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Categories: Laser theory, Optics, Physics.

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, 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 $\vb{E}^{+}(t) = \vb{E}_0^{+} e^{-i \omega t}$, 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 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, 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 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 $\Psi_n$ adjusts itself 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 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, 2010, American Physical Society.

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