Radiation pressure and stabilityof interferometric gravitational-wave detectors

Vijay Chickarmane, Sanjeev V. Dhurandhar, Roland Barillet, Patrice Hello, andJean-Yves Vinet

The effect of radiation pressure on the stability of Fabry–Perot cavities with hanging mirrors is inves-tigated. Such cavities will form an integral part of the laser interferometric gravitational-wave detec-tors that are being constructed around the globe. The mirrors are hung by means of a pendulumsuspension and are locked by servo controls. We assume a realistic servo-control transfer function thatsatisfies the standard stability criteria. We find that for positive offsets from the resonance of the cavitythe system is stable. However, we show that for negative offsets instabilities can occur, although theservo system has the effect of increasing the instability threshold, compared with the nonservoed case.Conditions for stability are finally given, involving the finesse of the cavity, the input power, the mass ofthe mirrors, the servo gain, and the phase detuning from perfect resonance. Gravitational-wave detec-tors with arm cavities having a finesse as low as approximately 200 could exhibit instabilities. Someimplications for the locking of these detectors are finally given. © 1998 Optical Society of America

OCIS codes: 120.0120, 000.2780, 350.5610, 040.0040.

1. Introduction

Considerable progress has been made over the pastfew decades in the development of laser interferomet-ric gravitational-wave detectors. These detectors,built on the lines of large Michelson interferometers,are sensitive to length changes in the two arms,which may be due to incident gravitational waves.1,2

Each arm of the detector comprises a Fabry–Perotcavity made of a corner mirror and an end mirrorsuspended as a pendulum. An incident gravita-tional wave modulates the phase acquired by thepropagating beam of light in the two arms of theinterferometer in antiphase. This can be observedas a time-dependent variation in the intensity at thephotodetector. Various noise sources plague the in-

V. Chickarmane and S. V. Dhurandhar are with the Inter Uni-versity Centre for Astronomy and Astrophysics, Postbag 4,Ganeshkind, Pune 411007, India. The following are with theVirgo Collaboration: R. Barillet, Laboratoire de l’Horloge Atom-ique, Universite Paris-Sud, Orsay 91405, France; P. Hello, Labo-ratoire de l’Accelerateur Lineaire, Universite Paris-Sud, Batiment208, Orsay 91405, France; and J.-Y. Vinet, Laboratoire d’OptiqueAppliquee, Ecole Polytechnique, Ecole Nationale Superieure deTechniques Avancees, Palaiseau 91761, France.

Received 3 June 1997; revised manuscript received 3 November1997.

0003-6935y98y153236-10$15.00y0© 1998 Optical Society of America

3236 APPLIED OPTICS y Vol. 37, No. 15 y 20 May 1998

terferometer, such as the seismic noise, thermalnoise, and shot noise.3 But in the high-frequencyregime of the bandwidth of the detector it is the pho-ton shot noise that dominates the noise spectrum.The photon shot noise falls off inversely as the squareroot of the power, and therefore for this purpose highinput powers are conducive for obtaining high sensi-tivities. The power-recycling technique4,5 serves toincrease the laser power of the light in the inter-ferometer and hence makes it more sensitive. As aresult of the recycling technique, the incident powersare of the order of kilowatts and could easily increaseto a few hundred kilowatts or even a megawatt in theadvanced detectors envisaged. The high powers in-cident on the mirrors can disturb the delicate balancein these highly-tuned sensitive instruments andcould either drive the cavity out of resonance or pro-duce time-dependent effects such as instabilities.Thermal absorption of laser power within the mirrorcan heat up the mirror, deforming it and effectivelychanging the length of the cavity. If the mirrors arefreely hanging, the effect of the radiation pressure onthe mirrors is important and must be taken into ac-count in the design of the interferometric system.

This problem has already been considered by sev-eral authors. Dorsel et al.6 first studied experimen-tally the occurrence of multistability caused by theradiation pressure force. The basics of multistabil-ity were then established theoretically in Ref. 7. Us-

ing more rigorous techniques, Deruelle andTourrenc8,9 and Aguirregabiria and Bel10 consideredthe stability of a cavity with freely suspended mirrorswith time-delay effects. They also did a numericalinvestigation of the mirror motion in the unstableregime. Meers and MacDonald11 also consideredthis problem in the context of proposed gravitationaldetectors that included recycling techniques. Theyincluded a crude servo system modeled simply as adamped harmonic oscillator, which they found couldstabilize the system under certain conditions. Herewe consider a realistic servo system for the large-scale detectors underway. The radiation pressurepushes on the mirrors, modifying the length of thecavity, which changes the incident power on the mir-rors, which in turn changes the radiation pressure.The coupling to the power is nonlinear, and moreoverthere are retardation effects; hence one can expectinstabilities.6,8,12 In this paper we consider only theeffects of radiation pressure pushing on the mirrors.However, thermal effects are also relevant when highpowers are employed since the mirror can be de-formed owing to thermoelastic effects.13 It has beenshown that, although the operating point of the cav-ity can be altered, no instabilities in the form of os-cillations are observed.14

In the present analysis we include the effect of theservo system since the mirrors in the actual cavitiesare locked by the servo. We used the servo-controltransfer function given by Caron et al.15 in incorpo-rating the effect of the servo. We proceed as follows:First we obtain the transfer function connecting thetime-dependent length of the cavity to the power thatfollows these changes. The method we use in deriv-ing this last result is closely related to the one used inRef. 11. We find that, at frequencies much smallerthan the storage time scale of the cavity, the powerfollows essentially the static Fabry–Perot curve, andonly when the changes in the cavity length are rapidenough is the response nontrivial as given by thetransfer function. The instantaneous radiationpressure force is just proportional to the power as afunction of the detuning. The radiation pressureforce and the servo force act together on the sus-pended mirror. In effect, we obtain a feedback loopand thus a characteristic equation for the mirror dis-placement. The roots of this equation determine thestability of the system, and they essentially dependon the phase offset d of the operating point, the fi-nesse of the cavity, and the input power. We findthat, for d . 0 and for relevant parameters, the sys-tem is stable. For d , 0, but chosen within the line-width, for a given finesse and above a certain criticalpower the cavity becomes unstable. The criticalpower depends on the servo gain. If no servo is em-ployed the critical power is small but nonzero owingto the restoring force of the pendulum suspension.

2. Transfer Function Connecting Cavity Length to theStored Power

In this section we derive the relation between thechange in the stored power as the length of the cavity

varies in time. We consider only small changes inthe length of the cavity so that the relation is linearand can be expressed by means of a transfer function.To do this we first need to calculate the intracavityfields and from these the intracavity power.

We consider a single mirror, the end mirror, sus-pended as a pendulum, which is acted on by the ra-diation pressure, the gravity, and the servo forces.In the static limit the radiation pressure, the restor-ing force of the pendulum, and the servo force balanceeach other to produce an equilibrium position for themirror. We examine the stability of the systemabout the equilibrium position ~or one of the posi-tions! by considering how a small motion of the mir-ror affects the intracavity power. To do this we mustfirst investigate how the electric field changes as themirror is moved. Subsection 2.A deals with this is-sue.

A. Electric Fields

Let the length of the cavity be modulated at a par-ticular frequency v. Obtaining the response at asingle frequency is equivalent to calculating the im-pulse response of the cavity to a change in its length.This motion of the mirror modulates the phase of thelight in the cavity. If the incident laser frequency~carrier! is vl, then at least two sidebands are pro-duced at frequencies vl 6 v. These sidebands beatwith the carrier to give rise to an intensity changethat exerts time-dependent radiation pressure on theend mirror. Let the cavity be aligned along the xaxis, and let the corner-mirror and end-mirror am-plitude reflectivity be r0 and r1, respectively. Thetotal round-trip travel time for light is t 5 2lyc, wherel is the length of the cavity and c is the speed of light.The carrier wave ~vl 5 2pcyl! acquires a phase owingto a small motion dx~t! 5 x0 sin vt,

df~t! 5 2kdx~t! 5 2kx0 sin vt, (1)

where the wave vector k 5 2pyl. In Eq. ~1! of thesmall phase shift we consider the wavelength as aconstant, the mechanical frequency being small com-pared with the optical one, and the sidebands areconsidered to have the same wavelength. Referringto Fig. 1, we note that the equations for the electric

Fig. 1. Electric fields in a Fabry–Perot cavity with front-mirrorreflectivity r0 and a suspended end mirror with reflectivity r1.

20 May 1998 y Vol. 37, No. 15 y APPLIED OPTICS 3237

fields at the corner mirror are

E1~t! 5 t0 E0~t! 1 r0 E2~t!, (2)

E2~t! 5 r1 E1~t 2 t!exp@2idf~t 2 ty2!#. (3)

The sign convention for reflection is chosen such thata positive sign is associated with a reflection from theinner surfaces, whereas a negative sign is associatedwith the outer surfaces. In Eq. ~3! we see that theextra phase change acquired by the light ~owing tothe moving mirror!, which arrives at a certain time atthe corner mirror, depends on the motion of the mir-ror at a delayed time, the delay being precisely half

x 5

1t0

1 2 Re~2ivlt!

2t0eReF2iSvl 2v

2DtG[1 2 Re~2ivlt!#$1 2 Re[2i~vl 2 v!t%

t0eReF2iSvl 1v

2DtG@1 2 Re~2ivlt!#$1 2 Re@2i~vl 1 v!t#%

t0eReF2iSvl 2v

2DtG@1 2 Re~2ivlt!#$1 2 Re@2i~vl 2 v!t#%

t0

1 2 Re@2i~vl 2 v!t#0

2t0eReF2iSvl 1v

2DtG@1 2 Re~2ivlt!#$1 2 Re@2i~vl 1 v!t#%

0t0

1 2 Re@2i~vl 1 v!t#

2 .

(8)

the round-trip travel time. Substituting for E2 inEq. ~2!, we obtain

E1~t! 5 t0 E0~t! 1 r0 r1 E1~t 2 t!

3 exp@22ikx0 sin v~t 2 ty2!#. (4)

Taking Fourier transforms of the electric fields andassuming that the mirror motion is much smallerthan a wavelength, we expand to first order in e [kx0. Denoting R 5 r0r1, we obtain the followingequations at the carrier frequency and the two side-bands:

E1~vl! 5 t0E0~vl! 1 RE1~vl!exp~2ivlt!

2 eRE1~vl 2 v!expF2iSvl 2v

2DtG1 eRE1~vl 1 v!exp@2i~vl 1 vy2!t#,

E1~vl 2 v! 5 t0E0~vl 2 v! 1 RE1~vl 2 v!exp@2i~vl

2 v!t# 1 eRE1~vl!exp@2i~vl 2 vy2!t#,

E1~vl 1 v! 5 t0E0~vl 1 v! 1 RE1~vl 1 v!exp@2i~vl

1 v!t# 2 eRE1~vl!exp@2i~vl 1 vy2!t#. (5)

In Eqs. ~5! all sidebands higher than the first areneglected, since these come with the laser light con-

3238 APPLIED OPTICS y Vol. 37, No. 15 y 20 May 1998

tribution at O@~kx0!2# or higher. If Ek stands for thecolumn vector

F Ek~vl!Ek~vl 2 v0!Ek~vl 1 v0!

G 5 Ek, k 5 0, 1, (6)

then the above equations can be written in matrixform. We invert the matrix with the approximationkx0 ,, 1 2 R, i.e., the frequency shift equivalent tothe displacement of the mirror is much smaller thanthe linewidth of the cavity. @For high-finesse cavi-ties the FWHM is ;2~1 2 R!.# Thus

E1 5 xE0, (7)where

Since the input electric field E0 is just at the carrierfrequency, only the first column of the matrix is rel-evant and the intracavity field is a superposition ofthe fields at the carrier and the sideband frequencies.Thus

E1~t! 5 $x11 exp~ivl t! 1 x21 exp@i~vl 2 v!t#

1 x31 exp@i~vl 1 v!t#E0~vl!. (9)

B. Static Component of the Intensity

We can obtain the intensity of the laser beam at anypoint by taking the square modulus of the electricfield at that point. Equation ~9! gives the intracav-ity electric field E1 at the corner mirror. At the endmirror

E19~v! 5 E1~v!exp~2ivty2!, (10)

where the v could be vl, vl 6 v. The intensity can bebroken up as a time-dependent part Idc and a time-varying part dI~t!, which is due to the motion of themirror. The first term in Eq. ~9! gives Idc, where

Idc~d! 5 ux11E0~vl!exp~2ivlty2!u2

5t0

2uE0~vl!u2

1 2 2R cos~d! 1 R2 , (11)

where the phase offset vlt 5 d 1 2np was introduced.Obviously, near resonance d ,, 2p. Equation ~11! isthe usual Fabry–Perot curve with resonances at vltequal to multiples of 2p. The FWHM is .2~1 2 R!when R is close to unity. The finesse of the cavity is^ 5 p=Ry~1 2 R!. For Virgo the finesse is chosento be approximately 5017, which corresponds to R ;0.94, and for the Laser Interferometer Gravitational-Wave Observatory ~LIGO! the finesse is approxi-mately 20018 ~R ; 0.985!.

If we ignore time-delay effects and assume that theintracavity power variation follows the static curve@that is, the motion of the mirror is slow comparedwith the time scale set by the storage time of thecavity, tstor 5 ty~1 2 R!#, we can naively understandthe nature of the problem when the radiation force isincluded. The radiation pressure force is Frp 5 2IdcAeffyc, where the intracavity power is Pdc 5 IdcAeff,Aeff is the effective area of the laser beam, and c is thespeed of light, c . 3 3 108 ms21. The restoring forceof the pendulum is 2Mvp

2x, where x 5 dy2k and vpis the natural pendulum angular frequency. In Fig.2 we plot the total force Ftot and the correspondingpotential that acts on the end mirror as a function ofthe displacement of the end mirror ~for convenientparameters!:

Ftot 5 2IdcAeffyc 2 Mvp2x. (12)

Fig. 2. ~a! Schematic depicting the force experienced by the endmirror versus the displacement. ~b! The corresponding potential.Points ~1! and ~2! represent unstable and stable equilibrium, re-spectively.

In Fig. 2~a! the points where the curve cuts the x axisare the multistable points ~zero force!. The slope ofthe curve at these points determines the stability.For a positive slope any small displacement awayfrom the equilibrium tends to take the end mirrorfarther away, and therefore these points are unsta-ble; accordingly, the potential is maximal at thesepoints. For the points at which the slope is negativethe force decreases and hence the mirror is broughtback to equilibrium, and therefore these points arestable ~minimum of potential!. For a larger power inthe cavity more of these multistable points appear.As the finesse increases, the magnitude of the slopealso increases ~as the peak gets narrower! and thestability–instability becomes more pronounced.

C. Time-Dependent Component of the Intensity

For the time-varying part of the intensity we compute

dI~t! 5 x11*x21uE0u2 exp@2iv~t 2 ty2!#

1 x11*x31uE0u2 exp@iv~t 2 ty2!# 1 c.c. (13)

The Fourier transform dI~v! is then

dI~v! 5

2it02RuE0u2e sin d exp~2ids!

u1 2 Re~2id!u2$1 2 Re@2i~d 1 ds!#%$1 2 Re@i~d 2 ds!#%,

(14)

where vt 5 ds. The factor e 5 kx0 contains x0, whichcan be related to the Fourier transform of the positionof the mirror as dx~v! 5 x0y2i. We can obtain thepower simply by multiplying by the effective area Aeffand we can relate the Fourier component of the powerto that of the position by means of a transfer functionas

dP~v!

P05 K~v!dx~v!, (15)

where P0 5 AeffuE0u2 is the input power. The expres-sion for K~v! that uses Eqs. ~13!–~15! is

K~v! 5

24kRt02 sin d exp~2ids!

u1 2 Re~2id!u2$1 2 Re@2i~d 1 ds!#%$1 2 Re@i~d 2 ds!#%.

(16)

This transfer function is derived with the restric-tion that the frequency drift related to the mirrordisplacement is much smaller than the linewidth ofthe cavity, but there is no restriction on v. However,we are interested in stability problems at frequencieslying in the bandwidth of the detector, ;10–2000 Hz.In this regime we can obtain a simplifying approxi-mation of the transfer function that then becomes arational function. In Subsection 2.D we obtain thesimplified form.

20 May 1998 y Vol. 37, No. 15 y APPLIED OPTICS 3239

D. Low-Frequency Approximation of the TransferFunction

If we assume that vt ,, 1, i.e., the fluctuations havea frequency v much less than the free-spectral range,then, keeping terms to second order in v, we get

K~v! 5 2g~d!

f ~V!, (17)

where V is a dimensionless frequency and is definedin terms of the storage time tstor, or equivalently as afunction of R and t, as

V 5 vtstor 5 vty~1 2 R!. (18)

In Eq. ~17! g~d! is like the envelope of the intensitychange, whereas f ~V! is the dynamical part. Theenvelope is determined by the operating point d andis appreciable within only the peak of the static curve.We have

g~d! 5 F 4kRt02 sin d

~1 2 R!4~1 1 F sin2 dy2!GS 12R2 2 R cos dD ,

(19)

f ~V! 5 V02 1 2ilV 2 V2, (20)

where

V02 5

1 1 F sin2 dy22R2 2 R cos d

,

l 5R~cos d 2 R!

~1 2 R!~2R2 2 R cos d!,

F 54R

~1 2 R!2 . (21)

Inspecting f ~V!, we find that V0 is the frequency ofoscillation of the intensity change, whereas l is pro-portional to the damping.

When V ,, V0, we recover the static case. Then

K~V 5 0! 5 2F 4kRt02 sin d

~1 2 R!4~1 1 F sin2 dy2!2G ;1P0

dPdc

dx,

(22)

where Pdc 5 AeffIdc is the intracavity power.Hence we see that at low frequencies the intensity

varies linearly with length ~for small changes inlength!, with a proportionality constant being essen-tially the derivative of Idc, which is described by thestatic-curve equation ~11!. At frequencies compara-ble with V0, the transfer function resembles that of adamped harmonic oscillator with the strength of theresponse scaled by g~d!. This strength depends pri-marily on the operating point d and is significant forvalues of d within the peak of the static curve. Toget some idea of the numbers involved, we can use theVirgo finesse ~^ 5 50!, which corresponds to R 50.94. We can examine the case when the operatingpoint is such that half of the maximum power isstored in the cavity, then d ; 6~1 2 R! 5 6 .06. ~Forthe operational gravitational-wave detectors d is

3240 APPLIED OPTICS y Vol. 37, No. 15 y 20 May 1998

smaller, typically less than a tenth of the HWHM ofthe Airy peak.16,15! For these parameters we get V0; 1.51, l ; 1.1, and thus the period of oscillation andthe damping time are of the same order. The time-dependent intensity therefore oscillates like a criti-cally damped harmonic oscillator with the time scaleof the storage time tstor.

3. Coupling the Radiation Pressure to the MirrorMotion

In Section 2 we calculated the response of the powerto the time-varying change in the cavity length,which could be due to the motion of the mirror. Inthis section we perform the reverse computation oflinking the power through the radiation pressure tothe mirror motion and obtain a characteristic equa-tion for the motion of the mirror. The forces actingon the mirror are the radiation pressure force, thegravity force that tends to restore the pendulum toequilibrium, and the force exerted by the lockingservo loop installed to maintain the resonance. Wemust include the servo force in the model because noreal suspended Fabry–Perot cavity can even existwithout it, and it would be physically meaningless toconsider only a simple pendulum. The problemtherefore must be solved self-consistently with allthese relevant interactions taken into consideration.It is assumed that the static radiation pressure force~owing to Pdc! balances the spring-restoring force atsome point, and we solve the equation of motionaround this point.

The equation for the displacement of the mirror inthe time domain is

M~ x 1 gx 1 vp2x! 5 Frp~t! 1 Fservo~t!, (23)

where x~t! is the time-dependent displacement of themirror near the equilibrium point and M, g, and vpare the parameters of the pendulum suspension,namely, the effective mass, the damping coefficient,and the natural angular frequency, respectively. Itis more convenient at this juncture to switch over tothe complex frequency domain by means of theLaplace transform and perform the analysis with theLaplace variable s, which is linked to the real angularfrequency v by the relation s 5 iv. In Section 2 wediscussed the radiation pressure force. In Subsec-tion 3.A we discuss the servo control and the refer-ence transfer function given by Caron et al.15 in thecontext of the locking of the Virgo interferometer.

A. Servo Loop

Since the bandwidth of the detector is typically be-tween 10 Hz and a few kilohertz, one possible servocontrol should be that the motion of the mirror shouldbe free in this range of frequencies while below 10 Hzthe motion should be frozen. Note that it is not theonly choice; the bandwidth of the servo can be ex-tended to the detection band of gravitational waves,but in this case the reconstruction of thegravitational-wave signal from the error signals isslightly more complex. The first case was inten-

sively studied within the Virgo collaboration; that iswhy we use it here. Let G~s! be the transfer functionof the pendulum suspension and H~s! be the transferfunction of the servo. Let xfree denote the free mo-tion of the mirror, i.e., without the action of the servo,and let xout be the residual motion when the servo isconnected in a feedback loop. Since the servo mustalso encounter the pendulum suspension, the productof the transfer functions G~s!H~s! is actually effective~see Fig. 3!. We have the following relation betweenthe Laplace transforms:

xout 5xfree

1 1 GH, (24)

and the xfree is related to the external force Fext ~inour case the radiation pressure force! by the relation

xfree 5 GFext. (25)

Between the various forces we have the followingrelations:

Ftotal 5 Fext 1 Fservo 5xout

G5

Fext

1 1 GH. (26)

According to the requirement mentioned above, wemust have the gain GH ; 0 in the bandwidth of thedetector so that xout ; xfree. Second, below 10 Hz,GH should be large so that the residual motion of themirror is damped out, i.e., xout ,, xfree. Apart fromthis the transfer function is required to satisfy certainstability criteria, such as the Nyquist criterion and aphase margin of at least 45° at unit gain ~uGHu 5 1!.The reference transfer function passes all these cri-teria. We describe the function below:

G 51

M~s 2 s1!~s 2 s1*!, (27)

H 5 H0

~s 2 s2!~s 2 s2*!~s 2 s3!~s 2 s3*!

s2~s 1 2p!~s 2 s4!~s 2 s4*!, (28)

where the parameters have the following values: H05 2.24 3 109 kg s23, s1 5 3.74~21026 1 i! rad s21, s25 ~215 2 37 i! rad s21, s3 5 ~225 2 10 i! rad s21, s4

Fig. 3. Schematic indicating the servo loop: G, pendulum sus-pension transfer function; H, servo transfer function; Fext, the forcecaused by the radiation pressure; xout, the total displacement of themirror; xfree, the displacement of the mirror without the servo.

5 ~2550 2 500 i! rad s21. The transfer function hashigh gain at very low frequencies below 1 Hz, uGHu ;106, 107 and falls to unity at 22.2 Hz with a phasemargin of approximately 45.5°. At high frequenciesat approximately 1 kHz, the transfer function be-comes negligible at ;1023. The plot of the transferfunction was reproduced after Ref. 15 in Fig. 4. Weuse this transfer function in our analysis henceforth,keeping the servo gain H0 as a possible parameter.

B. Characteristic Equation and the Natural Frequenciesof the System

In this subsection we consider the full system of theradiation pressure force and the servo-control forceacting on the suspended mirror, with the position ofthe mirror and the motion of the mirror determiningthe instantaneous power and thus the radiation pres-sure. After Laplace transforming Eq. ~22! or alter-natively using Eqs. ~23!–~26!, where Fext is theradiation pressure force equal to 2dP~s!yc, we obtainfrom Eq. ~15!

Fext~s! 5 2K~s!

cxout~s!. (29)

Combining Eq. ~29! with Eqs. ~23! and ~24!, we get thecharacteristic equation for the system,

2c

K~s!G~s! 5 1 1 G~s!H~s!. (30)

With s 5 iv 5 iVyts, where ts is an arbitrary time, werationalize Eq. ~30! and write it in terms of theLaplace variable s for which time is measured inunits of ts. Although this is not strictly necessary, itavoids large numbers. ~ts has nothing to do with thestorage time of the cavity but was chosen only forconvenience.! Setting s 5 tss, we obtain the result-ing equation:

s2~s 1 s0!~s 2 s4!~s 2 s4*!srp 1 s2~s 2 s1!~s 2 s1*!

3 ~V02 1 2ls 1 s2!~s 1 s0!~s 2 s4!~s 2 s4*! 1 k~V0

2

1 2ls 1 s2!~s 2 s2!~s 2 s2*!~s 2 s3!

3 ~s 2 s3*! 5 0. (31)

Fig. 4. Plot of uGHu ~the product of the pendulum suspension andservo transfer functions! versus the frequency in hertz.

20 May 1998 y Vol. 37, No. 15 y APPLIED OPTICS 3241

The quantities appearing in Eqs. ~31! have the fol-lowing expressions:

srp 52P0 g~d!ts

2

Mc, (32)

k 5H0ts

3

M, (33)

s0 5 2pts, (34)

si 5 sits. (35)

We assumed here and in all the following that ts 54 3 1024 s in the numerical expressions. The sihave the following values: s1 ; 1.5~21029 1 i1023!,s2 ; 20.006 2 0.0148i, s3 ; 20.01 2 0.004i, s4 ;20.22 2 0.2i.

If we put k 5 0 in Eqs. ~31!–~35! we remove theaction of the servo system. The parameter k essen-tially gives the gain of the servo weighted by the massof the mirrors. We examine the k 5 0 case later asa matter of interest.

The characteristic equation has real coefficientsand is of degree 9 and has 9 complex roots, at leastone of them real. For the system to be stable wemust have the real parts of all the roots negative.However, the roots are functions of the parameters ofthe cavity. The most easily varied among these arethe input power P0, the operating point d, and thefinesse determined through R. For the given trans-fer function of the servo these parameters can bevaried within wide ranges, and the roots then mi-grate in the complex plane. We, however, do notstudy all the roots but investigate the migration ofonly two roots that have the tendency to cross overinto the right half of the s plane. To fix ideas we listthe values of the various quantities occurring in ourcalculations and also the roots for one typical set ofthe above parameters. For parameters correspond-ing to the Virgo example,17 namely, mass M 5 28 kg,R 5 0.94 ~finesse of 50!, input power P0 . 1 kW~corresponding to the power-recycling gain of approx-imately 100!, and phase offset d . 20.01, we haveg~d! . 21011 m21, srp . 9.3 3 1025, V0

2 . 1.24, l .1.13. The roots in the s plane are the following:21.36, 20.91, 20.037, 20.18 6 0.16i, 20.01 60.005i, 20.0077 6 0.021i. The following observa-tions can be made for this example:

~1! Since all the roots have negative real parts, thesystem is stable for these values of the parameters.

~2! The first two roots have moduli close to unity.This means that the power adjusts itself on the timescale of ts as the cavity length changes because of themotion of the mirror ~not surprisingly since ts waschosen to obtain numerical values near unity!.These roots are basically governed by the transferfunction K~s!.

~3! However, it is the last two roots that decidethe stability of the system, in the sense that increas-ing the input power or the finesse makes these roots

3242 APPLIED OPTICS y Vol. 37, No. 15 y 20 May 1998

migrate to the right half of the s plane. For exam-ple, Fig. 5 shows the behavior of the real parts of thelast two roots ~only those for which the real parts canbecome positive!, as the input power increases whenR 5 0.94 and d 5 20.01 are held fixed. We observethat it is only the last pair of the roots that cross overto the right half of the s plane. This is generally thecase as was borne out by extensive numerical com-putations that took into account various values of theparameters M, P0, R, H0, and d.

We therefore concentrate on these two roots and in-vestigate how these roots cross the imaginary axis ofthe s plane as the parameters are varied.

C. Condition for Stability

One must now examine Eq. ~31! for the roots crossingover to the right half of the s plane. We considerthree parameters, namely, the input power P0, thereflectivity R, and the operating point d ~plus eventu-ally the mass M and the servo gain H0!, and hence onecan expect regions in this three-dimensional parame-ter space that correspond to stability–instability.However, we show below that a function k of P0, R, andd can be defined, and the problem reduces to a com-parison of the value of this function with a criticalvalue of this parameter kcrit that depends on the servo-control transfer function. The function k~d, P0, R! isessentially the scaled power. From the characteristicequation we make the following observations:

~1! The parameters d, R, and P0 enter the coeffi-cients through the quantities srp, V0

2, and l; M ap-pears in srp and k, whereas H0 appears in only k.

~2! We find after extensive numerical scanning ofthe roots of Eq. ~31! that the crossover of the two rootsoccurs near iVco ; 60.023i ~for M . 28 kg!, and hencewhen we are looking for the roots near this region thelast two terms in the expression V0

2 1 2ls 1 s2 aresmall and can be neglected for this purpose. If the

Fig. 5. Migration of the real parts of the relevant roots toward thehalf-plane Re~s! . 0 as the input power increases. The otherparameters are fixed: H0 5 2.24 3 109 kg s23, M 5 28 kg, R 50.94, and d 5 20.01. Here instability arises for an input power ofapproximately 8 kW.

mass is less, the crossover occurs for a slightly smallervalue of Vco and the approximation is even better.Note that the servo gain has only a little influence onthe value of Vco but affects the parameters ~P0, d . . .!for which the crossover occurs. This brings down thedegree of the equation from 9 to 7. More importantly,it permits us to define the variable k as

k~d, P0, R! 5 srpyV02 5

2P0 g~d!ts2

V02Mc

. (36)

This lumps the three parameters of interest into k, andthe problem simplifies considerably. Since at the cross-over s is imaginary, V is real. We separate the real andimaginary parts of the characteristic equation, each ofwhich must vanish at the crossover of the roots. Weobtain the two equivalent equations in terms of the vari-able u 5 V2 and keep the first one, namely,

u3 2 @k 1 us4u2 2 2s0 Re~s4!#u2 1 @~us4u2 2 2s0 Re~s4!#k

2 2k@Re~s2! 1 Re~s3!#u 1 2k@Re~s3!us2u2

1 Re~s2!us3u2# 5 0. (37)

Note that the last ~constant! term is negative sincethe real parts of the si are all negative.

Since at the crossover s should be purely imagi-nary, u must be real and positive. The roots obvi-ously depend on only the one variable k. We findthat there exists a critical value kcrit such that a realpositive root exists when k . kcrit. In fact, in Eq. ~37!the cubic term can be neglected so that we have todeal with only a quadratic. Indeed, for the range ofinterest for the different parameters, kcrit is foundnumerically to be approximately 1024–1023 and theu3 term can be dropped since this term is approxi-mately 3 orders of magnitude smaller than the otherterms. In the resulting quadratic we obtain kcrit bysetting the discriminant equal to zero, and so weobtain a quadratic equation for kcrit. Setting thepositive equations

a4 5 us4u2 2 2s0 Re~s4!,

b23 5 22k@Re~s2! 1 Re~s3!#,

g23 5 22k@Re~s3!us2u2 1 Re~s2!us3u2#,

we derive for this equation

a42kcrit

212~a4b2322g23!kcrit 1 b232 2 4a4g23 5 0. (38)

Solving Eq. ~38!, we find generally two negative roots,kcrit and kcrit9, and label them such that kcrit . kcrit9.For example, in the case of relevant parameters forVirgo ~M 5 28 kg and H0 5 2.24 3 109 kg s23! we findkcrit . 26.18 3 1024 and kcrit9 . 23.34 3 1023. Wefind the same numerical values for constant ratios ofH0yM, since these two parameters enter in Eq. ~38!through k } H0yM. The quadratic form giving thediscriminant of the simplified Eq. ~37! as a function ofk has then a sign depending on the location of k withrespect to the two roots. As we want u to be real, thediscriminant must be positive, giving k $ kcrit or k #kcrit9. However, in the second case we find generally

two real solutions for u, but at least one of them isnegative, which is rejected since u 5 V2 is required tobe positive for the stability of the system. The stabil-ity condition is finally k $ kcrit. This condition is au-tomatically fulfilled when d . 0, since in this case g~d!. 0 and then k . 0, but it can be violated if d , 0.This immediately shows that for d . 0 the system isstable. Only when d , 0 can the system be unstable.

To examine the case of instability we write thefunction k in more convenient variables.

We set Q 5 1 2 R, d 5 aQ, since the operating pointis more conveniently given in terms of the FWHM .2Q in these variables. In terms of Q, the finesse ^ isgiven by

^ 5 pÎ1 2 QyQ . pyQ, if Q ,, 1.

Compared with unity we keep terms to only first order inQ since normally Q is small compared with unity. FromEq. ~36! we obtain the following for k~a, P0, Q!:

k~a, P0, Q! 5a~1 1 a2 2 3Q!~1 1 1.5Q!

~1 1 a2!3Q2

32pts2

lMcP0.

(39)

The condition for stability k . kcrit then becomes

2a~1 1 a2 2 3Q!~1 1 1.5Q!

~1 1 a2!3Q2 P0 ,lMc32p

ukcrituts

2 . (40)

The equality between the left and right terms in inequal-ity ~40! gives the equation of the curve separating thestable and unstable domains in the space of parameters.Given the finesse, the mass, and kcrit, we can write, forexample, the critical input power Pcrit above which insta-bility occurs as a function of the detuning d:

Pcrit 5lMc32p

ukcrituts

2

~1 1 a2!3Q2

a~1 1 a2 2 3Q!~1 1 1.5Q!, (41)

which can be recasted as

Pcrit 5 Pchar

~1 1 a2!3Q2

a~1 1 a2 2 3Q!~1 1 1.5Q!, (42)

where we defined the characteristic power Pchar by

Pchar 5lMc32p

ukcrituts

2

. 340S M28 kgDS l

1064 nmDS ukcritu6.18 3 1024D kW.

(43)

In the case of relevant parameters for Virgo ~M 5 28kg and l 5 1064 nm!17 and ukcritu . 6.18 3 1024, wehave Pchar . 340 kW. In the case of LIGO ~M 5 10.7kg and l 5 1064 nm!18 and for a similar servo trans-fer function as for Virgo, we find ukcritu . 2.85 3 1023

and we have then Pchar . 600 kW, whereas for a servotransfer function that is the same except that thegain is chosen so that the ratio H0yM is the same asin Virgo ~ukcritu . 6.18 3 1024!, we have Pchar . 130kW. Figure 6 is a plot of the critical input power Pcrit

20 May 1998 y Vol. 37, No. 15 y APPLIED OPTICS 3243

as a function of the phase detuning from resonance dfor various finesses in the case of Virgo mirrors ~M 528 kg!. Above each critical curve the system is un-stable and below it is stable. We first note the exis-tence of a minimum at d 5 Q, which is at HWHM ofthe Airy peak. This minimum can be explained bynoting that at HWHM the derivative of the powerwith respect to d is maximum, so the radiation pres-sure force is also maximal @see Eq. ~22!#. Second, wesee that, with a planned finesse of 50 and an inputpower of approximately 1 kW ~circulating power in-side the recycling cavity!, the arm cavities of Virgoare always stable whatever the operating point.However, increasing the power in the recycling cavityandyor increasing the finesse of the arm cavities canproduce instabilities. For example, for a operatingpoint at d 5 Qy10 ~tenth of the HWHM!, the Virgocavities can become unstable for a power in the recy-cling cavity of approximately 10 kW, corresponding toan improvement of a factor of =10 . 3 of the shotnoise, with respect to the nominal configuration ~1kW in the recycling cavity!; however, the operatingpoint is designed to be much closer to the top of theresonance peak ~typically udu , 1025!, so the Virgocavities should not exhibit instability in their normalregime. The case of LIGO does not seem more prob-lematic, as it would be in the nonservoed case, owingto the lighter mirrors. Figure 7 shows the criticalpower as a function of d for a cavity having a finesseof 200 and 10.7-kg mirrors as in LIGO and two servogains corresponding to ukcritu . 2.85 3 1023 ~samegain as in Virgo! and ukcritu . 6.18 3 1024 ~same ratiogain–mass as in Virgo!. Except for the gain, theservo system is assumed to be the same as in theVirgo case, although this could not be the case, owingto different designs for the seismic isolation and so todifferent sensitivities at low frequencies. However,

Fig. 6. Critical input power Pcrit ~in kilowatts! versus the cavityphase detuning d in the case of a Virgo-like configuration: H0 52.24 3 109 kg s23, M 5 28 kg, but for different finesses. With afinesse of 50 and a power in the recycling cavity of typically ap-proximately 1 kW, the Virgo arm cavities are always stable what-ever d is. We see also the influence of the finesse: if the finessewould have to be increased, everything else being constant, insta-bility could occur for a finesse above 100.

3244 APPLIED OPTICS y Vol. 37, No. 15 y 20 May 1998

we assume the servo-loop transfer function for LIGOto be the same as for Virgo merely to get an idea of theorder of magnitude for the various parameters. Ifthe servo gain is designed such that the gain–massratio is the same as in Virgo, the solid curve in Fig. 7shows instability domains for input powers as low asapproximately 100 W. In the case of LIGO, the cir-culating power inside the recycling power should beof the order of 300 W ~with a laser power of 6 W anda recycling power gain of approximately 50!,19 so thatan operating point near a tenth of the HWHM couldbe possibly unstable, but, again, it is far from theplanned operating point. However, an increase ofthe gain ~dashed curve! by a factor of less than 3allows stability to be recovered anyway. In anycase, in future improved versions of LIGO–Virgo un-stable domains could be theoretically removed by anincrease of the servo gain, but, in practice, this gaincannot be increased indefinitely because of limita-tions in the electronics. So we must keep in mindthe possible occurrence of instabilities for some rangeof operating points, depending on the optical config-uration and on the design of the servo loops. Fi-nally, we emphasize that, if the powers in LIGO–Virgo should have to be increased even moderately~say, by a factor of ;10! and even if the locking pointis close to the perfect resonance ~d 5 0! ~so that thestatic operating point is stable!, in the domain aroundthe HWHM that induces possible instabilities, sometrouble could occur during the lock acquisition whenthe detuning of the cavities is near the HWHM.Clearly, this can have serious implications for theprotolocking procedure, and this problem needs fur-ther investigation.

4. Conclusions

We have analyzed the effect of radiation pressure inlaser interferometric cavities with freely hanging mir-

Fig. 7. Critical input power Pcrit ~in kilowatts! versus the cavityphase detuning d in the case of a LIGO-like configuration: M 510.7 kg, R 5 0.985 ~finesse of approximately 200!. The two curvescorrespond to two servo-loop gains. Solid curve, same gain as inFig. 6: H0 5 2.24 3 109 kg s23. Dashed curve, H0 5 0.86 3 109

kg s23, corresponding to the same ratio H0yM as in Fig. 6. We seeclearly the action of the gain.

rors that are locked by a servo-control feedback loop.We have assumed the transfer function for the servogiven by Caron et al.15 that satisfies the usual stabilitycriteria. After performing the analysis we find that sta-bility depends on the input power, the finesse, the massof the mirrors, the servo gain, and the operating point ofthe cavity. We find that, for relevant parameters, insta-bility can only occur if d , 0 ~where d is the phase offsetfrom resonance!. For a given d , 0 and a given servogain, mass, and a certain value of the finesse, there is acritical input power over which the system becomes un-stable. The system tends to be unstable for highfinesses–high powers. We derive an analytic relationbetween the input power, finesse, mass, operating point,and a parameter depending on the gain–mass ratio,which decides between stable and unstable situations.

The main result is that for a finesse of 50 ~whichapplies to Virgo!, the cavity always exhibits stabilityfor recycling powers of a few kilowatts. For a finesseof 200 ~which applies to LIGO! with the same inputpowers ~for a few kilowatts!, we find that instabilitymay occur but far away from planned locking points.This can, however, have some implications for the de-sign of the servo-loop transfer function. It will there-fore be of importance to investigate the stability ofLIGO cavities with an appropriate servo transfer func-tion since such high powers ~and also higher! are ex-pected to be present in the advanced detectors of thefuture. Indeed, the shot noise is likely to limit thesensitivity of LIGO–Virgo at least above some hun-dreds of hertz, so an improvement in the sensitivity ofthese detectors in the future demands a decrease in theshot-noise level. However, the shot-noise level de-creases only as the square root of the power stored in therecycling cavity. This implies that, to reduce the shotnoise by 1 order of magnitude, we need to increase thepowers by 2 orders of magnitude. Clearly, the radiationpressure effects can play an important role in improvedversions of LIGO–Virgo, and particular care must be ded-icated especially to the design of the servo loops.

When the servo is switched off the time delayscannot be neglected, as emphasized in earlierstudies.8–10 Preliminary results from numericalsimulations, taking fully into account the delay ef-fects, suggest that even for d . 0 instabilities mayoccur. However, when we include the effect of theservo, it is expected that the servo restabilizes thesystem, at least for standard parameters forgravitational-wave detectors, as shown here. Thisstudy will be presented in a future publication.

We have not considered in this problem any tilt ofthe mirror caused by the radiation pressure. A smalltilt of the mirror introduces higher-order modes intothe cavity. The radiation pressure will now couplethese different modes. It will be therefore importantto investigate the nature of the coupling and possibleoscillations, if any. We also plan to investigate theeffects of input laser noise on the stability of the cavity.

This study has been supported by the Indo-FrenchCenter for Promotion of Advanced Research project1010-1.

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