‘Cis’ bound states of three localized pulses of the cubic–quintic CGL equation

J. Opt. A: Pure Appl. Opt. 8 (2006) 319–326 doi:10.1088/1464-4258/8/3/015

‘Cis’ bound states of three localized pulses of the cubic–quintic CGL equation

H Leblond, A Komarov

, M Salhi, A Haboucha and F Sanchez

Laboratoire POMA, UMR 6136, Universit´e d’Angers, 2 Boulevard Lavoisier, 49000 Angers, France

Received 27 June 2005, accepted for publication 31 January 2006 Published 27 February 2006

Online at

We investigate triplet bound states with a new symmetry, called ‘cis’, using the cubic–quintic CGL equation. We show that the leading term of the functional J

, which governs the evolution of the momentum of the solution to the CGL equation, vanishes for the cis-symmetry. Numerical investigations show that stable cis triplet bound states are solutions of the CGL equation. Quasi-stable cis-states are also found, and also a stable quasi-stationary asymmetrical triplet state. Then we show that it is possible to experimentally distinguish between the trans and cis triplet states, using either the optical spectrum or the collinear autocorrelation trace.

Keywords:

fibre lasers, solitons, bound states

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Due to the particle-like behaviour of solitons, interaction between them is currently being studied. The first studies were based on the cubic–quintic complex Ginzburg–Landau (CGL) equation. Bound states of two solitons have been known about for a fairly long time in the case of anomalous dispersion, using the perturbative [1] or the energy and momentum balance approach [2, 3], or pure numerical resolution of the equation, with a representation in the separation–dephasing plane [4]. More realistic models taking into account the two polarization components of the electric field have been recently developed [5–7]. Numerical simulations show the existence of bound states of two solitons in both the normal and the anomalous average dispersion regime. The interaction of a two-pulse bound state with a single pulse can lead to a three-pulse bound state, but merging and elastic collisions also arise [8].

The CGL equation is a model commonly used to describe pulse formation in fibre lasers, especially passively mode- locked ring-fibre lasers. A cubic CGL model was first derived in this frame by Haus [9]. The coefficients of the equation have been computed explicitly as functions of the physical parameters for the Yb-doped fibre laser [10], the Er-doped

1 Permanent address: Institute of Automation and Electrometry, Russian Academy of Sciences, Academician Koptyug Avenue, 1, 630090, Novosibirsk, Russia.

fibre laser [11], and the stretched-pulse laser [12]. A good agreement has been found with the experiments performed with the Yb-doped fibre, which has normal dispersion.

In the same experimental set-up, bound states have been observed [13]. They can be described theoretically using the quintic CGL, which admits two-pulse bounded states in the case of normal dispersion [13]. In fact, two-pulse bound states have been observed in different fibre laser configurations and with different doping ions and can therefore be viewed as a regular operating regime of passively mode-locked fibre lasers [5–7,13–16]. More recently, three-pulse bound states have also been observed on different fibre lasers [17, 18].

Theoretically, bound states of more than two pulses were first considered in [2, 3] using the quintic CGL equation.

In particular, bound states with three pulses of particular symmetry, owing to the time reversal invariance of the equation, have been found. The aim of this paper is to demonstrate the existence of triplet bound states with different symmetry from those of [2,3]. The analysis is based on the cubic–quintic CGL equation. In section2, we briefly recall the results of references [2, 3] where the authors predict a triplet bound state with a symmetry that we will call ‘trans’.

From the energy and momentum approach, we demonstrate that the symmetry ‘cis’ can also exist. Section3is devoted to numerical simulations. It is first demonstrated that a stable ‘cis’

bound state is a solution of the quintic CGL equation. Quasi- stable states are also considered. In section4we consider the experimental possibility to determine the symmetry of a triplet

bound state. It is shown that the collinear autocorrelation trace or the optical spectrum allows a conclusion on whether the symmetry is ‘trans’ or ‘cis’.

2. The ‘cis’ symmetry

Let us briefly recall the energy and momentum balance approach of [2, 3]. The cubic–quintic CGL equation is, in dimensionless units,

i∂ψ

∂z + D 2

∂²ψ

∂t² +ψ|ψ|²+νψ|ψ|⁴=ⁱδψ+ⁱβ∂²ψ

∂t² +iεψ|ψ|²+iµψ|ψ|⁴. (1) The pulse energy evolves withzaccording to

d dz

_+∞

−∞ |ψ|²dt=F[ψ], (2) where the functionalF[ψ] is given by

F[ψ]=2 _+∞

−∞

δ|ψ|²+ε|ψ|⁴+µ|ψ|⁶−β ∂ψ

∂t ²

dt.

(3) Similarly, the momentum evolves according to

d dz^Im

_+∞

−∞ ψ∂ψ^∗

∂t ^dt=J[ψ], (4) where the functionalJ[ψ] is given by

J[ψ^]=^{2 Im} _+∞

−∞

δ+ε|ψ|²+µ|ψ|⁴ ψ∂ψ^∗

∂t

−β∂ψ

∂t

∂²ψ^∗

∂t²

dt. ⁽⁵⁾

The stationary solutions satisfy

F[ψ] =0 and J[ψ] =0. (6)

In [2,3]

ψ=ψ0(t−ρ/2)+ψ0(t+ρ/2)e^iϕ, (7) was used as a trial function, where ψ0 is the soliton solution, computed numerically by resolution of the evolution equation (1). The two parameters ρ and ϕ represent thus the distance and the dephasing between the two pulses, respectively. The zeros ofFandJare represented in the plane (X,Y) = (ρ cosϕ, ρ sinϕ). Intersection points correspond to stationary states, whose stability must be studied, by computing numerically the trajectories in the(X,Y)plane.

In the case of three pulses, we use the trial function ψ=ψ0(t−ρ1)^eiϕ1 +ψ0(t)+ψ0(t+ρ2)^eiϕ2. ⁽⁸⁾ We have two separation distancesρ1, ρ2and two dephasings ϕ1, ϕ2, thus four degrees of freedom (see figure 1). We notice that the cubic–quintic CGL equation is invariant in the transformation t −→ −t, therefore we can envisage some symmetry hypothesis, such asρ1=ρ2andϕ1= ±ϕ2, which reduces the number of degrees of freedom to two, which would a priori allow us to generalize the above approach. In [2,3] the symmetryϕ1= −ϕ2, which will be referred to as ‘trans’, was

-π/ 2 π / 2

-3 π/ 2 -5 π/ 2

-100 -5 0 5 10

1 2

-8 -4 0 4 8

0 1 2

-10 -5 0 5 10

0

-8 -4 0 4 8

-π / 2

-π

ρ ρ ρ ρ

2 1 1

1

1 2

2

2

|ψ| |ψ|

arg(ψ) a arg(ψ)

t

t t

t

ϕ

ϕ ϕ ϕ

Figure 1. Definition of the separation distancesρ1, ρ2and two dephasingsϕ1, ϕ2. Left, the cis-symmetryϕ1=ϕ2; right, the trans-symmetryϕ1= −ϕ2. The left figure corresponds to the stable state S of figure2, the right one to the initial data a1a2in the same figure.

considered, and it was checked that, when two-pulse bound states exist, trans triplet bound states can be formed with the same separation and dephasing.

In fact, the symmetryϕ1 = ϕ2, which will be referred to as ‘cis’, lets the dominant interaction term in J[ψ]^vanish.

Therefore, as we will see below, (i) this symmetry is generally stable or very weakly unstable, (ii) stable bound states of three pulses with this symmetry may exist in the absence of two-pulse bound states, (iii) states of indifferent equilibrium, and bunches of three pulses, i.e. bound states with a non- defined phase difference, exist. On the other hand, the vanishing of J[ψ]prevents the energy–momentum approach from describing the triplet bound states.

2.1. J[ψ] =0 for the cis-symmetry

2.1.1. Estimates. Since the cubic–quintic CGL equation is invariant in the transformationt−→ −t, and according to the numerical results,ψ0is even. Further it decays exponentially as t tends to ±∞, say ψ0 ∼ Ae^−r|t|. We will set for shortening ψ0 = ψ0(t), ψ0± = ψ0(t ± ρ), and for

+∞

−∞ dt. Let us consider the integral I = ψ₀₋ⁿ ψ₀^∗p, with two positive integer exponents n and p. We split it into I = I₁+ I₂, where I₁ is the integral fort < ρ/2 and I₂is the integral fort > ρ/2. The latter can be written as I₂ =

+∞

0

ψ0

t−^ρ₂n ψ₀^∗

t+^ρ₂p

dt (witht = t −ρ/2), where the first factor is bounded, while the above estimation can be used for the second. Thus

I₂A^pe^−prρ2 _+∞

0

ψ0

t−ρ

2

^ndt∈^O e^−prρ2

, ⁽⁹⁾

since ₀^+∞ψ0

t−^ρ₂ⁿ_dtψ0ⁿ_Lⁿ <∞. In the same way we can prove that

I1∈O

e^−nrρ2

, (10)

and thus

ψ0ⁿ−ψ₀^∗p∈O

e^{−min(n,p)rρ2}

. (11)

In the same way, we have

ψ0ⁿ+ψ0^∗p∈^O

e^{−min(n,p)rρ2}

, ^and

ψ₀₊ⁿ ψ₀₋^{∗ p}∈O

e−min(n,p)rρ .

(12)

The same holds if one of theψ0is replaced by its derivative.

Further, it follows that integrals like ψ₀₊^mψ₀₋ⁿ ψ₀^∗pare at most O

e^−rρ

, ifmandnare at least unity.

2.1.2. Computation ofJ[ψ]. We expandJ[ψ]as

J[ψ] =δJ1+εJ2+µJ3+βJ4, (13) andJ[ψ0]asδJ₁⁰+εJ₂⁰+µJ₃⁰+βJ₄⁰.

The computation of J1 = 2 Im ψψt^∗, where the index t designates the derivative, is as follows. Using the expression (8) ofψ, and the estimate (12), we obtain

iJ₁=3iJ₁⁰+

(ψ0ψ0^∗−,te^−iϕ1+ψ0ψ0^∗+,te^−iϕ2 +ψ0−ψ_0,t^∗ e^iϕ1+ψ0+ψ_0,t^∗ e^iϕ2−cc)+O

e^−rρ

, (14) where cc holds for complex conjugate. Making translations, integrations by parts, and using the fact thatψ0is even (and ψ0,todd), (14) reduces to

J1=³J₁⁰+⁴(^sinϕ2−^sinϕ1)

ψ0ψ0−,t+^O e^−rρ

. ⁽¹⁵⁾ The computation of the termJ2, which corresponds to the cubic term in the CGL equation (1), follows the same lines.

The key of the computation is the use of the estimates (11)–

(12). Keeping only the dominant interaction terms, we get iJ2=3iJ₂⁰+ T_·,−+T_−,·+T_+,·+T_·,+−cc

+O e^−rρ

, (16) with

T_·,−=

ψ₀²ψ₀^∗ψ_0−,^∗ t+ψ₀²ψ₀₋^∗ ψ_0,^∗t

e^−iϕ1+2ψ0ψ0−ψ₀^∗ψ_0,^∗te^iϕ1. (17) The other terms are analogous, with obvious notations. The same transformations as above reduceJ₂to

J2=3J₂⁰+2(sinϕ2−sinϕ1)

(ψ₀²ψ₀^∗ψ_0−,t^∗ +ψ0²ψ₀₋^∗ ψ_0,t^∗ −²ψ0ψ0−ψ0^∗ψ_0,t^∗ +^cc)+^O

e^−rρ . ⁽¹⁸⁾ Computation of J₃andJ₄is fully analogous to that ofJ₂and

J₁respectively; we get J₃=3J₃⁰+2(sinϕ2−sinϕ1)

(ψ0³ψ0^∗

2ψ0^∗−,t

+2ψ₀³ψ₀^∗ψ₀₋^∗ ψ_0,^∗t−3ψ₀²ψ0−ψ₀^∗2ψ_0,^∗t+cc)+O e^−rρ

, (19) and

J4=3J₄⁰ + 4(sinϕ2−sinϕ1)

ψ0,ttψ_0−,t^∗ +O e^−rρ

. (20) Using the fact thatψ0is stationary, we haveJ[ψ0] = 0, and thus

J[ψ] =(sinϕ2−sinϕ1)I+O e^−rρ

(21)

-6 -4 -2 0 2 4 6 8

-6 -4 -2 0 2 4 6

ρcosϕ

ρsinϕ a

a b b

c c d d

e e f

f g g

2 1

1 2

12 1 2

1 2 1

2 1 2

S

Figure 2. Trajectories(ρ1, ϕ1)(solid line) and(ρ2, ϕ2)(dashed line) starting from different initial data having the cis-symmetry (c1c2,d1d2,e1e2,g1g2), the trans-symmetry (a1a2,b1b2), or no symmetry ( f1f2). All trajectories converge to the stable state represented by the point S. The parameters are

(ε, β, δ, µ, ν,D)=(0.4,0.5,−0.01,−0.05,0,1).

whereI is some functional ofψ0. Therefore, asϕ1 =ϕ2, or more generally sinϕ1=sinϕ2, we haveJ[ψ] ∈O

e^−rρ . Thus the dominant interaction term in the functionalJ[ψ] vanishes for the cis-symmetry. According to (4), this implies that the momentum evolves very slowly when the triplet state has the cis-symmetry. As a consequence, the complete evolution of the triplet state itself is considerably slowed down as the cis-symmetry is realized. Hence, stationary states with this symmetry can be expected to exist. Further, we can expect that a non-stationary state having the cis-symmetry will evolve very slowly, and in certain cases slowly enough that it could be detected experimentally as a bound state. In the same way, the development of an instability in the vicinity of a cis stationary state can be expected to be very slow, and the stationary state will be considered as quasi-stable, since its duration can be long enough to allow its experimental observation. Results in accordance with these expectations have been found numerically, and are shown in the following section.

3. Three-pulse states

3.1. A stable cis bound state

Figure 2 presents the trajectories in the (ρ cosϕ, ρ sinϕ) plane, starting from several different points, having or not the cis-symmetry. For given initial data, the propagation is solved numerically. The CGL equation (1) is transformed into

∂ψˆ

∂z =

δ+

iD 2 −β

4π²ω²

ψˆ

+F((ε+ⁱ) ψ|ψ|²+(µ+ⁱν) ψ|ψ|⁴), ⁽²²⁾ using the Fourier transform ψ(ω,ˆ z) = F(ψ(t,z)) = e^−2iπωtψ(t,z)^dt. The propagation inzis implemented using a standard fourth order Runge–Kutta algorithm. The nonlinear terms are computed, at each step in z (more precisely, at each intermediary step of the Runge–Kutta algorithm), by first computingψ =F⁻¹(ψ)ˆ by means of a standard fast Fourier

0

0

1000

1000

2000

2000

3000

3000

4000

4000

5000

5000 -2

-1 0 1 2 3 4

ϕ − ϕ

ϕ_lim

z

z

0 100 200 300 400

-2 -1 0 1 2 3 4

1 2

a

a

a

a

f f

f f

b b

b b 2 1

a a_{1 2}

1 2 1

ff 2 1

2

b b_{1 2}

2 1 2 1 2

z ϕ ϕ_{1 2}

1

ϕ ϕ_{1 2}

-2 0 2 4

-4 a)

b)

Figure 3. (a) Time evolution of the dephasingsϕ1andϕ2, corresponding to the trajectories of figure2which do not have the cis-symmetry. The inset is a zoom in on the beginning of the evolution. (b) Time evolution of the corresponding phase difference ϕ2−ϕ1. The parameters are the same as for figure2.

transform algorithm, and then applying the Fourier transform to the whole nonlinear term as written in formula (22). Notice that the use of the spectral method implies periodic boundary conditions. We use 4096 points in thet direction, with a total box length of 400. The step in the z direction is typically 0.002. The initial data are built using expression (8), in which we use asψ0a numerically computed soliton solution of the same equation. The location of the maxima is determined using a parabolic interpolation of the numerical solution, then the two dephasing(ϕ1, ϕ2) and the two separation distances (ρ1, ρ2)are computed. This yields two trajectories for each numerical solution,(ρ1, ϕ1)(solid lines) and(ρ2, ϕ2)(dashed lines). Both trajectories are superposed in the cis-symmetry. It is seen that all trajectories converge to the same stable state, labelledS. The set of parameters has already been considered in [2, 3], and no two-pulse bound state was found in that case. A stable three-pulse bound state in the cis-symmetry is found. Notice that the stable value of the dephasing is notπ/2, while this value appears frequently for two-bound pulses. The evolution of the dephasings is given in figure3, for the three initial data of figure 2which do not have the cis-symmetry (a,b, and f). It must be noticed that the cis-symmetry is reached rather quickly, while the stable value of the dephasing ϕlim 1.20 (=68.7^◦)is reached much slower. Once the cis- symmetry is realized, the evolution becomes very slow, which is closely related to the vanishing of the functionalJ[ψ]^shown in the previous section. The computation has been pursued very far, further than z = 20 000 for two of the examples presented (aandd), confirming the stability of the stateS.

0 1 2 3 4 5 6 7

x 10⁴ 3.6

3.65 3.7 3.75

0 1 2 3 4 5 6 7

x 10⁴ -6

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

-4 -2 0 2 4

z z

ϕ ρ

k k₂

h₂ h1

h h 1 2

k₁ k2 1

k k_{1 2} k k_{1 2}

h h₁₂ h h_{1 2} k₁ k2

x 10⁴ z

x 10⁴ z

ϕ − ϕ₂ 2

1 ρ − ρ1

0.1 0 - 0.1 0.6 0.4 0.2 0 - 0.2 a)

b)

c)

d)

Figure 4. (a) Time evolution of the separation timesρ1(solid lines) andρ2(dashed lines), and (b) of the dephasingsϕ1(solid lines) and ϕ2(dashed lines), starting with two different initial data, denoted by k1,k2and h1,h2. The dashed–dotted lines in (b) are the same as the curves k1,k2, translated for 2π. (c), (d) The differences between the separation times and dephasings presented in (a) and (b). An asymmetrical state with dephasings varying very slowly is reached for both initial data. The initial data h1,h2having the cis-symmetry goes through a quasi-stable state with this symmetry. The parameters are(ε, β, δ, µ, ν,D)=(1.8,0.5,−0.01,−0.05,0,1).

3.2. A quasi-stable cis-state and a stable asymmetric one Figure 4 shows the evolution of the pulses separation and dephasing, starting with a cis-symmetry, in a situation where a stable two-pulse bound state is known to exist, and also stable triplet states of the symmetry trans. The parameters considered have already been used in [2,3]. First we observe the formation of a metastable bunch having the cis-symmetry, for which the dephasing evolves linearly with time. After a variation of dephasing of a few more than 2π, this state disappears, to be replaced by an asymmetrical bunch, for which the dephasings still evolve linearly with time, but much more slowly, and deviate from the cis-symmetry. Notice that the evolution is very slow: the duration of the metastable state is about 6000 units ofz, which is a roundtrip in the cavity [19], while the asymmetrical state has been computed on a distance equivalent to 50 000 roundtrips, which would correspond to a real duration in the millisecond range. The fact that the cis-symmetry corresponds to slow evolution, even for non- stationary states, is confirmed by this computation. If the cavity length is about 10 m, the duration of the metastable state is about 0.2 ms, which is sufficient to observe it as a bound state from the experimental point of view. Therefore, this state can be considered as quasi-stable. The evolution of the asymmetrical state is even considerably slower: about 0.3 rad in 50 000 roundtrips, i.e. more than 1 ms. This asymmetrical state can thus be considered as quasi-stationary, and is completely stable.

0 500 1000 1500 2000 8

8.5 9 9.5 10

0 500 1000 1500 2000

-4 -2 0 2 4

z

z

0 500 1000 1500 2000

z

0 500 1000 1500 2000

z 1

0 - 0.5 0.5

0 1

-1 ϕ ρ

ϕ − ϕ₂ 2

1 ρ − ρ1

a)

b)

c)

d)

Figure 5. (a) Time evolution of the separation timesρ1(solid line) andρ2(dashed line), and (b) of the dephasingsϕ1(solid line) andϕ2

(dashed line), starting from initial data having the cis-symmetry. A metastable cis-state, of duration about 1000 roundtrips, is observed.

(c), (d) The differences between the separation times and dephasings presented in (a) and (b). The parameters are

(ε, β, δ, µ, ν,D)=(1,0.8,−0.1,−0.1,−0.01,−1).

-10 -5 0 5 10

-10 -5 0 5 10

ρcosϕ

ρsinϕ Q

ll 12

-10 -9 -8 -7

-2 -1 0

Q

Figure 6. Trajectories in the(ρcosθ, ρsinθ)plane, starting from an initial data l1l2having the cis-symmetry. The trajectories go through a metastable state Q, and then go out of the cis-symmetry. The inset is a zoom around the point Q. The parameters are

(ε, β, δ, µ, ν,D)=(1,0.8,−0.1,−0.1,−0.01,−1).

3.3. Weak instability of steady cis-states

For other values of the parameters, here normal dispersion and nonzero ν, an interesting kind of instability has been found. Figure5presents the time evolution of the separation and dephasings of a three-pulse state having initially the cis- symmetry; and figure 6the corresponding trajectories in the

0 500

1000

–15 –10 0 – 5

10 5 0 1 2 3

|ϕ|

t

z

Figure 7. Time evolution of a state of three pulses having initially the cis-symmetry: two pulses merge together. The parameters are (ε, β, δ, µ, ν,D)=(1,0.8,−0.1,−0.1,−0.01,−1).

(ρcosϕ, ρsinϕ)plane. An instability occurs, passing through some unstable steady state, which can be considered as quasi- stable since the trajectory stays close to it during about 1000 roundtrips in the cavity. Once again, according to the analytical result of section2, the evolution of the cis-state is very slow.

Hence the instability is very weak. Other unstable steady states exist for the same set of parameters, with much shorter duration in general. The instability can lead to the collapse of two of the pulses, as shown in figure7. The values of the separation and dephasing of the steady state are ρ1 = ρ2 = 7.583 and ϕ1 = ϕ2 = ¹.⁴⁰⁴ = ⁸⁰.^44◦ in the latter case, and its lifetime was about 150 roundtrips. We did not succeed in finding stable bound states of three pulses having the cis- symmetry in the case of normal dispersion and nonzeroν^{, but} our investigation cannot be exhaustive, due to the very large computation time required. For other values of the parameters, indifferent equilibrium states have been found; they will be further investigated in the future.

4. The cis- or trans-symmetry can be determined experimentally

4.1. Autocorrelation trace

Experimentally, the structure of a multiple-pulse bound state is determined by the autocorrelation trace. It is thus of great importance to identify a cis- or trans-symmetry on the autocorrelation trace. The collinear autocorrelation trace is proportional to the integral

(τ)= _+∞

−∞ |E(t)+E(t−τ)|^4dt, ⁽²³⁾ whereE(t) = ψe^iω0t is the complex optical field andτ the retardation time. In the case of the triplet bound state given by (8), assumingρ1=ρ2=ρ, and at the limit of large values of the separation timeρ, the peak values ofcan be computed as follows: we notice thatU = (E(t)+E(t−τ))^{e−iω0t is a} sum of six terms6

j=1ψj, where we set

ψ1=ψ0(t), ψ2=ψ0(t−ρ)e^iϕ1, ψ3=ψ0(t+ρ)e^iϕ2, ψ4=ψ0(t−τ)e^−iω0τ, ψ5=ψ0(t−ρ−τ)e^{i(ϕ1−ω0τ)},

ψ6=ψ0(t+ρ−τ)^{ei(ϕ2−ω0τ)}.

(24) Then the autocorrelation trace takes the form

(τ)=

j,k,l,m

_+∞

−∞ ψjψkψ_l^∗ψ_m^∗dt. (25) Sinceψ0is localized, the terms in the sum (25) are nonzero (at the limit of largeρ) in two cases only: (1) j = k =l =m, for anyτ, which yield a continuous background; (2) the four indices take exactly two different values, one belonging to {1,2,3}, the other to{4,5,6}, in this case the contribution is nonzero whenτ is close to 0,±ρor±2ρ. The value of the continuous background is 6I, withI= _−∞^+∞|ψ0|⁴dt.

Let us detail first the computation atτ =ρ, which is the most important one for our purpose. We have to consider the quadruplets j klm=1166, 2244, 1116, 2224, 4442, 6661, and their permutations. The phase is not always zero. Let us detail the computation for j klm =1166 and its permutations in the following table:

j klm exponent

1166 0+0−ϕ₂−ϕ₂ = −2ϕ₂ 1616 0+ϕ₂−0−ϕ₂ =0 1661 0+ϕ₂−ϕ₂+0=0 6611 ϕ₂+ϕ₂−0−0=2ϕ₂ 6161 ϕ₂+0−ϕ₂−0=0 6116 ϕ₂+0−0−ϕ₂ =0

where we have setϕ2 =ϕ2+ω0τ. Thus the sum of these terms is(⁴+^{2 cos 2}ϕ2)I. Using the same computation method for the other quadruplets, and including the background, we get (ρ)=(⁶+

4+^{2 cos 2}ϕ1

+

4+^{2 cos 2}ϕ2

+2×4 cosϕ1+2×4 cosϕ2)I, (26) withϕ₁=ϕ1−ω0τ. Formula (26) reduces to

(±ρ)=(¹⁰+⁴

cos²(ϕ1∓ω0τ)+^cos2(ϕ2±ω0τ) +8 [cos(ϕ1∓ω0τ)+cos(ϕ2±ω0τ)^])I. (27) We have taken into account the symmetry asρ−→ −ρ.

Since the pulse length is long with regard to the optical period, the quantityω0τ varies very quickly, even ifτ remains close enough to±ρthat formula (27) is valid. The peak value is thus the maximumM(±ρ)of the function given by (27) as τvaries. The minimumm(±ρ)of the same function gives the lower bound of the envelope of the autocorrelation trace about

±ρ. For the cis-symmetry, we setϕ1 =ϕ2=ϕand compute the extremal values by means of elementary mathematical analysis, to get

M(±ρ)= I

×





10−4 cos 2ϕ− ⁴

cos 2ϕ ^{if π3} < ϕ < ^2π₃, 14+4 cos 2ϕ+16|cosϕ| else,

(28) and

m(±ρ)=I(¹⁴+^{4 cos 2}ϕ−¹⁶|cosϕ|) . (29)

For the trans-symmetry, in contrast, the oscillations are always in phase. Indeed,ϕ1 = −ϕ2 = −ϕ, and henceϕ₁ = −ϕ₂=

−ϕ2−ω0τ, and the cosines are equal. Thus the peak value and the lower bound of the oscillations do not depend onϕ, and are M(±ρ)=34I, m(±ρ)=2I: the amplitude of oscillations is always larger for the trans-symmetry than for the cis one, except forϕ = ^{0 or}π, where the two symmetries become identical.

The peak values at τ = 0 can be computed in an analogous way. The quadruplets j klm that contribute at τ = 0 are, in addition to the continuous background, 1144,²²⁵⁵,³³⁶⁶,¹¹¹⁴,²²²⁵,³³³⁶,⁴⁴⁴¹,⁵⁵⁵²,6663 and the permutations. After computation of the phase factors, we get

(0)=6(3+cos 2ω0τ+4 cosω0τ)I=48I. (30) It contains no information about the dephasingsϕ1andϕ2.

At τ = ²ρ, the quadruplets to be considered are 2266,²²²⁶,6662, and the permutations. An analogous reasoning leads to

(±2ρ)=4

1+4 cos⁴

ϕ2−ϕ1∓ω0τ 2

I. (31) The fast oscillations cancel the information about the phases, and the autocorrelation trace oscillates between 4Iand 20I in any case.

The above relations give the peak values of the autocorrelation trace. Information about the phase symmetry is found in the first lateral peak, which is smaller for the cis- symmetry, and maximal for the trans-one.

Beside this, the non-collinear autocorrelation trace is proportional to the integral

nc(τ)= _+∞

−∞ |E(t)E(t−τ)|²dt. (32) An analogous computation yields, under the same assump- tions,

nc(⁰)=³I, nc(±ρ)=²I, nc(±²ρ)= I. ⁽³³⁾ This does not contain any information about the relative phase of the pulses.

Both collinear and non-collinear autocorrelation traces can be computed numerically. In figure 8, we give the traces corresponding to the stable cis-states found in section3.1, and to the stable trans-state found by Akhmediev et al [3]. It is seen that the collinear autocorrelation trace allows us to distinguish both symmetry cases.

4.2. Optical spectrum

Another way of characterizing experimentally the phase symmetry is to consider the optical spectrum, which, apart from a translation in the frequency domain, is the square modulus of the Fourier transform F[ψ] = ˆψ(δω) of the complex amplitude ψ. The Fourier transform of the test function given by (8) is easily computed, using the fact that F[ψ0(t+ρ)^]=e^iδωρψˆ0. We obtain

ψˆ²=ψˆ0²(3+2 cos(ϕ1+δωρ)

+^{2 cos}(ϕ2−δωρ)+^{2 cos}(ϕ1−ϕ2+²δωρ)). ⁽³⁴⁾

-20 -15 -10 -5 0 5 10 15 20 0

0.2 0.4 0.6 0.8 1

-6 -4 -2 0 2 4 6

0 0.2 0.4 0.6 0.8 1

τ

τ Φ/Φm

a) stable cis state

b) stable trans state Φ/Φm

Figure 8. Normalized autocorrelation traces of triplet bound states having either cis- or trans-symmetry. Solid line, collinear autocorrelation trace; dashed line, non-collinear one. (a) The stable state S of figure2; (b) the stable state obtained with the parameters (ε, β, δ, µ, ν,D)=(1.8,0.5,−0.01,−0.05,0,1)in [3].

For the cis-symmetry ϕ1 = ϕ2 = ϕ, it is seen that| ˆψ²| is an even function ofδω: the spectrum is symmetrical around the central frequency ω0. On the other hand, the extremal values of| ˆψ²|/| ˆψ₀²|are 5+4 cosϕ(main maxima), 5−4 cosϕ (secondary maxima), and sin²ϕ (minima). It thus presents a nonzero minimal value.

For the trans-symmetry, as in the case of the auto- correlation trace, the two phases involved, (ϕ1+δωρ) ^and (ϕ2−δωρ), coincide in absolute value, and ϕ corresponds only to a shift of the oscillating part of the spectrum with re- spect to the centre of the spectrum of the single pulse| ˆψ0²|^. Then the spectrum is not symmetrical any more. The extremal values of| ˆψ²|/| ˆψ0²|do not depend onϕ; they are 9 (main max- ima), 1 (secondary maxima), and 0 (minima), i.e. the values obtained forϕ =0 in the cis-symmetry, sinceϕ1=ϕ2=0 is both of cis- and trans-symmetry.

Numerically computed spectra are given in figure 9, for the stable cis-states found in section 3.1 and the stable trans-state found in [3], as before. Results demonstrate that the optical spectrum allows us to conclude on whether the symmetry is cis or trans. Experimentally stable bound states of three pulses have been reported in the ytterbium-doped fibre laser [17] and also in the erbium-doped fibre laser [18]. In both cases, the mode locking is obtained using the nonlinear polarization rotation technique. From the optical spectra, we can conclude that these states had the trans-symmetry.

5. Conclusion

In this paper we have investigated new triplet bound states with particular symmetry. The model used is based on the usual cubic–quintic CGL equation. In addition to the classical trans- symmetric states we have shown that stable cis triplet bound states are solutions of the CGL equation. In contrast with the trans-symmetry, the cis bound state exists for parameters for which no stable two-pulse bound state exists. We have shown analytically that the leading term of the functionalJ[ψ],

-2 0 2

0 0.2 0.4 0.6 0.8 1

-10 0 10

0 0.2 0.4 0.6 0.8 1

2

a) stable cis state b) stable trans state

δω δω

|ψ|^ |ψ|^²

Figure 9. Normalized optical spectra of triplet bound states having either cis- or trans-symmetry. (a) The stable state S of figure2, (b) the stable state obtained with the parameters

(ε, β, δ, µ, ν,D)=(1.8,0.5,−0.01,−0.05,0,1)in [3].

which governs the evolution of the momentum of the solution to the CGL equation, vanishes for the cis-symmetry. This feature is closely related to the fact that three pulse bunches which have the cis-symmetry evolve very slowly, and enhances the stability in any case, as seen for the stable state. Also, the instabilities are weak, and the unstable states behave as quasi-stable ones. We found this way quasi-stable cis-states, one of which appears in a situation where a trans-state is known to exist. In this latter situation, a stable quasi-stationary triplet state has been found, which is not symmetrical either in dephasing or in separation. An interesting question arises from theoretical predictions. Is it possible to experimentally distinguish between the trans and cis triplet states? The response is fortunately yes: the symmetry of the three- pulse bound states can be identified through either the optical spectrum or the collinear autocorrelation trace.

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