The LIGO Scientific Collaboration

# Point absorbers in Advanced LIGO

A. F. Brooks  LIGO, California Institute of Technology, Pasadena, CA 91125,
USA  G. Vajente  LIGO, California Institute of Technology, Pasadena, CA 91125,
USA  H. Yamamoto  LIGO, California Institute of Technology, Pasadena, CA
91125, USA  R. Abbott  LIGO, California Institute of Technology, Pasadena, CA
91125, USA  C. Adams  LIGO Livingston Observatory, Livingston, LA 70754, USA
R. X. Adhikari  LIGO, California Institute of Technology, Pasadena, CA 91125,
USA  A. Ananyeva  LIGO, California Institute of Technology, Pasadena, CA
91125, USA  S. Appert  LIGO, California Institute of Technology, Pasadena, CA
91125, USA  K. Arai  LIGO, California Institute of Technology, Pasadena, CA
91125, USA  J. S. Areeda  California State University Fullerton, Fullerton, CA
92831, USA  Y. Asali  Columbia University, New York, NY 10027, USA  S. M.
Aston  LIGO Livingston Observatory, Livingston, LA 70754, USA  C. Austin
Louisiana State University, Baton Rouge, LA 70803, USA  A. M. Baer
Christopher Newport University, Newport News, VA 23606, USA  M. Ball
University of Oregon, Eugene, OR 97403, USA  S. W. Ballmer  Syracuse
University, Syracuse, NY 13244, USA  S. Banagiri  University of Minnesota,
Minneapolis, MN 55455, USA  D. Barker  LIGO Hanford Observatory, Richland, WA
99352, USA  L. Barsotti  LIGO, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  J. Bartlett  LIGO Hanford Observatory, Richland, WA
99352, USA  B. K. Berger  Stanford University, Stanford, CA 94305, USA  J.
Betzwieser  LIGO Livingston Observatory, Livingston, LA 70754, USA  D.
Bhattacharjee  Missouri University of Science and Technology, Rolla, MO 65409,
USA  G. Billingsley  LIGO, California Institute of Technology, Pasadena, CA
91125, USA  S. Biscans  LIGO, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  LIGO, California Institute of Technology, Pasadena,
CA 91125, USA  C. D. Blair  LIGO Livingston Observatory, Livingston, LA 70754,
USA  R. M. Blair  LIGO Hanford Observatory, Richland, WA 99352, USA  N. Bode
Max Planck Institute for Gravitational Physics (Albert Einstein Institute),
D-30167 Hannover, Germany  Leibniz Universität Hannover, D-30167 Hannover,
Germany  P. Booker  Max Planck Institute for Gravitational Physics (Albert
Einstein Institute), D-30167 Hannover, Germany  Leibniz Universität Hannover,
D-30167 Hannover, Germany  R. Bork  LIGO, California Institute of Technology,
Pasadena, CA 91125, USA  A. Bramley  LIGO Livingston Observatory, Livingston,
LA 70754, USA  D. D. Brown  OzGrav, University of Adelaide, Adelaide, South
Australia 5005, Australia  A. Buikema  LIGO, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  C. Cahillane  LIGO, California Institute
of Technology, Pasadena, CA 91125, USA  K. C. Cannon  RESCEU, University of
Tokyo, Tokyo, 113-0033, Japan.  H. Cao  OzGrav, University of Adelaide,
Adelaide, South Australia 5005, Australia  X. Chen  OzGrav, University of
Western Australia, Crawley, Western Australia 6009, Australia  A. A. Ciobanu
OzGrav, University of Adelaide, Adelaide, South Australia 5005, Australia  F.
Clara  LIGO Hanford Observatory, Richland, WA 99352, USA  C. M. Compton  LIGO
Hanford Observatory, Richland, WA 99352, USA  S. J. Cooper  University of
Birmingham, Birmingham B15 2TT, UK  K. R. Corley  Columbia University, New
York, NY 10027, USA  S. T. Countryman  Columbia University, New York, NY
10027, USA  P. B. Covas  Universitat de les Illes Balears, IAC3—IEEC, E-07122
Palma de Mallorca, Spain  D. C. Coyne  LIGO, California Institute of
Technology, Pasadena, CA 91125, USA  L. E. H. Datrier  SUPA, University of
Glasgow, Glasgow G12 8QQ, UK  D. Davis  Syracuse University, Syracuse, NY
13244, USA  C. Di Fronzo  University of Birmingham, Birmingham B15 2TT, UK  K.
L. Dooley  Cardiff University, Cardiff CF24 3AA, UK  The University of
Mississippi, University, MS 38677, USA  J. C. Driggers  LIGO Hanford
Observatory, Richland, WA 99352, USA  P. Dupej  SUPA, University of Glasgow,
Glasgow G12 8QQ, UK  S. E. Dwyer  LIGO Hanford Observatory, Richland, WA
99352, USA  A. Effler  LIGO Livingston Observatory, Livingston, LA 70754, USA
T. Etzel  LIGO, California Institute of Technology, Pasadena, CA 91125, USA
M. Evans  LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  T. M. Evans  LIGO Livingston Observatory, Livingston, LA 70754, USA  J.
Feicht  LIGO, California Institute of Technology, Pasadena, CA 91125, USA  A.
Fernandez-Galiana  LIGO, Massachusetts Institute of Technology, Cambridge, MA
02139, USA  P. Fritschel  LIGO, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  V. V. Frolov  LIGO Livingston Observatory,
Livingston, LA 70754, USA  P. Fulda  University of Florida, Gainesville, FL
32611, USA  M. Fyffe  LIGO Livingston Observatory, Livingston, LA 70754, USA
J. A. Giaime  Louisiana State University, Baton Rouge, LA 70803, USA  LIGO
Livingston Observatory, Livingston, LA 70754, USA  K. D. Giardina  LIGO
Livingston Observatory, Livingston, LA 70754, USA  P. Godwin  The Pennsylvania
State University, University Park, PA 16802, USA  E. Goetz  Louisiana State
University, Baton Rouge, LA 70803, USA  Missouri University of Science and
Technology, Rolla, MO 65409, USA  S. Gras  LIGO, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  C. Gray  LIGO Hanford Observatory,
Richland, WA 99352, USA  R. Gray  SUPA, University of Glasgow, Glasgow G12
8QQ, UK  A. C. Green  University of Florida, Gainesville, FL 32611, USA  A.
Gupta  LIGO, California Institute of Technology, Pasadena, CA 91125, USA  E.
K. Gustafson  LIGO, California Institute of Technology, Pasadena, CA 91125,
USA  R. Gustafson  University of Michigan, Ann Arbor, MI 48109, USA  E. Hall
LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  J.
Hanks  LIGO Hanford Observatory, Richland, WA 99352, USA  J. Hanson  LIGO
Livingston Observatory, Livingston, LA 70754, USA  T. Hardwick  Louisiana
State University, Baton Rouge, LA 70803, USA  R. K. Hasskew  LIGO Livingston
Observatory, Livingston, LA 70754, USA  M. C. Heintze  LIGO Livingston
Observatory, Livingston, LA 70754, USA  A. F. Helmling-Cornell  University of
Oregon, Eugene, OR 97403, USA  N. A. Holland  OzGrav, Australian National
University, Canberra, Australian Capital Territory 0200, Australia  W. Jia
LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  J. D.
Jones  LIGO Hanford Observatory, Richland, WA 99352, USA  S. Kandhasamy
Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India  S.
Karki  University of Oregon, Eugene, OR 97403, USA  M. Kasprzack  LIGO,
California Institute of Technology, Pasadena, CA 91125, USA  K. Kawabe  LIGO
Hanford Observatory, Richland, WA 99352, USA  N. Kijbunchoo  OzGrav,
Australian National University, Canberra, Australian Capital Territory 0200,
Australia  P. J. King  LIGO Hanford Observatory, Richland, WA 99352, USA  J.
S. Kissel  LIGO Hanford Observatory, Richland, WA 99352, USA  Rahul Kumar
LIGO Hanford Observatory, Richland, WA 99352, USA  M. Landry  LIGO Hanford
Observatory, Richland, WA 99352, USA  B. B. Lane  LIGO, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  B. Lantz  Stanford
University, Stanford, CA 94305, USA  M. Laxen  LIGO Livingston Observatory,
Livingston, LA 70754, USA  Y. K. Lecoeuche  LIGO Hanford Observatory,
Richland, WA 99352, USA  J. Leviton  University of Michigan, Ann Arbor, MI
48109, USA  J. Liu  Max Planck Institute for Gravitational Physics (Albert
Einstein Institute), D-30167 Hannover, Germany  Leibniz Universität Hannover,
D-30167 Hannover, Germany  M. Lormand  LIGO Livingston Observatory,
Livingston, LA 70754, USA  A. P. Lundgren  University of Portsmouth,
Portsmouth, PO1 3FX, UK  R. Macas  Cardiff University, Cardiff CF24 3AA, UK
M. MacInnis  LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  D. M. Macleod  Cardiff University, Cardiff CF24 3AA, UK  G. L. Mansell
LIGO Hanford Observatory, Richland, WA 99352, USA  LIGO, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  S. Márka  Columbia
University, New York, NY 10027, USA  Z. Márka  Columbia University, New York,
NY 10027, USA  D. V. Martynov  University of Birmingham, Birmingham B15 2TT,
UK  K. Mason  LIGO, Massachusetts Institute of Technology, Cambridge, MA
02139, USA  T. J. Massinger  LIGO, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  F. Matichard  LIGO, California Institute of
Technology, Pasadena, CA 91125, USA  LIGO, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  N. Mavalvala  LIGO, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  R. McCarthy  LIGO Hanford
Observatory, Richland, WA 99352, USA  D. E. McClelland  OzGrav, Australian
National University, Canberra, Australian Capital Territory 0200, Australia
S. McCormick  LIGO Livingston Observatory, Livingston, LA 70754, USA  L.
McCuller  LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  J. McIver  LIGO, California Institute of Technology, Pasadena, CA 91125,
USA  T. McRae  OzGrav, Australian National University, Canberra, Australian
Capital Territory 0200, Australia  G. Mendell  LIGO Hanford Observatory,
Richland, WA 99352, USA  K. Merfeld  University of Oregon, Eugene, OR 97403,
USA  E. L. Merilh  LIGO Hanford Observatory, Richland, WA 99352, USA  F.
Meylahn  Max Planck Institute for Gravitational Physics (Albert Einstein
Institute), D-30167 Hannover, Germany  Leibniz Universität Hannover, D-30167
Hannover, Germany  T. Mistry  The University of Sheffield, Sheffield S10 2TN,
UK  R. Mittleman  LIGO, Massachusetts Institute of Technology, Cambridge, MA
02139, USA  G. Moreno  LIGO Hanford Observatory, Richland, WA 99352, USA  C.
M. Mow-Lowry  University of Birmingham, Birmingham B15 2TT, UK  S. Mozzon
University of Portsmouth, Portsmouth, PO1 3FX, UK  A. Mullavey  LIGO
Livingston Observatory, Livingston, LA 70754, USA  T. J. N. Nelson  LIGO
Livingston Observatory, Livingston, LA 70754, USA  P. Nguyen  University of
Oregon, Eugene, OR 97403, USA  L. K. Nuttall  University of Portsmouth,
Portsmouth, PO1 3FX, UK  J. Oberling  LIGO Hanford Observatory, Richland, WA
99352, USA  Richard J. Oram  LIGO Livingston Observatory, Livingston, LA
70754, USA  C. Osthelder  LIGO, California Institute of Technology, Pasadena,
CA 91125, USA  D. J. Ottaway  OzGrav, University of Adelaide, Adelaide, South
Australia 5005, Australia  H. Overmier  LIGO Livingston Observatory,
Livingston, LA 70754, USA  J. R. Palamos  University of Oregon, Eugene, OR
97403, USA  W. Parker  LIGO Livingston Observatory, Livingston, LA 70754, USA
Southern University and A&M College, Baton Rouge, LA 70813, USA  E. Payne
OzGrav, School of Physics & Astronomy, Monash University, Clayton 3800,
Victoria, Australia  A. Pele  LIGO Livingston Observatory, Livingston, LA
70754, USA  C. J. Perez  LIGO Hanford Observatory, Richland, WA 99352, USA  M.
Pirello  LIGO Hanford Observatory, Richland, WA 99352, USA  H. Radkins  LIGO
Hanford Observatory, Richland, WA 99352, USA  K. E. Ramirez  The University of
Texas Rio Grande Valley, Brownsville, TX 78520, USA  J. W. Richardson  LIGO,
California Institute of Technology, Pasadena, CA 91125, USA  K. Riles
University of Michigan, Ann Arbor, MI 48109, USA  N. A. Robertson  LIGO,
California Institute of Technology, Pasadena, CA 91125, USA  SUPA, University
of Glasgow, Glasgow G12 8QQ, UK  J. G. Rollins  LIGO, California Institute of
Technology, Pasadena, CA 91125, USA  C. L. Romel  LIGO Hanford Observatory,
Richland, WA 99352, USA  J. H. Romie  LIGO Livingston Observatory, Livingston,
LA 70754, USA  M. P. Ross  University of Washington, Seattle, WA 98195, USA
K. Ryan  LIGO Hanford Observatory, Richland, WA 99352, USA  T. Sadecki  LIGO
Hanford Observatory, Richland, WA 99352, USA  E. J. Sanchez  LIGO, California
Institute of Technology, Pasadena, CA 91125, USA  L. E. Sanchez  LIGO,
California Institute of Technology, Pasadena, CA 91125, USA  T. R. Saravanan
Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India  R.
L. Savage  LIGO Hanford Observatory, Richland, WA 99352, USA  D. Schaetzl
LIGO, California Institute of Technology, Pasadena, CA 91125, USA  R. Schnabel
Universität Hamburg, D-22761 Hamburg, Germany  R. M. S. Schofield  University
of Oregon, Eugene, OR 97403, USA  E. Schwartz  LIGO Livingston Observatory,
Livingston, LA 70754, USA  D. Sellers  LIGO Livingston Observatory,
Livingston, LA 70754, USA  T. Shaffer  LIGO Hanford Observatory, Richland, WA
99352, USA  D. Sigg  LIGO Hanford Observatory, Richland, WA 99352, USA  B. J.
J. Slagmolen  OzGrav, Australian National University, Canberra, Australian
Capital Territory 0200, Australia  J. R. Smith  California State University
Fullerton, Fullerton, CA 92831, USA  S. Soni  Louisiana State University,
Baton Rouge, LA 70803, USA  B. Sorazu  SUPA, University of Glasgow, Glasgow
G12 8QQ, UK  A. P. Spencer  SUPA, University of Glasgow, Glasgow G12 8QQ, UK
K. A. Strain  SUPA, University of Glasgow, Glasgow G12 8QQ, UK  L. Sun  LIGO,
California Institute of Technology, Pasadena, CA 91125, USA  M. J.
Szczepańczyk  University of Florida, Gainesville, FL 32611, USA  M. Thomas
LIGO Livingston Observatory, Livingston, LA 70754, USA  P. Thomas  LIGO
Hanford Observatory, Richland, WA 99352, USA  K. A. Thorne  LIGO Livingston
Observatory, Livingston, LA 70754, USA  K. Toland  SUPA, University of
Glasgow, Glasgow G12 8QQ, UK  C. I. Torrie  LIGO, California Institute of
Technology, Pasadena, CA 91125, USA  G. Traylor  LIGO Livingston Observatory,
Livingston, LA 70754, USA  M. Tse  LIGO, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  A. L. Urban  Louisiana State University,
Baton Rouge, LA 70803, USA  G. Valdes  Louisiana State University, Baton
Rouge, LA 70803, USA  D. C. Vander-Hyde  Syracuse University, Syracuse, NY
13244, USA  P. J. Veitch  OzGrav, University of Adelaide, Adelaide, South
Australia 5005, Australia  K. Venkateswara  University of Washington, Seattle,
WA 98195, USA  G. Venugopalan  LIGO, California Institute of Technology,
Pasadena, CA 91125, USA  A. D. Viets  Concordia University Wisconsin, 2800 N
Lake Shore Dr, Mequon, WI 53097, USA  T. Vo  Syracuse University, Syracuse, NY
13244, USA  C. Vorvick  LIGO Hanford Observatory, Richland, WA 99352, USA  M.
Wade  Kenyon College, Gambier, OH 43022, USA  R. L. Ward  OzGrav, Australian
National University, Canberra, Australian Capital Territory 0200, Australia
J. Warner  LIGO Hanford Observatory, Richland, WA 99352, USA  B. Weaver  LIGO
Hanford Observatory, Richland, WA 99352, USA  R. Weiss  LIGO, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  C. Whittle  LIGO,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  B. Willke
Leibniz Universität Hannover, D-30167 Hannover, Germany  Max Planck Institute
for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover,
Germany  C. C. Wipf  LIGO, California Institute of Technology, Pasadena, CA
91125, USA  L. Xiao  LIGO, California Institute of Technology, Pasadena, CA
91125, USA  Hang Yu  LIGO, Massachusetts Institute of Technology, Cambridge,
MA 02139, USA  Haocun Yu  LIGO, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  L. Zhang  LIGO, California Institute of Technology,
Pasadena, CA 91125, USA  M. E. Zucker  LIGO, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  LIGO, California Institute of
Technology, Pasadena, CA 91125, USA  J. Zweizig  LIGO, California Institute of
Technology, Pasadena, CA 91125, USA

###### Abstract

Small, highly absorbing points are randomly present on the surfaces of the
main interferometer optics in Advanced LIGO. The resulting nano-meter scale
thermo-elastic deformations and substrate lenses from these micron-scale
absorbers significantly reduces the sensitivity of the interferometer directly
though a reduction in the power-recycling gain and indirect interactions with
the feedback control system. We review the expected surface deformation from
point absorbers and provide a pedagogical description of the impact on power
build-up in second generation gravitational wave detectors (dual-recycled
Fabry-Perot Michelson interferometers). This analysis predicts that the power-
dependent reduction in interferometer performance will significantly degrade
maximum stored power by up to 50% and hence, limit GW sensitivity, but
suggests system wide corrections that can be implemented in current and future
GW detectors. This is particularly pressing given that future GW detectors
call for an order of magnitude more stored power than currently used in
Advanced LIGO in Observing Run 3. We briefly review strategies to mitigate the
effects of point absorbers in current and future GW wave detectors to maximize
the success of these enterprises.

##  I Introduction

The Advanced LIGO (aLIGO) gravitational wave (GW) detectors, in conjunction
with the Virgo GW detector, having completed their third observing run
(O3)O3CommissioningPRD , have reported detection of multiple gravitational
wave events GW150914etal ; O1O2Catalog  (10 binary black-hole mergers and 1
binary neutron star mergers) and issued 56 public alerts for detection
candidates 111https://gracedb.ligo.org/superevents/public/O3/. The aLIGO
detectors, illustrated in Figure 1, are high laser power, dual-recycled,
$4\text{\,}\mathrm{km}$ Fabry-Perot, Michelson interferometers operating at
$1064\text{\,}\mathrm{nm}$ aLIGO_Overview . Passing gravitational waves cause
a strain in space-time, resulting in a differential length variation of the
Fabry-Perot arms that yields a detectable intensity variation at the output
port of the interferometer.

The sensitivity of the interferometers is limited by a variety of technical
and fundamental sources. At frequencies above approximately
$100\text{\,}\mathrm{Hz}$ quantum noise, in the form of shot noise on the
photodetector, is the limiting noise source Corbitt_2004 . Quantum noise can
be decreased by increasing the input laser power injected into the
interferometer, thereby increasing the amount of stored power in the
interferometer arms. The design level for aLIGO is $125\text{\,}\mathrm{W}$
input power with a stored arm power of $750\text{\,}\mathrm{kW}$ per arm as
illustrated in Figure 1. During O3, Advanced LIGO routinely operated at input
power levels of $35\text{\,}\mathrm{W}$ $\\--$ $40\text{\,}\mathrm{W}$
O3CommissioningPRD .

Figure 1: Schematic diagram of Advanced LIGO showing the main optics, nominal
power levels and different optical frequencies present in the interferometer.
The detector is a Fabry-Perot Michelson, with power-recycling and signal-
recycling cavities. The power recycling cavity (PRC) and Fabry-Perot arm
cavities hold large amounts of stored power. At $125\text{\,}\mathrm{W}$ of
input laser power, the nominal design yields $5.2\text{\,}\mathrm{kW}$ stored
in the PRC and $750\text{\,}\mathrm{kW}$ in the arms. The output mode cleaner
(OMC) is designed to transmit only the fundamental spatial mode of the arms,
which carries the GW signal, and reject other spatial modes. Radio-frequency
sidebands are injected into the detector to measure and control the multiple
degrees of freedom created by the coupled resonant cavities.

Optical power is absorbed at the sub-ppm level in the surfaces of the main
LIGO optics, referred to as the Test Masses (ITMX, ITMY, ETMX and ETMY in
Figure 1). As stored power is increased, these optics are exposed to several
hundred kW of resonating power, they absorb several tens of mW causing thermo-
elastic deformation of the optical surfaces and thermo-refractive lenses in
the substrates PhysRevA.44.7022 ; HelloVinet_TE:90 ; HelloVinet:90_TL .

Absorption is classified as uniform and non-uniform. Uniform absorption is
characterized by a spatially invariant (or nearly invariant) absorption
coefficient across the high-reflectivity (HR) surface of the optic. In the
Advanced LIGO test masses, the unambiguously measured uniform absorption
values ranged over 500$\pm$250 ppb. In the case of uniform absorption, non-
uniform, low-spatial frequency thermal lenses HelloVinet:90_TL  and surface
deformations HelloVinet_TE:90  are well approximated by spherical wavefront
errors. Advanced LIGO contains a thermal compensation system (TCS) Brooks:16
\- not shown in Figure 1 \- with actuators to provide low-spatial correction
for (a) thermal lenses in the substrates and (b) curvature errors on the
surfaces of the test masses.

Figure 2: Hartmann sensor measurement of optical path distortion (thermo-
refractive plus thermo-elastic) from a single point absorber on H1-ITMX. Cold
reference taken at GPS time: $1\,180\,229\,513\text{\,}\mathrm{s}$, hot
measurement taken $3322\text{\,}\mathrm{s}$ later at GPS time:
$1\,180\,232\,835\text{\,}\mathrm{s}$. This measurement corresponds to
approximately $27\pm 2.5$ mW power absorbed in the point.

Non-uniform absorption is any form of absorption with high-spatial frequency
dependence (where ”high spatial frequency” refers to features that are
significantly smaller than the Gaussian beam diameter of the illuminating
laser beam). A salient example is a point-like absorber (a ”point absorber”):
a sub-millimeter scale, highly absorbing region on the surface of the test
mass.

Within aLIGO, Hartmann wavefront sensors (HWS) Brooks:16 ; Brooks:09  measure
the spatial distribution of the integrated thermo-refractive and thermo-
elastic deformations in the main LIGO optics induced by operation at high
power. Measurements performed in-situ have detected unambiguous evidence of
point absorbers on at least 5 of 8 observed test masses
222https://dcc.ligo.org/LIGO-G1900693. An example of these measurements is
shown in Figure 2. Additional forensics of the aLIGO optics (performed off-
site on uninstalled optics) revealed point absorbers on multiple optics,
including several optics never exposed to high laser power in the vacuum
system. A microscope image of a point absorber is shown in Figure 3.

Figure 3: Dark field microscope image of point absorber measured on an
Advanced LIGO optic (corresponding to the thermal lens measurement shown in
Figure 2). Also shown in Buikema et. al. O3CommissioningPRD .

The spatial resolution of the Hartmann wavefront sensor measurements is
approximately one sample every $7.5\text{\,}\mathrm{mm}$ in both transverse
dimensions. Hence we cannot resolve the features in the wavefront smaller than
this. However, due to thermal diffusion to scales larger than the spatial
resolution, we can infer the total power absorbed by fitting thermal lens
models to this data. Observed absorption values lie within the range $100-350$
ppb of the total incident power, however, the number of measurements is too
small to reliably describe the distribution of absorption values. The thermal
lens and surface deformations induced by non-uniform absorption and point
absorbers are characterized by features smaller than the incident laser beam
size (by high spatial frequencies) and, currently, aLIGO contains no high
spatial frequency thermal compensation system to correct for these effects.

The origin of the point absorbers in Advanced LIGO is currently under intense
investigation. The absorbers cannot be removed with standard cleaning
techniques and appear to be embedded in the coating. Initial spectroscopic
analysis of absorbing points on witness samples (coated in same coating runs
as aLIGO optics) show high concentrations of aluminum. Additionally, following
a recent inspection of an aLIGO optic, more point absorbers appeared on that
optic - apparently introduced during the inspection process. The nature of
these new contaminants is not clear. The full scope of point absorber
forensics is beyond the scope of this paper and will be addressed in a future
manuscript. Additionally, it is not clear if the presence of point absorbers
is unique to the large scale optics used in gravitational wave detectors or is
common in all dielectric layer precision optics.

As Vajente Vajente:14  described, non-uniform surface deformation of the
optics in a gravitational wave detector will scatter power into higher-order
spatial modes (HOM) in the interferometer. The Fabry-Perot arms cavities will
enhance and suppress different HOM and may, depending on cavity geometry and
other factors discussed here, resonantly extract power from the fundamental
mode, increasing the observed loss. We briefly review this in Section II.

Within this article, we explore the thermo-optical interaction of point
absorbers with a high-power, high-finesse, Michelson interferometer.
Observations of the interferometer response are reported in Section III and
projections for future impacts on Advanced LIGO are presented in Section IV.

To provide more context, extended background material is included in the
Appendices. The basic operation of aLIGO is described in Appendix A. The
effects of point absorbers on (a) just the thermal state of the optic are
described in Appendix B, (b) the simple laser/optic interaction in Appendix C
and (c) the resonant-cavity/laser/optic interaction in Appendix D. The impact
of substrate thermal lensing on the interferometer control systems (and
subsequent noise couplings) is examined in Section E.

##  II Modeling formalism

In a dual-recycled, Fabry-Perot Michelson interferometer, the power-recycling
gain (the ratio of stored laser power in the power-recycling cavity to input
laser power) is a good proxy for monitoring the average loss in the arms of
the interferometer. In the appendices of this manuscript, we describe a
formalism for modeling the effective change in power-recycling gain due to
point absorbers on the test masses. We summarize the main findings here and
interested readers are encouraged to review the appendices for further detail.

As shown in Equation 20, the surface deformation from a point absorber is
approximated by:

$\Delta s\approx\frac{\alpha\,P_{\mathrm{abs}}}{2\,\pi\,\kappa}\,f(r)$ (1)

where $f(r)$, the functional form of the radial spatial distortion from a
point absorber, is defined in Equation 21. The surface deformation is
proportional to the coefficient of thermal expansion, $\alpha$, the absorbed
power, $P_{\mathrm{abs}}$ in a point and inversely proportional to the thermal
conductivity, $\kappa$. When the fundamental mode in a Fabry-Perot cavity
reflects off a mirror distorted by the above surface deformation, it scatters
some of that field into higher order modes (HOM). The cavity resonantly
enhances/suppresses those modes as a function of the round-trip phase they
accumulate in the cavity. The loss of power from the fundamental mode to a
HOM, $\mathcal{L}_{mn}$, is approximated by Equation 32, which is reproduced
here:

$\mathcal{L}_{mn}=a_{00|mn}^{2}\,g_{mn}.$ (2)

We emphasize here that losses are a function of two main elements. The first
element, $a_{00|mn}$, defined in Equation 31, is single-bounce amplitude
scattering from the fundamental mode into the $mn-\mathrm{th}$ HOM when
reflected off the mirror with the surface deformation $\Delta s$ located at
position $\mathbf{r_{c}}$. The second element is the resonant
enhancement/suppression factor of the HOM for the Fabry-Perot cavity geometry
and accounting for additional phase delays experienced by HOM due to surface
polish errors at the edges of the mirrors, defined in Equation 33.

The average arm loss experienced by the fundamental mode is the sum of losses
across all modes other than the fundamental mode:

$\displaystyle\mathcal{L}_{\mathrm{A}}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\mathrm{nom}}+\frac{1}{2}\sum_{m,n}\mathcal{L}_{mn,X}+\mathcal{L}_{mn,Y}$
(3) $\displaystyle=$
$\displaystyle\mathcal{L}_{\mathrm{nom}}+b\,\left(\frac{P_{A}}{100\mathrm{kW}}\right)^{2}$
(4)

From this model, the power-recycling gain of the interferometer is summarized
in Equation 42, reproduced here:

$G_{P}=\left(\frac{t_{p}}{1-r_{p}\,\left(1-\frac{G_{A}}{2}\,\left[\mathcal{L}_{\mathrm{nom}}+b\,\left(\frac{P_{A}}{100\mathrm{kW}}\right)^{2}\right]\right)}\right)^{2}$
(5)

In Appendix D.3, we highlight all of the physical elements that contribute to
the total loss, either through single-bounce amplitude scattering,
$a_{00|mn}$, (specifically total absorbed power, material properties and
location of position absorber) or through the resonant enhancement/suppression
factor, $g_{mn}$, (namely clipping losses, cavity geometry and mirror polish
errors).

As shown in Appendix C.1, the loss into HOM is dependent only on the absorbed
power, and not the size or distribution of the point absorber. Additionally,
it is shown in Appendix C.2 that time-scale for scattering into higher order
modes decreases as mode-order increases - allowing us to use the time-
evolution of power-recycling gain as a proxy for the contribution of HOM to
loss in the arm.

##  III Observation of point absorber effects in Advanced LIGO

###  III.1 Optical gain vs power and position

Figure 4: Average LIGO power recycling gain versus arm power for the first 120
days of 2019. The Livingston data has been split into two epochs: pre- and
post-interferometer realignment. Arm power is the product of the measured
input laser power, the measured power recycling gain, the arm cavity optical
gain and the beam-splitter transmission. Models of the PRG are shown by dotted
yellow and green curves (assuming a $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$
diameter point absorber at $x=0$mm and a
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter point absorber at $x=0$mm,
respectively). The large range covered by the models indicates a strong
variation in the PRG behavior is expected depending on the exact configuration
of point absorbers.

In aLIGO, we observed the power recycling gain decay as a function of input
laser power, $P_{\mathrm{in}}$ and arm power. The latter is determined from
the product of the power-recycling gain, the input laser power, the arm cavity
optical gain and the beam-splitter transmission:

$P_{A}=0.5\,G_{P}\,G_{A}\,P_{\mathrm{in}}$ (6)

For O3, the average arm cavity optical gains, $G_{A}$, are 268 and 265 for
Hanford (LHO) and Livingston (LLO), respectively (for full details, see
Section VB in Buikema et al.O3CommissioningPRD ). Figure 4 shows the power-
recycling gain (averaged into $1\text{\,}\mathrm{W}$ bins) versus the input
laser power over a 120-day period in early 2019 for the two LIGO
interferometers, LLO and LHO, with two epochs plotted for the former. After an
initial commissioning period the Livingston interferometer was realigned to
move the arm mode by approximately $30\text{\,}\mathrm{mm}$ on ETMY 333D.
Martynov, LLO aLOG 43121, February 2019, https://alog.ligo-
la.caltech.edu/aLOG/ . The Livingston data reflects this and is split into two
epochs: pre- and post-interferometer realignment, red and blue data,
respectively. The optimum Hanford alignment is also shown by green data.

Using the data sets of the measured power recycling gain vs arm power,
$P_{A}$, we fit for $\mathcal{L}_{\mathrm{nom}}$ and $b$ in Equation 42,
yielding the following results:

Interferometer |  $\mathcal{L}_{\mathrm{nom}}$ (ppm) |  $b$ (ppm)
---|---|---
LHO | 68.0 | 1.0
LLO (pre-realign) | 60.2 | 6.4
LLO (post-realign) | 60.7 | 1.3

The new alignment, blue data in Figure 4, shows a $5\times$ reduction of
power-dependent loss, strongly illustrating the position-dependent arm loss
described in Section C.

Lastly, we have modeled maxima and minima for expected power recycling gain,
dotted green and yellow, respectively, calculated using the model represented
by Equations 3, 4 and 5, assuming absorption ranges of 100-350ppb, varying
positions on the test masses and the same nominal loss as LLO. We assume that
these curves represent the likely extremum of PRG behavior for a single point
absorber on an optic. From the range, it is reasonable to expect large
differences between hypothetical individual interferometer configurations.

###  III.2 Time evolution of PRG reduction

Figure 5: Time evolution of power recycling gain during a power-up at LLO.
Laser power and (scaled) arm power are indicated by the right-hand axis. Power
recycling gain is shown on the left-hand axis. Also shown are two SIS models
of the power-recycling gain assuming surface deformation on a single ETM from,
uniform (unif.) and a point absorber (pt), respectively. The time-evolution of
the surface deformation was modeled in COMSOL and scaled in amplitude to yield
a PRG of 40 at $t=\infty$.

We measured a rapid drop in the power recycling gain with a time scale on the
order of 200s. This is illustrated in Figure 5.

Using a combination of a COMSOL finite-element thermal model of point or
uniform absorption induced surface deformation on a test mass and a numerical
FFT model of the full interferometer (the Static Interferometer Simulation
(SIS) 444https://labcit.ligo.caltech.edu/ hiro/SIS/), we simulated the time
evolution of PRG as the interferometer was powered up for two cases. In one
case we assumed uniform absorption and in the other, we assumed a point
absorber located approximately $20\text{\,}\mathrm{mm}$ from the center of the
optic.

The LLO arm power, $P_{L}(t),$ is shown in Figure 5 (dashed red curve). In two
time-dependent finite-element analyses, we modeled surface deformations,
$\Delta s_{i}(x,y,t)$, from uniform absorption and from a point absorber as a
function of time, scaling the absorbed power by $P_{L}(t)$ in the two cases.

The magnitude of these surface deformations were scaled such that, when added
to an ETM in SIS, the steady-state power recycling gain was equal to 40, the
same value measured in the interferometer (dotted blue curve in Figure 5).
Finally, the SIS model was run on all times from $t=0$ to steady-state
($7200\text{\,}\mathrm{s}$) and predicted the power-recycling gain.

As the results show, the rapid drop in PRG is predicted by the point-absorber
SIS model. This is particularly striking when compared to the uniform
absorption SIS model that shows a time scale more than an order of magnitude
larger. The smaller drop in PRG around $1300\text{\,}\mathrm{s}$ is thought to
be due to slight misalignment.

Consistent with the time-scale analysis in Appendix C.2, this result is highly
suggestive that the majority of the power-recycling gain reduction that we see
can be attributed to point absorbers interacting with higher order modes
rather than uniform absorption.

###  III.3 Noise couplings from ITM thermal lens effects

We measured the coupling of the input laser relative intensity noise (RIN) to
differential arm motion (DARM) in aLIGO. At frequencies above
$500\text{\,}\mathrm{Hz}$, the coupling was approximately two orders of
magnitude higher than expected from simulations of an ideal interferometer, as
shown in Figure 6. However, simulations including ITM thermal lenses from
point absorbers modeled at a variety of different radii showed significant
increase in high frequency RIN coupling, sufficient to explain the excess
coupling observed in the interferometer.

Figure 6: RIN coupling models versus measurements from the LHO interferometer.
The ”ideal IFO” trace is for a perfectly symmetric IFO, therefore the coupling
is due to radiation pressure at low frequency and sidebands RIN at high
frequency. The other traces have the expected spherical lenses on the ITM
substrates, plus one point absorber map at different distances from the
center. Simulations were created using the MIST simulation tool Vajente_2013 .

Uncertainty in point absorber position creates a large uncertainty in the
exact coupling coefficient. Hence, the model can only reproduce the
qualitative behavior of the observed RIN coupling. Noise budget measurements
reported in Buikema et al.O3CommissioningPRD  show the absolute laser
intensity contribution to overall noise is not yet a limiting noise source.
Below $2\text{\,}\mathrm{kHz}$, it is at least $10\times$ lower than the
differential arm motion in both LIGO interferometers (Figure [2] in Buikema et
al.O3CommissioningPRD  explicitly shows this). However, the coupling, and
hence the noise contribution, is expected to increase as arm cavity power
increases in future.

##  IV Considerations for working with point absorbers in the future

Figure 7: Arm power versus input laser power. The dashed line shows the case
of constant optical gain. The data points show the measurements from the LIGO
sites. The yellow range shows predictions from the SIS model assuming uniform
absorption and a variety of point absorbers on the optic consistent. As the
effect of the point absorber is decreased, the uniform absorption becomes the
limiting factor preventing ideal build-up of arm power.

It is clear that point absorbers adversely affect the performance of the aLIGO
interferometers and that this effect increases as input power is increased. In
Figure 7 we plot the projected arm power versus input power assuming the most
optimistic scenario of solely a power-dependent reduction in power-recycling
gain. This projection corresponds to the three different scenarios in Figure 4
(O3 LHO (green), O3 LLO pre-realignment (red) and O3 LLO post-realignment
(blue)). Briefly, all scenarios predict a significant reduction in the maximum
stored power (and, hence, performance) for aLIGO.

Despite the improvement in PRG attained by repositioning the beam on the ETM,
the SIS model predicts a minimum 33% deficit in arm power build-up at nominal
operating power of $125\text{\,}\mathrm{W}$. Assuming no other adverse effects
to interferometer operation, we would expect a corresponding increase in the
shot noise floor of 15%$\\--$20% above the nominal Advanced LIGO noise floor.

Mitigating the adverse effects of point absorbers can be accomplished by
addressing all the features identified in Section D.3 and expressed by the
amplitude and gain terms in Equation 2.

Amplitude reduction strategies seek to limit or eliminate the single bounce
scattering term, $a_{00|mn}$, in Equation 2. The first, and ideal, scenario is
to eliminate absorbers at the source when they are introduced into the
coating: either through modifications to the coating process or active
elimination after coating. This is being actively researched within the LIGO
Scientific Collaboration. Beyond that, the simplest method for partial
reduction of amplitude scattering in-situ is re-positioning the interferometer
beam, as demonstrated at LLO. Additionally, scattering to specific HOM can be
reduced by actuating on those modes using, for example, a high-spatial
frequency corrector CHRAAC:2013 . Finally, surface deformation (and therefore
amplitude scattering) are minimized for different optic materials; for
example, the proposed use of cryogenic silicon in future interferometers
Voyager_2020 ; NEMO:2020  has the benefit of an, effectively, zero coefficient
of thermal expansion.

Resonant enhancement/suppression techniques seek to reduce the resonant gain
term, $g_{mn}$, in Equation 2. Future interferometers are free modify the
overall cavity design and g-factor with respect to HOM spacing to minimize the
effect from point absorbers. A more targeted option, actively being explored
for Advanced LIGO, is to deliberately polish surface errors into the edges of
the optic, such as those illustrated in Figure 14, creating a wider HOM free
region around the fundamental mode resonance Hiro2020 . In addition,
conceptual work has begun on active front surface thermal actuators that could
produce such surface errors dynamically - thus allowing for in-situ fine-
tuning of the HOM resonances around the fundamental mode.

##  V Conclusion

We have shown that point-like absorbers present a serious impediment to
obtaining full operating power in Advanced LIGO, primarily due to increased
arm losses as absorbed power increases. We reiterate that that errors in the
mirror surface profile, mirror aperture effects and higher order mode behavior
matter are crucial to understanding considering these losses. Future
gravitational wave detectors call for stored power approximately an order of
magnitude or more higher than O3 Miao_HF:2018 ; Martynov:2019_PRD ;
Page_HF_GW:2020 ; ET2010 ; Reitze2019Cosmic , with similar laser intensities
to full-power Advanced LIGO, so the existence of point absorbers and their
interaction with the optical fields of those detectors is of significant
importance.

We have explored the physics of cavity-optic-deformation to explain the
observed reduction in interferometer performance whilst, simultaneously,
highlighting those elements that are most crucial to mitigating the effects of
point absorbers in Advanced LIGO and future gravitational wave
interferometers.

##  VI Acknowledgements

The authors gratefully acknowledge the support of the United States National
Science Foundation (NSF) for the construction and operation of the LIGO
Laboratory and Advanced LIGO as well as the Science and Technology Facilities
Council (STFC) of the United Kingdom, and the Max-Planck-Society (MPS) for
support of the construction of Advanced LIGO. Additional support for Advanced
LIGO was provided by the Australian Research Council. The authors acknowledge
the LIGO Scientific Collaboration Fellows program for additional support.

LIGO was constructed by the California Institute of Technology and
Massachusetts Institute of Technology with funding from the National Science
Foundation, and operates under cooperative agreement PHY-1764464. Advanced
LIGO was built under award PHY-0823459. This paper carries LIGO Document
Number LIGO-P1900287.

##  Appendix A Advanced LIGO interferometer operation

To understand the observed interferometer behavior, it is helpful to provide
an overview of how the aLIGO interferometer works. This section contains a
description of the ideal operation of the Advanced LIGO interferometer.
Interested readers can find more details in other references aLIGO_Overview ;
aLIGO_ASC_Barsotti_2010 ; aLIGO_Staley_2014 ; aLIGO_Martynov_2016 ;
aLIGO_LSC_2016 ; aLIGO_O2_Driggers .

###  A.1 Detector sensitivity

As shown in Figure 1, Advanced LIGO is a suspended Michelson interferometer,
modified to maximize sensitivity to GW with a bandwidth of approximately
$20\text{\,}\mathrm{Hz}$ \- $2\text{\,}\mathrm{kHz}$. All modifications are
designed to either amplify the signal strength, reduce the contribution of
different noise sources or improve robustness and reliability.

A passing gravitational waves of amplitude, $h$, induces a strain in
spacetime, lengthening one interferometer arm and shortening the other by an
amount, $\Delta L$, where

$\Delta L=h\,L$ (7)

and $L$ is the interferometer arm length. This imposes phase fluctuations on
the laser beams present in those arms (GW audio sidebands) that are converted,
by the beamsplitter, into intensity fluctuations on the output side of the
Michelson interferometer.

As Equation 7 shows, the resulting change in arm length (and, hence, the
signal strength) is proportional to the arm length. To exploit this, the aLIGO
arms are $4\text{\,}\mathrm{km}$ long, substantially amplifying the signal for
a given strain relative to a short interferometer. The arms contain resonant
Fabry-Perot cavities that increase the average number of round-trips photons
experience in the arms, and the phase they accumulate, by the optical gain of
the arms, $G_{A}$. For the Fabry-Perot arm cavity, since the input and end
mirror transmissions are very low ($T_{i}=$ 1.4% and $T_{e}=5$ppm,
respectively), the following approximation holds:

$G_{A}\approx\frac{4}{T_{i}}$ (8)

resulting in a nominal phase amplification factor of approximately 280 for
aLIGO. (The optical gain is also approximated by $2\,F/\pi$, where $F$ is the
cavity finesse, approximately 440).

Ideally, the Fabry-Perot arms resonate the fundamental Gaussian spatial mode
(sometimes referred to as TEM00) and suppress higher order spatial modes
(HOM). Higher order modes can be described in terms of different bases, for
example, Hermite-Gauss (HG) or Laguerre-Gauss (LG) Siegman:Lasers .

In contrast to the limited ways to amplify the signal, there are a large
number of independent noise sources in the interferometer. The majority of
noise sources limit the sensitivity below approximately
$100\text{\,}\mathrm{Hz}$ Martynov_noise:15 ; aLIGO_O2_Driggers . However, at
high frequencies, an ideal detector is limited only by fundamental quantum
sensing noise (shot noise). The Signal-to-Noise Ratio (SNR) from shot noise is
proportional the square root of the power incident on the input (symmetric)
side beamsplitter. Thus, one of the the simplest ways to improve the detector
sensitivity is to increase the input laser power (up to
$125\text{\,}\mathrm{W}$ for aLIGO).

The incident power on the beamsplitter is further increased by adding a power
recycling mirror (PRM), see Figure 1, to return power coming out of the
interferometer back into it. This mirror forms another resonant cavity, named
the power recycling cavity. The coupled power-recycling and common arm
cavities are referred to as the common coupled cavity.

For the power recycling to maintain a high optical gain, the interferometer is
held close to a dark fringe on the output (anti-symmetric) side of the
beamsplitter, sending nearly 100% of the optical power on the beamsplitter
back to the PRM. The differential arm length is held slightly off minimum
(close to dark fringe) such the static amount of power at the AS port is held
constant to serve as a local oscillator for DC (homodyne) readout Hild_2009 .

The coupled power-recycling and arm cavities increase the optical gain such
that the power stored in the PRC and a single arm are approximately $50\times$
and $6000\times$ the input power, respectively (in fact, these numbers vary in
the real interferometer depending on losses in the arms and power-recycling
cavities).

The storage time of photons the arm cavity is similarly increased. Intensity
noise present on the light resonant in the cavity is filtered above
$1/\tau_{CC}$, where $\tau_{CC}$ is the coupled cavity storage time,
corresponding to a pole at a frequency of approximately
$0.6\text{\,}\mathrm{Hz}$ Martynov_noise:15 . This filtering is essential to
suppress the coupling of relative intensity noise (RIN) from the input laser
to the GW channel.

As shown in Figure 1, on the output side of the interferometer a recycling
mirror (SRM) is present, forming a resonant cavity with the test masses called
the signal recycling cavity (SRC). The SRC is held on anti-resonance, lowering
optical gain and storage time of difference signals in order to increase the
signal bandwidth of the interferometer (known as resonant signal extraction
RSE:2003 ). The combination of the signal-recycling and arm cavities is called
the differential coupled cavity.

Finally, the output of the interferometer is spatially filtered through an
output mode cleaner (OMC) to remove higher order spatial modes that carry no
GW signal, and strip off any residual radio-frequency (RF) control sidebands
from the laser, described below. At the output of the OMC the GW signal
sidebands beat with the static offset induced by the arm differential offset
to provide a power fluctuation that is detected by a pair of photodiodes.

###  A.2 Power-recycling gain

The power recycling gain, $G_{P}$, defined as the ratio of the stored power
inside the power recycling cavity to the input laser power incident on the
PRM, is a function of the reflectivities of the power-recycling mirror, input
test masses and end test masses and also a function of the losses in the arms
and the PRC. The amplitude reflectivities are denoted in lower-case, $r_{a}$,
and corresponding power reflectivities are denoted in upper-case,
$R_{a}=r_{a}^{2}$. The values are given in Table 1 in Appendix F.

For this analysis, it is convenient to consider the two arm cavities as an
aggregate (common) arm cavity and the power-recycling cavity as a three mirror
cavity made of the PRM, an average Fabry-Perot arm. The power recycling gain
is given by:

$G_{P}=\left(\frac{t_{p}}{1-r_{p}\,r_{FP}\,\left(1-\mathcal{L}_{P}/2\right)}\right)^{2}$
(9)

when the PRC is on resonance AndoThesis . The amplitude reflectivity of the
average arm cavity, $r_{FP}$ AndoThesis , is given by:

$\displaystyle r_{FP}$ $\displaystyle=$
$\displaystyle\frac{-r_{i}+r_{e}^{*}}{1-r_{i}\,r_{e}^{*}}$ (10)
$\displaystyle\approx$ $\displaystyle
1-2\frac{T_{e}}{T_{i}}-\frac{G_{A}\,\mathcal{L}_{A}}{2}$ (11)

when the arm cavities are on resonance and $T_{e},T_{i}<<1$. Combining
equations 9 and 11 yields an expanded form of the power-recycling gain,

$G_{P}\approx\left(\frac{t_{p}}{1-r_{p}\,\left(1-(G_{A}\,T_{e}-G_{A}\,\mathcal{L}_{A}-\mathcal{L}_{P})/2\right)}\right)^{2}$
(12)

in which the arm loss contribution is greater than the recycling cavity loss
by a factor of $G_{A}$, approximately 280 for aLIGO. Thus, the power-recycling
gain can be used as a proxy for losses, particularly arm losses, in the
interferometer.

###  A.3 Feedback control

In order for the interferometer to function in low noise, all of the
longitudinal degrees of freedom (DOF) in the interferometer must be sensed and
locked onto resonance for the main laser field (the GW carrier). This is
achieved by using the Pound-Drever Hall locking technique that adds pairs of
radio frequency sidebands onto the main laser PDH_1983  at
$\pm$$9\text{\,}\mathrm{MHz}$ and $\pm$$45\text{\,}\mathrm{MHz}$. As
illustrated in Figure 1, both pairs of sidebands are anti-resonant in the
Fabry-Perot arms, resonant in the PRC. The $45\text{\,}\mathrm{MHz}$ sidebands
are resonant in the SRC. Full details can be found in Martynov_noise:15 .

Since sidebands and carrier fields have different frequencies, their
propagation inside the meters- or km-long interferometer cavities produce
different phase shifts. The net result is that carrier and sideband fields
have different resonant conditions in the various interferometer cavities. In
particular, while the carrier is tuned to be resonant simultaneously in the
power recycling cavity and in the Fabry-Perot arm cavities, the sideband
frequency is chosen so that the corresponding fields are close to anti-
resonance in the arms and resonant in the power recycling cavity. In this way
the sidebands provide a phase reference which is independent of the arm cavity
motions, and suitable error signals can be extracted by demodulating the
signals produced by fast photodiodes at the same modulation frequency
Martynov_noise:15 .

Thus linear combinations of this beating of these sidebands and carrier,
measured on RF photodiodes at different locations and in different modulation
quadrature yield enough control signals to uniquely measure all the desired
degrees of freedom and, when fed back into a control system, keep the residual
longitudinal motions less than the required levels. The noise in the control
system must be kept very low lest the feedback loop inject it back into the
interferometer at a level which then limits the detector sensitivity.

In the interests of brevity, the above description of the feedback control
system is deliberately terse. A full description of the control is beyond the
scope of this text, references are provided HallThesis_2017 ; MartynovThesis .
However, one element key to the following discussion concerns the differential
arm motion (DARM): the degree of freedom that encodes the gravitational wave
signal. For the subsequent discussion, it is important to know that the
feedback control system is set up to maintain a constant DC power
($20\text{\,}\mathrm{mW}$) on the output photodiodes by feeding the DARM error
signal back to the differential arm length.

In summary, Advanced LIGO achieves good sensitivity at high frequencies by
having large amounts of stored power in those cavities and incident on the the
test masses while relying on a low-noise control system to keep the optical
cavities on resonance within minimal residual longitudinal motions.

##  Appendix B Thermal-elastic surface deformation from point absorbers

Consider a point-like absorber, absorbing within a diameter $2\,\omega$, on
the surface of an optic that is exposed to optical power, as illustrated in
Figure 8. We solve for the surface deformation inside and outside the
absorbing region.

In the specific case that the absorbing region is on the surface of an optic
within a LIGO arm cavity, the absorbed power, $P_{\mathrm{abs}}$, is
determined by the local intensity of the illuminating beam multiplied by the
absorbing area:

$P_{\mathrm{abs}}=\left(\pi\,\omega^{2}\right)\,\frac{2\,P_{\mathrm{A}}}{\pi\,\mathrm{w}^{2}}\,\exp\left[-\left(\frac{\mathrm{r_{c}}}{\mathrm{w}}\right)^{2}\right]$
(13)

where $P_{\mathrm{A}}$ is the laser power stored in the arm, $\mathrm{w}$ is
the Gaussian beam radius and $\mathrm{r_{c}}$ is the position of the absorbing
region relative to the center of the Gaussian beam. For simplicity, we have
assumed that the region is 100% absorbing.

Figure 8: Power flowing into a point-like region. The radiated power is
negligible compared to the conduction. Hence, all power is effectively
transferred by conduction. The far-field gradient is only determine by the
absorbed power and thermal conductivity.

###  B.1 Small and large scale surface deformation and time scales

####  B.1.1 Inside the absorbing region

Winkler et al. solved for the approximate surface deformation, $\Delta
s_{\omega}$, on the surface of an optic, within the absorbing region
PhysRevA.44.7022 , finding

$\Delta s_{\omega}=-\frac{\alpha}{4\,\pi\,\kappa}P_{\mathrm{abs}}$ (14)

where $\alpha$ is the coefficient of thermal expansion, $\kappa$ is the
thermal conductivity and $P_{\mathrm{abs}}$ is the power absorbed within the
absorbing region. The negative sign is purely convention based upon the
deformation expanding away from a mirror. Expressed as a quadratic function of
position:

$\Delta
s=-\frac{\alpha}{4\,\pi\,\kappa}P_{\mathrm{abs}}\left(\frac{r}{\omega}\right)^{2}.$
(15)

####  B.1.2 Outside the absorbing region

We solve for the approximate surface deformation for a half-infinite cylinder
(i.e. assuming radial and longitudinal boundaries are far away). We assume
only conduction and ignore power radiated from the front surface of the optic
- given that the power radiated from one beam radii is of the order $2\%-3\%$
of the total power radiated by the optic.

Ignoring boundary conditions and considering the region outside the radius of
the point absorber, the temperature distribution is solely governed by the
equation of thermal conductivity:

$\displaystyle P_{\mathrm{abs}}$ $\displaystyle=$
$\displaystyle-\kappa\,A\,\overrightarrow{\nabla}T$ (16)

where $A=2\,\pi\,r_{S}^{2}$, the area of a hemispherical shell in the optic,
where $r_{S}$ is the spherical radial coordinate ($\sqrt{x^{2}+y^{2}+z^{2}}$).
Solving in spherical coordinates, the temperature profile is given by

$\displaystyle T(r)$ $\displaystyle=$
$\displaystyle\frac{P_{\mathrm{abs}}}{2\,\pi\,\kappa\,r_{S}}$ (17)
$\displaystyle=$
$\displaystyle\frac{P_{\mathrm{abs}}}{2\,\pi\,\kappa\,\sqrt{r^{2}+z^{2}}}$
(18)

where $r=\sqrt{x^{2}+y^{2}}$ and $z$ are the polar radial and longitudinal
coordinates, respectively. The surface deformation is approximated by the
coefficient of thermal expansion, $\alpha$, multiplied by the integral of
temperature field:

$\displaystyle\Delta s$ $\displaystyle=$
$\displaystyle\alpha\,\int_{0}^{h}T(r)\,\mathrm{d}z$ (19)

where $h$ is the thickness of the optic. Combining this with equation 15 and
solving yields a generalized approximation for the surface deformation on the
optic:

$\Delta s\approx\frac{\alpha\,P_{\mathrm{abs}}}{2\,\pi\,\kappa}\,f(r)$ (20)

where

$f(r)=\begin{cases}-\frac{1}{2}\,\left(\frac{r}{\omega}\right)^{2}&r\leq\omega\\\
c_{0}+\log\left[\frac{h+|(r,h)|}{r}\right]&r>\omega\end{cases}$ (21)

where $c_{0}$ is a constant offset such that the two parts have the same value
at $r=\omega$:

$c_{0}=-\frac{1}{2}-\log\left[\frac{h+|(\omega,h)|}{\omega}\right]$ (22)

and where

$|(r,h)|=\sqrt{r^{2}+h^{2}}.$ (23)

Note that the surface deformation has been referenced to the center of the
point absorber such that the surface deformation is zero at that point.
Equation 20 shows that outside of the absorbing region, the large scale
surface deformation depends only on the absorbed power and not on the size or
distribution of the absorbing region. This is illustrated by a finite-element
model of surface deformation from point absorbers of different size but fixed
absolute absorbed power, shown in Figure 9. The surface deformation from
uniform absorption is shown for comparison.

Figure 9: Cross-section of modeled surface deformation from uniform and point
absorbers shown in linear (upper) and semilog (lower) plots. The red, yellow
and purple curves show surface deformation from point absorbers of different
physical size but with each absorbing $40\text{\,}\mathrm{mW}$. Uniform
absorption of $10\text{\,}\mathrm{mW}$ (blue) is shown for comparison. All
point absorber cases show identical surface deformation outside the absorbing
region.

####  B.1.3 Time scale

Finally, the time scale for a temperature distribution to form is governed by
the heat equation:

$\frac{\partial T}{\partial t}=D\,\left(\frac{\partial^{2}T}{\partial
x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial
z^{2}}\right)$ (24)

where $D$ is the thermal diffusivity of the mirror, given by:

$D=\frac{\kappa}{\rho\,c}$ (25)

where $\rho$ is the density, $c$ is the specific heat capacity and $\kappa$ is
the thermal conductivity. A cursory examination of the heat equation shows
that the time-scale for the temperature distribution to form over a given
spatial scale will be proportional to that spatial scale squared. For example,
for a given spatial scale, $r_{0}$, the time constant for the temperature
distribution (and thus surface deformation) to form is:

$\tau_{r_{0}}\propto\frac{r_{0}^{2}}{D}.$ (26)

Thus, time-evolution of surface deformation is an effective discriminator for
different spatial scales. We return to this idea in the next appendix where we
evaluate time evolution for specific spatial distribution of higher order
optical modes.

##  Appendix C Single-bounce deformation-induced scattering into higher order
modes

Consider the immediate effect of reflection of a fundamental Gaussian (TEM00)
optical field, of power $P$, from a deformed mirror surface - hereafter
referred to as the single-bounce reflection off the optical surface. The
distribution of this field is given by:

$E_{00}=\left(\frac{2\,P}{\pi\,\mathrm{w}^{2}}\right)^{1/2}\exp\left[-\left(\frac{r}{\mathrm{w}}\right)^{2}\right]$
(27)

Specifically, we consider the magnitude of scattering into higher order
optical modes and the characteristic time scale for such scattering. Note that
in the following discussion, we justify ignoring the curvature of the mirror
and the curvature of the incident Gaussian beam by assuming that these are
identical.

###  C.1 Scattering as a function of point absorber size and position

Upon reflection from a mirror with surface deformation, $\Delta s(x,y)$, the
scattering from the fundamental mode to the $mn$-th HOM is determined by the
overlap integral of the TEM00 mode with the higher order mode, $E_{mn}$, plus
the phase error, $i\,k\,\Delta s$, multiplied by a factor of 2 to account for
the double-pass upon reflection. Thus, the complex amplitude coefficient is:

$\displaystyle a_{00|mn}$ $\displaystyle=$
$\displaystyle\iint\mathrm{E_{mn}}e^{\mathrm{i}\,k\,(2\,\Delta
s)}\mathrm{E_{00}}\,\mathrm{d}x\,\mathrm{d}y$ (28) $\displaystyle\approx$
$\displaystyle\mathrm{i}\,2\,k\,\iint\mathrm{E_{mn}}\,\Delta
s\,\mathrm{E_{00}}\,\mathrm{d}x\,\mathrm{d}y$ (29)

where $k=2\pi/\lambda$ and the approximation is valid for deformations that
are significantly smaller than the wavelength of light. In the case of a point
absorber, we combine with Equation 20 and find:

$a_{00|mn}\approx\frac{\mathrm{i}\,2\,\alpha\,P_{\mathrm{abs}}}{\lambda\,\kappa}\iint\mathrm{E_{mn}}\,f(r)\,\mathrm{E_{00}}\,\mathrm{d}x\,\mathrm{d}y$
(30)

The amplitude scattering coefficient is proportional to the absorbed power,
$P_{\mathrm{abs}}$ and, hence, the power scattering coefficient,
$a_{00|mn}^{2}$, is proportional to the square of the absorbed power.

Given that Equation 20 establishes that surface deformation is largely
independent of point absorber size, we expect that scattering to HOM should be
similarly independent. To verify this, scattering into different HOM was
determined as a function of point absorber size whilst holding the total
absorber power constant. The results are illustrated in Figure 10 which shows
the single bounce scattering into higher order modes is indeed largely
invariant with point absorber size. It is only once the absorber approaches
the characteristic size of a node of the higher order mode that we begin to
see variation in the scattering coefficient.

Figure 10: Single-bounce scattering amplitude into higher order modes vs point
absorber diameter for a fixed absorption of 30mW in the steady-state case. For
the modes shown (up to order 120), there is no dependence on spot size for
points below approximately $5\text{\,}\mathrm{mm}$.

However, HOM have spatial dependence. Thus the scattering amplitude is a
function of the point absorber position. For a point absorber displaced from
the center of the Gaussian beam by an amount $\mathbf{r_{c}}$, the scattering
coefficient is:

$\displaystyle a_{00|mn}(\mathbf{r_{c}})$ $\displaystyle=$
$\displaystyle\mathrm{i}\,2\,k\iint\mathrm{E_{mn}}\Delta
s(\mathbf{r}-\mathbf{r_{c}})\mathrm{E_{00}}\,\mathrm{d}x\,\mathrm{d}y$ (31)

Scattering versus point absorber position (for a $10\text{\,}\mathrm{mW}$
point absorber after $3600\text{\,}\mathrm{s}$) is illustrated in Figure 11.
The scattering to even modes is maximized when the point absorber is in the
center of the TEM00 field, whilst the scattering to the odd modes have zero
scatter amplitude there. The scattering to odd modes is maximized at different
radii with a clustering between $15\text{\,}\mathrm{mm}$ and
$25\text{\,}\mathrm{mm}$.

Figure 11: Scattering amplitude into Hermite-Gauss, HG$i$0, mode versus
position from the center of the TEM00 field for a $10\text{\,}\mathrm{mW}$
point absorber after $3600\text{\,}\mathrm{s}$ of heating. Odd modes are
identified by solid lines; even modes are identified by dashed lines.

###  C.2 Time scale for scattering

Figure 12: Top: Normalized power scattered from the incident TEM00 field into
even HOM versus time for a point absorber at the center of the TEM00 mode
(solid) and for uniform absorption (dashed). Bottom: power scattered into
HOMs, relative to 2nd order mode [TEM20], for a point absorber (solid) and
uniform absorption (dashed).

The scattering from the fundamental mode into even HOMs is shown in Figure 12,
both for a point absorber and for uniform absorption. The upper panel shows
the power scattered into HOM versus time, normalized to the maximum scattering
value for each mode. The solid lines indicate the case of a point absorber at
the center of the fundamental mode, absorbing 10mW of incident power (although
the normalization renders the absolute absorbed power irrelevant). The dashed
lines show scattering from uniform absorption with an incident beam size of
$53\text{\,}\mathrm{mm}$. Note that both cases were chosen to be axially
symmetric about the center of the optic resulting in zero coupling to odd
order modes, hence only even order modes are shown.

The time scale to achieve 70% of the maximum scattering is strongly dependent
on the mode order and only loosely dependent on the nature of absorption
(uniform vs point absorber). This is understood given that (a) the time scale
for thermal diffusion is a function of spatial scale and (b) the
characteristic spatial scale of the nodes of HOMs are a function of mode order
(with higher order modes having smaller characteristic spatial scales). Hence
shorter time constants function as an indicator for stronger prevalence of
higher order modes.

Furthermore, the lower panel of Figure 12 shows the scattered into HOM from
the fundamental mode in a steady-state single-bounce (normalized to the power
scattered into the second order mode). Scattering from a
$10\text{\,}\mathrm{mW}$ point absorber is indicated by the solid line and
scattering from $10\text{\,}\mathrm{mW}$ of uniform absorption is indicated by
the dashed line. As might be expected, the scattering from a point absorber
contains substantially more HOM content (i.e. high spatial frequency content)
than uniform absorption, with approximately $50$ times more power scattered
into the 8th order mode in the point absorber case. Thus the stronger
prevalence of higher order modes for a point absorber should yield shorter
time constants for observed optical effects (than for uniform absorption). In
an optical cavity, the exact scaling of time constants will be dependent on
the resonant enhancement and suppression of higher order modes. This is
explored in the next appendix.

##  Appendix D Resonant optical interaction, excess losses and optical gain

In this section, we explore the interaction of power-dependent losses on the
stored power of a resonant cavity. Although we describe the several power-
dependent loss mechanisms that exist, the majority of this section focuses on
resonant enhancement and suppression of losses from higher order modes which
is the dominant loss mechanism. Other adverse effects from power dependent
losses are discussed in the following section, focusing on intensity noise
coupling.

Figure 13: Round-trip loss, $\mathrm{L}_{RT_{m+n}}$, vs mode-order for a
single round trip of higher order modes in the aLIGO arm cavities. The labels
for ”low”, ”intermediate” and ”high” spatial frequencies are loosely defined
and intended only as a guide to help associate different mode orders with
different loss mechanisms.

###  D.1 Loss mechanisms

We characterize power-dependent losses from point absorbers into the following
categories.

  * •

Wide-angle scattering to the beam tube. These losses include all power lost on
a single bounce of the carrier field from the surface of an optic that does
not make it to the test mass at the other end of the arm. For the context of
this discussion, we define these as scatter into higher order modes where the
round-trip clipping loss of that mode is greater than 50%. This is illustrated
in Figure 13, and with the somewhat arbitrary delineation at mode order
greater than 9 for aLIGO. Simulations show that the total power lost to these
modes is minimal and thus we do not consider these in detail here.

  * •

Resonant enhancement of losses from intermediate higher order modes. These
losses include scattering and resonant loss of power into higher order modes.
These higher order modes experience more clipping at the edges of the test
masses and hence more loss. In practice, these are due to low to medium
spatial frequencies. This constitutes the main source of arm loss and is
discussed in more detail in the following subsections.

  * •

Mode-matching losses.

    * –

Differential (contrast defect) losses. Differential mode changes between the
two arms will increase loss to the anti-symmetric port of the interferometer.

    * –

Common mode losses. Common mode changes in the arms will reduce the overlap
between the input laser mode and the resonant mode in the power-recycling
cavity.

The last two items, contrast defect losses and reflection losses are,
essentially, recycling cavity losses, $\mathcal{L}_{P}$ in Equation 12. When
we model the aLIGO Fabry-Perot cavities with point absorbers, the mode-
matching loss between the recycling cavities and the arms, $\mathcal{L}_{P}$,
are 3 to 11 times larger than arm cavity losses, $\mathcal{L}_{A}$, in
absolute terms. However, as indicated by Equation 12, the effect of arm losses
on power-recycling are enhanced by the arm-cavity optical gain (a factor of
280). Hence, the contribution of mode-mismatch between the two arms or arms
and the power-recycling cavity is 28 to 95 times smaller (i.e. are negligible
or at most a perturbation) relative to the round-trip losses in the arms.

Thus, we shall discuss resonant enhancement and suppression of losses from
intermediate higher order modes in more detail.

###  D.2 Scattering into non-resonant and partially resonant higher order
modes

The resonant enhancement and suppression of losses from the fundamental
optical mode due to intra-cavity scattering was previously solved by Vajente
Vajente:14 . In that context, the solution was determined for small static
deformations of mirror surfaces. We shall extend that analysis allowing for
power-dependent surface deformations from point absorbers and for large
surface aberrations present in real mirrors.

Equation (9) from Vajente calculates the fundamental mode amplitude inside an
optical cavity that includes resonant scattering to a HOM. From that equation,
the manuscript then determines the additional loss term from the resonant
scattering to a HOM in Equation (11) in that manuscript. Unfortunately, that
manuscript contains a minor typographical error in the printed version of
Equation (11) which does not affect its conclusions. We produce the correct
derivation below, split between Equations 32 and 33 and substituting some
symbols for our own. The power loss from the fundamental mode to the $mn$-th
higher order mode, $\mathcal{L}_{mn}$, is given by:

$\mathcal{L}_{mn}=a_{00|mn}^{2}\,g_{mn}$ (32)

where $g_{mn}$ is the resonant enhancement/suppression factor of the $mn$-th
higher order mode:

$g_{mn}=\frac{1-r_{1}^{\prime\,2}\,r_{2}^{\prime\,2}}{1+r_{1}^{\prime\,2}\,r_{2}^{\prime\,2}}\,\frac{1}{1-\frac{2\,r_{1}^{\prime}\,r_{2}^{\prime}}{1+r_{1}^{\prime\,2}\,r_{2}^{\prime\,2}}\,\cos\left[\Phi_{mn}\right]}$
(33)

where $r_{1^{\prime}}$ and $r_{2^{\prime}}$ are the modified amplitude
reflectivities, attenuated by for mode-dependent clipping losses, of the ITM
and ETM, respectively.

$r_{i}^{\prime}=r_{i}\,\sqrt{\frac{\int_{0}^{\Omega}\,r\,\left|E_{i,mn}\right|^{2}\mathrm{d}r}{\int_{0}^{\infty}\,r\,\left|E_{i,mn}\right|^{2}\mathrm{d}r}}$
(34)

where $\Omega$ is the radius of the mirror and $E_{i,mn}$ is the $mn$-th
higher order mode propagated to the $i$-th mirror.

In the denominator of Equation 33, the argument of the cosine function,
$\Phi_{mn}$, is an expression of the additional round-trip phase that higher
order modes accumulate relative to the fundamental mode. For an ideal cavity
with (a) infinite and (b) purely quadratic mirror surfaces, the nominal phase
shift, $\Phi_{mn}^{\mathrm{nom}}$, is given by:

$\Phi_{mn}^{\mathrm{nom}}=(m+n)\,\phi_{g}$ (35)

where $\phi_{g}$ is the round-trip Gouy phase shift Feng:01 , given by:

$\phi_{g}=2\,\arccos\left[\sqrt{\left(1-\frac{L}{\mathcal{R}_{\mathrm{ITM}}}\right)\,\left(1-\frac{L}{\mathcal{R}_{\mathrm{ETM}}}\right)}\right],$
(36)

where $\mathcal{R}_{\mathrm{ITM}}$ and $\mathcal{R}_{\mathrm{ETM}}$ are the
ITM and ETM radii of curvature (ROC), respectively.

Figure 14: Optic cross section of ETMY at LHO. Upper panel shows a cross
section of the surface of an ETM with a ROC of 2240m. The lower panel shows a
cross section of residual ETM surface deformation after the spherical power
term seen by the TEM00 mode is subtracted from the surface. For reference,
intensity cross-sections of nominal LG00 modes and LG31 modes are shown. The
zeroth order mode samples almost none of the residual distortion in the wings
while the 7th order mode samples a large fraction of it.

In a real interferometer such as Advanced LIGO, the mirror surfaces are (a)
finite in diameter and (b) deviate from purely quadratic. This is illustrated
in Figure 14 which shows a cross section of one of the LIGO ETMs in which the
edges of the mirror deviate from quadratic by approximately
$50\text{\,}\mathrm{nm}$. Higher order spatial modes with large spatial field
extent, such as the Laguerre-Gauss three-one mode (LG31), sample a larger
fraction of this edge effect, relative to the fundamental mode, and accumulate
an additional phase offset beyond the Gouy-phase expressed in Equation 35.
This mirror phase offset/correction, $\Delta\phi_{mn}$, is a function of the
mirror surface error, $S_{TM}$, and the spatial distribution of the HOM.
Derived from the scattering of a HOM into itself:

$\left<\mathrm{E}_{mn}|\mathrm{e}^{\mathrm{i}\,2\,k\,S_{TM}}|\mathrm{E}_{mn}\right>\approx
1+2\,\mathrm{i}\,\Delta\phi_{mn}$ (37)

where, $2\,\Delta\phi_{mn}$, the average phase accumulated by the mode is:

$\Delta\phi_{mn}=\frac{2\,\pi}{\lambda}\,\frac{\iint\left|\mathrm{E}_{mn}\right|^{2}\,S_{TM}\,\mathrm{d}x\,\mathrm{d}y}{\iint\left|\mathrm{E}_{mn}\right|^{2}\,\mathrm{d}x\,\mathrm{d}y}$
(38)

such that the round-trip phase accumulated by a higher order mode is given by:

$\Phi_{mn}=\Phi_{mn}^{\mathrm{nom}}+2\,\Delta\phi_{mn}$ (39)

where the leading factor of 2 is added to account for double-passing on
reflection from the mirror. Note that we define the zero value of the residual
surface error, $S_{TM}$, with such that $\Delta\phi_{00}=0$. The effect of the
additional phase offset was studied by Bond et al Bond2011  in the context of
a larger investigation exploring resonances higher order modes in optical
cavities to reduce the effect of coating thermal noise in gravitational wave
interferometers.

Figure 15: Top: modeled cavity scan of the LHOX aLIGO arm cavity including
ring heater effects (ITMROC = 1939.17m, ETMROC = 2239.65m), with ideal
quadratic finite diameter mirrors (blue) and with actual mirror surface
deformation included (red). A highly mismatched beam (composed of many cavity
HOM) was injected into the model to better highlight the location of the
different higher order modes: 6th order (dot-dash), 7th order (dotted).
Bottom: the deviation of the resonance location of the higher order modes
[expressed as a fraction of a free-spectral range (FSR)].

This residual surface error effect shifts the resonance location of modes from
the nominal locations (calculated solely from the Gouy phase). This is
illustrated in Figure 15. The upper panel shows simulated cavity scans of a
poorly matched beam into the LIGO-Hanford $4\text{\,}\mathrm{km}$ Fabry-Perot
X-arm cavity. The simulated beam is poorly matched in order to excite higher
order modes in the scan. The blue and red curves show results from purely
quadratic mirrors and with included surface figure errors, respectively. The
dot-dash and dotted lines show the 6th and 7th modes, respectively.

Several features of this figure are notable. Firstly, the full-width half-
maximum linewidths of the fitted 6th and 7th order modes are 1.75% and 6.45%
of an FSR, respectively. These are considerably wider than the linewidth of
the fundamental mode (0.24% of an FSR), due to the increased round-trip loss,
$\mathrm{L}_{RT_{m+n}}$, that HOMs experience from the finite mirror apertures
(illustrated in Figure 13).

Secondly, the actual resonance locations of the HOM have shifted relative to
the case of ideal quadratic mirrors by amounts
$\Delta\phi\left(E_{i},S_{\mathrm{TM}}\right)$ In the case of the 7th order
mode, the resonance has shifted past the fundamental mode resonance through to
the other side. Note that in the ideal quadratic mirror case (blue), there is
still an anomalous phase shift: the 7th order resonance has shifted
approximately 2% of an FSR relative to the resonance calculated from the
nominal Gouy phase (dashed purple). This is addressed further when we discuss
the limitations of the resonant analysis detail in Equations 32 through 39.
The bottom panel shows the difference in the resonance locations of the higher
order modes for the two cases.

The resonant enhancement/suppression (HOM gain factor $g_{mn}$) is shown for
the aLIGO arm cavity in Figure 16. It is notable that simply adding the real
mirror surfaces error, a cross-section of which is illustrated in Figure 14,
amplifies the loss by a factor of 3 for the 7-th order mode. However, this
effect is also dependent on the ring heater settings (more precisely, the
mirror ROC), as illustrated by the bottom panel of Figure 16 in which the
addition of the actual mirror surface reduces the overall gain of the 7-th
order mode when the ring heaters are not employed.

Figure 16: Higher order mode (HOM) gain factor for the LHOX aLIGO cavity
modeled for quadratic finite mirrors (i.e. both ITM and ETM are purely
quadratic) and including the actual mirror surfaces (both ITM and ETM surface
polishing maps are included). Top plot: with the ring heaters at O3 settings,
mirror radii of curvature, ITM = $1939.17\text{\,}\mathrm{m}$, ETM =
$2239.65\text{\,}\mathrm{m}$. Bottom plot: ring heaters off and mirrors at
manufactured ROC, ITM = $1940.3\text{\,}\mathrm{m}$, ETM =
$2244.2\text{\,}\mathrm{m}$.

Thus, having determined the total round-trip loss from the Vajente formalism
expressed in Equations 32 and 33 and accounting HOM round-trip phase
perturbations due to residual surface errors in the mirrors, expressed in
Equation 38, one can approximate the impact on the power recycling gain.

One final note on the applicability of this resonant analysis. There are
several implicit assumptions in the Vajente formalism that limit its
predictive ability. Specifically, we assume:

  * •

power is scattered between the fundamental and individual higher order modes
only, and not between different higher order modes. This assumption becomes
less valid for higher order modes with significant clipping at the edge of the
mirror. This manifests in Figure 15, the location of the 7th order mode for
the simulation of ideal quadratic mirrors (blue) shows an anomalous phase
shift to the left relative to the resonant location of the 7th order mode
calculated from the Gouy phase (purple). This is independent of the surface
figure of the mirrors. This anomalous phase shift is due to power from the 7th
order mode scattering into HOM (due to the aperture of the mirror),
propagating through a round-trip of the cavity, where it accumulates
additional Gouy phase, and then scattering back into the 7th order mode after
one (or more) round trips. The resonant analysis presented in this paper does
not account for this mode-mixing: that is, in a full mode-mixing analysis,
there will be an additional phase term in Equation 39 that accounts for this.

  * •

that the loss is the same for modes of the same order - as expressed in Figure
13. This is appropriate on average but the exact losses for different HOM will
depend on the overlap between the mirror shape and the individual mode shape.

  * •

that the HOM phase shift is the same for all HOM of the same order. In
reality, there will be small variations between modes. This will result in
mode-splitting, but the phase variations between different modes will be small
relative to the broad linewidths. This will manifest as a broadening of a
resonant peak of modes of a specific order.

Therefore, the formalism presented here is valid for rapid estimation of the
effects on power-recycling gain. The relative shift in resonance location due
to the surface figure can be computed using this formalism. However, the
absolute resonance locations of higher order modes should be treated carefully
as mode-mixing induced phase shifts are not accounted for in this formalism.
Specifically, the mode-mixing induced phase shift would need to be added to
Equation 39 for absolute calculations of gain. For more precise calculations,
more comprehensive numerical/analytic simulations are recommended. Suggested
examples are Finesse FinesseRef  code, a multi-modal analysis of optical
cavities and SIS (Static Interferometer Simulation), an FFT-based numerical
analysis of optical cavities SIS-T070039 .

It is worth emphasizing that the results in Figure 16 underscore that precise
ROC and mirror surface errors must be considered when determining the resonant
gain of HOM in the cavity. Indeed, numerical calculations by SIS show that the
O3 ETM surface maps suppress the gain of the 7th mode relative to the
spherical shape, and will suppress the round trip loss. This is used to design
the optimal surface shape of the O4 ETM maps to mitigate the effect of point
absorbers.

###  D.3 Power recycling gain and generalized arm loss.

Thus, to summarize, the full arm loss, $\mathcal{L}_{A}$, is dependent on the
elements described in Appendices B, C, D. Bringing all these concepts together
allows us to better estimate the power recycling gain, our proxy for overall
interferometer performance at high frequencies, in the presence of losses due
to point absorbers.

As expressed in Equation 32, the overall loss is a function of the single-
bounce amplitude scattering into HOM and the resonant enhancement or
suppression of those modes, the elements of which are summarized below:

  1. 1.

Single-bounce amplitude scattering depends upon:

    1. (a)

Total absorbed power: $P_{\mathrm{abs}}$, (Equation 20). For a LIGO arm cavity
this is, in turn, a product of the arm power, $P_{A}$, the incident laser beam
size, $\mathrm{w}$, the location of the point absorber, $\mathbf{r_{c}}$, and
the equivalent “100% absorbing area” of the point absorber, (Equation 13).

    2. (b)

Material properties: thermal conductivity, $\kappa$, and the coefficient of
thermal expansion, $\alpha$, (Equation 20).

    3. (c)

Location of the point absorber: $\mathbf{r_{c}}$, from the context of its
position relative to HOM field distribution, (Equation 31).

  2. 2.

Resonant enhancement or suppression depends upon:

    1. (a)

Clipping losses in HOM, $r_{i}^{\prime}$, a function of mode size and mirror
size (Equation 34).

    2. (b)

Nominal round-trip Gouy phase: $\phi_{g}$, a function of cavity geometry
(Equation 36)

    3. (c)

Additional round-trip phase perturbations from mirror surface errors:
$\Delta\phi_{mn}$ (Equation 38).

Which, when combined, averaged across the two interferometer arms and
accounting for nominal power loss, the total arm loss can be expressed as:

$\displaystyle\mathcal{L}_{\mathrm{A}}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\mathrm{nom}}+\frac{1}{2}\sum_{m,n}\mathcal{L}_{mn,X}+\mathcal{L}_{mn,Y}$
(40) $\displaystyle=$
$\displaystyle\mathcal{L}_{\mathrm{nom}}+b\,\left(\frac{P_{A}}{100\mathrm{kW}}\right)^{2}$
(41)

where $\mathcal{L}_{\mathrm{nom}}$ is the nominal arm loss and $b$ is the
coefficient for power-dependent loss encompassing the elements described
above. As shown in Equation 30, all the amplitude scattering coefficients
contain a common factor of absorbed power, $P_{\mathrm{abs}}$, which is, in
turn dependent on arm power, $P_{A}$. Thus, we can finally write the power-
recycling gain as a function of arm power:

$G_{P}=\left(\frac{t_{p}}{1-r_{p}\,\left(1-G_{A}\,\left[\mathcal{L}_{\mathrm{nom}}+b\,\left(\frac{P_{A}}{100\mathrm{kW}}\right)^{2}\right]\right)}\right)^{2}$
(42)

It is the $b$ term that must be minimized to maximize the power-recycling
gain.

One final comment regarding extensions to this analysis. This discussion
focused on a single absorber on a single optic. It is easily extended to
multiple point absorbers since, as shown in Equation 29, amplitude scattering
to HOM is linear with respect to surface deformations that are small compared
to the wavelength of light. Thus the effect of multiple point absorbers can be
determined by taking a linear sum of amplitude scattering coefficients from
individual points.

##  Appendix E Substrate thermal lens, noise couplings and impacts on control
systems

Power dependent losses from surface deformation are the main observed effect
of thermal effects point absorbers. As an example of secondary adverse effects
due to point absorbers (and ones which originate from substrate thermal
lenses), this appendix describes why we expect increased coupling of intensity
noise to the gravitational wave channel via the feedback control system.

###  E.1 Substrate thermal lens effect of carrier and sidebands

Due to the complexity of the Advanced LIGO length sensing and control system
aLIGO_Martynov_2016 ; aLIGO_ASC_Barsotti_2010 ; Staley:15 ; Mullavey:12 , we
limit discussion of the interaction of that system with point absorbers to the
case study of increased relative intensity noise coupling.

As described in Appendix A.3, a feedback control system, using frontal
modulation RF sidebands and the Pound-Drever-Hall (PDH) technique, keeps the
interferometer operating at the correct working point to ensures high
sensitivity to gravitational waves.

In the ideal scenario, only the carrier fundamental mode, $E{}^{C}_{00}$, and
the sideband fundamental mode, $E{}^{SB}_{00}$, beat against each other on
detection photodiodes to create PDH control signals. The diodes sees one large
PDH signal centered around the correct locking point.

Any optical path distortion (OPD) in the input test mass substrate affects
both the carrier and sideband fields in the single propagation through the
optics. If we assume that the field at the AR face of the ITM has a TEM00
Gaussian mode distribution, matched to the cavity resonant mode, then at the
lowest order the effect of the ITM distortion is to couple a fraction,
$\epsilon$, of this fundamental mode into a combination of higher order modes:

$E{}_{ITM}=\sqrt{1-|\epsilon|^{2}}E{}_{00}+\epsilon\,E{}_{HOM}$ (43)

This equation is valid for carrier and sideband fields equally. When
considering the reflection off the Fabry-Perot arm cavity, we have to
distinguish between the carrier (TEM00 mode is resonant and HOM are
antiresonant) and the sidebands (both TEM00 and HOM modes are antiresonant). A
field resonant in the arm cavity is reflected with a phase shift of $\pi$
radians, while a field antiresonant in the arm cavity is reflected with zero
phase shift. Therefore the fields coming back from the cavities, but before
passing through the ITM substrate, are

$\displaystyle E{}_{R}^{C}$ $\displaystyle=$
$\displaystyle-\sqrt{1-|\epsilon|^{2}}\,E{}_{00}^{C}+\epsilon\,E{}_{HOM}^{C}$
(44) $\displaystyle E{}_{R}^{SB}$ $\displaystyle=$
$\displaystyle\sqrt{1-|\epsilon|^{2}}\,E{}_{00}^{SB}+\epsilon\,E{}_{HOM}^{SB}$
(45)

The transmission through the ITM distorted substrate couples the same amount,
$\epsilon$, of TEM00 into the same high order mode as for the first pass. But
since the carrier reflected back has the opposite sign, at first order in the
OPD the carrier HOM field cancel outs. This is not the case for the sideband
fields.

Therefore the carrier is largely immune to the effect of the optical path
distortion caused by point absorbers while the sidebands are fully affected.

###  E.2 Impact on control system and elevated intensity noise coupling

Now, consider the input intensity noise carried on the laser fields. Only the
carrier fundamental mode, $E{}^{C}_{00}$, is resonant in the arm cavity and,
as such, is the only field to experience the intensity noise filtering effect
of that cavity (a pole around $0.6\text{\,}\mathrm{Hz}$ Martynov_noise:15 ).
The remaining fields, $E{}^{C}_{\mathrm{HOM}}$, $E{}^{SB}_{00}$ and
$E{}^{SB}_{\mathrm{HOM}}$, experience no optical suppression of intensity
noise.

The carrier HOM and fundamental fields do not directly beat against each other
and, hence, do not contribute to RIN coupling. However, as shown in Figure 1,
the output of the interferometer passes through an output mode cleaner (OMC).
Any mode-mismatch between the interferometer beam and the OMC AWC2020  will
allow some carrier HOM to mix with the fundamental and present a path for
intensity noise to couple to the detection photodiodes. This is one possible
way an OPD can increase the coupling of intensity noise (active wavefront
control are being commissioned to minimize this mode-mismatch Cao:20 ).

Additionally, indirect coupling of laser intensity noise is created due to the
interaction of carrier and sideband HOM with the feedback control systems.
These two fields will beat against each other creating additional RF PDH-like
signal on the control photodiodes. Due to Gouy phase differences, these are
offset from nominal, or zero, locking point.

A offset in the short Michelson control signal means that the IFO is operating
such that the two short Michelson arms are not ”equal” to get perfect
destructive interference. However, the differential arm (DARM) loop is locked
to keep the output power to be a constant value Fricke_2012 . When coupled
with the Michelson offset, the DARM loop is driven offset from the ”proper”
working point, so to produce another static carrier field imbalance that
cancel out the field created by the MICH offset. Although the static power
cancels out, the RIN on the two carrier fields do not cancel out, since one is
noisy (coming from PRC/MICH) and the other is clean (coming from arms). The
end result is an increase in intensity noise coupling through the dark port,
still mediated by TEM00 modes, and so not ”fixed” by the OMC.

##  Appendix F Table of parameters

Parameter | Symbol | Value
---|---|---
ETM transmission (nom.) |  $T_{e}$ ($t_{e}^{2}$) | 5 ppm
ITM transmission (nom.) |  $T_{i}$ ($t_{i}^{2}$) | 1.4%
PRM transmission (nom.) |  $T_{p}$ ($t_{p}^{2}$) | 3.0%
Arm optical gain (nom.) | $G_{A}$ |  $\approx$ 280
Arm optical gain (LHO) | $G^{H}_{A}$ | 268 O3CommissioningPRD
Arm optical gain (LLO) | $G^{L}_{A}$ | 265 O3CommissioningPRD
ITM TEM00 beam rad. | $\mathrm{w}_{i}$ | $53\text{\,}\mathrm{mm}$
ETM TEM00 beam rad. | $\mathrm{w}_{e}$ | $62\text{\,}\mathrm{mm}$
Test mass (TM) radius | $\Omega$ | $170\text{\,}\mathrm{mm}$
TM thickness | $h$ | $200\text{\,}\mathrm{mm}$
Thermal expansion coeff. | $\alpha$ | $0.55\text{\times}{10}^{-6}\text{\,}{\mathrm{K}}^{-1}$
Thermal conductivity | $\kappa$ | $1.38\text{\,}\mathrm{W}\text{\,}{\mathrm{m}}^{-1}\text{\,}{\mathrm{K}}^{-1}$
TM specific heat capacity | $c$ | $703\text{\,}\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}\text{\,}{\mathrm{K}}^{-1}$
TM density | $\rho$ | $2203\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$
Laser wavelength | $\lambda$ | $1064\text{\,}\mathrm{nm}$
Arm length | $L$ | $3995\text{\,}\mathrm{m}$
ITM ROC (nom.) | $\mathcal{R}_{ITM}$ | $1939\text{\,}\mathrm{m}$
ETM ROC (nom.) | $\mathcal{R}_{ETM}$ | $2245\text{\,}\mathrm{m}$
ITM ROC (LHOX wo. RH) | $\mathcal{R}^{H*}_{ITM}$ | $1940.3\text{\,}\mathrm{m}$
ETM ROC (LHOX wo. RH) | $\mathcal{R}^{H*}_{ETM}$ | $2244.2\text{\,}\mathrm{m}$
ITM ROC (LHOX w. RH) | $\mathcal{R}^{H}_{ITM}$ | $1939.17\text{\,}\mathrm{m}$
ETM ROC (LHOX w. RH) | $\mathcal{R}^{H}_{ETM}$ | $2239.65\text{\,}\mathrm{m}$
Table 1: Relevant parameters for aLIGO. The specific ROC for LHOX account
include the change due to the application of the TCS ring heater (RH).

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