Four-wave-mixing-induced turbulent spectral broadening in a long Raman fiber laser

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Babin et al. Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. B 1729

Four-wave-mixing-induced turbulent spectralbroadening in a long Raman fiber laser

Sergey A. Babin, Dmitriy V. Churkin,* Arsen E. Ismagulov, Sergey I. Kablukov, and Evgeny V. Podivilov

Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences,Novosibirsk, 630090 Russia

*Corresponding author: [email protected]

Received October 27, 2006; revised February 26, 2007; accepted March 2, 2007;posted March 15, 2007 (Doc. ID 76531); published July 19, 2007

We present a detailed analytical self-consistent theory based on wave kinetic equations that describes genera-tion spectrum and output power of a Raman fiber laser (RFL). It is shown both theoretically and experimen-tally that the quasi-degenerate four-wave mixing (FWM) between different longitudinal modes is the mainbroadening mechanism in the one-stage RFL at high powers. The shape and power dependence of the intrac-avity Stokes wave spectrum are in excellent quantitative agreement with predictions of the theory. FWM-induced stochasticity of the amplitude and the phase of each of the �106 longitudinal modes generated in theRFL cavity is an example of a light-wave turbulence in a fiber. © 2007 Optical Society of America

OCIS codes: 140.3550, 030.7060, 190.4380.

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. INTRODUCTIONaman fiber lasers (RFLs) are attractive light sourcesroviding almost any wavelength in the near-infrared re-ion [1]. They are widely used in WDM telecommunica-ion systems as multiwavelength signal and pumpources for distributed Raman amplifiers [2,3]. Raman fi-er lasers can also be applied in long-distance remoteensing [4], supercontinuum generation [5], pulse com-ression [6], and optical coherence tomography [7]. Re-ently, high-power yellow output at 589 nm has been gen-rated by frequency doubling of RFL radiation [8,9], thusonsiderably extending the range of RFL applications.he RFL spectral performance is of great importance forhe majority of applications, and especially for those thatequire frequency doubling. At the same time, it is wellnown that the RFL output spectrum is strongly broad-ned; however, a spectral broadening mechanism is notet well understood.

A RFL usually has a long cavity providing at the sameime high pump and Stokes wave intensities in the fiberore. That is why nonlinear effects could have a signifi-ant impact on the laser performance. Moreover, in ultra-ong RFLs, the nonlinear effects can be comparable withinear ones, making possible, for example, a quasi-losslessransmission [10]. Though the stimulated Raman scatter-ng itself is a basic nonlinear process important for RFLperation, it cannot explain the formation and broadeningf the generated RFL spectrum. On the other hand, dif-erent nonlinear processes, such as stimulated Brillouincattering (SBS) or four-wave mixing (FWM), can alsoead to the spectral broadening.

It has recently been shown that SBS generation thresh-ld is not reached in the medium-power RFL because ofow spectral density [11]. Therefore the SBS effect doesot change the RFL spectral performance. On the con-rary, several indications of the FWM influence on thepectrum of RFLs [12–15] and of erbium-doped fiber la-

0740-3224/07/081729-10/$15.00 © 2

ers [16] have been reported. It has also been shown thathe coupling among longitudinal modes of the RFL cavityust be taken into account [17], as generation in spectralings is observed under the threshold calculated within

he model of independent longitudinal modes generation.oreover, several authors [12,13] have promised to de-

elop and publish models that take into account FWM inFL. However, their work has not yet been finished, ow-

ng to the complexity of the involved processes. Only oneaper [18] reports on an attempt to explain theoreticallyhe RFL generation spectrum through the FWM interac-ion between Stokes wave longitudinal modes. But thisodel is semiempirical and is based on the arbitrarily

ostulated relation among phases of different modes.Since there has been no adequate theoretical descrip-

ion of RFL spectrum broadening until now, to the best ofur knowledge, we recently performed a special theoreti-al and experimental investigation. Our preliminary re-ults have been published in [19]. It has been shown thatn approach based on the wave-turbulence formalism ad-quately describes FWM interaction between multipleongitudinal modes generated in an RFL cavity. In theresent paper, we extend this approach and give a de-ailed description of the developed self-consistent theorys well as a more complete comparison with experimentalesults. Furthermore, the role of the nondegenerate FWMetween pump and Stokes wave modes in RFL spectrumroadening at low powers has been analyzed. The possiblenfluence of FWM on pump wave spectrum broadening islso discussed.

. EXPERIMENTe have studied spectral broadening of a Stokes wave

enerated in a long cavity on the example of a one-stageFL based on a phosphosilicate fiber [1] (Fig. 1). Thehosphosilicate fiber has a distinct isolated P O -related
2 5
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RfRrto=wcsobSpw

wwfcpgctTcsppspts

wsNs2wdpotsta

Fig. 1. One-stage RFL based on a phosphosilicate fiber.

Fscurve).

F(12) total intracavity Stokes wave power.

Fu

1730 J. Opt. Soc. Am. B/Vol. 24, No. 8 /August 2007 Babin et al.

aman gain peak with a large Stokes shift that is freerom complications induced by overlapping of differentaman gain peaks in germanosilicate fibers. The pumpadiation was generated in a 16 m long low-Q cavity ofhe ytterbium-doped fiber laser (YDFL) at the wavelengthf 1.06 �m. In the high-Q RFL cavity having length L370 m and formed by two fiber Bragg gratings (FBGs)ith peak reflectivities R1�R2�99%, the pump wave is

onverted to the first Stokes wave �1.234 �m� due to thetimulated Raman scattering process. The spectral profilef effective FBG losses �=−ln�R1R2� determined mainlyy transmission is shown in Fig. 2. The total intracavitytokes wave power increases almost linearly as the inputump power P0 increases (Fig. 3); remaining comparableith the input pump power within the range up to 3 W.By means of a specially designed intracavity coupler

ith extinction ratio 5:95 at 1.234 �m, intracavity Stokesave spectra were measured near the input FBG at dif-

erent pump powers. Since the spectral shape is compli-ated, we have performed more detailed analysis in com-arison with the first experiments [19]. Near theeneration threshold, the spectrum is quite narrow andonsists of several peaks, Fig. 4(a), which can be easily at-ributed to the minima of the effective FBG losses profile.he total spectral width is �0.2 nm, remaining almostonstant up to the Stokes wave power �0.5 W; see openquares in Fig. 5(a). At the same time, the width of eacheak is as small as 0.08 nm, and it grows with increasingower. The width of the left peak is marked by solidquares in the Fig. 5(a) in the low-power region. Theeaks merge together at Stokes wave powers �0.5 W, sohe total spectrum profile can be well characterized by aingle width at high powers.

At high Stokes wave power, the intracavity Stokesave spectrum is strongly broadened and has a rather

mooth shape with exponential wings (Figs. 4(b)–4(d)).evertheless, the ripples in the spectrum still exist corre-

ponding to the ripples in the FBG losses profile; see Fig.. Since the ripples in the spectrum change their shapeith increasing power, they strongly affect the depen-ence of the spectrum width versus Stokes wave power. Inreliminary studies [19], the width values were measurednly at a few power values, which was not enough to iden-ify the power dependence. Here much more detailedpectral measurements have been performed. The ex-racted Stokes wave spectral width is plotted in Fig. 5(a)s a function of power. The jumps in spectral width values

old: P0=0.4 W. b–d, Measured (dots) and calculated (solid curve)ut pump power P : b, P =1 W; c, P =2 W; and d, P =3 W.

ig. 2. Effective losses of FBGs forming the RFL cavity: Mea-urements (thin curve) and parabolic approximation (thick

ig. 3. Measured (�) and calculated (curve) upon Eqs. (11) and

ig. 4. a, Intracavity Stokes wave spectrum near the generation threshpon Eqs. (8) and (9) intracavity Stokes wave spectrum at different inp
0 0 0 0

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re attributable to the presence of ripples. The spectralower density at maximum increases very fast near theeneration threshold and is smoothly saturated wellbove the threshold (Fig. 5(b)).The value of the effective FBG transmission defined as

he ratio of the output Stokes wave power to the intrac-vity Stokes wave power near the output FBG has beenlso directly measured (Fig. 6). Owing to spectral broad-ning, the effective transmission grows with power. Note,hat the pump wave, which is essentially multimode, islso broadened during its propagation in a long Raman fi-er (Fig. 7).

. WAVE-TURBULENCE APPROACHo explain the spectral broadening in RFLs, we performedtheoretical analysis based on wave-kinetic equations

20] originally used to describe wave turbulence. Ashown below, it is possible to apply the wave-turbulencepproach since multiple waves (up to 106 longitudinalodes) interact with each other via multiple quasi-

egenerate FWM processes in a long RFL cavity and sincehe phases of the waves remain stochastic. It is confirmedy an analysis of the experimentally measured Stokesave RF spectrum [21]. Intermode beating peaks in theF spectrum are diffused, so the width D of the intermodeeating peak becomes of the order of the Stokes wave lon-itudinal modes spacing �. Such behavior makes it evi-ent that there are strong mechanisms that dephasetokes wave components during a round trip. These facts

ig. 5. a, Full width at 1/cosh�1�=0.648 level of the intracavixperiment (�, �) and calculation (curve) upon Eqs. (9) and (10

ig. 6. Measured (�) and calculated (solid curve) upon Eqs. (15)ffective transmission coefficient Teff.

llow us to use the method of averaging over many�104� longitudinal modes while deriving the wave-inetic equation, which describes the Stokes wave spec-rum. Note that experimentally measured Stokes wavepectra are also averaged as there are �104 longitudinalodes within the standard spectral resolution of0.01 nm. In addition, at high-power Stokes wave beingell above the generation threshold, we can neglect for

implicity the nondegenerate FWM processes that couplehe Stokes wave and the pump wave, i.e., neglect pumpave fluctuations.The Stokes wave electromagnetic field E in the cavity

f length L can be represented as the sum of copropagat-ng and counterpropagating waves running with the ve-ocity c,

E�z,t� =1

�2�E+�z,t�eik�ct−z� + E−�z,t�eik�ct+z�� + c.c.,

here the amplitude is normalized by the average totalntracavity Stokes wave power I�z , t�= �E2�z , t��, and±�z , t�= �E±�z , t��2 denotes the power of forward and back-ard propagating wave along the z axis of the fiber, �02�n0 /k is the wavelength at the FBGs reflection maxi-um, and c is the light speed in the fiber. The amplitude

ariations of the copropagated and counterpropagatedtokes waves E± obey an equation that includes disper-ion and nonlinear phase modulation owing to FWM be-ween different longitudinal modes in the Stokes waves;ee, e.g., [22],

�1

c

d

dt±

d

dz�E±�z,t� = �gRP�z� − �

2 �E+�z,t�

−i

2�I±�z,t� + 2I�z,t�E±�z,t� + i

d2E±�z,t�

dt2 , �1�

here P�z� is the mean (over the time) pump wave power,R is the Raman gain coefficient, � is the Stokes wave op-ical losses in the fiber, � is the nonlinear Kerr coefficientamplitude of FWM, i.e., the cross-phase modulation coef-cient), and = �1/2�dc−1/d� is the dispersion coefficient.Because the RFL has a high-Q cavity, we can represent

he Stokes wave as the sum of longitudinal cavity modes:

es wave spectrum. b, Spectral power density at its maximum.
ty Stok

wSiipTp

ed

wtqqw(

Te

wiap=fl

tn

wait

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w(�

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E±�z,t� =1

�2�

nEn�t�exp�in�t i�zn�exp�− i nt�, �2�

here �=2� /�rt is the frequency shift between adjacenttokes wave longitudinal modes, �rt=2L /c is the RFL cav-

ty round-trip time for the Stokes wave, n=c�n��2+�cIs a small frequency shift that takes into account the dis-ersion and the mean nonlinear phase shift, and �=� /L.he factor 1/�2 is chosen in order to normalize the totalower in the nth longitudinal mode to In= �En�2=In

++In−.

Let us rewrite the generalized nonlinear Schrödingerquation (1) in terms of amplitude En of the nth longitu-inal modes defined in Eq. (2),

�rt

dEn

dt−

1

2�g − �n�En�t� = −

i

2�L�

l�0En−l�t� �

m�0En−m�t�

�En−m−l* �t�exp�2iml�2ct�,

�3�

here the effective FBG losses �n=−lnR1��n�R2��n� forhe nth longitudinal mode that is generated at the fre-uency detuned by �n=n� from the FBG center fre-uency have been taken into account. The integral Stokesave round-trip amplification g depends on the averaged

over the fiber length) pump power P̄ as

g = 2gRP̄L − 2�L. �4�

he averaged pump power P̄ is defined by the followingxpression (see [23] for the details):

P̄ = P0

1 − exp�− 2�pL − 2�

�pgRLI�

��p +�

�pgRI�L

, �5�

here P0 is the input (regarding the RFL) pump power, �ps the pump wave optical losses in the fiber, and �p and �re the pump and Stokes wavelengths. Note that this ex-ression is valid within the limits of assumption I�z�const, which is well justified in our case of a highly re-ective cavity for the Stokes wave [23].The terms with l=0 and m=0 in Eq. (3) give the slip of

he carrier phase with respect to the envelope (the meanonlinear phase shift); thus they do not affect the Stokes

Fig. 7. Pump wave spectrum, a, before and, b, afte

ave spectrum. An FWM process between copropagatingnd counterpropagating waves leads to the terms oscillat-ng at beat frequency 2l� and should be omitted sincehey are nonresonant at l�0.

The right-hand side (RHS) of Eq.(3) for the RFL cavityength of 370 m contains �1012 different terms with ran-om amplitudes and phases in each term that lead to thetochastic (turbulent) evolution of the amplitude andhase of the longitudinal mode En. It is obvious that suchn extremely complex nonlinear evolution can be de-cribed neither analytically nor numerically. For an ad-quate description of the Stokes wave spectrum, onehould use a statistical description of the spectral powerensity instead of a dynamical description of the longitu-inal modes amplitudes En, i.e., to derive and to solve theave-kinetic equation. Toward this end, we follow theethods of Zakharov et al. [20], well developed andidely applied for the description of the wave turbulencef interacting acoustic waves, waves on fluid surfaces,pin waves, waves in plasma, etc. After rather complexalculations, which are partially presented in Appendix A,e derive from Eq. (3) the simplified wave-kinetic equa-

ion for the averaged time-independent Stokes wave spec-rum I��,

�� + �2L + �NLI�� = 2gRLP̄I�� +�NL

I2 � I��1�I��2�

�I��1 + �2 − ��d�1d�2, �6�

here �NL defines nonlinear FWM-induced losses [see Eq.A5)], which can be written in our experimental case4L /�2�4.5� in the following form:

�NL =�2

3

�IL

�1 + �8L�2/3�NL�2. �7�

ere I= I��d� is the total intracavity Stokes waveower. The round-trip dispersion-induced phase differ-nce between the longitudinal modes is substituted herey the phase difference averaged over the spectrum,�22L, where the mean-square spectral width �2 is de-ned in Eq. (A8); see Appendix A.

mission trough RFL (input pump power P0=1.5 W).
r trans

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. ANALYTICAL SOLUTION FOR THETOKES WAVE SPECTRUM ANDOMPARISON WITH EXPERIMENTquation (6) has a simple physical meaning. The left-and side (LHS) gives the Stokes wave attenuation, andhe RHS gives the Stokes wave amplification. The lasterm in the LHS is the nonlinear attenuation of thetokes wave longitudinal mode I�� owing to the scatter-

ng of this mode on the mode with frequency �1+�2−� tother longitudinal modes with frequencies �1 and �2. Theast term in the RHS describes the scattering of the

odes with frequencies �1 and �2 to the longitudinalodes with frequencies � and �1+�2−�. The amplitude

f the mode I�� is increasing in this quasi-degenerateWM process. So this term is a FWM-induced gain, and it

s the origin of the spectrum broadening. It is importanthat FWM-induced losses are homogeneous, i.e., do notepend on frequency, while FWM-induced gain is inhomo-eneous; i.e., depends on frequency. Note that the integralf the FWM-induced terms of Eq. (6) over � is zero, whichs equivalent to conservation of the total energy in FWMrocesses, i.e., the total FWM-induced losses are equal tohe total FWM-induced gain.

The solution to the integral equations (6) and (7), asell as the solution to their extended versions [Eqs.

A3)–(A6); see Appendix A], can be found numerically, butt is always of interest to solve them analytically. If theffective FBG losses profile is close to the parabolic form��=�0+�2�2 (which corresponds to the widely appliedaussian-shaped FBG’s reflection spectrum) (see Fig. 2),e find an analytical solution to Eq. (6) for the intracavitytokes wave spectrum,

I�� =2I

�� cosh�2�/��, �8�

here the spectral width � is

� =2

��2�NL

�2�9�

nd the spectral power density at maximum is

I�0� =2I

��. �10�

By calculating the mean-square spectral half-widthrom Eq. (A8) in Appendix A and Eq. (8), �2=�NL /2�2, andubstituting it into Eq. (7), we find

�NL =�2

3

�IL

�1 + �4L/3�2�2. �11�

From Eqs. (9) and (11) it follows that the spectral widthincreases as the square root of the Stokes wave power I.he spectral power density at maximum I�0� increaseslso as the square root of I; see Eqs. (10) and (11).The obtained results should be compared with the pre-

iminary results published in [19]. It has been shown [seeq. 5 of [19]] that the spectral width is proportional to theroduct of the intracavity Stokes wave power I and thequare root of the parameter �=� /3�rt. Under the assump-ion �=const, the spectral width should exhibit linear

rowth with power. However, it was noted that the experi-entally measured value of � decreases with increasing

ower. One of the most important results of the presentaper is the fact that the correlation time � is almost com-letely defined by the nonlinear losses; see Eq. (A6) of Ap-endix A. Thus we can rewrite �=2/ �3�NL� through theonlinear losses coefficient and then show that � is in-ersely proportional to the intracavity Stokes waveower. As a result, we have discovered the square-rootaw that governs the increase in the spectral width and inhe peak power as the Stokes wave power grows.

By substituting Eq. (8) for Eq. (6) and integrating over, one can obtain how the intracavity Stokes wave powerdepends on the Stokes wave power P0,

�0 +�NL�I�

2+ 2�L = 2gRP0

1 − exp�− 2�pL − 2�

�pgRLI�

�p +�

�pgRI

,

�12�

here Eq. (5) has been used. Equation (12) means thathe gain (RHS) and losses (LHS) are equal.

From Eqs. (11) and (12), one can plot the value of thentracavity Stokes wave power I using only fiber and FBGarameters: �p=2.5 dB/km (including lumped losses, i.e.,osses on intracavity coupler, excess losses on FBGs, andplice losses); �=2.5 dB/km (including lumped losses); theormal dispersion �13.3 nm−2 km−1; the nonlinear Kerroefficient ��3 km−1 W−1; the Raman gain coefficient gR1.3 km−1 W−1; the length L=370 m; and the fitted curva-

ure of the effective FBGs losses �2=4 nm−2 (see Fig. 2).here is very good agreement between the calculated andeasured intracavity Stokes wave power I (see Fig. 3).Finally, using Eqs. (8), (9), and (11), one can easily cal-

ulate the Stokes wave spectral profiles I�� at differentntracavity Stokes wave powers I (Figs. 4(b)–4(d)), as wells the spectral width and spectral power density valuesFig. 5). There is excellent agreement between the ana-ytical theory predictions and the experiment in all powerange except the threshold. Let us note that absolutely notting was performed, and only real fiber and FBG pa-ameters were used. The fluctuations in the experimen-ally measured profiles correspond to the ripples in theBG losses profile. So we can conclude that our theoryives the perfect analytical description for the RFL gen-ration spectrum. Thus the Stokes wave spectrum isroadened owing to the quasi-degenerate FWM betweenongitudinal modes of the Stokes wave.

It is also important that the main RFL characteristic,he output RFL power I1,2

out��=−ln�R1,2�I��, can be alsoasily calculated within the developed theory. In the casef identical Gaussian FBGs, −ln�R1,2��0+�2�2� /2, theunning Stokes wave power is I±��I�� /2, which re-ults in

I1,2out�� =

�0 + �2�2

2

I

�� cosh�2�/��. �13�

Note that for an adequate description of the RFL out-ut power, the effective transmission coefficient T of
eff

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T

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ach FBG has been introduced in a phenomenological wayarlier [17,23,24]. Now it is possible to find it theoreti-ally. The effective transmission coefficient for each FBGn the case of small losses is Teff��eff /2, where the meanffective FBG losses �eff can be derived from Eq. (13):

�eff �2�I1

out + I2out�

I=� ��I��d�/I = �0 +

�NL�I�

2.

�14�

hus

Teff � �NL/4, �15�

here �0�0 (see Fig. 2). The effective transmission coef-cient grows linearly with increasing Stokes wave powerwing to the Stokes wave spectrum broadening inducedy the quasi-degenerate FWM processes (see Fig. 6, solidine). However, there is some discrepancy at high powersetween the theoretical prediction and experimentallyeasured values that can be attributed to the difference

etween the mean intracavity Stokes wave power I andhe power I�z=0� measured near the input FBG; this dif-erence increases with increasing transmission. One moreeason is the difference between R1�� and R2�� thatrows with detuning �.

. DISCUSSIONhe obtained results allow us to reach agreement be-ween apparently contradictory experimental and theo-etical results on the Raman gain saturation and the gen-ration spectrum. The first fact is related to the questionbout the RFLs’ multiwavelength generation. It is wellnown that the multiwavelength generation is easilychievable in RFLs experimentally, see, for example,25,26]. Usually it is deemed that this possibility existsue to the inhomogeneous nature of the Raman gain spec-rum. Nevertheless, it has been recently experimentallyhown that Raman gain saturates homogeneously even atigh pump and Stokes wave powers [27]. On the otherand, because of homogeneous nature of the Raman gainne should expect single-frequency RFL generation in-tead of the multiwavelength one. However, single-requency operation has not been observed in RFLs to thisate.Second, it is well known that the Stokes wave spectrum

s fairly broad even near the generation threshold. Atigh power, the Stokes wave spectrum is well describedy the phenomenological model of the independent gen-ration of different longitudinal modes [24,28]. Theseacts do not agree directly with the homogeneous naturef the Raman gain saturation. Thus some inhomogeneousechanisms must exist. Their manifestations in RFLs

ave been reported by several authors recently15,17,29,30]. In [15,29], it is supposed that the multi-avelength generation is possible owing to the intralineWM-induced inhomogeneous losses.In the present paper, we have proved experimentally

nd theoretically that inhomogeneous mechanisms are in-eed based on FWM. Moreover, mixing between differentongitudinal modes of the Stokes wave (quasi-degenerateWM) induces both nonlinear losses and gain. However,

e have clearly shown that FWM-induced losses in ourase �4L /�2�4.5� have a homogeneous nature in spite ofssumptions of some papers [15,29]. At the same time, theWM-induced gain is an inhomogeneous gain. So multi-avelength generation in RFL can be achieved owing to

he inhomogeneous gain, in particular.The developed analytical theory has its applicability

imits. The quasi-degenerate FMW processes are efficientt high powers. However, near the generation thresholdhen the power of the Stokes wave is small, the contribu-

ion of quasi-degenerate FWM processes to the spectralroadening can also be small. At the same time, the RFLpectrum is broadened even near the generation thresh-ld; see Fig. 4(a). Thus, some other efficient spectralroadening mechanisms should exist at low powers.We suppose that at low powers the Stokes wave spec-

rum is broadened owing to nondegenerate FWM betweenhe Stokes wave and different longitudinal modes in theump wave generated in the multimode YDFL. In thisase, the spectral width of the Stokes wave should be pro-ortional to the spectral width of the effective FBG lossesrofile near its minimum and pump power fluctuationshat are determined by its own longitudinal mode beat-ng. This mechanism explains why the single-frequencyegime has not yet been achieved in the RFL. In the usualDFL-pumped RFL, there are nonzero fluctuations of theump wave power at the Stokes wave generation thresh-ld [21]. Therefore, the Stokes wave spectrum is broad-ned owing the nondegenerate FWM with different longi-udinal modes of the pump wave, and single-frequencyeneration is not possible. The low-power theory detailsre presented in Appendix B.The developed wave-turbulence approach for the de-

cription of the RFL spectrum can be useful for solvingther problems of fiber optics such as, for example, a mul-ilongitudinal mode wave propagation in a fiber [31]. Onean qualitatively affirm that the multimode pump waveropagating in a long RFL cavity is broadened by its ownuasi-degenerate FWM processes between different pumpave longitudinal modes. The experimental results con-rm this supposition; see Fig. 7. Based on the wave-urbulence approach of the wave-kinetic equation thatakes into account the existence of a small but nonzerohase correlation, it is possible to perform analyticalnalysis of the wave spectral broadening during propaga-ion in long fibers.

. CONCLUSIONhus we have developed an analytical theory that de-cribes the Stokes wave spectrum formation and broaden-ng in RFLs. This self-consistent theory takes into ac-ount FWM and complies with all experimental andheoretical results regarding the RFL generation and theaman gain saturation nature. It has been shown that

he main spectral broadening mechanism is the quasi-egenerate FWM between different Stokes wave longitu-inal modes. The spectral shape of the Stokes wave is de-ned mainly by the effective FBG losses profile. The exactnalytical solution for the RFL intracavity spectra formedy longitudinal modes interacting via the FWM processesults in a specific hyperbolic secant shape at high pow-

eamwrmte

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w

ai

Hfa

sw

w=Ht

it−


rs. The theory explains generation of longitudinal modest the spectral wings that must be forbidden within theodel of independent mode generation. The spectralidth of the Stokes wave is proportional to the square

oot of the intracavity Stokes wave power as well as theaximum spectral power densities. The predictions of the

heory are in excellent quantitative agreement with thexperimental data.

The perfect theoretical description of the generatedtokes wave spectrum became possible since the dynami-al description of �106 amplitudes and phases of longitu-inal modes that change their values stochastically (tur-ulent behavior) was discarded in favor of statisticalescription of the spectrum averaged over a large number�104� of longitudinal modes.

The turbulent behavior observed in RFL is a funda-ental effect that may be important in other fields of fi-

er optics and may have a significant impact on the per-ormance of practical fiber systems. We believe thatredicted simple square-root law for the spectral widthnd peak power density growth should be valid for otherypes of fiber lasers as well.

PPENDIX A: SPECTRAL BROADENINGELL ABOVE THE THRESHOLD

ultiplying Eq. (3) by En* and taking the real part, one

an obtain the equation describing how the power of theth Stokes wave longitudinal mode changes,

�rt

dIn

dt− �g − �n�In�t� = − Re�i�L �

l�0,m�0En−l�t�En−m�t�

�En−m−l* �t�En

*�t�exp�2iml�2ct�� .

�A1�

2 1 2

tptiwttph

Tc

[l

If one assumes that phases of different longitudinalodes are random (uncorrelated) then averaging of theHS of Eq. (A1) gives zero value as all terms include aandom phase difference. To take into account a phaseorrelation induced by the FWM processes, we calculatehe correction to En by integration of the RHS of Eq. (3),

�En�t� =− i�L

2 �−�

t

�l��0,m��0

En−l��t��En−m��t��En−m�−l�* �t��

�exp�2im�l��2ct��dt�

�rt. �A2�

n the next step, we substitute this correction (with indexchanged to n− l) instead of En−l in the RHS of Eq. (A1)

nd then instead of En−m, etc. As a result, we obtain theum of four terms, and after averaging them over �104

pectral components near the frequency �=n�, we derivewave-kinetic equation for the stationary Stokes wave

pectrum I��= �In� /�:

�rt

dI��

dt= g − ��I�� + SFWM�� = 0, �A3�

here FWM-induced terms are

SFWM�� = − �NLI��

+ ��L�2� I�� − �1�I�� − �2�I�� − �1 − �2�

�3�rt/��1 + �4�L/3�rt�2�12�2

2d�1d�2,

�A4�

nd nonlinear FWM-induced losses �NL have the follow-ng form:

�NL = ��L�2� I�� − �1� + I�� − �2�I�� − �1 − �2� − I�� − �1�I�� − �2�

�3�rt/��1 + �4�L/3�rt�2�12�2

2d�1d�2. �A5�

ere we assume an exponential decay of the correlationunction �El�t�El

*�t��=Ile−�t−t��/� with correlation time �,nd the Gaussian statistics for field En�t�.The exponential decay of the correlation function re-

ults in the Lorentzian shape of the peaks in the Stokesave intermode beating RF spectrum:

F�� = �n�0

DFn

�D2 + �� − n��2,

here Fn=�lIlIl+n, and D=2/�. So the correlation time �2/D can be found from the experimentally measuredWHM of the peaks in the RF intermode beating spec-

rum of the Stokes wave.The last term in the RHS of the Eq. (A4) describes an

ncrease of intensity in the mode with frequency � owingo scattering of modes with frequencies �−�1 and �� to modes with frequencies � and �−� −� , i.e., this

erm represents the FWM-induced nonlinear gain. Thehysical meaning of �NL is the round-trip nonlinear at-enuation coefficient of the Stokes wave with frequency �nduced by its scattering on other longitudinal modesith frequencies �−�1−�2. It is exactly the nonlinear at-

enuation that leads to exponential decay of the correla-ion function. The nonlinear attenuation increases withower, and completely determines the correlation time atigh Stokes power:

�rt

�� NL/2, �A6�

his relation makes Eqs. (A4), (A3), and (A5) self-onsistent.

It is relevant to note that the wave-kinetic equationEqs. (A3)–(A6)] is valid when the nonlinearity is muchess than the dispersion:

w

mspll

leSlw

Tzc

irh

eSgTbt

wsiIgft

p−a

AztScr

wstgt

p

f


�IL � 4L�2, �A7�

here �2 is a mean-square spectral half-width,

�2 =� �2I��d�

I. �A8�

Condition (A7) provides a phase mismatch between re-ote spectral components (longitudinal modes), as a re-

ult, the FWM-induced phase synchronization is sup-ressed owing to dispersion, and phases of differentongitudinal modes can be considered as weakly corre-ated.

By evaluating the integrals in Eqs. (A3)–(A5) at theimit of condition (A7), one can find from the wave-kineticquation the spectral width and the spectral power of thetokes wave if the spectrum is bell shaped. For simplicity,

et us examine the case of Gaussian-apodized FBGs inhich effective FBG losses have parabolic form:

�� = �0 + �2�2.

he integral over the frequency of SFWM�� is equal toero, which means that energy is conserved in FWM pro-esses. That is why from Eq. (A3) it follows that

� + � �2 = g = 2g LP̄ − 2�L. �A9�
0 2 R
a

w

poel

f

iwm

This equation has a simple physical meaning of equal-ty between the Stokes wave power gain and losses in aound trip. In particular, Eq. (A9) connects the spectralalf-width with the saturated gain g.To obtain one more relation, let us evaluate the nonlin-

ar losses �NL in the case of a bell-shaped intracavitytokes wave spectrum. The main contribution to the inte-ral [Eq. (A5)] over �1 accumulates near zero frequencies.he residual integral over �2 gives a logarithm boundedy the spectral width. As a result, we obtain a relation be-ween the nonlinear losses and the spectral half-width:

3

2�NL � B

��IL�2

4L�2ln�8L�2

3�NL�� IL � 4L�2,

�A10�

here B is a constant that accounts for the particularpectral shape. For example, the Gaussian shape resultsn B=0.7, and the hyperbolic secant shape gives B=0.9.n evaluation of nonlinear losses [Eq. (A10)], we have ne-lected small terms ��NL / �4L�2��1. Inclusion of therequency dependence of the nonlinear losses is beyondhe framework of this approximation.

It remains to obtain one more relation. For this pur-ose, let us take �=0 in Eq. (A3), which gives us g=�0SFWM�0� /I�0�. After substituting Eq. (A6) for Eqs. (A4)nd (A5), we find

SFWM�0�

I�0�= ��L�2� d�1d�2

I��1�I��2�� I��1 + �2�

I�0�+ 1� − I��1� + I��2�I��1 + �2�

�3�NL/2�1 + �8L/3�NL�2�12�2

2. �A11�

t �1=0 or �2=0, the integrand numerator takes theero value, therefore all spectral components give a con-ribution to the integral. For a bell-shaped intracavitytokes wave spectrum, we obtain in the approximation ofondition (A7) with the result of Eq. (A10) the followingelation:

g − �0 =SFWM�0�

I�0�=

3

2A�NL� �IL

4L�2�2

� �NL, �A12�

here A is a constant that accounts for the spectralhape, so A=0.3 for the Gaussian shape, and A=1.5 forhe hyperbolic secant shape. Equation (A12) connects to-ether the saturated gain g, the nonlinear losses �NL, andhe spectral half-width.

Finally, from Eqs. (A9), (A12), and (A10), we find ex-ressions for the spectral half-width,

�2 =�I

4�4L

�2AB ln�4LA

�2B ��1/4

�A13�

or the nonlinear attenuation

�NL =2

3�IL�B ln�4LA

�2B ��3/4�4LA

�2�−1/4

, �A14�

nd for the Stokes wave power I,

2gRP̄�I�L = 2�L + �0 + �IL�AB ln�4LA

�2B ��1/4�4L

�2�−3/4

,

�A15�

here the averaged pump power P̄�I� is defined by Eq. (5).It is important that the spectral width and the spectral

ower density depend equally according a square-root lawn the intracavity Stokes wave power I. Spectral broad-ning leads to increase in the output losses, which growinearly with I [the last term in Eq. (A15)].

It should be noted that the inequality [condition (A7)]or the RFL conditions is equivalent to the inequality

4L

�2� 1, �A16�

.e., is valid for a long cavity, a large normal dispersion, oride FBGs forming the cavity. In the performed experi-ent, the condition (A16) is fulfilled; 4L /� �4.5. Never-
2

t(ttig(tl

ALNiwehtpgnf

to

Hpte=

mwnmwb

oo

fr

Afs

nt

wpnp

Siisqtspdttw

tbtd

ATgApRCR(bf

R


heless, this parameter is not as large as for Eqs.A13)–(A15) to give us sufficient accuracy of the evalua-ion. On the other hand, in this case, we can substitutehe averaged over-the-spectrum values of frequencies �1,2

2

nstead of its real values in all denominators of the inte-rands in the RHS of Eqs. (A4) and (A5). In that way, Eqs.A3)–(A6) can be simplified to Eqs. (6) and (7) on the sta-ionary intracavity Stokes wave spectrum I��. This al-ows us to find the analytical solution [Eq. (8)].

PPENDIX B: SPECTRAL BROADENING ATOW POWERSear the Stokes wave generation threshold, the intracav-

ty Stokes wave power I is much smaller than the pumpave power P. So we can neglect the Stokes wave influ-nce on pump wave fluctuations �P�z , t�. On the otherand, we can also neglect the Stokes wave influence onhe Stokes wave phase modulation as compared with theump wave influence. Thus, we take into account nonde-enerate FWM involving the Stokes and pump waves buteglect the quasi-degenerate FWM processes between dif-erent Stokes wave longitudinal modes.

One can obtain equations on the amplitudes En�t� ofhe longitudinal modes that are connected with eachther by means of the pump wave power fluctuations:

�rt

dEn�t�

dt=

g − �n

2En�t� − 2i�L�

l�0pl�t�expil��c1

− ��tEn−l�t�. �B1�

ere we have made an expansion of fluctuations of theump wave running with speed c1 in the positive direc-ion inside the RFL cavity: �P�z , t�=�l�0eil��c1t−z�pl�t�. Thexpansion coefficients have the following form: pl�t�p−l

* �t�= 0L�P�z , t�e−il��c1t−z�dz /L.

Note that the coefficients pl�t� in Eq. (B1) have theeaning of the coupling coefficients of different Stokesave longitudinal modes. Therefore, the more compo-ents the pump wave fluctuations expansion includes, theore remote Stokes wave longitudinal modes are coupledith each other. Thus the Stokes wave spectrum shoulde broadened owing to the pump wave fluctuations.Multiplying Eq. (B1) by En

* and taking the real part,ne can obtain an equation on In��En�2, i.e., an equationn the Stokes wave spectrum component:

�rt

dIn

dt= �g − �n�In − 2�L�

l�0ipl�t�eil��c1−��tEn−l�t�En

*�t�

+ c.c.. �B2�

Repeating the derivation given in Appendix A, we per-orm direct integration of Eq. (B1), which gives us the cor-ection for the Stokes wave longitudinal mode amplitude:

�En�t� = − 2i�L�−�

t

�l��0

pl�t��eil��c1−��t�En−l��t��dt�

�rt.

fter that, we can substitute the integral representationsor the RHS of Eq. (B2), first instead of En−l, and then in-tead of E*, and after averaging over spectral compo-
n
ents, we obtain the equation on the Stokes wave spec-rum I��= �In /�� at frequency ��n�,

�rt

dI��

dt− g − ��I�� = 2

�1

�rt�2�L�2�

−�

�

d��

�Fp��I�� − �� − I��

1 + �1�c1/c − 1��2 ,

�B3�

here �1=2/D1 is the dephasing time that is inverselyroportional to the pump wave RF spectrum width D1ear the Stokes wave generation threshold, Fp�� is theump wave RF spectrum profile.Equation (B3) is a wave-kinetic equation near the

tokes wave generation threshold. The first term in thentegrand of the RHS describes the nonlinear FWM-nduced gain: a Stokes wave with some frequency �−��catters on the pump wave to the Stokes wave with fre-uency � and other pump wave. Such processes lead tohe Stokes wave spectral broadening. The second term de-cribes the losses induced by the nondegenerate FWMrocesses between the Stokes wave and different longitu-inal modes of the pump wave. These losses correspond tohe scattering of the Stokes wave longitudinal mode � onhe pump wave, so the intensity of the scattered Stokesave is reduced.From Eq. (B3), it follows that if there are nonzero fluc-

uations of the pump wave power, then the Stokes wave isroadened even near the generation threshold. Such spec-ral broadening owing to nondegenerate FWM processesoes not depend on the type of the gain saturation.

CKNOWLEDGMENTShe authors acknowledge financial support by the Inte-ration program of the Siberian Branch of the Russiancademy of Sciences, the governmental program of sup-ort for leading scientific schools and young scientists inussia, the programs of the Russian Academy of Science,ivilian Research and Development Foundation grantUP1-1509-NO-05; and the Fiber Optic Research Center

Moscow, Russia) for the supply of the phosphosilicate fi-er. We also thank V. V. Lebedev and A. M. Shalagin forruitful discussions.

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Documents

Four-wave-mixing-induced turbulent spectral broadening in a long Raman fiber laser