Effects of mirror birefringence and its fluctuations to laser interferometric
gravitational wave detectors
Yuta Michimura ,
1,2,3
,*
Haoyu Wang ,
2
Francisco Salces-Carcoba ,
1
Christopher Wipf,
1
Aidan Brooks ,
1
Koji Arai ,
1
and Rana X. Adhikari
1
1
LIGO Laboratory, California Institute of Technology, Pasadena, California 91125, USA
2
Research Center for the Early Universe (RESCEU), Graduate School of Science, University of Tokyo,
Tokyo 113-0033, Japan
3
PRESTO, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan
(Received 4 August 2023; accepted 4 December 2023; published 29 January 2024)
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
=
ffiffiffiffiffiffi
Hz
p
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.
DOI:
10.1103/PhysRevD.109.022009
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 gravita-
tional 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 Tele-
scope
[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 birefrin-
gence of mirror substrates and coatings could cause optical
losses due to depolarization of the light and cause degra-
dation 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 sus-
pension system. Birefringence measurements in silicon
mirrors have revealed that the amount of the static
birefringence is
Δ
n
∼
10
−
7
or less at laser wavelengths
of 1.55
[21]
and
2
μ
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 birefrin-
gence and optical losses from the depolarization. However,
recent measurement of thermal noises in crystalline
mirror coatings at cryogenic temperatures reported excess
*
yuta@caltech.edu
PHYSICAL REVIEW D
109,
022009 (2024)
Editors' Suggestion
2470-0010
=
2024
=
109(2)
=
022009(12)
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© 2024 American Physical Society
birefringent noise, which could limit the sensitivity of future
gravitationalwave detectors
[25]
.Theoreticalcalculationson
thermal fluctuations of birefringence in crystalline mirror
coatings have also revealed that the noise from these
fluctuations could be similar to Brownian noise
[26]
.Itis
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
–
31]
. These temporal
birefringence fluctuations could also limit optical cavity-
based axion dark matter searches using the birefringence
effect from axion-photon coupling
[32
–
36]
.
In this paper, we study the effects of birefringence and its
fluctuations to gravitational wave detectors based on the
Fabry-P ́
erot-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 inho-
mogeneous 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-P ́
erot 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
θ
with respect to that of ITM. The input beam is
linearly polarized, and its polarization is rotated by
θ
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.
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
⃗
E
in
¼ð
v
1
⃗
e
o
þ
v
2
⃗
e
e
Þ
E
in
¼ð
⃗
e
o
⃗
e
e
Þ
⃗
v
in
E
in
;
ð
1
Þ
where
⃗
e
o
and
⃗
e
e
are the unit vectors along with the
ITM ordinary and extraordinary axes, respectively, and
⃗
v
in
≡
ð
v
1
v
2
Þ
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
1
¼
t
1
0
0
t
1
e
−
i
1
2
Δ
φ
t
1
;
ð
2
Þ
where
Δ
φ
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
¼
r
j
0
0
r
j
e
−
i
Δ
φ
r
j
;
ð
3
Þ
where
Δ
φ
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
¼
1
is for
ITM and
j
¼
2
is for ETM. Also, the amplitude reflectivity
of ITM from the substrate side can be written as
S
1
¼
−
r
1
0
0
−
r
1
e
−
i
Δ
φ
s
1
;
ð
4
Þ
where
Δ
φ
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,
Δ
φ
t
1
¼
Δ
φ
r
1
þ
Δ
φ
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
ITM
input
polarization
ETM
extraordinary
axis
ordinary
axis
ordinary
axis
extraordinary
axis
Faraday
isolator
FIG. 1. The schematic of a Fabry-P ́
erot cavity with mirror
crystal axes and input beam polarization axis illustrated. With
respect to the ITM ordinary axis, the input polarization is rotated
by
θ
pol
and the ETM ordinary axis is rotated by
θ
.
YUTA MICHIMURA
et al.
PHYS. REV. D
109,
022009 (2024)
022009-2
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
≃
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 cal-
culate the polarization eigenmodes in the cavity and the
phase of the cavity reflected beam.
A. Polarization eigenmodes in the cavity
The electric field inside the cavity that propagates from
ITM to ETM can be written as
⃗
E
cav
¼ð
I
−
A
Þ
−
1
T
1
⃗
E
in
;
ð
5
Þ
with
I
being the identity matrix. Here,
A
≡
R
1
R
ð
−
θ
Þ
R
2
R
ð
θ
Þ
e
−
i
φ
;
ð
6
Þ
where
φ
¼
4
π
L=
λ
is the phase acquired in the cavity round-
trip, with
L
and
λ
being the cavity length and the laser
wavelength, respectively, and
R
ð
θ
Þ
≡
cos
θ
−
sin
θ
sin
θ
cos
θ
;
ð
7
Þ
with the derivation described in Appendix
A
. Note that
φ
includes phase acquired in the ITM and ETM reflection for
their ordinary axes. The resonant polarization mode is the
eigenvectors of
M
cav
≡
ð
I
−
A
Þ
−
1
T
1
:
ð
8
Þ
The cavity enhancement factors for each mode will be the
eigenvalues of
M
cav
.
When
θ
¼
0
, the ITM axes and the ETM axes are
aligned, and the eigenvectors will be
⃗
v
a
¼
1
0
;
⃗
v
b
¼
0
1
;
ð
9
Þ
which means that the resonant modes are linear polar-
izations along the ITM ordinary axis
⃗
e
o
and the extraor-
dinary axis
⃗
e
e
. The cavity enhancement factors will be
w
a
¼
t
1
1
−
r
1
r
2
e
−
i
φ
;w
b
¼
t
1
e
−
i
1
2
Δ
φ
t
1
1
−
r
1
r
2
e
−
i
ð
φ
þ
Δ
φ
r
1
þ
Δ
φ
r
2
Þ
:
ð
10
Þ
The resonant frequency difference between two eigenm-
odes therefore will be
Δ
ν
¼
Δ
φ
r
1
þ
Δ
φ
r
2
2
π
ν
FSR
;
ð
11
Þ
where
ν
FSR
¼
c=
ð
2
L
Þ
is the free spectral range of the
cavity.
When
θ
¼
π
=
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
¼
t
1
1
−
r
1
r
2
e
−
i
ð
φ
þ
Δ
φ
r
2
Þ
;w
b
¼
t
1
e
−
i
1
2
Δ
φ
t
1
1
−
r
1
r
2
e
−
i
ð
φ
þ
Δ
φ
r
1
Þ
:
ð
12
Þ
The resonant frequency difference between two eigenm-
odes therefore will be
Δ
ν
¼
Δ
φ
r
1
−
Δ
φ
r
2
2
π
ν
FSR
:
ð
13
Þ
Since we defined the ITM and ETM axes such that
Δ
φ
r
i
have the same sign for ITM and ETM, when
θ
¼
0
,the
phase difference between the axes are added and the resonant
frequency difference is maximized. When
θ
¼
π
=
2
,itis
minimized, as the phase difference is canceled. When
0
<
θ
<
π
=
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.,
Δ
φ
r
i
≪
2
π
=
F
, and when the
effect from the ITM substrate birefringence is small, i.e.,
Δ
φ
t
1
≪
Δ
φ
r
1
F
=
π
, the resonant frequency difference can
be calculated with
Δ
ν
≃
2
π
ð
arg
w
a
−
arg
w
b
Þ
F
ν
FSR
2
π
;
ð
14
Þ
at
φ
¼
0
, where
F
¼
π
ffiffiffiffiffiffiffiffiffi
r
1
r
2
p
1
−
r
1
r
2
ð
15
Þ
is the finesse of the cavity. This can be further approxi-
mated as
[39]
Δ
ν
≃
δ
EQ
2
π
ν
FSR
;
ð
16
Þ
where
δ
EQ
≡
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
Δ
φ
r
1
−
Δ
φ
r
2
Þ
2
þ
4
Δ
φ
r
1
Δ
φ
r
2
cos
2
θ
q
;
ð
17
Þ
when
δ
EQ
≪
1
, with the derivation described in
Appendix
B
. Also, the cavity eigenmodes are linear
polarizations approximated as
⃗
v
a
¼
cos
θ
EQ
sin
θ
EQ
;
⃗
v
b
¼
−
sin
θ
EQ
cos
θ
EQ
;
ð
18
Þ
EFFECTS OF MIRROR BIREFRINGENCE AND ITS
...
PHYS. REV. D
109,
022009 (2024)
022009-3
where the polarization angle is defined by
cos
2
θ
EQ
¼
Δ
φ
0
r
1
Δ
φ
r
2
þ
cos
2
θ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Δ
φ
0
r
1
Δ
φ
r
2
−
1
2
þ
4
Δ
φ
0
r
1
Δ
φ
r
2
cos
2
θ
r
;
ð
19
Þ
with
Δ
φ
0
r
1
≡
Δ
φ
r
1
þ
π
F
Δ
φ
t
1
:
ð
20
Þ
When
Δ
φ
0
r
1
≫
Δ
φ
r
2
,
θ
EQ
is equal to zero; when
Δ
φ
0
r
1
¼
Δ
φ
r
2
,
θ
EQ
is equal to
θ
=
2
; and when
Δ
φ
0
r
1
≪
Δ
φ
r
2
,
θ
EQ
is equal to
θ
. Note that the polarization
state resonating inside the cavity are elliptic polarizations
given by
R
1
T
1
⃗
v
a;b
=
ð
r
1
t
1
Þ
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
Λ
2
¼
1
−
j
⃗
v
a
·
⃗
v
in
j
2
:
ð
21
Þ
When the input beam is linearly polarized with the
polarization angle of
θ
pol
such that
⃗
v
in
¼
R
ð
θ
pol
Þ
1
0
¼
cos
θ
pol
sin
θ
pol
;
ð
22
Þ
Eq.
(21)
reduces to
Λ
2
¼
sin
2
ð
θ
EQ
−
θ
pol
Þ
:
ð
23
Þ
The mismatch will be less than 0.1% when
j
θ
EQ
−
θ
pol
j
is
smaller than 1.8°. 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
shows the polarization eigenmodes of the cavity
as a function of ETM rotation angle
θ
, calculated using
Eqs.
(16)
and
(19)
. As we have discussed earlier, the
resonant frequency difference will be the maximized at
θ
¼
0
and minimized at
θ
¼
π
=
2
. When
θ
¼
π
=
2
and
Δ
φ
r
1
¼
Δ
φ
r
2
, the phase difference between ordinary and
extraordinary axes is completely canceled, 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
θ
¼
0
and
θ
¼
π
=
2
. To minimize the
mismatch and to make the resonant frequency difference
large, aligning the ETM rotation such that
θ
¼
0
and
aligning the input polarization to one of the ITM axes
will be the optimal choice. The requirement on the align-
ment will be not severe, since the dependence on the ETM
rotation angle goes with
θ
2
at
θ
¼
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
⃗
E
0
cav
¼
R
ð
−
θ
Þ
R
2
R
ð
θ
Þ
e
−
i
φ
M
cav
⃗
E
in
ð
24
Þ
≡
M
0
cav
⃗
E
in
:
ð
25
Þ
The eigenvectors of
M
0
cav
are the same as those of
M
cav
within our approximations discussed above, but the cavity
enhancement factors will be slightly different. When
θ
¼
0
,
the cavity enhancement factors will be
w
0
a
¼
t
1
r
2
e
−
i
φ
1
−
r
1
r
2
e
−
i
φ
;w
0
b
¼
t
1
r
2
e
−
i
ð
φ
þ
1
2
Δ
φ
t
1
þ
Δ
φ
r
2
Þ
1
−
r
1
r
2
e
−
i
ð
φ
þ
Δ
φ
r
1
þ
Δ
φ
r
2
Þ
;
ð
26
Þ
FIG. 2. The polarization eigenmodes of a Fabry-P ́
erot cavity as
a function of ETM rotation angle
θ
. The top panel shows the
round-trip phase difference between the eigenmodes in the unit
of
Δ
φ
r
1
, i.e.,
2
π
Δ
ν
=
ð
ν
FSR
Δ
φ
r
1
Þ
, which is proportional to the
resonant frequency difference. The middle panel shows the
polarization angle of the eigenmodes
θ
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
Δ
φ
r
2
=
Δ
φ
r
1
ratios. Blue lines
for
Δ
φ
r
2
¼
0
case in the bottom two plots are zero.
YUTA MICHIMURA
et al.
PHYS. REV. D
109,
022009 (2024)
022009-4
and when
θ
¼
π
=
2
, those will be
w
0
a
¼
t
1
r
2
e
−
i
ð
φ
þ
Δ
φ
r
2
Þ
1
−
r
1
r
2
e
−
i
ð
φ
þ
Δ
φ
r
2
Þ
;w
0
b
¼
t
1
r
2
e
−
i
ð
φ
þ
1
2
Δ
φ
t
1
Þ
1
−
r
1
r
2
e
−
i
ð
φ
þ
Δ
φ
r
1
Þ
:
ð
27
Þ
Compared with
w
a
and
w
b
, those have extra phase
φ
from
the cavity round-trip and extra phase
Δ
φ
r
2
for the corre-
sponding axis for one additional reflection from ETM.
B. 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
⃗
E
refl
¼
M
refl
⃗
E
in
;
ð
28
Þ
where
M
refl
≡
S
1
þ
T
1
M
0
cav
:
ð
29
Þ
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
⃗
v
in
¼
a
⃗
v
0
a
þ
b
⃗
v
0
b
;
ð
30
Þ
the polarization component of the reflected beam is
M
refl
⃗
v
in
¼
a
ð
S
1
þ
w
0
a
T
1
Þ
⃗
v
0
a
þ
b
ð
S
1
þ
w
0
b
T
1
Þ
⃗
v
0
b
:
ð
31
Þ
Since the resonant condition of each eigenmode is gen-
erally different, it is generally
j
w
0
a
j
≠
j
w
0
b
j
. 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
ð
E
out
Þ¼
arg
ð
E
refl
k
Þ¼
arg
ð
E
in
M
refl
⃗
v
in
·
⃗
v
in
Þ
:
ð
32
Þ
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
refl
, and that for the ITM
extraordinary axis is the (2,2) component of
M
refl
.
Let us first consider the effects from ITM. If we set
Δ
φ
r
2
¼
0
and the input beam is linearly polarized with the
polarization angle of
θ
pol
as shown in Eq.
(22)
, the reflected
electric field in the polarization parallel to
⃗
v
in
and in the
orthogonal polarization will be
E
refl
k
E
in
¼
M
refl
⃗
v
in
·
⃗
v
in
¼ð
−
r
1
þ
w
0
a
t
1
Þ
cos
2
θ
pol
þ
−
r
1
e
−
i
Δ
φ
s
1
þ
w
0
b
t
1
e
−
i
1
2
Δ
φ
t
1
sin
2
θ
pol
;
ð
33
Þ
E
refl
⊥
E
in
¼
M
refl
⃗
v
in
·
R
ð
θ
pol
Þ
0
1
¼
ð
−
r
1
þ
w
0
a
t
1
Þ
−
−
r
1
e
−
i
Δ
φ
s
1
þ
w
0
b
t
1
e
−
i
1
2
Δ
φ
t
1
×
sin
ð
2
θ
pol
Þ
2
:
ð
34
Þ
These are similar to the electric fields of the bright
reflection port and the dark antisymmetric port for a
Fabry-P ́
erot-Michelson interferometer that has an unbal-
anced beam splitter.
The effects from the ETM birefringence can be calcu-
lated by setting
Δ
φ
s
1
¼
Δ
φ
t
1
¼
0
and replacing
Δ
φ
r
1
with
Δ
φ
r
2
and
θ
pol
with
θ
þ
θ
pol
. If we combine the effects from
ITM and ETM, the phase of the reflected beam around the
resonance can be approximated as
arg
E
refl
k
E
in
¼ð
Δ
φ
s
1
−
2
Δ
φ
t
1
Þ
sin
2
θ
pol
−
F
π
φ
þ
Δ
φ
r
1
sin
2
θ
pol
þ
Δ
φ
r
2
sin
2
ð
θ
þ
θ
pol
Þ
;
ð
35
Þ
with the approximation that
Δ
φ
r
i
≪
2
π
=
F
and
r
2
¼
1
.Itis
clear that both the ETM rotation angle
θ
and the input
beam polarization angle
θ
pol
change the phase of the cavity
reflected beam and will contribute to the phase noise,
unless
θ
pol
and
θ
þ
θ
pol
are either 0 or
π
=
2
, where the
effects are quadratic to these angles. The fluctuations of
phase differences between ordinary and extraordinary axes
also create phase noises, unless
θ
pol
and
θ
þ
θ
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.,
Δ
φ
r
i
≫
2
π
=
F
. If the static coating birefringence is high
such that one of the eigenmodes is out of resonance when the
other is resonant, only
Δ
φ
s
1
and
φ
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
EFFECTS OF MIRROR BIREFRINGENCE AND ITS
...
PHYS. REV. D
109,
022009 (2024)
022009-5
the interferometer parameters summarized in Table
I
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-P ́
erot Michelson
interferometer. Although different interferometers plan to
use different materials for the mirrors, discussions pre-
sented here do not depend on the choice of materials.
A. Phase noises from substrate birefringence
The phase changes from the ITM substrate birefringence
can be calculated from Eq.
(35)
by setting
Δ
φ
r
1
¼
Δ
φ
r
2
¼
0
and
Δ
φ
s
1
¼
Δ
φ
t
1
. In this case, Eq.
(35)
reduces to
arg
E
refl
k
E
in
¼
−
Δ
φ
s
1
sin
2
θ
pol
−
F
π
φ
:
ð
36
Þ
Therefore, the length noise couplings from the fluctuations
of
θ
pol
and
Δ
φ
s
1
can be calculated as
δ
L
δθ
pol
¼
λ
4
π
δ
½
arg
ð
E
refl
k
Þ
δθ
pol
δ
½
arg
ð
E
refl
k
Þ
δφ
−
1
¼
λ
4
F
Δ
φ
s
1
sin
2
θ
pol
;
ð
37
Þ
δ
L
δ
ð
Δ
φ
s
1
Þ
¼
−
λ
4
F
sin
2
θ
pol
:
ð
38
Þ
B. 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
Δ
φ
r
1
and
Δ
φ
r
2
contributes more to the phase of the
reflected beam, compared with the first term from
Δ
φ
s
1
and
Δ
φ
t
1
, since the phase acquired inside the cavity is enhanced
by a factor of
F
=
π
. The length noise couplings from the
fluctuations of
θ
pol
,
θ
, and
Δ
φ
r
i
can be calculated as
δ
L
δθ
pol
¼
λ
4
π
Δ
φ
r
1
sin
2
θ
pol
þ
Δ
φ
r
2
sin
½
2
ð
θ
þ
θ
pol
Þ
;
ð
39
Þ
δ
L
δθ
¼
λ
4
π
Δ
φ
r
2
sin
½
2
ð
θ
þ
θ
pol
Þ
;
ð
40
Þ
δ
L
δ
ð
Δ
φ
r
1
Þ
¼
−
λ
4
π
sin
2
θ
pol
;
ð
41
Þ
δ
L
δ
ð
Δ
φ
r
2
Þ
¼
−
λ
4
π
sin
2
ð
θ
þ
θ
pol
Þ
:
ð
42
Þ
C. Requirements on birefringence fluctuations
Noise couplings discussed above are nulled when
θ
pol
¼
0
and
θ
¼
0
. For KAGRA test masses, the sapphire
c
axis was aligned to the cylindrical plane of the test mass
within 0.1°
[20]
. For deriving the requirements to birefrin-
gence 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
θ
pol
¼
1
° and
θ
¼
1
°,
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
Δ
n
, and
Δ
φ
s
1
can be written using the mirror thickness
t
as
Δ
φ
s
1
¼
4
π
λ
Δ
nt:
ð
43
Þ
We used the static birefringence value of
Δ
n
¼
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
Δ
φ
r
i
¼
1
mrad, which is a typical measured value for
FIG. 3. The designed displacement sensitivity for different
gravitational wave detectors. The strain sensitivity data are taken
from Refs.
[40
–
42]
and corrected to displacement sensitivities
by removing frequency-dependent responses to gravitational
waves
[43]
.
TABLE I. Interferometer parameters of Advanced LIGO
(aLIGO), A
þ
, Voyager, Cosmic Explorer (CE), Einstein Tele-
scope Low Frequency (ET-LF), and ET High Frequency (ET-HF)
used for calculating requirements.
L
, arm length;
F
, arm finesse;
t
, ITM thickness;
λ
, laser wavelength.
L
F
t
λ
Reference
aLIGO
4 km
450 20 cm 1064 nm
[4]
A
þ
4 km
450 20 cm 1064 nm
[44]
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]
YUTA MICHIMURA
et al.
PHYS. REV. D
109,
022009 (2024)
022009-6
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
Δ
φ
r
j
, we used Eq.
(42)
, as this gives more stringent
requirement than Eq.
(41)
. All the requirements are divided
by
ffiffiffi
2
p
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 gravi-
tational wave detectors is on the order of
10
−
10
rad
=
ffiffiffiffiffiffi
Hz
p
.
We note that the requirements on
θ
pol
and
θ
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
=
ffiffiffiffiffiffi
Hz
p
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
=
ffiffiffiffiffiffi
Hz
p
for
the substrate and
10
−
10
rad
=
ffiffiffiffiffiffi
Hz
p
for the coating. Bire-
fringence at
10
−
8
rad
=
ffiffiffiffiffiffi
Hz
p
level can be feasibly evaluated
with shot noise limited interferometry at the laser power of
P
¼
10
mW level, as the shot noise limited phase sensi-
tivity of a Michelson interferometer is given by
φ
shot
¼
ffiffiffiffiffiffiffiffi
hc
2
λ
P
r
;
ð
44
Þ
where
h
is the Planck constant and
c
is the speed of light.
Evaluation of birefringence at
10
−
10
rad
=
ffiffiffiffiffiffi
Hz
p
level requires
10-W class laser or cavity enhancements. Measurements can
be done at relatively lower power compared with gravita-
tional wave detectors, as the phase noise from birefringence
is attenuated by sin
2
θ
and sin
2
ð
θ
þ
θ
pol
Þ
, by aligning the
polarization axis and the mirror crystal axes. In the evalu-
ation 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 gravita-
tional wave detector sites are on the order of
10
−
12
T
=
ffiffiffiffiffiffi
Hz
p
at 10 Hz
[46]
, and the Verdet constant for silicon is
15
rad
=
ð
T·m
Þ
[47]
. These give
10
−
11
rad
=
ffiffiffiffiffiffi
Hz
p
level of
Δ
φ
s
1
for mirror thicknesses in Table
I
, 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
FIG. 4. The requirements on birefringence fluctuations from
the axis rotations (top) and from the phase difference between
ordinary and extraordinary axes (middle) for different gravita-
tional wave detectors. The bottom plot shows the requirement on
the substrate birefringence converted from the phase difference
requirements on
Δ
φ
s
1
in the middle plot, assuming uniform
Δ
n
,
using Eq.
(43)
. The solid lines are for the substrate that have a
static birefringence of
Δ
n
¼
10
−
7
, and the dashed lines are for the
coating that have a static birefringence of
Δ
φ
r
i
¼
1
mrad. For
deriving these requirements, we assumed that the input beam
polarization and the ETM axes are aligned to the ITM axes to
θ
pol
¼
1
° and
θ
¼
1
°, and no safety margin is considered.
EFFECTS OF MIRROR BIREFRINGENCE AND ITS
...
PHYS. REV. D
109,
022009 (2024)
022009-7
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.
D. 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 fluctua-
tions in the orthogonal polarization, and it will be a noise
source when the output Faraday isolator has a finite
extinction ratio
ε
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
P
refl
⊥
j
res
P
in
≃
1
4
Δ
φ
s
1
−
2
Δ
φ
t
1
−
F
π
Δ
φ
r
1
2
sin
2
ð
2
θ
pol
Þ
;
ð
45
Þ
when the cavity is on resonance. Here,
P
in
¼j
E
in
j
2
is the
input power to the cavity, and we used that
r
2
¼
1
,
r
1
≃
1
,
and
t
2
1
¼
1
−
r
2
1
, 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.,
Δ
φ
r
i
≪
2
π
=
F
,
Δ
φ
s
1
≪
1
, and
Δ
φ
t
1
≪
1
.
As we can see from Eq.
(34)
, the orthogonal polarization
is vanished when there is no birefringence or
θ
pol
is not 0 or
π
=
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
refl
⊥
is always around
π
=
2
away
from the phase of
E
refl
k
. This means that the orthogonal
polarization in the cavity reflection is always in the quad-
rature phase with respect to the gravitational wave signal,
independent of the resonant condition of the cavity.
In the case of gravitational wave detectors, the antisym-
metric 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-P ́
erot-
Michelson interferometer adds extra phase difference
between two polarization axes due to
∼
45
° incident angle,
and the fringe will be slightly shifted.
To derive the requirements for the extinction ratio
ε
of
the output Faraday isolator, let us assume that the power of
the orthogonal polarization component at the antisymmet-
ric 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
ε
<
1
P
refl
⊥
j
res
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
hcP
LO
λ
r
;
ð
46
Þ
where
P
LO
is the power of the local oscillator beam at the
antisymmetric port. When the requirements for the bire-
fringence 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
ε
≲
ffiffiffiffiffiffiffiffi
P
LO
P
in
s
Δ
φ
s
1
−
2
Δ
φ
t
1
−
F
π
Δ
φ
r
1
−
1
:
ð
47
Þ
For gravitational wave detectors operating with dc readout
scheme
[48]
,
P
LO
and
P
in
are on the order of 10 mW and
10 kW for the power-recycled case, respectively. Assuming
that the birefringence terms
Δ
φ
s
1
,
Δ
φ
t
1
, and
Δ
φ
r
1
F
=
π
are
on the order of 1 rad, the requirement to the extinction ratio
will be
ε
≲
0
.
1%
. This means that the input Faraday
isolator and the output Faraday isolator have to be aligned
within 1.8°.
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 orthogo-
nal 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
P
refl
⊥
j
off
P
in
≃
1
4
ð
Δ
φ
s
1
Þ
2
sin
2
ð
2
θ
pol
Þ
;
ð
48
Þ
under the same approximations used to derive Eq.
(45)
.
So far, we have considered only the birefringence uni-
form over the substrate and the coating. When there is a
YUTA MICHIMURA
et al.
PHYS. REV. D
109,
022009 (2024)
022009-8
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
Δ
φ
HOM
s
1
. The power in the
higher order modes when the cavity is on resonance and out
of resonance will be
P
HOM
refl
⊥
j
res
P
in
≃
1
4
Δ
φ
HOM
s
1
−
Δ
φ
HOM
t
1
2
sin
2
ð
2
θ
pol
Þ
;
ð
49
Þ
P
HOM
refl
⊥
j
off
P
in
≃
1
4
Δ
φ
HOM
s
1
2
sin
2
ð
2
θ
pol
Þ
;
ð
50
Þ
respectively. Note that the coefficient for
Δ
φ
HOM
t
1
is 1, as
opposed to 2 for
Δ
φ
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
Δ
φ
r
1
¼
0
,
Δ
φ
s
1
¼
Δ
φ
t
1
, and
Δ
φ
HOM
s
1
¼
Δ
φ
HOM
t
1
. In this case, the amount of the funda-
mental 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]
.Itis
worth noting that the cavity reflected power in the main
polarization
P
refl
k
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 inhomo-
geneous birefringence.
For KAGRA sapphire ITM, the transmission wave-front
error difference between two polarizations was measured to
be around 60 nm in rms
[19,20]
, which corresponds to the
round-trip phase difference
Δ
φ
HOM
s
1
of 0.7 rad in rms. If we
attribute this all to inhomogeneous refractive index differ-
ence using Eq.
(43)
, this corresponds to
Δ
n
HOM
of
2
×
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]
Δ
n
¼
n
o
ð
n
2
o
−
n
2
e
Þ
ψ
2
n
2
e
;
ð
51
Þ
where
n
e
¼
1
.
747
and
n
o
¼
1
.
754
are the refractive indices
in the
c
axis and in axes orthogonal to the
c
axis,
respectively, and
ψ
≪
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
ψ
HOM
being 0.2° 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°
[20]
. This suggests that
θ
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
θ
pol
is around
π
=
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 bire-
fringence of ITM substrate always smaller than 0.1%,
Δ
φ
s
1
and
Δ
φ
HOM
s
1
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
θ
pol
¼
0
or
π
=
2
.
When considering the effect from the ITM coating
birefringence, we can set
Δ
φ
s
1
¼
Δ
φ
r
1
. However,
Δ
φ
s
1
is not exactly
Δ
φ
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
Δ
φ
s
1
¼
l
Δ
φ
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 reso-
nance, for
l<
0
.
5
. The fundamental transverse mode in the
orthogonal polarization increases for high finesse cavities
with
F
=
π
≫
1
.
The mode content in the orthogonal polarization from
the ETM coating birefringence can be obtained by replac-
ing
Δ
φ
r
1
with
Δ
φ
r
2
and
θ
pol
with
θ
þ
θ
pol
in Eqs.
(45)
,
(48)
,
(49)
, and
(50)
and by setting
Δ
φ
s
1
¼
Δ
φ
t
1
¼
0
,as
P
refl
⊥
j
res
P
in
≃
1
4
F
π
Δ
φ
r
2
2
sin
2
½
2
ð
θ
þ
θ
pol
Þ
;
ð
52
Þ
P
refl
⊥
j
off
P
in
≃
0
;
ð
53
Þ
P
HOM
refl
⊥
j
res
P
in
≃
0
;
ð
54
Þ
P
HOM
refl
⊥
j
off
P
in
≃
0
:
ð
55
Þ
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
EFFECTS OF MIRROR BIREFRINGENCE AND ITS
...
PHYS. REV. D
109,
022009 (2024)
022009-9
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)
,
j
θ
j
and
j
θ
þ
θ
pol
j
needs to be less than 1.8°, requiring the optical
loss from birefringence be less than 0.1%, when the
birefringence terms
Δ
φ
s
1
,
Δ
φ
t
1
, and
Δ
φ
r
j
F
=
π
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° to achieve the optical loss of
less than 0.1%.
V. CONCLUSIONS AND OUTLOOK
In this paper, we have discussed the effects of birefrin-
gence and its fluctuations in the mirror substrate and
coating for laser interferometric gravitational wave detec-
tors. 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 birefrin-
gence 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
=
ffiffiffiffiffiffi
Hz
p
at 100 Hz, respectively, for future gravi-
tational wave detectors with mirrors that have
Δ
n
¼
10
−
7
level of substrate birefringence and
Δ
φ
r
i
¼
1
mrad level of
coating birefringence. When the static coating birefrin-
gence is large such that the resonant frequency difference
between two polarization eigenmodes is larger than the
cavity linewidth, the requirements on the coating birefrin-
gence fluctuations will be relaxed. In addition, we have
derived the equations for estimating the amount of optical
losses due to depolarization from inhomogeneous birefrin-
gence 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 the amount of birefringence
and misorientation of axes to be small. We also assumed
two interferometer arms of gravitational wave detectors to
be close to symmetric. Detailed interferometer modeling
will be necessary to treat larger birefringence, misorienta-
tion 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.
ACKNOWLEDGMENTS
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.
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 reflec-
tion in Eq.
(28)
.
In the basis of ITM crystal axes, the amplitude reflec-
tivity of ETM can be written as
R
ð
−
θ
Þ
R
2
R
ð
θ
Þ
[38]
. The
rotation matrix
R
ð
θ
Þ
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
¼
R
1
R
ð
−
θ
Þ
R
2
R
ð
θ
Þ
e
−
i
φ
:
ð
A1
Þ
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
⃗
E
cav
¼
T
1
⃗
E
in
þ
AT
1
⃗
E
in
þ
A
2
T
1
⃗
E
in
þð
A2
Þ
¼
X
∞
n
¼
1
A
n
−
1
T
1
⃗
E
in
:
ð
A3
Þ
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
φ
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 intra-
cavity field transmitted through ITM. Therefore, it can be
written as
⃗
E
refl
¼
S
1
⃗
E
in
þ
T
1
⃗
E
0
cav
;
ð
A4
Þ
and Eq.
(28)
can be derived.
YUTA MICHIMURA
et al.
PHYS. REV. D
109,
022009 (2024)
022009-10
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
⃗
σ
, where
I
is
the identity matrix and
⃗
σ
¼
σ
o
⃗
e
o
þ
σ
e
⃗
e
e
þ
σ
z
⃗
e
z
is the Pauli
vector used to map rotations along the ordinary, extraor-
dinary, and cavity axis unit vectors, respectively. For
example, the Jones operator for a half-wave plate with
phase anisotropy
δ
oriented at an angle
θ
away from the
ordinary axis
⃗
e
o
may be written in this representation as
⃗
W
ð
δ
;
θ
Þ
·
⃗
σ
¼
cos
δ
2
I
−
i
sin
δ
2
ð
sin
2
θσ
o
−
cos
2
θσ
z
Þ
ð
B1
Þ
and reduced to cos
ð
δ
2
Þ
I
þ
i
sin
ð
δ
2
Þ
σ
z
when aligned with
⃗
e
o
(i.e.,
θ
¼
0
).
Following Ref.
[39]
, the equivalent wave plate aniso-
tropy
δ
EQ
comprises two wave plate operators with phase
anisotropies
δ
1
and
δ
2
oriented at
θ
1
¼
0
and
θ
2
¼
θ
WP
,
respectively. Then, the total operator
⃗
W
ð
δ
EQ
;
θ
EQ
Þ
·
⃗
σ
¼
⃗
W
ð
δ
1
;
0
Þ
·
⃗
σ
⃗
W
ð
δ
2
;
θ
WP
Þ
·
⃗
σ
ð
B2
Þ
can be constructed by the individual operators. After some
manipulation, we obtain an equation for each component
beginning with
cos
δ
EQ
2
¼
cos
δ
1
2
cos
δ
2
2
−
sin
δ
1
2
sin
δ
2
2
cos
2
θ
WP
ð
B3
Þ
from terms along
I
and then
sin
2
θ
EQ
sin
δ
EQ
2
¼
cos
δ
1
2
sin
δ
2
2
sin
2
θ
WP
ð
B4
Þ
for terms along
σ
o
,
sin
δ
1
2
sin
δ
2
2
sin
2
θ
WP
¼
0
ð
B5
Þ
from terms along
σ
e
, and
sin
δ
EQ
2
cos
2
θ
EQ
¼
sin
δ
1
2
cos
δ
2
2
þ
cos
δ
1
2
sin
δ
2
2
cos
2
θ
WP
ð
B6
Þ
for terms along
σ
z
.
Letting
δ
EQ
≪
1
,
δ
1
≪
1
, and
δ
2
≪
1
, we may expand
Eqs.
(B3)
–
(B6)
to second order in
δ
EQ
,
δ
1
, and
δ
2
, keeping
terms only of up to
O
ð
δ
2
Þ
. Then, Eq.
(B3)
becomes
1
−
δ
2
EQ
8
≈
1
−
δ
2
1
8
−
δ
2
2
8
−
δ
1
δ
2
4
cos
2
θ
WP
;
ð
B7
Þ
from which Eq.
(17)
may be easily derived and applied for
the cases discussed in the text when
δ
i
¼
Δ
φ
r
i
. Finally,
inserting
(17)
back into
(B6)
gives
cos
2
θ
EQ
≈
δ
1
þ
δ
2
cos
2
θ
WP
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
δ
2
1
−
δ
2
2
þ
4
δ
1
δ
2
cos
2
θ
WP
p
;
ð
B8
Þ
from which Eq.
(19)
may be easily derived and applied for
the cases described in text when
δ
i
¼
Δ
φ
r
i
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