Frequency management

6carrier-carrier beatnotes in the test-mass interferometers,

tm^o_ij,c= FT˙_ib_ij^c, (7.11a)

tm^e_ij,c= FT˙_ih

(p˙_ik−(ν0+ν_ik,c^o )(N^˙_ij^bl+N^˙_tm^ob_ij←ik))

−(p˙_ij−(ν0+ν_ij,c^o )(2(N_ij^∆−^Nij^δ) +N_tm^ob_ij←ij(τ))) +N^˙_tm^ro_ij,c− ^b

ijcq˙^e_i 1+q^o_i

i.

(7.11b)

6MPRs,

R_ij =FT_i

d^o_ij−(D_ijδτˆj−δτˆi) +N_ij^R

. (7.12)

7.2.1 Output in total frequency

In reality, we will not have access to the two beatnote components, but only the total phase or the total frequency. The simulation accounts for that by providing an additional output variable for each interferometer.

It is generated by first computing the photodiode signals in the usual two-variable decomposition,ν_PD,ij^o ,ν_PD,ij^e , as described in chapter5. These are then added to compute the total frequency of the photodiode signal,

νPD,ij=ν_PD,ij^o +ν_PD,ij^e . (7.13)

We then apply eq. (6.20) to directly resample this total frequency, using the total clock error q_i. This means we perform no first-order expansion of the clock errors for this case.

The resulting variable is then filtered and downsampled as described in section7.1. This is done to include any numerical artifacts the FIR filter might introduce into the frequency variable.

7.3 frequency management

As discussed in section3.6.2, all beatnote frequencies in LISA are controlled to fall in a range between 5 MHz and 25 MHz¹.

The problem of finding such frequency plans has recently been studied in [46], and exact solutions have been found. We will use these solutions as an input to the simulation.

In this section, we describe how we simulate the laser locking control loop in section7.3.1. We then list the frequency locking schemes available for LISA in its baseline configuration in section7.3.2.

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 MHz from the carrier beatnotes.

86 onboard processing

7.3.1 Laser locking

Laser locking is achieved by controlling the frequency of a laser such that a given beatnote frequencyνPD(τ)remains equal to a pre-programmed ref-erence valueν_PD,r(τ). We do not simulate the actual control loop here, but instead directly compute the correct offsets and fluctuations of the locked laser for the locking condition to be satisfied. We perform this simulation in the TPS.

We consider the frequency lock to be perfect². This means that the locking beatnote offset is

This is true with respect to the local clock in the THE, the frequency in the TPS will be different from the desired value!

exactly equal to the desired value, and the locked lasers fluctuations are chosen in such a way that they exactly cancel the fluctuations in the beatnote signal.

In terms of phase, the result of this control is that the measured beatnote phaseΦ_PD(τ)is controlled to be exactly equal to a pre-programmed reference valueΦ_PD,r(τ). The locked lasers phase drifts and fluctuations are given as

φ_l^o(τ) and φ^e_l(τ), (7.14)

while those of the reference laser are

φ_r^o(τ) and φ^e_r(τ). (7.15)

The locking condition is derived by solving

φ_PD^τi,o(τ) =φ^τ_PD,r^i,o (τ) and φ^τ_PD^i,e(τ) =φ_PD,r^τi,e (τ). (7.16) The laser control loop operates on data delivered by the phasemeter at a high frequency of 80 MHz³. As such, we simulate the locking before applying any filtering or downsampling.

As explained in section5.4.2, the phasemeter treats all frequencies as positive.

The sign of the beatnote contains the information about which of the two lasers has the higher frequency, such that it is an essential information for the control loop to work properly. Fortunately, it is possible to determine the beatnote polarity by actuating the locked laser with a known frequency offset and observing the change in the beatnote.

We can there assume that the beatnote polarity is known at all times, and model the photodiode signals total phase without it, as

Φ_PD(τ) =Φ_r(τ)−^Φl(τ). (7.17)

We end up with photodiode signals as given in section5.6, including readout noise terms,

φ_PD^o (τ) =φ_r^o(τ)−^φl^o(τ), (7.18) φ_PD^e (τ) =φ_r^e(τ)−φ^e_l(τ) +N_PD^ro(τ), (7.19)

2 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.

3 K. Yamamoto, AEI phasemeter team, personal communication May2021.

7.3 frequency management 87 The reference signalφ^τ_PD,rⁱ (τ)is determined by the phasemeter clock, which is derived from the onboard USO (see section6.1.1), such that it is a perfect phase ramp at the desired beatnote frequencyνPD,r when expressed in the THE. Our control loop is simulated in the TPS, and using eqs. (6.5) and (A.5), we obtain

φ^τ_PD,rⁱ (τ) =φ^τ_PD,r^ˆi (τˆ_i(τ)) =ν_PD,r(τ+q_i(τ)). (7.20) We decompose this into large phase drifts and small fluctuations,

φ^τ_PD,r^i,o (τ) =νPD,r(τ+q^o_i(τ)) and φ^τ_PD,r^i,e (τ) =νPD,rq^e_i(τ). (7.21) Substituting eq. (7.21) in eqs. (7.18) and (7.19), we find the locked laser phase drifts and fluctuations

φ_l^o(τ) =φ^o_r(τ)−^νPD,r(τ+q_i^o(τ)), (7.22) φ^e_l(τ) =φ^e_r(τ)−^νPD,rq^e_i(τ) +N_PD^ro (τ). (7.23) Taking the derivative of these equations yields equivalent expressions in frequency,

ν_l^o(τ) =ν_r^o(τ)−^νPD,r(1+q˙^o_i(τ)), (7.24) ν_l^e(τ) =ν_r^e(τ)−νPD,rq˙^e_i(τ) +N^˙_PD^ro(τ). (7.25) Note that these equations describe the locked laser at the photodiode. To properly simulate this effect, we need the locked lasers frequencyat the source, which we denote here as ¯ν_l(τ). We recall from eq. (5.57b) that we subtract a noise termn_B←A(τ)from the fluctuations during propagation, such that we have

¯

ν_l^e(τ) =ν^e_l(τ) + (ν0+ν_l^o(τ))n_B←A(τ)

=ν_r^e(τ)−^νPD,rq˙^e_i(τ) +N^˙_PD^ro(τ) + (ν0+ν_l^o(τ))n_B←A(τ)

(7.26)

for the locked lasers fluctuations at the source.

Note that in the current version of the code, we do not use the desired beatnote frequencies as input, but use the desired offset with respect to the primary laser instead. Inspecting eq. (7.24), we see that the two are simply related by a sign. This will be changed in a future version in order to allow an easier interface to the frequency plans provided in [46], which are formulated in terms of desired beatnote frequency.

7.3.2 Locking configurations

In total,5 of the6lasers in the constellation will be locked (directly or indi-rectly) to one primary laser. Each of the locked lasers is locked to either

88 onboard processing

Table 7.1:Definition of6fundamental lock-ing configurations, with laser12as pri-mary laser

Configuration LA12 LA23 LA31 LA13 LA32 LA21 N1-LA12 Primary Local Distant Local Local Distant N2-LA12 Primary Local Distant Local Distant Distant N3-LA12 Primary Local Local Local Distant Distant N4-LA12 Primary Distant Distant Local Local Distant N5-LA12 Primary Distant Distant Local Local Local N6-LA12 Primary Local Local Distant Distant Distant

• the adjacent laser, using the reference interferometer, so that eqs. (7.24) and (7.26) read

O_ij(τ) =ν_ref^o _ij←ik(τ)−^νrefij,r(1+q˙^o_i(τ)), (7.27a) p_ij(τ) =ν_ref^e _ij←ik(τ)−νrefij,rq˙^e_i(τ) +N^˙_ref^ro_ij(τ)

+ (ν0+O_ij(τ))N^˙_ref^ob_ij←ij, (7.27b)

• or to the distant laser, using the inter-spacecraft interferometer, such that we get

O_ij(τ) =ν_isc^o _ij←ji(τ)−^νiscij,r(1+q˙^o_i(τ)), (7.28a) p_ij(τ) =ν_isc^e _ij←ji(τ)−^νiscij,rq˙^e_i(τ) +N^˙_isc^ro_ij(τ)

+ (ν0+O_ij(τ))N^˙_isc^ob_ij←ij, (7.28b) where the indexk is again chosen such that the whole set fullfils{^i,j,k}= 1, 2, 3.

These expressions can be substituted into the phasemeter equations in sec-tion7.2to derive the phasemeter signals with locked lasers.

As discussed in [46], there are 6 distinct non-swapping locking topologies.

For each of them, we have the freedom to choose the primary laser, such that, in total, we have36possible locking configurations. We give the locking conditions for the 6 configurations with laser 12 as the primary laser in table7.1. The other30combinations can be deduced by applying permutations of the indices, as described in appendix A.1.

Notice that we use a different notation for the locking configuration than what is proposed in [46]. We ordered the different configurations first by considering the length of consecutive laser locks, and second by favoring those for which the adjacent lasers on the same spacecraft are locked together.

The two notations are related in table7.2.

Conf. In [46]

N1 N1c

N2 N2b

N3 N1a

N4 N2c

N5 N1b

N6 N2a

Table 7.2:Notation in this thesis converted to the one used in [46].

Im Dokument Von der Fakultät für Mathematik und Physik der Gottfried Wilhelm Leibniz Universität Hannover (Seite 111-114)