Solitons optiques à quelques cycles dans des guides

Solitons optiques à quelques cycles dans des guides couplés Hervé Leblond 1, Dumitru Mihalache 2, David Kremer 3, Said Terniche 1,4 1 Laboratoire de Photonique d Angers LϕA EA 4464, Université 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, Université d Angers. 4 Laboratoire Electronique Quantique, USTHB, Alger. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés1 Hulubei Nation / 29

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 Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés2 Hulubei Nation / 29

Solitary wave vs envelope solitons Envelope soliton: the usual optical soliton in the ps range Pulse duration L λ wavelength Typical model: NonLinear Schrödinger 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 Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés3 Hulubei Nation / 29

Solitary wave vs envelope solitons Envelope soliton: the usual optical soliton in the ps range Pulse duration L λ wavelength Typical model: NonLinear Schrödinger 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 Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés3 Hulubei Nation / 29

Solitary wave vs envelope solitons Envelope soliton: the usual optical soliton in the ps range Pulse duration L λ wavelength Typical model: NonLinear Schrödinger equation (NLS) Solitary wave soliton: the hydrodynamical soliton Few-cycle optical solitons: L λ A single oscillation Typical model: Korteweg-de Vries equation (KdV) The slowly varying envelope approximation is not valid Generalized NLS equation We seek a different approach based on KdV-type models Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés3 Hulubei Nation / 29

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 d 3 k 3 E 6 dω 3 ω=0 τ 3 6π nc χ(3) (ω; ω,ω, ω) E 3 ω=0 τ H. Leblond and F. Sanchez, Phys. Rev. A 67, 013804 (2003) Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés4 Hulubei Nation / 29

Waveguide description The evolution of the electric field E: In (1+1) dimensions: The modified Korteweg-de Vries (mkdv) equation ζ E + β 3 τ E + γ τ E 3 = 0 Nonlinear coefficient γ = 1 2nc χ(3), Dispersion parameter β = ( n ), 2c Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés5 Hulubei Nation / 29

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 3 τ E + γ τ E 3 V 2 τ 2 ξ Edτ = 0 Nonlinear coefficient γ = 1 2nc χ(3), Dispersion parameter β = ( n ), 2c Linear velocity: V = c n. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés5 Hulubei Nation / 29

Waveguide description The evolution of the electric field E: A waveguide: c g c x cladding core cladding The cubic generalized Kadomtsev-Petviashvili (CGKP) equation ζ E + β α 3 τ E + γ α τ E 3 V α 2 τ 2 ξ Edτ = 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 α), 2c Linear velocity: V α = c. n α Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés5 Hulubei Nation / 29

Waveguide description The evolution of the electric field E: A waveguide: c g c cladding core cladding The cubic generalized Kadomtsev-Petviashvili (CGKP) equation ζ E + β α τ 3 E + γ α τ E 3 + 1 τ E V τ α ξ 2 V α 2 Edτ = 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 α), 2c Linear velocity: V α = c. n α x Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés5 Hulubei Nation / 29

Waveguide description The evolution of the electric field E: A waveguide: c g c cladding core 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 t α x 2 udt 2 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 α), 2c Linear velocity: V α = c. n α Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés5 Hulubei Nation / 29 x

Nonlinear propagation in linear guide We solve the CGKP equation starting from u(x,t,z = 0) = A cos(ωt)f (x)e t2 /w 2, { cos(kx x), for x a, f (x) = Ce κ x is a linear mode profile, for x > a, 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. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés6 Hulubei Nation / 29

Guided wave profiles (Normalized so that the total power is 1. v 2 = 3, w = 2.) The pulse is less confined in nonlinear (blue, and red) than in linear (pink and cyan) regime. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés7 Hulubei Nation / 29

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. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés8 Hulubei Nation / 29

Two-cycle soliton of the nonlinear waveguide -2-1 x 0 1 2-150 -100-50 0 50 100 150 t B 2 B 1 = 1. H. Leblond and D. Mihalache, Phys. Rev. A 88, 023840 (2013) Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés9 Hulubei Nation / 29

2D waveguiding structure: two cores 1 and 2 and dielectric cladding The generalized Kadomtsev-Petviashvili (GKP) equation (dimensionless) z u = A α 3 t u + B α t u 3 + V α t u + w α 2 α = g in the cores 1 and 2, α = c in the cladding. t 2 x udt, Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 11 Nation / 29

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 (K c K g ) I 2 (R + S), 2ω 1 + I 1 Involve overlap integrals I 1 = f 1f 2 dx, I 2 = g 1 f 1 f 2 dx = g 2 f 1 f 2 dx ( g j dx is the integral over the core j = 1 or 2.) Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 12 Nation / 29

We seek for a solution as u = R(z)f 1 (x)e iϕ + S(z)f 2 (x)e iϕ, i.e., two interacting modes. We report it into the GKP equation and get after averaging on x: z R = z S = iw g (K c K g ) I 2 (R + S), 2ω 1 + I 1 The few-cycle pulse is expanded as a Fourier integral of such modes, u 1 = Re iϕ dω, u 2 = Se iϕ dω. 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 3 t u 1 + B t u 3 1 + V t u 1 + C t u 2 + D 3 t u 2, z u 2 = A 3 t u 2 + B t u 3 2 + V t u 2 + C t u 1 + D 3 t u 1, Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 13 Nation / 29

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 t 3 u 1 + B t u1 3 + V t u 1 +C t u 2 + D t 3 ( u 2 + E t 3u 2 1 u 2 + u2 3 ) z u 2 = A t 3 u 2 + B t u2 3 + V t u 2 +C t u 1 + D t 3 ( u 1 + E t 3u1 u 2 + u1 3 ) H. Leblond, and S. Terniche, Phys. Rev. A 93, 043839 (2016) Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 14 Nation / 29

Nonlinear coupling The complete final system is We evidence z u 1 = A t 3 u 1 + B t u1 3 + V t u 1 +C t u 2 + D t 3 ( u 2 + E t 3u 2 1 u 2 + u2 3 ) z u 2 = A t 3 u 2 + B t u2 3 + V t u 2 +C t u 1 + D t 3 ( u 1 + E t 3u1 u 2 + u1 3 ) 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) Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 14 Nation / 29

Nonlinear coupling The complete final system is We evidence z u 1 = A t 3 u 1 + B t u1 3 + V t u 1 +C t u 2 + D t 3 ( u 2 + E t 3u 2 1 u 2 + u2 3 ) z u 2 = A t 3 u 2 + B t u2 3 + V t u 2 +C t u 1 + D t 3 ( u 1 + E t 3u1 u 2 + u1 3 ) 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) eblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 14 Nation / 29

We assume a purely linear and non-dispersive coupling z u = t (u 3 ) 3 t u C t v, z v = t (v 3 ) 3 t 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. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 15 Nation / 29

A typical example of few-cycle vector soliton (Dotted lines: u, solid lines: v. Left: at z = 0, right: at z = 60. < A u >= 1.837). Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 16 Nation / 29

Evolution of soliton s maximum amplitude during propagation. max t ( u ) max t ( v ) 2 1.8 1.6 0.6 0.5 0.4 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 z Soliton with < A u >= 1.789. Two types of oscillations: Fast: phase - group velocity mismatch Slower: periodic energy exchange, as in linear regime. z Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 17 Nation / 29

Consider now the coupled equations in the linearized case. The monochromatic solutions are ( ) ( ) u A = e i(ωt+bω3z), v B With, due to coupling, A = u 0 cos cωz + iv 0 sin cωz, B = v 0 cos cωz + iu 0 sin cωz. The maximum amplitude and the power density of the wave oscillate with spatial frequency cω/π = σ 0 = 1.326. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 18 Nation / 29

Oscillations of the few-cycle vector solitons The energies E u = u 2 dt and E v = 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 0 = 1.326). Amplitudes of oscillations vs amplitude of field u 0.2 A u, A v, E u 0.15 0.1 0.05 0 1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86 <A u > black saltires: E u; blue stars: A u; red crosses: A v. Well fitted with E R A < A >, etc., with A = 1.854. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 19 Nation / 29

Oscillations of the few-cycle vector solitons The energies E u = u 2 dt and E v = 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 0.2 A u, A v, E u 0.15 0.1 0.05 0 1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86 <A u > black saltires: E u; blue stars: A u; red crosses: A v. Well fitted with E u R A 0 < A u >, etc., with A 0 = 1.854. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 19 Nation / 29

Evolution of the ratio v/u Almost constant vs t v 1.5 1 0.5 0-0.5-1 -1.5 1.5 1 0.5 u 0-0.5-1 -1.5 Soliton with < A u >= 1.855. -136-132-128-124 -120 t Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 20 Nation / 29

Evolution of the ratio v/u Or θ = arctan v u. Oscillates almost harmonically with z. Amplitudes of oscillations vs field u amplitude: θ, <θ> (degree) 16 14 12 10 8 6 4 2 0 1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86 <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 to Phys.Rev. A Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 21 Nation / 29

c g c g c g c g c g c g c g c x... n = 3 n = 2 n = 1 n = 0 n = 1 n = 2 n = 3... A set of coupled waveguides within the same model, as: Initial data z u n = a t (u 3 n) b 3 t u n c t (u n 1 + u n+1 ), ) u n (z = 0,t) = A 0 sin(ωt + ϕ) exp ( n2 x 2 t2 τ 2 ; We fix ϕ 0 = 0, x = 1, λ = 1, and we vary A 0 and τ. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 23 Nation / 29

Formation of a solitons from a Gaussian pulse Input -40 t -20 0 20 40-6 -4-2 0 2 4 6 n z=0, fwhm = 3.5. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 24 Nation / 29

Formation of a solitons from a Gaussian pulse Low amplitude output: diffraction and dispersion -40 t -20 0 20 40-6 -4-2 0 2 4 6 n z = 0.72, A 0 = 0.2, fwhm = 3.5. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 24 Nation / 29

Formation of a solitons from a Gaussian pulse High amplitude output: space-time localization -40 t -20 0 20 40-6 -4-2 0 2 4 6 n z = 288, A 0 = 2.06, fwhm = 3.5. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 24 Nation / 29

An energy threshold for soliton formation? Domain for soliton formation fwhm 7 6 5 4 3 2 1 1.5 2 2.5 3 3.5 A 0 Blue: soliton; red: dispersion-diffraction. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 25 Nation / 29

An energy threshold for soliton formation? Domain for soliton formation 16 15.5 A 0 2 fwhm 15 14.5 14 1.5 2 2.5 3 3.5 A 0 Blue: soliton; red: dispersion-diffraction. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 25 Nation / 29

Two kind of solitons: { breathing and fundamental. localized in space and time Breathing soliton: oscillating wave packet t 320 330 340 350 360 370 380 n -4-2 0 2 4 max u = 3.1801 Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 26 Nation / 29

Two kind of solitons: breathing and fundamental. { localized in space and time Breathing soliton: oscillating wave packet 3 2 u 1 0-1 -2 320 330 340 350 360 370 380 t max u = 3.1801 Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 26 Nation / 29

{ localized in space and time Fundamental soliton: single humped t 20 30 40 50 60 70 80 n -4-2 0 2 4 max u = 2.5667 Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 27 Nation / 29

{ localized in space and time Fundamental soliton: single humped u 0.5 0-0.5-1 -1.5-2 -2.5-3 20 30 40 50 60 70 80 max u = 2.5667 t Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 27 Nation / 29

Thank you for your attention. Leblond, Mihalache, Kremer, Terniche ( Laboratoire Propagation de Photonique de solitonsd Angers optiqueslϕa à quelques EA 4464, cycles Université dans des d Angers., guides d ondes Horia couplés Hulubei 28 Nation / 29