%SourceDoc ./tesis.tex
\section{Field Noise Correlation Functions}
\label{app:4B}

Here we list the nonvanishing correlation functions of the field
noise operators defined in Eq.~\eqref{eq:G-adia1}. Combining the correlation function in the Appendix \ref{app:3B} and the correlation functions \eqref{eq:f-2cf}, one obtains
\begin{subequations}
\begin{align}
\la G_{k}(t) G_{l}^{\dag}(t') \ra &= 2 \mathcal{D}_{kl}^{A} \la
\vect{L}_{k} | \vect{L}_{l} \ra\, \delta (t-t') \,,\\ 
\la G_{k}^{\dag}(t) G_{l}(t') \ra &= 2 \mathcal{D}_{kl}^{N} \la
\vect{L}_{l} | \vect{L}_{k} \ra\, \delta (t-t') \,,\\ 
\end{align}
\end{subequations}
where the diffusion constants are defined by
\begin{subequations}
\begin{align}
2 \mathcal{D}_{kl}^{A} &= \rmi (\go_{k} - \go_{l}^{*}) - \frac{V
  g^{2}}{2\gga_{\bot}^{2}} \bigl( \gga_{\|} (\Lambda - \mean{S_{z}}) - 4
\gga_{\bot} \mean{S_{0}} \bigr) \,, \\ 
2 \mathcal{D}_{kl}^{N} &= \frac{V g^{2}}{2\gga_{\bot}^{2}} \bigl(
\gga_{\|} (\Lambda - \mean{S_{z}}) + 4 \gga_{\bot} \mean{S_{1}} \bigr)\,.
\end{align}
\end{subequations}  

The correlation functions for the slowly varying noises in the
frequency domain $\tilde{G}_{k}(\go)$ are easily obtained from the
above results using $\tilde{G}_{k}(\go) = \int\! \rmd t \,
\rme^{-\rmi (\go - \go_{k})t} G_{k}(t)$. They read
\begin{subequations}
\label{eq:W-CF}
\begin{align}
\label{eq:WA-CF}
\la \tilde{G}_{k}(\go) \tilde{G}_{l}^{\dag}(-\go') \ra &= 4\pi \mathcal{D}_{kl}^{A}\la
\vect{L}_{k} | \vect{L}_{l} \ra\, \delta
(\go + \go' - \bar{\go}_{l}^{*} + \bar{\go}_{k}) \,,\\ 
\label{eq:WN-CF}
\la \tilde{G}_{k}^{\dag}(-\go) \tilde{G}_{l}(\go') \ra &= 4\pi \mathcal{D}_{kl}^{N}\la
\vect{L}_{l} | \vect{L}_{k} \ra\, \delta
(\go + \go' - \bar{\go}_{k}^{*} + \bar{\go}_{l}) \,.
\end{align}
\end{subequations}

\section{Lasing-Nonlasing Modes Correlations}
\label{app:4C}

Here we derive Eq.~\eqref{eq:ephidk}. The whole
calculation is based on the fact that both the laser phase
fluctuations and the amplitude of the below threshold modes are
Gaussian random variables. We star from
\begin{equation}
\la\exp(-\rmi\delta\phi(t)) \delta d_{k}(0)\ra = \sum_{l=0}^{+\infty}
\frac{(-\rmi)^{l}}{l!}  \la [\delta\phi(t)]^{l} \delta d_{k}(0) \ra.
\end{equation}
Employing Wick's theorem, one can show that only terms with odd values $l$ will contribute to the sum on the right hand side. Hence
\begin{equation}
\label{eq:4C1}
\la\exp(-\rmi\delta\phi(t)) \delta d_{k}(0)\ra = \sum_{n=0}^{+\infty}
\frac{(-\rmi)^{2n+1}}{(2n+1)!}  \la [\delta\phi(t)]^{2n+1} \delta
d_{k}(0) \ra.
\end{equation}
The correlation functions on the right side are evaluated upon summation of all the
$(2n+2)!/(2^{n+1} (n+1)!)$ possible contractions. For the correlation
function of order $2n+2$, this yields
\begin{equation}
\label{eq:4C2}
\la [\delta\phi(t)]^{2n+1} \delta d_{k}(0) \ra =
\frac{(2n+2)!}{2^{n}(n+1)!} \la \delta\phi^{2}(t) \ra^{n} [\la
  \delta\phi(t) \delta d_{k}(0)\ra + \la \delta d_{k}(0)
  \delta\phi(t)\ra ]\,.
\end{equation}
Substituting Eq.~\eqref{eq:4C2} into Eq.~\eqref{eq:4C1},
and after some straight forward calculation, one recovers Eq.~\eqref{eq:ephidk}.

\section{Off-Resonance Case: Henry Factor}
\label{app:4A}

In Sec.~\ref{sec:ES-SML} we assumed the laser mode to be on resonance
with the atomic medium. Here we address the modifications to our
results that arise when this assumption is dropped. As is evident from
Eqs.~\eqref{eq:xqua-2} and \eqref{eq:yqua-2}, in a detuned laser,
where $\text{Im}[W_{0}] \neq 0$, amplitude and phase fluctuations are
coupled. Hence, the evaluation of the phase fluctuations must take
into account the excess noise originating in the coupling to the
amplitude fluctuations. Following Lax \cite{Lax67}, we will show that
the consequences of detuning can be quantified by an excess noise
factor. That factor depends in a simple manner on the detuning,
provided the laser is operating far above threshold so that the
amplitude fluctuations can be neglected as compared with the
steady-state laser amplitude. This excess noise factor is
mathematically equivalent to the Henry factor \cite{Henr82} arising in
semiconductor laser, where the coupling between amplitude and phase
fluctuations also plays a significant role.

Let us then start by writing Eqs.~\eqref{eq:xqua-2} and
\eqref{eq:yqua-2} for the amplitude and phase quadrature components in
the way 
\begin{subequations}
\begin{align} 
\label{eq:xqua-3} 
\delta X_{0}(\go) &= \frac{\rmi}{2\go} \bigl(r \cos (\beta) \delta
S_{z}(\go) + \tilde{G}(\go) + \tilde{G}_{0}^{\dag}(-\go)\bigr)\,, \\
\label{eq:yqua-3} 
\delta Y_{0}(\go) &= \frac{1}{2\go} \bigl(\rmi r \sin (\beta)
\mean{d}_{0} \delta S_{z}(\go) + \tilde{G}(\go) -
\tilde{G}_{0}^{\dag}(-\go)\bigr)\,. 
\end{align} 
\end{subequations}
Thus amplitude and phase fluctuations are coupled through the atomic
population inversion fluctuations. Here the two new parameters $r$ and
$\beta$ are defined by
\begin{align}
r &= 2 \mean{d_{0}} |W_{0}|\,, \\ 
\beta &= \text{Arg}[W_{0}] =
\text{Arctan}\left( \frac{-(\nu - \bar{\gO}_{0})}{\gga_{\bot}} \right)
\,.  
\end{align}
We now define
\begin{equation}
\label{eq:eta}
\eta (\go) = \cos(\beta) \delta Y_{0}(\go) - \sin(\beta) \delta
X_{0}(\go)\,,
\end{equation}
which can be written explicitly in terms of the noise operators using
Eqs.~\eqref{eq:xqua-3} and \eqref{eq:yqua-3},
\begin{equation}
\eta(\go) = \frac{1}{2\go} \bigl( \rme^{-\rmi \beta}
\tilde{G}_{0}(\go) - \rme^{\rmi \beta} \tilde{G}_{0}^{\dagger}(\go)
\bigr) \,.
\end{equation}
We notice that the new variable $\eta$ is a linear combination of the
laser mode noise forces. Any dependence on the population inversion
fluctuation has been eliminated. For small phase fluctuations, one can
express them in terms of the phase quadrature component as $\delta
\phi(\go) = \delta Y_{0}(\go) / \mean{d_{0}}$. Using
Eq.~\eqref{eq:eta}, we may write the phase fluctuations as
\begin{equation} 
\label{eq:phi-eta}
\delta \phi (\go) = \frac{1}{\mean{d_{0}}
\cos(\beta)} \bigl( \eta(\go) + \sin (\beta) \delta X_{0}(\go) \bigr)
\simeq \frac{\eta(\go)}{\mean{d_{0}} \cos(\beta)}\,, 
\end{equation}
where the last relation holds for a laser far above threshold where
$\delta X_{0} \ll \mean{d_{0}}$. We can use the above equations to
evaluate the phase fluctuations diffusion coefficient
$[\delta\phi]_{\go}$, which ultimately determines the laser line
width. Following the same reasoning that led to Eq.~\eqref{eq:dphi-c},
one obtains, starting from Eq.~\eqref{eq:phi-eta}, the result
\begin{equation}
\label{eq:phi-dif}
[\delta\phi]_{\go} = \frac{K_{0}
\delta\go_{\text{ST}}}{\cos^{2}(\beta)\go^{2}} = \frac{K_{0} (1+\alpha^2)
\delta\go_{\text{ST}}}{\go^{2}}\,,
\end{equation}
where we have written $\cos^{2}(\beta) = (1+\alpha^{2})^{-1}$. Here,
$\alpha=(\nu-\bar{\gO}_{0})/\gga_{\bot}$ is the Henry parameter
\cite{Henr82}. According to Eq.~\eqref{eq:phi-dif} the effect on the
detuning on the laser linewidth is an enhancement by the factor
$1+\alpha^2$.

The total emission spectrum for a detuned laser is obtained following
the same reasoning that led to the emission spectrum
\eqref{eq:ColoredS}, however one should now use the following
correlation functions 
\begin{align}
\la \delta\phi^{2} (t)\ra &= \frac{1}{\cos^{2}(\beta)} \la
\delta\phi^{2} (t)\ra_{\text{tune}} \,, \\
\la \delta\phi(t) \delta d_{k}(0) \ra &= \frac{\rme^{\rmi \beta}}{\cos(\beta)} \la
\delta\phi(t) \delta d_{k}(0)\ra_{\text{tune}}\,, \\
\la \delta d_{k}(0) \delta\phi(t) \ra &= \frac{\rme^{\rmi \beta}}{\cos(\beta)} \la \delta d_{k}(0) \delta\phi(t) \ra_{\text{tune}}\,,
\end{align}
which now hold for a detuned laser and that are evaluated using the
phase fluctuations \eqref{eq:phi-eta}, and . Here $\la \dots
\ra_{\text{tune}}$ refers to the results obtained for the laser on
resonance in Sec.~\ref{sec:ES-SML}. The emission spectrum then reads
\begin{equation}
\begin{split}
\label{eq:ColoredSD}
\mathcal{S}(\go) &= \frac{\mean{I}_{\text{out}}}{\pi}
\frac{\delta\go/2}{ (\go -\mean{\gO}_{0})^{2} + \frac{1}{4}
  (\delta\go)^{2}} \\ &\quad \times \left\{1 - 8 \sum_{k\neq 0}
\text{Re} \left( \left[\frac{\frac{1}{2}\delta\go +\rmi (\go -
\bar{\gO}_{0})}{\frac{1}{2} \delta\go + \bar{\gk}_{k} -\rmi (\go -
\bar{\gO}_{k})} \right] \right.\right. \\ &\quad \times \left.\left.
\left[  \frac{\rmi (\go_{k} - \go_{0}^{*})
     (\mathcal{D}_{0k}^{N}+\mathcal{D}_{k0}^{A})\la\vect{R}_{0}|\vect{R}_{k}\ra
     \la\vect{L}_{k}|\vect{L}_{0}\ra \cos(\beta) \rme^{\rmi \beta}}{2\gk_{k} K_{0}
       (\mathcal{D}_{00}^{N}+\mathcal{D}_{00}^{A})}\right]
\right)\right\} \,,
\end{split}
\end{equation}
where the total linewidth is given by $\delta \go = K_{0}
\delta\go_{\text{ST}} / \cos^{2}(\beta)$, which includes the Petermann
enhancement factor $K_{0}$ and the Henry factor $1/\cos^{2}(\beta)$.