Ternary Quarter Wavelength Coatings for Gravitational Wave Detector Mirrors: Design Optimization via Exhaustive Search

Multimaterial optical coatings are a promising viable option to meet the challenging requirements (in terms of transmittance, absorbance and thermal noise) of next generation gravitational wave detector mirrors. In this paper we focus on ternary coatings consisting of quarter-wavelength thick layers, where a third material (H') is added to the two presently in use, namely Silica (L) and Titania-doped Tantala (H), featuring higher dielectric contrast (against Silica), and lower thermal noise (compared to Titania-doped Tantala), but higher optical losses. We seek the optimal material sequences, featuring minimal thermal (Brownian) noise under prescribed transmittance and absorbance constraints, by exhaustive simulation over all possible configurations, for different values (in a meaningful range) of the optical density and extinction coefficient of the third material. In all cases studied, the optimal designs consist of a stack of (H'|L) doublets topped by a stack of (H|L) doublets, confirming previous heuristic assumptions, and the achievable coating noise power spectral density reduction factor is \sim 0.5. The robustness of the found optimal designs against layer thickness deposition errors and uncertainties and/or fluctuations in the optical losses of the third material is also investigated. Possible margins for further thermal noise reduction by layer thickness optimization, and strategies to implement it, are discussed.


I. INTRODUCTION
The visibility distance of the currently operating interferometric detectors of gravitational waves is set by thermal (Brownian) noise in the highly-reflective dielectricmultilayer coated mirrors of their optical cavities [1].
During the last two decades, the quest for coating materials featuring large dielectric contrast, low optical absorption (and scattering), and low mechanical losses (directly related to thermal noise, in view of the fluctuationdissipation principle) has been quite intense.
None of them qualifies as a straight substitute for the materials currently in use, but a few of them are better in terms of some properties (e.g., optical density, and/or mechanical losses), while unfortunately worse regarding * pierro@unisannio.it others.
Amorphous silicon (aSi), in particular, has received much attention, in view of its large refractive index and limited mechanical losses, down to cryogenic temperatures [6]. Its optical losses, on the other hand, are appreciably larger than those of titania-doped tantala, and appear to be strongly dependent on the deposition technology [7]. Amorphous silicon has been indicated as a candidate coating material for third generation (3G) cryogenic detectors using crystalline silicon for the mirror substrate (presently made of fused silica), and a 1550nm laser source [8] [9] Silicon nitrides, (SiN x ) have also been proposed as potentially interesting materials, in view of their flexible stoichiometry, which allows one to tune their refractive index in a wide range of values; their ability to accommodate large substrates, via plasma-enhanced chemical vapor deposition (PECVD); and their fairly low mechanical loss angles, down to 10 −4 and below, at ambient as well as cryogenic temperatures [10], [11]. Their optical losses, though, exceed those of currently used materials [12].
In Refs. [13] and [14] it was first suggested to modify the simplest binary coating design consisting of alternating quarter-wavelength (QWL) thick layers of SiO 2 and TiO 2 :: Ta 2 O 5 , by using a third denser but optically lossier material in the high-index layers closest to the substrate, where the field intensity is usually low enough to make its relatively large optical losses irrelevant. This would allow one to reduce the total number of layers (and hence the coating thermal noise), in view of the larger optical contrast with silica and/or the lower mechanical losses compared with titania-doped tantala. The feasibility of aSi-based ternary coatings has been recently demonstrated [15].
Optical coatings using more than two lossy materials (m-ary coatings, with m > 2) have been studied for a long time (see, e.g. Ref. [16] for a review). However, the design constraints and requirements of the mirrors used by interferometric detectors of gravitational waves are peculiar, especially regarding the key figure of merit represented by thermal (Brownian) noise, making further analysis necessary.
In this paper we implement exhaustive simulations aimed at identifying the structure of the optimal ternary coating design yielding the lowest thermal noise under prescribed (upper) bounds on power transmittance and absorbance, without any a priori assumption, except that all layers are QWL.
The paper is organized as follows. In Section II we summarize the relevant modeling assumptions used; in Section III we introduce the exhaustive procedure used to find the optimum material sequences; in Section IV we present and discuss the simulations done referring to two putative third materials, and in Section V we present results obtained for realistic materials, namely, amorphous silicon and silicon nitrides, at both ambient and cryogenic temperatures. The structure of the coating design families that comply with the prescribed transmittance and absorbance constraints, the properties of the minimum thermal noise (optimal) ones, including robustness against layer thickness errors, and uncertainties or fluctuations in the third material extinction coefficient are illustrated. Possible margins for further thermal noise reduction by layer thickness optimization, and strategies to implement it, are also discussed. Conclusions follow in Section VI.

II. COATING MODEL
In this section we summarize the modeling assumptions used throughout this paper, based on the transmission matrix formalism (see, e.g. Refs. [17,18]) and the simplest thermal noise model introduced in Ref. [19].

A. Optical Modeling
Here, an exp(ı2πf 0 t) time-dependence is understood, f 0 being the laser light frequency and ı being the imaginary unit. The optical properties of a multilayer coating can be deduced from its characteristic matrix where N T is the total number of layers (numbered from the vacuum to the substrate as in Figure 1 and T m is the transmission matrix of the m-th layer. Assuming normal incidence [17], where λ 0 and d m being the light free-space wavelength and the m-th layer thickness, respectively, and n (m) r being the real refractive index and κ (m) being the extinction coefficient of the m-th layer material.
The coating is placed between two homogeneous non dissipative dielectric half-spaces with refractive indices n (0) and n S , respectively (see Figure 1). The bottom half-space is the substrate; the top is assumed to be the vacuum, with n (0) = 1.
The effective complex refractive index of the whole substrate-terminated coating is, which can be used to compute the (monochromatic plane wave, normal incidence) coating reflection coefficient Γ C and the power transmittance where P in is the power density flowing into the coating through the vacuum/coating interface and P + is the power density of the incident wave, where E inc is the (transverse) incident electric field at the vacuum/coating interface and Z 0 = µ 0 / 0 is the characteristic impedance of the vacuum. The average power density dissipated in the coating is the difference between P in and the power density flowing into the substrate P out . This latter can be computed as where Re(·) gives the real part of its argument and E (S) and H (S) are the (transverse) electric and magnetic fields at the coating/substrate interface, which are readily obtained from the fields E (0) = E inc (1 + Γ c ) and Z 0 H (0) = E inc (1 − Γ c ) at the vacuum/coating interface using the formula Accordingly, the coating absorbance is where is the fraction of the incident power leaking into the substrate. Note that eq. (11) entails the obvious condition τ C ≥ τ S .

B. Thermal Noise Modeling
The frequency-dependent power spectral density S (B) coat (f ) of the coating thermal noise can be written where f is the frequency, T is the (absolute) temperature, w is the (assumed Gaussian) laser-beam waist, and φ c is the coating loss angle. Neglecting higher-order terms stemming from subtler effects [20], this latter can be written [19] where is the specific loss angle (loss angle per unit thickness) of the material making the m-th layer, φ m and Y m being its mechanical loss angle and Young's modulus, respectively, and Y S being the Young's modulus of the substrate. According to eq. (13), lowering the temperature T would reduce thermal noise [21]. However, in many coating materials, including those currently in use (silica and titania-doped tantala), mechanical losses peak [22] [23] in the range of the cryogenic temperatures of interest for next-generation detectors such as the Einstein Tele-

III. EXHAUSTIVE SCRUTINY OF QWL TERNARY COATINGS
Exhaustive scrutiny of QWL ternary coatings consists in evaluating the performance (in terms of power transmittance, power absorbance and coating loss angle) of all (admissible) ternary coatings consisting of QWL layers made of three possible materials, henceforth denoted as L, H, and H', that comply with given transmittance and absorbance constraints, For coatings consisting of QWL layers, the matrices T m , m = 1, 2, . . . , N T , in (2) take the simple form [27] T For ternary coatings, n (m) can only take values in {n L , n H , n H }, and the corresponding single-layer matrices will be denoted as T L , T H and T H , respectively. Among all possible material sequences, those for which n (m) = n (m−1) for some m should be obviously discarded [28]. Accordingly, we are left with a total of N C = 3 × 2 N T −1 distinct acceptable ternary coatings consisting of N T QWL layers [29].
Knowledge of the matrix (1) yields the coating transmittance and absorbance, as shown in Sect. II.
In order to compute the coating thermal noise it is expedient to let: where φ L , φ H and φ H are the material mechanical loss angles. Hence, using (14) and (15) N L , N H and N H being the number of layers made of the L, H, and H materials, respectively. The above is a typical constrained optimization problem [30], and has combinatorial complexity. In order to keep the computational burden and computing times within acceptable limits, we use the backtracking strategy [31] to reduce the number of matrix multiplications.

IV. NUMERICAL EXPERIMENTS
In this Section we apply the above mentioned exhaustive scrutinizing procedure to ternary QWL coatings laid on a fused-silica substrate (assumed to be of infinite thickness), using SiO 2 and TiO 2 :: Ta 2 O 5 for the lowindex (L) and high-index (H) materials, respectively. For illustrative purposes, we shall consider first two putative candidates for the third (high-index) material (H ).
These will be referred to as Material-A and Material-B and represent two rather extreme paradigms, similar to those discussed in Ref. [13]. Numerical experiments based on realistic materials, namely, aSi and SiN x , are presented in Section V.
Material-A has the same mechanical losses as TiO 2 :: Ta 2 O 5 and a fairly higher refractive index, but it has larger optical losses; Material-B has the same refractive index as TiO 2 :: Ta 2 O 5 and fairly lower mechanical losses, but it has larger optical losses. The numerical values of the relevant properties of putative materials A and B are collected in Table-I. We consider different values of the extinction coefficient ranging from 10 −6 to 10 −4 .

Property SiO2
TiO2:: Ta2O5   We use the following bounds in the transmittance and absorbance constraints (16): and scale the loss angle of the various admissible solutions to that of a reference LIGO/Virgo-like design, consisting of N T = 36 alternating titania-doped-tantala/silica layers, for which

A. Ternary QWL Coatings Using Material-A
The set of all admissible ternary QWL coatings compliant with the prescribed transmittance and absorbance constraints (21) and using Material-A for H is collected in Table-II  These findings confirm the heuristic assumption first made in Refs. [13] and [14] about the structure of ternary QWL coatings yielding minimal noise under prescribed transmittance and absorbance constraints.

B. Ternary QWL Coatings Using Material-B
In the case of ternary QWL coatings using Material-B,, materials H and H are iso-refractive; hence the total number of high-low index doublets needed to satisfy the prescribed transmittance constraint remains fixed, irrespective of whether the high-index layers consist of the H or H material, and is the same as for the reference TiO 2 ::Ta 2 O 5 /SiO 2 binary coating. Hence,  [ppm] [ppm] It is seen that all sub-optimal, constraint-compliant admissible designs can be divided into distinct families, represented by the aligned markers in Figure 2  For each family, the number of distinct admissible designs, in square brackets in Figure 2 (b) is (slightly) less than the binomial coefficient due to the (relatively few) designs that do not fulfill the transmittance and absorbance constraints.
Similar to the previous case, the optimal design featuring the lowest thermal noise under the prescribed transmittance and absorbance constraints, consists of a stack of [H |L] doublets grown on top of the substrate, topped by another stack of [H|L] doublets, again confirming the ansatz in Refs. [13] and [14] .

C. Robustness
The optimal ternary QWL coatings are nicely robust against uncertainties in the value of the extinction coefficient κ H , as well as against unavoidable inaccuracies in the layers' thicknesses, due to technological limitations of the deposition process [32].
As an illustration, Figure 4 shows the distributions of coating transmittances and absorbances in 10 5 realizations of the optimal ternary QWL coating using Material-A.
Optimal ternary coating using Material-A  Similarly, Figure 5 shows the distributions of coating transmittance and absorbance in a sample of 10 5 realizations of the optimal ternary QWL coating using Material-B, assuming κ B to be random uniformly distributed in (0.5κ B , 1.5κ  Not unexpectedly, uncertainties in the extinction coefficient stemming from fluctuations in the deposition process, rather than systematic uncertainty in the nominal value, have a lesser effect, due to possible fluctuation compensation, resulting in narrower distributions of the coating transmittance and absorbance. Figure 6 shows the distributions of coating transmittance, absorbance, and loss angle (normalized to the value of the reference binary coating) in a sample of 10 5 realizations of (i) the optimal ternary QWL coating using Material-A for H , with κ A = 10 −4 [ Fig. 6(a), 6(c) and 6(e)], and (ii) the optimal ternary QWL coating using Material-B for H , with κ B = 10 −5 [Figs. 6(b), 6(d), and 6(f)], assuming the thicknesses of all layers to be independent random variables identically distributed uniformly around the nominal QWL thickness, in a symmetric interval of total width 2 nm.
We may conclude that the optimal ternary QWL designs are fairly robust against uncertainties in the extinction coefficient of the H material, and deposition-related thickness errors.

Optimal ternary coating using
Material-B with = 10 Optimal ternary coating using Material-A with = 10

D. Transmittance Spectra
We computed the transmittance spectra of the above optimal ternary QWL coatings. These are shown in Figs. 7 (a) and 7 (b) for coatings using materials A and B, respectively.
The spectra were computed neglecting chromatic dispersion, except in the neighborhood of the operating wavelength (λ 0 = 1064 nm), shown in the insets, where a linear approximation was used, The pertinent values of dnr dλ λ0 were taken from Ref. [33] [nm] [nm]
Within the limits of this model, no ripple is observed in the high-reflectance band. The shape of the transmission spectrum for the optimal coating using Material-A departs more markedly from the reference spectrum compared with that of the optimal coating using Material-B. The observed asymmetry of the lobes stems from the fact that the coating is piecewise homogeneous, consisting of two cascaded homogeneous QWL stacks.

E. Thickness Optimization
Thermal noise in binary coatings can be effectively reduced compared with the reference QWL-layer design by suitably reducing the total thickness of the mechanically noisier material(s), while increasing the total thickness of the other material(s) and the total number of doublets, so as to keep the coating transmittance unchanged [18].
Remarkably, the thickness-optimized binary coatings turn out to consist of almost identical stacked [H|L] doublets whose thickness is one half of the working wavelength (Bragg condition), the exception being represented by a few layers near the coating top and bottom [34], [35], [36], [37].
Implementing thickness optimization for ternary coatings is computationally demanding. In a full-blind exhaustive approach, each and any L layer should be allowed to take any thickness value in the range (λ/4, λ/2), and each and any H and H layer should be allowed to take any thickness in (0, λ/4), λ being the local wavelength. Even after suitable discretization of the above search intervals, the computational burden of an exhaustive search would become prohibitive for any meaningful value of N T .
A reasonable heuristic approach to thickness optimization of ternary QWL coatings may thus consist in optimizing the two binary QWL stacks that form the top and bottom parts of the optimal QWL ternary coatings, assuming each of them to consist of identical non-QWL Bragg doublets [38].
Here, to illustrate the possible margins of further loss angle reduction obtainable from thickness optimization, we content ourselves with computing the (normalized) coating loss angle, absorbance and power transmittance of the coatings obtained after modifying the optimal ternary QWL coatings found in the previous Sections and listed in Figure 3 by letting so as to preserve the Bragg character of the doublets, and adding a few [H|L] and/or [H L] layers as needed to maintain compliance with the transmittance and absorbance constraints. The transmittance, absorbance, and coating loss angle (normalized to the reference value) of the resulting thickness-tweaked coatings are collected in Table-III. --

V. REALISTIC MATERIALS
The structure and properties of the optimal coatings are the same if we consider realistic candidates for the third material, namely aSi, and SiN x . In this section we present results based on the previous analysis/simulation tools for optimal QWL ternary coatings operating at 290K, 120K, and 20K, based i) on (silica, titania-doped tantala and aSi) at 1550nm and ii) on (silica, titania-doped tantala and SiN x ) at 1064nm, using current available measurements/estimates of the actual (or fiducially achievable) material parameters, collected in Table IV. The coating Brownian noise power spectral density (PSD) reduction factor with respect to the reference advanced LIGO (aLIGO) and advanced Virgo (adVirgo) coatings currently in operation (at ambient temperature) is shown in Figure 8(a) and 8(b), respectively for different values of the extinction coefficient of the third material, in the range from 10 −5 to 10 −4 .
In calculating the Brownian noise PSD reduction factor we obviously include the temperature-dependent factor in eq. (13). We note in passing that reducing the PSD by a factor ρ corresponds to reducing the so called amplitude spectral density (or rms noise level) by a factor ρ 1/2 and to boosting the visibility distance by a factor ρ −1/2 . Remarkably, as seen from the figure, the thermal noise reduction factor is found to be almost linear in log 10 (κ H ). The achievable coating thermal noise PSD reduction factor is nicely large, especially at cryogenic temperatures, where it gets close to the Einstein Telescope [44] and Cosmic Explorer [45] requirements, and compares to the expected performance of crystalline coatings [46], while possibly posing less demanding technological challenges.

VI. CONCLUSIONS
In this paper we addressed the problem of designing a ternary optical coating consisting of QWL layers to achieve the minimum thermal (Brownian) noise under prescribed optical transmittance and absorbance constraints.
We first considered ternary coatings where two materials are those presently in use in the advanced LIGO and Virgo detectors, namely, SiO 2 and TiO 2 :: Ta 2 O 5 , featuring the best trade-off between optical contrast, optical losses and thermal noise so far; and the third material is one of two putative materials featuring, respectively, the same mechanical losses as TiO 2 :: Ta 2 O 5 and a higher refractive index, but larger optical losses (Material-A), or the same refractive index as TiO 2 :: Ta 2 O 5 and fairly lower mechanical losses, but larger optical losses (Material-B), allowing their extinction coefficient to range from 10 −6 to 10 −4 in both cases.
We performed an exhaustive search over all possible (and admissible, in the sense discussed in Sect. IV) configurations consisting of QWL layers, using backtracking for numerical efficiency and seeking the optimal designs yielding minimum thermal (Brownian) noise under prescribed upper bounds for transmittance and absorbance.
The main results of this study can be summarized as follows.
All found optimal designs, consist of a stack of [H |L] doublets grown on top of the substrate, topped by another stack of [H|L] doublets, confirming the ansatz in Refs. [13,14] .
They are nicely robust against deposition inaccuracies in the individual layer thicknesses and systematic uncertainties and/or fluctuations from layer to layer of the extinction coefficient. Their transmittance spectra satisfy the design constraints in the useful band.
We have further shown that a further improvement in performance can be achieved by using thinner layers of the noisier materials in each Bragg doublet. Exhaustive blind thickness optimization of ternary coatings appears to be computationally unaffordable, though, and the problem will be accordingly studied in a future paper following a heuristic approach.
Next we applied the same optimization strategy to realistic candidates for the third material, namely aSi and SiN x , operating at three different temperaures (290K, 120K, and 20K). In order to take into account the present uncertainties (and possible margins of improvement [39]) for the optical losses of these materials, we considered different values of their extinction coefficient in the range from 10 −6 to 10 −4 .
All optimal designs outperform significantly the reference binary solution consisting of alternating QWL layers made of silica and titania-doped tantala, in terms of Brownian thermal noise.
In particular, according to our simulations, QWL ternary coatings using either aSi or SiN x in addition to silica and titania-doped tantala may achieve an almost tenfold reduction in the coating thermal noise (power spectrum) level compared with second generation detector coatings operating at ambient temperature.