experimental and theoretical study of linewidth narrowing in brillouin fiber ring lasers

556 J. Opt. Soc. Am. B/Vol. 18, No. 4 /April 2001 Debut et al.

Experimental and theoretical study of linewidthnarrowing in Brillouin fiber ring lasers

Alexis Debut, Stephane Randoux, and Jaouad Zemmouri

Laboratoire de Physique des Lasers, Atomes et Molecules, Unite Mixte de Recherche 8523, Centre d’Etudes et deRecherches Lasers et Applications, Universite des Sciences et Technologies de Lille, F-59655 Villeneuve

d’Ascq Cedex, France

Received May 24, 2000; revised manuscript received November 27, 2000

In Brillouin fiber lasers, the phase fluctuations of the pump laser are transferred to the emitted Stokes fieldafter being strongly reduced. The result is a linewidth narrowing that we study both experimentally andtheoretically. We derive simple expressions to connect the linewidths of the waves interacting in the fiber,and we show that the magnitude of the narrowing effect depends only on the acoustic damping rate and thecavity loss rate. We successfully compare these theoretical predictions with experimental results obtained byrecording the response of a Brillouin fiber ring laser to frequency modulation of the pump field. © 2001 Op-tical Society of America

OCIS codes: 290.5900, 140.3560, 030.1640, 190.4370.

1. INTRODUCTIONIn the 1970’s, the realization of low-loss fibers led to theadvent of nonlinear fiber optics.1 Owing to its lowthreshold, stimulated Brillouin scattering (SBS) wasamong the first nonlinear effects observed in optical fi-bers. As is manifested by the appearance of a backward-propagating Stokes wave carrying most of the inputpower, SBS is considered an effect that limits the perfor-mance of optical transmission systems.2 However, SBSin fibers is not only detrimental, and the gain of the in-teraction can be used to advantage in amplifiers orlasers.3,4 The properties of these systems have been ex-tensively studied during the past two decades. In par-ticular, the Brillouin fiber laser is a light source whose be-havior is now well characterized. Its spatiotemporaldynamics is correctly described by a three-wave coherentmodel, and all its operating regimes are clearly listed anddelimited.5,6 The cw regime is of particular interest forpractical applications because the Brillouin laser thenemits radiation that is much more coherent than that ofthe pump laser. This property can be exploited, for in-stance, for sensing, and systems such as Brillouin fiber-optic gyroscopes are well known for their highsensitivity.7

It was experimentally shown in 1991 that the Stokeslinewidth can be several orders of magnitude narrowerthan the linewidth of the pump laser.8 This point hasbeen the object of theoretical investigations presented ina recent paper.9 This linewidth narrowing has beenstudied within the framework of the usual three-wavemodel of SBS. It has been shown that the phase noise ofthe pump laser is transferred to the emitted Stokes waveafter being strongly reduced and smoothed under thecombined influence of acoustic damping and cavity feed-back. Moreover, the magnitude of the narrowing effecthas been precisely quantified by the derivation of an ana-

0740-3224/2001/040556-12$15.00 ©

lytical relation connecting the FWHM of the Stokes line-width to that of the pump laser.

In the present paper we supplement the previous stud-ies by presenting additional experimental and theoreticalresults about the narrowing effect in Brillouin lasers. InSection 2 we compare the nature of noise sources in Bril-louin lasers with that of conventional lasers. This rela-tionship is of importance for integrating our theoreticalresults with the many works devoted to the study of theinfluence of noise on the spectral properties of lasers. Af-ter the roles of the various noise sources have been char-acterized, a detailed theoretical analysis of linewidth nar-rowing is presented in Section 3. In Subsection 3.A thecoherent three-wave model of SBS is recalled and the con-text of our study is defined. The approximations madeare presented and discussed in Subsection 3.B. The ex-pressions that connect the field phases are established inSubsection 3.C, and the analytical relations between thewidths of the field spectra are deduced in Subsection 3.D.Additional results related to the spectral properties of thethree waves interacting in the laser are also given. InSection 4 our theoretical predictions are compared withexperimental results. As was mentioned in Ref. 9, thenarrowing effect is so strong that the Stokes linewidthcan be 104 times narrower than the pump linewidth. Infact, this order of magnitude is already comparable withthe one measured in experiments performed in an all-fiber Brillouin laser.8 To summarize this experimentalwork briefly, the authors of Ref. 8 compare the beat spec-trum of two Brillouin lasers that oscillate independentlywith the beat spectrum of their pump lasers. Although itpermits a comparison of the magnitudes of the frequencyjitters that characterize the pump and the Brillouin la-sers, this method does not give a precise measurement ofthe intrinsic linewidth of each laser. Therefore the ratiobetween the FWHM’s of the Stokes and the pump line-

2001 Optical Society of America

Debut et al. Vol. 18, No. 4 /April 2001 /J. Opt. Soc. Am. B 557

widths is not precisely determined. Moreover, in the ex-periments presented in Ref. 8 the Stokes linewidth was sonarrow that measurement of the beat between the Bril-louin lasers was limited by the instrumental resolution.In the experiments described in Section 4 we measure themagnitude of the narrowing effect that characterizes aBrillouin fiber ring laser operating in a low-finesse reso-nator, which we achieve not by measuring each linewidthseparately but by taking into account the fact that theevolution of the phase of the Stokes wave is strongly con-nected to that of the pump laser.9 In our experiments werecord the response of a Brillouin fiber ring laser to a fre-quency modulation of the pump beam. The ratio be-tween the two linewidths is simply deduced from the ratiobetween the modulation indices that characterize theStokes and the pump waves. The experimental valuethus obtained is then compared with the theoretical valuethat describes the experimental setup used. Finally, Sec-tion 5 is devoted to a conclusion in which we summarizeour research.

2. INFLUENCE OF NOISE INCONVENTIONAL LASERS AND INBRILLOUIN LASERSSince the advent of lasers, the influence of noise on thespectral properties of the emitted light has been carefullyexamined, and a large body of literature is devoted to thissubject. Obviously our aim in this section is not topresent a review of this considerable work. Rather, weshall try to summarize the main results to emphasize theessential differences between the nature of the noise thatoccurs in conventional lasers and that in Brillouin lasers.

In conventional lasers, light amplification is obtainedfrom the inversion of an atomic or a molecular population.Spontaneous emission, which inevitably occurs in thegain medium, acts as a quantum-noise source that is re-sponsible for the existence of a lower limit to the laserlinewidth. For a single-mode low-loss fully inverted la-ser tuned to the center of a homogeneously broadenedgain profile, this quantum-limited linewidth was origi-nally calculated by Schawlow and Townes;10 it is of the or-der of several tens of megahertz for semiconductor lasersand falls into the subhertz domain for gas lasers. In thelatter case it is obviously drowned by so-called technicalnoise, (i.e., mechanical and thermal noise), that is due toenvironmental perturbations. Nowadays the expressionfor the laser linewidth has been generalized to include theinfluence of many effects such as intensity fluctuations,11

large output coupling,12–15 and population and polariza-tion dynamics.16,17 Quantum-mechanical treatments ofthe laser linewidth are available,18–20 but semiclassicalformalisms have also been developed to deal with thisproblem. In this case the effects of spontaneous emissionare phenomenologically included by addition of Langevinnoise terms in the laser equations of motion.15,21

Several models to describe the spectral properties of la-sers are available.22 For single-mode lasers the phase-diffusion model is commonly used.12,23 One considersthat the light generated through spontaneous emissiondisturbs both the amplitude and the phase of the intrac-avity field generated by stimulated emission. Except for

semiconductor lasers,24–26 the amplitude fluctuations areusually neglected, so only the phase of the optical field issignificantly perturbed by spontaneous emission. In fact,it undergoes a random walk characterized by a diffusionconstant D. The field spectrum is then a Lorentzian witha FWHM equal to D/2p.27

Spontaneous emission is not the only source of noise inconventional lasers. The noise characteristics of the out-coming radiation also depend on the fluctuations of thepump mechanism. In fact, the intensity noise of a con-ventional laser is strongly related to the noise of thepumping process.28 Recently this property was carefullyexamined because a reduction of the output intensity fluc-tuations below the shot-noise limit can be achieved bysuppression of the pump noise.29–33 However, the fluc-tuations of the pumping process have no influence on thephase of the electromagnetic field emitted by a conven-tional laser.34,35 In the particular case of opticallypumped lasers, the phase noise of the pump light does notcontribute to the output field noise of the laser.36 As wasexplained in Ref. 9, the situation is drastically differentfor Brillouin lasers, in which there is a strong correlationbetween the phase of the emitted wave and that of thepump laser. But the Brillouin laser is not the only lightsource that is affected by fluctuations of the pump phase.It is also lasers without inversion that may exhibit a line-width below the Schawlow–Townes limit.37–41 Theirlinewidth strongly depends on the pump bandwidth, andpump phase fluctuations can even lead to a transitionfrom lasing without inversion to lasing withinversion.42,43 The optical parametric oscillator is an-other kind of light source that is inherently sensitive tothe phase noise of the pump laser.36 In that system, themaximum attainable squeezing is indeed dependent onthe input laser bandwidth.44,45

After this brief summary of the influence of noise inconventional lasers, let us now consider the case of Bril-louin lasers. First, these lasers can be optically pumpedonly by light sources that have linewidths much narrowerthan the SBS gain profile. As the Brillouin gain curve istypically 100 MHz wide in silica fibers, the pump sourcesusually used are single-mode lasers. The pump beamdoes not generate a population inversion but induces den-sity variations inside the fiber through electrostriction.This procedure gives birth to a backscattered Stokes waveand an acoustic wave that grow from the acoustic noisethat exists inside the fiber at thermal equilibrium.46,47

The optical fiber can then be considered a gain mediumamplifying the light about the Stokes frequency, and onecan make a Brillouin laser simply by enclosing the fiberwithin a resonator. In that system, three waves are in-volved in a nonlinear interaction that strongly couplestheir amplitudes and phases. The noise characteristicsof the emitted Stokes light are therefore deeply connectedwith those of the pump laser. In particular, the phasenoise of the Brillouin laser is strongly related to the phasenoise of the pump laser. This phenomenon was studiedin a recent theoretical paper in which we showed that thephase fluctuations of the pump laser are transferred tothe emitted Stokes wave after being reduced andsmoothed.9 This effect is due to the combined influenceof acoustic damping and cavity feedback. By using the


phase-diffusion model described above and by comparingthe diffusion constants that characterize the pump laserand the Stokes wave, we showed that the Stokes line-width is much narrower than the pump linewidth. Themagnitude of the narrowing effect was precisely quanti-fied by the derivation of an analytical relation connectingthe FWHM of the Stokes linewidth with that of the pumplaser and shows that the ratio between the pump and theStokes linewidths is ;100 in a low-finesse cavity and ofthe order of 10,000 in a high-finesse cavity.

In most of the experiments, the linewidth of the pumplaser is of the order of a few tens of kilohertz. The rela-tion mentioned above shows that the linewidth of theStokes radiation can be of the order of a few hertz. Asthis linewidth is narrow, the validity of the approxima-tions that lead to this result must be questioned. In fact,we established the theoretical relation of Ref. 9 by assum-ing that the phase noise of the pump laser is the predomi-nant noise source. However, other noise sources are alsoable to alter the spectral properties of the Brillouin laser.For instance, the cavity length and the feedback efficiencyare fluctuating parameters whose influence must be ana-lyzed. The fluctuations of the cavity characteristics arisefrom mechanical and thermal noise, and their bandwidthis typically of the order of 100 Hz. They are responsiblefor slow variations of the laser intensity and for slowdrifts of the Stokes frequency, but they do not contributeto the intrinsic linewidth of the emitted radiation. If thefield spectrum is recorded by beating of the Brillouin laseragainst a stable reference laser,48 the mean frequency ofthe beat spectrum will drift on a time scale of the order of0.01 s. The maximum frequency deviation then dependson the mechanical and thermal stability of the resonator.A Brillouin fiber laser operating in a low-finesse resonatorusually includes an aerial arm yielding noticeable feed-back and length fluctuations. In that kind of system themagnitude of the frequency jitters is comparable with thefree spectral range (FSR) of the resonator (i.e., typically ofthe order of 10 MHz). In all-fiber Brillouin lasers the useof couplers offers good stability of the feedback efficiency,and stabilization techniques permit the magnitude of fre-quency jitters to be reduced to well below the cavity FSR.

Another kind of noise source also affects the temporalcoherence of Brillouin lasers. Slight random fluctuationsof density indeed occur all along the fiber at thermal equi-librium. This acoustic noise is responsible for the initia-tion of the SBS process, but it can also give rise to a sto-chastic dynamics in Brillouin generators when care istaken to eliminate feedback from the fiber ends.46 Its in-fluence is taken into account in the three-wave model byaddition of a Langevin noise term in the equation thatgoverns the evolution of the acoustic wave.47 As it ex-presses the effect of spontaneous scattering, the relativeimportance of this noise term is much less than that ofthe term representing stimulated scattering. However,one cannot consider spontaneous scattering a process thatonly initiates SBS. As was previously mentioned, its roleis indeed determinant in Brillouin generators in whichgain narrowing of the Stokes spectrum is observed.46

The acoustic noise also affects the behavior of highly mul-timode Brillouin fiber lasers by perturbing the stability oftheir pulsed emission.4 In single-mode Brillouin lasers,

acoustic noise plays a role analogous to that of spontane-ous emission in conventional lasers. Therefore it is re-sponsible for the existence of a lower limit to the Stokeslinewidth. This point was studied from different ap-proaches in Refs. 9 and 49.

In Ref. 49 the authors reported observing the lightspectrum backscattered by a Brillouin generator. In theabsence of any feedback from the fiber ends, the width ofthis spectrum is of the order of the SBS gain bandwidth.The authors then showed that even a small amount offeedback from the fiber ends is sufficient to induce consid-erable narrowing of the Stokes spectrum. This behaviorwas analyzed within the framework of the usual three-wave model of SBS. By neglecting the influence of pumpnoise, the authors established an analytical relation thatgives the Stokes power spectrum in the limit of an unde-pleted pump field. The spectrum’s characteristics de-pend on the strength of the acoustic noise but also on pa-rameters such as the interaction length and the boundaryreflectivities. Numerical calculations performed with pa-rameter values that are representative of commonlyfound Brillouin lasers show that the width of this spec-trum always falls in the subhertz domain.

The approach adopted in Ref. 9 is much more qualita-tive. By analogy with spontaneous emission, spontane-ous scattering is considered an effect that slightly per-turbes the amplitude and the phase of an acoustic wave.In particular, it leads to diffusion of the phase of theacoustic wave. The influence of this diffusion is nolonger negligible when it becomes comparable with thediffusion induced by the phase fluctuations of the pumplaser. By applying this analysis it is possible to showthat spontaneous scattering is no longer negligible forpump linewidths as narrow as 10 Hz and Stokes line-widths that fall into the subhertz domain.9 As for theSchawlow–Townes limit in conventional lasers, the fun-damental linewidth of a Brillouin laser is quite narrow.In the following theoretical analysis, we consider that thenoise that is due to spontaneous scattering does not con-tribute noticeably to the intrinsic linewidth of the Bril-louin laser. It is drowned out by other noise sources andin particular by the phase noise induced by the phasefluctuations of the pump laser.

3. THEORETICAL ANALYSISA. ModelOur theoretical analysis of linewidth narrowing entersthe framework of the usual three-wave model of SBS. Inthis model the single-mode fiber is considered an unidi-mensional medium in which two linearly polarized opticalwaves and an acoustic wave are coupled through electro-striction. As SBS is a narrow-bandwidth process, each ofthese waves can be regarded as being composed of a high-frequency carrier that is modulated (in both amplitudeand phase) by a slowly varying envelope. When we ne-glect the perturbative influence of the optical Kerr effectand the weak attenuation of the fiber, the dimensionlessequations that govern the spatiotemporal dynamics of theslowly varying envelopes read as

]t«p 1 ]z«p 5 2gB«s , (1a)


]t«s 2 ]z«s 5 gB* «p , (1b)

~1/bA!]tB 1 B 5 «p«s* . (1c)

«p , «s , and B represent, respectively, the complex ampli-tudes of pump, Stokes, and acoustic waves. Time t isnormalized to the transit time of the light inside the fiber.z is the space coordinate that is normalized to the fiberlength. The fields «p and «s are measured in units of themaximum pump field available at the entrance end of thefiber. g is the SBS coupling constant, and bA representsthe normalized damping rate of the acoustic wave. Fol-lowing the discussion of Section 2, the weak term that de-scribes spontaneous Brillouin scattering was neglected inEq. (1c). Transforming the complex amplitudes intomodulus phase form yields the following equations:

]tAp 1 ]zAp 5 2gAaAs cos u, (2a)

]tAs 2 ]zAs 5 gAaAp cos u, (2b)

~1/bA!]tAa 1 Aa 5 ApAs cos u, (2c)

]tfp 1 ]zfp 5 2g~AaAs /Ap!sin u, (2d)

]tfs 2 ]zfs 5 2g~AaAp /As!sin u, (2e)

~1/bA!]tfa 5 2~ApAs /Aa!sin u, (2f)

where u(z, t) 5 fs(z, t) 1 fa(z, t) 2 fp(z, t). Ai andf i are real functions that represent, respectively, the am-plitudes and the phases of the pump, the Stokes, and theacoustic waves (i 5 p, s, a).

Recall the equations that govern the SBS interaction;let us now precisely define the context of our analysis oflinewidth narrowing. We consider only Brillouin fiberring lasers operating in a single longitudinal mode. Thiscondition is satisfied when the FSR of the ring cavity is ofthe same order of magnitude as the FWHM DnB of theBrillouin gain curve. The laser operates in the Brillouinmirror regime, and the intensity of the backscatteredStokes wave is time independent.5,50 It is then possibleto study how the phase fluctuations of the pump laser aretransferred to the Stokes wave, or, in other words, to com-pare the field spectrum of the Brillouin laser with that ofthe pump laser. If the FSR of the ring cavity is muchnarrower than DnB , the Stokes emission becomespulsed4,6 and the width of the field spectrum is then com-parable with DnB . In our study we consider only Bril-louin fiber ring lasers in which pump recoupling isavoided by an intracavity isolator. In these conditions,Eqs. (2) must be completed by the boundary conditions

Ap~z 5 0, t! 5 m, (3a)

As~z 5 1, t! 5 RAs~z 5 0, t!, (3b)

fp~z 5 0, t! 5 f0~t!, (3c)

fs~z 5 1, t! 5 fs~z 5 0, t!, (3d)

where m is a dimensionless pump parameter and f0(t)represents the phase of the incident pump field. R is theamplitude feedback parameter that characterizes the ringcavity. As was already mentioned in Section 2, Brillouinfiber ring lasers are usually pumped by well-stabilizedsingle-mode lasers. In the phase-diffusion model com-monly used to describe the field emitted by this kind of

laser, the amplitude noise is neglected and the phase is afluctuating variable that undergoes a random walk.With our notation, m is time independent, whereas theevolution of f0 is governed by the stochastic Langevinequation

df0~t!

dt5 q~t!, (4)

in which q(t) is a d-correlated Gaussian noise of zeromean.27

B. ApproximationsTo describe the Brillouin mirror regime we can first as-sume that the field amplitudes are rigorously time inde-pendent. In these conditions the time derivatives can beeliminated from Eqs. (2a), (2b), and (2c), and u becomes atime-independent variable that is a function only of thestationary amplitudes. However, the coupling betweenthe field phases is then lost [see Eqs. (2d), (2e), and (2f)],and we can conclude that the previous approximation istoo rough. Although pump parameter m is time indepen-dent, we rather consider that the phase fluctuations of theincident pump field can induce slight variations of thevarious amplitudes about their steady states. In con-crete terms, let us write Eq. (2c) in the form

1

bA

1

Aa

]Aa

]t1 1 5

ApAs

Aacos u. (5)

To describe the Brillouin mirror we consider that

U 1

bA

1

Aa

]Aa

]tU ! 1. (6)

This means first that the relative fluctuations of the am-plitude of the acoustic wave must be extremely weak.Moreover, the bandwidth characteristic of these fluctua-tions must be narrower than the Brillouin bandwidth. Ifthe Brillouin laser is pumped by a single-mode laser witha linewidth much narrower than the Brillouin bandwidth,these two conditions are clearly always satisfied, as hasbeen confirmed by numerical simulations in which Eqs.(2) and (3) are integrated in the presence of phase noisegenerated by Eq. (4). For instance, the numerical resultspresented in Fig. 1 of Ref. 9 are typical, and they showthat the value of the term u(bAAa)21]Aa /]tu never ex-ceeds 1024. By taking into account Eqs. (5) and (6), weobtain a good approximation of the amplitude of theacoustic wave that is in fact quite close to its steady-stateprofile Aa(z):

Aa~z, t! . Aa~z! . Ap~z, t!As~z, t!cos u~z, t!. (7)

Let us emphasize that in expression (7) the functionsAp(z, t) and As(z, t) fluctuate slightly about the steadystate. As the SBS interaction is subjected to precise reso-nance and phase-matching conditions, u(z, t) is also aweakly fluctuating variable. By substituting Eq. (7) intoEqs. (2d), (2e), and (2f) we obtain

]tfp 1 ]zfp 5 2~ g/2!As2~z, t!sin@2u~z, t!#, (8a)

]tfs 2 ]zfs 5 2~ g/2!Ap2~z, t!sin@2u~z, t!#, (8b)

~1/bA!]tfa 5 2~1/2!sin@2u~z, t!#. (8c)


Equation (3d) indicates that the cavity is resonant forthe Stokes wave. This means that the center of the Bril-louin gain curve coincides with one of the resonator eigen-frequencies such that frequency-pulling effects areignored.50 In these conditions, the variable u(z, t) fluc-tuates about a stationary value that is uniformly equal tozero. As these fluctuations are weak, the sine functionscan be developed to the lowest order in u. Inasmuch asthe relative fluctuations of the amplitudes As(z, t) andAp(z, t) are weak, the temporal evolution of the right-hand sides of Eqs. (8a) and (8b) is governed by the fluc-tuations of the variable u. Therefore amplitudes As(z, t)and Ap(z, t) can be well approximated to their steady-state values, As(z) and Ap(z). Finally, the equationsthat govern the spatiotemporal evolution of the phasescan be written as

]tfp 1 ]zfp 5 2gAs2~z!u, (9a)

]tfs 2 ]zfs 5 2gAp2~z!u, (9b)

]tfa 5 2bAu. (9c)

If the ring cavity is not resonant, a detuning term mustbe added to the right-hand side of Eq. (3d). The variableu(z, t) then oscillates slightly about a constant value u0that depends on the detuning term, the cavity loss rate,and the acoustic damping.50 The above-applied treat-ment can be performed, but the sine functions must belinearized about u0 . The right-hand side of each of Eqs.(8) then consists of the sum of a time-independent termand a weakly fluctuating term. If only the time-independent term is taken into account, the phases lin-early vary with time. This term just describes the factthat the fields no longer evolve in the rotating frame usedto derive the three-wave model and that their frequenciesare shifted by pulling effects. The narrowing effect islinked only to the weakly fluctuating term, and it cantherefore be studied in a rotating frame, which permitsthe frequency-pulling effects to be ignored [Eqs. (9)].

When the Brillouin laser operates well above threshold,the functions Ap

2(z) and As2(z) do not differ strongly [see

Fig. 1(c) of Ref. 9], so the weak terms that appear on theright-hand sides of Eqs. (9a) and (9b) are of the same or-der of magnitude. However, they do not play the samerole in the two equations. In Eq. (9a) the influence of theterm gAs

2(z)u is only perturbative; term gAp2(z)u how-

ever, plays a determining role in Eq. (9b). To explainthis point we describe the mechanism of linewidth nar-rowing in a simple way. As is mentioned in Section 2,the linewidth of the light sources that are used to pumpBrillouin fiber lasers is typically of the order of 100 kHz.This means that the time that it takes for the phase todiffuse over 1 rad (i.e., the optical coherence time51) is onaverage of the order of 10 ms. The phase fluctuations ofthe pump source are seen first by the acoustic wave,whose response time is of the order of 10 ns. Thereforethe acoustic phase nearly follows adiabatically the spa-tiotemporal evolution imposed by the pump laser (see alsoFig. 1 of Ref. 9). As was mentioned above, the resonanceand phase-matching relations impose the condition that ube a weakly fluctuating variable. In these conditions thespatiotemporal variations of the Stokes phase are neces-

sarily much weaker than the spatiotemporal variations ofthe pump and acoustic phases. In other words, the co-herence time of the Brillouin laser is much greater thanthe coherence time of the pump laser.

After these qualitative considerations, we can now ana-lyze the role of the terms that appear on the right-handsides of Eqs. (9a) and (9b). Equation (9a) must satisfythe boundary condition given by Eq. (3c). In Eq. (3c),f0(t) is a source term that induces strong spatiotemporalvariations of fp . The weak term, gAs

2(z)u, is only per-turbative, and its influence can be neglected. The bound-ary condition verified by the Stokes phase [Eq. (3d)], how-ever, does not contain a source term. The spatiotemporalevolution of the Stokes phase is governed by the weakterm gAp

2(z)u whose influence cannot be neglected. Byconsidering that the phase of the pump wave remains un-disturbed in the interaction, we find that the solution ofEq. (9a) simply reads as

fp~z, t! 5 f0~t 2 z!. (10)

Let us note that the previous solution has been validatedby numerical simulations that clearly yielded similar re-sults for the three-wave model [Eqs. (2)] and for the inte-gration of Eqs. (9b), (9c), and (10).

By taking into account the analytical expression ofAp

2(z) [Eq. (7a) of Ref. 52] we finally reduce our problem tothe resolution of the set of equations

]fs

]t2

]fs

]z5

2gV

1 2 D exp~22gVz!~ fs 1 fa 2 fp!.

(11a)

]fa

]t5 2bA~ fs 1 fa 2 fp!, (11b)

where fp is given by Eq. (10). In Eq. (11a), V 5 Ap2(z

5 0) 2 As2(z 5 0), and the constant D reads as

D 5R2 2 exp~22gV!

~R2 2 1 !exp~22gV!. (12)

C. Analytical Determination of the Relations Betweenthe Field PhasesAccording to the boundary condition given by Eq. (3d), wemust seek a solution for fs(z, t) that can be written inthe form

fs~z, t! 5 (n52`

1`

Sn~t!exp~iknz!, (13)

with kn 5 2pn and Sn!(t) 5 S2n(t). After a Fourier

transformation of Eqs. (10), (11), and (13) and by usingthe orthonormality condition *0

1 exp(iknz)exp(2ikmz)dz5 dnm , we obtain

bA 1 i2pn

gV~n 2 m !Sm~n!

5 nf0~n!E0

1 exp 2 i~km 1 2pn!z

1 2 D exp~22gVz!dz


2 (n52`

1`

Sn~n!nE0

1 exp i~kn 2 km!z

1 2 D exp~22gVz!dz. (14)

f0(n) and Sm(n), respectively, represent the Fouriertransforms of f0(t) and Sm(t). As the phase of theStokes wave is weakly fluctuating, its spatial profile is al-most independent of z [see also Fig. 1(d) of Ref. 9], andfs(z, t) can be well approximated by S0(t). The influ-ence of the components of index n Þ 0 can thus be ne-glected in the discrete sum, and Eq. (14) yields simply

S0~n!

5g *0

1@V exp~2i2pnz!dz/1 2 D exp~22gVz!#

g*01@Vdz/1 2 D exp~22gVz!# 1 bA 1 i2pn

f0~n!.

(15)

Equation (15) shows that the phase of the emittedStokes wave is directly connected to the phase of thepump laser. The expression found is obviously muchmore complicated than that derived in Ref. 9. In thatcase, we considered that the laser does not operate farfrom its threshold, so pump depletion is negligible. Thefunction Ap

2(z) can thus be approximated to 2ln(R)/g,and the analytical calculation is then greatly simplified.However, the result given in Ref. 9 can also be found di-rectly from Eq. (15). To that purpose, we assume onlythat As

2(z) . 0. With this condition V is equal to2ln(R)/g and D is equal to zero. Elementary calculationsthen directly yield the relation already found in Ref. 9:

S0~n! 52ln R

bA 2 ln R 1 i2pn

exp~2ipn!sin pn

pnf0~n!.

(16)

Contrary to Eq. (15), the term that connects S0(n) tof0(n) in Eq. (16) can be factorized into two parts. Thefirst part shows that the phase fluctuations of the pumplaser are filtered and reduced before being transferred tothe emitted Stokes wave. The ratio K between the am-plitudes of the fluctuations that characterize the pumpand the Stokes phases is simply equal to (bA2 ln R)/(2ln R). The second part of the term shows that

a smoothing effect also occurs during the transfer of thephase noises. In Eq. (15) it is not possible to separate thetwo effects (filtering and smoothing) by a simple factoriza-tion. However, we can determine the ratio between theamplitudes of the phase fluctuations. This quantity is in-deed of direct interest for determination of the ratio be-tween the FWHM’s of the pump and the Stokes line-widths. We now show that it is equal to the value of Kgiven above not only near threshold but also whatever thevalue of m is.

Let us recall that the bandwidth that is characteristicof the phase fluctuations of the pump laser is typically ofthe order of 100 kHz. This value is approximately 100times smaller than that of the cavity FSR. For values ofn much lower than unity, we calculate an asymptotic ex-pansion of the function S0(n)/f0(n). First, we can sim-plify the integral that appears in the numerator of Eq.(15) by considering that the term exp(2i2pnz) is equal tounity at the interval of integration. Moreover, the termi2pn in the denominator can be neglected with respect to

bA . Lengthy but elementary calculations then showthat the integral *0

1@Vdz/1 2 D exp(22gVz)# is simplyequal to 2ln(R)/g. Finally, we find that the ratio K be-tween the amplitudes of the pump and Stokes fluctua-tions is (bA 2 ln R)/(2ln R) not only near threshold butalso whatever the incident pump power is. This impor-tant result can be explained by the fact that the functionfs(z, t) has been approximated by S0(t). As shown inEqs. (14) and (15), the problem is then greatly simplified.The integration with respect to z simply leads to the ap-pearance of the previous integral that in fact representsthe mean value of the pump intensity inside the fiber,*0

1 Ap2(z)dz. When laser oscillation is initiated, all the in-

put energy is transferred from the pump wave to theStokes wave. In these conditions the mean value of thepump intensity inside the fiber always remains that re-quired for initiation of the laser oscillation and is equal to2ln(R)/g. Ratio K depends only on this mean value andis therefore independent of the incident pump power. Ifthe influence of a few functions Sn(t) of order n Þ 0 weretaken into account, the problem would consist of thetreatment not of only one equation but of a set of equa-tions. Moreover, the determination of the coupling coef-ficients between the functions Sn would require the calcu-lation of complicated integrals [see Eq. (14)].Fortunately, the components S61 become comparablewith S0 only if the bandwidth that characteristic of thefluctuations of the pump laser becomes of the order of thecavity FSR. Therefore the approach that consists in ap-proximating fs(z, t) by S0(t) is suitable for most of thelasers that operate in the Brillouin mirror regime.

We can now use the above-obtained result to determinethe ratio K8 between the amplitudes of the fluctuations off0(t) and fa(z 5 0, t). A Fourier transformation of Eq.(11b) is first performed, and Eq. (15) is simply substitutedinto the relation obtained. By using a method analogousto that already applied, we easily find that K8 is equal to(bA 2 ln R)/bA . If L and c/n are, respectively, the fiberlength and the phase velocity of the light inside the fiber,bA is equal to pDnBnL/c. By using this expression wecan easily return to physical variables and show that

K 5 1 1 gA /Gc , (17a)

K8 5 1 1 Gc /gA , (17b)

where gA and Gc represent, respectively, the dampingrate of the acoustic wave and the cavity loss rate (gA5 pDnB , Gc 5 2c ln R/nL).

D. Field SpectraIf the model used to describe the spectral properties of thepump laser is the phase-diffusion model, the relevantquantity for the determination of the pump linewidth isthe variance sp

2(t) 5 ^@ f0(t) 2 f0(0)#2& that measuresthe evolution of the dispersion of the values taken by thepump phase. sp

2(t) is a linear function of time that di-rectly yields the FWHM Dnp of the Lorentzian pump spec-trum @ sp

2(t) 5 2pDnpt#.53 Similar functions can obvi-ously be introduced to characterize the acoustic andStokes phase noises $ss

2(t) 5 ^@ fs(0, t) 2 fs(0, 0)#2&,sa

2(t) 5 ^@ fa(0, t) 2 fa(0, 0)#2&%. As the phase fluc-


tuations of the Stokes and the acoustic waves are weakerthan that of the pump wave, their dispersions are alsoweaker and are given, respectively, by ss

2 5 sp2/K2 and

sa2 5 sp

2/K82. The FWHM’s Dns and Dna of the Stokesand acoustic spectra are then related to Dnp through theexpressions

Dns 5 Dnp /K2, (18a)

Dna 5 Dnp/K82. (18b)

This analytical result has been verified by numericalsimulations that were presented in Ref. 9. They arebased on a statistical treatment in which Eqs. (2) and (3)are integrated over an ensemble of 50,000 realizations ofthe random process f0(t). The variances that are char-acteristic of the various noises are evaluated at differenttimes, and the FWHM’s of the various spectra are de-duced from the data obtained. Only a slight relative dif-ference (of the order of a few percent) between numericaland analytical results has thus been found. The origin ofthis discrepancy has already been discussed in Ref. 9: Itis the fact that the influence of the terms Sn(t) (n Þ 0)has been neglected in the analytical treatment.

The pump and Stokes spectra presented in Ref. 9 wereplotted on a linear scale. Moreover, their maximum am-plitude was normalized to unity. This kind of represen-tation clearly illustrates the difference between theFWHM’s of the two spectra. However, its main disad-vantage is that it conceals the difference between the am-plitudes of the field spectra. Let us recall that the fieldspectrum is defined as the Fourier transform of the fieldautocorrelation function C(t).27,53 As all the amplitudesare time independent, this function reads as

C~t ! 5 Ai2^exp$if i~t 1 t ! 2 f i~t!%& (19)

where Ai and f i represent, respectively, the amplitudeand the phase of any of the three waves (i 5 p, s, a). Byperforming a calculation usually found in the literaturefor phase diffusing fields,27,53 one easily derives the ex-pression for the corresponding Lorentzian field spectrum:

Si~n! 5 Ai2

2pDn i

~pDn i!2 1 4p2n2 . (20)

For values of Dn i deduced from the numerical simula-tions presented in Ref. 9, the three field spectra are nowpresented on logarithmic scales in Fig. 1(a). The valuesused for the field amplitudes are those measured at z5 0 [see Fig. 1(c) of Ref. 9]. Three vertical lines plotted

in Fig. 1(a) permit the HWHM of each spectrum to be lo-cated easily. As K82 is equal to 1.19, the widths of thepump and the acoustic spectra do not differ widely. How-ever, the Stokes spectrum is clearly much narrower (K2

5 144), and its maximum amplitude is much greaterthan that of the pump spectrum. In fact, as the ampli-tude of the pump field is nearly equal to the amplitude ofthe Stokes field at z 5 0, the ratio between the maximumamplitudes of the two spectra is nearly equal to K2 [seeEq. (20)].

We can now adopt a simplified approach and see theBrillouin fiber ring laser as a system in which the inci-dent pump field and the emitted Stokes field represent,respectively, the input and the output quantities. In

these conditions the laser acts as a system that squeezesthe width and stretches the peak amplitude of the fieldspectra. If the input and output fields have the same am-plitude, the squeezing and stretching factors are identicaland equal K2. As shown in Fig. 1(b), the narrowing ef-fect becomes very strong when a good cavity is used (R5 0.974). The coefficient K2 is then 2 3 104, and the

Stokes linewidth is much narrower than the pump andacoustic linewidths, which are almost identical (K82

. 1). If the FWHM of the pump linewidth is of the or-der of 100 kHz, the linewidth of such a Brillouin laser isas narrow as a few hertz. A value of this order of mag-nitude was effectively measured in Ref. 54 by beating ofthe two first-order Stokes components emitted by an all-fiber Brillouin laser.

Let us finally note that the expression that connectsthe linewidth of the acoustic wave to that of the pump la-ser [Eq. 18(b)] can be written as

Dna 5gA

2

~gA2 1 Gc

2!Dnp . (21)

The factor that connects the two linewidths has exactlythe same structure as the correction to the Shawlow–Townes formula found in bad-cavity lasers.14 In thiscase the acoustic damping rate is obviously replaced bythe decay rate of the atomic polarization. In bad-cavitylasers, the deviation from the Shawlow–Townes formulaarises from the fact that the phase diffusion no longer re-mains a Markovian process governed by Eq. (4). Randomforce q(t) is not d correlated but has a memory that is aresult of the finite relaxation time of the atomicpolarization.17 In Brillouin fiber lasers the situation issimilar. The phase of the pump laser undergoes a ran-dom walk governed by Eq. (4), but, because the acousticresponse time is finite (.10 ns), the acoustic phase doesnot adiabatically follow the evolution of the pump phase(see Fig. 1 of Ref. 9). As in bad-cavity lasers, the phasediffusion of the acoustic wave is not a Markovian processand the relations that give the linewidths are thus simi-lar.

4. EXPERIMENTSThis section is devoted to the presentation of experimentsin which we measure the coefficient K that characterizesthe narrowing effect in a Brillouin fiber ring laser operat-

Fig. 1. Field spectra of pump, Stokes, and acoustic waves: (a)g 5 6.04, bA 5 10.93, R 5 0.36, K2 5 144, K82 5 1.19, (b)g 5 0.77, bA 5 3.64, R 5 0.974, K2 5 2 3 104, K82 5 1.015.The frequencies are measured in units of cavity FSR.


ing in a low-finesse cavity. The principle of the experi-ment is based on the existence of a deterministic connec-tion between the pump and the Stokes phases [Eqs. (15)and (16)]. If the phase of the pump laser is modulated,the phase is transformed into the phase of the emittedStokes wave, and the ratio between the two modulationamplitudes is simply K. This principle is illustrated inFig. 2 by numerical simulations in which a sinusoidalmodulation is applied to the phase f0 of the pump laser.The integrated equations are Eqs. (2) and (3), and the nu-merical values of the parameters that describe the Bril-louin laser are identical to those used in Ref. 9. The ratiobetween the amplitudes of the two sine functions pre-sented in Fig. 2 is equal to 12.01, whereas that deducedfrom Eq. (17a) is 11.70. The origin of this slight differ-ence was mentioned in Subsection 3.D and in Ref. 9. Itarises from the fact that we derived Eq. (17a) by assum-ing that the spatial profile that characterizes the phase ofthe Stokes wave is perfectly uniform. This approxima-tion is well justified when the characteristic frequency as-sociated with the evolution of the pump phase is muchweaker than the cavity FSR. In Fig. 2 this frequency is100 times lower than the cavity FSR, but that value is notyet sufficient to ensure that the phase profile of theStokes wave does not slightly deviate from uniformity.This explains the difference between the numerical andanalytical values.

In our experiments, the method previously depicted isused: The frequency of the pump laser is modulated, andthe modulation index that characterizes the Stokes waveis compared with that of the pump laser by use of the ex-perimental setup presented in Fig. 3. The pump sourceis a Ti:sapphire laser emitting a linearly polarized beamat 810 nm. Its output power is 2.4 W, and it is opticallyisolated from the Brillouin laser by a Faraday isolator.Frequency modulation of the pump beam is ensured by awaveform generator, a voltage-controlled oscillator(VCO), and an acousto-optic modulator (AOM). When asinusoidal voltage is applied to the VCO, it generates aconstant-amplitude, frequency-modulated rf signal. Af-ter amplification, this rf signal, oscillating at nRF5 181.7 MHz, is applied to the AOM that diffracts the in-

cident beam. The transmitted beam passes through theAOM without any deviation, and it carries most of the in-

Fig. 2. Numerical simulations: response of the Brillouin fiberring laser to a sinusoidal modulation of phase f0 of the pump la-ser (g 5 6.04, bA 5 10.93, R 5 0.36); fs is the phase of the emit-ted Stokes wave, and time t is normalized to the transit time ofthe light inside the fiber.

cident power (1.3 W). The transmitted optical field isneither amplitude modulated nor frequency modulated,and it reads simply as

Ep0~t ! 5 Ep0 sin~2pnpt !, (22)

where np is the frequency of the pump laser. The orien-tation of the AOM is adjusted such that the power of thediffracted beam is ;900 mW. The diffracted field is fre-quency modulated and reads as

Ep1~t ! 5 Ep1 sinF2p~np 1 nRF!t 1dnp

fsin~2pft !G ,

(23)

where f is the frequency modulation and dnp is the maxi-mum frequency deviation. These two parameters arecontrolled, respectively, by the frequency and the ampli-tude of the sinusoidal signal delivered by the waveformgenerator. As the frequency of the rf signal that is fed tothe AOM varies, we should note that the diffraction anglebetween the transmitted and diffracted beams alsovaries.55 However, this effect is neglected in our experi-ments because dnp will always remain much lower thannRF .

After reflection off a beam splitter (BS2, Fig. 3), the dif-fracted beam is focused into the fiber through a 203 mi-croscope objective. The reflectivity of the beam splitter is30%, and the coupling efficiency is ;40%; the power in-jected into the fiber is approximately of 110 mW. Beforebeing coupled into the fiber, the other beam is succes-sively reflected and transmitted by two beam splitters,BS1 and BS2. As the characteristics of these two beamsplitters are identical (reflection coefficient, 30%; trans-mission coefficient, 70%), the optical power injected isagain ;110 mW. Let us emphasize that the input end ofthe polarization-maintaining fiber is adjusted such thatone of its main axes is parallel to the polarization direc-tion that is common to the two incident beams. The out-put end of the fiber is then rotated such that the polariza-tion direction of the output light becomes parallel to the

Fig. 3. Schematic representation of the experimental setup.


polarization direction at the input end. In a ring resona-tor thus constructed, the light remains always linearlypolarized, which prevents the appearance of polarizationinstabilities56 and permits the introduction of a Faradayisolator inside the cavity. In these conditions, pump re-coupling is not possible, and the correspondence withtheory is greatly simplified.

We summarize our experiment as follows: Two pumpfields are injected into the fiber. Their amplitudes aretime independent and nearly identical. One of them isfrequency modulated, and the mean frequency differencebetween the two fields is nRF . At the working wave-length, nRF is approximately three times greater thanDnB , the FWHM of the homogeneously broadened Bril-louin gain curve. Therefore the gain profiles associatedwith the two pump fields do not significantly overlap. Inthese conditions, each of the two pump fields will interactwith its own Stokes wave and its own acoustic wave. Inother words, two independent Brillouin lasers will oper-ate in the same ring cavity. In the experiments pre-sented in Ref. 8, an analogous method was also used, but,in that case, two different lasers were used to pump thesame ring cavity, and neither of them was frequencymodulated. The main value of the method lies in the factthat the two Brillouin lasers are subject to identical fluc-tuations of the fiber length. Therefore their beat fre-quency does not drift strongly.

As the single-mode fiber used is 12 m long, the FSR ofthe ring cavity is approximately 17 MHz. This value is ofthe same order of magnitude as DnB (.60 MHz at 810nm), and the homogeneously broadened Brillouin line re-stricts the Stokes oscillation to a single longitudinalmode.5 In these conditions the intensity of each Stokesfield is time independent and can be monitored by a sili-con photodiode. As the photodiode’s bandwidth is 200MHz, its connection to the rf spectrum analyzer will allowus to record the beat spectrum between the two Brillouinlasers near nRF . Before describing the spectra that aretypically observed, let us add an important commentabout the experimental setup. A small part (4%) of thepump power focused at the input end of the fiber under-goes Fresnel reflection. In the aerial arm of the resona-tor, two weak pump fields copropagate with the twoStokes waves. They are reflected by beam splitter BS3(reflection coefficient, 4%) and detected by the photodiode.Therefore the beat spectra between the Brillouin lasersand between the pump beams are simultaneously re-corded on the rf spectrum analyzer, as is illustrated inFig. 4, which clearly shows that two separate spectra areobserved on each recording. The beat spectrum betweenthe pump fields can easily be identified. Its peak poweris approximately 20 dB below the peak power of theStokes beat. Before detailing the other differences be-tween the two spectra, we first explain why the beat fre-quency of the Brillouin lasers differs slightly (.2 MHz)from the beat frequency of the two pump fields.

First, a simple comparison of relations (22) and (23)shows that the beat spectrum between the pump fields iscentered precisely about nRF . In fact, the recorded powerspectrum Sp(n) is composed of discrete frequency compo-nents symmetrically distributed about nRF and can bewritten as

Sp~n! 5 ap (n52`

1`

uJn~mp!u2d ~n 2 nRF 2 nf !, (24)

where Jn represents a Bessel function of order n.57,58 apis a coefficient proportional to the photodiode efficiencyand to the fields Ep0 and Ep1 . mp 5 dnp /f is a dimen-sionless parameter defined as the modulation index of thepump field. As illustrated in Fig. 5, pump field Ep0 gen-erates a gain bandwidth downshifted by acoustic fre-quency na (.20 GHz at 810 nm) and centered about ns .For simplicity let us assume that ns coincides with one ofthe resonator eigenfrequencies. In these conditions,frequency-pulling effects do not exist in the correspondingBrillouin laser.50 However, as nRF is not a whole mul-tiple of the resonator FSR, the center ns8 of the other gainbandwidth does not coincide with one of the resonatoreigenfrequencies. Therefore the second Brillouin laseroscillates at a mean frequency nL that lies between ns8and the frequency of the closest eigenmode (see Fig. 5).The central frequency of the beat spectrum between theBrillouin lasers is then slightly different from nRF . Inthe results presented in Fig. 4, it is slightly greater thannRF , but the situation can be reversed if nRF is modified byan amount of the order of 10 MHz or if the pump wave-length is slightly shifted.

Frequency of modulation f is the frequency differencebetween two peaks of a given spectrum (Fig. 4). In ourexperiments, it is 200 kHz, a value that is approximately100 times lower than the cavity FSR. Experimental con-ditions are thus very close to those used for numerical

Fig. 4. In each of these experimental figures, the left spectrumis the beat spectrum between the pump fields and the right spec-trum is the beat spectrum between the Stokes fields. (a) Themodulation index of the pump field is mp 5 1.25, whereas that ofthe Stokes field is only ms 5 0.12 (mp /ms 5 10.4). (b) The sup-pression of the central component of the pump spectrum corre-sponds to an index mp equal to 2.4. The corresponding value ofms is then equal to 0.25 (mp /ms 5 9.6).

Fig. 5. Schematic representation of the various spectral compo-nents involved in the experiment (see text).


simulations presented in Fig. 2. As for the pump fields,the beat spectrum between the two Brillouin lasers iscomposed of discrete components, which indicates thatthe frequency modulation applied to one of the pumpfields is effectively transferred to the correspondingStokes field. However, the beat spectrum between theBrillouin lasers is narrower than the beat spectrum be-tween the pump fields. This shows qualitatively that themaximum frequency deviation dns of the modulatedStokes field is less than dnp . It is possible to find more-quantitative values by writing the power spectrum SB(n)of the Stokes beat in a form comparable to that of Sp(n):

SB~n! 5 aB (n52`

1`

uJn~ms!u2d ~n 2 nB 2 nf !. (25)

aB is a constant that is proportional to the Stokes fieldsand to the photodiode efficiency. nB is the central fre-quency of the beat spectrum between the Brillouin lasers.ms 5 dns /f is the modulation index of the Stokes field.As the weights of the spectral components are given bythe values assumed by Bessel functions, the modulationindexes mp and ms are easily deduced from the recordedspectra. For Fig. 4(b) the central component of the pumpspectrum is suppressed. The corresponding modulationindex mp is the first zero of Bessel function J0 and is2.4.58 For the beat spectrum between the Brillouin la-sers, the first harmonic is 18 dB below the fundamentalcomponent, and the corresponding value of ms is 0.25.The ratio Kexp 5 dnp /dns between the frequency deviationsis then 9.6. Despite all the precautions taken, the char-acteristics of the ring laser (mainly the reinjection rate)inevitably drift during the experiments. From one re-cording to another, the weight of the various spectralcomponents varies slightly, and the measurement of Kexpis never rigorously reproducible. In the particular case ofFig. 4(a) the value of mp has been changed to 1.25, andthe measured value of ms is 0.12. The correspondingvalue of Kexp is slightly different from that previouslyfound and is equal to 10.4. For values of mp rangingfrom 0.5 to 3.8 (3.8 corresponds to suppression of the firstharmonic), the value of Kexp was measured 14 times in afew minutes. The data obtained are dispersed about amean value Kexp of 10.37. The standard deviation thatcharacterizes the dispersion of the measurements is 1.94.We then use Student’s law to determine a confidence in-terval for Kexp , and we finally find that it lies inside therange [8.8; 11.9] with a probability of 99%.58

Let us now compare this experimental result with thetheoretical prediction. The expression of K previouslyfound [Eq. (17a)] can be reformulated in such a way thatthe parameters that characterize the experimental setupclearly appear (see Subsection 3.C). Then

K 5 1 22pDnBnL

c ln~R2!, (26)

where c is the velocity of light in vacuum. Refractive in-dex n of the fiber of length L 5 12 m is 1.45. These twoparameters are known with good accuracy. The mainorigin of the uncertainty in the value of K is the impreci-sion of the other parameters: the FWHM DnB of the Bril-louin gain curve and the power feedback efficiency R2

that characterizes the ring cavity. In all our previousworks we considered that DnB is equal to 60 MHz at 810nm. This value is obtained by interpolation from a mea-surement performed at 514 nm in bulk silica.59 In fibers,the inclusion of GeO2 in the core can alter the Brillouinbandwidth,3 and we estimate that the relative uncer-tainty of the previous value of DnB is 10%. The value ofR2 that is systematically used to describe our experimen-tal setup is 0.13 (R 5 0.36).9,50,52,56 However, we esti-mated that value by assuming that only one beam splitterwas inserted inside the cavity; it must be reevaluatedwith inclusion of the losses that are due to the secondbeam splitter. The value of R2 that describes the experi-mental setup of Fig. 3 is then 0.19 (R 5 0.3). As wementioned above, the feedback efficiency inevitably driftsduring the experiments, and we estimate that the relativeuncertainty in the value of R2 is 20%. Following the pre-vious estimations, we can reasonably consider that thetheoretical parameter K that describes our setup lies in-side the range [8.1; 12.1]. Therefore there is a strongoverlap between the uncertainty intervals that character-ize the measured value and the theoretically estimatedvalue. Despite a relative uncertainty of 20% in the valueof K, the agreement between theoretical and experimen-tal results is correct. It could be improved in further ex-periments performed in an all-fiber laser characterized bya well-defined feedback efficiency. However, the value ofthe feedback efficiency should not be too high. If the nar-rowing effect becomes too strong, it will indeed be difficultto observe the weak spectral components that correspondto frequency modulation of the Stokes field.

5. CONCLUSIONLinewidth narrowing in Brillouin fiber ring lasers hasbeen studied both experimentally and theoretically.First, we compared the nature of noise sources in Bril-louin lasers with that of conventional lasers. Unlike la-sers with population inversion, the Brillouin laser is alight source that is highly sensitive to the phase noise ofthe pump laser. This property is also found in systemssuch as optical parametric oscillators and can be attrib-uted to the strong coupling between the phases and theamplitudes of the three waves involved in the interaction.In Brillouin fiber lasers, the acoustic response time ismuch shorter than the characteristic time of the pump la-ser fluctuations. In these conditions the phase fluctua-tions of the acoustic wave are nearly identical to thephase fluctuations of the pump laser. Because of the con-ditions of phase matching and resonance that govern theSBS interaction, the phase noise of the Stokes wave ismuch weaker than that of the pump laser. This relation-ship was studied within the framework of the usual three-wave model of SBS. In particular, analytical relationsthat connect the linewidths of the three waves have beenderived. The ratio between the pump and the Stokeslinewidths was then measured in a Brillouin laser oper-ating in a low-finesse resonator. The principle of the ex-periment consists in recording the response of the Bril-louin laser to frequency modulation of the pump beam.The coefficient K that characterizes the magnitude of the


narrowing effect is equal to the ratio between the modu-lation indices measured for the Stokes and the pumpwaves. In our experiments a quantitative agreement towithin 20% was found between the measured value andthe value theoretically estimated. Obviously, this agree-ment could be improved in further experiments per-formed in an all-fiber Brillouin laser characterized by pa-rameters that drift weakly.

In the research described here we neglected the influ-ence of the acoustic noise that is due to fluctuations of fi-ber density that occur inside the fiber at thermal equilib-rium. Like spontaneous emission in a conventionallaser, this acoustic noise will be responsible for the exis-tence of a lower limit to the linewidth of a single-modeBrillouin laser. As we mentioned in Section 2, this fun-damental linewidth was calculated in Ref. 49 by neglect ofpump phase noise and pump depletion. Theoretical stud-ies are currently in progress in which we analyze the in-fluence of spontaneous scattering and of pump phasenoise on the temporal coherence of a Brillouin laser. Theresults should permit the domain of validity of Eqs. (18)to be precisely delimited.

Our study shows that the Brillouin laser can be used asa system that squeezes the field spectrum of incoming ra-diation. To that effect, we have considered that thephase of this radiation underwent a random walk. How-ever, the Brillouin laser is not sensitive to the nature ofthe noise. As long as the characteristic frequencies ofthis noise are lower than the laser’s FSR, the Brillouin la-ser will emit a Stokes field whose phase fluctuations aremuch weaker than those of the pump field. This resultcan be useful in reducing the technical noise of a lightsource and may be of particular interest for metrologicalapplications in which extremely narrow linewidths arerequired.60 However, careful stabilization of the fiberlength is then necessary to avoid mode hops of the Bril-louin laser.5,50 Therefore, additional studies must be un-dertaken to characterize precisely the influence of jittersof the fiber length on the laser linewidth.

ACKNOWLEDGMENTSThe Centre d’Etudes et de Recherches Lasers et Applica-tions is supported by the Ministere Charge de la Recher-che, the Region Nord/Pas de Calais, and the Fonds Eu-ropeen de Developpement Economique des Regions. Thisresearch was partially supported by European contract‘‘Intereg II Nord-Pas de Calais/Kent.’’

S. Randoux’s e-mail address is [email protected].

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experimental and theoretical study of linewidth narrowing in brillouin fiber ring lasers

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