Motion of solitons in CGL-type equations

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HAL Id: hal-02572701

https://hal.univ-angers.fr/hal-02572701

Preprint submitted on 13 May 2020

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Hervé Leblond, Foued Amrani, Alioune Niang, Boris Malomed, Valentin Besse

To cite this version:

Hervé Leblond, Foued Amrani, Alioune Niang, Boris Malomed, Valentin Besse. Motion of solitons in CGL-type equations. 2020. �hal-02572701�

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Motion of solitons in CGL-type equations

Herv´e Leblond¹, Foued Amrani¹, Alioune Niang¹, Boris Malomed², and Valentin Besse¹

1Laboratoire de Photonique d’Angers LϕA EA 4464, Universit´e d’Angers, France

2Department of Physical Electronics, Tel Aviv University, Israel

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Galilean invariance of CGL

The CGL equation is

∂E

∂z =δE+

β+iD 2

∂²E

∂t² + (ε+i)E|E|²+ (µ+iν)E|E|⁴

Ifβ = 0, and E₀(z,t) solution to CGL E =E₀(z,t−wz) expi

Dwt− Dw²/2 z is a solution moving at inverse speed w.

(4)

Galilean invariance of CGL

The CGL equation is

∂E

∂z =δE+

β+iD 2

∂²E

∂t² + (ε+i)E|E|²+ (µ+iν)E|E|⁴ E: electric field amplitude;

δ: net linear gain;

β: spectral gain bandwidth;

D =±1: dispersion;

ε: cubic nonlinear gain;

µ: quintic nonlinear gain;

ν: 4th-order nonlinear index;

z: number of round-trips;

Ifβ = 0, and E₀(z,t) solution to CGL E =E₀(z,t−wz) expi

H. Leblond, F. Amrani, A. Niang, B. Malomed, V. Besse Motion of solitons in CGL-type equations

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Galilean invariance of CGL

The CGL equation is

∂E

∂z =δE+

β+iD 2

∂²E

∂t² + (ε+i)E|E|²+ (µ+iν)E|E|⁴

Ifβ = 0, and E₀(z,t) solution to CGL

E =E₀(z,t−wz) expi

(6)

Galilean invariance of CGL

Ifβ = 0, and E0(z,t) solution to CGL

E =E0(z,t−wz) expi

t

z

-2 -1 0 1 2 3 4 5

0 0.5 1 1.5 2 2.5 3

(7)

β∂²E/∂t² breaks Galilean invariance and prevents any motion of the solitons.

Ifβ 6= 0: the moving soliton does not exist.

t

z

-2 -1 0 1 2 3 4 5

0 0.5 1 1.5 2 2.5 3

Atz= 0,E=E0expi∆ωt,β= 0.55 instead of 0

Strong breaking due to the limited gain bandwidth.

(8)

1 Motion induced by a continuous wave Crystal, liquid and gas of solitons Pulse motion due to gain dynamics Injected continuous wave

2 The finite bandwidth of gain as a viscous friction Analytical expression of the viscous friction Numerical validation of the approximation

3 Transverse mobility in the presence of periodic potential Fondamental soliton

Dipoles and vortices

(9)

Experiments in mode-locked fiber lasers:

=⇒ existence of “soliton gas”

A large number of solitons in motion.

Spectral bandwidth of gain is finite:β 6= 0 Is the motion it due to the cw component?

Try to inject cw.

∂E

∂z = δE+

β+iD 2

100 200 0 200 400 600 800 1000

Soliton gas.

∆ν₀ = 0.8.

(13)

Changing “states of matter” of solitons

t

z

-200

-100

0

100

2000 200 400 600 800 1000

t

z

-200

-100

0

100

2000 200 400 600 800 1000

t

z

-200

-100

0

100

2000 200 400 600 800 1000

Soliton crystal, Soliton liquid, Soliton gas.

∆ν0 = 1.2, ∆ν0 = 0.9, ∆ν0 = 0.8.

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Changing “states of matter” of solitons

t

z -200

-100

0

100

200 0200400600800 1000

t

z -200

-100

0

100

200 0200400600800 1000

t

z -200

-100

0

100

200 0200400600800 1000

The different regimes vs detuning ∆ω₀ and amplitude A=A_cw of injected cw

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For a single soliton, input of the formE =E0expi∆ω1t, varyingA, ∆ω₀ and ∆ω₁ =⇒ Pulse motion.

(16)

For a single soliton, input of the formE =E₀expi∆ω₁t, varyingA, ∆ω0 and ∆ω1 =⇒ Pulse motion.

The velocity depends on Aand ∆ω0 (injected cw), but not on ∆ω₁ (initial speed).

i.e. Speed is entirely determined by injected cw.

Soliton velocity vs ∆ν0= ∆ω0/2πforA= 0.004 (green dotted), 0.002 (red dashed), 0.001 (blue solid line).

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For a single soliton, input of the formE =E₀expi∆ω₁t, varyingA, ∆ω0 and ∆ω1 =⇒ Pulse motion.

Speed is entirely determined by injected cw.

(18)

For a single soliton, input of the formE =E0expi∆ω1t, varyingA, ∆ω₀ and ∆ω₁ =⇒ Pulse motion.

The velocity depends on Aand ∆ω0 (injected cw), but not on ∆ω₁ (initial speed).

i.e. Speed is entirely determined by injected cw.

Motion is restored but Galilean invariance is not.

(19)

“Brownian motion” induced by injected cw

Soliton velocity is fixed by the cw.

At higher amplitudes of injected cw:

More complex nonlinear interaction between cw and solitons.

=⇒ The amplitude of cw (radiation) varies with t.

A tiny variation of the cw componentin either amplitude or frequency

changes radically the soliton velocity

=⇒ apparently random variations of the the soliton speed.

=⇒ Erratic motion of solitons, and soliton gas.

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An integral term accounting for the fast gain dynamics

∂E

∂z =δE+

β+iD 2

∂²E

∂t² + (ε+i)E|E|²+ (µ+iν)E|E|⁴

−ΓE Z t

−∞

|E|²−<|E|² >

dt⁰

Represents the decrease of the population inversion

(and hence of gain)

when stimulated emission occurs.

(22)

A single pulse (or two-pulse)input is unstable:

New pulses form in front of the input (towardst<0), and quickly disappear.

Unstable pulse emission repeats all along the cavity,

=⇒ multi-pulse pattern Solitons move slowly

t

z

-100

-50

0

50

100

0 1000 2000 3000 4000 5000

Γ = 0.01

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For larger Γ, the instability increases:

pulses form and vanish faster.

Then, the pulse train does not stabilize any more:

pulses are created and vanish permanently

t

z

100 50 0 -50 -100

0 200 400 600 800 1000

Γ = 0.03

Generation and vanishing process =⇒ effective soliton motion

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Generation and vanishing process =⇒ effective soliton motion The inverse velocity w of this motion is very large

t

z

100 50 0 -50 -100

0 200 400 600 800 1000

Γ = 0.1.

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Generation and vanishing process =⇒ effective soliton motion The inverse velocity w of this motion is very large

t

z

100 50 0 -50 -100

0 200 400 600 800 1000

Γ = 0.1.

For high values of Γ, the moving soliton is unstable and vanishes.

Gain dynamics can induce soliton motion.

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Model including external injection,fast gain dynamics, andgain saturation.

∂E

∂z =

g₀

1+<|E|² > /I_s −r

E+

β+iD 2

∂²E

∂t² + (ε+i)E|E|²+ (µ+iν)E|E|⁴

−ΓE Z t

−∞

|E|²−<|E|²>

dt⁰+Aexp (−i∆ω₀t)

(28)

Gain saturation limits the number of solitons

=⇒ A liquid: condensed phase, which does not fill the box

t

z

50 40 30 20 10 0 -10 -20 -30 -40

0 10000 20000 30000 40000 50000

∆ν0= 0.1,A= 0.115, with velocity compensation,w=−0.05877.

Not a crystal: no phase-locking

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We can still have a soliton gas:

t

z

50 40 30 20 10 0 -10 -20 -30 -40

0 20000 40000 60000 80000 100000

∆ν0= 0.1,A= 0.120,w=−0.07040.

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Harmonic mode-locking

Equidistant solitons filling all the box,

stable state. When 0.125≤A≤0.133.

t

z

50 40 30 20 10 0 -10 -20 -30 -40

0 10000 20000 30000 40000 50000

∆ν0= 0.1,A= 0.130,w=−0.02755.

Consecutive pulses are phase-locked: a crystal, but the crystal length exactly matches the box length.

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In a three bunch pattern,

elastic interaction according to the Newton’s cradle scenario

t

z

50 40 30 20 10 0 -10 -20 -30 -40

0 100000 200000 300000 400000 500000

A= 0.130, ∆ν0= 0.5 andw=−0.0024.

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The CGL equation (u=E)

uz=δu+

β+iD 2

utt + (ε+i)u|u|²+ (µ+iν)u|u|⁴.

Moving solution:u =u₀(t−T,z)eⁱ^(ωt−kz) with u0(t,z) solution to CGL with β= 0, T =Vz, ω= ^V_D and k = ^V_2D².

Perturbative approach: Consider some small non zero β.

u a soliton solution, M =

Z +∞

−∞

|u|²dt: its mass, T =

Z +∞

−∞

t|u|²dt/M: position of its center of mass.

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uz=δu+

β+iD 2

Z +∞

−∞

|u|²dt: its mass, T =

Z +∞

−∞

(35)

uz=δu+

β+iD 2

Z +∞

−∞

|u|²dt: its mass,

T = Z +∞

−∞

(36)

The velocity of the pulse is then dT

dz = 1 M

Z

t(u_zu^∗+cc)dt

(cc: complex conjugate,uz=∂u/∂z).

Using the CGL equation:

dT dz = 1

M

I1+iD

2 I2+βI3

dt,

where

I1= 2 Z

t δ|u|²+ε|u|⁴+µ|u|⁶ dt, I2 =

Z

t(uttu^∗−cc)dt,

I₃ = Z

t(u_ttu^∗+cc)dt.

(37)

The velocity of the pulse is then dT

dz = 1 M

Z

t(u_zu^∗+cc)dt

(cc: complex conjugate,uz=∂u/∂z).

Using the CGL equation:

dT dz = 1

M

I1+iD

2 I2+βI3

dt,

where

I1= 2 Z

t δ|u|²+ε|u|⁴+µ|u|⁶ dt, I2 =

Z

t(uttu^∗−cc)dt,

I₃ = Z

t(u_ttu^∗+cc)dt.

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Assumption: u₀ is a symmetrical pulse, centered att =T,

consequently the functionu0(t⁰), witht⁰=t−T, is even.

Then

I1= 2 Z

t δ|u|²+ε|u|⁴+µ|u|⁶ dt, becomes

I₁ = 2T Z

δ|u₀|²+ε|u₀|⁴+µ|u₀|⁶ dt⁰.

and so on.

Then we can compute the acceleration d²T/dz²:

d²T dz² = iD

2M dI2

dz.

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Then

I1= 2 Z

I₁ = 2T Z

δ|u₀|²+ε|u₀|⁴+µ|u₀|⁶ dt⁰.

and so on.

d²T dz² = iD

2M dI2

dz.

(40)

Then

I1= 2 Z

I₁ = 2T Z

δ|u₀|²+ε|u₀|⁴+µ|u₀|⁶ dt⁰.

and so on.

d²T dz² = iD

2M dI2

dz.

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d²T dz² = iD

2M dI₂ dz.

Using integration by parts and parity we compute a set of integrals

Finally; we obtain the expression of the force F =Md²T/dz²: F =−4β

Z

|u_0t⁰|²dt⁰V

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d²T dz² = iD

2M dI₂ dz.

Using integration by parts and parity we compute a set of integrals

Finally; we obtain the expression of the force F =Md²T/dz²:

F =−4β Z

|u_0t⁰|²dt⁰V

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the equation of motion is

MdV

dz =F =−4β Z

|u_0t⁰|²dt⁰V,

Hence, the velocity evolves as V(z) =V(0)e^−λz with the decay rate

λ= 4β M

Z

|u_0t⁰|²dt⁰.

(44)

the equation of motion is

MdV

dz =F =−4β Z

|u_0t⁰|²dt⁰V,

Hence, the velocity evolves as V(z) =V(0)e^−λz with the decay rate

λ= 4β M

Z

|u_0t⁰|²dt⁰.

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(46)

An example of calculation

Initial velocityV0= 0.7 and gain bandwidth coefficientβ= 0.004.

White line: approximate analytical solution Good agreement with the numerical solution.

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We plot the characteristics of the pulse motion vs z

Logarithmic scale.

V:velocity;γ: acceleration andM: mass;F: force from above theory;

∆F/F: relative difference betweenF andMγ.

Parameters:ω= 1,β= 0.0124.

(48)

Effect of finite bandwidth of gain on CGL soliton

with anomalous dispersion

is equivalent to a viscous friction force,

if it is not too large.

To construct simplified models to describe CGL soliton interactions as forces between effective particles: The lack of Galilean invarianceof CGL was a major difficulty,

since the concept of force is based on it.

With our result, we can approach soliton interaction in aconservative frame.

Then, the finite bandwidth of gain could be treated as a phenomenological friction force.

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(2+1)-D spatial Ginzburg-Landau equation:

∂u

∂Z =

−δ+iV(X,Y) + i

2∇²_⊥+ (i+)|u|²−(iν+µ)|u|⁴

u,

∇²_⊥=∂²/∂X²+∂²/∂Y²: the paraxial diffraction

A periodic potential: V(X,Y) =−V₀[cos(2X) + cos(2Y)]

breaks Galilean invariance

δ = 0.4,ε= 1.85,µ= 1,ν = 0.1,V₀ = 1,

for which the quiescent fundamental soliton is stable.

(54)

Y

X

-3 -2 -1 0 1 2 3

|u(X,Y)|; |u(X)|atY = 0.

The stable fundamental soliton

(55)

Inputu =u₀exp (ik₀R),

withR= (X,Y), andk0= (k0cosθ,k0sinθ) (0≤θ≤π/4)

In an amplifier, the factor (ik0R)

represents a deviation of the wave vector kfrom theZ-axis.

Indeed, CGL is derived within the SVEA:

Either E =U(X,Y,Z −vT)e^i(k^x^X^+k^Y^Y^+k^Z^Z−ωT⁾+c.c., or E =u(X,Y,Z−vT)eⁱ^(k^Z^Z^−ωT)+c.c.,

with u(X,Y,Z −vT) =U(X,Y,Z−vT)e^i(k^X^X^+k^Y^Y^).

Equivalent ifk_X,k_Y are small enough.

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Fundamental soliton with initial kick

Ifk0 is small, the pulse oscillates in the potential site

X

Z

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

0 10 20 30 40 50 60 70 80 90

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

Z = 28.00

(61)

Y

X

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

-20 -15 -10-5 0 5 1015 20

Y

X

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

-20 -15 -10-5 0 5 10 1520

Y

X

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

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

Y

X

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

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

Z= 5.29 Z= 9.39 Z= 17.01 Z= 28.00

|u(X,Z)|atY = 0, fork0= 1.6878,θ= 0.

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

Z = 48.24.77

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Y

X

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

Z= 48.24.77, fork0= 1.694

An arrayed set of 5 fix + 1 moving solitons.

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The total number of emitted solitons first grows fast with k0.

It reaches a maximum of 5(6 with the initial one)

Then slowly goes down to 0 (the initial one only)

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For the highest k0, the soliton moves freely

X

Z

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

0 50 100150 200 250 300350 400 450

|u(X,Z)|atY = 0, transverse speed,k0= 2.1,θ= 0.

The soliton velocity increases, approaching a certain limit value

(71)

Periodic elastic collisions

A moving soliton with one fix soliton

X

Z

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

0 100 200 300 400 500 600

k0= 1.867,θ= 0.

An example of the Newton’s-cradle scenario

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1 moving and 5 fix solitons

5 soliton-pattern moving soliton

elastic collisions

change of direction

absorption

6 soliton-pattern

|u(X,Z)|atY = 0; k0= 1.693,θ= 0.

An quite complex interaction

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elastic collisions

change of direction

absorption

6 soliton-pattern 5 soliton-pattern

moving soliton

|u(X,Z)|atY = 0; k0= 1.693,θ= 0.

(74)

change of direction

absorption

6 soliton-pattern

elastic collisions 5 soliton-pattern

moving soliton

|u(X,Z)|atY = 0; k0= 1.693,θ= 0.

(75)

absorption

moving soliton

change of direction elastic collisions

|u(X,Z)|atY = 0; k0= 1.693,θ= 0.

(76)

moving soliton

elastic collisions

absorption

change of direction

|u(X,Z)|atY = 0; k0= 1.693,θ= 0.

(77)

change of direction 5 soliton-pattern

moving soliton

elastic collisions

6 soliton-pattern absorption

|u(X,Z)|atY = 0; k0= 1.693,θ= 0.

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The dipole

Y

X

-4 -3 -2 -1 0 1 2 3 4

-2 -1 0 1 2 3 4 5

Y

X

-4 -3 -2 -1 0 1 2 3 4

-2 -1 0 1 2 3 4 5

-1.5 -1 -0.5 0 0.5 1 1.5 2

Amplitude phase (in units ofπ)

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Moving the dipole

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

0 25 50 75 100 125 150 175 200

|u(X,Y,Z)|atY = 0, fork0= 1.884.

After several quasi-elastic elastic collisions, the moving dipole absorbs the stationary chain

(86)

Transient Newton’s cradle with clearing the obstacle

X

Z

-8 -6 -4 -2 0 2 4 6 8

58 58.5 59 59.5 60 60.5 61 61.5

|u(X,Y,Z)|atY = 0, fork0= 1.884.

After several quasi-elastic elastic collisions, the moving dipole absorbs the stationary chain

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Square-shaped (offsite-centered) vortex.

Y

X

-2 0 2 4 6

Y

X

-2 0 2 4 6

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Amplitude phase.

It is unstable

(88)

Moving the vortex

Y

X

-15 -10 -5 0 5 10 15

Amplitude atZ'300, fork0= 1.5, andθ=π/8, 5π/8, 9π/8, 13π/8.

A set of fundamental solitons is formed

For a clockwise rotating vortex, solitons form on the other line

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Moving the vortex

Y

X

-15 -10 -5 0 5 10 15

(90)

Moving the vortex

Y

X

-15 -10 -5 0 5 10 15

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Moving the vortex

Y

X

-15 -10 -5 0 5 10 15

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Moving the vortex

Y

X

-15 -10 -5 0 5 10 15

Y

X

-15 -10 -5 0 5 10 15

Y

X

-15 -10 -5 0 5 10 15

Y

X

-15 -10 -5 0 5 10 15

For a clockwise rotating vortex, solitons form on the other line The position of the soliton set depends of the direction of the kick with respect to vortex orientation

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