Laser locking and frequency planning

As mentioned in Local beams, each laser source is either frequency-locked to a resonant cavity or phase-locked to another laser source using a specific interferometric beatnote. In this section, we describe how we simulate these laser locking control loops. We then list the various locking configurations available for LISA in its baseline configuration.

Frequency planning

The beatnote frequencies that can be measured by the LISA phasemeters are limited to between \(5\,\mathrm{MHz}\) and \(25\,\mathrm{MHz}\)[1].

As a consequence, all beatnote frequencies need to be controlled to fall in this range, which is achieved by introducing pre-determined offset frequencies in the laser locking control loops. A set of these frequency offsets for all lasers over the whole mission duration is called a frequency plan.

The problem of finding such frequency plans has recently been studied ([22], [23]), and exact solutions have been found. We will use these solutions as an input to the simulation.

Locking condition

Offset frequency laser locking is achieved by controlling the frequency of a locked laser, such that a given beatnote frequency \(\nu_\text{PD}(\tau)\) remains equal to a pre-programmed reference value \(\nu_\text{plan}(\tau)\) provided in the frequency plan.

We do not simulate the actual control loop, but instead directly compute the correct frequency offsets and fluctuations of the locked laser for this locking condition to be satisfied. In reality, the locking control loops will have finite gain and bandwidth, such that the locking beatnotes can still contain out-of-band glitches and noise residuals. Here, we consider the frequency lock to be perfect. This means that the locking beatnote offset is exactly equal to the desired value.

Locking control loops run according to their local clocks, such that the locking condition is fulfilled in the local clock time frame \(\hat\tau_1\). In addition, the frequency-plan locking frequencies are interpreted as functions of the same local clock time frame. In terms of total phase, the result of this control is that the measured beatnote phase is controlled to be exactly equal to the frequency-plan phase,

\[\Phi^{\hat\tau_1}_\text{PD}(\tau) = \Phi_\text{plan}(\tau).\]

Note that the control loop operates on data delivered by the phasemeter at a high frequency of \(80\,\mathrm{MHz}\) (K. Yamamoto, personal communication, May 2021). As such, we simulate the locking before applying any filtering or downsampling.

The previous locking condition is expressed in the local time frame, but we really want to solve for it in the TPS. We can use the equations converting phase between time coordinates (3), giving the clock times (12) and describing the clock behavior as two variables (11), to relate the measured beatnote phase to its equivalent in the TPS,

\[\Phi^{\tau_1}_\text{PD}(\tau) = \Phi^{\hat\tau_1}_\text{PD}(\tau + q_1^o(\tau) + q_1^\epsilon(\tau) + \delta\hat\tau_{1,0}).\]

Using this result, we can write the locking condition as

\[\Phi^{\tau_1}_\text{PD}(\tau) = \Phi_\text{plan}(\tau + q_1^o(\tau) + q_1^\epsilon(\tau) + \delta\hat\tau_{1,0}).\]

We expand the previous equation to first order in \(q_1^\epsilon(\tau)\),

\[ \begin{align}\begin{aligned}\Phi^{\tau_1}_\text{PD}(\tau) &= \Phi_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0})\\&\qquad + \nu_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}) q_1^\epsilon(\tau).\end{aligned}\end{align} \]

The second-order term is proportional to the product \(q_1^\epsilon(\tau)^2 \dot\nu_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0})\) of the square of the clock fluctuations and the time derivative of the frequency-plan locking frequency. To evaluate the order of magnitude of this term, we compute the average clock time deviation [24] after a time corresponding to its saturation frequency; we find a value of the order of \(10^{-9}\,\mathrm{s}\). In addition, all currently available frequency plans verify \(\dot\nu_\text{plan}(\tau) < 3\,\mathrm{Hz/s}\). Therefore, we neglect terms of the order of \(3 \times 10^{-18}\) cycles, far below the \(\mu\)-cycle level of gravitational-wave signals.

From the previous expansion, one directly obtains the usual decomposition in phase drifts and fluctuations,

\[ \begin{align}\begin{aligned}\phi^{\tau_1, o}_\text{PD}(\tau) &= \Phi_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}),\\\phi^{\tau_1, \epsilon}_\text{PD}(\tau) &= \nu_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}) q_1^\epsilon(\tau).\end{aligned}\end{align} \]

The current implementation uses higher order splines computed over coarsely sampled anchor points (~ 1/day) to compute the frequency-plan locking frequencies. As a consequence, the latter are slowly varying, i.e., only consist in large out-of-band frequency offsets, such that \(\nu_\text{plan}(\tau) = \nu^o_\text{plan}(\tau)\). This also applies to the pre-programmed reference phase \(\Phi_\text{plan}(\tau) = \phi^o_\text{plan}(\tau)\).

We denote the (local) locked laser phase drifts and fluctuations as \(\phi_l^{\tau_1,o}(\tau)\) and \(\phi_l^{\tau_1,\epsilon}(\tau)\), and the (distant or adjacent) reference laser phase drifts and fluctuations as \(\phi_r^{\tau_1,o}(\tau)\) and \(\phi_r^{\tau_1,\epsilon}(\tau)\). Using equation (4), we have

\[ \begin{align}\begin{aligned}\phi^{\tau_1, o}_\text{PD}(\tau) &= \phi_r^{\tau_1,o}(\tau) - \phi_l^{\tau_1,o}(\tau),\\\phi^{\tau_1, \epsilon}_\text{PD}(\tau) &= \phi_r^{\tau_1,\epsilon}(\tau) - \phi_l^{\tau_1,\epsilon}(\tau) + N_\text{PD}^\text{ro}(\tau).\end{aligned}\end{align} \]

It is now straightforward to write the resulting locked laser phase drifts and fluctuations,

\[ \begin{align}\begin{aligned}\phi_l^{\tau_1,o}(\tau) &= \phi_r^{\tau_1,o}(\tau) - \Phi_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}),\\\phi_l^{\tau_1,\epsilon}(\tau) &= \phi_r^{\tau_1,\epsilon}(\tau) - \Phi_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}) q_1^\epsilon(\tau) + N_\text{PD}^\text{ro}(\tau).\end{aligned}\end{align} \]

For frequency, we start by taking the derivative of the locking condition and expand it once again in \(q_1^\epsilon\),

\[ \begin{align}\begin{aligned}\nu^{\tau_1}_\text{PD}(\tau) &= (1 + \dot{q}_1^o(\tau)) \nu_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0})\\&\qquad+ \nu_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}) \dot{q}_1^\epsilon(\tau)\\&\qquad+ (1 + \dot{q}_1^o(\tau) + \dot{q}_1^\epsilon(\tau)) \dot\nu_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}) q_1^\epsilon(\tau).\end{aligned}\end{align} \]

The last term of the previous equation has a similar form as the small correction neglected earlier [1]. Therefore, we will neglect it in the rest of this derivation.

Using the same two-variable decomposition along with the frequency equivalent,

\[ \begin{align}\begin{aligned}\nu^{\tau_1, o}_\text{PD}(\tau) &= \nu_r^{\tau_1,o}(\tau) - \nu_l^{\tau_1,o}(\tau),\\\nu^{\tau_1, \epsilon}_\text{PD}(\tau) &= \nu_r^{\tau_1,\epsilon}(\tau) - \nu_l^{\tau_1,\epsilon}(\tau) + \dot{N}_\text{PD}^\text{ro}(\tau).\end{aligned}\end{align} \]

We finally obtain the resulting locked laser frequency offset and fluctuations,

\[ \begin{align}\begin{aligned}\nu_l^{\tau_1,o}(\tau) &= \nu_r^{\tau_1,o}(\tau) - (1 + \dot{q}_1^o(\tau)) \nu_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}),\\\nu_l^{\tau_1,\epsilon}(\tau) &= \nu_r^{\tau_1,\epsilon}(\tau) - \nu_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}) \dot{q}_1^\epsilon(\tau) + \dot{N}_\text{PD}^\text{ro}(\tau).\end{aligned}\end{align} \]

Note that these equations describe the locked laser at the photodiode. To properly simulate this effect, we need the locked lasers frequency at the laser source, which we denote here as \(\bar\nu_l(\tau)\). In Local beams, we add to the local beam frequency fluctuations an optical path length noise term \(\dot N^\text{ob}_\text{PD}(\tau)\) during its propagation from the laser source to the photodiode. As a consequence, we have

\[ \begin{align}\begin{aligned}\bar{\nu}_l^\epsilon(\tau) &= \nu_r^\epsilon(\tau) - \nu_\text{plan}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}) \dot{q}_1^\epsilon(\tau)\\&\qquad + \dot N^\text{ro}_\text{PD}(\tau) - \frac{\nu_0}{c} \dot N^\text{ob}_\text{PD}(\tau)\end{aligned}\end{align} \]

for the local locked lasers fluctuations.

Locking configurations

Laser locking configurations

Fig. 8 Laser locking configurations for laser 12 as primary laser.

In total, 5 of the 6 lasers in the constellation are locked (directly or indirectly) to one primary laser. Each of the locked lasers is locked to either the adjacent laser, using the REF, so that the frequency and phase equations read

\[O_{12}(\tau) = \nu_{\text{ref}_{12} \leftarrow 13}^o(\tau) - (1 + \dot q_1^o(\tau)) \nu_{\text{fplan},\text{ref}_{12}}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}),\]

\[ \begin{align}\begin{aligned}\dot{p}_{12}(\tau) = \nu_{\text{ref}_{12} \leftarrow 13}^\epsilon(\tau) - \nu_{\text{fplan},\text{ref}_{12}}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}) \dot q_1^\epsilon(\tau)\\+ \dot N^\text{ro}_{\text{ref}_{12}}(\tau) - \frac{\nu_0}{c} \dot N^\text{ob}_{\text{ref}_{12}\leftarrow 12}(\tau).\end{aligned}\end{align} \]

Or it is locked to the distant laser, using the SCI, such that we get

\[O_{12}(\tau) = \nu_{\text{sci}_{12} \leftarrow 21}^o(\tau) - (1 + \dot q_1^o(\tau)) \nu_{\text{fplan},\text{sci}_{12}}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}),\]

\[ \begin{align}\begin{aligned}\dot{p}_{12}(\tau) = \nu_{\text{sci}_{12} \leftarrow 21}^\epsilon(\tau) - \nu_{\text{fplan},\text{sci}_{12}}(\tau + q_1^o(\tau) + \delta\hat\tau_{1,0}) \dot q_1^\epsilon(\tau)\\+ \dot N^\text{ro}_{\text{sci}_{12}}(\tau) - \frac{\nu_0}{c} \dot N^\text{ob}_{\text{sci}_{12}\leftarrow 12}(\tau).\end{aligned}\end{align} \]

These expressions can be substituted into the equations of Telemetered beatnote measurements to derive the telemetered beatnote measurements with locked lasers.

The LISA model presented here permits 6 distinct locking topologies. For each of them, we have the freedom to choose the primary laser, such that, in total, we have 36 possible locking configurations. We plot the 6 configurations with laser 12 as the primary laser in Fig. 8. The other 30 combinations can be deduced by applying permutations of the indices.

[1]

The exact frequency range remains to be defined. In addition, some margins are required for both the upper and lower bounds to account for the sideband beatnotes, which are offset by \(1\,\mathrm{MHz}\) from the carrier beatnotes.