# Effects of mirror birefringence and its fluctuations to
laser interferometric gravitational wave detectors

Yuta Michimura<EMAIL_ADDRESS>LIGO Laboratory, California Institute of
Technology, Pasadena, California 91125, USA  Research Center for the Early
Universe (RESCEU), Graduate School of Science, University of Tokyo, Tokyo
113-0033, Japan  PRESTO, Japan Science and Technology Agency (JST), Kawaguchi,
Saitama 332-0012, Japan  Haoyu Wang  Research Center for the Early Universe
(RESCEU), Graduate School of Science, University of Tokyo, Tokyo 113-0033,
Japan  Francisco Salces–Carcoba  LIGO Laboratory, California Institute of
Technology, Pasadena, California 91125, USA  Christopher Wipf  LIGO
Laboratory, California Institute of Technology, Pasadena, California 91125,
USA  Aidan Brooks  LIGO Laboratory, California Institute of Technology,
Pasadena, California 91125, USA  Koji Arai  LIGO Laboratory, California
Institute of Technology, Pasadena, California 91125, USA  Rana X Adhikari
LIGO Laboratory, California Institute of Technology, Pasadena, California
91125, USA

###### Abstract

Crystalline materials are promising candidates as substrates or high-
reflective coatings of mirrors to reduce thermal noises in future laser
interferometric gravitational wave detectors. However, birefringence of such
materials could degrade the sensitivity of gravitational wave detectors, not
only because it can introduce optical losses, but also because its
fluctuations create extra phase noise in the arm cavity reflected beam. In
this paper, we analytically estimate the effects of birefringence and its
fluctuations in the mirror substrate and coating for gravitational wave
detectors. Our calculations show that the requirements for the birefringence
fluctuations in silicon substrate and AlGaAs coating will be on the order of
$10^{-8}$ and $10^{-10}$ rad/$\sqrt{\rm Hz}$ at 100 Hz, respectively, for
future gravitational wave detectors. We also point out that optical cavity
response needs to be carefully taken into account to estimate optical losses
from depolarization.

##  I Introduction

The first detections of gravitational waves from binary black holes [1] and
binary neutron stars [2, 3] by Advanced LIGO [4] and Advanced Virgo [5]
inaugurated gravitational wave physics and astronomy. Improvements in the
sensitivity of these laser interferometric detectors in recent years enabled
routine detections and more precise binary parameter estimation [6]. Further
improvements in the astrophysical reach of these detectors will allow us to
study the origin of massive black holes, the neutron star equation of state,
alternative gravity theories, and cosmology.

The fundamental limitation to the sensitivity of these detectors at the most
sensitive frequency band is set by thermal vibrations of mirror surface [7].
KAGRA [8, 9] and other concepts of future gravitational wave detectors plan to
utilize cryogenic crystalline test mass mirrors for thermal noise reduction,
instead of fused silica mirrors at room temperature. KAGRA uses sapphire test
masses, and plan to cool them down to 22 K [10]. Voyager is an upgrade plan of
LIGO to use 123 K silicon to increase the astrophysical reach by a factor of
4–5 over Advanced LIGO design [11]. The next generation detectors such as
Einstein Telescope [12, 13] also plan to use silicon test masses at cryogenic
temperatures for the low frequency detectors, and Cosmic Explorer [14, 15]
considers using them for an upgrade. In addition, crystalline coatings such as
AlGaAs coating [16] and AlGaP coating [17] are considered as promising
candidates to reduce coating Brownian noise, instead of amorphous silica and
tantala coating.

Although crystalline materials are promising to reduce thermal noise, it has
been pointed out that slight birefringence of mirror substrates and coatings
could cause optical losses due to depolarization of the light, and cause
degradation of interferometric contrast [18]. The birefringence and its
inhomogeneity of sapphire input test masses of KAGRA were found to be higher
than expected [19, 20], and around 10% of power was lost on reflection due to
depolarization, when arm cavities are not on resonance [9]. Ideally,
crystalline silicon is a cubic crystal and optically isotropic, but could have
strain-induced birefringence from crystal dislocations and due to support in
the mirror suspension system. Birefringence measurements in silicon mirrors
have revealed that the amount of the static birefringence is $\Delta n\sim
10^{-7}$ or less at laser wavelengths of 1.55 [21] and 2 $\mu$m [22] at room
temperature, which satisfies the optical loss requirements for future
detectors. Also, previous cavity experiments using AlGaAs coatings reported
birefringence at 1 mrad level [16, 23, 24].

These past studies have focused on the static birefringence and optical losses
from the depolarization. However, recent measurement of thermal noises in
crystalline mirror coatings at cryogenic temperatures reported excess
birefringent noise, which could limit the sensitivity of future gravitational
wave detectors [25]. Theoretical calculations on thermal fluctuations of
birefringence in crystalline mirror coatings have also revealed that the noise
from these fluctuations could be similar to Brownian noise [26]. It is also
worth noting that experiments to search for vacuum magnetic birefringence,
such as PVLAS (Polarizzazione del Vuoto con LASer) and OVAL (Observing VAcuum
with Laser), have been suspected to be limited by thermal birefringence noise
of mirrors [27, 28, 29, 30, 31]. These temporal birefringence fluctuations
could also limit optical cavity based axion dark matter searches using the
birefringence effect from axion-photon coupling [32, 33, 34, 35, 36].

In this paper, we study the effects of birefringence and its fluctuations to
gravitational wave detectors based on the Fabry-Pérot-Michelson
interferometer. We show that the polarization axis and the crystal axes of arm
cavity mirrors need to be aligned to avoid optical losses and to reduce noises
from birefringence fluctuations. We also show that the cavity response to
birefringence needs to be correctly taken into account for estimating the
noises and the optical losses of arm cavities. We start by analytically
describing the cavity response to birefringence in Sec. II. In Sec. III, we
focus on noises from substrate birefringence and coating birefringence, and
derive requirements for their fluctuations for future gravitational wave
detectors. In Sec. IV, we expand our formulation to include spatial higher
order modes, and discuss power losses from inhomogeneous birefringence of the
substrate and the coating. Our conclusions and outlook are summarized in Sec.
V.

Throughout the paper, we use 0.1% as a requirement threshold for the optical
losses from polarization. In this way the optical losses from polarization
will be small enough, as future gravitational wave detector designs require
total optical loss to be less than 10% [37].

##  II Cavity response to birefringence

Let us consider a Fabry-Pérot cavity formed by an input test mass (ITM) and an
end test mass (ETM) mirrors as shown in Fig. 1. We consider birefringence of
ITM substrate, ITM high-reflective coating, and ETM high-reflective coating.
The ordinary axis of the ETM coating is rotated by $\theta$ with respect to
that of ITM. The input beam is linearly polarized and its polarization is
rotated by $\theta_{\rm pol}$ with respect to the ordinary axis of ITM. We
assume that the crystal axes of ITM substrate are aligned with those of its
coating. This will not affect the results of this paper, as we will treat the
substrate birefringence and the coating birefringence independently in the
following sections.

Figure 1:  The schematic of a Fabry-Pérot cavity with mirror crystal axes and
input beam polarization axis illustrated. With respect to the ITM ordinary
axis, the input polarization is rotated by $\theta_{\rm pol}$ and the ETM
ordinary axis is rotated by $\theta$.

For calculating the cavity response to birefringence, we can use the Jones
matrix formalism [38]. In the basis of ITM crystal axes, the electric field of
the input beam can be written as

$\vec{E}_{\rm in}=(v_{1}\vec{e}_{\rm o}+v_{2}\vec{e}_{\rm e})E_{\rm
in}=\begin{pmatrix}\vec{e}_{\rm o}&\vec{e}_{\rm e}\\\
\end{pmatrix}\vec{v}_{\rm in}E_{\rm in},$ (1)

where $\vec{e}_{\rm o}$ and $\vec{e}_{\rm e}$ are the unit vectors along with
the ITM ordinary and extraordinary axes, respectively, and $\vec{v}_{\rm
in}\equiv(v_{1}\ v_{2})^{T}$ is the unit vector representing the input
polarization.

We suppose the ITM substrate is lossless, and the amplitude reflectivity and
the amplitude transmission of the whole ITM is determined by the high-
reflective coating. Then the amplitude transmission of ITM can be written as

$T_{\rm 1}=\begin{pmatrix}t_{1}&0\\\ 0&t_{1}e^{-i\frac{1}{2}\Delta\phi_{\rm
t_{1}}}\\\ \end{pmatrix},$ (2)

where $\Delta\phi_{\rm t_{1}}/2$ is the phase difference between the ordinary
and extraordinary axes in the ITM transmission from both the substrate and the
coating birefringence, and $t_{1}$ is the amplitude transmission of ITM. Here,
we assumed that the amplitude transmission is the same for both axes.
Similarly, the amplitude reflectivity of ITM and ETM from the high-reflective
coating side can be written as

$R_{j}=\begin{pmatrix}r_{j}&0\\\ 0&r_{j}e^{-i\Delta\phi_{r_{j}}}\\\
\end{pmatrix},$ (3)

where $\Delta\phi_{{\rm r}_{j}}$ is the phase difference between the ordinary
and extraordinary axes in ITM and ETM reflection, and $r_{j}$ is the amplitude
reflectivity of ITM and ETM. $j=1$ is for ITM and $j=2$ is for ETM. Also, the
amplitude reflectivity of ITM from the substrate side can be written as

$S_{\rm 1}=\begin{pmatrix}-r_{1}&0\\\ 0&-r_{1}e^{-i\Delta\phi_{\rm s_{1}}}\\\
\end{pmatrix},$ (4)

where $\Delta\phi_{s_{1}}$ is the phase difference between the ordinary and
extraordinary axes in the ITM reflection from the substrate side. From the
energy conservation and the time reversal symmetry,
$\Delta\phi_{t_{1}}=\Delta\phi_{r_{1}}+\Delta\phi_{s_{1}}$. Here, we use the
convention that $r_{j}$ and $t_{1}$ are real, and the sign is flipped for
reflection from the ITM substrate side. We keep the coordinate axis to be the
same even if the propagation direction flips on mirror reflections, so that
the sign for both polarizations will be the same.

For arm cavities in gravitational wave detectors, $r_{1}$ and $r_{2}$ are
designed to be $r_{2}\simeq 1$, and $r_{1}<r_{2}$, such that almost all the
light is reflected back. From the phase of the cavity reflected beam, cavity
length changes from gravitational waves are read out. In the following
subsections, we calculate the polarization eigenmodes in the cavity, and the
phase of the cavity reflected beam.

###  II.1 Polarization eigenmodes in the cavity

The electric field inside the cavity that propagates from ITM to ETM can be
written as

$\vec{E}_{\rm cav}=\left(I-A\right)^{-1}T_{1}\vec{E}_{\rm in},$ (5)

with $I$ being the identity matrix. Here,

$A\equiv R_{1}R(-\theta)R_{2}R(\theta)e^{-i\phi},$ (6)

where $\phi=4\pi L/\lambda$ is the phase acquired in the cavity round-trip,
with $L$ and $\lambda$ being the cavity length and the laser wavelength,
respectively, and

$R(\theta)\equiv\begin{pmatrix}\cos{\theta}&-\sin{\theta}\\\
\sin{\theta}&\cos{\theta}\\\ \end{pmatrix},$ (7)

with the derivation described in Appendix A. Note that $\phi$ includes phase
acquired in the ITM and ETM reflection for their ordinary axes. The resonant
polarization mode is the eigenvectors of

$M_{\rm cav}\equiv\left(I-A\right)^{-1}T_{1}.$ (8)

The cavity enhancement factors for each mode will be the eigenvalues of
$M_{\rm cav}$.

When $\theta=0$, the ITM axes and the ETM axes are aligned, and the
eigenvectors will be

$\vec{v}_{a}=\begin{pmatrix}1\\\ 0\\\
\end{pmatrix},\qquad\vec{v}_{b}=\begin{pmatrix}0\\\ 1\\\ \end{pmatrix},$ (9)

which means that the resonant modes are linear polarizations along the ITM
ordinary axis $\vec{e}_{\rm o}$ and the extraordinary axis $\vec{e}_{\rm e}$.
The cavity enhancement factors will be

$w_{a}=\frac{t_{1}}{1-r_{1}r_{2}e^{-i\phi}},\qquad
w_{b}=\frac{t_{1}e^{-i\frac{1}{2}\Delta\phi_{\rm
t_{1}}}}{1-r_{1}r_{2}e^{-i(\phi+\Delta\phi_{\rm r_{1}}+\Delta\phi_{\rm
r_{2}})}}.$ (10)

The resonant frequency difference between two eigenmodes therefore will be

$\Delta\nu=\frac{\Delta\phi_{\rm r_{1}}+\Delta\phi_{\rm r_{2}}}{2\pi}\nu_{\rm
FSR},$ (11)

where $\nu_{\rm FSR}=c/(2L)$ is the free spectral range of the cavity.

When $\theta=\pi/2$, the ITM ordinary axis and the ETM extraordinary axis are
aligned, and the eigenvectors again will be the same as the ones given in Eq.
(9). The cavity enhancement factors will be

$w_{a}=\frac{t_{1}}{1-r_{1}r_{2}e^{-i(\phi+\Delta\phi_{\rm r_{2}})}},\quad
w_{b}=\frac{t_{1}e^{-i\frac{1}{2}\Delta\phi_{\rm
t_{1}}}}{1-r_{1}r_{2}e^{-i(\phi+\Delta\phi_{\rm r_{1}})}}.$ (12)

The resonant frequency difference between two eigenmodes therefore will be

$\Delta\nu=\frac{\Delta\phi_{\rm r_{1}}-\Delta\phi_{\rm r_{2}}}{2\pi}\nu_{\rm
FSR}.$ (13)

Since we defined the ITM and ETM axes such that $\Delta\phi_{{\rm r}_{i}}$
have the same sign for ITM and ETM, when $\theta=0$, the phase difference
between the axes are added and the resonant frequency difference is maximized.
When $\theta=\pi/2$, it is minimized, as the phase difference is canceled.
When $0<\theta<\pi/2$, the resonant frequency difference will be in between
the maximum and the minimum.

When the resonant frequency difference is smaller than the cavity linewidth,
i.e., $\Delta\phi_{{\rm r}_{i}}\ll 2\pi/\mathcal{F}$, and when the effect from
the ITM substrate birefringence is small, i.e., $\Delta\phi_{\rm
t_{1}}\ll\Delta\phi_{\rm r_{1}}\mathcal{F}/\pi$, the resonant frequency
difference can be calculated with

$\Delta\nu\simeq\frac{2\pi(\arg{w_{a}}-\arg{w_{b}})}{\mathcal{F}}\frac{\nu_{\rm
FSR}}{2\pi},$ (14)

at $\phi=0$, where

$\mathcal{F}=\frac{\pi\sqrt{r_{1}r_{2}}}{1-r_{1}r_{2}}$ (15)

is the finesse of the cavity. This can be further approximated as [39]

$\Delta\nu\simeq\frac{\delta_{\rm EQ}}{2\pi}\nu_{\rm FSR},$ (16)

where

$\delta_{\rm EQ}\equiv\sqrt{(\Delta\phi_{\rm r_{1}}-\Delta\phi_{\rm
r_{2}})^{2}+4\Delta\phi_{\rm r_{1}}\Delta\phi_{\rm r_{2}}\cos^{2}{\theta}},$
(17)

when $\delta_{\rm EQ}\ll 1$, with the derivation described in Appendix B.
Also, the cavity eigenmodes are linear polarizations approximated as

$\vec{v}_{a}=\begin{pmatrix}\cos{\theta_{\rm EQ}}\\\ \sin{\theta_{\rm EQ}}\\\
\end{pmatrix},\quad\vec{v}_{b}=\begin{pmatrix}-\sin{\theta_{\rm EQ}}\\\
\cos{\theta_{\rm EQ}}\\\ \end{pmatrix},$ (18)

where the polarization angle is defined by

$\cos{2\theta_{\rm
EQ}}=\frac{\displaystyle{\frac{\Delta\phi^{\prime}_{r_{1}}}{\Delta\phi_{\rm
r_{2}}}}+\cos{2\theta}}{\sqrt{\left(\displaystyle{\frac{\Delta\phi^{\prime}_{r_{1}}}{\Delta\phi_{\rm
r_{2}}}}-1\right)^{2}+4\displaystyle{\frac{\Delta\phi^{\prime}_{r_{1}}}{\Delta\phi_{\rm
r_{2}}}}\cos^{2}{\theta}}},$ (19)

with

$\Delta\phi^{\prime}_{r_{1}}\equiv\Delta\phi_{\rm
r_{1}}+\frac{\pi}{\mathcal{F}}\Delta\phi_{\rm t_{1}}.$ (20)

When $\Delta\phi^{\prime}_{r_{1}}\gg\Delta\phi_{r_{2}}$, $\theta_{\rm EQ}$ is
equal to zero, when $\Delta\phi^{\prime}_{r_{1}}=\Delta\phi_{r_{2}}$,
$\theta_{\rm EQ}$ is equal to $\theta/2$, and when
$\Delta\phi^{\prime}_{r_{1}}\ll\Delta\phi_{r_{2}}$, $\theta_{\rm EQ}$ is equal
to $\theta$. Note that the polarization state resonating inside the cavity are
elliptic polarizations given by $R_{1}T_{1}\vec{v}_{a,b}/(r_{1}t_{1})$, and
are different from linear polarizations given by Eq. (18).

The mismatch between the cavity polarization mode and the input beam
polarization can be calculated with

$\Lambda^{2}=1-\left|\vec{v}_{a}\cdot\vec{v}_{\rm in}\right|^{2}.$ (21)

When the input beam is linearly polarized with the polarization angle of
$\theta_{\rm pol}$ such that

$\vec{v}_{\rm in}=R(\theta_{\rm pol})\begin{pmatrix}1\\\ 0\\\
\end{pmatrix}=\begin{pmatrix}\cos{\theta_{\rm pol}}\\\ \sin{\theta_{\rm
pol}}\\\ \end{pmatrix},$ (22)

Eq. (21) reduces to

$\Lambda^{2}=\sin^{2}{(\theta_{\rm EQ}-\theta_{\rm pol})}.$ (23)

The mismatch will be less than than 0.1% when $|\theta_{\rm EQ}-\theta_{\rm
pol}|$ is smaller than $1.8^{\circ}$. For gravitational wave detectors, this
is required for both arm cavities. This means that the axes of two arm
cavities need to be aligned to the same degree. Note that mismatch do not
directly mean that there is a same amount of power loss. The actual power loss
also depend on the amount of birefringence, as we will discuss in Sec. IV.

Figure 2:  The polarization eigenmodes of a Fabry-Pérot cavity as a function
of ETM rotation angle $\theta$. The top panel shows the round-trip phase
difference between the eigenmodes in the unit of $\Delta\phi_{\rm r_{1}}$,
i.e., $2\pi\Delta\nu/(\nu_{\rm FSR}\Delta\phi_{\rm r_{1}})$, which is
proportional to the resonant frequency difference. The middle panel shows the
polarization angle of the eigenmodes $\theta_{\rm EQ}$ calculated using Eq.
(19). The bottom panel shows the mismatch of the input beam polarization to
the eigenmodes, when it is linear and aligned with ITM axes, calculated using
Eq. (21). Different colors of the lines correspond to different
$\Delta\phi_{\rm r_{2}}/\Delta\phi_{\rm r_{1}}$ ratios. Blue lines for
$\Delta\phi_{\rm r_{2}}=0$ case in the bottom two plots are zero.

Figure 2 shows the polarization eigenmodes of the cavity as a function of ETM
rotation angle $\theta$, calculated using Eqs. (16) and (19). As we have
discussed earlier, the resonant frequency difference will be the maximized at
$\theta=0$, and minimized at $\theta=\pi/2$. When $\theta=\pi/2$ and
$\Delta\phi_{\rm r_{1}}=\Delta\phi_{\rm r_{2}}$, the phase difference between
ordinary and extraordinary axes is completely cancelled, and two modes will be
degenerate. In this case, two linear polarizations and two circular
polarizations will be cavity eigenmodes, since two modes have the same
resonant frequency.

The bottom panel in Fig. 2 shows the mismatch calculated using Eq. (21),
assuming the input polarization is linear and aligned with either of the ITM
axes. The mismatch is nulled at $\theta=0$ and $\theta=\pi/2$. To minimize the
mismatch and to make the resonant frequency difference large, aligning the ETM
rotation such that $\theta=0$ and aligning the input polarization to one of
the ITM axes will be the optimal choice. The requirement on the alignment will
be not severe, since the dependence on the ETM rotation angle goes with
$\theta^{2}$ at $\theta=0$.

For deriving the cavity reflected beam, we need to calculate the electric
field inside the cavity that propagates from ETM to ITM. This can be written
as

$\displaystyle\vec{E}^{\prime}_{\rm cav}$ $\displaystyle=$ $\displaystyle
R(-\theta)R_{2}R(\theta)e^{-i\phi}M_{\rm cav}\vec{E}_{\rm in}$ (24)
$\displaystyle\equiv$ $\displaystyle M^{\prime}_{\rm cav}\vec{E}_{\rm in}.$
(25)

The eigenvectors of $M^{\prime}_{\rm cav}$ are the same as those of $M_{\rm
cav}$ within our approximations discussed above, but the cavity enhancement
factors will be slightly different. When $\theta=0$, the cavity enhancement
factors will be

$w^{\prime}_{a}=\frac{t_{1}r_{2}e^{-i\phi}}{1-r_{1}r_{2}e^{-i\phi}},\quad
w^{\prime}_{b}=\frac{t_{1}r_{2}e^{-i(\phi+\frac{1}{2}\Delta\phi_{\rm
t_{1}}+\Delta\phi_{\rm r_{2}}})}{1-r_{1}r_{2}e^{-i(\phi+\Delta\phi_{\rm
r_{1}}+\Delta\phi_{\rm r_{2}})}},$ (26)

and when $\theta=\pi/2$, those will be

$w^{\prime}_{a}=\frac{t_{1}r_{2}e^{-i(\phi+\Delta\phi_{\rm
r_{2}})}}{1-r_{1}r_{2}e^{-i(\phi+\Delta\phi_{\rm r_{2}})}},\quad
w^{\prime}_{b}=\frac{t_{1}r_{2}e^{-i(\phi+\frac{1}{2}\Delta\phi_{\rm
t_{1}}})}{1-r_{1}r_{2}e^{-i(\phi+\Delta\phi_{\rm r_{1}})}}.$ (27)

Compared with $w_{a}$ and $w_{b}$, those have extra phase $\phi$ from the
cavity round trip and extra phase $\Delta\phi_{\rm r_{2}}$ for the
corresponding axis for one additional reflection from ETM.

###  II.2 Phase of cavity reflected beam

The noises due to temporal fluctuations of birefringence will be imprinted in
the phase of the cavity reflected beam. The electric field of the cavity
reflection can be written as

$\vec{E}_{\rm refl}=M_{\rm refl}\vec{E}_{\rm in}$ (28)

where

$M_{\rm refl}\equiv S_{1}+T_{1}M^{\prime}_{\rm cav}.$ (29)

The first term corresponds to the prompt reflection from ITM, and the second
term is the ITM transmitted beam from the cavity circulating beam. In general,
when the input beam polarization component is

$\vec{v}_{\rm in}=a\vec{v}^{\prime}_{a}+b\vec{v}^{\prime}_{b},$ (30)

the polarization component of the reflected beam is

$M_{\rm refl}\vec{v}_{\rm
in}=a(S_{1}+w^{\prime}_{a}T_{1})\vec{v}^{\prime}_{a}+b(S_{1}+w^{\prime}_{b}T_{1})\vec{v}^{\prime}_{b}.$
(31)

Since the resonant condition of each eigenmode is generally different, it is
generally $|w^{\prime}_{a}|\neq|w^{\prime}_{b}|$. Therefore, the polarization
component of the cavity reflected beam will be different from the input
polarization.

When we use a Faraday isolator to extract the cavity reflection, we extract
the polarization component which is the same as the input polarization.
Therefore, the phase of the cavity reflected beam can be calculated with

$\arg{(E_{\rm out})}=\arg{(E_{\rm refl\parallel})}=\arg{(E_{\rm in}M_{\rm
refl}\vec{v}_{\rm in}\cdot\vec{v}_{\rm in})}.$ (32)

In the case when the input beam polarization is aligned to the ITM ordinary
axis, this reflected phase is the phase of the (1,1) component of $M_{\rm
refl}$, and that for the ITM extraordinary axis is the (2,2) component of
$M_{\rm refl}$.

Let us first consider the effects from ITM. If we set $\Delta\phi_{\rm
r_{2}}=0$ and the input beam is linearly polarized with the polarization angle
of $\theta_{\rm pol}$ as shown in Eq. (22), the reflected electric field in
the polarization parallel to $\vec{v}_{\rm in}$ and in the orthogonal
polarization will be

$\displaystyle\frac{E_{\rm refl\parallel}}{E_{\rm in}}$ $\displaystyle=$
$\displaystyle M_{\rm refl}\vec{v}_{\rm in}\cdot\vec{v}_{\rm in}$ (33)
$\displaystyle=$
$\displaystyle(-r_{1}+w_{a}^{\prime}t_{1})\cos^{2}{\theta_{\rm pol}}$
$\displaystyle+(-r_{1}e^{-i\Delta\phi_{\rm
s_{1}}}+w_{b}^{\prime}t_{1}e^{-i\frac{1}{2}\Delta\phi_{\rm
t_{1}}})\sin^{2}{\theta_{\rm pol}},$ $\displaystyle\frac{E_{\rm
refl\perp}}{E_{\rm in}}$ $\displaystyle=$ $\displaystyle M_{\rm
refl}\vec{v}_{\rm in}\cdot R(\theta_{\rm pol})\begin{pmatrix}0\\\ 1\\\
\end{pmatrix}$ (34) $\displaystyle=$
$\displaystyle\left[(-r_{1}+w_{a}^{\prime}t_{1})-(-r_{1}e^{-i\Delta\phi_{\rm
s_{1}}}+w_{b}^{\prime}t_{1}e^{-i\frac{1}{2}\Delta\phi_{\rm t_{1}}})\right]$
$\displaystyle\times\frac{\sin{(2\theta_{\rm pol})}}{2}.$

These are similar to the electric fields of the bright reflection port and the
dark antisymmetric port for a Fabry-Pérot-Michelson interferometer that has an
unbalanced beam splitter.

The effects from the ETM birefringence can be calculated by setting
$\Delta\phi_{\rm s_{1}}=\Delta\phi_{\rm t_{1}}=0$ and replacing
$\Delta\phi_{\rm r_{1}}$ with $\Delta\phi_{\rm r_{2}}$ and $\theta_{\rm pol}$
with $\theta+\theta_{\rm pol}$. If we combine the effects from ITM and ETM,
the phase of the reflected beam around the resonance can be approximated as

$\begin{split}\arg{\left(\frac{E_{\rm refl\parallel}}{E_{\rm
in}}\right)}&=(\Delta\phi_{\rm s_{1}}-2\Delta\phi_{\rm
t_{1}})\sin^{2}{\theta_{\rm pol}}\\\
&-\displaystyle{\frac{\mathcal{F}}{\pi}}\left[\phi+\Delta\phi_{\rm
r_{1}}\sin^{2}{\theta_{\rm pol}}\right.\\\ &+\left.\Delta\phi_{\rm
r_{2}}\sin^{2}{(\theta+\theta_{\rm pol})}\right],\end{split}$ (35)

with the approximation that $\Delta\phi_{{\rm r}_{i}}\ll 2\pi/\mathcal{F}$ and
$r_{2}=1$. It is clear that both the ETM rotation angle $\theta$ and the input
beam polarization angle $\theta_{\rm pol}$ change the phase of the cavity
reflected beam and will contribute to the phase noise, unless $\theta_{\rm
pol}$ and $\theta+\theta_{\rm pol}$ are either 0 or $\pi/2$, where the effects
are quadratic to these angles. The fluctuations of phase differences between
ordinary and extraordinary axes also create phase noises, unless $\theta_{\rm
pol}$ and $\theta+\theta_{\rm pol}$ are both 0.

It is worth noting that, even if we use this phase to lock the cavity, this
does not generally mean that the cavity is locked on resonance to one of its
polarization eigenmodes, as the cavity reflected beam contains the phase
fluctuations from both polarization eigenmodes. To avoid the mixing of phase
noises from two polarization eigenmodes, it is actually better to have higher
static coating birefringence, i.e., $\Delta\phi_{{\rm r}_{i}}\gg
2\pi/\mathcal{F}$. If the static coating birefringence is high such that one
of the eigenmodes is out of resonance when the other is resonant, only
$\Delta\phi_{\rm s_{1}}$ and $\phi$ terms remain in Eq. (35).

##  III Noises from birefringence

In this section, we calculate the phase noises from temporal fluctuations of
birefringence and derive the requirements for the current and future
gravitational wave detectors. For calculating the requirements, we have used
the interferometer parameters summarized in Table 1 and the displacement
sensitivity curves shown in Fig. 3. At the last part of this section, we also
discuss the noise from the amplitude fluctuations in the orthogonal
polarization at the antisymmetric port of the Fabry-Pérot Michelson
interferometer. Although different interferometers plan to use different
materials for the mirrors, discussions presented here do not depend on the
choice of materials.

Table 1:  Interferometer parameters of Advanced LIGO (aLIGO), A+, Voyager, Cosmic Explorer (CE), Einstein Telescope Low Frequency (ET-LF), and ET High Frequency (ET-HF) used for calculating requirements. $L$: arm length, $\mathcal{F}$: arm finesse, $t$: ITM thickness, $\lambda$: laser wavelength. | $L$ | $\mathcal{F}$ | $t$ | $\lambda$ | Ref.
---|---|---|---|---|---
aLIGO | 4 km | 450 | 20 cm | 1064 nm | [4]
A+ | 4 km | 450 | 20 cm | 1064 nm | [40]
Voyager | 4 km | 3000 | 55 cm | 2050 nm | [11]
CE | 40 km | 450 | 27.3 cm | 2050 nm | [15]
ET-LF | 10 km | 900 | 57 cm | 1550 nm | [13]
ET-HF | 10 km | 900 | 30 cm | 1064 nm | [13]
Figure 3:  The designed displacement sensitivity for different gravitational
wave detectors. The strain sensitivity data are taken from Refs. [41, 42, 43]
and corrected to displacement sensitivities by removing frequency-dependent
responses to gravitational waves [44].

###  III.1 Phase noises from substrate birefringence

The phase changes from the ITM substrate birefringence can be calculated from
Eq. (35) by setting $\Delta\phi_{\rm r_{1}}=\Delta\phi_{\rm r_{2}}=0$ and
$\Delta\phi_{\rm s_{1}}=\Delta\phi_{\rm t_{1}}$. In this case, Eq. (35)
reduces to

$\arg{\left(\frac{E_{\rm refl\parallel}}{E_{\rm in}}\right)}=-\Delta\phi_{\rm
s_{1}}\sin^{2}{\theta_{\rm pol}}-\displaystyle{\frac{\mathcal{F}}{\pi}}\phi.$
(36)

Therefore, the length noise couplings from the fluctuations of $\theta_{\rm
pol}$ and $\Delta\phi_{\rm s_{1}}$ can be calculated as

$\displaystyle\frac{\delta L}{\delta\theta_{\rm pol}}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{4\pi}\frac{\delta[\arg{(E_{\rm
refl\parallel})}]}{\delta\theta_{\rm pol}}\left(\frac{\delta[\arg{(E_{\rm
refl\parallel})}]}{\delta\phi}\right)^{-1}$ (37) $\displaystyle=$
$\displaystyle\frac{\lambda}{4\mathcal{F}}\Delta\phi_{\rm
s_{1}}\sin{2\theta_{\rm pol}},$ $\displaystyle\frac{\delta
L}{\delta(\Delta\phi_{\rm s_{1}})}$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{4\mathcal{F}}\sin^{2}{\theta_{\rm pol}}.$ (38)

###  III.2 Phase noises from coating birefringence

Next, we consider the phase changes from the coating birefringence. From Eq.
(35), it is clear that the second term from $\Delta\phi_{\rm r_{1}}$ and
$\Delta\phi_{\rm r_{2}}$ contributes more to the phase of the reflected beam,
compared with the first term from $\Delta\phi_{\rm s_{1}}$ and
$\Delta\phi_{\rm t_{1}}$, since the phase acquired inside the cavity is
enhanced by a factor of $\mathcal{F}/\pi$. The length noise couplings from the
fluctuations of $\theta_{\rm pol}$, $\theta$, and $\Delta\phi_{{\rm r}_{i}}$
can be calculated as

$\displaystyle\frac{\delta L}{\delta\theta_{\rm pol}}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{4\pi}[\Delta\phi_{\rm r_{1}}\sin{2\theta_{\rm
pol}}$ (39) $\displaystyle+\Delta\phi_{\rm r_{2}}\sin{[2(\theta+\theta_{\rm
pol})]}],$ $\displaystyle\frac{\delta L}{\delta\theta}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{4\pi}\Delta\phi_{\rm
r_{2}}\sin{[2(\theta+\theta_{\rm pol})]},$ (40) $\displaystyle\frac{\delta
L}{\delta(\Delta\phi_{\rm r_{1}})}$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{4\pi}\sin^{2}{\theta_{\rm pol}},$ (41)
$\displaystyle\frac{\delta L}{\delta(\Delta\phi_{\rm r_{2}})}$
$\displaystyle=$
$\displaystyle-\frac{\lambda}{4\pi}\sin^{2}{(\theta+\theta_{\rm pol})}.$ (42)

Figure 4:  The requirements on birefringence fluctuations from the axis
rotations (top) and from the phase difference between ordinary and
extraordinary axes (middle) for different gravitational wave detectors. The
bottom plot shows the requirement on the substrate birefringence converted
from the phase difference requirements on $\Delta\phi_{\rm s_{1}}$ in the
middle plot, assuming uniform $\Delta n$, using Eq. (43). The solid lines are
for the substrate that have a static birefringence of $\Delta n=10^{-7}$ and
the dashed lines are for the coating that have a static birefringence of
$\Delta\phi_{{\rm r}_{i}}=1$ mrad. For deriving these requirements, we assumed
that the input beam polarization and the ETM axes are aligned to the ITM axes
to $\theta_{\rm pol}=1^{\circ}$ and $\theta=1^{\circ}$, and no safety margin
is considered.

###  III.3 Requirements on birefringence fluctuations

Noise couplings discussed above are nulled when $\theta_{\rm pol}=0$ and
$\theta=0$. For KAGRA test masses, the sapphire $c$ axis was aligned to the
cylindrical plane of the test mass within $0.1^{\circ}$ [20]. For deriving the
requirements to birefringence fluctuations for the substrate and the coating,
we assume that the input beam polarization and the ETM axes are aligned to the
ITM axes to $\theta_{\rm pol}=1^{\circ}$ and $\theta=1^{\circ}$, respectively.

The solid lines in Fig. 4 show the derived requirements for the substrate
birefringence fluctuations. We assumed that the ITM substrate has uniform
birefringence $\Delta n$, and $\Delta\phi_{\rm s_{1}}$ can be written using
the mirror thickness $t$ as

$\Delta\phi_{\rm s_{1}}=\frac{4\pi}{\lambda}\Delta nt.$ (43)

We used the static birefringence value of $\Delta n=10^{-7}$, which is a
typical measured value for silicon [21, 22]. The dashed lines in Fig. 4 show
the derived requirements for the coating using the static birefringence value
of $\Delta\phi_{{\rm r}_{i}}=1$ mrad, which is a typical measured value for
AlGaAs coating [16, 23, 24]. The requirements do not change for other
materials when they have the same amount of static birefringence. For deriving
the requirement for $\Delta\phi_{\rm r_{j}}$, we used Eq. (42), as this gives
more stringent requirement than Eq. (41). All the requirements are divided by
$\sqrt{2}$ to take into account of birefringence noises between two arm
cavities to be incoherent, assuming both cavities have a similar level of
birefringence. The requirements will be relaxed for common effects in two
arms, such as the fluctuations in the input beam polarization angle and
birefringence induced by laser intensity fluctuations.

The requirements on the axis rotations for future gravitational wave detectors
is on the order of $10^{-10}$ rad/$\sqrt{\rm Hz}$. We note that the
requirements on $\theta_{\rm pol}$ and $\theta$ presented here are also the
requirements for the polarization fluctuation requirement for the input beam
and the roll motion of the mirrors. As for the roll motion of the mirrors, the
vertical seismic motion create less than $10^{-11}$ rad/$\sqrt{\rm Hz}$ level
of roll motion above 10 Hz for the Advanced LIGO suspensions, if we
conservatively assume that the coupling from vertical to roll motion is unity
[35, 45]. Therefore, the birefringence noise from the roll motion of the
mirrors is small enough.

The requirements on the phase differences between ordinary and extraordinary
axes for future gravitational wave detectors are on the order of $10^{-8}$
rad/$\sqrt{\rm Hz}$ for the substrate, and $10^{-10}$ rad/$\sqrt{\rm Hz}$ for
the coating. Birefringence at $10^{-8}$ rad/$\sqrt{\rm Hz}$ level can be
feasibly evaluated with shot noise limited interferometry at the laser power
of $P=10$ mW level, as the shot noise limited phase sensitivity of a Michelson
interferometer is given by

$\phi_{\rm shot}=\sqrt{\frac{hc}{2\lambda P}},$ (44)

where $h$ is Planck constant and $c$ is the speed of light. Evaluation of
birefringence at $10^{-10}$ rad/$\sqrt{\rm Hz}$ level requires 10-W class
laser or cavity enhancements. Measurements can be done at relatively lower
power compared with gravitational wave detectors, as the phase noise from
birefringence is attenuated by $\sin^{2}{\theta}$ and
$\sin^{2}{(\theta+\theta_{\rm pol})}$, by aligning the polarization axis and
the mirror crystal axes. In the evaluation setup, the phase noise can be
enhanced by intentionally misaligning the axes.

One of the possible sources of birefringence fluctuations is magnetic field
fluctuations due to Faraday effect. Measured magnetic field fluctuations at
various gravitational wave detector sites are on the order of $10^{-12}$
T/$\sqrt{\rm Hz}$ at 10 Hz [46], and the Verdet constant for silicon is 15
rad/(T$\cdot$m) [47]. These give $10^{-11}$ rad/$\sqrt{\rm Hz}$ level of
$\Delta\phi_{\rm s_{1}}$ for mirror thicknesses in Table 1, which is below the
requirements given above.

We note that, when deriving the requirements shown in Fig. 4, no safety margin
was considered. This means that the designed sensitivity will be fully limited
by one of the noises when that noise spectrum is the same as the requirement
curve, and all the other noises are negligibly small. To achieve the design
sensitivity, each noise should be negligibly smaller than the requirement,
e.g., by a factor of 10.

###  III.4 Amplitude noise at the antisymmetric port

So far, we have considered the phase noise in the arm cavity reflected beams
in gravitational wave detectors. In gravitational wave detectors, the
differential arm length caused by gravitational waves will be read out as the
interference fringe changes at the antisymmetric port. Birefringence
fluctuations will also create power fluctuations in the orthogonal
polarization, and it will be a noise source when the output Faraday isolator
has a finite extinction ratio $\epsilon$, and the orthogonal polarization is
not completely rejected. A slight misalignment of the axes between the input
Faraday isolator and the output Faraday isolator would also cause a finite
extinction ratio.

From Eq. (34), the power of the cavity reflected beam in the orthogonal
polarization from the birefringence in ITM can be written as

$\displaystyle\frac{\left.P_{\rm refl\perp}\right|_{\rm res}}{P_{\rm in}}$
$\displaystyle\simeq$ $\displaystyle\frac{1}{4}\left(\Delta\phi_{\rm
s_{1}}-2\Delta\phi_{\rm t_{1}}-\frac{\mathcal{F}}{\pi}\Delta\phi_{\rm
r_{1}}\right)^{2}\sin^{2}{(2\theta_{\rm pol})},$ (45)

when the cavity is on resonance. Here, $P_{\rm in}=|E_{\rm in}|^{2}$ is the
input power to the cavity, and we used that $r_{2}=1$, $r_{1}\simeq 1$ and
$t_{1}^{2}=1-r_{1}^{2}$, which are good approximations for arm cavities of
gravitational wave detectors. We also assumed that the amount of birefringence
is uniform and small, i.e., $\Delta\phi_{{\rm r}_{i}}\ll 2\pi/\mathcal{F}$,
$\Delta\phi_{\rm s_{1}}\ll 1$ and $\Delta\phi_{\rm t_{1}}\ll 1$.

As we can see from Eq. (34), the orthogonal polarization is vanished when
there is no birefringence, or $\theta_{\rm pol}$ is not 0 or $\pi/2$. The
orthogonal polarization component is generated from the reflected electric
field unbalance between two eigenmodes. Therefore, when the amount of
birefringence is small, the phase of $E_{\rm refl\perp}$ is always around
$\pi/2$ away from the phase of $E_{\rm refl\parallel}$. This means that the
orthogonal polarization in the cavity reflection is always in the quadrature
phase with respect to the gravitational wave signal, independent of the
resonant condition of the cavity.

In the case of gravitational wave detectors, the antisymmetric port therefore
will be at either the bright or the dark fringe for the orthogonal
polarization, when it is at the dark fringe for the main polarization. When
the both arms are completely symmetric and the amount of birefringence is the
same, the antisymmetric port will be at the bright fringe for the orthogonal
polarization. This is the same as the reason why the polarization signal from
axion dark matter is present at the antisymmetric port, as discussed in Ref.
[35]. In reality, the beam splitter in the Fabry-Pérot-Michelson
interferometer adds extra phase difference between two polarization axes due
to $\sim 45^{\circ}$ incident angle, and the fringe will be slightly shifted.

To derive the requirements for the extinction ratio $\epsilon$ of the output
Faraday isolator, let us assume that the power of the orthogonal polarization
component at the antisymmetric port can be roughly estimated from the power
from one of the arms. By requiring the power fluctuation from the orthogonal
polarization from one of the arms to be less than the shot noise of the local
oscillator beam in the main polarization, we can require

$\epsilon<\frac{1}{\left.P_{\rm refl\perp}\right|_{\rm
res}}\sqrt{\frac{2hcP_{\rm LO}}{\lambda}},$ (46)

where $P_{\rm LO}$ is the power of the local oscillator beam at the
antisymmetric port. When the requirements for the birefringence fluctuations
derived in the previous subsections are met, the noise from the birefringence
fluctuations are lower than the shot noise of the gravitational wave detector.
Therefore, the requirement can be rewritten as

$\epsilon\lesssim\sqrt{\frac{P_{\rm LO}}{P_{\rm in}}}\left(\Delta\phi_{\rm
s_{1}}-2\Delta\phi_{\rm t_{1}}-\frac{\mathcal{F}}{\pi}\Delta\phi_{\rm
r_{1}}\right)^{-1}.$ (47)

For gravitational wave detectors operating with DC readout scheme [48],
$P_{\rm LO}$ and $P_{\rm in}$ are on the order of 10 mW and 10 kW for the
power-recycled case, respectively. Assuming that the birefringence terms
$\Delta\phi_{\rm s_{1}}$, $\Delta\phi_{\rm t_{1}}$, and $\Delta\phi_{\rm
r_{1}}\mathcal{F}/\pi$ are on the order of 1 rad, the requirement to the
extinction ratio will be $\epsilon\lesssim 0.1\%$. This means that the input
Faraday isolator and the output Faraday isolator have to be aligned within
$1.8^{\circ}$.

##  IV Optical losses from inhomogeneous birefringence

Birefringence and its inhomogeneity in cavities create power losses from
depolarization. The mode content of the cavity reflected beam in the
orthogonal polarization will be different depending on the locations of
birefringence and the resonant condition of the cavity. In this section, we
discuss the power of cavity reflected beam in the orthogonal polarization to
estimate the optical loss.

To show that the different locations of birefringence create different mode
content, we first consider the effects from ITM, as we have considered in Eqs.
(33) and (34). From Eq. (34), the power losses to orthogonal polarization when
the cavity is out of resonance will be

$\frac{\left.P_{\rm refl\perp}\right|_{\rm off}}{P_{\rm
in}}\simeq\frac{1}{4}(\Delta\phi_{\rm s_{1}})^{2}\sin^{2}{(2\theta_{\rm
pol})},\\\ $ (48)

under the same approximations used to derive Eq. (45).

So far, we have considered only the birefringence uniform over the substrate
and the coating. When there is a perturbation from a uniform birefringence,
spatial higher order modes are generated. The amount of the higher order modes
in the orthogonal polarization can be estimated from inhomogeneous
birefringence $\Delta\phi_{\rm s_{1}}^{\rm HOM}$. The power in the higher
order modes when the cavity is on resonance and out of resonance will be

$\displaystyle\frac{\left.P_{\rm refl\perp}^{\rm HOM}\right|_{\rm res}}{P_{\rm
in}}$ $\displaystyle\simeq$ $\displaystyle\frac{1}{4}\left(\Delta\phi_{\rm
s_{1}}^{\rm HOM}-\Delta\phi_{\rm t_{1}}^{\rm
HOM}\right)^{2}\sin^{2}{(2\theta_{\rm pol})},$ (49)
$\displaystyle\frac{\left.P_{\rm refl\perp}^{\rm HOM}\right|_{\rm off}}{P_{\rm
in}}$ $\displaystyle\simeq$ $\displaystyle\frac{1}{4}(\Delta\phi_{\rm
s_{1}}^{\rm HOM})^{2}\sin^{2}{(2\theta_{\rm pol})},$ (50)

respectively. Note that the coefficient for $\Delta\phi_{\rm t_{1}}^{\rm HOM}$
is 1, as opposed to 2 for $\Delta\phi_{\rm t_{1}}$ in Eq. (45), since higher
order modes do not resonate in the cavity and higher order modes are generated
in the ITM transmission of the intracavity beam.

For considering the effect from the ITM substrate birefringence, we can set
$\Delta\phi_{\rm r_{1}}=0$, $\Delta\phi_{\rm s_{1}}=\Delta\phi_{\rm t_{1}}$
and $\Delta\phi_{\rm s_{1}}^{\rm HOM}=\Delta\phi_{\rm t_{1}}^{\rm HOM}$. In
this case, the amount of the fundamental transverse mode in the orthogonal
polarization stays the same when the cavity is out of resonance or on
resonance. However, the amount of higher order modes in the orthogonal
polarization is suppressed to the second order, as we can see from Eq. (49).
This is similar to the Lawrence effect for the thermal lensing of ITM [49]. It
is worth noting that the cavity reflected power in the main polarization
$P_{\rm refl\parallel}$ could increase when the cavity is on resonance due to
this effect, if the optical loss in the cavity is small compared with the
optical loss from inhomogeneous birefringence.

For KAGRA sapphire ITM, the transmission wavefront error difference between
two polarizations was measured to be around 60 nm in RMS [19, 20], which
corresponds to the round-trip phase difference $\Delta\phi_{\rm s_{1}}^{\rm
HOM}$ of 0.7 rad in RMS. If we attribute this all to inhomogeneous refractive
index difference using Eq. (43), this corresponds to $\Delta n^{\rm HOM}$ of
$2\times 10^{-7}$ in RMS, using the KAGRA sapphire mirror thickness being 15
cm and laser wavelength being 1064 nm. For sapphire, the amount of
birefringence along the $c$ axis can be calculated with [50]

$\Delta n=\frac{n_{o}(n_{o}^{2}-n_{e}^{2})\psi^{2}}{n_{e}^{2}},$ (51)

where $n_{e}=1.747$ and $n_{o}=1.754$ are the refractive indices in the $c$
axis and in axes orthogonal to the $c$ axis, respectively, and $\psi\ll 1$ is
the inclination of the light propagation direction with respect to the $c$
axis. Using this equation, the amount of birefringence observed in KAGRA can
be explained by $\psi^{\rm HOM}$ being $0.2^{\circ}$ in RMS. This is larger
than nominal orientation of the beam propagation axis with respect to the $c$
axis, which was aligned within $0.1^{\circ}$ [20]. This suggests that
$\theta_{\rm pol}$ is also inhomogeneous and uncontrolled.

Using Eq. (50), this inhomogeneous birefringence create power loss to
orthogonal polarization of around 10% when the arm cavity is out of resonance,
if $\theta_{\rm pol}$ is around $\pi/4$. This is consistent with the measured
value in KAGRA, as reported in Ref. [9]. The reduction of the power loss to
orthogonal polarization on resonance was also observed, which is consistent
with the Lawrence effect described above. In the KAGRA case, the power of the
orthogonal polarization inside the power recycling cavity was reduced by a
factor of 3 when the arm cavity was locked on resonance.

To make the optical loss due to inhomogeneous birefringence of ITM substrate
always smaller than 0.1%, $\Delta\phi_{\rm s_{1}}$ and $\Delta\phi_{\rm
s_{1}}^{\rm HOM}$ need to be smaller than 0.06 rad in RMS. Achieving this with
surface figuring alone could be challenging, as surface figuring cannot
compensate for the phase difference between two axes. This requirement can be
eased by aligning the input polarization axis to $\theta_{\rm pol}=0$ or
$\pi/2$.

When considering the effect from the ITM coating birefringence, we can set
$\Delta\phi_{\rm s_{1}}=\Delta\phi_{\rm r_{1}}$. However, $\Delta\phi_{\rm
s_{1}}$ is not exactly $\Delta\phi_{\rm t_{1}}$, as the penetration length for
the coating is different from the coating thickness. Therefore, the Lawrence
effect does not completely suppress the higher order modes. If we can set
$\Delta\phi_{\rm s_{1}}=l\Delta\phi_{\rm t_{1}}$, where $0<l<1$ is the ratio
of the penetration length over the coating thickness, the higher order modes
in the orthogonal polarization increase when the cavity is locked on
resonance, for $l<0.5$. The fundamental transverse mode in the orthogonal
polarization increases for high finesse cavities with $\mathcal{F}/\pi\gg 1$.

The mode content in the orthogonal polarization from the ETM coating
birefringence can be obtained by replacing $\Delta\phi_{\rm r_{1}}$ with
$\Delta\phi_{\rm r_{2}}$ and $\theta_{\rm pol}$ with $\theta+\theta_{\rm pol}$
in Eqs. (45), (48), (49), and (50) and by setting $\Delta\phi_{\rm
s_{1}}=\Delta\phi_{\rm t_{1}}=0$, as

$\displaystyle\frac{\left.P_{\rm refl\perp}\right|_{\rm res}}{P_{\rm in}}$
$\displaystyle\simeq$
$\displaystyle\frac{1}{4}\left(\frac{\mathcal{F}}{\pi}\Delta\phi_{\rm
r_{2}}\right)^{2}\sin^{2}{[2(\theta+\theta_{\rm pol})]},$ (52)
$\displaystyle\frac{\left.P_{\rm refl\perp}\right|_{\rm off}}{P_{\rm in}}$
$\displaystyle\simeq$ $\displaystyle 0,$ (53) $\displaystyle\frac{\left.P_{\rm
refl\perp}^{\rm HOM}\right|_{\rm res}}{P_{\rm in}}$ $\displaystyle\simeq$
$\displaystyle 0,$ (54) $\displaystyle\frac{\left.P_{\rm refl\perp}^{\rm
HOM}\right|_{\rm off}}{P_{\rm in}}$ $\displaystyle\simeq$ $\displaystyle 0.$
(55)

Therefore, as for the effects from the ETM coating birefringence, the power in
the orthogonal polarization increases when the cavity is locked on resonance,
and the fundamental transverse mode dominates, because the higher order modes
are suppressed in the cavity.

The discussions above highlights the fact that the optical losses from
birefringence needs to be correctly taken into account to measure the optical
losses in the arm cavity. It also suggests that, by measuring the mode content
of the beam in the orthogonal polarization when the cavity is out of resonance
and on resonance, we can estimate where the optical losses from birefringence
are mainly coming from.

Future gravitational wave detector designs call for 10 dB of detected
squeezing, requiring that the total optical loss be less than 10% [37]. From
Eqs. (45) and (52), $|\theta|$ and $|\theta+\theta_{\rm pol}|$ needs to be
less than $1.8^{\circ}$, requiring the optical loss from birefringence be less
than 0.1%, when the birefringence terms $\Delta\phi_{\rm s_{1}}$,
$\Delta\phi_{\rm t_{1}}$, and $\Delta\phi_{\rm r_{j}}\mathcal{F}/\pi$ are on
the order of 1 rad. Similar to the discussions around Eq. (23), the
polarization of the injected squeezed vacuum also needs to be aligned to less
than $1.8^{\circ}$ to achieve the optical loss of less than 0.1%.

##  V Conclusions and outlook

In this paper, we have discussed the effects of birefringence and its
fluctuations in the mirror substrate and coating for laser interferometric
gravitational wave detectors. We have shown that the polarization axis of the
beam and the crystal axes of mirrors need to be aligned to minimize the
optical losses and the noises from birefringence fluctuations. The optical
losses from birefringence can be feasibly reduced to less than 0.1%, when the
axes are aligned within a few degrees. We have also shown that the
requirements for the birefringence fluctuations in the substrate and the
coating will be on the order of $10^{-8}$ and $10^{-10}$ rad/$\sqrt{\rm Hz}$
at 100 Hz, respectively, for future gravitational wave detectors with mirrors
that have $\Delta n=10^{-7}$ level of substrate birefringence and
$\Delta\phi_{{\rm r}_{i}}=1$ mrad level of coating birefringence. When the
static coating birefringence is large such that the resonant frequency
difference between two polarization eigenmodes is larger than the cavity
linewidth, the requirements on the coating birefringence fluctuations will be
relaxed. In addition, we have derived the equations for estimating the amount
of optical losses due to depolarization from inhomogeneous birefringence of
mirror substrates and coatings. Our results provide the basic theory to study
the noises and optical losses from birefringence fluctuations of mirrors in
gravitational wave detectors.

In our model, we assumed that the amount of birefringence and misorientation
of axes to be small. We also assumed that two interferometer arms of
gravitational wave detectors to be close to symmetric. Detailed interferometer
modeling will be necessary to treat larger birefringence, misorientation of
axes, inhomogeneity of birefringence and axes orientations, and asymmetry
between two arms including birefringent beam splitter effects. These effects
would create classical radiation pressure noise, as intracavity power
fluctuates from birefringence fluctuations. Including the power and signal
recycling cavities to the model would also be important when these effects are
not negligible and the resonant condition in the recycling cavities is
different between polarizations. We leave these studies to future work.

###### Acknowledgements.

We thank Hiroki Fujimoto, Kevin Kuns, Stefan W. Ballmer, Valery Frolov and
Martin M. Fejer for insightful discussions. This work was supported by the
Gordon and Betty Moore Foundation, by the National Science Foundation under
Grant No. PHY-1912677, by JSPS KAKENHI Grant No. JP20H05854, and by JST PRESTO
Grant No. JPMJPR200B. F.S.–C. acknowledges support from the Barish–Weiss
postdoctoral fellowship. This paper carries LIGO DCC No. LIGO-P2300220 and JGW
Document No. JGW-P2315068.

##  Appendix A Derivation of electric fields

Here we derive the electric field inside the cavity in Eqs. (5) and (25), and
the electric field of the cavity reflection in Eq. (28).

In the basis of ITM crystal axes, the amplitude reflectivity of ETM can be
written as $R(-\theta)R_{2}R(\theta)$ [38]. The rotation matrix $R(\theta)$ is
necessary to take into account of the axes rotation between ITM and ETM.
Therefore, the Jones matrix for the cavity round-trip can be written as a
product of ITM reflection, ETM reflection, and the phase shift accumulated in
the round-trip as

$A=R_{1}R(-\theta)R_{2}R(\theta)e^{-i\phi}.$ (56)

The electric field inside the cavity that propagates from ITM to ETM is a sum
of the ITM transmitted field and its multiple reflections inside the cavity,
which can be written as

$\displaystyle\vec{E}_{\rm cav}$ $\displaystyle=$ $\displaystyle
T_{1}\vec{E}_{\rm in}+AT_{1}\vec{E}_{\rm in}+A^{2}T_{1}\vec{E}_{\rm
in}+\cdots$ (57) $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}A^{n-1}T_{1}\vec{E}_{\rm in}.$ (58)

This is a sum of an infinite geometric series, and Eq. (5) can be derived.

The electric field inside the cavity that propagates from ETM to ITM has an
additional reflection from ETM and phase $\phi$ from a cavity round-trip,
which lead to Eq. (25). The electric field of the cavity reflection is the sum
of the field reflected from ITM substrate side and the intracavity field
transmitted through ITM. Therefore, it can be written as

$\vec{E}_{\rm refl}=S_{1}\vec{E}_{\rm in}+T_{1}\vec{E}^{\prime}_{\rm cav},$
(59)

and Eq. (28) can be derived.

##  Appendix B Derivation of equivalent phase anisotropy

Here we derive the equivalent phase anisotropy in Eq. (17). We consider the
situation described in Ref. [39], where the phase anisotropy and relative
orientation of the birefringent cavity are captured by a single equivalent
Jones transformation. To simplify the notation, we write the Jones operators
in the Pauli basis spanned by $I$ and $\vec{\sigma}$ where $I$ is the identity
matrix and $\vec{\sigma}=\sigma_{\rm o}\vec{e}_{\rm o}+\sigma_{\rm
e}\vec{e}_{\rm e}+\sigma_{\rm z}\vec{e}_{\rm z}$ is the Pauli vector used to
map rotations along the ordinary, extraordinary, and cavity axis unit vectors,
respectively. For example, the Jones operator for a half-wave plate with phase
anisotropy $\delta$ oriented at an angle $\theta$ away from the ordinary axis
$\vec{e}_{\rm o}$ may be written in this representation as

$\vec{\mathcal{W}}(\delta,\theta)\cdot\vec{\sigma}=\cos\left(\frac{\delta}{2}\right){I}-i\sin\left(\frac{\delta}{2}\right)\left(\sin
2\theta{\sigma}_{\rm o}-\cos 2\theta{\sigma}_{\rm z}\right)$ (60)

and reduced to
$\cos\left(\frac{\delta}{2}\right)I+i\sin\left(\frac{\delta}{2}\right){\sigma}_{\rm
z}$ when aligned with $\vec{e}_{\rm o}$ (i.e. $\theta=0$).

Following Ref. [39], the equivalent wave plate anisotropy $\delta_{\rm EQ}$
comprises two wave plate operators with phase anisotropies $\delta_{1}$ and
$\delta_{2}$ oriented at $\theta_{1}=0$ and $\theta_{2}=\theta_{\rm WP}$
respectively. Then, the total operator

$\vec{\mathcal{W}}(\delta_{\rm EQ},\theta_{\rm
EQ})\cdot\vec{\sigma}=[\vec{\mathcal{W}}(\delta_{1},0)\cdot\vec{\sigma}][\vec{\mathcal{W}}(\delta_{2},\theta_{\rm
WP})\cdot\vec{\sigma}]$ (61)

can be constructed by the individual operators. After some manipulation, we
obtain an equation for each component beginning with

$\displaystyle\cos\left(\frac{\delta_{\rm EQ}}{2}\right)$
$\displaystyle=\cos\left(\frac{\delta_{1}}{2}\right)\cos\left(\frac{\delta_{2}}{2}\right)$
(62)
$\displaystyle-\sin\left(\frac{\delta_{1}}{2}\right)\sin\left(\frac{\delta_{2}}{2}\right)\cos
2\theta_{\rm WP}$

from terms along $I$, and then

$\displaystyle\sin 2\theta_{\rm EQ}\sin\left(\frac{\delta_{\rm
EQ}}{2}\right)=\cos\left(\frac{\delta_{1}}{2}\right)\sin\left(\frac{\delta_{2}}{2}\right)\sin
2\theta_{\rm WP}$ (63)

for terms along $\sigma_{\rm o}$,

$\displaystyle\sin\left(\frac{\delta_{1}}{2}\right)\sin\left(\frac{\delta_{2}}{2}\right)\sin
2\theta_{\rm WP}=0$ (64)

from terms along $\sigma_{\rm e}$, and

$\displaystyle\sin\left(\frac{\delta_{\rm EQ}}{2}\right)\cos 2\theta_{\rm EQ}$
$\displaystyle=\sin\left(\frac{\delta_{1}}{2}\right)\cos\left(\frac{\delta_{2}}{2}\right)$
(65)
$\displaystyle+\cos\left(\frac{\delta_{1}}{2}\right)\sin\left(\frac{\delta_{2}}{2}\right)\cos
2\theta_{\rm WP}$

for terms along $\sigma_{\rm z}$.

Letting $\delta_{\rm EQ}\ll 1$, $\delta_{1}\ll 1$, and $\delta_{2}\ll 1$, we
may expand Eqs (62)–(65) to second order in $\delta_{\rm EQ}$, $\delta_{1}$,
and $\delta_{2}$, keeping terms only of up to ${\mathcal{O}}(\delta^{2})$.
Then, Eq. (62) becomes

$1-\frac{\delta_{\rm EQ}^{2}}{8}\approx
1-\frac{\delta_{1}^{2}}{8}-\frac{\delta_{2}^{2}}{8}-\frac{\delta_{1}\delta_{2}}{4}\cos
2\theta_{\rm WP}.$ (66)

from which Eq. (17) may be easily derived and applied for the cases discussed
in the text when $\delta_{i}=\Delta\phi_{r_{i}}$. Finally, inserting (17) back
into (65) gives

$\displaystyle\cos 2\theta_{\rm EQ}$
$\displaystyle\approx\frac{\delta_{1}+\delta_{2}\cos 2\theta_{\rm
WP}}{\sqrt{\delta_{1}^{2}-\delta_{2}^{2}+4\delta_{1}\delta_{2}\cos
2\theta_{\rm WP}}}.$ (67)

from which Eq. (19) may be easily derived and applied for the cases described
in text when $\delta_{i}=\Delta\phi_{r_{i}}$.

## References

  * Abbott  _et al._ [2016] B. P. Abbott _et al._ (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016).
  * Abbott  _et al._ [2017a] B. P. Abbott _et al._ (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017a).
  * Abbott  _et al._ [2017b] B. P. Abbott _et al._ , The Astrophysical Journal Letters 848, L12 (2017b).
  * Aasi  _et al._ [2015] J. Aasi _et al._ (The LIGO Scientific Collaboration), Classical and Quantum Gravity 32, 074001 (2015).
  * Acernese  _et al._ [2015] F. Acernese _et al._ (Virgo Collaboration), Classical and Quantum Gravity 32, 024001 (2015).
  * Abbott  _et al._ [2020] B. P. Abbott _et al._ , Living Reviews in Relativity 23, 3 (2020).
  * Adhikari [2014] R. X. Adhikari, Rev. Mod. Phys. 86, 121 (2014).
  * Aso  _et al._ [2013] Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto (The KAGRA Collaboration), Phys. Rev. D 88, 043007 (2013).
  * Akutsu  _et al._ [2021] T. Akutsu _et al._ (The KAGRA Collaboration), Progress of Theoretical and Experimental Physics 2021, 05A101 (2021).
  * Michimura  _et al._ [2018] Y. Michimura, K. Komori, A. Nishizawa, H. Takeda, K. Nagano, Y. Enomoto, K. Hayama, K. Somiya, and M. Ando, Phys. Rev. D 97, 122003 (2018).
  * Adhikari  _et al._ [2020] R. X. Adhikari _et al._ , Classical and Quantum Gravity 37, 165003 (2020).
  * Punturo  _et al._ [2010] M. Punturo _et al._ , Classical and Quantum Gravity 27, 194002 (2010).
  * ET steering committee [2020] ET steering committee,  _ET design report update 2020_, Tech. Rep. ET-0007B-20 (2020).
  * Abbott  _et al._ [2017c] B. P. Abbott _et al._ (LIGO Scientific Collaboration), Classical and Quantum Gravity 34, 044001 (2017c).
  * Evans  _et al._ [2021] M. Evans, R. X. Adhikari, C. Afle, S. W. Ballmer, S. Biscoveanu, S. Borhanian, D. A. Brown, Y. Chen, R. Eisenstein, A. Gruson, A. Gupta, E. D. Hall, R. Huxford, B. Kamai, R. Kashyap, J. S. Kissel, K. Kuns, P. Landry, A. Lenon, G. Lovelace, L. McCuller, K. K. Y. Ng, A. H. Nitz, J. Read, B. S. Sathyaprakash, D. H. Shoemaker, B. J. J. Slagmolen, J. R. Smith, V. Srivastava, L. Sun, S. Vitale, and R. Weiss, A horizon study for cosmic explorer: Science, observatories, and community (2021), arXiv:2109.09882 [astro-ph.IM] .
  * Cole  _et al._ [2013] G. D. Cole, W. Zhang, M. J. Martin, J. Ye, and M. Aspelmeyer, Nature Photonics 7, 644 (2013).
  * Cumming  _et al._ [2015] A. V. Cumming, K. Craig, I. W. Martin, R. Bassiri, L. Cunningham, M. M. Fejer, J. S. Harris, K. Haughian, D. Heinert, B. Lantz, A. C. Lin, A. S. Markosyan, R. Nawrodt, R. Route, and S. Rowan, Classical and Quantum Gravity 32, 035002 (2015).
  * Winkler  _et al._ [1994] W. Winkler, A. Rüdiger, R. Schilling, K. Strain, and K. Danzmann, Optics Communications 112, 245 (1994).
  * Somiya  _et al._ [2019] K. Somiya, E. Hirose, and Y. Michimura, Phys. Rev. D 100, 082005 (2019).
  * Hirose  _et al._ [2020] E. Hirose, G. Billingsley, L. Zhang, H. Yamamoto, L. Pinard, C. Michel, D. Forest, B. Reichman, and M. Gross, Phys. Rev. Appl. 14, 014021 (2020).
  * Krüger  _et al._ [2015] C. Krüger, D. Heinert, A. Khalaidovski, J. Steinlechner, R. Nawrodt, R. Schnabel, and H. Lück, Classical and Quantum Gravity 33, 015012 (2015).
  * Jaberian Hamedan  _et al._ [2023] V. Jaberian Hamedan, A. Adam, C. Blair, L. Ju, and C. Zhao, Applied Physics Letters 122, 064101 (2023).
  * Winkler  _et al._ [2021] G. Winkler, L. W. Perner, G.-W. Truong, G. Zhao, D. Bachmann, A. S. Mayer, J. Fellinger, D. Follman, P. Heu, C. Deutsch, D. M. Bailey, H. Peelaers, S. Puchegger, A. J. Fleisher, G. D. Cole, and O. H. Heckl, Optica 8, 686 (2021).
  * Tanioka  _et al._ [2023] S. Tanioka, D. Vander-Hyde, G. D. Cole, S. D. Penn, and S. W. Ballmer, Phys. Rev. D 107, 022003 (2023).
  * Yu  _et al._ [2023] J. Yu, S. Häfner, T. Legero, S. Herbers, D. Nicolodi, C. Y. Ma, F. Riehle, U. Sterr, D. Kedar, J. M. Robinson, E. Oelker, and J. Ye, Phys. Rev. X 13, 041002 (2023).
  * Kryhin  _et al._ [2023] S. Kryhin, E. D. Hall, and V. Sudhir, Phys. Rev. D 107, 022001 (2023).
  * Bielsa  _et al._ [2009] F. Bielsa, A. Dupays, M. Fouché, R. Battesti, C. Robilliard, and C. Rizzo, Applied Physics B 97, 457 (2009).
  * Hollis  _et al._ [2019] H. Hollis, G. Alberts, D. B. Tanner, and G. Mueller, A laser heterodyne polarimeter for birefringence measurement (2019), arXiv:1710.03801 [physics.ins-det] .
  * Zavattini  _et al._ [2018] G. Zavattini, F. Della Valle, A. Ejlli, W.-T. Ni, U. Gastaldi, E. Milotti, R. Pengo, and G. Ruoso, The European Physical Journal C 78, 585 (2018).
  * Ejlli  _et al._ [2020] A. Ejlli, F. Della Valle, U. Gastaldi, G. Messineo, R. Pengo, G. Ruoso, and G. Zavattini, Physics Reports 871, 1 (2020), the PVLAS experiment: A 25 year effort to measure vacuum magnetic birefringence.
  * Kamioka [2020] S. Kamioka,  _Search For Vacuum Magnetic Birefringence with a high repetitive pulsed magnet_ , Ph.D. thesis, University of Tokyo (2020).
  * Obata  _et al._ [2018] I. Obata, T. Fujita, and Y. Michimura, Phys. Rev. Lett. 121, 161301 (2018).
  * Liu  _et al._ [2019] H. Liu, B. D. Elwood, M. Evans, and J. Thaler, Phys. Rev. D 100, 023548 (2019).
  * Martynov and Miao [2020] D. Martynov and H. Miao, Phys. Rev. D 101, 095034 (2020).
  * Nagano  _et al._ [2019] K. Nagano, T. Fujita, Y. Michimura, and I. Obata, Phys. Rev. Lett. 123, 111301 (2019).
  * Nagano  _et al._ [2021] K. Nagano, H. Nakatsuka, S. Morisaki, T. Fujita, Y. Michimura, and I. Obata, Phys. Rev. D 104, 062008 (2021).
  * Oelker  _et al._ [2016] E. Oelker, G. Mansell, M. Tse, J. Miller, F. Matichard, L. Barsotti, P. Fritschel, D. E. McClelland, M. Evans, and N. Mavalvala, Optica 3, 682 (2016).
  * Jones [1941] R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941).
  * Brandi  _et al._ [1997] F. Brandi, F. Della Valle, A. M. De Riva, P. Micossi, F. Perrone, C. Rizzo, G. Ruoso, and G. Zavattini, Applied Physics B 65, 351 (1997).
  * Miller  _et al._ [2015] J. Miller, L. Barsotti, S. Vitale, P. Fritschel, M. Evans, and D. Sigg, Phys. Rev. D 91, 062005 (2015).
  * Evans  _et al._ [2020] M. Evans, R. Sturani, S. Vitale, and E. Hall,  _Unofficial sensitivity curves (ASD) for aLIGO, Kagra, Virgo, Voyager, Cosmic Explorer, and Einstein Telescope_, Tech. Rep. LIGO-T1500293-v13 (2020).
  * Kuns  _et al._ [2023] K. Kuns, E. Hall, V. Srivastava, J. S. Send, M. Evans, P. Fritschel, L. McCuller, C. Wipf, and S. Ballmer,  _Cosmic Explorer Strain Sensitivity_, Tech. Rep. CE-T2000017-v6 (2023).
  * ET design team [2018] ET design team,  _ET-D sensitivity curve txt file_, Tech. Rep. ET-0000A-18 (2018).
  * Essick  _et al._ [2017] R. Essick, S. Vitale, and M. Evans, Phys. Rev. D 96, 084004 (2017).
  * Aston  _et al._ [2012] S. M. Aston, M. A. Barton, A. S. Bell, N. Beveridge, B. Bland, A. J. Brummitt, G. Cagnoli, C. A. Cantley, L. Carbone, A. V. Cumming, L. Cunningham, R. M. Cutler, R. J. S. Greenhalgh, G. D. Hammond, K. Haughian, T. M. Hayler, A. Heptonstall, J. Heefner, D. Hoyland, J. Hough, R. Jones, J. S. Kissel, R. Kumar, N. A. Lockerbie, D. Lodhia, I. W. Martin, P. G. Murray, J. O’Dell, M. V. Plissi, S. Reid, J. Romie, N. A. Robertson, S. Rowan, B. Shapiro, C. C. Speake, K. A. Strain, K. V. Tokmakov, C. Torrie, A. A. van Veggel, A. Vecchio, and I. Wilmut, Classical and Quantum Gravity 29, 235004 (2012).
  * Coughlin  _et al._ [2018] M. W. Coughlin, A. Cirone, P. Meyers, S. Atsuta, V. Boschi, A. Chincarini, N. L. Christensen, R. De Rosa, A. Effler, I. Fiori, M. Gołkowski, M. Guidry, J. Harms, K. Hayama, Y. Kataoka, J. Kubisz, A. Kulak, M. Laxen, A. Matas, J. Mlynarczyk, T. Ogawa, F. Paoletti, J. Salvador, R. Schofield, K. Somiya, and E. Thrane, Phys. Rev. D 97, 102007 (2018).
  * Snetkov and Yakovlev [2022] I. Snetkov and A. Yakovlev, Opt. Lett. 47, 1895 (2022).
  * Fricke  _et al._ [2012] T. T. Fricke, N. D. Smith-Lefebvre, R. Abbott, R. Adhikari, K. L. Dooley, M. Evans, P. Fritschel, V. V. Frolov, K. Kawabe, J. S. Kissel, B. J. J. Slagmolen, and S. J. Waldman, Classical and Quantum Gravity 29, 065005 (2012).
  * Lawrence [2003] R. C. Lawrence,  _Active wavefront correction in laser interferometric gravitational wave detectors_ , Ph.D. thesis, Massachusetts Institute of Technology (2003).
  * Tokunari  _et al._ [2006] M. Tokunari, H. Hayakawa, K. Yamamoto, T. Uchiyama, S. Miyoki, M. Ohashi, and K. Kuroda, Journal of Physics: Conference Series 32, 432 (2006).