2.4 GHz signal or from the 10 MHz USO (see section6.1.2). See sec-tion3.7for a detailed description of this
Overall, the time-correlation of the received code with the local copy allows a measurement of the receiver’s clock time at the event of reception of the modulated beam versus the emitter’s clock time at the event of emission, as sketched in fig.3.10. This quantity is called a pseudo-range, which depends on both the actual distance of the satellites and the errors in the satellite clocks.
Note that the PRN code has a finite length, and simply repeats itself after a
certain timespan. In the case of LISA, this repetition period is The code repetition period is expected to correspond to a few 100 km.
significantly shorter than the overall light travel time between the spacecraft [34]. The PRN measurement therefore only determines the pseudo-range up to a constant offset.
This ambiguity can be resolved by combining the PRN measurement with ground-based obervations from the ESA tracking stations, which are accurate enough to determine which repetition of the PRN is actually received.
3.7 frequency distribution system
As sketched in section3.3, the clocks available for LISA are not precise enough to allow GW detection without any correction. We outline here the hardware involved in generating and distributing the different timing signals needed to suppress clock errors below the requirements. The most relevant signals
40 the laser interferometer space antenna
are then modelled in chapter6, while the actual clock correction algorithm is given in chapter13.
Figure3.11summarizes the main electrical, optical and digital signal paths on one optical bench, with a particular focus on the clock signals generated by the frequency distribution system (FDS). The figure is based on the much more detailed description in [14] and [15]. All shown frequencies should be seen as placeholder values and are subject to change.
The USO provides a timing signal, which we assume to be at 10 MHz. This signal is up-converted to generate two electrical signals, one at 2.4 GHz and one at 2.401 GHz. These are used to drive the electro-optical modulator (EOM) on the left-hand and right-hand optical bench, respectively.
The EOMs convert this electrical signal into a phase modulation of the local laser beam. We will model this modulation in section5.2.2. In summary, this can be described by adding two sidebands which are seperated from the main carrier frequency by the modulation frequency, as shown in fig.3.12 below.
Figure 3.12: Spectrum of the modulated laser beam, from [55]. The PRN code is visible as modulation around the carrier. The two clock sidebands are seperated from the
carrier by 2.4 GHz. Note that all interferometers always interfere beams from a left-handed optical bench with those from a right-handed optical bench. In addition, the sidebands on the left- and right-handed optical benches are seperated by 1 MHz, which will ensure that the resulting sideband-sideband beatnotes will be offset from the carrier-carrier beatnote by 1 MHz7. This allows the phasemeter to track both sideband and carrier beatnotes independently. The spectrum of the beatnote is shown in fig.3.13.
Figure 3.13: Spectrum of the recorded beat-note, from [55]. Two sideband-beatnotes appear seperated by 1 MHz from the carrier beatnote. Both local and received PRN codes are visible in the spectrum.
In addition to producing the sideband, the 2.4 GHz signal is also used as timing reference from which other signals are derived. Most notably the 80 MHz phasemeter clock8and the 75 MHz pilot tone signal, both of which can be generated by integer frequency dividers by30and32, respectively. The
7 This 1 MHz offset has to be taken into account as margin around each beatnote frequency when designing the frequency plan, cf. section3.6.2
8 Since the phasemeter clock is not performance critical, it could also be directly synthesized from the 10 MHz USO signal.
3.7 frequency distribution system 41 80 MHz signal is used to drive most onboard processing of the phasemeter.
This includes the field programmable gate array (FPGA) clock used for digital signal processing (DSP) algorithms, the ADC used to sample the beatnotes, counters used to generate the PRN modulation signal as well as a timer providing timestamps for all phasemeter measurements.
The 75 MHz pilot tone, on the other hand, is the primary timing reference to which all interferometric measurements on one spacecraft are referred, as explained in section3.6.3. The conversion chain from the 2.4 GHz sideband to the 75 MHz pilot tone is therefore performance-critical: the pilot tone represents the clock to which our measurements are ultimately referred, while the sidebands are the only measurements we can use to correct errors in it.
Conversely, the 2.401 GHz sideband does not need to be perfectly stable with respect to the pilot tone. Instead, it can be referred to the 2.4 GHz signal by utilizing either the sideband-sideband beatnotes in the reference interferometers, or even a dedicated electrical measurement between the 2.4 GHz and the 2.401 GHz signals (not depicted in fig. 3.11).
Part II
L I S A I N S T R U M E N TA L M O D E L L I N G A N D S I M U L AT I O N S
I N T R O D U C T I O N
4
4.1 simulations for lisa
LISA is a complex instrument, which combines highly precise on-board measurements with sophisticated on-ground processing algorithms to achieve the required precision to detect GWs in space. Some aspects of the full LISA signal chain – such as the propagation of laser beams through 2.5×106km of free space – are very challenging to accurately reproduce in ground based experiments.
Still, numerous ground based verifications of different aspects of the LISA measurement chain exist, including verifications of the working principles of TDI [30,54,59,74,79]. The development of TDI and the associated noise reduction algorithms, however, relies heavily on analytical studies, guided and verified by numerical experiments.
One main aspect of this thesis work was to develop a more accurate physical model for the simulated data streams of LISA.
In particular, this applies to
• accounting for large frequency offsets in the modelling of beams,
• simulation of Doppler frequency shifts during beam propagation,
• inclusion of laser frequency locking in the simulation, and
• proper modelling and simulation of different clocks and their impact on the signal chain.
Besides allowing the development and testing of noise suppression algorithms, as discussed in part iii, these simulated data sets also allow studies on the expected LISA performance [39], as well as provide a more realistic basis for
other consortium activities, Seelisa-ldc.lal.
in2p3.frfor more information on the LDC!
such as the LISA data challenges (LDCs).
This simulation model uses the index and notation conventions summarized in appendix A.
This activity was performed in the context LISA data-processing group (LDPG) working group (WG)7of the LISA consortium. As such, most of the modelling presented in this part of the thesis (chapter4to chapter8) was developed in close collaboration with other members of the LISA consortium, in particular J.-B. Bayle, and is based on a technical note published internally as [16].
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46 introduction