%SourceDoc ./tesis.tex
\chapter{Laser with Overlapping Resonances: Single-Mode Operation}
\label{ch:single-mode}

Lasers in cavities with overlapping resonances exhibit unusual noise
properties. This fact, by now extensively confirmed by experiments
\cite{Hame90,Chen96,Eijk96}, was first noticed by Petermann
\cite{Pete79} in his semiclassical analysis of the fundamental
linewidth in gain-guided semiconductor lasers. This kind of lasers
exhibit a greatly enhanced linewidth as compared with the
Schawlow-Townes value; the associated excess noise factor was later
named after Petermann. Later, Siegman\cite{Sieg89a,Sieg89b} attributed
the origin of the Petermann excess noise factor to the
nonorthogonality of the modes in cavities with overlapping resonances,
showing that excess noise is the fingerprint of optical systems with
nonorthogonal modes. Although Siegman's semiclassical theory provides
a detailed description of excess noise, to the best of our knowledge,
a complete quantum laser theory of that phenomenon was lacking. We
address this problem in this chapter.

We use the laser Langevin equations for overlapping resonances derived
in \emph{Chapter~\ref{ch:LEQ}} to study the emission power spectrum of
a single-mode laser oscillator. Excess noise emerges in our equations
as a consequence of the correlations between noise forces
corresponding to different modes. Excess noise thus is, in essence, a
multi-mode phenomenon. In order to properly account for that
phenomenon we not only consider the lasing mode but also the below
threshold modes, retaining the full multi-mode character of the
problem. In that respect our approach differs from other studies
\cite{Sieg89b} of excess noise which neglect the below threshold modes
altogether arguing that they are overwhelmed by the intensity of the
lasing mode. We show that this argument, derived from the knowledge in
standard lasers, must be reformulated when applied to lasers with
overlapping resonances. Indeed, while in standard lasers contributions
to the emission spectrum arise only from the autocorrelation functions
of the modes, additional contributions are obtained in lasers with
overlapping resonances from the nonvanishing correlations between
different modes. The autocorrelation functions are proportional to the
mode intensities, so that modes below threshold typically contribute
negligible to the output intensity. The same holds for most of the
cross-correlation contributions, namely for those that involve two
below threshold modes. There are, however, cross-correlation functions
that involve the lasing mode and a below threshold mode, which are
therefore only linear in the below threshold mode amplitude. Such
correlation functions comprise the effects of the below threshold
modes on the lasing mode noise properties. In this chapter we show
that such contributions may introduce dramatic changes to the laser
spectrum lineshape.

Our evaluation of the emission spectrum proceed by first considering
the contributions from the lasing mode. We recover a Lorentzian
lineshape with a linewidth enhanced by the Petermann factor. Next we
evaluate the contributions from cross-correlations between the lasing
mode and the below threshold modes. These contributions to the
spectrum are show to be important above threshold, where they lead to
deviations of the lineshape from Lorentzian form. Such deviations are
most important when both below threshold modes are close to threshold
and have frequencies close to the laser frequency.  Our result for the
emission spectrum can be tested in unstable cavities, but is also
important for single-mode lasers in chaotic cavities and random media.


\section{Laser Equations}

In this chapter we are mainly interested in the effects that the
nonorthogonality of the open cavity modes may have on the power
spectrum of a single-mode laser oscillation. To exhibit these effects
most clearly, we shall introduce some approximations to the laser
model presented in \emph{Chapter \ref{ch:LEQ}}. The first
approximation is to assume the atomic population inversion to be
spatially uniform\footnote{This approximation is quite reasonable for
gas lasers in which the atoms, due to the rapid thermal motion,
diffuse fast enough to destroy spatial hole burning
\cite{Dutr99,Dutr00c}. The argument, however, should be used with
caution. In order to destroy hole burning the minimum atomic velocity
is given by $v \approx D/\gl \gg \gl \gga_{\bot}$, where $D$ is the
diffusion constant and $\gl$
the light wave length. If $v$ is too large, the motion of the atoms
may lead to an inhomogeneously broadened gain line due to the Doppler
effect, a situation not covered by our laser equations.} and replace it by its average over the cavity volume,
\begin{equation}
S_{z}(t)=\frac{1}{V} \int \! \rmd\vect{r} \, S_{z}(\vect{r},t).
\end{equation} 
Physically this approximation amounts to neglecting spatial hole
burning, thereby inhibiting mode competition and effectively
allowing only for single-mode laser operation above threshold
\cite{Hake85}. In addition, we also take the pumping to be isotropic
and uniform all over the cavity volume, $\gL(\vect{r})=\gL$. The major
simplification that these approximations yield is that the modes of
the load cavity, i.e., the modes of the cavity filled with the active
medium, coincide with the modes of the empty cavity for all values of
the pumping. The second approximation consist in restricting our study
to lasers for which the dipole relaxation rate of the active medium is
much larger than both the inversion relaxation rate and the
characteristic decay rate of the open cavity modes, $\gga_{\bot} \gg
\gga_{\|},
\bar{\gk}$; this is an approximation valid for most solid-state
lasers. The atomic polarization can thus be adiabatically eliminated
from the equations of motions
\eqref{eq:leqnonH}. As a consequence, the polarization
\begin{equation}
S_{-}(\vect{r},t) = \rmi \sum_{k} \frac{ g_{k}^{R}(\vect{r})
  d_{k}(t) S_{z}(t)}{ \gga_{\bot} + \rmi (\nu - \bar{\gO}_{k})}
+ \frac{1}{\gga_{\bot}} F_{\bot}(\vect{r},t),
\end{equation}
follows instantaneously the phase  of the cavity  field.  The
frequencies  $\bar{\gO}_{k}$   are  the real    part  of  the complex
oscillation  frequencies   $\bar{\go}_{k}  =  \bar{\gO}_{k} -   \rmi
\bar{\gk}_{k}$ of the cavity modes in the  presence  of  the active
medium; they are still to be determined. Substituting this expression
into the remaining equations for the field operators and the atomic
inversion, one obtains the simplified laser equations
\begin{subequations}
\label{eq:advtic}
\begin{align}
\label{eq:advtic-f}
\dot{d}_{k}(t) &= \bigl[ -\rmi\go_{k} + W_{k} S_{z}(t) \bigr] d_{k}(t)
+ G_{k}(t), \\
\label{eq:advtic-S}
\dot{S}_{z}(t) &= \gamma_{\|} (\gL - S_{z}(t)) - \frac{2}{V}
\sum_{kl} \bigl(W_{k}^{*} + W_{l} \bigr) \la \vect{R}_{k} |
\vect{R}_{l} \ra d^{\dagger}_{k}(t) d_{l}(t) S_{z}(t) + G_{\|}(t)\,,
\end{align}
\end{subequations}
where
\begin{equation}
 W_{k}= \frac{V g^{2}}{\bigl(\gga_{\bot}+\rmi (\nu - \bar{\gO}_{k})\bigr)}.
\end{equation}
The noise terms for the field amplitudes are now given by
\begin{subequations}
\begin{align}
\label{eq:G-adia1}
G_{k}(t) &= F_{k}(t) + F_{\bot k}(t)\,, \\ 
\intertext{with}
F_{\bot k}(t) &\equiv -\frac{\rmi}{\gga_{\bot}} \int \! d\vect{r}\,
g_{k}^{L*}(\vect{r}) F_{\bot}(\vect{r},t), 
\end{align}
\end{subequations}
and the Langevin forces for the population inversion are
\begin{equation} 
\label{eq:G-adia2}
G_{\|}(\vect{r},t) =
F_{\|}(\vect{r},t) + \frac{2\rmi}{V \gga_{\bot}} \sum_{k} \int \!
\rmd\vect{r}\, \bigl[g_{k}^{R*}(\vect{r}) d^{\dagger}_{k}(t)
  F_{\bot}(\vect{r},t) - g_{k}^{R}(\vect{r})
  F_{\bot}^{\dagger}(\vect{r},t) d_{k}(t) \bigr].
\end{equation}
These new noise operators are easily shown to still provide a noise
with a white spectrum. Their second moments are evaluated using the
definitions \eqref{eq:f-2cf}, \eqref{eq:G-adia1}, \eqref{eq:G-adia2},
and the correlations functions for the atomic noise forces listed in
Appendix \ref{app:3B}. The nonvanishing correlation functions for the field
noise forces $G_{k}(t)$ are given in the Appendix \ref{app:4B}.


\section{Single-Mode Laser Operation}

The semiclassical laser equations obtained from Eq.~\eqref{eq:advtic}
by averaging over the state of the system were shown by
Hacken\cite{Hake85} to have only a single stable steady--state
solution for all values of pumping above the threshold. This solution
corresponds to a selfsustained laser oscillation in a single-mode of
the open cavity. The existence of this single solution is a
consequence of neglecting spatial hole burning, as the lasing mode
deflects the atomic inversion uniformly over the cavity inhibiting
other modes from lasing. In what remains of this chapter, we use the
laser equations
\eqref{eq:advtic} to determine the emission spectrum in the lasing
regime.


\subsection{Steady-State Solution}

In the lasing regime only a single cavity mode, say the mode with
$k=0$, is above threshold, while all other modes, with $k\neq 0$,
remain below threshold. We decompose the field amplitudes and the
atomic population inversion into their classical steady-state value
and their quantum fluctuations\footnote{That decomposition is always
possible whenever the number of atoms inside the cavity is large
$\mathcal{N} \gg 1$ and the medium is distributed over the whole
cavity. Then, the field intensity and the atomic variables scale
proportional to $\mathcal{N}$ while the noise forces scale as
$\sqrt{\mathcal{N}}$. Accordingly the noise terms become much smaller
than the deterministic terms in the limit $\mathcal{N}\gg 1$
\cite{Davi96,Gard00}.},
\begin{subequations}
\begin{align}
d_{k}(t) &= ( \mean{d_{k}} + \delta d_{k}(t)) \rme^{-\rmi\bar{\go}_{k} t}, \\
S_{z}(t) &= \mean{S_{z}} + \delta S_{z}(t).
\end{align}
\end{subequations}
In a single-mode laser there is a selfsustained oscillation in the laser mode with frequency  $\bar{\go}_{0} = \bar{\gO}_{0}$. Sufficiently far above threshold its amplitude
is much in excess of the amplitudes in any of the other modes
that remain below threshold. The steady-state field amplitudes are given by $\mean{d_{0}}\neq0$, and $\mean{d_{k}}=0$
for $k\neq0$. Consequently, the atomic medium only ``sees'' the lasing
mode, and the classical steady-state solutions for the laser mode
amplitude and the atomic population satisfy the following equations
\begin{subequations}
\begin{align}
\label{eq:ss-cond1}
\bigl[\rmi (\bar{\gO}_{0} - \gO_{0}) - \gk_{0} + W_{0} \mean{S_{z}}
\bigr] \mean{d_{0}} &= 0, \\  
\label{eq:ss-cond2}
\gga_{\|} (\gL - \mean{S_{z}}) - \frac{4}{V}
  \mathcal{L}_{0} \mean{I}_{0} \mean{{S}_{z}} &= 0.
\end{align}
\end{subequations}
Here we have introduced the Lorentzian function
$\mathcal{L}_{0}=\text{Re}[W_{0}]$ and the steady-state field
intensity $\mean{I_{0}} = \la \vect{R}_{0} | \vect{R}_{0} \ra
|\mean{d_{0}}|^2$.  Separating the expression inside the brackets in
Eq.~\eqref{eq:ss-cond1} into its real an imaginary parts, we obtain
the steady-state value for the population inversion and the laser
oscillation frequency,
\begin{align}
\label{eq:Sz-st}
\mean{S_{z}} &= \frac{\gk_{0}}{\mathcal{L}_{0}} = \Lambda_{\text{th}}\,, \\
\bar{\gO}_{0} &= \frac{\gga_{\bot} \gO_{0} + \gk_{0} \nu}{\gga_{\bot} +
\gk_{0}} \simeq \gO_{0} + \frac{\gk_{0}}{\gga_{\bot}} \nu\,,
\end{align} 
which coincide with the threshold conditions Eqs.~\eqref{eq:km-th} and
\eqref{eq:lth} obtained in the study of the below
threshold problem in Sec.~\ref{sec:laser-bt}. The steady-state
solution for the field intensity follows from Eq.~\eqref{eq:ss-cond2},
\begin{equation}
\mean{I}_{0} = \frac{V \gga_{\|}}{4 \mathcal{L}_{0}} \left(
\frac{\gL}{\mean{{S}_{z}}} - 1 \right).
\end{equation}
We notice that the mean optical phase of the laser field does not
enter the steady-state solutions. This is related with the fact
that in the steady-state the optical phase of the laser mode is
randomly distributed between $0$ and $2\pi$. We can then conveniently
choose the arbitrary mean value of the phase to be equal to $0$, since
then the steady-state field amplitude $\mean{d_{0}}$ becomes real.


\subsection{Dynamics of the Field Fluctuations}

The equations of motion for the quantum fluctuations follow upon
linearization of the laser equations \eqref{eq:advtic} around the
steady--state solution. The result is
\begin{subequations}
\label{eq:fluctua-t}
\begin{align}
\label{eq:dkbelow}
\dot{\delta d_{k}}(t) &= \tilde{G}_{k}(t); \qquad (k\neq
0) \,,\\
\dot{\delta d_{0}}(t) &= W_{0} \mean{d_{0}} \delta S_{z}(t) +
\tilde{G}_{0}(t)\,, \\
\dot{\delta S_{z}}(t) &= -\gga_{\|}\frac{\Lambda}{\mean{S_{z}}} \delta
S_{z}(t) - \frac{2 \mean{d_{0}}\mean{S_{z}}}{V} \sum_{k} \left[
(W_{0}+W_{k}^{*}) \la \vect{R}_{k}|\vect{R}_{0} \ra \delta d_{k} +
\text{ H.c. } \right] + G_{\|}(t)\,.
\end{align}
\end{subequations} 
Here we used $\mean{d_{0}} = \mean{d_{0}^{*}}$ and defined
$\tilde{G}_{0}(t)= \rme^{\rmi \bar{\gO}_{0} t} G_{0}(t)$ and
$\tilde{G}_{k} (t) = \rme^{\rmi \bar{\go}_{k} t} G_{k}(t)$. We note
that the equations of motion for the field modes below threshold
decouple and can be solve immediately; they correspond to the
equations of motion of a linear amplifier studied in
Sec.~\ref{sec:laser-bt}. It follows that the complex oscillation
frequencies $\bar{\go}_{k}$ of the below threshold modes are solutions
of the equation $\rmi(\bar{\go}_{k} -\go_{k}) + W_{k}
\mean{S}_{z} =0$. The linear set of equations \eqref{eq:fluctua-t} can
be solved exactly by Fourier transformation. We define the field
fluctuation operators in the frequency domain through
\begin{equation}
\delta d_{k}(t) = \frac{1}{2\pi} \int\! \rmd\go\, \rme^{\rmi \go t}
\delta d_{k}(\go),
\end{equation}
and analogous for the population inversion operator and the noise
operators. For notational simplicity we again use the same symbol for both
members of the Fourier transformed pair. The new variables obey the
algebraic set of equations
\begin{subequations}
\begin{align}
\label{eq:dkfluct}
-\rmi \go \delta d_{k}(\go) &= \tilde{G}_{k}(\go) \,; \qquad (k\neq 0)\,, \\
\label{eq:fluct-w1}
-\rmi \go \delta d_{0}(\go) &= W_{0} \mean{d_{0}} \delta S_{z}(\go) +
\tilde{G}_{0} (\go) \,, \\
\label{eq:fluct-w2}
-\rmi \go \delta S_{z}(\go) &= -\gga_{\|}\frac{\Lambda}{\mean{S_{z}}}
\delta S_{z}(\go) \nonumber \\
& \quad- \frac{2 \mean{d_{0}} \mean{S_{0}}}{V} \sum_{k}
\left[ (W_{0} + W_{k}^{*}) \la \vect{R}_{k}|\vect{R}_{0} \ra \delta
d_{k}(\go) + \text{ H.c. } \right] + G_{\|}(\go)\,.
\end{align}
\end{subequations}
The solution of these coupled equations is easily obtained. The correlation functions
for the $\go$-dependent fluctuation forces, can be evaluated
using the relations given in Appendix \ref{app:3B}.
				

\subsection{Field Quadratures}

Since we are mainly interested on the laser field fluctuations, the
relevant quantities to be determined are the amplitude and phase quadrature
components of the lasing mode fluctuations. For our choice of phase for the field steady-state solution, these quadratures are defined by
\cite{Wall94,Kolo93,Davi96}
\begin{subequations}
\begin{align}
\delta X_{0}(\go) &= \frac{1}{2} \bigl(\delta d_{0}(\go) + \delta
d_{0}^{\dag}(-\go)\bigr)\, , \\
\delta Y_{0}(\go) &= \frac{1}{2\rmi} \bigl(\delta d_{0}(\go) - \delta
d_{0}^{\dag}(-\go)\bigr)\, .
\end{align}
\end{subequations}
Using Eq.~\eqref{eq:fluct-w1}, the quadrature components can be written as
\begin{subequations}
\begin{align}
\label{eq:xqua-2}
\delta X_{0}(\go) &= \frac{\rmi}{2\go} \bigl[2\mean{d}_{0} \text{Re}[W_{0}]
\delta S_{z}(\go) + \tilde{G}(\go) +
\tilde{G}_{0}^{\dag}(-\go)\bigr]\,, \\
\label{eq:yqua-2}
\delta Y_{0}(\go) &= \frac{1}{2\go} \bigl[2\rmi\mean{d}_{0} \text{Im}[W_{0}]
\delta S_{z}(\go) + \tilde{G}(\go) - \tilde{G}_{0}^{\dag}(-\go)\bigr]\,,
\end{align}
\end{subequations}
where we used that $\delta S_{z}(t)$ is a Hermitian operator, so that
$\delta S_{z}(\go) = \delta S_{z}^{\dag}(-\go)$. We notice that for a
detuned laser $\text{Im}[W_{0}] \neq 0$, and amplitude and phase
fluctuations couple. Sufficiently far above threshold the laser
amplitude stabilizes and its fluctuations become negligible, then the
laser field emission power spectrum is determined solely from the
phase fluctuations. For a detuned laser, due to the coupling to the
amplitude fluctuations, the phase fluctuations are enhanced
\cite{Lax67}. This yield the well-known enhancement factor
$(1+\alpha^2)$ in the emission linewidth. The dimensionless detuning
parameter $\alpha$ is defined as the ratio $(\nu -
\bar{\gO}_{0})/\gga_{\bot}$ and is formally equivalent \cite{Gold91c}
to the Henry parameter \cite{Henr82,Henr83} in semiconductor lasers;
the latter originates from the coupling between amplitude and phase
fluctuations induced by changes in the refractive index due to the
relaxation oscillations that follow a spontaneous emission event. The
complete effect of detuning between the laser and the atomic medium
can be encoded in the Henry factor. In what remains of the chapter we
assume the laser to be on resonance and defer the discussion of our
results for the detune case to the Appendix \ref{app:4A}.


\subsection{Phase Diffusion Coefficient}

Far above threshold the laser power spectrum is determined from the
phase fluctuations. Small phase fluctuations are related to the phase quadrature component of the field fluctuations by
\begin{equation}
\delta\phi(t) = \frac{1}{\mean{d_{0}}} \delta Y_{0}(t)\,,
\end{equation}
which, after using Eq.\eqref{eq:yqua-2}, for a laser with zero
detuning reduces to
\begin{equation}
\label{eq:phiw}
\delta\phi(\go) = \frac{1}{2 \go \mean{d_{0}}} \bigl(
\tilde{G}_{0}(\go) - \tilde{G}_{0}^{\dag}(-\go)\bigr)\,.
\end{equation}
The phase fluctuations are therefore linear superpositions of Gaussian random variables and thus behave like random Gaussian variables themselves. We notice that the phase fluctuations diverge when $\go \to 0$. This divergence is related to the phase
diffusion process encountered far above threshold. In order to see this, let us first consider the second order phase correlation function
\begin{equation}
\begin{split}
\la \delta\phi(\go) \delta\phi(\go') \ra &=-
\frac{1}{4\go\go' \mean{d_{0}^2}} \la \tilde{G}_{0} (\go)
  \tilde{G}_{0}^{\dag}(-\go') + \tilde{G}_{0}^{\dag} (-\go) \tilde{G}_{0} (\go')\ra \\
&= 2\pi [\delta\phi^2]_{\go} \delta(\go+\go')\,.
\end{split}
\end{equation}
Here, for the first line we used $\la \tilde{G}_{0} (\go) \tilde{G}_{0}
(\go') \ra = \la \tilde{G}_{0}^{\dag} (-\go) \tilde{G}_{0}^{\dag}
(-\go') \ra = 0$, and the second line follows from the correlations
functions \eqref{eq:W-CF}. The phase diffusion coefficient is given by
\begin{equation}
\begin{split}
\label{eq:dphi-c}
[\delta\phi^2]_{\go} &=\frac{\la \vect{L}_{0}|\vect{L}_{0}
  \ra}{4\go^{2}\mean{d_{0}^2}} (2\mathcal{D}_{00}^{A} +
2\mathcal{D}_{00}^{N}) \\ 
&= \frac{\la \vect{L}_{0}|\vect{L}_{0} \ra}{2\go^{2}\mean{d_{0}^2}}
\left(\gk_{0}+ \frac{V g^{2}}{\gga_{\bot}} (\mean{S_{1}} + \mean{S_{0}})\right).
\end{split}
\end{equation}
If we now use the steady-state solution \eqref{eq:Sz-st} for the
population inversion density, and the fact that $\mean{S_{z}} =
\mean{S_{1}} - \mean{S_{0}}$, the phase diffusion coefficient may be
written as
\begin{equation}
\label{eq:phi2}
[\delta\phi^2]_{\go} = \frac{K_{0} \delta\go_{\text{ST}}}{\go^2},
\end{equation}
where $\delta\go_{\text{ST}}$ is the Schawlow--Townes diffusion
coefficient \cite{Sarg74}
\begin{equation}
\label{eq:ST-for}
\delta\go_{\text{ST}} = \frac{V g^{2} \mean{S_{1}}}{\gga_{\bot}
  \mean{I_{0}}}\,.
\end{equation}
According to Eq.\eqref{eq:phi2} the diffusion coefficient for a laser
with nonorthogonal modes is enhanced as compared to a standard laser
with orthogonal modes by the Petermann factor $K_{0} = \la
\vect{L}_{0}|\vect{L}_{0} \ra \la
\vect{R}_{0}|\vect{R}_{0} \ra > 1 $. This enhancement has its origin
in the nonorthogonality of the open resonator modes, leading to a 
faster diffusion of the laser phase fluctuations. The diffusive
behavior of the phase fluctuations is obtained by calculating the time
dependent mean-square variance of the phase. One obtains
\begin{equation}
\label{eq:phi-var}
\begin{split}
\la \delta\phi^2(t)\ra &= \int\limits_{0}^{t} \! \rmd t_{1}
\int\limits_{0}^{t} \! \rmd t_{2} \, \la \dot{\delta\phi}(t_{1})
\dot{\delta\phi}(t_{2}) \ra \\
&= \frac{1}{2\pi} \int\limits_{0}^{t} \! \rmd t_{1}
\int\limits_{0}^{t} \! \rmd t_{2} \! \int \! \rmd\go \, \rme^{-\rmi \go
(t_{1}-t_{2})} \go^2 [\delta\phi^2]_{\go} \\
& = K_{0} \delta\go_{\text{ST}} |t|,
\end{split}
\end{equation}
which shows that the phase fluctuations grows linearly with time. As
we will see in the following section this linear increase of the phase fluctuations with time leads to a Lorentzian laser lineshape.


\section{Laser Emission Spectrum}
\label{sec:ES-SML}

We now turn to the evaluation of the emission spectrum for a single
mode laser in a cavity with overlapping resonances. The laser emission
spectrum is given by Eq.~\eqref{eq:PowerS}, in terms
of the Wienier--Khintchine theorem \cite{Mand95} for the output field. Using the input-output relations \eqref{eq:in-outt} in the absence of external illumination, we obtain the emission spectrum
\begin{equation}
\label{eq:ES-non}
\mathcal{S}(\go) = \frac{\rmi}{2\pi} \sum_{kl} (\go_{l} - \go_{k}^{*})
\la \vect{R}_{k} | \vect{R}_{l} \ra \int\limits_{-\infty}^{\infty} \!
\rmd t \, \rme^{-\rmi \go t} \la d_{k}^{\dag} (t) d_{l} (0) \ra \,,
\end{equation}
where we used the general identity 
\begin{equation}
\begin{split}
(T^{\dag}\mathcal{W} \mathcal{W}^{\dag} T)_{kl} &=
\frac{\rmi}{2\pi\hbar} \la \vect{R}_{k} |(\mathcal{H} -
\mathcal{H}^{\dag}) | \vect{R}_{l} \ra \\
&= \frac{\rmi}{2\pi} (\go_{l} - \go_{k}^{*}) \la \vect{R}_{k} |
\vect{R}_{l} \ra \,,
\end{split}
\end{equation}
which follows from the properties of the non-Hermitian operator
$\mathcal{H}$ and its right eigenvectors $\{|\vect{R}_{k} \ra \}$ with
complex eigenvalues $\go_{k}$.

Let us first discuss the above emission spectrum in the limit of
isolated resonances. In this limit the right eigenmodes become an
orthonormal set of functions and the result reduces to the emission
spectrum for almost closed cavities with contributions coming only
from the diagonal terms with $k=l$. In particular, for a single-mode
laser with an amplitude much larger than the amplitude of the
modes below threshold, we recovers a Lorentzian lineshape with a full
width at half maximum given by the Schawlow--Townes formula
\eqref{eq:ST-for} \cite{Scha58,Hake85,Mand95}.

For a laser with overlapping resonances the spectrum has a much richer
structure. As an example, we recall the emission spectrum
\eqref{eq:EES-la} for a linear amplifier in a cavity with large
outcoupling to the outside world. Due to the nondiagonal terms
accounting for correlations among the modes, this spectrum differs
dramatically from a sum of Lorentzians expected for isolated modes. In
the laser regime we expect the laser amplitude far exceeds the
amplitudes of the below threshold modes. Therefore, similarly to the case of
isolated resonances, we can neglect contributions which are
second order in the below threshold mode amplitudes. For overlapping
modes, however, contributions linear in the below threshold mode
amplitudes are present and contribute to the emission spectrum for a
laser not too far above threshold. Our main purpose in this chapter is
to study the effects that such contributions may have on the laser
lineshape. Correlations between different modes are the fingerprint of
overlapping modes, they imply that the excess noise found in systems
with nonorthogonal modes is spectrally colored
\cite{Lee98,Lee00a,Gran98,Gran99,Exte01}. In particular, correlations
between a single laser mode and modes below threshold have been
measured in cavities with overlapping resonances by Poizat, Chang and
Grangier \cite{Poiz00} and van der Lee et al \cite{Lee00}, and were
found to hamper the noise reduction on the laser amplitude. Taking
into account these contributions the laser emission spectrum is given
by
\begin{equation}
\label{eq:ES1}
\mathcal{S}(\go) = \mathcal{S}_{00}(\go) + \sum_{k\neq 0}
(\mathcal{S}_{0k}(\go) +\mathcal{S}_{k0}(\go))\,,
\end{equation}
where the diagonal and nondiagonal contributions are defined,
respectively, by
\begin{align}
\label{eq:ES-d}
\mathcal{S}_{00}(\go) &= \frac{\gk_{0}}{\pi}  \la \vect{R}_{0} | \vect{R}_{0} \ra
\int\limits_{-\infty}^{\infty}\! \rmd t \, \rme^{-\rmi \go t} \la
d_{0}^{\dag} (t) d_{0} (0) \ra \,, \\
\label{eq:ES-nd}
\mathcal{S}_{0k}(\go) &= (\go_{k} - \go_{0}^{*}) \la
\vect{R}_{0} | \vect{R}_{k} \ra \int\limits_{-\infty}^{\infty}\! \rmd
t \, \rme^{-\rmi \go t} \la d_{0}^{\dag} (t) d_{k} (0) \ra\,,
\end{align}
and $\mathcal{S}_{k0}(\go)$ is obtained from \eqref{eq:ES-nd} by
interchanging the indices $k \leftrightarrow 0$. We now turn to the
evaluation of the different contributions to the laser spectrum.


\subsection{Laser Linewidth}

We first consider Eq.~\eqref{eq:ES-d} yielding the dominant
contribution to the emission spectrum. For this calculation we follow
the usual steps for evaluating the laser lineshape for a single-mode
laser \cite{Sarg74,Hake84}. Far above threshold the laser amplitude
fluctuations are small compared with the laser amplitude steady-state
value; following Haken \cite{Hake84} we approximate the laser field
amplitude by
\begin{equation}
\label{eq:labove}
d_{0} = \mean{d_{0}} \exp\big(-\rmi [ \mean{\gO}_{0} t -
\delta\phi(t)]\big).
\end{equation}
Substitution into Eq.~\eqref{eq:ES-d} and using the Gaussian
properties of the laser phase fluctuations $\delta\phi(t)$, one obtains 
\begin{equation}
\mathcal{S}_{00} = \frac{\mean{I}_{\text{out}}}{2\pi}
\int\limits_{-\infty}^{\infty}\! \rmd t \, \exp\Bigl(-\rmi (\go -
\gO_{0}) t -\frac{1}{2} \la \delta\phi^{2}(t)\ra \Bigr),
\end{equation} 
where we introduced the mean outgoing laser photocurrent
$\mean{I}_{\text{out}} = 2\gk_{0} \mean{I_{0}}$. Using
Eq.~\eqref{eq:phi-var} for the mean-square variance of the phase we find
\begin{equation}
\begin{split}
\label{eq:Sdg}
\mathcal{S}_{00} &= \frac{\mean{I}_{\text{out}}}{2\pi}
\int\limits_{-\infty}^{\infty}\! \rmd t \, \exp\Bigl(-\rmi (\go -
\gO_{0}) t -\frac{1}{2} K_{0} \delta\go_{\text{ST}} |t| \Bigr) \\
&= \frac{\mean{I}_{\text{out}}}{\pi} \frac{K_{0}
\delta\go_{\text{ST}}/2}{ (\go -\mean{\gO}_{0})^{2} + \frac{1}{4} (K_{0}
\delta\go_{\text{ST}})^{2}} \,.
\end{split}
\end{equation} 
Thus, like for a standard laser, we recovered a laser line given by a
Lorentzian. In this case, however, the full width at half maximum is
given by $K_{0}\delta\go_{\text{ST}}$, which presents an enhancement
by a factor $K_{0}$ with respect to the standard linewidth
$\delta\go_{\text{ST}}$. In the limit of a cavity with orthogonal
modes the Petermann factor $K_{0}$ equals one and we recover the standard result. We will show below that
provided the enhancement of the laser linewidth is large enough
$(K_{0}\gg 1)$,  the total laser lineshape may differ from a Lorentzian.

Finally, let us mention that recently Lamprecht and Ritsch
\cite{Lamp02}, and Cheng and Siegman \cite{Chen03} proposed
alternative quantum laser theories within which the laser line-width
enhancement due to the excess noise can be derived. Although in these
theories the Langevin equations considered are similar to ours the
quantization field scheme on which they are based is completely
different to system-and-bath underlaying our theory.


\subsection{Cross Correlation Contributions}
\label{sec:CCC}

We now evaluate the contributions to the emission spectrum arising
from the correlation between the laser mode and the modes below
threshold. Substitution of Eq.~\eqref{eq:labove} into
Eq.~\eqref{eq:ES-nd} yields
\begin{equation}
\label{eq:S-cross}
\mathcal{S}_{0k}(\go) = \frac{\rmi}{2\pi} (\go_{k} - \go_{0}^{*})
\la\vect{R}_{0}|\vect{R}_{k}\ra \mean{d_{0}}\int\limits_{-\infty}^{\infty} \!
\rmd t \, \exp\bigl(-\rmi(\go-\bar{\gO}_{0})t\bigr) \la \exp(-\rmi
\delta\phi(t)) \delta d_{k}(0)\ra \,,
\end{equation} 
where we used the below threshold amplitudes $d_{k}(t) = \delta d_{k} \rme^{-\rmi \bar\go_{k} t}$. The correlation function $\la \exp(-\rmi \delta\phi(t))
\delta d_{k}(0)\ra$ can be evaluated using that both the laser phase
fluctuations as well as the below threshold mode amplitudes are driven
by Gaussian random noises. Hence phase fluctuations and below
threshold amplitudes are themselves Gaussian random
variables. Employing Wick's theorem, in Appendix \ref{app:4C} is shown
that the correlation function $\la
\exp(-\rmi \delta\phi(t)) \delta d_{k}(0)\ra$ reduces to
\begin{equation} 
\label{eq:ephidk}
\la\exp(-\rmi\delta\phi(t)) \delta d_{k}(0)\ra = -2\rmi \exp\Bigl(-\frac{1}{2}
\la \delta\phi^{2}(t) \ra \Bigr) [\la \delta\phi(t) \delta d_{k}(0) \ra + \la
\delta d_{k}(0)\delta\phi(t) \ra ]\,.
\end{equation}
It is left to evaluate the simpler correlation functions $\la
\delta\phi(t) \delta d_{k}(0) \ra$ and $\la \delta
d_{k}(0)\delta\phi(t) \ra$. To this end, we write
\begin{equation}
\label{eq:phidk}
\begin{split}
\la \delta\phi(t) \delta d_{k}(0) \ra &= \int_{0}^{t}\! \rmd t' \, \la
\dot{\delta\phi}(t') \delta d_{k}(0) \ra \\
&= \frac{1}{(2\pi)^{2}} \int_{0}^{t}\! \rmd t' \int \! \rmd \go \int
\! \rmd \go' \, \rme^{-\rmi\go t'} (-\rmi \go) \la \delta\phi(\go)
\delta d_{k}(\go') \ra\,.
\end{split}
\end{equation}
The remaining correlation function is easily computed using a Fourier
transformation and relations in Appendix \ref{app:4B}. The result is
\begin{equation}
\la \delta\phi(\go) \delta d_{k}(\go') \ra\ = -\frac{2\pi \rmi}{\mean{d_{0}}\go
\go'} \mathcal{D}_{0k}^{N} \la \vect{L}_{k} | \vect{L}_{0}\ra \,
\delta (\go + \go' -\bar{\gO}_{0} + \bar{\go}_{k} )\,.
\end{equation} 
Substituting the result into Eq.~\eqref{eq:phidk} and performing the
integrals, one obtains
\begin{equation}
\label{eq:phidk-N}
\la \delta\phi(t) \delta d_{k}(0) \ra = - \frac{\mathcal{D}_{0k}^{N}
\la L_{k} | L_{0} \ra}{\mean{d_{0}}(\bar{\gO}_{0} - \bar{\go}_{k})} \Theta (-t)
\bigl[ \rme^{-\rmi (\bar{\gO}_{0} - \bar{\go}_{k})t} - 1 \bigr]\,,
\end{equation}
where $\Theta (x)$ is the step function which is equal to one if $x>0$
and vanishes otherwise. In a similar fashion one can show
\begin{equation} 
\label{eq:phidk-A}
\la \delta d_{k}(0) \delta\phi(t) \ra = -
\frac{\mathcal{D}_{k0}^{A} \la L_{k} | L_{0}
\ra}{\mean{d_{0}}(\bar{\gO}_{0} - \bar{\go}_{k})} \Theta (-t) \bigl[
\rme^{-\rmi (\bar{\gO}_{0} - \bar{\go}_{k})t} - 1 \bigr]\,.
\end{equation}
Substituting Eq.~\eqref{eq:ephidk} into Eq.~\eqref{eq:S-cross}, and
performing the time integration using the results \eqref{eq:phidk-N} and
\eqref{eq:phidk-A}, one obtains the cross correlation contributions to
the emission spectrum 
\begin{equation}
\label{eq:Sndg}
\mathcal{S}_{0k}(\go) = -\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}{\pi \Bigl[\frac{1}{2} K_{0}
  \delta\go_{\text{ST}} -\rmi (\go - \bar{\gO}_{0})
  \Bigr]\Bigl[\frac{1}{2} K_{0} \delta\go_{\text{ST}} + \bar{\gk}_{k}
  -\rmi (\go - \bar{\gO}_{k}) \Bigr] }\,,
\end{equation}
where we have split the complex frequencies
$\bar{\go}_{k}=\bar{\gO}_{k} -\rmi \bar{\gk}_{k}$ into their real an
imaginary parts. A calculation following the same steps as the one presented for
$\mathcal{S}_{0k}(\go)$ yields for $\mathcal{S}_{k0}(\go)$,
\begin{equation}
\label{eq:Sndg2}
\mathcal{S}_{k0}(\go) = \mathcal{S}_{0k}^{*}(\go)\,.
\end{equation}

The cross correlations contributions to the laser power spectrum
\eqref{eq:Sndg} are the key result of this chapter. Their presence
is a consequence of the mode nonorthogonality and leads to changes in
the laser line, causing the spectrum to deviate from a
Lorentzian shape. The magnitude of these contributions is determined by two features. First, the numerator on the right hand side of
Eq.~\eqref{eq:Sndg} that depends on the correlation between the laser mode
and the $k\neq 0$ non-lasing mode: For strongly correlated modes the
product of overlap integrals $\la\vect{R}_{0}|\vect{R}_{k}\ra
\la\vect{L}_{k}|\vect{L}_{0}\ra$ is large. In contrast, modes with
small correlations have small spatial overlap, the extreme case being
orthogonal modes for which the overlap integral vanishes. Second,
their frequency dependence through the complex denominator. This
denominator allows for large values of the cross correlation terms at
frequencies different from that of the laser mode, introducing more
structure to the laser line provided the non-lasing modes are close to
(but below) threshold, with $\bar{\gk}_{k}$ not to large. In order to
discuss these features more clearly, it is convenient to write down
the complete power spectrum.


\subsection{Emission Power Spectrum}

The total emission power spectrum for a single-mode laser in a cavity
with overlapping resonances is obtained by substituting the results
\eqref{eq:Sdg}, \eqref{eq:Sndg}, and \eqref{eq:Sndg2} into
Eq.\eqref{eq:ES1}. The final expression takes the form
\begin{equation}
\begin{split}
\label{eq:ColoredS}
\mathcal{S}(\go) &= \frac{\mean{I}_{\text{out}}}{\pi}
\frac{\delta\go/2}{ (\go -\mean{\gO}_{0})^{2} +
  (\delta\go /2)^{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}{2\gk_{0} K_{0}
       (\mathcal{D}_{00}^{N}+\mathcal{D}_{00}^{A})}\right]
\right)\right\} \,,
\end{split}
\end{equation}
where the total linewidth is given by $\delta \go = K_{0}
\delta\go_{\text{ST}}$, which includes the Petermann enhancement
factor $K_{0}$.

The first term inside the curved brackets on the right hand side of
Eq.~\eqref{eq:ColoredS} accounts for the lasing mode contribution to
the power spectrum. It corresponds to a Lorentzian with linewidth
$\delta\go$ that incorporates the effects of excess noise. As
discussed in Sec.~\ref{sec:CCC}, this term is the dominant
contribution to spectrum provided the correlation between the lasing
mode and the non-lasing modes are negligible. In case of strong
correlations, due to the second term inside the brackets, the power
spectrum will deviate significantly from a Lorentzian provided some
non-lasing modes are close to (but below) threshold. Indeed, close to
threshold $\bar{\gk}_{k}\approx 0$ for non-lasing modes, and the
contribution arising from the cross correlations term is large if
$\go$ is close to the oscillation frequencies of the non-lasing
modes. Far above threshold, cross correlation contributions become
negligible. Then, $\delta\go$ is very small (see Eq.~\eqref{eq:ST-for}),
and the laser line reduces to a very sharp Lorentzian. Any deviation
arising from the cross correlation terms is suppressed by the small
values of the Lorentzian at the wings of the spectrum.


\section{Discussion and Outlook}

In this chapter we derived the power spectrum of a laser with
overlapping resonances in the regime of single-mode laser
oscillation. To this end, we used the quantum Langevin equations
derived in \emph{Chapter~\ref{ch:LEQ}}, and demonstrated that they can
account for a quantum description of the Petermann excess noise. Our
quantum description of phase diffusion accounts for the laser
linewidth enhancement by the Petermann factor. Additionally, we found
that excess noise plays a more significant role in the determination
of the laser spectrum. In contrast to the power spectrum of standard
lasers for which the only contribution arises from the lasing mode
autocorrelation, we showed that in lasers with overlapping resonances
cross-correlations exist between the lasing mode and close-by
non-lasing modes. Their contribution to the spectrum can become of the
same order as the lasing mode autocorrelation contribution and cannot
be neglected for pumpings no too far above threshold. The origin of
these contributions lays only in the nonorthogonal character of the
laser modes. Due to these contributions the laser line deviates from
the Lorentzian shape expected from standard laser theory.

So far most of the experimental and theoretical studies of excess
noise has been restricted to lasers in optical cavities with simple
geometries (see Sec.~\ref{sec:intr-Excess}). Overlapping modes,
however, are a generic property of random lasers with weak disorder,
and therefore one should expect excess noise to be at least partially
responsible for the complex structure displayed by the power spectra
of random lasers (see Fig.~\ref{fig:12}). Investigations in this
direction, to the best of our knowledge, have not been yet carried
out. We briefly discuss how random matrix theory, may be applied in
this context.

The power spectrum \eqref{eq:ColoredS} can be used to evaluate the
emission spectrum of chaotic lasers with overlapping resonances in the
regime of a single laser oscillation. For this some simplifications
are convenient. Since we expect the laser mode to be close to the
center of the amplification spectrum, i.e., to be almost at resonance,
the effective gain seen by it and by the close-by below threshold
modes can be assumed to be $V g^{2}\mean{S_{z}}/ \gga_{\bot} =
\gk_{0}$ which holds at the center of the amplification
line. Close to the center of the gain profile we may also neglect the
gain frequency pulling. Then the resonances of the filled cavity are
trivially related to the cold cavity resonances by
\begin{subequations}
\begin{align}
\bar{\gO}_{0} &= \gO_{0} \,, \\
\bar{\go}_{k} &= \go_{k} + \rmi \gk_{0} = \gO_{k} -\rmi (\gk_{k}
- \gk_{0}) \,.
\end{align}
\end{subequations}
Upon substitution into Eq.~\eqref{eq:ColoredS}, the power spectrum
simplifies and can be written in terms of the passive cavity
resonances,
\begin{equation}
\begin{split}
\label{eq:S-simpl}
\mathcal{S}(\go) &= \frac{\mean{I}_{\text{out}}}{\pi}
\frac{\delta\go/2}{ (\go - \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 -
\gO_{0})}{\frac{1}{2} \delta\go + (\gk_{k}-\gk_{0}) -\rmi (\go -
\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}{2\gk_{0} K_{0}
       (\mathcal{D}_{00}^{N}+\mathcal{D}_{00}^{A})}\right]
\right)\right\} \,,
\end{split}
\end{equation}
where the total linewidth is given by $\delta \go = K_{0}
\delta\go_{\text{ST}}$.  As a consequence of the chaotic scattering at
the resonator mirrors the resonator right and left modes become
chaotic, and both the resonances frequencies $\gO_{k}$ and widths
$\gk_{k}$ become random quantities. Their statistics can be obtained
from the random-matrix model for chaotic scattering \cite{Fyod97}: The
passive resonator dynamics is assumed to be generated by the
non-Hermitian random matrix $\mathcal{H}/\hbar$, with the form given
in Eq.~\eqref{eq:Heff0}. Its Hermitian part is taken from the
orthogonal Gaussian ensemble. Although, due to its complexity an
analytical treatment of the spectrum \eqref{eq:S-simpl} may not be
possible at present, a numerical random matrix study is feasible.  



