path: root/content/know/concept/laser-rate-equations/index.pdc

---
title: "Laser rate equations"
firstLetter: "L"
publishDate: 2022-03-16
categories:
- Physics
- Optics
- Laser theory

date: 2022-03-12T20:23:42+01:00
draft: false
markup: pandoc
---

# Laser rate equations

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 $D$:

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

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](/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 $\pdv*{\vb{P}_0^{+}\!}{t}$ 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](/know/concept/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](/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.