Passive harmonic mode-locking in a fiber laser with nonlinear polarization rotation
Andrey Komarov
1, Herve´ Leblond
*, Franc¸ois Sanchez
Laboratoire POMA, UMR 6136, Universite´ d’Angers, 2 Boulevard Lavoisier, 49000 Angers, France Received 2 February 2006; received in revised form 22 May 2006; accepted 7 June 2006
Abstract
On basis of numerical simulation and analytical treatment, we investigate the multiple pulse passive mode-locking of a fiber laser with nonlinear loss due to the nonlinear polarization rotation technique. Various additional mechanisms resulting in the ordering of ultrashort pulses inside the laser resonator and in the realization of harmonic passive mode-locking are analyzed. Among which are the active har- monic modulation of the intracavity loss or of the refractive index, the passive modulation of the index due to the inertial properties of intracavity elements, and the amplification modulation due to the depletion and the relaxation of the gain. The velocity of pulses relative to each other is determined, it results in attractive or repulsive pulse–pulse interaction, and allows to evaluate the conditions of ordering of the pulses inside the cavity.
2006 Elsevier B.V. All rights reserved.
PACS: 42.60.Fc; 42.65.Sf
1. Introduction
The development of ultrashort optical pulse sources with high-repetition-rate is a key element in high-speed optical communication[1,2]. Such high-repetition-rate pulse sources can be created on basis of harmonic passive mode- locked fiber lasers with the nonlinear polarization rotation technique. These lasers are reliable, compact, and can be pumped with commercially available semiconductor lasers.
The lasing regimes are determined by the loss nonlinearity and the depth of the loss modulation which are easily con- trolled through the orientation angles of intracavity phase plates. The multiple pulse passive mode-locking is a usual operating regime for these lasers [3]. In the case of har- monic passive mode-locking, the distances between all neighboring pulses in the intracavity pulse train are the
same. As a consequence such lasers produce high-repeti- tion-rate pulse train.
The mechanism of multiple pulse passive mode-locking with nonlinear polarization technique in fiber lasers is related to the decrease of nonlinear losses with increasing light intensity, in circumstances where the gain spectral bandwidth is finite and the generated pulses have a consid- erable frequency chirp related to the nonlinear refractive index. Generally, under multiple pulse passive mode-lock- ing, the pulses are randomly spaced in the laser cavity but well separated (bunch states). The mechanisms, which can organize the pulses within the cavity are weak in com- parison with the mechanism responsible for passive mode- locking. To arrange the pulses within the cavity, active techniques have been used, as amplitude or phase modula- tion[2,4]. However, this arrangement and the realization of harmonic passive mode-locking can arise spontaneously without any active control[5]. Various explanations have been suggested for such spontaneous harmonic passive mode-locking: acousto-optic modulation of the refractive index, interaction between pulses through continuum field, gain depletion and so on. It has been shown that
0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2006.06.012
* Corresponding author. Tel.: +33 241 735 386; fax: +33 241 735 216.
E-mail address:herve.leblond@univ-angers.fr(H. Leblond).
1 Permanent address: Institute of Automation and Electrometry, Russian Academy of Sciences, Acad. Koptyug Pr., 1, 630090 Novosibirsk, Russia.
www.elsevier.com/locate/optcom
acoustically induced pulse interaction can lead to regularly spaced bunches of pulses[6]. The deep investigation of the mechanisms that produce the ordering of the pulses in the cavity is of great current interest for both identification of reasons for harmonic passive mode-locking and develop- ment of methods of controlling the high repetition rate laser operation. The aim of this paper is to develop a simple theoretical model allowing to describe the dynamics and the organization of harmonic passive mode-locking in fiber lasers. Our model takes into account the optical Kerr non- linearity, the group-velocity dispersion, the saturable gain and the orientation of the intracavity phase plates. We ana- lyze various mechanisms which can be responsible for self- organization of pulses in the cavity: additional active amplitude and phase modulation of intracavity field, deple- tion and relaxation of the gain, inertial nonlinear refractive index with finite relaxation time and other. It is assumed that these additional mechanisms organizing pulses in the cavity are weak in comparison with the ones which are responsible for the mode-locking. Thus, ordering of pulses in the cavity can be studied through a perturbative approach. The inertial nonlinear refractive index and losses, which can be responsible for the pulse ordering are introduced phenomenologically in the theory. Concrete mechanisms responsible for these nonlinearities can be refined and complemented in future investigations. Our study has been performed on the basis of numerical simu- lation, analytical methods, and numerical estimates.
Although the influence of noise and the problem of the jit- ter are of great interest for practical applications, they will not be investigated in this paper essentially because it is out of the scope of this work. Indeed, such important problems deserve specific studies but they must be considered in a second step after the harmonic mode-locking is well under- stood and correctly modelled. The paper is organized as follows. In Section2, we present the setup and a numerical model, recall the previous results concerning the multiple pulsing behavior[7], and treat numerically the case of the active modulation. Section3is devoted to the proper aim of the paper, which is the case of a passive modulation.
Numerical simulations are first given, then we investigate analytically the nature of the interaction between pulses.
This allows to state about the conditions for pulse ordering inside the cavity.
2. Multiple pulse operation and active modulation
In the present section, we describe the setup and a numerical model. We recall briefly some previous results, and then study numerically the active harmonic modula- tion of the passive mode-locking.
2.1. Investigated model
The setup under investigation is shown inFig. 1 [8]. The passive mode-locking is realized through the nonlinear rotation of the polarization of the intracavity light wave.
In our computations we use parameters for an ytterbium- doped fiber laser [7]. For isotropic fibers this scheme involves all necessary elements for the control of nonlinear losses. After the polarizing isolator the electric field has a linear polarization. Such state of polarization does not experience polarization rotation in the fiber because the rotation angle is proportional to the area of the polariza- tion ellipse. Consequently, it is necessary to place a quarter wave plate 3 (a3 represents 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 inten- sity, the area of the polarization ellipse and the fiber length.
At the output of the fiber, 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 is a2). Then this elliptical polarization can be transformed into a linear one by the quarter wave plate 1 (the orientation angle isa1). In this sit- uation the losses for the central part of the pulse are min- imum while the wings undergo strong losses.
The normalized evolution equations for the field have the form[7]
oE
of¼ ðDrþiDiÞo2E
os2þ ðGþxþiqjEj2ÞE; ð1Þ Enþ1ðsÞ ¼ g½cosðp0Inþa0Þcosða1a3Þ
þi sinðp0Inþa0Þsinða1þa3ÞEnðsÞ; ð2Þ where E, s, f=z/L (L being the cavity length) are the dimensionless field amplitude, time coordinate and space one, respectively.Diis the group-velocity dispersion,Dris the frequency dispersion for the gain–loss,qis the normal- ized Kerr nonlinearity, g is the transmission coefficient of the polarizer, a0= 2a2a1a3, I=jEj2, p0= sin(2a3)/3.
The variableGis the saturable gain G¼ a
1þbR
Ids; ð3Þ
whereais the pumping parameter andbthe saturation one.
The saturation is determined by the total intracavity radi- ation energy according to the integration in Eq.(3), which is carried out on the whole round-trip period Ta. In our approach, the gain modulation during one cavity round- trip is ignored and all fragments of the pulse undergo the same amplification. In addition, using Eq. (3) we ignore
Polarizing Isolator
4 λ
4 λ
2 λ α1 α2
α3
Pump power
Fig. 1. Schematic representation of the laser setup.
the effects inducing the spike lasing operation. Depending on the specific mechanism of pulse ordering in the cavity under investigation, the variable x describes either the active modulation of loss or refractive index, or the modu- lation of the gain due to its depletion and relaxation, etc.
Eq. (1) is used for an analysis of the field evolution in a fiber with dispersion, gain, and Kerr nonlinearity. Eq.(2) describes the change of the field in the polarizer due to the joint action of the nonlinear polarization rotation in the fiber and the intracavity phase plates.
2.2. Previous numerical results: multistability
The numerical procedure starts from the evaluation of the electric field after passing through the Kerr medium, the phase plates and the polarizer, using Eq.(2). The result- ing electric field is then used as the input field to solve Eq.
(1)over the fiber lengthL, using a standard split-step Fou- rier algorithm. The computed output field is used as the new input for Eq.(2). This iterative procedure is repeated until a steady-state is achieved.
Multiple pulse operation in this system has been studied in detail in[7]. It has been shown that, when the pumping rate is increased, the number of pulses in the cavity increases. The dependence of the number of pulses relative to the pumping parameter a exhibits a strong hysteresis (Fig. 2). In the frame of the model Eqs.(1)–(3), the arrange- ment of the pulses in the cavity is random, as shown in Fig. 3.
2.3. Pulse ordering through active modulation
We show now that the ordering of pulses in the cavity can be achieved by means of an active amplitude or phase modulation. Such a modulation is accounted for by defin- ing the variablexinvolved by Eq.(1) as
x¼hrmaxsin 2ps Tm
; ð4Þ
whereh= 1 for the amplitude modulation andh= i (imag- inary unit) for the phase modulation,rmax is the depth of modulation, Tm is the modulation period. Harmonic mode-locking requiresTm=Ta/nwherenis an integer.
Fig. 4 shows the steady-state operation in the case of a weak additional time-dependent harmonic loss for the ordering of the pulses in the cavity. During a transient process the ultrashort pulses drift to the domains with minimum losses. In the established operation the pulses arrange themselves in these domains and they have the same parameters: the duration, the frequency chirp, the form, the peak intensity.
Notice that the mode-locking is achieved by means of a passive process, only the ordering uses active modulation.
Indeed, the loss modulation is weak and is not sufficient to achieve mode-locking. This is demonstrated in Fig. 5, in which the nonlinear loss has been suppressed, but the active modulation maintained: a continuous behavior is observed.
As mentioned above, the number of pulses in the cavity can be adjusted by varying the pumping parametera. In this case, some of the time domains with minimum losses are occupied by stationary pulses, others are free from pulses, as can be seen inFig. 4. The established states in the form of the train of pulses and free domains are stable.
The arrangement of pulses and free domains in the estab- lished intracavity train depends on initial conditions. This means that the operating regime of the laser is multistable.
Fig. 2. Multihysteresis dependence of the numberNof pulses in steady- state operation with respect to the pumping parametera. The parameters arermax= 0.01,a0=1,a1= 0.2,a3= 0.2,Di=1,Dr= 0.2.
100 300 500 1
2
3
0 3 6
0 3 6
0.6 1.5
2.5 τ
a I
Fig. 3. Distribution of the intensityIinside the cavity for steady-state operation versus the pumping parametera, without any active or passive mechanism of pulse ordering. The parameters are the same as inFig. 2.
100 300 500 1
2
3
0 3 6
0 3 6
0.6
1.5
2.5 τ
a I
Fig. 4. Active harmonic ordering of the pulses in the cavity. The same as Fig. 3, but with active harmonic modulation.
The case of the active phase modulation has also been considered, the results are analogous to that described above in the case of active gain modulation. Phase modu- lation corresponds to h= i in Eq. (4). We observed in our numerical simulations the ordering of pulses in the cavity.
Therefore, our results demonstrate that a small active external modulation of the amplitude or of the phase can be responsible for the pulse ordering in the cavity.
3. Attractive or repulsive pulse interaction
A very interesting question in the physics of harmonic passive mode-locking is related to the pulse interaction.
This point can be studied through the velocity of the move- ment of pulses in the cavity relative to each other. Such a mechanism has been pointed out in[11], from a rather phe- nomenological point of view. We study it here from an ana- lytical point of view, in the framework of the complex Ginzburg–Landau (CGL) equation, which is able to account for various types of laser setups. In order to solve this task we use approximations based on the following remark: The mechanisms of pulse ordering influence very weakly the pulse characteristics, as the duration, the form, the peak intensity, the frequency chirp. Without such mechanisms the system is in an indifferently stable equilib- rium state, with respect to the change of distances between pulses. As a result, weak mechanisms can order the locali- zation of ultrashort pulses within the cavity.
Before we develop the analytical theory, let us give some numerical results, established in the frame of the model presented in Section2(Eqs. (1)–(3)).
3.1. Passive modulation: numerical results
We consider now the intracavity ordering of pulses resulting from the passive modulation of either the net gain or the refractive index. In both cases, the passive modula- tion is taken into account using the evolution equation
sr
oX
osþ ðX h~aÞ ¼ X~bI; ð5Þ
wheresris the relaxation time,~bis the saturation parame- ter, andXrepresents the gain or the refractive index. For h= i, Eq.(5) describes the inertial nonlinear refractive in- dex. If h= 1 and sr,~a, ~b correspond to the parameters of the gain medium, then Eq. (5) describes the evolution of the gain. Ifh= 1 and~a<0, Eq.(5)accounts for a satura- ble absorber, if~a>0 for a darkening absorber. The latter can be, for instance, a reverse saturable absorber: its absorption increases indeed with intensity[9]. Immediately after the ultrashort pulse, the characteristic Xof the med- ium (the gain, the absorption or the refractive index) is modified. After that it relaxes. As a result, this characteris- tic is modulated along the round-trip period. As in the case of active modulation, this modulation produces the relative movement of intracavity pulses, resulting in pulse ordering along the cavity, and then leading to the harmonic passive mode-locking.
At each step forf, knowingI(s), we obtainX(s) from Eq.
(5). Then fromX(s) we determine its average value over one round-trip hXi ¼s1a Rsa
0 XðsÞds, where sa is the cavity length in units ofs. We deduce from it the modulation term x(s) =X(s) hXi. In the case of the gain the mean value hXicoincides withGfrom Eq.(3), if we ignore the effects resulting in the spike operation. In the case of the saturable absorber we ignore the mean value hXi since it does not influence the drift of pulses in the cavity and changes only slightly the value of the pumping parameter. For analogous reasons we ignore the mean value hXi in the case of the inertial nonlinear refractive index. Indeed, it modifies only unessentially the wave vector of intracavity light wave.
Because of long relaxation time for a solid-state gain medium, the gain modulation at round-trip period is weak and cannot produce the harmonic passive mode-locking [10,11]. However, the darkening absorber with suitable relaxation time placed in the laser resonator can induce a modulation of the net gain large enough to produce har- monic passive mode-locking. A semiconductor with excita- tion of electrons in conduction band by the laser field can be used as such a darkening absorber. Uncontrolled mech- anisms responsible for weak inertial nonlinear losses in the laser cavity are also possible. For example, these mecha- nisms can be related with non-lasing energy levels of the intracavity medium.
In the case of the gain modulation and of the darkening absorber modulation, the mechanism of pulse ordering is connected to the degradation of the net amplification just after the pulse and its recovery during the succeeding time interval between pulses. Such a type of modulation in the net amplification produces a repulsion between ultrashort pulses in the cavity. As a result, the identical pulses become uniformly distributed over the round-trip period, with the same distances between all neighboring pulses. Fig. 6 shows such an ordering of pulses in the cavity by modula- tion of the net gain due to additional darkening absorber.
100 300 500 1
2
3
0 3 6
0 3 6
0.6
1.5
2.5 a τ
I
Fig. 5. The mode-locking inFig. 4was still passive: a continuous behavior is observed in the absence of nonlinear losses, with the same active harmonic modulation. Distribution of the intensityIversus the pumping parametera. The parameters are the same as inFig. 2.
For both the depletion of the gain and the darkening absorber, the modulation is described by Eq. (5) with
~
a>0 and h= 1. As mentioned above, the mechanism of depletion of the gain for the pulse ordering is weak because of the very long relaxation time of the gain (100ls) [10,11]. However, it should be noticed that the upper lasing level consists of several sublevels. The time of excitation exchange between these sublevels is very fast, below the nanosecond range[14]. For example, let us assume a value of about a nanosecond. Then, if the cross-sections of the lasing transitions for these sublevels considerably differ, the situation is changed cardinally. In this case, the order- ing modulation will have a period in the nanosecond range and its efficiency can increase considerably.
In the case of a saturable absorber,~a<0,h= 1, the loss modulation produces a mutual attraction of the pulses in the cavity. The pulses can merge, thereafter new pulses form, and this unstable picture repeats again and again.
New pulses arise in the round-trip period just after an exist- ing pulse, where the net gain is larger in comparison with other fragments of axial period. No harmonic mode-lock- ing can be achieved in these conditions.
We have studied the transient process and established operation in the case of an inertial nonlinear refractive index (h= ±i). If h= i (the refractive index decreases just after the pulse) the mutual repulsion of ultrashort pulses is realized and the harmonic passive mode-locking is estab- lished. With the same parameters, the transient process was longer in factor of approximately 10 in comparison with the darkening absorber. If h=i (the refractive index increases just after the pulse) the mutual attraction of ultra- short pulses is realized and the harmonic passive mode- locking is impossible. The reasons producing the inertial nonlinear refractive index can be of a great variety and we do not discuss them here.
3.2. An analytical pulse solution
The relaxation process is described by Eq. (5). We assume that the relaxation timesris very large in compar- ison with the pulse duration and that the relative variation
of the gain or refractive index is small:X=X0+X1, with X1X0. Then Eq. (5) can be solved in a perturbative way.X0¼h~a andX1satisfies
sr
oX1
os ¼ X0~bI: ð6Þ
Eq.(6)solves as X1¼ X0~b
sr
Z s
Iðs0Þds0: ð7Þ
The lower limit of integration is chosen in such a way that the mean valuehX1iis zero; thenxcoincides withX1.
In a previous paper[12], we considered a model consti- tuted by Eq. (1), in which x was zero andG=G0 a con- stant, and Eq.(2). Averaging the nonlinear losses Eq. (2) over one cavity round-trip and using an approximation of small intensity we derived the cubic–quintic (CGL) equation. We follow here the same procedure, with x=X1defined by Eq.(7), and neglecting the quintic terms, and we obtain the following master equation:
oE
of¼ ðDrþiDiÞo2E
os2þ G0rþ ðpþiqÞjEj2A Z s
jEj2ds0
E;
ð8Þ wherer accounts for the linear losses, andA¼X0~b=sr. In order to cover all physical situations to be investigated, we writeAas any complex numberA=A0+ iA00: the real part A0 corresponds to the variations of the net gain, the imaginary partA00to that of the refractive index. Following Ref.[13]we seek for a solution of this equation of the form E¼E0exp iðdxsdkfÞ
ch1þiabðswfÞ ; ð9Þ
whereE0, dx,dk,a, b,ware the peak amplitude, the fre- quency shift, the wave vector correction, the frequency chirp, the inverse duration, and the correction to the in- verse of the velocity of the pulse, respectively. Substituting expression Eq. (9) into Eq. (8), and then identifying the coefficients of E, Etanhb(swf), and E/cosh2b(swf), we obtain three complex or six real algebraic equations for the pulse parametersE0,dx,dk,a,b, andw[13].
From these equations, we can compute the correction to the inverse of the pulse velocityw, as
w¼ jE0j2 b2
1
1þa2 A0 1aDi
Dr
þA00 aþDi
Dr
; ð10Þ
the frequency shiftdx, as dx¼abwþA00jE0j2=b
2bðaDiDrÞ ; ð11Þ
the wave vector correctiondk, as dk¼A00jE0j2
b 2ab2Dr ðð1a2Þb2dx2Þ; ð12Þ and the squared peak amplitude, as
jE0j2¼3b2ðD2rþD2iÞ a
qDrpDi: ð13Þ
Fig. 6. Effect of passive net gain modulation related with darkening absorption. Solid curve: the established distribution of the intensityIin the form of equally spaced pulses. Dashed curve: the initial distribution of the intensity. a0=1, a1= 0.2, a3= 0.3, Di=1, Dr= 0.2, a= 1.8,
~
a¼0:2,sr= 100.
The frequency chirpa satisfies a
2a2¼ qDrpDi
3ðpDrþqDiÞ: ð14Þ
Eq.(14)admits two solutions with opposite sign. According to Eq.(13)the one which has the same sign as the difference (qDrpDi) must be retained. An important remark about Eq.(13)is that the ratiojE0j2/b2=Udepends on the char- acteristicsp,q,Dr andDi of the cavity only.a is indeed a function of these quantities. The frequency shift dx can thus be reduced to
dx¼UðaA0A00Þ 2ð1þa2ÞDr
¼ 3aðaA0A00ÞðD2rþD2iÞ
2ð1þa2ÞDrðqDrpDiÞ: ð15Þ The inverse pulse duration b must satisfy the quadratic polynomial equation
b2ðð1a2ÞDr2aDiÞ A0bUþG0rDrdx2¼0:
ð16Þ The existence of the solution Eq.(9)rests then only on the existence of a real solutionbof Eq.(16), that is on the sign of
D¼A02U24FðG0rDrdx2Þ; ð17Þ which depends on the value of the gainG0, and where we have set
F ¼ ð1a2ÞDr2aDi: ð18Þ IfF> 0, thenD> 0 and the analytical solution Eq.(9)ex- ists forG0<Gth, where
Gth¼A02U2
4F þDrdx2: ð19Þ
IfF< 0, the solution exists forG0>Gth. Thus, the analyt- ical solution exists either above or below the gain threshold Gthdepending on the sign ofF. Notice that the stability of the analytical solution remains to be investigated.
According to Eq. (7), the total change of the net gain or the refractive index along the pulse is 2dX ¼ ARþ1
1 jEðs0Þj2ds0¼ 2AjE0j2=b. The key parameter of the theory is the modulation depthdX. Using this parame- ter we see that the order of magnitude of the correction to the inverse of velocitywcan be described aswdX/b.
3.3. The passive modulation of the net gain
Now we concentrate on the case of the modulation in the net gain: the depletion of the gain or the saturation of the darkening absorption. Hence, we assume that ~a,
~b>0, and h= 1 in Eq. (5). The linear gain G0 coincides withX0, and we will use the notationdX=dG. In this case, Eq.(10)transforms into the following expression
w¼ jE0j2 b2
G0~b sr
1aDi=Dr
1þa2 : ð20Þ
If we assume thatDi=1, q= 1, Dr> 0, p> 0, which is the case for the Yb-doped fiber laser operating in the
normal dispersion regime, then a> 0 and consequently w is negative. The pulse velocity is
V¼ 1 vgþw 1
’vgv2gw; ð21Þ
assumingw1/vg, and denoting byvgthe group velocity.
As a result, the correctiondV¼ v2gwof the pulse velocity is always positive. In this case, the inertial nonlinearity sup- presses the trailing edge of the pulse and amplifies the lead- ing one. For a localized pulse, the corresponding attenuation or amplification length has the order of magni- tude of ðdGÞ1¼ ðjE0j2G0~b=bsrÞ1 (2dG is the total change in the net gain along the pulse). Through this effect, the pulse moves for a distance which is around its length b1. As a result, the correction to the inverse velocity is wdG/b. The physical meaning of the last fraction in Eq. (20)is discussed below.
The correction is proportional to G0. As a result, because of slow relaxation the velocities of two closely- spaced pulses are different. As this takes place, for the first pulse having largerG0the velocity is larger than for the sec- ond. This effect acts as an effective repulsion between the pulses, which increases with decreasing distance between pulses. As a result, the initial train spreads over, filling the whole cavity with equally separated pulses (see Fig. 6). Notice that, in the expression Eq. (10) of w, all quantities are characteristics of the cavity or functions of them (as are U=jE0j2/b2 and a, see Eqs. (14) and (13)), excepted the gain parameter G0. This parameter thus gov- erns the variations of the velocity. Further, notice that the suppression of the trailing edge of the chirped pulse, and the amplification of the leading one, induce the fre- quency shift dx given by Eq. (15). The last fraction in Eq.(20)involves the group velocity dispersionDi, that con- siderably strengthens the effect of the velocity change.
Let us derive a formula suitable for numerical estimates.
Between two consecutive pulses the gain decreases fromG0
toG00’G0dG. It follows a change in the inverse velocity was
dw’wdG G0
¼ ðdGÞ2 G0b
1aDi=Dr
1þa2 : ð22Þ
After a time intervalDs, the distance between two initially closely-spaced pulses will beDf¼dVDs, where the correc- tion to the pulse velocity isdV¼ v2gdw. Thus
Ds¼ Df v2g
1 b
G0
dG20
1þa2
1aDi=Dr: ð23Þ
ForDf= 1 round-trip, it becomes, in physical units, DT ¼Ta
Tp G0 dG0
2
1 G0
1þa2
1aDi=Dr; ð24Þ
whereTais the round-trip duration andTpthe one of the pulse. If N pulses are present in the cavity, the distance to be considered is Df= 1/N. Further we write dG as dG0/N, where dG0 is the depletion in the net gain during
the complete round-trip. Then the time of ordering of pulses in the cavityDTbecomes
DT ¼TaN G0
Ta Tp
G0
dG0
2
1þa2
1aDi=Dr: ð25Þ
Notice that in Eq.(25), the gain coefficientG0is still dimen- sionless:G0is the gain factor for one round-trip.
Assume that the depletion of the gain dG0/G0is about Ta/sr, which is typically about 104, that the gain factor G01, and that the chirp spectral broadening is about 1/Tp, which leads toa1. A reasonable value for the ratio between group velocity dispersion and gain filtering is Di/Dr=5. Then we get DT/Ta0.3·108Ta/Tp, which is a very large value, so that this mechanism cannot be real- ized in experiment. However, if the upper lasing level con- sists of several sublevels with strongly different lasing transition sections then the time of the partial gain recovery is determined by the excitation exchange time between the principle lasing sublevel and the excitation storage sublevel.
This exchange time is very fast, below the nanosecond range [14], then the situation changes cardinally. In this case, dG0/G01, G0104, Ta5·108s, Tp5· 1012s and, consequently, DT1 s. In the case of the darkening absorber withdG0/G01,G0102, we obtain DT10 ms.
In the case of saturable absorption, i.e.,~a<0 in Eq.(5), the valueG0in Eq.(20)becomes negative and the mecha- nism of the attraction of pulses is realized. In this case, the order of pulses in the cavity and the harmonic passive mode-locking are impossible.
3.4. The passive modulation of the nonlinear refractive index
Eq. (5) accounts for the evolution of the nonlinear refractive index when h= i; thenX0¼i~a0, and A¼iA00¼ i~a~b=sr. We use the notations ~a¼ Dn0 and dX=idn.
The expression Eq. (10) of the correction to the inverse velocity becomes
w¼jE0j2 b2
Dn0~b sr
aþDi=Dr
1þa2 : ð26Þ
The term proportional toais connected with the following physical mechanism of velocity change. The frequency chirp yields an antisymmetric detuning from the central carrier frequency which corresponds to the maximum of amplification. This detuning is zero at the point of the pulse where the amplitude is maximal. The inertial nonlinear refractive index evolves monotonically along the pulse.
The value of this total change is 2dn¼2jE0j2Dn0~b=bsr, dn coincide with the first term in the expression Eq. (12) of the correction to the wavevectordk. This yields an addi- tional global frequency detuning of the pulse. Conse- quently, the point with the central carrier frequency is shifted from the center to one of the edges of the pulse, and the gain coefficients for the trailing and leading edges are different. This results in a mechanism of velocity change
analogous to the one described above at the beginning of Section3.3. If the refractive index decreases after the pulse, i.e.,~a>0 andDn0< 0, the contribution of this term towis negative and the corresponding correction to the pulse velocityVis positive. If in the contraryDn0> 0, the correc- tion to the pulse velocity is negative.
The term proportional to Diis connected with the fre- quency detuningdxdue to the inertial nonlinear refractive index and the frequency dispersion of the group velocity Di. ForDi=1, the corresponding correction to the pulse velocity V is negative in the case where~a>0, i.e., when the refractive index decreases after the pulse, and positive for~a<0.
The complete correction v2gw is positive when ~aðaþ Di=DrÞ>0, i.e., when Dn0(a+Di/Dr) < 0. In this case, as explained in the case of the gain modulation, the modula- tion of the velocity acts as an effective pulse repulsion, and the ordering of pulse in the cavity can be realized.
The time of the pulse ordering can be evaluated in a way analogous to the derivation of Eq.(25)in previous subsec- tion. We obtain
DT¼ TaN Dn0
Ta Tp
Dn0
dn 2
1þa2
aþDi=Dr: ð27Þ
4. Conclusion
We have developed a theoretical model which describes the harmonic passive mode-locking in fiber lasers. The model is simple and is not computer time consuming.
Our numerical simulations account for both the transient process and the steady-state for this type of laser operation.
This model has been reduced to a CGL type equation, modified to take the gain or refractive index relaxation into account. An analytical localized solution of this equation allowed us to determine the velocity of pulses relative to each other. These movements can be induced by several mechanisms, and result in the self-organization of pulses in the cavity, that is the harmonic passive mode-locking.
In addition, the analytical treatment allows to state about the conditions of realization of the passive harmonic mode-locking.
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