Vol. 6, No. 6/June 1989/J. Opt. Soc. Am. B 1167

Longitudinal acoustic modes and Brillouin-gain spectra forGeO2-doped-core single-mode fibers

Nori Shibata

NTT Laboratories, 1-2356, Take Yokosuka-shi, Kanagawa-ken, 238-03, Japan

Katsunari Okamoto*

NTT Laboratories, Atsugi-shi, Kanagawa-ken, 243-01, Japan

Yuji Azuma

NTT Laboratories, 1-2356, Take Yokosuka-shi, Kanagawa-ken, 238-03, Japan

Received September 12, 1988; accepted February 20, 1989

Guided longitudinal-acoustic modes, which give rise to Brillouin gain, are theoretically clarified for a single-modefiber with a GeO2 -doped core and pure-silica cladding. Longitudinal-acoustic Lom modes are found from thetheoretical analysis to interact with the electromagnetic field of the HE,, mode. Brillouin-gain spectra aremeasured for clarifying the theory. A few gain peaks in the Brillouin-gain spectra are successfully explained bytaking account of the phase-velocity characteristics of the Lo,, L0 2 and L03 modes guided in the GeO 2 -doped coreregion. Furthermore, Brillouin frequency shifts per unit dopant concentration for GeO2 and F are experimentallyobtained to confirm the dispersion characteristics of the guided acoustic modes. -The evaluated frequency shifts perunit dopant concentration are 107 and 356 MHz/mol% for GeO2 and F, respectively, at a wavelength of 1550 nm.

1. INTRODUCTION

Optical nonlinear effects, such as Brillouin scattering' andfour-wave mixing processes, 2 are likely to be a concern formultichannel coherent light-wave transmission systems3 -5

employing narrow-linewidth single-frequency lasers. Sincebit-error-rate degradation due to spontaneous Brillouinscattering has been observed in a light-wave transmissionsystem operating at a wavelength of 0.8 ,um,4 precise mea-surements of Brillouin-gain spectra and the correspondingfrequency shift are of great importance from the viewpointof system design in constructing bidirectional communica-tion networks and frequency-division multiplexed commu-nication networks.3 -5

Thomas et al. 6 were the first to present the theory ofnormal acoustic modes and Brillouin scattering in a single-mode optical fiber for Vore > VC1ad and Vcore > Vs1ad, whereVcore, Vlad, Vcore, and Vcld are the longitudinal and shearvelocities in the core and the cladding, respectively. Jen etal.7 analyzed leaky modes in weakly guiding fiber-acousticwaveguides, where Vclore < V1lad, Vcore VIcad, and the densi-ty condition Pcore Pclad for the core and the cladding.Dispersion behavior for lower-order leaky modes was nu-merically clarified, and the analysis is useful for investigat-ing the Brillouin-gain spectra 8 9 for commercially availableGeO2 -doped-core fibers that satisfy the above physical con-dition with respect to phase velocity and density. In accor-dance with the theoretical treatment,7 the identification oflongitudinal acoustic modes guided in the GeO2 -doped-coreregion has been experimentally clarified by Brillouin-gain-spectra measurement. 10 As has been shown experimentally,the longitudinal-acoustic fields of the Lo. and L2 n modes 7

can interact with the electromagnetic field of the HE,,mode. However, it has not been yet clarified theoreticallywhich guided acoustic modes interact dominantly with thelight wave. Therefore theoretical discussions concerningthe guided acoustic modes that give rise to optical gain for aprobe light wave are of great importance for understandingthe acoustic-mode behavior. For forward Brillouin scatter-ing, Shelby et al." have shown that acoustic modes responsi-ble for scattering light in the core of a single-mode fiber areradial Rom modes and mixed torsional-radial TR2m modes.

In this paper we discuss the Brillouin-gain spectra behav-ior theoretically and experimentally for the nominal single-mode fiber with a GeO2 -doped core and silica cladding. Inparticular, we show that the longitudinal L0 m mode can dom-inantly interact with the HE1 , mode. In addition, 13rillouinfrequency-shift coefficients per unit dopant concentrationat a wavelength of 1550 nm are measured for GeQ2 and F toclarify the dispersion characteristics of the longitudinal-acoustic modes.

2. THEORY

Under the conditions Vcore Vclad Vcore = Vclad, and Pcore =

Pclad, the transverse components u, and uo of an acousticdisplacement field are negligible compared with its axialcomponent u'.7 For this situation, all acoustic modes arepredominantly longitudinal and are designated Lnm; other-wise they are designated axial-radial Rom for n -= 0 andflexural F'nm for n> 1. Here the prime is used to differenti-ate between these modes. In principle, the origin ofacousto-optic interaction observed in Brillouin-gain spec-

0740-3224/89/061167-08$02.00 © 1989 Optical Society of America

Shibata et al.

1168 J. Opt. Soc. Am. B/Vol. 6, No. 6/June 1989

tra' 0 is due to the acoustic-wave behaviors of the axial-radialmodes Rom and the flexural modes F which were rigorous-ly analyzed by Thurston1 2 and by ArmenAkas. 13 In thissection we clarify which longitudinal-acoustic Lnm modesstrongly contribute to interaction with the electromagneticfield of the optical HE,, mode.

We consider small deviations of the dielectric tensor fromthe unperturbed dielectric tensor . In this case, electro-magnetic fields [, ft] of the wave scattered by the acousticwaves can be treated as perturbation from the unperturbedelectromagnetic fields [E, H]. Then the scattered fields areexpressed as

X z [E,(P) H,,(q)* -Hv(P) E (q)*]r dr dO

= I _) _ 6_ - cLq),j0 (S)I

(9)

where Cz, is the unit vector along the z direction, subscripts pand q select i and s, respectively, vA& is Kronecker's , and(co) is the 6 function. The acousto-optically induced fluctu-

ation be in the dielectric constant is expressed as

be = ( - ) = /2X(r, O)exp j(Qt - Kz) + c.c. (10)E(r, 0, z, t) = > [a,(z)E() + b,(z)Ev(s)],

v=1

2

H(r, 0, z, t) = I [av(z)Hv() + bv(z)Hv(s)],v=1

(1)

(2)

where subscripts i and s represent incident and scatteredwaves, respectively, v = 1 and v = 2 represent, respectively,two orthogonally polarized HE11x and HEllY modes, and avand b are expansion coefficients as a function of distance zalong an optical fiber. The eigenelectromagnetic modes inEqs. (1) and (2) are expressed as

E(') = ev(r, O)expU[wit - 3v(j)z]l

Hv(j) = hv(r, 0)expU[oit - v()zJI

Ev(S) = e(r, 0)expU[wot - v(s)z]

Hv(-) hv(r, 0)exptj[cot - (s)z]j

(v = 1, 2), (3a)

(v = 1, 2) (3b)

(v = 1, 2), (3c)

(v = 1, 2), (3d)

where and are angular frequency and propagation con-stants, respectively, andj = FE. Since the frequency shiftsof the light wave scattered due to thermally induced acousticwaves are small, I,3v(s)I i v(i)j is valid. Maxwell's equationsfor the perturbed electromagnetic field [, ] are written as

V X E =-,40 dt } (5)

atV X Hft dt- (6)

at

Substituting Eqs. (1) and (2) into Eqs. (5) and (6), we canobtain the following equations for the expansion coefficientsav and bv:

da + jQos 2 bv 2' EJ.(')- * -E)E (s)r dr d = dz JI(~)~srd=

v= 1

= 1, 2), (7)

01(i) dbl, 2 (27r,'l d y + j(s)Ya I | E,,(s)* * )Ev()r dr dO =

13(i)1 dz Z0 ~ EE/I rd

= 1, 2), (8)

where subscript selects 1 or 2. We note here that we deriveEqs. (7) and (8) by applying the orthogonal relationship asfollows:

The components of X(r, 0) are 6

X, = -eo 4(PllSrr + P12Soo + P12Szz),

XO = -eo 4(PllSrr + P12S00 + Pl2Sz),

X = -E0fn4p44 S 0,

where

S._ du,

dur 1 u0

So= -Or + - U

r° r r aduz

SZ = '

Sr 1 dur dUo uO

'° r da dr r

(11)

(12)

(13)

(14)

(15)

(16)

(17)

where Eo is the dielectric constant in free space, n is therefractive index of glass, Puj, P12, and p44 are the optical-strain coefficients, and Srr, Soo, and Sro are strain compo-nents induced by the acoustic waves.

For the forward-and backward-scattered light waves, re-spectively, 13(s) --#(i) and V(s) _-f,(i) are valid. Here, thepropagation direction of the forward-scattered wave is iden-tical to that of the incident wave. Equation (8) is rewrittenfor the forward- and backward-scattered waves as follows:

db ~~~2dz i 2 E a, expU((i + Q-ow)t-j

v=1

27 X [v() + K - ,,)]z} |:" | e*, * XevrdrdO

2_- av expUj(.i-Q- co)t-j2 V= 1~

v=l,2

X [i) - K - i3,)]z} f j e*, X*evrdrdO

(, = 1, 2), (18)

and

Shibata et al.

Vol. 6, No. 6/June 1989/J. Opt. Soc. Am. B 1169

2~~~~~2dz =I2 Ea exp7(wi + -ws)t -jv1l

X [,d3ji + K + (,,i)]zl J J e, . Xevrdrd0

2

+ j i E aexpU(w.i- -w)t-j2 1v=1

X [fl(i)-K _ I3,(i]z1 i: e*,, . X*evrdrd0

(A= 1, 2), (19)

where the superscripts (+) and (-) indicate forward- andbackward-scattered waves, respectively. The angular fre-quency ws of the scattered light wave is found from Eqs. (18)and (19) and written as

co = i QU (20)

This equation shows that the scattered light-wave frequencyws is shifted from the incident light-wave frequency wi by theacoustic-wave frequency . Next we consider only thesmaller frequency component of the scattered wave. Theupshifted case is a simple extension of this result.

When spontaneous Brillouin scattering is considered, thescattered light emission is very small, and we can assume theexpansion coefficient a,(z) constant for v = 1 or v = 2. Onthe other hand, the forward- and backward-scattered wavesare assumed to be zero for z < z and z > zo + L, respectively;that is, the perturbed electromagnetic field exists for thefiber portion from z = zo to z = zo + L. For simplicity, we setz = 0 and z = L a length of fiber. This leads to theboundary conditions expressed as

b A(+)(O) = 0,

bg,(-)() =-j () L Eav e,*

* X* (K = + ,¾,,)evr dr d (AL = 1, 2). (26)

In Eqs. (25) and (26) we can neglect the ez component be-cause Ie2I2/Ier12 - ezj2/eol2 A14 where A is the relativeindex difference between core and cladding and the nominalvalue of A is usually less than 1%. Hence the integrands inEqs. (25) and (26) are approximated as

e,,*X*ev X,*e,*ev,.+ Xor*(ey,*evo + e,lo*ev,)

(27)

We consider the case in which the HEllx mode is launchedinto an optical fiber. Hence the expansion coefficient a2 =

0. Equations (25) and (26) are simplified as follows:

b()(L) =-j -Lal J J e2* *X*(K = O)elrdrdO, (28)

2 o J e 2 * X*(K = 01 - 02)e 1rdrdO,

(29)

and

bl(-)(0)= -j - Lal2

b2-)(0) = -j - Lal2

(21)

2r pO

a J0 e1* - X*(K = 20 1 )elrdrdO,

(30)2rXJ0 | e 2 * X*(K = 3,B + 2)elrdrdO.

(31)

bA(-)(L) = 0. (22)

Application of the above boundary conditions yields thefollowing expansion coefficients of the scattered waves at ws= wx -a

=( 2 - Ei- a, fi exp[-(O -K- 0)z] dz

27r

X e X*evr dr d ( = 1, 2) (23)

and

bA (z) _= -i-i E av | exp[-j(#,l - K + f,)z] dz

r2, X

X Lo f e,, *X*evr drd0 1 = 1 2); (24)

then Eqs. (23) and (24) may be rewritten, inserting z = L andz = 0 for the respective forward- and backward-scatteredwaves, as follows:

bA(+)(L) = -i L E a| | e,,

- X* (K = lB - l,,)evrdrdO

In the above equations, b1(+)(L) and bl(-)(0) represent thescattered waves whose polarization direction is parallel tothe incident-wave polarization, while b2(+)(L) and b2(-)(0)

represent the scattered waves having a polarization direc-tion perpendicular to the incident-wave polarization. Theintegrands in Eqs. (28)-(31) are

el X*e1 = l/2 R2 (r)[(Xr,* + XO*) + (Xr* - XOO*)

X cos 20 - 2XrO* sin 20], (32)

e2* X*el = l/ 2 R 2(r)[(X,,* - XOO*) sin 20 + 2XrO* cos 20],

(33)

with

(Xrr* + XOO*) = -Eon4 [(pll + P12)(S,* + Soo*) + 2Pl2S2 z*]

-Unn(r)cos(nO -TO),

(Xrr* - XOO*) = -e0fn4(pll - P12)(S., + S00*)

= Vn,(r)cos(nO -TO),

(34a)

(34b)

XrO* = -E 0n4p 44S,,,* = Wn.(r)sin(nO - T0). (34c)

Considering backward-scattered waves, the coefficientsbl(-)(zo) and b2(-)(zo) are rewritten, using 1 = 12 = 13, as

Shibata et al.

-- X*e,.O*evo.

Wib2(+)(L = -i2La,

(A = 1, 2), (25)

1170 J. Opt. Soc. Am. B/Vol. 6, No. 6/June 1989

b 1(0) -j La, | el* X*(K = 20)e 1rdrdO2 fo o

( i 2al){[J 2 R 2 (r) Unm(r)rdr]

X [f| cos(nO - fo)d0] + [f 2 R2(r)Vnm(r)rdrl

X [J cos(nO - *o)cos 20d0]

- [ R 2(r) Wnm(r)rdr]

X [J sin(nO - To)sin 20d0]} (35)

and

b2 -)(0) = -ji -Lal f J e 2* - X*(K = 2#)e1rdrdOo o,

- ( Lai) [1 2 R (r)Vnm(r)rdr]

X [ cos(nO - 410 )sin 20dO]

- [ R2 (r) Wnm(r)rdr]

X [J sin(nO - TO)cos 2Od6]} (36)

where R(r) is radial component of the HE,, mode and isgiven for a step-index profile byl3

R(r) = AJo(ur/a) for r < a,

- AJ0 (u)K0 (wr/a)/K0 (w) for r > a. (37)

Here a is the core radius, Jo and Ko are a Bessel function anda modified Bessel function of the second kind, respectively,and u and w are parameters given in the form of u = a(k 2n,2

-#v)1/2 and w = a(gB2 - k2n22)1/2 as functions of the wave

number k, core refractive index nl, and cladding index n2.From the angle integrals in Eqs. (35) and (36), bl(-)(zo) andb2 (-)(zo) are nonzero for axial-radial modes (n = 0 and ' 10 =0) and flexural modes (n = 2). Therefore the Rom and F2m

modes can contribute to the backward-scattered light wavein an acousto-optic interaction. The expansion coefficientsfor the respective Rom and F2m modes are

bl( )(O) = -jr 2 Lal R2(r)Uo0 (r)rdrI,2 Lo J

b2H()= 0,

(38)

(39)

and

cobl(-)(0) = -j7r 4 La cos To

X {J R 2(r)[V 2.(r) - 2W 2m(r)Irdr} (40)

= -hr -i La, sin To4

X {J R 2(r)[V 2.(r) - 2W 2m(r)]rdr}. (41)

It is clearly found from Eqs. (38)-(41) that the scatteredelectromagnetic fields due to the respective Rom and F2mmodes are polarized and depolarized light waves, respective-ly. In accordance with the notation introduced by Armena-kas,13 the acoustic displacements for the respective Rom andF2m modes are

Ur = [df + Kgr(r)]exPU(Qt t-Kz)],

UO = 0 O

(42a)

(42b)

UZ -j[Kf(r) + d r + _]expU(Qt - Kz)] (42c)

and

Ur = [. + Kg,(r) + -Jcos(20 - 'Io)expU(Qt - Kz)],

(43a)

U0 = [ r ) + Kgr(r) - drjsin(20 - Fo)expU(Qt -Kz)](43b)

u, = jKfr) + Or +-r cos(20 - l1)expU(Qt -Kz)],

(43c)

where f(r), gr(r), and gz(r) are the longitudinal, shear-radial,and shear-axial acoustic-mode fields, respectively. FromEqs. (11)-(17), (42), and (43), Uom(r) and V2 (r) - 2W2m(r)in Eqs. (38), (40), and (41) are written as follows:

Uom(r) = -Eon {(Pll + Pl2) [dr2 r dr (dr r )J

- 2P12K(K + -- + P12 ( fdr r )

and

V2 (r) - 2W2m(r) = -on 4 (pll - P12)[ 2 d +

+ 2 dz+ 4 _ X2),r dr \r2FZ

where t and q are expressed as

2 2 - K2

VL2

and

2=Q2 - 221 V2

(44)

(45)

(46)

(47)

It is known that the general solutions of f(r), gr(r), and gz(r)are expressed as12

f(r) = AlJnQ(tr), (48)

Shibata et al.

r2 4-�2f

Vol. 6, No. 6/June 1989/J. Opt. Soc. Am. B 1171

g,(r) = BjJ 0 +1 (n1r), (49)

and

g,(r) = CJJ(n1 r) (50)

for 412 = (Q/VC1 e1)2 -K 2 > 0 and tii2 = (/Vcore) 2 - K 2 > 0 in

the range of 0 < r < a. Here Al, B1, and C, are constants.Equations (48)-(50) represent the Ron and F2 n modes for n= 0 and n _ 1, respectively. The functions gr(r) and gz(r)oscillate many times in the integration interval because 71aL 88 for a = 5 um, X = 1.55 Arm, and K = 2,6 and f(r)gradually changes for 0 < r < a because of (la 1 - 2.4around Q - KVcorel. Then the contribution due to thesetransverse shear modes can be neglected in the integrationsof Eqs. (38), (40), and (41). Hence we can focus on thelongitudinal-acoustic field f(r). U(r) and V2m(r) -2W2n(r) are approximated as

Uo.(r) 2e0n4p,2K2A1J 0Q(tr) (51)

and

V2 (r) - 2W2.(r) _ on4(pll - P12)Aj1 2JO(tr). (52)

The backward scattering strength can be defined by[bi(-)(O)la112 as a ratio of the scattered to the incident inten-sities. The ratio of scattering strength [bj(-)(0)/a1]n=2/[bj(-)(0)/a 11n=0o 2 for the Lom and L2 n modes can be approxi-mated as

=012~ Pl - 12 ,k 2 2{ [bl(-) (0)/al n=2/[bl(-) (0)/al n=0 ( 1 2)(2)

(53)

where [b-(-)(0)/a11n=2 = [b2 (-)(0)/a11n=2 is valid from Eqs.(40) and (41). The right-hand side of approximation (53)leads to 3 X 10-9 for values ofpjj = 0.12, P12 = 0.27, and 42/K2

8 x 10-4. This means that it is primarily the longitudinalLo. modes, whose origin is the axial-radial Ro. modes, thatare responsible for Brillouin gain for the counterpropagatingHE1, mode. Here we note that the shear acoustic-wavefields g,(r) and g,(r) can be neglected for a circularly cylin-drical waveguide structure consisting of a core surroundedby an infinitely thick cladding for Vcore ' Vad2 Vcore = Vd1ad,and Pcore = Pclad.7 In this case the transverse particle-dis-placement components Ur and uo are clearly found to benegligibly small compared with uz because df/dr << Kf(r) and(2/r)f(r) << Kf(r) in Eqs. (42) and (43). The dispersioncharacteristics of the Lo,, modes have been described indetail in Ref. 7. It is remarkable that the phase-velocitycharacteristics of the Loi modes are similar to those of theLP modes'5 guided in a step-index multimode fiber.

3. BRILLOUIN-GAIN SPECTRAMEASUREMENT

We prepared six kinds of fiber whose refractive-index pro-files and material compositions are shown in Fig. 1. Thefiber groups 1-3, 4, and 5 and 6 are used for measuring theBrillouin-shift deviation 5VB as a function of F concentra-tion, for evaluating 6VB per unit GeO2 concentration, and forclarifying the guided-mode properties of longitudinal acous-tic Lo,, modes, respectively.

The experimental setup for measuring the Brillouin-gainspectra is shown in Fig. 2. The pump and probe distributed-feedback lasers operating at X = 1550 nm serve as the twocounterpropagating waves in a fiber. Optical isolators areinserted between the lasers and the single-mode fiber. Afterthe pump wave passes through a mechanical chopper, thepump laser can deliver approximately 0.5 mW of opticalpower into the fiber. The probe laser power of 0.2 mW islaunched at the front end of the test fiber. The frequency

Refractive Index

9.0 mol%GeO2

2.8 mol%(I1.26 mol%F

___ -u-- 2.2 mol%F

3 4

- 1. 9 mol%F

5

Fig. 1. Refractive-index profiles for test single-mode fibers 1-6.

I6.8 moI%GeO2

3e0j

mol%F --------------- ___SiO2

Fiber 1 2 6

Shibata et al.

0.55

1172 J. Opt. Soc. Am. B/Vol. 6, No. 6/June 1989

FunctionGenerator A/I

EZZ~~~CIsolator

Fig. 2. Experimental arrangement for measuring Brillouin-gain spectra.at a wavelength of 1550 nm; PC, and PC2 are polarization controllers.

difference Af between the pump-laser frequency fpump andthe probe-laser frequency fprobe was adjusted to be approxi-mately 11 GHz. A power transfer from the chopped light ofthe pump laser into the frequency fprobe would appear as achopped signal at the optical power meter, and this signalcomponent was detected with a lock-in amplifier. The fre-quency fprobe was swept over a range of 3 GHz by modulatingthe injection current with a sawtooth waveform. The out-put waveform from the lock-in detector as a function of Afgives the Brillouin-gain spectra.

As examples of the measured Brillouin-gain spectra, thegain spectra for test fibers 1, 4, and 6 are shown in Fig. 3 asdashed, dotted, and solid curves, respectively. Strong andweak gain peaks, marked a and b, were observed for fiber 1,which had a pure-silica core and 2.2 mol % F-doped claddingat Af = -11.4 and-10.6 GHz, respectively. The two peaks aand b are due to scattering from longitudinal-acoustic pho-nons in the silica core and the F-doped cladding, respective-ly. The Brillouin-gain profile obtained here is similar tothat obtained for the silica-core and B2 03-doped-claddingfiber examined by Thomas et al.6 On the other hand, threedifferent gain peaks, marked c, d, and e, were clearly ob-served for the GeO2-doped-core and silica-cladding fiber 6.The Brillouin-gain-spectra behavior of fiber 4 is quite simi-lar to that of fiber 1 except that the gain-peak position isnear Af =-11.1 GHz. The discrepancy between the spectrafor fibers 1 (or 4) and 6 is due to the difference in thewaveguide structure for the longitudinal-acousticmodes.6 7"10 For fiber 6, which satisfies the condition Vlore <

Vl4 ad, the guided modes of longitudinal-acoustic waves canpropagate in the GeO 2 -doped-core region.7"10 Of the longi-tudinal-acoustic modes, the Lon modes dominate the contri-bution to the optical gain, as described in Section 2. Hencethe Brillouin frequency shift VB for each gain peak shouldreflect the dispersion characteristics of Loi modes. But theVB for fiber 6 does not correspond to UB for GeO 2 -doped bulk

Chopper

C IsolatorCoupler

Powerl Meter I

Lock-in I

X-Y IRecorder

Probe and pump sources are distributed-feedback lasers operating

glass. Therefore the Brillouin-shift coefficient 6VB,Geo, perunit GeO2 concentration for the bulk glass cannot be evalu-ated from the VB measured for fibers 5 and 6. The coeffi-cient 6VB,GeO2 can be obtained by using the fibers that satisfythe condition Vlore > Vlad. Fiber 4 with a 2.8 mol % GeO2-

A = 1550 nm ---- Fiber

--- Fiber 4

Fiber 6a

1it f

0 I E

-12 -l l-l

Frequency Difference f (GHz)Fig. 3. Brillouin-gain spectra for the test fibers 1, 4, and 6.

Shibata et al.

Vol. 6, No. 6/June 1989/J. Opt. Soc. Am. B 1173

doped core and 1.9 mol % F-doped cladding was used formeasuring 5VBGeO,. Since the frequency difference betweenthe gain peaks due to scattering from longitudinal-acousticphonons in the cores of fibers 1 and 4 was 299 MHz, then the6VB,GeO2 is evaluated to be 107 MHz/mol % at 1550-nm wave-length.

The coefficient 6VB,F for F-doped glass was obtained byutilizing fibers 1-3. The Brillouin gain G as a function of Afis shown in Fig. 4. The dotted and dashed curves representthe Brillouin-gain spectra for fiber 1 and the sequentiallyspliced transmission line obtained from the fibers 1-3. Thegain peaks marked f and g are due to scattering from longitu-dinal-acoustic phonons in the 0.55- and 1.26-mol % F-dopedcores, respectively. Figure 5 shows 6vB as a function of Fconcentration CF. The line fitted to the experimental datashown as open circles gives VB,F = 356 MHz/mol %, which ismuch greater than that found for 6

VB,GeO2-

4. DISCUSSION

Gain peaks c, d, and e shown in Fig. 3 were identified bytaking into account the phase velocity characteristics of thelongitudinal Lo, modes. Figure 6 illustrates the phase ve-locity Va as a function of the normalized frequency Qa/27r.7'10

The relationship between Va and VB is given by' 6

Va = (X/2n)VB. (54)

The phase velocity V.lad of acoustic waves in the pure-silicacladding is evaluated from the frequency shift vB measuredfor the pure-silica-core fiber 1, and the phase velocities V, inthe 6.8 and 9.0-mol % GeO 2-doped cores of the respectivefibers 5 and 6 are evaluated from the measured 5

VBGeO2. By

N

m5 1000

co

.° 800*5

4 600.<

M2 400

0 200

00 0.5 1 1.5 2 2.5

F-concentration CF (mol%)Fig. 5. Brillouin frequency-shift deviation bVB as a function of Fconcentration CF-

6100

6000i

0

A)

5900

5800

5700

5600

4-

.;

Z

-12 -11 -10

Frequency Difference Af (GHz)Fig. 4. Brillouin-gain spectra for the sequentially spliced transmis-sion line composed of the fibers 1-3.

5500

Vclad=6054 mlse

Vcore =5674m/se

Vcore =5548nVsec

0 10 20 30 40 50 60Normalized Frequency OaI2r (1O3pm-MHz)

Fig. 6. Phase-velocity characteristics for the longitudinal Lommodes. Open and filled circles are measured points.

using the acoustic frequency Q/27r instead of VB in Eq. (54),Eq. (54) is rewritten in the form V, = (X/2na)((Qa/27r). Thetwo lines A and B in Fig. 6 are drawn according to thisequation by substituting the respective core radii of a = 2.8and a = 2.5 jam for X = 1550 nm and n = 1.458 in fibers 5 and6, respectively. The dispersion curves shown in dotted anddashed lines cross the lines A and B at three points. Thethree crossing points give the acoustic phase velocity foreach Lon mode. Open and filled circles, marked (c, d, e) and(x, y, z), represent phase velocities evaluated experimentallyfrom the Brillouin-gain spectra measured for fibers 6 and 5,respectively. Here the measured points a and (c, d, e) in Fig.6 correspond to those in Fig. 3. It is clearly understood fromFig. 6 that the pairs (x, c), (y, d), and (z, e) correspond to thegain peaks caused by the acousto-optic interaction betweenthe HE,, mode and L0 3, L02, and Lo, modes, respectively.The theoretical analysis predicts the observed three Bril-louin-gain peaks in Fig. 3.

Here we discuss the reasons that the gain peaks in Fig. 3were not discovered before this study. One is presumablythat fibers with Vlore < V!Iad have not been used for Brillouinscattering measurements, and the other is that a single-mode fiber with a core of low GeO2 concentration has been

B A = 1550 nm

Mea. Cal.\ Fiber 5 -

X Fiber 6 0

Y\ %L 0 3

\d -. LO

L;-w -~~~~~~~~.02

-L.01

Shibata et al.

1174 J. Opt. Soc. Am. B/Vol. 6, No. 6/June 1989

used. As in Fig. 6, the frequency interval between the gainpeaks becomes small as Vcore approaches V!Iad. When weuse a commercially available single-mode fiber with an -3-mol % GeO2-doped core, the frequency interval is roughlyevaluated as -100 MHz. It is relatively difficult for thefiber to distinguish each resonance peak in the measuredBrillouin-gain spectra because the intrinsic Brillouinlinewidth for each resonance is approximately 50-100MHz.' 0 Therefore the use of a standard single-mode fiberfor measuring Brillouin-gain spectra precludes detection ofBrillouin-gain peaks, even if the fiber has a GeO2 -dopedcore. We believe that when the frequency interval betweenresonance peaks is small the guided acoustic-mode effect inthe gain spectra is obscured.

5. CONCLUSION

The guided acoustic modes, which contribute to the Bril-louin amplification, are now clarified on a theoretical basis.Longitudinal-acoustic Lo. modes, whose origin is due toaxial-radial Roi modes, and whose dispersion characteris-tics were evaluated from the modal analysis of the acousticwaves, are found to interact effectively with the HE,, mode.A few of the gain peaks in the Brillouin-gain spectra for theGeO 2-doped-core fibers are successfully explained by takingaccount of the dispersion characteristics of the Lo,, L02, andL03 modes, based on the experimental results of the respec-tive Brillouin-shift coefficients of 107 and 356 MHz/mol %for GeO2 and F.

Coherent light-wave communication networks should becarefully designed to avoid the gain peaks prevalent in theGeO2 -doped-core fibers available commercially for bidirec-tional frequency-division-multiplexed transmission sys-tems. Although the modulation bandwidth of the sourcesand the data transmission scheme remain the most impor-tant parameters, the theoretical and experimental researchdescribed here also conveys useful information on the per-formance evaluation of those light-wave transmission sys-tems that employ gain-introduced single-mode transmissionlines'7"18 utilizing stimulated Brillouin scattering.

ACKNOWLEDGMENTS

The authors express their sincere thanks to K. Nosu, M.Tateda, and T. Horiguchi for useful discussion, and to S.Shimada and H. Kimura for continuous encouragement.

* On leave from the Research Center for Advanced Sci-

ence and Technology, The University of Tokyo, 4-6-1 Ko-maba, Meguro-ku, Tokyo, 153, Japan.

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