Reducing Thermal Noise in Future Gravitational Wave Detectors by employing Khalili Etalons
Alexey G. Gurkovsky, Daniel Heiner, Stefan Hild, Ronny Nawrodt, Kentaro Somiya, Sergey P. Vyatchanin, Holger Wittel
aa r X i v : . [ g r- q c ] J u l Reducing Thermal Noise in Future Gravitational Wave Detectors by employingKhalili Etalons
Alexey G. Gurkovsky, Daniel Heinert, Stefan Hild, Ronny Nawrodt, Kentaro Somiya,
4, 5
Sergey P. Vyatchanin, and Holger Wittel Faculty of Physics, Moscow State University, Moscow, 119991 Russia Institut f¨ur Festk¨orperphysik, Friedrich-Schiller-Universit¨at Jena, D-07743 Jena, Germany SUPA, School of Physics and Astronomy, Institute for Gravitational Research,Glasgow University, Glasgow G12 8QQ, United Kingdom Waseda Institute for Advanced Study, 1-6-1 Nishiwaseda, Shinjuku, Tokyo 169-8050, Japan Interactive Research Center of Science, Tokyo Institute of Technology,2-12-1 Oh-okayama, Meguro, Tokyo 152-8551, Japan Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany (Dated: September 21, 2018)Reduction of thermal noise in dielectric mirror coatings is a key issue for the sensitivity improve-ment in second and third generation interferometric gravitational wave detectors. Replacing an endmirror of the interferometer by an anti-resonant cavity (a so-called Khalili cavity) has been proposedto realize the reduction of the overall thermal noise level. In this article we show that the use of aKhalili etalon, which requires less hardware than a Khalili cavity, yields still a significant reductionof thermal noise. We identify the optimum distribution of coating layers on the front and rear sur-faces of the etalon and compare the total noise budget with a conventional mirror. In addition webriefly discuss advantages and disadvantages of the Khalili etalon compared with the Khalili cavityin terms of technical aspects, such as interferometric length control and thermal lensing.
I. INTRODUCTION
The sensitivities of second-generation (AdvancedLIGO, Advanced VIRGO, GEO-HF, and LCGT) andthird-generation (Einstein Telescope) interferometricgravitational wave detectors will be partly limited bythermal fluctuations in the mirrors [1–5].The pioneering articles on this issue were dedicatedto the investigation of the mirror substrate fluctua-tion: Brownian thermal noise [6–8] and thermo-elasticnoise [9]. Fundamental thermal motion (Brownian mo-tion) of material atoms or molecules causes Browniannoise. Fundamental thermodynamic fluctuations of tem-perature lead to thermo-elastic noise through the ma-terial’s thermal expansion. Similarly thermo-refractivenoise [10, 11] is caused by temperature fluctuations lead-ing to fluctuations of the refractive index and thereforefluctuations of the optical path length inside the mate-rial. These results were obtained for the model of aninfinite test mass, i.e. the mirror was considered to be anelastic layer with infinite width and finite thickness. Allof these results were generalized for a finite-size mirrormodel [8, 12].Very soon the importance of mirror coating thermalnoise was realized as its parameters may differ consid-erably from the mirror substrate parameters. Despiteits low thickness, the very high loss angle of the mirrorcoating materials (usually SiO and Ta O ) makes thecoating Brownian noise the most significant one amongall kinds of mirror thermal noise [13–15]. The thermo-elastic noise of the coatings only has a small contribu-tion in the total noise budget [16, 17]. Later, Kimble [18]proposed the idea of thermal noise compensation whichwas explored carefully in [19, 20] for a particular case of thermo-elastic and thermo-refractive noise.The coating Brownian noise is still one of the maincontributions to the noise spectra of gravitational waveobservatories [13–15]. One of the most promising ap-proaches aimed to decrease its level was offered by Khalili[21] who proposed to replace the end mirror in the inter-ferometer arm with a short Fabry-P´erot cavity tuned toanti-resonance (see center panel of Figure 1). In prac-tice most light is reflected from only a few first layers(farest from substrate) and all others (located closer tosubstrate) only reflect a small part of light. However,since the thermal fluctuations are proportional to the to-tal thickness of the coating and the inner layers of coatingare the main contribution to the phase fluctuations of thereflected light, the transmittance of each mirror can behigher to realize the same reflectivity of the system as acompound end mirror. The total thickness of coatings insuch Khalili cavity mirrors is the same as the thicknessof the conventional mirror, while the Brownian noise ofthe end mirror of a Khalili cavity is significantly reduced[21] because the thickness fluctuations of the second endmirror coating (EEM in Figure 1) do not influence thefluctuations of the input mirror coating (IEM in Figure1). Moreover, using a rigidly controlled Khalili cavityallows a reduction of coating Brownian noise [22].One of the main problems in the Khalili cavity is toestablish a low-noise control of the mirror positions (seedetailed explanation in Sec. III). A potentially easier wayis to use a
Khalili etalon (KE) instead of a Khalili cavity(KC) or a simple conventional mirror (CM). The idea isto use a single mirror but to split the coating into twoparts (see right hand panel of Figure (EEM in Figure 1): the front coating (on the front substrate surface) featuresjust a few layers and the rear coating (on the rear sub-
Figure 1: Simplified schematic of an Advanced LIGO interferometer with conventional end mirrors, featuring all the coatinglayers on their front (left), with Khalili cavities as end mirrors (center) and with Khalili etalons, featuring only a few coatinglayers on the front surface and the majority of the coating layers on the rear surface (right). strate surface) consists of the rest of the required coatinglayers.The purpose of this article is to develop an idea ofthe Khalili etalon [23], to calculate the total mirror ther-mal noise arising in a KE and in a CM, and to comparethem. We investigate the idea of using a KE in the Ein-stein Telescope (ET) and Advanced LIGO (aLIGO). InSec. II we describe the mirror parameter optimizationprocedure, namely the optimal number of layer pairs inthe front coating. In Sec. II D we describe the detailsof the thermal noises arising in the KE and CM calcula-tions. Section III is dedicated to the problem of thermallensing which is much more important in a KE than ina CM. In Sec. IV we discuss the obtained results anddraw the conclusions. Finally, some calculation detailsare provided in the appendices A-B.
II. COATING OPTIMIZATION
The main idea of using a KE is to reduce the mirror’stotal thermal noise without reducing its reflectivity. By total thermal noise spectral density we mean the sum ofthe Brownian, thermo-elastic and thermo-refractive noisespectral densities. Coating Brownian noise is causedmostly by the fluctuations of the entire coating thick-ness. It would then seem evident that Brownian noisebe lower when the front coating contains less layers andhence the lowest noise be achieved for the coating totallydisplaced to the rear mirror surface. This is in princi-ple true but at the same time some other noises, such assubstrate thermo-refractive noise, rise dramatically caus-ing the total noise level to rise also. Moreover, the lesslayers one puts onto the front coating, the higher willbe the absorption in the substrate. So there has to bean optimum of how to best distribute the coating layersbetween the front and back surfaces in order to obtainminimal total thermal noise and not too much of absorp- tion in substrate. The aim of this section is to find thisoptimum configuration.
A. Thermal Noise calculation technique
The only way to find the optimal number of front coat-ing layers, N , is to compare the thermal noise for ev-ery N . This requires the calculation of the differentnoise contributions as functions of the front coating lay-ers number N . The most basic principles we used are:(i) the total number of Ta O and SiO layers, N , isfixed, i.e. we used the coating structure planned for bothET and aLIGO and modified it to fit the double coatingparadigm: 20 Ta O and 18 SiO quarter-wave layersplus the substrate (it is considered as an ordinary but“slightly” thicker coating layer) and plus two caps con-sisting of a half-wavelength SiO layer (for the CM itwould have been 20 Ta O layers and 19 SiO layers plusone cap); (ii) a quarter-wavelength Ta O layer and aquarter-wavelength SiO layer are alternately coated onthe front or rear surface so that there are always an oddnumber of front coating layers ( N = 1, 3, 5 etc.) and alsoan odd number of layers of the rear coating N = N − N (37, 35, 33 etc.). Please note that the substrate andcaps are not included in these numbers; (iii) the num-ber of layers of the front coating N is the argumentand the total thermal noise driven mirror displacement S (KE)total is the function of it; (iv) we consider only Brow-nian, thermo-elastic and thermo-refractive noises (beingthe most significant contributions), and (v) we used themirror of a finite-size cylinder, the model of which hasbeen developed in Ref. [8, 12, 15, 23], and calculatedall noises numerically using the fluctuation-dissipationtheorem (FDT) [7, 24, 25] as it is briefly described inSecs. II A 1-II C. The optimal number of front coatinglayers appeared to be N = 3, i.e. 2 Ta O layers and 1SiO layer plus a cap in the front coating and 18 Ta O layers and 17 SiO layers plus a cap in the rear coat-ing. With the technical feasibility taken into account(see Sec. II C), however, it turns out that the systemwith N = 5 layers on the front surface (i.e. 3 Ta O layers and 2 SiO layers plus a cap on the front mirrorsurface and 17 Ta O layers and 16 SiO layers plus acap on the rear surface) will be better and we analyzethe system with N = 5 in detail. In this case the mirrorthermal noise does not reach its minimum but it is onlyabout 5 % higher.
1. Brownian noise
The total coating thermal noise of the etalon will bethe sum of noise on the front surface and noise on theback surface: δx = ǫ δx + ǫ δx . (1)Here δx is the displacement of the front surface of themirror and δx the displacement of the boundary sur-face between the rear surface of the mirror substrate andthe coating on it. Considering the KE as a Fabry-P´erotcavity consisting of two mirrors with amplitude reflectiv-ities R (front coating) and R (rear coating) tuned toanti-resonance, one can calculate the coefficients ǫ and ǫ (see details in Ref. [23]): ǫ = R (cid:2) n s ) R R + R (cid:3) + R (1 − n s )(1 + R R ) ,ǫ = n s R (1 − R )(1 + R R ) . (2)Here n s is the substrate refractive index. Note that R and R are functions of the number of front and rearcoating layers. In particular, we have the following for-mulas for R and R as functions of the number of thefront coating layers N ( N = N − N is the number ofthe rear coating layers): R = 1 − n s n N − n N +11 n s n N − n N +11 , R = 1 − n s n N − N − n N − N +11 n s n N − N − n N − N +11 , (3)where n and n are Ta O and SiO coating layersrefractive indices.Hence, in order to calculate spectral density S ( ω ) ofthe displacement δx caused by thermal noise using theFDT, one has to apply the forces ǫ F e iωt and ǫ F e iωt to the front and rear coatings correspondingly and to cal-culate the total dissipated power [7, 24, 25]. For the cal-culation of the spectral density S B ( ω ) of Brownian coat-ing noise the dissipated power may be calculated through the elastic energy U ( k )B stored in each k -th layer (of thefront or rear coating): U ( k )B = π Z h k dz Z R (cid:16) E ( k ) rr T ( k ) rr + E ( k ) φφ T ( k ) φφ ++ E ( k ) zz T ( k ) zz + E ( k ) rz T ( k ) rz (cid:17) dr , (4)where E ij and T ij are the strain and stress tensor com-ponents (only the non-zero components are shown in theformula above), R is the mirror radius and h k is the thick-ness of the k -th layer. The components E ij and T ij arecalculated as it is described in detail in [23]. Then theBrownian noise spectral density may be evaluated as fol-lows: S B = 8 k B Tω N +2 X k =0 U ( k ) B φ k (5)where k B is Boltzmann’s constant, T is the absolute tem-perature and φ k is the loss angle describing structurallosses in the k -th layer. The sum is taken over all layers,i.e. N = 38 is the number of layers without the substrateand the caps. The total number of summands is there-fore N + 3 = 41; N = 38 layers in the front and rearcoatings, plus 2 layer-caps and 1 layer-substrate. So theindex k = 0 refers to the front coating cap, the indices k = 1 ÷ N refer to the front coating quarter wavelength(QWL) layers, the index k = N + 1 refers to the sub-strate, the indices k = N + 2 ÷ N + 1 refer to the rearcoating QWL layers and the index k = N + 2 refers tothe rear coating cap. There are N + 2 = 40 summands;the ones with k = N are to be considered for coatingBrownian noise and the one with index k = N + 1 is tobe considered for substrate Brownian noise: S (coat)B = 8 k B Tω N X k =0 U ( k ) B φ k + N +2 X k = N +2 U ( k ) B φ k ! , (6a) S (sub)B = 8 k B Tω U ( N +1) B φ N +1 . (6b)Here φ N +1 = φ s is the loss angle of the substrate, while φ and φ represent the loss angels of the Ta O andSiO layers, respectively. The values are presented inTable II.
2. Thermo-elastic noise
The thermo-elastic (TE) noise calculations for the sub-strate and for the coating is similar. In order to calculatethe dissipated power one should calculate the diagonalcomponents E jj of the strain tensor for each layer (in-cluding the substrate and the caps) and take the trace: θ ( k ) = E ( k ) rr + E ( k ) φφ + E ( k ) zz . Then one may find the power dissipated through the TEmechanism [12, 25]: W ( k )TE = 2 πκ k T (cid:18) Y k α k (1 − ν k ) C k ρ k (cid:19) Z h k dz Z R h ~ ∇ θ ( k ) i r dr , (7)where κ k is the thermal conductivity, C k is the thermalcapacity per unit volume, Y k is the Young’s modulus, ν k is the Poisson’s ration, α k is the thermal expansioncoefficient, ρ k is the density, and the index k denotesnumber of the layer. Therefore, the TE noise spectraldensity will simply be: S TE = 8 k B Tω N +2 X k =0 W ( k ) T E (8)Similar to the Brownian noise calculations, the sum-mands with the indices k = N + 1 are relevant for thecoating TE noise while the one with the index k = N + 1needs to be considered for the TE noise of the substrate: S (coat)TE = 8 k B Tω N X k =0 W ( k ) T E + N +2 X k = N +2 W ( k ) T E ! , (9a) S (sub)TE = 8 k B Tω W ( N +1) T E . (9b)
3. Thermo-refractive noise
TR noise originates from thermodynamic fluctuationsof the temperature δT in the substrate, producing phasefluctuations of the reflected wave phase via the tem-perature dependence of the substrate’s refraction index n s . Likewise, the phase fluctuations may be recalculatedinto effective fluctuations of mirror surface displacement δx = − ǫ β s hδT where the coefficient ǫ introduced in(2), characterizes the light amplitude circulating insidethe substrate and β s = dn s /dT is the thermo-optic coef-ficient of the substrate.We calculate the thermo-refractive (TR) noise in thesubstrate using the model of an infinitely large plane inthe transverse directions with thickness of h . The spec-tral density of the temperature fluctuations in this modelis shown in [26] see Eq. (E8): S T = 16 k B T κ s πρ s C s w ω h where w is the radius of the light spot (intensity decreaseswith distance r from center as ∼ e − r /w ) and the pa-rameters with subscript s refer to the substrate. The TRnoise spectral density for the substrate (recalculated todisplacement) becomes S (sub)TR = ǫ β s h S T = ǫ k B T β s κ s hπρ s C s w ω (10) Benthem and Levin have pointed out that some correc-tions should be applied to this formula. This correctionsare based on the account of the fact that light inside thearm froms the standing wave and not a traveling wave.We can rewrite Eq. (2) of Ref. [11] in a simpler form withonly the normal incidence and the circular beam beingconsidered: S (sub)TR = ǫ k B T β s κ s hπρ s C s w ω k w k p κ s /C s ρ s ω ) ) ! (11)In addition we have to consider the TR noise presentin the coatings [10]. For its estimate we use the followingformula S (coat)TR = 2 √ k B T β Λ π √ κ s ρ s C s w √ ω , (12) β eff = 14 β n + β n n − n (13)where Λ is the wavelength of light in vacuum, β eff is theaveraged thermo-optic coefficient of the entire coating,and β = dn /dT and β = dn /dT are the thermo-optic coefficients of Ta O and SiO layers, respectively.This formula is based on the assumption that only thefirst few layers contribute considerably to the thermo-refractive loss mechanism. It is obtained for a mirrorwith an infinite radial dimension and a finite height. Thismodel is valid with good accuracy for CM. However, forKE we use the same formula as an order-of-magnitudeestimation. B. Optimization results
In this subsection we present the results of our opti-mization process. Using the proposed parameters we ob-tained numerical estimates of all noise sources discussedabove for ET and aLIGO. All geometrical design param-eters for these interferometers are presented in Table I.The physical constants and material parameters are sum-marized in Table II.First of all, we analyze the spectral density of the dis-placement noise S (CM)total for a Khalili etalon (KE) as afunction of the number of front layers N . This noiseanalysis considers the sum of the noise sources listed inSec. II A: Brownian, TE and TR noises which are dividedinto a coating and a substrate contribution each. The KEtotal thermal noise is then compared to the results for aconventional mirror (CM) S (CM)total using a gain parameter G which is defined as: G = vuut S (CM)total ( ω ) S (KE)total ( ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω =2 π
100 s − . (14)This gain G has to be maximized in order to enhancethe detector sensitivity. In Fig. 2 we plot the gain G as a function of the number of front coating layers N for both, ET and aLIGO. Please recall that the numberof rear coating layers N = N − N is constrained bythe total number of coatings N = 38. One can see that PSfrag replacements ga i n G Number of front coating layers N Figure 2: Gain G (see (14)) in ET (blue stars) and aLIGO (redstars) as a function of the number of front coating layers N .The number of rear coating layers N results from the totalnumber of layers N = 38 minus the number of front layers N .Parameters used for this calculation are presented in TablesI and II. For ET a maximum gain of G = 1 .
72 appears for N = 3 front coating layers. For aLIGO the maximum gainof G = 2 .
16 is also obtained for N = 3. for both detectors, ET and aLIGO, the gain is obviouslymaximized for the case of N = 3 front coating layers.Then the maximum gain appears to be G = 1 .
72 for ETand G = 2 .
16 for aLIGO.
C. Absorption
Another important parameter to be taken into accountis the substrate absorption Ψ abs . It describes the portionof light energy which is absorbed in the substrate withrespect to the light incident to the mirror. The value ofΨ abs is accessible via the light power circulating insidethe substrate. We know that this light power inside theetalon will be a factor of1 − R (1 + R R ) lower than the light power incident to the mirror. Finally,the value Ψ abs may be evaluated using the absorptioncoefficient of the substrate material η (in ppm per cm)and the substrate thickness h (in cm):Ψ abs = 2 ηh − R (1 + R R ) . (15)Note the factor of 2 in front of the substrate thickness,which occurs as the light passes the substrate twice: once forward and once backward. Also keep in mind that thereflectivities R and R are functions of the number offront and rear coating layers — see formulas (3). PSfrag replacements s ub s t r a t e a b s o r p t i o n Ψ a b s , pp m Number of front coating layers N Figure 3: Substrate absorption Ψ abs in ET (blue stars) andaLIGO (red stars) as a function of the number of front coatinglayers N . The number of rear coating layers N results fromthe total number of layers N = 38 minus the number of frontlayers N . Parameters used for this calculation are presentedin Tables I and II. As Fig. 3 illustrates Ψ abs decreases exponentially withan increasing number of front coating layers. For theoptimum number of front coating layers N = 3 the sub-strate absorptions in ET and aLIGO equal 2 .
67 ppm and1 .
78 ppm, respectively. It seems reasonable to assumethat a loss coefficient of 1 ppm is admissible. In this casewe have to choose N = 5. Indeed, using formula (3)with N = 5 and coating parameters listed in Tables Iand II we obtain:Ψ (AL)abs ≃ .
90 ppm , Ψ (ET)abs ≃ .
35 ppm . (16)It means that for aLIGO (circulating power W =0 . W (AL)abs ≃ .
72 W,as for ET ( W = 3 MW) — W (ET)abs ≃ . N = 5.This choice allows the gain to be G = 1 .
67 for ET and G = 2 .
03 for aLIGO.
D. Thermal noise of the Khalili etalon
In Fig. 4 and 5 we present the thermal noise spec-trum of a KE including different noise sources for ETand aLIGO, respectively. For numerical estimates we usethe parameter data listed in Tables I and II. One clearlyrealizes that Brownian thermal noise dominates the mir-ror thermal noise at almost all the frequency range from1 Hz to 10 kHz.
Table I: Parameters used for the numerical calculation. w isthe radius of the laser beam (intensity decreases with distance r from the center as ∼ e − r /w ), R and h are radius andthickness of the cylindrical mirror, W is the power circulatingin the arm cavities of the interferometer.Parameter Einstein Telescope Advanced LIGO w , m 0 .
12 0 . R , m 0 .
31 0 . h , m 0 .
30 0 . W , MW 3 0 . N (CM) N (KE)
38 + 2 caps 38 + 2 capsTable II: Parameters used for the numerical calculations. T is the temperature, Λ is the optical wavelength in vacuum, η denotes the optical losses per unit length in the substrateand n is the refractive index. The remaining parameters areexplained in Sec. II A 1-II A 3.Parameter substrate Ta O layer SiO layer T , K 300 [17]Λ, m 1 . × − [17] η , ppm/m 25 - - n [17] 1.45 2.035 1.45 β , 1/K [17, 19] 8 × − . × − × − α , 1/K[17] 5 . × − . × − . × − ρ , kg/m [17] 2202 6850 2202 Y , Pa 72 × [17] 140 × [28] 72 × [17] ν . . . κ , W/K m[17] 1 .
38 33 1 . C , J/K kg[19] 746 306 746 φ × − [29] 2 × − [30] 4 × − [31] Seperately, we also present the numerical results for allnoise sources at a single frequency of f = 100 Hz in Ta-ble III. We choose this frequency as a round number lo-cated in the frequency range with the highest sensitivity.This frequency value has already been used previously inthis article in Sec. II B for numerical estimates.Brownian noise is the main object of our investigationsas it dominates the sensitivies of both detectors (ET andaLIGO) almost in the whole detection band. It can becalculated accurately using the model developed in [23].Shortly, a calculation method is presented in Sec. II A 1.A further inspection of the spectral noise sensitivity plotsreveals substrate Brownian noise to be the second impor-tant noise process. Thus, Brownian noise dominates overTE and TR noise absolutely and is the main factor lim-iting both interferometers sensitivities in the frequencydomain near 100 Hz.Taking the same noise sources into account as for our Figure 4: ET noise spectral densities for KE with a frontcoating of N = 5 layers (plus a cap) and a rear coatingof N = 33 layers (plus a cap). Parameters used for thiscalculation are presented in Table I and Table II. KE investigation (see Sec. II A 1-II A 3) we have calcu-lated the numerical values for the spectral noise densityof a corresponding CM. For clarity we do not presentthe single contributions but only the total mirror ther-mal noise in this section. Using thermal noise values forKE and CM ( S (KE)tot and S (CM)tot ) and the definition of thegain parameter G (14) we compare KE and CM to esti-mate the benefit of using a KE. For the spectral densitiescalculated in Table III we arrive at a gain of G ET = 1 . , for ET parameters , (17) G AL = 2 . , for aLIGO parameters . (18)The sligtly larger gain value for aLIGO parameters maybe qualitatively explained by the mirror geometry. Sothe ET mirror is more sensitive to membrane deforma-tions than the aLIGO mirror. This property allows to’transfer’ Brownian fluctuations from the rear coating tothe front surface more effectively. Indeed, the geometri-cal factor g = 2 R/h (the fraction of diameter to thicknessof mirror) for ET is larger than for aLIGO: g AL ≃ . , g ET ≃ . . (19)
1. Semi-qualitative consideration
In this subsection we would like to present a way tosimply estimate the gain. For an order of magnitudeestimate we may approximate the total thermal noiseby Brownian coating noise that prevails at all frequency
Figure 5: aLIGO noises spectral densities for KE with a frontcoating of N = 5 layers (plus a cap) and a rear coatingof N = 33 layers (plus a cap). Parameters used for thiscalculation are presented in Table I and Table II.Table III: Numerical values for all noise sources (Brownian,TE and TR noise of mirror substrate and coating) and totalmirror thermal noise at 100 Hz for ET and aLIGO. Parame-ters used for this calculation are presented in Tables I and II.The front coating of the KE consists of N = 5 layers (plus acap) while the rear coating consists of N = 33 layers (plus acap).noise spectral density Einstein Telescope Advanced LIGOKhalili Etalon (KE):coating Brown., m √ Hz . × − . × − substrate Brown., m √ Hz . × − . × − coating TR, m √ Hz . × − . × − substrate TR, m √ Hz . × − . × − substrate TE, m √ Hz . × − . × − coating TE, m √ Hz . × − . × − KE total, m √ Hz . × − . × − Conv. Mirror (CM):coating Brown., m √ Hz . × − . × − substrate Brown., m √ Hz . × − . × − coating TR, m √ Hz . × − . × − substrate TE, m √ Hz . × − . × − coating TE, m √ Hz . × − . × − CM total, m √ Hz . × − . × − ranges as we have seen. Moreover, the thickness of SiO layers and Ta O layers differs only about 40 % while theTa O loss angle is 5 times higher than the loss angle ofSiO . Therefore, we may very roughly approximate thetotal thermal noise with the sum of Ta O coating layersBrownian noise (recall that all of them are uncorrelated).In a first approximation one could assume that the frontcoating is responsible for the main contribution to thenoise level. Thus, the total thermal noise level should beproportional to the number of front coating Ta O layersonly. For a CM this number is Q CM = 20 and for a KE– Q frontKE = 3. The ratio of the Q values should representthe gain of a KE: G naive = s Q CM Q frontKE = r
203 = 2 . . (20)The estimated gain is larger compared to the accurateresults (17). It may be explained by the fact that we donot account for elastic coupling (through substrate) be-tween rear coating layers motion and front coating layersmotion, i.e. the displacement of the front coating due toa deformation of the rear coating layers. One could say,the rear coating layers motion is ’transferred’ to the frontcoating through the substrate. This coupling is moder-ated by the elastic properties of the latter.We introduce a transfer ratio p to account for this elas-tic coupling. The variable p ranges from 0 for a Khalilicavity (KC) to 1 for a CM or ’zero’-thickness substrate.We can calculate p using the simple model of a cylindricalmirror whose front and rear surface are covered by equallayers (same thickness and same elastic parameters). Letus apply a single force at the front surface and keep therear surface free of forces. One can calculate the elas-tic energies in the front layer U front and in the rear layer U rear . Obviously, the transfer ratio p may be calculatedas the ratio of both energies. This estimate gives: p ≡ U rear U front , p ET ≃ . , p AL ≃ .
086 (21)The ratio p AL for aLIGO is smaller than for ET. Againthis behaviour can be explained by the different geometryfactors g (see estimates (19)).Now instead of Eq. (20) we can state a more accurateformula for the gain estimate taking into account theelastic coupling p of the Q rearKE = 17 rear coating layers: G app = s Q CM Q frontKE + p × Q rearKE , (22a) G (ET)app ≃ . , G (AL)app ≃ . . (22b)We see that the approximated gain values coincide withthe accurate values (17) within an accuracy of about 5 %.Real gains in ET and aLIGO are lower than the expectedapproximated values (22) because of other noise sourcesthat were omitted here (Brownian substrate and Brown-ian coating of the SiO layers, substrate and coating TEand TR noise).Note that the elastic coupling does not take place ina KC where both coatings are mechanically separatedby vacuum. Consequently for both detectors, ET andaLIGO, the usage of a KC instead of a CM is expectedto show a gain value of G ≈ . E. Potential sensitivity improvements for futureGW interferometers
In this section we quantitatively analyse the overallsensitivity improvement potentially achievable by replac-ing the conventional end mirrors by KE in aLIGO andET. −24 −23 Frequency [Hz] S t r a i n [ / (cid:214) H z ] Quantum noiseCM: Coating Brownian noiseKE: Coating Brownian noiseCM: aLIGO Total noiseKE: aLIGO Total noise
Figure 6: Potential overall sensitivity improvement of aLIGOfor the use of KE, compared to conventional end mirrors:The binary neutron star inspiral range increases by 15 % from196.2 to 225.0 Mpc. This corresponds to a potential increasein the detected event rate for binary neutron star inspirals ofslightly above 50 %.
Figure 6 shows the potential sensitivity improvementof aLIGO for the use of KC as end mirrors. The sensitiv-ity curves have been created using the GWINC software[32] and for a signal recycling configuration that is opti-mised for the detection of binary neutron star inspirals(see configuration 2 in [33]). Only the two main noisecontributions are shown: quantum noise (black trace)and coating Brownian noise (sum of all test masses) (redtrace), as well as the total noise (blue traces). Pleasenote that all other relevant noise sources have been in-cluded in the calculations of the total noise traces, buthave been omitted from the plot for clarity. The dashedlines indicate the strain levels for the standard aLIGOdesign, while the solid lines show the potentially reducednoise levels originating from the application of KE as endtest masses, as described in this article. The main dif-ference between these two scenarios originates from thereduction of coating Brownian noise by a factor 2.18, asdescribed by the values in the right hand column of TableIII. Please note that thermal noise contributions from theinput mirrors stay identical for the two scenarios. The −25 −24 Frequency [Hz] S t r a i n [ / (cid:214) H z ] Quantum noiseCM: Coating Brownian noiseKE: Coating Brownian noiseCM: ET−HF Total noiseKE: ET−HF Total noise
Figure 7: Potential overall sensitivity improvement of the EThigh frequency detector (as described in [34]) for the use ofKEs, compared to conventional end mirrors: The binary neu-tron star inspiral range increases by 15 %, corresponding to apotential increase in the detected event rate for binary neu-tron star inspirals of 50 %. corresponding increase in the binary inspiral range (1.4solar masses, SNR of 8, averaged sky location) is about15 % and therefore yields an relative increase of the bi-nary neutron star inspiral event rate of about 50 %.Figure 7 shows the sensitivity improvement of a poten-tial ET high frequency detector as described in [34] forthe replacement of the conventional end mirrors by KEs.Following the values given in Table III we considered aflat coating Brownian noise reduction factor of 1.76 forthe end test masses, while again we assumed the ther-mal noise of the input test masses to stay constant. Thisyields an overall reduction of the total thermal noise ofall test masses of about 25 % and an increase in the ob-servatory sensitivity of up to 20 % in the most sensitivefrequency band between 50 and 400 Hz. We find an in-crease in the binary neutron star inspiral range of 15 %from 1593 to 1833 Mpc. This corresponds to an increasein the binary neutron star inspiral event rate of about50 %.
III. TECHNICAL FEASABILITY OF KHALILIETALONS FOR FUTURE GW OBSERVATORIESA. Required Hardware
Figure 1 shows the simplified schematics of an aLIGOor ET interferometer with different end mirror configu-rations. Replacing the conventional end mirrors by KCswould have a significant impact on the required hard-ware. Instead of a single end mirror suspended from asingle seismic isolation system per end mirror, in the caseof the KC two mirrors with two full seismic isolation sys-tems are required at the end of each arm cavity.This means that with KCs there are six (2x IM, 2xIEM, 2x EEM) instead of four optical elements (2x IM,2 EM), which require the maximal seismic isolation. Theconcept of the KE allows us to still significantly reducethe thermal noise contribution of the end mirrors, whilebeing compatible with the already available seismic isola-tion systems. Therefore, it is in principle possible to up-grade a 2nd or 3rd generation gravitational wave detectorby replacing conventional end mirrors by KEs withoutaltering or extending the vacuum systems and seimsicisolation systems.
B. Interferometric Sensing and Control
In addition to the reduced hardware requirements, themain advantage of the KEs with respect to KCs is thepotential simplification of several aspects related to theinterferometric sensing and control. Upgrading aLIGO oran ET interferometer from its standard configuration toemploy KCs increases the length degrees of freedom of themain interferometer from five (DARM (differential armlength), MICH (Michelson cavity length), SRCL (signalrecycling cavity length), CARM (common arm length),PRCL (power recycling cavity arm length)) to a total ofseven.It is worth mentioning that the additional two degreesof freedom actually have a very strong coupling to the dif-ferential arm length channel of the interferometer. Forthe example of the coating distribution discussed in thisarticle, the length of the KC needs to be stabilized withan accuracy of only a factor 10 less than what is requiredfor the main arm cavities. That means the length of theKC needs to be orders of magnitude more stable than forexample the differential arm length of the central Michel-son interferometer.In order to achieve this demanding stability of the KCone has to make use of highly dedicated readout andcontrol schemes. Special care needs to be taken to avoidintroducing potential control noise at the low frequencyend, which could potentially spoil the overall sensitivityof the gravitational wave detector.Substituting the KC, consisting of two individual mir-rors potentially encountering independent driven motion(for example seismic), by the proposed KE would ensurethat both relevant mirror surfaces would be rigidly cou-pled via the etalon substrate. Therefore, the length ofthe KE would be much less susceptible to seismic dis-turbances or gravity gradient noise, as compared to thelength of KC. Also in terms of potential control noise theKE is advantageous over the KC. In case of the KC themirror positions would have to be controlled by means ofcoil magnet actuators or electro-static actuators, whichcan potentially introduce feedback noise at frequencieswithin the detection band of the gravitational wave de-tector. In contrast the length of the KE can be lockedby controlling the etalon’s substrate temperature (usingthe temperature dependency of the index of refraction).Since the etalon substrate acts as a thermal low pass, the etalon length will be extremely constant for all frequen-cies within the detection band of the interferometer.However, not only the length sensing and control ishighly demanding in case of a KC, but also the align-ment sensing and control. Again the key point here isto find a high signal to noise error-signal and then ap-plying low noise feedback systems to keep the mirrors ofthe KC aligned in pitch and yaw. As the KC would berather short compared to the main arm cavities, the KCswould unfortunately feature a high mode degeneracy, i.e.it would not only be resonant for the desired TEM mode, but also for higher order modes, which would fur-ther increase the alignment requirements. Using a KEwould potentially allow us to transfer the alignment con-trol problem from the detector operation to the manu-facturing process of the etalon. If it would be possibleto manufacture an KE with sufficiently parallel front andback surface, we would not need to actively control therelative alignment of the KE surfaces during operation.The two parameters that would be most relevant are therelative curvature mismatch of the etalon front and rearsurfaces as well as the parallelism of the two surfaces.As we have shown in [35] the curvature mismatch is thedominating factor for the etalon’s performance. C. Thermal Lensing
In this section we will compare the thermal lensing [39]of the KC configuration to the one of a KE. In the caseof the KC we have the following absorption processes:(i) IEM front coating, (ii) IEM substrate, (iii) IEM anti-reflex coating on its rear surface and (iv) front coatingof EEM. In the case of the proposed KE the situationis pretty similar apart from process (iii), which does notexist.In the following we will show by means of FEM thatthe actual thermal lensing induced into the KE is of thesame order, but slightly smaller than in the case of theKC.The FEM used here treats the mirror as a substrate.The coatings and the laser beam are included as heatsources. For the reflective coatings we assume an absorp-tion of 0.5 ppm, and 1 ppm for the anti-reflective coatingin the KC. The FEM assumes an emissivity ǫ = 0 .
93, anambient temperature of 300 K and uses the parametersfrom Table I and II for the values of aLIGO. After com-puting the temperature and displacements of the finiteelements, the optical path difference (OPD) is derived.For the OPD we included the temperature dependenceof the refractive index (which is the dominant thermallensing effect in fused silica) and the expansion, while weomitted the elasto-optic effect. We also did not includesurface to surface radiation in the KC. This would makethe thermal lens worse, and is therefore safe to excludein order to make a conservative comparison between KCand KE. Fig. 8 shows the temperature distribution ina KC and a KE for the aLIGO parameters with N =5,0 Khalili cavity Khalili etalon
Temperature in the IEM (in K) Temperature in the EEM (in K) Temperature in the etalon (in K)
Figure 8: The temperature distribution in the mirrors of a KC (left) and a KE (right) for aLIGO parameters (with N =5) ascalculated via FEM. Please note the different color scales for the IEM and the EEM in the KC. −15 −10 −5 0 5 10 15−2−1.5−1−0.50 Distance from the optical axis in cm O P D i n µ m OPD cross section for a Khalili cavity −15 −10 −5 0 5 10 15−2−1.5−1−0.50
Distance from the optical axis in cm O P D i n µ m OPD cross sections in a Khalili etalon dn/dTdV/dTtotalthin lens fitf=2004mdn/dTdV/dTtotalthin lens fitf=1944m
Figure 9: The optical path delay (OPD) is computed in the FEM simulation, in order to determine the focal length of thethermal lens in the KC (left) and the KE (right) for aLIGO parameters. Shown are the total OPD, the OPD caused by thethermo-refractive effect (named dn/dT ) and the OPD due to the expansion of the substrate ( dV /dT ). The plots show also fitsof the OPD for an ideal thin lens. while Fig. 9 presents the corresponding OPD for a sin-gle pass due to the thermo-optic effect and expansion ofthe substrate, as well as a fit of an OPD that would becaused by an ideal thin lens. The fits are least squarefits, weighted by the beam intensity. For the aLIGO pa-rameters with N =5, the thermal lensing in the KC canbe described by a thermal lens with a focal length of f = 1944 m, while the OPD in the KE can be fitted bya thermal lens with a focal length of f = 2004 m. Therespective values for ET are a focal length of 1797 m forthe KC and 1838 m for KEs. It follows that the inducedthermal lensing in the KC and KE is of similar strength.As we have shown above, the thermal lensing for the KE is slightly weaker than for the KC.As one can see from the magnitude of the inducedthermal lensing, the compensation of this effect will beextremely challenging in both cases. Potential ways ofmitigating the thermal lensing could include innovativeapproaches such as radiative cooling [38] or pre-shapedmirror or etalon substrates, which feature the wrong cur-vature, when being cold, but develop the correct ’shape’when operated at the designed optical power. The com-pensation of thermal lensing has turned out to be morechallenging than anticipated in the first generation grav-itational wave detectors. Only, the practical experiencethat will be collected with the advanced detectors will1allow us to realistically judge the feasibility of KCs aswell as KEs. However, the main purpose of the ther-mal lensing analysis presented here was to show that thethermal lens will not be worse, but slightly better for theproposed KE compared to a KC. IV. CONCLUSION
In this article we have investigated the main thermalnoise sources arising in the mirrors of the two next gener-ation gravitational wave detectors: Advanced LIGO andEinstein Telescope. The thermal noise sources includeBrownian, thermo-elastic and thermo-refractive noise ofthe mirror coatings and the mirror substrate, amongwhich the Brownian coating noise is the largest. We ap-plied our model developed in [23] to study the idea ofthe Khalili etalon to decrease the coating thermal noiseand to improve the sensitivity. The optimum KE con-figuration minimizing the total thermal noise level wasfound to be with 2 Ta O layers and 1 SiO layer plusa cap in the front coating and with 18 Ta O layers and17 SiO layers plus a cap in the rear coating. However,since the substrate absorption in ET with such a configu-ration is 8 .
00 W, and that in aLIGO is 1 .
42 W, our choiceis not to use the optimal but a slightly different coatingdistribution with N = 5, i.e. 3 Ta O and 2 SiO lay-ers plus a cap on the front surface and 17 Ta O and16 SiO layers plus a cap on the rear surface. The ab-sorbed power in the substrate with such a configuration is4 . .
72 W for aLIGO. Such an absorptionis around 1 ppm which seems to be reasonable price forthe thermal noise enhancement. The total noise spectraldensity of ET and aLIGO can be improved by the factorsof 1 .
67 and 2 .
03, respectively, compared with the casesof conventional end mirrors . Moreover, we have checkedour numerical calculations with a very simple qualitativeconsideration designed to make an order of magnitudeestimation. This estimation shows an agreement of bet-ter than 5 percent with the exact numerical calculations.A use of KEs instead of conventional end mirrors wouldimprove the detection rate of the binary neutron star in-spirals with the future gravitational wave observatoriesby about 50 %.We also discussed the feasibility of the Khalili etaloncompared with that of the Khalili cavity. The KE is moreadvantageous in terms of the hardware requirements. Wealso compared the thermal lensing effects in the KE andin the KC and found that the former is slightly betterfor not having the anti-reflective coatings that KC con-tains on the rear surface of the front mirror. In fact, thethermal lensing problem in either case is quite severe andwe should explore a way to compensate the lensing effectwithout imposing excess noise.In this paper we assumed that the light is reflectedon the outer surface of each coating without taking intoaccount reflections from inner layers. A more accurateanalysis shown in Ref. [36] gives a value of coating Brow- nian noise slightly lower (about 10 %).Thermoelastic and thermo-refractive noises originatefrom a thermodynamical fluctuation of the temperatureand the correlation of the two noises can be non-trivialwith a certain set of parameters [19, 20]. In this pa-per, the correlation was ignored and we treated the twonoises individually, which is not a problem as one of themis much lower than the other in the case of KE (see Ta-ble III). It should be noted, however, that the optimalKE configuration could be determined in such a way thatthermoelastic noise and thermo-refractive noise be neg-atively compensated if the mechanical loss angles of thecoating materials were 10 times lower than the currentvalues.
Acknowledgments
This work has been performed with the support of theEuropean Commission under the Framework Programme7 (FP7) Capacities, project Einstein Telescope (ET) de-sign study (Grant Agreement 211743). A.G. Gurkovskyand S.P. Vyatchanin were supported by LIGO team fromCaltech and in part by NSF and Caltech grant PHY-0967049 and grant 08-02-00580 from Russian Founda-tion for Basic Research. D.Heinert and R.Nawrodt weresupported by the German Science Foundation (DFG) un-der contract SFB Transregio 7. S.Hild was supported bythe Science and Technology Facilities Council (STFC).H.Wittel was supported by the Max Planck Society.
Appendix A: Coefficients ǫ and ǫ calculation Let us consider KE as a Fabry-P´erot interferometerwith two mirrors (namely two reflective coatings) withamplitude transmittances T and T , and amplitude re-flectivities R = p − T and R = p − T . The mir-rors are separated by a medium with a refractive index n s , and the mean distance between the mirrors is L . Op-tical losses are equal to zero. The fluctuations of thecoordinates of the front and rear mirrors are reprensetedby x and y , respectively. The probe beam A is incident onthe front mirror (coating) and is partially reflected. Weare interested now in the reflected beam B . The weightcoefficients ǫ and ǫ represent how much the fluctuations x and y contribute to the reflected beam B , respectively.For a short cavity we can use a quasi-static approxi-mation – it means that the motion of the mirrors are suf-ficiently slow compared with the relaxation rate of thecavity. We assume that the optical path between themirrors is fixed to a quarter wavelength, i.e. e ikn s L = i .We can consider the cavity as a generalized mirror. Ob-viously, the reflectivity of the generalized mirror dependson the fluctuations x and y . However, for the reflectedbeam we have to include the motion of the generalizedmirror, that is, the common-mode motion of x and y .2The reflected beam shall be described as: B = e ikx A R ϑ − R − R R ϑ , ϑ = − e ikn s ( y − x ) (A1)We have already taken into account the fact that e ikn s L = i (cavity is tuned in the anti-resonance) and ϑ describesthe variation of cavity length due to the mirror fluctua-tion.The fluctuations x and y are small enough comparedwith the cavity length L so that we may expand (A1)into series over x and y . Keeping the linear terms only,we get B = − A R + R R R − ikA h x ǫ + y ǫ i , (A2) ǫ = R (1 − n s ) + R (cid:2) n s ) R R + R (cid:3)(cid:0) R R (cid:1) , (A3) ǫ = n s R (cid:0) − R (cid:1)(cid:0) R R (cid:1) , (A4)which have been introduced in Eq. (2). Appendix B: Coating reflectivities R and R calculations Let us first consider a CM with N altering layers ofTa O and SiO with the refractive indices n and n ,respectively, and the substrate with the refractive index n s .A multilayer coating consisting of layers with refractiveindices n i and lengths l i is described by the same formu-las for the transmission line consisting of ports with waveresistances 1 /n i and the distances l i [37]. It is convenientto describe the transmission line with impedances Z i andreflectivities R i . Impedance Z i of the i -th layer can sub-stitute the total impedance of all the layers between thislayer and the substrate, which does not affect the otherlayers. It is convenient to start the calculation from theboundary of the substrate and the N -th layer, and Z will be the equivalent impedance of all the mirror and R will be the reflectivity of the entire system.There is a recurrent formula for impedances and re-flectivities of neighboring layers (neighboring ports of thetransmission line): Z i = 1 n i +1 − R i +1 θ i R i +1 θ i , (B1) R i = 1 − n i Z i n i Z i , (B2)where θ i = e ikn i l i is the phase shift in the i -th layer.Using (B1) and (B2) one may easily get the recursiveformula Z i = 1 n i +1 n i +1 Z (0) i +1 (1 + θ i ) + (1 − θ i ) n i +1 Z (0) i +1 (1 − θ i ) + (1 + θ i ) (B3) . The substrate is considered as an infinite half-space,so its impedance is given by Z N = 1 /n N +1 = 1 /n s and hence its reflectivity is given by R N = (1 − n N /n N +1 ) / (1+ n N /n N +1 ) = ( n s − n N ) / ( n s + n N ). ThenEqs. (B3) and (B2) yields each Z i and R i . We are in-terested only in R . The thickness of each layer in thehigh-reflective coating is a quarter-wavelength (QWL),i.e. θ i = e ikn i l i = i . Then (B3) becomes: Z i = 1 n i +1 Z i +1 . (B4)Using (B4) one may easily get the chain (rememberthat N is odd and refractive indices alter so that n = n = · · · = n N = n , n = n = · · · = n N − = n ): Z N = 1 n s , (B5a) Z N − = n s n , (B5b) Z N − = n n s n , . . . (B5c) Z m = n s n ( N − − m n ( N +1) − m , (B5d) Z m − = n N − (2 m − n s n N − (2 m − , . . . (B5e) Z = 1 n s (cid:18) n n (cid:19) N − , (B5f) Z = n s n (cid:18) n n (cid:19) N − . (B5g)and thus the total coating reflectivity is: R = 1 − n n s n (cid:16) n n (cid:17) N − n n s n (cid:16) n n (cid:17) N − . (B6)The cap does not change the impedance Z . The lengthof the cap l c is a half-wavelength (HWL) so that e ikn c l c = −
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