%SourceDoc ./tesis.tex
\chapter{Introduction}
\label{ch:intro}

Random lasers are a novel class of nonlinear amplifiers realized in
disordered dielectrics with a dielectric function $\epsilon
(\vect{r})$ that varies randomly in space. In contrast to standard lasers
in which the confinement of light is achieved by means of mirrors,
random lasers are cavities without mirrors in which the feedback of
light results from multiple scattering in the random media (see
Fig.~\ref{fig:11}). Light amplification is provided by an active
optical medium. Due to chaotic scattering, light is trapped inside the
material for long enough times for the amplification to become
efficient. Laser oscillations emerge when the amplification exceeds
the loss rate due to escape from and absorption in the medium.

In this thesis we present a quantum theory for lasing in random
media. Our theory applies to random lasers as well as to chaotic laser
cavities, hereafter called chaotic lasers, in which randomness is the
consequence of the irregular shape of the cavity or of artificial
scatters inside of it. The two types of random media considered in
this thesis are shown in Fig.~\ref{fig:11}.
\begin{figure}[t]
	\begin{center}
		\includegraphics[scale=0.55]{./Figures/standard_vs_random.eps}
		\caption{\label{fig:11} Comparison between a standard
		laser and the two type of random lasers considered in
		this thesis.  In a standard laser a simple geometry of
		mirrors forms the cavity. The mirrors confined the
		light inside the resonator providing the feedback to
		produce laser. In a random laser there are no mirrors
		but due to multiple scattering the light is
		trapped inside the material long enough to give rise
		to laser. In a chaotic laser photons are confined
		inside the resonator by an irregular shaped
		configuration of mirrors long enough to ergodically
		explore the resonator volume.}  \end{center}
\end{figure}

In recent years several experiments have demonstrated random laser
action (for a review see Refs.~\cite{Cao03,Cao03a}). Important
contributions were made by H. Cao and her group in an extensive series
of experiments on amplifying random media using dielectric films
\cite{Cao98,Cao99,Cao00a}, disordered dielectric cluster
\cite{Cao00c}, and suspensions of scattering particles in dye
solutions \cite{Cao00b}. Figure~\ref{fig:12} reproduces measured
emission spectra of a suspension of zinkoxide microparticles in a dye
solution of rhodamine 640; the pump strength increases from bottom to
top. The dye solution is used as amplifying medium while the zinkoxide
particles act as scatterers. The density of scatterers determine the
strength of the disorder. The samples were optically pumped and the
emitted light was collected by a photodetector. For small pumping one
observes a smooth emission spectrum with a broad maximum due to
amplified spontaneous emission. However, at a critical pump the
spectrum changes dramatically. Depending on the disorder two different
behaviors are found: For weak disorder a single maximum is present but
its linewidth shrinks very fast (Fig.~\ref{fig:12}(a)). For strong
disorder, in contrast, a number of sharp peaks appear
(Fig.~\ref{fig:12}(b)). Both phenomena signal the onset of lasing. This
interpretation is confirmed by measurements of the photon statistics
\cite{Zach00,Cao01}. They show a crossover from thermal statistics
$P(n) \sim [\bar{n}/(1+\bar{n})]^n$ below threshold to a Poissonian
statistics $P(n) = [\bar{n}^n/n!]  \exp(-\bar{n})$ above threshold,
where $P(n)$ is the occupation probability of the lasing mode with $n$
photons, and $\bar{n}$ the mean photon number. The Poissonian
statistics is the result expected for a single-mode laser far above
threshold.
\begin{figure}[t]
	\begin{center}
    		\includegraphics[scale=.8]{./Figures/RL_spectrum.eps}
    		\caption{\label{fig:12} Emission spectra from
    		rhodamine 640 dye solution containing ZnO
    		nanoparticles as measured by Cao et al
    		\cite{Cao00b}. (a) Random laser with weak
    		disorder. The ZnO particle density is $\sim 3 \times
    		10^{11} \text{cm}^{-3}$. The incident pump-pulse
    		energy is (from bottom to top) 0.68, 1.5, 2.3, 3.3,
    		and 5.6 $\mu$J. The laser threshold corresponds to a
    		pump-pulse of $\sim 3 \,\mu\text{J}$. The upper inset
    		is the emission intensity at the peak wavelength
    		versus the pump pulse energy. The lower inset shows
    		the emission linewidth versus the pump pulse
    		energy. (b) Random laser with strong disorder. The ZnO
    		particle density is $\sim 1 \times 10^{12}
    		\text{cm}^{-3}$. The incident pump pulse energy is
    		(from bottom to top) 0.68, 1.1, 1.3, and 2.9
    		$\mu$J. The laser threshold corresponds to a
    		pump-pulse of $\sim 1\, \mu\text{J}$. The inset shows
    		the emission intensity versus the pump-pulse energy.}
    		\end{center}
\end{figure}

The above mentioned features of the emission spectra of random lasers cannot
be explained by the standard laser theory
\cite{Sarg74,Hake84,Sieg86}. The reasons are twofold: First, in random
lasers the resonant modes and their frequencies depend on the
statistical properties of the underlying random medium. Random
lasers, therefore, must be analyzed in a statistical fashion, in
contrast to standard lasers for which the simple mirrors geometry
allows for an explicit determination of the resonant modes and their
frequencies. Second, due to the absence of mirrors, random laser light
is only weakly confined. This gives rise to spectrally overlapping
modes. Standard lasers, in contrast, usually display well-isolated
modes with high-$Q$. In order to attain a complete understanding of
random lasers both effects, random scattering of light and mode
overlap, must be included in a theory of random lasers. In this thesis
we show how this can be done in the framework of a quantum theory of
random lasing. The following introduction provides a short review of
early theoretical ideas and recent semiclassical approaches to random
lasing. We discuss the consequences of mode overlap and summarize
previous approaches to the field quantization in media with
overlapping modes. The fact, that none of these approaches allows to
include random scattering of light partly motivated our work on random
lasers.


\section{Early Ideas}
\label{sec:intro-early}

The theoretical study of random lasing was initiated in a seminal
paper by Letokhov \cite{Leto68}. He introduced the concept of feedback
by diffusive scattering and stated the feasibility of random
lasing. Letokhov considered a disordered medium in which light
scatters irregularly. The disorder strength can be characterized by
the (transport) mean free path $l_{\text{t}}$. Light propagates
diffusively provided $L \gg l_{\text{t}} \gg \gl$, where $L$ is the
linear size of the medium and $\gl$ the wavelength of light. Assuming
a uniform linear gain, the spectral density of photons follows a
diffusion type of equation \cite{Leto68,Leto96},
\begin{equation}
\label{eq:diff}
\frac{\partial W(\vect{r},\go,t)}{\partial t} = D \nabla^2 W(\vect{r},\go,t) + \frac{v}{l_{\text{g}}} W(\vect{r},\go,t) \, ,
\end{equation}
where $v$ is the transport velocity of light inside the scattering
medium, $l_{\text{g}}$ is the gain length, and $D=vl_{\text{t}}/3$ the
diffusion constant. Note that $l_{\text{g}}$ is frequency dependent;
it reaches its maximum at the frequency of the laser
transition. Equation \eqref{eq:diff} must be complemented by boundary
conditions. Clearly, the density of photons vanishes far away from the
medium. At the boundary it cannot be zero as photons cross the
boundary upon emission from the sample. Exact solutions of the
emission problem for simple geometries show that the photon density,
linearly extrapolated from the boundary, vanishes at a distance of
order $l_{\text{t}}$ \cite{Ross99}. As $l_{\text{t}}$ is much smaller
than the system size $L$, we may impose the condition
$W(\vect{r},\omega,t)=0$ directly at the boundary, yielding a very
good approximation to the true boundary conditions. The general
solution to Eq.~\eqref{eq:diff} is then easily seen to be given by
\begin{equation}
\label{eq:pho-den}
W(\vect{r},\go,t) = \sum_{n} a_{n} \Psi_{n}(\vect{r}) \rme^{-(DB_{n}^{2} - v/l_{g})t} \,,
\end{equation}
where $\Psi_{n}(\vect{r})$ and $B_{n}$ are the eigenfunctions and
eigenvalues, respectively, of the Helm\-holtz equation
\begin{equation}
\nabla^{2} \Psi_{n}(\vect{r}) + B_{n}^{2} \Psi_{n}(\vect{r}) =0 \, ,
\end{equation}
with boundary conditions $\Psi_{n} = 0$ at the boundary of the
scattering medium.

The laser threshold is defined by the crossover of the photon density
in Eq.~\eqref{eq:pho-den} from a decaying solution to a solution that
increases exponentially in time. The threshold condition then follows
at once from Eq.~\eqref{eq:pho-den},
\begin{equation}
DB_{1}^{2} - v/l_{g} = 0 \, ,
\end{equation}
where $B_{1}\approx 1/L$ is the lowest eigenvalue. At the threshold
the typical path length $L_{\text{pat}} \sim vL^2/D$ of light
propagating inside the active material matches or exceeds the gain
length; then, on average, each emitted photon will generate another
photon before escaping from the medium. Based on this threshold
condition, Letokhov predicted a critical volume for the onset of
lasing
\begin{equation}
V_{\text{cr}} \approx L^{3} \approx \left(\frac{l_{\text{t}}l_{\text{g}}}{3} \right)^{3/2} \, .
\end{equation}
As the proliferation of light in amplifying random media is similar to
the generation of neutrons in nuclear reactors and atomic bombs, early
theoretical work coined the term photonic bomb.


\section{Lasing: Nonlinear Equations}
\label{sec:including}

Clearly, the exponential increase predicted by the diffusion equation
(\ref{eq:diff}) is not found in actual media. In reality, the gain
length does depend on the photon intensity: for large photon density
the gain is eventually depleted and $l_{\text{g}}$ increases. The
description of laser emission should combine the concept of diffusive
feedback with nonlinear gain saturation. Letokhov \cite{Leto96} and
John and Pang \cite{john96} implemented a set of nonlinear diffusive
equations for the photon density in the scattering medium with a
photon density-dependent gain length. Their theory proved to account
for the linewidth shrinking of the spectrum. Focusing on the spectral
line narrowing, Balachandran, Lawandy, and Moon \cite{Bala97} proposed
a laser model based on nonlinear rate equations for the inversion
population $N$ of a two-level atomic medium and the mean field
intensity $W(\go)$,
\begin{subequations}
\begin{align}
\frac{\rmd N(t)}{\rmd t} &= [1-N(t)] B_{p} \Lambda(t) - N(t) \int \!
\rmd \go \, B_{l}(\go) W(\go,t) - N(t)\Gamma \, , \\
\frac{\partial W(\go,t)}{\partial t} &= c[\gamma_{0}(\go)N(t) -
\gamma_{\text{th}}]W(\go,t)+\eta(\go)N(t)\,,
\end{align}
\end{subequations} 
where $\Lambda$ is the pump intensity, and $B_{p}$ and $B_{l}$ are the
Einstein coefficients for the pump and the laser emission,
respectively. The noise terms $N\Gamma$ and $\eta N$ were added
phenomenologically to mimic spontaneous emission. The set of equations
describe many phenomena that were found in early experiments
\cite{Lawa94,Sha94,Lawa95,Zhan95} on random media (see
Fig~\ref{fig:12}(a)): A threshold pumping with a much stronger
increase of the emission output above threshold, a collapse of the
emission linewidth at threshold, relaxation oscillations of the
emission intensity. In contrast to the experiments of Cao et al such
early experiments did not reveal quasi--discrete emission peaks.  The
respective devices are sometimes called lasers with non--resonant
feedback.


\section{Petermann Excess Noise}
\label{sec:intr-Excess}

The theoretical approaches to random lasing discussed above are based
on a classical description of the electromagnetic field. The laser
dynamics is then describe by semiclassical equations: They characterize
the mean values of the spectral photon density and the atomic
inversion. Quantum effects such as spontaneous emission and vacuum
fluctuations are taken into account phenomenologically by including
noise terms. In lasers, spontaneous emission noise is
ultimately responsible for the laser linewidth, below the laser
threshold it gives rise to amplified spontaneous emission. A proper
description of noise can only be provided by a quantum laser theory.

It has long been known that lasers with overlapping modes exhibit
unusual noise properties.  In his investigations on gain-guided
lasers, Petermann \cite{Pete79} found that these lasers exhibit a
greatly increased linewidth $\delta \go$, whose enhancement with
respect to the Schawlow--Townes limit $\delta\go_{\text{ST}}$
\cite{Scha58} can be described by an excess noise factor $K
\ge 1$ known as the Petermann factor,
\begin{equation}
\delta \go = K \delta \go_{\text{ST}}.
\end{equation}
Later, Siegman demonstrated \cite{Sieg89a,Sieg89b} that this effect
was not restricted to gain-guided lasers, but was present in all laser
cavities with spectrally overlapping modes. He noticed that due to
mode overlap the cavity modes do not form an orthogonal set but a
biorthogonal set. Thus, generically, one can associate to each cavity
resonance a left mode $\la L_{l} | $ and a right mode $|R_{l}\ra$. The
biorthogonality condition then reads $\la L_{l} | R_{k} \ra =
\delta_{lk}$.  Excess noise is generated as noise corresponding to
different cavity modes is correlated; Petermann excess noise is thus
intrinsically a multi-mode phenomenon. The net result, however, can be
observed in a single mode laser as an increase in the amplitude and
phase fluctuations by the Petermann factor. For the laser oscillation
in mode $l$, the Petermann factor takes the form
\begin{equation}
K_{l} = \la L_{l} | L_{l} \ra \la R_{l} | R_{l} \ra \geq 1 \, ;
\end{equation}
where use was made of the Schwartz' inequality.

Excess noise has been experimentally well established: It has been
detected in stable cavities with large losses
\cite{Hame89,Hame90,Lind98}, unstable resonators with nonorthogonal
transverse modes \cite{Chen96,Eijk96}, and lasers with nonorthogonal
polarization modes \cite{Lee97,Lee00a}. Despite of large theoretical
and experimental evidence for excess noise, a quantum theory of the
phenomenon has long been missing. The quantum theory of random laser
presented in this thesis provides also a quantum description of excess
noise. Recently, alternatively to our work \cite{Hack02} other quantum
approaches to excess noise have been reported \cite{Lamp02a,Chen03}.


\section{Field Quantization of Leaky Cavities}
\label{sec:intr-OC}

The development of a quantum theory of excess noise has been hampered
for a long time by the lack of a proper quantization method of the
electromagnetic field in open resonators. Conventionally the
electromagnetic field in a leaky cavity is quantized in terms of modes
of a \emph{closed} resonator, while the cavity losses are modeled by
coupling each mode to an external bath that provides both damping and
noise \cite{Hake84,Hake85}. That traditional formulation of the
system-and-bath model \cite{Seni59,Seni60,Gard00} assumes spectrally
isolated sharp resonator modes, and therefore applies successfully to
the high-Q modes of traditional laser cavities. However, that
description fails when the cavity losses become so large that modes
overlap. Mode overlap is expected, in particular, for unstable laser
cavities and mirrorless random lasers. For such lasers a quantization
scheme suitable for open laser resonators must be used. 

Developing appropriate quantization methods has been a fundamental
problem of quantum optics and has generated a substantial amount of
literature. In their classical paper \cite{Fox61}, Fox and Li
numerically studied open resonators taking into account diffractive
losses. They found that open resonators support a well-defined set of
modes which due to the losses generally differ from the eigenmodes of
closed cavities. In particular, they do not form an orthogonal
set. The Fox-Li modes correspond to the Gamov states \cite{Kuku89} of
scattering theory. Lang, Scully and Lamb \cite{Lang73} proposed the
first rigorous quantization procedure for the field in open optical
cavities. Their so-called modes-of-the-universe approach consist in
quantizing the eigenmodes of Maxwell's equations in the whole space
comprising the resonator and the external world. Their method has been
applied to a number of open resonators problems
\cite{Ujih75,Ujih77,Ujih78,Ley87,Feng90,Gea90,Gea90a,Knol91,Plan96}. Although
the modes-of-the-universe approach provides an exact quantum
description of the electromagnetic field, it makes no distinction
between the cavity and its environment. Therefore it fails to single
out explicit information about the dynamics of the field inside the
cavity.

In recent years, studies on excess noise renewed the interest in open
cavities and several new field quantization schemes were proposed.
They are either based on mode expansions or they altogether abandon
the idea of cavity modes.  The quasimodes of Dalton, Barnett, and
Knight \cite{Dalt99,Dalt99a} provide a complete basis of field modes
satisfying idealized boundary conditions at the resonator surface.
Quasimode expansions have been successfully applied in cavity quantum
electrodynamics
\cite{Dalt99b,Dalt99c,Dalt99d,Brow01,Brow01a}. Fundamentally, however,
the quasimode method suffers from the same problem as the
modes-of-the-universe approach: The quasimodes are defined throughout
the whole space and therefore provide no specific information about
the cavity field. In order to overcome this structural disadvantage,
methods involving nonorthogonal (Fox--Li) modes have been
developed. Most remarkable are the works by Young and collaborators
\cite{Ho98,Chin98} and  by Lamprecht and Ritsch
\cite{Lamp99,Lamp02,Lamp02a}. These last authors introduced an
\emph{ad hoc} master equation for open cavities in order to describe
the field damping. Since that equation was not derived from first
principles, its status remained unclear. Fox--Li expansions of both
the cavity and the external field were developed in
Refs.~\cite{Dutr00b,Brow02}. Both approaches claim to constitute a
starting point for a quantum theory of excess noise, however, to the
best of our knowledge, up to now none of them has been used for that
purpose.

Alternative approaches not based on mode expansions were developed by
Gruner and Welsch \cite{Grun96,Grun96a} and by Artoni and Loudon
\cite{Arto97}. Their quantization methods are based on Green functions
and allow to include complex, frequency and spatially dependent
dielectric functions. These approaches have been used in connection
with {\em linear} random media in the papers of Beenakker
\cite{Been98,Been99} (see Sec.~\ref{sec:intr-CB}). Since the method is
based on Green functions it can not be extended to include nonlinear
phenomena like lasers.

In this thesis we address the problem of the quantization of the
electromagnetic field in open optical cavities. Our approach provides
a first principle derivation of a system-and-bath type Hamiltonian. In
contrast to previous approaches, we derive microscopic expressions for
all mode frequencies and coupling amplitudes. The quantization
procedure allows for a microscopic description of atom-field
interactions. In connection with random matrix theory, our method
provides a basis for a statistical theory of random lasers.


\section{Linear Random Media}
\label{sec:intr-CB}

For weak pumping, below the laser threshold, random lasers
behave like linear random amplifiers. A comprehensive quantum theory
of light propagation in linear active random media was recently
presented by Beenakker and coworkers
\cite{Been98,Been99,Patr00c}. Based on a scattering quantization of
the electromagnetic field \cite{Grun96,Grun96a,Arto97}, they
implemented an input-output relation for the propagating field. The
emission intensity was then showed to be completely determined by the
system $S$ matrix. Since the medium is active the $S$ matrix is
generally not unitary, but becomes sub-unitary in absorbing media and
super-unitary in amplifying media (cf. Eq.~\eqref{eq:nonuni}). The
statistical properties of the emitted light follow from the
statistics of the $S$ matrix. When applied to random media, the
statistical properties of the $S$ matrix are known from random matrix
theory. A major prediction of the theory was the characterization of
the super-Poissonian noise of a random medium as corresponding to a
black body with a reduced number of degrees of freedom
\cite{Been98}. In addition Beenakker and co-workers computed the
photon noise spectrum \cite{Mish01} and investigated the effects of photon
localization \cite{Been00} in linear random media.  

Most noticeable, Beenakker's theory of light propagation naturally
included both the statistical nature of the random modes and the mode
overlapping due to the weak confinement of light in random media. It
allows for an investigation of the noise enhancement in linearly
amplifying media below the laser threshold, and for an statistical
treatment of the Petermann factor for linear disordered media
\cite{Frah00,Scho00,Patr00a}. The theory for
random lasing proposed in this thesis reduces to Beenakker's for
amplifying media below the lasing threshold.


\section{About this Thesis}

The physics of amplifying random media involves both chaotic
scattering and weak confinement of light. Theoretically, therefore,
methods and concepts coming from both quantum optics and quantum chaos
theory must be used. We study first the consequences of
chaotic scattering. To this end, we investigate in \emph{Chapter
\ref{ch:photo}} the photocount statistics of the radiation emitted
from an almost closed chaotic laser cavity in the regime of
single-mode lasing. Chaotic scattering causes strong mode-to-mode
fluctuations of the laser losses, giving rise to additional
fluctuations of the photocount statistics \emph{on top} of the quantum
fluctuations inherent to the quantized electromagnetic field. For
cavities with a small number of escape channels the distribution of the
mean photocount number is shown to be broad, revealing up to three
peaks.

In \emph{Chapter \ref{chap:1}} we address the quantization of the
electromagnetic field in weakly confining resonators. We present and
discuss an exact quantization procedure for \emph{open} cavities. Our
method is based on a technique from nuclear and condensed-matter
physics known as the Feshbach projection formalism \cite{Fesh62}.  We
rigorously derive a system-and-bath Hamiltonian for the
electromagnetic field starting from Maxwell's equations. The
Hamiltonian is shown to correctly describe damping and even
overdamping in open resonators (when the damping rates become of the
order of the oscillation frequencies). In the limit of small
outcoupling losses our field Hamiltonian reduces to the well-known
independent oscillator model of quantum optics. Randomness can be
naturally included in our formulation employing random matrix
theory. We conclude by illustrating the quantization method in a
series of frequently used optical resonators.

The dynamics of optical fields in open cavities is studied in
\emph{Chapter \ref{chp:F-D}}. For linear optical systems we show that
the field dynamics can be completely described by the $S$ matrix of
the open resonator which may include gain/loss contributions. Taking
into account the multi-mode nature of the field, we derive and clarify
the status of stochastic equations for the field dynamics in
resonators with overlapping resonances. Our equations provide for
a quantum picture of excess noise. Upon inclusion of an active medium,
our quantization technique constitutes the basis of our laser theory
for open lasers with overlapping modes.

A quantum theory for lasing in random media is formulated in
\emph{Chapter \ref{ch:LEQ}}. A set of quantum lasers equations
is derived that includes the atomic-field interactions on a
microscopic level. The noise properties of such lasers is seen to
differ from traditional lasers. To explore the differences we consider
the emission spectra of linear random media below the laser threshold,
and postpone a complete discussion on excess noise to \emph{Chapter
\ref{ch:single-mode}}.

Although the semiclassical picture of excess noise has been known for
some time, a quantum theory of it was lacking. In \emph{Chapter
\ref{ch:single-mode}} we show that our lasers equations provide a
quantum description of excess noise. We concentrate on a laser
operating above threshold in the regime of single-mode lasing. Our
theory not only accounts for the Petermann linewidth enhancement due
to excess noise, but predicts the deviation of the laser line
from a Lorentzian shape. We conclude with a discussion on the consequences
of such deviations in the emission spectra of random lasers.
