## Abstract

Optical losses degrade the sensitivity of laser interferometric instruments. They reduce the number of signal photons and introduce technical noise associated with diffuse light. In quantum-enhanced metrology, they break the entanglement between correlated photons. Such decoherence is one of the primary obstacles in achieving high levels of quantum noise reduction in precision metrology. In this work, we compare direct measurements of cavity and mirror losses in the Caltech 40 m gravitational-wave detector prototype interferometer with numerical estimates obtained from semi-analytic intra-cavity wavefront simulations using mirror surface profile maps. We show a unified approach to estimating the total loss in optical cavities (such as the LIGO gravitational detectors) that will lead towards the engineering of systems with minimum decoherence for quantum-enhanced precision metrology.

Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. INTRODUCTION

Laser interferometry is a pillar of modern precision measurement and plays
an integral role in the preparation, interrogation, and manipulation of
quantum systems, including entangled states of light and matter. Optical
losses pose both practical and fundamental limitations to instrument
performances. Absorption and scatter from optical components or subsystems
decrease classical signal levels and introduce application-dependent
technical noise. Optical losses also drive the decoherence of
non-classical photon states, which find application in quantum computing,
cryptography, and quantum-enhanced metrology. A typical example for the
latter is the use of *squeezed light* in
ground-based interferometric gravitational-wave (GW) detectors to reduce
photon shot noise [1]. We discuss
optical loss as a source of decoherence for quantum states in the context
of these quantum noise-limited instruments.

The ability to directly observe GWs is continuously evolving our understanding of the universe and its history. The number of observed mergers of binary systems of black holes and neutron stars is steadily growing [2,3], and keeps informing black hole population models, formation channels for compact objects, and parameter constraints on theories of gravity and the equation of state of highly degenerate nuclear environments. Improved sensitivity, for example, from a reduction of quantum noise, and detections of other, new source types promise to multiply this knowledge across the GW frequency spectrum.

Interferometric GW detectors are electromagnetic transducers for GWs that convert the strain of passing GWs on space–time to phase modulations of a laser carrier. The two Advanced LIGO (aLIGO) observatories [4], Advanced Virgo (AdVirgo) [5], KAGRA [6], and GEO600 [7] are ground-based detectors that are sensitive to signals in the audible frequency range. They all have similar optical topologies, using a combination of optical cavities to increase the number and storage time of photons interacting with passing GWs. Optical loss generates a variety of technical problems for the control of these detectors. It broadens cavity linewidths, affects the behavior of coupled-cavity systems, and backscattering can create transient signal glitches or pollute entire frequency bands [8–10]. Radiation pressure-induced opto-mechanical effects due to scattered fields may also become relevant. A thorough understanding of the magnitude and nature of optical loss will help anticipate and mitigate such effects.

During their third observation run (O3), the measured round trip losses of the Fabry–Perot (FP) arm cavities of the aLIGO detectors were in the range of 60–70 ppm. This is below the requirement of 75 ppm and not currently considered a limiting factor for the interferometer performance. The optical absorption of high-quality dielectric mirror coatings is below the ppm level, which makes scatter the dominant loss mechanism in interferometers [4,5]. The expected arm loss based on surface figure phase maps and large-angle scattering data is about 20–30 ppm less than the measured value. Part of this discrepancy stems from the scattering into the angular region ${\sim}0.1 - {1^ \circ}$, which is difficult to access experimentally without ambiguity, and there are no data available for the size and quality of the mirrors used in GW detectors. It is therefore desirable and important to have a set of techniques that can reliably, repeatably, and accurately characterize optical loss, so that mitigation efforts can be validated.

This work examines loss measurements at the Caltech 40 m interferometer (40 m for short), which is a small-scale prototype with the same principal optical topology as aLIGO, and compares them with simulations to verify these models for the purpose of extending the analysis to other optical systems. The paper is structured as follows: Section 2 provides necessary formulas and quantitative and qualitative description of the scattering and loss. Section 3 describes the optical measurements utilized to estimate the total optical cavity losses. Section 4 compares the simulations with the measurements and discusses the remaining discrepancies. Finally, Section 5 concludes the paper.

## 2. THEORY AND MODEL OF SCATTER LOSS

In a long baseline FP cavity, we wish to know the optical field distribution in the plane of the mirror, so that we can estimate the amount of power in the cavity using the field reflected from the mirror while also getting a good estimate of the power loss due to non-zero field values outside the mirror. To this end, we employ the Huygens–Fresnel integral

An eigenmode of Eq. (1) that is of particular interest in long-baseline FP cavities is the ${{\rm TEM}_{00}}$ Gaussian beam mode, given by

#### A. Surface Aberration and Scattering

The amplitude spectral density, $H({f_x},{f_y})$, of the mirror surface is calculated from the inverse Fourier transform of its surface profile $h({x_0},{y_0})$ via

When a Gaussian field is reflected by a mirror surface with height aberration $h$ and nominal radius of curvature ${R_0}$, the prompt reflected field at the mirror surface is

and the corresponding disturbance of the reflected field in the limit of small $h$, where $2kh \ll 1$, isWe use Eq. (1) to propagate Eq. (6) to the second mirror. The wavefront mismatch at $z$ will determine the amount of power scattered out of the cavity mode by the first mirror. The scattered field distribution in the $(x,y)$ plane displaced a distance $L$ from the mirror is

where $\beta = \lambda (z - {z_0})$. Here, the Fresnel integral has been recast as a convolution integral in frequency space, namely,The far field condition, satisfied by long baseline interferometer, means that the spatial coordinate and spatial frequency are related as follows:

Furthermore, the angular coordinate, ${\theta _{x,y}}$, is related to spatial frequency, ${\nu _{x,y}}$, and spatial wavelength, ${\lambda _{x,y}}$, of the reflecting field leaving the mirror surface via

The finite size of the mirrors defines a characteristic cutoff angle ${\theta _{\rm{cut}}} = a/2L$, where $a$ is the diameter of the mirror, and a corresponding spatial wavelength $\lambda _s^{\rm{cut}} = \lambda /{\theta _{\rm{cut}}}$ of surface deformations. All power scattered to angles larger than ${\theta _{\rm{cut}}}$ by surface deformations with length scales smaller than $\lambda _s^{\rm{cut}}$ is lost entirely, similar to the tail of the reflected fundamental Gaussian mode, which misses the mirror. At angles smaller than ${\theta _{\rm{cut}}}$, the overlap between impinging wavefront distortions and mirror surface needs to be evaluated to determine the power lost.

For FP Michelson interferometers such as aLIGO and the Caltech 40 m prototype schematically shown in Fig. 4, we can define for each arm cavity an input mirror (IM), where the laser beam enters the cavity, and an end mirror (EM). Important geometrical parameters for the FP arm cavities of aLIGO and the Caltech 40 m prototype that are used in this calculation are listed in Table 1 along with the parameters derived above. As can be seen, the large separation of the LIGO mirrors results in over a factor of 20 longer spatial cutoff wavelength over the 40 m’s.

#### B. BRDF as a Loss Estimation Metric

The bidirectional reflectance distribution function (BRDF) describes the angular dependence of the intensity of light reflected and scattered from the mirror surface [11], defined by the following formula:

where $\theta$ and $\phi$ are angles in spherical coordinates, with the usual relation to Cartesian coordinates, i.e., $x = L\sin (\theta)\cos (\phi)$ and $y = L\sin (\theta)\sin (\phi)$. Again, ${P_0}$ is the total power in the beam, while $L$ is the distance between the mirror and the point of measurement.The one-dimensional distribution used often for loss estimation is

which is fairly accurate when no explicit dependence on $\phi$ is present in the phase map.Direct BRDF measurements of a mirror are usually limited to above several degrees from the normal, because the small spatial separation from the incident beam hinders the measurement of the small scattered power. Therefore, the BRDF for small angles is best estimated using scattering calculations from the phase map $h({x_0},{y_0})$ of the mirror surface by numerical methods. The angular range of the resulting BRDF data is limited by the spatial resolution of the phase map data, which is typically of order zero (0.1 mm), so that the maximum scattering angle covered by phase map data, using Eq. (13), is ${\sim}{0.3^ \circ}$. This leaves a gap in the angular range between phase map estimations and direct BRDF measurements.

Figure 3 summarizes various BRDF measurements and illustrates this gap. The solid and dashed line plots for aLIGO and the 40 m show a sharp Gaussian towards $\theta \sim 0$, whose width is determined by the Gaussian beam size resonating in the cavity, and a tail structure based on the phase map data that is indicative of approximate power law scaling. Note that in all cases, the BRDF line plots cover the entire angular range spanned by the mirror at the other end of the cavity, marked by respective vertical lines in Fig. 3. For the aLIGO detectors, we have also added BRDF measurements at $\theta \simeq {0.003^ \circ}$ and $\theta \simeq {0.006^ \circ}$, taken from four photodiodes (PDs) on baffles that are located 20–40 cm from the center of the mirror [13]. As discussed in this reference, the power distribution is very sensitive to the PD location relative to the center of the mirror due to the mirror aperture clipping effect, and the measured powers vary by an order of magnitude, but are consistent with the estimate based on phase map data.

For the region outside the mirror corresponding to $\theta \gt {\theta _{\rm{cut}}}$ (cf. Table 1), a power law fit to the data points yields an angular dependence of ${\theta ^{- n}}$ with $n \sim 2$. Note that this loss estimation is much larger than the typical estimation using the Gaussian beam shape, with a contribution also coming from the clipping effect of the mirror. We also have sparse BRDF measurement data in the range of 1° to 10° collected from various view ports located along the beam tubes. Moreover, the BRDF measurement of a high reflecting mirror in the angular range larger than 10° was reported in [12]. These BRDF data and the aLIGO BRDF data using view ports connect smoothly.

Total integrated scattering (TIS) from the mirror surface in some angle range can be calculated from the BRDF as follows:

## 3. MEASUREMENT TECHNIQUES

The Caltech 40 m interferometer is a small-scale GW detector prototype instrument. It was designed to share its optical topology with the full-scale LIGO observatories, including FP cavity-enhanced Michelson arms, a power recycling cavity (PRC), a signal recycling cavity (SRC), and an input mode cleaner (IMC). The optical configuration of the 40 m is shown in a simplified manner in Fig. 4, which shows only the principal building blocks. Among similar instruments, the 40 m’s long arms and the comparatively large beam spot sizes on its input and EMs make the 40 m an important subject for investigating optical scatter. Its optics were produced to the same standard as full-size GW detector mirrors, bringing its optical configuration closer to LIGO than any other prototype instrument.

We investigate optical losses in the FP arm cavities of the 40 m to bridge the gap between modeled and measured optical scatter. For this we use a variety of complementary techniques, including:

- (i) profile maps of the mirror surfaces combined with wave propagation simulations (phase maps);
- (ii) TIS measurements as a function of position on the mirror surface;
- (iii) static optical cavity impedance measurement;
- (iv) power recycling gain constraints.

In the following section, we describe how these measurements are performed and what their systematic errors are, and determine if they can provide a complete picture of the scatter loss distribution.

#### A. Phase Map Measurements

A phase-shifting Fizeau interferometer (PSFI) [14] was used to measure the surface figure of our mirrors. In such optical profilometers, a laser beam reflected from the sample under test is interfered with a beam reflected from a reference flat. A phase map is then constructed by extracting the optical phase difference between the beams for every point on the surface from the interference pattern.

Phase maps for the 40 m mirrors (IM4, EM7) were recorded at the time of installation with no significant qualitative differences between the different optics. However, while indicating that phase maps of different optics can be used interchangeably for simulation purposes, these initial phase maps were found to not be accurate enough for the estimation of the arm loss, especially in the short spatial wavelength region.

To extend the spatial frequency range of our phase maps, new measurements were done using spare mirrors (EM6, IM1, IM2), which were polished and coated as part of the same batch as the installed optics by the same companies for each process. Unfortunately, the coated surface profiles in the long spatial wavelength region do show differences among mirrors, even though all were produced with the same polishing and coating process.

Since the currently installed mirrors cannot be remeasured in a PSFI without removing them from their suspension cages in the vacuum envelope, we synthesize their phase maps by combining low spatial frequency data from the original measurement with high spatial frequency data from the spare mirrors. This was done by blending the lowpass filtered original maps with highpass filtered new maps at a spatial crossover wavelength of 1.3 mm.

In a fashion similar to Eq. (15), 1D PSDs were computed from the 2D PSDs of the phase maps by averaging over the azimuthal angle. The PSDs of the newly measured phase maps of spare IMs, IM1, and IM2 are shown in Fig. 5(a). These maps were measured focusing on the short spatial wavelength region, $\lambda \lt 5\;{\rm mm}$. The comparison of the red (IM1) and orange (IM2) lines shows that the PSDs in ${\lambda _s} \lt 1.5\;{\rm mm}$ are the same. From this result, the same PSD can be assumed for the IM4, which is the one actually installed. The black line is a combination of the IM04 map (blue) original measurement with the IM1 new measurement. The cutoff for the hybridization is $\lambda = 1.3\;{\rm mm}$ (green dashed line).

A corresponding analysis was done for the EMs. Figure 5(b) is the comparison of PSDs of the old EM7 map (blue), new EM6 map (red), and the hybrid of these two (black) merged at $\lambda = 1.3\;{\rm mm}$.

These hybrid phase maps are used in a numerical simulation to simulate beam propagation [15] in a cavity. In the simulation, an input Gaussian laser beam, mode matched to the cavity, is bounced back and forth by the cavity mirrors loaded with phase maps until a steady state is reached. The beam propagation between the cavity mirrors is computed using Fourier optics methods. At steady state, the round trip loss is computed by integrating over the intensity of the beam outside the mirrors’ apertures. For the case of a 40 m cavity with mirrors loaded with the hybrid phase maps shown in Figs. 5(d) and 5(c), the round trip loss was computed to be $6 \pm 2 \;{\rm ppm}$ over the angular range accessible by this method.

#### B. Total Integrating Scatter Measurement

The TIS from an HR coating is measured using an integrating sphere. Figure 6 shows a schematic of the experimental setup.

The 1064 nm output of a JDSU NPRO laser is first sampled for a reference photodiode (PD-ref) and spatially filtered to generate a high-purity TEM00 mode. A sequence of a polarizer, quarter-wave plate, and chopper constrains its polarization and allows for lock-in detection. The collimated beam enters the top port of the integrating sphere, and a mode-matching lens is used to position the beam waist at the mirror’s HR surface. The reflected specular beam is aligned to exit the integrating sphere centered on the top port. The mirror is raster scanned, and the scatter from the HR mirror is measured by a signal photodiode (PD-sig) attached to the integrating sphere to obtain the TIS map. The beam diameter and the scanning step size are both 300 µm. The polar angle of scatter collection of the integrating sphere is defined by the sizes of the top and bottom ports, and the current setup has a collection range from 1.0° to 75°, corresponding to a bandwidth of spatial frequency 16 to $908\;{{\rm mm}^{- 1}}$. The calibration is carried out by placing a diffuse reflectance standard (Labsphere Spectralon) at the HR mirror position.

Figure 7 shows TIS measurements that were pereformed on four 40 m prototype interferometer spare mirrors (IM1, IM2, EM6, EM8). Figure 8 shows the histograms of the power fractions in Fig. 7. The histograms are peaked at ${\sim}7\;{\rm ppm} $ for the IMs and ${\sim}3\;{\rm ppm} $ for the EMs, and there is a long tail extending over 1000 ppm.

We calculate the total scatter loss in the arm cavities to angles between 1.0° and 75° by integrating the TIS map weighted by the arm cavity intensity profile ${| {{E_q}} |^2}$. The integral is evaluated by dividing the arm cavity beam area into small beamlets with an area of $\pi \omega _{{\rm TIS}}^2$ each, the area of the TIS beam. The scattered power from each beamlet is then ${\rm TIS} \times {| {{E_q}} |^2}\pi \omega _{{\rm TIS}}^2$ evaluated at the center of the beamlet. Summing over all these scattered powers and dividing by the total power in the beam ${P_0}$, we obtain the total scatter loss. Explicitly, the total scatter loss is given by

Doing this summation for each sample, we find that the scatter losses, to angles between 1.0° to 75°, are $10 \pm 1.7\;{\rm ppm}$, $8.6 \pm 0.4$, $4.6 \pm 0.03$, and $4.9 \pm 0.14$ in IM1, IM2, EM6, and EM8, respectively. The errors are calculated by shifting the beams randomly around the centers of the mirrors with a standard deviation of 1 mm and computing the standard deviation of the resulting losses.

#### C. Static Optical Impedance Measurement

Inferring cavity round trip losses from measuring either optical ringdown time constants or the linewidths of cavity resonances and accounting for the known mirror transmissivities has a significant caveat: the full uncertainty in the measured mirror transmissivities is directly transferred to the loss estimate. For GW detector arm cavities, which have moderately high finesse and are highly over-coupled [4–6], the uncertainty of the IM transmission is usually of the same order or larger than the expected losses.

In a measurement of the optical cavity impedance, also referred to as
the *DC method*, we compare the static
reflected power from the IM in two cases: the prompt reflection off
the IM when there is no arm cavity (because either the EM is
misaligned or the intra-cavity beam path is blocked), and when the arm
cavity is on resonance and locked to the laser carrier. The
combination of the two individual measurements allows to solve for the
cavity losses with reduced sensitivity to the uncertainty in IM
transmission. This method measures only the total optical loss in the
cavity and cannot distinguish absorption from scatter; however, the
optical absorption of the high-quality tantala-silica coatings used in
GW detectors and prototypes is of order 1 ppm or smaller and can be
safely neglected.

It carries similarities to the optical ringdown measurements carried
out in reflection described by Isogai *et
al.* [16], but is more
suited to the optical topology of a cavity-enhanced Michelson
interferometer, in which the IMC acts as a lowpass filter for any
modulations of the input laser. All modulators and shutters are
located upstream from the IMC, such that the fast switching of the
input power required for proper ringdown measurements would get
filtered by the IMC’s optical response. The IMC’s storage time is
longer than those of the arm cavities, such that the measured ringdown
time constants would be entirely dominated by the IMC, significantly
limiting the achievable accuracy on arm cavity effect estimation.

We consider the cavity reflection coefficient on resonance ${R_r}$ and off resonance ${R_m}$ and the coupling inefficiency of the input beam into the cavity. We find the expression for the ratio of the two reflected power measurements $\frac{{{P_r}}}{{{P_m}}}$ to be

where $\eta$ is the mode matching efficiency of the input beam to the arm, and $\gamma$ is the fraction of power in the carrier frequency after sideband modulation. The values for these parameters are given in Table 2. The expressions for ${R_r}$ and ${R_m}$ can be found in [16]. Assuming the loss contribution from the two cavity mirrors is the same, we extract the round trip loss from Eq. (18) numerically.For very low round trip losses and/or low finesse, such as in the case of the arm cavities in the 40 m, the reflected power ratio gets very close to one, which requires long integration times to get good enough resolution between the two power measurements. We repeat the arm lock/unlock cycle with re-alignment procedures in between to counteract drifts. During the measurement time, we track the changes to the optical system (pointing/alignment drifts, etc.) and correct for fluctuations in the input power by scaling the reflected power to the circulating power in the IMC. An example of such a measurement sequence is shown in Fig. 9. The top (bottom) graph shows the measurement sequence for the $y\;(x)$ arm of the 40 m. The two traces appear opposite in phase due to the fact that the $x$ and $y$ arm losses are on opposite sides of the $\frac{{{P_r}}}{{{P_1}}} = 1$ point. After initial measurements of the 40 m arms, which showed contamination with excess losses, the mirrors were cleaned with first contact solution [17]. Our best estimates for the arm cavity losses post-cleaning are $20 \pm 2.5\;{\rm ppm}$ for the $x$ cavity and $50 \pm 2.5\;{\rm ppm}$ for the $y$ cavity.

#### D. Power Recycling Gain Measurement

In power recycled interferometers, such as the 40 m interferometer prototype, the recycling gain, the ratio between the power circulating the recycling cavity and the incident power, is inversely proportional to the round trip loss in the arm cavities. For the ideal case, where the PRC is critically coupled, the power recycling gain $G$ is given by

where $F$ is the average arm cavity finesse, and $L$ is the average round trip loss in the arm cavities. The systematic errors in this technique can be quite small since the loss influence on the power recycling gain (PRG) is amplified by the finesse of the arm cavities. In the more realistic case where the PRC is not exactly critically coupled, the expression for $G$ becomes more complicated, but the insensitivity to the uncertainty in the IM transmission remains. The reason is that the PRG depends on the reflection coefficients from the arm cavities. The arm cavity loss appears in those coefficients multiplied by the finesse of the arm cavities [16], regardless of the details of the PRC.To obtain an estimation of the PRG, we measure the optical power transmitted through the arm cavities when the PRC is locked, ${P_{{\rm Locked}}}$, and unlocked, ${P_{{\rm Unlocked}}}$. The PRG is then

where ${T_{{\rm PRM}}}$ is the transmission through the power recycling mirror.One complication that arises is the uncertainty in the mode matching between the PRC and the arm cavities, which is hard to quantify. For the 40 m, we know that the mode matching between the input beam and arm cavities, measured when the PRC is unlocked, is 92% (Table 2). In our calculations, we assumed that the mode matching between the input beam and coupled cavitiy, which includes the PRC and the arm cavities, is the same. This assumption is consistent with the other loss and PRG measurements as can be seen below.

As mentioned before, the uncertainties in the PRG measurement can be quite small. Unfortunately, the optical loss in the 40 m’s PRC, which is required in the analysis, suffered a comparably large systematic uncertainty at the time of measurement. Due to coating stress, the folding mirrors of the 40 m PRC have unintended marginally convex HR surfaces, which drives the original PRC design into the geometrically unstable regime. Flipping the folding mirrors solved this stability issue, but as a result, the back-side AR coatings and mirror substrates are now part of the optical path in the PRC, which adds extra losses that are not fully characterized. Especially, the residual reflection of the AR coatings adds significantly to the PRC loss uncertainty. As a result, the uncertainty in the inferred arm cavity loss is quite big. Our best estimate for the arm cavity loss from the PRG measurement, after the mirrors were cleaned, is $37 \pm 16\;{\rm ppm}$, consistent with the average loss measured using the static optical impedance measurement.

The viability of the method can still be demonstrated. Figure 10 shows the theoretical dependence of PRG on the average arm cavity loss computed using a finesse model [18] that includes the losses in the PRC, their uncertainty, and the non-ideal mode matching between the PRC and arm cavities. On top of that plot, the average arm cavity loss, measured using the DC reflection method, and the measured PRG are shown as data points. The measurements before and after the cleaning of the mirrors in the 40 m arms using first contact are colored in orange and blue, respectively. The widths of the error bars indicate the $1\sigma$ uncertainty of these measurements. It can be seen that the intersections of these measurements agree quite well with the theoretical line, demonstrating the consistency of these measurement techniques.

## 4. COMPARISON OF MEASUREMENTS

In the previous sections, we discussed several direct (DC reflection, PRG) and indirect (mirror phase maps, total integrated scatter, scattering measurements) loss measurement techniques. The results of the comparison of these methods are summarized in Table 3. The measured round trip loss using the DC reflection method after the mirrors were cleaned is quite low, making it suitable for comparison with the indirect methods. As described in Section 3.D, it is consistent with the PRG measurement.

To compare the indirect measurement results to the direct ones, some highly non-trivial assumptions need to be made. First, we define scattering angles where less than 1 ppm is reflected back from the mirror at the other end of the cavity as high-angle scattering. For the 40 m arm cavities, the boundary is found to be ${\sim}{0.065^ \circ}$. At scattering angles above this boundary, it is safe to use TIS to measure the loss. For scattering angles below that boundary, cavity simulations using phase maps are needed to calculate the round trip loss (see Section 3.A). In our simulations, the maximum scattering angle is ${\sim}{0.1^ \circ}$, above the required high-angle cutoff. Second, for high-angle scattering loss, we use the measurements of other mirrors from the same vendor. The reason is that while specifics of the mirror coating greatly affect the low-angle scattering behavior, high-angle scattering is largely unchanged from that of the mirror substrate. Another consistency check can be made by comparing the TIS estimated from the BRDF data shown in Fig. 3 and the TIS measured separately using an integrating sphere. The results, as can be seen in Table 3, are consistent.

To obtain the round trip loss from indirect measurements, we sum together the following terms: low-angle loss, IM average TIS measurement, EM average TIS measurement, and ${2} \times$ TIS estimation from the BRDF ${(0.1^ \circ} \lt \theta { \lt 1^ \circ})$. The result sums up to $28 \pm 10\;{\rm ppm} $, which comes close to the direct measurement of $35 \pm 3\;{\rm ppm} $ but has a large uncertainty, largely due to the uncertainty in the ${0.1^ \circ} \lt \theta { \lt 1^ \circ}$ angular region. The careful measurement of that loss contribution should be a high priority for future studies.

To put our results in perspective, we plot in Fig. 11 the loss measurement per cavity length, the figure of merit for filter cavities [24], together with previous loss measurements of low-loss cavities [16,19–23,25] as a function of the equivalent confocal cavity length defined as

where $w$ is the average beam waist on the optics, and $\lambda$ is the wavelength of the input laser beam. The choice of equivalent confocal length instead of the actual cavity length is made to remove the dependence of the loss on the cavity geometry.## 5. CONCLUSION

We have shown here that our simulations accurately estimate the empirically measured optical losses in the large beam regime (laser beam spot sizes $\mathbin{\lower.3ex\hbox{$\buildrel \gt \over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} 1\;{\rm mm} $ radius), which is relevant for long optical cavities. In this regime, the losses are mainly due to figure errors of the mirror, rather than the microroughness. This work establishes that our simulations can be reliably utilized to predict the performance of large optical cavities, such as the ones used in GW detectors.

In the future, it will be important to make measurements in the small-angle regime to accurately assess the losses due to coherent beaming from multiple point defects.

## Funding

National Science Foundation (PHY-0823459, PHY-1764464); Science and Technology Facilities Council (ST/V001019/1, ST/V005693/1); Australian Research Council (CE170100004, DE210100550).

## Acknowledgment

We acknowledge support from GariLynn Billingsley and Liyuan Zhang for supplying mirror surface maps and TIS measurements, and the many valuable conversations we have had with the Optics Working Group in the LIGO Scientific Collaboration. This material is based upon work supported by the National Science Foundation’s (NSF’s) LIGO Laboratory, which is a major facility fully funded by the NSF.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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