path: root/source/know/concept/laser-rate-equations/index.md

---
title: "Laser rate equations"
sort_title: "Laser rate equations"
date: 2022-03-16
categories:
- Physics
- Optics
- Laser theory
layout: "concept"
---

The [Maxwell-Bloch equations](/know/concept/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](/know/concept/electric-field/) $$\vb{E}^{+}$$,
the induced polarization $$\vb{P}^{+}$$,
and the total population inversion $$D$$:

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

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. $$\ipdvn{2}{\vb{E}_0^{+}\!}{t} \approx 0$$ and $$\ipdvn{2}{\vb{P}_0^{+}\!}{t} \approx 0$$,
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](/know/concept/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 $$\ipdv{\vb{P}_0^{+}\!}{t}$$ 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 $$\ipdv{\vb{P}_0^{+}\!}{t} \approx 0$$, 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](/know/concept/salt-equation/))
represents the 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 $$\ipdv{\vb{P}_0^{+}}{t} \approx 0$$,
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 the aforementioned 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:

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

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

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

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.
The photon loss rate $$\gamma_p$$, the gain $$G$$,
and the carrier supply rate $$R_\mathrm{pump}$$
are defined like so:

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

With $$Q$$ being the cavity mode's quality factor.
The nonlinear coupling term $$G N_p N_e$$ represents
[stimulated emission](/know/concept/einstein-coefficients/),
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.



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