Thermoelastic effects at low temperatures and quantum limits in displacement measurements

9 Pages·0.21 MB·

by  M. Cerdonio

Thermoelastic effects at low temperatures and
quantum limits in displacement measurements
∗
M. Cerdonio and L. Conti
INFN Section of Padova and Department of Physics, University of Padova, Italy
A. Heidmann and M. Pinard
Laboratoire Kastler Brossel†, 4 place Jussieu, F75252 Paris, France
(September29, 2000)
materialswellinuseformirrorssubstratesasfusedsilica
The displacement fluctuations of mirrors in optomechan-
and sapphire, with km long Fabry-Perot cavities, which
ical devices, induced via thermal expansion by temperature
are characterized by laser beam spots of size r 1 cm
0
fluctuations due either to thermodynamic fluctuations or to ≃
1 and comparatively low finesse 100, and with char-
0 fluctuations in the photon absorption, can be made smaller acteristic frequencies f 100FH≃z, for the mechanical
0 thanquantumfluctuations,atthelowtemperatures,highre- ≃
motion to be monitored optically.
2 flectivities and high light powers needed to readout displace-
There are a number of situations, at variance with
ments at the standard quantum limit. The result is relevant
n the above, which are of interest for optomechanical de-
for the design of quantum limited gravitational-wave detec-
a
vices. Insuchsituationsoneorbothofthethermoelastic
J tors, both”interferometers” and”bars”, andforexperiments
to studydirectly mechanical motion in thequantumregime. fluctuations effects may be of concern, when one would
0
liketoreach,inthemeasurementsofsmalldisplacement,
1
PACS : 04.80.Nn, 05.40.-a,42.50.Lc the so called Standard Quantum Limit (SQL) [5,6]. Al-
2 readyforLIGOII,BGVseemstodiscourage,infavourof
v fusedsilica,theuseofsapphire,whichonthe otherhand
4 may be the material considered for cold mirrors in con-
0 I. INTRODUCTION nection with advanced configurations of interferometric
1
gravitational-wave detectors, under study as LCGT [7],
9
LIGOIII [4] and EURO [8].
0 InarecentpaperBraginskyet al[1],henceforthBGV,
TheBGVeffectswouldbeofconcernforverysensitive
0 consideredthenoiseininterferometricgravitational-wave
0 displacement sensors based on high-finesse Fabry-Perot
detectorsduetothermoelasticfluctuationsofthemirrors
/ cavities, to be used in connection with bar detectors of
c attached to the test masses of the interferometer. These
gravitational waves as dual cavity transducers [9], or to
q thermoelastic fluctuations have contributions from two
- study the quantum effects of radiation pressure [10–12].
independent processes, both acting via the thermal ex-
r In both cases the cavities are much shorter (less than
g pansivity of the mirror substrate material. The first one
v: is the thermodynamic fluctuations in temperature of the aomfeewterc,enthtiembeetaerm) tshpaontsinareasgmraavllietart(iornal-w1a0v−e2icnmte)r,fefir--
Xi body of the mirror substrate (these, in the approxima- nesses much larger ( > 105) and tem0p≃eratures as low
tion of small thermal expansion, are independent from as T < 1 K. It mayFap∼pear from BGV results that the
r thethermodynamicfluctuationsinvolume,whicharere-
a therm∼oelasticeffectswouldgenerateparticularlylargeef-
sponsible for the well studied thermal or brownian noise
fects, inasmuch the volume involved in the fluctuation
[2]). The second one is the photothermal temperature
processes would be correspondingly smaller.
fluctuations due to the fact that the number of photons
Forthesereasons,itisofinteresttoexplorewhatwould
absorbed by the mirror fluctuate.
be the behaviourofboth thermoelastic effects in the low
BGV results for the thermodynamic noise, obtained
temperatures and small beam spot regimes, where some
for half-infinite mirrors, have been extended to the case
BGV assumptions break down. In particular, the heat
of finite size mirrors [3], with particular reference to the
diffusion length l depends on the temperature and can
designofadvancedinterferometricgravitational-wavede- t
become largerthan the laserbeam spotdimensionr , so
tectors,suchasLIGOII[4]. Inbothcasesthecalculations 0
that the adiabatic approximation is no longer valid.
areconcernedwithmirrorsatroomtemperature,madeof
In Section II we give the essentials of the regime of
phonons and heat propagation, which establishes at low
enoughtemperatures,and we evaluate the thermoelastic
noises with a simple calculation, in relation to the beam
∗visiting at Ecole Normale Sup´erieure, Laboratoire Kastler spot size and to the frequencies at which the optome-
Brossel chanical device is most sensitive.
†Laboratoire de l’Universit´e Pierre et Marie Curie et de
In Sections III and IV we give an exact calculation of
l’Ecole Normale Sup´erieure associ´e au Centre National de la
boththermoelasticeffectsinthewholeregionofinterest,
RechercheScientifique
1
thatisforanyvalueoftheratiol /r . Theresults,under thickness z , and on the other hand much smaller than
t 0 c
theassumptionsoflowtemperatureregimeofSectionII, the beam spot radius r ,
0
would directly apply to actual mirrors for the quoted
optomechanical devices. We also relate in a general way zc <lt <r0. (3)
thephotothermalnoisetothedisplacementnoiseinduced
This is the basic BGV approximation, which gives that
on the mirror by the quantum fluctuations of radiation
thevolumeinvolvedinthefluctuatingthermalexpansion
pressure in the cavity.
effectisthefractionofmirrorsubstrateV l r2andthus
In Section V we discuss limitations and relevance of ≃ t 0
one has z l in Eq. (1).
our approach in the design of the SQL optomechanical ≃ t
This argument reproduces the essential features of
displacement devices.
BGV spectral density S [f] of photothermal displace-
z
ment noise, as one may write around the frequency f,
II. HEAT PROPAGATION AT LOW
∆z2 α 2 S
TEMPERATURES abs
S [f] , (4)
z ≃ f ≃ ρCr2 f2
(cid:18) 0(cid:19)
Let us assume the optomechanical device to work in
where S =h¯ω W is the spectral power noise of the
abs 0 abs
somefrequencyrangecenteredaroundafrequencyf and
absorbed light, with W = h¯ω n the average absorbed
abs 0
let us discuss the photothermal effect. We revisit the
light power. In fact we see that Eq. (4) is the same
calculation of BGV in the following way, so to use it to
finalBGVrelation(Eq.8 of[1]), apartfromaterm with
see the regime which sets up at low temperatures.
the Poisson ratio of the mirror material and numerical
The multilayers coating of the mirror absorbs a small
factors.
fraction of the light power and this induces an inhomo-
Thecondition(3)maybreakdownforsmallbeamspot
geneous increase of the temperature of the bulk. The
radius r or for low temperature T,as the thermal diffu-
0
absorbedpoweris a Poissondistributed randomvariable
sion length gets longer, either in the mirror substrate or
(the statistics of the absorbed photon will be discussed
in the mirror coating or in both.
inmoredetailsinsectionIV),andthesefluctuationslead
For mirrors substrates of crystalline materials, as
to thermal fluctuations in the bulk of the mirror. They
specifically sapphire, for a frequency f 1 kHz, the
are consequently responsible for fluctuations of the posi- ≃
thermal length l in the substrate at low temperature
ts
tion of the reflecting face of the mirror, via the thermal
gets of the order of 10 cm, to be compared with a room
expansion of the mirror material. temperature value l (300 K) 10−2 cm (see Table I).
ts
The r.m.s. displacement noise of the mirror end face ≃
Then at low temperature we rather have
∆z = zα∆T is found by evaluating the r.m.s. fluctua-
tion in temperature ∆T, in a volume V of the mirror of l >r , (5)
ts 0
thicknessz,linearthermalexpansioncoefficientα(T)and ∼
specific thermal capacity C(T), as the absorbed photon at all frequencies below some 1 kHz, both for mirrors
flux n fluctuates, of gravitational-wave interferometers (for which r0
≃
1.5 cm), and for optomechanical sensors (for which r
0
∆z =zα¯hω0∆n, (1) 310−2cm). Thisvalueforlts staysconstantinthewho≤le
ρCV region T 10 K, as crystalline materials follow Debye
T3 laws fo≤r α(T), C(T) and κ(T), and thus their ratios
where ¯hω is the energy per photon, ∆n = n/f is the are all independent of T.
0
r.m.s. poissonian fluctuation of the number of photons High reflection coatings are typically 40 layers one
p
absorbed over the time 1/f (n is the average absorbed quarter wavelength thick of alternating amorphous ma-
photon flux), and ρ is the density of the mirror material terials as TiO and SiO , with a total thickness z
2 2 c
(axis z is taken normal to the plane face of the mirror). 10−3 cm for Nd-Yag laser light. For such a coatin≃g,
At room temperature and for large beam spots, BGV the breakdown temperature for Eq. (3) is different for
conditions apply: the phonon mean free path and relax- mirrors of large-scale interferometers and for mirrors of
ation times are very small respectively in comparison to high-finesse cavities, because of the difference in r . For
0
the mirror coating thickness (where the photons create LIGOII mirrors for instance, taking SiO as the refer-
2
the phonons in the absorption process), and in compari- ence material for the coating, the thermal length l in
tc
son with the characteristic time 1/f. The thermal diffu- the coating is of the order of r only at very low tem-
0
sion length at frequency f is given by perature, T < 1 K. Amorphous silica films would have
l > 10−2 cm for T < 10 K, so that for high-finesse
tc
κ cavities we have l > r in the whole low temperature
l = , (2) tc 0
t ρCf region. ∼
r
Despite this difference, there are two features of rel-
where κ is the thermal conductivity. For a frequency f evance, which affect similarly the thermal behaviour of
around100Hz,lt isononehandlargerthanthecoating thecoating-substratecompositeinbothtypesofmirrors.
2
Accordingto table I, α z is at leastone orderof magni-
c c
Fused silica 300 K 10 K 1 K tudesmallerthanα l atlowtemperature(z 10−5m
α (K−1) 5.5 10−7 −2.6 10−7 −2.6 10−10 and l 0.1 m). Wsetscan then neglect the excp≃ansion of
ts
κ (W/m.K) 1.4 0.1 2 10−2 the coa≃ting over its thickness z and we find that the
c
C (J/Kg.K) 6.7 102 3 3 10−3 effect is dominated by the substrate properties,
λ (m) 8 10−10 8 10−8 9 10−6
α/κ (m/W) 3.9 10−7 2.6 10−6 1.3 10−8 S [f] αs 2 Sabs. (7)
lt (m) 310−5 1.2 10−4 1.7 10−3 z ≃(cid:18)ρsCslt2s(cid:19) f2
Thisistherelevantresultofourdiscussionofthermalbe-
haviour of the coating-substrate composite at low tem-
Sapphire 300 K 10 K 1 K
perature, in that now the temperature fluctuations in-
α (K−1) 5 10−6 5.8 10−10 5.8 10−13
volve comparatively large substrate volumes, instead of
κ (W/m.K) 40 4.3 103 4.3 the comparatively small coating volume, where the ac-
C (J/Kg.K) 7.9 102 8.9 10−2 8.9 10−5 tual absorption of photons occurs. Notice that, would
λ (m) 5 10−9 2.2 10−3 not this be the case, one would have of course very large
α/κ (m/W) 1.2 10−7 1.4 10−13 effects just concentratedinthe volume, the externalsur-
lt (m) 1.1 10−4 0.11 faceofwhichisthatwheredisplacementsaregoingtobe
measured at SQL sensitivities.
When we substitute in Eq. (7) the expression (2) for
TABLEI. Thermalpropertiesoffusedsilica(top)andsap-
the thermal length, we see a dramatic change of regime,
phire(bottom)atdifferenttemperatures. Thethermalexpan-
sion coefficient α, thermal conductivity κ, thermal capacity
α 2
C andphononmeanfreepathλarederivedfrom [1]atroom s
S [f] S , (8)
z abs
temperature,andfrom[13–15]atlowtemperatures. Thether- ≃(cid:18)κs(cid:19)
mal length lt at 1 kHz is obtained from Eq. (2).
where now the (substrate) thermal conductivity κ ap-
s
pears,insteadofthethermalcapacity,andthefrequency
In both cases we have that, at all low temperatures, the dependence has disappeared. In fact the system behaves
thermal length ltc stays longer than the coating thick- as in the low frequency region of a low pass filter, while,
ness, ltc > zc, and that the mean free path λs of the under BGV conditions, it was rather in the high fre-
phonons in the substrate is itself long, atleast a fraction quency region.
of cm [13]. In a coating of a SiO2 film even the phonon We develop in the following sections a rigorous cal-
mean free path λc will be largerthan 10−3 cm, andthus culation of the effects in the low temperature regime,
λc >zc, for T <1 K [16]. which gives in clear details the features grossly antici-
Let us then∼consider how the thermal regime changes patedaboveandwhichcanbedirectlyappliedtomirrors
at sufficiently low temperatures (T 10 K). The heat of interest for optomechanical devices, when the above
≤
delivered by the absorbed photons in the volume r02zc thermal behaviour of the coating-substrate system is re-
of the coating crosses to the substrate in a time smaller alized.
than1/f,asl >z . Fromthesubstrate,asλ >r ,the
tc c s 0
thermalphononstherebycreatedwillreenterthe∼coating,
heatingitupinevenshortertimeoverdistancesλ . This III. THERMODYNAMIC NOISE
s
happens because the acoustic mismatchbetweencoating
and substrate is small [17], when densities and sound
In this section we determine the thermodynamic noise
velocities are quite close. The substrate thus acts as a
without any assumption on the ratio l /r between the
t 0
thermal short for the coating in the plane of the mirror
thermal diffusion length in the substrate and the beam
endface: thecoatingandthesubstratewillbepractically
spot size. Our analysis is an extension of the procedure
isothermaloverdistancesoftheorderofthephononmean
developedbyLiuandThorne[3],butitisvalidevenwhen
free path λ in the substrate. Then the coating will con-
s the adiabatic approximation is not satisfied. According
tributetothethermoelasticfluctuationswithitsthermal
to Eq. (2), the condition l < r can actually be written
t 0
expansion coefficient αc, but following the thermal fluc- as a condition over the frequency, f > κ/ρCr2. Our
0
tuations of spectral density S of the substrate. On the
T treatmentisthusalsovalidforanangularfrequencyω =
other hand, at the frequency f, the volume of substrate
2πf smaller than the adiabatic limit ω defined as
c
involved in the fluctuating heating will be of the order
V l3 , where the thermal length is that in the sub- κ
stra≃te.tsSo,including both the coating andthe substrate, ωc = ρCr2. (9)
0
wewritenowforthedisplacementspectraldensityS [f]:
z
As shown in the previous section we neglect any ef-
S [f] (α z )2+(α l )2 S [f]. (6) fectsofthecoating,takingonlyintoaccountthethermal
z c c s ts T
≃
(cid:16) (cid:17)
3
propertiesofthesubstrateofthemirror. Wealsoneglect To calculate the rate of energy dissipation W , it
diss
any finite-size effects since we have shown that the vol- is necessary to solve a system of two coupled equations,
ume of substrate involved in the fluctuating heating is thefirstoneforthe displacementu(r,t)ateverypointr
usually smaller than the size of the mirror, even at low inside the substrate, and the second one for the tem-
temperature. We thus approximate the mirror as a half perature perturbation δT (r,t). As the time required
space, the coated plane face corresponding to the plane for sound to travel across the mirror is usually smaller
z =0 in cylindrical coordinates. thantheoscillationperiod2π/ω,wecanuseaquasistatic
The analysis is based on a general formulation of the approximation and deduce the displacement u from the
fluctuation-dissipation theorem, used by Levin [18] to equation of static stress balance [19],
compute the usual thermal noise (brownian motion) of
( .u)+(1 2σ) 2u= 2α(1+σ) δT, (14)
the mirrors in a gravitational-wave interferometer. We
∇ ∇ − ∇ − ∇
knowthatinaninterferometerorinahighfinesseFabry-
where σ is the Poisson ratio of the substrate (α is the
Perotcavity,thelightissensitivetothe normaldisplace-
linear thermal expansion coefficient). The temperature
mentu (z =0,r,t)ofthecoatedplanefaceofthemirror,
z perturbation δT evolves according to the thermal con-
spatially averaged over the beam profile. This averaged
ductivity equation [19],
displacement u is defined as
∂(δT) αET ∂( .u)
e−r2/r02 a2∆(δT)= − ∇ , (15)
u(t)=b d2r u (z =0,r,t) . (10) ∂t − ρC(1 2σ) ∂t
z πr2 −
Z 0
where a2 = κ/ρC and E is the Young modulus of the
To compbute the spectral density S [ω] of the displace- substrate (ρ is the density, C is the specific thermal ca-
u
mentuatagivenangularfrequencyω,wedeterminethe pacity).
mechanicalresponseofthe mirrorto a sinusoidallyoscil- The solutions of Eqs. (14) and (15) must also fulfill
b
latingbpressure. More precisely, we examine the effect of boundary conditions. If we approximate the mirror as
a pressure P (r,t) applied at every point r of the coated a half space, the temperature perturbation δT and the
face of the mirror with the same spatial profile as the stresstensorσ mustsatisfythefollowingboundarycon-
ij
optical beam, ditions on the coated plane face of the mirror,
P (r,t)= F0 e−r2/r02cos(ωt), (11) σzz(z =0,r,t) =P(r,t), (16a)
πr02 σzx(z =0,r,t) =σzy(z =0,r,t)=0, (16b)
where F0 is a constant force amplitude. We can com- ∂(δT)(z =0,r,t) =0. (16c)
pute the energy W dissipated by the mirror in re- ∂z
diss
sponse to this force, averaged over a period 2π/ω of the
The stress tensor σ is defined in presence of changes of
ij
pressureoscillations.Thefluctuation-dissipationtheorem
temperature as (see Eq. (6.2) of [19])
then states that the spectral density of the displacement
noise is given by E
σ = αδTδ +
ij ij
−(1 2σ)
8kBT Wdiss −
S [ω]= , (12)
u ω2 F2 E σ
0 + u + δ u , (17)
ij ij kk
(1+σ) (1 2σ)
" #
wherekB isthe Bboltzman’sconstant. This approachhas − Xk
been used by Levin to compute the brownian noise [18].
where the strain tensor u is equal to 1 ∂ui + ∂uj .
We are interested here in the thermodynamic noise so ij 2 ∂xj ∂xi
that W correspondsto the energydissipated by ther- Wesolveperturbativelythissystemofe(cid:16)quationsat(cid:17)the
diss
moelastic heat flow. first order in α. We first solve the static stress-balance
The rate of thermoelastic dissipation is given by the equation at the zeroth order in α, neglecting the tem-
followingexpression(firsttermofEq. (35.1)ofRef.[19]): perature term in the right part of Eq. (14) and in the
expression (17) of the stress tensor. The solution u(0) of
dS κ this equation is well known (paragraph 8 of [19]). We
W = T = d3r ( δT)2 , (13)
diss dt T ∇ then solve the thermal conductivity equation (15) using
(cid:28) (cid:29) (cid:28)Z (cid:29) asasourcetermthesolutionu(0) andweobtainthetem-
where the integral is on the entire volume of the mir- perature perturbation δT(1) in the first order in α. The
ror and the brackets ... stand for an average over the calculationofu(0),δT(1) andfinallyW isdoneinAp-
h i diss
oscillation period 2π/ω. δT is the temperature pertur- pendix A. Using the results of this appendix, we show
bation around the unperturbed value T, induced by the that S [ω] is equal to
u
oscillating pressure. W is then related to the time
diss
derivative dS/dt of the mirror’s entropy, which depends k T2
on the temperature gradient δT. b Su[ω]=32α2(1+σ)2 BρC I, (18)
∇
b
4
where the integral I is given by Ω = 1 is a cut-off frequency. For Ω > 1 the curve has
a slope equal to 2, whereas for Ω < 1 the curve has a
a2 k⊥2e−k⊥2r02/2 smaller slope of t−he orderof 1/2. In this low frequency
I = (2π)3 dkxdkydkzk2(a4k4+ω2), (19) range, the noise is smaller −than the one which would
Z
be obtained using the adiabatic approximation (dashed
with k2 =k2+k2 and k2 =k2 +k2. curve in figure 1).
⊥ x y ⊥ z
WecanexpressS [ω]asafunctionofanintegralJ[Ω]
u
which depends only on a dimensionless variable Ω equal
to ω/ω , where ω = a2/r2 corresponds to the adiabatic IV. PHOTOTHERMAL NOISE
c c 0
b
limit (see Eq. 9). We get
We now briefly examine the case of the photothermal
8 k T2r
S [ω]= α2(1+σ)2 B 0J[Ω], (20) noise which exhibits somewhat a similar frequency be-
u √2π ρCa2 haviour as the thermodynamic noise. We use the same
method as BGV [1] to calculate the spectral density
where J[Ω] is derived from the integral I by the trans-
b S [ω] due to this noise but we do not make any adia-
formation of variables u≡k⊥r0 and v ≡kzr0, bautic approximation so that the calculation is valid also
2 ∞ ∞ u3e−u2/2 fobr frequencies smaller than the adiabatic limit ωc. We
J[Ω]= du dv . then obtain:
rπ Z0 Z−∞ (u2+v2) (u2+v2)2+Ω2 2 ¯hω W
S [ω]= α2(1+σ)2 0 absK[Ω], (24)
(cid:16) (2(cid:17)1) u π2 (ρCa2)2
Whenω ω (i.e. Ω 1),wecanneglectu2+v2with where thbe integral K[Ω] is equal to
c
≫ ≫
respecttoΩinthedenominatoroftheintegral. J[Ω]can
then be calculated analytically and we obtain 1 ∞ ∞ u2e−u2/2 2
K[Ω]= du dv .
J[Ω 1]=1/Ω2. (22) (cid:12)(cid:12)π Z0 Z−∞ (u2+v2)(u2+v2+iΩ)(cid:12)(cid:12)
≫ (cid:12) (cid:12)
(cid:12) (cid:12)(25)
Using this result and the definition of Ω, we finally show (cid:12) (cid:12)
that S [ω] is equal to When ω ω the adiabatic approximation is valid and
u ≫ c
the result of BGV should be recovered. Indeed, when
Sub[ω ≫ωc]= √82πα2(1+σ)2 kBρCT2ωa22r3. (23) Ωcalc≫ula1t,ewaencaalyntinceaglllyecKt u[Ω2]+wvh2ichwittuhrnrsesopuetcttotobeΩeqaunadl
0
to 1/Ω2. Using the definition of Ω, we finally show that
Thisfobrmulaisidenticaltotheexpression(18)ofRef.[3] S [ω] is equal to
u
and to the expression (12) of BGV [1].
2 ¯hω W
S [ω ω ]= α2(1+σ)2 0 abs . (26)
b u ≫ c π2 (ρCr2ω)2
0
This forbmula is identical to Eq. (8) of BGV [1].
For low values of Ω, K[Ω] can be calculated numeri-
cally. The result is shown in figure 2. As in the case of
the thermodynamic effect (figure 1), Ω = 1 is a cut-off
frequency: for Ω > 1 the function has a slope equal to
2, whereas for Ω < 1 the function has a much smaller
−
slope and is almost constant.
This result is in perfect agreement with the simple
derivation made in section II. The spectral density (24)
can actually be written as
FIG. 1. Frequency dependence of the thermodynamic S [ω]= 2 (1+σ)2 α 2S K[Ω], (27)
noise. The frequency ω is normalized to the adiabatic limit u π2 κ abs
ωc. The dashed curve corresponds to the adiabatic approxi- (cid:16) (cid:17)
mation. which is similar to Eq. (8) at low frequency where
b
K[Ω] 1,apartfromatermwiththePoissonratio. The
≃
For small values of Ω, the integral J[Ω] can be com- frequency dependence of the photothermal noise then
puted numerically. The result is shown in figure 1 where corresponds to a low-pass filter, with a cut-off frequency
we have plotted J[Ω] as a function of Ω in logarithmic equal to the adiabatic limit ω . At low frequency, that
c
scale,forΩbetween10−5and105. Thisfigureshowsthat is when the thermal diffusion length l in the substrate
t
5
Ontheotherhand,theabsorbedphotonsalwayscorre-
spondstoapoissonianstatistics,evenifitisnotthecase
for the intracavity photons. The spectral power noise
S of the absorbed light is given by
abs
S =h¯ω W =A¯hω W , (30)
abs 0 abs 0 cav
where A is the absorption coefficient of the mirror (the
averagefluxofabsorbedphotonsisn=AN).Thiseffect
cannotbe understood within the frameworkof a corpus-
cular model in which the photon absorption is described
as a poissonian process: due to the super-poissonian
statistics of the intracavity photons, one would find
a super-poissonian statistics for the absorbed photons.
FIG.2. Frequency dependence of the photothermal noise.
One has to take into account the interferences between
Thefrequencyω isnormalizedtotheadiabaticlimit ωc. The
the intracavity field and the vacuum fluctuations asso-
dashed curvecorresponds to theadiabatic approximation.
ciated with the mirror losses. This can be done by us-
ing a simple model where the absorption is described as
becomes larger than the beam spot radius r , one has a a small transmission of the mirror and where the ab-
0
dramatic change of regime and the photothermal noise sorbedphotonsareidentifiedtothe photonstransmitted
is much smaller than the one which would be obtained by the mirror. One thus has a high-finesse cavity with
within the adiabaticapproximation(dashedcurvein fig- twoinput-outputportsanditiswellknownthatthepho-
ure 2). ton statistics of the light either reflected or transmitted
Anotherimportantpointfortherealizationofoptome- by such a cavity are always poissonian, for coherent or
chanicalsensorsworkingatthe quantumlevelis to com- vacuum incoming beams [21].
pare the photothermal noise to the displacements in- Eqs. (29) and (30) clearly show that both the radia-
duced by the quantum fluctuations of radiation pressure tion pressure effect and the photothermal noise are pro-
of light. Quantum effects can be made larger than the portional to the intracavity light power Wcav; however
usual thermal noise by decreasing the temperature and the displacements induced by radiationpressure havean
by increasing the light power. This would not be conve- extradependenceonthecavityfinesse . Thephotother-
F
nient for photothermal noise since the effect is propor- mal noise can thus become negligible as compared to
tional to the light power. BGV results show that for quantum effects for a high-finesse cavity.
sapphire at room temperature the photothermal noise is Toperformaquantitativecomparisonbetweenthetwo
of the same order as the standard quantum limit in an effects, we calculate the susceptibility χeff defined by
interferometer. In a high-finesse cavity, hopefully, these Eq. (28). We determine here the mechanical response
twoeffectsarerelatedtoquitedifferentphotonstatistics. associated with the internal degrees of freedom of the
As a matter of fact, the displacement u induced by the mirror, which are of interest for displacement sensors.
radiationpressureoftheintracavityfieldisrelatedtothe We thus ignore the radiation pressure effects associated
intracavity photon flux N, with the global motion of suspended mirrors, which are
b
the dominant contribution to SQL at low frequency in
u[ω]=χ [ω]P [ω]=2h¯kχ [ω]N[ω], (28) gravitational-waveinterferometers. We calculate the av-
eff rad eff
erage displacement u induced by the radiation pressure
where 2¯hk is the momentum exchange during a photon P , assuming the mirror is a half space (z 0). The
reflebction(kisthewavevectoroflight),andχeff isanef- norarmd al displacementbuz(z =0,r,t) of the coa≥ted face of
fective susceptibility describing the mechanical response the mirrorcan be deduced from the results of paragraph
of the mirror to the radiation pressure Prad [12]. The 8 of [19],
noise spectrum S [ω] induced by radiation pressure is
thus proportionaluto the spectralpowernoiseScav ofthe u (z =0,r,t)= 2h¯kN(t) 1 σ2 d2r′e−r′2/r02. (31)
intracavity light, which for a resonant cavity is [12]: z Eπ2r2 − r r′
b 0 Z | − |
(cid:0) (cid:1)
2 /π Using the definition (10) of u, we obtain
S [ω]= F ¯hω W , (29)
cav 1+(ω/ωcav)2 0 cav u(t)= 2h¯kN(t) 1 σ2 bd2rd2r′e−(r2+r′2)/r02. (32)
nwehsesreeaωncdavWis th=e ch¯aωviNty ibsatnhdewaidvtehra,gFe iinstrtahceavciatvyitlyighfit- Eπ3r04 (cid:0) − (cid:1)Z |r−r′|
power. At lcoawv frequ0ency (ω < ωcav) the intracavity Tofbhveairniatebglersaluca=nreasirl′yabnedcval=culra+tedr′.byWuesifinngalalyngewetset
photon flux corresponds to a su∼per-poissonian statistics, −
thenoisepowerbeinglargerthanthepoissonianspectral 1 σ2
χ [ω]= − , (33)
density by a factor 2 /π [20]. eff √2πEr
F 0
6
andthe noisespectrumS [ω]inducedbyradiationpres- (second table). The results show that ω is increased
u c
sure fluctuations is equal to when the temperature decreases (3 orders of magnitude
for fused silica and 6 orders of magnitude for sapphire
b 2
2 1 σ2 when the temperature is reduced from 300 K to 1 K).
Su[ω]= √(cid:0)2π−Ecr0(cid:1)! Scav[ω], (34) aIfdwiaebactoincsaipdperroaxitmypaitcioanl firsenqeuveenrcvyalωid/2foπrosafp1p0h0irHeazt, ltohwe
temperature, whereas it is valid for fused silica only for
where c is tbhe speed of light.
large beam spot size.
For all the displacement sensors consideredin this pa-
perthecharacteristicangularfrequencyωissmallerthan
the cavitybandwidth ωcav. The noise spectrum Su[ω] is r0=10−2 m Fused silica Sapphire
consequently independent of ω and equal to 300 K 1 K 300 K 1 K
2 b ωc/2π (Hz) 1.5 10−3 4.8 2 10−2 1.9 104
S [ω ω ]= 2 1−σ2 2F¯hω W . (35) Ω2J[Ω] 1 0.74 0.98 1.3 10−4
u ≪ cav √(cid:0)2πEcr0(cid:1)! π 0 cav S (m2/Hz) 2.7 10−42 3.4 10−45 1.5 10−39 2.6 10−49
u
S (adiabatic) 2.7 10−42 4.6 10−45 1.5 10−39 2 10−45
Thisbexpression shows that the radiation pressure effect u
depends on the mechanical characteristics of the sub- b
strate (E and σ) whereas the photothermal noise (Eq. rb0=10−4 m Fused silica Sapphire
27)dependsonthethermodynamiccharacteristicsofthe
300 K 1 K 300 K 1 K
substrateviatheratioα/κ. Atlowtemperature,K[Ω]is
ωc/2π (Hz) 15 4.8 104 2 102 1.9 108
of the order of1 and the ratio α/κ is constantand equal
to 1.4 10−13 m/W for sapphire (see Table I). E is equal Ω2J[Ω] 0.51 3.5 10−5 6.4 10−2 1.4 10−10
to 4 1011 J/m3 and σ is equal to 0.25 so that the ratio S (m2/Hz) 1.4 10−36 1.6 10−43 1 10−34 2.8 10−49
u
between the photothermal and radiation pressure noises S (adiabatic) 2.7 10−36 4.6 10−39 1.6 10−33 2 10−39
u
is of the order of
b
TABLEII. Results for fused silica and sapphire at differ-
Ar2 b
Spt/Srad 2.5 1014 0. (36) ent temperatures, for a frequency ω/2π = 100 Hz and for a
u u ≃ beam spot size r0 =1 cm (top) and 10−2 cm (bottom). The
F rd
thermodynamic noise S (3 lines) is reduced as compared
sFiozre ar01opfpm10−a4bbsmor,pbtainodn raatceav(iAty=fin1e0s−se6),Faobfe1a0m5,spthoet ltioneitss)vbayluaefoabcttoarinΩed2Jw[iΩtuh]i(n2nthdeliandeisa)b.aticapproximation(4th
photothermal noise is more than 4 orders of magnitude b
smallerthanthe radiationpressureeffectsofinternalde-
We first focus on the thermodynamic noise whose val-
grees of freedom of the mirror. The photothermal noise
ues calculatedfromEq. (20)areshowninthe thirdlines
is thus negligible as compared to quantum effects in op-
of tables II. The noise is smaller than the one which
tomechanical sensors.
would be obtained within the adiabatic approximation
Note thatthis is notthe case ingravitational-wavein-
terferometerswherer 10−2mand 100. Thepho- (last lines in tables II). The reduction factor, equal to
0 ≃ F ≃ 1/Ω2J[Ω], can be as large as 104 for sapphire at low
tothermalnoiseisthen2ordersofmagnitudelargerthan
temperature with r = 1 cm, and as large as 1010 for
thequantumnoiseofinternalmotion.However,theinter- 0
r =10−2 cm (second lines in tables II).
ferometerisnotexpectedtobesensitivetothisquantum 0
We immediately see the impact for gravitational-wave
noise since for suspended mirrors it is overwhelmed by
interferometers (r = 1 cm): for sapphire at low tem-
the quantum noise associated with external pendulum 0
perature, the thermodynamic noise is more than 4 or-
motion.
der of magnitude smaller than for fused silica, so that
the choice of the material at low temperature would be
just the opposite than, as in BGV, at room tempera-
V. DISCUSSION AND CONCLUSION
ture. Furthermore, the thermodynamic noise at 100 Hz
would be equal to 2.6 10−49 m2/Hz for sapphire at low
Wehaveshownthatboththermoelasticandphotother-
temperature, well below the noise at room temperature
malnoises havea frequency dependence whichlooks like (2.7 10−42 m2/Hz for fused silica). It is also well be-
a low-pass filter: below a cut-off frequency ω , these
c lowthe SQLlimitdue tothe externalpendulummotion,
noises are much smaller than the noise which would equalto 3.610−41 m2/Hz for amirrormassof30kg [1].
be obtained according to the 1/ω2 dependence at high-
Similarly for optomechanical systems with smaller
frequency. beam spot size (r < 10−2 cm), the thermodynamic
0
First lines in tables II give the values of the cut-off noisecanbemadeas∼smallas2.810−49 m2/Hz byusing
frequency ω /2π for fused silica and sapphire, and for
c sapphire at low temperature, to be compared to a noise
a beam spot size r0 of 1 cm (first table) and 10−2 cm larger than 10−36 m2/Hz at room temperature both for
7
sapphire and fused silica. It is worth noticing that this fies the details of the thermal link of the mirror to the
verylow value is partly due to the reductionfactor asso- main heat sink. Similar care should be taken if the ther-
ciated with the non adiabaticity which is of the order of mal length at the lowest frequency of interest is of the
1010. This noise can be compared to the SQL limit due orderofthe mirrorsize. Possiblyinboth casesthe effect
to the internal motion of the mirror, which is equal to would be smaller than predicted in this paper, because
¯h χ 10−42 m2/Hz [22]. At low temperature, the the characteristic times for thermal equilibrium in the
eff
| | ≃
thermodynamic noise is thus smaller than the SQL limit mirror volume would get even shorter. In any case, as
so that optomechanical sensors as in Refs. [9,10] would to quote just one instance, phonon mean free paths and
be able to get to the SQL limit. thermal conductivities (and thus thermal lengths), have
Let us note that the thermodynamic noise for sap- strong dependencies at low temperatures on the level of
phire at 1 K is mostly independent of the beam spot impurities, so that each experimental configuration may
size r : similar values are obtained for large spot sizes be a case per se.
0
(2.6 10−49 m2/Hz for r = 1 cm) and small ones In conclusion our results may be possibly of interest
0
(2.8 10−49 m2/Hz for r = 10−2 cm). This result is in two respects. First because they let see promising
0
due to the fact that, in contrast with fused silica, the the use of low temperatures both for gravitational-wave
adiabatic approximation is not valid for sapphire what- interferometers and for optomechanical devices. Second
ever the beam spot size is, as long as it is smaller than a because they may be of help to study, in actual experi-
fewcentimeters. Thenonadiabaticconditionω <ω can mentalconfigurationsforSQL conditions,the roleofthe
c
actually be written as r < κ/ρCω (see Eq.9). In this thermoelastic effects in respect to choices of substrate
0
non adiabatic regime, we have shown that J[Ω] evolves materials,workingtemperatures, cavityfinesses, mirrors
p
as 1/√Ω which is proportionalto √ω and then to 1/r . losses, beam spot sizes and laser powers.
c 0
Thethermodynamicnoiseisproportionaltor J[Ω](Eq.
0
20) and is then independent of r .
0
ACKNOWLEDGEMENTS
Similar results can be obtained for the photothermal
noise. We have shown that as long as the adiabatic con-
dition is not satisfied, the noise mostly depends on the After the completion of this work, we learned that re-
ratio α/κ (Eqs. 8 or 27). In particular it does not de- sults for the photothermal effect similar to those of sec-
pend on the frequency ω nor on the beam spot size r0. tion IV were obtained with a different method [23], that
For sapphire at low temperature and for an average ab- the problem of section III has been addressed heuristi-
sorbed power Wabs 1 W of Nd-Yag laser light, the cally with conclusions in part similar [24], and that a
photothermal noise is≃then of the order of 10−45 m2/Hz calculation of thermoelastic effects due to photon ab-
bothforinterferometersandoptomechanicalsensors,well sorption in the bulk of cryogenic crystallin cavities [15]
below the SQL limits of both the external and internal was also having similar features. We are very gratefulto
motions. VladimirBraginskyandSergeyVyatchaninfortheir pri-
Let us finally note that the results obtained above ap- vatecommunicationandtoSheilaRowanandtoStephan
ply in detail to an actual mirror system when the condi- Schiller for drawing attention to their unpublished re-
tionsdescribedinsectionIIarefulfilled. Inparticularthe sults. M. Pinard is grateful to INFN Legnaro National
interplay of the various characteristic lengths (phonon Laboratories for hospitality during a short visit within
meanfreepathandthermallengths atthe frequenciesof the E.U. programme ”TMR - Access to LSF” (Contract
interest, both in the substrate and in the coating, beam ERBFMGECT980110).
spot, coating thickness and mirror size) must in the end
allow,insometemperaturerange,thatthethermalprop-
erties of the substrate dominate. APPENDIX A
This may be noteasy to achieve andthus our analysis
may correspond to a somewhat idealized situation. At In this appendix, we calculate the spectral density
the lowest temperatures T 0.5 K, the phonon mean S [ω] (Eq. 18) of the spatially averaged displacement
free path in the coating get≤s of the order of its thick- uuinduced by the thermodynamic noise. We approxi-
ness (as for an amorphoussilica coating,see for example mate the mirror as an infinite half space (z 0). At the
[16]), while the time constants for phonon local equilib- zebroth order in the thermal expansion coeffi≥cient α, the
b
rium continue to stay smaller than 2π/ω both for the solution u(0) of the quasistatic stress balance equation
coating and the substrate. At first sight, it may appear (14) is given by a Green’s tensor (Eqs. (8.13) and (8.18)
thatthe equalizationofcoatingversussubstratetemper- of [19]). The pressure P(r,t) applied on the coated sur-
atures will be even more facilitated. However at some faceofthemirrorhasonlyacomponentalongthenormal
intermediate temperature below 10 K the phonon mean axisz,andweobtainthefollowingexpressionforthedis-
free path in substrates like sapphire may get so long to placement expansion Θ(0) = .u(0),
exceed the dimensions of the mirror. In this case a more ∇
ad hoc model has to be considered, in which one speci- 2(1+σ)(1 2σ)
Θ(0)(r,t) = − F cos(ωt)
0
− E ×
8
dkxdkye−41k⊥2r02−k⊥z+i(kxx+kyy), (37) S [ω]=32α2(1+σ)2 kBT2I. (45)
× (2π)2 u ρC
Z
with k⊥ = kx2+ky2. b
To calculqate the dissipated energy Wdiss it is useful to
analytically extend the pressure-induced expansion Θ(0)
for negative values of z in such a way that it is an even
function of z. Its spatial Fourier transform Θ(0)[k,t] is
then equal to
[1] V.B. Braginsky, M.L. Gorodetsky and S.P. Vyatchanin,
Θ(0)[k,t]=−4(1+σ)E(1−2σ)F0cos(ωt)kk⊥2e−41k⊥2r02, [2] SPeheysf.orLeitnt.stAan2ce64P,.R1.(1S9a9u9l)son, Phys. Rev. D 42, 2437
(1990)
(38)
[3] Y.T.LiuandK.S.Thorne,Phys. Rev. D,inpress(2000)
[4] E.Gustafsonetal,LSCwhitepaperondetectorresearch
with k2 =k2+k2+k2.
x y z and development,LIGO T990080-00-D (1999)
In the same way, we analytically extend the temper-
[5] C.M. Caves, Phys. Rev. D 23, 1693 (1981)
ature perturbation δT in the half space z 0 in such
a way that δT(1)(r,t) is an even function ≤of z. Using [6] V.B. Braginsky and F.Ya. Khalili, Quantum measure-
ment, edited by K.S. Thorne (Cambridge University
the Fourier transform of the thermal conductivity equa-
Press, 1992)
tion (15) and the expression (38) of Θ(0), we find that [7] K.Kurodaet al,Int. Journ. Mod. Phys. D8,537(1999)
δT(1)[k,t] is equal to
[8] A. Giazotto, privatecommunication
[9] L. Conti et al, Rev. Sci. Instrum. 69, 554 (1998)
δT(1)[k,t]=A[k]eiωt+c.c., (39)
[10] P.F. Cohadon, A. Heidmann and M. Pinard, Phys. Rev.
where the function A[k] is given by Lett. 83, 3174 (1999)
[11] M. Pinard, Y. Hadjar and A. Heidmann, Eur. Phys. J.
A[k]= 2(1+ρCσ)αT k2(ai2ωkk2⊥+iω)F0e−14k⊥2r02. (40) [12] AD.7H,e1i0d7m(a1n9n9,9)Y. Hadjar and M. Pinard, Appl. Phys. B
64, 173 (1997)
WenowdeterminethethermoelasticdissipationWdiss. [13] R.Berman,E.L.FosterandJ.M.Ziman,Proc. Roy.Soc.
The integral over z in Eq. (13) is limited to the volume A 231, 130 (1955)
ofthemirror(infinitehalfspacez 0). SinceδT(1)(r,t) [14] R.C. Zeller and R.O.Pohl, Phys. Rev. B4, 2029 (1971)
≥
is an even function of z, W can be written as [15] R. Liu, S. Schiller and R.L. Byer, Stanford University,
diss
internal report (1993) and references therein
κ 2
W = d3r δT(1) , (41) [16] P.D. Vu, Xiao Liu and R.O. Pohl, arXiv:cond-
diss
2T ∇ mat/0002413
(cid:28)Z (cid:16) (cid:17) (cid:29) [17] W.A. Little, Can. Journ. Phys. 37, 334 (1959)
where the spatial integration is in the whole space. Us-
[18] Yu.Levin, Phys. Rev. D 57, 659 (1998)
ingtheBessel-Persevalrelation,wecanexpressthedissi-
[19] L.D. Landau and E.M. Lifshitz, Theory of Elasticity,
patedenergyW asafunctionofthetemporalaverage
diss third edition (Pergamon, Oxford,1986)
of δT(1)[k,t] 2 which is equal to 2 A[k]2(Eq. 39): [20] M.C. Teich and B.E.A. Saleh, in Progress in Optics
| |
XXVI, edited by E. Wolf (North-Holland, Amsterdam,
(cid:12) (cid:12) κ d3k 2
(cid:12) W (cid:12)= k2 δT(1)[k,t] 1988)
diss 2T (2π)3 [21] S. Reynaud, A. Heidmann, E. Giacobino and C. Fabre,
κ Z d3k (cid:28)(cid:12)(cid:12) (cid:12)(cid:12) (cid:29) in Progress in Optics XXX, edited by E. Wolf (North-
= k2 A[(cid:12)k]2. (cid:12) (42) Holland, Amsterdam, 1992)
T (2π)3 | |
Z [22] M.T. Jaekel and S. Reynaud, Europhys. Lett. 13, 301
Using Eq. (40), we finally obtain (1990)
[23] V.B. Braginsky and S.P. Vyatchanin, private communi-
Wdiss = 4Tα2(1+σ)2ω2I, (43) cation
F2 ρC [24] S.Rowan,M.M.Fejer,E.Gustafson,R.Route,J.Hough,
0
G. Cagnoli and P. Sneddon,Aspen 2000 WinterConfer-
where the integral I is given by
enceonGravitationalWavesandtheirDetection(unpub-
I = d3k a2k⊥2 e−k⊥2r02/2. (44) lished)
(2π)3k2(a4k4+ω2)
Z
This expression allows to determine the spectral den-
sity S [ω] of the displacement u from the fluctuation-
u
dissipation theorem (Eq. 12). One gets the result given
in the text by Eq. (18):
b
b
9