- Email: [email protected]
Coupling of mirror tilts with earth gravitational field in long-baseline interferometric gravitational-wave detectors
17 April 2000
Physics Letters A 268 Ž2000. 235–240 www.elsevier.nlrlocaterphysleta
Coupling of mirror tilts with earth gravitational field in long-baseline interferometric gravitational-wave detectors E. Calloni a,b,) , L. Di Fiore b, G. Di Sciascio b, L. Milano a,b, L. Rosa b,c,1, C. Stornaiolo b a
Dipartimento di Scienze Fisiche, UniÕersita` di Napoli ‘Federico II’, Complesso UniÕersitario Monte S. Angelo, Õia Cinthia, I-80126, Napoli, Italy b Istituto Nazionale Fisica Nucleare, sez. Naples, Complesso UniÕersitario Monte S. Angelo, Õia Cinthia, I-80126, Naples, Italy c Institut fur ¨ Theoretische Physik UniÕersitat ¨ Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany Received 28 July 1999; received in revised form 9 March 2000; accepted 16 March 2000 Communicated by P.R. Holland
Abstract We derive the cavity optical axis length fluctuations induced by tilts of the cavity mirrors in presence of the gravitational field of the earth. By comparison with the typical tilt-induced noises we show that this potential source of noise is negligible for the first operation gravitational wave interferometers. The possibility of utilizing the effect to perform a measurement of the post-Newtonian prameter g in improved-operation interferometers is discussed. q 2000 Elsevier Science B.V. All rights reserved. PACS: 04.80.Nn; 04.80.-y Keywords: Interferometry; Gravitational Waves Detectors; Experimental Gravity
1. Introduction Within recent years great attention has been dedicated to the study and realization of long-baseline Fabry–Perot Michelson interferometers as gravitational wave detectors w1–3x. In these interferometers, in order to enhance the sensitivity to the gravitational waves, each arm of the Michelson interferometer is )
Corresponding author. Fax: q39-81-676346. E-mail address: [email protected] ŽE. Calloni.. 1 On leave from Dipartimento di Scienze Fisiche, Universita` di Napoli ‘Federico II’.
composed of a Fabry–Perot cavity, operating in reflection. Due to the extreme weakness of the expected signals a careful analysis has been made on all possible noise sources that can overcome the gravitational signal by perturbing the physical or optical cavity length; on the other hand this particular sensitivity has also suggested to investigate if there is the possibility of using these powerful instruments in other fields of physics too w4x. A particular noise source that has been long studied concerns the coupling of angular tilts of the cavity mirrors with interferometer’s imperfections: non-perfect centering of the optical axis on the two
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 2 1 4 - 0
E. Calloni et al.r Physics Letters A 268 (2000) 235–240
236
mirror rotation centers, input laser geometry fluctuations, interferometer asymmetries w5–8x. In this Letter we analyze the effect of the coupling of mirror tilts with gravitational earth field firstly as a potential source of noise for gravitational wave interferometers: by a simple comparison with other greater noises associated to mirror tilts Žwhich however do not limit the interferometer sensitivity. we show that in the first operation detectors this noise will be negligible. Then, by pointing out that the effect is proportional to the square of the cavity length, we pay attention to long-baseline interferometers to discuss the possibility, to be expected in alignment-improved interferometers, to experimentally detect and measure this relativistic effect. In particular we focus our attention on the possibility to measure the adimensional post-Newtonian parameter ŽPNP. g , which ‘parametrizes’ the gravitational deflection of light. Its present empirical value is w9,10x:
g s 1.000 " 0.001 .
In order to close the experimental ‘gap’ many interesting experiments have been proposed. Among them two major projects are of particular interest: Gravity Probe-B, a relativity gyroscope experiment, which is expected to improve the accuracy in g to 6 = 10y5 w12x and the European Space Agency GAIA project, which is now in an advanced state of planning. It consists of an astrometric interferometer at 10 microarcsec accuracy level, to cover 15 = 10 6 objects in the whole celestial plane w13x. This would allow Žas one of the scientific payoff. an accuracy in g between 10y6 –10y7 w14x. In this Letter we show that when considering first generation gravitational wave interferometers with an ad-hoc improved alignment operation or second generation interferometers, others tilts induced effect can be suppressed and the g parameter can be measured, in principle, with an accuracy level of interest for testing alternarive theories of gravitation and comparable with sensitivities expected in astronomical observations.
Ž 1. 2. Effects of tilts
Its theoretical value is unity in general relativity, while others metric theories of gravitation preview slightly different values. The Brans–Dicke theory, for example, which is nowadays one of the most valid ‘competitors’ of general relativity, contains, besides the metric tensor, a scalar field f and an arbitrary coupling constant v which is related to the parameter g by the relationship:
gs
1qv 2qv
.
Ž 2.
Consider a Fabry–Perot cavity of length L having the input mirror flat Žlike VIRGO. and the far mirror convex of radius R Ž R ) L., as shown in Fig. 1. In our case we are interested in ‘ vertical’ tilts, so that the angles u 1 and u 2 are the mirrors tilts, around the horizontal axis, from the ideal position, corresponding to the case of cavity axis lying on line joining the mirror centers of rotation. As pointed out by various authors w5,6x the geometrical effects on the cavity are a first order displacement, d1 and d 2 , of the optical
In addition, many scalar-tensor theories have been proposed, generalizing the original Brans–Dicke concepts. All these theories, which estimate the value of the scalar field from cosmological considerations, preview a discrepancy of g from unity of the order of: <1 y g < f 10y7 –10y5 ,
Ž 3.
depending on the particular metric theory adopted for the cosmological calculations w11x.
Fig. 1. Geometry of half-symmetric Fabry–Perot cavity consisting of a flat ŽM1. and a concave ŽM2. mirror; in the figure the angle u 1 is positive, while the angle u 2 is negative.
E. Calloni et al.r Physics Letters A 268 (2000) 235–240
axis on the mirror 1 and 2, respectively, and a second order variation of the length l of the cavity given by: d1 s 2 au 1 y 2 bu 2 ,
Ž 4.
d2 s 2 b Ž u 2 y u 1 . ,
Ž 5.
l s L q au 12 q bu 22 y 2 bu 1 u 2 ,
Ž 6.
where, for the half-symmetric cavity geometry illustrated:
Let’s suppose that the mirror 2 rotates by an amount u 2 . The geometrical effect on the cavity is a parallel displacement of the optical axis by the quantity d y s Ž R q a2 .u 2 , while the geometrical length l s L of the cavity remains unchanged to the first order. Then the total time t Ž u 2 ., to the first order, becomes:
tŽ u2 . s
L
H0 L
a s 12 Ž R q a2 y L . ,
Ž 7.
237
s c
1 q Ž 1 q g . U Ž y . dx
ž
1 q Ž 1 q g . Ž U Ž 0. q
Ey
dy
/
.
Ž 12 .
and:
b s 12 Ž R q a2 . .
Ž 8.
In order to evaluate the effect of the coupling of tilts with the earth field let’s suppose that initially the cavity is perfectly aligned: the optical length in presence of the gravitational earth field can be easily calculated using the post-Newtonian approximation w9,15x: 2
2
2
i
k
ds s Ž 1 y 2U . c dt y Ž 1 q 2g U . d i k dx dx s 0 , Ž 9. where U is the gravitational potential and we pay attention only to the g parameter. The total time t 0 spent by light to cover the Fabry–Perot cavity length, considering that U < 1, can be calculated, to the lower order, as:
In this way the time difference dt 2 s t Ž u 2 . y t 0 between the tilted and unperturbed optical axes can be written as:
dt 2 s y
L Ž 1 q g . gd y c
c2
s yL Ž 1 q g .
L
H0 L
s c
1 q Ž 1 q g . U Ž 0 . dx
1 q Ž 1 q g . U Ž 0. ,
Ž 10 .
which is equivalent to the known result that the velocity of light in a weak gravitational field in the post-Newtonian approximation Žretaining only the g depending terms. depends on the gravitational potential by the equation w15x: Õc s c 1 y Ž 1 q g . U Ž 0 . .
Ž 11 .
g c3
Ž R q a2 . u 2 ,
Ž 13 .
where g s y E U Ž y . E y is the gravitational acceleration; we observe then that there is a first order effect of the tilts, equivalent to a geometrical variation of the optical length dx 2 s cdt 2 equal to: dx 2 s yL Ž 1 q g .
t0 s
E UŽ y.
g c2
Ž R q a2 . u 2 s yD 2 u 2 . Ž 14 .
In case of Virgo Ž L s 3 km, R s 3.45 km. the equivalent ‘arm’ D 2 which translates the rotation into ‘length’ variation is D 2 s 2 = 10y9 meters. We notice that, R being proportional to L, the effect is generally proportional to the square of the cavity length. In the same way it can be calculated the effect of a rotation u 1 , of the mirror 1. In this case the optical axis moves by an amount d1 s Ž R q a 2 y L.u 1 on the first mirror and by the amount d2 s yŽ R q a2 .u 1 on the second mirror, and rotates by the angle
E. Calloni et al.r Physics Letters A 268 (2000) 235–240
238
u 1 with respect to the unperturbed one. The time t 1 can then be calculated by considering that to the first order the light path is the straight line y s yu 1 x; retaining only the term g of the PNP approximation we obtain: ds 2 s Ž 1 y 2U . c 2 dt 2 y Ž 1 q 2g . U Ž y Ž x . . Ž 1 q u 12 . dx 2s0 .
Ž 15 . Expanding as previously the gravitational potential ŽUŽ y Ž x .. s UŽ0. q gu 1 x . on the light path and integrating we obtain: ct 1 s L q Ž 1 q g . U Ž 0 . L y 12 L2
Ž1qg . g c2
u1 .
Ž 16 . So that the equivalent difference between the unperturbed and tilted cavity length becomes: dx 1 s 12 L2
Ž1qg . g c
2
u 1 s D1 u 1 .
Ž 17 .
In this case the equivalent Virgo ‘arm’ D 1 which translates the rotation into ‘length’ variation is D1 s 10y9 meters and, as we have seen for D 2 , depends on L2 ; this dependence makes it of particular interest to analyze the effect considering the long-baseline interferometers.
3. Comparison with typical mirror tilt noise in Fabry–Perot interferometers We observe that the effect of coupling of mirror tilts with the gravitational earth field is a first order effect while, as it is known, the length variation of a Fabry–Perot axis when mirrors rotate is a second order effect. This implies that for a sufficient alignment precision the gravitational coupling can overcome the other effects; in principle it could generate noise or, in the other hand, the coupling effect could be put in evidence and measure the g parameter. Due to the fact that the gravitational equivalent arm, in Virgo interferometer is of the order of 10y9 meters and considering that the sensitivity of the interferometer will be of the order of S s 10y1 9 mr 'Hz in a wide range of frequencies Žfrom
10 to 1 kHz. we see that in principle a small high-frequency sinusoidal tilt of amplitude u 1,2 f 10y1 0 rad could be detected and studied in a very short time Ž1 s.. Nevertheless we must consider that in reality Žat least for the first generation interferometers. the alignment of the cavity will not be perfect, and the real mechanical-geometrical arm, will probably be much greater than D s 10y9 meters. This can be seen considering that although Eq. Ž1. applies to any Fabry–Perot cavity, the application to gravitational wave interferometers permits an intuitive simplification: gravitational wave interferometers will monitor l Ž t . in the range above 10 Hz while typical spectrum of mirror rotations is dominated by static or slowly varying components, at frequency well below 1 Hz. In these conditions, in the range of interest for the gravitational wave detection, the Fourier transform of Eq. Ž1. is simply w6,7x: l˜Ž f . f d1ˆ e 1 Ž f . q d 2ˆ e 2 Ž f . ,
Ž 18 .
where d1ˆ and d 2ˆ can be approximated with the root mean square of long-term fluctuations of the arm lengths d1 , d 2 defined by Eqs. Ž2. and Ž3. this equation allows a direct comparison of the noise due to coupling of small, high-frequency mirror rotations with the gravitational field or with the slowly varying mirror angle position. Indeed the estimated arm due to long term fluctuation is of the order of d1ˆ f d 2ˆ f 10y5 meters, while the gravitational coupling arm is D f 10y9 meters; thus the coupling of mirror tilts with slowly varying angle position will be the predominant effect. It is interesting to note that Žat least in Virgo. this value will be due to the absence of an error signal of the position of the beam with respect to the perfect alignment situation; indeed the fluctuation of optical axis will be reduced to few micron with a suitable feed-back by aligning the optical axis with the input beam extracting the error signal with the Anderson technique w3,16x Žin order to prevent coupling with laser jitter w5,8x. without performing a continuous feed-back control with respect to the mirror centers of rotations, this situation being sufficient for the first generation interferometers. Nevertheless for further operations it can be interesting to note that if a suitable error signal was
E. Calloni et al.r Physics Letters A 268 (2000) 235–240
Fig. 2. Optical scheme of a recycled Michelson interferometer folded with Fabry–Perot cavities: RM s Recycling Mirror; BS s Beam Splitter; M1a ŽM1b. s input mirror of the cavity of the Michelson arm a Žb.; M2a Žm2b. s far mirror of the arm a Žb..
present the mechanical arms d1ˆ and d 2ˆ could be reduced. This would allow us both to lower the noise due to the coupling of the mirror angle fluctuations with off-center position of the optical axis and to measure the coupling with the gravitational earth field. In order to perform a first terrestrial measurement of the g parameter the arm should be reduced at least to d f 10y9 ; in order to improve the nowadays accuracy in the g measurement and to reach the sensitivities nowadays projected by astronomical observations w9,10x it should be reduced at the level better than d f 10y1 3. This is not impossible in principle: as an example let’s remind that the long baseline interferometers recycle the power by using the scheme briefly sketched in Fig. 2: the length of the equivalent cavity constituted by the recycling mirror and the Michelson interferometer is monitored and maintained on resonance by a suitable feed-back system. If a small high-frequency sinusoidal modulation of tilts e 1 is added, and there is a low frequency misalignment d1ˆ , the length l r of the recycling cavity will be affected by this geometrical coupling by the term l r s d1ˆ P e 1 , while the effect of the tilts-earth field coupling will be negligible, due to the short length of the cavity itself. Thus at the modulation frequency of e the signal will be useful for a suitable feed-back to reduce the arm d1ˆ . Considering that for the first generation interferometer the sensitivity to recycling cavity length fluctuations is estimated of the order of S f 10y1 8 mr 'Hz w17x and that the bandwidth of the feed-back can be of about 1 Hz, with a modulation amplitude of e s 10y7 rad a reduction up to d1ˆ s 10y1 1 meter can be possible. Thus we see that with
239
the first generation parameters Ževen if not in the first operation interferometers. there is the possibility of performing the first terrestrial measurement of g , with the still poor interesting accuracy of Dg gf 10y2 . In second-generation interferometers, higher sensitivities are expected, of the order of S f 10y1 8 mr 'Hz for the recycling cavity w17x, while the feed back bandwidth should be far less than unity for angular degrees of freedom. In these conditions, with a modulation depth of e s 10y6 rad Žwhich still does not lower the interferometer performances. the accuracy of Dg gf 10y5 could be reached, a level of interest for testing alternative theories of gravitation. 4. Results and conclusions The effect of coupling of Fabry–Perot cavity mirror tilts with gravitational earth field has been investigated. It is shown that for first generation gravitational wave interferometers the phase variation induced by this coupling is negligible with respect to other tilts-induced sources of noise. Nevertheless, when considering improved-operation or second generation interferometers, a pointing stability could be envisaged such that other tilts induced noise are suppressed. In this way a terrestrial measurement of the g post Newtonian parameter can be performed with a precision, in principle, of interest for testing some of the most accreditated alternative theories of gravitation. Acknowledgements We are particularly grateful to A. Brillet of Observatoire de la cote d’Azur, Nice, and P. Tourrenc of Universite` Pierre et Marie Curie, Paris, for suggestions and useful discussions, and to S. Verdoliva for careful reading the manuscript. References w1x D.G. Blair, The Detection of gravitational waves, Cambridge Universtity Press, Cambridge 1991. w2x A. Abramovici et al., Science 256 Ž1992. 325. w3x A. Brillet, A. Giazotto et al., The VIRGO Project, Final
240
w4x w5x w6x w7x w8x w9x w10x
E. Calloni et al.r Physics Letters A 268 (2000) 235–240 Conceptual Design of the French–Italian Interferometric Antenna for Gravitational Wave Detection, unpublished, 1992. V.N. Rudenko, Phys. Lett. A 223 Ž1996. 412. A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, H. Billing, ¨ K. Maischberger, Opt. Acta 28 Ž1981. 641. P. Hello, Thesis, Universite´ Paris-Sud, Orsay, 1990. S. Kawamura, M.E. Zucker, Appl. Opt. 33 Ž1994. 3912. F. Barone, E. Calloni, L. Di Fiore, A. Grado, P. Hello, L. Milano, G. Russo, Phys. Lett. A 217 Ž1996. 90. C.M. Will; Theory and Experiment in Gravitational Physics, Cambridge University Press, Cambridge, 1993. L. Iess, G. Giampieri, J.D. Anderson, B. Bertotti, Class. Quantum. Grav. 16 Ž1999. 1487.
w11x T. Damour, K. Nordtvedt, Phys. Rev. D 48 Ž1993. 3436. w12x V.L. Fitch D et al., Review of Gravity Probe-B, National Academy, Washington, DC, 1995. w13x F. de Felice, M.G. Lattanzi, A. Vecchiato, P.L. Bernacca, Astron. Astrophys. 332 Ž1998. 1133. w14x M.A.C. Perryman, L. Lindegreen, C. Touron, in: M.A.C. Perryman, F. van Leewen ŽEds.., The Scientific Goals of the GAIA mission, Proc. ESA Symp. Hypparcos, ESA SP-402 Ž1997.. w15x C. Moller, The Theory of Relativity, Oxford University Press, New Delhi, 1972. w16x D.Z. Anderson Appl. Opt. 23 Ž1984. 2994. w17x M. Barsuglia, private communication, 1999.
Physics Letters A 268 Ž2000. 235–240 www.elsevier.nlrlocaterphysleta
Coupling of mirror tilts with earth gravitational field in long-baseline interferometric gravitational-wave detectors E. Calloni a,b,) , L. Di Fiore b, G. Di Sciascio b, L. Milano a,b, L. Rosa b,c,1, C. Stornaiolo b a
Dipartimento di Scienze Fisiche, UniÕersita` di Napoli ‘Federico II’, Complesso UniÕersitario Monte S. Angelo, Õia Cinthia, I-80126, Napoli, Italy b Istituto Nazionale Fisica Nucleare, sez. Naples, Complesso UniÕersitario Monte S. Angelo, Õia Cinthia, I-80126, Naples, Italy c Institut fur ¨ Theoretische Physik UniÕersitat ¨ Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany Received 28 July 1999; received in revised form 9 March 2000; accepted 16 March 2000 Communicated by P.R. Holland
Abstract We derive the cavity optical axis length fluctuations induced by tilts of the cavity mirrors in presence of the gravitational field of the earth. By comparison with the typical tilt-induced noises we show that this potential source of noise is negligible for the first operation gravitational wave interferometers. The possibility of utilizing the effect to perform a measurement of the post-Newtonian prameter g in improved-operation interferometers is discussed. q 2000 Elsevier Science B.V. All rights reserved. PACS: 04.80.Nn; 04.80.-y Keywords: Interferometry; Gravitational Waves Detectors; Experimental Gravity
1. Introduction Within recent years great attention has been dedicated to the study and realization of long-baseline Fabry–Perot Michelson interferometers as gravitational wave detectors w1–3x. In these interferometers, in order to enhance the sensitivity to the gravitational waves, each arm of the Michelson interferometer is )
Corresponding author. Fax: q39-81-676346. E-mail address: [email protected] ŽE. Calloni.. 1 On leave from Dipartimento di Scienze Fisiche, Universita` di Napoli ‘Federico II’.
composed of a Fabry–Perot cavity, operating in reflection. Due to the extreme weakness of the expected signals a careful analysis has been made on all possible noise sources that can overcome the gravitational signal by perturbing the physical or optical cavity length; on the other hand this particular sensitivity has also suggested to investigate if there is the possibility of using these powerful instruments in other fields of physics too w4x. A particular noise source that has been long studied concerns the coupling of angular tilts of the cavity mirrors with interferometer’s imperfections: non-perfect centering of the optical axis on the two
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 2 1 4 - 0
E. Calloni et al.r Physics Letters A 268 (2000) 235–240
236
mirror rotation centers, input laser geometry fluctuations, interferometer asymmetries w5–8x. In this Letter we analyze the effect of the coupling of mirror tilts with gravitational earth field firstly as a potential source of noise for gravitational wave interferometers: by a simple comparison with other greater noises associated to mirror tilts Žwhich however do not limit the interferometer sensitivity. we show that in the first operation detectors this noise will be negligible. Then, by pointing out that the effect is proportional to the square of the cavity length, we pay attention to long-baseline interferometers to discuss the possibility, to be expected in alignment-improved interferometers, to experimentally detect and measure this relativistic effect. In particular we focus our attention on the possibility to measure the adimensional post-Newtonian parameter ŽPNP. g , which ‘parametrizes’ the gravitational deflection of light. Its present empirical value is w9,10x:
g s 1.000 " 0.001 .
In order to close the experimental ‘gap’ many interesting experiments have been proposed. Among them two major projects are of particular interest: Gravity Probe-B, a relativity gyroscope experiment, which is expected to improve the accuracy in g to 6 = 10y5 w12x and the European Space Agency GAIA project, which is now in an advanced state of planning. It consists of an astrometric interferometer at 10 microarcsec accuracy level, to cover 15 = 10 6 objects in the whole celestial plane w13x. This would allow Žas one of the scientific payoff. an accuracy in g between 10y6 –10y7 w14x. In this Letter we show that when considering first generation gravitational wave interferometers with an ad-hoc improved alignment operation or second generation interferometers, others tilts induced effect can be suppressed and the g parameter can be measured, in principle, with an accuracy level of interest for testing alternarive theories of gravitation and comparable with sensitivities expected in astronomical observations.
Ž 1. 2. Effects of tilts
Its theoretical value is unity in general relativity, while others metric theories of gravitation preview slightly different values. The Brans–Dicke theory, for example, which is nowadays one of the most valid ‘competitors’ of general relativity, contains, besides the metric tensor, a scalar field f and an arbitrary coupling constant v which is related to the parameter g by the relationship:
gs
1qv 2qv
.
Ž 2.
Consider a Fabry–Perot cavity of length L having the input mirror flat Žlike VIRGO. and the far mirror convex of radius R Ž R ) L., as shown in Fig. 1. In our case we are interested in ‘ vertical’ tilts, so that the angles u 1 and u 2 are the mirrors tilts, around the horizontal axis, from the ideal position, corresponding to the case of cavity axis lying on line joining the mirror centers of rotation. As pointed out by various authors w5,6x the geometrical effects on the cavity are a first order displacement, d1 and d 2 , of the optical
In addition, many scalar-tensor theories have been proposed, generalizing the original Brans–Dicke concepts. All these theories, which estimate the value of the scalar field from cosmological considerations, preview a discrepancy of g from unity of the order of: <1 y g < f 10y7 –10y5 ,
Ž 3.
depending on the particular metric theory adopted for the cosmological calculations w11x.
Fig. 1. Geometry of half-symmetric Fabry–Perot cavity consisting of a flat ŽM1. and a concave ŽM2. mirror; in the figure the angle u 1 is positive, while the angle u 2 is negative.
E. Calloni et al.r Physics Letters A 268 (2000) 235–240
axis on the mirror 1 and 2, respectively, and a second order variation of the length l of the cavity given by: d1 s 2 au 1 y 2 bu 2 ,
Ž 4.
d2 s 2 b Ž u 2 y u 1 . ,
Ž 5.
l s L q au 12 q bu 22 y 2 bu 1 u 2 ,
Ž 6.
where, for the half-symmetric cavity geometry illustrated:
Let’s suppose that the mirror 2 rotates by an amount u 2 . The geometrical effect on the cavity is a parallel displacement of the optical axis by the quantity d y s Ž R q a2 .u 2 , while the geometrical length l s L of the cavity remains unchanged to the first order. Then the total time t Ž u 2 ., to the first order, becomes:
tŽ u2 . s
L
H0 L
a s 12 Ž R q a2 y L . ,
Ž 7.
237
s c
1 q Ž 1 q g . U Ž y . dx
ž
1 q Ž 1 q g . Ž U Ž 0. q
Ey
dy
/
.
Ž 12 .
and:
b s 12 Ž R q a2 . .
Ž 8.
In order to evaluate the effect of the coupling of tilts with the earth field let’s suppose that initially the cavity is perfectly aligned: the optical length in presence of the gravitational earth field can be easily calculated using the post-Newtonian approximation w9,15x: 2
2
2
i
k
ds s Ž 1 y 2U . c dt y Ž 1 q 2g U . d i k dx dx s 0 , Ž 9. where U is the gravitational potential and we pay attention only to the g parameter. The total time t 0 spent by light to cover the Fabry–Perot cavity length, considering that U < 1, can be calculated, to the lower order, as:
In this way the time difference dt 2 s t Ž u 2 . y t 0 between the tilted and unperturbed optical axes can be written as:
dt 2 s y
L Ž 1 q g . gd y c
c2
s yL Ž 1 q g .
L
H0 L
s c
1 q Ž 1 q g . U Ž 0 . dx
1 q Ž 1 q g . U Ž 0. ,
Ž 10 .
which is equivalent to the known result that the velocity of light in a weak gravitational field in the post-Newtonian approximation Žretaining only the g depending terms. depends on the gravitational potential by the equation w15x: Õc s c 1 y Ž 1 q g . U Ž 0 . .
Ž 11 .
g c3
Ž R q a2 . u 2 ,
Ž 13 .
where g s y E U Ž y . E y is the gravitational acceleration; we observe then that there is a first order effect of the tilts, equivalent to a geometrical variation of the optical length dx 2 s cdt 2 equal to: dx 2 s yL Ž 1 q g .
t0 s
E UŽ y.
g c2
Ž R q a2 . u 2 s yD 2 u 2 . Ž 14 .
In case of Virgo Ž L s 3 km, R s 3.45 km. the equivalent ‘arm’ D 2 which translates the rotation into ‘length’ variation is D 2 s 2 = 10y9 meters. We notice that, R being proportional to L, the effect is generally proportional to the square of the cavity length. In the same way it can be calculated the effect of a rotation u 1 , of the mirror 1. In this case the optical axis moves by an amount d1 s Ž R q a 2 y L.u 1 on the first mirror and by the amount d2 s yŽ R q a2 .u 1 on the second mirror, and rotates by the angle
E. Calloni et al.r Physics Letters A 268 (2000) 235–240
238
u 1 with respect to the unperturbed one. The time t 1 can then be calculated by considering that to the first order the light path is the straight line y s yu 1 x; retaining only the term g of the PNP approximation we obtain: ds 2 s Ž 1 y 2U . c 2 dt 2 y Ž 1 q 2g . U Ž y Ž x . . Ž 1 q u 12 . dx 2s0 .
Ž 15 . Expanding as previously the gravitational potential ŽUŽ y Ž x .. s UŽ0. q gu 1 x . on the light path and integrating we obtain: ct 1 s L q Ž 1 q g . U Ž 0 . L y 12 L2
Ž1qg . g c2
u1 .
Ž 16 . So that the equivalent difference between the unperturbed and tilted cavity length becomes: dx 1 s 12 L2
Ž1qg . g c
2
u 1 s D1 u 1 .
Ž 17 .
In this case the equivalent Virgo ‘arm’ D 1 which translates the rotation into ‘length’ variation is D1 s 10y9 meters and, as we have seen for D 2 , depends on L2 ; this dependence makes it of particular interest to analyze the effect considering the long-baseline interferometers.
3. Comparison with typical mirror tilt noise in Fabry–Perot interferometers We observe that the effect of coupling of mirror tilts with the gravitational earth field is a first order effect while, as it is known, the length variation of a Fabry–Perot axis when mirrors rotate is a second order effect. This implies that for a sufficient alignment precision the gravitational coupling can overcome the other effects; in principle it could generate noise or, in the other hand, the coupling effect could be put in evidence and measure the g parameter. Due to the fact that the gravitational equivalent arm, in Virgo interferometer is of the order of 10y9 meters and considering that the sensitivity of the interferometer will be of the order of S s 10y1 9 mr 'Hz in a wide range of frequencies Žfrom
10 to 1 kHz. we see that in principle a small high-frequency sinusoidal tilt of amplitude u 1,2 f 10y1 0 rad could be detected and studied in a very short time Ž1 s.. Nevertheless we must consider that in reality Žat least for the first generation interferometers. the alignment of the cavity will not be perfect, and the real mechanical-geometrical arm, will probably be much greater than D s 10y9 meters. This can be seen considering that although Eq. Ž1. applies to any Fabry–Perot cavity, the application to gravitational wave interferometers permits an intuitive simplification: gravitational wave interferometers will monitor l Ž t . in the range above 10 Hz while typical spectrum of mirror rotations is dominated by static or slowly varying components, at frequency well below 1 Hz. In these conditions, in the range of interest for the gravitational wave detection, the Fourier transform of Eq. Ž1. is simply w6,7x: l˜Ž f . f d1ˆ e 1 Ž f . q d 2ˆ e 2 Ž f . ,
Ž 18 .
where d1ˆ and d 2ˆ can be approximated with the root mean square of long-term fluctuations of the arm lengths d1 , d 2 defined by Eqs. Ž2. and Ž3. this equation allows a direct comparison of the noise due to coupling of small, high-frequency mirror rotations with the gravitational field or with the slowly varying mirror angle position. Indeed the estimated arm due to long term fluctuation is of the order of d1ˆ f d 2ˆ f 10y5 meters, while the gravitational coupling arm is D f 10y9 meters; thus the coupling of mirror tilts with slowly varying angle position will be the predominant effect. It is interesting to note that Žat least in Virgo. this value will be due to the absence of an error signal of the position of the beam with respect to the perfect alignment situation; indeed the fluctuation of optical axis will be reduced to few micron with a suitable feed-back by aligning the optical axis with the input beam extracting the error signal with the Anderson technique w3,16x Žin order to prevent coupling with laser jitter w5,8x. without performing a continuous feed-back control with respect to the mirror centers of rotations, this situation being sufficient for the first generation interferometers. Nevertheless for further operations it can be interesting to note that if a suitable error signal was
E. Calloni et al.r Physics Letters A 268 (2000) 235–240
Fig. 2. Optical scheme of a recycled Michelson interferometer folded with Fabry–Perot cavities: RM s Recycling Mirror; BS s Beam Splitter; M1a ŽM1b. s input mirror of the cavity of the Michelson arm a Žb.; M2a Žm2b. s far mirror of the arm a Žb..
present the mechanical arms d1ˆ and d 2ˆ could be reduced. This would allow us both to lower the noise due to the coupling of the mirror angle fluctuations with off-center position of the optical axis and to measure the coupling with the gravitational earth field. In order to perform a first terrestrial measurement of the g parameter the arm should be reduced at least to d f 10y9 ; in order to improve the nowadays accuracy in the g measurement and to reach the sensitivities nowadays projected by astronomical observations w9,10x it should be reduced at the level better than d f 10y1 3. This is not impossible in principle: as an example let’s remind that the long baseline interferometers recycle the power by using the scheme briefly sketched in Fig. 2: the length of the equivalent cavity constituted by the recycling mirror and the Michelson interferometer is monitored and maintained on resonance by a suitable feed-back system. If a small high-frequency sinusoidal modulation of tilts e 1 is added, and there is a low frequency misalignment d1ˆ , the length l r of the recycling cavity will be affected by this geometrical coupling by the term l r s d1ˆ P e 1 , while the effect of the tilts-earth field coupling will be negligible, due to the short length of the cavity itself. Thus at the modulation frequency of e the signal will be useful for a suitable feed-back to reduce the arm d1ˆ . Considering that for the first generation interferometer the sensitivity to recycling cavity length fluctuations is estimated of the order of S f 10y1 8 mr 'Hz w17x and that the bandwidth of the feed-back can be of about 1 Hz, with a modulation amplitude of e s 10y7 rad a reduction up to d1ˆ s 10y1 1 meter can be possible. Thus we see that with
239
the first generation parameters Ževen if not in the first operation interferometers. there is the possibility of performing the first terrestrial measurement of g , with the still poor interesting accuracy of Dg gf 10y2 . In second-generation interferometers, higher sensitivities are expected, of the order of S f 10y1 8 mr 'Hz for the recycling cavity w17x, while the feed back bandwidth should be far less than unity for angular degrees of freedom. In these conditions, with a modulation depth of e s 10y6 rad Žwhich still does not lower the interferometer performances. the accuracy of Dg gf 10y5 could be reached, a level of interest for testing alternative theories of gravitation. 4. Results and conclusions The effect of coupling of Fabry–Perot cavity mirror tilts with gravitational earth field has been investigated. It is shown that for first generation gravitational wave interferometers the phase variation induced by this coupling is negligible with respect to other tilts-induced sources of noise. Nevertheless, when considering improved-operation or second generation interferometers, a pointing stability could be envisaged such that other tilts induced noise are suppressed. In this way a terrestrial measurement of the g post Newtonian parameter can be performed with a precision, in principle, of interest for testing some of the most accreditated alternative theories of gravitation. Acknowledgements We are particularly grateful to A. Brillet of Observatoire de la cote d’Azur, Nice, and P. Tourrenc of Universite` Pierre et Marie Curie, Paris, for suggestions and useful discussions, and to S. Verdoliva for careful reading the manuscript. References w1x D.G. Blair, The Detection of gravitational waves, Cambridge Universtity Press, Cambridge 1991. w2x A. Abramovici et al., Science 256 Ž1992. 325. w3x A. Brillet, A. Giazotto et al., The VIRGO Project, Final
240
w4x w5x w6x w7x w8x w9x w10x
E. Calloni et al.r Physics Letters A 268 (2000) 235–240 Conceptual Design of the French–Italian Interferometric Antenna for Gravitational Wave Detection, unpublished, 1992. V.N. Rudenko, Phys. Lett. A 223 Ž1996. 412. A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, H. Billing, ¨ K. Maischberger, Opt. Acta 28 Ž1981. 641. P. Hello, Thesis, Universite´ Paris-Sud, Orsay, 1990. S. Kawamura, M.E. Zucker, Appl. Opt. 33 Ž1994. 3912. F. Barone, E. Calloni, L. Di Fiore, A. Grado, P. Hello, L. Milano, G. Russo, Phys. Lett. A 217 Ž1996. 90. C.M. Will; Theory and Experiment in Gravitational Physics, Cambridge University Press, Cambridge, 1993. L. Iess, G. Giampieri, J.D. Anderson, B. Bertotti, Class. Quantum. Grav. 16 Ž1999. 1487.
w11x T. Damour, K. Nordtvedt, Phys. Rev. D 48 Ž1993. 3436. w12x V.L. Fitch D et al., Review of Gravity Probe-B, National Academy, Washington, DC, 1995. w13x F. de Felice, M.G. Lattanzi, A. Vecchiato, P.L. Bernacca, Astron. Astrophys. 332 Ž1998. 1133. w14x M.A.C. Perryman, L. Lindegreen, C. Touron, in: M.A.C. Perryman, F. van Leewen ŽEds.., The Scientific Goals of the GAIA mission, Proc. ESA Symp. Hypparcos, ESA SP-402 Ž1997.. w15x C. Moller, The Theory of Relativity, Oxford University Press, New Delhi, 1972. w16x D.Z. Anderson Appl. Opt. 23 Ž1984. 2994. w17x M. Barsuglia, private communication, 1999.