# Optimizing Gravitational-Wave Detector Design for Squeezed Light

Jonathan W. Richardson<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of California, Riverside, Riverside, CA 92521, USA
Swadha Pandey  Department of Physics, Indian Institute of Technology Kanpur,
Kanpur, UP 208016, India  Edita Bytyqi  Department of Applied Physics and
Applied Mathematics, Columbia University, New York, NY 10027, USA  Tega Edo
LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
Rana X. Adhikari  LIGO Laboratory, California Institute of Technology,
Pasadena, CA 91125, USA

###### Abstract

Achieving the quantum noise targets of third-generation detectors will require
10 dB of squeezed-light enhancement as well as megawatt laser power in the
interferometer arms—both of which require unprecedented control of the
internal optical losses. In this work, we present a novel optimization
approach to gravitational-wave detector design aimed at maximizing the
robustness to common, yet unavoidable, optical fabrication and installation
errors, which have caused significant loss in Advanced LIGO. As a proof of
concept, we employ these techniques to perform a two-part optimization of the
LIGO A+ design. First, we optimize the arm cavities for reduced scattering
loss in the presence of point absorbers, as currently limit the operating
power of Advanced LIGO. Then, we optimize the signal recycling cavity for
maximum squeezing performance, accounting for realistic errors in the
positions and radii of curvature of the optics. Our findings suggest that
these techniques can be leveraged to achieve substantially greater quantum
noise performance in current and future gravitational-wave detectors.

##  I Introduction

In the last six years, Advanced LIGO and Virgo have established gravitational
waves as a new observational probe of the Universe. With projected
improvements in gravitational-wave detector sensitivities, new tests of
gravity, cosmology, and dense nuclear matter will become possible within the
next decade. Higher sensitivity in the 200 Hz–1 kHz band will resolve the
ringdown radiation of newly coalesced black holes, detecting or constraining
potential quantum modifications at the event horizon [1, 2, 3]. Higher
sensitivity in the 1.5–5 kHz band will resolve binary neutron star mergers to
the moment of coalescence, illuminating the neutron star equation of state [4,
5]. More frequent detections of binary neutron star mergers will also enable
independent measurement of the Hubble constant to high precision [6],
addressing the growing tension between cosmic microwave background and local
distance ladder measurements [7].

The LIGO detectors are sensitive to gravitational waves in a broad frequency
band ranging from 20 Hz to 5 kHz. Across this band, the limiting source of
instrumental noise transitions from sensing and controls noise, below roughly
30 Hz, to Brownian noise of the dielectric optical coatings, up to roughly 200
Hz, and finally to quantum noise at higher frequencies (see [8] for a thorough
review of the LIGO noise budget). In laser interferometers, quantum noise
arises not from the positional uncertainties of the mirrors, but from the
quantization of the electromagnetic field used to interrogate their positions
[9, 10]. This effect, commonly described as “shot noise,” arises from ground-
state fluctuations of the vacuum field, which enter the interferometer and
beat with the circulating laser field. The interference of the two fields
produces intensity fluctuations which modulate the interferometer output
signal. These fluctuations also apply force to the mirrors via radiation
pressure, producing actual mirror displacements at low frequencies. Shot noise
can be reduced through two means: higher laser power in the interferometer,
which increases number of photons incident on the beamsplitter, and the
injection of squeezed quantum states of light. Both are critical to improving
the high-frequency sensitivity of gravitational-wave detectors.

In the third observing run, the Advanced LIGO detectors operated with roughly
250 kW of resonating power inside the arm cavities [8]—still only one third of
their 750 kW design power. Recent tests in both detectors have shown that as
the injected laser power is increased, the arm cavity optical gain severely
decays due to increasing internal loss [8]. The source of this loss has been
identified as sub-millimeter, highly absorbing defects in the optical coatings
known as _point absorbers_. In situ wavefront sensors have detected their
presence on at least four of the eight currently installed test masses [8,
11]. Point absorbers appear to originate during the coating deposition
process, although it is still not understood how these contaminants enter the
coating nor to what extent they can be eliminated. Each point absorber absorbs
roughly 80 ppb of the total incident power, or 20 mW when exposed to 250 kW.
The extremely localized heating induces a sharply peaked thermoelastic
deformation of the mirror surface, which scatters power into higher-order
spatial modes [11]. To achieve higher operating power, point absorber losses
must be mitigated.

Beginning also in the third observing run, squeezed light was injected into
both Advanced LIGO detectors [12]. Squeezed light allows for the engineering
of the electromagnetic vacuum state that enters the interferometer. Quantum
fluctuations of the vacuum field, initially distributed uniformly between the
amplitude and phase quadratures, are redistributed so that they are suppressed
in the phase quadrature, containing the gravitational-wave signal, and
amplified in the unsensed amplitude quadrature. In Advanced LIGO, squeezed
vacuum field is generated via degenerate optical parametric amplification [13]
and injected into the interferometer output port. During the third observing
run, a shot noise reduction factor of roughly 3 dB was achieved in each
detector [12].

Although the injected level of squeezing can be high, the observed level of
squeezing at the interferometer output depends on the amount of entanglement
remaining in the squeezed field. Losses within the detector lead directly to
decoherence of the squeezed state, limiting the quantum noise reduction [14].
Losses arise from scattering, imperfect transmissivity or reflectivity of
optics, photodetector quantum efficiency, and spatial mode-mismatch between
the optical cavities. For the interferometer, the largest source of loss is
mode-mismatch between the coupled laser cavities. For example, in the Advanced
LIGO detectors, the mode-matching loss between the arm cavities and the output
mode cleaner cavity alone is measured to be 10%. It has been demonstrated that
this loss can be attributed to practical, and largely irreducible, limitations
in the fabrication and hand-positioning of the interferometer optics [15]. For
third-generation detectors [16, 17], reducing the internal mode-matching
losses to $\sim 1\%$ levels is imperative.

In this work, we present a novel optimization approach to gravitational-wave
detector design. It is aimed at maximizing the robustness to common optical
fabrication and installation errors, which introduce losses that degrade the
optical gain and squeezing performance. Under this approach, design
performance is assessed and improved statistically, over thousands of trials
in which realistic random errors are assumed in the surface figures and
positions of the optics. As a proof of concept, we employ these techniques to
perform a two-part optimization of the LIGO A+ interferometers, planned to
become operational in 2025 [18]. First, in §II we modify the arm cavities for
reduced scattering loss in the presence of point absorbers. Then, in §III we
optimize the signal recycling cavity (SRC) for maximum squeezing performance,
accounting for realistic errors in the positions and radii of curvature of the
optics. Our findings suggest that these techniques can be leveraged to achieve
substantially greater quantum noise performance in current and future
gravitational-wave detectors. Finally, in §IV we summarize and discuss future
extensions of this work.

##  II Arm cavity design

(a)  (b)

Figure 1: Proposed surface profiles for the LIGO A+ input test masses (ITM;
left) and end test masses (ETM; right). In each panel, the total surface
figure (pink curve) is the sum of the polishing profile (blue curve) and the
optical coating nonuniformity (grey curve). Based on current fabrication
capabilities, the polishing slope is restricted to $\leq 2.5$ nm/mm, as shown
in the lower panels.

In the Advanced LIGO arm cavities, point absorbers on the mirror surfaces
disproportionately scatter power into 7th-order spatial modes. Although a
point-absorber-induced deformation scatters power into many higher-order modes
(HOM), the Fabry–Perot cavity resonantly enhances or suppresses each mode as a
function of the roundtrip phase it accumulates in the cavity. This effect was
first analyzed for static deformations by Vajente [19] and extended to power-
dependent surface deformations from point absorbers by Brooks et al. [11], who
showed that the power loss from the fundamental mode to the $mn$-th HOM is
approximately

$\mathcal{L}_{mn}=a_{00|mn}^{2}\,g_{mn}\;.$ (1)

The first term, $a_{00|mn}$, is the single-bounce amplitude scattering from
the fundamental mode to the $mn$-th HOM when reflected off the deformed mirror
surface. The second term, $g_{mn}$, is the optical gain of that HOM, which
depends on the cavity geometry and the actual (nonideal) surface profiles of
the two mirrors:

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

The factors $r_{1}^{\prime}$ and $r_{2}^{\prime}$ are the effective amplitude
reflectivities of the input test mass (ITM) and the end test mass (ETM),
respectively, accounting for mode-dependent clipping losses, and $\Phi_{mn}$
is the additional roundtrip phase that the HOM accumulates relative to the
fundamental mode. In the LIGO arm cavities, modes of order 7, by coincidence,
are nearly co-resonant with the fundamental mode, leading to optical gain
factors $g_{mn}$ up to 100 times larger than those for non-resonant modes.

Thus, to reduce point absorber losses and achieve higher operational power in
LIGO A+, our design objective is to fully eliminate mode co-resonances below
order 8 in the arm cavities. In principle, this could be achieved by adjusting
the arm cavity parameters (the arm length and the radii of curvature of the
test masses) for a more favorable transverse mode spacing. However, a
significant change of the cavity parameters is precluded by other operational
constraints. The 4 km arm length is constrained by the existing infrastructure
to approximately $\pm 2$ m of the current length. With only the radii of
curvature of the two mirrors free to vary, it is not possible to maintain the
current beam sizes on both optics. A smaller beam size results in increased
coating Brownian noise—unacceptable for the A+ design, which is already
thermal-noise-limited across its mid-frequency band [18]. A larger beam size,
on the other hand, results in unacceptably higher clipping losses inside the
arms and the signal recycling cavity. Thus, the problem is overconstrained
from the perspective of a standard cavity design approach using spherical
optics.

In this section, we demonstrate that by applying novel, nonspherical surface
profiles to the LIGO test masses, the mode 7 co-resonances can be eliminated
_without_ incurring any increase in coating thermal noise or clipping loss.
Our approach exploits the large difference in transverse spatial confinement
between the fundamental mode and 7th-order modes. Each mirror profile is
spherical in the central region, where fundamental mode power is concentrated,
but assumes a sharply nonspherical shape at the outermost radii, where the
incident power is almost purely in higher-order modes. We show that the outer
surface profile can be tailored to control the roundtrip phase $\Phi_{mn}$
(see Eq. 2) that an HOM accumulates relative to the fundamental mode. This
provides a means to suppress problematic higher-order modes while negligibly
altering the fundamental cavity mode, leading to a significant loss reduction
in the presence of scattering sources such as point absorbers.

###  II.1 Nonspherical test mass profiles

For mirror fabrication in the A+ era, LIGO has the ability to specify an
arbitrary (nonspherical) polishing figure. Internal discussions with optics
manufacturers have indicated that an arbitrary radial profile, subject to a
maximum slope of 2.5 nm/mm, could be produced with high confidence, with a
possibility that an even steeper polishing slope could be achieved. Fig. 1
shows our proposed surface profiles for the LIGO input test masses (ITM) and
end test masses (ETM). In each panel, the polishing figure (blue curve) both
compensates the expected nonuniformity of the optical coating (grey curve) and
adds a nonspherical edge component to produce the total surface figure (pink
curve). To remain within demonstrated fabrication limits, we restrict the
polishing slope to 2.5 nm/mm, as shown in the lower panels of Fig. 1. In A+, a
new coating material with improved thermal noise performance, $\rm
TiO_{2}$-doped $\rm GeO_{2}$ [20], is expected to replace the $\rm
TiO_{2}$-doped $\rm Ta_{2}O_{5}$ coatings used in Advanced LIGO [21, 22].
Accordingly, we estimate the A+ coating nonuniformity as the measured
nonuniformity of the LIGO O4 coating plume, which will be reused to produce
the A+ optics, multiplied by the relative coating thickness required to
achieve the same reflectivity with the new material (1.2 for the ITM and 1.5
for the ETM).

Figure 2: Optical gain $g_{mn}$ of the 7th-order Laguerre-Gauss (LG) modes in
the LIGO A+ arm cavities, as a function of frequency detuning from the
fundamental mode resonance. _Top:_ The nominal resonance locations with a
spherical test mass polish. _Bottom:_ The new resonance locations after
including the compensation polish shown in Fig. 1 (blue curves). In both
panels, coating absorption of 120 mW per test mass is assumed.

Fig. 2 illustrates how these profiles eliminate the arm cavity modal
degeneracy. First, for comparison, the top panel shows the nominal locations
of the 7th-order Laguerre-Gauss (LG) mode resonances, assuming a spherical
mirror polish. With higher cavity power (or coating absorptivity), the
resonances shift toward higher frequency due to the increasing residual
thermal deformation of the test masses. Although ring heaters compensate the
central heating due to uniform coating absorption, the ring heaters
“overcorrect” the mirror surface at large radii, resulting in a net profile
that steeply rises near the edge of the optic [23]. Here, we assume 120 mW of
coating absorption per test mass, corresponding to a cavity power of 400 kW
for absorptivity at the level of the Advanced LIGO coatings. The bottom panel
shows the new locations of the 7th-order mode resonances after including the
compensation polish shown in Fig. 1 (blue curves). The polishing profiles are
designed to shift the 7th-order mode resonances rightward, toward higher
frequency, where any degree of thermal distortion now strictly shifts them
_further away_ from co-resonance. This has significant implications for the
loss performance of the arm cavities, as discussed in the following section.

###  II.2 Loss performance improvement

We now assess the impact of our proposed compensation polish on the arm cavity
loss. For this, we consider two scenarios, with (“proposed”) and without
(“nominal”) our proposed modification. The “nominal” test mass profiles (with
spherical power subtracted) are equal to the grey curves in Fig. 1. The
“proposed” test mass profiles (again with spherical power subtracted) are
equal to the pink curves in Fig. 1. For each set of profiles, we perform
numerical simulations of an A+ arm cavity using SIS [24], an FFT-based optical
simulation package. The model includes all thermoelastic effects: (1) uniform
coating absorption and (2) optimal ring heater compensation, to maintain
constant mode-matching of the arm to the recycling cavities. Throughout, we
assume coating absorption at the average level of the Advanced LIGO test
masses, 0.3 ppm. We also assume a fixed high-angle scattering loss of 25 ppm
per optic. To account for realistic nonidealities, which could unequally
impact the two designs, all loss analyses are performed as Monte Carlo
simulations over 1000 trials with random beam miscenterings and surface
roughnesses. Beam miscenterings on the ITM and ETM are independently drawn
from a Gaussian distribution with zero mean and a standard deviation of 5 mm.
Surface roughness profiles are randomly generated by SIS with a power spectral
density chosen to match that of the current Advanced LIGO optics [25].

Figure 3: Arm cavity loss as a function of mode order. Each curve represents
the median loss in 1000 trials with random miscenterings and surface
roughnesses, averaged over all Laguerre-Gauss modes ${\rm LG}_{p,l}$ of order
$N=2p+|l|$. The shading represents the 16th and 84th percentiles of the loss
distributions over all trials and modes. The proposed mirror profiles achieve
significantly enhanced higher-order mode dissipation, with no increase in
fundamental mode loss.

First, we evaluate the baseline loss performance of both designs in the
absence of scattering sources. The aim of our compensated design is to achieve
the HOM frequency shifts outlined in §II.1 without worsening the roundtrip
loss of the fundamental mode. Fig. 3 shows the roundtrip arm loss under each
design as a function of mode order, at an arm power of 750 kW. The curves
represent the median loss values over all randomized trials, and averaged over
all modes ${\rm LG}_{p,l}$ of order $N=2p+|l|$. The shading represents the
16th and 84th percentiles of the loss distributions across all trials and
modes. Our results indicate that the proposed test mass profiles do not
increase loss in the fundamental mode, but they do significantly increase the
losses of HOMs above order 2. While the design objective in §II.1 was only to
shift resonance frequencies of HOMs, the larger dissipation of HOMs is an
added advantage that helps to further reduce their optical gain. Enhancing the
dissipation of certain HOMs may be also relevant for improving the damping of
parametric instabilities in gravitational-wave detectors [26]. For this
reason, we include a full breakdown of the dissipation per optical mode in
Appendix A.

Figure 4: Arm cavity loss distributions due to point absorbers, shown at three
different power levels. Each loss distribution represents 1000 trials with a
point absorber randomly positioned on each test mass. Uniform coating
absorption of 0.3 ppm per test mass, along with optimal ring heater
compensation, is assumed.

Next, we add random point absorbers to the Monte Carlo simulation and
reevaluate the loss performance of both designs. One point absorber is applied
to each test mass, randomly positioned in the central 150 mm diameter. The
radial and angular coordinates are drawn from uniform distributions, with the
radial distribution truncated at 75 mm and the angular distribution spanning
the full $360^{\circ}$. Point absorber phase maps are generated using the
analytic formalism for thermoelastic surface deformation from Brooks et al.
[11]. We assume a fixed absorptivity chosen so that, at a cavity power of 250
kW, a perfectly centered point absorber absorbs 20 mW of incident power. Fig.
4 shows the roundtrip loss distributions for the fundamental cavity mode under
each design, at three different arm power levels. We find the proposed
profiles statistically outperform the nominal profiles in all cases.

##  III Signal recycling cavity design

The single largest source of loss in the Advanced LIGO interferometers is
spatial mode-mismatch between optical cavities. Mode-mismatch arises from
unintended deviations of the as-built optical system from design. The two
folding mirrors of the signal recycling cavity (SRC) are known to be
especially sensitive to fabrication and installation errors. Even small
perturbations in the curvatures and positions of these optics can result in a
significant mode-mismatch with the arm cavities.

The impact of mode-mismatch internal to the interferometer on the observed
squeezing is difficult to model analytically. In LIGO, the fundamental optical
mode is squeezed in the modal basis defined by a parametric amplifier cavity,
which serves as the squeezing source. The cavities of the interferometer each
define their own basis of optical modes. As the squeezed state propagates
through the interferometer, it is transformed from the modal basis of the
squeezing source into the basis of each respective cavity. If the spatial
modes of the cavities are imperfectly matched, these basis transformations
must mix the optical modes. Since only the fundamental mode in the source
basis is squeezed, the higher-order modes carry standard vacuum. Thus, basis
mixing from mode-mismatch leads to loss. However, unlike dissipative losses,
each modal mixing is coherent and unitary—leading to complex interference
effects which can potentially increase squeezing losses. To date, the most
detailed analytical treatment of the coherent interactions of transverse modal
mixing on squeezed states is given by McCuller et al. [27].

In the present work, we use a numerical simulation to model the squeezing
degradation from internal mode-mismatches. The aim of this analysis is to
identify an SRC design in LIGO A+ that is maximally robust to common errors
which induce mode-mismatch. We identify the maximally error-tolerant design
through a numerical optimization procedure described in §III.1, which employs
an evolutionary search algorithm in a parameter space spanning all possible
SRC designs. The results of this optimization and a quantitative analysis of
the design performance are described in §III.2.

###  III.1 Optimization procedure

Our objective is to find the optimal values of the radii of curvature and
positions of the SRC mirrors, such that deviations from these nominal values
minimally degrade the squeezing level observed at the interferometer output.
To perform this search, we employ a global-best particle swarm optimization
(PSO) algorithm provided by the Pyswarms optimization toolkit [28]. PSO is an
evolutionary search algorithm designed to efficiently explore high-dimensional
parameter spaces. Initially, many “particles,” each, in our case, representing
a candidate optical design, are scattered around the parameter space. Each
particle has an associated velocity which determines its position in the
parameter space at the following iteration. Its velocity is determined by its
best known local position as well as the best positions discovered by other
particles. In this way, the entire swarm is iteratively guided toward the
global optimum.

The relative “goodness” of positions within the parameter space is quantified
by a cost function whose value the algorithm seeks to minimize. Primarily, our
cost function is designed to penalize SRC designs in which the observed
squeezing level is strongly sensitive to perturbations of the SRC parameters.
To evaluate the cost function at each particle’s position, at each iteration,
a Finesse optical simulation [29] is used to compute the partial derivatives
of the observed squeezing level with respect to small detunings of each
candidate SRC parameter. We detail the Finesse simulation in §III.1.1. Then,
in §III.1.2 we describe the set of optical parameters which we optimize, as
well as relevant parameter constraints. Finally, in §III.1.3 we detail the
construction of the cost function used to define the optimization objective.

####  III.1.1 Optical simulation

Figure 5: Optical configuration used for simulating a LIGO-like
interferometer. All of the distances and radii of curvature are fixed to the
nominal LIGO A+ design values, except for the signal recycling cavity
parameters which are indicated in blue.

In our optimization routine, the core compute engine is a Finesse simulation
[29] used to analyze the performance of a given SRC design. Finesse is a
modal-based optical simulation package widely used for modeling laser
cavities, whose modern user interface is provided by the Pykat [30] package.
To illustrate the optimization procedure, we adopt a toy interferometer model
based on the LIGO A+ design. Its optical layout is shown in Fig. 5. However,
for the purpose of this illustration, several simplifying departures from the
A+ design are made to reduce the computational cost and complexity:

  * •

Frequency-independent squeezing. Although A+ will use frequency-dependent
squeezing [18], for simplicity we assume a frequency-independent squeezing
angle. In principle, our routine can be extended to the frequency-dependent
case by jointly optimizing the error tolerance at multiple frequencies.

  * •

DC readout. For signal detection, a bright carrier field must present at the
interferometer output port. In Advanced LIGO, this is generated by offsetting
the differential arm length $\sim 1$ pm from a dark fringe. In A+, this
technique, known as “DC readout,” will be replaced by balanced homodye readout
[31]. As balanced homodyne readout adds considerable complexity, we use DC
readout in this simulation.

  * •

Output mode-matching. Inclusion of an output mode cleaner (OMC) adds
significant computational cost because, as SRC parameters are detuned, at
least two adaptive optics between the SRC and OMC must be continually re-
optimized to maintain the mode-matching of the OMC to the arm cavities.
Although such a re-tuning of the output mode-matching been previously
demonstrated [15], for the present simulation we omit the OMC and instead
assume a fixed output loss ranging from $5-20\%$.

The Finesse simulation starts from a “nominal” model (using a provided set of
SRC parameters), then individually detunes each SRC parameter from its design
value and computes the change in observed squeezing. The parameter detunings
introduce a spatial mode-mismatch between the SRC and the arm cavities.
Higher-order modes up to order 4 are tracked, which is sufficient given the
small size of the parameter detunings. Coincidentally, the mode-mismatch
shifts interferometer length degrees of freedom away from their nominal
operating points, as well as rotates the squeezing quadrature away from the
interferometer readout quadrature. In a real detector, these offsets are
zeroed by a combination of control servos and manual optimizations. Thus, it
is necessary to implement servos within the Finesse simulation to zero all
such “artifical” detunings.

To prevent length detunings, we incorporate DC servos for all five length
degrees of freedom: the common arm length, differential arm length, power
recycling cavity length, Michelson length, and signal recycling cavity length
[see, e.g., 32]. Linear error signals are constructed by injecting 9 MHz and
45 MHz phase modulation sidebands at the interferometer input and measuring
the demodulated fields at the symmetric port, antisymmetric port, and a pick-
off port inside the power recycling cavity. Every time an SRC parameter is
varied, we re-orthogonalize the sensing matrix. Then, to verify the new servo
points, we individually detune each length degree of freedom and verify its
error signal to be at a zero crossing.

To calculate the observed squeezing level for the detector, we inject a $-14$
dB squeezed vacuum source at the output of the SRC, as shown in Fig. 5. The
injected squeezing level is chosen to match that expected for LIGO A+. We then
rotate the squeezing angle so as to minimize the quantum (shot) noise level in
the interferometer readout channel at 1 kHz. The signal frequency of 1 kHz is
chosen to lie in LIGO’s high-frequency, shot-noise-dominated band, where
optomechanical interactions with the interferometer optics may be neglected.
The injected squeezed field is mode-matched to the interferometer arm
cavities, rather than to the low-finesse SRC, equivalently to the procedure in
use for the real detectors. Every time an SRC parameter is varied, we adjust
the input squeezing mode to recover the mode-matching to the arm cavities,
then retune the squeezing angle for maximum shot noise reduction.

####  III.1.2 Parameters and constraints

During optimization, we allow the lengths and mirror curvatures defining the
SRC to vary, while keeping the arm cavities and the power recycling cavity
fixed. There are thus six degrees of freedom, as indicated in blue lettering
in Fig. 5: the radii of curvature of the SR3, SR2, and SRM mirrors ($R_{\rm
SR3}$, $R_{\rm SR2}$, and $\rm R_{\rm SRM}$, respectively) and the distances
between ITMX/Y and SR3, SR3 and SR2, and SR2 and the SRM ($L_{\rm
ITM\mbox{-}SR3}$, $L_{\rm SR3\mbox{-}SR2}$, and $L_{\rm SR2\mbox{-}SRM}$,
respectively). As shown in Fig. 5, the distance between the ITMs and SR3 is
the sum of the distances between the ITMs and beamsplitter ($L_{\rm X1}$ and
$L_{\rm Y1}$) and the beamsplitter and SR3 ($L_{\rm BS\mbox{-}SR3}$). To avoid
changing the power recycling cavity mode, we allow only the $L_{\rm
BS\mbox{-}SR3}$ component to vary.

It is necessary to impose two constraints on the SRC parameters, as described
below. In effect, these constraints reduce the dimensionality of the
optimization problem from six to four.

##### Total length.

For length sensing and control of the SRC, Advanced LIGO relies on the
resonance of a 45 MHz phase modulation sideband in this cavity [33]. In order
to avoid requiring a major change in the control system, the 45 MHz sideband
must remain resonant in the redesigned cavity. Thus, we require the total SRC
length to remain fixed. This reduces the dimensionality of the optimization
problem from six to five, via the constraint

$L_{\rm SRC}=L_{\rm ITM\mbox{-}SR3}+L_{\rm SR3\mbox{-}SR2}+L_{\rm
SR2\mbox{-}SRM}$ (3)

where $L_{\rm SRC}=56.01$ m is the current SRC length.

##### Mode-matching.

In order to read out the interferometer signal field through the SRC, the SRC
must be mode-matched to the arm cavities. Thus, at the longitudinal location
of the ITM reflective surface, $z=z_{\rm ITM}$, we require that the beam
parameter of the SRC, $q_{\rm SRC}$, equal that of the arm cavities, $q_{\rm
arm}$:

$q_{\rm SRC}\left(z_{\rm ITM}\right)=q_{\rm arm}\left(z_{\rm ITM}\right)$ (4)

This mode-matching constraint implies that for one roundtrip traversal through
the SRC, starting from the ITM, the ABCD matrix [34] of the SRC must satisfy

$q_{\rm arm}\left(z_{\rm ITM}\right)=\frac{A\,q_{\rm arm}\left(z_{\rm
ITM}\right)+B}{C\,q_{\rm arm}\left(z_{\rm ITM}\right)+D}\;.$ (5)

Implicitly, the matrix elements $A$, $B$, $C$, and $D$ are functions of the
six SRC design parameters. We numerically solve Eq. 5 for $R_{\rm SR2}$ in
terms of the other five parameters, further reducing the dimensionality of the
optimization problem from five to four.

####  III.1.3 Cost function

The relative performance of competing optical designs is quantified by a cost
function, whose value the optimization procedure seeks to minimize. Unlike
classical optimization methods, PSO does not use the gradient of the cost
function and, thus, does not require it to be differentiable. This allows a
high degree of flexibility in construction of the cost function. Primarily,
our cost function is designed to penalize SRC designs in which the observed
squeezing level is strongly sensitive to perturbations of the SRC parameters
($C_{\rm SQZ}$). Several additional penalties are included to ensure that the
cavity is stable ($C_{\rm stable}$), low-loss (due to clipping on the optics;
$C_{\rm loss}$), and modally non-degenerate ($C_{\rm HOM}$). The total cost of
an SRC design is defined as

$\textsc{Cost}=C_{\rm SQZ}+C_{\rm stable}+C_{\rm loss}+C_{\rm HOM}\;.$ (6)

Each of the terms in Eq. 6 is described in detail below.

##### Squeezing sensitivity ($C_{\rm SQZ}$).

To quantify the sensitivity of the SRC design to real-world errors, we detune
each SRC parameter individually to estimate the partial derivatives of the
observed squeezing. Radii of curvature are detuned by $\Delta R=\pm 0.1\%$ and
lengths by $\Delta L=\pm 3$ mm. These detunings are chosen to reflect the best
achievable fabrication and hand-placement tolerances for LIGO optics,
respectively. The design is assigned a cost of

$C_{\rm SQZ}\propto\sum_{i}\left[\left|\frac{\Delta S_{i,+}}{\Delta
x_{i}}\right|+\left|\frac{\Delta S_{i,-}}{\Delta x_{i}}\right|\right]\;,$ (7)

where $\Delta S_{i,\pm}$ is the change in observed squeezing for a positive or
negative detuning $\pm\Delta x_{i}$ of the parameter $x_{i}$.

##### Cavity stability ($C_{\rm stable}$).

To ensure the SRC is a stable optical resonator, we penalize cavities whose
$g$ factor is close to the instability limit of $\pm 1$ [35]. Designs with
stability factors of $\left|g\right|>0.9$ are assigned a linearly increasing
cost of

$C_{\rm stable}\propto\left|g\right|\;.$ (8)

Designs with stability factors below this threshold are assigned a fixed cost
of $C_{\rm stable}=-1$.

##### Clipping losses ($C_{\rm loss}$).

A larger beam size on the SRM will result in higher clipping losses further
downstream on the output mode-matching optics, which are smaller in diameter.
We thus include a penalty to ensure the beam exiting the SRC does not become
significantly larger than its present size. At the longitudinal location of
the SRM, $z=z_{\rm SRM}$, designs with a Gaussian beam radius $w\left(z_{\rm
SRM}\right)>3$ mm are assigned a linearly increasing cost of

$C_{\rm loss}\propto w\left(z_{\rm SRM}\right)\;.$ (9)

Designs with beam sizes below this threshold are assigned a fixed cost of
$C_{\rm loss}=-1$.

##### Modal degeneracy ($C_{\rm HOM}$).

The presence of higher-order mode (HOM) co-resonances in the SRC can lead to a
resonant amplification of scattering losses through a process known as “mode
harming” [36]. To ensure the SRC is modally non-degenerate, we penalize HOM
co-resonances up to order 10. If the roundtrip Guoy phase of the cavity,
$\phi_{G}$, is within $\pm 10\%$ of $2\pi\,/\,n$ for any $n\in\\{1,2,..10\\}$,
the design is assigned a linearly increasing cost of

$C_{\rm HOM}\propto 1-\frac{\left|\phi_{G}-2\pi\,/\,n\right|}{2\pi\,/\,n}\;.$
(10)

Designs with no HOM co-resonances within this threshold are assigned a fixed
cost of $C_{\rm HOM}=-1$.

###  III.2 Squeezing performance improvement

Parameter | A+ Nominal | A+ Optimal
---|---|---
SR3 radius of curvature | 35.97 m | 60.24 m
SR2 radius of curvature | -6.41 m | -4.77 m
SRM radius of curvature | -5.69 m | -56.27 m
Beamsplitter to SR3 length | 19.37 m | 9.97 m
SR3 to SR2 length | 15.44 m | 28.56 m
SR2 to SRM length | 15.76 m | 12.04 m
Table 1: Nominal versus optimized signal recycling cavity parameters for LIGO
A+. Figure 6: Nominal versus optimized signal recycling cavity designs for
LIGO A+. Shown is the Gaussian beam size (top) and the accumulated Gouy phase
(bottom) along the cavity axis, from the ITM to the SRM. Of all the optics,
only the positions of SR2 and SR3 are allowed to vary.

In this section, we present the most error-tolerant SRC design identified by
our optimization routine and characterize its optical performance. Table 1
lists the optimized SRC parameter values compared to those for the nominal A+
design. The design differences are visualized in Fig. 6. The top panel shows
the Gaussian beam diameter along the cavity axis, from the ITM to the SRM. The
bottom panel shows the accumulated Gouy phase along the same path. As shown,
the optimization favors converging the beam more slowly, which is achieved
largely by increasing the separation between the SR2 and SR3 telescope
mirrors. For the reasons discussed in §III.1, the total SRC length is
constrained to remain the same, which fixes the position of the SRM in Fig. 6,
and a larger beam size at the SRM position is also strongly penalized.

To assess the competitiveness of this candidate design, we analyze its
squeezing performance statistically using a Monte Carlo method. With a fixed
level of injected squeezing, small random errors are added each of the SRC
parameters and the observed squeezing is computed for a large number of
trials. The resulting squeezing distributions provide a direct, quantitative
comparison of the performance of competing cavity designs. In detail, our
procedure is as follows:

  1. 1.

Assume realistic uncertainties in the curvatures and positions of the SRC
optics. We assume the uncertainties to be normally distributed with zero mean
and a standard deviation of 0.1% for radius of curvature errors and 3 mm for
position errors.

  2. 2.

Draw a set of random errors for all six SRC parameters listed in Table 1.

  3. 3.

Using the Finesse model described in §III.1.1, simulate the interferometer
with these perturbed parameters and compute the observed squeezing level. A
fixed injected squeezing level of $-14$ dB is assumed.

  4. 4.

Repeat the previous steps 2000 times, each time drawing a new set of random
parameter errors.

  5. 5.

Calculate the probability distribution of observed squeezing across all
trials.

Convergence testing with varying numbers of trials has found 2000 to
adequately sample the distribution.

Fig. 7 shows the result of this comparative performance analysis for the
nominal and optimized SRC designs. The three panels show the probability
distributions of observed squeezing under varying levels of readout loss,
ranging from 5% (top panel) to 20% (bottom panel). This readout loss accounts
for attenuation losses due to output mode-mismatch, Faraday isolator insertion
loss, optical pick-offs for diagnostic and control purposes, and photodiode
quantum inefficiency. At the beginning of LIGO A+, the readout losses are
expected to be similar to the bottommost panel. We find that, in the presence
of random optical errors, our optimization procedure results in a significant
narrowing of the distribution of possible squeezing outcomes. As shown, this
narrowing leads to a modest improvement in the median squeezing level and, at
95% confidence, a dramatic improvement in the worst possible outcome (black
vertical lines in Fig. 7).

The squeezing distributions of the nominal design exhibit a bimodality which
arises from two distinct parameter regimes. In each panel, the left peak
(corresponding to higher squeezing) overwhelmingly consists of cases in which
the SR3 radius of curvature is smaller than intended ($\Delta R_{\rm
SR3}<0.0\%$). On the other hand, the right peak (corresponding to lower
squeezing) overwhelmingly consists of large, positive errors in the SR3 radius
of curvature ($\Delta R_{\rm SR3}>+0.1\%$). We find no strong correlation in
the values of the other five SRC parameters between the two peaks. The strong
dependency on $\Delta R_{\rm SR3}$ is reduced but not completely eliminated by
the optimization process. In the squeezing distributions of the optimal
design, nearly all of the low-squeezing outliers arise from cases of very
large, positive error in the SR3 radius of curvature ($\Delta R_{\rm
SR3}\geq+0.2\%$).

To understand the extreme sensitivity to errors in the SR3 mirror, and how our
optimization process reduces it, we generate a “corner plot” by detuning
individual pairs of SRC parameters, as shown in Fig. 8. In each panel, the
lines represent iso-squeezing contours at which errors in the two parameters
degrade the observed squeezing by 3 dB, compared to the unperturbed case
(located at the origin). The contours for the nominal SRC design (solid blue
lines) and the optimized design (dashed red lines) are overlaid to allow a
direct comparison of the parameter sensitivities. A greater error tolerance
appears as an increase in the area enclosed by the iso-squeezing contour (that
is, larger parameter errors are required to produce the same degradation in
squeezing). As shown, the single largest improvement is a dramatic reduction
of sensitivity to errors in $R_{\rm SR3}$.

Figure 7: Probability distributions of the squeezing achieved with two
different signal recycling cavity (SRC) designs, in the presence of realistic
random optical errors. The three panels assume various levels of readout loss
ranging from 5% (top) to 20% (bottom). In each panel, the black vertical lines
indicate the difference in worst possible outcome, at 95% confidence, between
the two designs. Figure 8: Corner plot comparing the sensitivity to error in
pairs of parameters (see Table 1) between two signal recycling cavity (SRC)
designs. In each panel, the lines represent iso-squeezing contours at which
errors in the two parameters degrade the observed squeezing by 3 dB, compared
to the unperturbed case (located at the origin). A larger area enclosed by the
iso-squeezing contour of one design indicates a greater error tolerance.

##  IV Conclusions

The aim of this paper has been to demonstrate the promise of two novel,
complementary techniques in optical experiment design:

  1. 1.

Nonspherical mirror surfaces as solutions to otherwise overconstrained cavity
design problems.

  2. 2.

Statistics-guided cavity design for optimal robustness to real-world optical
errors.

As a proof of concept, we have performed a two-part optimization of the LIGO
A+ design, first modifying the arm cavities for reduced point-absorber-induced
loss (see §II) and then optimizing the SRC for maximum squeezing performance
(see §III). Our findings strongly suggest that these techniques can be
leveraged to achieve greater performance in current and future gravitational-
wave detectors. Both act to minimize internal optical losses, as is critical
to achieving megawatt-scale power and high levels of squeezing in third-
generation detectors.

Several caveats apply to the results presented herein, which will be the
target of future studies. In §II, the optimal test mass profiles depend
critically on the thermal state of the optics. Central heating due to
(uniform) coating absorption and thermal compensation applied by the ring
heaters induce thermoelastic surface deformations of the same magnitude as the
static polish, even at the outer radii. For the purpose of illustration, we
assume coating absorption at the same level as for the Advanced LIGO optics.
However, the absorptivity of the new coating materials targeted for LIGO A+,
$\rm TiO_{2}$-doped $\rm GeO_{2}$ [20], is not currently known. Given this
uncertainty, an equivalent active solution is possible in which an annular
heating pattern is projected onto the front surface of the test mass near the
edge, providing a tunable means of generating surfaces profiles similar to
those shown in Fig. 1.

In §III, an analytical or semi-analytical squeezing model is desirable to
cross-validate the numerical results presented in Fig. 7. In future work, we
aim to develop a semi-analytic mode-scattering model that will enable
squeezing calculations to be performed using the FFT-based optical simulation
SIS [24]. This will provide an important cross-check of the Finesse-based
models. In future work, we also aim to extend the complexity of the
optimization in several key ways:

  1. 1.

Include an OMC, to remove sideband power and non-artificially capture output
mode-matching losses.

  2. 2.

Jointly optimize the quantum noise performance at multiple frequencies, to
extend our procedure to frequency-dependent squeezing.

  3. 3.

Incorporate additional constraints to generate optical designs compatible with
the footprint of the existing LIGO vacuum chambers.

  4. 4.

Jointly optimize the power and signal recycling cavities, enabling potential
searches for “mode-healing” designs [36] in the presence of point absorbers.

Each of these extensions incurs a considerably higher computational cost than
the present work, due either to an increase in the dimensionality of the
optimization itself, or by requiring additional sub-optimizations (or
servoing) to be performed within each iteration of the optimization. Improved
algorithmic efficiency, together with highly parallelized simulations, will be
essential to achieving them at the scale of a full gravitational-wave
interferometer.

## Acknowledgments

We are grateful to GariLynn Billingsley for providing test mass mirror maps,
Hiro Yamamoto for guidance on using SIS, and Lee McCuller for helpful comments
on the squeezing modeling. We would also like to thank the LIGO Laboratory for
providing the resources with which to conduct this research, as well as the
LIGO SURF program, the National Science Foundation, and the California
Institute of Technology for sponsoring the project in part. 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 No. PHY-1764464. Advanced LIGO was built
under Grant No. PHY-0823459. This paper has LIGO Document Number
LIGO-P2100184.

##  Appendix A Arm Cavity Loss by Mode

Mode ${\rm LG}_{p,l}$ | Roundtrip arm loss
---|---
Order | $p$ | $|l|$ | A+ Nominal | A+ Proposed | Units
0 | 0 | 0 | $76^{+0.7}_{-1.8}$ | $62^{+0.5}_{-0.3}$ | ppm
1 | 0 | 1 | $159^{+4}_{-15}$ | $122^{+6}_{-7}$ | ppm
2 | 0 | 2 | $357^{+17}_{-66}$ | $649^{+48}_{-88}$ | ppm
2 | 1 | 0 | $597^{+32}_{-110}$ | $1092^{+81}_{-148}$ | ppm
3 | 0 | 3 | $793^{+47}_{-174}$ | $3165^{+220}_{-420}$ | ppm
3 | 1 | 1 | $1.7^{+0.1}_{-0.4}$ | $7.2^{+0.4}_{-0.9}$ | ppt
4 | 0 | 4 | $2.7^{+0.1}_{-0.4}$ | $15^{+0.6}_{-1.8}$ | ppt
4 | 1 | 2 | $7.6^{+0.3}_{-1.1}$ | $35^{+1}_{-4}$ | ppt
4 | 2 | 0 | $10^{+0.4}_{-1.5}$ | $44^{+1}_{-5}$ | ppt
5 | 0 | 5 | $10^{+0.3}_{-1.2}$ | $42^{+1}_{-5}$ | ppt
5 | 1 | 3 | $32^{+1}_{-4}$ | $92^{+2}_{-8}$ | ppt
5 | 2 | 1 | $49^{+2}_{-6}$ | $120^{+3}_{-10}$ | ppt
6 | 0 | 6 | $31^{+0.6}_{-2.8}$ | $80^{+2}_{-7}$ | ppt
6 | 1 | 4 | $94^{+2}_{-7}$ | $160^{+3}_{-11}$ | ppt
6 | 2 | 2 | $150^{+3}_{-10}$ | $209^{+3}_{-11}$ | ppt
6 | 3 | 0 | $171^{+3}_{-11}$ | $225^{+3}_{-11}$ | ppt
7 | 0 | 7 | $76^{+2}_{-6}$ | $134^{+2}_{-9}$ | ppt
7 | 1 | 5 | $210^{+4}_{-13}$ | $261^{+3}_{-9}$ | ppt
7 | 2 | 3 | $312^{+5}_{-13}$ | $342^{+3}_{-8}$ | ppt
7 | 3 | 1 | $360^{+6}_{-13}$ | $379^{+3}_{-9}$ | ppt
8 | 0 | 8 | $154^{+3}_{-10}$ | $224^{+2}_{-7}$ | ppt
8 | 1 | 6 | $348^{+5}_{-11}$ | $388^{+3}_{-8}$ | ppt
8 | 2 | 4 | $463^{+6}_{-12}$ | $472^{+3}_{-11}$ | ppt
8 | 3 | 2 | $512^{+8}_{-14}$ | $511^{+4}_{-12}$ | ppt
8 | 4 | 0 | $526^{+8}_{-15}$ | $522^{+4}_{-12}$ | ppt
Table 2: Roundtrip arm cavity loss for each Laguerre-Gauss mode up to order 8,
shown for both the nominal and proposed test mass profiles. The lower and
upper error bars represent the 16th and 84th percentiles of the loss
distributions, respectively (see §II.2).

Although the test mass profiles presented in §II.1 are designed to shift the
resonance frequencies of 7th-order modes (to reduce point absorber
scattering), they also achieve a significantly greater dissipation of mode
orders 2-7 as shown in Fig. 3. Enhancing the dissipation of certain modes may
be relevant for improving the damping of parametric instabilities in
gravitational-wave detectors [26]. Thus, in Table 2 we include a breakdown of
the roundtrip dissipation per optical mode.

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