## Abstract

The second generation of ground-based interferometric gravitational wave detectors are currently being built and installed. They are designed to be better in strain sensitivity by about a factor 10 with respect to the first generation. Light originating from the laser and following unintended paths, called stray light, has been a major problem during the commissioning of all of the first generation detectors. Indeed, stray light carries information about the phase of the emitting object. Therefore, in the next generation all the optics will be suspended in the vacuum in order to mitigate their associated stray light displacement noise. Despite this additional precaution, the challenging target sensitivity at low frequency which is partially limited by quantum radiation pressure combined with up-conversion effects, requires more detailed investigation. In this paper, we turn our attention to stray light originating from auxiliary optical benches. We use a dedicated formalism to compute the re-coupling of back-reflected and back-scattered light. We show, in particular, how much care should be taken in designing and setting requirements for the input bench optics.

© 2013 Optical Society of America

## 1. Introduction

Stray light may affect the performances of many optical systems. Such kind of issue often originates from electro-magnetic radiation external from the system itself. But, in some cases, the stray light is simply the light from the source of the system itself that is following a different path from the intended one. This can happen through various processes: residual reflection from anti-reflective coatings, scattering from imperfect mirror surfaces, scattering from surface defects (like dust, digs or scratches), scattering from the enclosure of the optical system, and diffraction from the limited aperture of the optics. These processes can actually play a major role and need to be taken into account when designing demanding optical systems such as instruments for astronomy [1] or interferometers for Gravitational Wave (GW) detection [2, 3].

Stray light in GW detectors has long been identified as a serious issue [4] and was taken into account in the design of the first generation of GW interferometers [5–7], but it remains a major concern for the Laser Interferometer Space Antenna [8], as well as for the second generation of ground based interferometers [9].

Indeed, all first generation GW interferometers have experienced problems during their commissioning phase [10]. By carrying out specific measurements, many of these issues have been identified as originating from the scattering or even the specular reflection of some optics at different ports of the interferometers [11–13]. To fix these issues, it was fundamental to calculate the amount of light re-coupling to the main interferometer laser beam from each single optic [14]. Besides, while the first generation stray light displacement noise was critical through the phase modulation of the stray light, the second generation will suffer more severe constraints also through the amplitude modulation of the stray light due to the radiation pressure effects as pointed out in [9]. In the same reference, the effect of noise up-conversion is underlined when high microseismic conditions are taken into account, setting even more stringent requirements. It is then crucial to calculate accurately the light which re-couples to the interferometer for every single optical element in order to set its optical requirements or even make important optical setup design choices.

In this paper we will present a complete formalism to calculate the amount of light that is scattered or reflected once by an optic and directly re-couples to the interferometer. The formalism does not apply in case of multiple scattering events: for this specific case, a detailed study can be fond in [5] for instance. This re-coupling depends on which port of the interferometer the considered element is installed in. It also depends on the geometrical re-coupling of the back-reflected light on the fundamental mode of the interferometer. The same is true also for the back-scattered light. This last contribution is particularly problematic to quantify as we lack a simple model. We will present also which parameters to take into account and how to calculate how the scatterer motion translates into displacement noise at the interferometer output. As the most critical elements will be those placed at normal incidence, we expect the lenses of the input and output mode matching telescopes of Advanced Virgo [15], which consist of a meniscus lens and two parabolic mirrors, to be particularly critical. We will therefore apply the formalism to calculate the displacement noise produced by the meniscus lens of the input mode matching telescope of Advanced Virgo, and discuss about its relevant optical requirements. Of course, the treatment presented in this article is general and the method may be applied to any component of the optical setup of the auxiliary optics of the interferometer, in order to check if its optical specification and seismic isolation are compliant or not with the target science sensitivity.

## 2. Recoupling of stray light emitted from the external benches-Formalism

#### 2.1. Coupling mechanism of stray light

The laser light which exits from one of the output ports of an interferometer will travel through several optical elements. These elements can couple back some light into the interferometer port due to back reflection or scattering. This back propagating light enters into the instrument and can recouple with the main mode, adding a spurious field to the main one. This field will be in general not constant, but it will contain phase and amplitude information linked to the optical element motion. Since the amount of stray light is typically very small, the additional noise to strain amplitude *h* will depend linearly on the amplitude of the field and we can write:

*f*is the fraction of the light power exiting one of the interferometer ports which is coupled back into the main mode and

_{r}*z*is the position of the reflecting object. It is not possible in general to assume that the object motion is small and therefore the coupling function

_{r}*G*is not linear in the displacement

*z*. To properly evaluate the impact of stray light to the detector sensitivity it is therefore necessary to compute the fraction

_{r}*f*and to fully understand the coupling function

_{r}*G*, including non-linear effects. In this section we describe how to characterize

*G*, while in the following ones we show a way to compute

*f*in the case of specular reflection (

_{r}*f*) and diffusion (

_{sp}*f*).

_{sc}In the following, we will note *E _{in}* the field inside the interferometer, at a reference point which is relevant to compute the coupling of additional stray light to the detector sensitivity.

*E*is the field exiting from one of the interferometer ports at the location of the stray light source.

_{out}*E*is the part of this field which is scattered or reflected back and recoupled with the main interferometer mode at the reference point where

_{r}*E*is computed.

_{in}The motion *z _{r}* of the stray light source will be encoded in the phase of the recoupled field

*E*

_{r}The motion of the stray light source can span over multiple wavelengths, therefore the phase can be large and wrapping many times around 2*π*. This introduces a large non linear behavior of the stray light noise coupling.

*E _{r}* will sum to the unperturbed field inside the interferometer

*E*:

_{in}*ϕ*

_{0}is a static phase which depends on the scattered position in the interferometer. If we are considering the input port of the interferometer, the

*E*field can be identified with the one corresponding to the input beam. When considering other ports which are not used to inject the main beam into the detector, one should carefully compute the coupling of the back-propagating field with the main one inside the interferometer. In this case the

_{in}*f*term must take into account the propagation of the field to the reference position inside the instrument. In both cases, the net effect is an additional phase and amplitude modulation of the interferometer field:

_{r}*ϕ*

_{0}:

*x*≃

*e*. Moreover, the exponential term depending on the phases has been splitted in the real and imaginary part. The dominant noise coupling can be phase or amplitude, depending on the static phase

^{x}*ϕ*

_{0}, or in other words depending on the microscopic position of the stay light source relative to the interferometer. If the motion of the scattering element is much larger than one wavelength, the distinction between

*ϕ*

_{0}and

*ϕ*(

_{r}*t*) is irrelevant and the noise coupling will proceed through both phase and amplitude.

The effect of stray light is therefore described by additional phase and amplitude noises at the interferometer considered port given by

*G*is completely encoded in the above equations, that can be used to compute the phase and amplitude noise that enters the interferometer at the considered port. The noise can then be propagated to the detector sensitivity using the linear response of the system to additional phase and amplitude noise sources placed at the reference position. In the worst case scenario the contribution of each of the two noise sources to the sensitivity is given by

*h̃*(

*f*) is the amplitude spectrum of the noise as measured in the sensitivity,

*ℱ*indicates the Fourier transform and

*T*(

*f*) is the linear response of the interferometer to phase or amplitude noise.

The interferometer transfer functions *T*(*f*) depend strongly on the optical configuration of the system. In particular, advanced gravitational wave interferometers are very sensitive to all asymmetries between the two arms: mirror radii of curvature, round trip losses and Fabry-Perot cavity finesse. Moreover, advanced detectors will feature a large amount of stored power (800 kW inside the km-long arm cavities) that will introduce significant radiation pressure effects: any field fluctuation can induce a force on the test masses, which significantly change the detector response [16]. For all these reasons it is mandatory to use detailed optical simulations to compute the response of the interferometer to phase and amplitude noise. An example for an interferometer with Advanced Virgo like parameters is given in Fig. 1. Moreover, the noise entering from some of the interferometer ports might be affected by the action of the feed-back control systems that are needed to actively maintain the detector at its working point. In the following sections of this paper we will focus our attention on stray light coming from the main input port of the interferometer. In this case the phase noise introduced by any stray light source will be suppressed by the frequency stabilization control [18] and the dominant contribution will be the amplitude noise. Indeed advanced detectors will implement a DC readout scheme [19], which is particularly sensitive to amplitude noise at the input.

In the commissioning of first generation detectors the main coupling path was through phase modulation of the main interferometer field. Therefore the *G* function was considered to be simply [12]

*k*was assumed to be a frequency independent coefficient describing the coupling of the back propagating field with the main one. This model worked well in the case of Virgo and Virgo+ [12, 20, 21]. In the case of Advanced Virgo, stray light noise will be characterized by the frequency dependent coupling factors $T\left(f\right)\sqrt{\frac{{f}_{r}{P}_{\mathit{out}}}{{P}_{\mathit{in}}}}$ for both amplitude and phase noises. We computed these factors using a simulation based on Optickle [22] and the results are shown in Fig. 1 supposing

*f*= 1. The unphysical case of no radiation pressure effects is also shown for comparison. We observe that these effects play a major role for frequencies below 100 Hz. In order to project displacement noise of a stray light source on detector sensitivity, we need to calculate the fraction of the light power

_{r}*f*which is coupled back into the main mode. In the following sections, we carry out the calculation both for scattered light

_{r}*f*and specular reflection

_{sc}*f*.

_{sp}#### 2.2. Calculation of scattered light recoupling f_{sc}

In this section, we explain how to calculate the fraction of scattered light f* _{sc}* recoupled to the main mode of the interferometer starting from the BRDF (Bidirectional Reflectance Distribution Function) of the diffusing optic considered. Scattered photons from the optics placed on the external benches are propagated inside the interferometer using geometrical optics and re-coupling is then calculated by numerical integration.

### 2.2.1. f_{sc} as a function of surface BRDFs

_{sc}

A single photon scattered from the diffusing surface can be described by its emitting position (r, *α*) and its normalized emitting direction (*ϕ* cos*θ*, *ϕ* sin*θ*, 1) (see Fig. 2).

Considering an radial symmetry around the optical axis, we define the boolean recoupling function *F*(*r*, *θ*, *ϕ*) which is equal to 1 if the photon is recoupled, 0 otherwise. The power recoupled from an element of surface dS of the diffusing object can be written:

*P*is the power incident on the surface dS:

_{i}*d*

^{2}

*Pi*=

*rI*(

*r*)d

*α*d

*r*. The total power recoupled

*P*is obtained by integration over the whole surface:

*π*comes from integration over

*α*, given the radial symmetry.

*f*is then obtained by the ratio P over the incoming power P

_{sc}_{0}:

### 2.2.2. Calculation of the recoupling function

In this section we explain how to calculate *F*(*r*, *θ*, *ϕ*). To do this, we carry out the propagation of the photon inside the optical system placed on the external bench. Then the photon is propagated through the interferometer cavities. The photon is considered resonant if it couples with both systems. It means that it should not go outside the different optical apertures.

In the case of the cavity propagation, the photon should make a number of round trips of about *F*/2*π*, where *F* is the cavity Finesse, without going outside the mirror coatings. As the coating size is set to about 6 times the Gaussian beam radius on the considered mirror, this criteria guarantees that the photon remains within the cavity mode. All the photons verifying this criteria are then considered to interfere constructively with the cavity field which represents the worst possible case. The sum of all these photons (given by Eq. (13)) may create a field that has the right phase and spatial distribution to re-couple in the most efficient way inside the interferometer. The quantity *f _{sc}* therefore provides an upper limit of the recoupling.

In order to check if a given photon emitted by the scattering surface (that we suppose here placed perpendicularly to the beam propagation, which is the worst case) superates the above criteria, we calculate its coordinates *x _{i}*,

*y*,

_{i}*z*in the plane of the optical component number i using geometrical optics and check if ${\left({x}_{i}^{2}+{y}_{i}^{2}\right)}^{\frac{1}{2}}<{a}_{i}$ where

_{i}*a*is the radius of this component.

_{i}More into details, a photon at any instant of its propagation can be described by its position and direction summarized in the 6-vector:

It can be easily shown that the propagation matrix of a free-space distance *d*, and the propagation matrix through a thin lens of focal *f*, respectively named *P*(*d*) and *L*(*f*) can be written:

*M*of type

_{i}*L*or

*P*. These matrices enable a fast calculation of the full propagation matrix

*R*on the element i. ${R}_{i}={\prod}_{j=1}^{i}{M}_{i}={M}_{i}{R}_{i-1}$. Then we have:

_{i}*a*is the optical aperture of element i,

_{i}*u*

_{0}the vector representing photons emitted by the diffusing surface and where the norm corresponds to the distance of the photon from the optical axis:

Knowing the BRDF of the diffusing surface and the recoupling function *F*(*r*, *θ*, *ϕ*), the fraction of scattered light f* _{sc}* recoupled to the main mode of the interferometer is calculated using Eq. (13) and Monte-Carlo numerical integration.

### 2.2.3. Example of calculation using AdV cavities

As said before, the recoupling f* _{sc}* obviously depends on the optical setup of the input or output optics, the geometry of the interferometer cavity and the BRDF of the optic considered. In order to draw some general consideration on the parameters influencing the recoupling, we consider here a simplified case presented in Fig. 3.

In this case, we calculate the quantity of light emitted by a diffusing element (Total Integrated
Scatter, TIS=1) placed on the symmetric port of the interferometer (ITF), before the
recycling mirror. We will suppose here that the surface is Lambertian, which means it has a constant
BRDF. This is a realistic supposition for all optics tilted by a macroscopic angle (about
5–10°) such as most of the mirrors or photodiodes. Indeed, in such cases, only large
angle scattering will contribute to cavity recoupling. In the case considered here: TIS=1,
we have BRDF=1/*π*. Using formalism presented in the previous
paragraph, we derive in Table 1 the propagation matrix of a
single photon emitted by the diffusing surface on ith surface.

Using *R _{i}* matrixes and the relative optical apertures, it is possible to obtain the function

*F*(

*r*,

*θ*,

*ϕ*) using Eq. (17) and then to calculate

*f*. Figure 4 shows the value of

_{sc}*f*as a function of distance of the diffusing object with respect to the PR mirror.

_{sc}We observe that the recoupling varies of a few orders of magnitude as a function of distance to PR mirror and reaches a maximum at waist location.

From this simplified example, we see that much care should be used on the external benches to place the most diffusing optics far from the waist location in order to lower recoupling. Furthermore, optical designs with small beam waists should be avoided. Indeed, the recoupling at the waist location can be approximated by the multiplication of the solid angle of the beam aperture with the BRDF [23]:

#### 2.3. Recoupling from reflection f_{sp}

Light back reflected towards the interferometer by specular reflection may also recouple to the interferometer mode and generate noise on dark fringe signal by the mechanisms explained in section 2.1. The fraction of recoupled light *f _{sp}* from an optical element with reflectivity

*α*is given by the overlap integral:

*φ*and

_{i}*φ*are respectively the incoming and reflected laser fields. We consider the situation described in Fig. 5 where the incoming beam has a Rayleigh range ${z}_{0}=\pi {\omega}_{0}^{2}/\lambda $ and the reflecting element is placed at a distance D from the waist position. As the reflecting element can be for example a lens, we consider that the surface has a curvature R and is slightly tilted with respect to the optical axis by an angle

_{R}*β*. Equation (20) can be computed analytically assuming the input mode is a pure gaussian beam:

## 3. Application to Interferometer mode matching telescope of AdV

The choice of a marginally stable recycling cavity configuration for Advanced Virgo [15] required the development of a very compact mode matching telescope with a large magnification: around 19 for the whole interferometer mode matching telescope since the beam radius has to be increased from 2.6mm up to 49mm. As no large optics could be placed on the suspended injection bench, this implied to achieve part of the magnification using a curved anti-reflective surface for the Power Recycling mirror. The setup chosen is detailed in Fig. 6. It consists of an afocal off-axis parabolic telescope made of 2 mirrors (PM_{1} and PM_{2}). Then a diverging lens (meniscus lens) is used in combination with the PR mirror to obtain an interferometer matching as good as 99.9% [24]. One of the main concerns of this configuration is that the meniscus lens is placed perpendicular to the laser beam and may be an important source of stray light that can spoil the interferometer sensitivity [11]. The polishing quality as well as the low reflectivity of this optic are therefore major issues. This will be studied in this section calculating the coupling factors by diffusion and reflection of the meniscus, and projecting the associated noise on the interferometer sensitivity using the formalism presented in section 2.

#### 3.1. Coupling factor f_{sc} for diffused light of the Meniscus

In order to calculate the recoupling f* _{sc}* of the Meniscus using Eq. (13), we need to know the BRDF of the surface which will be determined in the following from the polishing specifications.

The BRDF of an optical surface illuminated at normal incidence can be expressed from its Power Spectral Density (PSD) by [25]:

*and f*

_{x}*are the spacial frequencies of the surface roughness in cartesian coordinates. In the following, we will consider that the surface has an isotropic roughness and therefore, the function PSD(f*

_{y}*, f*

_{x}*) has a radial symmetry. The PSD can be therefore expressed in polar coordinates (f,*

_{y}*γ*), with $f=\sqrt{{f}_{x}^{2}+{f}_{y}^{2}}$ and

*γ*= arctan(

*f*/

_{y}*f*). It is then common [25] to work with a simplified isotropic PSD (PSD

_{x}*) obtained by integration over*

_{ISO}*γ*:

In order to reduce scattering issues, superpolishing will be required for this optical component. As shown by Church [26], highly finished optical surfaces can be described by fractal models and their Power Spectral Densities expressed by an inverse power-law of the form:

The value of n for Fused Silica is found in the literature to range from 1 to 1.5 [26–29] while K can be determined by integration over all surface spatial frequencies and expressed as a function of total surface roughness*σ*[25]: Using the PSD model given by Eq. (25), the BRDF can be expressed: As a single spacial frequency f will create a scattering at a single angle

*ϕ*, with

*f*= sin

*ϕ*/

*λ*, the BRDF can therefore be expressed as a function of scattering angle

*ϕ*:

Finally the BRDF of a curved surface will depend on its radius of curvature R and may be obtained by a variable change:

*. As the spectral index n is not a-priori known and will depends on the polisher, calculations were carried out for different values of n between 1 and 1.5. Supposing that the surface will present a total roughness*

_{sc}*σ*of 10 nm RMS, we derive the spectral strength K for each value of n using Eq. (26).

Results for the two surfaces of the meniscus lens (of radius R_{1}= −1.07
m and R_{2}= −3.195 m) are given in Table
2 supposing the use of an anti-reflection coating of 100 ppm. We observe that
f* _{sc}* hardly depends on the spectral index n.

#### 3.2. Coupling factor f_{sp} for back-reflected light of the Meniscus

The other important aspect to be studied is the amount of back-reflected light
*f _{sp}* coming from the meniscus lens surfaces that is recoupled into the
interferometer. In order to compute

*f*, we determined the parameters (waist size and position) of the beam incident on the 2 surfaces of the meniscus lens and we used Eq. (21). Figure 7 shows the parameters considered in this computation. The beam coming from the interferometer (black line of Fig. 7) has a waist

_{sp}*ω*

_{01}located at D

_{1}from the meniscus lens surface 1. Reflection of this beam on the first surface of the meniscus recombines with the interferometer with a coupling factor

*f*

_{sp1}. Transmission through the first surface of the meniscus lens then creates a beam of waist

*ω*

_{02}located at D

_{2}from the second surface. Reflection of this beam on the second surface of the meniscus lens may recombine to the interferometer main beam with a coupling factor

*f*

_{sp2}. Beam parameters used for calculation of

*f*

_{sp1}and

*f*

_{sp2}were computed using Zemax and are given in Table 3.

Considering that the reflectivity of the meniscus surfaces is only 100ppm (what we can expect from a good anti-reflective coating), we obtain for a perfectly aligned optic (*β* = 0) *f*_{sp1}=3.5.10^{−11} and *f*_{sp2}=5.2.10^{−10}.

Since in reality there is a small error on optical alignment, it may be interesting to study the variation of *f*_{sp1} and *f*_{sp2} with *β* which is shown on Fig. 8. We observe that *f*_{sp2} drops quickly when the misalignment is around 1 mrad. The difference between the *f*_{sp1} and *f*_{sp2} curves is due to the size of the waist reflected by the two surfaces: the bigger waist to recouple, the more dependent it will be on alignment with the incident beam. This property is useful for all flat optics placed on the external benches and placed on setups using a millimeter-scale beam size: their recoupling by reflection will become negligible for very small misalignment.

In the following, we will use these calculated values of back reflected and back diffused light recoupling to project the associated displacement noise of the meniscus lens on the interferometer sensitivity using the formalism presented in section 2.1.

#### 3.3. Noise projection on AdV sensitivity

Stray light from the telescope meniscus lens may spoil the interferometer sensitivity. The calculation carried out in the previous section showed that this noise will be dominated by back reflection of the second surface of the meniscus lens with a recoupling of *f*_{sp2}=5.2.10^{−10}. Calculation of displacement noise introduced by the meniscus lens will therefore take into account only back reflection.

To make the noise projection on the AdV sensitivity curve, we have to make an estimation of the telescope optics horizontal motion. For this purpose, we have used an upper limit of ground seismic motion in case of high micro-seismic activity as presented in [30]. This noise was filtered to take into account the effect of the super-attenuator. The low frequency ground motion has been filtered by a factor of 100 between 0.1Hz and 1Hz as one can expect from the performances of Virgo super-attenuator inertial damping system [31]. The suspension chain also filters out the horizontal noise using the working principle of a pendulum that is filtering the noise above its cut-off frequency. Thus, the linear spectral density of the suspension point displacement decreases following the law 1/*f*^{2}* ^{n}* where n is the number of attenuation stages [32]. The Virgo super-attenuator has been designed to start to filter out the noise above a few Hz. So in our model, we used 1Hz as the cut-off frequency. In Fig. 9, the spectral density of the estimated upper limit of horizontal displacement of the optics installed on a suspended bench is shown. We considered having 3 stages of horizontal filtering thus the slope of the filter above the cut-off frequency follows 1/

*f*

^{6}.

To make the noise projection, we used Eq. (7), Eq. (8) and the transfer function between phase noise and h and amplitude noise and h given in Fig. 1. In this case, we have not considered any asymmetries of the interferometer but all the work presented here can be easily redone considering ITF asymmetries by just recomputing the transfer function as explained in section 2.1. We have also considered that the laser power incident on the ITF is *P _{in}*=125 W and the power reflected by the ITF

*P*is 20% of the incident power

_{r}*P*. The projection of noise coming from stray light from INJ telescope optics on AdV sensitivity curve is given on Fig. 10.

_{in}As it can be seen in the noise projection given in Fig. 10, the amplitude noise is the most worrisome noise coming from the input port. Indeed, in the worst ground seismic conditions, which are plausible for less than 5% of the time, the sensitivity may be limited at 10Hz by stray light coming from the meniscus lens. Therefore particular care will have to be taken with the suspension system of the benches to be sure that this system is working with optimal performance and in particular that all the low frequency resonances are properly damped and controlled. As explained in sections 3.1 and 3.2, the quality of the anti-reflective coating deposited on the meniscus lens surfaces is also particularly important since this is reducing the amount of light coming from back-reflected and back-scattered light recoupled in the interferometer. In the present work, we presumed a coating reflectivity lower than 100ppm that is already close to the best that can be achieved. For what concerns the phase noise, on Fig. 10 we see that it is expected to be a factor of 10 smaller than the amplitude noise. Moreover, in the interferometer standard operational mode, the phase noise coming from the interferometer input port is canceled out by a frequency stabilization loop [18] and in consequence we can reduce by several orders of magnitude the phase noise projection on AdV sensitivity.

Finally, particular care should be taken designing the optical benches located at the other exit ports of the interferometer (dark port, end station ports) that have demonstrated to be the most critical ones in the first generation of detectors. The formalism presented in this paper could be used to compute in the same way the contribution of those ports to the Advanced detectors noise budget.

## 4. Conclusion

In summary, we have discussed how the stray light originating from auxiliary optics is a challenging issue for Advanced GW detectors for achieving target sensitivity at low frequency. This issue has to be taken into account at the design phase of these instruments as optics have limited performance in term of scattering and Anti-Reflective coating. We have proposed a complete formalism to calculate the displacement noise induced by the stray light originating from any optics, and applied this formalism to one of the critical components of AdV. We believe that other optical components at other ports could be even more critical. In such a case, if neither the optical setup can be changed, nor the optical performance of such a component can be improved, the control of the low frequency relative motion between the interferometer and the critical optical component should be further improved.

## Acknowledgments

The research activity of one of the authors (G.Vajente) has been partially supported by Regione Toscana (Italy) through the program POR CreO FSE 2007–2013 of the European Community, within the project n. 18113 (ISAV).

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