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:

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

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 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:

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

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, 2010, American Physical Society.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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