PHYSICAL REVIEW A 83, 043831 (2011)

Spectral sidebands and multipulse formation in passively mode-locked lasers

Rafi Weill, Alexander Bekker, Vladimir Smulakovsky, and Baruch FischerDepartment of Electrical Engineering, Technion-IIT, Haifa IL-32000, Israel

Omri GatRacah Institute of Physics, Hebrew University, Jerusalem IL-91904, Israel

(Received 24 January 2011; published 25 April 2011)

Pulses in passively mode-locked lasers are often accompanied by dispersive waves that form spectral sidebandsdue to spatial inhomogeneities in the laser cavity. Here we present an explicit calculation of the amplitude,frequency, and precise shape of the sidebands accompanying a solitonlike pulse. We then extend the study tothe global steady state of mode-locked lasers with a variable number of pulses, and present experimental resultsin a mode-locked fiber laser that confirm the theory. The strong correlation between the temporal width of thesidebands and the measured spacing between the pulses in a multipulse operation suggests that the sidebandshave an important role in the interpulse interaction.

DOI: 10.1103/PhysRevA.83.043831 PACS number(s): 42.55.Wd, 42.60.Fc, 42.55.Ah, 42.65.Sf

I. INTRODUCTION

Short pulses that scatter from periodic spatial inhomo-geneities in a dispersive medium emit radiation that interferesconstructively at resonant frequencies, forming spectral side-bands, sometimes called Kelly sidebands [1]. Several papers[1–3] studied the formation of dispersive waves by solitons andtheir propagation in optical fibers with periodically spacedamplifiers. The same mechanism leads to the formationof sidebands in mode-locked lasers, where inhomogeneitiesin the cavity act periodically on the pulse. Sidebands canalso be generated by periodic perturbations of cw [4,5].The latter type of sidebands arise because of a four-wave-mixing-induced modulational instability, in contrast withthe linear resonance responsible for the Kelly sidebands.Recently, sidebands induced by modulational instabilitieswere also discovered in mode-locked lasers that, unlikeKelly sidebands, occur only beyond a threshold pumpingintensity [6].

When the peak power of a single pulse exceeds thatof maximal transmissivity, the steady state of a passivelymode-locked laser tends to bifurcate into configurations wheretwo or more pulses run in the laser cavity simultaneously [7].Since the early experiments that demonstrated multipulsemode locking [8–10], it has been observed that the pulsesdisplay a very rich dynamics, often forming bunches, as aconsequence of complex interpulse interactions. The inter-est in the dispersive waves in mode-locked lasers, beyondtheir prominent effect on the pulse shape, arises becausethey have often been suggested as a means of interpulseinteraction in multipulse mode-locked lasers [11–14]. Here,we focus on the steady-state structure of the dispersivewaves and the corresponding Kelly sidebands formed by theradiation emitted from the pulses in multipulse mode-lockedlasers.

The peak clamping leading to multipulse steady states isoften modeled by adding a quintic term to the equations ofmotion. In former papers [15,16], we applied the statisticallight-mode dynamics (SLD) theory to this system, and showedthat multipulse mode locking is, in effect, a series of first-

order phase transitions. SLD uses the methods of statisticalphysics to analyze the dynamics of the interacting many-bodylight-mode system at an effective finite temperature generatedby cavity noise [17–19]. Here, we apply the SLD gainbalance method [20] to derive the multipulse steady stateswith dispersive waves of mode-locked lasers with cavityinhomogeneities.

Our theoretical analysis is based on the master equation ofmode-locked soliton lasers [21,22], with an additive noise term[23], where the inhomogeneities in the cyclic light propagationin the cavity are modeled by a periodic modulation of the gainand the saturable absorption. We first study the sidebands ina single-pulse steady state and show that, unlike free fibersidebands, the mode-locked-laser sidebands reach a steadystate with a well-defined bandwidth and a Lorentzian shape;in real time, the dispersive waves form a wide pedestal withexponentially decaying tails. In particular, we demonstratehow the overall phase of each sideband depends on the relativephase of the gain and loss modulations. The theory is firmlysupported by experimental observations in mode-locked fiberlasers.

Next, we derive a nonlinear equation for the global steadystate of the laser, which includes a number of pulses andtheir accompanying pedestals, by applying the gain balanceprinciple to the pulses, sidebands, and cw components ofthe waveform. We find that the sideband intensity and thepedestal width increase by a large factor when the pumping isincreased with a constant number of pulses, and then decreaseabruptly when another pulse is formed, so the properties ofsingle-pulse sidebands display an oscillatory, approximatelyperiodic, dependence on the pump power. These theoreticalresults are again favorably compared with experiments ina mode-locked fiber laser. We conclude by considering theimplication of our results on the nature of sideband-mediatedinteractions.

II. THEORETICAL MODEL

The mode-locking master-equation model for soliton lasers[21,22,24], where the dominant dynamical processes are

043831-11050-2947/2011/83(4)/043831(6) ©2011 American Physical Society

WEILL, BEKKER, SMULAKOVSKY, FISCHER, AND GAT PHYSICAL REVIEW A 83, 043831 (2011)

chromatic dispersion and Kerr nonlinearity, takes the form

i∂zψ = −∂2t ψ − 2|ψ |2ψ + ipg(z)g[ψ]

(ψ + γ ∂2

t ψ)

+ ips(z)(s(|ψ |2) − l)ψ + �(z,t). (1)

The first two terms on the right-hand side are the aforemen-tioned dispersion and Kerr nonlinearity, the coefficients ofwhich are nondimensionalized by an appropriate choice ofunits of time and power. The next term models the saturablegain of the laser amplifier. g[ψ] is the overall gain coefficient,and the square brackets signify that g depends on the entirewaveform ψ(t) rather than the instantaneous value of ψ only,and γ is the coefficient of parabolic spectral filtering. We willassume that the the gain saturation is slow compared to thecavity round-trip time tR , so g is determined by the overallpower P = ∫ tR/2

−tR/2 |ψ |2dt in the usual manner:

g[ψ] = g0

1 + P/Ps

, (2)

where g0 and Ps are the small signal gain and the saturationpower, respectively.

The next term on the right-hand side of the master equationmodels the fast saturable absorber with transmissivity s(|ψ |) −l, where l is the small signal loss and s(0) = 0 by definition.We do not assume a particular form for the transmissivityfunction other than that it increases linearly at zero, s ′(0) > 0.We assume, for simplicity, that the saturable absorber accountsfor all the losses in the cavity. The final term is a Gaussian whitenoise source with covariance

〈�∗(z,t)�(z′,t ′)〉 = 2T δ(z − z′)δ(t − t ′), (3)

where the constant T is the rate of internal and injected noisepower. As mentioned above, the noise is a significant factorin the determination of the steady state; we conjecture thatit is also an essential ingredient in the interpulse interaction.The sidebands are formed by the spatial inhomogeneity of thegain and loss processes in the laser, described by pg(z) andps(z), respectively, which are periodic functions of the cavityround-trip length L, normalized to 1

L

∫ L

0 dzpg,sdz = 1.The dominance of the dispersive effects means that the gain

and loss terms in Eq. (1) are proportional to a small parameter,and that the noise term is proportional to an independentsmall parameter. In spite of their smallness, the gain, loss, andnoise are the crucial terms for the mode-locking phenomenadiscussed here. At the same time, they also perturb the pulseproperties that are dominated by the dispersive terms; thesesmall perturbations will be neglected.

III. SINGLE-PULSE SIDEBANDS

We begin our analysis assuming conditions under whichthere is a single pulse in the cavity with fixed parametersin the steady state. Since the dominant terms in the masterequation are the dispersion and Kerr effect, the pulse waveformis approximately that of a nonlinear Schrodinger (NLS)soliton. Solitons are defined by four parameters: amplitudea, frequency, timing, and phase. The gain and loss terms in themaster equation fix the frequency to zero, and we can set the

timing and phase to zero by an appropriate choice of origin,so the soliton waveform is

ψp(t,z) = a sech(at)eia2z. (4)

The soliton waveform ψp is not an exact solution of themaster equation because of the gain, loss, and noise terms. Wetherefore look for a solution of the form

ψ(t,z) = ψp(t,z) + ψb(t,z) + ψc(t,z), (5)

which, in addition to the pulse waveform, consists of thesideband waveform ψb generated as a result of scattering of thepulses off the cavity inhomogeneities, and the continuum ψc

generated by the cavity noise. The three waveform componentshave different characteristic time scales: the sidebands arenarrow resonances, the temporal width of which is, as shownbelow, inversely proportional to the gain-loss small parameter.This width is large compared with the pulse width, but smallcompared with the cavity round-trip time, i.e., the scale of thecontinuum.

Both ψb and ψc have low peak power and can thereforebe analyzed by the linearized master equation, although theirtotal power

∫dt |ψb|2,

∫dt |ψc|2 can be of the order P . The

effect of noise is negligible on ψb, which therefore satisfiesthe equation

∂zψb = i(∂2t ψb + 4|ψp|2ψb + 2ψ2

pψ∗b

)+pg(z)g[ψ]

(1 + γ ∂2

t

)(ψp + ψb)

+pp(z)(s(|ψp|2) − l)(ψp + ψb). (6)

The right-hand side of Eq. (6) retains terms that are of higherorder of smallness than ψb; these terms are, in effect, notnegligible for |t | � a where ψp is itself small, and play acrucial role in the shaping of the sidebands, as shown below.

The discrete modes of the real linear Eq. (6) expresssmall variations of the pulse parameters [23,25], while ψb,which consists of radiation emitted by the pulse, is a linearcombination

ψb(t,z) = eia2z

∫dω

2π[αω(z)uω(t) + αω(z)∗vω(t)∗] (7)

of the first component of the scattering states of the linearoperator L,

L

(uω

vω

)= [−(i + gγ )ω2 − ia2]

(uω

vω

), (8)

L

(v∗

ω

u∗ω

)= [(i − gγ )ω2 + ia2]

(v∗

ω

u∗ω

), (9)

which acts on two-component wave functions as

L =(

A B

B∗ A∗

)(10)

with

A = (i + gγ )∂2t − ia2 + 4ia2sech2(at) + s[a2sech2(at)]

and B = 2iasech2(at). It will be argued below that we canapproximate the space-dependent linear operator acting on ψb

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SPECTRAL SIDEBANDS AND MULTIPULSE FORMATION . . . PHYSICAL REVIEW A 83, 043831 (2011)

in Eq. (6) by its space average L. Within this approximation,the coefficients defined in Eq. (7) evolve according to

∂zαω = [−(i + gγ )ω2 − ia2]αω + (gαω + βgω)pg(z)

+ (−lαω + βsω)ps(z), (11)

where βg and βs are the expansion coefficients of theforcing terms g(1 + γ ∂2

t )[a sech(at)] and {s[a2sech2(at)] − l}asech(at), respectively. As usual, the expansion coefficientsare extracted by the inner product with the adjoint eigenfunc-tions defined by

L†(

uω

vω

)= [(i − gγ )ω2 + ia2](ω2 + a2)

(uω

vω

), (12)

L†(

v∗ω

u∗ω

)= [−(i + gγ )ω2 − ia2]

(v∗

ω

u∗ω

)(13)

normalized so that∫dt[uω(t)∗uω′(t) + vω(t)∗vω′(t)] = 2πδ(ω − ω′), (14)

∫dt[vω(t)uω′(t) + uω(t)vω′(t)] = 0. (15)

Since, by assumption, the dispersive terms are dynamicallydominant, the eigenfunctions of L are close to the eigenfunc-tions of the linearized NLS equation [26]

uω(t) = eiωt

(1 − 2iωe−at

(ω + ia)2asech(at) + a2sech2(at)

(ω + ia)2

),

(16)

vω(t) = eiωt a2sech2(at)

(ω + ia)2, (17)

uω(t) = a(ω + ia)2

(ω − ia)2u−ω(−t), (18)

vω(t) = −a(ω + ia)2

(ω − ia)2v−ω(−t), (19)

so that

βgω = a

2π

(ω + ia)2

(ω − ia)2

∫dt

(1 + γ a2∂2

θ

)a sechθ

× e−iωt

(1 − 2iωe−atasech(at) + a2sech2(at)

(ω + ia)2

),

(20)

βsω = a

2π

(ω + ia)2

(ω − ia)2

∫dt{s[a2sech2(at)] − l}asech(at)

× e−iωt

(1 − 2iωe−atasech(at) + a2sech2(at)

(ω + ia)2

).

(21)

The solution of Eq. (11) is

αω(z) =∫ z

dz′e[−ω2(i+gγ )−ia2](z−z′)

× e∫ z

z′ dz′′[gpg (z′′)−lps (z′′)][βgωpg(z′) + βsωps(z′)]. (22)

As observed by Gordon and Kelly [1–3], the amplitudes αω

are driven resonantly if the nonlinear frequency shift a2 +ω2 is an integer multiple of the cavity-based wavelength 2π

L;

spectrally, therefore, the dispersive waves contain a discreteset of sidebands at frequencies ±ωn:

ωn =√

2π

Ln − a2, n = 1,2, . . . . (23)

The nth sideband is forced mainly by the nth harmonic ofthe gain and loss modulation functions pg,l , but also by thez dependence of the gain and loss terms in the exponent inEq. (22). The latter modulation is small, however, if we makethe simplifying assumption that the total net loss per roundtrip is small, so the z′ integration in Eq. (22) is effective overmany round trips. This is the usual assumption underlyingthe Haus master equation [21] and is also consistent withthe relative weakness of the gain and loss processes in thedynamics dominated by the chromatic dispersion and Kerrnonlinearity that is studied here.

In this case, the mean net loss in the z′′ integration in Eq. (22)dominates over the variable parts of the gain and loss, andthe latter may be neglected. The same assumption justifies theapproximation of the time-dependent linear operator in Eq. (6),the equation of motion for ψb, by the fixed operator L that wasused to derive Eq. (11).

It now follows that the dispersive wave amplitudes are

αω(z) =∫ z

dz′e[−i(a2+ω2)+g(1−γω2)−l](z−z′)

× [βgωpg(z′) + βsωps(z′)] (24)

and defining the nth Fourier components pg,n and ps,n of pg

and ps , respectively, the coefficients of the nth sideband in thesteady state are

αω,n(z) = (pg,nβgω + ps,nβsω) e− 2πL

inz

l + g(γω2 − 1) + i(ω2 − ω2

n

) . (25)

We wish to characterize the sidebands by their ordinaryFourier spectrum, obtainable from Eq. (7),

ψbn(ω,z) = eia2z

∫dω′

2π[αnω′(z)uω′ω + αn,ω′ (z)∗v∗

ω′ω], (26)

where uω′ω = ∫dt e−iωtuω′(t), and vω′ω is defined similarly.

As a transform of a rapidly varying function, u is wideband(its ω bandwidth for fixed ω′ is comparable with the solitonbandwidth), but it is also singular for ω = ω′. The smooth partof u generates a small deformation of the soliton waveformin Eq. (26), which is unimportant for the present purpose ofcharacterizing the sideband spectrum. We therefore focus onthe singular part uω′ω of uω′ω, which is determined by the ±∞asymptotes of uω′ [see Eq. (16)] to

uω′ω = 2πω2 − 1

(ω + ia)2δ(ω − ω′) + 4ωa

(ω + ia)2P

1

ω − ω′ , (27)

where P denotes principal part. vω′ω = 0 since vω(t) tends tozero exponentially for t → ±∞.

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WEILL, BEKKER, SMULAKOVSKY, FISCHER, AND GAT PHYSICAL REVIEW A 83, 043831 (2011)

1550 1560 1570−80

−60

−40

Wavelength (nm)

Inte

nsity

(ar

b. u

nits

)

1550 1560 1570−80

−60

−40

Wavelength (nm)

FIG. 1. Measured spectra of the soliton and its sidebands,showing the notch formation. The two figures correspond to twomeasurement locations in the same cavity that are separated by onequarter of cavity length. These measurements agree well with thetheory.

Now we can carry out the ω′ integration in Eq. (26) andobtain the spectrum of the nth sideband

ψb,n(ω) = (pg,nβgω + ps,nβsω) ei(a2− 2πL

n)z

l + g(γω2 − 1) + i(ω2 − ω2

n

) . (28)

The spectral width (half width at half maximum) ofthe sideband is given by δω = [l + g(γω2

n − 1)]/2ωn � 1;temporally, therefore, the dispersive wave is a pedestal of fre-quency ωn with an exponentially decaying envelope centeredat the pulse, the decay time scale of which is much wider thanthe soliton width.

Experimentally, the sidebands appear as a series of sharppeaks in the pulse spectrum on the background of the widesoliton spectrum (see Fig. 1). A distinctive feature of thesideband spectrum (28) is the dependence of the overall phaseof the sideband on the measurement position in the cavity; as aconsequence, the sideband spectrum interferes constructivelyor destructively with the soliton spectrum, depending on theplacement of the output coupler, exhibiting a sharp notchfeature for some placements as the result of destructiveinterference. Figures 1 and 2 show the comparison betweenthe measurement and calculation of the soliton and the firsttwo sidebands’ spectrum for two cavity positions. The goodagreement between theory and experiment, and the fact thatthe sidebands are observed for all pumping strengths thatallow mode locking, demonstrate that these are indeed Kellysidebands The main qualitative discrepancy between theory

−3 −2 −1 0 1 2 3−20

−10

0

10

20

Wavelength (arb. units)

Inte

nsity

(ar

b. u

nits

)

−3 −2 −1 0 1 2 3−20

−10

0

10

20

Wavelength (arb. units)

Inte

nsity

(ar

b. u

nits

)

FIG. 2. Theoretical calculation of the pulse and sidebandspectrum with a = 1,g = 0.1,γ = 0.01,l = 0.11,s = 0.1(|ψ |2 −|ψ4|),tRT = 10−3,L = 2.56,pg = eiπ/2, and ps = eiπ . The cavityposition in the left figure is z = L/5, and in the right is z = 0.

and experiment is the asymmetry between the left and rightsidebands, which is lacking in the theory. The most likelysource of this asymmetry is the asymmetry of the spectral netgain curve with respect to the band center [27] and the resultantfrequency pulling [28], which are not included in our modelfor simplicity.

IV. THE CONTINUUM COMPONENT

The continuum ψc is the dominant component of thewaveform for most of its temporal extent, where both ψp(t)and ψb(t) are negligible. Therefore, the nonlinearity andinteraction with the pulses are unimportant for its dynamics,and it is natural to express it in terms of the ordinary Fouriermodes ψc(ω) that satisfy

∂zψc= − iω2ψc + pg(z)g(1 − γω2)ψc−ps(z)lψc + �(ω,z).

(29)

Because the noise and continuum extend throughout theentire round-trip time, we must use discrete frequencies tolabel their Fourier transform. It then follows from Eq. (3) that�(ω,z) = ∫

dte−iωt�(t,z) has zero mean and correlation func-tion 〈�(ω,z)�∗(ω′,z′)〉 = 2T tRδ(z − z′)δω,ω′ . The solution ofEq. (29) is similar to that of (22), and, by the same argumentsthat lead to Eq. (24), we again approximate the integrand in theexponent by its mean value. The resulting expression impliesthat 〈ψc(ω)〉 = 0 and

〈ψc(ω)ψ∗c (ω′)〉 = tRT δω,ω′

l + g(γω2 − 1). (30)

V. GAIN BALANCE AND MULTIPULSE SIDEBANDS

In the analysis presented so far, the pulse parameters andthe overall saturated gain g were assumed fixed and given.In the steady state, these are variables determined along withthe number of pulses as a solution of the optical equation ofmotion [Eq. (1)]. Since the equation of motion is random,the result is a statistical steady state. In previous works[19,20], statistical light-mode dynamics (SLD) theory wasused to study this problem and applied to multipulse modelocking in [15,16]. The global mode-locking analysis deter-mines, in particular, the properties of the sidebands, and in thisway allows us to reach our goal of describing the width andpower of the sidebands as a function of the laser parameters.

Here we study the statistical steady state by the gain balancemethod, deriving equations of motion for the power in the threecomponents of the optical waveform, that is, the power Pp inthe pulse waveform that comprises zero or more pulses, thesidebands power Pb, and the continuum power Pc. The threewaveform components are characterized by well-separatedtime scales, and the total power can therefore be calculated asa sum of the powers of the individual components. One looksfor a steady state where the three components are subject to thesame saturated gain, and finally the gain itself is determinedself-consistently from Eq. (2).

We will assume that the multipulse waveform is of thesimplest kind [15], which is also the most commonly observedexperimentally, consisting of k pulses of equal amplitude a. Inthe equation of motion for the pulse amplitude, we may neglect

043831-4

SPECTRAL SIDEBANDS AND MULTIPULSE FORMATION . . . PHYSICAL REVIEW A 83, 043831 (2011)

the noise term in the master equation, and use the results ofsoliton perturbation theory [3,29] to write

da

dz= Re

∫dt ψ∗

p

{gpg(z)

(1 + γg∂

2t

)ψp

+ps(z)[s(|ψp|2) − l]ψp

}

= 2gpg(z)a

(1 − γ

3a2

)+ ps(z)[2sa(a) − 2l]a, (31)

where

sa(a) = 1

2a

∫s[a2sech2(at)]a2sech2(at) dt. (32)

The pulse amplitude changes periodically during its propaga-tion in the cavity; for our purposes, we need the mean pulseamplitude a(z) = 1

L

∫a(z + z′)dz′. Under the assumption of

weak gain and loss processes, the dynamics of a is obtained bythe space average of Eq. (31), which gives in the steady state

g

(1 − γ

3a2

)− l + sa(a) = 0. (33)

This equation, together with Pp = 2ka, determines the pulsepart of the gain balance.

Next, we calculate the power carried by the pedestals ofthe pulses. Each pulse generates a series of sidebands of theform given by Eq. (28). The sidebands power is dominated bythe leading n = ±1 sidebands; since the different pulses thatact as sources for the sidebands are not phase locked, we treatthem as incoherent sources, and accordingly calculate the totalsidebands power as the sum of the individual sideband powers.The resulting expression is

Pb = k

∣∣pg,1βgω1 + ps,1βsω1

∣∣2

2ω1[l + g

(γω2

1 − 1)] . (34)

Finally, the mean power of the the continuum is not changedby the presence of the sidebands, and is therefore given by theexpression derived in [30]

Pc = T tR

2√

gγ (l − g). (35)

We now substitute P = Pp + Pb + Pc into Eq. (2) andobtain a nonlinear equation for the saturated gain g, whichis easy to solve numerically. Once we know the value of g,we obtain the steady-state solution of the entire waveform.The steady-state equation can have several solutions withdifferent numbers of pulses k. In such cases, the laserwaveform can exist in several states [15,16] in a similar mannerto the existence of metastable phases in thermodynamicsystems undergoing first-order phase transitions. The laserthen exhibits hysteresis; its actual state depends on thehistory.

Figures 3 and 4 show the results of the calculation asthe small signal gain g0 is varied. For very low gain, thewaveform is pure continuum, and its energy increases withthe gain. When g0 is increased beyond a certain threshold,a pulse forms and along with it also a dispersive wave.Because of gain saturation, the net gain must abruptly decrease,and along with it also the continuum component. Furtherincrease of g0 will mainly increase the sidebands and the

0.2 0.4 0.6 0.8 1 1.20

2

4

6x 10

−3

Pea

k in

tens

ity

0.2 0.4 0.6 0.8 1 1.2

4

6

8

10x 10

−3

Wid

th

0.2 0.4 0.6 0.8 1 1.20

0.01

0.02

0.03

0.04

0.05

g0

Pb

0.2 0.4 0.6 0.8 1 1.2

0.08

0.1

0.12

g0

g

FIG. 3. (Color online) The theoretical calculated values of thesidebands peak intensity, width, and total power Pb along with thesaturated gain coefficient g as a function of the small signal gaing0. The same parameters as in Fig. 2 were used in the calculation,except that γ = 10−3 and Ps = 1. Two noise injection rate values areshown: T = 10−3 in the continuous black line and T = 3 × 10−3 inthe dashed red line. The number of pulses is indicated. All values aregiven in natural soliton units.

continuum components, and slightly change the pulse ampli-tude, until the second threshold is met; then, the continuumas well as the sidebands’ power abruptly decrease again.This process then continues periodically when further pulsecreation thresholds are reached. In addition, as the pumpingis increased between pulse creation thresholds, the saturatedgain increases, so the net loss decreases and the sidebandsbecome spectrally narrower and, accordingly, temporallywider.

The theoretical predictions of the global sideband charac-teristics agree well with experimental observation summarizedin Fig. 5, made on a fiber ring laser mode locked bypolarization rotation as described in [16]. In both graphs, thesideband width displays roughly periodic sawtooth behaviorand the total sideband energy displays a sawtooth behavior

0.4 0.8 1.20

4

8

12

g0

Pp

0.4 0.8 1.20.5

1.5

2.5

3.5

g0

Pc

5

1

2

3

4

5

4

3

2

1

FIG. 4. (Color online) The theoretical calculated values of thetotal pulse power Pp and continuum power Pc as a functionof the small signal gain g0. The same parameters are used as inFig. 3. The number of pulses is indicated. All values are given innatural soliton units.

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WEILL, BEKKER, SMULAKOVSKY, FISCHER, AND GAT PHYSICAL REVIEW A 83, 043831 (2011)

40 60 80 100 120 1400

2

4

6

8x 10

−4

peak

inte

nsity

(ar

b. u

nits

)

40 60 80 100 120 1400.05

0.1

0.15

0.2

0.25

wid

th (

nm)

40 60 80 100 120 1400

2

4

6

8x 10

−5

Pumping current (mA)

tota

l ene

rgy

(arb

. uni

ts)

40 60 80 100 120 1400

200

400

600

800

Pumping current (mA)

dist

ance

(ps

)

FIG. 5. The measured sideband spectrum characteristics: peakpower (in arbitrary units), spectral width, and total energy (in arbitraryunits), and the distance between adjacent pulses in states with two ormore pulses.

with an increasing linear bias. There is somewhat worseagreement in the peak intensity, but this quantity is sensitiveto interferometric enhancement and reduction as explainedabove.

In both theory and experiments, the spectral width of thesidebands narrows with increasing pumping between pulsecreation thresholds, implying a temporal widening of the

pedestals. This widening shows a striking correlation withthe increase of the interpulse distance, suggesting that thesidebands play a role in the interpulse interactions [12,13].

VI. CONCLUSIONS AND OUTLOOK

We presented a fundamental theory of the formation ofsidebands and their effect on the statistical steady state inmultipulse mode-locked soliton lasers, which explains themain experimental observations, including the dependence ofthe sideband spectrum on the measurement position in thecavity, the growth of the sideband energy and temporal widthwhen pumping is increased, and the abrupt attenuation of thesidebands when a new pulse is created in the cavity. We alsofound strong correlations between the temporal width of thesidebands and the spacing between adjacent pulses in pulsebunches, giving further evidence for the role of the sidebandsin interpulse interaction. However, the most natural conclusionfrom our observation is that the sidebands generate repulsiveinteractions, and that additional attractive interactions areneeded to explain the ubiquitous formation of pulse bunches.Moreover, the phase and timing jitter of the pulses leads usto conjecture that the mechanism of interaction is incoherent.We postpone the in-depth study of these questions to a futurepublication.

ACKNOWLEDGMENT

This research was supported by the Israel ScienceFoundation.

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