11institutetext: Shanghai Jiao Tong University, Shanghai, China
11email: {fcy3649, frank92, yitian_li, hehao<EMAIL_ADDRESS>
22institutetext: Sun Yat-sen University, Guangzhou, China
22email<EMAIL_ADDRESS>
# MaxGNR: A Dynamic Weight Strategy via Maximizing Gradient-to-Noise Ratio for
Multi-Task Learning

Caoyun Fan 11†† Wenqing Chen 22†† Jidong Tian 11 Yitian Li 11 Hao He 11**
Yaohui Jin 11

###### Abstract

When modeling related tasks in computer vision, Multi-Task Learning (MTL) can
outperform Single-Task Learning (STL) due to its ability to capture intrinsic
relatedness among tasks. However, MTL may encounter the insufficient training
problem, i.e., some tasks in MTL may encounter non-optimal situation compared
with STL. A series of studies point out that too much gradient noise would
lead to performance degradation in STL, however, in the MTL scenario, Inter-
Task Gradient Noise (ITGN) is an additional source of gradient noise for each
task, which can also affect the optimization process. In this paper, we point
out ITGN as a key factor leading to the insufficient training problem. We
define the Gradient-to-Noise Ratio (GNR) to measure the relative magnitude of
gradient noise and design the MaxGNR algorithm to alleviate the ITGN
interference of each task by maximizing the GNR of each task. We carefully
evaluate our MaxGNR algorithm on two standard image MTL datasets: NYUv2 and
Cityscapes. The results show that our algorithm outperforms the baselines
under identical experimental conditions.

###### Keywords:

Multi-Task Learning Gradient Noise Weight Strategy.

##  1 Introduction

44footnotetext: These authors contributed equally.

Deep learning [10] has achieved significant success in Multi-Task Learning
(MTL) in the field of computer vision. The multi-task model captures the
intrinsic correlation among tasks and also allows multiple inferences in one
single forward pass, so MTL could achieve the unity of high performance and
efficiency in the optimization process [24]. However, MTL may suffer from the
insufficient training problem [16], which means some tasks in MTL are not
optimal compared with the single-task solution.

Most deep learning models are trained to be optimum by the Stochastic Gradient
Descent (SGD) technique [1]: models are optimized by the loss gradient based
on a mini-batch randomly selected from all data. The gradient obtained by the
SGD algorithm has noise [2, 26], and many studies [14, 13] point out that the
gradient noise magnitude could affect the SGD generalization in Single-Task
Learning (STL). This viewpoint gives us a key insight: in the MTL scenario,
due to the joint learning of multiple tasks, the Inter-Task Gradient Noise
(ITGN) could also interfere with the optimization of specific tasks, which may
be the cause of the insufficient training problem.

Table 1: Percentage change in performance111Compared STL and MTL at the same settings. Details in Section 5.  of MTL relative to STL. NYUv2 and CityScapes are two one-to-many image datasets, where Seg., Dep., and SN. denote Segmentation task, Depth task, and Surface Normal task, respectively. Task | NYUv2 | CityScapes
---|---|---
Seg. | Dep. | SN. | Seg. | Dep.
performance |  8.3% $\uparrow$ |  6.9% $\uparrow$ |  9.7% $\downarrow$ |  1.1% $\uparrow$ |  13.2% $\downarrow$

(a) NYUv2

(b) CityScapes

Figure 1: Gradient Norm distribution on NYUv2 and CityScapes.

Following this insight, we analyze the performance of each task (Tab. 1) as
well as gradient norm distribution (Fig. 1) on two image MTL datasets. We find
that the variance of the gradient norm varies widely across tasks, which
implies that different tasks typically have widely varying gradient noise. At
the same time, small gradient tasks tend to suffer from performance
deterioration in MTL, which shows the relative disadvantage of small gradient
tasks in MTL. In this paper, we demonstrate that Inter-Task Gradient Noise
(ITGN) is one of the key optimization challenges in MTL (Section 3). In fact,
the gradient in MTL optimization contains the noises of all tasks, so ITGN is
an additional source of gradient noise compared to STL. As a result, some
tasks (especially those with small gradients) may suffer from performance
deterioration due to the effect of ITGN because the model has difficulty
converging to the optimal position based on the gradient when the gradient
noise is too high.

To alleviate the ITGN effect in MTL, we design a dynamic weight strategy
called MaxGNR. Specifically, we realize that the magnitude of gradients and
noise may vary greatly from task to task, and to quantitatively describe the
effect of noise on SGD optimization, we define the Gradient-to-Noise Ratio
(GNR) to measure the relative magnitude of gradient noise. Further, we explain
that maximizing the GNR of each task in the MTL scenario is a reasonable
method to mitigate ITGN interference (Section 4.1). Then, we propose the
momentum method to approximate the theoretical GNR to a computable expression
(Section 4.2), therefore, we can dynamically select the weights that maximize
GNR at each iteration (Section 4.3). As a result, the ITGN interference is
reduced through the MaxGNR algorithm, and the insufficient training problem
could be mitigated.

Furthermore, we extensively examine our MaxGNR algorithm on two standard image
MTL datasets: NYUv2 [22] and CityScapes [7], and the experimental results
reveal that the MaxGNR algorithm obtain superior performance than other
baselines under the same premise of other settings. Our contributions are:

  * •

We analyze the effect of gradient noise on MTL and connect the insufficient
training problem with ITGN interference.

  * •

We define the Gradient-to-Noise Ratio (GNR) to measure the relative magnitude
of gradient noise in MTL, and quantitatively describe the ITGN interference by
GNR.

  * •

We propose the MaxGNR algorithm, a dynamic weight strategy to alleviate ITGN
interference and experiments demonstrate the algorithm’s validity.

##  2 Related Work

###  2.1 Multi-Task Learning

There are mainly two aspects in the field of MTL: network architecture
improvement [31, 25] and optimization strategy development, aiming at feature
extraction and balance of performance in MTL, respectively. There have been
lots of studies on multi-task architecture improvements, some works focus on
the encoder structure [19, 20, 9, 17] and others focus on the decoder part
[27, 31, 25]. In this paper, we focus on optimization strategy development. A
major challenge of MTL is how to balance the performance of tasks in joint
learning [24]. To solve this challenge, researchers have proposed many
algorithms, here we divide these algorithms into two categories: _coarse-
grained_ and _fine-grained_. The coarse-grained algorithm is to design
optimization strategy according to the metrics of prediction, Uncertainty [6]
utilizes homeostatic uncertainty to balance each loss; Dynamic Weight
Averaging [17] tries to balance the decline rate of different task losses;
Dynamic Task Prioritization [11] focuses on the balance of key performance
indicators for multiple tasks; GLS [5] uses the geometric mean of task-
specific losses as the target loss. The fine-grained algorithm is based on the
gradient of model parameters, which can more accurately reflect the updating
direction in each iteration. In recent years, such algorithms have been
developed rapidly. GradNorm [4] stimulates the task-specific gradients to be
of similar magnitude; MGDA [21] regards MTL as multi-objective optimization;
PCGrad [28] cut the conflict gradient of different tasks. However, although
gradient noise is an important problem in STL, the effect of gradient noise in
MTL has never been considered.

###  2.2 Stochastic Gradient Descent

Stochastic Gradient Descent (SGD) is one of the standard methods [1] to
optimize machine learning models. It is originally proposed to make up for the
computational bottleneck of gradient descent (GD). Some studies focus on the
generation capability of SGD. [32] reports SGD outperforms GD, [13, 14] deduce
theoretically that small-batch SGD generalizes better than large-batch SGD.
The gradient obtained by the SGD algorithm has un-biased gradient noise [18].
To suppress the optimization oscillation caused by gradient noise, [23]
proposes to introduce the momentum method in the optimization process, and
many optimizers absorb the idea of momentum method, such as Adagrad [8],
Adadelta [29], Adam [15].

##  3 Effect of Gradient Noise

In this section, we analyze the source of gradient noise from the perspective
of SGD (Section 3.1), and analyze the effect of gradient noise on STL (Section
3.2) and MTL (Section 3.3).

###  3.1 Preliminaries of SGD

Formally, the design of the machine learning algorithm is based on data
$D=\left\\{\left(X_{i},Y_{i}\right)\right\\}$, the hypothesis function $F$,
and loss function $l$. Algorithms are designed to search the optimal parameter
$\\{F_{\theta}|\theta\in\Theta\subset\mathbb{R}^{d}\\}$ to get the lowest
expected risk $\mathcal{R}(\theta)$ under the loss function $l$, where
$\theta$ is the parameter of the hypothesis function and $d$ is the dimension
of the parameter. The expected risk $\mathcal{R}(\theta)$ cannot be obtained,
so stochastic Gradient Descent algorithm (SGD) [1] selects a mini-batch
$S=\left\\{\left(X_{i},Y_{i}\right)\right\\}_{\left|S\right|}$ independent and
identically distributed (i.i.d.) from the data $D$, and the expected risk
$\mathcal{R}(\theta)$ can be estimated by the empirical risk
$\widehat{\mathcal{R}}(\theta)$ calculated by mini-batch. In this paper,
$\nabla_{\theta}\mathcal{R}(\theta)$ and
$\nabla_{\theta}\widehat{\mathcal{R}}(\theta)$ are denoted as $g(\theta)$ and
$\widehat{g_{S}}(\theta)$, and $l(F_{\theta}(X_{i}),Y_{i})$ is expressed as
$l_{i}(\theta)$.

The gradient $\widehat{g_{S}}(\theta)$ obtained by SGD is stochastic, so the
gradient can be decomposed into the expected gradient and gradient noise. Many
works [18, 12] assume that the gradient noise belongs to the Gaussian class
due to the classical central limit theorem [18]. In this paper, we follow this
assumption in order to describe the effect of gradient noise on MTL
quantitatively.

###### Assumption 1

An individual data $\left(X_{i},Y_{i}\right)$ is selected independent and
identically distributed (i.i.d.) from data $D$, and its gradient
$\nabla_{\theta}l_{i}(\theta)$ obeys the Gaussian distribution:

$\nabla_{\theta}l_{i}(\theta)\sim\mathcal{N}(g(\theta),C)$ (1)

where $g(\theta)$ is the expected gradient, $C$ is the covariance matrix, and
is approximately constant for $\theta$, which is determined by data $D$ and
loss function $l$.

###  3.2 Gradient Noise in STL

Many studies [14, 13] emphasize the magnitude of gradient noise could effect
the generalization of SGD. In practice, noise magnitude can be easily adjusted
by batch size $\left|S\right|$ [12]. According to Assumption 1,
$\widehat{g_{S}}(\theta)$ can be simplified as:

$\begin{split}\widehat{g_{S}}(\theta)=&\frac{1}{\left|S\right|}\sum_{i=1}^{\left|S\right|}\nabla_{\theta}l_{i}(\theta)=g(\theta)+n_{g(\theta)}\\\
&\text{where}\quad
n_{g(\theta)}\sim\mathcal{N}(0,\frac{C}{\left|S\right|})\end{split}$ (2)

where $n_{g(\theta)}$ is the gradient noise. Because of the Positive Semi-
definite of $C$, the noise magnitude can be measured as:

$\mathbb{E}[\|n_{g(\theta)}\|^{2}]=tr{(C)}/\left|S\right|$ (3)

By adjusting the batch size $\left|S\right|$, we evaluated the impact of noise
magnitude on model performance. The result in Fig. 2(a) showed that the
performance can be maintained only in the appropriate noise magnitude range.
In fact, large noise would interfere with the convergence of the model, while
the model is unable to escape the local optimum with small noise, so the
effect of gradient noise on performance can be expressed as Fig. 2(b).

(a) Perf. vs. Batch Size

(b) Perf. vs. Noise

Figure 2: The effect of gradient noise magnitude on the performance of STL.
The noise magnitude is controlled by the batch size. We conduct experiments on
MNIST, FashionMNIST, and CIFAR10. As the batch size increases, performance
first increases, then stabilizes, and finally decreases on all three datasets.

###  3.3 Inter-Task Gradient Noise

In the multi-task SGD process, multiple tasks have their own loss functions
$\left\\{l^{k}\right\\}$, and Weighted Average Method (WAM) is a mainstream
method to combine multiple loss functions. Specifically, WAM can be expressed
as:

$\begin{split}L=\sum_{k=1}^{n}\omega_{k}l^{k}(\theta)\quad
s.t.\sum_{k=1}^{n}\omega_{k}=1\end{split}$ (4)

According to Assumption 1, the SGD gradient in MTL is expressed as follows:

$\begin{split}\widehat{g_{S}}(\theta)&=\sum_{k=1}^{n}\omega_{k}g^{k}(\theta)+\sum_{k=1}^{n}\omega_{k}n_{g^{k}(\theta)}\\\
\end{split}$ (5)

Based on Eq. 5, ITGN also contributes to task $i$’s optimization in the MTL
scenario, where $\text{ITGN}_{i}$ is as follows:

$\text{ITGN}_{i}=\sum_{k\neq
i}\omega_{k}n_{g^{k}(\theta)}\sim\mathcal{N}(0,\frac{\sum_{k\neq
i}\omega_{k}^{2}C^{k}}{\left|S\right|})$ (6)

From Eq. 3, we can infer that $\left\\{C^{k}\right\\}$ and $\left|S\right|$
control the magnitude of $\left\\{n_{g^{k}(\theta)}\right\\}$, although
multiple tasks share $\left|S\right|$, $\left\\{C^{k}\right\\}$ are task-
specific and determined by the intrinsic characteristics of tasks, so
$\left\\{n_{g^{k}(\theta)}\right\\}$ usually exist huge discrepancies, as
shown in Fig. 1. Therefore, ITGN can cause performance deterioration in MTL
(similar to the performance deterioration due to large noise in STL).

(a) Performance in STL

(b) Performance in MTL

(c) Design weights to maximize GNR

Figure 3: Illustration of MaxGNR algorithm. In the STL scenario, reasonable
settings (model, batch size, learning rate, etc.) can be selected to ensure
that the gradient noise is in the appropriate range, as in Fig. 3(a). In the
MTL scenario, ITGN is also a source of gradient noise that can affect the
optimization of specific tasks. Therefore, ITGN may cause task-specific
performance deterioration due to the large differences in ITGN, as is
illustrated in Fig. 3(b). Our MaxGNR algorithm attempts to alleviate the ITGN
interference as Fig. 3(c). According to Eq. 12, We can design appropriate
weights to maximize GNR for the purpose of mitigating ITGN interference.

Input: data $\\{X_{i},Y_{i}^{1},\dots,Y_{i}^{n}\\}_{|D|}$, learning rate
$\mu$.

Output: optimal network parameters $\theta$.

Initialize network parameters $\theta_{0}$ ;

for  _t = 0 to T_ do

select mini-batch data $S=\\{X_{i},Y_{i}^{1},\dots,Y_{i}^{n}\\}_{|S|}$;

compute each empirical risk $\\{\widehat{\mathcal{R}^{k}}(\theta_{t})\\}$;

compute the gradient of empirical risk
$\\{\nabla\widehat{\mathcal{R}^{k}}(\theta_{t})\\}$;

compute the momentum gradient $\\{m_{t}^{k}\\}$ and noise $\\{n_{t}^{k}\\}$ ;

select $\\{\omega_{t}^{k}\\}$ by
$\operatorname*{arg\,max}_{\\{\omega_{k}\\}}\left\\{\min\\{\text{GNR}_{k}(m_{t}^{k},\\{n_{t}^{k}\\})\\}\right\\}$
;

update network parameters
$\theta_{t+1}=\theta_{t}-\mu\nabla_{\theta_{t}}\sum_{k=1}^{n}\omega_{t}^{k}\widehat{\mathcal{R}^{k}}(\theta_{t})$
;

end for

Algorithm 1 MaxGNR algorithm

##  4 Method

We describe the necessity of controlling the noise magnitude range to maintain
excellent model performance in Section 3.2 as Fig. 3(a) and explain that ITGN
could also interfere with the specific task’s optimization in MTL in Section
3.3. In summary, the range of ITGN may vary greatly, so it can be inferred
that ITGN is likely to cause performance deterioration on some tasks as Fig.
3(b). In this section, we propose the MaxGNR algorithm, a dynamic weight
strategy to minimize the ITGN interference in MTL as Fig. 3(c).

Gradient-to-Noise Ratio (GNR) is the core concept of this paper, which is
defined to measure the relative magnitude of gradient noise (Section 4.1), and
it can be found that ITGN would reduce GNR compared to STL because ITGN is an
additional gradient noise source. So it is a viable method to mitigate ITGN
interference by maximizing GNR. To solve the dilemma that GNR is not
computable, we propose to estimate $\\{g^{k}(\theta)\\}$ and
$\left\\{n_{g^{k}(\theta)}\right\\}$ using the momentum method, and thus
estimate a computable GNR (Section 4.2). Eventually, we can dynamically select
the weights that maximize GNR in each iteration, thus achieving the purpose of
alleviating ITGN interference (Section 4.3). The whole process of the MaxGNR
algorithm is shown in Alg. 1.

###  4.1 Gradient-to-Noise Ratio

In order to measure the noise interference on stochastic gradient
optimization, it is necessary to define an appropriate metric to measure the
relative magnitude of gradient noise to the gradient.

###### Definition 1

In the process of SGD, for a specific task, the expected gradient is
$g(\theta)$, the random variable that could make $\widehat{g_{S}}(\theta)$
deviate from $g(\theta)$ is the gradient noise $n_{g(\theta)}$, we define
Gradient-to-Noise Ratio (GNR) as:

$\text{GNR}=\frac{\|g(\theta)\|^{2}}{\mathbb{E}[\|n_{g(\theta)}\|^{2}]}$ (7)

GNR reflects the relative magnitude of gradient noise to the gradient. The
homogeneity ensures that GNR is independent of weight in STL, and according to
Eq. 3, $\mathbb{E}[\|n_{g(\theta)}\|^{2}]$ is an approximate constant and easy
to represent.

There is only one source of gradient noise in STL. According to definition 7,
$\text{GNR}_{S}$ is represented as:

$\text{GNR}_{S}=\frac{\|g(\theta)\|^{2}}{\mathbb{E}[\|n_{g(\theta)}\|^{2}]}=\frac{\|g(\theta)\|^{2}}{tr{(C)}/\left|S\right|}$
(8)

In the MTL scenario, due to the ITGN interference, task $k$’s GNR is
represented as:

$\text{GNR}_{M}^{k}=\frac{\|\omega_{k}g^{k}(\theta)\|^{2}}{\mathbb{E}[\|\omega_{k}n_{g^{k}(\theta)}+\text{ITGN}_{k}\|^{2}]}$
(9)

According to Eq. 3 and Eq. 5, $\mathbb{E}[\|\text{ITGN}_{k}\|^{2}]$ could be
represented by $\\{C^{k}\\}$ and $|S|$, and due to the linear property of
matrix’s trace:

$tr(m\cdot A+n\cdot B)=m\cdot tr(A)+n\cdot tr(B)$ (10)

$\text{GNR}_{M}^{k}$ is eventually simplified as:

$\begin{split}\text{GNR}_{M}^{k}=\frac{\|\omega_{k}g^{k}(\theta)\|^{2}}{\sum_{k=1}^{n}\omega_{k}^{2}tr(C^{k})/\left|S\right|}\leq\text{GNR}_{S}^{k}\end{split}$
(11)

In general, hyperparameters with superior performance in STL would be selected
for MTL, so we assume that $\text{GNR}_{S}^{k}$ can be controlled as an
appropriate GNR. However, ITGN interference reduces the GNR of each task, if
$\text{ITGN}_{k}\gg n_{g^{k}(\theta)}$, $\text{GNR}_{M}^{k}$ could be much
smaller than $\text{GNR}_{S}^{k}$, which is likely to cause performance
deterioration of task $k$. Therefore, selecting weights $\\{\omega_{k}\\}$ to
maximize $\text{GNR}_{M}^{k}$ is a feasible method to alleviate ITGN
interference for task $k$. In the MTL scenario, to ensure that all tasks could
alleviate ITGN interference as much as possible, we should maximize
$\min\\{\text{GNR}_{M}^{k}\\}$ as:

$\\{\omega_{k}\\}=\operatorname*{arg\,max}_{\\{\omega_{k}\\}}\left\\{\frac{\min\left\\{\|\omega_{k}g^{k}(\theta)\|^{2}\right\\}}{\sum_{k=1}^{n}\omega_{k}^{2}tr(C^{k})/\left|S\right|}\right\\}$
(12)

###  4.2 Estimation of Expected Gradient

However, GNR is a theoretical concept and it is not realistic to calculate
$\left\\{g^{k}(\theta_{t})\right\\}$ and $\left\\{C^{k}\right\\}$, therefore,
we should design a method to estimate them. During the training process, we
can only get the empirical gradient $\\{\widehat{g_{S}}^{k}(\theta_{t})\\}$.
In this paper, we propose that the momentum method can be used to estimate the
expected gradient and gradient noise.

Momentum Method The momentum method is widely applied in various optimizers.
AdaGrad, AdaDelta, Adam, and so on adopt the momentum method as a technique to
stabilize gradient. Momentum is expressed as $m$, and we set
$m_{0}=\widehat{g_{S}}(\theta_{0})$. The basic process of the momentum method
is as follows:

$\begin{split}&m_{t}=\gamma\cdot
m_{t-1}+(1-\gamma)\cdot\widehat{g_{S}}(\theta_{t})\\\
&\theta_{t}=\theta_{t-1}-\mu\cdot m_{t}\end{split}$ (13)

where $\gamma$ is the decay rate, $\mu$ is the learning rate, both are
hyperparameters. It is an option to estimate the expected gradient by momentum
gradient, because the momentum gradient could reduce noise and stabilize
gradient. According to Eq. 2 and Eq. 13, the momentum gradient can be derived
as:

$\begin{split}m_{t}^{k}&\approx(1-\gamma)\sum_{i=1}^{t}\gamma^{t-i}\cdot
g^{k}(\theta_{i})+n_{m_{t}^{k}}\\\ &\text{where}\quad
n_{m_{t}^{k}}\sim\sqrt{\frac{1-\gamma}{1+\gamma}}\mathcal{N}(0,\frac{C}{\left|S\right|})\end{split}$
(14)

By comparing Eq. 2 and Eq. 14, it can be found that the momentum method
produces a contraction coefficient of $\sqrt{(1-\gamma)/(1+\gamma)}$ on noise
magnitude, the larger $\gamma$ is, the smaller the estimated noise magnitude
is. At the same time, the cost is that the momentum gradient $m_{t}^{k}$ is no
longer the un-biased estimation of $g^{k}(\theta_{t})$, the larger $\gamma$
is, the greater the estimation error is. So $\gamma$ becomes a trade-off
factor of the expected gradient estimation, if the estimation error of
$g^{k}(\theta_{t})$ is too large, the momentum method is also invalid.
Fortunately, parameter optimization is a long process, and the change of
parameters in each iteration is small compared to the parameters themselves.
Based on this fact, we make the following assumption:

###### Assumption 2

the expected gradient changes slowly and the expected gradients in adjacent
iterations are similar:

$g^{k}(\theta_{n})\approx g^{k}(\theta_{n+\Delta n}),\ \text{when $\Delta n$
is small}$ (15)

On the premise of Assumption 2, the estimation error of $g^{k}(\theta_{t})$ in
Eq. 14 can be reduced to a large extent, so it is reasonable to estimate
$g^{k}(\theta_{t})$ with $m^{k}(\theta_{t})$.

###  4.3 Weight Selection

According to the estimation method proposed by Eq. 14, the empirical gradient
of each task can be decomposed into momentum gradient $m^{k}(\theta)$ and
gradient noise $n_{g^{k}}(\theta)$. Therefore, we can estimate
$n_{g^{k}}(\theta)$:

$\begin{split}n_{g^{k}}(\theta)=\widehat{g_{S}}^{k}(\theta)-m^{k}(\theta)\\\
\end{split}$ (16)

To calculate the GNR of each task, we can replace $g^{k}(\theta)$ with
$m^{k}(\theta)$ according to Eq. 14 and Assumption 2, and although
$\mathbb{E}[\|n_{g^{k}(\theta)}\|^{2}]$ is approximately determined, each
gradient noise is a random variable, so it is reasonable to use
$\|\sum_{k=1}^{n}\omega_{k}n_{g^{k}(\theta)}\|^{2}$ to replace
$\sum_{k=1}^{n}\omega_{k}^{2}tr(C^{k})/\left|S\right|$. Therefore, the MaxGNR
algorithm can ultimately expressed as:

$\\{\omega_{k}\\}=\operatorname*{arg\,max}_{\\{\omega_{k}\\}}\left\\{\frac{\min\left\\{\|\omega_{k}m^{k}(\theta)\|^{2}\right\\}}{\|\sum_{k=1}^{n}\omega_{k}n_{g^{k}(\theta)}\|^{2}}\right\\}$
(17)

Because of the non-linearity of Eq. 17, a general analytical solution cannot
be found. We use the steepest descent method [3] to obtain the optimal
$\\{\omega_{k}\\}$ at each iteration. Compared with the number of parameters
in the neural network, the number of parameters in the steepest descent method
is the number of tasks, so the computation cost can be ignored. At the same
time, the computation cost of the steepest descent method is not sensitive to
the number of tasks, which is different from grid search.

##  5 Experiments

Table 2: Experiment results of different algorithms for NYUv2, and we split the table into coarse-grained and fine-grained algorithms. The bold represents the top2 scores. Algorithm | Segmentation | Depth | Surface Normal
---|---|---|---
mIoU $\uparrow$ | Pix Acc $\uparrow$ | Abs Err $\downarrow$ | Rel Err $\downarrow$ | Mean $\downarrow$ | Median $\downarrow$
single task | 26.15 | 53.27 | 0.6655 | 27.89 | 30.57 | 25.12
equal weights | 28.32 | 55.60 | 0.6196 | 27.61 | 31.80 | 26.82
_coarse-grained_ |  |  |  |  |  |
DWA | 28.43 | 56.08 | 0.6075 | 25.08 | 31.98 | 26.89
Uncertainty | 28.06 | 54.46 | 0.6059 | 25.69 | 31.49 | 26.32
_fine-grained_ |  |  |  |  |  |
GradNorm | 28.92 | 56.26 | 0.6057 | 24.06 | 31.66 | 26.49
MGDA | 22.38 | 49.59 | 0.7414 | 27.51 | 30.50 | 23.85
PCGrad | 28.72 | 56.17 | 0.6131 | 27.33 | 32.19 | 27.00
MaxGNR ($\gamma=0.5$) | 29.76 | 57.64 | 0.5943 | 24.35 | 29.89 | 24.38
MaxGNR ($\gamma=0.9$) | 29.48 | 56.81 | 0.5904 | 24.91 | 30.43 | 24.85

###  5.1 Experimental Settings

####  5.1.1 Datasets

We focused on one-to-many predictions datasets
$\\{X_{i},Y_{i}^{1},\dots,Y_{i}^{n}\\}_{|D|}$ [30] in this paper. We evaluated
the proposed MaxGNR algorithm on two image datasets: NYUv2 [22] and CityScapes
[7]. NYUv2 is a challenging indoor scene dataset in various room types
(bathrooms, living rooms, studies, etc.), and this dataset has three tasks:
13-class semantic segmentation, depth estimation, and surface normal
prediction. Although NYUv2 is a relatively small dataset (795 training, 654
test images), it contains both regression and classification tasks, making it
a good candidate for testing the robustness of different types of loss
functions. CityScapes is a high-resolution street-view images dataset, we
chose two sub-tasks for our experiment: semantic segmentation and depth
estimation. Compared to NYUv2, the street-view image contained in CityScapes
is relatively simpler, because the viewpoints and lighting conditions are
relatively fixed, and the appearance of each object class changes little in
shape.

####  5.1.2 Implementation Details

We implemented our experiments based on the network architecture in [17]. The
network architecture had an encoder-decoder structure that was homogeneous
across all tasks, where the encoder extracted the representation and the
decoder matched individual tasks. For the NYUv2 and CityScapes datasets, we
set the batch size to 2 and 128, respectively, and the learning rate of the
Adam optimizer to 1e-4. After training each model for 100/200 epochs, we
selected the best model on the training set for testing. In our experiment, we
set the learning rate of the steepest descent method to 1e-3, and
$\\{\omega_{k}\\}$ updated 100 steps in each iteration.

####  5.1.3 Evaluation Metrics

To compare the baselines and our method, we chose two metrics for each task.
Following the settings in [6, 17, 28], we chose mIoU and Pixel Accuracy for
Segmentation task, Absolute and Relative Error for Depth task, and Mean and
Median Angle Distance for Surface Normal task.

###  5.2 Baselines

In addition to equal weights and single-task models, the baselines can be
divided into two types: coarse-grained algorithms and fine-grained algorithms.
Coarse-grained algorithms included DWA [17] and Uncertainty [6], fine-grained
algorithms included GradNorm [4], MGDA [21] and PCGrad [28].

###  5.3 Main Results

Table 3: Experiment results of different algorithms for CityScapes.  Algorithm | Segmentation | Depth
---|---|---
mIoU $\uparrow$ | Pix Acc $\uparrow$ $\uparrow$ | Abs Err $\downarrow$ | Rel Err $\downarrow$
single task | 66.91 | 90.03 | 0.0153 | 35.85
equal weights | 67.63 | 90.09 | 0.0163 | 57.13
GradNorm | 67.20 | 90.76 | 0.0169 | 57.21
MGDA | 66.83 | 90.88 | 0.0170 | 47.14
PGGrad | 67.75 | 91.01 | 0.0164 | 54.95
MaxGNR ($\gamma=0.5$) | 68.01 | 91.07 | 0.0162 | 44.11
MaxGNR ($\gamma=0.7$) | 67.33 | 90.76 | 0.0160 | 42.48

Results on NYUv2 was shown in Tab. 2. Our MaxGNR algorithm outperformed the
baselines in almost all metrics. When specific tasks were analyzed, most
approaches outperformed the single-task model in the Depth task, and most
baselines performed marginally better than the single-task model in the
Segmentation task, while, in the Surface Normal task, almost all baselines
suffered from performance deterioration. It was important to note that MGDA,
which outperformed the single-task model in the Surface Normal task, cannot
balance the optimization performance on all tasks, resulting in MGDA’s bad
performance in the other two tasks. Our MaxGNR algorithm not only outperformed
the single-task model and baselines in the Segmentation and Depth tasks, but
it also performed well in the Surface Normal task and overcame the
insufficient training problem.

Results on CityScapes was shown in Tab. 3. In CityScapes, the performance of
the single-task model was similar to all MTL algorithms on the Segmentation
task (about 91%), while there was a serious performance deterioration in the
Depth task. We analyzed the performance of baselines and our algorithm in two
tasks. In the Segmentation task, our algorithm was not significantly better
than other baselines, but in the Depth task, our algorithm improved the
performance of the model and greatly narrowed the gap with the single-task
model.

###  5.4 Dynamic Change of Weights

(a) NYUv2

(b) CityScapes

Figure 4: Dynamic change of weights and the training process.

The dynamic weight strategy can assign appropriate weights to each task at
different training stages. We compared the weights of the different stages
obtained by MaxGNR and MGDA in Fig. 4. We found that the weight trends
obtained by the two algorithms were similar, but MGDA’s weights were more
extreme compared to MaxGNR’s weights. Specifically, in NYUv2, the weight
assignment of MGDA led the model to focus only on the Surface Normal task,
while MaxGNR’s assignment was intuitively more balanced. The experimental
results (MGDA in Tab. 2) showed that extreme weight assignments may lead to
insufficient training problems. The situation was similar in CityScapes. We
believed that the weight assignment difference came from the object being
balanced: MGDA attempted to balance each task’s gradient, while MaxGNR
attempted to balance each task’s GNR.

##  6 Discussion

###  6.1 Gradient Noise and Performance

(a) NYUv2

(b) CityScapes

Figure 5: Gradient Norm Distribution and GNR of each task in NYUv2 and
CityScapes.

We fixed the best models under equal weights and MaxGNR settings and counted
the distribution of the gradient norm for all data and calculated the current
GNR of the models based on the gradient norm distribution, as shown in Fig. 5.

We found that in the equal weights model, the gradients and noises of each
task differed significantly, and our algorithm clearly balanced the gradients
and noises of each task (Fig. 5 Left). Compared with $\text{GNR}_{S}$, the
$\text{GNR}_{M}$ of Surface Normal task in NYUv2 and Depth task in CityScapes
decreased most significantly in the equal weights model, and the performances
of these tasks also exhibited a huge decrease. Our MaxGNR algorithm balanced
each $\text{GNR}_{M}$ (Fig. 5 Right), by slightly reducing the
$\text{GNR}_{M}$ of some tasks (usually harmless), the tasks suffering from
performance deterioration would mitigate the interference of ITGN, and gained
performance improvements. The experimental results also showed that the model
in the MaxGNR setting had significant performance improvements in the Surface
Normal task in NYUv2 and the Depth task in CityScapes. At the same time, the
performance improvement of some tasks can help the model to get better
representation, which can improve the performance of the model in general.

###  6.2 The Paradox of Weight Design

In the field of weight design of MTL, researchers frequently constructed
optimization algorithms depending on the difficulties of the tasks, however,
the starting points of the design sometimes conflicted with each other. For
example, Dynamic Task Prioritization (DTP) [11] allocated “difficult” tasks
higher task-specific weights, while Uncertainty [6] assigned “easy” tasks
higher task-specific weights. [24] gave a qualitative explanation for this
conflict paradox: Uncertainty seemed to be more suitable for noisy labeled
data, while DTP was more suitable for clean ground-truth annotations. The
paradox can be described more clearly when we viewed this phenomenon from the
perspective of MaxGNR algorithm. According to the Eq. 12, there are two
factors that influence task’s difficulty: small expected gradient $g(\theta)$
and large variance $\frac{C}{\left|S\right|}$.

If the model’s upper limit is restricted by the small expected gradients of
some tasks, the weights of these tasks should be adjusted to expand the
expected gradients, similar to DTP. The noise part of Eq. 12 can be ignored
as:

$\\{\omega_{k}\\}=\operatorname*{arg\,max}_{\\{\omega_{k}\\}}\left\\{\min\left\\{\|\omega_{k}g^{k}(\theta)\|^{2}\right\\}\right\\}$
(18)

On the contrary, if the model’s upper limit is restricted by the large
variances of specific tasks, the weights of these tasks should be reduced,
similar to Uncertainty, to reduce the impact on other small gradient tasks. So
the gradient part of Eq. 12 can be ignored as:

$\\{\omega_{k}\\}=\operatorname*{arg\,max}_{\\{\omega_{k}\\}}\left\\{\frac{1}{\sum_{k=1}^{n}\omega_{k}^{2}tr(C^{k})/\left|S\right|}\right\\}$
(19)

In conclusion, the MaxGNR algorithm not only considers the magnitude of the
gradient but also considers the noise interference, which to some extent
unifies these two kinds of algorithms with distinct starting points.

##  7 Conclusion

In this work, we attributed the insufficient training problem in MTL to the
ITGN interference, and we proposed MaxGNR algorithm, a novel dynamic weight
strategy to alleviate this interference. Experiments verified the
effectiveness of our algorithm. Looking ahead, gradient noise in MTL is a new
field, and we hope to explore the influence of gradient noise on more complex
tasks. Besides, how to choose appropriate tasks for joint learning is an open
question, and the GNR framework may be a possible research direction.

####  7.0.1 Acknowledgements

This work was supported by the Shanghai Municipal Science and Technology Major
Project (2021SHZDZX0102), and the Shanghai Science and Technology Innovation
Action Plan (20511102600).

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# Tilt-to-length coupling in LISA Pathfinder: analytical modelling

Marie-Sophie Hartig<EMAIL_ADDRESS>Max Planck Institute for
Gravitational Physics (Albert Einstein Institute)  Institute for Gravitational
Physics of the Leibniz Universität Hannover  Gudrun Wanner
<EMAIL_ADDRESS>Institute for Gravitational Physics of the Leibniz
Universität Hannover  Max Planck Institute for Gravitational Physics (Albert
Einstein Institute)

###### Abstract

Tilt-to-length coupling was the limiting noise source in LISA Pathfinder
between 20 and 200 mHz before subtraction in post-processing. To prevent the
adding of sensing noise to the data by the subtraction process, the success of
this strategy depended on a previous direct noise reduction by test mass
alignment. The exact dependency of the level of tilt-to-length coupling on the
set-points of LISA Pathfinder’s test masses was not understood until the end
of the mission. Here, we present, for the first time, an analytical tilt-to-
length coupling model that describes the coupling noise changes due to the
realignments. We report on the different mechanisms, namely the lever arm and
piston effect as well as the coupling due to transmissive components, and how
they contribute to the full coupling. Further, we show that a pure geometric
model would not have been sufficient to describe the coupling in LISA
Pathfinder. Therefore, we model also the non-geometric tilt-to-length noise
contributions. For the resulting coupling coefficients of the full model, we
compute the expected error bars based on the known individual error sources.
Also, we validated the analytical model against numerical simulations. A
detailed study and thorough understanding of this noise are the basis for a
successful analysis of the LISA Pathfinder data with respect to tilt-to-length
coupling.

##  I Introduction

LISA Pathfinder (LPF) [1, 2, 3] was the technology demonstrator mission for
the Laser Interferometer Space Antenna (LISA) [4, 5, 6]. Launched in December
2015, it was in operation mode from March 2016 until its shutdown in July
2017. The main scientific measurement of LPF concerned the relative
acceleration of its two hosted cubic test masses in free fall. These were
situated in electrode housings at two opposite sites of an optical bench. The
interferometer onboard LPF measuring the distance changes and likewise the
relative accelerations of the test masses, was the x12-interferometer, see
Fig. 1. Since one of the interfering beams (red in this figure) was reflected
at each of the test masses, any angular or lateral jitter of either the
spacecraft (S/C) or the test masses coupled into the scientific measurement.
We refer to this noise as tilt-to-length (TTL) coupling [7, 8]. Without
suppression, it was the most significant noise source between 20 and 200 mHz
[2, 3, 9].

Two strategies were applied to remove the TTL coupling from the readout and to
consolidate the original noise model: realignments of the test masses for TTL
noise suppression and the subtraction of the noise by a dedicated fit model
[9, 10].

The realignment method alone was not sufficient since the applied realignments
relying on the analytical TTL models available during the mission did reduce
but not fully mitigate the coupling noise. On the other hand, the subtraction
method relied on an a-priori low magnitude of TTL coupling since the
subtraction added sensing noise at higher frequencies otherwise [9, 11].

Combined, both TTL mitigation strategies successfully removed the TTL noise.
However, the underlying mechanisms leading to the observed cross-coupling were
insufficiently modelled. Also, the model fitted for TTL subtraction did not
resolve the physical dependency of the jitter coupling on the alignment
parameters. Additionally, the fit had to be performed repeatedly to account
for environmental changes and long-term drifts. On this regard, we continued
investigating the TTL coupling problem after the end of the LPF mission and
evaluated an analytical model that now describes the TTL noise in LPF [12]. It
will be presented in the following sections.

Beyond LPF, TTL coupling is a major noise source in space-based
interferometers [13, 14, 15, 16]. Particularly in the case of LISA, this noise
source is widely discussed. To minimise the TTL noise in the LISA measurement
band, it is planned to be suppressed by design, alignment and subtraction in
post-processing. In that respect, a successful modelling of the TTL noise in
LPF is of particular interest. By this, we do not only validate the planned
suppression strategies but also boost our confidence in their effective
application.

Figure 1: The x12-interferometer onboard LPF. The two beams interfering in
this interferometer are shown in red (measurement beam) and blue (reference
beam). The measurement beam goes out of the optical bench setup passing a
window (WIN) twice and is reflected by the freely falling test masses inside
the electrode housings not shown here. This figure presents the principle
setup of the LPF optical bench. Please note that the component sizes might
slightly deviate from scale in this image.

Within this paper, we evaluate an updated TTL coupling model for LPF. We start
with explaining our modelling strategy and define it from the general TTL
coupling derivations in [7, 8] in Sec. II. The resulting analytical model, is
presented in Sec. III. There, we introduce the different contributing TTL
coupling mechanisms, which are the lever arm coupling, the piston coupling and
the coupling effects due to transmissive components. We also distinguish
between geometric an non-geometric TTL coupling. The analytical model is being
verified against numerical simulations in Sec. IV. Furthermore, we
characterise the stability of the coupling coefficients to uncertainties of
the input parameters in (Sec. V). In Sec. VI, we transform the presented
analytical description of the interferometric length signal changes into a
model describing the TTL effects on the $\Delta g$ measurements in LPF. This
transformation sets the base for a TTL coupling analysis on the LPF data [12].
In Sec. VII, we discuss our findings and highlight the differences between our
updated model and the TTL noise equations used at the time of the mission. All
results will be summarised in the final section (Sec. VIII).

##  II General Assumptions and Modelling Strategy

For the analytical derivation of the TTL coupling in LPF, we have interpreted
the coupling effects described in [7, 8] for the LPF setup. The distance
changes of the two test masses were there captured by the x12-interferometer,
see Fig. 1. It processes the differential phase changes of the interfering
stable reference beam (blue) and the measurement beam (red) after its
reflection at both test masses.

Several aspects of the TTL mechanisms on the LPF optical setup and the
computation scheme presented here have already been addressed in [7, 8]. In
LPF, the TTL coupling originated primarily from the lateral and angular jitter
of the S/C itself. The individual jitter of the two hosted test masses played
a minor role. For the analytic description of the effect, the S/C jitter can
be interpreted as a simultaneous jitter of the test masses relative the S/C’s
centre of mass and therefore refers to the setup with jittering test masses
introduced in [7, 8]. The occurring TTL coupling mechanisms split into the
lever arm, the piston and the transmissive components’ effect.

Furthermore, the two interfering beams in the x12-interferometer are assumed
to be perfect Gaussian beams. This condition holds since both beams are local
and little clipping and only weak ghost beams can be expected. Therefore, the
residual imperfection of the beams’ Gaussian shapes is assumed to be
negligible. Thus, the computation of the non-geometric part of the TTL
coupling in LISA Pathfinder follows the method explained in [17, 8].

For our evaluation, we chose to analyse the signal changes measured by the
A-diode of this interferometer (PD12A). The signals of the A- and B-diode
differ due to the number of transmissions of the measurement beam, the
slightly longer lever arm in the case of the B-diode and small differences of
the beam alignments at the respective detector surface. However, when
considering only S/C jitter, these differences are small since the alignment
angle of the beam after its reflection at the second test mass stays
unchanged. Thus, the only difference in the final signal originates from the
small tilt-dependent lateral shift of the beam path multiplying with the
nominal beam alignment angles at the detectors. Due to the small differences
of the coupling in both detectors, we restrict our analysis to the A-diode.

Despite the agreement of several preconditions of the LPF setup and the
analysis in [7, 8], the analytical TTL coupling model for LPF cannot simply be
evolved by inserting the system parameters into the presented equations. The
LPF setup is more complicated in the sense that it hosts two test masses. As
the measurement beam reflects first at the first test mass (TM1), any jitter
of this test mass will generate a beam walk on the second test mass (TM2) and
also changes the angle of incidence there. Consequently, the TTL effects due
to the jitters of both test masses are not independent of each other. The
analysis obtains additional complexity from both test masses being nominally
tilted (about their centre of mass). Therefore, the expected geometric path
length changes of the measurement beam and its beam walk on the detector was
computed independently from [7, 8] in Mathematica [18]. The computation
assumes the lever arm and piston effect of both test masses and the
transmission of the rotated measurement beam through the windows in between
the test masses and the optical bench.

The resulting equations are very complex due to the number of parameters
included. Thus, we will present in the following sections only the series
expression with inserted parameters (see App. A). Thereby, we use the
coordinate system shown in Fig. 2. The full equations are shown in [18].

Figure 2: The global coordinate system that we use within this paper. Its
centre is defined as the geometrical centre of the (non-shifted) optical
bench. The $xy$-plane is plane-parallel to the (non-rotated) optical bench
surface and the $z$-axis points upwards. Accordingly, all yaw angles
($\varphi$) describe in-plane rotations, while the pitch angles ($\eta$)
describe out-of-plane rotations. The coordinate system is defined with respect
to the nominal location and orientation of the optical bench, but not fixed to
it: any S/C jitter would change the optical bench alignment but not the
coordinate system.

##  III Modelling of the Longitudinal Path Length Signal Changes in LISA
Pathfinder

We present in this section the TTL coupling induced changes of the
longitudinal path length signal (LPS) signal [8, 17] at the A-diode of the
x12-interferometer (Fig. 1). This S/C jitter noise splits up into the lever
arm and the piston effect [7, 8]. Additionally, the windows between the
optical bench and the vacuum housing of the test masses add coupling [7, 8],
which is, in the case of LPF, smaller than the lever arm and the piston
effect.

###  III.1 The Lever Arm Effect

The lever arm effect describes the signal changes due to the tilt of the beam
axis after the reflection at either of the test masses [7]. This tilt scales
with the path length of the beam in between the two test masses and between
the second test mass and the detector. Furthermore, this tilt induces a beam
walk on the photodiode surface which adds non-geometric TTL coupling [8]. We
expand the resulting analytical equation in the tilt angles, insert the
parameters provided in App. A and discard negligible terms. This yields

$\displaystyle\begin{split}\text{LPS}_{\text{lever}}^{\text{LPF}}&\approx
45\cdot 10^{-6}\frac{\text{m}}{\text{rad}}\,\varphi_{\text{SC}}-10\cdot
10^{-6}\frac{\text{m}}{\text{rad}}\,\eta_{\text{SC}}\\\
&+0.747\frac{\text{m}}{\text{rad}^{2}}\,\hat{\varphi}_{1}\,\varphi_{\text{SC}}+0.711\frac{\text{m}}{\text{rad}^{2}}\,\hat{\varphi}_{2}\,\varphi_{\text{SC}}\\\
&+0.746\frac{\text{m}}{\text{rad}^{2}}\,\hat{\eta}_{1}\,\eta_{\text{SC}}+0.702\frac{\text{m}}{\text{rad}^{2}}\,\hat{\eta}_{2}\,\eta_{\text{SC}}\\\
&-0.729\frac{\text{m}}{\text{rad}^{2}}\,\varphi_{\text{SC}}^{2}-0.724\frac{\text{m}}{\text{rad}^{2}}\,\eta_{\text{SC}}^{2}\,.\end{split}$
(1)

The lever arm effect in LPF induces linear and quadratic TTL coupling in the
S/C angles $\varphi_{\text{SC}}$ (in-plane / yaw jitter) and
$\eta_{\text{SC}}$ (out-of-plane / pitch jitter). The coupling depends on the
absolute test mass alignment angles: $\hat{\varphi}_{i}$, $\hat{\eta}_{i}$,
where $i\in\\{1,2\\}$ indicates the considered test mass. See Fig. 2 for the
definition of the coordinate system and the orientation of the angles. It is
$\hat{\varphi}_{i}=0$, if the reflecting test mass surface is parallel to a
plane including the $y$-axis. The same applies to $\hat{\eta}_{i}=0$ and the
$z$-axis. All cross-plane effects have shown to be negligible in the lever arm
case.

###  III.2 The Piston Effect

The piston effect covers the LPS signal changes due to the translation of the
measurement beam’s reflection points at the test masses by the motion of their
reflective surfaces. For angular S/C jitter, the surface motion into the beam
path originates from the offset of the centre of rotation from the beam’s
reflection point [7, 8]. Furthermore, the lateral jitter of the S/C shifts the
surface into the beam paths if they are nominally tilted with respect to the
optical bench. It splits into the in-plane jitter $y_{\text{SC}}$ (i.e.
parallel to the $y$-axis in Fig. 2) and the (vertical) out-of-plane jitter
$z_{\text{SC}}$ (i.e. parallel to the $z$-axis in Fig. 2). We found

$\displaystyle\begin{split}\text{LPS}_{\text{piston}}^{\text{LPF}}&\approx
19\cdot 10^{-6}\frac{\text{m}}{\text{rad}}\,\varphi_{\text{SC}}+14\cdot
10^{-6}\frac{\text{m}}{\text{rad}}\,\eta_{\text{SC}}\\\
&+1.994\frac{1}{\text{rad}}\,\left[(-\hat{\varphi}_{1}+\hat{\varphi}_{2})\,y_{\text{SC}}+(\hat{\eta}_{1}-\hat{\eta}_{2})\,z_{\text{SC}}\right]\\\
&-0.318\frac{\text{m}}{\text{rad}^{2}}\,\hat{\varphi}_{1}\,\varphi_{\text{SC}}-0.340\frac{\text{m}}{\text{rad}^{2}}\,\hat{\varphi}_{2}\,\varphi_{\text{SC}}\\\
&-0.319\frac{\text{m}}{\text{rad}^{2}}\,\hat{\eta}_{1}\,\eta_{\text{SC}}-0.339\frac{\text{m}}{\text{rad}^{2}}\,\hat{\eta}_{2}\,\eta_{\text{SC}}\\\
&+0.010\frac{\text{m}}{\text{rad}^{2}}\,(-\hat{\varphi}_{1}+\hat{\varphi}_{2})\,\eta_{\text{SC}}\\\
&+0.329\frac{\text{m}}{\text{rad}^{2}}\,\varphi_{\text{SC}}^{2}+0.329\frac{\text{m}}{\text{rad}^{2}}\,\eta_{\text{SC}}^{2}\,.\end{split}$
(2)

Here, we see not only linear and quadratic TTL coupling terms but also a small
cross-plane term depending on the yaw alignment of the test mass and the pitch
jitter of the S/C. It depends on the vertical offset of the S/C’s centre of
mass from the beam’s reflection points and their in-plane incoming angles at
the test masses. There is no analogue cross-plane term for yaw jitter since
the multiplicative lateral offset is one order of magnitude and the out-of-
plane incoming angles are three orders of magnitude smaller (Tabs. 4, 5). For
details, compare with [7, Eq. (33)] or [18]. Like the lever arm effect, the
piston effect depends on the nominal angular alignment of the test masses.
Their lateral alignment, however, does not couple with the S/C jitter [11].

###  III.3 Refraction at the Windows

Along their path in between the test masses and between the second test mass
and the detector, the (tilted) beam passed the windows of the vacuum chambers
of the electrode housings (see Fig. 1). The laser beam refracts when entering
and exiting the window. The propagation distance and direction inside the
material depends on the beam’s angle of incidence on the window. Since this
incidence angle varies if the S/C jitters, the latter yielded additional TTL
coupling by the window. It is described by

$\displaystyle\begin{split}\text{LPS}_{\text{window}}^{\text{LPF}}&\approx-0.018\frac{\text{m}}{\text{rad}^{2}}\,\left[\hat{\varphi}_{2}\,\varphi_{\text{SC}}+\hat{\eta}_{2}\,\eta_{\text{SC}}\right]\\\
&+0.009\frac{\text{m}}{\text{rad}^{2}}\,\varphi_{\text{SC}}^{2}+0.009\frac{\text{m}}{\text{rad}^{2}}\,\eta_{\text{SC}}^{2}\,.\end{split}$
(3)

By comparing Eq. (3) with Eqs. (1) and (2), we see that the TTL contribution
of the windows is smaller than the lever arm and the piston effect. We compare
the three presented TTL coupling contributors in Sec. III.6.

###  III.4 The Full TTL Coupling in LPF

As stated above, the TTL coupling in LPF can be described via the lever arm
effect, the piston effect and the signal changes due to the transmissive
components along the beam path. The full TTL coupling noise is approximately
the sum of the respective Eqs. (1)-(3). Only small deviation appear when
computing the combined effect: The full geometric TTL coupling can simply be
evaluated by adding the geometric contributions of the three effects. The non-
geometric TTL coupling is then computed as described in [8] by taking into
account the beam walk on the photodiode accumulated due to the three coupling
mechanisms (sum of the beam walks due to the three effects). Since the beam
walk appears squared in the non-geometric formalism, cross-terms of the single
coupling effect add in the full analytical TTL coupling model. Despite these
cross-terms and hence the deviation of the final result from the sum of the
equations presented above is small, we state here the model computed for full,
combined effect for correctness:

$\displaystyle\begin{split}\text{LPS}^{\text{LPF}}&\approx 63\cdot
10^{-6}\frac{\text{m}}{\text{rad}}\,\varphi_{\text{SC}}+5\cdot
10^{-6}\frac{\text{m}}{\text{rad}}\,\eta_{\text{SC}}\\\
&+1.994\frac{1}{\text{rad}}\,\left[(-\hat{\varphi}_{1}+\hat{\varphi}_{2})\,y_{\text{SC}}+(\hat{\eta}_{1}-\hat{\eta}_{2})\,z_{\text{SC}}\right]\\\
&+0.419\frac{\text{m}}{\text{rad}^{2}}\,\hat{\varphi}_{1}\,\varphi_{\text{SC}}+0.362\frac{\text{m}}{\text{rad}^{2}}\,\hat{\varphi}_{2}\,\varphi_{\text{SC}}\\\
&+0.417\frac{\text{m}}{\text{rad}^{2}}\,\hat{\eta}_{1}\,\eta_{\text{SC}}+0.354\frac{\text{m}}{\text{rad}^{2}}\,\hat{\eta}_{2}\,\eta_{\text{SC}}\\\
&+0.010\frac{\text{m}}{\text{rad}^{2}}\,(-\hat{\varphi}_{1}+\hat{\varphi}_{2})\,\eta_{\text{SC}}\\\
&-0.390\frac{\text{m}}{\text{rad}^{2}}\,\varphi_{\text{SC}}^{2}-0.385\frac{\text{m}}{\text{rad}^{2}}\,\eta_{\text{SC}}^{2}\,.\end{split}$
(4) $\displaystyle\begin{split}&\equiv
C_{\varphi}\,\varphi_{\text{SC}}+C_{\eta}\,\eta_{\text{SC}}+C_{y}\,y_{\text{SC}}+C_{z}\,z_{\text{SC}}\\\
&+\mathcal{O}(\varphi^{2},\eta^{2})\,,\end{split}$ (5)

where $C_{i}$, $i\in\\{\varphi,\eta,y,z\\}$, are the coupling coefficients
scaling the linear S/C jitter in the respective degree of freedom.

Note that solely non-geometric effects that depend on the three coupling
mechanisms presented above add to the signal for S/C jitter. Further,
independent non-geometric coupling contributions presented in [8], e.g.
nominal offsets between the interfering beams, are not present here. These
would only couple with a dynamic change in the measurement beam’s incidence
angle at the detector. However, any S/C jitter-induced changes in the
direction of the measurement beam after its reflection at the first test mass
will cancel after its reflection at the second test mass. This is because,
from the perspective of the S/C, both test masses jitter synchronously about
its centre of rotation. Thus, the angular alignment of the measurement beam at
the detector remains constant.

In general, we find by Eq. (4) that the lateral and angular TTL coupling in
LPF depended strongly on the alignment of the test masses. Second-order
coupling occurs only due to angular S/C jitter in the respective two degrees
of freedom.

We show in Sec. IV that Eq. (4) is consistent with numerical simulations.
Furthermore, we can, for the first time reconstruct the performance changes
due to test mass realignments during the LPF mission [12]. In addition, the
coupling coefficients computed with Eq. (4) for mission data of TTL of a
particularly designed coupling experiment match in almost all cases their
fitted counterparts within their error bars (see [12] for further details.

###  III.5 TTL Noise Minimisation

Eq. (4) shows that the degree of S/C jitter coupling into the LPS signal in
LPF depended strongly on the angular alignment of the test masses. By
minimising Eq. (4) regarding the test mass angles $\hat{\varphi}_{1,2}$ and
$\hat{\eta}_{1,2}$, alignments can be defined that fully cancel the linear TTL
coupling due to S/C jitter.

However, from Eq. (4) we cannot gain an fixed alignment angles valid for the
full mission duration: First, as will show in Sec. V, the test mass
independent S/C jitter terms (first row of Eq. (4)) depend highly on the
stability of the optical setup and could therefore change during the mission.
Second, the measurement of the test masses’ angular alignment changes was
highly precise in LPF, while their true absolute alignments (tilt with respect
to the global coordinate system, see Fig. 2) were uncertain. Thus, test mass
realignment angles would have to be found that counteract both the unknown
initial alignment and the test mass independent linear coupling terms.

Within this paper, we define as the nominal case, the angular test mass
alignment that minimises the differential angle of the measurement and the
reference beam at the detectors. Precisely, the differential wavefront sensing
(DWS) [19, 20] angles of the interfering beams in the x1-interferometer
(measuring the relative alignment between the S/C and the first test mass) and
the x12-interferometer (Fig. 1). These angles are summarised in Tab. 1. They
have been computed based on the measurements of the optical setup of the LPF
flight model [21].

Table 1: Nominal angular alignments of the test masses. These angles have been defined to minimise the the DWS angles measured by the x1- and x12-interferometer. angle | [unit] | nominal realignment
---|---|---
$\hat{\varphi}_{1}$ | [$\upmu$rad] | 7.98
$\hat{\varphi}_{2}$ | [$\upmu$rad] | -60.7
$\hat{\eta}_{1}$ | [$\upmu$rad] | 13.0
$\hat{\eta}_{2}$ | [$\upmu$rad] | 0.355

###  III.6 Comparison of the Noise Contributors

In LPF, the TTL coupling originated from both angular and lateral S/C jitter
[12]. The lateral jitter contributed significant linear coupling noise. Its
magnitude depended only on the angular test mass alignment and the beam
orientation before the reflection [7, 18]. In the case of angular jitter
coupling, all three TTL coupling mechanisms presented above add to the full
measured TTL noise. Their contribution to the final signal is visualised in
Fig. 3. We find that the lever arm and the piston effect are the dominating
mechanisms for angular S/C jitter. The lever arm effect cannot be fully
mitigated by design. Consequently, a direct TTL coupling suppression (i.e. by
realignment) would only have been possible, when the lever arm and the
(angular) piston effect would counteract each other.

Figure 3: The TTL coupling for S/C rotations in yaw (top) and pitch (bottom)
and its contributors: the lever arm effect, the piston effect and the
transmissive components. Dashed curves: coupling if all test mass alignment
angles are set to zero. Solid curves: coupling considering nominal test mass
alignments (Tab. 1). All curves have been derived analytically. We see that
the coupling strongly depends on the test mass alignment. The lever arm and
the piston effect are the most significant noise contributors.

Besides the contributions of the single TTL coupling mechanisms, we compare
here the geometric and non-geometric signal contributions. As shown in Fig. 4,
both add significant coupling to the full signal. Therefore, neither can be
neglected when analytically describing the observed TTL coupling.

Figure 4: Comparison of the contributions of the geometric and the non-
geometric coupling to the full LPS signal in the case of yaw ($\varphi$) and
pitch ($\eta$) rotations. Dashed curves: coupling if all test mass alignment
angles are set to zero. Solid curves: coupling considering nominal test mass
alignments (Tab. 1). All signals have been derived analytically. For both
rotations, we see that the geometric as well as the non-geometric coupling
terms are relevant for describing the full TTL coupling in LPF.

In general, the TTL contributions depended highly on the test mass alignments.
While for small S/C jitter amplitudes (few nano-radians), the coupling can be
linearised, the slope significantly changes if the test masses were rotated.
Therefore, it was essential to compute the series expansion of the LPS signal
up to second orders of the small angles.

##  IV Verification of the Analytical Model

To verify the presented analytical description of TTL coupling (Eq. (4)), we
compared it with numerical simulations. This has been done via the the C++
library IfoCAD [22, 19, 23]. All relevant components of the LPF setup have
been implemented in IfoCAD, i.e., all optical components of the LPF optical
bench and the windows of the vacuum chambers surrounding the test masses. The
implementation copies the in-flight model of LPF [21]. The two beams are then
automatically traced through this setup and interfered at the detectors.

The angular alignment of the test masses in IfoCAD has been adjusted to the
angles defined in Tab. 1, i.e., the angles minimising the DWS angles in the
x1- and x12-interferometer.

The full TTL coupling for angular variations is shown in Fig. 5. We plot the
IfoCAD-derived coupling for angular variations up to 5 $\upmu$rad. This was
the injected jitter amplitude of several experiments during the LPF mission
[12]. The jitter in noise runs was significantly smaller. The plot
demonstrates that the coupling can be approximated by linear formalisms for
small jitter amplitudes.

In Fig. 6, we compare our analytical model (Eq. (4)) against the IfoCAD
result. The difference between both models is more than three orders of
magnitude smaller than the original coupling. The residual difference
originates from the series expansion of the analytical model and likely from
simplifying assumptions in the computation of the non-geometric coupling
contributions [8]. So, the deviations are more than one order of magnitude
smaller for the sole geometric coupling or the comparison of the evaluated and
simulated beam walk on the detector [18]. The shown small deviation validates
our analytical TTL coupling description.

Figure 5: The LPS computed by IfoCAD for S/C rotations around its centre of
mass. The computation considers the nominal test mass alignments (Tab. 1).
Figure 6: The deviation between the analytically derived LPS and the LPS
computed by IfoCAD for S/C rotations around its centre of mass. Both
computations considers the nominal test mass alignments (Tab. 1). The residual
is three orders of magnitude smaller than the signal itself. It originates
from simplifying assumptions in the evaluation of the analytical non-geometric
signal.

Lateral jitter coupling almost only originates from geometric TTL coupling
effects [8]. A comparison between the analytical model and the simulated
lateral jitter only showed deviation of the order of the machine accuracy
[11].

##  V Uncertainty of the Modelled Coefficients

The analytical model presented in this manuscript relies on our best knowledge
of the setup and beam parameters. However, the measurement accuracy of the
latter is technologically limited and the parameters can change during the
mission due to stresses and relaxation of the materials. In general, we assume
the following parameters to be subject to measurement inaccuracies or in-
flight changes of the optical system.

#### The lever arm lengths:

The lever arm lengths (compare setup parameters list in appendix, Tab. 4) have
been computed via IfoCAD. Their numbers rely on the accuracy of the position
parameters of the optical components inserted in the script. In flight, the
lever arm lengths are affected by distortions of the optical bench itself but
also by longitudinal displacements of the test masses. The corresponding
length variations would change the angular coupling coefficients $C_{\varphi}$
and $C_{\eta}$. However, we assume the overall changes of the lever arm length
to be several orders of magnitude smaller than its full length. Since a 1%
length change of a lever arm approximately translates into 2% change of the
angular coupling coefficients, this coupling coefficient change would also be
several magnitudes smaller than its absolute value. Therefore, lever arm
length uncertainties will be neglected in the following.

#### The longitudinal offsets between the reflection point at the test mass
and the centre of rotation:

The measurement of the longitudinal offset between the beam’s point of
reflection at the test masses and the satellite’s centre of mass is limited by
the determination of the exact position of the latter. The uncertainty of the
LPF’s centre of mass is $\pm$5 mm in all axes [24]. Furthermore, this centre
varies over the mission time due to fuel (cold gas) consumption. Based on this
consumption, we assume here a variation of the longitudinal parameter of the
S/C’s centre of mass of $-0.5\,\text{mm}$ … $+2.3\,\text{mm}$ [25]. Additional
changes of this offset, e.g. due to distortions of the optical bench, are
assumed to be negligible. The changes of the longitudinal offsets yield
roughly a third of the errors summarised in the following (Tab. 2).

#### The lateral offsets between the reflection point at the test mass and the
centre of rotation:

Neither lateral shifts of the test masses nor of the S/C have a measurable
effect on the offset between the point of reflection and the centre of
rotation. Thus, these offsets would only change if the path of the beam
hitting the test masses differs from its simulation with IfoCAD. Such
distortions of the beam direction can result from thermal stresses on the
optical system and significantly change the linear angular coupling terms (see
Fig. 7, further discussion in Para. e). In addition, the measurement of the
lateral position of the S/C’s centre of mass has an uncertainty of $\pm$5 mm
[24] and it further varies by $-0.5\,\text{mm}$ … $+3.8\,\text{mm}$ along the
$y$-axis and marginally along the $z$-axis due to the cold gas consumption
[25]. All these changes mostly affect the constant offset of the angular TTL
coupling coefficients. They couple with the test mass alignment only to a
negligible degree.

#### Window properties:

We have shown in Sec. III.6 that the windows in between the optical bench and
the test masses are small TTL contributors compared to the lever arm and the
piston effect. Therefore, small changes of their thickness, alignment or
refractive index would be negligible in the full TTL coupling estimate.

#### The angular beam alignment:

Changes in the beam alignment at their source or due to distortions of the
optical bench [26, 27] would, on the one hand, change the lateral position of
the reflection points on the test masses (see above). On the other hand, the
angles of incidence at the test masses change, which yields a small change of
the piston effect. In general, these changes couple linearly with the S/C
angular jitter. However, in-flight changes of the beam alignment due to
stresses and relaxations of the optics would be (partially) corrected by test
mass rotation due to the drag-free attitude control system (DFACS) [28, 29]
control loop, which keeps the DWS readout [19, 20] of the relative beam
alignments at the photodiodes stable. Due to the number of potential origins
of beam misalignments, their actual effect cannot be easily determined
mathematically and we neglect it in the overall analysis. For the special case
of a beam rotation at the fibre injector optical subassembly (FIOS) (e.g.
thermally induced [26]), we exemplary show the effect on the coupling
coefficients (uncorrected by test mass alignments) in Fig. 7. We see that
angular beam alignment changes can have a significant effect on the linear
angular TTL coupling coefficients.

Figure 7: The beam walk on the two test masses and the photodiode (left axis)
due to a yaw (top) and pitch (bottom) tilt of the beam at the FIOS, as well as
the resulting TTL coupling coefficient variations (right axis). In the given
range, the TTL coefficient $C_{\eta}$ changes by more than 10 $\mu$m/rad from
its nominal value. All numbers have been computed with IfoCAD.

#### The beam parameters:

The measurement accuracy of the waist size and the distance from waist of both
interfering beams is limited. The resulting uncertainties yield a possible
error of the non-geometric coupling estimate. According to [26], the
measurement of the beam’s waist size was very accurate ($w_{0m}=(542\pm
4)\,\upmu$m, $w_{0r}=(500\pm 8)\,\upmu$m) yielding a maximal error of the
Rayleigh range of 3.4%. The measurement of the distance between the waist and
the beam source ($z_{m}=(142\pm 19)$ mm, $z_{r}=(500\pm 8)$ mm) corresponds to
a measurement error of 3.6% for the distance from waist at the diode PD12A.

#### Arbitrary wavefront or detector errors:

Arbitrary imperfections of the beams’ wavefronts and the detector surface
alter the measured phase signal. These effects are not included in the
analytical model but can cause a small deviation between the analytically
predicted coupling and the photodiode readout.

To determine the minimal and maximal deviations of the coupling coefficients,
we have to insert the parameter variations characterised above into the
extended exact coupling equations. We restrict this analysis to the variations
that we clearly characterised with numbers (summarised in Tab. 3). See [18]
for further details about the computation. The described parameter changes or
measurement inaccuracies mainly affect the angular coupling coefficients (i.e.
multipliers of $\varphi_{\text{SC}}$ and $\eta_{\text{SC}}$). The lateral
jitter coupling is mostly described by geometric TTL coupling [7] and strongly
depends on the angular test mass alignment. Any other small changes of the
optical setup have a negligible effect on the lateral TTL coefficients (i.e.
multipliers of $y_{\text{SC}}$ and $z_{\text{SC}}$).

Table 2: Uncertainties of the analytical coefficients assuming errors in the measurement of the beam parameters and the centre of mass of the S/C. term | coeff. [m/rad2] | abs. error [m/rad2] | rel. error [%]
---|---|---|---
$\varphi_{\text{SC}}\,\hat{\varphi}_{1}$ | 0.419 | -0.034/+0.031 | -8.2/+7.5
$\varphi_{\text{SC}}\,\hat{\varphi}_{2}$ | 0.362 | -0.037/+0.039 | -10.0/+10.9
$\eta_{\text{SC}}\,\hat{\eta}_{1}$ | 0.417 | -0.033/+0.030 | -8.0/+7.2
$\eta_{\text{SC}}\,\hat{\eta}_{2}$ | 0.354 | -0.035/+0.038 | -9.9/+10.8
Table 3: The parameter uncertainties that we assume in our analytical model. The corresponding nominal values are given in Tabs. 4 and 6. The points of reflection (PoR) are defined with respect to the S/C centre of mass. parameter | uncertainty
---|---
longitudinal distance between the PoR at TM1 and the S/C centre of mass |  $-$7.3/+5.5 mm
lateral distance between the PoR at TM1 and the S/C centre of mass |  $-$5.5/+8.8 mm
vertical distance between the PoR at TM1 and the S/C centre of mass |  $\pm$5 mm
longitudinal distance between the PoR at TM2 and the S/C centre of mass |  $-$7.3/+5.5 mm
lateral distance between the PoR at TM2 and the S/C centre of mass |  $-$5.5/+8.8 mm
vertical distance between the PoR at TM2 and the S/C centre of mass |  $\pm$5 mm
Rayleigh range of the measurement beam |  $\pm$0.014 m
Rayleigh range of the reference beam |  $\pm$0.025 m
distance from waist of the measurement beam |  $\pm$0.019 m
distance from waist of the reference beam |  $\pm$0.008 m

For the test mass alignment dependent terms of the angular TTL coupling
coefficients we show the computed errors in Tab. 2. These are of particular
interest for the analysis of the TTL coupling during the LPF mission since
angular realignments of the test masses have been applied for TTL coupling
suppression. Thus, the errors in Tab. 2 show the accuracy of the estimate of
the test mass alignment dependency of the coupling noise. As discussed in the
paragraphs above, our knowledge of the position of the S/C’s centre of mass
and the beam parameters are the largest contributors to these terms.

We intentionally do not show here the uncertainties of the linear
coefficients, that are independent of the test mass alignments (i.e. the first
two terms in Eq. (4)). In LPF, we did not have access to the exact test mass
alignment angles
($\hat{\varphi}_{i}=\varphi_{0i}+\varphi_{i},\hat{\eta}_{i}=\eta_{0i}+\eta_{i}$)
but only to their changes ($\varphi_{i},\eta_{i}$). Thus, Eq. (4)) can be
split in

$\displaystyle\begin{split}&\,\text{LPS}^{\text{LPF}}(\varphi_{0i}+\varphi_{i},\eta_{0i}+\eta_{i})\\\
=&\,\text{LPS}^{\text{LPF}}(\varphi_{i0},\eta_{i0})+\text{LPS}^{\text{LPF}}(\varphi_{i},\eta_{i})\end{split}$
(6)

The unknown part of the test mass angle dependent coupling (first term in Eq.
6) would be assigned to the (no longer) test mass independent linear
coefficient. Their sum form the ‘constant offsets’ of the linear coupling
coefficients. The additive term is approximately of the same magnitude as the
other linear terms. Therefore, the exact value of the constant offset of the
linear coupling terms cannot be determined in practice.

##  VI Tilt-To-Length Noise in the $\Delta g$ Measurements

The analytical modelling presented so far, showed the TTL coupling
contribution to the length signal measured at the diode PD12A. To prove the
principle of laser interferometric gravitational wave measurements in space
with LPF, this length signal has been transformed to an estimate of the
distance changes of the test masses

$\displaystyle\Delta x=x_{1}-x_{2}\,,$ (7)

which is achieved by the multiplication with the factor
$1/(2\cos(\beta_{y2}))$. Correspondingly, the same correction is applied to
the TTL noise contribution. Note that the sign convention in Eq. (7) was
chosen in accordance with the $o_{12}$ definition in [30].

Furthermore, we take the second derivative of the TTL coupling in the S/C
jitter variables. This yields the noise contribution to the $\Delta g$
measurement, which is the central parameter describing the stability of the
test masses. Neglecting second- and higher-order terms, we find

$\displaystyle\Delta
g_{\text{xacc}}^{\text{ana}}=C_{\varphi}^{\text{ana}}\,\ddot{\varphi}_{\text{SC}}+C_{\eta}^{\text{ana}}\,\ddot{\eta}_{\text{SC}}+C_{y}^{\text{ana}}\,\ddot{y}_{\text{SC}}+C_{z}^{\text{ana}}\,\ddot{z}_{\text{SC}}$
(8)

with

$\displaystyle C_{\varphi}^{\text{ana}}=$
$\displaystyle\,C_{\varphi,0}+0.210^{+0.017}_{-0.016}\,\frac{\mathrm{m}}{\mathrm{rad}^{2}}\,\varphi_{1}+0.182^{+0.018}_{-0.020}\,\frac{\mathrm{m}}{\mathrm{rad}^{2}}\,\varphi_{2}$
(9)
$\displaystyle\begin{split}C_{\eta}^{\text{ana}}=&\,C_{\eta,0}+0.209^{+0.017}_{-0.015}\,\frac{\mathrm{m}}{\mathrm{rad}^{2}}\,\eta_{1}+0.177^{+0.018}_{-0.019}\,\frac{\mathrm{m}}{\mathrm{rad}^{2}}\,\eta_{2}\\\
&+0.005^{+0}_{-0}\,\frac{\mathrm{m}}{\mathrm{rad}^{2}}\,(-\varphi_{1}+\varphi_{2})\end{split}$
(10) $\displaystyle C_{y}^{\text{ana}}=$
$\displaystyle\,C_{y,0}+\,1.000^{+0}_{-0}\
\frac{1}{\text{rad}}\,(-\varphi_{1}+\varphi_{2})$ (11) $\displaystyle
C_{z}^{\text{ana}}=$ $\displaystyle\,C_{z,0}+\,1.000^{+0}_{-0}\
\frac{1}{\text{rad}}\,(\eta_{1}-\eta_{2})\,,$ (12)

where the offsets $C_{i,0},\,i\in\\{\varphi,\eta,y,z\\},$ depend on the setup
parameters (App. A). In the previous section, we touched upon the topic that
these offsets significantly changed with parameter changes. However, the
change rate was negligible in short time segments (days or a few weeks)
without experiments. Thus, we can assume the $C_{i,0}$ to be constant in the
data analysis of single time segments. Furthermore, since the exact test mass
alignment remained unknown, the TTL coupling depending on the nominal test
mass angles would also be assigned to these coefficients in the analysis. The
pitch and yaw angles in Eqs. (9)-(12) would therefore be read as differential
angular readouts. Note that the equations hold for both absolute and
differential angles depending on this interpretation.

Note further that we also discarded the second-order terms
($\varphi_{\text{SC}}^{2},\,\eta_{\text{SC}}^{2}$) in this representation.
These become small for small S/C jitter amplitudes. However, we kept the terms
$\varphi_{\text{SC}}\varphi_{i}$, $\eta_{\text{SC}}\eta_{i}$, with
$i\in\\{1,2\\}$, which are basically also second-order terms but linearised
for static test mass alignment angles. Since the latter can significantly
exceed the level of S/C jitter (compare e.g. Tab. 1), these terms are not
negligible.

We will use the model Eq. (8) in [12] for the analysis of the LPF data.

##  VII Applicability of the Results to the Mission

The direct reduction of the TTL coupling noise by the realignment of critical
components is a powerful TTL coupling noise suppression strategy. In LPF, the
alignment of the test masses largely affected the TTL noise level. Based on a
simple analytical TTL coupling model, the nominal test mass set-points have
been changed three times during the mission for TTL noise suppression.
However, it was only possible to reduce but not fully mitigate the cross-talk
making an subtraction of the residual noise necessary.

We think that the realignment would have been more successful with the
analytical TTL model that we present in this paper. In this section, we
discuss the usability of the new model and compare it to the models used
during the operation time of LPF.

###  VII.1 Comparison with Tilt-To-Length Models Used During the LISA
Pathfinder Mission

The first TTL models for LPF relied mainly on a simplified geometric piston
model for test mass rotations and gained more complexity during the mission.
However, these models did not sufficiently describe the TTL noise in LPF. The
new analytical model presented in this paper accounts for additional TTL
coupling mechanisms and all optical setup parameters. Besides the complexity,
we find three significant differences between this new model and the models
available during the LPF mission.

First, when interpreting the S/C jitter as test mass jitter, we have to model
the corresponding coupling for rotations about the S/C’s centre of mass
instead of the test mass centres. During the mission, the relevance of the
location of the centre of rotation was underestimated and the models
considered common-mode rotations of the test masses about their centre of mass
instead. The derivation of the new model presented in this paper showed that
this simplification was erroneous. Mainly the large longitudinal offset of the
respective centre of rotation adds additional coupling.

Second, lateral shifts of the test masses were found not to change the
distance between the S/C centre of rotation and the point of reflection at the
test masses. Consequently, the model presented here describes that lateral
test mass displacements do not change the level of TTL noise, unlike the
original models.

Third, non-geometric coupling effects have often been ignored in previous
models. However, as we show in Sec. III.6, they significantly contribute to
the TTL coupling noise.

###  VII.2 Usability of the New Analytical Model

Since the LPF mission ended in 2017, which was before this model was derived,
there was no direct application of the analytical model (Eq. (8)) to the
mission. However, the equations help understand the TTL coupling mechanisms
and their importance for LPF better. By this, we could learn why the initial
attempts of deriving the realignment angles showed inconsistencies with the
resulting noise changes, while later attempts were reducing the noise, but
could not fully suppress it. Further information and particularly a detailed
TTL coupling data analysis for LPF are given in [12].

##  VIII Conclusions

Within this work, we have presented an updated and most complete analytical
TTL coupling model for LPF. For this, we considered geometric as well as non-
geometric TTL coupling effects. The resulting model mainly differs from the
models available during the mission in the assumed centre of rotation and the
previously neglected non-geometric terms. An important consequence of the
change of the centre of rotation is that the updated TTL coupling models are
independent from lateral realignments of the test masses. We could show that
the updated model coincides with numerical simulations with considerable
accuracy. While we assumed exact setup parameters in the modelling and
simulation, they are subject to measurement uncertainties or long-term drifts
in experiment. Therefore, we characterised how these uncertainties affect the
stability of the derived coupling coefficients. This analysis yielded relative
errors of the coupling terms up to 10%. In the last step, we translated the
length signal formulation of the TTL coupling into an equation showing the TTL
coupling contribution to the $\Delta g$-measurements in LPF.

From the new analytical model, we can deduce three main conclusions. First,
our investigations have shown that the lever arm and the piston effect
(geometric and non-geometric contributions) were the main noise contributors
to the overall TTL coupling. Both added angular coupling noise of the same
magnitude. Hence, a reasonable mitigation of the angular TTL noise could only
be achieved when both effects counteract each other.

Second, the TTL coupling model consists of first- and second-order terms.
While a linear coupling model would, in general, not be sufficient for the
characterisation of this noise, it can, for jitter of small amplitude, be
linearised in the S/C jitter parameters. The resulting coupling coefficients
depend on both the setup and beam parameters as well as the test mass
alignments. If the case that second-order coupling becomes non-negligible due
to large jitter amplitudes, we have seen that only the angular jitter yields
second-order noise terms.

Last, we have shown that the geometric and non-geometric coupling
contributions both add significant TTL coupling noise to the full signal.
Therefore, a purely geometric modelling of the TTL noise would not be
sufficient, but both have to be considered in the TTL noise analysis.

The presented analytical model is an important basis for the investigation of
the TTL coupling measured in LPF and the corresponding lessons learned for
LISA. Thus, the model Eq. 8 will be applied to the LPF data in [12] and [27].
We show in [12] that it can be used to explain the TTL coupling behaviour in
experiment. This analysis holds as a further validation of the presented
model. In addition, it can be used to analyse the stability of the optical
setup itself. As discussed in [27], our analytical TTL coupling description
can relate the long-term and temperature-depended drifts of the coupling
coefficients to unconsidered test mass realignments and deformations of the
LPF optical bench.

## Acknowledgements

We thank Valerio Ferroni and Nikolaos Karnesis for valuable discussions. This
work was made possible by funds of both the German Space Agency, DLR, and the
Deutsche Forschungsgemeinschaft (DFG). We gratefully acknowledge the German
Space Agency, DLR and support by the Federal Ministry for Economic Affairs and
Climate Action based on a resolution of the German Bundestag (FKZ 50OQ0501,
FKZ 50OQ1601, and FKZ 50OQ1801). Likewise, we gratefully acknowledge the
Deutsche Forschungsgemeinschaft (DFG) for funding the Cluster of Excellence
QuantumFrontiers (EXC 2123, Project ID 390837967).

##  Appendix A Parameter list

The LPF setup and beam parameters substituted into the analytical TTL model
presented in Sec. III are summarised in Tabs. 4, 5 and 6. The beam parameters
have previously been published in [26]. The S/C centre of mass has been
defined in [29]. The other parameters have been extracted from a three-
dimensional LPF model implemented in IfoCAD, which is based on [21]. The
numbers are given for the case of non-rotated test masses. We further compute
all numbers for the case of interference at the A-diode of the
x12-interferometer (PD12A, see Fig. 1).

Table 4: Setup parameters. The distances between the beam’s point of reflection (PoR) at the TM and the respective centre of rotation are given here in the LPF coordinate system to avoid confusion. Mind that for computation as introduced in [7, 8], the signs of the longitudinal and lateral offset for TM2 would have to be inverted: The coordinate system is set there by the propagation direction of the incident beam, i.e., opposite to the LPF coordinate system. The beam offsets are defined as the offset of the point where the respective beam axis hits the surface of the A-diode of the x12-interferometer from the centre of that diode’s surface. setup parameter | value
---|---
lever arm length between both TMs neglecting the windows | 0.356 m
lever arm length between TM2 and the PD neglecting the window | 0.143 m
longitudinal distance between the PoR and centre of TM1 | -0.023 m
lateral distance between the PoR and centre of TM1 | -6.37 $\upmu$m
vertical distance between the PoR and centre of TM1 | 3.42 $\upmu$m
longitudinal distance between the PoR and centre of TM2 | 0.023 m
lateral distance between the PoR and centre of TM2 | -15.8 $\upmu$m
vertical distance between the PoR and centre of TM2 | 10.5 $\upmu$m
longitudinal distance between the PoR at TM1 and the S/C centre of mass | 0.160 m
lateral distance between the PoR at TM1 and the S/C centre of mass | -0.006 m
vertical distance between the PoR at TM1 and the S/C centre of mass | 0.063 m
longitudinal distance between the PoR at TM2 and the S/C centre of mass | -0.170 m
lateral distance between the PoR at TM2 and the S/C centre of mass | -0.006 m
vertical distance between the PoR at TM2 and the S/C centre of mass | 0.063 m
thickness of the windows | 6.05 mm
refraction index of the windows | 1.61
slit width of the PD | 45 $\upmu$m
horizontal measurement beam offset from the PD centre | -1.34 $\upmu$m
vertical measurement beam offset from the PD centre | 13.3 $\upmu$m
horizontal reference beam offset from the PD centre | -2.12 $\upmu$m
vertical reference beam offset from the PD centre | -2.26 $\upmu$m
Table 5: Beam alignment angles. The angular measurement beam alignment before hitting the TMs is provided by the propagation angles in the respective plane. In the case of the beam alignment angles with respect to the detector, we assume the A-diode of the x12-interferometer. beam angle | value
---|---
tilt of incoming beam at TM1: $xy$-plane | -0.0785 rad
tilt of incoming beam at TM2: $xy$-plane | 0.0786 rad
tilt of incoming beam at both TMs: $xz$-plane | -20 $\upmu$rad
yaw angle between the surface normal of the PD an the negative measurement beam direction (interpreted as PD tilt) | -165 $\upmu$rad
pitch angle between the surface normal of the PD an the negative measurement beam direction (interpreted as PD tilt) | -20 $\upmu$rad
yaw tilt of the reference beam at the PD | 125 $\upmu$rad
pitch tilt of the reference beam at the PD | 7 $\upmu$rad
Table 6: The beam parameters in LPF. The distances from waist are given at the point where the beam axis hits the surface of the A-diode of the x12-interferometer. beam parameter | value
---|---
wave number of the beams | 5.81$\cdot 10^{6}$m-1
Rayleigh range of the measurement beam | 0.867 m
Rayleigh range of the reference beam | 0.738 m
distance from waist of the measurement beam | 0.519 m
distance from waist of the reference beam | 0.594 m

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