Internal thermal noise in the LIGO test masses: A direct approach

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1 PHYSICAL EVIEW D VOLUME 57, NUMBE 2 15 JANUAY 1998 Internal thermal noise in the LIGO test masses: A direct approach Yu. Levin Theoretical Astrophysics, Caliornia Institute o Technology, Pasadena, Caliornia eceived 21 July 1997; published 22 December 1997 The internal thermal noise in LIGO s test masses is analyzed by a new technique, a direct application o the luctuation-dissipation theorem to LIGO s readout observable, x(t) longitudinal position o test-mass ace, weighted by laser beam s Gaussian proile. Previous analyses, which relied on a normal-mode decomposition o the test-mass motion, were valid only i the dissipation is uniormally distributed over the test-mass interior, and they converged reliably to a inal answer only when the beam size was a non-negligible raction o the test-mass cross section. This paper s direct analysis, by contrast, can handle inhomogeneous dissipation and arbitrary beam sizes. In the domain o validity o the previous analysis, the two methods give the same answer or S x ( ), the spectral density o thermal noise, to within expected accuracy. The new analysis predicts that thermal noise due to dissipation concentrated in the test mass s ront ace e.g., due to mirror coating scales as 1/r 2, by contrast with homogeneous dissipation, which scales as 1/r (r is the beam radius ; so surace dissipation could become signiicant or small beam sizes. S PACS number s : 4.8.Nn, 5.4. j I. INTODUCTION andom thermal luctuations are expected to be the dominant noise source or the irst intererometers in the Laser Intererometer Gravitational Wave Observatory LIGO at requencies between 35 and 1 Hz 1. This thermal noise is generally decomposed into a suspension thermal noise and an internal thermal noise or the test masses. The ormer can be traced back to the riction in the test mass pendular suspension system; the latter is due to internal damping inside the test masses themselves. Traditionally, thermal noise calculations have been based on a normal-mode expansion 2,3. However, Gonzalez and Saulson have also perormed an exact calculation o the suspension thermal noise by applying directly the luctuation-dissipation FD theorem 4 in its most general orm, due to Callan and Welton 5. The purpose o this paper is to use the general method o Gonzalez and Saulson to calculate the internal thermal noise also, 6 has a somewhat complementary to this paper treatment o the internal thermal noise. In Sec. II we analyze a general situation when a measuring device e.g., a laser intererometer monitors the displacement o the surace o a test mass whose internal degrees o reedom are in thermal equilibrium with each other. We develop a general ormalism or using the FD theorem to calculate the thermal noise in the most general surace readout quantity. In brie our method is as ollows. To work out the thermal noise at a particular requency, one should mentally apply pressure oscillating at this requency to the observed surace o the test mass. The spatial variation o this pressure should mimic that o the light beam intensity or example, in the case o a Gaussian beam this oscillating pressure has a Gaussian proile o the same width as the beam. The thermal noise is then given by S x 2k BT 2 2 W diss F 2, 1 where k B and T are Boltzmann s constant and the temperature o the mirror, respectively, F is the amplitude o the oscillating orce applied to the surace i.e., the pressure integrated over the surace, and W diss is the time-averaged power dissipated in the test mass when this oscillating pressure is applied. To demonstrate the computational power o this general approach, in Sec. III we consider the case o a cylindrical used silica test mass monitored by a circular Gaussian laser beam. For the case when the radius o the beam is much less than the size o the the test mass and the dissipation is uniormly distributed throughout the test-mass volume, we derive an analytical expression or the thermal noise c. Eq. 15 o Sec. III : S x 4k BT 3 E r 1 O r Here, E, and are the Poisson ratio, Young s modulus, and dissipational loss angle Eq. 11 o the test-mass material, r is the radius o the laser beam which is deined here as a radius at which the intensity o light is 1/e o the maximum intensity, is a characteristic size o the test mass, and I inthecase o a Gaussian beam. Putting numbers in Eqs. 1 and 2, we ind that our results are in agreement with those o aab and Gillespie 3, who used the more complicated and computationally involved method o normal-mode decomposition. It is interesting to note that as r / tends to zero, our simple analytical ormula becomes more precise, whereas the more complicated and computationally involved method o normal-mode decomposition requires summing over a larger number o modes and thus becomes computationally more expensive. Not only can the normal-mode decomposition be computationally expensive, it can also be misleading. We demonstrate this point in Sec. IV by considering a test mass which has a lossy surace, e.g., due to a lossy mirror coating. We estimate the contribution o the surace to the thermal noise using the general method o Sec. II, and show that it diers rom the estimate obtained by the method o normal modes /97/57 2 /659 5 /$ The American Physical Society

2 66 YU. LEVIN 57 which gives a result too small by a actor o at least r /). This breakdown o the normal-mode analysis will in general happen when the sources o riction are not distributed homogeneously over the test mass. The undamental reason is that in this case dierent normal modes can have a common Langevin driving orce which is not so i the deects are distributed homogeneously. Our analysis shows that thermal noise due to surace losses near the laser beam spot scales as S x ( ) 1/r 2, whereas thermal noise due to volume losses scales as 1/r. Correspondingly, or small beam spots the surace losses could become signiicant. To protect against this, it is important to keep the surace near the laser beam spot as ree o potential sources o riction as possible. II. GENEAL METHOD For concreteness, consider a situation where LIGO s laser beam is shining on the circular surace o one o LIGO s cylindrical test masses. The phase shit o the relected light contains inormation about the motion o the test-mass surace. The variable read out by this procedure can be written as x t r y r,t d 2 r. Here r is the transverse location o a point on the test-mass surace, and y(r,t) is the displacement o the boundary along the direction o the laser beam at point r and time t. The orm actor (r ) depends on the laser beam proile and is proportional to the laser light intensity at the point r 3 ; itis normalized by (r )d 2 r 1. The internal thermal noise o the test mass is deined as the luctuations in x(t), and our objective is to ind the spectral density S x ( ) o these luctuations. We assume that the test mass is in thermal equilibrium at temperature T. Callen and Welton s generalized luctuation-dissipation theorem 5 says that the spectral density o the luctuations o LIGO s readout variable x(t) is given by the ormula S x k BT e Y, where k B is Boltzman s constant and Y ( ) is a complex admittance associated with x(t). This complex admittance can be understood and computed as ollows. Introduce a special set o generalized coordinates or the test-mass degrees o reedom a set or which x is one o the coordinates. Since x is not the coordinate o a normal mode o the test mass, these generalized coordinates will not be the usual ones associated with normal modes. Apply to the test mass a generalized orce F(t) that drives the generalized momentum conjugate to x but does not drive any o the other generalized momenta. This generalized orce will show up as the ollowing interaction term in the test-mass Hamiltonian: H int F t x. 3 5 This driving orce, together with the test-mass internal elastic orces and internal dissipation, will generate a time evolution x(t) o the observable x. Denote by F( ) and x( ) the Fourier transorms o the arbitrary driving orce F(t) and the observable s response x(t). Then the admittance that appears in the thermal noise ormula, Eq. 4, is Y 2 ıx /F The physical nature o the driving orce F(t) can be deduced by inserting the deinition 3 o the observable x into the interaction Hamiltonian 5 : where H int P r y r,t d 2 r, P r,t F t r. From Eq. 7 we see that the generalized orce F(t) consists o a pressure P(r,t) Eq. 8 applied to the test-mass surace. Note that the spatial distribution o this pressure is the same as LIGO s laser beam intensity proile. The real part o the admittance, e Y ( ), describes the coupling o the test-mass dissipation to the observable x. We can see this most clearly by applying an oscillatory pressure P(r,t) F cos(2 t)(r ) to the test-mass ace. From the response ormula 6 we iner that the power W diss that this oscillatory pressure eeds into the test mass, and that the test mass then dissipates, is related to e Y ( ) by e Y 2W diss. 9 2 F Substituting Eq. 9 into Eq. 4, weget S x 2k BT 2 2 W diss F 2. 1 Equation 1 is the most important equation o this paper. Let us reemphasize its physical content: 1 Apply an oscillatory pressure P(r,t) F cos(2 t)(r ) to the ace o the test mass; 2 work out the average power W diss dissipated in the test mass under the action o this oscillatory pressure; 3 use F and W diss in Eq. 1 to calculate S x ( ). This procedure is dierent rom the one employed in previous calculations o internal thermal noise or the LIGO and VIGO test masses 2,3,7. The previous authors decomposed a test-mass motion into normal elastic modes; then they calculated the contribution o each mode to S x independently and added up these contributions. This method o normal-mode decomposition works ine in many cases, but it has two drawbacks. 1 The undamental assumption in this method is that dierent normal modes have independent Langevin orces. This assumption is correct only i the sources o riction are homogeneously distributed over the test-mass volume. It breaks down i the deects are more concentrated in one

3 57 INTENAL THEMAL NOISE IN THE LIGO TEST place than in others or example, when there is signiicant damping concentrated in the test-mass surace. We will return to this in Sec. IV. 2 For a small laser beam diameter the sum over normal modes converges very slowly, and so one has to sum over many modes, which may be computationally expensive. By contrast, using the new method described in this paper, one can write down a simple analytic expression or the lowrequency noise in the case o a narrow laser beam. In the next section we derive this expression and make comparison with the normal-mode decomposition results derived in 3. III. THEMAL NOISE DUE TO HOMOGENEOUSLY DISTIBUTED DAMPING Consider the case where all the riction in the test mass comes rom homogeneously distributed damping. It is conventional to characterize such riction by an imaginary part o the material s Young s modulus: E E 1 ı ; 11 ( ) is called the material s loss angle. It is suspected 8,2 that or used silica, which will be used in LIGO s test masses, might be independent o requency within LIGO s detection band but there is no evidence or such behavior o or high-quality resonators see 9 or some healthy scepticism. In this -independent case the damping is called structural. To calculate the thermal noise or homogeneous dissipation, we express W diss in Eq. 1 as W diss 2 U max, 12 where U max is the energy o elastic deormation at a moment when the test mass is maximally contracted or extended under the action o the oscillatory pressure o Eq. 8. LIGO s detection requencies 1 3 Hz are much lower than the eigenrequencies o the test-mass normal modes the lowest o which is 6 khz ; so we can assume constant, nonoscillating pressure P(r ) F (r ) when evaluating U max. In the case when the beam proile is Gaussian and the center o the light spot coincides with the center o the transverse coordinates, we have 1 r 2 e r2 2 /r, 13 r where r is the radius o the laser beam. When the characteristic size o the test mass is much greater than r,we can approximate the test mass as an ininite hal-space in order to ind U max. The Appendix uses elasticity theory to derive U max in this case c. Eq. A5 : U max F 2 2 E r I 1 O r, 14 where E and are the Young s modulus and Poisson ratio o the material, respectively, and I Here O(r /) is a correction due to the inite size o the cylinder. Putting Eqs. 14 and 12 into Eq. 1, one gets S x 4k BT 3 E r 1 O r 15 Below we take the numerical values 1 used by Gillespie and aab 3 : r 1.56 cm, E Pa, 16, 1 7, a mirror diameter o 25 cm, and the mirror length o 1 cm. Gillespie and aab, ater summing over the relevant 3 modes, get S x G 1 Hz m 2 /Hz. 16 Our analytical approximation 15 which should be valid to within 1% in this case gives S x 1 Hz m 2 /Hz. 17 Notice that our analytic expression in Eq. 15 gets more exact when r /, whereas, by contrast, the sum over modes converges more slowly and gets more complicated. The ratio r / may turn out to be o order unity in real experiments. In this case, Eq. 15 can only be used or order-o-magnitude estimates. To work out the exact value o the internal thermal noise, one would need to calculate U max numerically. We have done such a numerical computation using inite-element techniques. More speciically, we have used inite-element sotware called PDEASE2D version 3., which runs as part o MASCYMA Version 2.1, to solve the elasticity equations or the loaded mirror and to compute U max and, by virtue o Eqs. 12 and 1, S x. The exact answer or the mirror and light spot parameters given above is S x 1 Hz m 2 /Hz, 18 which is consistent better than expected with our analytical approximation. The purpose o the present section is to convince the reader that the method presented in this paper is correct and could be computationally cheaper than the normal-mode expansion. The next section concentrates on the cases where a direct application o the FD theorem can be crucial or getting the right results, and the method o normal-mode decomposition ails. IV. CASE OF SUFACE DAMPING In this section we study thermal noise due to surace losses caused, e.g., by inadequate polishing or by a lossy mirror coating. From Eq. 1 we see that the key quantity in the thermal noise calculation is the power dissipated in the test mass when an oscillating pressure is applied to the laser beam spot on the test-mass surace. The power dissipated at each point o the material is proportional to the square o the stress at this point. Most o the surace stress is in or near the spot to which the pressure is applied, and so 1 Note that our deinition o the beam radius location where intensity has allen to 1/e o its central value diers by 2 rom the beam radius o e. 3 location o 1/e amplitude allo.

4 662 YU. LEVIN 57 W diss F 2 2 r r 2 F r Thus the thermal noise due to the surace damping scales like S x boundary 1/r 2. 2 For comparison, the thermal noise due to bulk damping Eq. 15 scales as S x bulk 1/r. 21 Thus, as the spot size decreases, the thermal noise due to surace damping grows aster than that due to bulk damping. Contrast this conclusion with the intuition one gets rom normal-mode decomposition. There one is concerned with how much the surace contributes to the quality actors (Q s o the normal modes. For a typical mode the strain at the surace is at most o the same order as the characteristic strain inside the test mass likely, much less or the irst ew modes, because o the ree boundary condition. Thereore, one would presume that the surace contributes no more than some mode-independent raction o the test mass s Q s. In order o magnitude this raction should be the ratio o the power dissipated in the surace to that in the bulk i one applies an oscillating pressure uniormally to the whole surace, which in the context o our method corresponds to a beam radius o. Thereore the normal-mode estimate o the surace thermal noise is at least r / less than the correct value. Current experiments show that the mirror coating does not contribute signiicantly to the Q s o the test-mass normal modes. The conclusion commonly made is that coating is also not likely to contribute signiicantly to the internal thermal noise. The above analysis shows that this conclusion is not justiied and that there might be a signiicant contribution o the coating to the internal thermal noise, despite the act that Q s are not signiicantly changed. FIG. 1. Identical deects A and B create luctuating stress in dierent parts o the test mass. The stress created by deect A will inluence the phase shit o the laser beam readout more than the stress created by deect B, although both A and B make identical contributions to the Q s o the test-mass elastic modes. o the test mass. By conservation o momentum, the part o the test mass which is lighter will respond more to the random stress than the other part; thereore deect A will have a larger eect on the optical readout than B. Note that i the deects A and B are positioned symmetrically with respect to the center o the test mass, they will have the same eect on the Q s o all elastic modes we assume or simplicity that only one-dimensional longitudinal modes are present and all o them are either symmetric or antisymmetric with respect to the center. Thereore, the normal-mode decomposition applied to the test mass with just one deect A or B would give the same result or the thermal noise as read by the laser. Clearly, we have ound yet another illustration o the breakdown o the normal-mode decomposition. The considerations presented above lead to the ollowing advice or real experiments: Keep the neighborhood o the laser beam spot as clean o deects as possible. Not only does our direct application o the luctuationdissipation theorem have broader validity than the normalmode decomposition; it is also computationally simpler. In the case o homogeneous structural damping it yields a simple analytical expression or the internal thermal noise spectrum c. Eq. 15 : S x 4k BT 3 E r 1 O r 22 V. DISCUSSION AND CONCLUSION The normal-mode decomposition o the thermal noise is exact when the deects are distributed homogeneously through the volume o the test mass. However, as was shown explicitly in Sec. IV or the case o surace losses, when the deect distribution is not homogeneous, the normal-mode decomposition may be misleading, and a direct application o the luctuation-dissipation theorem is required. Thermal noise is ultimately linked to riction in the test mass; this riction is caused by various structural and otherwise deects. Those deects which are closer to the beam spot will contribute more to the thermal noise that is read out by the laser-beam phase shit. Although this act is a direct consequence o the ormalism developed in this paper, we would like to give an intuitive example in order to emphasize this point. Consider, or the sake o simplicity, a one-dimensional elastic test mass with two identical deects A and B, as shown on Fig. 1; A is closer to the beam spot than B. Each o these deects creates a random stress which pushes apart or pulls together the let and right relative to the deect parts This result is consistent with the numerical sum over modes done in e. 3 and is accurate when the radius o the laser beam is small relative to the size o the test mass, i.e., in the regime when the sum over modes converges especially slowly. When r / is not small, a numerical solution o the elasticity equations to deduce the dissipation power W diss, and thence the thermal noise 1, is straightorward and is probably also much simpler than perorming a sum over modes. ACKNOWLEDGMENTS This work would not have been possible without discussions and help rom Vladimir Braginsky, on Drever, Darrell Harrington, Nergis Mavalvala, Fred aab, Glenn Soberman, and Kip Thorne. In particular, Glenn Soberman suggested the method o integration in Eq. A4, and Kip Thorne careully reviewed the manuscript and made a ew signiicant corrections and suggestions. This work was supported in part by NSF Grant No. PHY

5 57 INTENAL THEMAL NOISE IN THE LIGO TEST APPENDIX: THE STAIN ENEGY IN A TEST MASS SUBJECTED TO A GAUSSIAN DISTIBUTED SUFACE PESSUE 1 G r,r, E r r A3 The objective o this appendix is to derive Eq. 14 o Sec. III or the energy o elastic strain in a cylindrical test mass when the pressure P(r ) F (r ) is applied to one o its circular aces. As was discussed in Sec. III, we can assume that the pressure is constant in time since LIGO s detection requencies are much lower than the lowest normalmode requency. For a circular laser beam with a Gaussian intensity proile (r ) is given by c. Eq r 2 e r2 2 /r, A1 r where we assume that the center o the light spot coincides with the center o the test-mass circular ace. I the radius o the laser beam r is small compared to the size o the test mass, we can approximate the test mass by an ininite elastic hal-space. Then our calculation o the elastic energy is correct up to a ractional accuracy o O(r /), where is the characteristic size o the test mass. Let y(r ) be the normal displacement o the surace at location r under the action o the pressure P(r ). In the linear approximation o small strains, y r G r,r P r d 2 r, A2 where G(r,r ) is a Green s unction. The calculation o G is a nontrivial albeit standard exercise in elasticity theory 1, which gives where is the Poisson ratio and E the Young s modulus o the material. The elastic energy stored in the material is U max 1 2 P r y r d 2 r F E r 2 E P r P r d 2 rd 2 r r r e r 2 r 2 2 /r r 2 r 2 2rr cos d2 rd 2 r, A4 where is the angle between r and r. The integral in the last term o Eq. A4 as was pointed out by Glenn Sobermann can be taken by introducing polar coordinates and : r cos, r sin. One then integrates out the radial part o the integrand and expands the remaining angular part in a power series with respect to cos ; termwise integration o this power series inally yields Eq. 14 up to a ractional error o O(r /) U max F 2 2 I, A5 E r where I 3/2 4n 1!! 4 1 n 1 2n!4 n 22. A6 2n It can be shown that i, instead o an ininite hal-space, we consider a inite cylindrical test mass, the leading ractional correction to the elastic energy is o the order O(r /). 1 A. Abramovici et al., Science 256, ; C. Baradaschia et al., Nucl. Instrum. Methods Phys. es. A 289, P.. Saulson, Phys. ev. D 42, A. Gillespie and F. aab, Phys. ev. D 52, G. I. Gonzalez and P.. Saulson, J. Acoust. Soc. Am. 96, H. B. Callen and T. A. Welton, Phys. ev. 83, N. Nakagawa et al., ev. Sci. Instrum. 68, F. Bondu and J. Y. Vinet, Phys. Lett. A 198, A. Gillespie and F. aab, Phys. Lett. A 178, V. B. Braginsky et al., Phys. Lett. A 218, , and reerences therein; private communication. 1 Equation 8.19 o L. D. Landau and E. M. Lishitz, Theory o Elasticity Pergamon, New York, 1986.

arxiv:gr-qc/ v1 4 Jul 1997

arxiv:gr-qc/ v1 4 Jul 1997 Internal thermal noise in the LIGO test masses : a direct approach. Yu. Levin Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125 (February 7, 2008) arxiv:gr-qc/9707013v1