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THE ANNUAL REPORT OF THE MSU GROUP (Jan.-Dec. 1999)
Contributors: V.B.Braginsky (P.I.), I.A.Bilenko, M.L.Gorodetsky, F.Ya.Khalili, V.P.Mitrofanov, K.V.Tokmakov, S.P.Vyatchanin, some part of the researchwas done in collaboration with the theoretical group of prof. K.S.Thorne

Contents

I

Summary
A The improvements of the Q-factor of the suspension modes . . . . . . . . B The development and the realization of methods of measurements of the excess noise in all-fused-silica bers . . . . . . . . . . . . . . . . . . . . . C The investigation of the sources of noises produced by electrostatic actuators D The analysis of new topologies and new meters for the advanced LIGO . .

2
2 3 4 4

II

Appendix
A B C D The improvement of the Q-factor of the suspension modes . . . . . . . . . A method for Measuring Small Vibrations of Optically Transparent Ob jects The investigation of the sources of noises produced by electrostatic actuators The analysis of new topologies and new meters for the advanced LIGO . . 1 The Energetic Quantum Limit . . . . . . . . . . . . . . . . . . . . . . 2 Low noise rigidity produced by pondermotive force . . . . . . . . . . . 3 Photo-thermal shot noise and thermodynamical uctuations in gravitational waveantennae . . . . . . . . . . . . . . . . . . . . . . . . . .

6
6 11 17 20 20 23 27

1


I. SUMMARY A. The improvements of the Q-factor of the suspension modes
During the 12 months period V.P.Mitrofanov and K.V.Tokmakov carried out a set of measurements of the quality factors Q of torsional-pendulum mode in all-fused-silica suspension of mirror's models. The measurements were performed in a new vacuum chamber whichwas manufactured one year ago. This chamber was designed specially to provide large mismatching of the impedances to reduce recoil losses. Special precautions were made to reduce the contamination of the ber by dust and by sedimentation of fused silica vapor on the ber during the welding process. The laser readout system which recorded the small decrease of the amplitude of the oscillations due to the damping was substantially improved up to the level of resolution which permits to measure the value of Q ' 108 with error less than 10% during time interval ' 102 hours. The main results of these measurements: a) Quality factor 1:7 108 10% was obtained in the torsional-pendulum mode (eigenfrequency 0.34 Hz) of the all-fused-silica bi lar suspension of 2 kg mirror's model. The bers in the suspension were welded to small bumps carved in the fused silica cylinder. b) Quality factor 2:3 108 10% was obtained in the same type of mode and suspension of 500 gram mirror's model (eigenfrequency 1.17 Hz). This model was backed in vacuum during 6 hours at t = 100 150 C . The bers in this pendulum were welded to fused silica cones which were attached to the cylinder by hydroxide-catalysis bonding. The cylinder of the pendulum with bonded cones was manufactured and provided to us by the Glasgow group (J.Hough and S.Rowan). Details of these experiments are presented in Appendix A.

2


B. The development and the realization of methods of measurements of the excess noise in all-fused-silica bers
The initial plan of MSU group from the beginning of this grant was to propose and to realize methods of registering of the excess noise in the violin modes of fused silica bers and nally to investigate this noise. During the work under the previous grant, the MSU group successfully demonstrated the existence of this noise in tungsten and steel wires and estimated the values of the perturbation of metric which this noise may mimic. The quality factors of the violin modes of steel and tungsten wires are 104 105 when for fused silica bers the quality-factor are 107 108 (see the reports of MSU group of the previous grant). Because the partial variance of the Brownian uctuations for the time interval of few p periods is proportional to 1= Q in the current grant the planned sensitivity of the meter has p to be not worse than 10;13 cm= Hz i.e. two orders better than that realized for metal wires. This tough task is aggravated by one important condition: to preserve the high Q-factors of the ber's violin modes it is not possible to spoil the ber's surface by glueing a mirror on it or by depositing any re ecting coating. After several unsuccessful attempts to implementvarious readout systems, which have to meet the above conditions, a new concept of such a readout system was proposed by I.A.Bilenko and M.L.Gorodetsky. The idea of the concept is to insert the ber across the optical beam within Fabry-Perot resonator. If the ratio of the ber's diameter to the wavelength is close to certain values then the displacement of the ber relatively to the resonator's mirror produces large phase shift in the output optical wave corresponding to a substantially high nesse. In the rst experimental test of this concept the already obtained nesse of such a meter exceeded 60. This value means that this meter may have the sensitivity p ' 10;15 cm= Hz if limited only by shot noise of 1mW laser pumping. Theoretical analysis of this concept is presented in the Appendix B.

3


C. The investigation of the sources of noises produced by electrostatic actuators
V.P.Mitrofanov and graduate student N.A.Styazhkina continued the study of the degradation of mechanical Q produced by the electrical eld applied to a model of the mirror. This e ect will inevitably appear if electrostatic actuator will be used for the positioning. The damping e ect was investigated in a model of electrical actuator which consisted of a set of strip electrodes alternatively at positive and negative electrical potentials. The measurements haveshown that for such an actuator there is a substantial damping besides the trivial e ects caused by losses in the resistivity in the external circuit of the voltage supply. It was observed that this additional damping depends strongly on the material of electrodes and the process of the electrode fabrication (see details in Appendix C).

D. The analysis of new topologies and new meters for the advanced LIGO
In this area several di erent results deserveto be mentioned. D-1. F.Ya.Khalili, V.B.Braginsky, M.L.Gorodetsky, K.S.Thorne have analyzed the socalled energetic quantum limit. In gravitational waveantennae for any topology due to the laws of quantum mechanics there exists minimal optical energy which had to be stored in the Fabry-Perot resonator depending on the chosen sensitivity of the antenna. If one wants to beat the standard quantum limit substantially then for standard topologies this minimal energy becomes unacceptably high. Intracavity readout schemes may substantially reduce this value of energy. In this case the problem of measurement is transposed to the local meter which records the response of intracavity test mass (see details in the Appendix D-1 and also in gr-qc/9907057). D-2. F.Ya.Khalili and V.B.Braginsky have done the analysis of \arti cal" mechanical rigiditywhich appears due to pondermotive forces in high nesse optical resonator when the pumping frequency is out of resonance. The analysis shows that this mechanical rigidity may have a very low level of noise which is of quantum origin. The usage of this rigidity 4


may substantially reduce technical problems in the implementation of the local meter in the intracavity readout system (see details in Appendix D-2 and in Phys.Lett.A, 257 (1999) 214. D-3. M.L. Gorodetsky, S.P.Vyatchanin and V.B. Braginsky,have done an analysis in detail of twophysical e ects which are playing importantroleinthe interferometric part of LIGO when the planned sensitivity is approaching the standard quantum limit. The rst e ect (photo-thermal shot noise e ect) is the uctuations of surface due to speci c shot noise which appears when optical photon, absorbed in the multilayer re ector, gives birth to a bunch of thermal phonons. The bunches produce uctuations of the temperature inside the bulk of the mirror. If the thermal expansion of the mirror is nonzero then these uctuations of temperature cause the uctuations of the mirror's surface. Numerical estimates show that one megawatt circulating power, which is planed for LIGO-II, is close to the upper value of acceptable power. The second e ect is of pure thermodynamical origin: the uctuations of temperature in the bulk of mirror (the entropy uctuations) together with nonzero thermal expansion factor are the source of independent uctuations of the mirror's surface. Numerical estimates show that for the present planned size of the mirror and for the size of the laser beam spot on it the sensitivity of LIGO-II antenna will not exceed SQL if sapphire will be used to manufacture the mirror. See details in Appendix D-3 and in Physics Letters A, A264 (1999) 1.

5


II. APPENDIX A. The improvement of the Q-factor of the suspension modes
A set of measurements of the quality factor Q of torsion-pendulum mode in all-fusedsilica suspension of the model of the mirrors was carried out during the 12 months period. The measurements were performed in a new vacuum chamber which was manufactured one year ago. This chamber was designed specially to provide large mismatching of the impedances to reduce recoil losses. We used our usual technique to suspend a 2-kg fused silica cylinder on two fused silica bers with a length of about 25 cm and a diameter of about 0.2 mm the bers were welded to the small bumps whichwere carved in the cylinder (see the Annual - 1998 Report of the MSU group). We paid special attention to improvements of various elements in the technique of fabrication of the suspension. Special precautions (screening of the ber in the process of welding, control of large (more than 10 micron) dust particles on the ber surface) were made to reduce the contamination of the bers bydust and by sedimentation of fused silica vapor on the ber during the welding process. The laser readout system which recorded the small changes of the amplitude due to the damping was substantially improved. The signal/noise ratio at the output of the system was increased approximately by a factor of three. It was possible to measure the value of Q ' 108 with error less than 10% during time interval about 102 hours. Free decay in the amplitude of the 2-kg model of the mirror is shown in Fig.1. An eigenfrequency of the torsion-pendulum mode was 0.34 Hz. The record value of the relaxation time 1:6 108s 10% was obtained for this mode whose loss properties are similar to those of the simple pendulum mode. The quality factor of Q =1:7 108 10% was achieved.

6


5.710E-2

5.705E-2

d ar , e d uti l p m a el g n A
5.695E-2 5.700E-2

5.690E-2

Y = -1.131 7E -006 * X + 0 .0 5 705 13 C o e f o f de term inatio n , R -square d = 0.99 783 4

5.685E-2 0 40 80 120 160 200

Time, hour

Fig1. Decay in the amplitude of angular motion of 2 kg fused silica pendulum fabricated by welding.

We have also examined dissipation for the other method of attaching the fused silica suspension bers to the mirrors. In this case the bers were welded to the fused quartz cones which were attached to the fused quartz cylinder using hydroxide-catalysis bonding technique. This method was investigated in University of Glasgow. The 0.5 kg fused silica cylinder with bonded cones was manufactured and provided to 7


us by the Glasgow group (J.Hogh and S.Rowan). The construction of such all-fused-silica bi lar suspension is shown in Fig. 2.

Welding

KOH bonding

Fig2. All fused silica pendulum constracted use KOHbonding/welding.

Free decay in the amplitude of 0.5 kg model of the mirror is shown in Fig.3. eigenfrequency of the torsional-pendulum mode was 1.17 Hz. This pendulum was out in vacuum for 6 hours at the temperature 100 150 C. The Q factor was to be 2:3 108 10%. This result permits to conclude that the pendulum mode 8

The baked found losses


associated with hydroxide-catalysis bonding are small in comparison with losses caused by other mechanisms.
9.52E-2

9.50E-2

d ar , e d uti l p m a le g n A
9.46E-2 9.48E-2

Y = -5.45485E-006 * X + 0.0952799 Coef of determination, R-squared = 0.994237

9.44E-2 0 40 80
time, hour

120

160

Fig3. Decay in the amplitude of angular motion of 0.5 kg fused silica pendulum with KOH bonded attachments.

The measured Q can be compared with the expected Q ' 1 109 , obtained from calculations based on the pendulum dilution factor and the measured loss angle of =1:4 10;7 for the fused silica bers. The discrepancy may be explained by incomplete elimination of possible sources of excess loss such as recoil losses, contact losses in indium gasket at mounting point, ber surface contamination and others. The goal of the further researchis 9


to reduce substantially the in uence of the additional mechanisms of losses in order to reach the Q-factor which is limited by the loss in material - fused silica.

References
1. K.Tokmakov, V.Mitrofanov, V.Braginsky,S.Rowan, J.Hough, Bi- lar pendulum mode Q-factor for silicate bonded pendulum, submitted to Proc. of the 3d. Amaldi Conference on Gravitational Waves and their Detection, 1999.

10


B. A method for Measuring Small Vibrations of Optically Transparent Ob jects
The currently designed laser interferometric detectors of gravitational waves LIGO (USA) have the potential sensitivity su cient not only for detecting radiation splashes arising as a result of the black holes collapse, and other astrophysical catastrophes but also for analyzing the dynamics of these processes 1]. In the rst variant (LIGO-I), the minimum detectable perturbance of the metric has a scale of h ' 10;21 and, in the next variants (LIGO-II and LIGO-III), it can attain 10;22 10;23 . To provide such a sensitivity, it is necessary that the variations in test-mass coordinates, which are caused by extraneous factors including thermal uctuations in the test masses themselves and in the systems of their suspension, would be negligible. As follows from the uctuation-dissipation theorem, the spectral density of the equilibrium thermal noise can be reduced by increasing the Q-factor for all modes of mechanical vibrations. The Q-factor for the string modes of steel wire suspensions used in the LIGO-I amounts to Qm ' 105 . Recentinvestigations haveshown that this value reaches 1 108 2] for quartz bers that makes it possible to expect the further increase in the sensitivity. This is explained by the fact that the root-mean square vibrational amplitude variations for times < Qm=!m , which are induced by thermal uctuations, are equal to kT xT ' m4Q ! (1) mm Here, k is the Boltzmann constant, T is temperature, m is the string e ective mass, and !m is the resonance frequency of the string fundamental mode. At the same time, the existence of an extra (excess) noise of the nonthermal origin is possible. The source of the excess noise may be the development of microcracks, the motion of dislocations and other defects in the suspension. The experiments carried out haveshown the presence of such a noise in tungsten wires 3] and steel wires 4] under high tension. The Q-factor of the string modes increases, hence, the e ect of thermal noise decreases, and the excess-noise intensity increases with tension. Therefore, for the optimum choice of the construction and the degree of suspension loading, it is necessary to have a possibility to detect the vibrations of the bers used in suspensions with a spectral coordinate resolution equal to
s

11


p 16 2 x! ' m Q kT 3 ' 4 10;15 cm= Hz: m !m
s

(2)

Here, m =1:1 10;3 g (a quartz ber 10 cm long and 100 m in diameter), !m =2 2:8 kHz (about 90% of the ultimate tension), and the averaging time is ' 2 =!m . Here we propose a method of the noncontact and nonperturbing coordinate measurements, which provides the desired sensitivity level, and present the results of a preliminary experimental test for a possibility of its realization. The method developed is going to be applied for investigating thermal and excess noise in quartz bers. Moreover, it can be used directly in a gravitational-wave detector for monitoring the suspension noise. We discuss also a possibility to observe e ects of the quantum theory of measurements 5] on the basis of the detector proposed. In order to measure the proper noise in suspension bers, it is necessary to satisfy several conditions. First, the measuring device must have a su cient sensitivity second, in the process of the measurements, the ber must be isolated from any external action including that of the measuring procedure in itself. Therefore, the noncontact methods are preferable because they require no connection of supplementary details (mirrors, plates of capacitive sensors, etc.). Since the dimension of the ber-sensor interaction domain is rather small for the bers used of 10 ; 200 m in diameter, it is di cult to obtain a high resolution by means of the most sensitive radio-frequency and microwave parametric transducers. In 4], an optical sensor based on the Michelson single-beam interferometer was used in which the surface of a metallic wire played the role of a mirror. It is possible to show that the ultimate possible sensitivityof such a sensor
s

x

! min

=4

h! W

(3)

where and ! are the wavelength and the frequency of the optical beam, and W is its power, does not allow us to attain desired value (2). The methods of the "knife-and-slit" type also provide nearly the same ultimate sensitivity. A higher sensitivity is provided by multi-beam interferometers, in particular, a FabryPerot interferometer. However, the re ection factor for a pure dielectric surface is low. At 12


the same time, depositing a well-re ecting mirror layer on the surface being measured is associated in our case with a damage of its structure, which decreases a high mechanical Q-factor of vibrations. The measurement method proposed is also based on using the Fabry-Perot cavity with a high nesse. However, it is free of the disadvantages indicated and provides a required sensitivity level. We consider a system consisting of the Fabry-Perot interferometer of a length L with mirrors having the re ection factor rm, (losses for scattering and absorption in these mirrors are ignored) and a dielectric body (it can be a plate, a lens, a microsphere, a ber, etc.) placed inside the interferometer (approximately, in the middle between the mirrors). Let the bodybecharacterized by amplitude factors of transmission, re ection, and losses tp, rp and ap:

jrpj2 + jtpj2 + japj2 =1:

(4)

In this case, the resonance frequency ! and, thus, the phase of the transientwave depends on the coordinate of the body. In the case of the optimum choice of the body displacement with respect to the center, wehave ! = 2!jrpj (5) x L (6) (x)= 1 ; r2 2jrpj a j2) sin(4 x= ): m (1 ;j p For a dielectric plate, the re ection factor rp depending on both the body thickness and the wavelength varies from 0 to rmax: n2 ; 1 rmax = n2 +1 (7) where n is the refractive index of the dielectric. Thus, ignoring losses for scattering and absorption in the process of the light re ection from the body and assuming that jaj2 2 1 ; rm, we obtain that the maximum sensitivity of the sensor proposed, which is limited only by the laser quantum noise, is 13


n2 +1 x! = 16F n2 ; 1 h! W

s

(8)

2 2 where F = =(1 ; rm) -is the cavity nesse. For the laser power W = 1 mW and rm = 0:99, we obtain that the spectral sensitivity in the case of the quartz plate (n = 1.45) p is x! ' 6 10;16cm= Hz. With increasing the re ection factor of the mirrors, the sensitivity increases but the problem of providing stabilityof the cavity mode arises. For the normal operation of the displacement sensor, it is necessary to provide the stability of the fundamental mode for the optical system formed by two mirrors and two re ecting surfaces of the dielectric body. This can be attained only in the case when all the surfaces coincide with the wave-front surfaces. If Rm and Rp are, respectively, the curvature radii for the mirrors and body surfaces, d is the plate thickness, and L d is the spacing between the mirrors, we arrive at the following conditions:

p Rm ' L 1+ 2dRL2; d 2

2

!

d< 2R

p

(9)

The most interesting case for many experimental applications (the measurement of motions of bers, microspheres, and drops), corresponds to the geometrical-optics limit and lies, unfortunately, at the stability boundary. However, it is possible to show that, if a stable con guration of main cavity mirrors is chosen, so that Rm ; L=2 L, the instability leads only to a small increase in the di raction losses of the cavity the term ap in formula (6) and to the corresponding insu cient decrease in the sensitivity (the estimate presented corresponds to the cylindrical geometry):

Rm ; L=2 2 cm: ap =2rp Rm
!

2

2

(10)

Thus, to conserve the sensitivity at the same level for the chosen parameter values, it is necessary to ful ll the relatively weak condition Rm ; L=2 < 0:1Rm . Toverify the method proposed, wehave carried out preliminary measurements. We used a Fabry-Perot interferometerhaving a 16 cm base with a plane-parallel glass plate 5 mm thick placed in the center of the interferometer. The frequency-tuned He-Ne laser served 14


as a pumping source. The measurements were carried out by observing resonances on the screen of an oscilloscope, whose horizontal-sweep voltage was used for the laser frequency control. In the interferometer, we used spherical dielectric mirrors with a radii of 50 cm 2 and re ection factors rm ' 0:99. Initially, the nesse F =280 20 was measured without the plate. Then, the plate was installed inside the interferometer. In the process of the measurements, the plate position along its axis was varied by the PZT drive. The nesse of the interferometer with the plate amounted to F =250 20. We measured the resonance frequency as a function of the plate longitudinal displacement. The maximum tuning ratio f = x ' 490M H z= m turns out to be approximately four times as low as that found from formula (5). The result obtained con rms the possibility of using the scheme proposed for investigating noise in quartz bers provided that the choice of the mirrors geometry is adequate. The decreased frequency-tuning ratio compared to the theoretical is explained by non-perfect in the adjustment of the optical system and di erence between the actual plate thickness and the optimum one for which rmax is attained. It should be noted that even in the case of using mirrors with the re ection factor R> (1 ; 1 10;4 ), it is necessary to takeinto account in calculations quantum e ects of interaction of a measuring device with an ob ject. These e ects manifest themselves as the energy transfer byquantum uctuations of the eld in the cavity to mechanical degrees of freedom. Thus, in the case of the same parameters as before, the amplitude incrementowing to this e ect amounts to for one period of vibrations xh ' ! Fm h!cW =4:4 10;13cm (11) 2 m which is comparable with thermal uctuations (1).
s

References
1. Abramovici A.A., et at., Phys. Lett. A, , vol. 218, 157 (1996) 2. Braginsky V.B., MitrofanovV.P., and Tokmakov K.V., Phys.- Dokl. (Engl. transl.), 40, 564 (1995) 15


3. Ageev A.Yu., Bilenko I.A., Braginsky V.B. and Vyatchanin, S.P., Phys. Lett. A,, 227, 159 (1997) 4. Ageev A.Yu., Bilenko LA., and Braginsky, V.B., Phys. Lett. A, 246, 479 (1998) 5. Braginsky V.B. and Khalili F.Ya., Quantum Measurements, Cambridge: Cambridge Univ. Press, 1992.

16


C. The investigation of the sources of noises produced by electrostatic actuators
The group continued the study of damping produced by electric eld applied to a model of the mirror. This e ect appears if electrostatic actuator is used for the positioning. The damping e ect was investigated in a model of actuator which consisted of a set of strip electrodes having alternatively positive and negative potentials which produced force on fused silica model of the mirror due to the inhomogeneous electric eld. Schematic of the electrostatic actuator is shown in Fig.4. The gold lm strip electrodes (length of 2 cm, width of 1 cm, spaced by 0.2 cm) were fabricated on the surface of fused silica block by a method of cathode sputtering. The gap between the pendulum and the electrodes was about 0.3 cm. Such actuator produced the force of the order of 1 dyn if the voltage 103 V was applied between the electrodes. The damping of the pendulum introduced by the trivial e ect caused by the losses in the resistor in the external circuit of the voltage supply was inessential. The measurements have shown that for such an actuator there is an additional damping proportional to the square of the applied voltage (see Fig. 5). We observed that this additional damping depends strongly on the material of electrodes, the process of the electrode fabrication and treatment. It was found also that when AC electric eld was applied to the electrodes the additional damping decreased with the frequency of the electric eld. For the frequencies higher than 150 Hz this damping was less than the error of measurements. In this way one may expect the signi cant reduction of the additional thermal noise produced by the electrostatic actuator at frequencies of operation of the gravitational wave detector. The detailed investigation of these problems is now in progress.

17


18


1E-7

8E-8

Q/ 1 g in p m a D

6E-8

4E-8

2E-8

0E+0 0E+0 2E+5 4E+5
Squared voltage, V^2

6E+5

8E+5

Fig.5 Damping produced by electrostatic actuator References
1. I.A.Elkin, V.P.Mitrofanov, Damping in the elecromechanical system caused by the processes in electrical subsystem, Moscow UniversityPhys. Bull. No 3, 31, 1999. 2. V.P.Mitrofanov, N.A.Styazhkina, In uence of surface water on the pendulum damping in an external electric eld, Phys. Lett. A, 256, 351, 1999. 19


D. The analysis of new topologies and new meters for the advanced LIGO
1. The Energetic Quantum Limit

The creation of new or the development of existing theoretical models are forcing the experimentalists to invent more sensitive methods of measurements. This trend, in particular, is evident in experiments with test masses in which the task of detecting the signal is the detection of a small force F (t) (an acceleration, a gradient of acceleration) acting on a macroscopic mass m. For example, at the second stage of the pro ject LIGO-II (Laser Interferometer Gravitational Wave Observatory 2]) the experimentalists will be confronted with the so-called Standard Quantum Limit of the sensitivity (SQL) for the force F (t) 3], and on the third stage this limit probably will be overcomed. The next serious limitation is the Energetic Quantum Limit. Gravitational wave in the interferometric antennae changes the phase of optical eld. In order to detect this phase shift, uncertainty of the phase should be su ciently small. Hence due to the uncertainty relation

E

h! 2

0

(12)

(!0 - is optical frequency), large uncertainty of the optical energy is required. This is not a peculiar property of interferometric meters only, but a consequence of a more general principle: In order to detect external action on the quantum ob ject, uncertainty ^ of the interaction Hamiltonian HI should be su ciently large 4]:
* Z

1

;1

HI (t)dt

2

+

h2 4

(13)

It can be shown, that this condition leads to very severe requirements for the optical energy E stored in the interferometer: 3 2 2 8 ;1 !signal 3 !0 M L nal E = M!sig! L = 2 102 erg 104gr 42 0 103s;1 4 105cm 2 1015s;1 (14)
!

20


where < 1 is the required sensitivity in the units of SQL, M is the test mass, !signal is the signal frequency, L is the arm length and !0 is the optical frequency. All known methods of overcoming the SQL for the case of position measurement of free test masses 5,6] apply additional restriction on the value of optical energy and pumping power. All of them require correlation between back action noise and output noise of the meter. Due to this requirement all these schemes are limited by the level of intrinsic losses of the e.m. resonator because uctuations corresponding to these losses can't correlate with the output noise of the meter. It can be shown that sensitivity of these methods cannot exceed the limit =
load intr
!1

=

4

(15)

where intr is the relaxation time whichcharacterize intrinsic losses in the e.m. resonator (i.e. absorption in the mirrors and beam-splitters and nonzero transmittance of end mirrors) and load is the relaxation time corresponding to coupling of the resonator with e.m. detector ;1 which should be close to !signal . Using the best known mirrors and the LIGO value of L = 4 105cm one can achieve the value of intr 10s. Hence if !signal ' 103s;3 then condition (15) may be satis ed only if 0:1. Further increase of sensitivity leads to necessity to decrease load. In this case
load

;1 !signal

(16)

and
2 2 E = M!6signal L (17) 16 !0 intr Such a very strong dependence of E on it practically impossible to obtain the value of

< 0:1 using standard topologies. A new principle of extracting of the information from gravitational-wave antenna was proposed in the article 7]. Instead of measuring the time phase shift of output optical wave it was suggested to detect directly the spatial phase shift of the optical eld in the antenna
21


using some QND-type method. Two practical realizations of this idea was considered in the articles 8,9]. The most promising of them is probably the \optical bar" scheme 8]. In this scheme optical elds in two arms of the antenna work as a mechanical springs with rigidity proportional to the optical energy E stored in each arm. They shift internal mirror placed into the resonator when gravitational wave shifts the end ones. This displacement may be measured relative to an additional reference mass which is not a ected by the optical eld in the antenna, using some of the measurement methods developed for solidstate gravitational-waveantennae. The value of the displacementmay be close to Lh(t)=2 if energy E is su ciently high:

E

m!

signal

3

2!

L2

0

(18)

where m - is the mass of internal mirror. Structure of this expression is similar to expression (14) for traditional scheme, but it contains the mass m of internal mirror instead of the masses of the end mirrors M , and, which is more important, it doesn't depend on . Hence in this case it is possible to rise sensitivitybeyond the level of SQL without necessityto rise E . If, for example, m =103gr and the values of all other parameters are the same as above, then E' 4 107erg.

References
1. V.B.Braginsky, M.L.Gorodetsky,F.Ya.Khalili, and K.S.Thorne, \Energetic Quantum Limit in large-scale interferometers", Proceedings of 3rd Amaldy Conference. 2. Abramovici et al, Science 256 (1992) 325. 3. V.B.Braginsky, F.Ya.Khalili, Phys. Letters A232 (1997) 340 V.B.Braginsky, M.L.Gorodetsky, F.Ya.Khalili, Phys. Letters A246 (1998) 485. 4. V.B.Braginsky,F.Ya.Khalili, "Quantum Measurement", Chap. IX, ed.by K.S.Thorne, Cambridge University Press, 1992, 5. S.P.Vyatchanin, E.A.Zubova, Physics letters A201 (1995) 269 6. V.B.Braginsky, M.L.Gorodetsky, F.Y.Khalili, K.S.Thorne, Phys. Rev D, in press 22


7. V.B.Braginsky, F.Ya.Khalili, Phys. Letters A218 (1996) 167 8. V.B.Braginsky, M.L.Gorodetsky,F.Ya.Khalili, Phys. Letters A232 (1997) 480 9. B.Braginsky, M.L.Gorodetsky,F.Ya.Khalili, Phys. Letters A246 (1998) 485 10. V.B.Braginsky, F.Ya.Khalili, Phys. Letters A257 (1999) 214.
2. Low noise rigidity produced by pondermotive force

It is evident that in case of intracavityschemes the focus of the problem shifts to the local meter of the position of the central mirror. Intracavityschemes require higher precision to exceed the SQL. During the last twenty ve years several possible methods of such a measurementwere proposed. The experimentalist hasachoice in this type of experiments: to detect F (t) acting on a 2 free mass m or on a mass coupled with rigidity K = m!m, in other words on an oscillator with eigen frequency !m which is close to !F - characteristic frequency of F (t). At rst glimpse the choice of free mass does look more attractive because it is easier to isolate the mass from the heat bath. The free mass and the oscillator "behave" as quantum ob jects if the following conditions for them are satis ed: 2kT
2

2kT QH:B

H:B :

h h

(19) (20)

:

where is the averaging time, k is Boltzmann constant, T is the temperature of the heatbath, H:B: is the relaxation time for the free mass, QH:B: is the quality factor of the oscillator. For T =300K and =10;2 sec it is necessary to reach H:B: ' 1010 sec: to satisfy condition (19). The value H:B: ' 108 sec was already reached 1], new methods of the "subtraction" of the heat-bath action on the free mass are also proposed. These methods permit to reach the equivalent H:B: ' 1010sec 2]. For the same values of T and it is necessary to have QH:B: 1012 (note that at room temperature the highest obtained Q ' 4 108 3]) to satisfy the condition (20). 23


If conditions (19,20) are satis ed then to circumvent the SQL it is necessary to use some methods of quantum measurements, for example Quantum-Non-Demolition methods (see 4]) or methods based on continuous monitoring of coordinate with the use of spectral or time domain features of noises of the meter 5,6]. All proposed schemes for a free mass based on these last two methods are in essence parametrical converters (the x(t) is transferred into the modulation of some parameter of an e.m. resonator coupled to the mass m). Hence they limited by the condition (15) for the relaxation times of the resonator. In some speci c cases of this class of meter even more strict limitations can exist. For example, in the case of QND speed meter 7, 8] it is necessary to have (!
signal intr

)

;1=

4

(21)

so if = 0:1 and !signal = 102s;1 then must be intr 102 s (which corresponds to the quality factor Qintr 1012 even in the case of microwave resonator). In contrast with the free mass in the case of harmonic oscillator it is possible to overcome the SQL without using the correlation of the meter noises because during free unitary evolution quantum state of the oscillator is periodic. This feature permits to evade the confrontations with conditions (15,21). Pondermotive force which acts on two mirrors of Fabry-Perot resonator (in microwaveor optical bandwidth) creates mechanical rigidity Kpond whichmay be large if the re ectivityof the mirrors is high and its losses are low and if this resonator is pumped with the frequency ! which is detuned from the eigen frequency !e = ! ; (see e.g. 9]). If the detuning is large: e 1 (this is necessary to get a lowlevel of noises, see below), then this rigidityis equal to
2 Kpond = m!m = 2l!e E = l22!e3W : 2 e

(22)

Here
e

= lF c 24

(23)


- is the relaxation time of the resonator, l is the distance between the mirrors, c is the speed of light, E -is the energy between the mirrors. On the other hand the nite value of F will produce a uctuating force which will act on the masses coupled to the mirrors. The spectral density of this \back action" force is equal to
e SB:A: = lh!2e E = l2h!( W)2 : 2 4 e e

(24)

The \quality" of rigiditymaybe characterized by the minimal detectable force in units of SQL. Wemay use the same parameter as in formula (15-21), but nowwetakeinto account only the noise from the rigidity. In this case
2

= SB:A2: = 1 : hm!m 2 e

(25)

Substituting this value of into the formula (22) results in

Kpond

= 16!e WcF 22

26

' 10 dyn=cm
10

W 4 erg =s 10

!

5 10

F

2 5

!e 2 10

15

6

: (26)

Thus for = 0:3 and W = 107 erg=s one may obtain the rigidity Kpond ' 1010dyn=cm which will \convert" a 104 gram mass into a mechanical oscillator with eigen frequency !m ' 103 s;1. For < 0:3 the value of W will be inaccessible high. However for smaller masses (e.g. m ' 10gr) this type of arti cal rigidity seems attractive. It is worth noting that this method may be realized only because very high values of F were recently obtained 10]. Another scheme with the pondermotive rigidity has been considered in the article 11]. If mirror with transmittance T is situated inside the Fabry-Perot resonator equidistantfrom two end mirrors with high nesse F then this internal mirror splits the eigen modes of the resonator into doublets whose frequencies are apart from each other by = cT : l where l is the distance between the internal mirror and the end ones. 25 (27)


If the upper frequency component of the doublet is excited then the pondermotive force which acts on the internal mirror will strongly depend on the displacement of the mirrors due to the redistribution of the e.m. energy E in the two parts of the resonator. In other words mechanical rigidity Kpond between the internal mirror and the end mirrors will be created. The value of Kpond may be presented as

K

pond

= 4l!e E = 4!ecW F ' 1010dyn=cm 2 2T

W 3 107erg=s

!

!e 2 10

15

10

F

6

10;2 : (28)

T

;

1

Here E is the energy stored in each half of the resonator. This estimate show that for the mass of the mirror m =104g the mechanical eigen frequency of the internal mirror will be equal to 103s;1 if the pumping power equals to W =3 107erg=s. This relatively modest value of W for such a large Kpond is the rst substantial advantage of this type of arti cal rigidity. Spectral density of the back action force in this case is equal to

SB:A: = 2h!e W l2 2
hence
2

(29)

SB:A = hm!2: = 2FT =1:5 10; m

4

10

T

;1
2

;

106

F

;

1

:

(30)

Thus the second advantage is a very low level if intrinsic quantum noise. This level is approximately equal to the level of noise of mechanical oscillator with quality factor Qm ' 2 1016 in the heat-bath with temperature 300K. The \fee" which experimentalist has to pay for such a large value of Kpond and small value of SB:A: is the level of symmetry in the positioning of inside mirror l=l and the phase di erence in the two arms of the resonator . whichmay be quite severe if the value of < 0:1 is required.

References
1. V.B.Braginsky, V.P.Mitrofanov, K.V.Tokmakov, Phys. Letters A218 (1996) 164. 26


2. V.B.Braginsky,Yu.I.Levin, S.P.Vyatchanin, Meas., Sci. and Techn., in press. 3. V.B.Braginsky, V.P.Mitrofanov, V.I.Panov, \Systems with small dissipation, ed. K.S.Thorne, Chicago Univ. Press, 1985. 4. V.B.Braginsky, F.Ya.Khalili, Rev. Mod. Phys. 62(1996) 1. 5. V.A.Syrtsev, F.Ya.Khalili, Sov. Phys. JETP 106 (1994) 744. 6. S.P.Vyatchanin, E.A. Zubova. Phys.Lett.A, 201 (1995) 269. 7. V.B.Braginsky, M.L.Gorodetsky, F.Y.Khalili, K.S.Thorne, Phys. Rev D, in press 8. V.B.Braginsky, F.Ya.Khalili, Phys.Lett. A147 (1990) 251 9. V.B.Braginsky, A.B.Manukin, M.Yu.Tichonov, Sov. Phys. JETP 58 (1970) 1550. 10. G. Rempe, R. Tompson, H.J.Kimble, Opt. Lett. 17 (1992) 363. 11. V.B.Braginsky, F.Ya.Khalili, Phys. Letters A232 (1997) 340 V.B.Braginsky, M.L.Gorodetsky, F.Ya.Khalili, Phys. Letters A246 (1998) 485.
3. Photo-thermal shot noise and thermodynamical uctuations in gravitational wave antennae

The aim of this Appendix is to present the results of the analysis of the two e ects which are "responsible" for random uctuations of temperature in the bulk of the mirrors and whichmay mimic at certain level the gravitational waveif 6= 0: photo-thermal shot noise (due to random absorption of optical photons in the surface layer of the mirror) and thermodynamical uctuations of temperature. As in both e ects we are interested in the uctuations of the coordinate averaged over the spot on mirror surface with radius r0 ' 1:5 cm of laser beam, which is much smaller than the radius of mirror ' 15 cm, we replace the mirror by half-space: 0 x< 1 1 < y < 1 1 < z < 1. We consider also for simplicity this half-space to be isotropic (the anisotropy of di erent constants for sapphire is not very large). The role of these two e ects for the condition to reach the standard quantum limit of frequency stabilitywas analyzed more than 20 years ago 1]. As at room temperature the power radiated from the surface (Stephan-Boltzmann law) 27


is muchlower than the heat exchange due to thermal conductivity(Fourier law), belowwe use the simpli ed boundary condition: @u(x y z t) =0 (31) @x x=0 where u is the deviation of temperature from the mean value T . For the calculations we use approximation in whichwe neglect the e ects of limited speed of sound but takeinto account thermal relaxation.

Photo-thermal shot noise in the mirrors of the antennae
The multilayer coating of the mirrors absorbs a fraction of power W of the optical beam, circulating between the mirrors of Fabry-Perot resonator. This fraction for the best existing today coatings has not been measured with good precision yet, but for estimates the usually accepted value of this fraction is approximately equal to 5 10;7 10;6 . The value of W su cient to reach h ' 10;22 (which is close to Standard Quantum Limit (SQL) of sensitivity) _ is approximately W ' 1 106 Watt=1013erg=sec or N0 ' 5 1024 photon/s. It is reasonable _ to estimate that each mirror absorbs Wabs ' 1Watt = 107 erg/s or N ' 5 1018 photon/s. Each absorbed optical photon having energy h!0 ' 2 10;12 erg gives birth to a bunch of approximately 50 thermal phonons. Each of these phonons has a relatively short free path l (for SiO2 and Al2O3 the values are lSiO2 ' 8 10;8 cm and lAl2O3 ' 5 10;7 cm. After a very short time interval which is a few times longer than l =vs ' 10;12 s (where vs is the speed of sound) these phonons will produce a local jump of temperature inside the coating (5 10) 10;4 cm thick. As wewanttoknowthe variance of temperature during the averaging time =!grav 5 10;3 s which is several orders longer than l =vs and as _ during this time a large number (N =!grav 2:5 1016) of photons are creating bunches of thermal phonons | it is reasonable to use the model of shot noise. As the length lT is much smaller than the radius of laser beam r0 ' 1:5 cm for semi-qualitative analysis one can use one dimensional model, described by thermal conductivity equation: @u(x t) ; a2 @ 2u(x t) = w 2 (x) (32) 2 @t @x2 C r0 28


@u(x t) =0 0 x< 1 @x x=0 hw(t) w(t0)i = h!0W0 (t ; t0) w(t)= Wabs ; W

(33)
0

W0 = hWabs(t)i

where a2 = =( C ), is thermal conductivity, is densityand C is speci c heat capacity. Assuming that displacement X1D (whichisof interest) is proportional to temperature u averaged along the axis x

X1D =

Z

1
0

dx u(x t):
1D 1

(34) : (35)

it is not di cult to obtain the spectral densityof X

1 STS1D(!)= 2 2 ( h!0W20)2 !2 : C r0

For the exact calculations for half-space (see details in 2]) weintroduce the displacement X averaged over the beam spot on the surface

X= 1 r

ZZ

2 0

1 ;1

dy dz vx(x =0 y z) e

;(y2 +z2 )=r

2 0

(36)

where vx(x =0 y z) is x-componentof vector of deformation on the surface, and nd the spectral densityof X : 1 STS (!)= 2 2 (1 + )2 ( h!0W20)2 !2 C r0 (37)

where is the Poisson coe cient. Here and below we use the fact that for materials at room temperature the thermal 2 relaxation time of the spot T ' r0 =a2 is very large:
1

We use \one-sided" spectral density, de ned only for positive frequencies, which may be calcuhX t X t
Z

lated from correlation function
S

( ) ( + )i using formula
1 ;1
d hX (t)X (t

X

(! )= 2

+ )i cos(! ):

29


!

a2 : 2 r0

(38)

It is easy to see that the exact solution (37) di ers from the approximate one (35) only byPoisson coe cient.

Thermodynamical uctuations of temperature, thermoelastic damping and surface uctuations
It is known from thermodynamics that the total variance of uctuations of temperature in volume V is described by the following simple formula:

T h T 2 i = CV

2

(39)

where is the Boltzmann constant. In classical solid state thermodynamics these uctuations are not correlated with uctuations of volume (which are responsible for the Brownian noise) but this is not the case for nonzero thermal expansion coe cient. To nd the in uence of the fraction of this type of uctuations in the vicinityof !grav on the vibration of the surface along the x-axis in direct waywe use the Langevin approach and introduce uctuational thermal sources F (~ t) added to the right part of the equation of thermal conductivity: r

@u ; a2 u = F (~ t): r @t

(40)

These sources should be normalized in a way to satisfy formula (39). By substituting the solution of this equation for the temperature u into the equation of elasticity and averaging displacement (36) over the spot on the surface, we nd spectral density of its uctuations in frequency range of interest (see details in 2]):
2 21 2 0! STD(!) ' p8 2(1 + )2 T a3 !2 if ra2 1: (41) C r0 2 We can also nd the spectral density of displacement using Fluctuation-Dissipation theorem (FDT) if we assume that the only dissipation mechanism in the mirror is thermoelastic damping. Elastic deformations in a solid body through thermal expansion lead to inhomogeneous distribution of temperature and hence to uxes of heat and losses of energy. To

30


calculate the uctuations in this approachwe should apply a periodic pressure p distributed over the beam spot on the surface:

p(y z t)= Fr02 e 0

2 ;(y2 +z2 )=r0 ei!t

(42)

and we should nd susceptibility = X=F0. After that in accordance with FDT the spectral density of displacement X must be proportional to imaginary part of susceptibility :

Sx(!)= 4 !T jIm( (!))j:

(43)

Calculations (see details in 2]) show that FDT gives the same result as formula (41). Therefore, we can conclude that thermodynamical uctuations of temperature are the physical source of uctuations deduced from FDT based on thermoelastic damping. Calculations of Im( ) are rather bulky but Zener suggested a very simple approach allowing to estimate thermoelastic loss angles and hence the imaginary parts of elastic coe cients. It allows to nd Im( ) only from formula for in zero order approximation:
(0)

(1 2 (!)= p ; ) 2 E r0

(44)

where E is the Young's modulus. The imaginary parts of elastic coe cients due to thermoelastic damping may be calculated in Zener's approach according to the formulas: Im( ) = S ; T ! T = 2 TE 1+ 2 1+ !2 T C T Im(E ) = ES ; ET ! T = 2 TE 2 E ET 1+ !2 T C

!T 1+ !2 !T 1+ !2

T T

2 2

(45)

where ET ES are correspondingly iso-thermic and adiabatic Young's moduluses (ET ' ES ' E ) and T S are iso-thermic and adiabatic Poisson coe cients ( T ' S ' ), T is the time of thermal relaxation of our spot. By substituting these values in (44) and 2 then in (43) and taking into account that T =2r0 =a2 1=! (this relaxation time maybe found relatively easy from the solution of the problem of thermal relaxation of the heated spot) we nally obtain the same formula (41). 31


It is really worth noting that all the three methods give precisely the same result.

Numerical estimates
Nowwewant to compare the uctuations of the mirrors' surface caused by the described above mechanisms with other known sources of noise in LIGO antennae. The main determining source of noise associated with the mirrors' material is usually described by the losses in the model of structural damping (we denote it as Brownian motion of the surface). In this model the angle of losses does not depend on frequency and analogously by substituting E = E (1 + i ) in (44) and then in (43) we obtain: (1 2 (46) SSD(!) ' 4 !T p ; ) : 2 E ro For simplicity here we neglected the unknown imaginary part of Poisson coe cient. The sensitivity of gravitational wave antenna to the perturbation of metric may be recalculated from noise spectral density of displacement X using the following formula: p2S h(!)= L X (47) where we used the fact that antenna has two arms (with length L) and in each arm only one mirror adds to noise (on end mirrors the uctuations are averaged over the larger beam spot). The LIGO-II antenna will reach the level of SQL, so we also compare the noise limited sensitivity to this limit in spectral form: 8 (48) h(!)= m!hL2 : 2 On gures 1 and 2 we plot the sensitivity limitations for the perturbation of metric caused by the described thermal mechanisms and SQL (formulas 37,41,46,47,48) for fused silica (Fig.6) and sapphire (Fig.7) mirrors. We used the following numerical values for the parameters of the two materials | fused silica and sapphire (we should note that the actual gures, whichvarious sapphire manufactures provide, signi cantly vary { tens of percents, especially for the thermal expansion coe cient):
s

32


! =2
Fused silica: =5:5 =2:2

100 s

;1

r0 =1:56 cm T =300 K
(49) (50)

!0 =2 1015 s

E =7:2
Sapphire: =5:0 =4:0

E =4

L =4 105 cm W0 =107erg=s erg 10;7 K;1 =1:4 105 cm s K g C =6:7 106 erg m =1:1 104 g cm3 gK erg 1011 cm3 =0:17 =5 10;8 erg 10;6 K;1 =4:0 106 cm s K g C =7:9 106 erg m =3 104 g cm3 gK erg 1012 3 =0:29 =3 10;9 : cm

;1

(51) (52)

From these two gures it is evident that the e ects associated with thermal expansion in no case may be neglected. Especially unfortunate result is obtained for sapphire { the noise from thermodynamical uctuations of temperature dominates for frequencies of interest and is several times larger than the SQL. While for fused silica the situation is the opposite. The SQL is one order larger than TD uctuations. It is important to note, that unlike structural damping for which angles of losses at low frequencies were not yet directly measured, our mechanism has fundamental nature and is calculated explicitely without model assumptions.

33


Fused Silica (SiO2)
10
-21

10

-22

1 2 3 4
1 2

-

Brownian SQL Thermodynamic Photo-thermal

h, (Hz

-1/2

)
10
-23

3 10
-24

4

10

-25

1

10

100

1000

f, (Hz)

Fig.6 The sensitivity limitations for the perturbation of metric caused by the SQL and by thermal uctuations in fused silica mirrors.

34


Sapphire (Al2O3)
10
-21

10

-22

1 2 3 4
3 2 4

-

Brownian SQL Thermodynamic Photo-thermal

h, (Hz

-1/2

)
10
-23

10

-24

1

10

-25

1

10

100

1000

f, (Hz)

Fig.7 The sensitivity limitations for the perturbation of metric caused bythe SQL andby thermal uctuations in sapphire mirrors.

CONCLUSION
The analysis and numerical estimates presented above for two e ects allow to give some recommendations for the strategy of upgrading laser interferometric gravitational waveantennae. 1. It is important to emphasize that contrastingly to the gravitational wave, both e ects 35


do not displace the centers of masses of the mirrors. Thus, in principle it is possible to subtract these e ects as well as usual Brownian uctuations (with certain level of accuracy) in a way similar to the procedure suggested for the subtraction of thermal noise in suspension bers 3]. However, till now to our knowledge nobody presented the scheme of such subtraction. 2. Potentially there exists a possibility to decrease substantially these e ects bychoosing materials for the mirrors with substantially smaller values of . There also exists the possibility to use special cuts of monocrystals without center of symmetry for which turns to zero (crystal quartz for example) as in secondary frequency standards. 3 3. Thermodynamic uctuations decrease with growing radius of the beam spot as r0=2 even stronger then for the Brownian uctuations. In this way, some optimization of the geometry of the resonators may help. 4. The "brute force" method for the improvement of the sensitivity by increasing the circulating optical power W does not look promising, taking into account photo-thermal shot noise. It seems that few mega-Watts is the upper limit for W .

References
1. V.B. Braginsky and S.P.Vyatchanin, Sov. Phys. JETP, 47, 433 (1978). 2. V.B. Braginsky, M. L. Gorodetsky and S.P.Vyatchanin, Physics Letters A, A264,1 (1999). 3. V.B. Braginsky,Yu. Levin and S.P.Vyatchanin, Meas. Sci. Technol., 10, 598 (1999).

36