Scattering Loss in Precision Metrology due to Mirror Roughness
Yehonathan Drori,
1
Johannes Eichholz,
1, 2
Tega Edo,
1, 3
Hiro Yamamoto,
1
Yutaro Enomoto,
1, 4
Gautam Venugopalan,
1
Koji Arai, and Rana X Adhikari
1
LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
2
Centre for Gravitational Astrophysics, College of Science,
The Australian National University, Canberra, ACT 2600, Australia
3
Department of Physics and Astronomy, The University of Sheffield, S3 7RH, UK
4
Department of Applied Physics, School of Engineering,
The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Optical losses degrade the sensitivity of laser interferometric instruments. They reduce the number
of signal photons and introduce technical noise associated with diffuse light. In quantum-enhanced
metrology, they break the entanglement between correlated photons. Such decoherence is one of the
primary obstacles in achieving high levels of quantum noise reduction in precision metrology.
In this work, we compare direct measurements of cavity and mirror losses in the Caltech 40m
gravitational-wave detector prototype interferometer with numerical estimates obtained from semi-
analytic intra-cavity wavefront simulations using mirror surface profile maps. We show a unified
approach to estimating the total loss in optical cavities (such as the LIGO gravitational detectors)
that will lead towards the engineering of systems with minimum decoherence for quantum-enhanced
precision metrology.
I. INTRODUCTION
Laser interferometry is a pillar of modern precision
measurement and plays an integral role in the prepara-
tion, interrogation, and manipulation of quantum systems,
including entangled states of light and matter. Optical
losses pose both practical and fundamental limitations to
instrument performances. Absorption and scatter from
optical components or subsystems decrease classical signal
levels and introduce application-dependent technical noise.
Optical losses also drive the decoherence of non-classical
photon states, which find application in quantum comput-
ing, cryptography, and quantum-enhanced metrology. A
typical example for the latter is the use of
squeezed light
in ground-based interferometric gravitational-wave (GW)
detectors to reduce photon shot noise [
1
]. We discuss op-
tical loss as a source of decoherence for quantum states in
the context of these quantum noise-limited instruments.
The ability to directly observe gravitational waves
(GWs) is continuously evolving our understanding of the
universe and its history. The number of observed merg-
ers of binary systems of black holes and neutron stars is
steadily growing [
2
,
3
], and keeps informing black hole
population models, formation channels for compact ob-
jects, and parameter constraints on theories of gravity
and the equation of state of highly degenerate nuclear
environments. Improved sensitivity, for example from a
reduction of quantum noise, and detections of other, new
source types promise to multiply this knowledge across
the GW frequency spectrum.
Interferometric GW detectors are electromagnetic trans-
ducers for GWs that convert the strain of passing GWs on
space-time to phase modulations of a laser carrier. The
two Advanced LIGO (Adv. LIGO) observatories [
4
], Ad-
vanced Virgo (AdVirgo) [
5
], KAGRA [
6
], and GEO600 [
7
]
are ground-based detectors that are sensitive to signals
in the audible frequency range. They all have similar
optical topologies, using a combination of optical cavi-
ties to increase the number and storage time of photons
interacting with passing GWs. Optical loss generates
a variety of technical problems for the control of these
detectors. It broadens cavity line widths, affects the
behavior of coupled-cavity systems, and back-scattering
can create transient signal glitches or pollute entire fre-
quency bands [
8
–
10
]. Radiation pressure-induced opto-
mechanical effects due to scattered fields may also become
relevant. A thorough understanding of the magnitude and
nature of optical loss will help anticipate and mitigate
such effects.
During their third observation run (O3), the measured
round-trip losses of the Fabry-Perot arm cavities of the
Adv. LIGO detectors were in the range 60 – 70 ppm. This
is below the requirement of 75 ppm and not currently
considered a limiting factor for the interferometer per-
formance. The optical absorption of the high-quality
dielectric mirror coatings is below the ppm level, which
makes scatter the dominant loss mechanism in the in-
terferometers. The expected arm loss based on surface
figure phase maps and large angle scattering data is about
20 – 30 ppm less than the measured value. Part of this
discrepancy stems from the scattering into the angular re-
gion
∼
0
.
1 – 1
◦
, which is difficult to access experimentally
without ambiguity and not covered by existing measure-
ments. It is therefore desirable and important to have a
set of techniques that can reliably, repeatably and accu-
rately characterize optical loss, so that mitigation efforts
can be validated.
This work examines loss measurements at the Caltech
40 m interferometer (40m), which is a small-scale proto-
type with the same principal optical topology as Adv.
LIGO, and compares them with simulations to verify
these models for the purpose of extending the analysis to
other optical systems. The paper is structured as follows:
Section II provides necessary formulas and quantitative
arXiv:2201.05640v1 [physics.optics] 14 Jan 2022
2
and qualitative description of the scattering and loss.
Section III describes the optical measurements utilized
to estimate the total optical cavity losses. Finally, Sec-
tion IV compares the simulations with the measurements
and discusses the remaining discrepancies.
II. THEORY AND MODEL OF SCATTER LOSS
In a long baseline Fabry-P ́erot (FP) cavity, we wish
to know the optical field distribution in the plane of the
mirror, so that we can estimate the amount of power in
the cavity using the field reflected from the mirror whilst
also getting a good estimate of the power loss due to
non-zero field values outside the mirror. To this end, we
employ the Huygens-Fresnel integral
E
(
x,y,z
) =
i
λ
∫∫
dx
0
dy
0
E
(
x
0
,y
0
,z
0
)
exp[
−
ikr
]
r
cos(
θ
)
,
(1)
where
E
(
x
0
,y
0
,z
0
) and
E
(
x,y,z
) are the source and sink
field distributions, respectively, which will become the
locations of the two mirrors, and we use the definitions
k
= 2
π/λ
,
r
2
=
ρ
2
+
L
2
,
ρ
2
= (
x
−
x
0
)
2
+ (
y
−
y
0
)
2
,
L
=
z
−
z
0
and
cos
(
θ
) =
L/r
. A pictorial representation
of this geometry is shown in Figure 1.
FIG. 1. An illustration of scattering using the Fresnel integral
in Eq. (1). The source field,
E
(
x
0
,y
0
,z
0
), derives from the
reflections off the surface of a curved mirror—with surface
aberration,
h
, defined as the deviation of the real mirror
surface from a spherical surface. The sink field,
E
(
x,y,z
) is
the weighted sum of the source field, with a weighting factor
that comprises two terms, a spherical wave,
exp
[
−
ikr
]
/r
, that
encodes amplitude and phase interference information from
the different locations on the mirror surface, and the obliquity
factor,
cos
(
θ
), that encodes the departure of the scattering
vector from the optical axis, in accordance with Huygens
principle.
An eigenmode of Eq. (1) that is of particular interest
in long-baseline FP cavities is the TEM
00
Gaussian beam
mode, given by
E
q
(
x,y,z
) =
√
2
P
0
πw
2
0
q
0
q
(
z
)
exp
[
−
ik
(
z
+
x
2
+
y
2
2
q
(
z
)
)]
,
(2)
where
P
0
is the total power in the beam,
w
0
is the beam
waist and
q
(
z
) is the complex beam parameter—given by
the expression
q
(
z
) =
z
−
z
w
+
iz
R
,
(3)
where
z
w
is the location of the beam waist,
z
R
is the
Rayleigh range of the beam and
q
0
=
iz
R
; Or equivalently
its reciprocal
1
q
(
z
)
=
1
R
(
z
)
−
i
λ
πw
2
(
z
)
.
(4)
which is parameterized by the radius of curvature (RoC),
R
(
z
), and the radius of the beam,
w
(
z
).
A. Surface Aberration and Scattering
The surface profile
h
(
x
0
,y
0
) of the mirror can be calcu-
lated from its Fourier transform
H
(
f
x
,f
y
):
h
(
x
0
,y
0
) =
∫∫
H
(
f
x
,f
y
) exp
[
−
i
2
π
(
x
0
f
x
+
y
0
f
y
)
]
df
x
df
y
(5)
and the two dimensional power spectral density (PSD) is
given by the expression
PSD
2
(
f
x
,f
y
)
≡|
H
(
f
x
,f
y
)
|
2
.
(6)
When a Gaussian field is reflected by a mirror surface
with height aberration
h
and nominal radius of curvature
R
0
, the disturbance of the reflected field
1
, in the limit of
small
h
where 2
kh
1, is
δE
(
x
0
,y
0
,z
0
)
'
2
ikh
exp
[
ik
x
2
0
+
y
2
0
R
0
]
E
q
(
x
0
,y
0
,z
0
)
.
(7)
Note that in a resonating cavity, the radius of curvature
of the beam impinging the mirror surface is
R
(
z
0
) =
R
0
.
We use Eq. (1) to propagate Eq. (7) to the second
mirror. The wavefront mismatch at
z
will determine the
amount of power scattered out of the cavity mode by the
first mirror. The scattered field distribution in the (
x,y
)
plane displaced a distance
L
from the mirror is
δE
(
x,y,z
) = 2
ik
·
M
(
β
−
1
x, β
−
1
y
)
,
(8)
where
β
=
λ
(
z
−
z
0
). Here, the Fresnel integral has
been recast as a convolution integral in frequency space,
namely:
M
(
ν
x
,ν
y
) =
η
∫∫
H
(
f
x
,f
y
)
G
q
(
ν
x
−
f
x
,ν
y
−
f
y
)
df
x
df
y
,
(9)
1
The prompt reflected field at the mirror surface is
E
(
x
0
, y
0
, z
0
) =
exp
[
2
ik
(
h
+ (
x
2
0
+
y
2
0
)
/
2
R
0
)]
E
q
(
x
0
, y
0
, z
0
).
3
where
η
(
ν
x
,ν
y
) = exp
[
−
iπβ
(
ν
2
x
+
ν
2
y
)
/
2
]
(10)
and
G
q
(
ν
x
,ν
y
) =
E
q
(
βν
x
, βν
y
, z
0
+
L
)
η
(
ν
x
, ν
y
)
.
(11)
The far field condition, satisfied by long baseline IFO,
means that spatial coordinate and spatial frequency are
related as follows:
x
=
βν
x
, y
=
βν
y
.
(12)
Furthermore, the angular coordinate,
θ
x,y
, is related to
spatial frequency,
ν
x,y
, and spatial wavelength,
λ
x,y
, of
the exitwave leaving the mirror surface via
sin(
θ
x,y
) =
λν
x,y
,
sin(
θ
x,y
) =
λ/λ
x,y
(13)
respectively. For small angles, this approximates to
θ
x,y
'
λν
x,y
, θ
x,y
'
λ/λ
x,y
.
(14)
Eqs. (8) to (13) show that an aberration with spatial
wavelength
λ
x,y
reflects the field towards the nominal
angular direction
θ
x,y
with broadening,
δθ
= 2
w
(
z
)
/L
, as
illustrated in Figure 2.
FIG. 2. Scattering by continuous aberration. In the fre-
quency domain, we have
H
(
f
x
,f
y
)
'
2
ikh
max
(
f
s
)
δ
(
f
2
x
−
f
2
s
)
for a continuous aberration of spatial frequency,
f
s
along the
x
−
direction. Thus, the field distribution in the sink plane is
the sum of two displaced versions of
E
q
with an amplitude
scaling of 2
ikh
max
.
The finite size of the mirrors defines a characteristic
cutoff angle
θ
cut
and a corresponding spatial wavelength
λ
cut
s
of surface deformations. All power scattered to angles
larger than
θ
cut
by surface deformations with length scales
smaller than
λ
cut
s
is lost entirely, similar to the tail of the
reflected fundamental Gaussian mode which misses the
mirror. At angles smaller than
θ
cut
, the overlap between
impinging wavefront distortions and mirror surface needs
to be evaluated to determine the power lost.
For FP Michelson interferometers such as aLIGO and
and the Caltech 40 m prototype schematically shown in
Figure 4 we can define for each arm cavity an input mirror
(IM), where the laser beam enters the cavity, and an end
Parameters
aLIGO
40 m
Cavity length (
L
)
3994.5 m
38 m
Mirror Aperture (
a
)
34 cm
7.5 cm
RoC of IM (
R
1
)
1934 m
flat
RoC of EM (
R
2
)
2245 m
57.5 m
Beam radius on IM (
w
1
)
5.3 cm
0.30 cm
Beam radius on EM (
w
2
)
6.2 cm
0.52 cm
θ
cut
(=
a/
2
L
)
0
.
0024
◦
0
.
057
◦
λ
cut
s
(=
λ/θ
cut
)
2.5 cm
1.1 mm
δP
m
/P
(
=
∑
j
exp
[
−
2
a
2
/w
2
j
])
0.3 ppm
10
−
39
ppm
TABLE I. Fabry-Perot cavity parameters for the aLIGO arms
and the 40 m arms. Here
a
is the diameter of the mirror and
w
j
is the half-width of the beam at the
j
th
mirror.
mirror (EM). Important geometrical parameters for the
FP arm cavities of aLIGO and the Caltech 40 m prototype
that are used in this calculation are listed in Table I along
with the parameters derived above. As can be seen, the
large separation of the LIGO mirrors results in over a
factor of 20 longer cutoff wavelength over the 40 m’s.
B. BRDF as a loss estimation metric
The bidirectional reflectance distribution function
(BRDF) describes the angular dependence of the intensity
of light reflected and scattered from the mirror surface
[11], defined by the following formula
BRDF
(
θ,φ
) =
L
2
P
0
cos(
θ
)
|
E
(
x,y,z
)
|
2
,
(15)
where
θ
and
φ
are angles in spherical coordinates, with
the usual relation to Cartesian coordinate i.e.
x
=
L
sin
(
θ
)
cos
(
φ
) and
y
=
L
sin
(
θ
)
sin
(
φ
). Again,
P
0
is the
total power in the beam whilst
L
is the distance between
the mirror and the point of measurement.
The one-dimensional distribution used often for loss
estimation is
BRDF
(
θ
) =
1
2
π
∫
2
π
0
dφ BRDF
(
θ,φ
)
,
(16)
which is fairly accurate when no explicit dependence on
φ
is present in the phase map.
Direct BRDF measurements of a mirror are usually
limited to above several degrees from the normal, be-
cause the small spatial separation from the incident beam
hinders the measurement of the small scattered power.
Therefore, the BRDF for small angles is best estimated
using scattering calculations from the phase map
h
(
x
0
,y
0
)
of the mirror surface by numerical methods. The angular
range of the resulting BRDF data is limited by the spatial
resolution of the phase map data, which is typically of
4
10
4
10
3
10
2
10
1
10
0
10
1
Angle [degree]
10
7
10
4
10
1
10
2
10
5
10
8
BRDF [1/str]
aLIGO IM BRDF from phase map
aLIGO EM BRDF from phase map
40 m IM BRDF from phase map
40 m EM BRDF from phase map
aLIGO IM Baffles PDs
aLIGO EM Baffles PDs
aLIGO EM Viewport 1 PDs
aLIGO EM Viewport 2 PDs
aLIGO IM Viewport 2 PDs
40 m CCD
High reflectivity mirror sample
5 ppm
aLIGO mirror radius / 4000 m
40 m mirror radius / 38 m
Rough fit to 8
10
4
2
10
2
10
1
10
0
10
1
10
2
Spatial Wavelength [mm]
FIG. 3. BRDF by various methods combined. Data colored
in blue belongs to scatter by the EM (evaluated at the IM
for small angles), and red by the IM (again, evaluated at the
respective EM for small angles). The traces ending with ’from
phase map’ are BRDF estimates based on the phase maps
of the LIGO Livingston Observatory (LLO) Y-arm and the
40 m prototype interferometer mirrors (cf. Section III B). The
cutoff angles at which scattered light completely misses the
other mirror are also marked for the aLIGO (blue) and 40 m
(orange) designs. The fitted line for the data out of the end
mirror has an angular dependence of
θ
−
2
. Measurements
of scattered light by photodiodes located at various points
along the LIGO arms are shown in boxes and triangles. PDs
located on the baffles surrounding the arm cavities detect
light scattered at low angles while PDs located at the view
ports near the aLIGO mirrors detect light at high angles. The
single scattering measurement done at the 40 m prototype
interferometer is shown as x (cf. Section III A). The BRDF
behavior measured for superpolished mirrors at high angles is
marked by a purple line together with a 5 ppm level in dashed
grey indicating the constant BRDF behavior at high angles
[12].
order
O
(0.1 mm), so that the maximum scattering angle
covered by phase map data, using Eq. (14), is
∼
0
.
3
◦
.
This leaves a gap in the angular range between phase map
estimations and direct BRDF measurements.
Figure 3 summarizes various BRDF measurements and
illustrates this gap. The solid and dashed line plots for
aLIGO and the 40 m show a sharp Gaussian towards
θ
∼
0,
whose width is determined by the Gaussian beam size
resonating in the cavity, and a tail structure based on the
phasemap data that is indicative of approximate power
law scaling. Note that in all cases, the BRDF line plots
cover the entire angular range spanned by the mirror at
the other end of the cavity, marked by respective vertical
lines in Figure 3. For the aLIGO detectors, we have also
added BRDF measurements at
θ
'
0
.
003
◦
and
θ
'
0
.
006
◦
,
taken from four photodetectors (PDs) on baffles that are
located 20 – 40 cm from the center of the mirror [
13
]. As
discussed in this reference, the power distribution is very
sensitive to the PD location relative to the center of the
mirror due to the mirror aperture clipping effect, and the
measured powers vary by an order of magnitude, but is
consistent with the estimate based on phasemap data.
The region outside of the mirror corresponding to
θ >
θ
cut
(cf. Table I), a power law fit to the data points yields
an angular dependence of
θ
−
n
with
n
∼
2. Note that this
loss estimation is much larger than the typical estimation
using the Gaussian beam shape, with a contribution also
coming from the clipping effect of the mirror. We also
have sparse BRDF measurement data in the range of 1
◦
to 10
◦
collected from various view ports located along the
beam tubes. Moreover, BRDF measurement of a high
reflecting mirror in the angular range larger than 10
◦
degrees was reported in [
12
]. This BRDF data and the
aLIGO BRDF data using view ports connect smoothly.
Total integrated scattering (TIS) from the mirror sur-
face in some angle range can be calculated from the BRDF
as follows:
TIS = 2
π
∫
θ
2
θ
1
BRDF(
θ
) cos(
θ
) sin(
θ
)
dθ
(17)
The integral of the fitted line in Figure 3 between 1
◦
and 75
◦
is 8 ppm. This angular range corresponds to the
range captured by seperate TIS measurements done on
our mirrors. More details can be found in Section III C.
III. MEASUREMENT TECHNIQUES
The Caltech 40 m interferometer, 40 m for short, is a
small-scale GW detector prototype instrument. It was
designed to share its optical topology with the full-scale
LIGO observatories, including FP cavity-enhanced Michel-
son arms, a power recycling cavity (PRC), a signal recy-
cling cavity (SRC), and an input mode cleaner (IMC).
The optical configuration of the 40 m is shown in simpli-
fied manner in Figure 4, which shows only the principle
building blocks. Among similar instruments, the 40 m’s
long arms and the comparatively large beam spot sizes
on its input and end mirrors make the 40 m an important
subject for investigating optical scatter. Its optics were
produced to the same standard as full-size GW detector
mirrors, bringing its optical configuration closer to LIGO
than any other prototype instrument.
We investigate optical losses in the FP arm cavities
of the 40 m to bridge the gap between modelled and
measured optical scatter. For this we use a variety of
complementary techniques, including:
(i) Direct scattered light measurement by CCD
(ii)
Profile maps of the mirror surfaces combined with
wave propagation simulations (phase maps)
(iii)
Total Integrated Scatter (TIS) measurements as a
function of position on the mirror surface
5
X-Arm
Y-Arm
Beam-
Splitter
Input
Mirror
X-Arm
End
Mirrror
X-Arm
Input
Mirror
Y-Arm
End
Mirror
Y-Arm
Power
Recycling
Mirror
X-Arm
Power
Monitor
Y-Arm
Power
Monitor
Output
Power
Monitor
Pre-Stabilised
Input Laser
FIG. 4. Optical configuration of the Caltech 40 m prototype
interferometer. The pre-stabilized laser is sent into the Fabry
Perot Michelson interferometer and measured at the output
mode monitor. The power recycling mirror (PRM) forms the
impedance-matched power recycling cavity (PRC) with the
rest of the interferometer. The arm cavities, formed by IMs
and EMs increase the interaction time between stored photons
and passing GWs in the full-scale instruments. The power
in the arm cavities is monitored by PDs measuring the light
transmission through the EMs.
(iv) Static optical cavity impedance measurement
(v) Power recycling gain constraints
In the following section we describe how these measure-
ments are performed, what their systematic errors are,
and determine if they can provide a complete picture of
the scatter loss distribution.
A. Direct scattered light measurement with
cameras
An in-situ measurement of the total integrated scatter
at the Caltech 40 m prototype is not possible due to the
geometry of the vacuum envelope. However, the arm
cavity beam spots on the mirrors are visible from a small
number of viewports with a direct line of sight. These
allow a direct assessment of the scatter amplitude at
certain angles from the beam axis in particular directions,
determined by the relative positions and orientations of
the mirrors and viewports.
We performed a proof or principle measurement to ob-
tain an independent estimate of the intra-cavity scatter
using this approach. For this, we calibrated the pixel
scale of a standard CCD Gig-E camera at relevant combi-
nations of sensor gain and exposure time settings against
a precision optical power meter. The light source for
the calibration process was an power-monitored diode
laser with the same wavelength as the main laser. The
L
d
μ
FIG. 5. Evaluating large angle scatter of a suspended mirror
with a CCD camera. Setup used to image large angle scatter
(
θ
∼
50
°
) from the beam spot on the mirror into a CCD camera
using a
d
= 5 cm diameter lens. The distance from the image
to the lens is
L
= 65 cm.
calibration allows us to calculate the total optical power
captured by the CCD sensor from its cumulative pixel
values.
At the time of this study, only a single viewport on the
vacuum chamber housing the EM on the X arm (EMX)
was practical and available for this investigation. The
given setting allowed us to place a focusing lens at a
distance of 65 cm at an angle close to 50
◦
. The lens has
a standard AR-coating, such that power loss to reflec-
tions can be neglected, and a clear aperture of 2 inches
for photon collection. The arrangement is schematically
shown in Figure 5. The lens’ focal length and distance
from the camera was chosen such that it images the beam
spot on the mirror onto the CCD chip. We spatially filter
the recorded CCD image with a digital aperture to reject
stray light from parasitic light sources.
In addition to spatial image filtering and to isolate the
pixel readings of the beam spot and remove all traces of
ambient light scatter, we repeatedly turned the input laser
feed into the interferometer on and off with a mechanical
shutter and recorded exposures of the mirror with and
without beam illumination. Subsequent exposures were
thus taken in similar conditions, with the presence of the
beam spot being the only difference, allowing us to use
image subtraction to isolate the scattered power.
In total we measure the power scattered onto the CCD
to be
(193
.
9
±
2
.
2) nW
in the non-power recycled state
with misaligned PRM. We use this measurement to esti-
mate the bidirectional reflectance distribution function
(BRDF discussed in Section II B). Comparing this scat-
tered power to
10 W
of circulating power in the arm cavity
– witnessed by the EM transmission – we calculate the
6
BRDF of the 40 m’s EMX at
50
°
to be
(6
.
2
±
1
.
0) ppm
/
sr
.
This estimate was included in the compilation of BRDF
measurements in Figure 3.
The uncertainty in the BRDF result stems mainly from
the uncertainty in the exact distance between beam spot
to the focusing lens, which is estimated to be 5 cm.
B. Phase map measurements
A phase-shifting Fizeau inteferometer (PSFI) [
14
] was
used to measure the surface figure of our mirrors. In
such optical profilometers, a laser beam reflected from
the sample under test is interfered with a beam reflected
from a reference flat. A phasemap is then constructed
by extracting the optical phase difference between the
beams for every point on the surface from the interference
pattern.
Phase maps for the 40 m mirrors (IM4, EM7) were
recorded at the time of installation with no significant
qualitative differences between the different optics. How-
ever, while indicating that phase maps of different optics
can be used interchangeably for simulation purposes, these
initial phase maps were found to not be accurate enough
for the estimation of the arm loss, especially in the short
spatial wavelength region.
To extend the spatial frequency range of our phase
maps, new measurements were done using spare mirrors
(EM6, IM1, IM2), which were polished and coated as
part of the same batch as the installed optics by the
same companies for each process. Unfortunately, the
coated surface profiles in the long wavelength region do
show differences between mirrors, even though all were
produced with the same polishing and coating process.
Since the currently installed mirrors cannot be remea-
sured in a PSFI without removing them from their sus-
pension cages in the vacuum envelope, we synthesize their
phase maps by combining low spatial frequency data from
the original measurement with high spatial frequency data
from the spare mirrors. This was done by blending the
lowpass filtered original maps with high pass filtered new
maps at a cross-over wavelength of 1.3 mm.
In similar fashion to Eq. (16), 1D PSDs were computed
from the 2D PSDs of the phase maps by averaging over
the azimuthal angle. The PSDs of the newly measured
phase maps of spare IMs - IM1 and IM2 are shown in
Figure 6a. These maps were measured focusing in the
short wavelength region,
λ <
5 mm
. The comparison
of the red (IM1) and orange (IM2) lines show that the
PSDs in
λ
s
<
1
.
5 mm
are the same. From this result, the
same PSD can be assumed for the IM4, which is the one
actually installed. The black line is a combination of the
ITM04 map (blue) original measurement with the IM1
new measurement. The cutoff for the hybridization is
λ
= 1
.
3 mm (green dashed line).
A corresponding analysis was done for the EMs. Fig-
ure 6b is the comparison of PSDs of old EM7 map (blue),
new EM6 map (red) and the hybrid of these two (black)
10
1
10
0
Spatial frequency (1/mm)
10
4
10
3
10
2
10
1
10
0
10
1
10
2
PSD (nm^2 mm)
IM1 (New)
IM2 (New)
IM4 (Old)
IM1 & 4 Hybrid
(a)
10
1
10
0
Spatial frequency (1/mm)
10
4
10
3
10
2
10
1
10
0
10
1
10
2
PSD (nm^2 mm)
EM6 (New)
EM7 (Old)
EM7 & 6 Hybrid
(b)
(c)
(d)
FIG. 6. PSDs of (a) IMs. (b) EMs. Green dashed lines mark
the spatial frequency corresponding to
λ
=
1
.
3 mm
, where the
old and new maps were hybridized. Hybrid phase maps of (c)
IM1 and IM4 (d) EM6 and EM7.
merged at
λ
= 1
.
3 mm.
These hybrid phase maps are used in a numerical sim-
ulation to simulate beam propagation [
15
] in a cavity.
In the simulation, an input Gaussian laser beam, mode
matched to the cavity, is bounced back and forth by the
cavity mirrors loaded with phase maps until a steady-state
is reached. The beam propagation between the cavity mir-
rors is computed using Fourier optics methods. At steady
state, the roundtrip loss is computed by integrating over
the intensity of the beam outside the mirrors’ apertures.
For the case of a 40 m cavity with mirrors loaded with
the hybrid phase maps shown in Figure 6d and Figure 6c,
the roundtrip loss was computed to be
(6
±
2) ppm
over
the angular range accessible by this method.
C. Total Integrating Scatter Measurement
The total integrated scattering (TIS) from a HR coating
is measured using an integrating sphere. Figure 7 shows
a schematic of the experimental setup.
The 1064 nm output of a JDSU NPRO laser is first
sampled for a reference photodiode (PD-ref) and spa-
tially filtered to generate a high-purity TEM00 mode. A
sequence of polarizer, quarter wave-plate and chopper
constrains its polarization and allows for lock-in detec-
tion. The collimated beam enters the top port of the
integrating sphere and a mode-matching lens is used to
position the beam waist at the mirror’s HR surface. The
reflected specular beam is aligned to exit the integrating
7
FIG. 7. A schematic of the measurement setup of total inte-
grated scattering. Details of the setup are described in the
main text. (PD) Photodiode (BS) Beamsplitter (L) Lens (P)
Polarizer (
λ/
4) Quarter wave-plate (M) Mirror (HR) High
reflection side.
sphere centered on the top port. The mirror is raster
scanned and the scatter from HR mirror is measured by
a signal photodiode (PD-sig) attached to the integrating
sphere to obtain the TIS map. The beam diameter and
the scanning step size are both
300
μ
m
. The polar angle
of scatter collection of the integrating sphere is defined by
the sizes of the top and bottom ports, the current set-up
has a collection range from
1
.
0
°
to
75
°
, corresponding to
a bandwidth of spatial frequency 16 to
908 mm
−
1
. The
calibration is carried out by placing a diffuse reflectance
standard (Labsphere Spectralon) at the HR mirror posi-
tion.
Figure 8 shows TIS measurements that were pereformed
on four 40 m prototype interferometer spare mirrors (IM1,
IM2, EM6, EM8). Figure 9 shows the histograms of the
power fractions in Figure 8. The histograms are peaked
at
∼
7 ppm for the IMs and
∼
3 ppm for the EMs, and
there is a long tail extending over 1000 ppm.
We calculate the total scatter loss in the arm cavities
to angles between
1
.
0
°
and
75
°
in the following way. A
Gaussian beam, the same size as in the 40 m arm cavities,
is impinging on the center of the mirror. Each point in the
TIS map induces a scatter loss to the part of the Gaussian
beam that overlaps with it. The losses are then summed
incoherently to obtain the total scatter loss. Since we are
summing over large angle range interference effects are
heavily suppressed.
Explicitly, the total scatter loss is given by
Loss =
1
P
0
∑
n
TIS[
x
n
,y
n
]
|
E
q
[
x
n
,y
n
]
|
2
πω
2
TIS
,
(18)
where
TIS
[
x
n
,y
n
] are the TIS measurements data points,
E
q
[
x
n
,y
n
] is the normalized electric field distribution of
a Gaussian beam the size of the beam impinging on
that mirror and
ω
TIS
is the TIS beam waist. Doing
this summation for each sample we find that the scatter
(a)
(b)
(c)
(d)
FIG. 8. Reflected power fraction measured by a laser for (a)
IM1 and (b) IM2. (c) EM6 (d) EM8
10
1
10
2
10
3
HR Scattering [ppm]
10
1
10
3
Counts
IM1
(a)
10
1
10
2
10
3
HR Scattering [ppm]
10
1
10
3
Counts
IM2
(b)
10
1
10
2
10
3
HR Scattering [ppm]
10
1
10
3
Counts
EM6
(c)
10
1
10
2
HR Scattering [ppm]
10
1
10
3
Counts
EM8
(d)
FIG. 9. Populations of scattered power fraction in (a) Fig-
ure 8a, (b) Figure 8b, (c) Figure 8c and (d) Figure 8d.
losses, to angles between
1
.
0
°
to
75
°
, are
(10
.
0
±
1
.
7) ppm
,
(8
.
6
±
0
.
4)
,
(4
.
60
±
0
.
03) a
nd
(4
.
90
±
0
.
14) i
n IM1, IM2,
EM6 and EM8, respectively. The errors are calculated by
shifting the beams randomly around the centers of the mir-
rors with an standard deviation of
1 mm
and computing
the standard deviation of the resulting losses.
D. Static Optical Impedance Measurement
Inferring cavity round-trip losses from measuring either
optical ringdown time constants or the linewidths of cavity
8
resonances and accounting for the known mirror transmis-
sivities has a significant caveat: The full uncertainty in the
measured mirror transmissivities is directly transferred to
the loss estimate. For GW detector arm cavities, which
have moderately high finesse and are highly over-coupled,
the uncertainty of the IM transmission is usually on the
same order or larger than the expected losses.
In a measurement of the optical cavity impedance, also
referred to as the
DC-Method
, we compare the static
reflected power from the IM in two cases: the prompt
reflection off the IM when there is no arm cavity (either
because the EM is misaligned or because the intra-cavity
beam path is blocked), and when the arm cavity is on
resonance and locked to the laser carrier. The combination
of the two individual measurements allows to solve for the
cavity losses with reduced sensitivity to the uncertainty
in IM transmission. This method only measures total
optical loss in the cavity and cannot distinguish absorption
from scatter, however, the optical absorption of the high-
quality tantala-silica coatings used in GW detectors and
prototypes is of order
1 ppm
or smaller and can be safely
neglected.
It carries similarities to the optical ringdown measure-
ments that are carried out in reflection described by Isogai
et al. [
16
], but is more suited to the optical topology of a
cavity-enhanced Michelson interferometer, in which the
IMC acts as a lowpass filter for any modulations of the
input laser. All modulators and shutters are located up-
stream from the IMC, such that the fast switching of
the input power that is required for proper ringdown
measurements would get filtered by the IMC’s optical
response. The IMC’s storage time is longer than those of
the arm cavities, such that the measured ringdown time
constants would be entirely dominated by the IMC, sig-
nificantly limiting the achievable accuracy on arm cavity
effect estimation.
We consider the cavity reflection coefficient on reso-
nance
R
r
and off resonance
R
m
and the coupling ineffi-
ciency of input beam into the cavity. We find the expres-
sion for the ratio of the two reflected power measurements
P
r
P
m
to be
P
r
P
m
=
1
−
ηγ
(1
−
R
r
)
R
m
,
(19)
η
is the mode matching efficiency of input beam to the
arm,
γ
is the fraction of power in the carrier frequency
after sideband modulation. The values for these param-
eters are given in Table II. The expressions for
R
r
and
R
m
can be found in [
16
]. Assuming the loss contribution
from the two cavity mirrors is the same we extract the
roundtrip loss from Eq. (19) numerically.
For very low roundtrip losses, such as in the case of
the arm cavities in the 40 m, the reflected power ratio
gets very close to 1 which requires long integration times
to get good enough resolution between the two power
measurements. We repeat the arm lock/unlock cycle with
re-alignment procedures in between to counteract drifts.
During the measurement time we track the changes to the
0
20
40
60
80
100
120
Time [s]
105.5
106.0
106.5
107.0
107.5
108.0
Sequential Data YARM
Segment Average YARM
0
20
40
60
80
100
120
Time [s]
105.5
106.0
106.5
107.0
107.5
108.0
Sequential Data XARM
Segment Average XARM
Reflected Power [a.u.]
FIG. 10. DC reflection measurement sequence. The mea-
sured reflected power is shown as a function of time during
which the measured cavity is locked and unlocked periodically.
Top(bottom) figure shows the measurement sequence for the
Y(X) arm of the 40 m. The square wave lines show the average
at each measurement segment.
Parameter
Nominal Value
Uncertainty
IM Transmission (T1)
1
.
38 %
±
0
.
01 %
EM Transmission (T2)
13
.
7 ppm
±
3 ppm
Mode matching efficiency (
η
)
92 %
±
5 %
Modulation depth @ 11 MHz
0.179
Modulation depth @ 55 MHz
0.226
Carrier frequency power fraction (
γ
)
95
.
92 %
TABLE II. 40 m intereferometer prototype arm cavity param-
eters used in Eq. (19).
optical system (pointing/alignment drifts etc) and correct
for fluctuations in the input power by scaling the reflected
power to the circulating power in the IMC. An example
of such a measurement sequence is shown in Figure 10.
The top (bottom) graph shows the measurement sequence
for the Y(X) arm of the 40 m. The two traces appear
opposite in phase due to the fact that the X and Y arm
losses are on opposite sides of the
P
r
P
1
= 1 point. After
initial measurements of the 40m arms, which showed
contamination with excess losses, the mirrors were cleaned
with First Contact solution. Our best estimates for the
arm cavity losses post-cleaning are
(20
.
0
±
2
.
5) ppm
for
the X cavity and (50
.
0
±
2
.
5) ppm for the Y cavity.
E. Power Recycling Gain Measurement
In power recycled interferometers, such as the 40 m
interferometer prototype, the recycling gain, the ratio