Propagation de solitons optiques à quelques cycles dans des guides d’ondes couplés

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Propagation de solitons optiques à quelques cycles dans des guides d’ondes couplés

Hervé Leblond, Dumitru Mihalache, David Kremer, Said Terniche

To cite this version:

Hervé Leblond, Dumitru Mihalache, David Kremer, Said Terniche. Propagation de solitons optiques à quelques cycles dans des guides d’ondes couplés. ICOPA 4, 2016, Bordeaux, France. �hal-02564155�

Solitons optiques ` a quelques cycles dans des guides coupl´ es

Herv´e Leblond¹, Dumitru Mihalache², David Kremer³, Said Terniche^1,4

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

2Horia Hulubei National Institute for Physics and Nuclear Engineering, and Academy of Romanian Scientists, Bucharest.

3Laboratoire MOLTECH-Anjou, CNRS UMR 6200, Universit´e d’Angers.

4Laboratoire Electronique Quantique, USTHB, Alger.

1 Waveguiding of a few-cycle pulse How to model it

Nonlinear widening of the linear guided modes

2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms

Few-cycle optical solitons in linearly coupled waveguides

3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind of solitons

Solitary wave vs envelope solitons

Envelope soliton: the usual optical soliton in the ps range

Pulse durationLλwavelength Typical model: NonLinear Schr¨odinger equation (NLS)

It is a soliton if it propagates without deformation on DL, due to nonlinearity.

In linear regime: spread out by dispersion.

Solitary wave soliton: the hydrodynamical soliton A single oscillation

Typical model: Korteweg-de Vries equation (KdV)

Few-cycle optical solitons:L∼λ

The slowly varying envelope approximation is not valid Generalized NLS equation

Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d’Angers LϕA EA 4464, Universit´Propagation de solitons optiques `a quelques cycles dans des guides d’ondes coupl´e d’Angers., Horia Hulubei National Institute for Physics and Nuclear Engineering, and Academy of Romanian Scientists, Bucharest., Laboratoire MOLTECH-Anjou, CNRS UMR 6200, Universit´es3 / 29 e d’Angers., Laboratoire Electronique Quantique, USTHB, Alger. )

Solitary wave vs envelope solitons

Envelope soliton: the usual optical soliton in the ps range

Pulse durationLλwavelength Typical model: NonLinear Schr¨odinger equation (NLS) It is a soliton if it propagates without deformation on DL, due to nonlinearity.

In linear regime: spread out by dispersion.

Solitary wave soliton: the hydrodynamical soliton A single oscillation

Typical model: Korteweg-de Vries equation (KdV)

Few-cycle optical solitons:L∼λ

The slowly varying envelope approximation is not valid Generalized NLS equation

We seek a different approach based on KdV-type models

Solitary wave vs envelope solitons

Envelope soliton: the usual optical soliton in the ps range

Pulse durationLλwavelength Typical model: NonLinear Schr¨odinger equation (NLS) Solitary wave soliton: the hydrodynamical soliton

A single oscillation

Typical model: Korteweg-de Vries equation (KdV)

Few-cycle optical solitons:L∼λ

The slowly varying envelope approximation is not valid Generalized NLS equation

We seek a different approach based on KdV-type models

The mKdV model

A two-level model with resonance frequency ω UV transition only, with (1/τ_p)ω

1/ τ_p ω

=⇒ Long-wave approximation

modified Korteweg-de Vries (mKdV) equation

∂E

∂ζ = 1 6

d³k dω³ ω=0

∂³E

∂τ³ − 6π

ncχ⁽³⁾(ω;ω,ω,−ω) ω=0

∂E³

∂τ

H. Leblond and F. Sanchez,Phys. Rev. A67, 013804 (2003)

Waveguide description

The evolution of the electric field E: In (1+1) dimensions:

−→ The modified Korteweg-de Vries (mKdV) equation

∂_ζE+β∂_τ³E +γ∂_τE³= 0 Nonlinear coefficient γ = 1

2ncχ⁽³⁾, Dispersion parameter β= (−n⁰⁰)

2c ,

Waveguide description

The evolution of the electric field E: We generalize to (2+1) dimensions:

−→ The cubic generalized Kadomtsev-Petviashvili (CGKP) equation

∂ζE+β∂_τ³E +γ∂τE³−V 2

Z τ

∂_ξ²Edτ⁰ = 0 Nonlinear coefficient γ = 1

2ncχ⁽³⁾, Dispersion parameter β= (−n⁰⁰)

2c , Linear velocity:V = c

n.

Waveguide description

The evolution of the electric field E:

A waveguide:

c g c x

core

cladding cladding

−→ The cubic generalized Kadomtsev-Petviashvili (CGKP) equation

∂_ζE+β_α∂_τ³E +γ_α∂_τE³ −V_α 2

Z τ

∂_ξ²Edτ⁰ = 0 with α=g in the core andα=c in the cladding.

Nonlinear coefficient γ_α= 1 2n_αcχ⁽³⁾_α , Dispersion parameter β_α = (−n⁰⁰_α)

2c , Linear velocity:Vα = c

nα

.

Waveguide description

The evolution of the electric field E:

A waveguide:

c g c x

core

cladding cladding

−→ The cubic generalized Kadomtsev-Petviashvili (CGKP) equation

∂_ζE+β_α∂_τ³E +γ_α∂_τE³+ 1

V_α∂_τE−Vα

2 Z _τ

∂_ξ²Edτ⁰ = 0 with α=g in the core andα=c in the cladding.

Velocities : V_g <V_c

Nonlinear coefficient γα= 1 2n_αcχ⁽³⁾_α , Dispersion parameter βα = (−n⁰⁰_α)

2c , Linear velocity:Vα = c

n_α.

Waveguide description

The evolution of the electric field E:

A waveguide:

c g c x

core

cladding cladding

−→ The cubic generalized Kadomtsev-Petviashvili (CGKP) equation In dimensionless form:

∂zu =Aα∂_t³u+Bα∂tu³+vα∂tu+Wα

2 Z t

∂_x²udt⁰ with α=g in the core andα=c in the cladding.

Relative inverse velocities : v_g >v_c Nonlinear coefficient γα= 1

2n_αcχ⁽³⁾_α , Dispersion parameter βα = (−n⁰⁰_α)

2c , Linear velocity:Vα = c

nα

.

Nonlinear propagation in linear guide

We solve the CGKP equation starting from

u(x,t,z = 0) =Acos(ωt)f(x)e^−t2/w2, f(x) =

cos(kxx), for |x| ≤a,

Ce^−κ|x|, for |x|>a, is a linear mode profile Normalized coefficients A₁=A₂=B₁ =B₂ =W₁=W₂ = 1,

−→ we assume that

- Temporal compression occurs

- Spatial defocusing occurs, (else it collapses!)

- Dispersion and nonlinearity are identical in core and cladding.

Guided wave profiles

(Normalized so that the total power is 1.v2= 3,w= 2.)

The pulse is less confined in nonlinear (blue, and red)

than in linear (pink and cyan)regime.

Nonlinear waveguide

Wave guided and confined by using nonlinear velocity:

a higher nonlinear coefficient in the cladding than that in the core.

Guided profiles of the nonlinear waveguide. Normalized so that the total power is 1.

Two-cycle soliton of the nonlinear waveguide

x

t

-2

-1

0

1

2

-150 -100 -50 0 50 100 150

B2−B1= 1.

H. Leblond and D. Mihalache,Phys. Rev. A88, 023840 (2013)

1 Waveguiding of a few-cycle pulse How to model it

Nonlinear widening of the linear guided modes

2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms

Few-cycle optical solitons in linearly coupled waveguides

3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind of solitons

2D waveguiding structure: two cores 1 and 2 and dielectric cladding

The generalized Kadomtsev-Petviashvili (GKP) equation (dimensionless)

∂_zu =A_α∂_t³u+B_α∂_tu³+V_α∂_tu+w_α 2

Z t

∂_x²udt, α=g in the cores 1 and 2,α=c in the cladding.

We seek for a solution as

u =R(t,z)f₁(x)e^iϕ+S(t,z)f₂(x)e^iϕ, i.e., two interacting modes.

∗fj, (j = 1, 2): linear mode profiles of individual guides,

∗R(t,z),S(t,z): longitudinal wave profiles,

∗ϕ=ωt−βz.

We report it into the GKP equation and get after averaging on x:

∂zR=∂zS = iw_g 2ω

(K_c−K_g)I₂ 1 +I1

(R+S), Involve overlap integrals I1 =R∞

−∞f1f2dx,I2 =R

g1f1f2dx =R

g2f1f2dx

(“R

g_j·dx” is the integral over the corej= 1 or 2.)

We seek for a solution as

u =R(t,z)f₁(x)e^iϕ+S(t,z)f₂(x)e^iϕ, i.e., two interacting modes.

∗fj, (j = 1, 2): linear mode profiles of individual guides,

∗R(t,z),S(t,z): longitudinal wave profiles,

∗ϕ=ωt−βz.

We report it into the GKP equation and get after averaging on x:

∂zR=∂zS = iw_g 2ω

(K_c−K_g)I₂ 1 +I1

(R+S), Involve overlap integrals I1 =R∞

−∞f1f2dx,I2 =R

g1f1f2dx =R

g2f1f2dx

(“R

g_j·dx” is the integral over the corej= 1 or 2.)

We seek for a solution as

u =R(t,z)f₁(x)e^iϕ+S(t,z)f₂(x)e^iϕ, i.e., two interacting modes.

∗fj, (j = 1, 2): linear mode profiles of individual guides,

∗R(t,z),S(t,z): longitudinal wave profiles,

∗ϕ=ωt−βz.

We report it into the GKP equation and get after averaging on x:

∂zR=∂zS = iw_g 2ω

(K_c−K_g)I₂ 1 +I1

(R+S), Involve overlap integrals I1 =R∞

−∞f1f2dx,I2 =R

g1f1f2dx =R

g2f1f2dx

(“R

g_j·dx” is the integral over the corej= 1 or 2.)

We seek for a solution as

u=R(z)f1(x)e^iϕ+S(z)f2(x)e^iϕ, i.e., two interacting modes.

We report it into the GKP equation and get after averaging on x:

∂zR=∂zS = iw_g 2ω

(K_c−K_g)I₂ 1 +I1

(R+S),

The few-cycle pulse is expanded as a Fourier integral of such modes, u₁=R

Re^iϕdω,u₂ =R

Se^iϕdω.

We report∂zR and∂zS into ∂zu1, and get the linear coupling terms.

Finally, we get the system of two coupled modified Korteweg-de Vries (mKdV) equations

∂_zu₁=A∂_t³u₁+B∂_tu₁³+V∂_tu₁+C∂_tu₂+D∂³_tu₂,

∂_zu₂=A∂_t³u₂+B∂_tu₂³+V∂_tu₂+C∂_tu₁+D∂³_tu₁,

We seek for a solution as

u=R(z)f1(x)e^iϕ+S(z)f2(x)e^iϕ, i.e., two interacting modes.

We report it into the GKP equation and get after averaging on x:

∂zR=∂zS = iw_g 2ω

(K_c−K_g)I₂ 1 +I1

(R+S),

The few-cycle pulse is expanded as a Fourier integral of such modes, u₁=R

Re^iϕdω,u₂ =R

Se^iϕdω.

We report∂zR and∂zS into ∂zu1, and get the linear coupling terms.

Finally, we get the system of two coupled modified Korteweg-de Vries (mKdV) equations

∂_zu₁=A∂_t³u₁+B∂_tu₁³+V∂_tu₁+C∂_tu₂+D∂³_tu₂,

∂_zu₂=A∂_t³u₂+B∂_tu₂³+V∂_tu₂+C∂_tu₁+D∂³_tu₁,

We seek for a solution as

u=R(z)f1(x)e^iϕ+S(z)f2(x)e^iϕ, i.e., two interacting modes.

We report it into the GKP equation and get after averaging on x:

∂zR=∂zS = iw_g 2ω

(K_c−K_g)I₂ 1 +I1

(R+S),

The few-cycle pulse is expanded as a Fourier integral of such modes, u₁=R

Re^iϕdω,u₂ =R

Se^iϕdω.

We report∂zR and∂zS into ∂zu1, and get the linear coupling terms.

Finally, we get the system of two coupled modified Korteweg-de Vries (mKdV) equations

∂_zu₁=A∂_t³u₁+B∂_tu₁³+V∂_tu₁+C∂_tu₂+D∂³_tu₂,

∂_zu₂=A∂_t³u₂+B∂_tu₂³+V∂_tu₂+C∂_tu₁+D∂³_tu₁,

We seek for a solution as

u=R(z)f1(x)e^iϕ+S(z)f2(x)e^iϕ, i.e., two interacting modes.

We report it into the GKP equation and get after averaging on x:

∂zR=∂zS = iw_g 2ω

(K_c−K_g)I₂ 1 +I1

(R+S),

The few-cycle pulse is expanded as a Fourier integral of such modes, u₁=R

Re^iϕdω,u₂ =R

Se^iϕdω.

We report∂zR and∂zS into ∂zu1, and get the linear coupling terms.

Finally, we get the system of two coupled modified Korteweg-de Vries (mKdV) equations

∂_zu₁=A∂_t³u₁+B∂_tu₁³+V∂_tu₁+C∂_tu₂+D∂³_tu₂,

∂_zu₂=A∂_t³u₂+B∂_tu₂³+V∂_tu₂+C∂_tu₁+D∂³_tu₁,

Nonlinear coupling

An analogous procedure, treating the nonlinear term as a perturbation, allows to derive the nonlinear coupling terms The complete final system is

∂_zu₁ = A∂³_tu₁+B∂_tu³₁+V∂_tu₁

+C∂tu2+D∂_t³u2+E∂t 3u₁²u2+u₂³

∂_zu₂ = A∂³_tu₂+B∂_tu³₂+V∂_tu₂

+C∂_tu₁+D∂_t³u₁+E∂_t 3u₁u₂+u₁³

H. Leblond, and S. Terniche,Phys. Rev. A93, 043839 (2016)

Nonlinear coupling

The complete final system is

∂zu1 = A∂³_tu1+B∂tu³₁+V∂tu1

+C∂tu2+D∂_t³u2+E∂t 3u₁²u2+u₂³

∂zu2 = A∂³_tu2+B∂tu³₂+V∂tu2

+C∂tu1+D∂_t³u1+E∂t 3u1u2+u₁³ We evidence

a standardlinear couplingterm,

a linearcouplingtermbased on dispersion, anonlinear couplingterm

H. Leblond, and S. Terniche,Phys. Rev. A93, 043839 (2016)

Nonlinear coupling

The complete final system is

∂zu1 = A∂³_tu1+B∂tu³₁+V∂tu1

+C∂tu2+D∂_t³u2+E∂t 3u₁²u2+u₂³

∂zu2 = A∂³_tu2+B∂tu³₂+V∂tu2

+C∂tu1+D∂_t³u1+E∂t 3u1u2+u₁³ We evidence

a standardlinear couplingterm,

a linearcouplingtermbased on dispersion, anonlinear couplingterm

H. Leblond, and S. Terniche,Phys. Rev. A93, 043839 (2016)

Nonlinear coupling

The complete final system is

∂zu1 = A∂³_tu1+B∂tu³₁+V∂tu1

+C∂tu2+D∂_t³u2+E∂t 3u₁²u2+u₂³

∂zu2 = A∂³_tu2+B∂tu³₂+V∂tu2

+C∂tu1+D∂_t³u1+E∂t 3u1u2+u₁³ We evidence

a standardlinear couplingterm,

a linearcouplingtermbased on dispersion, anonlinear couplingterm

H. Leblond, and S. Terniche,Phys. Rev. A93, 043839 (2016)

Nonlinear coupling

The complete final system is

∂zu1 = A∂³_tu1+B∂tu³₁+V∂tu1

+C∂tu2+D∂_t³u2+E∂t 3u₁²u2+u₂³

∂zu2 = A∂³_tu2+B∂tu³₂+V∂tu2

+C∂tu1+D∂_t³u1+E∂t 3u1u2+u₁³ We evidence

a standardlinear couplingterm,

a linearcouplingtermbased on dispersion, anonlinear couplingterm

H. Leblond, and S. Terniche,Phys. Rev. A93, 043839 (2016)

We assume a purely linear and non-dispersive coupling

∂zu = −∂_t(u³)−∂_t³u−C∂tv,

∂_zv = −∂_t(v³)−∂_t³v−C∂_tu,

We look for stationary states (vector solitons) in this model The ”stationary” states oscillate witht andz: .

few-cycle solitons arebreathers.

Atypical example of few-cycle vector soliton

(Dotted lines:u, solid lines:v. Left: atz= 0, right: atz= 60.<Au>= 1.837).

Evolution of soliton’s maximum amplitude during propagation.

1.6 1.8 2

0 1 2 3 4 5 6 7 8

maxt(|u|)

z

0.4 0.5 0.6

0 1 2 3 4 5 6 7 8

maxt(|v|)

z

Soliton with<Au>= 1.789.

Two types of oscillations:

• Fast: phase - group velocity mismatch

• Slower: periodic energy exchange, as in linear regime.

Consider now the coupled equations in thelinearized case.

Themonochromaticsolutions are u

v

= A

B

e^{−i(ωt+bω3z)},

With, due to coupling,

A=u0coscωz+iv0sincωz, B=v0coscωz+iu0sincωz.

The maximum amplitude and the power density of the wave oscillate with spatial frequencycω/π=σ₀ = 1.326.

Consider now the coupled equations in thelinearized case.

Themonochromaticsolutions are u

v

= A

B

e^{−i(ωt+bω3z)}, With, due to coupling,

A=u0coscωz+iv0sincωz, B=v0coscωz+iu0sincωz.

The maximum amplitude and the power density of the wave oscillate with spatial frequencycω/π=σ₀ = 1.326.

Consider now the coupled equations in thelinearized case.

Themonochromaticsolutions are u

v

= A

B

e^{−i(ωt+bω3z)}, With, due to coupling,

A=u0coscωz+iv0sincωz, B=v0coscωz+iu0sincωz.

The maximum amplitude and the power density of the wave oscillate with spatial frequencycω/π=σ₀ = 1.326.

Consider now the coupled equations in thelinearized case.

Themonochromaticsolutions are u

v

= A

B

e^{−i(ωt+bω3z)}, With, due to coupling,

A=u0coscωz+iv0sincωz, B=v0coscωz+iu0sincωz.

The maximum amplitude and the power density of the wave oscillate with spatial frequencycω/π=σ₀ = 1.326.

Oscillations of the few-cycle vector solitons

The energies Eu=R

u²dt andEv =R

v²dt oscillate

almost harmonically, as E_u =<E_u >+∆E_usin(2πσ_az+φ_E,u), The same for Au= maxt(|u|) andAv = maxt(|v|)

Spatialfrequency σa∈[1.06,1.17], increasing with<Au >.

(linear:σ0= 1.326).

Amplitudes of oscillations vs amplitude of fieldu

0 0.05 0.1 0.15 0.2

1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86

∆ Au, ∆ Av, ∆ Eu

<A_u>

black saltires: ∆Eu; blue stars: ∆Au; red crosses: ∆Av.

Well fitted with∆E 'Rp

A −<A >, etc., with A = 1.854.

Oscillations of the few-cycle vector solitons

The energies Eu=R

u²dt andEv =R

v²dt oscillate

almost harmonically, as E_u =<E_u >+∆E_usin(2πσ_az+φ_E,u), The same for Au= maxt(|u|) andAv = maxt(|v|)

Spatialfrequency σa∈[1.06,1.17], increasing with<Au >.

(linear:σ0= 1.326).

Amplitudes of oscillations vs amplitude of fieldu

0 0.05 0.1 0.15 0.2

1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86

∆ Au, ∆ Av, ∆ Eu

<A_u>

black saltires: ∆Eu; blue stars: ∆Au; red crosses: ∆Av.

Well fitted with∆E 'Rp

A −<A >, etc., with A = 1.854.

Oscillations of the few-cycle vector solitons

The energies Eu=R

u²dt andEv =R

v²dt oscillate

almost harmonically, as E_u =<E_u >+∆E_usin(2πσ_az+φ_E,u), The same for Au= maxt(|u|) andAv = maxt(|v|)

Spatialfrequency σa∈[1.06,1.17], increasing with<Au >.

(linear:σ0= 1.326).

Amplitudes of oscillations vs amplitude of fieldu

0 0.05 0.1 0.15 0.2

1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86

∆ Au, ∆ Av, ∆ Eu

<A_u>

black saltires: ∆Eu; blue stars: ∆Au; red crosses: ∆Av.

Well fitted with∆E 'Rp

A −<A >, etc., with A = 1.854.

Oscillations of the few-cycle vector solitons

The energies Eu=R

u²dt andEv =R

v²dt oscillate

almost harmonically, as E_u =<E_u >+∆E_usin(2πσ_az+φ_E,u), The same for Au= maxt(|u|) andAv = maxt(|v|)

Spatialfrequency σa∈[1.06,1.17], increasing with<Au >.

(linear:σ0= 1.326).

Amplitudes of oscillations vs amplitude of fieldu

0 0.05 0.1 0.15 0.2

1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86

∆ Au, ∆ Av, ∆ Eu

<A_u>

black saltires: ∆Eu; blue stars: ∆Au; red crosses: ∆Av.

Well fitted with∆E 'Rp

A −<A >, etc., with A = 1.854.

Oscillations of the few-cycle vector solitons

The energies Eu=R

u²dt andEv =R

v²dt oscillate

almost harmonically, as E_u =<E_u >+∆E_usin(2πσ_az+φ_E,u), Amplitudes of oscillations vs amplitude of fieldu

0 0.05 0.1 0.15 0.2

1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86

∆ Au, ∆ Av, ∆ Eu

<A_u>

black saltires: ∆Eu; blue stars: ∆Au; red crosses: ∆Av.

Well fitted with∆Eu'Rp

A0−<Au>, etc., with A0= 1.854.

Evolution of the ratio v /u

Almost constant vs t

-136-132-128-124-120

t

-1 -1.5 0 -0.5 1 0.5

1.5

u

-1.5 -1 -0.5 0 0.5 1 1.5

v

Soliton with<Au>= 1.855.

Evolution of the ratio v /u

Orθ= arctan^v_u.

Oscillates almost harmonically withz.

Amplitudes of oscillations vs fieldu amplitude:

0 2 4 6 8 10 12 14 16

1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86

∆θ, <θ> (degree)

<A_u>

Black line: mean value< θ >; green line: ∆θ.

Crosses: raw numerical data; solid lines: linear or parabolic fits.

S. Terniche, H. Leblond, D. Mihalache, and A. Kellou, submitted toPhys.Rev. A

1 Waveguiding of a few-cycle pulse How to model it

Nonlinear widening of the linear guided modes

2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms

Few-cycle optical solitons in linearly coupled waveguides

3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind of solitons

g

n = 2

c g c x

n = 3

c g

n = ­2

c g

n = ­1

c g

n = 0

c g

n = 1

c g

n = ­3

c

... ...

Aset of coupled waveguides within the same model, as:

∂_zu_n=−a∂_t(u_n³)−b∂_t³u_n−c∂_t(un−1+u_n+1), Initial data

un(z = 0,t) =A0sin(ωt+ϕ) exp

−n² x² − t²

τ²

; We fix ϕ₀ = 0,x = 1,λ= 1, and we vary A₀ andτ.

Formation of a solitons from a Gaussian pulse

Input

-40

-20

0

20

40

-6 -4 -2 0 2 4 6

t

n

z=0, fwhm= 3.5.

Formation of a solitons from a Gaussian pulse

Low amplitude output: diffraction and dispersion

-40

-20

0

20

40

-6 -4 -2 0 2 4 6

t

n

z= 0.72,A0= 0.2, fwhm= 3.5.

Formation of a solitons from a Gaussian pulse

High amplitude output: space-time localization

-40

-20

0

20

40

-6 -4 -2 0 2 4 6

t

n

z= 288,A0= 2.06,fwhm= 3.5.

An energy threshold for soliton formation?

Domain for soliton formation

1 2 3 4 5 6 7

1.5 2 2.5 3 3.5

fwhm

A

Blue: soliton; red: dispersion-diffraction.

An energy threshold for soliton formation?

Domain for soliton formation

14 14.5 15 15.5 16

1.5 2 2.5 3 3.5

A

fwhm

A

Blue: soliton; red: dispersion-diffraction.

Two kind of solitons: breathing and fundamental.

Breathing soliton:

localized in space and time oscillating wave packet

320 330 340 350 360 370

380 -4 -2 0 2 4

t

n

max|u|= 3.1801

Two kind of solitons: breathing and fundamental.

Breathing soliton:

localized in space and time oscillating wave packet

-2 -1 0 1 2 3

320 330 340 350 360 370 380

u

t

max|u|= 3.1801

Fundamental soliton:

localized in space and time single humped

20 30 40 50 60 70

80 -4 -2 0 2 4

t

n

max|u|= 2.5667

Fundamental soliton:

localized in space and time single humped

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

20 30 40 50 60 70 80

u

t

max|u|= 2.5667

Thank you for your attention.

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