Optical simulation

In this section, we derive the model for the generation and propagation of the laser beams, as well as their interference at the photodiodes.

Optical bench design

As illustrated in Fig. 1, each spacecraft hosts two optical benches. We usually refer to one optical bench as the local optical bench; the other optical bench hosted by the same spacecraft as the adjacent optical bench; we call the distant optical bench the one situated on the spacecraft exchanging light with the local optical bench. Each optical bench is associated with a laser source, a GRS containing a free-falling test mass, and telescope to send and collect light to and from distant spacecraft.

Laser beams are combined in 3 different heterodyne interferometers. The SCI mixes the local beam with the distant beam (coming from the distant optical bench); the TMI mixes the local and adjacent beams, after it has bounced on the local test mass; and the REF mixes the local and adjacent beams without interaction with the test mass. Fig. 3 gives an overview of the optical bench 12.

Optical design

Fig. 3 Schematics of the optical design implemented in the simulation, along with notations for the laser beam and beatnote total phases, here for MOSA 12.

In reality, each single interferometer output is implemented using redundant balanced detection with four QPDs. We do not simulate balanced detection, and only consider a single data stream for each interferometer. Additional readouts related to the laser beams alignment, such as the DWS, are not included in the model presented here. We are currently working to implement them in the simulation by propagating additional independent variables representing the different beam tilts (however, TTL effects in the SCI are implemented).

Laser beam model

Simple laser beam

We use a number of assumptions to model the information carried by the electromagnetic field of a laser beam (in all generality, these are two 3-vector fields).

We work in the plane-wave approximation, and assume that any effects due to wavefront imperfections can be modeled as equivalent longitudinal path length variations. In addition, we neglect effects related to the fields polarization, and assume that the waves propagate in a perfect vacuum, such that we only model the scalar electric field amplitude (we do not model the magnetic field amplitude, as it is completely determined by the electrical field amplitude [8]).

At any fixed point inside a spacecraft, the complex amplitude of the electromagnetic field associated with a laser beam can be written as

\[E(\tau) = E_0(\tau) e^{i 2\pi \Phi(\tau)},\]

where \(\Phi(\tau)\) is the total phase in units of cycles.

LISA ultimately measures phase differences, such that we do not simulate the field amplitude term \(E_0\), but only the phase \(\Phi(\tau)\). We expect couplings between the field amplitude and the measured phase difference (e.g., the relative intensity noise [9]). We currently do not model these effects, but assume that they can be modeled as equivalent phase noise in the final readout.

Phase or frequency?

The optical frequency of the lasers is around \(\nu_0=281.6\,\mathrm{THz}\), such that the total phase increases quickly with time. This makes using it challenging for numerical simulations, as any variable representing the total phase will either numerically overflow when using fixed-point arithmetic, or eventually suffer an unacceptable loss of precision when using floating-point arithmetic (for a precision better than a micro-cycle, a 64-bit integer representing the total phase will overflow every 0.07 s.).

To avoid these issues, we simulate frequencies instead of phase (given by \(\nu = \dot{\Phi}\), since we express the phase in units of cycles), which are controlled to remain at the same order of magnitude during the whole mission duration. However, modeling the propagation of laser beams is often easier in phase. Therefore, we will derive most of the equations of this paper both in units of phase and frequency.

Two-variable decomposition

In LISA, effects on the laser beams come into play at completely different timescales and dynamic ranges. On the one hand, some effects modulate the frequency of our beams on a time scale of the orbital revolution around the Sun, which lies well outside our measurement frequency band (below \(10^{-4}\,\mathrm{Hz}\)). These effects tend to have large dynamic ranges; for instance, the Doppler shifts caused by the relative spacecraft motion can fluctuate by several megahertz over the mission duration.

On the other hand, we want to track small phase or frequency fluctuations within our measurement band (from \(10^{-4}\,\mathrm{Hz}\) and up to \(1\,\mathrm{Hz}\)), caused by gravitational-wave signals and instrumental noises. These fluctuations have a much smaller amplitude. The laser noise being the dominant effect, causing the heterodyne beatnote frequency to shift by about a few tens of Hertz, while gravitational waves typically cause frequency shifts of a few hundreds of nanohertz.

To address this problem, we model these different effects independently. We decompose the laser beam frequency into one constant and two variables,

\[\nu(\tau) = \nu_0 + \nu^o(\tau) + \nu^\epsilon(\tau),\]

The large frequency offsets \(\nu^o(\tau)\) are used to represent frequency-plan offsets and Doppler shifts, both on the order of megahertz, as well as the gigahertz sideband frequency offsets. The small frequency fluctuations \(\nu^\epsilon(\tau)\), on the other hand, are used to describe gravitational signals and noises. A simple laser beam would therefore be entirely represented by the couple \(\{\nu^o(\tau), \nu^\epsilon(\tau)\}\).

Alternatively, we can express the previous equation in phase units by writing the total phase as

\[\Phi(\tau) = \nu_0 \tau + \phi^o(\tau) + \phi^\epsilon(\tau) + \phi^0,\]

where the definitions of large phase drifts \(\phi^o(\tau)\) and small phase fluctuations \(\phi^\epsilon(\tau)\) follow from the previous equation,

\[\nu^o(\tau) = \dot{\phi}^o(\tau), \qquad \nu^\epsilon(\tau) = \dot{\phi}^\epsilon(\tau)\]

As we simulate frequencies, we do not track the initial phase of the laser beam \(\phi^0 \in [0, 2\pi]\) in the following.

Let us stress that this decomposition is entirely artificial. In reality, we will only have access to the total phase or frequency. Therefore, to produce data representative of the real instrument telemetry, we always compute the total phase or frequency as the final simulation output.

Warning

Rewrite the equations in phase keeping track of some initial phase offsets (which in the end will be correlated in the photodiode signals). They might be of interest for ranging using beatnote phases.

Modulated beams

In LISA, laser beams are phase-modulated using a gigahertz signal derived from the local clock. The electric field reads

\[E(\tau) = E_0 e^{i 2\pi \Phi_\text{c}(\tau)} e^{i m \cos(2 \pi \Phi_m(\tau))},\]

where \(m\) is the modulation depth; \(\Phi_\text{c}(\tau)\) is the total phase of the carrier, and \(\Phi_m(\tau)\) is the total phase of the modulating signal, both expressed in cycles.

The Jacobi-Anger expansion lets us write the previous expression using Bessel functions. Because the modulation depth \(m \approx 0.15\) is small [10], we can further expand the result to first order in \(m\) (newer values are greater, yet still \(m < 1\)) and write the complex field amplitude as the sum

\[E(\tau) \approx E_0 \left(e^{i 2\pi \Phi_\text{c}(\tau)} + i \frac{m}{2} \left[e^{i 2\pi \Phi_{\text{sb}^+}(\tau)} + e^{i 2\pi \Phi_{\text{sb}^-}(\tau)}\right]\right),\]

where we have defined the upper and lower sideband phases,

\[\begin{split}\Phi_{\text{sb}^+}(\tau) = \Phi_\text{c}(\tau) + \Phi_m(\tau) \\ \Phi_{\text{sb}^-}(\tau) = \Phi_\text{c}(\tau) - \Phi_m(\tau)\end{split}\]

The modulated laser beam can then be written as the superposition of carrier, upper sideband, and lower sideband,

\[E(\tau) \approx E_\text{c}(\tau) + E_{\text{sb}^+}(\tau) + E_{\text{sb}^-}(\tau)\]

For the purpose of our simulation, the information content of the upper and lower sidebands are almost identical (one difference is that they lie at a different frequencies, and are thus affected differently by Doppler shifts). We make the assumption that they can be combined in such a way that we can treat them as one signal. Therefore, we only simulate the upper sideband. For clarity, we drop the sign in all sideband indices and simply use sb when we refer to the upper sideband.

We apply the same two-variable decomposition to the sideband total frequency. Ultimately, each modulated laser beam is then implemented using 4 quantities,

\[\nu(\tau) \equiv \{\nu_\text{c}^o(\tau), \nu_\text{c}^\epsilon(\tau), \nu_\text{sb}^o(\tau), \nu_\text{sb}^\epsilon(\tau)\},\]

where \(\nu_\text{c}^o\) and \(\nu_\text{sb}^o\) are the carrier and sideband frequency offsets, respectively, and \(\nu_\text{c}^\epsilon\) and \(\nu_\text{sb}^\epsilon\) are the carrier and sideband frequency fluctuations.

Warning

Study upper and lower sideband properly. It is not obvious how they should be combined to recover the exact same formulas we used for clock calibration with a factor of sqrt2. In particular, we might want to use both sidebands to recover the modulated signal, instead of one sideband - carrier, depending on residual readout noise. The different Doppler shifts for the different sidebands also have to be looked at carefully for these computations. Also consider the effect of increased value for the modulation depth.

Local beams

Local beam at laser source

As illustrated in Fig. 3, optical bench 12 has an associated laser source. We call local beam the modulated beam produced by this laser source. We denote the total phase and frequency of the carrier as \(\Phi_{12,\text{c}}(\tau)\) and \(\nu_{12,\text{c}}(\tau)\), respectively. Similarly, the sideband total phase and frequency are denoted as \(\Phi_{12,\text{sb}}(\tau)\) and \(\nu_{12,\text{sb}}(\tau)\). All these signals are functions of the TPS \(\tau_1\).

The total phase \(\Phi_{12,\text{c}}(\tau) = \nu_0 \tau + \phi_{12,\text{c}}^o(\tau) + \phi_{12,\text{c}}^\epsilon(\tau)\) of the carrier is decomposed in terms of drifts and fluctuations, with

\[ \begin{align}\begin{aligned}\phi_{12,\text{c}}^o(\tau) &= \int_{\tau_{1,0}}^{\tau} O_{12}(\tau') \, \textrm{d} \tau'\\\phi_{12,\text{c}}^\epsilon(\tau) &= p_{12}(\tau)\end{aligned}\end{align} \]

where \(O_{12}(\tau)\) is the carrier frequency offset for this laser source with respect to the central frequency \(\nu_0\), and \(p_{12}(\tau)\) is the laser source phase fluctuations expressed in cycles.

As explained in Laser locking and frequency planning, \(p_{ij}(\tau)\) can either describe the noise \(N^p_{ij}(\tau)\) of a cavity-stabilized laser or the fluctuations resulting from an offset frequency lock. Likewise, \(O_{12}(\tau)\) is either set as an offset from the nominal frequency[1] or computed based on the locking conditions.

In terms of frequency, we simply have

\[ \begin{align}\begin{aligned}\nu_{12,\text{c}}^o(\tau) &= O_{12}(\tau)\\\nu_{12,\text{c}}^\epsilon(\tau) &= \dot p_{12}(\tau)\end{aligned}\end{align} \]

Let us now look at the sideband, which is derived from the local clock. As described in detail in Phase readout, frequency distribution, and clock error modelling, the modulating signal inherits any USO timing errors \(q_1\), such that we have

\[\Phi_{12,m}(\tau) = \nu_{12}^m \cdot (\tau + q_1^o(\tau) + q_1^\epsilon(\tau) + M_{12}(\tau))\]

for the total phase of the modulating signal. Here, \(\nu_{12}^m = 2.4\,\mathrm{GHz}\) is the constant nominal frequency of the modulating signal on optical bench 12. We use the same modulation frequency for all optical benches indexed cyclically (12, 23, 31), while the remaining ones (13, 32, 21) are instead at \(2.401\,\mathrm{GHz}\). The modulation noise term \(M_{12}(\tau)\) accounts for any additional imperfections (either in the electrical frequency conversion to \(2.4\,\mathrm{GHz}\) or the optical modulation).

The total phase of the modulating signal can then be decomposed into

\[ \begin{align}\begin{aligned}\phi_{12}^o(\tau) &= \nu_{12}^m \cdot (\tau + q_1^o(\tau))\\\phi_{12,m}^\epsilon(\tau) &= \nu_{12}^m \cdot (q_1^\epsilon(\tau) + M_{12}(\tau))\end{aligned}\end{align} \]

Inserting these terms in the sideband phase decomposition, we get the phase and frequency offsets and fluctuations for the local sideband,

\[ \begin{align}\begin{aligned}\phi_{12,\text{sb}}^o(\tau) &= \int_{\tau_{1,0}}^{\tau} O_{12}(\tau') \, \mathrm{d} \tau' + \nu_{12}^m (\tau + q_1^o(\tau))\\\phi_{12,\text{sb}}^\epsilon(\tau) &= p_{12}(\tau) + \nu_{12}^m (q_1^\epsilon(\tau) + M_{12}(\tau))\end{aligned}\end{align} \]

and

\[ \begin{align}\begin{aligned}\nu_{12,\text{sb}}^o(\tau) &= O_{12}(\tau) + \nu_{12}^m (1 + \dot q_1^o(\tau))\\\nu_{12,\text{sb}}^\epsilon(\tau) &= \dot p_{12}(\tau) + \nu_{12}^m ( \dot q_1(\tau) + \dot M_{12}(\tau) )\end{aligned}\end{align} \]

Note that there is only one clock per spacecraft, such that we use the same \(q_1\) for sideband beams on both optical benches on spacecraft 1.

Local beams at the interspacecraft and reference interferometer photodiodes

As shown in Fig. 3, local beams propagate in the local optical bench 12 and interfere at the SCI, TMI, and REF photodiodes. In our simulations, we neglect any phase term due to the on-board propagation time. However, all beams pick up a generic optical path length noise term \(N^\text{ob}(\tau)\) (different for each interferometer), which models all optical path length variations due to, e.g., jitters of optical components in the path of the laser beams. By convention, we choose that a positive value of the optical path length noise term corresponds to a decrease in the actual optical path length, which in turn corresponds to a positive shift in phase or frequency.

Therefore, we write the phase drifts and fluctuations of the local beams at the SCI and REF photodiodes (valid for both carriers and sidebands with \(k \in \{\text{c, sb}\}\)) as

\[ \begin{align}\begin{aligned}\phi^o_{\text{sci/ref}_{12} \leftarrow 12,k}(\tau) &= \phi^o_{12,k}(\tau),\\\phi^\epsilon_{\text{sci/ref}_{12} \leftarrow 12,k}(\tau) &= \phi^\epsilon_{12,k}(\tau) + \frac{\nu_0}{c} N^\text{ob}_{\text{sci/ref}_{12} \leftarrow 12,k}(\tau).\end{aligned}\end{align} \]

Equivalently, the frequency offsets and fluctuations of the same beams read

\[ \begin{align}\begin{aligned}\nu^o_{\text{sci/ref}_{12} \leftarrow 12,k}(\tau) &= \nu^o_{12,k}(\tau),\\\nu^\epsilon_{\text{sci/ref}_{12} \leftarrow 12,k}(\tau) &= \nu^\epsilon_{12,k}(\tau) + \frac{\nu_0}{c} \dot{N}^\text{ob}_{\text{sci/ref}_{12} \leftarrow 12,k}(\tau).\end{aligned}\end{align} \]

Local beam at the test-mass interferometer photodiode

The local beam reflects off the test mass before impinging on the TMI photodiode. As a consequence, it couples to the test-mass motion.

In reality, the motion of the test mass and spacecraft will be coupled by the DFACS. The spacecraft motion is expected to be suppressed in on-ground processing [10]. For our purposes, we simply assume that the spacecraft (and the associated optical benches) perfectly follows a geodesic.

The laser beam then picks up an additional noise term \(N^\delta_{23}(\tau)\) due to any deviation in the motion of the test-mass from geodesic, caused by spurious forces. This noise represents the movement of the test mass towards the measuring optical bench, such that a positive value corresponds to a decrease in path length (see Fig. 4), and thus a positive phase shift. The noise term enter with a factor 2, since the beam travels to the test mass and back.

Test-mass motion

Fig. 4 Definition of line-of-sight displacement of the test mass. Positive values of \(N_{12}^\delta\) indicate a motion towards the measuring optical bench.

Therefore, at the TMI photodiode, the phase components of the local beam read (again for both carriers and sidebands with \(k \in \{\text{c, sb}\}\))

\[ \begin{align}\begin{aligned}\phi^o_{\text{tmi}_{12} \leftarrow 12,k}(\tau) &= \phi^o_{12,k}(\tau)\\\phi^\epsilon_{\text{tmi}_{12} \leftarrow 12,k}(\tau) &= \phi^\epsilon_{12,k}(\tau) + \frac{\nu_0}{c} \left(N^\text{ob}_{\text{tmi}_{12} \leftarrow 12,k}(\tau) + 2 N^\delta_{12}(\tau)\right)\end{aligned}\end{align} \]

while the frequency offsets and fluctuations read

\[ \begin{align}\begin{aligned}\nu^o_{\text{tmi}_{12} \leftarrow 12,k}(\tau) &= \nu^o_{12,k}(\tau)\\\nu^\epsilon_{\text{tmi}_{12} \leftarrow 12,k}(\tau) &= \nu^\epsilon_{12,k}(\tau) + \frac{\nu_0}{c} \left(\dot{N}^\text{ob}_{\text{tmi}_{12} \leftarrow 12,k}(\tau) + 2 \dot{N}^\delta_{12}(\tau)\right)\end{aligned}\end{align} \]

Warning

Small inconsistencies between documentation and the code regarding the scaling of the noises with the central frequency, with or without including offsets, in the two sections above. For example, \(\dot{N}^\text{ob}_{\text{tmi}_{12} \leftarrow 12,k}\) is scaled by \(\nu_0\), whereas \(2 \dot{N}^\delta_{12}\) (which is the same noise for carrier and sideband) is scaled by \(\nu_0+\nu_{0,k}^o\).

Adjacent beams

In this section, we study the propagation of a modulated laser beam generated by laser source 13 (attached to the adjacent optical bench), which travels through the optical fiber to the local optical bench 12, to finally interfere on the TMI and REF photodiodes (see Fig. 3). We call it adjacent beam. We express all phase and frequency quantities as functions of the TPS \(\tau_1\).

Similarly to local beams, we neglect the propagation time for the adjacent beams, and model fluctuations in the optical path length by a noise term \(N^\text{ob}(\tau)\). We model any nonreciprocal noise terms related to the propagation through the optical fibres by the backlink noise term \(N^\text{bl}_{12 \leftarrow 13}(\tau)\), expressed as an equivalent path length change.

Therefore, the phase drifts and fluctuations of adjacent beams (carrier and sideband, denoted again with \(k\) as above) at the TMI and REF photodiodes read

\[ \begin{align}\begin{aligned}\phi^o_{\text{tmi/ref}_{12} \leftarrow 13,k}(\tau) &= \phi^o_{13,k}(\tau)\\\phi^\epsilon_{\text{tmi/ref}_{12} \leftarrow 13,k}(\tau) &= \phi^\epsilon_{13,k}(\tau) + \frac{\nu_0}{c} N^\text{bl}_{12 \leftarrow 13}(\tau) + \frac{\nu_0}{c} N^\text{ob}_{\text{tmi/ref}_{12} \leftarrow 13,k}\end{aligned}\end{align} \]

where \(\phi^o_{13}(\tau)\) and \(\phi^\epsilon_{13}(\tau)\) are the phase drifts and fluctuations of the laser beam produced by laser source 13, respectively. The equivalent frequency quantities are

\[ \begin{align}\begin{aligned}\nu^o_{\text{tmi/ref}_{12} \leftarrow 13,k}(\tau) &= \nu^o_{13,k}(\tau)\\\nu^\epsilon_{\text{tmi/ref}_{12} \leftarrow 13,k}(\tau) &= \nu^\epsilon_{13,k}(\tau) + \frac{\nu_0}{c} \dot{N}^\text{bl}_{12 \leftarrow 13}(\tau) + \frac{\nu_0}{c} \dot{N}^\text{ob}_{\text{tmi/ref}_{12} \leftarrow 13,k}\end{aligned}\end{align} \]

Warning

There is currently no distinction between carrier and sidebands for the OMS and backlink noises which are only scaled by the central frequency.

Distant beams

Finally, we study the propagation of a modulated laser beam generated by laser source 21 (attached to the distant optical bench), which travels roughly 2.5 million kilometers in free space before it reaches the local optical bench 12. This distant beam eventually interferes on the SCI photodiode, see Fig. 3.

Interspacecraft propagation

As described in Laser beam model, modulated beams are represented as the superposition of simple beams, each treated independently. Consequently, the same propagation equations apply to both carrier and sideband beams.

We shall derive the expression of a simple laser beam’s phase \(\Phi_{12\leftarrow 21}(\tau)\) and frequency \(\nu_{12\leftarrow 21}(\tau)\) measured on receiver optical bench 12 (expressed in comoving time coordinate \(\tau_1\)) as a function of the same beam’s phase \(\Phi_{21}(\tau)\) and frequency \(\nu_{21}(\tau)\) measured on emitter optical bench 21 (expressed in comoving time coordinate \(\tau_2\)). We write

\[\Phi_{12 \leftarrow 21}(\tau) = \Phi_{21}(\tau - d_{12}(\tau)),\]

where \(d_{12}(\tau)\) is the PPR, which includes not only the light time of flight, but also conversions between reference frames associated to \(\tau_1\) and \(\tau_2\).

Since we model small in-band and large out-of-band effects independently, we need to decompose the PPR in a similar manner. We define \(d_{12}^o(\tau)\) as slowly varying PPR offsets (e.g., due to constant path lengths and variations in orbital motion, relativistic effects, and coordinate transformations), and \(d_{12}^\epsilon(\tau)\) as small in-band PPR fluctuations.

In our simulation, we only consider the effect of gravitational waves and tilt-to-length coupling, and neglect any other small in-band fluctuations of the PPRs (such as variations of the interplanetary medium optical index). Therefore, if \(H_{12}(\tau)\) denotes the integrated fluctuations of the PPR due to gravitational waves measured on MOSA 12 and \(\mathrm{TTL}_{12}(\tau)\) the total TTL contributions, we have \(d_{12}^\epsilon(\tau) = -H_{12}(\tau) + \mathrm{TTL}_{12}(\tau)\). The total PPR now reads

\[d_{12}(\tau) = d_{12}^o(\tau) - H_{12}(\tau) + \mathrm{TTL}_{12}(\tau).\]

Note that, by convention, we define \(d_{12}^\epsilon(\tau)\) to contain \(H_{12}(\tau)\) with a minus sign, such that a positive fluctuation corresponds to a decrease in the PPR. This is in line with the conventions adopted in the LISA DDPC, where the GW-induced fractional Doppler shift corresponds to \(y_{12}(\tau) \equiv \dot H_{12}(\tau)\) in our notation.

Warning

Give a clearer definition of the PPR and its decomposition. Since we don’t simulate dynamics, we’re measuring here the test mass to test mass distance (somehow a spacecraft is just concentrated in a test mass). Any pathlength noise (optical bench or telescope) should be “differential”, i.e., measured in one ifo and not the other so that it remains when we combine the two.

Applying this decomposition, we have

\[ \begin{align}\begin{aligned}\Phi_{12 \leftarrow 21}(\tau) ={}& \nu_0 \cdot (\tau - d_{12}^o(\tau) + H_{12}(\tau) - \mathrm{TTL}_{12}(\tau))\\&+ \phi_{21}^o(\tau - d_{12}^o(\tau) + H_{12}(\tau) - \mathrm{TTL}_{12}(\tau)) + \phi_{21}^\epsilon(\tau - d_{12}^o(\tau) + H_{12}(\tau) - \mathrm{TTL}_{12}(\tau))\end{aligned}\end{align} \]

We expand the previous equation to first order in both the small fluctuations \(H_{12}(\tau)\), \(\mathrm{TTL}_{12}(\tau)\), and neglect any second-order cross terms with \(\nu_{21}^\epsilon(\tau)\), yielding

\[ \begin{align}\begin{aligned}\Phi_{12 \leftarrow 21}(\tau) ={}& \nu_0 \cdot (\tau - d_{12}^o(\tau) + H_{12}(\tau) - \mathrm{TTL}_{12}(\tau)) + \phi_{21}^o(\tau - d_{12}^o(\tau))\\& + \nu_{21}^o(\tau - d_{12}^o(\tau)) H_{12}(\tau) - \nu_{21}^o(\tau - d_{12}^o(\tau)) \mathrm{TTL}_{12}(\tau) + \phi_{21}^\epsilon(\tau - d_{12}^o(\tau))\end{aligned}\end{align} \]

We can again write the previous quantity as the sum of large phase drifts and small phase fluctuations, \(\Phi_{12 \leftarrow 21}(\tau) = \nu_0 \tau + \phi_{12 \leftarrow 21}^o(\tau) + \phi_{12 \leftarrow 21}^\epsilon(\tau)\), with

\[ \begin{align}\begin{aligned}\phi_{12 \leftarrow 21}^o(\tau) &= \phi_{21}^o(\tau - d_{12}^o(\tau)) - \nu_0 d_{12}^o(\tau)\\\phi_{12 \leftarrow 21}^\epsilon(\tau) &= \phi_{21}^\epsilon(\tau - d_{12}^o(\tau)) + [\nu_0 + \nu_{21}^o(\tau - d_{12}^o(\tau))] (H_{12}(\tau) - \mathrm{TTL}_{12}(\tau)).\end{aligned}\end{align} \]

We write the equivalent instantaneous frequency \(\nu_{12 \leftarrow 21}(\tau) = \nu_0 + \nu_{12 \leftarrow 21}^o(\tau) + \nu_{12 \leftarrow 21}^\epsilon(\tau)\) as the sum of a large frequency offsets and small frequency fluctuations,

\[ \begin{align}\begin{aligned}\nu_{12 \leftarrow 21}^o(\tau) &= \nu_{21}^o(\tau - d_{12}^o(\tau))(1 - \dot{d}_{12}^o(\tau)) - \nu_0\dot{d}_{12}^o (\tau)\\\nu_{12 \leftarrow 21}^\epsilon(\tau) &= \nu_{21}^\epsilon(\tau - d_{12}^o(\tau)) (1 - \dot{d}_{12}^o(\tau)) + [\nu_0 + \nu_{21}^o(\tau - d_{12}^o(\tau))] (\dot H_{12}(\tau) - \dot{\mathrm{TTL}}_{12}(\tau)).\end{aligned}\end{align} \]

Here, we have neglected first-order terms in \(\dot \nu_{21}^o (H_{12}(\tau)+\mathrm{TTL}_{12}(\tau))\), so these equations are only valid if the laser frequency is evolving slowly.

Warning

At the moment, we don’t add a contribution of optical pathlength noise in the simulation in the propagation, but have a noise term in the optical bench. We also don’t account for constant glspl{ppr} onboard the optical bench. To be evaluated what the impact of those is. In the long run, it is probably best to add those terms directly to the beams. This needs input from optical bench experts.

Distant beams at the interspacecraft interferometer photodiode

The received distant beam propagates inside the optical bench to interfere with the local beam at the SCI photodiode. As for the other beams, we only add a generic optical path length noise term \(N^\text{ob}_{\text{sci}_{12}\leftarrow 21}(\tau)\).

We write the phase drifts and fluctuations of the distant beam at the SCI photodiode (valid for both carrier and sideband) as

\[ \begin{align}\begin{aligned}\phi^o_{\text{sci}_{12} \leftarrow 21}(\tau) &= \phi_{21}^o(\tau - d_{12}^o(\tau)) - \nu_0 d_{12}^o(\tau)\\\phi^\epsilon_{\text{sci}_{12} \leftarrow 21}(\tau) &= \phi_{21}^\epsilon(\tau - d_{12}^o(\tau)) + [\nu_0 + \nu_{21}^o(\tau - d_{12}^o(\tau))] (H_{12}(\tau) - \mathrm{TTL}_{12}(\tau)) + \frac{\nu_0}{c} N^\text{ob}_{\text{sci}_{12} \leftarrow 21}(\tau)\end{aligned}\end{align} \]

Equivalently, the frequency offsets and fluctuations read

\[ \begin{align}\begin{aligned}\nu^o_{\text{sci}_{12} \leftarrow 21}(\tau) &= \nu_{21}^o(\tau - d_{12}^o(\tau)) (1 - \dot{d}_{12}^o(\tau)) - \nu_0 \dot{d}_{12}^o(\tau)\\\nu^\epsilon_{\text{sci}_{12} \leftarrow 21}(\tau) &= \nu_{21}^\epsilon(\tau - d_{12}^o(\tau)) (1 - \dot{d}_{12}^o(\tau)) + [\nu_0 + \nu_{21}^o(\tau - d_{12}^o(\tau))] (\dot H_{12}(\tau) - \dot{\mathrm{TTL}}_{12}(\tau)) + \frac{\nu_0}{c} \dot{N}^\text{ob}_{\text{sci}_{12} \leftarrow 21}(\tau)\end{aligned}\end{align} \]

Warning

The code and the text are currently not very explicit what is an OB noise and what is OMS noise and how they relate to the noise terms in the equations here. For example, here in the documentation both beams involved in the interference have an OB noise term, while in the simulation code we only add one generic OMS noise term to the photodiode signal.

Interferometers

Beatnote for simple beams

Using definitions given above, let us write the complex amplitude for two simple beams 1 and 2 interfering at a photodiode,

\[ \begin{align}\begin{aligned}\mathcal{E}_1(\tau) &= E_{1,0}(\tau) e^{i 2\pi \Phi_1(\tau)}\\\mathcal{E}_2(\tau) &= E_{2,0}(\tau) e^{i 2\pi \Phi_2(\tau)}\end{aligned}\end{align} \]

We ignore any effects due to spatial dimensions of the beam or the photodiode, and assume that such effects will be modeled as either an equivalent phase error in the readout signal, or as an independent quantity (for example, DWS could be modeled as a direct measurement of beam tilt angles, with all beam angles represented by independent variables).

The power of the total electromagnetic field measured near the photodiode is

\[P(\tau) \propto |\mathcal{E}_1(\tau) + \mathcal{E}_2(\tau)|^2.\]

Substituting the expressions of the two beams yields

\[P(\tau) \propto |E_{1,0}(\tau)|^2 + |E_{2,0}(\tau)|^2 + 2 E_{1,0}(\tau) E_{2,0}(\tau) \cos(2\pi (\Phi_1(\tau) - \Phi_2(\tau))).\]

The power near the photodiode has an oscillating component with a total phase of \(\Phi_\text{PD}(\tau) = \Phi_1(\tau) - \Phi_2(\tau)\). We call this signal the beatnote.

Let us use the two-variable representation described above,

\[ \begin{align}\begin{aligned}\Phi_1(\tau) &= \nu_0 \tau + \phi^o_1(\tau) + \phi^\epsilon_1(\tau)\\\Phi_2(\tau) &= \nu_0 \tau + \phi^o_2(\tau) + \phi^\epsilon_2(\tau)\end{aligned}\end{align} \]

to express the total phase of the beatnote as the sum of large phase drifts and small phase fluctuations,

(4)\[\Phi_\text{PD}(\tau) = \phi^o_\text{PD}(\tau) + \phi^\epsilon_\text{PD}(\tau)\]

with

\[ \begin{align}\begin{aligned}\phi^o_\text{PD}(\tau) &= \phi^o_1(\tau) - \phi^o_2(\tau)\\\phi^\epsilon_\text{PD}(\tau) &= \phi^\epsilon_1(\tau) - \phi^\epsilon_2(\tau).\end{aligned}\end{align} \]

We simulate the equivalent instantaneous frequency defined as \(\nu_\text{PD}(\tau) = \dot{\Phi}_\text{PD}(\tau)\). It can be written as

\[\nu_\text{PD}(\tau) = \nu^o_\text{PD}(\tau) + \nu^\epsilon_\text{PD}(\tau)\]

where the beatnote frequency offsets \(\nu^o_\text{PD}(\tau)\) and the beatnote frequency fluctuations \(\nu^\epsilon_\text{PD}(\tau)\) are defined by

\[ \begin{align}\begin{aligned}\nu^o_\text{PD}(\tau) &= \nu^o_1(\tau) - \nu^o_2(\tau)\\\nu^\epsilon_\text{PD}(\tau) &= \nu^\epsilon_1(\tau) - \nu^\epsilon_2(\tau)\end{aligned}\end{align} \]

Beatnote polarity

A closer look at the above equations shows that we do not have direct access to the total phase of the beatnote \(\Phi_\text{PD}(\tau)\), but only measure its cosine value. Therefore, the total phase can only be known up to a sign and a multiple of \(2 \pi\).

Physically, this sign ambiguity corresponds to the fact that the electrical signal does not contain any information about which of the two interfering laser beams is of higher frequency. In practice, however, the beatnote polarity can be determined at all times by applying a known frequency offset on the local laser beam and observing the resulting change in the beatnote frequency. In addition, once all lasers are locked, the beatnote polarities can simply be read from the frequency plan, as described in Laser locking and frequency planning.

Therefore, we do not include the beatnote polarity ambiguity in our optical models, and we will instead assume that it is solved directly by the phasemeter, or in a first processing step on ground.

Beatnotes for modulated beams

We now study the electromagnetic field of two interfering modulated beams, labeled \(k=1,2\). As derived above, we write both modulated beams as the sum of three independent simple beams, namely the carriers and the upper and lower sidebands,

\[\mathcal{E}_{k}(\tau) = \mathcal{E}_{k,\text{c}}(\tau) + \mathcal{E}_{k,\text{sb}^+}(\tau) + \mathcal{E}_{k,\text{sb}^-}(\tau)\]

with total phases

\[ \begin{align}\begin{aligned}\Phi_{k,\text{c}}(\tau) &= \nu_0 \tau + \Phi_{k,\text{c}}^o(\tau) + \Phi_{k,\text{c}}^\epsilon(\tau)\\\Phi_{k,\text{sb}^+}(\tau) &= \nu_0 \tau + \Phi_{k,\text{sb}^+}^o(\tau) + \Phi_{k,\text{sb}^+}^\epsilon(\tau)\\\Phi_{k,\text{sb}^-}(\tau) &= \nu_0 \tau + \Phi_{k,\text{sb}^-}^o(\tau) + \Phi_{k,\text{sb}^-}^\epsilon(\tau)\end{aligned}\end{align} \]

or the equivalent instantaneous frequencies

\[ \begin{align}\begin{aligned}\nu_{k,\text{c}}(\tau) &= \nu_0 + \nu_{k,\text{c}}^o(\tau) + \nu_{k,\text{c}}^\epsilon(\tau)\\\nu_{k,\text{sb}^+}(\tau) &= \nu_0 + \nu_{k,\text{sb}^+}^o(\tau) + \nu_{k,\text{sb}^+}^\epsilon(\tau)\\\nu_{k,\text{sb}^-}(\tau) &= \nu_0 + \nu_{k,\text{sb}^-}^o(\tau) + \nu_{k,\text{sb}^-}^\epsilon(\tau)\end{aligned}\end{align} \]

The total power at the photodiode reads

\[|\mathcal{E}_1(\tau) + \mathcal{E}_2(\tau)|^2 = \left| \mathcal{E}_{1,\text{c}}(\tau) + \mathcal{E}_{1,\text{sb}^+}(\tau) + \mathcal{E}_{1,\text{sb}^-}(\tau) + \mathcal{E}_{2,\text{c}}(\tau) + \mathcal{E}_{2,\text{sb}^+}(\tau) + \mathcal{E}_{2,\text{sb}^-}(\tau) \right|^2\]

Expanding this expression yields cross terms between all 6 terms, which correspond to beatnotes at their difference frequencies.

Because the sidebands are modulated at a frequency of about \(2.4\,\mathrm{GHz}\), most of these beatnote frequencies lie far outside of the phasemeters measurement bandwidth (approximately \(5\,\mathrm{MHz}\) to \(25\,\mathrm{MHz}\)).

Only three beatnotes lie inside this region:

• The carrier-carrier beatnote,

  \[ \begin{align}\begin{aligned}\Phi_{\text{PD},\text{c}}(\tau) &= \Phi_{1,\text{c}}(\tau) - \Phi_{2,\text{c}}(\tau)\\\nu_{\text{PD},\text{c}}(\tau) &= \nu_{1,\text{c}}(\tau) - \nu_{2,\text{c}}(\tau)\end{aligned}\end{align} \]

• The upper sideband-upper sideband beatnote,

  \[ \begin{align}\begin{aligned}\Phi_{\text{PD},\text{sb}^+}(\tau) &= \Phi_{1,\text{sb}^+}(\tau) - \Phi_{2,\text{sb}^+}(\tau)\\\nu_{\text{PD},\text{sb}^+}(\tau) &= \nu_{1,\text{sb}^+}(\tau) - \nu_{2,\text{sb}^+}(\tau)\end{aligned}\end{align} \]

• The lower sideband-lower sideband beatnote,

  \[ \begin{align}\begin{aligned}\Phi_{\text{PD},\text{sb}^-}(\tau) &= \Phi_{1,\text{sb}^-}(\tau) - \Phi_{2,\text{sb}^-}(\tau)\\\nu_{\text{PD},\text{sb}^-}(\tau) &= \nu_{1,\text{sb}^-}(\tau) - \nu_{2,\text{sb}^-}(\tau)\end{aligned}\end{align} \]

Because the sidebands of the lasers on MOSAs 12, 23, and 31 (respectively 13, 32, and 21) are offset by \(2.4\,\mathrm{GHz}\) (respectively, \(2.401\,\mathrm{GHz}\)), and because we always interfere beams from both types of MOSA, these three beatnotes will always be offset by \(1\,\mathrm{MHz}\). Therefore, they can be tracked individually by the phasemeter.

Each of these beatnote frequencies can be decomposed again as a sum of large frequency offsets and small fluctuations, and we recover equations similar to those above. Therefore, the carrier and sideband parts of a modulated laser beam can be implemented as three distinct beams in the simulation, from which we form three beatnotes.

As described in the previous sections, we only include the carrier and upper-sideband laser beams in our model; as a consequence, we only compute the carrier-carrier and the upper sideband-upper sideband beatnotes.

Interspacecraft, test-mass, and reference interferometer beatnotes

To obtain the beatnote phases (or frequencies) measured by the SCI, TMI, and REF, we can substitute in the previous equations the phases (or frequencies) of the interfering beams.

As discussed above, the beatnote polarities are arbitrary. As a convention, we will always write the beatnote phase (and frequencies) as the difference of the distant or adjacent beam phase (or frequency) and the local beam phase (or frequency),

(5)\[ \begin{align}\begin{aligned}\phi(\tau) &= \phi_\text{distant/adjacent}(\tau) - \phi_\text{local}(\tau)\\\nu(\tau) &= \nu_\text{distant/adjacent}(\tau) - \nu_\text{local}(\tau)\end{aligned}\end{align} \]

Following the optical-bench design of Fig. 3, we have the following beatnote phase offsets and fluctuations, for both carriers and sidebands,

\[ \begin{align}\begin{aligned}\phi_{\text{sci}_{12}}(\tau) &= \phi_{\text{sci}_{12} \leftarrow 21}(\tau) - \phi_{\text{sci}_{12} \leftarrow 12}(\tau)\\\phi_{\text{tmi}_{12}}(\tau) &= \phi_{\text{tmi}_{12} \leftarrow 13}(\tau) - \phi_{\text{tmi}_{12} \leftarrow 12}(\tau)\\\phi_{\text{ref}_{12}}(\tau) &= \phi_{\text{ref}_{12} \leftarrow 13}(\tau) - \phi_{\text{ref}_{12} \leftarrow 12}(\tau)\end{aligned}\end{align} \]

and similarly for beatnote frequencies.

Note that this corresponds to a decrease of the beatnote frequency for an increase of the optical path length. Consequently, the beatnote phase decreases if the optical pathlength increases. To obtain a calibrated length readout signal (sometimes called longitudinal pathlength signal, LPS which will be denoted also as \(x\) in the equation below and in Sec. Tilt-To-Length (TTL)), it is, therefore, necessary to account for the sign and use

(6)\[\text{LPS}_{ij} (t) = x_{ij}(t) = - \frac{\lambda}{2\pi} \Phi_{ij}(t)\,,\]

where \(\lambda\) is the laser central wavelength.

Tilt-To-Length (TTL)

Warning

This section is being updated to make notations consistent (with this section and aux. measurements section).

In this paragraph, we introduce the tilt-to-length coupling noise as implemented. Both its local and remote effects are included in the interspacecraft propagation of the beams, as explained in Interspacecraft propagation.

Contrary to literature (e.g. [11] , [12]), we do not consider phase changes due to lateral jitter nor constant angular offset here. Instead, we refer to TTL only as a phase change that is caused by angular jitter.

We define here TTL at interferometer \(ij,\) (with \(i,j \in \{1,2,3\}, i \ne j\)) as the change in the measured longitudinal optical path length due to rotational jitters:

(7)\[ \begin{align}\begin{aligned}N^{\text{TTL}}_{\text{IFO}_{ij}} &= N^{\text{TTL}}_{\text{IFO}_{ij}\leftarrow ij} + N^{\text{TTL}}_{\text{IFO}_{ij}\leftarrow ji}\\N^{\text{TTL}}_{\text{IFO}_{ij}\leftarrow ij} &= C_{\phi} N^{\text{jitter},\phi}_{\text{total}_{ij}} + C_{\eta} N^{\text{jitter},\eta}_{\text{total}_{ij}},\\N^{\text{TTL}}_{\text{IFO}_{ij}\leftarrow ji} &= C_{\phi} N^{\text{jitter},\phi}_{\text{total}_{ji}} + C_{\eta} N^{\text{jitter},\eta}_{\text{total}_{ji}},\end{aligned}\end{align} \]

where \(N^{\text{TTL}}_{\text{IFO}_{ij}}\) is given in units of lengths, while \(\phi\) (corresponding to yaw) and \(\eta\) (correpsonding to pitch) jitters are given in units of radian per root hertz. These are coupling via the coupling coefficients \(C_{\phi/\eta}\) to the longitudinal readout. With this definition, a positive coupling coefficient and a positive jitter lead to an increase in the meausred optical path length.

Note that \(\eta\) here refers to an angle, and not to the definition of the TDI intermediate variable \(\eta\). We describe these jitters (see next paragraph) in the spacecraft co-moving frame, as defined in Spacecraft co-moving reference frame and [13].

We simulate the spurious angular motion, or angular jitter, of the spacecraft and MOSAs around their nominal attitudes. For spacecraft, we model a jitter in all three angular degrees of freedom (yaw \(N^{\text{jitter},\phi}_{\text{SC}_i}\), pitch \(N^{\text{jitter},\eta}_{\text{SC}_i}\), roll \(N^{\text{jitter},\theta}_{\text{SC}_i}\)). For MOSAs, we model a jitter in the yaw \(N^{\text{jitter},\phi}_{\text{MOSA}_{ij}}\) and the pitch \(N^{\text{jitter},\eta}_{\text{MOSA}_{ij}}\) degrees of freedom corresponding to their opening angle, in the frame of the hosting spacecraft. These angular jitters are treated like noises and are defined in the the spacecraft co-moving frame.

Geometrical arguments (see also derivation in [13], equation 27a+b) yield the total MOSA angular jitter (in the co-moving SC frame) in the yaw degree of freedom:

(8)\[N^{\text{jitter},\phi}_{\text{total}_{ij}} = N^{\text{jitter},\phi}_{\text{SC}_i} + N^{\text{jitter},\phi}_{\text{MOSA}_{ij}},\]

while the total MOSA angular jitter (in the co-moving SC frame) in the pitch degree of freedom reads (with opening angle \(\omega_{ij}\) of MOSA \(ij\), compare also Fig. 2):

(9)\[N^{\text{jitter},\eta}_{\text{total}_{ij}} = \cos(\omega_{ij}) N^{\text{jitter},\eta}_{\text{SC}_i} - \sin(\omega_{ij}) N^{\text{jitter},\theta}_{\text{SC}_i} + N^{\text{jitter},\eta}_{\text{MOSA}_{ij}}.\]

The total tilt-to-length approximated by the linearized coupling coefficient and angular jitters are written as observed at the photodiode (i.e., reference frame of the optical bench), where the coupling coefficients are multiplied by the quantity measured in free space by a magnification factor.

(7) and (9), (8) are valid for all science and test-mass interferometers. Please see [13] for more information on the coupling in the individual interferometers and their combinations in a single link.

We present below the important description of TTL in the SCI, as it is currently implemented. Note that the current TMI simulation does not include TTL effects.

TTL in the SCI

In equation (5), the interferometric phase measurement is defined as the difference between the distant and the local beam phases. In the SCI, the TTL effects are imprinted on the Rx (received) beam, so the \(\textit{distant beam}\). However, these TTL effects on the Rx beam contain contributions from receiver MOSA and spacecraft jitter (referred to as Rx-terms) and transmitter MOSA and spacecraft jitter (Tx terms). These are summed in the Rx-beam phase, such that the total TTL noise in the SCI at the received time is given by

\[\begin{align} N^{\text{TTL}}_{ij}(t_r) ={}& N^\text{TTL, local}_{ij}(t_r) + N^\text{TTL,remote}_{ji}(t_r-L_{ij}^{o(t_r)}) \end{align}\]

(10)\[= C_{ij\eta\mathrm{Rx}} \, \eta_{ij}(t_r) + C_{ij\varphi\mathrm{Rx}} \, \varphi_{ij}(t_r) + \mathbf{D}_{ij} C_{ji\eta\mathrm{Tx}} \,\eta_{{ji}}(t_r) + \mathbf{D}_{ij} C_{ji\varphi\mathrm{Tx}}\, \varphi_{{ji}}(t_r)\]

Here, we denote a delay in units of time by \(L_{ij}^o(t_r)\). The delay operator (see equation (13)) is named \(\mathbf{D}_{ij}\) and \(t_r\) is the time of reception.

Note that the distant TTL contribution is taken at the emission time and indexed by \(ji\), as it is driven by the state of the emitting system at emission. The local term is taken at the current (i.e., reception) time and involves the receiving system.

In this model, we consider 24 coefficients due to 6 involved MOSA, which each jitter along two degrees of freedom, resulting in 12 TTL coefficients. Due to the Rx and Tx coupling effects, there are a total of 24 (= 12 Rx and 12 Tx) coefficients. We define these TTL coefficients as \(C_{ijmn}\) with \(i,j = 1, 2, 3\), \(m = \eta,\phi\) and \(n = \mathrm{Rx}, \mathrm{Tx}\), following [14], which we expect to vary over time. However, this variation is expected to be slow, such that we do not explicitly state the time dependency in (10). The coefficient values can either be fixed to some default value or manually set by the user. If a TTL model is needed in units of radian, \(N^{TTL}_{ij}(t_r)\) is converted according to

\[\Phi_{re,ij}^\mathrm{TTL} (t_r) = - \frac{2 \pi}{\lambda} N^\mathrm{TTL}_{ij}(t_r).\]

Consequently, \(\Phi_{re,ij}^\mathrm{TTL} (t_r)\) is the additional phase due to TTL of the phase measurement as defined in (5). The subscript \(re\) therefore denotes the difference between emitter and receiver.

In simulators, TTL is often implemented as an additional delay. When this is done, an intrinsic assumption is performed. TTL generally comprises two different effects: angular jitter induced optical pathlength changes referred to as \(\textit{geometric}\) TTL and additional phase changes due to wavefront properties and the detection process, referred to as \(\textit{non-geometric}\) TTL (see, e.g. [11]).

When TTL is expressed an additional delay, effectively only optical pathlength changes are considered. We can phrase a corresponding model starting from

\[\Phi_{re}(t_r) = \Phi_e\left(t_r-L_{re}(t_r)\right)-\Phi_r(t_r),\]

considering MOSA \(ij\) and substituting the delay time \(L_{re}\) by \(L_{ij}^o(t_r) - L^\mathrm{TTL}_{ij}(t_r)\) to obtain

\[\Phi_{re}(t_r) = \Phi_{ji} \left(t_r-L_{ij}^o(t_r) - L^\mathrm{TTL}_{ij}(t_r)\right) - \Phi_{ij}(t_r).\]

Since the delay due to TTL, denoted \(L^\mathrm{TTL}_{ij}\), is very small against the delay from one SC to the next, we can rewrite this to

\[\Phi_{re}(t_r) \approx \Phi_{ji} \left(t_r-L_{ij}^o(t_r)\right) - 2\pi\nu_0 L^\mathrm{TTL}_{ij}(t_r) - \Phi_{ij}(t_r),\]

with \(\nu_0\) as the nominal laser frequency. Similar to (5), this equation is given in radian.

A positive TTL coupling thus leads to a phase reduction by \(\nu_0 L^\mathrm{TTL}\). That means, if the length increases, the phase changes by \(\Delta\Phi\) as

\[\Delta\Phi = -2\pi \nu_0 L^\mathrm{TTL}_{ij}(t_r).\]

Combining the two previous expressions and considering the minus sign in the conversion to units of lengths according to (6) leads to

\[N^\mathrm{TTL}_{ij}(t_r) = \lambda\nu_0 L ^\mathrm{TTL}_{ij}(t_r) =- \frac{\lambda}{2\pi} \Delta \Phi,\]

which is the extra path length noise due to TTL in units of meters.

Both in the simulation and in the production of level one data, we rely on an approximate frequency version of (10):

\[\nu_{\mathrm{TTL}_{ij}} = -\frac{\nu_0}{c} (C_{ij\varphi\mathrm{Rx}} \dot\varphi_{ij}^\mathrm{DWS} +C_{ij\eta\mathrm{Rx}} \dot\eta_{ij}^\mathrm{DWS} + C_{ji\varphi\mathrm{Tx}} \mathbf{\dot{D}}_{ij} \dot\varphi_{ji}^\mathrm{DWS} + C_{ji\eta\mathrm{Tx}} \mathbf{\dot{D}}_{ij} \dot\eta_{ji}^\mathrm{DWS}),\]

where \(\dot\varphi_{ij}^\mathrm{DWS}\) and \(\dot\eta_{ij}^\mathrm{DWS}\) are the SC and MOSA angular velocities in units of rad/s, \(\mathbf{\dot{D}}_{ij}\) are the Doppler-delay operators (applied on frequency data).

[1]

In the current mission baseline, there is no way to measure or set the absolute laser frequency with high precision. Therefore, the values set in the simulation cannot be accessed in reality.