# On the noise effect of test mass surface roughness in spaceborne
gravitational wave detectors

Hao Yan  Center for gravitational experiment, MOE Key Laboratory of
Fundamental Physical Quantities Measurements, The School of Physics, Huazhong
University of Science and Technology, Wuhan 430074, China  Haixing Miao  State
Key Laboratory of Low Dimensional Quantum Physics, Department of Physics,
Tsinghua University, Beijing, China  Shun Wang  Center for gravitational
experiment, MOE Key Laboratory of Fundamental Physical Quantities
Measurements, The School of Physics, Huazhong University of Science and
Technology, Wuhan 430074, China  Yiqiu Ma<EMAIL_ADDRESS>Center for
gravitational experiment, MOE Key Laboratory of Fundamental Physical
Quantities Measurements, The School of Physics, Huazhong University of Science
and Technology, Wuhan 430074, China  Zebing Zhou<EMAIL_ADDRESS>Center for
gravitational experiment, MOE Key Laboratory of Fundamental Physical
Quantities Measurements, The School of Physics, Huazhong University of Science
and Technology, Wuhan 430074, China

###### Abstract

Spaceborne gravitational wave detection mission has a demanding requirement
for the precision of displacement sensing, which is conducted by the
interaction between the laser field and test mass. However, due to the
roughness of the reflecting surface of the test mass, the displacement
measurement along the sensitive axis suffers a coupling error caused by the
residue motion of other degrees of freedom. In this article, we model the
coupling of the test mass residue random motion to the displacement sensing
along the sensitive axis and derived an analytical formula of the required
precision of the surface error for the spaceborne gravitational wave
detectors. Our result shows that for the test masses in the LISA pathfinder,
this coupling error will not contaminate the picometer displacement sensing.

##  I Introduction

Spaceborne laser interferometer gravitational wave detectors such as LISA
Danzmann and the LISA study team (1996), Tianqin Luo  _et al._ (2016), and
Taiji Hu and Wu (2017) are targetted on detecting the gravitational wave
sources at the mili-Hertz range in the 2030s. The configuration of these
gravitational wave detectors consists of three satellites that are connected
by laser links. Each satellite consists of two independent subsystems in a
60-degree configuration, each of which contains an interferometry system
(optical bench, laser source, telescope, etc.) and a gravitational reference
sensor (GRS) with test mass inside, as shown in Fig. 1. Two key components
inside these satellites are the test masses which serve as inertia references,
and also the interferometer bench which measures the tiny displacement of the
test mass. The detailed interaction process between the test mass and the
laser field will affect gravitational wave detection.

Figure 1:  Schematic diagram of a satellite payload assembly in a spaceborne
gravitational wave detector, where the laser field (ideally) couples to the
test mass geodesic motion, which is preserved by the inertial sensor and drag-
free controller.

As an inertial reference, the cubic Au-Pt alloy test mass is surrounded by a
conducting electrostatic shield with electrodes that are used for simultaneous
capacitive position sensing and electrostatic force actuation Carbone  _et
al._ (2003). The drag-free and attitude control system (DFACS) Armano  _et
al._ (2016) ensures the very high dynamic stability of spacecraft and test
masses, in particular along the sensitive axis along which the laser field
gets reflected. For the other non-sensitive axis, the motion is controlled
while introducing non-negligible disturbance noise, which causes beam walking
and jitter effects by reflecting from the test mass surface as we shall
discuss in this work.

The surface roughness is an intrinsic property of the test mass originated
from the manufacturing process Walsh  _et al._ (1999a); Elson and Bennett
(1995); Bhushan (2000); Lyukshin  _et al._ (2021); Butler (2013). On one hand,
this surface roughness will introduce stray light and result in nonlinear
errors in interferometry Nardello and Lintz (2021); Sasso  _et al._ (2019),
which have been well studied Church (1988); Vinet  _et al._ (1996, 1997);
Ottaway  _et al._ (2012); Bennett and Mattsson (1999). On the other hand, the
surface roughness will distort the wavefront of the reflected laser field.
Furthermore, with the residue random motion perpendicular to the sensitive
axis, such a wavefront distortion due to the surface roughness will not be
time-independent, which could contribute noise to the phase measurement by the
local interferometer in the satellite. Analyzing such a noise effect is
interesting since it couples several figures of merits of the satellite
components such as the precision of the test mass manufacturing,
interferometric phase measurement, and the residue motion of the inertia
sensor.

This work is devoted to an in-depth analysis of how surface roughness
contributes to displacement measurement in the spaceborne gravitational wave
detector. The interaction details of the residue test mass motion, surface
roughness, and the light field are analyzed, and an analytical expression for
the requirement of the surface roughness error given the picometer
displacement sensing requirement is derived. Our results show that for the
level of residue motion of the test masses in the LISA pathfinder, the surface
roughness will not contribute significantly to the sensitivity of the
picometer displacement sensing as long as the test mass manufacturing process
limits the surface roughness error within about $100$ nm. This is not a strict
requirement with the current state-of-art technology.

Figure 2:  Schematic diagram of the interferometric sensing of the test-mass
displacement.

This paper is organized as follows. In Section II, theoretical modeling of the
test mass surface roughness is presented. Section III discusses how it couples
to the light field and distorts the wavefront. Then we discuss its impact on
the interferometric phase measurement of the noise in Section IV. The summary
and further discussions are presented in the last section.

##  II Modelling the surface roughness of a test mass

The measured surface roughness is modelled by a two-dimensional height
function $h(x,y)$, which is describes the random distribution of the surface
height along the sensitive axis. In this work, we assume there is no cross-
correlation between the $x$ and $y$ directions: $h(x,y)=h_{x}(x)h_{y}(y)$. The
spatial correlation function the surface roughness is given by:

$C_{hh}(\mathbf{x}_{\perp}-\mathbf{x}^{\prime}_{\perp})=\overline{h(\mathbf{x}_{\perp})h(\mathbf{x}^{\prime}_{\perp})},$
(1)

where $\mathbf{x}_{\perp}=(x,y)$ and the overline represents the ensemble
average. Therefore the correlation spectrum is:

$C_{hh}(\mathbf{k}_{\perp})=\int
d^{2}\mathbf{x}_{\perp}C_{hh}(\mathbf{x}_{\perp})e^{i\mathbf{k}_{\perp}\cdot\mathbf{x}_{\perp}},$
(2)

where $\mathbf{x}_{\perp}=(k_{x},k_{y})$. Since $h(x,y)=h_{x}(x)h_{y}(y)$, the
correlation spectrum is also separable:

$C_{hh}(k_{x},k_{y})=\int dxe^{ik_{x}x}C^{x}_{hh}(x)\int
dye^{ik_{y}y}C^{y}_{hh}(y)$ (3)

where $C^{x}_{hh}(x)=\overline{h_{x}(x)h_{x}(0)}$ (the same for
$C^{y}_{hh}(y)$). In our modelling, we assume that the surface roughness power
spectral densities (PSD) of a test mass as:

$C^{x}_{hh}(k_{x})=\frac{\mathcal{C}}{1+(k_{x}/k_{\Lambda})^{2}},$ (4)

where the $\mathcal{C}=\,{\rm 0.3\,\mu m^{2}}$ is the spectrum amplitude and
the $k_{\Lambda}=1\,{\rm mm^{-1}}$ is the spectrum turning frequency. The
effective surface roughness is Khodnevych (2020)

$\begin{split}\sigma_{\lambda}\approx&\left[\int^{1/\lambda}_{1/d}\int^{1/\lambda}_{1/d}dk_{x}dk_{y}C_{hh}(k_{x},k_{y})\right]^{1/2}\\\
=&\xi\pi\mathcal{C}k_{\Lambda},\\\ \end{split}$ (5)

where $\xi$ is a dimensionless coefficient that depends on the upper and lower
limits of integration (in this paper, we take $\xi\approx 1$). $\lambda$ is
the measurement laser wavelength, and $d$ is the area of the test mass surface
under measurement.

We construct the surface roughness by multiplying the ASD (amplitude spectral
density) by a random phase spectral density before transforming back to
coordinate space to obtain the root-mean-square value, similar to Hong  _et
al._ (2011); Drori  _et al._ (2022). Fig. 3 illustrates the surface roughness
power spectral densities (PSD) and a typical surface curve. Considering the
dimensions of the test mass (a few centimeters) and the diameter of the
measurement laser beam (about one millimeter), we take the spatial wavelength
in the range $25\,{\rm\mu m}-50\,{\rm mm}$ (see the horizontal coordinates of
the power spectral density). Fig. 4 illustrates a typical two-dimension
surface map with a roughness rms of about $1\,{\rm nm}$ generated by the
simulation.

Figure 3:  Upper panel: Power spectral densities of uncoated mirror surface
roughness. The red line is the PSD model [see Eq. (4)], and the black line is
a simulated PSD created to generate the random maps used in our work. Lower
panel: 1-D surface curve of a typical map.

In real manufacturing process, the surface roughness of the test mass
manufactured by advanced machining technologies can be suppressed down to a
few nanometers according to the current state-of-art.

Figure 4: Two-dimensional distribution of the surface roughness of the test
masses.

##  III Light field scattered by the surface roughness

###  III.1 Hermite–Gaussian Modes

The spatial electric field distribution of a freely-propagating light field is
described by the Hermite-Gaussian mode:

$\begin{split}\frac{E_{nm}(x,y,z)}{E_{0}}=H_{n}\left(\frac{\sqrt{2}x}{w(z)}\right)H_{m}\left(\frac{\sqrt{2}y}{w(z)}\right){\rm
exp}\left[-\frac{x^{2}+y^{2}}{w^{2}(z)}\right]\\\ \frac{w_{0}}{w(z)}{\rm
exp}\left(-i\left[kz-(n+m+1){\rm
arctan}\frac{z}{z_{R}}+\frac{k(x^{2}+y^{2})}{2R(z)}\right]\right),\end{split}$
(6)

where $H_{m}(x)$ is the Hermitian function, $w(z)$ is the diameter of the
cross-section of the light field as a function of propagation distance,
$z_{R}$ is the Reighley length and $R(z)=\sqrt{w^{2}(z)+z_{R}^{2}}$. Two
examples of the HG mode is plotted in Fig. 5. All the Hermitian-Gaussian (HG)
modes forms a Hilbert space and a general freely-propagating light field mode
(normalized by $E_{0}$) can be expanded as:

$\begin{split}&\psi(x,y,z)=\langle x,y,z|\psi\rangle=\sum_{j}a_{j}\langle
x,y,z|{\rm HG}\rangle_{j}\end{split},$ (7)

where $|{\rm HG}\rangle_{j}$ is the eigenvector representing the HG mode with
$j=(m,n)$ and $a_{j}$ is the expansion coefficient. The state-vector of the
general light field is $|\psi\rangle=\sum_{j}a_{j}|{\rm HG}\rangle_{j}$. The
$\langle x|{\rm HG}\rangle_{j}$ is the $j_{\rm th}$ HG mode in the coordinate
representation, which is a compact form of Eq. (6). The HG mode is a separable
function and for notational simplicity later on, we will break the
$x$-dependent part and $y$-dependent part of the HG mode by writing $|{\rm
HG}_{j}\rangle=|{\rm HG}^{x}_{j}\rangle|{\rm HG}^{y}_{j}\rangle$

Figure 5:  The Hermite Gaussian transversal spatial function of $\rm{HG}_{00}$
and $\rm{HG}_{01}$ modes.

Usually the incoming signal field is assumed to be a pure HG mode with
$n=m=0$. Reflecting by the imperfect test mass surface, the outgoing field
will be a combination of all different HG modes where the expansion
coefficient $a_{j}$ is determined by the mode shape. When the surface
roughness effect is small, it only perturbs the outgoing field so that
$|a_{j}|\ll|a_{0}|$ for $j\neq(0,0)$. The phase measurement by a heterodyne
detector superpose the reflected light field with a local oscillator with
$H_{00}(x)$ mode:

$\begin{split}|\psi_{m}\rangle=&a_{\rm LO}|{\rm
HG}\rangle_{00}+|\psi\rangle_{\rm ref}\\\ =&a_{\rm LO}|{\rm
HG}\rangle_{00}+a_{0}|{\rm HG}\rangle_{00}+\sum_{j\neq(0,0)}a_{j}|{\rm
HG}\rangle_{j}\end{split},$ (8)

with the intensity which is proportional to the photo-electric current:

$I_{m}=\eta E_{0}^{2}(\psi_{m}|\psi_{m})\approx\eta E_{0}^{2}|a_{\rm
LO}+a_{0}|^{2},$ (9)

where the inner product here is defined as:

$(A|B)\equiv\int d^{2}x_{\perp}\langle A|x\rangle\langle x|B\rangle,$ (10)

and the $\eta$ is the optical-electronical power conversion rate. The integral
is performed over the transversal $(x,y)$ plane. The physical meaning can be
read out from dimensional analysis: the energy density flux is $\sim
c\epsilon_{0}|E|^{2}\propto(\psi_{m}|\psi_{m})$, integration this energy
density flux over the transversal plane leads to a quantity with dimension of
power, which is the power of light absorbed by the photodetector.

###  III.2 Dynamical coupling with the surface roughness

The incoming field reflected from the surface dynamically couples to the test
mass motion. The surface roughness $h(x,y)$ couples to the light field as:

$\psi_{\rm ref}(x,y,z)=r\psi_{\rm in}(x,y,z_{0}+h(x,y,t)),$ (11)

where $r$ is the amplitude reflectivity and $h(x,y,t)$ is the surface
roughness sensed by the laser field. This $h(x,y,t)$ is a random number since
the transvesal motion is random. This coupling can be illustrated by the
Gaussian mode under the paraxial approximation

$\begin{split}\psi_{\rm
ref}(x,y,z)=r[1-2ikh(x,y,t)]H_{00}(x,y,z_{0})\end{split}$ (12)

where we have assumed the zero-th order flat surface located at $z=z_{0}$, and
the small roughness allows the approximation ${\rm exp}[-2ikh(x,y)]\approx
1-2ikh(x,y)$. Moreover, the $h(x,y,z,t)H_{00}(x,y,z_{0})$ can be expanded in
terms of all HG normal modes.

During the heterodyne detection, as we have discussed in the Section III.A,
there will be superposition of the local oscillator and the signal field and
the final photo-electric current at the photo-detector is given by:

$I_{m}=\eta E_{0}^{2}(\psi_{m}|\psi_{m})=\eta E_{0}^{2}\int
d^{2}x_{\perp}\langle\psi_{m}|x_{\perp}\rangle\langle
x_{\perp}|\psi_{m}\rangle,$ (13)

where the trivial propagation in the z-direction is neglected for simplicity.
The perturbation of the surface roughness to the measured photo-electric
current $I_{m}$ is given as:

$\delta_{h}I_{m}=\eta 2krE_{0}E_{\rm
LO}\left(\frac{w_{0}}{w_{D}}\right)^{2}\int d^{2}x_{\perp}{\rm
exp}\left[-2\frac{x^{2}+y^{2}}{w^{2}_{D}}\right]h(x,y,t),$ (14)

where $w_{D}$ is the diameter of the Gaussian optical beam at the detector
position.

The above result can be re-written using the effective optical path variation
defined as:

$h_{\rm eff}(t)=\frac{4}{\pi w_{D}^{2}}\int d^{2}x_{\perp}{\rm
exp}\left[-2\frac{x^{2}+y^{2}}{w^{2}_{D}}\right]h(x,y,t),$ (15)

so that $\delta_{h}I_{m}=\eta krE_{0}E_{\rm LO}h_{\rm eff}(t)\pi w_{D}^{2}/2$.
For small transversal displacements, we have:

$\begin{split}&h(x,y,t)=h(x+\delta x_{c}(t))h(y+\delta y_{c}(t))\\\ &\approx
h_{x}(x)h_{y}(y)+h_{x}(x)\partial_{y}h_{y}(y)\delta
y_{c}(t)+h_{y}(y)\partial_{x}h_{x}(x)\delta x_{c}(t),\end{split}$ (16)

where the first term $h_{x}(x)h_{y}(y)$ is a negligible time-independent DC
source. Therefore we have the coupling of the test mass transversal center of
mass motion to the phase error as:

$\begin{split}\delta h_{\rm eff}(t)=&\frac{4}{\pi w_{D}^{2}}\int
d^{2}x_{\perp}{\rm exp}\left[-2\frac{x^{2}+y^{2}}{w^{2}_{D}}\right]\\\
&\left[h_{x}(x)\frac{\partial h_{y}(y)}{\partial y}\delta
y_{c}(t)+h_{y}(y)\frac{\partial h_{x}(x)}{\partial x}\delta
x_{c}(t)\right],\end{split}$ (17)

which has a simpler form after integration by parts:

$\begin{split}\delta h_{\rm eff}(t)=&\langle{\rm
HG}^{y}_{01}|h_{y}\rangle\langle{\rm HG}^{x}_{00}|h_{x}\rangle\delta
y_{c}(t)\\\ &+\langle{\rm HG}^{x}_{01}|h_{x}\rangle\langle{\rm
HG}^{y}_{00}|h_{y}\rangle\delta x_{c}(t),\end{split}$ (18)

where the $\langle x|h\rangle\equiv h_{x}(x)$. Clearly, in this case the noise
can be written as:

$\begin{split}S_{h_{\rm eff}}(\Omega)=&|\langle{\rm
HG}^{y}_{01}|h_{y}\rangle\langle{\rm HG}^{x}_{00}|h_{x}\rangle|^{2}S_{\delta
y_{c}}(\Omega)\\\ &+|\langle{\rm HG}^{x}_{01}|h_{x}\rangle\langle{\rm
HG}^{y}_{00}|h_{y}\rangle|^{2}S_{\delta x_{c}}(\Omega).\end{split}$ (19)

Therefore the roughness-light conversion coefficient $\langle{\rm
HG}^{y}_{01}|h_{y}\rangle\langle{\rm HG}^{x}_{00}|h_{x}\rangle$ is the key to
obtain the displacement noise level along the sensitive axis. In Fig. 6, we
present the absolute value of the conversion coefficient when the light spot
is located at different positions of a surface with roughness distribution in
Fig. 4. As we shall see later, this figure is very useful for estimate the
test mass manufacture requirements. For a given test mass with the light spot
precisely on the geometrical center of the test mass surface, what matters is
the value at $(x=0,y=0)$, which is equal to $3\times 10^{-7}$ in the exemplary
Fig. 6. For LISA pathfinder Armano  _et al._ (2019, 2016), the error of the
transversal residue motion is $\delta x_{c}\sim 10^{-8}\,{\rm m}/\sqrt{{\rm
Hz}}$, which corresponds to the displacement sensing noise around $\delta
h_{\rm eff}\sim 3\times 10^{-15}\,{\rm m}/\sqrt{{\rm Hz}}$. Certainly it will
not contribute a serious noise to the detector sensitivity.

Figure 6:  Upper panel: filter function extended over the light spot. Lower
panel: Two-dimensional distribution of the roughness-light conversion
coefficient $|\langle{\rm HG}^{y}_{01}|h_{y}\rangle\langle{\rm
HG}^{x}_{00}|h_{x}\rangle|$, where in the main text we only give the result
for $\delta h_{\rm eff}$ of the geometric center of the test mass. The result
is based on the original surface roughness distribution in Fig. 4.

###  III.3 The surface roughness couples with imperfections of incoming light

Surface roughness has another effect on the phase error, which comes from the
imperfections of the incoming light field. Let us suppose the incoming light
contains other HG modes due to the distortion of the optical element in the
optical path Mueller (2005); Barsotti  _et al._ (2010):

$|\psi\rangle_{\rm in}=|{\rm HG}_{00}\rangle+\sum_{j\neq(0,0)}a_{j}|{\rm
HG}_{j}\rangle.$ (20)

Reflected from the rough surface, $|{\rm HG}_{01}\rangle$ can be converted
back into the $|{\rm HG}_{00}\rangle$ mode in a stochastic way so that there
will be a contamination of the phase measurement. Fig. 7 shows schematically
how the higher-order HG mode component in the incoming field being converted
to $|{\rm HG}_{00}\rangle$ through reflecting by a rough surface.

Figure 7: Higher-order component in the incoming light field is converted back
to the fundamental Gaussian mode via interaction with the surface roughness.

The modulation of the surface roughness to the $|{\rm HG}_{01}\rangle$ in the
incoming field can be written as:

$\sum_{j\neq(0,0)}a_{j}\langle x_{\perp}|{\rm
HG}_{j}\rangle\rightarrow\sum_{j\neq(0,0)}a_{j}(1+2ik\hat{h})|{\rm
HG}_{j}\rangle,$ (21)

where the randomly perturbed z-directional roughness (denoted as operator
$\hat{h}$) can have a non-zero overlap with the $|{\rm HG}_{00}\rangle$. This
is important since the local oscillator of the heterodyne detector is a $|{\rm
HG}_{00}\rangle$ mode, therefore its superposition with the perturbed field
can be written as:

$a_{\rm LO}|{\rm HG}\rangle_{00}+ik\sum_{j\neq(0,0)}a_{j}\hat{h}|{\rm
HG}\rangle_{j},$ (22)

and the corresponding perturbation to the photo-electric current is:

$\begin{split}&\delta^{\prime}_{h}I_{m}={\rm
Re}\left[2ik\sum_{j\neq(0,0)}a_{\rm LO}^{*}a_{j}({\rm HG}_{00}|\hat{h}|{\rm
HG}_{j})\right]\\\ &={\rm Re}\left[2ik\sum_{j\neq(0,0)}a_{\rm
LO}^{*}a_{j}\int\frac{d^{2}x_{\perp}}{\pi w_{D}^{2}}\langle{\rm
HG}_{00}|x_{\perp}\rangle h(x,y)\langle x_{\perp}|{\rm
HG}_{j}\rangle\right].\end{split}$ (23)

We can similarly construct an effective optical path variation in this case,
defined as:

$\begin{split}&\delta^{\prime}h_{\rm eff}=\\\ &{\rm
Re}\left[2ik\sum_{j\neq(0,0)}a_{\rm LO}^{*}a_{j}\int\frac{d^{2}x_{\perp}}{\pi
w_{D}^{2}}\langle{\rm HG}_{00}|x_{\perp}\rangle h(x,y)\langle x_{\perp}|{\rm
HG}_{j}\rangle\right].\end{split}$ (24)

The time-dependence of $\delta^{\prime}h_{\rm eff}$ may have two origins. (1)
The imperfections of the optical element in the interferometer may be time-
dependent so that $a_{j}=a_{j}(t)$, which directly couples with a time-
independent $h(x,y)$ roughness, contributing to the phase error:

$\begin{split}&\delta^{a_{1}}h_{\rm eff}(\Omega)=\\\ &{\rm
Re}\left[2ik\sum_{j\neq(0,0)}a_{\rm
LO}^{*}a_{j}(\Omega)\int\frac{d^{2}x_{\perp}}{\pi w_{D}^{2}}\langle{\rm
HG}_{00}|x_{\perp}\rangle h(x,y)\langle x_{\perp}|{\rm
HG}_{j}\rangle\right].\end{split}$ (25)

Supposing the HG01 mode dominates the random optical imperfections, the noise
spectrum can be written as:

$S^{a_{1}}_{h_{\rm eff}}(\Omega)=\left|\int\frac{d^{2}x_{\perp}}{\pi
w_{D}^{2}}\langle{\rm HG}_{00}|x_{\perp}\rangle h(x,y)\langle x_{\perp}|{\rm
HG}_{01}\rangle\right|^{2}S_{a_{1}a_{1}}(\Omega).$ (26)

With this formula, the picometer displacement sensing accuracy will set a
requirement to the imperfections of the incoming light field, as we will
discuss in the next section.

(2) Similar to the subsection B, the $h(x,y)$ is fluctuation due to the
transversal residue center of mass motion of the test mass:

$\begin{split}\delta^{\prime}h_{\rm eff}(\Omega)={\rm
Re}&\left[2ik\sum_{j\neq(0,0)}a_{\rm LO}^{*}a_{j}\int\frac{d^{2}x_{\perp}}{\pi
w_{D}^{2}}\langle{\rm HG}_{00}|x_{\perp}\rangle\langle x_{\perp}|{\rm
HG}_{j}\rangle\right.\\\ &\left.\frac{\partial h(x,y)}{\partial y}\delta
y_{c}(\Omega)+\frac{\partial h(x,y)}{\partial x}\delta
x_{c}(\Omega)\right],\end{split}$ (27)

with the corresponding noise spectrum as:

$\begin{split}S_{h^{\prime}_{\rm eff}}&(\Omega)=\\\
&\frac{2a^{2}_{1}}{w^{2}_{D}}\left|\int\frac{d^{2}x_{\perp}}{\pi
w_{D}^{2}}\langle{\rm HG}_{00}|x_{\perp}\rangle\langle x_{\perp}|{\rm
HG}_{01}\rangle\frac{\partial h(x,y)}{\partial y}\right|^{2}S_{\delta
x_{c}}(\Omega)\\\
+&\frac{2a^{2}_{1}}{w^{2}_{D}}\left|\int\frac{d^{2}x_{\perp}}{\pi
w_{D}^{2}}\langle{\rm HG}_{00}|x_{\perp}\rangle\langle x_{\perp}|{\rm
HG}_{01}\rangle\frac{\partial h(x,y)}{\partial x}\right|^{2}S_{\delta
y_{c}}(\Omega),\end{split}$ (28)

where the conversion coefficient is plotted in Fig. 8. Since the
interferometer in the GW satellite has a precise control of the beam, the
$a_{j}$ is typically small thereby the $\delta^{\prime}h_{\rm eff}$ should be
a secondary effect compared with the first one.

Figure 8:  Conversion coefficient between the surface roughness and the
imperfections of incoming light $\sqrt{S_{h^{\prime}_{\rm
eff}}/a^{2}_{1}S_{\delta x_{c}}}$.

##  IV Noise level and Test mass manufacture requirements

After analyzing the details of the coupling between the light field and the
surface roughness, this section devotes to discussing the noise level and the
corresponding requirements of the test mass manufacture. This discussion is
divided into two parts: (1) for a given test mass with a fixed and known
surface roughness distribution, we want to estimate the displacement sensing
noise level; (2) for a general manufacturing process, which corresponds to an
ensemble of surface roughness realizations, we want to discuss how the noise
level estimation put a requirement on the test mass manufacturing process.

Firstly, for a given test mass with known surface roughness distribution
$h(x,y)$, it is straightforward to calculate the phase error through the
spectrum of $\delta h_{\rm eff}$, $\delta h^{a_{1}}_{\rm eff}$, and $\delta
h^{\prime}_{\rm eff}$, which is shown in Fig. 9.

Figure 9:  Amplitude spectral densities (ASD) of non-sensitive axis
displacement of LISA Pathfinder Armano  _et al._ (2019).

Secondly, for setting a requirement on the test mass manufacturing process, we
should assume an ensemble of test mass realization to mimic the uncertainty of
the manufacturing process. Each test mass in this ensemble interacts with the
laser field where the light spot is on the geometrical center of one surface
of the test mass cubic. Since these ensemble test masses have a different
realization of the surface roughness, the reflected light field will be
modulated in a different and stochastic way. To give an indicator of surface
manufacture, we need to compute an ensemble average of the phase error over
all surface roughness realizations. According to the _ergodic theorem_ , this
ensemble average is equivalent to the average over different spot center
points on a sufficiently large surface plane, which allows us to make use of
the result given in the previous section.

The error calculated using ergodic theorem mathematically can be represented
as:

$\begin{split}\overline{\delta h^{2}_{\rm
eff}(\Omega)}&=\frac{1}{N}\sum_{j}\delta h^{2}_{\rm eff}[\Omega,h_{j}(x_{\rm
spot})]\\\ &=\frac{1}{A}\int d^{2}x_{\perp c}\delta h^{2}_{\rm
eff}[\Omega,h_{j}(x_{\perp c})],\end{split}$ (29)

where $x_{\perp c}$ is the coordinate of the light spot center at the surface
plane, and $\delta h_{\rm eff}[\Omega,h_{j}(x_{\perp c})]$ is already obtained
in Fig. 6. The gravitational wave detection mission requires that:

$\Delta\delta h_{\rm eff}(\Omega)=\sqrt{\overline{\delta h^{2}_{\rm
eff}(\Omega)}}\leq 1\,{\rm pm}/\sqrt{{\rm Hz}},$ (30)

from which a requirement on the surface roughness can be derived as follows.

Let us take the noise in Eq. (17) as an example. Substituting Eq. (17) into
Eq. (29) and after some straightfoward algebra, the result can be written in
terms of the relationship among the displacement sensing error, surface
roughtness spectral amplitude and the test mass residue motion error:

$\overline{\delta h^{2}_{\rm eff}}(\Omega)=\mathcal{C}^{2}H_{\delta
x_{c}}\overline{\delta x^{2}_{c}}(\Omega)+\mathcal{C}^{2}H_{\delta
y_{c}}\overline{\delta y^{2}_{c}}(\Omega),$ (31)

where the coefficients $H_{\delta x_{c}}$ and $H_{\delta y_{c}}$ are given by:

$\begin{split}H_{\delta x_{c}}=&\int\frac{d^{2}x_{\perp
c}}{A}d^{2}k_{\perp}\frac{2}{\pi
w_{D}^{2}}\frac{1}{1+(k_{x}/k_{\Lambda})^{2}}\frac{1}{1+(k_{y}/k_{\Lambda})^{2}}\\\
&|\mathcal{F}_{k_{x}}[\langle x|{\rm
HG}^{x}_{01}(x_{c})\rangle]\mathcal{F}_{k_{y}}[\langle y|{\rm
HG}^{y}_{00}(y_{c})\rangle]|^{2}\\\ H_{\delta y_{c}}=&\int\frac{d^{2}x_{\perp
c}}{A}d^{2}k_{\perp}\frac{2}{\pi
w_{D}^{2}}\frac{1}{1+(k_{x}/k_{\Lambda})^{2}}\frac{1}{1+(k_{y}/k_{\Lambda})^{2}}\\\
&|\mathcal{F}_{k_{y}}[\langle y|{\rm
HG}^{y}_{01}(y_{c})\rangle]\mathcal{F}_{k_{x}}[\langle x|{\rm
HG}^{x}_{00}(x_{c})\rangle]|^{2},\end{split}$ (32)

in which $\mathcal{F}_{k}[...]$ is the Fourier transformation defined as:

$\mathcal{F}_{k}[f(x)]\equiv\int dxe^{ikx}f(x),$ (33)

and $|{\rm HG}^{x}_{00/01}(x_{c})\rangle$ means the HG modes $|{\rm
HG}^{x}_{00/01}\rangle$ centered at $x_{c}$.

$\begin{split}H_{\delta x_{c}}=H_{\delta
y_{c}}=&\frac{\sqrt{2\pi}k_{\Lambda}^{3}}{w_{D}}e^{-\frac{k_{\Lambda}^{2}w_{D}^{2}}{2}}{\rm
Erfc}(\frac{k_{\Lambda}w_{D}}{\sqrt{2}})\\\
&\left[1-\sqrt{\frac{\pi}{2}}k_{\Lambda}w_{D}e^{-\frac{k_{\Lambda}^{2}w_{D}^{2}}{2}}{\rm
Erfc}(\frac{k_{\Lambda}w_{D}}{\sqrt{2}})\right],\end{split}$ (34)

where ${\rm Erfc}(...)$ is the complementary error function. Finally we have:

$\sigma_{\lambda}=\xi\pi\mathcal{C}k_{\Lambda}\leq\left[\frac{k_{\Lambda}^{2}\overline{\delta
h^{2}_{\rm eff}}(\Omega)}{H_{\delta x_{c}}[\overline{\delta
x^{2}_{c}}(\Omega)+\overline{\delta y^{2}_{c}}(\Omega)]}\right]^{1/2},$ (35)

or equivalently

$\overline{\delta x^{2}_{c}}(\Omega)+\overline{\delta
y^{2}_{c}}(\Omega)\leq\frac{k_{\Lambda}^{2}\overline{\delta h^{2}_{\rm
eff}}}{\pi^{2}\sigma^{2}_{\lambda}H_{\delta x_{c}}}.$ (36)

These formulae set a constraint on the surface roughness once the displacement
sensing error and the transversal residue motion error are given, or a
constraint on the precision for the residue motion control on the insensitive
axis with a given surface roughness. For the space-borne gravitational wave
detection mission, the LISA pathfinder residue motion $\delta x_{c}\sim\delta
y_{c}\sim 10^{-8}\,{\rm m}/\sqrt{{\rm Hz}}$ Armano  _et al._ (2019), the
surface roughness error must be lower than $100\,{\rm nm}$ with $H_{\delta
x_{c}}\sim 10^{12}\,{\rm m^{-4}}$. Moreover, for the fluctuation due to the
imperfections of the incoming light field, the formula Eq. (26) leads to a
requirement for the jitter of the incoming light, where we have
$\sqrt{S_{a_{1}a_{1}}}<10^{-2}/\sqrt{{\rm Hz}}$ for a picometer displacement
sensing accuracy.

##  V Discussion and Conclusions

In this work, we have investigated the effects of test mass surface roughness
in the spaceborne gravitational wave detectors, which was not shown much in
the previous works. It is worth mentioning that for ground-based gravitational
wave detectors such as LIGO/VIRGO, the surface roughness of the suspended
mirrors also impacts the detector sensitivity, which has been studied in many
works Bondarescu  _et al._ (2008); Bondarescu and Thorne (2006); Walsh  _et
al._ (1999b); Ast  _et al._ (2021); Tao  _et al._ (2021). A comparison between
the surface roughness effect on space-borne and ground-based detectors is
worth a discussion.

In the ground-based gravitational wave detector, two test mass mirrors form a
Fabri-Perot arm cavity, which enhances the detector’s sensitivity. The
detector sensitivity is dominated by quantum noise at the interested frequency
band, and it is quantum optically enhanced by injecting the squeezed light
Aasi  _et al._ (2013); Schnabel (2017). Therefore achieving a full-coherent
cavity is the key to the quantum metrology in the ground-based detector.
Surface roughness can scatter the cavity field from the fundamental Gaussian
mode into other higher-order modes, thereby contributing to an optical loss
that can degrade the coherence of the quantum light field. These effect has
been discussed by Drori  _et al._ (2022). Moreover, the mirror surface
roughness can also affect the interferometer designs based on injecting high-
order mode laser, which targets mitigating the thermal noise effect. In these
designs, it is found that the optical scattering by the mirror surface
roughness will seriously affect the interferometer operating with higher-order
Laguerre-Gaussian laser modes, while higher-order Hermit-Gaussian fields could
be useful with future designs Ast  _et al._ (2021); Heinze  _et al._ (2022).
In summary, ground-based detectors are more concerned about the effect of
mirror surface roughness on static optical loss.

Similarly, note that the wavefront distortion generated by the surface
scattering also contributes to a scattering loss as in the ground-based
detector case. However, since the interferometric sensing of the test mass
motion in a LISA-type space-borne detector does not involve the optical cavity
structure and quantum metrology, such a scattering loss will not alter the
shot noise floor. Differently, our work emphasized the dynamic side of this
effect. The residue random motion of the test mass along the non-sensitive
axis couples to the surface roughness, and finally affects the laser field
along the sensitive direction or the fluctuating component of the incoming
field that couples to the Gaussian mode via surface roughness.

In this paper, we have discussed the noise mechanism due to the dynamic
coupling of the test mass surface roughness with the light field. Our
conclusion is that with the residue motion of the test mass along the non-
sensitive axis in the LISA pathfinder, the nano-meter scale surface roughness
only contributes a $\sim 0.01\,{\rm pm}/\sqrt{\rm Hz}$ noise level to the
displacement sensing along the sensitive axis at around $10\,{\rm mHz}$. As
long as the test mass manufacturing state of the art can reduce the surface
roughness below the 100 nm level, a $10\,{\rm nm}/\sqrt{\rm Hz}$ residue
motion would not significantly affect the gravitational wave detection.

###### Acknowledgements.

We thank Dr. Ting Hong for useful discussions. H. Y. is supported by National
Natural Science Foundation of China (Grant No. 12105375). Y. M. is supported
by the start-up funding provided by Huazhong University of Science and
Technology. S. W. is supported by National Natural Science Foundations of
China (Grant No. U20A2077) and National Key R&D Program of China
(2021YFC2202300). H. M. is supported by the State Key Laboratory of Low
Dimensional Quantum Physics and the start-up fund from Tsinghua University. Z.
Z. is supported by National Natural Science Foundations of China (Grant Nos.
11727814 and 11975105).

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