Suspensions Thermal Noise in LIGO Gravitational Wave Detector

Gabriela GonzalezCenter for Gravitational Physics and Geometry, Department of Physics,

The Pennsylvania State University, 104 Davey Lab, University Park, PA 16802.

We present a calculation of the maximum sensitivity achievable by the LIGO Gravita-tional wave detector in construction, due to limiting thermal noise of its suspensions. Wepresent a method to calculate thermal noise that allows the prediction of the suspensionthermal noise in all its 6 degrees of freedom, from the energy dissipation due to the elastic-ity of the suspension wires. We show how this approach encompasses and explains previousways to approximate the thermal noise limit in gravitational waver detectors. We show howthis approach can be extended to more complicated suspensions to be used in future LIGOdetectors.

I. INTRODUCTION

Thermal noise is a fundamental limit to the sen-sitivity of gravitational wave detectors, such as theones being built in the use by the LIGO project [1].Thermal noise is associated with sources of energydissipation [2], following the Fluctuation-DissipationTheorem. Thermal noise comes in at least two im-portant kinds: one due to the brownian motion ofthe mirrors, associated with the losses in the mir-rors’ material; and another due to the suspensionof the mirrors, due to the losses in the wires’ ma-terial. The limits following from these assumptions(losses due to elastic properties of materials) are alower limit to the noise in the detector, since theremay always be other sources of energy dissipation inimperfect clamps, mirror attachments, etc. But thecorrect calculation of the thermal noise limit is es-sential to the design of detectors and diagnostics ofthe already-built detectors. We will deal in this arti-cle with thermal noise of suspensions (not of internalmodes of the mirrors themselves), and assume onlylosses due to the elasticity of the suspension wires.

The calculation of thermal noise can be done inseveral ways [3], [4], [5], [6], [7]. All of these followthe Fluctuation-Dissipation Theorem (FDT), but acomplication arises because in suspensions there aretwo sources of energy (gravitational and elastic), butonly one of them is “lossy” (elastic energy). More-over, the losses in the suspension wires are associ-ated with their bending, and seems to be localizedat the top and bottom of the wires. The ways toinclude these features into the thermal noise calcu-lations are different enough that they have led tosome confusion among the gravitational wave com-munity. Also, attention has been paid mostly to

the horizontal motion of the suspension, althoughall modes (angular, transverse, and vertical) appearto some degree into the detector’s noise. We presenta method to calculate thermal noise that allows theprediction of the suspension thermal noise in all its6 degrees of freedom, from the energy dissipationdue to the elasticity of the suspension wires. Wealso show how the contributions of thermal noise indifferent directions can be sensed by the interferom-eter through the laser beam position and direction.The results will follow from the consideration of thecoupled equations of the suspension and the contin-uous wire, first presented in [3] for just the horizon-tal degree of freedom. We show how this approachencompasses and explains previous ways to approx-imate the thermal noise limit in gravitational wavedetectors. We show how this approach can be ex-tended to more complicated suspensions to be usedin future LIGO detectors. To our knowledge, this isthe first time the thermal noise of angular degrees offreedom is presented, and that all suspension degreesof freedom are calculated in an unified approach.

Since the full treatment of the problem is some-what involved, we present first the problem withoutconsidering the elasticity of the wire, but adding asecond, lossy, energy source to the gravitational en-ergy in the treatment of the mechanical pendulum,and introduce the concepts of “dilution factors”, and“effective” quality factors. We also start with oneand two-degrees of freedom suspensions instead of6-dof. With these tools, most of the issues can beclearly presented and then we follow to the full treat-ment of the LIGO suspensions, presenting the impli-cations for LIGO.

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II. SIMPLE PENDULUM CASES: DILUTIONFACTORS, COUPLED MODES, EFFECTIVE

QUALITY FACTORS.

The full treatment of this case, considering theelastic coupling of the wire to the suspension, waspresented in [3]. Here, we will present the simpler“mechanical” treatment of this case, which will in-troduce the concepts of “dilution factors”, and mea-sured vs. effective quality factors.

A. A simple oscillator with a dissipative energysource

We first recapitulate the calculation of thermalnoise in the simplest case, a suspended point mass.The potential energy is PE = (1/2)Kx2 and FK =−dV/dx = −Kx. The kinetic energy is KE =(1/2)Mx2. The admittance to an external force Fextis given by

Y ≡ iωx

Fext= iω

1K −Mω2

The admittance has a pole at the system eigen-frequency w2

0 = K/M . If K is real, the resonancehas an infinite amplitude and zero width. If thespring constant has an imaginary part representingan energy loss, K → K(1 + iφ), then the amplitude|Y (ω0)| is finite, and the peak has a width deter-mined by the complex part of the eigenfrequencyω2

0 = (K/M)(1+ iφ). The width of the peak is char-acterized with a quality factor Q = 1/φ, and it isusually measured from the free decay time τ of thenatural oscillation at the frequency ω0: Q = ω0τ/2.

The thermal noise is proportional to the real partof the admittance, and thus to φ:

<[Y ] =wKφ

(K −Mω2)2 +K2φ2

We are usually interested in frequencies well aboveω0, since the pendulum frequency w0/2π in gravita-tional wave detectors is usually below 1 Hz, and thedetectors have their maximum sensitivity at 100 Hz.At those frequencies, the thermal noise is

x2(ω) =4kBT0

ω2<[Y ] ∼ 4kBT0ω

20φ

Mω5(1)

This how we see that the measured decay of thependulum mode can be used to predict the suspen-sion thermal noise at gravitational wave frequencies.Some beautiful examples of these difficult measure-ments and their use for gravitational wave detectorsare presented in [8], for example.

B. A pendulum with two energy sources: thedilution factor.

Next, we consider a suspended point mass, butwe now assume there two sources of energy, gravi-tational and elastic, each with its own spring con-stant. The potential energy is then V = Vg + Ve =(1/2)(Kg +Ke)x2, and

<[Y ] =ω(Kgφg +Keφe)

((Kg +Ke)−Mω2)2 + (Kg +Ke)2φ2

If we assume that Kg � Ke, then

<[Y ] ∼ ω(Kgφg +Keφe)(Kg −Mω2)2 +K2

gφ2

and at high frequencies

x2(ω) =4kBT0

ω2<[Y ] ∼ 4kBT0(Kgφg +Keφe)

M2ω5

If φg = 0 (“gravity is lossless”), or at leastKgφg � Keφe, then

4kBT0ω20(Kg/Ke)φeMω5

where ω20 ∼ Kg/M . We see that is the same ex-

pression as if we had just one energy source with acomplex spring constant K = Kg(1 + i(Ke/Kg)φe).This is why we call the factor Ke/Kg the “dilutionfactor”: the elastic energy is the one contributingthe loss factor to the otherwise loss-free Kg, but“diluted” by the small factor Ke/Kg. The dilutionfactor is also equal to the ratio of elastic energy togravitational energy Ke/Kg = Ve/Vg. The conceptof a dilution factor is very useful because it is usuallyeasier to measure the loss factor φe associated withthe elastic spring constant than the quality factor ofthe pendulum mode. This is because Ke is usuallya function of the complex Young modulus E, andthe imaginary part of the Young modulus is easilymeasurable for most fiber materials, and can even befound in tables of material properties. (Of course,there are subtleties to this argument, in particularwith thermoelastic or surface losses [9], but we areassuming the minimum material loss).

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C. A point mass suspended from an anelasticwire: calculating the dilution factor

This case is a particular case of the one treated in[3], and here we just mention it to present the ap-proach taken to the full problem, and present somenew relevant aspects.

We want to include the elasticity of the wire inthe equations of motion, so we treat the suspensionwire as an elastic beam, and then we have pendulumdegree of freedom x, plus the wire’s infinite degreesof freedom w(s) of transverse motion. We define acoordinate s, that starts at the top of the wire s = 0,and ends at the attachment point to the mirror,s = L. Correspondingly, we will have an eigenfre-quency, associated with the pendulum mode, and aninfinite series of “violin” modes. The potential en-ergy is PE = (1/2)

∫ L0ρ(w′(s))2ds, and the kinetic

energy is KE = (1/2)∫ L0 ρ(w(s))2ds + (1/2)Mx2.

The solutions to the wire equation of motion, withboundary conditions w(0) = 0 and w(L) = xare w(s) = x sin(ks)/ sin(kL), and the equationof motion for the mass M subject to an externalforce F is F = −Mω2x + Tw′(L) = −Mω2x +(T/L)x(kL/ tan(kL)). The admittance

Y = iωx

F= iω

T

L

tan kL/kL1− ω2(ML/T )(tankL/kL)

has a pole at the pendulum frequency w2p ∼ T/ML,

where kL � 1, and an infinite number of poles atthe violin mode frequencies, at frequencies wn ∼(T/ρL2)1/2(1 + (ρL/M)n2π2), where kL = nπ +√

(ρL/M)/(nπ).The spring “constant” associated with the

wire and gravity’s restoring force K =(T/L)(kL/ tan(kL)) is in fact a function of fre-quency, although it is the usual constant T/Lfor frequencies below the violin modes, wherekL � 1. At frequencies above the first vio-lin mode, the spring function is not even posi-tive definite, or finite. The function K is realat all frequencies because we haven’t added anysource of energy loss yet. We introduce energyloss in the system by adding the wire elastic en-ergy to the system, and then assuming a com-plex Young modulus. The potential energy is nowPE = (T/2)(

∫ L0 w′2(s)ds) + (EI/2)

∫ L0 w′′2(s)ds.

The equation of motion for the wire is

Td2w(s)ds2

− EI d4w(s)ds4

+ ρω2w(s) = 0,

a fourth order equation with boundary conditionsw(0) = 0, w′(0) = 0, and w(L) = x. The wire slopeat the bottom, w′(L), is a free parameter (since weare assuming a point mass), and the variation ofthe Lagrangian with respect to w′(L) provides thefourth boundary condition for the wire: w′′(L) = 0.We can find an exact solution for the wire shape as afunction of x, trigonometric functions of ks, and hy-perbolic functions of kes, where k, ke are solutions toTκ2−EIκ4+ρω2 = 0 which approximate at low fre-quencies the perfect string wavenumber, k2 ∼ ρω2/Tand a constant “elastic” wavenumber , k2

e ∼ T/EI

[3]. The distance ∆ =√EI/T is the characteristic

elastic distance over which the wire bends, especiallyat top and bottom clamps. In LIGO test mass sus-pensions, ∆ ∼2mm, a small fraction of L = 0.45m.The approximations k2 ∼ ρω2/T and k2

e ∼ T/EIare valid for frequencies that satisfy ω2 � T 2/4EIρ,about 12 kHz for LIGO, so we will use them in theremainder of this article. It is also equivalent tok∆ � 1.

We also use an approximate solution for the wireshape, good to order e−L/∆(∼ 10−99 (!) for LIGO):

w(s) = A (sin(ks)− k∆cos(ks))

+k∆(Ae−s/∆ +Be−(L−s)/∆

)(2)

The coefficients A,B are functions of x and k, andthus, functions of frequency:

A =x

sin(kL)− k∆cos(kL)B = xk∆.

In the limit ∆ → 0, we recover the perfectwire solution, w(s) = x sin(ks)/sin(kL). The ra-tio B/A measures how much more (or less) the wirebends at the bottom than at the top. The elas-tic energy is well approximated by the contribu-tion of the exponential terms in the wire shape, attop and bottom: PEe = (1/2)

∫EI(w′′(s))2ds =

(1/2)T∆2∫(w′′(s))2ds ∼ (1/2)Tk2L(A2 + B2). At

low frequencies where kL � 1, the ratio B/A ∼∆/L� 1, indicating that the wire bends much moreat the top than at the bottom (recall this is a pointmass).

The equation of motion for the mass when thereis an external force F is

F = −Mω2x+ Tw′(L)− EIw′′′(L)∼ −Mω2x+ Tw′(L)

∼ −Mω2x+(

kT

tankL1 + 2k∆tankL

1− (k∆)/(tan kL)

)x (3)

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The ratio of the elastic force to the gravitationalforce, EIw′′′(L)/Tw′(L), is of order k∆ � 1, andthus it was dropped. If we now consider ∆ complex,then the spring function

K =kT

tan kL1 + 2k∆tan kL

1− (k∆)/(tan kL)

is also complex, and the admittance will have a non-zero real part. At frequencies below the violin modeswhere kL � 1, we have K ∼ (T/L)(1 + ∆/L), anexpression that suggests a split between a real grav-itational spring constant Kg = T/L and a complexelastic spring constant Ke = T∆/L2. However, thisdistinction can only be done in the approximation∆/L � 1, and low frequencies kL � 1. In general,however, we cannot strictly derive separate gravi-tational and elastic spring constants from their re-spective potential energy expressions: notice thatthe total, complex spring constant ∼ T/(L−∆) wasderived from the variation of the gravitational poten-tial energy term, which becomes complex because weuse a wire shape involving the complex distance ∆,satisfying the boundary conditions.

Where the approximations ∆/L � 1, kL � 1are valid, we can consider the case of two separatespring constants and thus a “dilution factor” for thependulum loss, Ke/Kg = ∆/L ∼ 1/232, where thenumerical value corresponds to LIGO parameters inAppendix 1. However, if we numerically calculatethe exact pendulum mode quality factor, we get1/Qp = 461/φ: this would be the measured Q froma decay time of the pendulum mode, if there are noextra losses. Did we make a “factor of 2” mistake?In fact, this factor of 2 has haunted some people inthe community (including myself) [10], but there is asimple explanation. Since the elastic complex springconstant is proportional to ∆, and ∆ is proportionalto the square root of the Young modulus, then whenwe make E complex E → E(1 + iφ), we get Ke ∼T∆/L2 → (T∆/L2)(1 + iφ/2). That is, we get anextra dilution factor of two between the wire loss φand the pendulum loss: φp = (Kg/Ke)=Ke/<Kel =(Kg/Kel)φ/2 = (∆/2L)φ ∼ φ/464, very close to theactual value. This teaches us that if the spring con-stant of the dissipative force is not just proportionalto E, we will get correction factors ∂logKe/∂logE.

The thermal noise below the violin modes is wellapproximated by the thermal noise of a simple os-cillator, as in Eqn.1, with natural eigenfrequencyw2

0 = g/L and loss φ0 = ∆φ/(2L). We then callφ0 = ∆φ/(2L) the “effective” loss, in this case equalto the pendulum loss (but we will see this is notalways the case).

We saw that the complex spring constant T/(∆−L)was split into gravitational T/L and elasticT∆/L2 components. However both were derivedfrom the gravitational force Fg = −Tw′(L), sincethe elastic force, Fel = EIw′′′(L), was negligible.It is because the wire shape w(s) is different dueto elasticity, that the function Tw′(L) is differentfrom the pure gravitational expression Tx/L. Theway we split gravitational and elastic contributionsto the spring constant and then got a dilution factor,is only valid at low frequencies. So the argument weposed in the previous section about a dilution factorapplied to the calculation in the thermal noise in thegravitational wave band is in priciple not applicablehere, especially when taking into account that thetotal force was contributed by the variation of justthe gravitational potential energy, with the elastic-ity in the wire shape. However, using the wire shapewithout low frequency approximations, we can nu-merically evaluate the integrals that make up the po-tential and elastic energies (using a real ∆), and com-pare the ratio Vel/Vg =

∫T (w′)2ds/

∫EI(w′′)2ds

with the “low frequency” dilution factor ∆/L. Weshow the calculation of elastic and gravitational po-tential energies, and their ratio, in Fig.1. At low fre-quencies, the ratio is constant, and equal to ∆/2L:this is the dilution factor between the wire loss φand the pendulum loss φp, also the one to use for asimple-oscillator approximation of the thermal noise.It is not the ratio of the “gravitational” and “elas-tic” spring constants at low frequencies, but as weexplained, we had no reason to expect that, sinceVe/Vg 6= Ke/Kg. At higher frequencies, the ra-tio Vel/Vg is not constant, and it gives correctlythe dilution factors for the quality factors of theviolin modes. Notice that the loss at the violinmodes increases with mode number, as noted in [3]:φn = (∆/L)(1+n2π2∆/L) = (∆/L)(1+ρL∆ω2

n/T ),and this anharmonic behavior is well followed by theenergy ratio.

In summary, the concept of dilution factor isstrictly true only when the total restoring force canbe split into two forces, one lossless and one dissi-pative, both represented with spring constants. Inthe general case, if we can only split the potentialenergy into two terms, one lossless and another dis-sipative, then the ratio of the energies calculated asa function of driving frequency is the exact dilution“factor”. Moreover, this ratio can be calculated as afunction of frequency, and then we get the differentdilution factors for all the modes in the system. Thisis an important lesson that also we will use more ex-tensively in suspensions with more coupled degrees

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of freedom.

D. A suspended extended mass: coupleddegrees of freedom and observed thermal noise

We now consider an extended mass instead of apoint mass, with a single generic dissipative energysource. The pendulum motion is described with thehorizontal displacement of its center of mass x, andthe pitch angle of the mass, θ, as in the side viewof a LIGO test mass, in Fig.7. The kinetic energy isKE = (1/2)(Mx2 + Jθ2). Instead of a spring con-stant, we have a 2x2 spring matrix. The potentialenergy is

PE =T

2(Lα2 + hθ2)

=12(Kxxx

2 + 2Kxθxθ +Kθθθ2) (4)

where α = (x+ hθ)/L and θ are the normal coordi-nates. The point x+hθ is the point where the wire isattached to the mirror, and the angle α is the anglethe wire makes with the vertical. If we only considergravitational forces, Kxx = T/L, Kxθ = Th/L andKθθ = Th(L+ h)/L. However, we will assume thatthe elements of the spring matrix can be complex,and each has its own different imaginary part. Theeigenfrequencies are the solutions to the equation(Kxx −Mω2)(Kθθ − Jω2)−K2

xθ = 0, or

2MJω2± = (MKθθ + JKxx)

± ((MKθθ − JKxx)2 + 4MJK2

xθ

)1/2(5)

In order to calculate x2(ω) (the Brownian motionof the center of mass), we need to calculate the ad-mittance x/F to a horizontal force applied at thecenter of mass. In order to calculate θ2(ω), we needthe admittance θ/N to a torque applied around thepitch axis. If the spring constants are complex, thenthe admittances are complex and we can calculatetheir real parts, and the thermal noise determinedby them:

x2(ω) =4kBT0

ω2<

[iω

1− Jω2/Kθθ

Kxx(1− ω2/ω2+)(1 − ω2/ω2−)

]

θ2(ω) =4kBT0

ω2<

[iω

1−Mω2/Kxx

Kθθ(1− ω2/ω2+)(1 − ω2/ω2−)

]

where the eigenfrequencies are now complex: ω2± →ω2±(1 + iφ±). The quality factors measurable from

the free decay of each of the eigenfrequencies areQ± = ω±τ±/2.

At frequencies larger than any of the eigenfrequen-cies, we obtain

x2(ω) ∼ 4kBT0

Kxxω5((ω2

+φ+ + ω2−φ−)− φθθ(Kθθ/J))

θ2(ω) ∼ 4kBT0

Kθθω5((ω2

+φ+ + ω2−φ−)− φxx(Kxx/M)) (6)

The system may have the two eigenfrequenciesclose in value if h(L + h) ∼ J/M (see Fig.2), butfor h � J/ML, we have ω2

− ∼ Kθθ/J = Th/J andω2

+ ∼ Kxx/ML2 = T/ML; and for h(L+h)� J/M ,ω2

+ ∼ Kθθ/J = Th/J and ω2− ∼ Kxx/ML2 =

T/ML. In both limits, two terms cancel in thesum of loss factors in the formulas above (ω2

−φ− ∼φθθKθθ/J for small h, for example) and we see that

x2(ω) ≈ 4kBT0ω2pφxx

Mw5and θ2(ω) ≈ 4kBT0ω

2θφθθ

Jω5

Thus, even though it is a coupled system, ther-mal noise in x is always associated mostly toKxx(1 + iφxx) and the pendulum eigenfrequencyω2p = Kxx/M ∼ T/ML; and thermal noise in θ is

always associated with Kθθ(1 + iφθθ) and the pitcheigenfrequency ω2

θ = Kθθ/J ∼ Th/J . For both de-grees of freedom x and θ, we obtain the thermalnoise of single-dof systems. Unfortunately, neitherlimit (small or large h with respect to J/ML) appliesto the suspension parameters in LIGO test masses,and, more importantly, even though the approxima-tion for the eigenfrequencies is relatively good formost values of h, the approximation we used for thelosses is not (Fig.2). The measurable quality factorsgive us φ±, but we need φxx and φθθ to use in thethermal noise of the pendulum, and these cannot beprecisely calculated from φ± unless we know φxθ, ora way to relate it to the other loss factors. We willdo this in the next section, using the elasticity of thewire.

Notice that the forces and torques we have usedto calculate the admittances Yx and Yθ, will eachproduce both displacement and rotation of the pen-dulum. This means that the thermal noise in dis-placement and angle are not uncorrelated. This canbe exploited to find a point other than the centerof mass where the laser beam in the interferome-ter would be sensing less displacement thermal noisethan at the center of the mirror, as was done follow-ing a somewhat different logic in [5]. If we wereto calculate the thermal noise at a point a distance

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d above the center of mass, we then need to cal-culate the admittance of the velocity of that point(iω(x + dθ) = iωχ) to a horizontal force applied atthat point. The equations of motion are

F = (Kxx −Mω2)x+Kxθθ

Fd = Kxθx+ (Kθθ − Jyω2)θ

and then the thermal noise is

χ2(ω) =4kBT0

ω2<

[iω

(x+ dθ)F

]

=4kBT0

ω2< [

Yxx + d2Yθθ + 2dYxθ]

= x2(ω) + d2θ2(ω) + 2d4kBT0

ω2<(Yxθ) (7)

where Yxx is the admittance of x to a pure force F ,Yθθ is the admittance of θ to a pure torque N , andYxθ is the admittance of a displacement x to a puretorque N , equal to the admittance of θ to a pureforce F . There is an optimal distance d below thecenter of mass for which the thermal noise χ2(ω) isa minimum: this distance is d = −<(Yxθ)/<(Yθθ).The resulting thermal noise is

χ2min(ω) = x2(ω)− 4kBT0

ω2

(<Yxθ)2<Yθθ

which is less than the thermal noise x2(ω) observedat the center of mass. However, the expression ob-tained for the distance d is frequency-dependent:that means we have to choose a frequency at whichto optimize the sampling point.

Summarizing, we have shown that whenever thereare coupled motions, the thermal noise sensed at apoint whose position depends on both coordinates isnot the sum in quadrature of the two thermal noise(x2 and d2θ2(ω) in our case), but a combination thatdepends on the “cross-admittance”. Moreover, thethermal noise of each degree of freedom cannot ingeneral be calculated just from the measured qual-ity factors if the modes are coupled to each otherstrongly enough.

E. A 2-DOF pendulum suspended from acontinuum wire

We add to the previous 2-DOF pendulum a con-tinuum wire, to be able to add the losses due to thewire’s elasticity, and calculate modal and effectivequality factors, as well as the point on the mirror atwhich we can sense the minimum thermal noise.

If we add elasticity to the problem, as in [3], thepotential energy is PE = (T/2)(

∫ L0 w′2(s)ds+hθ2)+

(EI/2)∫ L0w′′2(s)ds. The boundary conditions for

the wire equation are w(0) = 0, w(L) = x+hθ, andw′(0) = 0, w′(L) = −θ. The equations of motionfor the pendulum are

−Mω2x+ Tw′(L)− EIw′′′(L) = F

−Jω2θ + EI(w′′(L) + hw′′′(L)) = N (8)

In order to complete the equations of motion ofthe pendulum, we need the shape of the wire at thebottom end. For this, we use the shape given bythe expression in Eqn.2, but this time the top andbottom weights are given by

A =x+ (h+ ∆)θ

D

B =1kD

(kx(cos(kL) + k∆sin(kL))

+θ(sin(kL) + k(h−∆) cos(kL))

with D = sin(kL)− 2k∆cos(kL) and ∆ =√EI/T

as we used earlier. With the shape known, we canwrite the equations for x, θ with a spring matrix:

(Kxx −Mω2)x+Kxθθ = F

Kθxx+ (Kθθ − Jyω2)θ = N

where the spring functions are

Kxx = Tk(cos(kL) + k∆sin(kL))/DKθθ = T (h+ ∆)(sin(kL) + k(h−∆) cos(kL))/DKxθ = Kθx = Tk(h+ ∆)(cos(kL) + k∆sin(kL))/D (9)

At this point, even though the expressions arecomplicated, we can calculate the complex admit-tances Yxx = iωx/F, Yθθ = iωθ/N using a com-plex E and ∆. The analytical expressions for theadmittances are quite involved, but we can alwayscalculate numerically the thermal noise associatedwith any set of parameters. We can also calculatethe widths of the peaks in the admittance, whichwould correspond to measurable quality factors forthe pendulum, pitch, and violin modes. The plotspresented in Figs. 2 and 3 were calculated usingthese solutions.

Fig.2 shows that the frequency and quality factorof the pendulum and pitch modes vary significantlywith the pitch distance h. At frequencies close to thependulum eigenfrequencies, the thermal noise spec-tral densities show peaks at both frequencies. How-ever, at higher frequencies, the thermal noise x2(ω)

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can always be approximated by the thermal noiseof a simple oscilaltor as in Eqn.1, with an “effec-tive” quality factor that fits the amplitude at highfrequencies to the position of the single peak. Wecan similarly define an effective quality factor forthe thermal noise in θ2(ω). We show in Fig.3 theactual and approximated thermal noise, with theircorresponding effective quality factors found to fitbest at 50 Hz. The effective quality factor at 50Hz can be calculated as a function of pitch distance,and we show this calculation in Fig. 2. The effec-tive pitch quality factor is well approximated, forany pitch distance, by the measurable pitch qualityfactor, while the pendulum effective quality factor isclose to the measurable quality factor of the pendu-lum mode only at very small, or very large pitchdistances. For the LIGO pitch distance of 8mm,the measurable pendulum quality factor is 10 timeslower than the effective quality factor, and wouldthen give a pessimistic estimate of thermal noise am-plitude.

Low frequency approximation. At low frequencies,where kL� 1, we have expressions that can help usunderstand how the elasticity loss factor contributesto the effective quality factors, as well as to the pen-dulum and pitch modes. We trade this gain in sim-plicity for the loss of expressions valid at or aboveviolin mode resonances.

The low frequency limit of the spring constants inEqns. 9 is

Kxx = T/(L− 2∆)Kθθ = T (h+ ∆)(L + h−∆)/(L− 2∆)Kθx = Kxθ = T (h+ ∆)/(L− 2∆).

If we assume ∆ has an imaginary part relatedto the material φ: ∆ → ∆(1 + iφ/2), then we getcomplex spring constants. If we use these complexspring constants in Eq.5 we can calculate the lossfactors of the pendulum and pitch mode, φp andφθ; and if we use them in Eqns. 6 we can getthe effective quality factors. Using h/L � 1 and∆/L � 1, we get Qxeff = 1/φxx ∼ ∆φ/L andQθeff = 1/φθθ ∼ ∆φ/2(h + ∆). This representsa “dilution factor” in displacement of ∆/L and inpitch of ∆/2(h+ ∆). These approximations fit verywell the values shown in Fig. 2.

At low frequencies, the equations of motion forx, θ can be derived from a potential energy

PEkL�1 =T

2

((x+ (h+ ∆)θ)2

L− 2∆+ (h+ ∆)θ2

).

Using a complex ∆, this gives us the complex spring

constants that we can use to get mode and effec-tive quality factors. We would like to break upthis potential energy into a gravitational part andan elastic part, corresponding to a real gravitationalspring constant (independent of ∆) and a lossy elas-tic spring constant. We know that if we take thelimit ∆ → 0 in PEkL�1, we obtain the regular po-tential energy PEg in Eq.4 for a 2 DOF pendulumwithout elasticity. Thus, we are tempted to say thatthe elastic energy is the remainder, proportional to∆, and thus having a complex spring constant whenwe consider a complex ∆. According to this argu-ment, we get Kgxx = T/L and Kexx = 2T∆/L2.This would then give us a dilution factor for the dis-placement loss Ke/Kg ∼ 2∆/L. This is the factorby which the imaginary part of Kel is diluted; how-ever, as explained before, we pick up another factorof two due to Ke being proportional to

√E. Thus,

the dilution factor between the effective quality fac-tor and the wire quality factor is ∆/L.

Energy Ratios and the Dilution Factor.We haveseen that there is another way of identifying the“gravitational” and “elastic” terms in the poten-tial energy, using the actual potential energy expres-sions from which the equations of motion were de-rived: PEg = (T/2)(

∫ L0w′2(s)ds+hθ2) and PEel =

(EI/2)∫ L0w′′2(s)ds. For any given applied force, or

torque, we can solve the equations of motion for x, θand w(s). (We don’t need to invoke a low frequencyapproximation to do this calculation.) Then, if wecalculate the ratio of elastic to gravitational poten-tial energy for a unit applied force, we get a func-tion which is frequency dependent, and is equal tothe dilution factors for the pitch mode at the pitchfrequency, for the pendulum mode at the pendulumfrequency, and for the effective quality factor at highfrequencies, ∆/L. We show the energy values andratios for different values of the pitch distance in Fig.4.

Potential Energy Densities.There is another inter-esting calculation we can do with the solution ob-tained for the wire shape, and that is to find outwhere in the wire the elastic potential energy is con-centrated. In other words, we want to find a rela-tionship between the variation of the dilution fac-tor with frequency, and the curvature of the wire,mostly at the top and bottom clamps. Since boththe gravitational energy and the elastic energy in-volve integrals over the wire length, we can defineenergy densities along the wire, and calculate a cu-mulative integral from top to bottom. The gravi-tational potential energy has also a term Thθ2/2:

7

we define a ratio R = Thθ2/(∫ L0w′2ds), indicat-

ing the relative contribution of this “pitch” term.From Fig.5, we observe that the gravitational po-tential energy density is distributed quite homoge-nously along the wire, even at the first violin mode.However, the pitch term, which can be considereda “bottom” contribution, contributes most of thegravitational energy when the system is excited atthe pitch eigenfrequency, but also in several othercases at the pendulum frequency and at low frequen-cies. The elastic energy density is concentrated attop and bottom portions of length 2∆. At low fre-quencies, the top contributes the most; at the pen-dulum eigenfrequency, the relative contributions de-pend strongly on h, but the bottom contributes atleast half the energy; at the pitch eigenfrequency, thebottom contributes more than 99% of the energy; athigher frequencies, including the violin modes, topand bottom contribute equally.

Motion of points away from center of mass. Wediscussed previously how it was possible to find apoint whose thermal noise displacement was smallerthan the thermal noise displacement of the center ofmass. Now that we have expressions for the com-plex spring functions, we can find the optimal pointand discuss the differences. The cross admittance inEq.7 is

Yxθ = iωx

N= iω

θ

F

= − Kxθ

(Kxx −Mω2)(Kθθ − Jω2)

∼ − Kxθ

MJω4

and then the optimal point (otimized at frequen-cies in the gravitational wave band, above pendulummodes) is d0 = −<(Yxθ)/<(Yθθ) ∼ J/ML. Noticethat even though the optimal distance was deducedfrom the thermal noise expressions, which all involveloss factors, the optimal distance only depends onmechanical parameters. As first explained in [5], theinterpretation of this distance is that when the pen-dulum is pushed at that point by a horizontal force,the wire doesn’t bend at the bottom clamp, produc-ing less losses. The fact that we can recover theresult from the FDT is another manifestation of thedeep relationship between thermal fluctuations andenergy dissipation. We show in Fig6 the dependenceof the thermal noise at 50 Hz on the point probedby the laser beam on the mirror, and the ratio of thethermal noise for d = 0 and d = d0 at all frequen-cies. As expected, since the integrated rms has to bethe the same for any distance at which we sense the

motion, the fact that the spectral density is smallerat 50 Hz if d = d0 means that the noise will beincreased at some other frequencies: this happensmainly at frequencies below the pendulum modes.

There are many lessons to be learned from this ex-ercise, but perhaps the most important one is thatthe explicit solutions to the equations of motion havemany different important results:

• using the solutions to calculate the elastic andgravitational potential energies allows us tocalculate a “dilution function” of frequency,equal to the dilution factor at each of the res-onant modes of the system, as well as to themost important effective dilution factor at fre-quencies in the gravitational wave band;

• we can calculate energy densities along thewire to identify the portions of the wire mostresponsible for the energy loss and thus thethermal noise;

• we can use low frequency approximations tofind out expressions for dilution factors thatcan be found using other methods, explainingin this way subtleties like factors of two;

• we can calculate the admittance of an arbi-trary point in the mirror surface to the driv-ing force, and thus find out improvements ordegradation of observed noise due to beammisalignments.

III. THE LIGO SUSPENSIONS: THERMALNOISE OF ALL DEGREES OF FREEDOM

We will now calculate the solutions to the equa-tions of motion for the six degrees of freedom of aLIGO suspended test mass, and then use the solu-tions to calculate the thermal nosie of all degrees offreedom, as well as the observed tehrmal noise in thegravitational wave detector.

The mirrors at LIGO are suspended by a singlewire looping around the cylindrical mass, attachedat the top at a distance smaller than the mirror di-ameter, to provide a low yaw eigenfrequency. Thisis equivalent to having a mass suspended by twowires, attached slightly above the horizontal planewhere the center of mass is. The mirror’s 6 degreesof freedom are the longitudinal and transverse hor-izontal x and y, and the vertical z, displacementsof the center of mass; the pitch θ and yaw φ ro-tations around the y and z axis, respectively, and

8

the roll ψ around the longitudinal x axis. We showthe coordinate system used and the relevant dimen-sions in Fig.7. The parameters used in the calcula-tion presented are those for LIGO test mass suspen-sions (Large Optics Suspensions). The mass of thecylindrical mirror is 10.3 Kg, the diameter is 25cm,and the thickness 10cm. The cylindrical wires aremade of steel with density ρ = 7.8× 103kg/m3 and0.62mm diameter. We assumed a complex Youngmodulus E = 2.1× 1011(1 + 10−3i)kg/m2. The ver-tical distance between the center of mass and thetop clamps is l = 45cm, the wires are attached tothe mass a distance h = 8.2mm above the centerof mass. The distance between the top attachmentpoints is 2a = 33.3mm. (In the previous exampleswhere one wire was used, we assumed the same wirematerial and the same test mirror, but we used a0.88mm radius, so the stress in the wires remainedconstant.)

Each wire element has displacement in a 2-dimensional plane transverse to the wire, ~wi⊥(s) anda longitudinal displacement along the wire, wi‖(s).The kinetic energy is given by

KE =12(∑i=1,2

∫ L

0

ρ| ~wi(s)|2ds

+M(x2 + y2 + z2) + Jxψ2 + Jy θ

2 + Jzφ2)

The potential energy is given by the sum of the ax-ial strain energy and the bending (transverse) strainenergy in each wire:

PEi =12

∫ L

0

ds(Twi

′⊥

2 + EA(w′i‖)2)

+12

∫ L

0

dsEI (w′′i ⊥)2

plus the energy involved in rotating the mass:

PEM = T (h cosα(θ2 + ψ2)− b sinα(φ2 + ψ2)).

The wires will be attached at the top (s = 0)at the coordinates ~wi = (0,±a, l), where l is thevertical distance of the top support from the equi-librium position of the center of mass. The wires’transverse slopes at the top will be zero. At thebottom the wires are clamped to the mass a dis-tance h above the center of mass, and a distance 2bon the y-direction between the wires on each side ofthe mass. The angle α = arctan((b − a)/(l − h)) isthe angle at which the wires are slanted from top tobottom when looking at the mass along the opticalaxis. If b = a, the wires hang vertically. The length

of the wires is L2 = (b − a)2 + (l − h)2. The ten-sion in each wire is T = Mg cosα/2. The positionof the bottom attachments when the mirror is mov-ing with a motion described by (x, y, z, θ, φ, ψ) arewi(L) = (x+hθ±bφ, y−hψ, z±bψ), and the slopesat the bottom are w′i(L) = (−θ, ψ, 0).

If we express the wire transverse and longitudi-nal displacements in the x, y, z coordinate system,we have wi‖ = −wz cosα ± wy sinα, and wi⊥ =√wi2x + (wiy cosα± wiz sinα)2 and the equations

of motion become non-linear. In order to keep theproblem simple, without losing any degree of free-dom, we will then consider two different cases: (i)the wire only has displacements in the x direction,and the mirror moves in x, θ, φ degrees of freedom;and (ii) the wire only has displacements in the y, zdirections, and the mirror moves in y, z, ψ degrees offreedom. We analyze these cases separately.

A. Longitudinal, Pitch and Yaw Thermal Noise

The boundary conditions for the wires at the topare zero displacements and slopes, and at the bot-tom attachment to the mass, w⊥i(L) = x + hθ ±φ,w′⊥i(L) = −θ.

We combine the wires’ transverse displacementsinto w±(s) = (w1(s) ± w2(s))/2, then the bound-ary conditions at the bottom are w+(L) = x −hθ, w′+(L) = θ, w−(L) = bφ, w′−(L) = 0. The solu-tions to the wire equations of motion and the bound-ary conditions are (up to order e−L/∆) will then bew±(s) as in Eqn.2, with

A− = φb/D

B− = −φb(cos(kL) + k∆sin(kL))/DA+ = (x+ (h+ ∆)θ)/DB+ = −∆(kx(cos(kL) + k∆sin(kL))

+θ(sin(kL) + k(h−∆) cos(kL)))/D

where D = sin(kL) − 2k∆cos(kL). The equationsfor the mass transverse dof subject to a force Fx andtorques Ny, Nz, are

Fx = −Mω2x− 2EIw′′′+ (L) + 2Tθ

Ny = −Jyω2θ + 2EI(w′′+(L) + hw′′′+ (L)

)+2Thθ(1− cosα)

Nz = −Jzω2φ− 2EIbw′′′− (L)− 2Tbφ sinα (10)

We see that the combination w+(s) is associatedwith the x, θ degrees of freedom just as for the single

9

wire case, while the combination w−(s) is associatedwith the yaw degree of freedom φ. This is easily un-derstood when imagining the wires moving back andforth “in phase” (w− = 0), producing displacementand pitch but nt yaw; while if they move back andforth in opposition (w+ = 0), then the only effectis into mirror’s yaw. Thus, we can solve the equa-tions for w−(s) and φ separately from the equationsfor ω+(s), x and θ: we will do so in the next parap-graphs.

Yaw angular thermal noise. The admittanceof yaw φ to a torque Nz is

Yφ = iω1

Kφ − 2Tb sinα− Jzω2

with Kφ = (2Tb2/L)(kL)(cos(kL) +k∆sin(kL))/(sin(kL) − 2k∆cos(kL)). As usual,when ∆ =

√EI/T is complex, the admittance is

complex and using the fluctuation-dissipation theo-rem we obtain φ2(f) = (4kBT0)<(Yφ)/ω2.

The effect of the tilted wires with α = (b−a)/L >0 is to lower the restoring force, and thus the res-onance frequency: in LIGO suspensions, the fre-quency is 0.48 Hz instead of 1.32 Hz if b = a.However, since sinα 6= 0 decreases the gravita-tional restoring force but not the elastic force, thedilution factor ∼ Ke/Kg increases, and so doesthe thermal noise. At frequencies where kL � 1,Kφ ∼ (2Tb2/L)(1 + 2∆/L). The thermal noise atfrequencies below the violin modes, where kL � 1,is well approximated by the thermal noise of a singleoscillator with resonance frequency w2

φ = 2Tab/JLand quality factor Qφ = (L/∆)(a/b)(1/φw), whereE → E(1 + iφw). Thus, the “dilution factor” is(∆/L)(b/a) (=1/72 for LIGO parameters, whereb/a = 7.5). As in the case of a simple pendulum,this dilution factor is half of the ratio of the elasticspring constant Ke = (2Tb2/L)(2∆/L) to the grav-itational spring constant Kg = 2Tab/L, because theelastic spring constant has an “extra” dilution factorof 2: if E → E(1 + iφw), then ∆ → ∆(1 + iφw/2)andKe → Ke(1+iφw/2). However, the ratio of elas-tic potential energy

∫ L0EI(w′′⊥)2ds to gravitational

potential energy Vg =∫ L0T (w′⊥)2ds − Tb sinαφ2 is

equal to the “right” dilution factor (b/a)(∆/L) ∼1/44 at low frequencies, as shown in Fig. 8. Theenergy ratio also gives us the right dilution factor atthe violin frequencies (2∆/L).

The yaw angular thermal noise may be seen inthe detectors’ gravitational wave signal if the beamhits a mirror at distance d to either side of the cen-ter of mass, or if it hits the mirror in a direction

an angle γ away from longitudinal. Consideringboth cases, the sensed thermal noise will be givenby χ2(ω) = (d cos γ +H sin γ/2)2φ2(ω), where H isthe thickness of the mirror:

χ2(ω) = (d cos γ + (H/2) sin γ)2φ2(ω)

∼ 4kBTω5

(d cos γ + (H/2) sin γ)22Tb2

JxL

∆Lφ

where the approximation is valid between the pendu-lum mode and the first violin mode, 1Hz-50Hz. At160 Hz, where the maximum sensitivity of

√h2(f) =

2.5× 10−23/√

Hz is expected, the yaw thermal noiseis

√φ2(160Hz) = 2.9 × 10−19rad/

√Hz. If the yaw

thermal nose is to be kept an order of magnitudebelow the dominant noise source, then it is requiredthat d ≤1cm and γ ≤ 10o.

Notice that the mirror will always be aligned nor-mal to the laser beam to make the optical cavitiesresonant; however, what matters is the beam direc-tion with respect to the coordinate system defined bythe local vertical and the plane defined by the mirrorin equilibrium. Presumably there will be forces ap-plied to align the mirror, but in principle they haveno effect on the response of the mirror to an osculla-tory driving force such as the one we imagine in thebeam’s direction, to calculate the admittance. Thusthe requirement on γ ≤ 10o is on the position of themirror when there are no bias forces acting, with re-spect to the ultimate direction of the beam. Thebeam’s direction must be within 1µrad of the nor-mal to the aligned mirror to keep the beam alignedon mirrors 4km apart, but that doesn’t mean thatthe mirror has not been biased by less than 10o toget it to the final position.

Pitch and displacement thermal noise. Wenow solve the equations for w+(s), θ and x. Theequations for these degrees of freedom in Eqns. 10are exactly the same as for the pendulum suspendedon a single wire (Eqn. 8), except for the addition ofa softening term to the torque equation, due to thetilted wires; and factors of two due to the two wires(with about half the tension) instead of a single wire.The extra term in the torque equation is a negligiblecontribution to the real part of Kθθ, at the level of1% for LIGO parameters. Therefore, the conclusionswe obtained, with respect to the optimization of thebeam location on the mirror, and the difference be-tween effective and measurable quality factors, areequally valid here. The spring constants we obtainedin Eqns. 9 involve now a factor 2T = Mg cosα, in-stead of T = Mg for a single wire. The elastic dis-tance is however determined by the tension in each

10

wire, ∆ =√EI/T =

√2EI/Mg cosα. To keep the

stress in the wires constant, the cross section areaof a single supporting wire is twice the area of twoeach of two supporting wires. Thus, the effectivequality factor determining the thermal noise for thedisplacement thermal noise x2(f) has a smaller di-lution factor of ∆/L = 1/326, instead of 1/231 for asingle wire. This is the well-know effect of reducingthermal noise by increasing the number the wires.The dilution factor for pitch is 1/14, considerablyhigher than the dilution factor for yaw, 1/44.

The displacement thermal noise at 160 Hz is√x2(160Hz) = 1.1× 10−20m/

√Hz, limiting the de-

tector sensitivity to h = 5.6 × 10−24/√

Hz. This isexpected to be lower than the thermal noise due tothe internal modes of the mirror mass, not consid-ered here [11]. The pendulum thermal noise couldbe reduced by a factor

√2, or about 40%, if the

beam spot was positioned at the optimal positionon the mirror. Since pendulum thermal noise is notthe dominant source noise, but the detectors’ shotnoise would increase due to diffraction losses, it isnot advisable for LIGO to proceed this way. How-ever, these considerations should be taken into ac-count for future detectors, where thermal noise maybe a severe limitation at low frequencies.

The pitch angular noise at 160 Hz, is√θ2(f) =

8.9 × 10−19rad√

Hz. Its contribution to the sensedmotion has to take into account the coupling withdisplacement, and we will do this in detail inthe lastsection.

B. Vertical, transverse displacement and roll

We are now concerned with the mirrormotion in its y, z and ψ degrees of free-dom. The potential energy is PEi =(1/2)

∫ L0

(T (w′i⊥)2 + EI(w′′i ⊥)2 + EA(w′i‖)

2)ds for

each wire, plus T (h cosα − b sinα)ψ2. Notice thatdue to the tilting of the wires, the “transverse” w⊥and “axial” w‖ directions are not y and z, but rota-tions of these directions by the wire tilt angle α. Wedefine wi⊥ for each wire pointing “out” (and thusin opposite directions if α = 0), and wi‖ pointingdown along the wire, from top to bottom.

The boundary conditions at top are wi⊥(0) =0, w′i⊥(0) = 0, wi‖(0) = 0, and at the bottom,wi⊥(L) = ±(y cosα − dψ) + z sinα, w′i⊥(L) = ±ψ,and wi‖(L) = ±(y sinα − cψ) − z cosα, where wedefined two new distances c = h sinα + b cosα andd = h cosα − b sinα. If we define as earlier, sums

and differences of the two wires shape functions,w±(s) = (w1(s) ± w2(s))/2, then the equations ofmotion for the mirror degrees of freedom, when sub-ject to external forces Fy, Fz and a torque Nx, are

Fy = −Mω2y + 2(Tw′−⊥(L)− EIw′′′−⊥(L)) cosα

+2EAw′−‖ sinα

Fz = −Mω2z + 2(Tw′+⊥(L)− EIw′′′+⊥(L)) sinα

−2EAw′+‖ cosα

Nx = −Jxω2ψ − 2EI(dw′′′−⊥(L) + w′′−⊥(L))

+2EAcw′−‖(L)

The solution for the wires’ transverse motionw±⊥(s) satisfying the boundary conditions up to or-der e−L/∆ and k∆ are of the same form as in Eqn.2, with top and bottom weights equal to

A− = (y cosα− (d+ ∆)ψ)/DB− = −((y cosα− dψ)(cos(kL) + k∆sin(kL))

−ψ(sin(kL)− k∆cos(kL))/k)/DA+ = z sinα/DB+ = −z sinα(cos(kL) + k∆sin(kL))/D (11)

The axial wire motion is

w±‖ = w±‖(L)sin(kzs)sin(kzL)

(12)

where w−‖(L) = y sinα − cψ and w+‖(L) =−z cosα. The wavenumber functions are k2 =ρω2/T , k2

z = ρω2/EA.Even though the tilting of the wires produces more

complicated formulas than in the pendulum-pitch-yaw case, the equations for the vertical motion de-couple from the equations from the transverse pen-dulum displacement and roll, similar to yaw decou-pling from pendulum and pitch. As before, if thewires move in phase (transverse or axially or both),they produce only vertical motion; but if they movein opposition, they produce side to side motion plusrotation around the optical axis. We analyze the twodecoupled systems separately.

Vertical thermal noise. Once we have solvedthe wire shape (from Eqns2,11, and 12), we can writethe equation of motion of the wire vertical displace-ment as

Fz = −Mω2z+2(EA

L

kzL

tan(kzL)cos2 α

+Tkcos(kL) + k∆sin(kL)sin(kL)− 2k∆cos(kL)

sin2 α

)z

11

or

Fz = −Mω2z + (KT sin2 α+KE cos2 α)z

where we defined KT as the spring constant that wasused in the pendulum-pitch case, and KE ∼ 2EA/Lthe spring constant of the wire. For LIGO parame-ters, and for usual wires, KT /KE � 1.

If the wires are not tilted and α = 0, we recoverthe simple case of vertical modes of a mirror hang-ing on a single wire. The restoring force is elasticand proportional to E, so there is no dilution fac-tor. The term added because of the wire tilting isa gravitational restoring force, much smaller thanthe elastic restoring force. Since it is also mostlyreal when considering a complex Young modulus E,it will not change significatively the loss terms, andthus the thermal noise. The wire tilting does add,however, the violin modes to the vertical motion,and it slightly decreases (by a factor cosα = 0.97)the frequency of the lowest vertical mode.

The vertical thermal noise at 160 Hz is√z2(f) =

3.1 × 10−18m/√

Hz, 260 times the pendulum ther-mal noise. This is due to the lower quality factor,and the higher mode frequency. However, verticalnoise is sensed in the gravitational wave interferom-eter through the angle of the laser beam and thenormal to the mirror surface, which is not less thanthe Earth’s curvature over 4km, (0.6 mrad). At theminimum coupling (0.3 mrad for each mirror in the4km cavity), the contribution due to vertical ther-mal noise is 10% of the pendulum thermal noise. Inadvanced detectors, vertical modes are going to be atlower frequencies due to soft vertical supports, likein the suspensions used in the GEO600 interferom-eter, but the ratio of quality factors is just the me-chanical dilution factor, so the contribution of verti-cal thermal noise to sensed motion will be of order3×10−4×√

L/∆×fz/fx ∼ 3×10−4×(EL2/Mg)1/4,not necessarily a small number!

Side pendulum and roll. The side motion androll of the pendulum are not expected to appearin the interferometer signal, but it is usually thecase that at least the high quality-factor resonancesdo appear through imperfect optic alignment. Theequations for the system y, ψ can be written as

(Kyy −Mω2)y −Kyψψ = Fy

(Kψψ − Jxω2)ψ −Kψyy = Nx

with

Kyy = KT cos2 α+KE sin2 α

Kψψ = KEc2 +KTd(d+ ∆)

+2T (d+ ∆)sin(kL)− k∆cos(kL)sin(kL)− 2k∆cos(kL)

Kyψ = Kψy = KT (d+ ∆) cosα+KEc sinα (13)

Within the (very good) approximationKT /KE �1, we can prove that the eigenfrequencies are w2

y =KT /M, fy = 0.75 Hz and ω2

ψ = KEb2/J, fψ = 19

Hz. The corresponding loss factors are φy ∼ ∆φ/2Land φψ ∼ φ.

If the wires are perfectly straight, the thermalnoise of the side-to-side pendulum motion below theviolin modes y2

0(f) is well approximated by that ofa simple oscillator with eigenfrequency ωy and a di-lution factor ∆/2L. The thermal noise of the rollangular motion ψ2(f) does not depend much onthe wire tilt, and is well approximated (below vi-olin modes) by the thermal noise of a simple oscilla-tor with eigenfrequency w2

r = 2EA/JxL and qualityfactor equal to the free wire’s quality factor (thereis no dilution factor).

For any small wire tilt, however, the spring con-stant Kyy in Eqn.13 has a large contribution ofKE, and thus the fluctuations increase as y2(f) ∼y20(f) cos2 α+ (Jx/Mb)2ψ2(f) sin2 α, as seen in Fig.

9. Since the roll eigenfrequency is higher than theside pendulum eigenfrequency, and its quality factoris lower, this is a significant increase in the thermalnoise spectral density ψ2(f), about a factor of 100for LIGO parameters. However, the gravitationalwave detectors are mostly immune to the roll degreeof freedom, as we will see later.

Also, the tilting of the wires introduces vio-lin modes harmonics of f (2)

n =√

(EA/ρ)/(2L) ∼n×6kHz apart from the usual harmonics f

(1)n =√

(T/ρ)/(2L) ∼ n× 300Hz.The violin modes that are most visible in the roll

thermal noise are the harmonics of√

(EA/ρ)/(2L)(and are strictly the only ones present if the wiresare not tilted).

C. Violin Modes

We have explored the relationship between qualityfactors of pendulum modes and the “effective” qual-ity factor needed to predict the thermal noise of anygiven degree of freedom (seen only in sensitive in-terferometers). For the most important longitudinalmotion, we’ve seen that the effective quality factor isapproximately equal to QwL/∆, where Qw = 1/φwis the quality factor of the free wire, related to the

12

imaginary part of the Young modulus. Under idealconditions where the pendulum mode is far awayfrom the pitch mode in frequency, its quality factoris close to the effective quality factor, but as we haveseen, the errors may be as large as 50%.

The violin modes, approximately equal to fn =n√T/ρ/2L, appear in the horizontal motion of the

pendulum in both directions, along the optical axisand transverse to it. The violin modes show someanharmonicity, as pointed in [3],with the frequen-cies slightly higher than n

√T/ρ/2L, and the quality

factors degrading with mode number. The complexeigenfrequencies are the solutions to the equation

kL = nπ + arctan(2k∆/(1− (k∆)2))

with k2 = ρω2/T . This means that if we measurethe quality factors of violin modes, and they followthe predicted anharmonic behavior in both direc-tions, we can assume the losses are only limited bywire losses. We can then predict the thermal noise inthe gravitational wave band with more confidence,having also consistency checks with the quality fac-tors measured at the pendulum modes. The thermalamplitude of the peaks at the violin modes follows asimple 1/f5/2 law, corrected by the change in longi-tudinal Qs. We show all these features in Fig.12.

D. Total Pendulum Thermal Noise in LIGO I

The right way to calculate the total thermal noiseobserved in the interferometer signal is to calcu-late the pendulum response to an applied oscillatingforce in the direction of the laser beam. The pen-dulum responds to a force in all its six degrees offreedom, but the motion we are sensitive to is themotion projected on the laser beam’s direction.

If the laser beam is horizontal, and its directionpasses through the mirror center of mass, it will besensitive to only longitudinal displacement and pitchmotion. If the beam is not horizontal, and for ex-ample is tilted up or down by an angle γ, (but stillgoing through the center of mass), we imagine an ap-plied force F applied in the beam’s direction, witha horizontal component F cos γ and a vertical com-ponent F sin γ. The motion we are interested in isx cos γ+z sinγ, since it is the direction sensed by thelaser beam. The admittance we need to calculate isthen Y = iω(x cos γ+z sinγ)/F = iω((x/Fx) cos γ+(z/Fz) sin γ) = iω(Yx cos2 γ + Yz sin2 γ), where Yxis the response of the horizontal displacement to ahorizontal force and Yz is the vertical response to

a vertical force. A force applied in the beam’s di-rection, with magnitude F0, will have componentsin all 3 axes, and torques around all 3 axes too:~F = F0(αx, αy, αz) and ~N = F0(Dψ , Dθ, Dψ). Themotion we are interested is, in general, χ = αxx +αyy + αzz + Dθθ + Dφφ + Dψψ. If the motion inall 6-DOF was uncoupled, then each dof responds tojust one component of the force or the torque, andthe admittance we need would be just the sum of theadmittances, each weighted by the square of a fac-tor αi or a distance Di. However, only z and φ aredecoupled dof from the rest, and x, θ and y, ψ formtwo coupled systems, for which we need to solve theresponse to a forces and torques together.

We define an admittance Yxθ = iω(x/Ny) as theadmittance of displacement x to an applied torqueNx, and Yθx = iω(θ/Fx) as the admittance of pitchto an applied horizontal force , and similar quanti-ties Yyψ, Yψ,y. Then, the total thermal noise sensedby the laser beam is

χ2(f) = 4kBT0ω2 <(

α2xYxx + αxDθ(Yxθ + Yθx) +D2

θYθθ

+α2yYyy + αyDψ(Yyψ + Yψy) +D2

ψYψψ

+α2zYzz +D2

φYφφ)

or

χ2(f) = α2xx

2(f) +D2θθ

2(f) + αxDθ4kBT0

ω2<(Yxθ + Yθx)

+α2yy

2(f) +D2ψψ

2(f)

+αyDψ4kBT0

ω2<(Yyψ + Yψy)

+α2zz

2(f) +D2φφ

2(f)

so it is a weighted sum in quadrature of the ther-mal noise of different degrees of freedom, plus somecross-terms. These terms may be negative, so it ispossible to choose an optimal set of parameters tominimize the sensed motion, as shown in [5]. Theweighting factors and distances are

αx = cos(γy) cos(γz)αy = sin(γy) cos(γz)αz = sin(γz)Dθ = R sin(γz) + dz cos(γy) cos(γz)Dφ = cos(γz)(dy cos(γy)−R sin(γy))Dψ = dy sin(γz) + dz sin(γy) cos(γz)

Typical distances dy, dz are few mm at most, andtypical angles γyare in the order of microradians,since an angle of such magnitude produces displace-ments of the order of millimeters at the beam at the

13

other end of the arm, 4km away. However, the angleγz cannot be less than half the arm length divided bythe curvature of Earth, or γz ≥ 3 × 10−4. We showin Fig13 the thermal noise sensed by a beam withdy = dz = 5mm , γz = 3× 10−4 and γy = 1µrad.

IV. RESULTS AND CONCLUSIONS

We have shown several general results, and thencalculated predicted thermal motions for LIGO sus-pensions. First, we showed that if there is a sin-gle oscillator with two sources of potential energy,one with a spring constant Kg and a dominant realpart, and the other with a complex spring constantKe, with a dominant loss factor, then the dilutionfactor Ke/Kg gives us the ratio between the oscil-lator’s quality factor (determining its thermal noisespectral density) and the loss factor ofKe. However,the elastic loss φe = =Ke/<Ke might itself have asmall dilution factor with respect to the loss factorof the Young modulus, for example if Ke ∼

√E,

then there is a dilution factor of 1/2.We also show that when two or more degrees of

freedom are coupled, the measurable quality factorsat resonance may not be as useful to predict the“effective” quality factor used in the thermal noisespectral density, unless the eigenfrequencies are farfrom each other.

We showed that by using approximations of or-der e−L/∆, we can easily obtain wire shapes andequations of motion for the 6 degrees of freedomof the mirror, as a function of applied oscillatingforces and torques. We think that this method willbe most useful when applied to multiple pendulumsystems such as those used in GEO600 and plannedfor advanced LIGO detectors [1,12]. However, evenin the simple pendulum case this approach allowsus to calculate the gravitational and elastic poten-tial energy as a linear energy density along the wire,and the total energy. We showed that, for an ap-plied horizontal force, the gravitational potential en-ergy ((1/2)

∫Tw′2ds) is homogeneously distributed

along the wires, while the elastic potential energy((1/2)

∫EI(w′′)2ds) is concentrated at the top and

bottom, but in different proportions depending onthe frequency of the applied force (Fig. 5. We alsocalculate the ratio of total elastic potential energy togravitational energy using the solutions for the wireshape, and show that this function of frequency cor-responds to the dilution factor for the eigenmodeloss factors as well as for the effective quality fac-tor that allows us to calculate the thermal noise at

gravitational wave frequencies (see Figs.1,4,8).Applying our calculation of wire shapes and equa-

tions of motion that include elasticity to LIGO sus-pensions, we show in Figs 10 and 11 the resultingspectral densities of displacement and angular de-grees of freedom of the mirror. More importantly,we show in Fig13 the resulting contribution of pen-dulum thermal noise to the LIGO sensitivity curve,assuming small misalignments in the sensing laserbeam (5mm away from center of mass, 1µrad awayfrom horizontal). As expected, the displacement de-gree of freedom is the one that dominates the con-tribution, but pitch noise contributes significantly(29% at 100 Hz) if the beam is 5mm above center.But this is not added in quadrature to the horizontalnoise, which makes 81%: the coupled displacement-pitch thermal motion makes up 99% of the totalthermal noise. A misplacement below the center ofmass will reduce the observed thermal noise, as firstnoted in [5]. The contribution of yaw thermal noise(11% at 100Hz) is smaller than that of pitch, butvery comparable. The contribution of vertical noisedue to the 4km length of the interferometer is 8%at 100 Hz. The side and roll motions are coupled,but the roll contribution dominates (due to the largeangle γz and the assumed dz = 5mm) and is 0.7%,much smaller than the contributions of pitch, yawand vertical degrees of freedom. When added inquadrature, the total thermal noise is 23% higherthan the contribution of just the horizontal thermalnoise. However, if the vertical misplacement of thebeam is 5mm below the center of mass, instead ofabove, the total contribution of thermal noise is 89%of the horizontal thermal noise of the center mass.

V. ACKNOWLEDGMENTS

Much of this work was motivated by many discus-sions held with Jim Hough, Sheila Rowan and PeterSaulson, and I am very glad to thank their insights.I want to especially thank P. Saulson for carefullyreading the original manuscript and making impor-tant suggestions. I also want to thank P. Fritscheland Mike Zucker, who first asked about thermalnoise of angular modes in pendulums. This workwas supported by NSF grants 9870032 and 9973783,and by The Pennsylvania State University.

14

[1] Recent reviews of the status of interferometric grav-itational waves detectors around the world can befound in the Proceedings of the Third EdoardoAmaldi Conference on Gravitational Waves, inpress: M. Coles, The Status of LIGO; F. Marion,The status of the VIRGO Experiment; H. Luck, P.Aufmuth, O.S. Brozek et al.,The status of GEO600;M. Ando, K. Tsubono, TAMA Project: Design andcurrent status; and D. McClelland, M.B. Gray et al,Status of Australian Consortium for Interferomet-ric Gravitational Astronomy. Electronic files can befound at http://131.215.125.172/info/paperindex/

[2] P.R. Saulson, Phys. Rev. D 40, 2437 (1990).[3] G.I. Gonzalez and P.R. Saulson, J. Acoust. Soc. Am.

96,207 (1994).[4] J. Logan, J.Hough and N. Robertson, Phys. Lett. A

183,145 (1993).

[5] V.B. Braginsky, Y. Levin, S. Vyatchanin, Mea-sur.Sci.Tech. 10,598 (1999).

[6] A. Gillespie and F. Raab, Phys. Lett. A 190, 213(1994).

[7] Matthew Hussman, Ph.D. Thesis, Uiversity of Glas-gow, 2000.

[8] S. Rowan et al., Physics Letters A, 233,303 (1997).[9] A. Gretarsson et al., preprint gr-qc/9912057.

[10] Jim Hough, Peter Saulson, Mitrofanov (personalcommunication); Damping Dilution Factor for apendulum in an interferometric gravitational wavedetector, G. Cagnoli et al., preprint.

[11] A. Gillespie and F. Raab, Phys. Rev. D 52, 577(1995).

[12] Abramovici et. al, Science 256,325 (1992). See alsoinformation on http://www.ligo.caltech.edu

15

10−15

10−10

10−5

100

Potential energy

V(N

m)

10−1

100

101

102

103

0

2

4

6

8x 10

−3

f(Hz)

dilu

tion

fact

or

ElasticGravitational

Vel/Vg∆/2L(∆/L)(1+ρ L∆ω2/T)eigenfrequencies

FIG. 1. Top figure: Gravitational and potential energies for a suspended point mass as a function of frequencyf , when excited by a sinusoidal force wiht frequency f . Bottom figure: ratio of energies, considered as a “dilutionfactor”. The stars indicate the factor φ/Qi, where φ is the wire loss factor and Qi is the eigenmode quality factor.

16

0

0.2

0.4

0.6

0.8

1

1.2

freq

(H

z)

0 2 4 6 8 10 12 14 16 18 2010

3

104

105

106

h(mm)

Q

FIG. 2. Eigenfrequencies and associated loss factors for a pendulum suspended on one wire. Quality factors andcorresponding eigenfrequencies are represented with the same symbol (points or stars). The lines in the top figure

are given by expressions for the pendulum frequency fp =√

g/l/(2π) (solid) and pitch frequency fθ =√

Th/J/(2π)(dashed). The lines in the bottom figure correspond to the effective quality factors for displacement (solid) and forpitch (dashed). Approximate expressions for the effective quality factors are Qp ∼ L/∆φ and Qθ ∼ 2(h + ∆)/∆φ,respectively.

17

10−1

100

101

102

103

10−20

10−15

m/r

tHz

Displacement Thermal Noisew/Qeff=2.3e5

10−1

100

101

102

103

10−20

10−15

10−10

freq (Hz)

rad/

rtH

z

Pitch Thermal Noisew/Qeff=1.0e4

FIG. 3. Displacement (top) and pitch (bottom) thermal noise of a single-wire pendulum, calculated without ap-proximations (solid line), and approximated with a single mode with an effective Q (dotted line). The pitch distanceused was 8.3 mm

18

10−5

100

Pot

entia

l Ene

rgy(

Nm

)

h=8.2mm

10−1

100

101

10−3

10−2

10−1

100

f(Hz)

dilu

tion

fact

or

ElasticGravitationalpendulum resonancepitch resonance

h=2mmh=8.2mmh=20mmQ

effφ ~ ∆/L

FIG. 4. At the top, we plot the elastic and gravitational potential energies for a pitch distance h=8.2mm. In thebottom figure, we plot the ratio of elastic and gravitational energy for three different values of the pitch distance.The stars represent the dilution factor of the pendulum mode, at the pendulum frequency; the circles are the dilutionfactors of the pitch mode, at the pitch eigenfrequency; and the solid horizontal line is the dilution factor of theeffective quality factor at 50 Hz.

19

∫0s w’2(s) ds / ∫

0L w’2(s) ds

f=0.

3 H

z

R=7.1, 3.3, 2.80

50

100

∫0s w’’2(s) ds / ∫

0L w’’2(s) ds

f=f p∼

0.75

Hz

R=0.74, 44, 820

50

100

f=f θ∈

[0.4

Hz,

3H

z]

103, 190, 1810

50

100

f=50

Hz

R<10−6

0

50

100

0 0.1 0.2 0.3 0.4

1st v

iolin

mod

e

wire coordinate (m)

R<10−10

0 0.1 0.2 0.3 0.40

50

100

wire coordinate (m)

h=2mmh=8mmh=20mm

FIG. 5. The figure shows cumulative potential energies along the wire (top is s = 0, bottom is s = 0.45m), whenthe pendulum is excited by a force a the center of mass of frequency f . The energies are shown for three differentpitch distances in each graph: h=2mm (dashed), h=8mm (thick points) and h=20mm (dotted). The energies are alsoplotted for five different frequencies: f = 0.3 Hz, below all eigenmodes; f = fp orpendulum frequency, approximately

0.75 Hz; f = fθ =√

Th/J/(2π) = (0.46, 0.69, 1.12)Hz for h = (2, 8, 20)mm respectively; f=50Hz, in the gravitationalwave band; and f = 319Hz, the first violin mode frequency.

20

−25 −20 −15 −10 −5 0 51

2

3

4x 10

−19

d(mm)

asd@

50H

z)(m

/rtH

z)

d=J/ML=−10.5 mmoptimal d=−9.8 mm

10−16

10−14

10−12

asd

(m/r

tHz)

*f5/

2

d=0mmd=−10mm

0.1 1 10 50 100 10000.5

1

1.5

asd

ratio

freq (Hz)FIG. 6. In the top figure, we plot the thermal noise amplitude spectral density (ASD) at 50 Hz, as a function of

the distance d at which the laser beam is positioned. In the middle figure, we plot the thermal noise ASD times f2/5

obtained for d = 0 and for the optimal distance for f = 50Hz. In the bottom figure, we plot the ratio of the twocurves in the middle figure.

21

FIG. 7. LIGO pendulum suspension, with a single wire loop, attached slightly above the center of mass and angledtowards the center.

22

10−10

10−5

100

105

Potential energyV

(Nm

)

ElasticGravitational

10−1

100

101

102

103

0.005

0.01

0.015

0.02

0.025

0.03

f(Hz)

dilu

tion

fact

or

Vel/Vg(b/a)(∆/L)(2∆/L)(1+Lρ∆ω2/2T)

FIG. 8. Potential energies and dilution factor for LIGO yaw degree of freedom

23

10−1

100

101

102

103

104

10−20

10−15

10−10

m/r

tHz

10−1

100

101

102

103

104

10−20

10−15

10−10

Hertz

rad/

rtH

z

a/b=1 (straight wires)a/b=1.1a/b=0.13 (LIGO I)

FIG. 9. Thermal noise of pendulum side and roll motion, for different values of the distance between the wiresat the top. The wires are tilted with respect to the vertical direction at an angle sinα = (b − a)/L, where L is thelength of the wires.

24

10−1

100

101

102

103

104

10−22

10−20

10−18

10−16

10−14

10−12

10−10

10−8

Hertz

m/r

tHz

horizontal longitudinalhorizontal transversevertical

FIG. 10. Thermal noise of the three mirror’s displacement degrees of freedom.

25

10−1

100

101

102

103

104

10−22

10−20

10−18

10−16

10−14

10−12

10−10

10−8

Hertz

rad/

rtH

z

pitchyawroll

FIG. 11. Thermal noise of the three mirror’s angular degrees of freedom.

26

0.95

1

1.05

Fn/n

F1

100

101

102

103

Qn

Qn

Qeff

(50Hz)/2

0 5 10 15 20 25 3010

−20

10−15

m/H

z1/2

mode number

Pendulum−Pitch motionSide−Roll motion∝ f5/2

FIG. 12. In the top figure, we show the frequencies of the resonant violin modes, scaled by fn = n√

T/rho/2L,for longitudinal and transverse modes. In the middle figure, we show the corresponding quality factors. The solidline is Qeff/2 ∼ L/2∆. In the bottom figure, we show the amplitude of longitudinal and transverse thermal noiseat the violin modes. The solid line shows a 1/f5/2 fall off from the first peak.

27

10−1

100

101

102

103

104

10−24

10−22

10−20

10−18

10−16

10−14

10−12

10−10

Hertz

m/r

tHz

SRDxpitchyawzrollytotal

FIG. 13. Thermal noise sensed by a laser beam 5mm away from center, and 1rad away from horizontal. The“SRD” curve is the expected sensitivity of LIGO I. The individual degrees of freedom are plotted in their order ofcontribution at 100 Hz.

28