# Enhanced Frequency noise suppression for LISA by combining cavity and arm
locking control systems

Jobin Thomas Valliyakalayil1<EMAIL_ADDRESS>Andrew J. H.
Sutton1 Robert E. Spero2 Daniel A. Shaddock1 Kirk McKenzie1 1\. Centre for
Gravitational Astrophysics, Australian National University
2\. Jet Propulsion Laboratory, California Institute of Technology

###### Abstract

This paper presents a novel method for laser frequency stabilisation in the
Laser Interferometer Space Antenna (LISA) mission by locking a laser to two
stable length references – the arms of the interferometer and an on-board
optical cavity. The two references are digitally fused using carefully
designed control systems, attempting minimal or no changes to the baseline
LISA mission hardware. The interferometer arm(s) provides the most stable
reference available in the LISA science band (0.1 mHz - 1 Hz), while the
cavity sensor’s wide-band and linear readout enables additional control system
gain below and above the LISA band. The main technical issue with this dual
sensor approach is the undesirable slow laser frequency pulling which couples
into the control system with the imperfect knowledge of the Doppler shift of
the light due to relative spacecraft motion along the LISA arm. This paper
outlines requirements on the Doppler shift knowledge to maintain the cavity
well within the resonance when activating the fused control system. Two
Doppler shift estimation methods are presented that use the already on-board
measurements, the inter-spacecraft interferometer link (the main science
measurement), and the absolute inter-spacecraft laser ranging system. Both
methods reach the required precision after a few thousand seconds of
measurement integration. The paper demonstrates an approach to initialise and
engage the proposed laser stabilization system, starting from free-running
laser and ending with the dual sensor frequency control system. The results
show that the technique lowers the residual laser frequency noise in the LISA
science band by over 3 orders of magnitude: from 30 Hz/$\sqrt{\textrm{Hz}}$ to
as low as 7 mHz/$\sqrt{\textrm{Hz}}$, potentially allowing the requirements on
Time-Delay-Interferometry (TDI) to be relaxed - possibly to the point where
first-generation TDI may be sufficient.

††preprint: APS/123-QED

##  I Introduction

The Laser Interferometer Space Antenna (LISA) mission is a space-based
gravitational wave detector, proposed and planned jointly by ESA, NASA and a
European consortium [1, 2]. The LISA mission consists of three spacecrafts in
a triangular formation with 2.5 million kilometer arms. The spacecraft
exchange laser beams, employing heterodyne interferometry to measure
displacement between spacecraft to detect gravitational waves in the frequency
band between 0.1 mHz to 1 Hz, a frequency band inaccessible to ground-based
detectors. The displacement sensitivity goal should be less than 10
pm/$\sqrt{\textrm{Hz}}$ for each arm-link to reach a strain sensitivity of
10-21/$\sqrt{\textrm{Hz}}$ [1].

Stabilisation of laser frequency noise is critical for LISA to meet the design
sensitivity, as it is indistinguishable from displacement along a single link
[2]. LISA will lock the laser to a fixed-length ultra-stable optical cavity as
reference using the Pound-Drever-Hall (PDH) technique [3]. A similar technique
has been recently demonstrated using the laser instrument on the GRACE Follow-
On mission [4] demonstrating the required cavity performance. However
stabilisation alone cannot realise the LISA sensitivity requirement, hence
Time Delay Interferometry (TDI) [5, 6, 7] will be employed. TDI is a post-
processing scheme that synthesizes laser frequency noise free measurement
variables by forming two-beam, equal arm-length interferometer combinations
using algebraic combinations of delayed link displacement measurements.
Velocity correcting TDI combinations, also known as second-generation TDI [8],
and the Sagnac combination [9] overcome the effect of spacecraft motion during
the light travel time. In addition to TDI, laser stabilisation is required to
a level of 282 Hz/$\sqrt{\textrm{Hz}}$ for second-generation TDI and to 1
Hz/$\sqrt{\textrm{Hz}}$ for first-generation TDI [10, 11]. While second-
generation TDI is currently being subject to detailed analysis to ensure all
signal and noise propagation is understood [12], experimental validation of
TDI [7] on the ground is extremely challenging. A supplementary method of
laser stabilisation, such as Arm Locking, will provide risk reduction against
unknowns and margin to link performance.

LISA’s inter-satellite link is designed (and required) to provide an intrinsic
length stability below the $\sim 10~{}\textrm{pm}/\sqrt{\textrm{Hz}}$ noise
floor in order to detect gravitational waves [1]. Hence, by design, the LISA
arm itself provides a highly stable length reference over frequencies of
interest. Arm Locking was proposed by Sheard et al. [13] to use the arm itself
for laser frequency noise stabilization and to provide further margin for TDI
operation, which can be demonstrated by the GRACE-FO mission as proposed in
[14]. Later a dual arm sensor was developed [16, 15] that utilised the two arm
lengths with each spacecraft, and allowed more freedom in the design of the
controller by moving the nulls to larger frequencies. This sensor was prone to
Doppler pulling [17], whereby on every light round trip time (16.7 s for
single arm locking, significantly less for dual arm locking), the laser
frequency experienced a shift, equal to the error in the knowledge of the
Doppler shifts. Doppler pulling was reduced with the help of modified-dual arm
locking [10, 18], that combined both common and dual arm locking sensors.
However none of these arm locking systems were compatible with the LISA
hardware baseline: cavity stabilization to a fixed-length resonator. To date,
to maintain compatibility of arm-locking with an optical cavity the approaches
investigated required a tunable cavity length [19] or employ modulation-
frequency that could be tuned [20].

This paper re-examines a combination of cavity locking (using the baseline
fixed resonator length) and arm locking, and shows

  * •

Suppression of laser frequency noise below the requirement-level cavity noise
(over 3 orders of magnitude of suppression),

  * •

Successful acquisition and operation of cavity lock in the presence of Doppler
pulling, so long as a prior Doppler estimate is available.

We emphasise that this combination relaxes the TDI suppression requirement
with minimal or no change to the optical or electronic hardware on board the
spacecraft - relying mostly on an FPGA upload to existing digital hardware
and/or a flight software update. Even if arm locking is not considered as
baseline for laser frequency stabilisation, this implementation could allow
for deployment after the launch of the spacecraft. The difficulty faced for
this hardware simplification is the requirement for high accuracy of knowledge
of Doppler shift. This paper studies this resultant laser frequency pulling
and outlines the requirements for robust lock acquisition and stable operation
thereafter.

The paper is divided into 7 sections. Section II discusses the system model
used for analysis and simulations, including the noise propagation through the
system. Section III discusses Doppler pulling with respect to the current
model and IV explores the convergence times for several Doppler estimation
techniques to measure the Doppler shifts. Section V presents the controller
design for the combined arm and cavity references, while Section VI discusses
a Simulink model used to verify the system in the time domain. Section VII
looks into some insights for utilising the combination of sensors, while
Section VIII gives the conclusions and possible future scope of this
technique.

##  II Model

Figure 1 is the schematic diagram of the model that includes the main sensors,
controllers, and the noise sources. All the equations in this paper are
derived with respect to this model. The laser frequency noise, from the laser
source in Spacecraft 1, serves as the main input and is fed into an arm sensor
and the cavity(PDH) sensor. The output of these sensors, including the noise
contributions, are given to individual controllers. The output from the two
controllers are summed and fed back to the laser source and adjusts the laser
frequency through PZT and thermal actuators, and thus create a feedback
system. The notations and formalism used in this paper follow McKenzie et al.
[10].

###  II.1 Sensors

From previous arm locking analysis, the frequency noise sensors can be styled
as a single arm sensor [13], common arm sensor [16], difference arm sensor
[16], dual arm sensor [16] or modified-dual arm sensor [10]. In this paper,
the common arm sensor is used, in which the average displacement measurements
from two interferometer arms are used to measure laser frequency noise.For
simplicity a single arm sensor could also be chosen, and we expect only minor
modifications to the results. We approximate the high gain transponder systems
of the other spacecrafts to act as a active retro-reflector, that transponds
time-delayed laser phase, and hence simplify computations. The common arm
locking transfer function from laser frequency to sensor (displacement) output
is given by;

$P_{+}(s)={S_{+}}\begin{bmatrix}P_{12}(s)\\\ P_{13}(s)\end{bmatrix},\hskip
28.45274pt{S_{+}}=\begin{bmatrix}1&1\end{bmatrix}.$ (1)

$S_{+}$ refers to the sensor matrix that utilises the individual arm responses
[10]. The frequency response of the individual arms is given by

$P_{12}(s)=1-e^{-2s\tau_{12}},\hskip 14.22636ptP_{13}(s)=1-e^{-2s\tau_{13}},$
(2)

where $\tau_{12}$ and $\tau_{13}$ are the times for the laser-light to
traverse arm 1 (from Spacecraft 1 to 2) and arm 2 (from Spacecraft 1 to 3) and
relates to the arm lengths, $L_{1}$ and $L_{2}$ as $\tau_{12}=L_{1}/c$ and
$\tau_{13}=L_{2}/c$ (c is the speed of light). For simplicity, the arm lengths
are assumed to be constant and symmetric within the round trip time of the
laser and thus have the approximations $\tau_{21}$ = $\tau_{12}$ and
$\tau_{31}$ = $\tau_{13}$. The parameter $s$ refers to the complex variable in
Laplace domain. Equation 1 can be also written analytically as

Figure 1: Simplified model of the feedback system using Spacecraft 1 and 3 of
LISA. Spacecraft 2 is not shown in this model, but can be considered symmetric
to Spacecraft 3. $\tau_{13}$ and $\tau_{31}$ are the travel times taken for
the laser to traverse the separation between Spacecraft 3 and 1. The laser
frequency noise, $\nu_{L_{i}}$, the clock noise, $\nu_{clock;ij}$, the shot
noise, $\nu_{shot;ij}$, the spacecraft motion noise, $\nu_{SC;ij}$, the
Doppler shifts, $\nu_{D;ij}$, the cavity noise, $\nu_{cavity}$ are the noise
sources that are considered in this model. The indices i and j takes the
values of 1, 2 or 3 referring to the different spacecrafts. The red trace
outlines the path of the laser in free space, while the black trace of the
path are in electrical/digital hardware.

$P_{+}(s)=2(1-\cos({\omega\Delta\tau})e^{-s\bar{\tau}}),$ (3)

where $\bar{\tau}=\tau_{12}+\tau_{13}$ and $\Delta\tau=\tau_{12}-\tau_{13}$.
Here the value of $\bar{\tau}=16.67$ s and the value of $\Delta\tau=0.083$ s.
This is with consideration the average length of the arms to be 2.5 million
kms and the maximum difference in the lengths to be 1% of that length [21].
This approximation may not be valid during the entire LISA mission at which
the phase margin may decrease by 1$\degree$ in the worst case scenario, when
the arm lengths are equally matched, which has negligible effect on the
results in this paper. Prior to the controllers, the sensors are scaled with a
normalisation factor of $\frac{1}{2}$.

The cavity PDH locking sensor is modelled as a low-pass filter whose cut-off
frequency, $f_{c}$, is at the cavity half-width half-max(HWHM) frequency for a
high finesse cavity [22]. With $f_{c}$ referring to the cavity pole, the PDH
sensor can be shown as

$P_{\textrm{pdh}}(s)=\frac{D_{0}}{1+\frac{s}{2\pi f_{c}}}.$ (4)

$D_{0}$ is a gain that is dependent on the length of the cavity, the
reflective and loss coefficients of the mirrors and the power modulation
coefficients in the PDH scheme [22]. We normalise the PDH sensor gain
$D_{0}=2$ to be compatible with the arm sensor (In practice, this will require
scaling the error signal prior to the controller). For LISA, the cavity on the
spacecraft is expected to have a length of 7.77 cm with a free spectral range
of approximately 2 GHz, similar to the cavity that is used in GRACE Follow-On
mission [4, 23]. Hence for a finesse of 10,000, the HWHM frequency can be
computed to be approximately $f_{c}=100$ kHz.

Figure 2: Bode plot of the arm sensor and the PDH sensor. The arm sensor is
modelled as the common sensor considered with $\bar{\tau}=16.67$ s and
$\Delta\tau=0.08335$ s. The arm sensor has nulls at multiples of 60 mHz where
the phase goes from -90$\degree$ to 90$\degree$. The difference in the arm
lengths is reflected in the null depths. The PDH sensor has a flat response
with the corner frequency at approximately 100 kHz, and a phase of $-6\degree$
at 10 kHz

The bode plot of both sensors are shown in Figure 2. The basic design of the
controller presented in Section V relies on the arm sensor being dominant
(higher gain) in the LISA science band ($10^{-4}$ Hz to 1 Hz) since it offers
the best frequency stability. Outside the LISA science band the controller
design emphasizes the cavity sensor; at frequencies below $10^{-4}$ Hz to
limit the laser frequency pulling due to imperfect Doppler shift knowledge,
and above 1 Hz to increase the control system’s phase margin at high
frequencies and through the nulls of the common arm sensor. The unity gain
frequency of the controller is selected at approximately 10 kHz, a decade
lower than the cavity HWHM frequency, to utilise the flat response of the PDH
sensor.

###  II.2 TDI Noise Requirements

The amount of laser frequency noise that TDI can suppress is limited by the
time synchronization errors [5]. Dedicated inter-spacecraft ranging, using
pseudo random noise modulation [24, 25], is proposed to provide a spacecraft
ranging error of 1 m and with an allowed residual noise floor of 2
pm/$\sqrt{\textrm{Hz}}$, the laser noise requirement before second-generation
TDI is applied [11] is

$\tilde{\nu}_{\textrm{TDI-2}}(f)=282\frac{\textrm{Hz}}{\sqrt{\textrm{Hz}}}.\sqrt{1+\left(\frac{2\hskip
1.42271pt\textrm{mHz}}{f}\right)^{4}}.$ (5)

First-generation TDI on the other hand does not correct for errors in the
lengths due to the relative velocity between the spacecrafts. If the ranging
is considered to be constrained by relative velocity of $\pm 5$ ms-1 [1]
between the spacecrafts, then in the worst case scenario, both the spacecrafts
would experience a maximum relative velocity of 10 ms-1, resulting in a change
in range of 167 m in one round-trip of the laser (around 16.67 s). Thus, the
laser noise suppression requirement prior to first-generation TDI is

$\tilde{\nu}_{\textrm{TDI-1}}(f)=1.7\frac{\textrm{Hz}}{\sqrt{\textrm{Hz}}}.\sqrt{1+\left(\frac{2\hskip
1.42271pt\textrm{mHz}}{f}\right)^{4}}.$ (6)

While second-generation TDI is a conceptual next step from first-generation,
the noise couplings of different noise sources in second-generation TDI may be
more complex due to the time-varying arm-lengths [12, 26].

As a goal, this paper designs an arm-locking/cavity control system with
ability to reduce the laser frequency noise to below the required level for
application of first-generation TDI. This additional noise reduction results
in additional margin for the the required TDI suppression and could be seen as
risk reduction for second-generation TDI, and if needed may mean that first-
generation TDI would be sufficient.

###  II.3  Noise propagation

This subsection explains the propagation of the various noise sources as shown
in Figure 1. The noise sources, notations, and transfer functions follows the
previous arm locking literature [13, 16, 17, 10] and are described in Appendix
A. In the following analysis, we use the notation of laser frequency noise
(Hz/$\sqrt{\textrm{Hz}}$) as opposed to laser phase noise
(cycles/$\sqrt{\textrm{Hz}}$) in other related papers and drop the Laplace
‘(s)’ notation from the noise models and controllers in this subsection. For
instance, $\nu_{O}(s)$ will be denoted as $\nu_{O}$.

The laser frequency noise, $\nu_{L}$, is added at the laser source, and
propagates through the interferometer and the cavity experiencing the transfer
functions in Equations 1 and 4. The requirement-level cavity noise,
$\nu_{cavity}$, gets added into the system as the base stabilisation [2]
provided by PDH locking and hence will propagate when the PDH error is taken.
This requirement-level cavity noise will be dominated by effects such as
readout noise and Brownian thermal noise. Shot noise, $\nu_{shot;ij}$, is
added at photodetector in Spacecraft i, when interfered with a laser from
Spacecraft j. There are four independent shot noise contributions that are
added at the primary spacecraft. Clock noise couples in each of the phasemeter
reading between spacecraft 1 and spacecrafts 2 and 3, totalling to four terms
of clock noise, $\nu_{clock;ij}$, with the beat-note frequency being the
maximum value (25 MHz) [1]. The spacecraft motion noise, $\nu_{SC;ij}$, get
coupled into the system whenever there is a change in the link between two
spacecrafts and hence eight terms are contributed by this noise [10]. The
total noise propagation through the entire open loop system at point O in
Figure 1 can be shown as:

$\begin{split}\nu_{O}&=G_{1}S_{+}\begin{bmatrix}N_{L}+N_{SN}+N_{CN}+N_{SCN}\end{bmatrix}\\\
&+G_{2}P_{\textrm{pdh}}\nu_{L}+G_{2}P_{\textrm{pdh}}\nu_{cavity},\end{split}$
(7)

where $N_{L}$ is the laser noise sensed at the primary spacecraft
photodetectors, $N_{SN}$ is the shot noise, $N_{CN}$ is the clock noise, and
$N_{SCN}$ is the spacecraft motion noise. These are given by

$\begin{split}N_{L}&=\begin{bmatrix}P_{12}\nu_{L}\\\
P_{13}\nu_{L}\end{bmatrix}\\\
N_{SN}&=\begin{bmatrix}\nu_{shot;12}+\nu_{shot;21}e^{-s\tau_{12}}\\\
\nu_{shot;13}+\nu_{shot;31}e^{-s\tau_{13}}\end{bmatrix}\\\
N_{CN}&=\begin{bmatrix}\nu_{clock;12}+\nu_{clock;21}e^{-s\tau_{12}}\\\
\nu_{clock;13}+\nu_{clock;31}e^{-s\tau_{13}}\end{bmatrix}\\\
N_{SCN}&=\begin{bmatrix}-\nu_{SC;12}\left(1+e^{-2s\tau_{12}}\right)-2\nu_{SC;21}e^{-s\tau_{12}}\\\
-\nu_{SC;13}\left(1+e^{-2s\tau_{13}}\right)-2\nu_{SC;31}e^{-s\tau_{13}}\end{bmatrix}.\\\
\end{split}$ (8)

G1 and G2 are controllers used for each of the arm sensor and cavity sensor,
respectively. Using equations 7 and 8, with transfer functions 23, 28, 31, 36,
40, the complete closed loop equation at point C can be written as

$\begin{split}&\nu_{C}=\nu_{L}-\nu_{O}\\\
=&\frac{\nu_{L}}{1+G_{1}P_{+}+G_{2}P_{\textrm{pdh}}}-\frac{\nu_{cavity}G_{2}P_{\textrm{pdh}}}{1+G_{1}P_{+}+G_{2}P_{\textrm{pdh}}}\\\
-&\frac{G_{1}}{1+G_{1}P_{+}+G_{2}P_{\textrm{pdh}}}S_{+}\begin{bmatrix}N_{SN}+N_{CN}+N_{SCN}\end{bmatrix}.\\\
\end{split}$ (9)

The noise budget after engaging the hybrid control system is plotted in Figure
3 using controllers for each sensor (covered in detail in Section V). The
black dashed line plots the pre-TDI requirements based on the LISA sensitivity
goal (Equation 6). It can be seen that the requirement-level cavity noise
(shown as red dashed lines) is suppressed by over 3 orders at 10 mHz (shown as
pink line) and is the dominant noise in the noise budget. At the null
frequencies (multiples of 60 mHz), the noise sources are not suppressed. The
other noise sources are coupled into the system when the arm is dominant. At
very low frequencies, the cavity noise is dominant and follows the LISA
requirement level.

Figure 3: Noise budget using the noise models and the transfer functions along
with the LISA requirements. The main contributing noise source is the
requirement-level cavity noise that is suppressed from engaging the arm
locking controller (pink trace) and within the first-generation TDI
requirement (dashed black line) in the science band.

The suppression function of the laser frequency noise, the first term in
Equation 9, and the suppression function of the requirement-level cavity
noise, the second term in the equation, is plotted in Figure 4, with the
controller discussed in Section V. This figure shows there is significant gain
for laser frequency at all frequencies below 10 Hz and the suppression of
cavity noise across the LISA science band.

Figure 4: Suppression function of the system with respect to laser frequency
noise and requirement-level cavity noise. The laser frequency noise is
stabilised by both the arm and the cavity while the cavity noise suppression
is provided by the arm in the LISA band

##  III Doppler pulling and propagation in LISA

As the LISA constellation orbits around the Sun, the spacecraft follow
independent orbits and the arm lengths are expected to change by upto 1% [21].
The relative movement of the spacecraft cause Doppler shifts in the laser
frequency given by $\nu_{d}$(t) = v(t)/$\lambda$, where v(t) is the relative
velocity at which the spacecraft move and $\lambda$ is the wavelength of the
laser (1064 nm). The maximum expected relative velocity between the
spacecrafts is 5 ms-1 [1], producing a one-way Doppler shift of approximately
5 MHz. As the lasers on the end station spacecrafts are phase locked to the
laser on the primary spacecraft, the laser beam on returning back to the
primary spacecraft will have twice the one-way Doppler shifts. The sum of the
Doppler shifts will be taken when the phase is given to the common arm sensor.
If the one-way Doppler shift of the laser travelling from spacecrafts i to j
is $\nu_{D;ij}$, the effective Doppler shifts in the system can be shown as
[10]

$\nu_{D,+}(t)=\nu_{D;12}(t)+\nu_{D;21}(t)+\nu_{D;13}(t)+\nu_{D;31}(t).$ (10)

Based on orbital dynamics of LISA [21, 27], we use a toy model that
approximates the common Doppler shifts to be a sum of sinusoids with half-year
and one year periods as shown below:

$\nu_{D,+}(t)=\nu_{1}\sin(\omega_{1}t+\phi_{1})+\nu_{2}\sin(\omega_{2}t+\phi_{2}).$
(11)

$\nu_{1}$ and $\nu_{2}$ are amplitudes of the two sinusoids of frequencies
$\omega_{1}$ and $\omega_{2}$, along with phase shifts $\phi_{1}$ and
$\phi_{2}$, respectively. This model can represent a smooth orbit evolution
with (approximately) correct dynamics over the $\sim$1 month time frame for
arm locking turn-on transient relevant to this paper. Pulling of the laser
frequency in arm locking is a known issue [10, 17] that arises because the arm
locking sensors have zero response at DC. On one hand, the cavity sensor can
sense this Doppler pulling and allows correction at long timescales (
$>10^{4}$ s). Conversely, the cavity sensor has finite linear range and
limited gain in the LISA band, hence pulling must be minimised to ensure
cavity lock is maintained. For this paper we consider this number to be $\pm
20$ kHz, a fifth of the HWHM frequency (100 kHz).

An offset to the cavity lock point is undesirable as it introduces coupling
from other noise sources, such as laser intensity noise. This section will
demonstrate that, with the controller from Section V, laser frequency pulling
reaches a maximum of 20 kHz at lock acquisition within the model framework. In
laboratory testing using a cavity and system with parameters similar to LISA,
the residual laser frequency noise was seen to increase up to level of 100
Hz/$\sqrt{\textrm{Hz}}$ level with a 10 kHz offset off resonance on a PDH-
stabilised laser with the setup in [28]. Note: this noise would be still
suppressed by the arm locking controller by the magnitude shown in Figure 4.

###  III.1 Lock Acquisition Timeline

This section will focus on the potential lock acquisition timeline for LISA to
employ arm locking while restricting the transient Doppler pulling caused by
the feedback of the controller. For implementation of arm-locking, precise
Doppler information is required for real-time subtraction from the phasemeter
signal before feeding back to the laser. Section IV discusses two methods to
measure the Doppler shift using only the hardware included in the baseline
mission. The two techniques are 1) to use the inter-satellite range signal and
average the beat note frequency for a sufficient time, and 2) use the inter-
spacecraft absolute ranging system whose baseline deploys a Pseudo Random Code
on the inter-satellite inter-spacecraft link [24, 25]. Depending on the noise
performance of the cavity, either the cavity or the ranging readouts can be
used for estimation. If the residual noise of the cavity is at requirements
level, the estimates need at least 50000 s ($\sim$ 13.9 hours) of integration
time to get the necessary accuracy. Instead, the absolute ranging system based
on PRN ranging is relied on, requiring only 6300 s ($\sim$ 1.75 hours) of data
to get a sufficiently accurate estimate of the Doppler trends. Cavity
performance that approach the thermal limit [4, 29] will significantly reduce
residual noise, allowing Doppler parameter estimates to converge within 1500 s
($\sim$ 0.5 hour) using the interferometer response only. These estimates
(using either approach), can populate the phasemeter with a model estimate of
the Doppler shifts based on Section III.2.

###  III.2 Arm Locking Acquisition

When the arm locking controller is enabled on the primary spacecraft
(Spacecraft 1) to stabilise the laser, estimates of the heterodyne frequency,
including Doppler shifts, are subtracted from the phasemeter readings. Any
error in Doppler shift estimate will represent a bias that cannot be
suppressed by the arm locking controller. Accordingly, such a bias excites a
transient response in the arm locking system and results in the frequency
pulling behavior documented previously [10, 17]. Importantly, the introduction
of the Fabry-Pérot (FP) cavity estimation provides necessary information to
limit frequency pulling. This section describes the model for this transient
behaviour. If the errors in Doppler shifts are small, then the transient laser
frequency pulling can also be reduced to within the allowable limits of the
cavity. The turn on transient of the controller can be represented by the
closed-loop step response of the error between the actual and estimated
Doppler shifts, and computed as

$\nu_{C}(t)=L^{-1}\left(\frac{\left[\nu_{D;+}(s)-\left(\nu_{D;est}(s)\right)\right]V(s)}{s}\right),$
(12)

$V(s)=\frac{-G_{1}(s)}{1+G_{1}(s)P_{+}(s)+G_{2}(s)P_{pdh}(s)}.$ (13)

$\nu_{D;est}(s)$ is an estimate of the common Doppler frequency. The term in
square brackets consist of the error in estimation of Doppler shifts. $L^{-1}$
represents the inverse Laplace transform function of the system and V(s) is
the transfer function of the Doppler shifts, with a similar analysis in
Section II.3.

Table 1: Parameter requirements for orbital knowledge in order to meet lock acquisition conditions using the polynomial model with the controller shown in Fig 14. Each parameter’s error limit is checked in combination with the errors of other parameters in Monte Carlo simulations. The achievable levels are cross-checked with estimation using Fabry-Pérot (FP) cavity estimation in [10] and PRN ranging, or Thermal Noise limited (TNL) cavity estimation, as described in Section IV. Parameter | Actual/Max | Max error | Fractional | Estimation
---|---|---|---|---
| Value | tolerance ($\pm$) | change | methods
$\nu_{0}$ | 12 MHz | 10 Hz | 8.33 x $10^{-7}$ | FP cavity estimation/ TNL cavity estimation
$\gamma_{0}$ | 4 Hz/s | 60 $\mu$Hz/s | 1.5 x $10^{-4}$ | PRN ranging/ TNL cavity estimation
$\alpha_{0}$ | -1.2 $\mu$Hz/s2 | 5 nHz/s2 | 4.17 x $10^{-3}$ | PRN ranging/ TNL cavity estimation

A simplification of the Doppler shifts can be done for smaller timescales,
where the Doppler shifts at that instant can be modelled as a second order
polynomial equation shown below [10]:

$\nu_{D;est}(t)=\left(\nu_{0;+}+\gamma_{0;+}t+\frac{\alpha_{0;+}t^{2}}{2}\right),$
(14)

where $\nu_{0;+}$, $\gamma_{0;+}$ and $\alpha_{0;+}$ are the estimates of the
Doppler shift, the first derivative of Doppler shift (Doppler rate) and second
derivative of Doppler shift (Doppler acceleration) at the instant when the
controller is just turned on. The error limits, based on the results of Fabry-
Pérot cavity estimation for 200 s given in [10],were sufficient due to the
wide limits afforded for a free running laser (upto 10 MHz). For this work, we
require that the worst case Doppler pulling be bounded by $\pm 20$ kHz.

The toy model values of $\phi_{1}$ and $\phi_{2}$ in Equation 11 determine the
point in the orbit paths at which laser lock is acquired. At the point of
higher Doppler rate, we expect the model to have more stringent error
requirements compared to locking at points where the Doppler rate have
relatively lower amplitude. Figure 5 show the Doppler pulling when locking at
different orbital points using a polynomial model with the error bounds given
in Table 1. The traces are shown to be restricted within $\pm$20 kHz and thus
provide us the flexibility to engage the arm locking controller at any point
in the spacecraft orbit, without causing any significant Doppler pulling. The
poor model fit (due to the difference in Equation 11 and 14) results in an
intrinsic ‘un-modelled’ Doppler error propagated into the control system. Even
with ideal parameter knowledge, the polynomial model is incapable of fully
resolving Doppler evolution and will result in a Doppler pulling of up to 17
kHz (relative to the assumed sinusoidal orbit description in Equation 11).

Figure 5: Frequency deviation from cavity line center due to Doppler-error-
pulling at lock acquisition. The estimate is calculated using the polynomial
functions in Equation 14. The different traces corresponds to the lock
acquisition Doppler pulling at different set points in the orbits with the
error bounds in the model, given in Table 1. The maximum Doppler pulling
corresponds to the point in the orbits with the maximum Doppler rate.

Adaptation of a more complex Doppler model can theoretically eliminate this
source of systematic estimation error and associated intrinsic Doppler pulling
component. However, the complexity and the number of variables that need to be
estimated requires further analysis. A preliminary analysis of the sinusoidal
model (along with the error bounds) is given in Appendix B, based on the toy
model in Equation 11.

Further investigations were done to test the error bounds of the Doppler
parameter estimates by running Monte-Carlo simulation (100 runs). With the
orbital set-point selected to give the maximum pulling, Figure 6 shows that
the polynomial model and associated derived error bounds in Table 1 can
restrict laser frequency pulling to within the $\pm$20 kHz requirement. After
25 days, the pulling would reach steady state that is limited by $\pm 27$ Hz.

During the steady state period, we expect the spacecraft to undergo orbits
corrections or engage spacecraft thrusters, which would come as a step or ramp
function in the Doppler error. Analysis on this disturbance in the steady
state of the Doppler pulling indicate an excitation of a transient response in
the control loop comparable to initial lock acquisition. If the frequency
deviations are within the requirements shown in Table 1 for $\nu_{0}$ and
$\gamma_{0}$, the laser will maintain lock with the cavity.

Figure 6: Monte Carlo simulations of frequency deviation from cavity line
center due to Doppler-error-pulling at lock acquisition. The estimate is
calculated using the polynomial functions in Equation 14. The black traces
shows the various Monte Carlo simulations with the dashed traces showing the
pulling with the maximum error limits, given in the Table 1. The maximum
pulling is limited to 20 kHz within the line-width, not breaking the resonance
of the cavity.

##  IV Doppler Frequency Estimation

The previous section presented details of the Doppler knowledge requirements
at controller activation. This section presents the convergence time required
to estimate the Doppler shifts with sufficient accuracy. The methods presented
here require nominal LISA measurements, without arm locking engaged, for a few
thousand seconds to allow Doppler shift estimate of sufficient accuracy. Once
sufficient Doppler shift knowledge is achieved, the arm locking controller can
be activated, and the associated improvement in laser frequency noise
suppression achieved. The two methods of measurement considered are: 1) LISA
phasemeter measurement containing the nominal interferometer response using
cavity-stabilised laser, 2) using the LISA baseline inter-spacecraft range
measurement [24, 25]. The required estimation times are determined based upon
the observation time over which the weighted Allan variance of the residual
noise reduces to below the required RMS level.

The PRN ranging system uses a Pseudo Random Binary Sequence (PRBS) modulation
to time stamp the outgoing light with a clock-like signal as it leaves the
spacecraft, providing a reference with which the clocks can be aligned and the
distance can be measured between the spacecrafts. This is limited by PRBS code
noise and shot noise, with a residual displacement noise expected of order
$\sim$0.1 m/$\sqrt{\textrm{Hz}}$ [24, 25].

The FP cavity estimation method described in Appendix A of [10], has a
residual noise derived from the cavity noise coupled through an open arm
sensor. For this calculation, we make two estimates of Doppler error based on
two levels of cavity residual errors: 1) the cavity residual given in Equation
29, which is a conservative estimate, 2) the cavity residual such as in GRACE-
FO [4] or in [29], that approach the cavity thermal noise limit (TNL) [30],
the best possible case (in reality, the cavity will likely be between these
two bounds), modelled as

$\nu_{cavity}\approx\nu_{thermal\\_limit}=\frac{0.1}{\sqrt{f}}\hskip
8.5359pt\textrm{Hz}/\sqrt{\textrm{Hz}}.$ (15)

The effective residual displacement noise can be derived as:

$\begin{split}\Delta
x_{residual}(s)&=\frac{\Delta\phi_{cavity}(s)*P_{+}(s)*\lambda}{2}\\\
&=\frac{\nu_{cavity}(s)*P_{+}(s)*\lambda}{2s}.\end{split}$ (16)

The estimator performance is predicted based upon the expected weighted Allan
deviation over time. The Allan deviation was derived from the time domain
using numerical simulation of the relevant noise spectra. To mitigate the
effects of temporal leakage, we apply a windowed and overlapped Allan variance
estimation from [31]. The following equation captures the operation of
windowing in the overlapped Allan variance formula,

Figure 7: Allan deviation for the velocity measurement between the
spacecrafts. The dashed line indicates a Doppler constant requirement of 10
Hz, corresponding to 10.64 $\mu$m/s. The PRN ranging scheme has a slope of
-3/2 while the requirement-level cavity-arm sensor has a slope of -1/2 till
around 500 s and then rolls up with slope of 1/2. Thermal-limited cavity-arm
sensor has a slope of -1 after 8.335 s. The time required for the deviation to
reach the error limit is 411 s for PRN while the cavity-sensor can reach it
under 30 s and thermal-limited cavity sensor can always realise this
requirement (within 1 s). Figure 8: Allan deviation for the acceleration
measurement between the spacecrafts. The dashed line indicates a Doppler rate
requirement of 60 $\mu$Hz/s, corresponding to 64 pm/s2. The PRN ranging scheme
has a slope of -5/2 while the requirement-level cavity-arm sensor has a slope
of -3/2 till 500 s and then rolls off with -1/2. The thermal-limited cavity-
arm sensor has a slope of -1 till 8.335 s and rolls off with -2. The time
required for the deviation to reach the error limit of 60 $\mu$Hz/s is 4311 s
for PRN while the cavity-sensor requires 50010 s. The thermal-limited cavity
sensor can estimate the Doppler rate to the required level within 370 s.
Figure 9: Allan deviation for the jerk measurement between the spacecrafts.
The dashed line indicates a Doppler acceleration requirement of 5 nHz/s2,
corresponding to 5.32 fm/s3. The PRN ranging scheme has a slope of -7/2 while
the requirement-level cavity-arm sensor has a slope of -5/2 till 500 s and
then rolls off with -3/2. The thermal-limited cavity-arm sensor has a slope of
-2 till 8.335 s and rolls off with -3. The time required for the deviation to
reach the error limit is 6270 s for PRN while the cavity-sensor requires 17740
s. The thermal-limited cavity sensor can estimate the parameter to the
required level within 1200 s.

$\sigma^{2}_{y}(\tau)=\frac{\sum\limits_{j=1}^{M-3m+2}\\{\sum\limits_{i=j}^{j+m-1}\left(w_{m}(i)\left[y(i+m)-y(i)\right]\right)\\}^{2}}{2m^{2}(M-2m+1)},$
(17)

where m=$\tau/\tau_{0}$, $\tau_{0}$ is the sampling rate of the measurement
data, y,of length, M. $w_{m}$ is the window function of length, m,
corresponding to each $\tau$, the averaging time. The window function used in
this paper is the normalised Blackman-Harris filter giving the correct Allan
deviation slopes for the different higher noise slopes. Figures 7, 8 and 9
showcase the Allan deviation (after normalisation) for each parameter (Doppler
constant, rate and acceleration) using the residual displacement sensitivity
as above in time domain. The data points highlighted in each figure are the
minimum times that is required for the noise to be averaged to meet the error
limits in Table 1.

Table 2: Minimum averaging time required using different estimation methods to reduce the Doppler terms to the error limits in Table 1.  Doppler | Inter-spacecraft link |  PRN |  Required
---|---|---|---
term |  Thermal |  Requirement |  absolute |  precision
|  limited |  level |  ranging |  (Doppler)
|  Cavity |  Cavity |  |  trends)
Constant |  1 s |  30 s |  411 s |  10 Hz
Rate |  370 s |  50010 s |  4311 s |  60 $\mu$Hz/s
Acceleration |  1200 s |  17740 s |  6270 s |  5 nHz/s2

From the Figures 7, 8 and 9, it can be observed for that the thermal-noise-
limited cavity-arm sensor (yellow trace) can be used for estimating all the
parameters within a short timescale ($<$ 0.5 hour). The PRN (blue trace) would
perform better than the cavity, if the residual noise is near the LISA pre-
stabilisation limit (red trace).

If the cavity does not reach the thermal-noise-limit, the cavity can still be
employed for determining the Doppler constant parameter, while the PRN can be
used for determining the higher derivatives of the Doppler shifts (which
necessitate longer integration times for estimation). The requirement-level
cavity-arm sensor is limited by the random walk function at lower frequencies,
and after 500 s has more residual noise. This estimation needs to be employed
every time the laser acquires lock with the spacecrafts and can be used only
in the beginning of the laser lock for estimating the constant Doppler shift.
The PRN technique would be ideal for determining the other Doppler parameters
due to the residual being white noise and hence over time, the measurement
errors reduce over time, by the square root of time, for velocity, and more
quickly for acceleration and jerk.

##  V Controller Design

From the descriptions, transfer functions, and the effects of noises on the
output, and the LISA goals, the main requirements for the controllers can be
derived. However additional requirements were also included to make the
controller more robust and perform better during the mission lifetime.

  1. 1.

The phase margin at unity gain crossings must be more than $30\degree$ (open
loop phase within $\pm 150\degree$). Additionally, the same condition is
applied for cross-over frequencies between arm and cavity.

  2. 2.

The arm locking control system is to have at least 15 times the gain of the
cavity control system at $10^{-4}$ Hz and at least 100 times the FP cavity
gain at 1 Hz. These are soft conditions to prioritise the bandwidth of the arm
locking sensor within the LISA science band.

Keeping these conditions in mind, the controller design can be split in two
parts.

### Controller 1: Arm Locking Controller

This controller is used with the arm locking sensor, and is used mainly to
control bandwidth of the Arm sensor, shown as G1 in Figure 1. This controller
can be further split in three stages as follows:

Stage I: Stage I is primarily a controller with a fractional slope of 2.3 and
a unity gain frequency at 13.63 kHz. The integrator scales the sensor
information and provides feedback back to the laser source at lower
frequencies. This provides the necessary suppression below LISA requirements
and the slope ensures that the cavity will dominate the response at higher
frequencies. The implementation of the controller is done using a sum cascade
of low-pass filters as shown in Table 5 in Appendix B.

Stage II: Stage II consists of a cascade of 7 high pass filters in the low
frequency band ($<$ 10 $\mu$Hz), to ensure the cavity dominates at lower
frequencies, allowing a reduction of the Doppler pulling induced by the arm
sensor. The different high-pass filters allow for a smooth transition of the
arm and the cavity in terms of the phase stability of the system. The combined
response gives 7 x 20 db = 140 db/decade of suppression of Doppler shifts at
very low frequencies ($<0.1\ \mu$Hz). Implementation and optimization of such
a filter structure is a topic for future work, with a particularly focus
required upon stability and the partitioning of the controller between
software and firmware.

Figure 10: Open loop gain of the controller. The UGF is at around 11 kHz and
the phase margin is within 30$\degree$. The arm sensor gain is dominant from
$\sim$10 $\mu$Hz to 1 kHz, while the cavity is dominant outside this region.
The blended sensor must maintain sufficient phase margin where the cavity and
arm-locking gain are equal to prevent noise amplification, such as the
approximately 40$\degree$ achieved at the low cross-over point ($\sim$10
$\mu$Hz). Phase margin of $\sim 20\degree$ is obtained around the nulls of the
arm-locking sensor, where the arm-locking gain crosses the cavity gain curve,
resulting in an increased sensitivity to cavity noise in this region.

Stage III: Stage III is a lag compensator to decrease phase of the arm
controller and thus allow the phase margin at the lower cross-over frequency
between the arm and the cavity to be within the requirements.

The combined response for Controller 1 is shown below with the values in Table
3:

$\begin{split}G_{1}(s)=&\left(\frac{g_{1}}{s}\right)^{2.3}\left(\frac{s}{s+p_{h_{1}}}\right)^{5}\left(\frac{s}{s+p_{h_{2}}}\right)^{2}\\\
&\left(g_{lc}\left(\frac{s+z_{lc}}{s+p_{lc}}\right)\right).\end{split}$ (18)

### Controller 2: Cavity Controller

This controller is used along with the cavity sensor, as shown as G2 in Figure
1. Within the LISA band, the controller gain follows a fractional f-3/2 slope
with a UGF of 7.32 kHz. The positive slope can make sure that the cavity does
not dominate arm locking within the LISA band but dominate at higher
frequencies, while at lower frequencies the cavity will have a larger gain due
to the high pass filters in Controller 1.

$G_{2}(s)=\left(\frac{g_{2}}{s}\right)^{1.5}.$ (19)

The implementation of the controller is done using a sum cascade of low-pass
filters as shown in Table 6 in Appendix B.

Table 3: Values for the gain, poles and zeros for the controllers as shown in Equations 18 and 19 Parameter | Value
---|---
$g_{1}$ | 2$\pi$ x 1.36 x $10^{4}$
$g_{2}$ | 2$\pi$ x 7.32 x $10^{3}$
$p_{h_{1}}$ | 2$\pi$ x 1.29x$10^{-6}$ rad/s
$p_{h_{2}}$ | 2$\pi$ x 1.16 x $10^{-3}$ rad/s
$z_{lc}$ | 2$\pi$ x $10^{-4}$ rad/s
$p_{lc}$ | 2$\pi$ x 4.5 x $10^{-6}$ rad/s
$g_{lc}$ | 0.045
Figure 11: Simulink Model illustrating the combination of arm and PDH locking.
The purple boxes contain the noise sources considered in this model while the
green boxes contain the controllers for each of the sensor. The UGF of the
combined controllers is 500 Hz, with the sampling frequency at 10 kHz.

##  VI Simulink Model

Time domain simulation for a representative control system was performed for
model validation. Due to computational limitations, simulations of the long (1
month) Doppler transients with fast control loops (UGF at 10 kHz) was
prohibitive. Instead, a simplified problem was studied to validate the
modelling procedure utilized for LISA predictions. The proposed controller
architecture was tested using a discretized Simulink time-domain simulation
shown in Figure 11. In the analysis, the round trip time is taken to be 1 s
with the controller unity gain frequency scaled down to 500 Hz. Results are
obtained by running a discrete solver for fixed step size at the sample rate
of 10 kHz.

The challenge of computationally exhaustive algebraic loop at higher sampling
frequencies was solved by added an explicit pipeline delay, converging the
loop within a time step, an accommodation needed to run the simulation. This
posed another problem as the phase introduced by this delay led to instability
of the controller at unity gain frequency. The additional phase introduced by
the delay can be measured as

$\phi_{\textrm{delay}}(f)=2\pi f\tau_{\textrm{delay}},$ (20)

where $\tau_{\textrm{delay}}=1/f_{\textrm{sampling}}$, the sampling frequency.
To ensure the delay’s contribution does not affect model conclusions, the
sampling rate should be at least a factor of 20 higher than the unity gain
frequency. Hence for UGF of 10 kHz, a sampling frequency of 200 kHz was
required. This will allow at most only $18\degree$ of phase variation at the
UGF and hence be less prone to instability as the phase margin is $30\degree$.
For the purpose of simulation and evaluation, instead of having a higher
sampling frequency, the gains of the controllers were scaled down using a
overall scale factor to get a lower unity gain frequency. Thus, the final
model was designed with controller of unity gain frequency at 500 Hz with a
sampling frequency of 10 kHz. In this modified control system, the arm is
dominant from 0.2 mHz till 200 Hz, while the cavity is dominant in the
remaining frequency band.

The Doppler shifts were added to the system as errors in the different Doppler
trends similar to the analysis in [10]. The resultant pulling will be
increased by the shorter round trip and at the same time, reduced due to a
lower arm controller bandwidth. The Simulink data is compared with the
predictive Doppler pulling model used in Section III using the modified
control system.

Figure 12: Noise Budget using the same noise models and transfer functions in
Section II.3. The controller has a reduced unity gain frequency at 500 Hz and
the sampling frequency for this simulation is 10 kHz and has been decimated to
100 Hz. Similar to Figure 3, the main contributing noise source is the
requirement-level cavity noise (pink trace) with the arm arm controller
engaged at lower frequencies.

Figure 12 shows the noise spectrum using the time-domain data from the
Simulink model. Similar to the analytical noise budget in Figure 3, the main
contribution is given by the requirement-level cavity noise that is suppressed
after engaging the arm controller. The other noise sources couple in a similar
way to in Figure 3. Figure 13 shows the Doppler pulling of the scaled system
with the associated analytical model. The close agreement between analytic and
simulated noise models provide validation of the analytical modelling approach
in Sections II and III.

Figure 13: Doppler pulling of the system with an errors in the Doppler terms
as $\nu_{0}$ = 1.682491 Hz, $\gamma_{0}$ = 0.084615 Hz/s and $\alpha_{0}$ = -
0.37239 nHz/s2. The controller has a reduced unity gain frequency at 500 Hz
and the sampling frequency for this simulation is 10 kHz and has been
decimated to 100 Hz. The predictive model is able to match exactly the system
performance from Simulink

##  VII Discussions

This paper has proposed a hybrid arm- and cavity- locking scheme for LISA. The
demonstrated highlights are;

  1. 1.

The hybrid control system increases the noise suppression by 3 orders of
magnitude over most of the LISA science band (a factor of 4500 at 10 mHz).
This large suppression can be considered as risk-reduction against TDI, which
continues to be one of LISA’s main areas of active research, either by
providing margin over baseline second-generation TDI or allowing the
deployments of simpler first-generation scheme. The noise reduction can
increase the margin for the baseline second-generation TDI or be sufficient to
deploy first-generation TDI - potentially offering simplifications in the
processing (Section II).

  2. 2.

A lock acquisition scheme is proposed that address the transient behaviour
when engaging the arm locking controller in the presence of Doppler shifts.
This was a limiting factor for the applicability of previous arm locking
schemes. In pre-stabilisation integration with arm locking [11], the cavity
control system uses a frequency-tunable or sideband locking PDH scheme,
supported by the arm locking controller. This allows the Doppler pulling in
the controller (up to maximum of 10 MHz) to persist due to the tunable
frequency. In this paper, we utilise a standard PDH locking system with a
fixed-length optical resonator and have stringent restrictions on the Doppler
pulling. In Section III, the laser frequency pulling is shown to be restricted
to $\pm 20$ kHz, significantly smaller than the line-width of the cavity (100
kHz), maintaining the lock on the cavity. This level of frequency pulling may
introduce a small, transient degradation in cavity noise performance, as
offsetting the cavity error signal from resonance can couple intensity noise
into the readout scheme - but this noise will be suppressed by the arm locking
control system. To achieve a suitable level of frequency pulling at start up,
we need to estimate the Doppler shifts with high accuracy and populate them in
the arm locking control system.

  3. 3.

Two estimation methods for high accuracy estimates of Doppler shifts using on-
board measurements have been proposed in Section IV; 1) averaging the inter-
spacecraft link phase readout and 2)using the PRN ranging measurements. For a
thermal-noise-limited cavity, the Doppler parameter estimates can be obtained
after an integration time of at least 1200 s. If the cavity is at requirement-
level performance, the integration time needed is 50,000 s. PRN measurements
can be used for estimation, requiring at least 6300 s of data. These
integration times are estimates based on the residual noise spectra and will
require further analysis.

  4. 4.

The scheme proposed here is compatible with the LISA baseline design,
requiring firmware updates for the laser control system with minimal or no
changes to the hardware. Previous work involving pre-stabilisation integration
with arm locking have been described in [11, 17], and can provide more
suppression compared to the combination in this paper. But compared to these
previous approaches, which require hardware changes to the optical cavity or
locking system, the proposed control scheme may require minor changes for the
summation of the feedback electronics of both the sensors depending on the
final implementation of the system. We expect the digital resources required
would be larger than a simple cavity locker or phase locker, but by a small
factor (less than a factor of 2 larger in our initial estimate).

Lastly, the Doppler estimation described here are indicative only, and thus we
acknowledge possible estimation methods involving complex control systems. For
example, a scheme utilizing the PDH sensors could be envisioned to iteratively
suppress visible pulling during lock acquisition phase, at the expense of
possible ’in-band’ perturbations. However the time-constants of such a scheme
needs to be carefully analysed for stability and may require a thorough
understanding of the LISA orbits. With further research, such schemes could
potentially serve as alternatives for lock acquisition for arm locking.

##  VIII Conclusions

This paper showed the control system dynamics and the Simulink model of a new
method that combines the locking of both LISA arms and a Fabry-Pérot cavity to
enhance the suppression of the laser frequency noise. From the noise budget,
the noise coupling into the interferometer response is predominantly
contributed by the requirement-level cavity noise, below which clock and
spacecraft motion noise will be dominant. The suppression of requirement-level
cavity noise upto 3 orders, demonstrated herein is sufficient for first-
generation TDI to meet the requirements of LISA (Figure 3), thereby reducing
the complexity of post-processing computations.
This paper also proposes solutions to improve the Doppler shift estimation
that is required for lock acquisition, by estimating the Doppler trends using
inter-spacecraft link and/or inter-spacecraft ranging measurements. The sensor
used in this paper comprises of simple architecture of common arm sensor, and
fixed cavity. The future scope could be to explore more complex schemes of
these two sensors [19, 20, 16, 10], that should help in better suppression
while maintaining low Doppler pulling. Experimental verification of this
combination would be useful in solidifying the theoretical work.

## Acknowledgement

The authors acknowledge that this research was conducted with support from the
Australian Research Council Centre of Excellence for Gravitational Wave
Discovery (OzGrav), through project number CE170100004. Kirk McKenzie’s
contribution was partially supported by a contract from Jet Propulsion
Laboratory, California Institute of Technology.

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##  Appendix A Noise Sources

The noise sources, considered in this paper, are summarised as follows:

  1. 1.

Laser frequency noise: The equation used for simulating the laser frequency
noise in a free running NPRO laser can be given as [10]

$\nu_{L_{i}}(f)=\frac{30000\hskip 2.27626pt\textrm{Hz}}{f}\hskip
2.84544pt\textrm{Hz}/\sqrt{\textrm{Hz }}.$ (21)

Here i can be 1, 2 or 3, referring to the laser frequency noise in the laser
sources in the individual spacecrafts. Although there are noises sourced from
Spacecrafts 2 and 3, they get suppressed as a result of the high gain
approximation of the transponders in the spacecrafts. As a result, in this
paper, the notation $\nu_{L}$ will refer to $\nu_{L_{1}}$, as Spacecraft 1 is
considered the primary spacecraft. From the model in Figure 1, the output (at
point C) due to the laser frequency noise can be written as

$\begin{split}\nu_{C;L}(s)=&\nu_{L}(s)-\nu_{C;L}(s)G_{1}(s)P_{+}(s)\\\
&-\nu_{C;L}(s)G_{2}(s)P_{\textrm{pdh}}(s),\\\ \end{split}$ (22)
$\begin{split}LN(s)&=\frac{\nu_{C;L}(s)}{\nu_{L}(s)}\\\
&=\frac{1}{1+G_{1}(s)P_{+}(s)+G_{2}(s)P_{\textrm{pdh}}(s)}\end{split}.$ (23)

Here G1(s) and G2(s) are controllers of the arm sensor and the cavity sensor,
respectively, described in Section V. It can be seen that both the sensors
contribute to the suppression of the laser frequency noise.

  2. 2.

Shot noise: The shot noise occurs due to quantum fluctuations of the laser and
is observed at the event when the laser strikes the photo detector. It can be
modelled as shown below [10]:

${\theta_{shot;ij}(s)=\sqrt{\frac{\hbar c}{2\pi}\frac{1}{\lambda P}}}\hskip
2.84544pt\textrm{cycles}/\sqrt{\textrm{Hz}}.$ (24)

The equation refers to the shot noise arising due to laser from Spacecraft j
being detected at Spacecraft i. As per the equation, if the laser has more
optical power, P, the amount of shot noise would be lower and vice versa.
Typical value for shot noise for the application of LISA can be estimated
using the parameters and is found to be 6.9 $\mu$cycles/$\sqrt{\textrm{Hz}}$
[1].

$\theta_{shot;ij}(s)=6.9\hskip 2.84544pt\textrm{x}\hskip
2.84544pt10^{-6}\hskip 2.84544pt\textrm{cycles}/\sqrt{\textrm{Hz}},$ (25)
$\nu_{shot;ij}(s)=(6.9\hskip 2.84544pt\textrm{x}\hskip
2.84544pt10^{-6}).s\hskip 2.84544pt\textrm{Hz}/\sqrt{\textrm{Hz}}.$ (26)

The contribution from each shot noise can be shown as

$\begin{split}\nu_{C;shot}(s)=&-\nu_{shot;ij}(s)G_{1}(s)-\nu_{C;shot}G_{1}(s)P_{+}(s)-\\\
&\nu_{C;shot}(s)G_{2}(s)P_{\textrm{pdh}}(s)\end{split},$ (27)
$\begin{split}SN(s)=&\frac{\nu_{C;shot}(s)}{\nu_{shot;ij}(s)}\\\
=&\frac{-G_{1}(s)}{1+G_{1}(s)P_{+}(s)+G_{2}(s)P_{\textrm{pdh}}(s)}\end{split}.$
(28)

Shot noise can be seen mainly in the frequency bands where arm is dominant but
gets suppressed otherwise. However, the amount of shot noise that couples into
the system is very low compared to other noise sources and can be seen as the
limits of noise suppression.

  3. 3.

Cavity Noise: For LISA, during pre-stabilisation period, the laser frequency
noise contribution can be reduced to a residual amount if the cavity is made
using low-loss mirrors using ULE or Zerodur [2, 32]. This residual noise
serves as a limitation of the stability provided by the cavity.

$\nu_{cavity}(s)=30\left(\sqrt{1+\left(\frac{2\hskip
2.27626pt\textrm{mHz}}{f}\right)^{4}}\right)\hskip
2.84544pt\textrm{Hz}/\sqrt{\textrm{Hz}}.$ (29)

The contribution due to the cavity noise can be shown as

$\begin{split}\nu_{C;cavity}(s)=&-\nu_{cavity}(s)G_{2}(s)P_{\textrm{pdh}}(s)\\\
&-\nu_{C;cavity}\left(G_{1}(s)P_{+}(s)+G_{2}(s)P_{\textrm{pdh}}(s)\right)\end{split},$
(30) $\begin{split}TN(s)=&\frac{\nu_{C;cavity}(s)}{\nu_{cavity}(s)}\\\
=&\frac{-G_{2}(s)P_{\textrm{pdh}}(s)}{1+G_{1}(s)P_{+}(s)+G_{2}(s)P_{\textrm{pdh}}(s)}\end{split}.$
(31)

The transfer function for the cavity noise is given by TN(s), one can see the
cavity noise is suppressed by the arm locking control system.

  4. 4.

Clock Noise : It is caused by the clock signal on board the spacecraft, that
is utilised for phase meter measurements [10], and is dependent on the beat
note frequency. The following equations describe the clock noise in LISA.

$C_{i}(f)=\frac{\tilde{y_{i}}(f)}{2\pi f}\hskip
2.84544pt\textrm{cycles}/\sqrt{\textrm{Hz}}.$ (32)

Here the value of $\tilde{y_{i}}$ is 2.4 x ${10^{-12}}/{\sqrt{f}}$
$1/\sqrt{\textrm{Hz}}$ and represents the fractional fluctuations of the clock
that is used. The value of i can be 1, 2 or 3, corresponding to the clock in
each individual phasemeter from Spacecraft 1, 2 or 3.

$\phi_{clock;ij}(f)=\Delta_{ij}C_{i}(f)\hskip
2.84544pt\textrm{cycles}/\sqrt{\textrm{Hz}},$ (33)
$\nu_{clock;ij}(f)=\Delta_{ij}.C_{i}(f).s\hskip
2.84544pt\textrm{Hz}/\sqrt{\textrm{Hz}}.$ (34)

The beat note frequency, $\Delta_{ij}$ between two Spacecrafts i and j, is
given a value of 30 MHz, assuming the worst-case scenario of Doppler pulling
between the spacecrafts to be 5 MHz ($\pm$5 MHz) and a maximum heterodyne
measurement offset of 25 MHz between two spacecrafts [1]. The corresponding
noise response due to clock noise can be computed as

$\begin{split}\nu_{C;clock}(s)=-&\nu_{clock;ij}(s)G_{2}(s)-\nu_{C;clock}G_{1}(s)P_{+}(s)\\\
-&\nu_{C;clock}(s)G_{2}(s)P_{\textrm{pdh}}(s),\end{split}$ (35)
$\begin{split}CN(s)=&\frac{\nu_{C;clock}(s)}{\nu_{clock;ij}(s)}\\\
=&\frac{-G_{1}(s)}{1+G_{1}(s)P_{+}(s)+G_{2}(s)P_{\textrm{pdh}}(s)}\end{split}.$
(36)

The clock noises would be correlated if the same clock source is used for
phase meter measurement and uncorrelated if there are separate clock sources.

  5. 5.

Spacecraft Motion noise: This noise is generated when the spacecraft follows
the proof masses to retain drag free operation and cause inter-spacecraft
jitters and is given by: [17]

$\Delta\tilde{X_{ij}}(s)=1.5\sqrt{1+\left(\frac{8\hskip
2.27626pt\textrm{mHz}}{f}\right)^{4}}\hskip
2.84544pt\textrm{nm}/\sqrt{\textrm{Hz}},$ (37)

$\phi_{SC;ij}(s)=\frac{\Delta\tilde{X_{ij}}(s)}{\lambda}\hskip
2.84544pt\textrm{cycles}/\sqrt{\textrm{Hz}},$ (38)

$\nu_{SC;ij}(s)=\phi_{SC;ij}(s).s\hskip
2.84544pt\textrm{Hz}/\sqrt{\textrm{Hz}},$ (39)

where $\lambda$ is the laser wavelength and is 1064nm for LISA. The
corresponding noise response due to spacecraft motion can be computed as

$\begin{split}SCN(s)=&\frac{\nu_{C;SC}(s)}{\nu_{SC;ij}(s)}\\\
=&\frac{-G_{1}(s)}{1+G_{1}(s)P_{+}(s)+G_{2}(s)P_{\textrm{pdh}}(s)}\end{split}.$
(40)

The spacecraft motion noise, $\nu_{SC;ij}$ refers to the jitter between
Spacecraft i and j, to match the proof-mass in spacecraft i. Both the clock
noise and spacecraft motion have the same transfer function and hence, reaches
the limit in the LISA science band while getting suppressed when the cavity is
dominant.

The total spacecraft motion noise propagated through the system can be shown
as below:

$\begin{split}\nu_{SC}(s)=&-\nu_{SC;12}(s)[1+e^{-2s\tau_{12}}]-2[\nu_{SC;21}(s)e^{-s\tau_{12}}]\\\
&-\nu_{SC;13}(s)[1+e^{-2s\tau_{13}}]-2[\nu_{SC;31}(s)e^{-s\tau_{13}}].\end{split}$
(41)

  6. 6.

Digitization noise: Digital hardware used for phasemeters and controllers
couple noise into the system, due to analog-digital conversions and precision
of integer arithmetic. But the contribution is very small of the order of
10-10 [17], and is neglected for noise budget in this paper.

##  Appendix B Sinusoidal spacecraft separation orbital model

With respect to the orbital dynamics of LISA that is described by the model in
Equation 11, the frequency pulling at lock acquisition can be reduced further
if the Doppler shifts are modelled as a combination of sinusoids of half-year
and a year periods, rather than the polynomial model of Doppler shifts in
Section III. The sinusoidal model is:

$\nu_{D;est}(t)=\nu_{0;+}+\gamma_{0;+}t+\iint\alpha(t)dt^{\prime}dt,$ (42)

where

$\alpha(t)=\hat{\alpha_{1}}\sin(\hat{\omega_{1}}t+\hat{\phi_{1}})+\hat{\alpha_{2}}\sin(\hat{\omega_{2}}t+\hat{\phi_{2}}).$
(43)

Here $\nu_{0;+}$ and $\gamma_{0;+}$ are the estimates of the Doppler shift and
the first derivative of Doppler shift (Doppler rate) at the instant when the
controller is just turned on. $\hat{\alpha_{1}}$ and $\hat{\alpha_{2}}$ are
amplitude estimates of the two sinusoids of frequency estimates
$\hat{\omega_{1}}$ and $\hat{\omega_{2}}$, along with phase shift estimates
$\hat{\phi_{1}}$ and $\hat{\phi_{2}}$, that provides the Doppler acceleration
of the system.

Table 4: Parameter requirements for orbital knowledge in order to meet lock acquisition conditions with the controller shown in Fig 10. Each parameter’s error limit is checked in combination with the errors of other parameters in Monte Carlo simulations. The achievable levels are cross-checked with estimation using Fabry-Pérot (FP) cavity estimation in [10] and PRN ranging, or Thermal Noise Limited (TNL) cavity estimation as described in Section IV. The other parameters require more information using orbital analysis before or during the commissioning of LISA. Parameter | Actual/Max | Max error | Fractional | Estimation
---|---|---|---|---
| Value | tolerance ($\pm$) | change | methods
$\nu_{0}$ | 12 MHz | 10 Hz | 8.33 x $10^{-7}$ | FP cavity estimation/ TNL estimation
$\gamma_{0}$ | 4 Hz/s | 60 $\mu$Hz/s | 2.5 x $10^{-4}$ | PRN ranging/ TNL estimation
$\alpha_{1}$ | -1 $\mu$Hz/s2 | 20 nHz/s2 | 2 x $10^{-2}$ | PRN ranging/ TNL estimation
$\alpha_{2}$ | 0.25 $\mu$Hz/s2 | 20 nHz/s2 | 8 x $10^{-2}$ | PRN ranging/TNL estimation
$f_{1}$ | 63.4 nHz | 0.1 nHz | 1.57 x $10^{-3}$ | Orbital dynamics
$f_{2}$ | 31.7 nHz | 0.1 nHz | 3.15 x $10^{-3}$ | Orbital dynamics
$\phi_{1}$ | 2$\pi$ rad | 10 $\mu$rad | 1.6 x $10^{-6}$ | Orbital dynamics
$\phi_{2}$ | 2$\pi$ rad | 10 $\mu$rad | 1.6 x $10^{-6}$ | Orbital dynamics

The maximum allowed values for the error in the Doppler shift parameters are
given in Table 4 for the sinusoidal orbital model. The values were obtained by
Monte Carlo simulations that select outputs that maintain less than $\pm$20
kHz deviation of laser frequency at lock acquisition. The results of the
simulations are shown in Figure 14. Figure 15 shows the spread of the
sinusoidal model, at the worst-case orbit phase, based on the parameters in
Table 4. The sinusoidal model is more accurate than the polynomial model (but
more complex). If parameters estimates for the orbital model are better than

Figure 14: Frequency deviation from cavity line center due to Doppler-error-
pulling at lock acquisition. The estimate is calculated using the sinusoid
functions in Equation 42 for estimation. The black traces shows the different
Monte Carlo simulations with the dashed traces showing the pulling with the
maximum error limits, given in the Table 4. The maximum pulling is 20 kHz and
the HWHM linewidth of the cavity is 100 kHz, maintaining the cavity lock.

those in Table 4, using the sinusoidal orbital model can lead to a smaller
frequency deviation at lock acquisition than is possible to achieve with the
polynomial model, which is limited by the intrinsic model error. Doppler shift
parameter estimation shown in Section IV will be able to converge to the
required precision of Table IV.

Alternatively, FP cavity estimation with LISA requirement-level performance,
described in [10] can be used for the estimation for $\nu_{0}$ while the PRN
ranging can be be used for estimating $\gamma_{0}$, $\hat{\alpha_{1}}$ and
$\hat{\alpha_{2}}$, similar to the analysis in Section IV. We expect the
parameters $\hat{f_{1}}$, $\hat{f_{2}}$, $\hat{\phi_{1}}$ and $\hat{\phi_{2}}$
to be derived from orbital models of LISA described analytically or measured
during commission period. During the transient of the lock acquisition, the
cavity is detuned up to 20 kHz. Though it is still in the linear regime, the
optimal noise performance may be compromised to some extent, discussed
earlier.

Figure 15: Doppler shifts with the estimation models using the sinusoid
functions in Equation 42 for estimation. With the red dotted trace showing the
actual Doppler shifts in Equation 11, the black traces shows the various Monte
Carlo orbital models for different errors in the parameters with the dashed
lines trace showing the worst case, given in the Table 4.

##  Appendix C Implementation of Controllers

For ease of implementation, stage I of controller 1 was implemented as a sum
of multiple low pass filters with appropriate gains. The poles and gains of
the low pass structure are given in Table 5

$G_{1;I}(s)=\frac{g_{0}}{s^{2}}\sum_{i=1}^{13}\frac{g_{i}}{s+p_{i}}.$ (44)

Table 5: Parameters of the gains and poles for the Low-pass filter cascade to implement $1/{s^{0.3}}$ in Equation 18 Index | Pole (pl) | Gain(gl)
---|---|---
number (l) |  |
0 |  | (2$\pi$ x 1.36 x $10^{4}$ )2.3 x 12.31
1 | 2$\pi$ x 5 x $10^{-6}$ | 2x $10^{-7}$
2 | 2$\pi$ x 1.6 x $10^{-4}$ | 2 x $10^{-4}$
3 | 2$\pi$ x 4 x $10^{-4}$ | 5 x $10^{-4}$
4 | 2$\pi$ x 5 x $10^{-3}$ | 5 x $10^{-3}$
5 | 2$\pi$ x 7.5 x $10^{-2}$ | 3 x $10^{-2}$
6 | 2$\pi$ x 0.5 | 8 x $10^{-2}$
7 | 2$\pi$ x 4 | 0.5
8 | 2$\pi$ x 50 | 2.5
9 | 2$\pi$ x 400 | 12
10 | 2$\pi$ x 6000 | 110
11 | 2$\pi$ x 1.5 x $10^{5}$ | 1500
12 | 2$\pi$ x 2 x $10^{6}$ | 7000
13 | 2$\pi$ x $10^{-10}$ | $10^{-4}$

A similar approach was done to implement Controller 2 using the values in
Table 6

$G_{2}(s)=\frac{g_{0}}{s}\sum_{i=1}^{13}\frac{g_{i}}{s+p_{i}}.$ (45)

Table 6: Parameters of the gains and poles for the Low-pass filter cascade to implement $1/{s^{0.5}}$ in Equation 19 Index | Pole (pl) | Gain(gl)
---|---|---
number (l) |  |
0 |  | 34 x (2$\pi$ x 7.32 x $10^{3}$ )0.5
1 | 2$\pi$ x 3 x $10^{-5}$ | 3.5 x $10^{-4}$
2 | 2$\pi$ x 3 x $10^{-4}$ | 8.82 x $10^{-4}$
3 | 2$\pi$ x 3 x $10^{-3}$ | 3.12 x $10^{-3}$
4 | 2$\pi$ x 3 x $10^{-2}$ | 8.82 x $10^{-3}$
5 | 2$\pi$ x 0.3 | 3.12 x $10^{-2}$
6 | 2$\pi$ x 3 | 8.82 x $10^{-2}$
7 | 2$\pi$ x 30 | 0.312
8 | 2$\pi$ x 3 x $10^{2}$ | 0.882
9 | 2$\pi$ x 3 x $10^{3}$ | 3.12
10 | 2$\pi$ x 3 x $10^{4}$ | 8.82
11 | 2$\pi$ x 3 x $10^{5}$ | 31.2
12 | 2$\pi$ x 3 x $10^{6}$ | 88.2
13 | 2$\pi$ x $10^{-10}$ | $10^{-4}$