The Real Housewives of Atlanta fashionista partied with Cardi B in style.
Cardi B had Los Angeles turnt up last night. The rapper officially dropped her highly anticipated Fashion Nova collection on November 14 and in honor of the occasion, she hosted a mega bash. The event drew out tons of celebs including Khloe Kardashian, Iggy Azalea, Christina Milian, and Bravo's very own ATL fashionista, NeNe Leakes. That's right honey, NeNe partied with Cardi.
The Real Housewives of Atlanta OG looked fabulous in a green Adidas tracksuit, which she dressed up with black heels and a snakeskin Gucci clutch (very athleisure chic). She wore her hair in a sleek low ponytail, and her skin was glowing. Okkurrr.
Not only did she catch Cardi B kill it on stage, but she also shared a moment with the Bronx native at the party. Cardi B sported a gold two-piece set from her new collection.
Seriously, this is the friendship we never knew we needed.
We can only imagine the tea these two queens are spilling in this photo. Oh, to be a fly on that wall.
Not surprisingly, Cardi B's line sold out within hours. But don't fret, the rapper shared that it will be restocked soon. In the meantime, you can check out her designs here.


text:
Alfred "Fredi" Breunig (* 5. Juni 1959 in Königshofen, Landkreis Rhön-Grabfeld) ist ein deutscher Kabarettist, Glossist und Mundart-Autor. Er ist Preisträger des Frankenwürfels 2013 und Träger des Bayerischen Verdienstordens.

Leben
Fredi Breunig wuchs gemeinsam mit zwei Schwestern in Großeibstadt auf. Er besuchte die Staatliche Realschule in Bad Königshofen im Grabfeld und erlernte den Beruf des Physiklaboranten. Im Rahmen der betrieblichen Praxis bildete er sich zum Qualitätsmanager und -auditor weiter und ist heute als Bereichsleiter Technischer Kundenservice bei einer Firma angestellt, die Arzneimittelverpackungen aus Glas herstellt.

Fredi Breunig ist verheiratet und hat zwei Kinder. Seit 1996 lebt er mit seiner Familie in Salz.

Kabarett und Mundart
Nebenberuflich startete Fredi Breunig bereits Anfang der 1980er Jahre seine Karriere als Kabarettist. Erste Bühnenerfahrungen sammelte er bei Auftritten in Großeibstadt als Mitglied der dortigen Theatergruppe sowie den "Abschter Fosenöchtern", einem fränkischen Faschingsverein.
Zusammen mit seinem Bühnenkollegen Martin Wachenbrönner als "Gotthold" und ihm selbst als "Eustach" absolvierte er weit mehr als 600 Faschingsauftritte in ganz Unterfranken und darüber hinaus.

Landesweite Bekanntheit erlangte Breunig durch seine Auftritte bei der TV-Sendung Fastnacht in Franken im Jahr 2010 (zusammen mit Martin Wachenbrönner) und zuletzt 2016. Diese quotenstärkste Sendung aller dritten Programme wird alljährlich vom Bayerischen Fernsehen als Live-Sendung ausgestrahlt.
Seit 2006 war er auch wiederholt Gast bei der "Närrischen Weinprobe", die vom Bayerischen Fernsehen jeweils in Zusammenarbeit mit dem Fastnacht-Verband Franken e.V. im Staatlichen Hofkeller zu Würzburg aufgezeichnet und ausgestrahlt wird.
Daneben wurde er 2013 auch zu "Franken sucht den Supernarr" und weiteren TV-Auftritten eingeladen.

Als Solokabarettist machte sich Breunig ab 2006 einen Namen in der Kabarettszene.
2011 startete er mit seinem Soloprogramm "Döff doss doss?" Zwischenzeitlich gibt es ein zweites Programm mit dem Titel "Wallich nedd!" Breunig ist damit seit 2013 unterwegs.

Seit Oktober 2007 schreibt Fredi Breunig für die Main-Post eine vierzehntäglich erscheinende Glosse mit dem Titel "Gotthold & Eustach". Breunig betrachtet darin das Zeitgeschehen und beleuchtet in typisch ost-unterfränkischer Mundart Dinge und Ereignisse, die ihm auffallen.
Ebenfalls seit 2007 schlüpft Fredi Breunig in der Fastenzeit jeweils in Burglauer in die Rolle des Fastenpredigers "Bruder Elisäus" und unterhält so in der Rudi-Erhard-Halle fast 600 Gäste beim "Fränkischen Politiker-Derbläggn" nach dem Vorbild der bekannten Veranstaltung in München auf dem Nockherberg.

Eine feste Institution sind mittlerweile die kabarettistischen Frühschoppen, die Fredi Breunig zweimal im Jahr in Wargolshausen sowie gelegentlich im Großen Kursaal in Bad Bocklet anbietet.
Im Jahr 2013 moderierte Fredi Breunig in Bad Neustadt a. d. Saale im Rahmen des "Neuschter Sommerfestivals" den "1. Neuschter Mundart Grooh Brie". Nach dem Vorbild des Eurovision Song Contest wurde mit der Gruppe "Spilk" dabei der Premierensieger dieses lustigen Mundart-Wettbewerbs gekürt.

In unregelmäßigen Abständen nimmt Fredi Breunig als Akteur an sogenannten "Mundart-Rallyes" teil. Dabei reisen vier Mundart-Kabarettisten reihum zu vier Veranstaltungsorten. Das Publikum kann jeweils sitzenbleiben und bekommt so an einem Abend im Stundenrhythmus vier Akteure mit unterschiedlichen Beiträgen geboten.
Nach wie vor ist Fredi Breunig auch aktives Mitglied der Theatergruppe Großeibstadt. Fast immer übernimmt er dabei eine Hauptrolle und steht so seit 30 Jahren von Mitte Oktober bis Mitte November auf der Bühne.

Fredi Breunig wurde im November 2013 mit dem "Frankenwürfel" der drei fränkischen Regierungspräsidenten ausgezeichnet.
Am 8. Januar 2023 wurde ihm in Wargolshausen "als Zeichen ehrender und dankbarer Anerkennung für hervorragende Verdienste um den Freistaat Bayern und das bayerische Volk" durch den Ministerpräsidenten Dr. Markus Söder der Bayerische Verdienstorden verliehen.

Literatur
 Dorit Schatz: Promis, Prunk und freche Witze: 25 Jahre Fastnacht in Franken, Nürnberg 2011, ISBN 978-3-941282-42-1

Weblinks

 Offizielle Website des Künstlers
 Offizielle Seite des Kabarettisten-Duo Breunig & Wachenbrönner
 Auftritte im Bayerischen Fernsehen / BR-Mediathek

Einzelnachweise

Kabarettist (Deutschland)
Deutscher
Geboren 1959
Mann

redpajama_set_name: RedPajamaWikipedia


1.)I would love to be able to preview images Flipped Horizontal in Bridge. You can in 3rd party psd viewers but I would love to have that built in bridge as it's something I do quit often in photoshop to check for imperfections.
2.)Another great feature would be able to preview your image in grayscale. Just to quickly check the values of an image or a group of images.


We will be hosting a team for the Whole Life Challenge beginning May 20th. Please check out the website or come to one of the coaches for more information regarding the challenge. This will be a great way to get shore up all other aspects of your life, outside of the confines of the gym. This challenge will be encompassing for all levels of fitness, and skill levels. We encourage ALL of our members to get involved and encourage your friends and family members to join as well. Please check out the website for further information!


text:
Phenacolepas granocostata is een slakkensoort uit de familie van de Phenacolepadidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1868 door Pease.

Phenacolepadidae


There is no other guaranteed green or blue Quest or crafting item that beats these for a resto shammy, even if you look at cloth or leather items.
Have a look at wowheads itemisation listing. Interestingly, a properly socketed Auchenia Bracers is almost as good.


text:
Історична Англія (, офіційно в Комісія історичних будівель і пам'яток Англії, ) є виконавчим позавідомчим державним органом  Британського уряду, який фінансується Департаментом у справах культури, ЗМІ і спорту (DCMS). Його завданням є захист історичного середовища Англії шляхом збереження та ведення списків історичних будівель і консультування центральних і місцевих органів влади.

Примітки

Посилання

Організації Великої Британії
Уряд Великої Британії
Архітектура Великої Британії
Культурна спадщина Великої Британії
Археологічні організації
Реставрація
Засновані у Великій Британії 1984


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text:
Home»Pollution»Carbon
Advocates, lawmakers push $30B fund to fight climate change
By Kate Lisa November 17, 2022 Carbon No Comments4 Mins Read
Advocates and lawmakers rallied in eight cities Wednesday to unveil legislation and invest billions of dollars for the state to meet its legally required reduction of greenhouse gas emissions and carbon footprint under the Climate Act.
They're pushing for lawmakers to prioritize several bills in NY Renews' Climate, Jobs and Justice Package to help the state transition to a green economy, protect air and water, create jobs and more to satisfy its climate goals.
"The CLCPA that passed has very aggressive goals for New York state, but the funding for those changes and the staggering crisis our state is facing due to climate change is unbelievable, so that state's going to need more revenues," NYPIRG Executive Director Blair Horner said during Wednesday's rally outside the state Capitol in Albany.
"When you have 70-degree weather in Albany, NY in November, that ought to send a message to folks that we are facing a very real climate crisis," @PhilSteck says at a rally for a Climate, Jobs & Justice the Capitol legislative pkg to fully fund & implement NY's Climate Act. pic.twitter.com/ENLHV0KXNX
— Kate Lisa (@KaitlynnLisa) November 16, 2022
The package of bills includes measures to close tax loopholes benefiting polluters and give state agencies the power to reduce utility companies' emissions, as well as a plan to redevelop fossil fuel sites and create other programs and incentives.
A priority is the Climate Change Superfund Act, which would charge oil companies and their shareholders $30 billion over the next decade to reverse the climate crisis in New York.
"Companies should learn something that we all learned in kindergarten — you make a mess, you clean it up," Horner said. "And they are responsible for the mess that we're in now."
It's estimated New York will pay $10 billion per year to deal with the effects of climate change by the year 2050. The U.S. Army Corps of Engineers estimates it will cost $52 billion to protect the New York Harbor from the impacts of climate change.
The legislation would place the burden on fossil fuel companies, removing the burden from taxpayers.
"They are spending money now to subsidize polluters because what happens is, when there's pollution and the costs of that pollution are remediated by the state of New York, the voters are paying for all that with their tax dollars," said Assemblyman Phil Steck, a Democrat from Colonie who co-sponsors most of the proposals. "So we need to be more proactive on this issue."
People of all ages at the rally begged lawmakers to prioritize environmental issues when they return to Albany.
Some even floated raising taxes on the wealthy to foot the bill. Horner said advocates are working to estimate how much that increase should be, and to have more conversations with Gov. Kathy Hochul by the end of the year.
People of color and those who live in low-income areas are more likely to live and work in places with higher levels of greenhouse gases, chemicals and pollutants and less access to clean water.
"I'm sick of people who look like me, people who grew up where I grew up, being on the receiving end of bad policy and greed, breathing polluted air, drinking tainted water and being exposed to toxic fields and chemicals, giving our babies asthma and our elders cancer, destroying our ecosystem and killing off the little biodiversity we have," said Tabetha Wilson, a resident of Albany's South End who spoke Wednesday.
Steck has long fought for climate justice and related funding in the state budget.
He says several of his colleagues support climate measures on the surface, but few are serious about action.
The Democrat said he's ready to discuss the urgency with colleagues in his own party and across the aisle, after Republicans flipped several seats in the Legislature in last week's election.
"The idea that we can't be doing any new initiatives on climate, I think that's a mistake," Steck said.
The state's economic future is uncertain amid high inflation and a potential downturn, giving lawmakers and New Yorkers pause at the hefty bill of addressing climate change.
Steck argues it's an issue that should be properly addressed and funded in the next budget.
"Before you talk about taxes, you have to see what the total package is," he said. "And it's a little premature to say what's the best way to get to the end goal?"
Businesses in the fossil fuel industry saw some of their greatest profits during pandemic, with Chevron and ExxonMobil seeing combined profits for the first quarter of 2022 of over $11.8 billion while charging incredibly high prices and not accepting responsibility of high costs paid by taxpayers.
APP Capital Tonight APP NY State of Politics APP Top Stories Capital Tonight Kate Lisa new york state news NY State of Politics NY State of Politics Blog Politics Top Stories VOD
Kate Lisa

redpajama_set_name: RedPajamaCommonCrawl


Includes children's zoo. General information, news, special events, education programs and volunteer opportunities.
Garden in Boise providing an aesthetic and educational environment containing significant plant collections and a variety of specialty gardens.


text:
\section{Introduction}

The laser has been a critical enabler of the modern discipline of quantum optics via its highly monochromatic and intense emission
properties.  The physical understanding of steady-state laser emission has presented challenges since the early days
after its invention, as first semiclassical \cite{haken_nonlinear_1963,lamb_theory_1964} and then fully quantum laser theories were developed \cite{lax_quantum_1966,scully_quantum_1967,graham_quantum_1968,graham_functional_1970}.
Although these
theories were able to capture most of the relevant physics, they tended to incorporate some phenomenological
or approximate steps due to the complexity of dealing with the full non-linear and space-dependent
semiclassical or quantum laser equations. Over forty years ago, several pioneers of early laser theory, Lang, Scully and Lamb,
addressed
some of the weak points in the theory in a classic paper entitled,``Why is the Laser Line so Narrow?~A Theory of Single
Quasi-mode Laser Operation'' \cite{lang_why_1973}.  This work was aimed at a fundamental challenge in laser theory, dealing with the
openness of the laser system, which invalidated standard Hermitian modal decompositions appropriate for closed systems.
Prior to this work essentially all theories were framed in terms of  ``quasi-modes'' of the type studied by Fox and Li and
were based on phenomenological modal equations with cavity damping inserted to represent the outcoupling of the modes \cite{fox_resonant_1961}.  The
goal of Lang \textit{et al.}\ was to go beyond these quasi-mode equations for the
laser field amplitudes to a more fundamental description which included both the cavity and exterior regions
explicitly (i.e.~a model of ``the universe''), and then work
back from the modes of the universe to the quasi-modes of the cavity region alone.
Using this approach the authors (1) ``hope[d] to understand the mechanism leading to the extreme monochromaticity of the
laser radiation'', and (2) ``to investigate the sense in which the present calculation, which does not include the cavity dissipation
as a phenomological loss mechanism, still leads to and implies a fluctuation-dissipation theorem'' \cite{lang_why_1973}.

While this approach to understanding the laser equations was groundbreaking at the time, there has been much progress in
laser theory in the subsequent years and it seemed to us a good time to re-examine these questions from a modern
perspective.  One reason for this is that during the past decade
a new theoretical framework has been developed, known as the steady-state \textit{ab initio} laser theory (SALT), which
treats the openness of an arbitrary cavity exactly within the semiclassical theory \cite{tureci06,tureci07,tureci08,ge10}, and thus
has provided new insights into lasing phenomena.  The resulting non-linear SALT equations
for steady-state lasing can be solved efficiently using a non-Hermitian basis set method and have been shown to agree to very
high accuracy with brute force simulations of the atomic lasing equations \cite{ge08,cerjan12}, including even N-level models with multiple
independent lasing transitions \cite{cerjan_csalt_2015}.  The theory can also be generalized to treat injected signals (I-SALT), as will be relevant
below \cite{cerjan_isalt_2014}.  Moreover, recently Pick \textit{et al.}\ have added Langevin noise to the SALT equations (N-SALT) and, using the SALT steady-state
lasing modes as a starting point, calculated the quantum-limited laser linewidth analytically \cite{pick_linewidth_2015}.  Their analytic results
have very recently been confirmed, again by direct simulations of the lasing equations, but with appropriate noise terms
included \cite{cerjan_noise_2015}.  It should be said that many aspects of the results obtained through SALT and N-SALT were anticipated by prior
theoretical approaches, a partial list of which includes the work by Spencer and Lamb \cite{lamb72}, Fu and Haken \cite{haken87,haken88,haken91}, and
Henry \cite{henry86}; however SALT and N-SALT appear to be the most general, flexible and
inclusive formalism yet developed for steady-state lasing. It treats the fields over the entire ``universe'' from the outset,
but in a different manner from Lang \textit{et al.}, and does not arrive at modal equations with damping in the conventional sense.
We will now examine the questions raised by Lang \textit{et al.}\ from the
perspective of SALT and N-SALT.

\section{Lasers as Scattering Systems}
\subsection{Motivation and definition of the problem}
Most early lasers emitted from lasing modes which were concentrated along some optical axis, e.g.~the axis joining the
centers of two mirrors or the geometric or index profile defining a waveguide or fiber, and so there was a tendency to
develop laser theories focused on one-dimensional or axially symmetric equations, and this is overwhelmingly the type of
formulation one finds in text books \cite{haken_LT,siegman,sargent}.  However in the past two decades a great variety of laser systems have been developed
with complex two or three-dimensional cavities, such as microdisks and microspheres \cite{levi_microdisk_1992,mccall_1992,collot_microsphere_1993},
micro-toroids \cite{armani_toroid_2003}, deformed (chaotic) disk cavities \cite{gmachl98,lacey_directional_2003,redding_dcav_pnas_2015},
photonic crystal defect-based and surface emitting lasers \cite{painter_two-dimensional_1999,meier_laser_1999,imada_coherent_1999}
and random lasers \cite{cao_random_1999,cao_microlaser_2000,polson_random_2001,bahoura_determination_2003,van_der_molen_spatial_2007,wiersma_light_2001,wu_random_2006}.  To describe such devices it was necessary to
formulate laser theory in a manner that could both describe, and be implemented computationally, for cavities with an arbitrary geometry,
and to include the full Maxwell wave equation, not its axial approximation.

Moreover, some of the new lasers (e.g.~random lasers) are extremely open systems, not even cavities in the usual sense; nor
do they always emit well-behaved narrow beams of light but rather generate complex emission patterns in two and three dimensions.
Thus it was crucial to find a description which did not over-simplify the spatial properties of the cavities and emission patterns,
and which hence treated the cavity boundary and outcoupling exactly.
The solution to this challenge provided by SALT is to describe the laser as a scattering system.  Any laser consists of two essential elements: a cavity
region, given as a linear dielectric function, $\varepsilon_c (\bx)$, which describes the scattering and confinement of photons,
either injected externally or internally generated, and a gain region, containing the medium which will be pumped to provide
optical gain (sometimes cavity and gain medium essentially coincide).
To include all cases the ``cavity'' will refer simply to a singly connected region large enough to enclose all the regions in which
$\varepsilon_c (\bx) \neq 1$; and for simplicity we will assume the gain medium is fully contained within this region as depicted in Fig.~\ref{fig:cavity}.

\begin{figure}[t!]
\centering
\includegraphics[width=0.3\textwidth]{cavity_drawing.eps}
\caption{Schematic of lasers as scattering systems:
a singly connected region surrounded by a last scattering surface
containing an irregularly shaped cavity defined by a linear dielectric function $\varepsilon_c \ne 1$ and
a non-linear gain region. Any incoming signals (blue arrows) are amplified and scattered by the cavity to outgoing states (red arrows);
lasing corresponds to outgoing solutions without any input, and is only possible at discrete frequencies for a given cavity
\label{fig:cavity}}
\end{figure}

The laser so defined can function both as an amplifier of input radiation (when the gain medium is inverted) and as a self-organized
oscillator/source of radiation in the absence of input, above the first threshold for lasing.  From this point of view the laser is a
scattering system which both scatters and amplifies input radiation, and also, at discrete frequencies, scatters specific internally generated
modes of outgoing radiation when sufficiently pumped.  However unlike linear scattering systems, the laser is subject to non-linear saturation
effects which are fundamental to its stability and must be taken into account.

Other non-linear effects within lasers, such as four-wave mixing, can destabilize their simple harmonic time-response, but these effects
are neglected in SALT, and also (typically) in the earlier laser theories which motivated the work by Lang \textit{et al}.~\cite{lang_why_1973}.  Leaving out such effects is
equivalent to assuming a stationary inversion density, giving rise to a gain susceptibility which does not mix frequencies.  This assumption
is always valid for single-mode lasing and amplification, which will be the focus of this work; it is also valid under certain rather general conditions
for multimode lasing in microlasers, and SALT has been shown to apply quantitatively to such cases \cite{ge08,cerjan12,cerjan_csalt_2015,cerjan_isalt_2014}.

Before proceeding, it should be mentioned that the input/output theory developed by Collett and Gardiner (and others)
also considers the cavity to be a scattering element \cite{collett_squeezing_1984,gardiner_input_1985} and is able
to treat the full quantum operator properties of the scattered fields.  This approach has certainly contributed
to our understanding of open quantum systems and the role played by the reservoirs and noise processes.
However, while there has been substantial
effort expended to describe lasing within this framework  \cite{glauber_quantum_1991,hackenbroich_field_2002,viviescas_field_2003,hackenbroich_quantum_2003},
it remains an open problem how to include the strong non-linearities present in active lasing systems in this
formalism; whereas at the semiclassical level this is done quite accurately by SALT.

\subsection{Scattering in the SALT formulation}

Initially we focus on the single-mode lasing case, as was done by Lang \textit{et al.}, so that the positive frequency component of the electric field will be given by:
\begin{equation}
{\bf E}^{(+)}(\bx,t) = {\bf \Psi} (\bx) e^{-i\omega t},
\end{equation}
where the {\it real} number, $\omega$, is
either the input radiation frequency (for the amplifier) or the unknown frequency of the laser oscillator, which must be
self-consistently determined.   ${\bf \Psi} (\bx) $ is the unknown spatial lasing (or amplified) mode, not assumed to be
equal to any passive cavity mode or quasi-mode and is also self-consistently determined; such lasing/amplifier modes {\it describe the solution over all space}, not just inside the cavity.

Assuming stationary inversion and neglecting quantum and thermal fluctuations, we can describe ${\bf \Psi} (\bx)$ for an
arbitrary laser by a non-linear wave equation of the form:
\begin{equation}
[{\mathbf \nabla \times \nabla \times} + (\varepsilon_c +  4\pi \chi_g) k^2] {\bf \Psi} (\bx) = 0, \label{eq:maxWave}
\end{equation}
where $k=\omega/c$, and $\chi_g$ is the complex, frequency-dependent saturable gain susceptibility, which can be calculated
for an arbitrary N-level atomic gain medium in steady-state \cite{cerjan12}.  For the case of a two-level gain medium it takes the form
\begin{equation}
\chi_g(\omega,\bx) = \frac{\frac{1}{4\pi}D_0 \gamma_\perp}{(i\gamma_\perp + (\omega - \omega_a))(1 + \Gamma(\omega-\omega_a)|{\bf \Psi} (\bx)|^2) }, \label{eq:chiDef}
\end{equation}
where $D_0$ is the pump parameter (proportional to the inversion), $\gamma_\perp$ is the dephasing rate of the polarization,
$\omega_a$ is the atomic resonance frequency, and $\Gamma (\omega-\omega_a)$ is a Lorentzian centered at $\omega_a$.  $D_0$ and ${\bf \Psi}$
are measured here in appropriate dimensionless units \cite{ge08}, and this form assumes that there is only one mode, either lasing
or amplified, active in the medium. (This is the only case we will consider explicitly here, but we will review the more general multimode SALT formalism in Appendix \ref{app:SALT}). The
non-linear gain susceptibility represents the polarization of the gain medium and is the source of the amplification and (above threshold)
of laser emission.  The lack of any Langevin force terms in Eq.~(\ref{eq:maxWave}) is a result of the semiclassical approximation, in which the
operators for the electromagnetic and polarization fields are replaced by their average values, eliminating the effects of quantum and thermal
fluctuations.  Such terms are taken into account in N-SALT, which will be discussed briefly in Sec.~\ref{sec:noise}, but were
neglected (intentionally) in the work of Lang \textit{et al.}

Assuming the system is pumped below the first lasing threshold and the inputs are small enough to generate negligible saturation of the
gain, this equation defines a linear non-unitary scattering (S-)matrix for input fields. Depending upon the geometry there are different
natural basis states describing input and output modes, ${\bf H}_{\alpha}^{(-)}(\bx)$, ${\bf H}_{\alpha}^{(+)}(\bx)$, from and to infinity, i.e.~sufficiently far away from the cavity region,
\begin{align}
{\bf \Psi}(\bx) =& \sum_\alpha  [c_\alpha {\bf H}_{\alpha}^{(-)}(\bx) + d_\alpha {\bf H}_{\alpha}^{(+)}(\bx)]  \\
=& \sum_{\alpha, \beta}  [c_\alpha ({\bf H}_{\alpha}^{(-)}(\bx) + S_{\alpha,\beta} {\bf H}_{\beta}^{(+)}(\bx))]
\end{align}

If the pump is strong enough to invert the medium and create gain at the input frequency, the S-matrix will be amplifying, with
eigenvalues greater than unity.  When functioning as an amplifier the boundary conditions are familiar: an input wave from infinity
interacts with the cavity plus gain medium, scattering according to the appropriate dielectric or metallic boundary conditions in
different regions of the cavity and giving rise to an outgoing wave which differs from passive scattering in that it is amplified
and more photon flux emerges from the scattering region than enters it. The case of large inputs leading to a saturated gain and the
resulting non-linear S-matrix can be handled by I-SALT and will be included in the discussion below.

As noted, In this formulation there is no distinction between cavity modes and modes of the universe,
every amplified mode exists inside and outside the cavity as a steady-state scattering solution.
However we now consider to the more challenging and subtle case of laser oscillation in the scattering picture.

\section{Quasi-modes, Resonances and Quasi-bound States \label{sec:poles}}

Any scattering region of finite extent with a real potential or dielectric function, and hence with a unitary S-matrix,
will have associated with it resonant solutions; i.e.~stationary solutions at discrete
complex frequencies which are purely outgoing.  It is easiest to think of these as evolving out of the bound
states of the system at real frequencies, $\tilde{\omega}_n$, which would occur if the region were surrounded by a perfect lossless
reflecting barrier.  As the reflectivity of the fictitious barrier is reduced from unity to zero these real frequency solutions,
corresponding to standing waves, evolve into complex frequency solutions with a substantial outgoing wave component (and no incoming
wave from infinity). Due to the purely outgoing boundary conditions it can be shown that the imaginary part of the frequency must be negative, and
the S-matrix, continued to complex frequencies, will become infinite with simple poles at $\omega_n - i \gamma_n/2, (\gamma_n > 0)$.
Hence these purely outgoing solutions are discrete and countably infinite, in one to one correspondence with the bound states of the closed cavity.

In the asymptotic region the frequency and wavevector of the solutions are linked by the dispersion relation and
the sign of $\gamma_n$ will cause the corresponding spatial solution to grow exponentially at infinity and carry a diverging outgoing
photon flux. Hence these solutions are not physically realizable, but have physical significance because the S-matrix will have a resonant
response near the real values $\omega_n$, with a bandwidth given by $\gamma_n$ (assuming well-separated high finesse resonances).
The corresponding spatial solutions are referred to by various terms: quasi-modes, quasi-bound states and resonances. Historically the use of
multiple terms seems to have arisen from the transition from 3D confinement, as e.g.~in a microwave resonator, where the connection to bound
states is more apparent, to open, axial mirror-based resonators, where the resonances perpendicular to the mirror axis are essentially absent,
and the Fox-Li method was developed to determine the high $Q$ resonances. However there is no fundamental difference between the various types
of outgoing solutions: they all correspond to poles of the passive cavity S-matrix.

Such states also are very well-known from quantum scattering theory (where they are usually referred to as quasi-bound states), and are often
considered to be physical states because if an initial condition is given corresponding to the quasi-bound state truncated to the ``cavity region'',
it will leak out to infinity with a rate given by $\gamma_n$ (this property was explicitly demonstrated for the one-dimensional cavity treated by Lang \textit{et al.}).
However in this context we would like to emphasize that the true quasi-bound states, including the exponentially growing component at infinity,
are not physically realizable steady-state solutions and cannot correspond to the lasing mode.
Also, since they occur at the poles of the S-matrix for the passive cavity, they can only
describe the properties of the passive cavity, independent of the gain medium and spectrum.  Thus using ``quasi-mode'' also as a term
for the lasing mode in the presence of gain can, and in our view often has, led to confusion.

\section{Threshold Lasing Modes and Constant Flux States \label{sec:tlm}}

However, as described above, the properties of the laser/amplifier are not governed
by a unitary S-matrix once the complex gain susceptibility is taken
into account.  This complex susceptibility violates the Hermiticity of the wave equation even for the closed cavity and invalidates the
standard modal decomposition.  But it makes possible purely outgoing solutions at real frequencies, which hence are physical solutions
over all of space.  The non-equilibrium steady-state gain susceptibility depends on the pump level, with its imaginary part changing sign
from positive (absorbing) to negative (amplifying) at the transparency point, at which the upper and lower lasing levels are equally
occupied and the susceptibility vanishes. At this pump value the poles of the cavity with gain coincide again with those of the passive
cavity. As the pump is increased beyond the transparency point, all of the passive cavity poles continue to exist, but typically they
move up in the complex plane towards the real axis \cite{ge10,chong12,pillay_2014}, at a rate that depends on their position in the gain spectrum.
The
S-matrix of the amplifier can be measured for real frequencies as the pump increases, and the height of the resonances corresponding
to each pole increases (the transmitted flux is greater than the incident flux due to amplification),
while their width decreases (gain narrowing), because the effective loss rate is proportional to the
imaginary part of the pole frequency, which is decreasing.  If there is no input to the amplifier then there is still no emitted field
and no indication within semiclassical theory of the approach to lasing (although when quantum fluctuations are included there will
be enhanced noise, amplified spontaneous emission (ASE), near the gain center).  If lasing is achieved, one of the poles of the S-matrix reaches
the real axis, gain completely balances loss, and we have a zero width resonance on the real axis.
This process is demonstrated in Fig.~\ref{fig:poles} for a non-trivial two-dimensional cavity,
a chaotic ``D-shaped" dielectric cavity \cite{redding_dcav_pnas_2015}, where each pole is seen to move
towards the real axis an amount proportional its distance from the center of the gain curve and one of the poles
experiences enough gain to reach threshold.

\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth]{poles_D.eps}
\caption{Simulations for a D-shaped laser cavity with $R=4\mu m$, $r_0 = .5 R$, $n=3.5$ as shown in the
bottom schematic, with a uniform two-level atomic
gain medium characterized by $\lambda_a = 1\mu m$ and $\gamma_\perp = 10nm$. (Top) Plot of the Lorentzian gain profile
for the atomic medium. (Bottom) Plot of the motion of the poles of the S-matrix as the D-shaped cavity
is pumped to the first lasing threshold. The pole corresponding to the first lasing mode is seen to reach the
real axis (marked with a red circle), and the spatial profile of the lasing mode at threshold is shown
just above.
\label{fig:poles}}
\end{figure}

It first may seem that we are back at the familiar situation of true bound states, which correspond to poles on the real axis, but
this is not so.  In the former case there is no photon gain and one simply has a decaying standing wave pattern, trapped in some region
of space. Here we have a different type of object, which we call a ``threshold lasing mode'' (TLM).
This type of solution never occurs in Hermitian theories, a purely outgoing solution at real frequency; it reflects the fundamentally
non-Hermitian nature of the electromagnetic wave equations with a complex susceptibility, which is an effective theory in which the
matter degrees of freedom are hidden.  Such a solution is characterized by a steady-state photon flux out to infinity, but with no
unphysical growth of intensity {\it outside} the cavity and an emission pattern corresponding to the eigenvector of the S-matrix
with a diverging eigenvalue. It is an example of a constant flux (CF) state: photons are generated within the cavity but are conserved
outside the cavity, hence corresponding to a true physical lasing solution \cite{tureci06,tureci08,ge10}.  In Appendix \ref{app:SALT} we will show that each TLM is just one member of a
complete biorthogonal basis set of CF states which forms a natural basis set for describing the non-linear lasing solution above threshold.

However we still have a conceptual problem: the linear S-matrix for this amplifier is now infinite at the lasing frequency.
Within linear theory any input would be infinitely amplified, violating conservation of energy.
The resolution of this dilemma is
that infinitesimally above threshold the saturating non-linearity kicks in and stabilizes the solution.  In fact the steady-state lasing
amplitude rises smoothly from zero and has no jump at threshold, just a discontinuous derivative.  Even though formally the S-matrix has
a pole fixed on the real axis above threshold, any input wave at the lasing frequency simply increases the gain saturation and doesn't
lead to a runaway of the output. Strictly speaking, any such input actually forces the pole off the real axis, as we will show with an
example in Sec.~\ref{sec:example}.  Hence the non-linearity is essential for stablizing the linear CF solution.  However, the actual solution
near threshold is accurately described by the single TLM calculated from the linear wave equation with gain, since the saturation term
is not strong enough to change the spatial form of the solution significantly.

Returning to the fundamental question of modes of the universe vs. quasi-modes of the cavity, we summarize the SALT perspective.
\begin{itemize}
\item  If one treats lasers as unbounded scattering systems from the outset, all solutions are defined both inside and outside the
cavity region and there is no distinction between cavity modes and modes of the universe.
\item If the laser is functioning as an amplifier, the solutions are simply amplified scattering states determined by the boundary conditions
in the cavity region and the degree of gain at the input frequency.
\item If the laser is functioning as a self-organized oscillator the solutions are purely outgoing states corresponding to an
eigenvector of the linear scattering matrix with diverging eigenvalue due to the existence of a pole on the real axis.  Such
solutions require the presence of the saturating non-linearity to be physically stable.
\item For any given cavity and gain susceptibility such purely outgoing states can only occur at discrete frequencies because they
arise from moving the discrete poles of the passive cavity onto the real axis.  Within semiclassical theory (neglecting
quantum and thermal fluctuations of the fields) these solutions have precisely zero linewidth.
\end{itemize}
We reserve the term ``lasing mode'' for these special discrete solutions.

In summary then, our response to the question ``why is the laser linewidth so narrow?''~is that indeed this {\it is} due to the extreme limit of
gain narrowing: only at discrete frequencies can the linear gain diverge and generate self-organized oscillation.  As we
will show in the next section, this zero-width resonance cannot be observed with an input signal, i.e.~in amplification mode.
However these perfectly monochromatic purely outgoing solutions can be found by SALT and have been
verified by brute force FDTD simulations of the laser equations for many examples \cite{ge10,cerjan12,cerjan_csalt_2015}.
Such infinitely sharp lines are not of course obervable in true laser emission due to quantum
fluctuations as we will discuss in Sec.~\ref{sec:noise}. However the smallness of these quantum effects lead to the extraordinary
narrowness of the resulting lines, often many orders of magnitude narrower than the passive cavity resonance linewidths.

The effects of quantum fluctuations were intentionally not included in the work of Lang \textit{et al.}, who thus also found a delta function
laser line within semiclassical theory.  In their approach they find that this extreme narrowing of the passive cavity
resonance can be regarded as a ``mode-locking effect'' in which all of the modes of the universe within the passive cavity
resonance linewidth contribute to one lasing quasi-mode with a single frequency. This point of view emerges from a limiting
procedure in which a finite system with quantized electromagnetic modes is studied and at the end the system size is
taken to infinity in the presence of small fictitious absorption to damp out reflections from the boundaries as they tend to
infinity.  The conditions for this procedure to reproduce the correct infinite system behavior are discussed in Sec.~\ref{sec:Modes} below.
We note that Lang \textit{et al.}\ expressed dissatisfaction with invoking gain narrowing to explain the laser linewidth, because
in their view such arguments (at the time) did not take into account the non-linearity of the problem.
However the conceptual framework of SALT/I-SALT, outlined here, {\it does} take
non-linearity into account, essentially exactly, for single-mode lasing. We will now illustrate the previous concepts with
a concrete one-dimensional example, very similar to the one studied by Lang \textit{et al.}, in the process making all of our statements above mathematically precise.

\section{Lasing modes as zero-width resonances: A simple example \label{sec:example}}

The example studied by Lang \textit{et al.}\ was a one-dimensional cavity of length $L + l$, with perfectly reflecting mirrors
at $x=l,-L$ and a partially reflecting delta function mirror at $x=0$, as shown in Fig.~\ref{fig:langCav}. The ``cavity'' is the region $[0,l]$ and the ``universe'' is
the region, $[-L,0]$, where in treating lasing a small uniform absorption is added over all space, the limit $L \to \infty$ is taken,
and lasing emission is in a single direction towards $x =  - \infty$.
For such a system the passive cavity will have a reflection coefficient of modulus unity at all frequencies,
and resonant reflection will be distinguished only by a long delay time in reflection and an associated build-up of field intensity in the cavity.
We prefer to make a small modification of the Lang \textit{et al.}\ example and
study a symmetric two sided partially transmitting (Fabry-Perot) cavity with laser emission to $x = \pm \infty$, and with passive cavity
resonances distinguished by enhanced transmission, reaching unity at the center, and with width, $\gamma_c$. As with
the Lang \textit{et al.}\ example, the electromagnetic part of the problem will be described by the scalar Helmoltz equation with gain.
However, unlike Lang \textit{et al.}, no limiting procedure is applied; our system is infinite from the start and no fictitious absorption is
added to avoid reflections from perfect mirrors far away from the cavity.  Hence, consistent with our scattering approach,
we start from continuous solutions for the
passive cavity $+$ ``universe'', and will impose outgoing boundary conditions at infinity on the lasing solutions.

\begin{figure}[t!]
\centering
\includegraphics[width=0.3\textwidth]{lang_cavity.eps}
\caption{Schematic showing the system studied by Lang \textit{et al.}\ \cite{lang_why_1973}, which
contains two regions, the cavity and the much larger ``universe'' (which also includes the cavity).  Damping
is added to the external region to prevent reflections from the boundary at $x=-L$, and as discussed below, this
must be done by a particular limiting procedure in order to obtain the correct result for the laser.
\label{fig:langCav}}
\end{figure}

We begin by reviewing the properties of such a one-dimensional optical cavity. In Fig.~\ref{fig:passive}
we plot the transmitted intensity of an incident signal through the passive cavity at frequency $\omega_{\IN}$.
As expected
we observe a series of resonances of the cavity, spaced by the free spectral range, $\Delta = \pi/cL$,
with a width given by the passive cavity decay rate $\gamma_c = -c/(2L) \ln(R^2)$, where $L$ is the length of the cavity.
The incident and reflected waves on the left side of the cavity satisfy a continuity boundary condition,
with both incoming and outgoing components, whereas
the transmitted wave leaving the right side of the cavity also is continuous but in addition is purely outgoing.

Any purely outgoing monochromatic wave is described
mathematically by the Sommerfeld radiation condition \cite{sommerfeld},
\begin{equation}
\lim_{|x| \to \infty} |x|^{\frac{d-1}{2}} \left( \partial_{|x|} - i k\right) \psi(x) = 0, \label{eq:SRC}
\end{equation}
in which $\hat x = \vec x / |x|$, $\omega = c k$, and $d$ is the dimensionality of the system.
In one dimension the external waves are simple plane waves (and not e.g.~cylindrical or spherical
waves) and the Sommerfeld condition can be imposed anywhere outside the boundary;
thus at the boundary
on the right side of the cavity the field must be an outgoing plane wave,
satisfying,
\begin{equation}
\left[\left( \partial_{x} - i k\right) \psi(x)\right]_{x=L} = 0.
\end{equation}

\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth]{passive.eps}
\caption{(Top) Plot of the transmitted intensity of an injected signal with frequency $\omega_{\IN}$,
into a passive, two-sided cavity with partially reflecting mirrors ($R=0.8$) on both ends, as
indicated in the schematic. (Bottom) Plot of the poles, $\omegap$ of the scattering matrix of the same
cavity, corresponding to purely outgoing solutions at complex frequency.
The real freuency resonances of the cavity, where complete transmission is seen,
are related to the poles of the cavity.
\label{fig:passive}}
\end{figure}

As noted, lasing would correspond to a solution satisfying a purely outgoing boundary condition
on {\it both} sides of the cavity (or in all directions in higher dimension).  If we attempt to impose this boundary condition
on the passive cavity we find that no solutions are possible at real frequencies, but there are discrete periodically
spaced solutions at
\begin{equation}
\omega_m = \frac{\pi}{2cL} + m \Delta - i \gamma_c/2 \label{eq:QBeq}
\end{equation}
as shown in Fig.~\ref{fig:passive}.  The discreteness of
the frequencies is a direct consequence of the Sommerfeld condition which can only be satisfied when there is constructive
interference in the cavity.  The fact that all the resonance frequencies have the same imaginary part is not generic, but is
particular to one-dimensional cavities of this type. As noted above, the presence of these purely outgoing (pole) solutions leads to
resonant transmission on the real axis centered at $\re[\omega_m]$ with width $\gamma_c$.  Applying the Sommerfeld condition
to the outgoing solutions $e^{\pm ikx}$, and the relation $\omega = ck$ implies that $|\Psi (x)|^2 \sim e^{+\gamma_c |x|}$ at $\pm
\infty$, the unphysical divergence we alluded to above.

In order to find normalizable physical solutions which are purely outgoing (i.e.~lasing solutions) we need to
add the effect of the gain medium to the cavity dielectric function in the form of $\chi_g$ as defined in Eq.~(\ref{eq:chiDef})
above.  For this simple one dimensional scalar model (similar to Lang \textit{et al.}), below threshold, the electric field satisfies
\begin{equation}
[\partial_x^2 + (nk)^2 ] \Psi (x) = 0,
\end{equation}
with the Sommerfeld boundary condition,
where now the index of refraction $n = \sqrt{\varepsilon_c + 4\pi \chi_g} = n_1 - in_2$, and the gain non-linearity
is neglected, because we are below threshold and these are not yet physically allowed solutions (they are simply modified active cavity resonances).
Note that the solutions inside the cavity here take the form $A \cos (n x) + B \sin (nx)$, i.e.~they are trigonometric functions
of a {\it complex} argument.  This will remain true at threshold and is a key point which holds more generally:
the electric fields for the physical system with
gain cannot be purely real; if they were real they would conserve photon flux and describe true standing waves, whereas lasing
or amplified modes must have running wave components. Any model which finds real spatial solutions for lasing or amplified modes has made
an approximation. Such approximations have often been made in laser theory, including by Lang \textit{et al.}\
\cite{lang_why_1973,baseia_semiclassical_1984,penaforte_quantum_1984,dutra_spontaneous_1996} (see further discussion
in Sec.~\ref{sec:Modes}).

\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth]{active_plot1.eps}
\caption{(Top) Plot of the transmitted intensity of an injected signal as a function of frequency, $\omega_{\IN}$, illustrating line narrowing with
increased gain.  $I_{\IN} = I_0 = 0.01$, and the two-sided cavity has partially reflecting mirrors ($R=0.8$) at both ends, (see
schematic). The pump on the cavity is increased from the transparency point, $D_0 = 0$ (red),
to just above the first lasing threshold, $D_0 = 0.0040$ (blue), calculated using I-SALT \cite{cerjan_isalt_2014}.
As can be seen, the width of the cavity
resonance narrows during this process. The slight shift of the resonance is due to line pulling from
central atomic transition frequency, $\omega_a = 60$, with $\gamma_\perp = 2$. At the transparency point this
system is identical to that shown in Fig.~\ref{fig:passive}, and the outermost red curve seen here is identical
to that from Fig.~\ref{fig:passive}. (Bottom) Plot of the
motion of the pole of the associated cavity resonance, $\omegap$, through the complex plane as the pump is increased. This is also calculated using
I-SALT, and thus includes the saturation of the gain medium due to the
injected signal, which prevents the pole from reaching the real axis even though the final pump
value is above the first non-interacting modal threshold. For the specific realization of the
motion of the pole shown, the injected field is taken at $\omega_{\IN} = 60.25$, which corresponds
to the peak in the blue curve. The pole is also seen to drift due to line pulling. Frequencies
and rates are reported in units of $c/L$.
\label{fig:active1}}
\end{figure}

When we add the gain medium to the cavity and pump above the transparency point
$(D_0 > 0)$ the imaginary part of $\chi_g$ corresponds to amplification.  This
has an immediate effect, causing the poles to move upwards, as discussed earlier and as shown for this specific cavity in Fig.~\ref{fig:active1}.
While the pole solutions are still not physically realizable, we can observe a change
in the cavity properties via the behavior of the transmission resonances at real frequencies.  To calculate this effect accurately with
a finite input wave we must now use the full saturable gain susceptibility of Eq.~(\ref{eq:chiDef}).  The simulations
shown here are performed using I-SALT which provides a nearly exact description of
the steady-state single-frequency response of an active optical cavity to injected signals and any self-generated lasing
signals \cite{cerjan_isalt_2014}, including the spatial degrees of freedom. SALT and I-SALT are reviewed in more detail in Appendix~\ref{app:SALT}.

As the pole rises towards the real axis the amplified resonance narrows.  Due to the frequency-dependent saturation
of the gain medium, as already noted, injected signals with different frequencies will induce
somewhat different trajectories for the pole in order to maximize the effective gain
 \cite{cerjan_isalt_2014}. These are essentially line-pulling or pushing effects of the gain
medium, which for a single frequency are always towards the atomic transition frequency.  In the active cavity with dispersion,
the precise width of the transmission resonance can no longer be defined as the distance of the
pole from the real axis, as probing different frequencies will yield
slightly different distances of the pole from the real axis. These line-pulling effects lead to the mild
asymmetry in the series of curves shown in Fig.~\ref{fig:active1}.

The bluest (and narrowest) curve
shown in Fig.~\ref{fig:active1} corresponds to a pump value just {\it above} the non-interacting
first lasing threshold. This may seem initially puzzling; wasn't the lasing threshold the point at which
the pole reaches the real axis?  That definition only applies when there is no input signal.  With
an input signal, even at the lasing frequency, the saturation of the gain medium prevents the pole
from reaching the real axis, keeping the response finite, and the amplifed resonance retains a non-zero width.

\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth]{active_plot2.eps}
\caption{(Top) Plot of the transmitted intensity of an injected signal as a function of frequency $\omega_{\IN}$,
into the active, two-sided cavity with partially reflecting mirrors ($R=0.8$) on both ends, as shown
in Fig.~\ref{fig:active1}. The pump on the cavity is held fixed just above the first lasing
threshold, $D_0 = 0.0040$, while the incident intensity is decreased from $I_{\IN}= I_0 =0.01$ (blue),
to $I_{\IN} = 0$ (red), where the cavity is operating as a free-running laser, calculated using I-SALT \cite{cerjan_isalt_2014}. As can be seen,
the width of the cavity resonance continues to narrow during this process, finally resulting
in the lasing signal with infinitesimal width.
Note, the outermost blue curve shown here is the same as that shown in Fig.~\ref{fig:active1}.
(Bottom) Plot of the motion of the pole of the associated cavity resonance, $\omegap$,
through the complex plane as the injected signal is turned off for the specific realization of
$\omega_{\IN} = 60.25$. This is also calculated using
I-SALT, and thus includes the saturation of the gain medium due to the
injected signal, which prevents the pole from reaching the real axis until the injected
signal is turned off. For different values of $\omega_{\IN}$ (not shown) the pole is able
to reach the real axis before the injected signal is fully turned off.
\label{fig:active2}}
\end{figure}

To see the true zero-width semiclassical lasing solution we slowly turn {\it off} the incident signal
with the pump held above the
first lasing threshold.  With this protocol we see that the transmission resonance width
continues to narrow,
as shown in Fig.~\ref{fig:active2}, while the pole continues
to move towards the real axis.
As the injected signal is turned off, the pole reaches
the real axis and the cavity begins to undergo self-organized lasing action,
shown as the reddest curve in the top panel of Fig.~\ref{fig:active2}.
This curve has precisely zero linewidth; no radiation comes out of the cavity except at this single frequency
without input waves
(assuming effectively zero temperature in the cavity and neglecting quantum fluctuations until
Sec.~\ref{sec:noise}).  This effect can never be fully achieved through gain-narrowing of an amplified input signal,
for the reasons we described above: gain saturation forces all poles off the real axis.  Hence this
effect does require correct treatment of the non-linearity as stated by Lang \textit{et al.}  However we have
shown here that it is possible to treat the non-linearity exactly at each stage and explain the
zero width of the laser line quantitatively.

The model we have treated here is essentially the same as that treated by Lang \textit{et al.}\ except
for the pedagogical difference of choosing a two-sided cavity in which one can study transmission resonances;
in the next section we will study exactly the model of Lang \textit{et al.}\ by a different but equivalent method.
As noted, in the standard SALT approach the system is infinite from the outset and there
is no dissipation outside the cavity, which is in our view the most natural model for a real laser system.
In contrast, Lang \textit{et al.}\ place their cavity in a much larger box which
eventually will be taken to infinite size, and add dissipation external to the cavity, intended to damp unphysical reflections
from the boundaries of the larger fictional box.  We will discuss the effects of this difference in the next section.

\section{Lasing modes and the Limiting Absorption Principle. \label{sec:Modes}}

While it is clear that lasing modes should satisfy the purely outgoing (Sommerfeld) boundary condition,
this leads to a rather unfamiliar description of the spatial behavior of the lasing modes in terms of
functions that are not power orthogonal, but instead satisfy a biorthogonality condition \cite{siegman,hamel_nonorthogonality_1989,hamel_observation_1990}.
Basis functions of this type, the CF states \cite{tureci06,tureci07,tureci08,ge10}, are central to the standard computational algorithm of SALT.
The approach taken by Lang \textit{et al.}\ of beginning with a finite system and adding damping to
remove the effect of the boundaries of the system is an alternative which
has evolved into the well-developed computational technique of surrounding a cavity or region of emitters with
a perfectly-matched (damping) layer (PML). When done correctly this can reproduce the imposition of Sommerfeld conditions
to arbitrary accuracy.  In fact, newer computational methods for solving the SALT equations \cite{esterhazy14} make use of PMLs and do not impose
the Sommerfeld boundary conditions in the straightforward manner of matching to outgoing solutions at a last scattering surface as did the
earlier methods of satisfying the SALT equations
\cite{tureci08}. The PML approach gives essentially perfect agreement with the Sommerfeld approach to SALT.

However, as noted above, the correct lasing solutions {\it must} be complex functions in space, so as to
represent the non-zero photon flux, which grows inside the cavity and remains constant outside.
For the simple 1D cavities studied here and by Lang \textit{et al.}, this means that the spatial solutions inside a constant
index cavity must be trigonometric functions of complex argument, corresponding to exponential growth towards
the mirrors as well as oscillation.  However in the work of Lang \textit{et al.}\ they find a lasing mode of the form
$E(x,t) \approx C_0e^{(\alpha - i\omega) t} \sin [k_0 (x-l)]$ for $ 0 < x <l$, where $\alpha$ is the difference between gain
and loss, $\omega$ is the lasing frequency, $C_0$ is a constant, and $k_0 = \Omega/c$ is the real wavevector
corresponding to the passive cavity resonance frequency, $\Omega$.  Since the spatial part of this function is real, it
does not capture the photon flux through the partially transmitting mirror and is only an approximation to the complex
sine wave solution which is correct for this one-sided cavity.

In our view the origin of this problem is the neglect of the Limiting Absorption Principle \cite{ignatowsky_reflexion_1905,eidus_LAP_1962,schulenberger_limiting_1971},
which gives the correct mathematical procedure for recovering the Sommerfeld radiation condition from a finite system.  Lang \textit{et al.}\ solved the
spatial part of the problem for finite $L$, finding the quantized Hermitian basis states for the whole passive system (``modes
of the universe'') and then addressing the lasing problem within this Hermitian basis set.  Effectively the spatial
part of the problem is fixed at finite $L$, and a small absorption, $\gamma$, is subsequently added everywhere in the universe so as to eliminate
the effect of reflection from the fictitious perfect mirror at $x=-L$.  For this procedure to work one needs to solve
the spatial problem {\it with} the finite damping, and carefully take the limit $L \to \infty$ prior to $\gamma \to 0$; this requirement
is known as the Limiting Absorption Principle (LAP). From a
practical computational point of view the LAP means that one needs to solve the {\it non-Hermitian}
spatial problem with Dirichlet (or other linear homogeous) boundary conditions {\it and}
finite damping, while maintaining the condition $\gamma k L \gg 1$.  The solutions so obtained have only exponentially small sensitivity
to the presence of the boundary and reproduce to high accuracy those obtained by actually imposing directly the Sommerfeld
conditions.  Hence they are significantly different from the Hermitian solutions calculated before the damping is added (or when the limits
are taken in the opposite order), which
can carry no net flux as they are bound states.  The modern PML techniques are a means for introducing {\it non-uniform}
absorption which is impedance matched so that reflections are negligible even when $L$ is {\it not} very large and one
reproduces the Sommerfeld solutions with a much smaller and manageable computational cell.  But the principle is basically
the same: in order to get the correct spatial solutions one must treat the non-Hermitian problem with damping from the
outset.

To illustrate this explicitly we briefly analyze the Lang \textit{et al.}\ system from this point of view.
The system is the one-sided cavity between $[0,l]$ attached to the ``universe'' between $[-L,0]$, as shown in Fig.~\ref{fig:langCav}, with
perfect mirrors (Dirichlet boundary conditions), $E(l) = E(-L) = 0$,
where $E(x,t)$ is the electric field of the system. Before adding the gain and considering
lasing, we explicitly include the weakly absorbing material uniformly in the universe, and focus
on the region external to the cavity. The wave equation in this region is written as,
\begin{equation}
\left[ \partial_x^2 - \frac{1 + i \gamma}{c^2} \partial_t^2 \right] E(x,t) = 0, \;\;\; -L \le x < 0.
\end{equation}
We expand the field inside of the system in terms of a set of basis functions,
$U_k(x)$, so that $E(x,t) = \sum_k U_k(x) e^{-i \omega t}$, and
solve for the form of these basis functions in the universe region as,
\begin{multline}
U_k(x) = c_{1,k} e^{ikx - \frac{\gamma}{2}kx} + c_{2,k} e^{-ikx +\frac{\gamma}{2}kx}, \\ -L \le x \le 0, \label{eq:Uksol}
\end{multline}
where we have expanded the exponent for $\gamma \ll 1$.
The coefficients $c_{1,k}$ and $c_{2,k}$ and the quantized values of $k$ are set by the boundary conditions $U_k(-L)$, $U_k(l) = 0$,
and the matching conditions at $x=0$, and will vary with $L$, $l$ and the transparency of the mirror at the origin.

We first apply the LAP to obtain the Sommerfeld solution for this problem, taking the limit
$L \to \infty$ prior to $\gamma \to 0$.
Performing the limits in this order on the solution of $U_k(x)$, Eq.~(\ref{eq:Uksol}), one finds
that to satisfy the Dirichlet boundary condition, $U_k(-L) = 0$, we must choose $c_{1,k} = 0$,
as the term multiplying this coefficient is diverging as $-L \to -\infty$ for finite $\gamma$. Then, with the
spatial limit taken, we can safely reduce the absorption in the problem to zero,
recovering
\begin{equation}
U_k(x) = c_{2,k} e^{-ikx}, \;\;\;\; -\infty \le x \le 0,
\end{equation}
finding that all of the solutions of our problem are purely outgoing from the cavity,
consistent with the Sommerfeld radiation condition, Eq.~(\ref{eq:SRC}). However, if these two limits are taken in the opposite order,
setting the absorption to zero before letting the length increase to infinity,
we find that
\begin{equation}
\tilde U_k(x) = b_k \sin(k(x+L)), \;\;\;\; -L \le x \le 0, \label{eq:Ukwrong}
\end{equation}
which is exactly the result of Lang \textit{et al.}\ (outside the cavity region).  With this result the matching to the cavity allows for real sine wave solutions,
whereas with the outgoing solutions only complex sine waves are allowed in the cavity.  The complex sines provide the outgoing
flux at the partially transmitting mirror which is needed for continuity with the field outside the cavity.

These conclusions can be further reinforced by directly solving for the eigenfunctions of
the finite system studied by Lang \textit{et al.}, shown in the three panels of Fig.~\ref{fig:langFig}.
As $L$ is increased, the modes of this finite system separate into
two sets. One set consists of discrete isolated solutions with smaller imaginary frequency (damping); the number of such solutions
does not change with increasing $L$. The
second set has larger damping, and gets denser and denser, forming a psuedo-continuum as $L$ is increased.
The first set corresponds
precisely to the complex resonance frequencies of the passive cavity, as
can be confirmed by plotting the known resonances of the cavity given by Eq.~(\ref{eq:QBeq}),
shown in Fig.~\ref{fig:langFig}(c).  The spatial functions corresponding to the passive resonances are identical to the
resonant solutions of the infinite system without damping {\it inside the cavity}, as there is no loss there, and
the boundary condition at the mirror is the same.  Outside the mirror these solutions experience a damping not
present in the infinite system and hence decay slowly as they propagate outwards.  On the scale shown in
the top panel of Fig.~\ref{fig:langFig2} one sees only that the outgoing boundary condition is satisfield and not the
weak decay. In contrast, the spatial solutions corresponding to the dense set of damped modes of the universe
can be seen in the bottom panel of Fig.~\ref{fig:langFig2}; they form standing waves outside the cavity, with small
leakage into the cavity, and do not satisfy the purely outgoing conditions. Hence simply discarding the latter and
keeping the first set is equivalent to finding the Sommerfeld solutions of the passive problem without damping, in the
vicinity of the cavity.

Finally, we can make contact with the SALT approach by adding gain (taken to be uniform for convenience) in the cavity region
while leaving the damping present outside the cavity.  The added gain has a much bigger effect on the cavity resonance solutions
which have high amplitude there, than on the external standing wave solutions.  For large enough gain
one of the cavity resonances can reach the real axis and will then correspond to a
threshold lasing mode for uniform gain, as shown in Fig.~\ref{fig:langFig}c (green crosses).
This is the essence of the method adopted in Ref.~\cite{esterhazy14}
except in that work a PML damping layer is used for higher computational efficiency and the saturating non-linearity is
taken into account. Thus the Lang \textit{et al.}\ method agrees with
the SALT approach if care is taken to apply the limits in the right order.

\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{lang_fig.eps}
\caption{Plot of the resonances of the finite system with external damping depicted in Fig.~\ref{fig:langCav} for three
different values of $L$, $L=25l$, $L=100l$, and $L=200l$, from top to bottom, except that the optical confinement
is instead provided by a passive cavity dielectric of $n = 6$ in the cavity, rather than a partially
reflecting mirror. In the external region an absorption of $\gamma = 0.003$ is present. In (c) the isolated solutions of the
problem with damping are compared to the resonances
of the same cavity subject to the Sommerfeld boundary condition on an infinite system without damping, given by Eq.~(\ref{eq:QBeq})
(red circles). Perfect agreement is found as discussed in the text.
(Note that due to the presence of the passive cavity dielectric constant
the right side of Eq.~(\ref{eq:QBeq}) must be multiplied by $1/n$ in making the comparison).
The green crosses shown in (c) show the
locations of the resonances when uniform gain is added to the cavity to find the threshold lasing mode, $n = 6 - .005i$.
As can be seen, the poles corresponding to resonances of the cavity have moved towards the real axis,with the rightmost
one reaching the lasing threshold, while those corresponding to the resonances of the universe have stayed relatively fixed,
as discussed in the text.  This is thus an alternative method to implement the Sommerfeld conditions for lasing modes.
\label{fig:langFig}}
\end{figure}

\begin{figure}[t!]
\centering
\includegraphics[width=0.4\textwidth]{lang_fig2.eps}
\caption{Plot of the modal amplitudes for a small region near the cavity-universe boundary for two modes,
one corresponding to a resonance of the cavity (top), and the other corresponding to a resonance of the external universe (bottom).
As can be seen, the cavity mode is a nearly pure running wave after it exits the cavity.
\label{fig:langFig2}}
\end{figure}

\section{Quantum limited laser linewidth: an alternative form \label{sec:noise}}

Since this paper is devoted to the ``SALT perspective'' on the laser linewidth, we include here
a section on the finite laser linewidth due to quantum fluctuations, which has recently been
addressed within two different generalizations of SALT, one based on input-output theory \cite{chong12,pillay_2014}
and another based on non-linear perturbation theory around the SALT solution \cite{pick_linewidth_2015}.
The fundamental origin of a finite quantum linewidth for the laser is the presence of spontaneous emission,
which leads to random jumps in the time-dependent phase of the output field.  The resulting phase diffusion is linear in
time and the phase diffusion coefficient determines the intrinsic linewidth. The basic physical understanding of this
goes back to the
seminal work of Schawlow-Townes \cite{schawlow_1958}, which was subsequently substantially extended by Lax, Henry, Petermann and many others.

Traditionally, the Schawlow-Townes (ST) linewidth is written in terms of the cavity
decay rate, $\gamma_c$, as
\begin{equation}
\delta \omega_{ST} = \frac{\hbar \omega_0 \gamma_c^2}{2 P}n_{sp}, \label{eq:STlw}
\end{equation}
where $\omega_0$ is the lasing frequency and $P$ is the total output power.  Here we have chosen to include
 in the ST formula the ``incomplete'' inversion factor, $n_{sp}$, which is sometimes treated as a correction to it
($n_{sp} \geq 1$ measures the degree of inversion of the laser frequency and tends to unity when the lasing transition is completely inverted,
the limit assumed by Schawlow and Townes in their original work) \cite{lax_conf_1966}.

As discussed extensively above, semiclassical laser theory neglects quantum effects such as spontaneous emission
and leads to a delta function linewidth.
In order to go beyond this one needs to treat all three fields, the electromagnetic field,
the polarization, and the inversion as quantum operators and consider their higher moments.  The laser linewidth will be
obtained from the correlation function of the electric field near the lasing frequency. It is well known
that the effects of these quantum fluctuations can be captured by adding appropriate Langevin (random) noise terms
to the equations of motion for the three fields \cite{haken}.  These noise terms will be related to the dissipation and
amplification in the system by the appropriate fluctuation-dissipation relations.  This was one of the questions raised
in the original work by Lang \textit{et al.}  It is now understood that semiclassical theory violates these relations but they
are restored by including the Langevin terms. The recent work by Pick \textit{et al.}\ \cite{pick_linewidth_2015} provides the most
complete treatment of these noise terms and the formalism they develop is referrred to as N-SALT (SALT + noise).

Pick \textit{et al.}\ begin with the time-dependent wave equation for the field and add the appropriate noise term
\begin{equation}
[{\bf \nabla \times \nabla \times  E} (\bx,t)  + \varepsilon_c \ddot{\mathbf{E}}(\bx,t)] = 4 \pi \ddot{\mathbf{P}} +  \tilde{\mathbf{F}}_{tot} ,
\end{equation}
in which $\tilde{\mathbf{F}}_{tot} (\bx,t) $ is the total effective noise driving the field in the cavity in units of electric field.  Since this random drive is broadband, it is no
longer possible to assume a monochromatic steady-state exists as we do in semiclassical theory.  However only the noise
near the lasing frequency is strongly amplified and by expanding the field around the SALT steady-state solution one can derive
non-linear coupled mode type equations which permit \textit{ab initio} calculation of the relevant correlation functions and
hence of the most general form of the laser linewidth yet derived \cite{pick_linewidth_2015}.  This form includes all of the important
corrections to the ST formula: the correction due to relaxation oscillations ($\alpha$ factor) \cite{henry82,henry86}, due to openness of the cavity
(Petermann) factor \cite{petermann_calculated_1979,haus_excess_1985,siegman_excess_1989,hamel_nonorthogonality_1989,hamel_observation_1990},
due to dispersion (bad-cavity factor) \cite{lax_conf_1966,kolobov_role_1993,kuppens_quantum-limited_1994,vanexter_theory_1995,kuppens_evidence_1995}, and corrections not previously studied due to the spatial
variation of $n_{sp}$ \cite{lax_conf_1966}.  We will not describe this formalism in any more detail here, but will briefly reproduce the essential
result at the end of this section.

However, in this section we would like to make a different conceptual point, not discussed in the earlier papers, relating to the
interpretation of the Schawlow-Townes formula.  It seems initially puzzling that the noise due to spontaneous
emission in the gain medium should lead to the standard ST formula, which is expressed only in terms of the
passive cavity linewidth, $\gamma_c$, the total output power, and the lasing frequency.  One would expect
the spontaneous emission rate to appear as a fundamental quantity, not the cavity decay rate; and here we will show that the ST linewidth can
(and perhaps should) be written in an equivalent form where this is the case.

The total Langevin force, $F_{tot} (t)$, in units of the quantized field amplitude operator, $\tilde{\mathbf{F}}_{tot} = i\sqrt{2\pi \hbar \omega} F_{tot}(t) \mathbf{\Psi}_0(\bx)$,
has a direct contribution from the amplitude fluctuations of the field, $F_a (t)$, due to quantum
fluctuations from outside the cavity, but has also has
indirect contributions from Langevin forces driving the atomic polarization and inversion, $F_{p}$ and $F_d$ respectively \cite{haken},
\begin{equation}
F_{tot}(t) = F_{a}(t) + \left(\frac{1}{\omega_0 - \omega_a + i\gamma_\perp}\right) \sum_\alpha g_\alpha F_{p}(t)
\end{equation}
where the summation runs over all of the atoms in the cavity at locations $x_\alpha$ and
\begin{equation}
g_\alpha = i \boldsymbol{\theta}\cdot \mathbf{\Psi}_0(\mathbf{x}_\alpha) \sqrt{\frac{2\pi \omega_0}{\hbar}},
\end{equation}
is the dipole coupling element for the quantized electric field.
$\boldsymbol{\theta}$ is the dipole matrix element for the lasing transition, and $\mathbf{\Psi}_0(x)$ is the normalized mode profile, $\int |\mathbf{\Psi}_0|^2 dx = 1$, of the electric field at frequency $\omega_0$.
The correlation function for the total effective Langevin force can then be calculated \cite{haken} as,
\begin{multline}
\langle F_{tot}^\dagger(t) F_{tot}(t')\rangle = \delta(t-t') \bigg(\gamma_c n_{th} \\
 +  \sum_\alpha \frac{|g_\alpha|^2}{\gamma_\perp^2} \left[\gamma_\perp(1 + \langle d_\alpha \rangle) + \frac{\gamma_\parallel}{2}(d_0 - \langle d_\alpha \rangle)\right]\bigg), \label{eq:Ftot}
\end{multline}
in which $n_{th}$ is the number of thermal photons at the lasing frequency, $\gamma_\perp$
is the dephasing rate of the polarization,
and $d_\alpha$ is the inversion of the $\alpha$th atom, which would relax to the steady-state
value of $d_0$ in the absence of an electric field at the rate $\gamma_\parallel$.
We have also simplified by assuming the lasing signal to be on resonance with the atomic transition,
$\omega_0 = \omega_a$, and that the dipole matrix element and field are co-linear, $\boldsymbol{\theta}\cdot \mathbf{\Psi}_0 = \theta \Psi_0$.

The first term in Eq.~(\ref{eq:Ftot}) corresponds
to the contribution from thermal and zero point fluctuations from outside the cavity to the field inside the cavity;
as such it is controlled by the decay rate of the cavity, $\gamma_c$.  The second and third terms
stem from spontaneous emission events leading to the dephasing of the atomic polarization, and these terms
depend on the parameters of the gain medium. For optical frequencies, the number of thermal photons
present is negligible, so the direct field fluctuations do not have any effect on resulting linewidth.
Moreover, for most lasers (class A and B) \cite{ohtsubo}, $\gamma_\parallel \ll \gamma_\perp$,
so that only the second term in Eq.~(\ref{eq:Ftot}) remains,
\begin{equation}
\langle F_{tot}^\dagger(t) F_{tot}(t')\rangle = \delta(t-t')
\frac{1}{\gamma_\perp} \sum_\alpha |g_\alpha|^2 (1 + \langle d_\alpha \rangle).
\end{equation}
Assuming that the atoms are uniformly distributed over the cavity,
the sum over the atoms in the cavity can be converted into an integral,
\begin{multline}
\langle F_{tot}^\dagger(t) F_{tot}(t')\rangle = \delta(t-t')
\frac{2|g'|^2}{\gamma_\perp} \\ \times \int \left(\frac{N_2(x)}{N_2(x) - N_1(x)}\right) D(x) |\Psi_0(x)|^2 dx, \label{eq:Ftot2}
\end{multline}
where $N_2(x)$ ($N_1(x)$) is the density of atoms in the upper (lower) lasing level,
and $g_\alpha = g' \Psi_0(x_\alpha)$.  Equation~(\ref{eq:Ftot2}) now expresses the Langevin force in terms of
the incomplete inversion factor \cite{lax_conf_1966},
 \begin{equation}
n_{sp}(x) = \frac{N_2(x)}{N_2(x) - N_1(x)}. \label{eq:incomp}
\end{equation}

Traditionally at this step the atomic
populations are approximated to be uniform in space which clamps the inversion
to its value at the lasing threshold, $D(x) \approx D_0 = D_0^{(thr)}$ \cite{haken}.
In this approximation, the integral in Eq.~(\ref{eq:Ftot2}) becomes just $D_0^{(thr)} n_{sp}$, and
can be related back to the cavity decay rate by
\begin{equation}
\frac{D_0^{(thr)} |g'|^2}{\gamma_\perp} = \frac{\gamma_c}{2}, \label{eq:gcApprox}
\end{equation}
which states that at the lasing threshold, gain must equal loss.
Using this approximation contributes one factor of $\gamma_c$ in the standard ST linewidth
formula, Eq.~(\ref{eq:STlw}). However this is a rather crude approximation.  Due to spatial hole-burning the atomic populations generally
vary substantially in space, and as a consequence the overall inversion is not clamped at the threshold
value.

Using Eq.~(\ref{eq:gcApprox}) obscures the fact that the physical
origin of the noise is spontaneous emission in the gain medium. It then is more natural to
relate the linewidth to the rate of {\it spontaneous emission}, $\gamma_{sp}$, which can be calculated
from the rate of stimulated emission, $\gamma_{SE}$ \cite{cerjan_csalt_2015}, as
\begin{equation}
\gamma_{sp}(x) = \frac{\gamma_{SE}(x)}{2 n_p} = \frac{2 |g'|^2}{\gamma_\perp} |\Psi_0(x)|^2,
\end{equation}
where $n_p$ is the number of photons in the steady-state lasing field, and is related to the intensity by $I_0 = 2\pi \hbar \omega_0 n_p$.
Thus, we can rewrite Eq.~(\ref{eq:Ftot2}) in terms of the rate of spontaneous emission, keeping the full space
dependence of the problem, as
\begin{equation}
\langle F_{tot}^\dagger(t) F_{tot}(t')\rangle = \delta(t-t')
\int \gamma_{sp}(x) D(x) n_{sp}(x) dx. \label{eq:Ftot3}
\end{equation}

As noted above, the coefficient of the Langevin noise determines the laser linewidth by
standard manipulations.  Using the more correct form of this coefficient above leads to
a quantum-limited linewidth,
\begin{equation}
\delta \omega = \frac{\int \gamma_{sp}(x) D(x) n_{sp}(x) dx}{2 n_p}.
\end{equation}
Typically, the next step is to relate this to the total output power of the cavity via
\begin{equation}
P_{ST} = \hbar \omega \gamma_c n_p,
\end{equation}
which is also only an approximate relation, neglecting spatial variations of the intensity.
Using both of the approximations mentioned above and some other simplifications valid
for high $Q$ cavities (and neglecting the amplitude contribution contained in the $\alpha$
factor), one then arrives at the standard ST linewidth form, Eq.~(\ref{eq:STlw}).

However, the actual power emitted can be calculated using the true spatially-varying
fields using Poynting's theorem \cite{jackson}, as
\begin{equation}
P = \frac{I_0 \omega_0}{2\pi} \int_c \im[-\varepsilon] |\Psi_0(x)|^2 dx.
\end{equation}
Using specific form of the imaginary part of the dielectric function for a two-level gain medium
(and using real units, not SALT units), the linewidth can then be written as
\begin{equation}
\delta \omega = \frac{\hbar \omega_0}{2 P} \int \gamma_{sp}(x) D(x) n_{sp}(x) dx \int \gamma_{sp}(x) D(x) dx. \label{eq:LW}
\end{equation}

Here we have derived a more physically intuitive and accurate form of the laser linewidth formula in which
the critical role of spontaneous emission in the gain medium leaps out:~the linewidth is determined by the
spatially-varying, mode-dependent spontaneous emission rate and by the steady-state saturated inversion in
the presence of spatial hole-burning. Only if one makes the approximation of $\int \gamma_{sp} D dx \approx \gamma_c$
does this reduce to the standard Schawlow-Townes formula, and in making this substitution one tends to hide
the true origin of the linewidth, which is spontaneous emission.

While the above argument captures the basic physics of the ST formula in an improved form, it is not
a rigorous general result, as is the formula derived by Pick \textit{et al.}\ \cite{pick_linewidth_2015}, which includes
essentially all known corrections in a more accurate form.  However the N-SALT linewidth does have the same general features
as our simpler result, Eq.~(\ref{eq:LW}), and for a single lasing mode can be written as \cite{pick_linewidth_2015},
\begin{multline}
\delta \omega_{\textrm{N-SALT}} = \frac{\hbar \omega_0}{2 P} (1 + \tilde{\alpha}^2) \\
\times \frac{\omega_0^2\int \im[\varepsilon] |\boldsymbol{\Psi}_0|^2 d\mathbf{x} \int \im[\varepsilon]\frac{N_2(\mathbf{x})}{D(\mathbf{x})}|\boldsymbol{\Psi}_0|^2 d\mathbf{x}}
{\left|\int \boldsymbol{\Psi}_0^2 \left(\varepsilon + \frac{\omega_0}{2} \frac{d\varepsilon}{d\omega}|_{\omega_0} \right) d\mathbf{x} \right|^2}, \label{eq:NSALT}
\end{multline}
where $\boldsymbol{\Psi}_0$ is the normalized SALT lasing mode, and $\tilde{\alpha}$ is a generalization of the Lax-Henry $\alpha$ factor.
Very recently this formula has been quantitatively confirmed using first-principles numerical simulations \cite{cerjan_noise_2015}
for laser parameters where it predicts significant departures from the standard (corrected) form of the Schawlow-Townes formula.
Note that in Eq.~(\ref{eq:NSALT}) the cavity decay rate is completely absent, while the imaginary part
of the response of the gain medium plays a critical role.

\section{Conclusion}

In summary we have presented a scattering framework for semiclassical laser theory and showed
how precisely it leads to zero laser linewidth, due to the discreteness of the poles of the scattering
matrix. This approach treats the full system as infinite from the beginning and does not introduce a
finite ``universe" and impose a limiting procedure and damping to recover the infinite system result,
although such an approach is a valid alternative method if the limiting absorption principle is observed.
We were inspired to analyze this question by the classic work of Lang \textit{et al.}\, which introduced
the notion of ``modes of the universe".  Our scattering point of view underlies recently developed \textit{ab initio} laser
theory (SALT), which leads to efficient computational tools for calculating steady-state lasing properties.
A generalization of the \textit{ab initio} theory to treat noise allows accurate calculation of
the intrinsic linewidth due to quantum fluctuations when one goes beyond semiclassical laser theory.
We emphasize that the quantum-limited linewidth depends directly on the properties of the
gain medium in the active lasing state, and it may be useful to write it in a form which does not
involve the passive cavity decay rate, but rather emphasizes the spontaneous emission rate.

\section*{Acknowledgements}
We acknowledge partial support by NSF grant DMR-1307632.  ADS thanks Marlan Scully
for drawing our attention to the work of Lang \textit{et al.}
We thank Steven Johnson and Adi Pick for very useful conversations about the laser
linewidth, noise effects, Perfectly Match Layers and the limiting absorption principle.
We would also like to thank Li Ge for stimulating discussions on methods for finding
the poles of the S-matrix for two-dimensional cavities.

redpajama_set_name: RedPajamaArXiv


text:
Le classement mondial de snooker 2016-2017 est celui des joueurs professionnels de snooker qualifiés pour la saison 2016-2017 (5 mai 2016 au  mai 2017).

Règlement
Les classements fonctionnent comme une liste roulante sur deux ans. Les points acquis pour chaque tournoi d'il y a deux ans sont retranchés et ceux acquis lors du même tournoi de la saison en cours sont ajoutés. La table suivante présente les diverses révisions du classement qui ont été utilisées pour déterminer les sélections des divers tournois.

Notes et références

Notes

Références

2016-2017
Classements 2016-2017
Classements 2016-2017


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