GLOBAL OPTIMIZATION OF MULTILAYER DIELECTRIC COATINGS FOR PRECISION MEASUREMENTS
7.4 Case studies and results
10−5 10−4 10−3 10−2 10−1 100
RorT,p-pol
400 600 800 1000 1200 1400 1600
Wavelength [nm]
10−5 10−4 10−3 10−2 10−1 100
RorT,s-pol
(a)
980 1000 1020 1040 1060 1080 1100 1120
Wavelength [nm]
20 40 60 80 100
Transmission[ppm]
θi= 40 θi= 40, measured θi= 45 θi= 45, measured
(b)
0.4 0.8 1.2 1.6 2.0
Rp−pol 532nm[%]
Ns= 100000 σLn= 0.005 σHn = 0.005 σL= 0.005 σθi= 0.005
20 40 60 80 100 Tp−pol1064nm[ppm]
0.4 0.8 1.2 1.6 2.0
Rs−pol 532nm[%]
0.4 0.8 1.2 1.6 2.0 Rp−pol532nm[%]
0.4 0.8 1.2 1.6 2.0 Rs−pol532nm[%]
(c)
Figure 7.3: Performance of an optimized harmonic separator coating design. Fig- ure 7.3(a) indicates the wavelengths of interest, λ1 (orange) and λ2 (green), with dashed vertical lines. Figure7.3(b)shows the measured performance of a harmonic separator fabricated with layer thicknesses generated using the optimization routine described in the text. Figure7.3(c)shows the robustness of the design choice of 20 layer pairs in the dielectric stack to small variations in assumed model parameters.
frequency range. For SiO2/Ta2O5coatings of the type used on the aLIGO optics, the mechanical loss angle,φdielectricis≈4×larger for Ta2O5than for SiO2[122], and so the total thickness of the latter in a given coating design dominates the thermal noise contribution. Hence, the design should be optimized for minimum thermal noise, while still meeting other requirements.
Finally, the circulating power in these Fabry-Pérot cavities is expected to beO(1 MW) during high power operation. The coating should be designed with a safety factor such that it is not damaged under these conditions. One possible damage mechanism is that residual particulate matter on the optic’s surface gets burnt into the coating.
In order to protect against this, the coating has to be optimized to have minimum surface electric field.
With these requirements as inputs to the optimization problem, we ran the particle swarm and sensitivity analysis and obtained a set of layer thicknesses. Figure 7.4 show the performance of the optimized coating, and compares it to the commonly used "quarter-wave stack" HR coating design.
20 40 60 80 100
R532nm[%]
Ns= 100000 σnL= 0.005 σnH= 0.005 σL= 0.005
1.8 2.0 2.2
STO[zm/√ Hz]
6.6 6.8 7.0 7.2
SBr[zm/√ Hz]
2 4 6 8 10
T1064nm[ppm]
1 2 3 4 5
~ ESurface
[V/m]
20 40 60 80 100 R532nm[%] 1
.8 2.0
2.2 STO[zm/√
Hz]
6.6 6.8
7.0 7.2 SBr[zm/√
Hz]
1 2 3 4 5
E~Surface[V/m]
Figure 7.4: Performance of an optimized aLIGO ETM coating design. Here, we compare the robustness of the optimized design (red) and the naive quarter- wave design (blue) to small variations in assumed model parameters. The superior performance of the optimized coating, with respect to the design goals at both wavelengths, is evident.
7.4.3 Crystalline HR coatings
As mentioned in Section7.2.3, alternative dielectric materials are being considered in an effort to reduce the coating Brownian noise, and hence, improve the sensi- tivity of laser interferometric GW detectors. One promising alternative is to use crystalline dielectrics consisting of alternating layers of Al0.92Ga0.08As (low-index material) and GaAs (high-index material), which has been shown to yield up to tenfold reduction in Brownian noise, relative to SiO2/Ta2O5 coatings with com- parable reflectivity. However, the overall thermal noise has to take into account both the Brownian noise contribution, as well as the Thermo-Optic noise contri- bution. The latter can be minimized by coherent cancellation of thermorefractive and thermoelastic effects. The optimizer is encouraged to favor solutions that have this cancellation effect by including a penalty for the TO noise at a representative
0.2 0.4 0.6 0.8 1.0
STO[zm/√ Hz]
Ns= 50000 σLn= 0.005 σHn = 0.005 σL= 0.005
2.23 2.25 2.27
SBr[zm/√ Hz]
1 2 3 4 5
~ ESurface
[V/m]
2 4 6 8 10
T1064nm[ppm]
0.6 0.8 1.0 1.2
αT[ppm]
0.2 0.4 0.6 0.8 1.0 STO[zm/√
Hz]
2.23 2.25 2.27 SBr[zm/√
Hz]
1 2 3 4 5
E~Surface[V/m]
0.6 0.8 1.0 1.2 αT[ppm]
Figure 7.5: Sensitivity analysis of the optimized coating’s performance.
frequency (we choose 100 Hz) in the cost function.
An additional consideration for future laser interferometric GW detectors, which are expected to be operated at cryogenic temperatures for thermal noise improvements [123], is that absorption in the mirror coating must be sufficiently small as to allow efficient radiative heat extraction and maintain the temperature of the optic at 123 K.
While the absorption is not explicitly included in the cost function that is minimized by PSO, we include it in the MC sensitivity analysis, and confirm that the likelihood of it lying within the acceptable range of ≤ 1ppm is high, even if there are small deviations in assumed model parameters. The overall performance of the optimized coating is shown in Figure7.5. Figure7.6 shows the variation of the electric field inside the coating, and also the layer thickness profile.
0.0 0.2 0.4
Normalized|E(z)|2
0 1 2 3 4 5 6 7
Distance from air interface,z[µm]
0 50 100 150
Physicallayerthickness[nm] GaAsAlGaAs
Figure 7.6: Right top panel: Square of electric field inside coating, normalized to the incident electric field, |E(z)
E+o
|2. The dashed vertical lines indicate boundaries between low- and high-index materials. Right bottom panel: Physical thickness profile of the individual dielectric layers.
7.4.4 Corner plots
In this chapter, corner plots are used extensively to visualize the ensemble of samples from MCMC sampling. So I will briefly remark on why they are a powerful visual- ization tool. The plot is made by arranging 2D histograms of samples, marginalizing over the other dimensions of the parameter space. The set of such 2D histograms are then arranged in a matrix - but of course, such a matrix would be symmetric, with the upper triangular plots contributing no new information (the correlations between parameters θ1 and θ2 are independent of the order in which they are considered).
Finally, the topmost row in each column marginalizes overallother dimensions, to make a 1D histogram of the posterior distribution of a particular parameter in the MCMC ensemble. In the case of the inverse problem described in Section 7.3.2, it is useful to overlay the prior distributions of parameters used to draw samples.
The difference between these two histograms would indicate how informative the measurement is - if they are nearly identical, then the measurement with which the inference is being made adds very little information to our prior assumptions, sig-
nalling the need for more data to make any meaningful conclusions. As can be seen in Figure7.5, the 2D histogramscan reveal correlations between the variables being studied. When strong correlations exist between a pair of parameters, the density of points will be high along flattened ellipses (e.g. bottom left plot). Conversely, when the correlation is weaker, the density will tend to be more randomly (and evenly) distributed (e.g. the 2D histogram betweenT1064 nm andE®surface). The inset in thee upper right part of the corner plots presented in this chapter indicate the number of samples drawn in the MCMC sampling phase, as well as the assumed fractional uncertainties in various model parameters.