Categories: Laser theory, Optics, Physics.

The Maxwell-Bloch equations (MBEs) give a fundamental description of light-matter interaction for a two-level quantum system for the purposes of laser theory. They govern the electric field \(\vb{E}^{+}\), the induced polarization \(\vb{P}^{+}\), and the total population inversion \(D\):

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

Where \(n\) is the background medium’s refractive index, \(\omega_0\) the two-level system’s gap resonance frequency, \(|g| \equiv |\matrixel{e}{\vu{x}}{g}|\) the transition dipole moment, \(\gamma_\perp\) and \(\gamma_\parallel\) empirical decay rates, and \(D_0\) the equilibrium inversion. Note that \(\vb{E}^{-} = (\vb{E}^{+})^*\).

Let us make the following ansatz, where \(\vb{E}_0^{+}\) and \(\vb{P}_0^{+}\) are slowly-varying envelopes of a plane wave with angular frequency \(\omega \approx \omega_0\):

\[\begin{aligned} \vb{E}^{+}(\vb{r}, t) = \frac{1}{2} \vb{E}_0^{+}(\vb{r}, t) \: e^{-i \omega t} \qquad \qquad \vb{P}^{+}(\vb{r}, t) = \frac{1}{2} \vb{P}_0^{+}(\vb{r}, t) \: e^{-i \omega t} \end{aligned}\]

We insert this into the first MBE, and assume that \(\vb{E}_0^{+}\) and \(\vb{P}_0^{+}\) vary so slowly that their second-order derivatives are negligible, i.e. \(\pdv*[2]{\vb{E}_0^{+}\!}{t} \approx 0\) and \(\pdv*[2]{\vb{P}_0^{+}\!}{t} \approx 0\), giving:

\[\begin{aligned} \mu_0 \bigg( i 2 \omega \pdv{\vb{P}_0^{+}}{t} + \omega^2 \vb{P}_0^{+} \bigg) = \nabla \cross \nabla \cross \vb{E}_0^{+} - \frac{n^2}{c^2} \bigg( i 2 \omega \pdv{\vb{E}_0^{+}}{t} + \omega^2 \vb{E}_0^{+} \bigg) \end{aligned}\]

To get rid of the double curl, consider the time-independent electromagnetic wave equation, where \(\Omega\) is an eigenfrequency of the optical cavity in which lasing will occur:

\[\begin{aligned} \nabla \cross \nabla \cross \vb{E}_0^{+} = \frac{n^2}{c^2} \Omega^2 \vb{E}_0^{+} \end{aligned}\]

For simplicity, we restrict ourselves to a single-mode laser, where there is only one \(\Omega\) and \(\vb{E}_0^{+}\) to care about. Substituting the above equation into the first MBE yields:

\[\begin{aligned} i 2 \omega \pdv{\vb{P}_0^{+}}{t} + \omega^2 \vb{P}_0^{+} = \varepsilon_0 n^2 \bigg( (\Omega^2 - \omega^2) \vb{E}_0^{+} - i 2 \omega \pdv{\vb{E}_0^{+}}{t} \bigg) \end{aligned}\]

Where we used \(1 / c^2 = \mu_0 \varepsilon_0\). Assuming the light is more or less on-resonance \(\omega \approx \Omega\), we can approximate \(\Omega^2 \!-\! \omega^2 \approx 2 \omega (\Omega \!-\! \omega)\), so:

\[\begin{aligned} i 2 \pdv{\vb{P}_0^{+}}{t} + \omega \vb{P}_0^{+} = \varepsilon_0 n^2 \bigg( 2 (\Omega - \omega) \vb{E}_0^{+} - i 2 \pdv{\vb{E}_0^{+}}{t} \bigg) \end{aligned}\]

Moving on to the second MBE, inserting the ansatz \(\vb{P}^{+} = \vb{P}_0^{+} e^{-i \omega t} / 2\) leads to:

\[\begin{aligned} \pdv{\vb{P}_0^{+}}{t} = - \Big( \gamma_\perp + i (\omega_0 - \omega) \Big) \vb{P}_0^{+} - \frac{i |g|^2}{\hbar} \vb{E}_0^{+} D \end{aligned}\]

Typically, \(\gamma_\perp\) is much larger than the rate of any other decay process, in which case \(\pdv*{\vb{P}_0^{+}\!}{t}\) is negligible compared to \(\gamma_\perp \vb{P}_0^{+}\). Effectively, this means that the polarization \(\vb{P}_0^{+}\) near-instantly follows the electric field \(\vb{E}^{+}\!\). Setting \(\pdv*{\vb{P}_0^{+}\!}{t} \approx 0\), the second MBE becomes:

\[\begin{aligned} \vb{P}^{+} = -\frac{i |g|^2}{\hbar (\gamma_\perp + i (\omega_0 \!-\! \omega))} \vb{E}^{+} D = \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}^{+} D \end{aligned}\]

Where the Lorentzian gain curve \(\gamma(\omega)\) (which also appears in the SALT equation) represents a laser’s preferred spectrum for amplification, and is defined like so:

\[\begin{aligned} \gamma(\omega) \equiv \frac{\gamma_\perp}{(\omega - \omega_0) + i \gamma_\perp} \end{aligned}\]

Note that \(\gamma(\omega)\) satisfies the following relation, which will be useful to us later:

\[\begin{aligned} \gamma^*(\omega) - \gamma(\omega) = \frac{\gamma_\perp (i \gamma_\perp + i \gamma_\perp)}{(\omega - \omega_0)^2 + \gamma_\perp^2} = i 2 |\gamma(\omega)|^2 \end{aligned}\]

Returning to the first MBE with \(\pdv*{\vb{P}_0^{+}\!}{t} \approx 0\), we substitute the above expression for \(\vb{P}_0^{+}\):

\[\begin{aligned} \pdv{\vb{E}_0^{+}}{t} &= i (\omega - \Omega) \vb{E}_0^{+} + i \frac{\omega}{2 \varepsilon_0 n^2} \vb{P}_0^{+} \\ &= i (\omega - \Omega) \vb{E}_0^{+} + i \frac{|g|^2 \omega \gamma(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} \vb{E}_0^{+} D \end{aligned}\]

Next, we insert our ansatz for \(\vb{E}^{+}\!\) and \(\vb{P}^{+}\!\) into the third MBE, and rewrite \(\vb{P}_0^{+}\) as above. Using our identity for \(\gamma(\omega)\), and the fact that \(\vb{E}_0^{+} \cdot \vb{E}_0^{-} = |\vb{E}|^2\), we find:

\[\begin{aligned} \pdv{D}{t} &= \gamma_\parallel (D_0 - D) + \frac{i}{2 \hbar} \Big( \frac{|g|^2 \gamma^*(\omega)}{\hbar \gamma_\perp} \vb{E}_0^{-} D \cdot \vb{E}_0^{+} - \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}_0^{+} D \cdot \vb{E}_0^{-} \Big) \\ &= \gamma_\parallel (D_0 - D) + \frac{i |g|^2}{2 \hbar^2 \gamma_\perp} \Big( \gamma^*(\omega) - \gamma(\omega) \Big) |\vb{E}|^2 D \\ &= \gamma_\parallel (D_0 - D) - \frac{|g|^2}{\hbar^2 \gamma_\perp} |\gamma(\omega)|^2 |\vb{E}|^2 D \end{aligned}\]

This is the prototype of the first laser rate equation. However, in order to have a practical set, we need an equation for \(|\vb{E}|^2\), which we can obtain using the first MBE:

\[\begin{aligned} \pdv{|\vb{E}|^2}{t} &= \vb{E}_0^{+} \pdv{\vb{E}_0^{-}}{t} + \vb{E}_0^{-} \pdv{\vb{E}_0^{+}}{t} \\ &= -i (\omega - \Omega^*) |\vb{E}|^2 - i \frac{|g|^2 \omega \gamma^*(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} |\vb{E}|^2 D + i (\omega - \Omega) |\vb{E}|^2 + i \frac{|g|^2 \omega \gamma(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} |\vb{E}|^2 D \\ &= i (\Omega^* - \Omega) |\vb{E}|^2 + i \frac{|g|^2 \omega}{2 \hbar \varepsilon_0 \gamma_\perp n^2} \Big(\gamma(\omega) - \gamma^*(\omega)\Big) |\vb{E}|^2 D \\ &= 2 \Im(\Omega) |\vb{E}|^2 + \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 |\vb{E}|^2 D \end{aligned}\]

Where \(\Im(\Omega) < 0\) represents the fact that the laser cavity is leaky. We now have the **laser rate equations**, although they are still in an unidiomatic form:

\[\begin{aligned} \boxed{ \begin{aligned} \pdv{|\vb{E}|^2}{t} &= 2 \Im(\Omega) |\vb{E}|^2 + \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 |\vb{E}|^2 D \\ \pdv{D}{t} &= \gamma_\parallel (D_0 - D) - \frac{|g|^2}{\hbar^2 \gamma_\perp} |\gamma(\omega)|^2 |\vb{E}|^2 D \end{aligned} } \end{aligned}\]

To rewrite this, we replace \(|\vb{E}|^2\) with the photon number \(N_p\) as follows, with \(U = \varepsilon_0 n^2 |\vb{E}|^2 / 2\) being the energy density of the light:

\[\begin{aligned} N_{p} = \frac{U}{\hbar \omega} = \frac{\varepsilon_0 n^2}{2 \hbar \omega} |\vb{E}|^2 \end{aligned}\]

Furthermore, consider the definition of the inversion \(D\): because a photon emission annihilates an electron-hole pair, it reduces \(D\) by \(2\). Since lasing is only possible for \(D > 0\), we can replace \(D\) with the conduction band’s electron population \(N_e\), which is reduced by \(1\) whenever a photon is emitted. The laser rate equations then take the following standard form:

\[\begin{aligned} \boxed{ \begin{aligned} \pdv{N_p}{t} &= - \gamma_p N_p + G N_p N_e \\ \pdv{N_e}{t} &= R_\mathrm{pump} - \gamma_e N_e - G N_p N_e \end{aligned} } \end{aligned}\]

Where \(\gamma_e\) is a redefinition of \(\gamma_\parallel\) depending on the electron decay processes, and the photon loss rate \(\gamma_p\), the gain \(G\), and the carrier supply rate \(R_\mathrm{pump}\) are defined like so:

\[\begin{aligned} \gamma_p = - 2 \Im(\Omega) = \frac{Q}{\Re(\Omega)} \qquad \quad G \equiv \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 \qquad \quad R_\mathrm{pump} \equiv \gamma_\parallel D_0 \end{aligned}\]

With \(Q\) being the cavity mode’s quality factor. The nonlinear coupling term \(G N_p N_e\) represents stimulated emission, which is the key to lasing.

To understand the behaviour of a laser, consider these equations in a steady state, i.e. where \(N_p\) and \(N_e\) are constant in \(t\):

\[\begin{aligned} 0 &= - \gamma_p N_p + G N_p N_e \\ 0 &= R_\mathrm{pump} - \gamma_e N_e - G N_p N_e \end{aligned}\]

In addition to the trivial solution \(N_p = 0\), we can also have \(N_p > 0\). Isolating \(N_p\)’s equation for \(N_e\) and inserting that into \(N_e\)’s equation, we find:

\[\begin{aligned} N_e = \frac{\gamma_p}{G} \qquad \implies \qquad \boxed{ N_p = \frac{1}{\gamma_p} \bigg( R_\mathrm{pump} - \frac{\gamma_e \gamma_p}{G} \bigg) } \end{aligned}\]

The quantity \(R_\mathrm{thr} \equiv \gamma_e \gamma_p / G\) is called the **lasing threshold**: if \(R_\mathrm{pump} \ge R_\mathrm{thr}\), the laser is active, meaning that \(N_p\) is big enough to cause a “chain reaction” of stimulated emission that consumes all surplus carriers to maintain a steady state.

The point is that \(N_e\) is independent of the electron supply \(R_\mathrm{pump}\), because all additional electrons are almost immediately annihilated by stimulated emission. Consequently \(N_p\) increases linearly as \(R_\mathrm{pump}\) is raised, at a much steeper slope than would be possible below threshold. The output of the cavity is proportional to \(N_p\), so the brightness is also linear.

Unfortunately, by deriving the laser rate equations from the MBEs, we lost some interesting and important effects, most notably spontaneous emission, which is needed for \(N_p\) to grow if \(R_\mathrm{pump}\) is below threshold.

For this reason, the laser rate equations are typically presented in a more empirical form, which “bookkeeps” the processes affecting \(N_p\) and \(N_e\). Consider the following example:

\[\begin{aligned} \boxed{ \begin{aligned} \pdv{N_p}{t} &= - (\gamma_\mathrm{out} + \gamma_\mathrm{abs} + \gamma_\mathrm{loss}) N_p + \gamma_\mathrm{spon} N_e + G_\mathrm{stim} N_p N_e \\ \pdv{N_e}{t} &= R_\mathrm{pump} + \gamma_\mathrm{abs} N_p - (\gamma_\mathrm{spon} + \gamma_\mathrm{n.r.} + \gamma_\mathrm{leak}) N_e - G_\mathrm{stim} N_p N_e \end{aligned} } \end{aligned}\]

Where \(\gamma_\mathrm{out}\) represents the cavity’s usable output, \(\gamma_\mathrm{abs}\) the medium’s absorption, \(\gamma_\mathrm{loss}\) scattering losses, \(\gamma_\mathrm{spon}\) spontaneous emission, \(\gamma_\mathrm{n.r.}\) non-radiative electron-hole recombination, and \(\gamma_\mathrm{leak}\) the fact that some carriers leak away before they can be used for emission.

Unsurprisingly, this form is much harder to analyze, but more accurately describes the dynamics inside a laser. To make matters even worse, many of these decay rates depend on \(N_p\) or \(N_e\), so solutions can only be obtained numerically.

- D. Meschede,
*Optics, light and lasers*, Wiley. - L.A. Coldren, S.W. Corzine, M.L. Mašanović,
*Diode lasers and photonic integrated circuits*, 2nd edition, Wiley.

© Marcus R.A. Newman, a.k.a. "Prefetch".
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