J. Opt. A: Pure Appl. Opt. 9 (2007) 1149–1156 doi:10.1088/1464-4258/9/12/007
Spectral-selective management of
dissipative solitons in passive mode-locked fibre lasers
Andrey Komarov
1,2, Konstantin Komarov
2, Herv´e Leblond
1and Franc¸ois Sanchez
11Laboratoire POMA, UMR 6136, Universit´e d’Angers, 2 Bd Lavoisier, F-49000 Angers, France
2Institute of Automation and Electrometry, Russian Academy of Sciences, Academician Koptyug Prospekt 1, 630090 Novosibirsk, Russia
Received 13 July 2007, accepted for publication 30 October 2007 Published 16 November 2007
Online at
stacks.iop.org/JOptA/9/1149 AbstractWe study properties of dissipative solitons in passive mode-locked fibre lasers due to additional intracavity narrow spectral selection. Such selection allows us to control the soliton velocity, the nature of the interaction between solitons (attraction or repulsion), to create a bistable state in which the velocity of a single soliton can have two different values, to realize an elastic particle-like collision between single solitons. In the investigated laser system, the soliton interaction distance can be considerably larger than the soliton duration, which allows the realization of passive harmonic
mode-locking.
Keywords:
fiber laser, dissipative soliton, soliton interaction
1. Introduction
Passive mode-locked fibre lasers exploiting nonlinear polariza- tion rotation technique have unique potentialities. They are re- liable, compact, and can be conveniently pumped with com- mercially available semiconductor lasers. The mode-locking is realized by fast, practically inertia-free nonlinear losses, for which the depth of the modulation and the saturating intensity are easily controlled through the orientation angles of intracav- ity phase plates. The great variety of operating regimes is an important feature of this type of laser. Indeed, these lasers have demonstrated bistability between continuous wave (cw) and mode-locking regimes, spike operation and Q-switching [1–3].
They can operate either with a single pulse in the laser cav- ity or in a multiple pulse regime, in connection with the effect of quantization of intracavity radiation into individual identical solitons [2,4–6]. The repetition period of the output beam can be different from the resonator round-trip time but equal to a multiple of it [7]. Multiple pulse regimes are multistable: the number of pulses in an established operation depends on initial conditions [2,5]. The dependence of the number of pulses on pumping and on orientation angles exhibits a hysteresis phe- nomena [8]. The type of soliton interaction plays a crucial role in the established multiple pulse regime [9–12]. In the case
of pulse attraction, bound solitons structures can form. It was found that asymmetric structures of bound solitons move rela- tive to the isolated pulses [13]. If a long-distance mechanism of repulsion is present, the regime of harmonic passive mode- locking is established; then the pulses in the laser cavity are equidistant [14]. If the repulsion distance between solitons is short, bunches of pulses are formed; in this case the phase of the individual solitons are not correlated. It has been shown by means of a quintic Swift–Hohenberg model that the complex- ity of the spectral response has an important influence on the interaction [15].
In this paper we propose and analyse a powerful technique which allows us to control the type of soliton interaction inside the laser cavity. It is based on the introduction of an additional narrow spectral selection of the intracavity radiation.
In section2we present the model describing the considered laser set-up. Section 3 is devoted to the presentation of numerical results obtained by means of this model, which simulate the soliton interaction in such systems. We first demonstrate the simultaneous generation of ultrashort pulses and of cw radiation between them. The possibility of achieving long-distance soliton interaction by means of such cw radiation is investigated in a first stage. Thereafter we introduce the additional spectral selection of radiation inside the laser
Figure 1. Schematic representation of the investigated laser. The ring laser resonator consists of the fibre gain medium, the polarizer, two quarter-wave plates and one half-wave plate.α1,α2andα3are the orientation angles of the phase plates.
resonator, and investigate its effect on the soliton interaction.
It is found that the nature, attractive or repulsive, of the interaction depends on the detuning of the selector frequency with respect to the centre of the spectral gain band. The interaction distance can be considerably larger than the soliton duration, which is of great importance for the realization of harmonic passive mode-locking. The lasing regimes in both cases of soliton attraction and repulsion are studied. On the basis of numerical simulation we further show that, in the investigated systems, the soliton velocity can be a two-valued function of the selector detuning parameter, which means that individual solitons can have different velocities and can move relative to each other. As a direct consequence of this movement, we also present elastic collisions of individual solitons. Section4 is devoted to a discussion of the results obtained.
2. Laser equations
We consider a fibre laser in which the passive mode-locking is realized through nonlinear polarization rotation for an intracavity light wave. In our investigation we use typical parameters for an ytterbium-doped fibre laser. Figure1shows the studied laser system, which is described in detail in [5]. For isotropic fibres this scheme involves all necessary elements for the control of nonlinear losses. After the polarizing isolator the electric field has a linear polarization. Such a state of polarization does not experience polarization rotation in the fibre because the rotation angle is proportional to the area of the polarization ellipse. Consequently, it is necessary to place a quarter-wave plate 3 (α3represents the orientation angle of one eigenaxis of the plate with respect to the laboratory frame). The rotation of the polarization ellipse resulting from the optical Kerr nonlinearity is proportional to the light intensity, the area of the polarization ellipse and the fibre length. At the output of the fibre, the direction of the elliptical polarization of the central part of the pulse can be rotated towards the passing axis of the polarizer by the half-wave plate 2 (the orientation angle isα2). Then this elliptical polarization can be transformed into a linear one by the quarter-wave plate 1 (the orientation angle is α1). In this situation the losses for the central part of the pulse are minimum while the wings undergo strong losses.
The resulting model assumes that the nonlinear losses, due to the Kerr nonlinearity combined with the phase plates and
the polarizer, are localized, while the gain and group velocity dispersion are distributed along the fibre (full details can be found in [5]). In dimensionless form, the final set of equations governing the evolution of the electric field amplitudeE(ζ, τ) is
∂E
∂ζ =(Dr+iDi)∂2E
∂τ2 +(G+iq|E|2)E, (1) En+1(τ)= −η[cos(p In+α)cos(α1−α3)
+i sin(p In+α)sin(α1+α3)]En(τ) (2) τis a time coordinate expressed in unitsδt=√
β2L/2, where β2 is the second-order group-velocity dispersion of the fibre and L is the fibre length. ζ is the normalized propagation distance (ζvaries from 0 to 1 in a round-trip).nis the discrete number of passes of the field through the laser resonator. Dr
and Di are the frequency dispersions for the gain–loss and for the refractive index, respectively, andq is proportional to the second-order nonlinear refractive index. In the numerical simulations we will consider that the dispersionDris constant.
This assumption is correct if the changes in the gain are small or if some additional spectral selector (which is not related to the narrow band selection used for the soliton interaction management) decreases considerably the frequency width of the intracavity radiation.
The termGin the second parenthesis in equation (1) is the saturable gain. The saturation is determined by the total energy of the intracavity radiation, according to
G= a
1+b
|E|2dτ, (3) where the integration is carried out on the whole round-trip period,ais the pumping parameter andbis the saturation one.
The second term in this parenthesis is connected with the Kerr nonlinearity of the fibre, as mentioned above. Equation (2) determines the relation between the time distribution of the field before and after itsnth pass through the polarizer.ηis the transmission coefficient of the intracavity polarizer. The values α1,α3andα2are orientation angles of the quarter-wave plates 1, 3 and of the half-wave plate 2, respectively. The parameters α,Iand pare determined by the relationsα=2α2−α1−α3, I = |E|2andp=sin(2α3)/3. The amplitudeE(τ)is subject to periodic boundary conditions with a period equal to the round-trip one.
When the additional spectral selection is introduced, equation (1) is replaced with [12,17]
∂E
∂ζ =D+N, (4)
in which the nonlinear part is
N =(G+iq|E|2)E, (5) and the dispersion part is written in the Fourier domain as
Dˆ=
−k2(Dr+iDi)+h
1
1+ 2(k−k0)2 −1
E(k),ˆ (6) where E(k)ˆ is the Fourier transform of the electric field amplitudeE.
For each round-trip, equation (1) or (4) is solved by means of a standard split-step Fourier method [16]. Then the field
Figure 2. Net spectral gain–loss profile with (solid line) and without (dashed line) additional frequency-dependent narrow band loss.
E(τ)is transformed according to equation (2) which accounts for the nonlinear losses in the polarizer, and the result being used as initial data for the next round-trip.
The additional spectral-dependent loss included in equation (6) is determined by a Lorentz profile: is the inverse half-width of the profile and his its height. For the spectral component with the wavevectork = k0, the additional losses are zero. They increase up toh when the wavevectork goes away from k0. The resulting spectral profile for the gain–
lossσ =σ(k)is described by the real part of the expression in the square brackets in equation (6) and it is presented in figure 2. With k0 = 0 this spectral profile is asymmetric.
The technical realization of the optical device achieving this filtering is outside the scope of this paper. This analysis does not take into account any accompanying phase shift from the spectral filter. The role of the latter is discussed in section4.
Because of technical difficulties, only a small part of the resonator length is involved in the numerical modelling. This part is considerably smaller than the real resonator length. We assume that all the ultrashort pulses present in the cavity are located inside the computation box, and that any other part of the cavity is filled by cw radiation only. The computation box is large enough so that interaction between a short pulse and its replica due to periodic boundary conditions can be neglected. The trapping of the radiation emitted by a pulse in the computation box is of no importance since the system is not conservative. On the other hand, interaction between
solitons depends mainly on the distance between them and on their parameters, so that the size of the computation box has little effect. This statement has been checked by numerical experiments with different box lengths. The remaining consequences of this technical point are taken into account in the interpretation.
3. Results of numerical simulation
Numerical simulation has been performed for typical parameters of a Yb-doped fibre laser with a normal group velocity dispersion [3]: Di = −0.5,q = 0.3,b = 0.017.
Figure3demonstrates a remarkable lasing regime: the stable coexistence of a stationary ultrashort pulse and cw radiation in the cavity. The figure shows both the intensity I = |E(τ)|2 and the spectrum densityIk = | ˆE(k)|2in the cavity. Here the additional spectral selection is absent. The sharp high peak in the spectral distribution corresponds to the cw component of the field. The stabilization of this combination of a low- amplitude cw radiation and an ultrashort pulse is due to the following properties of the nonlinear losses: (i) for a weak field, the nonlinear losses increase with increasing intensity. As a result the birth of pulses from the cw component of the field is suppressed. (ii) For moderate intensities, the nonlinear losses decrease with increasing intensity. This avoids the spreading out of the short pulses into the cw radiation. (iii) For large intensity the nonlinear losses increase again with increasing intensity. This circumstance prevents the short pulses from collapsing. This way the simultaneous generation of ultrashort pulse and cw component of the radiation is realized. In the investigated case the intensity-dependent losses are determined by equation (2). A dependence of the nonlinear losses with respect to the intensity possessing the required properties (i), (ii) and (iii) is achieved by means of an adequate choice of the orientation angles of the phase platesα1,α2andα3.
The cw component may act as very long soliton wings with a length comparable to that of the resonator. This offers an opportunity of achieving long-range interactions between solitons. However, our attempt to perform such long-distance interaction of intracavity solitons failed. Figure4shows the coexistence of two solitons and cw field component between them in an established regime. Although the solitons contact
(a) (b)
Figure 3. Coexistence of stationary single soliton and cw radiation in established operation (hybrid passive mode-locking). (a) Field distribution in laser cavity. (b) Spectrum distribution. a=1.8, Dr=0.2,α= −0.1, α1= −1.9, α3=0.2. Any additional spectral selection is absent (h=0).
Figure 4. Two-soliton hybrid passive mode-locking. The pulses do not move relative to each other (additional spectral selection is absent). a=2.6; other parameters are the same as in the case of figure3.
Figure 5. Repulsion of solitons in a laser cavity with additional spectral selection. The two solitons move in the ring laser (the point 102.4 is identical to the point 0). Initially they are close together (white pulses). After some transient process, the pulses in the ring laser cavity are equidistant (grey pulses). a=3, h=0.2, =20π, k0=0.2. Other parameters are the same as in figure3.
through cw component, they do not move relative to each other any more. Small oscillating structures in figure4move toward the solitons. The cw component manifests itself rather as an independent fragment of intracavity radiation than as wings of solitons. A further independent structure element in the intracavity radiation is observed: the gap which can be seen
Figure 7. The distance between two repulsing solitonsδτas a function of the number of resonator passesζ. The length of the computation box is equal to 409.6. a=2.8, =100π(solid curve),
=20π(dashed curve); other parameters and initial conditions are the same as in figure5.
betweenτ =80 and 100 is stable and hence can be considered as a dark soliton. It is asymmetric and moves between the bright solitons, refracting successively from the right edge of the right pulse and from the left edge of the left pulse (the field in the cavity is subject to periodic boundary conditions).
The picture of soliton interaction is drastically changed if the spectral narrow-band selection of radiation in the cavity is added. In this case an efficient interaction between solitons occurs, which is either attractive or repulsive depending on the detuning, namely on the selector frequency detuningk0
from the centre of the spectral gain band. Figure 5 shows the repulsion of solitons in the laser cavity. The initial data consists of two peaks, which are not solitons: their frequency chirp is smaller. The evolution leads to an enlargement of the spectrum which, due to finite gain bandwidth, reduces the peak intensity. At the same time, repulsive interaction occurs and, in the stationary regime, two pulses are located at opposite points of the computation ring. Figure 6models the realization of harmonic passive mode-locking (the operation with equidistant intervals between pulses in the cavity) due to the repulsion of solitons. During the process, the level of the intensity between solitons is about 10−4times the peak soliton intensity. Figure7 shows the increase of the distance between two pulses due to their repulsion. The final distance depends on the bandwidth of
(a) (b)
Figure 6. Modelling of the regime of harmonic passive mode-locking due to soliton repulsion. (a) Initial distribution of radiation in the ring cavity. (b) Final stationary distribution. a=4.65; other parameters are the same as in figure5.
(a) (b)
Figure 8. Attraction of solitons. (a) The figure shows the initial distribution of the field. (b) Because of the attraction between pulses the regime of bound solitons is achieved after a transient process. The detuning is k0=0.3. The other parameters are the same as in figure5.
Figure 9. Particle-like elastic collision of two solitons. The distribution of the field in the cavity I =I(τ )for different numbers ζ of round-trips in the resonator. a=2.9, =200π, k0=0.3. The other parameters are the same as in figure5.
the additional spectral selection. For =100π(solid curve) the final distance between the pulses is half of the length of the computation box. For a wider spectral selector (dashed curve), it is smaller than this value. In this computation the final distance between solitons is of the order of 100 soliton durations.
Figure8shows a long-distance attraction between pulses.
The laser parameters are the same as in figure6. The difference is only in the frequency detuning k0. The initial pulses are located at opposite points of the numerical resonator ring.
Because of the attraction, these pulses approach each other and, after some transient process, the solitons become bounded.
Between the two pulses of the bound state appears a structure oscillating in space and constant in time. This structure is shown in figure 8(b) on an enlarged scale. The level of the intensity of the cw wave is here of the order of 10−5times the peak soliton intensity.
For certain laser parameters the soliton interaction is elastic, particle-like, with exchange of momentum. Figure9 demonstrates this type of interaction of solitons. Before the collision the soliton on the left is moving while the one on the right is at rest. The level of the intensity between the solitons is of the order of 10−3times the peak soliton intensity. After
Figure 10. Dependence of the change in the soliton velocityδvon the frequency detuning k0of the additional narrow spectral selector.
Arrows←−−→and−→←−identify the regions of k0where the interaction is either repulsive or attractive, respectively.
α1= −1.7, α3=0.2, α= −0.1, Dr=0.2, h=0.2, =20π, a=1.8. The pulse interaction in the two-soliton operation was investigated with a=2.6.
the elastic collision, the solitons exchange their velocities, as do particles under head-on elastic collision. This figure shows the possibility for an individual soliton to have two different values of the velocity with the same laser parameters. This is related to a small difference in the central frequency of the two solitons.
Figure10shows the variations of the soliton velocityδv with respect to the frequency detuningk0. The value δv = dτ/dζ is defined as the ratio of the additional shift dτ of a pulse along the round-trip period, due to the narrow selector, to the corresponding variation of the normalized propagation distance dζ. The dependenceδv = δv(k0) with increasing k0 shows damped oscillations. For k0 > 0.6 the spectral transmission band of the selector goes out from the pulse spectrum. In this case all spectral components of radiation experience the same losses. No asymmetric deformation of the pulse spectrum arises. Accordingly, the change in the velocity disappears. The velocity dependence δv(k0) is an odd functionδv(−k0) = −δv(k0). Increasing the pumping rate a, stable two-pulse operation is realized. Initial pulses have Gaussian form. The distance between them was about two or three durations of the final stable pulse. Repulsion
Figure 11. Spectrum of a single pulse in established operation.
a=1.8, k0=0.2; other parameters are the same as in figure10.
between pulses arises whenk0belongs to the domains marked by divergent arrows in figure 10. Notice that, fora = 2.6, the parameters of each pulse were the same as in the case of single-pulse operation with a = 1.8. In the regions marked by convergent arrows, pulse attraction is achieved, which yields a stable bound soliton regime (see figure8). We check numerically that the character of the soliton interaction (attraction or repulsion) is not changed when changing the sign ofk0. This follows from the invariance of the model in the transformk−→ −k,k0−→ −k0, and the spectral symmetry of the soliton in the absence of the additional spectral selector.
For certain values of the laser parameters, passive mode- locking becomes unstable for smallk0[12].
The numerical simulation shows that, for the same values of the laser parameters as used in figure 10, and for small distances between solitons, the attraction is always realized.
We performed the following numerical experiment. An initial valuek0was chosen in the region of attraction (see figure10).
After a transient process the regime of bound solitons was established (of the type shown in figure 8). Then the value ofk0was changed from the region of attraction to the region of repulsion. However, the bound soliton regime was retained.
For certain values ofk0 belonging to the repulsion region, a structure oscillating in time appears between the two pulses.
Thus, fork0belonging to the repulsion region the bistability is realized: the distance between solitons in the final stationary regime depends on the initial distance between them. This situation presents some analogy with the interaction of protons in an atomic nucleus. If the distance between protons is small then the bound state of protons is realized because of the short-distance attraction due to the strong interaction. In the opposite case the protons fly apart because of the long-distance Coulomb repulsion.
Figure11presents the spectrum of a single pulse Ik =
| ˆE(k)|2, and figure12the analogous spectrum in bistable two- pulse operation. The narrow peak in the spectrum corresponds to the maximum of transmission of the narrow band spectral selector. In the spectrum is seen a broadband structure, which is strongly influenced by the narrow band spectral peak due to nonlinear effects. Changing k0 modifies the spectrum: however, it does not induce any visible change in the time dependenceI = I(τ). When the broadband spectral structure displaces in accordance with the narrow band peak position, the average frequency of the pulse oscillates. Indeed,
Figure 12. Spectra of single stationary pulses corresponding to the bistable operation with k0=0.4. Parameters are the same as in figure10, except Dr =0.1.
Figure 13. Dependence of the change in the soliton velocityδv versus the frequency detuning of the additional narrow band spectral selector k0. Other parameters of the laser system are the same as in figure12.
the location of the maxima and minima of the broadband spectral structure in the soliton spectrum influences the value of the average frequency of the pulse. A shift of the selector frequency k0 induces a corresponding shift of the oscillating broadband spectral structure. In accordance with the frequency dispersion of group velocity, such a shift of the oscillating broadband spectral structure induces the oscillating dependence of the pulse velocity shown in figure10.
We also investigated the change in the dependenceδv= δv(k0) with respect to the dispersion of gain–loss. The dependence shown in figure 10 was obtained with Dr = 0.2. The dependence corresponding to a wider spectral gain bandwidth, namelyDr =0.1, is presented in figure13. From the increase of the gain bandwidth, we observe a qualitative change in the behaviour: bistability arises. With the same laser parameters the pulse velocity can take two different values depending on the initial conditions. Spectra of these pulses for k0 = 0.4 are given in figure 12: it is seen that the central frequency of the two pulses are different, which leads to different pulse velocities. With two pulses in the laser cavity, and in domains of velocity bistability, we have observed different regimes. Both pulses can have the same velocity equal to one or the other value. The velocities of these pulses can differ, then the pulses move relative to each other.
Generally the peak amplitudes of these pulses differ, because of the difference in their central frequencies, since they are
submitted to the spectral dispersion of gains and losses. For a small difference in the peak amplitudes, we have observed elastic collisions of the pulses. For a larger difference in the amplitudes, the weakest pulse is absorbed by the strongest one. The resulting pulse then falls into chaotic dynamics, i.e. its amplitude varies randomly from one round-trip to the other [7,20].
We also investigated the phase relation for amplitudes of interacting solitons. During the transient process the phase differences between the interacting solitons are not conserved, whatever their initial value is. In the case of the pulse attraction shown in figure 8, during the transient process, the phase difference between the left and right pulses increases monotonically. It becomes 3π/2 in the final stable regime. In the case of the soliton repulsion shown in figure5, during the transient process the phase difference between the left and right pulses decreases monotonically. In the final stable stationary regime it is equal to zero. With the same laser parameters, but closer initial pulses, a bound soliton regime is obtained. In this case too, any initial phase difference becomes zero after a transient process. Hence the numerical computations show that the phase difference in the initial pulses does not influence the type (either attractive or repulsive) of pulse interaction. The type of the interaction is determined by parameters of the laser system, but not by the initial phases of the pulses.
4. Discussion
Using the model yielded by equations (1)–(3), which does not involve any additional spectral selection, we observed only attraction of dissipative solitons, realized with certain laser parameters. A direct consequence of this attraction was the emergence of bound solitons. We obtained symmetric and asymmetric bound solitons, moving relative to each other.
However, it was not possible to obtain a regime of repulsing solitons. The interaction of solitons is determined by phase relations of their wings which in their turn are determined by fine mechanisms of nonlinear self-action of ultrashort pulses in the laser resonator. The control of these phase relations due to the nonlinear self-action is a complicated task. We used another approach, connected with the formation of soliton wings by means of a spectral selection of intracavity radiation.
The additional narrow spectral band selector induces a narrow peak in the spectrum of an individual pulse (see figure11). Such a peak implies the production of long-distance wings of the pulse. Indeed, from the uncertainty relation the wing length δτ is inversely proportional to the peak spectral bandwidth δk (i.e.δτ ∝ 1/δk). As a result, the mechanism of long-distance soliton interaction is achieved. The phase of a spectral component of an individual soliton periodically evolves when increasing its wavenumber k. The frequency filter selects the spectral component with the number k0. Its phase and frequency determines the phase and the frequency of the soliton wings. Accordingly, the phase relations and the type of soliton interaction (attraction or repulsion) also oscillate when increasing k0, as seen from figure10. In the case of figure13, the number of oscillations of the velocityδv is two times higher than in figure 10. Notice that the pulse length in the former case (figure 13) is approximately twice its value in the latter (figure10), while the spectrum width is
approximately the same. That is, the number of oscillations is proportional to the pulse length if other conditions are identical.
From figure7it is seen that, for a narrower band selector (solid curve), the separation distance between pulses in the stationary state is larger than for a wider band (dashed curve).
The physical mechanism of this feature is as follows. A narrower band selector induces a narrower spectral peak in the spectrum of the individual pulse (see figure11), and hence longer soliton wings, and, accordingly, longer-distance soliton repulsion. The phenomenon of soliton repulsion is of interest for the achievement of harmonic passive mode-locking, as presented in figure 6. To increase the interaction distance, in order to obtain harmonic mode-locking, it is necessary to decrease the spectral selective bandwidth. To increase the interaction strength it is necessary to increase the wing intensity. This can be reached by increasing the spectral selective contrast, i.e. the parameterh in equation (6). The possibility of achieving harmonic passive mode-locking by means of an additional intracavity narrow spectral selector is a very important result, because it opens up a new way to construct ultrashort pulse sources with high repetition- rate, which is in turn a key element in high-speed optical communication [18].
The bistability found, realized for certain values of k0
(see figure 12), allows us to explain an elastic collision of dissipative solitons which arises for certain values of the parameters of the laser system (figure 9). Indeed, let us consider a two-pulse regime for which the pulses are situated in one of each of the two different stable states (it is bistable).
Accordingly, the pulses have different velocities and move relative to each other. Let us further assume that the pulses are approaching each other. When the distance between them becomes small enough, the pulses are deformed. Such a perturbation, which increases with decreasing separation distance, eventually transfers them into alternative states, due to bistability. When such a transfer occurs, the pulses exchange their parameters, and among them, their velocities. After that, the initially approaching pulses start moving away. Such a behaviour is shown as an elastic head-on collision of particles, in which they exchange their velocity and momentum.
It should be pointed out that the coexistence of cw radiation and ultrashort pulses in a passive mode-locked laser has been experimentally observed by the authors of the paper [19]. In this reference experimental spectra presenting a narrow band peak due to an additional frequency selection, as in figure11, were also presented. The broadband structure in the spectrum, as seen in figure12, was experimentally detected in [20].
In our previous analysis any phase shift caused by the narrow spectral filter was dropped. In additional numerical experiments taking such a phase shift into account, we have observed the same phenomena as above. The corresponding term for equation (6) is determined from a Kramers–Kronig relation through a Lorentzian amplitude function in this equation. The Kramers–Kronig relations hold for both gain and loss media. Usually the amplitude and phase transfer functions are determined from the computation of the radiation transmitted through the filter considered. Questions about optimization of filters and additional phenomena arising from filter phase shift call for further investigation.
5. Conclusion
We have proposed a way to control the interaction of dissipative solitons in fibre lasers. It is based on an additional narrow spectral selection of intracavity radiation.
Such selection allows us to realize the long distance wings of intracavity dissipative solitons with control of both their phases and their frequencies. As an important result, the type of interaction (attraction or repulsion) can be managed.
The interaction type depends on the detuning of the central frequency of the selector from the centre of the spectral gain band. The elastic collision of single solitons is found in lasers with a wide spectral gain bandwidth. Bistability has been shown, connected with the dependence of the pulse velocity with respect to initial condition. This bistability allows us to explain different velocities of individual pulses and an elastic collision of individual dissipative solitons. We presented the qualitative picture of peculiarities of an interaction of pulses in the investigated laser system. Among these peculiarities, there arises the relation between the spectral bandwidth of the selector transmission and the distance of interaction between pulses, the dependence of the soliton velocity with respect to the frequency detuning of the selector transmission from the centre of a gain band, and so on. The results obtained are of great interest to control the operating regime of fibre lasers through the control of the interaction of intracavity solitons, among which is the harmonic passive mode-locking. They can also be of importance to control the interaction of ultrashort pulses in fibre communications lines.
Acknowledgment
This research was supported by a Marie Curie International Fellowship within the 6th European Community Framework Programme (N 039942-PMLFL).
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