Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation

Transcription

1 PHYSICAL REVIEW E 80, Chirped dissipative solitons of the omplex ubi-quinti nonlinear Ginzburg-Landau equation V. L. Kalashnikov* Institut für Photonik, TU Wien, Gusshausstrasse 27/387, A-1040 Vienna, Austria Reeived 5 August 2009; published 15 Otober 2009 Approximate analytial hirped solitary pulse hirped dissipative soliton solutions of the one-dimensional omplex ubi-quinti nonlinear Ginzburg-Landau equation are obtained. These solutions are stable and highly aurate under ondition of domination of a normal dispersion over a spetral dissipation. The parametri spae of the solitons is three-dimensional, that makes theirs to be easily traeable within a whole range of the equation parameters. Saling properties of the hirped dissipative solitons are highly interesting for appliations in the field of high-energy ultrafast laser physis. DOI: /PhysRevE PACS number s : Tg, Re I. INTRODUCTION The omplex nonlinear Ginzburg-Landau equation CGLE has so wide a sope of appliations that the onept of the world of the Ginzburg-Landau equation 1 is not an exaggeration. The CGLE demonstrates its effetiveness in quantum optis, modeling of Bose-Einstein ondensation, ondensate-matter physis, study of nonequilibrium phenomena and nonlinear dynamis, quantum mehanis of selforganizing dissipative systems, and quantum field theory. In optis and laser physis, the CGLE provides an adequate desription of mode-loked osillators and pulse propagation in fibers 2,3. The CGLE is multiparameter and not integrable in a general form. As a result, an analysis of a multitude of its solutions requires extensive numerial simulations. The exat analytial solutions are known only for a few of ases, when they represent the solitary waves dissipative solitons and some algebrai relations on the parameters of equation are imposed 2,4. As a rule, one presumes some lass of funtional expressions to onstrut the solution. As a result of suh presupposition, the solutions outside a given lass are missed. In priniple, the missed solutions an be revealed on basis of the algebrai nonperturbative tehniques 4,5, whih, nevertheless, need a lot of omputer algebra. These hallenges stimulate interest in the approximate methods of integration of the CGLE. The perturbative method has allowed obtaining the dissipative soliton solutions of the redued and omplete ubiquinti CGLE in the limits of small or large dispersion and self-phase modulation SPM 6,7. Another approximate approah is to redue an infinite-dimensional in terms of degrees of freedom problem to finite-dimensional one on basis of, for instane, the method of moments. This allows traing an evolution of a finite set of the trial solution parameters 8. However, some physially interesting setors of the CGLE allow an approximation without any funtional onstraints imposed on the solution or/and the equation parameters. Moreover, dimensionality of the parametri spae orresponding to suh a solution an be redued in omparison *kalashnikov@tuwien.a.at with the parametri spae of the CGLE that makes the solution under onsideration to be easily traeable. A physially important setor, whih permits an approximate analysis, is represented by the hirped solitary pulse solutions, or the hirped dissipative solitons CDSs of the CGLE. The CDS exists in both anomalous and normal dispersion ranges It is very important that the CDS is energy salable 8,10 12 and an be onsidered as a model of femtoseond laser pulses with about of and overmirojoule energies 13,14. Energy salability of the CDS results from its strething aused by a large hirp. Hene, the CDS with large energy has a redued peak power that provides its stability 15,16. Simultaneously, a large hirp leads to spetral extra-broadening so that the CDS beomes to be ompressible down to a few of tens of femtoseonds 11,13,15. The mehanism of the CDS formation is a omposite balane of phase and dissipative effets 8. The first effet is a balane of phase ontributions from the SPM and the timedependent phase affeted by a normal dispersion 17. That is possible if a soliton is hirped, but this effet alone does not provide a soliton stability. The CDS stability an be provided by a balane between the nonlinear gain and the spetral dissipation 9,17,18. A large hirp of the CDS allows two main approximations: i soliton strething admits the adiabati approximation, and ii fast phase variation allows applying the stationary phase method in Fourier domain 15,16,20. As a result, the CDS of the redued ubi-quinti one-dimensional CGLE i.e., the CGLE with a Kerr s-type SPM and a ubiquinti nonlinear gain; see the lassifiation in 19 15,20 and the generalized one-dimensional CGLE i.e., the CGLE with a Kerr s-type SPM and a perfetly saturable loss 16 an be represented analytially as the two-parametri solitary pulse solution without any restritions on its funtional form as well as on the equation parameters ertainly, within the sope of approximations under onsideration, see below. Here, the extension of this approximate tehnique to the omplete ubi-quinti one-dimensional CGLE i.e., the CGLE with both ubi-quinti SPM and ubi-quinti nonlinear gain is presented. It is shown, that the CDS is the three-parametri solution with five types of the trunated spetral profiles: i finger-, ii paraboli-, and iii flat-top, as well as iv onave and v onave-onvex ones. The regions of existene and stability of the CDS are analyzed /2009/80 4 / The Amerian Physial Soiety

2 V. L. KALASHNIKOV PHYSICAL REVIEW E 80, systematially within a whole parametri range obeying the ondition of domination of a normal dispersion over a spetral dissipation. The obtained results are validated on basis of numerial solution of the CGLE and ompared with the existing results of extensive numerial simulations of the CGLE. II. CDS OF THE CUBIC-QUINTIC CGLE Let the CGLE be written down in the following form 15,20 : z 2 a z,t = a z,t + + i t2a z,t + i a z,t 2 a z,t + i a z,t 4 a z,t. 1 Here, z is the propagation longitudinal oordinate, whih an be the propagation distane in a fiber, or the avity round-trip in a laser osillator, for instane; t is the transverse oordinate, whih an be, for instane, the loal time for a propagating laser pulse 2. The omplex slowly varying field amplitude a z,t is hosen so that a 2 has a dimension of instant power. The first term on right-hand side of Eq. 1 desribes an ation of net-loss with the parameter. In the general ase, this parameter is energy-dependent i.e., it depends on a z,t 2 dt, see 15 and has to be positive to provide the vauum stability i.e., the subritial range of Eq. 1 is under onsideration. The seond term desribes a spetral dissipation is the squared inverse bandwidth of spetral filter and a dispersion is the dispersion oeffiient. Positivity negativity of orresponds to normal anomalous dispersion. The third term results from a ontribution of ubi nonlinearity, whih is a sum of ontributions from the nonlinear gain or the self-amplitude modulation SAM defined by the parameter 0 and from the SPM with the parameter. Only the fousing SPM with 0 will be onsidered below. A higher-order quinti nonlinearity defines the fourth term in Eq. 1. Its real part desribes the SAM saturation 0 provides stability of a desired solution against ollapse, while a orretion to the ubi SPM an be both enhaning 0 and saturating 0. To find the CDS solution, let s make the traveling wave redution in Eq. 1 by means of the ansatz a z,t = P t exp i t iqz, 2 where P t is the instant power, whih defines a CDS envelope; t is the phase, and q is the phase shift due to a slip of the arrier phase with respet to an envelope 3. Below, we shall onsider only the setor of CGLE, where a normal dispersion prevails over a spetral dissipation, that is 0 6. This assumption is well-grounded for both broadband solid-state 11,15,16 and fiber 16,21,22 laser osillators operating in the all-normal dispersion ANDi regime. But the numerial analysis demonstrates 16, that even the ase of 0 e.g., a thin-disk solid-state osillator 14 an be desribed adequately in the framework of the analytial approah under onsideration. The adiabati approximation T allows obtaining from Eqs. 1 and 2 2 = q P P 2, dp P t dt + d = P 1 P 2, dt where d t /dt is the instant frequeny. Sine the first equation in Eq. 3 is quadrati in P, there are two branhes of solution. However, it is reasonable to onfine oneself to the branh, whih has the limit 0 this limit has been onsidered in 15,20. Then, one has P = q Sine P 0 by definition, there is the maximum frequeny deviation from the arrier frequeny: 2 =q/. Thus, the seond equation in Eq. 3 and Eq. 4 lead, after some algebra 23, to + 2 A d A = dt 4 2 A A, A A A = The singularity points of Eq. 5 impose the restritions on the value = b + 4 b b a 1+ b 1+ b b b 32a b, 6 where three ontrol parameters are a /, b /, and /. These three parameters define the parametri dimensionality of the CDS. Equations 4 and 6 allow obtaining the CDS peak power has to be equal to 0 in Eq. 4 for this aim. It is onvenient to use the following normalizations: t =t / /, 2 = 2 /, 2 = 2 /, and P = P. For the dimensionless energy, one has E =E / /. Hereinafter, these normalization will be implied and the primes will be omitted. The expressions for the dimensionless quantities are

3 CHIRPED DISSIPATIVE SOLITONS OF THE COMPLEX PHYSICAL REVIEW E 80, A = , b P = b b 1, 2 = d = dt b b b b a b2 4 1 A 2 b +1 A A A b + A 1 A a b +1 b +1, 3 9b 32a b Sine the phase t of the CDS an be treated as a rapidly varying funtion of time in the limit of, one may apply the method of stationary phase to the Fourier image of a t 20. As a result, the expression for the CDS spetral profile is p e 2 A 1 A 1 b H 2 2 A A 1 a + b + b b b , 8 d 2 where e dt P t exp i t i t, H x is the Heaviside s funtion and one has to replae by in A given by Eq. 7. The CDS energy an be obtained from Eq. 8 by integration: E= p. This value an be related to the energy E of a solution of the linearized version of Eq. 1 through the saturated net-loss parameter : E/E 1, where d /de E=E 15. Suh a relation an be usable, for instane, to define the CDS parameters from those of a laser osillator 15,16. Thus, the approximate tehnique under onsideration allows representing the CDS parameters, its spetral and temporal profiles as well as energy from a few of algebrai expressions 7 and 8, single first-order ordinary differential Eq. 7 and numerial integration of Eq. 8. Sine the CDS is three-parametri, suh an approximation allows easily traing the soliton harateristis within a broad range of the CGLE parameters. It is important that the absolute values of the CGLE parameters are not relevant in ontrast to their relations presented by the parameters a, b, and. This allows a unified viewpoint at the diverse systems obeying the CGLE 16,24. III. MASTER DIAGRAM AND REGIONS OF THE CDS EXISTENCE Equation 7 demonstrates that there exist two branhes of the CDS orresponding to two signs before square root in the expression for. As will be shown below, suh a division into two branhes is physially meaningful. In aordane with the sign in Eq. 7, we shall denote these branhes as the positive + and negative ones. One has note, that only the branh has a limit for, 0 the Shrödinger limit 20. A. Positive branh of the CDS Regions of the + branh existene are shown in Fig. 1 on the plane a for the different b. Zero value of a orresponds to a marginally stable CDS. The existene regions are maximally broad in this ase. The restritions on the parameters are 0 2, and b 0 the SPM is unsaturable or b /3 4/3 the SPM is saturable. One has to resemble, that the derease of b means a growth of ontribution of the quinti SPM. The physial meaning of maximum is that there is a minimum dispersion or a maximum spetral dissipation, whih provides the CDS existene. A new view on the CDS results from a onsideration of hirp at the soliton enter i.e., at t=0 t=0 d = a b2 dt =0 4 1 A = b A =0. 9 Here, the hirp is normalized to /. The analysis demonstrates that the hirp beomes negative, when the parameter reahes some minimum value the lower borders of the hathed regions in Fig. 1. Suh a hirp orresponds to a spike on onstant bakground with lim t, lim t P=onst 0. These solutions will be not onsidered hereafter and the hirp positivity will be admitted as the additional riterion of the CDS existene. This riterion agrees with the analytial results of 10 in the limit of / 1 and

4 V. L. KALASHNIKOV PHYSICAL REVIEW E 80, FIG. 1. Regions hathed of the + branh existene. b / =20 blak solid urves, vertial hathing, 0.1 light gray dotted urves, horizontal hathing, and 2.5 gray dashed urves, horizontal hathing. / 1. The appropriate zero asymptoti lim t P=0 of the solutions analyzed in 10 exists only if 0 3 / 1 + /. Here, is defined as the parameter in the phase profile ansatz t = ln P t, whih is used in 10. As a result, there is some minimum i.e., maximum normal dispersion or minimum spetral dissipation for the + branh Fig. 1. This minimum tends to zero, when the positive b dereases Fig. 1. For b 0, the CDS existene range squeezes, when b approahes 2 Fig. 1. Ifb 2, the positively hirped CDS has a paraboli temporal profile and lim t =, lim t P=0 is some finite interval of loal time. We will not onsider suh an inverted CDS hereafter. The a growth, if it results from the inrease, enhanes the soliton stability against a vauum destabilization. However, the existene regions shrink along the parametri oordinate with suh a growth Fig. 1. FIG. 2. Borders of regions of the branh existene blak solid urve for b / =20, light gray solid urve for b=0.1, and gray solid urve for b= 2.5, points for b= 2. Open irles orrespond to the region border from Ref. 6. The existene regions lie below the orresponding borders. This means that the branhes oexist within the regions of their existene in the a,b, -parametri spae. C. Master diagram Representation of the existene regions in the form of Figs. 1 and 2 is awkward in some way, beause the a parameter an be energy-dependent. As a result, the branhes do not oexist as they differ in energy. It is more onvenient to use a representation on the plane E for the different b. Suh a representation will be alled the master diagram. The E value an be easily related to the experimentally ontrollable parameter E see Se. II. The master diagram for the CDS is shown in Fig. 3 for the ase of vanishing quinti SPM b 1 15,24. The solid urve shows the border of the CDS existene a=0. Above B. Negative branh of the CDS Regions of the branh existene are shown in Fig. 2 on the plane a for the different b. The CDS exists within the interval 0 2, whih squeezes with a. Sine this branh has a Shrödinger limit, suh a squeezing an be obtained on the basis of the perturbative method 6. Then, the existene region for b 1 is 2 4 6a/5 open irles in Fig One an see, that the approximation of 6 is quite aurate, when a 1 i.e., in the low-energy limit. The limiting a is defined by the hard exitation ondition a 1/4 6. The existene region shrinks with a growing positive quinti SPM i.e., when b 0 tends to zero and strethes with a negative quinti SPM verging toward b = 2. There are no negative hirp solutions for this branh. There exist the positive hirp solutions for 2 and b 2, but they are the spikes on bakground. One an see that the upper in the diretion borders of the regions oinide for the positive and negative branhes. FIG. 3. The master diagram for b / 1. There exists no CDS within the hathed region. Solid urve orresponds to a / =0. Dashed urve divides the regions, where the + and branhes exist. Crosses irles orrespond to the + branh for a=0.01. The points and orrespond to the parameters of the numerial solutions presented in Figs. 8 and 9 by open irles. The open squares indiate the numerially obtained stability border =0.04, =

5 CHIRPED DISSIPATIVE SOLITONS OF THE COMPLEX FIG. 4. The master diagram for b / =0.2. There exists no CDS within the hathed region. Blak solid urve orresponds to a / =0. Blak dashed urve divides the regions, where the + and branhes exist. Gray dashed dotted urve orresponds to the + branh for a=0.01. The point orresponds to the parameters of the numerial solution presented in Fig. 9 by open squares. The open squares indiate the numerially obtained stability border =0.04, =0.5. The points and orrespond to the analytial spetra presented in Fig. 12 =0.8 and 1.5, respetively; / =6.25, =0.002, =0.1 Ref. 12. The point is the branh ounterpart of. PHYSICAL REVIEW E 80, this border, the vauum of Eq. 1 is unstable hathed region. The dashed urve divides the existene regions for the + and branhes the branhes merge along this urve. Crosses irles represent the urve along whih there exists the + branh for some fixed value of a so-alled the isogain urve. The master diagram is interrelated with the existene regions in Figs. 1 and 2. The point of intersetion of isogain with the dashed urve defines the maximum value of in Figs. 1 and 2 for the orresponding a. Sine the + branh isogain has a nonzero asymptoti for E, there is the nonzero minimum, whih onfines the + branh region for a fixed a in Fig. 1. The branh has a zero asymptoti for E. Hene, the branh extends down to =0. The master diagram reveals four signifiant differenes between the branhes. The first one is that the branh has lower energy than the + branh for a fixed. The seond differene is that the + branh isogain has nonzero asymptoti for E. In this sense, the + branh is energy salable, that is its energy growth does not require a substantial hange of. The branh is not energy salable, that is its energy growth needs a substantial derease of e.g., owing to a dispersion growth 16. The third differene is that the + branh verges on =0 within a whole range of E. The fourth differene is that the branh has a Shrödinger limit, 0. Growth of the positive quinti SPM i.e., b 0 narrows the existene region Fig. 4. This means that smaller is required to provide the CDS existene for some E. That is, sine the positive quinti SPM means an enhanement of the SPM with power, a SPM enhanement has to be ompensated, for instane, by a dispersion inrease 1/. One an see from Fig. 4, that the + branh region narrows substantially with b 0 b 0 within a whole range of E. The situation is opposite, when the quinti SPM is negative. The existene range widens and a larger i.e., smaller dispersion provides the CDS existene for some E. The + branh region widens, as well. However, it is important to remember, that the range of, where the CDS with a fixed a exists, is defined by the differene between i the point of intersetion of the isogain with the boundary between the + and branhes and ii the isogain asymptoti for E. As a result, the range of, where some isogain exists, an be narrow in spite of the fat that a whole range of the + branh widens Figs. 1 and 5. The reversed situation, when a whole existene range is narrow, but the range of for some isogain is broad, is possible for b 0 Figs. 1 and 4. IV. CDS PROFILE, SPECTRUM, AND PARAMETERS FIG. 5. The master diagram for b / = 5. There exists no CDS within the hathed region. Blak solid urve orresponds to a / =0. Blak dashed urve divides the regions, where the + and branhes exist. Gray dashed dotted urve orresponds to the + branh for a=0.01. The point orresponds to the parameters of the numerial solution presented in Fig. 8 by open squares. The open squares indiate the numerially obtained stability border =0.04, =0.5. Figure 6 shows the frequeny deviations and the CDS profiles relating to the + branh see Eq. 7 for the different b. One an see, that the derease of positive b redues a soliton energy blak solid vs gray urves in Fig. 6 for the fixed and a. That agrees with Figs. 3 and 4, where the isogain shifts toward smaller energies for a fixed, when the positive b tends to zero. Sine a power dereases, a hirp d /dt dereases, too blak vs gray dashed urves in Fig. 6. In the ase of b 0, the dependene of on t beomes loitering light gray dashed urve in Fig. 6. As a onsequene, the CDS profile beomes flat-top light gray solid urve in Fig. 6. The energy inreases for a fixed a in agreement with a shift in the isogain toward larger energies in Fig. 5. The frequeny deviations and the CDS profiles for the branh are shown in Fig. 7. Out of the boundary between the branhes, the CDS relating to the branh has lower energy and power than its + ounterpart. Correspondingly, a hirp is lower, as well. Growth of the positive quinti SPM i.e., b

6 V. L. KALASHNIKOV PHYSICAL REVIEW E 80, FIG. 6. The + branh CDS profiles solid urves and frequeny deviations dashed urves for the different b /. / =1, a / =0.01. FIG. 8. The + branh CDS spetra for the different b /. / =1, a / =0.01. Solid, dotted, and dashed urves orrespond to analytial spetra. Gray irles orrespond to the numerial spetrum at the point in Fig. 3 =0.5, / =25, =0.04,E=820 / 2. Gray squares orrespond to the numerial spetrum at the point in Fig. 5 =0.5, / =25, =0.04,E=2350 / 2. 0 inreases the soliton energy, power and hirp blak vs gray urves in Fig. 7. Growth of the negative quinti SPM dereases the CDS energy, power and hirp. However, suh a derease is omparatively small and, therefore, it is not shown in Fig. 7. The CDS spetra are presented in Figs. 8 and 9. As has been shown in Setion II, the spetra are trunated at some frequeny. There are the following types of spetral profiles: i flat-top solid urve in Fig. 9, ii onvex solid urve in Fig. 8 and dotted urve in Fig. 9, iii fingerlike dotted urve in Fig. 8, and iv onave dashed urves in Figs. 8 and 9. All these types are widely presented in laser experiments and numerial simulations 11 13,15,16,21. One an see, that, as a rule, the CDS spetra relating to the + branh Fig. 8 are broader than those relating to the branh Fig. 9. The ause of this differene is a smaller hirp for the branh CDS. The spetrum narrows widens with an approah of positive b to zero for the + branh in aordane with a derease inrease of the CDS hirp. When the positive quinti SPM inreases b 0, onave spetra appear. In ontrast to the model of 22, the soure of suh spetra is not the self-amplifying SAM i.e., the negative in Eq. 1 but solely the positive quinti SPM 24.As a result, the onave spetrum solution of Se. II is stable against ollapse. It is important to note that, a verging of b toward zero for the + branh as well as a transition to the branh redue hirp. This an violate a validity of the method of stationary phase see Se. II. As a result, the spetrum edges beome smooth see Se. V. As was mentioned earlier, the + branh does not vanish along the urve of =0. This urve orresponds to marginal stability against a vauum exitation and the CDS has a broadest spetrum here. The dependene of half-width of suh a spetrum on the parameter for the different b are shown in Fig. 10. In the absene of the quinti SPM b 1, the dependene is symmetri relatively =1, where the FIG. 7. The branh CDS profiles solid urves and frequeny deviations dashed urves for the different b /. / =1, a / =0.01. FIG. 9. The branh CDS spetra for the different b /. / =1, a / =0.01. Solid, dotted, and dashed urves orrespond to the analytial spetra. Gray irles orrespond to the numerial spetra at the point in Fig. 3 two oiniding numerial profiles are defined by =0.1 ; / =30 and 40; =0.033 and ; E=280 / 2 and E=430 / 2, respetively. Gray squares orrespond to the numerial spetrum at the point in Fig. 4 =0.1, / =42, =0.024,E=600 /

7 CHIRPED DISSIPATIVE SOLITONS OF THE COMPLEX FIG. 10. The CDS spetral half-widths for the + branh and the different b / ; a / =0. spetral width is maximum. The maximum lowers rises and shifts toward =0 =2 for the positive negative b 0 Fig. 10. When b 4.5, the + branh disappears for a=0 and =1. Figure 11 demonstrates the dependenies of the spetral half-width on a for a varied b and a fixed. As a result of larger energy and hirp, the + branh solid urves has a larger spetral width, whih dereases with a and the positive b verging toward zero. The region of the + branh existene shortens with b 0 also, see Fig. 1. A negative b expands the + branh region toward a larger a. However, suh a region is disonneted with a=0, if b 4.5 for =1 see the region for b= 2.5 in Fig. 1. The existene of this minimum providing the CDS with a=0 is a result of asymptotial behavior of the zero isogain in Fig. 5. Physially, absene of the limit a 0 an mean that suh a CDS is not able to develop from the vauum of Eq. 1. Figure 11 demonstrates that the + branh disappears ompletely, when b 2. The branh has a lower spetral width, whih inreases with a Fig. 11, dashed urves. There exists some maximum a for a fixed, where both branhes merge. FIG. 11. The CDS spetral half-widths for the + solid urves and dashed urves branhes in dependene on a / for the different b / and / =1. PHYSICAL REVIEW E 80, V. NUMERICAL SIMULATION OF THE CDS The above obtained approximate solution for the CDS has to be verified numerially. With this purpose, a symmetrized split-step Fourier method is used for numerial solving of Eq. 1. The temporal greed ontains 2 16 points, and the nonlinear propagation is simulated in the time domain using a fourth-order Runge-Kutta method. Total propagation distane onsists of 10 4 steps. One step equals to one avity round-trip for an osillator or one nonlinear length for a fiber. The simulations demonstrate, that the neessary fator providing the CDS stability is a dependene of on E. Suh a dependene is hosen to be in the form presented in Se. II the parameter equals to Then, the E parameter in a master diagram an be easily replaed by the E one, but the differene between E and E is small and, therefore, insignifiant for further onsideration. The simulated spetra of the CDS are shown by open irles and squares in Figs. 8 and 9 for the parameters a, b, and orresponding to the points,,, and in Figs The agreement between the analytial and numerial results is perfet. Moreover, the numerial results demonstrate that the CDS is really three-parametri and its parameters sale in aordane with the rules of Se. II. This means that the normalized parameters and profiles of the CDSs are idential for the idential sets of a,b,. For instane, two-parametri sets: i b=20, a=0.01, / =30, =0.1, E =280 / 2, =0.033 e.g., a 100 nj Ti:sapphire osillator with =2.5 fs 2 and =4.55 MW 1 15 ; and ii / =40, E =430 / 2, =0.025 orrespond to the single point in Fig. 3. This is the branh, and the analytial solid urve as well as numerial gray open irles profiles oinide in Fig. 9. A single differene between the numerial and analytial spetra is that the former ones have gently smoothed edges. One has noted, that a salability of the CDS resembles the property of a true soliton, whih is a solution with no fixed parameters 4. The numerially obtained stability borders are shown in Figs. 3 5 by open squares. The stability ondition is 0, that provides a vauum stability. One an see that both analytial and numerial borders oinide. It is of interest to ompare the analytial results with the numerial ones presented in 10,12. There is a differene between the parametri setors onsidered in 10,12 and in this work. The ase of 0 lies beyond a validity of the analytial model under onsideration, whih requires. If approahes and then tends to zero as well as if prevails over, the spetrum edges beome smooth 12,16 rather than trunated. The saling rules of Se. II and the requirement of 2 an get broken in this ase 25. Nevertheless, i strong salability of E with, as well as both ii existene of maximum and iii minimum providing a stable soliton suggest that the solutions analyzed in 10,12 belong to the + branh here we onsider only normal dispersions. In onformity with 10, the stable CDS exists within the region of normal dispersion 0 for both positive and negative Figs. 1 and 2. A fast disappearane of the CDS with the inrease of b 0 10 is the harateristi feature of the + branh Fig. 1. As expeted, the CDS of 12 belongs to the + branh the points and in Fig. 4. The orresponding analytial spe

8 V. L. KALASHNIKOV PHYSICAL REVIEW E 80, one gray dashed urve in Fig. 12. Suh spetra have been studied numerially in 12 for. VI. CONCLUSION FIG. 12. The analytial CDS spetra for the + branh at the points =0.8 ; blak solid urve and =1.5 ; gray dashed urve in Fig. 4. b=0.2, / =6.25, =0.002, and =0.1. tra are shown in Fig. 12 p for the blak solid urve is resaled for onveniene. Both analytial spetra math with the numerial ones in Fig. 3 of 12 with the exeption of the smoothed edges for the latter owing to. Suh a smoothing enhanes for the branh the point in Fig. 4 beause a hirp is lower for this branh. When, the spetrum is onave blak solid urve in Fig. 12 like that in Fig. 8 for b=0.2. When exeeds, the new type of a spetral shape appears: the onave-onvex In onlusion, approximate hirped solitary pulse solutions of the ubi-quinti nonlinear CGLE have been onstruted analytially under ondition of domination of a dispersion over a spetral dissipation. The solutions are threeparametri and easily traeable within a whole parametri spae, whih has been represented in the form of the master diagrams. The solutions are divided into two branhes, whih differ in their energies and saling properties. It is found, that the hirped dissipative solitons under onsideration have trunated spetra with the onave, onvex, and onave-onvex tops. Numerial analysis and omparisons with the existing results have demonstrated, that the approximate analytial solutions are stable and highly aurate. The obtained results are of interest, in partiular, for a development of both solid-state and fiber laser osillators aimed to a generation of femtoseond pulses with over-mirojoule energy. ACKNOWLEDGMENTS Author thanks Boris Malomed for pointing out the solutions of Refs. 6,7. This work was supported by the Austrian Fonds zur Förderung der wissenshaftlihen Forshung FWF Projet No. P I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, N. N. Akhmediev and A. Ankiewiz, Solitons: Nonlinear Pulses and Beams Chapman and Hall, London, Few-Cyle Laser Pulse Generation and its Appliations, edited by F. X. Kärtner Springer-Verlag, Berlin, Dissipative Solitons, edited by N. N. Akhmediev and A. Ankiewiz Springer-Verlag, Berlin, Heidelberg, The Painlevé Property. One Century Later, edited by R. Conte Springer-Verlag, New York, B. A. Malomed, Physia D 29, B. A. Malomed and A. A. Nepomnyashhy, Phys. Rev. A 42, Dissipative Solitons: From Optis to Biology and Mediine, edited by N. N. Akhmediev and A. Ankiewiz Springer- Verlag, Berlin, Heidelberg, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, J. Opt. So. Am. B 8, J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, Phys. Rev. E 55, V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf and A. Apolonski, New J. Phys. 7, N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, Phys. Lett. A 372, S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, New J. Phys. 7, G. Palmer, M. Shultze, M. Siegel, M. Emons, U. Bünting, and U. Morgner, Opt. Lett. 33, V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, Appl. Phys. B: Lasers Opt. 83, V. L. Kalashnikov and A. Apolonski, Phys. Rev. A 79, V. L. Kalashnikov, A. Fernández, and A. Apolonski, Opt. Express 16, B. Protor, E. Westwig, and F. Wise, Opt. Lett. 18, A. Biswas and S. Konar, Introdution to Non-Kerr Law Optial Solitons Chapman & Hall, Boa Raton, FL, E. Podivilov and V. L. Kalashnikov, JETP Lett. 82, A. Chong, W. H. Ronninger, and F. W. Wise, J. Opt. So. Am. B 25, W. H. Renninger, A. Chong, and F. W. Wise, Phys. Rev. A 77, V. L. Kalashnikov, Maple worksheet unpublished info.tuwien.a.at/kalashnikov/ncgle2.html. 24 V. L. Kalashnikov, Pro. SPIE 7354, 73540T Perturbative analysis of the limit of,, 0 is presented in Ref. 7, where the solitonlike kink-antikink solutions of Eq. 1 have been found