Arm-Locking Frequency Noise Suppression for Astronomical Middle-Frequency
Interferometric Gravitational Observatory

Jun Nian1 and Wei-Tou Ni1,2

1 International Centre for Theoretical Physics Asia-Pacific,
University of Chinese Academy of Sciences, 100190 Beijing, China
2 Innovation Academy of Precision Measurement Science and Technology (APM),
Chinese Academy of Sciences, 430071 Wuhan, China

<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract
For space gravitational wave (GW) detection, arm locking is a proposal useful
in decreasing the frequency noise of the laser sources for current developing
space missions LISA and TAIJI/TIANQIN. In this paper, we study the application
of arm locking to the Astronomical Middle-frequency Interferometric
Gravitational Observatory (AMIGO) to decrease the frequency noise of laser
sources. For AMIGO, the arm-locking technique can suppress the laser frequency
noise by three orders of magnitude. The advantage of this is to make the
auxiliary noise assignment for AMIGO easier and more relaxed. For the first-
generation TDI configuration, the laser frequency noise contribution is
already below the core noise contribution. For the simple Michelson TDI
configuration (X0), the arm locking makes the acceleration-thrust scheme, the
delay-line scheme, or the combined scheme easier to implement. Within a
relatively short period of less than a day (compared to less than twenty days
for LISA/TAIJI), the Doppler frequency pulling can be efficiently reduced to
within $\pm$ 0.001 Hz and does not affect the mission duty cycle much.

###### Contents

  1. 1 Introduction
  2. 2 Noise Characterization
  3. 3 Single Arm Locking
  4. 4 Double Arm Locking
  5. 5 Combined PDH Cavity and Double Arm Locking
  6. 6 Doppler Frequency Pulling
  7. 7 Discussion

##  1 Introduction

LIGO/Virgo/KAGRA (LVK) collaborations have reported the laser-interferometric
detection of high-frequency ($\sim$ 10$\,-\,$1000 Hz) gravitational waves
(GWs) emitted from more than 90 compact binary mergers, i.e., stellar mass
binary black holes (stellar BBH mergers), binary neutron star (BNS) mergers,
and neutron star black hole (NSBH) mergers [1, 2, 3]. In the low-frequency
band, LISA and TAIJI/TIANQIN are actively developing and preparing laser-
interferometric space missions for the GW sources of massive binary black
holes, etc. The mid-frequency band is between these two bands (0.1 Hz$\,-\,$10
Hz). DECIGO and a number of earth-based and other space-borne detectors are
proposed for their GW sources (for a review of present status, see, e.g.,
[4]). According to a recent study [5] (see also [6]), the joint observation of
LISA/TAIJI, AMIGO, and CE/ET with a connected binary inspiral template through
mHz, deci-Hz, and deca-Hz frequencies, will be able to enhance the parameter
estimation by two orders of magnitude or more. These enhancements will
strengthen the distinguishability of various GW source models, the precision
of determination of cosmological models and Hubble constant, and the co-
evolution of star formation, black holes, and galaxies. The sky localization
will be unique for a significant number of sources to realize the
corresponding redshift measurement. This will make a 0.1% accuracy/precision
cosmological model feasible to serve as a strong basis for cosmology and
cosmological astrophysics. Each frequency band is crucial in reaching this
important achievement.

In the original white-light Michelson interferometry, the two interferometric
paths (two arms) must be almost equal to have fringes [7]. With the light of a
very narrow line, the match can be relaxed a bit (e.g., [8, 9]). For laser
light, the interferometry goes further, and the fringes would manifest for a
difference of optical paths within the laser coherent length. For Nd:YAG laser
of 1 kHz linewidth, the two optical length paths can differ by 100 km and
still be coherent to interfere with each other. However, the closer the path
lengths are matched, the clearer the fringes are (less noise). In space,
because of the large distances involved, at the receiving spacecraft (S/C), we
have to phase lock the local laser oscillator to the weak incoming beam to
transmit to another S/C or back; the interferometry measures the final phase
of interference of the two chosen paths. The final phase noise
$\delta\varphi_{\text{interference}}$ of interference at the receiving S/C is

$\displaystyle\delta\varphi_{\text{interference}}$
$\displaystyle=2\pi\cdot\delta\nu(f)\cdot\Delta L/c+\text{phase locking
noise(s)}+\text{timing noise(s)}$ $\displaystyle\quad+\text{signals \& other
noises accrued along the two paths}\,,$ (1.1)

where $\delta\nu(f)$ is the frequency noise of the laser source at frequency
$f$, $\Delta L$ is the optical path length difference of the two chosen paths,
and $c$ is the light velocity. Hence, to decrease the interference phase noise
$\delta\varphi_{\text{interference}}$, we have to either decrease the
frequency noise of the laser source or decrease the path length difference, or
both. After the vast distance traveled, the light received by the telescope in
the other S/C is attenuated greatly and needs amplification to go to another
S/C. The amplification method uses a local laser to phase-lock to the incoming
weak laser light. Hence, the phase information is transmitted in contiguous
propagation, whether in homodyne or with a known frequency offset. The
measured phase by the phasemeter is recorded with a time tag for later
propagation identification in the data analysis. This time-tagged tracing
after the recording is called time-delay interferometry (TDI). The first-
generation Michelson TDI is usually called X, Y, and Z TDIs. We call the
original (zeroth generation) Michelson topology the X0, Y0, and Z0 TDIs. Both
phase-locking and time-tagging contribute to noises. The TTL (tilt-to-length)
noise, etc., are included in the other noises.

For the present assumed laser source frequency noise of $\delta\nu(f)\leq
30\,\text{Hz}/\text{Hz}^{1/2}$, the interference phase noise requirements of
LISA and TAIJI, the requirements of the path length difference $\Delta L$ are
$\Delta L_{\text{LISA}}\leq 25\,\text{m}$ and $\Delta L_{\text{TAIJI}}\leq
30\,\text{m}$, respectively [10]. For the same laser source frequency noise
requirement, the requirement of the path length difference is proportional to
the arm length. For basically geodesic orbits of LISA, the variation of 2.5 Gm
arm length can be 1%, i.e., 25 Mm. Hence, for the simple (original) Michelson,
the path length match can be one million times off the requirement. For TAIJI,
it can be 1 million times off, too. For two paths of TDI X configuration of
LISA, the maximum path length difference can be less than $\pm 240\,\text{m}$
($\pm 800\,\text{ns}$) for a mission time of 2200 days [10]. For two paths of
TDI X configuration of TAIJI, the maximum path length difference can be less
than $\pm 375\,\text{m}$ ($\pm 1250\,\text{ns}$) for a mission time of 2200
days. For two paths of the second generation TDI X1 configuration of LISA, the
maximum path length difference can be less than $\pm 2.7\,\text{mm}$ ($\pm
9\,\text{ps}$) for 2200 days of mission time. For two paths of second
generation TDI X1 configuration of LISA, the maximum path length difference
can be less than $\pm 4.8\,\text{mm}$ ($\pm 16\,\text{ps}$) for 2200 days of
mission time. For simple Michelson, i.e., the TDI X0 configuration and other
first-generation TDIs of LISA and TAIJI compiled in [10], the path length
requirements are not satisfied. However, for the second-generation TDIs
compiled in [10], the path length requirements are well satisfied.

Since the phase noise $\delta\varphi_{\text{interference}}$ of interference is
also proportional to the frequency noise of the laser source, the other way to
save the TDI X1 configuration for LISA and TAIJI for GW detection is to
decrease the frequency noise $\delta\nu(f)$ of the laser sources. This can be
done by arm locking or using a more stable local laser system.

In this paper, we work out the arm locking schemes for the Astrodynamical
Middle-frequency Interferometric Gravitational wave Observatory (AMIGO) [11,
12, 13]. We use the AMIGO-S-8-12deg earth-like solar orbit trailing earth from
8 degrees to 12 degrees studied in [12]. For the assumed laser source
frequency noise of $\delta\nu(f)\leq 30\,\text{Hz}/\text{Hz}^{1/2}$ and the
interference phase noise requirements of AMIGO, the requirement of the path
length difference is $\Delta L_{\text{AMIGO}}\leq 0.1\,\text{m}$ [12]. For the
same laser source frequency noise requirement, the requirement of the path
length difference is proportional to the arm length. For basically geodesic
orbits of AMIGO, the arm length variation can be 1%, i.e., 100 km, for 20 deg
behind the earth or 20 deg leading the earth orbit configuration. For the
AMIGO-S-8-12deg earth-like solar orbit trailing earth from 8 degrees to 12
degrees, the variation could be several times higher. Hence, for simple
Michelson, the path length match can be several million times off. For two
paths of the TDI X configuration of AMIGO, the maximum path length difference
can be less than $\pm 6\,\text{mm}$ ($\pm 20\,\text{ps}$) for a mission time
of 600 days. For the TDI X configuration and other non-Sagnac first-generation
TDIs of AMIGO compiled in [12], the path length requirement is satisfied (for
the Sagnac TDI configurations, an extra path delay is needed for matching).

In Ref. [12], the authors have studied the various orbit configurations, their
Doppler velocities, and TDIs. Here, we use the AMIGO-S-8-12deg earth-like
solar orbit formation with 8$\,-\,$12 degrees behind the earth for 600 days
with initial conditions given in Column 3 of Tab. 1 of [12] as an example. The
arm length variations, the Doppler velocities, the formation angles, and the
angle between the barycenter of S/C and earth in 600 days are shown in Fig. 1.

Figure 1: Variations of the arm lengths, the Doppler velocities, the formation
angles, and the angle between the barycenter of S/C and earth in 600 days for
the AMIGO-S-8-12deg behind earth formation (from Ref. [12]).

Doppler velocities for neighboring arms are less than 35 mm/s. Path length
differences of X0, Y0, Z0, X, Y, and Z TDIs and X+Y+Z are listed below in Tab.
1. As we can see, the requirement on $\Delta L$ for the first-generation TDIs
is well satisfied by the orbit configurations (X, Y, Z, and their sum).
According to Eq. (1), the effective timing/tracking noise should also satisfy
this requirement. If we set the effective timing/tracing noise to be below 100
ps, it would be OK. As to the phase-locking noise requirement, it is also well
satisfied.

TDI | Min | Max | Average
---|---|---|---
X0 (arm1 - arm2) | several hundred km | several hundred km |
Y0 (arm2 - arm3) | several hundred km | several hundred km |
Z0 (arm3 - arm1) | several hundred km | several hundred km |
X |  $-19$ ps |  $14$ ps |  $7$ ps
Y |  $-15$ ps |  $14$ ps |  $6$ ps
Z |  $-12$ ps |  $16$ ps |  $6$ ps
X + Y + Z |  $-0.01$ ps |  $0.12$ ps |  $0.06$ ps
Requirement on $\Delta L$ |  $330$ ps |  $330$ ps |
Table 1: Minimum and maximum of path length differences for the Michelson
configurations X0, Y0, and Z0 TDIs, together with the minimum, maximum, and
average of the first-generation X, Y, and Z TDI configurations plus those of
the sum X+Y+Z are listed (from Ref. [12]).

This paper is organized as follows. Sec. 2 lists and discusses various noises
assumed for this study. Sec. 3 focuses on the single arm locking stabilization
for AMIGO and AMIGO-5. Sec. 4 works out the double arm locking stabilization
for AMIGO and AMIGO-5. Sec. 5 presents the combined PDH cavity locking and
double arm locking stabilization for AMIGO and AMIGO-5. Sec. 6 considers the
Doppler frequency pulling for this case. Sec. 7 discusses various options in
the context of mission implementation and what will be achieved in the network
observations.

##  2 Noise Characterization

In this section, we set the noise requirements for AMIGO and list the noise
models considered in this paper.

For the acceleration noise requirement, we set

$S_{a}^{1/2}(f)\leq 3\times
10^{-15}\left[1+\left(\frac{f}{0.3\,\text{Hz}}\right)^{4}\right]^{1/2}\frac{\text{m}}{\text{s}^{2}\,\text{Hz}^{1/2}}\,,\quad(10\,\text{mHz}<f<10\,\text{Hz})\,,$
(2.1)

for b-AMIGO, AMIGO and E-AMIGO. N.B., LISA Pathfinder (LPF) has already
demonstrated the following requirement in the frequency range
$20\,\mu\text{Hz}-0.03\,\text{Hz}$ in Feb 2017,

$S_{a}(f)\leq 9\times
10^{-30}\left[1+\left(\frac{10^{-4}\,\text{Hz}}{f}\right)^{2}+16\,\left(\frac{2\times
10^{-5}\,\text{Hz}}{f}\right)^{10}\right]\frac{\text{m}^{2}}{\text{s}^{4}\,\text{Hz}}\,,\quad(20\,\mu\text{Hz}-0.03\,\text{Hz})\,.$
(2.2)

With the achieved requirement (2.2) of LISA LPF, our requirement (2.1) is
already satisfied in $10\,\text{mHz}$ to $30\,\text{mHz}$ frequency band. For
the $0.03\,\text{Hz}$ to $10\,\text{Hz}$ band, the flat requirement
$S_{a}^{1/2}(f)\leq 3\times
10^{-15}\,\text{m}\,\text{s}^{-2}\,\text{Hz}^{-1/2}$ should be no more
difficult to implement. However, we relax this by a blue factor with a corner
frequency $[1+(f/0.3\,\text{Hz})^{4}]^{1/2}$, since there is already an
antenna factor in sensitivity, and the blue factor will not affect the
sensitivity significantly.

For the laser metrology noise, we set

  * •

Baseline (b-AMIGO):

$S_{\text{AMIGOp}}\leq 1.4\times
10^{-28}\,\frac{\text{m}^{2}}{\text{Hz}}\,,\quad S_{\text{AMIGOp}}^{1/2}\leq
12\,\frac{\text{fm}}{\text{Hz}^{1/2}}\,,\quad(10\,\text{mHz}<f<10\,\text{Hz})\,;$
(2.3)

  * •

Design Goal (AMIGO):

$S_{\text{AMIGOp}}\leq 0.14\times
10^{-28}\,\frac{\text{m}^{2}}{\text{Hz}}\,,\quad S_{\text{AMIGOp}}^{1/2}\leq
3.8\,\frac{\text{fm}}{\text{Hz}^{1/2}}\,,\quad(10\,\text{mHz}<f<10\,\text{Hz})\,;$
(2.4)

  * •

Enhanced Goal (e-AMIGO):

$S_{\text{AMIGOp}}\leq 0.0025\times
10^{-28}\,\frac{\text{m}^{2}}{\text{Hz}}\,,\quad S_{\text{AMIGOp}}^{1/2}\leq
0.5\,\frac{\text{fm}}{\text{Hz}^{1/2}}\,,\quad(10\,\text{mHz}<f<10\,\text{Hz})\,.$
(2.5)

As to laser frequency noise, we use the same criterion as LISA/TAIJI, but in
the $10\,\text{mHz}<f<10\,\text{Hz}$ band:

$\delta\nu(f)\leq
30\,\frac{\text{Hz}}{\text{Hz}^{1/2}}\,,\quad(10\,\text{mHz}<f<10\,\text{Hz})\,.$
(2.6)

This is the residual laser frequency noise after the pre-stabilization, e.g.,
through the PDH cavity discussed below. As a comparison, the stabilized laser
system of the GRACE follow-on has the following laser frequency noise [14,
15]:

$\displaystyle\delta\nu(f)$ $\displaystyle\approx
0.4\,\left(\frac{1\,\text{Hz}}{f}\right)\left[1+\left(\frac{1\,\text{mHz}}{f}\right)^{2}\right]^{3/4}\frac{\text{Hz}}{\text{Hz}^{1/2}}$
$\displaystyle\approx
0.4\,\left(\frac{1\,\text{Hz}}{f}\right)\frac{\text{Hz}}{\text{Hz}^{1/2}}\,.$
(2.7)

More explicitly, we list a few noises considered in this paper.

  * •

Laser frequency noise:

The laser frequency noise for both AMIGO and AMIGO-5 is given by the same
model as LISA:

$\nu_{L}\left(f\right)=\frac{30000\
\text{Hz}}{f}\frac{\text{Hz}}{\sqrt{\text{Hz}}}\ .$ (2.8)

This expression characterizes the typical laser frequency noise in a free-
running non-planar ring oscillator laser. In contrast, (2.6) is for the laser
frequency noise after pre-stabilization.

  * •

Shot noise:

We can compare the design goal for the laser metrology noise of AMIGO given by
(2.4) to the value of LISA, $10\,\text{pm}/\text{Hz}^{1/2}$. The shot noise
for AMIGO can then be obtained by rescaling the shot noise model of LISA:

$\nu_{shot}(f)=(3.8\times 10^{-4})\cdot(6.9\times 10^{-6})\cdot\frac{2\pi
if}{1\,\text{Hz}}\,\frac{\text{Hz}}{\sqrt{\text{Hz}}}\,.$ (2.9)

Similarly, the shot noise for AMIGO-5 is

$\nu_{shot}(f)=(1.9\times 10^{-3})\cdot(6.9\times 10^{-6})\cdot\frac{2\pi
if}{1\,\text{Hz}}\,\frac{\text{Hz}}{\sqrt{\text{Hz}}}\,.$ (2.10)

  * •

Clock noise:

We take the clock noise for both AMIGO and AMIGO-5 to be the same as LISA:

$\nu_{clock}\left(f\right)=(30\ \text{MHz})\cdot
C_{i}\left(f\right)\cdot\frac{2\pi
if}{1\,\text{Hz}}\,\frac{\text{Hz}}{\sqrt{\text{Hz}}}\ ,$ (2.11)

where $C_{i}\left(f\right)$ is given by

$C_{i}\left(f\right)=\frac{2.4\times{10}^{-12}}{2\ \pi\
\left(f/(1\,\text{Hz})\right)^{3/2}}\,\frac{1}{\text{Hz}}\ .$ (2.12)

  * •

Spacecraft motion noise:

The spacecraft motion noise for both AMIGO and AMIGO-5 has the same form as
LISA:

$\nu_{SC}\left(f\right)=\phi_{SC}\left(f\right)\cdot 2\pi i\cdot
f\,\frac{\text{Hz}}{\sqrt{\text{Hz}}}\ ,$ (2.13)

where $\phi_{SC}\left(f\right)$ is given by

$\phi_{SC}\left(f\right)=\frac{1.5\ \sqrt{1+\left(8\ \text{mHz}/f\right)^{4}}\
\text{nm}}{1064\ \text{nm}}\ \frac{1}{\text{Hz}}\ .$ (2.14)

  * •

PDH cavity noise:

The Pound-Drever-Hall (PDH) cavity can be used in the laser pre-stabilization,
but it also introduces noise for both AMIGO and AMIGO-5 is given by the same
form as LISA:

$\nu_{cavity}\left(f\right)=30\ \sqrt{1+\left(\frac{2\
\text{mHz}}{f}\right)^{4}}\frac{\text{Hz}}{\sqrt{\text{Hz}}}\ .$ (2.15)

##  3 Single Arm Locking

In space-based gravitational wave detection, several spacecrafts form an
interferometer, and the distances between each two of them are the arms of the
interferometer. Compared to the laser frequency noise, the arm lengths are
relatively more stable in this case. Following the idea in [16] (see also [17,
15]), we can use the stability of the arm length to suppress the laser
frequency noise. This technique is called arm locking, and single arm locking
denotes the model where only one arm is considered.

The basic idea of arm locking is constructing a feedback control system by
sending the amplified signal back into the original laser after it travels
around the arm once. In this section, we consider the single arm locking with
the feedback system shown in Fig. 2 and discuss how the noises mentioned in
the previous section are suppressed after the arm locking. For the single arm
locking, the close-loop laser noise at Point $B$ after the arm locking is
given by [16]:

$\mathrm{\Phi}_{\text{single}}=\frac{P_{1}}{1+L_{1}}-\frac{G_{1}\
P_{2}}{\left(1+G_{2}\right)\left(1+L_{1}\right)}-\frac{G_{1}G_{2}N_{2}}{\left(1+G_{2}\right)\left(1+L_{1}\right)}-\frac{G_{1}\
N_{1}}{1+L_{1}}+\frac{G_{1}\ \mathrm{\Phi}_{gw}}{1+L_{1}},$ (3.1)

where $L_{1}=\left(1-G_{2}\ e^{-s\,\tau}/(1+G_{2})\right)G_{1}$. In this
expression, $P_{1,2}$ and $N_{1,2}$ are the laser frequency noises and the
shot noises on Spacecraft 1 and 2, respectively, while $\mathrm{\Phi}_{gw}$
denotes the gravitational wave signal. We can tune the arm locking controllers
$G_{1,2}$ to adjust the final noise $\mathrm{\Phi}_{\text{single}}$ after the
single arm locking.

Figure 2: The feedback system of the single arm locking (see also [16])

Depending on the choices of the controllers $G_{1,2}$, there are two classes
of the single arm locking: (i) the symmetric controller solution when
$G_{1}=G_{2}$; (ii) the asymmetric controller solution when $G_{1}\neq G_{2}$.

With the choice of $G_{1}$ and $G_{2}$ shown in Tab. 2, we obtain the noises
after the single arm locking for AMIGO (arm length 10,000 km) and AMIGO-5 (arm
length 50,000 km). The results are shown in Fig. 3 $-$ Fig. 6. In these
figures, the free laser noise (after pre-stabilization) and the requirement of
the first-generation TDI are denoted by the red line and the black dotted
line, respectively. The orange, green, and blue lines indicate the laser
frequency noise after the single arm locking, the shot noise, and the total
noise, respectively.

| $G_{1}[f]$ | $G_{2}[f]$
---|---|---
Symmetric | $10^{5}/\sqrt{f\,[\text{Hz}]}$ | $10^{5}/\sqrt{f\,[\text{Hz}]}$
Asymmetric | $10^{5}/\sqrt{f\,[\text{Hz}]}$ | $10^{8}/\sqrt{f\,[\text{Hz}]}$
Table 2: The choices of the controllers $G_{1,2}$ for the single arm locking
Figure 3: The results of the single arm locking with symmetric controllers for
AMIGO Figure 4: The results of the single arm locking with asymmetric
controllers for AMIGO Figure 5: The results of the single arm locking with
symmetric controllers for AMIGO-5 Figure 6: The results of the single arm
locking with asymmetric controllers for AMIGO-5

After the single arm locking, we see that the laser frequency noise can be
suppressed for 2 to 4 orders of magnitude in the mid-frequency range (10
mHz$\,-\,$10 Hz).

##  4 Double Arm Locking

In gravitational wave detection, there are multiple arms in the configuration.
We can combine the signals from different arms before feeding them back to the
original laser. This can be done by an arm-locking sensor, which can be viewed
as a linear operation of the signals from two arms (see Fig. 7):

$\phi_{double}\left(\omega\right)=\mathbf{S}\left(\begin{matrix}\phi_{13}\left(\omega\right)\\\
\phi_{12}\left(\omega\right)\\\ \end{matrix}\right).$ (4.1)

When two arms participate in the feedback control system, this way of arm
locking is called double arm locking [18] (see also [17, 15]). Depending on
the linear combinations of the signals from the two arms, the double arm
locking includes several types (common, dual, modified dual, etc.).

Figure 7: The feedback system of the double arm locking (see also [18])

For simplicity, in this paper, we only consider the common double arm locking,
which is just taking the sum of the signals from two arms, i.e.,
$\mathbf{S}_{+}=(1,\ \ 1)$. Using the common double arm locking, the closed-
loop laser noise after the arm locking is given by [18]:

$\nu_{O}=\frac{\nu_{L}}{1+G_{1}\ P_{+}\ }-\frac{G_{1}\ \mathbf{S}_{+}\
{(N}_{SN}+N_{CN}+N_{SCN})}{1+G_{1}\ P_{+}\ }\ ,$ (4.2)

where $\nu_{L}$ is the laser frequency noise, $N_{SN}$, $N_{CN}$, $N_{SCN}$
denote the shot noise, the clock noise, and the spacecraft motion noise (the
optical bench displacement noise), respectively. The choices of the
controllers $G_{1,2}$ and the frequency response $P_{+}$ are given in Tabs. 3
and 4.

$G_{1}[s]$ | $\left(\frac{g_{1}}{s}\right)^{2.3}\ \left(\frac{s}{s+p_{h_{1}}}\right)^{5}\ \left(\frac{s}{s+p_{h_{2}}}\right)^{2}\ \left(g_{l_{c}}\left(\frac{s+z_{l_{c}}}{s+p_{l_{c}}}\right)\right)$
---|---
$G_{2}[s]$ | $\left(g_{2}/s\right)^{1.5}$
$P_{+}[\omega]$ | $2\ \left[1-\cos{\left(\Delta\tau\ \omega\right)}\ e^{-i\ \omega\ \bar{\tau}}\right]\qquad(\bar{\tau}\equiv\tau_{12}+\tau_{13},\ \ \Delta\tau\equiv\tau_{12}-\tau_{13})$
Table 3: The controllers for the common double arm locking  Parameter | Value
---|---
$g_{1}$ | $2\pi\times 0.68\times{10}^{5}$
$g_{2}$ | $2\pi\times 7.32\times{10}^{3}$
$p_{h_{1}}$ |  $2\pi\times 3.87\times{10}^{-5}$ rad/s
$p_{h_{2}}$ |  $2\pi\times 1.16\times{10}^{-2}$ rad/s
$z_{l_{c}}$ |  $2\pi\times{10}^{-5}$ rad/s
$p_{l_{c}}$ |  $2\pi\times 1.35\times{10}^{-4}$ rad/s
$g_{l_{c}}$ | $0.045$
Table 4: The parameters for the common double arm locking

The results after the double arm locking are shown in Figs. 8 and 9. Besides
the free laser noise (red line) and the first-generation TDI requirement
(black dotted line), we also include the laser frequency noise after the
double arm locking (orange line), the shot noise (green line), the clock noise
(violet line), the spacecraft motion noise (brown line), and the total noise
(blue line). We assume the average arm lengths for AMIGO and AMIGO-5 are
10,000 km and 50,000 km, respectively, and the difference between the two arms
can be up to 1%.

Figure 8: The results of the common double arm locking for AMIGO Figure 9: The
results of the common double arm locking for AMIGO-5

As we can see from Fig. 8 and 9, using the common double arm locking, the
total noise can be suppressed to meet the requirement of the first-generation
TDI in the mid-frequency range.

##  5 Combined PDH Cavity and Double Arm Locking

On top of the double arm locking, we can introduce a Pound-Drever-Hall (PDH)
cavity to enhance the laser frequency noise suppression. In the presence of
the PDH cavity, the closed-loop laser noise after the arm locking is given by
[19]:

$\nu_{PDH}=\frac{\nu_{L}}{1+G_{1}\ P_{+}+G_{2}\ P_{pdh}}-\frac{G_{2}\ P_{pdh}\
\nu_{cavity}}{1+G_{1}\ P_{+}+G_{2}\ P_{pdh}}-\frac{G_{1}\ \mathbf{S}_{+}\
{(N}_{SN}+N_{CN}+N_{SCN})}{1+G_{1}\ P_{+}+G_{2}\ P_{pdh}}\,,$ (5.1)

where the transfer function of the PDH cavity is given by

$P_{pdh\ }=\frac{D_{0}}{1+\frac{s}{2\ \pi\ f_{c}}}$ (5.2)

with a dimensionless constant $D_{0}$ and $f_{c}\approx 100\ kHz$. We use the
same controllers $G_{1,2}$ with the same parameters as in the common double
arm locking discussed in the previous section.

The results after the common double arm locking enhanced by the PDH cavity are
shown in Figs. 10 and 11. The convention of the lines is the same as the
double arm locking case in the previous section. As before, we assume the
average arm lengths for AMIGO and AMIGO-5 are 10,000 km and 50,000 km,
respectively, and the difference between the two arms can be up to 1%.

Figure 10: The results of the common double arm locking with PDH cavity for
AMIGO Figure 11: The results of the common double arm locking with PDH cavity
for AMIGO-5

From Figs. 10 and 11, we see that the common double arm locking enhanced by
the PDH cavity can efficiently suppress the total noise to meet the
requirement of the first-generation TDI in the mid-frequency range.

##  6 Doppler Frequency Pulling

Suppose that the genuine Doppler frequency noise can be described by a
sinusoidal model:

${\nu_{D,+}\left(t\right)=\nu}_{1}\ \sin{(\omega_{1}\
t+\phi_{1})}+\nu_{2}\sin{(\omega_{2}\ t+\phi_{2})}.$ (6.1)

Before the arm locking is turned on, we can estimate the Doppler frequency
using the following approximation:

$\nu_{D;est\ }\left(t\right)=\nu_{0;+\ }+\gamma_{0;+}\
t+\int_{0}^{t}\int_{0}^{t^{\prime}}\alpha\left(t^{\prime\prime}\right)\
dt^{\prime\prime}\ dt^{\prime}\,,$ (6.2)

where

$\alpha\left(t\right)=\alpha_{1}\ \sin{({\hat{\omega}}_{1}\
t+{\hat{\phi}}_{1})}+\alpha_{2}\sin{({\hat{\omega}}_{2}\ t+{\hat{\phi}}_{2})}\
.$ (6.3)

For AMIGO and AMIGO-5, we show the Doppler frequency noise model and its
estimate in Fig. 12 with the parameters in Tab. 5, where we pick up a part of
the early stage of Doppler velocity in Fig. 1.

Figure 12: The early-time Doppler frequency noise model for AMIGO and AMIGO-5 Parameter | Value
---|---
$\nu_{1}$ | $5.75807\times{10}^{4}$
$\nu_{2}$ | $5.49284\times{10}^{4}$
$\omega_{1}$ and $\hat{\omega}_{1}$ |  $2\pi\times 63.4\times{10}^{-9}$ rad/s
$\omega_{2}$ and $\hat{\omega}_{2}$ |  $2\pi\times 31.7\times{10}^{-9}$ rad/s
$\phi_{1}$ |  $-1.62513$ rad
$\phi_{2}$ |  $2.09707$ rad
$\nu_{0;+}$ | $-{10}^{4}$
$\gamma_{0;+}$ | $-0.0067413$
$\alpha_{1}$ | $9.13725\times{10}^{-9}$
$\alpha_{2}$ | $-2.17909\times{10}^{-9}$
${\hat{\phi}}_{1}$ |  $1.51646$ rad
${\hat{\phi}}_{2}$ |  $2.09707$ rad
Table 5: The parameters in the Doppler frequency noise model

When we turn on the arm locking, it takes some time for the approximation to
converge to the genuine Doppler frequency noise. To see this transient effect,
we compute the following quantity in the time domain as an inverse Laplace
transform

$\nu_{C}\left(t\right)=L^{-1}\
\left(\frac{\left[\nu_{D;+}\left(s\right)-\nu_{D;est\ }(s)\right]\
V(s)}{s}\right)\,,$ (6.4)

which is called the Doppler frequency pulling [18, 17, 19]. The arm locking
transfer function $V(s)$ is given by

$V(s)=-\frac{G_{1}(s)}{1+G_{1}\ P_{+}+G_{2}\ P_{pdh}}\,,$ (6.5)

which depends on the arm length and other parameters of the arm locking. Here,
we adopt the same choice of controllers $G_{1,2}$ and $P_{pdh}$ as the
combined double arm locking with PDH cavity discussed in the previous
sections. For AMIGO and AMIGO-5, the results of the Doppler frequency pulling
are plotted in Figs. 13 and 14. We can repeat the same steps and analyze the
frequency pulling for later-time Doppler noises, and the results are shown in
Figs. 15$\,-\,$20.

Figure 13: The result of the early-time Doppler frequency pulling for AMIGO
Figure 14: The result of the early-time Doppler frequency pulling for AMIGO-5
Figure 15: The intermediate-time Doppler frequency noise model for AMIGO and
AMIGO-5 Figure 16: The result of the intermediate-time Doppler frequency
pulling for AMIGO Figure 17: The result of the intermediate-time Doppler
frequency pulling for AMIGO-5 Figure 18: The late-time Doppler frequency noise
model for AMIGO and AMIGO-5 Figure 19: The result of the late-time Doppler
frequency pulling for AMIGO Figure 20: The result of the late-time Doppler
frequency pulling for AMIGO-5

We see that the transient times for AMIGO and AMIGO-5 are both roughly $\sim$
1.0 day, and the maximal magnitude ranges from $0.1$ Hz to $10^{7}$ Hz,
depending on the stages of the Doppler velocities. After the transient time,
the pulling will reach a steady state limited to $\pm$ 0.001 Hz. Hence, the
arm locking technique can efficiently reduce noise caused by the Doppler
frequency pulling within a relatively short period of time.

##  7 Discussion

In this paper, we study the arm locking technique applied to mid-frequency
gravitational wave detection. For AMIGO, the first-generation TDI requirement
is already satisfied by more than two orders of magnitude. For the original
Michelson topology (zeroth generation) TDIs, there is still a gap of around
six orders of magnitude. We look for arm locking to come to the rescue or
partial rescue. To find the efficiency of the arm locking technique, we have
analyzed the suppression of various noises in the frequency domain. In both
cases, with various concrete configurations, we have seen that the arm locking
can suppress the laser frequency noise by about three orders of magnitudes.
This way, the gap can be bridged by three orders of magnitude. If the thrust-
acceleration method is used for minimizing the inequality of the arm lengths,
the requirement on the thrust and inertia sensor/accelerometer is relaxed
quite a bit [12]. We have considered the frequency pulling caused by the
Doppler noise. By optimizing the parameters, we can reduce the transient time
to $\sim$ 0.5 days for AMIGO and $\sim$ 1.0 day for AMIGO-5, demonstrating the
practical feasibility of using arm locking to reduce laser-frequency noise for
mid-frequency gravitational wave detection.

## Acknowledgments

This work is supported by the National Key Research and Development Program of
China (Grant No. 2021YFC2201901) and in part by the NSFC under grant No.
12147103. J.N. would like to thank the International Centre for Theoretical
Physics (ICTP) and the Korea Institute for Advanced Study (KIAS) for the warm
hospitality during the final stage of this work.

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