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THE ANNUAL REPORT OF THE MSU GROUP (Jan.-Dec. 2000)
Contributors: V.B.Braginsky (P.I.), I.A.Bilenko, M.L.Gorodetsky, F.Ya.Khalili, V.P.Mitrofanov, K.V.Tokmakov, S.P.Vyatchanin

Contents

I

Summary
A The reduction of the test mass oscillation damping caused by electrostatic actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B The development and realization of methods of the excess noise measurement in the all fused silica ber . . . . . . . . . . . . . . . . . . . . . . . . C The analysis of the thermo-refractive noise in the gravitational waveantenna D On the possibility to subtract the Brownian uctuation in the mirror . . . E The scheme of measurement of thermo-refractive noise . . . . . . . . . . . F The analysis of the frequency uctuations of nonlinear origin in selfsustained optical oscillators . . . . . . . . . . . . . . . . . . . . . . . . . G The analysis of the sensitivity of the discrete sampling variation measurement

3
3 3 4 5 5 6 6

II

APPENDIXES Appendix A Damping of the test mass oscillations caused bymultistrip elec-

8

trostatic actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Appendix C Thermo-refractive noise in gravitational waveantennae . . . . . 19 Appendix D. The problem of compensation of internal mechanical noise in test mass of gravitational waveantennae . . . . . . . . . . . . . . . . . . . 28 Appendix E. On the possibility to measure thermo-refractive noise in microspheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1


Appendix F Frequency uctuations of nonlinear origin in self-sustained optical
oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Appendix G The Discrete Sampling Variation Measurement . . . . . . . . . 69

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I. SUMMARY A. The reduction of the test mass oscillation damping caused by electrostatic actuator.
During the year 2000 V.Mitrofanov, K.Tokmakov and graduate student N.Styazhkina completed the study of the test mass oscillation damping produced by electrostatic actuators. The MSU group believes that usage of actuator directly applied to the mirror is inevitable. Thus the damping e ect and correspondingly the uctuating force acting on the mirror may be a serious obstacle to reach the planned sensitivity of the antenna. In the years 1998 and 1999 of the current grant the damping e ect was observed and measured. The measurements have shown that this e ect is a big one (see the MSU group reports of the years 1998 and 1999) and that it will substantially reduce the sensitivity. In the year 2000 di erent sources of damping were carefully analyzed and measured. Several sources were substantially reduced by appropriate choice of the actuators parameters. After this reduction it turned out that the key source of damping is associated with the property of the modelg of fused silica mirror. Special procedure of the ame treatment of this surface was proposed and tested. This treatment allowed to reduce the damping e ect of the pendulum mode to the level smaller that 3 10;9when the electrostatic actuator provided the dc force 10;4 N. ~ This value of damping e ect is smaller than the lowest damping measured till now in the test mass suspension. (See details in Appendix A).

B. The development and realization of methods of the excess noise measurementin the all fused silica ber
The MSU group considers the problem of the excess noise in the high quality factor violin and pendulum modes in all fused silica suspension as one of the most important factors. In other words the unknown value of this noise may de ne the "fee" which has to be paid as some decrease of the sensitivityof the antenna and which cannot be precalculated. 3


In 1998 and 1999 dr. Bilenko and his students have proposed and tested two designs of the excess noise meter based on optical microspheres which promised to meet two important conditions: no any contaminations on the ber's surface and high nesse in the optical readout systems (see previous annual reports). Both design failed to satisfy the second condition. In the year 2000 it was decided to get rid of microspheres and use a Fabry-Perot resonator (optical meter) with a small at mirror (4 2 1mm3) made of pure fused silica welded in the middle of the tested ber. The surface of this mirror had multilayer coating with high re ectivity. Two short fused silica sticks were welded to the opposite "corners" of the mirror before coating. The "free" ends of these sticks will be used for welding the fused silica bers which noise will be measured. This design already was tested and the nesse p higher than 40 was obtained, whichallow to reach sensitivity at the level 3 10;13cm= Hz. This design guarantees that the coating will not be damaged bythe welding of bers. The assembling of all installation is in progress now, and the measurements of the violin mode noise will start in the close future.

C. The analysis of the thermo-refractive noise in the gravitational waveantenna
In the year 1999 the MSU group has reported the results of the analysis of two new sources of noise whichmay seriously change the antenna's sensitivity. In essence these sources are of nonlinear origin: namely due to nonzero value of the thermal expansion coe cient T = (1=l) dl=dT and due to the uctuations of the temperature (thermodynamical temperature uctuations in equilibrium and photo-thermal shot noise). The analysis has shown that these two sources of noise may decrease substantially the antenna's sensitivity(Physics Letters A, A264 (1999) 1 ). In the year 2000 the analysis of another source of noise was carried out by M. Gorodetsky and S. Vyatchanin (Physics Letters A, A 271 (2000) 303). It is similar by origin by previous ones: the same uctuations of temperature combined with nonzero value of the temperature dependence T =(1=n) dn=dT of the refraction index n in the mirror's coating are producing 4


e ective uctuating change of the distance between the mirrors because a fraction of e.m. eld is inside the mirror's coating. For usually used TiO2 or Ta2O5 in the multilayer coating the value of T is larger than T for SiO2, and the value of noise produced by thermorefractive e ect may "give contribution" to the total noise budget comparable with SQL (see Appendix C).

D. On the possibility to subtract the Brownian uctuation in the mirror
It was shown by S. P. Vyatchanin and undergraduate student S. E. Strigin (Physics Letters A, A272 (2000) 143) that the same thermo-refractive noise does not permit to measure the Brownian uctuations in the mirror itself with su cient accuracy in order to use this value for subtraction of this noise from the movement of the mirror's centers of mass (see part II of Appendix D).

E. The scheme of measurement of thermo-refractivenoise
M. Gorodetsky has proposed and analyzed a simple scheme of experimentwhichallows to measure the thermo-refractive noise in fused silica. The basic idea is to use as a test ob ject whispering gallery mode in fused silica microsphere. In this type of optical resonator (with the diameter D ' 10;2 cm) the main part of e.m. eld is inside the fused silica (in contrast with Fabri-Perot resonator). If the frequency of pumping wave is tuned to the slope of the resonance curve (the quality factor of the mode may exceed 109), the the thermo-refractive noise will be converted into random modulations of the output optic power. If the optimal mode of the resonator is used then the spectral density of the modulation of frequency will be at the level: 3=4 1000s;1 !1=4 1 q ;12 100 m p S !=! ( ) ' 4 10 (1) D Hz where is the frequency of analysis of uctuations. This value may be observed if technical problems of modes' identi cation are solved (see details in Appendix E). 5


F. The analysis of the frequency uctuations of nonlinear origin in self-sustained optical oscillators
To the best knowledge of MSU group members the theoretical analysis of the role of nonlinear origin noises in the pumping oscillator was not carried out till now. This analysis was done by S. P. Vyatchanin in the frame of this grant (Physics Letters A, accepted for publication). In the analysis the nonzero values of T and T were taken into account as well as back action noise. The net result of this analysis is that the thermo-refractive noise in NdYAG laser will dominate and that with 10 meter long reference cavity the pumping p laser mayhave the frequency uctuations at the level 10;20 1= Hz. This numerical estimate (based on certain parameters of the laser and the reference cavity) shows that the revision of the frequency noise of the pumping laser should be done as well as the condition for the level of symmetry in the two arms of the antenna (see Appendix F).

G. The analysis of the sensitivity of the discrete sampling variation measurement
According to the plan of researches the MSU group continued the analysis of di erent schemes and methods of measurement which may allow to circumvent the SQL value of the antenna sensitivity. In the year 2000 S.Danilishin, F.Khalili and S.Vyatchanin have completed the analysis of a new scheme of quantum meter. The rst key task in this analysis was to eliminate the most inconvenient feature of the variation quantum measurement: the necessity to know apriori the arrival time and the shape of the signal because the hardware setup of the antenna depends on both of them. This problem is solved. The basic feature of the new scheme of variation measurement is the discrete sampling: the procedure of the measurement is divided into many short intervals (shorter than the characteristic period of the signal). Thus the only necessary a priory knowledge is the upper frequency of the signal. The \fee" which has to be paid for this improvement is the rise of the amount of the energy pumped into the meter. It has to 6


be 720= 4 7:4 times bigger than in traditional quantum variation meter (see details in Appendix G).

7


II. APPENDIXES Appendix A. Damping of the test mass oscillations caused bymultistrip electrostatic actuator
Introduction

Laser interferometric gravitational wave detectors are now under construction by several groups around the world. The detectors test masses are the fused silica mirrors suspended by thin bers to isolate them from perturbative forces. The mirrors' position and orientation are controlled by the low-noise servosystems to maintain the interferometer at the proper operating point. Magnetic actuators applied to produce the control forces can create relatively high excess noise 1]. As an alternative to the magnetic actuators, electrostatic systems using electrical forces were developed 2,3]. The multistrip electrostatic actuator that does not use a conductive coating on the mirror is the most promising one. It consists of series of plain strip electrodes alternatively at positive and negative potentials whichare placed near the mirror surface. The detailed analysis of forces produced by the multistrip electrostatic actuator was given in 4]. Another very important feature of the actuator is damping of the test mass oscillations caused by this actuator because in accordance with the uctuation-dissipation theorem the excess damping of the test mass within the gravitational wave detector operating range of frequencies is a source of additional thermal noise. The articles devoted to the study of damping in torsion pendulums being under the action of the electrostatic force applied between an electrode placed on the pendulum bob and a nearby electrode were published recently 5{7]. This damping was found to depend on properties of the electrodes surfaces. In this article we present an analysis and measurements of the excess damping in the bi lar allfused silica torsion pendulum which results from the electric eld produced bythe multistrip electrostatic actuator. 8


Experimental set-up

The experimental arrangementis shown in Fig.1. The torsion pendulum is a 0.5 kg fused silica cylinder, 7 cm in length and 6.5 cm in diameter suspended by two fused silica bers 25 cm in length and 200 m in diameter. We used the cylinder with two fused silica cones that were hydroxide-catalysis bonded to the surface of the optical at polished along the length of the cylinder. The fused silica cylinder with the bonded cones was manufactured and provided to us by S.Rowan and J.Hough from the University of Glasgow 8]. The pendulum damping associated with hydroxide-catalysis bonding was found not to exceed 4 10;9 9]. The fused silica suspension bers were welded to the cones. Top ends of the suspension bers were welded to a fused silica disk that was attached through an indium gasket to the cover of the vacuum chamber rigidly fastened to a concrete wall. The chamber was evacuated to a pressure of about 10;7 Torr to minimizeany damping of the pendulum modes due to the presence of residual gas. The torsion mode of the pendulum was used to measure damping caused by the electric eld of the electrostatic actuator. This mode has the Q-factor of about 108 due to the pendulum damping dilution factor 10]. Unlike the swing pendulum mode, there is no need to have a too large gap between the face of the cylinder and the actuator plate when using this mode. The amplitude of the swing motion for our pendulum must exceed approximately 5 mm in order to exclude the seismic perturbation e ect on the Q measurement because the free decay change of the amplitude must be signi cantly greater than the change induced by seismic excitation of the pendulum. The electrostatic actuator plate was mounted parallel to the end face of the suspended cylinder with a separating gap of 2 ; 3 mm. It was the fused silica plate, 5 cm in length, 3 cm in width and 1 cm in thickness. Two sets of gold strips were sputter-deposited on the polished surface of this plate. Each strip had the width of 4 mm and was separated from the next one by the 3 mm gap. One set of the strips was grounded. The voltage was applied to the other set of the strips. The center of the actuator plate was displaced from the center 9


of the cylinder by 1.5 cm to excite the torsion motion of the pendulum. The amplitude of the torsion oscillation of the bi lar pendulum was monitored by the optical sensor that converted amplitude into the time interval measured by a counter. The laser beam to be re ected from the end face of the suspended cylinder was directed to the pair of 1 mm slits, with photodiodes placed behind them. Torsion oscillations of the pendulum resulted in a sequential pass of the light beam through the slits. The pulse signal was generated, whose duration determined the pendulum amplitude. The Q-factor was calculated from the measured decay time of free oscillations. The electrostatic actuator was also used to excite the torsion oscillation of the pendulum. The ac exciting voltage at the frequency of the torsion mode (approximately 1.14 Hz) was added to the dc bias and was applied to electrodes of the actuator. When the amplitude of the torsion motion was excited to the appropriate level of 0:05 rad the exciting voltage was switched o and the oscillation of the pendulum were allowed to decay freely. Electrodes of the actuator were used also to monitor electrostatic charges sitting on the end face of the fused silica cylinder. In this case the electrodes were connected to the high impedance operation ampli er AD 549. The motion of the charged cylinder induced the charge on the actuator electrodes which resulted in the change of the ampli er input voltage. This system did not allow us to nd the true magnitude and the charge distribution on the suspended fused silica cylinder end face. We could monitor however the charge total magnitude relativechange and estimate roughly this magnitude. It was possible to change the electrostatic charge on the pendulum by switching on the ion pump for some chosen time interval (we used only turbomolecular pump to evacuate the chamber). Electrostatic charging of the fused silica pendulum due to ultraviolet radiation produced by an ion pump was searched in 11]. The electrostatic charge monitoring and its reduction by the means of the ion pump allowed us to be sure that the electric charge on the end face of the fused silica cylinder did not in uence the measured damping.

10


Results of the measurements

In order to calibrate the actuator the ac voltage at the resonance frequency of the torsion mode was applied to the electrodes. The alternating torque produced by the actuator resulted in the change of the pendulum torsion amplitude whichwas measured. Basing on these results, we calculated the relation between the permanent torque Me and the dc voltage U applied to the actuator as: Me = aU 2 where a was found to be 4 10;12 N m=V 2. The torque Me depends on the value of the gap between the actuator plate and the end face of the cylinder. Consequently, the actuator introduces the (negative) torsion sti ness Ke dMe =d (where is the angle of the torsion motion) in addition to the own pendulum torsion sti ness K . The ratio Ke =K can be found from the equation Ke =K 2 !=! valid when j !j ! where ! is the natural frequency of the pendulum torsion mode and ! is the change in this frequency caused by the actuator. Fig. 2 shows the relativevariation of the pendulum natural frequency as a function of the square of the dc voltage applied to the actuator. The energy loss in the pendulum caused by the electric eld of the actuator can be described through considering Ke as a complex number Ke(1 ; i ). An imaginary part is the parameter suitable to quantify the level of dissipation in the actuator. Then, the damping Q;1 of the pendulum associated with the electric eld of the actuator can be e expressed as Q;1 jKej =K for jKe j K and 1. This expression allows to separate e out the dependence of Q;1 on the pendulum and the actuator parameters. e The damping Q;1 was determined in the experiments as a di erence between reciprocals e of the pendulum torsion mode Q-factors measured with and without the dc voltage applied to the actuator. The quality factor Q0 of the pendulum in the absence of the electric eld produced by the actuator was found to be (7:5 0:7) 107. The experiments have shown that the damping Q;1 of the pendulum associated with e the electric eld of the multistrip electrostatic actuator strongly depended on properties of the suspended fused silica cylinder end surface. Fig. 3 shows the measured damping Q;1 e 11


as a function of the dc voltage U applied to the actuator. Curve(2)was obtained for the mechanically polished end face of the fused silica cylinder cleaned in acetone and ethanol before evacuating the vacuum chamber with the pendulum in it. This curveis well tted by U 2 dependence. Baking the chamber with the pendulum in it at the temperature of 110 C during 7 hours did not change the damping caused by the actuator. Curve( )shows Q;1 measured after the thermal treatment of the fused silica cylinder end e face by the ame of an oxygen/natural gas torch. This treatmentwas done by applying the ame directly to the surface of the cylinder and moving the ame back and forth across the surface. The duration of the ame treatmentwas about 10 minutes. Immediately afterwards the vacuum chamber with the pendulum in it was evacuated. The repeat ame treatment of the cylinder end face resulted in additional reduction of the damping Q;1 presented by e curve( ). After that the chamber was opened and the end face of the cylinder was covered with a sheet of paper wetted bywater. After ve hours the chamber was evacuated again. Accomplishing this procedure we did not observedachange in the damping within the limits of experimental error. The ame treatment of the fused silica cylinder end face has changed some special feature of the pendulum damping caused by the electric eld of the actuator. Atlowvoltages the excess damping was not observed at the level within the resolution of the measurement, i.e. Q;1 3 10;9 . This range of voltages became larger and ranged up to 760 V after the e repeat procedure of the ame treatment. Atthis voltage applied to electrodes the actuator produced the torque of about 2:3 10;6 Nm acting on the fused silica cylinder. For the average moment arm of about 1:5 10;2 m this torque corresponded to the force in the order of 10;4 N. At the higher voltages the measured damping Q;1 weakly depended on e voltage. It was found that the damping Q;1 did not change depending on polarity of the applied e voltage. The other important feature of the pendulum damping caused by the electric eld of the electrostatic actuator is that it decreases if the ac electric voltage is applied instead of the dc voltage. In the case of the ac voltage with the frequency of 100 Hz and higher the 12


excess damping did not exceed 3 10;9 over all range of U used in our experiment.
Discussion

The damping of the pendulum caused by the electrostatic actuator is evidently associated with the electric eld that varies as the pendulum oscillates. In general case damping of a mechanical oscillator due to an electric eld may be caused by the number of di erent mechanisms, for example, the damping associated with eddy currents in the electrodes, the damping from the electric- eld-induced coupling between the pendulum and the electrode, as well as the damping due to Joule loss in the electric circuit of the voltage source. The pendulum losses in our set-up caused by these mechanisms were small enough to be neglected. The most importantlossmechanisms could be associated with the surface of the dielectric cylinder as well as with the surfaces of the metal electrodes. The damping associated with metal electrodes may be caused by electron transitions between local surface states in the oxide and adsorbed layers or polarization of these layers under the action of the electric eld. Gold coated electrodes were found to provide minimal losses 6,7,12]. Basing on these works, wemay conclude that the losses associated with the gold electrodes of the actuator in our experiments has given a small contribution to the observed damping caused by the actuator. The observed damping is evidently associated with electric losses on the surface of the suspended fused silica cylinder. Mechanical polishing of a fused silica sample results in the formation of the surface layer consisting of products of hydrolysis of SiO2 and having the porous structure that can adsorb water 13]. The ame treatment is likely to remove this layer. It is interesting to compare our results with those presented in 14]. The authors have found ame polishing had reduced surface mechanical losses in a fused silica sample and had allowed them to reachthe lowest measured at the room temperature value of the intrinsic mechanical loss in fused silica. The procedure of ame polishing was nearly identical to 13


the ame treatment described in this article. The warming-up was not so deep in our case. Nevertheless the ame treatment reduced surface electric losses in the fused silica cylinder. The results reported here show that the damping of fused silica test mass oscillation caused by the multistrip electrostatic actuator producing forces of the order of 10;4 Ncan be reduced to the level of less than 3 10;9 . This value is lower than the lowest damping measured up to now in the pendulum mode of a prototype fused silica test mass suspension for useininterferometricgravitational wave detectors.

14


FIGURES
Vacuum chamber Fused silica fibers Electrostatic actuator plate
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Fused silica cylinder Laser slits and photodiodes High voltage amplifier Drive oscillator DЯ voltage High impedance
amplifier

Counter Computer

Chart

recorder

FIG. 1. Experimental arrangement used to measure damping caused by the electrostatic actuator.
-/,
&
10-4

$

"

" &
U2, 105 V2



$

FIG. 2. Relativevariation of the pendulum natural frequency ; !=! as a function of the square of dc voltage U 2 applied to the actuator.

15


Qe-1, 10-8



&

"

" &
U, V



$

Figure 3. Variation of the pendulum damping Qe-1 as a function of dc voltage U applied to the actuator. (Ъ) ­ before the flame treatment; (Ю) ­ after the first flame treatment; (·) ­ after the repeat flame treatment.

16


REFERENCES
1] A. Abramovici, W. Althouse, J. Camp, D. Durance, J. A. Giame, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. J. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R.Vogt, R. Weiss, S. Whitcomb and M. Zucker, Phys. Lett. A 218 (1996) 157. 2] P. S. Linsay and D. H. Shoemaker, Rev. Sci. Instrum. 53 (1982)1014. 3] A. Cadez and A.Abramovici, J.Phys. E 21 (1988) 453. 4] S. Grasso, C. Altucci, F. Barone, V. Ragozzino, S. Solimeno, Pham-Tu, J. { Y. Vinet, R. Abbate, Phys.Let.A 244 (1998) 360. 5] V. P. Mitrofanov, N. A. Styazhkina, Phys.Lett.A 256 (1999) 351. 6] C. C. Speake, R. S. Davis, T. J. Quinn and S. J. Richman, Phys. Lett. A 263 (1999) 219. 7] E. Willemenot and P.Touboul, Rev.Sci.Instrum. 71 (2000) 302. 8] S. Rowan, S. M. Twyford, J. Hough, D.-H. Gwo and R. Route, Phys. Lett. A 246 (1998) 471. 9] K. Tokmakov, V. Mitrofanov, V. Braginsky,S.Rowan and J. Hough, in Gravitational Waves, proceedings of the Third Edoardo Amaldi Conference ed. S. Meshkov, Melville NY: American Institute of Physics, 2000, p.445. 10] V. B. Braginsky,V. P. Mitrofanov, K. V. Tokmakov, Phys.Lett.A 218 (1996) 164. 11] S. Rowan, S. Twyford, R. Hutchins and J. Hough, Class.Quantum.Grav. 14 (1997) 1537. 12] V. P. Mitrofanov and N. A. Styazhkina, Rev.Sci.Instrum. 71 (2000) 3905. 13] B. S. Lunin, S. N. Torbin, M. N. Danchevskaya and I. V. Batov, Moscow Univ. Chemistry Bulletin 35 (1994) 24, Allerton Press, Inc. 14] S. D. Penn, G. M. Harry, A. M. Gretarsson, S. E. Kittelberger, P. R. Saulson, J. 17


J. Schiller, J. R. Smith and S. O. Swords, Syracuse University Gravitational Physics Preprint 2000/8-11.

18


Appendix C. Thermo-refractive noise in gravitational waveantennae
Introduction

Wehaveshown in our previous article 4] that thermodynamical uctuations of temperature in mirrors (test masses) of LIGO-type gravitational waveantenna 1,2] are transformed due to the thermal expansion coe cient = (1=l)(dl=dT ) into additional (thermoelastic) noise which may be a serious "barrier" limiting sensitivity. This noise is caused in fact by random uctuations of the coordinate averaged over the mirror's surface. The spectral density of this random coordinate displacementmay be presented for in nite test mass in the following form 1: 22 2 TE Sx (!)= p8 T ( C(12 +3 !)2 : (2) ) r0 2 Here is the Boltzmann constant, T is temperature, is Poison ratio, is thermal conductivity, is density and C is speci c heat capacity, r0 is the radius of the spot of laser beam over which the averaging of the uctuations is performed. This noise is of nonlinear origin as the nonzero value of is due to the anharmonisity of the lattice. The goal of this article is to present the results of the analysis of another additional (and also of nonlinear origin) e ect whichmay be comparable with other known noise mechanisms limiting the sensitivity. Qualitatively this e ect is may be understood in the following way. The laser beam \extracts" the information not only about the change of the length of the antenna produced by gravitational wave but also about the uctuations of position of mirrors' surfaces averaged over the beam spot. These uctuations lead to phase noise in the re ected optical eld. However the phase noise may be produced by another e ect. High re ectivity of the mirrors is provided by multilayer coatings which consist of alternating
1

This result was re ned for the case of nite test masses by Yu. T. Liu and K. S. Thorne 6].

However the di erence from our calculation is only several percents for the planned sizes of test masses, and hence we use here much more compact expression (2).

19


sequences of quarter-wavelength dielectric layers having refraction indices n1 and n2. The most frequently used pairs of layers are TiO2 ; SiO2 and Ta2O5 ; SiO2. While re ecting * the optical wave "penetrates" in the coating on certain depth. This depth is of the order of the optical thickness of one pair of layers l < 1 . If the values of n1 and n2 depend on temperature T (thermo-refractivefactor = dn=dT is nonzero) then thermodynamical uctuations of temperature lead to uctuations of optical thickness of these layers and hence to the phase noise in the re ected wave. Though the thickness l of the working layer is small, the coe cient is usually signi cantly larger than (both have the same dimensions). For fused silica (SiO2) =5 10;7 K ;1 and =1:45 10;5 K ;1 (i.e. 30 times larger than ). This phase noise may be evidently easily recalculated in terms of equivalent uctuations of the surface and consequently compared with the spectral sensitivityof the antenna. Wehave analyzed also the photo-thermal refractive shot noise: due to random absorption of optical photons, the random uctuations of temperature in the surface layer of the mirror take place, producing uctuations of refractive indices of the coating and therefore phase uctuations of re ected lightwave (this e ect is similar to photo-thermal shot noise, analyzed in 4]). However, this e ect is numerically much smaller than thermo-refractive noise | that is whywe do not present here the detailed analysis of it.
Thermo-refractive noise

The theory of re ection of light from multilayer dielectrical coating is well known (see for example 5]). Using traditional approach we may recalculate the phase shift into equivalent displacement x of mirror (see Appendix C1):

x= 4
e

= ;u e 2 2 2 = n4(n12+ n12)2 : 1 ; n2

(3) (4)

Here u is the uctuation of averaged temperature, 1 = dn1=dT , 2 = dn2=dT . It is important to note, that e ective coating thickness is much smaller than the characteristic 20


length of di usive heat transfer: l a= ! (a is temperature conductivity, ! is the frequency of observation which is of order 100Hz for laser gravitational waveantenna). Therefore we may consider in our calculations that uctuations of temperature are correlated in the layers. To calculate thermodynamical uctuations of temperature u(~ t) in the surface layers r we use Langevin approach and introduce uctuational thermal sources F (~ t) added to the r right part of the equation of thermal conductivity:

p

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

a2 = C :

(5)

This approach was described and veri ed in 4] (see 4] we replace the mirror by half-space: ;1 < x < 1 the boundary condition of thermo-isolation on surface z spectrum of temperature uctuations: Z 1 d~ ! kd u(! ~ u(~ t)= r k ;1 (2 )4 ~ u(! ~ )= 2F~(k2 !) k a (k) + i! 2 hF (~ !)F (~ 1 !1)i = 2( T )2 (2 )4j~ j2 k k k C

all the details over there). As in ;1 < y < 1 0 z < 1 with = 0. We may now calculate the )e
i!t+i~ r k~

(6) (7)

(! ; !1)

(8)

(kx ; kx1) (ky ; ky1) (kz ; kz1 )+ (kz + kz1 )]:

2 The thermodynamical uctuations of temperature u averaged over the volume V = r0l may be presented in the following form: 1 Z 1 dxdy Z 1 dz u(~ t) e;(x2+y2 )=r0 e;z=l = 2 u = r2 l r 0 0 ;1 Z 1 d~ ! F (~ !)ei!t 2 2 2 kd k ;(kx +ky )r0 =4 1 = (9) 4 a2j~ j2 + i! e 1 ; ikz l ;1 (2 ) k

From this expression and from (8) we nd immediately the spectral density Su(!) of uctuations of the averaged temperature: 21


2 T 2 Z 1 2 k? dk? Z 1 dkz Su(!)= 2 ( C )2 (2 )2 ;1 2 0 2 kz2 + k? 22 ;k? r0 =2 (1 + 1) = 4 (k 2 + k 2 )2 + ! 2 e a pz ? 1+ kz2l2 22 ' r2p! T C 0

(10)

2 2 2 Here k? = kx + ky . The rst term 2 appears because as in 4] we use \one-sided" spectral density, de ned only for positive frequencies, which is connected with the correlation function 1 hu(t)u(t + )i by the formula Su (!) = 2 R;1 d hu(t)u(t + )i cos(! ): The term (1 + 1) appears due to two -functions in square brackets in (8). For the frequency of observation ! ' 2 100 s;1 characteristic length a=p! ' 50 (we used for the estimates constants p p!=a we for fused silica), so that l a= ! r0. Taking into account that k? ' 1=r0 may consider the rst denominator as constant while integrating over k? . In the same way kz ' 1=l p!=a and while integrating over kz wemay consider the second denominator as unity. Itisinteresting that in this model the spectral density Su (!) does not depend on l. This spectral densitymay be recalculated to the spectral density of equivalent uctuations of surface displacement to compare it with other known sources of noise:

TD Sx

2 2 ef f 2 T (!)= r2p! C 0

p

2

(11)

22


Noises in gravitational wave antennae

10

-22

1 2 3 4 5
-23

-

-1/2

)

thermorefractive (TiO2) thermorefractive (Ta2O5) SQL (m=30 kg) thermoellastic (Al2O3) brownian (SiO2)
5

10

h, (Hz

4 3 2 10
-24

1

10

100

1000

f, (Hz)

FIG. 3. Comparison of SQL-limited sensitivity with di erent sources of noise in gravitational waveantennae: thermo-refractive, Brownian (dominating in fused silica mirrors) and thermo-elastic (dominating in sapphire mirrors).
Numerical estimates

For the numerical estimates we assumed that the multilayer coating consists from alternating pairs of layers: TiO2 (n1 =2:2) and SiO2 (n2 =1:45), or Ta2O5 (n1 =2:2) and SiO2 * (n2 =1:45). The values of for TiO2 and for Ta2O5 were found in 9]. Wewantnow to compare the thermo-refractive uctuations with thermoelastic noise (2) and noise associated with the mirrors' material losses described in the model of structural 23


damping 10] (we denote it as Brownian motion of the surface). In this model the angle of losses does not depend on frequency and for its spectral density the following formula is valid for in nite test mass 8,4,6]: (1 2 B (12) Sx (!) ' 4 !T p ; ) 2 E ro where E is Young modulus, and is Poison ratio. The spectrsal sensitivity of gravitational wave antenna to the perturbation of metric h(!)may be recalculated from noise spectral density of displacement x using the following formula: q 2(Sx r01 (!)+ Sx r02 (!)) h(!)= (13) L where we used the fact that antenna has two arms (with length L) with two mirrors the uctuations on which are averaged over the radii r01 and r02. The LIGO-II antenna will approach the level of SQL, so we also compare the noise limited sensitivity to this limit in spectral form 9]: s 8 (14) hSQL (!)= m!hL2 : 2 For the calculations we used the set of parameters given in Appendix C2 (the same as in 4]) plus 9] *

r01 =3:6= 2cm r02 =4:6= 2cm n1 =2:2 n1 =2:2 n2 =1:45
2 2

p

p

=4 10;5 K =6 10;5 K
1

;1 ;1

(TiO2) (Ta2O5)
;1

=1:5 10;5 K

(SiO2)

We used gures from 9] for ion plating method only, for other methods of deposition the value of may be two times larger. In gure 1 we plot all previously known noises 4] together with the new one. We see that thermorefractive noise limitation is close to SQL * for the frequences near 200 Hz. 24


Conclusion

Summing up, wemaysay that thermo-refractive e ect is not small and it must be seriously considered in interferometricgravitational antennae (pro jects LIGO-II and especially LIGO-III, where overcomming of the SQL is planned). It is also important that this effect depends slower on the radii of the beam-spots than thermo-elastic noise and thus may become dominating for larger r0 planned in LIGO-II and LIGO-III.
Appendix C1. Coe cient of re ection

In this appendix we give the calculation of coe cient of re ection of light wave from multilayer coatings consisting of in nite sequences of pairs of quarter-wavelength dielectrical layers n1 and n2. Let the refraction index of odd layers uctuates on n1 and the refraction index of even layers on n2. One may reformulate this problem into the problem for distributed long line 5]. The equivalent impedance Z of this system of layers may be deduced using the following statement: the addition of twolayers does not change the value of Z . Voltage V2 and current I2 at the end of second layer may be found from input voltage V0 and current I0 using transformation matrix M ( 5], formula (3.9.27)): 1 0 01 01 B V2 C C B BC B C = M B V0 C M = B M11 M12 C A @ @A @A M21 M22 I0 I2 n M11 =cos 1 cos 2 ; n1 sin 1 sin 2 2 ! sin 1 cos 2 + sin 2 cos 1 M12 = ;i n1 n2 M21 = ;i (n2 sin 2 cos 1 + n1 sin 1 cos 2) n M22 =cos 1 cos 2 ; n2 sin 1 sin 2 1 0 1 '2 + '1 n1 ; n2 i n1 n2 C B C M 'B @ A n2 i (n2'1 + n1'2) ; n1 25


Here we take into account that for quarter-wavelength layers 1 = =2 + '1 =2+ '2 and therefore one may use approximation sin 1 ' 1 sin 2 ' 1 cos ;'1 cos 2 ';'2. Now we put that I0 = YV0 and I2 = YV2 (Y = 1=Z is generalized conductivity o sequence of layers) and obtain two equations: n V2 = V0 ; n1 + iY '2 + '1 n1 n2 2 n YV2 = V0 i (n2'1 + n1'2) ; Y n2 :
1

2 1

=

'

f the (15) (16) (17) (18)

Solving these equations we nd conductivity Y and re ection coe cient K : n Y ';i n2 1n2 2 (n2'1 + n1'2) 1 ; n2 n1n K = Y ; 1 ';1 ; 2i n2 ; 2 2 (n2'1 + n1'2) Y +1 n
1 2

From this point it is easy to obtain (3,4), assuming '2 = 2 nn '1 = 2 nn1 1 2
Appendix C2. Parameters

2

! =2

100 s

;1

T =300 K
=1:06
;1

m ==3 104 g
Fused silica: =5:5 10 g =2:2 cm3 E =7:2 10 =5:0 10 g =4:0 cm3 E =4 1012
;7

L =4 105 cm

11

Sapphire:

erg =1:4 105 cm s K erg C =6:7 106 gK erg =0:17 =5 10;8 3 cm K K
;1

;6

erg =4:0 106 cm s K erg C =7:9 106 gK erg =0:29 =3 10;9 : 3 cm 26


REFERENCES
1] V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, Physics Letters A 264, 1 (1999) cond-mat/9912139 2] A. Abramovici et al., Science 256(1992)325. 3] A. Abramovici et al.,Phys.Letters.A 218, 157 (1996). 4] Yu. T. Liu and K. S. Thorne, submited to Phys. Rev. D. 5] S. Solimeno, B. Crosignani and P. Diporto, Guiding, Di raction and Con nement of Optical Radiation, Academic Press, 1986. 6] P. R. Saulson, Rhys. Rev. D, 42, 2437 (1990) G. I. Gonzalez and P. R. Saulson, J. Acoust. Soc. Am., 96, 207 (1994). 7] F. Bondu, P. Hello, Jean-Yves Vinet, Physics Letters A 246, 227 (1998). 8] V. B. Braginsky, M. L. Gorodetsky, F. Ya. Khalili and K. S. Thorne, Report at Third Amaldi Conference, Caltech,July,1999. 9] Thin lms for optical systems (for the International Journal of Optoelectronics), Ed: F Flory, Marcel Dekker Inc, ISBN 0 8247 96333 0, 1995.

27


Appendix D. The problem of compensation of internal mechanical noise in test mass of gravitational waveantennae
Introduction

Fluctuations of re ecting surface of test mass relatively its center of mass (the internal mechanical noise in test mass) is one of key problems in the full scale terrestrial interferometric gravitational waveantennae (pro jects LIGO-II, VIRGO, GEO-600, TAMA), which have to be solved in order to achieve in year 2005 the planned sensitivity in units of amplitude of the perturbation of metric at the level h ' 2 10;23 which corresponds to the Standard Quantum Limit (SQL). At this level the quantum features of macroscopic body behavior became essential. Fused silica used as a material for rst stage of antenna (LIGO-I, VIRGO) 1,2] has a disadvantage | relatively large Brownian internal noise. We will call as Brownian the uctuations which are usually calculated from phenomenological model of structural losses in the material in order to distinguish them from thermoelastic noise. There was some optimism after the proposals to use sintetic sapphire instead of fused silica because of Brownian noise in sapphire is much less. Unfortunately the thermoelastic noise happens to be too large in sapphire | it was demonstrated in 4] for model of in nite test mass. The physical origin of thermoelastic noise is fundamental thermodynamical uctuations of temperature which causes variation of test mass shape due to thermal expansion. This result was re ned byYu. T. Liu and K. S. Thorne 6] for nite test mass (the di erence between spectral densities of internal noise for in nite and nite size test mass does not increase several percents for planned sizes of test mass). In this paper we discuss "the last line of defense" against internal noise | the possibility of compensation. It is not a new idea: in 5] there was demonstrated the possibility of compensation of suspension noise of test mass (the uctuations of test mass position caused by thermal noise in suspension ber) using additional measurement of horizontal displacement of ber averaged by proper way over its length and subsequent subtraction from antenna 28


interferometer readout. In order to realize compensation of internal mechanical noise wemust have the possibility to measure independently the coordinate of test mass surface Zspot averaged over laser beam crossection with radius r0: ZR Z2 (19) Zspot = 1 2 0 rdr 0 d e;r2 =r02 vz (r )jz=0 r0 where vz (r ) { is the uctuational displacement of test mass surface in normal direction to surface at point with coordinates r and . Below we use cylindrical coordinate system which axis coincides with axis of cylindrical test mass having radius R and height H , face surface has coordinate z = 0 and back surface | z = H . Let us imagine that wehave ideal position meter which error of measurement is negligible small. Then the key problem for compensation is to de ne the support body,i.e. body relatively to which position we measure the coordinate Zspot. The best variantwould be to measure relatively center of mass of our test mass. However center of mass is inaccessible for measurement. If we measure the spot coordinate relatively some support body we measure the di erence Z = Zspot ; Zbody between coordinate of spot and coordinate Zbody of support body. Therefore additional noise caused by uctuations of support body position is inevitably introduced. Indeed instead of measurement the value Zspot one measures another value Z and the accuracy of compensation will be de ned by di erence between these values, i.e. by Zbody. So one can proclaim as Archimed that for absolute compensation the "point of support" is necessary even if the ideal meter is available. In this paper we discuss one of the possibilities, which could be realized easier by experiment. It is measurement of the di erence Z = Zspot ; Zback between spot coordinate Zspot and coordinate 1 ZR Z2 Zback = R2 rdr d vz (r )jz=H
0 0

averaged over back surface of test mass. It is known that the larger the square of averaging the smaller the uctuations of averaged coordinate. That is why we consider coordinate Zback averaged all over the backwall of the test mass. 29


In section II we discuss how much can be the value of compensation if ideal position meter is used for measurement Z . We show that using compensation one can overcome SQL several times. However it is worth underlining that result numbers given in table I express the point of view of extreme optimism, who assumes that experimentalist can create ideal meter with any required properties. Unfortunately the reality is considerably more tough. Being optimists in section II we try to become realists in section II where we consider the variant of meter for measurement Z using Fabri-Perot interferometer inside the body of test mass ( g. 4) and show that in practice parasitic thermorefractive e ect does not allowto control di erence Z . This e ect consists in thermodynamical uctuations of temperature which causes uctuations of refractiveindex n due to its dependence on temperature T . Due to relatively large value dn=dT for fused silica and sapphire (the more suitable materials for test mass) the readout of meter will give little information about uctuations of di erence Z but mainly about uctuations of temperature, averaged over volume of meter beam. This scheme of meter for compensation is obvious and we hope that our negative result will be useful for further discussion and investigation on this problem. For calculation of thermoelastic and Brownian noise of coordinate Zspot we use the results of 4] obtained in approximation of in nite test mass.
The Value of Compensation

Using Fluctuation-Dissipation Theorem (FDT) one can calculate spectral density Sspot(!) of displacement Zspot 10,7,8,4,6]. In this approach one should apply imaginary periodic pressure p distributed over the beam spot on the surface:

p(t)= Fr02 e; 0

2 r2 =r0 ei!t

(20)

and to calculate the power Wloss of losses averaged over period 2 =!. Then the spectral density can be determined byformula 30


Sspot(!)= 8kB TWdiss : F02!2

(21)

One can also calculate separately the spectral density Sback(!). It is naturally to de ne coe cient K of relative compensation as the following v u S (!) u Kcomp = t Sspot (!) : (22) back It is also useful to compare the spectral density Sback(!) with the spectral density corresponding to SQL (which is planned to approach in LIGO II) and calculate the coe cient (see 4,9]): v u S SQL(!) s u 4 KSQL = t S (!) = m!2S h (!) (23) back back where h is Plank constantand m is the mass of test body. Belowwe consider thermoelastic and Brownian noises.
Thermoelastic noise

Using results of 4] one can write down the expression for spectral density of thermoelastic noise of spot for in nite test mass:
TE Sspot(!)= p8 2

T

22

(1 + )2 : 3 ( C )2 r0 !2

(24)

Here is Botzmann constant, T is temperature, is coe cient of thermal expansion, is Poison ratio, is thermal conductivity, is density and C is speci c heat capacity. This formula is valid in adiabatic approximation
2 ( C )r0!

1

(25)

which takes place for parameters of possible materials for test mass in LIGO. Using FDT approach one can calculate the similar expression for spectral density of thermoelastic noise for backwall (see Appendix D1): 31


S

TE back(

22 !)= ( 4C )2 T 2H! R

2

(26)

and calculate the coe cients for v u TE = u Kcomp t v u TE = u KSQL t s =

thermoelastic noise: v TE (! ) u p2 (1 Sspot = u t TE Sback(!) s S SQL(!) = h( C TE 2T Sback(!) h C2 : 2 T2

+ )2R2H 3 r0 )2 R2H = 2m

(27)

(28)

In last equalitywe used that m =

R2 H .
Brownian noise

The spectral density of Brownian noise for in nite test mass was obtained in 8,4,6] using FDT:
B Sspot

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

(29)

where E is Young modulus, is loss angle. Using FDT one can calculate the similar expression for spectral density of Brownian noise for backwall (see Appendix D2) and coe cients Kcomp and KSQL :

S

B back

(!)= 4 !T 3 HR2 E v u S B (! ) u B Kcomp = t Sspot (!) = B back v u SQL u B = t S! KSQL B Sback(!) = s 3E = h H 2! T

vp u 3 (1 ; 2)R u tp 2 r0H s h 3 E R2 = T mH !
:

(30)
2

(31)

(32)

32


Numerical Estimates

We use the same numerical values of parameters as in 4]:

! =2

100 s

;1

T =300 K:
;1

(33) (34)

Fused silica (SiO2): =5:5 10;7 K =2:2g=cm3 =1:4 105 erg=(cm s K)

C =6:7 106 erg=(g K) m =1:1 104 g R =12:5cm H =10:2cm
=5 10
;8

E =7:2 1011 erg=cm3 =0:17 ! 1 dn =1 10;5 K;1: = n dT Sapphire (Al2O3):
=5:0 10;6 K
;1

=4:0g=cm3

(35)

=4:0 106 erg=(cm s K)

C =7:9 106 erg=(g K) m =3 104 g R =14:0cm H =12:2cm
=3 10
;9

E =4 1012 erg=cm3 =0:29 ! 1 dn =0:7 10;5 K;1: = n dT
For these parameters one can obtain the estimates given in table I. From table I one can see that SQL can be overcome by 2.7 times for fused silica and by 3.9 times for sapphire. It is also natural that thermoelastic noise dominating in sapphire can be easier compensated due to its more strong dependence on spot radius r0 (compare (24) and (29)). 33


Interferometric Compensation Meter

One of possible meters for independent control of averaged coordinate of spot Zspot on test mass surface is shown on g. (4). Several Fabri-Perot interferometers are necessary in order to gather information on possible larger surface of back wall. It is di cult to use beam with radius r00 more than several millimeters (in opposite case the interferometer mode became unstable). Therefore several dozens of inner interferometric beams must be used. Let us assume that experimentalist have possibility to create such device.

FIG. 4. Interferometric meter for independent control of averaged coordinate of laser spot (on left side of test mass), consisting of several Fabri-Perot interferometers placed inside test mass. Only two inner interferometer beams are shown.

Having analyzed this meter we found that parasitic thermorefractive e ect plays an important role. This e ect consists in thermodynamical uctuations of temperature T which causes uctuations of refractive index n of test mass body due to its dependence on temperature (nonzero coe cient dn=dT ) and consequently leads to phase uctuations of wave propagating in interferometer. Indeed the phase ' of output beam, for example, of central 34


interferometer (on g. 4 its beam is horizontal) will take out information about sum:

' n(Z

F:spot

;Z

B:spot

dn )+ H dT T

(36)

where ZF:spot and ZB:spot are averaged coordinates on face and back surfaces of test mass correspondingly (averaging over crossection is made with weight gaussian function: 2 1 ;r2 =r00 ), H is the width of test mass, T is thermodynamical uctuations of temper2 r00 e ature averaged over beam volume. Using natural condition r00 > H we also assume that uctuations of ZF:spot and ZB:spot are not correlated. Then the value of parasitic in uence of thermorefractive e ect can be estimated by coe cient: s ! 1 dn : 2 Sspot = n dT (37) ATR = (H )2 S
TD

Here Sspot is the spectral density of beam spot, expressed byformulas (24, 29) for thermoelastic and Brownian noise correspondingly,and STD is the spectral density of thermodynamical uctuations of temperature averaged over beam volume. The last spectral density can be calculated using the model of in nite layer with width H in order not to account the boundary condition on lateral area. For calculation the thermodynamical uctuations of temperature u(~ t) one can use r Lahgevin approach and introduce uctuational thermal sources F (~ t) added to the right r part of the equation of thermal conductivity 4]:

@u ; r @t C u = F (~ t) h F (~ t)F (~0 t0) i = ;F02 (~ ; ~0) (t ; t0) r r rr 2 F02 =2 ( TC )2

(38) (39)

where is Laplace operator, (~)and (t) are spatial and time delta-functions correspondr ingly. As a result one can obtain formula for spectral density STD of thermodynamical uctuations of temperature averaged over beam volume (see Appendix D3):

T2 STD(!)=2 ( C )2H r41 !2 : 00
35

(40)


The expressions for coe cients A tained using (24,29,37):
TE TR

TR

for thermoelastic and Brownian noises can be ob-

A =(1 + vp u u B = t4 ATR

rp r 00 ) 42 H 3 2 (1 ; 2)( C )2 !r00 : T EH 2

(41)

In table II the numerical estimates for these coe cients are given. All coe cients ATR are much less than unity. This means that thermorefractive e ect strongly masks the useful information about uctuations of averaged coordinate of laser spot on test mass surface.

36


TABLE I. Coe cients K

comp

TABLES (22) and KSQL (23) for thermoelastic noise (superscript TE ) and

for Brownian noise (superscript B ) for fused silica and sapphire using parameters (33,34,35). Fused silica Sapphire 3.0 14.2 3.6 430.2 2.7
TR

r0 (cm) TE Kcomp B Kcomp TE KSQL B KSQL

1.5 40.2 5.1

1.5 54.3 4.5 14.1 3.9
TE

3.0 19.2 3.2

TABLE II. The coe cients A

for thermoelastic noise (superscript

) and for Brownian

noise (superscript B ) for fused silica and sapphire using parameters (33,34,35). Fused silica

r00 ATE TR AB TR r00 ATE TR AB TR

0:15 cm 0:024 0:051 Sapphire 0:15 cm 0:32 0:01

0:30 cm 0:034 0:14 0:30 cm 0:46 0:027

37


Conclusion

As we said in Introduction the results of section II collected table I re ect the extreme optimistic points of view. Unfortunately nobody knows how to realize such compensation measurement. In section II we demonstrate that parasitic thermorefractive e ect happens to be considerable value and it prevents from considered design of "inner" compensation meter. One of the alternatives way is to construct device controlling the distance between laser spot on test mass and averaged coordinate of additional support body. In this case one can create many beams inteferometer, but thermorefractive e ect will not make di culties because each beam can propagate in free space. However the problem of high quality suspension of additional body with mass not less than test mass is a separate and rather complicated experimental one.
Appendix D1. Thermoelastic Noise

In this appendix we calculate the formula (26). In according to FDT one must apply force F0 alternating with frequency ! which is homogeneously distributed over cylinder base (i.e. it makes the constant pressure F0= R2) and calculate the value of losses averaged over the period. Therefore we must solve the system of equations of elasticity and thermoconductivity: (1 ; ) grad div ~ ; 1 ; 2 v 2(1 + (1 + ) ~ ~z F0(12; 2 ) + r T e R HE E zz jz=H = 1;2 F0 zz jz=0 = ; 2 R
zr jz=0 H

v ) rot rot ~ = T jz=H E T jz + 1;2 zr jr=R =0 Tj
r=R

(42) (43)
=0

=0

rr jr=R

E = 1;2
38

(44)


@ T ;a @t @T @z
Zero Approximation

2

z=0

v T = ; C (1E T2 ) @ div ~ ; @t a2 = C @T =0 @r r=R =0: H

(45)

We solve this system by method of perturbation. In zero approximation we nd deformation ~0 formally putting = 0. The solution for vector of displacement ~0 v v problem is known (wego into noninertial system and face the problem for cylinder of arti cial gravity | see e.g. sec. 1.7 in 10]): ! r2 ; z + z 2 F0 v0z = r2 E 2H 2H F z v0r = r20E Hr 1 ; H 2 z div ~ = ; F0(1R;E ) 1 ; H : v 2

eld of of this in eld (46) (47) (48)

The axis of cylindrical system of coordinate coincides with axis of cylinder and bases of cylinder have coordinates z =0 H . it First Approximation Substituting ~0 in the right part of equation (45) we nd the temperature eld T1, v appearing due to elastic deformations (it is the rst approximation proportional to ). We are to solve the following problem in twosteps:

i! T1 ; a @ T1 @z
~ i! T1 ; a ~ @ T1 @z

2

z=0

2

z=0

ET T1 = ; C (1 ; 2 ) i! div ~0 v V T =0 @@r 1 =0 H r =R ET ~ T1 = ; C (1 ; 2 ) div ~0 + T1 v V ~ T1 =0 ET @ div ~0 v = CV (1 ; 2 ) @z z=0 H H
39


~ @ T1 @r

r=R

ET v = C (1 ; 2 ) @ div ~0 @r V

r =R

=0:

~ The solution for T1 can be found in the following form: ~ T1 = F02 R D0 = s H

T D es0 z + G e;s0 z 0 0 CV p e;s0 H i! s0 = a ;s0 H ) 0 (1 + e 1 G0 = ; s H (1 + e;s0 H ) : 0

Let us calculate auxilary dimentionless integral, which will be used below. 1 ZH z I0 = H 0 dz D0es0z + G0 e;smz ;1+ H = 2 3 1 41 ; 2 1 ; e;s0 H 5 ' 1 : = (s0H )2 s0H (1 + e;s0 H ) (s0H )2 In last equalitywe use condition js0H j
Second Approximation

(49)

1.

~ Now one can substitute r T1 in the right part of equation (42). Then in principle one can nd the second approximation ~2 to the eld of deformations (it is proportional to 2) v and then after extraction of imaginary part of ~2 it is possible to nd energy of losses Eloss v per period. However we have the possibilityto make calculation procedure much easier and not to sovle elastic problem for ~2. Note that rightpart in(42)is iquivalentto volume force v E ~ ~ f2 = ; (1 ; 2 ) r T1: E n p ~2 = (1 ; 2 ) T1 ~ acting normally to surface of cylinder (~ is external unit normal, see (42 - 44)). The imagin ~ nary parts of f2 and p2 are the small forces of friction and one can nd power Wdiss of losses ~ by calculation the power of these forces:
40 Plus wehave pressure


Z Wdiss = ; ! = 2 V ! = ; 2(1 ;E 2 2(1 + = ; 2F0 C

dV (~0 v Z )= V )2! 2
V

Z ~ f2)+ dS (~0~2) = vp
dV div ~0 T1 = v~ T H = (I ) : R2 0
S

(50)

Substituting (50) into (21) we obtain the formula (26).
Appendix D2. Brownian noise

In this Appendix we calculate the formula (30). Using formulas for displacement vector ~0 obtained in Appendix II one can calculate v elastic energy E as work, performed byvolume and surface forces: = Ef + EF : Z H=2 Z R F =1 dz 2 r dr R20H v0z = 2 ;H=2 0 ! 2 R2 0H = FR2E 1 ; 8H 2 6 ZR EF = ; 1 0 2 rdr F02 v0z jz=H=2 = 2 R F02H R2 = R2E 8H 2 2 0 E = 6FRHE : 2 Then assuming that W
diss

E Ef

(51)

= E ! and substituting it for (21) one can obtain (30).

Appendix D3. TD uctuations of temperature

In this Appendix we calculate the formula (40). The TD uctuations of temperature u(~ t) one can nd in spectral form: r ZZ Z 1 dkx dky d! u(~ t)= r ;1 (2 )3 X un(kx ky !)ei!t;ikx x;iky y cos bn z
n

41


k ) un (kx ky !)= i!Fn(a2x(bky+!k2 2 +n? 2 2 2 k? = kx + ky bn =

)

n: H

Using (39) one can nd correlators of spectral expantion of the uctuational thermal sourses:
2 h Fn (kx ky !)Fn1 (kx1 ky1 !1) i = 2( T )2 (2 )3 C 2 2 k? + b2 l nn1 (kx ; kx1 ) (ky ; ky1 ) (! ; !1) n

(52)

Now one can calculate the spectral density Su (!) of 2 averaged over volume V = r00H along axis z: 1 Z H ZZ 1 u(t)= r2 H 0 dz 0 dx dy u(~ t r 00 2 2 T 2 Z Z 1 dkxdky e; r002k? Su(!)= ( C )2H ;1 (2 )2 Z 1 k? dk? r00 k? 2 2 T2 ;2 = ( C )2H 2e !2 0 T2 ' (2 C )2H r41 !2 :
00

uctuations of temperature u(t), )e
; x2r+y2 2 00

2 k? 4 !2 + a4k? = 2 k? 4 + a4k? '

(53)
2

The last expression, obtained in adiabatic approximation (!r00)=a with (40).

1, obviously coincides

42


REFERENCES
1] A. Abramovici et al., Science 256 (1992) 325. 2] A. Abramovici et al., Physics Letters A218, 157 (1996). 3] V. B. Braginsky, M. L. Gorodetsky and S. P.Vyatchanin, Physics Letters A, A264,1 (1999). 4] Yu. T. Liu and K. S. Thorne, submited to Phys. Rev. D (available in archive xxx.lanl.gov: gr-qc/0002055) 5] V. B. Braginsky,Yu. Levin and S. P.Vyatchanin, Meas. Sci. Technol., 10, 598 (1999). 6] P. R. Saulson, Rhys. Rev. D, 42, 2437 (1990) G. I. Gonzalez and P. R. Saulson, J. Acoust. Soc. Am., 96, 207 (1994). 7] Yu. Levin, Phys. Rev. D, 57, 659-663 (1998). 8] F. Bondu, P. Hello, Jean-Yves Vinet, Physics Letters A 246, 227 (1998). 9] V. B. Braginsky, M. L, Gorodetsky, F. Ya. Khalili and K. S. Thorne, Report at Third Amaldi Conference, Caltech,July,1999. 10] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, third edition (Pergamon, Oxford, 1986).

43


Appendix E. On the possibility to measure thermo-refractive noise in microspheres
Introduction

Optical microresonators with whispering gallery modes when made of pure fused silica have a unique combination of high quality factor (of the order of 1010) and small size (hundreds of microns) 1{3]. These features in combination with ease of production naturally lead to the usage of these resonators as external cavities for diode laser stabilization, discriminators and passive lters. However very small size of resonators have the drawback. It is shown below that thermodynamical uctuations of temperature in the mode-volume of the resonators may not be neglected. These uctuations lead due to thermal dependence of refractive index to its uctuations and hence to the trembling of the resonance frequency and depending on the tuning to phase or amplitude noise of the output light-wave. This e ect is di erent and uncorrelated with the known e ect of thermodynamical uctuation of density leading to Rayleigh scattering. The e ect of thermodynamical uctuations of temperature in the bulk of the mirrors on the sensitivityof Fabry-Perot etalon was analyzed for the rst time in 4]. The thermodynamical uctuations of temperature in this case are acting in twoways 1) due to coe cient of thermal expansion they produce uctuations of the surfaces of the mirrors (thermoelastic noise) 2) due to coe cient of thermal refraction they produce uctuations of phase in the wave re ected from multilayer coating whichmay also be recalculated to the equivalent uctuations of the surface (thermo-refractive noise). It was shown that these e ects will restrict the sensitivity of laser gravitational wave antenna 6] that are being constructed now and should be seriously taken into account on the next stages of this pro ject. The idea of the e ect follows from the well known thermodynamical equation for the variance of temperature uctuation u in the volume V :

T hu2i = C V
where T is temperature of the heat-bath, 44

2

(54)

is the Boltzmann constant, is density, and


C speci c heat capacity. By substituting in this equation parameters for fused silica = 2:2 g=cm3, C = 6:7 106 erg=(g K) and e ectivevolume of eld localization of the most q localized mode in the microsphere with V ' 10;9 cm3 we obtain the value of hu2i' 30 K whichincombination with the coe cient of thermal refraction dn=dT =1:45 10;5 K ;1 lead to rather pronounced e ect of relative resonance frequency uctuations !=! 3 10;9 . This value is even larger than the linewidthes of resonances achievable in microspheres. I shall not analyze thermoelastic noise here, as the coe cient of thermal expansion = 5:5 10;7 K;1 is su ciently smaller in fused silica than dn=dT . It is evident, however, that to nd spectral properties of this noise, more rigorous analysis is required, taking into account peculiar eld distribution of whispering gallery modes.
Thermodynamical uctuations of temperature

In 4,5] the new approach for the analysis of thermodynamical uctuations of temperature was developed - the method of uctuational uctuational thermal sources F (~ t) substituted r to the right part of the equation of thermal conductivity

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

(55)

where a2 = =( C ) and is thermal conductivity. This approach is analogous to the Langevin approach with uctuational forces in equations of dynamics. It was shown that if the proper normalization of the sources is used:
22 hF (~ t)F (~0 t0)i = 2 TC a r2 (~ ; ~0) (t ; t0) r r rr

(56)

this approach leads to correct results which satisfy Fluctuation-Dissipation theorem (FDT). In particular it was shown that thermoelastic noise is associated through FDT with thermoelastic damping.

45


Simple estimate of the spectrum of uctuations in microspheres

Variations of refractive index n in dielectric cavity lead to the change of resonance frequencies. To nd this change one may use variational approach. If the perturbed waveequation has the form: 0 ~ ~ E + +2n n !2E =0 (57) c2 dn ~ Where E is the electric eld in the cavity, 0 = n2 is dielectric susceptibility, n = dT u is ~ the variation of refractive index due to uctuations of temperature u. If E0 is the orthoR~~ r normalized eld distribution of an eigenmode of the unperturbed cavity ( EiEj d~ = ij ) and ! = !0 + ! is the frequency shift, then after multiplication of this vector equation on ~ complex conjugate vector E0 and integration over the whole volume, neglecting the terms of the second order we obtain: ! = ; 1 Z jE 2j nd~ = ; 1 dn u ~ (58) !0 nV 0 r n dT where u is the temperature deviation averaged over the mode volume. For simplicity of calculations below I restrict myself with only fundamental whispering-gallery mode TE``1 in microspheres of radius R, which has the smallest volume of localization. The eld distribution of this mode may be approximated as follows: )' 8 > 4( `)1=4 e;` cos2 =2+i` < j`(knr)=j` (knR)for r R p > n n2 ; 1a3=2 : e; (r;R) for r> R n knR ' ` +1=2+ 1:8558(` +1=2)1=3 ; p 2 (59) n ;1 However, even this approximation is too complex for further evaluation, so below I shall use the following Gaussian approximation of radial dependence: 2 (r;R )2 `1=4 ^ (r ) ' p2bR e; 2b20 ; ` cos2 =2 +i` e 0 knR0 ' ` +1=2+ 0:71(` +1=2)1=3

~ E(r

~ ) ' E (r

knb ' 0:81(` +1=2)1=3:
46

(60)


This approximation describes rather adequately the distribution of optical energy inside the resonator. Moreover as it will be found below because of the small depth of the eld (parameter b, radial distribution practically does not in uence on the frequency uctuations. It is now necessary to calculate thermodynamical uctuations of temperature in the sphere. In 5] the spectral density of these uctuations near the boundary of half-space was found: Z1 F ( ~ ) ~ r u(~ t)= a2 2 + i ei t+i~~ d d)4 r (2 ;1 22 0 0 h F ( ~ )F ( ~ 0 0) i =(2 )4 2a CT 2 ( ? ; ?)+ ( ? + ?)] ( ~k ; ~k0 ) ( ; 0) (61) where ? is the componentof wave-vector of uctuations normal to the surface and ~k are components, parallel to it, a2 = =( C ), and is thermal conductivity. We may use this result for the evaluations by putting ? = r as we are interested in the uctuations near the surface of the microsphere and the relaxation time of the whole microsphere T R2=a2 2:5ms (for R 50 mis much longer than the usual times of interest. Rigorous integration in spherical coordinates is rather complex here, so . for the rst estimate one may signi cantly simplify the problem byintroducing local orthogonal coordinates near the surface with axes z normal to the surface, axes y orthogonal to the equatorial plane and x, normal to it. The whole torus of the mode near the surface of the sphere is substituted in this approachby a stripe under the surface of the half-space: 2 (z;z )2 2 `1=4 ^y (x y z) ' p2bR e; 2b20 ; 2xd2 ; 2yw2 e 0 p p d = R0= ` knz0 ' 1:14(` +1=2)1=3 w ' 2 R0 (62) The thermodynamical uctuations of temperature u averaged over the volume V = 2 2Rdb maybe presented in the following form: 1 Z1 Z1 Z1 u(~ t) e;x2=w2 e;y2 =d2 e;(z;b)2 =l2 dzdxdy u= V r 0 ;1;1 Z1 d~ ! F (~ !)ei!t 2 2 k kd 2 (63) e;kx w =4e;ky d2=4e;kz2b2 =4 ' (2 )4 2 ~ 2 a jkj + i! ;1 47


the components of wave-vector adding to the p!=a wemay neglect the term and x y also substitute complex exponentby unity: p 2 (65) 2 ' 4a3 p ! 2p dn 2 Tp ` 2 ( )= S !=! (66) 2 n 2 R2 C dT Substituting these parameters we obtain for the resonator at =0:63 m: 3=4 1000s;1 !1=4 1 ;11 50 m p (67) S !=! ( ) ' 10 R Hz This estimate should be valid for the range of frequencies 103 s;1 < < 105 s;1. Comparing this estimate with the quality factors already achieved in microspheres Q 109 1010 we see, that this value may be measured, and on the slope of the resonance curve the uctuations of the output power in the narrow band of several Hertz mayreach several percents ( W=W = Q !=!). In order this noise could be registered, it should dominate other noises. The fundamental 2 limitation here is the shot noise of laser with spectral density of uctuations of power SW = h!W , W is the power of optical light, registered on the detector. However, from the estimate W > Q2h! ' 2 10;5 erg/s (68) S 2!=! where I used a moderate value for microspheres of Q =108 , it follows that this limitation is absolutely inaccential here. The Schawlow-Townes quantum limit of frequency uctuations in laser eld is also not important here, 48

As w d b and due to the exponents, integral, satisfy the condition: z y x p e; z2b2=2. If z0=a 1. In this waywemay Z1 d z 2 z 4 ;1 2 a4 z +

Nowwemay calculate the following averaged value B ( )= hu(t)u(t+ )i and (correlation function of relative frequency uctuations) and from Wiener-Hinchin theorem the one-sided (hence additional factor 2) spectral density S 2!=! ( ) of relative uctuations of frequency in this approach: 11 2 2 dn !2 Z Z Z1 2(1 + e2i z z0 ) 2 2 2 ; x w2 =2; y d2 =2; z b2 =2 d x d y d z (64) 2 ( ) ' 4a T S !=! 2 C dT 4 4+ 2 e n (2 )3 ;1;1;1 a


W > Q2h! 2 L S !=

!

(69)

even if a laser with cold cavity quality-factor QL few orders lower than Q =108 is used for measurement. Summing up, the thermo-refractive noise in microspheres may be registered and estimated if the technical problems associated with appropriate whispering-gallery mode excitation and identi cation are solved.

49


REFERENCES
1] V.B. Braginsky, M.L. Gorodetsky, V.S. Ilchenko, Phys. Lett. A137, (1989) 393. 2] M.L. Gorodetsky, V.S. Ilchenko, A.A. Savchenko, \Ultimate Q of optical microsphere resonators", { Opt. Lett. 21 (1996) 453. 3] D. W. Vernooy,V. S.Ilchenko, H. Mabuchi, E. W. Streed and H. J. Kimble, Opt. Lett. 23 (1998) 247. 4] V.B. Braginsky, M.L. Gorodetsky and S.P.Vyatchanin, \Thermodynamical uctuations and photo-thermal shot noise in gravitational wave antennae", { Phys. Lett. A264 (1999) 1. 5] V.B. Braginsky, M.L. Gorodetsky and S.P. Vyatchanin, \Thermo-refractive noise in gravitational waveantennae", { Phys. Lett. A271 (2000) 303. 6] A. Abramovici et al., Science 256 (1992) 325.

50


Appendix F. Frequency uctuations of nonlinear origin in self-sustained optical oscillators
Introduction

The impressiveachievements in quantum optical and spectroscopic experiments during the last two decades in many cases have its origin in the invention and implementation of selfsustained optical oscillators (lasers) with very small frequency uctuations. The record high accuracy obtained in the comparison of the values of eigen frequencies of two Fabri-Perot optical resonators performed in the gravitational waveantennae prototype 1,2], is reasonable to be mentioned as a good example of such an achievement. The possibility to detect the relative di erence between the two eigenfrequencies at the level of !0=!0 ' 3 10;19 with averaging time ' 10;2 sec was demonstrated in this experiment. There is another "source" of achievements in this area of experimental physics: the steady rise of the optical mirrors nesse F . The obtained few years ago value of F ' 2 106 3] permits to realize in the table-top experiment an optical Fabri-Perot resonator with the eigenmode quality p factor Q ' 1013. Using this value of Q the Townes limit !0=!0 ' 1=(Q N ) "permits" to measure the level of relative frequency uctuations ' 10;21 if only N =1016 optical photons will be "spend". It is worth noting that nobody has formulated any fundamental limit for the value of F yet. Recently M. L. Gorodetsky together with the authors of this article 4,5] and Yu. T. Liu with K. S. Thorne 6] have analyzed the random deviations of the eigen frequency !r of Fabri-Perot resonator produced by thermodynamic uctuations of temperature T and by the temperature uctuations appearing due to the random absorption of optical photons in the resonator mirrors. These e ects are in essence of nonlinear origin: namely due to thermal dl expansion, characterized bycoe cient = 1 dT , and thermorefractivity (dependence of the l 1 dn refraction index n(T ) on temperature T )which can be characterized by coe cient = n dT ). The results presented in these articles show that the thermal uctuations in the mirrors may 51


be a serious obstacle when experimentalist wants to reach or even to overcome Standard Quantum Limit (SQL) of frequency. The goal of this article is to present the results of the analysis of random uctuations of the frequency in the output radiation of a selfsustained optical oscillator (laser) which are of similar nonlinear origin as in passive Fabri-Perot resonator. All estimates presented in text are given for parameters listed in Appendix F1.
AnalyzedModel of Laser

For the analysis we have chosen the laser scheme in which the solid body acts as the amplifying part. The inversion of population in certain optical transition in this solid body is provided by the radiation of the nearby set of photodiodes. One of this laser types (based on Nd:YAG) has a very narrow line width, very high e ciency of photodiode-to-output radiation conversion and high mean output power W (several tens or even hundreds of Watts). The output radiation wavelength mean value in this laser type is =1:06 7,8]. This laser is being used nowinthe Laser Interferometer Gravitational wave Observatory (the LIGO pro ject) 1,2,9] in the rst stage (LIGO-I) and is planned to be used in the second stage (LIGO-II) when the value of W will be at the level of 100 W ' 1021 photon=sec. The phase di erence between oscillations in twoFabri-Perot resonators pumped bysucha laser is expected to be measured with the error of ' 10;10 rad. The scheme of laser whichwe analyze is presented at g. 3: the resonator AB ,containing solid state active media (shown by dashed line on the g. 3), is coupled with reference cavity CD. We assume that: 1. All the mirrors have no losses and the mirror A is ideally re ecting one. Thus the power W is irradiated only through the mirror D. 2. The mirrors A, B and the solid state active media are rigidly assembled and the length of solid state active media are equal to the distance between mirrors A and B . 52


3. The nesses F of the mirrors B , C and D are assumed to be equal for simplicity,but the length lAB of AB is much smaller than the length lCD of CD. Thus the resonator AB cavity decay rate (bandwidth) AB is much larger than the resonator CD cavity decayrate CD ( AB = CD = lCD =2lAB 1). 4. The resonators AB and CD are optimally coupled and their mean frequencies !0 coincide. Therefore the output frequency mean value is very close to the CD resonance frequency and the frequency random deviations have to be substantially reduced due to the high quality Q = !=(2 CD )of the CD. 5. The solid state active media have negligible losses near the operating wavelength , and we also assume the random distortions of this solid body due to Brownian motion in itself are possible not to be taken into account.
A A B B
1

C C

D D1 D
2

B2

l

AB

l

CD

FIG. 5. The laser scheme. Resonator formed by the mirrors A and B contains the active media shown by the dashed line. The reference cavity is formed by the mirrors C and D. The mirror A is fully re ecting thus the power is irradiated only through the mirror D.

Under these assumptions we can write down the general expression for the output radiation frequency uctuations spectral density S! ( ) for the laser working far above the threshold2:
2

We use \one-sided" spectral density, de ned only for positive frequencies, which may be calcu-

53


S! ( 2 !0 S0! ( 2 !0 S !( 2 !0

) = S0!( 2 !0" ) ' h!0 W8 )= 1 < !2 :
0

) + S !( ) 2 !0 2# CD 2 1 !0 + 2 !0 9 ! = lAB 2 S + S !CD : !AB 2l
CD

(70) (71) (72)

See details of calculations in Appendix F2. Here is the observation frequency. The rst term S0!( ) in (70) describes the uctuations caused byvacuum uctuations (penetrating through mirror D) and spontaneous emission in active media: in case CD it is practically the Townes formula in which the cavity decay rate CD of reference cavity plays the decisive role and in opposite case when CD this term corresponds to the phase uctuations '2 ' 1=(4N ) of the wave irradiated through the mirror D where N = W =(h!0)is the mean number of photons irradiated during the time . The second term S ! describes the resonators AB and CD frequency uctuations: S !AB and S !CD are spectral densities of eigen frequencies uctuations of these resonators (the resonator AB eigen frequency uctuations contribution is suppressed by the factor 2llAB ). CD The rst term (S0! ( )) gives the ma jor limit which for parameters listed in Appendix F1 is of the order: q S0! ( ) (73) ' 1:1 10;21 p1 : !0 Hz In the following sections we analyze several intrinsic e ects which de ne the value of second term (S ! ( )).
lated from correlation function hX (t)X (t + )i using formula

SX ( ) = 2

Z

1 ;1

d hX (t)X (t + )i cos( ):

54


Thermorefractive Fluctuations in Laser Resonator

Consider that the only sources of frequency uctuations are the temperature uctuations which together with nonzero produce random changes of optical length in AB active media and thus produce the AB resonance frequency random changes. (In this section we assume that the positions of all the mirrors A B C and D do not uctuate.) These frequency uctuations will be substantially reduced in accordance with formula (72) due to the assumed small ratio lAB =lCD value. In this section we do not consider the contribution of the temperature uctuations through thermoexpansion (coe cient ) because these e ects are relatively small due to the small values of lAB =lCD and . There are twotypes of temperature uctuations in the discussed scheme. The rst ones are thermodynamic (TD) uctuations (see details in 4,5] and references therein). These ones are of pure classical origin in thermal equilibrium. The second ones are of quantum origin: the optical photons from the pumping photodiodes being absorbed in the laser solid body produces random local jumps of temperature | photo-thermal shot noise (SN). We assume that radius r0 of light beam in solid state active media is much smaller than cross dimensions of active media and for calculations one can expand the cross dimensions of media to in nity. Therefore wehave the problem to calculate the uctuations of temperature in cylindrical volume with radius r0 and length lAB in the in nite layer with the width lAB (the cylinder axis is perpendicular to the layer). The results of calculations for photo-thermal shot noise (noted by superscript SN) and the temperature thermodynamic uctuations (noted by superscript TD)may be presented in the following form: q SN S !AB ( ) s = (C )2h!20lW V 2 (74) !0 r0 AB q lAB S SNAB ( ) ' 2:2 10;21 p1 ! 2lCD q !0 Hz v TD ( ) u S !AB u 42 = t (C )2 T 4l 2 (75) !0 r0 AB 55


l

AB CD

l

q S

TD !AB !0

()

' 1:8 10

;20

p1 :
Hz

Here is thermal conductivity, is densityand C is speci c heat capacity of active media, is Boltzman constant, T is absolute temperature, V is e ective volume, W is absorbed power (all over belowwe assume power absorbed in active media is equal to optical power irradiated by laser for simplicity). Here we also use the adiabatic approximation 2 =( C r0 ). For numerical estimates we used material parameters of nondoped Nd:YAG. Details of calculations are presented in Appendix F3. We see that thermorefractive e ect caused by the temperature TD uctuations increases considerably the frequency uctuations | its contribution is about 16 times larger than Townes limit (compare (75) and (73)).
Fluctuations of Reference Cavity

SQL of Frequency

The Standard Quantum Limit (SQL) existance for self-sustained oscillator was predicted more that 20 years ago 12]. This limit origin is very simple: the output optical radiation "brings out" the information about the coordinate (in our scheme the frequency is linearly connected with the distance between the mirrors C and D). During the continuous coordinate measurement process the momentum of the masses (in our case they are the mirrors C and D) have to be inevitably perturbed. In other words, the Heisenberg uncertainty principle has to be ful lled. The momentum perturbation origin is well known: these are the random "kicks" of optical photons on the mirrors. This e ect in essence is nonlinear because it is created by the uctuations of the ponderomotive pressure of the optical eld inside the Fabri-Perot resonator. If the mirrors C and D can be regarded as free equal masses mC = mD (the eigen frequency of the mechanical suspension is much smaller than

56


the observation frequency ) then the frequency uctuations SQL is equal to q SQL s S !CD ( 0) = m l82h 2 ' 1:4 10;21 p1 !0 Hz C CD 0 whichisachieved at the optimal output power Wopt:
2 mC 2 0 Wopt = 16 ! cF 0 2 2 0

3

(76)

' 12 Watt:

(77)

where c0 is the velocity of light. These formulae are obtained in approximation CD however for opposite case CD it di ers by factor about unity only. It is important that frequency SQL (76) decreases when the distance lCD increases, whereas optimal power Wopt does not depend on the length lCD and depends on the mirror nesse F . It is worth to note that the limit (76) is valid only for speci c frequency 0 de ning the optimal power (77). In general case for arbitrary frequency and power W the frequency uctuations are bigger: q PM v 4! S !CD ( ) u 4h u 0 t 2 2 Wopt + W = ml : !0 Wopt C CD 0 W
Mirrors C and D Surface Fluctuations

Unfortunately, apart from mentioned sources of noise there are additional ones which change the e ective distance lCD : 1. Thermodynamic temperature uctuations cause the mirror surface uctuations due to nonzero thermal expansion coe cient . These uctuations are also called as thermoelastic noise.
3

The similar AB ponderomotive pressure uctuations are substantially smaller and were not taken

into account in previous section because the mirrors A, B and active media had been assumed to be rigidly assembled, and the lowest mechanical eigen frequency of suchaset is much higher than . The frequency uctuations SQL of such a set (without reference cavity) was analyzed in 12].

57


2. Thermal shot noise temperature uctuations cause the mirror surface uctuations due to nonzero thermal expansion coe cient . 3. The noise associated with the mirrors' material is inevitably related to the mechanical losses which can be originated by di erentphysical mechanisms. We use the model of structural damping and denote it as Brownian motion of the surface. In this model the loss angle does not depend on frequency 10]. These e ects are analyzed in details in 4] and we give only nal formulae denoted by superscripts TD and B for thermodynamic uctuations and Brownian uctuations correspondingly (the thermal shot noise e ect is substantially smaller and we do not present formula for it): S TDCD ( ) p 2 T 2(1 + )2 ! = 16 ( C )2r3 2l2 (78) 2 !0 2 0 CD v u S TD ( ) u CD Fused silica: t !!2 ' 6:6 10;23 p1 Hz 0 v u S TD ( ) u CD Sapphire: t !!2 ' 1:6 10;20 p1 Hz 0 S B!CD ( ) p8 T (1 ; 2) (79) 2 2 !0 ' v 2 r0 ElCD u SB ( ) u CD Fused silica: t !!2 ' 5:4 10;21 p1 Hz 0 v u SB ( ) u CD Sapphire: t !!2 ' 5:4 10;22 p1 : Hz 0 Here is the Poison ratio and E is the Young modulus. For thermodynamic uctuations we 2 also use adiabatic approximation =( C r0 ). The estimations are given for twokinds of mirror material: fused silica and sapphire. We see that these uctuations are not small and may exceed the SQL uctuations for chosen numerical parameters.
Thermorefractive Noise in Mirrors

Thermodynamic temperature uctuations together with photothermal shot noise also originate the thermorefractive uctuations: the mirror coating optical layers e ective re58


fractive indexes uctuations lead to these layers optical thickness uctuations and hence to the phase noise in the re ected wave. The detailed calculations of these e ects are given in 5] and below we write down only the nal formulae for thermorefractive e ect caused by thermodynamic temperature uctuations (we do not present formula for thermorefractive shot noise because of its smaller numerical value):

S

TR !CD 2 !0

()
e

2 2 2p 2 T = r2l2 ef f C 0 CD n2 1 + n2 2 2 = 4(n2 ; n12) 1 2

p

2

(80) = 2!0c : 0 (81)

Here 1 and 2 are thermorefractive co correspondingly. For often used pairs the following estimates: q TR S !CD ( ) ' 1:4 !0 q TR S !CD ( ) ' 2:5 !
0

e cients for layers with refractive indexes n1 and n2 of layers TiO2 - SiO2 and Ta2O5 - SiO2 we obtain 10;23 p1 Hz 10;23 p1 Hz (TiO2 ; SiO2) (Ta2O5 ; SiO2)

This e ect seems to be small enough for presented parameters however it has weak deq p pendence S TRCD ( ) 1= 4 on observation frequency and can be signi cant for the ! frequencies above 1 kHz.
Conclusion

We see that for used parameters thertmorefractive uctuations in laser resonator makes the largest contribution into frequency instability. The frequency instability caused by the e ects considered in this article is inversely proportional to distance lCD and therefore their negative in uence can be suppressed by increasing the length of the reference cavity. The e ects analyzed in this article can be analytically calculated. However, there are a lot of other processes responsible for additional frequency instability,whichmay provide 59


substantial contribution especially within the band of observation frequency near 100 Hz. They are usually called as "1=f " noise or excess noise. For example, as "the candidate" responsible for such process, we maypoint to the random jumps of vacancies or the birth of dislocations in the solid ob jects. Unfortunately, there are no reliable theoretical model for such processes whichwould allow to obtain analytical formulae and numerical estimates. Thus the presented in this paper analysis may o er to experimentalists only the lower limits of the frequency instability. For real experiments this value has to be larger. In the same time none of the noise sources analyzed in this article may be called as the fundamental one. This statement is also correct for the frequency SQL: the ponderomotive nonlinearity(which is equivalent to positive cubic nonlinearity of optical material) was emphasized in 13] and in principle can be compensated by a nonlinearity with opposite sign in solid. In this case the output radiation will "bring out" information not only about the coordinate, and the frequency deviation may be smaller. However, this potential possibility has not been seriously investigated yet. We also think that the analysis presented abovemay be useful for the nal choice of the optimal LIGO topology. In the ideal case when both arms of the LIGO interferometer are identical the laser frequency uctuations are completely subtracted in output signal. For real case the symmetry level of two interferometer arms mayplay an important role. The requirements for symmetry level became more tough when the power recycling is used as it was planned in the LIGO-II and the LIGO-III.
Appendix F1. Parameters

For estimates we used the following parameters:

!0 =2 1015 s T = 300 K
=2

;1

W =10 W r0 =0:5cm
60

mc =10 kg
;1

100 s


lAB =30 cm l

CD

=103 cm
;1

2 V =2 ( r0 lAB ) ' 50 cm3 F =3 103 CD = 2F lc ' 1:5 104 s CD Nd:YAG:

C =1:4 107 erg=(g K )
=1:4 106 erg cm=(s K ) =0:7 10;5 (K ) Fused silica:
;1

=4:55 g=cm

3

C =6:7 107 erg=(g K )
=1:4 105 erg cm=(s K )

=2:2g=cm =0:17
;1

3

E =7:2 1011erg/cm3
=5 10 Sapphire:
;8

=5:5 10;7 (K )

C =7:9 106 erg=(g K )
=4:0 106 erg cm=(s K )

=4:0g=cm =0:29 =3 10;9 :

3

E =4 1012 erg/cm3
=5:0 10;6 (K )
;1

The material parameters (C ) are given for fused silica, sapphire 4], and nondoped YAG 11]. For estimates of the thermorefractive e ect we used the following optical parameters for the mirror coatings layers:

n1 =2:2 n1 =2:2 n2 =1:45

1 1

=4 10;5 K =6 10;5 K
2

;1

(TiO2) (Ta2O5)

;1

=1:5 10;5 K;1 (SiO2): 61


Appendix F2. Fluctuations of frequency

In this Appendix we present the calculations of general formula (70) for frequency uctuations. Below we use the complex amplitudes (A B C D shown on gure 5) related to the electric eld E and the mean power W by the following equations s 2 2 E = A(t) h!0 e;i!0t +h:c: W = Sc04hE i = h!0A Sc 2
0

where S is the cross-section of the light beam, c0 is the velocityof light. Complex amplitude (for example A(t)) is written below as a sum of the constant mean amplitude (denoted by capital letter with zero superscript) and the uctuation part (denoted by small letter):
t A(t)= e;i!0s(A0 + a(t)) + h:c: Z1 (82) a(t)= ;1 1+ ! a( ) e;i t d 0 h i a( ) a( 0)] = 0 a( ) a+( 0) = ( ; 0) h i a(t) a+(t0) =2 (t ; t0): (83) q We assume below that !0 and drop the term 1+ !0 under the integral in (82). We write down the equation for the complex amplitude A describing the eld within the resonator AB : s 2) A = c0 AB (B + e ) _ A +( ; + kjAj (84) lAB 2 sp +iA !AB :

Here and k are the constants describing negative nonlinear losses, esp is the additional noise caused byspontaneous emission in the active media. The last term describes the uctuations of the resonator frequency. For the complex amplitude C (in the resonator CD) we have the following equation: s _ + CD C = c0 CD (D2 + B1)+ iB3 !CD : (85) C 2l
CD

62


These equations have to be supplemented by the boundary conditions on the mirrors B C and D:

p p p p B2 = ;C T ; B1 1 ; T p p D1 = ;D2 1 ; T + C T:
B1 = ;B2 1 ; T + A T
Here T is the transparency coe cient (remind that the mirrors B , C and D have the same transparency). Now we consider every amplitude as a sum of the mean constant value and the small uctuation part. For the mean amplitudes wehave: s p 0 0 0 A0 = C 0 = ; D1 = B1 = A0 T B2 =0: k For the uctuation components we use linear approximation keeping the terms p and assuming 1 ; T ' 1: s + )= c0 AB (b + e )+ iA0 ! @ta +( ; )(a + a 2 sp AB l s AB 0 c_ + CD c = c2l CD (d2 + b1)+ iC 0 !mc

p

T

b1 = ;b2 ; a T

p

b2 = c T ; b

p

CD

1

d1 = ;d2 + c T:

p

ph Fourier transform of the output radiation phase component D1 ( ) may be obtained after long but simple calculations:

ph 2 2 2 ph D1 ( ) = ( CD +2 AB )+ CD (2 AB )+ i ) D;i( )) + ( CD ; i )( CD +2 AB ( 2 CD ph + ;i ( AB +2 ) Esp ( ) +

D0 + (;i1 ) ( + ph D1 = d1 +i d1 2
CD

CD

AB !AB CD + 2 !CD AB CD +2 AB ) ( CD +2 AB ) + ph ph D2 = d2 +i d2 Esp = esp 2

!

(86)

+ e+ sp : 2i

Assuming that

AB

ph the expression for D1 ( ) can be simpli ed resulting in:

63


; i ) Dph ( ) + CD E ph ( ) + 2 ;i sp ! 0 D1 + (;i ) !AB 2 CD + !CD AB ph ph Assume the spectral density SD2( ) of D2 uctuations corresponds to the vacuum uctuph ph ations and is equal to the spectral density SEsp( ) of spontaneous noise Esp :
ph D1 ( ) ' i( CD ph ph hD2 (t)D2 (t0)i = 24 (t ; t0) ph SD2( )=2

Z

1

ph SEsp( ) =

;1 ph SD2(

ph ph dt e;i t hC3 (0)C3 (t)i =

)= :

Then the output radiation phase uctuations spectral density can be obtained as follows: ! ph SD2( ) = h!0 2 CD 2 +1 + (87) S'( ) = jD0j2 2W +
2 AB
CD 2

S

!AB 2

+S

!CD

:

Using (87) the formula (70) can be easy obtained.
Appendix F3. Thermorefractive uctuations in in nite layer

In this Appendix we present calculations for the laser resonator AB thermorefractive uctuations formulae (74, 75.
Thermal Shot Noise

For calculation of temperature u uctuations in in nite layer with thermoisolated boundary that caused by shot noise uctuations, we have the equation of thermal conductivity (for this case z-axis is perpendicular to the layer):

C @tu ;

u = w(~ t) r @z ujz=h =0
64

(88) (89)

@z ujz=0 =0

hw(~ t)w(~0 t0)i = h!0 W0 (t ; t0) (~ ; ~0): r r rr


Here W0 is averaged power absorbed in a unit volume. We assume that W0 = W=V where W is the total averaged power absorbed in whole volume V of the laser active media. We write the solution as a Fourier-series expansion Z 1 dkxdky d! X 1 cos(bnz) u(t ~? z)= r ;1 (2 )3 n=0 ~ ei!t;ikxx;iky y C wn (!2 k?) ) (a2k + i! bn = l n a2 = C AB ~? = ~xx + ~y y ~ ? = ~xkx + ~y ky re e ke e

hwn(! ~ ? )wn0 (!0 ~ ?)i = h!0 W0 k k0 2 (! ; !0) 2 ; 0 n nn0 (2 )2 (~ ? ; ~ ? ): k k0 l
AB

We are interested in the uctuations of temperature u averaged over the beam volume: Z lAB dz Z 1 dx dy 2 ;(x2 +y2 )=r0 u(t ~) u(t)= r 2e lAB ;1 r0 0 Z 1 dkx dky 2 2 2 e;(ky +kz )r0 =4 u(!)= ;1 (2 )2 w0(! ~ ? ) k 2k 2 + i! ) C (a ? h! W hu(!)u(!0)i =2 (! ; !0) (C 02l 0 ) AB 2 Z 1 dkx dky e;k?r02 =2 4 a4k? + !2 : ;1 (2 )2
2 Using adiabatic approximation ! a2=r0 the term a4k4 in the last fraction denominator TS can be dropped. Then the expression for spectral density Su (!) of averaged temperature u will be the following: Z 1 dkxdky 2 2 TS ;k? r0 =2 = Su (!) ' (C2h!0W20!2 lAB ) (2 )2 e ;1 h!0 W (90) = (C )2 r2l V !2 0 AB

The formula (74) can be easily obtained from (90).
TD Temperature Fluctuations

65


For the TD temperature u uctuations calculation in in nite layer we use the Langevin approach, i. e. we add uctuation forces into the thermal conductivity equation right side (see details in 4]):

C @tu ; @z uj

u = F (~ t) r
z=0

(91)
=lAB

=0

@z ujz
2

=0: (~ ; ~0): rr

(92)

hF (~ t)F (~0 t0)i = ;2 T r r

(t ; t0)

We write the solution as a Fourier-series expansion Z 1 dkx dky d! X 1 cos(bnz) u(t ~? z)= r ;1 (2 )3 n=0 ~ ei!t;ikx x;iky y C Fn(!2 k? ) ) (a2k + i! bn = l n a2 = C AB ~? = ~xx + ~y y ~ ? = ~xkx + ~y ky re e ke e

(93)

hFn(! ~ ? )Fn0 (!0 ~ ? )i = k k0 =2 T 2 2 (! ; !0) 2 ; 0 n lAB 2 (~ ; ~ 0 ) (b2 + k 2 ): (2 ) k? k? n ?

nn0

(94)

We are interested in uctuations of temperature u averaged over the beam volume: Z lAB =2 dz Z 1 dx dy 2 2 2 ;(x +y )=r0 u(t ~) r u(t)= 2e ;lAB =2 lAB ;1 r0 Z 1 dkx dky u(!)= ;1 (2 )2 2 0 ~? e;(ky+kz2 )r02=4 C Fa(!2 k+ )i!) ( 2k? 2 hu(!)u(!0)i =2 (! ; !0) (2 T 2l C ) AB 2 Z 1 dkx dky k2 e;k? r02=2 ? 4 a4k? + !2 : ;1 (2 )2 2 Using adiabatic approximation ! a2=r0 the term a4k4 in the last fraction denominator TD can be dropped. Then the expression for spectral density Su (!)ofaveraged temperature u will be the following: 66


4 T 2 Z 1 k? dk? e S (!) ' (C )2l AB ;1 2 42 = (C )2 T 4l !2 : r0 AB
TD u

2 22 ;k? r0 =2 k?

!2 =

(95) (96)

The formula (75) can be easily obtained from (96).

67


REFERENCES
1] A. Abramovici et al., Science 256 (1992) 325. 2] A. Abramovici et al., Physics Letters A 218 (1996) 157. 3] G. Rempe, R. Thompson, Y. J. Kimble, Optics Letters, 17 (1992) 362 4] V. B. Braginsky, M. L. Gorodetsky,and S. P.Vyatchanin, Physics Letters A264 (1999) 1 cond-mat/9912139. 5] V. B. Braginsky, M. L. Gorodetsky,and S. P.Vyatchanin, Physics Letters A271 (2000) 303. 6] Yu. T. Liu and K. S. Thorne, submitted to Phys. Rev. D. 7] B. Zhou, T. J. Kane, G. J. Dixon and R. L. Byer, Optics Letters 10 (1985) 62. 8] T. J. Kane and R. L. Byer, Optics Letters 10 (1985) 65. 9] W. Wiechnann, T. J. Kane, D. Haserot, F. Adams, G. Tuong, J. D. Kmetec, Proceedings of CLEO'98 (1998) 432. 10] P. R. Saulson, Rhys. Rev. D, 42 (1990) 2437 G. I. Gonzalez and P. R. Saulson, J. Acoust. Soc. Am., 96 (1994) 207. 11] F.Bondu, Report on Aspen Winter Conference on Gravitational Waves and their Detection,Feb 20-26, 2000. 12] V. B. Braginsky and S. P.Vyatchanin, Zh. Eksp. i Teor. Fiz., 74 (1978) 828, English translation: Sov. Phys. JETP, 47 (1978) 433. 13] V. B. Braginsky,V. I.Panov and S. P.Vyatchanin, Doklady AN SSSR, 247 (1979) 583 English translation: Physics-Doklady, 24 (1979) No. 7.

68


Appendix G. The Discrete Sampling Variation Measurement
Introduction

It is common knowledge that the sensitivity of traditionally designed position meters, including interferometric meters used in the large-scale gravitational wave detectors, is limited by the Standard Quantum Limit (SQL) 1]. One of the most promising ways of evading the SQL is the variation quantum measurement 2] because it requires minimal modi cations in the interferometric meters hardware setup only. This method makes it possible to eliminate the output signal uctuations caused by the back action of the meter simply by proper modulation of the local oscillator phase LO. Unfortunately the function LO (t) used in the variation measurement depends on the signal shape and arrival time. This dependance of the meter hardware setup on the signal shape is the main disadvantage of the variation measurement. Because of this disadvantage the variation measurement in its original version can be used for detection of determenistic signals only. In the article 2] the modi ed version of the above mentioned procedure had been proposed and considered in brief. It allows to circumvent this disadvantage and makes it possible to monitor the signal shape. This method is based on the signal approximation by series of short rectangular \slices" and periodical applying the variation measurement procedure tted to such a rectangular pulse. We propose to name this procedure \discrete sampling variation measurement"(DSVM). In this paper we consider this method in details, applying to the cases of free mass and harmonic oscillator. In the next section we review brie y the variation measurement as it was proposed in the original works 4,3] and introduce some useful notations. In the section II we describe the discrete sampling variation measurement and in the section II we compare its sensitivity with the Standard Quantum Limit and the Energetic Quantum Limit.

69


The Variation Measurement

Any position meter can be reduced to the simple abstract scheme presented on Fig.6. Its output signal x(t) can be written as a sum ~

x(t)= x(t)+ x uct(t) ~

(97)

where x(t) is the test body \real" position and x uct(t) is noise added by the meter. On the other hand due to the uncertainty relation the meter perturbs the test body motion by the random back action force F uct(t). Hence the value of x(t) presents a sum of three components: the test ob ject unperturbed motion xinit, response on external classical force Fsignal which should be detected and response on back action force:

x(t)= xinit(t)+ D;1 F

signal

(t)+ F

uct

(t)]

(98)

where D is the linear di erential operator describing the dynamics of the test ob ject. To take an example, for an oscillator with mass m and eigenfrequency !m . d2 2 D = m dt2 + m!m (99) In the case of the interferometric meters F uct(t) is produced by the optical pumping power shot noise. The noise x uct(t) is the mix of the output optical beam amplitude and phase uctuations with the weights depending on the local oscillator phase LO. In this article we will limit ourselves for simplicity by the case when both of these noises can be considered as white ones. This condition becomes invalid only for very long interferometers using the signal-recycling technique. It can be shown that the main results of our consideration are held true in this case too. In the case of the interferometric meter with the resonant pumping spectral densities of the noises x uct(t) and F uct(t), and its cross spectral density are equal to

Sx = S
F

SxF =

hL2 16QE sin2 LO = 4hQE L2 ; h cot LO 2

(100)

70


where L is the interferometer arm length, E is the energy stored in it, Q is the optical resonator quality factor. It is easily to see that they satisfy the uncertainty relation
2 2 SxSF ; SxF = h 4

(101)

which is a general propertyof any position meter see 5]]. It is convenient to rewrite the noise x uct as

x
where

uct

= x(0) + aF uct

uct

(102)

a = SxF SF

(103)

and x(0) is the part of x uct uncorrelated with the back action noise. In the case of the uct interferometric meter this noise is produced by the output beam phase uctuations. In this case its spectral density is equal to

h2 = hL2 S = 4S 16QE F
(0) x

(104)

and

L2 cot a = ; 8QE

LO

:

(105)

It is important to note that a depends on time if LO is time-dependant. The output signal x(t) should be processed in order to optimally extract information ~ about Fsignal. This signal processing can be represented as a two-stage process. At the rst stage the operator D is applied to x(t) in order to eliminate term xinit: ~ ~ F (t)= Dx(t)= F ~ where
signal(

t)+ F

noise

(t)

(106)

Fnoise(t)= F uct(t)+ Dx uct(t)
71

(107)


~ is the total noise. At the second stage F (t)is integrated with optimally chosen lter function v(t) which gives the signal required parameter estimation of the required parameter, for example, for the force amplitude A: Z1 Z1 ~ ~ A= v(t)F (t) dt = v(t) Fsignal(t)+ F uct(t)+ Dx(0) (t)+ Da(t)F uct(t)] dt : (108) uct
;1 ;1

It is easy to show that if a(t) satis es the equation

a(t)Dv(t)+ v(t)=0

(109)

then two terms in (108) containing the back action force F uct(t) are compensate each other.4 Hence the measurement precision in this case is limited by the noise x(0) only: uct Z 2 = S (0) 1 Dv (t)]2 dt : ( A) (110) x
;1

This is the basic principle of the variation measurement which permits in concept to obtain (0) any necessary sensitivityby reducing the value of Sx .
The discrete sampling variation measurement

Suppose that a priori information is available for the experimentalist that the signal spectrum is limited by some value of !max. In this case small \slice" of the signal with ;1 duration < !max can be approximated by rectangular pulse with the same duration. This allows to use variation measurement with the function a(t) tted to this rectangular pulse and measure the mean value of the signal over this interval. Repeating this procedure periodically it is possible to reconstruct \slice" by \slice" the signal shape. The precision of such procedure is not limited by the SQL due to using variation measurement. On the other hand, the hardware setup here does not depend on the signal shape. The law for LO (t)
4

Strictly speaking this is valid only if the operator D is \hermitian", i.e there is no dissipation in

the probe ob ject. We shall consider here this case only.

72


can be obtained simply by periodical repetition of a function corresponding to rectangular signal pulse. So consider the Fsignal(t) force mean value, F0, measurement over short time interval ; =2 t =2. We suppose the test ob ject to be a harmonic oscillator, so the operator D is equal to (99). Particular case of a free mass can be easily obtained by putting !m =0. If the variation technique is used and back action noise is compensated then the measurement error is equal to Z (0) =2 ( F )2 = Sx ; =2 Dv(t)]2 dt : (111) It can be shown that optimal lter function v(t)must satisfy the following equation

D2v(t) = const
with the normalization condition

(112)

Z

=2

; =2

v(t) dt =1 dv(t) dt

(113)

and the boundary conditions

v(t)
The solution is equal to

t= =2

=0

t= =2

=0 :
!m 2

(114)

! and the corresponding function a(t) is equal to !m +sin !m ; 4sin !m cos !mt 1 2 a(t)= ; m!2 !m + sin !m ; 2 sin !m + !m cos !m cos !m t ; 2 sin m 2 2 2

v(t)= !m

!m +sin !

m

; 2 sin

!m 2 m (!m

+ !m cos !m cos !mt ; 2sin 2 2 +sin !m ) ; 4(1 ; cos !m )

!m t sin !mt

(115)

!m 2

: !m t sin !m t (116)

Substitution of this function v(t) into formula (111) gives the the measurement error value:
22 ( F )2 = 180m 5h k(!m ) SF

(117)

73


where

x5 x k(x)= 720 x(x +si(n This function is plotted on Fig.7. In the particular case of a free mass (!m 2 v(t)= 30(t

+sin x) x) ; 4(1 ; cos x)] : =0) ; 2=4)
5

(118)

2

(119) (120)

t2 2 4)2 a(t)= ; 21 (6t2; ==2 m ;2
and

22 ( F )2 = 180m 5h : (121) SF It should be noted that singularities in the function a(t) do not prevent realization of the described procedure. They correspond simply to values of LO equal to 0 see equation (105)]. The graphics of the functions v(t)and LO(t) for the free mass are presented at Fig.8 and Fig.9. The corresponding graphics for the oscillator are almost the same if the parameter !m is chosen in the optimal way,

!

m

:

(122)

Every repetition of this procedure on time intervals =2 t 3 =2, 3 =2 t 5 =2 and so on gives an estimation for the signal force mean value over the corresponding interval Fj signal: ~ Fj = Fj + F where j =0 1 2 :::,
j noise

(123)

Fj =

Z

=2

; =2

v(t)F

signal(

t ; j ) dt

(124)

and Fj noise are the uncorrelated random values with the variance (111). The values (123) ~ form vector fFj g that approximate the signal force. 74


Comparison with the SQL and the EQL

In this section we will limit ourselves by the case of a free mass only because the gain in the sensitivitywhich can be obtained by using an oscillator is not very signi cant (see Fig.7). In order to compare proposed procedure with traditional meters we should return back to continuous representation. We have assumed Fsignal to vary slowly during the time . ~ This allows to approximate the vector fFj g by the continuous function ~ F (t)= Fsignal(t)+ FDSVM(t) where FDSVM(t) presents a noise produced by the measurement errors Fj density is equal to
22 SDSVM = ( F )2 = 180m 4h SF noise

(125) . Its spectral (126)

The Standard Quantum Limit usually de ned as the ultimate sensitivity of the ordinary position meter, i.e position meter with white and non-correlated noises x uct and Ff luct. For such a meter the spectral density of the total noise is equal to
2 m2 4 SSQL( ) = SF + h 4S : F

(127)

Minimum of this expression for anygiven observation frequency is achieved if

SF = S
and is equal to

SQL F

hm 2

2

(128)

S

SQL

= hm 2 :

(129)

The ratio of the spectral densities (126) and (129) is equal to
2 DS V M

SQ SDSVM = 180mh = 360 SF L : SSQL SF 2 4 ( )4 SF

(130)

75


The value of SF for the parametric position meters (including interferometric ones) is proportional to the energy stored in them see (100)]. Hence the last formula may be presented as
2 DS V M

= ( 360 4 ESQ )E

L

(131)

where ESQL is the energy which is necessary to achievethe level of the SQL and E is the energy actual value. As it is known from the digital signal processing theory, a signal can be restored correctly if the sampling frequency is at least two times larger than the signal bandwidth. In accordance with this principle we suppose that = In this case
2 DS V M max

:
3:7 ESQL : E

(132)

= 360 ESQL 4 E

(133)

Conclusion

Precision of the proposed method is de ned by the sampling period . The less is this value, the better temporal resolution can be obtained. On the other hand, the pumping energy required to obtain given level of sensitivity rises as ;4 with reducing . Nevertheless if the value of is chosen wisely then a good balance can be achieved. It is useful to compare the sensitivity of the proposed method (133) with the Energetic Quantum Limit 5,6] which presents the ultimate limit of the sensitivity for anygiven energy E:
2 EQL

SQ = E2EL :

(134)

The variation measurement traditional form sensitivity is de ned by this formula too. Comparing the values (133) and (134) it is possible to conclude that the \price" for the hardware 76


setup independance on the signal shape is the pumping energy 720= 4 7:4 times higher q than for the usual variation measurement, or sensitivityabout 720= 4 2:7 times lower for the same pumping energy value. It is worth noting that it can be reasonable to use the harmonic oscillator as a probe system because in this case the given sensitivity can be obtained using the pumping energy approximately 0:7 times less than for a free probe mass. It should be noted also that for our opinion the more sophisticated expansion of the signal can be constructed, whichallows variation measurement with periodical precalculated law for the local oscillator phase time-dependance. The wavelet technology looks especially promising here.

77


x= x+x ~

uct

The meter

x
-

u

Fsignal -

F

uct

m

FIG. 6. Abstract scheme of the position meter

2 1.8 1.6 1.4 1.2 1

( F )2 osc ( F )2: m: f

0.8 0.6 0.4 0.2 0 0

1

2

3

4

5

6

!

m

FIG. 7. Measurement errors for the harmonic oscillator relative to the one for the free mass

78


0.2 0 -0.2 -0.4 -0.6 -0.8

v(t)

-1 -1.2 -1.4 -1.6 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t=
FIG. 8. Optimal lter function for the free test mass

79


2 1.8 1.6 1.4 1.2 1

LO

(t)

0.8 0.6 0.4 0.2 0 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t=
FIG. 9. Phase of local oscillator for the free test mass ( =1=3)

80


REFERENCES
1] V.B.Braginsky,Yu.I.Vorontsov, K.S.Thorne, Science 209 (1980) 547. 2] S.P.Vyatchanin, Phys.Lett. A239 (1998) 201. 3] S.P.Vyatchanin, E.A.Zubova, Phys.Lett. A201 (1995) 269. 4] S.P.Vyatchanin, A.Yu.Lavrenov, Phys.Lett. A231 (1997) 38. 5] V.B.Braginsky,F.Ya.Khalili, \Quantum measurement",ed. by K.S.Thorne, Cambridge Univ. Press, 1992. 6] V.B.Braginsky, M.L.Gorodetsky, F.Ya.Khalili and K.S.Thorne, \Energetic Quantum Limit in Large-Scale Interferomters", proceedings of Third Edoardo Amaldi Conference, Pasadina, 1999.

81