the angle between the magnetic field and the linear repumper polarization when dark states are destabilized using $B=50\;\mu$T \[solid (black) curve\] and $B=100\;\mu$T \[dashed (red) curve\]. $\Omega_\mathrm{c} = \Omega_\mathrm{r} = \Gamma$, $\delta_\mathrm{c} = -\Gamma/2$, $\delta_\mathrm{r} = \Gamma/2$, $\gamma_{g,m}=\Gamma/10$, and cooling laser $\sigma$ polarized.](Fig07-repump-angle){width=".85\columnwidth"}

Polarization modulation \[sec:polarizationmodulation\]
======================================================

Dark state destabilization using an external magnetic field has been discussed in Sec. \[sec:dark\_states\]. However, there are applications where a strong magnetic field is not desired, in particular ion clocks, where typically fields of only a few microteslas are used.

A simple and commonly used way to modulate the repumper polarization is to pass the beam through an electro-optical modulator (EOM) at a 45$^\circ$ angle to the optical axis. The resulting polarization (with beam direction as QA) is $$\begin{aligned}
\mathbf{u}_\text{EOM}(t) &=& \frac{1}{2}\left[(1+i e^{-i\varphi_\text{EOM}(t)}) \mathbf{u}_{+1} + (1-i e^{-i\varphi_\text{EOM}(t)}) \mathbf{u}_{-1} \right] \nonumber \\
&=& -\frac{i}{\sqrt{2}} (e^{-i\varphi_\text{EOM}(t)} \mathbf{u}_x + \mathbf{u}_y). \label{eq:EOM-pol}\end{aligned}$$ where the phase retardation is [@Berkeland2002a] $$\label{eq:EOM-ret}
\varphi_\text{EOM}(t) = \frac{1}{2} \Phi_\text{EOM} ( 1-\cos{\omega_\text{EOM}t}).$$ If the modulation frequency $\omega_\text{EOM}=0$ or the modulation amplitude $\Phi_\text{EOM}=0$, the resulting polarization is the stationary linear polarization incident on the EOM, $(\mathbf{u}_x + \mathbf{u}_y)/\sqrt{2}$. In the following, we consider only the EOM modulation technique. Modulation techniques using acousto-optic modulators (AOMs) give 