{\text{mix}}(0) = \frac{1}{2}\\
%   g^{(2)}_{\text{unc}}(0) \geq 3 +\frac{1}{1-2 g^{(2X)}_{\text{mix}}(0)}, g^{(2X)}_{\text{mix}}(0) > \frac{1}{2}\\
%   \end{aligned}
% \end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experiment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In our experiment,
we measure interferometric photon correlations of light 
emitted from a temperature-stabilised distributed feedback laser diode 
with a central wavelength around 780\,nm.


% The laser diode was temperature stabilised to 21.506$^\circ C$ precision.
% and driven by a laser driver with mA precision.
%The interferometric photon correlation measurements were performed
%over a range of $I_L$ around its threshold, covering a region where the laser diode is expected to emit a statistical mixture of coherent and incoherent light, 
%and a region above threshold where coherent light emission at two frequencies were observed.

The setup is shown in Fig.~\ref{fig:setup}. Interferometric photon
correlations are obtained from an asymmetric Mach-Zehnder interferometer, formed by 50-50 fibre beamsplitters
%for splitting and recombination of the beam,
and a propagation delay $\Delta$ of about 900\,ns  through a 180\,m long
single mode optical fibre in one of the arms.
Photoevents at each output port of the interferometer were detected with
actively quenched silicon single photon avalanche photo diodes (APD). 
%with a timing resolution lower than 50\,ps.
The detected photoevents were time-stamped using a timetagger with a
resolution of 2\,ns for an integration time $T$.

The correlation function
$g^{(2X)}$ is extracted through histogramming all time differences $t_2-t_1$
between detection event pairs in the inverval $T$ numerically\DIFdelbegin \DIFdel{, which allows for a clean normalization. 
}\DIFdelend \DIFaddbegin \DIFadd{. 
%DIF >  which allows for a clean normalization. 
}\DIFaddend The resulting correlation is 
fitted to a two-sided exponential function, 
\begin{equation}
 g^{(2X)}(t_2 - t_1) = 1 -  A\cdot\exp\left(-\frac{\abs{t_2 - t_1}}{\tau_c}\right)\,,
 \label{eqn:fit}
\end{equation} 
where $\tau_c$ is the characteristic time constant of the coherent light,
and $A$ is the amplitude of the dip.
The value of $g^{(2X)}(0)$ is the extracted from the fit as $1-A$.
Examples of measured correlation functions and corresponding fits for
different laser powers are shown in Fig.~\ref{fig:sample_plots}.
\begin{figure}
    \begin{center}
    \includegraphics[width=\linewidth]{./figures/sample_g2x_plots/sample_plots.eps}    
    \end{center}
  \caption{Interferometric photon correlations $g^{(2X)}$ for different
    laser currents $I_L$, extracted from a histogram of photodetector time
    differences (green symbols). The error range at a specific time bin
    indicates an expected uncertainty according to a Poissonian counting
    statistics. The black solid lines show a fit to  Eqn.~\ref{eqn:fit},
    resulting in values for $A$ (from top to bottom) of
    $-0.0006\pm0.0003$, $0.326\pm0.008$, $0.455\pm0.002$, respectively. } 
  \label{fig:sample_plots}
\end{figure}

% From Eqn.~\ref{eqn:extract}, the corresponding fraction of coherent light is
% simply given by
% \begin{equation}
%   \rho=2A\,.
%   \label{eqn:cfraction}
% \end{equation}
%is extracted using Eqn.~\ref{eqn:extract}.
% the value at zero-time difference $g^{(2X)}(0)$ is obt
% The  is then calculated from $g^{(2X)}(0)$



%%%%%%%%%%%%%%%%%%%%
\subsection{Tranisition from incoherent to coherent light}
%%%%%%%%%%%%%%%%%%%%
A transition from incoherent to coherent emission
is expected as the laser current is increased 
across the lasing threshold of the laser.
We identify the lasing threshold of a laser diode $I_{L,th}$, 
by measuring the steepest increase of optical power
with the laser current (see
Fig~\ref{fig:power_vs_current}). 
For our diode, we find $I_{L,th}=37$\,mA. 
\begin{figure}
    \begin{center}
    \includegraphics[width=\linewidth]{./figures/power_vs_current/power_vs_current.eps}    
    \end{center}
  \caption{Measured laser power against laser current $I_L$. The sharpest change in
    current was measured at $I_{L,th}=37$\,mA, indicating the threshold current (dashed
    line). 
  %Purple, right axis: Second-order difference quotient of power with respect to current,
  } 
  \label{fig:power_vs_current}
\end{figure}

To observe the transition from incoherent to coherent emisssion,
we measure
the fraction of \DIFdelbegin \DIFdel{coherent }\DIFdelend light intensity 
of the brightest coherent component \DIFdelbegin \DIFdel{in the light field }\DIFdelend $\rho$ 
%DIF >  in the light field
,
at different laser current $I_{L}$
across the lasing threshold,
extracted from $g^{(2X)}$ measurements.
The interferometric photon correlation measurements as a function of the laser current 
\DIFdelbegin \DIFdel{around the lasing threshold }\DIFdelend %DIF >  around the lasing threshold 
are shown in
Fig.~\ref{fig:fraction_vs_current} (top part). 
The amplitude of the dip is then extracted by fitting these
measurements to Eqn.~\ref{eqn:fit},
from which the upper bound and lower bound of 
the fraction of \DIFdelbegin \DIFdel{coherent }\DIFdelend light from the most dominant coherent emission $\rho$ 
is extracted
(see Fig.~\ref{fig:fraction_vs_current},
middle part).

From the fit,
the fraction $\rho$ of coherent emission 
remains near 0 when operated below threshold.
Above the lasing threshold at 37\,mA,
$\rho$ increases with $I_L$
% , increasing trend in 
% the fraction of coherent light emitted from the laser diode
in a phase-transition manner,
reaching $\rho=0.986$ ($90\%$ confidence interval: $0.982$ to $0.989$) at
$I_L=120$\,mA.
This agrees with the expectation that the emission of the laser diode is 
increasingly dominated by stimulated emission past the lasing
threshold\cite{Haug1967, Haken1981}.

Near the lasing threshold,
a tight upper and lower bound of $\rho$ is observed,
in agreement with the expectation that
emission of light with a \DIFdelbegin \DIFdel{statistical }\DIFdelend %DIF > statistical
mixture of coherent and incoherent light
from the laser diode is expected when operating it near its lasing
threshold \cite{Haug1967, Haken1981}. 


\begin{figure}
    \begin{center}
    \includegraphics[width=\linewidth]{./figures/general_trend/general_trend_g2x_tau_c_rho_vs_current.eps}
    \end{center}
  \caption{Top:
      Interferometric photon corrrelations $g^{(2X)}$ for different \DIFdelbeginFL \DIFdelFL{drive
      }\DIFdelendFL \DIFaddbeginFL \DIFaddFL{operating
      }\DIFaddendFL currents $I_L$. 
      Middle: Corresponding upper bound of
      fraction $\rho$ of coherent light (red) extracted via 
      Eqn.~\ref{eqn:fraction_upper_bound_A},
      and the lower bound (blue) extracted via Eqn.~\ref{eqn:fraction_lower_bounds_A}
      from $g^{(2X)}(0)$.
      The dip in $\rho$ is a result of emission at multiple \DIFdelbeginFL \DIFdelFL{chop }\DIFdelendFL \DIFaddbeginFL \DIFaddFL{chip }\DIFaddendFL modes as explained in Section~\ref{subs:mixture}. 
      The inset shows the extracted bounds for $\rho$ at finer steps of laser current near the lasing threshold.
      Bottom: Coherence time of coherent light $\tau_{c}$ extracted from $g^{(2X)}$. 
      The dashed line indicates the threshold current $I_{L,th}=37$\,mA.
    % , where a sharp change in  optical power is observed.
  } 
  \label{fig:fraction_vs_current}
\end{figure}


% In the proximity of this injection current, 
% we expect a possibility that the emitted light from the laser diode to be a statistical mixture of stimulated and spontaneous emission.

% Sample interferometric photon correlation measurements of the emitted light for
% the laser current $I_L$ above and below the threshold are shown in Fig.~\ref{fig:sample_plots}.

% \begin{figure}
%     \begin{center}
%     \includegraphics[width=\linewidth]{./figures/g2x_vs_current/g2x_rho_vs_current.eps}    
%     \end{center}
%   \caption{Top: Interferometric photon corrrelations $g^{(2X)}$, measured for a
%     range of laser currents around the lasing threshold. Bottom: Corresponding upper bound of
%     fraction $\rho$ of coherent light (red) extracted via 
%     Eqn.~\ref{eqn:fraction_upper_bound_A},
%     and the lower bound (blue) extracted via Eqn.~\ref{eqn:fraction_lower_bounds_A}
%     from $g^{(2X)}(0)$.
%     The dashed line indicates the lasing threshold at $I_{L,th}= 37$\,mA.
%     % , where a sharp change in  optical power is observed.
%   } 
%   \label{fig:fraction_vs_current}
% \end{figure}


The coherence time of the coherent light $\tau_{c}$ can also be extracted
from fitting $g^{(2X)}$ measurements
to Eqn.~\ref{eqn:fit} (bottom Fig.~\ref{fig:fraction_vs_current}\DIFaddbegin \DIFadd{)}\DIFaddend . 
We observe that the coherence time increases 
with the current after the threshold current, 
before reaching a steady value between 300 to 350\,ns.
The increase of coherence time correspond to a narrowing of the emission linewidth,
agrees with predictions from laser theory,
that line narrowing is expected with an increase in pumping of the laser 
(here an increase in laser current)\cite{Haken1981}.
An oscillation of the coherence time is also observed starting at \DIFdelbegin \DIFdel{the about }\DIFdelend 66\,mA, 
with a periodicity of about 6\,mA.


%%%%%%%%%%%%
\subsection{Light statistics near a mode hop}\label{subs:mixture}
% {\bf suggestion for section title: Light statistics near a mode hop}
%%%%%%%%%%%%%
% We demonstrate our technique on stimulated emission at multiple frequencies.
Above the lasing threshold, the laser oscillates at different longitudinal
modes for different laser currents. The technique to extract the fraction of
coherent and incoherent light allows to investigate the behavior also in the
\DIFdelbegin \DIFdel{transition regime between }\DIFdelend \DIFaddbegin \DIFadd{regime where }\DIFaddend oscillation on different longitudinal lasing modes
\DIFaddbegin \DIFadd{is observed}\DIFaddend .

\DIFdelbegin \DIFdel{For this,
we }\DIFdelend \DIFaddbegin \DIFadd{In measuring the proportion of coherent light $\rho$ for different operating currents,
we observed a reduction in $\rho$ over a range of currents above the lasing threshold (Fig.~\ref{fig:fraction_vs_current} middle). 
To further investigate,
%DIF >  For this, 
we }\DIFaddend measured the spectrum of light emitted by the laser
diode \DIFdelbegin \DIFdel{at different currents
 above the lasing threshold 
}\DIFdelend \DIFaddbegin \DIFadd{over this range of currents
 %DIF >  above the lasing threshold 
}\DIFaddend %defined by the point of sharpest increase in its power as its operating current is increased.
%The spectrum of the laser emission was measured
with an optical spectrum analyser based on a Michelson interferometer with a
spectral resolution of 2\,GHz (Bristol 771B-NIR).
\DIFdelbegin \DIFdel{The laser diode emitted light into two distinct narrow spectral bands 
with a changing power ratioin a diode current range between }\DIFdelend \DIFaddbegin \DIFadd{At laser currents between }\DIFaddend 49.0\,mA and 52.4\,\DIFdelbegin \DIFdel{mA}\DIFdelend \DIFaddbegin \DIFadd{mA,
two distinct narrow spectral bands of light were emitted by the laser diode
with a varying power ratio,
suggesting emission in two chip modes}\DIFaddend . 
Outside this window, only one of the modes was present.
Below 49.0\,mA, the laser emission was centered around 780.07\,nm, above
52.4\,mA around 780.34\,nm,
% The linewidths of both chip modes are found to be about 2\,GHz suggesting that
% the spectral measurements were limited by the spectral resolution of 2\,GHz of
% the spectrum analyser. {\bf CK: I think we should remove this sencence, as we
  % know from the coherence time measurement that the line width is
  % significantly shorter than that, so a reader will not learn anything from
  % mentioning this mesaurement.}

\begin{figure}
    \begin{center}
    \includegraphics[width=\linewidth]{./figures/two_mode_vs_current/bristol_rho_vs_current.eps}    
    \end{center}
  \caption{Different chip modes of the laser diode are excited for different
    currents, resulting in a reduction of the $g^{(2X)}$ signature in a mode
    competition regime. Top: Power ratios $r_{\alpha,\beta}$ as a function of
    current for the chip modes $\alpha$ and $\beta$ emitting in narrow bands
    around 780.07\,nm (solid squares) and 780.34\,nm (hollow circles), respectively.  Bottom:
    Upper bound of
    fraction $\rho$ of coherent light (red) extracted via 
    Eqn.~\ref{eqn:fraction_upper_bound_A},
    and the lower bound (blue) extracted via Eqn.~\ref{eqn:fraction_lower_bounds_A}
    from $g^{(2X)}(0)$.
  } \label{fig:two_mode_vs_current}
\end{figure}

The power fractions $r_{\alpha, \beta}$ of these two chip modes $\alpha$ and
$\beta$ \DIFaddbegin \DIFadd{emitting respectively at powers $P_{\alpha,\beta}$ }\DIFaddend near this transition,
\begin{equation}
  r_{\alpha,\beta}=\frac{P_{\alpha,\beta}}{P_{\alpha} + P_{\beta}}\,,
 \label{eqn:ratio}
\end{equation} 
undergo a \DIFdelbegin \DIFdel{nearly linear transition 
}\DIFdelend \DIFaddbegin \DIFadd{transition 
from a chip mode with emission wavelength centered 
around 780.07\,nm to another chip mode
with centered around 780.34\,nm }\DIFaddend (see top traces of
Fig.~\ref{fig:two_mode_vs_current}).

We measured $g^{(2X)}$ in the same transition regime \DIFdelbegin \DIFdel{and extract }\DIFdelend \DIFaddbegin \DIFadd{at finer current steps and extracted }\DIFaddend the
proportion of coherent light from the \DIFdelbegin \DIFdel{most dominant }\DIFdelend \DIFaddbegin \DIFadd{brightest }\DIFaddend coherent emission 
$\rho$ as described above (see
Fig.~\ref{fig:two_mode_vs_current}, bottom trace). 
In the transition regime, the fraction $\rho$ of coherent emission extracted
\DIFdelbegin \DIFdel{this way }\DIFdelend \DIFaddbegin \DIFadd{from $g^{(2X)}$ }\DIFaddend decreases, when there is emission at multiple chip modes, 
and increases again when the emission approaches a single chip mode. 
\DIFdelbegin \DIFdel{This }\DIFdelend \DIFaddbegin \DIFadd{Based on our model of the light field in Eqn.~\ref{eqn:mixture_field},
this }\DIFaddend can be interpreted as the light of one emission band 
being uncorrelated to the light of the other emission band,
although the light in each band is coherent with itself. 

%%%%%%%%%%%%%%%%%%%%
% \subsection{Coherent emission trend below and above lasing threshold}
%%%%%%%%%%%%%%%%%%%%


% \begin{figure}
%     \begin{center}
%     \includegraphics[width=\linewidth]{./figures/general_trend/general_trend_g2x_rho_vs_current.eps}
%     \end{center}
%   \caption{Top: Interferometric photon corrrelations $g^{(2X)}$, measured from 1 to 120\,mA, in 1\,mA steps. Bottom: Corresponding upper bound of
%     fraction $\rho$ of coherent light (red) extracted via 
%     Eqn.~\ref{eqn:fraction_upper_bound_A},
%     and the lower bound (blue) extracted via Eqn.~\ref{eqn:fraction_lower_bounds_A}
%     from $g^{(2X)}(0)$.
%     The dip in $\rho$ is a result of emission at multiple chip modes as explained in Section~\ref{subs:mixture}. 
%     The dotted line indicates the lasing threshold of 37\,mA.
%     % , where a sharp change in  optical power is observed.
%   } 
%   \label{fig:general_fraction_vs_current}
% \end{figure}

% Bristol spectrum Data is in C:\Users\nicho\Google Drive\Storage\Academic\Lab_Projects\thermal_compression\group_meetings\DFB-0780-080_measurements\2022-09-08_fibre_cross_normal_g2


%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%

We presented a method to extract the proportion of coherent light emitted by a laser diode without the use of spectral filters,
using interferometric photon correlations.
% at different injection currents from interferometric photon correlation measurements.
As a demonstration, we measured
interferometric photon correlations of light emitted from a laser diode over a range of operating currents\DIFdelbegin \DIFdel{near the lasing threshold,
 }\DIFdelend \DIFaddbegin \DIFadd{,
 %DIF >  near the lasing threshold,
}\DIFaddend and extracted the proportion of \DIFdelbegin \DIFdel{coherent light }\DIFdelend \DIFaddbegin \DIFadd{light intensity }\DIFaddend emitted from the brightest coherent emission,
showing an increase in proportion of coherent light emission as the operating current was increased past the lasing threshold.
% statistical mixture of coherent light with light from spontaneous emission
% is emitted.
We also used this technique to characterize the coherence of emission in a
\DIFdelbegin \DIFdel{transition regime }\DIFdelend \DIFaddbegin \DIFadd{mode hop }\DIFaddend between longitudinal modes above the lasing threshold,
and find a reduction of the fraction of coherent light there, suggesting that
the two longitudinal modes can be viewed as independent and mutually
incoherent coherent emissions.
Apart from the characterisation of lasers, 
this method 
% that quantifies the proportion of coherent light in a light source by sampling the light at low intensities 
may also be useful in practical applications of some continuous-variable quantum key distribution protocols~\cite{Lodewyck2005,Weedbrook2012},
where the noise of a coherent state source such as a laser,
may need to be characterised~\cite{Shen2009,Usenko2010,Shen2011}.


% APPPENDIX

\appendix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Interferometric photon-correlation for a mixture of light fields}\label{app:g2x_mixture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We show here in further detail the derivation of Eqn.\ref{eqn:g2x_mixture_extractable},
by calculating the interferometric photon correlation 
using the model light field in Eqn.~\ref{eqn:mixture_field}.

The evaluation of $g^{(2X)}$ via Eqn.~\ref{eqn:expansion} 
requires the conventional second-order photon correlation function \DIFdelbegin \DIFdel{$g^{(2)}(t_1 -t_2)=
\langle {E^{*}(t_{1})E^{*}(t_{2}+\Delta)E(t_{2})E(t_{1}+\Delta)} \rangle$}\DIFdelend \DIFaddbegin \DIFadd{$g^{(2)}(t_1 -t_2)=
\langle {E^{*}(t_{1})E^{*}(t_{2})E(t_{2})E(t_{1})} \rangle$}\DIFaddend ,
and the last \DIFdelbegin \DIFdel{terms }%DIFDELCMD < {\bf %%%
\DIFdel{(CK: how many?)}%DIFDELCMD < } %%%
\DIFdelend \DIFaddbegin \DIFadd{two terms }\DIFaddend in Eqn.~\ref{eqn:expansion} containing negative signs.
%
For the light field mixture Eqn.~\ref{eqn:mixture_field},
its conventional second-order photon correlation function is given by
\begin{equation}
  \DIFdelbegin %DIFDELCMD < \begin{aligned}
%DIFDELCMD <       g_{\text{mix}}^{(2)}(t_{2}-t_{1})=\\
%DIFDELCMD <      =  &\rho^{2}\,g_{\text{coh}}^{(2)}(t_{2}-t_{1}) + (1-\rho)^{2}\,g_{\text{unc}}^{(2)}(t_{2}-t_{1}) \\
%DIFDELCMD <     & + 2\rho(1-\rho) \left[1 + \Re[g_{\text{coh}}^{(1)}(t_{2}-t_{1})\,g_{\text{unc}}^{(1)*}(t_{2}-t_{1})\right] \\
%DIFDELCMD <   \end{aligned}%%%
\DIFdelend \DIFaddbegin \begin{aligned}
      g_{\text{mix}}^{(2)}&(t_{2}-t_{1})=\\
     =  &\rho^{2}\,g_{\text{coh}}^{(2)}(t_{2}-t_{1}) + (1-\rho)^{2}\,g_{\text{unc}}^{(2)}(t_{2}-t_{1}) \\
    & + 2\rho(1-\rho) \left[1 + \Re[g_{\text{coh}}^{(1)}(t_{2}-t_{1})\,g_{\text{unc}}^{(1)*}(t_{2}-t_{1})\right] \\
  \end{aligned}\DIFaddend ,
  \label{eqn:g2_mixture}
% \end{split}
\end{equation}
where $g^{(1)}$ is the first-order field correlation function for the respective component light fields, 
$g^{(1)*}$ its complex conjugate,
and $\Re[\cdots]$ extracts the real part of its argument.

The last terms in Eqn.~\ref{eqn:expansion} can be written as
\begin{equation}
  \begin{aligned}
  &\langle {E_{\text{mix}}^{*}(t_{1})E_{\text{mix}}^{*}(t_{2}+\Delta)E_{\text{mix}}(t_{2})E_{\text{mix}}(t_{1}+\Delta)} \rangle \\
        =  &\rho^{2}\,\abs{g_{\text{coh}}^{(1)}(t_{2}-t_{1})}^{2} +
        (1-\rho)^{2}\abs{g_{\text{unc}}^{(1)}(t_{2}-t_{1})}^{2} \\
        + &2\rho(1-\rho)\,\Re[g_{\text{coh}}^{(1)}(t_{2}-t_{1})\,g_{\text{unc}}^{(1)*}(t_{2}-t_{1})] \\
        + &2\rho(1-\rho)\,\Re[g_{\text{coh}}^{(1)}(\Delta)\,g_{\text{unc}}^{(1)*}(\Delta)]\\
        %      +  &\rho^{2}\,\abs{g_{\text{coh}}^{(1)}(\Delta)}^{2} +
        %      (1-\rho)^{2}\abs{g_{\text{unc}}^{(1)}(\Delta)}^{2} \\
  \end{aligned},
  \label{eqn:cross_terms_mixture}
% \end{split}
\end{equation}
where $g^{(1)}(\Delta)\approx0$ 
for our experimental situation of the propagation delay $\Delta$ significantly larger than the coherence times of the respective light sources.
Note that all terms in Eqn.~\ref{eqn:cross_terms_mixture} are
real-valued.  
%its complex conjugate equals to itself.
% and hence the last expression in Eqn.~\ref{eqn:expansion} 
% also evaluates to Eqn.\ref{eqn:cross_terms_mixture}. 

With this, the interferometric photon correlation at
zero time difference is given by
\begin{equation}
  \DIFdelbegin %DIFDELCMD < \begin{aligned}
%DIFDELCMD <     g_{\text{mix}}^{(2X)}(0)=\\
%DIFDELCMD <          \frac{1}{4}[&g_{\text{mix}}^{(2)}(\Delta)+ g_{\text{mix}}^{(2)}(-\Delta)\\ 
%DIFDELCMD <         &+2(\rho^{2}\,g_{\text{coh}}^{(2)}(0) + (1-\rho)^{2}\,g_{\text{unc}}^{(2)}(0) + 2\rho(1-\rho)) \\
%DIFDELCMD <         &-2(\rho^{2}\,\abs{g_{\text{coh}}^{(1)}(0)}^{2} + (1-\rho)^{2}\abs{g_{\text{unc}}^{(1)}(0)}^{2})]\,.
%DIFDELCMD <   \end{aligned}%%%
\DIFdelend \DIFaddbegin \begin{aligned}
    g_{\text{mix}}^{(2X)}&(0)=\\
         = \frac{1}{4}[&g_{\text{mix}}^{(2)}(\Delta)+ g_{\text{mix}}^{(2)}(-\Delta)\\ 
        &+2(\rho^{2}\,g_{\text{coh}}^{(2)}(0) + (1-\rho)^{2}\,g_{\text{unc}}^{(2)}(0) + 2\rho(1-\rho)) \\
        &-2(\rho^{2}\,\abs{g_{\text{coh}}^{(1)}(0)}^{2} + (1-\rho)^{2}\abs{g_{\text{unc}}^{(1)}(0)}^{2})]\,.
  \end{aligned}\DIFaddend 
  \label{eqn:g2x_mixture_raw}
% \end{split}
\end{equation}

We make further assumptions that 
the propagation delay in the interferometer $\Delta$ is significantly longer than the
coherence time scale of the light source, 
such that $g_{\text{mix}}^{(2)}(\pm\Delta)\approx1$,
the interferometer has good visibility,
such that \DIFdelbegin \DIFdel{$g^{(1)}(0)\approx1$}\DIFdelend \DIFaddbegin \DIFadd{$\abs{g^{(1)}(0)}\approx1$}\DIFaddend ,
and for the second order photon correlation of the coherent light field is $g^{(2)}_\text{
coh}(0)=1$. 
The evaluation of these assumptions in Eqn.~\ref{eqn:g2x_mixture_raw}
leads to the relationship shown in Eqn.~\ref{eqn:g2x_mixture_extractable}.


%%%%%%%%%%%%%%%%%%%%%%%%
\section{Boundaries of physically meaningful combinations of interferometric
  correlations in a mixture}\label{app:boundaries}
%%%%%%%%%%%%%%%%%%%%%%%%
Assuming a binary mixture of the light field as per
Eqn.~\ref{eqn:mixture_field}, the interferometric correlation of the mixture,
$g_{\text{mix}}^{(2X)}(0)$, and the conventional second order correlation of
the incoherent light, $g_{\text{unc}}^{(2)}(0)$, at zero time difference are
constrained by relation Eqn.~\ref{eqn:g2x_mixture_extractable}. Further
assuming the physical requirement $0 \leq \rho \leq 1$ for the coherent light
fraction gives a lower bound for $g_{\text{unc}}^{(2)}(0)$,
\begin{equation}
        g_{\text{unc}}^{(2)}(0) \geq  \begin{cases}0\,,  &g_{\text{mix}}^{(2X)}(0) \leq \frac{2}{3}\\
                        3 +\frac{1}{1 - 2 g_{\text{mix}}^{(2X)}(0) }, &g_{\text{mix}}^{(2X)}(0) \in[\frac{2}{3},1]  \\
                        2g_{\text{mix}}^{(2X)}(0)\, &g_{\text{mix}}^{(2X)}(0) \geq 1 \end{cases}\,.
\label{eqn:lowerbound_g2}
\end{equation}
For $g_{\text{mix}}^{(2X)}(0) \in [0,\frac{1}{2})$, there is an upper bound
for $g_{\text{unc}}^{(2)}(0)$,
\begin{equation}
    g_{\text{unc}}^{(2)}(0) \leq 2g_{\text{mix}}^{(2X)}(0)\,.
 \label{eqn:upperbound_g2}
\end{equation}


%%%%%%%%%%%%%%%%%%%%%%%%
\section{Error propagation from fitting of $g^{(2X)}$ measurement}
%%%%%%%%%%%%%%%%%%%%%%%%
Standard techniques in propagation of uncertainties are not used,
as the expressions in Eqn.~\ref{eqn:fraction_upper_bound}-\ref{eqn:fraction_lower_bounds} 
would lead to indefinite values of uncertainties at some values of $A$.
We thus extract the upper and lower bounds of $\rho$ 
by performing a change in variables from the probability density of $A$.

The probability density of $A$ is assumed to be a normal distribution 
with a mean value and standard deviation 
respectively the value and uncertainty of $A$ extracted 
from the curve fitting of $g^{(2X)}$ to Eqn.~\ref{eqn:fit}.

The probability densities describing the upper and lower bounds of $\rho$ 
is obtained from a change of variable from $A$ by rewriting
Eqn.~\ref{eqn:fraction_upper_bound}-\ref{eqn:fraction_lower_bounds}
in terms of $A$.
The transfomation of the upper bound is
\begin{equation}
     \rho = \sqrt{2A},
 \label{eqn:fraction_upper_bound_A}
\end{equation}
and the lower bound,
\begin{equation}
    \rho =  \begin{cases}
    2A, &\text{for}\, 0 \leq A \leq \frac{1}{4}\\
    \frac{1}{2}+\frac{1}{2}\sqrt{4A-1}, &\text{for} \, \frac{1}{4} \leq A \leq \frac{1}{2}
    \end{cases}.
 \label{eqn:fraction_lower_bounds_A}
\end{equation}

We exclude non-physical values of $\rho$ 
by setting the probability density outside the domain $0 \leq \rho \leq 1$ to $0$\DIFdelbegin \DIFdel{,
to 0.
}\DIFdelend \DIFaddbegin \DIFadd{.
}\DIFaddend The probability density is renormalised by dividing over its integral.
From these probability densities of the upper bound and lower bound of $\rho$,
we compute the expectation of $\rho$ and its $90\%$ confidence interval,
which would be reported respectively as the data points
and errorbars in plots which contains measurements of $\rho$.

\bibliography{references}

% \appendix
% \section{Derivation of interferometric photon correlations for a mixture}

% \begin{equation}
%   \begin{aligned}
%   &g^{(2X)}(t_{2}-t_{1}) \\
%         =  \frac{1}{4}[&2\,g^{(2)}(t_{2}-t_{1}) + g^{(2)}(t_{2}-t_{1}+\Delta)+g^{(2)}(t_{2}-t_{1}-\Delta) \\
%         - & 2\,\Re(\langle {E^{*}(t_{1})E^{*}(t_{2}+\Delta)E(t_{2})E(t_{1}+\Delta)} \rangle)]\\
%   \end{aligned},
%   \label{eqn:expansion_2}
% % \end{split}
% \end{equation}
% where the operator $\Re$ extracts the real part of its argument.

\end{document}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:


% ANNOTATED
% \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}
% UNANNOTATED 

% 4-in-1 Plot
% \begin{figure}
%     \begin{center}
%     \includegraphics[width=\linewidth]{./figures/4_in_1_plot/four_in_one_vs_current.png}    
%     \end{center}
%   \caption{(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{figure}
