Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hbar.phys.msu.ru/articles/ligorep03.pdf
Дата изменения: Fri Dec 17 00:00:00 2004
Дата индексирования: Mon Oct 1 20:08:59 2012
Кодировка:

THE ANNUAL REPORT OF THE MSU GROUP (Jan.-Dec. 2003)
Contributors: V.B.Braginsky (P.I.), I.A.Bilenko,.Ya.Khalili, V.P.Mitrofanov, 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)

Contents
Summary 2 1 Exp erimental researches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 Investigation of effects asso ciated with variation of electric charge on a fused silica test mass . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Measurement of thermal and excess noise in high-Q mo des of all fused silica susp ension . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Measurements of the Q factor of all fused silica p endulum with thin susp ending fib ers and small masses. . . . . . . . . . . . . . . . . . . 4 2 Theoretical researches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Thermo elastic noise in the mirror coating . . . . . . . . . . . . . . . 4 2.2 Low pumping energy mo de of the "optical bars"/ "optical lever" top ologies of gravitational-wave antennae . . . . . . . . . . . . . . . 5 2.3 Sensitivity limitations in optical sp eed meter top ology of gravitationalwave antennae. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Collab oration b etween MSU group and LIGO Lab, LSC and K.S.Thorne's group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

App endixes

9

A Investigation of effects asso ciated with variation of electric charge on a fused silica test mass 10 B Mechanical losses in thin fused silica fib ers C Thermo dynamical fluctuations in optical mirror coatings D Measurements of the optical mirror coating prop erties 20 27 48

E Low pumping energy mo de of the "optical bars"/"optical lever" top ologies of gravitational-wave antennae 52 F Sensitivity limitations in optical sp eed meter top ology of gravitationalwave antennae. 70

1

Summary
1
1.1

Experimental researches
Investigation of effects asso ciated with variation of electric charge on a fused silica test mass

V.P.Mitrofanov, K.V.Tokmakov and p ostgraduate student L.G.Prokhorov in collab oration with Phil Willems (Caltech) continued long-term measurements of variations of electric charge lo cated on the fused silica test mass prototyp e. The mirrors can trap and store electric charges, for example, due to adsorption-desorption pro cesses that o ccur on their surfaces, due to background radiation and cosmic rays, and other pro cesses. Variation of electric charge can pro duce additional fluctuating force, which acts on the mirror due to electrostatic interaction with surfaces of surrounding b o dies of susp ension and control structures. In this way variation of electric charge could b e a p otential source of excess noise. A multistrip capacitive prob e connected with high imp edance electrometric amplifier was used to monitor the electrostatic charge lo cated on the end face of fused silica cylinder susp ended as a bifilar torsion p endulum. The high relaxation time of this p endulum (2.2 в 107 s) allowed the long-term measurement of electric charge after initial excitation of the oscillation amplitude. The amplitude of the cylinder torsion oscillation was monitored by means of the optical sensor. The cosmic ray detectors were installed around the vacuum chamb er to search for coincidences with changes of the signal from the prob e. They consisted of 11 plastic scintillator paddles (180 в 18 в 0.8 cm3 ) supplied by photomultipliers, readout and data pro cessing electronics. The cosmic ray detectors allowed to select cosmic ray shower events. Eight new runs of measurements (each run lasted of ab out one month) have b een carried out this year. Two typ es of b ehavior of the system "fused silica p endulum nearby electro des" were found. The first b ehavior was characterized by slow decrease of the p endulum amplitude corresp onding to Q = (6 В 8) в 107 and slow change of the averaged voltage from the prob e corresp onding to the charging of the test mass with a rate of the order of 104 e/cm2 day. The second b ehavior was characterized by the presence of jumps of the amplitude and prob e voltage in which these values changed several orders faster than in the first case. The jump represented strong change of the prob e voltage up to 100 times. Usually it consisted of several step changes during several hours. The jumps of the prob e signal were synchronous with the jumps of the p endulum oscillation amplitude. Decrease of the amplitude in the pro cess of jump corresp onded to the damping value Q-1 of order of 10-4 . This value is almost four orders larger than for the usual damping of this p endulum. The state of the system changes after the first jump. The rep eated increase of the p endulum amplitude by means of the electrostatic excitation resulted in a new cascade 2

of jumps of the p endulum amplitude synchronous with jumps of the prob e voltage. This "sp oiled" state of the system transferred to the original state with relaxation time of the order of one month. The "sp oiled" state was observed very rarely (in 4 runs during two years of observation). We have not found statistically significant coincidences b etween large signals from the cosmic ray detectors and jumps of the amplitude and the prob e voltage in the "sp oiled" state of the system. But we can not exclude the coincidence b etween the first jump in the "original" state and high intensity signal from the cosmic ray detector caused by cosmic ray showers. We also can not exclude the coincidence b etween jumps and signals from the cosmic ray detector caused by low energy particles passing lo cally through the surface layer of the test mass near the capacitive prob e. We did not yet know conditions of formation of the "sp oiled" state. The observed effects as well as their p ossible role in the generation of excess noise in gravitational wave detectors are worth further in-depth investigation. See details in App endix A and in [1].

1.2

Measurement of thermal and excess noise in high-Q mo des of all fused silica susp ension

Different kinds of random mechanical oscillations in violin mo des of the susp ension may give substantial contribution to the total noise budget b ecause these oscillations act directly on the test masses. In previous grants this kind of noise in steel wires (used in initial LIGO) was registered and analyzed in-depth. Those measurements have shown the existence of the ordinary Brownian comp onent and also of the excess noise additional to the Brownian one. The last one dep ends sharply on the stress in wires. The violin Q's in those exp eriments were of the order of 5 в 104 , the sensitivit of optical readout meter y (based on Michelson interferometer scheme) was 2 в 10-11 cm/ Hz near the frequency of observation 2 KHz. During the first part of the year 2003 I.A.Bilenko and his p ostgraduate student N.Yu.Markova have completed several improvements of the Fabry-Perot cavity based installation, prepared for the measurement of thermal and excess noise in high-Q mo des of all fused silica susp ension prototyp es. The anti-seismic filter now includes multispring stage for vertical degree of freedom (eigen frequency is ab out 2 Hz) and vibration damp er used samarium-cobalt magnets. New ultra low noise preamplifier was placed inside the vacuum chamb er, nearby the photo detector. Some mo difications of the mechanical parts were made in order to reduce the stress in junctions. As a result, the sensitivity has reached the value of 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 the requirements declared in the grant prop osal. In order to check the measurement system for the presence of internal excess noise sources series of test records have b een done. For these tests sample fib er with a small mirror was replaced by a large fixed mirror. The analysis of the obtained data shows the absence of any significant excess noise. In April 2003 the recording of mechanical noise in the high quality mo des of the fused silica fib ers has b een started. To date 15 different samples have b een tested, ab out 65 hours of records have b een obtained. Applied strain varied from 3% to 15% of breaking p oint. Signal-to-noise ratio in the 10 Hz bandwidth (near the mo de frequency) varied from 10 to 100 dep ending on the applied stress. Measured quality factors varied from 5 в 106 to 2 в 107 . After the application of vetoing pro cedure based on the seismic and 3

other control channels information, the distribution of the amplitude variation app eared to b e pure exp onential. All data in these measurements were sampled with the rate of = 0.2 s, that allows to resolve the amplitude variations A A в 2 / = 2 в 10-2 A (here A is a mean value of amplitude of thermal fluctuations). Value of the variation shows that only Brownian noise was observed. During the second part of the year 2003 the data abscission part of the installation has b een mo dified in order to sample data with rate = 0.05 s. It may allow to resolve weaker excess noise. We also added the second feedback lo op to the interferometer stabilization scheme in order to suppress a 2 Hz p eak originated from the internal mo de of the antiseismic filter. Mechanical noise may b e much stronger during the very first time interval after the load is applied to the fib er, but we were able to start records at least 20-30 hours later, when the chamb er had b een closed and pump ed out. Now, an electrically driven manipulator which can, one time only, add a makeweight in-vitro has b een designed and tested. The noise records now restarted using renovated facilities. Preliminary data obtained at the stress 10% of breaking p oint confirmed the results mentioned ab ove.

1.3

Measurements of the Q factor of all fused silica p endulum with thin susp ending fib ers and small masses.

I.A.Bilenko and his p ostgraduate student S.L.Lurie have carried out a set of measurements of the Q factors of p endulum and violin mo des of the small masses (from 1 to 520 mg ) with fib ers with diameters from 1.5 to 40 µm. These tests were p erformed for the planned table top scale QND exp eriment. The obtained values of effective mechanical loss angle show noticeably sharp er growth with lower diameters than it might b e exp ected while extrap olating known results of research done for thicker fib ers. For the finest fib er 1.5 ± 0.5 µm diameter loaded with 1.01 ± 0.2 mg the p endulum mo de quality factor was equal to (1.9 ± 0.2) · 107. For fib er 2.5 ± 0.5 µm diameter loaded with 245 ± 7 mg the fundamental violin mo de quality factor was equal to (9.2±0.9)·106 . These values are sufficient for the planned QND exp eriments. Now the measurements of Q factor for the torsion oscillation of the p endulums are p erformed. See details in App endix B and in [2].

2
2.1

Theoretical researches
Thermo elastic noise in the mirror coating

During the previous grant memb ers of MSU group V.B.Braginsky, M.L.Goro detsky and S.P.Vyatchanin have carried out an in-depth analysis of the role of thermo elastic and thermorefractive noises in the mirrors. The practical consequence of this analysis was the reconsideration of the appropriate value of the laser sp ot diameter at the mirror: instead of 3 cm the diameter of the sp ot will b e 10 В 15 cm. In this analysis the thermo elastic noise in the mirror coating was ignored. During the last year of the current grant V.B.Braginsky and S.P.Vyatchanin in collab oration with M.Fejer from MIT have analyzed the contribution of the thermo elastic noise in the coating into the total noise budget. This analysis has shown that the numerical value of the thermal expansion co efficient of Ta2 O5 (material used in the coating) is playing a crucial role. This circumstance has stimulated an exp eriment p erformed by V.B.Braginsky and A.A.Samoilenko. The value of thermal expansion co efficient Ta2 O5 4

obtained and theoretical analysis p ermit to predict that thermo elastic noise in the coatings used to day will limit the sensitivity of Advanced LIGO (in units of metric variation) b etween 0.6 в 10-24 Hz-1/2 and 1.4 в 10-24 Hz-1/2 . These values indicate that taking into account only this typ e of noise the SQL may b e reached in Advanced LIGO. At the same time higher sensitivity (b etter than SQL) may b e obtained only with substantial changes in the coating technology. At present time V.B.Braginsky and S.P.Vyatchanin analyze the alternative p ossibilities which could help to decrease the sensitivity restriction pro duced by thermo elastic noise in mirror coating. See details in App endixes C,D and in [3],[4].

2.2

Low pumping energy mo de of the "optical bars"/ "optical lever" top ologies of gravitational-wave antennae

The "optical bars"/"optical lever" intracavity top ologies of gravitational-wave antennae allow to obtain sensitivity b etter that the Standard Quantum Limit while keeping the optical pumping energy in the antenna relatively low. In particular, they allow to overcome the Energetic Quantum Limit E= E
SQL 2 2 2

.

Here E is the minimal optical energy which have to b e stored in the interferometer arms in order to obtain given sensitivity, is the ratio of the signal amplitude which can b e detected to the amplitude corresp onding to the Standard Quantum Limit (the smaller is the b etter is the sensitivity), and is the squeezing factor ( = 1 for the optical field coherent quantum state and < 1 for the squeezed state), ESQL is the optimal energy for the SQL-limited interferometric antenna. In the case of the Advanced LIGO values of the antenna parameters ESQL 40 J. Corresp onding circulating optical pumping p ower is equal to WSQL 0.75 MWt. In the intracavity top ologies the necessary non-classical quantum state of the optical field [factor ] is generated automatically and therefore the pumping energy do es not dep end directly on the required sensitivity. In the "optical bars"/"optical lever" schemes the optical field acts as two rigid springs lo cated b etween the antenna end mirrors and additional lo cal mirror placed b etween them. If optical energy exceeds some threshold value of Ethres then these springs are rigid enough to provide the signal displacement of the lo cal mirror equal to the displacement of the end mirrors. In this case the sensitivity do es not dep end on the optical energy and is defined by the sensitivity of the lo cal meter which monitors the lo cal mirror p osition. However, in case of a simple lo cal meter with non-correlated measurement noise and back-action noise the threshold energy is rather large and close to ESQL . At the same time if a lo cal meter with cross-correlated noise is used then the threshold energy can b e substantially lower than ESQL . F.Ya.Khalili analyzed this cross-correlation regime in detail. It was shown that if SQL-limited lo cal meter is used then sensitivity of the antenna will b e limited by the Standard Quantum Limit for heavy end mirrors. Pumping energy in this case has to b e several times smaller than ESQL . Typical energy (for the Advanced LIGO values of parameters) can b e equal to E 0.25ESQL . The lo cal mirror mass in this case has to b e equal to m 0.25g. Substantially b etter results can b e obtained if QND lo cal meter, for example Strob oscopicVariation meter is used. In this case the pumping energy is limited by internal losses in the optical cavities only, and sensitivity can b e several times b etter than the SQL. 5

Estimates show that for values of parameters available for contemp orary and planned gravitational-wave antennae, sensitivity ab out one order of magnitude b etter than the Standard Quantum Limit can b e obtained using the pumping energy ab out one order of magnitude smaller than energy required in traditional top ology in order to obtain the Standard Quantum Limit level of sensitivity. Values of the lo cal mirror mass in this case can vary in wide range ( 10g В 10Kg) and should b e chosen taking technological reasons into consideration. See details in App endix E and in [5].

2.3

Sensitivity limitations in optical sp eed meter top ology of gravitational-wave antennae.

Sensitivity of the third-generation interferometric gravitational wave detectors is planned to b e higher than Standard Quantum Limit (SQL). To overcome the SQL one needs to monitor not the co ordinate of the detector prob e b o dy as in contemp orary detectors but the observable that is not p erturb ed by the measurement, i. e. QND observable. It had b een shown that this can b e p erformed using measurement of velo city of a test mass instead of its p osition. The device that measures ob ject velo city is agreed to b e called quantum speed meter. Later several p ossible implementations of this idea were prop osed and analyzed. Post-graduate student S.L.Danilishin analyzed the sp eed meter scheme with resp ect to optical losses in interferometer elements. Several different regimes, b oth wide-band and narrow-band ones, are considered. It was shown, that in the narrow-band case sensitivity ab out ten times b etter than the SQL can b e obtained. Ab out the same sensitivity can b e obtained in wide-band regime, if frequency-dep endent phase of the lo cal oscillator is used. Using the frequency-indep endent phase of the lo cal oscillator, sensitivity gain of ab out 2 В 3 can b e achieved in wide band. In addition, a p ossible role of the signal recycling mirror (SRM) which is prop osed to use in the Advanced LIGO pro ject was analyzed. It is well known that in the traditional top ologies (based on the p osition measurement) sensitivity dep ends up on the pro duct of transmittances of SRM and ITM (input mirrors of Fabry-Perot cavities). It is shown that for the sp eedmeter top ologies there is no such a symmetry. For given value of the optical p ower circulating in the interferometer arms, the b est sensitivity can b e obtained without SRM at all. See details in App endix F and in [7]

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, and have given seminars ab out recent MSU researches: V.B.Braginsky (twice p er year), V.P.Mitrofanov (once p er year), S.P.Vyatchanin (once p er year), F.Ya.Khalili (once p er year). Those Caltech visits have b een timed to coincide with LSC meetings, and the MSU visitors have attended those LSC meetings where they have discussed ongoing and completed research with colleagues from other groups, and gave presentations in LSC working group sessions. The total numb er of seminars by V.B.Braginsky, V.P.Mitrofanov, S.P.Vyatchanin, and F.Ya.Khalili is 4 at Caltech and is 5 at LSC meetings. 6

V.P.Mitrofanov and K.V.Tokmakov have visited University of Glasgow and had fruitful discussions with the memb ers of Glasgow group ab out various problems of the mirrors fused silica susp ensions and mechanisms of the energy dissipation. F.Ya Khalili, S.P.Vyatchanin and p ostgraduate student S.L.Danilishin have useful three days discussions ab out p ossible designs of QND gravitational-wave antennae with memb ers of GEO-600 group. F.Ya Khalili and S.P.Vyatchanin gave 2 talks each at GEO600 group seminar. S.L.Danilishin have attended LSC meeting in Hannover and gave presentation at LSC meeting and 2 talks at GEO-600 group seminar. The memb ers of MSU group in the collab oration with the memb er of LIGO Lab P.Willems (Caltech) continued long-term measurements of variations of electric charge lo cated on fused silica test mass prototyp e and search for correlations b etween these variations and cosmic ray shower events. The joint article is in press in Class. Quantum Grav. The memb ers of MSU group in the collab oration with M.Fejer from MIT derived the calculations of thermo elastic noise in mirror coating. Namely M.Fejer had p ointed out the error in S.P.Vyatchanin's calculations for thermo elastic noise in monolayer on substrate, later S.P.Vyatchanin had p ointed out the error in M.Fejer's calculations for thermo elastic noise in multilayer coating on substrate. This mutual checking was very fruitful in spite of noticed errors were the minor ones. V.B.Braginsky, M.L.Goro detsky, F.Ya.Khalili, A.B.Matsko, K.S.Thorne and S.P.Vyatchanin published article [6], where they have shown that photon shot noise and radiationpressure back-action noise are the sole forms of quantum noise in interferometric gravitational wave detectors that op erate near or b elow the standard quantum limit, if one filters the interferometer output appropriately. No additional noise arises from the test masses' initial quantum state or from reduction of the test-mass state due to measurement of the interferometer output or from the uncertainty principle asso ciated with the test-mass state.

7

Bibliography
[1] V.P.Mitrofanov, L.G.Prokhorov, K.V.Tokmakov and P.Willems, Class. Quantum Grav., in press. [2] I.A.Bilenko, V.B.Braginsky, S.L.Lourie, Class. Quantum Grav., in press. [3] V.B.Braginsky, S.P.Vyatchanin, mat/0302617 (2003). [4] V.B.Braginsky, A.A. Samoilenko, qc/0304100v1 (2003). Physics Letters A 312, 244, 2003; arXiv:condPhysics Letters A 315, 175, 2003; arXiv:grK.S.Thorne and

[5] V.B.Braginsky, M.L.Goro detsky, F.Ya.Khalili, A.B.Matsko, S.P.Vyatchanin, Physical Review D 67, 082001 (2003). [6] F.Ya.Khalili, Physics Letters A 317, 169 (2003). [7] S. L. Danilishin, arXiv:gr-qc/0312016 (2003).

8

Appendixes

9

Appendix A Investigation of effects associated with variation of electric charge on a fused silica test mass
Variation of electric charge on dielectric mirrors can pro duce excess noise in interferometric gravitational wave detectors. Charge lo cated on the end face of a fused silica cylinder susp ended as a torsion p endulum was measured using a multistrip capacitive prob e. Two typ es of b ehavior of this system were found in long-term measurements of the free decay p endulum amplitude and voltage from the prob e. The first b ehavior was characterized by the slow decrease of the p endulum amplitude corresp onding to Q = (6 - 8) в 10 7 and the slow change of the averaged voltage from the prob e corresp onding to the charging of the test mass with a rate of order of 104 e/cm2 day . The second b ehavior is characterized by the presence of jumps of the amplitude and the prob e voltage in which these values changed by several orders faster than in the first case. We did not find coincidences b etween these jumps and signals from plastic scintillators installed around the vacuum chamb er to detect cosmic ray showers.

Introduction
Long baseline laser interferometric gravitational wave detectors have b egun taking science data [1, 2, 3, 4]. Simultaneously, large efforts are b eing applied to identify and reduce noises of different origins existing in the various elements of detectors in order to increase the detectors' sensitivity. The fused silica mirrors of the interferometric detectors susp ended in high vacuum are electrically very well isolated from the environment. They can trap and store electric charges, for example, due to adsorption-desorption pro cesses that o ccur on their surface, due to background radiation and cosmic rays and other processes. Charges on the mirror interacting with surfaces of surrounding susp ension and control structures can degrade the different mo des' Q-factors of the susp ended mirror and increase the thermal noise [5, 6, 7]. Also, accidental variation of the electric charge may b e a p ossible source of additional fluctuating forces acting on the mirror. The electrostatic charging of the test mass p oses the most significant disturbance to the LISA space gravitational wave detector as well [8]. In previous work we rep orted ab out observation of rare large jumps of electric charge detected by means of a capacitive prob e installed near a susp ended fused silica test mass [9]. In this work we fo cus on the study of the system b ehavior after the jump.

10

Experimental set-up
The key element of the exp erimental set-up was the high Q bifilar all-fused silica torsion p endulum with the capacitive prob e installed nearby. A 0.5k g p olished fused silica cylinder with a diameter of 6.5cm was susp ended by two fused silica fib ers 25cm in length and 200µm in diameter which were welded to two cones. The cones were attached to the cylinder using the silicate b onding technique [10]. A schematic of the exp erimental arrangement is shown in Fig.1. The top ends of the susp ension fib ers were welded to a fused silica disk that was attached to the vacuum chamb er cover through an indium gasket. The chamb er was evacuated to a pressure of less than 10-7 T or r by means of the turb omolecular pump. We did not use an ion pump b ecause it could cause charging of the susp ended mass by UV radiation [5]. A multistrip capacitive prob e was used to monitor the electrostatic charge lo cated on the end face of the cylinder. Electro des of the prob e were placed on a fused silica plate, 5cm in length, 3cm in width and 2cm in thickness. The plate was mounted parallel to the end face of the susp ended cylinder with a separating gap of ab out 3mm. Two sets of gold strips were sputter-dep osited on the p olished surface of the plate. Each strip had a width of 4mm and was separated from nearby strips by 3mm gaps. One set of strips was grounded. The other set of strips was connected to the high imp edance electrometric amplifier based on the op erational amplifier AD 549. The motion of the charged cylinder induced charge variations on the prob e electro des that pro duced the amplifier input voltage change. We used the free torsion oscillation of the cylinder b ecause the torsion mo de of a bifilar p endulum has a very high quality factor Q due to the p endulum damping dilution factor [11]. The natural frequency of the torsion mo de was found to b e 1.14H z with the Q 8 в 107 . This mo de relaxation time of ab out 2.2 в 107 s allowed us to monitor the charge in the pro cess of the torsion oscillation free decay over long times. We could not find the true value and distribution of the charge lo cated on the end face of the fused silica cylinder and in the bulk close to the end face using this prob e. But we could roughly estimate the value of the charge density given the parameters of the prob e and the amplitude of the cylinder oscillation, assuming the charge distribution on the end face of the fused silica cylinder to b e uniform. The amplitude of the cylinder torsion oscillation was monitored using an optical sensor. It converted the oscillation amplitude into a time interval measured by a counter. An analogous sensor is describ ed in [12]. The cosmic ray detectors were installed around the vacuum chamb er to search for coincidences with changes of the signal from the prob e. They consisted of 11 plastic scintillator paddles (180 в 18 в 0.8cm3 ) monitored by photomultipliers, readout and data pro cessing electronics (see Fig. 2). The cosmic ray detectors allowed us to select cosmic ray shower events.

Result of measurements
Twelve runs of measurements (each run lasted ab out one month) were carried out over 1.5 years. If the initial charge on the cylinder was to o large we reduced it by means of a high-voltage electric discharge in the residual gas. After the electric charge reduction the chamb er was evacuated by the turb omolecular pump to a pressure of less than 10-7 Torr. Before each run we excited torsional oscillations of the p endulum up to the angular amplitude of ab out 0.07r ad by applying dc (300V ) voltage and ac (100V ) excitation voltage at the p endulum resonance frequency to the electro des of the capacitive prob e. In 11

this case the prob e was used as an actuator. After the excitation the prob e was connected to the electrometric amplifier in order to measure the electric charge. The amplifier output ac voltage was prop ortional to the electric charge on the test mass. The output signal was sent to a 100H z data acquisition system. The signal amplitude was averaged over a time of ab out 70 s. The data were stored on the disk. At the op erating range of amplitudes of the cylinder the amplifier output voltage was also approximately prop ortional to the amplitude of the cylinder oscillation that decreased in the pro cess of free decay. This change of the amplitude was taken into account by intro ducing a correction factor into the output signal. The correction factor was prop ortional to the ratio of the current amplitude to the initial amplitude of the cylinder oscillation. In the most of runs a slow change of the voltage from the prob e corresp onding to negative charging with an approximate rate of 104 e/cm2 day (assuming a uniform distribution of charge on the end face of the cylinder) was observed. The resolution in these measurements was limited by fluctuations of the prob e voltage due mainly to p endulum amplitude fluctuations asso ciated with seismic noise. In several runs we observed large jumps of signal from the prob e. We had four such events in 12 runs of measurements. A jump was a strong change of the prob e voltage of up to two orders of magnitude. Usually it consisted of several step changes during several hours. The jumps of the prob e signal were synchronous with jumps of the p endulum oscillation amplitude. The decrease of the amplitude in the pro cess of a jump corresp onded to a damping Q-1 of order of 10-4 . This value is almost four orders larger than the usual damping of this p endulum. Therefore we called "jumps" such changes of the amplitude and the signal from the prob e. The resolution of details of the jumps was determined by the averaging time, which was ab out 70s. These results were describ ed in our previous work [9]. In the last run we investigated the b ehavior of the p endulum after the jump in detail. This run had some features. During more than one month of observation in this run we found no large changes of the p endulum amplitude and the prob e voltage. After that we increased the p endulum oscillation amplitude to 0.07r ad by applying electrostatic excitation at the p endulum frequency. Then DC voltage 1500V was applied over 14 hours. After that the voltage was switched off and we monitored the p endulum amplitude free decay. Five hours later the jump of the p endulum amplitude o ccurred. Unfortunately, the electric charge detector and the cosmic ray detectors were switched off this time. The next day measurement of the prob e voltage have shown the increase of those one of ab out 15 times in comparison with the value measured b efore the jump of the amplitude. Results of the subsequent measurements of the p endulum amplitude and the prob e voltage are presented in Fig. 3. One can see that the jumps of the amplitude and the prob e voltage ceased when the p endulum amplitude decreased to the some value AT . The free decay of the p endulum amplitude approximately corresp onded to the usual p endulum Q-factor to ok place. The rep eated increase of the p endulum amplitude by means of the electrostatic excitation resulted in a new cascade of jumps of the p endulum amplitude synchronous with jumps of the prob e voltage. Note that the amplitude always decreased in the pro cess of the jump. The prob e voltage could b oth increase and decrease in the pro cess of the jump. Such a picture was observed after the subsequent increases of the p endulum amplitude. But the value of the threshold amplitude AT increased with time. Also we registered signals from the cosmic ray detectors in order to search for the coincidence b etween time of the b eginning of the amplitude jump and cosmic ray shower events. A fragment of the record of the p endulum amplitude and the prob e voltage together with the signal from the cosmic ray detector exceeding some threshold is shown 12

in Fig. 4. We have not found statistically significant coincidences b etween the large signals from the cosmic ray detectors and jumps of the p endulum amplitude and the prob e signal.

Discussion
We interpret results of our measurements in the following manner. Free oscillating fused silica test mass susp ended in high vacuum and nearby electro des form the electromechanical system due to the electrostatic interaction of charge lo cated on the test mass with the conductive surface of electro des. This system can b e found in two different states. In the first state, which can b e called "original" we observed a smo oth decrease of the p endulum oscillation amplitude in the pro cess of free decay with the Q-factor of ab out (6 - 8) в 10 7 . Also the averaged voltage from the capacitive prob e (taking into account the amplitude correction factor) changed weakly due to the charging of the test mass. The increase of the amplitude by means of electrostatic excitation did not change b ehavior of the system. In the second state, which can b e called "sp oiled" we observed a cascade of jumps (much more fast changes) of the p endulum amplitude. In the pro cess of jump the amplitude decreased according to the Q-factor reduced to 104 . Large changes of voltage from the prob e synchronous with jumps of the amplitude were observed as well. These jumps to ok place when the p endulum amplitude A was larger than the some threshold value AT . When A < AT the b ehavior of the system was the same as in the "original" state. The increase of the p endulum amplitude A > AT by means of the electrostatic excitation results in a new cascade of jumps of the amplitude and the voltage from the prob e. The threshold p endulum amplitude is likely asso ciated with the threshold electric field formed by the charge lo cated on the test mass and the charge induced on the electro des. This field stimulates electron transitions b etween lo cal surface states in the surface layer of the fused silica test mass. The "sp oiled" state of the system was observed very rare. It was found after the first large jump of the p endulum amplitude and the prob e voltage when the oscillating test mass was in vacuum during some time. But we do not yet know conditions of its formation. For example, application of high DC voltage to the electrostatic plate do es not likely stimulate the jumps. The "sp oiled" state was unstable. The threshold amplitude AT increased with time elapsed from the first jump. Jumps of the prob e voltage decreased. The "sp oiled" state transferred to the "original" state. The characteristic time of this transfer was of the order of one month. We supp ose that prop erties of the electronic structure of the fused silica test mass surface play the key role in the observed effects. Note that the b ehavior of the system in the "sp oiled" state is very nonlinear. This embarrasses the extrap olation its prop erties to the test masses of gravitational wave detectors, which op erate with very small amplitudes and at the higher frequencies. We have not found statistically significant coincidences b etween large signal from the cosmic ray detectors and jumps of the amplitude and the prob e voltage in the "sp oiled" state of the system. But we cannot exclude the coincidence b etween the first jump in the "original" state and the high intensity signal from the cosmic ray detector caused by cosmic ray showers. Also we cannot exclude the coincidence b etween jumps and signals from the cosmic ray detector caused by low energy particles passing lo cally through the surface layer of the test mass near the capacitive prob e.

13

Conclusion
We used the high-Q bifilar torsion p endulum and the capacitive prob e in order to search variations of electrostatic charges lo cated on the fused silica p endulum b ob b ecause this system combines the simplicity and the go o d sensitivity. But its b ehavior was found to b e more interesting and complicated. The observed effects as well as their p ossible role in the generation of excess noise in gravitational wave detectors are worth the further detail investigation.

Acknowledgments
The authors are grateful to V.B.Braginsky for supp ort, useful discussion and valuable advice, S.Rowan and J. Hough for the fused silica test mass with cones attached by means of the silicate b onding technique. This work was supp orted by the Russian Ministry of Science and by the National Science Foundation, USA, under grant No PHY-0098715.

14

Figures

Figure A.1: Schematic of the p endulum with the capacitive prob e.

15

Figure A.2: Schematic of the cosmic ray scintillator detector system.

16

Figure A.3: Time dep endence of the p endulum amplitude (a) and the voltage from the prob e (b). 17

Figure A.4: Fragment of record of the p endulum amplitude, the voltage from the prob e and the cosmic ray detector signal.

18

Bibliography
[1] A. Abramovici et al, Science 265 (1992) 325 [2] B. Willke et al., Class.Quantum Grav. 19 (2002) 1377. [3] F. Acernese et al., Class. Quantum Grav. 19 (2002) 1421. [4] M. Ando et al., Class. Quantum Grav. 19 (2002) 1409. [5] S. Rowan, S.M. Twyford, R. Hutchins and J. Hough, Class. Quantum Grav. 14 (1997) 1537. [6] V.P. Mitrofanov, N.A. Styazhkina, K.V. Tokmakov, Phys. Lett. A 278 (2000) 25. [7] M.J. Mortonson, C.C. Vassiliou, D.J. Ottaway, D.H. Sho emaker, G.M. Harry, Rev. Sci. Instrum. 74 (2003) 4840; http://arxiv.org/abs/gr-qc/0308001. [8] H.M. Araujo, A. Howard, D. Davidge, T.J. Sumner, Gravitational-Wave Detection, Mike Cruise, Peter Saulson, Editors, Pro ceedings of SPIE 4856 (2003) 55. [9] V.P. Mitrofanov, L.G. Prokhorov, K.V. Tokmakov, Phys. Lett. A 300 (2002) 370. [10] S. Rowan, S.M. Twyford, J. Hough, D.-H. Gwo, R. Route, Phys. Lett. A 246 (1998) 471. [11] V.B. Braginsky, V.P. Mitrofanov and K.V. Tokmakov, Phys. Lett. A 218 (1996) 164. [12] I.A. Bilenko, V.P. Mitrofanov and O.A. Okhrimenko, IET 36 (1993) 785.

19

Appendix B Mechanical losses in thin fused silica fibers
Intracavity top ology of the readout system for LIGO I I I pro ject and table-top QND mechanical measurements under development require using small prob e masses and susp ensions with very low level of internal losses. A go o d choice is to use thin fused silica fib ers similar to LIGO I I mirrors susp ensions. Mechanical losses of silica fib ers are investigated in this work through the study of quality factor dep endence on diameter for p endulum and violin mo des of oscillations with diameters ranging from 1.5 to 40 µm. The estimated values of effective mechanical loss angle show noticeably greater growth with lower diameters than it might b e exp ected while extrap olating known results of research done for thicker fib ers.

Introduction
The sensitivity level that should b e reached in the next stages of LIGO pro ject ([1]) will b e close to standard quantum limit (SQL). In order to overcome it, a sp ecial intracavity top ology of the readout system is suggested ([2]). This scheme requires small prob e masses and susp ensions with very low level of internal losses. Similar requirements should b e satisfied for table-top QND mechanical measurement ([3]). It is a go o d choice to use thin fused silica fib ers similar to pro jected LIGO I I mirrors susp ensions b ecause of high quality factors already achieved with silica fib ers ([4]). However, a small prob e mass (1 mg - 10 g ) will require very thin fused silica fib ers (1 - 40 µm) to b e used as susp ensions. It was not clear whether such fib ers would have the necessary quality factor b ecause the problem of determining fib er's surface influence on the mechanical dissipation remains unsolved. ~ The dissipation is usually describ ed using complex Young's mo dule Y as following: ~ Y = Y (1 + i) (B.1)

where Y is the real Young's mo dule describing rigidity of the material and is called mechanical loss angle. The previous studies ([5]) indicate that mechanical loss angle for fused silica fib ers evaluated from exp eriment can b e approximated by: (d) = 3.5 · 10-8 (1 + 1.336 · 10-3 m ) d (B.2)

here d is the diameter of the fib er given in meters. Mechanical loss angles were estimated by measuring the quality factors of b ending mo des and the obtained values agree well with ones calculated from (B.2) in the range of 40 - 3000 µm. 20

Turbomolecular pump

Fiber

Probe mass

Laser

Screen

Electric discharge pump

Figure B.1: Exp erimental design scheme for measuring p endulum mo des ring down time.

Experimental setup and evaluation basics
In our exp eriment we have measured the relaxation times for p endulum and violin mo des of oscillations with fused silica fib ers ranging in diameter from 1.5 to 40 µm and prob e masses ranging from 1 to 520 mg . Our samples were made of two different kinds of fused silica: Russian brand KS4V and Heraeus Co. Suprasil 312. Fib ers were pro duced in two stages: during the first one fib ers (11-15 µm thick) were made from two silica ro ds (1-2 mm thick), melt together in the flame at 1700 o C and quickly stretched outside the flame. During the second stage fib er with load attached was susp ended on a supp ort and gently warmed up until it started to stretch to diameters b elow 10 µm. With prob e masses b elow 10 mg we had to use additional technical loads to comp ensate the surface tension forces of warm silica. The technical load was removed after the fabrication pro cess was finished. After pro ducing the fib ers were set inside the vacuum chamb er as so on as p ossible to prevent the fib er surface hydratation and contamination. It usually takes no more than 5-10 minutes to put the fib er inside the chamb er and pump out to 10-3 T or r . The fib er quality and uniformity control, measurements of length and diameter were p erformed after the relaxation time had b een recorded. In each series of measurements it had b een ensured that damping due to residual gas pressure, recoil losses, and radiation pressure did not affect the measured quality factors. Two slightly different installations were used. To record the co ordinate of prob e mass and amplitude of p endulum mo de oscillations we have used a simple shadow metho d: a defo cused laser b eam lighted the prob e mass through the input window and then gathered on the screen 2 meters away from chamb er (See Fig. 1). The optical magnification of the system is close to 10 and the 10% variation of the amplitude could b e seen with mere eyes. The typical eigenfrequency of the p endulum was ab out 3H z , initial amplitude 21

Turbomolecular pump Photodetector

Laser Electrostatic exciter

Electric discharge pump

Figure B.2: Exp erimental design scheme for measuring violin mo des ring down time.

excited by shaking of vacuum chamb er was 2 mm. The p endulum was welded to a pin prepared on a 600 g silica cub e, firmly attached to the heavy flange of the chamb er. As a result, all internal mo des of the supp ort were substantially higher than the p endulum eigenfrequency and the value of recoil losses was negligible. For violin mo des we have used a knife-and-slot metho d (see Fig.2). Laser b eam was fo cused on the edge of the fib er near the middle of it. Behind the fib er we have placed a photo detector. Amplified and filtered signal from photo detector was observed by oscilloscop e and sp ectrum analyzer. In this setup the upp er 600 g silica cub e was susp ended from the flange by four steel wires 15 cm long each. Thus, the p endulum mo de frequency of supp ort ( 1 H z ) was substantially lower than fib er violin mo des (900 H z 3.5 k H z ) which allowed to avoid recoil losses influence. Also, two electro des, flat and ro d-like, were placed close to the fib er to create the non-uniform electric field for fib er excitation. The registration of the exp onential decay usually to ok 2-5 days. During this p erio d pressure in the chamb er was kept at the level b elow 1 · 10-7 T or r . We also to ok prop er account of the small amplitude variations caused by overlay and b eating of different mo des of oscillation. Sp ecial attention was paid to the p ossible electric charge fluctuation on the fib er. Overall, we have measured p endulum mo des for 15 samples and violin mo des for 13 samples. Estimating the value of mechanical loss angle with p endulum and violin mo des requires to consider the influence of gravitational force resulting in app earance of dilution factor for p endulum and violin mo des. We therefore use a term "effective mechanical loss angle" for value describing losses in the fib ers which can b e estimated as: ~ k·Q
-1

(B.3)

here k is the dilution factor for selected mo de: kv for violin mo de and kp for p endulum mo de; Q is the quality factor. The quality factor is calculated from the ring down time 22

1E-4

Mechanical loss angle

Russaian brand KS4V pendulum modes Results of Gretarsson & Harry G&H approximation, = 3.5E-8(1+1.336E-3 m/d) Refined approximation = 3.5E-8(1+(1.336E-3 m/d)+(6.2E-4 m /d ))
2 2

1E-5

1E-6

1E-7

1

10

100

1000

Diameter, µm

Figure B.3: Mechanical loss angle diameter dep endence as estimated from p endulum mo des.

: Since the expression for n-th violin mo de frequency taking the contribution of b ending rigidity into consideration is the following:
v

Q = · /2

n

=

n L

T 2 {1 + µ L

YI n2 2 Y I + [4 + ] } T 2 T L2

(B.4)

where T is tension force, L is fib er length, I is the moment of fib er cross-section inertia. Dilution factor for violin mo des kv can b e evaluated from (B.4) by substituting the expression for frequency in the quality factor, yielding: k
v

L 2

T YI

(B.5)

The expression for dilution factor for p endulum mo des kp is obtained from similar simple calculations: T kp 2 · L (B.6) YI We would like to p oint out again that values of effective mechanical loss angle obtained from (B.3, B.5, B.6) describ e mechanical losses not only in the bulk material of the fib er but in the surface, to o.

Experimental results
The values of effective mechanical loss angle estimated from measurements of quality factor are presented in Figures B.3 and B.4. The distinction b etween mechanical loss angles 23

Russian brand KS4V violin modes Suprasil 312 violin modes Refined approximation =

Mechanical loss angle

10

-4

3.5E-8(1+(1.336E-3 m/d)+(6.2E-4 m /d ))

2

2

10

-5

10

-6

1

10

Diameter, µm

Figure B.4: Mechanical loss angle diameter dep endence as estimated from violin mo des.

obtained from measuring the ring down time of p endulum mo des and the extrap olated values is greater than the accuracy of measurement for diameters b elow 5 µm. We regard this result as evidence of the additional surface losses contribution and suggest a refined approximation containing a higher-order term. Distinction b etween mechanical loss angles obtained from measuring the ring down time of violin mo des and the extrap olated values is greater than the accuracy of measurement for diameters b elow 20 µm. The different b ehaviour of p endulum and violin mo des might b e due to frequency dep endence of surface losses. We could not observe the frequency dep endence in p endulum mo des due to very small variation of eigenfrequency (2.5-3.3 H z ) yet the fundamental violin mo de frequency of fused silica fib ers changed due to its thickness more than threefold (900-3500 H z ). The mechanical losses of Suprasil 312 are noticeably lower than that of KS4V. For the finest fib ers we have obtained the following results. For fib er 1.5 ± 0.5 µm thick loaded with 1.01 ± 0.2 mg the p endulum mo de quality factor was equal to (1.9 ± 0.2) · 10 7 . For fib er 2.5 ± 0.5 µm thick loaded with 245 ± 7 mg the fundamental violin mo de quality factor was equal to (9.2 ± 0.9) · 106 .

Conclusion
We have measured the relaxation times of fused silica fib ers p endulum and violin mo des with diameters 1.5 - 40 µm. The estimated values of effective mechanical loss angle show noticeably sharp er growth with lower diameters than it might b e exp ected while extrap olating known results of research done for thicker fib ers. Yet the obtained results of measuring the relaxation time prove that using thin fused silica fib ers for table-top QND measurements is p ossible.

24

Acknowledgements
We would like to thank Prof. V. P. Mitrofanov for numerous helpful advices on the prop erties of fused silica and welding techniques. This work was supp orted partially by NSF grant PHY0098715, by Caltech Inst. of Quantum Information and by Russian Ministry of Industry and Science.

25

Bibliography
[1] Barish, B. and Weiss, R., Phys. To day, 52, 44-50 (1999). [2] F.Ya.Khalili Phys. Letters A 298, 308 (2002). [3] V.B.Braginsky, F.Ya.Khalili, P.S.Volikov Phys. Letters A 287, 3138 (2001). [4] P.Willems, V.Sannibale, J.Weel, V.Mitrofanov, Phys. Letters A 297, 37 (2002) [5] Andri M. Gretarsson, Gregory M. Harry, Rev. Sci. Instrum 70 (10): 4081-4087 Oct 1999

26

Appendix C Thermodynamical fluctuations in optical mirror coatings
Thermo dynamical (TD) fluctuations of temp erature in mirrors may pro duce surface fluctuations not only through thermal expansion in mirror b o dy [1] but also through thermal expansion in mirror coating. We analyze the last "surface" effect which can b e larger than the first "volume" one due to larger thermal expansion co efficient of coating material and smaller effective volume. In particular, these fluctuations may b e imp ortant in laser interferometric gravitational antennae.

Introduction
Outstanding exp erimental achievements in quantum optics and high resolution sp ectroscopy within recent four decades are substantially due to creation and development of new technologies. The coating of mirrors, i.e. the dep osition of many thin dielectrical layers on the mirror surface is likely to b e one of the most imp ortant technology in this part of exp erimental physics. To day this technology provides the reflectivity R of such coating very close to unity (at the b eginning of laser era the value of (1 - R) was ab out several p ercent while at present (1 - R) 10-6 [2, 3]). There is reason to hop e that in the not to o distant future the value of (1 - R) will b e close to 10-9 . This improvement in particular means that measurement using squeezed quantum states will b ecome routine as the squeezed state lifetime is longer if losses are smaller. The rise of squeezing factor provides the p ossibility to increase sensitivity with the same numb er of "used" photons [4, 5]. The value of (1 - R) is of great concern in terrestrial gravitational wave antennae (pro jects LIGO and VIRGO, e.g. see [6, 7, 8]). These antennae can b e regarded as extremely sensitive sp ectrometers which goal is to detect very small deviations FP /FP of one Fabry-Perot optical resonator mo de eigen frequency in comparison with another one. The amplitude of these frequency relative deviations which differs from the metric p erturbations amplitude only by the factor of two, is quite small. The planned resolution for the stage LIGO-I is FP /FP 10-21 , and for the stage LIGO-I I is one order smaller. The ab ove considerations show that the value of (1 - R) decrease is imp ortant for further development of many optical measuring devices typ es. However, to our knowledge there is no deep analysis of p ossible additional noises which can app ear sp ecifically due to coating. In this article we present the analysis of a noise source in the coated mirror, which deserves serious attention. Thermo dynamical (TD) fluctuations of temp erature in mirror coating can pro duce additional fluctuations of surface through thermal expansion

27

d6

9 -

T

2 TD

=

kB T 2 3 C rT

r

0

r

T
3 r0 3 rT

C

N

?

Figure C.1: Illustration for qualitative consideration of temp erature thermo dynamic fluctuations pro ducing the thermo elastic noise in mirror coating. This noise may b e larger than analogous one in mirror b o dy due to larger coating materials thermal expansion co efficient and smaller effective volume.

Semi-qualitative consideration
"Bulk" TD fluctuations. We have shown in [1] that thermo dynamical fluctuations of temp erature in mirrors are transformed due to thermal expansion co efficient b = (1/l)(dl/dT ) into additional noise which can b e a serious "barrier" limiting sensitivity. This noise (it is also called as thermo elastic noise) pro duces random fluctuational "ripples" on surface. This noise has a nonlinear origin as nonzero value of b is due to the lattice anharmonicity. Our main interest here is fluctuations of averaged displacement X (t) of mirror face surface. It is the normal displacement vz of the mirror face averaged over the b eam sp ot Gaussian p ower profile: 1 X= r
2 0 0 0 2

e

-r 2 /r

2 0

vz (r, ) r ddr,

(C.1)

Ї (r0 is the b eam radius). The sp ectral density of displacement X can b e presented for half infinite medium as follows: 2 8 b kB T 2 (1 + )2 TD Sbulk ( ) = . (C.2) 23 CV r0 2 2 Here kB is the Bolzmann constant, T is temp erature, is Poisson ratio, is thermal conductivity and CV is sp ecific heat capacity p er unit volume. This result has b een refined for the case of finite sized mirror by Liu and Thorne [12], however, the variation from our result is only several p ercent for typical mirror sizes, and hence we use more compact expression (C.2) here. This physical result can b e illustrated using the following qualitative consideration. We consider the surface fluctuations averaged over the sp ot with radius r0 which is larger than the characteristic diffusive heat transfer length rT rT = , CV 28 r
T

r0 ,

where is the characteristic frequency (for LIGO it is ab out 2 в 100 s-1 ). Since -3 in fused silica rT 3.9 в 10 cm and in sapphire rT 1.4 в 10-2 cm the condition rT r0 6 cm is fulfilled (material parameters are presented in App endix C4). 3 We know that in volume rT the length variation due to temp erature TD fluctuations T is ab out kB T 2 . xT = b T rT b rT 3 CV rT The TD temp erature fluctuations in such volumes can b e considered as indep endent ones. 3 3 The numb er of such volumes that contribute to surface fluctuations is ab out N r0 /rT , and hence the displacement X averaged over the sp ot with radius r0 consists of these indep endently fluctuating volumes displacements sum, and is approximately equal to X xT N b r
T

kB T 2 . 3 CV r0

(C.3)

TD Sbulk (correct to the multiplier Comparing (C.3) with (C.2) we can see that X of ab out unity) if one assumes . This fact confirms that our semi-qualitative consideration is correct.

TD fluctuations in thin layer. One can consider the surface fluctuations of half space covered by thin layer, pro duced by TD temp erature fluctuations as consisting of two parts: "bulk" part dep ending on mirror b o dy thermal expansion co efficient b (which has b een already calculated in [1]) and "surface" part due to temp erature fluctuations in layer with thickness d and effective thermal expansion co efficient = l - b (here l is thermal expansion co efficient of layer material). It is imp ortant to note that layer thickness d is much smaller than the diffusive heat transfer characteristic length rT (for optical coating d 10 µm) and therefore the following condition is valid: r0 rT d (C.4) Therefore we may consider temp erature fluctuations in our layer to b e the same as in layer with thickness rT . It means that TD temp erature fluctuations in layer does not dep end on thickness d. In this case we prop ose the following semi-qualitative consideration: X
d

T d, kB T 2 в 3 CV rT d kB T 2 . 2 CV r0 rT
T 2 rT = 2 r0

(C.5) kB T 2 , 2 CV r0 rT (C.6) (C.7) ( ) of

T X

d

Formula (C.6) is in go o d agreement with rigorous formula for sp ectral density S averaged temp erature T in layer obtained in [9] 2kB T 2 2kB T 2 ST ( ) , = 2 2 CV r0 rT r0 C and one can estimate the sp ectral density S sp ot with radius r0 as following
TD est layer 22 TD est layer

(C.8)

( ) of surface fluctuations averaged over

22 2 d kB T 2 S ( ) d ST ( ) = . (C.9) 2 C r0 rT Below we will show this formula to give the result correct to the multiplier of ab out unity. 29

Analytical results
Using Langevin approach [1] (see also App endix C1) we have calculated the field of temp erature TD fluctuations approximating mirror as a half infinite medium with assumption that all material parameters of layer and half space are identical with exeption of thermal expantion co efficients -- b for half space and l for layer. These temp erature fluctuations pro duce surface deformations of half infinite elastic space covered by thin layer. These deformations can b e calculated using elastic equation (C.28) (presented in App endix C1). We use quasistatic approximation for elastic problem b ecause time sound requires to travel across the distance r0 is much smaller than characteristic time 1/ . We also assume the layer to b e homogeneous. We can divide our problem into two problems: a) The "bulk" problem. The surface fluctuations are pro duced by TD temp erature fluctuations in half space covered by layer with different elastic constants. The layer and half space have the same thermal expansion co efficient b . It corresp onds to the problem solved in [1]. b) The "surface" problem. The surface fluctuations are also pro duced by TD temp erature fluctuations in half space covered by layer. The effective layer thermal expansion co efficient = l - b = 0, and there is no thermal expansion in half space. The qualitative considerations in previous section make clear that TD temp erature fluctuations pro ducing fluctuations in "surface" and "bulk" problems do es not virtually correlate and can b e calculated indep endently. For "surface" problem we assume that additional elasticity pro duced by layer is much smaller than mirror b o dy bulk elasticity (due to small thickness of the layer) and we take into account only stresses pro duced by layer due to nonuniform temp erature distribution u(r). Then one can take equation (C.28) and just substitute into right part the term u d u (x - ), = l - b . (C.10) (C.11)

(Recall that here we assume identity of material constants for layer and half space.) As a result we obtain sp ectral density of surface fluctuations (see details in App endix C1): 4 2(1 + )2 2 d2 kB T 2 TD Slayer ( ) = . (C.12) 2 r0 CV Here is Poisson ration. Note that this formula differs from semi-qualitative one (C.9) only by multiplier of ab out unity. TD est TD The fact that Slayer ( ) > Slayer . ( ) can b e qualitatively explained by the sp eculation that thermal expansion in layer also causes stresses in mirror b o dy under layer: the material under layer additionally "swells up" ("distend"). The similar effect one can observe when heated bimetallic plate is b ending. We have generalized the formula (C.12) for the case of different material parameters of layer and substrate deriving it from fluctuation-dissipation theorem (FDT) (see details in App endix C2): 4 2(1 + b )2 2 d2 kB T 2 TD Slayer ( ) = , (C.13) 2 r0 b CV ,b = b , (C.14) l CV , l + в (C.15) =- CV , b 2b El (1 - 2b ) 1 + l + . в (1 - l )(1 + b ) Eb (1 - l ) 30

Here Eb , El are Young mo dulus, subscripts b and l refer to parameters of half space and layer corresp ondingly. This result was earlier obtained in [10]. We can further generalize the result (C.13 - C.15) for multilayer coating with alternate material constants which differs from material constants of mirror b o dy. Let the multilayer coating consists of N alternating sequences of quarter-wavelength dielectric layers. Every o dd layer has the refraction index n1 , thickness d1 = /4n1 ( is the incident light wavelength). Every even layer has parameters n2 , d2 = /4n2 corresp ondingly. Then formulas (C.13, C.15) can b e used with the total coating thickness and effective expantion co efficient calculated using the following formulas (see details in App endix C2): = b d 1 b d 2 1 + d1 + d 2 d1 + d 2 d = N (d1 + d2 )
2

(C.16) (C.17)

Here factors 1 and 2 are calculated using (C.15) with substitution material parameters for o dd and even layers corresp ondingly. It should b e emphasized that negative influence of thermo elastic fluctuations in layer in finite sized mirror can b e larger than in half infinite space mo del due to "bimetallic" effect: the temp erature fluctuations in layer should additionally cause mirror b end through thermal expansion. Using FDT approach develop ed by Liu and Thorne [12] we have calculated numerically the following co efficient C
fsm

=

S S

TD, finite test mass layer TD, infinite test mass layer

.

(C.18)

For design planned in LIGO-I I [11] the test mass is manufactured from fused silica with T a2 O5 + S iO2 coating and has following parameters R = 19.4 cm, H = 11.5 cm, r0 = 6 cm. For this case we estimate Cfsm 1.56 (see details of rather comb ersome calculations in App endix C3). But if R value of CFTM may b e substantialy larger. H then the

Numerical estimates
High reflectivity of mirrors is provided by multilayer coating which consists of alternating sequences of quarter-wavelength dielectric layers having refraction indices n1 and n2 . The frequently used pairs are T a2 O5 (n1 2.1) and S iO2 (n2 1.45): namely these coatings are used in LIGO [21]. The total typical for high finesse mirror numb er of layer pairs is N 19 (this value we used for estimates b elow) so that total thickness of T a2 O5 + S iO2 coating is ab out N (d1 + d2 ) 6 в 10-4 cm. To estimate TD temp erature fluctuations in interferometric coating of mirror we use formula (C.13) with substitutions (C.14 - C.17). To obtain numerical estimates for these typ es of fluctuations it is necessary to have the value of for thin layer of T a2 O5 . In the existing publications we have found very wide range: from Ta2 O5 -(4.43 ± 0.05) в 10-5 K-1 [13] (the scheme of measurement is describ ed in [14]) to Ta2 O5 3.6 в 10-6 K-1 (without measurement error, unfortunately) [15] and Ta2 O5 (5 ± 2) в 10-6 K-1 [16]. It can b e explained not only by p ossible errors of exp eriment but also by the fact that prop erties of tantalum p entoxide may b e strongly dep ends on pro cedure of layer dep osition on substrate [17]. So we use for estimate the value Ta2 O5 5 в 10-6 K-1 b ecause coating 31

measured in pap er [16] was dep osited on thin fused silica plates by the same technology as coating on LIGO mirrors. The other material parameters are listed in App endix C4. It is also useful to present estimates for thermorefractive noise [9]: TD temp erature fluctuations pro duce fluctuations of coating layers refraction indices n1 and n2 due to its dep endence on temp erature: 1 = dn1 = 0, dT 2 = dn2 = 0, dT

Refraction indices fluctuations pro duce in turn fluctuations of phase of the wave reflecting from mirror. This noise can b e recalculated into equivalent noise displacement of mirror1 : 22 2 2 T TD , (C.19) Strefr ( ) = 2 r0 C = n1 n2 (1 + 2 ) . 4(n2 - n2 ) 1 2 (C.20)

For estimates we use "optimistic" (i.e. smallest) value of Ta2 O5 2.3 в 10-6 K-1 [18] (for example, [13] rep orts unexp ectedly high value Ta2 O5 1.4 в 10-4 K-1 ). We also want to estimate the noise asso ciated with the mirror material losses describ ed in the mo del of structural damping [19], b elow we denote it as Brownian motion of the surface. In this mo del the angle of losses do es not dep end on frequency and the following formula is valid for its sp ectral density in the infinite medium [20, 1, 12]:
B Sx ( )

4kB T (1 - 2 ) . 2 E r0

(C.21)

Note that we have used the new value of 0.5 в 10-8 for the loss angle in very pure fused silica [24]. The gravitational wave antenna sp ectral sensitivity to the p erturbation of metric h( ) may b e recalculated from the displacement X noise sp ectral density using the following formula: N0 Sx ( ) (C.22) h( ) = L where we use N0 = 4 due to the fact that antenna has two arms (with length L) with two mirrors in each one. However, for TD layer fluctuations we use N0 2 b ecause in LIGO interferometer arms the Fabry-Perot resonators end test mass (ETM) coating TD fluctuations are considerably larger than ones in the input test mass (ITM) coating due to the coating thickness in ETM is approximately 6 times larger [21]. The LIGO-I I antenna will approach the level of SQL so we also compare the noise limiting sensitivity with this limit in sp ectral form [22]: h
SQL

=

8 . m 2 L2

(C.23)

For estimate we use the set of parameters for fused silica mirror with T a2 O5 + S iO2 coating using the parameters listed in App endix A4. The results are presented in fig. C.2. We can see that in fused silica mirror the TD bulk fluctuations are negligibly small but TD fluctuations in layer of interferometric coating are ab out Brownian one and only 2 times smaller than standard quantum limit at 2 в 100 s-1 . Note that thermo elastic
1

Here we present formula correcting an error in formula (3) in [9]

32

TD bulk h, 1/ H z 10
-23

Fused silica with T a2 O5 + S iO SQL Brown

2

10

-24

TD layer TD refr TD bulk

10-25 10

100

1000 frequency, Hz

Figure C.2: Dep endence of different noises on frequency in gravitational wave antenna with fused silica mirror and T a2 O5 + S iO2 coating presented in terms of dimensionless metric (C.22). Notations: "SQL" -- hSQL ( ) (C.23), "Brown" -- Brownian fluctuations caused by structure losses and describ ed by formula (C.21), "TD layer" - surface fluctuations caused by TD temp erature fluctuations (thermo elastic) (C.12 C.16) in coating, "TD bulk" -- the same fluctuations in the mirror volume (C.2), "TD refr" -- thermorefractive noise (C.19). fluctuations in layer weakly dep ends on frequency -1/4 (see (C.12)) and for frequencies 2 в 300 s-1 these fluctuations dominate completely. For comparison we present in fig. C.3 the estimates for sapphire mirror with the same T a2 O5 + S iO2 coating. We can see that for sapphire mirror the contribution of TD noise in layer is smaller and only at > 2 в 103 s-1 it is close to SQL. We can see that TD fluctuations in the same coating T a2 O5 + S iO2 on sapphire and on fused silica differs approximately by 3 times. It can b e explaied by the fact that factor s CS (denominator in (C.12)) for sapphire is ab out two orders larger than for fused silica. Note that equivalent thermal expansion co efficients describ ed by formulas (C.14 C.16) are practically the same
Ta2 O5 +SiO SiO2 Ta2 O5 +SiO Al2 O3
2 2

2.8 в 10

-6

K-1 , K-1 ,

-2.8 в 10

-6

It is also worth noting that surface fluctuations caused by TD temp erature fluctuations (thermo elastic and thermorefractive) in layer (coating) dep ends on laser sp ot radius -3/2 - weaker ( r0 1 ) than "bulk" noise ( r0 ).

Conclusion
The slow dep endence of interferometric layer (coating) TD noise on frequency -1/4 and - on the radius of the b eam sp ot r0 1 is worth noting. We can see that TD noise in interferometric layer is close to the standard quantum limit and it may b e an obstacle for interferometric gravitational antennae (pro jects LIGO-I I and esp ecially LIGO-I I I). The resume of this pap er may b e formulated in the following way: it is imp ortant to measure in situ the value of effective thermal expansion co efficient for interferometric multilayer coatings of high quality mirrors for presision measurements.

33

TD bulk h, 1/ H z 10
-23

Sapphire with T a2 O5 + S iO TD bulk SQL Brown TD layer TD refr

2

10

-24

10-25 10

100

1000 frequency, Hz

Figure C.3: Dep endence of different noises on frequency in gravitational wave antenna with sapphire mirror and T a2 O5 + S iO2 coating in terms of dimensionless metric (C.22). Notations are the same as in fig.C.2

Acknowledgements
We are grateful for fruitful dicussions with M. Fejer p ointing out on incorrrectness of several formulas in previous version of this pap er. We are pleased to thank Helena Armandula, Garilynn Billingsley, Sheila Rowan and Naci Inci providing priceless information ab out material constants for optical coating. We are also grateful to F. Ya. Khalili and V. P. Mitrofanov for helpful discussions. This work was supp orted in part by NSF and Caltech grant PHY0098715, by Russian Ministry of Industry and Science and by Russian Foundation of Basic Researches.

C1. TD fluctuations of surface of half space. Langevin approach
Here we present the derivation of formula (C.12) using Langevin approach. We calculate the surface fluctuations (averaged over the sp ot with radius r0 ) of half-space covered by layer with thickness d. The surface fluctuations are originated by TD temp erature fluctuations which in turn pro duces deformations due to non-zero thermal expantion. We consider the case when thermal expantion co efficient is non-zero only inside layer and other thermal and elastic parameters are the same in layer and half-space. We also assume that following strong unequalities fulfills: d r
T

r0 ,

rT =

C

34

TD temp erature fluctuations
We use Langevin approach prop osed by [1] using thermo conductivity equation for halfspace z > 0 with thermoisolated surface with fluctuational forces F in right part: u - a2 u = F (r, t), a2 = , t C kB T 2 F (r, t)F (r1 , t1 ) = 2 (t - t1 ) в (C )2 в (x - x1 ) (y - y1 ) в (z - z1 ) + (z + z1 ) . (C.24) (C.25)

Here we intro duce mirror fluctuational forces (last term in square brackets) in order to reformulate problem for the whole space. After space and time Fourier transform we can write down the solution

u(r, t) =
-

dk d u( , k)e (2 )4 ,

i t+ik r )

, (C.26)

u( , k) = F (k , )F (k1 , 1 )

F (k , ) a2 (k )2 + i

2 F0 T

2 0

22 = F0 T0 (2 )4 |k |2 ( - 1 ) в в (kx - kx1 ) (ky - ky1 ) в в (kz - kz 1 ) + (kz + kz 1 ) . kB T 2 =2 (C )2

(C.27)

Equation of elasticity
The problem of elastic deformations v for half-space covered by thin layer caused by thermal expansion due to TD temp erature fluctuations u can b e describ ed by equation [23, 25] with zero stresses z z , yz , xz on free surface z = 0: 1- в grad div v 1+ 1 - 2 - 2(1 + ) E z z = 1 - 2 1 - rot rot v = u, - (C.29) (C.30) (C.28)

Using condition d

yz

= =

xz

vx vy vz + + + x y z 1 - 2 vz -u + = 0, 1 + z z =0 vz vy E = 0, + 2(1 + ) y z z =0 E vx vz + = 0. 2(1 + ) z x z =0

r0 we can present the right part of (C.28) as following u d 35 u (z - ) (C.31)

One can find solution as sum (see for example sec. 3.4 in [25]): v = v function fulfills Poison equation (it is derived from (C.28)): = 1+ d u (z - ) 1-
(a)

(a)

+

. Here

(C.32)

without b oundary condition and function v the following b oundary conditions:
zz

fulfills to (C.28) with zero right part and

=

xz

yz

E 1+ E =- (1 + E =- (1 +

2 2 +2 y2 x 2 , ) x z z =0 2 . ) y z z =0

,
z =0

(C.33)

By this way for function v (a) we obtain the problem of half space deformation with stresses (C.33) on b oundary. We are interesting of surface displacement along z axis averaged over sp ot with radius r0 = 2/0 in center of co ordinate system: Ї X (t) = 1 r
2 0

( + vza) z - x2 + y 2 . в exp - 2 r0 dy dz

z =0

в

(C.34)

Solution of Poison equation (C.32) is known: (1 + ) (x, y , z ) = - 4 (1 - ) в
-

dx dy в
2

(C.35) .

2u(x , y , z )

(x - x )2 + (y - y )2 + z

Here we add solution by mirror source ( (x + )) -- it pro duces the multiplier 2 in last fraction. Such function makes zero contribution into surface displacement (see (C.34). As addition the tangent stresses xz and yz in b oundary conditions (C.33) are also b ecome zero. Only normal pressure is non-zero and is equal to:
z =0 zz

=-

dE (yy + xx ) в 2 (1 - ) dkx dky dkz d в в (2 )4 - в F (k )T ( ) a |k | + i
2 2

e

i t+iky y +ikx x

в

в в

- iky (y -y )+ikx (x -x)

dy dx в

e

(y - y )2 + (x - x ) 36

2

We take the last integral (over dy dx ) going into cylindric co ordinate system with notation 2 2 k = ky + kx (using formulas 2.5.24.1 in [26] and 2.12.4.28 in [27]):

I1 (k ) = =
0

dy dx
- 2 0 zz

e

iky (y -y )+ikx (x -x) 2

d

(y - y )2 + (x - x ) eik r cos 2 r dr = k r2
-

=

Gathering we obtain expression for
z =0 zz

(y , x) =

dE (1 - ) в
2 2

dkx dky dkz d в (2 )4 (C.36)

k F (k )T ( ) a |k | + i
(a) 2

e

i t+iky y +ikx x

Now we can write down the expression for v v
(a) z

at z = 0 (see sec. 8 in [23]):
z =0 z z (y , x ) . (y - y )2 + (x - x )2

=

1- E

dy dx
-

Substituting it into (C.34) we obtain: Ї X (t) = 2d(1 + ) в F (k )T ( ) (a2 |k |2 + i )
-

0 i t-

dkx dky dkz d в (2 )4

(C.37)

e

22 k r0 4

One can write down the time correlation function (using (C.34)): Ї Ї B ( ) = X (t)X (t + ) = d i 22 2 = 4 d (1 + ) eв - 2 22 dkz k dk 2F0 T0 k 2 -k e в (2 )2 a4 |k |4 + 2 - 0 Now we can write down one side sp ectral density: 4 22 (1 + )2 2 d2 F0 T0 в SX ( ) = 2 2 2 (k + kz ) dkz k dk в e 2 2 (a4 (k + kz )2 + 2 ) - 0

(C.38)
22 r0

/2

(C.39)
22 -k r0 /2

2 2 Using condition a2 /r0 (and assuming that k 2 kz ) we can easy take integral over k then over kz and obtain the formula (C.12). Note that formula (C.39) can b e used for calculation SX with arbitrary value :

SX ( ) = I
X

4 22 (1 + )2 2 d2 F0 T0 в IX 2 (x2 + y ) dx dy 1 e = 2 r0 a4 - 0 (x2 + y )2 + b4 ) 2 r 0 a4 2 2 r0 , = 4a4
0

(C.40)
-y

= (C.41)

= b
4

y + y 2 + b4 e (y 2 + b4 )

-y

dy ,

37

C2. FDT approach calculations of thermo-elastic noise in interferometric coating
Here we derive formula (C.12) using fluctuation-dissipation theorem (FDT)[28, 29] by the following thought exp eriment: We imagine applying a sinusoidally oscillating pressure, P =F e
o -r 2 /r
2 o

r

2 o

e

i t

(C.42)

to one face half-infinite mass covered by layer. Here Fo is a constant force amplitude, is the angular frequency at which one wants to know the sp ectral density of thermal noise, and the pressure distribution (C.42) has precisely the same spatial profile as that of the Ї generalized co ordinate X , whose thermal noise SX ( ) one wishes to compute. Ї The oscillating pressure P feeds energy into the test mass, where it gets dissipated by thermo elastic heat flow. Computing the rate of this energy dissipation, Wdiss , averaged over the p erio d 2 / of the pressure oscillations we can just write down (in according with fluctuation-dissipation theorem) the sp ectral density of the noise SX ( ) is given by Ї SX ( ) = Ї 8kB T Wdiss 2 Fo 2 (C.43)

The rate Wdiss of thermo elastic dissipation is given by the following standard expression (first term of Eq. (35.1) in [23]): W
diss

=

( u)2 r d dr dz T

.

(C.44)

Here the integral is over the entire test-mass interior using cylindric co ordinates; T is the unp erturb ed temp erature of the test-mass material and u is the temp erature p erturbation pro duced by the oscillating pressure, is the material's co efficient of thermal conductivity, and . . . denotes an average over the pressure's oscillation p erio d 2 / (in practice it gives just a simple factor ( eit )2 = 1/2). The computation of the oscillating temp erature p erturbation is made fairly simple by two well-justified approximations [1]: First: Quasistatic approximation. We can approximate the oscillations of stress and strain in the test mass, induced by the oscillating pressure P , as quasistatic. Second: We assume that the following strong unequality is valid d r
T

ro ,

rT =

, C

here rT is length of diffusive heat transfer and is the density and C is the sp ecific heat . This means that, when computing the oscillating temp erature distribution, we can approximate the oscillations of temp erature as adiabatic in transversal direction (tangent to surface) but non-addiabatic in normal to surface direction. The quasistatic approximation p ermits us, at any moment of time t, to compute the test mass's internal displacement field v , and most imp ortantly its expansion = div v , from the equations of static stress balance (Eq. (7.4) in [23]) (1 - 2 )
2

(C.45)

v+ 38

(

· v) = 0

(C.46)

(where is the Poisson ratio), with the b oundary condition that the normal pressure on the test-mass face b e P (r, t) [Eq. (C.42)] and that all other non-tangential stresses vanish at the test-mass surface. Ones has b een computed, we can evaluate the temp erature p erturbation u from the thermo conductivity equation t - a 2 u = -E T t , C (1 - 2 ) a2 = C (C.47)

Here E is Young's mo dulus. It worth to underline that we can not use addiabatic approach in (C.47) due to small thickness of layer. This temp erature p erturbation u can then b e plugged into Eq. (C.44) to obtain the dissipation Wdiss as an integral over the gradient of the expansion. This Wdiss can b e inserted into Eq. (C.43) to obtain the sp ectral density of thermo elastic noise.

Elastic problem
Elastic infinite half space. We can assume that layer do es not influece on deformations in substrate due to its small thickness. Then we can use the solution to the quasistatic stress-balance equation (C.46) given by a Green's-function expression (see (8.18) in [23]) with Fx = Fy = 0, Fz = P (r ), integrated over the surface of the test mass: Below we will need the expression for (b) = div u. Using results of [1] one can calculate longitudial and transversal parts of (b) separately:
(b)

= (x ux + y uy ) =-

z =0

=

(C.48) (C.49) (C.50)

(b)

2(1 + b )(1 - 2b )P , Eb (1 + b )(1 - 2b ) P = z uz |z =0 = - Eb

Layer. For solution of thermal problem we will need to calculate expansion (l) in layer. We assume that deformations of layer in transversal plane are the same as in substrate (l) (b) (the same natural assumption was done in [10]), i.e. = |z =0 . One can use equation (5.13) for stress in [23] for calculation
zz (l)

uz z :

(l)

-P =

El в (1 + l )(1 - 2l )
(f ) zz

(C.51)
(f ) yy

в (1 - l )u

+ l (u

(f ) xx

+u
|

).

(s)

z =0

Using this equation one can find u u
(l) zz

(l) zz

and full expansion l in layer: l Yl , Yb Yl (1 - 2l ) + Yb -
b

Y

l

b

P 1 Yl (1 - l ) P =- 1 Yl (1 - l ) Eb = (1 + b )(1 - 2 =-

(C.52) , (C.53)

)

,

Yl =

El (1 + l )(1 - 2l )

39

Thermal problem
The expansion in layer and in substrate are the source of heat with p ower p er unit volume (see for example sec. 31 in [23]) which are equal to Wl inside layer and to Wb in substrate Wl = -l El T i l , (1 - 2l ) Wb = -b Eb T i b . (1 - 2b )

It seems that we can divide our problem into two problems: First: The "bulk" problem. In half space there is p ower source Wb . It corresp onds to the problem solved in [1]. Second: The "surface" problem. There is p ower source only inside layer. Below we consider only second problem. We have thermal conductivity equation in layer with constant right part. Using conditions (C.4) one can simplify thermal conductivity equations to one dimension form as following b 2 1- Tb = 0, (C.54) i Cb z l 2 Tl = w , (C.55) 1- i Cl z b E b T b -l El T l + = (1 - 2l ) Cl (1 - 2b ) Cb 2(1 + b ) T b P , =- Cb l C b 1 + l = -1 + в b Cl 2(1 - l )(1 + b ) El (1 - 2b )(1 + b ) в 1+ Eb (1 + l ) w=

(C.56) (C.57)

We have condition of thermal isolation at z = 0 and condition of continuity of temp erature and heat flux at z = d. So we can find the solution of system (C.54, C.55) as following T T
l

= A - B cosh l z , =

b

C= b b C = A= B= R=

1 + i Cl l = , l 2 1 + i Cb C e-b (z -d) , b = , b 2 A - B cosh l d, l l B sinh l d, w R sinh l d w, C = , R sinh l d + cosh l d w , R sinh l d + cosh l d l l b b

(C.58) (C.59) (C.60) (C.61) (C.62)

Due to conditions (C.4) and hence |g d| T T
l

1 we can simplify espressions for temp erature: (C.63) (C.64)

b

w (1 - cosh l z + Rl d cosh l z ), w l l2 d -b (z -d) e . b b 40

We are interesting only in temp erature distribution in infinite space b ecause volume of layer pro duces small contribution into total budget of dissipated energy: W
diss 2 Cl2 2 F0 = (1 + b ) 2 в 2 Cb r0 2 d2 T в 2 C b b 2 2 b

(C.65)

Using this result and (C.43) one can write down sp ectral density 4 2(1 + b )2 kB T 2 2 2 eff d , SX ( ) = 2 r 0 C b b Cl eff = b , = . Cb

(C.66) (C.67)

Generalization for multilayer coating
We can generalize this result for multilayer coating consisting of N alternating sequences of quarter-wavelength dielectric layers. Every o dd layer has the refraction index n 1 , thickness d1 = /4n1 ( is the incident light wavelength), its parameters we mark by subscript 1 . Every even layer has parameters n2 , d2 = /4n2 , its parameters we mark by subscript 2 . It can b e easy shown that the equivalent thermal expansion co efficient eff of total coating and the total thickness d of coating are the following: s d 2 s d 1 1 + 2 , d1 + d 2 d1 + d 2 d = N (d1 + d2 ), C f 1 f 1 1 = - + в Cb 2b 1 + f 1 Ef 1 (1 - 2b ) в , + (1 - f 1 )(1 + b ) Eb (1 - f 1 ) C f 2 f 2 + в 2 = - Cb 2b 1 + f 2 Ef 2 (1 - 2b ) в . + (1 - f 2 )(1 + b ) Eb (1 - f 2 )
eff

=

(C.68) (C.69) (C.70)

(C.71)

And the sp ectral density SX ( ) is defined by substitution (C.68) into formula (C.66). We have checked it by solution of thermal problem and calculation Wdiss for one pair of layers, for two and three pairs of layers.

C3. Finite sized mirror
Our calculations for finite sized mirror are based on results of Liu and Thorne [12]. The order of calculation steps is the following. First step: we calculate total expansion s and (s) (s) its tranversal part in substrate. Second step: we substitute calculated value into (C.51) and calculate and then f . Third step: we substitute the values s and f into thermal cunductivity equations (C.54 and C.55). Forth step: we calculate dissipated energy Wdiss .
(f )

41

Calculations of s . We use the results and notations of [12]. The radius of mirror is R, its height is H .
s

= F0 e +

i t

-

p0 (1 - 2s ) 2(1 - 2s )c0 + + Es Es

(C.72)

[km Am + Bm ] J0 (km r ) ,
m

= p
0

E , (1 - 2 )(1 + ) 1 , R2 c0 =

µ=

E , 2(1 + )

(C.73) (C.74)

=

6 R2 H2

m=1

J0 (m )pm , 2 m

Here m are the m-th ro ot of equation J1 (x) = 0, J0 (x), J1 (x) are Bessel functions of zero and first orders. p A B
m m m 22 m exp(-km r0 /4) , km = , = 2 J 2 ( ) R 0 m R = Am |z =0 = m + m , µs (m - m ) = dz Bm (z )|z =0 = km - (s + 2µs ) -km (m + m ), µs (m - m ) = [km Am + Bm ] = km , (s + 2µs ) = exp(-2km H ) 1 - Qm + 2km H Qm pm (s + 2µs ) = 2 km µs (s + µs ) (1 - Qm )2 - 4km H 2 Qm pm (s + 2µs )Qm 1 - Qm + 2km H = 2 km µs (s + µs ) (1 - Qm )2 - 4km H 2 Qm + qm Qm + µs (1 - Qm ) -pm = 2 2km µs (s + µs ) (1 - Qm )2 - 4km H 2 Qm - qm - µs (1 - Qm ) -pm Qm , = 2 2km µs (s + µs ) (1 - Qm )2 - 4km H 2 Qm 2 = 2km H 2 (s + µs ) ± 2µs km H, (1 - Qm )2 pm =- 2 (s + µs ) (1 - Qm )2 - 4km H 2 Qm

D Q

m

m m

q D

m

m

m ± m m

Calculations of .
(s)

(s)

= F0 e +

i t

2s p0 2(1 - s )c0 + + Es Es km Am |
z =0

(C.75) (C.76) (C.77)
2

J0 (km r ) ,

m

km A

m

= km (m + m ) = - в

pm (1 + s ) в Es 2 (1 - Qm )2 (1 - 2s ) + 4Qm km H 2 (1 - Qm )2 - 4km H 2 Qm 42

.

Calculations of expansion f in layer. Using (C.51) we obtain: u
zz f

=

= uz z + =

-P (1 + f )(1 - 2f ) - Ef (1 - f )
(s)

(s)

=

f , 1 - f 1 - 2f 1 - f

(C.78)

-P (1 + f )(1 - 2f ) + Ef (1 - f )

(s)

(C.79)

It is useful to present pressure P as series and rewrite f as following; P = F0 e
f i t

p0 +
m=1 i t

pm J0 (km r ) , p
0

(C.80)

1 - 2f = F0 e 1 - f +

2s (1 + f ) + - Es Ef pm J0 (km r )K
m

2(1 - s )c0 - Es

,

(C.81) (C.82)

m=1

K

m

(1 + f ) (1 + s ) = + в Ef Es в
2 (1 - Qm )2 (1 - 2s ) + 4Qm km H 2 (1 - Qm )2 - 4km H 2 Q m 2

Calculations of right part in thermal conductivity equation. Using (C.55) we obtain the expresion for w : s E s T s -f Ef T f + , (1 - 2f ) Cf (1 - 2s ) Cs Cf w = s T F0 eit в Cf f 1 + f 2s Ef в -p0 + - Cs s (1 - f )Es 1 - f f Ef (1 - s ) Cf - +2c0 + Cs s Es (1 - f ) w=

(C.83) (C.84) +

+
m=1

pm J0 (km r )L

m

,

L

m

=

f (1 + f ) 1 + s - + s (1 - f ) 1 - f (1 + s )(1 - Qm )2 + в 2 (1 - Qm )2 - 4km H 2 Qm f Ef (1 - 2s ) 1 2Cf в + - s Es (1 - f ) 1 - f Cs

(C.85)

43

Calculations of dissipated energy W W
fsm diss

diss

. Using (C.44) we obtain: (C.86) - Cf + Cs
2

2 F 2 s 0 2 2 R f 1 P= s 1 = +S
1

T 2 d2 P, 2 C ss + f 2s Ef - - f (1 - f )E

s

S S

1

12R = H2

2

Cf f Ef (1 - s ) - Cs s Es (1 - f ) e-km r0 /4 , 2 m J0 (m )
2 2

+ S2 , (C.87) (C.88)

m=1
2

2

=
m=1

e-km r0 /2 2 L. 2 J0 (m ) m

2

Here we used the (nonstandard) orthogonality relations:
R

J0 (km r ) J0 (kn r ) r dr =
0 R

R2 2 J (m ) 20

mn

,

J0 (km r ) r dr = 0.
0

Now one can calculate numerically the co efficient C =
fsm Wdiss = Wdiss 2 r0 P 2R2 (1 + s )2

fsm

2

(C.89)

using (C.86) and (C.65). We numerically estimate Cftm for R = 19.4 cm, H = 11.5 cm, r0 = 6 cm and for T a2 O5 + S iO2 coating on fused silica substrate: C
ftm

1.56

44

C4. Parameters
= 2 в 100 s-1 , T = 300 K, r0 6 cm m = 3 в 104 g, = 1.06 µ, L = 4 в 105 cm; ; Fused silica: 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 erg E = 7.2 в 1011 3 , = 0.17, cm = 0.5 в 10-8 [24]; dE erg = -1.5 в 108 . dT K cm3 dn = 1.5 · 10-5 K-1 , n = 1.45, dT Sapphire: erg , = 5.0 в 10-6 K-1 , = 4.0 в 106 cm s K g erg = 4.0 , C = 7.9 в 106 , 3 cm gK erg E = 4 в 1012 , = 0.29, = 3 в 10-9 , 3 cm erg dE = -4 в 108 , dT K cm3 n = 1.76, ( = 1 µm [32] Tantal p entoxide T a5 O5 = -4.4 · 10-5 K-1 [13], = 3.6 · 10-6 K-1 [15], = (5 ± 1) · 10-5 K-1 [16] erg E = 1.4 в 1012 [30], = 0.23, [30] cm3 erg g = 4.6 в 106 , = 6.85 , cm3 cm sK erg , C = 3.06 в 106 gK n = 2.1, dn = 1.21 · 10-4 K-1 ( 1.4µm) [13] dT dn = 2.3 · 10-6 K-1 ( 0.63µm)[18], dT dn = 4.7 · 10-6 K-1 [31] dT

45

Bibliography
[1] V. B. Braginsky, M. L. Goro detsky, and S. P. Vyatchanin, Physics Letters A 264, 1 (1999); cond-mat/9912139; [2] G. Remp e et al, Opt.Letters, 17, 363 (1992). [3] H. J. Kimble, private communication. [4] V. B. Braginsky and F. Ya. Khalili, Quantum Measurement, ed. by K.S. Thorne, Cambridge Univ. Press, 1992. [5] V. B. Braginsky and F. Ya. Khalili, Rev. Mo d. Physics, 68, 1 (1996). [6] A. Abramovici et al., Science 256, 326 (1992). [7] A. Abramovici et al., Phys.Letters.A 218, 157 (1996). [8] V. B. Braginsky, Physics "Usp ekhi", 43, 691 (2000). [9] V. B. Braginsky, M. L. Goro detsky, and S. P. Vyatchanin, Physics Letters A 271, 303-307 (2000). [10] M. M. Fejer, S. Rowan et al tb d, "Thermo elastic dissipation in inhomogenious media: loss measurement and displacement noise . . . ", manuscript, 2003 [11] http://www.ligo.caltech.edu/ligo2/ scripts/12refdes.htm [12] Yu. T. Liu and K. S. Thorne, Phys. Rev. D62, 122002 (2000). [13] M. N. Inci, Simultaneous measurements of thermal optical and linear thermal expansion co efficients of T a2 O5 films, ICO 19, 25-30 August 2002, Firenze, Italy. [14] G. Gulsen and M. N. Inci, Optical Materials 18, 373 (2002). [15] C.-L. Tien, C.-C. Jaing, C.-C. Lee and K.-P. Chuang, J. Mod. Opt., 47, 1681 (2000). [16] V. B. Braginsky and A. A. Samoilenko, submitted to Physics Letters A. [17] Stan Whitcomb, private communication. [18] A.K. Chu, H.C. Lin, W.H. Cheng J. Electro. Mater., 26, 889 (1997). [19] P. R. Saulson, Phys. Rev. D, 42, 2437 (1990); G. I. Gonzalez and P. R. Saulson, J. Acoust. Soc. Am., 96, 207 (1994). [20] F. Bondu, P. Hello, Jean-Yves Vinet, Physics Letters A 246, 227 (1998). 46

[21] K. Srinivasan, Coating strain induced distortion in LIGO optics, LIGO do cument T-970176-00, available on www.ligo.caltech.edu. [22] V. B. Braginsky, M. L. Goro detsky, F. Ya. Khalili and K. S. Thorne, Report at Third Amaldi Conference, Caltech, July, 1999. [23] L. D. Landau and E. M. Lifshitz Theory of Elasticity, Nauka, Moscow, 1980, (translation: Pergamon, Oxford, 1986) [24] A. Ageev, S. Penn ageev@physics.syr.edu and P.Saulson, Private communication. E-mail:

[25] B. A. Boley and J. H. Weiner Theory of thermal stresses, J.Wiley and sons, 1960, [26] A. P. Prudnikov, Yu. A. Brychkov and O. B. Marychev, Integrals and Sums. Moscow, Nauka, 1983. [27] A. P. Prudnikov, Yu. A. Brychkov and O. B. Marychev, Integrals and Sums. Special Functions. Moscow, Nauka, 1983. [28] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951). [29] Yu. Levin, Phys. Rev. D 57, 659 (1998). [30] P.J. Martin et al Thin solid films 239 181-185 (1994) [31] M.A. Scob ey and P. Stupik, 37-th Annual Thechnical Conference Proceedings, 47-52 (1994). [32] Physical parameters, ed. I.S. Grigorev and E.Z. Meilikhova, Moscow, 1991 (in russian)

47

Appendix D Measurements of the optical mirror coating properties
The results of measurement of optical mirror coating are presented. These results indicate that Standard Quantum Limit of sensitivity can b e reached in the second stage of LIGO pro ject if it is limited by thermo elastic noise in the coating only.

Introduction.
The sensitivity of terrestrial laser interferometer gravitational wave antennae [1] crucially dep ends on how the analysis of different kind of noises in parts of the set up, esp ecially in the optical mirrors (test masses), is in-depth. Few years ago the analysis of the mirrors thermo elastic and thermorefractive noises "contribution" to the total noise budget was carried out [2, 3, 4]. The net result of this analysis is: these typ es of noises can make a significant "contribution" to the total noise if wrong material will b e chosen for the mirrors. The analysis has also shown that fused silica has a serious advantage b ecause of very small thermal expansion co efficient Si O2 5 в 10-7 K-1 . At the same time the role of the mirrors coating has not b een taken into account in this analysis. The coating of the mirror provides very imp ortant parameter of the mirror: the difference b etween the reflectivity R and unity. The smaller is the value of (1 - R), the higher is the sensitivity of the antenna. It is worth noting that the smallness of (1 - R) also defines the sensitivity in many high resolution sp ectroscopical measurements. The analysis of the role of thermo elastic noise pro duced due to the coating has b een carried out recently by S.P. Vyatchanin and by one of this article authors and indep endently by M. Fejer and S. Rowan [5, 6]. The analytical formulas for the values of sp ectral densities of the noises in this publications are practically the same. The only difference is in dimensionless multiplier which dep ends on the ratio of the substrate (the mirror) and coating material constants. But substantial difference in [5] and [6] is in the numerical estimate of the sp ectral densities of the noise. This difference app ears b ecause the numerical value of the thermal expansion factor of tantalum p entoxide Ta2 O5 in [5] has b een b orrowed from the exp eriment by Inci [7]: Ta2 O5 = [-4.43 ± 0.05] в 10-5 K-1 . This value is unexp ectedly negative and large in contrast to [6], where the value of Ta2 O5 = +3.6 в 10-6 K-1 has b een b orrowed from the measurement by Tien et al. [8] (p ositive value but without confidence limits). Due to this difference the predictions of the value of the noise produced by thermo elastic effect in [5] and [6] differs by one order. It is worth noting that this difference b etween these indep endent measurements may not b e due to a mistake (or mistakes) of exp eriment but due to "pathological" prop erties of thin tantalum p entoxide 48

layers. This harsh description of that solid b elongs to S. Witcomb [9], who mentioned that the prop erties of Ta2 O5 layer substantially dep end on the chosen pro cedure of the layer dep osition, and one may exp ect even mechanical and optical anisotropy of this typ e of layer. These circumstances are the motivation for the exp eriments we have carried out and which are describ ed b elow.

The procedures and results of measurements.
To measure the value of thermal expansion factor of tantalum p entoxide Ta2 O5 we used a well known metho d (see e.g. [8, 10]). This metho d is based on the measurement of b ending of thin substrate plate (made of Si O2 in our case) with coating, when plate is heated. The b ending app ears due to the difference Ta2 O5 - Si O2 . If one end of the plate, which length is l, is rigidly attached to massive platform (which is not distorted by change of temp erature) then one can observe the displacement z of the plate free end due to the tension b etween the coating and the substrate that app ears if the plate temp erature changes by T . The value of Ta2 O5 may b e calculated using the following formula which can b e easily derived from formulas (30c) and (32b) of [10]:
Si O
2

-

Ta2 O

5

1 ESi O2 (1 - Ta2 O5 )d2 O2 Si = 3 ETa2 O5 (1 - Si O2 )dTa2 O5 l

2

z T

(D.1)

where ESi O2 is the plate Young mo dulus, ETa2 O5 is the Ta2 O5 Young mo dulus, dTa2 O5 is the total thickness of Ta2 O5 coating, dSi O2 is the thickness of the plate, Ta2 O5 and Si O2 are the Poisson's ratios of Ta2 O5 and plate solid resp ectively. To repro duce the coating for a real mirror manufactured for the LIGO antenna in our tests we have prepared 100 µm flat fused silica plates which were very well p olished and have the shap e of strip es 2 cm long and 1 cm wide. A set of such thin fused silica platesstrip es was placed on a big relatively thick fused silica mini platform. The very ends of each strip e were rigidly attached to the platform. Due to the care of Dr. H. Armandula this platform (with the plates-strip es on it) was placed into the same vacuum chamb er together with big End Mirror for LIGO (10 cm thick, 25 cm in diameter), where the dep osition of the multilayer coating was p erformed. Our fused silica plates-strip es "received" the same 19 pairs of quarter-wavelength Ta2 O5 + Si O2 layers. Thus our plates-strip es were covered with the same coating (which has the thickness of 5.5 µm and the total thickness of Ta2 O5 2.2 µm) as the big mirror. This coating provides the value of (1 - R) 1 в 10-5 . When the pro cess of dep osition was over and the plates-strip es were detached from the platform we have got the first (unexp ected) exp erimental result: all plates-strip es were b ent. The measured value of the curvature radius of these plates-strip es was 11 cm (in the plane of long axis of the strip es). This b ending remained unchanged many weeks after the detachment from the platform. The main elements of the measuring set up are the two massive fused silica rectangular blo cks which were rigidly attached to a heavy optical b ench. A plate-strip e was rigidly clamp ed b etween two blo cks. To evade damage due to the clamping a 15 µm teflon film was inserted b etween plate-strip e and the surfaces of fused silica blo cks. Power stabilized He-Ne laser, two lenses and photo dio de were installed on the b ench. This simple optical system p ermitted to measure 0.1 µm displacement of the plate-strip e edge ("knife and slot" technique). Small b ox around a plate-strip e p ermitted to heat the plate-strip e by a slow stream of heated air. 49

Four sets of measurements with two different plates were carried out. The measurements with each plate were p erformed two times, second time` with the plate flipp ed up side down. The value of [z /T ] was obtained from the directly measured dep endence of the edge displacement at the free end of plate-strip e on temp erature. By using formula erg (D.1) with parameters Si O2 = 5.5 в 10-7 K-1 , ESi O2 = 7.2 в 1011 cm3 , Si O2 = 0.17, erg ETa2 O5 = 1.4 в 1012 cm3 , Ta2 O5 = 0.23 and measured [z /T ] the linear thermal expansion co efficient Ta2 O5 was estimated. Its value is equal to:
Ta2 O
5

= [5 ± 1] в 10

-6

K-1 .

Conclusion.
The obtained value of Ta2 O5 is evidently closer to the one obtained by Tien et al. [8] then to the one obtained by Inci [7]. In the same time our value is almost two times larger then the smaller value [8]. In addition we have to note that our value of Ta2 O5 is based not only on the measured value of z but also on assumption that the values of ESi O2 and ETa2 O5 (measured by other exp erimentalists) are correct. Taking into account that these values may b e substantially different in thin layers and that the pro cess of dep osition may also cause changes of the Young mo duli values (see e. g. [11]), we may conclude that the value of Ta2 O5 is b etween 3 в 10-6 K-1 and 7 в 10-6 K-1 . The only "remedy" to solve this problem is the direct measurement of the mirror with coating surface fluctuations sp ectral density. At the same time the limits (from 3 в 10-6 K-1 to 7 в 10-6 K-1 ) p ermit to cancel the p essimistic prediction for the sp ectral density of the thermo elastic noise in the coating [5] which was based on Inci's value Ta2 O5 = [-4.43 ± 0.05] в 10-5 K-1 . For example, the p otentially achievable sensitivity for LIGO-I I near the frequency of observation f 102 Hz may not b e at the level of Sh 1.5 в 10-23 1/ Hz (as in [5]) but b etween 0.6 в 10-24 1/ Hz Sh 1.4 в 10-24 1/ Hz. It means that the pro jected sensitivity of LIGO-I I I (b etter than the Standard Quantum Limit) will b e evidently not p ossible due to this noise. To the ab ove conclusive remarks we have to add that the "unpleasant discovery" i.e. the observation of the "frozen" b ending of the plates-strip es with coating, which indicates the existence of mechanical stresses in the coating has to b e regarded as strong argument in favor of direct measurement of the noises in coating, esp ecially the measurement of 1/f comp onent. The authors of this article are expressing their sincere gratitude to S.P. Vyatchanin and H. Armandula for their advice and direct help. This work was supp orted by NSF grant PHY-0098-715 and by Russian Ministry of Industry and Science.

50

Bibliography
[1] Abramovici A.A. et al., LIGO: The laser interferometer gravitational-wave observatory. Science, v. 252, 1992, p. 325 [2] V.B. Braginsky, M.L. Goro detsky, S.P. Vyatchanin, Phys. Lett. A 264, 1 (1999). [3] Yu.T. Liu, K.S. Thorne, Phys. Rev. D62, 122002 (2002). [4] V.B. Braginsky, M.L. Goro detsky, S.P. Vyatchanin, Phys. Lett. A 271, 303-307 (2000). [5] V.B. Braginsky, S.P. Vyatchanin, Thermo dynamical fluctuation in optical mirror coatings. arXiv:cond-mat10302617, 3 Mar 2003. [6] M.Fejer, S.Rowan rep ort at LSC meeting, Livingstone, March 18-21, 2003. [7] Inci M.N. Simultaneous measurements of thermal optical and linear thermal expansion co efficients of Ta2 O5 films, ICO 19, 25-30 August 2002, Firenze, Italy. [8] Tien C.L. et al., Simultaneous determination of the thermal expansion co efficient and the elastic mo dulus of Ta2 O5 thin film using phase shifting interferometry., J. of Mo d. Opt., 2000, v. 47, no. 10, 1681-1691. [9] S.Whitcomb private communication. [10] P.H. Townsend et al., Elastic relationship in layered comp osite media with approximation for case of thin films on a thick substrate. J. Appl. Phys., v.62 (11), 1987, p.4438-4444. [11] Martin P.J. et al., Mechanical and optical prop erties of thin films of tantalum oxide dep osited by ion-assisted dep osition, Mat. Res. So c. Symp. Pro c., v. 308, 1993, p.583-588.

51

Appendix E Low pumping energy mode of the "optical bars"/"optical lever" topologies of gravitational-wave antennae
The "optical bars"/"optical lever" top ologies of gravitational-wave antennae allow to obtain sensitivity b etter that the Standard Quantum Limit while keeping the optical pumping energy in the antenna relatively low. Element of the crucial imp ortance in these schemes is the lo cal meter which monitors the lo cal test mirror p osition. Using cross-correlation of this meter backaction noise and its measurement noise it is p ossible to further decrease the optical pumping energy. In this case the pumping energy minimal value will b e limited by the internal losses in the antenna only. Estimates show that for values of parameters available for contemp orary and planned gravitational-wave antennae, sensitivity ab out one order of magnitude b etter than the Standard Quantum Limit can b e obtained using the pumping energy ab out one order of magnitude smaller energy than is required in the traditional top ology in order to obtain the the Standard Quantum Limit level of sensitivity.

Introduction
First generation large-scale laser interferometric gravitational-wave antennae [1, 2] are b eing placed into op eration nowadays [3]. The second generation of laser gravitational-wave antennae development is under way concurrently [4]. Sensitivity of these second generation antennae will b e close to the Standard Quantum Limit (SQL) [5] that is characteristic sensitivity level where the measurement noise (i.e. the shot noise in the laser interferometric devices) and the back-action noise (i.e. the radiation pressure noise) contribute equal parts to the measurement error: S
SQL h

() =

4 . M 2 L2

(E.1)

SQL Here Sh () is the double-sided sp ectral density of the equivalent noise for the dimensionless amplitude of the metrics p erturbation h(t), is the mean frequency of the gravitational-wave signal, M is the mass of the interferometer mirrors, and L is the length of the interferometer arms. Further improvement of the sensitivity will require to use the Quantum Non-Demolition (QND) measurement metho ds [6, 7, 8] which allow to eliminate the part pro duced by the back-action noise from the meter output. Several p ossible design of QND laser

52

gravitational-wave antennae have b een prop osed already. They can b e divided into three main groups. The first group is based on the fact that the value of the SQL dep ends on the nature of the test ob ject. In particular, a harmonic oscillator provides b etter sensitivity in the narrow band close to its resonance frequency than a free mass do es, even if the same SQL-limited meter is used in b oth cases [9]. It was shown in articles [10, 11] that in the signal-recycling configuration of the interferometric gravitational-wave antennae an optical rigidity can b e created rather easily that will turn test masses into mechanical oscillators. Moreover, this optical rigidity has sp ecific sp ectral dep endence which allows to obtain sensitivity b etter than the SQL for b oth free mass and ordinary harmonic oscillator. Using this metho d it is p ossible to obtain sensitivity a few times b etter than the SQL for a free mass in a relatively wide band [10, 11] or "dive" deep b elow the SQL in a narrow band [12]. It is necessary to note that b oth regimes require ab out the same optical pumping energy as standard SQL-limited schemes. It is the author's opinion that these relatively simple metho ds could (and should) b e implemented already in the second generation of laser gravitational-wave antennae. The second group of metho ds requires more substantial mo difications of the laser gravitational-wave antenna top ology which convert it into a QND device. Examples of this approach are: interferometer with mo dified input and/or output optics, which implements the sp ectral variational measurement [13] and different implementations of the quantum speedmeter scheme [14, 15, 16, 17, 18]. In principle, they allow to obtain arbitrarily high sensitivity. In practice, however, they "suffer" from very high optical p ower circulating in the interferometer arms, which also dep ends sharply on the required sensitivity. It can b e shown (see [19]) that the optical energy has not to b e smaller than the value of E= E
SQL 2 2 2

(E.2)

where is the ratio of the signal amplitude which can b e detected to the amplitude corresp onding to the Standard Quantum Limit (the smaller is the b etter is the sensitivity), and is the squeezing factor ( = 1 for the optical field coherent quantum state and < 1 for the squeezed state), M L 2 3 (E.3) 2o is the optimal energy for the SQL-limited interferometric antenna, o is the pumping frequency, and L is the length of the interferometer arms 1 . Formula (E.3) is valid in the wide-band regime where the bandwidth . In the narrow-band regime the energy can b e reduced by the factor of /. If, for example, M = 40 Kg, L = 4 Km, = 2 в 100 s-1 and o = 2 в 1015 s-1 (these values corresp ond to the prop osed gravitation-wave antenna LIGO-I I [4]) then ESQL 40 J. Corresp onding circulating optical pumping p ower is equal to E
SQL

=

Numerical factors in formulae (E.1, E.2, E.3) correspond to the case of the standard Michelson-- Fabry-Perot topology with four mirrors having equal masses M , which was analyzed in detail in article [13]. Note also the factor 1/2 in formula (E.2), which arises for the back-action noise that makes up half of the net noise in the SQL-limited detectors, is eliminated from the output signal of the QND detectors. It allows to obtain sensitivity equal to the SQL ( = 1) using to times smaller energy or, put it otherwise, w to obtain sensitivity 2 times better than the SQL ( = 1/ 2) using E = ESQL .

1

53

A

g replacements

C

B

Figure E.1: The "optical bars" intracavity scheme c ESQL 0.75 MWt . (E.4) 4L It is p ossible to conclude that the feasibility of these second group metho ds dep ends crucially on the exp erimental progress in preparation of highly squeezed quantum states (with 1). The third and the most radical approach, intracavity readout scheme, was prop osed in the article [20]. It was prop osed to measure directly the redistribution of the optical energy inside the antenna in a QND way (without absorption of the optical quanta) instead of monitoring output light b eam outside the antenna using photo detectors. In this case the necessary non-classical quantum state of the optical field [factor in the formula (E.2)] is generated automatically and therefore the pumping energy do es not dep end directly on the required sensitivity. In the article [21] a p ossible implementation of the gravitational-wave antenna with intracavity detection, so-called "optical bars" scheme, was prop osed (see FIG. E.1). In this scheme end mirrors A, B, and an additional lo cal mirror C form two Fabry-Perot cavities coupled by means of the mirror C amplitude transmittance T . Relatively weak external pumping (not shown in the picture) is necessary in order to comp ensate internal losses in the optical elements and to supp ort steady value of the optical energy in the cavities. The optical field acts here as a two rigid springs with one lo cated b etween the mirrors A and C, while the second one (L-shap ed) lo cated b etween the mirrors B and C. The rigidity of these springs is equal to W
SQL

=

K= where

2o E , L2 B

(E.5)

cT . (E.6) L is the sloshing frequency of the system of two coupled cavities AC and BC [strictly sp eaking, K is equal to the asymptotic value of the rigidity at B ; see formula (E.52)]. Due to these springs displacement of the end mirrors A and B caused by the gravitational wave pro duces displacement of the lo cal mirror C, which can b e detected, for B = 54

A

g replacements A' C B' B

Figure E.2: The "optical lever" intracavity scheme example, by a small-scale optical interferometric meter which monitors p osition of the mirror C relative to reference mass placed outside the optical pumping field. It was shown in the article [21] that if optical energy exceeds some threshold value of Ethres then these springs are rigid enough to provide the signal displacement of the lo cal mirror C equal to the displacement of the end mirrors. In this case the sensitivity do es not dep end on the optical energy and is defined by the sensitivity of the lo cal meter only (compare with formula (E.2) which contains the factor -2 ). However, it was concluded in the article [21] that the threshold energy have to b e rather large and close to E SQL . In the article [22] an improved version of the "optical bars" scheme was considered. It differs from the original "optical bar" scheme by two additional mirrors A' and B' (see FIG. E.2) which turn the antenna arms into two Fabry-Perot cavities AA' and BB' similar to the standard Fabry-Perot -- Michelson top ology of the contemp orary gravitationalwave antennae. This scheme was called the "optical lever" b ecause it can provide significant gain in the signal displacement of the lo cal mirror similar to the gain which can b e obtained using ordinary mechanical lever with unequal arms. The value of this gain is equal to 2 F, (E.7) where F is the finesse of the Fabry-Perot cavities AA' and BB'. It was shown in the article [22] that in all other asp ects the "optical lever" scheme is identical to the "optical bars" one but in the former one the lo cal mirror C transmittance T have to b e times larger, and its mass have to b e 2 times smaller. Due to this scaling of mass the gain in the signal displacement by itself do es not allow to overcome the SQL as the SQL value increases exactly in the same prop ortion. But it allows to use less sensitive lo cal p osition meter and increases the signal-to-noise ratio for miscellaneous noises of non-quantum origin. No means of reducing the optical p ower b elow the level of ESQL were discussed in the article [22]. At the same time different regimes of the "optical bars" scheme were considered in brief in the article [23], and it was mentioned that if a lo cal meter with cross-correlated measurement noise and back-action noise is used then the threshold energy can b e substantially lower than the ESQL (see subsection 4.2 of that article). It is evident that this = 55

M
1 2

F

1 2

K

m

1 2

K

M
1 2

grav

F

grav

x

1

y

x

2

Figure E.3: Mechanical mo del conclusion is also valid for the "optical lever" scheme. In the current article this cross-correlation regime is analyzed in detail.

Mechanical model
Consider the mechanical mo del shown in FIG. E.3. Here masses M corresp ond to the end mirrors of the gravitational-wave antenna (compare with FIG. E.1), mass m corresp onds to the lo cal mirror, and springs K/2 corresp ond to the optical rigidity. Equal signal forces Ё Mh (E.8) 2 2 acts on b oth end masses M , and the goal is to detect the signal by monitoring the lo cal mass m state. Supp ose that conditions for the "intermediate case" of the article [23] are fulfilled, namely, the lo cal mass m is small, F
grav

=

m

2

K,

(E.9)

where is the signal frequency and the rigidity K is also relatively small: K 2M 2 , (E.10)

Due to condition (E.9) the signal displacement of the lo cal mass will b e equal to the signal displacement of the end ones. However, due to condition (E.10) the Standard Quantum Limit for the p osition of the lo cal test mass ySQL will b e much larger than the Standard Quantum Limit for the p ositions of the heavy end masses . (E.11) M 2 Really, let ymeas b e the measurement precision provided by the lo cal meter. Due to the uncertainty relation p erturbation of the mass m momentum in this case will b e equal to x
SQL

p

p ert

=

2y

.
meas

(E.12)

It corresp onds to the random force with the uncertainty equal to F
p ert

=

2y

meas

,

(E.13)

which pro duces the additional random displacement of the lo cal test mass y
p ert

=

2K y 56

meas

.

(E.14)

Therefore, the sum error will b e equal to
2

y =

(y

meas

)2 +

K y

meas

.

(E.15)

Minimum of this expression is equal to , (E.16) 2K and this value is M 2 /K 1 times larger than the Standard Quantum Limit (E.11). Due to this consideration it was concluded in the article [21] that in order to obtain sensitivity close to the xSQL it is necessary to use strong rigidity: y
SQL

=

K

M 2 ,

(E.17)

and therefore, large pumping energy. Consider, however, the situation more precisely. If condition (E.9) is fulfilled then equations of motion (in the sp ectral representation) of the system shown in FIG. E.3 lo oks like: -2M 2 x() + K x() = K y () + Fgrav () , ^ ^ ^ ^ K y () = K x() + Fmeter () , ^ ^ ^ where x = (x1 + x2 )/2 and Fmeter is the back-action force of the lo cal meter. It follows from these equations that the output signal of the meter is equal to: ^ ^ () + Fmeter () Fmeter () + + ymeter () , ^ (E.20) -2M 2 K where ymeter is the measurement noise which determines the measurement error ymeas . ^ It is easy to see that if the measurement noise contains the part prop ortional to the back-action force: y () = ~ F
grav

(E.18) (E.19)

ymeter = y ^

(0) meter

-

F

meter

K

,

(E.21)

^ then the main back-action term Fmeter /K in the equation (E.20) vanishes: y () = ~ F
grav

^ Of course strong p erturbation Fmeter /K still exists in this case but the meter do es not "see" it as it is masked by the correlated with Fmeter part of ymeter . Equation (E.22) corresp onds exactly to the output signal of the p osition meter with the (0) measurement noise ymeter attached directly to a test mass 2M . In particular, if an ordinary ^ interferometric p osition meter with constant pumping p ower and time- and frequencyindep endent phase is used, then sensitivity of such a scheme will b e limited by the SQL (E.11) even if the rigidities are much smaller than M 2 , as long as condition (E.9) is fulfilled.

^ () + Fmeter () (0) + ymeter () . ^ 2 -2M

(E.22)

57

Sensitivity Limits
We will consider here only two main factors which limit the sensitivity of the scheme b eing considered: optical losses and noises of the lo cal meter putting aside numerous sensitivity limits which are common for all top ologies of the laser gravitational-wave antennae (in particular, miscellaneous internal noises in the mirrors and mirror susp ensions). In this case output signal of this scheme normalized as an equivalent gravitational-wave signal can b e presented as ~ h=h+h
loss

+h

meter

,

(E.23)

where h is the actual gravitational-wave signal, hloss is the noise which arises due to the optical losses, and hmeter is the one created by the lo cal meter fluctuations. These noises are calculated b elow.

Optical losses
Taking into account formulae (E.64), (E.56), sp ectral density of the noise h presented as S
loss h loss

can b e

2 L2 , (E.24) o E where is the Fabry-Perot cavities damping rate. Ratio of this sp ectral density to the sp ectral density (E.1) which corresp onds to the SQL is equal to 2 =
2 loss loss Sh ESQL M 2 = SQL = = . 2o E E Sh

(E.25)

This formula resembles formula (E.3) b ecause in b oth these formulae the larger is ratio E /ESQL the b etter is the sensitivity. At the same time formula (E.25) contains the factor / which in principle can b e made very small. In particular, mo dern achievements in fabrication of high-quality mirrors p ermits to obtain 1 s-1 10-3 . Therefore, even, say, for E 0.1ESQL sensitivity of ab out one order of magnitude b etter than the SQL, loss 0.1 could b e achieved.

SQL-limited lo cal meter
Supp ose that a SQL-limited p osition meter with noise ymeter and back-action noise Fmeter are used small-scale optical interferometer with the length Supp ose also that these noises are cross-correlated ymeter = ymeter - ^ ^
2

frequency-indep endent measurement as the lo cal meter. For example, a l L can b e used as such a meter. [compare with formula (E.21)]: (E.26)

(0)

^ 2M Fmeter , 2M + m K

The last term in formula (E.25) looks a bit misleading because it seems that the smaller is the worse is sensitivity. It is easy to see that it is not the case because ESQL also depends on and the worst case takes place at the highest signal frequency. Therefore, it is for this frequency all estimates should be done.

58

10

S
1 1

PSfrag replacements
3 1

meter () h asympt (SQL h

S

)

2

/

SQL

Figure E.4: Sp ectral density of the noise of the SQL-limited lo cal meter for different values of the optical energy E : 1 -- asymptotic curve for E ; 2 -- the optimal energy; 3 -- the energy is to o small. This simple frequency-indep endent cross-correlation can b e created rather easily by using a homo dine detector with the fixed lo cal oscillator phase 3 . Sp ectral densities of the measurement noise Sy and the back-action noise SF satisfy the uncertainty relation
2

, 4 and their ratio dep ends on the optical pumping energy E Sy SF =
2

(E.27)
lo cal

in the meter:

SF 8QElocal = , (E.28) Sy l2 where Q is the quality factor of the lo cal meter cavity. meter The sp ectral density Sh () of the noise hmeter for this meter has rather sophisticated sp ectral dep endence [see formulae (E.68), (E.69)] which allows in principle to obtain sensitivity b etter than the SQL in some narrow band. In this pap er, however, wide-band optimization only will b e considered. It is easy to see that if the pumping energy E is very large (E ESQL ) then the lo cal meter output is close to the output of the meter attached directly to the test mass 2M + m Ё with the signal force M Lh acting on this mass. Sp ectral density of the lo cal meter noise hmeter in this case is equal to S
3

asympt h

SF 1 () = 2 + L M 2 4
(0)

2M + m M

2

S

y

2M + m . M M 2 L2

(E.29)

In principle, the noises ymeter and Fmeter can also be made cross-correlated by this method. It can be shown, however, that this additional cross-correlation does not provide any significant advantages so we do not consider this possibility here.

59

(see curve 1 in FIG. E.4). The rightmost part of this formula corresp onds to the SQL for the considered scheme which, as well as the SQL (E.1), is reached at some sp ecific frequency = SQL only. In the case considered here this frequency is equal to
SQL

=

4

1 SF = (2M + m)2 Sx

8QElocal . (2M + m)l2

(E.30)

It should b e noted that if m M then the SQL (E.29) corresp onds to 2 times b etter sensitivity than the SQL for traditional schemes (E.1). This gain is obtained b ecause two mirrors M are required for this scheme instead of four ones as in traditional schemes. At the same time if E is to o small then sp ectral density (E.69a) of back-action noise increases sharply at high frequencies; however, some narrow-band gain in sensitivity can b e obtained in this regime (see curve 3 in FIG. E.4). Reasonable intermediate value of E can b e chosen using, for example, the following asympt meter algorithm: (i) require that Sh () do es not exceeds Sh () all over the sp ectral range of interest, up to some given frequency max (see curve 2); (ii) with resp ect to this requirement set E as small as p ossible; in any case it have to b e much smaller than ESQL . In addition condition (E.60) which is necessary for the system dynamic stability (see app endix D of the article [21]) is also should b e taken into account. It is easy to show that these requirement can b e satisfied only if the sloshing frequency is large,
B

max

,

(E.31)

and in this case the optimal values are equal to 4 2M m = 12M max , 2M + m 4 B 3 M L 2 4 max max E= = 3ESQL . 2 o B B m

(E.32) (E.33)

Supp ose, for example, that T 0.1, L = 4 Km, and therefore B 7.5 в 103 s-1 . In this case if max = 2 в 102 s-1 , then E 0.25ESQL and m m 25 g. In principle, more transparent mirror C can b e used, which allows further decrease of E . However, E dep ends on B as -1 only, while m dep ends as -4 . Therefore, B B the smaller values of E corresp ond to very small (sub-gram) values of the mass m. It is unclear, if such a small mirror could tolerate tens of the kilowatts of the optical p ower reflecting from it and hundreds of watts passing through it.

QND lo cal meter
The small-scale optical interferometer discussed in the previous subsection can b e converted into a QND meter by using, for example, so-called Strob oscopic-Variation Measurement (SVM) technique (see articles [24, 25, 26]). This metho d p ermits to filter out the back-action noise by using p erio dic mo dulation of the lo cal oscillator phase and/or the pumping energy Elocal with frequency which has to b e higher than the upp er frequency of the gravitational-wave signal. For small-scale interferometers this mo dulation can b e implemented rather easily. The residual noise sp ectral density will b e prop ortional to the sp ectral density Sy of the meter measurement noise yfluct and in principle can b e made arbitrarily small [see 60

formulae (E.69b, E.70)] by reducing Sy (i.e. by increasing the lo cal meter sensitivity). However, for technological reasons it is useful to provide the value of the lo cal mirror signal displacement ygrav as large as p ossible [see formula (E.66)]. An optimization algorithm similar to one considered in the previous section can b e used here. Start with the simple quasi-static (low-frequency) case, when m 2 K . In this case the signal displacement is equal to M Lh . 2M + m and corresp onding measurement noise is equal to y
grav

=

(E.34)

S

asympt meas

1 =2 L

2M + m M

2

Sy .

(E.35)

asympt Require now that Smeas () do es not exceeds Smeas for all frequencies within the range 0 max . It is shown b elow that in this case the pumping energy has to satisfy the following inequality:

Ek

m L 2 B 2o

2 max

(E.36)

where k is a numerical factor which varies from 1/8 when B = max to 1/2 when B max [see formula (E.73)]. This inequality together with the stability condition (E.60) can b e rewritten as the following condition for the mass m : 1 4 max B
3

Values of m and B in formula (E.36) can vary in wide range and should b e chosen considering technological reasons. In principle, heavy lo cal mirror C with low transmittance T (i.e low sloshing frequency) as well as relatively small one with high transmittance T (i.e high sloshing frequency) can b e used. Supp ose that E = 0.1ESQL . This value lo oks like the reasonable one due to limitation caused by the internal losses. Supp ose also that M = 40 Kg and max = 2 в 100 s-1 . In this case typical numerical examples are the following. Heavy lo cal mirror: T = 0.01 , B 750 s-1 , 10 Kg m 16 Kg ; small lo cal mirror: T = 0.1 , B 7500 s-1 , 10 g m 700 g . Of course all intermediate values b etween these two examples are also p ossible.

E

E
SQL

m 1 max E . M k B ESQL

(E.37)

(E.38)

(E.39)

61

The "optical lever" scheme
In principle, the same idea of the lo cal meter with cross-correlated noise can b e used in order to reduce optical pumping energy in the "optical lever" scheme to o. However, due to technological limitations in the case of SQL-limited lo cal meter only mo dest advantages can b e obtained. Really, the sloshing frequency in the "optical lever" scheme is equal to 7.5 в 104 s-1 cT < . (E.40) L (it is supp osed that L = 4K m). At the same time it follows from the formula (E.33) that the sloshing frequency have to b e equal to B = 3ESQL ESQL max 2 в 104 s-1 . (E.41) E E (it is supp osed that max = 2 в 102 s-1 ). Due to these limitations it is imp ossible to obtain significant gain in the signal displacement using the pumping energy E < ESQL . On the other hand, in the case of a QND lo cal meter low sloshing frequency B max can b e used [see formulae (E.38)] which makes it p ossible obtain the gain up to B = cT 102 . (E.42) Lmax the "optical lever" scheme have to b e 2 time smaller It follows from the estimates (E.38) that it has to b e not seems unrealistic one as the optical p ower falling of and for the case of E 0.1ESQL will b e equal to =

The lo cal mirror mass in the case of than the figures of formula (E.38). equal to ab out 1 g. This value do es on it will b e reduced by the factor ab out few hundred watts.

Conclusion
In the authors opinion, regime of the optical bars/optical lever intracavity top ologies considered in this pap er lo oks rather promising for implementing in the third generation of gravitational-wave antennae. It allows to obtain sensitivity b etter than the SQL and it can do it using rather mo derate value of the optical pumping energy: just tens of kilowatts of circulating p ower, instead of megawatts or tens of megawatts. At the same time, b efore the implementation of this metho d, several issues of a technological origin have to b e solved. Some of them are common for all prop osed top ologies of the laser gravitational-wave antennae: the problem of an internal noises of different origin in the test masses (see, for example, pap ers [27, 28, 29, 30]) is the most notable one, and some are sp ecific for the intracavity top ologies. It seems that the most imp ortant of them is the design of the lo cal meter, and it is evident that this device has to b e explored intensively, b oth exp erimentally and theoretically b efore the decisions ab out the design of the third generation laser gravitational-wave antennae will b e made.

Acknowledgments
Author thanks V.B.Braginsky, S.L.Danilishin, M.L.Goro detsky and S.P.Vyatchanin for useful remarks. This pap er was supp orted in part by NSF and Caltech grant #PHY0098715, by the Russian Foundation for Basic Research, and by the Russian Ministry of Industry and Science. 62

PSfrag replacements

1 2

F

grav

M x
1

m y

M

1 2

F

grav

x

2

Figure E.5: Two cavities mo del

System Dynamics
In this app endix we will consider a simplified mo del of the system similar to one used in the original pap er [21]. Replace the Fabry-Perot cavities AC and BC (see FIG. E.1) by two coupled single-mo de e.m. cavities which eigenfrequencies dep end on the p ositions on the mirrors, see FIG. E.5 Equations of motion for this system lo oks like: Ё q1 (t) + 2 q1 (t) + ^ ^ x1 (t) - y (t) ^ ^ 1+ L x2 (t) - y (t) ^ ^ L
2

2 o

q1 (t) + o B q2 (t) = ^ ^
2

o [U

pump

^ (t) + U1 (t)] , (E.43a)

Ё q2 (t) + 2 q2 (t) + ^ ^

2 o

1-

q2 (t) + o B q1 (t) = ^ ^

o [U

pump

^ (t) + U2 (t)] , (E.43b)

o q1 (t) Fgrav (t) ^2 Ё M x1 (t) = - ^ + , L 2 o q2 (t) Fgrav (t) ^2 Ё M x2 (t) = ^ + , L 2 o 2 Ё ^ q1 (t) - q2 (t) + Fmeter (t) , ^ ^2 my (t) = ^ L

(E.43c) (E.43d) (E.43e)

where q1,2 are the generalized co ordinates of the cavities, Upump is the pumping voltage, ^ B is the sloshing frequency which is prop ortional to the coupling of the cavities, is the ^ damping rate of the cavities, U1,2 are the corresp onding fluctuational voltages, x1,2 are ^ the p ositions of the masses M , y is the p osition of the mass m, Fgrav is the signal force, ^ ^ Fmeter is the back-action force of the lo cal meter which monitors variable y (not shown in the picture). Let intro duce new variables then q1 (t) ± q2 (t) ^ ^ , 2 x1 (t) + x2 (t) ^ ^ x(t) = ^ , 2 ^ ^ U1 (t) ± U2 (t) ^ U± (t) = , 2 x1 (t) - x2 (t) ^ ^ ^ X (t) = . 2

q± (t) = ^

(E.44a) (E.44b)

For these variables we obtain: 2 2 ^ 2 Ё ^ ^ ^^ q+ (t) + 2 q+ (t) + + q+ (t) + o X (t)q+ (t) + [x(t) - y (t)]q- (t) ^ ^ ^ L o ^ [ 2 Upump (t) + U+ (t)] , = 63

(E.45a)

2 2 Ё q- (t) + 2 q- (t) + - q- (t) + ^ ^ ^ L

o ^^ U- (t) , (E.45b) [x(t) - y (t)]q+ (t) + X (t)q- (t) = ^ ^^ 2o Ё 2M x(t) = - ^ q+ (t)q- (t) + Fgrav (t) , ^ ^ (E.45c) L o 2 Ё ^ [q (t) + q- (t)] , ^ ^2 (E.45d) 2M X (t) = - L+ 2o Ё ^ q+ (t)q- (t) + Fmeter (t) , ^ ^ (E.45e) my (t) = ^ L

2 o

where ± = o ± B /2. Supp ose that the pumping frequency is equal to + and amplitude of the pumping field in the mo de "+" is equal to q0 . Keeping only linear in q0 term in the right parts of the equations (E.45), these equation can b e rewritten as: q+ (t) = q0 cos + t , o ^ 2 2 q0 2 Ё q- (t) + 2 q- (t) + -q- (t) = ^ ^ ^ ^ ^ U- (t) + o [y (t) - x(t)] cos + t , L 2o q0 Ё 2M x(t) = - ^ q- (t) cos + t + Fgrav (t) , ^ L 2o q0 Ё ^ q- (t) cos + t + Fmeter (t) . ^ my (t) = ^ L Using then the rotation p olarization approximation: q- (t) = qc (t) cos + t + qs (t) sin + t , ^ ^ ^ q- (t) = + (qc (t) cos + t + qs (t) sin + t) , ^ ^ ^ ^ ^ ^ U- (t) = Uc (t) cos + t + Us (t) sin + t , (E.47a) (E.47b) (E.47c) (E.46a) (E.46b) (E.46c) (E.46d)

we obtain a simple linear equations set which is convenient for sp ectral representation: ^ Us () , 2 ^ Uc () o q0 + [y () - x()] , ^ ^ -B qc () + (i + )qs () = ^ ^ 2 L o q0 qc (t) + Fgrav () , ^ -2M 2 x() = - ^ L o q0 ^ -m2 y () = ^ qc () + Fmeter () . ^ L (i + )qc () + B qs () = - ^ ^ From the first two equations we obtain: ^ ^ (i + )Us () + B Uc () o B q0 1 + [y () - x()] , ^ ^ D () 2 L

(E.48a) (E.48b) (E.48c) (E.48d)

qc () = - ^ where

(E.49)

D () = (i + )2 + 2 . B Substitution of this value of qc into the last two equations of (E.48) gives: 64

(E.50)

^ [-2M 2 + K()]x() - K()y () = Floss () + Fgrav () , ^ ^ ^ ^ [-m2 + K()]y () - K()x() = -Floss () + Fmeter () , ^ ^ where K() = is the complex p ondermotive rigidity, K 2 2o E B B = L2 D () D () o q 2
2 0

(E.51a) (E.51b)

(E.52)

E= is the pumping energy and ^ Floss () =

(E.53)

is the fluctuational force which arises due to losses in the cavities. Sp ectral densities of ^ Uc,s (t) are equal to S therefore, sp ectral density of F S
loss U
c,s

o q 0 (i + )Us () + B Uc () 2LD ()

(E.54)

= 4 ,

(E.55)

(t) is equal to =
2 2 o E 2 + 2 + B . L2 |D ()|2

F loss

(E.56)

It follows from the equations (E.51), that the p osition of the lo cal mass is equal to y () = y ^ where y
grav grav

() +

^ ^ [K() - 2M 2 ]Fmeter () + 2M 2 Floss () , -(2M + m)2 [K() - m 2 ] M K() Lh() 2M + m K() - m 2 m =

(E.57)

() =

(E.58)

is the signal displacement and

2M m . (E.59) 2M + m It have to b e noted also that as it was shown in the article [21] this system is dynamically unstable. If m 2 B , (E.60) 4 then this instability is relatively small and can b e rather easily suppressed by a feedback system. In the opp osite case, however, very strong asynchronous instability arises (see app endix D of the article [21]) which makes the scheme virtually useless. Therefore, condition (E.60) has to b e considered as necessary one. K

65

The output signal
The output signal of the lo cal meter is equal to y () = y () + ymeter () , ~ ^ ^ where ymeter () is the measurement noise. ^ If the meter noises are cross-correlated, see formula (E.26), then 1 -(2M + m) [K() - m 2 ] 2M m 4 ^ K() (0) ^ -1 - K() + 2M 2 Fmeter () + 2M 2 Floss () + ymeter () , ^ K K (E.62)
grav

(E.61)

y () = y ~ в

() +

2

It is convenient to present this expression as follows: y () = ~ where M K()L 2M + m K() - m ^ h
2

^ h() + h

loss

^ () + h

meter

() ,

(E.63)

loss

() = -

is the equivalent noise pro duced by the optical losses, and 1 L 1 1 1 -2 - 2 M K K() 2m 2 ^ Fmeter () K K() 2M + m 2m 2 + 1- M K() +

2Floss () K()L

(E.64)

^ h

meter

() =

-

ymeter () ^

(E.65)

is the equivalent noise of the meter (b oth these noises are normalized as an equivalent fluctuational gravitational-wave signals). 2 Taking into account that in order to obtain loss < 1 [see formula (E.25)] the optical losses have to b e small, , expressions (E.58), (E.65) can b e slightly simplified: y
grav

() =

M 2M + m

Lh() , m 2 (2 - 2 ) B 1- K 2 B

(E.66)

^ h

meter

() =

1 L

-

1 2m 2 (2 - 2 ) ^ 22 B + Fmeter () - M 2 K 2 K 2 2 B B m 2 (2 - 2 ) (0) 2M + m B 1- + ymeter () ^ M K 2 B

(E.67)

If a SQL-limited lo cal meter is used, then sp ectral density of this noise can b e presented as a sum S
meter h

() = S

B.A.

() + S

meas

() ,

(E.68)

66

where 1 1 22 2m 2 (2 - 2 ) B - - + 2 2 2 2 2 L M K B KB 1 L2 2M + m M
2 2

S

B.A.

() =

SF ,
2

(E.69a) (E.69b)

S

meas

() =

1-

m 2 (2 - 2 ) B K 2 B

S

y

^ and SF and Sy are sp ectral densities of the meter back-action noise Fmeter and its measurement noise ymeter (). ^

Noise optimization for a QND local meter
In the case of a QND lo cal meter the back-action noise can b e filtered out by some means and sensitivity is limited by the measurement noise only: S
meter h

() = S

meas

() .

(E.70)
max

Require that Smeas () has not to exceed Smeas (0) for all frequencies 0 this case the following condition has to b e fulfilled for all these frequencies: 1- m 2 (2 - 2 ) B 1. K 2 B

. In

(E.71)

The solution of this inequality can b e presented as follows: K k m where 2 1 B , 8 max k= 1 1 - max 2 B
2 max 2 max

,

(E.72)

B 2
max

.

2

max

(E.73)

, B >

67

Bibliography
[1] K.S.Thorne, 300 Years of Gravitation, Cambridge University Press, 1987. [2] A.Abramovici et al, Science 256, 325 (1992). [3] See the recent information at http://www.ligo.caltech.edu. [4] E.Gustafson, D.Sho emaker, K.A.Strain and R.Weiss, LSC White pap er on detector research and development, LIGO Do cument Numb er T990080-00-D, 1999, (LIGO Do cument Numb er T990080-00-D, www.ligo.caltech.edu/do cs/T/T990080-00.p df ). [5] V.B.Braginsky, Sov. Phys. JETP 26, 831 (1968). [6] V.B.Braginsky,Yu.I.Vorontsov, F.Ya.Khalili, Sov. Phys. JETP 46, 705 (1977). [7] Thorne K.S., R.W.P.Drever, C.M.Caves, M.Zimmerman, V.D.Sandb erg, Physical Review Letters 40, 667 (1978). [8] Braginsky V.B., Yu.I.Vorontsov, and K.S.Thorne, Science 209, 547 (1980). [9] V.B.Braginsky, F.Ya.Khalili, Physics Letters A 257, 241 (1999). [10] A.Buonanno, Y.Chen, Physical Review D 64, 042006 (2001). [11] A.Buonanno, Y.Chen, Physical Review D 65, 042001 (2002). [12] F.Ya.Khalili, Physics Letters A 288, 251 (2001). [13] H.J.Kimble, Yu.Levin, A.B.Matsko, K.S.Thorne and S.P.Vyatchanin, Physical Review D 65, 022002 (2002). [14] V.B.Braginsky, M.L.Goro detsky F.Ya.Khalili and K.S.Thorne, Physical Review D 61, 4002 (2000). [15] P.Purdue, arXiv:gr-qc/0111042 (2001). [16] P.Purdue, Y.Chen, arXiv:gr-qc/0208049 (2002). [17] Y.Chen, arXiv:gr-qc/0208051 (2002). [18] F.Ya.Khalili, arXiv:gr-gc/0211088 (2002). [19] V.B.Braginsky, M.L.Goro detsky, F.Ya.Khalili and K.S.Thorne, Energetic quantum limit in large-scale interferometers, in Gravitational waves. Third Edoardo Amaldi Conference, Pasadena, California 12-16 July, ed. S.Meshkov, Melvil le NY:AIP Conf. Proc. 523, pages 180189, 2000. [20] V. B. Braginsky, F. Ya. Khalili, Physics Letters A 218, 167 (1996). 68

[21] V.B.Braginsky, M.L.Goro detsky, F.Ya.Khalili, Physics Letters A 232, 340 (1997). [22] F.Ya.Khalili, Physics Letters A 298, 308 (2002). [23] V.B.Braginsky, M.L.Goro detsky, F.Ya.Khalili, Physics Letters A 246, 485 (1998). [24] S. P. Vyatchanin, Physics Letters A 239, 201 (1998). [25] S. L. Danilishin, F. Ya. Khalili and S. P. Vyatchanin, Physics Letters A 278, 123 (2000). [26] S.L.Danilishin, F.Ya.Khalili, Physics Letters A 300, 547 (2002). [27] V.B.Braginsky, M.L.Goro detsky and S.P.Vyatchanin, (1999). Physics Letters A 264, 1

[28] V.B.Braginsky, M.L.Goro detsky, S.P.Vyatchanin, Physics Letters A 271, 303 (2000). [29] Yuk Tung Liu, Kip S. Thorne, Physical Review D 62, 122002 (2000). [30] V.B.Braginsky, S.P.Vyatchanin, arXiv:cond-mat/0302617 (2003).

69

Appendix F Sensitivity limitations in optical speed meter topology of gravitational-wave antennae.
The p ossible design of QND gravitational-wave detector based on sp eed meter principle is considered with resp ect to optical losses. The detailed analysis of sp eed meter interferometer is p erformed and the ultimate sensitivity that can b e achieved is calculated. It is shown that unlike the p osition meter signal-recycling can hardly b e implemented in sp eed meter top ology to replace the arm cavities as it is done in signal-recycled detectors, such as GEO 600. It is also shown that sp eed meter can b eat the Standard Quantum Limit (SQL) by the factor of 3 in relatively wide frequency band, and by the factor of 10 in narrow band. For wide band detection sp eed meter requires quite reasonable amount of circulating p ower 1 MW. The advantage of the considered scheme is that it can b e implemented with minimal changes in the current optical layout of LIGO interferometer.

Introduction
It is a well known fact that in order to detect gravitational wave very high precision devices are needed. The current technological progress in this area of knowledge suggests that detector will achieve the level of sensitivity where its quantum b ehavior will play the main role. Sensitivity of the third-generation interferometric gravitational wave detectors is planned to b e higher than Standard Quantum Limit (SQL) [1, 2]. Therefore, to overcome the SQL one needs to monitor not the co ordinate of the detector prob e b o dy, as in contemp orary detectors, but the observable that is not p erturb ed by the measurement. Such an observable was called Quantum Non-Demolition (QND) observable [3, 4, 5, 6]. One needs to monitor this kind of observable b ecause, if there were no force acting up on the prob e ob ject the observable value will remain unp erturb ed after the measurement. It means that there is no back action and, therefore, no SQL, limiting the sensitivity. In order to detect external action on the ob ject it is reasonable to measure its integral of motion that is a QND-observable at the same time. For the free mass, its momentum can b e that QND observable, but the realization of momentum measurement is not an easy task. In the article [7] it was prop osed to measure velo city of a free mass instead of momentum. Though it is not a QND observable and is p erturb ed during the measurement, it returns to the initial value after the measurement, and therefore can b e used to b eat the SQL. The device that measures ob ject velo city is agreed to b e called sp eed meter. The first realization of gravitational-wave detector based on sp eed meter principle was 70

prop osed in article [7]. In this article authors suggested to measure the velo city of the free mass placed in the gravitational wave field. The analysis of the scheme has shown that measuring of velo city instead of b o dy co ordinate allows to cancel the back action noise and, therefore, significantly increase scheme sensitivity. The measuring system was presented by two coupled microwave cavities which coupling constant had to b e chosen so that phase shift prop ortional to co ordinate was fully comp ensated, while the output signal contained information ab out the prob e mass velo city only. Later, in article [8] the p ossibility to apply sp eed measurement technique to interferometric gravitational-wave detectors has b een analyzed. It was suggested to attach to the prob e b o dies of the detector small rigid Fabry-Perot cavities. These cavities, fully transparent for light at certain frequency when immobile, intro duce phase shift to the output light due to Doppler effect. Measuring this phase shift it is p ossible to measure prob e b o dy velo city. It was shown in this article that sp eed meter can p otentially b eat the SQL if pumping p ower is larger than one for SQL limited p osition meter: W > WSQL . The realization of sp eed meter based on two microwave coupled resonators, suggested in [7] was considered in article [9]. This sp eed meter was prop osed to b e attached to the detector prob e b o dy to measure its velo city. It was demonstrated that it is feasible, with current technology, to construct such a sp eed meter that b eats the SQL in a wide frequency band by a factor of 2. It was also prop osed a p ossible design of sp eed meter for optical frequency band. This design demands construction of four large scale cavities instead of two as in traditional LIGO detector. The disadvantage of this scheme, common for all sp eed meters, is the extremely high p ower circulating in cavities. The further comprehensive analysis of the scheme prop osed in [9] was carried in [10]. It was shown that in principle the interferometric sp eed meter can b eat the gravitationalwave standard quantum limit (SQL) by an arbitrarily large amount, over an arbitrarily wide range of frequencies, and can do so without the use of squeezed vacuum or any auxiliary filter cavities at the interferometer's input or output. However, in practice, to reach or b eat the SQL, this sp ecific sp eed meter requires exorbitantly high input light p ower. The influence of losses on the sp eed meter sensitivity is also analyzed and it was shown that optical losses in considered scheme influence the sensitivity at low frequencies. In articles [11, 12] more practical schemes of large scale interferometric sp eed meters based on Sagnac effect [13] were analyzed and prop osed for use as third generation LIGO detectors p ossible design. It should b e noted that use of Sagnac interferometers in gravitational-wave detection was suggested earlier in the works [14, 15, 16, 17]. In [11, 12] it was shown that to lower the value of circulating p ower one should use the squeezed vacuum input with squeezing factor of 0.1, and variational output detection [18]. Then it is p ossible to b eat the SQL by the factor of 10. The analyzed schemes are based on using either three large scale Fabry-Perot cavities [11] or ring cavities and optical delay lines [12]. The comprehensive analysis of the ab ove mentioned schemes including optical losses had b een carried out. Another variant of Sagnac based sp eed meter was prop osed in [19]. This design requires little changes in initial LIGO equipment and seems to b e a go o d candidate for implementation. The scheme of interferometer prop osed in the article mentioned ab ove is considered in this pap er. We analyze here how this meter will b ehave when there are optical losses in interferometer mirrors, and obtain the optimal value of circulating p ower and how it dep ends on cavity parameters.

71

PSfrag replacements

x

n

g

n

PRM SRM

be a n bw bn aw ae bs a s

g

e

x

e

Figure F.1: Simplified scheme of sp eed meter.

Simple Sagnac speed meter scheme with optical losses
Input-output relations for simple sp eed meter scheme
In this section we will analyze the action of simple sp eed meter scheme based on Sagnac effect to estimate how optical losses affect its sensitivity. This scheme is the mo del that is quite easy for analysis and yet contains all features sp ecific for sp eed meter. It differs from real, complicated schemes only by the fact that there is no b ending up of noise curve at high frequencies that arises due to the finite bandwidth of Fabry-Perot cavities b eing used in real sp eed meter interferometers. Therefore, it seems convenient to investigate this simple scheme b efore analyzing complicated sp eed meters designed for LIGO. The scheme we want to consider is presented in Fig. F.1. Its action can b e describ ed as follows: the light b eam from laser enters the scheme from the "western" side of the figure and is divided by the b eam-splitter into two b eams propagating in "northern" and "eastern" directions corresp ondingly. Each b eam is reflected sequentially from two movable end mirrors. Between these reflections b eams are reflected from central mirror. The end mirrors we supp ose to b e ideally reflective, while the central mirror has the amplitude reflectivity r and transmittance t. We supp ose that all optical losses are concentrated in the central mirror and therefore can b e expressed by a single parameter t. It can b e shown that for gravitational-wave detectors this approximation is fulfilled pretty well. After three ab ove mentioned reflections the light b eams return to the b eam-splitter where they are mixed up. We supp ose that arm lengths are chosen so that all the light p ower returns to the laser. Then photo-detector lo cated in the "southern" part of the figure will register only vacuum oscillations whose phase is mo dulated by the end mirrors motion. The input light can b e presented as a sum of classical pumping wave with frequency and sideband quantum oscillations. The electric field strength of incident wave can b e written as

^ E (t) = (0 ) Ae

-io t

+A e

io t

+
0

( ) a( )e ^

-i t

+ a ( )e ^

i t

d , 2

(F.1)

72

where A is classical wave quadrature amplitude, a( ) is quantum fluctuations sideband ^ 2 is normalization factor, and A is the b eam cross section. op erator, and ( ) = Ac Using this formalism we can write down input-output relations for the scheme. The incident wave quadrature amplitude and fluctuations we will denote as A and aw as they ^ come from the "western" side of the figure. Zero oscillations entering the scheme from the "southern" side we will denote as as . After the b eam-splitter we will have two light ^ b eams propagating in "northern" and "eastern" directions. These b eams values we will mark by indices "n" and "e" corresp ondingly. Moreover, there is a transparent central mirror that is the source of additional noises that are denoted as gn and ge (see Fig. F.1). ^ ^ Let denote the b eam that travel in the interferometer arm for the first time as "primary" b eam, while the same b eam that has left its first arm and entered the second one we will denote as "secondary" b eam. Due to absorption the p owers of "primary" and 2 "secondary" b eams relate to each other as Wsecondary /Wprimary = r 2 = 1 - loss , where loss is the interferometer absorption co efficient. This co efficient is quite small and in practice can b e neglected for the values of classical p owers but, in principle, there is an opp ortunity to comp ensate these losses by intro ducing some additional p ower into arms. This opp ortunity can b e taken into account by assuming Wsecondary /Wprimary = 2 , where 2 is some co efficient that is equal to 1 - loss without additional pumping, and can b e larger with it. We will assume that = 1 as optical losses in considered schemes are supp osed to b e very small (loss 10-5 ). Then, output wave can b e written as (see b elow) ^s = b where
input

^ input as

+

^ loss gs

+K
4i

simple x-

,

(F.2)

= ir e

is the co efficient that characterizes input fluctuations,
loss

= -ite

2i

is the co efficient that characterizes fluctuations arising due to losses, K
simple

= -2( )A(ir e

i( +3 )

- e

i( + )

),

is the simple scheme coupling constant, where ( ) =

/c, is the time light needs to xe - x n is the difference come from the b eam-splitter to the end mirror, and x- ( - ) = 2 ige - gn ^ ^ of the end mirrors co ordinates in frequency domain. Here gs = ^ stands for the 2 loss noises recalculated to output.

Quantum noise sp ectral density
In order to evaluate the scheme sensitivity we need to calculate sp ectral density of total quantum noise. Supp ose mirrors dynamics can b e describ ed by free b o dy equation of motion, i. e. mx = F . Ё Here F = FGW + Ff luct , where FGW is the gravitational-wave force measured, Ff luct is the fluctuational force that arises due to radiation pressure of the incident light, and x = xGW + xf luct is the mirror displacement that consists of displacement due to gravitational 73

10

0

P
SQL
10
-1

simple

=1

S/S

PSfrag replacements

P
10
-2

simple

= 10 = 100

P

simple

10

-3

10

-3

10

-2

10

-1

10

0

Figure F.2: Typical curves for 2 = S/S
SQL

at different P

simple

.

wave action and noisy part that arises due to incident light phase fluctuations. The total noise sp ectral density then can b e written as S () = SF () + m2 4 Sx () - 2m
2

([SxF ()] ,

(F.3)

where is observation frequency, and Sx (), SF (), SxF () are sp ectral densities of fluctuational mirror displacement xf luct , fluctuational radiation pressure force Ff luct , and their cross-correlation corresp ondingly. In our sp ecific case these sp ectral densities are equal to SF () = Sx () = 8 W · (1 - r cos(2 )) , c2 (F.4.1) (F.4.2)

c2 1 , · 2 2 - 2r cos(2 ) 16 W sin 1 + r

SxF () = - cot . (F.4.3) 2 Here W is the pumping p ower at the end mirrors, and is the homo dyne angle that allows to minimize the value of (F.3) significantly. This angle is chosen so that one measures not the amplitude or phase quadrature comp onent but their mixture. This principle provides the basis for variational measurement technique [18, 20]. Here is one of the optimization parameters that allows to overcome the SQL b eing chosen in the prop er way. Supp ose that signal varies slowly compared to the scheme characteristic time , and, therefore 1, (F.5.1) 2 , (F.5.2) where = L/c, and L is the distance b etween the central and end mirrors. These assumptions we will call narrow-band approximation. Later we will see that for real detectors this approximation works pretty well. Using this approximation we can rewrite (F.4) as: SF () = Sx () = 8 W · (1 - r + 2r ( )2 ) , c2 (F.6.1) (F.6.2)

1 c2 , · 2 16 W sin (1 - r )2 + 4r ( )2 74

(F.6.3) SxF () = - cot . 2 If we substitute the obtained expressions to (F.3) and divide by the value of radiation pressure noise SQL sp ectral density for the free mass SSQL = m2 , we will obtain the factor by which sp eed meter b eats the SQL: 2 = 2(1 + cot2 ) 1 Psimple ( )2 · (1 - r + 2r ( )2 ) + + 2 cot , 2 ( )2 Psimple (1 - r )2 + 4r ( )2 (F.7)

16 W 2 where Psimple = . mc2 The main goal of optimization is to b eat the SQL in the frequency band that is as wide as p ossible. First of all we should find the optimal value of cot and Psimple . These parameters should not dep end on frequency therefore we will optimize them at high frequencies, i. e. . This optimization will result in the following values: cot = -2r P and b eing substituted to (F.7) will pro duce
2 HF simple

,

(F.8)

1 4r P
simple

.

(F.9)

Obviously, the rise of 2 at low frequencies is determined by the radiation pressure sp ectral density as all other items in (F.3) are prop ortional to 2 and 4 and, therefore, can not influence at low frequencies. Hence we see that optical losses lead to decrease of sp eed meter sensitivity when observation frequency is small. It is useful to calculate the value of parameter Psimple that provides minimum of 2 at defined frequency . It can b e shown that for 1 - r 1 this value is equal to P
opt simple

2(1 - r )

.

(F.10)

One can also readily obtain that minimal frequency where sp eed meter sensitivity is equal to the SQL, i. e. where 2 = 1 is fulfilled, is defined by the expression:
min

Psimple 2(1 - r ) , 4Psimple - 1

(F.11)

where r is central mirror amplitude reflectivity co efficient that characterizes scheme optical losses. Here we supp ose t 1. Comparison of expressions (F.9) and (F.11) shows that one needs to increase optical p ower (Psimple ) in order to obtain high sensitivity, while to have wide frequency band and increase the sensitivity at low frequencies it is necessary to have low value of pumping p ower to decrease radiation pressure noise. In Fig. F.2 several curves are presented that demonstrates how 2 dep ends on frequency at different values of Psimple . Chain lines are for ideal case where losses are equal to zero (r = 1).

Signal-recycled speed meter vs. signal-recycled position meter
In this section we will consider semi-qualitatively the influence of signal recycling mirror on sensitivity of traditional p osition meter and sp eed meter. Let us consider the two 75

x PSfrag replacements N

2

x PSfrag replacements

2

FP

N

FP

ITM

BS ITM out

N

FP

ITM BS x
1

N
RM

FP

in NS RM SRM

in NS SRM

x ITM out

1

Figure F.3: Signal-recycled p osition meter

Figure F.4: Signal-recycled sp eed meter

schemes presented in Fig. F.3 and Fig. F.4. Supp ose TF P 1/NF P and TS RM 1/NS RM are transmittances of arm cavities input mirrors (ITM) and signal-recycling mirror (SRM) corresp ondingly. Here NF P shows how many times light b eam is reflected from the end mirrors until it leaves the cavity, and NS RM is the numb er of light b eam reflections from the SRM b efore leaving the whole scheme. The first numb er characterizes Fabry-Perot half-bandwidth, while the second one represents signal-recycling cavity half-bandwidth. Let b e the time light needs to travel from one cavity mirror to another. Supp ose also mirrors motion is uniform and can b e expressed during light storage time ( NF P NS RM < GW ) by formula GW, max xi = x
i, 0

+ vi t ,

where subscript i = 1, 2 denote the interferometer arm, xi, 0 are the end mirrors initial p osition, and vi are their velo cities. In the case of p osition meter light path is the following. After leaving the b eam-splitter light enters b oth cavities, it passes from one arm cavity mirror to another 2NF P times, then it is reflected from the SRM and returns to the cavity. This cycle rep eats NS RM times until the light leaves the scheme and get to the homo dyne detector. Output b eam phase shift in our simple case can b e written as
sig nal

2 N c

FP

N

Ї S RM x-

,

(F.12)

where x- = x1 - x2 is the mean end mirrors relative displacement during the storage Ї Ї Ї time, and c is the sp eed of light. As for the sp eed meter, the light path in this scheme differs from the previous one. Light that leaves, for example, the first cavity go es not to the b eam-splitter and then to the SRM, but to the second cavity where it exp eriences NF P reflections from the end mirror in addition to those in the first cavity. Then it is reflected from the SRM and so on for NS RM times until it leaves the scheme. It can b e shown that in this sp ecific case the output b eam phase shift will b e written as
sig nal

2 N c

S RM

N

2 Ї F P v-

,

(F.13)

where v- = v1 - v2 is the relative velo city of the end mirrors. Ї Ї Ї 76

We can readily see that p osition meter and sp eed meter output signals dep end on arm cavities (NF P ) and signal recycling cavity (NS RM ) parameters in different way. It can b e shown that this difference significantly influence the sensitivity of these schemes. Let us calculate the amount of circulating optical p ower necessary to achieve the level of SQL in each of the ab ove mentioned schemes. The output signal field quadrature comp onent asignal for b oth schemes is prop ortional to the pro duct a
sig nal

A

sig nal

,

where A is the input radiation field quadrature amplitude. Therefore sp ectral density of phase fluctuations can b e expressed in terms of sp ectral density of asignal as S = Sa . |A|2

The expression for Sa can b e readily obtained and will b e equal to 1 Sa = , 4 2 and |A| can b e represented in terms of p ower circulating in arms as N F P N S Then for S one will have the following expression S = |A|2 = W .
RM

NF P NS RM . (F.14) 4W On the other hand, for p osition meter this sp ectral density can b e expressed in terms of caused by radiation shot noise mirror displacement sp ectral density corresp onding to the S SQL Sx QL = m2 as S
PM

=

2 4 N

2 FP c2

N

2 S RM

S

S QL x

=

22 4 NF P N mc2 2

2 S RM

,

(F.15)

where is some fixed observation frequency, m is the mirror mass, and for sp eed meter in terms of caused by radiation shot noise mirror velo city sp ectral density corresp onding S to the SQL Sv QL = m as
2 4 2 2 4 2 4 2 NF P NS RM S QL 4 2 NF P NS RM Sv = . (F.16) c2 mc2 Substituting (F.14) into (F.15) and (F.16) we will obtain the following values of circulating p owers for b oth schemes considered:

S

SM

=

W

PM

=

mc2 2 , 16NF P NS RM

W

SM

=

16N

3 FP

mc NS

2 RM

2

.

(F.17)

If we divide WS M by WP M we will obtain the factor of by which sp eed meter circulating p ower necessary to achieve the SQL is larger than one for the p osition meter WS M 1 2 = = NS RM , (F.18) WP M (NF P )2 where we have taken into account that light storage time should b e much smaller than p erio d of gravitational wave, i. e. NF P NS RM / . Obviously, the ab ove estimates show that sp eed meter with signal recycling hardly can b e considered as the b est variant for implementation as QND-meter, as it requires much more circulating p ower than signal-recycled p osition meter with the same parameters. More precise calculations that prove the ab ove simple considerations are p erformed b elow. 77

x Tn
...............................

...............................

be Tbn c an Tce c

/4
o

b

(90o ) e (0o ) n

Tb(90 n c
(0 Tce c
o

)

a

)

d d d

bnE (0o ) aeE
'o

(90o )

ae 6cn ? cw a
w

PBS
'

(0 ) c' n (90o ) be

ў ў ў ў ў

bE n
'

aE e c be

n '

.. .. . . . . . . .. ..

.. .. . . . . . . . . . .

x

E
e

6 cs ?as

an E c
e

/4

Figure F.5: Sp eed meter interferometer

Optical losses in speed meter interferometers with FabryPerot cavities in arms.
Sp eed meter interferometer action and output signal
Let us consider the scheme of gravitational-wave detector based on sp eed meter principle presented in Fig. F.5. This interferometer differs from the traditional LIGO interferometers by additional p olarization b eam-splitter (PBS) and two quarter-wave plates (/4). Quarter-wave plates are needed to transform input light p olarization from linear (0 and 90 ) to circular one ( and ) during each pass. After reflecting from one of 4 km long Fabry-Perot (FP) cavities light p olarization varies over 90 , and light b eam is reflected from the PBS to the second FP cavity. Hence, the light b eam passes not only one FP cavity, but b oth cavities consequently. As there are two b eams that propagate in opp osite directions (in clo ckwise and counter-clo ckwise directions corresp ondingly) the scheme output do es not contain any information ab out the symmetric mechanical mo de of the mirrors, i. e. ab out the sum of end mirrors displacements. This mo de is not coupled with gravitational-wave signal and its presence can significantly lower the sensitivity, and prevent from b eating the SQL. The output signal of this scheme can b e found in the same manner as it was done in [19]. The only problem is to include additional noises, arising due to internal losses in optical elements to calculations. To do so, supp ose each mirror to b e a source of two indep endent additional noises, that stand for interaction of electromagnetic radiation 78

with mirror medium excitations. This approach is based on the Huttner-Barnett scheme [21] of electromagnetic field quantization in linear lossy dielectrics. Using this approach one can obtain input-output relations for any optical device with losses (see [22, 23]). Let us use the same notations as ab ove for the scheme parameters. In order to distinguish values that corresp ond to inner and end mirror we will denote the first ones by subscript 1 and the second ones by subscript 2 . Each Fabry-Perot has the following parameters: mirrors reflectivity, transmittance, and absorption co efficients are equal to -r1 , -r2 , it1 , 0, and ia1 , ia2 corresp ondingly, and cavity length is L. The end mirrors transmittance we supp ose to b e equal to zero b ecause there is no difference whether light run out or is absorb ed in the end mirror. Supp ose that classical pumping wave amplitude is A and sideband fluctuations operator is aw . Supp ose also zero oscillations that enter the scheme from the "southern" ^ side are describ ed by op erator as . We will also need op erators gs11 , gs12 , gs21 , and gs22 to ^ ^ ^ ^ ^ describ e noises due to internal losses in FP mirrors (the second numerical subscript stands for the numb er of noise that arises in the mirror due to internal losses). Sup erscripts I and I I denote what noises arise during the first and the second reflection corresp ondingly. Parameter we assume to b e equal to unity, where has the same meaning as in simple scheme and is equal to the ratio of classical amplitudes inside the cavity during the second and the first reflection corresp ondingly. Finally, we are able to write down the output sideband fluctuations op erator cs that ^ contains information ab out the mirrors movements, and therefore ab out the gravitational wave force: cs = ^ 1 [iB 2 ( )aI ( ) - e2i r2 t1 a1 gs11 ( ) + ei t1 a2 gs21 ( ))- ^s ^I ^I L2 ( ) 1 - iL( )(e2i r2 t1 a1 gwI11 ( ) - ei t1 a2 gwI21 ( ))] - KS M x- ( - ) . (F.19) ^I ^I
2i

where L( ) = r1 r2 e

- 1, B1 ( ) = r1 - e K
SM

2i

2 r2 (r1 + t2 ), and 1 i

=-

(B1 ( ) - L( ))E ( )e L2 ( )

r2 t

1

is the coupling constant of the sp eed meter interferometer, E is the classical amplitude of pumping field near the movable mirror after the first reflection.

Sp eed meter sp ectral densities and sensitivity
Here we will obtain the expressions for radiation pressure noise, shot noise and their crosscorrelation sp ectral densities. We will supp ose again that the same additional pumping pro cedure is taking place. Here we will use the same approximation that are defined by (F.5). It is useful to evaluate if this approximation is valid for LIGO. Parameters for LIGO interferometer are the following: = 1.77 · 1015 s-1 , = 1.33 · 10
-5

s,

m = 40 kg ,

L = 4 · 103 m .

If we supp ose that gravitational signal upp er frequency is ab out = 103 s-1 then there remain no doubts that for LIGO condition of narrow-band approximation applicability is fulfilled as = 1.33 · 10-2 1. It seems reasonable to intro duce the following notations: 1 = t2 1 , 4 (F.20.1)

79

= 1 + 2 = = Using this notations one can obtain:

a2 + a 2 1 2 , 4

(F.20.2) (F.20.3)

1 - r 1 r2 = 1 + . 2

B1 ( - ) = 2 ( - 21 + i) , L( - ) = -2 ( + i) .

(F.21.1) (F.21.2)

Taking all of the ab ove into account one will obtain the following expressions for sp ectral densities (see derivation b elow) Sx () = SF () = ( 2 + 2 )2 L2 , 32 W 21 sin2 (( - 1 )2 + 2 ) 8 W ( 2 + 2 ) - 1 ( 2 - 2 ) , L2 ( 2 + 2 )2 S
xF

(F.22.1)

(F.22.2)

= - cot , (F.22.3) 2 Where is the homo dyne angle, and W is the pumping p ower at the end mirrors. One can obtain that the minimum force that can b e measured by sp eed meter presented in Fig. F.5 dep ends on the total measurement noise sp ectral density. This sp ectral density can b e expressed by the formula (F.3) but one exception: m in this formula is equal to one fourth of the real end mirror mass. The multiplier 1/4 app ears when we supp ose not only end mirrors to b e movable, but also the inner ones. Now we can write down the expression for 2 = SSM /SSQL , where SSM is the total quantum noise of the sp eed meter that is calculated in accordance with (F.3), and SSQL = m2 is the SQL sp ectral density for fluctuational force: 8 W ( 2 + 2 ) - 1 ( 2 - 2 ) ( 2 + 2 )2 mL2 2 + + cot . mL2 2 ( 2 + 2 )2 32 W 21 sin2 (( - 1 )2 + 2 ) (F.23) This value characterizes scheme sensitivity. Our goal is to have this value as small as p ossible. In the next subsection we will p erform two p ossible optimizations of 2 , corresp ondingly in narrow and wide frequency bands. 2 =

Optimization of

2

Narrow-band optimization. Let optimize (F.23) at some fixed frequency. The p ossible situation when such optimization can b e useful is the detection of gravitational radiation emitted by quasi-mono chromatic sources. Compact quikly rotating neutron stars, i. e. pulsars, may b e the example of such sources. Gravitational radiation from pulsars is quasy-mono chromatic and relatively weak so it is crucial to have high sensitivity in narrow frequency band to detect these sources. So it is convenient to find the minimum of 2 at the source main frequency . If we supp ose that optical losses are small enough, i. e. 0 then the minimal value of will b e equal to
min

=

4

=

4

,

(F.24)

80

10

1

10

1

g replacements
10
-1

PSfrag replacements
10
-1

0

10

0

10

0

10

10

-1

10

-2

10

-1

10

0

10

1

= / Figure F.6: Plot of = SS M /SS QL , optimized at some fixed frequency . For given = 103 s-1 and fixed optical losses contribution to the total half-bandwidth = 1 s-1 one should take the following parameters to obtain the b est sensitivity at given frequency: total half-bandwidth = , circulating p ower W =
mL2 8
3

= / Figure F.7: Plots of for circulating optical p ower W = 1 MW and several fixed homo dyne angles (thin blue curves; the curve with rightmost minimum corresp onds to the least angle). Bold black curve corresp onds to frequency-dep endant homo dyne angle.

27 MW, and cot = - -3 · 10-2 . The minimal in that case will b e equal to 4 0.18.

and is reached at

opt

= , optical circulating p ower W
opt

opt

=

mL2 8

3

, homo dyne

(see calculations b elow for detail). For = 1 s-1 it will b e equal to 0.18. For LIGO interferometer optical p ower necessary to reach the ab ove value of is equal to Wopt 27 MW. Unfortunately, at frequencies different from the value of is much worse. Function b ehaviour at different frequencies is presented in Fig. F.6. We can see that significant gain in sensitivity compared to the SQL can b e achieved in a very narrow frequency band of ab out several tens of hertz. Thus, this regime of sp eed meter op eration that we prefer to call "narrow-band" regime can b e used to b eat the SQL by significant amount only in narrow band near some arbitrarily chosen frequency for the purp oses of weak quasy-mono chromatic sources detection. angle defined by formula cot =- Wide-band optimization. Contrary to the narrow-band case, considered ab ove, the vast ma jority of gravitational wave sources either radiate in relatively wide frequency band, or their main frequency is unknown. In b oth cases it is necessary to p erform wideband detection pro cedure. This problem can not b e solved as easy as the previous one, b ecause there are no criteria what should b e the frequency bandwidth value and what sensitivity should b e reached within the ranges of this band. As there is no clarity in this question it seems convenient to represent the variety of different p ossible regimes of 81

10

1

10

1

g replacements
10
-1

PSfrag replacements
10
-2

-1 0 1

10

0

10

0

10

10

10

10

-1

10

-2

10

-1

10

0

10

1

= / Figure F.8: Plots of for circulating optical p ower W = 3 MW and several fixed homo dyne angles (thin blue curves; the curve with rightmost minimum corresp onds to the least angle). Bold black curve corresp onds to frequency-dep endant homo dyne angle.

= / Figure F.9: Plots of for circulating optical p ower W = 10 MW and several fixed homo dyne angles (thin blue curves; the curve with rightmost minimum corresp onds to the least angle). Bold black curve corresp onds to frequency-dep endant homo dyne angle.

op eration that the sp eed meter is capable of. Therefore we will supp ose that optical p ower W circulating in the arms of interferometer and resonators half-bandwidth are fixed. Then varying homo dyne angle it is p ossible to obtain variety of different sensitivity curves. These curves for three different values of circulating optical p ower W = 1, 3, and 10 MW are presented by thin blue curves in Figures F.7, F.8, and F.9 corresp ondingly. Varying homo dyne angle it is p ossible to reach sensitivity even three times b etter than SQL in relatively wide frequency band. It is also p ossible to cho ose the frequency where the b est sensitivity is reached by changing the homo dyne angle, i. e. increase of leads to sensitivity curve offset into the lower frequencies domain. It should b e also noted that the increase of circulating p ower in this regime leads to reduction of frequency band where is less than 1, therefore it is reasonable to use the mo derate values of optical p owers. Using the technique of frequency-dep endant variational readout suggested in article [24] it is p ossible to increase the sensitivity of sp eed meter significantly in wide frequency band. The achievable sensitivities in this case are represented in the same figures by b old black curves. Here is the dep endence of () that allows to obtain the ab ove mentioned results: 2 ( 2 + 2 )2 mL2 · () = - arctan . (F.25) 32 W 1 (2 + 2 )

Conclusion
In this section we will sum up the results of our consideration. The following conclusions can b e made: · The attempts to increase sp eed-meter sensitivity by intro ducing signal-recycling mirror (SRM) or replace arm cavities by this mirror will not b e successful, b ecause contrary to the p osition meter, sp eed meter sensitivity dep ends up on the transmittances of arm cavities input mirrors (ITM) and SRM not symmetrically. The 82

influence of ITM is considerably greater than SRM influence b ecause in sp eed meter light b eam passes consequently through b oth cavities b efore it is reflected from SRM. Therefore, it is convenient to use arm cavities in gravitational-wave detectors based on sp eed-meter principle. · Sp eed meter top ology of gravitational-wave antennae allows to achieve sensitivity ab out three times b etter than SQL in relatively wide frequency band preserving optical p ower circulating in the arms at reasonable level of 1 MW. Moreover, its sensitivity can b e improved and its frequency band can b e significantly increased if one applies variational readout technique. · The ultimate sensitivity sp eed meter is capable of is defined by two factors, optical losses and bandwidth of arm cavities and, therefore, circulating p ower. This sensitivity can b e expressed in terms of these factors as = h h
S QL

=

4

,

where h is the metric variation that can b e measured by sp eed meter, hS QL is the standard quantum limit for h, is the optical losses contribution to the total interferometer half-bandwidth . However, high sensitivity (small ) requires large amount of circulating optical p ower (ab out tens of megawatts) and can b e achieved in relatively narrow frequency band. In our opinion, the b est op eration mo de for sp eed meter is the wide-band regime with frequency dep endant readout.

Acknowledgements
Author would like to thank V. B. Strigin for helpful discussions and and Caltech grant #PHY0098715, Russian Ministry of Industry and Braginsky, F. Ya. Khalili, S. P. Vyatchanin, and S. E. countenance. This pap er is supp orted in part by NSF by the Russian Foundation for Basic Research, and by Science.

Derivation of spectral densities for simple case
Input-output relations derivation
Here we will obtain the formulae for quantum noise sp ectral densities, presented by expressions (F.4). The input light can b e describ ed by the formula (F.1). Input amplitudes are the following (Subscripts "w, e, s, n" are for light b eams corresp onding to the consequent part of the scheme. For example, Aw means classical amplitude of light b eam propagating in "western" direction.): Aw = A , and corresp onding sideband op erators are ^ aw , as . ^ (F.27) As = 0 , (F.26)

As we have mentioned ab ove we are able to intro duce additional pumping through the central mirror. This additional pumping increases light p ower in the scheme and do es not create additional noises, therefore, it increases scheme sensitivity. To describ e it we 83

will intro duce complex parameter = ei = 2C /A that is equal to the ratio of light amplitude at the end mirrors with additional pumping(C = A/ 2 + Aadd ) and light amplitude after the b eam splitter (A/ 2) (that is equal to the amplitude at the end mirror provided that central mirror is ideally reflecting, i. e. in lossless case). Taking all ab ove into account one can easily obtain that classical output of the scheme is equal to Bs = 0 , and corresp onding sideband output is equal to ^s ( ) = iras ( )e b ^ - 2( )A(ir e
4i

(F.28)

- i

i( +3 )

-gn ( ) + ige ( ) 2i ^ ^ e- 2 - ei( +) )x- ( - ) =

^ input as

+

^ loss gs

+K

simple x-

, (F.29)

see (F.2) for notations.

Radiation pressure noise sp ectral density SF ()
The radiation pressure force corresp onding to system mo de x- is equal to: ^ ^ ^ F = Fe - Fn = 2

( )A [(ie
0

i( - )

+ e

-i

re

i(3 - )

)as ( )- ^
i( - )t

-

2 e

-i i( - )

e

-gn ( ) + ige ( ) ^ ^ ]e 2

d + h. c. , (F.30) 2

^ ^ where Fe and Fn are the radiation pressure fluctuational forces acting up on "eastern" and "northern" movable mirrors corresp ondingly. In order to calculate radiation pressure sp ectral density one should calculate symmetric correlation function: BF (t - t ) = 1 ^^ ^ ^ 0|(F (t)F (t ) + F (t )F (t))|0 , 2 (F.31)

where |0 -- is the radiation field ground state. If we calculate this value we will obtain: BF (t - t ) = where |F ( )|2 = 4 2 2 ( )|A|2 (|(e
i( - )

1 2

0

|F ( )|2e

i( - )(t-t )

d + 2

0

|F ( )|2e

-i( - )(t-t )

d 2

(F.32)

- i ei r e

i(3 - )

)|2 + 2 2 ) .

In order to obtain this result one should take into account that 0|a( )a ( )|0 = 2 ( - ). The radiation pressure sp ectral density can b e defined ^^ as: 1 16 W 1/2 + 2 /2 - r sin cos 2 dt = (S ()+S (-)) = · . 2 c2 1 + 2 - (F.33) In order to provide sp eed meter mo de of op eration we need to set = /2, then one will obtain that 8 W 1 + 2 - 2 r cos(2 ) · . (F.34) SF () = c2 1 + 2 SF () = BF (t)e
-it

84

Shot noise sp ectral density Sx ()
The output signal of the scheme is mixed up with lo cal oscillator wave in order detect phase shift due to end mirrors displacement in the optimal way. This mixed radiation enters the homo dyne detector. The photo-current of detector is prop ortional to the following time-averaged value: ^ Ip.d. (t) 2Ebs (t) cos( t +
LO

)=
0

=

^s ( )e b

i( - )t+i

LO

d + h. c. = 2
it

K ()(x ^
-

f luct

() + x- ())e

d , (F.35) 2

where LO is the lo cal oscillator phase, r () is the noise op erator which sp ectral density ^ is equal to unity, and K () is equal to: K () = - where =
LO

4

3/2

c

|A|

e

-3i

[r sin + e

2i

cos( + )] ,

(F.36)

+ arg A. Fluctuations of co ordinate are describ ed by the op erator x ^
f luct

() =

r () ^ . K ()

(F.37)

At last, assuming = /2, the co ordinate noise sp ectral density is equal to c2 1 + 2 Sx () = · , 32 W sin2 r 2 + 2 - 2r cos(2 ) (F.38)

Cross-correlation sp ectral density SxF ()
Now we know everything to calculate the cross-correlation sp ectral density S xF (). In order to do it one should find the cross-correlation function of (F.30) and xf luct (). Using ^ the same algorithm as in previous subsections one will obtain that SxF () = - cot . 2 (F.39)

Power- and signal-recycled speed meter interferometer sensitivity. Precise analysis
Input-output relations for p ower- and signal-recycled sp eed-meter interferometer.
In this subsection we will analyze the scheme of sp eed meter interferometer with p ower (PRM) and signal (SRM) recycling mirrors installed (See Fig. F.11). Here we will confirm by exact calculations the results of qualitative consideration presented ab ove. Let consider the influence of additional optical elements on the input-output relations. Figure F.10 represents the situation common for PRM and SRM. Beams a and b represent light entering and leaving the b eam-splitter, and b eams a and b stand for light, entering and leaving the PRM/SRM. The length L of recycling cavity should b e chosen in the way that provides the maximal values of circulating p ower in case of PRM, and signal sidebands in case of SRM. 85

L a b a a

b b PRM/SRM Figure F.10: Power recycling/signal recycling mirrors PSfrag replacements x
n

gn PRM be an bw bn g aw ae bs a s

e

x

e

SRM Figure F.11: Power- and signal-recycled sp eed meter interferometer.

Power recycling mirror. Power recycling influences the value of circulating p ower only, as quantum fluctuations from the laser that are influenced by the PRM, return back to the laser due to dark p ort tuning of interferometer and do not contribute to the output signal. Interferometer circulating p ower dep ends up on the quadrature amplitude A of the b eam a as W = |A|2 .

Then we need to express A in terms of A , the amplitude of the b eam a entering the PRM. One can write down the following equations: A = it
PR

A -r

PR

B, A,

A = Ae

-i i

PR

, ,

(F.40a) (F.40b) (F.40c)

B = it

PR

B -r

PR

B = Be )A ,

PR

where tP R and rP R are PRM transmittance and reflectivity, P R = L/c, and loss represents losses in the entire scheme (we will neglect these losses in this calculation as loss is sufficiently small). Solving these equations leads to the following expression: A= itP R eiP R 1 + irP R e2i
PR

B = i(1 -

loss

A.
PR

(F.41) = /4, (F.42)

In order to maximize circulating p ower one should tune PR-cavity so that then we will obtain that 4W t2 R W P , WP R = 2 (1 - rP R ) t2 R P 86

where W = |A |, and W

PR

= |A|2 .

Signal recycling mirror. In the case of signal recycling cavity only quantum fluctuations transformation is worth to b e examined. As the interferometer is tuned so that output p ort is kept "dark", classical amplitude of the b eam leaving the BS is equal to zero. Now a and b stand for fluctuations entering and leaving the BS, while a and b stand for fluctuations entering and leaving the SRM. Therefore we write down the following equations: ^ = it b
SR

e

i

SR

^-r b

SR

a, ^

ae ^

-i

SR

= it

SR

a -r ^

SR

e

i

SR

^, b

(F.43)

where tS R and rS R are SRM transmittance and reflectivity, S R = L/c. In these equations we should supp ose a as , ^ ^s , and the equation that gives the relation ^ ^b b b etween these op erators is represented by expression (F.2). Solution of these equations gives us the following expression for PR&SR sp eed meter output signal: ^ =- r bs where K
SR SR

+

t2 R e2iS R input S 1 + rS R input e2iS =

R

as + ^

itS R eiS R loss 1 + rS R input e2i K ,

SR

gs + K ^

S R x-

,

(F.44)

itS R eiS R loss 1 + rS R input e2i

SR

simple

input , loss , and Ksimple are defined in (F.2). The value of S R is chosen so that output signal is maximal, i. e. KS R max. Therefore, signal recycling cavity should b e tuned in the way to provide S R = /4. Now we are able to calculate sp ectral densities of radiation pressure and shot noises and their cross-correlation sp ectral density.

Radiation pressure noise sp ectral density SF ()
In order to calculate radiation pressure noise sp ectral density for p ower- and signalrecycled sp eed meter interferometer one needs to replace op erator as in formula (F.30) of ^ previous subsection by op erator as that can b e expressed in terms of as and gs as ^ ^ ^ as = ^ it
SR

e

i

SR

as - r ^

S R loss

e

2i

1+r

SR

gs - rS R e ^ input e2iS R
SR

2i

SR

K

simple x-

.

(F.45)

One can obtain that radiation pressure noise sp ectral density in case of p ower- and signalrecycled sp eed meter interferometer is equal to 16 W SF () = c2
PR

( + S R )( S R + 2 ) · . ( + S R )2 + 42

(F.46)

Shot noise sp ectral density Sx ()
To obtain the expression for shot noise sp ectral density in the case of signal-recycled interferometer we should use the same pro cedure as ab ove but for one exception. We should substitute value of Ksimple by KS R . To account for p ower recycling we should also write down WP R instead of W . After all we shall obtain the following formula: Sx () = c 64 W
2

PR

( + S R )2 + 42 . S R ( 2 + 2 ) sin2 · 87

(F.47)

Cross-correlation sp ectral density SxF ()
Cross-correlation sp ectral density in this case is the same as in previous section and is common for all considered interferometric schemes with homo dyne detection: SxF () = - cot . 2 (F.48)

Power- and signal-recycled sp eed meter sensitivity
Total noise of the PR&SR sp eed meter is describ ed by the same formula as (F.3). Sp ectral densities of noises in this particular case can b e obtained from formulae (F.46), (F.47), and (F.48). Being substituted to (F.3) and divided by SS QL = m2 these formulae will give us the expression for the factor 2 by which one can b eat the SQL using PR&SR sp eed meter: 2 = P where P
SR

(S R + )( S R + 2 ) 1 + cot2 2 (( + S R )2 + 42 ) · + + cot , 2 (( + S R )2 + 42 ) 4PS R S R ( 2 + 2 ) =

(F.49)

16 WP R 1-r ,= is the interferometer half-bandwidth part due to 2 mc 2 1 - rS R is the part of half-bandwidth due to signal-recycling optical losses and S R = 2 mirror. We can now optimize this expression at some fixed frequency with resp ect to homo dyne angle , and circulating p ower WP R . The optimal circulating p ower Wopt and homo dyne angle opt for considered scheme are equal to:
SR

W

opt

mc2 32

1+4

2 2 S R

2 , S R

opt

- arctan

S R , 2

(F.50)

and the minimal value of that can b e achieved at frequency is
opt
4

S R . 2

(F.51)

We can see that to b eat the SQL considerably using signal-recycled sp eed meter one needs to decrease internal losses. Decreasing S R (or increasing rS R ) will also increase the scheme sensitivity at given frequency but at the sacrifice of sensitivity at other frequencies. Let us estimate the optimal circulating p ower that is necessary to obtain 0.1 . Let substitute the following parameters: = 103 s
-1

, m = 5 kg , L = 600 m , = 1.77 · 10

15

s

-1

, 1 - r = 10

-5

,
SR

(F.52) = 40 s
-1

that are typical for GEO 600 interferometer. To obtain opt = 0.1 one needs (1 - rS R = 1.6 · 10-4 ) and circulating p ower of the order of W
opt

= 10

11

W.

(F.53)

We can see that considered scheme of p ower- and signal-recycled sp eed meter can b e hardly implemented in gravitational wave detection, as it requires enormous amount of optical p ower and its sensitivity is high only in very narrow frequency band. This result confirms the statement we have made ab ove, i. e. in order to achieve the same sensitivity as p osition meter with Fabry-Perot cavities in arms, sp eed meter requires highly reflecting SRM to b e used and therefore enormous amount of pumping p ower is needed. Taking it into account it is reasonable to conclude that the scheme with FabryPerot cavities in b oth arms without SRM lo oks like the b est candidate for implementing as sp eed meter interferometer. 88

-r1 , it1 , ia1 d1 a1 b
1

-r2 , 0, ia c2 d L = c
2

2

c

1

x

Figure F.12: Fabry-Perot cavity with movable mirror

Derivation of spectral densities for speed meter interferometer with Fabry-Perot cavities in arms
Fabry-Perot input-output relations
To obtain input-output (IO) relations for the scheme presented in Fig. F.5 one needs to know IO relations for Fabry-Perot cavity with movable end mirror. The case of two movable mirrors will give the same result but when investigating cavity dynamics one should replace end mirror displacement x by the difference x1 - x2 of each mirror displacements, and mirrors masses should b e taken one half of real value to account for changes in radiation pressure forces. Consider FP cavity presented in Fig. F.12. We supp ose end mirror to have zero transmittance for the same reason as in subsection F. Supp ose we know classical amplitude of internal field C1 and input quantum fluctuations a1 of input light. To obtain expressions ^ for internal and output fields one needs to solve the following equations: ^1 ( ) = -r1 a1 ( ) + it1 c1 ( ) + ia1 g12 ( ) , b ^ ^ ^ ^ d1 ( ) = it1 a1 ( ) - r1 c1 ( ) + ia1 g11 ( ) , ^ ^ ^ ^ d2 ( ) = -r2 c2 ( ) + ia2 g21 ( ) - 2i( )r2 C2 x( - ) , ^ ^ ^ c2 ( ) = d1 ( )e ^ The solution can b e written as: ^1 = b 1 [B ( )a1 ( ) - e2i r2 t1 a1 g11 ( )+ ^ ^ L( ) + ia1 L( )g12 ( ) + ei t1 a2 g21 ( ) - 2( )e ^ ^
i i

(F.54)

,

^ d2 ( ) = c1 ( )e ^

-i

.

i

r2 t1 C2 x( - )] , (F.55.1)
2i

c2 = ^

1 [-ie L( )

t1 a1 ( ) - ie ^

i

a1 g11 ( ) + ie ^

2i

r1 a2 g21 ( ) - 2i( )e ^

r1 r2 C2 x( - )] , (F.55.2)

where

1 ^ d2 = [ie L( )

i

t1 r2 a1 ( ) + ie ^

i

r2 a1 g11 ( ) - ia2 g21 ( ) + 2i( )r2C2 x( - )] , (F.55.3) ^ ^ - 1, B ( ) = r1 - e
2i 2 r2 (r1 + t2 ) . 1

L( ) = r1 r2 e

2i

(F.56)

Expressions (F.55) represent the FP-cavity input-output relations.

89

Radiation pressure noise in FP-cavity
In this subsection we will obtain the expression for radiation pressure force acting up on the movable mirror of the FP-cavity. In accordance with formula for radiation pressure we can write down the following: ^ ^ Wc2 + W ^ Fr.p. = c
d
2

=

^2 ^2 E c 2 + E d2 A. 4

(F.57)

Taking into account expressions (F.55) one can obtain: ^ F (t) = F +
0 ^ ( )[C2 c2 ( ) + D2 d2 ( )]e ^ i( - )t

d + h. c. = 2 g21 ( )]e ^
i( - )t

= F +
0

( )[Fa1 a1 ( ) + F ^

g

11

g11 ( ) + F ^

g

21

d + h. c. , (F.58) 2

where F
a
1

-

2it1 ei C2 , L( )

F

g

11

-

2ia1 ei C2 , L( )

F

ia2 (e
g
21

2i

+ 1)C2 , L( )

(F.59)

2 2 |C2 |2 |C2 |2 (1 + r2 ) is the constant classical radiation pressure force. F = c c Items that corresp ond to other noise op erators are neglected as they are small compared to the ab ove values.

Input-output relations derivation
Now we can write down IO relations for our sp eed meter scheme. One can notice that light b eam b etween input and output moments is sequentially reflected from two FP cavities. Therefore, sideband op erator that describ es output b eam for the first cavity at the same time describ es input b eam for the second one. Moreover, the b eam that enters cavity for the first time and the b eam that enters cavity b eing once reflected do not interact as they have different p olarizations ( and in our case). Then, b eams that leaves the scheme falling to the b eam splitter from the "north" and "east" are characterized by sideband op erators cn and ce that can b e expressed in terms of input b eams op erators as ^ ^ cn = ^ 1 [B ( )^I ( )-e bn L( )
2i

r2 t1 a1 ge11 ( )+e ^I I

i

t1 a2 ge21 ( )+i 2( )e ^I I

i

r2 t1 F xe ( - )] , (F.60.1)

ce = ^

1 [B ( )^I ( )-e be L( )

2i

r2 t1 a1 gnI ( )+e ^I 11

i

t1 a2 gnI ( )- 2( )e ^I 21

i

r2 t1 F xn ( - )] , (F.60.2)

b1 where F is the amplitude inside the cavity during second reflection, ^I n and ^I e are defined b1 I I I by formula (F.55.1) with replacing index 1 by 1n and 1e (sup erscript means first reflection and sup erscript I I means second reflection from FP cavity), and C2 by En = iE / 2 and Ee = -E / 2, where where E is the amplitude inside the cavity during the first reflection. Here we also intro duced additional pumping to comp ensate energy losses due to absorption in FP cavities. We supp ose that this pumping should feed the main b eam b efore it enters 90

the second cavity, then parameter = Fe /Ee = Fn /En . Now we are able to calculate icn - ce ^ ^ output b eam sideband op erator cs = ^ 2 1 ^I ^I [iB 2 ( )aI ( ) + iB ( )(e2i r2 t1 a1 gs11 ( ) - ei t1 a2 gs21 ( ))- ^s L ( ) -iL( )(e2i r2 t1 a1 gwI11 ( )-ei t1 a2 gwI21 ( ))-(B1 ( )- L( ))E ( )ei r2 t1 x- ( - )] . ^I ^I (F.61) cs = ^
2

where x- = op erator).

n + ie ^ ^ xn - x e e + in ^ ^ , s = - ^ , w = - ^ (here stands for any b osonic ^ 2 2 2

Radiation pressure noise sp ectral density SF ()
In order to calculate the radiation pressure noise for the sp eed meter as a whole we need to use the expression for radiation pressure fluctuational force acting up on the end mirror of Fabry-Perot cavity, presented in subsection F by formula (F.58). As in simple scheme one can present fluctuational force acting up on each of FP-cavity end mirrors as a sum of two indep endent items: ^ ^ ^ Fe = Fe + Fe , ^ ^ ^ and Fn = Fn + Fn , (F.62)

and the net force acting up on the scheme can b e presented as ^ ^ ^ F =F +F , where ^ ^ ^ F = Fn - Fe ,
0

(F.63) (F.64)

^ ^ ^ and F = Fn - Fe .

Let write down the explicit form of these expressions ^ F (t) = F + 2 ( )[Fas as ( ) + F ^
g ,I I gs11 ^

s11

( ) + F

g

s21

,I I gs21 ^

( )]e

i( - )t

d + h. c. , 2 (F.65.1)

where F

|E |2 is the corresp onding classical radiation pressure force, c F
,I
s11

a

s

2t1 ei E , L( ) F
g ,I
s21

(F.65.2)
2i

F and ^ F (t) = F + 2

g

2a1 ei E , L( )

-

ia2 (e

+ 1)E , L( )
,I I gs21 ^

(F.65.3)

0

^ ( )[Fas as ( ) + F +F
g
s11

g

s11

,I I gs11 ^

( ) + F
s21

g

s21

( )+ d + h. c. , (F.66.1) 2

, II II gs11 ^

( ) + F

g

, II II gs21 ^

( )]e

i( - )t

91

where F

2 |E |2 is the corresp onding classical radiation pressure force, c F
a
s

-

2t1 ei E B ( ) · , L( ) L( ) F
g ,I
s21

(F.66.2)

F

g

,I
s11

2a1 ei E t1 a1 e2i · , L( ) L( ) F
g , II
s11

ia2 (e
, II
s21

2i

-

Sp ectral density of radiation pressure noise can b e calculated using the same technique as used ab ove, i. e. 1 SF () = (SF ( - ) + SF ( + )) , (F.67) 2 where SF ( - ) = 4 2 2 ( - )(|Fas + Fas |2 + |F
g ,I
s11

2a1 ei E , L( )

F

g

-

ia2 (e

+ 1) E t1 a2 ei · , L( ) L( )
2i

(F.66.3) (F.66.4)

+ 1) E , L( )

+F

g

,I 2
s11

+ |F

| + |F
g
s11

g

,I
s21

+F

g

,I 2
s21

, II 2

| + |F

g

, II 2
s21

|+

| ) , (F.68)

If one uses formulae (F.5) and (F.20), then it is easy to obtain that radiation pressure noise sp ectral density in narrow-band approximation is defined by the following expression: SF () = where W = 8 W (1 + 2 )( 2 + 2 ) - 21 ( 2 - 2 ) , L2 (1 + 2 )( 2 + 2 )2 (F.69)

|E |2 (1 + 2 ) is the light p ower at the end mirror. 8

Shot noise sp ectral density Sx ()
Shot noise sp ectral density can b e obtained in the same manner as ab ove. Here K () = - where = expression: Sx () = 2i
3/2

|E | e-i r2 t1 [B1 ( - ) sin - L( - ) sin( + )] , c L2 ( - )

(F.70)

LO

+ arg C , = | |, = arg , and sp ectral density is defined by the following (F.71)

c2 (1 + 2 )|L( - )|2 = . 2 |K ()|2 32 W r2 t2 |B1 ( - ) sin - L( - ) sin( + )|2 1

Using the narrow band approximation defined by (F.5) and (F.20) the following expression for Sx can b e written if one supp ose = 0: Sx () = where L = c . L2 (1 + 2 )( 2 + 2 )2 , 32 W 1 sin2 (((1 + ) - 21 )2 + (1 + )2 2 ) (F.72)

92

Real scheme cross-correlation sp ectral density S

xF

It can b e shown that cross-correlation sp ectral density for the real sp eed meter scheme with optical losses is the same as for the ideal one, i. e. S
xF

= - cot . 2

(F.73)

The ab ove expression do es not dep end on frequency and is the same in narrow band approximation.

Narrow-band optimization of

2

In this section we will p erform the optimization of sp eed meter interferometer sensitivity at some fixed given frequency . In this connection it seems convenient to intro duce new dimensionless variables = to rewrite (F.23) as 2 = where a= 1 + A2 1 Pa + b +A, 2 P b= 2 (1 + 2 )2 . 2(1 - )(2 + 2 ) (F.75) , = - 1 =, P= 16 W , mL2 3 A = cot , (F.74)

+ 2 (2 - ) , 2 (1 + 2 )2

(F.76)

where a and b should b e taken at frequency . Being substituted to (F.75) these expressions will turn it to 2 = ab - 1 . (F.78) The second step is the minimization of the ab ove expression with resp ect to . Obviously, to obtain the optimal value of it is necessary to solve the following equation K = 0, where K = ab. Here we should rememb er that = / and = / , then the ab ove equation will b e trasformed to K K + = 0, that after simplification will b e written as 2 + 2 - 1 = The p ositive solution of this equation is
opt

Optimizing (F.75) with resp ect to P and A one can readily show that it reaches minimum at b 1 P= , and A = - , (F.77) ab - 1 ab - 1

2 +2 -1 = 0. 2

(F.79)

=+

2 + 93

2

,

(F.80)

where the last approximate equality corresp onds to the case of small losses . Substituting the obtained results to (F.78) with resp ect to the case of small losses, one will have + 2 - 3 4 4 min = , (F.81) = 4 2 - 3 1-+ that is the same expression that is presented in formula (F.24).

94

Bibliography
[1] V.B.Braginsky, Sov. Phys. JETP 26, 831 (1968). [2] Yu.I.Vorontsov, F.Ya.Khalili, Moscow Univ. Phys. Bull. 17, 205 (1976). [3] V.B.Braginsky,Yu.I.Vorontsov, F.Ya.Khalili, Sov. Phys. JETP 46, 705 (1977). [4] K.S.Thorne, R.W.P.Drever, C.M.Caves, M.Zimmerman, V.D.Sandb erg, Physical Review Letters 40, 867 (1978). [5] Y. I. Vorontsov, Theory and Methods of Macroscopic Measurements, NAUKA PUBLISHERS, 1989. [6] V.B.Braginsky, F.Ya.Khalili, Quantum Measurement, Cambridge University Press, 1992. [7] V. B. Braginsky, F. Ya. Khalili, Physics Letters A 147, 251 (1990). [8] F. Ya. Khalili, Yu. Levin, Physical Review D 54, 4735 (1996). [9] V.B.Braginsky, M.L.Goro detsky F.Ya.Khalili and K.S.Thorne, Physical Review D 61, 4002 (2000). [10] P. Purdue, Physical Review D 66, 022001 (2002). [11] Y. Chen, P. Purdue, Physical Review D 66, 122004 (2002). [12] Y. Chen, Physical Review D 67, 122004 (2003). [13] G. Sagnac, C. R. Acad. Sci. 95, 1410 (1913). [14] R. W. P. Drever, in Gravitational radiation, edited by N. Deruelle and T. Piram, (North Holland, Amsterdam, 1983). [15] K-X.Sun, M.M.Fejer, E.Gustafson, D.Sho emaker, and R.L.Byer, Physical Review Letters 76, 3055 (1996). [16] P.Beyersdorf, M.M.Fejer, and R.L.Byer, Optics Letters 24, 1112 (1999). [17] S.Traeger, P.Beyersdorf, L.Go ddard, E.Gustafson, M.M.Fejer, and R.L.Byer, Optics Letters 25, 722 (2000). [18] S. P. Vyatchanin, E. A. Zub ova, Physics Letters A 201, 269 (1995). [19] F.Ya.Khalili, arXiv:gr-qc/0211088 (2002). [20] S. P. Vyatchanin and A. Yu. Lavrenov, Physics Letters A 231, 38 (1997). 95

[21] B. Huttner, S. M. Barnett, Physical Review A 46, 4306 (1992). [22] T. Gruner and D.-G. Welsch, Physical Review A 51, 3246 (1995). [23] L. KnЁ S. Scheel, E. Schmidt, D. -G. Welsch, and A. V. Chizhov, Physical Review oll, A 59, 4716 (1999). [24] H.J.Kimble, Yu.Levin, A.B.Matsko, K.S.Thorne and S.P.Vyatchanin, Physical Review D 65, 022002 (2002).

96