1. Introduction

Scattered light in high-sensitivity interferometers is an often discussed and well known issue, and one of the main limitations of the sensitivity of such instruments [1–3]. Especially after the recent detection of gravitational waves from two merging black holes [4], it is clear to what sensitivities one has to reach to join a global network of gravitational wave detectors. Thus, our efforts are focused on increasing the sensitivity of KAGRA, and the reduction of stray-light noise is a main part of these efforts.

The reason for scattering is the interaction of light with surfaces having irregularities, which can be characterized by a root mean square (rms) roughness, σ. Though the specific behavior of light that hits a surface with a random roughness is more complicated to describe and follows statistical laws in the same way as the surface, it can be described statistically with surface-height distributions and corresponding auto-covariance functions. According to Harvey et al. (2012) [5], the total integrated scattering (TIS), the ratio of the scattered power to the total reflected power, can be written as a function of σ, being a factor in the Gaussian-like exponent of the TIS and defining the total amount of scattering for a given wavelength and angle of incidence (AOI).

As gravitational wave detectors based on interferometry like KAGRA use many different optical elements [2, 3, 6], scattering from these elements is an issue almost everywhere. Even though the optics – mirrors especially – are well polished and smooth, scattering in general cannot be avoided. This is due to the nature of surface treatments (like polishing) or the randomness of environmental conditions during the fabrication of materials (e.g., temperature differences on a microscopic scale). For a high-sensitivity interferometer, this can become an issue. Stray light that is guided back into the main laser beam by, e.g., a mirror, can affect the final measurement when its phase information carries frequencies which are different from the main beam [7]. These frequencies come from surfaces other than the mirror, which can further scatter the primary scattered light from the mirror. Back-scattered light can reach the mirrors a second time and couple back into the main laser beam, now carrying phase information from those surfaces.

As previous works by Flanagan and Thorne [8] or Vinet et al. [7] pointed out, such recombined scattered light can harm the sensitivity of gravitational wave detectors, especially when the recombination takes place at the main mirrors inside the cavities of such instruments. Proper control of scattering is thus mandatory. Such control can be achieved either by reducing the power of the scattered light (for instance by using absorbing coatings on the surfaces) or by reducing the phase distortions (by suspending the scattering surfaces), but is in any case connected to an increase of cost and time, which are both limited in the construction of gravitational wave detectors. A key aspect is, thus, always a proper understanding of how the scattered light influences the sensitivity and whether or not a better control of scattering is necessary.

In previous works concentrating on the problem of scattering in gravitational wave detectors, it was always assumed that the mirror is just a smooth surface with small irregularities and that the distribution of scattered light from it is just determined by the surface’s power spectral density (PSD) [7, 8]. However, the mirrors that are used in KAGRA and other gravitational wave detectors like LIGO (USA) and VIRGO (Italy) have multilayer coatings. Instead of being just a single bulk of polished metal and having only one surface, there are many interfaces between layers of different dielectric materials (usually silica and tantala) which interact with the incoming light. Multilayer coatings are needed to get very accurate reflectivities at certain wavelengths of the incident light, and ultra-high sensitivity interferometers like KAGRA usually use them [9]. However, multilayer coatings lead to multiple reflections of scattered light within the coating and thus to positive and negative interferences at different scattering angles [10–13]. The angular distribution of the scattered light differs distinctly from that of pure surface scattering. These coatings also influence the polarization of the scattered light. For a comprehensive study on the effects of scattering in gravitational wave detectors, it is thus important to take into account also the multilayer coatings. Therefore, in this work we will apply the well known theory of scattering in multiple layers by Elson et al. and Bousquet et al. [10, 11] and compare it with a simplified GHS (generalized Harvey-Shack) theory for single-surface scattering similar to what has been previously used in the field of gravitational wave detectors. Moreover, we will show that the differences between these two cases affect the overall power that recombines with the main beam, especially when the scattering surface is very close to the mirror and that the assumption of a single-surface mirror is often not reasonable enough for the calculation of the scattering noise in the sensitivity of KAGRA.

As a basic example, we apply the theories to the so-called recoil masses of the power- and signal-recycling mirrors (PR and SR) and the beam splitter (BS) of KAGRA. Recoil masses are a part of the seismic vibration isolation system of the mirrors. They are additional structures that allow control of each mirror’s position by electromagnetic actuators without influencing other parts of the vibration isolation system. The recoil masses in KAGRA surround the mirror and have nearly the same center of mass to further compensate forces coming from the mirror actuation [14]. Due to the limited data available so far for the mirrors that will be used in KAGRA, we could concentrate our work only on a few chosen ones. In Fig. 1, diagrams of these mirrors are presented: (i) the PR and SR and (ii) the BS mirror. The advantage of taking these as examples is that, first, we have already performed simulations on the scattering of recoil masses in KAGRA and thus the models are already existing and well understood. Second, the recoil masses are very sensitive to the distribution of scattered light of the mirror due to their closeness to it. Differences in scattering distributions are therefore more easy to see.

 

Fig. 1 Cross-sectional and front views of the PR/SR-mirrors (left) and the BS-mirror (right) with their respective recoil masses. The diameter of the respective mirror is given in mm in the front view.

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The structure of the paper is as follows. In the second section, we will give an overview of the theory of scattering, focusing on the needs of this work. In the process, we treat the theory of surface characterization and how stray light may influence the sensitivity of interferometers like KAGRA. The third section is focused on the spatial scattering distributions of the PR, SR, and BS mirrors according to the theory. In addition to that, we present actual results of simulations showing the influence of back-scattered light from the mirror’s recoil masses on the sensitivity of KAGRA. Sections four and five are for the discussion and the conclusions.

2. Theoretical approach

2.1. Scattering on a mirror

2.1.1. Bidirectional reflection distribution function (BRDF)

If a laser hits the surface of a mirror, a small portion of the light will be scattered due to the mirror’s surface properties (for high-quality mirrors, TIS ∼ 10⁻⁴–10⁻⁵). The angular distribution of scattered light, which is described by the BRDF, depends on the individual surface structure of the mirror and thus is hard to describe in general (for a more basic definition of the BRDF, see appendix A).

It is possible to model those structures with the aid of the two-dimensional surface power spectral density (PSD) which can be calculated by taking the Fourier transform of the profile of a given surface [15]. For a simple surface profile, the BRDF can then be calculated with the aid of different models (all using the PSD of the respective surface). Examples for scattering models that were often used to estimate the scattering profile of mirrors in the field of gravitational wave detectors are the theories of Rayleigh-Rice (RR), Beckmann-Kirchhoff, or GHS, which all give the amount of scattered light as a function of the incident angle of a given light field under different constraints [5, 16–18]. For smooth surfaces, where σ is considerably smaller than the wavelength of the incident light [5, 19, 20], usually the RR theory is used. In contrast to it, the GHS theory in its smooth surface approximation predicts non-vanishing BRDF toward 90° latitude but is for smaller angles quite similar to RR [18, 19, 21]. As it is the purpose of this paper to analyze the differences of single-surface scattering and multilayer scattering on the sensitivity of KAGRA when we include structures like recoil masses, which can be reached only under very wide scattering angles, we decided to use the GHS model in its smooth surface approximation rather than the RR theory. With this model we have the opportunity to calculate in a worst-case scenario where at latitudes close to 90° a higher fraction of light will reach the recoil mass. According to Krywonos et al. [19], this is a more realistic scenario for single-surfaces in general although other authors claim that the RR theory is actually more accurate for smooth surfaces [18]. For our case (smooth polished surface), we will refer to the GHS model as

In order to treat multilayer-scattering, we will use the first-order vector scattering theory that has been developed by Elson et al. and Bousquet et al. [10, 11]. This model has been shown to be remarkably accurate [10, 22, 23] and can be taken also for non-coated surfaces (number of interfaces is one) which results into an expression similar to the RR theory. In their theory, the surface inhomogeneities of the substrate material (in the case of KAGRA’s PR, SR and BS mirrors it is fused silica) and all layers are treated independently from each other. Moreover, as we do not have information on the PSD of each layer, and we just want to give a simplified scenario, we assume the whole PSD of the substrate is transferred to each layer. This is reasonable insofar as high-energetic deposition processes used for coatings usually produce highly correlated layers [10, 13, 22].

According to Bousquet et al. [11], the BRDF of scattering on a (s − 1)-layer coating (where i, j = 0…s are the indices of the interfaces) can be written as

Regarding the scattering angles θ_sc and ϕ_sc, C_i is a specific constant for the i-th layer in which λ, θ_in as well as the refractive index and the respective thickness are summarized [11, 13]. The parameter f_pol depends on the polarization of the incoming and scattered light and is given by Bousquet et al. [11].

γ_i,j is the Fourier transform of the autocorrelation function (i = j) or the cross-correlation function (i ≠ j) of the surface roughness [11]. However, as mentioned above, we will assume only the substrate PSD (γ_s,s) as transferred to each layer, taking into account full correlation between all interfaces. Be z the dimension of the substrate’s normal in case of an idealy flat surface. The two spatial frequencies f_x and f_y, which appear due to the Fourier transformation, can be expressed for both models in terms of θ_in, θ_sc, and ϕ_sc:

Note: the representations of the BRDF given above describe the angular distribution of scattering but not the specular reflection itself.

2.1.2. PSD

The PSD for a given surface is generally two-dimensional. However, as most of the mirrors’ surfaces are isotropic, a one-dimensional PSD is usually used instead (the one-dimensional PSD is just the average of the two-dimensional one along one of the axes). It is possible (assuming isotropy) to reconstruct the two-dimensional PSD with the aid of a fitting model, called the K-correlation or ABC-model [5]. We will use it in a slightly modified version as

The parameters A, B, and C are variable and depend on the individual surface structure. A has the dimension of PSD_1−D which is given here in nm²mm, while B can be interpreted as the correlation length of the surface irregularities and will be given here in mm. C has no unit and gives a measure for the slope of the PSD with increasing spatial frequency, as usually found for polished surfaces. The respective parameters for the mirrors that we are using as examples are given in Table 1.

Table 1. ABC parameters of the fits on the combined PSD_1−D curves (see text for details).

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Due to the connection of the spatial frequency with the scattering angles as given in Eq. (3) (which follows from the grating equation), only part of the PSD is responsible for scattering. There is only a “relevant” surface roughness that produces scattering, which is not an intrinsic surface characteristic but depends upon wavelength and incident angle [5, 17]. For normal incidence, e.g., those spatial frequencies greater than 1/λ produce only evanescent waves and are irrelevant for scattering. For arbitrary incident angles, the absolute value of f should be smaller than 1/λ to count for scattering.

Measurements on the surfaces of the mirrors’ substrates have shown their isotropy. Hence, the assumption of such a simple PSD is reasonable. However, due to the fact that a surface is not infinitely expanded and a measurement of its surface structure cannot show infinitesimal small irregularities, a measured PSD, as shown in Fig. 2, is always limited at both small and large spatial frequencies. Thus, predictions of the BRDF are also generally limited to certain angular distributions around the scattering event (usually for very small and very large latitudes). Measured PSDs of the mirrors that we are using here to model the scattering are shown in Fig. 2, together with measurements from two Virgo mirrors done at the Laboratoire des Matériaux Avancés (LMA) in Lyon. We used the combined datasets to create a fitted PSD according to the ABC-model.

 

Fig. 2 Comparison of the one-dimensional PSDs of the mirrors which were investigated in this paper and two measurements done for VIRGO mirrors at the LMA. Additionally, the fit done for the BS is drawn.

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2.2. Scattering in transmission

In the same way as in reflection, scattering happens also in transmission. To take account of it, we will give the respective equations for the bidirectional transmission distribution function (BTDF) for the two cases of scattering on smooth surfaces and multiple layers of dielectric material. The GHS in transmission can be written as

n₀ and n_m are the refractive indexes of the medium of the incident and transmitted light, respectively. In case of the multilayers, Bousquet et al. [11] have given a similar expression as in Eq. (2):

The spatial frequencies f_x and f_y must now be written as

Here, n_s and n₀ are the refractive indexes of the substrate and the surrounding medium, respectively. f_pol is again the polarization dependent factor as mentioned above.

2.3. Influence of scattering on the sensitivity of KAGRA

The phase shift of the laser light (λ = 1.064 μm) in the cavities of an interferometer like KAGRA is what is eventually going to be measured to search for gravitational waves [24, 25]. A noisy phase fluctuation carried by stray light disturbs this signal. Phase noise in scattered light comes from motions of, e.g., the recoil mass relative to the mirror, and is mainly caused by seismic motions. The strain of space time caused by a gravitational wave h(f) leads to a phase shift of the laser [25]. Conversely, the noisy part of the shift can be described in terms of an effective h(f) [3, 8, 24].

After Aso (2014) [6], for any mirror in KAGRA, the effective strain noise is the product of the square root of the total scattered power that recombines with the main beam, the phase noise, and a coupling coefficient named G. This function depends on the strain frequency and is different for each mirror. G tells how much the injected scattered light will add noise to the gravitational wave readout channel of the interferometer. [26]. It can be determined by simulating the effect of injected noise from various positions on the output of an interferometer. Computer models, like “Optickle” [27] as has been used also by Aso [6], can be used for this. Then, the transfer function between the injection ports and the signal ports can be calculated. What makes the G-factor differ from mirror to mirror is the strain-noise effect of the injection of scattered light at different locations within the interferometer. This effect is complex and not trivial. However, we will refer to it only in a very simplified version by omitting the frequency dependence. As our interest is focused only on the effect of the scattering coming from multilayer coatings, we take its maximum value in the observation band, which will be the value at 10 Hz. The G-factor is then for all three mirrors of the order of 10⁻¹³ W^−1/2rad⁻¹ (see Table 2 for the exact values for each mirror).

Table 2. List of the mirrors, their beam waist parameter w_m, the solid angle toward it, the laser power incoming on each mirror, and the related G-factor. The reason for the doubled entries is that the mirrors face different solid angles ΔΩ_l for incident and reflected beams.

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Defining h_rec(f) as the influence of the scattered light coming from the recoil mass on h(f), it is

P_sssc is the power of the recombined radiation that appears, e.g., by scattering on the mirror, followed by back-scattering on the recoil masses, and eventually scattering again on the mirror in the direction of the incoming laser (see the appendix A for a more comprehensive description). Φ(f) is the spectral density of the scattered light phase fluctuations at a frequency f.

Φ(f) is related to the seismic noise ξ(f) of the ground, which is transmitted to the recoil mass via the respective suspension and causes (in the worst case) a relative motion between mirror and recoil mass $\widehat{\xi }\left(f\right)$. This term is determined by the appropriate transfer function, TF, of the suspension in the horizontal or vertical direction. In frequency space, $\widehat{\xi }$ can be easily calculated by multiplying the TF and ξ(f) [8, 25]. Thus, the spectral density of the scattered light’s phase fluctuations would become $\mathrm{\Phi }\left(f\right)=4\pi /\lambda \cdot \text{TF}\left(f\right)\cdot \widehat{\xi }\left(f\right)$. However, the exact solution of the influence of scattered light on the gravitational wave strain is actually derived from the sine of the time-dependent phase distortions. Often, these distortions are small enough (compared to the wavelength) to ignore the sine. But for certain frequencies where the TF shows resonances, the phase distortions may become large enough to differ significantly from the approach. Therefore, we have to calculate the “up-converted” noise spectrum to achieve the exact values [8, 28].

There are two possibilities for doing an up-conversion. One is using the time-dependent data of the seismic noise itself and calculating the spectrum of sin $\widehat{\xi }\left(t\right)$. The other one is using the noise spectrum and converting it by using the method of Flanagan and Thorne [8]. Their method is relatively simple and useful but has limitations in real calculations using computers, especially when the original noise spectrum reaches values ≳100 · 4π/λ. In Fig. 3 the spectrum of the phase fluctuations caused by ξ(f) as measured in the Kamioka mine where KAGRA is currently built is shown together with the transferred phase-noise spectra of PR and SR mirrors and their respective up-conversions. The low-frequency peaks of up-converted spectra are suppressed and decrease in intensity while toward higher frequencies the intensity is increasing. In the up-converted spectra, the horizontal vibration for the PR mirrors shows the biggest of these “bulges”. The reason for this is the relatively strong resonance peak at ~0.42 Hz of the transfer function, which amplifies the noise in this frequency by more than two orders of magnitude.

 

Fig. 3 Spectral density of the seismic phase noise of the Kamioka mine (black) and the respective noise of the PR, SR, and BS mirrors, resulting from multiplying the TF with the seismic noise (see text). The phase noise horizontal to the ground is drawn in the left figure; the vertical one is drawn in the right figure. For comparability and illustration, the up-conversion (see text) of the respective phase-noise spectra is also shown.

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As can be seen, at maximum (in our case for the PR suspension), ϕ reaches a value of the order of 10⁻¹² rad·Hz^−1/2 at 10 Hz and is further decreasing with increasing frequency. With our assumption TIS ~100 ppm for the mirror, this means already that we can give an upper limit for h_rec in the case of the PR mirrors of the order of 10⁻²² Hz^−1/2 when we assume that all scattered light coming from the mirror is scattered back and injected into the main beam (the upper limit of the other mirrors lies six to nine orders of magnitude lower). This upper limit is already equal to the goal sensitivity for KAGRA at this frequency (~10⁻²² Hz^−1/2). Therefore, in case of back-scattering, the suspended recoil masses probably do not have an effect on the measurements that will be done for KAGRA, regardless what kind of BRDF_m is given. However, non-suspended elements on the other side will have an upper limit two orders of magnitude higher than with the suspension, and for other mirrors the G-factor can be even higher and thus the upper limit may increase accordingly. Also, these estimations are valid only for KAGRA. More advanced gravitational wave detectors (e.g., the Einstein Telescope) will have an even higher sensitivity. In these cases the particular scattering distribution will become important, as peaks in the BRDF_m may reach surfaces within the chamber and therefore increased back-scattered power can reach the mirror for the recombination with the main beam.

3. Scattering distributions and calculation of P_sssc

3.1. Mirror parameter

In the previous section, we have given a general description of how the power of the scattered light couples back into the main beam and how it affects the sensitivity of a gravitational wave detector in terms of an effective strain h. Since each mirror faces different solid angles ΔΩ_l depending on the incident or reflected beam, we have to deal with 10 different values for ΔΩ_l.

In Table 2 the mirrors together with their respective parameters as well as the power of the laser incoming to each mirror, and ΔΩ_l are listed. The beam size on the mirror, w_m, refers to the radius of a Gaussian beam at which the intensity has dropped to 1/e² of the value right in the center. Thus, it is a measure of the beam’s diameter, as within this radius, approximately 86.4% of the beam’s power is concentrated. Limitations to the beam’s extension perpendicular to its propagation axis are maintained with baffles for KAGRA.

3.2. Simulating back-scattering

The theoretical derivation presented above is a very general way of calculating back-coupled scattered light. Because of the complex shape of the recoil mass (or the structures within a chamber in general) and the fact that we need rapid results, however, we decided not to solve the problem analytically. Instead, we calculated the resulting power of the scattered light with a numerical analysis tool for optical processes, called LightTools.

LightTools uses Monte-Carlo simulation procedures to calculate the optical paths of a large number of virtual optical rays (ray-tracing). Depending on the complexity of the problem, simulations with approximately 2 – 20 million rays have been performed. For interactions with surfaces, the properties of the surface specified via a BRDF given as input are taken into account.

The light source was set to be the scattered light distribution of the mirrors, calculated from the respective BRDF as given in Eq. (1) and (2). Thus, for normal incidence (PR, SR), the source was set to be a circular surface lying on the mirror, while for non-normal incidence (BS), the source was set to be an elliptical surface. The size of the sources was determined by the clear aperture which defines the amount of light “seen” by each mirror. It has been set to be 2.7 · w_m, so that theoretically ca. 99.99995% of the beam’s power can reach the mirror. Each mirror is subjected to the spatial power distribution of a Gaussian beam (see Table 2 for the parameters). In the simulations we focused on S-polarized light, which means that the incoming light beam is polarized perpendicular to the plane of incidence. This is reasonable as KAGRA also uses this polarization for the laser. However, as LightTools cannot distinguish the state of polarization during scattering events, the coupling back into the main laser beam is a superposition of the power of S- and P-polarized light. The power distributions as well as the scattered light distribution for the light source, and the analysis regarding h_rec, have been calculated outside LightTools.

The recoil masses that will be used for KAGRA are made of 6V-4Al-titanium alloy (with roughness corresponding to a simple polishing) for which the BRDF was measured in order to use the data in LightTools and perform a realistic simulation. As a measure for comparison, we performed also simulations assuming the surface of the recoil mass being a perfect Lambertian scatterer (perfect white surface).

As a result of the simulations, a mesh of the back-scattered power P_ssc per solid angle is created as a function of latitude θ and longitude ϕ. As the measured PSD is always limited at very low and very high frequencies, we have no information about the TIS. Therefore, we set the total hemispherical scattering for all mirrors to 10⁻⁴ as representative of high-performance mirrors [22, 29]. Thereby, we ignored a solid angle of 0.012 sr around the specular reflection in order to account for real measurements of TIS which cannot collect the scattering at very low angles.

3.3. Results for the PR and SR mirrors

The PR and SR mirrors are designed to reflect most of the light (~99.995 – 99.9995%) at 1.064 μm wavelength specularly under near normal incidence. The substrate for these mirrors is fused silica (SiO₂), while the multilayer coating is made of tantala (Ta₂O₅) and fused silica layers each with a thickness of λ/4. To reach such reflectivities, we need 14 and 17 layer-pairs of tantala and silica, with the first and the last layer made of tantala. Partial design data were given by the company which did the coating but the number of layers has been calculated according to the final properties of the mirror after Henderson (1978) [30]. The scattering maps for the respective designs have been calculated with the aid of Eq. (2). The results in terms of the scattering probability are given in Fig. 4 for the PR and SR mirrors together with the respective values for the GHS model, both normalized to 100 ppm total integrated scattering. We give the scattering profiles along the plane of incidence (POI) as well as perpendicular to it. While the GHS model in our case predicts isotropic scattering, the scattering due to multilayer coatings depends on the longitude even at normal incidence. Another difference is the structure of the profile. While the GHS model predicts a smooth, decreasing function with increasing θ_sc according to the PSD, the scattering due to multiple layers shows some weak features at θ ~ 65° and 80° (for PR3 and SR3) and θ ~ 60° and 75° (for PR2 and SR2). The scattering is increased between θ ~ 30° and θ ~ 85° by up to a factor of 2 compared to the GHS model. On the other side, toward latitudes smaller than 30° and latitudes wider than 85°, the GHS model shows slightly increased scattering. Due to the strong similarities in the structure of the multilayer coating for the PR and SR mirrors, the scattering distributions are also generally very similar between both PR and SR mirrors. Differences may be found only in the absolute values of the scattering probability, which is mainly due to different PSDs.

 

Fig. 4 Calculated BRDF · cos θ_sc (scattering probability density) as a function of the scattering latitude θ according to Eqs. (2) and (1) for the PR and SR mirrors by normal incidence. In case of the multilayers, we give also the profile perpendicular to the plane of incidence (POI).

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Fig. 5 The paths of light for the PR/SR mirrors (left) and the BS (right). While the (near) normal incidence for the PR/SR mirrors creates only single (rotational symmetric) scattering distributions, the AOI of 45° for the BS and its 50% transparency give rise to 8 different asymmetric scattering distributions, considering multilayer coatings.

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The results of the calculated values dP_sssc/dΩ_l are given in Table 3 for both the multilayer model and the GHS model for the two cases of a recoil mass made of a Lambertian scatterer and of titanium. The results for the multilayer coatings are in any case larger by a factor of up to 2. This may be explained by the slightly increased scattering probability of the multiple layers at wide scattering angles compared to the GHS model. However, the differences are not very large and for the recoil masses, using the GHS model seems to be sufficient to describe the scattering by reflection of the PR and SR mirrors.

Table 3. Values for $\frac{d{P}_{\text{sssc}}}{d{\mathrm{\Omega }}_{\mathrm{l}}}$ in W/sr for the PR and SR mirrors calculated from the outcome of the simulations for the two models and recoil masses made of a Lambertian scatterer and titanium. The AOI is zero for all mirrors.

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3.4. Results for the beam splitter

Unlike the PR and SR mirrors, the BS mirror is 50% transparent to the power of the incoming light (under 45° AOI) on its high-reflective side (HR), which faces the PR mirrors. Thus, not only reflective but also transmissive scattering has to be taken into account. Moreover, the anti-reflective (AR) side is another source of scattering, in reflection and transmission, which cannot be described realistically with our simplified GHS model as it requires a reflectance as a factor to get an absolute number for the BRDF. Therefore, for the multilayer coating model, eight different processes have to be distinguished here: scattering in reflection and transmission for the HR and AR side in two directions. However, for the GHS model only the processes on the HR side were investigated. To get an impression of the scattering profiles, the cross sectional views of the scattering probability distribution along the POI are displayed in Figs. 6 and 7 for both models, each of them normalized to a TIS of 100 ppm. Obviously, the number of features in the multilayer model and the overall structure of the scattering distribution is very different from case to case. The smoothest distribution is found for the reflection and transmission of light coming from outside to either side of the BS. Note that the curves in the Figs. 6 and 7 are only the distributions along the POI and can thus give only an impression of the basic structure of the whole scattering map. That is the reason why in the “In → HR → Out” diagram, the GHS model lies for most of θ above the multilayer model although both curves are normalized: the two features in the multilayer model for θ ≥ 50° actually extend over a very wide range in ϕ with the result that the TIS is the same for both models.

 

Fig. 6 Cross sectional view of the scattering probability-density distribution along the POI for the four cases of scattering on the HR surface of the BS mirror. The AOI is 45° for the two cases in the upper panel and 29.187° for the lower panel. Positive values for θ indicate scattering in the direction of the incoming light along the POI (ϕ = 0°).

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Fig. 7 Cross sectional view of the scattering probability distribution along the POI for the four cases of scattering on the AR surface of the BS mirror. The AOI is 45° for the two cases in the upper panel and 29.187° for the lower panel. Positive values for θ indicate scattering in the direction of the incoming light along the POI (ϕ = 0°).

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The coating on both sides of the beam splitter consists of the same materials as the PR and SR mirrors: tantala and silica. Yet, the number of layers and their thickness is different. As for the HR side (coming from outside), we find a tantala-silica-tantala coating and for the AR side (again when coming from outside) a silica-tantala-silica-tantala coating. We assumed the design of the HR-coating to be [0.93 | 0.68 | 0.82]·λ and the design of the AR-coating [1 | 0.17 | 0.49 | 0.8]·· λ (left side faces vacuum; right side faces the substrate). Of course, we can not guarantee that this is completely accurate for our BS but for now it will be taken as reasonable enough for our calculations.

Due to the transmission properties and the non-zero AOI of the BS, the calculation and illustration of the results becomes a little bit more complex than for the PR and SR mirrors. The back-scattered light can couple into both the transmitted and the reflected main beam. Also, we have two additional possibilities for a coupling back into the main beam as there are now two main beams hitting the BS. Therefore, each entry for dP_sssc/dΩ_l is the sum of all four different possibilities of coupling. Those values are given in Table 4 for all eight cases of reflection and transmission, each of the two cases of having a Lambertian and a titanium recoil mass. The scattering distributions for the model that was used in LightTools have been calculated using Eqs. (2), (7), and (1) for the main-beam directions of 45° from outside and ~29° from inside the BS body (each with respect to the mirror’s surface normal; see Fig. 5). KAGRA is using silica as the BS substrate material (index of refraction: ~ 1.45). In the table the different cases of reflection and transmission are designated according to the view one would see when traveling with the respective incident main beam. For example, for the simple reflection on the HR side when the beam hits the BS from outside, the designation would be “out → HR → out”, meaning the beam comes from outside toward the HR side and is reflected. Then, the angular distribution of the BRDF for that specific case has been taken to run the simulation.

Table 4. Values for $\frac{d{P}_{\text{sssc}}}{d{\mathrm{\Omega }}_1}$ in W/sr for the different scattering cases on the BS mirror calculated from the outcome of the simulations for the two models and recoil masses made of a Lambertian scatterer and titanium.

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The relation between the two models is more complex than in case of the PR and SR mirrors. As a general tendency, the multilayer coating model leads to more scattering power being recombined with the main beam than the GHS model (by factors between 2 and 100). However, there are also cases where the GHS model shows slightly higher values (“out → HR → in”). As mentioned above, a direct comparison between the two models is possible only for the HR side of the BS mirror. Recombining power coming from back-scattering caused by the AR side can be handled only with the multilayer coating model. For both the HR and AR side however, by far the highest amounts of recombining back-scattered light is generated when the beam enters the BS (Out→Out), as well as for internal reflection (In→In). This is reasonable as the scattered light in those cases encounters a larger surface of the recoil mass which surrounds the mirror. The highest value (multilayer coating model) is reached by the reflection on the inner side of the HR side (“in → HR → in”) and using a Lambertian scatterer as recoil mass, which leads to 1.44 · 10⁻⁷ W/sr back-coupled intensity.

4. Discussion

4.1. PR and SR mirrors

For the recoil masses, and under normal incidence on the mirror, we have seen that in the general case the multilayer coating model has larger values of back-scattered power recombining with the main beam compared with the GHS model, even though the multilayer model tends to give smaller BRDF-values toward 90° latitude. However, the differences are small and hence the scattering of the mirror can be well approximated with the simpler GHS model. Care must be taken for the back side of the PR and SR mirror. We did not calculate the scattering on this side as we have no information on the structure of the coating there and we firstly wanted to concentrate or efforts on the main surfaces of the mirrors but a small (but still reasonable) amount of light will pass through the mirror and eventually hit the back side, which is usually AR-coated and oblique. As can be learned from the scattering on the BS mirror, light that leaves the substrate through coated surfaces under non-normal incidence may cause additional strong scattering features. Another point that was not investigated so far is the inner scattering due to the transmission of the beam and the following back-scattering on the inner surfaces of the recoil mass. Also learned from the BS mirror, the inner back-scattering can be orders of magnitude higher than on the outer sides of a mirror. However, due to the small amount of the actual back-scattered power and the suspensions, we do not expect a big impact on the sensitivity of KAGRA at all.

Important differences between the models are found in the scattering distributions as given in Fig. 4. We see that the multilayer coatings are the cause of some interference structures which can trigger the scattering in certain angular ranges, as is the case for both the SR and PR mirrors for θ ranging from 40 to 80°. Mirrors with other coating designs are very likely to show much different distributions as given for the mirrors above, and may have even stronger features. In contrast, the GHS model assumes a smoother distribution of the TIS over the angular space. Therefore, in the general case, one should take care about surfaces that are close to the mirrors and that lie in the focus of those scattering features, especially when these surfaces belong to non-suspended elements. The subsequent back-scattering can lead to substantial differences from the anticipated effect of scattering when only the GHS model is used.

4.2. BS mirror

The more interesting effects of scattering on multilayer coatings were found for the BS mirror. Here, we have to deal with non-normal incidence, which leads to even bigger differences in the scattering distribution compared to a single-surface approach. Also, with the BS mirror and the available specifications, we were able to estimate the scattering from the AR surface, which is basically impossible with the simplified single-surface approach usually used in the field of gravitational wave detectors. Moreover, we have also estimated the back-scattering of the recoil mass for scattering happen within the BS mirror. But as mentioned above, although the recombining power is by far larger than for scattering happening on the outer surfaces of the mirror, there is no actual harm for the sensitivity of KAGRA as long as the recoil mass is properly suspended.

A possible threat to the sensitivity is rather the transmissive scattering peaks between 45 – 80° latitude produced by the AR surface when the beam is going outside the BS (see the right graph in the bottom panel of Fig. 7). In our simulation these peaks did not have any influence as they are not entirely hitting the recoil mass. Moreover, we have to assume that they eventually hit the inner walls of the chamber which are not suspended. Of course, the absolute magnitude of the scattering probability of these features depends on the TIS of the AR surface but taking only these features into account, they are responsible for more than 66% of the TIS (for comparison, in the GHS-model on the HR side this range accounts for 29% of the TIS). The same effect can be seen on the HR surface for the outgoing beam (right graph in the bottom panel of Fig. 6). Here also, several scattering features are visible for latitudes between 45 – 80°, which represent actually 78% of the entire TIS. It seems that for both sides of the BS mirror one should pay additional attention to those scattering features. Unfortunately, we have not yet performed simulations including the inner walls of the BS chamber to see how big the effect of the scattering features really is. However, if these features become too large, additional baffles around the BS (or other mirrors) may be necessary to absorb them.

It becomes apparent that in all scattering cases given in the table, the Lambertian scatterer leads to larger values of the recombined power than the titanium, independent of the model used. Yet, for different scattering cases one cannot simply follow the same ratio of back-coupled power between a Lambertian and titanium scatterer. Each case has its own specific influence on the scattering and thus on dP_sssc/dΩ_l, beginning with a unique scattering profile for each side of the BS surface. However, also for titanium, the case “in → HR → in” leads to the highest values of dP_sssc/dΩ_l of 1.21 · 10⁻⁷ W/sr. The smallest values for Lambertian recoil masses are reached for the case of reflective scattering on the AR side when the main beam comes from outside (dP_sssc/dΩ : 2.07·· 10⁻¹² W/sr). However, the smallest value for a titanium recoil mass is reached when the beam leaves the AR side from inside (dP_sssc/dΩ_l : 2.96 · 10⁻¹² W/sr).

At this point we mention again that we have assumed a full correlation of the interfaces within the coatings itself. Especially for the PSD at higher frequencies, where wide-angle scattering is created, it may not entirely reasonable to assume the coating will just copy the substrate’s PSD. On the other hand, a coating might also hide high-frequency roughness and will thus lead to smoother surfaces. A more general theory, as given for instance in Amra et al. [13, 31], would assume a kind of bandpass function for each layer, allowing only the inhomogeneities at specific spatial frequencies of one layer to be transferred to the next higher one. Applying this theory may be done in a future, more comprehensive investigation. What we also did not take into account was the scattering of light within the materials itself (bulk scattering in coating and substrate). It has been pointed out that bulk scattering in usual coatings and on substrates with roughnesses larger than 0.5 nm is at least one order of magnitude lower than the surface scattering [12, 32, 33]. The investigated mirrors here have substrate roughnesses in the range of 0.3 to 1 nm and bulk inhomogeneities either in the coating or in the substrate may play a role at least for those angular ranges where the scattering is below 10⁻⁶ [32]. This too must be considered in a more comprehensive study if one is going to understand scattering and all related issues in gravitational wave detectors like KAGRA. We note also that in all models, we have ignored subsequent scattering of the scattered light as we think this will not have an important impact on the outcome of the simulations. Also, for the BS mirror, the sum of the recombined power density given for each case of the inner scattering is calculated only for a recombination on either the HR or the AR surface.

5. Conclusions

The main concern of this paper is to show the influence and the effects of multilayer coatings on the scattering profile of mirrors that are used for KAGRA. We think that in the field of gravitational wave detectors, where this issue has not been found suitable attention so far, it is of great importance to point out the differences between single-surface scattering and multilayer scattering. Especially for non-normal incidences and AR coated surfaces, the additional features that appear in the scattering distributions are not negligible and may have impacts on the goal sensitivity of KAGRA. The same holds true for future detectors having even tighter sensitivity constraints. In this paper we neither investigated the back surfaces of the PR and SR mirrors, which are also AR coated, nor included bulk scattering into our considerations. Actually, we expect some additional impact if we include these factors into our investigations and they will be considered in a future work on this topic.

As we have seen, the influence of the back-scattered light coming from the recoil masses is negligible for the PR, SR, and BS mirrors for KAGRA. In all cases, we have a safety factor of several orders of magnitude relative to KAGRA’s goal sensitivity. This is mainly due to the suspension of the recoil masses, which, on the other side, due to the inavoidable resonant frequencies and their up-conversion, produces a bulge appearing at around 2 Hz in all strain noise spectra (see Fig. 3). We have seen also that the usage of titanium generally tend to reduce the back-scattered power from recoil masses compared to a Lambertian scatterer. But, the actual power must be calculated seperately for each case as no analytical dependence can be anticipated generally.

The amount of scattered light that recombines with the main beam is strongly dependent on many different parameters like the PSD of the mirror’s substrate, the coating of the mirror, the inner structure of the used materials, and the surface structure of the (in our example) recoil masses. Thus, this amount is hard to predict even when there are only small changes to the parameters. We therefore hope that this work will serve as a set of tools for further calculations (or simulations) on the general issue of scattering on mirrors or having structures close to sensitive optical parts in interferometers and how they influence back-scattering and back-coupling. We also hope that in future works dealing with optical topics of gravitational wave detectors the multilayer coatings are taken into account as they are an additional important factor in the calculation of scattering and all related issues.

Appendix A: Basic notes about the BRDF

The surface scattering (as well as specular reflection) from an area dA_sc can be described in terms of the BRDF, defined as

(A1)$$\text{BRDF}=\frac{\partial L_{\text{sc}}({\theta }_{\text{in}},{\varphi }_{\text{in}};{\theta }_{\text{sc}},{\varphi }_{\text{sc}})}{\partial E_{\text{in}}({\theta }_{\text{in}},{\varphi }_{\text{in}})}.$$

Here, L_sc is the radiance of reflected/scattered light and E_in the irradiance of the incoming light (plane wave) with power P_in; θ_in/ϕ_in and θ_sc/ϕ_sc are the incident and scattered latitude/longitude in spherical coordinates as shown in Fig. 8. Both radiance and irradiance depend on the specific location of the scattering event on the surface and are defined as

(A2)$$L_{\text{sc}}=\frac{{\partial }^2{P}_{\text{sc}}}{\partial A_{\text{sc}}\cdot \cos {\theta }_{\text{sc}}\cdot \partial {\mathrm{\Omega }}_{\text{sc}}},E_i=\frac{\partial P_{\text{in}}}{\partial A_{\text{sc}}},$$

where P_in and P_sc are the power of the incident and the scattered light, respectively. The term ∂A_sc · cos θ_sc is the effective radiating area an observer would see at the scattering angle θ_sc. Note that due to the relations

(A3)$$d{\mathrm{\Omega }}_{\text{sc}}=d{A}_{\mathrm{c}}\cos {\theta }_{\text{sc}}\cdot r^{-2}d{\mathrm{\Omega }}_{\mathrm{c}}=d{A}_{\text{sc}}\cos {\theta }_{\text{sc}}\cdot r^{-2},$$

for a source (sc) – in our case the scattering surface – and a collector (c) – or detector – at a distance r from each other, L_sc in Eq. (A2) can also be expressed in terms of the power received by the collector. Ω_sc is the solid angle toward the detector surface, and conversely, Ω_c is the solid angle from the detector toward the surface of the source [17].

 

Fig. 8 Diagram of a scattering event with the principal geometrical quantities that are used in this paper. ϕ_in is set to be 0 without loss of generality.

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Fig. 9 Simplified diagram of the events that are discussed in this paper, with nomenclature used for the calculations. The blue arrows represent scattered light from the mirror while the red ones represent scattered light from the recoil mass. The red-dotted arrow is scattered light from the recoil mass that hits the laser-spot on the mirror and can couple back into the main beam.

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Appendix B: Scattering on recoil masses and back-coupling

If a laser hits a mirror, it will produce scattered light during the reflection of the laser beam. If a part of this scattered power ($P_{\text{sc}}$) hits the recoil mass, the twice-scattered light ($P_{\text{ssc}}$) from the recoil mass may reach the area where the laser hits the mirror and scatter a third time to give a portion ($P_{\text{sssc}}$) back toward the laser because of the scattering of the mirror (see Fig. 9).

According to Eq. (A1), we can formulate the radiance of this back-coupled light as

(B1)$$\begin{array}{l}d{L}_{\text{sssc}}={\text{BSDF}}_{\mathrm{m}}\cdot d{E}_{\text{ssc}}\\ d{L}_{\text{sssc}}={\text{BSDF}}_{\mathrm{m}}\cdot L_{\text{ssc}}\cos {\theta }_{\text{sc}}d{\mathrm{\Omega }}_{\text{sc}}.\end{array}$$

Here, E_ssc is the irradiance coming from the recoil mass, which is at the same time the radiance from it multiplied by the cosine of the scattering latitude as seen from the mirror (θ_sc) and the partial of the corresponding scattering solid angle. BSDF means bidirectional scattering distribution function and includes BRDF and BTDF.

L_ssc is given by Eqs. (A2) and (A3)

(B2)$$L_{\text{ssc}}=\frac{{\partial }^2{P}_{\text{ssc}}}{\partial A_{\mathrm{m}}\cdot \cos {\theta }_{\text{sc}}\cdot \partial {\mathrm{\Omega }}_{\text{sc}}},$$

leading to

(B3)$$d{L}_{\text{sssc}}={\text{BSDF}}_{\mathrm{m}}\cdot \frac{{\partial }^2{P}_{\text{ssc}}}{\partial A_{\mathrm{m}}\partial {\mathrm{\Omega }}_{\text{sc}}}d{\mathrm{\Omega }}_{\text{sc}}.$$

This is what we need for the calculation of the power P_sssc that recombines with the laser beam. It is only that power necessary to calculate the effect of differently scattered light on the sensitivity of a mirror. From Eq. (A2), P_sssc is the integral of L_sssc · cos θ_sc after dA_m and dΩ_l, the solid angle toward the waist of the laser beam. The used simulation program (a commercial product named LightTools) is providing us already ∂P_ssc/∂Ω_sc, which means that we do not have to integrate after dA_m anymore. The BRDF (and the BTDF) are obained from the calculations above.

Funding

MEXT; JSPS Leading-edge Research Infrastructure Program; JSPS Grant-in-Aid for Specially Promoted Research (26000005); MEXT Grant-in-Aid for Scientific Research on Innovative Areas (24103005); JSPS Core-to-Core Program, A. Advanced Research Networks; the joint research program of the Institute for Cosmic Ray Research, University of Tokyo; Computing Infrastructure Project of KISTI-GSDC in Korea

Acknowledgments

We would like to thank Mark Barton for a helpful proofreading of the text and many welcome suggestions to improve the paper as well as the Advanced Technology Center (ATC) of NAOJ for their technical asstistance. Furthermore, we would like to thank the two referees for their suggestions and helpful hints.

References and links

1. Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto, “Interferometer design of the kagra gravitational wave detector,” Phys. Rev. D 88, 043007 (2013). [CrossRef]  

2. D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015). [CrossRef]  

3. K. Kuroda, “Ground-based gravitational-wave detectors,” Int. J. Mod. Phys. D 24, 1530032 (2015). [CrossRef]  

4. LIGO Scientific Collaboration and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016). [CrossRef]   [PubMed]  

5. J. E. Harvey, S. Schröder, N. Choi, and A. Duparre, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012). [CrossRef]  

6. Y. Aso, Y. Michimura, and K. Somiya, “Kagra main interferometer design document,” KAGRA internal document JGW-T1200913-v6 (2014).

7. J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, and E. Tournie, “Scattered light noise in gravitational wave interferometric detectors: A statistical approach,” Phys. Rev. D 56, 6085–6095 (1997). [CrossRef]  

8. E. E. Flanagan and K. S. Thorne, “Noise due to light scattering in interferometric gravitational wave detectors. i: Handbook of formulae, and their derivations,” LIGO internal document (2011).

9. V. Mitrofanov, S. Chao, H. wei Pan, L.-C. Kuo, G. Cole, J. Degallaix, and B. Willke, “Technology for the next gravitational wave detectors,” Sci. China Phys. Mech. Astron. 58, 120404 (2015). [CrossRef]  

10. J. Elson, J. Rahn, and J. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980). [CrossRef]   [PubMed]  

11. P. Bousquet, F. Flory, and P. Rouche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. A 71, 1115–1124 (1981). [CrossRef]  

12. C. Amra, G. Albrand, and P. Roche, “Theory and application of antiscattering single layers: antiscattering antireflection coatings,” Appl. Opt. 25, 2695–2702 (1986). [CrossRef]   [PubMed]  

13. C. Amra, C. Grezes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. 32, 5492–5503 (1993). [CrossRef]   [PubMed]  

14. F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016). [CrossRef]   [PubMed]  

15. E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71–86 (1991). [CrossRef]  

16. E. L. Church and P. Z. Takacs, “Surface scattering,” in Handbook of Optics: Fundamentals, Techniques, and Design, M. Bass, ed.(McGraw-Hill, 1994), Vol. 1.

17. A. Krywonos, “Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction,” Ph.D. thesis (University of Central Florida2000).

18. S. Schröder, A. Duparre, L. Coriand, A. Tuennermann, D. H. Penalver, and J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Express 19, 9820–9835 (2011). [CrossRef]   [PubMed]  

19. A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surface with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A , 28, 1121–1139 (2011). [CrossRef]  

20. A. von Finck, T. Herffurth, S. Schröder, A. Duparre, and S. Sinzinger, “Characterization of optical coatings using a multisource table-top scatterometer,” Appl. Opt. 53, 259–269 (2014). [CrossRef]  

21. N. Choi and J. E. Harvey, “Numerical validation of the generalized harvey-shack surface scatter theory,” Opt. Eng. 52, 115103 (2013). [CrossRef]  

22. S. Maure, G. Albrand, and C. Amra, “Low-level scattering and localized defects,” Appl. Opt. 35, 5572–5582 (1996). [CrossRef]  

23. M. Zerrad, S. Liukaityte, M. Lequime, and C. Amra, “Light scattered by optical coatings: numerical predictions and comparison to experiment for a global analysis,” Appl. Opt. 55, 9680–9687 (2016). [CrossRef]   [PubMed]  

24. K. Kuroda, W.-T. Ni, and W.-P. Pan, “Gravitational waves: Classification, methods of detection, sensitivities and sources,” Int. J. Mod. Phys. D 24, 1530031 (2015). [CrossRef]  

25. P. R. Saulson, ed., Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, 1994), 1st ed. [CrossRef]  

26. H. Yamamoto, “Transfer functions of scattered light in advligo coc,” LIGO internal document T060073-x0 (2006).

27. M. Evans, “Optickle,” LIGO internal document T070260 (2007).

28. D. J. Ottaway, P. Fritschel, and S. J. Waldman, “Impact of upconverted scattered light on advanced interferometric gravitational wave detectors,” Opt. Express 20, 8329–8336 (2012). [CrossRef]   [PubMed]  

29. B. A. Sones, “The optimization and analytical characterization of super cavity mirrors for use in the single atom laser experiment,” Ph.D. thesis (1997).

30. A. R. Henderson, “The design of non-polarizing beam splitters,” Thin Solid Films 51, 339–347 (1978). [CrossRef]  

31. C. Amra, “Light scattering from multilayer optics. i. tools of investigation,” J. Opt. Soc. Am. A , 11, 197–210 (1994). [CrossRef]  

32. C. Amra, J. Apfel, and E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992). [CrossRef]   [PubMed]  

33. A. Duparre and S. Kassam, “Relation between light scattering and the microstructure of optical thin films,” Appl. Opt. 32, 5475–5480 (1993). [CrossRef]   [PubMed]