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

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

Herv´ e Leblond ¹ , Dumitru Mihalache ² , David Kremer ³ , Said Terniche ^1,4

1

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

2

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

3

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

4

Laboratoire 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 duration L λ wavelength Typical model: NonLinear Schr¨ odinger equation (NLS)

It is a soliton if it propagates without deformation on D L, 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 duration L λ wavelength Typical model: NonLinear Schr¨ odinger equation (NLS) It is a soliton if it propagates without deformation on D L, 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 duration L λ wavelength 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

ω

= ⇒ Long-wave approximation

modified Korteweg-de Vries (mKdV) equation

∂E

∂ζ = 1 6

d ³ k d ω ³ ω=0

∂ ³ E

∂τ ³ − 6π

nc χ ⁽³⁾ (ω; ω,ω, − ω) ω=0

∂E ³

∂τ

H. Leblond and F. Sanchez, Phys. Rev. A 67, 013804 (2003)

Waveguide description

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

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

∂ _ζ E + β∂ _τ ³ E + γ∂ _τ E ³ = 0 Nonlinear coefficient γ = 1

2nc χ ⁽³⁾ , Dispersion parameter β = (−n ⁰⁰ )

2c ,

Waveguide description

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

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

∂ ζ E + β∂ _τ ³ E + γ∂ τ E ³ − V 2

Z τ

∂ _ξ ² Ed τ ⁰ = 0 Nonlinear coefficient γ = 1

2nc χ ⁽³⁾ , Dispersion parameter β = (−n ⁰⁰ )

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

∂ _ζ E + β _α ∂ _τ ³ E + γ _α ∂ _τ E ³ − V _α 2

Z τ

∂ _ξ ² Ed τ ⁰ = 0 with α = g in the core and α = c in the cladding.

Nonlinear coefficient γ _α = 1 2n _α c χ ⁽³⁾ _α , Dispersion parameter β _α = (−n ⁰⁰ _α )

2c , Linear velocity: V α = 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

∂ _ζ E + β _α ∂ _τ ³ E + γ _α ∂ _τ E ³ + 1

V _α ∂ _τ E − V α

2 Z _τ

∂ _ξ ² Ed τ ⁰ = 0 with α = g in the core and α = c in the cladding.

Velocities : V _g < V _c

Nonlinear coefficient γ α = 1 2n _α c χ ⁽³⁾ _α , Dispersion parameter β α = (−n ⁰⁰ _α )

2c , Linear velocity: V α = 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:

∂ z u = A α ∂ _t ³ u + B α ∂ t u ³ + v α ∂ t u + W α

2 Z t

∂ _x ² udt ⁰ with α = g in the core and α = c in the cladding.

Relative inverse velocities : v _g > v _c Nonlinear coefficient γ α = 1

2n _α c χ ⁽³⁾ _α , Dispersion parameter β α = (−n ⁰⁰ _α )

2c , Linear velocity: V α = c

n α

.

Nonlinear propagation in linear guide

We solve the CGKP equation starting from

u(x,t,z = 0) = A cos(ωt)f (x)e ^−t

^/w

, f (x) =

cos(k x x), for |x| ≤ a,

Ce ^−κ|x| , for |x| > a, is a linear mode profile Normalized coefficients A ₁ = A ₂ = B ₁ = B ₂ = W ₁ = W ₂ = 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. v

= 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

B

− B

= 1.

H. Leblond and D. Mihalache, Phys. Rev. A 88, 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)

∂ _z u = A _α ∂ _t ³ u + B _α ∂ _t u ³ + V _α ∂ _t u + w _α 2

Z t

∂ _x ² udt,

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

We seek for a solution as

u = R(t,z )f ₁ (x)e ^iϕ + S (t,z)f ₂ (x)e ^iϕ , i.e., two interacting modes.

∗ f j , (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:

∂ z R = ∂ z S = iw _g 2ω

(K _c − K _g ) I ₂ 1 + I 1

(R + S ) , Involve overlap integrals I 1 = R ∞

−∞ f 1 f 2 dx , I 2 = R

g

f 1 f 2 dx = R

g

f 1 f 2 dx

(“ R

g_j

·dx” is the integral over the core j = 1 or 2.)

We seek for a solution as

u = R(t,z )f ₁ (x)e ^iϕ + S (t,z)f ₂ (x)e ^iϕ , i.e., two interacting modes.

∗ f j , (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:

∂ z R = ∂ z S = iw _g 2ω

(K _c − K _g ) I ₂ 1 + I 1

(R + S ) , Involve overlap integrals I 1 = R ∞

−∞ f 1 f 2 dx , I 2 = R

g

f 1 f 2 dx = R

g

f 1 f 2 dx

(“ R

g_j

·dx” is the integral over the core j = 1 or 2.)

We seek for a solution as

u = R(t,z )f ₁ (x)e ^iϕ + S (t,z)f ₂ (x)e ^iϕ , i.e., two interacting modes.

∗ f j , (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:

∂ z R = ∂ z S = iw _g 2ω

(K _c − K _g ) I ₂ 1 + I 1

(R + S ) , Involve overlap integrals I 1 = R ∞

−∞ f 1 f 2 dx , I 2 = R

g

f 1 f 2 dx = R

g

f 1 f 2 dx

(“ R

g_j

·dx” is the integral over the core j = 1 or 2.)

We seek for a solution as

u = R(z )f 1 (x)e ^iϕ + S (z )f 2 (x)e ^{i ϕ} , i.e., two interacting modes.

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

∂ z R = ∂ z S = iw _g 2ω

(K _c − K _g ) I ₂ 1 + I 1

(R + S ) ,

The few-cycle pulse is expanded as a Fourier integral of such modes, u ₁ = R

Re ^iϕ d ω, u ₂ = R

Se ^{i ϕ} d ω.

We report ∂ z R and ∂ z S into ∂ z u 1 , and get the linear coupling terms.

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

∂ _z u ₁ = A∂ _t ³ u ₁ + B∂ _t u ₁ ³ + V ∂ _t u ₁ + C ∂ _t u ₂ + D∂ ³ _t u ₂ ,

∂ _z u ₂ = A∂ _t ³ u ₂ + B∂ _t u ₂ ³ + V ∂ _t u ₂ + C ∂ _t u ₁ + D∂ ³ _t u ₁ ,

We seek for a solution as

u = R(z )f 1 (x)e ^iϕ + S (z )f 2 (x)e ^{i ϕ} , i.e., two interacting modes.

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

∂ z R = ∂ z S = iw _g 2ω

(K _c − K _g ) I ₂ 1 + I 1

(R + S ) ,

The few-cycle pulse is expanded as a Fourier integral of such modes, u ₁ = R

Re ^iϕ d ω, u ₂ = R

Se ^{i ϕ} d ω.

We report ∂ z R and ∂ z S into ∂ z u 1 , and get the linear coupling terms.

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

∂ _z u ₁ = A∂ _t ³ u ₁ + B∂ _t u ₁ ³ + V ∂ _t u ₁ + C ∂ _t u ₂ + D∂ ³ _t u ₂ ,

∂ _z u ₂ = A∂ _t ³ u ₂ + B∂ _t u ₂ ³ + V ∂ _t u ₂ + C ∂ _t u ₁ + D∂ ³ _t u ₁ ,

We seek for a solution as

u = R(z )f 1 (x)e ^iϕ + S (z )f 2 (x)e ^{i ϕ} , i.e., two interacting modes.

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

∂ z R = ∂ z S = iw _g 2ω

(K _c − K _g ) I ₂ 1 + I 1

(R + S ) ,

The few-cycle pulse is expanded as a Fourier integral of such modes, u ₁ = R

Re ^iϕ d ω, u ₂ = R

Se ^{i ϕ} d ω.

We report ∂ z R and ∂ z S into ∂ z u 1 , and get the linear coupling terms.

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

∂ _z u ₁ = A∂ _t ³ u ₁ + B∂ _t u ₁ ³ + V ∂ _t u ₁ + C ∂ _t u ₂ + D∂ ³ _t u ₂ ,

∂ _z u ₂ = A∂ _t ³ u ₂ + B∂ _t u ₂ ³ + V ∂ _t u ₂ + C ∂ _t u ₁ + D∂ ³ _t u ₁ ,

We seek for a solution as

u = R(z )f 1 (x)e ^iϕ + S (z )f 2 (x)e ^{i ϕ} , i.e., two interacting modes.

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

∂ z R = ∂ z S = iw _g 2ω

(K _c − K _g ) I ₂ 1 + I 1

(R + S ) ,

The few-cycle pulse is expanded as a Fourier integral of such modes, u ₁ = R

Re ^iϕ d ω, u ₂ = R

Se ^{i ϕ} d ω.

We report ∂ z R and ∂ z S into ∂ z u 1 , and get the linear coupling terms.

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

∂ _z u ₁ = A∂ _t ³ u ₁ + B∂ _t u ₁ ³ + V ∂ _t u ₁ + C ∂ _t u ₂ + D∂ ³ _t u ₂ ,

∂ _z u ₂ = A∂ _t ³ u ₂ + B∂ _t u ₂ ³ + V ∂ _t u ₂ + C ∂ _t u ₁ + D∂ ³ _t u ₁ ,

Nonlinear coupling

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

∂ _z u ₁ = A∂ ³ _t u ₁ + B∂ _t u ³ ₁ + V ∂ _t u ₁

+C ∂ t u 2 + D ∂ _t ³ u 2 + E∂ t 3u ₁ ² u 2 + u ₂ ³

∂ _z u ₂ = A∂ ³ _t u ₂ + B∂ _t u ³ ₂ + V ∂ _t u ₂

+C ∂ _t u ₁ + D ∂ _t ³ u ₁ + E∂ _t 3u ₁ u ₂ + u ₁ ³

H. Leblond, and S. Terniche, Phys. Rev. A 93, 043839 (2016)

Nonlinear coupling

The complete final system is

∂ z u 1 = A∂ ³ _t u 1 + B∂ t u ³ ₁ + V ∂ t u 1

+C ∂ t u 2 + D ∂ _t ³ u 2 + E∂ t 3u ₁ ² u 2 + u ₂ ³

∂ z u 2 = A∂ ³ _t u 2 + B∂ t u ³ ₂ + V ∂ t u 2

+C ∂ t u 1 + D ∂ _t ³ u 1 + E∂ t 3u 1 u 2 + u ₁ ³ We evidence

a standard linear coupling term,

a linear coupling term based on dispersion, a nonlinear coupling term

H. Leblond, and S. Terniche, Phys. Rev. A 93, 043839 (2016)

Nonlinear coupling

The complete final system is

∂ z u 1 = A∂ ³ _t u 1 + B∂ t u ³ ₁ + V ∂ t u 1

+C ∂ t u 2 + D ∂ _t ³ u 2 + E∂ t 3u ₁ ² u 2 + u ₂ ³

∂ z u 2 = A∂ ³ _t u 2 + B∂ t u ³ ₂ + V ∂ t u 2

+C ∂ t u 1 + D ∂ _t ³ u 1 + E∂ t 3u 1 u 2 + u ₁ ³ We evidence

a standard linear coupling term,

a linear coupling term based on dispersion, a nonlinear coupling term

H. Leblond, and S. Terniche, Phys. Rev. A 93, 043839 (2016)

Nonlinear coupling

The complete final system is

∂ z u 1 = A∂ ³ _t u 1 + B∂ t u ³ ₁ + V ∂ t u 1

+C ∂ t u 2 + D ∂ _t ³ u 2 + E∂ t 3u ₁ ² u 2 + u ₂ ³

∂ z u 2 = A∂ ³ _t u 2 + B∂ t u ³ ₂ + V ∂ t u 2

+C ∂ t u 1 + D ∂ _t ³ u 1 + E∂ t 3u 1 u 2 + u ₁ ³ We evidence

a standard linear coupling term,

a linear coupling term based on dispersion, a nonlinear coupling term

H. Leblond, and S. Terniche, Phys. Rev. A 93, 043839 (2016)

Nonlinear coupling

The complete final system is

∂ z u 1 = A∂ ³ _t u 1 + B∂ t u ³ ₁ + V ∂ t u 1

+C ∂ t u 2 + D ∂ _t ³ u 2 + E∂ t 3u ₁ ² u 2 + u ₂ ³

∂ z u 2 = A∂ ³ _t u 2 + B∂ t u ³ ₂ + V ∂ t u 2

+C ∂ t u 1 + D ∂ _t ³ u 1 + E∂ t 3u 1 u 2 + u ₁ ³ We evidence

a standard linear coupling term,

a linear coupling term based on dispersion, a nonlinear coupling term

H. Leblond, and S. Terniche, Phys. Rev. A 93, 043839 (2016)

We assume a purely linear and non-dispersive coupling

∂ z u = −∂ _t (u ³ ) − ∂ _t ³ u − C ∂ t v ,

∂ _z v = −∂ _t (v ³ ) − ∂ _t ³ v − C ∂ _t u,

We look for stationary states (vector solitons) in this model The ”stationary” states oscillate with t and z : .

few-cycle solitons are breathers.

A typical example of few-cycle vector soliton

(Dotted lines: u, solid lines: v. Left: at z = 0, right: at z = 60. < A

>= 1.837).

Evolution of soliton’s maximum amplitude during propagation.

1.6 1.8 2

0 1 2 3 4 5 6 7 8

max t (| u |)

z

0.4 0.5 0.6

0 1 2 3 4 5 6 7 8

max t (| v |)

z

Soliton with < A

>= 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 the linearized case.

The monochromatic solutions are u

v

= A

B

e ^−i(ωt+bω

^z) ,

With, due to coupling,

A = u 0 cos c ωz + iv 0 sin c ωz , B = v 0 cos cωz + iu 0 sin c ωz .

The maximum amplitude and the power density of the wave oscillate

with spatial frequency c ω/π = σ ₀ = 1.326.

Consider now the coupled equations in the linearized case.

The monochromatic solutions are u

v

= A

B

e ^−i(ωt+bω

^z) , With, due to coupling,

A = u 0 cos c ωz + iv 0 sin c ωz , B = v 0 cos cωz + iu 0 sin c ωz .

The maximum amplitude and the power density of the wave oscillate

with spatial frequency c ω/π = σ ₀ = 1.326.

Consider now the coupled equations in the linearized case.

The monochromatic solutions are u

v

= A

B

e ^−i(ωt+bω

^z) , With, due to coupling,

A = u 0 cos c ωz + iv 0 sin c ωz , B = v 0 cos cωz + iu 0 sin c ωz .

The maximum amplitude and the power density of the wave oscillate

with spatial frequency c ω/π = σ ₀ = 1.326.

Consider now the coupled equations in the linearized case.

The monochromatic solutions are u

v

= A

B

e ^−i(ωt+bω

^z) , With, due to coupling,

A = u 0 cos c ωz + iv 0 sin c ωz , B = v 0 cos cωz + iu 0 sin c ωz .

The maximum amplitude and the power density of the wave oscillate

with spatial frequency c ω/π = σ ₀ = 1.326.

Oscillations of the few-cycle vector solitons

The energies E u = R

u ² dt and E v = R

v ² dt oscillate

almost harmonically, as E _u =< E _u > +∆E _u sin(2πσ _a z + φ _E,u ), The same for A u = max t (|u|) and A v = max t (|v|)

Spatial frequency σ a ∈ [1.06,1.17], increasing with < A u >.

(linear: σ

= 1.326) .

Amplitudes of oscillations vs amplitude of field u

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

∆ A

, ∆ A

, ∆ E

<A

>

black saltires: ∆E

; blue stars: ∆A

; red crosses: ∆A

.

Well fitted with ∆E ' R p

A − < A >, etc., with A = 1.854.

Oscillations of the few-cycle vector solitons

The energies E u = R

u ² dt and E v = R

v ² dt oscillate

almost harmonically, as E _u =< E _u > +∆E _u sin(2πσ _a z + φ _E,u ), The same for A u = max t (|u|) and A v = max t (|v|)

Spatial frequency σ a ∈ [1.06,1.17], increasing with < A u >.

(linear: σ

= 1.326) .

Amplitudes of oscillations vs amplitude of field u

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

∆ A

, ∆ A

, ∆ E

<A

>

black saltires: ∆E

; blue stars: ∆A

; red crosses: ∆A

.

Well fitted with ∆E ' R p

A − < A >, etc., with A = 1.854.

Oscillations of the few-cycle vector solitons

The energies E u = R

u ² dt and E v = R

v ² dt oscillate

almost harmonically, as E _u =< E _u > +∆E _u sin(2πσ _a z + φ _E,u ), The same for A u = max t (|u|) and A v = max t (|v|)

Spatial frequency σ a ∈ [1.06,1.17], increasing with < A u >.

(linear: σ

= 1.326) .

Amplitudes of oscillations vs amplitude of field u

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

∆ A

, ∆ A

, ∆ E

<A

>

black saltires: ∆E

; blue stars: ∆A

; red crosses: ∆A

.

Well fitted with ∆E ' R p

A − < A >, etc., with A = 1.854.

Oscillations of the few-cycle vector solitons

The energies E u = R

u ² dt and E v = R

v ² dt oscillate

almost harmonically, as E _u =< E _u > +∆E _u sin(2πσ _a z + φ _E,u ), The same for A u = max t (|u|) and A v = max t (|v|)

Spatial frequency σ a ∈ [1.06,1.17], increasing with < A u >.

(linear: σ

= 1.326) .

Amplitudes of oscillations vs amplitude of field u

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

∆ A

, ∆ A

, ∆ E

<A

>

black saltires: ∆E

; blue stars: ∆A

; red crosses: ∆A

.

Well fitted with ∆E ' R p

A − < A >, etc., with A = 1.854.

Oscillations of the few-cycle vector solitons

The energies E u = R

u ² dt and E v = R

v ² dt oscillate

almost harmonically, as E _u =< E _u > +∆E _u sin(2πσ _a z + φ _E,u ), Amplitudes of oscillations vs amplitude of field u

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

∆ A

, ∆ A

, ∆ E

<A

>

black saltires: ∆E

; blue stars: ∆A

; red crosses: ∆A

.

Well fitted with ∆E u ' R p

A 0 − < A u >, etc., with A 0 = 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 < A

>= 1.855.

Evolution of the ratio v /u

Or θ = arctan ^v _u .

Oscillates almost harmonically with z .

Amplitudes of oscillations vs field u 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

∆

, <

>

<A

>

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 to Phys.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

... ...

A set of coupled waveguides within the same model, as:

∂ _z u _n = −a∂ _t (u _n ³ ) − b∂ _t ³ u _n − c ∂ _t (u n−1 + u _n+1 ) , Initial data

u n (z = 0,t) = A 0 sin(ωt + ϕ) exp

− n ² x ² − t ²

τ ²

;

We fix ϕ ₀ = 0, x = 1, λ = 1, and we vary A ₀ and τ .

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, A

= 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, A

= 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 0

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 0 2 fwhm

A 0

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.

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