stimulated brillouin scattering in a dual-clad fiber amplifier

Nathan A. Brilliant Vol. 19, No. 11 /November 2002 /J. Opt. Soc. Am. B 2551

Stimulated Brillouin scattering in a dual-cladfiber amplifier

Nathan A. Brilliant*

U.S. Air Force Research Laboratory, Directed Energy Directorate, 3550 Aberdeen SE, Kirtland Air Force Base,New Mexico 87117

Received October 12, 2001; revised manuscript received March 25, 2002

Experimental measurements are presented of the stimulated Brillouin scattering (SBS) threshold in a dual-clad fiber amplifier with a single-mode core and an approximation is explored for calculation of the SBS thresh-old and numerical solutions for the coupled differential equations. Good agreement is shown between mod-eled and experimental data. © 2002 Optical Society of America

OCIS codes: 060.2320, 060.4370, 140.3510.

.1. INTRODUCTIONDual-clad fiber lasers and amplifiers are an attractive so-lution for high-power devices with diffraction-limited out-put. They also offer the potential for rugged packagingand easy thermal management. However, since single fi-ber elements are unlikely to exceed the few hundred wattrange, very high-power devices would require some sort ofbeam combination. A coherent combination requiresnarrow line sources, which is problematic in typicalsingle-mode dual-clad fiber amplifiers because they mustbe long to absorb all the pump light. The high fieldscaused by confinement to the single-mode core and longfibers make the gain for nonlinear processes quite high.For amplifiers with seed source linewidths well below thestimulated Brillouin scattering (SBS) bandwidth, powerconversion into the Stokes can be quite efficient and lim-its the power extractable by the seed. In addition, SBSarises from noise that causes severe amplitude and phasemodulation of the amplified signal.

There have been many recent results that demonstratehigh power1 and wide tunability2 in dual-clad fiber lasers.There have also been many results for dual-clad fiberamplifiers.3,4 These results have either been in short fi-bers or have had broad bandwidths. Not one reportedSBS. Hofer et al.5 reported a promising result (20 W andno SBS) in a large-mode area dual-clad fiber seeded witha single-frequency laser. The large core allows forshorter fibers and lower core intensities, which reduce thegain for SBS.

Most experimental observations of SBS have beenmade of long passive fiber with signal powers in the mil-liwatt regime. Some experiments have been made insingle cladding erbium-doped amplifiers.6,7 However,these were for pulsed systems with high peak power.There have been only a few published results for SBSthresholds in continuous-wave amplifiers with narrow-linewidth seeds. Zawischa et al.8 presented a 30-m,single-frequency amplifier that was limited to 5.5 W bySBS. Their results are difficult to compare directly withthe results presented in this paper since the fiber geom-etries are different. However, this study confirms that

0740-3224/2002/112551-07$15.00 ©

SBS limits power to a few watts for single-mode, single-frequency fiber amplifiers.

Rate equation models for fiber amplifiers arenumerous.9–13 Models for amplifiers with SBS are alsoavailable. Most of these models14–16 do not include theinversion; they treat the gain as distributed and unsatur-able. The model from Zhang and O’Reilly17 includes thesteady-state population inversion. However, theypresent results for Brillouin scattering of the pump light,not the signal. For a dual-clad amplifier, scattering ofthe pump is not significant, since the pump sources arebroadband and the intensities are low. Moore18 provideda time-dependent model for SBS in fiber amplifiers. Thismodel works with field amplitudes as opposed to powerand includes the effects of population inversion and phaseconjugation. Moore did not, however, apply the model tothe specific problem of predicting SBS thresholds.

Here I present an analysis of the dependence of theSBS threshold on fiber length. I calculate the thresholdby numerically solving the differential equations forpump, signal, and Stokes power and the rate equation forthe population inversion. I also find the threshold by ap-proximating the inversion gain as a constant distributedgain. Both predicted thresholds agree with experimen-tally measured thresholds for two fiber lengths.

2. NUMERICAL MODELAn amplifier seeded with a narrow-line source can be de-scribed by a modified rate equation model that includesdifferential equations for the pump, signal, Stokes, andpopulation inversion. The model used for the data in thispaper is based on the models of Marcerou et al.,9 Zhangand O’Reilly,17 and Oron and Hardy.12,13 My model dif-fers from these models in that it includes the effects ofSBS for the propagating signal and also includes multipleorders of Stokes radiation. Most of the papers writtenabout SBS use a different nomenclature from what I usein this paper. For SBS in passive fiber, the beam thatcauses the SBS is referred to as the pump. I use signal to

2002 Optical Society of America

2552 J. Opt. Soc. Am. B/Vol. 19, No. 11 /November 2002 Nathan A. Brilliant

describe the beam that initiates SBS. The pump in thispaper refers to the diode light that is used to invert thedopant ions.

The differential equation that describes the interactionof light with the population inversion is

dPi6

dz5 6@~ s i

e 1 s ia!n2 2 s i

a#N0G iPi6 7 a iPi

6.

(1)

The upper index indicates direction of propagation, andthe lower index identifies the beam (pump, signal, etc.).P is the power, N0 is the number density of the rare earthdopant ions in the core; G is the mode overlap of theguided mode with the dopant; s i

e and s ia are the emis-

sion and absorption cross sections; n2 is the fractional in-version, the ratio of inverted ions to total ions, and variesbetween 0 and 1; and a is a scattering loss coefficient.

Equation (1) is adequate for the pump but not for thesignal or the Stokes. For the signal and the Stokeswaves, there are additional terms in the differential equa-tion

6gBGBPi6~Pi21

7 2 Pi117! 6 gSB~Pi21

7 2 Pi6!, (2)

where gB is the SBS gain. There is a well-known line-width dependence for the SBS gain. According to Licht-man and Friesem,19 the SBS gain decreases when the sig-nal linewidth increases according to

gB 5 gB81

1 1 Dn i /DnB, (3)

where gB8 is the maximum gain, DnB is the intrinsic line-width of the Brillouin process, gB8 is of the order of 53 10211 m/W, and DnB is approximately 50 MHz. How-

ever, there can be wide variation in both values depend-ing on the core dopants.20 When Dn i ! DnB , the Bril-louin gain is nearly constant at its maximum value. IfDn i @ DnB , then gB 5 gB8DnB /Dn i . The modeled andexperimental data in this paper fall into the first categoryin which the seed linewidth is much smaller than theBrillouin bandwidth. The lower indices, i 1 1 and i2 1, refer to the different orders of SBS. There is a fun-damental beam that has the highest frequency. Thefirst-order Stokes is shifted to a lower frequency by DnS .The second Stokes is shifted from the first order by an-other DnS . SBS can generate a series of lines, eachshifted by DnS from its neighbor. One-order Stokes gainspower from the next lower order and loses power to thenext highest order. The index i 1 1 refers to the nexthigher-order Stokes; i 2 1 refers to the next lower-orderStokes. SBS is initiated by a spontaneous process. Toaccount for the spontaneous process, I assume that thereis one noise photon in the next higher order. This noisephoton interacts through the SBS process and movespower from the lower order into the higher order. Thegain for the spontaneous process is gSB 5 gBGBPN ,where PN is the power associated with one photon withinthe beam linewidth and is given by PN 5 hnDn i and Dn iis the linewidth of the beam that is being scattered. Thecomplete description for a beam interaction with the in-version and through SBS is found by addition of the ex-pression in Eq. (2) to the right-hand side of Eq. (1). For a

Gaussian approximation to the guided mode, GB5 1/(pv2), where v is the radius at which the intensityhas dropped by 1/e2. GB

21 is also referred to as the ef-fective area.

For clarity, the complete set of differential equations foran amplifier with pump, signal, and the first-order Stokesis

dP12

dz5 2@~ s1

e 1 s1a!n2 2 s1

a#N0G1P12 1 a1P1

2,

(4)

dP21

dz5 1@~ s2

e 1 s2a!n2 2 s2

a#N0G2P21 2 a2P2

1

2 gBGBP21P3

2 2 gSBP21, (5)

dP32

dz5 2@~ s3

e 1 s3a!n3 2 s3

a#N0G3P32 1 a3P3

1

2 gBGBP21P3

2 2 gSBP21. (6)

Index 1 marks the pump, 2 is the signal, and 3 is theStokes. Note that the upper index on the pump is minussince it is injected at z 5 L, where L is the full length ofthe fiber. The seed is injected at z 5 0. The first Stokesalso counterpropagates; it starts with no power at z 5 Land grows as it moves toward z 5 0. In the steady state,the normalized inversion is

n2 5

(i

s iaG i~Pi

1 1 Pi2!~Ahn i!

21

1

t21 (

i~ s i

e 1 s ia!G i~Pi

1 1 Pi2!~Ahn i!

21

,

(7)

where A is the area of the core and t2 is the upper-statelifetime.

One can find the performance of an amplifier by nu-merically solving the coupled differential equations forpump, signal, and Stokes powers. Since there are bound-ary conditions at both fiber ends, a simple integration isnot sufficient to find a solution. The modeled data in thispaper were calculated by implementing a shooting21 algo-rithm with a globally convergent Newton–Raphson22 al-gorithm. The boundary condition for the pump requiresthat the pump power at z 5 L equal the power injectedinto the pump cladding. The first-order Stokes powerand all the odd-order Stokes powers must be 0 at z 5 L.

Figure 1 shows modeled data for a 50-m amplifierpumped with 10 W of 915-nm light and seeded with 1 mWof 1100-nm light. The seed is injected at z 5 0; thepump is injected at z 5 L. The signal linewidth is 1kHz, which is much smaller than the Brillouin bandwidthof 50 Mhz, making the SBS gain its maximum value.The data were calculated assuming that the passive scat-tering for all the waves is zero. The fiber geometry is thesame as the fiber in the experiment: 225-mm, 0.22-NAcladding, and 8-mm, 0.1-NA core. The number density ofthe rare earth dopant is 0.9 3 1026 m23. The upper-state lifetime and absorption and emission cross sectionsare from Patel.23 The signal and the pump counter-propagate. The pump power, not shown in Fig. 1, decays


exponentially from right to left. The population inver-sion does not follow an exponential decay from right toleft because of strong saturation by the signal and even-order Stokes waves. Because of the large first-orderStokes, the signal undergoes strong attenuation in thefirst half of the fiber; there is gain only after approxi-mately 20 m. This attenuation near z 5 0 is not due toground-state absorption since the population inversion inthis region is above the minimum necessary for transpar-ency at the signal wavelength. The first-order Stokesinitiates from the signal near z 5 L and undergoes mod-est gain until it reaches the middle of the fiber. The first-order Stokes does not reach an appreciable power leveluntil after it passes through the region of high gain. Asthe first-order Stokes reaches the left-hand side of the fi-ber, it experiences lower gain because there is low popu-lation inversion and there is not much power in the signalto transfer through the Brillouin process. Although theStokes starts from the nonlinear interaction with the sig-nal, it does not receive much power from the signal. Thesecond-order Stokes is initiated by the first order near z5 0. As with the other beams, the greatest gain is obvi-ous near the center of the fiber where the inversion islarge. SBS limits the extraction by the signal becausethe Stokes steals the population inversion and makes thesignal less able to compete for the gain. Four orders ofSBS have appreciable power and contribute to the satu-ration of the population. All the beams undergo stronggain near 25 m because of the large population inversion.

3. STIMULATED BRILLOUIN SCATTERINGTHRESHOLDThe threshold signal power at which SBS begins to ariseis difficult to ascertain, especially in the presence of gain.There is a threshold condition for low-loss passivefibers.24 There is also a commonly quoted threshold forfiber lasers and amplifiers, which is 40 W m. The originof this rule of thumb is not presented in the literature, butit is likely based on the passive fiber threshold. Theanalysis in this section follows that of Pannell et al.16 inthat it assumes that the gain per unit length is constant.Pannell et al. concentrated mostly on the dependence of

Fig. 1. Modeled data that depict the evolution of signal powerand the power for multiple Stokes orders along the fiber lengthfor 10 W of pump power. Power is read from the left scale. Thefractional inversion is read from the right scale.

the threshold on gain. Here, I emphasize the dependenceof the SBS threshold on the fiber length.

Consider a simple situation in which there are twofields and the gain per unit length is constant. Althoughthis approximation is not accurate, it allows an analyticsolution for the differential equations. The equationsthat describe the forward-propagating signal (Ps) and thebackpropagating Stokes (PB) are based on Eqs. (1) and (2)and are given by

dPs

dz5 gPs 2 gBGBPsPB , (8)

dPB

dz5 2gPB 2 gBGBPsPB . (9)

where g is the gain coefficient, gB is the Brillouin gain co-efficient, and GB is the Brillouin overlap. The signal isinjected at z 5 0. The Stokes begins at z 5 L and in-creases as z get smaller. Assuming that the Stokes doesnot rise enough to affect the signal, the signal evolution isstraightforward:

Ps~z ! 5 Ps~0 !exp~ gz !. (10)

After substituting Eq. (10) into Eq. (9) one can solve Eq.(9) to determine

lnFPB~L !

PB~0 !G 5 2gL 2

gBGB

gPs~0 !@exp~ gL ! 2 1#.

(11)

Equation (11) can be recast in a different form to providegreater insight into the threshold dependence on fiberlength. Ps

th(L) is the output signal power at threshold.Let P 5 Ps

th(L)L, where PN is the power of one noisephoton from Ref. 24, G 5 exp( gL) is the integrated gain,h is a number that sets the Stokes-to-signal ratio that de-fines the threshold, and PB(0) 5 hPs

th(L). Equation(11) becomes

F lnS hP

PNL D 2 ln~G !G ln~G !

gBGB

1

1 2 G21 5 P. (12)

Fig. 2. Graphic solution to Eq. (12) that shows the insensitivityof the SBS threshold to the fiber length. The solid line repre-sents the right-hand side of the equation. The dashed, dotted,and dash–dot curves represent the left-hand side of the equationfor lengths of 5, 50, and 500 m.


The analysis here uses h 5 0.01. Equation (12) can eas-ily be solved numerically in a high-level language such asMathematica to determine the SBS threshold power.Modeled data and the rule of thumb lead us to expect aproduct law dependence for the SBS threshold. Figure 2provides some insight by showing the graphic solution ofEq. (12) for three different lengths. The solid line repre-sents the right-hand side of Eq. (12). The dotted curverepresents the left-hand side of Eq. (12) for L 5 50 m.The dashed curve and the dash–dot curve represent theleft-hand side for 5 and 500 m, respectively. The solutionfor 5 m and the solution for 500 m differ by approximately10% from the solution for 50 m. An order of magnitudechange in the length results in a fractional change P.The threshold power varies strongly with length, butpower times length does not change dramatically. There-fore, measurement of the SBS threshold for one length offiber will provide an accurate estimate for other fiberlengths as well. The threshold power-length product isinsensitive to the length because the length appears in-side the natural logarithm on the left-hand side of Eq.(12). The data in Fig. 2 were calculated for parameterstaken from the SBS measurement experiment: G ; 10,gB 5 5 3 10211 mW21, and GB

21 5 6.6 3 10211 m2. Al-though P is somewhat insensitive to the length, it has astrong dependence on both the SBS gain and the SBSoverlap. For seed linewidths larger than the Brillouinbandwidth, the gain is proportional to the reciprocal ofthe seed linewidth. Therefore, P is linearly dependenton the seed linewidth. For narrow-line sources, thethreshold is independent of the seed linewidth. P alsorises linearly with the effective area.

4. AMPLIFIER CONFIGURATIONFigure 3 shows the experiment in which the seed consistsof a fiber distributed feedback laser and a series of ampli-fiers. The pump is a 40-W source that exits from a 200-mm, 0.22-NA fiber. The amplifier fiber is single mode,dual clad, and doped with ytterbium. The pump claddingis approximately 225 mm in diameter with a 0.4 NA; thecore is approximately 8 mm in diameter with a 0.1 NA.Both ends are angle polished to 8° to minimize reflectionsand feedback, which would exacerbate instability andself-pulsing. Since the fiber is not polarization maintain-ing, part of the fiber is wound into polarization paddles tocorrect the output polarization. The signal and the pump

Fig. 3. Experimental apparatus. The slanted dotted lineshows the path of the signal and the SBS. The detector is eitheran OSA or a 40-GHz photodetector.

light are separated by a dichroic mirror. There are twoFaraday isolators between the seed and the amplifierwhich are necessary to prevent SBS from the main ampli-fier from affecting the last amplifier in the seed chain.Each isolator is tunable in wavelength. However, achiev-ing maximum isolation can be difficult. Two isolatorsguarantee high isolation even if the tuning of the indi-vidual isolators is not perfect. There is a 90% transmis-sion mirror between the isolators and the amplifier. Thismirror was used to pick off some of the counterpropagat-ing light for examination. The light reflected from thismirror can be sent to an optical spectrum analyzer (OSA),a monochromator, a 40-GHz detector, or a powermeter.After addition of all the optics, there was up to 300 mW oflinearly polarized seed light available at the input of theamplifier. The coupling of the seed light into the fibercore was of the order of 50%. The amplified signal wasalso sampled with a powermeter and the OSA.

5. EXPERIMENTAL RESULTSThe experiment successfully measured the SBS thresholdfor two fiber lengths. The presence of SBS was confirmedby the optical spectrum and radio frequency (rf) spectrumof the counterpropagating light. Figure 4 shows the am-plifier performance for a 50-m fiber. The data aregrouped into two sets differentiated by the position of thepolarization paddles. For the first set—squares, up tri-angles, and right triangles—the paddles were adjusted tomaximize the forward-propagating power. The forward-propagating power (up triangles) rises linearly to a maxi-mum of approximately 2 W. For greater pump power, theforward power declines, and the backward SBS power(right triangles) rises. The second data set—circles,down triangles, and left triangles—the paddles were setto maximize the backpropagating light. For this set, theamplified signal (down triangles) rises to approximately1.2 W, where it levels off and the SBS (left triangles) takesoff. With additional pump light, the forward-propagatingpower increases again. This increase comes from thesecond-order Stokes generated from the first order. Forboth sets of data, the total power extracted (circles andsquares) rises with a constant slope, which is the same asthe slope of the signal for low pump power. All the pre-vious discussions of SBS have assumed that the signaland the Stokes have the same polarization. Fiber is no-torious for changing the polarization state; even thoughthe seed is linearly polarized, the output of the amplifiercould be elliptically polarized. The SBS gain dependsstrongly on the length of the fiber over which the signaland the Stokes are polarized in the same direction. Theinitial polarization of the Stokes is determined by the po-larization of the seed at z 5 L. As it propagates towardz 5 0, the Stokes polarization can change with respect tothe polarization of the seed. By adjusting the paddles,one can maximize the length of the fiber for which theseed and the Stokes have nearly the same polarization.This leads to a lower SBS threshold. Conversely, one canminimize the overlap in polarization leading to a higherthreshold. The maximum nonlinear gain occurs whenthe signal and the Stokes are linearly polarized in thesame direction at every point in the fiber. This condition


provides a lower bound on the SBS threshold. I made asimilar measurement for a 75-m fiber with the expectedresults: SBS occurred for a lower signal power. Thedata for the 75-m fiber are shown in Fig. 5. The 75-mdata show the same behavior with respect to polarization.The experiment with a 25-m fiber resulted in no meaning-ful data. The peak gain was at a much shorter wave-length than the seed. Amplified spontaneous emission(ASE) and self-pulsing became a problem before I couldmeasure the SBS threshold.

The counterpropagating light need not necessarily becaused by SBS. It could be the result of strong ASE.However, the spectrum of the back propagating lightshows two resolved peaks. The first peak is signal re-flected from an optic or coupled into the counterpropagat-ing mode by Rayleigh scattering. The peak at the lowerwavelength was shifted by approximately 0.05 nm. At1.1 mm, a shift of 0.05 nm is approximately 12 GHz. This

Fig. 4. Amplifier performance for a 50-m fiber. The circles,down triangles, and left triangles represent the total, signal, andStokes powers when the polarization paddles are adjusted formaximum SBS gain. The squares, up triangles, and right tri-angles represent the minimum SBS gain.

Fig. 5. Amplifier performance for a 75-m fiber. The circles,down triangles, and left triangles represent the total, signal, andStokes powers when the polarization paddles are adjusted formaximum SBS gain. The squares, up triangles, and right tri-angles represent the minimum SBS gain.

is the correct order of magnitude for SBS, however, theprecision of this estimate is limited by the resolution ofthe OSA. Even with no pump power, the OSA spectrumshows a small amount of power in the redshifted peak, in-dicating that the unamplified seed has enough power togenerate some small amount of Stokes light. As thepump increases, the power in the SBS peak increases at amuch faster rate than the power in the signal peak.Since a shift of 0.05 nm is near the resolution limit of theOSA, the rf spectrum of the beat note is a better measure-ment. The beat note measured at the high-speed detec-tor, shown in Fig. 6, shows a strong peak in the expected16-GHz range. The earlier estimate of 12 GHz is inaccu-rate because of the lack of resolution of the OSA. Thetwo sets of data in Fig. 6 were taken at different times,with different resolution settings on the rf spectrum ana-lyzer. They both represent typical results in their fre-quency ranges. In Fig. 4, the forward power for the sec-ond data set rises at approximately 8 W of pump powerbecause of the second-order Stokes. The presence of thesecond-order Stokes is confirmed by the OSA that showsthe peak for the fundamental and a second peak shiftedby 0.1 nm. Again, this wavelength shift is near the reso-lution limit of the instrument. The rf spectrum analyzerdid not have sufficient bandwidth to measure the beatnote between the fundamental and the second Stokes.The amplitude fluctuations at these pump powers werelarge enough to make fiber damage a concern. As a re-sult, I cannot make any statements about the thresholdfor the second-order Stokes.

The thresholds measured for the 50- and the 75-m fibercompare well with the model and with the prediction fromthe constant-gain approximation. Figure 7 shows theprediction (solid curve), the modeled data (up trianglesand dotted curve), and the experimental data (circles).The modeled data and constant gain prediction should bea lower bound on the threshold since it assumes that theStokes and the signal have the same polarization. Theinset in Fig. 7 has an expanded scale and shows that thelower of the two points at both lengths lie nearly on theline for the constant-gain prediction. The points for lowpolarization overlap lie significantly above the lower

Fig. 6. Spectra (rf) of the counterpropagating light from the 40-GHz detector. The low-frequency spectrum is the amplitudenoise caused by SBS. The high-frequency peak is the beat notebetween the signal and the Stokes.


bound. The modeled data are also close to the experi-mental points. A suitable change in input parameterscould easily bring the modeled data into agreement withthe experiment. In any case, the model inputs used aregB 5 5 3 10211 mW21 and DnB 5 50 MHz. The datafrom Nikles et al.20 indicate that the gain and the line-width for SBS can vary widely depending on the concen-tration and the nature of the core dopants. As far as Iknow, no data for Yb-doped fibers have been published.

Neither the numerical model nor the constant-gain ap-proximation deal with amplifier dynamics in the presenceof SBS. However, the amplifier output becomes noisynear the SBS threshold. The low-frequency part of the rfspectrum, shown in Fig. 6, has a substantial peak near1.5 MHz and smaller peaks at the harmonics. The am-plitude modulation is small, only a few percent, at theSBS threshold. However, the amplitude fluctuations be-come much larger, 30–40%, as the Stokes rises. Thereare many papers that address the dynamics of SBS in fi-bers. One explanation for these low-frequency dynamicsis relaxation oscillations of the Stokes wave.25,26 Thesepapers indicate that pulses should occur with a period de-termined by the time for one round trip in the fiber. Forthe 50-m fiber, the frequency should be around 2.1 MHz.Since the SBS arises from thermal fluctuations, the noisein the output of the amplifier could be described by sto-chastic dynamics.27,28 The data in these papers are dif-ficult to interpret, since there is little regularity to thepulses. However, the data in Ref. 27 seem to indicatepulse periods of a few round-trip times; the data in Ref. 28indicate several pulses per round trip. Another possibil-ity is chaos.29,30 In these papers the authors predicted astrong component at the round-trip time or faster. It isalso possible that the noise is not related to the SBS atall. The modeled data in Ref. 18 show fluctuations atvarious time scales of the order of a single transit time totens of transit times. Reference 31 reports experimentalresults on the noise in a Yb-doped amplifier. They showa distinct signal in the 60-kHz region. The observed be-havior does not seem to fit into any of these theories. An-

Fig. 7. Comparison of theoretical (solid curve), modeled (tri-angles and dotted line), and experimental (circles) SBS thresh-olds. Inset: the same data on an expanded scale. The experi-mental points agree well with both the theoretical and the modeldata.

other likely explanation is that the SBS perturbs thepopulation inversion by some small amount, which thenaffects the gain and the amplitude of the output.

The noise at the SBS threshold is a system level con-cern. Noise such as this in an amplifier leg makes coher-ent combination difficult. The noise becomes muchworse when the second order varies above threshold.The amplitude fluctuations become large and represent apossible source of damage to the fiber. A principal resultin Ref. 18 is that the amplitude fluctuations are reducedby seeding the Stokes with a second laser. However, re-ducing the amplitude noise does not alter the fact thatSBS reduces the power extracted by the seed. The datain this experiment are not sufficient for a detailed analy-sis of the dynamics associated with SBS.

6. SUMMARYI have determined the SBS threshold by solving thecoupled differential equations and by making the approxi-mation that the gain is uniform and by solving the equa-tions analytically. I have shown that the SBS thresholdis well approximated by a product law. I have experi-mentally measured the SBS threshold for an amplifierseeded with a source whose bandwidth is far smaller thanthe SBS bandwidth; the measured threshold agrees withthe theoretical predictions. However, the threshold israther flat over the range of fiber lengths tested. Be-cause of ASE and self-pulsing, I could not test shorter fi-bers. The next step is to examine methods to suppressSBS. Expanding the mode diameter,5 shortening the fi-ber, and spreading the spectrum4 have been shown towork to some degree, allowing tens of watts of single-frequency, single-mode light. More research is requiredto apply these techniques to amplifiers and phased arraysat the hundreds of watt and kilowatt power levels.

*Present address, Coherent Technologies, Inc., Boulder,Colorado; e-mail address, [email protected]

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stimulated brillouin scattering in a dual-clad fiber amplifier

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