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\section{Maser Theory}  \label{sec:maser_theory}  All maser emission arises when photons of specific frequencies incite an excited molecule to emit a photon of the same frequency. This causes a cascade of photons to be emitted within a region of excited molecules, resulting in exponential growth of the intensity of a ray of light. The process itself requires only a population inversion between two states in a molecule, which is provided by some pumping mechanism. In order to be detectable, the masering molecules must also be coherent in velocity (within the thermal width) to achieve a significant gain over the path length of the ray \citep{lo2005}. This section explores the theory of maser emission based on these two requirements. A significant portion thorough treatment  of modern maser theory is presented and derived can be found  in a series of papers byMoshe  Elitzur \citep{Elitzur_1990,Elitzur_1990_paperI,Elitzur_1990_paperII,Elitzur_1991}. \subsection{Population Inversion \& Radiative Transfer}  \label{sub:pop_inverse_rad_trans}  In this section, I consider a two-level system of molecular energy levels and derive this the  exponential growth in the ray intensity in terms of the emitting region's physical properties, and the properties of the emitting molecule. It should be noted that a proper treatment of any stimulated emission process requires a minimum of three energy levels, as is shown in the thorough derivations by \cite{Elitzur_1992} and \cite{Gray_2009}. Here, I largely follow the simplified derivations presented in Chapter 4 of \citet{Elitzur_1992} \cite{elitzur1992_text}  and Chapter 14 of \citet{stahler_palla_2004}, which implicitly treat levels unassociated with the maser transition as generic gain and loss rates. Let there be an upper and lower states, state,  such that the maser results from the transition from the upper to the lower. From the Boltzmann distribution, the excitation temperature is defined by \begin{equation}  \label{eq:level_pops}  \frac{n_u}{n_{l}} = \frac{g_{u}}{g_{l}} \mathrm{exp}\left( \frac{\Delta E}{kT_{\mathrm{ex}}} \right)  \end{equation}  where  $n_u$ and $n_l$ are the number densities in the respective upper and lower states, $g_u$ and $g_l$ are the degeneracies of those states, and $\Delta E$ is the transition energy between them. The population levels become inverted when $n_\mathrm{u}/g_\mathrm{u} \gt n_\mathrm{l}/g_\mathrm{l}$, which corresponds to a negative $T_\mathrm{ex}$. Maintaining an inverted population requires an energy source, referred to as the {\it pump} (see XXX ADD SUBSECTION XXX). (\S\ref{sub:pumping}).  Transistions in the microwave and radio require the smallest pump action, since $n_\mathrm{u}/g_\mathrm{u} \approx n_\mathrm{l}g_\mathrm{l}$ n_\mathrm{l}/g_\mathrm{l}$  for kinetic temperatures of a few hundred kelvin, even when in thermal equilibrium. The role of the pump is to excite molecules into the upper and lower states of the maser transition, and is typically achieved through a collisional or radiative process (see XXX add subsection XXX). process.  Without regard for the details of how the pump achieves this, let $P_\mathrm{u}$ and $P_\mathrm{l}$ be the rate per unit volume at which the pump adds molecules to the upper and lower states from other states of the molecule, respectively. Both states may also decay or be excited further into other energy states. Since this must depend on the population in the upper and lower levels, these loss rates per unit volume are $n_u\Gamma_u$ and $n_l\Gamma_l$, where $\Gamma$ is the decay/excitation rate in each level. The level populations of the upper and lower states also depend on the Einstein A and B coefficients, representing spontaneous decay and absorption between the states, in the presence of a radiation field, $\bar{J}$, the mean intensity. Finally, the molecules may collide with other molecules, giving collisional excitation ($\gamma_{lu}$) and deexcitation ($\gamma_{ul}$) rates. A schematic of the processes discussed here are presented in Figure \ref{fig:energy_diagram}. While observed masers may vary on time-scales ranging from days to months \citep[e.g., ][\S XXX]{Elitzur_1992_review}, \citep[e.g.,]{Elitzur_1992},  I make the assumption that the maser mechanism is generally in a steady-state. The steady state populations of the upper and lower levels are then given by, \begin{equation}  \label{eq:ss_pops}  0 = P_u - n_u \Gamma_u - \left( n_u B_{ul} - n_l B_{lu} \right)\bar{J} - \left( n_u \gamma_{ul} - n_l \gamma{lu} \right) n_{\mathrm{tot}} - n_u A_{ul} \\ 
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\end{equation}  Here, $n_{\mathrm{tot}}$ is the number density of the background gas. The $B$-coefficients are related by $g_lB_{lu}=g_uB_{ul}$, and the collisional coefficients approximately follow $\gamma_{ul}g_u \approx \gamma_{lu}g_l$, since $T_{\mathrm{kin}} \gg \Delta E / k_{\mathrm{B}}$ for the relevant energy transitions for masers \citep{stahler_palla_2004}.  The relevant quantity for understanding the radiative transfer of a maser is the {\it degree of inversion} of the population levels. To derive this, \citet{stahler_palla_2004} make two simplifying assumptions: the loss rates are equal ($\Gamma=\Gamma_u=\Gamma_l$), and, in practice, spontaneous emission is negligible compared to the loss rates for the transitions of interest($\Gamma \gg A_{ul}$) \citep{Elitzur_1992}. \citep{elitzur1992_text}.  Assuming equal loss rates does not change the form of the final result of the derivation. Using these assumptions, and the aforementioned relation, the population inversion can be described in terms of the properties of the two states: \begin{equation}  \label{eq:pop_inverse_full}  \Delta n \equiv \frac{n_u}{g_u} - \frac{n_l}{g_l} \\ 
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\label{eq:sat_intensity}  \bar{J}_s \equiv \frac{\Gamma + \left( 1 + g_u/g_l \right)\gamma_{ul}n_{\mathrm{tot}}}{\left( 1 + g_u/g_l \right)B_{ul}}  \end{equation}  Equation \ref{eq:pop_inverse} shows that the population inversion is not affected greatly by radiation ($\bar{J}$), until it exceeds $\bar{J}_s$ and quickly declines. This makes the maser mechanism self-limiting, since $\bar{J}$ is amplified by the process. The value of $\bar{J}_s$, as will be shown below, sets the maximum intensity that the maser mechanism can produce, and distinguishes between the unsaturated and saturated regimes \citep{Elitzur_1992}. \citep{elitzur1992_text}.  For spectral line emission, the emission and absorption coefficients are related to the Einstein $A$ and $B$ values of the transition: $j_\nu = h \nu_0 n_u A_{ul}\phi(\nu)$ and $\alpha_\nu = h\nu_0 \left( n_l B_{lu} - n_u B_{ul} \right)\phi(\nu)/4\pi$, respectively. Here, $\phi(\nu)$ is a normalized Doppler profile, corresponding to the ambient temperature, and $\nu_0$ is the central frequency. The radiative transfer equation then has a form of,  \begin{equation} 
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\label{eq:rad_trans}  \frac{dI}{ds} = \frac{h\nu_0}{4\pi \Delta \nu} \left[ \frac{\Delta n^{\circ} g_u B_{ul}}{1 + \bar{J}/\bar{J}_s} I + n_u A_{ul} \right]  \end{equation}  where $I \equiv \int I_{\nu}\phi(\nu)d\nu$, and $\Delta \nu \equiv \int I_{\nu}d\nu/I$ is the effective bandwidth. To solve this equation, the forms of $\bar{J}$ and $n_u$ must be specified. Masers are tightly beamed (see \S XXX REF SUBSECTION XXX), (\S\ref{sub:beaming}),  so $\bar{J} \approx I\Delta\Omega/4\pi$, assuming $I$ is constant across $\Delta\Omega$. Then, $\bar{J}$ may be eliminated from Equation \ref{eq:rad_trans}, since $\bar{J}/\bar{J}_s = I/I_s$. The second term in Equation \ref{eq:rad_trans}, containing $n_u$, arises due to spontaneous emission ($A_{ul}$). This contribution will be significantly smaller than the stimulated emission, however it plays the vital role for the maser process in the absence of background radiation. Because of this, I follow \citet{stahler_palla_2004} in approximating $n_u \approx n_u^{\circ}$ to be the population level in the upper state in the absence of radiation. These assumptions simplify Equation \ref{eq:rad_trans} into \begin{equation}  \label{eq:final_rad_trans}  \frac{dI}{ds} = \frac{I}{L (1+I/I_s)} + \beta \frac{I_s}{L} 
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\label{eq:sat_solution}  i(s) = \frac{s-s_{T}}{L}  \end{equation}  The transition to a saturated maser occurs at $s_T$, where $i(s)=1$. At that point, $i_0$ is negligible. Although the intensity growth of a saturated maser is linear, this regime represents the maximum efficiency that the pumping process can produce \citep{Elitzur_1992}. \citep{elitzur1992_text}.  It represents the steady-state between the energy exhausted by stimulated emission and the energy provided by the pumping mechanism. An example of the gain in both regimes, and the transition between them, is shown in Figure \ref{fig:maser_lwidth}. \subsection{Line Narrowing}  \label{sub:line_narrow}  Observations of masers have shown linewidths narrower than the expected thermal linewidths \citep[][e.g.,]{Elitzur_1992,stahler_palla_2004}. \citep[e.g.,]{Elitzur_1992,stahler_palla_2004}.  This is due to the exponential amplification in the unsaturated regime. Assume that the line follows a Doppler profile, so the absorption coefficient is expressed as \begin{equation}  \label{eq:doppler_profile}  \alpha_\nu = \alpha_0 \exp\left( -\frac{(\nu - \nu_0)^2}{\Delta\nu_D^2} \right) 
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\label{eq:line_narrow}  \Delta\nu^\star = \frac{\Delta\nu_D}{\sqrt{\alpha_0 s}}  \end{equation}  This result, as shown in \citet{stahler_palla_2004}, shows that the maser linewidth decreases with increasing path length. However, this result was derived using an unsaturated maser. A maser saturates from the inside out of it's line profile, since the greatest intensity is at the center. Thus, the center saturates before the wings of the profile do. The profile must then broaden, since the unsaturated wings are growing exponentially. This effect sets a limit on the narrowing of the line \citep{Elitzur_1992,stahler_palla_2004}. \citep{elitzur1992_text,stahler_palla_2004}.  When the maser is completely saturated, it maintains a Doppler linewidth. These properties are demonstrated in Figure \ref{fig:maser_lwidth}. \subsection{Beaming}  \label{sub:beaming} 
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\subsection{Pumping Mechanisms}  \label{sub:pumping}  Stimulated emission requires an energy source to maintain the population inversion. The non-LTE conditions in the ISM are ideal for stimulated emission since they require the least amount of energy from a pump to create a population inversion. The pump mechanisms broadly fall into two categories: radiative and collisional. The environment and the characteristics of an energy transition also play a key role in how the pump mechanism achieves a population inversion. Table \ref{subsub:rad_pump} \ref{tab:maser_props}  shows typical properties of the regions where mega-maser maser  emission is capable of occurring. This list is not exhaustive for masers in general, as many more species have been observed as galactic masers \citep[][e.g., ]{Elitzur_1992_review}. \citep[e.g., ]{Elitzur_1992}.  \subsubsection{Collisional Pumping}  \label{subsub:coll_pump}  Collisionally pumped (Type I) masers occur in higher-density regions, where the molecules responsible for the masers are excited from collisions primarily with H$_2$. In order to maintain the inversion through collisions: (1) the spontaneous decay from higher energy states into the upper maser level must be faster than into the lower maser level, or (2) the spontaneous decay of the upper maser level must be slower than the lower maser level \citep{Goldreich_1974}. However, the transition will thermalize beyond the critical number densities ($n_{\mathrm{H}}$ Table \ref{subsub:rad_pump}), setting an upper limit on the energy transfer to the masing transition. The H$_2$O and SiO transitions in Table \ref{subsub:rad_pump} are examples of masers that are typically collisionally excited \citep{Elitzur_1992}. Note that both of these transitions have particularly high critical number densities. These masers are associated with circumstellar material driven by stellar winds around late-type stars and outflows from embedded young stars \citep{Elitzur_1992}. The energy source in these cases is the bulk gas motion \citep{stahler_palla_2004}. Extragalactic H$_2$O mega-masers are thought to behave similarly to circumstellar masers, however they are coined as {\it circumnuclear} masers since the energy source is the AGN \citep[see \S\ref{sub:h2o_props}][e.g.,]{lo2005}. (\S\ref{sub:h2o_props}).  \subsubsection{Radiative Pumping}  \label{subsub:rad_pump}  Radiatively pumped (Type II) masers rely primarily on absorption of IR photons, and correlations are observed between the IR luminosity and maser luminosity \citep{darling2002_paperIII}. Successful use of this pump generally relies on having more transitionsin the masing molecule  into the upper maser state than the lower one when IR photons are absorbed \citep{lo2005}. Inverted populations levels in the 18 cm line quadruplet of OH result from radiative pumping \citep[][see \S\ref{sec:OH} for more on the OH transitions]{Elitzur_1992, lo2005}. (\S\ref{sec:OH}).  Galactic OH masers are primarily associated with the expanding shells of HII regions and SNRs, and are observed to correlate with the IR luminosity \citep{Elitzur_1992}. This correlation is also observed in OH mega-masers \citep{darling2002_paperIII}, which arise in extreme starbursts. \begin{table}   \begin{tabular}{ c c c c c c } 
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\subsection{Polarization}  \label{sub:polarization}  Many masers, both galactic and extra-galactic, are observed to be at least partially polarized. From \citet{Elitzur_1992}, the electric field of maser radiation is confined to a plane wave (two degrees of freedom) in the presence of a magnetic field. This occurs since only waves with the correct phase difference are capable of being amplified, and therefore observed (see also \citet{Goldreich_1973,Elitzur_1991}). The phase difference here is set by the angle between the magnetic field and direction of propagation of the wave \citep{Elitzur_1992}. This theory explains why para-magnetic species (such as OH) are typically highly polarized (both linearly and circularly). polarized.  However, there is significant difficulty in extending the theory theoretical predictions  to include time variability, large velocity gradients, and Faraday rotation \citep{Elitzur_1992}. The Zeeman splitting in mega-masers has been used explore the role of magnetic fields in extreme starburst environments (see \S\ref{sub:oh_zeeman}).