Dinda et al. Vol. 16, No. 5 /May 1999/J. Opt. Soc. Am. B 757

Demonstration of stimulated-Raman-scatteringsuppression in optical fibers

in a multifrequency pumping configuration

P. Tchofo Dinda, S. Wabnitz, and E. Coquet

Laboratoire de physique de l’Universite de Bourgogne, avenue Alain Savary, B.P. 47870, 21078 Dijon Cedex, France

T. Sylvestre, H. Maillotte, and E. Lantz

Laboratoire d’Optique P.M. Duffieux, Unite Mixte de Recherche, Centre National de la Recherche Scientifique,Universite de Franche-Comte No 6603, 25030 Besancon Cedex, France

Received August 17, 1998; revised manuscript received November 17, 1998

We analyze the stimulated-Raman-scattering-(SRS) process induced by a linearly polarized multifrequencypump field in a normally dispersive single-mode fiber. We show, by theoretical analysis and numerical simu-lations, that the SRS process may be either controlled by switching all the generated Stokes radiations to thelowest-frequency pump or suppressed for all the frequency components of the pump field. The suppressionprocess is achieved by an appropriate choice of the frequency separation between the pumps and a particularpower distribution among the frequency components of the pump field. We present experimental spectrashowing the effectiveness of this suppression process for a dual-frequency pumping configuration. © 1999Optical Society of America [S0740-3224(99)01704-X]

OCIS codes: 190.0190, 190.5650, 190.5890.

1. INTRODUCTIONStimulated Raman scattering (SRS) is a well-known non-linear process in which a part of the energy propagatingat a given frequency is converted into one or severaldownshifted Stokes beams through the vibrational modesof the material.1–5 In optical fibers this phenomenonmay be detrimental to long-haul transmissions and mul-tichannel communications.3 For example, the RamanStokes spectral band affects the spectral evolution of in-tense pulses through cross-phase modulation,6 which alsocauses phenomena such as the distortion of spectralchirps in fiber pulse compressors.7 Another well-knownRaman phenomenon in nonlinear fiber optics is the soli-ton decay and self-frequency shift in femtosecondtransmissions.8 To avoid detrimental effects of SRS in fi-bers, several mechanisms for suppressing the first-orderRaman Stokes radiation have been suggested.9–16 One ofthose mechanisms is the parametric suppression of SRS,in which a four-wave mixing (FWM) process involvingparametric Stokes and anti-Stokes sidebands inhibits thegrowth of the ordinary Raman Stokes radiation.9–13 Onthe other hand, polarization switching of SRS can beachieved in high-birefringence fibers by use of a mecha-nism that is not associated with FWM but rather with theorthogonal component of the Raman gain.14 Further-more, recent experiments showed that wavelengthswitching of SRS may be achieved by use of a mechanismassociated with the parallel component of the Ramangain.15 Nevertheless, the general drawback of the abovemethods9–15 is that they allow for SRS suppressionfor only one frequency component of the pump field. Anearlier study proposed that the beating between twopulses propagating at different frequencies in an optical

0740-3224/99/050757-11$15.00 ©

fiber induces a temporal index Bragg grating that may in-hibit the growth of the Raman Stokes waves.16 Recently,an advance was made when the total suppression of thefirst-order Raman Stokes radiation in a dual-frequencypumping experiment was obtained.17

Here we analyze control processes, including wave-length switching and the suppression of SRS in a multi-frequency pumping configuration. To avoid the genera-tion of unwanted parametric sidebands, we consider thesituation in which all frequency components of the pumpfield are linearly polarized in the normal dispersion re-gime of a single-mode fiber. In such a multifrequencypumping configuration, the phenomenology of SRS be-comes more complicated than in a typical single-frequency pumping case. Indeed, the Raman-activatedenergy-transfer processes between the frequency compo-nents of the pump modify the ordinary process of SRS (inwhich an intense light wave at frequency v generates adownshifted Stokes beam at frequency v 2 VR , wherethe Raman frequency shift, VR ' 440 cm21, is deter-mined by the vibrational modes of silica glass.1,2,5) So tofacilitate the control of SRS we use a pump field that ismade up of several frequency components regularlyspaced by Dv. In this configuration, whenever the pump-frequency spacing is approximately 2VR , i.e., twice theRaman frequency shift, the Raman Stokes beam associ-ated with each pump (except the lowest-frequency pump)appears at a frequency located about the maximum of theRaman anti-Stokes depletion that is induced by the pres-ence of the nearest pump of lower frequency. In otherwords, each Raman Stokes radiation, except the one asso-ciated with the lowest-frequency pump, is then subject tosimultaneous effects of Raman Stokes amplification and

1999 Optical Society of America

758 J. Opt. Soc. Am. B/Vol. 16, No. 5 /May 1999 Dinda et al.

anti-Stokes depletion. We show in this research that theresulting wave dynamics is extremely sensitive to the in-put power distribution among the frequency componentsof the pump field. In particular, we show that when thefrequency components of the pump field are launched inthe fiber with the same input power, the Raman Stokeslight is suppressed for all components except the lowest-frequency pump. This partial suppression of SRS leadsto a complete wavelength switching of all the generatedStokes radiations to the one induced by the lowest-frequency pump. Moreover, we demonstrate that a suit-able choice of the power distribution among the pump fre-quencies allows for SRS suppression of the Raman Stokesradiations for all the frequency components of the pumpfield. We refer to this process as being the total suppres-sion of SRS. We present experimental spectra that showthe effectiveness of these SRS suppression processes inthe case of a dual-frequency pumping configuration.

The paper is organized as follows: in Section 2 wepresent the general theoretical model, in Section 3 wepresent numerical simulations, in Section 4 we analyzeexperimental spectra, and in Section 5 we give some con-cluding remarks.

2. THEORETICAL MODELLet us consider a linearly polarized pump field made up ofN frequency components, P1 ,P2 ,..., PN , with frequen-cies vPj

, j 5 1, N, where vPj. vPj11

. The highest-(lowest-) frequency pump is P1 (PN). To facilitate thedemonstration of SRS suppression we use a simplifiedanalysis in which each Raman Stokes beam is replaced bya single wave at a frequency detuning that corresponds tothe condition of peak parallel Raman gain, i.e., VR5 440 cm21. In this context, we consider a system inwhich the total electric field E consists of N 5 3 pumpwaves vPj

, j 5 1, 2, 3 (with vP1. vP2

. vP3), and their

associated Raman Stokes waves Sj , with frequency vSj

5 vPj2 VR . In this case the third-order nonlinear po-

larization may be written as2

P~t ! 5 sE~t !E~t !E~t ! 1 E~t !E2`

t

dssR~t 2 s !E~s !E~s !,

(1)

where s and sR represent the instantaneous and nonin-stantaneous (Raman) parts of the fiber response, respec-tively. Hereafter we use the parallel component of theRaman susceptibility, xR(v) [ sR/2. Figure 1 shows theRaman susceptibility as it is obtained from the fused-quartz results of Ref. 2, which we have normalized toR@xR(0)# 5 1. Hereafter, R(X) and I(X) designate thereal and imaginary parts of X, respectively. In Eq. (1)the total electric field reads as

E 51

2 (j51

3

(Q5P,S

EQj1 c.c.

51

2 (j51

3

(Q5P,S

1

aNQj

AQj~z !cQj

~x, y !

3 exp@i~kQjz 2 vQj

t !# 1 c.c., (2)

where c.c. denotes complex conjugation, cQjis the trans-

verse field distribution of the LP01 mode evaluated at fre-quency vQj

, and AQjis the slowly varying amplitude.

The normalization parameters NQj[ **c2

Qjdxdy and a

[ nce0c/2, where nc is the refractive index of the fibercore, are defined such that the power is given by Qj[ uAQj

u2, with Q 5 P,S. Inserting Eq. (2) into Eq. (1)yields explicitly the component of the polarization at thefrequency vP1

of the lowest-frequency pump:

P~vP1! 5

1

2 F H(j51

3

(Q5S,P

hQjQj* P1uEQj

u2EP1J1 hP2S2* S1

exp~2iDk3z !EP2ES2

* ES1

1 hP3S3* S1exp~2iDk1z !EP3

ES3* ES1

1 hS2S3* P2exp~2iDk5z !ES2

ES3* EP2

1 hP2P3* P2exp~2iDk6z !EP2

2 EP3*

1 @hS1P2* S1exp~2iDk4z !ES1

2 EP2*

1 hS2P3* S1exp~2iDk2z !ES2

EP3* ES1

#d~2VR!G1 c.c., (3)

where d(2VR) 5 0 or d(2VR) 5 1 depending on whetherDv Þ 2VR or Dv 5 2VR ,

hQj Qj* Qj[ 3s/4 1 xR~0 !, Q 5 S, P, j 5 1, 3, (4a)

hQjQj* Gl[ 3s/2 1 xR~0 ! 1 xR~vQj

2 vGl!,

Q, G 5 S, P, l 5 1, 3, Qj Þ Gl , (4b)

Fig. 1. Plots showing the frequency dependence of the parallelcomponents of the Raman susceptibility: solid curve, imaginarypart I(xR); dotted curve, real part R(xR). The largest (small-est) dashed vertical lines indicate the gain at frequency detun-ings of 550 cm21 and 888 cm21.

Dinda et al. Vol. 16, No. 5 /May 1999/J. Opt. Soc. Am. B 759

hP2S2* S1[ 3s/2 1 xR~2VR! 1 xR~2Dv!,

hP3S3* S1[ 3s/2 1 xR~2VR! 1 xR~22Dv!, (4c)

hS2S3* P2[ 3s/2 1 xR~2Dv! 1 xR~2Dv 2 VR!,

hP2P3* P2[ 3s/4 1 xR~2Dv!, (4d)

hS1P2* S1[ 3s/4 1 xR~2Dv 1 VR!,

hS2P3* S15 3s/2 1 xR~22Dv 1 VR! 1 xR~2Dv 1 VR!.

(4e)

The other frequency components of the polarization areimmediately obtainable from Eqs. (1) and (2) by the sameprocedure. The propagation of the six waves of Eq. (2) isruled by a set of coherently coupled-amplitude equations(CCE) that may be formally written as follows:

]AQj

]z5 igQj (

U,V,W(

l,m,nHQjUlVmWn

3 exp~iDkz !AUlAVm

* AWn, (5)

where gQj[ vQj

n2 /(cjQjQjQjQj) is the nonlinear coeffi-

cient, n2 is the fiber nonlinear refractive index, andjQjQjQjQj

is the effective core area, whereas the factorexp(iDkz) 5 1 or Þ1 depending on whether Dk [ kUl

2 kVm1 kWn

2 kQjis zero or nonzero. Those phase

mismatches Dk are explicitly expressed as follows:

Dk1 [ kP12 kP3

1 kS32 kS1

,

Dk2 [ kP11 kP3

2 kS12 kS2

, (6a)

Dk3 [ kP12 kP2

1 kS22 kS1

,

Dk4 [ kP11 kP2

2 2kS1, (6b)

Dk5 5 Dk1 1 Dk2 2 Dk4 ,

Dk6 5 Dk2 1 Dk3 2 Dk4 ,

Dk7 5 Dk4 2 Dk1 , (6c)

Dk8 5 Dk2 2 Dk4 , Dk9 5 Dk4 2 Dk3 ,

Dk10 5 Dk3 2 Dk5 , (6d)

Dk11 5 Dk2 2 Dk3 , Dk12 5 Dk1 2 Dk3 ,

Dk13 5 Dk2 2 Dk1 . (6e)

In Eq. (5) the coefficient

HQjUlVmWn[ hUlVmWn

jQjQjQjQj

jQjUlVmWn

,

~U, V, W ! 5 S, P, ~l, m, n ! 5 1, N, (7)

measures the importance of the coupling between the in-teracting waves (Qj , Ul , Vm , Wn). The overlapping ofthe transverse fields of the interacting waves is given byj21

QjUlVmWn, where

jQjUlVmWn[

~NQjNUlNVmNWn

!1/2

EEcQjcUl

cVmcWn

dxdy

,

~U, V, W ! 5 S, P, ~l, m, n ! 5 1, N. (8)

Setting AQj5 (Qj)

1/2 exp(ifQj) leads to the following set

of coupled-mode equations:

]Qj

]z5 AQj

]AQj*

]z1 AQj

*]AQj

]z, (9a)

]fQj

]z5

i2Qj

FAQj

]AQj*

]z2 AQj

*]AQj

]zG . (9b)

The substitution of Eq. (5) into Eqs. (9) leads to coupledequations for the powers and phases of the beams. Forexample, we obtain

]P1

]z5 22gP1H(

j51

3

(Q5S,P

I~HQjQ* jP1P1!QjP1

1 @2R~HP2S* 2S1P1!sin u3

1 I~HP2S* 2S1P1!cos u3#AS1S2P1P2

1 @2R~HP3S* 3S1P1!sin u1

1 I~HP3S* 3S1P1!cos u1#AS1S3P1P3

1 @2R~HS2S* 3P2P1!sin u5

1 I~HS2S* 3P2P1!cos u5#AS2S3P2P1

1 @2R~HP2P* 3P2P1!sin u6

1 I~HP2P* 3P2P1!cos u6#P2AP1P3

1 @@2R~HS2P* 3S1P1!sin u2

1 I~HS2P* 3S1P1!cos u2#AS1S2P3P1

1 @2R~HS1P* 2S1P1!sin u4

1 I~HS1P* 2S1P1!cos u4#S1AP1P2#d~2VR!J ,

(10)

]u1

]z5 Dk1 1

]fP1

]z2

]fP3

]z1

]fS3

]z2

]fS1

]z, (11a)

]u2

]z5 Dk2 1

]fP1

]z1

]fP3

]z2

]fS1

]z2

]fS2

]z, (11b)

]u3

]z5 Dk3 1

]fP1

]z2

]fP2

]z1

]fS2

]z2

]fS1

]z, (11c)

]u4

]z5 Dk4 1

]fP1

]z1

]fP2

]z2 2

]fS1

]z, (11d)

where

760 J. Opt. Soc. Am. B/Vol. 16, No. 5 /May 1999 Dinda et al.

Fig. 2. Plot showing the coherence lengths 2p/Dkj versus Dv. The coherence length corresponding to a phase mismatch Dkj is indi-cated in the figures by the numerical j. The vertical dashed lines indicate the experimental operating condition considered in thepresent paper: Dv 5 888 cm21, lP1

5 508 nm.

u1 [ zDk1 1 fP12 fP3

1 fS32 fS1

, (12a)

u2 [ zDk2 1 fP11 fP3

2 fS12 fS2

, (12b)

u3 [ zDk3 1 fP12 fP2

1 fS22 fS1

, (12c)

u4 [ zDk4 1 fP11 fP2

2 2fS1, (12d)

u5 [ u1 1 u2 2 u4 , (12e)

u6 [ u2 1 u3 2 u4 . (12f )

Similar coupled-mode equations for the other frequencycomponents of the electric field can be easily obtained bythe same procedure. Note that the above coupled-modeequations involve coherent coupling terms (i.e., FWMterms) resulting from the phase mismatches Dkj in Eqs.(6). Figure 2 shows their coherence lengths 2p/Dkj as afunction of the pump-frequency spacing Dv. In Fig. 2each coherence length does not exceed 32 mm for all Dvoutside a very small region (,0.5 cm21) around Dv5 2VR 5 880 cm21. Thus it appears that the coherentcoupling terms play a role only in this very small fre-quency range. The experimental condition considered inthis research, which is indicated in Fig. 2 by the vertical-dashed lines, is Dv 5 888 cm21, for which the coherencelengths are extremely small (,32 mm) when comparedwith the fiber length L 5 3 m. As a consequence, unlessoperating under very special conditions of exact phasematching, in most practical situations, such as those con-sidered in this research, one can ignore the influence ofcoherent coupling terms. A noticeable exception is the

case Dv . 440 cm21, where the lowest-frequency Stokeswave is seeded by the coherent terms at the very begin-ning of the fiber. This case was already treated in Ref.18 and will not be further considered here. Note thatonly two types of susceptibility coefficient in Eqs. (4) areto be considered when coherent coupling terms are ne-glected:

hQjQj* Qj

[ 3s/4 1 xR~0 !, Q 5 S, P, j 5 1, N,(13)

hQjQj* Gl[ 3s/2 1 xR~0 ! 1 xR~vQj

2 vGl!,

G 5 S, P, l 5 1, N, Qj Þ Gl . (14)

Then, setting AQj5 (Qj)

1/2 and using the coupled-amplitude equation (5), we obtain the following incoher-ently coupled equation:

]Qj

]z5 22gQj (

l51

N

(G5S,P

I~HGlGlQjQj!GlQj ,

Q 5 S, P, j 5 1, N. (15)

Equation (15) shows that the imaginary part of a givennonlinear coupling coefficient, say I(HGlGlQjQj

), governs aunilateral Raman-induced transfer of energy from Gl(Qj)to Qj(Gl) in the case when vGl

. vQj(vGl

, vQj). In

other words, the Raman nonlinearity causes energy-transfer processes from higher- to lower-frequency waves.

Dinda et al. Vol. 16, No. 5 /May 1999/J. Opt. Soc. Am. B 761

In general, the wave dynamics result from the combinedeffects of all coupling coefficients. Hereafter we system-atically use the CCE whenever N < 3 and use the inco-herently coupled equation for N . 3. Indeed, from N5 4 upward, derivation of the CCE becomes extremelytedious.

3. SIMULATIONSIn the extreme case when Dv @ VR , the Raman couplingbetween the frequency components of the pump becomesso weak that a sufficiently large input power Pj(z 5 0)leads to an ordinary process of SRS, which generates aRaman Stokes wave Sj . Such a situation is illustratedin Figs. 3(a), 3(b), and 3(c), in which the total input powerPT is equally distributed among the N pumps. That is,each input power Pj(0) is equal to a mean power Pm(0),which is, hereafter, systematically chosen to be above thestimulated Raman threshold. Throughout this paper werefer to the ‘‘stimulated Raman threshold’’ as being theminimum pump power required for generating an appre-ciable power of Raman Stokes radiation, compared with

Fig. 3. Theoretical variation of normalized pump and Stokespowers Qj /Pm(0), Q 5 S, P, j 5 1, N, versus propagation coor-dinate z for Dv 5 1760 cm21 and Pj 5 Pm(0) 5 150 W. (a) N5 2, (b) N 5 3, and (c) N 5 4. Cross, P1 ; barred cross, S1 ;square, P2 ; square on cross, S2 ; diamond, P3 ; diamond on plus,S3 ; plus sign, P4 ; barred plus, S4 . vP1

5 590.55 THz.

the signal-to-noise ratio, at the fiber output. Figure 3was obtained by numerical solution of the coupled-modeequations for Dv 5 4VR 5 1760 cm21. An importantfeature in Fig. 3 is that although Pj(z 5 0) 5 Pm(0) forall pumps Pj , the wave dynamics and the resulting out-put Stokes powers are clearly different. We attributethis difference to two main effects that act in opposite di-rections: First, the Raman-activated energy-transferprocesses between the pumps favor the growth of thelower-frequency Stokes waves. The second effect is theoverlapping of the transverse fields for each pump Pj andits associated Raman Stokes wave Sj , j21

PjPjSjSj,which

can significantly vary over a relatively large spectrum(here the whole spectrum ranges from vSN

up to vP1).

The curve in Fig. 4(a) shows that this transverse overlap-ping decreases as the pump frequency decreases, thuscausing a drop in the Raman gain. Typically,j21

PPSS/j21P0P0S0S0

with vP05 600 THz decreases by

;40% in the range 475 THz < vP < 600 THz [see Fig.4(a)]. On the other hand, Fig. 4(b), which shows(vS /vS0

)(j21PPSS/j21

P0P0S0S0versus vP , indicates that

the functional dependence of the Raman gain upon theStokes frequency vS contributes to further reduction ofthe gain at low frequency. Thus the Raman-induced en-ergy transfers between the pumps favor the growth of thelower-frequency Stokes wave, whereas the transverse-field-overlapping effects favor the higher-frequencyStokes waves. For Dv 5 4VR 5 1760 cm21, the wholespectrum occupied by the 2N waves is so large that thetransverse-field-overlapping effects become more promi-nent than the Raman-induced energy-transfer effects,

Fig. 4. Plot showing the frequency dependence of the transverseoverlapping of a pump vP with its associated Raman Stokeswave vS [ vP 2 VR : (a) j21

PPSS/j21P0P0S0S0

versus vP ; (b)(vS /vS0

)(j21PPSS/j21

P0P0S0S0) versus vP . vP0

5 600 THz.

762 J. Opt. Soc. Am. B/Vol. 16, No. 5 /May 1999 Dinda et al.

which leads to a dominant Stokes power at high fre-quency: Sj(L) . Sj11(L) (L is the fiber length). At thisstage one can already notice that the higher the frequencyvPj

, the larger the depletion of the pump Pj at the fiberoutput. This feature plays an important role, as we willshow below. But the most important result to be empha-sized here is that the suppression of SRS cannot beachieved in the extreme case where the pump-frequencyspacing is too large (see Fig. 3).

The other extreme case corresponds to the situation inwhich Dv is relatively small. This situation occurs whenDv falls in the spectral region of high Raman gain: 0, Dv , 1.5VR 5 660 cm21 (see Fig. 1). In this caseeach pump Pj (except P1) falls in the spectral region ofhigh Raman gain induced by the adjacent pump of higher

frequency, i.e., Pj21 . The pumps are then stronglycoupled by the Raman nonlinearity, which causes astrong transfer of energy from higher-frequency pumps tothe lowest-frequency pump PN through the intermediatepumps. The resulting wave dynamics depend on the dis-tribution of the input power among the pumps, as illus-trated in Fig. 5, that we obtained for Dv 5 550 cm21 (notethat the largest dashed-vertical line in Fig. 1, which cor-responds to 550 cm21, falls precisely in a spectral regionof relatively large Raman gain). We see in Figs. 5(a),5(b), and 5(c) that when the total input power is equallydistributed among the N frequency components of thepump field, the pump PN is strongly and continually am-plified until the generation of the Stokes SN occurs. Al-though all Stokes waves except SN are suppressed, the

Fig. 5. Theoretical variation of normalized pump and Stokes powers Qj /Pm(0), Q 5 S, P, j 5 1, N, versus propagation coordinate zfor Dv 5 550 cm21. (a) N 5 2, Pj(0) 5 Pm(0); (b) N 5 3, Pj(0) 5 Pm(0); (c) N 5 4, Pj(0) 5 Pm(0); (d) N 5 2, P1(0) 5 1.83Pm(0),P2(0) 5 0.17Pm(0); (e) N 5 3, P1(0) 5 1.84Pm(0), P2(0) 5 Pm(0), P3(0) 5 0.16Pm(0); (f ) N 5 4, P1(0) 5 1.81Pm(0), P2(0)5 P3(0) 5 Pm(0), P4(0) 5 0.19Pm(0). Cross, P1 ; barred cross, S1 ; square, P2 ; square on cross, S2 ; diamond, P3 ; diamond on plus,S3 ; plus sign, P4 ; barred plus, S4 . vP1

5 590.55 THz.

Dinda et al. Vol. 16, No. 5 /May 1999/J. Opt. Soc. Am. B 763

dynamics lead to a strong depletion of the whole pumpfield. The total amount of power transferred to thelowest-frequency Stokes wave SN is so strong that if wewere to include the cascade process in our model forhigher-order Raman Stokes waves, we would expect thoseStokes waves to be generated. Such a Raman cascadeprocess tends to amplify the whole SRS process instead ofsuppressing it. Moreover, Figs. 5(d), 5(e), and 5(f ), whichwe obtained by choosing an input power distributionamong the pumps that minimizes the total power of thegenerated Stokes waves, demonstrate that one cannotachieve total suppression of SRS in the extreme case ofstrong Raman coupling between the pumps. The out-standing feature in the extreme case shown in Figs. 5(d),5(e), and 5(f) is the strong amplification of the lowest-frequency pump PN . Its output power can be very higheven when its initial power is chosen well below the Ra-man threshold.

When the pump-frequency spacing is chosen betweenthe two extreme cases mentioned above, i.e., 1.5VR, Dv ! 4VR , the Raman coupling between the fre-quency components of the pump becomes weak but non-negligible. In this case the wave dynamics are governedmainly by energy-transfer processes involving each waveQj and its nearest and next-nearest neighbors in the fre-quency domain: the two nearest Raman waves locatedon both of its sides, as well as the two nearest pumpwaves on both of its sides. Interactions with longerranges can indeed be neglected simply because of the lim-ited width of the region of high Raman gain (see Fig. 1).So under such circumstances one expects that thehighest-frequency pump P1 is always depleted through astrong transfer of energy to S1 and a slight transfer toP2 , whereas any scattering of a pump Pj , j 5 2, N 2 1,into its Stokes photons Sj is also subject to a further flowof energy into the neighboring pump Pj11 (and a slighttransfer to Sj11). A particularly interesting situation oc-curs when Dv is approximately twice the Raman fre-quency shift (2VR . 880 cm21). Indeed, each Stokescomponent Sj of the pump Pj (except SN) becomes theanti-Stokes component of the neighboring pump Pj11 , asSj appears at a frequency detuning at which the pumpPj11 causes the maximum Raman anti-Stokes absorption.The wave Sj is therefore subject to simultaneous effects ofRaman Stokes amplification (from pump Pj) and deple-tion (due to Pj11). These two effects may cancel eachother because of the antisymmetry of the imaginary partof the Raman susceptibility (see Fig. 1). One may there-fore expect that essentially no Raman Stokes radiation Sjis generated, except for SN , which is not subject todepletion.15 In fact, the resulting behavior dependsstrongly on the power distribution among the N pumps:

(i) When the total power is equally distributed amongthe N pumps, all Raman Stokes waves, except the lowest-frequency Stokes wave SN , are suppressed. This wave-coupling behavior, hereafter referred to as the ‘‘partialsuppression of SRS,’’ is illustrated in Figs. 6(a), 6(b), and6(c), which we obtained from the coupled-mode equationsfor Dv 5 2VR 5 888 cm21 and N equal to 2, 3, and 4, re-spectively. Note that in Fig. 6 the output power of SN de-creases as N increases as a result of the transverse-field-

overlapping effects mentioned above. Furthermore, thepump P1 is continually depleted, whereas the pump PN iscontinually amplified until the generation of SN occurs.

(ii) Thus it comes out from the above discussion that apump-frequency spacing Dv . 2VR ensures the suppres-sion of all Stokes radiations Sj from j 5 1 to j 5 N 2 1.Only the Stokes SN is generated when the pump power isequally distributed among the pumps at the fiber input(see Fig. 6). Our strategy for suppressing all the RamanStokes radiations, including SN , is based on the fact thata SRS process that grows from spontaneous noise re-quires a minimum propagation distance to develop sig-nificantly. It is therefore obvious that there always ex-ists a first section of the fiber length in which the amountof power in the Raman Stokes waves is negligible, andthere the wave dynamics are essentially governed by theenergy-transfer processes between the pumps. In thisfirst section of the fiber length, each intermediate pump,i.e., Pj with j 5 2, N 2 1, is subject to simultaneous ef-fects of Raman amplification and depletion, each inducedby one of the two adjacent pumps. These effects partiallycancel each other, which in general causes ultimately only

Fig. 6. Plots showing the partial suppression of SRS. Theoret-ical variation of normalized pump and Stokes powers Qj /Pm(0),Q 5 S,P, j 5 1,N, versus propagation coordinate z for Dv5 888 cm21 and (a) N 5 2, (b) N 5 3, and (c) N 5 4. Cross,P1 ; barred cross, S1 ; square, P2 ; square on cross, S2 ; diamond,P3 ; diamond on plus, S3 ; plus sign, P4 ; barred plus, S4 . vP1

5 590.55 THz.

764 J. Opt. Soc. Am. B/Vol. 16, No. 5 /May 1999 Dinda et al.

slight variations in the intermediate-pump powers.Quite in contrast, the highest- (lowest-) frequency pumpP1(PN), which has only P2(PN21) as adjacent pump, iscontinually depleted (amplified) owing to that neighbor-ing pump. Consequently PN grows much quicker thanall the other pumps, while P1 decreases, in the first sec-tion of the fiber length where SRS is still negligible. So,although there is no direct Raman coupling between P1and PN (except for N 5 2), a large part of the energy lostby P1 is actually transferred to PN by the intermediatepumps, which leads to the generation of SN when Pj(0)5 Pm(0), j 5 1, N. In this context, our strategy to in-hibit the growth of all the Raman Stokes radiations (in-cluding SN) while maintaining the mean power Pm(0)above the Raman threshold is the following: The fre-quency components of the pump that will be undergoing

amplification (depletion) in the first section of the fiberlength are launched with smaller (larger) input powers tocounterbalance in advance the energy-transfer effects.In practice, the input power for each of the intermediatepumps is set identical to the mean value of the inputpump power:

Pj~0 ! 5 Pm~0 ! [ PT /N, j 5 2, N 2 1. (16)

Then PN(0) is lowered below Pm(0) to minimize thepower of the Raman Stokes radiation SN . Meanwhile,P1(0) is raised by the same amount above Pm(0), whichdoes not change the value of the mean power Pm(0):

@PN~0 ! 1 P1~0 !#/2 5 Pm~0 ! 5 PT /N. (17)

Fig. 7. Plots showing the total suppression of SRS. Theoretical variation of normalized pump and Stokes powers Qj /Pm(0), Q5 S,P, j 5 1,N, versus propagation coordinate z for Dv 5 888 cm21 and (a) N 5 2, (b) N 5 3, and (c) N 5 4; and for Dv5 840 cm21 and (d) N 5 2, (e) N 5 3, and (f ) N 5 4. Cross, P1 ; barred cross, S1 ; square, P2 ; square on cross, S2 ; diamond, P3 ;diamond on plus, S3 ; plus sign, P4 ; barred plus, S4 . vP1

5 590.55 THz.

Dinda et al. Vol. 16, No. 5 /May 1999/J. Opt. Soc. Am. B 765

Thus for a given power Pm(0), we determine the appro-priate distribution of the input pump power by minimiz-ing the following quantity with respect to P1 (or, equiva-lently, PN):

ST~Dv! [

(j51

N

Sj@Dv, Pj~0 !#

(j51

N

Sj@Dv 5 4VR , Pj~0 ! 5 Pm~0 !#

. (18)

In Eq. (18) ST measures the total power of the Stokes ra-diations in a situation where the Raman-coupling be-tween the pumps is nonnegligible, related to the totalpower of the Stokes radiations for Dv 5 4VR and Pj(0)5 Pm(0), i.e., when the Raman coupling is negligible.However, this minimization procedure may lead to a largedispersion of the output pump powers [see Figs. 5(d), 5(e),and 5(f )], which is not convenient for multichannel fibertransmissions. To achieve SRS suppression with a mini-mum dispersion of the output powers, we instead deter-mine the optimum power distribution that minimizes thequantity

T [ ST 1 DP, (19)

where

DP [1

N (j51

N

@Pj~L ! 2 Pm~L !#2/Pm2~0 ! (20)

measures the deviation of the output pump powers withrespect to the mean value of the output power Pm(L).Furthermore, we emphasize that this procedure for deter-mining the input power distribution for SRS suppression,subject to the relation given in Eqs. (16) and (17), is anoptimization problem with the single free parameterP1(0) [or, equivalently, PN(0)]. Consequently, the powerP1(0) may be used as an effective control parameter forthe SRS suppression process. Figures 7(a), 7(b), and 7(c)show typical examples of total suppression of SRS forDv 5 888 cm21 and N equal to 2, 3, and 4, respectively.In Fig. 7 we find that the total power of the generatedStokes waves, (Sj /Pm(0) 5 0.14 [Fig. 7(a)], (Sj /Pm(0)5 0.096 [Fig. 7(b)], and (Sj /Pm(0) 5 0.042 [Fig. 7(c)],represent only '25% of the total power of the Stokeswaves that are generated when all the pumps arelaunched with the same power [compare Fig. 6]. Notethat (Sj /Pm(0) can even be lowered well below 25%through the above optimization procedure, but this reduc-tion leads to a larger dispersion between the output pumppower values. Furthermore, the total power of theStokes radiation in Fig. 7 does not exceed 7% of the totalinput pump power: (Sj(N 5 2) 5 7%PT , (Sj(N 5 3)5 3%PT , and (Sj(N 5 4) 5 1%PT . Thus, the resultsappearing in Fig. 7 demonstrate the efficiency of themethod for optimization of the input power distributionfor achievement of the total suppression of SRS, which wepresented above. Furthermore, an important feature tonote in Figs. 7(a), 7(b), and 7(c) is that (Sj decreases as Nincreases. This decrease is due to the fact that theRaman-activated energy-transfer effects favor (throughthe amplification of PN) the generation of SN and thereby

act in a way that tends to restrict the whole SRS to thelowest-frequency pump. Quite in contrast, thetransverse-field-overlapping effects, which enter stronglyinto play from N 5 3 upward, reduce the effective Ramangain for lower-frequency Stokes waves. The total sup-pression of SRS results from balance of these two compet-ing effects. This balance is easier to achieve for large N,when one suitably chooses the power distribution amongthe pumps, as we mentioned above. On the other hand,we see in Figs. 7(d), 7(e), and 7(f ), which we obtained forDv 5 840 cm21, that SRS suppression is still obtainableas the pump-frequency spacing decreases, with the samegeneral features as for Dv 5 888 cm21 but with less effi-ciency, and a larger dispersion between the output pumppower values.

4. EXPERIMENTSAs the first step to assess the efficiency of the abovemethod for SRS suppression, we have implemented dual-frequency pumping experiments in which Dv5 888 cm21 (N 5 2). The experimental setup is shownin Fig. 8. The pump beam P1 was a 40-ps stretchedpump pulse, with frequency vP1

5 590.5 THz (508 nm),produced by a tunable optical parametric generator–amplifier (OPG–OPA) pumped by the third harmonic of apulsed mode-locked Nd:YAG laser operating at 1.064 mmat a rate of 10 Hz. The same laser source was used toproduce the frequency-doubled pump beam P2 , a 35-pspump pulse with frequency vP2

5 563.9 THz (532 nm).The two linearly polarized pumps were launched synchro-nously along the fast axis of a single-mode high-birefringence fiber (with a linear birefringence of Dn5 3 3 1024). This launching condition indeed preventsthe growth of parasitic parametric sidebands, as no FWM(or polarization instability) can occur for this polarizationaxis of a normally dispersive high-birefringence fiber.The fiber length was L 5 3 m. After propagationthrough the fiber, the output intensity was spectrallyanalyzed with a Littrow-grating spectrometer (0.75 m,

Fig. 8. Schematic diagram of the experimental apparatus. M,mirrors; B.S, beam splitter; O1 ,O2, microscope objectives; P’s,Glan polarizers; l/2’s, half-wave plates; DL, prism delay-line forpulse synchronization. Polarizations of pumps P1 and P2 areparallel to the fast axis of the fiber.

766 J. Opt. Soc. Am. B/Vol. 16, No. 5 /May 1999 Dinda et al.

600 lines/mm) with a spectral resolution of 200 GHz and asingle-shot CCD camera. In the experiments, first we de-termined a mean power Pm(0) situated above the thresh-old power for the generation of the Raman Stokes beamsS1 and S2 (by setting P32j 5 0 and then raising thepower Pj until we obtain the generation of Sj , j 5 1, 2).As we can see in Fig. 9, the mean power Pm(0) 5 415 Wis well above the threshold power above which a single-frequency pumping with any of the frequency componentsof the pump field (P1 or P2) leads to the generation of aRaman Stokes radiation (S1 or S2). Now Fig. 10(a)shows that when the two pumps are launched simulta-neously with the same input power, P1(0) 5 P2(0)5 415 W, we obtain the suppression of S1 but not of S2 ,that is, the partial suppression of SRS,15 in agreementwith the theoretical analysis. To obtain the total sup-pression of all the Raman Stokes radiations, we raise thepower P1(0) above Pm(0) [P2(0) is lowered below Pm(0)],while keeping @P1(0) 1 P2(0)#/2 5 Pm(0) 5 415 W as in-dicated in Eq. (17), until the suppression occurs. We seein Fig. 10(b) that no significant power of Raman Stokesradiations is scattered, in agreement with the predictionof the theoretical analysis [compare Fig. 7(a)]. Note thatthe comparison here is only qualitative because of the ap-proximate nature of our theoretical model, which replacesall waves generated in the Raman-gain band by a singlewave at a frequency detuning from the pump that corre-sponds to the condition of peak parallel Raman gain.Moreover, group-velocity-dispersion effects have to betaken into account for a better fitting of the experimentalresults.

On the other hand, we see in Fig. 9 that each pump (P1and P2) is spectrally broadened by self-phase modulationand, in addition, undergoes an asymmetric up-frequencychirp. This chirp is also present in the spectral band ofthe pump P2 in Fig. 10(a). This phenomenon is due tocross-phase modulation between the pump pulses and theRaman pulses, and depletion of the pumps. As the Ra-man pulses propagate faster in the normal dispersion re-gime, the Stokes side of the pump spectrum is altered.An interesting feature in Fig. 10 is that this spectral

Fig. 9. Experimental output spectra measured under single-frequency pumping conditions (a) P1(0) 5 415 W and P2(0)5 0 W and (b) P1(0) 5 0 W and P2(0) 5 415 W. vP1

5 590.5 THz, vP25 563.9 THz.

asymmetry almost disappears from the pump P1 in Fig.10(a) and from both pumps in Fig. 10(b), owing to the SRSsuppression. Such a property could be useful in pulsecompression experiments for preserving the linearity ofSPM-induced spectral chirps at higher values of the non-linear phase-shifts, to yield better compression ratios.

5. DISCUSSION AND CONCLUSIONWe have demonstrated in this research that SRS can bepartially or totally suppressed for all the frequency com-ponents of a linearly polarized high-power pump fieldpropagating in a single-mode fiber. This nonparametricSRS suppression mechanism is based on the spectral an-tisymmetry of the Raman susceptibility. But the totalsuppression of SRS requires, in addition, careful controlof the power distribution among the pumps at the fiber in-put. Indeed, we have shown that optimum conditions forthe total suppression of SRS are achieved by choosing notonly a frequency separation between the pumps close to880 cm21, i.e., twice the Raman frequency shift, but alsoan appropriate distribution of the total input pump poweramong the frequency components of the pump field. Wehave found that the input power distribution that mini-mizes the total power of the Raman Stokes radiation atthe fiber output must favor the higher-frequency compo-nents (at the fiber input). We have presented experimen-tal spectra that show the effectiveness of the partial andthe total suppression processes in a dual-frequency pump-ing configuration. The experimental demonstration ofSRS suppression for a larger number of frequency compo-nents (N > 3) remains to be done. The excellent resultsthat we have obtained for N 5 2 is a promising step inthis direction.

Furthermore, some other problems remain to be consid-ered, among which is the one related to the pump-frequency spacing Dv required for the SRS suppression.The present study, which uses a linearly polarized multi-frequency pump field, leads to a relatively large pump-

Fig. 10. Experimental output spectra measured under dual-frequency pumping conditions. (a) Partial suppression of SRS(with equal pump powers): P1(0) 5 P2(0) 5 415 W. (b) Totalsuppression of SRS (with unbalanced pump powers): P1(0)5 460 W, P2(0) 5 370 W. vP1

5 590.5 THz, vP25 563.9

THz.

Dinda et al. Vol. 16, No. 5 /May 1999/J. Opt. Soc. Am. B 767

frequency spacing: Dv . 880 cm21. There exist at leasttwo lines of research that may be followed for reducingDv. One of them could be the modification of the nonin-stantaneous part of the fiber response (i.e., the Raman-gain function) by using a specific fiber with some dopingmaterials that induce a prominent peak parallel Ramangain at a frequency shift much less that 440 cm21. In-deed, it has been demonstrated that some oxide glassesused in the fiber fabrication can significantly modify theRaman frequency shift.19 Another way for reducing Dvcould be the use of a pump field with frequency compo-nents alternately polarized along the fast and slow axes ofa birefringent fiber. In such a configuration two consecu-tive pumps are orthogonally polarized, so that the pump-frequency spacing can be chosen in a region known as thecritical gap, in which the parametric FWM gainvanishes.20 Then the value of Dv could be arbitrarilytuned by changing the linear birefringence of the fiber,while the orthogonal component of the Raman gain wouldensure a slight Raman coupling between the pumps,14 asis required for SRS suppression.

Finally, the following general features result from thepresent study as well as from previous relatedstudies12,14,15,17 in the context of control processes for SRSin optical fibers. The processes for suppressing SRS inoptical fibers have been applied so far to single-frequencypumping or polarization-switching configurations. Insingle-frequency pumping conditions, it appears that theonly way to achieve SRS suppression is parametric sup-pression. In this situation parametric FWM does causeSRS suppression but at the expense of the generation ofparametric sidebands. When the pump field containsmore than one frequency component, the phenomenologyof SRS is more complex because of the Raman-inducednonlinear coupling between the frequency components ofthe pump field. Such a Raman coupling leads to energytransfers between the pumps. Moreover, when a RamanStokes wave is created, the action of this wave should alsobe considered in the energy-transfer processes. Our gen-eral strategy for obtaining the total suppression of SRS ina multifrequency pumping system is based on the factthat a SRS process growing from spontaneous noise re-quires a minimum fiber length to develop significantly.As a result, there always exists a first section of the fiberlength in which the amount of power in the Raman Stokeswaves is negligible, and there the wave dynamics are es-sentially governed by the energy-transfer processes be-tween the frequency components of the pump. Thoseenergy-transfer processes that take place in the first sec-tion of the fiber determine the importance of the wholeSRS process. So the basic idea for achievement of the to-tal suppression of SRS is the following: first, one identi-fies by theoretical analysis the relative importance of theenergy-transfer processes in the first section of the fiberlength. Then the pump-frequency components that willbe undergoing amplification (depletion) in this first sec-tion of the fiber length should be launched in the fiberwith smaller (larger) input powers to counterbalance inadvance the energy-transfer effects.

The main contribution of the present and previousstudies12,14,15,17,18 is the demonstration that the first-order Raman Stokes radiation induced by a multifre-quency pumping can be precisely controlled, through asuitable choice of the pump frequency spacing and thepower distribution among the frequency-components ofthe pump field. It turns out that one may either enhanceor reduce the SRS process, obtain wavelength or polariza-tion switching of SRS,14,15,18 and even achieve a total sup-pression of SRS.

ACKNOWLEDGMENTSFinancial support from the Centre National de la Recher-che Scientifique, the Ministere de la Recherche et de laTechnologie, and the Conseil Regional de Bourgogne isgratefully acknowledged.

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