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THE ANNUAL REPORT OF THE MSU GROUP (Jan.-Dec. 2004)
Contributors: V.B.Braginsky (P.I.), I.A.Bilenko,.Ya.Khalili, V.P.Mitrofanov, A.V.Stepanov, K.V.Tokmakov, S.P.Vyatchanin. The researches were supported by NSF grant "Low noise suspensions and readout systems for LIGO" (PHY-0098715 July 2001--June 2004) (PHY-0353775 July 2004 June 2007)

Contents
Summary 1 Exp erimental researches . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The investigation of thermal and excess noise in the high-Q mo des of all fused silica susp ension for the Advanced LIGO test masses. 1.2 The investigation of thermal and excess noise in the fused silica plates with multilayer reflective coating. . . . . . . . . . . . . . . 1.3 The investigation of effects asso ciated with electrical charging of fused silica test masses . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Nonstationary electrical charge distribution on the fused silica bifilar p endulum and its effect on the mechanical Q-factor . . . . . . 2 Theoretical researches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Corner reflectors in the main Fabry-Perot cavities . . . . . . . . . 2.2 Double mirror test mass . . . . . . . . . . . . . . . . . . . . . . . 3 Collab oration b etween MSU group and LIGO Lab, LSC and K.S.Thorne's group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Publication of MSU group memb ers in 2004 . . . . . . . . . . . . . . . . 5 Publication of MSU group memb ers in collab orations with LIGO Lab and LSC in 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 3 3 4 5 5 6 7 9 9

. . . . . . . . . . .

App endixes

10

A The investigation of thermal and non-thermal noises in Advanced LIGO susp ension prototyp e. 11 B Nonstationary electrical charge distribution on the fused silica bifilar p endulum and its effect on the mechanical Q 23 C Corner reflectors and Quantum-Non-Demolition Measurements in gravitational wave antennae 29

D Reducing the mirrors coating noise in laser gravitational-wave antennae by means of double mirrors 54

1

Summary
1
1.1

Experimental researches
The investigation of thermal and excess noise in the high-Q mo des of all fused silica susp ension for the Advanced LIGO test masses.

The program of these investigations was formulated several years ago after the discovery made by MSU group of the excess noise in steel wires [1]. During previous years (2000-2003) new technique of measurements of Brownian and excess noise in fused silica fib ers used in the susp ension was prop osed and realized by I.A.Bilenko and his students. This technique allowed to monitor the vibrations of thin fused silica fib ers with sensitiv ity Sx = 3 в 10-13 cm/ Hz near the frequency 1 kHz and Sx < 9 в 10-14 cm/ Hz at frequencies higher than 2 kHz (these values even slightly exceed requirements declared in the grant prop osal). With this sensitivity it is p ossible to resolve the energy innovation on chosen mo de during the time interval t = 0.1 c that corresp onds to 1% of k T . The analysis shows, that the future improvement isn't p ossible unless the principal and costly mo difications will b e made. No any significant excess noise for 10 different samples tested under stress varies from 3% to 15% of breaking value was observed. During the January-June of the 2004 I.A.Bilenko concentrated his efforts on the measurements on the samples under higher stress. Finally, 4 series of records on the samples under stress > 40% from breaking value was obtained. During the June-Decemb er of the 2004 all the obtained data was evaluated using the universal algorithm. Part of the records was excluded from the investigation due to insufficient sensitivity of the readout system at the time of the records. Total duration of the refined records is ab out 90 hours obtained on 9 fused silica samples loaded from 4% to 50% of breaking stress. At the level limited by the sensitivity of our apparatus only Brownian noise was registered. This result may b e regarded as a part of approval of the Advanced LIGO susp ension design. Complete rep ort on the excess noise program is in the App endix A. The final pap er ab out the retrieval of the excess noise in the stressed fib ers will b e submitted to Physics Letters after the LIGO approval.

2

1.2

The investigation of thermal and excess noise in the fused silica plates with multilayer reflective coating.

A new program of exp erimental researches was stimulated by the analysis of the contribution of the thermo elastic noise in the coating into the total noise budget p erformed by V.B.Braginsky and S.P.Vyatchanin in 2003 [2] as well as the consequent direct measurements of the thermal expansion co efficient Ta2 O5 p erformed by A.A.Samoilenko [3]. These analysis and measurements have shown that thermo elastic noise may prevent to reach the sensitivity of the Advanced LIGO antenna b etter than SQL. On the other hand the metho d used for the Ta2 O5 measurement based on the b ending of thin SiO2 plates under heating demonstrated that the sum of Brownian, thermo elastic and, probably, excess noise comp onents would b e not to o difficult to register. This approach lo oks promising b ecause in essence it will b e an examination of the "b ehaviour" of thermal and nonthermal fluctuations in the reflective coating identical to the coatings which will b e used in Advanced LIGO. I.A.Bilenko and A.V.Stepanov started the implementation of this exp erimental program. At present a new vacuum chamb er for this program is prepared and tested as well as antiseismic isolation for the optical mini-table. The shadow sensor based on the He-Ne laser was prepared for preliminary measurement of the quality factor of the thin fused silica plates coated with reflecting multilayer. We also prepared comp onents for the optical sensor based on Fabry-Perot vity pump ed ca -14 by stabilized Nd:YAG laser. The pro jected sensitivity is 5 в 10 cm/ Hz at 500 Hz. The Pound-Drever scheme and other optic parts are under development now.

1.3

The investigation of effects asso ciated with electrical charging of fused silica test masses

V.P. Mitrofanov, K.V. Tokmakov and p ostgraduate student L.G. Prokhorov in the collaboration with Phil Willems (Caltech) in 2004 continued the investigation of effects asso ciated with charging of fused silica test mass. During 2003, in first version of the set up the vertically oriented plate with capacitive prob e electro des connected to high imp edance amplifier was placed in parallel with the end face of susp ended fused silica cylinder so that the separation gap was of ab out 3 mm. The initial amplitude of torsion oscillations of the cilinder was ab out 2 mm. Nevertheless p endulum could swing far enough to b e very close to the plate with electro des due to the lo cal seismic exitation. During January June 2004 V.P. Mitrofanov and his colleagues carried out the search of variation of the charge lo cated on the cylinder when the provoked touching of the plate with electro des by the cylinder to ok place. It was found that the provoked touching resulted in the jump-like changes of the p endulum amplitude and the prob e signal (which was prop ortional to the electrical charge on the cylinder) analogously to the events observed in 2003. But there was at least two distinctions in the b ehaviour of the system "p endulum-electro des" in case of the provoked touching: 1) changes of the amplitude and the prob e signal were faster than in case of events observed earlier; 2) admission of a small p ortion of air into vacuum chamb er resulted in the significant drop of the prob e signal, whereas after events the prob e signal could b e reduced only by electrical discharge in p o or vacuum. (See details in [4] and [5]). 3

1.4

Nonstationary electrical charge distribution on the fused silica bifilar p endulum and its effect on the mechanical Qfactor

V.P.Mitrofanov, K.V.Tokmakov and p ostgraduate student L.G.Prokhorov continued the investigation of effects asso ciated with charging of the fused silica test mass. In the previous measurements which were carried out in MSU and the other groups the electrical charge lo cated on the test mass was in the state close to the stationary state. This year the MSU group b egan to study the dissipation in the case when the electrical charge was in nonstationary one. A numb er of new exp erimental data was obtained. The b ehavior of electrical charges lo cated on a 0.5 kg fused silica cylinder susp ended on two fused silica fib ers in vacuum of ab out 10-7 Torr was investigated. Several mo difications of the previous version of the set up were done. The fused silica plate with gold electro des dep osited on it b eing horizontally oriented was placed under the cylinder with the separation gap of ab out 1 mm. These electro des were the nearest ob jects electrical charges could interact with. The charge lo cated on the cylinder was measured by means of the capacitive prob e. This prob e (2 cm2 copp er plate connected with the electrometric amplifier) was placed at the side of the cylinder at a distance of 2 cm so that it monitored some effective electrical charge lo cated on the cylinder. The average value of charge density on the fused silica cylinder was of the order of -12 10 C/cm2 (i.e. 107 electrons/cm2 ). In this case the Q of the bifilar torsion mo de was ab out 6 в 107 . This value of Q was lower than those obtained for the p endulum when nothing was close enough to the susp ended cylinder (Q 9 в 107 ). So it is p ossible to supp ose that the reduction of the p endulum Q was asso ciated with electrical charges lo cated on the fused silica cylinder and their interaction with the electro des under the p endulum. The disturbance in the distribution of electrical charges on the p endulum was pro duced by two ways. The first one was realized by means of touching the fused silica cylinder by a metal wire. A sp ecial manipulator was used. The lo cal dep osition of the charge o ccurred in the p oint of touching (1.5 cm b elow of the center of the cylinder) due to the contact electrification. After this the dep osited charge spread over the cylinder due to diffusion. Another way to disturb the charge distribution was to create electric field applying high voltage to the electro des, which were under the p endulum. After the voltage 1400 V was switched on during 24 hours one could observe the relaxation of the voltage from the prob e as well as in the first case. After the transient in the first several hours the decay of the voltage with the relaxation time of the order of 200 hours was recorded in b oth cases. After the disturbances of the charge distribution it was observed that the b ehavior of the p endulum amplitude free decay had some feature. In the first case the rate of the free decay of the p endulum amplitude remained invariable in limits of the uncertainty of the measurements asso ciated mainly with the seismic noise. In the second case after the voltage was switched off the rate of the free decay of the p endulum amplitude during approximately 200 hours was lower than b efore the application of the high voltage. This change in the rate was to o large and lasted to o long to b e asso ciated with seismic fluctuations. After that the rate of the decay returned back to the initial value. Such b ehavior can b e interpreted as if it was asso ciated with the increase of the p endulum Q from ab out 6 в 107 up to (1 В 2) в 108 . The uncertainty in the last value was caused by the uncertainty of the approximation of the p endulum amplitude time dep endence in 4

this range. Within these limits the rate of the amplitude decrease was different in various runs of measurements. After the contact electrification the relaxation of the charge distribution was observed but the change of the p endulum Q was not detected. This can b e explained by the large distance b etween the p oint of the contact and the electro des. Four runs of the measurements were carried out. Every run had duration from 600 hours to 1600 hours. The b ehaviors of the free decay of the amplitude describ ed ab ove were observed in every run. The transient observed in the first several hours after the disturbances of the charge distribution needs more detailed investigation. The measurements are in progress. It is necessary to find the dep endence of these effects on duration of the high voltage application and on the other factors. The aim of the research is to identify the mechanism of losses and noise asso ciated with charging of the test masses (see details in App endix B). Also V. P. Mitrofanov and his colleagues b egan to test the new setup for the search of the spatial charge distribution over the surface of fused silica samples and other dielectric materials which might b e used in the test masses. The setup has a go o d p otential sensitivity but a numb er of parasitic effects need to b e eliminated.

2

Theoretical researches

It was emphasized in section 1.2 of this rep ort that theoretical analysis and direct measurements p erformed by the memb ers of MSU group have shown that thermo elastic noise in multi-layer dielectrical coating of the mirrors initially planned in advanced LIGO pro ject will not p ermit to reach the sensitivity substantially b etter than standard quantum limit (SQL) (see publications [2, 3] and MSU group 2003 annual rep ort). This motivated not only the start of a new exp erimental program describ ed in section 1.2, but also the searches of new typ es of reflective mirrors. Two typ es of such new mirrors were prop osed and analyzed. The net result of these analysis shows that using any of two prop osed typ es it is p ossible to circumvent SQL sensitivity with a factor ab out ten if thermo elastic noise is the only limiting factor.

2.1

Corner reflectors in the main Fabry-Perot cavities

The key idea of this version prop osed by V. B. Braginsky and S. P. Vyatchanin is based on the concept of corner reflectors (CR) -- tri-hedral or two-hedral prisms. These CR may play the role of high finesse reflectors instead of cylindrical shap e mirrors with multi-layer dielectric coating. To make stable the chosen optical mo de it is necessary to make the "fo ot" of such a CR with finite value of curvature. To evade the reflection of the light from the "fo ot" (and thus to exclude the creation of additional set of optical mo des in the cavity) it is necessary to cover the "fo ot" with relatively small numb er (2-4 pairs) of anti-reflective layers. Taking into account that usual reflective coating has 20 - 40 pairs of layers, it is reasonable to exp ect that with such a thin anti-reflective coating the thermo elastic noise in it will b e ab out one order smaller. Pure fused silica (S iO2 ) is an appropriate glass to manufacture corner reflectors b ecause its refractive index nSiO2 = 1.45. This value is sufficiently large to provide total internal reflection (due to Snellius law) b oth for two or three facets CR: nSiO2 > 2 (two facets) and nSiO2 > 3/2 (three facets). V. B. Braginsky and S. P. Vyatchanin have

5

carried out in depth analysis for the conditions which have to b e fulfilled to implement CR: 1. the conditions for the displacements of one CR relatively to the second CR; 2. the conditions for the tilt angle of one CR; 3. the conditions for the exp osure angle (angle b etween facets); 4. the fundamental difractional losses on the edge where two facets meet. The analysis shows that to day existing technology of mechanical manufacturing and p ositioning allow to satisfy these conditions. The only "fee" which has to b e paid for such a version of Fabry-Perot cavity is the necessity to use non-Gaussian distribution of the light over cross section of b eam -- so called mesa-b eams [6, 7]. The origin of this "fee" is thermo-refractive noise in the bulk of CR (this noise originates from thermo dynamic temp erature fluctuations which pro duced the phase fluctuations in reflected wave due to dep endence of refractive index on temp erature). There is also one p otential p ossibility to evade the usage of CR: to find such a glass for multi-layer coating which has the value of thermo-refractive index close to the value nTa2 O5 2.1 but with thermal expansion factor close to the value of fused silica thermal expansion. See details in App endix C and in [8].

2.2

Double mirror test mass

An alternative concept of a Fabry-Perot cavity with substantially reduced thermo elastic noise in the mirrors coating was prop osed by F.Ya.Khalili. Key idea of this elegant concept is to use two indep endently susp ended mirrors separated by relatively small distance l = 30 - 1000 cm instead of a single high-reflective mirror in the end of the Fabry-Perot cavity. These mirrors may have the same cylindrical shap e and masses as it is planned in Advanced LIGO. The mirrors "working" surfaces have to b e parallel to each other. In this double mirror test mass the first surface which "meet" the incident b eam has to have relatively small reflectivity, e.g. 1 - R1 10-2 . The second surface of the first mirror has to have thin antireflective coating. The third surface (the first of the second mirror) has to have high reflectivity, e.g. 1 - R2 10-5 . With these numerical values, the total reflectivity of such a double mirror will b e defined by the following relation: (1 - R1 )(1 - R2 + 2A) 10-7 , (.1) 4 if the first mirror absorption A 10-5 and the optical distance b etween the mirrors' reflective surfaces 1-R 1 (.2) 4 (i.e the anti-resonance condition, is the wavelength and N is an integer). It have to b e emphasized, that in this example only ab out 0.25% of the light p ower will circulate b etween the two mirrors. l= N+ 6

Thermo elastic noise in the coating in this double mirrors system will b e substantially reduced b ecause the main contribution may b e from the first surface of the first mirror (which reflects most of the light p ower), but this surface is ab out 3 В 5 times thinner than it is planned for the Advanced LIGO ETM mirrors, and corresp ondingly the contribution of the thermo elastic noise in this surface to the total noise budget will b e reduced with the same factor. The remaining small fraction of the light p ower will b e reflected from the second mirror. This fraction of the p ower will b e substantially "contaminated" by the thermo elastic noise in the coating of the second mirror (which will have thick multilayers reflector). However, its part is the sum noise from two mirrors will b e negligible small in comparison with the part of the first mirror. This new concept has another advantage: the mass of the second mirror may b e substantially smaller than the mass of the first one (in the ab ove numerical example it may b e 106 times smaller). This reduction of the second mass will not increase the value of the SQL. See details in App endix D and in [9].

3

Collaboration between MSU group and LIGO Lab, LSC and K.S.Thorne's group

Memb ers of the MSU group have visited Caltech for research collab oration and extensive discussions: V.B.Braginsky -- two times, V.P.Mitrofanov, S.P.Vyatchanin and F.Ya.Khalili -- once. V.P.Mitrofanov has given a talk at the LSC meeting at Livingston Observatory; S.P.Vyatchanin and F.Ya.Khalili -- at the LSC meetng at Hanford observatory. V.P.Mitrofanov have visited University of Glasgow. He gave a talk at the seminar and to ok part in exp erimental work in the lab oratory and had fruitful discussions with memb ers of Glasgow group ab out various mechanisms of dissipation in the test masses of LIGO pro ject. The memb ers of MSU group in the collab oration with P.Willems (LIGO, Caltech) continued long lasting measurements of variations of electrical charge lo cated on the fused silica test mass prototyp e and searched for correlations b etween these variations and cosmic ray shower events. The total numb er of seminars and presentations by V.B.Braginsky, V.P.Mitrofanov, S.P.Vyatchanin, and F.Ya.Khalili is 3 at Caltech and is 3 at LSC meetings.

7

References
[1] I.A.Bilenko, A.Yu.Ageev, V.B.Braginsky , Physics Letters A 246, 479 (1998). [2] V.B.Braginsky, S.P.Vyatchanin, Physics Letters A 312, 169 (2003). [3] V.B.Braginsky, A.A. Samoilenko, Physics Letters A 315, 175 (2003). [4] V.Mitrofanov, L.Prokhorov, K.Tokmakov and P.Willems, Class. Quantum Grav. 21, 1083 (2004). [5] V.Mitrofanov, L.Prokhorov, K.Tokmakov and P.Willems, Talk at LSC meeting, Louisiana, March 15-18, (2004). [6] E. d'Ambrosio, R. O'Shaughnessy, S. Strigin, K.Thorne and S. Vyatchanin, Reducing Thermo elastic Noise in Gravitational-Wave Interferometers by Flattening the Light Beams, submitted to Phys. Rev. D, available as file b eamreshap e020903.p df at http://www.cco.caltech.edu/kip/ftp/ [7] O'Shaughnessy, S. Strigin and S. Vyatchanin, The implifications of Mexican-hat mirrors: calculations of thetmo elastic noise and interferometer sensitivity to p erturbation for Mexican-hat mirror prop osal for advanced LIGO, submitted to Phys. Rev. D. [8] V.B.Braginsky, S.P.Vyatchanin, Physics Letters A 324, 345 (2004). [9] F.Ya.Khalili, arXiv:gr-qc/0406071 (2004).

8

4

Publication of MSU group members in 2004
1. B. B. Braginsky and S. P. Vyatchanin, "Corner reflectors and quantum-non-demolition measurements in gravitational wave antennae", Physics Letters A324 (2004) 345-360, arXiv:cond-mat/0402650. 2. I. A. Bilenko, V. V. Braginsky, S. L. Lourie, "Mechanical losses in thin fused silica fib ers", Class. Quantum Gravity, 21 (2004) 1231. 3. M. L. Goro detsky, I. S. Grudinin, "Fundamental thermal fluctuations in microspheres", Journal of Optical Siciety of America B21 (2004) 697. 4. F.Ya.Khalili, "Reducing the mirrors coating noise in laser gravitational-wave antennae by means of double mirrors", accepted for publication in Physics Letters A, arXiv:gr-qc/0406071. 5. V.Mitrofanov, L.Prokhorov, K.Tokmakov and P.Willems, "Investigation of effects asso ciated with variation of electric charge on a fused silica test mass", Class. Quantum Grav. 21, 1083 (2004).

5

Publication of MSU group members in collaborations with LIGO Lab and LSC in 2004
1. B. Abb ott et al. (LSC including I. A. Bilenko, V. B. Braginsky, V. P. Mitrofanov, S.P. Vyatchanin) "Detector Description and Performance for the First Coincidence Observations Between LIGO and GEO" Nuclear Instrum and Methods in Physics Research" A517 (2004) 145. 2. B. Abb ott et al. (LSC including I. A. Bilenko, V. B. Braginsky, V. P. Mitrofanov, S.P. Vyatchanin), "First Upp er Limits from LIGO on GW Bursts", Phys. Rev. D69 (2004) 102001. 3. B. Abb ott et al. (LSC including I. A. Bilenko, V. B. Braginsky, V. P. Mitrofanov, S.P. Vyatchanin), "Setting Upp er Limits on the Strength of Perio dic GW from PSR J1939 + 2134 Using the First Science Data from the GEO600 and LIGO Detectors", Phys. Rev. D69 (2004) 082004. 4. B. Abb ott et al. (LSC including I. A. Bilenko, V. B. Braginsky, V. P. Mitrofanov, S.P. Vyatchanin), "Analysis of LIGO Data for GW from Binary Neutron Stars", Phys. Rev. D69 (2004) 122001. 5. B. Abb ott et al. (LSC including I. A. Bilenko, V. B. Braginsky, V. P. Mitrofanov, S.P. Vyatchanin), "Analysis of LIGO Data for Sto chastic GW", Phys. Rev. D69 (2004) 122004.

9

Appendixes

10

Appendix A The investigation of thermal and non-thermal noises in Advanced LIGO suspension prototype.
The results is rep orted. from 105 to found in fib of exp erimental investigation of the mechanical noise pro duced by fused silica fib ers The b est achieved resolution is ab out 0.01 k T for the mo des with quality factor 107 . Over the 90 hours of observation no excess (additional to Brownian) noise was ers loaded at 4% В 50% from breaking stress.

Introduction
The planned sensitivity in the second generation of the Laser Interferometer Gravitational wave Observatory (Advanced LIGO) has to b e close to the value of the amplitude of the p erturbation of metric h 10-22 (from the frequency 100 Hz to 1 kHz and in the bandwidth ab out 100 Hz) [1]. This value of h corresp onds to the amplitude of oscillation of antenna's mirror relatively another one ab out xgr 2 в 10-17 cm.These values of h and xgr are approximately ten times smaller than the recently achieved sensitively in one of the two op erating LIGO antennae (Initial LIGO), see [2]. The planned gain in the sensitivity means that all kinds of noises in the antennae have to b e corresp ondingly reduced. Starting from 1991 several groups of researchers in LIGO Scientific Collab oration (LSC) and several researches from LIGO Lab were steady searching new typ es of material for the mirrors and new typ es of mirror's susp ensions which allow to increase substantially the mechanical quality factors Q in all mo des of mirror itself and mirrors susp ension. The Fluctuation-Dissipation Theorem (FDT) predicts that to obtain a gain of one order in sensitivity it is necessary to increase the values of Q by two orders. Several groups of researches which b elong to LSC and LIGO Lab obtained very promising value of Q factors in the mo dels of mirrors and in the susp ension mo des using extremely pure fused silica. The Q of p endulum mo de and violin mo des of such susp ension turned out to exceed 108 at ro om temp erature [3]-[7]. These values are approximately three orders higher than the Q in p endulum and violin mo des in the steel wires susp ension which are used at present in Initial LIGO [8]. Thus, "from the p oint of view of FDT" it is reasonable to exp ect at least one order of gain in the sensitivity. But FDT may predict the values of thermal noise in thermal equilibrium only, not the so-called excess noise (of 11

fused silica block

mirror fiber p in s

levers

Main load

Figure A.1: Samples supp ort design

non-thermal origin). This noise (or noises) particulary app ears due to the transformations of free energy accumulated in solids from the static elastic form (typical free energy in solid is ab out 106 erg/g) to the kinetic (oscillatory) one. The probability of these transformations ("leakages") increases when the mechanical stress is applied. Such an effect was observed in steel wires which are used in the Initial LIGO susp ension. Noise bursts were observed in some well stressed samples [9]. If, as planned, the violin mo de's quality factor of Advanced LIGO susp ension fib ers will b e Qviol 108 , then, according to FDT, the rms variation of energy stored in this -1/2 4 в 10-18 erg at ro om temp erature. mo de over a half of p erio d has to b e E 2k T Qviol This value is 1021 times (!) smaller than the total free energy stored in one milligram fib er. Thus even very small "leakage" of free energy into violin mo de may effectively "warm" it and degrade the antenna sensitivity. The ab ove arguments motivated investigations describ ed in this article in which author's goal was to measure the noises in high Q mo des of fused silica susp ension prototyp e resolving small fraction of k T - mean energy of thermal equilibrium oscillations.

Experimental setup
To achieve the sensitivity mentioned ab ove on the fused silica fib er susp ension prototyp e the installation used in [9] was substantially redesigned. One of the key elements of the installation was heavy fused silica blo ck (see Figure A.1) with a rigid horizontal cantilever with a pin carved from it. Upp er end of the fib ers were welded to this pin. The lower end

12

Figure A.2: Violin-like (left) and mirror-swinging (right) mo des of the sample.

of the fib er were welded to a pin on a fused silica piece (main load, ab out 200 g). This piece was connected to the blo ck by two thin horizontal levers. The stretching rigidity of the levers was high enough to prevent p endulum oscillations of the samples while b ending one was small as compared to the longitudinal rigidity of the fib ers. Nearby the blo ck there was an electrically driven manipulator which allowed to place a makeweight (additional load) automatically and observe the noise right after it. The diameter of tested fib ers was from 50 µm to 180 µm, total length was 9.2 cm. Another key element of the installation was an optical sensor based on a Fabry-Perot cavity. We have develop ed a technique of welding a small (approx. 4 в 2 в 0.5 mm) diamond shap ed mirror with high reflective multilyer coating on it which allows to keep the quite high quality factor of the violin-like mo de (oscillation of the fib er with the lump ed mass in the middle; in our case the masses of fib ers were smaller then of the mirror). We also monitor a mirror-swinging mo de of the mirror (see figure A.2) which has a quality factor > 107 . The Fabry-Perot cavity was formed by fixed spherical mirror (front) and mirror welded into the sample (end). Length of the cavity was 1.2 cm. The reflection co efficient of mirrors was R = 0.97, measured finesse varies from 50 to 100 for different samples. The schematic diagram of the installation is shown on figure A.3. A He-Ne single frequency 2 mW stabilized laser (Mellies-Grot STP-903) has b een used as a pump source. The light was passed through the pair of Faradey isolators, electro-optic mo dulator (EOM), b eam splitter to the single mo de waveguide input. Fraction of the light was used by amplitude noise suppression feedback chain (inclusive photo detector, amplifier and EOM) and also by the amplitude spikes monitor (see b elow). By the waveguide the light was passed inside the vacuum chamb er, where, through the mo de matching lens, illuminated the cavity. Behind the end mirror of the cavity a detector connected to twin low noise preamplifier was placed. Outputs of the preamplifier were connected to the pair of lo ck-in amplifiers (Stanford research SR810) pre-tuned to the selected mo des frequencies and drift comp ensation feedback chain (inclusive band pass filters, amplifier and PZT drive attached to the front mirror mount). This chain allows to adjust the interferometer so the pump frequency was tuned to the slop e of resonance curve. At this regime p ower

13

11 13 15 8 3 1 2 2

12

4

5

67 14

8 8

9 9

2x SR810

10 SR765

Figure A.3: The installation schematic diagram: 1-laser; 2-Faradey isolators; 3-EO mo dulator ; 4-wave guide launch ; 5-mo de matching lens; 6-front mirror; 7-sample with the mirror welded; 8-photo detectors; 9-preamplifiers; 10-drift comp ensation feedback; 11amplitude noise suppression feedback;12-anti-seismic susp ension; 13-seismic shadow sensor; 14-makeweight manipulator; 15-vacuum chamb er.

14

1E-9

Sx(F),cm/Sqrt(Hz)

1E-10

1E-11

1E-12

1E-13 0 500 1000 1500 2000 2500 3000

F, Hz

Figure A.4: Typical mirror oscillation sp ectrum. Sp ectrum analyzer binwidth was 4 Hz. Peaks at 450 Hz and 2130 Hz corresp onds to violin-like and mirror-swinging mo des, resp ectively.

collected by the detector was 100 µW . For the finesse of the cavity F = 100 and quantum effectiveness of the detector q = 0.5 the ultimate sensitivity was: S
min x

2F

= 2.5 в 10 qW

-14

cm Hz

(A.1)

cm The lowest noise flo or achieved in the exp eriments corresp onds to Sx 9 в 10-14 H z which is 3 times higher than this shot noise limit (see figure A.4). We consider that the main contribution in this value and in the rise of the noise sp ectrum density in the 0.5 - 1.5 kHz band mainly is from the residual laser phase noise. A new anti-seismic multistage susp ension allowed to reduce noises induced by the environment vibrations. It consists of heavy (10 kg) brass disks susp ended on gaunt viton rings (frequencies of stretching mo des - 2.2 Hz, ro cking mo de 1.2 Hz). In this installation we pay sp ecial attention to any p ossible external effects which may stimulate or mimic the excess noise in the tested fib ers. In order to veto such an events we monitor microseismic near the vacuum chamb er using a laser shadow sensor, pump laser p ower and the signal in the drift comp ensation feedback. The subsidiary ADC's of the SR810 have b een used. All signals (amplitudes of two selected mo des and outputs of 3 monitor channels) were sampled at 100 1/s rate and passed to PC via IEEE488 bus.

Measurements and results
We presented results obtained on 9 fused silica samples loaded from 4% to 50% of breaking stress. The frequencies of oscillation on the violin-like mo de v were from 400 15

Hz to 750 Hz for different samples and its amplitudes corresp onded to Brownian noise within 30% accuracy (for the calibration we used a harmonic signal applied to the PZT drive): Ї A= 2k T = 2 mv 2 в 4 в 10-14 er g 1 в 10-2 g (2 в 500 H z )
2

1 в 10

-9

cm

(A.2)

here m mm + 0.5 mf , mm is a mass of the the welded mirror, mf - mass of the fib er. This outcome confirms our previous results [10]. Short time variations were in a go o d agreement with [11]: Ї Ї 2t (A.3) A2 A2 t here is a relaxation time for this mo de. We cho ose the value of measurement (averaging) time t = 0.1 s. It was difficult to make the same comparison accurate for the mirror-swinging mo de, b ecause of strong dep endence of sensor conversion co efficient on the p osition of the reflection p oint on the mirror with resp ect to the rotation axes. Hence, an b earable assumption that the average amplitude measured for this mo de also corresp onds to the thermal motion wasn't checked. Nevertheless we pro cessed all data for this mo de on the same way as for violin-like one lo oking for non-Gaussian addition to the Gaussian distribution of the amplitude.

16

Test set Nr.

Sample co de

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Q2 Q4 Q5 Q6 Q9 Q9 Q11 Q11 Q11 Q11 Q20 Q20 Q21 Q25 Q25

Diameter and applied load(% of breaking stress for this sample) 180 µm, 4% 120 µm, 8% 90 µm, 15% 70 µm, 19% 85 µm, 16% 85 µm, 19% 70 70 70 70 75 µm, µm, µm, µm, µm, 24% 29% 35% 42% 20%

75 µm, 21% 60 µm, 33% 50 µm, 50% 50 µm, > 50%

15 16 17

Mo de freq. [H z ] and typ e 1087m 762v 2319m 1538m 1932m 748v 1980m 759v 2197m 1600m 1747m 1946m 2083m 624v 1852m 627v 1861m 450v 2130m 404v 1811m 413v 1860m

Quality factor

Obs. time (sec)

Noise flo or

Nr. of candidate events 62 7 48 26 0 17 1 0 0 0 0 0 2 0 5 0 28 2 3 0 11 0

Nr. of excess events proved 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 3.4 1 7.4 2 1.4 1 1.9 1.3 1.4 1.3 9 5 8.2 8.5 8.0 8.5 1.3 5 1 8.5

в в в в в в в в в в в в в в в в в в в в в

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

6 6 7 6 7 6 7 6 7 7 7 6 6 5 6 5 6 5 6 5 6

17000 7450 31300 14400 23600 6700 5050 33450 16750 18250 16500 15850 2400 2400 4800 4800 28200 28200 8750 8750 14550 14550

5 1.5 4 4 1 3 5 5 2 2 3 6 3 5 3 5 2 6 1 9 1 9

в в в в в в в в в в в в в в в в в в в в в в

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

-2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2

Table A.1: Results summary. Index v means violin-like mo de, m - mirror-swinging mo de. The noise flo or was determined as a ratio of minimal amplitude variation resolvable in the bandwidth 10 Hz near the mo de frequency to the mean amplitude obtained for this mo de (for the violin-like mo des this amplitude corresp onds to k T with accuracy b etter than 30%). -- test set started right after the makeweight was loaded automatically. -- approximate value, sample got broken.

17

The results of the noise measurement are presented in the Table A.1. Each record was from 1500 sec to 4000 sec long. We joined the records made for the same sample under the identical conditions in a set. Initially we were able to monitor the amplitude of oscillation on one mo de at the time only, so for some samples (Q4 and Q9) the pairs of sets presented, one contains records made on the violin-like mo de, another - on the mirrorswinging mo de. Later, when the second SR810 lo ck-in amplifier was installed, amplitudes of b oth mo des were stored simultaneously, thus set numb er 13 and subsequent contain information ab out the noise in two mo des obtained at the same time. Some samples were tested under different stresses. These results are represented as a separated sets. For the samples Q9 and Q11 load has b een increasing manually, step by step, each time vacuum chamb er was op en and pump ed out again. In the case of samples Q20 and Q22 the electrically driven manipulator describ ed ab ove has b een used. Usually, the breaking stress value was measured for each sample when the measurements for it had b een completed. The sample Q25 is an exception as it broke off right during the record (hence, the applied stress was ab out to breaking value). The value of applied stress in the table for this sample is an approximate estimation. It is worth to note that many samples were broken during the preparation, esp ecially when we were trying to apply high (ab out 50 % of breaking stress) load. The sensitivity to the variation of the amplitude of the selected mo des varies from set to set due to the difference of the cavity finesse, dep endence of the background noise on the frequency and optic adjustment imp erfection. We characterize it by the noise flo or value. It was determined as a ratio of minimal amplitude variation resolvable in the bandwidth 10 Hz (corresp onded to 0.1 s measurement time) near the mo de frequency to the mean squared amplitude obtained for this mo de (for the violin-like mo des this amplitude corresp onded to k T with metrological accuracy b etter than 30%). Each stored record were pro cessed as follows. First, the time intervals when the seismic overcome a predefined limit were excluded. Then the amplitude variations At = Ai - Ai-1 over the unity time interval (time of measurement, in our case t = 0.1 s) was obtained and r ms amplitude variation has b een calculated: A=
2 t

(At ) N0

2

(A.4)

here N0 - total numb er of unity intervals. For the pure thermal motion the distribution of amplitude variations should b e close to normal [11]: 1 A2 exp(- t2 ) P [At ] = 2 2 and = A2 . The exp ected numb er of intervals where X At < X + X is: t N (X, X ) = N0 X P [X + X/2] (A.6) (A.5)

(it is assumed, that X X ). 2 All p oints where At > 3A2 were considered as candidate events. Then, if a spike was t registered by any monitor channel simultaneously with a candidate event, this event was excluded from the future consideration. We also excluded consequent pairs of candidates which corresp onds to single laid-down amplitude value and caused by environment electric 18

1800 1600 1400

20 18 16 14

Number of events

Number of events
-10 -5 0 5 10

1200 1000 800 600 400 200 0 -200

12 10 8 6 4 2 0 -10 -5 0 5 10

Amplitude variation,

Amplitude variation,

Figure A.5: Left: Histogram for the amplitude variations for one record made for the sample Q21. Right: enlarged b ottom part.

19

1

2

3

4

Energy innovation threshold

Figure A.6: Cumulative histogram for the energy innovation values (sample Q21). The 2 total numb er of events when i overcomes a threshold is plotted as a function of the threshold value. The deviation from the straight line (that represents exp onential distribution) is insignificant, hence there are no evidence of excess noise presence. 5 p oints at 2 the right end corresp onds to only one event when i > 4

spikes. Then the data were plotted as a histogram. The typical example is on the figure A.5 (-10 < X < 10 , X = /10, N0 = 45986). The dash lines b ound area there N should b e with the probability P > 0.99 for the given N0 . It is clear, that there are no significant excessive events in the example. The similar results were observed for all records. Total duration of the refined records were ab out 20 hours for violin-like mo de and ab out 70 hours for mirror-swinging mo de. Quite often the value named energy innovation [12] is used for representation of a noise in the oscillatory mo de:
2 i [A1i - A1 i-1

]2 + [A2i - A2

i-1

]

2

(A.7)

here A1i = Ai sin(i ), A2i = Ai cos(i ) are quadrature amplitudes, Ai and i - amplitude and phase, resp ectively. In fact, is sensitive to the variation of the oscillator phase as well as its energy. During the first part of our measurements only the values of Ai were stored. Later the p ossibility to store a pair of quadrature amplitudes instead was implemented. On the figure A.6 the cumulative histogram for the energy innovation obtained for one record made on the sample Q21 is presented for comparison purp ose.

Conclusion
The results describ ed ab ove demonstrated absence of excess noise in the stressed fused silica fib ers at the achieved sensitivity level. From our p oint of view this can b e regarded 20

as a supp ort of the Advanced LIGO susp ension design. Though our samples were only mo dels of this susp ension having smaller diameters and extra mirrors inside, one can extrap olate these results to estimate the corresp onding amplitude of the Advanced LIGO mirror oscillations: mm Ї 1 в 10 AL A 2 t/ в ML
-9

cm

2 · 0.1s/600s в

0.02 g 1 в 10 4 в 104 g

-17

cm (A.8)

here ML is a mass of the Advanced LIGO mirror. The ultimate displacement sensitivity in this exp eriment was improved by two orders of magnitude as compared to [9] that allows us to use 10 times smaller measurement time and reach the amplitude variation resolution which corresp onds 1%k T . To investigate the noise in the susp ension with b etter resolution b oth the quality factor of violin-like mo de and sensitivity of the readout system should b e improved.

21

References
[1] Barish, B. and Weiss, R., Phys. To day, 52 (1999) 44. [2] B.Abb ott, R.Abb ott, R.Adhikari, et al. Nucl. Instrum. and Metho ds in Phys. Research A:517 (2004) 154. [3] V.B.Braginsky, V.P.Mitrofanov, S.P.Vyatchanin. Rev. Sci. Instrum, 65 (1994) 3771. [4] A.M.Gretarsson, G.M.Harry, P.R.Saulson, S.D.Penn, S.Rowan, G.Cagnoi, Phys.Lett. A 270 (2000) 108. W.J.Startin, J.Hough,

[5] V.B.Braginsky, V.P.Mitrofanov, K.V.Tokmakov, Phys. Lett. A 218 (1996) 164. [6] P.Willems, V.Sannibale, J.Weel, V.Mitrofanov, Phys. Letters A 297 (2002) 37. [7] N.A.Rob ertson, G.Cagnoli, D.R.M.Cro oks, E.Elliffe, J.E.Faller, P.Fritschel, S.Gossler, A.Grant, A.Heptonstall, J.Hough, H.Luck, R.Mittleman, M.PerreurLloyd, M.V.Plissi, S.Rowan, D.H.Sho emaker, P.H.Sneddon, K.A.Starin, C.I.Torrie, H.Ward and P.Willems, Class.Quantum. Grav. 19 (2002) 4043. [8] A.Gillespie and F.Raab, Phys.Lett. A 190 (1994) 213. [9] A.Yu.Ageev, I.A.Bilenko, V.B.Braginsky. Phys. Lett. A, 246 (1998) 479. [10] I.A.Bilenko, S.L.Lourie Phys. Lett. A, 305 (2002) 31. [11] V.B.Braginsky and A.B.Manukin, Measurements of Weak Forces in Physical Exp eriments, Univ. of Chicago Press, 1977. [12] A.M.Gretarsson and P.R.Saulson in press. [13] A.M.Gretarsson, G.M.Harry, Rev. Sci. Instrum 70 (1999) 4081.

22

Appendix B Nonstationary electrical charge distribution on the fused silica bifilar pendulum and its effect on the mechanical Q
. Effect of electrical charging of the fused silica test mass on the mechanical Q of various mo des has b een observed earlier [1, 2, 3, 4]. Degradation in the Q is asso ciated with interaction b etween charges sitting on the test masses and surrounding b o dies, for example, due to Coulomb force. In the previous measurements we have found that the losses asso ciated with charging dep ended not only on the value of the charge but probably on the state of the charge (for example, the character and distribution of infill of traps, which catch electrical charges). So far there is no go o d theory which describ es the losses asso ciated with charging of the test masses. We continued to study dissipation caused by the charging. In the previous measurements the charge lo cated on the test mass was in the state close to the stationary state. This year we b egan to study dissipation in the case when the charge was in the nonstationary state. The Figures presented b elow illustrate the main results of research which were describ ed in the Summary.

23

Manipulator

Optical sensor

To electrometer 1400 V

Figure B.1: Schematic of the all fused silica p endulum with additional arrangements used to investigate effects asso ciated with electrical charging of the p endulum b ob: the optical sensor for measurement of the bifilar torsion mo de amplitude; the plate with electro des for creation of the electric field; the capacitive prob e for monitoring of electrical charge; the manipulator for contact electrification of the cylinder.

24

2 .2

2

1400 V Amplitude, a.u.

2
1 .8

1 .6

1

1 .4 0 400 800 1200 1600

Time, hours

Figure B.2: Free decay of the p endulum amplitude (curve 1). Free decay of the p endulum amplitude in case when the high voltage was applied to the electro des (curve 2 - application of the voltage during 24 hours is shown by arrows).

25

0.018

0.016

Votage f rom probe

0.014

0.012

0. 01 0 100 200 300 400

Time, hours

Figure B.3: Time dep endence of the prob e voltage after electrical charge dep osition produced by means of the contact electrification.

26

0 .0 1 0 .8 1400 V

1
0 .0 0 9

2
0 .7

Voltage f rom probe

Amplitude, a.u.

0 .0 0 8

0 .6

0 .0 0 7

0 .5

0 .0 0 6

0 .0 0 5 400 600 800 1000 1200 1400

Time, hours

Figure B.4: Fragment of free decay of the p endulum amplitude (run #3) in case when the high voltage was applied to the electro des (curve 1 - application of the voltage is shown by arrows). Time dep endence of the prob e voltage (curve 2).

27

References
[1] S.Rowan, S.M.Twyford, R.Hutchins, and J.Hough, Class. Quantum Grav. 14, 1537 (1997). [2] V.P.Mitrofanov, N.A.Styazhkina, K.V.Tokmakov, Physics Letters A 278, 25 (2000). [3] M.J.Mortonson, C.C.Vassiliou, D.J.Ottaway, D.H.Sho emaker, and G.M.Harry, Rev. Sci. Instrum. 74 4840 (2003). [4] V.Mitrofanov, L.Prokhorov, K.Tokmakov and P.Willems, Class. Quantum Grav. 21, 1083 (2004).

28

Appendix C Corner reflectors and Quantum-Non-Demolition Measurements in gravitational wave antennae
We prop ose Fabry-Perot cavity with corner reflectors instead of spherical mirrors to reduce the contribution of thermo elastic noise in the coating which is relatively large for spherical mirrors and which prevents the sensitivity b etter than Standard Quantum Limit (SQL) from b eing achieved in laser gravitational wave antenna. We demonstrate that thermo-refractive noise in corner reflector (CR) is substantially smaller than SQL. We show that the distortion of main mo de of cavity with CR caused by tilt and displacement of one reflector is smaller than for cavity with spherical mirrors. We also consider the distortion caused by small nonp erp endicularity of corner facets and by optical inhomogeneity of fused silica which is prop osed as a material for corner reflectors.

Introduction
The existing to-day's multi-layer dielectric coating on optical mirrors allows to realize very high resolution exp eriments (see e.g. [1]). The reflectivity R in the b est optical coating has reached the level of (1 - R) 10-6 [1, 2, 3] (commercially available (1 - R) 10-5 ), and there are many reasons to exp ect that further improvement of coating technologies will p ermit to obtain the value of (1 - R) 10-9 . With the value of (1 - R) 10-5 it is p ossible to realize the ring down time F P 1 sec in 4 km long Fabry-Perot (FP) resonators which are the basic elements in laser interferometer gravitational wave antennae (pro ject LIGO [4, 5]). This relatively large value of F P p ermits to have relatively small value of the ratio of av /F P 7 в 10-2 (if the averaging time av 5 в 10-3 sec). This ratio is the limit for the squeezing factor which may b e obtained if QND pro cedure of measurement in such FP resonator is used [6, 7]. Such a pro cedure will allow to circumvent the Standard Quantum Limit (SQL) of sensitivity (see details in [8]). Few years ago the role of the thermo elastic noise in the bulk of the mirrors was analyzed [9]. This analysis has shown that if the laser b eam sp ot size on the mirror surface is sufficiently large then the small value of the thermal expansion co efficient S iO2 5 в 10-7 K -1 of fused silica will p ermit to circumvent the SQL sensitivity by the factor of 29

0.1 (if the thermo elastic noise is the only source of noises). The consequent analysis of thermo elastic noise in the coating itself unfortunately predicts that the limit of sensitivity will b e close to the SQL of sensitivity [10, 11, 12, 13]. The origin of this obstacle is relatively big numerical value of thermal expansion co efficient T a2 O5 5 в 10-6 K -1 of amorphous T a2 O5 [13] which is used in the b est coatings as well as relatively big numb er of layers (usually 20-40) which is necessary to have small value of (1 - R). For LIGO pro ject these limitations may b e illustrated by the following numerical values. The SQL sensitivity of detectable amplitude of the p erturbation of the metric is equal to [14] 8 m 2 L

S

S QL h

( ) =

2

2 в 10

-24

Hz

-1/2

,

(C.1)

where m = 40 kg is mass of test mass, L = 4 km is distance b etween them, = 2 в 100 s-1 is observation frequency. Here and b elow the estimates are calculated for numerical parameters listed in App endix C. At the same time, according to the measurement [13], the limit of sensitivity of such an antenna only due to the thermo elastic noise in the multi-layer T a2 O5 + S iO2 coating on S iO2 substrate has to b e b etween [10] S
T D coat h

( )

0.6 В 1.4 в 10

-24

Hz

-1/2

(C.2)

The goal of this article is to present the analysis of another version of optical FP cavity where the "contribution" of the thermo elastic noise in coating is substantially reduced. The key idea of this version is based on concept of corner optical reflector (tri-hedral or two-hedral prism). These typ es of reflectors were well known among the jewelers at least from 16-th century (see e.g. autobiography by Benvenutto Cellini [15]). In the 70-s of the previous century corner reflectors (CR) installed on the Mo on allowed to test the principle of equivalence for the gravitational defect of mass by laser ranging [16]. Here we prop ose to substitute mirrors with finite value of surface curvature (see fig.C.1 a) by 3 facets (fig.C.1 b) or 2 facets (see fig.C.1 c) corner reflectors (CR) manufactured from fused silica. For the same radius Rb of laser b eam the mass of CR is ab out the same value as cylindrical mirrors (with height ab out equal to radius of cylinder) esp ecially if idle parts of CR is removed. For example for 2 facets CR the size of fo ot surface has to b e ab out 45 в 45 cm2 with total test mass ab out 40 kg (the same as planned in advanced LIGO). In the prop osed scheme the stability of optical mo de is provided by surfaces lens shaping of each CR fo ot (as shown in fig.C.2a). For reflectors manufactured from fused silica the total internal reflection inside the reflectors is p ossible b ecause refraction index nS iO2 = 1.45 is large enough (due to Snellius law): nS iO2 2/3 > 1 for 3 facets reflector or nS iO2 1/2 > 1 for 2 facets reflector. In section C we consider the mo des of ideal cavity (fig.C.2) and the distortion of the main mo de structure caused by small p erturbations of different kinds: tilt angle (fig. C.3), displacement x of one reflector (fig. C.4), exp ose angle (fig.C.5). In section C we compare these p erturbations for cavities with spherical mirrors and with corner reflectors and give numerical estimates for the particular case of laser b eam radius Rb 6 cm (intensity of b eam decreases as e-1 at distance Rb from center) which is planned for Advanced LIGO. Distortions of mo de are undesirable in high accuracy sp ectroscopic measurements b ecause they may pro duce additional noise. For example, in 30

b)
2

+

a) PSfrag replacements a) g replacements g replacements c) a) b) + 2

2

+

Figure C.1: We prop ose to replace mirrors with finite value of surface curvature (a) by 3 facets "triprism" typ e CR (b) or 2 facets "ro of" typ e CR (c). laser gravitational wave antenna the light b eams from two indep endently p erturb ed FP cavities (placed in each arm of Michelson interferometer) will not pro duce completely zero field at the dark p ort after the b eam splitter. It is equivalent to additional noise at the dark p ort. In section C we consider the different sources of optical losses of CR and show that they can b e at level (1 - R) 10-5 . In cavity with CRs it is necessary to use nevertheless relatively thin anti-reflective coatings (2 4 layers) on lense shap e fo ot. It has to b e done to keep the value (1 - R) at the level 10-5 . Because this coating is substabtialy thiner than typical high reflective one (20 40 layers) used in curved mirrors, thermo elastic noise may b e depressed by the factor 10 i.e. ab out one order less than SQL (the estimate (C.2) is given for 38 layers). The "fee" for use of CR is the additional thermo-refractive noise [17] (fluctuations of temp erature pro duce the fluctuations of refractive index) b ecause light b eam is traveling inside the corner reflector. However we will show in section C that it is several times smaller than SQL for reflectors manufactured from fused silica. Imp ortant that this thermo-refractive noise rapidly decreases with increasing of radius Rb of b eam sp ot as - Rb 2 . In cavity with CR the thermo elastic noise in the facets remains, but as mentioned ab ove if the reflectors are manufactured from fused silica and the b eam sp ot is large enough (see e.g. [10]) then it is p ossible to circumvent SQL.

The Distortions of main mode in FP cavity with CR
Firstly, we consider FP cavity with two identical p erfect corner cub e reflectors with three reflecting facets (see fig.C.1b): i) the corner angles b etween the facets are exactly equal 31

to /2; ii) the top p oints of the reflectors are lo cated exactly on common optical axis; iii) the "fo ots" of the reflectors have slight curvatures (shap e of a lens surface as shown in fig. C.2a) and they are p erp endicular to the axis. Then we consider the distortion of mo de in this cavity caused by different p erturbations.

FP resonator with p erfect CR
We can consider that each CR consists of reflector with plane fo ot surface together with spherical lens as shown in fig.C.2b. The CR pro duces mirror transformation, i.e. light b eam which enters the reflector in p oint C is transformed into the b eam leaving the reflector in p oint C . Using Fresnel integral one can obtain the integral equations for calculations of eigenmo de distribution: e e
ik L

~ G0 (r1 , r2 ) 2 (r2 ) dr2 = 1 (r1 ), ~ G0 (r1 , r2 ) 1 (r2 ) dr1 = 2 (r2 ), dr1 = dx1 dy1 , dr2 = dx2 dy2 .

(C.3) (C.4)

ik L

Here functions 1 (r1 ) and 2 (r2 ) describ e distribution of complex field amplitude emitted from imagine flat fo ot surfaces of reflector 1 (left) and reflector 2 (right) corresp ondingly just under lenses in planes AA and B B . L is the optical path b etween reflectors (including path inside the reflector). Evidently phase fronts coincide with planes AA and ~ B B so that phases of functions 1 and 2 are constants. The notation 1 (r1 ) means the "mirror" transformation (pro duced by 3 facets CR) relative to the optical axis 1 : ~ 1 (r1 ) = 1 (-r1 ), The kernel G0 is: i i (r1 -r2 ) -h 2 G0 (r1 , r2 ) = - e 2 2 r1 r2 h1 = 2 , h2 = 2 2 rh rh
,,
2 1

~ ~ 2 (r1 ) = 2 (-r1 )
«

(r1 )-h2 (r2 )

,

(C.5) (C.6)

Here we use the dimensionless transversal co ordinates r1 and r2 (at planes AA and B B ) which can b e expressed in terms of physical co ordinates R1 and R2 as r1 = R1 , b r2 = R2 , b b= L , k

where k is wave vector, h1 and h2 are additional phase shifts pro duced by spherical lenses at each reflector fo ot. It is easy to see that for spherical lenses the set of eigenmo des mn 1 , mn and their eigenvalues mn (b elow we assume 00 = 1) of our FP cavity are 2
~ For 2 facets reflector we have mirror transformation only relatively x coordinate: 1 (x, y )) = 1 (-x, y )
1

32

a) g replacements

A x y z A

B

B A

b)

C

C A Figure C.2: a). The p erfect alignment of CR assembling cavity. b). Each reflector can b e regarded as reflector with plane "fo ot" plus a lens. describ ed by generalized Gauss-Hermite functions:
mn 1 mn 2

= m (x)n (y ), = (-1) r
L -m+n

(C.7) ~ =
mn 1 m

mn 1

(C.8) x rL в (C.9)

m (x) =

1

2 m!

m

H

x2 , 2 2rL 1 , mn = e2i(m+n) , = arctan 2 2r0 r2 1 2 2 h1 (r ) = h2 (r ) = 2 , 2rL = 2r0 + 2 2rh 2r0 2 22 2 2 2rh = (2r0 ) + 1 = 2rL 2r0 . в exp -i(m + 1/2) -

(C.10) (C.11)

Here Hm (t) is the Hermite p olynomial of the order m, r0 and rL are the radii of b eam in the waist and at the lens corresp ondingly. It is useful to write down the expressions for r0 , rL , rh using g -parameter [18] (R -- is the radius of wave front curvature (in cm) just after the propagation of b eam through the lens outside of the reflector): g = 1-
2 rL =

L , R

2 r0 =

1 2 ,

2 Rb = b2

1 1-g
2

1+g , 1-g 1 R 2 = , rh = L 1-g

(C.12) (C.13) (C.14)

sin 2 = g ,

cos 2 =

1 - g2.

We are interested in the main mo de 00 (x, y ) of resonator (amplitude distributions of 1 left and right reflectors obviously coincide with each other for the main mo de: 00 (x, y ) = 1 00 (x, y )). 2 33

a) g replacements l b) x x z l )

Figure C.3: a). Small tilt of the left CR around its head. b) This tilt is equivalent to untilted reflector and displaced lens.

Distortion due to the Tilt of CR
Here we consider the main mo de 00 (x, y ) p erturb ed due to tilt misalignment shown in 1 fig. C.3a. We expand the p erturb ed main mo de into series over the set of unp erturb ed mo des limiting ourselves to the lowest (dip ole) approximation: 1 00 (x1 , y1 ) 00 (x2 , y2 ) 2
tilt 00 (x1 , y1 ) - 1 10 (x1 , y1 ), 1 1 00 2 tilt 1

(C.15) (C.16)

(x2 , y2 ) +

(x2 , y2 ).

10 2

The tilt of CR around its head through a small angle of can b e considered as untilted reflector with lens displaced a small distance x l p erp endicular to optical axis (l is the dimensionless distance b etween the fo ot and head of CR, it is illustrated in fig. C.3b). For this case the p erturbations of the main mo de can b e describ ed by dip ole co efficients defined in (C.15, C.16): |
tilt 1

tilt 1

tilt 1

|

g (1 - g ) , 2 (1 - g 2 )3/.4 (1 - g ) cot - i l , 2 2 (1 - g 2 )1/4 l 4 1 - g 2 1+g l

(C.17) (C.18) (C.19)

See details of calculations in App endix C. Note that distortion pro duced by tilt of two facets CR around p erp endicular axes (angle on fig. C.1c) can b e describ ed by the same formulas as tilt of spherical mirror.

The Distortion due to the Displacement of CR
One CR can b e displaced by a small distance x so that optical axes of reflectors do not coincide with each other as it is shown in fig. C.4. For this case the p erturbations of the main mo de can b e describ ed by dip ole co efficients defined by formulae (C.15, C.16).

34

x g replacements x z

y Figure C.4: One CR is displaced relative to another one. a) g replacements b) c)

2

+

Figure C.5: (a). Exp ose p erturbation: the angle b etween facets of corner reflector differs from direct angle by a small value of . It pro duces the transformation of incident plane wave front (b) into a "broken" front of reflected wave (c). Denoting the dip ole co efficients as
displ 1 displ 1

and

displ 1

one can obtain: +
4

=

displ 1

- x 22

4

-2ig

1-g

2

1-g

2

(C.20)

See details of calculations in App endix C.

The Distortion of Exp ose Angle
Here we consider the case when, for example the left reflector (2-hedral prism shown in fig. C.1c) has a non-p erfect p erp endicular facets, so that exp ose angle b etween them differs from /2 by a small angle of (fig. C.5a). Then the plane front of incident wave after reflection from the reflector is transformed into a broken surface consisting of two plane parts declined to the incident wave front by an angle of = 2 as shown in fig. C.5b,c. This statement is also correct for tri-hedral reflectors (shown in fig. C.1b) with the exception of numerical factor: if only one facet is declined by angle of from the normal p osition (and other two facets are non-p erturb ed) the angle will b e equal to = 2 2/3. Again we can expand the p erturb ed main mo de over the set of unp erturb ed mo des of ideal cavity keeping only the lowest first-order non-vanishing term of expansion: 00 1 (x1 , y1 ) 00 2 (x2 , y2 ) 00 (x1 , y1 ) + 1 (x2 , y2 ) +
00 2 exp ose 2 exp ose 2

20 (x1 , y1 ), 1 (x2 , y2 )
20 2

(C.21) (C.22)

35

(due to the symmetry of this kind p erturbation the dip ole term is null). Calculation gives the following value for 2 : |
exp ose 2

4 2 b
exp ose 2

i L
4

1-g

2

в

g+i ig

exp ose 2

| = |

|

See details in App endix C.

L 4 2 b g (1 - g 2 )

1-g
3/4

1-g

22

2

(C.23) (C.24)

Comparison of FP cavities with CR and with spherical mirrors
Recall that uncontrollable p erturbations of the mo de pro duce additional noise: in laser interferometer gravitational antenna the signal at dark p ort will contain additional noise with p ower prop ortional to the square of distortion co efficients ||2 , | |2 . In this section we compare numerically the distortion of the main mo de in traditional FP cavity with spherical mirrors (SM cavity) with FP cavity assembled by CR (CR cavity). The distortion of the main mo de in SM cavity caused by small displacement and tilt of one mirror can b e also describ ed by co efficients 1 of expansion (C.15)[20, 21]: L 1 , (C.25) 2 )3/4 b 2 (1 - g (1 - g )1/4 displ 1 , Sph = x (C.26) 2(1 + g )3/4 For estimates for b oth SM and CR cavity we use the parameters of cavity prop osed for Advanced LIGO [20]:
tilt, Sph 1

=

rL = 6/2.6

2.3,

g = 0.982.

(C.27)

These parameters corresp ond to the radius of laser b eam Rb at the reflector surface of ab out Rb 6 cm. Assuming additionally that dimension length l from fo ot to top is equal to l = 20 cm, i.e. 20 cm 7.7 , l= b and dimension displacement x = b x we obtain the following estimates for SM cavity: and for CR cavity:
tilt, CR 1 tilt, SM 1

disp, SM 1

, 10-8 x = 0.0059 0.1 cm = 0.013
-7

(C.28) (C.29)

= 1.2 в 10 = 0.06

10-8

,

(C.30) (C.31) (C.32)

displ, CR 1

exp ose 2

x , 0.1 cm . = 0.11 10-6 36

We see that CR cavity is substantially more stable to tilt and less stable to displacement than SM cavity. However, the total requirements for SM cavity lo oks more tough if one compares the estimates (C.28 and C.31). Indeed to keep control of tilt in SM cavity with accuracy 10-8 rad (see (C.28)) one has to op erate the p ositioning system with accuracy ab out lb 2 в 10-6 cm. But the same level of displacement distortion in CR cavity (see (C.31)) one can obtain by displacement control with much lower accuracy: 1 mm only (!). The requirement for an exp ose angle in CR cavity lo oks also acceptable: for accuracy of manufacturing 3в10-7 (commercially available prisms have ±1в10-5 ) and hence = 2 2/3 4.9в10-7 we have to take into account that three angles b etween facets (increasing factor 3) in each of two tri-hedral reflector (one more increasing factor 2) may b e indep endently p erturb ed (so the total increasing factor is equal to 3 в 2 = 6): expose 2 exp 2, i ose 6 2 0.13 Optical inhomogeneity is one more source of p erturbation which is sp ecific to CR: the refraction index of fused silica (which CR is manufactured from) changes over the value of n 2 в 10-7 along the length l 10 cm [22]. To estimate negative influence of this effect we can consider the mo del task using the fact that distance scale l of refraction index p erturbation is ab out the dimension bl of the reflector. Let one half of corner reflector has a p erturb ed refraction index n + n(x) which dep ends on a transversal co ordinate x only: n(x) = n = - n 1 - n = 0
x l

In this case the b eam after reflection from such a reflector will have a broken wave front as shown in fig. C.5c with angle n = (C.34) 4n Hence one can estimate the value of p erturbation due to inhomogeneity using formula (C.24). It is obvious that for p erturbation with another dep endence of n on space co ordinates the formula (C.34) must change this estimate by the factor of ab out unity or slightly larger. So for estimates we use the 4 times greater value of than (C.34). In that case for two reflector with indep endently p erturb ed refraction index (factor 2) we obtain: 2 n inhomo = 2 в 10-7 , 2 0.011 (C.35) n The two last kinds of p erturbations (exp ose angle and inhomogeneity) dep end only on manufacturing pro cedure and there is a hop e they can b e decreased due to the improvement of manufacturing culture.

if x > 0, if x 0

(C.33)

The optical losses
The loss co efficient for CR cavity must b e small -- ab out 10 ppm. We consider the following sources of losses. 37

Fundamental losses on edge are pro duced by diffraction on edge where two facets meet. Qualitatively it can b e describ ed as two surface waves outside CR (b ounded with waves inside due to complete internal reflection) meet at edge pro ducing diffractional scattering (we acknowledge to F.Ya.Fhalili p ointed out the existtence of this kind losses). To our b est knowledge nob o dy p erformed rigorous analysis of this problem. So we prop ose the consideration to estimate this effect: (a) using the formulas for complex co efficient of reflection for plane wave from plane infinite b oundary b etween two media for the case of internal reflection (see e.g. [23]) one can construct the solution inside CR; (b) using the b oundary condition one can obtain the fields along outside surface of CR; (c) applying Green's formula one can calculate the radiation field in far wave zone and total diffractional p ower. The most vulnerable for critics item of this consideration is (a) -- applying the formulas for infinite b oundary to corner configuration. We apply this consideration for the case of incident wave p olarized along edge of CR of "ro of" typ e (fig. C.1c). For this particular case (with obvious assumption that magnetic p ermittivity µ = 1) the all field comp onents of constructed solution are smo oth inside CR and on outside surface CR. Our calculation gives the following loss co efficient: (1 - R)
d

0.4 Rb

0.7 в 10

-5

(C.36)

where is the optical wavelength, (see details in App endix C). Note that our consideration of incident wave p olarized p erp endicular to the edge of CR allows to construct smo oth solution inside CR but this solution on the outside surface will have break of comp onents of electrical fields at the edge. Thus to get a reliable confirmation of the approximatetive estimate (C.36) it is necessary either to find a rigorous analytical solution of this problem or to p erform straightforward numerical calculation. The losses on non-p erfect edge is pro duced by scattering of the plane optical wave on the non-p erfect "ridges" where two facets meet. The edge of facets intersection with uncontrollable width of s 0.5 µm will pro duce optical losses which can b e roughly estimated as following: (1 - R)
non-p erfect

s Rb

0.8 в 10
FP

-5

In other words it satisfies our initial condition to obtain

1s.

Optical losses of material. Internal optical losses in purified fused silica at the level loss 0.5 ppm/cm [19] give the loss co efficient ab out (1 - R)
opt.loss

loss

2bl

0.8 в 10

-5

Losses in anti reflective coating. As we mentioned in Intro duction it will b e necessary to use anti reflective coating on the b ottom surfaces of CR. Calculations which we omit here shows that to keep the value of (1 - R) at the level 10-5 it is sufficient to use 2-4 anti reflective layers of coating.

38

Thermo-refractive noise
The origin of thermo-refractive noise is thermo dynamic (TD) fluctuations of temp erature which pro duce fluctuations of phase of light traveling inside the CR through dep endence of refractive index n on temp erature T : = dn/dT = 0 [17]. One can estimate TD temp erature fluctuations using the mo del of infinite layer with width lc (0 z lc ). We additionally assume that layer is in vacuum and its b oth surfaces are thermally isolated (thermal radiation in accordance with Stefan-Boltzmann law is so small that this assumption is quite correct). If light (Gaussian b eam) with radius Rb travels through the layer p erp endicular to its surface the fluctuations of light phase during time will b e defined 2 by TD temp erature fluctuations u averaged over the cylinder Rb lc : = k lc u Ї
2

,

k=

2 .

(C.37)

Subscript means that we are interested in variation of temp erature during observation time . The total variation of TD temp erature fluctuations is equal to u Ї
2

=

kB T 2 2 C Rb l

c

where kB is the Boltzmann constant, is density, and C is the sp ecific heat capacity. Then the variation of temp erature over the small time (adiabatic approximation) must 2 b e ab out u2 Ї u2 в / where = C Rb / is thermal relaxation time of our Ї cylinder through lateral surface (base surfaces of cylinder are thermo isolated), is thermal conductivity. Here we assume that . This result can b e rewritten in form u Ї
2

=

kB T 2 в 2 C Rb lc

2 rT 2 Rb

,

rT =

C

Rb , l

where rT is thermal diffusive length for the time . Equating u2 Ї Su ( ) we can obtain the estimate for sp ectral density Su ( ) of Ї Ї averaged temp erature putting 1/ . The accurate expressions for these sp ectral densities Su ( ) and S ( ) of temp erature fluctuations and phase fluctuations corresp ondingly are the following Su ( ) Ї S ( )
R
2

4kB T 2 1 , 2 l R4 2 (C ) c b 1 4 2 k 2 lc kB T 2 4 (C )2 Rb

(C.38)
2

(C.39)

b for adiabatic case, i.e. for . C One can easy check that our estimate differs from accurate expression (C.38) for Su ( ) only by the factor of ab out unity. In App endix C we present derivation of general Ї expression for adiabatic and non-adiabatic cases. The formulae (C.38, C.39) allows to recalculate thermo-refractive fluctuations into the fluctuations of dimensionless metric h (which usually describ es the sensitivity of laser gravitational antennae): h = /(k L), where L is cavity length. It is useful to estimate

39

its sp ectral density Sh ( ) for parameters of laser gravitational antenna (advanced LIGO) presented in App endix C and lc = 10 cm: Sh ( ) 0.5 в 10
-24

Hz

-1/2

.

(C.40)

It is ab out 4 times smaller than the sensitivity of Standard Quantum Limit (C.1) which is planned to achieve in Advanced LIGO [19]. Imp ortant that this thermo-refractive noise - rapidly decreases with increase of b eam radius: Sh ( ) Rb 2 . Thus the using so called "mesa-shap ed" b eams [20, 21] (having flat distribution in the center and fall to zero more quickly than Gaussian distribution at the edges) with larger radius will allow to decrease thermo-refractive noise by several times. For example using the 45 в 45 cm2 fo ot of CR and Rb = 10 cm with mesa shap e distribution of the intensity of light in the b eam one may exp ect the gain of sensitivity for Sh ( ) approximately one order b etter than S
S QL h

( ).

Conclusion
The presented analysis of FP cavity with two CR (instead of mirrors) has to b e regarded as an example of cavity in which thermo elastic noise in the coating may b e substantially decreased and which p ermits to circumvent substantially the SQL of sensitivity and also to have (1 - R) 10-5 . We have shown that CR cavity is considerably more stable than cavity with spherical mirrors relative to tilt and displacement distortion. The distortion due to exp ose angle of CR is not so small but it dep ends only on manufacturing pro cedure, and there is a hop e to decrease it due to improvement of manufacturing culture. There do es exist another argument in favor of using CR in FP resonators. The very recent measurements p erformed by LIGO collab orators from University of Glasgow, Stanford University, Iowa State University, Syracuse University and LIGO Lab have shown that multi-layer coating also decreases the quality factors of mirrors internal mo des [24]. This effect may substantially increase the Brownian comp onent of the noise in the mirror itself and thus decrease the sensitivity of LIGO antennae (see details on Brownian moise in coating in [25, 26, 27, 28, 29, 30]). At the same time it is likely that there do es exist other version of cavity free from thermo elastic noise in coating, probably more easy to implement which evidently deserves similar in-depth analysis. One of the "candidates" is a cavity with unusual reflective coating: each layer of it has to have the same small value of thermal expansion as fused silica. Unfortunately the technology which may provide it is not yet invented. In the ab ove analysis we have limited ourselves to the calculations of the cavity properties itselves and did not discuss the coupling of cavity with pumping laser and readout system. This analysis has to b e done esp ecially b ecause the readout system may b e an intra-cavity one [8] and b ecause sp ecial attention has to b e paid to the p ossible sp ecific deformation of the main mo de distribution in FP cavity with CR. One of several p ossible ways to realize the coupling of the mo de with pumping source may b e based on the existence of the evanescent optical field "outside" the surface of the facets. In this case it will b e evidently necessary to use very thin dielectric grating on the surface of the facet. We have not also discuss the p olarization characteristics of CR. For example it is known that the phase shift of wave reflected from plane surface dep ends on p olarization [23]) and hence the FP cavity with 2-facets CR will have slightly different eigen frequencies for waves 40

with p olarization along and p erp endicular to edge of CR. The additional problem to b e analyzed is the p olarization characteristics of 3-facets CR. It seems that the CR cavity for Advanced LIGO have to b e used not for mo des with Gaussian distribution of p ower over the cross section but for so called "mesa-shap ed" [20, 21] mo de with flat distribution in the center and fall to zero more quickly than Gaussian distribution at the edges. For "mesa-shap ed" mo des the profile of "lenses" (h1 and h2 in C.5) on the fo ot of the corner reflector must have sp ecial dep endence on radius calculated in [20, 21]. It is worth noting that the discussed in this pap er features of CR cavity may b e useful not only for the gravitational wave antennae but also in other high resolution sp ectroscopic exp eriments where the low level of optical eigenmo de fluctuations is imp ortant.

Tilt of corner reflection
The tilt of one reflector through angle (see fig.C.3a) can b e considered as displacement of lens on distance x l (see fig.C.3b) p erp endicular to axis. Then the set of integral equations (C.3, C.2) for eigen mo de have the same form with replacing kernel G0 G1 : G
1

G

0

1-

ix1 x 2 rh

.

(C.41)

The p erturb ed main mo des 00 (x1 , y1 ), 00 (x2 , y2 ) we find as expansion into series 1 2 over eigen functions of resonator with p erfectly p ositioned reflectors: 00 (x1 , y1 ) = 0 (y1 ) 1
m

(-1)m m m (x1 ) , m m (x2 ) .

(C.42) (C.43)
0

(x2 , y2 ) = 0 (y2 )
00 2 m

After substitution these expansions into set (C.3, C.2) replacing kernel G obtain:
m

G1 we (C.44)

m (-1)m m m (x2 )- -e
ik L

G0 (r1 , r2 )

ix1 x 2 rh

в

в =
m

m

(-1)m m m (x1 ) dx1 =

(-1)m m m (x2 ),
ik L

m

m m m (x1 ) - e в ix1 x 2 rh

G0 (r1 , r2 )в

(C.45)

m m (x2 ) dx2 =
m

=
m

m m (x1 )

41

Here is eigen value of p erturb ed main mo de. After multiplying equation (C.44) by m0 (x2 ) and integrating we obtain: I
m
0

m0 ,0

= m0 ,0 m0 + Im0 , i x rL = 2 m0 ,0 в 2 rh в m0 -1 m0 +
m
0

(C.46)

m0 +1

m0 + 1 (C.47)

After multiplying equation (C.45) by m0 J
m
0

(x1 ) and integrating we obtain:

m0 ,0

= m0 ,0 m0 + Jm0 ,0 , i x rL = 2 m0 -1 m0 -1 m0 + 2 rh m0 +1 m0 +1 m0 + 1

Now we can rewrite equations (C.46,C.47) for different m0 taking in mind that 0 , 1, 1 1 = O ( x), 2 2 = O ( x2 ), . . . : i x rL m0 = 0, 0 0 + 2 0 1 = 0 , 2 rh i x rL 0 = 0 0 + 2 1 1 , 2 rh 0 0 1, = 0 + O ( x2 ), m0 = 1, 1 1 - -i x rL 2 1 , 2 rh -i x rL 2 0 , - 1 + 1 1 2 rh 2 -i x rL 1 + 2 0 22 1 , 2 2 r h 1 - 0 -i x rL 1 , 1 2 2 r h 1 - 0
1 2

0

(C.48) (C.49) (C.50) (C.51) (C.52) (C.53) (C.54)

m0 = 2, 2 2 -

- 2 + 2
2

2

-i x rL 2 2 2 1 , 2 rh -i x rL 2 2 1 1 , 2 rh 2 1 + 1 0 2 2 - 2
2 0 1 1

(C.55) (C.56)

-i x r 2 rh

L

,

(C.57)

2

-i x rL 2 0 1 + 1 2 rh 2 - 2 2 0

,

(C.58)

Rewriting values 1 and 1 using g -parameter (C.12 C.14) one can obtain formulas (C.17 C.19). 42

Displacement of CR
Let the right corner is displaced by value x in transversal direction (see fig. C.4). Then the integral equations for p erturb ed eigen mo de is the following e
ik L

G1 (x1 , y1 , x2 , y2 ) 1 (x1 , y1 ) dx1 dy1 = = 2 ( x - x2 , -y2 ),

(C.59) (C.60) (C.61) (C.62) (C.63) (C.64) (C.65)

e

ik L

G1 (x1 , y1 , x2 , y2 ) 2 (x2 , y2 ) dx2 dy2 = = 1 (- x - x1 , -y1 ),

1 (- x - x1 , -y1 )

1 (-x1 , -y1 )- - x x1 1 (-x1 - y1 ), 2 ( x - x2 , -y2 ) 2 (-x2 , -y2 )+ + x x2 2 (-x2 - y2 ), G1 (x1 , y1 , x2 , y2 ) G0 (x1 , y1 , x2 , y2 )в в 1 - i h1 + h
2

,

x1 x -x2 x , h2 . (C.66) 2 2 2rh 2rh We find p erturb ed main mo de distributions 00 (x1 , y1 ), 00 (x2 , y2 ) as expansion into 1 2 series over eigen functions of resonator with p erfectly p ositioned reflectors: h
1

00 (x1 , y1 ) = 0 (y1 ) 1
m

(-1)m m m (x1 ) m m (x2 ).

(C.67) (C.68)

(x2 , y2 ) = 0 (y2 )
m

00 2

After substitution these expansions into (C.59 - C.65) we obtain:
m m,0

(-1)m m m (x2 ) - e i(x1 - x2 ) x 2 2rh
m

ik L

G0 (r1 , r2 )в

в =
m

m

(-1)m m m (x1 ) dx1 = (C.69)

(-1) m m (x2 )+
m

+ x

(-1)m m (x2 m (x2 )) , (x1 ) - e
ik L

m

m,0 m m

G0 (r1 , r2 )в m m (x2 ) dx2 =
m

(C.70)

в =
m

i(x1 - x2 ) x 2 2rh

m m (x1 ) - x 43

m (x1 m (x1 )) ,
m

After multiplying equation (C.69) by I
m0 ,0 m0 ,0 m0 ,0

m

0

(x2 ) and integrating we obtain:
m0 ,0

+I

m0 ,0

=

i x rl = 2 m0 ,0 - m0 -1,0 2 2 rh + m0 ,0 - m0 +1,0 m0 + 1 =- x rL
m0 +1

+ Jp0 ,0 , m0

(C.71)
m0 -1

+ ,

m0 +1

J

m0 ,0

m0 + 1 - 2

m0 -1

m0 2

.

After multiplying equation (C.70) by I
m0 ,0 m0 ,0 m0 ,0

m

0

(x1 ) and integrating we obtain: =
m0 ,0

+I

m0 ,0

J

m0 ,0

i x rl = 2 m0 ,0 - m0 -1,0 0 -1 + 2 2 rh + m0 ,0 - m0 +1,0 m0 + 1m0 +1 , - x = m0 +1 m0 + 1 - m0 -1 m0 . 2 rL

+ Jp0 ,0 , m0 m

(C.72)

Here we use the rule: co efficients m = 0 and m = 0 if m < 0. Substituting Im0 ,0 and Jm0 ,0 into (C.71) and substituting Im0 ,0 and Jm0 ,0 into (C.72) we obtain two equations. They may b e transformed from one to other by replacement m m and vice versa m m . So assuming that m = m we can solve only one equation:
m0 ,0 m
0

i x rL + 2в 2 2 rh
m0 ,0

(C.73)

в

-

+ =
m
0

m0 ,0

m0 m0 -1 + - m0 +1,0 m0 + 1m
m0 -1,0 m0 +1

0

+1

x - 2 rL

m0 + 1 -

m0 -1

m

0

,

We assume that 0,0 = 1, = 0,0 + = 1 + , 0 1, 1 x, 2 x2 , . . . . Putting m0 = 0 in (C.73) we obtain x2 . And putting m0 = 1 in (C.73) we find
1

x -0 2r

L

2 1 irL + 2 2rh 1 -

(C.74)
1,0

Using (C.12 C.14) one can rewrite this formula in form (C.20).

44

Table C.1: Numerical values of co efficients F m0 = 0 m=0 m=2 1/ 1/ 2 m=4 m=6 -1/(2 6 ) 1/(4 5 )

m0 ,m

for low indices

Expose perturbation of CR
For this case the equations for eigen mo des calculations are the following in this section we do not mark by distribution function of p erturb ed mo de: e
ik L

G0 (r1 , r2 ) 2 (r2 ) dr

2

(C.75)

~ 1 (r1 ) 1 - ib k |x1 | , e
ik L

G0 (r1 , r2 ) 1 (r2 ) dr2 =

~ 2 (r2 ).

We find the solutions as expansion 1 (x1 , y1 ) = 0 (y1 )
m

(-1)m m m (x1 ) m m (x2 ).

(C.76) (C.77)

2 (x2 , y2 ) = 0 (y2 )
m

Substituting them into equations (C.75, C.76) we obtain
m m,0 m m

(x1 ) =

(C.78)

=
m

m m (x1 ) 1 - ib k |x1 | ,
m m,0

(-1)m m m (x2 ) =

m

(-1)m m m (x2 )

(C.79)

From last equation we obtain

m,0 m and substitute it into (C.78). After multiplying obtained equation by grating over dx1 we get: m =
2 m,0 m
0

m

0

(x1 ) and inte(C.80) (C.81)

= 2 =
-

m

0

- ib 2 k r
m
0

L m

m F

m0 ,m

,

F

m0 ,m

|x| m (x)

(x) dx

We have tabulated co efficients Fm,m0 -- result is presented in Table C.1. Assuming that 0,0 = 1 and 0 1, 1 , 2 , · · · 1 we see that this system can b e divided by two indep endent subsystems: one for o dd indices and another one -- for even

45

x H g replacements E H y E H E H EE H

Figure C.6: Plane wave traveling and reflecting from dielectric corner reflector with angle b etween facets /2. The axis z is directed upward and p erp endicular to plane of figure. Vector E is directed along z -axis. indices. Odd indices can b e put zero and for even indices we have
2

1+

2

4

iL rL , b iL k rL , 2 b 1 - e-8i iL k rL , 2 6 b 1 - e-16i

(C.82) (C.83) (C.84)

We see that all co efficients 2 , 4 , · · · , i.e. they have the same order over . However 5/4 the convergence seems to take place due to decreasing the co efficients Fm0,0 1/m0 with m0 . These expressions can b e rewritten using g -parameter in form (C.23, C.24).

Diffractional losses on edge
Here we write down the calculations to obtain estimate (C.36). We consider the mono chromatic plane wave p olarized along z -axis traveling and reflecting from dielectric CR with angle b etween facets /2 (see fig. C.6). Incident wave: E
z inc

H = kE ,

= E0 exp - i t - ik n sin x - ik n cos y , µ = 1,

Below I drop the multiplier e-it . Condition of internal reflection is fulfilled on b oth facets: n cos > 1, n sin > 1. We assume that magnetic p ermittivity µ = 1 as in 2 [23] and hence n = ( dielectric p ermittivity). Field inside CR. We use Fresnel formulas for light wave reflection from plane b oundary b etween dielectric and vacuum using complex reflection co efficient R [23] (for the case of complete internal reflection): R ( ) = ncos - i ncos + i 46 n2 sin2 - 1 n2 sin2 - 1

z g replacements y

x

2

+

r

Figure C.7: where is incident angle. The sum field after two reflections from b oth facets inside dielectric is the following: Ez = E
0

e

-ikx x-iky y

+ R ()e

+ikx x-iky y

+

(C.85)

+ R ( /2 - )R ()e+ikx x+iky y , k = , kx = k n cos , ky = k n sin (C.86) c The first terms in brackets describ e the incident wave, the second and third terms -- the waves reflected from planes (y = 0) and (x = 0) corresp ondingly, the last term describ es wave double reflected from b oth planes. We simplify formula (C.85) and calculate magnetic field: Ez = 4E0 e tan x = cos(kx x - x ) cos(ky y - y ), n2 sin2 - 1 n2 cos2 - 1 , tan y = n cos n sin
-i(x +y )

+ R ( /2 - )e

-ikx x+iky y

+

(C.87)

(C.88) The field on outside surface of CR Now we can write down the expressions for fields outside CR in planes x = 0 - , y = 0 - using b oundary conditions -- continuity of tangent comp onent of E and normal comp onent µH . Then the expressions for fields are the following:
x=0 z x Hx =0 x Hy =0 y Ez =0 y Hx =0 y Hy =0

E

= 4E0 e

-i(x +y )

= 4i nE0 sin e = 4E0 e
-i(x +y )

-i(x +y )

cos(x ) cos(ky y - y ),
-i(x +y )

= i4nE0 cos e

cos(x ) sin(ky y - y ),

= -4inE0 sin e

cos(kx x - x ) cos(y ),
-i(x +y ) -i(x +y )

sin(x ) cos(ky y - y ), cos(kx x - x ) sin(y ), sin(kx x - x ) cos(y )
2

= -4inE0 cos e

We take in mind that plane wave is limited by Gaussian multiplier b = exp - a2 [x sin - y cos ]2 + z 47

with small parameter a, or more precisely: a

k.

Radiation field. We can apply diffraction Green formula to Ez and calculate it in far wave zone in direction characterized by angles , and distance |r - r | (see fig.C.7): Ez (r ) = G(R) = n G(R) Ez n G(R) - G(R)n E eikR , 4 R R = |r - r| 1 , k
z

dr,

(C.89) (C.90)

n(r - r ) . (C.91) R Here r is radius-vector of observation p oint, r is radius-vector of p oint on surface of integration, dr is element of integration surface, n is normal to surface of integration. The result of calculations is the following: -ik G(R) в Ez (R) = iE0 e e-i(x +y ) в R -k2 sin2 /4a2 e в в Ix + I a
ik r f (k ) (k ) x 0

(C.92)
y

,

Ix = k b

dx cos(kx x - x )e

ik xcos -a2 x2 sin2

e ,

bx n cos sin x - i cos cos n2 cos2 - cos2 Iy = k b
y 0

x

(C.93)
-a2 y 2 cos2

dy cos(ky y - y ) e

ik y sin

e

by n sin sin y - i sin cos n2 sin2 - sin2 Ab ove we used auxiliary formulas: Ic (k ) =
0

y

(C.94)

cos k x e sin k x e

-a2 x

2

dx (k ), dx 1 , k if a

if a k,

k,

(C.95)

Is (k ) =
0

-a2 x

2

Power of diffraction losses W
/2 - /2

W=
- /2

can b e calculated as following 2 E0 c 2 |Ez |2 c 2 R cos d d = в в A, 2 2 3 ka (C.96)

A=
- /2

Ix + Iy )2 d

We have to compare the value W with total p ower W0 of incident wave
2 W 2a E0 c , (1 - R)d = = 2 в A, 2 2a W0 k Replacing notations a 1/ 2 Rb and calculating numerically integral A n = 1.45 (S i O2 ) and = /4 we finally obtain the estimate (C.36).

W0 =

(C.97) 32.8 for

48

TD temperature fluctuations in thermo-isolated layer
We have thermal conductivity equation for temp erature u(r, t) in infinite layer with width l (0 z l) with fluctuating force in right part [9] and the following b oundary conditions: u - a2 u = F (r, t), t u(r , t) = 0, z z =0, l 2 kB T 2 (r - r ) (t - t ), F (r, t)F (r , t ) = - (C )2 where a2 = /C , kB is Boltzmann constant, is Dirac delta function. We find solution as series: u(r, t) =
-

(C.98) (C.99) (C.100)

dkx dky d (2 )3

n

un (kx , ky , )в

в eit-ikx x-iky y cos bn z , Fn (kx , ky , ) , un (kx , ky , ) = 2 i + a2 (b2 + k ) n n 2 2 2 bn = , k = k x + k y , l un (kx , ky , ) =
l

dx dy dt e
0,n

-i t+ikx x+iky y

-

в

в

dz
0

2- l

cos bn z u(x, y , z , t),

We find correlation functions of co efficients Fn (kx , ky , ): F
n,n
1

= Fn (kx , ky , )Fn1 (kx1 , ky1 , 1 ) = 2(2 )3 kB T 2 2 2 - 0,n n,n1 в k + b 2 = n 2 (C ) l в (kx - kx1 ) (ky - ky1 ) ( - 1 ).

2 Ї We are interested in temp erature u(t, x0 , y0 ), averaged over volume V = Rb l along axis parallel to axis z with transversal co ordinates x0 and y0 , and also its correlation

49

function u(t, 0, 0)u(t + ), x0 , y Ї Ї

0

with sp ectral density Su ( ): Ї 1 2 Rb l
l

u(t, x0 , y0 ) = Ї

dz
0 -

-

dx dy в =
(x-x0 )2 +(y -y0 )2 R2 b

в u(r, t)e
l

(x-x0 )2 +(y -y0 )2 R2 b

=
0

dz l

в

-

dx dy e 2 Rb 0 dkx dky d (2 )3
i t-ikx x-iky y

-

в

n

un (kx , ky , )в

в cos bn z e = вe

=

- i t-ikx x0

2 dkx dky d - R2 k b e 4в (2 )3 -iky y0 u0 (kx , ky , ),

(C.101)

Bu ( ) = u(t, 0, 0)u(t + , x0 , y0 ) = Ї Ї Ї 2kB T 2 1 dkx dky d = в 2l (C ) (2 )3 - 2 R2 k k 2 ei b в e- 2 -ikx x0 -iky y0 2 4 4 = + a k 1 kB T 2 e = 2 2 (C ) Rb l(1 + 2a2 /Rb )
i 0 -

2 x2 +y0 0 2(R2 +2a2 ) b

,

(C.102)

Su ( ) = 2 Ї =

d e l

u(t, 0, 0)u(t + , 0, 0) = Ї Ї k dk e 2
-
2 R2 k b 2

- 4kB T 2 (C )2

2 k 2 + a4 k

4

(C.103)

Making following substitutions =
22 k Rb , 2

w=

2 Rb , 2 a2

a2 =

C

(C.104)

one can express the sp ectral density using exp onential integrals: Su ( ) = Ї
d kB T 2 e 2 2 + 2 C l a 0 w kB T 2 = в 2 C l a2 -iw -

=

в eiw Ei1 (iw ) + e Ein (x) =
1

e

-x t

dt tn

Ei1 (-iw ) ,

50

For particular cases this formula can b e simplified: Su ( )| Ї Su ( )| Ї
w 1

w

1

4kB T 2 1 , 4 (C )2 l Rb 2 2 4kB T 2 rT в 2. 2 C Rb l Rb

(C.105) (C.106)

The formulas (C.105 and C.106) refer to non-adiabatic and adiabatic cases corresp ondingly.

Parameters
For our estimates we used the following parameters, material parameters corresp ond to fused silica. = 2 в 100 s-1 , = 1.064 µm, b = 2.3 cm, Rb 6 cm m = 4 в 104 g, L = 4 в 105 cm; T = 300 K,

erg , = 5.5 в 10-7 K-1 , = 1.4 в 105 cm s K g erg = 2.2 , C = 6.7 в 106 , 3 cm gK dn = 1.5 · 10-5 K-1 n = 1.45, = dT

51

References
[1] J. Ye, D. E. Verno oy and H. J. Kimble, "Trapping of single atoms in cavity QED", quant-ph/9908007. [2] H. J. Kimble, private communication. [3] G. Remp e et al, Opt.Letters, 17, 363 (1992). [4] A. Abramovici et al, Science 256, 326 (1992). [5] A. Abramovici et al, Phys.Letters. A218, 157 (1996). [6] V. B. Braginsky and F. Ya. Khalili, Quantum Measurement, ed. by K.S. Thorne, Cambridge Univ. Press, 1992. [7] V. B. Braginsky and F. Ya. Khalili, Rev. Mod. Physics, 68, 1 (1996). [8] F. Ya. Khalili, Physics Letters A317, 169 (2003) [9] V. B. Braginsky, M. L. Goro detsky, and S. P. Vyatchanin, Physics Letters A 264, 1 (1999); cond-mat/9912139; [10] V. B. Braginsky and S. P. Vyatchanin, Physics Letters A312, 244 (2003); arXiv: cond-mat/0302617, [11] G. Cagnoli, D. Cro oks, M. M. Fejer, Gregg Harry, Jim Hough, Norio Nakagawa, Steve Penn, Roger Route, Sheila Rowan, P. Sneddon, LIGO do cument: G030195-00, (2003) [12] M. M. Fejer, S.Rowan, D. Cro oks, P. Sheddon, G. Harry, J. Hough, S. Penn, to b e published in Phys.Rev.D. [13] V.B.Braginsky, A.A. Samoilenko, Physics letters A315, 175 (2003), arXiv:grqc/0304100v1. [14] V. B. Braginsky, M. L. Goro detsky, F. Ya. Khalili and K. S. Thorne, Report at Third Amaldi Conference, Caltech, July, 1999. [15] Benvenuto Cellini, Autobiography, Penguin Bo oks Ltd, 1956. [16] Izwin I. Shapiro et al, Phys. Rev. Lett. 36, 555 (1976); J. G. Williams et al, Phys. Rev. Lett. 36, 551 (1976). [17] V. B. Braginsky, M. L. Goro detsky, and S. P. Vyatchanin, Physics Letters A 271, 303-307 (2000) 52

[18] A. E. Siegman, Lasers, Univ. Science Bo ok, 1996, ch. 19 [19] LIGO-I I conceptual pro ject b o ok. LIGO do cument M990288-A1, available on www.ligo.caltech edu. [20] E. d'Ambrosio, R. O'Shaughnessy, S. Strigin, K.Thorne and S. Vyatchanin, Reducing Thermo elastic Noise in Gravitational-Wave Interferometers by Flattening the Light Beams, submitted to Phys. Rev. D, available as file b eamreshap e020903.p df at http://www.cco.caltech.edu/kip/ftp/ [21] O'Shaughnessy, S. Strigin and S. Vyatchanin, The implifications of Mexican-hat mirrors: calculations of thetmo elastic noise and interferometer sensitivity to p erturbation for Mexican-hat mirror prop osal for advanced LIGO, submitted to Phys. Rev. D. [22] Garrilynn Billingsley, private communication. [23] L. D. Landau and E. M. Lifshitz, Electro dinamics of continious media, Moscow, 1982. sec. 86. [24] D.R.Cro oks, P.Sneddon, G.Gagnoli, J.Hough, S.Rowan, M.M.Fejer, E.Gustavson, R.Route, N.Nakagawa, G.M.Harry, A.M.Gretarsson, to b e published in Classical and Quantum Gravity. [25] D.R.Cro oks et al, Classic and Quantum Gravity, 19, 4229, (2002). [26] G. Harry et al, et al, Classic and Quantum Gravity, 19, 897, (2002). [27] S. Penn et al, Classic and Quantum Gravity, 20, 2917, (2003). [28] N. Nakagawa et al, Phys. Rev. D, 65, 102001, (2002). [29] Numata eat al Phys.Rev.Lett. 91, 260602 (2202). [30] E. Black et al, submitted to Phys.Rev.Lett.

53

Appendix D Reducing the mirrors coating noise in laser gravitational-wave antennae by means of double mirrors
Recent researches show that the fluctuations of the dielectric mirrors coating thickness can intro duce a substantial part of the future laser gravitational-wave antennae total noise budget. These fluctuations are esp ecially large in the high-reflectivity end mirrors of the Fabry-Perot cavities which are b eing used in the laser gravitational-wave antennae. We show here that the influence of these fluctuations can b e substantially decreased by using additional short Fabry-Perot cavities, tuned in anti-resonance instead of the end mirrors.

Introduction
One of the basis comp onents of laser gravitational-wave antennae [1, 2, 3] are highreflectivity mirrors with multilayer dielectric coating. Recent researches [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] have shown that fluctuations of the coating thickness pro duced by, in particular, Brownian and thermo elastic noise in a coating, can intro duce substantial part of the total noise budget of the future laser gravitational-wave antennae. For example, estimates, done in [9] show that the thermo elastic noise value can b e close to the Standard Quantum Limit (SQL) [14] which corresp onds to the sensitivity level of the Advanced LIGO pro ject [3] or even can exceed it in some frequency range. For this reason it was prop osed in [15] to replace end mirrors by coatingless corner reflectors. It was shown in this article that by using these reflectors, it is p ossible, in principle, to obtain sensitivity much b etter than the SQL. However, the corner reflectors require substantial redesign of the gravitational-wave antennae core optics and susp ension system. At the same time, the value of the mirror surface fluctuations dep ends on the numb er of dielectric layers which form the coating. It can b e explained in the following way. The most of the light is reflected from the first couple of the layers. At the same time, fluctuations of the mirror surface are created by the thickness fluctuations of all underlying layers, and the larger is the layers numb er, the larger is the surface noise. Therefore, the surface fluctuations are relatively small for the input mirrors (ITM) of the Fabry-Perot cavities of the laser gravitational-wave antennae with only a few coating layers and 1 - R 10-2 (R is the mirror p ower reflectivity), and is considerably larger 54

PSfrag replacements

ITM

IETM

EETM

L = 4 Km

l

10 m

Figure D.1: Schematic layout of a Fabry-Perot cavity with double mirror system instead of the end mirror: ITM and IETM are similar mo derate reflective mirrors; EETM is a high-reflective one. ETM

PSfrag replacements

Figure D.2: The double reflector based on a single mirror. for the end mirrors (ETM) with coating layers numb er 40 and 1 - R 10-5 . In this pap er another, less radical way of reducing the coating noise, exploiting this feature, is prop osed. It is based on the use of an additional short Fabry-Perot cavity instead of the end mirror (see Fig. D.1). It should b e tuned in anti-resonance, i.e its optical length l should b e close to l = (N + 1/4), where is a wavelength. The back side of the first mirror have to have a few layers of an antireflection coating. It can b e shown that in this case reflectivity of this cavity will b e defined by the following equation: (1 - R1 )(1 - R2 ) , (D.1) 4 where R1,2 are the reflectivities of the first (EETM on Fig. D.1) and the second (IETM) mirrors. Phase shift in the reflected b eam pro duced by small variations y in p osition of the second mirror reflecting surface relative to the first one will b e equal to 1-R 1 - R1 в 2k y , (D.2) 4 where k = 2 / is a wave numb er. It is supp osed for simplicity that there is no absorption in the first mirror material; more general formulae are presented b elow. It follows from these formulae that the first mirror can have a mo derate value of reflectivity and, therefore, a small numb er of coating layers. In particular, it can b e identical to the input mirror of the main Fabry-Perot cavity (ITM). At the same time, influence of the coating noise of the second (very-high-reflective) mirror will b e suppressed by a factor of (1 - R1 )/4, which can b e as small as 10-2 В 10-3 . 55

PSfrag replacements IETM a b
1

EETM

1

a b

a b

0

a b

2

0

2

Figure D.3: The double mirror reflector. In principle, another design of the double reflector is p ossible, which consists of one mirror only, see Fig. D.2. Both surfaces of this mirror have to have reflective coatings: the thin one on the face side and the thick one on the back side. In this case the additional Fabry-Perot cavity is created inside this mirror. However, in this case thermo elastic fluctuations of the the back surface coating will b end the mirror and thus will create unacceptable large mechanical fluctuations of the face surface. Estimates show that using this design, it p ossible to reduce the face surface fluctuations by factor 3 only [16]. So the design with two mechanical ly isolated reflectors only will b e considered here. In the next section more detail analysis of this system is presented.

Analysis of the double-mirror reflector
The rightmost part of Fig. D.1 is presented in Fig. D.3, where the following notation is used: a, b are the amplitudes of the incident and reflected waves for the first mirror, resp ectively; a0 , b0 are the amplitudes of the waves traveling in the left and right directions, resp ectively, just b ehind the first mirror coating; a1 , b1 are the same for the waves just b ehind the first mirror itself; a2 , b2 are the amplitudes of the incident and reflected waves for the second mirror, resp ectively. These amplitudes satisfy the following equations: a0 a1 a2 b b0 b1 b2 where: 56 = = = = = = = -R1 T0 a0 a1 , -R1 T 0 b1 b2 , -R2 b0 + iT1 a , + A 1 na , a + iT1 b0 , + A 0 nb , a 2 + A 2 n2 , (D.3a) (D.3b) (D.3c) (D.3d) (D.3e) (D.3f ) (D.3g)

na , nb , n2 are indep endent zero-p oint oscillations generated in the first (na , nb ) and the second (n2 ) mirrors; = eikl1 , where l1 is the distance b etween the first mirror back surface and the second mirror; -R1 and iT1 are the amplitude reflectivity and transmittance of the first mirror coat2 2 ing, resp ectively, R1 + T1 = 1; T0 and A0 are the amplitude transmittance and absorption of the first mirror bulk, resp ectively, |T0 |2 + A2 = 1; 0 -R2 and A2 are the amplitude reflectivity and absorption of the second mirror, re2 sp ectively, R2 + A2 = 1. 2 R1 , T1 , A0 , R2 , A2 are real values; T0 is a complex one, its argument corresp onds to the phase shift in the first mirror bulk. Here we do not consider absorption in the first mirror coating for two reasons: (i) it is relatively small and (ii) it exists b oth in traditional one-mirror reflectors and in the one considered here, and the main goal of this short article is to emphasize the differences b etween these two typ es of reflectors. It follows from equations (D.3) that the reflected b eam amplitude is equal to b=
2 (R2 T0 2 - R1 )a - iR2 A0 T0 T1 2 na + iA2 T0 T1 n2 + iA0 T1 nb . 2 1 - R 1 R 2 T0 2

(D.4)

This solution can b e presented in the following form: ~ b = Ra + An , where

(D.5)

is the equivalent complex reflection factor for the scheme considered, A= is its equivalent absorption factor, and n=
2 T1 1 - R2 |T0 |4 2 |1 - R1 R2 T0 2 |

2 R 2 T0 2 - R 1 ~ R= 2 1 - R 1 R 2 T0 2

(D.6)

(D.7)

is the sum noise normalized as zero-p oint fluctuations. As mentioned ab ove, this system should b e tuned in anti-resonance: l where N is an integer and y 1 arg T0 = k k . In this case
ik y

1 -iR2 A0 T0 T1 2 na + iA2 T0 T1 n2 + iA0 T1 n 2 A 1 - R 1 R 2 T0 2

b

(D.8)

N+

1 2

+y,

(D.9)

and

T0 = i(-1)N |T0 |e

i(-1)N (1 + ik y ) ,

(D.10)

~ R -Rei , 57

(D.11)

where R =1- and (1 - R1 )(1 - R2 |T0 |2 ) , 1 + R1 R2 |T0 |2 (D.12)

is the phase shift pro duced by the deviation y in the distance l. Supp ose that factors T1 , A0 , A2 are small. In this case 1-R (1 - R1 )(1 - R2 + A2 ) 0 , 2 (1 - R1 )k y . (D.14) (D.15)

2 2R2 |T0 |2 T1 ky (R2 |T0 |2 + R1 )(1 + R1 R2 |T0 |2 )

(D.13)

Using p ower reflection and absorption factors instead of the amplitude ones: R = R2 , 2 R1,2 = R1,2 , A0 = A2 , 0 equations (D.14), (D.15) can b e rewritten as follows: 1-R (1 - R1 )(1 - R2 + 2A0 ) , 4 1 - R1 ky . 2 (D.19) (D.20) (D.16) (D.17) (D.18)

Conclusion
The main goal of this short article is just to claim the idea, so the detailed design of the additional cavity is not presented here. However, the following imp ortant topics have to b e discussed in brief. The first one concerns the optimal value of the IETM mirror reflectivity. The smaller is 1 - R1 , the larger is suppression factor for the EETM mirror surface noises; at the same time, the larger is the IETM mirror coating noise. The rigorous optimization requires exact knowledge of the coating noise dep endence on the coating layers numb er. A crude estimate based in the exp onential dep endence of the IETM mirror transmittance T1 1 - R1 on the coating layers numb er gives that the optimal transmittance value is relatively large, T1 10-1 . On the other hand, smaller values of the IETM mirror transmittance, down to the input (ITM) mirror transmittance TITM are also acceptable. Therefore, identical ITM and IETM mirrors can b e used. In the Advanced LIGO interferometer, the input mirrors transmittance will b e equal to TITM 5 в 10-3 , and its bulk absorption will b e equal to AITM 10-5 [17]. Using such mirror as an IETM mirror in the scheme prop osed in this article, and mirror with commercially available value of 1 - R2 10-5 as an EETM mirror, it is p ossible to create 58

a double-mirror reflector with 1 - R < 10-6 and suppression factor for the EETM surface 1 - R1 fluctuations 10-3 . 4 The second issue concerns the optical p ower circulating through the IETM mirror. It is 4 easy to show using equations (D.3), that it is 103 times smaller that the p ower 1 - R1 circulating in the main cavities. In the Advanced LIGO top ology, it will b e approximately equal to the p ower circulating through the ITM mirrors and the b eamsplitter (ab out 1 KW). It is necessary to note also that y in the calculations presented ab ove includes not only coating noise of the EETM mirror but all p ossible kinds of its surface fluctuations, including ones caused by Brownian and thermo elastic fluctuations in this mirror bulk, Brownian fluctuation in its susp ension, seismic noise as well as the mirror quantum fluctuations. This feature simplifies greatly the EETM mirror design b ecause the requirements for all these noise sources can b e reduced by a factor of (1 - R1 )/4. In particular, the SQL value m
2

for this mirror (m is its mass and is the 1-R 4
1 -1

observation frequency) can b e larger by a factor of b e, in principle,
-2

. Therefore, its mass can

1 - R1 106 times smaller than for the main (ITM and IETM) 4 mirrors. Of course, such a small mirror hardly can b e used in the real interferometer. This estimates shows only that the quantum noise do es not imp ose any practical limitation on the EETM mirror mass.

59

References
[1] A.Abramovici et al, Science 256, 325 (1992). [2] A.Abramovici et al, Physics Letters A 218, 157 (1996). [3] E.Gustafson, D.Sho emaker, K.A.Strain and R.Weiss, LSC White pap er on detector research and development, 1999, LIGO Do cument T990080-00-D (www.ligo.caltech.edu/do cs/T/T990080-00.p df ). [4] Yu.Levin, Physical Review D 57, 659 (1998). [5] D.Cro oks et al, Classical and Quantum Gravity 19, 883 (2002). [6] G.M.Harry et al, Classical and Quantum Gravity 19, 897 (2002). [7] N.Nakagawa, A.M.Gretarsson, E.K.Gustafson and M.M.Fejer, Physical Review D 65, 102001 (2002). [8] S.D.Penn et al, Classical and Quantum Gravity 20, 2917 (2003). [9] V.B.Braginsky, S.P.Vyatchanin, Physics Letters A 312, 169 (2003). [10] V.B.Braginsky, A.A. Samoilenko, Physics Letters A 315, 175 (2003). [11] G.Cagnoli et al, LIGO Do cument G0301195-00 (2003). [12] M.M.Fejer et al, arXiv:gr-qc/0402034 (2004). [13] G.M.Harry et al, LIGO Do cument P040023-00 (2004). [14] V.B.Braginsky et al, Physical Review D 67, 082001 (2003). [15] V.B.Braginsky, S.P.Vyatchanin, arXive:cond-mat/0402650 (2004). [16] S.P.Vyatchanin, private communication. [17] www.ligo.caltech.edu/AdvLIGO

60