# Geometric tilt-to-length coupling in precision interferometry: mechanisms
and analytical descriptions

Marie-Sophie Hartig, Sönke Schuster, and Gudrun Wanner  Max Planck Institute
for Gravitational Physics (Albert Einstein Institute) and Institute for
Gravitational Physics of the Leibniz Universität Hannover, Callinstrasse 38,
30165 Hannover, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>,

###### Abstract

Tilt-to-length coupling is a technical term for the cross-coupling of angular
or lateral jitter into an interferometric phase signal. It is an important
noise source in precision interferometers and originates either from changes
in the optical path lengths or from wavefront and clipping effects. Within
this paper, we focus on geometric TTL coupling and categorize it into a number
of different mechanisms for which we give analytic expressions. We then show
that this geometric description is not always sufficient to predict the TTL
coupling noise within an interferometer. We, therefore, discuss how
understanding the geometric effects allows TTL noise reduction already by
smart design choices. Additionally, they can be used to counteract the total
measured TTL noise in a system. The presented content applies to a large
variety of precision interferometers, including space gravitational wave
detectors like LISA.

  * January 2022

Keywords: tilt-to-length coupling, optical cross talk, interferometric noise
sources, laser interferometry, space interferometry, LISA, gravitational wave
detection

##  1 Introduction

Precision laser interferometers often share a number of common noise sources,
such as laser frequency noise, electronic readout noise, thermal noise, stray
light and cross talk. There are numerous types of cross-talk since this term
generally describes that a certain signal $s$ is picked up unintentionally by
a sensor not built for sensing $s$. In this paper, we investigate a particular
type of sensing cross-talk: the pickup of angular and lateral motion of a
reflective component within the interferometer by the measured interferometric
phase that is designed to measure longitudinal motion only. To some extent,
also tilt and lateral shifts of the beam’s origin itself are analysed. In
principle, the interferometric phase should only read longitudinal distance
variations between two reference points. Neither lateral motion nor tilts
should be sensed by the interferometric phase. Yet, due to many reasons we
discuss below, the motion in any degree of freedom usually couples to a
certain extent into the phase readout, an effect to which we refer to as tilt-
to-length (TTL) coupling. The TTL coupling is a considerable noise source in
high precision laser interferometry [1, 2, 3, 4, 5, 6, 7]. In LISA Pathfinder,
TTL was visible as a ‘bump’ in the noise spectrum at frequencies between
$20\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}200\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$
[8, 4, 9, 10] and was reduced by realignment and subtraction in data post-
processing [11]. In the second generation Gravity Recovery And Climate
Experiment (GRACE Follow-On) [12, 13, 5], TTL coupling was considered as one
of the highest noise contributors after laser frequency noise in the Laser
Ranging Interferometer (LRI). In flight, the TTL noise was then shown to lie
within the requirements [5]. Furthermore, TTL coupling is of particular
interest in the future space gravitational wave detector LISA [14, 15, 16, 17,
6, 7] where it is one of the most significant noise sources, and a variety of
measures are being taken to suppress it optimally. Also, in other space
gravitation wave detectors like Taiji and TianQin [18, 19, 20, 21], TTL
coupling will be a considerable noise source.

Within this paper, we systematically investigate a variety of TTL coupling
mechanisms. We thereby focus on phase changes originating from alterations in
the optical path length (OPL) of a laser beam, i.e. geometric effects. These
geometric TTL effects are categorised, described analytically and then
classified as first- or second-order effects. Besides the geometric TTL, there
exists non-geometric TTL, i.e. phase changes in the interferometric readout
due to variations in orthogonal degrees of freedom for which the OPL does not
change. Non-geometric effects originate from wavefront and detector properties
and can be described analytically only in special cases. In this work, we will
demonstrate the relevance of these effects and describe how the full signal
computes on an example. We will discuss these non-geometric TTL effects in
detail in a follow-up paper [22] but discuss here how the knowledge of the
geometric TTL can be used to mitigate also non-geometric TTL effects. The
concepts discussed throughout this paper are fundamental and therefore
independent of the application. They can be used in any laboratory experiment
as well as in preparation for space missions such as LISA.

We introduce in Sec. 2 the different systems we consider and how angular and
lateral jitter cause OPL changes. In Sec. 3 we categorise and model the
various geometric TTL effects and reduce the equation for typical applications
to first or second-order effects. In Sec. 4 we show exemplary systems, where a
geometric TTL description is insufficient and fails to describe the
interferometric phase readout. We summarise all effects in Sec. 5, list them
for a typical special case and discuss how well-understood effects can be used
to reduce the total TTL within a system. Finally, we give a conclusion in Sec.
6.

##  2 TTL in different systems due to angular and lateral jitter

Tilt-to-length coupling can occur in any type of interferometer. Only for
initial illustration purposes, we assume a Michelson interferometer as
depicted in Fig. 1. Here, the incident laser beam is split into two arms,
which we call here a reference arm and a measurement arm. The reference beam,
depicted in blue, reflects from a hypothetically perfectly aligned plane
mirror, transmits through the beam splitter and impinges orthogonally and
centred on a photodetector. The measurement beam reflects on a tiltable
mirror, reflects then from the beam splitter and impinges at arbitrary angles
and possibly off-centred on the detector.

Figure 1: Michelson interferometer to illustrate angular cross-coupling – the
measurement beam is reflected from a tilting mirror, while the reference beam
does not change. This results in a misalignment of the two beams and a phase
change of the measurement beam when impinging on the detector. Figure 2:
Geometrical cross-coupling splitting into a ‘lever arm effect’ and ‘piston
effect’. Tilting the mirror yields path length changes due to the rotation of
the beams propagation axis (lever arm, path from $P_{4}$ to $P_{\text{PD,1}}$)
and the displacement of the reflecting surface (piston, path from $P_{2}$ to
$P_{1}$ to $P_{3}$). Thereby, $d_{\text{long}}$ and $d_{\text{lat}}$ define
the longitudinal and lateral distance between the point of reflection and the
centre of rotation. Both are positive here. Mirroring them around the nominal
mirror surface ($d_{\text{long}}$) or orthogonal axis ($d_{\text{lat}}$)
respectively, switches their sign and inverts the corresponding TTL effect.
Arrows pointing clockwise indicate negative angles.

This image is, however, more complex than needed for simulations. In
particular, for describing geometric effects, we can reduce the image to only
the tiltable mirror and the photodiode, and even suppress the reference beam
since it is constant. The result, including a description of the alignment
parameters and the occurring TTL effects, is depicted in Fig. 2. There the
beam nominally impinges on the mirror with angle $\beta$ at point
$\text{P}_{1}$, is reflected and hits the photodiode at
$\text{P}_{\text{PD,0}}$. We thereby assume $-90^{\circ}<\beta<90^{\circ}$ to
ensure that the beam impinges on the mirror’s front surface. If the mirror
then rotates by an angle $\varphi$ around an arbitrary centre of rotation, the
beam is reflected instead at point $\text{P}_{2}$ and hits the photodiode at
$\text{P}_{\text{PD}}$. This alternative beam path has a different path length
than in the nominal case, which means the OPL is angle-dependent. We,
therefore, expect that any angular jitter of the mirror would result in phase
noise. This type of effect is the most obvious TTL contribution: geometric TTL
coupling due to angular jitter of a component resulting in OPL variations.
Also, static mirror tilts can result in OPL changes if the mirror jitters
laterally, as illustrated in Fig. 3. This effect is strongly related to the
angular jitter case, as described in Sec. 3.1 below, such that we also
categorise this effect as TTL.

Figure 3: Sensing cross-talk due to lateral jitter of a mirror. Left: For a
perfect mirror (no surface roughness or defects), the shown mirror motion does
not affect the beam path, and there is no cross-talk. Right: The lateral
jitter of the tilted mirror alters the beam path, resulting in a path length
change. The same principle applies to a tilted receiving system and laterally
jittering detector surface.

One alternative mechanism of TTL coupling is that not a reflecting surface is
rotating, but instead, the receiving system rotates. This can happen, for
instance, when a beam propagates from one optical bench to another, and the
entire receiving bench rotates or is subject to angular jitter. In space
interferometry, this occurs when the receiving spacecraft rotates relative to
the incident wavefront (e.g. in the gravity recovery mission GRACE Follow-On,
or in space gravitational wave detectors such as LISA, Taiji or TianQin). This
scenario is depicted in Fig. 4. We see that a rotation of the system can move
the photodiode position with respect to the beam and therefore change the
beam’s OPL. Like in the case of a reflection from a tilted mirror (see. Fig.
3), lateral jitter can cause additional TTL coupling even if the angle
indicated in Fig. 4 is static. We will quantify and describe the different
kinds of TTL effects below.

Figure 4: Geometrical cross-coupling due to angular jitter of the receiving
system (grey open box) with respect to the incoming beam (red trace). In the
untilted case, the beam originating from point $P_{0}$ hits the detector in
point $P_{1}$. When the receiver rotates by the angle $\varphi_{\text{RS}}$,
the beam hits the detector in point $P_{2}$ instead. The distances
$d_{\text{long}}$ and $d_{\text{lat}}$ define the longitudinal and lateral
distances between the nominal point of incidence $P_{1}$ and the centre of
rotation. Both are positive in this figure. Switching their signs, i.e.
placing the centre of rotation behind the photodiode ($d_{\text{long}}<0$) and
below the beam axis ($d_{\text{lat}}<0$), switches also the sign of the path
length changes.

##  3 Geometric TTL coupling effects

Geometric TTL coupling describes the path length changes of the beams as
depicted in Fig. 2 and 4. In both figures, the nominal beam propagates from
point $P_{0}$ to point $P_{1}$, where it hits a non-tilted surface. In Fig. 2,
the beam gets further reflected and propagates to point $P_{\text{PD,0}}$ on
the detector surface. In both cases, this nominal path serves us as a
reference. We compare the optical path lengths (OPL) of this reference case
with the optical path length the beam accumulates in the case of a tilted
mirror or setup, respectively. We can then define the optical path length
difference (OPD) as the magnitude of the change computed by the difference of
the OPL in the tilted and the nominal case. We will derive the OPD for the
case of a rotating mirror in Sec. 3.1 and describe the other case of the
rotating setup in Sec. 3.2. Furthermore, we will extend our analysis to a
broader set of cases in Sec. 3.4, e.g. we will investigate OPDs due to
transmissive components along the beam path, additional misalignments and
three-dimensional cases. Additionally, we discuss there the relation between
the case of a rotating reflecting surface and a rotating receiving system.

###  3.1 Geometric TTL coupling effects for a reflection at a mirror

We consider first the setup depicted in Fig 2. Let us first assume that the
mirror rotates by an angle $\varphi$ around the reflection point
$\text{P}_{1}$. The beam will then follow the dashed path and hit the
photodiode at $\text{P}_{\text{PD,1}}$ instead. In comparison to the reference
path, the beam propagates now an additional distance given by the geometrical
distance between the points $\text{P}_{4}$ and $\text{P}_{\text{PD,1}}$. We
call this geometric effect the lever arm effect.
If the mirror does not rotate around the reflection point, but around an
arbitrary centre of rotation that differs from the reflection point
longitudinally by $d_{\text{long}}$ and laterally by $d_{\text{lat}}$, the
beam reflects instead at point $\text{P}_{2}$ and hits the photodiode at
$\text{P}_{\text{PD}}$. In this case, the distance between the points
$\text{P}_{3}$ and $\text{P}_{\text{PD,}1}$ is identical to the distance
between $\text{P}_{2}$ and $\text{P}_{\text{PD}}$. Therefore, the only
additional path length change is given by the distances between
$\text{P}_{2}$, $\text{P}_{1}$ and $\text{P}_{3}$. We define this additional
path length change due to an arbitrary centre of rotation as piston effect,
since the reflecting surface moves in and out of the beam like a piston.

By assigning a position vector $\vec{p}_{i}$ to each point Pi, we can express
the beams accumulated optical path length between its defined origin $P_{0}$
and the incidence point P${}_{\text{PD,0}}$ on the photodiode as

$\text{OPL}(\varphi=0)=|\vec{p}_{1}-\vec{p}_{0}|+|\vec{p}_{\text{PD,0}}-\vec{p}_{1}|\,,$
(1)

likewise the OPL in the rotated case can be expressed by

$\text{OPL}(\varphi)=|\vec{p}_{2}-\vec{p}_{0}|+|\vec{p}_{\text{PD}}-\vec{p}_{2}|\,.$
(2)

While Eq. (2) is naturally valid for all angles $\varphi$ including zero, we
deliberately define here Eq. (1) independently for a clearer description of
the equations below.
We are now interested in the optical path length change due to the rotation,
so we compute the optical path length difference (OPD):

$\text{OPD}=\text{OPL}(\varphi)-\text{OPL}(\varphi=0)\,.$ (3)

From Fig. 2 we can see that the OPD is the sum of the lever arm and piston
effect and naturally independent of the choice of the starting point P0:

OPD
$\displaystyle=\left(|\vec{p}_{\text{PD,1}}-\vec{p}_{4}|\right)+\left(|\vec{p}_{2}-\vec{p}_{1}|+|\vec{p}_{\text{3}}-\vec{p}_{1}|\right)$
(4) $\displaystyle=:\text{OPD}_{\text{lever}}+\text{OPD}_{\text{piston}}\,.$
(5)

We use the fact that the distance between P1 and P${}_{\text{PD,0}}$ is
identical to the distance between P1 and P4, and find

$\displaystyle\text{OPD}_{\text{lever}}$
$\displaystyle=|\vec{p}_{\text{PD,1}}-\vec{p}_{1}|-|\vec{p}_{\text{PD,0}}-\vec{p}_{1}|$
(6) $\displaystyle\text{OPD}_{\text{lever}}^{\text{2D}}$
$\displaystyle=d_{\text{lever}}\left[\sec(2\varphi-\varphi_{\text{PD}})\cos(\varphi_{\text{PD}})-1\right]\,.$
(7)

For the piston effect, only absolute length changes are described by Eq. (5),
however, the path length can increase or decrease in comparison to the non
tilted case $\varphi=0$. If we account for this sign, we find:

$\displaystyle\text{OPD}_{\text{piston}}^{\text{2D}}=\,$ $\displaystyle
2\sec(2\varphi-\varphi_{\text{PD}})\cos(\beta+\varphi-\varphi_{\text{PD}})$
(8)
$\displaystyle\cdot\left\\{-d_{\text{lat}}\sin(\varphi)+d_{\text{long}}\left[1-\cos(\varphi)\right]\right\\}\,.$

Here, $d_{\text{long}}$ and $d_{\text{lat}}$ are displacements of the centre
of rotation from the nominal reflection point P1 in longitudinal (i.e. here
orthogonal to the reflecting surface) and lateral direction (here parallel to
the reflecting surface). As before, $d_{\text{long}}$ is defined to be
positive if the incoming beam passes the centre of rotation first and negative
otherwise. Further, $d_{\text{lat}}$ is positive if the beam passes this
centre to the left. Given the setup of Fig. 2, both distances are positive. We
assume that the interferometer is set to measure mirror displacements in the
longitudinal direction, while motion in the lateral direction or angular
jitter are by definition cross-talk - and we will focus only on this cross-
talk.

So far, we have defined variables without time dependency, but have implicitly
assumed that $\varphi=\varphi(t)$ (i.e. the mirror performs angular jitter),
while there is no lateral motion (i.e. $d_{\text{lat}}=\text{const.}$).
Likewise, we could assume that the mirror jitters in lateral direction such
that $d_{\text{lat}}=d_{\text{lat}}(t)$. In the latter case, the piston effect
describes now lateral jitter coupling. Since the very same equation for the
piston effect is being used and only interpreted differently, we do categorize
lateral jitter coupling as a TTL effect. This is particularly visible in Fig.
3 and also Eq. (8). If the mirror was ideally aligned ($\varphi=0$), the
lateral jitter would not couple.
Equations (7) and (8) are valid both for small and large angles $\varphi$. In
our typical applications, we have a very small angular jitter, and the system
is also well aligned. A Taylor series to second-order around zero is therefore
useful for many applications:

$\displaystyle\text{OPD}_{\text{lever}}^{\text{2D}}$
$\displaystyle\approx\leavevmode\nobreak\
2d_{\text{lever}}\left[\varphi^{2}-\varphi\varphi_{\text{PD}}\right]$ (9)
$\displaystyle\text{OPD}_{\text{piston}}^{\text{2D}}$
$\displaystyle\approx-2d_{\text{lat}}\left[\cos(\beta)+\varphi_{\text{PD}}\sin(\beta)\right]\varphi$
(10)
$\displaystyle\;+\left[2d_{\text{lat}}\sin(\beta)+d_{\text{long}}\cos(\beta)\right]\varphi^{2}\,.$

Especially for the case of normal incidence or nearly normal incidence
($\beta\approxeq 0,\varphi_{\text{PD}}\approxeq 0$), the equations reduce to

$\displaystyle\text{OPD}_{\text{lever}}^{\text{2D}}$
$\displaystyle\approx\leavevmode\nobreak\ 2d_{\text{lever}}\varphi^{2}$ (11)
$\displaystyle\text{OPD}_{\text{piston}}^{\text{2D}}$
$\displaystyle\approx-2d_{\text{lat}}\varphi+d_{\text{long}}\varphi^{2}\,.$
(12)

In this special case, we find that the lever arm effect is purely second-
order, while the piston effect splits into a first-order due to the lateral
displacement and a second-order due to the longitudinal displacement of the
component’s centre of rotation with respect to the beam reflection point on
the component.
This finding is of particular relevance for suppressing TTL in typical optical
setups: Since these two geometric effects are known and well understood as
described above, they can be used to counteract any unknown TTL effect,
originating, for instance, from the non-geometric coupling. We will further
discuss this in Sec. 5.

###  3.2 Geometric TTL coupling effects for rotating systems

Let us now assume a case of a freely propagating laser beam that is perfectly
stable and does not jitter in any degree of freedom. This beam is then
incident on a jittering receiving system, e.g. an optical table in a
laboratory setup or a remote satellite in a space mission like GRACE-FO, LISA,
Taiji or TianQin. We now assume the receiving system (i.e. optical bench or
satellite) to jitter relative to the incident beam. In this case, the
reference beam as well as all other possibly existent components such as
mirrors, beamsplitters and photodiodes move perfectly synchronously, such that
the OPL of the reference beam is constant. However, the photodiode then moves
in and out of the received beam, as indicated in Fig. 4, resulting in an OPL
change, and therefore a geometric TTL coupling.

The geometric TTL coupling can easily be calculated by comparing the path
lengths of the beam for the non-rotated and the rotated case. While the OPL
for the nominal, non-rotated case is given by

$\displaystyle\text{OPL}_{\text{RS}}(\varphi_{\text{RS}}=0)=|\vec{p}_{1}-\vec{p}_{0}|\,,$
(13)

the OPL for the rotated case, i.e. $\varphi_{\text{RS}}\neq 0$, is in Fig. 4
shorter, namely

$\displaystyle\text{OPL}_{\text{RS}}(\varphi_{\text{RS}}=0)=|\vec{p}_{2}-\vec{p}_{0}|\,.$
(14)

The OPD is then given by the difference between the rotated and the non-
rotated case, i.e.

$\displaystyle\text{OPD}_{\text{RS}}=|\vec{p}_{2}-\vec{p}_{0}|-|\vec{p}_{1}-\vec{p}_{0}|\,.$
(15)

This OPD can be described analytically by the rotation angle
$\varphi_{\text{RS}}$, and the distance between the point of detection and the
centre of rotation. The absolute longitudinal difference, $d_{\text{long}}$,
is in this paper defined by the distance between both points parallel to the
nominal beam propagation axis. Meanwhile, the sign of $d_{\text{long}}$ is
positive if the centre of rotation lies before the point of detection and
negative otherwise. Analogously, the absolute lateral difference,
$d_{\text{lat}}$, is given by the distance between both points along the axis
orthogonal to the beam propagation axis. We define $d_{\text{lat}}$ to be
positive if the beam “sees” it at the right-hand side and negative otherwise,
e.g. like in the case depicted in Fig. 4. With this, the OPD is given by

$\displaystyle\text{OPD}_{\text{RS}}^{\text{2D}}=\sec(\varphi_{\text{RS}})\left\\{-d_{\text{lat}}\sin(\varphi_{\text{RS}})+d_{\text{long}}\left[1-\cos(\varphi_{\text{RS}})\right]\right\\}\,,$
(16)

or, allowing the photodiode to be tilted by an angle $\varphi_{\text{PD}}$
like in the case of a rotating mirror (compare Fig. 2), we have

$\displaystyle\text{OPD}_{\text{RS}}^{\text{2D}}$
$\displaystyle=-d_{\text{lat}}\sec(\varphi_{\text{PD}})\sec(\varphi_{\text{RS}}+\varphi_{\text{PD}})\sin(\varphi_{\text{RS}})$
(17)
$\displaystyle\,+d_{\text{long}}\left[\cos(\varphi_{\text{PD}})\sec(\varphi_{\text{RS}}+\varphi_{\text{PD}})-1\right]\\}\,.$

Assuming small rotation angles $\varphi_{\text{RS}}$, we can Taylor expand
this equation and find

$\displaystyle\text{OPD}_{\text{RS}}^{\text{2D}}\approx-
d_{\text{lat}}\,\varphi_{\text{RS}}+d_{\text{long}}\,\varphi_{\text{RS}}^{2}/2\,$
(18)

or, having a tilted photodiode,

$\displaystyle\text{OPD}_{\text{RS}}^{\text{2D}}\approx-
d_{\text{lat}}\,\varphi_{\text{RS}}+d_{\text{long}}\left(\varphi_{\text{RS}}^{2}/2+\varphi_{\text{RS}}\varphi_{\text{PD}}\right).$
(19)

Hence, the lateral distance between the beam axis and the centre of rotation
makes a linear TTL coupling and the longitudinal distance adds a second-order
coupling term. This means that particularly lateral displacements between the
centre of rotation and the beam axis should be avoided. Ideally, the centre of
rotation should lie in the measurement beam incidence point on the photodiode.
In that case, the geometric TTL described here would vanish entirely.

The equations presented for the receiving system have so far been derived for
angular jitter coupling, only. Contrary to the setup with a jittering mirror,
lateral jitter does not couple into the geometric length readout in the
receiver case. This is due to the definition of the lateral jitter axis, which
is an axis fixed to the receiving system. In Fig. 4 it is parallel to the
detector surface. Thus, the measurement beam would walk along the detector
surface, but its length would stay unchanged. However, if the photodiode
happens to be misaligned by an angle $\varphi_{\text{PD}}$ with respect to the
receiver coordinate system, we find for lateral receiving system jitter
$y_{\text{RS}}$ also linear TTL coupling:

$\displaystyle\text{OPD}_{\text{RS}}^{\text{2D}}$
$\displaystyle=y_{\text{RS}}\,\frac{\sin(\varphi_{\text{PD}})}{\cos(\varphi_{\text{RS}}+\varphi_{\text{PD}})}$
(20) $\displaystyle\approx y_{\text{RS}}\,\varphi_{\text{PD}}\,.$ (21)

###  3.3 Comparison of mirror and system rotation

We now want to compare the geometric TTL coupling in the case of a mirror
rotation featuring a beam of normal incidence at the mirror (Eq. (12)) with
the case of a system rotation (Eq. (18)). The two equations are identical,
except for a factor of two. This is not surprising since in the first case,
the rotation of the system effectively causes the detector surface to move
into (or out of) the beam, while in the system rotation case, the mirror
surface moves into (or out of) the beam in the very same way. The factor of
two is then coming from the fact that the mirror surface reflects the beam,
such that the beam propagates the shortened (or additional) length twice.
However, in the case of the system rotation, the detector absorbs the beam,
such that the path length change is not doubled.

###  3.4 Extension for broader set of applications

We will now extend the geometric TTL equations introduced above for a wider
set of applications. We will start with the effect of plane-parallel
transmissive components along the beam path in Sec. 3.4.1. Then, we discuss in
Sec. 3.4.2 how the equations need to be adapted if the setup is located in a
medium other than vacuum. In Sec. 3.4.3 we describe the effect of initial
misalignment and then discuss in Sec. 3.4.4 the relevance of the photodiode
angle $\varphi_{\text{PD}}$. Finally, in Sec. 3.4.5 we extend the equations
for jitter in three-dimensional space.

####  3.4.1 Additional plane-parallel transmissive components

For a more complete theory, we introduce transmissive optical components (tc)
with plane-parallel surfaces, e.g. optical windows, beam splitters or beam
combiners, as shown in Fig. 5. For receiver jitter, we generally need to
consider all components along the beam path. However, in the case of
reflection at a jittering mirror, we can ignore all transmissive and
reflective components prior to the mirror because these change only the
initial conditions (i.e. incidence angle $\beta$ and incidence point $P_{2}$)
of the reference case. Their effect can therefore be accounted for by a simple
adjustment of the parameters $\beta$ and $d_{\text{lat}}$, which does not need
to be particularly modelled. On the contrary, components along the beam path
after the reflection from the jittering mirror directly affect the TTL
coupling.

For all considered components, we assume a thickness $t_{\text{BS},i}$, a
refractive index $n_{\text{BS},i}$ and that their surface normal in a two-
dimensional setup is rotated by an angle $\varphi_{\text{BS},i}$ against the
beam direction of the initial beam. Hence, the beams hit the transmissive
component under an angle $\varphi_{\text{BS}}-\varphi_{m}$, where
$\varphi_{m}$ defines the orientation of the beam, i.e.
$\varphi_{m}\rightarrow 2\varphi$ in the case of a mirror rotation and
$\varphi_{m}\rightarrow-\varphi_{\text{RS}}$ in the case of a rotating system.
While the beam travels with the speed of light through vacuum, it slows down
when entering the component $i$. Thus its path length within the component
scales with $n_{\text{BS},i}$ and seen by the photodiode the point of
reflection seems to be further away.

Figure 5: Optical path length changes due to a transmissive optical component:
Instead of propagating the distance
$|\vec{p}_{\mathrm{tc1}}-\vec{p}_{\mathrm{tc0}}|$ through vacuum (dashed path)
the beam propagates from $P_{\mathrm{tc0}}$ to $P_{\mathrm{tc1^{\prime}}}$
through the medium of the component and additionally from
$P_{\mathrm{tc1^{\prime}}}$ to $P_{\mathrm{tc2^{\prime}}}$ through vacuum
(solid path). The beam direction within the medium is defined by the angle
$\psi_{\text{BS}}=\arcsin(\sin(\varphi_{\text{BS},i}-\varphi_{m})/n_{\text{BS},i})$,
where $\varphi_{\text{BS}}$ defines the alignment of the component with
respect to the nominal beam and $\varphi_{m}$ the additional tilt of the beam,
i.e. $\varphi_{m}\rightarrow 2\varphi$ in the case of a mirror rotation and
$\varphi_{m}\rightarrow-\varphi_{\text{RS}}$ in the case of a rotating
receiver.

Additional to the path length scaling, the direction of the beam changes when
it enters a component as shown in Fig. 5. The new beam direction relative to
the surface normal vector of the component $i$ is given by the angle of
refraction

$\displaystyle\psi_{\text{BS}}=\arcsin\left(\frac{\sin(\varphi_{\text{BS},i}-\varphi_{m})}{n_{\text{BS},i}}\right)\,.$
(22)

When leaving the component, the beam continues propagating in its original
direction. The additional optical path length by each transmissive component
is then given by

$\text{OPL}_{\text{tc}}=n_{\text{BS}}\,|\vec{p}_{\mathrm{tc1^{\prime}}}-\vec{p}_{\mathrm{tc0}}|+|\vec{p}_{\mathrm{tc2^{\prime}}}-\vec{p}_{\mathrm{tc1^{\prime}}}|-|\vec{p}_{\mathrm{tc1}}-\vec{p}_{\mathrm{tc0}}|\,,$
(23)

which is the new path length inside the respective component
$|\vec{p}_{\mathrm{tc1^{\prime}}}-\vec{p}_{\mathrm{tc0}}|$ scaled with its
refractive index $n_{\mathrm{BS}}$ plus the distance between the point
$P_{\mathrm{tc1^{\prime}}}$ where the beam leaves the component and the
intersection point $P_{\mathrm{tc2^{\prime}}}$ between the new beam path and
the line parallel to the detector (which also intersect with the original exit
point $P_{\mathrm{tc1}}$), i.e.
$|\vec{p}_{\mathrm{tc2^{\prime}}}-\vec{p}_{\mathrm{tc1^{\prime}}}|$. From that
we have to subtract the original path length
$|\vec{p}_{\mathrm{tc1}}-\vec{p}_{\mathrm{tc0}}|$. We can evaluate Eq. (23)
for $N$ transmissive components

$\displaystyle\text{OPL}_{\text{tc}}^{\text{2D}}$
$\displaystyle=\sum_{i=1}^{N}n_{\text{BS},i}$
$\displaystyle\,|\vec{p}_{\mathrm{tc1^{\prime}},i}-\vec{p}_{\mathrm{tc0},i}|-|\vec{p}_{\mathrm{tc1},i}-\vec{p}_{\mathrm{tc0},i}|+|\vec{p}_{\mathrm{tc2^{\prime}},i}-\vec{p}_{\mathrm{tc1^{\prime}},i}|$
(25) $\displaystyle=\sum_{i=1}^{N}\,t_{\text{BS},i}$
$\displaystyle\Big{\\{}\,n_{\text{BS},i}^{2}\left[n_{\text{BS},i}^{2}-\sin^{2}(\varphi_{m}-\varphi_{\text{BS},i})\right]^{-1/2}-\sec(\varphi_{m}-\varphi_{\text{BS},i})$
$\displaystyle\
+\sin(\varphi_{\text{BS},i}-\varphi_{\text{PD}})\,\sec(\varphi_{m}-\varphi_{\text{PD}})$
$\displaystyle\ \>\cdot\Big{[}\sin(\varphi_{m}-\varphi_{\text{BS},i})\
\left[n_{\text{BS},i}^{2}-\sin^{2}(\varphi_{m}-\varphi_{\text{BS},i})\right]^{-1/2}$
$\displaystyle\quad\ -\tan(\varphi_{m}-\varphi_{\text{BS},i})\Big{]}\Big{\\}}$
$\displaystyle\text{OPD}_{\text{tc}}^{\text{2D}}$
$\displaystyle\approx\sum_{i=1}^{N}\frac{t_{\text{BS},i}}{2}$
$\displaystyle\Big{\\{}n_{\text{BS},i}^{2}\cos^{2}(\varphi_{\text{BS},i})\left[(n_{\text{BS},i}^{2}-\sin^{2}(\varphi_{\text{BS},i})\right]^{-3/2}$
(26) $\displaystyle\
-\sin^{2}(\varphi_{\text{BS},i})\left[n_{\text{BS},i}^{2}-\sin^{2}(\varphi_{\text{BS},i})\right]^{-1/2}$
$\displaystyle\
-\cos(\varphi_{\text{BS},i})\Big{\\}}\left(\varphi^{2}_{m}-2\varphi_{m}\,\varphi_{\text{PD}}\right)\,.$

According to Eq. (26) the optical cross-talk due to transmissive components is
a second-order effect. For a few thin components and rather small angles
$\varphi_{\text{BS},i}$ we find it to be smaller than the lever arm effect.
Nevertheless, it should not be neglected when high accuracy is required.
The geometric TTL effects described above and the OPL change due to
transmissive components, as it was defined in this section, are independent of
one another. We, therefore, get the full geometric coupling by adding
OPD${}_{\text{tc}}$ to the previously found TTL effects, namely

$\displaystyle\text{OPD}_{\text{MRT}}$
$\displaystyle=\text{OPD}_{\text{lever}}+\text{OPD}_{\text{piston}}+\text{OPD}_{\text{tc}}\,,$
(27) $\displaystyle\text{OPD}_{\text{RST}}$
$\displaystyle=\text{OPD}_{\text{RS}}+\text{OPD}_{\text{tc}}\,.$ (28)

####  3.4.2 Experiments in air or other surrounding medium

The above equations of the lever arm and piston effect,
$\text{OPL}_{\text{lever}}$ and $\text{OPL}_{\text{piston}}$, as well as the
analytic description for rotating systems, describe the changes in the OPL
while assuming that the beams propagate through vacuum. If this is not the
case, we need to multiply all found OPLs in Sec. 3.2 and Sec. 3.1 by the
medium’s refractive index $n$. For transmissive components, we replace
$n_{\text{BS},i}$ in equations (25) and (26) by $n_{\text{BS},i}/n$ and
multiply the full path length change by $n$. This additional factor originates
from the lower speed of light in a medium than in a vacuum resulting in an
increase of the light travelling time in the medium, which then results in the
larger phase change. Therefore, the beam paths appear longer when detected at
the photodiode unless it is accounted for by a dedicated setting or
calibration of the phasemeter.

####  3.4.3 Additional misalignment

In sections 3.1 and 3.4.1 we compared the initial beam path length with the
path length of a beam that got reflected at a misaligned mirror. In
experiments, we often face the case that the mirror $M_{\text{tilt}}$ is
nominally misaligned and we are interested in path length changes to this
nominal case. This can be easily implemented into our formulas by substituting
$\varphi\rightarrow\varphi^{\prime}+\varphi_{0}$ and
$d_{\text{lat}}\rightarrow d_{\text{lat}}^{\prime}+y_{0}$, where $\varphi_{0}$
denotes the nominal tilt and $y_{0}$ the nominal lateral shift of the mirror.
All OPDs are then defined as the difference of the OPL for an arbitrary angle
$\varphi^{\prime}$ minus the OPL at a nominal angle, i.e. at
$\varphi^{\prime}=0$:

$\text{OPD}=\text{OPL}(\varphi^{\prime})-\text{OPL}(\varphi^{\prime}=0)\;.$
(29)

Angular misalignment can likewise happen in the case of a rotating system (see
Sec. 3.2). Due to experimental tolerances, the system is likely slightly
misaligned with respect to the incoming beam. The equations in Sec. 3.2 can
then be adapted analogously.

####  3.4.4 Relevance of the photodiode angle

The OPD depends on the photodiode angle via

$\displaystyle\text{OPD}_{\text{MRT}}(\varphi_{\text{PD}})-\text{OPD}_{\text{MRT}}(\varphi_{\text{PD}}=0)$
$\displaystyle=$
$\displaystyle\Bigg{\\{}\sum_{i=1}^{N}t_{\text{BS},i}\Big{\\{}n_{\text{BS},i}^{2}\cos^{2}(\varphi_{\text{BS},i})\left[(n_{\text{BS},i}^{2}-\sin^{2}(\varphi_{\text{BS},i})\right]^{-3/2}$
$\displaystyle\qquad\qquad\
-\sin^{2}(\varphi_{\text{BS},i})\left[n_{\text{BS},i}^{2}-\sin^{2}(\varphi_{\text{BS},i})\right]^{-1/2}-\cos(\varphi_{\text{BS},i})\Big{\\}}$
$\displaystyle\
+d_{\text{lever}}+d_{\text{lat}}\,\sin(\beta)\Bigg{\\}}\left(-2\varphi\,\varphi_{\text{PD}}\right)\,,$
$\displaystyle\text{OPD}_{\text{RST}}(\varphi_{\text{PD}})-\text{OPD}_{\text{RST}}(\varphi_{\text{PD}}=0)$
$\displaystyle=$
$\displaystyle\Bigg{\\{}\sum_{i=1}^{N}t_{\text{BS},i}\Big{\\{}n_{\text{BS},i}^{2}\cos^{2}(\varphi_{\text{BS},i})\left[(n_{\text{BS},i}^{2}-\sin^{2}(\varphi_{\text{BS},i})\right]^{-3/2}$
(31) $\displaystyle\qquad\qquad\
-\sin^{2}(\varphi_{\text{BS},i})\left[n_{\text{BS},i}^{2}-\sin^{2}(\varphi_{\text{BS},i})\right]^{-1/2}-\cos(\varphi_{\text{BS},i})\Big{\\}}$
$\displaystyle\
+d_{\text{long}}\Bigg{\\}}\left(\varphi_{\text{RS}}\,\varphi_{\text{PD}}\right)\,.$

This is natural due to the geometric nature of the OPD. However, the tilt of
the detector affects how the wavefronts of both beams, the measurement and
reference beam, are being detected. We will, therefore, show in [22] that the
non-geometric effects contribute further terms that cancel the detector
orientation from the total interferometric readout. All terms containing
$\varphi_{\text{PD}}$ should therefore be handled with care: they do exist in
the OPD but do not affect the total interferometric readout. It can,
therefore, be an advantage to neglect the photodiode angle
$\varphi_{\text{PD}}$=0 in simulations, even in cases where the corresponding
experiment comprises large angles.

####  3.4.5 Three-dimensional case

So far, we only considered a two-dimensional setup, which can then be applied
in 3D setups for changes in a specific plane. If angular jitter occurs in two
orthogonal planes, one can simply add up the effect of the individual planes,
provided that we have beams with normal incidence and the jitter is small and
therefore describable by a first-order series expansion. If the jitter causing
the TTL has a larger amplitude, effects occurring in one plane also affect the
orthogonal plane. This can, for instance, be seen in the lever arm effect, see
Eq. (10). There, a rotation in the orthogonal plane would affect the length of
the now three-dimensional lever arm $d_{\text{lever}}$, which results in a
product term of the involved angles, i.e. $\varphi\cdot\eta$.

As illustrated in Fig. 2 and Fig. 4, all equations given so far were
describing motion and alignment in the $xy$-plane. The angle $\varphi$ (or
$\varphi_{\text{RS}}$ respectively) is, therefore, a yaw angle originating
from a rotation around the $z$-axis. The orthogonal pitch rotation $\eta$ (or
$\eta_{\text{RS}}$) is then a rotation around the $y$-axis. The cross-coupling
between yaw and pitch makes the equations significantly more complex. We,
therefore, series expand the expressions for the 3D geometric effects to
second-order and find:

$\displaystyle\text{OPD}_{\text{lever}}^{\text{3D}}$
$\displaystyle=d_{\text{lever}}\Bigg{[}\frac{2\cos^{2}(\beta_{z})(\varphi^{2}-\varphi\varphi_{\text{PD}})+2\cos^{2}(\beta_{y})(\eta^{2}-\eta\eta_{\text{PD}})}{\cos^{2}(\beta_{y})+\cos^{2}(\beta_{z})\sin^{2}(\beta_{y})}$
(32)
$\displaystyle\;+\frac{\sin(2\beta_{y})\sin(2\beta_{z})\,\varphi\,\eta}{\cos^{2}(\beta_{y})+\cos^{2}(\beta_{z})\sin^{2}(\beta_{y})}\Bigg{]}$
$\displaystyle\text{OPD}_{\text{piston}}^{\text{3D}}$
$\displaystyle=2\left(-d_{\text{lat}}\,\varphi+d_{\text{vert}}\,\eta\right){\hbox{$\left[\vbox
to20.5pt{}\right.\n@space$}}\frac{\cos(\beta_{y})\cos(\beta_{z})}{\sqrt{\cos^{2}(\beta_{y})+\cos^{2}(\beta_{z})\sin^{2}(\beta_{y})}}$
(33)
$\displaystyle\;-\frac{\sin(\beta_{y})\cos(\beta_{z})(\varphi-\varphi_{\text{PD}})+\sin(\beta_{z})\cos(\beta_{y})(\eta-\eta_{\text{PD}})}{\sqrt{\cos^{2}(\beta_{y})+\cos^{2}(\beta_{z})\sin^{2}(\beta_{y})}}{\hbox{$\left]\vbox
to20.5pt{}\right.\n@space$}}$
$\displaystyle+d_{\text{long}}\,\frac{\cos(\beta_{y})\cos(\beta_{z})\,\left(\varphi^{2}+\eta^{2}\right)}{\sqrt{\cos^{2}(\beta_{y})+\cos^{2}(\beta_{z})\sin^{2}(\beta_{y})}}$
$\displaystyle\text{OPD}_{\text{RS}}^{\text{3D}}$
$\displaystyle=-d_{\text{lat}}\,\varphi_{\text{RS}}+d_{\text{vert}}\,\eta_{\text{RS}}+d_{\text{long}}\,\left(\frac{\varphi_{\text{RS}}^{2}+\eta_{\text{RS}}^{2}}{2}\right)\,,$
(34)

where $\beta_{y},\beta_{z}$ are the projection-angles of the incoming beam at
the mirror into the $xy$\- and $xz$-plane, respectively. The different signs
of the $d_{\text{lat}}$\- and $d_{\text{vert}}$-terms arise from the
orientations of the pitch and yaw angles.

Photodiode tilts in the $xz$-plane are indicated by $\eta_{\text{PD}}$, and
the vertical difference between the point of incidence and the point of
reflection is given by $d_{\text{vert}}$. Naturally, all three-dimensional
equations reduce to their two-dimensional counterparts if the parameters in
the respective orthogonal plane are set to zero.

Another special case arises if the nominal beams hit the test mass
orthogonally, i.e. $\beta_{y,z}=0$. Here, all cross-plane couplings vanish and
the resulting signal equals the sum of the two two-dimensional equations
applying the same assumption. We get

$\displaystyle\text{OPD}_{\text{lever}}^{\text{3D}}$
$\displaystyle=\,d_{\text{lever}}\left[2(\varphi^{2}-\varphi\varphi_{\text{PD}})+2(\eta^{2}-\eta\eta_{\text{PD}})\right]$
(35) $\displaystyle\text{OPD}_{\text{piston}}^{\text{3D}}$
$\displaystyle=\,d_{\text{long}}\left(\varphi^{2}+\eta^{2}\right)+2\left(-d_{\text{lat}}\,\varphi+d_{\text{vert}}\,\eta\right)$
(36)

Also, transmissive components affect the beam in all degrees of freedom. In a
three-dimensional setup, we can now additionally consider the components to be
rotated in yaw by $\varphi_{\text{BS}}$ and in pitch by $\eta_{\text{BS}}$.
Hence their 3D-equations become very long such that we summarise here only the
principle behaviour. In the case of normal incidence
($\varphi_{\text{BS}}=\eta_{\text{BS}}=0$), the transmissive components add a
second-order effect. For non-normal incidence, an additional linear effect
occurs, see Fig. 6. This linear effect is typically small, such that the
second-order effects stay dominant.

Figure 6: Optical path length difference of a beam, that got tilted in pitch
($\eta_{\text{RS}}$) and yaw ($\varphi_{\text{RS}}$), passing a transmissive
optical component. We set $t_{\text{tc}}=10\,$mm, $n_{\text{tc}}=2$ and
rotated the component with respect to the beam by
$\varphi_{\text{tc}}=0.2\,\pi$ and $\eta_{\text{tc}}=0.13\,\pi$. While the
second-order OPD is dominant, the color grid indicates, that the tilt of the
transmissive component also adds linear TTL coupling.

##  4 Exemplary setups showing limitations of geometrical estimates

The cross-coupling mechanisms explained so far originate from variations of
the beam propagation axis that can be described geometrically. In order to
treat these effects as introduced in Sec. 3.1, we did assume that the laser
beams are classical rays. This assumption, however, is often insufficient
since it neglects all wavefront properties. We will show in the examples below
that the expected TTL crosstalk can significantly deviate from geometrically
expected values if the laser light is instead described as fundamental
Gaussian mode.

###  4.1 Rotation around the centre of curvature

Assume the setup previously published in [23, 24] and illustrated in Fig. 7.
There, we consider two identical Gaussian beams, i.e. with the same intensity
distribution and phase fronts. We rotate one of the beams (measurement beam)
with respect to the other around a pivot that has a longitudinal distance to
the detector that is equal to the radius of curvature of the wavefront on the
detector.

Figure 7: A Gaussian beam is tilted around a pivot that coincides with the
centre of the wavefront curvature at the detector position. While the beam is
tilted, the phase distribution on the detector stays unchanged. Figure 8:
Simulated path length signals (i.e. LPS and OPD) compared to the expected
analytic TTL coupling in the scenario shown in Fig. 7, i.e. the lever arm
coupling if the rotation is induced by a rotating test mass. Hence the beam
angle $\varphi_{m}$ depends on the mirror tilt via $\varphi_{m}=2\varphi$. The
expected lever arm TTL coupling cannot be observed in the simulation due to
the special conditions of this scenario (rotation without change of the
wavefront). In this simulation, we assumed identical Gaussian beams with a
waist radius $w_{0}=$1 mm, distance from waist at detector $z=z_{R}$, lever
arm $d_{\text{lever}}=2z_{R}$, single element photodiode with radius 5 mm.

In this setup, a lever arm between the point of rotation and detector should
lead to significant cross-coupling. However, in this scenario, a tilt of the
measurement beam does not change the phase distribution on the detector since
the wavefront is mapped onto itself during the rotation. That means only the
intensity distribution on the detector plane changes during the rotation. If,
however, the reference wavefront curvature equals the measurement wavefront
curvature, then the phase difference is zero in every detector point such that
zero TTL coupling occurs despite the rotation of the beam axis.

To demonstrate that, in fact, no cross-coupling is present in this scenario,
we numerically compute the variations of the longitudinal pathlength sensing
(LPS) signal, i.e. the interferometric phase signal processed by the detector
converted to units of lengths. In the computation of this signal, not only the
OPD but also wavefront and detector properties are being considered (see [24]
and [22] for details on the numerical computation of this signal). This was
done using the simulation tool IfoCAD [24, 25]. The obtained LPS signal is
shown in Fig. 8. We compare it with the numerically derived OPD, which
corresponds to the analytic expression of the lever arm cross-coupling Eq.
(7). Unlike the geometric description, the simulated LPS signal shows no
cross-coupling, just as expected, demonstrating that the pure geometric
description of this scenario is not sufficient. The resulting TTL coupling can
only be correctly estimated when non-geometric effects are considered as well.

###  4.2 Identical fundamental Gaussian beams with centre of rotation on the
beam axis

Figure 9: Simulated path length signal compared to the expected lever arm TTL
coupling in the scenario with two equal Gaussian beams and a large single
element detector. The beam angle equals two times the mirror tilt. The TTL
coupling vanishes due to symmetric phase cancellations. The simulation
parameters were: waist radius $w_{0}=$1 mm, distance from waist at the
detector $z=100\,$mm, lever arm $d_{\text{lever}}=100\,$mm, detector radius
100 mm.

The cross-coupling also vanishes for two identical Gaussian beams with one
beam tilted around a pivot that has now an arbitrary longitudinal but no
lateral distance to the detector. In this case, we discover a lever arm effect
and not necessarily equal wavefronts. However, we find no cross-coupling as
illustrated in Fig. 9 and introduced in [26]. We conclude that a non-geometric
cross-coupling mechanism caused by the tilt dependent beam walk on the
detector cancels the expected lever arm cross-coupling. A general analytic
description of the non-geometric effect is given in the next section.
Additionally, an intuitive approach to understand the cancellation is given in
A.

##  5 Summary geometric TTL coupling and TTL mitigation strategies

In this paper, we have introduced different geometric TTL coupling effects and
derived equations that describe whether they add primarily linear, quadratic
or mixed terms to the OPD. All effects are summarized in Tab. 1 for
convenience. Lateral jitter coupling appears as a linear effect only in the
case of mirror reflection:

$\displaystyle\text{OPD}_{\text{piston,lat}}^{\text{2D}}(d_{\text{lat}}(t))$
$\displaystyle\approx-2d_{\text{lat}}(t)\,\varphi\,,$ (37)
$\displaystyle\text{OPD}_{\text{RS,lat}}^{\text{2D}}$ $\displaystyle=0\,.$
(38)

For angular jitter TTL we have the following linear TTL effects:

$\displaystyle\text{OPD}_{\text{piston,lat}}^{\text{2D}}(\varphi(t))$
$\displaystyle\approx-2d_{\text{lat}}\,\varphi(t)\,,$ (39)
$\displaystyle\text{OPD}_{\text{RS,lat}}^{\text{2D}}(\varphi_{\text{RS}}(t))$
$\displaystyle\approx-d_{\text{lat}}\,\varphi_{\text{RS}}(t)\,,$ (40)

which are caused by lateral displacements of the centre of rotation with
respect to the principle beam axis. Second-order TTL effects are:

$\displaystyle\text{OPD}_{\text{lever}}^{\text{2D}}$ $\displaystyle\approx
2d_{\text{lever}}\,\varphi^{2}\,,$ (41)
$\displaystyle\text{OPD}_{\text{piston,long}}^{\text{2D}}$
$\displaystyle\approx d_{\text{long}}\,\varphi^{2}\,,$ (42)
$\displaystyle\text{OPD}_{\text{RS,long}}^{\text{2D}}$
$\displaystyle\approx\frac{1}{2}d_{\text{long}}\,\varphi_{\text{RS}}^{2}\;,$
(43)

which all originate from longitudinal displacements of the centre of rotation
with respect to either the beam incidence point on the reflecting component
($d_{\text{long}}$), or the photodiode ($d_{\text{long}}$,
$d_{\text{lever}}$). An additional second-order TTL effect is caused by
transmissive components:

$\displaystyle\text{OPD}_{\text{tc}}^{\text{2D}}$
$\displaystyle\approx\sum_{i=1}^{N}\frac{t_{\text{BS},i}}{2}$
$\displaystyle\Big{\\{}n_{\text{BS},i}^{2}\cos^{2}(\varphi_{\text{BS},i})\left[(n_{\text{BS},i}^{2}-\sin^{2}(\varphi_{\text{BS},i})\right]^{-3/2}$
(44) $\displaystyle\
-\sin^{2}(\varphi_{\text{BS},i})\left[n_{\text{BS},i}^{2}-\sin^{2}(\varphi_{\text{BS},i})\right]^{-1/2}$
$\displaystyle\ -\cos(\varphi_{\text{BS},i})\Big{\\}}\,\varphi^{2}_{m}\,,$

with $\varphi_{m}=2\varphi$ for components between the mirror and the detector
and $\varphi_{m}=-\varphi_{\text{RS}}$ for components along the beam path
inside a receiving system with angular jitter. The knowledge about these
first- and second-order couplings can be used to mitigate the geometric
effects by smart design choices, for instance, by aligning the beam axis to
the centre of rotation ($d_{\text{lat}}=0$) or by imaging the centre of
rotation to the photodiode centre, resulting in
$d_{\text{long}}=d_{\text{lever}}=0$. Lateral jitter coupling is minimized in
the case of a mirror reflection by an angular adjustment to make $\varphi=0$.
Moreover, the given equations can be used to counteract other TTL effects,
i.e. the non-geometric TTL effects, which we will introduce in our follow-up
paper [22], that are more difficult to describe or mitigate. In that case, the
total TTL of the system is measured, i.e. the sum of all geometric and non-
geometric effects. The total linear effects can then be minimized by adjusting
either of the lateral alignment parameters, for instance, by adjusting the
beam axis. The sum of all second-order effects is in the next step minimized
by adjusting either of the longitudinal parameters, for instance, by
displacing the photodiode longitudinally. Such mitigation was experimentally
used, for instance, in [1, 27].

Table 1:  Overview of the different geometric cross coupling mechanisms for a beam with normal incidence and no photodiode tilt, i.e. $\beta=\varphi_{\text{PD}}=0$. For each effect we give a short description and the general behaviour (approximated), like linear, quadratic or mixed with respect to the tilt angle. |  |  Cross coupling mechanism | Name |  General behaviour | Eq. | Sec. |  Description
---|---|---|---|---|---|---
lateral |  piston effect | $\text{OPD}_{\text{piston}}$ |  linear | (37) | 3.1 |  If the mirror is initially rotated, further lateral shifts generate TTL coupling.
|  receiver jitter | $\text{OPD}_{\text{RS}}$ |  zero | (38) | 3.2 |  Lateral jitter of the receiver does not cause geometric TTL coupling.
angular |  lever arm effect | $\text{OPD}_{\text{lever}}$ |  quadratic | (41) | 3.1 |  Longitudinal offsets between rotation point and detector lead to variations in the propagation distance to the detector.
|  piston effect | $\text{OPD}_{\text{piston}}$ |  |  | 3.1 |  Offsets between the points of rotation and reflection at a component’s surface generate movement of this surface.
|  $-$ longitudinal |  |  quadratic | (42) |
|  $-$ lateral |  |  linear | (39) |  |
|  setup jitter | $\text{OPD}_{\text{RS}}$ |  |  | 3.2 |  Offsets between the points of rotation and detection generate movement of the detector surface relative to the beam axis.
|  $-$ longit. CoR |  |  quadratic | (43) |
|  $-$ lateral CoR |  |  linear | (40) |
|  transmissive components | $\text{OPD}_{\text{tc}}$ |  quadratic | (44) | 3.4.1 |  Transmissive optical components between the mirror and the detector induce a variation of the beams’ optical path length and a beam displacement on the detectors surface.

##  6 Conclusion

Within this paper, we have discussed how and why angular and lateral jitter
causes phase noise in laser interferometers. This tilt-to-length (TTL) noise
can often be characterized by changes in the optical path length of a beam.
These changes can be fully characterized by geometric effects except for the
naturally necessary scaling by refractive indices. Within this paper, we have
focused on these geometric TTL effects. We have categorized different coupling
mechanisms and analytically derived their contribution to the interferometric
path length signal from trigonometric relationships.

We have introduced TTL coupling noise for two different cases: jitter of a
mirror within the interferometer and jitter of the receiving system. Though
the described effects are independent of the context, they apply, for
instance, for the LISA mission and other space interferometers. In that
context, mirror jitter corresponds to the case observed in test mass
interferometers, while receiver jitter corresponds to long arm interferometers
and effectively describes satellite jitter relative to the received wavefront.

Besides the geometric TTL contributions, there exists non-geometric TTL
coupling originating from the wavefront properties of the interfering beams as
well as from detector characteristics. We have presented two examples
demonstrating the relevance of the non-geometric TTL contributions. However,
these non-geometric effects are significantly harder to describe than their
geometric counterparts and are the subject of a follow-up paper [22].

The TTL coupling within a system will always be the sum of all present non-
geometric and geometric TTL contributions. Nevertheless, knowing the
properties of the geometric TTL contributions alone can still help suppress
TTL noise in precision laser interferometers in two different ways.

On the one hand, the geometric effects can a priori be suppressed by smart
design choices. For the suppression of angular jitter coupling, it is, for
instance, advisable to particularly match the lateral position of the centre
of rotation and the reflection point on the jittering mirror (or test mass)
with high precision because the residual offset causes a linear effect (Eq.
(39)). Using imaging optics as described in [1, 27] can significantly reduce
the TTL coupling. Such imaging systems image the centre of rotation to the
photodiode centre, which results in $d_{\text{lever}}=d_{\text{long}}=0$ and
minimize the lever arm effect (Eq. (41) or Eq. (43), respectively). Also, the
TTL from lateral jitter can be reduced by respective design choices. Here,
angular misalignments of the mirror need to be minimized (Eq. (37). Therefore,
the effects described here allow for a TTL suppression by smartly designing
the interferometric layout.

On the other hand, the known geometrical TTL mechanism can be used to
counteract the measured TTL noise in an experiment. From experimental data,
one cannot necessarily tell apart the geometric and non-geometric effects.
Instead, the total TTL noise is being measured. In some cases, one can then
use Tab. 1 to counteract the total measured TTL. For instance, one can
laterally adjust the reflection point on the jittering mirror to balance the
sum of all existent first-order effects. This could potentially mean that the
lateral piston is then intentionally being increased. Likewise, if the sum of
all second-order effects is sufficiently small and if space in the
experimental layout permits, one can, for instance, shift the detector
longitudinally (altering the magnitude of the lever arm effect) in order to
minimize the observed second-order TTL noise.

In summary, we have introduced and analytically described a variety of
geometric TTL coupling effects. These can add significant noise to the signal
of laser interferometers. The mechanisms described here allow the suppression
of TTL noise by a dedicated design optimization, as well as in the
experimental setup by a dedicated realignment to counteract the observed TTL
noise.

We thank Gerhard Heinzel for valuable discussions. This work was made possible
by funds of both the Deutsche Forschungsgemeinschaft (DFG) and the German
Space Agency, DLR. We gratefully acknowledge the Deutsche
Forschungsgemeinschaft (DFG) for funding the Sonderforschungsbereich (SFB
1128: geo-Q) “Relativistic Geodesy and Gravimetry with Quantum Sensors”,
project A05 and all work contributions to this paper made by Sönke Schuster.
Furthermore, we acknowledge DFG for funding the Clusters of Excellence
PhoenixD (EXC 2122, Project ID 390833453) and QuantumFrontiers (EXC 2123,
Project ID 390837967). Likewise, we gratefully acknowledge the German Space
Agency, DLR and support by the Federal Ministry for Economic Affairs and
Energy based on a resolution of the German Bundestag (FKZ 50OQ1801). Finally,
we would like to acknowledge the Max Planck Society (MPG) for supporting the
framework LEGACY on low-frequency gravitational wave astronomy, a cooperation
between the Chinese Academy of Sciences (CAS) and the MPG (M.IF.A.QOP18098).

##  Appendix A Signal cancellation for two identical Gaussian beams

In Sec 4.2, we have shown a case where the total interferometric longitudinal
pathlength sensing (LPS) signal vanishes despite a considerable OPD response.
This setup consisted of two identical Gaussian beams, with one rotating around
a pivot located at an arbitrary point along the beam axis and a large single
element photodiode that detects both incident beams without clipping. In this
section, we show in a small angle approximation that this LPS signal vanishes
due to the symmetry of the interfering wavefronts on the detector surface. We
do this in two steps: In A.1, we show the symmetry of the amplitude profile of
the interference pattern, in A.2 the symmetry of the phase difference of the
interfering beams. We thereby show that for every point $P_{1}$ on the
detector, there exists a point $P_{2}$ which is found by mirroring $P_{1}$ on
the symmetry plane. In $P_{2}$, the interference pattern has the same
amplitude as in $P_{1}$, but an inverse phase. The LPS signal then vanishes
because it is a signal that is integrated over all detector points.

###  A.1 Symmetry of the amplitude profile

In the considered setup, we rotate the measurement beam around a point along
its propagation axis, while the reference beam is not tilted
($\varphi_{r}=0$). Let the distance between the point of rotation and the
point of incidence, i.e. the point where the ray describing the beam axis
impinges on the detector, be given by $d_{\text{lever}}$. For
$d_{\text{lever}}\neq 0$, the point of incidence gets shifted along the
photodiode surface when rotating the beam by $\varphi_{m}$. Thus the
measurement beam’s new point of incidence varies by

$\displaystyle x_{im}\approx-d_{\text{lever}}\,\varphi_{m}\,,$ (45)

while the reference beam stays constant ($x_{ir}=0$). The amplitude profile of
the interference pattern $A(x,y)$, defined according to [28] by

$\displaystyle
A(x,y)=\frac{1}{w(z_{m})w(z_{r})}\exp\left(\frac{-r_{m}^{2}(x,y)}{w^{2}(z_{m})}\right)\exp\left(\frac{-r_{r}^{2}(x,y)}{w^{2}(z_{r})}\right)\,,$
(46)

is then symmetric to the plane defined by $x=x_{c}$. Here, $w(z_{m,r})$
defines the spot radius of the measurement and reference beam on the detector
surface, and $r_{m,r}$ is the cylindrical coordinate of the respective beam.
Furthermore, $x_{c}$ is the mean value between the $x$-coordinates of the two
incidence points

$x_{c}=\frac{x_{im}-x_{ir}}{2}\approx-\frac{1}{2}d_{\text{lever}}\,\varphi_{m}\;.$
(47)

The cylindrical coordinates of the beams when impinging on the detector are
given by

$\displaystyle r_{m}^{2}$
$\displaystyle=(x-x_{im})^{2}\cos^{2}(\varphi_{m})+y^{2}\,,$ (48)
$\displaystyle r_{r}^{2}$ $\displaystyle=x^{2}+y^{2}\,.$ (49)

The coordinate transformation, performed here on the $x$ and $y$ coordinates,
naturally also needs to be performed on the $z$-coordinate [28], resulting in

$\displaystyle z_{m}$ $\displaystyle\approx
z+\frac{1}{2}d_{\text{lever}}\varphi_{m}^{2}-(x-x_{im})\,\varphi_{m}$ (50)
$\displaystyle z_{r}$ $\displaystyle=z=\text{const}\,.$ (51)

Substituting accordingly $z_{m,r}$, $r_{m,r}$ in Eq. (46) reveals the
symmetry:

$\displaystyle
A(-\frac{1}{2}\,d_{\text{lever}}+x,y)=A(-\frac{1}{2}\,d_{\text{lever}}-x,y)\,.$
(52)

###  A.2 Symmetry of the phase difference and cancellation on the detector

Next, we examine the local phase differences. We will show that this phase
difference is zero at the centre between the points of incidence, while its
absolute value increases and decreases symmetrically with increasing distance
from this point. Since the phase relates to a length $l$ via
$\phi=2\pi\,l/\lambda$, where $\lambda$ is the wavelength of the beam, we can
analogously investigate length changes here. Therefore, we study the beams’
properties and geometric relationships further. For simplicity, we can reduce
our analysis to the plane in which the rotation of the measurement beam gets
applied. The phase differences in an orthogonal plane, i.e. for a constant
$x$-value, are in a small angle approximation constant. First, we investigate
the Gaussian beam properties. We know that Gaussian beams provide the same
phase along their wavefronts. For our analysis, we propagate the beams until
their wavefronts hit the point of interest $P$ at the detector, see Fig. 10.

Figure 10:  Properties of a tilted measurement and a non-tilted reference beam
whose wavefronts are intersecting a point $P$ at the detector surface. The
paths of the beam rays are described by the solid grey lines. The measurement
beam got rotated by an angle $\varphi_{m}$ (negative in this figure) around
the centre of rotation (CoR). Left figure: The beams propagate the distances
$d_{m,r}$ from the CoR until their wavefronts hit the point $P$. This distance
differs from the $d_{\text{lever}}$, i.e. the distance between the CoR and the
detector, by $\delta_{m,r}$. Right figure: The propagation distance of the
beams can also be defined via their radii of curvature $R_{m,r}$. We
additionally define the auxiliary lengths $\varepsilon_{m,r}$ as the distances
between the CoR and the beam’s centres of wavefront curvature (CoCm,r). If
$R_{m,r}$ are greater than $d_{\text{lever}}$, the lengths $\varepsilon_{m,r}$
are negative. For every investigated point $P$, there can be found an angle
$\psi$ defining its position in relation to the reference beam axis. Figure
11:  Investigation of the two beams’ local phase differences in the two points
$P_{1}$ (left figure) and $P_{2}$ (right figure). In both figures, the beam
rays are shown as solid grey lines, where the non-tilted right line belongs to
the reference and the left line belongs to the measurements beam. $P_{1}$ and
$P_{2}$ lie symmetrically around the centre between the beams’ points of
incidence. The points are defined via an angle $\psi$ with respect to the
beams’ axes. The respective radii of curvature are denoted with $R_{i;m,r}$
and likewise the auxiliary lengths are tagged $\varepsilon_{i;m,r}$. Figure
12:  Investigation of the two beams’ local phase differences in the centre
between their points of incidence, i.e. for $x=x_{c}$. The measurement (red)
and the reference beam (blue) are symmetric with respect to the bisecting line
from the centre of rotation to $x_{c}$. Therefore, it is $R_{m}=R_{r}$ and
$\varepsilon_{m}=\varepsilon_{r}$.

The beams have then propagated a distance $d_{m,r}$. We compare this distance
with the lever arm $d_{\text{lever}}$, which is the distance the beam axis
propagates from the CoR to the diode if $\varphi_{m}=0$. We can then show the
symmetry of the wavefront using the difference
$d_{m,r}-d_{\text{lever}}=\delta_{m,r}$. This distance $\delta_{m,r}$ can also
be expressed as function of the radii of curvature $R_{m,r}$ and the auxiliary
lengths $\varepsilon_{m,r}$:

$\displaystyle\delta_{m,r}=R_{m,r}+\varepsilon_{m,r}-d_{\text{lever}}\,,$ (53)

as visible in Fig. 10. We will therefore derive expressions for these
quantities ($R_{m,r}$ and $\varepsilon_{m,r}$) in the following.

The radii of curvature are defined via

$R_{m,r}=z_{m,r}\left(1+\left(\frac{z_{0}}{z_{m,r}}\right)^{2}\right)\,,$ (54)

where $z_{0}$ denotes the Rayleigh range. The distances from the waists
($z_{m,r}$) can be related to $\delta_{m,r}$ via

$z_{m,r}=z+\delta_{m,r}\,,$ (55)

where $z$ is the distance from waist in the nominal non-tilted case, defined
for the point of incidence at the detector. We can now substitute $z_{m,r}$ in
Eq. 54. Using that $\delta_{m,r}\ll 1$ holds for the typical cases of
interest, the radii of curvature can be series expanded to

$R_{m,r}\approx R+\delta_{m,r}\left(2-\frac{R}{z}\right)\,.$ (56)

Here,

$\displaystyle R=z\left(1+\left(\frac{z_{0}}{z}\right)^{2}\right)$ (57)

is the radius of curvature at the point of incidence of a non-tilted beam.

In the next step, we define the auxiliary lengths $\varepsilon_{m,r}$. As
shown in Fig. 10, they describe the distance between the centre of rotation
(CoR) and the respective centre of wavefront curvature (CoCm,r).

Any point on the detector can be described by an angle $\psi$ in relation to
the initial point of incidence as shown in the right-hand sketch of Fig. 10.
There, the point $P$ of interest, the centre of rotation and the centres of
curvature span triangles. By geometric relations (law of cosines) we then know
that

$\displaystyle R_{m,r}^{2}=$ $\displaystyle\
d_{\text{lever}}^{2}\sec^{2}(\psi)+\varepsilon_{m,r}^{2}$ (58)
$\displaystyle-2d_{\text{lever}}\sec(\psi)\,\varepsilon_{m,r}\cos(\varphi_{m,r}-\psi)\;.$

We have now derived expressions for $R_{m,r}$, $\varepsilon_{m,r}$,
$\delta_{m,r}$ (i.e. (53),(56),(58)) and can solve these for $\delta_{m,r}$.
In order to show the symmetry of the phase difference of the interfering beams
and the cancellation of the LPS signal, we derive $\delta_{m,r}$ for points
$P_{1}$ and $P_{2}$ that are symmetric around the point $x_{c}$, see Fig. 11.
In the next step, we then argue that the deviations $\delta_{m,r}$ are equal
in the symmetry point $x_{c}$, such that the difference is zero.

We solve the system of equations (53),(56),(58) for $\delta_{m,r}$ studying
two points on the detector that possess in a good approximation the same
distance to the center of the points of incidence, i.e.
$\psi_{1}=\varphi_{m}-\psi$ and $\psi_{2}=\psi$. We get

$\displaystyle\delta_{1r}$
$\displaystyle\approx\frac{d_{\text{lever}}^{2}}{2R}(\varphi_{m}-\psi)^{2}\,,$
(59) $\displaystyle\delta_{1m}$
$\displaystyle\approx\frac{d_{\text{lever}}}{2}(\varphi_{m}^{2}-2\varphi_{m}\psi)+\frac{d_{\text{lever}}^{2}}{2R}\psi^{2}\,,$
(60) $\displaystyle\delta_{2r}$
$\displaystyle\approx\frac{d_{\text{lever}}^{2}}{2R}\psi^{2}\,,$ (61)
$\displaystyle\delta_{2m}$
$\displaystyle\approx\frac{d_{\text{lever}}(d_{\text{lever}}-R)}{2R}(\varphi_{m}^{2}-2\varphi_{m}\psi)+\frac{d_{\text{lever}}^{2}}{2R}\psi^{2}\,,$
(62)

and therefore

$\displaystyle\delta_{1m}-\delta_{1r}$
$\displaystyle=-\frac{d_{\text{lever}}(d_{\text{lever}}-R)}{2R}(\varphi_{m}^{2}-2\varphi_{m}\psi)$
(63) $\displaystyle=-(\delta_{2m}-\delta_{2r})\,.$ (64)

This shows that the phase differences between measurement and reference beam
are identical in the symmetric points $P_{1}$ and $P_{2}$ but with inverse
signs.

In the centre point $x_{c}$, which can be described via $\psi=\varphi_{m}/2$,
we find a phase difference of zero in Eq. (64). This special case is
additionally visualized in Fig. 12. In this figure, we see that the line
connecting the CoR with $x_{c}$ is (for small angles) the bisector. For
symmetry reasons, the wavefronts that are intersecting along this bisecting
line must belong to beams having propagated the same distance. It follows
$z_{m}=z+\delta_{m}=z+\delta_{r}=z_{r}$ and hence $\delta_{m}-\delta_{r}=0$.

Taking together our findings, we have now shown that the amplitude profile of
the interference pattern is symmetric to the plane defined by $x=x_{c}$, where
$x_{c}$ is the centre between the points of incidence along the $x$-axis.
Furthermore, the phase differences at points symmetrically arranged to this
plane have identical magnitude but inverse signs. In the LPS signal, the
amplitude scales the phase difference. Furthermore, the interferometric LPS
signal experimentally originates from the complete detector surface, which
mathematically is described by an integral over all detector points (see e.g.
[24, 28]). Therefore, the shown symmetries result in a cancellation of the LPS
signal.

A detailed mathematical derivation of the LPS signal for the setup described
here and the cancellation in the integral is additionally shown in [22].

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