Figure 3.8:Sketch of the principle behind DWS, from [48]. A tilt of the incoming beams causes a small phase shift between the signals in the up-per and lower quad-rants, which can be readout using DPLLs.
C D B A
Photocurrent
Time QPD
for transmission to earth in multiple steps. See [36] for more information on DPLLs and related functionality of the phasemeter.
Note that the PIR and PA represent the full dynamic range of the input signal, including the MHz offsets in the frequency and corresponding phase ramps of millions of cycles per second. The desired gravitational wave signals cause µcycle phase fluctuations in these signals, which need to be recovered in further processing steps on ground, see chapter8for more details.
3.6 auxilliary phasemeter functions
In addition to the main phase readout, the phasemeter will provide a number of additional auxilliary functions and measurements.
3.6.1 Differential wavefront sensing
The main purpose of the LISA phase measurement is to determine the longi-tudinal pathlength change between the test masses located in the local and distant spacecraft.
In addition, it will also provide a precise measurement of the angles between the local and incoming beams, using a method called differential wavefront sensing (DWS). See fig.3.8for a schematic overview.
Instead of using a single photodiode at each interferometer output, this method uses a so-called quadrant photo diode (QPD), whose sensitive area is split into4quadrants of equal area.
Any tilt between the phase front of the incoming beam and that of the local beam will create a slight phase shift in the signals recorded in the different quadrants. By tracking the phase of each quadrant individually, using one DPLL for each of them, it is possible to reconstruct the tilt of the incoming wavefronts from the relative phase between the four readouts. [48] contains a good summary of this method, as well as an efficient algorithm of combining the signals from the different quadrants to recover both the relative beam tilts as well as the longitudinal signal.
36 the laser interferometer space antenna
This readout will be used as input to the DFACS, as well as for on-ground cal-ibrations which compensate for residual tilt-to-length (TTL) couplings.
3.6.2 Laser locking and frequency planning
The overall available bandwidth for tracking the different interferometers will be limited to a range of approximately5 5 MHz to 25 MHz. Therefore, we must ensure that any two lasers entering into an interferometer must have a frequency difference within this range.
This can be achieved by offset-frequency locking all lasers in the constellation to one primary laser used as reference, utilizing additional control loops.
Each of them uses the frequency output of the DPLL as error signal to adjust the frequency of one of the lasers entering the interferometer. The result is that the locked laser’s frequency is adjusted in such a way that the measured beatnote is at exactly the desired value.
As described above, each spacecraft carries two local lasers and receives light from two remote spacecraft. In principle, either of the local lasers could therefore be locked to:
(a) The adjacent beam,
(b) the distant beam arriving on the same optical bench, or (c) the distant beam arriving on the adjacent optical bench.
Mixing options (a) and (c) is referred to as ’frequency-swap’, and not consid-ered as the LISA baseline. Therefore, each laser will be locked using either (a) or (b), utilizing an interferometric readout of the optical bench on which it acts as local laser. See [46] for further information.
The set of offset frequencies of all locked lasers for the whole mission duration is called a frequency plan, and its computation is non-trivial. We can count the number of frequencies we have to determine: All reference and test-mass interferometers on the same spacecraft use the same two laser beams, thus they end up
We will later introduce a sign convention such that the two reference beatnotes on the same spacecraft have opposite signs, cf.
section5.4.2.
at the same frequency. So there are a total of three reference interferometer beatnote frequencies for the three spacecraft. In addition, each ISC interferometer uses a different set of beams, giving an additional6 beatnote frequencies. Therefore, we have to find as set of5offsets for the5 locked lasers to control the value of9different beatnote frequencies.
This is further complicated by the fact that the laser beams pick up Doppler shifts during propagation between the spacecraft, such that the frequency plan has to adjust over time to compensate for changes in the orbits. In addition, to avoid cross-talk between different interferometer channels, no two frequencies on the same spacecraft should have the same value at the
5 For example, the photoreceivers only have a limited bandwidth which limits at high frequen-cies, while some noise sources like relative intensity noise (RIN) are expected to become limiting at low frequencies. In addition, the aliased pilot tone will appear at 5 MHz, such that all other signals should be above this frequency.
3.6 auxilliary phasemeter functions 37
Figure 3.9:Example frequency plan for four years, data pro-vided by G. Heinzel. 5 laser offset frequencies are controlled such that all9beatnotes fall within a range of
±5 MHz to±25 MHz, indicated by black lines. We plot all9 beatnote frequencies, legend omitted for clarity.
0 0.5 1 1.5 2 2.5 3 3.5 4
−20
−10 0 10 20
Time in years
BeatnotefrequencyinMHz
Evolution of beatnote frequencies over the mission duration
same time - including some margin to account for the additional sideband beatnotes required for the clock correction (cf. section3.7).
This topic, in particular the computation of a frequency plan given a set of constraints on the beatnote frequencies, can be solved exactly, as discussed in detail in [46]. An example frequency plan is depicted in fig.3.9, showing that it is possible to control all frequencies to fall within the desired range. This frequency plan is provided by G. Heinzel and based on the same orbits used for the illustration in fig.3.3.
We will include a model for laser locking as well as an overview of the different possible locking configurations first presented in [46] in our simulation model, see section7.3. In addition, we will study the impact laser locking has on the INREP in section12.2.
3.6.3 Pilot tone correction
As described above, the analog photoreceiver signal is digitized at the input to the phasemeter, using an ADC. This ADC uses the on-board USO as timing reference, therefore inheriting its timing jitters. However, the ADC itself will also contribute its own timing jitter on top of those of the sampling clock, at a level incompatible with the stringent noise requirements.
The planned solution is to superimpose a copy of the pilot tone on each ADC channel. The phase of this pilot tone gets tracked with a dedicated DPLL. Since the pilot tone is very stable and at a well known frequency, we can predict it’s phase evolution. Comparing the recovered phase of the pilot tone with its expected phase then allows an accurate measurement6 of
6 The pilot tone is intentionally generated at a relatively high frequency of 75 MHz to be more sensitive to timing jitter of the ADC. This is above the phasemeters Nyquist frequency of 40 MHz, such that the pilot tone appears as a signal aliased to 5 MHz.
38 the laser interferometer space antenna with a local copy, al-lowing a measurement of the time delay∆T.
PRNRX
PRNlocal
∆T
the additional ADC jitter, up to errors in the pilot tone itself. This phase measurement can then be used to correct for the ADC jitter
These are typically tracking one carrier and two sideband beatnotes.
in all other DPLLs which are processing the same time series. Therefore, the pilot tone should be seen as the clock defining the onboard time for the phasemeter measurements.
3.6.4 Absolute ranging
As discussed above, the LISA metrology system will be able to measure relative distance fluctuations at pm precision over timescales of about 1 s to 1000 s. Constructing the laser-noise-suppressing TDI combinations out of the raw phase measurements, however, needs additional information on the absolute separation between the spacecraft. More precisely, as we will discuss in chapter13, the application of TDI requires knowledge of a combination of the physical light travel time and of the desynchronization between the3 independent spacecraft clocks.
This information can be obtained by
This is realized using a low-power phase modulation, cf. [34]
and [15] for more technical details.
imprinting a unique pseudo random noise (PRN) code on each laser beam. On the one hand, these binary codes are perfectly pre-determined - they are generated from a fixed sequence of ones and zeros which is known on each spacecraft. On the other hand, they are ’random’ in the sense that their auto-correlation function approximates a delta distribution. In addition, they are orthogonal, meaning that the cross-correlation between any two of these codes is vanishing.
The absolute ranging measurement relies on these correlation properties to determine the delays experienced by the beams as they propagate between the spacecraft. When the beam is sent, the sending laser’s PRN code is imprinted on it. This requires a conversion from the digital code to an analog signal, which in turn requires referencing to the local clock. Therefore, the PRN code that is actually imprinted on the laser will inherit any timing errors of the sending spacecraft’s clock.
The beam then propagates to the distant spacecraft, where it is received with a time delay. The local phasemeter on the receiving spacecraft records the phase difference between the incoming and the local beam, in which the PRN code is visible as a step-wise modulation. This modulation pattern is correlated to a local copy of the code in a delay-locked loop (DLL). Any errors of the receiving spacecrafts USO will now also affect this correlation, since the phasemeter uses it as timing reference.