path: root/source/know/concept/salt-equation/index.md

---
title: "SALT equation"
sort_title: "Salt equation" # sic
date: 2022-02-07
categories:
- Physics
- Optics
- Laser theory
layout: "concept"
---

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:

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

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** $$\ipdv{D}{t} = 0$$:

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

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 active 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:

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

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

If $$\Imag(k_n) < 0$$, the $$n$$th mode's amplitude decays with time, so it acts as an LED:
it emits photons without any significant light amplification taking place.
Upon gradually increasing the pump $$D_0$$ in the active region,
all $$\Imag(k_n)$$ become less negative,
until one hits the real axis $$\Imag(k_n) = 0$$,
at which point that mode starts *lasing*:
its Light gets Amplified by [Stimulated Emission](/know/concept/einstein-coefficients/) of Radiation (LASER).
After that, $$D_0$$ can be increased even further until some other $$k_n$$ become real,
so there are multiple active modes competing for charge carriers.

Below threshold (i.e. before any mode is lasing), the problem is linear in $$\Psi_n$$,
but above threshold it is nonlinear via $$h(\vb{x})$$.
Then the amplitude of $$\Psi_n$$ gets adjusted
such that its respective $$k_n$$ never leaves the real axis.
Once a mode is lasing, hole burning makes it harder for any other modes to activate,
since they modes must compete for the carrier supply $$D_0$$.



## 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.