2201.09482.pdf

Optimizing Gravitational-Wave Detector Design for Squeezed Light

Jonathan W. Richardson,

∗

Swadha Pandey,

Edita Bytyqi,

Tega Edo,

and Rana X. Adhikari

Department of Physics and Astronomy, University of California, Riverside, Riverside, CA 92521, USA

Department of Physics, Indian Institute of Technology Kanpur, Kanpur, UP 208016, India

Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA

LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA

(Dated: May 31, 2022)

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 grav-

ity, cosmology, and dense nuclear matter will become pos-

sible 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 [

–

]. 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 [

]. More

frequent detections of binary neutron star mergers will

also enable independent measurement of the Hubble con-

stant to high precision [

], 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 [

] 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 [

]. 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 fluc-

tuations which modulate the interferometer output signal.

∗

jonathan.richardson@ucr.edu

These fluctuations also apply force to the mirrors via ra-

diation 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 detec-

tors operated with roughly 250 kW of resonating power

inside the arm cavities [

]—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 [

]. The source of this loss has been identified

as sub-millimeter, highly absorbing defects in the opti-

cal 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 ab-

sorbers appear to originate during the coating deposition

process, although it is still not understood how these con-

taminants 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 [

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 [

Squeezed light allows for the engineering of the elec-

tromagnetic vacuum state that enters the interferome-

ter. 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

arXiv:2201.09482v2 [astro-ph.IM] 28 May 2022

quadrature. In Advanced LIGO, squeezed vacuum field

is generated via degenerate optical parametric amplifica-

tion [

] 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 [

]. Losses arise from scattering, imper-

fect 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 fabrica-

tion and hand-positioning of the interferometer optics [

For third-generation detectors [

], reducing the inter-

nal mode-matching losses to

∼

1% levels is imperative.

In this work, we present a novel optimization approach

to gravitational-wave detector design. It is aimed at maxi-

mizing 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 [

]. 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 real-

istic 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 de-

tectors. Finally, in §IV we summarize and discuss future

extensions of this work.

II. ARM CAVITY DESIGN

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 [

]

and extended to power-dependent surface deformations

from point absorbers by Brooks

et al.

[

], who showed

that the power loss from the fundamental mode to the

-th HOM is approximately

L

(1)

The first term,

, is the single-bounce amplitude scat-

tering from the fundamental mode to the

-th HOM

when reflected off the deformed mirror surface. The sec-

ond term,

, is the optical gain of that HOM, which

depends on the cavity geometry and the actual (nonideal)

surface profiles of the two mirrors:

−

′

−

′

cos

[

]

(2)

The factors

′

and

′

are the effective amplitude reflectiv-

ities of the input test mass (ITM) and the end test mass

(ETM), respectively, accounting for mode-dependent clip-

ping losses, and

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

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 ad-

justing 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 ex-

isting infrastructure to approximately

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 in-

creased coating Brownian noise—unacceptable for the A+

design, which is already thermal-noise-limited across its

mid-frequency band [

]. A larger beam size, on the other

hand, results in unacceptably higher clipping losses inside

the arms and the signal recycling cavity. Thus, the prob-

lem 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 re-

gion, 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

(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

−

100

−

100

Surface Height [nm]

A+ ITM

Coating

Polish

Total Surface

100

125

150

Radius [mm]

Slope

[nm/mm]

Maximum Polishing Slope

−

100

−

100

Surface Height [nm]

A+ ETM

Coating

Polish

Total Surface

100

125

150

Radius [mm]

Slope

[nm/mm]

Maximum Polishing Slope

FIG. 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

≤

2.5 nm/mm, as shown in the lower panels.

cavity mode, leading to a significant loss reduction in the

presence of scattering sources such as point absorbers.

A. Nonspherical test mass profiles

For mirror fabrication in the A+ era, LIGO has the abil-

ity to specify an arbitrary (nonspherical) polishing figure.

Internal discussions with optics manufacturers have in-

dicated that an arbitrary radial profile, subject to a max-

imum slope of 2.5 nm/mm, could be produced with high

confidence, with a possibility that an even steeper polish-

ing 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 nonspher-

ical 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, TiO

doped GeO

[

], is expected to replace the TiO

-doped

coatings used in Advanced LIGO [

]. Accord-

ingly, 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).

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 spher-

ical mirror polish. With higher cavity power (or coating

−

Optical Gain

A+ Nominal

Frequency / FSR

−

Optical Gain

A+ Proposed

FIG. 2. Optical gain

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 com-

pensation polish shown in Fig. 1 (blue curves). In both panels,

coating absorption of 120 mW per test mass is assumed.

absorptivity), the resonances shift toward higher frequency

due to the increasing residual thermal deformation of the

test masses. Although ring heaters compensate the cen-

tral heating due to uniform coating absorption, the ring

heaters “overcorrect” the mirror surface at large radii, re-

sulting in a net profile that steeply rises near the edge of

the optic [

]. Here, we assume 120 mW of coating ab-

sorption 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 loca-

tions 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 im-

plications for the loss performance of the arm cavities, as

discussed in the following section.

B. 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 nu-

merical simulations of an A+ arm cavity using

SIS

[

], an

FFT-based optical simulation package. The model includes

all thermoelastic effects: (1) uniform coating absorption

and (2) optimal ring heater compensation, to maintain con-

stant 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 indepen-

dently drawn from a Gaussian distribution with zero mean

and a standard deviation of 5 mm. Surface roughness pro-

files are randomly generated by

SIS

with a power spectral

density chosen to match that of the current Advanced

LIGO optics [25].

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 A 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 me-

dian loss values over all randomized trials, and averaged

over all modes

of order

. The shading

represents the 16th and 84th percentiles of the loss distri-

butions 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

Mode Order

−

Roundtrip Arm Loss

A+ Nominal, 750 kW, 0.3 ppm

A+ Proposed, 750 kW, 0.3 ppm

FIG. 3. Arm cavity loss as a function of mode order. Each curve

represents the median loss in 1000 trials with random miscen-

terings and surface roughnesses, averaged over all Laguerre-

Gauss modes

of order

. The shading represents

the 16th and 84th percentiles of the loss distributions over

all trials and modes. The proposed mirror profiles achieve sig-

nificantly enhanced higher-order mode dissipation, with no

increase in fundamental mode loss.

the losses of HOMs above order 2. While the design ob-

jective in §II A was only to shift resonance frequencies of

HOMs, the larger dissipation of HOMs is an added ad-

vantage that helps to further reduce their optical gain.

Enhancing the dissipation of certain HOMs may be also

relevant for improving the damping of parametric instabil-

ities in gravitational-wave detectors [

]. For this reason,

we include a full breakdown of the dissipation per optical

mode in Appendix A.

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

◦

Point absorber phase maps are generated using the ana-

lytic formalism for thermoelastic surface deformation from

Brooks et al. [

]. 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 fundamen-

tal 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 devia-

100 kW

A+ Nominal

A+ Proposed

95% confidence (A+ Nominal)

95% confidence (A+ Proposed)

400 kW

A+ Nominal

A+ Proposed

95% confidence (A+ Nominal)

95% confidence (A+ Proposed)

100

110

120

130

140

150

160

170

180

Roundtrip Arm Loss (ppm)

750 kW

A+ Nominal

A+ Proposed

95% confidence (A+ Nominal)

95% confidence (A+ Proposed)

Probability Density

FIG. 4. Arm cavity loss distributions due to point absorbers,

shown at three different power levels. Each loss distribu-

tion 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 compen-

sation, is assumed.

tions 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 instal-

lation 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 interfer-

ometer 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 am-

plifier 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 inter-

ference 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 com-

mon errors which induce mode-mismatch. We identify the

maximally error-tolerant design through a numerical op-

timization procedure described in §III A, 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 B.

A. 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 parti-

cle swarm optimization (PSO) algorithm provided by the

Pyswarms optimization toolkit [

]. PSO is an evolution-

ary 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 parame-

ter 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

INESSE

optical

simulation [

] is used to compute the partial derivatives

of the observed squeezing level with respect to small de-

tunings of each candidate SRC parameter. We detail the

INESSE

simulation in §III A 1. Then, in §III A 2 we de-

scribe the set of optical parameters which we optimize, as

well as relevant parameter constraints. Finally, in §III A 3

we detail the construction of the cost function used to de-

fine the optimization objective.

1. Optical simulation

In our optimization routine, the core compute engine

is a

INESSE

simulation [

] used to analyze the perfor-

mance of a given SRC design.

INESSE

is a modal-based

optical simulation package widely used for modeling laser

cavities, whose modern user interface is provided by the

YKAT

[

] package. To illustrate the optimization proce-

dure, we adopt a toy interferometer model based on the

LIGO A+ design. Its optical layout is shown in Fig. 5.

X-Arm

Cavity

Signal Recycling

Cavity

SR3

SR2

SRM

ETMXITMX

BS-SR3

SR2-SRM

SR3-SR2

Squeezing

Input

Y-Arm

Cavity

ETMY

ITMY

Power Recycling

Cavity

PR3

PR2

PRM

Laser

SR3

SR2

SRM

FIG. 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.

However, for the purpose of this illustration, several sim-

plifying departures from the A+ design are made to reduce

the computational cost and complexity:

•

Frequency-independent squeezing.

Although A+ will

use frequency-dependent squeezing [

], for simplic-

ity 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

∼

1 pm from a dark fringe.

In A+, this technique, known as “DC readout,” will

be replaced by balanced homodye readout [

]. As

balanced homodyne readout adds considerable com-

plexity, 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 [

], for the present simu-

lation we omit the OMC and instead assume a fixed

output loss ranging from 5

−

20%.

The

INESSE

simulation starts from a “nominal” model

(using a provided set of SRC parameters), then individu-

ally detunes each SRC parameter from its design value

and computes the change in observed squeezing. The pa-

rameter detunings introduce a spatial mode-mismatch be-

tween 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 inter-

ferometer 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

INESSE

simulation to zero all such

“artifical” detunings.

To prevent length detunings, we incorporate DC ser-

vos for all five length degrees of freedom: the common

arm length, differential arm length, power recycling cav-

ity length, Michelson length, and signal recycling cavity

length [see, e.g.,

]. Linear error signals are constructed

by injecting 9 MHz and 45 MHz phase modulation side-

bands at the interferometer input and measuring the

demodulated fields at the symmetric port, antisymmet-

ric 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.

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 (

SR3

SR2

, and R

SRM

, respectively) and

the distances between ITMX/Y and SR3, SR3 and SR2,

and SR2 and the SRM (

ITM

SR3

SR2

, and

SR2

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 (

and

) and the beamsplitter

and SR3 (

SR3

). To avoid changing the power recycling

cavity mode, we allow only the

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.

a. 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 [

]. 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

SRC

ITM

SR3

SR2

SRM

(3)

where

SRC

01 m is the current SRC length.

b. Mode-matching.

In order to read out the interfer-

ometer signal field through the SRC, the SRC must be

mode-matched to the arm cavities. Thus, at the longitu-

dinal location of the ITM reflective surface,

ITM

, we

require that the beam parameter of the SRC,

SRC

, equal

that of the arm cavities,

arm

SRC

(

ITM

)

arm

(

ITM

)

(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

arm

(

ITM

)

A q

arm

(

ITM

)

C q

arm

(

ITM

)

(5)

Implicitly, the matrix elements

, and

are func-

tions of the six SRC design parameters. We numerically

solve Eq. 5 for

SR2

in terms of the other five parameters,

further reducing the dimensionality of the optimization

problem from five to four.

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 optimiza-

tion 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 pa-

rameters (

SQZ

). Several additional penalties are included

to ensure that the cavity is stable (

stable

), low-loss (due to

clipping on the optics;

loss

), and modally non-degenerate

(

HOM

). The total cost of an SRC design is defined as

OST

SQZ

stable

loss

HOM

(6)

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

a. Squeezing sensitivity (

SQZ

To quantify the sen-

sitivity 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

∆

=±

1% and lengths by

∆

=±

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

SQZ

∝

∑

[

∣

∆

∣

∆

−

∆

∣

]

(7)

where

∆

is the change in observed squeezing for a

positive or negative detuning

∆

of the parameter

b. Cavity stability (

stable

To ensure the SRC is a

stable optical resonator, we penalize cavities whose

fac-

tor is close to the instability limit of

1 [

]. Designs

with stability factors of

9 are assigned a linearly

increasing cost of

stable

∝

(8)

Designs with stability factors below this threshold are

assigned a fixed cost of

stable

=−

c. Clipping losses (

loss

A larger beam size on the

SRM will result in higher clipping losses further down-

stream 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,

SRM

, designs with a Gaussian beam radius

(

SRM

)

3 mm are assigned a linearly increasing cost of

loss

∝

(

SRM

)

(9)

Designs with beam sizes below this threshold are assigned

a fixed cost of

loss

=−

d. Modal degeneracy (

HOM

The presence of higher-

order mode (HOM) co-resonances in the SRC can lead to a

resonant amplification of scattering losses through a pro-

cess known as “mode harming” [

]. To ensure the SRC is

modally non-degenerate, we penalize HOM co-resonances

up to order 10. If the roundtrip Guoy phase of the cav-

ity,

, is within

10% of 2

for any

∈

{

, ..

}

, the

design is assigned a linearly increasing cost of

HOM

∝

−

∣

−

∣

(10)

Designs with no HOM co-resonances within this threshold

are assigned a fixed cost of

HOM

=−

B. Squeezing performance improvement

In this section, we present the most error-tolerant SRC

design identified by our optimization routine and charac-

terize its optical performance. Table I lists the optimized

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 I. Nominal versus optimized signal recycling cavity

parameters for LIGO A+.

-0.06

-0.04

-0.02

0.02

0.04

0.06

Beam Size (m)

ITM

SRM

SR3

SR2

SR3

SR2

A+ Nominal

A+ Optimal

Distance along Cavity Axis (m)

-5

Gouy Phase (deg)

ITM

SRM

SR3

SR2

SR3

SR2

FIG. 6. Nominal versus optimized signal recycling cavity de-

signs 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.

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 A, 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 squeez-

ing, small random errors are added each of the SRC pa-

rameters and the observed squeezing is computed for a

large number of trials. The resulting squeezing distri-

butions provide a direct, quantitative comparison of the

performance of competing cavity designs. In detail, our

procedure is as follows:

Assume realistic uncertainties in the curvatures and

positions of the SRC optics. We assume the uncer-

tainties to be normally distributed with zero mean

and a standard deviation of 0.1% for radius of curva-

ture errors and 3 mm for position errors.

Draw a set of random errors for all six SRC parame-

ters listed in Table I.

Using the

INESSE

model described in §III A 1, simu-

late the interferometer with these perturbed param-

eters and compute the observed squeezing level. A

fixed injected squeezing level of

−

14 dB is assumed.

Repeat the previous steps 2000 times, each time

drawing a new set of random parameter errors.

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 quan-

tum 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 narrow-

ing 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 dra-

matic improvement in the

worst

possible outcome (black

vertical lines in Fig. 7).

The squeezing distributions of the nominal design ex-

hibit a bimodality which arises from two distinct param-

eter regimes. In each panel, the left peak (correspond-

ing to higher squeezing) overwhelmingly consists of cases

in which the SR3 radius of curvature is smaller than in-

tended (

∆

SR3

0%). On the other hand, the right peak

(corresponding to lower squeezing) overwhelmingly con-

sists of large, positive errors in the SR3 radius of curva-

ture (

∆

SR3

1%). We find no strong correlation in

the values of the other five SRC parameters between the

two peaks. The strong dependency on

∆

SR3

is reduced

but not completely eliminated by the optimization pro-

cess. 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

(

∆

SR3

≥+

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 in-

crease 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

SR3

IV. CONCLUSIONS

The aim of this paper has been to demonstrate the

promise of two novel, complementary techniques in op-

tical experiment design:

Nonspherical mirror surfaces

as solutions to other-

wise overconstrained cavity design problems.

Statistics-guided cavity design

for optimal robust-

ness to real-world optical errors.

As a proof of concept, we have performed a two-part opti-

mization 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 per-

formance (see §III). Our findings strongly suggest that

these techniques can be leveraged to achieve greater per-

formance in current and future gravitational-wave detec-

tors. 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 op-

timal test mass profiles depend critically on the thermal

state of the optics. Central heating due to (uniform) coat-

ing 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 mate-

rials targeted for LIGO A+, TiO

-doped GeO

[

], is not

currently known. Given this uncertainty, an equivalent

5% Readout Loss

-0.91dB : 95% confidence (A+ Nominal)

-9.36dB : 95% confidence (A+ Optimal)

Squeezing distribution (A+ Nominal)

Squeezing distribution (A+ Optimal)

Probability Density

10% Readout Loss

-0.89dB

-7.90dB

−

Observed Squeezing Level (dB)

20% Readout Loss

-0.67dB

-5.93dB

FIG. 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.

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 pre-

sented 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

[

]. This will provide an important cross-

check of the

INESSE

-based models. In future work, we

also aim to extend the complexity of the optimization in

several key ways: