---
title: "SALT equation"
firstLetter: "S"
publishDate: 2022-02-07
categories:
- Physics
- Optics
- Laser theory

date: 2022-01-20T22:01:48+01:00
draft: false
markup: pandoc
---

# SALT equation

The **steady-state *ab initio* laser theory** (SALT) is
a theoretical description of lasers, whose mode-centric approach
makes it especially appropriate for microscopically small lasers.

Consider the [Maxwell-Bloch equations](/know/concept/maxwell-bloch-equations/),
governing the complex polarization
vector $\vb{P}^{+}$ and the scalar population inversion $D$ of a set of
active atoms (or quantum dots) embedded in a passive linear background
material with refractive index $c / v$.
The system is affected by a driving [electric field](/know/concept/electric-field/)
$\vb{E}^{+}(t) = \vb{E}_0^{+} e^{-i \omega t}$,
such that the set of equations is:

$$\begin{aligned}
    - \mu_0 \pdv[2]{\vb{P}^{+}}{t}
    &= \nabla \cross \nabla \cross \vb{E}^{+} + \frac{1}{v^2} \pdv[2]{\vb{E}^{+}}{t}
    \\
    \pdv{\vb{P}^{+}}{t}
    &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+}
    - \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} D
    \\
    \pdv{D}{t}
    &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big)
\end{aligned}$$

Where $\hbar \omega_0$ is the band gap of the active atoms,
and $\gamma_\perp$ and $\gamma_\parallel$ are relaxation rates
of the atoms' polarization and population inversion, respectively.
$D_0$ is the equilibrium inversion, i.e. the value of $D$ if there is no lasing.
Note that $D_0$ also represents the pump,
and both $D_0$ and $v$ depend on position $\vb{x}$.
Finally, the transition dipole matrix elements $\vb{p}_0^{-}$ and $\vb{p}_0^{+}$ are given by:

$$\begin{aligned}
    \vb{p}_0^{-}
    \equiv q \matrixel{e}{\vu{x}}{g}
    \qquad \qquad
    \vb{p}_0^{+}
    \equiv q \matrixel{g}{\vu{x}}{e}
    = (\vb{p}_0^{-})^*
\end{aligned}$$

With $q < 0$ the electron charge, $\vu{x}$ the quantum position operator,
and $\ket{g}$ and $\ket{e}$ respectively
the ground state and first excitation of the active atoms.

We start by assuming that the cavity has $N$ quasinormal modes $\Psi_n$,
each with a corresponding polarization $\vb{p}_n$ of the active matter.
Note that this ansatz already suggests
that the interactions between the modes are limited:

$$\begin{aligned}
    \vb{E}^{+}(\vb{x}, t)
    = \sum_{n = 1}^N \Psi_n(\vb{x}) \: e^{- i \omega_n t}
    \qquad \qquad
    \vb{P}^{+}(\vb{x}, t)
    = \sum_{n = 1}^N \vb{p}_n(\vb{x}) \: e^{- i \omega_n t}
\end{aligned}$$

Using the modes' linear independence to treat each term of the summation individually,
the first two Maxwell-Bloch equations turn into, respectively:

$$\begin{aligned}
    \mu_0 \omega_n^2 \vb{p}_n
    &= \nabla \cross \nabla \cross \Psi_n - \frac{1}{v^2} \omega_n^2 \Psi_n
    \\
    i \omega_n \vb{p}_n
    &= \big( i \omega_0 + \gamma_\perp \big) \vb{p}_n
    + \frac{i}{\hbar} \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n \: D
\end{aligned}$$

With being $\vb{p}_0^{+} \vb{p}_0^{-}$ a dyadic product.
Isolating the latter equation for $\vb{p}_n$ gives us:

$$\begin{aligned}
    \vb{p}_n
    &= \frac{\big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n \: D}{\hbar \big((\omega_n - \omega_0) + i \gamma_\perp\big)}
    = \frac{\gamma(\omega_n) D}{\hbar \gamma_\perp} \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n
\end{aligned}$$

Where we have defined the Lorentzian gain curve $\gamma(\omega_n)$ as follows,
which represents the laser's preferred frequencies for amplification:

$$\begin{aligned}
    \gamma(\omega_n)
    \equiv \frac{\gamma_\perp}{(\omega_n - \omega_0) + i \gamma_\perp}
\end{aligned}$$

Inserting this expression for $\vb{p}_n$
into the first Maxwell-Bloch equation yields
the prototypical form of the SALT equation,
where we still need to replace $D$ with known quantities:

$$\begin{aligned}
    0
    &= \bigg( \nabla \cross \nabla \cross - \, \omega_n^2 \frac{1}{v^2}
    - \omega_n^2 \frac{\mu_0 \gamma(\omega_n) D}{\hbar \gamma_\perp} (\vb{p}_0^{+} \vb{p}_0^{-}) \cdot \bigg) \Psi_n 
\end{aligned}$$

To rewrite $D$, we turn to its (Maxwell-Bloch) equation of motion,
making the crucial **stationary inversion approximation** $\pdv*{D}{t} = 0$:

$$\begin{aligned}
    D
    &= D_0 + \frac{i 2}{\hbar \gamma_\parallel} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big)
\end{aligned}$$

This is the most aggressive approximation we will make:
it removes all definite phase relations between modes,
and effectively eliminates time as a variable.
We insert our ansatz for $\vb{E}^{+}$ and $\vb{P}^{+}$,
and point out that only excited lasing modes contribute to $D$:

$$\begin{aligned}
    D
    &= D_0 + \frac{i 2}{\hbar \gamma_\parallel} \sum_{\nu, \mu}^\mathrm{active}
    \bigg( \vb{p}_\nu^* \cdot \Psi_\mu e^{i (\omega_\nu - \omega_\mu) t}
    - \vb{p}_\nu \cdot \Psi_\mu^* e^{i (\omega_\mu - \omega_\nu) t} \bigg)
\end{aligned}$$

Here, we make the [rotating wave approximation](/know/concept/rotating-wave-approximation/)
to neglect all terms where $\nu \neq \mu$
on the basis that they oscillate too quickly,
leaving only $\nu = \mu$:

$$\begin{aligned}
    D
    &= D_0 + \frac{i 2}{\hbar \gamma_\parallel} \sum_{\nu}^\mathrm{act.}
    \bigg( \vb{p}_\nu^* \cdot \Psi_\nu - \vb{p}_\nu \cdot \Psi_\nu^* \bigg)
\end{aligned}$$

Inserting our earlier equation for $\vb{p}_n$
and using the fact that $\vb{p}_0^{+} = (\vb{p}_0^{-})^*$ leads us to:

$$\begin{aligned}
    D
    &= D_0 + \frac{i 2 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
    \bigg( \gamma^*(\omega_\nu) \big(\vb{p}_0^{+} \vb{p}_0^{-}\big)^* \!\cdot\! \Psi_\nu^* \cdot \Psi_\nu
    - \gamma(\omega_\nu) \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \!\cdot\! \Psi_\nu \cdot \Psi_\nu^* \bigg)
    \\
    &= D_0 + \frac{i 2 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
    \bigg( \gamma^*(\omega_\nu) \big(\vb{p}_0^{+} \cdot \Psi_\nu^*\big) \vb{p}_0^{-} \cdot \Psi_\nu
    - \gamma(\omega_\nu) \big(\vb{p}_0^{-} \cdot \Psi_\nu\big) \vb{p}_0^{+} \cdot \Psi_\nu^* \bigg)
    \\
    &= D_0 + \frac{i 2 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
    \Big( \gamma^*(\omega_\nu) - \gamma(\omega_\nu) \Big) \big|\vb{p}_0^{-} \cdot \Psi_\nu\big|^2
\end{aligned}$$

By putting the terms on a common denominator, it is easily shown that:

$$\begin{aligned}
    \gamma^*(\omega_\nu) - \gamma(\omega_\nu)
    &= \frac{\gamma_\perp ((\omega_\nu - \omega_0) + i \gamma_\perp)}{(\omega_\nu - \omega_0)^2 + \gamma_\perp^2}
    - \frac{\gamma_\perp ((\omega_\nu - \omega_0) - i \gamma_\perp)}{(\omega_\nu - \omega_0)^2 + \gamma_\perp^2}
    \\
    &= \frac{\gamma_\perp (i \gamma_\perp + i \gamma_\perp)}{(\omega_\nu - \omega_0)^2 + \gamma_\perp^2}
    = i 2 \big|\gamma(\omega_\nu)\big|^2
\end{aligned}$$

Inserting this into our equation for $D$ gives the following expression:

$$\begin{aligned}
    D
    &= D_0 - \frac{4 D}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
    \Big|\gamma(\omega_\nu) \vb{p}_0^{-} \cdot \Psi_\nu\Big|^2
\end{aligned}$$

We then properly isolate this for $D$ to get its final form, namely:

$$\begin{aligned}
    D
    &= D_0 \bigg( 1 + \frac{4}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
    \Big|\gamma(\omega_\nu) \vb{p}_0^{-} \cdot \Psi_\nu\Big|^2 \bigg)^{-1}
\end{aligned}$$

Substituting this into the prototypical SALT equation from earlier
yields the most general form of the **SALT equation**,
upon which the theory is built:

$$\begin{aligned}
    \boxed{
        0
        = \bigg( \nabla \cross \nabla \cross
        -\,\omega_n^2 \bigg[ \frac{1}{v^2(\vb{x})} + \frac{\mu_0 \gamma(\omega_n)}{\hbar \gamma_\perp}
        \frac{D_0(\vb{x})}{1 + h(\vb{x})} (\vb{p}_0^{+} \vb{p}_0^{-}) \cdot \bigg] \bigg) \Psi_n(\vb{x})
    }
\end{aligned}$$

Where we have defined **spatial hole burning** function $h(\vb{x})$ like so,
representing the depletion of the supply of charge
carriers as they are consumed by the active lasing modes:

$$\begin{aligned}
    \boxed{
        h(\vb{x})
        \equiv \frac{4}{\hbar^2 \gamma_\parallel \gamma_\perp} \sum_{\nu}^\mathrm{act.}
        \Big|\gamma(\omega_\nu) \vb{p}_0^{-} \cdot \Psi_\nu(\vb{x})\Big|^2
    }
\end{aligned}$$

Many authors assume that $\vb{p}_0^- \parallel \Psi_n$,
so that only its amplitude $|g|^2 \equiv \vb{p}_0^{+} \cdot \vb{p}_0^{-}$ matters.
In that case, they often non-dimensionalize $D$ and $\Psi_n$
by dividing out the units $d_c$ and $e_c$:

$$\begin{aligned}
    \tilde{\Psi}_n
    \equiv \frac{\Psi_n}{e_c}
    \qquad
    e_c
    \equiv \frac{\hbar \sqrt{\gamma_\parallel \gamma_\perp}}{2 |g|}
    \qquad \qquad
    \tilde{D}
    \equiv \frac{D}{d_c}
    \qquad
    d_c
    \equiv \frac{\varepsilon_0 \hbar \gamma_\perp}{|g|^2}
\end{aligned}$$

And then the SALT equation and hole burning function $h$ are reduced to the following,
where the vacuum wavenumber $k_n = \omega_n / c$:

$$\begin{aligned}
    0
    = \bigg( \nabla \cross \nabla \cross -\,k_n^2 \bigg[ \varepsilon_r
    + \gamma(c k_n) \frac{\tilde{D}_0}{1 + h} \bigg] \bigg) \tilde{\Psi}_n
    \qquad
    h(\vb{x})
    = \sum_{\nu}^\mathrm{act.} \Big|\gamma(c k_\nu) \tilde{\Psi}_\nu(\vb{x})\Big|^2
\end{aligned}$$


In addition, some papers only consider 1D or 2D *transverse magnetic* (TM) modes,
in which case the fields are scalars. Using the vector identity

$$\begin{aligned}
    \nabla \cross \nabla \cross \Psi
    = \nabla (\nabla \cdot \Psi) - \nabla^2 \Psi
\end{aligned}$$

Where $\nabla \cdot \Psi = 0$ thanks to [Gauss' law](/know/concept/maxwells-equations/),
so we get an even further simplified SALT equation:

$$\begin{aligned}
    0
    = \bigg( \nabla^2 +\,k_n^2 \bigg[ \varepsilon_r
    + \gamma(c k_n) \frac{\tilde{D}_0}{1 + h} \bigg] \bigg) \tilde{\Psi}_n
\end{aligned}$$

The challenge is to solve this equation for a given $\varepsilon_r(\vb{x})$ and $D_0(\vb{x})$,
with the boundary condition that $\Psi_n$ is a plane wave at infinity,
i.e. that there is light leaving the cavity.

If $k_n$ has a negative imaginary part, then that mode is behaving as an LED.
Gradually increasing the pump $D_0$ in a chosen region
causes the $k_n$'s imaginary parts become less negative,
until one of them hits the real axis, at which point that mode starts lasing.
After that, $D_0$ can be increased even further until some other $k_n$ become real.

Below threshold (i.e. before any mode is lasing), the problem is linear in $\Psi_n$,
but above threshold it is nonlinear, and the amplitude of $\Psi_n$ is adjusted
such that the corresponding $k_n$ never leaves the real axis.
When any mode is lasing, hole burning makes it harder for other modes to activate,
since it effectively reduces the pump $D_0$.


## References
1.  L. Ge, Y.D. Chong, A.D. Stone,
    [Steady-state *ab initio* laser theory: generalizations and analytic results](http://dx.doi.org/10.1103/PhysRevA.82.063824),
    2010, American Physical Society.